AECL-8490

ATOMIC ENERGY » Ä S ö L'ENERGIE ATOMIQUE

OF CANADA LIMITED ^ S M f DU CANADA LIMITÉE

THERMAL-NEUTRON SCATTERING LENGTHS AND CROSSSECTIONS FOR CONDENSED-MATTER RESEARCH

Tableau des parcours de diffusion des neutrons thermiques etdes sections efficaces pour la recherche relative

aux matières condensées

by

VF. SEARS

Chalk River Nuclear Laboratories Laboratoires nucléaires de Chalk River

Chalk Miver, Ontario

June 1 984 juin

ATOMIC ENERGY OF CANADA LIMITED

THERMAL-NEUTRON SCATTERING LENGTHS AND CROSS SECTIONS

FOR CONDENSED-MATTER RESEARCH

V.F. SEARS

Theoretical Physics Branch

Chalk River Nuclear LaboratoriesChalk River, Ontario, Canada ROJ 1J0

June, 1984

AECL-849O

L'ENERGIE ATOMIQUE DU CANADA, LIMITEE

Tableau des parcours de diffusion des neutrons thermiques et

des sections efficaces pour la recherche relative

aux matières condensées

par

V.F. Sears

Résumé

Un tableau des parcours de diffusion des neutrons thermiqueset des sections efficaces des éléments et de leurs isotopes estréalisé et les résultats sont présentés sous la forme requise pourles applications dans la recherche relative aux matières condensées.Le tableau est aussi complet et aussi logique que le permettentles meilleures données actuellement disponibles. Pour les nuclêidesà forte absorption nous obtenons également, pour la première fois,les parties imaginaires non seulement des parcours de diffusioncohérente mais aussi des parcours de diffusion incohérente. Parailleurs, nous cherchons à clarifier la confusion qui existe dansla littérature en ce qui concerne le signe de la partie imaginairedu parcours complexe de diffusion.

Département de physique théoriqueLaboratoires nucléaires de Chalk RiverChalk River, Ontario, Canada KOJ 1J0

Juin 1984

AECL-8490

ATOMIC ENERGY OF CANADA LIMITED

THERMAL-NEUTRON SCATTERING LENGTHS AND CROSS SECTIONS

FOR CONDENSED-MATTER RESEARCH

V.F. SEARS

Abstract

A table of thermal-neutron scattering lengths and cross

sections of the elements and their isotopes is constructed and the results

are presented in the form needed for applications to condensed-matter

research. The table is as complete and consistent as the best currently

available data permits. For strongly absorbing nuclides we also obtain

for the first time the imaginary parts of not only the coherent scattering

lengths but also of the incoherent scattering lengths. We also call

attention to and clarify the confusion that exists in the literature con-

cerning the sign of the imaginary part of the complex scattering length.

Theoretical Physics Branch

Chalk River Nuclear LaboratoriesChalk River, Ontario, Canada KOJ 1J0

June, 1984

AECL-8490

C O N T E N T S

1. INTRODUCTION 1

2. THEORETICAL BACKGROUND 2

2.1 Scattering Amplitude 2

2.2 Scattering Length 2

2.3 Bound Scattering Length 4

2.4 Spin-dependent Scattering Length 5

2.5 Experimental Determination of Scattering Lengths 7

2.6 Isotope Incoherence 8

2.7 Scattering Lengths in Crystallography 9

3. CALCULATIONS AND DISCUSSION 11

3.1 Calculations 11

3.2 Scattering Cross Sections 11

3.3 Absorption Cross Sections 12

Acknowledgements 13

References 13

Table I. Bound Scattering Lengths and Cross Sections of the

Elements and Their Isotopes 15

Table II. Spin Versus Isotope Incoherence 31

Table III. Radiative-capture Resonances 32

Table IV. Charged-particle Reactions 33

Figures 34

1. INTRODUCTION

The application of thermal-neutron scattering to the study

of the structure and dynamics of condensed matter requires a knowledge of

the scattering lengths and the corresponding scattering and absorption

cross sections of the elements- In some cases values for the individual

isotopes are needed as well. This information is required to obtain an

absolute normalization of the scattered-neutron distributions, to calcu-

late unit-cell structure factors in neutron crystallography, and to cor-

rect for effects such as absorption, self-shielding, extinction, multiple

scattering, incoherent scattering, an.d detector efficiency.

The development of modern neutron optical techniques during

the past fifteen years has produced a dramatic increase in the accuracy

with which scattering lengths can be measured.1~3 This is illustrated

in Fig. 1 which shows the average percentage error in the bound coherent

scattering length of a selected group of twenty elements as a function oftime. The open circles are the values estimated by the authors at the

time of measurement while the filled circles are the actual errors

determined by a comparison with current values.

The aim of the work described in the present report is to

use the best current values of the neutron scattering lengths,^ supple-

mented where necessary with available scattering and absorption cross sec-

tion data, » in order to obtain as complete and consistent a set of

thermal-neutron scattering lengths and cross sections as possible, and to

present the results in the most convenient form for the analysis of ther-

mal-neutron scattering experiments on condensed matter.

The emphasis here is on consistency. Our main task hasbeen to try to reconcile discrepancies that often exist between the data

for an element and its individual isotopes or between the measured values

of the scattering lengths and the scattering cross sections. For the

strongly absorbing nuclides we have also obtained for the first time the

imaginary parts of not only the coherent scattering lengths but also the

incoherent scattering lengths.

A preliminary report on some of this work was published

previously1-3 and an updated version of Table I in the present report

will eventually be published in Volume C of the new edition of the Inter-

national Tables for Crystallography. The present report is intended to

provide additional theoretical background and computational details that

will not be included there because of space limitations.

- 2 -

2. THEORETICAL BACKGROUND2.1 Scattering Amplitude

Consider the collision of a thermal neutron with a free

nucleus. The wave function that describes the relative motion in the

center-of-nass system has the asymptotic form*-*"*-"

<H?) •> exp(iic-r) + f( 9)exp(ikr)/r as r ->• ». (1)

Here.r denotes the position of the neutron relative to the nucleus, k the

incident-neutron wave vector, and f(0) the scattering amplitude. The dif-

ferential scattering cross section is then given by

do /dQ = )f( 9)2 (2)

where dQ = sin9d9d<t> is an element of solid angle in which ( 9, ()>) are the

polar angles that define the direction of r relative to k. The total

scattering cross section is therefore given by

a = / |f(0)|2 dO. (3)S I I

The total collision cross section, including both scattering andabsorption, is given by the optical theorem,-"-'>•*•"

at = °s + °a = ÏT Im[f (0)] . (4)

2.2 Scattering Length

i g

The scattering amplitude has a partial-wave expansion

of the form

f ( 9 ) = 2 Î k ^ (2*+l)[exp(2iôp-l] PA(cose), (5)

in which 5^ is the phase shift of the Urth partial wave and P^(cos9)

is the 1-th order Legendre polynomial.

The phase shift 6^ is a function of k and is proportional

to k x as k •+ 0. For thermal neutrons only the 1 = 0 (s-wave) term in

(5) is appreciable so that

f(9) = ~ [exp(2iôo)-l] = (kcotô^ik)"1. (6)

- 3 -

Since 6O is an odd function of k we can expand^ > 21

kcotôQ = - I + I rgk2 + 0(k4), (7)

in which a is the scattering length with the Fermi sign convention andrg is the effective range parameter. Hence,

f(9) = -a + ika2 + 0(k2), (8)

in which the k2 term contains the effective range correction. When the2

higher-order partial waves are included the k term also contains ananisotropic p-wave contribution.

The scattering length a is in general complex,

a - a'-ia", (9)

so that

a = 4n)a|2[l - 2ka" + 0(k2)], (10)

and

a = ~ a" [1 - 2ka" + 0(k2)]. (11)3 K.

It will be noted that, while a' may be either positive or negative, a" isalways positive.

In general, la - 5 fm and k = 2 Ä" for thermal neutronsso that Ikal = 10" . Also, for most nuclides, <% = q. so that a"/a' =ka1 =10 and ka" = (ka1) = 10" . For strongly absorbing nuclides,where cra = lO^q, (say) and a" = a

1, we note that ka" = 10" which isstill much less than unity. Thus, for «ill practical purposes,

% = M * ! » <12>and

a = ~ a". (13)a k v '

In the above discussion we have tacitly assumed that thereare no (n,y) resonances at thermal neutron energies. The effect of suchresonances, which occur for only a few heavy nuclides such as Cd, is

* 1Â - 0.1 nm

that the scattering amplitude acquires a resonance correction of the

Breit-Wigner type15»22»23 so that, in effect,

r /2ka = R + FVÏT7T •

Here R is the potential-scattering length, EQ the energy of the reson-

ance, r = r_+rY the total width, and E = Cnk) 2/2 \x the incident-neutron

energy in the center-of-mass system, \i being the reduced mass. The values

of the resonance parameters are tabulated in Refs. 11 and 12.

2«3 Bound Scattering Length

The scattering and absorption of thermal neutrons in con-

densed matter is most conveniently discussed in terms of the bound

scattering length,2^1

b = b'-ib" = {^) a, (15)

where A is the nucleus/neutron mass ratio which is approximately equal tothe mass number of the nucleus.

The scattering cross section for a single 'strongly' boundnucleus is given by2-"»26

a = 4n(b|2, (16)

so that

. 2a (bound) = pli) a (free). (17)S A S

The absorption cross section is given by

aa = ££ b", (18)

where ko is the incident-neutron wave vector in the laboratory system.

Since k/k0 = (A+l)/A i t follows that27

a (bound) = a (free). (19)3 3

- 5 -

Except for those few nuclides that have an (n, -) resonance

at thermal-neutron energies the scattering length b is independent of kj

so that o is constant and a is inversely proportional to kg and,

hence, to the velocity, v, of the incident neutron- As a result, the

expression (18) is usually called the 1/v-law. Absorption cross sections

are conventionally tabulated^»9»11»12 for v = 2200 m/s which corresponds

to ko = 3.4942 Ä" . Scattering lengths are measured in units of 'fm' and

cross sections in units of 'barns' where 1 barn = 100 fm . Finally, it

will be noted that the scattering cross sections that are listed in theft Q 11 1 *?

'barn book'"»''-Li>iz are the free-atom cross sections required for

nuclear physics work whereas those given in Table I of the present report

are the bound-atom cross sections needed for condensed-matter

applications•

2.4 Spin-dependent Scattering Length

Up to this point we have ignored the possible effect ofnuclear spin. When this effect is taken into account the scattering

TO

length becomes spin-dependent,

(20)

Here bc and b i are the nominal bound coherent and incoherent

scattering lengths, s is the neutron spin and I the nuclear spin. Tb~

total spin is J = I + s and, since s = 1/2, we see that J - I±1/2. Hence,the bound scattering lengths for given values of J are:

1/2b+ = bc + [1/(1+1) ] '

(21)

= bc -

Conversely,

bc " S+b+

(22)

- 6 -

in which g+ are the statistical weight factors,

1+1 I

8+ " 2Ï+T • 8- * 2Î+T '

with the normalization property

g+ + g_ - 1. (24)

The bound scattering and absorption cross sections are given by

'>, (25)a, = 4n < b

and

aa = !T <b">> ( 2 6 )

a KQ

where the brackets <•••> denote the appropriate s t a t i s t i ca l average overthe neutron and nuclear spins. If the neutrons or the nuclei are

unpolaiized, so that <s> = 0 or <I> = 0, then

bc = <b>, (27)

and

% = ac + alf (28)

in which a is the bound coherent scattering cross section,

» ( 2 9 )

and a. the bound incoherent scattering cross section,

q. = 4it|bJ2 = 4^+g_|b+-b_|

2. (30)

Alternatively,

os = g+as(+) + g . a / - ) , (31)

where

±:> l | 2 (32)

- 7 -

The absorption cross section is given by

= ai h" = ^ + ("

where

J.±) = _4ii b „ (34ja kQ ±

2.5 Experimental Determination of Scattering Lengths

The real part of the bound coherent scattering length, b',

can be measured by various kinds of neutron optical methods:

(1) transmission,

(2) mirror reflection,

(3) prism refraction,

(4) Bragg reflection in powders,(5) small-angle scattering,

(6) Christiansen filter,

(7) neutron gravity refractometer,

(8) Pendellosung interference,

(9) neutron interferometry.

The direct experimental determination of the real part of the bound inco-

herent scattering length, b', requires the use of polarized neutrons

and oriented nuclei. The two principal methods are:(1) polarized-neutron diffraction,3°

31

(2) pseudo-magnetic method.

The imaginary parts of the scattering lengths are signifi-

cant only for large absorption and, as a rule, the absorption is large for

only one spin state of the compound nucleus. For nuclides such as Cd,

' Gd, or Gd, where the absorption is large only in the J •» 1+1/2

state, we have b/| = 0 so that from (22),

(35)

and, hence,

i it ^o

'I' -TZ° • (36)

- 8 -

Thus, the measured value of a determines both b" and b".

On the other hand, for nuclides such as ^e or 10B, where

the absorption is large only in the J » 1-1/2 state, we have b^ - 0 so

that from (22),

and, .hence,

)V - -F- c . (38)

Again, the measured value of a determines both b" and b". Note that in3. O X

this case b" is negative whereas in the previous case it was positive. On

the other hand, b" is always positive.

For nuclides such as 6Li, where the absorption is large for

both the J = 1+1/2 states, we get

K. K.

c 4n a 4 it LO+ a -

•jc+) _ J-

a J

(39)

In this case the measured value of a determines only b", and a and(-) a C a

a must be measured separately as in Ref. 33 in order to obtain b".a (

Here b" is positive or negative depending on the relative values of av

and o^~\a

2.6 Isotope Incoherence

Equations (25), (26), and (27) also apply to a mixture of

isotopes if the brackets <•••> are understood to denote an average over

both the spin and isotope distributions. * Thus, if C a is the frac-

tion of isotopes of type a, so that

I C • 1, (40)a

then

- 9 -

With

a = } C a ,s % a a,s

a - y C a , (41)a L a a,a

b = y C b- £ a <x,c

ac = 47t |bJ2 , (42)

as before, it then follows that

°8 " °c + V <43>

where

a. = a. (spin) + a. (isotope), (44)

in which the contribution from spin incoherence is given by

al(Spin) - X V M - 4 x | c J b M | 2 . (45)

and that from isotope incoherence is given by

^( i so tope) - 4* l C a C ß | b a > c - b ß ) J2 . (46)

<x<ß

2.7 Scattering Lengths An Crystallography

There is great confusion in the literature concerning thesign of the imaginary part of b. In Refs. 1, 9, 35-38 the theoretical

relations imply that Im(b) < 0 whereas the experimentally-determined

values all have Im(b) > 0. This discrepancy is removed in Refs. 11 and 12

by arbitrarily changing thp. sign of Im(b) in the theoretical relations and

in Ref. 10 by arbitrarily changing the sign of Im(b) in the experimen-

tally-determined values.

- 10 -

The origin13'39 of this confusion lies in the fact that

the conventional way of representing complex waves is different in cry-

stallography from what it is in quantum mechanics- The Schro'dinger equa-

tion, iti(J » H4>, requires that a complex plane wave be expressed as

* - exp[i(i?«r - cot)] . (47)

To express it in the alternative form,

(p = exp[-i(ic-r - ut) ] , (48)

the Schrtfdinger equation would have to be written in the unorthodox form

-ifi£ = H<|>. On the other hand, the electric field that describes X rays

satisfies a wave equation that is of second order in r and t and, hence,

can be expressed either in the form (47) or in the form (48). By conven-

tion, crystallographers always use the form (48). One consequence of this

is _hat the X-ray scattering amplitude corresponds to the complex conju-

gate of the neutron scattering amplitude.

When crystallogiaphers determine complex scattering lengthsby means of neutron diffraction measurements they usually analyse their

results using standard X-ray diffraction formulae. The 'scattering

length' which they obtain is then the complex conjugate of the conven-

tional quantum mechanical quantity. This is why there is a difference in

sign between the 'measured1 values of Im(b) and the conventional Breit-

Wigner formula for Im(b).

In une present report we follow conventional quantum

mechanics so that Im(b) < 0. Thus, b" and b" are positive. Since b. isc x x

not a scattering length per se, b£ may, of course, be either positive or

negative as discussed in Sec. 2.5.

- 11 -

3. CALCULATIONS AND DISCUSSION

3.1 Calculations

The bound scattering lengths and cross sections of the ele-

ments and their isotopes are lifted in Table I. The. basic procedure used

to compile this table was to take the best current values of the scatter-

ing lengths10 and to compute from them a consistent set of scattering

cross sections. The trailing digits in parentheses give the standard

errors and were calculated from the errors in the input data on the basis

of the statistical theory of error propagation. ^ The imaginary parts

of the scattering lengths of the strongly absorbing elements and isotopes

were calculated from the measured absorption cross sections ' as des-

cribed in Sec. 2.5. Those isotopes that have an (n,y) resonance at ther-

mal-neutron energies are indicated by an asterisk.

In cases where the scattering lengths have not yet been

measured directly the reverse procedure was used. That is, the measured11 12values of the scattering cross sections ' were used to calculate the

corresponding scattering lengths. The relations (41) were used where

necessary to fill in gaps in the table. For about 25% of the elements

these relations indicated inconsistencies in the data. In such casesappropriate adjustments in the values of some of the quantities were made.

In almost all cases the adjustments were comparable with the stated accur-

acies. Finally, for some elements it was necessary to estimate the scat-

tering lengths of one or two isotopes In order to be able to complete the

table. Such estimates are indicated by the letter 'E' and were only made

for isotopes of low natural abundance so that the estimated values willhave only a marginal effect on the final results.

3.2 Scattering Cross Sections

Figure 2 shows the distribution of bound scattering and

absorption cross sections over the periodic table. It is seen that aQis typically 1 to 10 barns whereas a± <_ 1 barn as a rule.

The incoherent scattering cross section vanishes identi-

cally for an element that consists of a single isotope with 1 = 0 . The

only example of this is Th. By the same token 0j is usually very small

for an element that consists almost entirely of a single isotope with

1 = 0 . There are many examples of this, notably, He, C, 0, and U. An

exception is Ar which has relatively large incoherent scattering in spite40

of the fact that it is 99.6% Ar. The reason for this is the abnormally

large coherent scattering cross section of 36Ar.

- 12 -

Table II shows the separate contributions of spin and iso-tope incoherence to the total bound incoherent scattering cross section of

a selected group of elements listed in order of increasing isotope inco-

herence. Spin incoherence dominates for elements that consist entirely

(Na, Co) or almost entirely (H,V) of a single isotope with I * 0. H and V

are unique in having o » ac> This results from the fact that b+ and

b_ are of opposite sign for these nuclides. On the other hand, isotope

incoherence dominates for elements that consist entirely (Ar) or almost

entirely (Ti.W.Ni) of isotopes with I =» 0.

3.3 Absorption Cross Sections

It is seen in Fig. 2 that there is a much larger variationover the periodic table in a than in q,. The only absorption pro-

cesses that occur with thermal neutrons are:(n,y) reactions (radiative capture),

(n,p) reactions,

(.n,a) reactions,

(n,f) reactions (fission).The absorption cross section is therefore of the form

a = a + a + a + a c . (49)a y p a f v

It is clear that aa can vanish identically only if all four cross sec-

tions on the right-hand side of (49) equal zero. Among the naturally

occurring nuclides this happens only for " He. There are, however, many

radionuclides, such as H, for which aa = 0.

For most nuclides the only non-vanishing contribution tooa comes from radiative capture and the value of a_ ranges from

1.0 x io~h barns for 160 to 2.7 x 106 barns for l Xe. Table III liststhe radiative-capture cross sections and resonance energies of those

naturally-occuring nuclides that have an (n,y) resonance at thermal-neu-tron energies. These nuclides are indicated by an asterisk in Table I.

For most nuclides charged-particle reactions are not ener-getically allowed while, for the remainder, the reactions are inhibited by

the Coulomb barrier so that the cross sections tend to be very small. For

example, oa = 8.2 x 10~5 barns for 155Gd. Table IV lists those nuclides

that have appreciable cross sections for charged-particle reactions with

thermal neutrons. The only nuclides with large cross sections for

charged-particle reactions are He, i , °B, and the fissile actinide

isotopes.

- 13 -

Acknowledgements

The author is grateful to L. Koester, H. Rauch, and S.F.

Mughabghab for making available copies of Refs. 10 and 12 prior to

publication. A helpful correspondence with H. Glâ'ttli is also gratefully

acknowledged.

References

1. L. Koester, Neutron Scattering Lengths and Fundamental Neutron

Interactions, Springer Tracts in Modern Physics J5£, 1 (1977).

2. V.F. Sears, Phys. Rep. 82_, 1 (1982).

3. A.G. Klein and S.A. Werner, Rep. Prog. Phys. 4£, 259 (1983).

4. J.R. Dunning, G.B. Pegram, G.A. Fink, and D.P. Mitchell, Phys. Rev.

48, 265 (1935).

5. M. Goldhaber and G.H. Briggs, Proc. Roy. Soc. A162, 127 (1937).

6. E. Fermi and L. Marshall, Phys. Rev. 71» 6 6 6 (1947).

7. C G . Shull and E.O. Wollan, Phys. Rev. 81_, 527 (1951).

8. J.R. Stehn, M.D. Goldberg, B.A. Magurno, and R. Wiener-Chasman,

Neutron Cross Sections, Brookhaven National Laboratory Report No.

BNL 325, second edition, 1964.9. S,F. Mughabghab and D.I. Garber, Neutron Cross Sections, Brookhaven

National Laboratory Report No. BNL 325, third edition, 1973.10. L. Koester and H. Rauch, Neutron Scattering Lengths, IAEA Report No.

2517/RB, second edition, 1983.

11. S.F. Mughabghab, M. Divadeenam, and N.E. Holden, Neutron Cross

Sections, Vol. 1, Part A, Z = 1-60 (Academic Press, New York, 1981).12. S.F. Mughabghab, Neutron Cross Sections, Vol. 1, Part B, Z = 61-100

(Academic Press, New York, 1984).

13. V.F. Sears, Bound Coherent and Incoherent Thermal Neutron Scattering

Cross Sections of the Elements, Atomic Energy of Canada Limited

Report No. AECL-7980, 1982.

14. L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).

15. L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Addison-Wesley,

Reading, Mass., 1958).16. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964).

17. E. Feenberg, Phys. Rev. 40, 40 (1932).18. R.G. Newton, Am. J . Phys. 44_, 639 (1976).19. H. Faxen and J . Holtsmark, Z. Phys. 45_, 307 (1927).

20. J.M. Blat t and J.D. Jackson, Phys. Rev. Tb_, 18 (1949).21 . H.A. Bethe, Phys. Rev. 76, 38 (1949).

- 14 -

22. G. Breit and E. Wigner, Phys. Rev. 49_, 519 (1936).23. J.E. Lynn, Theory of Neutron Resonance Reactions (Clarendon Press,

Oxford, 1968).24. W. Marshall and S.W. Lovesey, Theory of Thermal Neutron Scattering

(Clarendon Press, Oxford, 1971).25. E. Fermi, Ric. Sei. ]_, 13 (1936).

26. V.F. Sears, Can. J. Phys. 56, 1261 (1978).

27. W.E. Lamb, Phys. Rev. 5^, 187 (1937).

28. E. Amaldi, 0. D'Agostino, E. Fermi, B. Pontecorvo, F. Rasetti, and

"E. Segrè, Proc. Roy. Soc. A14£, 522 (1935).

29. J. Schwinger and E. Teller, Phys. Rev. _52 286 (1937).

30. C G . Shull and R.P. Ferrier, Phys. Rev. Lett. 1£, 295 (1963).

31- A. Abragam, G.L. Bacchella, H. Gla'ttli, P. Meriel, M. Pinot, and

J. Piesvaux, Phys. Rev. Lett. Zl, 776 (1973).

32. V.F. Sears and F.C. Khanna, Phys. Lett. 56B, 1 (1975).33. H. Gla'ttli, A. Abragam, G.L. Bacchella, M. Fourmond, P. MéYiel,

J. Piesvaux, and M. Pinot, Phys. Rev. Lett. 40_, 748 (1978).34. G.C. Wick, Phys. Z. 3£, 689 (1937).

35. S.W. Peterson and H.G. Smith, Phys. Rev. Lett. £, 7 (1961).

36. S.W. Peterson and H.G. Smith, J. Phys. Soc. Japan, Yj_, Supl. B-II,

335 (1962).37. H.G. Smith and S.W. Peterson, J. de Phys. 25^ 615 (1964).

38. G.E. Bacon, Neutron Diffraction, third edition (Clarendon Press,

Oxford, 1975).

39. G.P. Felcher and S.W. Peterson, Acta Cryst. A33 , 76 (1975).40.. H.D. Young, Statistical Treatment of Experimental Data (McGraw-Hill,

New York, 1962).

- 15 -

Table I. Bound scattering lengths, b, in fm and cross sections, o, in

barns (1 barn = 100 fm ) of the elements and their isotopes.

col.l element

col.2 Z atomic number

col.3 A mass number

col.4 I(n) spin (parity) of the nuclear ground state

col.5 C (%) natural abundance. (For radioisotopes the half-life is

given instead.)

col.6 b bound coherent scattering length

col.7 b. bound incoherent scattering length

col.8 a bound coherent scattering cross section

col.9 a. bound incoherent scattering cross section

col.10 a total bound scattering cross section

S

col-11 ö absorption cross section for 2200 m/s neutrons

Note: Because of the limitations of the FORTRAN character set, 'bc' is

denoted by 'B(C)1, 'ac' by 'S(C)', years by 'A' instead of 'a', etc.

EX K P ) PCT B ( C ) S I C ) SIS) S ( A )

HE

LI

BE

1

2

3

3

4

6

7

9

10

11

12

13

14

15

16

17

18

19

l/2(+)

1( + )

l/2(+)

1/2!+)

0( + )

l(+>

3/2(-)

3/2(-)

3< + )

3/2(-)

0( + )

l/2(-)

1< + )

1/2I-)

0( + )

5/2(+)

0(+,

1/2I+)

99.985

0.015

(12.32 A)

0.00014

99.99986

7.5

92.5

100

20.0

80.0

98.90

1.10

99.63

0.37

99.762,

0.038

0.200

100

-3 .

-3.

6.

4.

3.

5.

3-.

-1.

i*o:-2.

7.

i*o!

i*î:6.

6.

6.

6.

9.

9.

6.

5.

5.

5.

5.

5.

7409(11)

7423(12)

674(6)

94(B)

26(3)

74(7)483(2)

26(3)

90(3)

0(1)261(1)

2211)

79(1)

30(4)213(2)

1(4)067(3)

65(4)

6484(13)

6535(14)

19(9)

36(2)

37(2)

44(3)

805(4)

,805(5)

78(12)

.84(7)

.654(12)

25.217(6)

4.033(32)

0.00(37)

-1.8(6)+1*2.568(3)

0

-1.79(24)+1*0.257(11)

-2.49(5)

0.20(2)

-4.7(3)+1*1.232(3)

-1.31(17)

0

-0.52(9)

1.98(17)

-0.02(2)

0

0.17(6)

0

-0.082(9)

1.7586(10)

1.7599111)

5.597(10)

3.07(10)

1.34(2)

4.42(10)

1.34(2)

0.454(14)

0.51(5.'

0.619(6)

7.63(2)

3.54(5)

0.14(2)

5.56(7)

5.554(2)

5.363(2)

4.81(14)

11.01(5)

11.03(5)

5.21(5)

4.235(6)

4.235(71

4.2(2)

4.29(10)

4.017(17)

79.90(4)

79.91(4)

2.04(3)

0.00(2)

0.00

1.2(3)

0

0.91(3)

0.41(11)

0.78(3)

0.005(1)

1.70(12)

3.0(4)

0.22(6)

0.001(4)

0

0.034(12)

0.49(10)

0.49(8)

0.00005(10)

0.000(9)

0

0.004(3)

0

0.0008(2)

81.66(4)

81.67(4)

7.64(3)

3.07(10)

1.34(2)

5.6(3)

1.34(2)

1.36(3)

0.92(12)

1.40(3)

7.64(2)

5.24(11)

3.1(4)

5.78(9)

5.555(3)

5.563(2)

4.84(14)

11.50(9)

11.52(9)

5.21(5)

4.235(7)

4.235(7)

4.2(2)

4.29(10)

4.018(17)

0.3326(7)

0.3326(7)

0.000519(7)

0

0.00747(1)

5333.(7.)

0

70.5(3)

940.(4.)

0.045413)

0.0076(8)

767.(8.)

3837.(9.)

0.0055(33)

0.00350(7)

0.00353(7)

0.00137(4)

1.90(3)

1.91(3)

0.000024(8)

0.00019(2)

0.00010(2)

0.236(10)

0.00016(1)

0.0096(5)

EL

NE

NA

MG

AL

SI

P

S

CL

AR

Z

10

11

12

13

14

15

16

17

18

A

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

36

35

37

36

38

40

KP)

0( + )

3/2(+)

0( + )

3/2(+)

0( + )

5/2(+)

0(+)

5,2(*>

0( + )

1/2I+,

0( + )

1/2«*,

O( + )

3/2(+)

0( + )

0( + )

3/2(+)

3/21+)

0( + )

0(+)

0( + )

PCT

90.51

0.27

9.22

100

78.99

10.00

11.01

100

92.23

4.67

3.10

100

95.02

0.75

4.21

0.02

75.77

24.23

0.337

0.063

99.600

B(C)

4.547(11)

4.610(12)

6.66(19)

3.87(1)

3.63(2)

5.375(4)

5.68(2)

3.62(14)

4.92(15)

3.449(5)

4.149(1)

4.106(6)

4.7(1)

4.58(8)

5.13(1)

2.847(1)

2.804(2)

4.74(19)

3.48(3)

3.(1.) E

9.5792(8)

11.66(2)

3.08(6)

1.909(6)

24.90(7)

3.5(3.5)

1.83(5)

B(I)

0

0.63(12)

0

3,59(3)

0

0.9(3)

0

0.26(1)

0

-1.1(2)

0

0.22(7)

0

1.5(1.5)

0

0

6.0(2)

0.02(5)

0

0

0

S(C)

2.598(13)

2.671(14)

5.6(3)

1.88(1)

1.66(2)

3.631(5)

4.05(3)

1.65(13)

3.04(19)

1.495(4)

2.163(1)

2.119(6)

2.78(12)

2.64(9)

3.307(13)

1.0186(7)

0.9880(14)

2.8(2)

1.52(3)

1.1(8)

11.531(2)

17.08(6)

1.19(5)

0.45B(3)

77.9(4)

1.5(3.1)

0.42(2)

S U ,

0.008(18,

0

0.05(2,

0

1.62(3)

0.077(6,

0

0.10(7)

0

0.0085(7)

0.015(2)

0

0.15(6,

0

0.006(4)

0.007(5)

0

0.3(6)

0

0

5.2(2)

4.5(3)

0.0001(3)

0.22(2)

0

0

0

SIS)

2.606(13)

2.671(14)

5.7(3,

1.88(1)

3.28(4)

3.708(8,

4.05(3)

1.75(15)

3.04(19)

1.504(4)

2.178(2)

2.119(6)

2.93(13)

2.64(S,

3.313(14,

1.026(5)

0.9880(14)

3.1(6)

1.52(3)

1.1(8)

16.7(2)

21.6(3,

1.19(5)

0.68(2)

77.9(4)

1.5(3.1)

0.42(2,

S(A)

0.039(4)

0.037(4,

0.67(11)

0.046(6)

0.530(5,

0.063(3,

0.051(5)

0.19(3,

0.0382(8)

0.231(3)

0.171(3)

0.177(5)

0.101(14)

0.107(2,

0.172(6,

0.53(1,

0.54(4,

0.54(4,

0.227(5,

0.15(3)

33.5(3)

44.1(4)

0.433(6)

0.675(9)

5.2(5,

0.6(2,

0.660(10,

EL

K

CA

se

TI

V

CR

MN

Z

19

20

21

22

23

24

25

A

39

40

41

40

42

43

44

46

48

45

46

47

48

49

50

50

51

50

52

53

54

55

I(P)

3/2(+)

4(-)

3/2(+)

01 + )

O( + )

7/2(-)

O( + )

0(+)

0( + )

7/2(-)

O(+)

5/2(-)

0(+)

7/2(-)

O( + )

6( + )

7/2(-)

O( + )

0( + )

3/21-)

0( + )

5/2I-)

PCT

93.258

0.012

6. 730

96.941

0.647

0.135

2.086

0.004

0.187

100

8.2

7.4

73.8

5.4

5.2

0.250

99.750

4.35

83.79

9.50

2.36

100

B(C)

3.71(2)

3.79(2)

3.(1.) E

2.58(6) .LT.

4.90(3)

4.99(3)

. 3.15(20)

0.2(0.2)

1.8<1>

2.55(25)

1.5(2)

12.29(11)

-3.3012)

4.73(6)

3.49(12)

-5.84(2)

1.00(5)

5.93(8)

-0.3824(12)

7.6(7)

-0.4024(21)

3.635(7)

-4.50(5)

4.920(10)

-4.20(3)

4.55(10)

-3.73(2)

81 I)

1.4(3)

1.(1.) E

0.2

0

0

3.(2.1 E

0

0

0

-6.0(3)

0

-3.5(2)

0

5.1(2)

0

2.(2.) E

6.419(10)

0

0

6.86(11)

0

1.79(4)

S

1.

1 .

1.

0.

3.

3.

1.

0.

0.

0.

0.

19.

1.

2.

1.

4.

0.

4.

0.

7.

0.

1.

2.

3

2.

2,

1

(Ci

7 3(2)

81(2-

1(8)

84(4) .LT.

02(4)

13(4)

25(16)

005(10)

41(5)

82(16)

28(8)

0(3)

37(2)

Bl( 7)

53(11)

29(3)

13(1)

42(12)

.0184(1)

3'1.4)

0203(2)

.660(6)

.54(6)

.042(12)

.22(3)

.60(11)

.75(2)

S( I)

0.25c10)

0.25111)

0.1(3)

0.00b

0.03(6)

0

0

1.1(1.5)

0

0

0

4.5(5)

2.67(4)

0

1.54(18)

0

3.27(26)

0

5.187(16)

0.5(1.0)

5.178(16)

1.83(2)

0

0

5.91(19)

0

0.40(2)

S(Si

r. 98 ( ic >

:.06i) î )

1.2(91

û.84l4 )

3.05(4)

3.13(4)

1.25(16)

1.1(1.5)

0.41(5)

0.82(16)

0.28(8)

23.5(6)

4.04(3)

2.81(7)

3.1(2)

4.2913)

3.4(3)

4.42(12)

5.205(16)

7.8(1.0)

5.198(16)

3.49(2)

2.54(6)

3.042(12)

8. 13(19)

2.60(11)

2.15(3)

s

2

2.

35.

1 .

0.

0.

0.

6.

0.

0.

1.

27.

6.

0.

1.

7.

2.

0.

5.

60.

4.

3.

15.

0.

18.

0.

13

(A)

1(1)

1(2)

(8. )

461 3)

43(2)

41(2)

68( 7)

2(6)

88( 51

74( 7 )

09(14)

2(2)

09<13.

59(18)

7(2)

04(25)

2(3)

17913)

08(4)

(40. )

9(1)

.07(8)

.9(2)

.76(6)

.2(1.5)

.36(4)

.3(2)

KP) PCT B(C) BU) S(C) S(S) S<A>

FE 26

CO 27

NI 28

CU 29

ZN 30

GA 31

CE 32

5456

5 7

5 8

0( + )0(+)

1/21-)

0( + )

5 .

9 1 .

2 .

0 .

8

7

2

3

59

58

60

61

62

64

63

65

64

66

67

66

70

69

71

70

72

73

74

76

7/2(-)

0( +

0(

0(

0(

5/2I-)

3/2I-)

9/2I+)

0(

100

68.27

26.10

1.13

3.59

0.91

69.17

30.83

48.6

27.9

4.1

18.8

0.6

60.1

39.9

20.5

27.4

7.8

36.5

7.8

9.54(6)

4.2(1)

10.03(7)

2.3(1)

15.(7.)

2.50(3)

10.3(1)

14.4(1)

2.8(1)

7.60(6)

-8.7(2)

-0.38(7)

7.718(4)

6.43(15)

10.61(19)

5.680(5)

5.23(10)

6.01(12)

7.64(15)

6.05(12)

6.(1.) E

7.288(10)

X

X

8.1929(17)

9.5(4)

8.8(4)

3.2(1.3)

7.9(2)

9.(1, ) E

0

û

2.(2.) E

0

-6.2(2)

0

0

4.0(3)

0

0

0.22(2)

1.79110)

0

0

2.(2.) E

0

0

0

0

2.(2.) E

0

0

11.44114)

2.2(1)

U.64(18)

0.66(6)

28.(26.)

0.79(2)

13.3(3)

26.1(4)

0.99(7)

7.26(11)

9.5(4)

0.018(7)

7.486(8)

5.2(2)

14.1(5)

4.054(7)

3.44(13)

4.54(18)

7.3(3)

4.60(18)

4.5(1.5)

6.67(2)

X

X

8.435(4)

11.3(1.0)

9.7(9)

1.3(1.0)

7.8(4)

10.(2.)

0 . 3 9 ( 3 )0

0

0 . 5 ( 1 . 0 )

0

4 . 8 ( 3 )

5 . 2 ( 4 )

0

0

2 .0 (3 )

0

0

0 .52 (4 )

0 .0061(11)

0 .40 (5 )

0 .077(7)

0

0

0 . 5 ( 1 . 0 )

0

0

0 . 0 ( 2 )

X

X

0 .17(6)

0

0

0 . 5 ( 1 . 0 )

0

0

11 .83(14)2 . 2 ( 1 )

12 .64(18)

1 . 2 ( 1 . 0 )

2 8 . ( 2 6 . )

5 .6 (3 )

1 8 . 5 ( 3 )

2 6 . 1 ( 4 )0 .99 (7 )

9 . 3 ( 3 )

9 . 5 ( 4 )

0 .018(7)

8 .01(4)

5 . 2 ( 2 )

14 .5 (5 )

4 .131(10)

3.44113)

4 .54 (18 )

7 . 8 ( 1 . 0 )

4 .60 (18)

4 . 5 ( 1 . 5 )

6 . 7 ( 2 )

XX

8.60(6)

1 1 . 3 ( 1 . 0 1

9 . 7 ( 9 )

1 .8 (1 .4 )

7 .8 (4 )

1 0 . ( 2 . )

2 .56 (3 )2 .25(18)

2 .59(14)

2 .48(301

1.28(5)

37 .18 (6 )

4 . 4 9 ( 1 6 )

4 . 6 ( 3 )

2 . 9 ( 2 )

2 .5 (8 )

14 .5 (3 )1.52(3)

3 .78 (2 )

4 . 5 0 ( 2 )

2 .17 (3 )

1.11(2)

0 .76 (2 )

0 .85 (20 )

6 . 8 ( 8 )

1.1(1)

0 .092(5 )

2 . 9 ( 1 )1 .68(7)

4 . 7 1 ( 2 3 )

2 . 3 ( 2 )

3 .43 (2 )

0 . 9 8 ( 9 )

l b . ( 2 . )

0 . 5 1 ( 8 )

0 . 1 5 ( 2 )

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3/2(+)

0( + )

0( + )

l/2(-)

0( + )

3/2(-)

0( + )

0( + )

l/2<+)

l/2(+)

0(+)

0( + )

l/2(-)

0( + )

9/2(-)

PCT

37.3

62.7

0.01

0.79

32.9

33.8

25. 3

7.2

100

0.2

10.1

17.0

23.1

13.2

29.6

6.8

29.524

70.476

1.4

24.1

22.1

52.4

100

BCC)

10.6(3)

XX

9.63(5)

9.(1.)

9.9(5)

10.55(8)

8.91(9)

9.89(8)

7.83(10)

7.63(6)

12.66(2)

30.3(1.0)-1*0.86(5)

X16.9(4)

-1*0.60(1)

X

X

X

X

8.785(10)

6.99(16)

9.54(10)

9.4003(14)

9.5(5) E

8.8(5)

9.46(5)

9.65(23)

8.5256(14)

Bill

X

X

0

0

0

-1.0(2)

0

0

-1.69(9)

0

0

15.6(7)

0

X0

0

0.38(8)

-0.13(3)

0

0

0.09(17)

0

0.239(10)

SIC)

14.1(8)

X

X

11.65(12)

10.(2.)

12.3(1.2)

14.0(2)

10.0(2)

12.3(2)

7.7(2)

7.32(12)

20.14(6)

115.(8.)

X35.9(1.7)

X

X

X

X

9.70(2)

6.1(3)

11.4(2)

11.104(3)

11.3(1.2)

9.7(1.1)

11.25(12)

11.7(6)

9.134(3)

SU)

0.2(2.9)

XX

0.13(16)

0

0

0

0.13(5)

0

0

0.36(4)

6.7(1)

0

0

30.5(2.6)

0

X0

0

0.14(17)

0.018(8)

0.0021(10)

0.0030(7)

0

0

0.001(4)

0

0.0072(6)

SIS)

14.3(2.8)

X

X

11.78(11)

10.(2.)

12.3(1.2)

14.0(2)

10.1(2)

12.3(2)

7.7(2)

7.6B(13)

26.8(1)

115.(8.)

X66.4(2.0)

X

X

X

X

9.84(17)

6.1(3)

11.4(2)

11.107(3)

11.3(1.2)

9.7(1.1)

11.25(12)

11.7(6)

9.141(3)

SIA)

425.3(2.4)

954.(10.)

111.(5.)

10.3(3)

800.(70.)

10.0(2.5)

1.2(4)

27.5(1.2)

0.72(4)

3.7(2)

98.65(9)

372.3(4.0)

3080.(180.)

2.0(3)

2150.(46.)

.LT. 60.

7.8(2.0)

4.9(2)

0.43(10)

3.43(6)

11.4(2)

0.107(18)

0.171(2)

0.66(7)

0.0306(8)

0.712(10)

0.00049(3)

0.0338(7)

I

- 29 -

intn

57

(18

).9

07

m

OB

S

.9(3

1(1

100

.3(9

5(1

680

in

.3(1

089

15

).8

82

m m en i-H (No — — — co(T> CO m r-4 CO in in es

r* i-t co

Op* IN f> iD r^rH — ^ . w O* r-i *t CO ^

ao o CJ en oo

m in o in

*

< S

t - l

tn

m

mU

r-H

i-H

CO

inW

ID — — —

« m con n fiIN N (V

co en on m -«•(M N N

S E S

- 30 -

<to

uto

u

- 31 -

Table II. Contributions of spin and isotope incoherence to the totalbound incoherent scattering cross section of a selected group

of elements listed in order of increasing isotope incoherence.

The cross sections are in barns.

element o^spin) a. (isotope) 0 (isotope)/a.

Na

Co

H

V

Li

Cl

Cr

Ti

W

Ni

Ar

1.62

A.8

79.90

5.19

0.91

5.2

1.83

2.67

2.00

5.2

0.22

1.62

4 . 8

79.90

5.17

0.75

3.4

0.56

0.29

0.01

0.023

0

0

0

0.00

0.02

0.16

1.8

1.27

2.38

1.99

5.2

0.22

0

0

0.00

0.004

0.18

0.35

0.69

0.89

0.995

1.00

1

- 32 -

Table III. Radiative-capture cross sections and resonance energies ofthose naturally-occuring nuclides that have an (n, y) resonance

11 19at thermal-neutron energies. »

nuclide a Eg

(barns) (meV)

113Cd 20 600 178.

Sm 40 140 97.3

151Eu 9 200 321.

155Gd 60 900 26.8

157Gd 254 000 31.4

176Lu 2 065 141.3

- 33 -

Table IV. Absorption cross sections (in barns) for those nuclides that

have significant charged-particle reactions with thermal

neutrons.11»12

nuclide

3He

6Li

ioB

1 7 0

33s35 C A

233u

2 3 5U

2 3 8 p u

239Pu

aa

5333.

940.

3837.

1.91

0.236

0.54

44.1

3 5 .

580.

681.

558.

1017.

0.000031

0.0385

0 .5

0.075

0.00054

0.35

43.6

30 .

4 9 .

9 7 .

540.

269.

aP

5333.

0

0

1.83

0

0.002

0.489

4 .4

0

0

0

0

• \ +

0

940.

3836.

0

0.235

0.19

0

0.39

0

0

0

0

a f

0

0

0

0

0

0

0

0

531.

584.

17.9

748.

OCoococ

Fig. 1. Average percentage error in the bound coherent scattering length

of a selected group of twenty elements as a function of time.

The open circles are the values estimated by the authors at the

time of measurement*"10 and the filled circles are the actual

errors determined by a comparison with current values.10

- 35 -

100

80 —

6 0 -

1*0-

2 0 -

—

[OHESCA"

:RENfTER

TiNG

—

100

- 80 -

- 60 —

20 -

I I ITOTAL

_ SCATTERING

•Â ""

i

I

I I.001 .01 .1 1 10 100 1000 .001 .01 .1 1 10 100 1000

LU100

LU Q nLJ 80ceLUQ.

60

20

i i rINCOHERENT

h SCATTERING

t . J.001 .01 .1 1 10 100 1000

100

80

60

2 0 -

ABSORPTION

i t e ? - : , - l . l \,.001 .01 .1 1 10 100 1000

CROSS SECTION (BARNS)

Fig. 2. Distribution of bound scattering and absorption cross sectionsover the periodic table.

ISS\ 006 7 - 036?

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