advances in gamma ray resonant scattering and absorption: long-lived isomeric nuclear states

Springer Tracts in Modern Physics 261

Advances in Gamma Ray Resonant Scattering and AbsorptionLong-Lived Isomeric Nuclear States

Andrey V. Davydov

Springer Tracts in Modern Physics

Volume 261

Honorary Editor

G. Höhler, Karlsruhe, Germany

Series editors

Atsushi Fujimori, Tokyo, JapanJohann H. Kühn, Karlsruhe, GermanyThomas Müller, Karlsruhe, GermanyFrank Steiner, Ulm, GermanyWilliam C. Stwalley, Storrs, CT, USAJoachim E. Trümper, Garching, GermanyPeter Wölfle, Karlsruhe, GermanyUlrike Woggon, Berlin, Germany

Springer Tracts in Modern Physics

Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics ofcurrent interest in physics. The following fields are emphasized: Elementary Particle Physics,Condensed Matter Physics, Light Matter Interaction, Atomic and Molecular Physics, ComplexSystems, Fundamental Astrophysics.

Suitable reviews of other fields can also be accepted. The Editors encourage prospectiveauthors to correspond with them in advance of submitting a manuscript. For reviews of topicsbelonging to the above mentioned fields, they should address the responsible Editor as listedbelow.

Special offer: For all clients with a print standing order we offer free access to theelectronic volumes of the Series published in the current year.

Elementary Particle Physics, EditorsJohann H. KühnInstitut für Theoretische TeilchenphysikKarlsruhe Institut für Technologie KITPostfach 69 8076049 Karlsruhe, GermanyPhone: +49 (7 21) 6 08 33 72Fax: +49 (7 21) 37 07 26Email: [email protected]/*jk1

Thomas MüllerInstitut für Experimentelle KernphysikKarlsruhe Institut für Technologie KITPostfach 69 8076049 Karlsruhe, GermanyPhone: +49 (7 21) 6 08 35 24Fax: +49 (7 21) 6 07 26 21Email: [email protected]

Fundamental Astrophysics, EditorJoachim E. TrümperMax-Planck-Institut für Extraterrestrische

PhysikPostfach 13 1285741 Garching, GermanyPhone: +49 (89) 30 00 35 59Fax: +49 (89) 30 00 33 15Email: [email protected]/index.html

Solid State and Optical PhysicsUlrike WoggonInstitut für Optik und Atomare PhysikTechnische Universität BerlinStraße des 17. Juni 13510623 Berlin, GermanyPhone: +49 (30) 314 78921Fax: +49 (30) 314 21079Email: [email protected]

Condensed Matter Physics, EditorsAtsushi FujimoriEditor for The Pacific RimDepartment of PhysicsUniversity of Tokyo7-3-1 Hongo, Bunkyo-kuTokyo 113-0033, JapanEmail: [email protected]://wyvern.phys.s.u-tokyo.ac.jp/

welcome_en.html

Peter WölfleInstitut für Theorie der Kondensierten MaterieKarlsruhe Institut für Technologie KITPostfach 69 8076049 Karlsruhe, GermanyPhone: +49 (7 21) 6 08 35 90Phone: +49 (7 21) 6 08 77 79Email: [email protected]

Complex Systems, EditorFrank SteinerInstitut für Theoretische PhysikUniversität UlmAlbert-Einstein-Allee 1189069 Ulm, GermanyPhone: +49 (7 31) 5 02 29 10Fax: +49 (7 31) 5 02 29 24Email: [email protected]/theo/qc/group.html

Atomic, Molecular and Optical PhysicsWilliam C. StwalleyUniversity of ConnecticutDepartment of Physics2152 Hillside Road, U-3046Storrs, CT 06269-3046, USAPhone: +1 (860) 486 4924Fax: +1 (860) 486 3346Email: [email protected]/faculty/stwalley.html

More information about this series at http://www.springer.com/series/426

http://www-ttp.physik.uni-karlsruhe.de/~jk

http://www-ekp.physik.uni-karlsruhe.de

http://www.mpe-garching.mpg.de/index.html

http://www.ioap.tu-berlin.de

http://wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html

http://wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html

http://www-tkm.physik.uni-karlsruhe.de

http://www.physik.uni-ulm.de/theo/qc/group.html

http://www-phys.uconn.edu/faculty/stwalley.html

http://www.springer.com/series/426

Andrey V. Davydov

Advances in Gamma RayResonant Scatteringand AbsorptionLong-Lived Isomeric Nuclear States

123

Andrey V. DavydovGroup N 216Institute for Theoretical

and Experimental PhysicsMoscowRussia

ISSN 0081-3869 ISSN 1615-0430 (electronic)ISBN 978-3-319-10523-9 ISBN 978-3-319-10524-6 (eBook)DOI 10.1007/978-3-319-10524-6

Library of Congress Control Number: 2014947133

Springer Cham Heidelberg New York Dordrecht London

© Springer International Publishing Switzerland 2015

Copyright for English edition only, other than that the copyright is with PhysMathLit, Moscow.

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publisher’s location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book was written by an experimenter. It summarizes the results of my fifty-year work at A.I. Alikhanov Institute of Theoretical and Experimental Physics onproblems of gamma ray interaction with nuclei. I tried to reveal the physicalmeaning of these results, making the exposition as simple as possible and some-times resorting to arguments and derivations that could seem insufficiently strict, atleast to orthodox theorists. The main part of the book addresses the problem ofstudying resonant gamma ray absorption and scattering by nuclei. These processes,which are essentially the simplest nuclear reactions, permit, if studied profoundly,revealing very interesting special features that are inherent in phenomena of gammaray emission and absorption by nuclei, and which are seemingly of a generalcharacter. It is noteworthy that the concepts of the nature of the photon that areprevalent among the physics community are inaccurate in many respects, evensometimes erroneous. In particular, the assignment of a well-defined frequency ν toa photon of energy E = hν is an approximation because a monochromatic harmonicoscillation is infinite in time, but by no means does a photon, which is produced atspecific instant, exist limited during time, ending up in absorption inside a detectoror in some substance. This means that the Fourier frequency spectrum of a photonmust have a finite width. Also, opinions on the particle-wave duality of the photondiffer widely. Recently, an article of the present author where resonant gamma rayscattering on nuclei was considered and where a photon was shown to manifest aspatial and a time extent in this process was rejected by an authoritative Russianphysics journal on the basis of reviewer’s evaluation. The argument of the reviewerwas that the photon is a particle because it experiences photo-absorption even invery finely dispersed powders, and therefore cannot have extensive dimensions. Ofcourse, the statement of the reviewer that, in processes like the photoelectric effect,photons behave as almost quasi point objects, not displaying wave properties, iscorrect. The same reveals in the behavior of photons in Compton scattering byelectrons. However, the other processes exist in which the photon interacts withmatter behaving itself like a wave of macroscopic size, not showing any particleproperty. In the monograph by Robert Wood “Physical Optics” [1], there is adescription of an experiment where one observes light diffraction at a grating 3 cm

v

long and measures the resolution of the grating. After covering half the grating witha screen, the resolution becomes lower by a factor of two. Since a stationarydiffraction pattern arises owing to the interference of a photon with itself (theinterference between of two photons cannot lead to a stationary pattern because of astochastic character of the phase difference), this means that, under conditions of theexperiment being discussed, photons have a size not smaller than 3 cm. Braggscattering in crystals is yet another process of this type, but, here, it is gamma rayphotons rather than optical photons behave as extended waves. In this process, eachphoton interacts with all crystal atoms within its absorption length, exhibiting noparticle properties. A very convincing example is provided by an experiment of agroup headed by V.K. Voitovetsky [2], where gamma rays of the 181Ta nuclidewere transmitted between the cogs of a rotating gear, the shape of the detectedgamma line being measured with the aid of a Mössbauer spectrometer. It was foundthat, at a large number of gear revolutions per unit time such that the gap betweenthe cogs traverses the gamma beam within 0.1 of the mean lifetime of source nucleiin the excited state, the measured width of the Mössbauer gamma line was muchlarger than that in the case of a very slow rotation of the gear. This obviouslyindicated that the gear cogs interrupted the spatially extended wave train of aphoton because wave trains shorter than natural ones corresponded to gamma linesof width larger than the natural width. We would like to emphasize that, in nophysics process, a photon demonstrates its wave and particle properties simulta-neously—either the former or the latter. After being involved in Bragg scattering ina crystal, a photon is recorded by a detector in an event of photo-absorption orCompton scattering; that is, the photon behaves as a particle that lost completely thewave properties that it has just revealed. However, this does not mean that the wavetransformed into a particle immediately after Bragg scattering. If, instead of adetector, one places a second crystal on the path of a photon that experienced Braggscattering, and if the Bragg conditions hold in this crystal, then the photon would beable to undergo Bragg scattering once again with a sizable probability—that is, toexhibit anew its wave properties. At the same time, a photon that has shown particleproperties in an event of Compton scattering in a detector can thereupon interact ina wave manner with a crystal (under Bragg conditions other than those in the firstcase, because the photon energy changed after scattering), transforming from aparticle into a wave again. The question of how and why such transformationsoccur is one of the most mysterious in modern physics.

The ensuing exposition is organized as follows. In the first chapter, we considertheoretically the process of resonant gamma ray scattering by nuclei. We areinterested in a question of how the angular distribution of resonantly scatteredgamma rays depends on the perturbing action of magnetic fields. Solving thisparticular and seemingly trivial problem, we arrive at conclusions that give suffi-cient grounds to take a fresh look at some special features of processes involvinggamma ray emission and absorption by nuclei. In the second chapter, we describeexperiments performed by our group and devoted to measuring unperturbed andmagnetic-field-perturbed angular distributions (ADs) of resonantly scattered gammarays of 182W and 191Ir. Those experiments confirmed the prediction of the theory

vi Preface

that the result of perturbing ADs depends on the width of the spectrum of gammarays incident to a resonant gamma ray scatterer. At the end of this chapter, we showthat important conclusions follows from the theoretical and experimental datadescribed in it: the mean lifetime of nuclei in an excited state depends on the modeof its excitation, and processes of gamma ray emission and absorption by nucleihave a protracted character. In the third chapter, we consider in detail the problemof gamma resonant excitation of long-lived isomeric states of nuclei. Experimentalinvestigations of this problem revealed a glaring contradiction between present-daytheoretical predictions, which require, among other things, that the Mósbauergamma line emitted in the decay of 109Ag nuclei that were in the isomeric excitedstate characterized by an energy of 88.03 keV and a mean lifetime of 57 s must bebroadened by five to six orders of magnitude in relation to the natural width, and theexperimental results of three research groups (including ours), which obtained dataindicating that the relative broadening of this gamma line does not exceed one totwo orders of magnitude. So small a broadening of the Mósbauer gamma line of the109mAg isomer permitted implementing the idea of a gravitational gamma spec-trometer and directly measuring the profile of the Mósbauer gamma resonance inthis isomer. The use of a traditional Mósbauer spectrometer for this purpose istechnically impossible because this would require creating a device capable ofmoving a gamma source with respect to the absorber at a velocity of about 10-12

cm/s; that is, it would be necessary to push it forward over a distance per secondnearly equal to the diameter of the silver-atomic nucleus, and to measure simul-taneously this velocity by some method. The principle of operation of the gravi-tational gamma spectrometer based on the 109mAg isomer is described in the fourthchapter. Its resolution is about eight orders of magnitude higher than the resolutionof usual Mósbauer spectrometers employing gamma rays of the 57Fe nuclide. In thenext chapter, we describe our experiments devoted to exploring the resonantscattering of annihilation photons by nuclei, whereupon (in the last chapter) weshow how one can use this phenomenon to study the shape of Fermi surfaces inmetals.

Some other experiments performed by our group with gamma rays are discussedat the end of this book along with the ideas of experiments that have yet to beconducted.

Some of the experiments described here were performed by methods that seemobsolete from the modern point of view, but I deemed it necessary to tell aboutthem because they were an inalienable link in the chain of experiments that led toimportant conclusions both in what is concerned with the dependence of the meanlifetime of nuclei in an excited state on the method of excitation and in what isconcerned with the duration of nuclear radiative processes.

One comment on the notation used is in order. Vector quantities appearing insome equations are printed in boldface.

Preface vii

Acknowledgments

It is my pleasant duty to record here the benefit of working over many tens of yearsside by side with Yury Denisovich Bayukov, Yury Nikolaevich Isaev, and MarkMikhailovich Korotkov, who are members of our research group and who made aninvaluable contribution to the implementation of the experiments described in thisbook. I am also indebted to my son Professor V.A. Davydov for valuable adviceand help in solving some mathematical problems, and to my second son Anton, mywife Nina Mikhailovna, and to the scientist from our group Yuri B. Novozhilov fortheir very valuable help in the preparation of this book for printing.

I would also thank the assistance of ITEP library A.A. Alekhina, E.V. Sandrakova,and O.M. Kuz’mina, the assistance of JINR library V.M. Smirnova and the head oflibrary of Physical Faculty of Moscow State University V.M. Zuev for their help inthe search for translation in English versions of the papers from Russian journals.

Special thanks are due to Professor F.S. Dzheparov; the corresponding membersof Russian Academy of Sciences Yu.G. Abov, M.V. Danilov, and B.L. Ioffe; andAcademician L.B. Okun. Over many years, I have had the opportunity of discussingwith them problems considered in this book. I am grateful to Doctor N.V. Lazarevfor his interest in the work on this book and for his help in translating it.

I will always mourn the untimely death of the collaborators of our group VladilenGrigor’evich Alpatov, Gavriil Romanovich Kartashov, Vadim MikhailovichSamoylov, Galina Eugen’evna Bizina, Mikhail Georgievich Gavrilov, GennadiyVictorovich Rotter, and Yury Ivanovich Nekrasov and cherish memory of theirselfless work, which ensured the success of our experiments.

I recall with gratitude my first supervisor Professor N.A. Burgov, whointroduced me in the realms of resonant gamma ray scattering, and the firstdirector of ITEP Academician A.I. Alikhanov, whose permanent attention to ourwork and support were invaluable. I nourish warmest recollections of ProfessorA.L. Suvorov, who was ITEP’s director until his untimely death in 2005. Hissupport of our investigations was a great help to us all, and his attitude to mepersonally was highly benevolent.

A.V. Davydov

References

1. R.W. Wood, Physical Optics (The MacMillan Company, New York, 1934)2. V.K. Voitovetsky, I.L. Korsunsky, Yu.F. Pazhin et al., Phys. At. Nucl. 38, 394 (1983)

viii Preface

Contents

1 Theory of the Resonant Scattering of Gamma Rays by Nucleiin a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 ADRSG Function for the Case Where the Magnetic Field

Is Perpendicular to the Plane of Gamma-Ray Scattering . . . . . . . 21.3 Angular Distribution in the Case Where Magnetic-Field

Directions Are Distributed Chaotically Over the ScattererVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Some Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Experimental Study of Resonant Gamma-Ray Scattering . . . . . . . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Measurement of the Angular Distribution of 100.1 keV Gamma

Rays Resonantly Scattered by 182W Nuclei . . . . . . . . . . . . . . . . 362.3 Measurement of the Magnetic Moment of the 182W Nucleus

in the 2+ Excited State at 100.1 keV . . . . . . . . . . . . . . . . . . . . 422.4 Measurement of the Unperturbed Angular Distribution

of Gamma Rays Resonantly Scattered by 191Ir Nuclei . . . . . . . . 472.5 Measurements of Magnetic-Field-Perturbed Angular

Distributions of 129.4 keV Gamma Rays ResonantlyScattered by 191Ir Nuclei in an Ir–Fe Alloy . . . . . . . . . . . . . . . . 52

2.6 Some Special Features of Gamma-Radiation Processesas Suggested by the Foregoing Analysis . . . . . . . . . . . . . . . . . . 69

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Problem of the Resonant Excitation of Long-Lived NuclearIsomeric States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.1 Small Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2 Physical Reasons Behind Expected Difficulties in Performing

Mössbauer Experiments with Long-Lived Isomers . . . . . . . . . . . 80

ix

http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1#Sec1











http://dx.doi.org/10.1007/978-3-319-10524-6_1#Bib1

http://dx.doi.org/10.1007/978-3-319-10524-6_2

http://dx.doi.org/10.1007/978-3-319-10524-6_2

























http://dx.doi.org/10.1007/978-3-319-10524-6_3

http://dx.doi.org/10.1007/978-3-319-10524-6_3

http://dx.doi.org/10.1007/978-3-319-10524-6_3






3.3 Early Experiments Performed at ITEP to Study the MössbauerExcitation of Long-Lived Isomeric States of 107Ag and 109AgNuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.4 Influence of the Direction of the Magnetic Field in Whichthe Silver Gamma Source Is Placed on the Probabilityfor Resonant Self-absorption of 109mAg-Isomer GammaRays in It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.5 Foreign Experiments Devoted to the Observationof Resonant Self-absorption of 109mAg-Isomer Gamma Raysin Metallic Silver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.6 Experiments of Our ITEP Group Performed in the Last Yearswith the 109mAg Isomer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4 Fundamentals of Gravitational Gamma Spectrometry . . . . . . . . . . 1274.1 Design of a Gravitational Gamma Spectrometer Based

on the 109mAg Isomer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2 Experiments Performed at ITEP with the Aid of a Gravitational

Gamma Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5 Nuclear Resonant Scattering of Annihilation Photons. . . . . . . . . . . 1415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2 Expected Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.3 Description of Our Experiments. . . . . . . . . . . . . . . . . . . . . . . . 1495.4 Data on the Cross Sections for the Rayleigh Scattering

of Gamma Rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.5 First Experiment Aimed at Observing Nuclear Resonant

Scattering of Annihilation Photons . . . . . . . . . . . . . . . . . . . . . . 1575.6 Second Experiment in Which the Nuclear Resonant

Scattering of Annihilation Photons was Observed . . . . . . . . . . . 1615.7 Cross Section for the Resonant Scattering of Annihilation

Photons by 106Pd Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.8 Further Ways Toward Refining Upon the Method

for Observing the Process Under Discussion . . . . . . . . . . . . . . . 168References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6 Small Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1 Manifestations of the Binding Energy of Electrons of Scattering

Atoms in the Spectra of Scattered Gamma Radiation . . . . . . . . . 1736.2 Application of Resonant Gamma Ray Scattering

to Determining the Magnetic Moment of the 65Cu Nucleusin the Excited State at 1115.5 KeV . . . . . . . . . . . . . . . . . . . . . 176

x Contents























http://dx.doi.org/10.1007/978-3-319-10524-6_4

http://dx.doi.org/10.1007/978-3-319-10524-6_4









http://dx.doi.org/10.1007/978-3-319-10524-6_5

http://dx.doi.org/10.1007/978-3-319-10524-6_5
























http://dx.doi.org/10.1007/978-3-319-10524-6_6

http://dx.doi.org/10.1007/978-3-319-10524-6_6









6.3 On the Possibility of Applying the Nuclear Resonant Scatteringof Annihilation Photons to Studying the Shape of FermiSurfaces in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.4 Nuclear Resonant Scattering of Annihilation Photonsand Problem of the Tunguska Event. . . . . . . . . . . . . . . . . . . . . 184

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Contents xi









Chapter 1Theory of the Resonant Scatteringof Gamma Rays by Nuclei in a MagneticField

1.1 Introduction

In this chapter, we consider the problem of magnetic-field-induced perturbation ofthe angular distribution of resonantly scattered gamma rays. Originally, interest inthis problem arose in connection with the possibility of employing a magnetic fieldperturbing the angular distribution of resonantly scattered gamma rays (ADRSG) todetermine magnetic moments of nuclei in excited states that are intermediate inresonant-scattering processes. At first glance, this method is similar to the widelyused method for determining magnetic moments of excited nuclei by measuringmagnetic-field-perturbed angular correlations of two photons sequentially emittedby a nucleus [1]. Indeed, functions that describe ADRSG and angular correlationsfor a two-photon cascade (ACG) are identical in the absence of a perturbing field,provided that the spins of initial, intermediate, and final states are identical for alltransitions in the two cases and so are the parameters of multipole-mixing ratios. Itseemed natural to extend the identity of the descriptions to the same processesoccurring in a perturbing magnetic field. In particular, the results of the experimentsreported in [2–4] were considered from this point of view. However, it was firstshown in [5] that, in the particular case of a pure E2 transition between 0+ and 2+

levels, the result of the perturbation introduced by a magnetic field in ADRSGdepends substantially on the hierarchy of the natural width of the excited nuclearlevel and the characteristic width of the spectral distribution of exciting (resonantlyscattered) gamma rays. In [6], this result was generalized to the case of arbitrarymixed transitions. The final results of these two studies refer to the case where thehyperfine-interaction energy μH (here, μ is the magnetic moment of the excitednucleus and H is the strength of the magnetic field affecting the nucleus) is small inrelation to the natural width Г of the excited nuclear state. The most general results,free from constraints on the hyperfine-interaction energy, were obtained by ourgroup in [7] for the case where the magnetic field is perpendicular to the gamma-rayscattering plane. Also given there is an expression for the ADRSG function in the

© Springer International Publishing Switzerland 2015A.V. Davydov, Advances in Gamma Ray Resonant Scattering and Absorption,Springer Tracts in Modern Physics 261, DOI 10.1007/978-3-319-10524-6_1

1

case where magnetic-field directions are distributed chaotically over nuclei of thesample that scatters gamma rays [8].

Below, a derivation of expressions for ADRSG functions is given for the firsttwo cases on the basis of the method that we used in [7, 8]. First, we follow thecomputations from [6] and then go over to deducing general expressions. Our lineof reasoning is basically the same as that in [1]. Constant factors that appear inintermediate computations, but which do not affect the form of the angular distri-bution, will be discarded without mentioning this in each specific case.

1.2 ADRSG Function for the Case Where the MagneticField Is Perpendicular to the Plane of Gamma-RayScattering

We represent the time-dependent ADRSG function in the form

W q1; q2; tð Þ ¼ Rmimf

Aif q1; q2; tð Þ A�if ðq1; q2; tÞ ð1:1Þ

where q1 and q2 are the wave vectors of, respectively, the incident and the scatteredphoton and

Aif q1; q2; tð Þ ¼ Rmamb

If mf H2j jI mb� �

I mb K tð Þj jI mah i I ma H1j jIi mih i: ð1:2Þ

The quantities appearing in the summand in (1.2) include the following:I ma H1j jIi mih i, which is the matrix element of the operator H1 for the first

transition in the resonant-scattering process (the transition that corresponds tophoton absorption and in which the nucleus goes over from the state of spin Ii andmagnetic quantum number mi to the state where these quantum numbers are I andma, respectively);

I mb K tð Þj j I mah i, which is the matrix element of the nuclear transition from theexcited state of spin I and magnetic quantum number ma to the other sublevel of thisstate where the magnetic quantum number is mb. This transition occurs under theeffect of the perturbation operator Λ(t) which, in the case being considered, has theform

K tð Þ ¼ e�iKt�h : ð1:3Þ

In the reference frame where the quantization axis is aligned with the magnetic-field strength vector H, the Hamiltonian K appearing in (1.3) and representing theinteraction of the nucleus with the magnetic field is given by

K ¼ �ðlHÞ ¼ �glNHm ¼ Xm�h; ð1:4Þ

2 1 Theory of the Resonant Scattering of Gamma Rays…

where μ is the magnetic moment of the excited nucleus, g is the g-factor of thenuclear excited state, μN is the nuclear magneton, m is the quantum number of thenuclear-spin projection onto the quantization axis z, and Ω is the Larmor frequencyof nuclear-spin precession in the magnetic field.

The summand in (1.2) also involves t he matrix element If mf H2j j I mb� �

of theoperator H2 of the second transition occurring in the resonant-scattering process andcorresponding to photon emission from the nucleus, whereupon the nucleus returnsto the ground-state sublevel where the magnetic quantum number is mf. Obviously,If = Ii in the resonant-scattering process.

Let us first consider the matrix element of photon emission. Since the photon-emission process in resonant scattering does not differ from the analogous processin cascade gamma-ray emission, one may borrow the expression for the matrixelement of the final transition from the theory of angular correlations of sequentiallyemitted photons. This expression is given in [1] [Eq. (19.59)]. In our case, it has theform

If mf H2j j I mb� � ¼ R

L2 l2 M2 p2�1ð Þ�IfþL2�mb

If L2 I

mf M2 �mb

�� 0 r2 L2 l2 p2jh i

� If L2 p2k kI� �D

L�2M2l2

z ! q2ð Þð1:5Þ

Here, the factor 0 r2 L2 l2 p2jh i is the eigenfunction corresponding to theeigenvalues L2, μ2, and π2 of the operators of, respectively, the angular momentum,its projection on the quantization axis, and parity in the reference frame where thequantization axis coincides with the direction of the photon wave vector q2. Uponthe multiplication of this function by the rotation matrix DL2�

M2l2z ! q2ð Þ; it trans-

forms into its counterpart in a reference frame where the quantization axis z has anarbitrary direction. The corresponding Euler angles are arguments of the D-func-tions. The factor If L2 p2k k I� �

is the reduced matrix element of the transitionoperator. Further, M2 and μ2 are the quantum numbers of the total-angular-momentum projections onto the quantization axes z and q2, respectively;If L2 Imf M2 �mb

�� is the Wigner 3j coefficient, which is determined by the quantum

numbers appearing in it and representing the angular momenta and their projec-tions; π2 is a parity of the radiation wave function; and σ2 is the photon spin.

The expression for the matrix element corresponding to photon absorption may bederived in following way. We represent the matrix element of our interest in the form

I ma H1j jIi mih i ¼ I ma H1j jIi mih q1 r1i¼ R

L1M1p1L1 M1 p1h j q1 r1i I ma H1j jIi mi L1 M1 p1h i ð1:6Þ

where σ1 is the spin of the photon to be absorbed.

1.2 ADRSG Function for the Case… 3

We first transform the second factor in the summand in (1.6) as in [9]

I ma H1j jIi mi L1 M1 p1h i ¼ Ii mi L1 M1 p1 H1j jI mah i

then isolate the reduced matrix element Ii Lipik kIh i in it as

Ii mi L1 M1 p1 H1j jI mah i ¼ �1ð ÞIiþL1�maffiffiffiffiffiffiffiffiffiffiffiffiffi2I þ 1

p Ii L1 Imi M1 �ma

�� Ii L1 p1k kIh i

ð1:7Þ

The eigenfunction L1 M1 p1h j q1 r1i associated with a reference frame featuringan arbitrary quantization axis z will be transformed [10] as

L1 M1 p1h jq1 r1i ¼ q1r1 L1 M1 p1j i�h .

At the same time, we have

q1h r1 L1M1p1j i� ¼Xl1

0r1h jL1l1p1i�DL1M1l1

z ! q1ð Þ ð1:8Þ

From Eqs. (1.6)–(1.8), it follows that

I ma H1j jIi mih i ¼ RL1M1l1p1

�1ð Þ�IiþL1�maIi L1 I

mi M1 �ma

�� 0 r1h L1 l1 p1j i�

� Ii L1 p1k kIh iDL1M1l1

z ! q1ð Þð1:9Þ

The hypothesis of parity conservation in strong and electromagnetic interactionsdoes not contradict modern experimental data; therefore, we can retain only oneterm in the sums over π1 and π2 in expressions (1.5), (1.6) and (1.9) and henceforthavoid employing summation over the parity quantum number.

Let us perform the Fourier transform the functions Aif(q1, q2, t) in the referenceframe where the quantization axis z coincides with the direction of the magnetic-field strength vector (ω is the frequency of the photon to be absorbed):

Uif q1; q2;xð Þ�Z10

Aif q1; q2; tð Þeðix�C2�hÞ tdt ¼

Z10

Rm

If mf H2j jI m� �I m H1j jIi mih iei mXþxþiC

2�hð Þ th idt

¼ i Rm

If mf H2j jIm� �I m H1j jIi mih i

xþ mXþ iC2�h

� � :

ð1:10Þ


In this reference frame, the magnetic quantum number of the intermediate statedoes not change under the effect of the magnetic-perturbation operator; therefore,we have

ma ¼ mb ¼ m:

The quantity Г appearing in (1.10) is the natural width of the excited nuclearstate. Following [6], one can represent the correlation function in the form

W q1; q2ð Þ ¼ Rmimf

Z1�1

fi xð Þj j2 Rr1r2

Uif r1; r2ð Þ�� 2� dx ð1:11Þ

We emphasize that we consider only the angular dependence of the correlationfunction, assuming that the polarization of gamma rays is not measured and that theinitial gamma radiation is not polarized.

The function fi(ω) appearing in expression (1.11) describes the frequency dis-tribution of radiation to be absorbed. Following (1.8), we set it to

fi xð Þ ¼ C1

x� sþei�h þ i D2�h

; ð1:12Þ

where C1 is a dimensional normalization constant; below, we omit its numericalpart. This frequency distribution corresponds to the Lorentzian gamma line formwith a width Δ. In (1.12), s is the summed isomeric and Doppler shifts of thegamma line, while εi is the energy of the hyperfine interaction of scatterer nuclei inthe ground state. The photon energy determined by the frequency ω is reckonedfrom the position of the non split resonance.

Omitting, as usual, constant factors, which do not affect the form of the angulardistribution, we represent the correlation function as

W q1; q2ð Þ ¼ S1S2 Rmimf mm0

Z1�1

fi xð Þj j2

� If mf H2j jI m� �I m H1j jIi mih i If mf H2j jI m0� ��

I m0 H1j jIi mih i�xþ mXþ iC

2�h

� �xþ m0X� iC

2�h

� � dx

ð1:13Þ

The symbols S1 and S2 denote summation over unobserved gamma raypolarizations.


The integral with respect to the frequency in (1.13) has the form

J ¼Z1�1

dx

x� sþei�h

� �2þ D2

4�h2

h ixþ mXþ iC

2�h

� �xþ m0X� iC

2�h

� � ð1:14Þ

Evaluating this integral by means of residue theory, we obtain

J¼CþD�h þi m0 � mð ÞX

CþD2�h

� �2þ sþei�h þmX

� �sþei�h þm0X

� �þ i CþD2�h

� �m0 � mð ÞX

h iC�h þ i m0 � mð ÞX �

ð1:15Þ

The energy εi of the hyperfine interaction of a nucleus in the ground state with amagnetic field depends on the ground-state spin of the nucleus, Ii; on the magneticquantum number of the sublevel being considered, mi; on the magnetic-fieldstrength, H; and on the magnetic moment of the nucleus in the ground state, μi.Specifically, we have

ei ¼ �miliHIi

: ð1:16Þ

We will now consider individually the sums of products of matrix elementsassociated with the processes of gamma ray absorption and emission. We recall thatthe expression for J involves the magnetic quantum number mi, which determinesthe energy of the hyperfine interaction of a nucleus in the ground state with amagnetic field according to Eq. (1.16). Therefore, one cannot separate J from thematrix elements of the absorption operator in carrying out summation over mi.

Let us introduce the following notation

T1¼S1Xmi

JhIm H1j jIimiihIm0 H1j jIimii� ð1:17Þ

T2¼S2Xmf

hIf mf H2j jImihIf mf H2j jIm0i� ð1:18Þ

First, we transform expression (1.17) as

T1 ¼ S1P

mi;L1;L01;M1

01;l1;l

01

Jð�1Þ�IiþL1�m Ii L1 Imi M1 �m

�� 0r1h jL1l1p1i�

Ii L1p1 Ikkh iDL1M0

1l1z ! q1ð Þ �1ð Þ�IiþL01�m0 Ii L01 I

mi M 01 �m0

�� 0r01 L01l

01p1

�� Ii L01p1 Ik��

DL0�1M 0

1l01

z ! q1ð Þ

ð1:19Þ


Using Eqs. (19.11), (19.13) and (19.17) from [1], we can recast the product ofthe D-functions in (1.19) into the form

DL1M1l1

z ! q1ð Þ DL0�1M0

1l01

z ! q1ð Þ ¼Xk1

�1ð ÞM1�l1�s1�N1 2k1 þ 1ð Þ

� L1 L01 k1M1 �M0

1 �N1

�� L1 L01 k1l1 �l01 �s1

��Dk1

N1s1 z ! q1ð Þ

ð1:20Þ

The summation index k1 runs through the integers from L1 � L01�� to L1 þ L01

�� ,and the symbols N and τ stand for the sums M þM0 and lþ l0, respectively.

Expression (1.19) now takes the form

T1 ¼ S1X

mik1L1M1l1L01M

01l

01

J �1ð Þ�2IiþL1þL01�m�m0þM1�l1 2k1 þ 1ð Þ Ii L1 I

mi M1 �m

��

� Ii L01 I

mi M01 �m0

�� L1 L01 k1M1 �M0

1 �N1

�� L1 L01 k1l1 �l01 �s1

��h0r1 L1l1p1

�� 0r01

��L01l01p1� �Ii L01p1�� I� ��

Ii L1p1k kIh iDk1N1s1 z ! q1ð Þ

ð1:21Þ

Let us introduce the radiation parameters CksðL; L0Þ; defined by the relation [1]

CksðL; L0Þ ¼ SXll0

ð�1ÞL�l ffiffiffiffiffiffiffiffiffiffiffiffiffi2k þ 1

p L L0 k

l �l0 �s

�� h0r Llpj i�h0r0 L0l0pj i

ð1:22Þ

Substitution of radiation parameters in (1.21) gives

T1 ¼X

mik1N1L1M1L01M01s1

Jð�1Þ�2IiþL01�m�m0þM1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k1 þ 1

pCk1s1ðL1; L01Þ

� Ii L1 I

mi M1 �m

�� Ii L01 I

mi M01 �m0

�� L1 L01 k1M1 �M0

1 �N1

��

� hIi L1p1k kIihIi L01p1�� Ii�Dk1

N1s1 z ! q1ð Þ ð1:23Þ


Similar transformations can be performed for T2 as well. The final expressioncan be substantially simplified in this case upon summation over mf because thisquantum number is not involved in J. We have

T2 ¼ S2 Rmf L2M2l2L

02M

02l

02

�1ð Þ�2IfþL2þL02�m�m0 If L2 I

mf M2�m

�� If L02

I

mf M 02

�m0

��

� 0r2 L2l2p2jh i 0r02 L02l02p2

�� If L2p2k kI� �

If L02p2�� I� ��DL2

M2l2� z ! q2ð Þ DL02

M02l

02

z ! q2ð Þ

¼ S2P

mf ;L2;M2;l2L02;M

02;l

02

ð�1Þ�2IfþL2þL02�m�m

0þM2�l2P

K2;N2;s2

If L2I

mf M2�m

��If L

02

I

mf M02

�m0

��

� L2 L02k2

M2 �M02

�N2

�� L2 L02

k2

l2 �l02�s2

�� 2k2 þ 1ð Þ 0r2 L2l2p2jh i 0r02 L02l

02p2

�� hIf L2p2k kIihIf L02p2

�� Ii�Dk2N2s2

�

¼P �1ð Þ�2IfþL02�m�m0þM2

mf L2M2L02M02

P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k2 þ 1

pC�k2s2 L2; L02

� �k2N2s2

If L2 I

mf M2�m

�� If L02

I

mf M02

�m0

��

� L2 L02k2

M2 �M02

�N2

��hIf L2p2k kIihIf L02p2

�� Ii�Dk2N2s2

�: ð1:24Þ

The sum of the products of three 3J coefficients in (1.24) can be simplified byusing the well-known relation [1]

Xm4m5m6

ð�1ÞJ4þJ5þJ6þm4þm5þm6J1 J5 J6m1 m5 �m6

�� J4 J2 J6�m4 m2 m6

�� J4 J5 J3m4 �m5 m3

��

¼ J1 J2 J3m1 m2 m3

�� J1 J2 J3

J4 J5 J6

� ;

ð1:25Þ

whereJ1 J2 J3J4 J5 J6

� is a Wigner 6J coefficient.

Upon setting

J1 ¼ I; m1 ¼ m

J2 ¼ I; m2 ¼ �m0

J3 ¼ k2; m3 ¼ �N2

J4¼L02; m4¼�M02

J5 ¼ L2; m5 ¼ �M2

J6 ¼ If ; m6 ¼ mf ;


we arrive at

If L2 I

mf M2 �m

�� If L02 I

mf M02 �m0

�� L2 L02 k2M2 �M0

2 �N2

��

¼ �1ð ÞL2þL02þk2 I L2 Ifm �M2 �mf

�� L02 I IfM0

2 �m0 mf

�� L02 L2 k2�M0

2 M2 �N2

��

ð1:26Þ

One can readily verify that the three 3J coefficients in (1.26) correspond exactlyto the required combination of 3J coefficients in (1.25). Substituting (1.26) into(1.24) and composing the necessary phase factor by adding and subtracting thecorresponding quantities in the exponent of (–1), one obtains

T2 ¼X

L2L02k2N2s2

ð�1ÞIfþL02�mþk2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k2 þ 1

pC�k2s2ðL2; L02Þ

I I k2m �m0 �N2

��

� I I k2L02 L2 If

� hIf L2p2k kIihIf L02p2

�� Ii�Dk2N2s2

� z ! q2ð Þ ð1:27Þ

We will now make several general comments concerning expressions (1.23) and(1.27).

1. If one deals, as in our case, only with a measurement of the correlation ofradiation-propagation directions, then the summation indices k1,2 may take onlyeven values and one must set the indices τ1,2 to zero [1].

2. If interactions that are responsible for nuclear gamma transitions are invariantunder time reversal (this is compatible with all experiments performed thus far),then the reduced matrix elements must be real-valued.

3. The radiation parameters Cks L; L0ð Þ can be made to be real-valued by properlychoosing the efficiency matrix that is used to perform summation over unob-served polarizations [1]. Therefore, we suppress below the signs of complexconjugation on Cks L; L0ð Þ and on the reduced matrix elements.

If the gamma transition being considered is a mixture of electric and magnetictransitions having the same parity, then summation over L2 and L02 reduces totaking into account the following combinations of these quantum numbers:

L2 L02L2 L2L2 + 1 L2L2 L2 + 1

L2 + 1 L2 + 1


For a mixture of E2 and M1 multipoles, L2 = 1 in each combination.In the case of mixed transitions, the explicit form of the expression for Ck0 L; L0ð Þ

is given in [1]:

Ck0 L; L0ð Þ¼ �1ð ÞL�1 2Lþ 1ð Þ1=2 2L0 þ 1ð Þ1=2 2k þ 1ð Þ1=2 L L0 k1 �1 0

�� ð1:28Þ

Substituting Eq. (1.28) into (1.27) and isolating the part depending onL2 and L02, we obtain

T2 ¼Xk2N2

�1ð Þ�If�mþk2 ð2k2 þ 1Þ I I k2m �m0 �N2

��Dk2�

N20 z ! q2ð Þ � T 02; ð1:29Þ

where

T 02 ¼

XL2L02

�1ð ÞL02þL2�1 2L2 þ 1ð Þ1=2 2L02 þ 1� �1=2 L2 L02 k2

1 �1 0

��

� I I k2L02 L2 If

� If L2p2k kI� �

If L02p2�� I� �� ð1:30Þ

We now transform the product of the reduced matrix elements [1]:

If L2p2k kI� �If L02p2�� I� �� ¼ I L2p2k kIf

� � �1ð ÞIf�IþL2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2If þ 12I þ 1

r

� I L02p2�� If� �� 1ð ÞIf�IþL02

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2If þ 12I þ 1

r ð1:31Þ

Substituting Eq. (1.31) into (1.30), we arrive at

T 02 ¼

XL2L02

�1ð Þ2If�2I�12If þ 12I þ 1

2L2 þ 1ð Þ1=2 2L02 þ 1� �1=2

� L2 L02 k21 �1 0

�� I I k2

L02 L2 If

� I L2p2k kIf� �

I L02p2�� If� �� ð1:32Þ

Summation over L2 and L02 leads to the following expression for T 02

T 02 ¼ �1ð Þ2If�2I�1 2If þ 1

� �I L2p2k kIf� �2

2I þ 1ð Þ 2L2 þ 1ð Þ L2 L2 k21 �1 0

�� I I k2

L2 L2 If

� �

þ 2 2L2 þ 1ð Þ1=2 2L2 þ 3ð Þ1=2 L2 L2 þ 1 k21 �1 0

�� I I k2

L2 þ 1 L2 If

� d2

þ 2L2 þ 3ð Þ L2 þ 1 L2 þ 1 k21 �1 0

�� I I k2

L2 þ 1 L2 þ 1 If

� d22

ð1:33Þ


Here, d2 ¼ I L2þ1 p2k kIfh iI L2 p2k kIfh i is the multipole-mixing parameter in the gamma tran-

sition being considered. Expression (1.33) may be substantially simplified byintroducing in it the coefficients F defined by the relation [1]

Fk LL0II 0ð Þ ¼ �1ð ÞIþI 0�1 2Lþ 1ð Þ 2L0 þ 1ð Þ 2I 0 þ 1ð Þ 2k þ 1ð Þ½ �1=2

� L L0 k

1 �1 0

�� L L0 k

I 0 I 0 I

� ð1:34Þ

Tables of numerical values of the coefficients F are given, for example, in [1].Substituting Eq. (1.34) into (1.33) and (1.33) into (1.29), discarding constant

factors, and taking into account the fact that k2 is an even number, we obtain

T2 ¼Xk2N2

�1ð Þ�m 2k2 þ 1ð Þ1=2 I I k2m �m0 �N2

�� Fk2 L2L2If I

� �þ 2d2Fk2 L2L2 þ 1 If I

� �þ d22Fk2 L2 þ 1 L2 þ 1 If I� ��

Dk2N20

� z ! q2ð Þð1:35Þ

Unfortunately, one cannot reduce the expression for T1 to a similar compactform and therefore has to deal with the following cumbersome expression obtainedafter summation over L1 and L01

T1 ¼X

mik1N1M1M01

Jð�1Þ2Ii�m�m0þM1ð2k1 þ 1Þ ð2L1 þ 1Þ L1 L1 k11 �1 0

��

�

� Ii L1 I

mi M1 �m

�� Ii L1 I

mi M01 �m

�� L1 L1 k1M1 �M 0

1 �N1

��þ d1 2L1 þ 1ð Þ1=2 2L1 þ 3ð Þ1=2

� L1 L1 þ 1 k11 �1 0

�� Ii L1 I

mi M1 �m

��

Ii L1 þ 1 I

mi M 01 �m0

�� L1 L1 þ 1 k1M1 �M0

1 �N1

��

þ Ii L1 þ 1 I

mi M1 �m

�� Ii L1 I

mi M01 �m0

�� L1 þ 1 L1 k1M1 �M 0

1 �N1

��

þ d21 2L1 þ 3ð Þ L1 þ 1 L1 þ 1 k11 �1 0

�� Ii L1 þ 1 I

mi M1 �m

�� Ii L1 þ 1 I

mi M01 �m0

��

� L1 þ 1 L1 þ 1 k1M1 �M 0

1 �N1

��Dk1

N10 ðz ! q1Þ

ð1:36Þ

If one deals with a mixture of E2 and M1 multipoles, then it is necessary to setthe quantum numbers L2 and L1 in Eqs. (1.35) and (1.36), respectively, to 1. In thecase of resonant gamma-ray scattering, δ1 = δ2 = δ (for our choice of form for thereduced matrix elements).


The expression for the ADRSG function now assumes the form

Wðq1; q2Þ ¼X

mimm0k1N1k2N2M1M01

J �1ð Þ�miþN1 2k1 þ 1ð Þ 2k2 þ 1ð Þ1=2

� 2L1 þ 1ð ÞL1 L1 k1

1 �1 0

��

(Ii L1 I

mi M1 �m

��Ii L1 I

mi M01 �m0

��L1 L1 k1

M1 �M01 �N1

��

þ d 2L1 þ 1ð Þ1=2 2L1 þ 3ð Þ1=2L1 L1 þ 1 k1

1 �1 0

��

Ii L1 I

mi M1 �m

��Ii L1 þ 1 I

mi M01 �m0

��

"

�L1 L1 þ 1 k1

M1 �M01 �N1

��þ

Ii L1 þ 1 I

mi M1 �m

��Ii L1 I

mi M01 �m0

��L1 þ 1 L1 k1

M1 �M01 �N1

��#

þ d2 2L1 þ 3ð ÞL1 þ 1 L1 þ 1 k1

1 �1 0

��Ii L1 þ 1 I

mi M1 �m

��Ii L1 þ 1 I

mi M01 �m0

��

�L1 þ 1 L1 þ 1 k1

M1 �M01 �N1

��)

I I k2

m �m0 �N2

�� Fk2 L2L2If I

� �þ 2dFk2 L2L2 þ 1 If I� �

þ d2Fk2 L2 þ 1 L2 þ 1 If I� ��

Dk1N10 z ! q1ð ÞDk2

N20� z ! q2ð Þ

ð1:37Þ

From Eqs. (1.35) and (1.37), it follows that N1 = N2. Indeed, the 3J coefficientI I k2m �m0 �N2

�� appearing in (1.35) yields N2 ¼ m� m0. At the same time, the

3J coefficientsIi L1 Imi M1 �m

��; Ii L1 I

m M01 �m0

��; L1 L1 k1

M1 �M01 �N1

�� and the

analogous 3J coefficients appearing in the remaining terms in T1 lead to the rela-tions N1 ¼ M1 �M0

1; M01 ¼ m0 � mi; M1 ¼ m� mi: Therefore, N1 ¼ m� m0, that

is N1 ¼ N2 ¼ N.We now write the expressions for D functions appearing in (1.37) in accordance

with [1] as

Dk1N0 z ! q1ð Þ ¼ 4p

2k1 þ 1

� �1=2YN�k1 h1;u1ð Þ ð1:38Þ

Dk2N0

� z ! q2ð Þ ¼ 4p2k2 þ 1

� �1=2YNk2 h2;u2ð Þ ð1:39Þ

The angles θ1, θ2, φ1, and φ2 are shown in Fig. 1.1.We now use the following definition of the spherical harmonics YN

k h;uð Þ [1]:

YNk h;uð Þ¼VN

k PNk coshð Þ eiNu ð1:40Þ


Here

VNk ¼ �1ð ÞN 2k þ 1ð Þ k � Nj jð Þ!

4p k þ Nj jð Þ! �1=2

; ð1:41Þ

and PNk coshð Þ is an associated Legendre function of the first kind. It can be

expressed in terms of the ordinary Legendre polynomials Pk coshð Þ as [11]

PNk coshð Þ¼ 1� cos2h

� � Nj j=2 d Nj j

d coshð Þ Nj j Pk coshð Þ

Fig. 1.1 Angles θ1, φ1 and θ2, φ2 determining the directions of the wave vectors q1 and q2 of,respectively, initial and scattered photons with respect to the coordinate system associated with thequantization axis Z


Thus, one can represent the product of D functions appearing in (1.37) in theform

Dk1N0 z ! q1ð Þ Dk2

N0� z ! q2ð Þ ¼ 4p

2k1 þ 1

� �1=2 4p2k2 þ 1

� �1=2�1ð Þ2N 2k1 þ 1ð Þ k1 � Nj jð Þ!

4p k1 þ Nj jð Þ! �1=2

� 2k2 þ 1ð Þ k2 � Nj jð Þ!4p k2 þ Nj jð Þ!

�1=2PNk1 cosh1ð ÞPN

k2 cosh2ð Þ e�iNu1 eiNu2

¼ k1 � Nj jð Þ! k2 � Nj jð Þ!k1 þ Nj jð Þ! k2 þ Nj jð Þ!

�1=2PNk1 cosh1ð ÞPN

k2 cosh2ð Þ eiN u2�u1ð Þ

ð1:42Þ

This is equivalent to

Dk1N0 z ! q1ð Þ Dk2

N0� z ! q2ð Þ ¼ k1 � Nj jð Þ! k2 � Nj jð Þ!

k1 þ Nj jð Þ! k2 þ Nj jð Þ! �1=2� PN

k1 cos h1ð ÞPNk2 cos h2ð Þ cos N /2 � /1ð Þ½ � þ i sin N /2 � /1ð Þ½ �f g

ð1:43Þ

Let us separate the real and imaginary parts in expression (1.15) for J. SettingJ = A + iB, we arrive at

A¼ CþDð Þ�h

C�h

Cþ D2�h

� �2

þC�h

sþ ei�h

þ mX� � sþ ei

�hþ m0X

� �� N2X2 Cþ D

2�h

� ��"(

þ N2X2 Cþ Dð Þ 3Cþ Dð Þ4�h2

þ sþ ei�h

þ mX� � sþ ei

�hþ m0X

� �io Z�1 ð1:44Þ

B¼� NXC�h

Cþ D2�h

� �2

þC�h

sþ ei�h

þ mX� � sþ ei

�hþ m0X

� �� N2X2 Cþ D

2�h

� �(

� Cþ D�h

� �Cþ Dð Þ 3Cþ Dð Þ

4�h2þ sþ ei

�hþ mX

� � sþ ei�h

þ m0X� �io

Z�1

ð1:45Þ

where

Z¼ C�h

Cþ D2�h

� �2

þC�h

sþ ei�h

þ mX� � sþ ei

�hþ m0X

� �� N2X2 Cþ D

2�h

� �" #2

þ N2X2 Cþ Dð Þ 3Cþ Dð Þ4�h2

þ sþ ei�h

þ mX� � sþ ei

�hþ m0X

� � �2ð1:46Þ

The dimensionless character of the functions A and B is ensured by the squareddimensional unity appearing in them as a factor and originating from the constantC1 in Eq. (1.12).


We now represent the function W(q1,q2) in the form

W q1; q2ð Þ ¼X

mimm0k1k2NM1M01

P Aþ iBð Þ cos N u2 � u1ð Þ½ � þ isin N u2 � u1ð Þ½ �f g

ð1:47Þ

All factors that are not involved in J or in the bracketed expression are includedin P. The imaginary part of the angular-distribution function must be zero; that is,the following condition must hold

Im W q1; q2ð Þ½ � ¼X

mimm0k1k2NM1M01

P A sin N u2 � u1

� � �þ B cos N u2 � u1ð Þ½ �� ¼ 0

ð1:48Þ

We will prove that this is indeed so. From an analysis of Eqs. (1.44)–(1.46), onecan see that A is an even function of N, while B is an odd function of this quantity.Therefore, the braced expression on the right-hand side of (1.48) is an odd functionof N. There are terms of two types in the factor P. First, there are symmetric (withrespect to L1) groups of factors of the form

Ii L1 Imi M1 �m

�� Ii L1 Imi M0

1 �m0

�� L1 L1 k1M1 �M0

1 �N

�� I I k2m �m0 �N

�� ð1:49Þ

Among a great number of different combinations of these factors correspondingto positive and negative values of N, there are inevitably pairs of combinationsdiffering by the interchange of numerical values of m and m0 and of M and M0.Thus, a specific combination in (1.49) can be associated with the following com-bination of 3 J coefficients:

Ii L1 Imi M0

1 �m0

�� Ii L1 Imi M1 �m

�� L1 L1 k1M0

1 �M1 N

�� I I k2m0 �m N

�� ð1:50Þ

Obviously, expression (1.50) coincides with expression (1.49) because the lasttwo factors in these combinations differ by the phase factor�1ð Þ4L1þ2k1þ4Iþ2k2¼ þ1:Second, P contains groups of factors that are not symmetric with respect to L1,

but which appear there in the form of symmetric pairs:

Ii L1 I

mi M1 �m

��

Ii L1 þ 1 I

mi M01 �m0

�� L1 L1 þ 1 k1M1 �M0

1 �N

��

þ Ii L1 þ 1 I

mi M1 �m

�� Ii L1 I

mi M01 �m0

�� L1 þ 1 L1 k1M1 �M0

1 �N

��

I I k2m �m0 �N

��

ð1:51Þ


It is easy to prove that the replacement of M1 by M01 and of m by m0, and vice

versa, and of N by –N does not change the value of (1.51). Thus, the factor P has astructure such that summation over N, m, m0, M1, and M0

1 in (1.48) does indeed leadto the vanishing of (1.48) owing to the fact that Asin N u2 � u1ð Þ½ � þBcos N u2 � u1ð Þ½ � is an odd function of N.

The real part of W(q1, q2) has the form

Re W q1; q2ð Þ½ � ¼ W q1; q2ð Þ¼ R

mimm0k1k2NM1M01

P A cos N /2 � /1ð Þ½ � � B sin N /2 � /1ð Þ½ �f g

ð1:52Þ

If the quantization axis z coinciding in direction with the magnetic-field strengthvector H is orthogonal to the gamma-ray scattering plane, then cosθ1 = cosθ2 = 0,with the result that, in (1.42), there remains only the dependence on the anglebetween the directions of the photon wave vectors q1 and q2; in the case beingconsidered, this angle is equal to the difference of the angles φ2 and φ1.

Including the immaterial factor �1ð ÞIi , we then obtain

W q1; q2ð Þ ¼X

mimm0k1k2NM1M01

�1ð ÞIi�miþN 2k1 þ 1ð Þ 2k2 þ 1ð Þ1=2 2L1 þ 1ð Þf

� L1 L1 k11 �1 0

�� Ii L1 I

mi M1 �m

�� Ii L1 I

mi M01 �m0

�� L1 L1 k1M1 �M 0

1 �N

��þ d 2L1 þ 1ð Þ1=2

� 2L1 þ 3ð Þ1=2 L1 L1 þ 1 k11 �1 0

�� Ii L1 I

mi M1 �m

��

Ii L1 þ 1 I

mi M01 �m0

�� L1 L1 þ 1 k1M1 �M0

1 �N

��

þ Ii L1 þ 1 I

mi M1 �m

�� Ii L1 I

mi M01 �m0

�� L1 þ 1 L1 k1M1 �M0

1 �N

��þ d2 2L1 þ 3ð Þ L1 þ 1 L1 þ 1 k1

1 �1 0

��

� Ii L1 þ 1 I

mi M1 �m

�� Ii L1 þ 1 I

mi M01 �m0

�� L1 þ 1 L1 þ 1 k1M1 �M0

1 �N

��

I I k2m �m0 �N

��

� Fk2 L2L2If I� �þ 2dFk2 L2L2 þ 1If I

� �þ d2Fk2 L2 þ 1L2 þ 1If I� � �

� k1 � Nj jð Þ ! k2 � Nj jð Þ !k1 þ Nj jð Þ ! k2 þ Nj jð Þ !

�1=2PNk1 0ð ÞPN

k2 0ð Þ

� Acos N u2 � u1ð Þ½ � � Bsin N u2 � u1ð Þ½ �f gð1:53Þ

The associated Legendre functions have the explicit form [11]

PNk ðxÞ¼

1� x2ð Þ Nj j=22kk!

dkþ Nj j

dxkþ Nj j x2 � 1� �k

: ð1:54Þ

If we are dealing with a transition that is a mixture of E2 and M1 multipoles,then k1 and k2 may take the values of 0, 2, and 4 [1]. The expressions for PN

k xð Þ inall possible cases corresponding to these values of k1 and k2 are presented inTable 1.1. One can see that, for all odd N, the values of PN

k 0ð Þ are zero. Thus, wesee that, in the case where the quantization axis z is perpendicular to the plane


spanned by the vectors q1 and q2, only terms corresponding to even values ofN survive in (1.53) because of the properties of associated Legendre functions.Therefore, one may omit the phase factor (–1)N in (1.53).

1.3 Angular Distribution in the Case Where Magnetic-FieldDirections Are Distributed Chaotically Overthe Scatterer Volume

We now proceed to derive a formula that would describe W(q1, q2) for the casewhere the magnetic field has a random orientation at different points of the scattererwith respect to the gamma-ray scattering plane and a strength of the same mag-nitude. This case arises if one employs, for a scatterer, a multidomain ferromagneticsample whose magnetization does not have a specific direction. If the quantizationaxis is aligned, as before, with the direction perpendicular to the scattering plane,then the magnetic field is oriented at random with respect to this axis. In this case,the magnetic-interaction operator changes, in general, the magnetic quantumnumber of the intermediate nuclear state. The matrix element of this operator cannow be written in the form [1]

Table 1.1 Associated legendre polynomials as a function of quantum numbers k and N

k N PNk xð Þ PN

k 0ð Þ0 0 P0

0 xð Þ ¼ 1 1

2 2 P22 xð Þ ¼ 3ð1� x2Þ 3

1 P12 xð Þ ¼ 3x 1� x2ð Þ1=2 0

0 P02 xð Þ ¼ 1=2ð Þ 3x2 � 1ð Þ –1/2

–1 P�12 ¼ 3x 1� x2ð Þ1=2 0

–2 P�22 xð Þ ¼ 3 1� x2ð Þ 3

4 4 P44 xð Þ ¼ 105 1 - x2ð Þ2 105

3 P34 xð Þ ¼ 105x 1� x2ð Þ3=2 0

2 P24 xð Þ ¼ 1=2ð Þ 1� x2ð Þ 105x2 � 15ð Þ –15/2

1 P41 xð Þ ¼ 1=2ð Þ 1� x2ð Þ1=2 35x2 � 15ð Þx 0

0 P40 ¼ 1=8ð Þ 35x4 � 3x2 þ 3ð Þ 3/8

–1 P4�1 xð Þ ¼ 1=8ð Þ 1� x2ð Þ�1=2 7x4 � 10x2 þ 3ð Þx 0

–2 P�24 xð Þ ¼ 1=2ð Þ 1� x2ð Þ 105x2 � 15ð Þ –15/2

–3 P�34 xð Þ ¼ 105x 1� x2ð Þ3=2 0

–4 P�44 xð Þ ¼ 105x 1� x2ð Þ2 105


hImb e�iKt�h

�� Imai¼Xn

DInmb

� a; b; 0ð Þ e�iEnt�h DI

nmaa; b; 0ð Þ ð1:55Þ

The Euler angles α and β are shown in Fig. 1.2; En¼Xn�h.The matrix element Uif (q1, q2, ω) has the form

Uif ðq1; q2;xÞ ¼Z10

Aif q1; q2; tð Þe ix�C2�hð Þt dt

¼i Rmamb

If mf H2j jI mb� �

I ma H1j jIimih iDI�nmb

a; b; 0ð ÞDInma

a; b; 0ð Þxþ nXþ iC

2�h

ð1:56Þ

The expression for the angular distribution can be written as

W q1; q2ð Þ ¼ S1S2X

mimf mambm0am

0bnn

0JhIf mf H2j jImbihIf mf H2j jIm0

bi�

� hIma H1j jIimiihI m0a H1j jIimii�DI

nmb

�DInma

DIn0m0

bDI

n0m0a

�ð1:57Þ

Here, J is given by formula (1.15) where m is replaced by n, while m′ is replacedby n′. The sum of the products hIf mf H2j jImbihIf mf H2j jIm0

bi� of the matrix elementscan be evaluated by means of the procedure identical to that which was applied to

Fig. 1.2 Euler angles α and βdetermining the direction ofthe magnetic-field strengthvector H with respect to thereference frame associatedwith the quantization axis


expression (1.18). As a result, we have [upon the inclusion of the immaterial factorof �1ð ÞIf ]S2

XIf mf H2j jI mb� �

If mf H2j jI m0b

� ��mf

¼X

�1ð Þ�mb�2IþIf 2k2 þ 1ð Þ1=2k2N2

I I k2mb �m0

b �N2

��

� Fk2 L2L2If I� �þ 2dFk2 L2L2 þ 1If I

� �þ d2Fk2 L2 þ 1L2 þ 1If I� � �

Dk2N20

� z ! q2ð Þ

ð1:58ÞFor S1

Pmi

JhIma H1j jIimiihIm0a H1j jIimii�, one can write an expression similar to

(1.36), but it differs from (1.36) by the replacement of m by ma and the replacementof m0 by m0

a everywhere, with the exception of J. In J, m and m0 are replaced byn and n0, respectively.

The expression for the correlation function now assumes the form

Wðq1; q2Þ ¼P

mi;ma;m0a;mb;m0

b;

M1;M01;k1;N1;k2;N2;n;n0

�1ð Þ�ma�m0aþmbþM1�If J 2k1 þ 1ð Þ 2k2 þ 1ð Þ1=2

� 2L1 þ 1ð ÞIi L1

I

mi M1 �ma

��

8<:

Ii L1I

mi M01 �m0

a

��L1 L1

k1

1 �1 0

��

�L1 L1

k1

M1 �M01 �N1

��þ d 2L1 þ 3ð Þ1=2 2L1 þ 1ð Þ1=2

L1 L1 þ 1 k1

1 �1 0

��

�Ii L1 þ 1 I

mi M1 �ma

��

24 Ii L1

I

mi M01 �m0

a

��L1 þ 1 L1

k1

M1 �M01 �N1

��

þIi L1

I

mi M1 �ma

��Ii L1 þ 1 I

mi M01 �m0

a

��L1 L1 þ 1 k1

M1 �M01 �N1

35

þd2 2L1 þ 3ð ÞL1 þ 1 L1 þ 1 k1

1 �1 0

��Ii L1 þ 1 I

mi M1 �ma

��Ii L1 þ 1 I

mi M01 �m0

a

��

�L1 þ 1 L1 þ 1 k1

M1 �M01 �N1

��9=;Dk1

N10 z ! q1ð ÞI I k2

m0b �mb N2

�� Fk2 L2L2If I

� �

þ2k2 L2L2 þ 1 If I� �þ d2Fk2 L2 þ 1L2 þ 1 If I

� ��Dk2

N20�ðz ! q2ÞDI

nmaa; b; 0ð ÞDI

nmb

� a;b; 0ð ÞDIn0m0

a

� a;b; 0ð ÞDIn0m0

ba; b; 0ð Þ ð1:59Þ

Here, we have taken into account the fact that �1ð ÞIf�mb¼ �1ð Þ�Ifþmb .

1.3 Angular Distribution in the Case… 19

We will now transform the product of the last four D functions. First, we isolatefactors containing mb and m0

b and sum them over these quantum numbers, fol-lowing [1]. This yields

Xmbm0

b

�1ð ÞmbI I k2m0

b �mb N2

��DI

nmb

�DIn0m0

b¼ �1ð Þn

Xmbm0

b

I I k2m0

b �mb N2

��

� DI�n�mb

DIn0m0

b¼ �1ð Þn I I k2

n0 �n p2

��Dk2

p2N2

�

ð1:60Þ

We then have

DInma

DI�n0m0

a¼ �1ð Þn0�m0

aDInma

DI�n0�m0

a¼ �1ð Þn0�m0

a�p1�r1

�Xv

2vþ 1ð Þ I I v

n �n0 p1

�� I I v

ma �m0a �r1

��Dv

p1r1

ð1:61Þ

From the last 3J coefficient, one can see that r1 ¼ ma � m0a: At the same time,

the relation N1 ¼ ma � m0a follows from the 3J coefficients appearing in the term

with δ2 in the braced expression on the right-hand side of (1.59). This means thatr1 = N1. A further transformation of (1.61) gives

DInma

DI�n0m0

a¼ �1ð Þn�ma

Xv

2vþ 1ð Þ I I vn0 �n p1

�� I I vm0

a �ma N1

��Dv

p1N1

ð1:62Þ

In order to take into account a chaotic character of the orientation of the mag-netic-field strength vector with respect to the quantization axis z, it is necessary toaverage Eq. (1.59) over the Euler angles α and β, which are the arguments of thelast four D functions in (1.59). This averaging reduces to integrating the productDv

p1N1Dk2

p2N2

�: with respect to the Euler angles.Owing to the orthogonality of the D functions, we have

14p

Z2p0

Zp0

Dvp1N1

Dk2p2N2

�sinbdadb¼ 12 2k2 þ 1ð Þ dvk2dp1p2dN1N2 ð1:63Þ


We now introduce the following notation: p1 = p2 = p, N1 = N2 = N, and χ = k2.In (1.59), we isolate factors appearing in (1.60) and (1.62) and also the factorsJ,Dk1

N10 h1;u1ð Þ/Dk2N20

� h2;u2ð Þ and transform them taking in account (1.63) and(1.43):

We have

J �1ð ÞnI I k2n0 �n p

��2 I I k2m0

a �ma N

�� 1ð Þ n�maD

k1N0 h1;/1ð ÞDk2 �

N0 h2;/2ð Þ

¼ ðAþ iBÞ �1ð ÞmaI I k2n0 �n p

��2 I I k2m0

a �ma N

�� k1 � Nj jð Þ! k2 � Nj jð Þ!

k1 þ Nj jð Þ! k2 þ Nj jð Þ! �1=2

� PNk1 0ð ÞPN

k2 0ð Þ cos N u2 � u1ð Þ½ � � i sin N u2 � u1ð Þ½ �f g

¼ �1ð ÞmaI I k2n0 �n p

��2 I I k2m0

a �ma N

�� k1 � Nj jð Þ! k2 � Nj jð Þ!

k1 þ Nj jð Þ! k2 þ Nj jð Þ! �1=2

PNk1 0ð ÞPN

k2 0ð Þ

� A cos N u2 � u1ð Þ½ �f g � B sin N u2 � u1ð Þ½ � þ i B cos N u2 � u1ð Þ½ �f g

þ A sin N u2 � u1ð Þ½ �ð1:64Þ

Here, we have used the circumstance that (n – ma) is an integer. As a result,�1ð Þn�ma¼ �1ð Þma�n: Neither A nor B now depends on N. However, they dependon n and n′; B is an odd function of n� n0ð Þ, while A is an even function of thisdifference. In performing summation over n and n′, one can single out pairs of termsthat differ by the interchange of n and n′. Under this replacement, the square of the

3J coefficientI I k2n0 �n p

�� does not change; A does not change either, but

B changes sign. As a result, terms of the following form are canceled upon sum-mation over n and n′:

Upon summation over N, the second terms appearing in the imaginary part of(1.64),—Asin N u2 � u1ð Þ½ �, also vanish for the same reasons as those for which thewhole imaginary part of expression (1.47) disappeared. It follows that, after sum-mation over n, n′, and N, the contribution of (1.64) to (1.59) is determined by onlyone term in the braced expression—namely, by the product 6. Therefore, theangular distribution function in (1.59) can now be represented in the form

1.3 Angular Distribution in the Case… 21

W q1; q2ð Þ ¼X

mimam0aM1M0

1k1k2Npnn0�1ð Þ�m0

aþM1�If 2k1 þ 1ð Þ 2k2 þ 1ð Þ1=2

� 2L1 þ 1ð ÞIi L1 I

mi M1 �ma

��

(L1 L1 k1

1 �1 0

��Ii L1 I

mi M01 �m0

a

��L1 L1 k1

M1 �M01 �N

��

þ d 2L1 þ 3ð Þ1=2 2L1 þ 1ð Þ1=2L1 L1 þ 1 k1

1 �1 0

��

Ii L1 þ 1 I

mi M1 �ma

��Ii L1 I

mi M01 �m0

a

��

"

�L1 þ 1 L1 k1

M1 �M 01 �N

��þ

Ii L1 I

mi M1 �ma

��Ii L1 þ 1 I

mi M01 �m0

a

��L1 L1 þ 1 k1

M1 �M 01 �N

��#

þ d2 2L1 þ 3ð ÞL1 þ 1 L1 þ 1 k1

1 �1 0

��Ii L1 þ 1 I

mi M1 �ma

��Ii L1 þ 1 I

mi M01 �m0

a

��

�L1 þ 1 L1 þ 1 k1

M1 �M 01 �N

��)

� Fk2 L2L2If I� �þ 2dFk2 L2L2 þ 1 If I

� �þ d2Fk2 L2 þ 1 L2 þ 1 If I� ��

� k1 � Nj jð Þ! k2 � Nj jð Þ!k1 þ Nj jð Þ! k2 þ Nj jð Þ!

�1=2PNk1 0ð ÞPN

k2 0ð ÞI I k2

n0 �n p

��2 I I k2

m0a �ma N

��

� Acos N u2 � u1ð Þ½ �ð1:65Þ

If one considers a mixture of E2 and M1 multipoles, then L1 = L2 = 1 in thisexpression.

It should be borne in mind that Eqs. (1.53) and (1.65) are strictly speakingapplicable to the cases of infinitely thin scatterers, but, in practice, they can be usedif gamma-ray resonant absorption in the substance of a scatterer is weak in relationto total absorption. Otherwise, the gamma-line form in the case of Mössbauerresonant scattering changes as gamma rays penetrate into the interior of the scat-terer: the gamma line broadens and the ratio Δ/Г increases. Under such conditions,expressions (1.53) and (1.65) must be represented in the form of integrals that aretaken over the scatterer thickness and in which the change in the gamma spectrumas gamma rays penetrate into the interior of the scatterer must be taken into account.

The angular distribution of scattered gamma rays that is described by expression(1.65) is symmetric with respect to an angle of u2�u1 ¼ p=2. The application ofmagnetic fields having chaotic directions does not lead to the rotation of thegamma-ray angular distribution but only weakens its anisotropy.

1.4 Some Particular Cases

(a) Absence of magnetic fields.

In this case, both Ω and εi in (1.53) are equal to zero; as a result, B vanishes,while A reduces to a constant, which does not affect the form of the angulardistribution. It now proves to be possible to sum factors associated with the firsttransition over mi, M1, and M0

1 by means of the procedure applied above in


transforming T2. In order to simplify the procedure, we go back and, before per-forming summation over L1 and L01, recast T1 into the form

T1 ¼X

k1N1L1L01

�1ð Þ�IiþL01�m 2k1 þ 1ð Þ1=2Ck10 L1; L01

� � I I k1m �m0 �N1

�� I I k1

L01 L1 Ii

�

� hIi L1p1k kIihIi L01p1�� IiDk1

N10 z ! q1ð Þð1:66Þ

Using Eq. (1.28), performing summation over L1 and L01, and going over to theF coefficients, we obtain for T1 an expression similar to (1.35); that is,

T1 ¼Xk1N1

�1ð ÞI�m 2k1 þ 1ð Þ1=2 I I k1m �m0 �N1

�� Fk1 L1L1IiIð Þ þ 2dFk1 L1 þ 1 L1IiIð Þf

þ d2Fk1 L1 þ 1L1 þ 1 IiIð ÞgDk1N10 z ! q1ð Þ

ð1:67Þ

The angular distribution now has the form

Wðq1; q2Þ¼ P

mm0T1T2¼

Pk1k2N1N2

mm0

�1ð Þ�2m 2k1 þ 1ð Þ1=2 2k2 þ 1ð Þ1=2 ð1:68Þ

From the 3j coefficients appearing in Eq. (1.68), it follows that N1 = N2 = N. The

expression 2k1 þ 1ð Þ1=2 appears as a factor in each term in the first brackets [see

Eq. (1.34)]. We represent these brackets as the product 2k1 þ 1ð Þ1=2U k1; L1;ðL01; Ii; IÞ. After that, one can combine inside of (1.68) the factors

2k1 þ 1ð Þ I I k1m �m0 �N1

�� I I k2m �m0 �N2

�� and thereupon perform summation

over m and m/for them, omitting the phase factor (–1)−2m because 2 m is an integernumber that is either even for all values of m or odd for all terms. Therefore, thephase factor has the same value for all values of m in the set being considered.Because of the orthogonality of the 3j coefficients, we have (see [1])

Xmm0

2k1 þ 1ð Þ I I k1m �m0 �N1

�� I I k2m �m0 �N2

��¼dk1k2dN1N2 ð1:69Þ

We then arrive at

W q1; q2ð Þ ¼XkN

2k þ 1ð Þ1=2U k; L1; L01; Ii; I

� �Fk L2L2If I� �þ

2dFk L2L2 þ 1 If I� �

þ d2Fk L2 þ 1 L2 þ 1 If I� ��

DkN0 z ! q1ð Þ Dk

N0� z ! q2ð Þ

ð1:70Þ

1.4 Some Particular Cases 23

Combining 2k þ 1ð Þ1=2 with the function U k; L1; L01; Ii; I� �

and considering thatIi = If and L1 = L2, we obtain

W q1; q2ð Þ ¼XkN

fFkðLLIiIÞ þ 2dFkðLLþ 1IiIÞ þ d2FkðLþ 1Lþ 1IiIÞg2

� DkN0 z ! q1ð Þ Dk

N0� z ! q2ð Þ

ð1:71Þ

We now transform the product of D functions and sum it over N (see [1]). Wehave

DkN0

� z ! q2ð Þ ¼ Dk0N q2 ! zð ÞX

N

DkN0 z ! q1ð Þ Dk

0N q2 ! zð Þ ¼ Dk00 q2 ! q1ð Þ ¼ Pk coshð Þ ð1:72Þ

where Pk(cosθ) is a Legendre polynomial with h ¼ u2�u1 if the magnetic-fieldstrength vectorH is parallel to the quantization axis z and is perpendicular to q1 and q2.

Introducing the notation

Fk LLIiIð Þ þ 2dFk LLþ 1 IiIð Þ þ d2Fk Lþ 1 Lþ 1 IiIð Þ � 2 ¼ Akk; ð1:73Þ

we obtain a well-known expression that represents the unperturbed ADRSGfunction and which coincides with the unperturbed angular-correlation function fortwo sequentially emitted photons:

W hð Þ ¼Xk

AkkPk coshð Þ ð1:74Þ

(b) Nuclei of a scatterer in a weak magnetic field perpendicular to the gamma-rayscattering plane.

We consider a magnetic field as a weak one if X�h � C and ei � C. SettingA ¼ a cos g and B ¼ a sin g, where α is a constant, one can recast the expressionA cosNh � B sinNh h ¼ /2 � /1ð Þ appearing in (1.53) into the form

A cosNh� B sinNh ¼ a cos dþ Nhð Þ; ð1:75Þ

In the case of a weak field, we have

tgd � d ffi NX�hCþ Dð Þ CþDð Þ 3CþDð Þ

4 þ s2h i

� C CþDð Þ24 � Cs2

n oC CþDð Þ2

4 þ Cs2h i

Cþ Dð Þð1:76Þ


Expression (1.53) can now be represented in the form

W q1; q2ð Þ ¼X

mimm0k1��k2N M1 M0

1

P acos Nhþ dð Þ ð1:77Þ

Here, P includes, as before, all factors that do not appear in J. Omitting theunimportant factor α, we obtain

Wðq1; q2Þ ¼X

mimm0k1k2N M1 M0

1

Pcos N hþ Dhð Þ½ �; ð1:78Þ

where

Dh ¼ X�hCþ Dð Þ CþDð Þ 3CþDð Þ

4 þ s2h i


n oC CþDð Þ2

4 þ Cs2h i

Cþ Dð Þð1:79Þ

If there is no magnetic field, then Ω = 0, with the result that expression (1.78)reduces to

W q1; q2ð Þ ¼X

mimm0k1k2N M1 M0

1

P cos Nhð Þ ð1:80Þ

A comparison of expressions (1.80) and (1.78) shows that the application of aweak magnetic field leads to the rotation of the ADRSG rosette through an angle Δθthat is determined by expression (1.79). The direction of this rotation is determinedby the sign of the magnetic moment of an excited nucleus and by the direction ofthe magnetic field, which is perpendicular to the scattering plane. Experimentsdevoted to measuring ADRSG perturbed by a magnetic field do not measure, foreach nucleus emitting a photon, the time within which this nucleus remains in theexcited state. If the measurement of the gamma-ray intensity at a given scatteringangle spans a time period that is substantially longer than the mean lifetime ofnuclei in an excited state (in fact, this is always so), then the final picture of theangular distribution turns out to be nearly identical to that in the case of an infinitelylong measurement. Actually, this picture is the result of a physical averaging ofrosette rotation angles for individual photon-emission events over the lifetimesof corresponding nuclei in excited states. Therefore, the measured angle of rotationof the angular distribution must be equal to the product of the Larmor frequency


and the average lifetime of the nucleus in the excited state (tav) under conditions ofa given experiment. From Eq. (1.79), it follows that this time is given by

tav ¼ �hCþ Dð Þ CþDð Þ 3CþDð Þ

4 þ s2h i


n oC CþDð Þ2

4 þ Cs2h i

Cþ Dð Þð1:81Þ

This formula shows that the average lifetime of nuclei in the excited statepopulated upon gamma-ray resonant scattering depends not only on the naturalwidth Г of the level to be excited but also on the characteristic width Δ of thespectrum of exciting gamma rays and on the shift s of the source emission line withrespect to the scatterer absorption line. In other words, this time is determined bythe conditions of nuclear excitation in a given experiment. Let the shift s in (1.81)be equal to zero. We then have

tav ¼ ð2Cþ DÞðCþ DÞ

�hC

¼ 2Cþ Dð ÞCþ Dð Þ s ð1:82Þ

where τ is the mean lifetime of a nucleus in an excited state for an exponential law

of decay of the excited state: e�t=s ¼ e�Ct=�h. This law is valid if the energyspectrum of emitted photons that was measured over an infinitely long time intervalafter the population of the excited state is a Lorentzian line of width Г.

If D C (excitation by radiation of wide spectrum), then it follows from (1.79)that

tav ¼ s ð1:83Þ

At Δ = Γ, which corresponds to Mössbauer gamma-ray resonant absorption, wehave

tav ¼ 1; 5s: ð1:84Þ

If, at last, D C (excitation by a quasimonochromatic line), then

tav ¼ 2s: ð1:85Þ

Theoretical data on the dependence of the angle of rotation of the ADRSGrosette on the ratio of Г to Δ were first obtained by Goebel and McVoy [5], whoconsidered a particular case of gamma resonant excitation of a 2+ nuclear level via apure E2 transition—more specifically, the 0+→2+ transition. In particular, theyobtained the following expression for the angle of ADRSG rotation in the case ofzero shift between the emission and absorption lines: Dh ¼ Xs 2CþD

CþD . Moreover,those authors considered the question of whether the effect of the ratio Δ/Г on theangular distribution depends on the mode of gamma-ray emission: with a nuclear


recoil or without it. They arrived at the conclusion that the difference between thetwo modes is insignificant. It is noteworthy that one could obtain this answer apriori, knowing that the recoil of the emitting nucleus arises after the lapse of thetime within which the nucleus being considered remains in the excited state. In [6],the conclusions drawn in [5] were extended to nuclear transitions of arbitrarymultipolarity, including mixed transitions. However, the statement that the Δ/Γ ands dependence of the mean lifetime of the nucleus in the excited state plays adecisive role was not presented explicitly either in first or in the second of thosestudies. The most general consideration of this problem was given by our group in[7], and this consideration was used as a basis for deriving the above expressionsfor the ADRSG functions.

Let us now address the question of how the shift s affects the value of tav. In thecase where Δ ≫ Γ—that is, in the case of the excitation of nuclei by gammaradiation of very wide spectrum—the shift s has no physical meaning. In the casewhere Δ = Γ, Eq. (1.78) leads to the expression

tav ¼ 3þ s=Cð Þ2

2 1þ s=Cð Þ2h i s; ð1:86Þ

For Δ ≪ Γ, we have

tav ¼ 2C2=4

s2 þ C2=4s ð1:87Þ

One can see that, in the case of the Mössbauer excitation of nuclei [formula(1.86)], their average lifetime changes from 1.5τ to 0.5τ as the shift s changes from0 to infinity. In the second case (excitation by gamma rays of very narrow spectrumin relation to Г), the average lifetime changes from 2τ to 0 over the same interval ofthe shift s as in the first case. The plots representing the s dependence of tav for thesetwo cases are given in Fig. 1.3.

(c) Nuclei of a scatterer in weak magnetic fields having the same strength anddirections chaotically distributed over the volume of a scatterer.

In this case, we must address expression (1.65). If we set ei ¼ s ¼ 0 for thesake of simplicity, then it is possible to perform summation over mi, M1, M1′ ma,and ma′ in (1.65), whereupon we arrive at (in the function A given by expression(1.44), m goes over to n, m′ goes over to n′, and N goes over to n − n′)

W q1; q2ð Þ ¼XkNpnn0

2k þ 1ð Þ Fk LLIiIð Þ þ 2dFk LLþ 1 IiIð Þ þ d2Fk Lþ 1 Lþ 1 IiIð Þ � 2

� k � Nj jð Þ !K þ Nj jð Þ ! PN

k 0ð Þ� �2 I I k

n0 �n p

�� 2Acos N u2 � u1ð Þ½ �

ð1:88Þ


After performing respective computations, one can prove that the transition fromthe case of Δ ≫ Γ to the case of Δ = Γ leads to an increase in the perturbation of theangular distribution, and this manifests itself in a decrease in its anisotropy.

It is interesting to compare expression (1.87) with the result of solving the classicalproblem of the transformation of an electromagnetic oscillation by a resonance filter.The present author was acquainted with this solution by the late Dr. V.N. Andreev.Let A(ω) be the frequency-dependent oscillation amplitude at the input of the filter,whose frequency characteristic is B xð Þeiu xð Þ, where B(ω) and φ(ω) are the amplitudecharacteristic and the phase characteristic, respectively. The time Fourier transformof the frequency-spectrum amplitude at the filter output has the form

F tð Þ �Z1�1

A xð ÞB xð Þeiu xð Þeixtdx:

If the frequency distribution A xð Þ is substantially narrower than B xð Þ, then, inthe frequency interval where A xð Þ differs significantly from zero, the function B xð Þcan be treated as a constant and can therefore be factored outside the integral sign.We then have

F tð Þ � BZ1�1

A xð Þeix tþu xð Þxð Þdx:

The phase characteristic of the resonance filter has the form

u xð Þ ¼ �arctgc=2

x0 � x

Fig. 1.3 Lifetime of anucleus in an excited state asa function of the shift S of theexciting


where x0 is the resonance frequency and γ is the resonance width. Suppose thatA xð Þ � 1

x�x1þiD2with D � c.

We now expand u xð Þ in a Taylor series in the vicinity of x ¼ x1. Owing to thesmallness of the spectrum width at the filter input, we can restrict ourselves to thefirst two terms:

u xð Þ ¼ u x1ð Þ þ dudx x¼x1j x� x1ð Þ þ � � � ¼ C �

c2x

x1 � x0ð Þ2þ c2

4

:

Here, the constant C is

C ¼ u x1ð Þ þcx12


4

:

We then have

F tð Þ � BeiCZ1�1

A xð Þeix t�

c2

x1�x0ð Þ2þc24

�dx:

The time Fourier transform of the spectrum amplitude at the filter input is givenby

f tð Þ �Z1�1

A xð Þeixtdx

Thus, the role of the filter in the time transformation of the primary narrowspectral line reduces to delaying the signal by the time

Dt ¼c2


4

:

Going over from frequencies to energies (by multiplying the correspondingquantities by �h), one obtains

Dt ¼C2

2

E1 � E0ð Þ2þ C2

4

� �hC; where C ¼ c�h:

This formula coincides with that in (1.87). Therefore, the quasimonochromaticoscillation time delay created by a classical resonance filter is similar to the resultthat a quantum-mechanical consideration gives for the mean lifetime of a nucleus inan excited state after the resonant absorption of gamma rays with a narrow spectrum


on the scale of Г. So far does the similarity extend in the behavior of these classicaland quantum-mechanical resonance systems, which seem very different ones.

At an early stage of our theoretical analysis of resonant gamma-ray scattering,when we had not yet arrived at the conclusion that the mean lifetime of a nucleus inan excited state depends on the mode of excitation, we asked Dr. E.B. Bogomol’ny,who then worked at ITEP, to help us to understand the reasons behind the aston-ishing dependence of the result of the perturbation induced by a magnetic field inADRSG on the width of the spectrum of photons to be scattered. Bogomol’nyexplained in the following way the difference between the cases of nuclear exci-tation by gamma rays of wide and narrow spectrum. In the process where gammarays whose spectral distribution is characterized by a width Δ are absorbed by anucleus, the duration of the absorption process is determined by the ratio ħ/Δ. IfΔ ≫ Г, then ħ/Δ ≪ τ. Therefore, the processes of gamma-ray emission andabsorption are separated in time in a resonance-scattering event. The total proba-bility w of such an event is proportional to the sum of the probabilities for photonemission within all time intervals after the population of an excited state; that is,

w �Z1�h=D

A tð Þj j2dt ð1:89Þ

where A(t) is the amplitude for the emission process occurring after the lapse of atime t from the beginning of photon absorption.

If Δ * Г or Δ < Г, then one cannot separate in time photon-absorption andphoton-emission processes, so that it is illegitimate to consider the possible radi-ation events after different time intervals from the beginning of absorption asindependent events. In this case, we have

w�Z�h=D0

A tð Þdt��

��2

ð1:90Þ

Our approach is equivalent in its limiting cases of Δ ≫ Г and Δ * Г to the lineof reasoning leading to Eqs. (1.89) and (1.90). From Eqs. (1.10) and (1.11), one canobtain

W hð Þ ¼Z1�1

fi xð Þj j2dxZ10

Aif h; tð Þ e ix�C2�hð Þ tdt

Z10

A�if h; t0ð Þ e �ix�C

2�hð Þ t0dt0: ð1:91Þ


We isolate here an integral with respect to ω:

Ix ¼Z1�1

fi xð Þj j2eix t�t0ð Þdx ð1:92Þ

Taking expression (1.12) for fi(ω) and setting sþei�h ¼ c0, we obtain

Setting c0 ¼ 0 for the sake of simplicity, we have

Ix �Z1�1

Deixðt�t0Þdxx� c0 þ iD

2�h

� �x� c0 � iD

2�h

� � ð1:93Þ

We now perform integration in (1.93) in a complex plane by using the inte-gration contour closed in the half-plane of positive values of Imω if t � t0 [ 0 andin the half-plane of negative values of Imω if t � t0\ 0. We obtain

Ix ¼ 2p eic0 t�t0ð Þ � A ð1:94Þ

where A ¼ e�D2�h t�t0ð Þ if t [ t0 and A ¼ e

D2�h t�t0ð Þ if t\ t0.

W hð Þ ¼ 2pZZ

t[ t0e�

D2�h t�t0ð Þe�

C2�h tþt0ð ÞAif h; tð ÞA�

if h; t0ð Þdtdt0

þ 2pZZ

t\t0e

D2�h t�t0ð Þe�

C2�h tþt0ð ÞAif h; tð ÞA�

if h; t0ð Þdtdt0 :ð1:95Þ

Owing to the appearance of the factors e�C�ht and e�

C�ht0in the integrands, a

dominant contribution to W(θ) comes from the time intervals determined by theinequalities

0 � t � �h=C;

0� t0 � �h=C:

The following inequality also holds:

0\ t � t0j j � �h=C

If Δ ≫ Γ, then the main role is played by time intervals for which the followingrelation holds:

t � t0 � �hD

� �hC


A transition to a very wide spectrum means that Δ→∞, in which case t � t0 !0 or t ! t0:

In the first integral on the right-hand side of (1.95), we replace t by t0 in e�C2�h tþt0ð Þ

and in Aif h; tð Þ. In the corresponding terms of the second integral, we replace t0 by tand, after that, integrate the first expression with respect to t and the secondexpression with respect to t0. As a result, we obtain

W hð Þ ¼Z10

e�C2�ht

0� �2

Aif h; t0ð Þ�� 2dt0 þ Z10

e�C2�ht

� �2Aif h; tð Þ�� 2dt

¼ 2Z10

e�C2�ht

� �2Aif h; tð Þ�� 2dt

ð1:96Þ

If Δ ≪ Γ, then e�D2�h t�t0ð Þ � 1 because t � t0ð Þ � �h

C and DC � 1. Therefore, we

have

WðHÞ ¼Z10

Z10

e�C2�hðtþt0ÞAif ðH; tÞ A�

if ðH; t0Þdtdt0

¼Z10

e�C2�htAif h; tð Þdt

Z10

e�C2�ht

0A�if h; t0ð Þdt0 ¼

Z10

e�C2�htAif h; tð Þdt

��2

:

ð1:97Þ

Obviously, expressions (1.96) and (1.97) are very close to expressions (1.89)and (1.90). Thus, the difference between the perturbations of ADRSG in the case ofwide and narrow spectra of photons that undergo scattering stems from the dif-ference in the dependence on the time of photon-absorption and photon-emissionprocesses in these two cases. It is noteworthy that herein does the concept of aprotracted character of nuclear radiative processes manifest itself as a self-evidentone.

References

1. Frauenfelder G, Steffen R (1965) Angular correlations. In: Siegbahn K (ed) Alfa-, beta- andgamma-ray spectroscopy, vol 2. North-Holland Publishing Company, Amsterdam

2. R.F. Metzger, Nucl. Phys. 27, 612 (1961)3. Davydov AV (1970) The results of the experiment by A.V. Davydov and O.N. Sorokin

adduced in the lecture “resonant scattering of gamma rays”. In: Proceedings of 5-th winterschool on nuclear theory and physics of high energy of A.F. Ioffe LPTI, part II, Leningrad,p 263 (in Russian)

4. Y.W. Chow, L. Grodzins, P. Barrett, Phys Rev Lett 15, 369 (1965)5. Goebel KCJ, McVoy W (1966) Phys Rev 148:1021


6. H. Eicher, Z Phys 212, 176 (1968)7. Vishnevsky IN, Davtdov V, Lobov GA, Povsun VI (1974) Preprint ITEP-70, Moscow, 1974

(in Russian)8. A.V. Davydov, G.A. Lobov, Bull Acad Sci USSR Phys 45, 11 (1981)9. Baldin AM, Gol’danskii VI, Rozenthal IL (1961) Kinematics of nuclear reactions. Pergamon

Press, New York10. Dirac PAM (1958) The principles of quantum mechanics. Clarendon Press, Oxford11. Korn GA, Korn TM (1968) Mathematical handbook. McGraw-Hill Book Company, New

York

References 33

Chapter 2Experimental Study of ResonantGamma-Ray Scattering

2.1 Introduction

In 1961, Metzger published an article [1] in which he described an experimentdevoted to studying “classic” (without the use of the Mössbauer effect) resonantgamma-ray scattering on 57Fe nuclei. He measured the magnetic moment of the 56Fenucleus in the first excited state, whose spin–parity is 2+, by using the internalmagnetic field in an annular iron scatterer to perturb the angular distribution (AD) ofresonantly scattered gamma rays. There arose the question of whether it is possible touse the Mössbauer scattering of gamma rays for the same purpose of observingcorresponding magnetic-field-perturbed ADs of resonantly scattered gamma rays.By that time, it has not yet been clear (at least, to the present author) that, at the firststage of Mössbauer gamma-ray scattering, the nucleus to be excited does not ofcourse experience recoil, nor do violations of the other degrees of freedom occur.Such violations could lead to AD distortions that are difficult to take into account.

In order to obtain an answer to this question, it was necessary to perform anexperiment aimed at measuring unperturbed ADs of gamma rays undergoingMössbauer resonant scattering by nuclei entering into the composition of a sub-stance in which internal magnetic or nonuniform electric fields could not affectADs. Therefore, a nonmagnetic substance that has a cubic crystal lattice must be thescatterer in such an experiment.

Tungsten in which 182W nuclei must be excited was chosen for this goal. Thesuccess of this experiment [2] permitted us to perform a whole series of investi-gations of both unperturbed and magnetic-field perturbed angular distributions ofresonantly scattered gamma rays.

Before presenting these experimental data, it is worthwhile to note that Mössbauergamma-ray scattering is a particular case where the character of processes of gamma-ray absorption and emission is such that the AD of gamma rays scattered by nuclei ofa solid body may remain unperturbed even at average lifetimes of excited nuclei aslong as 10–9–10–7 s. It is well known that unperturbed angular correlations of


35

sequentially emitted photons are usually observed in the case of liquid or gaseousrather than in the case of solid gamma sources. This special feature of Mössbauerscattering is directly related to the fact that, owing to the absence of recoil, thenucleus that absorbed a primary photon remains at its site of the crystal latticethroughout its lifetime in the excited state. At the same time, a nucleus that expe-rienced recoil in the case of “classic” resonant gamma-ray scattering and whichmoves in the substance of the scatterer undergoes collisions with its atoms, with theresult that the quantum numbers of the excited state change. If the scattering nucleusis at a site of the cubic lattice of a nonmagnetic crystal, then, naturally, neitherelectric-quadrupole nor magnetic-dipole interaction affects this nucleus since thereare no internal fields at this site.

2.2 Measurement of the Angular Distribution of 100.1 keVGamma Rays Resonantly Scattered by 182W Nuclei

The isotope 182W was chosen for the first experiment for the following reasons:

(a) The Mössbauer effect for the 100.1 keV gamma line had already beenobserved by that time with this isotope in traditional transmission geometry[3–6].

(b) The effect observed in the studies indicated above was rather large at liquid-nitrogen temperature, and this rendered the experiment simpler since there wasno need for employing liquid helium.

(c) Metallic tungsten is not a ferromagnet and has a cubic body-centered crystallattice.

(d) It is convenient to fabricate gamma sources irradiating metallic tantalum withreactor neutrons.

The decay scheme for the parent nuclide 182Ta produced upon neutronabsorption by a 181Ta nucleus is shown in Fig. 2.1 [7]. It was proposed to observethe resonant excitation of the 2+ level of the 182W nucleus at 100.1 keV. In thiscase, the process of resonant gamma-ray scattering proceeds via the sequence of0+→2+ and 2+→0+ E2 transitions. The corresponding AD of resonantly scatteredphotons is highly anisotropic. Its shape, which is represented by the solid line inFig. 2.5, is described by the expression

W hð Þ ¼ 1þ 0:3571P2 coshð Þ þ 1:1429P4 coshð Þ; ð2:1Þ

where P2(cosθ) and P4(cosθ) are Legendre polynomials.The layout of the experimental setup is shown in Fig. 2.2. A scatterer in the form of

a metallic-tungsten plate 0.5 mm thick was arranged in the narrowed part of a Sty-rofoam container, which may be filled with liquid nitrogen. The thickness of thecontainer walls at the scatterer position was about 15 mm. This was sufficient forpreventing the formation of ice at the container walls throughout the measurement time

36 2 Experimental Study of Resonant Gamma-Ray Scattering

and for thereby avoiding the additional absorption and scattering of initial and scatteredgamma rays in ice. The position of the scatterer in the narrow part of the containerpermitted minimizing th e contribution of gamma-ray scattering by liquid nitrogen.

A gamma source was made from a tantalum foil 0.1 mm thick in the form of adisk 1 cm in diameter irradiated with thermal reactor neutrons at a flux density ofabout 2 × 1013 cm2 s–1 for about 100 h. The source was clutched between aluminumplates and fastened to a Plexiglas holder, which was rigidly connected to a movingcoil of an electromagnetic vibrator. The lower ends of the aluminum plates wereimmersed in liquid nitrogen filling the Styrofoam cuvette. The source position wasoutside liquid nitrogen and was close to its surface, and the source temperature wasclose to its boiling temperature. The electromagnetic vibrator was an electromagnetwith a hollow cylindrical iron yoke inside which there was a magnetizing coil on airon rod joined to the back wall of the yoke. In the front part of the yoke, there was anannular clearance between it and the central rod, and an alternating-current-carryingcoil put on the central rod could oscillate along the magnet axis. The averageposition of the coil was fixed in the axial direction by two springs. The vibrator,together with the gamma source and the cuvette filled with liquid nitrogen, wereclosed from the outside by a lead layer of thickness sufficient for reducing theexternal-radiation level to a permissible value. The initial gamma-ray beam came tothe scatterer through a lead collimator. Scattered gamma rays were detected by a

Fig. 2.1 Scheme of β–decay of 182Ta nucleus and of subsequent gamma transitions in daughternucleus 182W according to [7]

2.2 Measurement of the Angular Distribution of 100.1 keV Gamma Rays… 37

scintillation counter based on a NaI(Tl) crystal. The detector was placed inside a lead“house” with a window 40 mm in diameter. The collimator, scatterer, and detectorwere arranged in such a way that, at all scattering angles, the perpendicular to thescatterer plane was the bisector of the angle complementary to the scattering angle.In the case of this geometry and under the condition that the scatterer thickness dwasso large that one could treat the scatterer as that which was indefinitely thick forgamma rays of energy interesting to us, the absorption of gamma rays penetratinginto the scatterer and going backward after a scattering event was independent of thescattering angle and was therefore immaterial in processing the results of the mea-surements. Indeed, the number of photons having an energy in the interval betweenE and E + dE and undergoing scattering at an angle of h ¼ 180o � 2h0 in the layer ofthickness dx at a depth x is given by (see Fig. 2.3)

Fig. 2.2 Layout of the setup for measuring unperturbed angular distributions of resonantlyscattered gamma rays. The inset shows the scheme of fastening of the liquid-nitrogen-cooledgamma source to the movable coil of the electromagnetic vibrator. The following notation is usedhere: (1) protective lead screen of the scintillation detector, (2) body of the scintillation detector,(3) X-ray filters, (4) scatterer, (5) Styrofoam container, (6) liquid nitrogen, (7) gamma source,(8) lead “house” with a collimator for the source and vibrator, (9) electromagnetic vibrator,(10) magnetizing coil of the vibrator, (11) iron yoke of the vibrator, (12) movable coil of thevibrator, (13) gamma source, (14) cold finger of the source (the thin aluminum plate pressing thesource to the cold finger is not shown), (15) liquid nitrogen, and (16) Styrofoam cuvette


dN E; xð Þ ¼ N Eð Þ e�2m rtþ0:5arra Eð Þ½ � xcosh0

drrs E; hð ÞdX

amdxdEcosh0

: ð2:2Þ

Here, N Eð ÞdE is the number of photons that have an energy between E andE + dE and which are incident to a 1 cm2 area orthogonal to the gamma-beam axis(we neglect the beam divergence within the scatterer thickness), ν is the number ofscatterer atoms per 1 cm3, a is the relative fraction of atoms whose nuclei canresonantly scatter gamma rays, σt is the total nonresonant-absorption cross sectionin the scatterer material for the gamma rays in question, σra(E) is the energy-

dependent total cross section for resonant gamma-ray absorption, drrs E;hð ÞdX is the

differential cross section for the resonant scattering of gamma rays with energy E atan angle θ into a solid-angle unit, and h0 is half the angle complementary to thescattering angle. The coefficient of 0.5 in front of aσra(E) in the bracketedexpression in the exponent is associated with the fact that the weakening of thegamma-ray intensity because of resonant absorption must be taken into accountonly before the scattering event. The gamma rays escaping from the scatterer areable to undergo secondary resonant absorption to a smaller degree because of a lowprobability to be scattered without recoil in primary scattering. the scattererthickness), ν is the number of scatterer atoms per 1 cm3, a is the relative fraction ofatoms whose nuclei can resonantly scatter gamma rays, σt is the total nonresonant-absorption cross section in the scatterer material for the gamma rays in question,σra(E) is the energy-dependent total cross section for resonant gamma-ray

absorption, drrs E;hð ÞdX is the differential cross section for the resonant scattering of

gamma rays with energy E at an angle θ into a solid-angle unit, and h0 is half the

Fig. 2.3 Geometry that waschosen for our experimentaimed at studying gamma-rayscattering and in which theperpendicular to the scattererplane is the bisector of theangle complementary to thescattering angle θ. Here, q1and q2 are the wave vectorsof, respectively, the initial andthe scattered photon


angle complementary to the scattering angle. The coefficient of 0.5 in front ofaσra(E) in the bracketed expression in the exponent is associated with the fact thatthe weakening of the gamma-ray intensity because of resonant absorption must betaken into account only before the scattering event. The gamma rays escaping fromthe scatterer are able to undergo secondary resonant absorption to a smaller degreebecause of a low probability to be scattered without recoil in primary scattering.

Integration of expression (2.2) with respect to x yields

dN Eð Þ ¼ N Eð ÞdE drrs E; hð ÞdX

a2 rt þ 0:5arra Eð Þ½ � 1� e

�2m rtþ0:5rra Eð Þ½ �dcosh0

n o: ð2:3Þ

For a rather thick scatterer, the exponential factor is very small, in which case the

angular dependence of dN(E) remainsnly in drrs E; hð ÞdX —that is, it is the true AD of

resonantly scattered gamma rays that determines it.The measurement of the AD of resonantly scattered gamma rays consisted in

determining, for each of five chosen scattering angles (90o, 112o, 127o, 141.5o, and150o), the difference of the number of counts in the scintillation counter for the casewhere the source was at rest and the case where it performed oscillations ofamplitude 0.5 mm at a frequency of 50 Hz. Under these conditions, the maximumvelocity of source motion was 15.7 cm/s, which was 150 times as high as thevelocity that is necessary for shifting the exciting gamma line with respect to theresonance position by its natural width.

Erbium, gadolinium, and holmium filters were used to reduce the background oftungsten and tantalum X-rays. Because of the smallness of resonant gamma-rayabsorption in the scatterer in relation to the total nonresonant absorption, the con-tribution of Rayleigh scattering was nearly identical in the two measurement modes(source at rest versus moving source). After the subtraction of the background, theremaining difference effect corresponding to the Mössbauer resonant scattering of100.1 keV gamma rays was 2–4 % (depending on the scattering angle) of the numberof counts for the gamma source at rest. A correction associated with the change in theeffective average position of the oscillating source in relation to its position at rest wasintroduced in the numbers of counts corresponding to the resonant-scattering effect.This correction (about 0.5 % of the average number of counts) was determinedexperimentally from the change in the number of counts in the analyzer channelscorresponding to the energy of scattered gamma rays, about 185 keV, upon goingover from the source-at-rest to the oscillating-source mode. The background wassubtracted simultaneously. Figure 2.4 shows the spectrum of detector pulses that wasobtained by means of the channel-by-channel subtraction of the numbers of countscorresponding to the measurement where the vibrator was switched on from theircounterparts measured with the gamma source at rest. The peak of the total absorptionof resonantly scattered gamma rays with energy 100.1 keV is clearly seen.

The results of the measurements are shown in Fig. 2.5. The scale along theordinate for the smooth curve representing the unperturbed angular distributionW(θ) calculated by formula (1.89) was found by the least squares method for one


http://dx.doi.org/10.1007/978-3-319-10524-6_1

Fig. 2.4 Fragment of the amplitude spectrum of pulses induced in the NaI(Tl) scintillationdetector by scattered 182W gamma rays for the expected position of the 100.1 keV gamma line. Itwas obtained by means of the channel-by-channel subtraction of the number of counts in the modewhere the vibrator was switched on from the number of counts in the mode where the vibrator wasswitched off

Fig. 2.5 Unperturbed angulardistribution of 100.1 keVgamma rays resonantlyscattered by 182W nuclei. Thesolid curve represents thetheoretical unperturbed-angular distribution matchedwith experimental data bydetermining a singleparameter (ordinate scale) viaa least squares fit


parameter to be determined. For four degrees of freedom that are present in thiscase, the value of the χ2 criterion is 2.8. Therefore, the measured AD agrees withinthe experimental errors with the unperturbed AD calculated for the set of nuclear-level spins being considered. The results of this experiment were published in [2].This experiment showed that, by using the Mössbauer effect, one could observe, atleast in some cases, the unperturbed AD of resonantly scattered gamma rays in thecase of comparatively long average lifetimes of nuclear states to be excited. TheADs in question may be highly anisotropic, as in the case of 182W, and this is afavorable circumstance for experiments aimed at measuring the magnetic momentsof such nuclei in corresponding excited states.

2.3 Measurement of the Magnetic Moment of the 182WNucleus in the 2+ Excited State at 100.1 keV

In order to check the possibility of measuring nuclear magnetic moments by usingmagnetic-field-perturbed ADs of gamma rays that experienced Mössbauer resonantscattering, we undertaken an experiment aimed at measuring, by this method, themagnetic moment of the 182W nucleus in the already known excited state at100.1 keV. The layout of the setup used in that experiment is shown in Fig. 2.6.A gamma source in the form of a tantalum-foil disk 0.1 mm thick and 18 mm indiameter was irradiated with thermal reactor neutrons and was soldered after thatinto a thin-walled brass ampule. The tightly sealed gamma source was placed into aStyrofoam cuvette filled with liquid nitrogen. This cuvette was mounted on the

Fig. 2.6 Layout of the setup for measuring the magnetic moment of the 182W nucleus in theexcited state at 100.1 keV: (1) scintillation detector based on a NaI(Tl) crystal, (2) protective leadscreens, (3) Styrofoam container, (4) liquid nitrogen, (5) scatterer, (6) coil of the electromagnet, (7)pole piece of the magnet, (8) electromagnetic vibrator, (9) yoke of the magnet, (10) gamma source,and (11) cuvette filled with liquid nitrogen


movable rod of an electromagnetic vibrator and could be driven in oscillatorymotion, together with the gamma source, at a frequency of 50 Hz and with anamplitude of 0.5 mm. This was sufficient for substantially violating resonanceconditions and for suppressing the resonant-scattering effect to a level below 2 % ofthe maximum possible level in the case of the source at rest. The vibrator and thecuvette with the gamma source were placed inside a lead “house.” A narrow beamof gamma rays from the source came out through a cylindrical collimator. A 0.6-mm-thick scatterer from metallic tungsten of natural isotopic composition wasplaced in the extension of the Styrofoam container filled with liquid nitrogen. Thisextension was pushed in the gap between the pole pieces of the electromagnetenergized from a stabilized rectifier and used to create, in the scatterer region(19 mm in height and 22 mm in width), a constant magnetic field of strength 13,500Oe and uniformity not poorer than 2 %. Scattered gamma rays were detected by aNaI(Tl) scintillation detector surrounded by a triple magnetic-shield layer (two ironcylinders 3 mm thick each separated by air gaps and a cylinder 2 mm thick from aPermalloy tape 0.2 mm thick annealed in hydrogen after fabrication. From theoutside, the detector was shielded by a lead layer about 5 cm in thickness. Theaverage scattering angle was 109.5o. Effects of resonant gamma-ray scattering weremeasured for two opposite directions of the magnetic field as the difference of thenumbers of counts for the oscillating gamma source and the gamma source at restunder the condition that both the source and the scatterer were at liquid-nitrogentemperature. Corrections for the shift of the source center upon the transition fromthe state at rest to the state of motion were introduced in the data that we obtained.In order to compare the experimental results with the theoretical expression for theangular distribution, it is convenient to represent these results in the form

R0 ¼ 2Nþ � N�

Nþ þ N� ; ð2:4Þ

where N+ and N– are the numbers of counts for the liquid-nitrogen-cooled sourceand scatterer (numbers of resonantly scattered gamma rays) in the cases of,respectively, the positive and the negative direction of the magnetic field. Inexpression (1.92), it was necessary to introduce a correction that would take intoaccount the possible influence of the change in the magnetic-field direction ondetector operation. This correction, R00, was determined experimentally as a quantitythat was analogous to R0, but which was measured under conditions where both thescatterer and the gamma source at rest were at room temperature. In that case, therewas virtually no resonant scattering.

Therefore, data on the magnetic moment of the 182W nucleus in the excited stateat 100.1 keV must be extracted from the quantity R ¼ R0�R00, which was measuredto be R = –0.153 ± 0.031. The results of that study, which was performed before theappearance of the article quoted in [8], were treated according to the theory thatdescribed perturbed angular correlations of sequentially emitted photons withoutallowance for the ratio of the width Δ of the spectrum of resonantly scatteredgamma rays to the natural width Г of the nuclear level to be excited. We now

2.3 Measurement of the Magnetic Moment… 43

http://dx.doi.org/10.1007/978-3-319-10524-6_1

reproduce here this line of reasoning and obtain, on its basis, the g-factor value forthe 182W nucleus in the excited state of interest and, after that, introduce, in thisvalue, the correction associated with the full theory described in the first chapter,whereupon we compare both values with the results of various experiments.

Expression (2.1) for the angular distribution of resonantly scattered gamma rayscan be written in a simpler form; that is,

W hð Þ ¼ 1� 3cos2hþ 4cos4h: ð2:5Þ

Denoting by N +(θ) [N –(θ)] the number of photons recorded by the detector andresonantly scattered at an angle θ in the case of a positive (negative) direction of themagnetic field and assuming that the measurement time was much longer than themean lifetime of the nucleus in the excited state, we obtain

N� hð Þ ¼Z10

e�t=sW h� Xtð Þdt; ð2:6Þ

where τ is the mean lifetime of nuclei in the excited state (it is related to the naturalwidth Г of this state by the equation τΓ = ћ) and Ω is the Larmor frequency ofnuclear-spin precession in the magnetic field.

Substituting Eq (2.5) into Eq (2.6) and performing integration, one obtains

N� hð Þ ¼ s 1þ 0; 5cos2h1

1þ 4X2s2� 0; 5sin2h

2Xs

1þ 4X2s2þ 0; 5cos4h

1

1þ 16X2s2�

�0; 5sin4h

4Xs

1þ 16X2s2

�:

ð2:7Þ

From here, it follows that R can be represented in the form

R ¼ 2Nþ hð Þ � N� hð ÞNþ hð Þ þ N� hð Þ ¼ �

4Xs sin2h1þ4X2s2

þ 2sin4h1þ16X2s2

h i2þ cos2h

1þ4X2s2þ cos4h

1þ16X2s2

: ð2:8Þ

In the case being considered, θ = 109.5°. Substituting the numerical values of thetrigonometric functions involved, we obtain

R ¼ �4Xs 1;9562

1þ16X2s2� 0;6293

1þ4X2s2

h i2� 0;7771

1þ4X2s2þ 0;2079

1þ16X2s2

: ð2:9Þ

A direct comparison of expression (2.9) with the above experimentally measuredvalue of R leads to the following value of Ωτ:

Xs ¼ �4:28� 0:95ð Þ � 10�2: ð2:10Þ


This value must be corrected for the finiteness of the solid angles within whichgamma radiation is incident to the scatterer and, after a scattering event, to thedetector. This correction can be evaluated by comparing the value of R calculated atan angle of θ = 109.5° with the R value obtained upon averaging over all scatteringangles allowed by setup geometry—that is, by the dimensions of the gamma source,detector, and working area of the scatterer, as well as by the corresponding dis-tances. In the case being considered, one may neglect, for the sake of simplicity, thequantities 4 and 16 Ω2τ2 against unity in the denominators of the correspondingterms in expression (2.9). After the introduction of this correction, Ωτ proved to be

Xs ¼ �4:45� 1:00ð Þ � 10�2: ð2:11Þ

The half-life of the 182W nucleus in the excited state at 100.1 keVwas measured inmany studies. A compendium of the results obtained to 1966 inclusive was given in[9], the result averaged over all studies being T1/2 = (1.37 ± 0.01) × 10

–9 s. An updatedresult averaged over studies performed before September 1974 was given in [10]:T1/2 = (1.38 ± 0.02) × 10–9 s. Although it is quite surprising that the accuracy indetermining this quantity became worse in the course of time, either value isacceptable for us. Taking the value of T1/2 = (1.38 ± 0.02) × 10–9 s, one obtainsτ = 1.99 ± 0.03 ms, which leads to Γ = (0.530 ± 0.008) × 10–18

erg = (3.31 ± 0.05) × 10−7 eV. Substituting this value of τ into (2.11), one obtains forΩthe value of –(2.24 ± 0.50) × 107 s–1. From here, one arrives at the g-factor value of0.346 ± 0.077. The natural width of the excited state in units of the relative velocity ofthe Mössbauer gamma source and the absorber is (0.0992 ± 0.0015) cm/s.

Direct measurements of the real width of the 100.1 keV gamma line in Mössbauerexperiments in transmission geometry were performed in several studies quoted in[5, 6, 11–15]. Those articles either present the directly observed experimentalMössbauer resonance width, which is equal to the sum of the width of the gamma-source (metallic tantalum irradiated with reactor neutrons) emission line, Γs, and thewidth of the resonant-absorber (tungsten metal) absorption line, Γa, or other datafrom which one can extract the value of Γs + Γa. The average of this sum over theresults of the quoted studies is 0.2170 ± 0.0046 cm/s. Under the assumption that thewidth Δ of the Mössbauer gamma line emitted by metallic tantalum is half this value,one obtains the value of 1.094 ± 0.025 for Δ/Γ.

As a matter of fact, the above value of Ωτ = (–4.45 ± 1.00) × 10−2 is equal to theangle Δθ through which the AD of resonantly scattered gamma rays was rotated inthe experiment being discussed. According to the correct theory of magnetic-field-

perturbed ADs, this angle is 2CþDð ÞCþDð Þ Xs rather than Ωτ. In our case, we then have

Δθ = 1478 Ωτ. This means that the value correctly determined from the experimentin question for the g-factor of the 182W nucleus in the 2+ excited state at 100.1 keVis (0.346 ± 0.077)/1.478 = 0.234 ± 0.052. These measurements, performed by thepresent author together with Sorokin, were not published separately. Their resultsformed the content of Sorokin’s thesis completed in 1964 and presented for adiploma. Later, these results were included by the present author in his lecture


delivered at the Winter School in Physics at the Ioffe Leningrad Institute for Physicsand Technology in 1970 [16].

Let us now compare the results that we obtained with data of investigationsperformed by different methods. The g-factor values for the first excited 2+ state ofthe 182W nucleus are given in Fig. 2.7 according to studies reported before thecompletion of our experiment described immediately above. The results of the firstthree studies deviate strongly from one another and contradict the majority of morerecent data. Starting from fourth study (the respective references are given in thecaption under Fig. 2.7), however, the g-factor values are quite consistent, and onecan use them to calculate of the average g-factor value. It turned out to be0.258 ± 0.006. The horizontal shaded band in Fig. 2.7 shows this average value andthe error in it. They were calculated by using data numbered by integers in the range

Fig. 2.7 Measured values of the g-factor of the 182W nucleus in the excited state at 100.1 keVfrom (1) [24], (2) [25], (3) [26], (4) [27], (5) [28], (6) [29], (7) [30], (8) [31], (9) [32], (10) [33],(11) [15], (12) [16], and (13) [18]. The g-factor values obtained upon the treatment of data on thebasis of the theory that takes into account the ratio of the width of the spectrum of gamma rays thatexperienced resonant scattering to the natural width of the excited nuclear level are shown by thedotted lines under points 12 and 13. The average g-factor value and the error in it according tocalculations based on the g-factor values corresponding to points 4–11 are represented by thehorizontal shaded band


from 4 to 11. The g-factor value that we obtained by treating the experimental dataaccording to the theory of angular correlations of sequentially emitted photons –thatis, without taking into account the ratio Δ/Γ—is represented by point 12. The valuededuced by using the correct theory and the error in this value are shown by thedotted line below point 12. One can see that this value agrees much better with theaverage g-factor value than the value represented by point 12. Later, I becameaware of yet another study, that which was reported in [17] and according to whichthe g-factor of the state of the 182W nucleus at 100.1 keV is 0.264 ± 0.006. Takinginto account this value, together with the data in Fig. 2.7, we arrive at the averageg-factor value of 0.261 ± 0.004, which slightly raises the shaded band in Fig. 2.7.

In 1965, there appeared the article of Chow et al. [18], who reported on anexperiment where they used a method similar to ours to measure the g-factor for the182W, 186W, 186Os, and 188Os nuclei. Although those authors treated their datawithin old strategies—that is, without taking into account the ratio Δ/Γ—theyestimated the g-factor for the 182W nucleus at 0.233 ± 0.027 (point 13 in Fig. 2.7),which is close to the average result (in Fig. 2.7) of the studies performed by othermethods, which obviously did not require taking into account the ratio Δ/Γ. If onecorrects the result from [18] in accordance with requirements of the correct theory,then the corresponding point (0.159 ± 0.018) appears to be considerably lower thanthe average g-factor value (in Fig. 2.7, this point is shown by the dotted line belowpoint 13). The situation around the other three g-factors measured in [18] is similar,but, for osmium isotopes, especially for 188Os, there is some excess of the measuredvalues over the g-factor values obtained by averaging the results of other studies.

It does not seem possible to explain this situation on the basis of informationpresented in [18]. There are no reasons to criticize that experiment. However, one ofauthors of [18] recommended, in his more recent review article [19], to addressthese data with caution because they were not reproduced. In addition to their mainresults for the first excited 2+ states of 182, 186W and 186, 188Os, the authors of [18]also presented the g-factor value for the 192Os nucleus in the 2+ state at 206 keVaccording to a measurement in a separate experiment (judging by the energy of thislevel, without the use of the Mössbauer effect). This result turned out to be at oddswith more recent measurements performed by Goldring et al. [20], who indicatedthat the authors of [18] informed them that their result for 192Os was erroneous.However, it remains unclear whether this also applies to the other results from [18]—in particular, the results for 182W.

2.4 Measurement of the Unperturbed Angular Distributionof Gamma Rays Resonantly Scattered by 191Ir Nuclei

After successful (from our point of view) experiments with 182W, we beganstudying gamma-ray resonant scattering by 191Ir nuclei that was accompanied bythe excitation of the 5/2 level at 129.4 keV. This is the same nucleus and the sametransition as those studied by R. Mössbauer himself when he discovered in 1958 the


phenomenon of recoilless gamma-ray emission and absorption [21, 22]. The ADsof resonantly scattered gamma rays of the 191Ir nucleus had not been measured bythe time when we began our study. In the course of our work, however, thereappeared the article of F. Wittmann [23], who reported on the measurement of ADsfor this nuclide. Those measurements were performed for scattering angles notlarger than 90o, but our measurements were made for scattering angles from 90o to150o. As will be seen below, our results turned out to be in good agreement withdata from [23], slightly surpassing them in statistical accuracy.

The ADs of 129.4 keV gamma rays experiencing Mössbauer scattering by 191Irnuclei were measured at the same setup as that which was used in the experimentswith 182W [2]. A gamma source was a round pellet 1 cm in diameter from a mixturecontaining 150 mg of metallic osmium and 350 mg of aluminum powder added inorder to increase the mechanical strength of the pellet. The pellet was irradiated for 2weeks in a flux of reactor thermal neutrons with a density of about 4 × 1013 n/cm2s inan evacuated quartz ampule. The source must be in a tightly sealed cuvette duringthe irradiation because, in the case of irradiation in open air at elevated temperature,there arises the OsO4 compound, which is volatile.

The decay scheme for the radioactive isotope 191Os is shown in Fig. 2.8. Theintensity of the 82 keV transition is very small in this case, and one uses the parentisotope 191Pt as the source material in order to study the Mössbauer effect with82 keV gamma rays. The 129.4 keV transition is a mixed E2 + M1 transition. Theunperturbed angular distribution of gamma rays that experienced resonant scatter-ing accompanied by the excitation of the 129.4 keV level can be represented in theform

Fig. 2.8 Scheme of 191Osdecay and of subsequenttransitions in the daughternucleus 191Ir according to[36]. The total intensitiesof the γ1, γ2, γ3, and γ4transitions (gammaradiation + internal-conversion electrons) pereach 100 decay events are100, 99.4, 0.36, and 0.36,respectively


W hð Þ ¼X

k¼0;2;4

BkB0kPk coshð Þ ¼

Xk¼0;2;4

AkPk coshð Þ; ð2:12Þ

where Pk(cosθ) are Legendre polynomials, as before, and the coefficients Bk and B0k

are given by

Bk ¼ Fk L1L1IiIð Þ þ 2d1Fk L1L1 þ 1 IiIð Þ þ d21Fk L1 þ 1 L1 þ 1 IiIð Þ; ð2:13Þ

B0k ¼ Fk L2L2If I

� � þ 2ad2Fk L2L2 þ 1 If I� � þ d22Fk L2 þ 1 L2 þ 1 If I

� �: ð2:14Þ

The functions Fk are determined by expression (1.34).In the case of resonant gamma-ray scattering, we may set δ1 = δ2 = δ if, in

accordance with the formalism of Dolginov [34], we assume that a = –1. In theformalism adopted in [23] (and also in the study that was reported in [35] and whichwas used as a basis for our computations in Chap. 1), a = + 1 and the coefficients Bk

and B0k are identical. Historically, the results of our AD measurement for 191Ir were

treated on the basis of Dolginov’s formalism. As a result, the signs of the multipole-mixture parameter δ for the 129.4 keV transition in our study and in [34] turned outto be different. This circumstance was highlighted in our article quoted in [36] anddevoted to describing those experiments.

The numerical values of Fk are such that the products of the coefficientsBk and B0

k in Dolginov’s formalism are given by

B0B00 ¼ 1þ d2

� �2; ð2:15Þ

B2B02 ¼ 0:374 þ 1:898d � 0:191d2

� �2; ð2:16Þ

B4B04 ¼ 0:497 d4: ð2:17Þ

After dividing both sides of Eq. (2.12) by the constant B0B00 ¼ 1þ d2

� �2, we

can recast this equation into the form

W hð Þ ¼ 1þ A2P2 coshð Þ þ A4P4 coshð Þ; ð2:18Þ

where

A2 ¼ B2B02

B0B00; A4 ¼ B4B0

4

B0B00; ð2:19Þ

The procedure for measuring ADs was identical to that in the case of 182W. Thenumber of resonantly scattered photons was determined as the difference of thenumber of counts in the NaI(Tl) scintillation counter in the cases where the gammasource was at rest and where it oscillated under the influence of the electromagneticvibrator at a root-mean-square velocity sufficiently high for the disregard of residual

2.4 Measurement of the Unperturbed Angular Distribution… 49

http://dx.doi.org/10.1007/978-3-319-10524-6_2_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

resonant scattering to be legitimate. The geometry of the setup, resonant andnonresonant gamma-ray absorption in the scatterer, and the secondary (Compton)scattering of gamma rays that first experienced Mössbauer scattering were takeninto account in the treatment of the data that we obtained. The mathematical aspectof this treatment consisted in selecting, by the maximum-likelihood method, valuesfor the coefficients A2 and A4 in expression (2.18) such at which the results of takinginto account the aforementioned factors provided the best agreement between themeasured and calculated angular distributions. These computations led to the fol-lowing form of the angular distribution:

W hð Þ ¼ 1þ 0:901 � 0:041ð ÞP2 coshð Þ þ �0:042 � 0:053ð ÞP4 coshð Þ: ð2:20Þ

The graph of this function calculated on the basis of experimental data withallowance for the geometry of the experiment and for the absorption of initial andscattered gamma rays in the scatterer is shown in Fig. 2.9 along with the measuredvalues of this function. In order to extract the value of the multipole-mixtureparameter δ from these data, it is necessary to construct graphs representing thedependence of the coefficients A2 and A4 on the relative fraction of the E2 multipolein the total intensity of the 129.4 keV transition. Obviously, we have

I E2ð ÞI E2ð Þ þ I M1ð Þ ¼ d2

d2 þ 1: ð2:21Þ

Fig. 2.9 Unperturbed angulardistribution of 129.4 keVgamma rays resonantlyscattered by 191Ir. Theexperimental errors aresmaller than the size of theopen circles representingexperimental data. The solidline was obtained withallowance for the requiredcorrections to the data


This ratio changes from 0 to 1 as δ changes from 0 to 1. In Fig. 2.10, the ratiod2

d2þ1is plotted along the abscissa, while the coefficients A2 and A4 calculated by

formulas (2.19) with allowance for Eqs. (2.15)–(2.17) are plotted along theordinate.

In this figure, there are two curves for A2, one for positive and the other fornegative values of δ. In the same figure, the values found experimentally for thecoefficients A2 and A4 are represented by the horizontal dotted lines, while theconfidence intervals for these quantities at a 68 % C.L. are shown by the shadedbands. From the intersections of these bands with the theoretical curves, it unam-biguously follows that the multipole-mixture parameter for the 129.4 keV transitionin the 191Ir nucleus is +0.398 ± 0.020. In [23], a value of �0:36þ0:04

�0:01 was obtainedfor δ (the reasons for the difference in sign were explained above). Therefore, thetwo studies in question gave consistent results. It should be noted that the δ valuemeasured in our study was included, as the most precise one, in the tables presentedin [37] as an adopted characteristic of the 129.4 keV level of the 191Ir nucleus.

Fig. 2.10 Values foundexperimentally for thecoefficients A2 and A4 (thesevalues and the errors in themare shown by the shadedbands), which determine theunperturbed angulardistribution of 129.4 keVgamma rays resonantlyscattered by 191Ir nuclei,along with the results oftheoretical calculations

2.4 Measurement of the Unperturbed Angular Distribution… 51

2.5 Measurements of Magnetic-Field-Perturbed AngularDistributions of 129.4 keV Gamma Rays ResonantlyScattered by 191Ir Nuclei in an Ir–Fe Alloy

The main objective of these experiments was to obtain quite convincing data on therole that the hierarchy of the characteristic width of the spectrum of exciting gammaradiation and the natural width of the excited nuclear level plays in experimentsdevoted to studying the magnetic perturbation of the angular distribution of reso-nantly scattered gamma rays. A direct check of Eqs. (1.53) and (1.65) must consistin measuring magnetic-field-perturbed ADs for identical nuclei using excitinggamma rays whose spectra have different widths. In principle, this could beachieved by measuring ADs first with the aid of Mössbauer scattering and then withthe aid of “classic” resonant gamma-ray scattering—that is, by using first a cooledgamma source and then a heated one. However, it is very difficult to perform thissequence of experiments because nuclei that may be well excited by means of theMössbauer effect are usually characterized by a very small cross section for“classic” resonant scattering. At the same time, nuclear transitions that are the mostsuitable for observing “classic” resonant scattering are absolutely useless forMössbauer experiments because of a high energy of respective photons. It istherefore more convenient to compare the results of measuring, through theobservation of the perturbation of ADs with the aid of the Mössbauer effect, themagnetic moment of any nucleus in an excited state with data obtained for thismagnetic moment by totally different methods. Nuclei of 191Ir are very convenientfor such an experiment for the following reasons:

1. If use is made of metallic iridium as a scatterer material and of metallic osmiumas a gamma-source material, then the Mössbauer effect is observed very welleven at liquid-nitrogen temperature.

2. The unperturbed AD of 129.4 keV resonantly scattered gamma rays is quiteanisotropic (see above).

3. Very high strengths of the internal magnetic field can be obtained in the case ofemploying Ir–Fe ferromagnetic alloys.

4. Data on the average lifetime of 191Ir nuclei in an excited state at 129.4 keV areavailable from [38–41]. The natural width of this state can be deduced fromthese data.

5. There are a number of studies [3, 22, 42, 43] devoted to the Mössbauer effect for191Ir gamma rays of energy 129.4 keV. Information about the width of thegamma line of exciting radiation can be extracted from their results.

6. Finally, there are some studies that were performed without using the Mössbauereffect and in which consistent data on the magnetic moment of this nucleus in thestate at 129.4 keV were obtained.

The layout of the setup used in the experiment being discussed is shown inFig. 2.11. A gamma source in the form of a pellet 15 mm in diameter consistedof 400 mg of metallic-osmium powder mixed with 600 mg of aluminum powder.


http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

This pellet was irradiated for 2 weeks in a thermal-neutron flux of density about2 × 1013 n/cm2s. After irradiation, the pellet was tightly sealed in a brass ampulewith a thin (0.2 mm) front wall. The ampule containing the gamma source wasscrewed on a spring membrane from beryllium bronze by means of a rod connectedto the movable coil of an electromagnetic vibrator. This membrane was mounted onthe front face of a horizontal pipe soldered in the lower part of a rectangular brasscontainer for liquid nitrogen. This container was placed in a Styrofoam thermalscreen. A detailed structure of the container is shown in Fig. 2.12. In order to coolbetter the gamma source, which did not have a direct contact with liquid nitrogen,the pipe that removed nitrogen vapor from the container went through the volumeof liquid nitrogen from the top of the container to its lower part and came outsidenear the gamma source. Upon switching on the vibrator, the gamma source couldexecute nearly sinusoidal oscillations at a frequency of 59 Hz with an amplitude ofup to 2.5 mm, which corresponded to the highest velocity value of 78.5 cm/s (inorder to shift the Mössbauer gamma line of energy 129.4 keV by its natural width, itwas necessary that the source velocity with respect to the absorber be 0.87 cm/s).The entire source–vibrator assembly was covered with a radiation screen formed byblocks from Pb and a W–Cu alloy. The photon beam went out through a leadcollimator of rectangular cross section that broadened gradually outward in thehorizontal plane. The scatterer of gamma rays was a plate from an Ir–Fe alloycontaining 7 wt.% of iridium of natural isotopic composition. The dimensions of the

Fig. 2.11 Layout of the setupfor measuring perturbedangular distributions ofresonantly scattered gammarays (top view): (1) Styrofoamcontainer, (2) electromagnetpole piece, (3) protective“house” formed by blocksfrom lead and a W-Cu alloy,(4) brass container, (5)Styrofoam thermal screen, (6)electromagnetic vibrator, (7)liquid nitrogen, (8) Ge(Li)detector, (9) scatterer, (10)gamma source, (11) liquidnitrogen, and (12) rubbermembrane

2.5 Measurements of Magnetic-Field-Perturbed Angular Distributions… 53

plate were 46 × 30 × 1 mm3. The alloy for the scatterer was fabricated by meltingiridium and iron in a nitrogen atmosphere. The sample produced in this way wasrepeatedly rolled at gradually increasing temperature until the required thicknesswas achieved. The fabricated plate was polished and cut to the required dimensions,whereupon it was annealed in a vacuum at a temperature of 800 °C for 8 h, themode of cooling being stepwise. After that, the plate was etched in a mixture of

Fig. 2.12 Container for cooling the gamma source used in the experiments devoted to measuringperturbed angular distributions of resonantly scattered gamma rays of 191Ir: (1) plug of the pipe forpouring liquid nitrogen, (2) pipe for pouring liquid nitrogen, (3) pipe for removing nitrogen vapor,(4) membrane from beryllium bronze, (5) gamma source, (6) Styrofoam thermal screen, (7) brassbody of the container, (8) liquid nitrogen, (9) brass pipe soldered in the container body, (10) rodconnecting the gamma source to the movable coil of the electromagnetic vibrator, and (11) rubbermembrane


hydrochloric and nitric acids. An X-ray study revealed the cubic body-centeredstructure of the alloy. The scatterer was mounted inside a Styrofoam container filledwith liquid nitrogen. The back side of the scatterer was directly washed by a liquid-nitrogen layer about 3 mm in thickness. The container central part, where thescatterer was placed, was arranged in between the poles of a small electromagnetthat created a field of strength about 1 kOe. A dedicated experiment aimed atmeasuring the alloy magnetization curve revealed that this was sufficient formagnetizing the scatterer in a specific direction.

Scattered gamma rays were recorded by a Ge(Li) detector belonging to a coaxialtype and having an operating volume of about 25 cm3. The detector was coveredwith a lead layer from the top and from all sides, with the exception of the sidefacing the scatterer. A copper filter was used to reduce the detector counting rateassociated with X rays generated in the scatterer. The amplitude spectra of pulsesfrom the detector were measured by a Nokia LP 4840 analyzer. The measurementof each spectrum lasted 10 min. For each scattering angle, the spectrum wasmeasured with a vibrator first in the off mode and then in the on mode. In the courseof the measurements, the detector remained immobile with respect to the electro-magnet and the scatterer, and this ruled out the possibility of variations in the effectof the dissipated magnetic field on its operation. The scattering angle could bechanged in the range from 78o to 143o by rotating the movable platform on whichthe gamma source and vibrator were placed together with the collimator and withthe radiation screen. The measurements were performed for seven values of thescattering angle, the above range of angles being scanned repeatedly in the directand reversed directions at a fixed direction of the magnetic field. In all, about 600spectra were measured for each direction of the magnetic field. Figure 2.13 showsparts of the spectra obtained for the scattering angle of 117.8o (the sign of the fieldaffecting iridium nuclei is negative). The open circles represent results for thegamma source at rest, while the closed circles (shown in the region of the total-absorption peak for the 129.4 keV gamma line) refer to the oscillating source.A negative sign corresponds to the applied magnetic field that is directed downwardwith respect to the scattering plane shown in Fig. 2.11. The internal field acting oniridium nuclei is directed upward in this case. The sign indicated here changes inresponse to the reversal of the applied-field direction and serves for discriminatingbetween the two directions of the scatterer magnetization. This sign should not beconfused with a minus sign stably assigned to the internal magnetic field at thepositions of iridium nuclei in the Ir–Fe alloy. The meaning of the latter is that thedirection of this field is opposite to the direction of the applied magnetizing field.

In the computer- assisted treatment of the measured spectra, the regions near thetotal-absorption peak for the 129.4 keV gamma line were described by analyticfunctions depending on several parameters (six to eight for different scatteringangles). The very total-absorption peak was assumed to have the instrumental lineshape measured in an individual experiment involving a direct irradiation of thedetector with gamma rays of 191Ir. The most probable parameter values weredetermined by using the standard code for minimizing the χ2 criterion. The positionof the total-absorption peak for the 129.4 keV gamma line in terms of the number of


analyzer channels was one of the parameters to be determined. In accordance withthe value of this parameter, we evaluated the number of channels, which, in generalwas a fractional number, but which always corresponded to the same energyinterval in which one sums the number of counts associated with the central part ofthe total adsorption peak (within its FWHM value). Moreover, we calculated thetotal areas of the total-absorption peaks and the sums of the numbers of counts incontrol channels lying well above the total-absorption-peak on the energy scale.The control-interval width expressed in terms of channel numbers also depended onthe peak position, but it was always on the same energy scale. The ratio of the sumsof the numbers of counts in the control channels at a fixed scattering angle for themeasurements with the gamma source at rest and with the oscillating gamma sourceserved, after averaging over all measurements, for correcting the numbers of countsat the total absorption peak for the 129.4 keV gamma line with allowance for thepossibility that the source-to-scatterer distance may be somewhat different in thetwo measurement modes (oscillating gamma source versus gamma source at rest).This correction turned out to be very small: the average value of the ratio of thenumbers of counts in the control channels for the two measurement modes was1.00178 ± 0.00043, and we can attribute a significant part of this value to a decreasein the source activity within the time between the starts of two successive mea-surements in accordance with the usual exponential law of 191Os decay.

The number of resonantly scattered photons was determined for each scatteringangle as the difference of the sums of the numbers of counts in the aforementioned

Fig. 2.13 Typical form of those sections of the spectrum of scattered 191Ir gamma rays thatcontain the total-absorption peak for 129.4 keV gamma rays according to measurements with a6Ge(Li) detector. The scattering angle was 117.8°. The sign of the magnetic field acting on iridiumnuclei is negative. The open circles stand for data obtained by using the gamma source at rest,while the closed circles (which are shown only in the region of the total-absorption peak for the129.4 keV gamma line) represent data for the oscillating source. The scale of the errors is indicatedby the vertical bars at some points


intervals of equally wide (in energy) channels in the area of the 129.4 keV peak inmeasurements with the vibrator in the off and on modes. The contribution ofRayleigh scattering may be considered to be identical in these two cases becauseresonant gamma-ray absorption in the scatterer is small in relation to the totalabsorption and cannot lead to a decrease in the fraction of gamma rays undergoingRayleigh scattering, as would be the case under conditions of strong resonantabsorption [44, 45]. For the same reason, the width of the Mössbauer gamma lineshows virtually no increase as radiation penetrates into the interior of the scatterer.Employing the difference of the computer-calculated total peak areas as a measureof the number of resonantly scattered photons proved to be less advantageous if onewants to minimize inaccuracies because an additional subtraction of one large valuefrom another does in fact arise in that case (one first subtracts the computer-calculated pedestal from the sum of the total numbers of counts and then takes thedifference of the peak areas obtained upon performing the first procedure). How-ever, the calculated areas of the total-absorption peaks were used to determineindependently the 191Os half-life by considering the decrease in these areas over themeasurement time (about 10 days for either sign of the magnetic field). This half-life was estimated at 14.60 ± 0.43 d. Together with results of the other studies [46],this value led to an average half-life value of T1/2(

191Os) = 15.34 ± 0.32 d, whichwas used to introduce corrections for the decay of the source in averaging theresults of the measurements.

A further treatment of the experimental results was performed by followingmethod. First, the perturbed angular distributions of resonantly scattered gammarays were calculated by formula (1.53) for either of the two magnetic-field direc-tions and for twelve presumed values of the g-factor of the 191Ir nucleus in theexcited state at 129.4 keV from 0.14 to 0.25. Employing these functions and takinginto account setup geometry, gamma-ray absorption in the scatterer, and thedependence of the detection efficiency for gamma rays on geometric conditions oftheir arrival at the detector, we thereupon calculated the angular distributions ofresonantly scattered gamma rays for each g-factor value. Finally, we determined thesought g-factor by comparing the calculated and measured ADs.

In order to calculate perturbed ADs, it is necessary to know, in addition to theg-factor of the nucleus in the excited state being considered, the strength of theinternal magnetic field acting on iridium nuclei in the Ir–Fe alloy, as well as thequantities τ, Δ/Γ, and δ and the g-factor of the 191Ir nucleus in the ground state.Figure 2.14 shows data from [47–55] on the dependence of the strength of theinternal magnetic field at nuclei of iridium on its concentration in the Ir–Fe alloy attemperatures in the range of T ≤ 4.2 K. It can be seen that this dependence issatisfactorily described by a linear law. In [48], the magnetic field was measured atroom temperature. The H value associated in Fig. 2.14 with that study was rescaledby us to the temperature of T = 4.2 K in accordance with data from [56] on thetemperature dependence of the magnetization of pure iron. The least squaresmethod leads to the following result (recall that a minus sign means that thedirection of the internal field is opposite to the direction of the magnetizing field):


http://dx.doi.org/10.1007/978-3-319-10524-6_1

H ¼ � 1409:1 � 4:3ð Þ� 0:73 � 0:46ð Þ � CIrkOe: ð2:22Þ

Here, the iridium concentration is expressed in atomic-percent units. In our case,formula (2.22) gives H = –1414.2 ± 5.4 kOe. In this value, it is necessary tointroduce a correction associated with the fact that the experiment was performed atliquid-nitrogen temperature, while expression (2.22) corresponds to temperatures inthe range of T ≤ 4.2 К. In order to introduce this correction, we employed data on thetemperature dependence of the internal magnetic field at iron nuclei in pure iron [56],

0

-500

-1000

-1500

0 5 10

Content of iridium atoms in Ir-Fe alloy (percintage)

Stre

ngth

of

inne

r m

agne

tic f

ield

on

Ir n

ucle

iin

the

Ir-F

e al

loys

15

1

2 2

2

2

2 2 2 2 2

66

5

43

Fig. 2.14 Data on the internal-magnetic-field strength at iridium nuclei in an Ir–Fe alloy as afunction of the iridium concentration (in percent): (1) averaged result of [47–49, 55]; (2) data from[50]; (3) averaged result of [51, 52]; and data from (4) [47], (5) [53], and (6) [54]. The errors in thedata from [50] are so small that they cannot be shown in this figure. The solid line represents theleast squares fit of a linear function to the data in the figure


assuming that this dependence survives for the internal field at nuclei of a smalladmixture of iridium in iron. This assumption is based on published data on thetemperature dependence of internal magnetic fields for a whole series of admixedatoms in ferromagnetic substances [57] and also on the influence of admixtures onthe temperature dependence of the field at nuclei of ferromagnetic substances [58].From these data, it can be seen that a deviation of the temperature dependence of thefield at admixed nuclei from the respective dependence of the field at nuclei of a pureferromagnetic material, if any, is observed only at temperatures higher than (0.2–0.3)TCurie for this ferromagnetic material and is not greater in these cases than severalpercent.

Therefore, an anomalous temperature dependence of the internal magnetic field atiridium nuclei in iron should not be expected in our case (T = 77 К). Existingexperimental data on iridium nuclei admixed to nickel [44] support this point of view.

After introducing the temperature correction, we obtain H = –1411.0 ± 5.4 кOe.It is necessary to subtract the external- magnetizing-field strength of 1 kOe from thisvalue since the external field is opposite in direction to the internal field. Ultimately,we have H = –1410.0 ± 5.4 kOe.

The averaging of data from studies performed without employing the Mössbauereffect [38–41] leads to the following value of the mean lifetime of the 191Ir nucleus inthe excited state at 129.4 keV: τ = (1.753 ± 0.085) × 10−10 s. This value and theresults obtained by measuring the width of the 129.4 keV Mössbauer gamma line[3, 22, 42, 43] give Δ/Γ = 1.351 ± 0.050. The parameter characterizing the mixture ofE2 and M1 multipoles for the 129.4 keV transition was evaluated by averaging theresults quoted in [37] and obtained in several studies, including our study reported in[36]. The resulting multipole-mixture parameter is δ = –0.4020 ± 0.0038. Inaccordance with [59], the g-factor value for the ground state of the 191Ir nucleus wastaken to be g0 = 0.09687 ± 0.00040. On the basis of data from [60], one can neglectthe isomeric shift between the emission line of a source from metallic osmium andthe absorption line of a scatterer, which is a plate from an Ir–Fe alloy.

With the aid of precisely these constants, the aforementioned perturbed angulardistributions were calculated for 12 values of the g-factor for the 191Ir nucleus in thestate under investigation and for two directions of the magnetic field acting oniridium nuclei. Each such function was computed at 116 scattering angles, thisbeing done with a step of one degree in the interval from 60o to 167o. After that,these functions were used to calculate the ADs of scattered gamma rays, and theresults could be compared directly with measured ADs. This was done withallowance for setup geometry—that is, the dimensions of the source, scatterer, anddetector and the distances between them and for gamma-ray absorption as theypenetrate into the interior of the scatterer and escape from it. Moreover, we tookinto account the dependence of the efficiency of the coaxial germanium detector onthe geometric conditions of the arrival of photons at it. This was dictated by theneed for taking into account the absorption of gamma rays in the outer “dead” layerof the detector and their passage through the inoperative region of the inner core inthe body of the detector. All this required computing, for each of seven scatteringangles for which we measured the intensity of scattered gamma rays, sevenfold


integrals with respect to two coordinates in the gamma-source plane, three coor-dinates of scatterer-volume elements, and two coordinates of the points at whichphotons hit the detector. The g-factor value for the 191Ir nucleus in the excited stateunder study was determined individually for either magnetic-field direction byminimizing the χ2 criterion in comparing measured ADs with a set of 12 computedADs. We varied two parameters: the g-factor value and the ordinate scale, whichwas used to level the measured and calculated ADs. For the two signs of themagnetic field, we obtained the following values of the g-factor:

for the case where the external field is positive.

g ¼ 0:206 � 0:035

for the case where the external field is negative and

g ¼ 0:195 � 0:017

The average result is

g ¼ 0:197 � 0:015

The errors indicated here are associated exclusively with the statistics of thenumber of counts. Upon taking into account the errors in τ, H, δ, Δ/Γ, and g0, thefinal result becomes

g ¼ 0:197 � 0:018: ð2:23Þ

The results of the measurements and calculations are shown in Fig. 2.15. Thetheoretical angular distributions calculated by formula (1.53) for the g-factor valuesobtained in the way outlined above are represented by the solid curves. Thesecurves are slightly asymmetric with respect to an angle of θ = 90°, because themeasured g-factor values are different in these two cases. The dashed lines stand forthe angular distributions calculated with allowance for setup geometry, gamma-rayabsorption in the scatterer, and the geometric dependence of the detector efficiency.The degree of their agreement with experimental data is determined by thefollowing values of the χ2 criterion:

for curve 1a; v2 ¼ 7:90;

for curve 2a; v2 ¼ 5:95:

The number of the degrees of freedom is five in each case—that is, the expectedvalue of the criterion must satisfy the condition χ2 = 5.0 ± 3.2. Therefore, thedescription of experimental data by the curves calculated with the above values ofthe parameter g is quite satisfactory.


http://dx.doi.org/10.1007/978-3-319-10524-6_1

The results of three studies that were performed before the completion of ourexperiment without employing the Mössbauer effect and in which the g-factor wasmeasured for the 191Ir nucleus in the excited state of our interest are given inTable 2.1. In order to average these data and to compare the averaged result with theg-factor value that we measured, it is necessary to rescale these three g-factor valuesto the same average lifetime that we adopted for the 191Ir nucleus in the excitedstate. This is necessary because experimental data give the quantity Ωτ = –gμNHτ/ћrather than the g-factor proper, so that the resulting g-factor value depends on thevalue adopted for τ. The averaging of rescaled g-factor values yields

Fig. 2.15 Results of measurements and calculations for magnetic-field-perturbed angulardistributions of 129.4 keV gamma rays resonantly scattered by 191Ir nuclei in an Ir–Fe alloy.Curves 1 and 1a and the open circles refer to a positive sign of the external magnetic field, while 6the remaining data correspond to its negative sign. The dashed curves 1a and 2a represent the mostprobable angular distributions calculated on the basis of experimental data with allowance for thegeometry of the experiment, gamma-ray absorption in the scatterer and the detector efficiency. Thesolid curves 1 and 2 were calculated by formula (1.53) with the optimum g-factor values that weobtained for the two signs of the magnetizing field. The normalizations are different for the pair ofcurves 1 and 2 and the pair of curves 1a and 2a. The arrows indicate the expected positions of theminima of the angular distributions for the case where their rotation angles are calculated by theformula Δθ = Ωτ


http://dx.doi.org/10.1007/978-3-319-10524-6_1

g ¼ 0:203 � 0:015: ð2:24Þ

One can see that this value agrees well with our result, there more so as, there isno need in this comparison, for taking into account the error in τ since it enters intoboth values under comparison. Good agreement between the values in (2.23) and(2.24) indicates that our method for studying the magnetic hyperfine interaction ofexcited nuclei on the basis of the magnetic-field-induced perturbation of the AD ofgamma rays resonantly scattered via the Mössbauer effect gives correct results if, intreating experimental data, one takes into account the ratio Δ/Г, as is required by thetheory expounded in Chap. 1.

The results of this experiment are a direct experimental indication that, under theexcitation conditions prevalent in the case being considered, the mean duration ofthe precession of excited 191Ir nuclei in a magnetic field, or, what is evidently thesame, the mean time within which this nucleus remains in the excited state, exceedssubstantially the mean lifetime of excited nuclei, τ, that is determined by the relationτ = ћ/Γ. In order to prove this, we can compare the observed (in our experiment)shift of the minimum of the perturbed AD in relation to the position of the anal-ogous minimum in the unperturbed AD (see Fig. 2.15) with its counterpart fol-lowing from the usual theory of angular correlations, which does not take intoaccount the role of the width of the spectrum of exciting gamma rays. The aver-aging of the g-factor value for the 129.4 keV state in (2.24) with the value obtainedin our experiment leads to the following result:

gav: ¼ 0:201 � 0:012: ð2:25Þ

From here, it follows that, under conditions of our experiment, the Larmorfrequency of the precession of excited nuclei was

Table 2.1 The g-factor values of the 129.4 keV state of 191Ir nuclide measured without ofMössbauer effect

References Method ofmeasurement

τ value adopted by theauthors for the level at129.4 keV

g-factor value obtainedfor the 191Ir nucleus inthe state at 129.4 keV

Avida et al. [67] Perturbed angulardistribution of gammarays after Coulombexcitation

(1.89 ± 0.14) × 10−10 s 0.23 ± 0.04

Owens et al. [55] Perturbed angularcorrelation ofsequentially emittedgamma rays

(1.44 ± 0.10) × 10−10 s 0.22 ± 0.02

Il’khamdzhanovet al. [68]

The same 1.82 × 10−10 s 0.24 ± 0.03


http://dx.doi.org/10.1007/978-3-319-10524-6_1

X ¼ �glNH=�h ¼ ð1:324� 0:074Þ � 109 s�1 ð2:26Þ

The errors in the quantities g and H were taken into account here.According to the theory of angular correlations that does not take into account

the role of the ratio Δ/Γ, the angular shift of the minimum of the curve representingthe perturbed AD with respect to its position in the case where there is no magneticfield must be equal to Ωτ. In the case being considered, this yields

Dh ¼ Xs ¼ 1:324 � 0:074ð Þ � 109 � 1:753 � 0:085ð Þ � 10�10

¼ 0:232 � 0:017 rad ¼ 13:29 � 0:96�;ð2:27Þ

but the actual shifts of the AD minima (they were determined from the solid curvesin Fig. 2.15) exceed the above value of Δθ by a factor of 1.40 ± 0.11. The actuallifetime of the nuclei in the excited state exceeds by the same factor the quantity τunder conditions of the experiment being discussed. The above theory of perturbedADs predicts, on the basis of the known ratio Δ/Γ (which is 1.351 ± 0.05 in the casebeing considered), a value of 1.425 ± 0.009 for the average lifetime of excitednuclei.

Thus, our experiments confirm fully the conclusion of the theory of perturbedADs that the average lifetime of nuclei in an excited state depends on the form ofthe spectrum of exciting gamma rays. The results of this study were presented inour articles quoted in [61, 62].

It was indicated in [63] (p. 160) that, after the publication of our study in [61],the theoretical prediction that the result of perturbing, by a magnetic field, the ADof resonantly scattered gamma rays depends on the width of the spectrum of gammarays undergoing scattering were also confirmed in [64] (to be published), but Icould not find that publication. It was not mentioned in the last review of data on191Ir nuclide [65] either.

Experimental data obtained in [66] by using gamma rays of the 57Fe nuclide alsofurnish an explicit piece of evidence confirming the above conclusion that thelifetime of a nucleus in an excited state depends on excitation conditions—inparticular, on the shift of the exciting Mössbauer gamma line with respect to theposition of the excited-nucleus resonance. Among other things, the dependence ofthe number of counts in the resonance detector on the time t that passed from theopening of a fast shatter for the Mössbauer gamma line of energy 14.4 keV wasmeasured in that study. This dependence must be described by a function of theform 1—e–t/τ, where τ is the average lifetime of a nucleus in an excited state.

The measurements were performed at the shifts s of the exciting gamma linewith respect to the absorption line that were equal to 0, Г and 2.5 Г (Г is the naturalwidth of the state to be excited). The results of these measurements are presented inFig. 2.16. The second and third columns of Table 2.2 give, respectively, the τ valuesobtained by the authors of [66] from experimental data and the τ values calculatedby formula (1.86). One can see that the τ values obtained experimentally are veryclose to those expected theoretically.


http://dx.doi.org/10.1007/978-3-319-10524-6_1

Fig. 2.16 Experimental data from [66] that confirm the above theoretical conclusions (Eq. 1.86)on the dependence of the average time within which the nucleus remains in the excited state, τ0

*, onthe energy shift of the exciting Mössbauer line with respect to the position of the resonance ofabsorber nuclei. The numbers of counts in the resonance detector for 14.4 keV 57Fe gamma raysare plotted along the ordinate versus the time that passed from the instant at which the fast shatterwas opened for these photons. The detector records photoelectrons created in the interactions of allgamma rays emitted by the source with detector atoms and conversion electrons emitted by 57Featoms in the detector after the resonant absorption of 14.4 keV photons. The right scale corre-sponds to the total counting rate in the detector, while the left scale corresponds to the countingrate for conversion electrons alone. The coordinates of the points corresponding to the time τ0

* thatelapsed from the instant at which the shatter was opened are indicated by the dashed lines and thesymbol τ0

*


http://dx.doi.org/10.1007/978-3-319-10524-6_1

The difference of the experimental and calculated values, which exceeds slightlythe errors fotheshift s equal to 0 and Γ, is most likely due to some unrevealedsystematic uncertainties. It is noteworthy, however, that the authors of [66] interprettheir results from a different point of view, assuming that the theory relating theaverage lifetime of a nucleus in an excited state to the form of the spectrum ofexciting gamma rays is correct in the case of a “stationary” measurement regime inwhich our experiments with 182W and 191Ir were apparently performed, but it isinapplicable in the case of “nonstationary” regime. However, our “stationary”regime is such that the mean time interval between the instants at which photons hitthe detector exceeds considerably the duration of the process of resonant photonabsorption in the scatterer; therefore, it does not differ in dynamics from the pro-cesses that occur after the opening of the shatter in the experiments described in[66]. We note that the decay of absorber nuclei that was observed by the authors of[66] and which was characterized by the (e�t=s) law natural for the nuclei inquestion irrespective of the shift of the source gamma line is possible only under thecondition that the distribution of excitation energies of absorber nuclei has, in allcases, a Lorentzian shape with a width Г, but this may be so only if the spectrum ofexciting gamma rays has a width exceeding Г considerably. Under conditions of theexperiment being discussed, this may mean that the main part of absorber nucleithat proved to be in an excited state after the interruption of the gamma beam wasexcited by photons whose wave trains were cut off (shortened) by shatter-operationevents.

Let us now consider the question of possible deviations from the parallelism ofthe strength vector of the internal magnetic field at nuclei of some admixed atoms ina ferromagnetic material and the strength vector of the external magnetic fieldmagnetizing this ferromagnet [69]. The assumption that there may exist suchdeviations from the parallelism (antiparallelism in the case of a negative internalfield) in question stemmed from the fact that, in a number of experiments, thehyperfine-interaction energies measured by using internal magnetic fields turned outto be systematically lower than the values that followed from experiments per-formed with the same nuclei by using external magnetic fields and nonmagneticsubstances. Such data fitted satisfactorily in the scheme according to which thestrength vectors of the internal magnetic field at nuclei of admixed atoms in apolycrystalline ferromagnet lie along the generatrices of a cone whose axis isaligned with the strength vector of the external magnetic field. In some cases, halfof the cone opening angle (or the angle between the directions of the internal andexternal fields) may be 25o–30o at a magnetizing-external-field strength of about1 kOe. Although such a magnetic-field strength was sufficient for magnetizing the

Table 2.2 Thevalues of meanlifetime of the 14.4 keV state of57Fe nuclide independence onthe shift s of exciting gammaline measured in [66] andcalculated by formula (1.86)

s τexp.(ns) τcalc. (ns)0 230 ± 10 211.5

Г 160 ± 15 141

2.5Г 78 ± 15 90


http://dx.doi.org/10.1007/978-3-319-10524-6_1

ferromagnet almost completely, it was not sufficient for orienting the internal fieldat nuclei of admixed atoms along the direction of the external field. A nearlycomplete alignment of the field directions (or their nearly perfect antiparallelism inthe case of a negative internal field) was reached at an external-field strength as highas about 15 kOe. A qualitative model that is intended for describing the emergenceof this phenomenon, but which is not claimed to be highly precise, was proposed in[70]. According to this model, the introduction of an alien atom in the crystal latticeof a ferromagnet generates (mainly because of the difference in the size betweenthis atom and host-material atoms) additional intracrystalline interactions (in par-ticular, magnetostriction interaction), with the result that the magnetization of thesubstance in the immediate vicinity of the admixed atom differs from the averagemagnetization of the whole block of the substance. A deviation from the parallelismof the internal magnetic field at the nucleus of the admixed atom and the externalfield is a manifestation of an unsaturated magnetization of the substance in thevicinity of the admixed atom. All experiments in which indications of this phe-nomenon were obtained were performed under such conditions that the nuclei understudy occurred in a ferromagnet as a result of preceding nuclear processes (forexample, the penetration of recoil nuclei after Coulomb excitation or the formationof excited nuclei after beta decay). Under conditions of our experiment, there wereno transient processes associated with the appearance of the nucleus under study inthe ferromagnet immediately before the event of detection of the experimentallyobserved effect (in our case, resonant photon scattering). Iridium atoms wereintroduced in iron at the stage of manufacturing the alloy and were at their pre-assigned places in the course of the experiment (at crystal-lattice sites, as wasshown via an X-ray analysis of the scatterer). They did not experience recoil orexchange phonons with the crystal lattice because resonant absorption proceededvia the Mössbauer effect. Unfortunately, no experiments that could show thepresence or absence of an unsaturated magnetization of the substance in the vicinityof admixed atoms under such conditions have been performed thus far. If this effectexists (it can exist in our case because iridium atoms differ in size from iron atoms),then allowance for it must lead to some increase in the g-factor value measured byus for the 191Ir nucleus in the state at 129.4 keV. At the same time, the conclusionthat the time within which the nucleus remains in the excited state increases inrelation to τ remains valid because this conclusion follows from the observedpositions of the minima in the AD of resonantly scattered gamma rays: thesepositions differ from those that would correspond to the case where the averagelifetime of the nucleus in the excited state is τ.

In order to estimate roughly the change in the g-factor value upon taking intoaccount the deviation from the parallelism of the internal and external fields,we consider qualitatively the influence of magnetic fields not parallel to the normal tothe scattering plane on the AD of resonantly scattered gamma rays. In Fig. 2.17, q1and q2 are the wave vectors of, respectively, exciting and scattered photons and z isthe direction of the normal to the scattering plane. The strength vector H of theinternal magnetic field acting on the scattering nucleus is decomposed into twocomponents:Hvert. andHhoriz. The first of them causes the usual perturbation of ADs,


which is described by formula (1.53), while the second causes an additional pre-cession of the nuclear spin about its direction. Owing to this precession, photons that,in the absence of a magnetic field, would be emitted by nuclei situated near the pointO, for example, along the straight line OA lying outside the scattering plane S shownin Fig. 2.17 (or along the straight line OB in the case of the different precessiondirection) will find their way to the detector. The intensities of such gamma rays aredetermined by the scattering angle θ′,which exceeds slightly the angle θ. In responseto the reversal of the sign of the magnetizing field, the vector H also changesdirection. The precession of nuclear spins about the axis aligned with Hhoriz also hasan opposite direction, with the result that gamma rays that, in the absence of a field,would be emitted along the straight line OB find their way to the detector. Theirintensity is determined by the same angle θ′ because the angle of AD rotation doesnot depend on the sign ofΩ. Thus, the horizontal component of the internal magneticfield acts identically in the two cases: it changes the intensity of resonantly scatteredgamma rays as if the scattering angle increases slightly in relation to its adoptedgeometric value. As a result, theΩ value measured for one sign of the external field isslightly overstated; for its opposite sign, it is understated by nearly the same value.The averaging of the results obtained by measuring ADs for two signs of the externalfield must lead to a nearly complete elimination of the effect of the horizontalmagnetic-field component. If the vectors H lie at the generatrices of the cone whoseaxis coincides with z, then the horizontal projections of these vectors fill uniformlythe circle whose center is at the point O. It is quite obvious that this set of horizontalprojections will remain unchanged after the reversal of the external-field direction.The effect of this set of projections, whatever it may be, must vanish upon averagingthe results of the measurements for two signs of the external field. Therefore,a decrease in the effective strength of the magnetic field perturbing ADs andappearing in expression (1.53) is the only important consequence of the deviationfrom the parallelism of the internal and external fields—namely,Heff =Hvert =Hcosβ.

Fig. 2.17 Scheme clarifying the effect that a magnetic field not perpendicular to the scatteringplane exerts on the angular distribution of resonantly scattered gamma rays. The explanations tothe figure are given in the main body of the text


http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

Since the g-factor and Heff. appear in Ω as factors, which cannot be determinedindividually under conditions of the experiment being discussed, a decrease in one ofthem implies the respective increase in the other if Ω remains invariable.

If one accepts that, under conditions of our experiment, the angle β is about 25o

at Hext = 1 kOe [71], then the g-factor value appears to be 0.217 ± 0.020 instead of0.197 ± 0.018 (2.13). The result that one obtains by taking the average over thestudies reported in [55, 67, 68] and with which we compared our g-factor value willalso change because the authors of [55, 68] also employed an Ir–Fe alloy, so that itis also necessary to introduce a correction in their data that would take into accountthe deviation from the parallelism of the internal and external fields. After theintroduction of this correction, the average result of [55, 67, 68] becomesg = 0.230 ± 0.010. Thus, good agreement between the two values under comparisonsurvives after taking into account the deviation from the parallelism of the internaland external fields at iridium nuclei.

Over the time that elapsed from the completion of the aforementioned experi-ments with the Ir nuclide, there appeared several publications [72–75] that reportedon new results of measurements of both the average lifetime of this nucleus in theexcited state at 129.4 keV and the g-factor of this state. However, these results lookquite contradictory. The tables of nuclear magnetic moments in [72] present a valueof T1/2 = (1.29 ± 0.04) × 10–10 s as the recommended half-life of the state of the191Ir nucleus at 129.4 keV [from here, τ = (1.86 ± 0.06) × 10–10 s] and two valuesfor the magnetic moment of this state: 0.450 ± 0.023 and 0.485 ± 0.044 nuclearmagnetons (the latter is our result obtained in [62] and rescaled with allowancefor the T1/2 value adopted in those tables). The average magnetic moment followingfrom these data is 0.458 ± 0.020 nuclear magnetons, which corresponds to ag-factor value of 0.183 ± 0.008.

In [73], the average lifetime of the 191Ir nucleus in the excited state of interest wasdetermined by measuring the dependence of the ratio of the intensity of the gammaline shifted via the Doppler effect to the intensity of the unshifted line on the meanfree path of recoil nuclei after Coulomb excitation. It turned out to be(1.776 ± 0.060) × 10–10 s, which is very close to the value that we used. The g-factorof the 129.4 keV state was measured in that study by two methods: by using theperturbation of the angular distribution of gamma rays by the transient magnetic fieldacting on 191Ir nuclei that, after undergoing Coulomb excitation, traverse the mag-netized iron and by using the perturbation of the angular distribution of gamma raysby the transient magnetic field acting on 191Ir nuclei that, after undergoing Coulombexcitation, traverse the magnetized iron and by using the perturbation of the angulardistribution of gamma rays by the static magnetic field acting on nuclei stopped inmagnetized iron. The resulting value was g = 0.180 ± 0.009 in the first case andg = 0.172 ± 0.013 in the second case. It is likely that corrections for the deviationfrom the parallelism of the internal and the magnetizing field must be introduced inthese values. One can see that these g-factor values are close to the value obtained inour experiment. However, a different research group, who used a similar method,reported [74, 75] data that were in a glaring contradiction with the results of almostall of the studies performed by the usual methods involving the application of static


magnetic fields. Specifically, g-factor values of 0.342 ± 0.024 and 0.322 ± 0.022 arepresented in [74, 75], respectively, for the 129.4 keV state of the 191Ir nucleus. Thevalue obtained in [75] for the average lifetime of the 191Ir nucleus in this excited stateis (1.268 ± 0.023) × 10–10 s, which contradicts the results of the majority of previousstudies, including the result presented in [73]. It turns out that the situation isunfavorable not only in the Danish kingdom. I am inclined to the opinion that dataobtained under stationary conditions are more reliable since the values of effectivemagnetic fields are quite certain in that case and since average lifetimes of excitednuclei are then measured by the reliable method of γ–γ and β–γ coincidences. Ourdata agree well with data obtained by these methods.

2.6 Some Special Features of Gamma-Radiation Processesas Suggested by the Foregoing Analysis

Stepanov and Zipenyuk indicate in [76], referring to the book by Migdal [77], thatthe duration of the process of photon emission from a nucleus [and, hence, of theprocess of resonant photon absorption by a nucleus (author)] is about , where

is the gamma-radiation wavelength divided by 2π and c is the speed of light. Forgamma rays of energy about 100 keV, this quantity is about 3 × 10–20 s. We willnow show that there are reasons to doubt that the duration of such processes is soshort. For this, we calculate the average lifetime of nuclei in the excited statepopulated after Mössbauer resonant photon absorption, assuming that the time ofabsorption is much shorter than τ = ћ/Γ. For this purpose, one can make use of thequantum-mechanical Krylov–Fock theorem [78], which states that the decay lawfor an excited quantum system is determined completely by the excitation-energydistribution in the initial state. In the case of an“ideal” Mössbauer resonant gamma-ray scattering, the emission and absorption gamma lines are characterized naturalwidths, are not shifted with respect to each other, and are described by theLorentzian function

F Ec� � � C2�

4

Ec � E0� �2 þC2�

4; ð2:28Þ

where Eγ is the photon energy and E0 is the position of the resonance center. Theexcitation-energy distribution W(Eγ) of nuclei excited upon the resonant absorptionof such photons is proportional to the product of two functions in (2.28); that is,

W Ec� � � C2�

4� �2

Ec � E0� �2 þC2�

4h i2 : ð2:29Þ


This function is no longer a Lorentzian function; therefore, one cannot expectthat the decay of an excited nuclear state will follow the exponential law. Thespectrum of photons emitted by excited nuclei will have the same form. One canprove that the FWHM of the gamma line described by expression (2.29) is smallerthan C by a factor of about 1.5. From this fact alone, it follows that the averagelifetime of nuclei in an excited state, tav, must be longer than τ by a factor of about1.5. But from the rigorous theory described above, it follows, as we have alreadyseen, that the relation tav = 1.5 τ holds exactly. According to the Krylov–Focktheorem, the probability L(t) for an excited system to remain undecayed after thelapse of the time t from its creation is determined by the excitation-energy distri-bution W(Eγ) formed upon the emergence of the excitation within a time short inrelation to τ and is given by

L tð Þ ¼Z

e�iEc�h tW Ec

� �dE

2

: ð2:30Þ

If we take expression (2.20) for W(Eγ), it is necessary to calculate the modulussquared of the integral

I ¼ C2

4

�2 Z1�1

e�iEc�h t dEc

Ec � E0� �2 þC2�

4h i2: ð2:31Þ

The integral is evaluated by taking the residue at the second-order pole in thelower half-plane. Omitting factors independent of t, one obtains

L tð Þ � 2 þ Ct�h

�2

e�Ct�h : ð2:32Þ

The number of decays per unit time at the instant t is determined by the timederivative of the function L(t):

dL tð Þdt

¼ �C2t

�h22þ Ct

�h

�e�

Ct�h : ð2:33Þ

The average lifetime of nuclei in an excited state turns out to be

tav ¼

R1�1

t dL tð Þdt dt

R1�1

dL tð Þdt dt

¼ 2:5�hC¼ 2:5s ð2:34Þ


instead of 1.5 τ, a result that follows from the width of the gamma line emitted bythe scatterer; from the theory of magnetic-field-perturbed angular distributions ofresonantly scattered gamma rays; and, finally, from the experimentally measuredangle of rotation of the rosette of such an angular distribution of resonantly scat-tered 191Ir gamma rays. This was first indicated in our article quoted in [79]. It isnoteworthy that the experimental value obtained in our study for tav differs by tenstandard deviations from the tav value given by the Krylov–Fock theorem. Thisresult is likely to be sufficient for disproving the statement that, in the process of“ideal” Mössbauer resonant scattering, the nucleus involved absorbs (and, hence,emits) a photon within a time that is short in relation to τ. As a matter of fact, theresult in question indicates that an excitation-energy distribution of width C=1.5cannot be formed within a time shorter than 1.5�h=C. Evidently, one has to abandonthe point of view according to which it is sufficient for nuclei to remain “passively”in the excited state within the time s ¼ �h=C, on average, for the formation of agamma-ray spectrum with a width C. Instead, one has to accept the concept of anexcited nucleus as a generator of electromagnetic oscillations that acts within thewhole time over which the nucleus remains in the excited state. Below, we willshow that there are other physics factors supporting the concept of a protractedcharacter of nuclear radiative processes.

As for our experiments devoted to measuring the magnetic-field- perturbed ADsof resonantly scattered gamma rays of 191Ir, it follows from them that, if theMössbauer gamma line exciting nuclei has a natural width, the average lifetime ofthese nuclei in the respective excited state is 1.5 τ, as the theory requires. Thismeans that the transition of the nucleus from the ground to an excited state and theinverse transition last a short time in relation to τ. Thus, we arrive at the conclusionthat, although nuclear transitions proceed via photon absorption and emission, thetime scales of these transitions and radiative processes are drastically different. Atthe onset of gamma-wave interaction with a nucleus, the latter goes over quickly toan excited state, acquiring its spin, parity, and magnetic moment, but the energy ofthis state has a large uncertainty over the initial period of gamma-wave absorption.This is because the excitation-energy distribution at this instant must be determinedby the frequency spectrum (Fourier integral) of that part of gammaradiation wavetrain which had time to act on the nucleus by this instant. The longer the time withinwhich the wave train acts on the nucleus, the narrower the excitation-energy dis-tribution and, accordingly, the higher the degree to which the energy of the excitedstate turns out to be determined. The broadness of the excitation-energy distributionwithin the initial period of the gamma-wave-absorption process must stem from ahigh probability of photon emission over this period. Owing to this enhancement ofthe decay, the value 1.5 τ appears instead of the value 2.5 τ required by theKrylov–Fock theorem. It is noteworthy that, according to the Krylov–Fock theo-rem, the form prescribed by this theorem for the initial excitation-energy distri-bution remains invariable in the course of time—only its intensity changes. In ourmodel, the form of this distribution must also change: it becomes narrower in thecourse of the protracted process of gamma-wave interaction with the nucleus.

2.6 Some Special Features of Gamma-Radiation Processes… 71

Under these conditions, the Krylov–Fock theorem cannot describe correctly theprocess of decay of an excited nuclear state. This theorem is valid in the case wherethe distribution W(Eγ) arises within a time that is much shorter than the charac-teristic time of decay of the respective excited state; moreover, the form of thedistribution arising at the initial instant must not change under any external effectsdisturbing the nuclei during the decay process. Such conditions arise, for example,in experiments aimed at observing magnetic-field-perturbed ADs of resonantlyscattered bremsstrahlung photons that are generated by decelerated electrons. In thatcase, the wave trains of the radiation to be absorbed are very short, while the photonspectrum is wide; as a result, one can consider this spectrum in the region of thenuclear resonance as an energy-independent constant. As a result, the excitation-energy distribution is described by the Lorentzian function (2.28) and is formedwithin a time interval short in relation to τ. For this case, expression (1.82) forΔ >> Γ and the Krylov–Fock theorem give identical values, equal to τ, for theaverage time within which nuclei remain in the excited state.

It is noteworthy that, by definition, the values of tav that are obtained fromexperiments devoted to measuring magnetic-field-perturbed ADs of resonantlyscattered gamma rays and from the Krylov–Fock theorem have the same meaning.In either case, one averages the lifetime of a nucleus over the time dependence ofthe photon-emission probability. In the first case, one averages the angle of rotationof the AD rosette, Δθ = Ωt (t is the lifetime of a specific nucleus in the excitedstate), with respect to its unperturbed position, and this is equivalent to the aver-aging of t. In the second case, one averages the value of t directly. Naturally, theuncertainty principle plays a decisive role in the nuclear radiative processes beingconsidered: one cannot create the excitation-energy distribution of characteristicwidth C by acting on nuclei within a time shorter than �h=C, nor is it possible toobtain a gamma line of width C unless each excited nucleus emits a gamma wavetrain within τ = ħ/Г, on average.

In connection with the foregoing, we will try to estimate the scale of theuncertainty in the spatial position of a photon. From two well-known relationsCs ¼ �h; and Dp � Dx �h, it follows that CsDp� Dx. For the photon, we haveDp ¼ DE

c , where E is its energy and c is the speed of light. Obviously, the uncer-tainty ΔE in the photon energy must be set to C. We then have Cs C

c � Dx orΔx * cτ. This means that the uncertainty in the photon coordinate x is on the sameorder of magnitude as the length of the wave train emitted by the nucleus involvedover its average lifetime in the excited state. It is noteworthy that one cannotinterpret this estimate of Δx as a consequence of the uncertainty in the photon-emission instant within τ, assuming that this process is short-term. In the abovederivation, the instant of the nuclear transition to the excited state was not fixed inany way, so that this instant cannot be used to reckon time from it. Moreover, τ canalso be determined experimentally without fixing of this instant, as was done in ourexperiments discussed above and devoted to studying the magnetic-field-perturbedADs of resonantly scattered 191Ir gamma rays. Unfortunately there is no until nowthe explanation why the wave train of a photon lose suddenly wave properties and


http://dx.doi.org/10.1007/978-3-319-10524-6_1

behaves as a like point particle, for example in the Compton scattering. Here is aninteresting picture of a similar process not giving however its cause and mechanism.Imagine the two-dimensional plane with zero thickness on which the fragment ofstraight line is moving. Suddenly this fragment turns so that its ends go out in thethird dimension. The point is left on the plane—the place of the fragment inter-section with it. The “wave” turned into “particle” and this happened during the zerotime independently on the speed of the fragment rotation.

We now consider the Fourier frequency spectrum of an extremely short wavetrain of a photon as yet another argument in favor of the statement that gamma-rayemission and absorption are protracted processes. If one admits, in accordance withMigdal’s treatment, that the time of photon emission is , then the length of thecorresponding wave train must be about 1/6 of the period T; that is, it must have theform sinωot, where 0 < t<T/2π. Calculating the Fourrier integral for a signal ofduration equal to one period of oscillations, we obtain

f xð Þ ¼ 12p

ZT0

sinx0t eixtdt ¼ 1

2p

ZT0

eix0t � e�ix0t

2ieixtdt ¼ 1

4pi

ZT0

ei x0þxð Þt � e�i x0�xð Þ th i

dt

¼ � 14p

e2pixx0 � 1

xþ x0þ e2pi

xx0 � 1

x� x0

" #¼ e2pi

xx0 � 12p

� x0

x2 � x20:

ð2:35Þ

The squared modulus of this amplitude is

f xð Þj j2¼ 12p2x2

0

1

xx0

� �2�1

� �2 1� cos 2pxx0

�¼

x20 1� cos 2p x

x0

� �2p2 x2

0 � x2� �2 : ð2:36Þ

The graph of this function is shown in Fig. 2.18. The width of the resultingfrequency spectrum is many times larger than the natural width of nearly anygamma line emitted by excited nuclei. If, for example, one locates the Mössbauergamma line of 57Fe (14.4 keV) at the point ω/ω0 = 1, then its natural width is about3.3 × 10–13 on the scale of Fig. 2.18.

There may arise the question of whether the manifestation of a spatial and a timeextension of photons is due to the possibility of discriminating between photons bytheir origin. First of all, we note that, in detecting a single photon, one cannot revealits wave properties. In order to observe the interference and diffraction patterns andto establish the resonant character of photon interaction with nuclei, one has todetect a large number of photons. At the same time, corpuscular properties of aphoton such as its momentum and energy and the coordinates of the vertices of itsinteraction with the detector material can be determined upon detection by theCompton effect or by the photoabsorption of a single photon. It can be stated thatphotons of different origin cannot be distinguished in processes where their cor-puscular properties are revealed. At the same time, photons of different origin are


distinguishable by their wave properties, but this distinction manifests itself only inan ensemble of photons—one cannot notice them in detecting single photons. Forexample, electron bremsstrahlung followed by nuclear absorption creates a naturalLorentzian distribution of excitation energies, which leads to a natural averagelifetime of nuclei in the excited state and to the exponential decay law. However,the irradiation of the same nuclei with Mössbauer photons resonant for them cre-ates, as was shown above, an excitation-energy distribution that is 1.5 times nar-rower, which is associated with the corresponding increase in the average lifetimeof excited nuclei. We note that the need for detecting many microscopic objects inorder to reveal their wave properties permits pinpointing the category (wave versuscorpuscular) to which one measured characteristic of a microscopic object oranother belongs. For example, the particle spin, which cannot be determined indetecting a single particle, must be classed with wave characteristics of the object.Indeed, it is the spin orientation that determines the particle-beam polarization,which is of course a wave characteristic.

We now revisit expression (1.87), which refers to the case where a nucleus isexcited by gamma rays whose spectrum has a characteristic width Δ much smallerthan the natural width Γ of the excited nuclear level. Obviously, the average lifetimeof excited nuclei emitting such a narrow gamma line (of course, these nuclei are notthose that are excited) is Δ/Γ times as long as τ = ћ/Γ. It seems that wave trainscorresponding to photons of such a narrow spectrum must act on the nuclei to beexcited over the time ћ/Δ >> ћ/Γ, on average. From Eq. (1.87), it follows, however,that this time does not exceed 2ћ/Γ, becoming still shorter in the presence of a shiftof the exciting gamma line with respect to the absorption line. This means that theexcited nucleus terminates the process of resonant interaction with the excitingphoton earlier than, as may appear, the completion of the action of a long wave train

Fig. 2.18 Fourier integralIF(ω) for a single period of thesinusoid sin(ωot). The verticalstraight line ω/ωo = 1represents symbolically theMössbauer gamma line ofenergy ћωo. If it was thegamma line of the Mössbauernuclide 57Fe, then its FWHMvalue would be about3.1 × 10−13 on the scale ofthis figure


http://dx.doi.org/10.1007/978-3-319-10524-6_1

http://dx.doi.org/10.1007/978-3-319-10524-6_1

on the nucleus must occur. With it all, however, the nucleus absorbs completely theenergy of the exciting photon. This is because the energy of a photon is determinedby its frequency rather than by the number of oscillations in the corresponding wavetrain. A mysterious disappearance of the unabsorbed wave-train residue is one ofthe manifestations of the so-called collapse of the photon wave function—one ofthe most enigmatic phenomena of quantum physics.

Let us imagine an experiment in which scatterer nuclei are excited by gammarays of a uniform continuous spectrum. Nuclei excited under these conditions emitphotons forming a gamma line of natural width because, in this case, there arises anexcitation-energy distribution of the Lorentzian shape, which is natural for thenuclei being considered, so that the decay of excited states will be exponential.

Exponential gamma decay occurs in those cases where the nuclei involved areexcited by gamma rays whose wave trains are short and the frequency distributionsare wide. This situation may be exemplified by the excitation of nuclei by electronbremsstrahlung, as well as by Coulomb excitation.

Pick out from the excited nuclei ensemble the group of nuclei with exit actionenergies lying in a narrow area around the energy Eγ. Evidently these nuclei wereexcited by resonant absorption of gamma rays related to the such narrow area ofenergy. Therefore the mean life time for nuclei of this group must be determinate beformula (1.87). To obtain the mean life time for total ensemble of excited nuclei onehas to average the quantity related to narrow line over the excitation energy dis-tribution of the ensemble. If this distribution has f Lorentzian shape we obtain

tav ¼

R1�1

C2=4

E0�Ecð Þ2þC2=4

�C2=2

E0�Ecð Þ2þC2=4

s dEc

R1�1

C2=4

E0�Ecð Þ2þC2=4

dEc

¼ s: ð2:37Þ

This means that the energy of a photon plays a significant role in this procedure,but that its frequency spectrum does not affect the result.

In a similar way, one can calculate the average lifetime of excited nuclei for thecase of “ideal” Mössbauer resonant gamma-ray scattering as well—that is, for aprocess in which the emission and absorption lines have the Lorentzian shape withnatural width and in which there is no shift between them. In this case, the dis-tribution of excitation energies is determined by the product of two identical

Lorentzian functions. For tav, we then have

tav ¼

R1�1

C2=4

E0�Ecð Þ2þC2=4

" #2

�C2=2

E0�Ecð Þ2þC2=4

s dEc

R1�1

C2=4

E0�Ecð Þ2þC2=4

" #2

dEc

¼ 1:5s: ð2:38Þ


http://dx.doi.org/10.1007/978-3-319-10524-6_1

Thus, a recipe for calculating the average lifetime of nuclei in an excited statehave been obtained for the case where the excitation-energy distribution W(Eγ) isknown. It is the following:

tav ¼

R1�1

W Ec� � �

C2=2

E0�Ecð Þ2þC2=4

s dEc

R1�1

W Ec� �

dEc

: ð2:39Þ

Of course, the spectrum of gamma rays emitted by nuclei of a resonant scattereris identical in all cases to the excitation-energy distribution WðEcÞ.

References

1. R.F. Metzger, Nucl. Phys. 27, 612 (1961)2. A.V. Davydov, V.P. Selyutin, Bull. Acad. Sci. USSR: Phys. 27, 875 (1963)3. L.L. Lee, L. Meyer-Schutzmeister, J.P. Schiffer, D. Vincent, Phys. Rev. Lett. 3, 223 (1959)4. A. Bussiere de Nercy, M. Langevin, M. Spigel, Comp. Rend. 250, 1031 (1960)5. E. Kankeleit, Z. Physik 164, 442 (1961)6. O.I. Sumbaev, A.I. Smirnov, V.S. Zykov, Soviet JETP 42, 115 (1962). (in Russian)7. Radionuclide Transformations—Energy and Intensity of Emissions. ICRP Publication 38

(Pergamon Press, Oxford, 1983), 182Ta8. K.C.J. Goebel, W. McVoy, Phys. Rev. 148, 1021 (1966)9. N.A. Voinova, B.S. Dzhelepov, Isobaric Nuclei with Mass Number A = 182 (Nauka,

Leningrad, 1968). (in Russian)10. M.R. Schmorak, Nucl. Data Sheets 14, 559 (1975)11. E.A. Phillips, L.Grodzins, in Perturbed Angular Correlations, ed. by E. Karlsson, E. Matthias,

K. Siegbahn, (North-Holland Publishing Cmp, Amsterdam, 1964), p, 29412. D.A. Agresti, E. Kankeleit, B. Persson, Phys. Rev. 155, 1342 (1967)13. N. Sikazono, H. Takekoshi, T. Shoji, J. Phys. Soc. Jpn 20, 271 (1965)14. S.G. Cohen, N.A. Blum, Y.W. Chow et al., Phys. Rev. Lett. 16, 322 (1966)15. R.B. Frankel, Y.W. Chow, L. Grodzins, J. Wulff, Phys. Rev. 186, 381 (1969)16. A.V. Davydov and O.N. Sorokin, in The Results of the Experiment. Adduced in the lecture by

A.V. Davydov Resonant Scattering of Gamma Rays, ed. by A.F. Ioffe, Proceedings of 5-thwinter school on nuclear theory and physics of high energy of LPTI, part II, Leningrad (1970),p. 263 (in Russian)

17. J.A. Cameron, L. Keszthelyi, G. Mezei et al., Can. J. Phys. 50, 736 (1972)18. Y.W. Chow, L. Grodzins, P. Barrett, Phys. Rev. Lett. 15, 369 (1965)19. L. Grodzins, Ann. Rev. Nucl. Sci. 18, 291 (1968)20. G. Goldring, R. Kalish, H. Spehl, Nucl. Phys. 80, 33 (1966)21. R.L. Mössbauer, Z. Physik 151, 124 (1958)22. R.L. Mössbauer, Z. Naturforsch. 14a, 211 (1959)23. F. Wittmann, Z. Naturforsch. 19a, 1409 (1964)24. G.H.R.Kegel, Thesis MIT. Quoted in [28]25. H. Korner, J. Radeloff, E. Bodenstedt, Z. Physik 172, 279 (1963)26. G. Goldring, Z. Vager, Phys. Rev. 127, 929 (1962)27. V.A. Klyucharev, A.K. Valter, I.I. Zalyubovskii, V.D. Afanasiev, Soviet JETP 17, #3, (1963)


28. R.P. Scharenberg, J.D. Kurfess, G. Schilling et al., Nucl. Phys. 58, 658 (1964)29. W. Ebert, O. Klepper, H. Spehl, Nucl. Phys. 73, 217 (1965)30. B. Persson, H. Blumberg, D. Agresti, Phys. Rev. 170, 1066 (1968)31. I. Ben Zvi, P. Gilad, M.B. Goldberg et al., Nucl. Phys. A151, 401 (1970)32. D.A. Garber, M. Behar, W.M.C. King et al., Phys. Rev. C9, 2399 (1974)33. A.M. Muminov, Abstract of a physics-mathematics candidate dissertation, Tashkent, 1972. (in

Russian)34. A.Z. Dolginov, Gamma-rays (Publishing house of Academy of Sciences of USSR, 1961),

p. 523 (in Russian)35. G. Frauenfelder, R. Steffen, Angular Correlations, in Alfa-, Beta- and Gamma-Ray

spectroscopy, ed. by K. Siegbahn (North-Holland Publishing Company, Amsterdam, 1965).V.2

36. A.V. Davydov, G.R. Kartashov, Y.V. Khrudev, Phys. Nucl. A 7, 447 (1968)37. K.S. Krane, At. Data Nucl. Data Tables # 2,137 (1976)38. B.V. Narasimha Rao, S. Jnanananda, Proc. Phys. Soc. 87, part 2, 455 (1966)39. E.E. Berlovich, Yu.K. Gusev, V.V. Il’in, M.K. Nikitin, Soviet JETP 43, 1625 (1962) (in

Russian)40. J. Lindskog, T. Sundström, P. Sparrman, Z. Physik 170, 347 (1962)41. S.G. Malmskog, A. Bäcklin, Ark. f. Fysik 39, 411 (1969)42. P.P. Craig, J.C. Dash, A.D. McGuire et al., Phys. Rev. Lett. 3, 221 (1959)43. P. Steiner, E. Gerdau, W. Hautsch, D. Steenken, Z. Physik 221, 281 (1969)44. B. Balko, G.R. Hoy, Phys. Rev. B 10, 36 (1974)45. W. Meisel, L. Keszthelyi, Hyperfine Interact. 3, 413 (1977)46. M.B. Lewis, Nucl. Data Sheets 9, 479 (1973)47. F. Wagner, G. Kaindl, P. Kienle et al., Z. Physik 207, 500 (1967)48. S. Gustaffsson, K. Johansson, E. Karlsson et al., Ark. f. Fysik 34, 169 (1967)49. F. Wagner, W. Potzel, in Hyperfine Interactions in Excited Nuclei, ed. by G. Goldring, R.

Kalish, V. II, (Gordon and Breach Science Publishers, New York-London-Paris, 1971), p. 68150. R.L. Mössbauer, M. Lengsfeld, W. Von Lieres et al., Z. Naturforsch. 26a, 343 (1971)51. J.A. Cameron, J.A. Campbell, J.F. Compton et al., Nucl. Phys. 59, 475 (1964)52. U. Atzmoni, E.R. Bauminger, D. Lebenbaum et al., Phys. Rev. 163, 314 (1967)53. M. Kontani, J. Itoh, J. Phys. Soc. Jpn 22, 345 (1967)54. V. Kogan, V.D. Kul’kov, L.P. Nikitin et al. Soviet JETP 18, #2 (1964)55. W.R. Owens, B.I. Robinson, S. Jha, Phys. Rev. 185, 1555 (1969)56. P.C. Riedli, Phys. Rev. B 8, 5243 (1973)57. K. Johansson, E. Karlsson, L.O. Norlin, International Conference on Hyperfine Interactions

Studied in Nuclear Reactions and Decay (Uppsala, Sweden, 1974) (Contributed papers.Upplands Grafiska AB. Uppsala, 1974), p. 164

58. I. Vincze, Solid State Commun. 10, 341 (1972)59. A. Narath, Phys. Rev. 165, 506 (1968)60. F. Wagner, J. Klöckner, N.J. Körner et al., Phys. Lett. 25B, 253 (1967)61. A.V. Davydov, M.M. Korotkov, P.I. Romasheva, JETP Lett. 31, 560 (1980)62. A.V. Davydov, M.M. Korotkov, P.I. Romasheva, Bull. Acad. Sci. USSR: Phys 44, 1 (1980)63. F.E. Wagner, Hyperfine Interact. 13, 149 (1983)64. R. Ehehalt, E. Nolte, F.E.Wagner, H.J. Körner (To be published)65. V.R. Vanin, N.L. Maidana, R.M. Castro et al., Nucl. Data Sheets 108, 2393 (2007)66. G.V. Smirnov, Y.V. Shvyd’ko, Soviet Phys. JETP 68, 444 (1989)67. R. Avida, I. Ben Zvi, P. Gilad et al., Nucl. Phys. A147, 200 (1970)68. N.F. Il’khamdzhanov, P.S. Radzhapov, K.T. Salimbaev, Izvestia Acad. Sci. USSR, Ser. Phys.

Math. 4, 79 (1973) (in Russian)69. K.S. Krane, W.A. Steyert, Phys. Rev. C 9, 2063 (1974)70. A. Aharoni, Phys. Rev. B 2, 3794 (1970)71. K.S. Krane, B.T. Murdoch, W.A. Steyert, Phys. Rev. Lett. 30, 321 (1973)72. P. Raghavan, At. Data Nucl. Data Tables 42, 189 (1989)

References 77

73. W.R. Kölbl, J. Billowes, J. Burde et al., Nucl. Phys. A456, 349 (1986)74. A.E. Stuchbery, S.S. Anderssen, E. Bezakova, Hyperfine Interact. 97/98, 479 (1996)75. E. Bezakova, A.E. Stuchbery, H.H. Bolotin et al., Nucl. Phys. A669, 241 (2000)76. A.V. Stepanov, YuM Tsipenyuk, Phys. Usp. 43, 79 (2000)77. A.B. Migdal, Qualitative Methods in Quantum Theory (W.A. Benjamin Inc, London,

Amsterdam, Don Mils. Ontario, Sydney, Tokyo, 1977)78. N.S. Krylov, V.A. Fock, Soviet JETP 17, 93 (1947). (in Russian)79. A.V. Davydov, P.I. Romasheva, Bull. Acad. Sci. USSR: Phys. 45, 58 (1981)


Chapter 3Problem of the Resonant Excitationof Long-Lived Nuclear Isomeric States

3.1 Small Introduction

Shortly after R. Mössbauer had discovered the phenomenon of recoilless gamma-ray emission and absorption by nuclei, the present author noticed that both silverisotopes have comparatively low-lying levels of rather long mean lifetime. Therearose the impression that these isomeric states could be excited with the aid of theMössbauer effect according to an unusual method that would employ their longlifetimes. It seemed possible to perform an experiment in which one first manu-factures a rather strong gamma source and irradiates, with its gamma rays, a res-onant absorber for several half-lives at low temperature and thereupon removes thesource quickly behind a radiation screen, places the irradiated absorber at a detector,and observes the gamma activity induced in it. Of course, it was necessary to tunethe detector to recording gamma rays of required energy and, in addition, to verifywhether the absorber activity decreases in accordance with the half-life of theisomeric state being studied. The experiment seemed simple and promised to besuccessful. However, the first attempt aimed at observing the gamma activation of asilver absorber and made almost immediately after the idea of such an experimentarose failed. A piece of palladium foil was irradiated with thermal neutrons at theITEP heavy-water reactor in order to create in it a beta activity of the 109Pd nuclide(T1/2 = 13.427 h). Upon the decay of these nuclei, there arise nuclei of 109Ag in theexcited isomeric state at 88.03 keV. We irradiated a silver plate with gamma rays ofthis source, whose activity was higher than 1 Ci, at liquid-nitrogen temperature andtried to observe its gamma activity with aid of a scintillation counter. The failure ofthat experiment was due to several reasons, which became clear to us somewhatlater. The point is that, in contrast to silver, palladium is paramagnetic; therefore themicro-spectrum of gamma rays emitted by a palladium gamma source does notensure resonance conditions for gamma-ray absorption in diamagnetic silver. Ofcourse, there could exist an isomeric shift of gamma-transition energies betweensilver nuclei created in the palladium source and the silver absorber. Moreover, the


79

probability for the recoilless absorption of 88.03 keV gamma rays in metallic silverat liquid-nitrogen temperature is as low as 0.0076 (it is 0.0535 at liquid-heliumtemperature). This means that the probability for the Mössbauer absorption of109Ag gamma rays in metallic silver is about 50 times lower at liquid-nitrogen thanat liquid-helium temperature. Later, the question of excitation conditions for long-lived nuclear isomeric states was studied in detail by the present author togetherwith the late professor N.A. Burgov, and some pieces of valuable advice were givento them by I.Ya. Pomeranchuk and L.D. Landau in the course of that study. Itbecame clear that there were a number of reasons that could hinder a successfulimplementation of Mössbauer experiments with long-lived isomers.

3.2 Physical Reasons Behind Expected Difficultiesin Performing Mössbauer Experiments with Long-Lived Isomers

In accordance with the generally accepted point of view, there are several reasonshindering the implementation of Mössbauer experiments with very narrow nuclearlevels. We first consider reasons that could seemingly lead to the broadening ofgamma lines of long-lived isomers, which is to be accompanied by a decrease in theresonant-absorption cross section. They are the following:

1. The interaction of the magnetic moments μe of excited nuclei with the magneticmoments μg of surrounding nuclei in the ground state broadens both theemission line of a source and the absorption line of a resonant absorber. As amatter of fact, this interaction is the usual interaction of a magnetic moment withthe magnetic field created another magnetic moment.

The Zeeman effect, to which such an interaction leads, must give a smearedpattern instead of a finite number of hyperfine-structure emission and absorptionlines because of different and time-dependent orientations of nuclear magneticmoments and because of different distances between the nuclei.

The energy of interaction between the magnetic moments μe and μg of nucleiin, respectively, an excited and the ground state can be calculated by the generalformula for the energy of magnetic dipole–dipole interaction [1]; that is,

U ¼ d1d2ð ÞR2�3 d1Rð Þ d2Rð Þ� �=R5; ð3:1Þ

where d1 and d2 are the vectors of two interacting dipole magnetic moments andR is the point-to-point vector between the centers of the systems that have theabove two magnetic moments (the dimensions of the systems are assumed to besmall in relation to R). The maximum value of U corresponds to the case whered1 and d2 are antiparallel and are aligned along the straight line coinciding withR. In this case, we have U = Umax. = 2d1d2/R

3; in our notation, this becomesUmax: ¼ 2lelg=R

2, where μe and μg are the magnetic moments of a nucleus in,

80 3 Problem of the Resonant Excitation of Long-Lived Nuclear…

respectively, an excited and the ground state. Using this expression and thevalues of μe and μg from [2, 3] and setting R = 2.6 × 10−8 cm, we estimate U at0.99 × 10−12 eV. Since the energy U changes at random from –Umax. to +Umax.,this hyperfine interaction alone must lead to the broadening of Mössbauergamma lines by five orders of magnitude and to a decrease in the cross sectionfor resonant gamma-ray absorption in the same proportion.

2. The interaction of magnetic moments of excited and ground-state nuclei withmagnetic moments of paramagnetic-admixture atoms must also broaden theresonant gamma line, but one can in principle reduce this broadening to anyrequired value by increasing the purity of the substances applied in fabricatingthe gamma source and absorber. For this broadening to be not larger than thevalue generated by the reasons indicated in the preceding item, the relativecontent of paramagnetic-admixture atoms must not be greater than 10−5.

3. The interaction between magnetic moments of nuclei and those of conductionelectrons in metallic samples must also broaden the resonant gamma line.However, the number of electrons involved in the interaction with nuclei ismodest because, at a temperature T, only a small part of the electrons, about kT/EF (EF is the Fermi energy), are able to exchange energy portions of about kT.At T = 4.2 K and EF = 7 eV, we have kT/EF = 5 × 10−5. It follows that theresonant-gamma-line broadening caused by this mechanism is on the same orderof magnitude as that which is generated by paramagnetic admixtures of relativeconcentration about 10−5. Clearly, this mechanism becomes inoperative if onedeals with nonmetallic samples.

4. The relative vibrational motion of nuclei emitting and absorbing gamma raysalso broadens the resonant gamma line. If a nucleus moving at a velocity v emitsa photon at an angle θ with respect of the direction of its motion and if theenergy of this photon in the center-of-mass frame is Eγ, then the photon energyE in the laboratory frame is Eγ − E = E cos θ × v/c to a rather high degree ofprecision.If one accepts that ΔEγ = Eγ−E ≤ 10−12 eV, then, at Eγ ≈ 105 eV, velocityv must be not higher then 3 × 10−7 cm/s. This corresponds to the maximumpermissible amplitude value of the relative velocity of nuclei emitting andabsorbing a photon if one requires to keep the acoustic broadening of the gammaresonance below 10−12 eV. Since the wavelengths of sound and ultrasoundvibrations in solid bodies lie between about 10 cm and 100 m for ordinaryfrequencies (10–104 Hz) and since the distance between the emitting andabsorbing nuclei may be smaller than 1 mm, one can ensure, upon joining thegamma source and absorber into a united rigid system, fulfillment of conditionsunder which the vibration-phase difference between emitting and absorbingnuclei is much less than π for the whole frequency range. This permits sub-stantially relaxing the requirements on the level of acoustic vibrations.

5. The broadening of a Mössbauer gamma line may be caused by the presence ofthe electric quadrupole interaction of nuclear quadrupole moments with non-uniform electric fields in the source and absorber substances under the condition

3.2 Physical Reasons Behind Expected Difficulties in Performing… 81

that the gradients of these fields are not identical in different parts of the samples(if the electric-field gradient is constant over the sample volume, the gamma linesplits into a finite number of components). In perfect cubic crystals, there is noquadrupole interaction. If, however, the crystal lattice is distorted by the pres-ence of mechanical or radiation defects or by the presence of admixtures,nonuniform electric fields may appear. They can be partly removed by annealingsamples.

6. The temperature-dependent broadening of gamma lines was first consideredtheoretically in [4] and later in [5]. At very low temperatures, this broadeninggrows in proportion to temperature raised to the seventh power. Estimates of therelative temperature-dependent broadening of the 88.03 keV 109mAg gammaline are presented in Table 3.1. Upon going over from 4.2 to 25 K, the prob-ability for the recoilless emission (absorption) of a 88.03 keV photon in metallicsilver changes by a value as small as 20 %, but it is necessary to work at 4.2 K inorder to exclude the temperature-dependent broadening of the gamma line.

7. Yet another mechanism of the broadening of a Mössbauer gamma line may bedue to a nonuniform isomeric shift of the energy of this line because of anirregular distribution of chemical admixtures or because of a structural non-uniformity over the sample volume. This situation is possible, for example, inmanufacturing a gamma source by implanting in silver, via thermal diffusion, aparent nuclide, such as 109Cd, whose decay leads to the formation of excited109Ag nuclei emitting 88.03 keV gamma rays.

Apart from factors leading to gamma-resonance broadening, there are severalreasons causing a shift of the emission line with respect to the absorption line.These include the following:

1. The difference in the electron density at nuclei of the source and the absorbergenerates an isomeric shift [6], which is

DE ¼ 45p Ze2R2S0

DRR

wa 0ð Þj j2� ws 0ð Þj j2� �

; ð3:2Þ

where Z is the charge number of the nucleus being considered; R is its ground-state radius; ΔR is the change in the nuclear radius upon the transition to theexcited isomeric state; wa 0ð Þj j2 and ws 0ð Þj j2 are the electron densities at thenuclei of the absorber and the source, respectively; and S′ is a coefficient thattakes into account relativistic effects associated with the atomic-electronvelocity. This coefficient must be introduced for Z > 20 if use is made of thenonrelativistic wave functions wa and ws.

Table 3.1 Estimates of therelative temperature-dependentbroadening of the88.03 keV 109m Ag gamma line

Temperature (K) Relative gamma-line broadening (ΔΓ/Γnat)4.2 3 × 10−5

13 0.082

25 7.94


The substances of the gamma source and absorber were always identical inour experiments. It follows that, in the case where the source and the absorberare two separate samples, there may arise an isomeric shift of the emission linewith respect to the absorption line because of their different treatment after theseparation from the initial metallic sample. First of all, distinctions arise whenone of blanks (would-be gamma source) is subjected to mechanical, thermal, orradiation action in preparing and implementing irradiation at a cyclotron or areactor. For example, cyclotron irradiation with protons or deuterons introduceslarge amounts of hydrogen or deuterium in the metal (approximately 1019 atomsper layer about 0.1 mm in thickness and about 1–2 cm2 in area). Moreover,radioactive cadmium-isotope atoms produced in the course of irradiation appearin sizable concentrations as well (about 10−5 in the target region that the protonor deuteron beam hits). Finally, cyclotron irradiation leads to the formation of alarge amount of heavy radiation defects. In our practice, there was a case wherea whole layer of silver separated from a cyclotron target—evidently at theboundary of proton ranges. With the aid of annealing, one can reduce to someextent the concentration of hydrogen atoms and the amount of radiation defectsand make, owing to thermal diffusion, the distribution of cadmium atoms moreuniform. In dealing with the 107mAg isomer, however, a short half-life of theparent nuclide 107Cd (6.49 h) gives no way to anneal samples for more thanseveral hours, so that one can hardly expect a complete elimination of theisomeric shift of the emission gamma line in the source with respect to theabsorption line in the sample not subjected to any action.

In manufacturing gamma sources by means of the reactor irradiation of, forexample, samples from a silver–palladium diamagnetic alloy, changes in thechemical composition (in particular, because of the formation of cadmium witha relative concentration of about 10−6–10−5 in the reaction107Ag + n → 108Ag → (β decay) → 108Cd) and the appearance of radiationdefects also occur. In this case, it is therefore natural to expect an isomeric shift,which will be nonuniform in just the same way as in the case of cyclotronirradiation.

The isomeric shift associated with the change in the chemical compositioncan be roughly estimated by using the results presented in [7], where the iso-meric shift of the 77 keV 197Au gamma line was studied as a function of theconcentrations of Ag and Pd in Ag–Au and Pd–Au systems. It was shown that,for concentrations of these elements in the range between 0 (of course, this is notan exact zero) and 90 %, the isomeric shifts change in direct proportion to therespective concentrations. We assume, even though this assumption is not verywell substantiated, that a similar proportionality of the isomeric shift to thecadmium concentration also applies in the case of Ag–Cd systems at very low

cadmium concentrations. Taking the values of wAg 0ð Þ�� 2 ¼ 0:53� 1026 1�cm3 ;

wCd 0ð Þj j2¼ 0:76� 1026 1�cm3 , and S0 ¼ 2:12 from [8] for, respectively, the

electron density at Ag nuclei, the electron density at Cd nuclei, and the coef-

ficient that takes into account relativistic effects and setting DR=R ffi 10�4,


we obtain an isomeric shift of DEj j ffi 0:6� 10�7 eV between the resonancelines of almost pure Ag and Cd. For the case where the relative concentration ofcadmium in silver is about 10−5, we then have DEj j ffi 6� 10�13 eV, which issomewhat smaller than the gamma-resonance width expected for the abovemechanisms of gamma-line broadening.

We note, however, that experiments devoted to studying the Mössbauer self-absorption of gamma rays of the 109mAg isomer and described below indicatethat, in the case where the relative concentration of cadmium in silver is about10−6–10−5, the isomeric shift is not as large as that given immediately above.

2. We have indicated above that the Earth’s gravity may affect the gamma reso-nances of silver isotopes. We will now consider this issue in more detail. Thenonuniformity of the gravitational field near the Earth’s surface leads to the shiftof the energy of the Mössbauer gamma resonance by DEc ¼ Ec

gHc2 [9], where

H is the difference of the vertical coordinates of emitting and absorbing nuclei,g is the acceleration due to gravity, Eγ is the nuclear-transition energy, and c isthe speed of light. The gamma-resonance shift of 10−12 eV corresponds toH ≈ 10 cm. At the same time, a value of H ≈ 10−4 cm is sufficient for shiftingthe gamma resonance by the natural gamma-line width (about 10−17 eV). Thismeans that if it were possible to remove all reasons for the broadening of theMössbauer gamma line, then one could observe the resonant absorption of thesegamma rays only in a horizontal beam of very small divergence.

3. Owing to the second-order Doppler effect, the temperature difference betweenthe regions of nuclei emitting and absorbing gamma rays leads to the followingresult for the shift dEγ of the emission line with respect to the absorption line[10]:

ooT

dE0

E0

� ¼ � CL

2Mc2ð3:3Þ

Here, E0 is an energy of energy of resonance, oE0 absolute value of the shift, CL

is the molar lattice heat capacity of the gamma-source substance, M is themolecular weight of this substance, and T is temperature. According to theDebye theory of heat capacity [11], the temperature dependence of CL in thelow-temperature region (T ≪ θD, where θD is the Debye temperature) is givenby the expression

CL ¼ 1944 n T=hDð Þ3 J/mol K ð3:4Þ

where n is the number of atoms in the molecule of the substance beingconsidered.

From last two formulas, it follows that, at T = 4.2 K, the temperature dif-ference of 1 K between the source and absorber leads to ΔEγ/Eγ = 6.565 × 10−19 K−1 for 107Ag and to ΔEγ/Eγ = 6.4446 × 10−19 K−1 for 109Ag.The respective temperature gamma-line shifts are


DEc107Ag � ¼ 6:11� 10�14 eV=K;

DEc109Ag � ¼ 5:67� 10�14 eV=K;

Of course, values accessible in practice for the temperature difference betweenthe regions of gamma-ray emission and absorption are much smaller than 1 K,especially when one combines the source and absorber into one sample andobserves gamma-ray self-absorption. However, the temperature differencebetween the regions of nuclei emitting and absorbing gamma lines that is dic-tated by the requirement that, at a temperature of 4.2 K, the temperature shift notexceed the natural width of silver-isomer gamma lines (about 10−17 eV) mustnot be greater than a value of about 1.6 × 10−4 K, and this is quite an effort toreach this.

There are few nuclides in nature with which one can perform such experi-ments with a hope for some kind of a success. These nuclides and their prop-erties of interest in connection with the problem being discussed are indicated inTable 3.2. From the data presented in this table, it follows that, for the casewhere the emission and absorption gamma lines are not broadened, the crosssection for resonant gamma-ray absorption at the silver-source and silver-absorber temperature of 4.2 K (rescaled to pure isotope) is 30.7 b for 107mAg-isomer gamma rays and 66.1 b for 109mAg-isomer gamma rays. At the sametime, the cross sections for the nonresonant (electron) absorption of silver-isomer gamma rays in metallic silver are 306 and 347 b for 107mAg and 109mAg,respectively. Among other things, this means that metallic silver is not appro-priate for employing it as a working material for a gamma laser on the basis ofthese isomers. One needs materials in which the probabilities for recoillessgamma-ray emission would be higher than 0.107 for 107mAg gamma rays andhigher than about 0.123 for 109mAg gamma rays. Possibly, larger values wouldbe obtained by using substances such as AgO2 and Ag2O2.

Among the nuclides listed in Table 3.2, 109Ag and 103Rh are the mostappropriate for employing them in gamma-resonance spectroscopy. Metallicrhodium is good because it has a high Debye temperature; it seems that, owingto a low energy of gamma rays of the 103mRh isomer, this would make itpossible to perform Mössbauer experiments at room temperature. However,metallic rhodium is paramagnetic, and this may become a factor that wouldprevent one from employing rhodium in the form of a metal. It is necessary toselect a diamagnetic rhodium compound and to determine its Debye tempera-ture. Moreover, rhodium gamma radiation is strongly converted, so that high-activity gamma sources would be required for experiments with it. As regards tothe 189Os nuclide, experiments with it are complicated, despite its favorableproperties making it possible to work at room temperature, by the fact that theinternal-conversion coefficient for osmium gamma transition is still larger thanthat for rhodium. One would need a gamma source of enormous activity in orderto obtain the required gamma intensity, and experiments with this nuclide mayprove to be impossible for this reason.


Tab

le3.2

Someprop

ertiesof

thenu

clei

which

areon

interestfortheexperimentswith

thelong

-lived

isom

ers

Excited

nucleus

Energy

ofisom

eric

level(keV

)

Meanlifetim

eof

nuclei

intheisom

eric

state

Relative

naturalwidth

ofthereso-nance

Coefficient

ofinternal

conversion

ofγ-transitio

n

Nucleus

ofgammasource

Gam

ma-source-productionreactio

nCalculatedprobability

ofrecoillessγ-ray

absorptio

n

103 R

h39.750

±0.007

80.970

±0.014min

3.4.10

−24

α K+α L

=1147.

103 Pd

103 R

h(p,n)

103 Pd

0.465Room

T°

α K=137±1.9

107 A

g93.13±0.03

63s

1.1.10

−22

α K+L+M

=20.3

107 C

d10

7 Ag(p,n)

107 C

d0.034Tem

perature

4.2K

α K=9.5±1.0

107 A

g(d,2n)107Cd

109 A

g88.033

±0.030

57.13±0.29

s1.3.10

−22

α tot.=26.4

±0,3

109 C

d10

8 Cd(n,γ)

109 C

d0.0535

Tem

perature

4.2K

109 Pd

108 Pd(n,γ)

109 Pd

α K=11.4

±0.3

109 A

g(p,n)

109 C

d18

9 Os

30.80±0.04

8.66

±0.14

h6.83

×10

−25

α>3000

189 Ir

189 O

s(p,n)

189 Ir

0.86

Room

T°


3.3 Early Experiments Performed at ITEP to Studythe Mössbauer Excitation of Long-Lived IsomericStates of 107Ag and 109Ag Nuclei

The schemes of the population and decay of long-lived isomeric states of silverisotopes are shown in Fig. 3.1. In our first experiments, we observed the excitationof 93.1 keV state of the isotope 107Ag with a mean lifetime of 63 s. For thoseexperiments, we designed and constructed an experimental setup in which acryostat making it possible to cool, to a temperature close to 4.2 K, the gammasource and absorber arranged within the cryostat was the main part. The layout ofthe cryostat is shown in Fig. 3.2. In the upper part of the cryostat, there was avertical tube that could be separated from the vacuum volume of the cryostat bymeans of a sluice. Inside the tube, a copper box was suspended on a rubber cord2 m long, which was attached to a nylon-6 thread 60 cm long entering into thecooled space of the cryostat. Two silver plates wrapped in an aluminum foil andtightly pressed to each other were placed inside the box under a cover with a springthat permitted opening the box very quickly. One of the plates was a gamma source(target irradiated at a cyclotron), while the other was the absorber of photonsemitted by the first plate. Both plates were made from the same silver sample ofpurity 99.999 %. For the first experiments, the source plates were irradiated at thecyclotron of Kurchatov Institute of Atomic Energy with 17 MeV protons for10–12 h at a proton current of about 70 μA. For later experiments, the targets were

107

1/2

Cd

107Ag 109Ag

T = 6.49 h

109Pd 109

1/2

CdT = 464 h

e-capturee-capture β

93.1 keVτ = 63 secαt = 20.4

88.03 keVτ = 57 secαt = 26.7

7/2+

1/2–

7/2+

1/2–

1/2T = 13.46 h

Fig. 3.1 Decay schemes for parent nuclides and daughter silver isotopes arising in excitedisomeric states. Here, τ is the mean lifetime of the nuclei in an excited state, while αt is the totalcoefficient of internal conversion of gamma radiation

3.3 Early Experiments Performed at ITEP to Study the Mössbauer… 87

irradiated with protons (in some cases, with deuterons) at the cyclotron of theInstitute of Physics and Power Engineering (Obninsk), the energy being 22 MeV.

The results of the first successful experiments [12] devoted to the excitation ofthe 107Ag isomeric state at 93.1 keV are shown in Fig. 3.3. These data are thesummary of the results obtained in 18 series of absorber activation. One can see thatthe decrease in the gamma activity with time is consistent with the exponential lawof decay of this isomeric state.

The observed effect of absorber activation is small: it corresponds to thedetection of only four to five photons within the first minute of the measurements,the background being nearly at the same level. We now estimate the cross sectionfor the Mössbauer absorption of 107mAg gamma rays in metallic silver at liquid-helium temperature. We have

Fig. 3.2 Layout of theexperimental setup used in thefirst experiments devoted togamma-resonance excitationof 107Ag nuclei (details ofsecondary importance,including the girder supportare not shown): (1) rod with ahandle for lifting the boxcontaining the gamma sourceand absorber, (2) vacuumgasket of the rod, (3)immobile vertical tube (thescale is not exact), (4) rubbercord, (5) cover of the sluice(conventionally), (6) blocks,(7) nylon-6 thread, (8) thin-walled vertical tube of thecryostat, (9) guides alongwhich the slide blocks move,(10) counterbalances, (11)liquid nitrogen, (12) liquidhelium, (13) copper box withan easily opened spring cover,(14) gamma source andabsorber wrapped separatelyin an aluminum foil, and (15)cryostat body


r ¼ p�k22Ie þ 12I0 þ 1

a1

1þ atf 2 ¼ 15:9 b ð3:5Þ

where �k is the gamma-radiation wavelength divided by 2π; Ie and I0 are the spins of,respectively, an excited and the ground state of the nucleus being considered; a isthe relative fraction of the isotope whose level is excited; αt is the total coefficient ofinternal conversion of the nuclear transition; and f is the probability for therecoilless gamma-ray emission (absorption). In order to obtain the resonance-photon-scattering cross section, which is in fact measured in the experimentsdescribed here, one must multiply the value given by Eq. (3.5) by 1

1þat. This leads to

σres.scat. = 0.746 b. If the above theoretical concepts concerning the broadening of aMössbauer gamma line by a factor 105 are correct, then we must reduce the value

Fig. 3.3 Results of the first five successful experiments (18 series of absorber activation) devotedto the gamma-resonance excitation of 107Ag nuclei. The shaded band shows the averagebackground and the error in it. Their values were calculated by using the numbers of countsstarting from the fifth minute after switching on the detector


obtained for the cross section by the same factor. Therefore, the expected value ofthe cross section for the resonant scattering of 107mAg gamma rays turns out to be7.46 × 10−30 cm2 in the case of a broadened but not split gamma line. We note thatthe Earth’s magnetic field must cause the Zeeman splitting of this line, and theeffect of this splitting on the resonant-scattering cross section depends on the anglebetween the vector of the magnetic-field strength and the direction in which thegamma rays are detected. For the case of the 109mAg isomer, this issue was analyzedin our study reported in [13] and will be considered in detail in the next section. Theexperiments described in the present section did not determine even the averagevalue of this angle. Therefore, the factor by which the cross section for resonantabsorption (and, hence, for resonant scattering) decreases because of gamma-linesplitting may lie, according to [13], between 0.266 and 0.087. The average result ofdetermining this cross section from the data of 18 series of resonant-absorberactivation in Fig. 3.3 is (0.74 ± 0.20) × 10−30 cm2, which is one-tenth as large as thevalue expected for the unsplit gamma line. This difference may be partly explainedby the effect of the Earth’s magnetic field.

Test experiments were also performed. The irradiation of a silver absorber atroom temperature did not lead to its activation. In the second test experiment(performed on the recommendation of Academician A.I. Alikhanov), a coppersample served as an absorber instead of a silver plate, but the temperature condi-tions were identical to those in the experiments with a silver absorber. The objectiveof that experiment was to prove that gamma-active silver atoms, which could findtheir way to the surface of the source wrapped in an aluminum foil not hermeticallyare not transferred by air whirlwinds upon the opening of the cryostat to theabsorber surface. That experiment did not show the appearance of a gamma activityof the irradiated sample either. One has to recognize that, with allowance for therole of the geomagnetic field, there is no contradiction between the expected andmeasured effects. However, it remains unclear whether the observed decrease in thecross section is due to gamma-line broadening or to the isomeric shift.

The next series of experiments with the 107mAg isomer [14] was performed byusing an upgraded setup that permitted employing both scintillation counters and agermanium detector for gamma rays. The layout of this setup in the version thatemploys scintillation counters is shown in Fig. 3.4. A gamma source was placed inthe lower part of the setup and was cooled with liquid helium poured in the cryostatwith the aid of a gaseous helium coolant filling the central vertical tube of the setup.In working with scintillation counters, a resonant absorber wrapped in an aluminumfoil was lowered on its suspension device to the source and was pressed to it withaid of a controlled mechanism. After 3 min of activation, the absorber was lifted byan electric motor to the upper part of the setup and was inserted into the gapbetween two scintillation detectors, the tube was closed by a lead shutter, and theinduced gamma activity was detected. In employing the Ge(Li) detector manu-factured by our group, we mounted it on a cold horizontal copper finger. One end ofthe cold finger had a good thermal contact with the cryostat screen cooled withliquid nitrogen. Its second end entered into the central vertical tube, and the planarGe(Li) detector was on it in the horizontal position. The resonant absorber from


silver was mounted slightly higher than the detector but at a temperature close to4.2 K. The gamma source was lowered onto the absorber from upper part of thesetup. During the activation of the absorber, the Ge(Li) detector was exposed toirradiation from a very strong source of gamma rays. As became clear from separateexperiments, this irradiation led to a temporary deterioration of the detector effi-ciency, but its gradual restoration occurred within several minutes. Figure 3.5shows the course of this restoration according to measurements with a weak gammasource placed near the detector.

Fig. 3.4 Second experimental setup for studying the gamma-resonance excitation of isomericstates of silver isotopes: (1) sheave of the electric motor, (2) lead, (3) lead, (4) resonant absorber,(5) scintillation detectors, (7) support of the upper part of the setup, (8) liquid nitrogen, (9) liquidhelium, (10) site of the vacuum-sealed joint of the upper and lower parts of the setup, (11) cryostatbody, (12) gamma source, (13) truck permitting to move the cryostat from under the upper part ofthe setup, (14) sandy pillow, and (15) concrete


The statistical significance of the results was substantially higher in thoseexperiments, but the very effect of the resonant activation of 109Ag nuclei turned outto be small, as before. The results of the experiments with the Ge(Li) detector areshown in Fig. 3.6. Yet another series of experiments was performed with the109mAg isomer [15]. The gamma sources for those experiments were fabricated byirradiating samples from a silver–palladium alloy with thermal neutrons.

Absorbers were fabricated from the same alloy. The required parent nuclide 109Pd,whose beta decay resulted in the creation of 109Ag nuclei in the excited state at88.03 keV, originated from the reaction 108Pd(n,γ)109Pd. Alloys of silver and palla-dium are diamagnetic at silver concentrations close to 50 % and above this value. Itfollows that, in contrast to what we have in pure paramagnetic palladium, a stronggamma-line broadening in excess of the above broadening caused by the dipole–-dipole interaction of nuclear magnetic moments cannot arise in such alloys.

2500

2000

15000 5 10

t, min.

Cou

nt n

umbe

r pe

r m

in.

Fig. 3.5 Gradual restoration of the efficiency of the Ge(Li) detector after the irradiation of it withgamma rays from a closely situated strong gamma source


The results of ten experiments with samples from a silver–palladium alloy are pre-sented in Fig. 3.7 in terms of the number of counts above the background over the firstminute of the measurements per activation event and per source-activity unit (1 Ci).Although the activation effect is very small in this case as well, it is distinctly visible.As might have been expected, the activation effect for paramagnetic samples is notseen beyond the errors, which are rather small. Of course, the experiment in questionalso gives no way to pinpoint the reason behind a small value of the activation effect:broadening versus the isomeric shift. This question becomes especially acute in viewof the appearance of a number of studies performed by several groups (including ourgroup) by quite a different method. All experiments made thus far by this newmethodindicate that the broadening of the Mössbauer gamma line of the 109mAg isomer, ifany, is insignificant—one to two orders of magnitude rather than five orders ofmagnitude. Below, we will describe these experiments and their results.

1500C

ount

num

bers

1000

0 5 10 15

t, min.

Fig. 3.6 Summed result of four experiments devoted to studying the resonant excitation of the107Ag nuclear isomeric state with the aid of the second experimental setup involving a Ge(Li)detector. Corrections that take into account the gradual restoration of the efficiency of the detectorafter the completion of the irradiation acting on it and on the absorber were introduced


3.4 Influence of the Direction of the Magnetic Fieldin Which the Silver Gamma Source Is Placedon the Probability for Resonant Self-absorptionof 109mAg-Isomer Gamma Rays in It

This issue was analyzed by our group in [16]. As was indicated above, the externalmagnetic field splits the gamma line of 109mAg isomer into 14 components owing tothe Zeeman effect. The corresponding scheme of allowed electromagnetic transi-tions between the Zeeman sublevels of the 7/2+ and 1/2− states is shown in Fig. 3.8.

Fig. 3.7 Results of 10 experiments aimed at observing the gamma- resonance excitation of theisomeric state of the 109Ag nucleus by using samples from Ag–Pd alloys. Shown in this figure arethe numbers of counts above the background within the first minute of the measurements peractivation event and per source-activity unit (1 Ci)


Since these components are separated by energy gaps exceeding the naturalgamma-line width by a factor of about 106, each component of the emissionspectrum may be resonantly absorbed only within the width of the correspondingabsorption-spectrum component (under the condition that the emitting andabsorbing nuclei are in the same magnetic field). It is well known [17] that theintensities of the Zeeman components depend on the angle θ between the emitted-

= +7/2 mi

+5/2

+3/2

+1/2

–1/2

–3/2

–5/2

–7/2

–1/2

+1/2

mf =

Fig. 3.8 Scheme of allowed gamma transitions from the 7/2+ isomeric state of the 109Ag nucleus

3.4 Influence of the Direction of the Magnetic Field… 95

photon momentum and the strength vector of the applied magnetic field. As a resultof this the probability of resonant gamma ray absorption depends on this angle.Explain this by the following simple example. Let the source emits the singlenon-split gamma ray line with intensity equal to 1 and the resonant absorber has anabsorption line coinciding with energy of source gamma ray and having the sameintensity equal to 1. In this case the effect of resonant absorption which is pro-portional to the product of these intensities, that is to 1. If both lines are split intotwo components each then the effect of resonant absorption would be equal to thesum

0:5� 0:5þ 0:5� 0:5 ¼ 0:5

Therefore the splitting of both gamma lines into two components leads to thedecrease of the resonant absorption effect by two times.

It will be shown below that the corresponding angular dependences can beexpressed in terms of combinations of spherical harmonics. The intensity of aspecific component of the Zeeman hyperfine structure of the gamma spectrum isdetermined by the product of a Clebsch–Gordan coefficient that depends on thequantum numbers of the initial and final sublevels and the function expressing theangular dependence in question (see below). The probability for the process inwhich the emission of a photon associated with a given component of the hyperfinestructure is followed by resonant absorption is proportional to the square of thisproduct. If the process of resonant self-absorption is observed by using a detectorwhose resolution is insufficient for separating individual components of thehyperfine structure, then the measured resonant-absorption probability is propor-tional to the sum of the probabilities for the absorption of all components—that is,to the sum of the squares of the above products.

The probability for the emission of a photon with a momentum k by a nucleus inthe direction specified by the angle θ with respect to the quantization axis alignedwith the strength vector of the magnetic field is given by [17, p. 408]

W hð Þ ¼Xmi;mf

f mið ÞG J; Ii; If ;mi;mf ;M �

FJ;M¼mi�mf hð Þ ð3:6Þ

where I is the spin of the nucleus being considered;m is its projection; J is the angularmomentum carried away by the emitted photon;M is its projection; the indices i andf refer to, respectively, the initial and the final state of the nucleus; f(mi) is thepopulation of the sublevels for which the nuclear-spin projection is mi (in our case, ithas the same value for any value of mi); and

G J; Ii; If ;mi;mfm;M � ¼ 0:5 p k 2J þ 1ð Þ fhj jAk J;Mð Þ jds ij ij2:

Here, Aλ is the vector potential of electromagnetic radiation (λ is the multipo-larity of this radiation—E3 in our case). The quantity FJ,M has the form


FJ;M ¼ RP¼�1;þ1

DJM;P

�� 2

and satisfies the symmetry condition FJ,M = FJ,–M (P is the projection of the photonmomentum onto the symmetry axis of the nucleus).

We now use the representation of the Wigner function DJM;P in terms of spherical

harmonics,

YJ;M H;uð Þ ¼ �1ð ÞM 2J þ 1ð Þ J� Mj jð Þ!=4p J þ Mj jð Þf !Þ1=2PMJ cos hð Þ eiMu;

(PMJ cos hð Þ is an associated Legendre polynomial) and obtain an expression for FJ,M

in the form of a polynomial in YJ,M(θ,φ) (see, for example, [18]),

FJ;M hð Þ ¼ 4p 2M2 YJ;M h;uð Þ�� 2þ J �Mð Þ J þM þ 1ð Þ YJ;Mþ1 h;uð Þ�� 2nþ J þMð Þ J �M þ 1ð Þ YJ;M�1 h;/ð Þ�� 2o = J J þ 1ð Þ½ �:

ð3:7Þ

In our case, where individual components of the hyperfine structure are verywidely spaced, the angular distribution of photons emitted in a Zeeman componentupon the transition between sublevels characterized by the magnetic quantumnumber mi and mf is determined by the product of the function FJ,M(θ) and theweight equal to the square of the corresponding Clebsch–Gordan coefficient,

Ii;mi; J;M If ;mf

�� 2. At the values of Ii = 7/2, If = 1/2, and J = 3, which cor-respond to the case of the 109mAg isomer, these weights are linear functions of M:(4 − M)/7 for mf = 1/2 and (4 + M)/7 for mf = –1/2 [19].

The dependences of the weights on M are presented in Fig. 3.9. Consideringphoton absorption by a nucleus under the symmetry condition FJ,M = FJ,–M as aprocess that is inverse to emission and which is determined by the matrix elementcomplex conjugate to that of emission, we obtain the following expression for theangular dependence characteristic of the whole radiation process followed byabsorption for the Zeeman component whose magnetic quantum numbers are mi

and mf:

Imimf MðHÞ� Ii;mi; J;M If ;mf

�� 4 FJ;M hð Þ� �2:

For the sum of all Zeeman components, the angular dependence of the intensityof this process has the form

IR hð Þ ¼ RmimfM

Ii;mi; J;M If ;mf

�� 4 FJ;M hð Þ� �2: ð3:8Þ

Substituting into Eq. (3.8) the required values of the Clebsch–Gordan coefficientsand expression (3.7) for the function FJ,M(θ) at the specific angular-momentum andangular-momentum-projection values corresponding to the problem at hand, we


arrive at a final expression for the angular dependence characteristic of the process inwhich the Mössbauer emission of a 109mAg photon is followed by its absorption inthe same sample. Omitting a common factor determining the radial part of therespective matrix element, we have

IR hð Þ� 25 18 Y33j j2þ6 Y32j j2n o2

þ20 8 Y32j j2þ6 Y33j j2þ10 Y31j j2n o2

þ 17n2 Y31j j2þ10 Y32j j2þþ 12 Y30j j2

o2þ 8 24 Y31j j2

n o2:

ð3:9Þ

Fig. 3.9 Zeeman structure of 7/2+ → 1/2− gamma transitions. The height of each vertical-linesegment represents the weight of the respective component and is proportional to the square of theClebsch–Gordan coefficient corresponding to it. The component energy reckoned from the energyof the unsplit gamma line and denoted by Eγ is plotted along the abscissa


The graph of this function is shown in Fig. 3.10. It can be seen that the resonant-absorption probability is maximal in the cases where photons are emitted parallel orantiparallel to the magnetic-field direction.

If there is no magnetic field, then all components of the Zeeman hyperfinestructure merge together into a single line. As a result, all 14 components of theabsorption spectrum may be involved in the resonant absorption of photons cor-responding to one of these components of the emission spectrum. This must nat-urally lead to an increase in the resonant-absorption probability in relation to itsvalue attainable in the presence of a magnetic field. In this case, expression (3.8)reduces to

IR;H¼0 � RmimfM Ii;mi; J;M If ;mf

�� 2FJ M hð Þ�� 2: ð3:10Þ

By virtue of the properties of the spherical harmonics YJ,m(θ,φ) appearing on theright-hand side of (3.10), this function is independent of θ. We now comparethe probabilities of this process for the case where a magnetic field is absent and forthe case where a magnetic field is present and θ = 0. Only the terms involvingY30(θ,φ), which contain powers of cos θ and which do not contain powers of sin θ,make nonzero contributions in the second case. The functions F3,1(θ) and F3,–1(θ)

1.00

θ , degrees

θP ( )

0.75

0.50

0.25

0 45 90 135 180

Fig. 3.10 Factor IΣ(θ) proportional to the probability of the resonant absorption of 109mAg-isomergamma rays versus the angle θ between the strength vector of the magnetic field and the directionin which the photons are detected


appearing in the terms of expression (3.8) have these properties. Taking this intoaccount, we obtain

IR h ¼ 0ð Þ=IR;H¼0 ¼ 2 � 5=7ð Þ2 F3;1 0ð Þ�� 2þ2 � 3=7ð Þ2� �

=2 � 5=7ð ÞF3;1 0ð Þ

þ 2 � 3=7�F3;�1 0ð Þ �2

:

With allowance for the condition FJ,M = FJ,–M, we have

IR h ¼ 0ð ÞIR;H¼0

¼ 1764

Thus, the resonant absorption of gamma rays detected in any direction in theabsence of a magnetic field is larger by a factor of 64/17 than the absorption in ahorizontal gamma beam parallel (antiparallel) to the magnetic-field direction. Toensure such conditions, it is necessary to reduce the geomagnetic field at least byseven orders of magnitude, but this is extremely difficult.

3.5 Foreign Experiments Devoted to the Observationof Resonant Self-absorption of 109mAg-Isomer GammaRays in Metallic Silver

In 1979, W. Wildner and U. Gonser published an article [20] in which they reportedthat they observed the Mössbauer resonant absorption of 109mAg-isomer gammarays, employing a method totally different from our method. They prepared agamma source in the form of a plate that was made from single-crystal silver and inwhich the parent nuclide 109Cd was introduced by means of thermal diffusion. Theregime of thermal-diffusion annealing was chosen in such a way that the averagedepth of penetration of cadmium atoms was 0.125 mm. The temperature depen-dence of the intensity of gamma rays escaping from the plate was measured. Uponthe cooling of the gamma source from room temperature to 77 K, the compaction ofsilver occurred, which led to an increase in its linear coefficient of gamma-raynonresonant absorption. For this case, calculations predicted a decrease of 0.74 %in the yield of 88.03 keV gamma rays. The experiment confirmed this degree ofweakening of the intensity of gamma rays. In response to a further cooling of thegamma source to 4.2 K, silver continued undergoing compaction but to a muchsmaller degree because its coefficient of linear expansion (contraction) decreases asthe temperature becomes lower. However, respective experimental data (seeFig. 3.11) showed that the weakening of detected gamma rays exceeded substan-tially the value expected on the basis of well-known data on the temperaturedependence of silver contraction. The authors of [20] attributed the observed excessof self-absorption to the Mössbauer effect, knowing that the cross section for theresonant absorption of 109mAg gamma rays that is associated with it must increase


nearly by a factor of 50 upon going over from 77 to 4.2 K. The magnitude of theexcess attributed to the Mössbauer effect permitted estimating the broadening of theMössbauer gamma line. The respective factor proved to be 30 (the error was notindicated). However, the authors of that study did not explain the reason for theabsence of gamma-line broadening because of dipole–dipole interaction.

Several years later, there appeared publications of a research group from theUnited States of America [21–23], who reported the results of similar experiments.The results of the first experiments of this American group are shown in Fig. 3.12.In contrast to their German colleagues, American researchers measured the tem-perature dependence of the ratio of the intensities of X-rays and gamma raysemitted by a source. To some extent, this permitted improving the stability of theresults of measurements and reducing the effect of possible deformations of theexperimental setup on these results. An enhancement of the observed effect, albeitvery weak, should additionally be revealed, because not only does the process ofresonant gamma-ray self-absorption lead to the weakening of the gamma ray-yield,but it also causes a small increase in the X-ray yield owing to the internal con-version of gamma rays emitted by nuclei after the resonant absorption of primarygamma rays. The first experiments of that group yielded data that confirmed theresults of the German group: the broadening factors turned out to be 16 and 24. Inthe last of their published articles [23], the American group presented somewhatless convincing results whose errors were as large as 50 %. According to theestimate obtained in that study, the gamma-line broadening factor is 100. It isnoteworthy that the American group does not explain the absence of broadeningdue to magnetic dipole-dipole interaction either.

Fig. 3.11 Results of the experiments reported in [20]. The displayed lines represent the timedependences of the numbers of detector counts according to measurements at (upper line) roomtemperature, (middle line) liquid-nitrogen temperature, and (lower line) liquid-helium temperature

3.5 Foreign Experiments Devoted to the Observation of Resonant… 101

Since we had no grounds at that time to doubt the inevitability of a largebroadening of the 109mAg Mössbauer gamma line because of the dipole–dipoleinteraction of nuclear magnetic moments, our impression was that the resultsreported in [20] are paradoxical and, of course, call for checking them. No testexperiments performed by the authors themselves were mentioned in [20]. There-fore, we could not rule out the possibility of explaining the observed excessweakening of the gamma-ray intensity upon cooling the gamma source to 4.2 K bycryostat deformations, which shift slightly the intensity upon cooling the gammasource to 4.2 K by cryostat deformations, which shift slightly the source away fromthe detector or by a previously unknown phase transition that could occur in silverbetween 77 and 4.2 K and upon which the silver density increased. In order tocheck the second assumption, we performed experiments aimed at measuring thetransmission of 109mAg (88.03 keV), 155Gd (86.5 and 105.3 keV), and 57Fe (122and 136 keV) gamma rays through silver. The experiments in question were carriedout in a vertical geometry. In the case being considered, this excluded the

Fig. 3.12 Results of the firsttwo experiments of theAmerican group [21, 22]. Inthe upper (A) and lower(B) parts of the figure, whichshow data from the differentexperiments, the lower curvesrepresent the calculatedtemperature dependence ofthe ratio of the intensities ofX-ray and gamma radiationdetected in the absence ofresonant self-absorption ofgamma rays in the silversource. The upper curvescorrespond to calculationsperformed under theassumption that theMössbauer self-absorption ofgamma rays occurs and agreeoptimally with the displayedexperimental data


possibility of the resonant absorption of silver gamma rays at a level higher than0.01 %. Those experiments showed that there was no anomalous absorption of88.03 and 86.5 keV gamma rays at absorber temperatures of 4.2, 77 and 293 K. Theresults of those experiments were reported in [24, 25].

3.6 Experiments of Our ITEP Group Performed in the LastYears with the 109mAg Isomer

We have undertaken experiments aimed at revealing the resonant self-absorption of109mAg-isomer gamma rays in silver samples containing the parent nuclide 109Cdintroduced by means of thermal diffusion, employing the effect of not only tem-perature but also gravity and the magnetic-field direction on the gamma resonance.For this purpose, we created an experimental setup that we show in Fig. 3.13(layout) and Fig. 3.14 (general view). A small-size flow-type cryostat that permittedcooling a gamma source placed in it to liquid-nitrogen or liquid-helium temperaturewas the main part of the setup. In the cryostat, there was, in addition to the mainsilver gamma source, a control nonresonant gamma source made from 57Cointroduced in a copper foil in the first experiments and from 241Am in more recentexperiments. Gamma rays emitted from the sources in the horizontal and verticaldirections were detected by two germanium detectors, and the signals from themwere transferred to a common amplitude analyzer (Nokia LP 4900B) through ablock of memory-group selection. A pair of Helmholtz coils was mounted coaxiallywith the cryostat in order to compensate for the vertical component of the geo-magnetic field at the gamma-source locus. Through glass windows of the cryostat,one could keep track of the position of the cryostat cooled volume with the aid of atheodolite and measure its deformation-induced shifts to a precision of about20 μm. Using a laser beam reflected from a small mirror mounted on this volume,one could trace the rotation of the source plane due to deformations of the internalparts of the cryostat and estimate the angle of this rotation. The experiments wereperformed at room temperature and then at 77 and 4.2 K. We measured theintensities of the 109mAg gamma line and the gamma line of the control gammasource and calculated the ratio R of these intensities. Upon the transition from roomtemperature to 77 K, the gamma-line intensities decreased because of the com-paction of the materials of the sources and their covers. It will be shown below thatthe compaction of the aluminum covers of the sources affects only slightly thegamma-ray yields. Therefore, the temperature dependence of R was governedpredominantly by the behavior of silver and was virtually unaffected by very weakchanges in the intensity of americium gamma rays with temperature. By using theobserved decrease in the intensity of silver gamma rays because of cooling to 77 Kand knowing the thoroughly studied temperature dependence of the coefficient oflinear expansion of silver in the region of low temperatures [26], one could cal-culate the decrease in this intensity in response to further cooling the sources down

3.5 Foreign Experiments Devoted to the Observation of Resonant… 103

to liquid-helium temperature. The effect of resonant absorption manifested itself atliquid-helium temperature as an excess decrease in the intensity of silver gammarays below the value expected on the basis of data on the compaction of the sourcematerial. The effect of cooling on the yield of gamma rays from the control gammasource was much smaller than that for silver gamma rays because the control sourcewas very thin: in most experiments, it was a piece of chromatographic paperimpregnated with aqueous solution of americium-241 nitrate, dried, and glued upinto an aluminum foil with a cold-resistant glue. It is noteworthy that the use of theratio R instead of the gamma-line intensities themselves in order to reveal resonantgamma-ray absorption reduces substantially the influence of deformations thatcould arise in the internal parts of the cryostat upon the change in the temperature ofthese parts. For example, the shift of the source by 1 mm with respect to thedetector positioned at a distance of positioned in such a way that one of them is at adistance of 30 cm from the detector would change by only 0.004 % in response tothe same shift. The drift of electronics also has a less pronounced effect on R thanon the detected intensities of individual lines.

Fig. 3.13 Layout of the final version of the setup for observing the Mössbauer resonantabsorption of 109mAg-isomer gamma rays (the scale is not exact): (1) thin-walled tube on which thehelium volume is hanged (it is simultaneously used for the admission of cryogenic liquids, (2) tubefor releasing helium vapors, (3) quartz spacers preventing the bending of tube 1, (4) thermal screenof helium volume, (5) cryostat body (6) glass window, (7) tube twisted as a helix for cooling thethermal screen by evaporated helium, (8) mirror, (9) antideformation rests, (10) helium volume,(11) light source, (12) Helmholtz coils, (13) gamma sources, (14) HPGe detectors, and (15)support (Fig. 3.14)


Fig. 3.14 General view of the setup for observation of resonance absorption of 109mAg gammarays

3.6 Experiments of Our ITEP Group Performed in the Last Years… 105

The first experiment [27] was performed when the setup was not yet equippedwith Helmholtz coils or with gadgets for tracing the possible deformations ofinternal parts of the cryostat. We determined changes in the ratios R of gamma-rayintensities for the 88.03 keV gamma line of 109mAg and the 122 keV gamma line of57Fe (control source on the basis of 57Co) for detectors of horizontal and verticalgamma beams upon the transition from the gamma-source temperature of 77–12 K(we could not cool a silver source below this temperature). They turned out to bethe following:

for a horizontal gamma beam,R 77 Kð Þ � R 12 Kð Þ

R 77 Kð Þ ¼ 0:00064� 0:00044;

for a vertical gamma beam,R 77 Kð Þ � R 12 Kð Þ

R 77 Kð Þ ¼ �0:00047 � 0:00051:

The first of these values agrees with data reported in [20], where the excess self-absorption of silver gamma rays was 0.1 %, but, in the second case, the measuredresult differs by three standard deviations from a value that would correspond to0.1 %. Of course, the results of this first experiment cannot be considered as areliable observation of the Mössbauer absorption of silver gamma rays in a hori-zontal gamma beam and as a simultaneous indication that it is absent in a verticaldirection. The latter could be attributed to the gravitational suppression of reso-nance conditions. However, one should recognize that these results are compatiblewith this pattern.

The next experiment [28] was performed under considerably improved condi-tions. There appeared using a theodolite, and we mounted compensating Helmholtzcoils. Instead of a control source on the basis of 57Co, which revealed ferromagneticproperties, we arranged a source from 241Am, which emitted 59.54 keV gammarays. This was done in order to avoid distortions of the geomagnetic field in theregion of the gamma sources by the influence of a ferromagnetic sample. It shouldbe noted that, to this end in view, the cryostat itself and the details of the setup inthe vicinity of it, including fastening bolts and nuts, were made from nonmagneticmaterials. As yet another means for revealing the resonant absorption of silvergamma rays, we initially intended to switch on and off the Helmholtz coils atregular time interval, since this would lead to the change in the direction of thegeomagnetifield acting on the gamma sources from a natural direction for coilsswitched off (in Moscow, the geomagnetic-field strength is directed downward at anangle of 70° with respect to the horizontal plane) to the horizontal direction.According to [16], the resonant-absorption probability must increase by a factor of2.5 upon switching on the Helmholtz coils. However, it turned out before the startof the experiment being discussed that the activity of the silver gamma source thatwe fabricated earlier decreased because of the natural decay of 109Cd. Under suchconditions, we could not perform the experiment within a reasonable time in theregime of switching coils. Therefore, all measurements were conducted with thecoils in the on mode—that is, under conditions most favorable for observing theabsorption of silver gamma rays in a horizontal gamma beam. After the introduction


of corrections for the compaction of the source materials and for the measureddeformation-induced shifts of the sources with respect to the detectors, the results ofour measurements turned out to be the following:

for a horizontal direction,DRR

¼ R 77 Kð Þ � R 4:2 Kð ÞR 77 Kð Þ ¼ 0:00397� 0:00069;

for a vertical direction,DRR

¼ �0:00093 � 0:00067.

A negative value of ΔR/R for a vertical gamma beam somewhat beyond thestatistical errors is explained by a small rotation of the gamma-source plane becauseof the deformation of the suspension system for the helium volume of the cryostatupon cooling the sources from 77 to 4.2 K. After the introduction of the corre-sponding correction, the value of ΔR/R for a horizontal gamma beam was taken tobe 0.00300 ± 0.00096. For a vertical gamma beam, the corresponding value iscompatible with zero within the errors.

In order to obtain quantitative data on the cross section for the resonantabsorption of silver gamma rays, one must first determine the form of the distri-bution of implanted atoms of the parent nuclide 109Cd over the thickness of thesilver gamma source. In manufacturing sources, we followed the recommendationsof Doctor V.N. Kaigorodov (Institute of Metal Physics, RAS, Ekaterinburg). Ithank him for a consultation that he kindly gave to one member of our group. Silvergamma sources were manufactured in the following way [29]. Silver plates 25 mmin diameter and about 1 mm in thickness cut from a cylindrical silver single crystalby using an electroerosion cutting machine [the cylinder axis corresponded to the(100) crystal direction] were etched in an aqueous solution of CrO3 and CaCl2 takenin amounts of 23.0 and 6.5 % of the solution mass. Slightly bent upon cutting them,the plates were straightened by annealing at 700 °C for 3 h. The radioactive isotope109Cd was precipitated onto both flat sides of a plate by means of electrolysis influoroplastic bath with a platinum anode. The electrolyte used was a normalsolution of HCl containing 109Cd in the concentration corresponding to 10−4 of amolar solution. In order to prevent the release of hydrogen at the cathode (thiswould hinder the precipitation of cadmium), pH of the solution was maintained at alevel of 8−9 by adding an ammonia solution. The activity of 109Cd in the solutionwas 185–370 MBq in different cases. It usually proved to be possible to precipitatehalf of this amount onto silver. In order to implement diffusion annealing, a silverplate, together with cadmium precipitated onto it, was placed into a quartz ampule,which was evacuated to a pressure of 10−4 Torr and was unsoldered. Annealing wasperformed in a horizontal tubular furnace. In manufacturing the first gammasources, the furnace was warmed at a rate of 2 °C per minute from 20 to 750 °C, thetemperature being maintained with a precision of 0.5 °C. Later, we found that onecan conduct warming in such a way that the rate of growth of the temperature ischaracteristic of our furnace in the mode where the power corresponds from theoutset to a final temperature of 700 to 750 °C. The annealing was performed forseveral days (91.4 h in the first case). Cadmium was likely to enter into silver from


the vapor phase. This was suggested, for example, by the following fact. In man-ufacturing the first gamma source, we precipitated cadmium by electrolysis ontoonly one side of the silver plate, trying to prevent the arrival of cadmium at theother side by closing the plate from both sides by flat quartz plates. Nonetheless,cadmium diffused into silver from both sides of the plate in identical amounts, asour measurements of the radiation intensities showed (with a precision better than1 %). It is interesting to note that the quartz ampule in which the silver plate was inthe course of annealing was very weakly soiled with 109Cd activity. Silver placed inthe ampule acted in annealing process as a pump, absorbing both cadmium vaporand cadmium that was at its surface. The average depth of penetration of cadmiuminto silver was 125 μm in the first gamma source that we manufactured. In order todetermine this average depth, we developed a nondestructive method based on acomparison of the ratios of the yields of 109mAg gamma rays and X-rays from silverfor a given metallic source and a very thin source fabricated by impregnatingfiltered paper with an aqueous solution of 109Cd chloride. In such a thin source,neither silver X-rays nor 88.03 keV gamma rays are observed. At the same time,these two types of radiation are absorbed in silver differently because of a largedifference between the respective absorption coefficients. In our measurements ofthe ratios of the intensities of gamma rays and X radiation, the detector was placedat a large distance from the gamma sources. Therefore, it can be assumed thatphotons detected in this experiment are emitted in the direction perpendicular to thesource plane. Suppose that, in recording thin-source radiations, the detector useddetermined the intensity of a specific X-ray line (for example, Kα line), Ix1, and,simultaneously, the intensity of the 88.03 keV gamma line, Iγ1. The ratio of theseintensities, R1 = Ix1/Iγ1, is determined by the actual ratio of the yields of X-rays andsilver gamma rays per event of the decay of the 109mAg excited state and by therespective detection efficiencies for these radiations. If there were no absorption ofradiations in the silver source, then the detector would obviously give the sameintensity-ratio value for this source as well. In an actual case, this ratio will beIx1 e�le d�zð Þ=Ic1 e�lc d�zð Þ for emitting nuclei situated at the depth d−z (d is the platethickness) reckoned from the plate surface facing the detector. In the case beingconsidered, the quantities Ix1 and Iγ1 may be interpreted as the products of theintensities of corresponding radiations per event of the decay of the 109mAg iso-meric state and the corresponding detection efficiencies of the detector used (withallowance for geometric factors). If cadmium diffused into silver from only one sideof the plate, then the distribution of cadmium over the silver thickness after the endof diffusion annealing would have the form

NCd zð Þ � e�bz2 ;

where b = 1/4 Dt; here, D is the diffusion coefficient and t is the duration of thediffusion process. The coordinate z is reckoned from the plate surface onto whichcadmium was precipitated in the direction orthogonal to it.


Since, in our case, there are two flat fronts of cadmium diffusion into silver thatmove toward each other, the distribution of cadmium after the end of diffusionannealing along the z direction perpendicular to the plate plane can approximatelybe represented in the form

NCdðzÞ ¼ Const:� e�bz2 þ e�bðd�zÞ2h i

: ð3:11Þ

For a distribution of the form e�bz2 , the average depth at which cadmium lies insilver is given by

zav ¼R10 ze�bz2dzR10 e�bz2dz

¼ 1=2b12

ffiffipb

p ¼ 1ffiffiffiffiffiffibp

p : ð3:12Þ

By exactly solving the problem of two-sided diffusion [30], one obtains thefollowing distribution of atoms of the diffused element:

N zð Þ ¼ Const 1þ 2X1k¼2

e�p2Dk2

l2t cos

kpzl

� " #: ð3:13Þ

Here, l is the plate thickness, D is the diffusion coefficient, and t is the durationof the diffusion process. Summation in (3.13) is performed over even values ofk. Under our conditions, the series in (3.13) converges rather fast, and it is sufficientto retain the k = 2 and 4 terms. A comparison of the distributions N(z) calculated byformulas (3.11) and (3.13) for specific conditions shows that their difference is verysmall. Therefore, one may use the simpler expression (3.11) without anylimitations.

In order to calculate the ratio R2 of the intensities of X-ray and gamma radiation,it is necessary to determine the following ratio of integrals:

R d0 NCd zð ÞIx1e�lx d�zð ÞdzR d0 NCd zð ÞIc1e�lc d�zð Þdz

¼ Ix1Ic1

�R d0 NCd zð Þe�lx d�zð ÞdzR d0 NCd zð Þe�lc d�zð Þdz

¼ R2: ð3:14Þ

The value of b appearing in expression (3.11) for NCd(z) and, hence, zav can bedetermined by comparing the results of calculations by formula (3.14) for differentb values with the measured value of R2. Knowing the value of b and, hence, thedistribution determined according to Eq. (3.11) for cadmium atoms in the silver plate,one can now first find how the gamma-ray yield must decrease because of the com-paction of silver as the temperature becomes lower and second calculate, for specificgeometric conditions of our experiment, the relative value of resonant gamma-rayself-absorption in the source substance as a function the cross section for this process.Let us start from the first issue. The temperature dependences of the coefficients oflinear expansion of silver and aluminum are presented in Fig. 3.15. Those data weretaken from Tables 3.2 and 17 in [26]. If, upon the reduction of temperature, the linear


dimensions of the source decrease by a factor of n in all directions, then the number ofatoms per unit volume increases by a factor of n3, while the length l over whichgamma-ray absorption occurs decreases by a factor of n. As a result, the dimen-sionless exponent in the exponential function that takes into account absorption, μl (μis the absorption coefficient), must increase by a factor n2. In response to a smallchange in the temperature from T to T−dT, the length l changes by the quantity

dl ¼ �la Tð ÞdT ; ð3:15Þ

where α(T) is the temperature-dependent coefficient of linear expansion. IntegratingEq. (3.15) with

ln l1=l2ð Þ ¼ �ZT2T1

a Tð Þ dT: ð3:16Þ

Fig. 3.15 Coefficients oflinear expansion of aluminumand silver versus temperature(according to data presentedin [24])


Since T2 < T1, we have

l2 ¼ l1 e�RT2T1

a Tð ÞdT��

��: ð3:17Þ

For the product μl, we accordingly obtain

ðllÞ2 ¼ ðllÞ1e2RT2T1

a Tð ÞdT��

��: ð3:18Þ

The integral in the exponent in the exponential function on the right-hand side ofEq. (3.18) can be evaluated with a precision sufficient for us by using the graphs inFig. 3.12. For silver, the integral corresponding to the change in the temperaturefrom 293 to 77 K is 0.00368; for a further decrease in the temperature from 77 to4.2 K, it is 0.0004357. For aluminum, the corresponding integrals are 0.003967 and0.0002647.

Let us consider several particular cases. If radioactive atoms are uniformlydistributed over the volume of a silver plate of thickness d, then the gamma-rayintensity Iγ recorded by the detector separated from the source by a distance muchlarger than the source dimensions is given by

Ic ¼Ne 1� e�ld

�ld

; ð3:19Þ

where N is the total number of photons emitted by the source over the time ofmeasurement of Iγ and ε is the detector efficiency including the geometric factor.For 88.03 keV gamma rays, the absorption coefficient μ at room temperature is21.535 cm−1 (interpolation of data from [31]). If d = 1 mm, then Iγ = 0.41272 Nε atT = 293 K. Upon cooling the source to liquid-nitrogen temperature, the product μd

increases in accordance with the factor e2R77K

293K

a Tð ÞdT��

�� ¼ e2�0:00368 ¼ 1:0073872. Thequantity Iγ turns out to be 0.41056 Nε, the relative change in Iγ being 0.523 %. Uponcooling the source further to 4.2 K, the recorded gamma-ray intensity reduces to0.41030 Nε. The change in Iγ with respect to the intensity value detected at theliquid-nitrogen temperature is 0.0633 %.

In the treatment of experimental data for which statistical errors exceed 10 %,one can make use of a simplified model of the distribution of emitting atoms in thesource material, assuming that they lie in two thin layers spaced from the platesurfaces by the average depth lav of cadmium penetration into silver. For thisdistribution, the detected gamma-ray intensity is


Ic ¼ Ne2

e�l d�lavð Þ þ e�l lavh i

: ð3:20Þ

If lav = 0.15 mm and d = 1 mm, then we have Iγ = 0.44418 Nε at roomtemperature, Iγ = 0.44223 Nε at 77 K, and Iγ = 0.44200 Nε at 4.2 K. The relativechange in the gamma-ray intensity is 0.439 % upon the transition from 293 to 77 Kand 0.052 % upon cooling the source from 77 to 4.2 K. It should be noted that thesimplified model in question can be used in calculating the yield of silver gammarays but not in calculating the yield of X-radiation, for which the coefficient ofabsorption is significantly larger than that for gamma rays. In calculating the X-rayyield, one must use the actual distribution of cadmium in silver.

In conclusion, we consider precisely this case of the actual distribution ofcadmium atoms that arises in silver owing to two-sided thermal diffusion and whichhas the form in (135). In this case, the detected gamma-ray intensity is given by

Ic ¼ CNeZd

0

e�bx2 þ e�bðd�xÞ2h i

e�l d�xð Þdx; ð3:21Þ

where C is a normalization factor determined by the condition

CZ10

e�bx2dx ¼ CZ10

e�b d�xð Þ2 ¼ 1=2: ð3:22Þ

At lav = 0.15 mm, b is 14.147 mm−2 according to (3.12). If, in addition,d = 1 mm, then C = 2.122. A numerical computation of the integral in (3.21) leadsto the Iγ values of 0.45559, 0.45368, and 0.45347 for the source temperatures of293, 77, and 4.2 K, respectively. One can see that these values differ by not morethan 3 % from the corresponding values calculated by means of the simplifiedformula (3.8). The relative changes in Iγ upon the transitions from the temperatureof 293 K to 77 K and from the temperature of 77 K to 4.2 K are 0.419 and0.0463 %, respectively.

We now address the question of how the intensity of gamma rays of the controlsource made from 241Am nitrate changes upon cooling it because of the compactionof its aluminum cover, which is 0.1 mm thick. The absorption coefficient for59.54 keV 241Am gamma rays in aluminum at room temperature is 0.735 cm−1

(interpolation of data from [31]). Therefore, 0.992677 of americium gamma-rayintensity that would be recorded by the detector in the absence of the aluminumcover traverses an aluminum layer 0.1 mm thick. This value will decrease to0.992648 upon the transition to liquid-nitrogen temperature and will become0.992646 upon cooling the source further to 4.2 K. The relative decrease in thegamma-ray intensity is 0.0029 % upon going over to liquid-nitrogen temperatureand 0.0002 % upon going over to liquid-helium temperature. Both these values are


rather small in relation to the errors of our experiments, and this permits neglectingthe effect of compaction of the aluminum cover of the americium source on thedetected intensity of gamma rays emitted by this source.

We now proceed to consider our method for determining the cross section forresonant gamma-ray absorption, employing experimental data corresponding toconditions where gamma rays are detected in the horizontal and vertical directions.The geometry of the experiment with a horizontal gamma beam is shown, withoutpreserving a scale, in Fig. 3.16. A planar high-purity germanium (HPGe) detectordetects gamma rays and X-ray radiation finding their way to it through a restrictivewindow. We assume that all gamma rays having identical energies and hitting thewindow of the detector are detected in it with the same efficiency. Let the z axisshown in Fig. 3.16 lie in the horizontal plane perpendicular to the silver plate of thesource and to the rectangular detector window. The entrance window of the detectoris at a distance z0 from the origin of coordinates. We consider the absorption ofphotons in the source material that were emitted by atoms situated in the source-volume element dV1(x1, y1, z1) = dx1·dy1·dz1 toward the area element dS2 = dx2·dy2of the detector window. The trigonometric functions that determine the angles α, β,and γ are the following:

y

gamma source

detector windowc

z

1

3

2

a

b

x

α

γβ

Fig. 3.16 Geometry of the experiment performed at a horizontal direction of the gamma-beamaxis. The scale is not preserved. A photon flies into the detector window along the straight line 1.The straight line 2 is the projection of line 1 onto the horizontal plane. The straight line 3 isparallel to the z axis


tan a ¼ x2 � x1z0 � z1

; ð3:23Þ

tan b ¼ y2 � y1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz0 � z1ð Þ2þ x2 � x1ð Þ2

q ; ð3:24Þ

tan c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � x1ð Þ2þ y2 � y1ð Þ2

qz0 � z1

: ð3:25Þ

Photons emitted at a nonzero angle β with respect to the horizontal planeundergo an ever increasing shift in the vertical direction as the distance from theemission point of (x1, y1, z1) becomes larger. This shift is given by

h zabsð Þ ¼ y2 � y1ð Þ zabs � z1ð Þz2 � z1

; ð3:26Þ

where zabs is the coordinate of the point of possible resonant gamma-ray absorption.For this reason, the cross section for the resonant absorption of these photonsdecreases gradually owing to the increase in the gravitational shift of the gammaresonance. While ordinary electron nonresonant photon absorption is determined byan exponential function with a negative exponent equal to the product of the usualabsorption coefficient μe and the length of the photon path in the source material, (d−z1)/cos γ, nuclear resonant absorption is determined by an integral along thephoton path, the integrand being an exponential function whose exponent decreasesgradually in magnitude because of the decrease in the resonant-absorption crosssection. This cross section for gamma rays whose spectrum features a Mössbauercomponent in the form of a Lorentzian line with a real width under conditions of theexperiment being discussed, Creal, is given by

Ec � ¼ rres:abs:

k2

2p� 2Je þ 12J0 þ 1

� a1þ at

� Cecm:

Cpear:f 2

C2pear:=4

Ec � E0 �2þC2

pear:=4

¼ r0k� C2

pear:=4

Ec � E0 �2þC2

pear:=4;

ð3:27Þ

where λ is the gamma-radiation wavelength; Je and J0 are the spins of, respectively,an excited and the ground state of the nucleus; a is the fraction of that isotope in thenatural mixture which resonantly absorbs gamma rays; αt is the total coefficient ofthe internal conversion of gamma radiation; Cnat: is the natural width of the gammaline; Creal is its real width; f is the probability for recoilless photon emission(absorption); Eγ is the photon energy; E0 is the position of the gamma-resonancecenter on the energy scale, and k = Γreal/Гnat. is the factor of broadening of the


Mössbauer gamma line. We assume that the real width of the Mössbauer absorptionline is also equal to Creal.

The spectrum of source gamma rays that is normalized to unity and which isshifted by S in energy with respect of the center of the gamma resonance ofabsorbing nuclei has the form

N Ec � ¼ 2

pCpear:� C2

pear:=4

Ec � E0 � S �2þC2

pear:=4: ð3:28Þ

Averaging the cross section in (3.16) over the gamma-ray spectrum (3.17), weobtain

r ¼ r02k

� 1

1þ S2=C2pear:

¼ r02k

� 1

1þ S2=C2ecm:k

2: ð3:29Þ

In the case of the 109mAg isomer, the gravitational shift of the gamma resonance,ΔEγ = Eγgh/c

2, is

DEc ¼ 0:153472 � 10�24h erg; ð3:30Þ

where the difference h of the vertical coordinates of emitting and absorbing nuclei isexpressed in centimeters.

Substituting expression (3.19) for S into Eq. (3.20) and considering that Cnat: for109mAg is 1.16 × 10−17 eV = 1.856 × 10−29 erg, we obtain

r ¼ r02k

� 1

1þ 0:683758� 108 h=kð Þ2 : ð3:31Þ

We now revisit Fig. 3.16. The absorption in the source material of a photonoriginating from the point (x1, y1, z1) and hitting the restrictive window of thedetector at the point (x2, y2) is determined by the factor

exp �l d�z1ð Þ=cos c½ � � exp �Zd

z1

r zabsð Þm dzabs= cos c24

35; ð3:32Þ

where zabs is the coordinate of the point of possible resonant photon absorption,σ(zabs) is given by Eq. (3.31) under the condition that h is given by Eq. (3.26), m isthe number of silver atoms in 1 cm3 of the gamma-source material, and μ is thecoefficient of nonresonant (electron) gamma-ray absorption in silver.

With allowance for Eqs. (3.26), (3.31) and (3.32), we now write an expressionfor the number dN of photons originating from 109mAg in the volume elementdV1 = dx1dy1dz1 and hitting the element dS2 = dx2dy2 of the detector-window area


under the conditions where the gamma source is at liquid-helium temperature.We have

dN � e�bz21 þ e�b d�z1ð Þ2h i dS2 cos3 c

4p z0 � z1ð Þ2 exp � l d � z1ð Þcos c

��

exp � r0m2k

Zd

z1

dzabs

1þ 0:683758 � 108 y2�y1ð Þ zabs�z1ð Þk z2�z1ð Þ

h i2� �8>><>>:

9>>=>>;dV1:

ð3:33Þ

The integral in the exponent of the last exponential function can be evaluatedanalytically, whereupon expression (3.32) assumes the form

dN� e�bz21 þ e�b d�z1ð Þ2h i dS2 cos3 c

4p z0 � z1ð Þ2 exp � l d � z1ð Þcos c

��

exp � r0m z2 � z1ð Þ1:653794� 104 y2 � y1ð Þ arctg 0:826897 � 104

y2 � y1ð Þ d � z1ð Þk z2 � z1ð Þ

� �dV1:

ð3:34Þ

In order to obtain a quantity that is proportional to the gamma-ray intensityrecorded by the detector at the source temperature of 4.2 K, it is necessary tointegrate expression (3.34) over the source volume and over the detector-windowarea.

The expressions for the gamma-ray intensity detected at room temperature and at77 K differ from (3.34) by the absence of the last exponential function and by aslightly smaller value of μ in accordance with a silver density that is lower than thatat liquid-helium temperature.

In the case of a vertical gamma beam, the situation is somewhat simpler. A weakdivergence of this beam affects only slightly both ordinary and resonant gamma-rayabsorption. Therefore, one can estimate the resonant-absorption effect, assumingthat photons are emitted in the strictly vertical direction. Let us consider a hori-zontally arranged source in the form of a plate of thickness d where atoms aredistributed according to Eq. (3.11) under the condition that the z axis is directeddownward. In this case, the gamma-ray intensity Iγ detected at liquid-helium tem-perature has the form

Ic �Zd

0

C e�bz2 þ e�b d�zð Þ2h i

e�le d�zð Þe�

Rdz

r zabsð Þmres dzabs8><>:

9>=>;dz: ð3:35Þ

In this formula, the dependence of Iγ on k is contained in the expression forσ(zabs).


The corresponding expressions referring to room temperature and to liquid-nitrogen temperature do not contain the last exponential function in (3.35).

Using Eqs. (3.34) and (3.35), we calculated the quantitiesDIcIc

¼ Ic 77 Kð Þ�Ic 4:2 Kð ÞIc 77 Kð Þ

versus the factor k of broadening of the Mössbauer gamma line. If the weak tem-perature dependence of the intensity of americium gamma rays is neglected, which is

legitimate, as was shown above, then the values of the ratios DIcIc

may be directly

compared with the measured values ofDRR

¼ R 77 Kð Þ�R 4:2 Kð ÞR 77 Kð Þ , where R is the ratio of

the measured intensities of 109mAg and 241Am gamma rays. Using the results of thiscomparison, one can then evaluate k.

Figure 3.17 shows calculated values of DRR versus the gamma-line broadening

factor k for horizontal and vertical gamma beams. The shaded band in this figurerepresents the value obtained for this ratio in the experiment reported in [28] and therespective error for a horizontal gamma beam. The reason for the weak dependenceof the ratio DR

R on k for a vertical gamma beam is that the k-fold decrease in the crosssection for resonant absorption is accompanied by a k-fold increase in the

Fig. 3.17 Results of the experiment aimed at observing the Mössbauer resonant absorption of109mAg-isomer gamma rays and described in [26]. Here, we plotted the DR

R values obtained (andcalculated) for a horizontal gamma beam along the left ordinate and their counterparts for a verticalgamma beam along the right ordinate. The solid curves 1 and 2 represents the calculated ratio DR

R asa function of the broadening factor k for, respectively, a horizontal and a vertical gamma beam.The shaded band shows the DR

R value, together with the error in it, found experimentally for ahorizontal beam. The measurements were performed with the Helmholtz coils in the on mode


photon-path length over which this absorption is possible in the source substance.One can see that the DR

R value measured for a horizontal gamma beam correspondsto values of the broadening factor for the Mössbauer gamma line between 1 and 3.

If the experiment reported in [28] was performed under the same conditions asthe experiments of the aforementioned German and American groups ([20–23])—that is, without the compensation of the vertical component of the Earth’s magneticfield—and if the cross section for the resonant absorption of 109mAg gamma rays foremission and absorption lines not split by a magnetic field was used in treatingthese results, then, as was indicated in [28], the value obtained for the broadeningfactor would be 35þ19

�10, which is consistent with the results reported in [20–23].Later, several experiments were additionally performed with newly fabricated

gamma sources. Those experiments already involved changing, at regular timeintervals, the direction of the geomagnetic field acting on the gamma sources. In theexperiment reported in [32] and supported by INTAS (under project no. 97-31566),use was made of a single-crystal gamma source 0.74 mm in thickness, for which themean depth of 109Cd penetration was 0.160 ± 0.012 mm according to our mea-surement. After the introduction of all corrections, the relative value of the reso-nant-absorption effect for 109mAg gamma rays in a horizontal gamma beam uponcooling the source from 77 to 4.2 K turned out to be 0.00116 ± 0.00080. At thesame time, the measured relative change in DR

R upon switching on and off Helmholtzcoils was 0.00113 ± 0.00075. Since, according to [16], this last value is 60 % of thetotal relative resonant-absorption effect, the total effect in question is0.00188 ± 0.00125. By combining the two results, we obtain 0.00137 ± 0.00067 forthe relative value of Mössbauer resonant absorption.

The following data were obtained in that experiment for a vertical gamma beam(measurements in the two beams were simultaneous): the change in DR

R upon coolingthe source from 77 to 4.2 K turned out to be 0.00062 ± 0.00061; in the experimentinvolving the change in the geomagnetic-field direction, it was 0.00006 ± 0.00068.Thus, we see that, in this experiment, all three factors (change in the temperature,change in the magnetic-field direction, and the effect of gravity) furnished indica-tions that resonant gamma-ray absorption occurred in a horizontal gamma beam butthat there was no such effect beyond the errors in a vertical gamma beam. For thefactor k of broadening of the Mössbauer gamma line, the value corresponding toresonant silver-gamma-ray absorption measured in that experiment was determineby means of the same procedure as that which was used for the value of k in [28].The result was 3:4þ2:6

�1:4.Two other experiments [33] performed with a source from polycrystalline high-

purity silver in one of them and with a single-crystal source in the other did notfurnish data suggesting that the value of the effect of resonant gamma-rayabsorption was at a level corresponding to the change in the gamma-ray intensityupon cooling the source from 77 to 4.2 K. If the resonant-absorption effect wasabsent for some reason, then the photon intensity would nevertheless show adecrease because of the compaction of silver. Instead, a small increase in thegamma-ray intensity with decreasing temperature was observed in those cases.


Moreover, no deformation-induced shift of the source toward the detector wasfound at a level that could be sufficient for such an increase. In all probability, ananomalous temperature dependence of the gamma-ray yield was associated with themethod used in those experiments to fasten the silver gamma sources to the wall ofthe cryostat volume cooled with liquid helium. The sources were soldered with acadmium–bismuth alloy to a rather thin copper wall of the helium volume, and abimetallic pair was formed by the source and this wall. The coefficients of linearexpansion of silver and copper differ sizably. They are, respectively, 18.9 and 16.7(in units of 10−6 K−1) at room temperature, 10.64 and 6.52 at 80 K, and 0.0177 and0.009 at 5 K [34]. The bimetallism-induced deformation of the sources could not benoticed with aid of our equipment because it did not lead to a shift of the heliumvolume as a discrete unit. It is noteworthy, however, that, in the steady-statetemperature regime, when there are no reasons for the emergence of furtherdeformations of the cryostat or the gamma sources themselves, it is quite possible toobtain data on resonant gamma-ray absorption in experiments where the directionof the external magnetic field is changed at regular time intervals. The effect of thischange must reveal itself at liquid-helium temperature for 109mAg gamma rays in ahorizontal gamma beam. It must not be observed at room temperature of the gammasource or at 77 K. This effect must be negligible in the measurements in a verticalgamma beam. Moreover, this must not affect the measured intensity of americiumgamma rays under any conditions. Of course, the last comments are correct only inthe case where the detectors used record gamma rays with an efficiency unaffectedby switching on and off the Helmholtz coils. The corresponding data were obtainedin measuring the intensities of 88.03 keV (109mAg) and 59.54 keV (241Am) gammarays and silver X-rays in the on and off modes of the Helmholtz coils at roomtemperature and at 77 K—that is, under conditions such that there was virtually noresonant absorption of silver gamma rays. The results of those measurements aregiven in Table 3.3. In the second, third, and fourth rows of this table, we present the

Table 3.3 The rations of the intensity of gamma rays 88.03 keV of 109mAg, 59.54 keV of 241Amand 109Ag X-rays measured in the on and the off modes of the Helmholz coils at room and liquidnitrogen temperatures

Ratios of radiation intensitiesmeasured in the on and offmodes of the Helmholtz coilsand corresponding ratios of Rvalues

Results of measurementsperformed with the detectorfor a horizontal beam ofradiations

Results of measurementsperformed with the detectorfor a vertical beam ofradiations

I Agð ÞI Agð Þ

1.00007 ± 0.00011 0.99972 ± 0.00013

I Amð ÞI Amð Þ

0.99993 ± 0.00008 0.99995 ± 0.00007

IX�ray

IX�ray

1.00005 ± 0.00014 1.00017 ± 0.00015

R

R1.00015 ± 0.00015 0.99981 ± 0.00014


ratios of the intensities measured for 109mAg and 241Am gamma rays and silverX-rays in the on and off modes of the Helmholtz coils. The intensities measured inthe mode where the compensating magnetic field is switched on are marked withasterisks. The values of R = I(Ag)/I(Am) are given in the last row of Table 3.3. Onecan see that the quoted values, which refer to the detector for a horizontal gammabeam parallel to the plane of the Helmholtz coils, are compatible with unity withinthe errors. However, the detector for a vertical gamma beam showed the presence ofan explicit effect of switching on the Helmholtz coils on the detection efficiency for88.03 keV gamma rays; concurrently, there was no commensurate effect on thenumbers of counts associated with 59.54 keV americium gamma rays and 22 and25 keV silver X-rays. A direct influence of the change in the magnetic-field strengthand direction on the germanium detector for gamma rays may underlie one possibleexplanation of this effect. The results of studying the effect of strong magnetic fieldson the work of a planar Ge(Li) detector in which the direction of the electric fieldwas perpendicular to the magnetic-field direction are described in [35]. The peak oftotal gamma-ray absorption showed a catastrophic decrease in response to anincrease in the magnetic-field strength. Specifically, the amplitude of the peakdecreased by a factor greater than ten at the field strength of 60,000 Oe. The authorsof [35] explained this effect by an increase in the path length for carriers in ger-manium in the presence of a magnetic field and the corresponding increase in theloss of carriers on the way to collecting electrodes. We cannot rule out the possi-bility that a weak reduction of the detection efficiency for 88.03 keV gamma rays inour experiments with a detector for a vertical gamma beam may also be due to suchan increase. However, the magnetic-field strength H decreased rather than increasedupon switching on the Helmholtz coils. Therefore, the effect of a decrease in thedetector efficiency was more likely due to a small change in the magnetic-fielddirection rather than to a change in H. The possible reason behind the fact that thisphenomenon was observed for silver gamma rays and was missing (within theerrors) for americium gamma rays and for X-rays is that the mean path of88.03 keV gamma rays is longer than the mean path of 59.54 keV gamma rays(X rays) by a factor of 3.2 (several tens). It follows that the path lengths of carrierstraveling toward the front electrode of the planar detector are significantly longer indetecting 88.03 keV gamma rays than the path lengths corresponding to 59.54 keVgamma rays and X-rays. Therefore, the carriers that are collected at the frontelectrode are lost in larger amounts than carriers of opposite charge, which travel tothe rear electrode. Since a negative potential was applied to the front electrode ofthe detector used, holes were lost in the detector material more frequently thanelectrons. An alternative interpretation assumes an increase in the carrier-collectiontime and a decrease in the signal amplitude because of a change in the hierarchy ofthis time and the time constants of the detector electronics circuits. The absence of asignificant effect of the on–off switching of the Helmholtz coils on the efficiency ofthe detector for a horizontal gamma beam is due to a smaller variation in thegeomagnetic field at the detector locus upon this switching.

In experiments performed at a constant temperature, in which case there are noreasons for deformation-induced shifts of the gamma sources, one can in principle


reveal the effect of resonant gamma-ray absorption not only by the change inR upon changing the magnetic-field direction but also directly by the change in thedetected intensity of 109mAg gamma rays. Yet, the effect of the possible drift of theelectronics is not excluded in this case. For this reason, data on the effect of thechange in the magnetic-field direction were obtained separately for the ratios R andfor the intensities of 109mAg and 241Am gamma lines. Figure 3.18 shows the resultsof an experiment with a polycrystalline gamma source 0.5 mm thick, the meandepth of cadmium penetration into it being 0.128 mm on both sides. One can seethat the change in the magnetic-field direction affects the detected intensity of silvergamma rays in a horizontal gamma beam at liquid-helium temperature but does notaffect it at elevated temperatures. The change in the magnetic-field direction doesnot affect the intensity of americium gamma rays under any conditions. In the caseof a vertical gamma beam, we did not find any manifestations of resonant gamma-ray absorption, as might have been expected.

The next figure (Fig. 3.19) shows similar results obtained with a single-crystalsilver source. At first glance, it may seem that these data are less convincing than the

Fig. 3.18 Results of experiments performed with a polycrystalline silver gamma source and witha control source from 241Am and reported in [33]. Plotted along the ordinate are the ratios ofradiation intensities measured individually for silver and americium gamma lines in the on and offmodes of the Helmholtz coils compensating for the vertical component of the geomagnetic field.Under the bracket signs marked with R, we show the ratios of the quantities R determined in the onand off modes of the Helmholtz coils and defined as the ratios of silver- and americium-gamma-rayintensities (double ratios) measured simultaneously by the same detector. The open circlesrepresent the averaged results of measurements at room temperature and at 77 K. The closedcircles correspond to liquid-helium temperature


results of the experiment with a polycrystalline source. Indeed, the error bar for theratio of the intensities of silver gamma rays that was measured at liquid-heliumtemperature touches the line corresponding to unity.However,we note that the ratio ofamericium-gamma-ray intensities measured simultaneously with silver-gamma ray-intensities by the same detector went sizably upward above unity. The simultaneousshifts of the ratios of gamma-ray intensities in the same direction for the main and forthe control gamma source aremost probably due to drift phenomena in the electronics.

In the ratios of R values obtained in the on and off modes of the Helmholtz coils,the effect of the resonant absorption of silver gamma rays manifests itself quiteclearly. This is one of the examples demonstrating that, in experiments of the typebeing discussed, it is much more reliable to reveal resonant absorption by studyingR values than by directly employing the intensities of the corresponding gammalines, even though the errors are naturally greater in the former case.

In our article devoted to describing these experiments [33], we presented thefollowing values of the broadening factor for the 109mAg Mössbauer lines:

for a polycrystalline gamma source, k ¼ 22þ25�8 ;

for a single-crystal gamma source, k ¼ 21þ13�6 .

However, errors were later found in evaluating these broadening factors. Thecorrected k values are the following:

for a polycrystalline source, k = 6.8 ± 3.4;for a single-crystal source, k = 6.9 ± 2.5.

Fig. 3.19 Results of the experiment reported in [33] and performed with a single-crystal gammasource and a control source from 241Am. The notation here is identical to that in Fig. 3.18


Thus, all of the experiments described above (four foreign and five ours) yieldeddata that are indicative to some extent of the smallness of broadening of the 109mAgMössbauer gamma line. This suggests that nuclei emit and absorb gamma rays notwithin a time as short as �k=c (�k=c is the gamma-radiation wavelength divided by 2πand c is the speed of light), as A.B. Migdal wrote in [36], but within a time muchlonger than the characteristic time of a local change in the dipole–dipole interactionenergy, or, most probably, within a time commensurate with the mean lifetime ofnuclei in the respective excited state (see [37–39] for a discussion on this issue).Within the photon-emission time as short as �k=c, the theoretically predicted largebroadening would be emission as �k=c the theoretically predicted large broadeningwould be inevitable because the energy of dipole-dipole interaction may be con-sidered to be invariable over such a short time. In this case, different nuclei wouldexperience Zeeman splitting under the effect of magnetic fields having differentstrengths. As a result, the photon energies would be distributed over the range ofthese splittings. If a photon is emitted within a “long” time, then an as-yet-unknownmechanism averaging the dipole–dipole interaction energy in the photon-emissionprocess to a very small value changing in magnitude and sign may be operative, orthe nuclei involved are insensitive to these quickly changing perturbations withinthe time of the “prolonged” processes of gamma-ray emission and absorption. Onecan in principle find out experimentally which of these hypotheses is correct. Ifseveral experiments aimed at determining the broadening factor for a Mössbauergamma line are performed with the same gamma source, then different values ofthis factor must be obtained in the case where the mechanism of averaging dipole-dipole interactions is operative. This is because the mean value of a quantityundergoing random fluctuations must fluctuate itself. But if the absence of a largebroadening is due to the insensitivity of nuclei to magnetic effects changing withtime in the course of a nuclear radiative processes, then all experiments with thesame gamma source must give identical values for the broadening factor.

We note, however, that there are serious theoretical objections against thehypothesis that there occurs averaging of dipole–dipole interaction [40]. We canadd the following to this: combined with the hypothesis that the system formed by anucleus and the gamma wave emitted (absorbed) by it does not interact with thesurrounding medium before the completion of the radiation (absorption) process,the concept of a protracted character of photon emission leads to the conclusion thatit is impossible to detect the respective photon before the end of its emission. Inother words, the emitted wave cannot manifest itself before this instant in thedetector material as a particle. If, however, there was the possibility of detecting aphoton before the end of its emission, then it would be impossible to obtainexperimentally a gamma line of natural width. One must then admit that, in thiscase, the interruption of the photon-emission process by the photon-detection eventmay occur at any instant after the start of this process—on average, at the instantcorresponding to the lapse of half the mean lifetime of the nucleus in the excitedstate from the start of photon emission. As a result, the minimum observablegamma-line width would be the doubled natural width. However, lines of width


close to the natural width are observed in some Mössbauer experiments. Moreover,it would be impossible to determine correctly the half-lives of nuclear excited statesin experiments employing delayed gamma–gamma coincidences.

Small values obtained for the broadening factor in all experiments performed bythe method described above may be due to a nonuniform isomeric shift associatedwith an irregular distribution of cadmium in silver and to the high-frequency part ofthe noise from boiling helium. One cannot rule out either the possibility of thesituation where part of silver nuclei that emerged after the decay of cadmium nucleiappear to be at interstitials of the crystal lattice. A quadrupole splitting could manifestitself in this case in the spectrum of photons emitted by these nuclei, and this may bethe reason for a severalfold decrease in the resonant-absorption cross section. Itshould be noted that the distinction between the broadening-factor values obtained inthe experiments reported in [28, 32] and the experiment reported in [33] may beassociated with the different treatment of the experimental results: the divergence of agamma beamwas taken into account in thefirst two cases, but it was disregarded in theexperiment reported in [33], as well as in experiments described in the next chapter.

References

1. L.D. Landau, E.M. Lifshitz, Classical Theory of Fields (Addison Wesley, Cambridge, MA,1951)

2. C.M. Lederer, V.S.Shirley (eds.), Table of Isotopes, 7th edn. (Wiley, New York, 1978)3. P.B. Sogo, C.D. Jeffries, Phys. Rev. 93, 174 (1954)4. M.Y. Kagan, Soviet JETP 20, 1 (1965)5. H.H.F. Wegener, Z. Physik A281, 183 (1977)6. V.I. Goldanskii, E.F. Makarov, in Chemical Applications of Mössbauer Spectroscopy, ed. by

V.I. Goldanskii, R.N. Herber (Academic, New York, 1968)7. L.D. Roberts, R.L. Becker, F.E. Obenshain, J.O. Thomson, Phys. Rev. A 895, 137 (1965)8. A. Shirley, Rev. Mod. Phys. 36, part II, 339 (1964)9. G.K. Wertheim. Mössbauer Effect (Academic, New York, 1964)10. R.V. Pound, G.A. Rebka, Phys. Rev. Lett. 4, 274 (1960)11. W. Keesom, N. Pearlman, in Kaltephysik. Handbuch der Physik, B. XIV–XV, (Springer,

Berlin, 1956)12. G.E. Bizina, A.G. Beda, N.A. Burgov, A.V. Davydov, Soviet JETP 18, 5 (1964)13. A.V. Davydov, Y.N. Isaev, V.M. Samoylov, Bull. Acad. Sci USSR, Phys. 61, 1747 (1997)14. A.G. Beda, G.E. Bizina, A.V. Davydov, in Problems of Nuclear Physics and Physics of

Elementary Particles (Nauka, Moscow, 1975) p. 209 (in Russian)15. V.G. Alpatov, A.G. Beda, G.E. Bizina et al., in Proceedings of the International Conference

on Mössbauer Spectroscopy, Bucharest, Romania, 1977, vol 1, contributed papers, p. 43 (InRussian)

16. G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill Book Company, New York,1968)

17. A.S. Davydov, Theory of Atomic Nucleus (Fizmatgiz, Moscow, 1958) (in Russian)18. A.T. Levon, O.F. Nemez, Electromagnetic Moments of the Excited and Radioactive Nuclei

(Naukova Dumka, Kiev, 1989), p. 213 (in Russian)19. A.I. Akhiezer, V.B. Berestetskii, Quantum Electrodynamics, (Interscience, New York, 1965),

p. 66


20. W. Wildner, U. Gonser, J. Phys. Coll. Suppl. 40(C2), 47 (1979)21. G.R. Hoy, R.D. Taylor, J. Quant. Spectrosc. Radiat. Transf. 40, 763 (1988)22. R.D. Taylor, G.R. Hoy, SPIE 875, 126 (1988)23. S. Rezaie-Serej, G.R. Hoy, R.D. Taylor, Laser Phys. 5, 240 (1995)24. V.G. Alpatov, G.E. Bizina, A.V. Davydov et al., Voprosy Tochnosti v yadernoi spektroskopii

(Questions of Precision in Nuclear Spectroscopy) (Institute of Physics, Academy of Sciencesof LietSSR, Vilnius, 1984), p. 15 (in Russian)

25. V.G. Alpatov, G.E. Bizina, A.V. Davydov et al., Preprint ITEP No 130 (Moscow, 1984) (inRussian)

26. S.I. Novikova, Thermal Expanding of Solids (Nauka, Moscow, 1974). (in Russian)27. V.G. Alpatov, A.V. Davydov, G.R. Kartashov et al., Meas. Tech. 38, 341 (1995)28. V.G. Alpatov, Y.D. Bayukov, V.M. Gelis et al., Laser Phys. 10, 952 (2000)29. V.G. Alpatov, G.E. Bizina, A.V. Davydov et al., Meas. Tech. 37, 101 (1994)30. B.M. Budak, A.A. Samarskii, A.N. Tikhonov, Collection of Problems on Mathematical

Physics (Fizmatlit, Moscow, 2003), p. 289 (in Russian), (Dover Publication, New York, 1988)31. E. Storm, H. Israel, Photon Cross Sections from 0.001 to 100 MeV for Elements 1 Through

100 (Scientific Laboratory, Los Alamos, New Mexico, 1967)32. V.G. Alpatov, Y.D. Bayukov, A.V. Davydov et al., Final report on the INTAS project No 97-

3156633. V.G. Alpatov, Y.D. Bayukov, A.V. Davydov et al., Laser Phys. 15, 1680 (2005)34. I.S. Grigor’ev, E.Z. Melikhov (eds.), Physical Quantities (handbook) (Energoatomizdat,

Moscow, 1991), p. 223 (in Russian)35. P. Ganner, H. Rauch, Nucl. Instr. Meth. 76, 295 (1969)36. A.B. Migdal, Qualitative Methods in Quantum Theory (W.A. Benjamin, London, 1977)37. A.V. Davydov, Phys. At. Nucl. 66, 2113 (2003)38. A.V. Davydov, Phys. At. Nucl. 70, 1182 (2007)39. A.V. Davydov, Phys. At. Nucl. 74, 11 (2011)40. F.S. Dzheparov, D.V.L’vov, E.V. Sil’vacheva, J. Surf. Invest. X-ray, Synchrotron Neutron

Tech 3, 47 (2009)

References 125

Chapter 4Fundamentals of Gravitational GammaSpectrometry

4.1 Design of a Gravitational Gamma Spectrometer Basedon the 109mAg Isomer

A small value of the broadening factor for the 109mAg Mössbauer gamma linepermits developing quite a new line of research in gamma spectroscopy—gravi-tational gamma spectrometry, whose resolution may turn out to be eight to tenorders of magnitude higher than that of ordinary Mössbauer spectrometers thatemploy gamma rays of the 57Fe nuclide. A gravitational gamma spectrometer wasdesigned and created for the first time by our group at ITEP [1]. We present itslayout in Fig. 4.1 and its general view in Fig. 4.2. A support, together with acryostat and a pair of Helmholtz coils mounted on it, is placed on a flat platform thatmay be rotated about a horizontal axis through angles of up to 30° in both direc-tions. Using the Helmholtz coils, one can compensate for the geomagnetic-fieldcomponent parallel to the cryostat axis (that is, perpendicular to the platform plane)at the site of gamma sources placed inside the cryostat. For the gamma sources, weused a single-crystal silver plate with the parent nuclide 109Cd introduced in it fromboth sides by means of thermal diffusion and two thin control sources from 241Am.The americium gamma sources were situated on both sides of the silver plate andwere tightly pressed to it. In the working state, the sources were directly washed byliquid helium. Two planar HPGe detectors that measured the intensities of gammabeams parallel to the platform plane that were emitted from the sources werearranged on the opposite sides of the cryostat. Since the geomagnetic-field com-ponent parallel to the cryostat axis changes upon inclining the platform, it wasnecessary to change accordingly the current through through the Helmholtz coils inorder to compensate for this component. The values of the current for differentangular positions of the platform were determined in experiments that employed asmall-size magnetic-field-strength indicator that permitted verifying, to a rather highdegree of precision, that the current through the Helmholtz coils was such at whichthe component to be compensated was close to zero (the residual fluctuations of this


127

component did not exceed 0.1 % of its uncompensated value). The principle ofoperation of the gravitational gamma spectrometer is clarified by Fig. 4.3, in whichc the cross section the silver-plate gamma source in the position where the directionof the detected gamma beam is deflected through some angle from the horizontalplane is shown. For a photon emitted at point A, the difference H of the verticalcoordinates between the emission point and the point of possible resonantabsorption (B) increases as the photon moves to the detector. This leads to a gradualdecrease in the resonant-absorption cross section because of the growing gravita-tional shift of the gamma-resonance energy. The larger the angle of deflection, thestronger this effect; at a rather large deflection angle, resonant gamma-rayabsorption turns out to be possible only over such a short segment of the photon

Fig. 4.1 Layout of the gravitational gamma spectrometer: (1) cryostat, (2) germanium detectors,(3) rotatable platform, (4) platform pivot, (5) Helmholz coils, (6) gamma sources, and (7) supportof the cryostat and Helmholz coils

128 4 Fundamentals of Gravitational Gamma Spectrometry

Fig. 4.2 General view of the gravitational gamma spectrometer

Fig. 4.3 Illustration clarifying the principle of operation of a gravitational gamma spectrometer.The cross section of a silver gamma source in the position where the gamma beam is deflectedfrom the horizontal direction is shown; A is the point of emission of a photon, B is the point of itspossible resonant absorption; and H is the difference of the vertical coordinates of points A andB. This difference grows with increasing distance between these points

4.1 Design of a Gravitational Gamma Spectrometer Based on the 109mAg Isomer 129

path in the source material that the resonant-absorption effect becomes unobserv-able in practice. If one measures the intensity of 109mAg gamma rays for thegamma-beam direction strongly deflected from the horizontal plane and, continuingmeasurements, gradually reduces the deflection angle to zero and thereupon makesthe platform rotate in the same direction, so that the deflection angle increases, thenit would be natural to expect that the number of counts is minimal for a horizontaldirection of the gamma beam, smoothly and symmetrically growing with respect tothis minimum as the deflection angle becomes larger.

The difference of the vertical positions between the photon-emission pointA(x1, y1, z1) and the resonant-absorption point B(x2, y2, z2) can be expressed interms of the angle θ of deflection of the spectrometer platform from the horizontalplane as (see Fig. 4.4)

H ¼ zabs � z1cosw

sin h� wð Þ; ð4:1Þ

Fig. 4.4 Scheme clarifying the meaning of the geometric quantities appearing in Eqs. (4.1)–(4.3).Shown in the figure are the projections of the horizontal plane (S1) and the detector-window plane(S2) onto the (y, z) plane of the gamma-source cross section (inclined rectangle of width d). Thescale is not preserved. By A(x1, y1, z1) and B(xabs, yabs, zabs), we denote, respectively, the point atwhich the emission of a photon occurred and the point of its possible resonant absorption. Thez axis is parallel to the axis of a weakly divergent gamma beam connecting the centers(intersections of the diagonals) of the gamma source and the detector window. By φ and ψ, wedenote, respectively, the angle of deflection of the beam axis from the horizontal plane and theangle between the projections onto the (y, z) plane of the straight line parallel to the z axis and thestraight line along which the photon originating from point A and passing through point B movesto the detector window


where w ¼ arctg y2�y1z2�z1

. The number of photons originating from the gamma-sourcevolume element dV1 = dx1 dy1 dz1 with coordinates x1, y1, and z1 and hitting thedetector-window area element dS2 = dx2 dy2 with coordinates x2 and y2 is pro-portional to

dN � e�bz21 þ e�b b�z1ð Þ2h i dS2 cos3 c

4p z0 � z1ð Þ2 e�l d�z1ð Þcos c

� e

�r0m2k

Rdz1

dzabs

1þ0;683758 � 108zabs�z1ð Þ sin h�wð Þ

k cosw

h i2

dz1: ð4:2Þ

With the exception of the angles θ and ψ introduced immediately above, thequantities appearing in expression (4.2) are identical to those in expression (3.33).The angle γ is determined by formula (3.25).

After performing integration in the exponent of the last exponential function inexpression (4.2), we obtain

dN ¼ e�bz1 þ e�b d�z1ð Þh i dS2 cos3 c

4p z0 � z1ð Þ2 e�l d�z1ð Þ

cos c

� e� r0m cosw

1;653793�104 sin h�wð Þarctg0;826897�104 d�z1ð Þ sin h�wð Þ

k cosw

h idz1 ð4:3Þ

Integration of expression (4.3) over the source volume (that is, with respect tothe coordinates x1, y1, and z1) and over the detector-window area (that is, withrespect to the coordinates x2 and y2) gives a quantity that is proportional to the totalnumber of detected photons for a given angle of the inclination of the platform.Simultaneously, the divergence of the gamma beam is taken into account. After thecalculation of the angular dependences of the number of counts for different valuesof the broadening factor k, one must find (from the minimum of the χ2 criterion) thatvalue which describes optimally respective experimental data. This procedureinvolves varying two parameters: the factor k itself and the coefficient of propor-tionality between the experimental and calculated data. If there arises a doubt as towhether the beam position at an angle of θ = 0 in fact differs from the horizontalposition, then it is necessary to add a small angle of inclination of the platform as athird parameter to be varied.

If it is legitimate to neglect the gamma-beam divergence, the description of theangular dependence of the number of counts becomes simpler; that is,

N �Zd

0

e�bz2 þ e�b d�zð Þ2h i

e�l d�zð Þ e�Qdz; ð4:4Þ

4.1 Design of a Gravitational Gamma Spectrometer Based on the 109mAg Isomer 131

http://dx.doi.org/10.1007/978-3-319-10524-6_3

http://dx.doi.org/10.1007/978-3-319-10524-6_3

where

Q ¼ 5:963923� 10�5

sin harctg

0:826897 � 104 d � zð Þ sin hk

� �: ð4:5Þ

In [1], we considered, among other things, the case where the main gammasource of a gravitational gamma spectrometer is a silver plate 1 mm thick in whichwe introduced the parent nuclide 109Cd from one side, the mean depth of itspenetration being 0.15 mm. In order to simplify calculations, we assumed that allcadmium atoms lie in a thin layer at this depth and that the dimensions of the sourceand the detector are small in relation to the distance between them. This permitsneglecting the divergence of the gamma beam. In this case, the number of photonsrecorded by the detector for a gamma beam deflected through an angle θ is pro-portional to the following quantity (the linear dimensions are in centimeters):

N ¼ N0 expð�0,085lÞ expZ0;0850

r0m2k

dx

1þ 0,683759� 108 x sin hk

� �2h i24

35

¼ N0 exp �0,085lð Þ r0m2

arctg 702,826sin hk

� �: ð4:6Þ

The notation here is identical to that in (4.5). With the aid of this expression, wecalculated the angular dependence of the probability for the resonant absorption of88.03-keV gamma rays in a gamma source of the type being considered for severalvalues of the broadening factor k. The results are presented in Fig. 4.5. All curvesare normalized to the same value at θ = 0. However, it should be recalled that theresonant-absorption cross section is in inverse proportion to k. Using the curves inFig. 4.5, one can find the gamma-beam-deflection angle θ1/2 at which the resonant-absorption probability decreases by a factor of two in relation to its maximum valueas a function of the gamma-line broadening factor k. This dependence, which isshown in Fig. 4.6, turns out to be very simple in the case being considered; that is,

h1=2 ¼ 0:197k;

where θ1/2 is measured in degrees.

4.2 Experiments Performed at ITEP with the Aidof a Gravitational Gamma Spectrometer

In our first experiment performed with a gravitational gamma spectrometer, themeasurements were conducted at gamma-beam deflection angles of +7°, +3°, +1°,0°,−1°,−3°, and −7°. A positive sign corresponds to the rise above the horizontal


Fig. 4.5 Decrease in the intensity of 109mAg gamma rays, Δiγ, because of resonant self-absorptionin silver as a function of the gamma-beam deflection angle α for several values of the broadeningfactor k for the Mössbauer gamma line, which are indicated on the corresponding curves. Thethickness of the silver layer that photons traverse is 0.85 mm. The curves were normalized to thesame value at α = 0. However, it should be borne in mind that the real values of ΔIγ are in inverseproportion to k

Fig. 4.6 Deflection angle α1/2corresponding to a twofolddecrease in resonant gamma-ray absorption as a function ofthe broadening factor k for theMössbauer gamma line

4.2 Experiments Performed at ITEP... 133

plane of the detector, which we assign number 1. We present the results obtained inthis experiment with the detector in question in Fig. 4.7, which, in the upper part(A), shows data (ratios R+ of the numbers of counts for gamma lines of Ag and Am)obtained at a temperature 4.2 K and with the Helmholtz coils in the on mode—thatis, under conditions optimum for the observation of resonant gamma-ray self-absorption in the source substance. The middle part of the figure (B) gives the

Fig. 4.7 Results of the first experiment with a gravitational gamma spectrometer: PanelA shows theratio R+ of the number of counts in detector no. 1 for Ag and Am gamma lines as a function of thegamma-beam deflection angle α in the on mode of the Helmholtz coils. The solid line 1, which wascalculated by formula (4.4), describes optimally this dependence; it corresponds to k= 11 and to the χ2

criterion equal to 0.655 per degree of freedom. Curve 2 was calculated for the value of χ2 = 1. Itcorresponds to the value of k = 7.7. The dashed straight line shows the calculated level of the ratio Rfor the case where there is no resonant absorption. Panel B shows the analogous dependence for theratioR−measured in the offmode of theHelmholtz coils. PanelC presents the results ofmeasuring theangular dependence of R+ at room temperature of the gamma sources. The shaded band represents thenumber of counts that was averaged over all angular positions and the error in it


results obtained from measurements at the same temperature but in the off mode ofthe Helmholtz coils (R−).

Since the geomagnetic field had a natural direction in that case (in Moscow, itpoints downward,0 forming an angle of 70° with the horizon), the cross section forthe resonant absorption of silver gamma rays must be smaller by a factor of 2.5 thanits value in the preceding case. The third part of the figure (C) shows the results of themeasurements conducted at room temperature of the gamma sources in the on modeof the Helmholtz coils. A resonant-absorption effect must be absent in that case.Indeed, no reduction of the number of counts for a horizontal direction of gammabeams was observed. The shaded band in Fig. 4.7c represents the number of countsthat was averaged over all angular positions and the error in it. The χ2 criterion,whose value makes it possible to assess the extent to which this average value isconsistent with the hypothesis that the intensity of detected gamma radiation isidentical for all angular positions is 0.85 per degree of freedom among six ones in thecase being considered. The smooth curve 1 in Fig. 4.7a was calculated by formula(4.4). It describes optimally the experimental data being considered (the value of theχ2 criterion per degree of freedom among six ones is 0.655) and corresponds to abroadening factor of 11�22:5

�5:0 for the Mössbauer gamma line of silver. The smoothcurve 2 is the result of the calculation by formula (4.4), in which case χ2 = 1, thebroadening factor then being 7.7. Visually, curve 2 seems to describe experimentaldata better than curve 1, but this is an erroneous impression. However, either curvegives quite an acceptable description of the experimental data. The dashed horizontalstraight line in Fig. 4.7a corresponds to the R+ value for the case where there is noresonant absorption of silver gamma rays. In our articles quoted in [2, 3] and devotedto describing the experiment being discussed, we present results obtained by adifferent, simplified, method for determining the broadening factor. We calculatedthe difference of the ratio R+ of the numbers of counts for Ag and Am gamma linesthat corresponds to the so-called basic line determined by the average value of R+ fordeflection angles of ±7° and ±3° and the value of R+ for the horizontal gamma-beamposition. Using this difference, we calculated the effective cross section for theresonant absorption of silver gamma rays and compared it with the computed valuesof the cross section for the gamma line that did not undergo broadening. Thebroadening factor was determined as the ratio of the cross section calculated in theabsence of broadening to the cross section found experimentally.

As a matter of fact, the above method for determining the basic line correspondsto the assumption that there is no manifestations of resonant absorption at thegamma-beam deflection angles of ±7° and ±3°, but this is not true. At the level ofaccuracy of the experiment being discussed, it turned out, however, that thebroadening-factor value indicated in the aforementioned publications (6:3þ5:2

�1:9) andthe broadening-factor value presented above and calculated more correctly agreewithin the errors. Figure 4.8 shows the angular dependence of the number of countsfor the gamma line of the control gamma source (241Am) according to measure-ments at the temperature of 4.2 K. One can see that there is no reduction of thenumber of counts for the horizontal position of the gamma beam.


Unfortunately, the results obtained by measuring the angular dependence of R+

with the second detector turned out to be less informative. To all appearances, thegamma-beam axis was deflected from the direction parallel to the platform planethrough an angle somewhat larger than one degree—that is, it was directed to theregion of inclination angles where the measurements were not performed. In thisgamma beam, resonant absorption manifested itself only in a weak reduction of R+

at an inclination angle of +1°.The second experiment with a gravitational gamma spectrometer [4] was carried

out without any changes in the structure and geometry of the setup in relation to thefirst experiment. We employed the same gamma sources, which remained intactbetween the two experiments. Only the measurement procedure was slightlymodified by adding the angular positions at ±0.67° and at ±0.33°. Therefore, thesecond experiment may be considered as a continuation of the first one. In thecourse of the measurements, there arose some troubles in detecting americiumgamma rays. To all appearances, the americium gamma sources, which were fab-ricated 2.5 years ago by impregnating chromatographic paper with a solution ofamericium nitrate, were partly destroyed. Under a chemical effect and under theinfluence of intense irradiation with americium alpha particles, the paper partlydecomposed. Americium-containing paper particles that emerged upon this beganchaotically moving inside the aluminum covers in response to changes in theinclination angle of the spectrometer platform and in response to unavoidablepushes caused by the on–off switches of its drive. As a result, the number of countsexperienced unpredictable changes exceeding severalfold the statistical errors of themeasurements. For this reason, we will consider only the results concerning silvergamma rays. It should be noted that the analogous damage of the calibrating

Fig. 4.8 Results of measurements with detector no. 1 of the intensity of 241Am gamma rays as afunction of the gamma-beam deflection angle (in degrees) according to the first experiment with agravitational gamma spectrometer. The statistical standard deviations are equal to the radii of thecircles


americium gamma source fabricated at approximately the same time and accordingto the same technology was established quite obviously. The number of counts thatwas measured with detector no. 1 for the 109mAg gamma line at the gamma-sourcetemperature of 4.2 K is shown in Fig. 4.9 as a function of the gamma-beamdeflection angle. Figure 4.9a corresponds to the on mode of the Helmholz coilscompensating for the geomagnetic-field component perpendicular to the gamma-beam axis. These conditions are the most favorable for observing resonantabsorption. The data in Fig. 4.9b were obtained for a natural direction of thegeomagnetic field. According to the measurements of this field that were performedjust before the start of the experiment being discussed, its strength vector pointeddownward at our laboratory, forming an angle of 83° at that time with the horizontalplane. Under these conditions, the cross section for the resonant absorption of silvergamma rays was smaller by a factor of 2.06 than the value corresponding to themost favorable conditions. Thus, we can clearly see the effect of the change in themagnetic-field direction, and this is one of the most convincing pieces of evidencethat we do indeed observe the resonant absorption of silver gamma rays. Thesmooth curves in Fig. 4.9a, b were calculated by formula (4.4). They describe

Fig. 4.9 Intensity of 109mAggamma rays that wasmeasured in the secondexperiment with agravitational gammaspectrometer as a function ofthe gamma-beam deflectionangle at liquid-heliumtemperature in the (a) on and(b) off modes of theHelmholtz coils. Themeasurements wereperformed with detector no. 1


optimally the experimental results (in calculating the curve in Fig. 4.9b, we did nottake into account the point corresponding to an inclination angle of +7° andstrongly deviating from the general regularity). These curves correspond to thefollowing values of the broadening factor k and the χ2 criterion: for Fig. 4.9a,k = 7:0þ7:3

�2:5 and χ2 = 0.62 per degree of freedom among 10 ones, while, for the curvein Fig. 4.9b, k = 8 with greater errors and χ2 = 0.64. Figure 4.10 shows dataobtained at room temperature of the gamma sources. Figure 4.10a, b corresponds tomeasurements performed in the on (off) mode of the Helmholtz coils. The shadedbands represent the values obtained by averaging the numbers of counts over allangular positions and the errors in these values. The values of the χ2 criterion thatmake it possible to assess the degree to which the description of respective angulardependences in terms of these average values under the assumption that therecorded gamma-ray intensities are identical for all inclination angles is reliable are1.04 for Fig. 4.10a and 1.57 for Fig. 5.7b per degree of freedom among 10 ones. Asin the previous experiment, detector no. 2 showed but a slight reduction of thenumber of counts at freedom among 10 ones, while, for the curve in Fig. 4.9b, k = 8

Fig. 4.10 Intensities of 109mAg gamma rays versus the gamma-beam deflection angle at roomtemperature according to measurements in the second experiment with a gravitational gammaspectrometer in the (a) on and (b) off modes of the Helmholtz coils. The shaded bands representthe average values of the numbers of counts and the errors in these values


http://dx.doi.org/10.1007/978-3-319-10524-6_5

with greater errors and χ2 = 0.64. Figure 4.10 shows data obtained at room tem-perature of the gamma sources. Figure 4.10a, b corresponds to measurementsperformed in the on (off) mode of the Helmholtz coils. The shaded bands representthe values obtained by averaging the numbers of counts over all angular positionsand the errors in these values. The values of the χ2 criterion that make it possible toassess the degree to which the description of respective angular dependences interms of these average values under the assumption that the recorded gamma-rayintensities are identical for all inclination angles is reliable are 1.04 for Fig. 4.10aand 1.57 for Fig. 5.7b per degree of freedom among 10 ones. As in the previousexperiment, detector no. 2 showed but a slight reduction of the number of counts atan inclination angle of +1°. This circumstance is quite understandable becausedetector no. 2 was not shifted from its place after the first experiment. Therefore, theentire body of experimental data obtained in the two experiments with a gravita-tional gamma spectrometer gives sufficient grounds to state that we did indeedobserve the resonant absorption of 109mAg-isomer gamma rays and determined thewidth of the gamma resonance, albeit with a low precision. The value found for thiswidth indicates that we reached a gamma-spectrometer resolution that is eightorders of magnitude higher than that characteristic of Mössbauer spectrometersworking with the 57Fe nuclide. This gives sufficient grounds to plan experimentsthat were proposed previously, but which either did not reach the stage of imple-mentation or did not give expected results because of an insufficient resolution ofaccessible instruments. We imply here experiments such as those that aim atsearches for the anisotropy of inertia [5] and at the detection of gravitational waves[6]. The experiments that we have thus far performed with a gravitational gammaspectrometer confirmed once again the conclusion of previous studies that, incontrast to predictions of contemporary theory, dipole–dipole interaction does notlead to a large broadening of the Mössbauer gamma line of the 109mAg isomer.There is still no convincing explanation for the absence of this broadening. Onemay only consider some hypothetical reasons for this phenomenon. It is worthrecalling once again that the absence of broadening is incompatible with thestatement that a photon is emitted within a time of about ƛ/c (see above).

To conclude this chapter, we consider some possibilities for improving the effi-ciency of the gravitational gamma spectrometer. A transition to manufacturing agamma source from pure or almost pure 109Ag isotope will permit doubling theobserved effect of resonant absorption. We cannot rule out the possibility that theapplication of silver oxides instead of metallic silver will lead to a substantialincrease in the probability for recoilless gamma-ray emission (absorption), asoccurred in the case of 119Sn gamma rays. In this case, however, additional studieswill be necessary in order to develop a method for implanting cadmium into a silverhost material in such a way that cadmium would appear there in the form of an oxide,because it is possible that silver atoms arising upon the decay of cadmium nucleiintroduced in the silver lattice in the form of atoms would emit photons with a largeisomeric shift with respect to the line of absorbing silver nuclei in silver-oxidemolecules. The creation of conditions under which the Earth’s magnetic field wouldbe reduced by about seven orders of magnitude in the region of the silver gamma


http://dx.doi.org/10.1007/978-3-319-10524-6_5

source would be the most radical but, at the same time, very complicated method forenhancing the effect of resonant gamma-ray absorption. The emission and absorp-tion gamma lines would not be split in this case. As a result, the cross section forresonant absorption would increase by a factor of 64/17 [7]. It is possible to design agravitational gamma spectrometer where the inclination angle of the gamma beamundergoes automatic and unceasing changes accompanied by the synchronouschange in analyzer channels recording the intensities of source gamma rays, as inMössbauer spectrometers working in the regime of unceasingly changing speed ofthe source with respect to the absorber. It would very interesting to study the pos-sibility of creating a gravitational gamma spectrometer on the basis of the 103mRhisomer. The energy of the isomeric state of this nuclide is 39.75 keV, and the half-lifeof this state is 56.12 min [8]. The respective mean lifetime of the nucleus in theexcited state is 80.964 min. This means that the natural width of the 103mRh isomer issmaller by a factor of 85 than the corresponding value for the 109mAg gamma line.The Debye temperature of metallic rhodium is 480 K [9]; taken together with a ratherlow photon energy, this yields a value of 0.465 for the probability of recoillessemission (absorption) at room temperature. The possibility to working at roomtemperature would of course be a great advantage in relation to what we have for the109mAg isomer. Unfortunately, rhodium is paramagnetic, and it is not clear atthe present time whether this paramagnetism would affect the broadening factor forthe rhodium Mössbauer gamma line. A very large coefficient of internal conversionfor 39.75 keV gamma rays (as large as about 1430) is yet another drawback ofrhodium. Nonetheless, one may cherish a hope for creating conditions under whichthe effect of resonant absorption would exceed values char acteristic of experimentswith 109mAg at least by one order of magnitude. It is noteworthy that, in 2005, agroup of Chinese physicists published an article [10] where they stated that theywere able to observe a manifestation of the Mössbauer effect in a sample of metallicrhodium, exciting nuclear isomeric states by bremsstrahlung. However, no confir-mation of this result has appeared to date, as far as I know.

References

1. V.G. Alpatov, YuD Bayukov, A.V. Davydov et al., Meas. Tech. 48, 194 (2005)2. V.G. Alpatov, YuD Bayukov, A.V. Davydov et al., Laser Phys. 17, 1067 (2007)3. V.G. Alpatov, YuD Bayukov, A.V. Davydov et al., Phys. At. Nucl. 71, 1156 (2008)4. YuD Bayukov, A.V. Davydov, YuN Isaev et al., JETP Lett. 90, 499 (2009)5. C.W. Shervin, H. Frauenfelder, E.L. Garwin et al., Phys. Rev. Lett. 4, 399 (1960)6. W. Kaufmann, Nature 227, 157 (1970)7. A.V. Davydov, YuN Isaev, V.M. Samoylov, Bulleten Acad. Sci. USSR Phys. 61, 1747 (1997)8. Radionuclide Transformations—Energy and Intensity of Emissions. ICRP Publication 38.

Pergamon Press, 1983, 103mRh9. P. Boolchand, J. Quant. Spectrosc. Radiat. Transf. 40, 777 (1988)10. Cheng Yau, Xia Bing, Liu Yi-Nong, Jin Qing-Xiu, Chin. Phys. Lett. 22, 2530 (2005)


Chapter 5Nuclear Resonant Scatteringof Annihilation Photons

5.1 Introduction

There are several nuclei (see Table 5.1) having excited states at energies in thevicinity of 511 keV. This circumstance permits conducting experiments aimed atobserving resonant scattering on these nuclei that is experienced by photons orig-inating from positron annihilation in matter. The experiments reported in [1–8] andaimed at studying the shapes of the energy spectra of annihilation photons showthat the widths of respective photon lines reach 3–3.5 keV; therefore, the spectrumof annihilation photons may overlap the nuclear levels in question. The τ valuesquoted in Table 5.1 were calculated by using the half-lives indicated in the cor-responding articles.

We performed experiments devoted to observing resonant scattering by 106Pdnuclei that is experienced by photons created in positron annihilation in copper.

5.2 Expected Cross Section

The cross section for this process can be estimated most straightforwardly on the basisof an elementary model of the positron-annihilation phenomenon in metals. Weassume that, in the overwhelming majority of cases, a positron that appeared in a metalundergoes annihilation there together with one of the conduction electrons. Indeed, themean time of positron thermalization is about 10−12 s [9]. At the same time, the meanlifetime of positrons in metals with respect to two-photon annihilation exceeds greatlythis value, falling within the range of 10−10–10−9 s. This means that, immediatelybefore annihilation, the majority of positrons have a thermal energy and cannotapproach bound electrons of atomic shells to undergo annihilation with them, becausepositively charged ions of the crystal lattice repel such positrons. In contrast to elec-trons, positrons obey a Maxwellian rather than a Fermi energy distribution because oftheir low concentration in metals. The Pauli exclusion principle forbids this in the case


141

of electrons. Therefore, the form of the energy spectrum of annihilation photons isdetermined largely by the character of motion of conduction electrons in a metalbecause the velocities of these electrons exceed greatly, on average, the velocities ofthermalized positrons. It follows that, before annihilation, an electron–positron pairowes its center-of-mass velocity to the electron velocity exclusively.

In order to deduce a formula for the energy spectrum of annihilation photons, weassume that positrons of negligible energy undergo annihilation only with con-duction electrons having a Fermi energy distribution. In order to simplify relevantcalculations, we disregard the effect of the crystal lattice on the shape of the Fermisurface, assuming that it is spherical.

If an annihilation photon emitted in the z direction is detected, then its energyE is obviously related to the projection vz of the velocity of the center of mass of theannihilating particle pair onto the z axis by the equation

E ffi Ec0 þ Ec0

vzc; ð5:1Þ

where Ec0 is the energy of a photon in the annihilation of a pair at rest. Therefore, it isnecessary to find the distribution of annihilating pairs with respect to the projectionsof their center-of-mass velocities onto the z axis. We now apply the laws of energyand momentum conservation to the process of electron–positron annihilation(see Fig. 5.1). We have

Ec1 þ Ec2 ¼ 2mc2 þ p2

2m; ð5:2Þ

Ec1

c�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEc2

c

� �2

� p2 � p2z� �s��

�� ¼ pz; ð5:3Þ

Table 5.1 Properties of the some nuclei having the levels with energy near 511 keV

Nucleus Relative content inthe natural mixtureof isotopes (percent)

Energy of the excited state(keV)

Mean lifetime τof nuclei in theexcited state (s)

71Ga 39.8 [33] 511.5 ± 0.1 [33] 1.98 × 10−12 [34]

0.19 × 10−9 [33]106Pd 27.33 [35] 511.76 ± 0.08 (averaging of

data from [14, 17, 32,36–41]

(16.31 ± 0.87) × 10−12 [32]

165Ho 100 [35] 515, 472 [42] (24.5 ± 4.3) × 10−12 [42]177Hf 18.5 [35] 508.1 ± 0.1 [43]187Re 62.93 [44] 511.65 ± 0.05 [45] (23.1 ± 8.7) × 10−12 [45]239Pu 511.8 [46]

511.84 [34]

142 5 Nuclear Resonant Scattering of Annihilation Photons

where Ec1and Ec2 are the energies of two annihilation photons, the first of themmoving along the z axis in the positive direction; p is the electron momentum; pz isits projection onto the z axis; m is the electron mass; and c is the speed of light.Obviously, neither p nor pz must exceed in magnitude the electron Fermimomentum pF—the value of electron momentum corresponding to upper limit of

Fermi distribution. Because the Fermi energy EF ¼ p2F2m falls within the range

between 5 and 10 eV, then p2

2m � 2mc2 and p2ð Þ � p2z � Ec2c2 . Therefore, Eqs. (3.31)

and (3.32) can besubstantially simplified upon neglecting the quantity p2

2m against

2mc2 in the first of them and the difference p2 � p2z� �

againstEc2c

� 2in the second.

After that, we have

Ec1 þ Ec2 ffi 2mc2; ð5:4Þ

Ec1

c� Ec2

cffi pz: ð5:5Þ

From these equations, it follows that

Ec1 ffi mc2 þ cpz

2: ð5:6Þ

One can see that, in our model, the spectrum of annihilation radiation extendsover an interval of width cpzmax

2 ¼ cpF2 on either side of the mean value Ecmean ¼ mc2:

Therefore, the total width of the spectrum is D ¼ cpF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mc2EF

p: From here, it

follows that, for the majority of metals, the value of Δ is about 2.2 to 3.2 keV.

Fig. 5.1 Addition of momenta in an event of the annihilation of an electron—positron pair in thecase where the positron momentum is negligible against the electron momentum. Here, P is theelectron momentum, while pz is the projection of this momentum onto the z axis, which iscoincident with the direction of emission of one of the annihilation photons, that of momentumpc1 pc1 ¼ Ec1=c� �

. The momentum of the second annihilation photon is pγ2 = Eγ2/c

5.2 Expected Cross Section 143

http://dx.doi.org/10.1007/978-3-319-10524-6_3

We now proceed to derive an expression for the distribution of conductionelectrons with respect to the projections of their momenta onto the z direction. If wedeal with a metallic sample of volume V containing N (total number) conduction

electrons whose Fermi energy is EF ¼ p2F2m (we assume that the conduction-electron

effective mass is m), then the phase space for these electrons is

Vphase ¼ 43p p3F V : ð5:7Þ

The volume of an elementary cell in the phase space is h3 [10]. In such a cell,two electrons may be distinguished by their spin orientations. Hence, the totalnumber of Fermi gas electrons in the metallic sample being considered is

N ¼ 2� 43p p3F

Vh3

¼ 8p 2mð Þ3=2E3=2F V

3h3: ð5:8Þ

The number of electrons N(E)dE with energy between E and E + dE is obtainedby multiplying the product of the density of states ρ(E) and dE by the probabilitythat the state being considered is occupied. This probability is determined by theFermi distribution, which, according to [11], has the form

w Eð Þ ¼ 1

eE�EFkT þ 1

: ð5:9Þ

At T = 0 K, this distribution is w(E) = 1 for E < EF and w(E) = 0 for E > EF.These relations survive up to temperatures of several hundred K units (deviationsbecome observable at energies differing from EF by several kT units). In our case,we therefore have

N Eð Þ dE ¼ q Eð Þ dE: ð5:10Þ

According to [12], we have

q Eð Þ dE ¼ 4pV2mh2

� �3=2 ffiffiffiffiE

pdE: ð5:11Þ

Since the condition

N Eð Þ dE ¼ N pð Þ dp

must be satisfied, the number of electrons whose momenta lie in the intervalbetween p and p + dp is given by


N pð Þ dp ¼ N Eð Þ dEdp

dp ¼ 4pV2mh2

� �3=2 pffiffiffiffiffiffi2m

p pmdp ¼ 8pV

h3p2dp: ð5:12Þ

Let f p; pzð Þ dpz be the probability that an electron of momentum p has theprojection of this momentum onto the z axis in the range between pz and p z + dpz.Obviously, the total number of electrons with momentum projections from thisinterval is then given by

N pzð Þ dpz ¼ZpFpz

N pð Þ f p; pzð Þ dp

264

375dpz: ð5:13Þ

The function f(p, pz) can be determined in the following way. The number ofelectrons whose momenta fall within the range from p to p + dp is proportional tothe volume of a spherical layer in momentum space, the inner radius and thicknessof this layer being, respectively, p and dp (see Fig. 5.2). But the number N(p, pz) ofelectrons belonging to this layer and having momentum projections onto the z axisin the interval between pz and pz + dpz is proportional to the volume of the sphericalring obtained by cutting the spherical layer in question with two planes that areperpendicular to the z axis and which are separated by the interval dpz. The plane

Fig. 5.2 Scheme explaining the determination of that part of conduction electrons whosemomentum projections onto a specific z axis lie in the range between pz and pz + dpz (for metalswhose Fermi surface is spherical)


closest to the center of the sphere is at the distance pz from it. In Fig. 5.2, the crosssection of this ring by the plane of the figure is hatched. The volume of the ring is

p acð Þ2�p bcð Þ2j k

dpz ¼ p pþ dpð Þ2�p2zj k

� p p2 � p2z� �n o

dpz ffi 2p p dp dpz:

ð5:14Þ

Therefore, we have

f p; pzð Þ dpz ¼ N p; pzð Þ dpzN pð Þ dp ¼ 2p p dp dpz

4p p2dp¼ dpz

2p: ð5:15Þ

Substituting Eqs. (5.12) and (5.15) into (5.13), we obtain

N pzð Þ dpz ¼ 4p Vh3

dpz

ZpFpz

p dp ¼ 2p Vdpzh3

p2F � p2z� � ð5:16Þ

From (5.8), it follows that

2p Vh3

¼ 3N4p3F

: ð5:17Þ

Therefore, we have

N pzð Þ dpz ¼ 34N

1pF

� p2zp3F

� �dpz: ð5:18Þ

But the probability that an electron has a momentum projection onto the z axis inthe interval between pz and pz þ dpz is

w pzð Þ dpz ¼ N pzð ÞdpzN

¼ 34

1pF

� p2zp3F

� �dpz: ð5:19Þ

We will now check the normalization of this probability:

2ZpF0

34

1pF

� p2zp3F

� �dpz ¼ 3

21pF

ZpF0

dpz � 1p3F

ZpF0

p2z dpz

24

35 ¼ 3

21� 1

3

� �¼ 1:

ð5:20Þ

The factor of 2 in front of the integral in Eq. (5.20) takes into account the factthat pz may be both positive and negative. Because the energy of an annihilationphoton emitted in the z direction is related to pz by Eq. (5.6), one can obtain from


Eq. (5.19) an expression for the probability that this photon has an energy in theinterval between Eγ and Eγ + dEγ. Since the relation

w pzð Þ dpz ¼ w Ec� �

dEc ð5:21Þ

must hold and since pz ¼ 2Ec

c � 2mc, so that dpz ¼ 2dEc

c , we have

w Ec� �

dEc ¼ 32D

1� 4Ec � mc2

D

� �2" #

dEc; ð5:22Þ

where D ¼ cpF .Figure 5.3 shows the spectrum of annihilation photons according to calculations

for the case of Δ = 3 keV (this corresponds to EF = 8.8 eV).We now proceed to derive a formula for the cross section for the resonant

scattering of annihilation photons by nuclei that have a level at an energy E0 closeto mc2. If deexciting transitions from this level go mainly to the ground state and if,in addition, the coefficient of internal conversion for these transitions is small, thenthe cross section for the resonant scattering of photons with energy Eγ is given by

ð5:23Þ

Fig. 5.3 Spectrum of annihilation photons that was calculated by formula (5.11) for the case ofΔ = 3 keV (EF = 8.8 eV)


where is the scattered-radiation wavelength divided by 2p; Je and J0 are,respectively, the excited- and the ground-state spins; and Er is the resonance energy,which differs from E0 by the energy of recoil experienced by the nucleus as itabsorbs a photon,

Er ffi E0 þE2c

2Mc2ð5:24Þ

(M is the mass of the nucleus).If a resonant scatterer is irradiated with a photon flux whose spectral distribution

is specified by a function denoted by N(Eγ) and normalized to unity, then theresonant-scattering cross section averaged over this distribution is given by

rcpeo:pe3: ¼Z10

rpe3: Ec� �

N Ec� �

dEc: ð5:25Þ

Under the condition of a small change in N Ec� �

with energy in the region whererpe3: Ec

� �differs markedly from zero, one can factor N Ec

� �outside the integral sign

and, with the aid of expression (5.23), reduce Eq. (5.25) to the form

rcpeo:pe3: ¼ k2

4gCN Erð Þ; ð5:26Þ

where g ¼ 2Jeþ12J0þ1.

Since the width of the spectrum of annihilation photons, D ¼ cpF ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mc2EF

p,

lies in the interval between 2.5 and 3 keV for the majority of metals and since thedistinction between E0 and Er is about 2 eV even for the lightest of the nucleipresented in Table 5.1, 71Ga, one may neglect the difference between E0 and Er insubstituting, into Eq. (5.26), the value of N(Er) obtained from expression (5.22) andultimately recast Eq. (5.26) into the form

rcpeo:pe3: ¼ 38k2g

CD

1� 4E0 � mc2

D

� �2" #

: ð5:27Þ

Among the nuclides listed in Table 5.1, we selected the 106Pd nuclide for ourexperiments. The position of the level of interest and its width were known betterfor this nucleus than for the remaining ones. Moreover, it was important that theground- and excited-state spins were 0 and 2, respectively. Therefore, the statisticalfactor g appearing in expression (5.27) for the cross section for the process beingconsidered is equal to five—this is nearly the largest value for which one may hopein these experiments.

Table 5.2 presents data on the energy of the level of our interest for the 106Pdnucleus that were known to the present author at the time of the experiment with


this nuclide. Data nos. 10–13 were published after the completion of our experi-ments. It is noteworthy that the value measured by the authors of [13] differs bothfrom the most precisely measured energy of the level in earlier experiments [14]and from the later result quoted in [15], the respective error corridors not over-lapping each other. We also note that the authors of [16] give two values for thislevel of 106Pd, 511.52 and 511.8 keV, without commenting on this and withoutindicating errors. For the energy of the level, we therefore used in our study thevalue obtained by averaging values nos. 1 to 9 from Table 5.2. The result was511.76 ± 0.08 keV. According to [17], the mean lifetime of the 106Pd nucleus in thisstate is (16.31 ± 0.87) × 10−12 s, whence it follows that the natural width of the levelis Γ = (4.04 ± 0.22) × 10−5 eV. Using these data and setting Δ = 3 keV, we then findthat expression (5.27) leads to the following estimate for the resonant-scatteringcross section averaged over the spectrum of annihilation photons:

ravres ¼ 1:08� 0:24ð Þ � 10�27cm2: ð5:28Þ

5.3 Description of Our Experiments

The most suitable method for experimentally determining the cross section for theresonant scattering of annihilation photons consists in measuring the spectra ofradiation emitted by an annihilation-photon source and a source of gamma rays withan energy close to 511 keV and scattered by two scatterers close in atomic numberZ, one of them being a resonant scatterer for annihilation radiation. Therefore, onehas to measure four spectra. As will be shown below, the cross section for the

Table 5.2 Experimental dataon the energy of the 2+ stateof 106Pd

No. Energy of the first 2+ state (keV) References

1 511.6 ± 0.3 [17, 36]

2 511.77 ± 0.20 [37]

3 510.8 ± 0.9 [38]

4 511.6 ± 0.5 [39]

5 511.8 ± 0.1 [14]

6 511.7 ± 0.3 [32]

7 511.8 ± 0.8 [40]

8 511.0 ± 1.0 [41]

9 512.0 ± 1.0 [41]

10 511.52 ± 0.03 [13]

11 511.52 The error is not indicated [16]

12 511.8 The error is not indicated [16]

13 511.81 ± 0.08 [15]


resonant scattering of annihilation photons can be derived from these data if thecross section for Rayleigh scattering of gamma rays whose energy is close to511 keV, the energy dependence of this cross section, and its dependence on theatomic number of the scatterer are known. Figure 5.4 illustrates the geometry of ourexperiment. Annihilation photons or gamma rays from a source (1) pass a colli-mator (2) and hit a scatterer (3) arranged in such a way that a normal to it formsangles of α and β with respect to the directions of, respectively, the incident and thescattered radiation beam. After traversing a filter (4), photons scattered at an angle θfind their way to a detector (5).

Fig. 5.4 Geometry of the experiment aimed at observing resonant annihilation-photon scatteringon nuclei. Explanations are given in the main body of the text


Let us now introduce the following notation:N1 is the number of detector counts per unit time for the case where use is made

of a resonant scatterer and nonresonant photonsN2 is the analogous quantity for a nonresonant scatterer and nonresonant

radiationN3 is the same for a nonresonant scatterer and a source of annihilation

photonsAnnihilation photonsN4 is the same for a resonant scatterer and annihilation photonsμ1 is usual (electron) linear coefficient of absorption of nonresonant photons in

a resonant scattererμ2 is the analogous quantity for the absorption of the same photons in a

nonresonant scattererμ3 is the coefficient of absorption of annihilation photons in a nonresonant

scattererμ4 is the analogous quantity for the absorption of annihilation photons in a

resonant scattererd1 is the thickness of a resonant scattererd2 is the thickness of a nonresonant scattererε1 is the detector efficiency for nonresonant gamma raysε2 is the detector efficiency for annihilation photonsν1 is the number of atoms in 1 cm3 of a resonant scattererResonant scattererν2 is number of atoms in 1 cm3 of a nonresonant scattererS is the area of the detector surface to which scattered photons are incidentN01 is the number of photons emitted by a nonresonant source per unit timeN02 is the same for the source of annihilation photonsdr1dX

is the differential cross section for the elastic scattering of nonresonantgamma rays by atoms of a resonant scatterer (in our case, one may neglect alltypes of elastic nonresonant scattering, with the exception of Rayleighscattering)

dr2dX

is the same for atoms of a nonresonant scattererdr3dX

is the same for annihilation photons and atoms of a nonresonant scattererdr4dX

is the differential cross section for non-nuclear elastic scattering ofannihilation photons by atoms of a resonant scatterer (that is, the crosssection for Rayleigh scattering)

drres:dX

is the differential cross section for the resonant scattering of annihilationphotons by nuclei of resonant-scatterer atoms

a is the relative content of that isotope in the natural mixture of isotopes of aresonant scatterer which has a resonance level.

It should be noted that the quantities N1, N2, N3, and N4 are determined as theareas of the elastic-scattering peaks observed in the spectra of scattered radiation.These peaks manifest themselves in the amplitude spectra of detector pulses at thatplace where the peak of the total absorption of initial gamma or annihilationradiation must be.

5.3 Description of Our Experiments 151

One may write

N1 ¼N01

dr1dX se1m1

4pR21R

22 cos a

Zd10

e�l1x1

cos aþ 1cos bð Þ dx

¼ 1� e�l1d11

cos aþ 1cos bð Þ� N01

dr1dX se1m1

8pl1R21R

22 cos a

1cos a þ 1

cos b

� ;ð5:29Þ

where R1 and R2 are, respectively, the distance from the source to the scatterer andthe distance from the scatterer to the detector and the x axis is orthogonal to thescatterer plane.

In a similar way, we obtain

N2 ¼ 1� e�l2d21


dr2dX se1m2

8pl2R21R

22 cos a

1cos a þ 1

cos b

� ; ð5:30Þ

N3 ¼ 1� e�l3d21


dr3dX se2m2

8pl3R21R

22 cos a

1cos a þ 1

cos b

� ; ð5:31Þ

N4 ¼ 1� e�l4d11


dr4dX þ a drres

dX

� �se2m1

8pl4R21R

22 cos a

1cos a þ 1

cos b

� ; ð5:32Þ

whence it follows that

N2

N1¼

dr2dX m2l1 1� e�l2d2

1cos aþ 1

cos bð Þ� dr1dX m1l2 1� e�l1d1

1cos aþ 1

cos bð Þ� ð5:33Þ

and

N3

N4¼

dr3dX m2l4 1� e�l3d2

1cos aþ 1

cos bð Þ� dr4dX 1þ a

drresdXdr4dX

� �m1l3 1� e�l4d1

1cos aþ 1

cos bð Þ� : ð5:34Þ

It will be shown below that the differential cross section for the Rayleighscattering at an angle θ of gamma rays with energy E by atoms of atomic numberz can be described by the empirical formula


dr h;E; zð ÞdX

¼ A � B E; hð Þ � C z; hð Þ: ð5:35Þ

At a fixed scattering angle θ, the functions B(E) and C(z) can be represented in asimple form—Em and zn, respectively—over rather wide ranges (see below).

Substituting Eq. (5.24) into Eqs. (5.22) and (5.23), we obtain

N2

N1¼

C z2ð Þm2l1 1� e�l2d21

cos aþ 1cos bð Þ�

C z1ð Þm1l2 1� e�l1d11

cos aþ 1cos bð Þ� ; ð5:36Þ

N3

N4¼

Cðz2Þm2l4 1� e�l3d21

cos aþ 1cos bð Þ�

Cðz1Þ 1þ adrresdXdr4dX

� �m1l3 1� e�l4d1

1cos aþ 1

cos bð Þ� : ð5:37Þ

The energy dependences of the Rayleigh cross sections were excluded fromhere. From Eqs. (5.36) and (5.37), it follows that

N2N4

N1N3¼ D 1þ a

drresdXdr4dX

!; ð5:38Þ

where

D ¼l1l3 1� e�l2d2

1cos aþ 1

cos bð Þ� 1� e�l4d1

1cos aþ 1

cos bð Þ� l2l4 1� e�l1d1

1cos aþ 1

cos bð Þ� 1� e�l3d2

1cos aþ 1

cos bð Þ� ð5:39Þ

From here, it follows that

drresdX

¼ dr4dX

N2N4

DN1N3� 1

�� 1a

ð5:40Þ

By appropriately choosing the scatterer thicknesses and the angles α and β, onecan make the quantity D very close to unity. Equation (5.40) is correct only if theradiation spectrum of the annihilation-photon source does not contain gamma raysof energy higher than 1.022 MeV. If such gamma rays are present, then 511 keVphotons may additionally arise via the mechanism of pair production by thesegamma rays that is followed by positron annihilation in the scatterer (I am gratefulto Yu.K. Shubnyi, who indicated that it is necessary to take this effect into account).From an analysis of this process, one can deduce a formula that describes thedifferential cross section for the resonant scattering of annihilation photons in thecase where the source spectrum contains one gamma line whose energy is higher than


1.02 MeV and whose intensity relative to the positron-annihilation intensity is k. Spe-cifically, we have

drresdX

¼ 1adr4dX

N2N4

N1N3A� B

� �; ð5:41Þ

where

A ¼ 1� e�l31

cos aþ 1cos bð Þd2

l31

cos a þ 1cos b

� þ k

dr03�dX 1� e

� l03

cos aþl03

cos b

� d2

" #

dr3=dXl03

cos a þ l3cos b

� 8>>>><>>>>:

9>>>>=>>>>;

�1� e�l1d1

1cos aþ 1

cos bð Þh il2l4

1cos a þ 1

cos b

� 1� e�l2d2

1cos aþ 1

cos bð Þh i1� e�l4d1

1cos aþ 1

cos bð Þh i ;ð5:42Þ

B ¼ k

dr04�dX 1� e

� l04

cos aþl4cos b

� d1

" #l4

1cos a þ 1

cos b

� dr4=dX

l04cos a þ l4

cos b

� 1� e�l4d1

1cos aþ 1

cos bð Þh i þ 1: ð5:43Þ

The quantities appearing in these expressions in addition to those specifiedabove are the following:dr03dX

is the differential cross section for the annihilation-photon yield from theprocess of pair creation in a nonresonant scatterer

dr04dX

is the analogous quantity for a resonant scattererResonant scatterer

l03 is the linear coefficient of absorption of gamma rays in nonresonant-scatterersubstance that are able to create pairs in this substance

l04 is the analogous quantity for the substance of a resonant scatterer

As was indicated above, a resonant scatterer for our experiments was made frommetallic palladium of natural isotopic composition. This scatterer had the shape of asquare plate 4 mm thick, the length of the square side being 160 mm. As a non-resonant scatterer, we used a silver plate having the same area and a thickness of4.5 mm. Annihilation-photon sources were made from a copper foil in the form ofsquare plates having a thickness of 0.1 mm and a mass of 0.5 g. These plates wereirradiated at the ITEP heavy-water reactor in a thermal-neutron flux of density about3 × 1013 1/cm2s for 23 h. As radioactive substance of the second gamma source, weused the 181Hf nuclide produced by irradiating hafnium dioxide of natural isotopiccomposition with reactor neutrons. Hafnium-dioxide powder was pressed into around pellet 15 mm in diameter. The pellet was wrapped in an aluminum foil andwas irradiated with neutrons in this form. After irradiation, the pellet extracted from


the foil was soldered in the brass container whose walls had a thickness of 0.1 mm.In the gamma spectrum of 181Ta produced upon 181Hf beta decay, there is a 488keV gamma line, which we used in our experiments. Figure 5.5 shows the decayscheme for the isotope 64Cu. One can see that its radiation spectrum contains a1345.9 keV gamma line of intensity 0.0311 relative to the intensity of the positrondecay of 64Cu [18, 19]. Although two annihilation photons are produced in thescatterer in each pair-creation event induced by respective gamma rays, which isfollowed by positron annihilation, these two photons must not be taken into accountbecause two photons are also produced in the source upon positron annihilation andbecause the ratio of the annihilation-radiation intensities for these two processes isin fact contained in expression (5.30).

5.4 Data on the Cross Sections for the Rayleigh Scatteringof Gamma Rays

From Eqs. (5.29) and (5.30), one can see that, in order to determine drresdX , it is

necessary to know the differential cross section for Rayleigh scattering at least forone source–scatterer combination of four ones (for annihilation photons and pal-ladium in the case being considered). At the time when the experiments describedhere were performed, the most detailed data on differential cross sections forRayleigh scattering in the energy region of our interest were those that wereobtained in [20, 21]. In the first of those studies, the differential cross sections weremeasured in the angular range of 45°–135° for 662- and 279 keV gamma rays forthe following scatterers: zinc, molybdenum, tin, neodymium, tantalum, and lead.The authors of that study presented a half-empirical method for calculating thesecross sections, which, in almost all of the cases, leads to agreement with experi-mental data within not more than 6 %. Only for an energy of 279 keV and z > 70

Fig. 5.5 Scheme of decay of the 64Cu nucleus according to [18]


does the deviation from experimental results become as large as 12 % for angles inexcess of 90°. In [21], the same authors present cross sections calculated for eightgamma-ray energies in the range between 145 and 750 keV. By interpolating thesedata, one can obtain comparatively reliable data on cross sections for the Rayleighscattering of gamma rays with an energy of about 500 keV by silver and palladiumatoms. Figure 5.6 shows the results obtained by interpolating the energy depen-dences of the differential cross sections for Rayleigh scattering at scattering anglesof 105° and 120° for scatterers whose atomic numbers z are 42 and 50. At values ofthe gamma-ray energy Eγ in the region of our interest, these dependences are welldescribed by functions of the form

drRdX

� E�kc : ð5:44Þ

If Eγ is expressed here in keV units, then k = 5.39 for z = 42 and k = 5.38 forz = 50. From here, it follows that, for Ag and Pd atoms (z = 47 and 46) inclusive,one can adopt the energy dependence of the cross section in the form (5.33) with anexponent of k = 5.385.

In Fig. 5.7, the dependences of drRdX on the scatterer atomic number z for 482- and

511 keV gamma rays at scattering angles of 110° and 120° are shown according toresults also obtained by interpolating data from [21]. In the z range of our interest,

Fig. 5.6 Dependence of differential cross sections for the Rayleigh scattering of gamma rays ontheir energy E for the scatterers whose atomic numbers are z = 42 (1, 3) and 50 (2, 4)


these dependences are close to linear ones in terms of log drRdX

� �and log z plotted

along the coordinate axes. This means that they can described as

drRdX

� zp: ð5:45Þ

The exponent p is quite large; it is 6.22 at Eγ = 511 keV and θ = 120° anddepends only slightly on the gamma-ray energy.

Therefore, the real dependences of drRdX on Eγ and on z differ strongly from what is

given by the Franz formula [22]. The cross-section values also differ severalfoldfrom those given by the Franz formula in the regions of energies and scattereratomic numbers of our interest.

The interpolation procedure that we used leads to differential cross sections forRayleigh scattering with errors that we estimate at 10 % (6 % is the error of thecalculations performed by the authors of [21], and the remaining part is due to theinterpolation). Table 5.3 gives the values for the cross sections of interest at scat-tering angles of 110° and 120°.

5.5 First Experiment Aimed at Observing NuclearResonant Scattering of Annihilation Photons

The experiment being discussed was performed at the setup that was used previ-ously in the experiments devoted to measuring unperturbed angular distributions ofgamma rays resonantly scattered by 182W and 191Ir nuclei. At the beginning of that

Fig. 5.7 Dependence of differential cross sections for Rayleigh scattering on the scatterer atomicnumber z for the gamma-ray energies of 482 keV (1, 2) and 511 keV (3, 4)

5.4 Data on the Cross Sections for the Rayleigh Scattering of Gamma Rays 157

experiment, we had not yet had germanium detectors that would have been suffi-ciently large for efficiently detecting scattered gamma rays of energy about0.5 MeV; therefore, we employed a scintillation counter based on a NaI(Tl) crystal40 mm in diameter and in height. Pulses from a photomultiplier tube were amplifiedand were fed to an AI-100 amplitude analyzer. The front face of the counter wascovered with a lead layer 6 mm thick in order to reduce additionally the workloadof the scintillation detector and electronics circuit due to pulses from low-energygamma rays.

The measurements were conducted alternately with scatterers from Pd and Ag,which were changed in the experiments with sources from 64Cu every five to tenminutes. Since the half-life of 181Hf is considerably longer than that of 64Cu, thereplacements of the scatterers in the experiments with 181Hf were rarer (every20 min). In either case, the scattering angle was to 120°.

The total spectrum of pulses from scattered radiation for a nearly monoenergeticinitial gamma beam contains an intense peak that is due to the Compton scatteringof gamma rays by free and quasifree electrons and which goes over to a longdecreasing tail extending to the region of high energies. At the end of this tail, thereis a small peak of elastic processes, including Rayleigh scattering and nuclearresonant scattering. Figure 5.8 shows that part of this spectrum which is recorded bya multichannel analyzer after the low-energy part of the spectrum is cut off by anintegral discriminator. The smoothly decreasing part of the spectrum is due pri-marily to the incoherent scattering of gamma rays by bound electrons [23] andpartly to bremsstrahlung generated by fast electrons arising in the scatterer underthe effect of the initial gamma-ray beam.

The ratios of the numbers of counts in channels for scattered-radiation spectrameasured for the scatterers from Pd and Ag are shown in Fig. 5.9 according tocalculations for the cases where 181Hf (I) and 64Cu (II) were used as radiationsources. One can see that there is a peak on curve II in the vicinity of 511 keV; atthe same time, there is no peak on curve I at 482 keV. We interpreted this result asan indication of the presence of an additional mechanism of elastic gamma-rayscattering in palladium and identified this mechanism with nuclear resonant scat-tering. We note, however, that the energy dependence of the ratios in questionrepresented by the curves in Fig. 5.9 differs somewhat from that which might havebeen expected a priori (it was M.G. Gavrilov who attracted our attention to thisdifference). Indeed, the differential cross sections for Rayleigh scattering is much

Table 5.3 Cross-sections of Rayleigh scattering of Gamma rays with energies of interest

Scatteringangle

Gamma-ray energy(keV)

Scatterer Differential cross section for Rayleighscattering (mb/sterad)

110° 511 Pd 0.219 ± 0.022

110° 511 Ag 0.250 ± 0.025

120° 511 Pd 0.190 ± 0.019

120° 511 Ag 0.217 ± 0.022


more strongly dependent on the scatterer atomic number z than the cross sectionsfor incoherent scattering by bound electrons, which makes a dominant contributionto the formation of a smoothly decreasing (with energy) continuous spectrum lyingbetween the peaks of Compton and elastic scattering:

drRdX

zp; p[ 6 see aboveð ÞRincoh z:

[23].Since zAg > zPd, a dip must appear in the region of the elastic-process peak on the

curve representing NPd/NAg because Rayleigh scattering, which is more intense inthe case of a silver scatterer, also contributes in this energy region in addition toincoherent scattering. But on the curve for the source from 64Cu, a decrease in thisdip, its filling, or a maximum similar to that in Fig. 5.9(I) must appear, dependingon the resonant-scattering cross section for annihilation photons. The curve inFig. 5.9(II) seems to show some trend toward a decrease in NPd/NAg near 482 keV,

Fig. 5.8 Fragment of the amplitude spectrum of pulses from annihilation radiation scattered bypalladium and detected by a scintillation counter

5.5 First Experiment Aimed at Observing Nuclear Resonant Scattering… 159

but, in view of overly large experimental errors, the definitive conclusion on thepresence of this decrease would be premature. This circumstance henceforth servedas one of the reasons for which we repeated these measurements at a substantiallyupgraded setup.

The areas of the peaks corresponding to elastic processes were determined byapproximating measured spectra by functions of the form

y ¼ a1 þ a2xþ a3 x� a4ð Þ2e� x�a4ð Þ2a5 þ a6e� x�a7ð Þ2a8 : ð5:46Þ

The first three terms on the right-hand side of this equation describe a smoothlydecreasing background, which, in the experiments with a scintillation detector, canbe approximated rather well by a Gaussian distribution. The experimental data werecompared with the function in (5.46) on the basis of the least squares method withthe aid of a computer, the coefficients a7 and a8 being determined beforehand inexperiments with calibration sources from 64Cu and 181Hf. Knowing the parametersa6, a7, and a8, one can determine the area under the corresponding Gaussian curve.The calculations led to the following values for the quantities appearing inEqs. (5.40) and (5.41):

Fig. 5.9 Ratios of the numbers of counts in analyzer channels for the spectra of radiation scatteredby Pd and Ag scatterers in the cases of the (1) 64Cu source (annihilation photons) and (2) 181Hfsource (gamma rays)


N2=N1 ¼ 1:130� 0:012;

N4=N3 ¼ 0:970� 0:012:

At the time when we published the results of our first experiment [24], we did nothave at our disposal sufficient data on differential cross sections for Rayleigh scat-tering and therefore had to use the aforementioned Franz formula to determine them.Moreover, we treated our data without taking into account contributions of detectedannihilation photons that arise in the scatterers under the effect of 1345.9 keV gammarays from the copper source.

Because of these two circumstances, an exaggerated value of 0.27 ± 0.05 mb/srwas obtained for drres

dX (120°). In performing the second experiment devoted toobserving the resonant scattering of annihilation photons, we took into account theeffect of the 1345.9 keV gamma line.

Below, we will show that the results of the two experiments are compatible andpresent the value averaged over the data of the two experiments as the cross sectionfor the resonant scattering of annihilation photons by 106Pd nuclei.

5.6 Second Experiment in Which the Nuclear ResonantScattering of Annihilation Photons was Observed

This experiment was performed at a setup substantially improved by replacing thescintillation counter by a coaxial Ge(Li) detector of working volume about 25 cm3.This permitted sharply improving conditions for selecting the peak of elastic-scattering processes in the scattered-radiation spectrum, despite the fact that,because of a high counting workload of the detector, its resolution was not high—the FWHM of the peak at 511 keV was about 8 keV. The setup was placed in adifferent lodging, where, there was virtually no background of annihilation photonsnot stemming from the sources used. The sources and scatterers used had the samedimensions and shapes as in the first experiment. The measurement procedure wasthe same, but we changed the scattering angle to about 110° (in the first experiment,it was 120°) with the aim of improving the protection of the detector.

A fragment of the spectrum of pulses from the Ge(Li) detector recording radi-ation scattered by palladium is shown in Fig. 5.10 for the case where radioactivecopper was a source. A comparison with Fig. 5.8 shows clearly advantages of thenew setup. In Fig. 5.11, the ratios of the numbers of counts in channels are given forscatterers from palladium and silver for both sources (64Cu and 181Hf). As mighthave been expected, the graph corresponding to the scattering of hafnium gammarays exhibits a dip in the region corresponding to the energy of the gamma line inquestion (482 keV). The graph for annihilation photons does not show such a dip at511 keV, and this is indicative of the contribution from the resonant scattering ofthese photons.

5.5 First Experiment Aimed at Observing Nuclear Resonant Scattering… 161

In order to separate the peak of elastic-scattering processes from the decreasingpart of the spectrum, the latter was approximated by an analytic curve depending onseveral parameters, but the part associated with the peak was not included in theapproximation. The parameters of the curve were determined by minimizing the χ2

criterion with the aid of a computer. It was less convenient to determine the peakareas by approximating the entire spectrum by an analytic curve depending onmany parameters because the peaks of elastic-scattering processes differed stronglyin shape from Gaussian distributions and required introducing an overly largenumber of parameters in order to describe them.

It is interesting that attempts at selecting an optimum shape of curves that woulddescribe the decreasing parts of the spectra led to revealing a previously unobserved

4×104

511 keVCou

nt n

umbe

rs

3×104

2×104

1×104

0 25 50

Channel number of analyzer

75 100

Fig. 5.10 Fragment of the amplitude spectrum of pulses from scattered annihilation radiationrecorded by the Ge(Li) detector. We used 64Cu as a source (annihilation photons) and palladium asa scatterer


peculiar structure in the spectra of incoherently scattered gamma rays. This will beconsidered below in more detail.

Finally, the decreasing parts of the spectra were satisfactorily described by thesums of smoothly decreasing functions and small oscillating terms. An evaluationof the areas of the peaks of elastic-scattering processes by simply subtracting thecalculated decreasing distributions from the corresponding parts of the measuredspectra leads to an overly large error in the expression N2N4

N1N3appearing in Eq. (5.30).

In order to reduce the error, one can rely on the assumption that the peaks corre-sponding to elastic-scattering processes are identical in shape for each pair of thespectra associated with the same source. This assumption is justified by the fact thatthe instrumental gamma-line width of our detector is considerably larger than thewidth of the annihilation-photon spectrum, which is reproduced upon Rayleighscattering and, to a still greater extent, exceeds the Doppler width of the spectrum ofresonantly scattered radiation. As a result, lines belonging to the measured spectraand referring to different scattering processes are similar to each other. Therefore,one can evaluate the ratios N2/N1 and N4/N3 by averaging, with proper weights, thecorresponding ratios of the peak ordinates obtained for each channel of the spec-trum. These calculations yielded

Fig. 5.11 Ratios of the numbers of counts in individual analyzer channels according tomeasurements with a Ge(Li) detector for the spectra of radiation scattered by Pd and Ag scatterersin the cases of sources from (a) 181Hf and (b) 64Cu (annihilation photons). The energy scales aredifferent in these two cases. The straight lines represent least squares fits (the areas of the peaks ofelastic-scattering processes, 15 channels in each case, did not take part in the respectiveapproximations)

5.6 Second Experiment in Which the Nuclear Resonant Scattering… 163

N2

N1¼ 1:129� 0:013;

N3

N4¼ 1:057� 0:010:

5.7 Cross Section for the Resonant Scattering of AnnihilationPhotons by 106Pd Nuclei

In order to determine the resonant-scattering cross section by formula (5.41), oneneeds, in addition to differential cross sections presented in Table 5.3 for Rayleighscattering, the cross sections for pair production by 1345.9 keV gamma rays insilver and palladium. More precisely, it is necessary to know the differential crosssections for the yield of annihilation radiation created in the scatterer under theeffect of the aforementioned gamma rays. Since annihilation photons arising in this

process have an isotropic angular distribution, the differential cross sections dr03dX and

dr04dX appearing in (5.30) are obtained by dividing the total cross sections for pairproduction in the corresponding scatterers by 4π.

Unfortunately, the data that existed at the time of those experiments for pair-production cross sections (see, for example, [25] ) gave no way to determine, withthe required precision, these cross sections for 1345.9 keV gamma rays. There wereno rather simple and reliable methods for calculating the pair-production crosssection in this energy region. The available formulas for these cross sections [26]were derived either for relativistic energies of the components of product pairs orfor their threshold production. Therefore, we had to determine the contribution ofannihilation radiation associated with 1345.9 keV gamma rays by means of anindividual dedicated experiment. This simple experiment consisted in measuringthe yield of annihilation radiation from the scatterer under the effect of radiationsfrom the copper source in the presence and in the absence of a lead absorber on thepath of the initial beam of these radiations (I am grateful to Dr. A.G. Beda, whorecommended to use this simple method). Owing to a large difference in thecoefficients of absorption of primary annihilation photons and 1345.9 keV gammarays, we were able to separate the contributions of the two radiations to the for-mation of the 511 keV gamma line in the scattered-photon spectrum.

For a silver scatterer 0.475 cm thick located at the same position as the scatterersin the second experiment, we found that, among annihilation photons, the part thatoriginated from pair creation by gamma rays is 0.197 ± 0.027 relative to the totalyield of annihilation radiation. Using this value and relying on the value presentedin Table 5.3 for the differential cross section for the Rayleigh scattering of anni-hilation photons by silver atoms, we estimate the expected cross section for paircreation in silver as


r03 ¼ 20� 4 mb:

The analogous result obtained for palladium by reducing this value in accor-dance with the factor z2 is

r04 ¼ 19:2� 3:8 mb:

These values are substantially smaller than those that one could expect on thebasis of a smooth extrapolation of the values calculated in [25]. According tocalculations by formula (5.41) that employ data on the cross section for Rayleighscattering (Table 5.3) and on the cross section for pair production by 1345.9 keVgamma rays, the differential cross section for the resonant scattering of annihilationphotons by 106Pd nuclei is

drresdX

ðh ¼ 120Þ ¼ 0:067� 0:017 mb/sr

in the first experiment and

drresdX

ðh ¼ 110Þ ¼ 0:050� 0:016 mb/sr

in the second experiment.In order to obtain from here total cross sections for resonant scattering and to use

them as a basis for determining the width of the excited state of the 106Pd nucleus at511.76 keV and, hence, its mean lifetime in this state, it is necessary to invoke dataon the angular distribution of gamma rays resonantly scattered by nuclei whoseground- and excited-state spins are, respectively, 0 and 2 [27]. We have alreadyconsidered such a distribution in connection with measuring it in the experimentwith 182W. It has the form

W hð Þ ¼ 1þ 0:3571 P2 cos hð Þ þ 1:1429 P4 cos hð Þ; ð5:47Þ

where Pk is a Legendre polynomial of order k.Recasting expression (5.47) into a simpler form, we arrive at

W hð Þ ¼ 1� 3 cos2 hþ 4 cos4 h: ð5:48Þ

The differential cross section is related to the angular distribution W(θ) by theequation

dr hð ÞdX

¼ A �W hð Þ: ð5:49Þ

5.7 Cross Section for the Resonant Scattering of Annihilation Photons… 165

The total cross section is given by

rtot ¼Z

drdX

dX: ð5:50Þ

Therefore, we have

rtot ¼ AZ

W hð Þ dX ¼ 2pAZp0

W hð Þ sin h dh: ð5:51Þ

Using expression (5.48), we find that, in our case, the total cross section is

rtot ¼ 16p5

A: ð5:52Þ

The coefficient A is determined from Eq. (5.49) on the basis of measured values ofdrresdX and the angular distribution calculated for the same angles. We finally obtain

rtot ¼ 1:35� 0:34 mb

for the first experiment and

rtot ¼ 0:72� 0:23 mb

for the second experiment.The value obtained for the total cross section for resonant scattering upon

averaging over the results of the two experiments in question is

rtot ¼ 0:92� 0:19 mb:

According to the χ2 test, the probability for obtaining, in two independentexperiments, the above values of the total cross section for resonant scattering is12 %. Therefore, the cross-section values obtained in the two experiments arecompatible. The results of the second experiment were published in [28].

The application of formula (5.51) to determining rtot on the basis of drresdX and

expression (5.48) for W(θ) is justified only in the case where there is no hyperfineinteractions, which could perturb the function W(θ). Although metallic palladiumhas a cubic face-centered lattice in which there should be no electric quadrupoleinteraction with excited 106Pd nuclei, such an interaction is possible in principle fornuclei that experienced recoil in an event of resonant photon absorption and provedto be off a crystal-lattice site. Under the assumption that the perturbation of theangular distribution by the electric quadrupole interaction is maximum possible(hard-core case), the calculation of the total cross section for resonant scattering onthe basis of the measured differential cross section yields


rtot:perturbedrtot:unperturbed

¼ 0:891

at θ = 110° and

rtot:perturbedrtot:unperturbed

¼ 0:654

at θ = 120°.Thus, the disregard of the perturbation of the angular distribution may in principle

lead to an error in the total cross section for resonant scattering as large as 11 % in theexperiment at θ = 110° and as large as 35 % in the experiment at θ = 120°. It turnsout, however, that, because of a small value of the mean lifetime of the 106Pd nucleusin the excited state (τ * 10−11 s), the perturbation is estimated at a value muchsmaller than that at which the angular distribution is described by the hard-corefunction. By way of example, we indicate that, if one sets the electric quadrupolemoment of the 106Pd nucleus in the excited state to 0.56 × 10−24 cm2 ([29, p. 406])and assumes that the effective gradient of the electric field is 1018 V/cm2, theattenuation coefficient for the anisotropy of the angular distribution of resonantlyscattered photons, Gkk [30], is compatible with unity within the accuracy importantfor us. Therefore, there are sufficient grounds to treat this angular distribution as anunperturbed one.

The form of the annihilation-radiation spectrum was studied, for example, in[31], by using a beta spectrometer of high resolution, the authors of that studymeasured the spectrum of photoelectrons knocked out by annihilation photons froma thin gold radiator. They showed that the spectral distribution of annihilationphotons emitted upon positron annihilation in copper and brass differs stronglyfrom the predictions of an elementary annihilation-process model, according towhich positrons undergo annihilation with conduction electrons having the Fermimomentum distribution. The measured spectrum turned out to be substantiallywider than the spectrum expected on the basis of this model, suggesting a signif-icant role of processes in which positrons undergo annihilation with electrons ofatomic shells. The form of the spectrum measured in [31] is described to a precisionsufficient for our goals by the expression

N E�c

� ¼ 0:375 expð0:01573 E�

c

�� 0:49405E�2c þ 0:06144 E�

c

�� 3Þ1=keV; ð5:53Þ

where the energy of annihilation photons is expressed in keV units and is reckoned

from 511 keV. The function N E�c

� given by expression (5.53) is normalized to

unity under the assumption that N E�c

� differs from zero in the energy range of

–4.65 keV ≤ E�c ≤ + 4.65 keV (E�

c = Ec – 511 keV).

5.7 Cross Section for the Resonant Scattering of Annihilation Photons… 167

The shape of this spectrum is shown in Fig. 5.12, and the position of the resonantlevel of the 106Pd nucleus is indicated there.

By using Eq. (5.26), expression (5.53) for N(Er), and the value that we found forthe total cross section for the resonant scattering of annihilation photons, we candetermine the natural width of the excited level of the 106Pd nucleus. The result is

C ¼ rtotk2

42J1þ1ð Þ2J0þ1ð ÞN Erð Þ

¼ 4:26� 0:91ð Þ � 10�5eV:

From here, we obtain the following value for the mean lifetime of the 106Pdnucleus in the excited state being considered: s ¼ �h

C ¼ 15:5� 3:3ð Þ � 10�12 s.It should be recalled that this value of τ was calculated by using the Γ value

averaged over two experiments. It agrees well with the value measured in [32] bymeans of the Coulomb excitation of 106Pd nuclei [τ = (16.31 ± 0.87) × 10−12 s]. Theagreement between the two τ values obtained by the radically different methodsindicates that our experiments and our treatment of their results were quite correct.

5.8 Further Ways Toward Refining Upon the Methodfor Observing the Process Under Discussion

The presence of more intense Rayleigh scattering of annihilation photons is themain hindrance to clearly separating the effect of the nuclear resonant scattering ofthese photons. At first glance, it seems that, in the spectrum of detected radiation,

Fig. 5.12 Spectrum of annihilation radiation created upon the annihilation of electron–positronpairs in copper according to measurements reported in [31]


the separation of the total-photon-absorption peaks associated with Rayleigh andresonant scattering is impossible in principle. However, there are distinctionsbetween the special features of these two scattering processes and between theproperties of scattered radiations. At least some of these distinctions can be used toenhance the effect of our interest. They include the following:

1. The polarization of radiation that experienced Rayleigh scattering must inprinciple be different from the polarization of radiation that experienced reso-nant scattering.

2. On average, the time of the resonant-scattering process is about τ – the meanlifetime of scatterer nuclei in the excited state. For 106Pd, this time is about 10−11

s. The time of Rayleigh scattering is much shorter (about 10−18 s).3. The angular distributions of two types of scattered radiation differ strongly from

each other. As the scattering angle increases in the backward hemisphere, theintensity of Rayleigh scattering decreases [21], while the intensity of resonantscattering increases [see Eq. (5.48)].

4. Only one isotope participates in resonant scattering, but all isotopes take part inRayleigh scattering.

5. The energy micro-spectra of photons that experienced Rayleigh scattering differfrom the energy micro-spectra of photons that experienced resonant scattering.While the spectrum of photons that experienced Rayleigh scattering reproducesclosely the initial spectrum of annihilation radiation and therefore has a width ofabout 3 keV, the micro-spectrum of resonantly scattered photons is a narrow line

whose width is equal to the Doppler width DD ¼ Ec

ffiffiffiffiffiffi2kTMc2

q(where k is the

Boltzmann constant, T is the absolute temperature of the scatterer, and M is themass of the scatterer nucleus). For 106Pd at room temperature, the Doppler widthis 0.39 eV. Therefore, the shape of the spectrum of detected radiation thatexperienced resonant scattering is determined exclusively by the shape of thedetector instrumental line.

For purely technological reasons, it is hardly possible to use the first two items inthe above list to separate Rayleigh and resonant scattering. However, the remainingthree distinctions may be used even at the present time. After improving the pro-tection of the detector from direct radiation emitted by the source, it would becomepossible to go over to scattering angles of about 150°. Owing to the distinctionbetween the angular distributions of photons that underwent Rayleigh and resonantscattering, this must lead to an increase of about 35 % in the fraction of resonantlyscattered photons. The application of a scatterer enriched in the isotope 106Pd to themaximum possible degree would increase the contribution of resonant scattering tothe total-absorption peak associated with elastic-scattering processes by a factor of3.5. A palladium mass of about several tens of grams would suffice upon appro-priately fitting setup geometry. Germanium detectors that have a resolution of about1 keV and a high efficiency for gamma rays of energy about 500 keV are availableat the present time. By means of quite a straightforward mathematical treatment ofspectra measured by such a detector, one can separate the peak associated with the

5.8 Further Ways Toward Refining Upon the Method for Observing 169

resonant scattering of annihilation photons from the wider energy distribution ofphotons that experienced Rayleigh scattering.

The phenomenon of resonant scattering of annihilation photons can be used instudying the shape of Fermi surfaces in metals and alloys. In the next chapter, thispossibility will be discussed in more detail.

References

1. G. Murrey, Phys. Lett 24B, 268 (1967)2. K. Rama Reddy, R.A. Carrigan, Jr., S. De-Benedetti, R.B. Sutton. Bull. Am. Phys. Soc. 12, 74

(1967)3. H.P. Hotz, J.M. Mathiesen, J.P. Hurley, Phys. Rev. 170, 351 (1068)4. H. Van den Berg, M.K. Ramaswami, Nuovo Cimento 57B, 521 (1968)5. S. Charalambous, M. Charaladas, Sp Dedoussi, Phys. Lett 59A, 235 (1976)6. M.K. Georgieva, M.A. Misheva, GKh Tumbev, Bulg. J. Phys 5, 574 (1978)7. M. Alatalo, H. Kauppinen, K. Saarinen et al., Phys. Rev. B 51, 4176 (1995)8. P. Asoka-Kumar, M. Alatalo, V.J. Ghosh et al., Phys. Rev. Lett. 77, 2097 (1996)9. V.L. Sedov, Phys. Usp. 11, 163 (1968)10. Ya.I. Frenkel. Statistical Physics. Gos. (Technick and Theoretical Publisher, Moscow-

Leningrad, 1933) (in Russian)11. L.D. Landau, E.M. Lifshitz. Statistical Physics, vol 5, 3rd edn. Butterworth-Heinemann,

Oxford, 1980)12. J.E. Mayer, M. Goeppert-Mayer. Statistical Mechanics, 2nd edn. (Wiley, New-York, 1977)13. G.A. Shevelev, A.G. Troizkaya, Program and Abstracts of Reports of XXV Conference on

Nuclear Spectroscopy and Atomic Nucleus Structure. (Leningrad, Nauka, Leningrad branch,1975), p. 78 (in Russian)

14. H.W. Taylor, N. Neff, J.D. King, Nucl. Phys A106, 49 (1968)15. S.T. Hsue, H.H. Hsu, F.K. Wohn et al., Phys. Rev. C 12, 582 (1975)16. G.A. Shevelev, A.G. Troizkaya, V.M. Kartashov, G.P. Chursin. Ibid., p. 7917. W.G. Smith, Phys. Rev. 131, 351 (1963)18. L. Auble. Nucl. Data Sheets A = 64 12, 305 (1974)19. Radionuclide Transformations—Energy and Intensity of Emissions. ICRP Publication 38.

Pergamon Press, 1983, 64Cu20. F. Smend, M. Schumacher, T. Borchert, Nucl. Phys A213, 309 (1973)21. F. Smend, M. Schumacher, T. Borchert, Nucl. Phys A223, 423 (1974)22. K.J. Malmfors, in Alfa-, Beta- and Gamma-Ray Spectroscopy. ed. by K.A.I. Siegbahn (North-

Holland Publishing Company, Amsterdam, 1955)23. Sh. Davisson, in Alfa-, Beta- and Gamma-Ray Spectroscopy, vol 1. ed. by K.A.I. Siegbahn.

(North-Holland Publishing Company, Amsterdam, 1965)24. A.V. Davydov, G.R. Kartashov, I.N. Vishnevsky, Phys. Lett 30B, 616 (1969)25. E. Storm, H. Israel, Photon Cross Sections from 0.001 to 100 MeV for Elements 1 Through

100. (Los Alamos Scientific Laboratory, New Mexico, 1967)26. A.I. Akhiezer, V.B. Berestetskii, Quantum Electrodynamics (Nauka, Moscow, 1969). (In

Russian)27. S.P. Lloyd, Phys. Rev. 83, 716 (1951)28. M.G. Gavrilov, A.V. Davydov, M.M. Korotkov, Phys. At. Nucl. 25, 131 (1977)29. A.T. Levon, O.F. Nemez, Electromagnetic Moments of the Excited and Radioactive Nuclei.

(Naukova Dumka, Kiev, 1989), p. 213 (in Russian)30. G. Frauenfelder, R. Steffen, in Alfa-, Beta- and Gamma-Ray spectroscopy, ed. by K. Siegbahn.

Angular correlations, vol 2 (North-Holland Publishing Company, Amsterdam, 1965)


31. I.N. Vishnevsky, V.I. Gavrilyuk, V.T. Kupryashkin et al., in Prikladnaya YadernayaSpektroskopiya (Applied Nuclear Spectroscopy), issue 3, Atomizdat, 1972, p. 262 (in Russian)

32. R.L. Robinson, F.K. McCowan, P.H. Stelson et al., Nucl. Phys. A 124, 553 (1969)33. K.R. Alvar, Nucl. Data Sheets 10, 205 (1973)34. C.M. Lederer, V.S. Shirley (eds.), Table of Isotopes, 7th edn. (Wiley, New York, 1978)

(Chichester-Brisbana-Toronto)35. B.S. Dzhelepov, L.K. Peker, V.O. Sergeev. Decay Schemes of Radioactive Nuclei. –A >100.

(Publishing House of Academy of Sciences of the USSR, Moscow-Leningrad, 1963) (inRussian)

36. W.G. Smith, Phys. Rev. 122, 1600 (1961)37. W. Scheuer, T. Suter, P. Reyes-Suter, E. Aasa, Nucl. Phys 54, 221 (1964)38. J.K. Temperlay, A.A. Temperlay, Nucl. Phys A101, 641 (1967)39. P. Venugopala, R.W. Fink, Nucl. Phys A103, 385 (1967)40. C. Marsol, G. Ardisson, Rev. Roum. Phys. 18, 1101 (1973)41. K.D. Strutz. Z. Phys. Bd201, 20 (1967)42. A. Buyrn, Nucl. Data Sheets 11, 189 (1974)43. Y.A. Ellis, B. Harmatz, Nucl. Data Sheets 16, 135 (1975)44. A.V. Davydov, V.P. Selyutin, Bulleten Acad. Sci. USSR Phys. 27, 875 (1963)45. Y.A. Ellis, Nucl. Data Sheets 14, 347 (1975)46. A. Artna-Cohen, Nucl. Data Sheets 6, 577 (1971)

References 171

Chapter 6Small Addition

This chapter contains a description of several experiments already performed andsome proposals for experiments that have not yet been carried out, but which are ofinterest from the point of view of the present author.

6.1 Manifestations of the Binding Energy of Electronsof Scattering Atoms in the Spectra of Scattered GammaRadiation

In our experiments aimed at observing resonant scattering of annihilation photonsby 106Pd nuclei, a mathematical description of the background in the pulse spectrumof a detector (especially a scintillation detector) under the peak associated withRayleigh and resonant scattering of these photons (and, of course, in the spectrumof scattered gamma rays from the control gamma source) was a serious problem. Inorder to obtain a curve that would describe this background, we had to use ratherlarge sections of the spectra on the left and on the right of the total-absorptionpeaks, extrapolating and matching them under these peaks. In the course of thisprocedure, we revealed an irregularity in each measured spectrum at energies lowerthan peak energy in the form of a smeared step not admitting an explanation interms of statistical fluctuations. According to rough estimates, the position of thisirregularity corresponded to the energy equal to the difference of the energy of theprimary photon to be scattered and the binding energy of K-shell electrons in thepalladium atom (and, of course, the silver atom). In this connection, we undertook aseries of experiments devoted to measuring the spectra of gamma rays emitted bythree nuclides and scattered by samples from three elements [1], employing gammarays of 198Au (412 keV), 181Hf (482 keV), and 137Cs (661.6 keV). Gamma sourcesof activity 0.3–1 Ci were placed in a lead “house” with a collimator, which, at theoutlet, yielded a gamma-ray beam fully covered by scatterer plates. The angulardivergence of the beam in the scattering plane was about 15°. For scatterers, usewas made of plates from palladium, tungsten, and lead of natural isotopic com-position. Scattered gamma rays were recorded by a coaxial Ge(Li) detector of


173

working volume about 25 cm3 in the cases of the first two gamma sources and about50 cm3 in the case of the third gamma source, the average scattering angle being90°. In order to reduce the detector workload associated with soft gamma rays andX rays, filters from lead 3 to 5 mm thick and cadmium and copper foils 0.3 mmthick each were placed in front of the inlet window of the lead screen protecting thedetector. The results of the measurements are shown in Fig. 6.1. One can see that, inall cases, the energy dependence of the number of counts shows an irregularity in

Fig. 6.1 Spectra of scattered gamma rays of 198Au (412 keV), 181Hf (482 keV), and 137Cs(661.6 keV). Here, the scatterers were metallic Pd, W, and Pb (indicated in the figures); Qk(Pd),Qk(W), and Qk(Pb) are the binding energies of K-shell electrons in the atoms of these elements;and QL(Pb) is the mean binding energy of L-shell electrons in lead. The arrows indicate theenergies of primary gamma rays and the energies that differ from the primary energies by thebinding energies of corresponding electrons

174 6 Small Addition

the form of a smeared step spaced from the energy of primary gamma rays by thebinding energy of K-shell electrons, QK, in scatterer atoms. In the case of a leadscatterer, similar steps are likely to be present at the scattered-gamma-ray energy ofEγ − QL as well, where QL is the mean binding energy of L-shell electrons. Theenergy regions where one observes the irregularities in question are inaccessible tophotons scattered by free electrons. They correspond to an interaction in which theprimary photon loses an energy much lower than that in the Compton effect.Incoherent scattering by bound electrons that is accompanied by the escape of areleased electron from the atom involved is one of the processes leading to theappearance of radiation in this energy region. The minimum energy loss of a photonin a scattering event belonging to this type must be very close to the binding energyof scattering electron (indistinguishable from it under conditions of our experi-ment). Evidently it is impossible to transfer an energy less than QK to a K-shellelectron. A simple interpretation of the spectra shown in Fig. 6.1 follows from here.Photons that experienced scattering on K-shell electrons cannot appear in theenergy interval between Eγ − QK and Eγ. This interval of the spectrum correspondsto gamma rays scattered incoherently by electrons of higher lying shells.A jumplike change in the intensity of scattered gamma radiation at the lowerboundary of the interval in question indicates that the K-shell electrons begin takingpart in the scattering process. In a similar way, one can explain the origin of lesspronounced irregularities at (Eγ − QL) in the spectra measured for a lead scatterer.

It should be noted that photoelectrons arising in the scatterers under the effect ofincident gamma rays generate bremsstrahlung indistinguishable under conditions ofour experiment from the scattered gamma rays of the same energy. It seems that thisradiation can make a contribution to the measured gamma rays of the same energy.It seems that this radiation can make a contribution to the measured spectra thatresembles the observed steps. This is because the endpoint energy of the brems-strahlung spectrum is also equal to Eγ − Qi, where Qi is the electron binding energyin the ith shell.

Data on the bremsstrahlung yield [2], on the cross section for the photoelectriceffect in the energy region of our interest [3], and on the specific energy loss ofphotoelectrons in matter [4] permitted estimating the contribution of bremsstrah-lung to the formation of the irregularities observed in the spectra. The ratio of theyield of bremsstrahlung in the interval of 10 keV from the endpoint energy of itsspectrum to the yield of gamma rays that experienced Rayleigh scattering, forwhich data were obtained by interpolating the results reported in [5, 6], was cal-culated for the lead scatterer and Eγ = 482 keV. It turned out that the result was notmore than 0.1 of its counterpart observed experimentally (in calculating therespective ratio on the basis of experimental data, we assumed that the elastic-scattering peak is due exclusively to the Rayleigh scattering of gamma rays, andthis is true under our conditions). Despite a low accuracy of this estimate (about40 %), we can state quite confidently that bremsstrahlung makes but a small con-tribution to the formation of the irregularities that we found.

6.1 Manifestations of the Binding Energy of Electrons… 175

6.2 Application of Resonant Gamma Ray Scatteringto Determining the Magnetic Moment of the 65CuNucleus in the Excited State at 1115.5 KeV

This experiment is of interest for a number of reasons. First, a nontrivial methodwas used in it to reach resonance conditions in the non-Mössbauer case on the basisof the Coulomb fragmentation of molecules, which was used for the first time byF. R. Metzger for this purpose [7, 8]. Second, the excited state at 1115.5 keV has avery short mean lifetime: the averaging of the results of three studies [8–10] wherethe authors observed resonant gamma-ray scattering leads to a τ value of(4.28 ± 0.17) × 10−13 s; with allowance for the result reported in [11][τ = (6.5 ± 1.6) × 10−13 s], the value of τ turns out to be (4.30 ± 0.17) × 10−13 s. Forso short a mean lifetime of the nucleus in the excited state, only owing to a fortunatecoincidence of circumstances can one try to measure the magnetic moment of thisstate by perturbing the angular distribution of resonantly scattered gamma rays.Third, an original idea of creating simple conditions for observing resonant gamma-ray scattering with the aid of the Coulomb fragmentation of gamma-source mole-cules arose in performing this experiment. The decay scheme for the 65Zn nuclide isshown in Fig. 6.2.

The phenomenon of Coulomb fragmentation consists in the following. Radio-active decay of a nucleus via electron capture leads as a rule to multiparticle Augerprocesses [12] after which the atom involved turns out to be in a strongly ionized

Fig. 6.2 Decay scheme forthe 65Zn nuclide


state. If this atom enters into the composition of a molecule, then part of its chargesare distributed among the other atoms of the molecule. Likely charged atomsexperience Coulomb repulsion leading to the disintegration of the molecule. Theemerging fragments fly away, acquiring considerable velocities. If the mean life-time of the nucleus of an atomic fragment in the excited state populated afterelectron capture by the parent nucleus is so short that it can emit a photon before acollision with another atom, then the photon flying toward the resonant absorbermay receive, owing to the Doppler effect, an energy addition sufficient for com-pensating the energy loss upon recoil in emission and absorption events.This provides the possibility of observing resonant photon scattering. It was shownby F.R. Metzger that, in the case of 65Cu nuclei, the application of gaseous gammasources formed by ZnCl2 and ZnI2 molecules with the parent nuclide 65Zn andobtained by means of high-temperature heating of corresponding salts leads to anextremely large resonant-photon-scattering effect, which is many times as great asthe level of counting for a cold gamma source, in which case its substance is in asolid state, where the average time between the emergence of a fragment and itscollision with a neighboring atom becomes significantly shorter. It is interesting thatresonance conditions become more favorable as the halogen-atom mass increases.This is due to an increase in the fraction of the kinetic energy of emerging copperatoms and to the corresponding broadening of the range of velocities of copperatoms appearing as fragments.

The method used in the experiment being discussed is similar to the method thatwe used in measuring the magnetic moment of the 182W nucleus and consists inperturbing the angular distribution of resonantly scattered gamma rays by a mag-netic field. The ground- and excited-state spins of the 65Cu nucleus are 3/2 and 5/2,respectively, according to data reported in [7–9], and the parameter δ of the mixtureof E2 and M1 multipoles is −0.437 ± 0.015 according to [8]. At the above values ofthe spins and multipole-mixture parameter, the angular distribution of resonantlyscattered gamma rays, which, in the case of the mixture of E2 and M1 multipoles,has the general form W(θ) = 1 + A2P2(cos) + A4P4(cosθ), where θ is the scatteringangle reckoned from the primary-gamma-beam axis in the counterclockwisedirection and P2 and P4 are the respective Legendre polynomials, reduces to

WðhÞ ¼ 1þ ð0:98� 0:02ÞP2 þ ð0:0139� 0:0137ÞP4: ð6:1Þ

If a magnetic field acts on a nucleus that scatters a photon, it perturbs the angulardistribution of scattered gamma rays. According to semiclassical concepts, theprecession of the nuclear spin about the magnetic-field direction at the Larmorfrequency ωL = −μH/Iħ occurs for a time t within which the nucleus remains in anexcited state. If the quantity μH is small in relation to the natural width Γ of thisexcited state, then the rotation of the angular-distribution rosette through the angleΔθ = ωLt with respect of its unperturbed position will be the result of the pertur-bation, and the angular distribution will assume the form

6.2 Application of Resonant Gamma Ray Scattering to Determining… 177

W h;H; tð Þ ¼ W h�xLtð Þ: ð6:2Þ

If, in each angular position, the angular distribution of scattered gamma rays ismeasured over the time much longer than the mean lifetime of the nucleus in theexcited state, then the angle of rotation of the angular-distribution rosette is ωLτ, andexpression (6.2) becomes

W h;H; t � sð Þ ¼ W h�xLsð Þ: ð6:3Þ

If the detector records gamma rays scattered at an angle θ in the plane per-pendicular to the direction of the applied magnetic field at two opposite directionsof this field, then it is convenient to compare the experimentally determined valueof the ratio

R ¼ W h;H�ð Þ �W h;Hþð ÞW h;H�ð Þ þW h;Hþð Þ ¼ W h� -Lsð Þ �W hþ -Lsð Þ

W h� -Lsð Þ þW hþ -Lsð Þ ð6:4Þ

with the theoretical result obtained by using Eq. (6.1), which involves the soughtquantity ωLτ.

The possibility of employing the Coulomb fragmentation of gamma-sourcemolecules and the availability of a convenient substance for manufacturing a res-onant scatterer—namely, the ferromagnetic Heusler alloy CuMnAl (65.4 at.% ofCu, 16.5 at.% of Mn, and 18.1 at.% of Al), in which the internal magnetic field ofstrength 212.5 ± 0.5 kOe acts on nuclei [13]—are circumstances favorable forperforming the experiment being discussed. Figure 6.3 shows the spectra that weobtained in [14] for gamma rays scattered by a sample from the above alloy,employing the 65ZnI2 salt placed in an evacuated quartz ampoule as the gamma-source substance. The ampoule was within an electric heater that made it possible toheat it to a temperature of 800 °C, at which the salt evaporated in the ampoule. Onecan see that a very large effect of resonant scattering of 1115.5 keV 65Cu gammarays arose upon the transition of the salt into a gaseous state and that there is nosuch effect at temperatures below 100 °C.

Four scintillation detectors based on NaI(Tl) crystals 70 mm in diameter and100 mm in thickness were used in the experimental setup to record gamma raysscattered at angles of ±135°, +77°, and −67°. The experimental results are given inTable 6.1. Their averaging yielded a value of 4.45 ± 0.92 nuclear magnetons for themagnetic moment of the 65Cu nucleus in the excited state at 1115.5 keV.

A leading role in performing that work belonged to P.I. Romasheva. She alsoproposed an original idea to observe resonant gamma-ray scattering by employingthe Coulomb fragmentation of molecules dissolved in liquids such that not all of themolecules experience electrolytic dissociation. Together with V.M. Novikov, sheperformed experiments with gamma sources manufactured from the radioactive65ZnI2 salt dissolved in ethyl alcohol and in glycerin [15]. In either case, the effect


of the resonant scattering of 1115.5 keV gamma rays was present in experimentswith a bronze scatterer, but it was absent in experiments with a zinc scatterer. Dataobtained in those experiments are given in Figs. 6.4 and 6.5. For the sake ofcomparison, the spectrum of gamma rays scattered by zinc is shown in Fig. 6.6. Itshould be no resonant scattering in that case. Indeed, the spectra obtained with thesources from the 65ZnI2 solid salt and from the salt dissolved in alcohol are nearlyindistinguishable.

Although the resonant-scattering effect observed in those experiments is twoorders of magnitude smaller than that observed with a gaseous gamma source, theachieved simplification of the method is of great importance and, upon a furtherdevelopment, may lead to important applied results.

Fig. 6.3 The spectra of 65Cugamma rays scattered byHeusler alloy for the (closedcircles) gaseous and (opencircles) solid states of thegamma source made from the65ZnI2 salt


Table 6.1 Results of the measurements of the magnetic moment of 65Cu nucleus in the state withenergy 1.1155 MeV

Number of detector θ, degrees Rexp. ± ΔRexp. μ ± Δμ, nucl. Magn.

1 +135 −1.05 ± 0.35 4.96 ± 1.70

2 −135 +1.00 ± 0.36 4.80 ± 1.80

3 −67 –0.88 ± 0.44 3.96 ± 2.00

4 +77 +0.68 ± 0.38 3.62 ± 2.05

Fig. 6.4 Spectra of gammarays scattered by a bronzescatterer. The closed circlesrepresent data obtained with agamma source in the form ofa solution of 65ZnI2 in ethylalcohol. The crosses stand forthe results of measurementswith a solid-salt source


Fig. 6.5 Spectra of 65Cugamma rays scattered by abronze scatterer. The closedcircles and crosses representthe results of measurementswith a source from,respectively, a 65ZnI2 saltdissolved in glycerin and asolid salt. The open circlesstand for the background

Fig. 6.6 Spectra of 65Cugamma rays scattered by azinc scatterer. The notationfor the points is identical tothat in Fig. 6.5


6.3 On the Possibility of Applying the Nuclear ResonantScattering of Annihilation Photons to Studyingthe Shape of Fermi Surfaces in Metals

If the source of annihilation photons—that is, substance in which the annihilation ofpositrons occurs—is a single crystal of the metal under study, then the probabilitythat a photon emitted in a given direction has an energy permitting it to be reso-nantly scattered by a nucleus of an appropriate scatterer depends on the crystalorientation with respect to this direction under the condition that the Fermi surfaceof the source substance is not spherical. We will clarify this by considering a coppersingle crystal as an example. Figure 6.7 shows the Fermi surface of copper inaccordance with [16].

One can see that it has a shape that is rather close to spherical shape, but whichdiffers from it by the presence of characteristic “necks.” If a copper single crystalused as a source of annihilation radiation is oriented in such a way that photonsemitted in the [17] direction are incident to the scatterer (see Fig. 6.7a), then thatpart of all these photons which has an energy required for resonant absorption byscatterer nuclei is proportional to the volume of the flat layer that has a thicknessdPz, which is perpendicular to the [17] direction, and which lies at the distancePz � 2E0

c � 2mc from the center of the Fermi surface [see Eq. (182)]. The thicknessof this layer, dPz, depends only on the Doppler width ΔD of the resonance-absorption line of the scatterer, namely,

dPz � 2DD

c� Pz:

Since, for the chosen substance of the scatterer, dPz remains constant irrespec-tive of the orientation of the source crystal, one can assume that the number ofannihilation photons capable of undergoing resonant scattering on scatterer nuclei(and we are interested precisely in this number) is proportional to the area of thecross section obtained by cutting the body bounded by the Fermi surface with theplane S that is perpendicular to the photon-emission direction and which is at thedistance Pz from the center of this body. As the crystal rotates about the [18] axisperpendicular to the figure plane, there inevitably comes an instant at which theplane S intersects one or several “necks” (see Fig. 6.7b). The area of the sectionobtained by cutting the body bounded by the Fermi surface with the plane S willincrease under this condition, and the number of counts in the detector recordingscattered photons will increase accordingly. If one measures the detector countingrate as a function of the angle of rotation of the source single crystal about the [18]axis, there must arise a picture belonging to the same type as that in Fig. 6.8. It isnatural that similar (but not always identical) curves would be obtained upon therotation of the crystal about any other axis. By changing rotation axes, one can get acomprehensive idea of the shape of the Fermi surface for a given sample. Forcopper, the situation is somewhat simplified by the circumstance that a copper


http://dx.doi.org/10.1007/978-3-319-10524-6_5

single crystal irradiated with neutrons becomes itself a source of positrons. Instudying other substances, it may turn out to be necessary to use a separate positronsource for each axis of rotation. In that case, the sample under study is cut from asingle crystal in the form of a tube with known crystallographic directions of axesand with a wall thickness sufficient for the total absorption of positrons emitted by asource placed inside the tube. Figure 6.9 shows the layout an experimental setup fordealing with such samples. The need for manufacturing a new sample for each newrotation axis is a drawback of substances that cannot become a positron source uponirradiation with neutrons or in reactions involving charged particles. Mathemati-cally, the problem at hand reduces to determining the surface of a body on the basisof an indefinitely large number of areas of its sections by planes spaced from somepoint (“center” of the body) by the same distance. For a Fermi surface of any shape,the point from which one reckons values of electron momenta is this center.

Fig. 6.7 Diagram employing the example of the copper Fermi surface to explain why the numberof annihilation photons capable of experiencing nuclear resonant scattering may depend on therotation of the source crystal about the axis perpendicular to the photon-emission direction

Fig. 6.8 Possible form of thedependence of the number ofcounts for resonantlyscattered annihilation photonson the angle of rotation of thecrystal in which annihilationoccurs

6.3 On the Possibility of Applying the Nuclear Resonant Scattering… 183

6.4 Nuclear Resonant Scattering of Annihilation Photonsand Problem of the Tunguska Event

This section, where the reasoning does not purport to be absolutely strict, serves forentertaining the tired reader rather than for providing him with a guide to action.The problem in question involves a great many unclear circumstances requiringadditional studies. The question to be discussed is that of experimentally verifyingthe hypothesis that the Tunguska meteorite is a piece of antimatter. If there are orescontaining palladium, gallium, or rhenium in the region where the meteorite fell,then structural changes could occur in these ores because of their irradiation with anintense flux of annihilation radiation. By way of example, we indicate that, if thebinding of palladium, gallium, or rhenium atoms in their chemical compounds isweak, then resonant scattering of annihilation photons could lead to the selectiverelease of 71Ga, 106Pd, and 187Re atoms from the molecules containing them, withthe result that they replenished the corresponding metal fractions. We note that thecontribution of Rayleigh scattering to this process is greater than the respectivecontribution of resonant scattering because, in the first case, the momentum transferto the atom involved is higher—that is, because photon scattering does not lead tothe population of an excited nuclear state. Moreover, the cross section for theRayleigh scattering of annihilation photons is larger than the cross section forresonant scattering (see Sects. 5.4 and 5.7). However, Rayleigh scattering must leadto the same degree of release for all isotopes (the isotope-mass dependence of thebinding energy has a weak effect on the probability for this process, but this effect isnot of a resonance character). Therefore, the expected effect consists in an excesscontent of the aforementioned isotopes in the respective metal fractions. The pos-sibility of performing such an investigation depends on several circumstances. First,the presence of respective ores in the region where the meteorite fell is necessary.

Fig. 6.9 Layout of the setup for studying the shape of Fermi surfaces in metals by using thenuclear resonant scattering of annihilation photons: (1) single crystal cut out in the form of a tube,(2) source of positrons, (3) crystal holder, (4) lead protection from radiation, (5) scatterer, and (6)detector for scattered radiation (by convention)


http://dx.doi.org/10.1007/978-3-319-10524-6_5

http://dx.doi.org/10.1007/978-3-319-10524-6_5

Second, the binding energies of Ga, Pd, and Re atoms must be sufficiently small incorresponding molecules for the knockout of the isotopes in question from thesemolecules to be possible upon the resonant scattering of 511 keV photons. Even forthe lightest of these isotopes, 71Ga, the recoil energy upon annihilation-photonabsorption by a nucleus is as small as 1.975 eV. Painstaking efforts and greatexpenses are required for solving these problems. The organization of a geologicalexpedition aimed at this purpose is hardly possible. One may only hope for aparticipation in an expedition working in that region with different intents, but thisis of course problematic. The creation of gamma-ray sources not emitting photonsof energy in excess of 511 keV is required for experimentally testing the possibilityof the process involving the release of atoms of the aforementioned elements fromcorresponding molecules. Therefore, sources usually used in experiments involvingannihilation photons and fabricated from 22Na or 64Cu are inappropriate because thespectra of both of these sources involve gamma lines of energy higher than 1 MeV(the application of filters that have different coefficients of absorption for two typesof radiation from these sources opens some possibilities). It is also difficult to workin these realms with electron bremsstrahlung because the form of its spectrum givesno way to obtain 511 keV photons in sufficient amounts without an admixture ofhigher energy photons. It would be preferable to have a source of low-energypositrons, but its creation is a difficult challenge. In my opinion, there is thereforelittle hope that work in this direction would begin in the near future.

References

1. M.G. Gavrilov, A.V. Davydov, JETP Lett. 21, 267 (1975)2. J.W. Motz, Phys. Rev. 100, 1560 (1955)3. K.W. Seemann, Bull. Am. Phys. Soc. 1, 198 (1956)4. G. Knop, V. Paul, in Alfa-, Beta- and Gamma-Spectroscopy, ed. by K. Ziegbahn (North-

Holland Publishing Company, Amsterdam, 1965)5. Radionuclide Transformations—Energy and Intensity of Emissions (ICRP Publication 38,

Pergamon Press, 1983) 64Cu6. F. Smend, M. Schumacher, T. Borchert, Nucl. Phys. A213, 309 (1973)7. F.R. Metzger, Phys. Rev. Lett. 18, 434 (1967)8. F.R. Metzger, Phys. Rev. 171, 1257 (1968)9. G.B. Beard, Phys. Rev. 135, B577 (1964)10. M.A. Eswaran, H. Gove, A. Litherplandt, C. Brande, Phys. Lett. 8, 52 (1964)11. D.I. Kaipov, R.B. Begzhanov, A.V. Kuz’minov, Yu.K. Shubnyi. Soviet JETP 17, # 6, (1963)12. A. Snell, in Alfa-, Beta- and Gamma-ray Spectroscopy, ed by K.A.I. Siegbahn, vol. 2 (North-

Holland Publishing Company, Amsterdam, 1965)13. K. Sugibuchi, K. Endo, J. Phys. Chem. Solids 25, 1217 (1964)14. A.V. Davydov, V.M. Novikov, P.I. Romasheva, Bulleten of academy of sciences of the

USSR. Physics, 43, 110 (1979)15. V.M. Novikov, P.I. Romasheva, Collection of papers “Prikladnaya Yadernaya

Spektroscopiya (Applied Nuclear Spectroscopy), issue 7 (Moscow, Atomizdat, 1977),p. 238 (in Russian)

6.4 Nuclear Resonant Scattering of Annihilation Photons and Problem… 185

16. K. Fujiwara, O. Sueoka, J. Phys. Soc. Japan 21, 1947 (1966)17. V.G. Alpatov, G.E. Bizina, A.V. Davydov et al. Voprosy Tochnosti v yadernoi spektroskopii

(Questions of Precision in Nuclear Spectroscopy). Vilnus, Institute of Physics, Academy ofSciences of LietSSR (1984), p. 15 (in Russian)

18. V.G. Alpatov, G.E. Bizina, A.V. Davydov et al. Preprint ITEP No 130 (Moscow, 1984)(in Russian)


Conclusion

This small book presents a description of the results of studies performed overmany years by our research group, which, in the best period, included 15 physicistsand laboratory assistants and technicians, but which now diminished to fivescientists without any subsidiary personnel. Nonetheless, we were able to createexperimental devices, even in the last, especially hard, years, relying on our ownefforts exclusively, and to obtain unique results by using this equipment. Here, Iwould like to remind briefly the main that we have done and which I have describedin this book. First, this is, of course, a series of studies devoted to exploringmagnetic-field-perturbed angular distributions of resonantly scattered gamma rays.We were able to prove experimentally the correctness of the theoretical predictionsaccording to which the result of such a perturbation depends on the shape of thespectrum of scattered gamma rays and explained this effect by the dependence ofthe mean lifetime of the participant nucleus in an excited state on this shape. Adetailed analysis of this situation led to the conclusion that nuclear processes ofgamma-ray emission and absorption are of a protracted character. This point ofview, albeit possibly in a less explicit form, existed earlier (see the argument of Dr.E.B. Bogomol’nyi above in explaining the difference between the excitation ofnuclei by gamma rays of narrow and wide spectrum), but it turned out to be quiteunexpected for the overwhelming majority of physicists (and not onlyexperimenters) with whom I discussed this problem.

The results of our experiments devoted to the gamma-resonant excitation oflong-lived isomeric states of nuclei proved to be even more important. Untilrecently, the common point of view was (has been to date for many) that theminimum gamma-line width accessible to measurement is about 10−13 to 10−12 eV.Even in diamagnetic substances, the widths of narrower gamma lines shouldincrease up to such values because of the dipole-dipole interaction of nuclearmagnetic moments with the magnetic moments of neighboring nuclei andconduction electrons. Paradoxically as it might seem, our experiments with the109mAg isomer showed that there is no such broadening. Because one can hardlybelieve that quantum electrodynamics, which is a highly reliable theory, could leadto an incorrect result in this case, it only remains to think that some as-yet-unknownspecial features of nuclear radiative processes are responsible for this. An attempt atexplaining this situation by a possible averaging of dipole-dipole interaction energy,

© Springer International Publishing Switzerland 2015A.V. Davydov, Advances in Gamma Ray Resonant Scattering and Absorption,Springer Tracts in Modern Physics 261, DOI 10.1007/978-3-319-10524-6

187

which changes quickly in magnitude and sign, over the mean lifetime of the excitedstate of the nucleus involved raised serious objections of theorists. The secondhypothesis that we proposed was that the radiating nucleus and the gamma waveemitted by it are both insensitive to external effects as long as the radiative processlasts. In support of this hypothesis, we put forward the argument that, if this was notso, it would be impossible to observe gamma lines of natural width, but onesometimes observes them. Of course, a much more profound analysis is required forexplaining this anomaly conclusively.

It is also worth mentioning that our group designed and manufactured agravitational gamma spectrometer, which is an instrument belonging to quite a newtype and which is simple in underlying idea and in design and was made fromimprovised materials. By using this spectrometer, we were able to measure theshape of the gamma resonance in the long-lived 109mAg isomer, thereby improvingthe resolving power of gamma spectrometry by about eight orders of magnitude inrelation to that of Mössbauer spectrometers dealing with gamma rays of the 57Fenuclide. This result confirms fully earlier data demonstrating that the 109mAgMössbauer gamma line does not undergo a large broadening that could be causedby dipole-dipole interaction.

Our observation of resonant annihilation-photon scattering on nuclei is the mostimportant point in the remaining part of the book since this process permitsdeveloping a new method for studying Fermi surfaces of metals. The discovery ofmanifestations of binding energies of atomic electrons in the spectra of scatteredgamma radiation is yet another important result.

Last but not least, I gratefully acknowledge that some experiments with the109mAg isomer received support from the Russian Foundation for Basic Researchand from INTAS.

188 Conclusion

Index

AAbsorption coefficients, 108Absorption operator, 6Acoustic broadening, 81Amplitude for the emission process, 30Angular distribution, 23, 25, 43, 48, 52, 166,

167, 177Angular distribution function, 15Angular distribution of resonantly scattered

gamma rays, 44Angular-correlation function, 24Angular-distribution function, 15Annealing, 108Annihilation photon, 141, 146, 149, 155,

161, 167, 182Antimatter, 184Associated Legendre function, 13, 16Associated Legendre polynomial, 97Atomic number, 156

BBackground, 93, 160, 161Beta decay, 155Binding energy, 173, 175Boltzmann constant, 169Bremsstrahlung, 158, 175, 185Broadening factor, 123, 127, 131, 132, 135,

138Broadening of gamma lines, 80Broadening of a Mössbauer gamma line, 82,

84, 94

CCoefficient of absorption, 151, 164, 185Coefficient of internal conversion, 147Coefficients of linear expansion, 106, 111, 119

Compaction, 113Compaction of the materials, 106Compton effect, 175Conduction electrons, 141, 144, 167Correlation function, 5, 19Coulomb fragmentation, 176, 178Cross section, 114Cryostat, 87, 107Crystallographic directions, 183

DD-functions, 7, 14, 20, 24Decay, 109Detector efficiency, 112Differential cross section, 152, 153, 154, 161,

165Diffusion, 109Diffusion annealing, 108Diffusion coefficient, 110Dipole-dipole, 123Dipole-dipole interactions, 124Doppler effect, 177

EE2 and M1 multipoles, 10, 11, 22Earth’s magnetic field, 118Efficiency, 59, 60, 120Efficiency matrix, 9Eigenfunction, 3Elastic-scattering peaks, 151Electric quadrupole interaction, 166Electrolytic dissociation, 178Electromagnetic oscillation, 28Electron density, 84Electron momentum, 143Excited state, 25, 27, 36, 62, 93

© Springer International Publishing Switzerland 2015A.V. Davydov, Advances in Gamma Ray Resonant Scattering and Absorption,Springer Tracts in Modern Physics 261, DOI 10.1007/978-3-319-10524-6

189

Exciting gamma radiation, 52Exciting gamma rays, 62

FFermi energy, 81, 143Fermi surface, 142, 170, 182Fourier transform, 28, 29Frequency characteristic, 28Frequency distribution, 28

GGamma beam, 106, 107, 117, 118, 122, 124,

128, 131Gamma line, 22Gamma ray absorption, 6Gamma resonance, 139Gamma source, 36, 37, 40, 42, 43, 45, 49,

52, 53, 81–83, 88, 101, 103, 106 ,116, 135

Gamma transitions, 9Gamma-beam, 39Gamma-beam divergence, 131Gamma-line broadening, 102Gamma-ray absorption, 57Gamma-ray intensity, 112, 116, 119Gamma-ray resonant absorption, 22Gamma-ray self-absorption, 85Gamma-ray spectrum, 115Gamma-ray yield, 111Gaussian distributions, 162Geomagnetic field, 90, 135Germanium detector, 90g-factor, 45–47, 57, 59, 60Gravitational gamma spectrometer, 132, 136,

139Gravitational gamma spectrometry, 127Gravitational shift, 128Gravitational shift of the gamma resonance,

115Gravitational suppression of resonance condi-

tions, 107Gravitational waves, 139Gravity, 84Ground state, 6

HHalf-life, 45, 141Hamiltonian, 2Helmholtz coils, 104, 107, 120, 123, 127,

135Holes, 121

Hyperfine interaction, 81Hyperfine structure, 96

IInclination angle, 136Incoherently scattered gamma rays, 163Interaction, 123Internal magnetic field, 35, 55, 58Internal-conversion coefficient, 87Interstitials of the crystal lattice, 124Irradiation, 83, 90, 154Isomeric shift, 83, 139Isomeric state, 140Isotope, 151

J3J coefficients, 9, 20, 23

KK-shell, 175

LLarmor frequency, 25, 177Lattice heat capacity, 84Legendre polynomials, 24, 49Liquid nitrogen, 36, 53, 92Liquid-helium temperature, 114Long-lived nuclear isomeric states, 80Lorentzian gamma line form, 5

MMagnetic field, 2Magnetic hyperfine interaction, 62Magnetic moment, 35, 42, 52, 178Magnetic moments of nuclei, 1Magnetic quantum number, 6, 97Magnetic-field strength, 16, 20Matrix element, 2, 17Maximum-likelihood method, 50Mean lifetime, 26, 59, 149, 165, 169Mean lifetime of a nucleus, 29, 44Mean lifetime of nuclei in an excited state, 25Metzger, R.F., 35Momentum projections, 145, 146Momentum space, 145Mössbauer absorption, 106Mössbauer effect, 36, 48, 52, 59, 79, 102Mössbauer emission, 99Mössbauer excitation of nuclei, 27

190 Index

Mössbauer gamma line, 57, 90, 118Mössbauer resonance width, 45Mössbauer resonant absorption, 118Mössbauer resonant scattering, 42Mössbauer scattering of gamma rays, 35Multipolarity, 97Multipole-mixing parameter, 11, 50, 177Multipole-mixing ratios, 1

NNarrow spectral line, 29Natural width, 44, 52, 124Natural width of the excited nuclear state, 5,

168Nihilation photons, 143Nuclear radius, 82Nuclear resonant scattering, 158Nuclide, 148

OOrthogonality, 23Oscillation amplitude, 28

PPair production, 153, 165Pair-production cross section, 164Parent nuclide, 93Parity, 4Peak of elastic processes, 158Peak of elastic-scattering processes, 161Peak of the total absorption, 151Perturbation of the angular distribution, 28Perturbed angular distributions of resonantly

scattered gamma rays, 57Phase characteristic, 28Phase space, 144Photoelectric effect, 175Photomultiplier tube, 158Polarization of radiation, 169Positron annihilation, 141Positron thermalization, 141Precipitation, 108Protracted character of nuclear radiative pro-

cesses, 32

QQuadrupole interaction, 82Quantization axis, 16, 17, 20Quantum number, 8, 36

Quasimonochromatic line, 26

RRadiation parameters, 7Radioactive atoms, 111Rayleigh scattering, 40, 57, 152, 155, 157, 158,

161, 164, 168, 175, 184Real width, 45Recoil, 36, 166Recoil energy, 185Recoil of the emitting nucleus, 27Recoilless gamma-ray emission, 48, 79, 85, 90Reduced matrix element, 4, 10, 11Resonance filter, 28, 29Resonance frequency, 29Resonant absorber, 80Resonant absorption, 103Resonant absorption of gamma rays, 100Resonant gamma-ray absorption, 39, 85, 114,

117, 119, 122, 128, 140Resonant gamma-ray scattering, 30, 35, 176Resonant scatterer, 148Resonant scattering, 90, 169Resonant scattering of annihilation photons,

173Resonant-absorption cross section, 115, 124,

132Resonant-absorption probability, 100Resonantly scattered photons, 56Resonant-scattering, 1Resonant-scattering cross section, 149, 164Rotation matrix, 3

SScatterer, 17, 22, 36, 38, 39, 43, 45, 50, 55, 59Scattering angles, 25, 38, 48, 55, 153, 156Scattering plane, 1, 25Scintillation counter, 40, 79, 158Single crystal, 182Single-crystal gamma source, 118Spectral distribution, 30Spectrum of annihilation photon, 147, 148, 153Spectrum of annihilation radiation, 143, 169Spherical harmonics, 12Spin, 3Statistical factor, 148

TTaylor series, 29Thermal-diffusion, 112

Index 191

Thermal-diffusion annealing, 101Thermal-neutron flux, 154Total-absorption peaks, 169, 173Total cross section, 166Tunguska meteorite, 184

VVery wide spectrum, 32

WWave vectors, 2Wavelength, 115Weak magnetic fields, 25, 27

Wigner 3J coefficient, 3Wigner 6J coefficient, 8

XX-ray, 40, 55, 103, 108, 110, 112, 114, 120,

121χ2 criterion, 42, 55, 60, 131, 135, 162

ZZeeman component, 98Zeeman hyperfine structure, 100Zeeman splitting, 90, 123

192 Index

advances in gamma ray resonant scattering and absorption: long-lived isomeric nuclear states

Documents