pairing correlation in atomic nuclei under extreme conditions1156741/fulltext01.pdf · the pairing...

Pairing correlation in atomic nucleiunder extreme conditions

Sara Asiyeh Changizi

Department of PhysicsKTH Royal Institute of Technology

Albanova November 2017

ii

TRITA-FYS 2017: 69ISSN 0280-316XISRN KTH/FYS/–17:69—SEISBN 978-91-7729-537-2

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges tilloffentlig granskning för avläggande av teknologie doktorsexamen i fysik fredagen den 24novemeber klockan 14:30 i FA32, Roslagstullsbacken 21, Albanova Universitetscentrum,Stockholm.© Sara Asiyeh Changizi, november 2017Tryck: Universitetsservice US AB

KTH School of Engineering ScienceSE-100 44 Stockholm

SWEDEN

"The real voyage of discovery consists not in seeking new landscapes,but in having new eyes."

Marcel Proust

I would like to dedicate this thesis to my loving Michael andone of the great minds of our time who left us much too soon Maryam Mirzakhani.

Acknowledgements

I have met some great physicists during my Ph.D. time. Prof. Ramon Wyss and Prof. RobertoLiotta are two of those who I have learned a lot from them. They have shown me the beautyof physics, the beauty of understanding the meaning of abstract ideas, and the physics behindthe mathematics formulas. Together with my dedicated supervisor Dr. Chong Qi, who taughtme a lot including the research work ethic and how to push forward in research, encouragingme to come up with my ideas.

Dr. Farnaz Ghazimoradi who brought me into the era of Ph.D. studies and inspired meto join the Theoretical Nuclear Physics course. Dr. Sara Bartot, a dear friend, who inspiresme every day and makes my days full of joy. Dr. Maria Doncel who helped me with herencouragement and kindness during the hard times of Ph.D. There have been a lot of peoplein Nuclear Physics group during this period that inspired me a lot. However, there is notspace enough here to mention all their names, but I am deeply grateful to them. I also wantto acknowledge KTH and the Department of Physics, which facilitated and made this workpossible.

My dear husband Michael who with his deep passion for cutting-edge scientific solutionsand sometimes "crazy" ideas has made my life brighter.

My adorable son William, to whom I owe all my happiness. The happiness I could notimagine was ever even possible.

Sara A. Changizi,Stockholm, Nov. 7, 2017

Abstract

The pairing correlation has long been recognized as the dominant many-body correlationbeyond the nuclear mean field. Pairing plays an essential role in many nuclear phenomenaincluding the occurrence of a systematic odd-even staggering (OES) of the nuclear bindingenergy. Pairing interaction plays, in particular, a significant role in the stability of weaklybound nuclei. Therefore it has been one of the most critical topics in nuclear physicsbecause of the weakly bound structure of all newly discovered nuclei. Beyond the lineof stability, pairing interaction is not a residual interaction anymore. Its strength can beof the same order of magnitude as the mean field. In this thesis we have focused on thepairing interaction in atomic nuclei under extreme conditions to investigate the structure ofloosely bound nuclei near the dripline; to probe and globally assess different outcomes ofvarious zero-range pairing interactions and their effect on the loosely bound low angularmomentum states. How much can density-dependence in zero-range pairing interactionnuclei affect the line of stability? Calculations predict that pure density-dependent pairinginteraction so-called surface interaction enhances the collectivity and gives stronger neutronpairing gap in nuclei far from stability, while, the density dependence pairing interaction doesnot affect the gap for bound nuclei as much. The odd-even staggering of nuclear bindingenergies has been investigated to estimate the empirical pairing gap. A 3-point formula ∆

(3)C

(12 [B(N,Z)+B(N −2,Z)−2B(N −1,Z)]) is advocated in this thesis, which we believe it is

more suitable to measure the magnitude of pairing gap in even-even nuclei. The strength of∆(3)C can be a good indicator of the two-particle spatial correlation. One-quasiparticle energies

and binding energy for those odd nuclei, which can be approximated by spherical symmetry,are calculated to obtain binding energy odd-even staggering (OES) in all known semi-magiceven-even nuclei. The pairing strength is fitted globally to all available data on the OESof semi-magic nuclei with Z ≥ 8. The difference between different zero-range density-dependent pairing interactions reduces with this global fitted parameter. The differencebetween the mean pairing gap and the OES gets larger as we get closer to the dripline. At theend of the thesis, a simple model has been developed which shows that when the mean-fieldbecomes shallower the odd-even staggering of charge radii is reduced.

viii

This thesis covers results that are not included in the three published papers and somecomplementary works on the subject.

Sammanfattning

Parkopplingen i atomkärnan anses vara den dominerande mångkropparskorrelationen efterkärnmedelfältet. Parkoppling i atomkärnan spelar en viktig roll i många kärnfenomen,inklusive förekomsten av en systematisk udda-jämn förskjutning (OES) av bindningsen-ergin. Hos svagt bundna atomkärnor däremot spelar parkopplingen en signifikant roll förstabiliteten. Det gäller alla de nyupptäckta neutronrika atomkärnor som karakteriseras aven svag bindningsenergi där således parkopplingen blir ett kritiskt fenomen. När vi läm-nar stabilitetslinjen och närmar oss linjen av spontan neutronsönderfall är parkopplingeninte längre obetydlig, utan tvärtom spelar en viktig roll för stabiliteten hos atomkärnan.Dess styrka kan vara av samma storleksordning som medelfältet. I denna avhandling harvi fokuserat på parkopplingen i atomkärnor under extrema villkor i syfte att undersökastrukturen hos löst bundna atomkärnor nära linjen för neutronsönderfall; att utforska ochglobalt bedöma resultaten av olika parkopplingar baserad på en deltakraft (räckvidd noll)och deras effekt på löst bundna tillstånd med lågt rörelsemängdsmoment. Hur mycket kan etttäthetsberoende i parkopplingen påverka stabilitetslinjen? Beräkningarna förutsäger att rendensitetsberoende växelverkan för parkoppling, så kallad ytväxelverkan, ökar kollektiviteteni atomkärnan och ger ett större pargap för neutroner i kärnor långt ifrån stabilitetslinjen,medan densitetsberoende parkoppling påverkar inte gapet för bundna kärnor i samma om-fattning. Udda-jämn-spridning av bindningsenergier har undersökts för att hitta storlekenpå parkopplingens gap. En 3-punktsformel ∆

(3)C (1

2 [B(N,Z)+B(N −2,Z)−2B(N −1,Z)])förespråkas i denna avhandling, som vi anser vara mera lämplig för att mäta storleken på par-gapet i jämn-jämna kärnor. Storleken på ∆

(3)C kan vara en bra indikator på rumskorrelationen

mellan två nukleoner. En-kvasipartikelenergi för de udda kärnorna, som kan approximerasmed sfärisk symmetri, beräknas för att erhålla udda-jämn spridning (OES) med avseendepå bindningsenergin för alla kända semi-magiska jämn-jämna kärnor. Parkopplingsstyrkanär globalt anpassat med all tillgänglig data på OES vad gäller semi-magiska kärnor medZ ≥ 8. Skillnaden mellan olika täthetsberoende parkopplingar med vår växelverkan minskarmed dessa globalt anpassade parametrar. Skillnaden mellan det teoretiska genomsnittligaparkopplingsgapet och OES blir större när vi kommer närmare tröskeln för atomkärnans

x

stabilitet. I slutet av avhandlingen har en schematisk modell utvecklats som, genom attgöra medelfältet grundare, reducerar de udda-jämnt spridda laddningsradierna. Avhandlin-gen innehåller dessutom resultat som inte ingår i de tre publicerade artiklarna samt någrakompletterande arbeten om ämnet.

List of Publications

This thesis is mainly based on the first three papers.I. Changizi, S. A., Qi, C., and Wyss, R. (2015). Empirical pairing gaps, shell effects,

and di-neutron spatial correlation in neutron-rich nuclei. Nuclear Physics A 940 210. Idea,execution, calculations and first draft of the manuscript are done mostly by the first author.

II. Changizi, S. A. and Qi, C. (2015). Density dependence of the pairing interactionand pairing correlation in unstable nuclei. Phys. Rev. C, 91 024305. Idea, execution, allcalculations and writing are done mostly by the first author.

III. Changizi, S. A. and Qi. C. (2016). Odd–even staggering in neutron drip line nuclei.Nuclear Physics A 951 97. Idea, execution, calculations and first draft of the manuscript aredone mostly by the first author.

IV. Zheying Wu, S. A. Changizi, and Chong Qi (2016) Empirical residual neutron-protoninteraction in odd-odd nuclei. Phys. Rev. C 93 034334. Idea, some parts of the writing andfinal analysis of the paper is done by the second author.

Table of contents

List of figures xv

List of tables xix

1 Introduction 1

2 Empirical pairing gaps and odd-even staggering in nuclear binding energies 52.1 Pairing gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Binding energy odd-even staggering . . . . . . . . . . . . . . . . . . . . . 6

3 The Hartree-Fock-Bogoliubov approach 113.1 Hartree-Fock approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 BCS approach for pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Hartree-Fock-Bogoliubov equation in coordinate space . . . . . . . . . . . 18

3.4.1 Continuum treatment . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Canonical states . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 Choices of the pairing interaction . . . . . . . . . . . . . . . . . . 203.4.4 Two-particle wave function . . . . . . . . . . . . . . . . . . . . . . 213.4.5 Odd-A nuclei and the blocking effect . . . . . . . . . . . . . . . . 223.4.6 Varying external potential . . . . . . . . . . . . . . . . . . . . . . 23

4 Numerical calculations and physical results 254.1 Systematic calculations of pairing gaps for all even-even nuclei . . . . . . . 254.2 Effective forces in the particle-hole and particle-particle channel . . . . . . 32

4.2.1 Proton pairing gap over nuclear landscape . . . . . . . . . . . . . . 364.3 Even-even nuclei in a spherical box . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 General calculation for xenon isotopes . . . . . . . . . . . . . . . . 394.3.2 Weakly bound isotopes of nickel . . . . . . . . . . . . . . . . . . . 41

xiv Table of contents

4.4 Calculations with varying mean-field . . . . . . . . . . . . . . . . . . . . . 474.4.1 Charge radii of nickel isotopes . . . . . . . . . . . . . . . . . . . . 49

4.5 Two-particle wave functions . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Summary 55

References 57

Appendix A The Hartree-Fock approach 63

Appendix B BCS equation 65

Appendix C Hartree-Fock-Bogoliubov equation 67

List of figures

2.1 The absolute value of neutron binding energy odd-even staggering in MeV.Red squares are for even-A nuclei, blue and green squares show odd-A nucleiwith odd number of neutrons and odd number of protons respectively. . . . 8

2.2 The absolute value of proton odd-even staggering in MeV. Red squares arefor even-A nuclei, blue and green squares show odd-A nuclei with oddnumber of neutrons and odd number of protons respectively. . . . . . . . . 9

3.1 The density-dependant average pairing gap ⟨∆⟩ with different box radii and N 203.2 The model of clustering two nucleons . . . . . . . . . . . . . . . . . . . . 21

4.1 Experimental neutron OES-parameters ∆(3)C (N) in MeV for known even-even

nuclei in different isotopic chains. The solid line is fitted to N. . . . . . . . 264.2 Experimental proton OES-parameters ∆

(3)C (Z) in MeV for known even-even

nuclei in isotonic chain. The solid line is fitted to Z. . . . . . . . . . . . . . 264.3 Experimental proton OES-parameters (in MeV) with four different formulae

for all known even-even nuclei. The dashed line is fitted curve as function of Z. 284.4 Proton pairing gaps calculated for ∆(3) (top left), ∆

(3)C (top right), ∆(4) (bottom

left) and ∆(5) (bottom right) for all known even-even nuclei. All ∆z ≥ 1.7has been set to 1.7 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Proton pairing gaps calculated for ∆(3) (top left), ∆(3)C (top right), ∆(4) (bottom

left) and ∆(5) (bottom right) for all known even-even nuclei with the newestmass table from 2016. All ∆z ≥ 2.5 has been set to 2.5 MeV. . . . . . . . . 30

4.6 Difference between the proton pairing gaps derived from different OESformulas and ∆

(3)C in MeV. The right bottom figure shows the proton pairing

gap for Z = N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.7 Difference between the proton pairing gaps ∆(3)−∆

(3)C presented in MeV. . 31

xvi List of figures

4.8 HFBTHO calculations with SLy5 parameterization in the mean-field andmixed pairing interaction for the Fermi level λn (top right panel), two-neutronseparation energy S2n = B(Z,N −2)−B(Z −N) (top left panel), mean neu-tron pairing gap ∆n (bottom left panel) and the deformation β (bottom rightpanel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.9 HFBTHO calculations with SIII parameterization in the mean-field andmixed pairing interaction for the Fermi level λn (top right panel), two-neutronseparation energy S2n = B(Z,N −2)−B(Z −N) (top left panel), mean neu-tron pairing gap ∆n (bottom left panel) and the deformation β (bottom rightpanel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.10 HFBTHO calculations with SKM parameterization in the mean-field andmixed pairing interaction for the Fermi level λn (top right panel), two-neutronseparation energy S2n = B(Z,N −2)−B(Z −N) (top left panel), mean neu-tron pairing gap ∆n (bottom left panel) and the deformation β (bottom rightpanel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.11 HFBTHO calculations with SLy4 parameterization in the mean-field andmixed pairing interaction for the Fermi level λn (top right panel), two-neutronseparation energy S2n = B(Z,N −2)−B(Z −N) (top left panel), mean neu-tron pairing gap ∆n (bottom left panel) and the deformation β (bottom rightpanel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.12 Difference between binding energies obtained from various Skryme forceparameterizations (SLy5, SKM, SLy4, and SIII) and experimental values foreven-even nuclei Z < 52 and N < 84. The line shows the N = Z nuclei. . . 35

4.13 Proton pairing gap ∆z obtained from various Skyrme force parameterizations.More nuclei are calculated in case of SLy4 parameters. . . . . . . . . . . . 36

4.14 HFBTHO calculations with SLy4 parameterization in the mean-field andmixed pairing interaction for the proton pairing gap ∆z . . . . . . . . . . . . 37

4.15 The convergence of binding energy. . . . . . . . . . . . . . . . . . . . . . 384.16 Theoretical and experimental neutron pairing gaps for xenon isotopes. Left

column: neutron pairing gap with jmax = (15/2,17/2) for proton and neu-tron, respectively. Right column: neutron pairing gap with jmax = 25/2 forboth proton and neutron. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.17 Changes in the equivalent energy 3s1/2 and its occupation probability withtuning SLy4 parameters t0, t1 and t2 with different density-dependent pairinginteractions for nickel isotopes . . . . . . . . . . . . . . . . . . . . . . . . 43

List of figures xvii

4.18 Self-consistent local neutron field at left and neutron pairing field at right in84Ni calculated with different values of t0, t1 and t2 of SLy4 parameterizationand volume pairing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.19 Self-consistent neutron density as a function of R at left and neutron pairingdensity at right in 84Ni calculated with different values of t0, t1 and t2 of SLy4parameterization and volume pairing. . . . . . . . . . . . . . . . . . . . . 44

4.20 As same as figure 4.18 with mixed pairing. . . . . . . . . . . . . . . . . . . 454.21 As same as figure 4.19 with mixed pairing. . . . . . . . . . . . . . . . . . 454.22 As same as figure 4.18 with surface pairing. . . . . . . . . . . . . . . . . . 464.23 As same as figure 4.19 with surface pairing. . . . . . . . . . . . . . . . . . 464.24 As same as figure 4.23 with surface pairing (different scale). . . . . . . . . 474.25 Single particle energies of 84Ni in Skyrme HF calculations without pairing

with varying external mean-field at the top. The middle and lower panelscorrespond to Skyrme HFB calculations with the surface and volume pairinginteractions, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.26 Charge radii of nickel (solid line) with respect to different one-quasiparticleblocking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.27 Charge radii of nickel (solid line) with respect to different one-quasiparticleblocking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.28 Charge radii of 83Ni (solid line) and 85Ni (dashed line) with different orbitalsbeing blocked in comparison to that of the ground state of 84Ni (stars line). 51

4.29 Two-particle wave function Ψ(2) for 80Ni top row and 82Ni bottom row.Volume, mixed and surface interaction are shown at the left, center and rightcolumn, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.30 Two-particle wave function Ψ(2) for 88 Ni with volume interaction at the top,mixed in the middle and surface interaction at the bottom. Notice that thescale is different. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.31 Two-particle wave function Ψ(2) for 22O right column and 26O left row.Volume, surface and mixed interaction are shown at the top, middle andbottom row, respectively. Notice that the scale is different. . . . . . . . . . 54

List of tables

4.1 Isotopes with ∆(3)C (N) and ∆

(3)C (Z)≥ 2.0 MeV. The values are in MeV. . . . 27

4.2 The fitting parameters of empirical neutron and proton pairing gaps foreven–even nuclei to bAm with different OES formulae as functions of A, with95% confidence bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 The fitting parameters of empirical neutron and proton pairing gaps foreven–even nuclei to bNm and bZm with different OES formulae as functionsof N and Z (with 95% confidence bounds). . . . . . . . . . . . . . . . . . . 28

4.4 Table over Fermi level λn in [MeV ], energy level of the single-particle stateof 3s1/2 in [MeV ], and the occupational probability for 80 82 88Ni. . . . . . 52

Chapter 1

Introduction

One of the main purposes of nuclear physics is to understand the structure of nuclei andits underlying nuclear forces. Considerable progress has been made in nuclear physics dueto new opportunities for producing a variety of isotopes by using radioactive beams. Theenormous opportunity of the accessibility to a wider range of unstable nuclei makes it bothrelevant and necessary to study the structure of exotic nuclei. From the proton-rich region,where we have observed proton emitters that involve the penetration of protons through theCoulomb barrier [18, 43, 47] and more recently [14, 62, 45]; to the neutron-rich nuclei [77],where the nuclear surface is diffused, gives rise to neutron halos (e.g. 8He, 11Li and 11Beisotopes [54, 70, 57]), and many other exotic phenomena.

Many nuclei close to the dripline may not be reproduced easily in experiments. However,in nuclear astrophysics, there is a need for knowing the physical observables of such exoticnuclei to understand the origin of matter [44, 49]. This understanding is only possible bydeveloping sophisticated models that can predict the nuclear structure of the excising isotopeswith a small systematic error in comparison to experimental data.

To solve the many-body problem in nuclear physics one can start from the basic degreesof freedom in the nucleus. In Refs. [7, 8], there are direct solutions to lattice QuantumChromodynamics (QCD) for very light isotopes of He with a large systematic error. The AbInitio approach, which assumes realistic bare N-N and 3N forces among nucleons, aims tosolve the Schrödinger equation for interacting nucleons numerically in an exact way. Hence,one can assess the theoretical error bars [5]. By including symmetries and setting constraints,one can reduce the size of the Hamiltonian and obtain a solution for light and medium-sizenuclei. The medium-mass nuclei up to 60 has been calculated by the coupled-cluster (CC)method [39] or the self-consistent Green’s function [6]. However, the Ab Initio methods arestill in their early stages and have limitations due to the large matrix dimensions necessary forheavier nuclei. The Shell model with configuration interaction is another tool that can model

2 Introduction

nuclear structures, compute measurable observables, and which describes well many lowenergy nuclear states. These single-particle states have been used extensively to construct arepresentation that would describe many-particle states. In the Shell model one assumes aninert core of independent particles and a few correlated valence nucleons. The Shell modelis also dependant on the model space and has a difficulty with huge matrices, even thoughone can tackle the size of exponentially growing matrices in the Shell model by differentapproaches, for instance, using Monte Carlo techniques [48, 58]. The pairing interaction,which is extremely important for exotic nuclei, is not included profoundly in the Shell modeland it lacks by design.

Mean-field theory can profit from distinctive characteristics of the nuclear force. It iswell established that the nuclear force is very short range. As a consequence, the nucleonsmove along orbits with large mean-free paths. One has thus arrived at the very first simplemicroscopic nuclear treatment, namely that the dynamics of nuclei may be understood withinthe framework of a Fermi gas model. The Shell model warrants the picture of independentparticles moves in an averaged potential. A group of independent particles is confined ina quantum potential, i.e., a so-called mean-field. To construct the mean-field one can takeadvantage of an effective interaction in a self-consistent manner (SCMF) [9]. In SCMFtheory one looks for this averaged potential and tries to explain it by simple nucleon-nucleoninteraction. In another word, it produces the Shell model potential microscopically. At themost fundamental level, the Ab Initio theories [51] with their fully microscopic methodscannot reach the heavier nuclei due to the scale of computation. On the other hand, theHartree-Fock approach can produce the nuclear bulk properties over the whole nuclear chart.For instance in Ref. [69] they produce the nuclear masses with an RMS error of about700 KeV.

The ground state of open-shell nuclei calculated by Hartree-Fock is mostly deformed.However, by including pairing into account, this deformation will be reduced, and one canreproduce the experimental results better. Pairing effect is especially crucial for driplinenuclei since the contribution from mean-field is small. Thus, the pairing gap, even with itssmall value, can play a significant role in nuclear deformation and binding energy, which canmake the nuclei bound [40]. One of the most popular models that generalize the mean-fieldapproach with pairing field is the Hartree-Fock-Bogoliubov approach.

The closeness of bound states to the continuum in loosely bound and unbound nucleimakes it important to have a right model, which can discretize the continuum properly [59].Continuum plays an essential role for dripline nuclei [27, 11, 80]. Neutron-rich nuclei areweakly bound and, hence, can be extended spatially. With the right model that can simulate

3

this property one can reproduce the right particle continuum. Continuum has a large effecton weakly bound nuclei such as neutron-rich Li isotopes [76, 33, 53].

To solve the HFB-equation one can use zero-range or finite-range forces, for example,the so-called Gogny forces. Solving the HFB coupled-equations is much easier with theformer force. The Gogny force with its finite range properties makes the HFB-equation intocoupled integro-differential equations that can to be solved in a harmonic oscillator basis[22]. Harmonic oscillator basis has a major deficiency when discretizing the continuum. Asystematic calculation for deformed nuclei at dripline with an improved harmonic oscillatorin respect to the continuum is performed in Ref. [75].

In this thesis, the investigation on pairing effect is performed in coordinate space in spher-ical symmetry from bound to dripline nuclei. This geometry with its sole and straightforwarddependency on the nuclear radius will help us to understand the physics more clearly. Inthis work, we have focused extensively on the study of density-dependent of the pairinginteraction. A discussion about this has started from Ref. [29].

The effective interaction in the particle-hole channel to obtain the mean-field is theSkyrme interactions in this work, in which the time-odd part of the energy functional isexcluded. The even term, which contains the time-even density, describes the stationaryeven-even nuclei satisfactory. Properties of the effective Skyrme force is well described inRef. [9].

There is an excellent interest in analyzing the theoretical uncertainty of nuclear propertiesat dripline [46, 2, 1]. The extreme extrapolation of different models will cause significantdifferences in the obtained results. Interactions that may describe the structure of stablenuclei well may not be applicable for unstable nuclei, and give a different picture of reality.Hence, one needs to have a good overview of the theoretical error. Systematic calculationswith various models can shed light on the strengths of each approach [30]. One may needto look more into why certain kinds of interaction were introduced in the first place andmodify them accordingly. Hence, we have performed a systematic comparison betweendifferent Skyrme parametrizations and pairing interactions to analyze the consequences ofeach interaction.

By analyzing the density-dependent pairing interaction in the Hartree-Fock-Bogoliubovapproach, one can see the feedback of this effect in both p-h and p-p channels. For instance,having a pairing correlation in the model, a pair of particles can scatter to the particlecontinuum and contribute to the pairing energy of the ground state. In Ref. [28] hasdemonstrated that the size of the neutron halo differs drastically by the density dependence ofthe pairing interaction. One may state that the physics of weakly bound nuclei may vary fromstable ones and form a new many-body system (low-density neutron matter). The extreme

4 Introduction

extrapolation into the dripline region makes it difficult to draw any convincing conclusions.Thus, extensive uncertainty analyses are unavoidable.

Pairing gap is not a direct observable. On the other hand, one can probe the pairinginteraction through the odd-even staggering of binding energies. There are several formulae,which have been advocated to give a fingerprint to the pairing gap. Hence, we have discussedthis topic comprehensively in this work.

Overview

The first part of this thesis is mainly focused on empirical pairing gaps, the odd-evenstaggering of binding energies, and various formulae that exhibit this phenomenon. Adetailed description of the theory of Hartree-Fock-Bogoliubov is given in Chapter III. TheHartree-Fock and BCS model are also briefly explained for the sake of completeness. Thenumerical and physical results are presented in Chapter IV. In this chapter, a number ofnuclear properties of some spherical nuclei, for instance, nickel and xenon isotopes havebeen investigated. Two-particle wave functions of 22O, 26O and other isotopes are plotted aswell. Finally, a summary is given in the last chapter.

Chapter 2

Empirical pairing gaps and odd-evenstaggering in nuclear binding energies

One outstanding feature in nuclear spectroscopy is the abrupt changes in binding energy asone goes from a nucleus with an even number of neutrons (or protons) to its neighbor withan odd number of equivalent nucleons. This feature is known as odd-even mass staggering(OES). Here we will discuss the OES to understand its relation to pairing gap. The questionswe want to answer is how to evaluate the pairing gap qualitatively and quantitatively from itsbinding energies?

2.1 Pairing gap

Bohr and Mottelson applied the BCS concept in nuclear physics [17] after Bardeen, Cooper,and Schrieffer explained a theory of superconductivity in metal, that is the existence of lowerstates than normal ones, which makes the metal super-conductive at lower temperatures.Electrons near the Fermi surface can overcome the repulsiveness of coulomb force and forma pair. Bohr and Mottelson present the idea that such a mechanism can also exist in atomicnuclei. There is overwhelming evidence pointing to the existence of strong pairing correlationin atomic nuclei. For instance, a small value of the nuclear moment of inertia in nuclei canbe explained by this effect to some extent. In all even-even nuclei, J = 0+ (two particleswith identical quantum numbers but opposite spin projection) is the lowest energy state. Thelow-lying level densities of even-even nuclei are much smaller than those of the odd ones.Hence, there should be a correlation that lowers the ground states.

In the BCS picture, two nucleons with the same quantum numbers except for the pro-jection of their spin on the same axis interact attractively, which gives rise to lower states.

6 Empirical pairing gaps and odd-even staggering in nuclear binding energies

There will be a gap between the normal state and the lower one due to this pairing of the twoparticles.

⟨ j1 j2J|Vpair| j3 j4J⟩ (2.1)

Two-particles interact not only in their orbit, but also they can scatter to other orbits. Bothdiagonal and non-diagonal matrix elements 2.1 will play a roll in the whole picture, which itcalls pairing collectivity. If pairing interaction is only important for pairs in the same state,then all the matrix elements of 2.1 except the diagonal one will be zero.

There are several questions one may ask. For instance, is pairing interaction only activewhen the pair overlaps spatially? What is the form of such interaction? Is it more efficientaround the surface of the nuclei and those particles close to the Fermi level or the pairinginteraction is involved in the whole volume of nuclei? Does pairing interaction only exist inbound nuclei or does extend beyond dripline? To get closer to the answers of these questions,we need to find a way to measure the pairing interaction quantitatively.

2.2 Binding energy odd-even staggering

What is the origin of odd-even staggering of binding energies in nuclear physics and what isthe physical mechanism behind it? The discussion has been comprehensive and pointed outthis phenomenon extensively. Primarily, it is attributed to pairing effect. Binding energiesgive us some essential information about the atomic nuclear structure, the net impact ofnuclear forces. However, it is very challenging to obtain a more detailed view of the nucleichart from these energies. It is desirable to maximise the information we can get from suchfundamental data. The very first question is how to see the effect of pairing effectivelyby only investigating systematic of binding energies, before going into the more in-depthmicroscopic study.

Various formulae have been used to exhibit the odd-even mass staggering by a parameter∆ in isotopes and isotones. For instance the very well-known formula Eq. (2.2) [16] forisotopes, where B is the binding energy and Sn is the one-neutron separation energy.

∆(3)(N) =−1

2[B(N −1,Z)+B(N +1,Z)−2B(N,Z)]

=−12[Sn(N +1,Z)−Sn(N,Z)] (2.2)

2.2 Binding energy odd-even staggering 7

There are several other OES-formulae such as the four and five-points formulae Eqs.2.3 and 2.4 (see Refs. [16, 71] and [52, 55, 32]), that are averaging the result of the former3-points over neighbouring odd-A nuclei.

∆(4)(N) =

14[−B(N +1,Z)+3B(N,Z)

−3B(N −1,Z)+B(N −2,Z)]

=12[∆(3)(N)+∆

(3)C (N)]. (2.3)

∆(5)(N) =

18[B(N +2,Z)−4B(N +1,Z)

+6B(N,Z)−4B(N −1,Z)+B(N −2,Z)]

=14[∆

(3)C (N +2)+2∆

(3)(N)+∆(3)C (N)]. (2.4)

They are in general a Taylor expansion of the nuclear mass in nucleon number differences[52].For the neutron OES, they keep the Z constant and varies the N around neighboring nuclei.The proton OES can be expressed in the same way, by keeping N constant and varying Z.

We use a simple 3-point formula 2.5 to obtain the mass parameter ∆ for even-even nuclei.Figure 2.1 shows the OES values for all nuclei from the lightest nuclei to the heaviest onein sequence with both 3-points formulae mentioned above. The figures are plotted with thenuclear binding energies extracted from Refs.[4, 79].

∆(3)C (N) =

12[Sn(N,Z)−Sn(N −1,Z)]

=12[B(N,Z)+B(N −2,Z)−2B(N −1,Z)] (2.5)

One may notice that both produce same overall results. However, ∆(3)C gathers the even-

A nuclei and odd-A nuclei more clearly. They are more concentrated in an area than bedispersed. This grouping may arise from the fact that with ∆

(3)C formula we can see, which

smooth part of the mean field is still contributing to 3-points OES formula uniformly, andthe quick varying component of the mean-field has been reduced efficiently. Hence, only bycutting the uniform part, we can get closer to the real value of pairing gap for these typesof nuclei. Fig. 2.2 presents the proton pairing gap for both 3-points formulae for all knownnuclei.

8 Empirical pairing gaps and odd-even staggering in nuclear binding energies

0 200 400 6000

0.5

1

1.5

2

∆(3) (N)

0 200 400 6000

0.5

1

1.5

2

∆(3)

C(N

)Fig. 2.1 The absolute value of neutron binding energy odd-even staggering in MeV. Redsquares are for even-A nuclei, blue and green squares show odd-A nuclei with odd numberof neutrons and odd number of protons respectively.

One should bear in mind that the pairing gap is not a direct observable. However, differentOES formulae give a reasonably good indication of its value. Many favorable and unfavorablesituations make the choice of OES formula for every case cumbersome. For instance, the5-point formula takes away the smooth varying part of the mean-field very efficiently [32].However, the availability of as many as five nuclei results in less experimental OES (such as570 measured ∆

(3)C (N) with 516 for ∆(5)(N)). Especially close to dripline, that obtaining new

binding energies are challenging, and pairing gap plays an important role in understandingnew physics. Furthermore, one cannot be sure if pairing contribution has also been averagedout. The ∆

(3)C formula is not adequate to give the pairing gap for odd-nuclei since around

Fermi level the splitting of the single-particle spectrum of the odd nucleon cannot be reduced[71]. Additionally, as long as nuclei with N and N − 2 have similar deformation, ∆

(3)C for

even-even nuclei works fine. In this sense, we have chosen to compare the pairing gap with∆(3)C (N).

2.2 Binding energy odd-even staggering 9

0 200 400 6000

0.5

1

1.5

2

∆(3) (Z)

0 200 400 6000

0.5

1

1.5

2

∆(3)

C(Z)

Fig. 2.2 The absolute value of proton odd-even staggering in MeV. Red squares are foreven-A nuclei, blue and green squares show odd-A nuclei with odd number of neutrons andodd number of protons respectively.

Chapter 3

The Hartree-Fock-Bogoliubov approach

It is challenging to deal with the many-body problem in nuclear physics. Already at the mostbasic step in this task, namely the determination of the nuclear force acting among nucleons,there are significant challenges. The relationship between nucleon-nucleon interaction andthe single-particle picture is still not well understood. In fact, one can only, with great effort,solve the Quantum Chromodynamics (QCD) equations corresponding to the most simpleof all systems in nuclear physics, namely the free nucleon itself, and this was performednumerically only. It is therefore not surprising that in the context of many-nucleon physics oneintroduces ad-hoc interactions among nucleons, which are supposed to efficiently reproducethe effects induced by the many degrees of freedom determining the corresponding QCDinteraction. However, even assuming that this effective interaction is somehow knownprecisely, the description of the relevant nuclear spectrum becomes a formidable task. Thisindicates that one has to perform drastic approximations, which are consistent with thephysics of the problem to proceed further in the analysis and evaluation of nuclear properties.To achieve this one can profit from particular characteristics of the nuclear force.

The single-particle states have been used extensively to construct the representation thatwould describe many-particle states. Such descriptions have been obtained in a numberof ways, but the most significant in relation to this thesis is the HF procedure describedin the following. We have also seen that very simple pairing interaction leads to the BCSapproximation. For stable bound nuclei the BCS approximation provides results that agreereasonably well with experimental data. But far from the nuclear line of stability, near thedriplines, one needs to enlarge the single-particle representation including states that woulddescribe the influence of the pairing field upon unstable states, and for that, the BCS modelis not adequate. This is what the HFB representation provides; a proper description thatgeneralizes the single-particle states with pairing interaction.

12 The Hartree-Fock-Bogoliubov approach

To understand the HFB theory, I introduce the most general Hartree-Fock method brieflyfollowing BCS theory. In this chapter, I add the Hartree-Fock method and the BCS approachfor reasons of consistency. Furthermore, I present Hartree-Fock-Bogoliubov in general andfocus within a space-coordinate system in particular. This thesis tries to shed light on thephysical meaning of canonical states and shows the two-particle wave-function on this basis.I present a blocking-method in a spherical box having a space-coordinate system followedby a toy-model that introduces an external potential in the particle-hole channel.

3.1 Hartree-Fock approach

The most primitive starting point in solving a many-body state is to consider a system ofnon-interacting particles, where the following Hamiltonian H describes exactly this.

H =A

∑i=1

hi (3.1)

where hi is a Hamiltonian for each individual particle i in the system of A-numbered particles.Hartree-Fock is a variational method that gives an approximation of the lowest energy interms of quasi-gas. There is a clear picture of the particles and holes. The variational methodproduces a set of equations, which is solved iteratively. In a computational sense, the HFmethod starts with an initial guess, provides a solution, and then takes the answer repeatedlyas an input until it converges into the final result or else fails to converge. The trial wavefunction in the variational method can simply be the direct product of all single-particle statesin a complete orthogonal basis ν as follow:

Φ(1, . . . ,A) = φν1(1)φν2(2) . . .φνA(A) (3.2)

Which is called a Hartree approximation. One can apply the variational method with this trialwave function and obtain an approximated lowest energy. The two independent particles arenot correlated. However, due to the Pauli principle, one needs to make sure that the total wavefunction for A-nucleon system is anti-symmetrized. This anti-symmetrization introduces astrong correlation between two particles. By using the Slater determinant, one ensures thatthe permutation of the two nuclei results in an exchange of the total wave function sign.

Φ(1 . . .A) =1√A !

∣∣∣∣∣∣∣φν1(1) φν2(1) . . . φνA(1)

......

...φν1(A) φν2(A) . . . φνA(A)

∣∣∣∣∣∣∣ (3.3)

3.1 Hartree-Fock approach 13

Where, Φ(1 . . .A) is the A-nucleon wave function, φν j(i) is the wave function of the i-thnucleon in j-state. Depending on the basis, j can be different sets of quantum numbers. Thefactor 1√

A !normalizes the wave function Φ.

One can introduce the many-body state in the occupation-number representation:

|Φ⟩= |n1,n2, . . . ,nν , . . .⟩ (3.4)

where for fermions nν is 0 or 1. In Hartree-Fock approach, a set of Slater determinantsΦ(1, . . . ,A) is chosen to be the A-body wave function:

|ΦHF⟩=A

∏i=1

a+νi|0⟩ (3.5)

Operators a+ and a create/annihilate a particle in a given single-particle state in the basis ofν . They correspond to the single-particle wave function φνk(i), which are the eigenfunctionsof the single particle Hamiltonian hi. Hence, the energy of the system within this set is to beminimized.

In this representation, the one-body operator e.g. the kinetic energy that changes the statei and keeps the other states unchanged is constructed as follow

t = ∑i j

ti ja+i a j (3.6)

and the two-body operator that changes the two states i and j simultaneously as:

V =12 ∑

i jklvi jkl a+i a+j al ak (3.7)

One can define the Hamiltonian for a many-body system in occupation-number representationas follows:

H = ∑k1k2

tk1k2 a+k1ak2 +

14 ∑

k1k2k3k4

vk1k2,k3k4 a+k1a+k2

ak4 ak3 (3.8)

where indices ki cover all degrees of freedom of all available single particle states. v is theantisymmetric two-body interaction matrix element. To determine the ground state energyand the wave function ΦHF , we can vary the wave function |δΦ⟩= ε a+m ai|Φ⟩ with m > Aand i ≤ A. We assume the set of wave functions is orthonormal ⟨Φ|Φ⟩= 1. The variational


equation is written:

⟨δΦ|H|Φ⟩= ε∗⟨Φ|a+i amH|Φ⟩= 0 (3.9)

Inserting 3.8 into 3.9 one obtain [66]

hmi = tmi +A

∑j=1

(vm ji j − vm j ji) = 0 (3.10)

hkk′ = tkk′ +A

∑i=1

vkik′i = ekδkk′ (3.11)

The equation 3.10 requires that the states between occupied and unoccupied vanishes.This is the essence of Hartree-Fock equations, where h for when particles and holes aremixed, vanishes (diagonal h). By this requirement, we get the eigenvalue problem in Eq.3.11. The first term t is kinetic energy and the second term is the self-consistent field,which is a direct result of the Pauli principle. We can apply the HF method and use aphenomenological interaction with density dependence like Skyrme forces and obtain variousnuclear observables over the entire nuclei chart.

3.2 BCS approach for pairing

Including the pairing effect will give rise to even lower states: the quasi-bound states. Statesthat are not purely occupied or empty. There is a combination of states that a pair willbe available in. Different pairs will be found in different mixtures of single-particle states.Minimizing the energy of this sea of pairs gives the lowest energy. However, in HFB themixing of single-particle states and making the quasi-particle states is more evolved, and it isnot only for conjugate pairs.

BCS is an approximation to the ground state of the nuclei, derived from a variationalmethod. The trial wave function |BCS⟩ to be varied is the following:

|BCS⟩= ∏k>0

(uk + vka+k a+k )|−⟩ (3.12)

∝ |−⟩+ ∑k>0

vk

uka+k a+k |−⟩+ 1

2 ∑kk′>0

vkvk′

ukuk′a+k a+k a+k′a

+k′|−⟩+ . . . (3.13)

3.3 The Bogoliubov transformation 15

The state presented in 3.12 is a product of pairs of single-particle levels(k, k) with theprobability of v2

k and u2k , in which the paired level (k, k) is occupied and empty, respectively.

k is the conjugate state of k. The product is rewritten in 3.13 and the expansion shows thatBCS state is a superposition of different number of pairs and not related exactly to the numberof particles. Using this trial wave function 3.12 and varying one of the parameters vk oruk (vk and uk are dependent through normalization requirement |uk|2 + |vk|2 = 1|), one canminimize the auxiliary Hamiltonian H

′= H −λN, where H is 3.8 with trial wave function

|BCS⟩ and λ is the Lagrange multiplier to conserve the average particle number. Hence, onecan obtain BCS equations as follow [66]:

v2k =

12

1− ek√e2

k +∆2

u2k =

12

1+ek√

e2k +∆2

(3.14)

where ∆ is defines as

∆k =− ∑k′>0

vkkk′k′uk′vk′ (3.15)

3.3 The Bogoliubov transformation

In the following section I try to show that in contrast to other methods, HFB will includepairing interaction profoundly at the microscopic level, and is already incorporated in themean-field. Bogoliubov suggested in Ref. [15] the use of especial transformation so-calledBogoliubov transformation, thus, the pairing effect close to the Fermi level is considerable,and it is not only a small perturbation. For dripline nuclei, the pairing gap and Fermi level areboth at comparable sizes and equally important. Bogoliubov transformation 3.16 enables us totransfer the BCS basis, i.e. pairwise interacting particles, into non-interacting quasi-particles,which are the superposition of both particles and holes; an approximation to the exact solutionof the many-body problem defined by Migdal and Landau. Bogoliubov transformation 3.16is a linear combination of particle creation a+ and annihilation operators a in the vacuum ofquasi-particle operators |−⟩. A simple relationship between quasi-particles operators andbare particle/hole operators is the following:

α+k = uka+k − vkak (3.16)

α+k = uka+k + vkak


One can recover particle-hole states as in HF approach from this transformation, by settinguk = 1,vk = 0 for k above the Fermi level and uk = 0,vk = 1 for k below the Fermi level.Hence, the quasi-particle operators become the bare particle/hole operators.

To unify the Hartree-Fock states and BCS states into one generalized state with gener-alized single-particle wave functions, the so-called HFB wave functions, matrix W defineswith the above transformation as following [66]:(

β

β+

)=

(U+ V+

V T UT

)(c

c+

)W =

(U V ∗

V U∗

)(3.17)

Operators β+ and β create and annihilate quasi-particles with some general particle operatorsc+k , ck. U,V are the two components of quasi-particle wave function, which are chosen in away that β+ and β meet the anti-commutation relation. For instance if βk for all k over thewhole configuration space applies onto a |BCS⟩ state, it will vanish (βk|BCS⟩= 0), whichmeans that |BCS⟩ is considered as vacuum in HFB theory and contains no quasi-particles.

One can obtain the approximated lowest energy of the ground state |Φ0⟩ of Hamiltonian3.8 by using a Bogoliubov transformation 3.17 and variational method. In HFB the particlesand holes are mixed, and for that reason the particle number is not conserved. Hence, theparticle number operator N = ∑i c+i ci sets as a constraint with a Lagrangian multiplier λ asH −λ N. Transforming Hamiltonian 3.8 according Bogoliubov formalism 3.17 one yields[66]:

H −λ N = H0 + H11 + H20 + H40 + H31 + H22︸︷︷︸Neglected

(3.18)

The indices of Hi j are the number of creation/annihilation operators in the respective term,for instance H0 is the value of the vacuum energy with no quasi-particle operators, and H11

is one quasi particle-quasi hole excitation energy. The other terms with four quasi-particleoperators may be ignored. Varying this as introduced in [66] to uniquely describe |Φ0⟩, oneobtains the requirement H20 = 0. One can conclude that HFB in general is diagonalizing thefollowing super-matrix in a quasi-particle basis:

(H11 H20

−H∗20 −H∗

11

)(3.19)

3.3 The Bogoliubov transformation 17

−→ In the space of particle operator cl , c+k : H =

(h−λ ∆

−∆∗ −(h−λ )∗

)(3.20)

Where, H11 includes the sum of terms of operators β+k βk and H20 includes the sum of terms

of β+k β

+k +βkβk.

With density-depended forces one can obtain a reasonably good spectrum of singleparticle states, which can be used as a building block for HFB. In this analogy, we canobtain the energies as functional of density matrix ρ and pairing tensor κ . The definition ofmean-field h and pairing field ∆ are as following [66]:

h = t +Γ, Γkl = ∑i j

vk jliρi j, ∆kl =12 ∑

i jvkli jκi j (3.21)

where ρ and κ can define the wave function |Φ0⟩ uniquely by the two components U and V :

ρi j = ⟨Φ0|c+j ci|Φ0⟩= (V ∗V T )i j (3.22)

κi j = ⟨Φ0|c jci|Φ0⟩= (V ∗UT )i j (3.23)

and the generalized density matrix R:

R =

(ρ κ

−κ∗ 1−ρ∗

)(3.24)

One can rewrite the Hamiltonian in this notation and use the variational method on⟨Φ|H |Φ⟩ with respect to ρ and κ [66], Hartree-Fock-Bogoliubov is obtained once again:

[H ,R ] = 0 (3.25)

Which means one can diagonalize H and R at the same time in coefficients Uk,Vk:(h−λ ∆

−∆∗ −(h−λ )∗

)(Uk

Vk

)=

(Uk

Vk

)·Ek (3.26)

Here, we will have independent quasi-particles moving in a potential which contains bothmean-field generated from all single quasi-particles plus the pairing interaction between suchparticles.


3.4 Hartree-Fock-Bogoliubov equation in coordinate space

In coordinate space, once we discretize the quasi-particle space in a spherical box withradius R and using spherical symmetry, we can solve a set of HFB differential equationswith Skyrme force in one dimension. In coordinate space, the correct treatment of continuummakes it possible to find out more about the asymptotic tail of quasi-particle wave functions.The solution in the coordinate space for spherical symmetry is completely formulated in Ref[24]. In this section, we explain some essential concepts for this work.

3.4.1 Continuum treatment

As mentioned in the introduction harmonic oscillator basis is not adequate for nuclei nearthe drip. They lack a good description of coupling to the continuum. Wave functions inHarmonic oscillator basis are local. Hence, they cannot describe the non-locality and thelong tail of neutron wave function in the proximity of the particle continuum.

In coordinate space, the wave function is represented by its spatial coordinate, whichfacilitates the coupling to the particle continuum. With assuming spherical symmetry thewave function for neutron or proton represents only by its radial coordinate r and the spinprojection σ =±1

2 as following:

a+rσ = ∑k

ψ∗k (rσ)a+k and inverse a+k = ∑

σ

∫d3rψk(rσ)a+rσ (3.27)

where ψk is the wave function of the kth single-particle state and form a complete orthonormalbasis.The requirement for these wave functions is that they vanish at large distances. Theenergy spectrum E is discrete for En <−λ and continuous for E >−λ . In a spherical box,one can discretized the particle continuum by requiring that wave functions vanish at the boxradius Rbox. Large Rbox can simulate the real continuum very well [30], and the summationover discrete E can replace the integral for the continuous E very well (see Eqs. 3.28 and3.29).

ρ(rσ ,r′σ ′) = ∑0<En<−λ

V (En,rσ)V ∗(En,r′σ ′)+∫

∞

−λ

dn(E)V (En,rσ)V ∗(En,r′σ ′)(3.28)

ρ(rσ ,r′σ ′) =− ∑0<En<−λ

V (En,rσ)U∗(En,r′σ ′)−∫

∞

−λ

dn(E)V (En,rσ)U∗(En,r′σ ′)(3.29)

The integral over the energies can be replaced by a sum, and an energy cut-off will beintroduced for practical reason.

3.4 Hartree-Fock-Bogoliubov equation in coordinate space 19

3.4.2 Canonical states

We have used canonical states in the calculation of two-particle wave functions to describe aclustering feature. Here we try to look closer into their physical meaning and how to obtainthem. The Bogoliubov states are separable BCS-like states within a canonical transformation.The canonical states of the HFB states are virtual creations of particles and holes. These statesare localized in space in the sense that they do not have their asymptotic tail as quasi-particlestates. They are the eigenstates of a density matrix, which is entirely defined by the lowercomponent of HFB solution V.

Hence, diagonalizing the general density matrix in HFB yield the canonical states. [30]∫d3r′∑

σ ′ρ(rσ ,r′σ ′)ϕα(r′σ ′) =V 2

α ϕα(rσ) (3.30)

where r, σ represent the coordinate vector and spin projection ±12 . The index α is the

set of quantum numbers n, l, j for only one type of particle in our work. ϕα(rσ) is thecanonical wave function for the set of quantum numbers α . One can represent the completeand orthogonal sets of wave function ϕk in some arbitrary complete and orthogonal set ofsingle-particle wave functions φk with corresponding operators ck,c+k as follow:

ϕk = ∑l

Dlkφl → a+k = ∑l

Dlkc+l (3.31)

This is a transformation among single-particle wave functions themselves and the unitarytransformation D will not change the anti-commutation relations for fermions. Densitymatrix ρ will be diagonal in the basis of a+k ,ak. This basis define the quasi-particle vacuumcompletely as well.

Natural orbits represent a basis, which the density matrix is diagonal in it. This meansthat they are equivalent to canonical states [30]. In other words, a canonical basis representsthe natural states of the independent quasi-particle many-body states. To produce a BCS-liketype of wave function from HFB, one can use the canonical wave function obtained from thediagonalization of one-body density matrix ρ [3]. To get the canonical basis, one needs tosolve a complete HFB equation first and then construct the diagonalized density matrix.

There should not be a major difference between the properties of canonical basis andordinary transferred quasi-particles basis of the system in the particle-hole channel[25]. Ifthere is a significant difference between these two, there may be some interesting physicalfeature in the Hamiltonian that has lifted the degeneracy.


R0

R

h 0i

h i

N0 N0

(a) Same nuclei with different R

R R

h 0i

h i

N0 N

(b) Different nuclei with same R

Fig. 3.1 The density-dependant average pairing gap ⟨∆⟩ with different box radii and N

3.4.3 Choices of the pairing interaction

The pairing interaction used in particle-particle channel is a zero-range one, which has thefollowing formula [21]:

Vpair(r,r′) =V0

(1−η

ρ(r)ρ0

)δ (r− r′) (3.32)

where V0 is the pairing strength and can be obtained by fitting to the OES parameter for adesirable isotope. ρ(r) is the iso-scalar nucleonic density and ρ0 = 0.16 f m−3

One can vary η to change the density dependence of the pairing interaction. By settingη = 1 (η = 0), we will have pure (none) density-dependent pairing interaction, also calledsurface interaction (volume interaction), in the particle-particle channel. The average of theseinteractions is called mixed pairing with η = 0.5.

Since the HFB equations are solved in a discretized box with radii R, one should realizethe effect of the box radii R on the average pairing gap ⟨∆⟩. A schematic picture is given inFig 3.1 to show this. By increasing R, the average gap ⟨∆⟩, which is dependent on the density,will decrease concerning the boundary condition of the two-component single-quasiparticleHFB wave function φHFB(E,Rbox) = 0 [78]. There will be the same effect with increasingthe number of particle in a box with a fixed R. The continuum wave functions will be pushedinto smaller region closed to the surface of the nucleus and make the density-dependentpairing gap ⟨∆⟩ larger.

In this work we use two different theoretical pairing gaps to compare with the OESparameters, the pairing gap for the lowest canonical state ∆LCS [50] and the mean gap ∆mean


R

R

θ12

2

1

Fig. 3.2 The model of clustering two nucleons

[24].

∆LCS = ⟨ϕLCS|∆|ϕLCS⟩ (3.33)

∆mean =− 1A

Tr(∆ρ) (3.34)

where in equation 3.33, ∆LCS is the diagonal matrix elements of particle-particle mean-field,and ∆mean is the average pairing gap of a given A-nucleus.

3.4.4 Two-particle wave function

By computing the two-particle wave function, the spatial structure of the valence neutronscan help us to analyze the clustering feature of such particles. A major part of a microscopicdescription of clustering lies on the coupling of single-particle wave functions with thecontinuum. Since the continuum part of a single particle wave function is formulatedproperly in HFB within coordinate-space, one can obtain a correct description for suchclusters. A simple cluster of two nucleons is considered here, which the model of the clusterdescription is shown in figure 3.2. Two neutrons (1) and (2) are at the same distance R fromthe core, and θ12 is the angle between the two neutrons.

The formula of this simple clustering may be written as stated in Ref. [64, 63]:

Ψ(2)(r1,r2,θ12) =

14π

∑pq

√2 jp +1

2δlplqδ jp jqXpqφp(r1)φp(r2)Plp(cosθ12) (3.35)

Xpq = upvq (3.36)

where φ is the canonical single-particle wave function obtained from the results of HFBcalculations. Plp is the Legendre polynomial, and Xpq is the expansion coefficient, whichcorresponds to the product of variational parameters upvq within the HFB approach. The


summation over p and q includes the different configurations of Ψ(2). In this work, I obtainΨ(2) as a function of θ12 and radius r1 = r2, which is chosen to be 9 fm. Hence, one canobtain the two-neutron wave function |Ψ2ν(r,r,θ)| for some exciting nuclei to analyze theeffect of different pairing interactions.

3.4.5 Odd-A nuclei and the blocking effect

To obtain OES and the magnitude of pairing for the even semi-magic nuclei we need tocalculate the binding energy of the neighboring odd nuclei. Hence, in this section, we try toexplain a method to obtain a wave function for odd-nuclei in the HFB approach.

We use a phenomenological blocking scheme in coordinate-space by assuming equal-filling. As mentioned before, the HFB states has the same structure of BCS states within thecanonical states. The ground state of odd nuclei is constructed in this basis by a one-quasi-particle excitation as following:

α+k1|BCS⟩= a+k1 ∏

k =k1

(uk + vka+k a+k )|−⟩ (3.37)

α+k1|BCS⟩= a+k1

∏k =k1

(uk + vka+k a+k )|−⟩

Or directly by Bogolibov transformation:

βδ = ∑k

U∗kδ

ck +V ∗kδ

c+k (3.38)

β+δ= ∑

kVkδ ck +Vkδ c+k (3.39)

From the definitions of the density matrix ρ and pairing tensor κ 3.23 one gets [61, 12]:

ρδ

kk′ = (V ∗V T )kk′ +(UkδU∗k′δ −V ∗

kδVk′δ ) (3.40)

κδ

kk′ = (V ∗UT )kk′ +(UkδV ∗k′δ −V ∗

kδUk′δ ) (3.41)

Where the index δ denotes the state that is blocked in every iteration. In this work, wehave omitted the time-odd field, and to preserve time-reversal symmetry by equal-fillingapproximation, a half of nucleon will place in a given orbital δ and the other half will occupythe time-reversed state. By neglecting time-odd channel, equal-filling approximation andexact blocking will be equivalent for one-quasi particle excitation [73]. Depending on whichstate δ is to be blocked, one can obtain HFB energies and the occupation probabilities. In this


simple blocking scheme, we consider several blocking possibilities of δ to find out whichone gives the minimum energy.

3.4.6 Varying external potential

In this part, we are interested in exploring the loose binding effect within a simple toy modelby adding a slightly repulsive external potential, V (r), as a perturbation on top of the standardSkyrme functional and solve the HFB equation self-consistently. By changing the strengthof the potential, one can then show explicitly how the single-particle wave functions andpairing gaps respond when the total mean-field gets shallower.

We consider an external potential of volume type as follows:

V 1(r) = V 10 f (r) (3.42)

where f (r) = 1/(1+ e(r−R)/a) and V 10 is the coupling strength. R is the nuclear radius

as determined self-consistently from the HFB calculation, and the diffuseness is taken asa = 0.6 fm. As mentioned above, we will consider only the case of V (r) being added as aperturbation on top of the HF field Γ. There are other cases, e.g., with V (r)>> Γ(r), which,however, may be of less interest in nuclear physics than in cold atom physics. It should alsobe pointed out that, with the repulsive perturbation potential thus chosen, it will effectivelydecrease the depth of the total mean-field and will not generate any artificial barrier at thesurface. If the potential of volume type is chosen, it will also affect the single-particleenergies of the deep-lying states. However, the single-particle wave functions for those stateswill not be noticeably influenced.

Chapter 4

Numerical calculations and physicalresults

4.1 Systematic calculations of pairing gaps for all even-evennuclei

I start this chapter by presenting two figures over the empirical neutron pairing gaps ∆(3)C (N)

and empirical proton pairing gaps ∆(3)C (Z) for all even-even nuclei obtained from Eq. 2.5.

The data is taken from the 2012-mass table [4] and the recent results for calcium [79] (seeFigs. 4.1 and 4.2). They are plotted in isotopic and isotonic chains, respectively. There areseveral major dips in Fig. 4.1, which some of them occurs right after a filled major shell atN = 30,52,84 and other two dips at N = 16 and 126. The smallest value is reached by 208Pbfollowed by 54Ca. The other low value is for N = 58 at 98Zr.

Proton pairing gap ∆(3)C (Z) is shown in Fig. 4.2, which have also some major dips, for

instance at Z = 16,30,40,52 and 82. Generally, the ∆(3)C increases as mid-shell fills with

neutrons and protons. There are few nuclei with ∆(3)C (N) and ∆

(3)C (Z) larger than 2 MeV. In

the table 4.1, the proton and neutron pairing gaps ∆(3)C larger than 2 MeV are included. Some

of the isotopes have both large neutron pairing gaps and proton pairing gap for instance 8Be,10Be, 10C and 12C.

In Figs. 4.1 and 4.2, the fitted curve of pairing gap to the number of neutron and proton isalso included. The numerical values of the fitting parameters are included in Tabs. 4.2 and 4.3.Additionally, we have included the fitting parameter of these theoretical gaps to mass-numberA in Tabs. 4.2 and 4.3. As mentioned in Refs. [41, 37], A-dependence should be weakerthan 12A− 1

2 for both neutron and proton gaps. ∆(3)C shows the weakest A-dependency among

the other OES-formulae and the fitted parameters are close to each other for both proton and

26 Numerical calculations and physical results

20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

N

∆3 C

Fig. 4.1 Experimental neutron OES-parameters ∆(3)C (N) in MeV for known even-even nuclei

in different isotopic chains. The solid line is fitted to N.

20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

Z

∆3 C

Fig. 4.2 Experimental proton OES-parameters ∆(3)C (Z) in MeV for known even-even nuclei

in isotonic chain. The solid line is fitted to Z.

4.1 Systematic calculations of pairing gaps for all even-even nuclei 27

Table 4.1 Isotopes with ∆(3)C (N) and ∆

(3)C (Z)≥ 2.0 MeV. The values are in MeV.

Isotope ∆(3)C (N) Isotope ∆

(3)C (Z)

8Be 4.11 8Be 3.6410Be 2.57 10Be 2.8510C 3.53 12Be 3.612C 2.8 10C 2.114O 3.15 12C 2.3620Ne 2.6 14C 2.5122Mg 2.3 16C 2.1326Si 2.02 18C 2.5944Ti 2.0 20Ne 2.42

22Ne 2.0756Ti 2.21

Table 4.2 The fitting parameters of empirical neutron and proton pairing gaps for even–evennuclei to bAm with different OES formulae as functions of A, with 95% confidence bounds.

Formula b m Formula b m

∆(3)(N) 13.43±1.38 −0.48±0.03 ∆(3)(Z) 14.45±1.14 −0.45±0.02∆(3)C (N) 4.03±0.46 −0.28±0.03 ∆

(3)C (Z) 4.0±0.44 −0.29±0.02

∆(4)(N) 7.37±0.75 −0.38±0.02 ∆(4)(Z) 8.50±0.72 −0.39±0.02∆(5)(N) 6.93±0.7 −0.37±0.02 ∆(5)(Z) 7.87±0.65 −0.37±0.02

neutron pairing gap compared to other formulas. As Tab. 4.2 presents, fitted parameters inthe case of ∆(4) and ∆(5) are close to each other, however, not as quite as one gets for ∆

(3)C .

Fig. 4.3 visualizes these fitted parameters for proton pairing gap for different OES-formulas.In Fig. 4.4, the results of different proton OES-parameters obtained from formulae 2.2,

2.5, 2.3 and 2.4 are plotted over N and Z to overview the proton pairing gap trend over theknown nuclear landscape. There are 543 measured proton pairing gap ∆

(3)C compared to 518

measured ∆(3), 484 measured ∆(4) and only 461 measured ∆(5). The pairing gaps larger than1.7 MeV are set to 1.7 MeV for better comparison with neutron pairing gap in Ref. [20]. Allproton pairing gap formulae except ∆

(3)C expression have sudden peaks at major closed shells,

which is more apparent in Figs. 4.5 and 4.6.Almost all nuclei with Z ≤ 50 have a proton pairing gap ∆(3) equal or larger than 1.7 MeV,

and those between 50 and 82 have a rather large proton pairing gap (see Fig. 4.4). The protonpairing gaps ∆

(3)C are overly smaller and follow the same trend as the other OES-formulae

except at the major closed proton shells. Proton pairing gaps for heavy nuclei are slightlylarger than neutron pairing gaps (see Fig. 1 in Ref. [20]) for conventional 3, 4 and 5-points


Table 4.3 The fitting parameters of empirical neutron and proton pairing gaps for even–evennuclei to bNm and bZm with different OES formulae as functions of N and Z (with 95%confidence bounds).

Formula b m Formula b m

∆(3)(N) 10.62±0.8 −0.50±0.04 ∆(3)(Z) 10.27±0.65 −0.46±0.02∆(3)C (N) 3.66±0.33 −0.30±0.02 ∆

(3)C (Z) 3.56±0.33 −0.33±0.03

∆(4)(N) 6.25±0.49 −0.39±0.02 ∆(4)(Z) 6.75±0.46 −0.41±0.02∆(5)(N) 5.90±0.45 −0.38±0.04 ∆(5)(Z) 6.30±0.42 −0.40±0.02

0

1

2

3

4

5

6

Proton Gap − ∆(3)

0

1

2

3

4

5

6


C

0 20 40 60 80 1000

1

2

3

4

5

6

Z


0 20 40 60 80 1000

1

2

3

4

5

6


Z

Fig. 4.3 Experimental proton OES-parameters (in MeV) with four different formulae for allknown even-even nuclei. The dashed line is fitted curve as function of Z.


0

20

40

60

80

100

∆(3)

Z

0

0.5

1

1.5

∆(3)

C

0

0.5

1

1.5

0 50 100 1500

20

40

60

80

100

N

∆(4)

Z

0

0.5

1

1.5

0 50 100 150

N

∆(5)

0

0.5

1

1.5

Fig. 4.4 Proton pairing gaps calculated for ∆(3) (top left), ∆(3)C (top right), ∆(4) (bottom left)

and ∆(5) (bottom right) for all known even-even nuclei. All ∆z ≥ 1.7 has been set to 1.7 MeV.

formulas. However, in case of ∆(3)C the difference between proton and neutron gaps are not as

much. For the sake of better comparison and color distinguishing, we have also included Fig.4.5, which all pairing gaps larger than 2.5 MeV has been approximated to 2.5 MeV. Thisfigure is obtained from the new mass table from 2016 and includes 553 measured protonpairing gap ∆

(3)C compared to 536 measured ∆(3), 499 measured ∆(4) and only 471 measured

∆(5). The larger magnitude of OES in the three following formulae can be attributed to thecontribution from the mean-field to the pairing gap, which is not desirable in the measurementof pairing gap. As it is mentioned in Refs. [71, 26], the mean-field from particle-hole channelvaries quickly from an even nucleus to the odd one. Hence, this effect will be minimized inthe odd-mass system with ∆(3)-formula. Hence, the last proton Z +1 will contribute largelyto the ∆(3), ∆(4) and ∆(5) in even-mass system. In the two latter cases, this contributionis averaged out effectively; however, the contribution from pairing in the particle-particlechannel will also be averaged over 4 and 5 sequencing nuclei as well.

In Fig. 4.6 all the differences between various OES parameters and ∆(3)C are shown. The

major shortcoming of ∆(3)C is the uniform contribution of Coulomb field to the value of pairing

gap. The proton pairing gap for N = Z nuclei is plotted in the bottom left part of Fig. 4.6.In these kinds of nuclei, there is an important interaction between neutron and proton thatone cannot ignore, so-called Wigner energy, which enhance the binding energies for N = Znuclei [72]. As the plot shows the proton pairing gap is the lowest for the case of ∆

(3)C , while


0

20

40

60

80

100

∆(3)

Z

0

0.5

1

1.5

2

2.5

∆(3)

C

0

0.5

1

1.5

2

2.5

0 50 100 1500

20

40

60

80

100

N

∆(4)

Z

0

0.5

1

1.5

2

2.5

0 50 100 150

N

∆(5)

0

0.5

1

1.5

2

2.5

Fig. 4.5 Proton pairing gaps calculated for ∆(3) (top left), ∆(3)C (top right), ∆(4) (bottom left)

and ∆(5) (bottom right) for all known even-even nuclei with the newest mass table from 2016.All ∆z ≥ 2.5 has been set to 2.5 MeV.

∆(4) and ∆(5) give uniformly larger and almost same values for the gap, systematically. ∆(3)C

gaps follow nicely the average trend, which can indicate the pairing gap does not include asignificant contribution from Wigner energy. There are some enhancements for a few of ∆

(3)C ,

which can be due to significant nucleus shape changes from Z to the Z −2 system.As Fig. 4.7 shows as long as Z changes inside a single shell the differences between ∆(3)

and ∆(3)C is uniformly small, unless at closed shells, for which there is an abrupt difference.

The uniform part of the difference is mainly due to the contribution from the Coulomb fieldto proton pairing gaps [19].

As indicated above, the ordinary 3, 4 and 5-points formulas contain a contribution fromsingle particle energy in the particle-hole channel that manifests itself in the closed shellnuclei and may persist for the other nuclei. In the same manner, ∆

(3)C is not appropriated to

evaluate the pairing gap in odd-N and odd-Z nuclei.


−0.5

0

0.5

1

1.5

2

2.5

3

∆(3)

−∆(3)

C

−0.5

0

0.5

1

1.5

2

2.5

3

∆(4)

−∆(3)

C

0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

2.5

3

Z

∆(5)

−∆(3)

C

0 10 20 30 400

1

2

3

4

5

6

7

Z

∆(3)

∆(3)

C

∆(4)

∆(5)

Fig. 4.6 Difference between the proton pairing gaps derived from different OES formulasand ∆

(3)C in MeV. The right bottom figure shows the proton pairing gap for Z = N.

20 25 30 35 40 45 50 550

0.5

1

1.5

2

2.5

3

Z

∆3−

∆3 C

16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

Fig. 4.7 Difference between the proton pairing gaps ∆(3)−∆(3)C presented in MeV.


4.2 Effective forces in the particle-hole and particle-particlechannel

In this section, we have included the calculations performed with the DFT solver HFBTHO[74]. HFBTHO solves the HFB equations with Skyrme functionals in the axially deformedharmonic oscillator and transformed basis. To cover different nuclear matter characteristics,and their overall effect on pairing gaps, we exploit four parameterizations of Skyrme func-tionals as SIII, SLy4, SLy5, and SKM with this DFT solver. In this code, the time-reversalsymmetry is assumed as in the HFBRAD-solver [10]. Our investigation is restricted toeven-even nuclei for simplicity. To be sure that we obtain the minimum binding energies,we initiate our calculations with two starting points for basis deformation in opposite signsand choose the one, which generates the larger binding energy. The oscillator basis has 25major shells, which is sufficient to produce the right deformation for light and medium-sizednuclei. The interaction in the particle-particle channel is the mixed pairing, and the strengthof pairing interaction has kept constant for all parameterizations for the sake of consistency.The parameter in the particle-particle channel is fitted for SLy4 parameterization to reproducethe pairing gap of 1.31 MeV for 120Sn. The quasi-particle energy cut-off is set to 60 MeV.Pairing correlations can couple states, which are located high in the particle continuum. Thisenergy cut-off is enough to exhaust this coupling [30].

The results for nuclei with Z < 52 and N < 84 are presented in Figs. 4.8, 4.9 and 4.10.They compare the two-neutron separation energy S2n, the Fermi level λn, the mean neutronpairing gap ∆n, and the quadrupole deformation β2 with interactions SLy5, SIII, and SKM.There are only a few dripline nuclei in these calculations. However, a larger calculation withSLy4 force is shown in Fig. 4.11, which includes nuclei close and beyond dripline.

The two-neutron separation energy S2n and Fermi levels λ follow the same trend forthese parameterizations, qualitatively. There are more discrepancies for ∆n. Since the pairingstrength is fitted for SLy4, only this parameterization produces an averaged neutron gapof 1 MeV magnitude. For other parameterizations, neutron gap is a slightly overestimated.Reducing the strength of V0 in neutron pairing channel will adjust the quantity of the gap.Generally, all of these figures 4.8, 4.9 and 4.10 show that the gap increases for open-shellnuclei and decreases sharply at neutron magic numbers. There is no obvious trend at majorproton closed shells.

We obtain low values for neutron pairing gap for N = 40 (see Fig. 4.11), which is notpresented in other parameterizations. The form of pairing interaction is same for all thesecalculations; however, the behavior of pairing gaps are not the same. It is concluded thatthe pairing gap is sensitive to the functional used in the particle-hole channel, which is also

4.2 Effective forces in the particle-hole and particle-particle channel 33

0

10

20

30

40

50

S2n

Z

0

5

10

15

20

25

30

λ

−25

−20

−15

−10

−5

0

0 20 40 60 800

10

20

30

40

50

N

∆n

Z

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80

N

β2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 4.8 HFBTHO calculations with SLy5 parameterization in the mean-field and mixedpairing interaction for the Fermi level λn (top right panel), two-neutron separation energyS2n = B(Z,N − 2)−B(Z −N) (top left panel), mean neutron pairing gap ∆n (bottom leftpanel) and the deformation β (bottom right panel).

0

10

20

30

40

50

S2n

Z

0

5

10

15

20

25

30

λ

−25

−20

−15

−10

−5

0

0 20 40 60 800

10

20

30

40

50

N

∆n

Z

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80

N

β2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 4.9 HFBTHO calculations with SIII parameterization in the mean-field and mixedpairing interaction for the Fermi level λn (top right panel), two-neutron separation energyS2n = B(Z,N − 2)−B(Z −N) (top left panel), mean neutron pairing gap ∆n (bottom leftpanel) and the deformation β (bottom right panel).


0

10

20

30

40

50

S2n

Z

0

5

10

15

20

25

30

λ

−25

−20

−15

−10

−5

0

0 20 40 60 800

10

20

30

40

50

N

∆n

Z

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80

N

β2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 4.10 HFBTHO calculations with SKM parameterization in the mean-field and mixedpairing interaction for the Fermi level λn (top right panel), two-neutron separation energyS2n = B(Z,N − 2)−B(Z −N) (top left panel), mean neutron pairing gap ∆n (bottom leftpanel) and the deformation β (bottom right panel).

0

10

20

30

40

50

S2n

Z

0

5

10

15

20

25

30

λ

−25

−20

−15

−10

−5

0

0 20 40 60 800

10

20

30

40

50

N

∆n

Z

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80

N

β2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 4.11 HFBTHO calculations with SLy4 parameterization in the mean-field and mixedpairing interaction for the Fermi level λn (top right panel), two-neutron separation energyS2n = B(Z,N − 2)−B(Z −N) (top left panel), mean neutron pairing gap ∆n (bottom leftpanel) and the deformation β (bottom right panel).

4.2 Effective forces in the particle-hole and particle-particle channel 35

0 20 40 60 800

20

40

N

(a)

Z

0 20 40 60 800

20

40

N

(b)

Z

−5 0 5 10

0 20 40 60 800

20

40

N

(c)

Z

0 20 40 60 800

20

40

N

(d)

Z

Fig. 4.12 Difference between binding energies obtained from various Skryme force param-eterizations (SLy5, SKM, SLy4, and SIII) and experimental values for even-even nucleiZ < 52 and N < 84. The line shows the N = Z nuclei.

stated in Ref. [13]. This sensitivity can be traced back to the description of the underlyingshell structure of the mean-field [2], which is one of the sources of uncertainty of pairingcalculations in atomic nuclei.

The deformation β2 varies with the Skyrme forces (see bottom right panel in Figs. 4.8,4.9, 4.10,and 4.11), and the shape of the ground state alters from spherical to deformed indifferent parameterizations [65]. SLy5 and SLy4 predictions are more similar to each otherand are in quite good agreement with the result from HFB approach with the Gogny force[23].

Fig. 4.12 shows the differences between the experimental binding energies and the onesobtained from several Skyrme force parameterizations. All of them are underestimating thebinding energy of N = Z nuclei. SKM force predicts the binding energies of N = Z nucleithe best. However, as we get close to dripline, the error gets larger compared to the otherforces. Since the pairing force is fitted only for the case of SLy4 force parameters, its resultsperform more satisfactory.


0 20 40 60 800

20

40

N

SIII

Z

0 20 40 60 800

20

40

N

SKM

Z

0 1 2

0 20 40 60 800

20

40

N

SLy5

Z

0 20 40 60 800

20

40

N

SLy4Z

Fig. 4.13 Proton pairing gap ∆z obtained from various Skyrme force parameterizations. Morenuclei are calculated in case of SLy4 parameters.

4.2.1 Proton pairing gap over nuclear landscape

Proton pairing gaps ∆z for different mean-field functionals are presented in Fig. 4.13, whichare obtained with same numerical inputs as it mentioned earlier in this section. The sametrend for proton pairing gap is noticeable here as we did for neutron pairing gap ∆n. Thesensitivity to the functional in case of proton pairing gap is more evident than neutron gaps.The enhancement of the ∆z for mid-shell nuclei can be reduced by decreasing the pairingstrength V0 and fitting it individually for each set of parameters.

A systematic calculation for even-even nuclei with the same assumption for HFBTHOas above is shown in Fig. 4.14. This figure shows the results for proton pairing gap ∆z

over the entire nuclear landscape from stable nuclei to the proton, neutron dripline, andbeyond. Proton pairing gaps have smaller values than neutron pairing gaps because thepairing strength V0 is only fitted to ∆n, and we do not take into account isospin dependencyin the zero-range pairing interaction. A larger V0 in the proton pairing channel than neutronpairing channel is required to adjust this underestimation [13]. ∆z decreases sharply at proton

4.3 Even-even nuclei in a spherical box 37

0 50 100 150 200 250 3000

20

40

60

80

100

120

N

Z

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 4.14 HFBTHO calculations with SLy4 parameterization in the mean-field and mixedpairing interaction for the proton pairing gap ∆z .

closed shells except for a few numbers of nuclei with almost half-full neutron shells at Z = 82.They gain larger values for mid-shell nuclei as the neutron pairing gap did.

4.3 Even-even nuclei in a spherical box

HFB-solver HFBRAD [10] has been used extensively in this work to provide solutions forthe single quasi-particle wave functions in the coordinate-space in a spherical box. In theparticle-hole channel, we used the Skyrme functional with SLy4 parameterization. Differentzero-ranged density-dependent pairing interactions are used in the particle-particle channel(see the interaction given in 3.32). The strength of pairing interaction V0 is fitted individuallyfor each interaction to neutron pairing gap ∆

(3)C of 120Sn. These calculations are performed in

a spherical box with radii R = 30 fm. The quasi-particle energy cut-off is set to 60 MeV.Fig. 4.15 shows the convergence of binding energies obtained from HFBRAD for two

nuclei 138Xe and 175Sn using mixed pairing interaction. The left figure presents the bindingenergies of different blocked one-quasi particle states as a function of jmax. As figure 4.15shows at jmax = 25/2, the binding energy has converged. Hence, maximal spin jmax isset to 25/2, however, for heavier nuclei it is up to jmax = 39/2, and for very light nucleijmax = 21/2 to avoid numerical instability. The main dependence on jmax is mainly noticeablefor binding energies calculated with surface pairing interaction.


20 30 40

1144.5

1145

1145.5

1146

jmax

175Sn

Bin

din

g E

ner

gy [

MeV

]

f7/2

h9/2

p1/2

i13/2

20 30 401147.5

1147.6

1147.7

1147.8

1147.9

1148

1148.1

1148.2

1148.3

jmax

138Xe

Fig. 4.15 The convergence of binding energy.


4.3.1 General calculation for xenon isotopes

Two theoretical neutron pairing gaps ∆mean and ∆LCS (see Eq. 3.34) are compared withexperimental pairing gap from Eq. 2.5 in Fig. 4.16. We have performed calculations forstable xenon isotopes and those close to dripline that have small deformation and can beapproximated by a spherical model. The left column shows the calculations with spin up tojmax = 15/2 for proton and 17/2 for neutron and the right column with jmax equal to 25/2for both neutron and proton. Even though the binding energy is not completely converged forjmax = 17/2 and 15/2, the complication of increasing the value of maximal spin is still thesame for different zero-range pairing interactions. Increasing maximal spin will increase thegaps for density-dependent pairing interaction, especially ∆LCS. The noticeable differencebetween the two theoretical gaps is enhanced even more by pure density-dependent pairinginteraction. The immense difference between jmax = (15/2,17/2) and jmax = (25/2,25/2),especially for ∆LCS is due to the density-dependent and mixing of more number of states withcase of jmax = 25/2. However, these two theoretical pairing gaps are almost equal for theother two interactions. Systematically, the gaps obtained from surface interactions are largerfor dripline nuclei than the stable isotopes.

Only analyzing the converged solutions at the right column in Fig. 4.16, volume in-teraction is slightly underestimated for neutron-rich xenon isotopes, whereas the mixedinteraction corrects this underestimation. The observed ∆

(3)C can be reproduced by vol-

ume and mixed pairing. Surface interaction is slightly underestimated for stable isotopes.However, it increases notably in the neutron-rich region as the other ones do not show thistendency. The overestimation enhances this by including more quasi-particle states withjmax = (25/2,25/2) as mentioned earlier.


75 80 85 900

0.5

1

1.5

2

[MeV]

(a) Volume pairing75 80 85 90

0

0.5

1

1.5

2

∆

LCS∆

mean ∆(3)

(b) Volume pairing

75 80 85 900

0.5

1

1.5

2

[MeV]

(c) Mixed pairing75 80 85 90

0

0.5

1

1.5

2

(d) Mixed pairing

75 80 85 900

0.5

1

1.5

2

N

[MeV]

(e) Surface pairing

75 80 85 900

0.5

1

1.5

2

N

(f) Surface pairing

Fig. 4.16 Theoretical and experimental neutron pairing gaps for xenon isotopes. Left column:neutron pairing gap with jmax = (15/2,17/2) for proton and neutron, respectively. Rightcolumn: neutron pairing gap with jmax = 25/2 for both proton and neutron.


4.3.2 Weakly bound isotopes of nickel

In order to model the problem of mean-field changes in the particle-hole channel andunderstand its complication and feedback together with different pairing interactions forweakly bound nuclei, we try to tune some of the parameters of SLy4 set (t0, t1, and t2). Theseparameters are the most general ones in the Skyrme parameterizations and are enough tosimulate the deepness of the mean-field (see left columns of Figs. 4.18, 4.20, and 4.22). Theother parameters in the HFB calculations have kept constant. To see how the equivalentenergy of 3s1/2 in canonical basis for neutron-rich nickel isotopes and its occupation numberwill change, we alter the parameters in an interval of −4% to 3%.

The fields and the densities in the spherical case are only dependent on radial coordinates.Hence, one can easier compare qualitatively the contributions of different components ofthe two channels (p-h and p-p channel) together. The behavior of the mean-field in thep-h channel has been discussed in several references, for instance in Ref. [38]. They havediscussed the behavior of p-h potential with Hartree-Fock calculation. Thus, no pairinginteraction is included. In Ref. [25], they have performed HFB-calculation with SkP force(same parameterization of forces in both p-h channel and p-p channel) to discuss the shellstructure of exotic nuclei. Here, we have focused on the neutron-rich isotopes of nickel, andspecifically on the loosely bound 3s1/2. The responses of the l = 0 and 1 single-particleorbitals to a shallowing mean-field have been known and are re-studied recently from differentperspectives (see, e.g., Ref. [42]). In the present work, different density-dependent pairingsare adopted in this schematic model to see the effect of the self-consistency of differentpairing interactions and mean-field together on the loosely bound s1/2.

The changes of the equivalent energy of s1/2 and its occupation number for neutron-richisotopes of nickel are shown in Fig. 4.17. The occupational probability of s1/2 in 84Ni is themost sensitive one among these isotopes to the changing parameters of mean-fields and therespective pairing interaction. The occupational probabilities of s1/2 increase for 86Ni and88Ni by a shallower mean-field, which differs from the other isotopes.

Figs. 4.18, 4.19, 4.20, 4.21, 4.22, and 4.23 show how the underlying neutron densities andfields for 84Ni change concerning the tuning of the parameters and their respective pairinginteractions. The different characters of these pairing interactions are well shown in pairingfields (right column of these figures). The volume pairing is very weak for this isotope andonly by a very shallow mean-field, volume pairing will contribute to overall neutron pairingfield and density (see Fig. 4.18). The pairing field is its largest at the origin and decreasestowards the surface. The pairing field follows the same form as the pairing density. Surfacepairing field yields a peak at the surface of the nuclei as it is shown in figure 4.22, as one mayexpect for this pairing interaction. The pairing field gets weaker as the neutron mean-field


gets deeper. Fig. 4.24 shows the densities in an ordinary scale to show the diffuseness and thepeak, which neutron pairing density yields at the surface. As mentioned before, the pairingfield follows the form of neutron pairing density.

Volume pairing is the least sensitive to the properties of the functional because of itsnon-density-dependent properties. The pure density-dependent pairing interaction mixes thestates more profoundly. The occupation numbers (see Fig. 4.17f) display the mixture ofthe quasi-particle states. The level density increases and more states will contribute to thecollectivity of the nucleus.


−4 −2 0 2−1.5

−1

−0.5

0

0.5

single

par

ticl

e en

ergy 3

s 1/2

(a) Volume pairing−4 −2 0 20

0.2

0.4

0.6

0.8

1

Occ

upat

ion n

um

ber

N52N54N56N58N60

(b) Volume pairing

−4 −2 0 2−1.5

−1

−0.5

0

0.5

single

par

ticl

e en

ergy 3

s 1/2

(c) Mixed pairing−4 −2 0 20

0.2

0.4

0.6

0.8

1

Occ

upat

ion n

um

ber

(d) Mixed pairing

−4 −2 0 2−1.5

−1

−0.5

0

0.5

SLY4 parameters changes in %

single

par

ticl

e en

ergy 3

s 1/2

(e) Surface pairing

−4 −2 0 20

0.5

1

SLY4 parameters changes in %

Occ

upat

ion n

um

ber

(f) Surface pairing

Fig. 4.17 Changes in the equivalent energy 3s1/2 and its occupation probability with tuningSLy4 parameters t0, t1 and t2 with different density-dependent pairing interactions for nickelisotopes


0 5 10 15

−80

−60

−40

−20

0

R [fm]

Par

ticl

e−h

ole

fie

ld [

MeV

]

0 5 10 15−4

−3

−2

−1

0

R [fm]

Pai

rin

g f

ield

[M

eV]

−4−3−2−101234

Fig. 4.18 Self-consistent local neutron field at left and neutron pairing field at right in 84Nicalculated with different values of t0, t1 and t2 of SLy4 parameterization and volume pairing.

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Sin

gle

−neu

tron d

enis

ty [

fm−

3]

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Neu

tron p

airi

ng d

enis

ty [

fm−

3]

Fig. 4.19 Self-consistent neutron density as a function of R at left and neutron pairing densityat right in 84Ni calculated with different values of t0, t1 and t2 of SLy4 parameterization andvolume pairing.


0 5 10 15

−80

−60

−40

−20

0

R [fm]

Par

ticl

e−h

ole

fie

ld [

MeV

]

0 5 10 15−4

−3

−2

−1

0

R [fm]

Pai

rin

g f

ield

[M

eV]

Fig. 4.20 As same as figure 4.18 with mixed pairing.

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Sin

gle

−neu

tron d

enis

ty [

fm−

3]

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Neu

tron p

airi

ng d

enis

ty [

fm−

3]

Fig. 4.21 As same as figure 4.19 with mixed pairing.


0 5 10 15

−80

−60

−40

−20

0

R [fm]

Par

ticl

e−h

ole

fie

ld [

MeV

]

0 5 10 15−4

−3

−2

−1

0

R [fm]

Pai

rin

g f

ield

[M

eV]

Fig. 4.22 As same as figure 4.18 with surface pairing.

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Sin

gle

−neu

tron d

enis

ty [

fm−

3]

0 10 20 3010

−20

10−15

10−10

10−5

100

R [fm]

Neu

tron p

airi

ng d

enis

ty [

fm−

3]

Fig. 4.23 As same as figure 4.19 with surface pairing.

4.4 Calculations with varying mean-field 47

0 5 100

0.05

0.1

R [fm]

Sin

gle

−n

eutr

on

den

isty

[fm

−3]

0 5 100

0.005

0.01

0.015

0.02

0.025

R [fm]

Neu

tro

n p

airi

ng

den

isty

[fm

−3]

Fig. 4.24 As same as figure 4.23 with surface pairing (different scale).

4.4 Calculations with varying mean-field

The present section, a continuation of the previous work, is an attempt to study the responsesof the l = 0 and 1 single-particle orbitals to a shallowing mean-field and how they are affectedby the pairing correlation. However, this time we will add a varying term to the mean-fieldand solve the HFB-equations with this additional term self-consistently in a way, that issimilar to the study of the response of open quantum many-body systems like cold atomsto the varying external environment even though one is not able to vary directly the nuclearmean-field potential. We have chosen nickel isotopes as a typical example.

Firstly we have performed pure Skyrme HF-calculations for the nucleus 84Ni, which ispredicted to be bound with two bound quasi-particle orbitals s1/2 and d5/2. A varying externalpotential is considered as defined in Sec. 3.4.6. A positive value for the coupling constantsmeans that the external potential is repulsive and makes the total mean-field shallower. Asit can be seen from the Fig. 4.25, the s1/2 loses its energy in a way that is significantlyslower than the other d5/2 and g9/2 orbitals. A crossing between the s1/2 and d5/2 orbitalsis then expected. In particular, the g9/2 orbital will be shifted up dramatically when thepotential gets shallower. The calculations give very similar results in agreement with therecent Woods-Saxon [42] and self-consistent HF [20] calculations.

As seen in Fig. 4.25 (the top figure), the system turns unbound with V 10 ∼ 5 MeV in the

Skyrme HF calculations. Two calculations with the surface (at the middle) and the volume (at


-6

-3

0

0

1

2

0 2 4 6 80

1

2

s1/2

d5/2

g9/2

Esp

[MeV

]

s1/2

p1/2

p3/2

d3/2

d5/2

| |

E qp [M

eV]

V10

Fig. 4.25 Single particle energies of 84Ni in Skyrme HF calculations without pairing withvarying external mean-field at the top. The middle and lower panels correspond to SkyrmeHFB calculations with the surface and volume pairing interactions, respectively.

4.4 Calculations with varying mean-field 49

the bottom) pairing interactions are plotted in the figure. The position of the threshold is notaffected much by the inclusion of the pairing correlation. As discussed in a recent systematiccalculation [20], calculations on the pairing correlation properties of neutron dripline nucleiare susceptible to the form of the pairing interaction. This dependence can also be seen fromFig. 4.25. Even though, the system is bound for calculations with V 1

0 < 5 MeV, the paircontent of the wave function can be very different. There is no bound quasi-particle statefor calculations with the surface pairing, whereas the s1/2 and d5/2 quasi-particle orbitalsare both predicted to be bound in HFB calculations with the SLy4 force and volume pairinginteraction. In addition, the occupancy of s1/2 increases from 0 to 0.55 for calculationswith V 1

0 < 5 MeV and volume interaction, while for the surface pairing interaction thiss1/2 occupancy decreases from 0.15 to 0.06. However, the other s1/2 states in continuumwill contribute to pairing and their occupational probability increase with surface pairinginteraction. There is a substantial decrease in the chemical potential λ when the mean-fieldgets shallower. As a result, the s1/2 orbital may become the only bound quasi-particle statefor specific values of V 1

0 with volume pairing interaction.

4.4.1 Charge radii of nickel isotopes

The odd-even staggering of charge radius is very hard to replicate globally with conventionalparameterization of nuclear forces within DFT except for Fayans functional [35, 34], whichcan describe the staggering of the radii even for the peculiar case of calcium isotopes [36, 34].However, one cannot replicate the larger charge radii of, the heavier neutron-rich calciumisotopes [67]. In this section, we try to investigate if the proper description of the continuumwill affect the charge radii in spherical nuclei. A recent study about the influence of pairinginteraction on the radius of neutron-rich nuclei with appropriate treatment of continuum bythe Green’s function techniques is performed in Ref. [82]. In this section, I calculate thecharge radii of the semi-magic isotopes of nickel for both even and odd mass-number fromneutron-deficient to neutron-rich isotope. The maximal spin is included up to jmax = 25/2 forboth neutron and proton. The calculations are performed with Skyrme force with the set ofSLy4 parameters and mixed pairing interaction. We have blocked each possible quasi-particlestates in the respective major shell to obtain the largest binding energy. Figs 4.26 and 4.27show the changes in charge radii along the isotopic chain of nickel. In these figures, variousone-quasi particle states are blocked in major shell N = 3 and N = 4.

In Fig. 4.28 we calculated the charge radii of the odd-A isotopes 83,85Ni and comparedthem to that of the even-even isotope 84Ni with varying external field. A large incrementin the charge radii of all three nuclei is seen when the mean-field gets shallower. Althoughthere are staggering of charge radii for each blocking (see Figs. 4.26 and 4.27), a clear OES


30 40 50

3.8

3.9

4

f5/2

blocked

N

Rch

[fm

]

30 40 50

3.8

3.9

4

f7/2

blocked

N

Rch

[fm

]

30 40 50

3.8

3.9

4

g9/2

blocked

N

Rch

[fm

]

30 40 50

3.8

3.9

4

p1/2

blocked

N

Rch

[fm

]

30 40 50

3.8

3.9

4

p3/2

blocked

N

Rch

[fm

]

Fig. 4.26 Charge radii of nickel (solid line) with respect to different one-quasiparticleblocking.

50 60 70 804

4.05

4.1

4.15

4.2

s1/2

blocked

N

Rch

[fm

]

50 60 70 804

4.05

4.1

4.15

4.2

d5/2

blocked

N

Rch

[fm

]

50 60 70 804

4.05

4.1

4.15

4.2

d3/2

blocked

N

Rch

[fm

]

50 60 70 804

4.05

4.1

4.15

4.2

g7/2

blocked

N

Rch

[fm

]

50 60 70 804

4.05

4.1

4.15

4.2

h11/2

blocked

N

Rch

[fm

]

Fig. 4.27 Charge radii of nickel (solid line) with respect to different one-quasiparticleblocking.

4.5 Two-particle wave functions 51

0 2 4 6 8 104,02

4,04

4,06

4,08

4,10

4,12

Rch

[fm

]

V10

s1/2

d3/2

d5/2

g7/2

h11/2

84Ni

Fig. 4.28 Charge radii of 83Ni (solid line) and 85Ni (dashed line) with different orbitals beingblocked in comparison to that of the ground state of 84Ni (stars line).

of radii with right blocking of the quasi-particle state is not present. However, as Fig. 4.28shows, there is a reduction in charge radii staggering between 85Ni to 84Ni compared to 83Nito 84Ni. By shallowing the mean-field, one can affect the value of staggering. Thus, thedifference between the charge radii of 84,85Ni reduces significantly when the mean-field getsshallower. The odd-even staggering as observed in the charge radii systematics may vanishwhen the systems become unbound.

4.5 Two-particle wave functions

Current investigation aims at the assessment of different density-dependent pairing inter-actions in the particle-particle channel of Hartree-Fock-Bogoliubov in a different aspect.We investigate the neutron pairing correlation near the dripline from the viewpoint of thedi-neutron correlation. The assumption of the HFB-calculation is as same as mentionedearlier (see Sec. 4.2). The two-neutron wave function is based on the results of canonicalwave functions from Hartree-Fock-Bogoliubov in coordinate representation in a sphericalbox. In this way, the solution will contain the continuum part of the single-particle spectrum.

The result of the square of two-neutron wave function |Ψ(r,r,θ)|2 for different paringinteractions for two nickel isotopes 80Ni and 82Ni are shown in Fig. 4.29, and for 88Ni in Fig.4.30. Furthermore, the Fermi levels, the energy level of canonical single particle state 3s1/2

and its occupation number of these isotopes are listed in Tab. 4.4 for each isotope. As onemay have anticipated, a neutron pair will exhibit collectivity while they have θ = 0◦, and


0501001500

1

2

3

4

x 10−3

θ

(a) 80Ni

0501001500

1

2

3

4

x 10−3

θ

(b) 80Ni

0501001500

1

2

3

4

x 10−3

θ

(c) 80Ni

0501001500

1

2

3

4

x 10−3

θ

(d) 82Ni

0501001500

1

2

3

4

x 10−3

θ

(e) 82Ni

0501001500

1

2

3

4

x 10−3

θ

(f) 82Ni

Fig. 4.29 Two-particle wave function Ψ(2) for 80Ni top row and 82Ni bottom row. Volume,mixed and surface interaction are shown at the left, center and right column, respectively.

Isotopes 80Ni 82Ni 88NiInteraction λn 3s1/2 v2 λn 3s1/2 v2 λn 3s1/2 v2

Volume −1.76 0.21 1% −1.68 −0.17 2% −0.15 −1.09 99%Mixed −1.79 0.07 1% −1.66 −0.27 3% −0.18 −1.08 97%Surface −2.04 0.10 4% −1.66 −0.27 11% −0.45 −1.08 72%

Table 4.4 Table over Fermi level λn in [MeV ], energy level of the single-particle state of 3s1/2

in [MeV ], and the occupational probability for 80 82 88Ni.

non-collectivity behavior while they are separated by θ = 180◦. Figs. 4.29 and 4.30 showclearly that density-dependent pairing interaction gives rise to stronger di-neutron correlationthan a non-density-dependent interaction.

Fig. 4.31 shows the same concept for oxygen isotopes 22O and 26O from a differentview. The two-neutron spatial correlation indicates the reduction of pairing collectivity fromsurface interaction to volume interaction by one order of magnitude.

4.5 Two-particle wave functions 53

0501001500

1

2

3

4

5

6x 10

−4

θ

(a) 88Ni

0501001500

0.5

1

1.5x 10

−3

θ

(b) 88Ni

0501001500

2

4

6

8x 10

−3

θ

(c) 88Ni

Fig. 4.30 Two-particle wave function Ψ(2) for 88 Ni with volume interaction at the top, mixedin the middle and surface interaction at the bottom. Notice that the scale is different.


0 2 4 6 8 10 120

50

100

150

R

θ

0

0.2

0.4

0.6

0.8

1

1.2

x 10−4

0 2 4 6 8 10 120

50

100

150

R

θ

0

1

2

3

4

5

x 10−4

0 2 4 6 8 10 120

50

100

150

R

θ

0.5

1

1.5

2

x 10−3

0 2 4 6 8 10 120

50

100

150

R

θ

0

0.5

1

1.5

2

x 10−3

0 2 4 6 8 10 120

50

100

150

R

θ

0

2

4

6

8x 10

−4

(a) 22O

0 2 4 6 8 10 120

50

100

150

R

θ

0

1

2

3

4

5

6

7

x 10−4

(b) 26O

Fig. 4.31 Two-particle wave function Ψ(2) for 22O right column and 26O left row. Volume,surface and mixed interaction are shown at the top, middle and bottom row, respectively.Notice that the scale is different.

Chapter 5

Summary

In this thesis, I have studied the effect of the density-dependence of the zero-range pairinginteractions on the nuclei close to dripline systematically. We review the pairing propertiesof spherical nuclei in coordinate space with HFB-calculation. Both even-even nuclei andodd-mass nuclei with this symmetry in the coordinate system have been investigated. Fordeformed systems and global calculations over the whole nuclear chart, I have performedcalculations in deformed harmonic oscillator basis. We made a detailed analysis of exper-imental and theoretical pairing gaps for all semi-magic nuclei from the stable isotopes toneutron-rich ones in the coordinate system and harmonic oscillator basis. Surface interactionpredicts a significantly stronger pairing gap than volume and mixed pairing. By fittingthe strength of various zero-range pairing interactions globally to all available ∆

(3)C with

Z ≥ 8, the overestimation of pairing gap has reduced. A global calculation for both protonand neutron pairing gap over the entire nuclear landscape has been performed. Numerouscalculations have been performed to study the relationship between pairing interaction andOES of binding energies. Various OES-formulae have been investigated to see which formulacan provide a good measure of the pairing gap in even-even nuclei. We demonstrate thestrength of ∆

(3)C , which can indicate the two-particle spatial correlation. The weakening of

∆(3)C can point to a weak di-neutron correlation in some neutron rich-nuclei.

We use several toy models with an empirical correction to complement the pairing effect.The occupational probability and the energy of the loosely bound s1/2 compared consistentlyfor each zero-range pairing interaction.

I benchmark the equal-filling blocking in the coordinate system with HFBRAD code withHFBTHO. Hence, the one-quasiparticle states for all the neighboring odd-A isotopes witheven-even semi-magic isotopes could be calculated with three different zero-range pairingforces.

56 Summary

Charge radii in neutron-rich isotopes of nickel have been investigated. The odd-evenstaggering of charge radii is not reproduced; however, we could show that altering the mean-field has a profound effect on the OES of charge radii. As the system gets unbound, thestaggering reduces.

I conclude that at dripline one should be careful of using density dependent pairinginteraction. Different zero-range pairing interactions affect the properties of loosely boundnuclei, for instance, pairing gap and collectivity. The various density dependence pairinginteraction gives rise to two different theoretical pairing gaps ∆LCS and ∆mean for driplinenuclei. The model-dependent shell structure in the underlying mean-field in density functionaltheory present an uncertainty that one should mainly be considered in HFB-calculations. Itrepresents a larger uncertainty for the results of pairing calculations with density-dependentinteraction.

It may be useful to mention the limitation of this work for not including any finiterange interactions in the p-h channel and p-p channel (for this treatment see Ref. [22]).The neutron-proton interaction was absent. However, it has been studied thoroughly in[81, 68, 56]. Additionally, the shortcoming in the calculation of deformed nuclei [60], andthe absence of odd terms in the mean-fields, which is discussed extensively in Ref. [31].

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Appendix A

The Hartree-Fock approach

Schrödinger equation: Hψ = Eψ , where E is the exact solution and ψ are eigenstates of H,which forms a complete basis with normalized ψ and ⟨ψm|ψn⟩ = δmn. The approximatedsolution of φ can be expanded on this basis as φ = ∑n cnψn.

The variational method provides an approximated solution to the ground state of Hamil-tonian H:

δ E =δ ⟨φ |H|φ⟩⟨φ |φ⟩

= 0 (A.1)

Some properties of the Slater determinant:- Having two particles in a same state make the two rows of the determinant identical,

hence, the determinant vanishes. This is the direct result of Pauli principle.- Changing two rows of the determinant changes the sign of it. This makes sure that total

wave function is anti symmetrized.- The single particle states cannot be linear dependent, otherwise the determinant will

vanishes. We need a complete set of single particle wave functions which are not lineardependant.

- Slater determinant is the sum of orthonormal terms ⟨Φ|Φ⟩= N !- Slater represents independent identical particles. The definition of systems wave func-

tion comes originally from the addition of Hamiltonians of independent particles, however, inthe process the exchange term makes the particles strongly correlated due to Pauli principle.

Anti-commutation relation for two fermion wave functions changes the sign, whenevertwo fermions exchange their coordinate:

{a(r),a(r′)+}= δr,r′ (A.2)

Appendix B

BCS equation

Pure pairing force many-body Hamiltonian with BCS description:

H = ∑eka+k ak −G ∑kk′>0

a+k a+−ka−k′ ak′ (B.1)

Gap equation:

∆ =G2 ∑

k>0

∆√e2 +∆2

(B.2)

where ek = ek −λ , and G is a constant pairing force. We assume that ek and G are known.In spherical basis k represents the angular momentum projection, and −k is the time-

reversed state. The pair of (k,−k) is coupled by constant force G as shown in Eq. B.1. ∆ inthis case is not dependent on k and it is constant for all paired level. uk and vk have sameexpressions as Eq. 3.14. Constant ∆ is mainly depended on the single particle energies of ek.To evaluate BCS equations, one fits the pairing force G to experimental data.

Appendix C

Hartree-Fock-Bogoliubov equation

The two-component of HFB wave functions are U and V . Requirement for U and V fromRef. [66]:

UV++V ∗UT = 0 (C.1)

One can replace the pairing tensor κ with pairing density matrix ρ which is Hermitian forthe case of time-reversal symmetry [30, 24] as follow:

ρ(r) =∞

∑Eα

V (Eα ,rσ)V ∗(Eα ,r′σ ′) (C.2)

ρ(r) =−∞

∑Eα

V (Eα ,rσ)U∗(Eα ,r′σ ′) (C.3)

In order to obtain eqvuivalent energy εk, one can apply the BCS formula Eq. 3.14 for v2k

as following:

Nk =12(1− εk −λ

Ek)→ εk = λ +Ek(2Nk −1) (C.4)

Where Ek is the quasiparticle energy obtained from HFB-calculation and λ is the Fermilevel.

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