Abstract

Experimental and Numerical Analysis of Mixed-Symmetry States and Large Boson

Systems

Robert James Casperson

2010

An investigation into the role of the hexadecapole degree of freedom in weakly-

collective vibrational nuclei is described. The interacting boson model-2 is used to attempt

to describe the structure of 94Mo, but certain key features of a low-lying mixed-symmetry

state seem to lie outside the model space. Shell model calculations indicated that the

hexadecapole degree of freedom plays a role, and calculations using g-bosons were per-

formed to attempt to describe the anomalous features. These calculations produced an

excellent fit for 94Mo, and showed that the hexadecapole degree of freedom plays a role in

the low-lying 4+ mixed-symmetry state of that nucleus.

To continue the experimental search for the hexadecapole degree of freedom in

other weakly-collective nuclei, an experiment was performed at WNSL at Yale University.

The nucleus 140Nd was produced using a proton beam, in order to populate 4+ states of

interest, and many new states were identified during the analysis. Several new multipole

mixing ratios are found for transitions that were observed, and candidates for the 4+

mixed-symmetry states are identified. Future calculations using the shell model should

provide insight into whether the hexadecapole degree of freedom plays a role in 140 Nd.

The interacting boson model is used to study quantum phase transitional behav-

ior in collective nuclei. The software ibar is developed for the purpose of understanding

the effects of finite system size on the characteristics of quantum phase transitions in nu-

clei. The behavior of electric monopole (E0) transition strengths in transitional nuclei is

introduced, and an investigation into the theoretical behavior of this transition for large

system sizes is described. The features of the E0 transition strength curves are identified

and understood using wavefunction analysis. Finally, the details of those wavefunctions

provide an interesting opportunity for relating the algebraic quantum numbers to the

geometric variables (3 and 7.

Experimental and Numerical Analysis of Mixed-Symmetry States and Large Boson Sys tems

A Dissertation Presented to the Faculty of the Graduate School

of Yale University

in Candidacy for the Degree of Doctor of Philosophy

by

Robert James Casperson

Dissertation Director: Professor Volker Werner

May 2010

UMI Number: 3415010

All rights reserved

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Contents

List of Figures viii

List of Tables x

Acknowledgements xi

Part I Introduction

1 Introduction 2 1.1 Nuclear shell model 2 1.2 Collective structure 3 1.3 Interacting boson model 3 1.4 The present work 4

Part II The hexadecapole degree of freedom in the interacting boson model

2 The interacting boson model 7 2.1 sd-interacting boson model-1 7

2.1.1 Calculations in the interacting boson model 12 2.1.2 Describing nuclei with the interacting boson model 13

2.2 sd-interacting boson model-2 13 2.2.1 The F-spin quantum number 15 2.2.2 Hamiltonian and operators in the sd-IBM-2 16 2.2.3 Calculations in the sd-IBM-2 17 2.2.4 Identifying mixed-symmetry states 19

2.3 Experimental examples of mixed-symmetry states 22 2.4 Simple sd-IBM-2 fit for 94Mo 23

3 The hexadecapole degree of freedom in 9 4Mo 28 3.1 The sdg-interacting boson model-1 29 3.2 The sdg-interacting boson model-2 32 3.3 Calculations in the sdg-IBM-2 34 3.4 Fitting 94Mo using the sdg-IBM-2 35

3.4.1 Vibrational fit for 94Mo 36

v

3.4.2 Deformed gamma-soft fit for 94Mo 38 3.4.3 Transitional fit for 94Mo 41

3.5 E4 transition strengths 48 3.6 Discussion 51

Part III The search for 4 + mixed-symmetry states in 1 4 0 N d

4 Experimental techniques 54 4.1 Selecting a nucleus near N=82 54

4.1.1 Past experiments on 140Nd 56 4.1.2 Producing 140Nd 56

4.2 Experimental setup 57 4.2.1 YRAST Ball detector array 58 4.2.2 Gamma-ray coincidence 60 4.2.3 Angular correlation techniques 61 4.2.4 Measuring branching ratios 65 4.2.5 Internal conversion coefficients 67 4.2.6 Angular correction to branching ratios 69

4.3 Gamma-ray spectrum analysis using spectre 69 4.3.1 Choice of gamma-ray peak fit function 69 4.3.2 Fitting algorithm 71 4.3.3 Parameters and uncertainties 72 4.3.4 Compton scattering and the 2D matrix view 73

5 Experimental results 77 5.1 New states and transitions in 140Nd 81 5.2 Multipole mixing ratios between 4+ states 95 5.3 Discussion 97

Part IV Quantum phase transitions and large boson sys tems

6 Quantum phase transitions in nuclei 101 6.1 Quantum phase transitions in nuclei 102 6.2 Phase transitions in the interacting boson model 103 6.3 Basis States 107 6.4 Wigner-Eckart theorem 108 6.5 Hamiltonian 109 6.6 Reduced Matrix Elements 113

6.6.1 Multiplicity of States 113 6.6.2 Isoscalar factors 114 6.6.3 Arbitrary precision 114

6.7 Transition matrix elements 117 6.8 System size 117

vi

7 Electric monopole transition strengths in ibar 119 7.1 EO transition strengths in transitional nuclei 119

7.1.1 Alternative interpretation for large EO strengths 120 7.2 EO transition strengths for large boson numbers 123 7.3 Features of EO strengths between U(5) and SU(3) 124 7.4 Wavefunctions of 0+ states between U(5) and SU(3) 127 7.5 Interpretation of the wavefunctions 129 7.6 Visualizing the full 0+ wavefunctions 130 7.7 Amplitude of the EO features with boson number 132 7.8 Discussion 134

Part V Conclusions

8 Conclusions 138

Appendices A ibar manual 143

A.l Hamiltonian 143 A.2 Transition operators 144 A.3 Input file 145 A.4 Output files 150 A.5 Installation 151 A.6 Example files 152

vii

List of Figures

2.1 Spectrum with harmonic form of U(5) symmetry for 6 bosons 9 2.2 Spectrum with 0(6) symmetry for 6 bosons 10 2.3 Spectrum with SU(3) symmetry for 6 bosons 11 2.4 Casten triangle with the three symmetries of the sd-IBM-1 14 2.5 Group algebras for the sd-IBM-2 14 2.6 Geometric illustration of collective motion in mixed-symmetry states. . . . 15 2.7 Illustration of states in a two boson Hamiltonian 18 2.8 sd-IBM-2 fit to 94Mo in the Uw+l/(5) symmetry. 24 2.9 sd-IBM-2 fit to 94Mo in the 0^+^(6) symmetry 26

3.1 Lattice of algebras for the sdg-IBM-1 30 3.2 sdg-IBM-2 fit to 94Mo using vibrational Hamiltonian 37 3.3 sdg-IBM-2 fit to 94Mo using deformed gamma-soft Hamiltonian 39 3.4 sdg-IBM-2 final fit to 94Mo using transitional Hamiltonian 42 3.5 sdg-IBM-2 final fit to 94Mo using transitional Hamiltonian 43 3.6 Plot of E4 transitions strengths from 4 + states as a function of (3 50

4.1 Nuclear chart in region around 140Nd 55 4.2 Photo of the YRAST Ball detector array 59 4.3 Example of transitions in 140Nd with similar energies 62 4.4 Example angular correlation pattern from 140Nd 66 4.5 Coincidence gates used when determining branching ratios 68 4.6 Correlation between parameters for two possible peak shapes for fitting. . . 71 4.7 View of gamma matrix in spec t re , highlighting Compton scattering. . . . 74 4.8 Pull view of 1095 keV gate in spec t r e 76 4.9 Zoomed in view around 214 keV in the 1095 keV gate in spec t r e 76

5.1 Transitions from the lower energy states in the observed 140Nd level scheme. 82 5.2 Transitions from the higher energy states in the observed 140Nd level scheme. 83 5.3 Total projection from 140Nd gamma matrix 84 5.4 Angular correlation pattern from the 2264 keV 4 + state in 140Nd 85 5.5 Fitting the 1937 keV peak in the 774 keV gate 91 5.6 Multipole mixing ratios of transitions between 4 + states in 140Nd 96

6.1 Illustration of first and second order phase transitional behavior 102 6.2 Energy potential surface between U(5) and SU(3) 104 6.3 Energy potential surface between U(5) and 0(6) 105

viii

6.4 Casten triangle with shape coexistence region shown 106

7.1 Comparison of select nuclei with sd-IBM-1 EO transition strength curve. . . 121 7.2 EO transition strengths as a function of neutron number 121 7.3 Surface plot of EO behavior for 10 bosons in the sd-IBM-1 122 7.4 Plot of EO transition strengths for 100 bosons in the sd-IBM-1 123 7.5 Plot of EO transition strengths for 10, 40, 100, and 400 bosons in the sd-

IBM-1 between U(5) and 0(6). 124 7.6 Plot of E0 transition strengths for 10, 40, 100, and 400 bosons in the sd-

IBM-1 between U(5) and SU(3) 125 7.7 Plot of E0 transition strengths for 400 bosons in the sd-IBM-1 between

U(5) and SU(3) 126 7.8 Plot of the energies of the first three 0 + states in the sd-IBM-1 between

U(5) and SU(3) 126 7.9 Plot of the wavefunction components scaled by ^/n^ for 400 bosons in the

sd-IBM-lbetween U(5) and SU(3) 128 7.10 Surface plot of 2D 0 + wavefunction components with parameters n^ and

n A for C = 0.85 131 7.11 Plot of E0 strength of peak feature between U(5) and SU(3) for a range of

N 133 7.12 Plot of E0 strength of peak feature using a different definition of the E0

operator between U(5) and SU(3) for a range of N 133 7.13 Plot of E0 strength in the SU(3) limit for a range of N using a different

definition of the E0 operator 134

ix

List of Tables

3.1 Dominant components of wavefunctions in shell model calculations for 94Mo 29 3.2 Comparison of 94Mo levels with sdg-IBM-2 calculation 46 3.3 Comparison of E2 transition strengths from 94Mo with sdg-IBM-2 calculation. 47 3.4 Comparison of Ml transition strengths from 94Mo with sdg-IBM-2 calcu-

lation 48 3.5 Boson composition in wavefunctions of O1" and 4 + states from sdg-IBM-2

calculation 49 3.6 Calculated E4 transition strengths for 94Mo using the sdg-IBM-2 49

4.1 YRAST Ball angle groups for in-beam measurement 63

5.1 Table of results from 140Nd experiment 78

7.1 Percent overlap of the wavefunctions near the avoided crossing at transition point between U(5) and SU(3) 130

x

Acknowledgements

The work described in this dissertation could not have been accomplished without the help and support of a large group of people. I am also grateful to have worked at the lab itself, as there are few facilities that can provide the hands on experience I have gained while at WNSL.

Volker Werner, you have been a great advisor, and I appreciate the time you have spent answering my questions. The many discussions we have had throughout the years led to the research topics contained in this dissertation, and I am grateful for the advice that you have provided.

I would like to thank my whole committee for being willing to read this disser-tation, and for providing feedback on the content. Rick Casten, you shaped the lab into what it was during my time at WNSL, and your class was a fantastic introduction to the field. Francesco Iachello, the model you developed has significantly influenced the field, and extensions of the model clearly inspired the various topics contained in this disserta-tion. Peter Parker, thank you for the support you have provided. Paddy Regan, thank you for the opportunity to participate in the Legnaro experiment.

I appreciate the advice and support from Andreas Heinz during the experiments performed at the lab. Thanks to Zvi Berant and Alex Wolf, for introducing me to g-factor measurements and experimental work with superconducting magnets.

I am grateful to the staff at WNSL: Karen DeFelice, Paula Farnsworth, Mary Anne Shultz, John Baris, Jeff Ashenfelter, Walter Garnett, Frank Lopez, Sal DeFrancisco, Sam Ezeokoli, Craig Miller, Tom Barker, and Dick Wagner. You keep the lab running and have provided valuable assistance during my time as a student.

Thanks to Liz Williams, Ryan Winkler, Tan Ahn, Russ Terry, Mike Bunce, Dave McCarthy, Linus Bettermann, and Raphael Chevrier for the fun and enlightening conver-sations at the lab. Thanks to all of the other students, postdocs, and visitors at the lab for helping out with the experiments.

I am grateful to my parents, Lee and Susan Casperson, and my entire family for the love and support they have provided throughout my life. None of this would have been possible without your help and encouragement.

Becca, thank you for your unfailing patience, love, and support. I cannot express how lucky I am to have you in my life.

This work was supported by the US Department of Energy, under grant number DE-FG02-91ER-40609, and by Yale University.

xi

Part I

Introduction

i

Chapter 1

Introduction

The atomic nucleus is an incredibly complex system, where dozens and sometimes

hundreds of particles interact in extremely complicated ways. Each nucleon is made up of

three quarks that interact via the strong force. The residual strong force is responsible for

the short ranged attractive nuclear force that holds the nucleus together, and an additional

Coulomb interaction between protons provides a repulsive force. It is clear that with such

a complex system, a single model that describes all features of nuclei and includes all

nuclear interactions will be impossible to implement. As such, identifying the underly-

ing symmetries, the important degrees of freedom, and the most relevant interactions is

extremely important for understanding the general behavior of the nucleus.

1.1 Nuclear shell model

One important first step that was made by Jensen and Mayer was the develop-

ment of the nuclear shell model [1, 2]. In atomic systems, a central Coulomb potential is

provided by the protons in the nucleus. Electrons are fermions, and as they are added to

the system, they fill up electron orbits. With certain specific numbers of electrons, the

orbits form closed shells that no longer interact as strongly with other atoms, or additional

outer electrons. The nuclear system was found to behave in a similar way, but unlike the

Coulomb central potential that exists in the atoms, the nuclear central potential is instead

generated by the nucleons themselves.

This assumption about the formation of shells in nuclei dramatically simplifies

any attempts to model the structure of excited states in nuclei. The general energy spacing

of possible proton and neutron orbits can be roughly predicted using a three-dimensional

quantum Harmonic oscillator, an L2 interaction, and a spin-orbit coupling term. Con-

structing a basis out of the most likely configurations for the nucleons to occupy, and

2

applying the relevant interactions, can in many cases reproduce the structure of the low-

lying excited states of nuclei.

The main restriction with such a model is that even though the vastly complex

system of the nucleus was dramatically simplified, it is still too complex to model nuclei

with a large number of valence nucleons. The shell model is primarily applicable to nuclei

that lie near closed shells, but as more valence nucleons are added to the system, an

interesting type of behavior called collective motion arises.

1.2 Collective structure

Nuclei that have closed proton or neutron shells have a spherical shape. As

valence nucleons are added to the system, the shape remains spherical, but becomes softer,

and vibrational structure is visible in the excited states of such nuclei. This softening of

the spherical shape is the onset of collectivity, where collective motion refers to the valence

nucleons moving together as a whole.

Excited states in nuclei decay to lower energy states via gamma-decay, where

a gamma-ray of a particular multipolarity is emitted from the nucleus. The vibrational

structure that appears at the onset of collectivity is a quadrupole oscillation around a

spherical equilibrium shape, and the transitions that occur as a vibrational state decays to

a more spherical state is typically an electric quadrupole (E2) transition. Strong transitions

of this type are one of the signatures of collective behavior.

As even more valence nucleons are added to the system, a quadrupole deformed

equilibrium shape becomes energetically favorable in nuclei. This deformed shape is asso-

ciated with rotational structure in the excited states in such nuclei. The geometric nature

of the transition from spherical to deformed makes using a geometric model a clear choice,

and such a model was developed by Bohr and Mottelson [3, 4], An algebraic model whose

symmetries represent the different geometric states of collective nuclei will be one of the

focuses in this work, but the geometric variables 0 and 7 from the Bohr-Mottelson model

will be referenced on occasion.

1.3 Interacting boson model

The interacting boson model is an algebraic collective model that has a micro-

scopic foundation in the shell model [5, 6]. As mentioned above, one of the keys to

3

understanding nuclei is isolating the important interactions and degrees of freedom. The

interacting boson model forgoes some of the single particle structure of the shell model,

and focuses on the L = 0 and L — 2 couplings that play a dominant role in the low-lying

states in even-even collective nuclei [7]. The symmetries in this model allow successful de-

scriptions of vibrational, axially-symmetric deformed, and deformed gamma-soft collective

structure.

When the proton-neutron degree of freedom is included in the interacting bo-

son model, an additional class of states called mixed-symmetry states are allowed. When

compared to their symmetric counterparts, these states have a negative phase factor be-

tween the proton and neutron boson components of the wavefunction. The experimental

signatures for these mixed-symmetry states are strong Ml transitions to symmetric states.

The interacting boson model is a useful framework for the study of quantum phase

transitions in nuclei. By using the method of coherent states, the algebraic structure of

states in this model can be related to geometric variables (3 and 7 [8, 9]. With this

formalism, an energy potential surface for the ground state can be found, which can help

illustrate the transition from spherical to deformed between the symmetries. The behavior

of the minima in that energy potential surface shows that both first and second order phase

transitions should occur in the model.

1.4 The present work

The results and discussion in the following chapters fit into two categories, both of

which relate to transitions that occur as nuclei become more collective. The first addresses

the relevant degrees of freedom in vibrational nuclei at the onset of collectivity. The second

is about the transition from spherical to deformed, and the signatures of phase transitions

in finite systems.

In Part II, the interacting boson model is described in detail, and this model will

be an important tool in all parts of this work. The experimental search for low-lying mixed-

symmetry states is described, and one nucleus with well-observed mixed-symmetry states

is discussed in particular. Fits using the sd-IBM-2 are shown to not adequately describe

certain key features of 94Mo, and the hexadecapole degree of freedom is considered in

detail. Using this expanded model, the best fit for 94Mo using an interacting boson model

is shown, and details about the strength of certain interactions of the Hamiltonian are

considered.

4

In Part III, the experimental search for the hexadecapole degree of freedom in

other weakly-collective nuclei is continued in an experiment performed at WNSL at Yale

University. The nucleus 140Nd was produced using a proton beam, in order to populate 4+

states of interest, and many new states were identified during the analysis. Details about

the energies and spin assignments of the new states are discussed in detail. Several new

multipole mixing ratios are found for transitions that were observed. Candidates for the 4+

mixed-symmetry states are described, but without lifetimes for the states, Ml transition

strengths cannot be determined. Future experiments that are needed for determining these

lifetimes are briefly discussed.

In Part IV, the study of phase transitional behavior in nuclei is discussed in

detail. The interacting boson model provides an excellent framework for studying the

behavior of nuclear observables at varying system sizes, and motivation is given for software

that can scale to large system sizes. The software ibar is developed for that purpose,

and it allows calculations for up to 400 bosons. The behavior of electric monopole (EO)

transition strengths in transitional nuclei is introduced, and an investigation into the

theoretical behavior of this transition for large system sizes is described. The features

of the EO transition strength curves are identified and understood using wavefunction

analysis. Finally, the details of those wavefunctions provide an interesting opportunity for

relating the algebraic quantum numbers to the geometric variables j3 and 7.

5

Part II

The hexadecapole degree of freedom in the interacting boson

model

6

Chapter 2

The interacting boson model

One of the most fundamental models in nuclear structure is the shell model,

which was developed by Jensen and Mayer [1, 2]. It is very useful for describing nuclei

with a small number of valence nucleons, but as one moves away from closed shells, and

collectivity takes hold, the model-space becomes much too large for calculations to be

possible even on modern computers. At low energies in the shell model for even-even

nuclei, pairs of identical valence nucleons occupy the same orbits, with the pairs coupling

to L = 0 at the lowest energy and L = 2 at a higher energy. Many other configurations

are possible, but at low energies, truncating the model space to include only those two-

particle configurations leads to an interesting model that has a much smaller model space,

and wide applicability. This model is the interacting boson model (IBM) [5, 6].

2.1 sd-interacting boson model-1

The interacting boson model, which can be treated as a vast truncation of the

shell model [10], is an algebraic model that can be used to describe collective motion in

even-even nuclei. It is primarily successful at low energies, where states are less likely

to be influenced by single-particle excitations. The states are constructed from s and

d-bosons, which correspond to pairs of valence nucleons coupling to L = 0 and L = 2

respectively [7]. Higher angular momentum excitations are possible, but tend to occur at

higher energies, and do not play a role in the low-energy states. It is unclear how high

an energy these higher-spin bosons would come into play, however, and the possibility of

including g-bosons in the model space is explored in Chapter 3.

Beyond only allowing s and d-bosons in the model space, the simplest form of this

model does not distinguish between bosons constructed from pairs of protons or neutrons.

The model with these specific restrictions is called the sd-IBM-1. The sd-IBM-2 refers

7

to an extra degree of freedom where the model distinguishes between proton and neutron

bosons, and this possibility will be considered in a later section of this chapter.

States and operators in the sd-IBM-1 are constructed from s and d-boson creation

and annihilation operators. The s-bosons are defined by L = 0, which means they have one

magnetic substate. For d-bosons, L = 2, so they have five magnetic substates. The model

constructed from these particles therefore has six dimensions, and the algebraic group

that describes them is U(6). The operators are s^, d]Iy s, and dfl, where n = — 2 . . . 2, and

they satisfy the Bose commutation relations. In order to calculate matrix elements in the

sd-IBM-1, the operators should be spherical tensors. The creation operators s^ and dj,

satisfy this condition, but the annihilation operators need to be modified.

S = s = ( - 1 Yd^ (2.1)

Tensor products of these spherical tensors can be used to construct a Hamiltonian for the

model. The number of bosons in the model space is defined by the total number of pairs

of valence particles and holes in the nucleus, and that number of bosons is conserved by

the Hamiltonian. Although the relative numbers of s and d-bosons can change, the total

number remains the same. Another restriction on the Hamiltonian is that it contains

only one and two-body terms. Higher order terms are possible, but typically are not

necessary, and are removed from the Hamiltonian. One last restriction is that terms in

the Hamiltonian couple to a total angular momentum of 0, so that angular momentum is

conserved, and the Hamiltonian can be written in block diagonal form with respect to L.

As previously mentioned, the sd-IBM-1 is described by the U(6) group algebra.

The subgroups of particular interest, U(5), 0(6), and SU(3), are called dynamical symme-

tries, and are physically relevant to a discussion of nuclear structure. Before getting into

these symmetry limits, here is a simple Hamiltonian that can be used to describe all three

limits [11, 12]:

# C Q F = c ( ( 1 - C )h d - ^ Q * • Q * ) (2.2)

where c is a scale factor, hd = • d. is the d-boson number operator, and Qx =

+ x[d^d}^ is the quadrupole operator. The Hamiltonian is labelled CQF

[11], because it refers to the consistent-Q formalism, which means that the E2 transition

operator is defined in terms of the same quadrupole operator:

T(E2) = eBQx (2.3)

8

(nd,0) (nd,1) (nd,2) (nd-2,0) (nd-2,1) (nd-4,0) (nd-6,0)

6 ]2+10+9i+ T6>+ _6+ £_3.+ 8_+ _6+ _5+_4+ _2.+ i .+ i .+

5 J. l£+JL+ iL+ .§ .+ .£.+ i . + i . + j6+ _4+ _3_+ £ 2_+

3 3_+ 0.+ 2_+

2 f l+i.+

2+

£ U(5)

Figure 2.1: Spectrum with harmonic form of U(5) symmetry for 6 bosons. Uses the parameters c = 1 and C = 0 in the Hamiltonian from Eq. 2.2. The quantum numbers in parentheses are (v,n&).

where eg refers to the effective boson charge. In the Hamiltonian, £ can take on values

from 0 to 1, and x c a n have values from — to 0. The factor of AN in front of the

quadrupole term is there so that the transition from vibrational structure to rotational

occurs at approximately C — Each of the symmetry limits of the sd-IBM-1 is described

by a chain of groups, which can be used to define the quantum numbers for the construction

of a basis.

U(5) group algebra

The U(5) algebra is described by the chain

U(6) D U(5) D 0(5) D 0(3) D 0(2) (2.4)

U(6) is characterized by the total boson quantum number N, U(5) by the number of d-bosons in a state nrj, 0(5) by the seniority v, 0(3) by the angular momentum L, and 0(2) by the magnetic substate quantum number ttil- The seniority quantum number v can also be replaced by n^, which is the number of pairs of d-bosons, where the pairs couple to angular momentum 0. 0(5) is not fully reducible with respect to 0(3), so an additional quantum number is needed to fully characterize the chain, and this quantum number will be riA- It refers to the number of d-boson triplets, where the triplets couple to angular

9

(2 0) ( 0 ' 0 )

(6,0) (6,1) (6,2) M % % £ + 12+10+9+ 8+ r 6+ 6 + 4 + B + 0+ i i i l -

1 ••

0 1 -

2 J- £ £ £ T0+_8+ 2_+

4 + 2 +

2+

8_+ _5_+ £ £

£££ £ 4+ 2+

2+

0*

2+

2" 0(6)

Figure 2.2: Spectrum with 0(6) symmetry for 6 bosons. An L • L term is left out of the Hamiltonian, which would also be part of the 0(6) symmetry. Uses the parameters c = 1, C = 1) and x = 0 in the Hamiltonian from Eq. 2.2. The quantum numbers in parentheses are (u,va)-

momentum 0, but it could be defined in many other ways.

The U(5) limit can be used to describe vibrational motion in nuclei, and in terms

of Hcqf in Eq. 2.2, it has the parameter £ = 0, which leaves only the d-boson number

operator n^. Other terms are available in the U(5) limit, but only those that appear in Eq.

2.2 were kept, x c a n have values from — to 0, but this only effects the E2 transition

operator. For each state in the U(5) limit, n,i is a good quantum number. If x = 0>

then E2 transitions can only occur between states whose number of d-bosons differ by 1.

If x 0, then there will be a matrix element between states with the same number of

bosons. This vibrational structure can be seen in Fig. 2.1, where a sample calculation

with N = 6 is shown.

0 ( 6 ) group algebra

The 0(6) algebra is described by the chain

U(6) D 0(6) D 0(5) D 0(3) D 0(2) (2.5)

10

(12,0) (8,2) (4,4) (0,6) (6,0) (2,2) (0,0)

4+

3 12+

2 10+

8+

2 + Z o 0 + -

2 0 "

SU(3)

Figure 2.3: Spectrum with SU(3) symmetry for 6 bosons. Uses the parameters c = 1, £ = 1, and x = — in the Hamiltonian from Eq. 2.2. The quantum numbers in parentheses are

U(6) is again characterized by the total boson quantum number N, 0(6) by the quantum

number a, 0(5) by the quantum number r , 0(3) by angular momentum L, and 0(2)

by the magnetic substate quantum number mi . 0(5) is again not fully reducible with

respect to 0(3), so an additional quantum number is needed. For this group chain, the

extra quantum number is called

The 0(6) limit can be used to describe gamma-soft rotational motion in nuclei.

This can be thought of as a rotating deformed nucleus, where the nucleus is free to oscillate

in the direction of the axis of rotation. In the Hamiltonian given in Eq. 2.2, the parameters

for this symmetry limit are ( = 1 and x = 0. The gamma-soft structure can be seen in

Fig. 2.2, where a calculation for the symmetry limit 0(6) is shown for N = 6. Similarities

can be seen between the structure of U(5) and 0(6), but additional underlying rotational

structure is visible in 0(6).

SU(3) group algebra

(A,M).

The SU(3) algebra is described by the chain

*7(6) D SU(3) D 0(3) D 0(2) (2 .6)

11

U(6) is characterized by the total boson quantum number N, SU(3) by the quantum

numbers A and /u, 0(3) by angular momentum L, and 0(2) by the magnetic substate

quantum number m^. SU(3) is not fully reducible with respect to 0(3), so an additional

quantum number is needed. For this group chain, the extra quantum number is K.

The SU(3) limit can be used to describe rotational motion in nuclei. This can

be thought of geometrically as a rigid deformed nucleus. In the Hamiltonian given in Eq.

2.2, the parameters for this symmetry limit are £ = 1 and X = — A n illustration of

this symmetry limit when N = 6 can be seen in Fig. 2.3. The band structure expected in

a rigid rotor can clearly be seen in the figure.

2.1.1 Calculations in the interacting boson model

Each of the three group chains provide a good method for enumerating the basis

states in the sd-IBM-1. When performing a calculation in this model, matrix elements

of the Hamiltonian must be calculated for all combinations of basis states in the model

space. The Hamiltonian is written in terms of s and d-boson creation and annihilation

operators, so it makes sense to use a basis that has the numbers of s and d-bosons as good

quantum numbers. This is only true in the U(5) symmetry limit, so a U(5) basis is ideal.

The angular momentum quantum number L is a good quantum number for sd-

IBM-1 Hamiltonians, due to each term in the Hamiltonian coupling to angular momentum

0. This causes the matrix to be block diagonal, and each angular momentum L can be

calculated independently. The magnetic substate quantum number tul is always a good

quantum number in this model, and by using the Wigner-Eckart theorem, reduced matrix

elements of the Hamiltonian can be calculated.

Tensor products in the Hamiltonian are typically expanded into sums of prod-

ucts of reduced matrix elements of the d-boson creation operator. These reduced matrix

elements are directly related to coefficients of fractional parentage from the coupling of

identical bosons, and the coefficients of fractional parentage are derived ahead of time,

and stored in tables. Finally, the Hamiltonian is diagonalized, to determine the energies

and wavefunctions of the system. Additional details about how sd-IBM-1 calculations are

performed numerically can be found in Chapter 6.

12

2.1.2 Describing nuclei wi th the interacting boson model

The sd-IBM-1 is a spectrum generating model, and results from calculations can

be compared to the level schemes of nuclei. If the energy of the state in a calculation is

scaled to the energy of the 2f state in an even-even nucleus, the Hamiltonian from Eq. 2.2

has essentially two parameters, and a two parameter fit for the nucleus can be performed.

Those two parameters span the space between the three dynamical symmetries of the sd-

IBM-1, and therefore can describe a large range of structures in nuclei. This type of fit is

an excellent way to characterize the basic structure of collective nuclei [13]. A simple first

step in this process is to look at the ratio R4/2 — E(2^), as a vibrational nucleus

has .R4/2 = 2, and a rotational nucleus has R4/2 — 3.33.

One way of visualizing the symmetry limits of the sd-IBM-1, as well as the

parameter space that lie between them, is with the Casten triangle [14]. An illustration

of this symmetry triangle along with the parameter behavior can be seen in Fig. 2.4.

In later sections of this chapter, as well as in the following chapter, additional degrees

of freedom will be added back into the interacting boson model. These extra degrees of

freedom result in an excess of parameters in the Hamiltonian, and steps need to be taken to

keep the parameter space reasonable. In most cases, simply remaining as close as possible

to analogs of these three symmetries will be a significant help in keeping the number of

parameters down. For this work, fits that were performed with these expanded models

have been done while keeping the symmetry triangle in mind.

2.2 sd-interacting boson model-2

In the sd-IBM-1, the system of bosons is constructed by coupling pairs of protons and neutrons to bosons of L = 0 and L — 2, essentially treating protons and neutrons as identical particles. The IBM is considered to be a truncation of the nuclear shell model, but the shell model has independent shell structure for protons and neutrons, so a logical next step would be to treat proton bosons and neutron bosons as distinguishable. The expanded model that does this is the sd-interacting boson model-2 (sd-IBM-2).

There are a few different ways to treat the group algebra that combines the 1^(6) and U„(6) algebras, but in the scope of this document, the scheme shown in Fig. 2.5 will be used. With this scheme the basis for protons and neutrons are enumerated independently up to 0,r(3) and 0„(3), which are characterized by the total angular momentum of the

13

0(6)

Figure 2.4: The Casten triangle with the three symmetries of the sd-IBM-1. The range of the two parameters can be seen along the sides of the triangle.

l y s ) - c y s ) \

\Jk(6) - S i y 3 ) 0 ^ ( 3 )

X / 0 ^ ( 6 ) - 0 ^ ( 5 )

O K + V (3) - 0 3 t + v ( 2 )

U v (5) — O v (5 ) \

Uv(6) — SUV(3) * O v (3 )

N / O v ( 6 ) — O v (5 )

Figure 2.5: One scheme for the group algebras of the sd-IBM-2.

14

t

I

0 Symmetric

) Mi Mixed-Symmetry

Vibrator Rotor

Figure 2.6: Geometric illustration of collective motion in mixed-symmetry states.

proton bosons and neutron bosons respectively. The two chains then link with 0^+^(3),

which is characterized by the quantum number for the total angular momentum of the

combined bases. The magnetic substate quantum number from 0^+^(2) refers to the

magnetic substates of the combined angular momentum.

In the sd-IBM-2, proton and neutron bosons are treated independently, and this

results iu an additional degree of freedom, which essentially can be thought, of as a phase

factor between proton and neutron components of the wavefunction. When all of the

proton and neutron bosons in the system are in phase, the state is considered to be a

symmetric state. These tend to appear at lower energies in the system, and are analogous

to the states that are found in the sd-IBM-1. When some of the proton and neutron

bosons in the system are out of phase, an additional class of states appear that are called

mixed-symmetry states [7, 15, 16]. These states can be illustrated geometrically with some

of the proton and neutron bosons oscillating or rotating out of phase, and an example of

this can be seen in Fig. 2.6.

2.2.1 The F-spin quantum number

With the extra degree of freedom in the sd-IBM-2, and the new class of states that result from it, it makes sense to classify these states according a quantum number called F-spin. With the group chains as illustrated in Fig. 2.5, F-spin will not necessarily be a good quantum number, but in other, less computationally practical schemes for the group algebra, F-spin can be a quantum number in the construction of the basis. Even for

15

the group chain used in calculations, choosing the right Hamiltonian can conserve F-spin.

For most of the remaining discussion about the interacting boson model, this will be the

case. By evaluating the matrix element of the F-spin operator, jF2, states can be classified

as either symmetric or mixed-symmetry states.

The F-spin operator is constructed in the following way [17]:

F+ = slsv + 4 • du (2.7)

F~ = + 4 • dn (2.8)

F° = - 4§„ + 4 • 4 - 4 • du (2.9)

F2 = F+F~ + F°F° - F° (2.10)

The F2 operator can be directly evaluated as a two-body matrix element of states from an

sd-IBM-2 calculation, which gives information about the proton-neutron boson symmetry

of the state. When F-spin is a good quantum number for a Hamiltonian, the maximum

value it can have is Fm a x = + N„), and this corresponds to a symmetric state.

For a mixed-symmetry state with one proton boson and one neutron boson out of phase,

F = Fm a x — 1. The minimum value that F can have is Fm;n = ^(A^ — Nv\.

2.2.2 Hamil tonian and operators in the sd-IBM-2

As was visible in the previous section, operators in the sd-IBM-2 are constructed

from s and d-boson operators for both protons and neutrons. The number of possible two-

body Hamiltonians that could be constructed with such a variety of operators is huge, as

there are only a small number of restrictions: the total number of bosons is conserved, the

total angular momentum is conserved, and the terms have only one-body and two-body

operators. To keep the number of parameters down, the discussion here will be restricted

to a Hamiltonian that is analogous to the sd-IBM-1 Hamiltonian given in Eq. 2.2. One

modification to that Hamiltonian is the inclusion of a term called the Majorana operator.

H = c({ 1 - 0(nd7r + hdv) + Ql) • 0 & + Qi) + A M ) (2.11)

ndp = 4 • dp (2.12)

Qp = + [4s~p](2) + x[dW2) (2-13)

M = Fm a x(Fm a x + 1 ) - F 2 (2.14)

16

where Fm a x = + Nu) is a conserved value in the sd-IBM-2. Matrix elements of F2

have a smaller value when a state is less symmetric, so M has the effect of increasing the

energy of mixed-symmetry states. In the Hamiltonian given in Eq. 2.11, F-spin is a good

quantum number, so each state has a pure F-spin that can be used to identify the proton-

neutron boson symmetry of the state. With F-spin being conserved in this Hamiltonian,

adjusting A will have no effect on symmetric states with F — Fmax.

The E2 transition operator in the sd-IBM-2 is similar to the E2 operator in the

sd-IBM-1, but has both a proton and a neutron term rather than just one term.

T(E2) = eBTQ* + eBvQ* (2.15)

where and cbu are the effective boson charges for the proton and neutron bosons

respectively. The Ml transition operator in the sd-IBM-2 is defined as the following:

T(Ml) = ^ (frL* + gvLv) (2.16)

where gn and gu are the boson g-factors for the proton and neutron bosons respectively.

The Lp operators are the angular momentum operators for the proton and neutron, and

are defined to be L„ The simplest illustration of mixed-symmetry states in sd-IBM-2 is with a 11^+^(5)

Hamiltonian like the one shown in Fig. 2.7. The symmetric states are shown on the left,

and have the same vibrational structure that a two boson U(5) Hamiltonian in the sd-

IBM-1 would have. The mixed-symmetry states are shown on the right. A one d-boson

mixed-symmetry state can be seen at an energy of e + 2A, and 1 + and 3+ two d-boson

mixed-symmetry states can be seen at an energy of 2e + 2A. The wavefunctions for all

states are listed to the right of the levels, and the wavefunctions for the one d-boson 2+

states illustrate the phase difference for symmetric and mixed-symmetry states.

The symmetric one d-boson 2+ state had a wavefunction of MdUl + sUl) |o>,

and the mixed-symmetry one d-boson 2+ state has a wavefunction of - ^ (d l s t — st<^t)|0). The difference between these is that there is minus sign between the proton and neutron & operators in the mixed-symmetry state. If protons and neutrons were exchanged in that state, the net result would be that the wavefunction flipped sign. 2.2.3 Calculations in the sd-IBM-2

Just like in the sd-IBM-1, the group chain that provides the best method for

enumerating basis states includes the U(5) groups. The basis is constructed from the

17

Un+V{5) Hamiltonian H = e(hd„ + hd J + A M

2 e -

0 A

M =N„ = 1

2 e + 2A ••

e + 2 A - -i+ 0H

[ 4 4 ] ( L ) | o >

2L - L ( 4 4 + 4 4 ) | o )

— 4 4 l o )

Symmetric o A

H [ 4 4 ] W | 0 >

2 1 J _ V2

( 4 4 - 4 4 ) 1 0 )

Mixed-Symmetry

Figure 2.7: Illustration of states in a two boson 11^+^(5) Hamiltonian using the sd-IBM-2. F-spin is a good quantum number for this Hamiltonian, and the Majorana operator simply shifts the mixed-symmetry states up or down by the parameter A.

18

quantum numbers n v n , i ia , , and Ln for the protons, and n<f„, vu, and Lu for the

neutrons. These two sub-bases are then coupled together into a total angular momentum

L. For a system with a large number of proton and neutron bosons, even L = 0 can have

an extremely large basis, because it will contain states with Ln = Lv, where Ln and Lu

could potentially be very large.

In the sd-IBM-1, the s and s operators do not play a significant role in the

angular momentum decoupling of the tensor products in the Hamiltonian, due to s-bosons

having L = 0. This leaves the d-boson operators, which is the only type of particle, so the

decoupling reduces to a set of sums of reduced matrix elements and 6-j symbols [17]. In the

sd-IBM-2, the decoupling is significantly more complicated because the tensor products

contain two types of particles with L = 2. The first step is to decouple the proton and

neutron operators, which results in a sum of reduced matrix elements of the proton tensor

products, reduced matrix elements of the neutron tensor products, and 9-j symbols. The

reduced matrix elements of proton and neutron tensor products can then be decoupled

in the same way the d-boson tensor products in the sd-IBM-1 were handled. Additional

details about calculating the d-boson tensor products in the sd-IBM-1 can be found in

Chapter 6.

2.2.4 Identifying mixed-symmetry states

In the sd-IBM-2, matrix elements of the F2 operator can be directly calculated

in order to evaluate the proton-neutron boson symmetry of each state. Experimentally,

such matrix elements are not observable, so other signatures for mixed-symmetry states

are important to identify. In the sd-IBM-1, off-diagonal Ml transition matrix elements

vanished, due to the L operator referring to total angular momentum, which is a good

quantum number for each state. This means that L only effects diagonal terms in the

Hamiltonian. The Ml operator in the sd-IBM-2 is constructed from a sum of L^ and L„,

which can be seen in Eq. 2.16. To help illustrate how the Ml operator behaves in the

IBM-2, it can be rewritten in the following way:

T(M1) = ^ ^ ((<?, + g„)(Lv + L„) + (g, - gv){Lir - L„)) (2.17)

The term with (L^ + L„) refers to the total angular momentum, which is a good quantum

number for each state, and therefore only affects diagonal terms of the Hamiltonian. That

leaves the term with (Ln — Lv) as the only part that contributes to Ml transition matrix

19

elements between states. The operators Ln and Lu conserve the underlying U(5) quantum

numbers, so the (Lw — Lv) term effectively creates a phase difference between proton and

neutron bosons, and can change the F-spin of a state. The matrix elements of the Ml

transition operator can be large between states of different F-spin, and an example using

the U7r-fi/(5) Hamiltonian from Fig. 2.7 will help illustrate this. The Ml transition matrix

elements between states with F = Fmax vanish, because the phase difference created by

the (Lw — Lu) term of the operator causes the terms in the matrix element to completely

cancel.

This discussion will focus specifically on 2+ states from that N„ = Nu = 1 system.

There are a total of three possible 2+ states, that are constructed from the following basis

states: In^n^n^iL^); n^n^n^L^ L)

121) >+

= |000(0); 100(2); 2)

|2+) = |100(2);000(0);2)

|2+) = |100(2) ;100(2) ;2) (2.18)

The Hamiltonian from Fig. 2.7 is defined as the following:

H = e(ndlt + hdv) + AM (2.19)

As an example, assume that e = 1 and A = The Hamiltonian matrix will be the

following: 2'

! H = . 1 5 Q

4 4 (2.20)

V 0 0 2 /

The diagonal elements of the first two 2+ states have the same value, which indicates that

the two states will be strongly mixed, and pushed apart by the off-diagonal interaction

matrix elements that result from M. The following are the energies, eigenvectors, and

F-spins of the 2+ states after diagonalizing:

E f 1 \

3 2

V 2 J

' A l

V2 V o

i V2 1

0

o \

0

1 1

F = ( 1 \

0

V i J

(2.21)

The eigenvectors ipi and ip2 in Eq. 2.21 are both constructed from equal parts of the |2^)

and 2^) basis states from Eq. 2.18, with only a phase factor distinguishing the two. The

20

matrix element of (L^ — Lv) between these two states will be large, because the phase factor

between Ln and L„ cancels out the phase factor between the proton-neutron components

of the two eigenvectors.

The (L^—Lu) term of the Ml operator only contributes when g^ ^ gu. Physically,

gn is much larger than gu, and for most of the calculations in this work, the g-factors have

been set to gn = 1 and g„ — 0. This is not necessarily the best choice, but it reduces

the number of parameters needed for a fit. For the rest of the chapter, assume that

f (Ml) = The Ml transition strength from 2 \ to 2f in the previous example

was B(M 1; —> 2 f ) = 0.3581 fi2N. The unit /ujy is the nuclear magneton squared, and it

is a standard unit for Ml transition strengths. Ml transitions on the order of 0.5 /ifj are

signatures of mixed-symmetry.

To briefly illustrate the effect of F-spin breaking, assume a slightly different

Hamiltonian:

H = + <Lvhd„ + AM (2.22)

where e^ = g, e„ = 1, and A = In matrix form, with the same basis from Eq. 2.18, the

Hamiltonian is

H ( \ - i 0 \

I 0 (2.23)

V 0 0 I J The diagonal elements of the first two 2+ states no longer have the same value, so they

cannot fully mix during diagonalization. The amount that they do mix ends up depending

on the strength of the Majorana operator, although as one moves away from a vibrational

Hamiltonian, other interaction matrix elements will contribute to the mixing. Due to the

incomplete mixing of the first two 2+ states, F-spin for those two states will no longer be

pure. In other words, F will not be 0 or 1 for those two states.

( 0.646 \ / 0.924 0.383 0.000 \ / 0.899 \

E = 1.354

\ 1.500 )

F = 0.237

\ 1.000

(2.24) 0.383 -0.924 0.000

\ 0.000 0.000 1.000 J

Having unequal contributions to the n ^ and n ^ terms in the Hamiltonian caused F-spin

mixing, which from the perspective of the chosen basis, was simply a decreased mixing

between the basis states. For reference, B(M1] 2% —> 2+) = 0.1790 for this calculation,

which is half as large as the B(M1) from the F-spin symmetric Hamiltonian.

21

To summarize this discussion, Ml transitions can be observed between different

states in the sd-IBM-2, due to unequal contribution of proton and neutron terms in the

Ml operator. These transitions occur between states of similar structure, with a simple

example being basis states with opposite proton-neutron structure split into states of

different F-spin during diagonalization, due to strong mixing. The strong Ml transitions

between these states of different F-spin makes Ml transitions an excellent observable to

use when identifying mixed-symmetry states experimentally.

2.3 Experimental examples of mixed-symmetry states

The theoretical framework of the sd-IBM-2 suggests the existence of mixed-

symmetry states, which are collective states with a phase difference between proton and

neutron bosons. One of the most striking examples of this new type of collective states

is the mixed-symmetry state in 811^+^(3) symmetry limit. This state can be un-

derstood geometrically as proton and neutron bosons with a deformed ellipsoid shape

contra-oscillating. This type of motion is often called a scissors mode, and an illustration

of this can be seen in the lower-right panel of Fig. 2.6. This 1 + mixed-symmetry state was

first observed experimentally in the deformed nucleus 156Gd [18], by looking for a strong

Ml transition from a 3 MeV 1 + state to the ground state.

Since that first observation, the experimental search for inixed-symmetry states

has continued, with success being found in some vibrational nuclei [19, 20, 21, 22, 23, 24,

25, 26, 27, 28, 29, 30]. Unlike the 1+ mixed-symmetry state found in 156Gd, the mixed-

symmetry states in vibrational nuclei can be found at around 2 MeV. These vibrational

nuclei tend to be found near closed shells, and are weakly collective as a result. The mixed-

symmetry states found in such nuclei are particularly interesting, because they can provide

additional information about the transition from the single-particle structure found in the

shell model to more collective structure.

One excellent example of mixed-symmetry excitations in a vibrational nucleus is 94MO. It has been thoroughly studied, and the one-boson mixed-symmetry 2+ state, as

well as the two-boson mixed-symmetry 1+ , 2+ , 3+ , and 4+ states have been well observed.

Details about the experimental identification of mixed-symmetry states, as well as the

determination of Ml transition strengths can be found in Chapter 4. The nucleus 94Mo

will be the focus of the rest of this theoretical investigation, and all experimental values

have been derived from a comprehensive paper about that nucleus [26].

22

2.4 Simple sd-IBM-2 fit for 94Mo

Having found a nucleus with vibrational structure and well observed mixed-

symmetry states, it makes sense to attempt to fit the nucleus using the 11^+^(5) symmetry

limit of the sd-IBM-2. The Hamiltonian for this symmetry is the following:

H = e(ndK+ndv) + XM (2.25)

This Hamiltonian can be described using the more general Hamiltonian found in Eq. 2.11,

which is analogous to the sd-IBM-1 CQF Hamiltonian. The system describing 94Mo has 4

proton bosons and 1 neutron boson, because 100Sn was chosen as the core for this nucleus,

rather than 88Sr or 90Zr. This choice moderately changes some transition strengths, but

does not significantly alter any energies.

A simple fit was done using this Hamiltonian, and the result of the fit can be seen

Fig. 2.8. The fits are organized in a way that the experimental levels are shown on the left

side, and the theoretical calculations are shown on the right side. The states are organized

according to angular momentum, with increasing angular momentum to the right. The

y-axis is the energy of the states. All states shown in magenta are symmetric states, and

the states shown in green are mixed-symmetry states. This convention will be maintained

for all further theoretical discussion. For the experimental levels, only states that were

previously identified as mixed-symmetry states by observation of strong Ml transitions are

shown in green. It is possible that other low-lying states have mixed-symmetry, but simply

have not been identified yet. The transitions in the top figure are Ml transition strengths,

and the transitions in the bottom figure are E2 transition strengths. The strength of the

transitions is represented by the thickness of the lines between states.

The calculations for this fit were performed in an sd-IBM-2 called ibar2, for the

convenience of integrating figure-generating routines. They could just as well have been

performed in npbos [31], which is the standard code for sd-IBM-2 calculations, e was set to

0.871 MeV in order to match the 2f energy to the experimentally observed energy. A was

set to 0.430 MeV, in order to place the mixed-symmetry states at appropriate energies.

As mentioned previously, the g-factors in the Ml transition operator have been set to

gn = 1 and gv = 0, and will remain that way for all fits. The effective boson charge for

the neutron has been set to 0, but the proton boson charge has been scaled such that

the calculated B(E2;2f —> 0^) matches the experimental value of 15.4 W.u.. For this

calculation, esw = 2.2 VW.u..

23

94 Mo M1 Transitions Ujt+V(5)

T ^ 3 *

2

1

° 0+ 1+ 2+ 3+ 4+ 5+ 6+ ° 0+ 1+ 2+ 3+ 4+ 5+ 6+

94 Mo I W 5 ) E2 Transitions

r_ 3 2

1

° 0+ 1+ 2+ 3+ 4+ 5+ 6+ ° 0+ 1+ 2+ 3+ 4+ 5+ 6+

Experiment sd-IBM-2

Figure 2.8: sd-IBM-2 fit to 94Mo in the 11^+^(5) symmetry. The Hamiltonian is from Eq. 2.25, and the parameters are e = 0.871 MeV, A = 0.430 MeV, and eB„ = 2.20 \/W.u.. Dashed transitions represent upper bounds for the transition strengths. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

24

In Fig. 2.8, it can be seen that the general distribution of the states resembles the

experimental levels, but there are some significant details that differ. In the calculation,

the and 6+ states show up at too high of an energy. The dashed transitions that

appear in the figure represent upper bounds for the transition strength. A calculated E2

transition is visible between O^ and but there is no experimentally observed transition

between those two states. However, the lifetime of the O2" is not yet known, so the E2

strength simply cannot be determined. The Ml transition strengths overall are too weak

in this vibrational limit. The Ml transition from 23" to 2f should have 0.56 n2N, and in

the calculation it was found to be 0.23 )J2N.

The most obvious discrepancy between experiment and fit is the Ml transition

from the 4j~ state that was observed experimentally. The transition has a strength of

1.23 (J-%, but in the calculation, the 4 + mixed-symmetry state is found at a much higher

energy, and has a significantly weaker Ml transition strength of 0.29 [i2N. Although the

Hamiltonian chosen for this fit is extremely simple, there are no obvious choices of addi-

tional terms that would push the 4+ mixed-symmetry state from 2.817 MeV down to the

experimentally observed energy of 2.294 MeV. Similarly, no obvious choice for additional

terms would increase the Ml transition strength by a factor of 4. This 4 + state and its

corresponding Ml transition will be discussed further later on, and is the motivation for

the discussion in the next chapter. For now though, it is worthwhile to look at one other

fit for 94MO using the sd-IBM-2.

This second fit will use a ()„•+„(6) Hamiltonian in the sd-IBM-2, to see if there

are any signatures for gamma-soft structure in 94Mo. The Hamiltonian used for this fit is

the following:

H = + Qi) • (Q* + Qu) + A M (2.26)

A simple fit was done using this second Hamiltonian, and the result can be seen

in Fig. 2.9. The parameters were k = —0.315 MeV and A = —0.010 MeV, where k

was selected such that the energy for the theory and experiment match. A more

conventional choice would be to scale to the 2f energy, but this way was chosen so that

more states and transition strengths would be visible in the figure. This choice is somewhat

misleading, because a lot of structure information is contained within the R4/2, and scaling

to the masks the effect. The choice of A was a compromise between the 2+ mixed-

symmetry states energies and the 1 + and 3 + energies. For this calculation, — 1.64

VWm..

25

9 4

3

2

1

0

Mo M1 Transitions •MMpNi

0 n + v ( 6 )

0+ 1+ 2+ 3+ 4+ 5+ 6+ 0

0+ 1+ 2+ 3+ 4+ 5+ 6+

9 4 Mo

3

2

1

0

\ /V V • \ /

A:-

0+ 1+ 2+ 3+ 4+ 5+ 6+

Experiment

E2 Transitions

r_ 3 2

1

0

0jt+v(6)

/ / y

y-

0+ 1+ 2+ 3+ 4+ 5+ 6+

sd-IBM-2

Figure 2.9: sd-IBM-2 fit to 94Mo in the ()„•+„ (6) symmetry. The Hamiltonian is from Eq. 2.26, and the parameters are re = -0.315 MeV, A = -0.010 MeV, and eBl[ = 1.64 V'W.u.. Dashed transitions represent upper bounds for the transition strengths. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

26

The first observation about the fit is that the the density of levels is too low. Many

of the 4+ states are much too high in energy, and the energies of the mixed-symmetry states

do not match the pattern in 94Mo. The E2 transitions among the 2 + and 0 + states at first

glance appears to an improvement over the U7r+t/(5) case, but that is simply because the

O2" state is missing in the calculation. Observing the state in gamma-ray coincidences

is challenging, because the —> transition has about the same energy as the 2f —> 0+

transition. This state is fairly important for describing the structure of the nucleus, but

unfortunately the transition information simply is not known yet. Assuming the O^ state

is collective, and has a gamma transition to the 2f state, it appears as though 0^+^(6)

falls seriously short of describing 94Mo.

Another clear problem with the fit is in the Ml transition strengths. The calcu-

lated Ml strength of 2g~ to 2f is 0.30 fj?N, which is smaller than the experimental value

of 0.56 h2n. The calculated 4 + mixed-symmetry state again lies much too high in energy,

and has a transition strength 0.33 fJ2N, which is significantly less than the observed value

of 1.23 (J?N.

Although these two fits used extremely simple Hamiltonians, there are some

specific features of the nucleus that cannot be adequately described even with significantly

more complex Hamiltonians, without sacrificing the structure of the rest of the fit. The

most obvious example is the extremely low-lying mixed-symmetry 4 + state, and the 1.23

li2N Ml transition from 4 t to 4j~. This feature seems to lie outside the scope of sd-IBM-

2. The original choice of d-bosons in the interacting boson model was based on L = 2

excitations playing a significant role in low-lying collective states. The model space can be

expanded to include hexadecapole excitations, which have L = 4, by including g-bosons

in the IBM-2. This is a logical choice, because identical particles with j > | couple to

L — 0, L = 2, and then L = 4 in order of energy. Unlocking the hexadecapole degree of

freedom is the topic of Chapter 3.

27

Chapter 3

The hexadecapole degree of freedom in 94Mo

The microscopic justification for the use of the interacting boson model on col-

lective nuclei is that the states in the IBM represent a truncation of the shell model. At

the lowest energy in collective even-even nuclei, pairs of nucleons couple to L = 0, and

at higher energies the pairs couple to L = 2. The inclusion of g-bosons in the interacting

boson model implies that on a microscopic level, pairs of nucleons are coupling to L = 4.

As was discussed at the end of Chapter 2, the motivation for unlocking the hexadecapole

degree of freedom is to help explain the behavior of the low-lying 4+ states in 94Mo. Specif-

ically, the goal is to gain an understanding of why the 4 + mixed-symmetry state lies at

such a low energy, and also why there is such a strong Ml transition between the and

4, states. As a reminder for the reader, B(M 1; 4+ -> 4+) = 1.23 in 94Mo.

As mentioned in Chapter 2, weakly collective nuclei provide an insight into the

transition of nuclei, from the single-particle behavior of the shell model, to the collective

motion that occurs when even-even nuclei have a large number of valence particles. In

nuclei like 94Mo, both single-particle and collective models can be tested and compared.

Shell model fits [32] have been performed to better understand the structure of low-lying

states in 94Mo, and these have yielded interesting results with regard to the unique behavior

of the 4 + states observed in this nucleus.

The shell model calculations were performed using 88Sr as the core [32], and the

two neutrons could be distributed among five single particle orbitals, the lowest being

2(4/2. This choice of core differs from the IBM-2 calculations performed in this work,

where 100Sn has been chosen. In the IBM-2 calculations, the transition strengths are

moderately affected by the choice of core, but there is no significant change in energies. In

the shell model calculations, the four protons had the 2p1(/2 and I59/2 orbital available. The

n(2p1/2) orbital was 1 MeV lower than 7r(l<79/2), and in many of the main components of the

28

Component 4i+ 4?

^ 9 / 2 ) 0 x ^ 5 / 2 ) 4 20% 23%

^ 9 / 2 ) 4 x ^ 5 / 2 ) 0 13% 21%

Table 3.1: Dominant components of wavefunctions of and 4% during shell model cal-culations for 94MO [32], Components are given in percentages, and the two configurations listed are the largest components in both wavefunctions.

wavefunctions, two protons were in the 7r(2p1/2) orbital coupling to zero, which effectively

left two protons available. The dominant components of the 4^ and 4^ wavefunctions are

shown in Table 3.1.

The wavefunctions of the 4+ and 4 j states in shell model calculations of 94Mo

have significant components where two particles couple to L = 4. The configuration

n{g29/2)o x has two neutrons in the 2d5/2 orbital coupling to L = 4, and the

configuration (59/2)4 x ^(^5/2)0 has two protons in the lgg/2 orbital coupling to L = 4.

Looking back at the microscopic justification for the interacting boson model, and the

construction of the s and d-bosons, this seems like an excellent justification for including

g-bosons in IBM-2 model space. The remaining question is whether the shell model cal-

culations observed a strong Ml transition between 4% and 4 f . The calculated strength

was B(M\\4~2 —> 4+) = 1.79 n2N, which confirms that the inclusion of the hexadecapole

degree of freedom in IBM-2 could help describe 94Mo.

3.1 The sdg-interacting boson model-1

Adding a g-boson to the sd-IBM-2 is a fairly complicated step, and the simpler

system of the sdg-IBM-1 will be useful for discussing the general properties of the g-boson.

The sdg-IBM-1 is not entirely relevant for the investigation of the 4+ states in 94Mo,

because without the proton-neutron degree of freedom, there are no mixed-symmetry

states. This model can be of interest for the discussion of E4 transition strengths, due to

the E4 operator that can be defined with the inclusion of g-bosons [33]. An E4 operator

can be defined in the sd-IBM-1 including [d^d]^, but such an operator does not change

the number of d-bosons, and cannot easily account for large E4 transitions from 4 + states

to the ground state.

The sdg-IBM-1 has a fairly complex lattice of algebras, which can be seen in

29

Figure 3.1: Lattice of algebras for the sdg-IBM-1. Adapted from a paper about the symmetries of sdg-IBM-1 [34].

Fig. 3.1. The chains all start with the U(15) algebra, which has 15 dimensions: one

for the s-boson, 5 for the d-boson, and 9 for the g-boson. There are three dynamical

symmetries that can be considered analogous to the three sd-IBM-1 symmetries: 0(15)

represents deformed gamma-soft motion and is analogous to 0(6) in the sd-IBM-1, the

U(14) symmetry represents vibrational motion and is analogous to U(5) in the sd-IBM-1,

and SU(3) represents a rigid rotor, and is analogous to SU(3) in the sd-IBM-1. The new

operators in the sdg-IBM-1 are gl and g^ where fi = —4... 4. The operators should be

made into spherical tensors so that reduced matrix elements can be calculated in this

30

model. The annihilation operator must be modified in the following way:

~9ll = ( - 1 Y g ^ (3.1)

The terms in the Hamiltonian can include gt and g, and there is a huge variety of possible

two body terms that can be constructed from s^, s, S, d, and g. Symmetry limits

are typically a good place to start with the construction of a Hamiltonian, but if the

spectra of those symmetries do not describe nuclei well, a multipole Hamiltonian is a

physically instructive system to use. To extend the CQF Hamiltonian from Eq. 2.2 into

this extended model, some g-boson terms should be added. This necessarily increases the

number of parameters, but describing the most physical variety from the fewest parameters

is part of the art of constructing a Hamiltonian for fitting nuclei. One possible Hamiltonian

is the following:

H = c ((1 - C)(nd + an9) - ^ Q • Qj

Q = [ S t J + dfs](2) + 0[Sg + s fd] (2) (3.2)

This Hamiltonian has four parameters, and one is simply an energy scale factor. The pa-

rameter C changes the Hamiltonian from vibrational to deformed gamma-soft by switching

from (hj + ahg) to Q Q. The parameter a represents the energy of the g-boson excitation.

It is assumed that the g-boson has more energy than a d-boson excitation, but beyond

that, a is a variable parameter. In the quadrupole term, ft determines the strength of the

d-g interaction. The E2 and Ml transition operators in the sdg-IBM-1 can be defined to

be the following:

T(E2) = eBQ

f (Ml) = ^ g L (3.3)

where Q is defined in Eq. 3.2. The angular momentum operator is defined to be L =

VT0[dtJ]M + i/BO^gjW [35], In the transition operators, es is the effective boson charge,

and g refers to the boson g-factor.

In the E2 transition operator, the decision to use the same Q as in the Hamilto-

nian was done to minimize the number of parameters during a fit. Another option would

be to give the s-d and d-g terms of the E2 operator separate parameters. In the Ml

transition operator, the g-factor for the d and g-bosons were defined to be the same, but

another option would be to make them independent. In the sd-IBM-1, no Ml transitions

31

are observed between different states because L is diagonal in the Hamiltonian. In the

sdg-IBM-1, if the d and g-bosons are given independent g-factors, then Ml transitions are

not necessarily forbidden:

t(Ml) = ^ { g d L d + ggLg)

f (Ml) = y^L ((ft, + ga)(Ld + Lg) + (gd - gg)(Ld - Lg)) (3.4)

The (Ld + Lg) term is diagonal in the Hamiltonian, but the (Ld — Lg) term can induce transitions. Experimental Mis between proton-neutron symmetric states can be compared to model calculations in order to derive gd and gg [36]. Comparing Ml strengths to model calculations in this way is difficult for 94Mo, because it is not clear which expanded model or Hamiltonian best describes it. The g-factors for d and g-bosons will therefore be kept equal in further calculations.

3.2 The sdg-interacting boson model-2

Adding the proton-neutron degree of freedom into the sdg-interacting boson model produces the sdg-IBM-2, which is the model that will be used to fit 94Mo. There are many different ways of merging the proton and neutron algebra lattices. Merging 11^(15) and U„(15) into Uff+I/(15) produces group chains where F-spin is a good quantum number. This is not necessarily convenient for enumerating basis states computationally, but the Casimir operators of such a group chain can be used to construct a Hamiltonian that has good F-spin.

Analogous to the step from sd-IBM-1 to sd-IBM-2, switching to the sdg-IBM-2 results in twice as many boson operators: one set of proton boson creation and annihilation operators, and one set of neutron boson creation and annihilation operators. The Hamilto-nian has an extremely large number of potential two-body terms, but the simplest option is to convert an sdg-IBM-1 Hamiltonian into an F-spin symmetric sdg-IBM-2 Hamiltonian. Majorana terms should be added as well, to adjust the energies of mixed-symmetry states. This process converts the Hamiltonian in Eq. 3.2 to the following:

H = c ^(1 - C)(nd„ + hdv + a(hg7t + ngJ)) - Q„ + Q„) • (Q* + Q„)

+A sdMsd + AS9MSS) (3.5)

Qp = [sjdp + d & p ) + m\~9p + g\dp]W (3.6)

32

ndp = s\sp ndp = d j • dp hgp = 5p • 3p (3.7)

= + nSv + nd„ + ndi/)(nSn + nSv + ndv + ndv)

+ 2 (nSvr + nsu + nd„ + ndu) + -(nSw - nSl/ + ndir - ndl/)

- - nsv + no, ~ ndv){nSir - ns„ + ndv - nd„)

+ (3% + dl • du)(slsn + dt • dn) (3.8)

Msg = ^(nSir + nav + n9ir + n9u)(n s-„ + nsi, + ngn + ) ! / x 1/ x

+ + + + + ~ n s " + n g « ~ n 9»>

~ \(ns„ - na„ + n97T - ngi/)(nSir - nSu + n9n - n9u)

+ (4S„ + g\ • gv) (4«7r + " 5tt) (3.9)

This Hamiltonian has 6 parameters, where one is simply an energy scale factor. The new

terms in the Hamiltonian are two Majorana operators, Msd and Msg, and they are con-

structed in much the same way that M was constructed in Chapter 2. The full Majorana

operator would be constructed with s, d, and g operators using the generators F+, F~, and F°:

F+ = s{sv + 4 • dv + g\ • gu

F~ = slsw + dl-d7r+gl-gn

F° = n8w - n,„ + ndir - ndv + n9n - n9„

M — + 1) - F+F~ — F°F° + F° (3.10) 2 2

where N is the total number of bosons in the system, and is a good quantum number. This

full Majorana operator shifts the energy of all F = Fm a x — 1 states equally, regardless of

the d and g-boson components of the wavefunctions. Adjusting mixed-symmetry states in

a way that depends on d and g-boson content is a useful option, so Msd are Msg are used

in place of M

The transition operators in the sdg-IBM-2 are similar to those in the sdg-IBM-1:

f (£2) = eB„Q* + eBvQv

f (Ml) = ^ + gvL„) (3.11)

33

where Qp is defined in the same way that it appears in Eq. 3.6. The operator Lp is defined

to be Lp = / I 0 K U P ] ( 1 ) + The effective boson charges of the E2 operator

are defined in such a way that e ^ = 0 and e ^ is scaled to reproduce the experimental

B(E2;2f —> 0+). If the E2 operator is split into (Qw + Q„) and (Qn — Qv), one finds

that the E2 transition strength from 2 " to 0+ depends on the (Qn + Qv) term. The E2

transition strength from the mixed-symmetry 2+ state to 0j" depends on the (Qn — Qu)

term, so by comparing the experimental values of those two transitions, one can solve for

both eBT, and eB„- As will be seen later on, the ratio is already reproduced

well with esu = 0, so that boson charge will be left at 0.

The Ml operator behaves the same way it did in the sd-IBM-2. The (Lv + Lv)

component of the operator is diagonal in the Hamiltonian and does not contribute to

transitions between states. The (Ln — Lv) component allows large Ml transition strengths

between states of different F-spin. As discussed in Section 3.1, the g-factors for d and

g-bosons are set equal, so there will be no Ml transitions between states of F = Fmax.

The proton and neutron g-factors are set to gn = 1 and gu = 0.

There is the possibility of large E4 transitions from 4+ states to 0 + states due to

the hexadecapole degree of freedom, so it is worth defining the E4 operator. Due to a lack

of E4 data in 94Mo, E4 transition strengths will only be considered once a final fit for the

nucleus is found.

T(£4) = evi[sl~g^ + g l ~ s ^ + e„ 2 [d \d„p + e ^ f e + «/R]{ 4 ) + e^[g lg n }^

+ [atgv + gtavp + e„2 [4d„](4> + e„3 [d% + gfa]™ + e„4 \gtgvp (3.12)

The 8 parameters in this operator seem excessive, and some assumptions can be made

to simplify it. As was seen in the E2 operator, the effective boson charge for the proton

boson contributes the most to the E2 transition strengths. As a first test, the neutron

parameters can be set to 0, which leaves four parameters. Any further assumptions should

be based on the relative s, d, and g-boson content of the ground state, as the E4 transition

strengths that will be considered are B(S4;4+ —> 0*). This will be discussed further in

Section 3.5.

3.3 Calculations in the sdg-IBM-2

Enumerating the basis states in the sdg-IBM-2 is very similar to how basis states

in the sd-IBM-2 were enumerated. The difference is that the Up(5) d-boson basis and

34

the Up(9) g-boson basis are not entirely independent. The total number of proton and

neutron bosons must be the same for each basis state, so when the Up(5) and Up(9) bases

are coupled together, the Nn and quantum numbers must be enforced. The total

proton basis and total neutron basis are then coupled together in the same way that they

were in in the sd-IBM-2, with LV and L„ coupling into the total angular momentum L.

All calculations that follow in this Chapter were performed in a new code called

ArbModel that was recently developed by S. Heinze [37]. The software is actually much

more general than the sdg-IBM-2, allowing bosons of many different spins, and many

types. Optimizing a code this general is extremely difficult however, so calculations with

a large boson space can take a long time. The calculations here for 94Mo assume Nn = 4

and N„ = 1, with no limit on the number of g-bosons aside from the total number of

bosons in the calculation.

3.4 Fitting 94Mo using the sdg-IBM-2

The Hamiltonian and operators that will be used in these calculations for 94Mo

has already been described in detail, but it is worth considering one of the dynamical

symmetries of the model first. The symmetry that was first considered is 0^+^(15), which

has the following group chain:

U„(15) ® U„(15) D U t + v(15) D 0 ^ ( 1 5 ) D 0W+I/(14) D 0T + I /(5) D 0 ^ ( 3 ) (3.13)

The Hamiltonian that describes this symmetry in terms of Casimir operators is the fol-

lowing:

H = AI(%(LF7R+Y(15))+A2C2(07R+(/(15))+A302(0

(3.14)

Such a Hamiltonian is difficult to understand in a physically intuitive way. The

C^UN+VQS)) operator acts as a Majorana operator, and C2{ON+L/(3)) refers to the L • L

term that appears in multipole Hamiltonians, but the three other terms are more difficult

to understand.

When calculations are performed with this Hamiltonian, nearly all low-lying

states have significant g-boson content, due to strong interactions between s, d, and g-

bosons. It is possible to adjust the parameters in a way that makes the lowest state of

each spin have approximately the experimental energy, but the level distribution much

35

too dense, and strong Ml transitions occur between states in a way that has not been

observed experimentally. Overall, this Hamiltonian is difficult to work with and does not

appear to describe 94Mo. The large g-boson content in all low-lying states contradicts the

seemingly vibrational structure of 94Mo. The \Jn+l/(lA) symmetry could also be consid-

ered, but it seems worthwhile to just use the multipole Hamiltonian defined in Eq. 3.5

with parameters that remove the quadrupole interaction.

3.4.1 Vibrational fit for 9 4 Mo

To create a vibrational fit for 94Mo, the quadrupole interaction in Eq. 3.5 should

be removed. By setting the parameter £ to 0, the Hamiltonian becomes the following:

H = c (ndic + ndv + a(nSir + hgu) + \adMsd + XsgMs^j (3.15)

The parameter c is an overall scale factor, and directly controls the energy of the d-bosons. The energy of the g-bosons is ca, and using the analogy of two identical fermions coupling in the shell model, the L = 4 coupling is expected to have a larger energy than the L = 2 coupling. The Majorana operators represent a proton-neutron interaction, which was shown in the example calculation in Section 2.2.4. The operator Msd specifically refers to an interaction between proton and neutron d-bosons, and the Msg refers to an interaction between proton and neutron g-bosons.

The fit can be seen in Fig. 3.2, and uses the following parameters: c = 0.871 MeV, a = 1.8, Xsd — 0.16, and \ d g — 0.25. The effective proton boson charge was set to ea^ = 2.19 VW.u.. The Hamiltonian has no interactions between d and g-bosons, and vibrational structure of the two types of bosons is apparent. The first 4+ state is the symmetric g-boson state. Directly above that is the two d-boson 4+ states, at the same energy as the 0+ and 2+ states. Then there is the one g-boson 4+ mixed-symmetry state. At 2.44 MeV are five symmetric states formed from one g-boson and one d-boson. At 2.61 MeV are the three d-boson symmetric states.

The Ml transition from the 2+ mixed-symmetry state had a strength of B(M 1; 2J -> 2 f ) = 0.47 /j.%, which is close to the experimental value of 0.56 (J?N. The Ml transition from the 4+ mixed-symmetry state had a strength of B(M 1; —> 4]*") = 1.56 jj?N) which is larger than the experimental value of 1.23 but indicates that the inclusion of g-bosons might reproduce the strong Ml of interest in 94Mo.

The fit to 94MO in Fig. 3.2 clearly does not match the experimental energy levels of the nucleus very well, but the details are instructive. For example, the state is

36

9 4 M O M1 Transitions Vibrational

94Mo E2 Transitions Vibrational

Experiment sdg-IBM-2

Figure 3.2: sdg-IBM-2 fit to 94Mo using vibrational Hamiltonian. The Hamiltonian is from Eq. 3.15, and the parameters are c = 0.871 MeV, a = 1.8, Xsd = 0.16, Xdg = 0.25, eBw = 2.19 VW.u., and (3 = 1. Dashed transitions represent upper bounds for the transition strengths. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

37

constructed from two d-bosons, and no g-bosons. The 4+ and 4^ both have one g-boson

and no d-bosons, so the strong Ml of interest is between the mixed-symmetry one g-boson

4 + state and the symmetric one g-boson 4+ state. Therefore, the two d-boson 4j" will need

to be pushed higher in energy in order to describe 94Mo. This can be done by using an

d-g interaction term, which is available in the Q operator from Eq. 3.5.

Discussing the E2 transition strengths is complicated, because the E2 operator

has an extra parameter of its own, (3. This parameter is contained in the Q operator, and

the energy fit does not help select an appropriate value. If the d-g interaction in the E2

operator is disabled, no E2 transition occurs between 4* and 2^ due to the boson content

of those states. When a quadrupole term is included in the Hamiltonian, the (3 from the

Hamiltonian can be used in the transition operator, but for this case (3 = 1 was chosen as

a compromise between s-d and the d-g interactions. It can be seen in Fig. 3.2 that the E2

transition from and 2f is still much too weak, but a d-g interaction in the Hamiltonian

would help resolve this. The next Hamiltonian choice will use the quadrupole term rather

than the vibrational terms.

3.4.2 Deformed gamma-soft fit for 9 4 Mo

The next choice emphasizes the quadrupole term of Hamiltonian from Eq. 3.5,

by setting the parameter £ = 1. This results in the following Hamiltonian:

H = C + Qv) • (Qn + Qv) + AsdMsd + XsgMsg^j (3.16)

Qp = \s\dp + djgp](2) + (3[d% + g\dp]W (3.17)

The quadrupole term geometrically corresponds to deformation and rotational motion,

in contrast to the boson number operators in previous Hamiltonian corresponding to vi-

brational motion. The Majorana operators have the same behavior as they did in the

vibrational Hamiltonian. The parameter c is a scale factor for the energy of the states,

and (3 defines the strength of the d-g interaction in the quadrupole operator. As with

the ()„•+„ (6) fit from Chapter 2, the energies will be scaled so that the calculated 4+

energy matches the experimental value. This can be misleading, and 2+ is a more conven-

tional choice, but this is done to make it easier to compare the calculated results to the

experimental values in the figure.

The fit can be seen in Fig. 3.3, and uses the following parameters: c = 8.007

MeV, (3 = 0.39, Xsd = —0.0115, Xdg = —0.0010. The effective proton boson charge was set

38

94 Mo M1 Transitions Gamma-soft

7 ~ _ 3

1

0+ 1+ 2+ 3+ 4+ 5+ 6+ 0 0+ 1+ 2+ 3+ 4+ 5+ 6+

94Mo E2 Transitions Gamma-soft

2

1

0

/-X • il '"/

0+ 1+ 2+ 3+ 4+ 5+ 6+

Experiment 0+ 1+ 2+ 3+ 4+ 5+ 6+

sdg-IBM-2

Figure 3.3: sdg-IBM-2 fit to 94Mo using deformed gamma-soft Hamiltonian. The Hamil-tonian is from Eq. 3.16, and the parameters are c = 8.007 MeV, (3 = 0.39, Xsd = —0.0115, Adg = —0.0010, and = 1-63 VW.u.. Dashed transitions represent upper bounds for the transition strengths. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

39

eB-n = 1-63 \/W.u.. The second 0 + state appears at much to high an energy, which is a

serious problem with this fit. The 1 + and 3+ states are too low, and the 6+ appears slightly

too high in energy. Besides these discrepancies, the relative ordering of the symmetric and

mixed-symmetry 2 + and 4 + states appears to be reasonable.

The Ml transition from to 2+ has a strength of 0.68 which is close to

the experimental value of 0.57 /i2N. In the vibrational fit, this Ml was slightly too small

which indicates that somewhere between these two fits could have the correct transition

strength. The Ml transition from 4J to has a strength of 0.56 /i2N) which is smaller

than the experimental value of 1.23 In the vibrational fit, this Ml strength was too

large which again indicates that somewhere between these two fits could have the correct

transition strength.

The sheer number of E2 and Ml transitions that appear in the calculation seem

too large, but it is important to keep in mind that even if a transition strength has not

been identified experimentally, it could still exist. The measurement of these transition

strengths require information about the states and transitions that can be difficult to

measure. Without a lifetime for the state, and multipole mixing ratios for the transitions,

the transition strengths cannot be calculated, and such transitions are left out of the fit.

Basically, the spiderweb appearance of the calculated E2 and Ml transition strengths in

the fit should not automatically be cause for concern. If the transition strengths deviate

significantly from experimentally known values, then that is a shortcoming of the fit.

The E2 transition from the 4 + to 2+ state is stronger than that found in the

vibrational fit, which confirms that the interaction term resolved the issue with relative

boson content of those states. The interaction term may have been too strong though,

as the spreading of g-boson content to other 4+ states seemed to have decreased the

Ml strength of interest. It is important to note that unlike the 0^+^(15) symmetry

that was tested first, the ground state in this calculation has very little g-boson content.

The quadrupole interaction did give it a moderate d-boson component, but without a

Hamiltonian term that couples to angular momentum 4, it is difficult to mix in g-bosons.

The deformed gamma-soft calculation using the Hamiltonian from Eq. 3.16 saw

several improvements over the vibrational fit, but seemed to cause some problems as well.

By looking at the Ml transitions and level energies, it seems as though a Hamiltonian

that spans the space between vibrational and deformed gamma-soft would be useful. The

logical next step is be to go back to the general Hamiltonian from Eq. 3.5.

40

3.4.3 Transitional fit for 9 4 Mo

Now that the features of the vibrational and deformed gamma-soft Hamiltonians

are better understood, the transitional Hamiltonian from Eq. 3.5 will be used to create a

final fit for 94Mo.

H = c({ I - C )(hd„ + hdv + a(ng„ + hgJ)) - ^{Q* + Q„) • (Qn + Qv)

+A sdMsd + KgMsg) (3.18)

Qp = [8\dp + Spsp}™ + 0[d% + g\dp\^ (3.19)

The terms all represent what they did in the vibrational and deformed gamma-soft Hamil-tonians, but now the parameter £ will allow the Hamiltonian to transition between those two limits. There are six-parameters in total, and one is simply a scale factor. The factor of 4N in front of the quadrupole term scales the quadrupole interaction such that the transition from vibrational to rotational motion occurs at approximately £ = 0.5. With-out this factor, that transition point would depend on the total number of bosons in the system, but with it, £ can be used as a gauge on the general structure of the fit.

The final fit for 94Mo can be seen in Figs. 3.4 and 3.5. Fig. 3.4 shows the cal-culated results in the same form that has been used previously. Fig. 3.5 only shows tran-sitions that were observed experimentally. Determining E2 and Ml transition strengths experimentally can be difficult, as they depend on values like the lifetime of the states, and the multipolarity of the transitions. If a transition strength is not known, it does not necessarily mean it is extremely small, just that it has not been determined. Transition strengths that were observed to be very small are compared to the calculated results in Fig. 3.5.

The overall scale factor was c = 3.534 MeV, and was determined by scaling the calculated results so that the 2f energy matched the experimental value. The parameter £ was found to be 0.64, which shows that both the vibrational and quadrupole terms in the Hamiltonian played an important role. The Majorana parameters were set to Xsd = 0.050 and \ d g = 0.016. The g-boson relative energy parameter a is 1.40, which is a reasonable value. The d-g interaction term in the quadrupole operator has the parameter /? = 1.86, which seems a bit large when compared to the SU7r-(-£/(3) limit of the sdg-IBM-2, where the ratio of the d-g component to the s-d component of Q is approximately 1.29 [33]. However, it is difficult to relate a symmetry like that to Eq. 3.18, because the Hamiltonian is simply

41

94Mo M1 Transitions Transitional

94Mo E2 Transitions Transitional

Experiment sdg-IBM-2

Figure 3.4: sdg-IBM-2 final fit to 94Mo using transitional Hamiltonian. All calculated Ml and E2 transitions between visible states are shown. The Hamiltonian is from Eq. 3.18, and the parameters are c = 3.534 MeV, C = 0.64, a = 1.40, 0 = 1.86, \ s d = 0.016, Xdg = 0.050, and e ^ = 1.83 \/W.u.. Dashed transitions represent upper bounds for the transition strengths. The Oj" and 0^ states were at very similar energies, and have been artifically split for clarity. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

42

94 Mo

3

2

1

0

M1 Transitions Transitional"

7" 3

1

0 0+ 1+ 2+ 3+ 4+ 5+ 6+ 0+ 1+ 2+ 3+ 4+ 5+ 6+

9 4 M O E2 Transitions Transitional*

3

2 N \ / / • /

0 1 0 0+ 1+ 2+ 3+ 4+ 5+ 6+ 0+ 1+ 2+ 3+ 4+ 5+ 6+

Experiment sdg-IBM-2

Figure 3.5: sdg-IBM-2 final fit to 94Mo using transitional Hamiltonian. The asterisk indicates that only the calculated Ml and E2 transitions that were observed experimentally are shown. The Hamiltonian is from Eq. 3.18, and the parameters are c = 3.534 MeV, C = 0.64, a = 1.40, 0 = 1.86, \ s d = 0.016, Xdg = 0.050, and eBn = 1-83 VW.u.. Dashed transitions represent upper bounds for the transition strengths. The O3" and O4 states were at very similar energies, and have been artifically split for clarity. Symmetric states are magenta, and mixed-symmetry states are green. Additional details about this figure can be found in the text. 94Mo data adapted from a summary paper [26].

43

a generalized version of the sd-IBM-1 Hamiltonian from Eq. 2.2, and is not necessarily a

Hamiltonian that spans the symmetries of the 11^+^(15) group.

Looking at Fig. 3.5, the overall energy distribution of the states is a significant

improvement over the the vibrational and deformed gamma-soft fits. The O3" and O4" states

were at very similar energies during the fit, and were artifically split to make the source

of transitions clearer. There appears to be a missing 2+ state in the calculation which

corresponds to the 24 state in the experiment, but it may also be due to the higher lying

2+ states being pushed too high in energy by the s-d interaction in the quadrupole term.

The energies of the O2", O3", and seem to support this second possibility, as they appear

too high in energy as well. The calculated 1+ state appears at a similar energy to the

l * experimental state, but that experimental state cannot easily be described by a boson

model like the sdg-IBM-2. The calculated 1+ state should instead be compared to the

experimental state, which is higher in energy.

The 3 + and 4 + states look good, although the energies of the 4 + states were

given a high priority during the fit. The calculated 4 " state is strongly mixed with the 4+

state due to the strong d-g interaction in the quadrupole term. If the d-g interaction is set

to j3 = 0, the is at a similar energy to 2^, and has a large d-boson component in the

wavefunction, while the 4 f state has a large g-boson component. A large d-g interaction

pushes those two states apart and mixed the d-g components of the two states. With the

chosen Hamiltonian, a large interaction like this appears to be the only way to push the

state up in energy.

The two Ml transitions of particular interest look excellent. The transition from

2g" to has a strength of 0.55 fi^ which is very similar to the experimental value of

0.56 /i^f. The transition from to 4^ has a strength of 0.98 fi]y which is similar to the

experimental value of 1.23 fj?N. The experimental value of 1.23 fi2N has an uncertainty of

0.20 p?N, so the calculated value is almost within the uncertainty of the experiment value.

The large d-g interaction that pushed the 4+ and 4;j" states apart effectively reduced

the transition strength between 4 j and 4+, because the g-boson content of the state

decreased. If the d-g interaction is reduced, the transition becomes stronger, but the

general energy fit becomes worse.

The calculated E2 transition from 4+ to 2f has a strength of 25.8 W.u., which

is in good agreement with the experimental value of 26.0 W.u.. As was seen in the

vibrational fit, the E2 strength 4 " to 2+ depends on the d-g interaction in Q, so this

44

agreement supports the large d-g interaction in the fit. The E2 transition from 2J to

was calculated to be 23.8 W.u. which seems like a significant deviation from the

experimental value of W.u., although the uncertainty of the experimental value is

large.

The most clear deviation of E2 strengths is the transition from Ot to which

has an experimental value of 104^1 W.u.. The calculated strength is 26 W.u. which is a

factor of 4 smaller, and not a good match even when considering the uncertainty. There

is no clear way to have l ^ l l > 6 in the sdg-IBM-2 using a similar Hamiltonian. In O\HJ2;2j —>0 )

an N=5, sd-IBM-1, U(5) system, ) 'J.—-M = 1.8, although as N —* oo, the ratio goes J D y r / Z f l ^ — • U ^ J

to 3. Even over-inflating the d-g interaction in the E2 operator would not help because

the g-boson component of the O3 state is not very large.

A table with a comparison of experimental and calculated levels is shown in

Table 3.2. The calculated s, d, and g-boson wavefunction components are included. A

comparison of experimental and calculated E2 transition strengths is shown in Table 3.3.

A comparison of experimental and calculated Ml transition strengths is shown in Table

3.4.

In Tables 3.3 and 3.4, there are several cases where small transitions were observed

experimentally, but the sdg-IBM-2 calculation had a transition strength of 0. In many of

these instances, the discrepancy can be explained in terms of F-spin mixing. To keep

calculations simple, and to minimize the number of parameters, the Hamiltonians chosen

have had F-spin as a good quantum number. This does not necessarily happen in nature,

and proton-neutron symmetry can get mixed between states of similar energy. As an

example, the experimental Ml transition of interest is B(M 1; —> 4*) = 1.23 n2N. The

Ml transition strength from the 43" is —> 4+) = 0.23 (j?N. This nonzero value

is likely due to F-spin mixing between the and 4^ states. The calculated Ml strength

from is 0, but if an F-spin mixing was introduced to the Hamiltonian, the transition

would fragment from the 4^ to Af transition.

This situation can also be seen with the 2 \ to 2f Ml transition. Due to F-spin

symmetry, this transition was not observed in the sdg-IBM-2, but in the experiment an

Ml of 0.026tooi6 y N w a s observed. F-spin mixing between 2+ states likely caused the

fragmentation. This affects E2 transition strengths as well, and the E2 from 3^ to is

an example of it. The experimentally observed transition strength of 0.91J'g W.u. is not

seen in the calculation but can be explained with this effect.

45

Table 3.2: Comparison of 94Mo levels with sdg-IBM-2 calculation using transitional Hamil-

tonian. s, d, and g-boson calculated wavefunction components are included. 94Mo data

adapted from a summary paper [26].

Energy (MeV) Components

Levels Calculated Experiment ns rid, ng

o t 0 . 0 0 0 0 . 0 0 0 4.8 0.2 0.0

0+ 2.073 1.742 3.0 1.9 0.1

03+ 3.205 2.781 1.9 3.0 0.1

0? 3.308 2.8 1.8 0.4

11 2.987 3.129a 2.7 1.9 0.4

21 0.871 0.871 3.8 1.2 0.0 2 + 2 1.974 1.864 2.8 2.1 0.1 2 + 3 2.239 2.067 3.7 1.1 0.2 2 + 4 2.953 2.393 2.7 1.6 0.7 2 + 5 3.323 2.870 2.8 1.9 0.3

2e+ 3.379 2.993 3.0 1.1 0.8

3 f 2.849 2.805 2.6 1.7 0.7

3 2 + 3.107 2.965 2.8 2.0 0.2

CO

3.193 3.082b 2.8 1.2 1.0

1.594 1.573 3.3 1.2 0.5

2.263 2.295 3.8 0.2 1.0

2.481 2.564 3.4 1.0 0.5

44+ 2.724 2.768 2.6 1.7 0.7

3.226 3.201c 2.9 1.1 1.0

3.137 3.243b 2.8 1.2 1.0

5+ 3.183 2.9 1.1 1.0

2.453 2.423 2.6 1.7 0.7

3.227 2.872 2.9 1.2 1.0

"Mixed-symmetry 1 + state listed, rather than bSpin and parity tentative, with parity listed. cSpin tentative, no parity listed.

46

Table 3.3: Comparison of E2 transition strengths from 94Mo with sdg-IBM-2 calculation

using transitional Hamiltonian. E2 transitions strengths are given in units of W.u.. 94Mo

data adapted from a summary paper [26].

Transition Calculated Experiment

B(E2 B{E2 B{E2 B(E2 B{E2 B(E2 B{E2 B{E2 B(E2 B(E2 B{E2 B(E2

o3+-

1+ ' - m s

1+ ms

1+

2 + -

2+ _ 2

2 + -2

2 + -3

2+ _ 5

2 + -5

2f)

Of Of 2f Of 2f 2f 2+ 2+

•2J

2+ 2+ 2

2+ 3

£(£2 ;3 f

B(£2; 3f

B(E2- 3+

B(E2-,3$ B(E2;3+

B(E2-,3} B(E2\ 4+

B(E2-4+ B{E2-4+ B(E2~, —> 2f

B(£2 ;6+-+4+

£(£2; 6+ 6f

4f 2f 2f 2f

26 .12

1 .88

0 19.64

15.40

0 23.75 2.25 1.56

0

17.10

0

17.84

7.83

2.41

0

14.79

0

25.75

1.37

2.06

0 31.67

0

104±a 0.721^1

0.99±g;g

< 27

15.5(2)

0.33(11)

2 . 2 0 ( 2 1 )

0 . 4 0 ^ g j 4

o.4of° :9

< 140

0.22(7)

0.55(10) 19(3)

9.2(13) n Q+0-5

0 9+1'6

58(14)

9.5^23

5

0-1418:8

26.0 t t i

5-9i H

1.1(3)

4-9^5

< 160

< 34

47

Table 3.4: Comparison of Ml transition strengths from 94Mo with sdg-IBM-2 calculation

using transitional Hamiltonian. Ml transitions strengths are given in units of [i2N. 94Mo

data adapted from a summary paper [26].

Transition Calculated Experiment

B(M 1 B(M1 B(Ml B(M1 B(M1 B(M1 B(Ml B{M1 B(M1 B(Ml B(Ml B(M1

1+ xms

1+

1 + '-•ms

1 +

2 + -2

2 + -3

2 + -5

2 + -5

2 + -

3+ -2

3+ -"2 3+ -2

->2+)

-23+)

2?) 2 f )

2?) 2^) 23+)

2 f )

2^) 23+)

B(Ml;3+->4+) B(Afl;4+-*4+) B(Ml;4+-»4+) B(Ml;6+->6+)

0.05

0 0.84

0 0

0.55

0 0.19

0 0

0.42

0

0.14 0.98

0 1.17

0.16(1)

0.012(3)

0.44(3)

< 0.05 0 02fi+0041 u.uzo_0 0 1 6

0.56(5) fl 0017+0 0010 u-uui'-0.0012

0.27(3)

< 0.16

o o n f i + 0 0 0 3 U.UUO_o 004

0.24(3)

0 021+0016

0.09(2)

0.075(10)

1.23(20)

0.23(6)

< 0.62

3.5 E4 transition strengths

The possibility of calculating E4 transition strengths was mentioned in Section

3.2. The E4 operator in sdg-IBM-2 has eight parameters, but after setting the effective

neutron boson charges to 0, only four parameters remain:

f (£4) = e7ri [s% + + eW2 [4<k](4) + e ^ [d% + + e7r4 [<4^](4) (3-20)

The E4 transitions from 4+ to are the least difficult to determine experimentally. By focusing on those transitions specifically, some assumptions can be made about which

48

State Energy Components

(MeV) ns nd ng

of 0.000 4.8 0.2 0.0

4f 1.594 3.3 1.2 0.5

4f 2.263 3.8 0.2 1.0

43+ 2.481 3.4 1.0 0.5

4f 2.724 2.6 1.7 0.7

4 3.226 2.9 1.1 1.0

Table 3.5: Boson composition in wavefunctions of Of and 4+ states from sdg-IBM-2 calculation.

Transition E4 Strength

5 ( £ 4 ; 4 f -- o f ) 1.540

B ( £ 4 ; 4 + - ->of) 0.711

B(E4-, 4+ -- o f ) 1.503

B(E4-, 4+ -- o f ) 0.000

5 ( £ 4 ; 4 + - - o f ) 0.000

Table 3.6: Calculated E4 transition strengths for 94Mo using the sdg-IBM-2. Only the [sign + g l s T r ] t e r m is included in the E4 operator. Transition strength is given in W.u., although the effective boson charge is arbitrary.

terms in the operator will actually affect the matrix element. The boson composition of

the Of and 4 + states from the final 94Mo fit can be seen in Table 3.5. All of the states

shown are mostly constructed from s-bosons, but the ground state in particular is 96%

s-bosons, and 4% d-bosons. There is not a large g-boson component in the ground state

because the Hamiltonian did not have a term with s and g-bosons coupling to angular

momentum 4.

With a 96% s-boson component in the ground state, the only term that will

significantly contribute to the transition matrix element is [s^g-n+g^s^]^. For the simplest

approximation, the parameters for all other terms can be set to 0. Without experimental

E4 transition strengths to compare to, it is difficult to select the effective boson charge,

49

0.5 1.0 1.5 2.0 H

Figure 3.6: Plot of E4 transitions strengths from 4 + states as a function of the d-g inter-action parameter (3.

so a default value of e^, = l v W .u. has been chosen. Using the final fit for 94Mo, the

calculated E4 transition strengths are shown in Table 3.6.

It is clear that B(EA\Af —• Of) is more than twice as large as B(E4\ 4^ —> 0j").

The E4 operator as defined is an interaction between s-boson components and g-boson

components of two states. Due to the strong d-g interaction between 4^ and 4^, the

d and g-boson components of those two states are strongly mixed. The spread of g-

boson components is what fragments the E4 strength between the two states. If the d-g

interaction is reduced by setting (3 to 0.93, rather than its final fit value of f3 = 1.86, the

E4 transition strengths in order of 4+ states become 2.33, 0.76, and 0.74 W.u. rather

than 1.54, 0.71, and 1.50 W.u. for the first three transitions. This shows that the relative

strengths of the B(E4\ 4^ —> 0+) and B(E4;4f —> 0^) can provide information about

the d-g interaction in Q. A plot of these two transition strengths as a function of the

d-g interaction parameter (3 can be seen in Fig. 3.6. Once experimental values for these

E4 transition strengths become available, they can be used to help better understand the

interactions in 94Mo.

Due to there being some d-boson component in the ground state wavefunction,

the [d+dn](4) does play a role in E4 transition strengths, but an extra parameter unfor-

tunately clouds the discussion about physical information that can be drawn from the

50

experimental values. When en2 is positive and large, the net effect is that the difference

B(EA; —» Of) — B(E4] 4^ —> Of) remains large even with a large f3. When e7T2 is nega-

tive and large enough however, the reverse happens. It is not clear what should be done

with this second term of the E4 operator.

3.6 Discussion

The purpose of this investigation was to better understand the interactions and

degrees of freedom that play a role in the nucleus 94Mo. The energy of the low-lying

4 + mixed-symmetry state, as well as the strong Ml transition from 4f to 4f were not

described well using the sd-IBM-2, and shell model calculations seemed to indicate that

the hexadecapole degree of freedom plays a role in those first two 4+ states. This was

applied to the IBM-2 by including g-bosons in the model space.

First fits using vibrational and deformed gamma-soft Hamiltonians did not de-

scribe the energy spacing in 94Mo, but both had features that were promising. The final

fit used a Hamiltonian that spanned a transitional path between these two limits, and it

described 94Mo well both in energies and transition strengths. The 2g~ to 2f Ml transition

had the correct strength, even though the Ml transition operator was free of parameters.

The 4f to 4f Ml transition, which was the transition that motivated the investigation,

was similarly well reproduced, and was almost within the uncertainty of the experimental

value.

The g-boson component of the 4f state was mixed into the 4g" state, which

fragmented some of the Ml transition strength between these 4 + states. The fragmentation

was caused by the d-g interaction in the quadrupole operator Q, and whether the large

strength of that interaction is appropriate remains an open question. The d-g interaction

also affects the distribution of the E4 transition strengths from the 4+ states to the ground

state, and experimental values for those transitions could provide further insight into that

interaction.

Regardless of the verdict on the d-g interaction, the calculations indicate that E4

transitions should occur between the 4 + and Of states, and also that the E4 transition

strength from the symmetric 4+ state should be larger than the transition strength from

the mixed-symmetry 4 + state. If the d-d interaction in the the E4 operator is large and

negative, then the roles of the 4f and 4 : | symmetric states could reverse, but there is no

clear physical reason for this to occur.

51

The shell model calculations justified the inclusion of the hexadecapole degree of

freedom in the IBM-2, and calculations in the sdg-IBM-2 using the six-parameter transi-

tional Hamiltonian has resulted in the best fit for 94Mo that has ever been found using an

interacting boson model. There were strong interactions between the d and g-bosons in

the final Hamiltonian, so it was not simply a matter of inserting a missing state of interest

using a higher-spin boson. The two d-boson state really did appear at a much higher

energy than it would in a vibrational sd-IBM-2 Hamiltonian, so this new degree of freedom

along with the d-g interaction successfully reproduced many features of 94Mo beyond the

strong Ml between the first two 4+ states.

With one successful example of the hexadecapole degree of freedom playing a

role in a low-lying mixed-symmetry state, it makes sense to apply the model to other

collective nuclei with vibrational structure. The experimental search for similar behavior

among 4 + states in weakly collective vibrational nuclei is the topic of Chapter 4. Looking

at vibrational nuclei is a good first step, as the shell model can be applied to such nuclei,

and hexadecapole excitations can be tested for directly. If more examples of g-boson

content in low-lying 4 + state are found, and the sdg-IBM-2 is successful in describing the

structure of those nuclei, then the model could be applied to more collective nuclei that

are difficult to describe using the shell model. The hexadecapole degree of freedom has

been observed to play a role in a low-lying mixed-symmetry state, and it will be interesting

to see how prevalent such behavior is, and whether it will be observed in nuclei that are

more collective than 94Mo.

52

Part III

The search for 4 +

mixed-symmetry states in 1 4 0 N d

53

Chapter 4

Experimental techniques

4.1 Selecting a nucleus near N = 8 2

In the discussion of the sdg-IBM-2 from the previous chapter, it was noted that

the strong Ml transition between the and 4f states in 94Mo was an indication of

possible hexadecapole content in those two 4+ states. The strong Ml transition alone was

not enough to confirm the presence of g-bosons, but by examining the entire set of low-

lying states and transitions using the sdg-IBM-2, it became clear that the model described 94Mo very well. The significant improvement in model fits that resulted from including

g-bosons in the IBM-2 showed that the hexadecapole degree of freedom does in fact play

a role in some low-lying nuclei near the N=50 shell closure.

With this confirmation of g-boson content in low-lying states near the N=50

shell closure, the question arises of how applicable the model is to other mass regions

and shell closures. To continue this investigation, the decision was made to examine a

nucleus near the N=82 shell closure. To keep the search as analogous as possible to the 94Mo investigation, only nuclei with N=80 were considered, as they are 2 neutrons away

from the closed shell. The three nuclei 136Ba, 138Ce, and 140Nd all satisfy that basic

requirement, and are also easily accessible with the instruments available at the Wright

Nuclear Structure Laboratory (WNSL) at Yale University.

The two nuclei 136Ba [19] and 138Ce [28] have both been the subject of investiga-

tions into the fragmentation of the Ml transition strength from first 2+ mixed-symmetry

state to 2 + fully symmetric state. These nuclei are stable and the low-energy region of

their level schemes have been thoroughly studied. Unlike 136Ba and 138Ce, 140Nd has a

half-life of 3.37 days [38] and information about the low-energy region of the level scheme

is fairly sparse [39]. As such, a lot can be learned during a study of 140Nd, and it is the

nucleus that was chosen to continue the g-boson investigation.

54

z 61

60 r ^sl

140Nd 142Nd ,43Nd "4Nd

59 141 Pr

58 ,36Ce 138Ce 1 4 0 C e "2Ce

57 138La 139La

56 134Ba 135Ba 136Ba 137Ba 138 Ba

55 i 3 3 C s i 3 5 C s

N 78 79 80 81 82 83 84

Figure 4.1: Nuclear chart in the region around 140Nd. A neutron shell closure can be seen at N=82, with 136Ba, 138Ce, and 140Nd all having two less neutrons. 140Nd is 10 protons past the Z=50 proton shell closure.

55

4.1.1 Past experiments on 1 4 0Nd

Many previous experiments have provided information about the nucleus 140Nd.

Three electron capture experiments were performed using coincidence techniques, which

yielded level energies, transition energies, and spin assignments to a reasonable accuracy

[40, 41, 42], Unfortunately, the data from many of these experiments is decades old, and

technological limitations of the instruments that were available at the time may have left

many of the states and transitions of this nucleus hidden. Also, due to beta decay selection

rules, 4+ states were not well populated from electron capture in 140Pm.

Other experiments utilized two-neutron transfer reactions to populate states in 140Nd [43]. Although this technique can be used to identify states that might not be

populated in an electron capture experiment, it provides no information about the gamma-

ray transitions between the states. A third category of experiments to probe this nucleus

used a fusion-evaporation reaction with a 48Ca beam to populate higher-spin states [44].

Such an experiment provides gamma-ray transition information, but tends to bypass low-

spin states of higher energy than the ground-state band.

4.1.2 Producing 1 4 0Nd

One of the primary objectives of this experiment is to identify new low-lying 4+

states in 140Nd. In order to do this, the nucleus must be produced in a way that either

populates these states directly, or populate states which cascade through these states.

Electron capture is not an optimal choice, as it populates specific states, and the gamma

decay back to the ground state may miss the 4+ states of interest. Another option is

to use fusion-evaporation, but as mentioned in the previous section, if the mass of the

beam nucleus is too large, high-spin states will be populated. Therefore, to have the most

confidence that the 4+ of interest are populated, a compound-nucleus reaction using a

proton beam was selected.

The isotope 140Nd (Neodymium) has 60 protons, so the target for this experiment

was chosen to be 141 Pr (Praseodymium), which has a 100% natural abundance. The

cross-sections for various reaction channels were calculated using cascade [45], and a

beam energy of 16 MeV was chosen to select for the 2n channel. This means that the

highest cross-section occurred for the channel where the 141Pr fused with the proton and

evaporated two neutrons away (i.e. produced 140Nd). When an excited nucleus has a large

excitation energy, it de-excites by evaporating nucleons [46]. Once the excitation energy

56

drops below the particle binding energy, gamma emission becomes the primary mode of

de-excitation.

4.2 Experimental setup

The proton beam used to produce 140Nd was generated using the ESTU tandem

Van de Graaff accelerator in WNSL at Yale University. Tandem accelerators benefit from

two stages of acceleration, and for heavier ions, this results in much larger energies than

a single stage Van de Graaff accelerator can produce. The first step in the process is

to add an electron to the atom being used as the beam, using an alkali metal such as

cesium. Even hydrogen, which naturally has only one electron, can be negatively ionized

for a short time. The anion is accelerated towards the positive terminal in the middle of

the tank, which at WNSL can be charged up to 20 MV. It then passes through a carbon

stripping foil, which removes several electrons, and the cation then accelerates down the

reverse potential. The H - ions produced at the ion source for this experiment only have

two electrons, so those two electrons are removed in the carbon foil. Due to the two stage

acceleration, for a 16 MeV proton beam, the positive terminal only needs to be charged

to 8 MV.

As mentioned in Section 4.1.2, the target was made from 141Pr, and it had a

thickness of 1 mg/cm2. Proton beams of this energy tend to generate an abundance of

neutrons when they encounter most materials, as the Coulomb barrier is fairly low for

proton reactions. In order to prevent a large sudden flux of neutrons on the gamma-ray

detectors, the beam current was kept at 1.4 enA. The effect of neutron damage on the

gamma-ray detectors used in this experiment is addressed in Section 4.3.1. The 1 mg/cm2

target was thick enough to stop the product nuclei that were produced during the reaction.

The target frame was thicker than desired, but due to time constraints it was

one of the only options. After performing the experiment, it was discovered that the

target frame was likely an alloy that included 56Fe, 48Ti, 52Cr, and 58Ni. The interaction

of the halo of the proton beam with these isotopes resulted in some contamination, but

energy overlaps with 140Nd were not a serious problem. Due to gain shifting between the

time of the experiment and when a 152 Eu energy calibration was taken, it was unclear

how accurate the post-calibration energies were. Transitions from 56Fe and 48Ti provided

a post-calibration energy correction, which made the transition energies found during

the experiment much more reliable. One additional consequence of the thickness of the

57

target frame was that the in-plane detectors were shadowed by it. This required a minor

correction to the detector efficiencies of the detectors that were placed perpendicular to

the beam line.

4.2 .1 Y R A S T Ball detector array

To measure the gamma-rays emitted from 140Nd as it de-excites, the YRAST Ball detector array [47] at WNSL was used. The array consisted of 8 high-purity Germanium (HPGe) clover detectors, positioned at various angles around the target chamber. The name clover comes from the arrangement of 4 cylinders of germanium contained within the aluminum case of the detector. The clovers used at WNSL are segmented further into a total of 8 segments per detector, but that additional segmentation was not used in the analysis of this experiment.

There are mainly three ways that gamma-rays interact with matter: photoabsorp-tion, Compton scattering, and pair production. During photoabsorption, the energy from the gamma-ray is converted to the kinetic energy of the electron. In Compton scattering, some of the energy of a photon is transferred to the electron, but the photon scatters away with the remaining energy. During pair-production, gamma-rays with energy greater than 1.022 MeV interact with matter and produce an electron-positron pair. The gamma-ray then scatters away with 1.022 MeV less energy, and the positron eventually annihilates with another electron. High-purity Germanium detectors are excellent gamma-ray spec-trometers due to the small energy gap (~0.7 eV) between the valence and conduction bands. Electrons produced during the three processes described above interact with many other valence electrons and eventually generate a large number of electrons proportional to the energy of the gamma-ray. Having more charge carriers produced during an interaction results in a higher resolution, and the statistical nature of the charge collection results in a peak shape that is approximately gaussian.

During a Compton scattered event, there is a possibility that that the scattered gamma-ray will leave the HPGe crystal, and the measured energy will be smaller than the original gamma-ray. This can create a large background of useless data, and it is important to minimize this effect as much as possible. Having a larger crystal decreases the probability of an escaping gamma-ray, and having adjacent crystals linked together in what is called addback decreases the effect further. Addback is a process where the gamma-rays in two adjacent crystals are assumed to originate from the same gamma-ray,

58

Figure 4.2: Photograph of the YRAST Ball array, as used during this experiment. Nine clover detectors are shown in the image, but the fifth in-plane detector had signal problems and was removed during analysis. The additional angle groups including it would have had insufficient statistics for it to be useful.

59

and the energy is added together. In this experiment, the clovers in YRAST Ball were set

up for two-fold addback.

To minimize the Compon scattered background further, each clover was sur-

rounded by a Bismuth Germanate shield (BGO), which vetoes events when it detects

gamma-rays. The probability of a gamma-ray interacting with matter scales with Z4, and

bismuth has 83 electrons, compared to the 32 electrons in germanium. Even a few cen-

timeters of this BGO crystal has a large probability of detecting escaping gamma-rays.

Even with these preventative measures, a small number of Compton scattered events will

still be measured, and this issue is explored further with relation to gamma-ray spectrum

analysis in Section 4.3.4.

The detectors in the YRAST Ball array were positioned with four detectors per-

pendicular to the beam line (in-plane detectors), two at forward angles, and two at back-

ward angles. The four in-plane detectors had 90° separation. The forward and backward

angled detectors were positioned 41.5° away from the beam line.

4.2.2 Gamma-ray coincidence

As mentioned previously, 140Nd has a half-life of 3.37 days, which is extremely long relative to the lifetimes of its excited states. Due to this difference in lifetimes, it is very probable that the nucleus will de-excite down to its ground state via gamma emission, long before it beta decays into 140Pr. With the amount of energy remaining in the nucleus after the (p,2n) reaction, it is also very likely that several transitions are required in order for it to return to its ground state. The near-simultaneous cascade of gamma-rays that result from this de-excitation are extremely useful for deducing the energies of states in the level scheme, and also analyzing angular correlations for the purpose of determining spin assignments.

Energies acquired from HPGe gamma-ray detectors are typically sorted into a histogram, and the histogram representing the whole experiment is fit with gaussian func-tions to determine energies and the areas of various peaks. Without 7-7 coincidence data, the task of building up a level scheme is extremely difficult, as one is limited to a one-dimensional spectrum. With 7-7 coincidence enabled, a two-dimensional spectrum becomes available, which provides information about whether specific gamma-rays were detected in the same cascade as other gamma-rays. Higher order spectra are possible for higher multiplicity events, but the efficiency of the array quickly becomes small at higher

60

multiplicities. For example, if the efficiency of one detector is e, then the efficiency of

a 7-7-7 event will be e3. In special cases of hard-to-place transitions, 7-7-7 coincidence

was used in the analysis of this experiment, and the spectrum took the form of a three-

dimensional cube. However, the number of events was often so low that the technique was

only relevant for a few cases.

In situations where two transitions have very similar energies, and both occur in

the same nucleus, it becomes absolutely essential to have 7-7 coincidence data available

in order to place the transitions and levels properly. During analysis of a two-dimensional

spectrum, gates are placed on specific peaks in the one-dimensional projection of the

matrix. This gate includes peak channels that are summed into a peak spectrum, and

background channels which are summed into the background spectrum. When the back-

ground spectrum is appropriately subtracted from the peak spectrum, the spectrum that

remains only includes peaks that are coincident with the initial gated peak. By looking

in various gates around the level scheme, specific transitions can be isolated, and in most

cases, even transitions with the same energy can be distinguished. This occurred several

times in the analysis of 140Nd, which will be seen in Chapter 4. An example of this can

be seen in the low-energy region of the 140Nd level scheme in Fig. 4.3.

4.2.3 Angular correlation techniques

A nucleus at rest in a material with no external electric or magnetic fields has no preferred orientation. If a gamma-ray is emitted from such a nucleus, the angular distribution of that gamma-ray will be isotropic. Unlike the initial state of the nucleus, the nucleus after such a transition will have a preferred orientation, and one can define what is called a quantization axis. Depending on the environment the nucleus is in, it could maintain such an alignment for a long time, or it could rapidly de-orientate. The distribution during the emission of a second gamma-ray can depend strongly on the alignment and spin of the nucleus. The dependence of the angular distribution on the spin of the states involved allows one to use these distributions to deduce the spin assignments of those states [48].

In experiments where the initial state of the nucleus has no preferred orientation, the angular distribution is measured as a function of the angle between the two detected gamma-rays. In an in-beam experiment such as the one performed for this work, the reaction itself aligns the nucleus, and defines a quantization axis. This alignment manifests

61

2264.4 / 2221.6

Figure 4.3: Low-energy region of the 140Nd level scheme. A 1490 keV and 1491 keV transition can be seen from the 1490 keV and 2264 keV states respectively. Situations like this occur several times in 140Nd, and gamma coincidence techniques are used to isolate the transitions.

62

Group 01 02 0 N

1 90 41.5 90 8

2 90 90 90 8

3 90 90 180 4

4 41.5 90 -90 8

5 41.5 90 0 8

6 41.5 138.5 0 8

7 41.5 138.5 180 4

8 90 41.5 0 8

Table 4.1: YRAST Ball angle groups as defined by three angles. 8\ and 62 are the angles between the beam line and each detector, and <f> is the azimuthal angle between the two detectors. N is the number of pairs of detectors in the angle group

itself in the distribution of magnetic substates, and the additional information requires that

the angle groups be defined not only by the angle between the two gamma-rays, but also

with the angles between the gamma-rays and the beam axis. With this definition of angle

groups, the arrangement of detectors in YRAST Ball provides eight unique groups, which

are defined in Table 4.1.

The general directional correlation function for oriented states is

W{6ue2,$)= Y , BXl{Jl)Ax^{Xl)AX2{X2)HXlXX2{el62^) (4.1) A1AA2

where 9\ is the angle between the first gamma-ray and the beam axis, 62 is the angle

between the second gamma-ray and the beam axis, and $ is the azimuthal angle between

the two gamma-rays [48]. BXl(J\) represents the orientation of the initial nuclear state,

and is defined in terms of the distribution of magnetic substates of the nuclear state after

the reaction:

Ji J3Al(Ji) = y / z h + i Y ( - l ) J l - m ( J i m J i - m | A i O ) — • ^ (4.2)

m=-Ji j r e ^ m'=—Ji

The distribution of magnetic substates is assumed to be gaussian, and the width of the

gaussian is defined by the parameter a [49]. The directional distribution coefficient for the

63

first gamma transition is

= TTd? (LLJ2Ji) + 2SiFtXl(LL'W + 6lFj?Xi(L'L'J2Jij) (4.3)

where L and L' are the possible multipolarities of the transition, and £1 is the multipole

mixing ratio. The generalized F-coefficients are defined as

F^2*1 (LL' J2J\) = y/{2Ji + 1)(2 J2 + 1)(2L + 1)(2V + 1)(2A + l)(2Ai + 1)

J2 L Ji

X V ( 2 A T H ) ( - 1 ) L ' + A + A 2 + 1 ' L V A

" 1 - 1 0 J2 V Jx

A2 A Ai

(4.4)

The directional distribution coefficient for the second gamma transition is

AX2(X2) = ^ ^ (^(LLJ3J2) + 252F^{LL'J3J2) + 522F°^(L'L'J3J2)) . (4.5)

The angular function H\ 1 \ \ 2 (9I9 2$) is defined by

A' 2A + 1 ( A - 9 ) ! ( A 2 - 9 ) !

q>0

x Pa9(cos(0i))Pa

92 (cos(02)) cos(?$) (4.6)

where A, Ai, and A2 refer to the order of the Legendre polynomials in the angular de-

pendence of the overall directional correlation function. For this work, the definition of

the multipole mixing ratio 8 has two definitions, which depend on the difference of spin

in the initial and final spins, \J2 — J\\. If the difference is 0 or 1, <5 represents the ratio

jjjj. If the difference is 2, 8 represents the ratio For example, if a pure Ml transition

occurs between 4£ and 4 f , then <5 = 0. A pure E2 transition between 4+ and 2f would

also be defined by 8 = 0. The multipole mixing ratios are extremely important for this

work, as they determine whether a particular transition is predominantly E2 or Ml. This

information is essential when attempting to identify mixed-symmetry states.

The formulas given in Eqs. 4.1 through 4.6 assume that the gamma-rays are

detected with point detectors, and therefore that the angles 0\, 02, and <£ refer to the

angle that the gamma-ray was detected. In reality, gamma-ray detectors such as HPGe

have a finite size, and the correlation function should be integrated over all of the angles

where gamma-rays are accepted. A convenient way of dealing with this complication is

to multiply the directional distribution coefficients by attenuation coefficients that are

64

calculated for the specific detectors used. This is possible because the main effect of a detector with finite size is to blur out the angular distribution. This attenuation has the largest effect on the higher order Legendre polynomials, as they have the strongest angular dependence.

The formulas above have been implemented in an angular correlation fitting program called corleone [50, 51]. The attenuation coefficients were calculated using a program called qc, which were integrated into corleone by modifying the angular corre-lation functions in the source code. After being provided with the experimental data, as well as the three possible values for spins in a given cascade, corleone can fit the magnetic substate distribution of the initial state, and the multipole mixing ratios of the gamma transitions. By looking at the fit patterns, one can select the most likely spin assignments, alignments, and multipole mixing ratios.

An example angular correlation pattern can be seen in Fig. 4.4. In the upper pattern, the alignment parameter a is left free during the x2 minimization. Throughout the analysis of 140Nd, a was observed, and it was noted that at higher energy transitions, a had values as low as 2.6, and for the lowest energy transitions it had a maximum value of 3.1. This is due to gradual deorientation as the nucleus deexcites to lower energies. Using this range of appropriate a values, it is simple to restrict the angular correlation fit to realistic alignment. The lower plot in Fig. 4.4 shows an angular correlation pattern using this restriction. It is clear in both patterns that a 4+ —> 2+ —> 0+ spin assignment describes the 1028 keV and 774 keV cascade best, but in situations that are less clear, restricting a to reasonable values can be a useful tool.

4.2.4 Measuring branching ratios

As mentioned previously, coincidence techniques are useful for building up the a

level scheme of a nucleus, and for angular correlation studies. An additional observable

that can be directly obtained from coincidence measurements is the relative branching

ratios of transitions from a given state. The gamma-ray branching ratio represents the

number of events from a particular state transitioning via gamma emission to a lower

energy state, divided by the number of events from the most probable transition from that

state. Typically branching ratios are listed with the most probable decay branch scaled

to 100, with the less probable transitions scaled by the same amount.

The HPGe gamma-ray detectors have an energy-dependent efficiency. In order

65

1028 keV and 774 keV cascade

02 41.5 90 90 90 90 138.5 1 38.5 41.5 <p 90 90 180 -90 0 0 180 0

1028 keV and 774 keV cascade

e2 41.5 90 90 90 90 138.5 138.5 41.5 (p 90 90 180 -90 0 0 180 0

Figure 4.4: Example angular correlation pattern from 140Nd. The upper pattern allows the alignment parameter a to take any value. The lower pattern restricts a to realistic values, which for this experiment are between a = 2.6 and a = 3.1.

66

to gain information about the structure of the nucleus, such equipment dependent factors

must be removed. The Germanium efficiency can be fit reasonably well using the following

function:

e(x) = ae-blos(x+ce~dx> (4.7)

For 7-7 coincidence data like that used in angular correlations and branching ratio mea-surements, a simulated doubles efficiency is used, by simply multiplying two efficiencies together. To determine the branching ratios from a given state there are two standard techniques. The simplest technique is to gate on a strong transition feeding into the state of interest. Then the branching ratio is proportional to the area of the transitions of inter-est in the gated spectrum, after correcting for the efficiency of the detectors. A diagram of this can be seen in Fig. 4.5a. The branching ratios of B, C, and D are determined directly from the coincidences A-B, A-C, and A-D.

The second technique is necessary when there are no strong transitions feeding into the state of interest. In this method, the gates need to be placed below the transitions of interest. The levels that the transitions of interest feed into typically have transitions down to lower energy states. The gates are placed on those lower transitions and the area of the transitions of interest are measured. By summing these areas together, the branching ratios from the transitions are found. In cases where the secondary transition has a low energy, the areas must be corrected for internal conversion before being summed together. An example illustration of which transitions are used is shown in Fig. 4.5b. For transition A, coincidence A-C and A-D are combined, and for transition B, coincidence B-E and B-F are combined.

4.2.5 Internal conversion coefficients

When a nucleus transitions from an excited state to a lower energy excited state, the energy loss typically occurs via gamma-ray emission. However, there is a certain probability that rather than emitting a gamma-ray, one of the atomic electrons will interact with the nucleus, directly taking the transition energy minus the binding energy of the electron as kinetic energy. These electrons are called internal conversion electrons. At higher energies, the probability of this occurring relative to gamma emission is extremely small. At lower energies, this internal conversion process competes strongly with gamma emission, and corrections to the measured gamma events need to be considered in certain situations.

67

Figure 4.5: Gating techniques used when determining branching ratios. In a), a coincidence between a feeding transition and the transitions of interest are used to determine areas. In b), transitions coincident from below the transitions of interest must be summed together to calculate the areas of the upper transitions.

The gamma-ray branching ratio B1 simply represents the probability that a

gamma-ray will be emitted along one branch relative to another. With this definition,

no internal conversion corrections need to be made to B7 when it is measured by gating

from above. When gating from below, however, the different possible cascade paths are

added together to approximate the number of events from the higher lying transition.

Because of the possibility that a lower energy transition will occur via internal conversion,

this results in a certain number of events from the higher lying transition that will not

be included in the sum, and this needs to be corrected for. Therefore, when gating from

below, internal conversion corrections must be made to the lower energy transitions before

added them together to determine the higher lying gamma-ray branching ratios.

Internal conversion coefficients can be calculated numerically, and the software

used to calculate them for this research is Brlcc [52]. The magnitude of the correction can

depend strongly on the multipolarity of the transitions involved. As a consequence of this,

certain low energy transitions have a very large internal conversion coefficient uncertainty,

when the multipolarity of the transition is unknown.

68

4.2.6 Angular correction to branching ratios

When measuring gamma-ray branching ratios, pairs of gamma-rays in coincidence

are considered, for both the situation when gating from above and when gating from below.

The gamma-ray branching ratios represent ratios of probabilities of gamma-rays being

emitted from a particular state, and the should not involve the angular distribution of the

gamma-ray. This means that the effect of any angular distribution needs to be removed

in order to determine the proper branching ratio. With a detector array with full angular

coverage, the effect would cancel out, and this complication could be ignored. In a detector

array like YRAST Ball, the effect can cause a deviation of approximately 5%, depending

on the correlation pattern. The experiment being described in this work has many states

and transitions with unknown spin assignment and multipolarities, which makes correcting

for the effect impossible. To account for the effect, a systematic uncertainty of 5% has

been added to all measured peak areas for branching ratio calculations.

4.3 Gamma-ray spectrum analysis using spectre

The experimental and numerical techniques described up until this point have all

assumed that the individuals performing the experiment and analysis have an appropriate

spectrum analysis software for fitting peaks, placing gates on matrices, and in general

identifying the useful information from the data. The author of this document felt that

the currently available tools were inadequate for the analysis of this experiment, and

created a software package called spectre.

4.3.1 Choice of gamma-ray peak fit function

The most important feature of gamma-ray spectrum analysis software is that it

gives the correct values and uncertainties; be they transition energies, peak areas, or energy

resolution at a particular energy. Aside from the choice of fitting algorithms, the most

fundamental factor in attempting to reach this goal is selecting the appropriate fit function.

The most obvious choice when working with a large HPGe crystal would be a gaussian

function, due to the statistical nature of the charge collection. However, in situations

where the crystal is exposed to a large flux of fast neutrons, the crystal forms trapping

centers, which cause a certain probability of stopping an electron as it travels through

the crystal. Assuming uniformly spread neutron damage to the crystal, the probability

69

distribution of energy lost during charge collection after gamma-ray photoabsorption is

exponential. It is most probable for full energy collection to occur, and the width of the

exponential distribution depends on the extent of neutron damage [53].

If the HPGe crystal had unlimited energy resolution, an exponential might have

been adequate to describe the peak shape. As mentioned in Section 4.2.1, the statistical

nature of the charge collection causes a gaussian shape for the resolution function. There-

fore, to have an appropriate peak shape that accounts for both the detector resolution

and neutron damage, a convolution of an exponential and gaussian are needed. Here is an

example function for this direct convolution:

/ (x) = ^ e r f c ( ^ + - ^ ) (4.8,

Most fitting algorithms for peak fitting with gaussian statistics use some form

of least-squares regression. With this technique, the squares of the differences between

the experimental values and the fit value are summed up, and that sum is minimized by

varying the parameters. In the convolution shown in Eq. 4.8, the parameters (3 and H

are strongly correlated, and the uncertainties of both parameters will be overestimated.

In certain situations, if the start parameters are not chosen very carefully, the fit will not

even converge. The sums of squares is often called x2 ;

i 1

where Xi and m refer to the experimental data, and <7i is the uncertainty of the each

data point y, [54]. Another consequence of this fit function is that the maximum of the

peak will move as a function of (3, This correlation will not necessarily destabilize the

minimization, but it does cause problems when looking at a total projection of multiple

detectors. When the position of the peak maximum depends on the value of (3, then if

different detectors have different amounts of neutron damage, the peak maximum of the

detectors will not align even after calibration. This will result in a strange total projection

peakshape, and an unreliable fit. To get around both problems, some analysis softwares

will add a symmetric gaussian at position c. This stabilizes the fit somewhat, but often

requires locking either (3 or H during the fit. Other programs will attach an exponential

to a symmetric gaussian to simulate the convolution, but there is no physical justification

for this choice.

In spectre , a different solution to the problem is used. An energy offset is added

to the function to align the parameter c with the peak maximum, and this is done with

70

- 4 - 2 2 4 - 4 - 2 2 4

Figure 4.6: Illustration of the two peak shapes given in Eqs. 4.8 and 4.10. Four possible values of the skew parameter (3 are used. For this example, the parameters are set to H — 1, a = 1, and c = 0. For a), the original convolution from Eq. 4.8 is shown, and a large correlation can be seen between the height of the peak and (3. To a lesser extent, a correlation can be seen between the x-position of the maximum and (3. For b), the modified convolution from Eq. 4.10 is shown, and those two correlations are vastly reduced.

a function 7{(3, a). There does not appear to be any analytical solution for the function

y(/3,cr), so a numerical algorithm is used. With the maximum value at position c, it is

then simple to lock the peak height to the parameter H. The final function looks similar

to the previous convolution, with just two modifications:

An illustration comparing this fit function with the plain convolution can be seen in Fig.

4.6. This function results in more reliable uncertainties and a much more stable fit.

represent the exact theoretical function for the peak shape, due to detectors having varying

resolution and neutron damage. In a projection like that, the peak should actually be

the sum of several very similar peak shapes. Due to angular correlation effects and low

statistics, such a peak shape is more trouble than it is worth. The peak shape given in

Eq. 4.10 is an excellent choice for gamma-ray spectrum analysis.

4.3.2 Fit t ing algorithm

Once a fit function is selected, an algorithm for minimizing x2 is needed. One

method is called a gradient search, and the gradient of %2 is used to select a direction for the

(4.10)

In the total projection of several detectors, this function will not necessarily

71

search [54]. This direction matches the direction of steepest descent of the x2 surface, and

will take the parameters close to the minimum of x2- This method unfortunately converges

slowly as it approaches the minimum. A second technique involves a parabolic expansion

of x2, and uses Newton's method to find the minimum of the expansion. This technique

is computational intensive, but rapidly converges to the minimum in a small number of

steps. The expansion technique will not converge if the x2 surface is not parabolic, or

more specifically if a diagonal element of the curvature matrix is negative.

A technique that takes advantage of both techniques, and avoids the weaknesses

of each is called the Levenburg-Marquardt minimization algorithm [55, 56]. The procedure

is similar to the parabolic expansion method, but scales the diagonal elements of the

curvature matrix by a factor (1 + A). When A is small, the expansion method dominates,

but if A is large, the gradient method plays a larger role. Every step, x2 is evaluated. If

the new x2 is smaller, then A is decreased, bringing the technique closer to the expansion

form. If the new x2 is larger, the surface is probably not very parabolic, and A is increased,

bringing the technique closer to the gradient form. There are still situations where the

fit will not converge, but if the initial parameters are reasonable, the fitting algorithm is

extremely stable.

4.3.3 Parameters and uncertainties

In gamma-ray spectrum analysis, it can be useful to lock parameters to specific

values, or to simply define parameters in separate peaks to be equal. If two peaks are at

very similar energy, the peak widths and the skew parameter (3 should essentially be the

same for both peaks. Often when using HPGe detectors, the peak itself can fit into as

few as 7 channels, and the algorithm will sometimes conclude that two overlapping peaks

have different widths or skew width, just for statistical reasons. The user knows this, and

it can be useful to define the widths of multiple peaks to be the same. In spectre this

is numerically done by dropping extra parameters from the parameter space by using a

group lookup table. For example, if three widths are grouped together, the first parameter

is kept, and the other two are dropped. When the other two are referenced during the

minimization, the value is redirected to the first value. This results in a smaller curvature

matrix, but complicates the procedure.

One benefit of the algorithm being so closely related to the parabolic expansion

form is that once the fit has converged, the covariance matrix of the parameters is simply

72

the inverse of the curvature matrix. The covariance matrix can then be used in the

calculation of the peak area uncertainty from the fit parameters.

It is important to note that the uncertainties of the fit parameters don't neces-

sarily represent the uncertainty of the physical values. For example, with the statistics

present in an experiment like the 140Nd experiment performed in this work, it is possible

to find an energy uncertainty of 10 eV for the peak energy. This does not represent the

actual uncertainty of the transition energy in the nucleus, but rather the uncertainty of

the position parameter in the spectrum. The spectra that are used to fit the peaks have all

had an energy calibration applied to them, and that calibration must be considered when

evaluating the uncertainties of transition energies. The uncertainties of the raw data from

the calibration source, as well as the uncertainty due to the calibration curve deviating

from the data points both play a role in the energy uncertainty. In this work, this is

treated as a systematic uncertainty, and a 0.1 keV uncertainty is added to each transition

energy after fitting.

4.3.4 C o m p t o n scattering and the 2 D matrix v iew

Looking past the physically justified fit function, parameter selection, and fitting

algorithm, there are some features in spectre that exist mainly for the convenience of the

user. The graphics display is hardware accelerated, and navigating and fitting spectra is

fluid. Past fits can easily be saved to disk and reloaded when the spectrum is next analyzed.

One feature that was mainly intended to provide the user with insight into different regions

of the 7-7 matrix has also identified some concerns that need to be considered when using a

Compton suppressed HPGe array like YRAST Ball. This feature is the 2D matrix viewer.

When a gamma-ray Compton scatters out of one detector and into another, it

can result in a 7-7 event where the sum energy of the two gamma-rays is equal to the

energy of the original gamma-ray. This appears in a 7-7 matrix as diagonal lines with

the x and y intersect equaling that energy. Without Compton suppression shielding, the

energy spread of these diagonal lines is very broad. If shielding is used, then it is unlikely

for a gamma-ray to escape from one detector and end up in an adjacent detector. It is

mainly only possible for a Compton coincidence to be observed if the scattered gamma-ray

ends up in the detector on the opposite side of the array. This narrow angular acceptance

results in a similarly narrow energy distribution in the scattered gamma rays. In fact, if

the Compton scattered event is precisely at an angle of 180°, then the recorded energy of

73

Figure 4.7: Image of a 140Nd 7 - 7 matrix as seen in s p e c t r e . A white line has been added to illustrate the trajectory of (£1, £2) from Eq. 4.11. The diagonal lines crossing through the white line represent strong transitions that have had their energy divided across two detectors, due to Compton scattering.

74

the two gamma rays should be

ec2Ei „ „ mec2Ei E l = m ^ ^ E , E 2 ~ E i ~ m'J+2Ei ( 4 1 1 )

where mec2 is the rest energy of the electron, and is approximately 511 keV. This precise

angular acceptance is unrealistic, but even with the fairly broad collimator of the BGO

shield for HPGe clovers, the energy distribution remains rather narrow for the diagonal

line. This narrow energy distribution is a problem when gating on the matrix, because

there is a probability that either energy E\ or E2 will be in coincidence with a different

gamma-ray from the same nucleus. Due to the narrow energy distribution of E\ and E2,

gating on either will result in a spectrum that includes coincidences with E{, rather than

the intended transition. A view of a 2D matrix in spec t r e can be seen in Fig. 4.7.

An example of this appears in the following chapter, but a brief description of the

problem will be shown here. A new transition of energy 1095 keV was observed in 140Nd.

In the gated spectrum of this transition, several transitions from 48Ti were observed, and

a 1095 keV transition has never been observed in 48Ti. The reason for this is that the

—> 2f transition in 4 8Ti has an energy of 1312 keV, and using the formula above gives

Compton scattered energies of 1098 keV and 214 keV. The contaminant transitions were

observed because of an artificial coincidence between the 1098 keV and other transitions

from 48Ti. As far as the author is aware, this is primarily a problem with Compton

suppressed arrays, when there are detectors on opposite sides of the array. Without the

Compton shielding, the diagonal lines would be significantly more broad in energy, and

false coincidences should not be as pronounced.

An image of this specific situation can be seen in Fig. 4.8. The Compton coinci-

dence event is due to the 1312 keV transition from 48Ti, so it was fairly simple to separate

the false data. If the coincidence had been due to a different transition in 140Nd, it could

lead to a confusion situation, where placing the transition properly in the level scheme

would be difficult. One problem that is visible when gating on a Compton scattered di-

agonal line is the feature shown at 214 keV in the 1095 keV gate from the 140Nd data.

This feature does not depend so much on the width of the energy distribution, so even

arrays without Compton suppression will have it. It is problematic if a real coincidence

lies on the diagonal line, as it will be difficult to fit when looking at a gated spectrum. An

illustration of this can be seen in Fig. 4.9.

75

File Fit View Matrix

w0Nd ndl40.mat - Gate X 1095

,40Nd

Compton Gate

511 keV

400 500 600 700 800 900 1000 1100 1200 1300

Figure 4.8: Full view of the 1095 keV gate in spectre . The Compton gate effect can be seen near 214 keV. The only two other features that should be visible in this gate are 774 keV and 1028 keV from 140Nd. Due to there being a strong Compton coincidence at 1095 keV from the 1312 keV transition in 48Ti, there are two additional lines from 48Ti.

File Fit View Matrix

Figure 4.9: Zoomed in view around 214 keV in the 1095 keV gate in spectre . This effect is typically simple to identify as being due to Compton coincidence, but it can lead to difficulties if a real concidence overlaps with this feature.

76

Chapter 5

Experimental results

Many new transitions and levels in 140Nd have been identified during the analysis

of this in-beam experiment. The events recorded using the YRAST Ball detector array

have a large statistical advantage over many of the past experiments that observed 7-7

coincidences. In addition, this is the first experiment to populate 140Nd using a (p,2n)

reaction, so many of the new states were likely more accessible due to spin and energy

distribution of the excited states. Over the course of this experiment, 67 new transitions

were identified, and 34 new levels were placed. Up to 10 of the new levels may have been

observed in a previous (p,t) experiment[43], but energy uncertainties made it difficult to

determine which level to relate the new results to.

All states and transitions that were observed in this experiment have been listed

in Table 5.1. The energies are all listed to one decimal point in keV. The seemingly large

uncertainties for the energy, which are on the order of 0.1 keV, are due to the energy

calibration. The fit parameters for energy nearly always had a much smaller uncertainty,

but the energy calibration polynomial fit introduces some uncertainty on its own. This

was treated as a 0.1 keV systematic uncertainty, and was added to each transition energy.

The branching ratios J?7 were determined using techniques described in Section 4.2.4.

Where possible, gating from above the level of interest was used. When that was not an

option, gating from below was used, and internal conversion was taken into account when

summing the peaks together.

Many of the listed spin assignments were determined using the angular correlation

techniques described in Section 4.2.3. The fits were performed using corleone. At higher

energies, the statistics were not large enough to adequately determine the spins of most of

the states. The multipole mixing ratios listed in Table 5.1 were determined using angular

correlation fits. Ratios listed for states with an uncertainty for the spin assignment should

77

be treated as even more uncertain.

The observed 140Nd level scheme is shown in Figs. 5.1 and 5.2. Due to the

large density of transitions, the image has been separated into two parts. Fig. 5.1 has

transitions from lower energy states, and Fig. 5.2 has transitions from the higher energy

states. Most states and transitions are new, but Table 5.1 can be used to identify which

states and transitions were specifically first observed in this experiment.

Table 5.1: Levels and transitions from 140Nd. All states and transitions listed were observed

in this experiment. Where possible, spin assignments J f , gamma-ray branching ratios B7,

and the multipole mixing ratios <5 are listed. Energies are all given in units of keV, and the

branching ratios are a percentage.

Ex T7T Ji Jf Ef E-y Bry 6{J? JJ)

773.6(1) 2+ 0+ 0 773.6(1) 100

1413.0(2) 0+ 2+ 773.6 639.4(1) 100

1490.0(1) 2+ 2+ 773.6 716.1(1) 93(8) -1.22(13)

0+ 0 1490.1(1) 100(6)

1802.0(2) 4+ 2+ 773.6 1028.4(1) 100 0.02(3)

1935.3(1) 3~ 2+ 1490.0 445.2(l)b 16(2)

2+ 773.6 1161.7(1) 100(7) 0.01(3)

2140.3(2) 2+ 2+ 773.6 1366.7(1) 100 -0.25(10)

2221.6(2) 7" 4+ 1802.0 419.6(1) 100

2264.4(l)a 4+c 4+ 1802.0 462.3(2)b 2.3(3)

2+ 773.6 1490.8(l)b 100(5) 0.01(4)

2276.0(1) 5~ 3" 1935.3 340.7(l)b 0.8(2)

4+ 1802.0 474.1(1) 100(6) 0.04(2)

2333.1(2) 2+ 2+ 773.6 1559.6(1) 100

2366.3(1) 6+ 5~ 2276.0 90.3(1) 25(2)

7~ 2221.6 144.6(1) 78(7)

4+ 1802.0 564.4(1) 100(6) -0.11(4)

2400.1(1) 4+ 2+ 1490.0 910.1(l)b 100(8) -0.03(7)

2+ 773.6 1626.5(l)b 69(7) 0.04(7)

2450.6(2)a (3)c 2+ 773.6 1677.0(l)b 100 -0.19(9)

78

140 Nd continued

Ex J? T7T Jf E-y 8(J? - JJ)

2480.0(l)a (5)c 5" 2276.0 204.0(l)b 96(9)

3~ 1935.3 544.8(l)b 97(10)

4+ 1802.0 678.0(l)b 100(7) 0.58(9)

2513.4(l)d (3)c 4+ 1802.0 711.2(l)b 19(4)

2+ 1490.0 1023.5(l)b 76(12)

2+ 773.6 1739.7(2)b 100(9) i 9 S + 0 ' 2 5

0.33

2520.6(l)a (4)c 4+ 2264.4 256.1(l)b 23(2) 0.11(20)

4+ 1802.0 718.6(l)b 29(3)

2+ 1490.0 1030.8(l)b 100(9)

2+ 773.6 1746.9(l)b 26(3)

2571.0(l)d 4e 3" 1935.3 635.7(l)b 100(6) -2-16181? 4+ 1802.0 769.0(l)b 78(7) 0.34(18)

2586.9(l)a (5)c 6+ 2366.3 220.5(l)b 49(5)

5~ 2276.0 310.9(l)b 31(3) 0.23(20)

4+ 2264.4 322.6(l)b 5.7(7)

4+ 1802.0 785.1(l)b 100(6) -0.13(4)

2612.1(1) (2+) 2+ 1490.0 1122.2(2) 30(4)

2+ 773.6 1838.4(1) 100(7)

2683.1(l)d ?e 4+ 2264.4 418.7(l)b 37(5)

2+ 773.6 1909.6(2)b 100(8)

2684.8(l)d ?e 3~ 1935.3 749.6(l)b 61(7)

4+ 1802.0 882.8(l)b 100(7)

2693.1(l)a ? 4+ 2264.4 428.7(l)b 9(1)

4+ 1802.0 891.2(l)b 100(7)

2+ 773.6 1919.5(2)b 6.2(9)

2707.7(l)a 5C 4+ 2400.1 307.7(2)b 2.6(4)

4+ 1802.0 905.8(l)b 100(6) 0.04(3)

2710.5(2)d ?e 2+ 773.6 1936.9(2)b 100

2714.9(2) 2+ 2+ 773.6 1941.3(1) 100

2725.9(2)a (3,4)c 2+ 773.6 1952.3(l)b 100

79

140 Nd continued

Ex J ? T7T Jf Ey By 5(Jf - J J)

2842.3(2) (6 + ,7~) 5~ 2276.0 566.3(1) 100(8)

7~ 2221.6 620.8(2)b 20(3)

2884.4(l)a (4 r 4+ 2400.1 484.2(2)b 7(1)

4+ 1802.0 1082.4(l)b 100(6) 0.03(16)

2896.7(l)a 5° 5" 2276.0 620.5(l)b 99(9)

3" 1935.3 961.6(l)b 32(3)

4+ 1802.0 1094.8(l)b 100(7) 0.08(6)

2912.1(l)a (5)c ? 2693.1 219.0(2)b 13(2)

5" 2276.0 636.1(l)b 89(9)

4+ 2264.4 647.7(l)b 100(7) - 0 - 4 8 1 ^

4+ 1802.0 1110.2(2)b 97(10)

2943.3(1) (6+) 5" 2276.0 667.3(1) 100(7) 2.81(21)

7~ 2221.6 721.7(1) 99(10)

2944.5(2)a (l,2)c 2+ 1490.0 1454.7(2)b 100(13)

0+ 1413.0 1531.4(2)b 56(9)

2950.1(2)a (5,6)c 4+ 1802.0 1148.1(l)b 100

3015.5(2) ?e 4+ 1802.0 1213.6(l)b 100

3024.4(l)a 6C (5) 2480.0 544.3(l)b 100(9)

5 _ 2276.0 748.4(l)b 38(4)

3061.6(2) 7" 6+ 2366.3 695.3(1) 100 _40+° 30

3077.7(3)a ? 2+ 773.6 2304.1(2)b 100

3140.0(2) ? 4+ 1802.0 1338.0(l)b 100

3185.2(2) 8+ 6+ 2366.3 818.9(1) 100

3193.3(2)a ? 4+ 1802.0 1391.4(2)b 100

3197.7(2)a ? (3) 2513.4 684.3(l)b 100

3210.6(3)d ? 2+ 773.6 2437.0(3)b 100

3228.7(l)a (7,8)c (6+) 2943.3 285.4(l)b 100(9)

(6+,7") 2842.3 386.4(l)b 38(5)

3239.0(2) 8 _ 7" 3061.6 177.2(1) 32(3)

7~ 2221.6 1017.7(1) 100(6)

80

140 Nd continued

Ex If T7T Jf Ef E*y 6(J? JJ)

3268.5(2)a ? 2+ 773.6 2494.9(2)b 100

3320.1(2)d ? 2+ 773.6 2546.5(2)b 100

3323.2(2)d (2,4)e 3- 1935.3 1388.0(l)b 100

3333.0(2)a ? 2+ 773.6 2559.4(2)b 100

3349.8(2)a ? 4+ 1802.0 1547.9(2)b 100

3387.9(2)d ?e (5,6) 2950.1 437.8(l)b 100

3453.6(2)a ? 6+ 2366.3 1087.3(2)b 100

3454.5(2) 9 - 8~ 3239.0 215.5(2) 100

3477.9(3)a ? 2+ 773.6 2704.3(3)b 100

3492.1(2)d ?e 3~ 1935.3 1556.9(2)b 100

3698.2(2)a ? 5- 2276.0 1422.2(2)b 100

"State not previously observed.

''Transition not previously observed. cSpin assignment not previously known. dState may have been previously observed in (p,t). eSpin assignment may have been previously determined in (p,t).

5.1 New states and transitions in 140Nd

Many of the new transitions observed in this experiment were difficult to place

in the level scheme of 140Nd. In some cases this was due to multiple transitions having

very similar energy, which made gating on the transition of interest difficult. Often this

was resolved by carefully considering the coincidence logic with the level scheme as it had

existed up until that point. After the analysis was completed, all transitions were checked

again in reference to the final level scheme to verify that the placement still made sense.

There were some cases where logic was not enough to place the level, and 7-7-7 had to

be used with two gates. This was performed using a previous version of spectre , which

81

Figure 5.1: States and transitions from the observed 140Nd level scheme. Only transitions from the lower energy states are shown.

82

I

1 4 0 Nd

Figure 5.2: States and transitions from the observed 140Nd level scheme. Only transitions from the higher energy states are shown.

83

Energy (keV)

Figure 5.3: Total projection from the 140Nd gamma matrix. The largest peaks have been labelled with their source. Contamination of 56Fe and 48Ti from the target frame material can be seen.

supported three-dimensional gamma cubes. The somewhat low 7-7-7 statistics made this

strategy fairly limited in application, but in a few cases it allowed placement of transitions

that otherwise could not have been determined.

In other cases, the Compton coincidences that occurred between detectors on

opposite sides of the clover array created some contamination in the 7-7 matrix. Most of

the time this was simply an inconvenience, where a gate had to be placed on part of a

large bump in the spectrum, in order to isolate a transition that was hidden on it. In other

cases, the Compton coincidences showed up in the gated spectrum as contamination, but

this was rare enough to not cause a serious problem. A more detailed discussion of this

issue can be found in Section 4.3.4.

In the following section, the coincidence reasoning used to place the new transi-

tions in the level scheme is described. Any additional difficulties due to similar energies

or contaminants in the matrix are described. Several of the transitions had clean gates

without complications, and the discussions for those transitions are kept brief. All fitting

was done using spectre , and for almost all fits of peaks with similar energies, the width

and skew parameters were locally linked together. An example of this can be seen in the

2711 keV state discussion. The total projection from the 140Nd 7-7 matrix can be seen in

Fig. 5.3, and the strongest contaminants are labelled.

84

1491 keVand 774 keV cascade

e2 41.5 90 90 90 90 138.5 138.5 41.5 <p 90 90 180 -90 0 0 180 0

Figure 5.4: Angular correlation pattern from the 2264 keV 4+ state in 140 Nd. The 4+ 2+ —> 0+ cascade clearly fits the experimental values most accurately.

Previously observed 1935 keV 3~ s tate

A 1935 keV state has been previously observed with a 1162 keV transition to the 774 keV 2f state. This transition and state were confirmed in this experiment, and a 445 keV transition to the 1490 keV state was observed for the first time. The 445 keV transition is strongly coincident with the 716 keV transition from the 774 keV 2f state, and was not coincident with the 1162 keV transition.

2264 keV 4+ state

A new state was observed at 2264, based on two new transitions. A 1491 keV

transition was observed strongly in coincidence with 774 keV. The energy of this transition

is different than the 1490 keV transition that is in coincidence with the 910 keV from the

new 2400 keV state. By looking in the gate of the new 256 keV transition that feeds

directly into the 2264 keV state, one sees a strong coincidence with 774 keV and 1491 keV,

which places the 1491 keV transition directly above the 774 keV 2f state.

A second transition of 462 keV was observed directly feeding into the 1802 keV

4+ state. The transition is weak, but is also observed in the 256 keV gate, making it

85

likely to belong to the same state. Angular correlations indicated a spin assignment of 4.

The transition from 4 —> 2f is likely E2, rather than M2, which gives the state a positive

parity. The angular correlation pattern from this state can be seen in Fig. 5.4.

Previously observed 2276 keV 5~ s tate

This state was previously observed with a 474 keV transition to the 1802 keV 4f state. A new transition of 341 keV to the 1935 keV 3 _ state has been observed. The transition is weak, but was observed to be coincident with 445 keV and 1162 keV. Gating on 340 keV itself was difficult due to 511 keV gamma-rays Compton scattering from one detector into a different detector. This scattering left a diagonal bump of coincident events at energies of 339 keV and 171 keV in the 7-7 matrix.

2400 keV 4 + s tate

A state was observed at 2400 keV, based on two new transitions. The first

transition of 910 keV was strongly coincident with only the 716 keV, 774 keV, and 1490 keV

transitions, which places it directly above the 1490 keV 2+ state. A second transition of

1626 keV was observed only in coincidence with the 774 keV 2f state. Angular correlation

analysis for both transitions confirmed a spin assignment of 4. The multipole mixing ratio

for both transitions was approximately 0, with a reasonable alignment of a as 3. The

transitions are probably E2, rather than M2, which gives the state a positive parity. A

state of 2400(2) keV was observed in a (p,t) experiment, with a spin assignment of 4+ .

That state is likely the same state as what was observed in this experiment.

2451 keV (3) s tate

A new state was observed at 2451 keV, based on a 1677 keV transition. This tran-

sition was found to be coincident with only the 774 keV transition. Angular correlations

were not conclusive, but a spin assignment of (3) seems likely.

2480 keV (5) state

A new state was observed at 2480 keV, based on three new transitions. The 204

keV transition was strongly in coincidence with 474 keV, 1028 keV, and 774 keV, which

places it directly above the 2276 keV 5~ state.

86

The 545 keV transition was difficult to place due to coincidence with a 544 keV

transition from the higher lying 3024 keV state. This transition was in coincidence with

445 keV and 1162 keV, which places it directly above the 1935 keV 3~ state.

The 678 keV transition was strongly in coincidence with 1028 keV, 774 keV, and

the higher lying 544 keV transition, which places it directly above the 1802 keV 4+ state.

2513 keV (3) s tate

A new state was observed at 2513 keV, based on three new transitions. A 711 keV transition was observed in strong coincidence with 1028 keV and 774 keV, which places it directly above the 1802 keV 4^ state.

A 1023 keV transition was observed in strong coincidence with 716 keV and 774 keV. Gating on 1023 directly was difficult due to the large tail from the 1028 keV transition in the matrix projection, but careful selection of gate channels provided enough information to verify the placement of the transition directly above the 1490 keV 2+ state.

A third transition of 1740 keV was observed only in coincidence with 774 keV and 684 keV. After observing that 684 keV is in coincidence with all three transitions from this state, it becomes clear that the 1740 keV feeds directly in the 774 keV 2f state, and that 684 keV feeds into the 2513 keV state.

Angular correlations give a likely spin assignment of (3), but a previous (p,t) experiment observed a 2514(3) keV 5~ state, which may or may not be the 2513 keV state observed in this experiment.

2521 keV (4) s tate

A new state was observed at 2521 keV, based on four new transitions. A new

256 keV transition is found to be in coincidence with 1491 keV, 774 keV, and 462 keV,

which indicates that it feeds into the 2264 keV state. The lack of a 716 keV transition in

the coincidence gate confirms that the 1490 keV state is not involved.

A 719 keV transition was observed in coincidence with 1028 keV and 774 keV,

but the presence of the strong 716 keV transition in the matrix projection made directly

gating on this transition difficult. The use of 7-7-7 coincidences with two gates in spectre

helped confirm the placement directly above the 1802 i f keV.

A 1031 keV transition was observed in coincidence with 1031 keV, but the pres-

ence of the strong 1028 keV transition in the matrix projection made a direct gate difficult.

87

7-7-7 coincidences with a 716 keV and 1030 keV gate helped confirm the placement directly

above the 1490 keV 2+ state.

A 1747 keV transition was observed in coincidence with 774 keV, placing it di-

rectly above the 774 keV state. Angular correlations from the 256 keV transition

indicates that a spin assignment of 4 is likely, but the statistics from other transitions

from this level were too low to confirm it.

2571 keV 4 s tate

This new state was observed based on two new transitions. The 636 keV transi-

tion was difficult to gate on due to an additional new 636 keV transition between the new

2912 keV level and the 2276 keV 5~ level. 7-7-7 coincidence with a gate on 1162 keV and

716 keV placed this transition directly above the 1935 keV 3~ state.

The 769 keV transition was in coincidence with 1028 keV and 774 keV. The

1028 keV coincidence required 7-7-7 coincidences to identify, due to 774 keV and 769 keV

both being strongly in coincidence with 1028 keV, and 774 keV being such a large peak.

Angular correlations indicate a spin assignment of 4, but fixing the alignment parameter

to a reasonable value was necessary for realistic fits of the 769 keV transition.

2587 keV (5) s tate

This new state was observed based on four new transitions. The 221 keV transi-

tion was in coincidence with 91 keV, 145 keV, 474 keV and 565 keV, all of which cascade

down from the 2366 keV 6+ state. This places the transition directly above the 2366 keV

6f state.

The 311 keV transition was observed in strong coincidence to 474 keV, 1028 keV,

and 774 keV, which places directly above the 2276 keV 5" state. A small amount of 141Nd

was produced in this experiment, and a 312 keV transition from that nucleus caused an

anomalous 1565 keV coincidence in the 311 keV gate.

The 323 keV transition was observed in strong coincidence with 1491 keV and

774 keV placing it directly above the new 2264 keV 4+ state. Gating on 323 keV was

difficult due to 511 keV gamma-rays Compton scattering from one detector into another

detector.

The 785 keV transition was in coincidence with only 774 keV and 1028 keV, which

resulted in a simple placement directly above the 1802 keV 4+ state. Angular correlations

88

with the 311 keV and 785 keV transitions indicated a spin assignment of 5, although the 221 keV transition preferred an assignment of 7 over 5. Two of the transitions from this state lead to a spin 4 state, which makes an assignment of 7 unlikely.

2683 keV state

This new state was observed based on two new transitions. The 419 keV transi-

tion was difficult to isolate due to the 420 keV transition from the 2222 keV 7~ isomer.

The 419 keV transition was observed in coincidence with 1490 keV, and 7-7-7 coincidences

using an additional 774 keV gate confirmed placement above the new 2264 keV 4+ state.

The 1910 keV transition was found to be only in coincidence with 774 keV, which

places it directly above the 774 keV state. A 2686(3) keV 4+ has been previously

observed in a (p,t) experiment, but the large energy uncertainty of that state makes it

unclear whether is refers to the new 2683 keV state from this experiment, or the new 2685

keV state.

2685 keV state

This new state was observed based on two new transitions. The 750 keV transi-

tion was difficult to isolate from the new 748 keV transition from the 3024 keV state to

2276 keV 5_ state. The transition was in strong coincidence with 445 keV and 1162 keV,

and by identifying all remaining transitions in the 750 keV gate as belonging to the 748

keV transition, it was determined that the 748 keV transition feeds directly into the 1935

keV 3~ state.

The 883 keV transition was in strong coincidence with only 1028 keV and 774

keV, placing it directly above the 1802 keV 4+ state. As mentioned in the 2683 keV state

section, a 2686(3) keV 4+ was previously observed in a (p,t) experiment. The large energy

uncertainty of that state makes it unclear whether it refers to the 2683 keV state from

this experiment, or the 2685 keV state.

2693 keV state

This new state was observed based on three new transitions. The 429 keV tran-

sition was in coincidence with 1490 keV and 774 keV, and was not in coincidence with 716

keV, which places it directly above the new 2264 keV 4+ state. The 891 keV transition

was only in coincidence with 1028 keV and 774 keV which places it above the 1802 keV 4+

89

state. The 1919 keV transition was found to only be in coincidence with 774 keV which

places it directly above the 774 keV state.

2708 keV 5 state

This new state was observed based on two new transitions. The 308 keV tran-

sition was observed in coincidence with the 910 keV transition from the 2400 keV state.

It was also in coincidence with the 1626 keV transition from the same 2400 keV state.

Placing a gate directly on 308 keV transition was not possible due to low statistics and

the nearby 311 keV transition.

A second transition of 906 keV was observed only in coincidence with 1028 keV

and 774 keV, placing it directly above the 1802 keV 4f state. Although the 910 keV

transition overlapped with 906 keV slightly, the 906 keV coincidences were easily resolved.

Angular correlations from the 906 keV transition indicates a spin assignment of 5.

2711 keV state

This new state was observed based on a new 1937 keV transition that feeds directly into the 774 keV 2f state. Although 1937 keV and 1941 keV transitions overlap in the matrix projection, both are solely in coincidence with 774 keV, which makes the placement of the 1937 keV transition clear.

This situation is a good example of when parameters like the width and skew should be linked between multiple peaks. A fit of this cluster of peaks in spectre can be seen in Fig. 5.5. If the parameters had not been linked, the 1936.9 keV transition would have been found to have an energy of 1938.1 keV.

A 2710(2) keV 2+ state was observed in a (p,t) experiment, but it is unclear whether it refers to the same state.

2726 keV (3,4) s tate

This new state was observed based on a 1952 keV transition that is only in

coincidence with 774 keV. This places it directly above the 774 keV 2f state. Angular

correlations found the spin assignment to be (3,4).

90

:ndl40.mat - Gate X 774

Energy: 1936.896(119) Width: 3.656(144) Skew: 2.191(120) Area: 4485(157)

Chi2: 0.855 J 2 ,

e Fit View Matrix

Figure 5.5: A fit of the 1937 keV peak in the 774 keV gate from spectre. The width parameter a and the skew parameter (3 are shared between the three peaks. This transition originated from the 2711 keV state.

Previously observed 2842 keV (6 + , 7~ ) s tate

This previously observed state was placed based on a 566 keV transition to the

2276 keV 5~ state. An additional transition with energy 621 keV was observed during

this experiment, but gating on the transition was not helpful, due to a strong 621 keV

transition from the 2884 keV state to the 2276 keV 5~ state. The 621 keV transition from

this 2842 keV state was observed by gating on a new 386 keV transition from the 3229

keV state. The 621 keV transition feeds directly into the 2222 keV 7~ isomer, so 386 keV

was the only observed transition in coincidence with it.

2884 keV (4) s tate

This new state was observed based on two new transitions. The 484 keV transi-tion was observed in coincidence with the two transitions from the 2400 keV state. Gating on 484 keV was not possible due to low statistics and a nearby 56Co contaminant. The 1082 keV transition was observed to be only in coincidence with 774 keV and 1028 keV which places it directly above the 1802 keV 4j" state. Angular correlations from the transitions in this experiment indicate a spin assignment of (4).

91

2897 keV 5 s tate

This new state was observed based on three new transitions. The 621 keV state

was found to be directly in coincidence with 474 keV, 1028 keV, and 774 keV, along with

a contaminant transition from 141Nd. This places the transition directly above the 2276

keV 5~ state. The 621 keV transition from the 2842 keV state did not affect the 621 keV

coincidence gate due to the small number of events, and because it directly feeds into the

isomer.

It was not possible to gate on the 962 keV transition due to energy triplet of

similar amplitudes at that energy. The transition was in coincidence with 1162 keV, 445

keV, and 774 keV, which places it directly above the 1935 keV 3_ state.

The 1095 keV transition is strongly in coincidence with 1028 keV and 774 keV.

Titanium contamination is visible in the 1095 keV gate due to the small energy distribution

of the Compton scattered diagonal smear in the gamma matrix. The 1312 keV line in 48Ti

splits most probably into 1097 keV and 215 keV gamma-rays, when scattering from one

detector to a different detector on the opposite side of the array. A more detailed discussion

of this situation can be found in Section 4.3.4.

Angular correlation analysis of this level gives a spin assignment of 5 for both

the 1095 keV and 621 keV transitions. The 962 keV transition did not have adequate

statistics.

2912 keV (5) s tate

This new state was observed based on four new transitions. The 219 keV tran-

sition was difficult to isolate due to the lower lying 221 keV transition between the 2587

keV state and the 2366 keV state. Gating on 891 keV yields a coincidence with 1028 keV,

774 keV, and 219 keV, which places the transition directly above the 2693 keV state.

The 636 keV transition was difficult to isolate due to the lower lying 636 keV

transition between the 2571 keV and 1935 keV states. Gating using 7-7-7 coincidences on

636 keV and 474 keV yields a spectrum with only 1028 keV and 774 keV, which indicates

a placement directly above the 2276 keV 5 - state.

The 648 keV transition was difficult to isolate due to contamination from both 52Cr and 141Pr. Gating using 7-7-7 coincidences on 1490 keV and 774 keV gives a strong

coincidence with the 648 keV line, which places the transition directly above the 2264 keV

4+ state.

92

The 1110 keV transition was strongly in coincidence with 1028 keV and 774 keV,

placing directly above the 1802 keV 4^ state. Angular correlations indicate that the spin

assignment is probably (5).

2945 keV (1,2) s tate

This new state was observed based on two new transitions. The 1455 keV tran-

sition was extremely difficult to isolate due to a 58Ni contaminant. Aside from the con-

taminant coincidences, the transition was also in coincidence with 716 keV and 1490 keV,

placing it directly above the 1490 keV 2+ state.

The 1531 keV transition was found to be in coincidence with 639 keV and 774

keV, placing it directly above the 1413 keV 0 + state. The state likely has a spin assignment

of (1,2).

2950 keV (5,6), 3016 keV, 3140 keV, 3193 keV, and 3350 keV states

All of these states were observed to be strongly in coincidence with 1028 keV and

774 keV, with no other strong coincidences. This places them directly above the 1802 keV

4+ state. The 2950 keV state is new, and angular correlations found a spin assignment of

(5,6). The 3016 keV was likely previously seen in a (p,t) experiment as a 3014(4) keV 4+

state. The 3140 keV state may have been previously observed in a (p,t) experiment as a

3136(4) keV 4+ state. The 3193 keV and 3350 keV states are both new.

3024 keV 6 s tate

This new state was observed based on two new transitions. The 748 keV transi-

tion was difficult to isolate due to the 750 keV transition from the 2685 keV state to the

1935 keV 3~ state. The 750 keV transition was in coincidence with 1162 keV and 774

keV, so the 474 keV and 1028 keV transitions that appear in the gate belong to the 748

keV coincidence. This places the transition directly above the 2276 keV 5~ state.

The 545 keV transition was isolated from the 544 keV transition from level it

feeds directly into, by looking at the 678 keV gate from the 2480 keV state. The 678

keV transition was found to only be in coincidence with 545 keV, 1028 keV, and 774 keV,

placing it directly above the 2480 keV state. Angular correlations from 544 keV gives a

spin assignment of 6.

93

3078 keV, 3211 keV, 3268 keV, 3320 keV, 3333 keV, and 3478 keV states

All of these states were observed to be strongly in coincidence with only 774

keV, placing them all directly above the 774 kev 2f state. Most of these states are new,

although the 3211 keV state may have been previously observed in a (p,t) experiment as a

3206(4) keV (2+) state. The 3320 keV state may have been observed in a (p,t) experiment

as a 3324(4) keV (2+ ,4+) state, but this could be true of the new 3323 keV state from

this experiment instead.

3198 keV s tate

This new state was observed based on a 684 keV transition to the 2513 keV

state. The transition is strongly in coincidence with the 711 keV, 1024 keV, and 1740 keV

transitions, which places it directly above the 2513 keV state.

3229 keV (7,8) s tate

This new state was observed based on two new transtions. The 285 keV transition

is strongly in coincidence with only transitions that lead from the 2943 keV state, placing

it directly above that state. The 386 keV transition is strongly in coincidence only with

transitions leading from the 2842 keV state, placing it directly above that state. Based on

angular correlations, this state likely has a spin assignment of (7,8).

3323 keV (2,4) and 3492 keV states

These two states were observed based on transitions directly to the 1935 keV 3~

state. Both the 1388 keV and 1557 keV transitions from these two states were found to

be only in coincidence with 1162 keV and 774 keV. The 3323 keV state was found to have

a spin assignment of (2,4), based on angular correlations. The 3492 keV state may be the

same as the 3493(5) keV 4+ state seen in a (p,t) experiment.

3388 keV state

This state was observed based on a 438 keV transition, which was found to be in coincidence with 1148 keV, 1028 keV, and 774 keV. This places it directly above the 2950 keV state. This state may be the same as the 3387(4) keV 2+ state observed in a (p,t) experiment.

94

3454 keV state

This state was observed based on a 1087 keV transition, which was found to be

in coincidence with the 90 keV, 145 keV, and 564 keV transitions, which places it directly

above the 2366 keV 6+ state.

3698 keV state

This state was observed based on a 1422 keV transition, which was found to be

strongly in coincidence with 474 keV, 1028 keV, and 774 keV. This places it directly above

the 2276 keV 5~ state.

5.2 Multipole mixing ratios between 4+ states

Although many new transitions in 140Nd were discovered over the course of the

experiment, one of the main goals of this research was specifically to identify potential 4+

mixed-symmetry states. To identify mixed-symmetry states, Ml transition strengths are

needed, which require the multipole mixing ratios of the transitions. Angular correlation

fits done using corleone provide the magnetic substate alignment parameter a as well

as the transition mixing ratios S. The mixing ratio discussion here will be limited to 4+

states and potential 4+s, to focus on the g-boson content topic. A summary of the 4+

states and the mixing ratios of the transitions between them can be seen in Fig. 5.6.

1802 keV state

This is the first 4+ state and its existence has been confirmed in many different

experiments.

2264 keV s tate

This state was first observed during this experiment based on a 1491 keV tran-

sition to 2+. As mentioned in the previous section, the multipolarity is likely E2 which

would give the state a positive parity. A small 462 keV transition was observed between

this state and the 1802 4^ keV state, but statistics were too low to obtain an acceptable

multipole mixing ratio.

95

2884.4*

1802.0

1500 A

256 keV: 5=0.11(20) 718 keV: 6=1.11^ 769 keV: 6=0.34 (18)

1082 keV: 6=0.03 (16)

Figure 5.6: Energies and multipole mixing ratios of transitions between potential 4+ states in 140Nd. The states labelled with asterisks were first observed during this experiment. All mixing ratios shown were measured in this experiment.

2400 keV state

This state was previously seen in a (p,t) experiment as a 4+ state, and angular

correlations from this state to two 2+ states confirm that a 4+ spin assignment is likely.

No transitions to other 4+ were observed from this state.

2521 keV state

This state was first observed in this experiment, and angular correlations found a

spin assignment of (4). Two strong transitions from this state were observed to 2+ states,

which means that if the spin really is 4, a positive parity is likely. Angular correlations for

the 256 keV transition from this state to the 2264 keV state give a multipole mixing

ratio of 0.11(20) which is predominantly Ml in nature. Angular correlations for the 718

keV transition from this state to the 1802 keV 4f state give a multipole mixing ratio of

I .IIIq 5°, which is approximately a 50/50 mix of Ml and E2.

96

2571 keV s tate

Angular correlations for this state found a spin assignment of 4. A state of energy

2575(3) keV was found during a (p,t) experiment, with a spin assignment of (4+,5~). If

this state is the same as the state observed in that experiment, it likely has positive

parity. Angular correlations for the 769 keV transition to the 1802 keV 4+ state had to be

performed in coincidence with 774 keV rather than 1028 keV due to the energy overlap of

the transitions, corleone can handle situations where a missing transition occurs between

two other transitions, and a multipole mixing ratio of 0.34(18) was found. This indicates

that the transition is predominantly Ml in nature.

2884 keV s tate

Angular correlations for this state found a spin assignment of 4, and the parity

of the state is unknown. The 1082 keV transition fed into the 1802 keV 4f state, and the

multipole mixing ratio for that transition is 0.03(16). If the parity of the state is positive,

then that transition is predominantly Ml in nature.

5.3 Discussion

The results found in this experiment are interesting, and vastly expand what was previously known about 140Nd. Many new low-lying states at moderate spin have been identified, including the state. Several new spin-assignments were made, but there is still a lot of room for improvement. In retrospect, the beam current could probably have been raised to improve the statistics of the experiment. Neutron damage is a serious consideration when using a proton beam, but if that were not an issue, the increased statistics would significantly improve the angular correlations. A larger array with more detectors and angle groups would also improve the angular correlations, but 8 HPGe clovers is already fairly substantial. An array like Gammasphere would see improved results, but the high fast-neutron flux would likely be unacceptable.

Several new multipole mixing ratios have been determined for transitions between 4+ states. The transition from i f to was found to be predominantly Ml in nature. The transition from 4f to 4f was found to be approximately half E2, and half Ml. The transition from 4$ to 4+ was also found to be mostly Ml. The transition from to 4^, which was the motivation for the hexadecapole degree of freedom in 94Mo, did not have

97

enough statistics to adequately determine a multipole mixing ratio. One possibility to

consider is that in the sdg-IBM-2, if the energy of the g-boson is large, then the states

with the hexadecapole degree of freedom will appear at a higher energy. As such, the

strong Ml between the mixed-symmetry and symmetric g-boson states would also occur

between states of higher energy.

Determining the multipole mixing ratios of transitions between 4 + states in 140Nd

is a great step towards identifying potential 4+ mixed-symmetry states. The next step in

the process is to determine the Ml transition strengths of transitions between 4 + states,

which requires a value for the lifetimes of those states. Mixed-symmetry states have

very short lifetimes, on the order of 100 fs. One convenient way of measuring lifetimes

is with the recoil distance method, which uses an experimental device called a plunger.

Unfortunately, such a device can measure lifetimes only down to the picosecond range.

An alternative technique is called the Doppler-shift attenuation method (DSAM)

[57], Highly energetic ions typically come to rest in a solid material in the time frame of a

picosecond. When a fast moving ion emits a gamma-ray, the energy of that gamma-ray is

Doppler-shifted. If the ion emits the gamma-ray as it is slowing down in the material, the

energy of the Dopper-shift will be attenuated. This results in a complicated peak shape

that can be analyzed to determine the time frame that the gamma-ray was emitted, and

therefore determine the lifetime of the state involved.

In order to measure the lifetimes of states using DSAM, the nucleus of interest, 140Nd, must be moving quickly. In this experiment, a proton beam was used in order to

populate a good range of 4+ states, but unfortunately, the product nucleus was moving too

slowly have any visible Doppler-shifting. A reaction with a heavier beam would result in

Doppler-shifting, but would probably not populate the states of interest for this research.

One alternative would be to use inverse kinematics, and accelerate 141Pr into hydrogen.

This is unfortunately extremely inconvenient due to the target being a gas, and it would

also require a higher energy accelerator than is available at WNSL. Using a target like

TiH2 would be more convenient due to the attenuation being stronger and having a solid

target.

Once the lifetimes are known, the Ml transition strengths of interest can be

determined, and any potential mixed-symmetry states could be identified. One feature

that really stood out in 94Mo was that the 4+ mixed-symmetry state was at such a low

energy, and that the Ml transition was strong. This feature was outside of the model

98

space of the sd-IBM-2, and an additional degree of freedom had to be unlocked in order to

properly describe the behavior. Until it is known whether there are strong Ml transition

between 4 + states, and which energies they occur at, any application of the sdg-IBM-2 to 140Nd would be premature.

One of the motivations for including g-bosons in the IBM-2 was that shell model

calculations determined that the hexadecapole degree of freedom played a role in the first

two 4+ states of 94Mo. One of the goals of this experiment was to provide information that

could be tested using the shell model, in order to determine whether a similar situation

occurs in 140Nd. Shell model calculations could be used to identify which states have a

large wavefunction component where two particles couple to angular momentum 4, and

also whether such states are observed at low energy. In 94Mo, the first two 4 + states had

a large L = 4 component, but it is possible that hexadecapole excitations might occur at

a higher energy in 140Nd. It will be interesting to see what such shell model calculations

reveal, and whether the hexadecapole degree of freedom will play a role in the low-lying

4+ states in 140Nd.

99

Part IV

Quantum phase transitions and large boson systems

100

Chapter 6

Quantum phase transitions in nuclei

In Chapter 2, the interacting boson model was introduced within the context of

generating spectra that could be compared to nuclei, in order to extract information about

the collective structure of those nuclei. Prom there, more degrees of freedom were added to

the model to describe behavior that had recently been observed, and seemed to lie outside

the scope of the sd-IBM-2. One topic that has recently been of interest in nuclear structure

is the study of quantum phase transitions in nuclei [58, 59, 60, 61, 62, 63, 64, 65, 66, 67].

The characteristics of such phase transitions can be studied by expanding the interacting

boson model in a way that has not been discussed in previous chapters, and this expansion

is to large boson numbers.

The number of bosons included in an sd-IBM-1 fit for a nucleus is determined

by how many valence particles or holes the nucleus is away from proton and neutron

shells. The standard proton and neutron shell closures occur at 2, 8, 20, 28, 50, 82, and

126 nucleons. Knowing these numbers, it might be difficult to imagine why a calculation

should ever require more than 22 bosons total. The answer to this is that large boson

calculations are not intended to generate spectra for comparing to experimental data, but

rather to understand the behavior of certain observables when the system has a large

number of particles. If such behavior is characteristic of a first or second order quantum

phase transition, which can only be determined by looking at how the behavior evolves

into an equivalent infinite-sized system, then the system can be scaled back down to a

realistic number of particles to see if any experimentally observed behavior in nuclei could

be characteristic of such a phase transition.

Before going into the details of large boson number calculations, and how they

apply to the study of quantum phase transitions in nuclei, it is worthwhile to introduce

quantum phase transitions in general.

101

First Order Second Order

Figure 6.1: Illustration of first and second order phase transitional behavior.

6.1 Quantum phase transitions in nuclei

The nuclei discussed in the first few chapters were weakly collective vibrational

nuclei. The collective structure of an even-even nucleus depends rather strongly on the

number of valence particles in the nucleus, where the number of valence particles is defined

by how far away from proton and neutron shell closures the nucleus is. When the nucleus

is near a closed shell, like in the cases of 94Mo and 140Nd, it has a small number of

valence nucleons, and is weakly collective. As the number of valence nucleons increases,

the structure changes from spherical with vibrational structure, to well-deformed, where

rotational structure is observed.

The discussion of a quantum phase transition in nuclei refers to the changes in the

equilibrium structure of the nucleus: the change from spherical to deformed. This differs

from thermal phase transitions, where the energy of the system controls the transition. A

phase transition occurs when a system observable changes rapidly as a control parameter

is changed. Such a system observable is called an order parameter. In the case of a thermal

phase transition, the control parameter is typically a value like pressure or temperature,

but in the case of nuclei the main control parameter available is the number of valence

nucleons. As discussed previously, the equilibrium shape of the nucleus strongly depends

on its collectivity, which is determined by the number of valence nucleons.

102

In an infinite system, phase transitions manifest as discontinuities in the order

parameter or one of its derivates. In a first order phase transition, the first derivative of the

system is discontinuous at the critical point, which is the value of the control parameter

where the transition occurs. In a second order phase transition, the second derivative

of the system is discontinuous at the critical point. These discontinuities are only really

discontinuities in the infinite particle limit. If the system is scaled back to a finite size,

they get smoothed out, and distinguishing between the two types of phase transition can

be difficult. An illustration showing possible behaviors for the order parameter can be seen

in Fig. 6.1. In the case of nuclei, where there are often only a few valence particles, the

smoothing can be extreme, and finding a good order parameter for distinguishing between

the two types of phase transition can be a challenge.

The quantum phase transition that is observed in nuclei occurs as the equilib-

rium shape of the nucleus changes from spherical to deformed, which corresponds to the

excitations changing from vibrational to rotational. The Hamiltonian for such a system

can take the following form:

H = (1 - C)#vib + C^rot (6.1)

The control parameter £ in the system allows the Hamiltonian to transition between the

two types of motion. The symmetries of the sd-IBM-1 can be used to represent the two

terms in this Hamiltonian, and such a transitional Hamiltonian can easily be constructed

in that model.

6.2 Phase transitions in the interacting boson model

One aspect of the sd-IBM-1 that has not yet been discussed in this work is

the coherent state formalism, which allows one to connect an algebraic system with the

geometric analog. The method involves constructing classical limits of operators that

belong to a compact Lie algebra, and it can be applied to quantum systems of identical

particles that are invariant under a symmetry group SU(N) [68, 69, 70]. In the case of

the sd-IBM-1, the U(6) algebra is directly related to the geometric parameters f3 and 7

from the Bohr Hamiltonian. A normalized projected coherent state can then be defined

in terms of the boson operators [71]:

\Nf3-y) = 1 N ( V + / 3 c o s ( 7 ) 4 + 4 = / ? s i n ( 7 ) ( 4 + 4 ) ) |0> (6-2)

x / M / T T ^ V V 2 J

103

U ( 5 ) - S U ( 3 )

Figure 6.2: Energy potential surface between U(5) and SU(3). Calculated for 400 bosons and 7 = 0. Cs, Cc and Ca refer to the spinodal, critical, and anti-spinodal points respec-tively.

An upper bound for the ground state energy of a U(6) Hamiltonian can be found by

minimizing the expectation value of Hamiltonian using the coherent state:

(Ar/37|£|jV/?7> E(N,(3,7) = (6.3)

By minimizing this energy, an energy potential surface can be found for the ground state of

the system. The Hamiltonian chosen for this research is one that spans the three dynamical

symmetries of the sd-IBM-1, and it was introduced in Eq. 2.2:

H = c((l-Qnd--£RQX-QX (6.4)

Qx = [at d p ) + [rftg](2) + xfrftrfp) (6.5)

With this Hamiltonian, the energy potential surface can be calculated in terms of the

geometric parameters (3 and 7, as well as the Hamiltonian parameters £ and x [8, 9, 25]:

N(32 C ( x 2 - 3 ) A N(N-1)C E(N,0,7) = l + f32 1 -

4AT-4NC + C/ (1 +(32)2(4N-4NC + C)

2/o4 x 4/5 — 4y cos(37) + ~x P (6.6)

This energy potential for the ground state is very instructive for the discussion of

quantum phase transitions in nuclei. Plotting the energy potential surface as a function of

the deformation parameter /3 reveals the nature of the transition from spherical deformed,

in the U(5)-0(6) region and in the U(5)-SU(3) region.

104

I

0.8

0.6

^ 0.4 cd. ^ 0.2

-0.2

-0.4

N=400

Figure 6.3: Energy potential surface between U(5) and 0(6). Calculated for 400 bosons and 7 = 0. (c refers to the critical point.

A plot of the energy surface between U(5)-SU(3) can be seen in Fig. 6.2. The important feature of this plot is the finite barrier between the minima at f3 = 0 and f3 > 0. The minimum at (3 = 0 refers to the spherical shape that the nucleus has near the U(5) symmetry. The minimum at (3 > 0 refers to the deformed shape that takes over near the SU(3) symmetry. The range of £ between ( s and ( a is the shape coexistence region, where there are two visible minima. The specific point Q is the critical point, which occurs where the two minima have the same energy. It is at this point that the nucleus transitions from spherical to deformed. In the coherent state formalism, £c has the following value:

28N Cc = (6.7) 56(N - 1) + x

2 (5 + 2N)

The value of £ where the deformed minimum first appears is called the spinodal

point, but there is not a good analytic expression for this value. The value £ where

the spherical minimum vanishes is called the anti-spinodal point, £a, and in the coherent

state formalism, it takes the following value:

4 N Ca (6.8) 8N + x2 - 8

The transition from spherical to deformed between U(5) and SU(3) is a first order

phase transition, due to the existence of the finite barrier between the two minima that

represent the two symmetries. This first order phase transition actually occurs for any

X / 0, but for x = 0 a second order phase transition occurs. The transition from U(5) to

0(6) is parameterized by x = 0; a n ( l a plot of the energy surface between U(5)-0(6) can

105

0(6)

Figure 6.4: Casten triangle with the first and second order phase transitions marked. The double lines roughly trace the spinodal and anti-spinodal points of the first order phase transition.

be seen in Fig. 6.3. The main feature to note is that no finite barrier is visible between

the two minima. The critical point is where the minimum moves from (3 = 0 to @ > 0.

The sd-IBM-1 is a finite system, where the number of particles in the system

can be chosen. The symmetry limits of the model describe the general types of collective

motion in nuclei very well, and a transitional Hamiltonian like in Eq. 6.4 can be used to

better understand the behavior of nuclei as they transition from spherical to deformed.

The Casten triangle, which has a symmetry limit at each corner, was introduced in Chapter

2 as a way of visualizing the parameter space between the symmetry limits [14]. With

this discussion of phase transitional behavior in the sd-IBM-1, the triangle can be used as

way of visualizing the first and second order phase transitions. The double lines in Fig.

6.4 roughly illustrates the path of the spinodal and anti-spinodal points of the first order

phase transition. The region between the two lines is the shape coexistence region.

Due to the definable size of the system, one can specify the amount of finite N

smoothing that takes place in the system. By taking the model into the realm of large

boson numbers, this smoothing can be reduced, and the behavior of a variety of observables

can be better understood. For example, in the large boson limit, an observable might show

characteristics of a first order phase transition between U(5) and SU(3), and characteristics

106

of a second order phase transition between U(5) and 0(6). The next step would be to

decrease the number of bosons to a realistic size in order to identify if the two types of

phase transition would be distinguishable in real nuclei.

Another possibility would be that an observable in a realistic system size vaguely

shows characteristics of a phase transition. Without increasing the system size, it would

be difficult to identify whether the observable is actually useful as an order parameter.

There is a possibility that such behavior remains smooth even in large N limit, and that

the observable is not useful for identifying phase transitional behavior in nuclei.

The only way to test different observables in this way is to actually run a cal-

culation in the sd-IBM-1 with a large number of bosons. Over the course of this work, a

code called ibar was written for this purpose, and it can currently handle calculations for

up to 400 bosons. A number of numerical difficulties had to be overcome that were not

present in smaller scale calculations, and the specifics of such difficulties are addressed in

the following section. The remainder of the chapter contains details on how to perform

sd-IBM-1 calculations in general, and also how to expand the system to a large number

of bosons. It should be noted that the standard sd-IBM-1 code is called phint , and it is

entirely satisfactory for calculations with a realistic number of bosons [72].

6.3 Basis States

As discussed in Chapter 2, the group chain that is most useful for enumerating

a basis for sd-IBM-1 calculations is the following:

17(6) D U{5) D 0(5) D 0(3) D 0(2) (6.9)

U(6) is characterized by the total boson quantum number N, U(5) by the number of d-

bosons in a state rid, 0(5) by the seniority v, 0(3) by the angular momentum L, and 0(2)

by the magnetic substate quantum number m^. The seniority quantum number v can also

be replaced by np, which is the number of pairs of d-bosons, where the pairs couple to

angular momentum 0. 0(5) is not fully reducible with respect to 0(3), so an additional

quantum number is needed to fully characterize the chain, and this quantum number will

be «a- It refers to the number of d-boson triplets, where the triplets couple to angular

momentum 0, but it could be defined in many other ways.

When constructing the set of basis states to be used for computing matrix ele-

ments of an sd-IBM-1 Hamiltonian, it is convenient to examine a single angular momentum

107

L at a time. The number of d-bosons, rid can take on all integer values from 0 to N. The

seniority quantum number v represents the seniority of the state and can have the following

values: [73]

v = nd, rid - 2, rid- 4, . . . , 0 if nd is even.

v = nd, rid-2, rid- 4, . . . , 1 if nd is odd. (6.10)

Using v and rid, an alternate quantum number np is defined: np = |(n<f — v). In order to

find the allowed values for the additional quantum number nA, the index A is used, which

takes on values between L^pJ and L. One additional restriction on A is that L = 2A — 1

is not allowed [17]. Using the allowed values of A, n^ is defined by the following.

nA = ^(nd - 2np - A) (6.11)

After calculating the possible values for nA, two final checks are made for every state in

the basis: nA > 0 and rid — 2np + 3ua + A.

6.4 Wigner-Eckart theorem

Before getting into the Hamiltonian itself, it is important to remark on the calcu-

lation of matrix elements of tensor products in general. All operators in the sd-IBM-1 are

constructed from s and d-boson operators. One of the benefits of these operators being

spherical tensors is that the Wigner-Eckart theorem can be applied to matrix elements of

tensor products of the operators. The Wigner-Eckart theorem separates matrix elements

into two parts: a reduced matrix elements which contains all physical information, and a

Clebsch-Gordan coefficient which contains the geometric information.

(ndnpnALm\T^\n'dn'pn'AL'm') = {~l)L~M f L k L' ) (ndnpnAL\\T^\\n'dn'pn'AL') \ — m Km I

(6.12)

The tensor products of boson operators within the reduced matrix elements can then be

decoupled in to sums of reduced matrix elements of the operators that made up the tensor

product. The 3-j symbol that appears in 6.12 can be directly related to Clebsch-Gordan

coefficients [17].

ji 32 h \ (-i)h-h-m3 1 ~ (jirn1j2m2\jij2h,-m3) (6.13) mi rri2 m3

108

In the sd-IBM-1, all terms in the Hamiltonian couple to angular momentum 0, which is

why there are no cross terms between states of different L. This makes both k and the

projection quantum number k equal to 0, and Eq. 6.12 reduces to the following:

(ndn0nALm\T^O)\n'dn'0n'ALm) = - ^ = 2 = ( n d n ^ n A L | | T 0( 0 ) | | n ^ n ^ L ) (6.14)

The value of the matrix element does not depend on the quantum number m, which is

why the matrix elements that appear in Eq. 6.24 have no dependence on m.

6.5 Hamiltonian

One form of sd-IBM-1 Hamiltonian which is often referred to as the consistent-

Q formalism (CQF) Hamiltonian was given in Eq. 2.2 [11, 12]. That Hamiltonian was

constructed specifically to transition between vibrational and rotational structure at ap-

proximately C = and a more general form that includes additional multipole terms is

the following:

H = e 0 + ehd + kQx • Qx + axL • L + a3?W . + a 4 T ( 4 ) • T ( 4 ) (6.15)

where k is sometimes referred to as a2. The name consistent-Q formalism implies that

the parameter x in the quadrupole operator will be equal to the x i n the E2 transition

operator. The operators in this Hamiltonian are defined in terms of s and d-boson creation

and annihilation operators:

hd = d) • d

Q x - Q x = ( V i p ) + [rftg](2) + x[</tJ](2)) . ( j s t J ] (2) + [ d t s 1(2) + x [ d td~](2))

T(3).f( 3) = [d^(3).[dtJ](3) f W • T (4 ) = [d^d]^ • [rftrfjW (6.16)

The alternate parameter £ and the scale factor c from Eq. 2.2 can be reintroduced into

the Hamiltonian using the following formulas:

e = c ( l - 0

109

The scalar products in the Hamiltonian be represented as tensor products using the fol-

lowing equation: [17]

V(K) . y(K) = (_!)*v /21( + 1(UM X y (6.18)

It is important to note that the d-boson creation and annihilation operators have angular

momentum 2. A tensor product of two $ or d operators can couple up to angular mo-

mentum 4. The scalar product implies a coupling to 0 angular momentum. The d-boson

number operator can be rewritten in the following way:

nd = S-d = V5[dfd](0) (6.19)

The s-boson operators have angular momentum 0, so a tensor product with two s or s^

operators will only couple to 0, and can be represented as a scalar product.

sf -5 = [st$](°) (6.20)

A tensor product of an s-boson operator s or s^ with a d-boson operator d or d) must

couple to angular momentum 2, and there can be no scalar product between the two. A

convenient formula for recoupling angular momentum can be used to convert the CQF

Hamiltonian into normal order form, which is more convenient for the eventual evaluation

of Hamiltonian matrix elements. Normal ordering simply means that all creation operators

are on the left side in the tensor product, and all annihilation operators are on the right

[74].

[d}d]^-[Sd\^ = 2-^[d}d\^ + {2k + l) Y | g 2 L } ^ ^ ' [ ^ ] { L ) ( 6 ' 2 1 )

The normal order Hamiltonian can be written in the following way:

H = c0 + c i ^ s p ) + c2v /5[c^dp) + c3

+ CsfdW]*4* • [ddp + C6 ( V d t p ) . [dsf) + [dtst](2) .

+ C7V5 (Vdt](0) . J55J(0) + [stst](0) . + Cg[dt5t](2) . f^](2)

+ C9[stst](0) • [ss](0) (6.22)

110

Once the CQF Hamiltonian in Eq. 6.15 is converted to normal order form, the coefficients

of the Hamiltonian will be the following:

Co = eo

ci = 5a2

7 9

c2 = e + 6ai + (1 + X + +

2 7 9 c3 = - 6 a i + % a2 - - a 3 + - a 4

3 2 4 1 8

c4 = - 3 a i - —x 0-2 + r a 3 + — a4 14 5 35 2 2 1 1 c5 = 4ai + - X a2 + — a3 + — a4

c6 = 2xa2

c7 = a2

c8 = 2a2

c9 = 0 (6.23)

This is the Hamiltonian form used in calculations, because the U(5) basis states can be

applied to calculate the matrix elements of this Hamiltonian [17].

{nsndn0nA\ [s^]'0' \nsndn0nA) = ns

{nsndnl3nA\ v^d^0) \nsndn0nA) = nd

(nsndn/3riA\ 5^^]^ • [dd\^ \nsndn0na) = nd(nd + 3) - v(v + 3)

<(ns + 2)(nd - 2)(n 0 - l )n A | VE[dd}^ • [sV] ( 0 ) \nsndnpnA) = y/nd{nd + 3) - v(v + 3)

x y/ns -f Wns + 2

(nsndnpn&\ V^t^d]^ • [s^]^ \nsndn0nA) = ndns

(nsndnpnA\ [ s V ] ^ • [ss](0) \nsndn^n^) = ns{ns - 1) (6.24)

The matrix elements of the three Hamiltonian terms associated with the parameters c3,

c4, and C5 can be found by relating those terms to the following Hamiltonian written in

terms of Casimir operators.

H = Eo + e'C^Uib)) + a ' (5)) - |Ci(E7(5))

+ 0 (-\C2(0(5)) + C2(U(5)) - Ci(£/(5))) + j Q c 2 ( 0 ( 3 ) ) - 6 ^ ( ^ ( 5 ) ) ) (6.25)

111

The eigenvalues of this Hamiltonian are

E0 + e'nd + a'^nd(nd - 1) + 0 (nd{nd + 3) - v(v + 3)) + 7 ' {L{L + 1) - 6n d ) (6.26)

The parameters a', /?', and 7' are related to the parameters C3, C4, and C5 by the following

transformation [6]:

4 3 = + -c5

ft = f <* - + | C 5

y = - i c 4 + yC5 (6.27)

By transforming the three [Sd^]^ • [dd]^ terms of the normal order Hamiltonian into

Casimir form, the matrix elements of the Casimir terms can be used in place of the matrix

elements of those three normal order terms.

The remaining matrix element of the normal order Hamiltonian is

{nsndrij3nAL\\[d^sty2) • [dd]^||n'5n';n^n'AL), which is calculated by fully decoupling the

reduced matrix element. The first step is decoupling the scalar product [17]:

(ndnpnAL\\T^ • U^\\n'dn'0n'AL) = £ ( -1 ) L"-L

The next step is to decouple the remaining [dd]^> reduced matrix element:

(ndnpnAL\\ (t™ X UM)™ ||n'dn'0n'AL') = (-l)^'+fe)^2fcTT £ nd'np<n'k'L"

^ * 1 (6.29) L L L I

At this point, the reduced matrix element (nsndnpnAL\\[d^sty2). [dd]^^n^n'jn'^n'^L) has

been decoupled into a sum of products of reduced matrix elements of the individual boson

operators, and 6-j symbols. The reduced matrix elements of the boson operators can be

related to the coefficients of fractional parentage of identical bosons.

112

6.6 Reduced Matrix Elements

The matrix elements of s and s are simple to evaluate:

n$,nfs — 1

(Lnsndnpn&\st\n'sridripn&L) = y/n'a + 1 <5ns,<+i (6.30)

The reduced matrix elements of d and dcan be related to each other:

(Lnsndn0nA\\d\\n'sn'dn'0n'AL') = (risridri0riAL'\\d^\\nsndn0n&L) (6.31)

The reduced matrix element {nsndn0nAL\\<$\\n'sn'dn'0n'AL') is found using a recursion for-

mula. An efficient algorithm that uses isoscalar factors for calculating these coefficients is

particularly well suited for identical boson systems with well-defined seniority [75]. The

procedure begins by first calculating the multiplicity of states for a given angular mo-

mentum L and seniority v. These can be found by solving for the coefficients of certain

Gaussian polynomials [76].

6.6.1 Multipl icity of States

The first step in calculating the multiplicity of states for a given angular momen-

tum L and seniority v is to solve for the coefficients pm of a Gaussian polynomial [76].

The allowed values of m range from 1 to nl, where n is the number of bosons, and I is the

angular momentum of each boson.

min(r,p—r) p Pm= Y1 S ~ Y , S

s=1 s=max(r,p—r)+1 s l m s | m

where p = 21 + 1, r = n, and s | m means that only terms where s is divisible by m are

included. Next, the coefficients cm are calculated recursively using pm.

j m— 1 Cm = / , Pm—scs m z—' s = 0

where co = 1. The multiplicity of states for n bosons and angular momentum L is —

c^ni-L-i)- Calculations in ibar require the multiplicty of states for a seniority v and

angular momentum L, which is calculated by C(vl_L)-C(vi_L_i)-C({v_2) l-L)+c{(

assuming v >2. For v < 2, the multiplicity is just C ( v i _ L ) — F o r a d-boson system,

I should be set to 2.

113

6.6.2 Isoscalar factors

Isoscalar factors (ISFs) are closely related to the required reduced matrix elements

of the single d-boson creation operator, and can solved for using the following recursive

formula [75]: ( 1 ) ( t , - l )

I a\L\ P(ai Li ai Lx L)

~ \JvP(a[ L[ ai U L)

where P(a[ L[ a\ L\ L) is defined by

P(a[ L\ ai Li L) = Sa>1SViM + {-l)L+L'^v - 1 ) ^ ( 2 ^ + l)(2Li + 1)

L {2L + l){N -A + 2v)

I OL2L2

assuming the following initial values:

(1) {v-2)

I CX2L2

(v-1)

= 1 (L = 0,2,4,..., 21)

This set of coefficients will be overcomplete, so Gram-Schmidt orthogonalization is used to

calculate the orthogonal ISFs. The a = 0 set of coefficients for a given L and v is accepted

without orthogonalization. For the a > 0 set of coefficients when there is a multiplicity

greater than 1, all previous sets of coefficients with the same L and v must be projected

out. If only zeros are left after projecting out sets of coefficients, then the new set was

not unique, and should be ignored. If non-zero values are left, then a new set of ISFs has

been found, and the whole set should be normalized.

This procedure uses the multiplicity of states to identify when all sets of ISFs

have been calculated. The relationship between reduced matrix elements of the d-boson

creation operator and ISFs is

(vvaL\\S\\{v - 1)( v — 1 )oi L') = \fv 1 (1) ( v - 1 )

2 a'L' (6.32)

6.6.3 Arbitrary precision

When using standard double-precision floating point arithmetic for calculating these reduced matrix elements, numerical errors begin to dominate in the coefficients

114

beyond 50 bosons, due to the procedure being recursive. At each recursion, the ISFs will

have fractionally less precision than the previous recursion, and eventually, not enough

precision will be left to orthogonalize the set. For the program ibar , an arbitrary precision

arithmetic library called GMP was used to calculate the coefficients to a user-defined

precision [77].

Beyond the precision loss from using a recursive formulas, the calculation of 6-j

symbols, which are essential ingredients in angular momentum coupling, can result in

significant precision loss. They can be calculated in the following way [17]:

= A(Ji J2 J3)A(JU2J3)A(ji J2h)A(jij2 J:i) ^ 1)1 (6'33)

where A(jij2j3) are the triangle coefficients:

V U 1 + J2 + J3 + 1 ) !

The factor f(t) in the denominator of Eq. 6.33 is a large product of factorials:

M = (t - Ji - J2 - Js)Kt - J i - h - J3)'.(t - ji -J2- j3y.(t - h - j2 - J3)! X (ji + h + Ji + J2- t)\{j2 + j3 + J2 + J3- t)\(j! + j3 + Ji + J3- t)\ (6.35)

where t takes on all integer values where the factorials have non-negative arguments.

Double-precision floating point numbers can have exponents that range from approxi-

mately 10 - 3 0 8 to 10308. A system with 400 d-bosons can couple to angular momentum

800, and 800! has about 1977 decimal digits, so beyond precision loss from alternating

signs in the t sum, there is some risk of overflow when multiplying the factorials. This

is resolved by taking the natural logarithm of the factorials, and adding those together,

instead of multiplying the factorials. As a simple example of this process:

In ^ = ln(a!) + ln(6!) - ln(c!) - In(d\) (6.36)

Also, the calculation of the factorial itself can be rewritten:

(a \ a

JJjfc) =5^1n(fc) (6.37) fc=i / fc=i

Once the factorials in the numerator and denominator of the sum in Eq. 6.33 have been

combined in the logarithm, an exponential is taken, and the terms are added together.

115

There is precision loss from the alternating signs, so calculating these coefficients with

high-precision is helpful. Logarithms and exponentials have not yet been added to GMP

at the time of this writing, so the exponentials were calculated in the form of a Taylor

expansion. The logarithms were calculated as the inverse of the exponential using Newton's

method. To avoid precision loss from these 6-j symbols in ibar , a lookup table is saved at

the same time that the reduced matrix elements of eft are calculated in high-precision.

The reduced matrix elements calculated by the described algorithm are ordered

by a quantum number a rather than In order to use these elements in a calculation,

the discussion about in Section 6.3 is used to define the relationship between and

a. Also, rather than storing every possible reduced matrix element (ndvaL\\d) Wn'^v' a' L'),

it is adequate to keep only (vvaL\\d)\\(v — l)(v — 1 )a'L'). The following relations define

the reduced matrix elements in terms of those where nd = v [17]:

{ndvaL\\S\\{nd - l)(t, - 1 )a'L') = y j " ' 1 * ^ 1 (waL\\<fl\\(v - 1)(« - 1 )a'L') (6.38)

(6.39)

With a procedure known for calculating all Hamiltonian matrix elements, the full Hamil-

tonian can be calculated. The basis can get quite large for system with a large number of

bosons. For example, the L = 20 Hamiltonian for 400 bosons is a 138007x138007 matrix,

and this would require 145.3 GB of storage to simply save the matrix if it was not com-

pressed. The Hamiltonian matrix is sparse, so ibar compresses the matrix in a way that

leaves out all zero matrix elements.

The next step is diagonalizing the Hamiltonian, and the library ARPACK is used

to iteratively diagonalize the Hamiltonian [78]. ARPACK uses an implicitly restarted

Lanczos method for diagonalization, and the library asks the user to perform the matrix

multiplication manually. This is fortunate, as there would be no way for ARPACK to

know about the custom matrix compression used in ibar .

For full diagonalization of a matrix, an iterative method like Lanczos is not de-

sirable, but full diagonalization is somewhat of a special case in ibar , in that the program

was designed for large boson calculations. Memory constraints make full diagonalization

of large boson systems unrealistic, due to the necessity of storing the full matrix in un-

compressed form. Solving for all eigenvectors in a small system should work well in ibar ,

116

but solving for all eigenvectors for a large system will be unrealistically slow, and the user

will likely run out of available memory.

6.7 Transition matrix elements

The E2 transition operator in ibar is defined in the following way:

T(E2) = eB ( V s ] ® + [ sW 2 ) + x [ d W 2 ) ) (6.40)

In the consistent Q-formalism, x of this operator is defined to be the same as the x that

appears in the Hamiltonian. Changing x to a different value is an option, however, so

this operator has up to two parameters. The reduced matrix elements of the operator are

calculated in a similar way that the Hamiltonian reduced matrix elements were calculated.

The difference is that the Hamiltonian matrix elements used basis states, where transition

operators use eigenstates of the Hamiltonian. This results in the evaluation of a large sum

of cross terms between basis states for each transition matrix element.

Labeling the eigenstates by |La) allows us to define the R(E2) and B(E2) in

terms of reduced matrix elements of the E2 operator:

R(E2-La -> L'a,) = (L'a'\\T^\\La)

B{E2-La L'a,) = -^(L'a'WT^WLa)2 (6.41)

Electric monopole transitions can also be calculated, and the operator is defined

in terms of the d-boson number operator:

T(E0) = f30nd (6.42)

The rid operator does not change any U(5) quantum numbers, so the only terms in the

evaluation that can be nonzero have the same basis state on either side of the reduced

matrix elements.

6.8 System size

The software ibar has been tested for up to 400 bosons. The calculation for d)

reduced matrix elements were performed with 300 decimal digits of precision, although 100

digits would probably have been adequate for 400 bosons. The calculation was stopped

at 400 bosons, due to memory and time constraints. The sets of reduced matrix elements

117

grow larger as more bosons are added, and just storing 400 bosons worth of binary format

reduced matrix elements in memory requires 500 MB of RAM. As a rough estimate, the

storage size for 600 bosons worth of reduced matrix elements would be around 1.6 GB,

and the calculation of such coefficients could take many weeks to accomplish.

118

Chapter 7

Electric monopole transition strengths in ibar

In Chapter 6 the concept of quantum phase transitions in nuclei was introduced,

and the framework of the interacting boson model was suggested as a means to probe

quantum phase transitional behavior. The code ibar was developed for the purpose of

performing large boson calculations in the sd-IBM-1, and it can be applied to the discussion

of characteristics of phase transitions in nuclei. The code ibar has been used in a number

of investigations into the behavior of the interacting boson model in large systems [79,

80, 81, 82, 83]. One observable that has been of recent interest in the discussion of

phase transitions in nuclei is electric monopole (E0) transition strengths. A comparison

of isotopic trends to the interacting boson model provided an interesting explanation for

sharp rises in E0 strengths in the transitional nuclei [84], This explanation can benefit

from an examination with large boson calculations, but a different interpretation for the

E0 strengths should first be introduced.

7.1 EO transition strengths in transitional nuclei

Electric monopole transitions are a type of transition in nuclei that does not

change the total angular momentum of the nucleus. Unlike E2 and Ml transitions, this

decay mode is forbidden for single gamma-rays. Photons are spin 1 particles, and carry

at least 1 unit of angular momentum away from the nucleus, so a monopole transition

with photons is not possible. Instead, this decay mode is typically detected by electrons

that carry energy away from the nucleus. The internal conversion corrections discussed in

Section 4.2.5 dealt with the same process of energy transfer to inner electrons, but for Ml

and E2 transitions instead.

119

The EO transition operator in radial form is the following:

T(£0) = J > r f (7.1) i

It is clear from the definition of this operator that the transition strength between two

states will strongly depend on the radial distribution of those two states. A survey of EO

transition strengths in a variety of nuclei has found that transitional nuclei can have large

EO strengths between the and 0+ states. One interpretation of this behavior is that

the transition strength is due to shape mixing with intruder states [85].

If an intruder state that was formed from configurations that include a higher

lying oscillator shell has a similar energy to the ground state, then the radial wavefunctions

of those two states will differ. Without an interaction between the two states, the EO

transition matrix element will be small. If the two states are strongly mixed however, the

wavefunctions can have a large radial overlap, and large EO transition matrix elements

are possible. This situation is an example of shape mixing, where spherical and deformed

configurations mix.

7.1.1 Alternative interpretation for large EO strengths

An alternative interpretation for large EO strengths in transitional nuclei can be

found in the interacting boson model [84]. The EO transition operator in the sd-IBM-1

can be defined as the following:

T(E0) = p0nd (7.2)

where nd is the d-boson number operator. The Hamiltonian used in the following discussion

is the same as the Hamiltonian that was introduced in Chapter 2:

H = c ( ( l - O n d - ^ Q X - Q X S ) (7.3)

QX = [ s t J ] ( 2 ) + [dt§](2) + x[rftJ](2) (7.4)

By fitting a variety of nuclei using the energy ratio i?4/2 = , and assuming that

X = — a value for £ could be found for those nuclei. Using a simple 10 boson calculation,

and plotting the experimental EO transition strengths with the curve, a trend becomes

apparent. Fig. 7.1 shows the sd-IBM-1 curve with the data. It is clear from the select

nuclei shown, that the EO transition strengths become large in transitional region, in a way

that parallels what is found in the sd-IBM-1. This interpretation of the large EO strengths

120

p2(E0; 0+2-*0+i

U(5) to SU(3) 140

[ 10 3 ] 120

100 fN °-80

60

40

20

0

| € > S r

: 1 1 M o 102 E3

; i t S m

: • G d / / ,<®'54 "—

A Z r

1514/ A152 / / A* 98 loop / /

J/ t / 1 5 0

9 6 0 / W

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Zeta

Figure 7.1: Comparison of select nuclei with sd-IBM-1 EO transition strength curve for 10 bosons. Figure directly adapted from [84], with most data coming from [85].

200-180 ^ 160-

'140-120-

-100-80^ 60-

S 40-<Q- 20-

0

o t

A=100-176

-Zr

Mo

R u JCd

G d

Sm rTe

i Er

- T e - M o - G d

- R u

- C d

-Sm - o — P d

- o — Z r

- • — E r

50 55 60 65 70 75 80 85 90 95 100 105 Neutron Number

Figure 7.2: E0 transition strengths as a function of neutron number. Figure is from [86], with data adapted from [87].

does not depend on shape mixing with intruder states, as the model does not know about

other oscillator shells.

It is worth noting that the E0 data shown in Fig. 7.1 is not the full set of data

available, even from that mass region. Unfortunately, a more complete set cannot be

shown in comparison to a simple sd-IBM-1 curve like in that figure, as too many other

121

Figure 7.3: Surface plot of EO behavior for 10 bosons in the sd-IBM-1. The EO strength shown is the sum p2(E0; —> 0^) + p2{E0] 0^ —> 0+) in order to account for EO frag-menting near the U(5)-0(6) leg.

factors play a role. The more complete set of data can be seen in Fig. 7.2.

Looking back at the sd-IBM-1 comparison in Fig. 7.1, some interesting features

should be noted in the EO curve. The transition strength becomes large in the transition

region, and seems to remain large even as £ goes to 1. There is not yet enough experimental

data to test out whether such behavior occurs in nature, and having EO strengths for a

chain of isotopes all the way out to strongly deformed would help in understanding the

trend. This feature cannot be removed from the sd-IBM-1, as it is built into the Q • Q

term of the Hamiltonian.

A surface plot showing the EO transition strength trends can be seen in Fig. 7.3.

The EO transition strengths for the to 0* transition have been added to the to Of

transition, to account for fragmenting of the E0 strength near the U(5)-0(6) leg of the

square. On the U(5)-0(6) leg itself, the OjJ" and O3" states actually cross, but the summing

resolves the issue. The important feature to note in this figure is that the E0 transition

strength remains large in the deformed limit, for both 0(6) and SU(3).

122

p 2 ( E 0 ; ( £ - O

Figure 7.4: Plot of EO transition strengths for 100 bosons in the sd-IBM-1. One curve shows the U(5)-0(6) leg of the triangle, and the other show U(6)-SU(3). For the U(5)-0(6) leg, the U(5) state crosses many 0 + states as £ increases, so the strength gets fragmented. The large EO strength is simply tracked along this leg, since the state is not always O^.

The behavior of the EO transition strength between spherical and deformed looks

similar to the way an order parameter for a first order phase transition behaves. To better

understand the features of the EO curve, it is useful to look into the large boson limit, and

those calculations will performed using ibar .

7.2 EO transition strengths for large boson numbers

The first test is to switch the system size to 100 bosons, and look at how the EO

transition strength curves look from U(5) to 0(6), and U(5) to SU(3). A plot showing

this result can be seen in Fig. 7.4. For the U(5)-0(6) curve, the U(5) 0^ state crosses

many higher lying 0 + states as £ increases, so the strength gets fragmented. The large

EO is simply tracked along this leg, and the state shown is not necessarily Oj. The curve

for U(5)-0(6) looks very much like the behavior of an order parameter for a second order

phase transition, which is shown in Fig. 6.1. Finite N effects are still visible however, as

the connection between the small EO on the left and the sharp rise on the right is still

123

P 2 ( E 0 ; O; * - < ) / N

0.3 N=r N=4 N=1 N=4

£ N 2

0.1

0.2

0.2 0.4 I

0.6 0.8 1.0

Figure 7.5: Plot of EO transition strengths for 10, 40, 100, and 400 bosons in the sd-IBM-1 between U(5) and 0(6). The E0 strengths have been scaled down by N in order to compare the curves.

smooth. Presumably, as N —> oo, that connection will become sharp. Calculated E0

strengths between the U(5) and 0(6) symmetries for up to 400 bosons can be seen in Fig.

The U(5)-SU(3) leg, which should show characteristics of a first order phase

transition, seems to have more complex behavior, and will be the focus of the remaining

discussion. In Fig. 7.4, a sharp rise in E0 strength is visible in the transition region between

U(5) and SU(3). This behavior is resembles the first order phase transition behavior from

Fig. 6.1, but an additional feature seems to appear in transition region, and it is worth

investigating this feature further.

7.3 Features of E0 strengths between U(5) and SU(3)

To show the evolution of the features between U(5) and SU(3), a plot was made

for the EO curve between those two symmetries for four different system sizes, and the

curves for these four system sizes can be seen in Fig. 7.6. What starts as a ripple near the

top of the 10 boson E0 curve appears to evolve into more of a peak as the boson number

is increased. A smooth shelf feature is visible in the 10 boson calculation, which becomes

7.5.

124

p2(E0; Oj—•Oj)/N

Figure 7.6: Plot of EO transition strengths for 10, 40, 100, and 400 bosons in the sd-IBM-1 between U(5) and SU(3). The E0 strengths have been scaled down by N in order to compare the curves.

more like a step as the system size increases. The behavior of this step in a large system

size appears to resemble the first order phase transitional behavior from Fig. 6.1, but this

does not explain the peak feature.

A zoomed in view of the peak and shelf feature in the 400 boson calculation can

be seen in Fig. 7.7. The peak feature no longer seems to rest on top of the shelf feature,

and continuing into the large boson limit, the peak feature and shelf feature may separate

entirely. The observed behavior suggests that the two features are due to different factors

in the calculation. Looking into the energies of the first three 0 + states gives a hint into

what may be occurring, and the energy curve can be seen in Fig. 7.8. The energies

shown are not shifted, which means the binding enegy of the system is not subtracted

out. Eigenvalues of an sd-IBM-1 Hamiltonian do not generally have E(O^) — 0, so the

calculated value is typically subtracted from all energies.

Showing the unshifted energies in Fig. 7.8 is helpful because it allows discussion

about avoided crossings in the three 0+ states. Comparing Figs. 7.7 and 7.8, it appears

as though the peak feature in the E0 curve occurs at the same £ that the and 0^

curves cross. Similarly, the shelf feature appears to form at the same £ that the 02 and

curves cross. To better understand why the crossings correlate to sharp changes in the

125

p2(EO; Oj-*Oj")/N

Figure 7.7: Plot of EO transition strengths for 400 bosons in the sd-IBM-1 between U(5) and SU(3).

-0 .6

-0 .7

- 0 . 8

-0 .9 E(0+n)

- 1 . 0

- 1 . 1

- 1 . 2 0.466 0.468 0.470 0.472 0.474 0.476 0.478

Unshifted Energies

Figure 7.8: Behavior of the energies of the first three 0 + states in the sd-IBM-1 between U(5) and SU(3) for 400 bosons. The energies shown are not shifted, which means the binding energy of the system is not subtracted out.

126

EO strength, it is worth looking into the wavefunctions of the 0 + states.

7.4 Wavefunctions of 0+ states between U(5) and SU(3)

In the sd-IBM-1, the quantum numbers that label the basis states are rid, np,

and L. The quantum number np refers to the number of pairs of d-boson, where the

pairs couple to angular momentum 0, and n& refers to the number of triplets of d-bosons,

where the triplets couple to angular momentum 0. For 0 + states specifically, there is some

redundancy in these three quantum numbers. For a given rid, it is clear from np what the

value of tia should be for a given basis state. This means that only two quantum numbers

are needed to label the non-zero components of 0 + states.

When plotting wavefunctions, ordering the components in the same way the basis

was enumerated can make interpreting the physics of the state difficult. For states in a 400

boson system, there are a significant number of non-zero components in the wavefunction,

so a good way of organizing the labels is needed. Having rid along the x-axis of the

plot is useful, because the geometric deformation parameter (3 is directly related to the

expectation value of the rid operator, using the coherent state formalism [8]:

M = Y T p ( 5)

Although components of the wavefunction where n A > 0 do play a role in the EO strength,

the higher look qualitatively similar to = 0, so only those component will be plotted.

In order to emphasize contributions to the E0 strength, rather than plot the wavefunction

component ip(rid,nA), the value ip(nd,riA)y/nd will be plotted instead.

The plot of the ^{rid,Q)^JrTd components for various £ can be seen in Fig. 7.9.

The first detail to note is that the features of the E0 transition strength curve do directly

correlate to the avoided crossings in the unshifted energies. In the component plots, it

is important to note that the smaller rid components correspond to spherical structure,

and larger rid components correspond to deformed structure. The interactions of these

components in the 0]1" and wavefunctions will help explain the behavior of the E0

strengths at the different values of

In plot 1 of Fig. 7.9, which corresponds to £ = 0.46800, the ground state has

mostly spherical structure, occupying the smaller rid components. The state is more

deformed, and its components are spread across higher rid• This value of £ is less than the

point where the first avoiding crossing occurs.

127

Unshifted Energies p 2 ( E 0 ; 0 ^ 0 t ) / N U(5)-SU(3)

N=400

0.474 0.476

60 80 100 120 140

Figure 7.9: Plot of the wavefunction components scaled by y/rid for 400 bosons in the sd-IBM-1 between U(5) and SU(3). Only nDelta — 0 components of ip(rid, nA)y/nd are shown. Unshifted energies are shown in the upper left, and E0 strengths are shown in the upper right. The red components represents the 0^ state, and the blue components represent the Of state. Due to the relationship between (rid) and (3, smaller components correspond to spherical structure, and larger correspond to deformed structure.

128

In plot 2, which corresponds to £ = 0.47116, the first two 0 + states have a large

wavefunction overlap from the avoided crossing. The spherical and deformed components

of the two wavefunctions have opposite phase, but the large overlap in the wavefunc-

tion components results in a large EO strength. The two wavefunctions are by definition

orthogonal, but the y/n^ factor which accounts for the EO operator gives the deformed

components a much larger weight in the matrix element.

In plot 3, which corresponds to ( — 0.47230, the ground state is more deformed,

and the Of state is more spherical. After the avoided crossing, the wavefunctions of

those two states have switched. There is the added factor of interactions that cause the

wavefunction of the Of state to overlap with the wavefunction of the Of state, as well as

all states gradually becoming more deformed due to the influence of ( in the Hamiltonian,

but the rapid change of the ground state from spherical to deformed is primarily due to

the avoided crossing.

In plot 4, which corresponds to £ = 0.47314, the wavefunction of the Of state

has a large overlap with the wavefunction of the Of state, due to the avoided crossing of

those two states. At that point, the Of is transitioning from spherical to deformed, which

means that both the Of and Of states have some deformed structure. Looking at the EO

transition strength curve, one can see that the Of state transitioning to deformed causes

the rise onto the shelf feature.

In plot 5, which corresponds to C = 0.47444, both the Of and Of states are

deformed, and the E0 transition strength nears the maximum of the shelf feature. There

are no additional energy crossings for the Of and Of states as they move on towards the

SU(3) limit. The large E0 strength results from the large overlap in the wavefunctions

and the y/n^ factor.

In plot 6, which corresponds to £ = 0.47700, the states looks the same as plot 5,

with the structure of the two 0+ states being reminiscent of Hermite-Gaussion functions.

As ( moves towards 1, the features in the wavefunction plots move towards higher

7.5 Interpretation of the wavefunctions

The concept of mixing is useful for understanding what occurs during an avoided

energy crossing. Usually when describing mixing between two states, the interaction is

between two basis states, and the amount of mixing can easily be quantified. In this

situation, the two states are eigenvectors of a vastly complex system with thousands of

129

Ota

0& 54% 44%

45% 50%

0 tc 1% 85%

0 1 97% 0%

Table 7.1: Percent overlap of the wavefunctions near the avoided crossing at the transition point between U(5) and SU(3) in the sd-IBM-1. 0+ states are from £ = 0.47, states are from £ = 0.47116, and 0+ states are from £ = 0.472.

interaction matrix elements. To make the discussion clearer, one can calculated the overlap

of the two states at different values of £.

The wavefunctions for the and states at £ = 0.47 will be used as the basis

for this pseudo two-state mixing calculation. The avoided crossing is at £ = 0.47116, which

will be one comparison value, and £ = 0.472 is beyond the crossing and will be the second

comparison value.

The results from this calculation can be seen in Table 7.1, and the labels for the

states are that 0+ states are from £ = 0.47, OjJ" states are from £ = 0.47116, and 0+ states

are from £ = 0.472. In the Oj and 0 ^ basis, the 0^ and are strong mixed. The

state has a large overlap with 0 ^ but not 0^ . The 0^. state has a large overlap with

but not O2"a. It is clear from this picture that the wavefunctions of the 0]1" and are

essentially reversed after the avoided crossing.

7.6 Visualizing the full 0+ wavefunctions

In the wavefunction discussion, the assumption was made that > 0 compo-

nents are qualitatively the same as the tia = 0 components, but it is worth investigating

how the n A > 0 components actually behave. In the wavefunction, the components appear

to have a (—1)"A phase factor, so an additional factor should be added in before attempt-

ing to visualize the wavefunction. The result can be then be plotted in a 3D surface plot,

and a series of such plots can be seen in Fig. 7.10.

It is clear that even though rid and n A are discrete quantum numbers, a contin-

uous behavior is apparent in the two-dimensional wavefunction. The system in fact bears

some resemblance to a two dimensional quantum harmonic oscillator. As one looks at the

130

0

U S 200

U(5)-SU(3) 400 bosons

£=0.85

o.i

/ 0.05

/ u * 220 240 -0.05

200 220

260 280 -0.1 240 / -0.05

260 280 ''"-0.1

0 .. -"

; o.i \

0.05 \

'" ? 0.1

0.05

0

} 0.1

/ 0.05

240

nd 260

280 ' -0.1

> 0.1

/ 0.05

Figure 7.10: Surface plot of 2D 0 + wavefunction components with parameters rid and n A

for £ = 0.85. Calculations were performed for 400 bosons, between the U(5) and SU(3) symmetries. The value of £ places the parameters on the deformed side of the first order phase transition.

131

higher lying excited states, more nodes appear in the rid and directions, and the

state is the first wavefunction with a node in the both directions. Although the calculation

shown in Fig. 7.10 is for £ = 0.85 rather than £ = 1, the states can be associated with

the SU(3) quantum numbers (A,fi). For at least the low lying states in this calculation,

the number of nodes along the rid-axis is equal to ^, and the number of nodes along the

riA-axis is . For higher energies, counting nodes can be difficult due to the discrete

quantum numbers along the axes.

The quantum number rid c a r i be directly related to the deformation parameter (3,

but the quantum number does not have quite as clear a geometric analog. If some way is

found to relate riA to (3 and 7, then it should be possible to derive the potential in terms of

/3 and 7 using these wavefunctions. One of the strengths of i ba r is that the wavefunctions

use the full range of double-precision, so taking the second derivative of the wavefunction

can yield a value of reasonable precision, even on the fringe of the wavefunction, where

it takes on small values. It does not have unlimited precision, however, so any calculated

potential would only be accurate relatively close to the wavefunction. Eventually, if a

relation between and (3 and 7 is found, it could be possible to derive a potential for

the shape coexistence region in terms of f3 and 7.

7.7 Amplitude of the EO features with boson number

The strength of the peak and shelf features visible in the U(5)-SU(3) EO transition

curve depend on the number of bosons in the system. By determining the height of the

peak feature, it is possible to plot the EO transition strength as a function of boson number

in the system. In Fig. 7.11, a plot of two curves can be seen. One curve is simply the

height of EO peak feature, and the second corrects for the shelf contributing to the EO

strength. Both curves can be fit well with a polynomial of only linear and cubic terms:

f(x) = ax + bxs. As N —> 00, the EO strength cannot exceed quadratic behavior, simply

because rid is a one body operator. At these smaller boson numbers, a cubic term is

present, but it should drop out as N becomes large.

To help illustrate this trend, the EO operator can be written in the following way:

T(EO') = (7.6)

A figure of the EO trends using this alteration can be seen in Fig. 7.12. To prevent

confusion, the transition strength is labelled with EO' rather than EO. In the figure, the

132

p2(E0;0^0|)

Figure 7.11: Plot of EO strength of peak feature between U(5) and SU(3) for a range of N. The shelf subtracted curve accounts for the contribution to the EO strength from the shelf feature. The uncorrected curve simply shows the EO strength.

p2(E0';0^0t)

100 200 300 400 N

Figure 7.12: Plot of E0 strength of peak feature using a different definition of the E0 operator between U(5) and SU(3) for a range of N. The shelf subtracted curve accounts for the contribution to the E0 strength from the shelf feature. The uncorrected curve simply plots the E0 strength.

133

p2(E0' ; (£-O

N

Figure 7.13: Plot of EO strength in the SU(3) limit for a range of N using a different definition of the EO operator.

uncorrected curve flattens out from 200 bosons onward, and the shelf subtracted curve is

linear from 200 bosons onward, but it seems to be flattening out. A figure of the EO trend

at the SU(3) limit can be seen in Fig. 7.13. This feature clearly does not have quadratic

behavior, and the strength is tending to 0 as N —> oo for the EO operator from Eq. 7.6.

One justification for the use of this alternate definition for the EO operator is based on the

value of the matrix element in the coherent state formalism [8]:

/ \ (1 7\ { n d ) = T+p ( 7 ' 7 )

The EO operator from geometric collective models are proportional to just /32, so it makes

sense to divide out the factor of N that is present in the coherent state matrix element

[88, 89].

7.8 Discussion

Quantum phase transitions in nuclei have been a topic of interest in recent years,

and understanding the characteristics of these phase transitions for a finite number of

particles is an important step in understanding the evolution of nuclei. The characteristics

of first and second order phase transitions are distinguishable in the large particle number

limit, but for smaller system sizes, it becomes unclear whether the two can be distinguished

for certain observables. The interacting boson model has provided an excellent framework

134

for understanding the transition from spherical to deformed nuclei. The path from the

U(5) to 0(6) limit crosses a second order phase transition, and the path from the U(5) to

the SU(3) limit crosses a first order phase transition. In order to determine whether certain

observables in a small system size actually correspond to phase transitional behavior in

the large boson limit, large boson number calculations must be performed.

The software ibar was written for that purpose, and it is currently capable of

performing calculations for up to 400 bosons. Numerous numerical hurdles had to be

overcome while writing the software, but the use of arbitrary precision libraries helped

reduce several of the numerical issues, ibar has been used in several theoretical probes

into the nature of quantum phase transitions in nuclei [79, 80, 81, 82, 83], and in this

work, it was applied to the behavior of EO transition strengths across the first and second

order phase transition.

A comparison of experimental data to a simple sd-IBM-1 calculation in [84]

showed a clear trend to larger EO strengths in the transition region, and in order to better

understand the features of that trend, the system size was increased. Crossing the sec-

ond order phase transition, the EO transition strength trend matched what was expected

for an order parameter for a second order phase transition. Across the first order phase

transition, the EO transition strength curve resembled the expected behavior of first order

phase transition order parameter, but an additional feature became more pronounced as

the boson number was increased. By looking at energy plots as a function of and by

analyzing the wavefunctions of the first two 0+ states, it is clear that before the first or-

der phase transition, the ground state is mostly spherical, and the 02 state is becoming

deformed. At the critical point, the two states have a large wavefunction overlap, which

results in a large EO strength. Immediately after the critical point, the ground state was

deformed, and the 0^ was mostly spherical.

It becomes clear from this wavefunction analysis that for the ground state, the

first order phase transition from spherical to deformed is due to an avoided crossing with

the 02 state. It is likely that avoided crossings like this are responsible for the observed

phase transitional behavior in other order parameters for the first order phase transition

between U(5) and SU(3). Without a large number of bosons, it is very difficult to have

nearly pure spherical and deformed states near the phase transition region, and therefore

it can be difficult to clearly identify the source of the transition in a given state. If the

system size could be increased further, the crossings and interactions between the various

135

states should become even clearer.

For 0+ states in particular, the wavefunction can be written in terms of the quan-

tum number nd and After applying an dependent phase factor, these wavefunctions

have been plotted in 3D form as a surface plot. The shapes of these 2D wavefunctions near

the deformed limit are very interesting and seem to resemble a two-dimensional quantum

harmonic oscillator. This physically makes sense for the (3 dimension, as the bottom of

the (3 potential should be fairly quadratic. It is not clear what the dimension repre-

sents in terms of the geometric variables (3 and 7. If a relationship between the quantum

numbers and the geometric variables can be found, then a potential in /3 and 7 could be

solved for locally using the numerical wavefunctions. Having a potential in (3 for the shape

coexistence region of the first order phase transition could provide some insight into how

the spherical and deformed minima behave.

The trends of EO strengths as a function of boson number raise some questions

about the EO operator itself. If the operator is divided by N, then the shelf feature will

vanish as N —> 00, and the peak feature should maintain a finite value. This would make

the shelf feature a finite N effect. The justification for the change to the EO operator

comes from a comparison of the coherent state matrix element of the nd operator to the

EO operator as written in the geometric collective model. No matter how the operator is

scaled, for realistic boson numbers, the sd-IBM-1 calculation has a large EO strength in the

deformed limit. Measuring EO transition strengths for deformed nuclei is experimentally

challenging, but having EO data for an isotopic chain that continues into the deformed

region would be invaluable for this discussion about large EO strengths in transitional

nuclei. It will be very interesting to see how such trends behave, and whether the data is

compatible with the simple sd-IBM-1 perspective on EO transition strengths.

136

Part V

Conclusions

137

Chapter 8

Conclusions

Nuclei are incredibly complex systems, and it is remarkable how well certain col-

lective models can describe the behavior of even-even nuclei. Single-particle models like

the shell model provide insights into the nucleon-nucleon interactions, and are particu-

larly useful when the system has a small number of valence nucleons. As the system size

increases, however, collectivity takes hold, and a different class of models become more

appropriate and accessible. As the system transitions from single-particle to weakly collec-

tive, vibrational structure becomes visible. With even more valence nucleons, the system

transitions from vibrational structure to rotational, which is geometrically characterized

by deformation in the nucleus.

Several topics were discussed in this work, and they all related to the transitions

that occur in nuclei as valence nucleons are added. The theoretical exploration of the

hexadecapole degree of freedom in low-lying mixed-symmetry states presented a situation

where behavior that was visible in a single-particle model could be explored in a collective

model, with the relevant degrees of freedom. The experimental investigation into 140Nd

continued the search for low-lying mixed-symmetry states with a hexadecapole degree

of freedom, and this weakly-collective nucleus could provide additional insight into the

onset of collectivity. The discussion of quantum phase transitions in nuclei addressed the

transition from vibrational spherical nuclei to rotational deformed nuclei, and the new

software that was developed for this research can help to understand the behavior of

observables in such a finite system.

Behavior that was observed in a mixed-symmetry state of 94Mo motivated an

investigation into the degrees of freedom that play a role in that nucleus. Calculations

were done using the sd-IBM-2, which included only L = 0 and L = 2 bosons, and it was

determined that the model would not adequately describe the energies of the 4 + states and

138

the strong Ml transition from the mixed-symmetry 4 + to 4^. Shell model calculations had

been performed to better understand the behavior of the mixed-symmetry states, and they

indicated that the hexadecapole degree of freedom could play a role in those 4 + states.

The possibility of including the hexadecapole degree of freedom in the interacting

boson model was discussed, and g-bosons were added to the model space of the IBM-

2. After selecting an appropriate Hamiltonian that kept the number of parameters low,

calculations indicated that both the vibrational term and quadrupole interactions play

a significant role in the structure of 94Mo. A transitional Hamiltonian was constructed,

and the calculations that resulted provided the best fit for 94Mo that has ever been found

with an interacting boson model. One result of particular interest was that the strong Ml

between 4j" and 4^ could be explained using the sdg-IBM-2, and that to achieve a good

energy distribution, a strong d-g interaction had to be included in the quadrupole term of

the Hamiltonian.

That strong d-g interaction influences the entire level-scheme, and shows that

the good fit for 94Mo was not achieved by simply dropping states of interest into the

distribution of s-d states by adding a higher energy boson. The interaction also influences

the relative strengths of certain E4 transitions, so future E4 data could help to verify

whether the current d-g interaction has an appropriate value. The sdg-IBM-2 calculations,

along with the original shell model inspiration, have shown that the hexadecapole degree

of freedom does in fact play a significant role in the 4+ mixed-symmetry state of 94Mo.

The investigation continued to an experiment with an analogously weakly-

collective nucleus, near the N=82 shell closure. The analysis of 140Nd yielded many new

states and transitions, including the 4% state. The reaction was selected to ensure the

population of 4 + states, and many 4+ states were observed. The multipole mixing ratio

for some transitions between 4+ states were identified, but the statistics from the to 4^

transition were too low to determine the multipole mixing ratio. To determine Ml transi-

tion strengths, lifetimes of the states must be known, and some techniques for measuring

the 4+ lifetimes were discussed.

Shell model calculations will be performed to better understand the structure of 140Nd, and it will be interesting to see if the hexadecapole degree of freedom plays a role in

low-lying states near the N=82 shell closure. Once lifetimes in 140Nd are known, the sdg-

IBM-2 can be applied to see whether a collective model that includes the hexadecapole

degree of freedom is able to describe the structure of the nucleus. It is worth noting

139

that even with the transitional sdg-IBM-2 Hamiltonian that was applied to 94Mo, the

g-boson states could be shifted to a higher energy. If it is found that the 4f state has no

hexadecapole component, a higher lying g-boson state could still be described with the

model.

In contrast to the inclusion of new types of bosons in the interacting boson model,

the quantum phase transition study expanded the model into large numbers of bosons.

Nuclei are a quantum system that undergo a change from spherical to deformed as nucleons

are added to the system. It is interesting to discuss this transformation in the context of

a quantum phase transition, but the finite number of particles present certain obstacles

when trying to understand the characteristics of the phase transition. By examining the

nucleus in a collective model like the sd-IBM-1, and by expanding the model space to a

large number of bosons, the effects of system size on the observables of interest can be

explored.

The software ibar was written for this purpose, and is now capable of calculating

with up to 400 bosons. The behavior of EO transition strengths in transitional nuclei is

a topic that has met some success with an sd-IBM-1 interpretation for small numbers of

bosons. To better understand the features of this sd-IBM-1 interpretation, the number

of particles was increased dramatically using ibar , and two separate features of the EO

transition strength curve between U(5) and SU(3) were identified. Using wavefunction

analysis, a peak feature in the EO curve was identified as being caused by the transition

from spherical to deformed in the ground state. This transition was caused by an avoided

crossing with the Of state, and after the crossing, the first two 0+ states had reversed

shapes.

The shelf feature that is visible in the low boson number calculations is caused

by the wavefunction overlap of the Of and Of states after an avoided crossing between the

two states. It is likely that avoided crossings like these are responsible for the observed

phase transitional behavior in other order parameters for the first order phase transition

between U(5) and SU(3). However, without the ability to calculate large boson systems,

these crossings get smoothed out, and the transition from spherical to deformed becomes

harder to identify. The code ibar should be useful in identifying the source of such

characteristics.

Two-dimensional surface plots of the 0 + wavefunctions provided new insight into

the relationship between the U(5) quantum numbers and the geometric parameters (3

140

and 7. The wavefunctions near the deformed limit resembled a two-dimensional quantum

harmonic oscillator, due to the quadratic nature of the minimum of the 0 potential. If

a relationship between n A and the geometric parameters can be found, the potential for

the shape-coexistence could locally be calculated in terms of 0 and 7 using the numerical

wavefunctions. Any expansion into larger numbers of bosons would make this even more

accurate, but such an expansion would be computationally challenging.

The studies described in this work relate to the degrees of freedom that play a

role at the onset of collectivity, and also to the behavior of nuclei during the transition

from spherical to deformed. Several new computational tools were discussed, and they

all showed promising results that warrant further investigation. New data was found that

could be used as an additional test in the study of hexadecapole excitations near closed

shells. It was the purpose of this work to describe and test some potentially interesting

degrees of freedom, and only future experiments will tell whether they are useful for

describing the structure of nuclei.

141

Appendices

142

Appendix A

ibar manual

The program i b a r performs interacting boson model (sd-IBM-1) calculations for

systems with up to 400 bosons. The program should be located in a directory called

iba r . For the program to function, this directory should contain an additional directory

called cf p which has 400 data files. The c f p directory should have a size of approximately

500 MB. Once compiled, the program is executed by entering i b a r ( i n p u t f i l e ) in the

command line, where ( i n p u t f i l e ) is replaced by the name of the input file. An example

input file should be packaged with the installation, and its name is example. The program

iba r should be executed from the directory it is stored in. The example input file, along

with the resulting output files can be found at the end of this chapter.

A . l Hamiltonian

One possible Hamiltonian in i ba r is called the consistent-Q formalism (CQF)

Hamiltonian.

HQQF = e0 + end + • fr + a^L • L + a3T<3> • T<3> + a 4 f <4> • T<4>

where k can also be referred to as 02. The operators are defined as follows:

nd = • d

Q^-Qx= (Vd](2) + [^](2) + XMW2)) • (Vd](2) + [<^](2) + XMW2))

L • L = lO^ty1) • [d^dp

T(4) . T(4) = [ d tJ](4) . [ d tJ](4)

In this Hamiltonian, k should have a negative value. For the SU(3) symmetry limit,

X should be equal to — w h i c h numerically is -1.32287565553230 for 15 digits. An

143

alternative to e and k is the £ and c parameterization. This is performed with the following

substitution:

e = c(l - C) c C

AN

Another alternative to e and k is the 77 and c parameterization. This is performed with

the following substitution:

e = crj

Eventually the CQF Hamiltonian is converted to normal order form, which just means that

the operators are recoupled so that the d) operators appear on the left, and d operators

appear on the right. The normal order Hamiltonian is written in the following way:

Hnorm = Cq + C l ^ s ] ^ + C 2 ^ [ d ) + C3[dU^ • [dd\<® + C4[dtdt](2) • [dd}{2)

+ C5[cftdft]W • [dd]W + C6 ([dtrft](2) . ^ ( 2 ) + [dtat]<2) . {dd}(2)j

+ C7V5 ([dtrft](0) . [5g](0) + [stst](0) . + Cg[dtst](2) . ^ ( 2 )

+ c9[Stst](o).[gs-](o)

The coefficients in the CQF Hamiltonian are converted to this form before calculation.

These cn coefficients can be set directly, but they will added to any CQF Hamiltonian

contributions. Setting the cn coefficients will not replace the previously entered CQF

coefficients, and setting these coefficients is a useful way to add an additional interaction

to the CQF Hamiltonian.

A. 2 Transition operators

The E2 transition operator in iba r is defined in the following way:

T(E2) = eg

If x is not set for the E2 operator, the value from the CQF Hamiltonian will be used. The

boson charge eg defaults to a value of 1. If the command reduced is set to t rue , then the

144

transition strength outputs will represent the following value, where a refers to the state

index:

R{E2;La —• L'a,) = (L' a'\\T^E2^\\La)

If the command reduced is set to f a l s e , then the transition strength outputs will be:

B{E2-La L'a,) = ^^(LWWT^WLa)2

This is the default output, as reduced is set to f a l s e by default. The EO operator is

defined in the following way:

T(E0) = (30nd

The behavior of the EO transition strengths are controlled by the parameter reduced in

the same way that the E2 transition strengths are. The scale factor defaults to a value

of 1.

A.3 Input file

To comment lines in the input file, begin the line with #. Blank lines and com-

mented lines should have no effect on the execution of ibar. The example input file shows a

preferred structure for organizing the input file. It is most convenient to have Hamiltonian

parameters appear at the start of the file, as they are typically changed most frequently.

Hamiltonian parameters

The parameters here are for setting the Hamiltonian. The values with a letter in

parentheses are supplied by the user: ( f ) refers to a floating point value, ( i ) refers to

an integer value, (s) refers to a string, and (b) refers to a boolean value, (b) should be

replaced by either t rue or f a l s e . Parentheses should not be included with any of the

input parameters mentioned here. This can be seen in the example input file shown at

the end of this chapter.

cqf eO ( f )

The value eo in HCQF- Has units of MeV.

cqf zeta ( f ) The value ( in Hqqf• This parameter will override k and a2 when set. This parameter

145

has no units,

cqf e t a ( f )

The value r] in Hqqf- This parameter will override k and a2 when set. This parameter

has no units.

cqf sca le ( f )

The value c from the alternative parameterizations of HCQF- This parameter can scale

both the C parameterization and the rj parameterization. Has units of MeV.

cqf eps i lon ( f )

The value e in HCQF- Has units of MeV.

cqf kappa ( f )

cqf a2 ( f )

The value k in Hcqf- This parameter should be negative. a2 is the same parameter. Has

units of MeV.

cqf chi ( f )

The value x i n #cqf - This parameter has no units. For the SU(3) symmetry limit, the

parameter should be -1.32287565553230.

cqf a l ( f )

The value AI in HCQF- Has units of MeV.

cqf a3 ( f )

The value <23 in HCQF- Has units of MeV.

cqf a4 ( f ) The value a4 in HCQF- Has units of MeV. norm cO ( f )

norm cl ( f )

146

norm c2 ( f )

norm c3 ( f )

norm c4 ( f )

norm c5 ( f )

norm c6 ( f )

norm c7 ( f )

norm c8 ( f )

norm c9 ( f )

Adds to the values cn in / f n o r m - H a s units of MeV.

General calculation parameters

These are the general calculation parameters for ibar.

boson ( i )

The number of d-bosons to be included in the calculation,

set L ( i ) ( i ) ( i ) . . .

A specific set of spins to be calculated. These values must be less than or equal to

2*boson. Any number of spins can be entered. Repeated spins will be ignored. This

command may be used multiple times.

range L ( i l ) (i2)

A range of spins from ( i l ) to (i2) to be calculated. These values must be less than or

equal to 2*boson. This command may be used multiple times. Repeated spins will be

ignored.

vector ( i )

The maximum number of energies and wavefunctions to be calculated for each spin. If the full basis size is smaller than this parameter, then the full basis size is used. This check is performed for each spin. This number applies to all spins to be calculated. This parameter will not affect the components of the wavefunction, as it simply determines how many energies and wavefunctions to calculate and display.

147

o u t f i l e (s)

Warning: There is no need to include this parameter if you want the output files to have

the same prefix as the input file. The default value is simply the input file name. This

parameter is the filename for calculation results. This name should not include a suffix, as

suffixes are added automatically. See the Section A.4 for more information. The o u t f i l e

filename should be set relative to the directory that ibar is stored.

verbose (b)

Specify whether output should be verbose. The state is f a l s e by default. If this command

is f a l s e , then ibar will run silently. Provides information about the calculation as it is

happening, as well as a large range of results.

reduced (b)

Specify whether reduced matrix elements of the transition operators should be output,

instead of the transition strengths. The state is f a l s e by default. If this parameter is

f a l s e , then ibar will output B(E2)s and B(E0)s. If this parameter is t rue , then ibar

will output R(E2)s and R(E0)s.

s h i f t e d (b)

Specify whether the binding energy of the ground state should be subtracted out of all

energies. The state is t r u e by default. If this parameter is f a l s e , then ibar will output

unshifted energies. If this parameter is t rue , then iba r will output shifted energies. This

parameter controls the behavior of the output energy data file as well.

vecsave (b)

Specify whether the wavefunctions should be saved to disk. The state is f a l s e by default.

If this parameter is t rue , then ibar will save the wavefunctions.

p r ec i se (b)

Specify whether values in output files should be displayed in 16-digit scientific notation,

or fixed-point with 8 digits beyond the decimal point. The state is f a l s e by default. If

this parameter is f a l s e , then ibar will output fixed-point values with 8 digits beyond

148

the decimal point. If this parameter is t rue , then ibar will output values in 16-digit

scientific notation. This setting only affects values in output files.

wavechop ( f )

The minimum value to be displayed in the wavefunction components of the eigenvector

output file. Setting is 0 by default, which allows wavefunction components for the entire

basis. For example, if ( f ) is 0.001, only components with a value greater than or equal

to 0.001 will be listed in the output file.

Transition commands

These commands are how the user specifies which transitions should be calculated. All

four of these commands may be repeated as many times as necessary.

set e2 ( i l ) (i2) (13) (i4)

Calculate the E2 transitions strength from the Lj = ( i l ) state to the L j = (i3) state.

The parameter (i2) is the state index for the Lj state, and (i4) is the state index for

the Lf state.

range e2 ( i l ) (i2) (i3) (i4) (i5) (i6)

Calculate the E2 transitions strength from the Lj = ( i l ) state to the Lj = ( i 4 ) state.

The range of the state index for the Lj state is from (i2) to ( i3) . The range of the state

index for the Lf state is from (i5) to (16).

set eO ( i l ) (i2) (i3) (14)

Calculate the E0 transitions strength from the Li = ( i l ) state to the Lf = ( i 3 ) state.

The parameter (i2) is the state index for the Li state, and (i4) is the state index for

the Lf state. Li must equal Lf for the calculation to be successful.

range eO ( i l ) (i2) (i3) (14) (i5) (16)

Calculate the E0 transitions strength from the Li = ( i l ) state to the Lf = ( i 4 ) state. The range of the state index for the Li state is from (i2) to ( i3) . The range of the state index for the Lf state is from (i5) to ( i6) . Lj must equal Lf for the calculation to be successful.

149

Transition parameters

These parameters control the behavior of the EO and E2 transition operators.

e2 scale ( f )

The value of the effective boson charge e^ in the E2 operator. The default value is 1.0.

e2 chi ( f )

The value of the parameter \ i n the E2 operator. If this is not set, the value defaults to

X from the CQF Hamiltonian. eO s c a l e ( f )

The value of the scale parameter /?o in the EO operator. The default value is 1.0.

A.4 Output files

The name of the output file is specified with the parameter o u t f i l e from the input file. If a filename is not provided, the name of the input file is used. Suffixes are added automatically, depending on the type of file being saved. A backup of the input file used during the calculation has a suffix of the form *. i in , and it is in text format.

The file containing energy eigenvalues has a suffix of the form *. ien, and the output is in text format. The output is a simple table, with text headings at the top to identify the columns. The columns are the angular momentum of the state L, the index of the state n, and the energy of the state E(L+). The first line of the file gives the binding energy of the ground state. This value is listed even if the unshifted energies are requested by setting the s h i f t e d parameter to f a l se .

The file containing E2 transition strengths has a suffix of the form *. ie2, and the output is in text format. The output is a simple table, with text headings at the top to identify the columns. The columns are the angular momentum of the first state L\, the index of the first state n\, the angular momentum of the second state L2, the index of the second state n2, and the transition strength. If the parameter reduced is set to t rue , then the last column will give the R(E2). Otherwise it will give the B(E2).

The file containing EO transition strengths has a suffix of the form *. ieO, and

150

the output is in text format. The output is a simple table, with text headings at the top

to identify the columns. The columns are the angular momentum of the first state L\,

the index of the first state ni , the angular momentum of the second state L2, the index of

the second state n2, and the transition strength. If the parameter reduced is set to t rue ,

then the last column will give the R(E0). Otherwise it will give the B(EQ).

The file containing wavefunction components has a suffix of the form * . ivc, and

the output is in text format. The minimum value of the allowed wavefunction components

is specified by the parameter wavechop. This value is displayed at the top of the file with

the word Threshold. Each state has a line identifying its angular momentum and index,

before the components are listed. Each wavefunction component has a line, and the start

of the wavefunction has text headings to identify the columns. The first four columns

label the basis state, and the last column is the wavefunction component. The basis state

columns are the number of d-bosons nj, the number of pairs of bosons coupled to 0 np, the

number of triples of d-bosons coupled to 0 nA, and the angular momentum of the state L.

A. 5 Installation

A R P A C K

The diagonalization functions in ibar require the fortran library ARPACK. This

library can be downloaded from http://www.caam.rice.edu/software/ARPACK/. The

file arpack96.tar.gz can be downloaded and extracted into the directory ARPACK.

The default build configuration is called SUN4, and a few changes need to be made to

ARmake. inc in order for it to compile on desktop computers.

Modifications to ARmake. inc:

Change home to the global path of the ARPACK directory.

Change MAKE to the path of make (probably /usr/bin/make).

Change FFLAGS by removing -cg89.

Change FC to whichever compiler you are using.

To build, use the command make l i b from the ARPACK directory. To install the li-

brary, use the following command from the same directory:

sudo mv libarpack_SUN4.a / u s r / l o c a l / l i b / l i b a r p a c k . a

151

ibar

The program ibar is written in C++ and requires g++ to compile. The

Makefile in the ibar directory shows which libraries are linked to after compiling.

The library / u s r / l o c a l / l i b / l i b a r p a c k . a is linked to, which is why the arguments

- L / u s r / l o c a l / l i b and - larpack are listed. A fortran runtime library must be linked

to, because g++ does not link to one automatically. If you are using g77 for the fortran

compiling, one option is l ibg2c.a . In the default Makefile, the argument - lg2c is used

to link to l ibg2c.a . If you are not using g77, you may need to find a different runtime

library.

A.6 Example files

example

### Input f i l e notes ### # cqf chi i s -1.32287565553230 fo r SU(3) # cqf kappa should be negative

# The maximum number of bosons i s 400

### Set t ings f o r Hamiltonian ###

cqf ze ta 1.0 cqf chi -1.32287565553230 cqf scale 1.0 ### General ca lcu la t ion s e t t i n g ###

boson 4 vector 10 range L 0 4 prec i se f a l s e verbose t rue vecsave t rue wavechop 0.000000005

### Transi t ion commands ###

se t e2 2 1 0 1 se t eO 0 2 0 1 range e2 4 1 2 2 1 2 range eO 2 1 2 2 1 2

152

### S e t t i n g s f o r t r a n s i t i o n s ###

e2 s c a l e 1 . 0 eO s c a l e 1 . 0

example.ien

Binding Energy: -2 .75000000 L n Energy 0 1 0 .00000000 0 2 1 .31250000 0 3 1 .87500000 0 4 2 .43750000 2 1 0 .14062500 2 2 1 .45312500 2 3 1 .45312500 2 4 2 .01562500 2 5 2 .57812500 3 1 1 .59375000 4 1 0 .46875000 4 2 1 .78125000 4 3 1 .78125000 4 4 2 .34375000

example. ieO

LI n l L2 n2 0 2 0 1 2 1 2 1 2 1 2 2 2 2 2 1 2 2 2 2

BEO 1.17551020 1.17959184 0.19619606 0.19619606 1.04774083

example. ie2

LI n l L2 n2 BE2 2 1 0 1 8, .80000000 4 1 2 1 11. .14285714 4 1 2 2 0. .00000000 4 2 2 1 0. .00000000 4 2 2 2 2. .44263547

example, ivc

Threshold: 5e-09

L: 0 n: : 1 nd nB nD L Component

0 0 0 0 0 .33333333 2 1 0 0 0 .73029674 3 0 1 0 -0 .45074894 4 2 0 0 0 .39036003

L: 0 n: : 2 nd nB nD L Component

0 0 0 0 0 .70272837 2 1 0 0 0 .19245009 3 0 1 0 0 .29695694 4 2 0 0 -0 .61721340

L: 0 n: : 3 nd nB nD L Component

0 0 0 0 0 .44444444 2 1 0 0 -0 .24343225 3 0 1 0 0 .52587376 4 2 0 0 0 .68313005

L: 0 n: : 4 nd nB nD L Component

0 0 0 0 -0 .44444444 2 1 0 0 0 .60858062 3 0 1 0 0 .65734220

L: 2 n: : 1 nd nB nD L Component

1 0 0 2 0 .49441323 2 0 0 2 -0 .45773771 3 1 0 2 0 .64733887 4 0 1 2 -0 .18401748 4 1 0 2 -0 .30515847

L: 2 n: : 2 nd nB nD L Component

1 0 0 2 0 .68746523 2 0 0 2 -0 .25437436 3 1 0 2 -0 .40489565 4 0 1 2 0 .31652149 4 1 0 2 0 .44559949

„: 2 n: : 3 nd nB nD L Component

1 0 0 2 0 .30869531 2 0 0 2 0 .67757155 3 1 0 2 0 .37807118 4 0 1 2 -0 .29555184 4 1 0 2 0.46402078

2 n: 4 nd nB nD L Component

1 0 0 2 0.37184890 2 0 0 2 0.51639778 3 1 0 2 -0 .12171612 4 0 1 2 0 .38059948 4 1 0 2 -0 .65984161

2 n: 5 nd nB nD L Component

1 0 0 2 -0 .22222222 3 1 0 2 0 .50917508 4 0 1 2 0 .79608094 4 1 0 2 0 .24002743

3 n: 1 nd nB nD L Component

3 0 0 3 1.00000000

4 n: 1 nd nB nD L Component

2 0 0 4 0 .67377166 3 0 0 4 -0 .57459440 4 0 0 4 -0 .11878277 4 1 0 4 0 .44918111

• : 4 n: 2 nd nB nD L Component

2 0 0 4 -0 .39709513 3 0 0 4 0 .18091628 4 0 0 4 0.19447916 4 1 0 4 0 .87850021

,: 4 n: 3 nd nB nD L Component

2 0 0 4 0 .58216855 3 0 0 4 0 .55376958

4 0 0 4 4 1 0 4

0 .59528441 -0 .00744964

.: 4 n: : 4 nd nB nD L Component

2 0 0 4 -0 .24343225 3 0 0 4 -0 .57089923 4 0 0 4 0 .76712278 4 1 0 4 -0 .16228817

156

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