Pairing correlation in atomic nuclei under extreme conditions

Pairing correlation in atomic nuclei under extreme conditions

Sara Asiyeh Changizi

Department of Physics

KTH Royal Institute of Technology

Albanova November 2017

ii

TRITA-FYS 2017: 69 ISSN 0280-316X

ISRN KTH/FYS/–17:69—SE ISBN 978-91-7729-537-2

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i fysik fredagen den 24 novemeber klockan 14:30 i FA32, Roslagstullsbacken 21, Albanova Universitetscentrum, Stockholm.

© Sara Asiyeh Changizi, november 2017 Tryck: Universitetsservice US AB

KTH School of Engineering Science

SE-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 and

one 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. Roberto Liotta are two of those who I have learned a lot from them. They have shown me the beauty of physics, the beauty of understanding the meaning of abstract ideas, and the physics behind the mathematics formulas. Together with my dedicated supervisor Dr. Chong Qi, who taught me a lot including the research work ethic and how to push forward in research, encouraging me to come up with my ideas.

Dr. Farnaz Ghazimoradi who brought me into the era of Ph.D. studies and inspired me to join the Theoretical Nuclear Physics course. Dr. Sara Bartot, a dear friend, who inspires me every day and makes my days full of joy. Dr. Maria Doncel who helped me with her encouragement and kindness during the hard times of Ph.D. There have been a lot of people in Nuclear Physics group during this period that inspired me a lot. However, there is not space enough here to mention all their names, but I am deeply grateful to them. I also want to acknowledge KTH and the Department of Physics, which facilitated and made this work possible.

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

My adorable son William, to whom I owe all my happiness. The happiness I could not imagine was ever even possible.

Sara A. Changizi,

Stockholm, Nov. 7, 2017

Abstract

The pairing correlation has long been recognized as the dominant many-body correlation beyond the nuclear mean field. Pairing plays an essential role in many nuclear phenomena including the occurrence of a systematic odd-even staggering (OES) of the nuclear binding energy. Pairing interaction plays, in particular, a significant role in the stability of weakly bound nuclei. Therefore it has been one of the most critical topics in nuclear physics because of the weakly bound structure of all newly discovered nuclei. Beyond the line of stability, pairing interaction is not a residual interaction anymore. Its strength can be of the same order of magnitude as the mean field. In this thesis we have focused on the pairing interaction in atomic nuclei under extreme conditions to investigate the structure of loosely bound nuclei near the dripline; to probe and globally assess different outcomes of various zero-range pairing interactions and their effect on the loosely bound low angular momentum states. How much can density-dependence in zero-range pairing interaction nuclei affect the line of stability? Calculations predict that pure density-dependent pairing interaction so-called surface interaction enhances the collectivity and gives stronger neutron pairing gap in nuclei far from stability, while, the density dependence pairing interaction does not affect the gap for bound nuclei as much. The odd-even staggering of nuclear binding energies has been investigated to estimate the empirical pairing gap. A 3-point formula ∆ _C ⁽³⁾ ( ¹ ₂ [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-magic

even-even nuclei. The pairing strength is fitted globally to all available data on the OES

of semi-magic nuclei with Z ≥ 8. The difference between different zero-range density-

dependent pairing interactions reduces with this global fitted parameter. The difference

between the mean pairing gap and the OES gets larger as we get closer to the dripline. At the

end of the thesis, a simple model has been developed which shows that when the mean-field

becomes 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 some

complementary works on the subject.

Sammanfattning

Parkopplingen i atomkärnan anses vara den dominerande mångkropparskorrelationen efter kärnmedelfältet. 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 för stabiliteten. Det gäller alla de nyupptäckta neutronrika atomkärnor som karakteriseras av en svag bindningsenergi där således parkopplingen blir ett kritiskt fenomen. När vi läm- nar stabilitetslinjen och närmar oss linjen av spontan neutronsönderfall är parkopplingen inte längre obetydlig, utan tvärtom spelar en viktig roll för stabiliteten hos atomkärnan.

Dess styrka kan vara av samma storleksordning som medelfältet. I denna avhandling har

vi fokuserat på parkopplingen i atomkärnor under extrema villkor i syfte att undersöka

strukturen hos löst bundna atomkärnor nära linjen för neutronsönderfall; att utforska och

globalt bedöma resultaten av olika parkopplingar baserad på en deltakraft (räckvidd noll)

och deras effekt på löst bundna tillstånd med lågt rörelsemängdsmoment. Hur mycket kan ett

täthetsberoende i parkopplingen påverka stabilitetslinjen? Beräkningarna förutsäger att ren

densitetsberoende växelverkan för parkoppling, så kallad ytväxelverkan, ökar kollektiviteten

i 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 storleken

på parkopplingens gap. En 3-punktsformel ∆ _C ⁽³⁾ ( 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 approximeras

med sfärisk symmetri, beräknas för att erhålla udda-jämn spridning (OES) med avseende

på bindningsenergin för alla kända semi-magiska jämn-jämna kärnor. Parkopplingsstyrkan

är globalt anpassat med all tillgänglig data på OES vad gäller semi-magiska kärnor med

Z ≥ 8. Skillnaden mellan olika täthetsberoende parkopplingar med vår växelverkan minskar

med dessa globalt anpassade parametrar. Skillnaden mellan det teoretiska genomsnittliga

parkopplingsgapet och OES blir större när vi kommer närmare tröskeln för atomkärnans

x

stabilitet. I slutet av avhandlingen har en schematisk modell utvecklats som, genom att

göra medelfältet grundare, reducerar de udda-jämnt spridda laddningsradierna. Avhandlin-

gen innehåller dessutom resultat som inte ingår i de tre publicerade artiklarna samt några

kompletterande 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 interaction and pairing correlation in unstable nuclei. Phys. Rev. C, 91 024305. Idea, execution, all calculations 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 are done mostly by the first author.

IV. Zheying Wu, S. A. Changizi, and Chong Qi (2016) Empirical residual neutron-proton

interaction in odd-odd nuclei. Phys. Rev. C 93 034334. Idea, some parts of the writing and

final 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 5

2.1 Pairing gap . . . . 5

2.2 Binding energy odd-even staggering . . . . 6

3 The Hartree-Fock-Bogoliubov approach 11 3.1 Hartree-Fock approach . . . . 12

3.2 BCS approach for pairing . . . . 14

3.3 The Bogoliubov transformation . . . . 15

3.4 Hartree-Fock-Bogoliubov equation in coordinate space . . . . 18

3.4.1 Continuum treatment . . . . 18

3.4.2 Canonical states . . . . 19

3.4.3 Choices of the pairing interaction . . . . 20

3.4.4 Two-particle wave function . . . . 21

3.4.5 Odd-A nuclei and the blocking effect . . . . 22

3.4.6 Varying external potential . . . . 23

4 Numerical calculations and physical results 25 4.1 Systematic calculations of pairing gaps for all even-even nuclei . . . . 25

4.2 Effective forces in the particle-hole and particle-particle channel . . . . 32

4.2.1 Proton pairing gap over nuclear landscape . . . . 36

4.3 Even-even nuclei in a spherical box . . . . 37

4.3.1 General calculation for xenon isotopes . . . . 39

4.3.2 Weakly bound isotopes of nickel . . . . 41

xiv Table of contents

4.4 Calculations with varying mean-field . . . . 47 4.4.1 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 nuclei with 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 are

for even-A nuclei, blue and green squares show odd-A nuclei with odd number of neutrons and odd number of protons respectively. . . . 9 3.1 The density-dependant average pairing gap ⟨∆⟩ with different box radii and N 20 3.2 The model of clustering two nucleons . . . . 21 4.1 Experimental neutron OES-parameters ∆ _C ⁽³⁾ (N) in MeV for known even-even

nuclei in different isotopic chains. The solid line is fitted to N. . . . 26 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. . . . 26 4.3 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. 28 4.4 Proton pairing gaps calculated for ∆ ⁽³⁾ (top left), ∆ _C ⁽³⁾ (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. . . . 29 4.5 Proton pairing gaps calculated for ∆ ⁽³⁾ (top left), ∆ _C ⁽³⁾ (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. . . . 30 4.6 Difference between the proton pairing gaps derived from different OES

formulas and ∆ ⁽³⁾ _C in MeV. The right bottom figure shows the proton pairing

gap for Z = N. . . . 31

4.7 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 and mixed pairing interaction for the Fermi level λ _n (top right panel), two-neutron separation energy S _2n = B(Z, N − 2) − B(Z − N) (top left panel), mean neu- tron pairing gap ∆ _n (bottom left panel) and the deformation β (bottom right panel). . . . 33 4.9 HFBTHO calculations with SIII parameterization in the mean-field and

mixed pairing interaction for the Fermi level λ _n (top right panel), two-neutron separation energy S _2n = B(Z, N − 2) − B(Z − N) (top left panel), mean neu- tron pairing gap ∆ _n (bottom left panel) and the deformation β (bottom right panel). . . . 33 4.10 HFBTHO calculations with SKM parameterization in the mean-field and

mixed pairing interaction for the Fermi level λ _n (top right panel), two-neutron separation energy S _2n = B(Z, N − 2) − B(Z − N) (top left panel), mean neu- tron pairing gap ∆ _n (bottom left panel) and the deformation β (bottom right panel). . . . 34 4.11 HFBTHO calculations with SLy4 parameterization in the mean-field and

mixed pairing interaction for the Fermi level λ _n (top right panel), two-neutron separation energy S _2n = B(Z, N − 2) − B(Z − N) (top left panel), mean neu- tron pairing gap ∆ _n (bottom left panel) and the deformation β (bottom right panel). . . . 34 4.12 Difference between binding energies obtained from various Skryme force

parameterizations (SLy5, SKM, SLy4, and SIII) and experimental values for even-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 and

mixed pairing interaction for the proton pairing gap ∆ _z . . . . 37 4.15 The convergence of binding energy. . . . 38 4.16 Theoretical and experimental neutron pairing gaps for xenon isotopes. Left

column: neutron pairing gap with j _max = (15/2, 17/2) for proton and neu- tron, respectively. Right column: neutron pairing gap with j _max = 25/2 for both proton and neutron. . . . 40 4.17 Changes in the equivalent energy 3s _1/2 and its occupation probability with

tuning SLy4 parameters t ₀ , t ₁ and t ₂ with different density-dependent pairing

interactions for nickel isotopes . . . . 43

List of figures xvii

4.18 Self-consistent local neutron field at left and neutron pairing field at right in

84 Ni calculated with different values of t ₀ ,t ₁ and t ₂ of SLy4 parameterization

and volume pairing. . . . 44

4.19 Self-consistent neutron density as a function of R at left and neutron pairing density at right in ⁸⁴ Ni calculated with different values of t ₀ ,t ₁ and t ₂ of SLy4 parameterization and volume pairing. . . . . 44

4.20 As same as figure 4.18 with mixed pairing. . . . 45

4.21 As same as figure 4.19 with mixed pairing. . . . . 45

4.22 As same as figure 4.18 with surface pairing. . . . 46

4.23 As same as figure 4.19 with surface pairing. . . . 46

4.24 As same as figure 4.23 with surface pairing (different scale). . . . 47

4.25 Single particle energies of ⁸⁴ Ni in Skyrme HF calculations without pairing with varying external mean-field at the top. The middle and lower panels correspond to Skyrme HFB calculations with the surface and volume pairing interactions, respectively. . . . 48

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

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

4.28 Charge radii of ⁸³ Ni (solid line) and ⁸⁵ Ni (dashed line) with different orbitals being blocked in comparison to that of the ground state of ⁸⁴ Ni (stars line). 51 4.29 Two-particle wave function Ψ ⁽²⁾ for ⁸⁰ Ni top row and ⁸² Ni bottom row. Volume, mixed and surface interaction are shown at the left, center and right column, 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 the scale is different. . . . 53

4.31 Two-particle wave function Ψ ⁽²⁾ for ²² O right column and ²⁶ O left row.

Volume, surface and mixed interaction are shown at the top, middle and

bottom row, respectively. Notice that the scale is different. . . . 54

List of tables

4.1 Isotopes with ∆ _C ⁽³⁾ (N) and ∆ _C ⁽³⁾ (Z) ≥ 2.0 MeV. The values are in MeV. . . . 27 4.2 The fitting parameters of empirical neutron and proton pairing gaps for

even–even nuclei to bA ^m with different OES formulae as functions of A, with 95% confidence bounds. . . . . 27 4.3 The fitting parameters of empirical neutron and proton pairing gaps for

even–even nuclei to bN ^m and bZ ^m with different OES formulae as functions of N and Z (with 95% confidence bounds). . . . 28 4.4 Table over Fermi level λ n in [MeV ], energy level of the single-particle state

of 3s _1/2 in [MeV ], and the occupational probability for ^{80 82 88} Ni. . . . . . 52

Chapter 1 Introduction

One of the main purposes of nuclear physics is to understand the structure of nuclei and its underlying nuclear forces. Considerable progress has been made in nuclear physics due to new opportunities for producing a variety of isotopes by using radioactive beams. The enormous opportunity of the accessibility to a wider range of unstable nuclei makes it both relevant 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 the Coulomb 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. ⁸ He, ¹¹ Li and ¹¹ Be isotopes [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 exotic nuclei to understand the origin of matter [44, 49]. This understanding is only possible by developing sophisticated models that can predict the nuclear structure of the excising isotopes with a small systematic error in comparison to experimental data.

To solve the many-body problem in nuclear physics one can start from the basic degrees

of freedom in the nucleus. In Refs. [7, 8], there are direct solutions to lattice Quantum

Chromodynamics (QCD) for very light isotopes of He with a large systematic error. The Ab

Initio approach, which assumes realistic bare N-N and 3N forces among nucleons, aims to

solve 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-size

nuclei. 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 are

still in their early stages and have limitations due to the large matrix dimensions necessary for

heavier 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 low energy nuclear states. These single-particle states have been used extensively to construct a representation that would describe many-particle states. In the Shell model one assumes an inert core of independent particles and a few correlated valence nucleons. The Shell model is also dependant on the model space and has a difficulty with huge matrices, even though one can tackle the size of exponentially growing matrices in the Shell model by different approaches, 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 model and it lacks by design.

Mean-field theory can profit from distinctive characteristics of the nuclear force. It is well established that the nuclear force is very short range. As a consequence, the nucleons move along orbits with large mean-free paths. One has thus arrived at the very first simple microscopic nuclear treatment, namely that the dynamics of nuclei may be understood within the framework of a Fermi gas model. The Shell model warrants the picture of independent particles moves in an averaged potential. A group of independent particles is confined in a quantum potential, i.e., a so-called mean-field. To construct the mean-field one can take advantage of an effective interaction in a self-consistent manner (SCMF) [9]. In SCMF theory one looks for this averaged potential and tries to explain it by simple nucleon-nucleon interaction. In another word, it produces the Shell model potential microscopically. At the most fundamental level, the Ab Initio theories [51] with their fully microscopic methods cannot reach the heavier nuclei due to the scale of computation. On the other hand, the Hartree-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 about 700 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 can reproduce the experimental results better. Pairing effect is especially crucial for dripline nuclei since the contribution from mean-field is small. Thus, the pairing gap, even with its small value, can play a significant role in nuclear deformation and binding energy, which can make the nuclei bound [40]. One of the most popular models that generalize the mean-field approach with pairing field is the Hartree-Fock-Bogoliubov approach.

The closeness of bound states to the continuum in loosely bound and unbound nuclei makes 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 are

weakly 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 effect on 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 the former force. The Gogny force with its finite range properties makes the HFB-equation into coupled 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. A systematic calculation for deformed nuclei at dripline with an improved harmonic oscillator in 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 straightforward dependency on the nuclear radius will help us to understand the physics more clearly. In this work, we have focused extensively on the study of density-dependent of the pairing interaction. A discussion about this has started from Ref. [29].

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

There is an excellent interest in analyzing the theoretical uncertainty of nuclear properties at dripline [46, 2, 1]. The extreme extrapolation of different models will cause significant differences in the obtained results. Interactions that may describe the structure of stable nuclei 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 calculations with various models can shed light on the strengths of each approach [30]. One may need to look more into why certain kinds of interaction were introduced in the first place and modify them accordingly. Hence, we have performed a systematic comparison between different Skyrme parametrizations and pairing interactions to analyze the consequences of each interaction.

By analyzing the density-dependent pairing interaction in the Hartree-Fock-Bogoliubov

approach, 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 particle

continuum and contribute to the pairing energy of the ground state. In Ref. [28] has

demonstrated that the size of the neutron halo differs drastically by the density dependence of

the pairing interaction. One may state that the physics of weakly bound nuclei may vary from

stable 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 pairing interaction 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 discussed this topic comprehensively in this work.

Overview

The first part of this thesis is mainly focused on empirical pairing gaps, the odd-even

staggering of binding energies, and various formulae that exhibit this phenomenon. A

detailed description of the theory of Hartree-Fock-Bogoliubov is given in Chapter III. The

Hartree-Fock and BCS model are also briefly explained for the sake of completeness. The

numerical and physical results are presented in Chapter IV. In this chapter, a number of

nuclear properties of some spherical nuclei, for instance, nickel and xenon isotopes have

been investigated. Two-particle wave functions of ²² O, ²⁶ O and other isotopes are plotted as

well. Finally, a summary is given in the last chapter.

Chapter 2

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

One outstanding feature in nuclear spectroscopy is the abrupt changes in binding energy as one goes from a nucleus with an even number of neutrons (or protons) to its neighbor with an 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 questions we want to answer is how to evaluate the pairing gap qualitatively and quantitatively from its binding 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 lower states 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 form a pair. Bohr and Mottelson present the idea that such a mechanism can also exist in atomic nuclei. There is overwhelming evidence pointing to the existence of strong pairing correlation in atomic nuclei. For instance, a small value of the nuclear moment of inertia in nuclei can be explained by this effect to some extent. In all even-even nuclei, J = 0 ⁺ (two particles with identical quantum numbers but opposite spin projection) is the lowest energy state. The low-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 two particles.

⟨ j ₁ j ₂ J|V _pair | j ₃ j ₄ J⟩ (2.1)

Two-particles interact not only in their orbit, but also they can scatter to other orbits. Both diagonal and non-diagonal matrix elements 2.1 will play a roll in the whole picture, which it calls 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 active when the pair overlaps spatially? What is the form of such interaction? Is it more efficient around the surface of the nuclei and those particles close to the Fermi level or the pairing interaction is involved in the whole volume of nuclei? Does pairing interaction only exist in bound 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 is the physical mechanism behind it? The discussion has been comprehensive and pointed out this phenomenon extensively. Primarily, it is attributed to pairing effect. Binding energies give us some essential information about the atomic nuclear structure, the net impact of nuclear forces. However, it is very challenging to obtain a more detailed view of the nuclei chart from these energies. It is desirable to maximise the information we can get from such fundamental data. The very first question is how to see the effect of pairing effectively by only investigating systematic of binding energies, before going into the more in-depth microscopic 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] for isotopes, where B is the binding energy and S _n is the one-neutron separation energy.

∆ ⁽³⁾ (N) = − 1

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

= − 1

2 [S _n (N + 1, Z) − S _n (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 former 3-points over neighbouring odd-A nuclei.

∆ ⁽⁴⁾ (N) = 1

4 [−B(N + 1, Z) + 3B(N, Z)

− 3B(N − 1, Z) + B(N − 2, Z)]

= 1

2 [∆ ⁽³⁾ (N) + ∆ _C ⁽³⁾ (N)]. (2.3)

∆ ⁽⁵⁾ (N) = 1

8 [B(N + 2, Z) − 4B(N + 1, Z)

+ 6B(N, Z) − 4B(N − 1, Z) + B(N − 2, Z)]

= 1

4 [∆ _C ⁽³⁾ (N + 2) + 2∆ ⁽³⁾ (N) + ∆ _C ⁽³⁾ (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 one in sequence with both 3-points formulae mentioned above. The figures are plotted with the nuclear binding energies extracted from Refs.[4, 79].

∆ _C ⁽³⁾ (N) = 1

2 [S _n (N, Z) − S _n (N − 1, Z)]

= 1

2 [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 be

dispersed. This grouping may arise from the fact that with ∆ _C ⁽³⁾ formula we can see, which

smooth part of the mean field is still contributing to 3-points OES formula uniformly, and

the quick varying component of the mean-field has been reduced efficiently. Hence, only by

cutting the uniform part, we can get closer to the real value of pairing gap for these types

of nuclei. Fig. 2.2 presents the proton pairing gap for both 3-points formulae for all known

nuclei.

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

0 200 400 600

0 0.5 1 1.5 2

∆ (3) (N)

0 200 400 600

0 0.5 1 1.5 2

∆ (3) C (N)

Fig. 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 nuclei with odd number of neutrons and odd number of protons respectively.

One should bear in mind that the pairing gap is not a direct observable. However, different OES formulae give a reasonably good indication of its value. Many favorable and unfavorable situations make the choice of OES formula for every case cumbersome. For instance, the 5-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 as 570 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 understanding new physics. Furthermore, one cannot be sure if pairing contribution has also been averaged out. 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, ∆ _C ⁽³⁾ for even-even nuclei works fine. In this sense, we have chosen to compare the pairing gap with

∆ _C ⁽³⁾ (N).

2.2 Binding energy odd-even staggering 9

0 200 400 600

0 0.5 1 1.5 2

∆ (3) (Z)

0 200 400 600

0 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 for

even-A nuclei, blue and green squares show odd-A nuclei with odd number of neutrons and

odd 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 most basic step in this task, namely the determination of the nuclear force acting among nucleons, there are significant challenges. The relationship between nucleon-nucleon interaction and the 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 simple of all systems in nuclear physics, namely the free nucleon itself, and this was performed numerically only. It is therefore not surprising that in the context of many-nucleon physics one introduces ad-hoc interactions among nucleons, which are supposed to efficiently reproduce the effects induced by the many degrees of freedom determining the corresponding QCD interaction. However, even assuming that this effective interaction is somehow known precisely, the description of the relevant nuclear spectrum becomes a formidable task. This indicates that one has to perform drastic approximations, which are consistent with the physics 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 that

would describe many-particle states. Such descriptions have been obtained in a number

of ways, but the most significant in relation to this thesis is the HF procedure described

in the following. We have also seen that very simple pairing interaction leads to the BCS

approximation. For stable bound nuclei the BCS approximation provides results that agree

reasonably well with experimental data. But far from the nuclear line of stability, near the

driplines, one needs to enlarge the single-particle representation including states that would

describe the influence of the pairing field upon unstable states, and for that, the BCS model

is not adequate. This is what the HFB representation provides; a proper description that

generalizes 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 briefly following BCS theory. In this chapter, I add the Hartree-Fock method and the BCS approach for reasons of consistency. Furthermore, I present Hartree-Fock-Bogoliubov in general and focus within a space-coordinate system in particular. This thesis tries to shed light on the physical 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 followed by 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 of non-interacting particles, where the following Hamiltonian ˆ H describes exactly this.

H ˆ =

A i=1 ∑

h ˆ _i (3.1)

where ˆh _i 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 in terms of quasi-gas. There is a clear picture of the particles and holes. The variational method produces a set of equations, which is solved iteratively. In a computational sense, the HF method starts with an initial guess, provides a solution, and then takes the answer repeatedly as an input until it converges into the final result or else fails to converge. The trial wave function in the variational method can simply be the direct product of all single-particle states in a complete orthogonal basis ν as follow:

Φ(1, . . . , A) = φ _ν

(1)φ ν

(2) . . . φ ν

(A) (3.2) Which is called a Hartree approximation. One can apply the variational method with this trial wave function and obtain an approximated lowest energy. The two independent particles are not correlated. However, due to the Pauli principle, one needs to make sure that the total wave function for A-nucleon system is anti-symmetrized. This anti-symmetrization introduces a strong correlation between two particles. By using the Slater determinant, one ensures that the permutation of the two nuclei results in an exchange of the total wave function sign.

Φ(1 . . . A) = 1

√ A !

φ _ν

(1) φ _ν

(1) . . . φ _ν

(1)

.. . .. . .. .

φ _ν

(A) φ ν

(A) . . . φ ν

(A)       

(3.3)

3.1 Hartree-Fock approach 13

Where, Φ(1 . . . A) is the A-nucleon wave function, φ _ν

(i) is the wave function of the i-th nucleon in j-state. Depending on the basis, j can be different sets of quantum numbers. The factor ^{√ 1}

A ! normalizes the wave function Φ.

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

|Φ⟩ = |n 1 , n ₂ , . . . , n _ν , . . . ⟩ (3.4) where for fermions n _ν is 0 or 1. In Hartree-Fock approach, a set of Slater determinants Φ(1, . . . , A) is chosen to be the A-body wave function:

|Φ ^HF ⟩ =

A

∏ i=1

ˆ a ⁺ _ν

i

|0⟩ (3.5)

Operators ˆ a ⁺ and ˆ a create/annihilate a particle in a given single-particle state in the basis of ν . They correspond to the single-particle wave function φ _ν

(i), which are the eigenfunctions of the single particle Hamiltonian ˆ h _i . Hence, the energy of the system within this set is to be minimized.

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

ˆt = ∑

i j

t _{i j} a ˆ ⁺ _i a ˆ _j (3.6)

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

V ˆ = 1 2 ∑

i jkl

v _{i jkl} a ˆ ⁺ _i a ˆ ⁺ _j a ˆ _l a ˆ _k (3.7)

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

H ˆ = ∑

k

k

t _k

_k

a ˆ ⁺ _k

1

a ˆ _k

+ 1

4 ∑

k

k

k

k

¯

v _k

_k

_,k

_k

a ˆ ⁺ _k

1

a ˆ ⁺ _k

2

a ˆ _k

a ˆ _k

(3.8)

where indices k _i cover all degrees of freedom of all available single particle states. ¯ v is the

antisymmetric two-body interaction matrix element. To determine the ground state energy

and the wave function Φ ^HF , we can vary the wave function |δ Φ⟩ = ε ˆ a ⁺ _m a ˆ _i |Φ⟩ with m > A

and i ≤ A. We assume the set of wave functions is orthonormal ⟨Φ|Φ⟩ = 1. The variational

14 The Hartree-Fock-Bogoliubov approach

equation is written:

⟨δ Φ| ˆ H |Φ⟩ = ε ^∗ ⟨Φ| ˆ a ⁺ _i a ˆ _m H ˆ |Φ⟩ = 0 (3.9) Inserting 3.8 into 3.9 one obtain [66]

h _mi = t _mi +

A

∑

j=1

(v _{m ji j} − v _{m j ji} ) = 0 (3.10)

h _kk

= t _kk

+

A i=1 ∑

¯

v _kik

i = e _k δ _kk

(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 are mixed, 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 a phenomenological interaction with density dependence like Skyrme forces and obtain various nuclear 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. States that are not purely occupied or empty. There is a combination of states that a pair will be 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 the mixing of single-particle states and making the quasi-particle states is more evolved, and it is not only for conjugate pairs.

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

|BCS⟩ = ∏

k>0

(u _k + v _k a ⁺ _k a ⁺ _¯k )|−⟩ (3.12)

∝ |−⟩ + ∑

k>0

v _k

u _k a ⁺ _k a ⁺ _¯k |−⟩ + 1

2 ∑

kk

>0

v _k v _k

u _k u _k

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 the probability of v ² _k and u ² _k , 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 that BCS state is a superposition of different number of pairs and not related exactly to the number of particles. Using this trial wave function 3.12 and varying one of the parameters v _k or u _k (v _k and u _k are dependent through normalization requirement |u _k | ² + |v _k | ² = 1|), one can minimize the auxiliary Hamiltonian ˆ H

= ˆ H − λ N, where ˆ H is 3.8 with trial wave function

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

v ² _k = 1 2



1 − e ˜ _k q

˜ e ² _k + ∆ ²



 u ² _k = 1 2



1 + e ˜ _k q

˜ e ² _k + ∆ ²



 (3.14)

where ∆ is defines as

∆ _k = − ∑

k

>0

¯

v _k¯kk

¯k

u _k

v _k

(3.15)

3.3 The Bogoliubov transformation

In the following section I try to show that in contrast to other methods, HFB will include pairing interaction profoundly at the microscopic level, and is already incorporated in the mean-field. Bogoliubov suggested in Ref. [15] the use of especial transformation so-called Bogoliubov 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 are both at comparable sizes and equally important. Bogoliubov transformation 3.16 enables us to transfer 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 solution of the many-body problem defined by Migdal and Landau. Bogoliubov transformation 3.16 is a linear combination of particle creation a ⁺ and annihilation operators a in the vacuum of quasi-particle operators |−⟩. A simple relationship between quasi-particles operators and bare particle/hole operators is the following:

α ˆ _k ⁺ = u _k a ˆ ⁺ _k − v _k a ˆ _¯k (3.16)

α ˆ _¯k ⁺ = u _k a ˆ ⁺ _¯k + v _k a ˆ _k

16 The Hartree-Fock-Bogoliubov approach

One can recover particle-hole states as in HF approach from this transformation, by setting u _k = 1, v _k = 0 for k above the Fermi level and u _k = 0, v _k = 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 defines with the above transformation as following [66]:

β ˆ β ˆ ⁺

!

= U ⁺ V ⁺ V ^T U ^T

! c ˆ ˆ c ⁺

!

W = U V ^∗

V U ^∗

!

(3.17)

Operators ˆ β ⁺ and ˆ β create and annihilate quasi-particles with some general particle operators ˆ

c ⁺ _k , ˆ c _k . U,V are the two components of quasi-particle wave function, which are chosen in a way that ˆ β ⁺ and ˆ β meet the anti-commutation relation. For instance if ˆ β _k for all k over the whole configuration space applies onto a |BCS⟩ state, it will vanish ( ˆ β _k |BCS⟩ = 0), which means 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 |Φ ₀ ⟩ of Hamiltonian 3.8 by using a Bogoliubov transformation 3.17 and variational method. In HFB the particles and holes are mixed, and for that reason the particle number is not conserved. Hence, the particle number operator ˆ N = ∑ i c ˆ ⁺ _i c ˆ _i sets as a constraint with a Lagrangian multiplier λ as H ˆ − λ ˆ N. Transforming Hamiltonian 3.8 according Bogoliubov formalism 3.17 one yields [66]:

H ˆ − λ ˆ N = ˆ H ₀ + ˆ H ₁₁ + ˆ H ₂₀ + ˆ H ₄₀ + ˆ H ₃₁ + ˆ H ₂₂

| {z }

Neglected

(3.18)

The indices of ˆ H _{i j} are the number of creation/annihilation operators in the respective term, for instance ˆ H ₀ is the value of the vacuum energy with no quasi-particle operators, and H ₁₁ is one quasi particle-quasi hole excitation energy. The other terms with four quasi-particle operators may be ignored. Varying this as introduced in [66] to uniquely describe |Φ ₀ ⟩, one obtains the requirement ˆ H ₂₀ = 0. One can conclude that HFB in general is diagonalizing the following super-matrix in a quasi-particle basis:

H ˆ ₁₁ H ˆ ₂₀

− ˆ H ₂₀ ^∗ − ˆ H ₁₁ ^∗

!

(3.19)

3.3 The Bogoliubov transformation 17

−→ In the space of particle operator c _l , c ⁺ _k : H = h − λ ∆

−∆ ^∗ −(h − λ ) ^∗

!

(3.20)

Where, H ₁₁ includes the sum of terms of operators β _k ⁺ β _¯k and H ₂₀ includes the sum of terms of β _k ⁺ β _¯k ⁺ + β _k β _¯k .

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

h = t + Γ, Γ _kl = ∑

i j

¯

v _{k jli} ρ _{i j} , ∆ _kl = 1 2 ∑

i j

¯

v _{kli j} κ _{i j} (3.21)

where ρ and κ can define the wave function |Φ 0 ⟩ uniquely by the two components U and V : ρ _{i j} = ⟨Φ 0 | ˆc ⁺ _j c ˆ _i |Φ 0 ⟩ = (V ^∗ V ^T ) _{i j} (3.22) κ i j = ⟨Φ 0 | ˆc _j c ˆ _i |Φ 0 ⟩ = (V ^∗ U ^T ) _{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| ˆ 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 U k ,V _k :

h − λ ∆

−∆ ^∗ −(h − λ ) ^∗

! U _k V _k

!

= U _k V _k

!

· E _k (3.26)

Here, we will have independent quasi-particles moving in a potential which contains both

mean-field generated from all single quasi-particles plus the pairing interaction between such

particles.

18 The Hartree-Fock-Bogoliubov approach

3.4 Hartree-Fock-Bogoliubov equation in coordinate space

In coordinate space, once we discretize the quasi-particle space in a spherical box with radius R and using spherical symmetry, we can solve a set of HFB differential equations with Skyrme force in one dimension. In coordinate space, the correct treatment of continuum makes 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 near the drip. They lack a good description of coupling to the continuum. Wave functions in Harmonic oscillator basis are local. Hence, they cannot describe the non-locality and the long tail of neutron wave function in the proximity of the particle continuum.

In coordinate space, the wave function is represented by its spatial coordinate, which facilitates the coupling to the particle continuum. With assuming spherical symmetry the wave function for neutron or proton represents only by its radial coordinate r and the spin projection σ = ± ¹ ₂ as following:

a ⁺ _rσ = ∑

k

ψ _k ^∗ (rσ )a ⁺ _k and inverse a ⁺ _k = ∑

σ

Z

d ³ rψ _k (rσ )a ⁺ _rσ (3.27)

where ψ _k is the wave function of the kth single-particle state and form a complete orthonormal basis.The requirement for these wave functions is that they vanish at large distances. The energy spectrum E is discrete for E _n < −λ and continuous for E > −λ . In a spherical box, one can discretized the particle continuum by requiring that wave functions vanish at the box radius R _box . Large R _box can simulate the real continuum very well [30], and the summation over discrete E can replace the integral for the continuous E very well (see Eqs. 3.28 and 3.29).

ρ (rσ , r ^′ σ ^′ ) = ∑

0<E

<−λ

V (E _n , rσ )V ^∗ (E _n , r ^′ σ ^′ ) + Z _∞

−λ dn(E)V (E _n , rσ )V ^∗ (E _n , r ^′ σ ^′ )(3.28) ρ (rσ , r ˜ ^′ σ ^′ ) = − ∑

0<E

<−λ

V (E _n , rσ )U ^∗ (E _n , r ^′ σ ^′ ) − Z _∞

−λ dn(E)V (E _n , rσ )U ^∗ (E _n , r ^′ σ ^′ )(3.29)

The integral over the energies can be replaced by a sum, and an energy cut-off will be

introduced 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 a clustering feature. Here we try to look closer into their physical meaning and how to obtain them. 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 states are localized in space in the sense that they do not have their asymptotic tail as quasi-particle states. They are the eigenstates of a density matrix, which is entirely defined by the lower component of HFB solution V.

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

Z

d ³ r ^′ ∑

σ

ρ (rσ , r ^′ σ ^′ )ϕ α (r ^′ σ ^′ ) = V _α ² ϕ _α (rσ ) (3.30)

where r, σ represent the coordinate vector and spin projection ± ¹ ₂ . The index α is the set of quantum numbers n, l, j for only one type of particle in our work. ϕ _α (rσ ) is the canonical wave function for the set of quantum numbers α. One can represent the complete and orthogonal sets of wave function ϕ _k in some arbitrary complete and orthogonal set of single-particle wave functions φ _k with corresponding operators c _k , c ⁺ _k as follow:

ϕ _k = ∑

l

D _lk φ _l → a ⁺ _k = ∑

l

D _lk c ⁺ _l (3.31)

This is a transformation among single-particle wave functions themselves and the unitary transformation D will not change the anti-commutation relations for fermions. Density matrix ρ will be diagonal in the basis of a ⁺ _k , a _k . This basis define the quasi-particle vacuum completely as well.

Natural orbits represent a basis, which the density matrix is diagonal in it. This means that they are equivalent to canonical states [30]. In other words, a canonical basis represents the natural states of the independent quasi-particle many-body states. To produce a BCS-like type of wave function from HFB, one can use the canonical wave function obtained from the diagonalization of one-body density matrix ρ [3]. To get the canonical basis, one needs to solve 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 and

ordinary transferred quasi-particles basis of the system in the particle-hole channel[25]. If

there is a significant difference between these two, there may be some interesting physical

feature in the Hamiltonian that has lifted the degeneracy.

20 The Hartree-Fock-Bogoliubov approach

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 the following formula [21]:

V _pair (r, r ^′ ) = V ₀



1 − η ρ (r) ρ ₀



δ (r − r ^′ ) (3.32)

where V ₀ is the pairing strength and can be obtained by fitting to the OES parameter for a desirable isotope. ρ(r) is the iso-scalar nucleonic density and ρ ₀ = 0.16 f m ⁻³

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

Since the HFB equations are solved in a discretized box with radii R, one should realize the effect of the box radii R on the average pairing gap ⟨∆⟩. A schematic picture is given in Fig 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-quasiparticle HFB wave function φ _HFB (E, R _box ) = 0 [78]. There will be the same effect with increasing the number of particle in a box with a fixed R. The continuum wave functions will be pushed into smaller region closed to the surface of the nucleus and make the density-dependent pairing gap ⟨∆⟩ larger.

In this work we use two different theoretical pairing gaps to compare with the OES

parameters, the pairing gap for the lowest canonical state ∆ _LCS [50] and the mean gap ∆ _mean

3.4 Hartree-Fock-Bogoliubov equation in coordinate space 21

R

R θ

2 1

Fig. 3.2 The model of clustering two nucleons

[24].

∆ _LCS = ⟨ϕ LCS |∆|ϕ LCS ⟩ (3.33)

∆ mean = − 1

A 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 neutrons can help us to analyze the clustering feature of such particles. A major part of a microscopic description of clustering lies on the coupling of single-particle wave functions with the continuum. Since the continuum part of a single particle wave function is formulated properly in HFB within coordinate-space, one can obtain a correct description for such clusters. A simple cluster of two nucleons is considered here, which the model of the cluster description is shown in figure 3.2. Two neutrons (1) and (2) are at the same distance R from the core, and θ 12 is the angle between the two neutrons.

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

Ψ ⁽²⁾ (r ₁ , r ₂ , θ 12 ) = 1

4π ∑

pq

r 2 j _p + 1

2 δ _l

_l

δ j

j

X _pq φ p (r ₁ )φ p (r ₂ )P _l

(cos θ 12 ) (3.35)

X _pq = u _p v _q (3.36)

where φ is the canonical single-particle wave function obtained from the results of HFB

calculations. P _l

is the Legendre polynomial, and X _pq is the expansion coefficient, which

corresponds to the product of variational parameters u _p v _q within the HFB approach. The

22 The Hartree-Fock-Bogoliubov approach

summation over p and q includes the different configurations of Ψ ⁽²⁾ . In this work, I obtain Ψ ⁽²⁾ as a function of θ ₁₂ and radius r ₁ = r ₂ , which is chosen to be 9 fm. Hence, one can obtain the two-neutron wave function |Ψ ^2ν (r, r, θ )| for some exciting nuclei to analyze the effect 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 to calculate the binding energy of the neighboring odd nuclei. Hence, in this section, we try to explain 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 the canonical states. The ground state of odd nuclei is constructed in this basis by a one-quasi- particle excitation as following:

α _k ⁺

1

|BCS⟩ = a ⁺ _k

1

∏

k̸=k

(u _k + v _k a ⁺ _k a ⁺ _¯k )|−⟩ (3.37) α _¯k ⁺

1

|BCS⟩ = a ⁺ _¯k

1

∏

k̸=k

(u _k + v _k a ⁺ _k a ⁺ _¯k )|−⟩

Or directly by Bogolibov transformation:

β _δ = ∑

k

U _kδ ^∗ c _k +V _kδ ^∗ c ⁺ _k (3.38)

β ⁺

δ = ∑

k

V _kδ c _k +V _kδ c ⁺ _k (3.39)

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

ρ _kk ^δ

= (V ^∗ V ^T ) _kk

+ (U _kδ U _k ^∗

δ −V _kδ ^∗ V _k

δ ) (3.40)

κ _kk ^δ

= (V ^∗ U ^T ) _kk

+ (U _kδ V _k ^∗

δ −V _kδ ^∗ U _k

δ ) (3.41)

Where the index δ denotes the state that is blocked in every iteration. In this work, we

have omitted the time-odd field, and to preserve time-reversal symmetry by equal-filling

approximation, a half of nucleon will place in a given orbital δ and the other half will occupy

the time-reversed state. By neglecting time-odd channel, equal-filling approximation and

exact blocking will be equivalent for one-quasi particle excitation [73]. Depending on which

state δ is to be blocked, one can obtain HFB energies and the occupation probabilities. In this