Applications of Effective Field Theory/Density Functional Theory to Ground-State Properties of Nuclei Far from Stability

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Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects

2003

Applications of effective field theory/density functional theory to ground-state properties of nuclei far from stability

Marco Antonio Huertas Chacon College of William & Mary - Arts & Sciences

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Recommended Citation Huertas Chacon, Marco Antonio, "Applications of effective field theory/density functional theory to ground-state properties of nuclei far from stability" (2003). Dissertations, Theses, and Masters Projects. Paper 1539623424. https://dx.doi.org/doi:10.21220/s2-0mjr-vd11

This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. APPLICATIONS OF EFFECTIVE FIELD THEORY/DENSITY FUNCTIONAL THEORY TO

GROUND-STATE PROPERTIES OF NUCLEI FAR FROM STABILITY

A Dissertation

Presented to

The Department of Physics

The College of William and Mary in Virginia

In Partial Fulfillment

Of the Requirements for the Degree of

Doctor of Philosophy

Marco Antonio Huertas Chacon

2003

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET

This dissertation is submitted in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy

Marco Antohjjo IHuertas Chacon

Approved, May 2003

D r/John D. Walecka

Dr. Henry Kr^kauer

Dr. Carl E. Carlson

Dr. K eith iriffioen -""X

Dr. Rocco Schiavilla (Thomas Jefferson National Accelerator Facility)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DEDICATION

To the memory of my father.

iii

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Page

Dedication iii

Acknowledgments v

List of Tables vi

List of Figures vii

Abstract xii

Introduction 2

Chapter I. Convergence of EFT/DFT approach for doubly-magic nuclei far from stability. 45

Chapter II. Applications to electroweak transition rates in selected nuclei far from stability. 72

Chapter III. Extensions to semi-magic nuclei far from stability. 96

Chapter IV. Conclusions 118

Appendix 120

Bibliography 172

V ita 178

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS

I want to thank my wife Irayda for all her love and support during these past years. For always

being there when I needed her and for helping me in difficult times.

I want to express my appreciation to Professor John Dirk Walecka, under whose supervision this

investigation was conducted, for his patient guidance and support throughout my years of study.

I am also indebted to the members of my thesis committee, who brought up interesting comments

and suggestions which improved the quality of the final work.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES

Table Page

1 Values of the different parameters in each parameter set used to solve the self-consistent

mean-field equations from [1, 9]. Here m s is in M eV ...... 58

2 Results of extracting the configuration energy from the experimental data on S2iV

using the pairing model described in the thesis ...... 70

3 Calculated /3-decay half-lives (in ms) of 1|oSn83. The quantity in parenthesis on the

column exp indicates the error (e.g. 10(2) means 10 ± 2) ...... 77

4 Calculated /3-decay half-lives (in ms) of ^gln^.The quantity in parenthesis on the

column exp indicates the error (e.g. 10(2) means 10 ± 2) ...... 77

5 Matrix elements for each of the multipole operators. The following notation is used

nik) = k(Xp - \ n)/2m and r]{k) — 1 — k2/(m l + q2), where q2 = k2 - uiq and k = |gj.

Here w0 is the total decay energy ...... 83

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES

Figure Page

1 Total binding energy of the selected nuclei. The left hand side shows two models with

the same level of approximation but fit to different data sets (see text). The right

hand side shows models with increasing level of approximation. NLC and Q1 have

the same level of approximation, as do Gl and G2 ...... 31

2 Percentage deviation of the total binding energy for even-even Sn-isotopes using Ql,

Gl and G2 parameter sets. The stable isotopes are indicated in the plot ...... 32

3 Level spectrum of isotones of 1foSng2 differing by one proton ...... 33

4 Calculated results of the /3-transition rates in the cases of particle-particle transitions

and the experimental values. The asterisk denotes an excited state ...... 35

5 Particle and hole states neighboring even-even N=28 isotone nuclei. Binding energies

are considered positive. Only nuclei with particle and hole states with respect to filled

j-shells have been plotted. The values of Z correspond to the even-even nuclei in the

middle row. Calculated F values are included and, unless indicated otherwise, are in

agreement with those of the experimental ground state. Where the experimental J77

value disagrees with the calculated one, or is uncertain, it is also included in the figure. 37

6 Particle and hole states neighboring even-even Z=28 isotopes. Binding energies are

considered positive. Only nuclei with particle and hole states with respect to calcu

lated filled j-shells have been plotted. The value of N corresponds to' the even-even

isotopes in the middle of row. Calculated J77 values are included and, unless indi

cated otherwise, are in agreement with the experimental ground states. Where there

is disagreement with the experimental J77 value, or it is uncertain, the experimen

tal ground-state J77 value is also indicated in the plot. In addition, in this case, the

lowest-lying excited state with the calculated J77 is also included ...... 38

vii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 Total binding energy of the selected nuclei. The left hand side shows two models with

the same level of approximation but fit to different data sets (see text). The right

hand side shows models with increasing level of approximation. NLC and Ql have

the same level of approximation as do Gl and G2 ...... 59

8 Percentage deviation of total binding energy of the four selected doubly-magic nuclei.

Parameter sets Ql, Gl and G2 are shown ...... 61

9 Comparison between experimental and calculated total binding energies for Sn-isotopes

using the G2 parameter set ...... 62

10 Percentage deviation of the total binding energy for Sn-isotopes using Ql, Gl and G2

parameter sets. The stable isotopes are indicated in the plot ...... 63

11 Level spectrum of isotopes of 1foSng2 differing by one neutron ...... 64

12 Level spectrum of isotones of 1foSn82 differing by one p ro to n ...... 64

13 Level spectrum of isotopes of ^ S n s o differing by one neutron ...... 65

14 Level spectrum of isotones of ^oSnso differing by one proton ...... 65

15 Level spectrum of isotopes of ll^iso differing by one neutron ...... 66

16 Level spectrum of isotones of 2|N i5o differing by one proton ...... 66

17 Comparison between single-particle level spectrum of 1|?Sbg2 and recent experimental

d a t a ...... 67

18 Comparison between single-particle level spectrum of ^fSnss with recent experimental

d a t a ...... 67

19 Proton and neutron densities for 1!oSn82 ...... 68

20 Proton and neutron densities for ^ S n 5 0 ...... 68

21 Comparison between neutron densities of ^gSn^ and ^ qShso...... 69

viii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Comparison between configuration energies for Sn-isotopes. The dashed line corre

sponds to calculated values using the G2 parameter set and the solid one is the value

extracted from the experimental data on two-neutron separation energies S2n using

the simple pairing force model described in the thesis ...... 71

23 Single-particle level spectrum of parent and daughter nuclei in the case of particle-

particle transitions. All energy levels are measured from the calculated or experimental

ground-state level, accordingly ...... 87

24 Single-hole level spectrum of parent and daughter nuclei in the case of hole-hole transi

tions. All energy levels are measured from the calculated or experimental ground-state

level, accordingly ...... 89

25 Upper and lower components of the initial and final nucleon wave functions. This cor

responds to a particle-particle transition proceeding from the ground-state of 1|oSn83

to the ground-state of 90

26 Upper and lower components of the initial and final nucleon wave functions. This cor

responds to a particle-particle transition proceeding from the ground-state of

to an excited state of 'IfSbga ...... 90

27 Upper and lower components of the initial and final nucleon wave functions. This

corresponds to a hole-hole transition proceeding from the ground-state of 1fgln82 to

the calculated ground-state of ^ S n s i ...... 91

28 Upper and lower components of the initial and final nucleon wave functions. This

corresponds to a hole-hole transition proceeding from the ground-state of ^ g h ^ to

an excited state of ^ S n si ...... 91

29 Calculated results of the /^-transition rates in the cases of particle-particle transitions

and the experimental values. The crosses indicate the calculated values, the filled

circles represent the experimental data ...... 92

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Calculated results of the /^-transition rates in the cases of hole-hole transitions and the

experimental values. The crosses indicate the calculated values, the arrows indicate

the upper limit given by experimental data ...... 93

31 Magnitude of the multipoles defined in Eqs. (59)-(62) for J=0 and J=l. These

multipoles are the dominant terms contributing to the particle-particle transition un

dergoing between the ground-states of ^Sngs and 1|fSb82- In this figure Mj = M j,

Lj = Cj, T j1 = T f and T^ag = 7jmas ...... 94

32 Calculated and experimental values of the total binding energy for even N=28 isotones.

Energies are positive and measured in MeV ...... 104

33 Calculated and experimental values of the total binding energy for even N=50 isotones.

Energies are positive and measured in MeV ...... 105

34 Calculated and experimental values of the total binding energy for even N=82 isotones.

Energies are positive and measured in MeV ...... 105

35 Calculated and experimental values of the total binding energy for even N=126 iso

tones. Energies are positive and measured in MeV ...... 106

36 Percentage difference between calculated and experimental values for all even isotones. 106

37 Particle and hole states neighboring even-even N=28 isotone nuclei. Binding energies

are considered positive. Only nuclei with particle and hole states with respect to filled

j-shells have been plotted. The values of Z correspond to the even-even nuclei in the

middle row. Calculated J,r values are included and, unless indicated otherwise, are in

agreement with those of the experimental ground state. Where the experimental J ’1'

value disagrees with the calculated one, or is uncertain, it is also included in the figure. 107

38 Same as in Fig. 37 for N=50 isotones ...... 108

39 Same as in Fig. 37 for N=82 isotones. The position of the (11/2-) metastable state of

147Tb has been shifted so it can be seen in this energy scale. It actually lays 50 keV

above the ground-state ...... 108

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Same as in Fig. 37 for N=126 isotones ...... 109

41 Calculated and experimental values of the total binding energy for even Z=28 isotopes.

Energies are positive and measured in MeV ...... 110

42 Calculated and experimental values of the total binding energy for even Z=5Q isotopes.

Energies are positive and measured in MeV ...... 110

43 Calculated and experimental values of the total binding energy for even Z=82 isotopes.

Energies are positive and measured in MeV ...... I l l

44 Percentage difference between calculated and experimental values for all even isotopes

considered ...... I l l

45 Particle and hole states neighboring even-even Z=28 isotopes. Binding energies are

considered positive. Only nuclei with particle and hole states with respect to calcu

lated filled j-shells have been plotted. The value of N corresponds to the even-even

isotopes in the middle of row. Calculated J1 values are included and, unless indi

cated otherwise, are in agreement with the experimental ground states. Where there

is disagreement with the experimental value, or it is uncertain, the experimen

tal ground-state value is also indicated in the plot. In addition, in this case, the

lowest-lying excited state with the calculated J 71' is also included ...... 112

46 Same as in Fig. 45 for Z=50 isotopes ...... 114

47 Same as in Fig. 45 for Z=82 isotopes ...... 115

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT

The applicability of a new approach, developed by Furnstahl, Serot and Tang, to describe the nu

clear many-body system is studied for nuclei far from stability. This approach combines elements of

effective field theory (EFT) and density functional theory (DFT) and has already been tested in the

stability region. In this thesis, several steps are taken to address the question of the applicability

of the EFT/DFT formulation to nuclei far from stability. First the convergence of the approach

is studied by applying it directly to selected doubly-magic nuclei far from stability. An indepen

dently developed code, which can incorporate various levels of approximation of the chiral effective

lagrangian, is used to solve the self-consistent relativistic Hartree equations. This program is used to

obtained results for ground-state properties such as binding energies, single-particle level structure,

and densities of the doubly-magic nuclei 132-100Sn and 48,78Ni. Second, the accuracy of the calcula

tion of the ground-state density is indirectly studied by using the single-particle (Kohn-Sham) wave

functions to calculate weak transitions. The weak currents used correspond to the Noether currents

derived from the effective lagrangian. The calculations use a general expression for single-particle

transition matrix elements, derived here, from which any semi-leptonic weak rate can be calculated.

Attention is focused on beta-decay processes in nuclei neighboring 132 Sn and the results are com

pared with available experimental data. Finally calculations are extended to determine total binding

energies and single-particle and single-hole binding energies, spins, and parities for isotones with

magic numbers N = 28, 50, 82, 126, and isotopes with Z = 28, 50, 82. The results of this thesis

indicate that the EFT/DFT formulation of Furnstahl, Serot and Tang is capable of reproducing the

relevant ground-state binding energies to within 1%, chemical-potentials with an average accuracy

of 10%, and ground-state spins and parities, at least if the neutron level density is not too high.

Xll

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPLICATIONS OF EFFECTIVE FIELD THEORY/DENSITY FUNCTIONAL THEORY TO

GROUND-STATE PROPERTIES OF NUCLEI FAR FROM STABILITY

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION

Atomic nuclei far from stability are the systems under investigation in this dissertation. In recent

years, the study of these nuclei has drawn significant attention from the nuclear physics community.

Nuclei far from stability play an important role in the nucleosynthesis of heavy elements, beyond

iron. During the explosion of Type-II supernovae, some heavy elements are produced as a result of

two competing processes of rapid neutron capture and /3-decay, a mechanism called the r-process.

Knowledge of the detailed structure of these nuclei, as well as the reaction cross-sections relevant

to these processes, is important to predict the rates of production of these heavy elements. Addi

tionally, new experimental facilities and the implementation of new techniques have given valuable

experimental evidence of several new nuclear phenomena, e.g. halo nuclei, neutron skins, and proton

radioactivity, to mention some. In addition to this, new generations of accelerators will make some

remote regions of the nuclear chart available for study, while experiments on other regions have

already produced enough reliable experimental data that can be compared with predictions from

several models.

In this work, a theoretical approach is used to study selected nuclei fax from stability, which are

used as a testing ground to investigate the range of applicability of this new approach to nuclear

structure. This formulation emerges by combining elements of effective field theory (EFT) and

density functional theory (DFT). This EFT/DFT approach to study the nuclear many-body system

has been pioneered by Furnstahl, Serot and Tang [1]. Although the EFT/DFT theory has already

shown satisfactory results for medium-heavy nuclei along the stability line, it has not been tested

outside this specific region of the nuclear chart.

This task is undertaken here by applying the EFT/DFT formulation to calculate ground-state

properties of some selected nuclei far from stability. The results of this study indicate that this

approach can reproduce such properties in a consistent way for all nuclei tested, thus validating its

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applicability outside the stability region.

In this introduction, a brief description of some properties of the atomic nucleus is given. Both

effective field theory and density functional theory are reviewed, and the EFT/DFT formulation to

describe the nuclear many-body problem is presented.

1 The atomic nucleus and the nuclear force.

The atomic nucleus is a complex system composed mainly of two types of particles: the proton

and the neutron, which together are called nucleons. These particles are fermions with an intrinsic

angular momentum, or spin, equal to h/2. In addition to spin, nucleons possess an additional degree

of freedom called isospin, which has similar mathematical properties as angular momentum. Nucleons

have an isospin value equal to 1/2. The proton and neutron have isospin third-components equal to

1/2 and -1/2, respectively. The proton and the neutron have almost the same mass, the neutron

being slightly heavier (« 1.3 MeV/c2). The total charge of an atomic nucleus comes solely from the

protons in it. Each proton has a charge equal to +e, where e is the magnitude of the electric charge

of the electron.

Nucleons interact through the nuclear force. In addition, protons also interact via the electromag

netic interaction. The nuclear force between nucleons has several properties, among these it is: a)

attractive, as can be appreciated from the fact that the deuteron, a two-nucleon system, is a bound

state, b) short-ranged, with an extension of a few fm (1 fm = 10~13 cm), c) charge-independent

which means that the force is the same between different pairs of nucleons regardless of their charge,

d) spin-dependent, and e) at very short distances, ss 0.4 to 0.5 fm, it has a repulsive core.

The exact analytical form of this force is still a m atter of theoretical investigation, and the pages of

the history of nuclear physics are filled with incredible theoretical efforts to describe and understand

it [2, 3]. The complexity of the task can be appreciated from the various dynamical effects observed

in nuclei. For instance, nuclei can have rotational and vibrational states, they can be deformed and

they have an internal structure similar to that of the atom. All these effects are the result of the

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dynamics generated by the nuclear force.

Nuclei are characterized by their mass number A and total number of protons Z. Since any atom

is electrically neutral, the number of electrons in any given atom equals the number of protons in

the nucleus. These electrons determine the chemical properties of the atom, thus different values of

Z correspond to different chemical elements. The mass number A equals the sum of protons (Z) and

neutrons (N) in the nucleus, i.e. A — Z + N, and the following notation is used here to label them:

where X stands for the name of the chemical element. An important characteristic of nuclei

is that for a given value of Z there are several nuclei with different numbers of neutrons. Such nuclei

are called isotopes. For example, 1|Og and O7 are two different isotopes of oxygen with 8 and 7

neutrons, respectively.

Protons repel each other due to their electric charges. Neutrons, being electrically neutral,

interact with the protons to produce a net attracting effect that makes the nucleus a bound system.

The number of neutrons in a given nucleus is large for large values of Z, and in general N>Z. On the

contrary, for small nuclei the number of neutrons is approximately equal to the number of protons.

Plotting the number of protons versus the number of neutrons for all nuclei gives rise to a graph

known as the nuclear chart. In such a graph one can observe how nuclei are distributed, and as the

size of nuclei increases, this distribution deviates from the Z = N line. An interesting question arises

by considering the maximum number of neutrons that a nucleus with fixed Z can hold. In principle

one would think that a nucleus with Z protons could hold any number of neutrons together once

it has overcome the electric repulsion due to the proton charge. The fact is that as the number of

neutrons grows in nuclei, they become less bound until the nucleus cannot hold the neutrons together

and they start dripping from it. In a similar fashion, for any given number of neutrons, N, there is

a limit to the number of protons a nucleus can have before it breaks down. Nuclei with the same

number of neutrons are called isotones. The exact location of the proton and neutron drip lines on

the nuclear chart is a mayor area of research pursued both at the experimental and theoretical levels.

Although neutrons inside nuclei help to work against the breakdown of the nucleus caused by

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the Coulomb repulsion, several isotopes are unstable and they become radioactive. Radioactivity is

the process by which a nucleus looses its excess energy by emitting it in the form of electromagnetic

radiation, or particle emission. There are several types of radioactivity characterized by the form in

which energy is released. The most common forms are:

1. a-decay: here the process results in the emission of a He nucleus, 2^ 2, called an a-particle.

This type of radiation is common in heavy elements such as Uranium.

2. /Udecay: here one type of nucleon is transformed into another by the emission of leptons. This

form of radioactivity is common in medium-sized nuclei and occurs when there is an excess of

either neutrons or protons in the nucleus.

3. 7-decay: this corresponds to the emission of highly energetic photons, typically of several MeV,

as a result of a transition from an excited state to the ground-state or to lower excited states

of the nucleus.

Nuclei that do not decay are called stable. These nuclei are located in the central region of the

nuclear chart, following very closely the line defined by Z = N. This region is referred as the valley

of stability. The stability it refers to is against /?-decay. Nuclei that lie far outside of this region are

therefore termed nuclei far from stability.

As already mentioned, nuclei show an internal structure similar to that observed in atoms. This

structure is manifested experimentally by observing that the binding energy of a nuclear system

does not change smoothly when nucleons are added. In fact there are sharp changes occurring at

specific values of the number of nucleons in the system. This special numbers of protons or neutrons

are called magic numbers. In the atomic case, the quantum mechanical picture indicates that the

internal level structure of electrons is arranged in energy shells, such that when one of these shells

is completely full the atom becomes almost chemically inert; i.e. the configuration is stable. The

closure of these atomic shells occurs in for the noble gases.

This shell picture can also be applied in the case of the atomic nucleus. The energy levels of

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neutrons and protons are grouped in sets called shells. These shells, when filled, give the nucleus a

particular stability. The observed sequence of magic numbers for either neutrons or protons is 2, 8,

20, 28, 50, 82 and 126. In this shell picture, this means that the first shell is filled with 2 nucleons,

(protons and neutrons levels are filled separately) the second shell is filled adding six more nucleons

until it reaches 8, and so on. If nuclei have both Z and N equal to any of these magic numbers the

nucleus is called doubly magic. Examples of these are: ^Og, f$Ca2o, 20^a28 and 1|oSnS2. If either

the proton or neutron number is equal to a magic number then the nucleus is called a semi-magic

nucleus.

The origin of these magic numbers is related to the dynamics of the nuclear system imposed

by the nuclear force. One of the earliest attempts to explain this property was to assume that the

nucleons inside a nucleus moved in a three dimensional central potential [2]. This central potential

can be considered as the collective, many-body effect of all the nucleons interacting with a single

one. Since the exact form of this potential is unknown, different functional forms were used based on

data of nucleon-nucleon scattering and on the relevant symmetries observed in nature and nuclear

phenomena.

If this central potential is spherically symmetric then the energy eigenstates are also eigenstates of

the total angular momentum, denoted here as f 2. Each energy level has a degeneracy equal to 2j + l,

where j is the total angular momentum quantum number. These levels correspond to the allowed

values of m = j-. Nucleons fill the single-particle energy levels according to the Pauli exclusion

principle until 2 j + 1 nucleons occupy each level. When a particular level of given j is full, nucleons

start filling the next one until all A nucleons in the nucleus are distributed among the lowest energy

levels. Proton and neutron levels are filled separately following this principle. Unfortunately, the

observed nuclear shell structure could not be entirely reproduced solely by the assumption of this

central potential. For instance, if the central potential is assumed to be a three-dimensional square

well with infinite walls, only the first two magic numbers, 2 and 8, could be reproduced [2]

The addition of a spin-orbit interaction term to the central potential modified the internal struc

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ture produced by the central potential only, shifting various energy levels according to their angular

momentum quantum numbers [4]. The effect of this extra term made some energy levels group

together in larger sets called major shells, which separated by energy gaps much larger than the

energy separation between individual levels in each set. The individual levels in a major shell, which

still have a degeneracy of 2j + 1, are called sub-shells. Each major shell, when completely filled with

protons or neutrons, reproduces the sequence of magic numbers observed in nature. Therefore, using

this model it is possible to interpret the magic numbers as the number of nucleons necessary to fill

all the levels in and below a major shell. The change in stability observed at these magic numbers

can be explained by observing that when a nucleon is added to a nucleus with a filled major shell,

it goes into the next higher one and thus is less tightly bound. The model just described is known

as th e single-particle shell model [4, 2],

Although the shell model can provide a remarkably accurate description of finite nuclear systems,

it does not determine the actual form of the central potential, nor specify how it might be obtained.

The first steps toward the resolution of this question came in 1935 with the meson theory of the

nuclear force, proposed by Yukawa [5], which links nuclear physics to relativistic quantum field

theory.

2 Meson theory of nuclear forces.

By an analogy with quantum electrodynamics [6], QED, in which the electromagnetic interaction

is pictured as the exchange of a photon between charged particles, the meson theory of the nuclear

force assumes a similar mechanism for the interaction between nucleons. In the latter, the exchanged

particle is a meson. Because the photon is a massless particle, the electromagnetic interaction has an

infinite range, as can be seen from Coulomb’s potential which decays with the inverse of the distance

between two charged particles. The nuclear force on the other hand, is short ranged (« few fm),

implying that the meson being exchanged between nucleons has a finite mass. This can be observed

by considering the potential generated by the exchange of a particle of mass m, in the limiting case

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of infinitely heavy sources, is [2] 1

Here g denotes the strength of the interaction. This potential decays exponentially with r and

thus is of finite range. If m is set equal to zero, which is the case of QED, then one recovers

Coulomb’s potential. The form of the potential given in the previous equation is known as the

Yukawa potential [5].

The exchange of mesons as the origin for the nuclear force has been experimentally verified.

There is evidence of a potential of a similar form arising from the exchange of one pion between

nucleons [7]. The pion, which is the lightest meson, a 140MeV, is responsible for the long-range

part of the nuclear interaction.

This formulation of the nuclear force gives rise to a more complex picture of the nucleus. In addi

tion to nucleons, there are mesons being exchanged continuously between them to hold the nucleus

together. These mesons then take part in the dynamics of the nuclear system. This implies that

the natural degrees of freedom needed to describe nuclear phenomena include baryons and mesons.

Other mesons besides the pion, should also be taken into account to describe nuclear phenomena

depending on the energy scale being considered. In the case of nuclei, where the characteristic en

ergy scale is given by the mass of the nucleon (« 1 GeV), the important degrees of freedom are

the nucleon, the pion and other lowest-lying mesons like the p, and the u. The strongly interacting

particles, nucleons and mesons, are called hadrons.

3 Relativistic mean-field theory.

The picture that emerges from the introduction of mesons as the quanta responsible for the nu

clear interaction requires that, the dynamics of the nuclear system be described by a many-body

formulation that incorporates these additional features. In addition, such a formulation should also

1 Here and henceforth we adopt conventional units where h = c = 1.

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satisfy the general principles of quantum mechanics, microscopic causality, Lorentz covariance, and

electromagnetic gauge invariance.

Since all these requirements are met by quantum field theories (QFT) the new description of the

nuclear system should be given in the form of a QFT that uses nucleons and the lightest mesons as

the relevant degrees of freedom.

A problem arises immediately when one considers this proposition: nucleons and mesons are

composite objects and with an internal structure. The underlying structure of these hadrons is

determined by still another quantum field theory called quantum chromodynamics or QCD [2, 8].

This quantum field theory governs the dynamics of quarks and gluons, which are the fundamental

physical objects that form hadrons. This underlying theory is the fundamental theory of the strong

interaction and is responsible for the structure and properties of hadrons and therefore, in principle,

also governs the dynamics of nuclear phenomena. The nuclear force should then be viewed as a

manifestation of the strong interaction between quarks.

Protons and neutrons are composed primarily of two types of quarks (u,d), which are bound by

the strong interaction mediated by gluons which play the same role as photons in the electrodynamic

interaction. It is natural then to conclude that a full understanding of the underlying dynamics of

nuclei should be sought in this theory of QCD. There are many practical difficulties in following this

path, in particular, the fact that the coupling of QCD is very strong at the energy scales relevant

to the vast majority of nuclear phenomena. This means that an expansion of any observable of the

nuclear system based on this coupling constant is destined to diverge and thus fail. This particular

problem is also present if hadronic degrees of freedom are used, since the observed coupling constants

turn out also to be large numbers. Another drawback from trying to describe nuclear phenomena

using QCD is confinement, which means that quarks are confined to the interior of hadrons and they

cannot be observed as free degrees of freedom at this energy scale.

All these limitations make it almost impractical to use QCD directly to describe the dynamics of

nuclear systems. Instead, one is forced to consider its description using hadrons as relevant degrees

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of freedom. A description based on these degrees of freedom is, in fact, more natural since these are

the observed quanta at the normal densities and energies of low-temperature nuclei.

One of the most relevant examples is given by quantum hadrodynamics or QHD, a quantum

field theory based model introduced in 1974 by Walecka [2, 9]. This model, based on a lagrangian

density constructed using the lowest-lying hadronic degrees of freedom, will serve as background to

understand some of the aspects involved in the formulation followed in this investigation.

In its simplest form, called QHD-I, two neutral meson fields are used to describe the nuclear force.

A scalar field, denoted by

Vp, describes a repulsive effect [9, 2]. Nucleons are introduced as a two-component isospinor field,

N, whose components are Dirac spinors corresponding to the proton and the neutron, respectively.

The lagrangian density for these fields takes the form 2

C(x) = N (ij^ - M + gsm + \d ,4 > d ^ - i m 2^ 2 - \ v llvV»v + (1)

where gv and gs stand for the vector and scalar coupling constants, M is the mass of the nucleon,

m s is the mass of the scalar field and mv is the mass of the vector field. The quantity is an

antisymmetric tensor defined as — dvV^.

A critical step is taken to solve this model. Instead of attempting to quantize the various fields

and perform calculations using the perturbation methods developed in quantum field theory, a non-

perturbative solution is sought in the form of coupled, local, non-linear differential equations. This

is done by first considering that for large sources (e.g. large number of nucleons) the meson fields

can be approximated by their expectation values, which are classical fields3:

-» (<}>) = 4>{x) (2)

(V») = d^Voix) (3)

2 The metric and conventions used are those of [1, 14] and differ from [2] 3The scalar field is ultimetly thought to represent correlated 2-pion exchange, and the vector field the exchange of

an a; meson [2]. The scalar and vector fields, and V^, are the only meson fields that develop a non-zero expectation

value in symmetric nuclear matter

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This approximation is known as relativistic mean-field theory RMFT [9, 2].

The equations of motion obtained from this new lagrangian density form a set of coupled, non

linear differential equations.

-V2$ + m2$ =9^Ps(x)

-V2W + m2W - 9Ip b {x )

{ il^ - gvjoVo - [M - gso]}N = 0 (4)

Here N still represents the nucleon field, which can now be expanded in terms of wave functions

satisfying this linearized Dirac equation. The nucleon wave functions are expressed in terms of Dirac

spinors of the form ( . \ iG a(r)$Ktn lpa(x) Ct (5) ^ r F j( r ) $ _ Km

where Ga(r) and Fa(r) are radial wave functions, is a spin-spherical harmonic, and Ct is an

isospinor with components t = 1/2 for the proton and t = - 1/2 for the neutron.

The sources on the r.h.s. of the meson equations are defined in terms of the above Dirac spinors

as (for filled shells)

occ Ps(x) = ]T^aO?) 1pa(x) a

OCC -«

= a occ

Pb (x) = ^«(f ) a 000 9 ' i 1 = (6) a

where occ means that the sum is over occupied levels only.

These equations are solved iteratively using a Green’s function method until a self-consistent

solution is found. Details are given in Appendix 1.

When these equations are solved for nuclear matter, or for finite nuclei, the classical mean-fields

turn out to be large (ss hundreds of MeV). The interplay between these two mean-fields has an

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important role in determining the details and structure of nuclear phenomena. For instance, the

attractive and repulsive nature of these fields, and their large magnitudes is such that:

1. the binding energy of nuclear matter, or finite nuclei, is obtained from the cancellation of these

two fields, while

2. the strong spin-orbit splitting in finite nuclei is obtained by their sum, with the remarkable

additional result of reproducing the shell model [2].

QHD-I, introduces two constants, which define the strength of the coupling between nucleons

and mesons, i.e.

per nucleon and density (or Fermi momentum) of nuclear matter. When this model is extended to

include additional mesons, such as an isovector pM-field and the electromagnetic interaction, it

can reproduce, in the RMF approximation, the bulk properties of finite nuclei [9, 2], One particularly

striking feature of this formulation is the fact that by fixing its phenomenological parameters from

bulk nuclear properties, it is able to reproduce the level structure observed in the shell model without

further recalibration of the parameters. Therefore this particular characteristic observed in finite

nuclei appears naturally in this relativistic description based on quantum field theory.

Relativistic mean-field models, like QHD-I, can account for a large number of bulk properties

of nuclei and there are different methods that have produced acceptable results. Recently, Toki, et

al. [10] used a relativistic mean-field approach to calculate properties of nuclei from the proton to

the neutron drip lines, extending the calculations to super-heavy nuclei. The agreement with known

experimental data shows how relativistic mean-field models can account for a variety of phenomena

in the nuclear many-body system. Toki and his co-workers, using a mean effective lagrangian, have

an extensive program underway on applications of this approach to astrophysics, in particular to

element formation in supernovas [11].

Although QHD is a very successful model to describe bulk nuclear phenomena in the relativistic

mean-field approximation, it encounters serious difficulties when it is written as a renormalizable

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model. Loop integrals that appear in calculations and that take into account the dynamics at

very short distances produce very large effects [12]. The requirement of renormalizability in this

case appears to be too stringent and limits the predictability of the model. Modern approaches

to renormalization, which make sense of effective theories written in term of low-lying degrees of

freedom, provide a new alternative to this situation. The inclusion of non-renormalizable terms allows

the model a better description of the short-distance phenomena typically explored by loop integrals.

This presents a new theoretical problem since by removing the requirement of renormalizability one

can, in principle, add any number of terms to the lagrangian density, and therefore a new approach

should be implemented to truncate this lagrangian to make the theory useful and retain its predictive

power. The solution to this is provided by effective field theory (EFT).

The mean-field approximation used in QHD has not been justified so far, except by the success

it has in describing bulk properties of nuclei. This ll.MF approximation is equivalent to the Hartree

formulation for many-body systems. This method considers the sum of “tadpole” diagrams and

describes in an average fashion the influence of the many-body system on the dynamics of a particle.

This means that each particle “feels” only the average effect of the rest of the system, and as such it

is possible to reduce the problem to the motion of a single-particle in an “effective” potential. The

main justification of this Hartree approximation comes from density functional theory (DFT). Here

it is found that the Hartree term is the dominant term in the expansion of an energy functional

which describes the many-body system.

The EFT/D FT approach followed here allows a new formulation of the nuclear many-body system

that, on one hand, extends the ideas behind the QHD model by including additional interaction

terms in the lagrangian density based on effective field theory principles, and on the other, gives a

strong theoretical justifications based on density functional theory for considering the meson fields

as classical fields.

It is to the description of the underlying principles behind each of these theories that the following

sections are dedicated.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Effective field theory.

The purpose of effective field theory is to provide a simple way of describing the dynamics of a theory

at the low energy limit [13]. It relys on two very important observations [9]:

1. Relativistic quantum field theory is the most efficient way to parametrize the observables of a

system consistent with analyticity, unitarity, causality, cluster decomposition, and symmetries.

As such, relativistic quantum field theory is not reserved only for elementary particles.

2. Most problems in physics include a typical energy scale or length. Only those degrees of freedom

of the system that can be excited at this energy scale, or that can resolve this characteristic

length, are important to describe the dynamics of the system.

Based on these observations, effective field theory establishes a general procedure for constructing a

quantum field theory that describes the dynamics of a system at some relevant energy scale.

Once a characteristic energy scale has been identified in the problem and the relevant degrees

of freedom chosen, they are introduced into the theory in the form of quantum fields from which

a lagrangian density is constructed. In many cases, there will be additional degrees of freedom

that cannot be excited at the energies relevant to the problem. The contributions coming from these

heavier fields are included in the theory in the form of coupling constants of various interaction terms

between the lighter fields. As mentioned before, the terms included in the effective lagrangian are not

restricted by renormalization requirements. When there is an underlying theory which describes the

system under investigation, but whose form is unknown at the energy considered in the problem, then

the symmetries of this underlying theory are included in the construction of the effective lagrangian.

To make the theory predictive and limit the number of interaction terms included, a procedure

should be developed to truncate the lagrangian density at any desired level in order to achieve a

certain degree of accuracy. Such a procedure, or power counting scheme, should be able to order

the terms in the effective lagrangian in terms of powers of some relevant expansion parameter of

the problem under investigation. It is an assumption of effective field theory that once this power

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counting scheme is found, the remaining coupling constants of the theory will be of order unity, a

characteristic referred to as “naturalness”. This condition of naturalness, which in general cannot be

proven a priori, must be verified by fitting the theory to experimental data. The resulting effective

lagrangian corresponds to the theory that describes the system at the characteristic energy scale.

Based on the above observations, one can understand why a description of the nuclear system is

possible based only on nucleons and the lowest-lying mesonic degrees of freedom. The typical energy

scale of nuclear phenomena is given by the mass of the nucleon (ss 1 GeV), and therefore heavier

mesons do not contribute significantly to the dynamics of the system since they are suppressed by

their large masses. The first observation above sheds some light on the success of the QHD model, as

well as other models, which have been formulated in a similar way. Since these approaches have been

developed from the framework of quantum field theory, they already correspond to the most general

way in which such a description is possible. Therefore, from the point of view of EFT, extensions

to the original ideas behind the QHD model, in the form of non-renormalizable terms, constitute a

formal expansion of an effective lagrangian density in terms of the relevant degrees of freedom.

As already pointed out above, QCD is the underlying theory of nuclear phenomena. As such,

some features of it should still be present in a description based on hadrons. These features, according

to effective field theory, are the symmetries of QCD, and they impose restrictions on the interaction

terms in the effective lagrangian.

The EFT/DFT approach used here follows the procedure described above to construct the most

general lagrangian density based on nucleons and the lowest-lying mesonic degrees of freedom, keeping

at the same time the symmetries of QCD. The details of this construction are considered next. The

most important aspect to review here are the symmetries of QCD, and in particular isospin and

chiral symmetry.

If the mass of the u and d quarks is exactly the same; i.e. m u = m^, then the pair of fields

describing these quarks transform according to elements of an SU(2) group, such that

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( \ u ij) = ip' = exp {—if ■ 0)ip \ d / where the components of r correspond to Pauli matrices and 6 is an arbitrary constant vector.

This field transformation is known as the isospin transformation. Invariance with respect to this

transformation gives rise to a conserved isovector current, according to Noether’s theorem.

Since there is a small mass difference between the u and d quarks, this symmetry is broken at the

QCD lagrangian level. This can be seen by rewriting the mass terms of these quarks in the following

way,

, j j rnu + m d(_ j rnu - m d j um uu + am dd — ------{uu H- dd) H------——---- (uu — dd) Z Z mu + md - mu~md r . . = ------(ipip) + ------(ipvnp) (7)

The last term breaks the symmetry unless both quark masses are equal.

Chiral symmetry is connected with the fact that massless fermions have a specific handedness.

In other words, a massless fermion moving with a given momentum can have its spin aligned (right-

handed) or anti-aligned with respect to it. Both chiral projections are linearly independent and a

massless fermion field ip can be written in terms of a “right-handed” and “left-handed” component,

ipR and %pi respectively.

1 1 ^ = 2 ^ +75)4’ + ^ -T sW ’

= Ip'R + lpL (8 )

The QCD lagrangian for massless quarks can be written in terms of these left and right-handed

fields, which do not mix and therefore can be considered as independent. The kinetic energy terms

for th e u and d quarks, for example, takes the form

iplnd^-tp = ipLlndpipL + tpR'y^df.tpR

and therefore the left and right handed quark fields have separate invariances. For instance, the

left handed fields transform according to tpi —> LipR while the right handed fields transform as

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V>r -> ityft, where L and i? are elements of a global SU(2) group. The complete symmetry group is

then denoted as SU (2)l x 517(2)#. An equivalent way to represent these transformation properties

is through a vector and axial-vector isospin transformations defined as

ip —t exp (—i6\r • f)ip

ip -¥ exp(-jdyi • T j^ip (9)

As will be explained below, at the nuclear level only the vector part of the symmetry group is still

present, while the axial part is slightly broken by the mass of the pion.

The chiral symmetry is broken at the QCD lagrangian level by the small, finite mass of the

quarks. Considering these masses to be equal, but finite, the mass term in the lagrangian becomes,

ipmip = m(ip + ^ rV ’l )

where this term breaks the chiral invariance if m ^ 0.

In the nuclear domain only the u and d quarks are relevant. This indicates that the fields formed

from the left and right-handed quarks will transform according to elements of the global group

517(2)/, x 517(2)#. Right-handed fields transform according to elements of 517(2)# and left-handed

fields according to those of 517(2)/,. The QCD lagrangian, written for the u and d quarks only,

is invariant under the isospin and chiral transformations and gives rise to conserved isovector and

axial-isovector currents in this massless limit.4 Because QCD is the underlying theory of the nuclear

dynamics, the conservation of both currents should also be manifested at the energy scales relevant

to finite nuclei. The fact that the u and d quarks are not massless breaks this chiral invariance at

the lagrangian level. At the nuclear level, chiral symmetry is also broken, but the mechanism by

which this occurs is called spontaneous symmetry breaking.

Evidence for the breaking of chiral symmetry is given in electroweak processes where it is observed

that although the vector component of the hadronic current is conserved, the axial one is not. Such

a behavior of the axial current can be seen from the fact that the pion decay constant, f„, given in

4These currents are conventionally referred to as simply the vector and axial-vector currents.

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the process it -> fiv^ is different from zero. This decay constant is related to the matrix element

describing this process, as can be seen from the following,

= -iUtfe-^Sab

where a and b denote isospin indices, qM is the four-momentum of the pion, and A% (x) is the

corresponding axial current. Therefore, by calculating the matrix element of using this last

expression one obtains,

<0|<9MA£(x)|7r6(a;)) = -S 1tm \e~%q’x Sab

showing that the divergence of the axial vector current cannot vanish completely.

The mechanism of spontaneous symmetry breaking is adequately illustrated by the a-model.

Here a chirally symmetric lagrangian density is written in terms of massless scalar (a), nucleon (xp)

and pion (if) fields [2, 14, 13]. The lagrangian density includes a potential of the form V (a2 + it2)

that describes the interactions Between the a and pion fields. This lagrangian is given by:

£ = xpij^d^xp + ^<9M7f ■ <9Mif + ^d M(7 ■ d^a

—gtp(

where all nucleons are massless. This lagrangian is invariant under global isospin transformations,

which in infinitesimal form are (e -> 0):

xp1 = xp — - t■ exp

ip = it + e x it (11)

This same lagrangian density is also invariant under chirial transformations. Their infinitesimal form

is:

£ xp' = ip -- t ■ e^xp

it' = 7r — ere

a’ — a + it ■ e (12)

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The last two terms leave the quantity a2 + n 2 invariant. Because there is an isospin and chiral

invariance of this lagrangian density, both axial and vector currents are conserved, as happens in the

case of massless quarks in the QCD lagrangian.

Now if the potential V (a2 + if2) takes the form V — j[{cr2 + if2) - v2]2, with A and v positive

constants, its minimum occurs at non-vanishing values of the fields, implying that in the vacuum

(minimum energy state) these can acquire an expectation value. Taking into account the fact that

pions are not massless, and that there is a small breaking of chiral symmetry at the lagrangian level

(coming from the small mass of the quarks), a term can be added to the lagrangian to incorporate

this effect. Thus the term —sa, where e is a small number, is added to the lagrangian, which modifies

the potential in such a way that now the pion field vanishes at the minimum and the a field develops

an expectation value denoted here as cro- Expanding the a field around its expectation value, it is

possible to define yet another field, 4>, through a — oo + 4>- Using this new o-field in the original

lagrangian density generates a mass term for the nucleon field, thus breaking the chiral symmetry.

The fact that the chiral symmetry is not realized in the vacuum of the model, although it is in the

original lagrangian, 5 illustrates the process of spontaneously symmetry breaking.

As a consequences of the addition of — ea to the original lagrangian density, the divergence of the

axial current, becomes proportional to the square of the mass of the pion (which in turn is

proportional to e) in accordance with the results obtained above using the argument of pion decay.

Therefore, only at vanishing pion mass is the above cr-model lagrangian density, with a potential

of the form discussed, invariant under chiral transformations and the axial-vector current exactly

conserved.

The effective lagrangian considered in the following calculations, incorporates this chiral symme

try in its exact chiral limit, i.e. = 0, and will impose restrictions on every interaction term used

in the effective lagrangian. Chiral symmetry breaking at the lagrangian level is then included with

a term explicitly proportional to to 2. The specific form in which this chiral symmetry is realized

5 This is exact if e = 0

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is discussed next, along with the transformation properties of the relevant degrees of freedom used.

The arguments become quite technical and are only reviewed briefly here. Details can be found

in [1, 9, 13],

The pion field is introduced as the phase of a chiral rotation of the identity matrix in isospin

space. A field £(x) is defined in the following way:

£(z) = exp(in(x)/U). (13)

where the pion field tt(x) is given by the expression ir(x) — | r • 7?

Nucleons, which are the heaviest fields entering this effective lagrangian, are included as isospinor

fields, N(x), where the upper and lower components correspond to the proton and neutron, respec

tively.

p(x) N = n(x) j

To include the contributions from a symmetry energy term, an isovector field pM is added to the

lagrangian. This field is defined by pM(x) = | r ■

Chiral symmetry is incorporated in a non-linear realization. 6 According to the formalism devel

oped by Callan, Coleman, Wess, and Zumino [15], the non-linear realization of the chiral symmetry

is defined by global transformations, L and R, of the subgroups SU(2)l and SU(2)r such that the

various fields defined above obey the transformation rules:

L®R: (14)

where

g(x) = L^(x)h^(x) - h(x)£(x)Rf (15)

p\k{x) = h{x)pu{x)h\x) , (16)

N '(x) = h(x)N(x) . (17)

6 An account of previous attempts to introduce chiral symmetry in the description of the nuclear many-body system

can be found in [2, 9].

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Here h(x) is a local 517(2) matrix defined by Eq.(15). Two additional fields, ap and vp are defined

in order to complete the necessary elements for this effective lagrangian. These fields have axial and

vector transformation properties and are given in terms of the £-field, defined above in Eq.( 13).

v, = + (18)

The terms in the effective lagrangian, which are constructed from all these fields, must be invariant

under the chiral symmetry transformations given above. As an example, consider a term of the form

N '^a 'N '

Using the transformation rules given above, the nucleon field transforms as

N* -> JVW (19)

N' -> hN (20)

while for the axial-vector field one obtaines

< + Zdntf) - K B )R [(d^ + ed^h)})

= (21)

The end result is that this interaction term is invariant under the chrial transformation.

N'^a'^N' ->• N^rfha^hN = N^a^N

In addition to the above fields, an electromagnetic field A p is included and the electromagnetic

structure of the nucleons is taken into account through their anomalous magnetic moments. These

are labeled Xp and A„ for proton and neutron respectively.

An isoscalar-vector field Vp is also included in this effective lagrangian to correctly reproduce

the bulk properties of nuclei. This field is treated as a chiral singlet and its mass is taken to be

equal to the co-meson mass. Finally, an isoscalar-scalar field , again a chiral singlet, is included to

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incorporate the observed mid-range N-N attraction, and its mass is determined by fitting the theory

to experimental results.

The terms in the effective lagrangian are organized in increasing powers of these fields and their

derivatives. An index v is assigned to each term, which is given by

n v — d + — + b

Here d denotes the number of derivatives, 7 n the number of nucleon fields, and b the number of

non-Goldstone boson fields included in the interaction term.

As already mentioned above, to be able to truncate the effective lagrangian at a given level it is

necessary to find a procedure that will arrange the terms according to some expansion parameter.

The expansion parameters that make this truncation possible are the ratios of the isoscalar, scalar and

vector fields and their derivatives to the nucleon mass. When these fields are treated as mean-fields,

these ratios are known to be small,

ffsd’O 9vVq i/o M ’ M ~ '

The method used here to arrange the terms comes from the rules of naive dimensional analysis

(NDA) devised by Georgi and Manohar [16,17]. The rules for a given term in the effective lagrangian

are

1. Include a factor of 1/ f-n for each strongly interacting term.

2. Assign an overall factor of f 2A.2, where A is an appropriate scale of the system.

3. Multiply by factors of 1/A to achieve dimensions of (mass)4.

Each interaction term in the lagrangian contains a coupling constant, which takes into account

the contributions from heavier degrees of freedom. It is a basic assumption that once the appropriate

7Derivatives of fermion fields are not counted here because they bring a factor of M, which invalidates the decreasing

nature of the contribution. Any such derivatives can be replaced using the equations of motion [1],

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scale has been extracted using NDA, the remaining dimensionless coefficients should be of order unity.

In the present application of this effective lagrangian, the scale A corresponds to the mass of the

nucleon, M (i.e. the chiral symmetry breaking scale). This assumption of naturalness, which can only

be verified after fitting the theory to experimental observables, was investigated in [1], There it was

concluded that using NDA as an organizing principle renders coefficients of order unity. Additional

details concerning the construction of the chiral effective lagrangian can also be found in [1],

To illustrate the end product of the procedure described above, only a portion of the mesonic

part of the complete effective lagrangian density is reproduced here.

r t \ 1 /1 , 9s^a _2j.2,1 , K39s

To summarize, the description of the nuclear many-body system is approached using an effective

field theory formulation. A lagrangian density is constructed using the lowest-lying hadronic degrees

of freedom (nucleons, pions, rho, 4>, u and the electromagnetic field), these are arranged in various

interaction terms in the effective lagrangian following the rules of naive dimensional analysis (NDA)

as an organizing principle. This procedure allows a truncation of the expansion at some arbitrary

order and guarantees that all non-redundant terms at the same level have already been taken into

account. To satisfy the restrictions imposed by chiral symmetry, the lagrangian density is made

invariant under a non-linear realization of this symmetry. Other symmetries imposed by QCD

are also incorporated. The heavier degrees of freedom, which have been integrated out, appear

as coupling constants in the theory. These coefficients, obtained by fitting the theory to nuclear

observables, are found to be of order unity.

5 Density functional theory

Density functional theory is an alternative formulation of the description of many-body systems of

fermions. It has found applications in atomic, molecular, solid state, and nuclear physics. Although

originally formulated for non-relativistic systems of electrons, it has been extended to relativistic

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ones [18], and also to describe th e nuclear m any-body system using the QHD model [19, 20, 21].

Here the important aspects of this theory are reviewed in its non-relativistic formulation. The nec

essary changes needed to describe relativistic systems will be indicated as needed. A more thorough

exposition of this topic can be found in [22, 18].

At the base of DFT is the Hohenberg-Kohn theorem [22]. This theorem establishes that there

is a one-to-one correspondence between a local, single-particle external potential Vo (a?) acting on a

closed system of interacting fermions and the ground-state density of the system, n(x).

The proof of this theorem follows by first establishing a map C between the set of all external

potentials V and the set of ground-state wave functions tf>. By construction, any element of V has

an element of C associated to it.

C : V -* $ (24)

A second map D can also be established between the setand the set of ground-state densities

N. Again here, by construction, any element of corresponds an element in J\f.

D : ¥ -> JV (25)

The theorem is then proven by showing that to any element of N there corresponds a unique

element in $, and also that to any element in $ there is only one element in V. This establishes

then that the maps are indeed one-to-one and as such they can be inverted.

From the inversion of map D alone, one concludes that the ground-state wave function can be

written as a functional of the density, i.e. \^[n(x)} ). A functional is. a function whose argument is

another function.

Several consequences follow from this:

1. The ground-state expectation value of any observable O is a unique functional of the exact

ground-state density.

($[n]| O |H/[n]) = 0[n]

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where O is an operator written in second-quantization. By applying this to the hamiltonian op

erator a unique energy functional of the ground-state density can, in principle, be constructed.

2. The exact ground-state density is found by minimization of the energy functional with respect

to the density. Therefore the lowest energy of the system, given a local external potential

Vq(x ), occurs when the density is equal to the ground-state density.

3. The energy functional of the system, denoted here as E[n], is composed of two terms. The first

one, denoted as Fh k [ti], is universal and independent of the form of the external potential.

The second one involves the interaction with the external potential.

E[n] = Tff_R-[n] + j dxv0(x)n(x)

From these, the Hohenberg-Kohn theorem can be restated in the following way: For any interact

ing system of fermions subjected to a local, single-particle external potential, there exists a unique

energy functional of the ground-state density.

In its relativistic form this theorem establishes that the energy functional depends not only on

the ground-state density of the system but also on the ground-state currents, i.e. Tpijj and $ 7^ .

The proof follows the same arguments given above.

A second important aspect underlying the DFT formulation is the Kohn-Sham scheme to find

the ground-state density and energy of the system. This procedure consists in reducing the interact

ing many-body system to a non-interacting system subject to an auxiliary, single-particle external

potential Vq(x ). The superscript s indicates a single-particle operator.

[Ts + (£)]<$ (x) = e*$(£)

where T s is the single-particle kinetic energy operator in quantum mechanics and, °(x) are the

single-particle wave functions with energy eigenvalues e*.

Due to the connection between the external potential and the ground-state density guaranteed by

the Hohenberg-Kohn theorem, the main claim of the Kohn-Sham scheme is that there exists a local

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single-particle external potential Vq (x) such that it gives rise to a non-interacting ground-state density

which is equal to the ground-state density of the full interacting system. The non-interacting ground-

state density is parameterized using the wave functions *■ (x) obtained from the above equation. This

is written as:

no(x) = ][]|<^(x)|2 i

The Kohn-Sham scheme shows formally how the many-body system of fermions can be reduced

to a single-particle problem. This reduction is not an approximation since the full interacting hamil-

tonian is to be used in the construction of the appropriate effective potential Vq(x). As a result of

this procedure the energy functional can be expressed as a sum of four terms:

E[n] = T s[n] + ^ J dx dx1 n{x)w{x^)n{^) 4- j dx v0(x)n(x) + E xc[n]

Here the first term is the non-interacting kinetic energy of the system, the first integral corresponds

to a Hartree-type interaction, the second integral is the contribution coming from an actual external

potential and the last term, which contains all the complexities of the many-body interactions, is

called the exchange-correlation energy.

From this decomposition of the energy functional one can observe that the Hartree approximation

is equivalent to neglecting the effects coming from the exchange-correlation term, and as such it

gives the dominant term in the expansion of the energy functional. Prom the point of view of density

functional theory, the success of the relativistic mean-field approach is now formally well understood.

Density functional theory then gives an alternative approach to the description of an interacting

many-body system of fermions. The Hohenberg-Kohn theorem guarantees that there is a unique

energy functional of the density. By minimization of this functional with respect to the density it is

possible to obtain the ground-state energy and density of the system. The Kohn-Sham scheme is a

procedure to implement this minimization, where the density of the system is parameterized using

the single-particle wave functions of an equivalent non-interacting system subject to a local, external

single-particle potential. This potential is chosen such that the ground-state density of the original

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interacting system is obtained.

6 The EFT/DFT formulation of the nuclear many-body sys

tem .

In the last two sections the principles underlying EFT and DFT have been discussed. Here the

connection between both theories in the EFT/DFT formulation is given [1].

The nuclear many-body system is described from the DFT point of view. It is based on an energy

functional, E[pg.s], of the ground-state densities of the system. Here the pg,s, denotes collectively the

various densities of the system. The exact energy functional for nuclear systems is unknown. This

can be resolved by constructing a lagrangian density following the prescriptions given by effective

field theory. From it, the energy functional is obtained by calculating the corresponding Hamiltonian

density in second quantization. Once this Hamiltonian is obtained, the meson-fields are replaced by

local, classical fields.

Formally the connection between E FT and DFT is made by considering the expansion of the

relativistic mean-field lagrangian as an expansion of the energy functional of the nuclear many-body

system in terms of nucleon densities and auxiliary classical meson fields. As described above, accord

ing to the principles of DFT, knowledge of the exact energy functional can be used, by optimizing

it with respect to the fermion densities and auxiliary potentials to calculate the exact ground-state

densities, energy, and chemical potential for the fully interacting many-fermion system [23]. Mini

mization of the energy functional leads to a set of coupled, local, non-linear differential equations for

the nucleon wave functions and meson fields, which are the equations actually solved here. Details

of how the resulting equations of motion are solved are given in Appendix 1. The inclusion of new

interaction terms in the lagrangian aims to give a better description of the energy functional, thus

going beyond the Hartree approximation and incorporating exchange-correlation effects.

In this way the EFT principles combined with DFT methods give an approach to the nuclear

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many-body problem where new contributions coming from the interacting fields can be incorporated

in a systematic and controlled way.

The complete lagrangian density, as well as the equations of motion derived from it are presented

in C hapter I.

7 Definition of the problem.

The EFT/DFT approach discussed in the last sections represents a novel way of describing the

nuclear many-body system. The theoretical contribution from density functional theory in the form

of the Hoenberg-Kohn theorem, guarantees that there exists a unique energy functional that describes

the dynamics of this many-fermion system in terms of densities and auxiliary classical potentials.

In addition, a formal procedure for constructing this energy functional is given based on principles

of effective field theory. Here, once the relevant degrees of freedom are identified they are used to

construct an effective lagrangian, which obeys the symmetries of QCD. This formulation allows a

systematic expansion of the energy functional in terms of the classical fields and the densities of the

system.

Using naive dimensional analysis as an organizing principle, and arranging the terms in the

effective lagrangian in powers of the meson and nucleon fields (and their derivatives), it is possible

to generate several expansions of the energy density with increasing levels of approximation. For

each, a set of coefficients, associated with the contributions of the heavier degrees of freedom in the

effective lagrangian, must be determined by fitting the resulting theory to nuclear observables. In

each case, the naturalness of these coefficients must be verified.

In [1], Furnstahl, Serot and Tang showed that the different sets of coefficients, or parameter sets,

obtained by fitting the EFT/DFT theory to properties of spherical nuclei in the valley of stability

region were indeed natural. This result confirms the basic assumption behind NDA proving that the

counting procedure can be applied successfully in the nuclear domain.

Furthermore, naive dimensional analysis indicates that the addition of more terms in the la-

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grangian density improves the results. This means that the expansion of the effective lagrangian, or

equivalently of the energy functional, is convergent.

In recent years, there has been an increased interest in the study of nuclei far from stability. New

experimental facilities and the implementation of new techniques are giving valuable experimental

evidence of several new nuclear phenomena. These facilities will make some remote regions of the

nuclear chart available for study, and some have already produced data that can be compared with

predictions from several models and theories. These nuclei are the ideal testing ground for new and

old approaches that describe the nuclear system.

It is in this context that this thesis is written. The satisfactory results of the EFT/DFT approach

to describe bulk nuclear properties in the stability region [1] motivate this investigation to explore

the range of validity of this theory. By doing so, it should be possible to verify if this description

of the nuclear many-body system can consistently reproduce properties of nuclei far from stability.

At the same time, it will indicate possible modifications that need to be implemented in order to

improve the predictability of the theory.

Based on these arguments, the main objective of the present investigation is then to apply the

EFT/DFT approach to calculate ground-state properties of some selected nuclei far

from stability and ascertain from these results the range of validity of the approach

outside the stability region.

Furthermore, the calculations are performed using the various effective lagrangians already tested

in the stability region, and for which the naturalness of the parameter sets has been verified. This

means that no new parametrization of these theories is attempted.

To perform the calculations, the coupled, non-linear, differential equations resulting from this

EFT/DFT formulation and minimization of the energy functional are solved numerically using

an independently developed code, which is reproduced in Appendix 1 of this thesis. The

solutions obtained from this program are in agreement with those published in [1], which was taken

as a bench mark before any calculations on nuclei far from stability was performed.

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To address the question posed in this investigation, several steps were taken. The first one was

to investigate the convergence of the approach when the level of approximation was increased. For

this, the total ground-state binding energy of some selected doubly-magic nuclei far from stability

was computed using the various parameter sets given in [1], The selected nuclei, 1foSns2, M-gSnso,

afNiao, and ^Niso, were chosen for their role in nucleosynthesis processes. To extend the comparison

between this EFT/DFT approach and some mean-field models, two more parameter sets, denoted

here as L2 and NLC, were used to compute the same observables. These additional sets where chosen

because they corresponded to similar levels of approximation as some of the parameter sets given in

[1] but were obtained by fitting different nuclei in the stability region [9]. All parameter sets, are

included in Table 1 in Chapter I.

When the calculated total binding energy of these selected doubly-magic nuclei is plotted against

the corresponding parameter set, see Fig.l, the results show convergence when the level of approx

imation is increased. In general, it is observed that there is a small improvement coming from

additional terms in the expansion of the effective lagrangian.

The convergence of higher levels of approximation can also be observed in Fig. 2, where calculation

of the total binding energy was extended to all the even-even Sn isotopes. In this plot, only a selection

of results is plotted corresponding to parameter sets Ql, G1 and G2 given in Table 1 (see page 58).

The quantity plotted on the vertical axis is the percentage deviation of the calculated results from

the experimental data. The deviation for the three sets Ql, G1 and G2 is in absolute value less than

1%. The energies plotted are negative and, in general, all three parameter sets give less binding

energy than observed. For all parameter sets shown in this figure, G1 gives overall the smallest

deviation. This implies that there is indeed an improved convergence overall in going from Ql to

the extended parameter sets Gl.

Once the convergence of the EFT/DFT approach had been verified, a second step was taken by

comparing the level structure of neighboring nuclei differing by a single particle or hole from the

doubly magic nuclei. In this case, a comparison between the predicted and experimental chemical

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4& .. • 28 20 i- ExB _ r > -400 -400 ©

> a if’O) -600 2X 0 -600 E x e r B ©c ® B3 BS9 W © A O) £c -800 „ E x p ______. ♦ 8 > 50 -800 T 3 ------+ ------♦ ------c ♦ XJ A 15 -* — ■ -1000 -1000 o 13^ E x p A 50 82

-1200 -1200 L2 W1 NLC Q1 Q2 G1 G2

Figure 1: Total binding energy of the selected nuclei. The left hand side shows two models with

the same level of approximation but fit to different data sets (see text). The right hand side shows

models with increasing level of approximation. NLC and Ql have the same level of approximation,

as do G1 and G2.

potentials of these neighboring nuclei is made. In Fig. 3 the core nucleus is plotted in the center and

is marked as 0+. To the left, the neighboring nucleus with one more particle and to the right the

nucleus with one less particle is shown. The ground-state energy, in either case is given relative to

the doubly-magic nucleus. The arrows in this plot indicate the position of the experimental data.

Similar calculations were done for nuclei neighboring the other selected doubly-magic nuclei with

similar results. The predicted quantum numbers of these neighboring nuclei, denoted as J’r, agree

with experimental evidence in almost all cases. The only exception occurs for 1|oSn81 where the

predicted quantum numbers for the ground state are l l / 2~ and the experimental data indicate a

3/2+. Although there is disagreement between the predicted and the measured quantum numbers,

the observed 11/2— state lies only 200 keV above the experimental ground-state.

The calculations of the nuclear properties using the DFT approach represent mean field values

and do not include explicitly the contributions of pairing interactions. Pairing effects can be observed

by calculating the two-neutron separation energy, S^n = BE(N + 2) — BE(N). The results obtained

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Sn-isotopes

LLJ CD

UJ Stable CD nuclei

# 2 9 1 S I ? -0.5 LU CD

Figure 2: Percentage deviation of the total binding energy for even-even Sn-isotopes using Ql, G1

and G2 parameter sets. The stable isotopes are indicated in the plot.

with the present formulation indicates that S2n is essentially constant for sub-shells of given j and

the level ordering for the set of even-even Sn nuclei considered is reproduced. Even though other

calculations use some type of pairing, the goal here was to determine the convergence of the results

for the optimal observables as new interaction terms were included. Additional results and more

details are given in Chapter 1. The results of this study have been published [24].

The agreement obtained for the binding energies of the selected nuclei discussed above, show that

the energy functional derived from the effective lagrangian of Furnstahl, Serot and Tang [1] is indeed

a good approximation to the true energy functional. According to DFT, the ground-state densities

obtained in each case should also be in good agreement with the true ground-state densities.

The ground-state densities are parametrized using the single-particle wave functions obtained by

solving the Kohn-Sham equations of the system. These correspond to energy eigenvalue equations

for a system of non-interacting particles subject to a local external potential. Except for the energy

eigenvalue close to the Fermi surface, all other energy eigenvalues have no direct physical meaning;

therefore if the single-particle (hole) energy at the Fermi surface agrees with experiment, it is assumed

that the corresponding wave function, as calculated from these equations, also represents a good

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3 3, > 2p /.i : V 20 | - 2 p V U / 9> -

Isotones 9/ o> 2^" 0 10- Exp.Value c 0 O) c — 0 + T3 O ' C 2d% %+ Lq 0 1 9 V = , V 7 -10 \ + . g0 Exp.Value 0 EC 133C h 1 3 2 - 131. -20 51 82 50 82 49 82

Figure 3: Level spectrum of isotones of 1foSn82 differing by one proton

approximation.

A second step taken in this investigation was to verify the validity of this statement. At the same

time it gave an indirect way to measure the accuracy to which the ground state density is reproduced.

The wave functions describing one particle outside a doubly-magic core, or one hole in a doubly-magic

core, are used to calculate various /3-transition rates. These wave functions represent the initial or

final nuclear states in these transitions. Because the EFT/DFT approach is formulated to reproduce

ground-state properties of the nuclear many-body system, the ideal case of study corresponds to

transitions going from ground-state to ground-state. Wave functions of excited states will be less

accurately described as these states lie farther away from the Fermi surface and their energies less

well reproduced.

Using available experimental data and the results obtained for the nuclei far from stability indi

cated above, a comparison between predicted and observed /3-decay transition rates was performed.

The selected transitions occur between neighboring nuclei of 1foSng2: ^ S n ^ and ^gSngi, 1fiSb82

and 1|gln82. Two types of transitions, particle-particle and hole-hole, were investigated to validate

the assumptions made above. In the first case, i.e. the particle-particle transition, attention was

placed on the /3-decay process ^gSn^ -»• 1|fS b82 +e~ + F e, while for the second type, the hole-hole

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case, the transition 14 9 I n 82 ->1|o S n 8i + e + ve was investigated. In both cases, ground-state to

ground-state transitions, as well as transitions from and to low-lying excited states, were considered.

To study these processes a general expression is developed here to describe any semi-leptonic

transition. This formulation incorporates the Dirac wave functions obtained from the Kohn-Sham

equations, which are then used to calculate the matrix elements of the appropriate electroweak

currents. To make the whole approach self-consistent, these currents are obtained directly from the

same effective lagrangian. They correspond to the leading order Noether currents [25, 2], and at

any level of approximation in the effective lagrangian, the axial-vector currents satisfy PCAC and

show a pion-pole dominance [25].8 The matrix elements of the currents thus constructed are used

in a general multipole expansion from which the final expression to describe any semi-leptonic weak

interaction is obtained. This formulation and the resulting equations developed here are discussed

in Chapter 2 of this thesis.

As already pointed out, predictions for /3-transitions rates between ground-states are expected to

agree better with the experimental evidence, while predictions for transitions involving excited states

should be less reliable. A verification of this assumption is obtained in the case of particle-particle

transitions occurring in the process ^ S n ^ -t 1|8Sbs2 + e~ + Ve. This can be observed in Fig. 4.

In this figure it is evident that the disagreement between predicted and calculated ^-transition

rates increases systematically when the transitions go between the ground-state of 1|oSng3 and higher

excited states of 1ffSb82, therefore suggesting that the ground-state wave functions are better de

scribed. A detailed account of the formulation followed here is presented in Chapter 2, along with

additional results. The results of this work have been submitted for publication and can be found

in [26].

The results obtained so far indicate that the EFT/DFT approach can be used successfully to

describe some ground-state properties of selected nuclei far from stability. Most of the calculations

®Here the leading one-body terms in the Noether currents have been retained. The effects of higher-order terms in

the current remain to be investigated.

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133 133

0.0001 ^7/2~* 1&7/2 2f7/2—>2d5/2 2f?/2—>2d3/2

Figure 4: Calculated results of the ^-transition rates in the cases of particle-particle transitions and

the experimental values. The asterisk denotes an excited state.

were performed on double-magic nuclei and their single-particle or single-hole neighbors. In addition

to these, the EFT/DFT approach was able to reproduce, within 1 %, the total binding energy of the

even-even Sn isotopes.

To further explore the range of validity of this approach, similar calculations were performed on

all isotones with magic numbers N= 28, 50, 82, 126 and isotopes with Z= 28, 82, 126, for which

experimental data was available. Again here, as was done before, the calculations centered on the

optimal observables DFT is designed to reproduce: ground-state binding energy and densities, as

well as chemical potential [23]. In general, it is seen that the predictions of the total ground-state

binding energy, for both isotones and isotopes, lie below the 1% level, which is consistent with the

results previously obtained. In addition to this, as the size of the nucleus increases, the disagreement

is reduced. The agreement between the results of the chemical potential with experimental data is

also consistent with previous results.

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Predictions for the ground-state J71' values of odd nuclei neighboring even-even ones, show different

levels of agreement with experimental values, which depend on whether the tested nuclei were semi

magic isotones and isotopes. The difference in both cases arises from the fact that when the number

of neutrons gets very large in the regime N>Z, the neutron level density gets very high and the

detailed ordering of the levels between major shells depends sensitively on the details of the various

densities. Therefore the best agreements were obtained for the isotone cases.

In the case of the isotopes, the disagreements between the predicted and experimental quantum

numbers of the ground-state are linked to small variations, sometimes as low as a few keV, in

the predictions of the ground-state energy relative to neighboring even-even nuclei. This type of

disagreement was already observed in the case of 1|oSn8i, and it is now better understood from the

analysis done on the larger set of nuclei. Some representative results are shown in Fig. 5 and Fig. 6

corresponding to the case of isotones and isotopes, respectively. Further details are given in Chapter

3 of this thesis. The results of this work have been submitted for publication and can be found

in [27].

In summary , the three major steps undertaken in this thesis to address the question of the validity

of the EFT/DFT approach outside the valley of stability are the following:

1. Study of convergence for doubly-magic nuclei far from stability.

2. Analysis of /3-transition rates using single-particle wave functions obtained from the Kohn-

Sham equations to test the accuracy of the wave functions and indirectly the ground-state

density.

3. Extensions to semi-magic isotone and isotope nuclei.

These have rendered impressive results which indicate that the energy functional derived from the

effective lagrangian developed by Furnstahl, Serot and Tang and fitted to nuclear properties of stable

nuclei, is capable of reproducing ground-state properties of nuclei outside the stability region. Thus

it appears that the EFT/DFT does indeed represent QCD in the nuclear domain.

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Particle states (Z+l) — Calculated • ■ ■ Experimental

3/2- Cu

7/2-

1/2+ C l # i

Hole states (Z-l)

Figure 5: Particle and hole states neighboring even-even N=28 isotone nuclei. Binding energies are

considered positive. Only nuclei with particle and hole states with respect to filled j-shells have been

plotted. The values of Z correspond to the even-even nuclei in the middle row. Calculated values

are included and, unless indicated otherwise, are in agreement with those of the experimental ground

state. Where the experimental value disagrees with the calculated one, or is uncertain, it is also

included in the figure.

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30 > Particle states (N+l) > bO 5/2- o w —^5/2- ]/2- c 3/2- :::\ .... ffl 3/2- (1/2-) w> 0 fl ’"3 3/2- •S -10 1/2- ffl 7/2- 5/2- •H -20

Figure 6: Particle and hole states neighboring even-even Z=28 isotopes. Binding energies are con

sidered positive. Only nuclei with particle and hole states with respect to calculated filled j-shells

have been plotted. The value of N corresponds to the even-even isotopes in the middle of row.

Calculated J71’ values are included and, unless indicated otherwise, are in agreement with the experi

mental ground states. Where there is disagreement with the experimental V value, or it is uncertain,

the experimental ground-state J77 value is also indicated in the plot. In addition, in this case, the

lowest-lying excited state with the calculated -V is also included.

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8 Review of other relevant work.

The results obtained from this EFT/DFT formulation support the new approach as an alternative

way to describe the nuclear many-body system for nuclei far from stability. Other formulations,

which have been applied to understand these nuclei, are described here.

Several authors have investigated different phenomena on nuclei far from stability. These include:

the formation of neutron and proton skins in nuclei near the drip-lines, the reduction observed in the

spin-orbit interaction close to the Fermi surface, and the realization of the pseudo-spin symmetry

in large nuclei [28, 29, 30]. Some of these studies use an approach based on relativistic mean-

held models including both particle-hole interactions, as well as pairing forces in the particle-particle

channel. The most effective and frequently used are the density dependent relativistic Hartree formu

lation (DDRH) [31, 28], the relativistic continuum Hartree Bogoliubov (RCHB) [29], the relativistic

H artree Bogoliubov (RHB) [32], the Hartree-Fock-Bogoliubov [33], and th e relativistic Dirac-Hartree-

Bogoliubov (DHB) [34]. These different approaches vary from each other in the form of the original

lagrangian density, the selected type of interaction for the particle-hole channel, or the inclusion

of direct contributions from pairing correlations. This last aspect constitutes a difference with the

present EFT/DFT formulation, which does not include pairing effects directly.

The particular form used for the pairing interaction varies in the above methods. Some use

phenomenological forces like the Gogny force [29, 32], or a density-dependent zero-range approxi

mation [31], to account for dynamics in the particle-particle channel. In both cases the results are

very similar and agree very well with experiment. In other cases [35, 31], the BCS approach has

been implemented using a constant pairing strength to take into account the effects of the contin

uum. All these initiatives use different types of interactions to describe the particle-hole and the

particle-particle channels.

A different situation is presented in RHB [32, 34], perhaps the most self-consistent approach,

where ph-interactions and pairing correlations are taken into account in a unified way by solving

self-consistently the RHB-equations simultaneously with the Klein-Gordon equations that describe

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the meson fields. This approach is very similar to the relativistic mean-held model given in QHD,

except that contributions to the energy functional coming from the pairing correlations are included

and the exact analytical form for the pairing force is derived by quantizing the theory.

Other relativistic mean held models (RMF), also being tested in these new regimes, use the a — oj

lagrangian density in which both linear and non-linear a interactions [32, 36] are considered. A direct

application of the QHD-II lagrangian (which includes tt and a in a renormalizable model [2])to study

doubly-magic Sn isotopes and calculations of single-particle level spectrum as well as nucleon densities

can be found in [37].

Shell-model calculations have also been used to understand particular phenomena in nuclei far

from stability. In [38], such calculations are performed using the CD-Bonn effective potential to

study the importance of the 1So and 3P2 partial waves in reproducing the constancy of the 2 f — 0]*"

level spacing which is apparent in Sb nuclei

Other authors have produced new parameter sets of known potentials, like the Nilsson [36] and

Groningen [31] potentials, by htting to new experimental data of binding energies, single-particle

levels, and nuclear radii for nuclei far from stability. Although this procedure is still an option in

the EFT/DFT formulation used here, all calculations here were performed using the parameters

obtained by fitting to experimental data along the stability line.

In the context of density functional theory, other models based on this principles have been

carried out using the QHD-I and QHD-II lagrangians [19, 20, 21]. The contributions from exchange-

correlations effects are taken directly by applying a local density approximation.

All the above examples show the increasing interest in studying nuclei far from stability from

various successful models. The approach followed here has also been used to study /3-transition rates

in a self-consistent way. The results indicate that for the selected transitions studied in nuclei far

from stability, the predicted values are well in agreement with experimental evidence and from what

is expected according to DFT.

A review of some recent /3-decay studies in nuclei far from stability is given next.

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Calculations of /3-decay properties, like the half-live (ri/2) and /3-dclayed neutron emission (Pra),

have been done using mainly gross theory, quasi-particle random phase approximation (pn-QRPA),

and the shell model. The first one has been used extensively in large-scale calculations to predict

both allowed and first-forbidden transitions using Q-values derived from mass formulas. Improved

calculations have also been done that include pairing and other shell effects. [39]

In addition to the previous, shell model calculations have also been implemented. Here multi

particle transition amplitudes have been incorporated directly from the model Hamiltonian. Such an

approach has the advantage of predicting the state and mass dependence of observed decay strengths,

making the results independent from any input of mass formulas. Calculations for proton-rich nuclei

have been attempted for the cases of sd-shell nuclei and can be found in [40].

The pn-QRPA model has also been used extensively in calculations of /3-decay observables. A

description of the formalism for even-even mother nuclei, as well as additional references, can be

found in [41], The extensions to odd systems and odd nuclei can be found in [42] and [43].

This last model has given results in essential agreements with experimental data. Some extensive

calculations [44, 45, 46] show an agreement with experiment within a factor of two. The calculations

included in [44] describe the /3-strength function by applying the pn-QRPA method with a Gamow-

Teller residual interaction, the strength of which has been fitted to experimentally known half-lives

of known isotopes for a fixed mass number A. In cases of very neutron-rich nuclei only allowed

Gamow-Teller transitions have been considered and the influence of first-forbidden transitions on

the half-lives of nuclei far from stability was neglected. The results obtained show an agreement

with experimental data for nuclei with short half-lives (<1 s) with an average deviation of 1.4.

In other calculations a specific set of nuclei, which play an important role during the r-process,

have been chosen to study /3-decay observables. In [47] an analysis of the ground-state properties and

T\ / 2 of nuclei close to 132Sn is performed using the Hartree-Fock-Bogoliubov method where pairing

correlations are described using the BCS approach (HFB+BCS). Pairing correlations are treated

using a constant strength, zero-range force and only allowed GT transitions were calculated. Using

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the finite Fermi-system theory the values were also determined and no additional mass formulas

were used.

Approaches using a density functional formulation have also been pursued. In [47] a density

functional describing the nuclear system was used with parameters fitted to stable nuclear properties.

Here, besides ground-state properties of magic nuclei, single-particle energies of 132Sn were used. A

similar study [48] was done focusing on nuclei lying in the path of the r-process. In this approach a

general method that combines the microscopic QRPA model for allowed GT /3-decay with statistical

gross theory of first-forbidden decays was used. The results showed that in calculations of half-lives

for a large range of nuclei the quantity In (Tcaic/ r exp) lied in the range 10-0.1.

In [49] a compilation of results of /3-decay properties has been gathered, which include the most

recent experimental data for half-lives and results from two models: 1) Kratz-Herrmann formula

(KHF) and 2) Macroscopic-microscopic QRPA.

Tables 3 and 4 in chapter 2 summarize the results obtained from the methods discussed above in

the particular cases of the nuclei 133Sn and 131In which were studied here. The best agreement for

the decay of 133Sn is given by R3, with a ratio of calculated to experimental half-lives, ti/2, equal

to 0.87 or a deviation of 13%. In the case of 131In, the best agreements are obtained by KHF and

R3 with a deviation of 22% and 18%, respectively.

In all the cases discussed above, the objective has been to reproduce the known experimental

/3-decay half-lives and predict values for other nuclei, especially those far from stability. Most of

the models fall in one of two categories [49]: those which give a mathematical expression (e.g. a

polynomial) for the quantity of interest and those based on an effective interaction. Models of the

first type have no direct link to the underlying nucleon-nucleon interaction and do not give additional

information regarding the nuclear single-particle wave functions. For models of the second type, these

approaches use different effective interactions fitted to better reproduce the experimental fi-decay

data.

In summary, one can observe from these examples the difficulty of having a formulation of the

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nuclear many-body system which is also capable of describing /3-decay transitions. Different models

can be used to reproduce experimental data, but the physics input is sometimes lost.

The EFT/DFT formulation of the nuclear many-body system allows a description of ground-

state properties of finite nuclei and, by applying the approach developed here to study semi-leptonic

processes using the appropriate electroweak currents, it also gives the framework to compute /?-

transition rates. The EFT/DFT approach is still a theory under development, but the results

obtained from its present status, particularly the possibility to extend it to describe nuclei far from

stability, indicate that the effective field theory and density functional theory approach provides a

consistent, predictive framework for dealing with QCD in the strong-coupling, nuclear physics regime.

9 Summary of new contributions.

The EFT/DFT formulation of Furnstahl, Serot and Tang is a novel approach to describe the nuclear

many-body system. It forms part of an ongoing effort to understand the nucleus, particularly new

phenomena occurring in regions of the nuclear chart far from the valley of stability. In its present

form, this formulation had only been tested only for stable nuclei with excellent results. Its validity

outside this region is the central objective of this thesis. The main contributions resulting from the

present work are the following:

1. The convergence with increasing order of the EFT/D FT formulation for nuclei far from stability

has been verified for the doubly magic nuclei 15oSn50, 1foSng2, 4fsNi2o and ^jNiso- The addition

of new interaction terms in the effective lagrangian, following the prescriptions given by naive

dimensional analysis, results in the convergence of the calculations to the observed experimental

values.

2. The EFT/DFT formulation can reproduce the ground-state total binding energy of semi-magic

nuclei far from stability with N = 28, 50, 82, 126 and Z = 28, 50 and 82, with an accuracy

of approximately 1%. Additionally, chemical potentials of odd nuclei neighboring these nuclei

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can be reproduced within 10% of experimental data.

3. A general expression for calculating any semi-leptonic process has been derived. The nu

cleon wave functions are written in terms of Dirac spinors and obtained from solving the

self-consistent Kohn-Sham equations. These Dirac wave functions are used in combination

with the electroweak currents obtained directly from the same effective lagrangian, giving the

whole approach a self-consistent character. Applications to calculate /3-transition rates in se

lected nuclei far from stability, i.e. 1||S n 51 and ^glnso, using this general expression, gave

results that suggest that the description the wave functions of particles in its ground-state is

better than for excited states. This provides (indirect) evidence that this approach indeed also

yields valid ground-state densities.

4. Using the present form of the EFT/DFT formulation, ground-state quantum numbers of odd

nuclei were accurately reproduced in the case of semi-magic isotones with N = 28, 50, 82 and

126 and semi-magic isotopes with Z = 28. In the case of heavy isotopes with Z = 50 and 82,

where N > Z, although the predictions of chemical potentials are in essential agreement with

experiment, the high neutron level density results in the fact that the EFT/DFT formulation

begins to lose its predictive power for the ground-state quantum numbers.

The EFT/DFT approach is still a theory under development, but the results obtained from its

present status, particularly the possibility to extend it to describe nuclei far from stability, indicate

th a t the effective field theory and density functional theory approach provides a consistent, predictive

framework for dealing with QCD in the strong-coupling, nuclear physics regime.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter I.

Extensions to semi-magic nuclei far from stability.

1 Introduction

The study of nuclei far from the /3-stability line has drawn the attention of nuclear physicists. The

implications of the structure and reaction cross-sections from these nuclei play an important role in

the description of nucleosynthesis processes such as the r-process, which takes place in the region of

neutron rich nuclei. In addition to this, these nuclei are the perfect place to test the validity of some

successful methods and theories, which have rendered satisfactory results along the stability line

for medium-heavy nuclei. The development of new experimental facilities and the implementation

of new techniques have already given experimental data for some of these, like the doubly-magic

nuclei 1foSn82 and 15oSn5o. Some of the new approaches being tested are those of relativistic mean-

held models. This thesis deals with a relativistic mean-held theory derived from the chiral effective

lagrangian of Furnstahl, Serot and Tang [1], This lagrangian, which incorporates the symmetries of

QCD, particularly a non-linear realization of chiral symmetry, is built as an effective held theory and

contains the lowest lying hadronic degrees of freedom as the main ingredients. Though it reproduces

the ground-state properties of stable doubly-magic nuclei, its predictability far from the /3-stability

region has not been tested, especially the convergence of the calculations as the number of effective

terms in the lagrangian increases.

Relativistic mean-held models can account for a large number of bulk properties of nuclei and

there are different methods that have produced acceptable results. Recently, Toki, et al. [10] used

a relativistic mean-held approach to calculate properties of nuclei from the proton to the neutron

drip lines, extending the calculations to super-heavy nuclei. The good agreement with known ex-

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46

perimental data shows how relativistic mean-field models can account for a variety of phenomena

in the nuclear many-body system. Toki and his co-workers, using a mean effective lagrangian, have

an extensive program underway on applications of this approach to astrophysics, in particular to

element formation in supernovas [11].

The idea of using effective field theories to describe the nuclear problem is an increasing field of

study, an account of recent results can be found in [50]. The relativistic mean-field approach to the

nuclear many-body problem has been successful in describing medium-size nuclei, and its success can

be understood under the perspective of effective field theory (EFT) and density functional theory

(DFT) [51, 9]. The success of previous relativistic mean-field models, like the original Walecka

model [2, 9] and its extensions to include nonlinear couplings of the a field, is now understood by

applying the ideas of naturalness from EFT [52]. A comparison of various mean-field models and an

account of their success was presented by Furnstahl, et al [53]. In that paper it was shown, using the

EFT approach and NDA, why various mean-field approaches have been successful and that the use

of different degrees of freedom to describe nuclear properties corresponds to different organizational

principles. Thus a description of the nuclear many-body system based on nucleon and meson fields,

as in this approach, or an expansion in nucleon densities, as in the Skyrme model, are equivalent.

The principles of EFT rely on the fact that there exist natural scales and in order to understand

a particular phenomenon it is enough to use probes that can resolve the dynamics at that scale.

These probes are the relevant degrees of freedom needed to describe the system. The most general

lagrangian density is built from these interacting fields paying attention to symmetries imposed by

an underlying theory (QCD in this case) and arranged by some organizing principle, like the naive

dimensional analysis (NDA) from Georgi and Manohar [16, 17]. The unknown dynamics, of heavier

degrees of freedom, is integrated out and appears as coupling constants in the theory. These are

determined by fitting the theory to known experimental data. The number of terms in this EFT

lagrangian is infinite and, in order to give predictive power to the theory, it is necessary to have an

organizing principle that groups together terms that contribute at the same level. In this way it is

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possible to truncate the expansion at some arbitrary order and be certain that all non-redundant

terms, of the same order, have already been taken into account.

The inclusion of new interaction terms in the lagrangian aims to give a better description of

the energy functional, thus going beyond the Hartree approximation and incorporating exchange-

correlation effects. The new terms are non-renormalizable, a distinction with standard RMF models,

which truncate the effective lagrangian at this level without any underlying physical justification. A

chiral effective lagrangian has been derived by Furnstahl, Serot, and Tang [1], which incorporates a

non-linear realization of chiral symmetry and uses the lowest lying hadronic levels: i) pions, whose

field appears as the phase of a chiral rotation of the identity matrix, ii) nucleons, described by

isospinor field N, iii) an isoscalar-vector field, whose mass is taken to be equal to the w-meson mass

and iv) an isoscalar-scalar field which simulates correlated two-pion exchange with an effective mass

around 500 MeV, and v) an isovector field, the p-rneson, to take into account the symmetry energy

in nuclei. By applying the relativistic mean-field approximation, a set of equations of motion can be

derived from the effective lagrangian in which the meson fields act as classical fields.

The connection to DFT is made by considering the expansion of the relativistic mean-field la

grangian as an expansion of the energy functional of the nuclear many-body problem in terms of

nucleon densities and auxiliary classical meson fields. According to the principles of DFT [22, 19],

knowledge of the exact energy functional can be used, by optimizing it with respect to the fermion

densities and auxiliary potentials, to calculate the exact ground-state scalar and vector densities,

energy, and chemical potential for the fully interacting many-fermion system [23]. These optimal

observables are used here to compare the experimental values with the calculated ones. Other ob

servables are not expected to be well reproduced, but a comparison with some of them is included

in this thesis in order to understand the extent to which the theory can be applied, as well as to

find new directions in which more work needs to be done. In this way the EFT principles combined

with DFT methods give an approach to the nuclear many-body problem that uses the simplicity of

solving the self-consistent Hartree equations in which new contributions coming from the interacting

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fields can be incorporated in a systematic and controlled way. The expansion parameters that make

the truncation of the effective lagrangian possible in the nuclear structure case are essentially the

ratios of mean isoscalar, scalar and vector fields to the nucleon mass (« 1/3) and the ratio of the

Fermi momentum to the nucleon mass (« 1/4).

In this chapter the convergence of the EFT/DFT approach to the nuclear many-body problem

is studied when extrapolated to selected doubly-magic nuclei far from the /3-stability region. The

study focuses at how well additional levels of approximation of the energy functional reproduce the

above ground-state properties of the selected, spherical, doubly-magic nuclei. In addition to this,

the effect of fitting a larger set of observables, in the mean, along the stability line is examined.

Attention is placed here on the four doubly-magic nuclei 1|02Sn82, ^S nso, ||N i20 and ™NiM cho

sen for their relevance to the nucleosynthesis processes. Here a comparison is made between the

predictions of EFT/DFT at various levels of approximation with the most recent experimental re

sults for these nuclei. Results are compared of nuclear binding energies for these nuclei and also

extend this analysis to neighboring nuclei that differ by one particle or one hole from these nuclei

thus comparing the values of the chemical potential. In both cases a remarkable agreement is found

with recent experimental data. In addition to this the convergence for single-particle levels, both

for neutrons and protons is investigated. The sensitivity of these calculated spectra to the various

levels of approximation in the EFT/DFT approach is examined, as new interaction terms come into

play. Nuclear densities are also shown for the “proton rich” nucleus ^S nso and the neutron rich one

1foSn82- These densities are in very good agreement with other calculations [32] using the relativistic

Hartree Bogoliubov equations. According to the EFT approach used here, the systematic inclusion

of interaction terms will approach the true energy functional, going beyond the Hartree level and

including exchange-correlation terms. Thus, here additional pairing interactions are included only

to the extent they are systematically incorporated through the additional terms in the EFT/DFT

approach and the fitting of the constants. The goal of the present work is to study how well the

current level of approximation in EFT/DFT reproduces the ground-state properties of the selected

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nuclei without explicit use of the pairing interactions.

The calculations are done using an independently developed code, which solves the self-consistent

Hartree equations at any level of approximation. The code has been tested successfully by repro

ducing previous calculations. The coupling constants, for each level of approximation, were obtained

by Furnstahl, et al. [1] by fitting in the mean to properties of stable doubly-magic nuclei leading to

different parameter sets according to the level of approximation [1, 9]. The results obtained here are

used and the same name for the parameter sets is kept, i.e. W l, Ql, Q2, G1 and G2 in increasing

level of approximation, respectively. In addition to these, two additional sets: L2 and NLC, are

included. The parameter set L2 corresponds to the same level of approximation as the set W l, it

contains no non-linear terms in the meson fields. The other parameter set, NLC, has the same level

of approximation as Ql. Both L2 and NLC were fit to describe different nuclear properties than

those corresponding to the sets W l, Ql, Q2, G1 and G2 9. They also differ from these in the number

of nuclei used to fit the parameters. The values of the constants used in this thesis are given in

Table 1, and the fitting procedure is discussed in more detail later.

Other approaches have been applied to studies of nuclei far from stability with various degrees

of success. Neutron and proton skin formation, the reduction of the spin-orbit interaction and

the realization of pseudo-spin symmetry are some of the phenomena studied recently by various

authors [28, 29, 30]. Shell-model calculations using the CD-Bonn effective potential modified by

the G-matrix and renormalized using the Q-box method have been used to study the importance

of the 1So and 3P2 partial waves in reproducing the constancy of the 2 f - Of level spacing which

is apparent in Sb nuclei [38]. Some of the methods that have been tested on these nuclei with

various degrees of success are the density dependent relativistic Hartree (DDRH) [31, 28], relativistic

continuum Hartree Bogoliubov (RCHB) [29], relativistic Hartree Bogoliubov (RHB) [32], Hartree-

Fock-Bogoliubov [33] and relativistic Dirac-Hartree-Bogoliubov (DHB) [34], In these methods pairing

correlations have been taken into account using a phenomenological force like the Gogny force [29, 32]

®The NLC set describes the deformation of selected medium-weight nuclei [62]

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or a density-dependent zero-range interaction [31]. Other authors [35, 31] used the BCS approach

with a constant pairing strength to take into account the effects of the continuum. Perhaps the

most self-consistent approach has been the use of the RHB, in which ph-interactions and pairing

correlations have been taken into account in a unified way by solving self-consistently the RHB-

equations with the Klein-Gordon equations that describe the meson fields [32, 34]. Other authors

have derived new param eter sets of known potentials, e.g. Nilsson [36], Groningen [31], to fit the

new experimental data of binding energies, single-particle levels,, and nuclear radii for these nuclei far

from stability. Relativistic mean field models (RMF), which have been successful in describing bulk

properties of stable nuclei, are being tested in these new regimes. Some of the models use the a — u

lagrangian with linear and non-linear a interactions [32, 36]. A pplications of D FT to the nuclear

many-body problem have been carried out using the QHD-I and QHD-II lagrangians [19, 20, 21].

Here the exchange-correlations are taken directly from the applications of many-body techniques

in infinite nuclear matter, but extensions to nuclei far from stability have not yet been performed

in this approach. Direct application of the QHD-II lagrangian to the doubly-magic Sn isotopes to

calclulate single-particle level spectrum and nucleon densities can be found in [37].

The combination of the fundamental idea of DFT and the realization that an approximation to

the exact energy functional can be derived using EFT methods is a new approach to the nuclear

many-body problem. As such, it is still in development. To the extent that a sufficient number

of terms are retained in the effective lagrangian to provide the correct energy functional, one has

actually developed the effective field theory for QCD at the nuclear density domain. Extensions to

other density regimes still have to be explored. In this chapter it is investigated how additional terms

in the effective lagrangian contribute to the results for ground-state properties of selected nuclei far

from stability. The EFT methods in this chiral effective lagrangian have already shown that the

coupling constants are all natural [1], therefore the relevant degrees of freedom have been correctly

taken into account and the NDA works. Knowing how well this EFT/DFT approach behaves at

these extreme cases of nuclei far from stability will give a firmer ground for the theory. Predictions

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for the properties of these nuclei is also, of course, of direct relevance to experiments, both ongoing

and planned [54], and to element formation in supernovas.

In this thesis it is observed that the increasing level of approximation approaches the total bind

ing energy of the selected nuclei. There is also very good agreement with the binding energies of

neighboring nuclei, which differ by one particle or one hole from the doubly-magic nucleus, i.e. the

chemical potential. The spins and parities of these nuclei, to the extent to which they have been

measured, are also correctly given. In addition, the calculated single-particle excitation spectra are

shown and the effects of additional levels of approximation is examined. Finally, the nucleon densities

for neutron rich and neutron deficient nuclei predicted with this approach, are shown.

2 Calculations

The chiral effective lagrangian used here was constructed using the relevant lowest lying hadronic

degrees of freedom [1]. The pion field is introduced as the phase of a chiral rotation of the identity

matrix in isospin space. A field £(x) is defined in the following way:

f(z ) = exp(iir(x)/fn). (26)

where the pion field tt(x ) is defined by ir(x) = | r • if

Nucleons are included as isospinor fields, N (x), in which the upper component corresponds to

the proton and the lower component to the neutron.

( p{x) \ N = (27) n(x) ^

To include a contribution to the symmetry energy of nuclear matter, a pM-field is added to the

lagrangian.

A non-linear realization of chiral symmetry is considered following the work of Callan, Coleman,

Wess, and Zumino [15]. A description of the incorporation of chiral symmetry in the nuclear many-

body lagrangian can be found in [2, 9]. In this formalism, the non-linear realization of the chiral

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symmetry is defined by a global transformation, L and R, of the subgroups SU(2)l and SU(2)r

such th a t

L ® R : (28)

where

$'(x) -- L£(x)ht(x) = h(x)^(x)Rj' , (29)

P'n(x) = h{x)pti{x)K! {x) , • (30)

N'(x) = h(x)N(x) . (31)

here p^(x) - \ t • pM.

In addition to the above fields, an electromagnetic field A M is included and the electromagnetic

structure of the nucleons has been taken into account. Their anomalous magnetic moments, labeled

as Ap and A„ for proton and neutron respectively, then enter. An isoscalar-vector field is needed

to correctly represent the bulk properties of nuclei. This field is treated as a chiral singlet and its

mass is taken to be equal to the w-meson mass. Finally, an isoscalar-scalar field 0, again a chiral

singlet, is included to incorporate the observed mid-range N-N attraction: Its mass is determined by

fitting the calculations to experimental results.

The various interaction terms present in the effective lagrangian have been arranged following the

NDA from Georgi and Manohar [16, 17]. Details concerning the construction of the chiral effective

lagrangian can be found in [1] where the full effective lagrangian is shown.

In order to solve the equations of motion derived from this lagrangian the meson fields are

treated as classical fields. In doing so the pseudo-scalar pion field does not contribute because it has

no expectation value. The interplay between these fields, particularly the scalar and vector fields,

accounts for the bulk properties of nuclei as is well described in [2, 9]. For spherically symmetric

systems, as is the case for double-magic nuclei, the spatial components of the vector field vanish.

The rho isovector field develops only the non-charged component, po. The resulting equations of

motion consist of a Dirac equation for the nucleons and a system of non-linear, coupled differential

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equations for the meson fields.

The nucleonic part of the lagrangian density is 10:

CN (x) = N W D v - M + gs)N - ^ N p ^ N - f-^ - N N

^FfxvNXo^N ~ ^ N 7 li(ps + pvrz)NdvF^ (32)

where D p = + igppp + igv Vp + %eAp( 1 + r 3).

The mesonic part is the following:

Cm (x) i ( l + a1^ ) d li4>d^-±tr(pfluPn

-^-F ^M npn + \v n + ^(i+m

+ ^ , C o 9v(VPV^)2 + ( 1 + 7 7p ^ W p t r i p ^ )

p ^ v p (33)

In the above lagrangian, is the scalar field, Vp the vector field, pp the isovector field rho defined

by pp = | f • Pfi and A p is the electromagnetic field.

From the above effective lagrangian density one obtains the Kohn-Sham equations and the aux

iliary equations for the potentials. To do so, the various mesonic and electromagnetic fields as

considered as local classical fields. Then the necessary equations are obtained by calculating the

Euler-Lagrange equations.11

The Dirac hamiltonian for the nucleon fields takes the form:

h(x) — - ia • V + W (x) + \ t3R(x) + /3(M - $(x)) + ^(1 + t3)A(x) JL £d

(34)

Here A = §AP(1 + r3) + §A„(1 — r3) and the numerical values used for the anomalous magnetic

moments are Xp = 1.793, Xn = —1.913.

10The metric and conventions used are those of [1, 14], and differ from [2]. u The subsequent equations are given in [1]. They were independently derived by the current author.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54

The mean meson fields are denoted by W — gv\o, $ = gs(f>o, R = gPbo, and A = eAo respectively.

The quantities f p, fv,/3s,/3v are parameters fit to experiment (see below). The meson field equations

become the following:

_V2$ + m2$ = q2pAx) - + — — ) + -^-(m + r?2— ) — IF2 + s 9sPs^x> M 2 3! M 2 M ' ' M g%

+ i f f * 2 +l ^ $)2 +2$V2$J+3 h ^ w)2 (35)

Here ps is the baryon Lorenz scalar density and gs. ms, k3, Kt, r/i, 772, gv, 9p, ot 1, 0:2 are again

parameters fit to experiment. Furthermore,

- V 2W + m 2vW = g2v[PB(x) + ~ V < p l ( x ) r ) ] - | ~ ) ^ r n 2vW

- J 7C0W 3 + g ( V $ ■ VW + $V2IF) - ^Pchg(x) (36)

Here ps is the baryon density, p rB is the baryon tensor density, pchg is the charge density and f v, Co

are parameters. In addition,

- V 2R + m2R = ^g2p[pz(x) + ^ ^ ■ ( /> s (*)?)] - VP~ m 2pR - ^ p c h g(x) (37)

Here p3 and pT are the isovector densities, f p, gp, gp are parameters and <7-,.=5.01, is the coupling

of the photon to the w-meson. Finally,

~ V 2A = e2pChg (x) (38)

where pChg is the charge density. To obtain the single-particle energy levels one has to solve the

Dirac equation

hlpa{x) = E a1pa(x) (39)

The Dirac solutions are expressed as,

f • \ pG a( r ) $ Km 1pa(x) Ct (40)

where the $ KTO is a spin spherical harmonic and the index t in Q indicates the isospin component

of the nucleon, this is equal to 1/2 for protons and -1/2 for neutrons. Inserting this into Eq.(39)

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and using Eq.(34) a set of two coupled first-order differential equations for the G and F functions is

obtained:

+ f ) ° a{r) ~ [Ea ~ Ul{r) + U^ Fa{r) ~ UsGa{r) = 0 (41)

{ i ~ 7 ) Fa[r) + [Ea ~ Ux{r) ~ U2^ Ga{r)+UM r) = 0 (42)

where the single-particle potentials are given by

Ui(r) ~ W (r)+taR(r) + (ta + -)A(r) + +2ta{3vW2A(r) (43)

U2{r) = M ~ $ (r) (44)

U3(r) = ~{fvW '{r)+ tafpR!{r)+A'{r){{\p + \n)l2 + ta{\p- \ n)]} (45)

Here the prime indicates a radial derivative; e.g. W '(r) = dW (r)/dr. Once the single-particle wave

functions are calculated, the various densities that appear on the r.h.s. of the meson equations can

be obtained. They are defined as follows:

occ - . .. *(*> = E (G>) - F«V)) («) a 000 O' i i » ( ■ ) = E - ^ r K w + ^ w ) m a

OCC . ^

Pb (*> = <48> a occ = E ^ f ^ ( 20 ( ^ ( r ) + F a2(r)) (49) a occ ^ . - /# (* ) = E ^ T - ( 2^ ) 2G“W F“ M (50) a

where the sum goes over the occupied orbitals. In addition to these, the charge density is composed

of two pieces: a direct nucleon charge density and the vector meson contribution

Pchg = Pd(x) + Pto (^) (51)

The direct part is given by

Pd(x) = PP(x) + ^ - ( p l( x ) r ) + ^ ( P sV2pB(x) + PvV2p3(x)} (52)

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and the vector meson contribution arising from the coupling of the neutral vector mesons to the

photon (VMD) takes the form

pm(x) = — V 2f? + (53) 579p ^9i9v

Here the point proton density pp and nucleon tensor density p \ are given by

Pp = 2 ^ b +Ps) (54)

OCC Pz = H ^ (x)iXP® • f Vv (x) (55) a

To solve the meson equations a Green’s function method is used, and is described in the Appendix.

The equations are iterated until a consistent solution for all the meson fields is obtained, that is, a

0^-order solution to the meson equations is obtained by neglecting all non-linear terms on the r.h.s.,

and then use this solution to obtain a l si-order solution, continuing until convergence is achieved.

Because NDA guarantees that each additional term on the rhs of Eqs. (35) - (38) is smaller than

the previous one, convergence is both expected and obtained.

The whole system of Eqs. (41), (42) and (35) - (38) is solved self-consistently until a global

convergence is reached.

The values of the various constants used in this paper are given in Table 1 with a label indicating

the parameter set. These values are taken from [1, 9]. The constants were obtained by fitting several

nuclear properties [1]. From the 16 constants in Table 1 the last 3 where determined by fitting to

electromagnetic properites of the nucleon. The sets labeled W l, Ql, Q2, G1 and G2 were obtained

using a weighted generalized ;\;2 defined by

2 __ Y ' \ ^ r Aexp y ( * ) ~ Ath y (* ) 2

A—' ' w(®) y(‘) i X vvX A exp

to fit a total of 29 observables listed as follows:

• The binding energies per nucleon E/B with W=0.15%;

i • The rms charge radii {r )^hg with W=0.2%;

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• the diffraction-minimum-sharp, d.m.s., radii Rdms with W=0.15%;

• The spin-orbit splittings A E so of the least-bound proton and neutrons with W=5% for 16 0,

15% for 208Pb, 25% for 40Ca and 48Ca, and 50% for 88Sr; 12

• The proton energy Ep(lhs/2) and the proton level splitting E p(2d3/2) — Ep(lh ii/2) in 208Pb

with W=5% and 25%, respectively;

• The surface-energy and symmetry-energy deviation coefficients Sa2 and £04 w ith a weight

W =0.08.

The parameter set L2 was obtained by fitting the constants to reproduce the infinite nuclear

matter properties [9]:

• Saturation density p% = 0.1484 fm~3.

• Binding energy per nucleon E/B = -15.75 MeV.

• Bulk symmetry energy = 35 MeV.

• rms charge radius of 40Co = 3.482 fm.

3 Results

In this section the results of the calculations of several ground-state properties of the nuclei 1|QSn82,

^Snso, 4|Ni20, and IfNiso are shown and also those of the neighboring nuclei differing by one

particle or one hole. Results for the total binding energy, are shown in Fig. 7

The plot is divided in two parts, which show different forms of the convergence of the models. On

the left are the results of total binding energy using the simplest lagrangian, i.e. with no non-linear

terms. The two parameter sets L2 and W l represent the result of fitting the constants of the model to

12Note that the spin-orbit splitting arises predominantly from the mean isoscalar scalar and vector fields that are

determined by the bulk properties [2, 9].

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TABLE 1: Values of the different parameters in each parameter set used to solve the self-consistent

m ean-held equations from [1, 9]. Here m s is in MeV.

C onstant L2 NLC W l Q l Q2 G1 G2

m s 520 500.8 566.26 504.6 509.6 506.7 520.3

al 109.63 95.11 138.93 103.6686 97.7699 97.39 110.16

al 190.43 148.93 203.97 164.6963 149.2 147.09 162.88

92p 65.23 74.99 95.551 77.9558 73.2256 77.033 89.936

Vi 0.07060 0.64992

m -0.96161 0.10975

1.9194 1.6582 1.7424 2.2067 3.2467

Ki -7.3923 -6.6045 -8.4836 -10.090 0.63152

Co -1.7750 3.5249 2.6416

vP -0.2722 0.3901

a x 1.8549 1.7234

a 2 1.788 -1.5798

fv 0.4316 0.6936

fp 3.7328 4.1328 4.2640 4.1572 3.8476

f t -0.38482 -0.10689 0.01181 0.02844 -0.09328

f t -0.54618 -0.26545 -0.18470 -0.24992 -0.45964

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9 4 a . . 28 20 - E X & > -400 -400

>* S D) Im. -600 S ' * ' 150 -600 E x p 9

-1200 ..... i ...... a, ..., ... -1200 L2 W1 NLC Q1 Q2 G1 G2

Figure 7: Total binding energy of the selected nuclei. The left hand side shows two models with

the same level of approximation but fit to different data sets (see text). The right hand side shows

models with increasing level of approximation. NLC and Ql have the same level of approximation

as do G1 and G2.

different nuclear properties. As described in the previous section, the W l set is fit to more data than

L2. The improvement of the results is clear. The experimental result is shown as a long-dash line in

the plot. The error bars are of the same size as the thickness of the line showing the experimental

value. The experimental results are taken from [55], except for ||N i2o where the value quoted in [56]

are used. On the right side of Fig. 7 are the results obtained from the more elaborate lagrangians,

which in general include different non-linear interactions as shown in the previous section (see Eqns.

(35) - (38)). The first two points, corresponding to the NLC and Ql parameter sets, represent the

same level of approximation of the lagrangian. Here, it can be seen again, that the use of an extended

set of observables in the fitting procedure dramatically improves the results. In this case, the Q l set

already gives an excellent result for all nuclei.

Moving to the right on this part of the plot, the effect of additional interaction terms in the

lagrangian can be seen. The inclusion of a fourth-order vector field term in the lagrangian, see

Eq.(36) and Table 1, gives a larger binding energy. This effect is further balanced by additional non-

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linear terms in the G1 and G2 sets. At this level, the sets G1 and G2 give almost the same results

and cannot be differentiated. Since both sets correspond to the same level of approximation, these

results show the sensitivity to the variations of the fitted constants. The same features are verified

for all the nuclei tested. Figure 8 shows the percentage deviation in estimating the total binding

energy of the selected nuclei. It is observed that it lies below approximately 1%, in all cases, and the

parameter set Ql gives the smallest deviation. Extending the parameter set to Gl and G2, while

validating the NDA and the assumption of "naturalness” as discussed in [1], does not significantly

improve the quality of the fit to this quantity for the nuclei shown. For the four doubly-magic nuclei

tested a better agreement is obtained in the case of the Sn-isotopes than for those of Ni.

For the Sn isotopes, the calculations of the total binding energy are extended to cover the entire

range of even-even nuclei. Figure 9 shows the results for the G2 parameter set for the isotopes

'ggSnso - 1foSn82. The percentage deviation in this extended case is shown in Figure 10. Here it

is observed that the deviation for the three sets Ql, Gl and G2 is in absolute value less than 1%.

In general, all three parameter sets give less binding energy than observed. For all parameter sets

shown in this figure, G l gives overall the smallest deviation, except for the two nuclei ^oSnso and

1foSn82, which lie at the extreme edges of the plot. This would imply that there is indeed improved

convergence overall in going from Ql to the extended parameter sets G l and G2, and Gl gives better

results than G2 over the entire range.

From this figure the effects of the accuracy of the isovector contributions to the effective la

grangian, can be observed. Following, for example, the G l results one can observe that in the center

of the plot there is essentially a constant deviation from the experimental values, while this increases

close to the edges of the isotope range. A study of the problem of the isovector contribution to the

description of finite nuclei has been presented recently by Furnstahl [61].

One can conclude that a more sophisticated lagrangian, fit to stable nuclear properties, can

reproduce ground-state total binding energies of nuclei far from stability.

It is important to note that even though the constants of the model were fit using only stable

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 8: Percentage deviation of total binding energy of the four selected doubly-magic nuclei.

Parameter sets Ql, Gl and G2 are shown.

doubly-magic nuclei, the results for binding energies of these nuclei far from stability are remarkably

good.

Figures 11 through 16 show the level structure of neighboring nuclei differing by a single particle

or hole, relative to the doubly-magic nuclei. They are separated in isotopes and isotones. The core

nucleus is plotted in the center of the plot marked with 0+. On the left are the neighboring nuclei

with one more particle. On the right the corresponding nucleus with one less particle is shown.

When available, the experimental values are indicated in the plot. These results were obtained using

the G2 parameter set, although the Gl set gives very similar ground state binding energies. In all

these cases there is excellent agreement with the experimental data [55].

The chiral effective lagrangian has been applied to calculate the ground-state properties of se

lected doubly-magic nuclei. The inclusion of additional interaction terms in the effective lagrangian,

consistent with NDA, gives results that approach asymptotically to the experimental value for the

total binding energy. These calculations were extended to neighboring nuclei, differing by one par

ticle or one hole, and determine the binding energy relative to the doubly-magic one and found an

overall agreement of better than 1% for the total binding energy of the doubly-magic nuclei and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62

-810 o G2 > ♦ Exp ©

' r ' - 9 i o a? cCD CD O) .E -1010 T 3 C m

-1110 98 102 106 110 114 118 122 126 130 134 A

Figure 9: Comparison between experimental and calculated total binding energies for Sn-isotopes

using the G2 parameter set.

better than 10% for the relative binding energy of the neighboring nuclei. In this latter case, the

spin and parities of the ground-states are correctly predicted.

Single-particle excitation spectra for the nuclei b^jSnss and 1ff Sbg2 have been measured by Hoff,

et al. [58] and by Sanchez-Vega et al. [59]. The level spectra for these nuclei is shown as a function of

parameter set in Figs. 17 and 18. Here results using only the G2, G l and Ql sets have been plotted

since these give the best results for the total binding energy of 1fpSn82. From the discussion of the

DFT approach one cannot expect these single-particle and single-hole excitation spectra to be well

reproduced, though it is observed that there is a slow convergence of the results with higher levels of

approximation. This might indicate the need for many more terms in the lagrangian. On the other

hand the excellent agreement shown in Fig 9 for the total binding energy, and Figs. 11 and 12 for the

ground-state properties (i.e. chemical potential leading to the neighboring single-particle and single

hole nuclei) suggests that the current level of approximation is enough to reproduce the ground-state

observables the theory is designed for. One can observe that increasing levels of approximation give

better results in the level spacing, e.g. Fig 17 though the ordering is essentially independent of the

level of approximation. Note that in the case of 1fjiSb82 results for the Ql parameter set are not

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63

Sn-isotopes

LU CO

LU Stable m nuclei

LU m

Figure 10: Percentage deviation of the total binding energy for Sn-isotopes using Ql, G1 and G2

parameter sets. The stable isotopes are indicated in the plot.

included for levels above 2d5/2- This is because the program used did not find those states to be

bound, a problem that is not present in the case of 1foSns3. This feature shows that since the higher

order terms included in the G1 and G2 parameter sets not only contribute to the bulk properties of

the nuclei but also improve the description of the single-particle spectra. In Figs. 19 and 20 the

results for the neutron and proton densities of ^Snsa and ^qSiiso are shown. In both cases the

density of neutrons is always higher in the interior of the nuclei. This is particularly interesting in the

case of 1goSn5o since there are an equal number of neutrons and protons implying that the protons

are pushed out, due to the Coulomb potential, forming a thin layer at the surface. In contrast, in

the neutron rich case the neutrons extend far out and form a neutron layer. In Fig. 21 the neutron

densities of these two doubly-magic Sn nuclei are compared. These results are in good agreement

with calculations made using other approaches [32].

The calculations of the nuclear properties using the DFT approach do not include the contri

butions of pairing interactions. Even though most modern calculations use some type of pairing,

as mentioned before, the goal of this study is to determine the convergence of the results for the

optimal observables as new interaction terms are included. Extending these calculations beyond

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 /2+ > 2d3/0 CD 3s1/"'z ■■==:■■ • 11.'2+ Isotopes 1h11, __ ••• 'z in /„• ■ __ E? '2/ • 0) 6 - c Exp.Value CD D) C T3 1h%. ®/-2 C 3pV w _ 0 + Id 0 - ■:%- ->*&- 7 . *-» '2 as Exp.Value

Figure 11: Level spectrum of isotopes of 1|oSn82 differing by one neutron.

> 2P% cd 2 0 - 2p/27 9, * .U* >. Isotones ° I D) /Z+ O 10 - Exp.Value c CD a> % 0H . 0+ C 2d% %+ 1q CD =

Figure 12: Level spectrum of isotones of xfoSn82 differing by one proton

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. > 3/ 0 2d%. .• 20 - 2p1/2. Isotopes 9,2 5^ ig%- /2+ g> 0 10 c 0 O) c § 0 — 0+ X 3 2d% ..%+ c l a l97/2/— '/'=/2+ | - 10- Exp.Value J9 i°ic„ 100„ " q _ 0 50 51 50 50 50 49 DC -20

Figure 13: Level spectrum of isotopes of 15oSn50 differing by one neutron.

> 0 15- Isotones E? 10 2p%. 0 /,V c 2p1/2 ' \__ ■' .* 9y + 0 « 9/ 0) 5 ig /2.. c 7 3 Exp.Value C 0 ---- 0+

0 > -5- 1°0Qrt 99, 0 SO 50 QC -10 49 n50

Figure 14: Level spectrum of isotones of 15oSn50 differing by one proton.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. > 15 - 0 2p%---- •..%- ^ 1 0 - 2p%----- uO) lu 0 Isotopes C , v 0 19%---- • CT> 5 - c Exp.Value T3 C 2d%. !a 3s1/, - , y 0 0- ■ 0 + .> 2d%.::5 • ■ y 0 /2+ 0 79k i. 78, DC ,5 . 28 51 28^*50 28^49

Figure 15: Level spectrum of isotopes of IgNiso differing by one neutron.

35 > 1d3/2-. 0 30 . _ A + 25 2s1/,-- • — — • • /g+ 20 H Isotones ------■•%- u03 0 15 C 0 10 O) c 5 ~a OH -----0+ c JD -5 2p\ A f 0 -10 2p% /A? > -15 0 Exp.Value 0 -20 79n 78.,■ DC -25 29 50 28 50 27<“'° 5 0

Figure 16: Level spectrum of isotones of jfNiso differing by one proton.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67

6 0 0 0 133Q. > 51 82 © 5 0 0 0 3sV2------.... 1h /2 ______2 4 0 0 0 o 2 d % ------::------© % 3 0 0 0 11/2- 2d % _ © ___ 3/2+ o 2000 ------.....______■■C © 5/2+ Q.g 1000 © ig % ------___ 7/2+ E 0 CO -1000 Q 1 G1 G 2 Exp

Figure 17: Comparison between single-particle level spectrum of 1|iSbg2 and recent experimental

d ata

^ 3 00 0 133.'a ,Sn,

5/2- •+—'E! 2000 O © 9/2- Q. Ct) © 1000 3/2- o '■c cd Q. Q 7/2- ® O) _c

^ -1000 Q1 G1 G 2 Exp

Figure 18: Comparison between single-particle level spectrum of with recent experimental

data

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68

0.08

CO E

£ 0.04 0 proton density O neutron density

r (fm)

Figure 19: Proton and neutron densities for 1|QSn82

0.08 — proton - - neutron

0.06

c 0.04

0.02

r (fm)

Figure 20: Proton and neutron densities for

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <1) "D 0.04 C u.O —‘ ZS CD 0.02 Z

r (fm)

Figure 21: Comparison between neutron densities of 1foSn82 and ^ S ^ o .

these observables gives less satisfactory results, as has been seen in the single-particle and single-hole

excitation spectra of neighboring nuclei, Figs 17 and 18. Although beyond the original purpose, it

is interesting to compare the experimental values of the two-neutron separation energy S2iv for the

series of nuclei ^oSnso - 1|oSn86 with results obtained with this approach. Results are given for G2

only, since G1 gives essentially the same values. To make a reasonable comparison, the configuration

energy contribution coming from the mean-fields is extracted by eliminating from the experimental

data the pairing contributions. To do this a simple solvable pairing model [60] is used. A j-dependent

strength Gj corresponding to a pure pairing force is introduced. The result of the model is that 52 jv

is related to the number of particles in the j-th sub-shell by the formula:

S2 N — 2eeff(j) + Gj(2j + 5 — 2 n)

where n is the number of particles in the j-th shell. Fitting the experimental data to this formula,

which indeed goes through all the experimental points, gives the values of the configuration energy

2ee£f(i) and Gj shown in Table 2.

To obtain the calculated configuration energy £eff{j) using the present approach, it was calculated

using two methods. In one case only one particle in the j-th sub-shell is considered, while the other

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70

TABLE 2: Results of extracting the configuration energy from the experimental data on S2 N using

the pairing model described in the thesis.

7 5 3 11 7 j 2 2 2 2 2

2eeffC0 21.88 18.51 16.27 13.49 5.86

Gj 0.2712 0.2415 0.2095 0.1210 0.03652

lower major shells and sub-shells where filled, and under these conditions the total binding was

determined. Extracting the contribution from the lower filled major shells and sub-shells, gave the

configuration energy. In the second case the j-th sub-shell was completely filled, and proceeded as

before by calculating the total binding energy and then subtract from it the contribution from all the

lower filled shells and sub-shells. This final result was averaged by dividing it by the total number

of neutrons in that j-th sub-shell, 2 j + 1. Both methods gave essentially the same result. Figure 22

shows the comparison between the extracted and calculated configuration energies. The plot shows

that, though not quantitatively, the present approach reproduces the level ordering.

The inclusion of pairing correlations is a subject that goes beyond the scope of the present study,

but it is recognized that some form of pairing interaction has to be included to obtain a more

complete description of nuclear observables. Work on this subject has shown that in the case of

standard RMF models the approach using the DHB seems to be the most straight forward way to

include such effects [34].

4 Conclusions

From the results shown in this chapter the following conclusions can be drawn:

1. As additional interaction terms are included in the effective lagrangian, following the prescrip

tions given by naive dimensional analysis , the calculated results of ground-state properties of

selected nuclei far from stability converge to the observed experimental values. The overall

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71

> 30 CD • S2n- ex p - 2 e ef((j)-exp - 2e 0(t(j)-theory

•— __a_

19 7/2 265/2 2d3/: 1h11/2

3 O) c 2f7/2 qo 0 L— ■ 1 —1------^ 96 100 104 108 112 116 120 124 128 132 136

Figure 22: Comparison between configuration energies for Sn-isotopes. The dashed line corresponds

to calculated values using the G2 parameter set and the solid one is the value extracted from the

experimental data on two-neutron separation energies S2 N using the simple pairing force model

described in the thesis.

agreement, averaged over all tested nuclei is better than 1% in the total binding energy. The

relative ground-state binding energy of neighboring nuclei, differing by one particle or one hole,

are well reproduced, with a relative error of better than 10%. The spin and parity of these

nuclei, where measured, is also correctly reproduced.

2. The effect of fitting the constants of the Chiral Effective Lagrangian to a large number of

observables greatly improves the results for the total binding energy of these nuclei far from

stability.

3. Nucleon densities calculated with this approach are in agreement with previous works.

4. The effective field theory and density functional theory approach provides a predictive frame

work for dealing with QCD in the strong-coupling, nuclear physics regime [23].

5. Additional interactions to reproduce the effect of pairing are needed to extend the theory to

reproduce additional ground-state properties.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter II.

Applications to electroweak transition rates in selected nuclei

far from stability.

1 Introduction

In the previous chapter the convergence of calculations to experimental results at different levels of

approximation from a new approach to the nuclear many-body problem was studied. This approach

combines the principles of Effective Field Theory (EFT) with Density Functional Theory (DFT).

The results of that work showed that the total binding energy of even-even Sn isotope nuclei could

be reproduced below the 1% level. In addition to this agreement for the total binding energy of

the doubly-magic nuclei 153o2Sn82, X5SSn5o, 28^50 and 28^ 20, the chemical potential for neighboring

nuclei, differing by one particle or hole from the doubly-magic ones, was also well reproduced below

the 10% level. The agreement in binding energies shows that the energy functional derived from

the effective lagrangian of Furnstahl, Serot and Tang [1] is indeed a good approximation and thus,

according to DFT, the ground-state densities obtained in each case are also a good approximation to

the true ground-state densities. Although in the previous chapter both proton and neutron densities

for 132 Sn and 100 Sn were presented, there has been no direct measurement of either of them and thus

the comparison with experiment has not yet been established directly.

In the Kohn-Sham approach, the ground-state density is constructed from single-particle wave

functions, obtained by solving the Kohn-Sham equations of the system. These equations are the

energy eigenvalue equations for a system of non-interacting particles subject to a local external

potential. Except for the energy eigenvalue close to the Fermi surface (i.e. the energy necessary to

extract one particle from the system) all other energy eigenvalues have no direct physical meaning.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73

If the single-particle (hole) energy at the Fermi surface agrees with experiment, one can assume that

the wave function associated with it must also be a good approximation.

In this chapter the validity of the last statement is studied, and in an indirect way the accuracy to

which the ground state density is reproduced. The wave functions describing one particle outside a

doubly-magic core or one hole in a doubly-magic core are used to calculate various ^-transition rates.

These wave functions represent the initial or final nuclear states (in coordinate representation) in

these transitions. Here, based on the above arguments, an accurate description of the wave function

is assumed to be guaranteed by the accuracy with which its binding energy is reproduced. Therefore

the ideal case of study corresponds to a ground-state to ground-state transition, since in these cases

there is a closer agreement with the experimental energy values. Wave functions of excited states will

be less accurate as these states lie farther away from the Fermi surface and thus their energy is not

well reproduced. Furthermore, it is expected that the particle-particle transition will give a cleaner

result since the description of a particle outside the doubly-magic core can be well approximated by a

single-particle wave function. The description of a hole, on the other hand, is more complicated since

it represents a more complex many-body state and its description by a single-particle wave function

might be expected to be less accurate. Using the results obtained in the last chapter and the

available experimental data, this chapter focuses on the core nucleus 1foSn82 and its single-particle

and single-hole neighbors ^ S n ^ , ^Snai, 1siSb82 and ^ In ^ . Using these nuclei, the two types

of transitions, particle-particle and hole-hole, can be investigated and compared with experimental

results to validate the observations made above. The first case, i.e. the particle-particle transition,

corresponds to the /3-decay process ^ S n ^ -> 1fiSbs2 + e~ + ve and the second one, the hole-

hole case, to the transition ^gln^-^fgSnsi + e~ + Ve. In both cases ground-state to ground-state

transitions, as well as transitions from and to low-lying excited states, are considered.

To study these processes a general expression describing semi-leptonic transitions has been ob

tained that incorporates the Dirac wave functions calculated by solving the Kohn-Sham equations

derived from the effective lagrangian [1], The upper and lower components of the Dirac spinors are

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74

then used to calculate the matrix elements of the appropriate electroweak currents. These currents

are obtained from the same effective lagrangian and correspond to the leading order Noether cur

rents [25, 2]. At any level of approximation in the effective lagrangian, the axial-vector currents

satisfy PCAC and show a pion-pole dominance [25]. The matrix elements of the currents thus con

structed are used in a general multipole expansion from which any semi-leptonic weak interaction

can be calculated [2]. The currents used here are one-body operators and as such the matrix elements

involve only contributions from single-particle dynamics. Other effects such as couplings to collective

modes are not included13.

This chapter focuses on /3-decay rates. Applications to other semi-leptonic processes are being

considered for future work. The calculated decay rates have been corrected for the screening effect

from the valence electrons in the daughter nucleus [63] and for the slowing down of the emitted

electrons due to the attractive Coulomb core (Fermi function). This last correction uses a relativistic

description of the electron and takes into account the effects of the size of the nucleus [63].

Calculations of /?-decay properties, like the half-live (r1(/2) and /3-delayed neutron emission (P„),

have been done extensively using mainly the following models: gross theory, quasi-particle ran

dom phase approximation (pn-QRPA) and shell model calculations. Gross theory has been used in

large-scale calculations where the discreetness of the final energy levels of the daughter nucleus are

smoothed out and different single-particle strength functions (Gaussian, modified Lorentz) are used

to calculate the /3-decay strength [64]. This approach has been used to calculate both allowed and

first-forbidden transitions using the Q-values from mass formulas as input. Improvements to this

model have been made in which pairing and other shell effects are taken into account [39]. Shell

model calculations are one of the most elaborate methods used. In this type of calculation it is

possible to incorporate multi-particle transition amplitudes, which are uniquely determined by the

specification of the model Hamiltonian. Such an approach has the advantage of predicting the state

and mass dependence of observed decay strengths making the results independent of any mass for

13These effects can be as large as 50% as discussed in [74] for the case of 28fPbi27.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75

mulas input. The disadvantage of this method lies in the large matrices that have to be computed

when the number of nucleons increases, therefore its applications has been limited to small nuclei.

A calculation for proton-rich nuclei has been attempted for the cases of sd-shell nuclei and can be

found in [40]. The third approach, pn-QRPA, has also been used in extensive calculations of /3-decay

observables. This approach can be considered to be between the shell model and gross theory. A

description of the formalism for even-even mother nuclei, as well as references, can be found in [41].

The extensions to odd systems and odd nuclei can be found in [42] and [43].

From the above models used to calculate /3-decay half-lives, QRPA has shown to give good

agreement with experimental results. Some extensive calculations have been performed [44, 45, 46]

showing an agreement with experiment within a factor of two. The calculations included in [44]

describe the /3-strength function by applying the pn-QRPA method with a Gamow-Teller residual

interaction, the strength of which is fitted to experimentally known half-lives of known isotopes for

a fixed mass number A. Pairing correlations are treated in the BCS model using a constant pairing

force and without taking into account the Pauli blocking. The proton and neutron gaps are equal

and are taken from the values of global systematics. In addition to this, spin-isospin ground-state

correlations are included. For very neutron-rich nuclei only allowed Gamow-Teller transitions are

considered and the influence of first-forbidden transitions on the half-lives of nuclei far from stability

is neglected. Nuclear deformations also are taken into account using the Nilsson model from which

the wave functions of the parent and daughter nuclei are calculated assuming the same ground-state

deformation for both. The main uncertainties in these calculations come from the mass formulas used

as input, which in general become less accurate for nuclei far from stability. The results obtained

show an agreement with experimental data for nuclei with short half-lives (<1 s) with an average

deviation of 1.4.

Other calculations have concentrated on some specific range of nuclei important for the r-process

such as those close to 132Sn, as in our case. In [47] an analysis of the ground-state properties and

TXj2 of nuclei close to 132Sn is performed using the Hartree-Fock-Bogoliubov plus BCS pairing ap

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76

proach (HFB+BCS). Although some studies indicate that the nuclei in this region are spherical,

the calculations included both spherical and deformed nuclei. Contributions from pairing using a

constant strength, zero-range force were also included. Again here, only allowed GT transitions were

calculated and the GT strength function was obtained using the method of a self-consistent treat

ment of the ground and excited states of even-even and odd-A superfluid nuclei solving QRPA-like

equations in the finite Fermi-system (FFS) theory. Using this method, the were also determined

and no additional mass formulas were used. A density functional describing the nuclear system was

used with parameters fitted to stable nuclear properties obtaining three different sets. One of them

was specialized to reproduce not only known ground-state properties of magic nuclei but also single

particle energies of 132Sn. Details and additional references are given in [47]. A similar study [48],

focused on nuclei in the r-process path, used a general method that combines the microscopic QRPA

model for allowed GT /3-decay with statistic gross theory of first-forbidden decays. In general, the

results show that there is a much better agreement with experiment when the first-forbidden transi

tions are included, especially for large values of t x/ 2 - When the calculated half-lives for a large range

of nuclei were compared with experimental results it was found that the quantity In (Tcaic/ r exp) lies

in the range 10-0.1.

Because of new data and improved methods to calculate /3-decay properties, there have been

efforts to compile the existing experimental, as well as calculated data. In [49] such a compilation

can be found which includes the most recent experimental data for half-lives as well as the results from

two different models: 1) Kratz-Herrmann formula (KHF) and 2) Macroscopic-microscopic QRPA.

Tables 3 and 4 summarize the results obtained from the methods discussed above in the particular

cases of the nuclei 133Sn and 131In. Here one can appreciate the level of agreement with experiment.

The best agreement for the decay of 133Sn is given by R3, with a ratio of calculated to experimental

half-lives, t x/ 2, equal to 0.87 or a deviation of 13%. In the case of 131In, the best agreements are

obtained by KHF and R3 with a deviation of 22% and 18%, respectively.

All these models aim to reproduce the known experimental /3-decay half-lives as well as predict

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77

TABLE 3: Calculated /3-decay half-lives (in ms) of ^gSn^. The quantity in parenthesis on the

column exp indicates the error (e.g. 10(2) means 10 ± 2).

1KHF 1QRPA-1 1QRPA-2 2DF2 2DF3 2 HI 2R2 2R3 3G T 3G T + ff 4 exp

362 9479 9479 9320 8200 823 10290 1260 - 1450(30)

4See [49], 2See [47], 3See [48], 4See [65, 66],

TABLE 4: Calculated /3-decay half-lives (in ms) of 13gIn82-The quantity in parenthesis on the column

exp indicates the error (e.g. 10(2) means 10 ± 2).

4 4KHF ^RPA-l 4QRPA-2 2DF2 2DF3 2R1 2R2 2R3 3G T 3G T + ff *exp

216 146 146 390 350 394 1470 332 147.1 1.39.2 280 ± 30

1See [49], 2 See [47], 3 See [48], 4 See [65, 66],

its value for other nuclei, especially those far from stability. As pointed out in [49], most of the

models can be grouped in two categories: those which give a mathematical expression (e.g. a

polynomial) for the quantity of interest and those based on an effective interaction. Models of the

first type have no direct link to the underlying nucleon-nucleon interaction and do not give additional

information regarding the nuclear single-particle wave functions. For models of the second type, these

approaches use different effective interactions fitted to better reproduce the experimental /3-decay

data. In other cases, because there is an overparametrization of the effective interaction, its relation

with the nucleon-nucleon interaction is also lost. Most of these models combine different approaches

and approximations to better reproduce the experimental data.

In general, it can be said that there is no consistent theory that by fixing its parameters once,

reproduces not only the masses but also the single-particle levels and (3-decay properties of nuclei.

In this chapter such a consistent approach is explored and compared with experimental data

for /3-transition rates of neighboring nuclei 132 Sn . This approach corresponds to a new theory

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78

that combines Density Functional Theory and Effective Field theory, where an energy functional is

constructed which is consistent with the symmetries of QCD and whose parameters are fitted to

properties of stable nuclei [1], The single-particle wave functions are obtained by solving the self-

consistent Kohn-Sham equations. In order to maintain a consistent approach, the correct electroweak

currents, needed to calculate the ^-transition rates, are derived from the same effective lagrangian.

These currents are the leading order Noether currents. The axial-vector current satisfies PCAC

at any order in the effective lagrangian [25]. A comparison with experimental data will give this

approach its validation and limitations as a calculational tool and will also test implications of

DFT regarding the single-particle wave functions used to construct the ground state density in the

Kohn-Sham approach.

The result for the ground-state to ground-state /3-transition rate between b^Sn^ to 1|13Sb82 gives

a ratio of calculated to experimental value of 1.47, or a deviation of 47% in the rate (24% in the

amplitude), a better result than for transitions to excited-states. This is in agreement with the

expectation that since the binding energy of the nuclei and the single-particle energy close to the

Fermi surface of the particle outside the core are well reproduced, then the single-particle wave-

function are also well reproduced. Transitions to excited states show a systematic deviation from

experimental data, again in agreement with the expectation about the wave functions of those excited

states. In the case of hole-hole transitions, studied here, the results are less reliable.

In the following sections a general expression for semi-leptonic transition rates in terms of a

multipole expansion and using Dirac wave functions is presented; then this formulation is used to

calculate ^-transition rates for selected nuclei and results are compared with the existing experimental

data; and finally, the conclusions drawn from this study are presented at the end of the chapter.

2 General Semi-leptonic processes

The calculation of /3-decay rates in this chapter is based on a general expression that can be applied

to any semi-leptonic process. In this section the derivation of that expression is discussed. Much of

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79

the material presented here follows [2].

The starting point is the interaction hamiltonian Hw which for low energy processes, like J3-

decay, is described by the semi-leptonic weak hamiltonian of the standard model. This interaction

hamiltonian is written down to first order in the weak constant, G. This implies that leptons are

treated to this order but the strong interaction part of the hamiltonian is treated to all orders. The

interaction hamiltonian is described by a current-current form 14

Hw = -^ = j d3xj^\x)M x) (56)

where x) is the lepton current and J ^ x ) the hadronic current. By taking matrix elements of

this hamiltonian between the initial and final states (lepton and hadron) the leptonic and hadronic

parts factorize and the matrix elements of the leptonic currents can be expressed by

= (s?)

where q = ke + kv is the momentum transferred in the process and ke, kv are the corresponding

electron and neutrino momenta. The quantities f\ and represent initial and final lepton states. In

addition, = (I, Uq) and l3 = I ■ q , where q is a unit vector in the direction of q, i.e. q/\q\.

Using the expression in Eq.(56) and making a multipole expansion of the hadronic current to

project irreducible tensor operators, one arrives at the following expression for the matrix elements.15

In this and the following equations the magnitude of q is defined by k = |

(f\Hw\i) = £VM 2j + i)(-i)J £h [ x r r m + T?-x(k)] V1 J> 1 A = ± l

+ £ v/4x(2J + !)(-»)■J [hCjoik) - loMJ0(k)}}\i) (58) j> o

The four multipole operators occurring in the previous expression are defined in the following way [2,

68 ]:

MjM(k) = J d3x{jj{kx)YjM(ttx)]J0(x) (59)

14For charge-changing semileptonic processes G± =Gcos 0(7 where G =1.0267xl0~5/mp and cosdc — 0.974. See [2] 15 Single nucleon matrix elements of the currents also contain form factors [1, 2] which are unimportant for the low

momentum transfer processes considered here.

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CJM(k) = £ J d3x{V [jj{kx)YJM(nx)}} • jf(5 ) (60)

-7~e 1 / J (*) = f / ^{V X [jj(fc*)%(nx)]} • J(5f) (61)

TTMS(k) = J d3x [jjik x ^fjM ] ■ J{x) (62)

The multipole operators have the vector- axial-vector (V-A) structure. The general form of the

hadronic current is given by

Jm = h + J.,5 (63)

and the different parts of the total current (vector and axial) used in this section are given by

J)i = 7 4 + ^AP2„,A"} (6 4 )

s = f

where Ap and An are the anomalous magnetic moments of the nucleon and dada stands for the

D’alambertian operator. Fa is taken as a constant, i.e. here only small momentum transfers are

considered, and its numerical value is -1.257. These currents are the leading currents as described

in [1, 25]. There are no additional contributions to the one-body current coming from higher order

terms in the effective lagrangian [25]. The next contribution to the current comes from two-body

effects which are estimated to be of the order of 0(g/m w), with q the momentum transfer. Since the

maximum value of q is of the order of a few MeV, this contribution corresponds to a few percent

correction. Additional contributions, from still higher order terms, are reduced by a factor of q/M,

w ith M the nucleon mass. These estimates do not take into account the spin structure of these two-

body currents, and the actual correction will depend on a detailed calculation, which lies beyond the

scope of the present work. 16

In order to evaluate the nuclear matrix elements of the four multipole operators it is necessary

to define the basis wave functions that are going to be used. For this the wave functions obtained

by solving the Kohn-Sham equations corresponding to the effective lagrangian that describes the

16See for example [67]

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nuclear many-body system are used. This lagrangian and the corresponding Kohn-Sham equations

can be found in chapter I [1].

At this point, the important aspect of the wave functions used is that they are described by Dirac

spinors and thus the contributions of both upper and lower components have to be taken into account

to all orders. Both currents and wave functions are used without a non-relativistic reduction. The

form of the Dirac spinors is given by ( \ 1 *Tr ( r ) n/i ^Km {^x ) Vt (66) ^ —T'(r)nK$ _ Km(flj,) J

where $ rem(0 a:) are spin spherical harmonics defined by

1 1 $rem(fi*) = 2-/ {0, (67) mims

and Xm, are two-component spinors. rjt is a two component isospinor describing either a proton or

a neutron. 17

The currents as well as the multipole operators have a matrix structure (two by two block matrix)

due to the Dirac matrices included in them, mixing the upper and lower components of the initial and

final nucleon wave functions. In what follows, the elements of the matrix form of these multipoles

play an important role and to keep track of them they will be labeled as follows.

( d J M hJM \

Ojm (68 ) \ bJM d{M j Using Eqs.(64) and (65) and the following matrix form for the Dirac 7 matrices (here the conventions

of [2] are used)

/ 0 —id \ (69) id 0 V 17The quantum number k is related to the orbital angular momentum quantum number, I, by the following relation:

I — k if k > 0, I = —(re + X) if re < 0. The magnitude of re is given by |re| = j + 1/2, where j is the total angular

momentum quantum number [2].

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f \ 1 0 0 = (70) 1 ° -1) then the elements of the multipole matrices can be obtained.

The matrix elements of the multipole operators shown in Eq.(58) will mix the upper and lower

components of the Dirac wave functions as shown in the following expression. Here all quantum

numbers referring to the final state are indicated by a prime. Those corresponding to the lower

components of the wave functions, either from initial or final states, are denoted by an underline.

Thus the Dirac wave function given in Eq.(66) is symbolically written as

^PnKm(^) (71)

Y 'lPn(l_l/2)jmj J

Combining this with the matrix given in Eq.(68) one obtains an expression for the matrix elements

of the multipole operators between initial and final nuclear states

(f\OjM\i) = (j'm'jlOjMljrrij)

= (n1 (I 11 / 2)j'm'j\d{M\n(l 1/2)jrrij) + {n'(/'l/2)j'm '|4M|n(/l/2)im,)

+i(nl(l'l/2)j'm'j \bJM\n(ll/2)jmj) - i(n'(('l/2)j'm ' |feJM|n(/l/2)jm,') (72)

where Ojm represents any of the multipole operators occurring in Eq.(58). By applying the Wigner-

Eckard theorem to the above matrix elements a reduced formed is obtained.

/ i \j-m j {j'm'j \OjM\jmj) = —7====-(j'm'jj - m j \j'jJ M ) x (j'\\Oj\\j) (73)

where the reduced matrix elements of Ojm are given by:

(f:40 \= {{n'n'mrwdiMnm + (n'on^rwdiMiim

+i((n'(l'l/2)j'\\bJ\\n(ll/2)j) - i(n'(I) 1/2)j'\\bJ\\n(ll/2)j))} (74)

Each of these reduced matrix elements is composed of a coefficient corresponding to the angular

momentum structure of the matrix element and a radial integral involving the radial wave functions,

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TABLE 5: Matrix elements for each of the multipole operators. The following notation is used

n(k) — k(Xp - Xn)/2 m and r](k) = 1 - k2/(m2 + q2), where q2 = k2 - ujq and k — |g|. Here w0 is the

total decay energy.

Multipole Operator d{M hJM

M j m M f d{M

M j m -i(wo/A:)[l-»7(fc)]FAS"j^ d(M Fa M y

L j m 0 d(M

T 5 VJM ir){k)FA E " " d(M 0

'Tel J M - d ( M t i f 1 rpel,5 1 J M %FA d(M 0

m m a g JM - d ( M s f

rp m a g , 5 JM v(k)FA d{M 0

G(r) and F(r) and spherical Bessel functions. The angular momentum coefficients are tabulated

and can be found in [68, 69]. The radial integrals were performed numerically.

Table 5 shows the form of each matrix element of every multipole operator defined by Eqs.(59)-

(62). Here the following definitions have been used/r(fc) = k(Xp—Xn)/2m and rj(k) = 1—k2/(m2+q2),

where q2 = k2 — Wq and k = \q\. The quantity wo is identified as the total decay energy.

Several new definitions taken from [68] have been used in this table. They are reproduced here

for completeness.

S f (fcr) = M fj{kx)-a

X j* (kx) = - i j^V x Mjj(fcaf)} ^

S j M = {XVMfikx)} -3

M f(kx)

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where

M f(k x ) = jj(kx)YjM(n x) (76) M ji (kx) = jJ{kx)Y}[1{nx)

All reduced matrix elements of the multipole operators are now given by combining Eq.(74) and

the expressions given in Table 5. Erom the derivation given in [2], one arrives at the following

expression which is general for any semi-leptonic process.

j> l

-ii(rx f)32i?e((i'||r/||i)(i'||77ag|br)]

+ E ^3 !0 ''ll^lli)l2 + W ol(i'll^^lli)l2 j> o (77)

The rest of this derivation, and the calculations done, concentrate on /^-transition processes. For

this particular semi-leptonic process the transition rate is given by

(78)

l e p t o n m'j mj s p i n s

where the e and v subscripts identify quantities related to the electron and the neutrino, respectively,

and wo is the total decay energy. In addition, V corresponds to the volume of quantization for the

lepton wave functions, 0 is the solid angle in the direction of emission of the lepton, either electron or

neutrino, 2 j +1 is the statistical factor corresponding to the initial nuclear state and rrij corresponds

to the projections of the total angular momentum of the initial and final nuclear states. Using

Eq.(77) and evaluating the spin sums (i.e. lepton traces) one arrives at the final expression that

describes /?-transition rates [2].

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{ E k k /iitfw + io-,ii7rgiii>i3)(i - («• m ■»)) j >i + q-{v- P)2Re((nTf\\j)(j'\\lJ&S\m }

+ Y^[(l-P-p + 2(q-m-vMj'\\ o +(i + /M ll2

-q • (P + 0)2Re((j'\\£,j\\j}(j'\\Mj\\j)*)]} (79)

where q = q/\q\, P = v/v and fi = k/e, q is the momentum transfer, v is the momentum of the

neutrino and v its energy, k is the momentum of the electron and e its energy.

This expression can be integrated over the corresponding phase-space to obtain the desired /?-

transition rate. All the nuclear structure input to this formula is embedded in the reduced matrix

elements of the multipole operators. These in turn are generally composed of four terms as given by

Eq.(74) and the expressions of Table 5. For the last part of this section a summary of the expressions

needed to evaluate these reduced matrix elements is included.

As mentioned above, all reduced matrix elements of the multipole operators are composed of two

factors: one corresponding to the angular momentum structure of the matrix elements and the other

corresponding to an integral over the initial and final wave functions weighted by spherical Bessel

functions. Their expressions are given in [69].

(n'(i'l/2)/||Mj(*f)||n(/l/2)j) = ( 4 ? r lj){n'l'j'\jj(p)\nlj) (80)

{n'(l'l/2)j'\\'Ej(kx)\\n(ll/2)j) = (4tt)~xf2Dj{l'j’-, lj)(n'l'j'\jj(p)\nlj) (81)

(n'(l'l/2)j'\\'S'j(kx)\\n(ll/2)j) =

(47r)-1 /2 {—J 1/2.Dj (l'j' ; Ij) (n'l'j' \jj+i (p) \nlj)

+(J + l ) 1/2-Z}J"(l'j1', lj)(n'l'(p)\nlj)} (82)

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(47r)-1/2{(J+l)1/2D+(i'j';Ii)

+J1/2Dj(l'j'}lj){n'l'j'\jj-i{p)\nlj)} (83)

In all these expressions jj{p) corresponds to a spherical Bessel function of order J and argument

p = kx. The coefficients Aj, Dj, D j and D j are tabulated in [68, 69] for a large, b u t limited

number of transitions. For coefficients of transitions not included in these tables, explicit expressions

of the above coefficients in terms of 3-j and 6-j symbols can be found in [68]. 18 The integral over

the initial and final radial wave functions is given by

pOQ (n'l!j'\6(p)\nlj) = dr u(r)n’K,0(p)v(r)nK (84) Jo

Here 6(p) stands for the appropriate spherical Bessel function and u(r) and v(r) for either the

upper or lower component of the Dirac wave function, i.e. either G(r) or F(r) of Eq.(66), of the

initial and final nucleon state.

3 Results

The calculations of ^-transition rates are done by numerical integration of Eq.(79) over the electron

and neutrino phase-space. These transition rates are very sensitive to the exact decay energy u>o-

Although the results obtained for total binding energies of the nuclei involved in this study agree

w ithin 1% of its experimental values, here the calculations of the decay rates are done using the

experimental decay energies so as to take into account the full phase-space available to the process.

This way, no additional uncertainties are introduced and the direct contributions coming from the

calculated nuclear wave functions can be accounted for. The decay energies used here are taken

from [65, 66]. The Dirac wave functions used in these calculations are solutions to the Kohn-Sham

equations derived from the effective lagrangian given in chapter I [1] using the G1 parameter set.

18Some coefficients for the hole-hole transition (lgg/2 )~~1 (lhxi / 2 )~ 1 where obtained using the explicit expressions

given in [68] and tabulated 3-j and 6-j coefficients [70].

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,-yvr 7/2- 133, ^ 7 /2 ...... - \ \ \ 50 83 > ^ 6 s \ \ \ Particle-particle \ \ \ P -transitions

4 - ^ 3 / 2 - — ... 004 in

- *§7/2...... ^ — 7 /2 + 51S b 82 .. . 1...... I ...... Calculated Experimental

Figure 23: Single-particle level spectrum of parent and daughter nuclei in the case of particle-particle

transitions. All energy levels are measured from the calculated or experimental ground-state level,

accordingly.

Two types of transitions are investigated in this paper: 1) particle-particle and 2) hole-hole. The

diagrammatic representation of the transitions studied here are shown in Fig. 23 for transitions of

the particle-particle type and in Fig. 24 for the hole-hole type. The level structure indicated on

the left side of each figure corresponds to the results obtained using the EFT/DFT approach. On

the right are the measured levels. In both cases the energy levels are measured with respect to the

ground-state level, either the calculated (left) or experimental (right). The results obtained give, for

the nuclei considered in the particle-particle transitions, the correct level ordering although not the

right splitting. This is in accord with DFT with regard to excited states. On the other hand, based

on the agreement obtained for the ground-state binding energies of the parent and daughter nuclei,

it is expected that the wave functions corresponding to these states, i.e. 2f7/2 and lg7/2 respectively,

are a good approximation. These particles are outside a filled core and in this approach they only

interact with it through the mean-fields. An assumption has been made here that these nuclei, with

one particle outside its core, can be described by spherically symmetric wave functions.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The approximation of an undistorted core neglects core polarization due to the valence parti

cle. Such an effect resolved the previous “magnetic moment problem”, whereby the single-particle

isoscalar convection current was reduced by a factor of M*/M [71, 76, 2], bringing the static moments

back to the Schmidt values. Here the weak currents are pure isovector and are derived consistently

from the same effective lagrangian that renders the energy functional and corresponding Kohn-Sham

equations. Corrections to the isovector currents coming from many-body effects and exchange cur

rents [25] remain to be investigated. 19

In the case of the nuclei considered here for hole-hole transitions, several complications emerge.

First, the level ordering of the daughter nucleus, b^Snsi is reproduced except for the (2d3/2)-1

state, which is the measured ground-state. The calculated ground-state corresponds to the lhu /2

level, which lies approximately 200 keV off the experimental value. Another complication arises

from the fact that there are no ground-state to ground-state transitions in the hole-hole case, so

a direct comparison with the particle-particle situation cannot be made. Here all transitions go

from or to excited states. Yet another complication comes from the assumption that it is possible to

approximate a hole state wave function by a single-particle wave function. The hole state corresponds

to an unfilled core, with one particle missing, and no interactions among the particles in the unfilled

core have been taken into account beyond the mean-field one. These three situations make it difficult

to compare the hole-hole type of transitions with experiment. Still the results, as will be shown

below, are not so different than other calculations and in general follow the pattern of behavior of

the experimental transition rates. The calculation of /^-transition rates using Eq.(79) employs the

initial and final nuclear wave functions. These wave functions enter into the calculation in a different

form depending on the type of transition, i.e. particle-particle or hole-hole. In the particle-particle

case the initial and final wave functions correspond with the initial and final nuclear states. For

example, for the ground-state to ground-state transition ^S n ss - t1!? Sb82 the initial and final wave

19Even though the operators included in the currents are transition operators, i.e. charge-changing, and these might

make coherent effects unimportant, there might still be an effect coming from core polarization because the wave

functions of the parent and daughter nuclei are not identical, as has been assumed here. See also [74].

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89

1 0 - 1/2- 131

Hole-hole (3 -transitions

.5/2+

..1/2+ 131

Calculated Experimental

Figure 24: Single-hole level spectrum of parent and daughter nuclei in the case of hole-hole transitions.

All energy levels are measured from the calculated or experimental ground-state level, accordingly.

function are 2f7/2 and lg7/2 respectively, see Fig. 23. Hole-hole transitions, on the contrary, can be

interpreted as a particle transitions going in the opposite direction [73]. Thus, for example, in the

transition 1|gln82 -+1| q Sn8i going from ground-state to the (lhu /2)_1 state, the initial and final

wave functions would be (lhllf/2) and (lgg/2) respectively. 20

Examples of the single-particle wave functions used in the calculation of /3-transition rates are

given in Fig. 25 through 28. In these figures the upper and lower components of the Dirac wave

functions, i.e. the G(r) and F(r) functions, are plotted simultaneously as a function of the radial

distance r, in Fermis.

The results of the calculation of /3-transition rates for the particle-particle type are shown in

20In the hole-hole case, there is an additional phase factor coming from exchanging the creation and destruc

tion operators in the matrix elements of the tensor operator. The states are characterized by their total angular

momentum quantum number j, therefore, if the initial and final states are denoted by Ij^1} and \jal), then the

matrix elements of the transition matrix element of a tensor operator Ojm will be given by : (jo” 1||Oj|b'^'1) =

-{-iy»+}*-J(jb\\Oj\\ja) [73],

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 25: Upper and lower components of the initial and final nucleon wave functions. This corre

sponds to a particle-particle transition proceeding from the ground-state of ^Sngs to the ground-

state of 1|f Sb 82-

- 0.2

-0.4

0 5 10 r(fm)

Figure 26: Upper and lower components of the initial and final nucleon wave functions. This cor

responds to a particle-particle transition proceeding from the ground-state of 1foSn83 to an excited

state of 1|jJSb82-

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91

- 0 .2 , r(fm)

Figure 27: Upper and lower components of the initial and final nucleon wave functions. This cor

responds to a hole-hole transition proceeding from the ground-state of ^gln^ to the calculated

ground-state of

-0.4

- 0.6

0 5 10 r(fm)

Figure 28: Upper and lower components of the initial and final nucleon wave functions. This corre

sponds to a hole-hole transition proceeding from the ground-state of 1|9lng2 to an excited state of

'lo'Snsi-

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3 0) 0) +-> 2 0.01 >> a o n

0.0001

^7/2""* ^7/2 ^7/2~*^3/

Figure 29: Calculated results of the /3-transition rates in the cases of particle-particle transitions and

the experimental values. The crosses indicate the calculated values, the filled circles represent the

experimental data.

Fig. 29. These are compared with experimental values for each of the transitions. Here the calculated

results are indicated with a cross. The ratio of calculated to experimental value for the ground-state

to ground-state transition is 1.47, or a deviation of 47%. The other transitions go from the ground-

state of the parent nucleus to an excited state in the daughter nucleus.

The results for these transitions show systematically a larger value of the decay rates compared

with the experimental ones. This agrees well with what is expected of DFT. Since the calculations

have been performed using the experimental decay energies, the only input from the EFT/DFT

approach at this point has been the single-particle Dirac wave functions. These, on the other hand,

are used to construct the nuclear ground-state densities and since this quantity is in principle well

reproduced, the result of the ground-state to ground-state transition serves as an indirect way of

checking it.

Returning to the results for the ground-state to ground-state transition in the particle-particle

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93

l r

t/3 0.1 r W -M

- 1 3 U 1 3 1 „

O 0.001 49 82” * 50 81 U Q 0 .0 0 0 1 r

le -0 5 (lg9/2H(2d5/2) (2p1/2)-K2d3/2)

Figure 30: Calculated results of the /3-transition rates in the cases of hole-hole transitions and the

experimental values. The crosses indicate the calculated values, the arrows indicate the upper limit

given by experimental data.

case, Fig. 31 shows the dominant multipoles as a function of the k — |g|. The values plotted

correspond to the square of the magnitude of the multipoles defined in Eqs. (59)-(62). The multipoles

plotted correspond to values of J = 0 and J = 1, with |.Mo(fc)|2 being the most dominant term.

From this figure it is also clear that at low momentum transfer the only non-vanishing multipoles

are |A/(o(^)|2) \Til(k)\2 and |A(fc)|2, in accordance with the analysis made in [2].

The results obtained in the hole-hole type of transitions are shown in Fig. 30. In this case, the

results are less satisfactory than in the particle-particle case. The experimental results are shown

here as arrows indicating upper limits to the decay rates. From the three cases calculated here, only

one lies within the experimentally determined range. This transition corresponds to a ground-state

to excited state hole-hole transition. The two other cases shown occur from ground-state to excited

state and excited to ground-state respectively. Their numerical results are very similar. In these

cases, there is no direct ground-state to ground-state transition which makes it difficult to compare

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94

0.292 « 0.291

<4* -2 0.287

.1 lM

0.01 0.03 0.04

Figure 31: Magnitude of the multipoles defined in Eqs. (59)-(62) for J=0 and J= l. These multi

poles are the dominant terms contributing to the particle-particle transition undergoing between the

ground-states of xf$Sn83 and 1fj,Sb82. In this figure Mj = Adj, Lj = £ j . T j1 = 771 and T™ag = 7(,mag.

with the particle-particle case.

4 Conclusions

This chapter shows the results of calculations of the /3-transition rates of nuclei close to 1|oSnig2-

These calculations were inspired from the success of the results obtained in a previous chapter dealing

with the application of the EFT/DFT approach to the nuclear many-body system to nuclei far from

stability. In reaching this goal several important results have been obtained.

1. A general expression to calculate any semi-leptonic process has been derived. The main equa

tions and definitions needed are given by Eqs. (74), (75)-(76), (77), (80)-(83) and Table 5.

The nucleon wave functions are the Dirac wave functions obtained from solving the Kohn-

Sham equations derived in the EFT/DFT approach and considers relativistic corrections to

all orders in the nucleon wave functions. Additionally, the single-particle electroweak currents

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used correspond to the leading Noether currents obtained directly from the same effective la-

grangian. In this sense, the calculation is self-consistent since all relevant elements have been

obtained directly from a single theory.

2. The assumption at the beginning of this paper regarding the view that the transitions between

ground-states are expected to be in better agreement than those involving excited states, can

at this point be considered only as a conjecture since the amount of evidence provided by the

two test cases studied here, while suggestive, does not conclusively support this view. A more

comprehensive study has to be undertaken that requires a whole set of systematic calculations

on a larger set of nuclei using the framework presented here. This goes beyond the present

scope of the paper.

Although first-forbidden single-particle beta transitions have been extensively studied in the

past, and a comprehensive analysis on this matter can be found in such basic texts as [74] and [75],

a contribution has been made to this subject: The nuclear dynamics (binding energies, chemical

potentials, predictions for ground-state J71- of odd nuclei, etc.), and the weak currents used to probe

the nuclei studied here, follow self-consistently from a single, covariant lagrangian that manifests

the spontaneously broken chiral symmetry of QCD. Furthermore the EFT/DFT approach has been

successfully extended from the original one of Furnstahl, Serot and Tang to nuclei far from stability.

All these are new contributions. In addition, general expressions are given for calculating any semi-

leptonic process within this framework. A theory capable of reliable extrapolation and predictions

of beta decay properties should be of interest for nuclear astrophysics.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter III.

Extensions to semi-magic nuclei far from stability.

1 Introduction.

In the first chapter the convergence of a new approach to the description of the nuclear many-body

system, based on principles of effective field theory (EFT) and density functional theory (DFT) [1],

was studied by applying it to selected nuclei far from stability. The results of the calculations showed

an overall agreement with experiment below 1% for the total binding energy of the doubly magic

nuclei 1|oSng2, ^Snso, 28^20 and ||N i50 as well as for the entire range of known even-even Sn

isotopes. In addition, calculations for the chemical potential of nuclei differing by one particle or one

hole from these doubly-magic ones, produced results with an overall agreement with experimental

values below the 10% level, and also correctly predicted the spin and parity of these nuclei. It was

also shown that the calculations converged to the experimental values as the level of approximation

of the effective lagrangian was increased. From the various levels of approximation used, it was found

that the ones with the highest levels reproduced better the properties of nuclei far from stability.

In addition to this, it was also observed that when this approach was tested to the whole range of.

even-even Sn isotopes the results obtained agreed equally well with the experimental values.

Based on the agreement of these results, a calculation of ^-transition rates was performed for

nuclei close to 1foSn82. The objective was to test, through a semi-leptonic process, the accuracy of

the single-particle Dirac wave functions obtained in the first chapter describing a particle or a hole

outside the filled core. The calculations where done using a current-current interaction hamiltonian

in which the lepton current is treated in first-order perturbation theory while the nuclear currents

are expanded in multipoles and treated to all orders. The nuclear electroweak currents used in these

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97

calculations are those derived directly from the same effective lagrangian describing the nuclear

many-body system. In this way the calculations were done based on a self-consistent approach.

Two different cases where explored in: particle-particle transitions ^S ngs —t 1|fSbs2 + e~ + Ve and

hole-hole transitions 1|gIn82- t 15oSn8i + e~ + V,.. The results obtained indicated that the particle-

particle transitions proceeding between ground states were reproduced to 47% while transitions to or

from excited states deviate systematically from the experimental values. This behavior was expected

based on the fact that ground-state observables (in this case the ground-state density and therefore

presumably the wave functions) are best reproduced on the basis of DFT. Results obtained in the

hole-hole case, in which all transitions start or end on an excited state, are less reliable in reproducing

the /3-transition rates. In any case, more studies are underway to understand this behavior.

Both chapters showed that this EFT/DFT approach can be used successfully to describe some

ground-state properties of the selected nuclei far from stability; this extends the applicability of the

approach beyond the valley of stability. To further explore the range of validity of this approach,

the calculations done in chapter I are extended here to all isotones with magic numbers N= 28, 50,

82, 126 and isotopes with Z= 28, 82, 126.

The calculations done in chapter I, which are extended here, are based on an energy functional

of the ground-state density E \pg,s]. By minimization of this energy functional with respect to the

density one obtains the ground-state energy and density of the nuclear system. The existence of

a unique functional of the density is guaranteed by the Hohenberg-Kohn theorem [22, 18]. This

minimization process can be accomplished using a procedure developed by Kohn and Sham in which

the ground-state density of a interacting fermion system is computed using the single-particle wave

functions obtained from an equivalent systems of non-interacting fermions subject to an appropriate

external potential, known as the Kohn-Sham potential [22, 18]. The basis for this procedure is

contained in the Hohenberg-Kohn theorem which can be stated by saying that there is a one-to-one

correspondence between the ground-state density of a system of interacting fermions subject to an

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98

external potential and the external potential 21.

Although the formulation based on DFT is straightforward, the exact energy functional is un

known, particularly in the nuclear system where the exact form of the nuclear interaction is still

a subject of intense research. To overcome this, an energy functional is constructed based on the

principles of effective field theory (EFT) [1]. The starting point is an effective lagrangian density

that satisfies the symmetries imposed by QCD, which is considered the underlying theory of the

nuclear dynamics. This lagrangian is built using the lowest lying hadronic degrees of freedom (as

well as the electromagnetic field). The result is the most general lagrangian containing (in principle

an infinite number of) interaction terms constructed using these fields and subject to the imposed

symmetries. The various interaction terms in this lagrangian are ordered in powers of the fields

and their derivatives. This expansion is equivalent to an expansion in the ratios of the fields to the

nucleon mass and the Fermi momentum to the nucleon mass. These expansion parameters are small:

« l/3 and « l/4 respectively. Naive Dimensional Analysis (NDA), which is a prescription by which

the various interaction terms are suppressed by powers of a relevant mass scale A (the nucleon mass

M in this case), indicates that the remaining coupling constants should be of order unity, a charac

teristic called ’’naturalness.” This natural size of the constants can only be verified after fitting the

theory to experimental data. This procedure was done in [1] where it is shown that the remaining

constants are indeed ’’natural.” There the constants were fit to ground-state properties of selected

spherical nuclei along the valley of stability. The effective lagrangian, as well as the corresponding

equations of motion obtained from it, are shown in the next section.

As has been mentioned before, this paper explores the extent of applicability of this EFT/DFT

approach. To understand the objectives of this work, one has to keep in mind that the coupling

constants of the effective lagrangian have been fit to properties of selected stable nuclei lying in the

/3-stability region. Therefore the results of chapter I, and the ones presented here, test how well this

21 In relativistic theories the energy functional is written in terms of more than one density: e.g. scalar densities ipij},

vector densities ijijpii, etc.

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theory reproduces ground-state properties outside the stability region. As indicated previously the

relevant observables which can be reproduced, based on DFT, are the ground-state binding energy

and chemical potential of the nuclear system, so this section concentrates on these quantities.

The calculations done here focus on nuclei with semi-magic number of neutrons, i.e. N = 28, 50,

82 and 126 or protons, i.e. Z = 28, 50 and 82. In general the predictions of the total ground-state

binding energy, for both isotones and isotopes, lie below the 1% level as was found before, and that

as the size of the nucleus increases, the error is reduced. The agreement between the results of the

chemical potential with experimental data is also consistent with the conclusions reached in chapter

I. In the case of the prediction of the ground-state J71' values of odd nuclei neighboring even-even ones,

the agreement with experimental values varies between isotones and isotopes. When the number of

neutrons gets very large in the regime N>Z, the level density gets very high and the detailed ordering

of the levels between major shells depends sensitively on the details of the various densities. This

small energy shifts (« tens keV) make the predictions of the ground-state quantum numbers less

accurate, though the chemical potentials are still relatively well described.

All nuclei selected for this study are essentially spherical [76], therefore the approximation of

spherically symmetric nuclei, made here, is assumed to be an accurate one. The calculations are

done using the most convergent parameter set of Chapter I, called Gl, which corresponds to the

most sophisticated effective lagrangian.

To summarize, this chapter extends the calculations made in chapter I to known semi-magic

isotones and isotopes using the Gl parameter set. In the following section the formulation of the

Kohn-Sham equations and potentials is reviewed, followed by a section on results. The chapter ends

with the conclusions and proposes new directions of investigation to further test this approach.

2 Formalism.

In this section an overview of the formalism of EFT/DFT is presented. The energy functional

that describes the dynamics of the nuclear system is obtained from an effective lagrangian. This

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is constructed using the relevant lowest lying hadronic degrees of freedom [1], which are the pion

and the rho fields (isovectors) and the nucleon field (isospinor). In addition to these, a scalar field

is included to reproduce the known short-range attraction of the nuclear force, and a vector field

(associated to the lo meson) that reproduces the mid-range repulsion of the nuclear interaction is

also included.

The pion field is introduced as the phase of a chiral rotation of the identity matrix in isospin

space, see Eq.(26) from chapter I. Nucleons are included as a two-component isospinor field N (x ).

The upper and lower components correspond to the proton and neutron fields, respectively. ( \ p{x) N (85) ^ «(*)

The pM-field has been added to the lagrangian to include contributions coming from the asymme

try between the number of protons and neutrons. In addition to the above fields, an electromagnetic

field A p is included and the electromagnetic structure of the nucleons is also taken into account.

Their anomalous magnetic moments, labeled as \ p and An for proton and neutron respectively, then

enter.

The effective lagrangian contains, in principle, an infinite number of terms. In order to make the

theory predictive, these terms have to be arranged according to some ordering scheme that takes into

account all equally contributing terms. Here the various interaction terms in the effective lagrangian

are arranged following the prescriptions of NDA [16, 17]. As mentioned before, all calculations

performed here will be done using the parameter set Gl which corresponds to the most sophisticated

lagrangian density of [1], and is reproduced in Table 1.

The equations describing the various meson and electromagnetic fields are reproduced here and

are the following:

2/K3 , «4 $ \ , g2s , . 2 -V*.+m*. = - q * q + 3 f S 1 + & ( *

+ + + <86)

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Here ps is the baryon Lorenz scalar density and gs , ms, k3, /c4, i)i, g2, gp, an, a2 are again

parameters fit to experiment. Furthermore,

- V 2VF + m 2vW = g2v[pB {x) + ~ V -{ p TB{x)v)) - (r/i + y - ^ r ) - ^ r m 2H r

- ^C o fF 3 + ^ ( V $ • V IF + $ V 2TF) -~ P c h g ( x ) (87)

Here pB is the baryon density, p|) is the baryon tensor density, pchg is the charge density and Co

are parameters. In addition,

- V 2f? + m 2pR = ^[psfc) + ^V -(p|’ (®)r)] - gp~ m 2pR - ^ p c hg(») (88)

Here p3 and p j are the isovector densities, f p, gp, gp axe parameters and g7=5.01, is the coupling

of the photon to the w-meson. Finally,

- V 2A = e2pchg(x) (89)

where pchg is the charge density. All parameters used in these equations are tabulated in Table 1

given in chapter I.

From the Dirac hamiltonian given in chapter I, Eq.(34) one can write down the energy eigenvalue

equation which will give the single-particle wave functions that are used to construct the ground-state

densities. The various mesons and electromagnetic fields play the role of the Kohn-Sham potentials.

The whole set of equations (86) - (89) and (90), below, form the Kohn-Sham equations.

hipa(x) = Eaif;a (x) (90)

To obtain the single-particle wave functions from (90), they are expressed in terms of Dirac

spherical wave functions: ( lzG a{r)$ Km \ lpa(x) = Ct (91)

where the $ Km is a spin spherical harmonic, (t is a two-component spinor and the index t is equal

to 1/2 for protons and -1/2 for neutrons. By inserting this form of the solution into Eq.(90) a set of

two coupled first-order differential equations for the G and F functions is obtained:

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{ t + 7 ) Ga{r) ~ [Ea ~ Ul{r) + U2^ F^ r) ~ U3° a(r) = 0 (92)

“ 7 ) Fa(r) + [Ea ~ Ul{r) ~ t/2(r)lGa(r) + U*Fa(r) = ° (93)

where the single-particle potentials are given by

^ (r) = W (r)+ taR(r) + (ta + ±)A(r) + ^ ( / 3 t + 2ta/3v )V2A(r) (94)

U2(r) = M — $(r) (95)

U3(r) = ~ { fv W '( r ) + tafpR'(r)+A'(r)[(Xp +Xn)/2 + ta(Xp- X n))} (96)

Here the prime indicates a radial derivative; e.g. W ‘(r) — dW(r)/dr. Once the single-particle wave

functions are calculated, the various densities appearing on the r.h.s. of the meson equations can be

obtained. They are defined by:

OCC q • -j '• M = £-feS-- * > > > <97> a occ

» ( ■ ) = a occ

< " > a occ ~ ^ /*(*) = £-f^-(2t«) (Gl(r)+F2a(r)) (100)

+ - a 4ixr2

where the sum goes over the occupied orbitals. In addition to these, the charge density is defined

as the sum of two pieces: a direct nucleon charge density /3d ( x ) and the vector meson contribution

Pro(x)

Pchg = Pd(x) + Pra(x) (102)

The direct part is given by

P d ( x ) = p„(s) + ^rV -(/°r(a;)f) + 7j^p[PsV2pB(x) + 0vV'2pa(x)] (103)

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and the vector meson contribution arising from the coupling of the neutral vector mesons to the

photon takes the form

(104)

Here the point proton density pp and nucleon tensor density f>[ are given by

(105) Pp - 2 (P b + P 3 )

OCC (106) a

All the above equations were derived from the effective lagrangian given in Eqs. (32) and (33) of

chapter I, and are the same as those used in [24] and reproduce those shown in [1]. To solve them

an independently developed program was written to solve these local, coupled, non-linear equations

(86)-(89) and Eqs. (92) and (93), where the appropriate densities are defined in Eqs. (97)-(105).

This program was used to obtain the results in chapter I and reproduces those shown in [1].

The meson equations are solved iteratively using a Greens function method. Because NBA

guarantees that each additional term on the rhs of Eqs. (86) - (89) is smaller than the previous one,

convergence is both expected and obtained.

The whole system of Eqs. (92), (93) and (86) - (89) is solved self-consistently until a global

convergence is reached.

3 Results.

The procedure described above to solve the Kohn-Sham equations to find the eigenvalues and wave

functions is performed using the parameter set G l given in [1], and which is reproduced in Table 1.

These equations are solved and used to calculate the ground-state total binding energy of the selected

nuclei 22. Before starting the discussion of the results, it is important to keep in mind that the

effective lagrangian on which these calculations are based, was fitted to ground-state properties of

22 The total binding energy of the system follows directly from the effective lagrangian and hamiltonian, see [1]

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> 500 N=28 isotones 4) Ni Zn Ge Fe I 1 8 Cr % Calculated S2450 Ti ® t Experimental <3 a Ca ffl M400 Ar •H T3 fi ffl 350 0 H Si

300 14 16 18 20 22 24 26 28 30 32 z

Figure 32: Calculated and experimental values of the total binding energy for even N=28 isotones.

Energies are positive and measured in MeV.

selected spherical nuclei in the stability region [1] and that there has been no refitting of these

parameters.

The first set of results shown correspond to the total binding energy of even-even isotones with

total neutron number, N, equal to 28, 50, 82 and 126. These are shown in Fig. 32 - Fig. 35. As can

noted from these figures, there is a very good agreement between the calculated and experimental

values, which are taken from [77]. In Fig. 36, the percentage deviation, or error, of the results is

shown. The overall agreement can be seen to improve as one goes to larger nuclei and, except for

the N=28 isotones, the percentage error is in general less than 0.5%.

The following set of figures, Fig. 37 - Fig. 40, show the predictions for the ground-state energies

of nuclei neighboring even-even isotones of given N. The energy differences are taken between the

odd nucleus and the closest even nucleus. These differences in energy, i.e. the magnitude of the

chemical potential, are calculated using the binding energies obtained in this EFT/DFT approach

and are compared with available experimental data. The even-even isotones shown in these plots,

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> Sn 0 Pd “ N=50 isotones Ru | s Mo g fl BOO- Zr $ >1 Sr 6 s? « Calculated Kr 0 S3 750 f- « Experimental W Se too ■S Ge "2 700 [- « .5 Zn ffl $ "cS o 650 h 28 30 32 34 36 38 40 42 44 46 48 50

Figure 33: Calculated and experimental values of the total binding energy for even N=50 isotones.

Energies are positive and measured in MeV.

1250

N=82 isotoees

•S 1200

C alculated W 1150

m 1100

1050 z

Figure 34: Calculated and experimental values of the total binding energy for even N=82 isotones.

Energies are positive and measured in MeV.

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1700 r

<0 N=126 isotones ® Calculated f Experimental

U Th « 60 Ra % 8 u 1660 - c Rn ® 4) Po 60 fl 1640 - Pb '0 •Hfl X Hg 16201- ® •wfl 0

80 82 90 92

Figure 35: Calculated and experimental values of the total binding energy for even N=126 isotones.

Energies are positive and measured in MeV.

Percentage deviation N=28 Isotones Isotones N=50 Isotones M -82 Isotones W N =126 Isotones /-\ea

a io w ° -0.5 ffl *O -1 1 o pH

Figure 36: Percentage difference between calculated and experimental values for all even isotones.

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Particle states (Z+l) — Calculated • - ■ Experimental 1/2 +

3/2- Cu

7/2-

Hole states (Z-l)

Figure 37: Particle and hole states neighboring even-even N=28 isotone nuclei. Binding energies

are considered positive. Only nuclei with particle and hole states with respect to filled j-shells have

been plotted. The values of Z correspond to the even-even nuclei in the middle row. Calculated Jw

values are included and, unless indicated otherwise, are in agreement with those of the experimental

ground state. Where the experimental J77 value disagrees with the calculated one, or is uncertain, it

is also included in the figure.

and which are located at the zero point of energy, correspond to nuclei with (calculated) closed

j-shells. A filled j-shell has an even number of particles, and .J71 equal to 0+, here only particles or

holes relative to filled j-shells contribute to the total angular momentum and parity of the nucleus.

This is precisely the argument given in the shell model [2]. Since these calculations are effectively

mean-field calculations, there is no explicit inclusion of pairing [24]. Thus states corresponding to

partially filled j-shells are not included in these plots.

From these plots one can appreciate the excellent agreement between the calculated and exper

imental values of this energy difference, or chemical potential. In addition to this, the predicted

ground-state quantum numbers of the odd nuclei agree with the experimental values. The only

disagreement in all the isotone cases considered occurs for 1g|Tb82, Fig. 39. Here the experimental

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20 Particle states (Z+l) > Cu 0 ' ---- 5/2- _ Calculated ss-' 3/2- Br ...... Experimental 10 M 1/2-JL u ■ - • _Nb_ 9 /2+ o • iH 9/2+ In ■0 C (9/2+) ffl JLl/2- 3/2-JL $ -10 5 / 2 - i L 1 (5/2-,3/2-)..... 0 a! Hole states (Z-l) i I I I I I I I ! 1 1 1 28 30 32 34 36 38 40 42 44 46 48 50 52

Figure 38: Same as in Fig. 37 for N=50 isotones.

20 /■“V Particle states (Z+l) > « 15 — Calculated ■ ■ ■ Experimental

7/2+

Pr 5/2+

I ° Sn "g T Gd t3 E u...... a -s 7/2+ w La £-10 ■a (9/2+) In f a 9/2+ Hole states (Z-l) -20 50 58 64

Figure 39: Same as in Fig. 37 for N=82 isotones. The position of the (11/2-) metastable state of

147Tb has been shifted so it can be seen in this energy scale. It actually lays 50 keV above the

ground-state.

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Particle-states (Z+l)

— Calculated ■ - ■ Experimental

9/2- JPL

S -10 1/2+ Hole-states (Z-l) 3/2+

Figure 40: Same as in Fig. 37 for N=126 isotones.

ground-state corresponds to a (1/2+) state while the predicted ground-state is 11/2-. The result is

not far from the metastable state (11/2-) which lies 50 keV above the true ground-state [78]. This is

indicated in Fig. 39, where the metastable state has been shifted in order to be seen on this energy

scale.

The next set of plots, Fig. 41 - Fig. 43, show the ground-state total binding energy of even-even

isotopes with Z= 28, 50 and 82. They are equivalent to the plots shown for the isotones. Here again

the predicted values agree overall with experiment below the 1% level as can be seen in Fig. 44. The

agreement between predicted and experimental data improves as one moves to larger nuclei.

The final set of plots, Fig. 45 - Fig. 47, correspond to predictions of the relative binding energy

of odd nuclei neighboring even-even isotopes of given Z, with respect to the binding energy of these

even-even nuclei. As in the equivalent plots for isotones, the even-even nuclei correspond to the

calculated filled j-shells and locate the zero-energy level.

Figure 45 shows the results for isotopes of Ni (Z=28). Here one can observe that overall, there is

a very good agreement with experiment, similar to that of the N=28 isotones. In the case of 2gNi33,

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700 > 0 Z=28 isotopes S 650 9 . 9 9 i u 8 w> ft & I 550 0 Calculated 0 s 8 t Experimental Sf 500 - % 1 % •S 450 § £ ■ « 3 400 0 e H 350 i i 1 1 1 1 1 1 1 1 1 1 1 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 N

Figure 41: Calculated and experimental values of the total binding energy for even Z=28 isotopes.

Energies are positive and measured in MeV.

1150r

> Z=50 Isotopes «• 0 1100b S •s 1C >5 60

w t>0 9 Calculated 950 - • Experimental I G ffl 900 - B 0 850b H

I . I i i 52 56 60 64 68 72 76 80 84 N

Figure 42: Calculated and experimental values of the total binding energy for even Z=50 isotopes.

Energies are positive and measured in MeV.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I l l

1700 > u Z=82 isotopes ^ 1650 .9 > ,1 6 0 0

0 Calculated f l 1550 « Experimental W . * 5 P 1500 .5 T3 .S 1450 S3 $ 1400 0 H 1350 _ L 100 104 112 116 120 124 128 N

Figure 43: Calculated and experimental values of the total binding energy for even Z=82 isotopes.

Energies are positive and measured in MeV.

Percentage deviation *-» Z=28 isotopes Isotopes *—a Z=50 isotopes ♦ Z=82 isotopes

-0.5

Figure 44: Percentage difference between calculated and experimental values for all even isotopes

considered.

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(C 30 Particle states (N+l) * — Calculated S 201 ■ - ■ Experimental M 5/2- -—^5/2- 1/2- 0) 101- G 3/2- ’\ ... ffl 3/2- (1/2-) «SP ol •H ”0 (1/2 -) S3 3/2- •H - 1 0 - ffl 7/2- 5/2- 1/2' 0 > -201 1—^ u Hole states (N-l) -30Lx x J L X 24 26 28 30 32 34 36 38 40 42 44 46 N

Figure 45: Particle and hole states neighboring even-even Z=28 isotopes. Binding energies are

considered positive. Only nuclei with particle and hole states with respect to calculated filled j-

shells have been plotted. The value of N corresponds to the even-even isotopes in the middle of

row. Calculated J’r values are included and, unless indicated otherwise, are in agreement with the

experimental ground states. Where there is disagreement with the experimental J ’1' value, or it is

uncertain, the experimental ground-state «P value is also indicated in the plot. In addition, in this

case, the lowest-lying excited state with the calculated J v is also included.

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there is a disagreement in the predicted quantum numbers of the ground-state. The experimental

value corresponds to a 3/2“ state, while the predicted one is a 5/2“ . There is an excited state

with J* value in agreement with the calculated ground-state that lies only 70 keV above it [79]. In

this figure, this separation in energy has been exaggerated to show it in the scale of the plot. The

disagreement in the location of the ground-state shows the sensitivity of the actual ordering of states

to the form of the central potential. One can understand this by looking at the effects of putting

additional neutrons into the system. There is a major shell closure at N=28.«If one more neutron

is added to the system it will go into the next j-shell which is j = 3/2 corresponding to a 2p3/2

level, as expected from the naive shell model 23. This level becomes completely filled at N=32, since

2 j -f 1 = 4, which is correctly predicted as can be seen in the plot. Adding one more particle to

this nucleus, should start filling the next j-shell, the lf5j/2 level, giving the ground state the quantum

numbers 5/2“ as predicted. Instead, it seems that it is energetically more favorable to promote one

particle from the 2p3/2 level to the lf5/2 leaving a hole. The two particles in the lf5/2 level do not

contribute to the angular momentum, which is solely determined by this hole in the 2p3/2. Adding

more neutrons fills the lf5/2 until it is completely full at N=38, since 2 j + 1 = 6, which again is

correctly predicted. After this, more neutrons just start filling the next level, 2pi/2, following the

ordering given by the naive shell model. Therefore the predictions follow very closely the way the

neutron levels are being filled and the fact that the disagreement at N=33 is small only shows how

sensitive the ordering is to the form of the central potential.

The exact ordering of energy levels has a larger effect when the level density is high and the levels

are very close together. This is precisely what happens in the case of isotopes with large values of

Z. For any given Z the density level of neutrons close to the Fermi surface is larger since in general

N>Z making the energy levels very close to each other. This can be appreciated in Fig. 46 where,

if there is a disagreement between the calculated and experimental ground-state value of J*, the

23The level ordering is sensitive to the actual form of the central potential, and here the one presented in [2], chapter

6 is used.

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20 Particle states (N+l) - ^ 15 — Calculated • Experimental * 5/2+ * 10 3/2 \3 /2 + 1/2+ 11/2- (50l-i 5/2(+j —r - n / 2- 0 5 1/2+ S3 3 3/2+ 7/2- ' W ! ° "d c -5 3/2+,5/2+ (11/2-k ' m -(7 /2 + k ™ + 5/2+-^** SJ -10 1/2+** ■a • (512+/'"' m 1/2+ 1 3 - 1 5 & Hole states (N -l). -20 I I I i i 58 64 68 70 82 N

Figure 46: Same as in Fig. 45 for Z=50 isotopes.

lowest-lying excited state with the calculated quantum numbers has also been plotted [80], Note

in particular the small (« tens of keV) energy differences between these states and each case and

the fact that there is always a partner with quantum numbers that agree with the ones predicted

for the ground-state. As for similar cases before, the energy difference has been exaggerated to be

appreciated at this scale.

A similar effect occurs in the case of the isotopes of Pb (Z=82). Figure 47 shows the filled j-shell

states closest to 2gf Pbi26 which is a major shell closure. Adding an extra neutron to this nucleus

will start filling the next j-shell, which according to the naive shell model is a 2g9/2 state. This is

precisely what is predicted here as can be seen in this figure for the particle state at N=126 (i.e.

N=127). Removing a neutron from 28fP bi26 corresponds to depleting the 3p!/2 state, as can be seen

in the figure, and again it is also correctly predicted. Since formed part of the nuclei used

to fit the parameters of the theory, it is expected that there will be an accurate description of the

neighboring nuclei, and in fact there is.

Removing one additional neutron completely depletes the j = 1/2 shell bringing the total number

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iT' 15 Particle states (N+l)

— Calculated 1/2- . . . Experimental

9/2+

3/2- 5/2- 3/2-

Hole states (N -l)'

Figure 47: Same as in Fig. 45 for Z=82 isotopes.

of neutrons to N=124, which is also observed occurring in this plot. According to [2] (see footnote

above) the next hole level would be a lii3/2 level, i.e. 13/2+, but the experimental evidence indicates

that this level is in fact a 5/2~. The reordering of the levels, with respect to the naive shell model,

can be understood as either a large splitting between the iQ i/ 2 and 3p3/2 levels, which brings the

3p!/2 level above the li13/2, or a considerable lowering of the lii3/2 which brings the li13/2 below

the 3pi/2- In either case removing one more neutron from this nucleus will start depleting the next

j-shell that could be a 3p3/2 or a 2f5/2. In Fig. 47 it is shown that this level is a 5/2“ in disagreement

with the calculated one, 3/2~. The closest excited state having this predicted quantum numbers

lays only 262 keV above the ground-state. This state is also plotted [81]. The energy difference has

been exaggerated in the figure, as in the previous plots. The reordering between the 2f5/2 and 3p3(/2,

shift the position of the real j-shell closures with respect to the calculated ones, as happened for the

Z=50 isotopes shown in Fig. 46; therefore they have not been included in this plot. Despite this

disagreement, the magnitude of the chemical potentials of these nuclei is relatively well described.

In summary, these results show that the binding energy of all even-even nuclei with semi-magic

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116

numbers of neutrons and protons, N = 28, 50, 82, 126 and Z = 28, 50, 82 are reproduced to 1%

or better in this EFT/DFT approach with parameters G1 fit along the valley of stability. With all

the isotones, the chemical potential of nuclei with a single proton particle or hole outside of filled

j-shells is well reproduced and the predicted value of for the ground-state of these nuclei agrees

with experiment.

When the number of neutrons gets very large in the regime where N > Z, the level density is very

high and the detailed ordering of the levels between major shells depends sensitively on the details

of the various densities. Thus, while the chemical potential (binding energies) are still relatively

well-described, the theory begins to lose its predictive power for the spin and parity of the ground

states of single neutron particle and hole nuclei neighboring those with filled j-shells.

4 Conclusions.

This work investigates the applicability of the EFT/DFT approach to reproduce total binding ener

gies and chemical potentials of semi-magic nuclei (N = 28, 50, 82, 126 and Z= 28, 50, 82) far from the

valley of stability. It extends the work done in chapter I, where the convergence of this EFT/DFT

approach was studied for doubly-magic nuclei. All calculations were performed using an indepen

dently developed program to solve the Kohn-Sham equations derived from the most sophisticated

effective lagrangian given in [1] with parameter set Gl.

From the figures given in the previous section, several conclusions are drawn:

1. The EFT/DFT approach can reproduce the total binding energies of even-even isotones and

isotopes with semi-magic numbers of neutrons and protons, respectively. The agreement is in

general better than 1 % where the best results are obtained for larger nuclei. This conclusion

is in agreement with the previous results obtained in chapter I.

2. The predictions for the magnitude of the chemical potentials for odd nuclei corresponding to

particle or holes neighboring filled j-shells agree overall with the experimental evidence. This

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117

agreement varies for different nuclei and on average lies between 10% and 20%.

3. Regarding the predictions of the ground-state quantum numbers of odd nuclei neighboring

even-even nuclei with filled j-shells, the results indicate that in the case of isotones these are

accurately reproduced; for heavy isotopes with N > Z, although the predictions of chemical

potentials are in essential agreement with experiment, the EFT/DFT approach begins to lose

its predictive power.

Overall, the use of this effective field theory approach for QCD and the interpretation of the

resulting relativistic Hartree equations through Kohn-Sham potentials and density functional the

ory [1] provides an excellent description of the ground-state properties of semi-magic nuclei when

extrapolated away from the valley of stability.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter IV.

Conclusions

The EFT/DFT formulation of Furnstahl, Serot and Tang is a novel approach to describe the nuclear

many-body system. It forms part of an ongoing effort to understand the nucleus, particularly new

phenomena occurring in regions of the nuclear chart far from the valley of stability. In its present

form, this formulation had been tested only for stable nuclei with excellent results. Its validity

outside this region is the central objective of this thesis. The main contributions resulting from the

present work are the following:

1. The convergence with increasing order of the EFT/D FT formulation for nuclei far from stability

has been verified for the doubly magic nuclei ^gS^o, 1fgSn82, 4|gNi2o and IfNiso- The addition

of new interaction terms in the effective lagrangian, following the prescriptions given by naive

dimensional analysis, results in the convergence of the calculations to the observed experimental

values.

2. The EFT/DFT formulation can reproduce the ground-state total binding energy of semi-magic

nuclei far from stability with N = 28, 50, 82, 126 and Z = 28, 50 and 82, with an accuracy

of approximately 1%. Additionally, chemical potentials of odd nuclei neighboring these nuclei

can be reproduced within 10% of experimental data.

3. A general expression for calculating any semi-leptonic process within this framework has been

derived 24. The nucleon wave functions are written in terms of Dirac spinors and obtained

from solving the self-consistent Kohn-Sham equations. These Dirac wave functions are used in

combination with the electroweak currents obtained directly from the same effective lagrangian,

24See equations (74), (75)-(76), (77), (80)-(83) and Table 5 in Chapter II.

118

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119

giving the whole approach a self-consistent character. Applications to calculate /^transition

rates in selected nuclei far from stability, i.e. Sn5i and 1|gIn5o, using this general expression,

gives results that suggest that the description of the wave functions of particles in the ground-

state is better than for excited states. This provides (indirect) evidence that this approach

indeed also yields valid ground-state densities.

4. Using the present form of the EFT/DFT formulation, ground-state quantum numbers of odd

nuclei are accurately reproduced in the case of semi-magic isotones with N = 28, 50, 82 and