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Quantum Chaos and Nuclear Spectra
presented by

Declan Mulhall

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6/01 cyclRC/DateDuopesvp. 1 5

QUANTUM CHAOS AND NUCLEAR SPECTRA

By

Declan Mulhall

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2002

ABSTRACT

QUANTUM CHAOS AND NUCLEAR SPECTRA

By

Declan Mulhall

The regularities in nuclear spectra, especially the spin-zero ground states of even-even
nuclei are usually ascribed to the effect of the nuclear pairing interaction. This is seen to
be an incomplete picture. Similar regularities are seen in the spectra of random two-body
Hamiltonians. The largely ignored role of geometric chaoticity is shown to be the driving
force behind this. A system of N spin- j fermions under the influence of a random two-body
angular momentum conserving interaction is studied. The spectra exhibited a propensity
for the ground states to have spin-zero or maximum spin. The role of pairing was eliminated
unambiguously as the cause. A statistical theory based on equilibrium statistical mechanics
was developed and an expression for the equilibrium energy derived. The theory was used

to successfully describe the salient features of the ensembles studied.

To my family William, Elizabeth, John, Gerard and Desmond Mulhall, and Jennifer

Greene.

iii

ACKNOWLEDGMENTS

I am very much indebted to my advisor, and my friend, Professor Vladimir chevinsky.
He was there for me when things were tough. His kindness and brilliance inspires awe in
those who know him. Shari Conroy, a gracious and thoughtful person. was a pleasure to
work around. I owe a debt of gratitude to Alexander Volya, who taught me the value and
joy of collaboration. Viktor Ccrovski kept me sane with our friendship, and accompanied
me on the American adventure. Thanks to B. Alex Brown and Mihai Horoi for invaluable

collaboration, and teaching me how to use OXBASH.

Lastly I want to acknowledge the best human being I ever met, Jennifer Greene, who

was with me through it all.

iv

Table of Contents

List of Tables vii
List of Figures viii
1 Introduction 1
2 Neutron Resonance Data 12
2.1 The Level Density ............................... 13

2.2 The Level Spacing Distribution ........................ 16

2.3 Spectral Rigidity: the A3(L) Statistic ..................... 18
2.4 The Reduced Width Distribution ....................... 19
2.4.1 A graphical search for missing levels ................ 21

2.4.2 Higher moments of the reduced widths ............... 22

3 Randomly Interacting Fermions 26
3.1 A Simple System ............................... 29

3.2 Properties of the Operators .......................... 31

3.3 The Spectra .................................. 34
3.3.1 The distribution of ground state spins ................ 35

3.3.2 The role of pairing .......................... 36

3.3.3 The yrast line ............................. 40

3.3.4 The ground state energy gap ..................... 42

3.3.5 The level density ........................... 44

3.3.6 The distribution of level spacings and off-diagonal matrix elements 46

3.3.7 Multipole moments and transition collectivity ............ 49

3.3.8 Pairing and fractional pair transfer collectivity ............ 55

4 A Statistical Theory 61
4.1 The Bosonic Approximation ......................... 61

4.2
4.3
4.4
4.5
4.6
4.7

Equilibrium Statistical Mechanics Approach .................
Ground State Spin ...............................
The Distribution of VL for JO 2 Jmm .....................
Ground State Energy .............................

Sin gle-Particle Occupation Numbers .....................

Moment of Inertia ...............................

5 Conclusions

6 Appendices
6.1 The Darwin-Fowler Method .........................
6.2 Sums Involving Clebsch-Gordan Coefficients ................
6.3 An expression for (H2) ............................
Bibliography

vi

62
67
70
71
72
73

76

79
79
82
84

87

' List of Tables

2.1

3.1

Data characteristics and results for or. The negative values for I? corre-
spond to situations where the minimization procedure failed, there is no

physical meaning to them. .......................... 15

The multiplicity Do, of J = 0 for various values of particle number, N, and
single—particle spin, j .............................. 36

vii

List of Figures

2.1

2.2

2.3

2.4

2.5

2.6

3.1

The level spacing distribution for each data set. For the 235U data only the
first 544 levels were included and an analysis was performed for the J : 3

and 4 resonances taken together and separately. ...............

The procedure for extracting or from P(s) was tested on GOE spectra (N :—
1000). The measured values ammured were systematically lower than the
actual values am, by approximately 0.05 . However the slope of the graph

is unity up to a value of (X = 0.42 ......................

The A3(L) statistic for the data sets. The solid lines in each plot are the
RMT results for spectra with the Poissonian level spacing distribution,
A3(L) = L/ 15, and for the GOE (lower logarithmic curve Eq. (2.4)). The
long-dashed line in each plot corresponds to the RMT prediction for a su—

 

. . ‘ . . 211+]
perposrtion of two sequences in the proportions (er+1)+(212+1)' The short-

dashed lines are the results for the data. In the 235U plot the lower dot-
ted lines are the A3(L) calculations for the J = 3 and 4 sequences sepa-

rately. The 235U data agrees well with the RMT result both for individual
sequences and their combination. The 233U data has a logarithmic A3(L)
which looks more rigid than one would expect from simple angular mo-

mentum considerations .............................

The distribution of reduced widths I0 for all four data sets. The dotted line

is P(x) = x2(x|v == 1) .............................

The minimum In 8((X1)) vs. (X1), (here I? is written as 0(1)). The left
hand panels correspond to the second (n = 2) moment of the data, and the

right hand panels correspond to the third (n = 3) moment of the data. . . . .

The values of or that minimized 5(amiml'?) for a given I? vs. the corre-

sponding T? (labelled as (X1) on the y-axis). The values of T? taken from
Fig. (2.5) were used here to get the values of or that best fit the data.

The left hand panels (a-c) show values of f0 (vertical axis) vs the single—
particle angular momentum j (horizontal axis). In the right-hand panels
f0 has been divided by Do, the fraction of the total dimension of the space

(ignoring the 2] +1 degeneracy) accounted for by J = 0 ..........

viii

17

20

22

24

25

3.2
3.3

3.4

3.5

3.6

3.7

3.8
3.9

3.10
3.11

3.12

3.13

3.14

3.15

3.16

3.17

Same as Fig. 3.1 but this time we show f,

The distribution fj of ground state Spins J0 for various systems of N = 4
particles. ................................... 39

The distribution f) of ground state spins J0 for various systems of N = 5
particles. ................................... 40

The distribution f; of ground state spins Jo for various systems of N = 6
particles. ................................... 41

f0 is shown for the N = 6, j : ]—,1— system. The cases where the pairing
interaction are random or zero are practically indistinguishable. ...... 42

The distribution of overlaps, eq. 3.18, of ground states with JO 2 0 with the
fully paired (seniority zero) state are shown as solid histograms. Panel (a)
corresponds to the ensemble with random V1./() and regular pairing, V1,:o =
—1. Panel (b) corresponds to the ensemble with all VL random. Dashed
histograms show the predictions for chaotic wave functions of dimension
d :2 2, panel (a), and d = 3, panel (b). .................... 43

The average yrast line for a representative system, N = 6, j = '71. ..... 44

The average yrast lines for all the N = 4 and 6 particle systems. The range
ofj starts from 11/2 and goes to 27/2 for N = 4, and 21/2 forN = 6. The
longer lines correspond to systems of higher j. Error bars for 3 panels have
been omitted for clarity, but are comparable to those of the JO 2 0 spectra. . 45

The ensemble average ground state energy gaps for all values of N and j. . 46

The mean value, (VL), of the subsets of {V1} vs. L that give JO 2 0 or
JO 2 Jmm, for all the N = 6 systems. The panels are labelled with the value
of the single-particle spin, j. On the left are the Jo = 0 data, and on the
right are the JO 2 1",.“ data ........................... 47

The standard deviation of the level density, p j (E), of states with total an-
gular momentum JTotal for three representative systems. .......... 48

The standard deviation of the level density p; (E) of states with total angu-
lar momentum 170,“, for all the systems. The values of j start at 11/2 and
increase by one as the lines get progressively longer. ............ 49

The fraction of systems with ground states spin JO 2 J vs J for fake spectra
where 0(J) is taken from the N = 6, j = 1 1/2 ensemble. .......... 50

The distribution of energies, p) (E), (right panels), and energy spacings,
P) (s), (left panels), with fixed J for three representative systems. ..... 51
The probability distribution P(H,-j) of the off-diagonal matrix elements of
the 3 x 3 J = 0 block of the N = 6, j = 11/2 Hamiltonian in the IJM) basis.

Except where the seniority basis is specified the basis has been randomly
chosen. Note the shoulders on P(H,-j) in the seniority basis .......... 52

The probability distribution P(H,-j) of the off-diagonal matrix elements of
the 2 x 2 J = 0 block of the N = 4, j = 11/2 Hamiltonian in the IJM) basis
rescaled according to the prescription in the text. .............. 53

ix

3.18

3.19

3.20

3.21

3.22

3.23

3.24

4.1

4.2

4.3

4.4

The probability distribution P(H,-j) of the off-diagonal matrix elements of
the 2 x 2 J =2 0 block of the N = 4, j = 1l/2 Hamiltonian in the IJM) basis
but this time rescaled with an alternative prescription to that of Pi g. 3.17. . 54

The contribution to the occupation numbers n,,, of the IS = 0,N = 4) state
from Slater determinants of seniority 0, straight line. The jagged lines
depict the decomposition of nm for [S 2 4,N = 4) into contributions from
seniority-4 (upper dotted line) and seniority-0 (lower dashed line) Slater
determinants; here j = 11/2. ......................... 55

The contribution to nm for the states S = 0, N = 6 (a), S = 4, N = 6 (b)
and S = 6, N = 6 (c) from Slater determinants of various seniority; here
j = 11/2. Note that for] = 0, nm is independent of m. ........... 56

The quantities SQ; L [J] and S], [J] are strongly correlated for different values
of J. This is just a geometric effect. The quantities are calculated for the
lowest J states. HereN = 4 and j = 11/2. .................. 57

The numerator of f p vs the denominator in eq. (3.31), evaluated for the

N = 4 and 6, j = 11/2 systems with Jo = 0. Pg creates a pair of particles
coupled to spin zero. It acts on N = 4, J = 0 ground states to make an N = 6.
J = 0 state which is overlapped with the ground state of the corresponding

N = 6 system. The operator PgPo counts the number of pairs in the N = 4
state. Note the structure is unchanged by setting VL = 0, again the effect is
geometric .................................... 58

The distribution of the angles made by the J = 0 g.s. in the 2-dimcnsional
space {IS = 0) , IS = 4)} for N = 4, 6. Panel a) corresponds to those points
on the straight line portion of Fig. 3.22 while panel b) is for those points
on the curved part. .............................. 59

The strong correlation between the angle the spin-0 ground states make
in the two-dimensional N = 4, J = 0 subspace, 0x174 and N = 6, J = 0
subspace, (IA/:6. The states are eigenstates of Hamiltonians with the same
interaction parameters. ............................ 60

ff for the bosonic approximation. The situation for N = 6, j = 11/2, is
shown. Here Ln,“ = 10, k = 6. The bosonic approximation gives the solid
line while the statistical distribution of of allowed values of total angular
momentum J, gives the dashed line. Note that f0 only slightly exceeds 1 / 6. 63

The cranking frequency 7 vs. M for Nle and j=27/2. Solid and dotted
lines correspond to exact and approximate values, respectively. This gives
an indication of the range of validity of the expansion that gives eq. (4.14). 66

fo, the fraction of ground states with J = O, and (an upper boundary for)
f [m for N = 4, and 6 for different j. Ensemble results; solid line. Statisti-
cal theory; dotted line. ............................ 70

The mean values of VL for the subsets of VL that result in Jo 2 Jmm ground
states. The theory, solid line, agrees with the ensemble average values,

dashed line. Here N = 4 and j = L5. ..................... 71

4.5

4.6

4.7

4.8

 

Ground state energies ED for Jo = 0,(a), and JO 2 J,mu,(b), vs. predictions of
the statistical model, eq. (4.16). Here N _--_. 6 and j = 17/2. The theoretical
energies are fit to a straight line, included in (a). ..............

The constant shift in the theoretical equilibrium energies of spin J vs. J for
the N = 6 j = 15/2 ensemble. ........................

The sin gle-particlc occupation numbers nm averaged over the lowest energy
spin J wave functions of the 1417 out of 2000 (70%) spectra with Jo = 0
(solid line), compared to the simplest theoretical expression (4.10) (dashed
line) ......................................

The moment of inertia for the N = 4 and 6 particle systems of a given j
plotted as a function of j. The positive values are for the JO 2 0 systems
and the negative values correspond to the JO 2 Jmm systems. ........

xi

73

74

Chapter 1

Introduction

Classical nonlinear dynamics became the poster-child of the physics community with the
advent of the computer. Numerical experiments, impossible without a computer, in the
1960’s led to some remarkable discoveries. The so called “Butterfly effect” of Lorentz [32]
arising from the sensitive dependence on initial conditions of a particular set of nonlinear
equations, and the subsequent “strange attractor” in phase space, and Feigenbaum’s work
on pitchfork bifurcations were among the exciting topics of this period. Laboratory stud-
ies of turbulence and convection currents and other nonlinear phenomena were producing
interesting results. The Russian physicist Chirikov, in his studies of the beam instabilities
in accelerators, established criteria of the exponential divergence of trajectories in Hamil-
tonian dynamics [10]. This was a boom time for the growing field of classical chaos, and
by the mid 1980’s the field had captured the imagination of the general public, as is ev-
idenced by the number of popular articles on the topic and the huge success of popular
books like Gleick’s Chaos: the making of a new science [21] which was short-listed for
both the Pulitzer prize and the national book award. The world of art, and kitsch, was af-
fected also. Peitgen [37] made an industry based on churning out images of fractals and

the by then ubiquitous Mandelbrot set [33].

The situation in quantum physics was quite different. The key question is what proper-

ties of a quantum system indicate that the corresponding classical system will be chaotic.
The quantities of interest in classical chaos have no clear definition in quantum mechanics
and are often meaningless, although the notion of a periodic orbit in phase space and a
Lyapunov exponent has some meaning in the mean field picture. Taking the classical limit
of a quantum system the operations lim 5 —> 0 . lim 1 —> 00 do not commute. Where is one
to get chaos from the linear Schrodinger equation? Tire notion of sensitive dependence to
initial conditions is not so obvious here as the uncertainty principle makes the notion of dis-
tance between trajectories blurry. The answer lies in statistical spectroscopy. This was first
studied in quantum systems by the Russian physicist Gurevich in 1939 when he investi-
gated the regularities of level spacings in nuclear spectra [23]. Although Bohr’s compound
nucleus description [8] is similar to what we call quantum chaos, one could argue that the
first application of the ideas of quantum chaos was in the derivation of Bethe’s level density
formula ['4‘] where a statistical approach was used for the structure of nuclear spin states.
What is clear is that the real pioneer of quantum chaos as it is known today was Wigner. He
proposed that as the theoretical nuclear Harrriltonian is too complex to be studied directly,
the local fluctuation properties of random matrices with the same global symmetries could
be studied. Modern random matrix theory was made a separate branch of science by the

works of Dyson.

The basic idea is simple. The exact nuclear Hamiltonian is unknown. Even for non-
interacting particles the level density is huge just from combinatorics; with interactions
exact calculations even in a reasonably truncated space quickly become impossible. In
lieu of the real Hamiltonian and assuming that the physical system is both time-reversal
invariant and complicated in any basis except for some exceptional ones which form a
manifold of measure zero, one can study an ensemble of random real Hamiltonians, the
distribution function of which is invariant under any orthogonal transformation (change of

basis). This leads us to the Gaussian Orthogonal Ensemble (GOE). The properties of the

GOE can be derived from P(H) = P(H’) where P(H) = ning,j(H,-,j) is the probability
of a given Hamiltonian and H’ = 0TH0 is the Hamiltonian in a different arbitrary basis,
with 0 being the transformation matrix and 07' its transpose. This requirement leads to a
Gaussian distribution of independent and uncorrelated matrix elements with the standard
deviation of the off—diagonal matrix elements being twice that of the diagonal elements. If
H is not time reversal invariant, for example a system of charges in an external magnetic
field, we come to the Gaussian Unitary Ensemble (GUE). Finally the Gaussian Symplectic
Ensemble (GSE) deals with systems with Kramers degeneracy, systems of an odd number
of spin-1/2 particles. The study of these ensembles is the main concern of Random Matrix

Theory (RMT) [34]. There are other ensembles of interest which will be mentioned later.

The secular, or long range, behavior of the level density is not the domain of RMT.
Observable quantities can exhibit variations within a few level spacings. These variations,
known as fluctuations, are the bread and butter of RMT. The similarities between the flue-
tuation properties of experimental data and those of the GOE lead us to accept RMT as a
working definition of quantum chaos: a quantum system is deemed chaotic if its spectra
exhibit the same local fluctuation properties as those of the appropriate Gaussian ensemble.
This definition frees us from the need to work backwards from classical mechanics. The
connection to classical mechanics is given by the famous Bohi gas conjecture [6]: “Spectra
of time reversal invariant systems whose classical analogs are chaotic show the same fluc-
tuation properties as the GOE For systems without time-reversal invariance GOE can be

replaced by GUE.

A different point of view is provided by one-body quantum systems. In this situation
one can take some classically chaotic systems like a Newtonian particle in a Sinai bil-
liard and look at an analog, for example a microwave cavity. This electromagnetic analog
is described by the Helmholtz equation which has a correspondence with the stationary

Schrodinger equation. Here the shape of the cavity (billiard) leads to chaotic trajectories,

playing the role of weak residual interactions in many-body quantum chaos. Much work

has been done in this area of analog systems [41], all in agreement with RMT.

As stated earlier the field of quantum chaos has its origins in nuclear physics. This
is no accident because while the spectra involved are very complex and level densities
high, individual states can be studied and statistical methods used, the so-called mesoscopic

situation. RMT demands a level spacing distribution well approximated by

-«1
P(s) 2170i?) (g)5i'l\/r_tsl3exp(—§-sz) (1.1)

where B = 1,2 and 4 for the orthogonal, unitary and sympleetic ensembles [34], respec-
tively. The case where B = 1 gives the famous Wigner distribution. Early studies of neutron

resonances agreed with RMT in this respect [7].

At high level densities even weak interactions can be effectively strong compared to the
local level spacing. In shell model calculations, parameterizing the interaction strength by
7t 6 [0,1] a given class of levels (a set of levels with the same quantum numbers) can be
studied as a function of it. A plot of E ( 7t) for all levels in the class shows a turbulent flow
of the energy levels [46] with many avoided crossings. At each avoided crossing or level
repulsion, the colliding states become mixed. After a few such collisions the states become
highly mixed. We are now in the domain of quantum chaos, and statistical quantities like
entropy, temperature, spreading width etc. become meaningful [46]. Some individual states
in the chaotic regime can be studied. High resolution neutron and proton scattering give us

details of quasistationary states for a brief energy range.

One may question the wisdom of comparing physical Hamiltonians to random matrices.
In physical systems a natural ordering of the basis is in energy. The two—body interaction
used in shell model calculations leads to sparse matrices as states distant in energy are
unconnected by the relatively weak interaction strengths. The Gaussian Ensembles on the

other hand are everywhere dense, allow all particles to interact through all possible n-body

interactions. Furthermore non—zero matrix elements in the physical system are generated
by relatively few numbers, or interaction parameters, and geometrical quantities, namely
Clebsch-Gordan coefficients. Research on the Two-Body Random Ensemble (TBRE) [43],
where only two-body interactions are allowed and the interaction parameters are random
Gaussian distributed quantities, still reproduced the fluctuation properties of the GOE. It is
important to note that RMT results for local level statistics are largely independent of the
precise distribution of the matrix elements. This is a direct consequence of the law of large

numbers, as each eigenvalue is a complicated function of these random matrix elements.

A decade after its inception, the applicability of RMT to nuclear spch‘a was an open is-
sue. In 1963 Dyson and Mehta lamented: “We would be very happy if we could report that
our theoretical model had been strikingly confirmed by the statistical analysis of neutron
capture levels. We would be even happier if we could report that our theoretical model had
been decisively contradicted ...Unfortunately our model is as yet neither proved nor dis-
proved” [12]. Almost two decades later Haq et al. performed a careful and sophisticated
analysis on an ensemble of high quality data from neutron and proton scattering experi-
ments [24]. They calculated a selection of RMT statistics and showed that RMT describes

nuclear spectra.

There has been a significant practical output from the theory of quantum chaos. Pos-
sibly the most elegant example of this is the statistical enhancement of parity violation in
nuclear physics, see the review by Flambaum and Gribakin [17]. The basic idea is a very
simple mechanism whereby the chaotic structure of eigenfunctions at high energies can
enhance the effect of a weak interaction. An informal account of this mechanism follows.
Suppose II = [NW/aim: is an operator that mixes states of different parity, for example

Is) and | p). Near the ground state

 

nsp . .
(plIlls) ~ E E ~ 1 in some units. (1.2)
p s

At high excitation energies the wave functions are chaotic, and Is) 2 )2in [i), with the same
for [p). Tire assumption of chaos means [Cg] ~ |Cipl ~ l/\/N where N is the number of

significant components in the wave function. Writing II in the second quantized form we

have
(plllala [ls
(plfllé‘) N .._______._v__v__)_
Ep_E5
_ngdq
—— Ep-ES .

where i and z" are simple states connected by the single-particle operator agave Already,
because of the high level density there is a huge enhancement from the small size of (Ep —
ES) ~ 1 /N. From the assumption of chaotic wave functions, [iC‘PCf is the incoherent
sum of N terms of order 1 /N, which is l/\/N These two effects combine to give an

enhancement of (l/x/N)/(1/N) = sqrtN.

Another practical application by Kilgus er al. [42] is in spectroscopy. In an exper-
iment with insufficient resolution, the assumption of a Wigner level spacing distribution
and Porter-Thomas distribution of multipole strengths makes possible a statistical analysis

to recover the missing strengths.

The complexity of a wave function

lW=Z¢W a»
k

in some physically preferred (unperturbed or shell model) basis, {[k) }, can be measured by

the basis-dependent Shannon (or information) entropy
1., = —}:wgrnwg ' (1.4)
k

where w}? = [Cfli Shannon entropy, being sensitive to Na, the number of significant com-
ponents of la), is a measure of the delocalization of the wave function in {[k)}. Izrailev

[28] defines the localization length of let) as the ensemble average of la 2 exp Ia. In the

GOE limit, because of orthogonality constraints, 10 = 0.48N, whereas a fully delocalized
wave function would have la 2 N. Zelevinsky et al. [46] showed that for shell model eigen-
states la approaches the GOE value, but does not reach it for realistic interactions. Being
basis-dependent, the Shannon entropy by itself is not an indication that a wave function is
chaotic. Collective excitations have coherent components, and their wave functions are not
chaotic. Enders et al. [14] used this point in their study of the level spacing distribution
P(s) = cxp(—s) of the scissors mode in heavy nuclei. Their observation of a Poissonian

distribution for P(s) lead them to conclude that this mode is collective in nature.

A useful quantity for analysis of wave functions in this situation is the strength func-
tion Fk(E ) [46] of a simple mode [k) or local density of states, LDOS, as it is known in
condensed matter physics,

m5) = waNE — 150,). (1.5)
(1

Experimental data do not resolve dense individual states, so Fk(E) is an important tool
for connecting experiment to the theory of quantum chaos. Frazier et al. [‘19] established
the generic shape of the strength function of shell model states: Gaussian at the center
with exponential tails. Suppressin g the interaction strength, they observed a return to the
Breit-Wigner shape. The transition between these two regimes was explained by them in
agreement with the theory of quantum chaos. The exponential tail of the strength function
was exploited by Horoi et al. [26] to develop a method of calculating low-lyin g shell model
levels. The method consists of the diagonalization of successfully larger truncations of the
Hamiltonian, and extrapolating the values of the lower energy levels, thus avoiding the

intense task of diagonalizing the full Hamiltonian.

This leads us to another very important application of quantum chaos. Multiple gi-
ant dipole resonances are analogs of zero sound in Fermi-liquids. In nuclear matter, the

energy of a single quantum is large compared to the temperature. The widths of pure n-

quanta states are estimated by the standard model of strength functions [7] as I‘M/1"] = n.
However, deviations from the standard model and Breit—Wigner shape are amplified in the
convolution of strength functions for multiple sequential excitations, leading to an almost-
Gaussian strength function. Now the widths are added in quadrature, and F,,/F[ = \/r_z, in

agreement with experiment.

Wang et al. ['45], in their study of the structure of the eigenstates and LDOS of a
schematic three-orbital shell model found excellent agreement with RMT. They went on to
derive analytic expressions for the observed exponential tail of both the eigenstates and the
strength function. It is pleasant to note that, in this instance, the classical counterpart of this
system is chaotic. An important development in the theory of chaotic wave functions was
the statistical theory of Flambaum and Izrailev [15]. Their theory, which considered finite
systems of interacting fermions, is based on the properties of chaotic eigenfunctions. A
partition function based on the average shape of the eigenstates was developed and from it

was derived an expression for the average occupation numbers of the sin gle-particle states.

Until recently, RMT dealt with correlations within a sequence of levels with the same
set of quantum numbers. The question of correlations between different sequences in the
spectra of random matrices has not been fully investigated. It was noticed by Johnson et
al. [29], [30], that a rotationally invariant random two-body Hamiltonian showed a surpris-
ing regularity in its spectra, namely J = 0 occurred as the spin of the ground state with a
huge probability. Furthermore they noticed that the gap to the next highest level was much
larger than expected. The mean yrast line, the mean value of the lowest spin-J energy, was
parabolic, and claims for vibrational/rotational bands were also made. These are prime
examples of inter—sequence correlations. Not surprisingly this work generated a flurry of
activity, with many daring suggestions as to the origin of this effect. The undisputed pres-
ence of a pairing force in the atomic nucleus was the previously accepted reason for the

observed spin-zero ground state of all even-even nuclei. Now it seemed that other, hitherto

unknown, effects could disguise themselves as pairing. Horoi er al. [25] compared random
and realistic interactions in the shell model and found that while the random interaction
generated an abundance of J"T = 0" 0 ground states, they had small overlaps with those of
the realistic interaction, and their B(E2) values were unreasonably small. In short, these
chaotic wave functions didn’t look similar to realistic nuclear wave functions. Clearly there

are many open questions regarding these phenomena.

The purpose of this dissertation is to use the ideas of quantum chaos to shed light on
nuclear structure. The main issue is that of the origin of the regularities in nuclear spectra
and in particular, the zero spin of the ground state of all even-even nuclei. Conventional
wisdom is that specific features of the nuclear interaction explain observed properties of the
spectra. This picture is incomplete. Even an ensemble of random Hamiltonians with just
global symmetries can have spectral regularities. Such a system is studied here, and after
analyzing numerical data and distilling all features of interest, a theory is developed based
on equilibrium statistical mechanics which succeeds in explaining many of the regularities.
Preceding this, however, is an analysis of neutron resonance data. In the guise of RMT, we
use quantum chaos to glean information from neutron resonance data. A more traditional

statistical analysis of the neutron resonance widths is also carried out.

Chapter 2 deals with the analysis of neutron resonance data. The data is from four
odd-A targets and consists of two sequences of levels with angular momentum J] and J2,
in proportions 0t and ( l — or). The goal is modest, simply to get an estimate of or. Naively,

from the 2] + l degeneracy of angular momentum states. one would expect

2J1+I
: . . 1.
a (21, +1)+(213+ 1) ( 6)

 

The level density is compared to the back-shifted Fermi- gas model, and the results suggest
that (1.6) is indeed sensible. Next, the level spacing distribution P(s) is calculated and a

value of 0t extracted. The results were poor. However, paydirt was hit with the more robust

spectral rigidity statistic, A3(L), where the data and theory agreed nicely. An analysis
of the resonance widths follows, where values of or are extracted from the second and
third moments of the distribution of widths, based on the assumption of a Porter-Thomas

distribution for a single sequence of widths.

Chapter 3 starts with a discussion of randomly interacting fermions, describing the
current state of research and posing the main question of the dissertation: How can a
rotationally invariant random interaction generate spectral regularities in the ensemble
average ? The stage is set for the answer with two illustrative examples. The system to
be studied, N fermions on a single level of angular momentum j is introduced next. The
fermions have angular momentum conserving pairwise random interactions. The striking
features of the spectra are described. These include: very high fractions, f0 and f ,W, of
spin zero and maximum ground state spin, Jo = 0 and 1mm, ; a parabolic yrast line with
a pronounced staggering for small values of total angular momentum Jm, disappearing for
higher values; aWigner P(s); non-Gaussian distribution of off diagonal Hamiltonian matrix
elements: a simple particle-hole structure in the low energy states; smoothly oscillating
single-particle occupancies; and finally some very strong correlations between the spin-

zero ground states of the j = 121, N z 4 and 6 systems.

In chapter 4, a first attempt is made to explain these features; pairs of fermions coupled
to angular momentum L are treated as spin-L bosons. An enhancement of f0 is observed,
but not of fjm. Then theory based on equilibrium statistical mechanics is developed,
with an expression for the equilibrium energy of states with a given angular momentum.
The empirical observations of chapter 3 are tackled by the theory. The distribution of
ground state spins is well explained, as is the distribution of interaction parameters that
give JO 2 Jmlu. The equilibrium energies of given angular momentum given by the theory
are beautifully correlated with the lowest energies for each value of Jm. differing only by a

constant negative shift in the theoretical energies which disappears for high values of JW.

10

The ground state wave functions are examined and the role of pairing is addressed. The
theoretical single-particle occupancies compare nicely with the empirical values, which
oscillate around them. The moment of inertia given by the theory is in excellent agreement

with the numerical data, as is the standard deviations of the total level densities.
Appendices are included containing derivations of expressions used in the text.
The main results of this thesis were published in the following:

o D. Mulhall, A. Volya and V.Zelevinsky, Geometric Chaoticity Leads to Ordered

Spectra for Randomly Interacting Fermions, Phys. Rev. Lett. 85 (2000) 4016.

o D. Mulhall, V. Zelevinsky and A. Volya, Spin Ordering of Nuclear Spectra from

Random Interactions, Acta Physica Polonica B, 32 (2001) 2491.

o D. Mulhall, A. Volya and V. Zelevinsky, Random Interactions: Shedding Light on

Nuclear Structure, Nuclear Physics A682 (2001) 229c-235c, 64 (2001 ).

0 V. Zelevinsky, D. Mulhall and A. Volya, Do We Understand the Role of Incoherent

Interactions in Many-body Physics .9, Physics of Atomic N uclei, 64 (2001) 579.

11

Chapter 2

Neutron Resonance Data

Amongst the first successes of RMT was in the 1970’s with its application to the Nuclear
Data Ensemble (N DE) [43], which was a collection of much of the neutron resonance data
available at the time. The data was scant, and a test of RMT on a superposition of two
sequences, or a decent statistical analysis of one long sequence of levels was impossible.
The current availability of higher quality neutron resonance data provides an excellent op-

portunity to test the predictions of quantum chaos.

The absorption cross—section of thermal neutrons, which are neutrons with energy range
0 ~ 10 keV, show narrow peaks with energy separations ~ 1 eV. The absorbtion peaks
broaden until eventually they overlap and become indistinguishable. This phenomenon was
the impetus for Bohr’s compound nucleus model [8], in which a resonance corresponds to
a single—particle configuration with all the kinetic energy concentrated in one neutron. This
configuration rapidly melts through residual interactions into a chaotic nuclear wave func-
tion. Thus, neutron resonances correspond to eigenvalues of the full nuclear Hamiltonian,
and neutron resonance data furnish us with a portion of the nuclear spectrum at high level

densities, where quantum chaos reigns.

The angular momentum of the resonance will be a sum of that of the incident neutron,

j,,, and the target nucleus, jN. For low energy neutrons, the orbital angular momentum will

12

be predominantly zero so jn = 1/2 and the total angular momentum jr can have one of two
values, j-I- = jN i 1 / 2. A sequence of neutron resonances, from experiments not sensitive
to polarization observables, on odd-A targets is a superposition of two pure sequences of
eigenvalues. A fraction 0t will have angular momentum j], and a fraction (1 — 0t.) will have

angular momentum jg, (see Eq. (1.6)).

With this in mind, data for 239Pu, 241Pu, 2331} and 235U neutron resonance reactions at
Los Alamos National Laboratory and RIKEN was analyzed. The data came in the form
of a list of triples {E ,J, F } containing the energy of a level, E, its spin assignment, J, and
its width, F. Our ambitions with the data was were modest, simply to extract the relative
density 0!. of spin states using the machinery of RMT. This analysis is described in the
remainder of this chapter. The level density was checked against some theoretical models.
The distribution P(s) of nearest-level spacings, s, was calculated and from it was extracted
0t. A A3( L) analysis was next performed, see the discussion in section 2.4 for a definition
of A3(L). An analysis of the reduced neutron widths To 2 F/ x/ITI yields information on the

quality of the data and its consistency with the Porter-Thomas distribution.

Spin assignments were randomly made by the experimentalists, with the exception of
the 544 235U levels in the energy interval 0 ——* 300 eV. Table 2.1 shows the characteristics

of the data sets.

2.1 The Level Density

The Bethe level density formula, based on the Fermi- gas model, remains the foundation for
interpreting experimental data. Although it has been elaborated to include states of a given
spin J, the assumptions remain the same, namely, independent particles i.e. no residual

interaction, equidistant spacing of single-particle states near the Fermi level, 8f, and the

13

validity of the saddle-point approximation. The end result is ??:

180(2J+1)(5(£)(E_J(1+1h2> i

met) — 1 21

_ 48 (630)25'

2 3 J 1+1 ‘3

is the single-particle level spacing and 8f re 40 MeV is the Fermi energy;

 

2a.
28f

Here go 2
I = éMR2 is the rigid body moment of inertia (M is the nuclear mass); and a nuclear radius
R of 1.25Ai fm is assumed, where A is the atomic number. This expression for the level
density consistently underestimates the level density calculated from the data, typically by

a factor of 10.

While the Bethe formula (2.1) captures the main features of the nuclear level density,
in particular the exph/E) dependence, it does not include odd—even effects of the mass
number A in its assumptions. An improved model was developed, based on the functional
form of (2.1) but with added free parameters chosen empirically to improve the fit to exper-
imental data. This model is based on the conventional shifted Fermi-gas model developed
by Newton and Cameron [1 l], [36], [9]. The key feature is that odd-even effects are ad-
dressed by means of a pairing energy shift where the excitation energy is measured from
a fictive ground state energy. The level spacing at the Fermi-level is parameterized by an
energy—dependent fitting parameter a. A further refinement is the back shifted Fermi-gas

model, where the fictive ground state energy is treated as an adjustable parameter A.

1 2,“ cxp(2a(E—A)%—J(J+1)/2o‘-’)
_24x/§ oai (E—A+t)

 

 

905.!) (2.2)

Salk/t

Heret is the thermodynamic temperature defined by (E — A) : at2 — t, and I is the ef-

. . . 2 . .
fectrve moment of mertra. I was calculated, and 62 = 1% z 0.015()A3t . The two empirical

14

239pu 241 Pu 233U 235U

 

jl , jg 0,1 2,3 2.3 34
size 825 237 184 544
or”, 0.25 0.42 0.42 0.44
am) 0.08 0.0 0.06 0.15
2"d moment

aw 0.5 0.5 0.05 0.43
it? 1.45 1 - 10 0.54
F, 0.55 1 1.59 1.35
3’dmomcnt

am 0.13 0.05 0.05 0.42
F, -0.7 1.18 12.21 0.65

--1
15¢

1.25 0.99 0.41 1.25

Table 2.1: Data characteristics and results for 01. The negative values for F? correspond to
situations where the minimization procedure failed, there is no physical meaning to them.

parameters, a and A were taken from [11], where two values of each are listed correspond-
ing to I equal to the rigid-body moment of inertia and half the ri gid-body moment of inertia.
Also, in lieu of actual data, the 242Pu values for a and A were used for the 240Pu data. This

expression does better, agreeing with the data to within a factor of two.
The rotational kinetic energy term in eq. )2.1) is

JJ+1r2

_(___)_’- m 10*1 MeV (2.3)
21

This is tiny compared to the typical neutron separation energy involved, so the ratio

p(E,J,) N 21, +1
p(E,Jg) "’ 212+1

 

 

(2.4)

in eq.(2.2), at the neutron separation gives no information about the mixing parameter, (1,

beyond the naive estimate can be extracted.

15

2.2 The Level Spacing Distribution

We attempted to extract the mixing parameter 0t from the level spacing distribution P(s).

The statistical arguments and the calculation of the level density suggest that at will be close

2,14 l
(yeti-1212+

 

to 1)‘ We expect the level spacing distribution P(s) to follow the predictions
of RMT. In our case we have a superposition of two pure sequences of levels. Again, in the
parlance of RMT, a pure sequence is a list of levels with the same quantum number, in this

case spin I. Here each sequence is assumed to have
713 _ 1t 2
P(s) 2 2S8 4‘ , (2.5)

the famous Wigner distribution. Two such sequences superimposed, with relative abun-

dances (x and B = (1 — on), are expected to have

I: , _ , , 2 NOW) 2 P(B-S‘)
P’ ((1,.s)—e(0ts)e(Bs)(0t . +B 61.138)

e(0t.s‘)

 

 

 

+2 QB ftas) film) (26)

e(Ot.s) e([3s)

where p(s) = gse"§52, f(s) : e“ 3‘52 and e(s) : l - Erf(iéis). The derivation of this
formula is very technical and can be found in [34]. In order to separate the local properties
of a spectrum from global (or secular) behavior, for example the mph/£73) increase in the
level density, the spectrum must be unfolded. In its simplest form, which is quite adequate
in this case, the unfolding process is simply a matter of expressing each nearest neighbor
spacing in terms of the local average. Accordingly, the set of level spacings {Si}, were
extracted from each spectrum, {15,}, by the following unfolding procedure,

Eur - E;
3“. : _——————. 2.7
S "E,_,% — E, g ( )

Thus {sf} is just the average spacing over an n~leve1 interval. The results were not so
sensitive to the value of n. The procedures were run for n = 5, 10, 20, 50 with no noticeable
difference in P(s), so a value of n = 10 was used. Since, from N levels you only get N — 2n

spaces, so a smaller value of n gives better statistics (as there are more spaces).

16

The spacings, {Si}, were binned and the counts per bin dea’“ were compared with
theoretical counts per bin. The theoretical count for a bin bound by s and s + ds is
h s ~i-ds
dN’ (a) = P'h((l, s)ds (2.8)
The difference between data and theory for a given value of (x is characterized by a number,
A(0t), which was minimized with respect to or.
2(a) : Emmi“ — dN;"( 11))?- (2.9)
i
The subscript i labels the bins. This scheme was verified for mixtures of GOE spectra and

worked well, as is seen in Fig. 2.2. However the values of 0t obtained from the data were

very low, see Table l.

0.75 1

w
v 0.5 figs: ..

 

0.25

 

Figure 2.1: The level spacing distribution for each data set. For the 235U data only the first
544 levels were included and an analysis was performed for the J = 3 and 4 resonances
taken together and separately.

17

0.5
0.4 i . '
E 0.3 ’ i a
a 1
5% f 1
0.2 E J
‘ 1
. l
0.1

l
l

. i .
, ___ ._4_ ‘|_. _4.__ _._*.A_.. _. . b-— -A... .

____A___

dab—0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
(X

 

lest

Figure 2.2: The procedure for extracting or from P(s) was tested on GOE spectra (N =
1000). The measured values amasured were systematically lower than the actual values

(1,“, by approximately 0.05 . However the slope of the graph is unity up to a value of
0!. = 0.42

2.3 Spectral Rigidity: the A3(L) Statistic

The level spacing distribution P(s) of the data gave limited information, but P(s) is sensitive
to missed and spurious levels. Another statistic, the spectral rigidity or A3(L) statistic, is
far more robust, being insensitive to missed and spurious levels. The spectral rigidity is
a measure of the extent that the unfolded spectrum deviates from a rigid, or picket fence,
spectrum. We know that chaotic dynamics lead to level repulsion, hence a rigid spectrum,
where the levels “crystalize” . The level of crystallization, or rigidity is measured by A3 (L),
which is just the mean deviation of the cumulative level number SM(E) = fAEmp(E’)dE’
from a straight line, over an energy interval of length L. The mean is taken with respect to

the position of the interval on the unfolded spectrum.

18

1 5.1. a I
A3(L)=(mm,,,,,z A (5M(E’)-—AE’——B)*dE) (2.10)

For regular dynamics where P(s) is Poissonian, P(s) -—- exp(—s), A3(Ll) : L/15. For
the GOE, with its Wigner distribution of level spacings, level repulsion means a smaller
deviation from a purely rigid spectrum, so A3(L) is linear for small L and then grows

logarithmically for large L,

1 5 ?-
A3(L)::: t; (lniQnL)+Y—Z_%)’ (2.11)

where y is Euler’s constant. RMT tells us that for a superposition of two pure sequences

mixed in proportion 0t and B = (1 — (1) we have A3(L) = A3(0tL) + A3([3L).

Calculations of the A3(L) statistic are shown in Fig. 2.3. Only for the 235U data is

211 1

115-1.)-L(2Jz+1)' The Pu data sets have a

 

the calculated value value of 0t consistent with (1
linear A3( L) which suggest a number of possibilities. Maybe the neutrons weren’t just 3-
wave so more than two spin assignments are required. There may have been missed or
spurious levels. There is an additional quantum number in deformed nuclei, namely the K
quantum number. If this is conserved there may be more subsequences, for example K =2 0
for 1,0, = 0 and 1,0, = 1, and K = 1 for JM, 2 1. This would result in 3 subsets, and the
fluctuation properties would be closer to those for the Poisson case. The 233U data looks
more rigid than one would expect for a mixture of J = 2 and J = 3 resonances, but it is not

clear why.

2.4 The Reduced Width Distribution

The picture of neutron resonances is based on Bohr’s idea of a compound nucleus. The
neutron is absorbed by the target nucleus. The combination of the target in its ground

state and the barely free neutron corresponds to one distribution of energy in the compound

19

0.8 ,-

0.6

      
  

  
   

5.. 0.4 .3; 1, :2:
(J ', , <1 ..........
0.2 ‘ ——— 0t=0.438
’ ____ 235U
o E. .2, .2,.~L-7. __ . ,-__; _ _-. ._w O .- a; 4.2- . .--_ ._ .__.' - _..,A L ..--
0 5 10 15 20 25 30 35 40 o 5 10 15 20 25 30 35 40
z.

— — - (120.417
241

—--— Pu

 
 
   

‘5‘- A..- ..._..-.--._.__._4 . -h.——.— ...._ A» g.__,_.

0 MW-.. -2 —~~- - 0 ,
0510152025303540 0510152025303540
L L

Figure 2.3: The A3(L) statistic for the data sets. The solid lines in each plot are the RMT
results for spectra with the Poissonian level spacing distribution, A3(L) = L/15, and for the
GOE (lower logarithmic curve Eq. (2.4)). The long-dashed line in each plot corresponds to

. . ‘ ‘. . . . . N145]
the RMT predrctron for a superposition of two sequences in the proportions (211 E 05* (1,2 + 1 ).

 

The short-dashed lines are the results for the data. In the 235U plot the lower dotted lines
are the A3(L) calculations for the J = 3 and 4 sequences separately. The 235U data agrees
well with the RMT result both for individual sequences and their combination. The 233U
data has a logarithmic A3(L) which looks more rigid than one would expect from simple
angular momentum considerations.

nucleus with all the kinetic energy concentrated in one neutron. The kinetic energy of
the incident neutron is rapidly spread out. In order to re-emit a neutron, all the kinetic
energy must again be concentrated in one particle. This corresponds to a tiny region in
phase space. The typical time needed for the excited target to revisit this tiny region is

the Poincare recurrence time, whic is quite long, corresponding to a narrow width for this

decay mode.

There are two aspects of the physics contained in F. On one hand, it is related to the

overlap of the target-neutron initial state with the compound nucleus states. The assump-

20

tion of quantum chaos means that F is proportional to the square of Hamiltonian matrix
elements, which have a Gaussian distribution. On the other hand, the local density of states
for a neutron in the continuum increases as fl, as d p ~ 1 /sqrtEdE. This kinematical

effect can be separated from the intrinsic (chaotic) dynamics by studying the reduced width

 

r“ = T/EF' If
P(l‘) = 1 “PF—if) (2.12)
m 20-
then
P(fO «13): ————1———exp(—£). (2.13)

W 2<rs> '
the famous Porter—Thomas distribution which is just a x3 distribution with v 2 1 degrees
of freedom. Each set {To} was scaled so that (1") 2: 1 as we are interested in fluctuations.
An immediate test on the quality of the data is described in 2.4.], where P(I'O) is examined

graphically. Then, in section 2.4.2, an attempt is made to extract the mixing parameters 0t

from each data set {If}.

2.4.1 A graphical search for missing levels

The or values extracted from the level spacings are too low compared with the Wigner
Surmise. If there were levels randomly deleted or spurious levels detected, or if there were
more than 2 sequences of levels in the data, then P(s) would move towards the Poisson
limit, and 0t would look higher then expected, not lower. But if especially close levels were

selectively missed, the spectrum could be made to look more rigid than it is.

The data was examined for evidence of missed levels._ A convenient method, after
Garrison [20], of testing the data for missed levels is to plot ln(xP(.r)) vs. ln(x). These plots
are sensitive to the number of narrow widths missed. Narrow levels appear everywhere in
the spectrum, as seen in Fig. 2.4, independent of their neighbors. The resulting histograms

tell us that if levels were missed, they were missed uniformly in I“. Small 1'0 were no more

21

likely to be missed than larger ones.

The assumption that v = 1 is not valid for such a superposition. However for the pur-
poses of Fig. 2.4 the difference is not significant. The data was analyzed using the maxi-

mum likelihood method. An array of graphs for the datasets is shown in Fig. 2.4.

 

 

 

r0 (Reducc‘d Width)

Figure 2.4: The distribution of reduced widths [0 for all four data sets. The dotted line is
P(x) = xztxlv = 1)

2.4.2 Higher moments of the reduced widths

The distribution of T0 values is a weighted sum of two Porter—Thomas distributions with
means T”: and fig,
P0“) = aPpi‘tc’f‘?) + <1 — armpit—s). (2.14)

22

The n’h moment X" of a distribution P(.r) of a random variable X is

(-t”) :- /w.r"P(.r)d.r. (2.15)

. 00

For the X2 distribution with v degrees of freedom and mean value .7? : l

e i (2.16)

 

(2.17)

The empirical moments of the data can be used to extract the best values of or, I? and

1,11

sz" r—(r’iT—l (ab—{Ho waif?) (2.18)

+
[01—-

A
101

After many false leads a procedure was found which gave encouraging. i.e. unam-
biguous, results. Writing the n’h moment of a data set I—‘T'Tdam and the theoretical value
calculated with lEq.(2.18) as Tmm, which depends on or and fix a function Noel?) was

defined the minimum of which occurred at the optimum values of or and T0;
8(a, f?) .—. 1137),“, —- Wm] (2.19)
The initial rescaling of the data to ensure F5 = 0 imposed the constraint
afi+(1—a)fi,:1, (2.20)

giving T3 as a function of I? and Ct.

5013??) was evaluated for on in the range (005,095) and F? in the range (0.3,2). For
each value of fit the value am” of 01 that minimizes 8 was selected and the corresponding
triples {Mamba—1'?) ,amm FOI} were saved. A plot offi1 vs. ln5(amg,,,F(13) was made and the

value of F?— that minimized ln5(0t,,,,-,,,I—?) was read of f , see Fig. 2.5. The value of 0cm,-n was

23

then read off a graph of 01mm vs. 7, see Fig. 2.6. The resulting values of (1 and F? for the
second and third moments of all the data sets are listed in Table 2.1. There are negative
values for F1) in this table. This happened when the minimization procedure failed. To
avoid this the condition 0 S or g 1 could be imposed. This would be a purely cosmetic

exercise. The important point is the failure of the procedure in these cases.

The negative values of T1) 2 in Table 2.1 show a breakdown of the method. Once again
the procedure worked well for the 235 U. For the other data sets no analysis worked. This
could be interpreted as an indictment of the quality of the data. These experiments are

notoriously difficult to perform.

 

 

 

 

O i ' ' 1* 1 KT" . r *— i o; *‘r-r-F—r—r—r-v—fr-flr—r ---- -----~--1
71: —5,~ mz-ZPun 2:Pun=3 ,
pg 2 ..
6% -10 l r“1|" ”WEN "a“ ll": ~10 1 1' II”!!! “I ‘. Mum?” a”
_ L. . 0 .1 7 _ "—
7‘: 1 l 2“Pu n- 2 J x/ - _2 2“PU 11:3
>5 -1.2 ." , *4 4 ;
- ’ \ x” i ' i
=0 —1.4 i" \ , 7 -—6 ._

 

 

 

 

A 0 ' ' " j 6 1
A_ l w z 1 4 ._ 213
E —5 t 213U 2 W 2 . 1 U n=3
°° -10 1 ‘ ‘1 o 1'
P i—e-re—ir—oq—wt-v—fi—fi-fi—i—o—j — —-~« ‘ .,.__1__.__2._..-_-___.___._ +- 3
,. _ ... 1 ... .
E —12L lwali 111““: ”ll ‘3: W ‘ . ”mm : m
as _ f— - g~ , , ., . .
'7 - ‘ . _. ', WA .._._.;..-2__.. 2.2.55;
3 'r 1mU n= —3 ‘i 1
a; 1 ' mild MMPW
00.250.50.7511.251..51752 0025050751125151752
<X1> <X1>

Figure 2.5: The minimum In 8((X1)) vs. (X1), (here I? is written as 0(1)). The left hand
panels correspond to the second (n = 2) moment of the data, and the right hand panels
correspond to the third (n = 3) moment of the data.

24

 

 

 

 

 

 

 

 

2 l ‘1, \V fi’ . 2 f \\--. ‘ “—r—
A 1.5 l 239PU n=2977 ‘ 1.5 [ zaspu ”=3 "
x’ 1 1" g --l 1 2 ,
v 1 r -
0.5 f H,,.-ar’ ” 41 0,5 F /,-/ - --
2 i—~—+—v--'r'~—*+ -~-—«--H-we~4+4—j§ 2 — -5. -4——-+—-~— ._ -—.—. 4
. i l 1 1,5 ‘1- \ .
A? 1 :5 E 241PU ”:2 j j 1 — 241P7 -=3 _ 7 7 i
x g .
v 1 1 '
0.5 1L 3 0.5 1'
22”“ “ r‘ r 5 “<7: 2: r‘ *
A 1.5 i _ 1.5 IL
F g f) 233
>6 1?. 233Un=2 § 11.. Un3
0.5 ,= 3 0.5
2 13—_\ 2 , x 12 '4. ~——-~ —— ~—
A 15 ; z,35U7r1=2“‘— ~_-_ ~ ~ ‘ 1'5 f ABSU n3 ‘ ‘*
>5; 1 -- . l T
0.5 1 , ,—~ 7 f 0.5 i ,m
2 4., . 4 2 ;--—\+—~~——4—-~ 4— _-
L 235 ‘ . L ’ - -__
A 1.5 U n=2~a -. ‘ 2_ ‘ 1.5 .. 235U ..=3 _ 2
x” 1 i." *1 1 l
v 1 ,_, — = '
0.5 1' I/ e“ f 05 :‘ , 7
o '_ 1 A ' A L _ 1 _. -
O 0 2 0 4 O 6 0 8 1 0 0 2 0 4 0 6 0 8 1
(X (I

Figure 2.6: The values of or that minimized 5((1,,,,-,,,T_(1’) for a given F?— vs. the corresponding

31 (labelled as (X1) on the y-axis). The values of T? taken from Fig. (2.5) were used here
to get the values of or that best fit the data.

25

Chapter 3

Randomly Interacting Fermions

The complexity of the spectra of many-body systems can originate from completely differ-
ent sources. There are dynamic and kinematic effects that are fundamentally different. For
example, even weak residual interactions at high level densities can give strong mixing of
the mean field configurations; this is a dynamic effect. On the other hand, the complexity
of states of good angular momentum forces us to treat the coupling of many single-particle
angular momenta as a pseudo-random process i.e. not random per se but so hopelessly com-
plicated that it must be treated as such. This kinematic effect leads to geometric chaoticity,
introducing immense complexity into our system without any interaction at all. The roles

of these disparate effects in shaping spectra are not easily distinguishable.

The standard methods used in understanding global features of nuclear spectra vary
with excitation energy. To include angular momentum, the early nuclear level density
formulae used this approximation of random coupling of angular momentum, reducing
the problem to one of a random walk in the space of the angular momentum projection.
Although the concept had not yet been invented, this was the earliest application of quantum
chaos. RMT clarified the role of the global symmetries of the Hamiltonian in shaping
the local features of the spectra. Its main successes have been in deriving expressions

for correlations between levels with fixed quantum numbers for the Gaussian ensembles.

26

While traditional studies of random spectra concentrate on correlations within such pure
sequences of levels, the important question of correlations between different sequences of
levels, for example levels with different angular momentum, in random systems has not
been thoroughly explored. These correlations could be quite rich in physical information.
For example a highly excited deformed nucleus will mix states of the same J. Higher
excited states very close in energy but differing in J by one or two unit will probably
have a very similar structure, if the energy difference is rotational in nature. This idea led
Mottelson to suggest the existence of the so-called rotational compound bands [40] which

were seen in the shell model calculations but not yet in nature.

Recently there has been much interest in the spectra of small many-body rotationally
invariant systems with a random two—body interaction, stimulated by the work of Johnson
et a1 [29], [30]. They examined such spectra and discovered strong correlations between
different I classes. The yrast energies (the lowest energy of spin-J) showed an increase
with J (J + 1 ), and a huge probability to have a spin zero ground state. The initial knee-jerk
interpretation was that the random interaction effectively had a strong pairing component.
They also investigated the distribution of energy gaps in the low—lying levels and claimed

evidence of “rotational/vibrational band structures”.

There are many interesting results in the flurry of research that ensued. Mulhall et
al. [35] discovered that in some cases there was also a huge abundance in the number of
maximum spin ground states. Furthermore they showed that when the pairing component
of the interaction is switched off the excessive appearance of spin zero and maximum spin
ground states is only slightly affected. All even-even nuclei have J =0 ground states, and
this fact is routinely explained by the formation of Cooper pairs. It has been an irresistible
temptation to reverse the rule “Cooper pairs give I = 0 ground states in Fermi systems of
even particle number”, and say that “J = 0 ground states in Fermi systems of even particle

number imply the existence of Cooper pairs”. This is a trap into which many a researcher

27

has fallen.

The formation of Cooper pairs is intimately related to time—reversal invariance of the
Hamiltonian. Bijker et al. [5] added a small time-reversal non—invariant component to
a random two-body Hamiltonian and observed an increase in the number of J 2 0 ground
states. These observations should put the role ofpairin g on a less extravagant footing. Zhao
et al. [47] developed an algorithm for predicting the distribution of ground state spins which
sampled the interaction space by taking different L—components individually and merged
the results. The intriguing aspect of this is, rather like Samuel Johnson’s reaction to a dog
walking on its hind legs, not so much that it does it well, but that it does it at all. It is not
clear even to the authors themselves why the algorithm works. Horoi et al. [25] compared
a realistic interaction with a random one in the shell model and found that although the
random interaction had a large propensity for J r: 0 ground states, the states themselves
had a small overlap with those of the realistic interaction. Kusnezov et al. [31] showed
that a system of random interacting bosons in the Interacting Boson Model (IBM) [27]
displayed the now familiar excess of J : 0 ground states and vibrational/rotational bands,
but on comparison with experimental nuclear data there are other physical features not
present in the random ensemble, thereby putting constraints on the random systems. Most
of the researchers worried about fermionic systems in the dilute limit where the single-
particle occupation numbers were small and Pauli blocking was not a strong factor. Santos
et al. [39] went beyond the dilute limit to show that the regular features of these random

spectra were largely independent of whether they were bosons or fermions.

Subsequent investigations found this to be a robust phenomenon with respect to the
choice of ensemble, being driven by the imposition of a global symmetry. An ensemble
is specified by fixing particle number N, single-particle spin j, and distribution of ran-
dom parameters. Early on in the drama, a reasonable physical explanation was proposed

by Zelevinsky et al. [35]. The effect was considered to be a consequence of geometric

28

chaoticity, the complexity of a many-body state that comes from making a state of good
angular momentum. The construction of an N-body wave function of angular momentum
J can be performed by coupling a pair of particles to spin-113, coupling this pair to a third
particle to get spin—J123 and so on. The choices for the intermediate spins, J12 etc., are
arbitrary and one scheme is related to another by a unitary transformation whose matrix el—
ements are 3n- j symbols. Again the reader is reminded that it is this immense complexity
of the UM) states in the M—scheme basis that leaves us with no other choice but to treat the

angular momentum coupling as pseudo-random.

To reiterate, the question can be stated as follows: How can a rotationally-invariant
random interaction generate predominantly spin-0 and marimtmz-spin ground states as
well as a smooth yrast line and other non-random features in the ensemble average. In
order to answer this question a randomly interacting system was chosen which is rich
enough to exhibit the regular features yet simple enough to tackle analytically. This ap-
proach stripped the question of all less important details and allowed us to concentrate on

the important aspects of the puzzle.

3.1 A Simple System

Before proceeding any further it is instructive to look at a few simple examples, the first
being an analog of the Hund rule in atomic physics. Consider a system of N pairwise
interacting spins with the Hamiltonian

H=Azsa-s,,=A[s°~—Ns(s+1)1. (3.1)
of!)

If the interaction strength A is a random variable with zero mean, then the ground state
of the system will have equal probabilities, f0 = fgm = 1 / 2, to have total spin S = 0 or

S = S,,m (antiferromagnetism or ferromagnetism).

The second example is the degenerate pairing model [38] where a similar situation takes

29

place. In this model the pair creation, P0, pair annihilation, Po, and particle number, N,
operators form an SU(2) quasispin algebra. Now the eigenenergy is simply proportional to
the pairing constant so that, for a random sign of this constant, the ground state quasispin
will take a value of 0 corresponding to an unpaired state of maximum seniority, or the
maximum possible value corresponding to a fully paired state of zero seniority, with each
possibility occurring on average in 50% of cases. For those readers unfamiliar with the
argot of pairing, the seniority of a state is simply the number of unpaired particles in that
state. In the Elliott SU(3) model [13], as well as in any model with a rotational spectrum,
the normal or inverted bands will happen with equal frequency if the moment of inertia

takes positive or negative values with equal probability.

Our system is one of N fermions on a single energy level of spin j which has a 2j +1
degeneracy in the single-particle energies. We are free to choose this system because the
regularities we are investigating are robust with respect to the choice of both system and
ensemble. Within this space, the general two-fermion rotationally invariant Hamiltonian
can be written as

H = ZVLPZAPLA, (3.2)
LA

where the pair operators for a pair of fermions coupled to spin L and projection A are
defined as
1 [A r + 1
:fi— ZCmrrarnanr PLA 2 E
mn

am annihilates a particle with sin gle~particle angular momentum j and projection m, while

PZA : XCnganam; (3.3)
mn

a; creates such a particle, C152 are the Clebsch-Gordan coefficients (LAljm; jn); and the
two-particle states l2;LA) = PZAIO) are properly normalized, (2;L’A’|2;LA) = BULB/(1A.
Because of Fermi statistics, only even values of L for the pair operators are allowed in
the single- j space, as interchanging the particles gives a phase of (— l )L 1. Two particles
with projections ml and mg coupled to angular momentum L interact with energy VI, and

leave with projections m3 and 1714 while the Clebsch-Gordan coefficients ensure conser-

30

vation of angular momentum at all times. There are j + % interaction parameters VL and
these numbers define our ensemble. The ensemble we chose was a uniform distribution of
VL between -1 and 1, which sets the energy scale. The property P(VL) : P(—V1,) makes
attraction and repulsion between particles in a spin-L pair equiprobable. Once again we
remind the reader that the results are robust with respect to this choice. In some cases we

used Gaussian ensembles, as they are easier to work with analytically.

3.2 Properties of the Operators

Let’s look at the properties of the pair operators. This will give useful relations which
will be crucial later in the game. In what follows, the fact that L, the angular momentum
of the pair, can be only an even integer, will be expressed by means of the factor @1. =

[1 +(—1)"}/2 if needed. Also, a factor \/2L+l is denoted g1,

Using the commutation relations and addition formulae for the single—particle cre-
ation/annihilation operators am and (1,7,, in appendix A we get the commutation relations

for pair operators of the quasiboson type,

r L’A’ LA r
[Pl/A’s PM} = 01.5I.’I.5AA' + 291.911 E , ij’anjn jmamam’- (3-4)
mm’n

This can be rewritten in terms of multipole (particle—hole) Operators as:

[PI.’A’2PZAl = @L5l.'1.5/\'A (3.5)
. L . 1 1
_2®[,®L’gl.gl.’2(*)K{ if! j ['2 }CII3\(LI A'(_)A fl’fxx-
KK

The multipole operators are

911ng : ZCfiEJ-na;awn(—)jw"a (3.6)

31

and inversely

a;,,a,, = [,ij ,,—( )1“ "MAX. (3.7)
KK
Here all values of K from 0 to 2 j are allowed. Tlre hermitian conjugate gives the standard

rule for the multipole moments,
an, : (—)‘<M,jf K. (3.8)

The monopole operator, K = 0, is proportional to the number of particles N,

N
91100— — 00— — — amam : — _’ (39)

g} m gj

The K = 1 component is proportional to the angular momentum operator.

 

 

.. 3 ..
9W : — _ , 1'. 3.10
“‘ ”(H—0121+” K ( )
The relevant commutator is
r r 1 L j .1 ' 1
UN

The operators {P1 A P QWKK} form a closed algebra under commutation, but its too

L~IAI

complicated to be practical. For example, for N = 6 fermions of spin j = 171 any pair of

particles can couple to L = 0,2,4,...8, 10 and K can be 0,1,2,3...,9,10, giving us 253

operators in our algebra.

The Hamiltonian can be identically transformed to the particle-hole ( p-h) channel where

it is expressed in terms of the multipole operators as

. 1 - .
H 2 EN — i ZVKflflr’KMKK' (3.12)
KK

The effective single-particle energy in eq. (3.12) is e = (1/2) [K W, and the interaction

matrix elements 17K in the p~h channel (K = 0, 1,...,2j :2 £2 — 1 can be both even and odd)

32

are related to the original matrix elements V1, in the particle—particle ( p— p) channel as

j j L ~
V,= . . V3 evan), 3.13
, 21.311 , K} 1 1 1 )
or, inversely,
~ g . . K
VK:(2K+1)£(2L+1){1. 1. }V,,. (3.14)
Levcn J J L

The transition between the p— p and p-h channels. the so-called Pandey transformation,
was studied from the dynamical viewpoint by Belyaev [2]. The channels are complemen-
tary in the following sense: the low L components of the p-p channel contribute mainly
to the high K components of the p-h channel, and vice versa. The pairing part of the

interaction, L = 0, corresponds to the sum of all multipole interactions,
1 , K ~
V0=-—§Z(*) VK- O”)
K

This complementarity explains the success of the popular interpolating model “pairing +
multipole—multipole interaction” where the effective interaction combines phenomenolog-
ically the lowest components in both channels. Note that the interaction parameters cannot
be assumed to have the same distributions, P(VL) 71$ P(VK). 17),, are functions of V1, in eq.

(3.14), and there are more of them.

The relationship between seniority and the pair-creation/annihilation operators gives us
a useful analytical tool. The seniority S of a state is the number of unpaired particles in that
state. We are usually interested in J = 0 states and will restrict our examples accordingly.
Denoting an N-particle seniority-S state by IS,N), possible spin-0 states for an N = 4 sys-
tem are [0,4) , [4,4). Notice that |2,4) is impossible because the two remaining unpaired
particles must couple to angular momentum zero to have J = 0. Similarly for an N = 6
particle system the possible J = 0 states are [0,6), [4, 6), [6, 6). P; and P0 do not change
seniority as they create/annihilate a spin-0 pair. A very elegant formalism called ”The Qua—

sispin Model”, described in [22] and [38], gives us some very useful relations for dealing

33

with seniority eigenstates. In the quasispin model operators are constructed which fulfill

the angular momentum commutation rules. Definin g

S"- : Z ail-nai- m

m>0
S : Z (1m(L m
m>0
1 -:- +
SO 2 5 Z (amam +(l_-,,,(1.,m —- l).
m>0

(3.16)

A simple check shows that these operators commute as the angular momentum operators,

1.1., J__, and J0. The operator S2 analogous to 12 is defined as
s2 = s s — so + 52-,

and has eigenvalues ,1, (o — S) (a —- s + 1 ) where o = 21' + 1. For the definition 61‘ pg and

P0 in eq. (3.3), the quasispin model gives,

 

(S,N+2|P315,N) = \/(Q—N—S)(N—S+ 2)

 

(S,N— 2110015110 2 \/(N——S)(§2—N—S+ 2)
(0,419,111,104) : [(0,43P510,2)13 = o.— 2
(441193110144): 0

[(6,6]P,J;|S,4)|2 = 0 (as S = 6 doesn’t exist for N = 4)
1<4161P31414>|3 = (12— 8)

(52—4).

NIWNl-d

[(0.611’3 19.4012 2

(3.17)

3.3 The Spectra

A body of empirical information was gathered from numerical simulations in which the

eigenvalues and in some cases the eigenvectors of many random Hamiltonians were cal-

34

culated. A wide range of systems, see Table (3.1), was examined in this way. Direct
diagonalization of the Hamiltonian in the m-schcme was used but as the dimensions got
large it was necessary to employ the shell model code OXBASH. What follows is a sum-
mary of the main features of the ensemble. Some of the figures make reference to “theory”.

This means the statistical theory developed in the next chapter.

Natations: The ground state spin will be denoted by Jo. The fraction of ground states
with spin-J is denoted f). D] is the multiplicity of the spin-value J (D) : 1 for all our

mar

systems).

3.3.1 The distribution of ground state spins

The strongest feature of the spectra was the huge incidence of ground states with total
angular momentum J = 0 (or J -: j in the odd-N systems) and J = J,,,,u., the maximum
allowed value, shown in Figs. 3.3—3.5. In the right hand panels of Fig. 3.1 (3.2) the

enhancement of J = 0 (J = Jmax) states is given as the ratio of f0 (f; ), the observed

fraction of spin 0 (Jmm) ground states, and D0 (D 1”,“), the fraction of allowed values of J
that are 0 (Jmm). This enhancement is clearly large. From the distribution of allowed values
of total angular momentum J for a given N and j it is clear that J : 0 and J = me account
for a tiny portion of the available states. Moreover, the enhancement of the J = j states for
the N =-— 5 system, shown in Fig. 3.4, is consistent with a valence spin j particle on a spin

0 core. It is counterintuitive that a random interaction would generate such a distribution.

One would expect something closer to the statistical distribution of allowed values of J.

For the N = 4 case there was a staggering of f0 as j varied which had a period of 3,
as shown in Fig. 3.1. Systems with different N were correlated in the sense that the local

minima of f0 (f,- in the N = 5 case) occurred at j z %, 3;, L29

for all values ofN. The origin
of this effect lies in the behavior of Do, the multiplicity of] = 0, which for N = 4 increases

by 1 each time j increases by 3, see Table (3.1).

35

 

j N=4 N26 N28 N210 N=12
3/2 1 0 0 0 0
5/2 1 1 0 0 0
7/2 2 1 1 0 0
9/2 2 2 1 1 0
11/2 2 3 2 1 1
13/2 3 4 4 2 1
15/2 3 6 7 6 3
17/2 3 8 12 12 8
19/2 4 10 20 24 20
21/2 4 13 31 52 52
23/2 4 16 47 97 127
25/2 5 20 71 177 291
27/2 5 24 102 319 639
29/2 5 29 144 540 1330
31/2 6 34 201 887 2634
33/2 6 40 272 1429 4998
35/2 6 47 365 2219 9113

Table 3.1: The multiplicity D0, of J : 0 for various values of particle number, N, and
single-particle spin, j.

3.3.2 The role of pairing

As mentioned in the introduction, the high values of f0 were initially assumed to be con-
nected in some way to pairing. There is no obvious dynamical reason for the regularities
seen in the spectra, yet pairing was assumed by many investigators as a possible culprit.
Switching off the pairing interaction by setting V0 2 0 has a very limited effect in the N = 6,
j = 11/2 system. As shown in Fig. 3.6, the value of f0 did decrease but only by a small
amount, from 59% to 55%. Setting V0 : —1 increased f0 by 25%, but this is hardly un-
expected, as the pairing force is now strong and attractive. Of course the J : Jmar state

contains no spin-0 pairs so isn’t directly influenced by the value of V0.

The effect could be kinematical. If the regularities in f0 are due to random spin cou-
pling, the ground state wave functions are expected to be chaotic and the coherent features

of a realistic ground state absent. A point of reference is the totally paired state,

 

0.11) 0f

36

01
O

i‘ i 60
5° (a) I - . i (b i 50
40 l: N=4 O . .- . O" N=4 . 4O 0
“- 30 i ‘ ‘ . . 30 :0
20 i . O O O _' 7V . I p. 20
10 m I. . 10
0 1 RH F2_,__+__.__,_I_‘_+ —4 i J .7+'_‘l_.m, __. 0
1 . , i - . 6
30 i (C) ' (d) o .
u-_20 ,' N_5 . . . . D . 4 Q:
10 , . i 0 ~c_ . 2
0 :7 A A 7'77 .. i i ‘ r . 0
i e 0', ‘ f p
60 i ( ) 131:6 ' 1 100
»—° | . i. . Q0
50 r st 1 .-» ' 50 *5
40 l— ~ 4- ..._LL_-L _, J t_ ._:.-

L.. L“; ....___.: 1.. _. __ _ -_ A. .. 2-..-.." _ .-.. __ _; _ .‘ . 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
l 1

Figure 3.1: The left hand panels (a-c) show values of f0 (vertical axis) vs the sin gle—particle
angular momentum j (horizontal axis). In the right-hand panels f0 has been divided by Do,
the fraction of the total dimension of the space (ignoring the 2] + 1 degeneracy) accounted
for by J = 0

seniority zero which would be the ground state for the pure pairing attractive interaction
corresponding to a choice of parameters Vn : — 1, VI./() : 0. For any set of the parame-
ters from the random interaction ensemble which leads to the zero ground state spin, we

calculate the overlap

x: 1(J=0,g.s.|0,p)13. (3.18)

The histogram of the distribution T(.r) is presented in Fig. 3.7(b) for the case of N = 6

particles on the j = 11/2 level. Clearly, the overlap is very low.

In the extreme chaotic limit the eigenfunctions are expected to be nearly random su-
perpositions of the basis states Ik) with the uncorrelated amplitudes C k constrained only by
the orthonormality requirements. The distribution function of the (real) components of a
given eigenfunction is the same as that of a multidimensional vector uniformly spread over
a unit sphere, ?({Ck}) oc 801C}? — 1). For the dimension (1, this leads to the distribution

function of any given component C, which corresponds to a projection of the eigenvector

37

 

 

 

 

 

 

 

 

 

 

r— - , . ‘ a . r ff 350
so 1 (a) .. ”=4 (b) 4o
50 [ l L N24 , . 30 .
=40 f- . .. g
J 30 l t J 209
20 i‘ C‘hk“‘ ,r'. a
10 L ‘O‘...,.. ‘ . ‘\. i . ./"/. 10‘}
o 5L#4_LTFW__L___.#_;_ _. i,__r_,r ‘ ‘1 m J f v n. r 0
25 ” (C) (d) . 125
20 r -k- ,— ' 4 100
k _. I N=5 . ' 3
:15 r- t r 3 75 §
E — t‘~ . | D
-"10 ~ ”‘5 " . t ,1 50Q
5 r- . y at .'/ A 25 w—E
o 1 l ‘hfi‘fi. *‘ 4 L A ; 0
25 f (e) L (1) . 250
20 L i I" N=6 ’- " 3 200 ~
315 * "“"“~.\\._ f . . '3 150 g
.210}. «..., 3 , {ioo\a
5 l - ‘ r. . 5 so 5
0 u I d _.3 ~__ #03 " i . 0
4 6 8 10 12 4 5 8 10 12

Figure 3.2: Same as Fig. 3.1 but this time we show f

"161

onto a specified direction in Hilbert space,

_ ltd/2)
— firttd— 1)/21

In the limit of (1 >> 1, eq. (3.19) gives the Gaussian distribution with (C 2) = 1 /d while the

£P(C) (1 — C3)(d"3)/3(~)(1 —C3). (3.19)

 

weights x = C2 are distributed according to the Porter-Thomas law. Complicated nuclear
shell model wave functions obtained in realistic calculations with no random parameters
are close to the chaotic limit. The typical d-dependence can be unfolded I l 8] into a regular
scheme of approximations, the so-called N-scaling, which allows one to classify various
processes going through the compound nucleus stage. The statistical enhancement of weak
interaction effects observed via parity non-conservation in neutron resonances, described
in chapter 1, and [16], is one of the most convincing illustration of statistical regularities

seen on the level of individual wave functions.

In our examples the dimensions (1 are typically not very large, but the statistical features
are brought about by random interactions. We can look at the distribution of overlaps (3.19)

as the representative of the distribution of components of eigenvectors along the pairing

38

 

   
 

   

  

10
0 5 10 150 5 10 15
4o {CT—‘fi—Tfirflu'P—M Mamm— H WT
A 30 (d)j=17/2

 

 

 

 

1"6 2‘0 30 '30

 

A30 ~ (9)i=23/2 (h)j=25/2 3 (i)j=27/2

 

 

O 10 20 30 O 10 20 30 4O 0 10 20 30 4O 50
J J J

Figure 3.3: The distribution f_, of ground state spins Jo for various systems ofN = 4 parti-
cles.

basis vector |O.p). Six particles on a j = 11/2 lcvcl give rise to d z: 3 states of spin J z 0.

In the chaotic limit, this would lead to ‘1’(C) = const, and therefore

1
Tde3(X) : m, 0 < x S 1. (3.20)

This distribution appears in the problem of pion multiplicity from a disordered chiral con-
densate [44], [1]. This prediction, the dashed histogram in Fig. 3.7(b), is in agreement
with the numerical results. Another situation is shown in Fig. 3.7(a) where the data, a solid
histogram, are taken for the ensemble which contains regular attractive pairing, V0 2 — 1,
plus uniformly random interactions in all channels with L > 0. The presence of pairing
generates a significant probability for the paired ground state, a peak at x 2 1. However, it
is possible to show that there exist an “antipaired” state with J = 0 but a vanishingly small

“probability” to be a ground state so that the space of the candidates for the ground state

39

 

 

 

 

Figure 3.4: The distribution f} of ground state spins J0 for various systems of N = 5 parti-
clcs.

position is effectively two-dimensional. For (1 : 2

l
(Pd,_7(x) : ——————, O < x S 1, (3.21)

' n\/.t(1—x)

which agrees qualitatively with the data, Fig. 3.7(a).
3.3.3 The yrast line

Until recently the question of correlations between different classes of levels in random
spectra has largely been ignored. RMT has little to say on this topic, yet it is a compelling
. question. Our ensemble exhibited such a correlation in the form of regular average yrast
lines. From the ensemble for each N -body system of spin-J particles, two groups of spectra
were extracted, those with Jo 2 0 ( or j in the N = 5 case) and JO 2 Jmax. From these groups

the average lowest energy for a given J was extracted. The results for each N , j system were

40

 

 

 

 

 

 

  
    

.
a
i
.
4
‘—‘~ ‘

‘.—_~

'_-- '

t
0 1o 15 20 25
t

 

 

 

 

v___.,_--.,_.._,-_ v ._.__v_. ... - ,_ ..Pfi___..__.v

(e)j=19/2

 

,. ‘f ......... .--l . ..- . J
1015 20 25 30 35400 101520253035404550
J J

O ‘57,

Figure 3.5: The distribution f; of ground state spins Jo for various systems of N = 6 parti-
cles.

fitted to the form

E(J_) _—. E0 + aJ(J+ 1 ). (3.22)

Fig. 3.8 shows the yrast line for a representative system, with the zero of the energy scale
taken as the ground state energy. Large fluctuations are especially apparent at small J, with
the J = 0 and 2 energies, which are on opposite sides of the parabola, while the higher I
energies are well approximated by it. This behavior is quite general, being present in all the
N = 4 and 6 particle systems. Fig. 3.9 refers to the ensemble average of all the six-particle

systems.

Any set of VL will create a mean field which will not, in general, be spherical. This,
and the fluctuation of the yrast levels about the parabola, mean that the low lying states are

connected by non-collective rotation.

41

 

 

 

~ 80
l
0 ~~o Random VL
o 60i- ""'Vo=-1
o\ 1 .- — -— 4 V0 = O
3 =1 ~ ' DISI. Of Allowed JTotal
O t
- 40
. l
20 i“. ,
l 2' ' ' V "'~ I I . . "i
0 L.*.g,:rb,_..._,.o_..__.w_ -_ -3
0 2 4 6 8 1O 12 14 16 18 20

J

Figure 3.6: f0 is shown for the N = 6, j = Q system. The cases where the pairing interac-
tion are random or zero are practically indistinguishable.

3.3.4 The ground state energy gap

In even-even nuclei the ground-state is pushed down with respect to neighboring nuclei.
This is a result of pairing, but it is not a signature of pairing. Pairing correlations lead to a

depressed ground state, but a depressed ground state does not imply pairing.

The energy difference between the ground-state and the first excited state, E2 — E1, is
the ground state gap. The distribution of the ground state gap was compared with the next
gap in the spectrum, E3 — E3. The resulting numbers for each ensemble were divided into
three groups corresponding to Jo = 0 (or j for the N = 5 cases), JO 2 Jmax, and the rest. The

gaps from the first two groups were divided by those of the last group so that they could all

be compared sensibly.

The result is that the gaps from the JO 2 0 (or J) sets are larger then those from the

10 71E 0, Jmax set, their ratio being consistently close to 2. An examination of the left panels

42

 

 

0.15 :——« ;——~~~—-._— . ......

 

 

 

 

 

 

 

 

 

 

 

 

65E (a)
0.1 E
1 "1 1
0'05 7 i
3: o 1‘: 1 ?4
0.. l
5 (b)
0.3 1
0.2 11
0.1 «
0 7 E 1 E 1:1: 11 [1W 511(11ng 1“ 13
0 0.2 0.4 0.6 0.8 1

Figure 3.7: The distribution of overlaps, eq. 3.18, of ground states with JO 2 O with the
fully paired (seniority zero) state are shown as solid histograms. Panel (a) corresponds to
the ensemble with random VL/O and regular pairing, V1130 = —1. Panel (b) corresponds
to the ensemble with all V1, random. Dashed histograms show the predictions for chaotic
wave functions of dimension d = 2, panel (a), and d = 3, panel (b).

of Fig. 3.11 suggests that while the pairing force is attractive for these spectra, it is still
quite small. Furthermore, the interaction parameters for low L tend to be small or negative.
The value of V0 is not the sole driving force behind the depression of the ground state
energy. For a given value of j, there are j — 1 / 2 two—body Slater-determinants with total
projection M = 0, and there are j— 5 / 2 with total projection M = :12. In general there are
(j — 1 / 2 — L) two-body Slater-determinants with total projection M = iL. This means that

a Jo == 0 state is more sensitive to lower-L interaction parameters as more low-L pairs can

be involved.

43

1 _
3 1 a) Jo_0 /

 

< ( EJ) Lowest>

1 b) Jo=Jmax \/\

__ 16 ____, "18"”

0 1. _ __ .. . .__ ._._ ______. ._. .. _______ ____.--_ ..._____ _._ ___-

o 2 4 6 8 1o 12 13,—
J

Figure 3.8: The average yrast line for a representative system, N = 6, j = 1—21.

The ratio of gaps from the JO 2 Jun,r sets to those from the Jo 75 0, 1",,“ sets is close to l;
the ground state is not pushed down significantly when JO 2 Jmax. The right panels of Fig.
3.11 indicate that the interaction parameters for high L are large and negative. The ground
state energy is not expected to be depressed here because, from simple combinatorics, the

Jmax states cannot form many hi gh-L pairs.

The behavior of the ground state gaps can thus be qualitatively explained an effect
of spin projection combinatorics. If this picture is accurate it is another very interesting

example of a correlation between different classes of states being generated by geometry.

3.3.5 The level density

In our model, we expect the level density for a given class of states with angular momen-
tum J to be Gaussian, centered at O, and with a standard deviation 0], as for the two-body

random ensemble (TBRE) [43]. Initially it was suggested that the source of the regularities

44

 

 

' 1' '3'
. ,
f "I 4
1' '.!":' u"'"'.ll‘|

,:ut..ot‘~‘

c) N=6 Jo=0

 

 

——».———1—-—.+--—-—-——1——+

 
 

d) N=6 J°=Jm

  

 

 

 

60

Figure 3.9: The average yrast lines for all the N z: 4 and 6 particle systems. The range
of j starts from 11/2 and goes to 27/2 for N : 4. and 21/2 for N = 6. The longer lines
correspond to systems of higher j. Error bars for 3 panels have been omitted for clarity, but
are comparable to those of the Jo :2 0 spectra.

in the ground state spin was the behavior of OJ for the different classes J. If 01 is dra-
matically larger for J = 0 and J = Jmax, then the ground state would be biased in favor of
these values, as the distribution of energies with these spins would stretch to very negative
values. Even if this was correct, it would be just a reformulation of the problem rather than
an explanation. The actual value of 0] was extracted from the ensemble by superimposing

all the levels of angular momentum J from the spectra, and fittin g the resulting normalized

distribution, p(E), to a Gaussian centered at 0.

Indeed, o; is larger f or J = O and J = Jmax classes, see Fig. 3.12, but the degeneracy, D J,
of these values of J is small, and the effect of the larger size of o J is not nearly large enough
to explain the distribution of Jo. To illustrate this the following experiment was performed.
“Spectra” with the appropriate J degeneracy, D J, were made, each of the energies generated

from a Gaussian distribution centered at 0 and with o = 0], taken from the N = 6, j = 11 / 2

45

 

 

 

 

 

 

;'3 3r
ug :.
.— 2t
”J l
v 5
13' . —
5:._ - an» M? _--- .....-” ._ _ -_
4i ————N=4
- l Jozdmax -» N=5
P5 3i ----N=6
LLIJ
— 2? .
g a .
1i
0 A _ __ __.. __-__.______ __t .__.__. _‘_ M~_..__._ M _ _4 -

Figure 3.10: The ensemble average ground state energy gaps for all values ofN and j.

ensemble. The resulting f] is shown in Fig. 3.14. It follows DJ closely until Jmax. fjm
is greater than the statistical weight of 1",,“ by a factor of 4. This is the closest similarity
to the ensemble result. The conclusion is that OJ by itself plays no perceivable role in the
predominance of spin—O ground states, but goes some way in explaining the high incidence

of Jmm ground states.

3.3.6 The distribution of level spacings and off-diagonal matrix ele-
ments

So far the ground state wave functions have exhibited chaos, and the fluctuations of the
yrast states suggested non-collective rotation. Now we come to another signature of chaos,
this time on the level of spectral fluctuations within a class of states, namely the level
spacing distribution, P(s), discussed in 2.2. To study P(s), a value of J was chosen with
large enough D _] to give a dozen or so spacings, the sequence was unfolded and a set of

spacings {s} extracted. The resulting distribution P(s) agreed well with the parameter—free

46

1 1 (award (b>J=13/2 $2?

0.5
<VL> 0

.-.“ ..._A,.
H

—o.5
_1

.i _ ._..» __L.-, -.. -L..__.L-__._L____‘~. .1 - H.. ,._. ..._ L... .-__.___.A_.._..-..__A_____.a_-. _._. _ A, .

1 “ (c)j=15/2 J=0 s (d)j=15/2 J=3O
0.5 Q—

l
L ," ‘ -L

<VL> 0 ;;1_1AA~AA, . ,,#51, A

#—v— -.-1,1.T._,,

—o.5 e
-1 L

_.+_—..___A ...__ a A ._.—..._. ._.. ._ ._ . ...—._.

(e)j=17/2 J=O

(t)j=17/2 J=36

 

Figure 3.] 1: The mean value, (V1,), of the subsets of {V1,} vs. L that give JO 2 0 or JO 2 Jmax,
for all the N = 6 systems. The panels are labelled with the value of the sin gle-particle spin,
j. On the left are the Jo = 0 data, and on the right are the Jo = 1,,“u data.

universal Wigner distribution (1.1), see Fig. 3.15.

The distribution of the off-diagonal matrix elements of the Hamiltonian is Gaussian,
as expected from RMT. If the basis is changed from the m-scheme to the IJM) basis, the
Hamiltonian is block-diagonal, and orthogonal invariance is lost. The off-diagonal matrix
elements in a given block no longer have a Gaussian distribution. In shell model calcula-
tions with realistic two—body interactions, the distribution of of f-diagonal matrix elements
is close to the Porter—Thomas form,

l~+»q

0t . .
new e ”w

q)

, 27(71— (3'23)

PJ(Hi/j)

Actually, this distribution arises in a variety of systems, including nuclear and atomic shell

models, and may be generic for these many-body interactions [46].

47

51 ...
.. .. .. ~—-«N=4i=25/2
' ' - - N=5j=15/2
4; . ~~~ N=6j=17/2
o 3?
2 ' ..r“‘°~

. _*00
. 0....N009.0H-900’0°
o

o O '. '
4 1_ 1 o . .0 0 ."j' .
. .7 O 0 ' V." "V . O ' .
0

.~“., ',

Figure 3.12: The standard deviation of the level density, p_, (E), of states with total angular
momentum how, for three representative systems.

Our Hamiltonians for the j : ’11/2,N : 4 and 6 systems have J = 0 blocks of 2 and 3
dimensions. respectively. The matrix elements in a randomly chosen basis were calculated
explicitly in terms of the interaction parameters {V1,}. To compare Hi“ from different sets

{VL}, they must be brought to the same scale. A reasonable scaling procedure is to divide

 

by \/Tr(H — (H))3. Statistically Tr((I-I)) z D17, Tr(H), where Do is the dimension of the

 

’13

J = 0 block. Our procedure is to divide each Hi.“ by \/Tr(H3) + (131(5) - Do )Tr(H)3. As
we are concerned with the J : 0 block of the Hamiltonian we have an additional quantum
number at our disposal, namely S, the seniority. An ordering of. the seniority basis is chosen
such that the diagonal elements H1] ,Hgg, and H33 correspond to S = 0, 2, and 6 respectively.
For N = 6 in the seniority basis only H13 survives, reason being that according to (3.17),

the spin-O block of H cannot mix S = 6 states with S = 0 or 4 states and consequently

the value of D0 must be changed to 2. The results for N = 6, Fig. 3.16, show that in

48

 

 

 

 

 

 

 

 

 

8 ;'
6 i
-=. .
b 4
2 l
o
4 .
f; 3
5 2
1 r
o it -~ +— — 1, —~—+—— ~—— —
2 L
3 1.5 \
b 1 w? \Z‘V‘AAAMAA/‘WA ‘
0.5 t N=4
o ‘A.._____r___.-- ,,,..,_l, ..1 3L 1 . ._.AA -1-,,._1_11.AA_..1-1_1_._. A ...
o 10 . 20 30 4o 50
J tot

Figure 3.13: The standard deviation of the level density pj (E) of states with total angular
momentum how, for all the systems. The values of j start at lI/2 and increase by one as
the lines get progressively longer.

the seniority basis P(ng) is far from Gaussian, it is more like a sum of two negative
exponential functions, and it is quite close to P(le ) for the random basis. In the random
basis P(H13) and P(H33) fall off rapidly, suggesting that the random basis is accidentally
close to the seniority basis where H13 and H33 are zero. In a three-dimensional system such
an accident is quite plausible. In the N = 4 system the distribution in both bases has two
peaks, a surprising result and most likely an artifact of the scaling procedure, see Fig. 3.17.
An alternative scaling procedure was tried where H13 was divided by W. This gave

a negative exponential distribution, with q = 0 and or 2 1.8 in (3.23), Fig. 3.18.

3.3.7 Multipole moments and transition collectivity

The structure of the low-lying states generated by random interactions is of interest. We

have already seen in section 3.3.2 that the spin—0 ground states are chaotic in nature, hav-

49

 

, —— Data
‘5 o.
’3 ‘ I‘
\ 1 1‘
2.! 10 l ’\ I \t I
U i ’l I, ‘ ’\\
m I l \ I \ I \
X— f I \ I ‘ I \
W. ‘ I ‘ I \ I \
I \ I H ‘I
l‘ ’ \ f v 1‘
, l \ I ‘l I \
1' I \\1 ‘1 I \
5 i I ‘
1‘ , \\
i‘ I \__
I \ ' \
l, ‘ l \
g I X-
I I \ I,
l \ /
0 A A AA A A AA __ A, _...1__A_-__ .- A A,
0 5 10 15 20

Figure 3.14: The fraction of systems with ground states spin J0 : J vs J for fake spectra
where o(J) is taken from the N =2 6, j = 11/2 ensemble.

ing none of the coherent properties of a physical ground state. Nor do the J : 0 ground
states have any of the characteristics of a paired condensate. One can ask if the low-lying
states are built of single-particle-hole excitations or if, for example, Slater determinants of
a particular seniority play a special role in the J = 0 ground states. With this in mind the

low energy wave functions were probed with multipole Operators, Mgr, defined by

zchKa *a (—)J" (3.24)

jm jn aam

and the spin-0 pair operators P; and P0. The multipole operators are creation-annihilation
operators for particle-hole excitations of a given angular momentum. Here we will define a
quantity, X K(i —» f), with which we can gauge the collectivity of an excited state in terms

of the reduced transition probability. The reduced transition probability between an initial

50

N=6 1.17/2 N36 ,—17/2

1 _._.A _.__..__ -. ___,..,- -._AA ---A..- __

0.2

0113)

011 r

 

0

 

-5 ‘ L1
NOS [HIS-’2

1 y 1 —--e~-—.—v—. 0.4 m .A1_1_, ———-——-~——- —-—— ———-~

 

3'; .4 ;
gray?“ J._.11/2 1. 0.3 r

. 027*

E
o
.8,
NM

0.1 i

 

0 1,1 -1 AA A__A._,11. _ _. ._---A; __ 1 .
-4 -2 o 2 4
N=4 F2512

0‘5 .._.—-~__V—-,..-._.__ -th -._.. .-L

 

' l
0.4 ; .

013

0113)

.2#
1

O
O

   

__ . ... _ .. : 0 111. _._..__1.-_. A- __ -.A,11._. _ . .
2 3 4 _25 -15 -os 05 1,5 25
Spacrng (5) Energy

Figure 3.15: The distribution ofenergies, pj (E), (right panels), and energy spacings, P} (s),
(left panels), with fixed J for three representative systems.

state Ii) and a final state If), i —-> f, of multipolarity K is defined as

2

. l i . .
3,,(1 ——» f) = Z |(fJfotMKK|zJ,M,-) , (3.25)
MfK
The average fluctuation of 911ng in the initial state Ii) is
Sui] = [are —+ f) = [(MMAMKKMQMM.» (3.26)

fJ/ K

The fractional collectivity of a given state, labelled f, is

_BK(i—’f).

X K(i —+ f) will be peaked around 1 if the final state consists of simple particle-hole excita-
tions of the initial state. However, if the lowest J = 4 state is connected to a J = 0 ground
state by [my zfiwfllefllfsz this would indicate a two particle two hole structure which

has more potential for collectivity. The N = 4. 6 j = 11/2 systems were thus analyzed, and

51

 

 

 

 

—— H1 .
2 r ........... H; {i
l
___ H23 ’13
1 t —— — H12 Seniority basis I: it
7
A 0 L /'3
z: ’ I? i
3:, 1, l / I:
.0 f I;
o ‘ /
E —3 r- '/ It" 1‘
t J J E 5
1 fl l ,t
-4 l , nfl _ Vt ';
i1' 1‘ H ' {“953 i
—5 ' ‘ F ‘ J lar-.53: I - ‘I
1 l\.-"-. F“.
ii “'1 i l _.§I\Jn'}l l ’1‘"
-—6 r 1 ‘ 3 ‘
_2 0 2

Figure 3.16: The probability distribution P(Hij) of the off-diagonal matrix elements of the
3 x 3 J = 0 block of the N = 6, j = 11/2 Hamiltonian in the IJM) basis. Except where
the seniority basis is specified the basis has been randomly chosen. Note the shoulders on
P(Hij) in the seniority basis.

there was no evidence of such a structure. The distribution of values of XK(i —> f) were

peaked at unity indicating that the excited states were simple particle-hole excitations.

The 12 and 2K MKKMlzK matrices were calculated for the N = 4,6 j = 11/2 systems.
The degeneracy of the J = 0 states, Do, is 2 for the N = 4 particle system, and 3 for the
N = 6 particle system. The J = 0 states that maximized and minimized S3[O] and S4[O]
were made by diagonalizing [K MKKMA in the J = 0 block. Having found these states,
the single-particle occupation numbers n", were extracted for each state. The contributions
to n,,, from Slater determinants of specific seniority were separated. It is immediately clear
that the states that maximize S3[0] and S4[0] consist mainly of seniority zero Slater deter-
minants, while the states that minimize Sg[0] and S4[()] mix all seniorities, see Figs. 3.19

and 3.20.

52

 

 

0 1.., A _ - .. ,,_. ; _ A_AA__é.._.-_ A _A A A AA A _
. l '
i ”an. t E * if"
f?“ —1 I i . ‘ I
I ‘ i
. , . .
Q —2 r .. .
9 ' ‘15.;
0" 3 J 1 I ‘NI .
_ '— f; I I E ‘
C r 1‘ :h I 'r; . I
—— L i. ‘ 1:1: ‘ . 9' . .5 : a!
'- ..1 gRandom Basrs I
_ l t .': E I . . . :' ' I.
4 . 1:? l“ i: ‘ —'—' Seniority baSlS E ;' l: I
., 3 . I g .
.giIII'. ' e s I I
I i ' a i i
_ L__,_ , , ALAAAA A A _1A _;
—2 —1 5 -1 —0 5 0 0 5 1 1 5 2
12

Figure 3.17: The probability distribution P(Hij) of the off-diagonal matrix elements of the
2 x 2 J = 0 block of the N = 4, j = 11 / 2 Hamiltonian in the IJM) basis rescaled according
to the prescription in the text.

To further examine the structure of the lowest J = 4 states we looked at the operator
r _ LA 1 r
QI.A;KK' “ Z CKK1;K’K3MKK1MK’K2 (3-28)
Krixz

which is just a pair of spin-K particle-hole excitations coupled to angular momentum L.

In analogy with (3.25) and (3.26), we define
2

Butt —> f) = 2 I <fJfAIQI,A10> , (3.29)
A

and

SQ;LIOI = X<OIQMQZAI0>- (3.30)
A

Now the same questions we asked about the single-particle-hole excitation nature of wave
functions can be asked for two-particle-two-hole excitations. When 84(0 —+ 4) compared

to BQ;4(0 ——> 4) we see a strong correlation. If one quantity is large, the other is small. This

53

_ -.~. random basis 1
' ------------ Seniority basis j;

 

 

 

H

12

Figure 3.18: The probability distribution P(Hij) of the off-diagonal matrix elements of the
2 x 2 J = 0 block of the N = 4, j = 11/2 Hamiltonian in the IJM) basis but this time
rescaled with an alternative prescription to that of Fig. 3.17.

is especially true of the N = 6 systems. An obvious interpretation of this is that a state can

have either one nature or another, but seldom combination of both.

Another geometric effect is that the pairs {59410154101} form an ellipse, see Fig.
3.21. This is to be expected as the J = 0 block is two dimensional, and the operators
[A Q4/‘Q4A’ and 2K MKMJK have different eigenvalues and eigenvectors. If we take the
two eigenvectors of [K MK 9%: as a basis, and call the corresponding eigenvalues M 1 and
M 2, any spin zero state of the system can be specified by an angle 9 in this space. The
eigenvectors of 2A QMQIA make an angle (I) with the axes, and have eigenvalues Q1 and
Q2, now the pairs {SQ;4 [0] S4 [0]} are just { M1 cosz(9) + M2 sin2(9), Q1 cos2(9 — (1)) +
Q2 sin2(9 —— (p) }, which is the equation for an ellipse parameterized by 9. The distribution
of points {SQ;4[2I, S4 [2]} can be explained by a similar argument for three dimensions as

there are three J = 2 states again see Fig. 3.21. One must exercise extreme caution when

54

 

 

0.4a # q— . ww-

0.3

.._~_'_..__. -——. .._.,
o
a
a
c
l
o
I
u
a
e
I
l
O

a .
. .‘ ."‘.
. ......... ‘.' ..' "
CE 0.2 ‘ ‘
I
l f f
1 !\ / \ / \ '
\ \ /
01 \\ // \ p——0\ / \ /
‘e \ / \ / V
\ o’/ \k //
\e’/ \‘e
0

 

_s_5_’4_3_2_1012345'”s

Figure 3.19: The contribution to the occupation numbers n,,, of the IS : 0,N = 4) state
from Slater determinants of seniority 0, straight line. The jagged lines depict the decompo-
sition of nm for IS = 4,N = 4) into contributions from seniority-4 (upper dotted line) and
seniority-0 (lower dashed line) Slater determinants; here j = 1 1/ 2.

interpreting combinations of single-particle operators. There can be striking correlations

begging to be misinterpreted as exciting physics instead of the subtle geometric effects that

they so often are.

3.3.8 Pairing and fractional pair transfer collectivity

Previous investigators ['29], [’30], tried to pinpoint the role of pairing in this problem in a
number of ways. So far we have seen that the pairing component of the random interaction,
in the sense of the magnitude and sign of V0, plays a minimal role. When set to 0 it doesn’t
impact the fraction of spectra with spin 0 ground states, Fig. 3.6(c). Furthermore the ground
state wave functions are close to the chaotic limit, they are not oriented in the Hilbert space

along the vector of the pure pairing wave function, Fig. 3.7. Following the example of

55

 

 

 

 

 

0.6 1 I -.. . ,; A , A
0.5 i; ---------------------------------------- b) c) 1
1 f '1
0.4 1 a) ____________
CE 0.3 I """""""" 3:0 «f— y ““““““““ 1
0.2 L ----- S=4 3‘» I
01 :_ 8:6 I} """"""""""""""""""""" :3 ........................................
o ' ; ~~~ - -

95—3—1 1 3 5 —5-3 —1 1 3 5‘ 245—341 1,131.5
m m m

Figure 3.20: The contribution to n,,, for the states S = (l, N = 6 (a), S = 4, N = 6 (b) and
S = 6, N = 6 (c) from Slater determinants of various seniority; here j = 11 / 2. Note that
for J = 0, nm is independent of m.

Ref.[29I, [?], we examine fp, the so-called fractional pair transfer collectivity. fp is defined

by
(N '* ZIPI)IN>2
(NIPSPOIN)

 

fp : (3.31)

The states IN) and IN + 2) are spin zero ground states ofN and N + 2 particle systems with
the same interaction parameters V1,, and single-particle spin j. The numerator is a mea-
sure of the extent that IN + 2) is built by adding a spin zero pair to IN). The denominator
counts the number of spin zero pairs in IN). A significant pairing interaction would favor
a high degree of pairing in the ground state, making the distribution of fp peaked at 1, and
it is. This was interpreted as a signature of pairing. However a plot of (N —— 2IP0IN)2 vs
(N IP61 POIN) reveals an interesting structure which may be a purely geometric effect, see
Fig. 3.22. The structure is unaffected by setting V0 2 O. In Fig. 3.22, the pair creation
operator Pg connects the spin zero multiplets of the N = 4 and 6 particles which have di-
mensions 2 and 3 respectively. Conversely one can examine the opposite case, connecting
the 6 particle state to a 4 particle state by the action of P0, which would leave the numer-
ator unchanged in (3.31, but the denominator would then be (6IPgP0I6). In this case the

structure in Fig. 3.22 is only slightly modified.

This can be explained by the geometric properties of P; which are summarized in

(3.17).

56

50 rm—H -._MM-u-__fi-___-_._mu___m___,m_fl

..........

"h . »-'~v ...-~-.
40 r
-’ I .
. J8; 0“ (...-I.- , I" .
K... 1' ’5‘
a "5"" 7 -
- c, ' .

 

A 30 !
2.. i=4
’q ..
Q.
10 J=2
_ J=O
O I_,_-__W7u ______ _J___.. - “A” *2.-. ~44
0 1O 2O 30

<JIM4M‘4|J>

Figure 3.21: The quantities SQ;LIJI and SLIJI are strongly correlated for different values of
J. This is just a geometric effect. The quantities are calculated for the lowest J states. Here
N=4andj= 11/2.

 

Writing the general N = 4, 6 spin-zero states I0)4 and IO)6 in the seniority basis, S ,N),
as ((XIO,4) + BI4,4)), and (yI(),6) + 8I4,6) +y|6,6)), with the additional constraint that

(13 + B3 2 l, and 73 + 53 +83 2 l, the plot Fig. 3.22 is the set of points
{6<0IP3I0>4 =100t2 +2? , 4<0IP3P0I0>4 =100t2 +2}. (3.32)

Note that for convenience we used PgPo ( 2 P0P; + 2) instead of POPS. The operator P0
cannot connect I6,6) to I4,4) so the value of e is either 0 or 1, which accounts for the
points on the horizontal axis in Fig. 3.22. The straight line portion of the plot corresponds
to CL = 'y. In principle if 0t and y are uncorrelated, the points would cover an area on
the plane. The states I0)4 and I0)(,,8.___._0 make angles 64 and 96 in their respective two-
dimensional planes. These angles were extracted from the wave functions and plotted
against each other and are highly correlated (as too are or and y), lying on a tangent to and

on the positive side of the line 04 = 66, see Fig. 3.24. When this curve is rotated by 325 so

57

as to lie tangent to the horizontal axis, it is, to a very high degree of accuracy, described
by A + Bcos((n04 + c) where A : 0.0970, 8 = 0.1028, c = 0.8901? and a) : 2.8288. The

reason for these phenomena, although qualitatively unexplained, are certainly geometric in

   

nature.
15 I‘m." a! “.--“,W - W .. W - WWI
N=4&6 j=1 1/2
,I
E"
N 10 ‘ ‘07" $5
T r .t
2 / a
+0. /
25’
z .
..Y. 5 r*
. v,.—.o
V0 random
0 I" “'—“"—"—“—"""O'I‘“"W & -l «hm--a«.~-.-mflu;.,-. ._.... ..-a-__._,-.;,£,:-+.m_
O 2 4 6 8 1 O 12
<N4IPP+IN4>

Figure 3.22: The numerator offp vs the denominator in eq. (3.31), evaluated for the N = 4

and 6, j : l l / 2 systems with JO 2 0. Pg creates a pair of particles coupled to spin zero. It
acts on N = 4, J = 0 ground states to make an N = 6, J = 0 state which is overlapped with
the ground state of the corresponding N = 6 system. The operator PgPo counts the number

of pairs in the N = 4 state. Note the structure is unchanged by setting VL = 0, again the
effect is geometric.

58

Fraction(%)
8

10?

 

 

 

15 I: I3,

 

10}

0 II EEII 2H: a . . . ..fllln

_1 .5 _r 40.5 o 0.5 1 1.5
0(rad)

Fraction(°/o)

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.23: The distribution of the angles made by the J = 0 g.s. in the 2-dimensional

space {IS = 0), IS = 4)} for N = 4, 6. Panel a) corresponds to those points on the straight
line portion of Fig. 3.22 while panel b) is for those points on the curved part.

59

 

 

 

—2 if —1 o' 1‘ 2

0

N=4

Figure 3.24: The strong correlation between the angle the Spin-0 ground states make in
the two-dimensional N = 4, J 2 0 subspace, 9~W4 and N = 6, J = 0 subspace, GNfig. The
states are eigenstates of Hamiltonians with the same interaction parameters.

60

Chapter 4

A Statistical Theory

The high incidence of J = O and J = Jmax ground states can be thought of as a result of
geometric chaoticity, i.e. the inherent complexity of an N-body system with fixed J. This
feature depends only weakly on the ensemble from which the random Hamiltonians are
taken, depending only on the geometry of the Hilbert space of the system. In other words
the effect is determined by the Fermi-statistics of the particles, and the values of N and j. In
the first illustrative example in Section (3.1), the analog of Hund’s rule in atomic physics,
the geometric factors of the system, S and s(s + 1), have been separated from the details
of the interaction, i.e. the random number A, in a very simple way. We will approximately
achieve a similar separation in our problem for the equilibrium energy of a state of angular
momentum J. The role of A will be replaced by a relatively simple function of N and j,

resulting from the geometric properties of the finite Fermi—system.

4.1 The Bosonic Approximation

One simple idea is related to the approximate realization of fermion pairs as bosons. The
term beyond the boson-like commutator in (3.4) is of order 31%. Here Q = 2 j + l is the
number of single-particle states, and N is the number of particles, so this term is small

for small mean occupancies. Particle-hole conjugation means its analog will be small for

61

near full occupation also. For intermediate occupation it is not small but almost constant.
This constant can be absorbed into PM and PZW by renormalizing them to get the nearly
bosonic operators 19M and blw. Thesc ‘bosons’ are pairs of fermions, and their number
and spin depend on the system (i.e. the set V1,). This simplifies the problem immensely.
Now our Hamiltonian is a sum of random energies, (01,, each of which are proportional to
V].

J

H = ZVLPZAPLA 2 2mm, (4.1)
LA LA

where m, is the number operator of the bosons with spin-L. In each realization of the
system, the fermion pairs form bosons which condense into the level with lowest energy,
ie. the lowest 03],. Once this happens, the total angular momentum I can be anywhere
from 0 to nbL where n}, is the number of bosons. We see that already L = 0 is singled out,
because in this case the only value J can have is 0, and this will happen I of the time, where
k = j + l / 2 is the number of values of L. For all other values, J = 0 accounts for a small

fraction of the total space.

In short the bosonic approximation singles out I _—_ 0 but doesn’t account for the full
effect. The values of f0 ~ 1 / (j + 1/2) are much lower than the observed values, and the
j behavior is wrong compared with that seen in, for example, Fig. 3.1. Also, meI is not

singled out, and its allowed value is too high. See Fig. 4.1.

4.2 Equilibrium Statistical Mechanics Approach

Our approach is based on equilibrium statistical mechanics which, in the case of degenerate
fermionic states, leads to the Fermi-Dirac distribution. The mean particle number with

projection m on an arbitrary quantization axis is

1
nm 2 ,
CXme —u) +1 ‘

 

(4.2)

62

20 .r a I . . . , . W

 

N=5

15 L=0,2,4,6,8,1O

 

 

 

J

Total

Figure 4.1: fj’ for the bosonic approximation. The situation forN = 6, j = 11 / 2, is shown.
Here Ln,“ = 10, k = 6. The bosonic approximation gives the solid line while the statistical
distribution of of allowed values of total angular momentum J, gives the dashed line. Note
that f0 only slightly exceeds 1 / 6.

under the constraints for total particle number N and total angular momentum projection

M ,
N = 212,", M = Zmnm. (4.3)
m m

The quantization axis will later be identified with the total angular momentum J. It is rea-
sonable to write this equation down and just start from there. However it can be derived
from very general assumptions using the Darwin—Fowler method. A detailed derivation
of this is included in Appendix (6.1). The quantities p(N,M), the chemical potential, and
7(N,M), the cranking frequency, are the Lagrange multipliers associated with the two con-

straints. The equilibrium energy for an N particle state with total projection M is

1 + r
<H> = ,- 2: vicrrca<a.a.a3a4>. (4.4)
L,A,l,2,3,4

63

where as before CIQA is the Clebsch-Gordan coefficient (L, AI j, m 1; j, I712). The assumption
of statistical equilibrium implies the off-diagonal matrix elements of am and (2;, and their
products make an incoherent contribution to the expectation value which sums to zero.
We introduce another approximation at this point by treating the single-particle occupation
numbers as uncorrelated, (mng) —> (n1)(n3). This is a key approximation and is based on

the assumption of quantum chaos, where the eigenstates are chaotic. Keeping N fixed, and

.I.
Q

m, (4.4) can be rewritten as

using the commutation rules for am and a

<H> = Z vrIcsz 2<ni><n2>. (4.5)

L,A,1,2

 

For M = 0 the single-particle occupancies nm are uniform,

l N .
:l+e‘Ho:§ I “2:214.” (46)

 

I10

where p0 = p(N,M = 0). For M % 0 and treating y as an expansion parameter,

 

 

1 1 1 2 2 2 2 3
anW :- 1+e_p(l—Ot(‘ym+-2-Tm )+0t TIN ). (4.7)
Here
e_”
= . 4.
0t 1+e““ ( 8)

Expanding the chemical potential, p(N,M) 2 pg + Au and using the constraint N = 2:," nm
we get

1 o o
#(MM)=uo+(Oto-§)T(m‘>+0(14)- (4.9)

64

The final expression for n,,, is

 

nm :1+e M)Il —0t0‘ym+
(10((10 — $an — <mz>w’2 (4.10)
+ao((2ao —1)(0to — I1<mz>m
—(0t8— ao+ IMP)?
+aotao — Iitmi —0to+ 51m“ — <m4>)

— (30L?) — 30to + %)(mz)((mz) — mz) 174].

(4.11)

where

e"”° Q—N
=——=1— . 4.12
l+e #0 Q ”0 ( )

 

(10 :
Thus the equilibrium energy for a state with N particles and total projection M is

3 N o
<H>~,M——= )3 VLICI? (firm

I.,A,1,2

 

ag'yzmlmg + 0(0(Oto — 2 )(m? + mg — 2(m2))y3

1
—(X(2)I(2(Io —1)(0to — %)(mz)(mlmg + "lg/711) —

(0t?) — Oto + )(mlm; + mgm‘m'y4

0|“

1 . 1
+ao<ao— 'Z'IIWti—Otoi“ 1—2-><mt+né—2<m“>)—

(3(13— 30:0 + I) (n12)(2(mz) — m? — ”1%)‘Y4I. (4.13)

We use M = 2m mnm to replace 7 to get

. M 3M
“N’M’ _ _N0(o<mz> " _(Q-—N)j(j+1)° (4’14)

 

It is important to note that 7 is treated as an expansion parameter, so this procedure is not

good for all values of 7. However it is valid for a wide range as seen in Fig. 4.2. Finally,

65

after using the explicit values for the sums, see Appendix (6.2), all this fits together to give
mm = —; (2L+ 1M (4.15)

‘7 3 o o
M-— + v -— '-
+ 2j4§23 2L:(2L 1) ,,(L 2) )

 

9(9—2N)'~’
. o 7 (2L+])VL
40.]8(§2—N)~N~§22 ;

>< (3L4 + 31.2 — 12j2L2 — 6j3 + 8j4) .

Here L2 = L(L+1), L4 = L2(L'+ l )2, with the same notation forj2 and j4.

1.5:. 1 _ . .W

 

 

0.5 I

 

.5 I r I r i r 3 r _I
—100 —50 O 50 100
M

Figure 4.2: The cranking frequency 7 vs. M for N=10 and j=27/2. Solid and dotted lines
correspond to exact and approximate values, respectively. This gives an indication of the
range of validity of the expansion that gives eq. (4.14).

Identifying <H>NIJ with <H>NIM___—_[ we have the main result of the statistical treatment,

(Hm = [(21.4—1)vl.(ho(L)+12:(L)J2 Haw“). (4.16)
L

66

where

ho(L) =

9.12...

._7

4.4L) = ————,,3,,. (L2 — 213)

Q
l

 

__ 9(52 2N)?-
174(L)— 40jx(-Q*N)2N2-QZ
x (3L4 +3L2 —12j2L2 — 6j3 +8j4) . (4.17)

Note that the J2 term in (4.16) can be written down directly from the K = 1 term in eq.

(3.6) as an effective spin-spin interaction i the particle-hole channel,

 

- j j 1
V :3 2L+1 . . V. 4.18
1 [gaff II J J L} L ( )
with
,- ,- 1 L3—212
. . = ., 4.1
I J 1 L} {223"- ( 9)

4.3 Ground State Spin

Given that the regular features of the spectra are independent of the choice of ensemble,
the distribution of equilibrium energies (4.16) can be evaluated analytically assuming a

Gaussian distribution for each interaction strength V1,

PLIVL) = 1 exp <— M) . (4.20)

.,
\ /27t0'i 2c5L

Our Hamiltonian in Eq. (4.16) has a form

 

(H) = const+aJZ+bJ4, (4.21)

where

a = Z VLaL, b = 2 VLbL, (4.22)
L L

67

and

3 ’3 ’7
= — 2L L“ — " , .

9(o——2N)2 4 7 .7 7 .4 .4
I = 2L 1 3L 3L‘—12”L“—6~ 8
)I. 40j8(Q—N)2NZQE ( + ) ( + J J + J )3

 

We will first examine the probability that this random system has given values of a and

b. This is given by

TIG, b) 2/5 (a —ZLVL(11) 5 (b — z V161,)“ PL(VL)(1VL . (4.24)
L L

This can be rewritten by using the Fourier representation of the delta function, 5(1) 2

(.71; . .
if (171 and performrng all integrals over VL to get

T,(a b): 2171:2/ dxdx'e W1 a” (4.25)

xnexp (——2( (la; +7t'b1) )20,‘ —i2( (V,) )(71111 +K'bL)).

The remaining two dimensional Gaussian integral over A and N is again relatively easy,

 

 

 

 

.——<>
where we have defined
A = Z afici, B = Z biog, D = 2,, aLbLoi, (4.27)
L L
and
5,, = a — )1; (V1,) a,,, a), = b — 2,, (V1,) 19L. (4.28)
Assuming (VL) = 0 we have
.——()

68

This is a two dimensional Gaussian distribution, the independent distribution of one

variable is also Gaussian

 

{P(a) : fdbfP(a,b) : I/ZlWfleflz/ZA. (4.30)

We first address the question of the probability that the ground state has spin zero. The
a 2 0 in Eq. (4.21) is an obvious necessary condition, and if b > O the statement is definitely
true. In the case b < 0 the curve (H) eventually goes down and there is a possibility to have
a ground state at the maximum possible value of Jmax : (j — (N — 1) / 2)N . Finally the
condition for the zero ground state can be expressed as a Z O and a + ngax > 0. The

probability of the ground state having spin zero is given by the integral over the region

P = ’P(a,b)dadb. (4.31)

a20,b> “wt/13,.“

In spherical coordinates a = r cos¢ and b = r sincI) the region of integration is

4 e 1— arcane/13...). 4/21. (4.32)
which gives
1 1 D+A/J~’~
P = — + — arctan ————m§I . 4.33
4 27C I x/AB—Dz ( )

Fig. 4.3 (top panels), dashed line, shows the smooth behavior of the probability fo as
a function of j for N = 4, and 6 particles. The solid line gives the results of the exact nu-
merical diagonalization. This simple statistical theory captures the excess of J = 0 ground

states but of course it does not explain the observed staggering effects.

It would be premature to conclude from (4.16) that the remaining 50% of cases have a
maximum spin ground state. Indeed, the expansion used for 'y in the derivation is not valid
for large angular momenta. However, the wave functions for the largest possible spins are
unique and can be constructed explicitly being independent of the interaction parameters.

Their energies can be compared to the energies of lower spins. In this a slight improvement

69

 

 

 

 

 

.._L A A L -_L.. A

o“ 2 4‘11'5I81‘012Ir4o 2 4 6 84(712‘14
1 1'

 

Figure 4.3: fo, the fraction of ground states with J = 0, and (an upper boundary for) fjm
for N = 4, and 6 for different j. Ensemble results; solid line. Statistical theory; dotted line.

in the results is obtained above for spin f0, this is taken into account in plotting Fig. 4.3,
and we also extract the upper boundary for the probability of the maximum ground state

spin mec, Fig. 4.3, (lower panels).

Thus, the statistical approach provides a natural qualitative explanation and reasonable
quantitative estimate for the dominance of ground states spins JO 2 0 and JO 2 Jmax in
our ensemble. Other non-statistical details of the picture, especially the non—monotonous

changes like the odd—even staggering of f0, require a more subtle approach.

4.4 The Distribution of VL for Jo : Jmax

The spin of the ground state in each particular realization is determined by the values of VL
in this realization. The subset of VL that gives Jo = 0 or J0 = Jmax has been discussed in

section 3.3.4. We can use (4.16) to calculate the mean values (VI) of those subsets that give

70

JO 2 Jmax. If we assume that the VL’s have a Gaussian distribution this can be achieved by
exact integration. In Fig. 4.4 the theoretical values for (VL) vs. L were compared with the
mean values of actual sets of random parameters V,, which led to JO 2 Jmux. The resulting

curve is in total agreement with the statistical theory.

 

 

 
 
   

 

E r
r :
_,o —0.2 I
A I
>‘J I
V —0.4

l
1

I o———4 Theory

*0-5 I k - - -1 ensemble

1

r
L

—O.8 WW _W_ m._....--___. n
O 2 4 6 8 1O 12 14

 

 

Figure 4.4: The mean values of V1, for the subsets of VL that result in Jo = Jmax ground
states. The theory, solid line, agrees with the ensemble average values, dashed line. Here
N = 4 and j = 122.

4.5 Ground State Energy

The equilibrium energies (4.16) are strongly correlated with the corresponding numerical
values. In a scatter plot of theoretical energies vs numerical energies for a specific value
of J, the points fall about a straight line, as in Fig. 4.5. Two subsets of the spectra were
selected, those with JO 2 0 and JO 2 Jmm. The slopes of the lines were independent of J
in both groups, having a value of 0.81 i 0.03. This is a direct result of treating the single-

particle occupation numbers nm as uncorrelated. When the exact correlated occupancies

71

are used to evaluate (4.5) the slopes become 0.98 :1: 0.03. The theoretical energies for a
given ensemble are shifted by a constant negative amount that decreases with J, as in Fig.
4.6. The shift was smaller for the Jo =2 Jmax set. Using the correlated occupation numbers
only reduced the magnitude of the shift by 15%. This nonstatistical effect is presumably
due to the regular part of the dynamics related to (V5). The simplest description of this part
can be reached with the aid of the boson expansion technique [3], and corresponds in fact
to boson pairing. For J0 : erm Fig. 4.5(b), the ground state energy (??) is in one-to-one

correspondence to the statistical predictions, although the slope differs slightly from unity

because of the use of the y—expansion.

 

 

 

 

1O i_—_T'H I ‘_—_”‘H"’ ”’“i 10 .
5 b)
o ._--_ “MI/fur
I i
mo —5 //
-10
—15 ?-
‘15 -5 5 15 A “—15 _5 5
<H>N’0 .—_ ho <H>NJ

’ MAX

Figure 4.5: Ground state energies E0 for Jo : 0,(a), and Jo : Jmax,(b), vs. predictions of
the statistical model, eq. (4.16). Here N = 6 and j = 17 / 2. The theoretical energies are fit
to a straight line, included in (a).

4.6 Single-Particle Occupation Numbers

The single—particle occupation numbers n,,, can be written in terms of the multipole opera-
tors using (3.7) as

(If-tram = ZCfISj—m(_)jhmMI-§Qa (434)
K

72

 

 

. ___._..___J__‘_l __ _.__- .__ _-

(5‘2"‘10 15 20 2530

 

0'5 {015 20 2‘5 30
J J

Figure 4.6: The constant shift in the theoretical equilibrium energies of spin J vs. J for the
N = 6 j = 15/2 ensemble.

where 0 S K S 2 j. The average nm oscillate about their theoretical values. These oscilla-
tions indicate a shell structure that is not captured by the statistical theory. This arises from
the effective deformation present in individual realizations and requires a self consistent
mean field approach. The occupation numbers exhibit an oscillatory behavior in Fig. 4.7,
which may follow from the similarity of the Clebsch-Gordan coefficient (KOIjm, j —— m)
and the Legendre polynomial PK(m/ \/j_(j—+T)). In the Legendre polynomial expansion of
nm for a wave function l], J) the coefficients of PK(.r) are zero for K > 2]. In other words,

the curves in Fig. 4.7 are precisely described by the first 2.1+ 1 Legendre polynomials.

4.7 Moment of Inertia

The effective moment of inertia can be estimated directly from the rotational term of the

equilibrium energy (4.16),

o 3 '3 o
Hm, .—_ AJ", A = ———.— (2L+1)VL(L~ — 2;”). (4.35)
492J4 Lén

The contribution to the moment of inertia by the pair interaction for a pair with spin L

changes its sign at L2 : 2j3, which corresponds to the absence of alignment of the paired

73

 

 

-_ _.' __1 . .._ ..- ._. _ .....__. ..___.‘ o L ._L___._ .L-_4_.-L..- .._.L._.___.'_._._-L_L__. --4

'.~7”—5“—‘3 —1”1 3 5 7 ' I—‘7I— 4—1 1 3 5 7

m m

Figure 4.7: The single—particle occupation numbers nm averaged over the lowest energy
spin J wave functions of the 1417 out of 2000 (70%) spectra with JO 2 0 (solid line),
compared to the simplest theoretical expression (4.10) (dashed line)

particles, as their spins are perpendicular to each other. At low L pairs are antialigned,
L2 < 2j2. If these pair states are attractive, V1, < 0, the contribution to the moment inertia is
positive, preferring a normal rotational spectrum with angular momentum increasing with
excitation energy. Contrary to that, at high L, the pairs are aligned, L2 > 2j3, so that the
attraction in such pairs leads to the negative contribution to the moment of inertia which
favors the bands with an inverted spin sequence. The moment of inertia was taken from the
coefficient of J2 in (4.35) via

I=—-. .
2A” (436)

and was nearly independent of N, in agreement with the statistical theory, Eq. (4.35).
Again this implies non-collectivity. What I does depend on is geometry, in the guise of j.
The calculated moment of inertia I is in excellent agreement with the ensemble results for

N = 4 and 6 and all values of j, see Fig. 4.8.

74

 

2000 .5 --.-—-+~ +w— : - 1 .

i o——* Data
1500 l o --------- 0 N24 Theory
o - - * N=6 Theory

 

 

1000 L
(U 1
rs fr
L l
g 500 1—
:
3 0 3r
c
<D
E —500 .5
0
E
-1000
—1500 i

_2000 L..- _-._._.__._.__-.; _a_*__._- _ -*___.__,_._._ _ . . ._ _...._ _u.__#_ .- ._ .._._

4.5 6.5 8.5 10.5 12.5 14.5
I

Figure 4.8: The moment of inertia for the N z 4 and 6 particle systems of a given j plotted
as a function of j. The positive values are for the JO 2 0 systems and the negative values
correspond to the JO 2 Jm, systems.

75

Chapter 5

Conclusions

The main purpose of this dissertation is to shed light on the origin of the regularities in
nuclear spectra, and the application of quantum chaos theory to nuclear spectra. In principle
the results are also applicable to atomic systems, quantum dots, and other mesoscopic

systems.

After some background on the theory of quantum chaos, a review of current research
on this and related problems was presented. An analysis of neutron resonance data in the
framework of Random Matrix Theory was described, where the problem of finding the
relative fractions of resonances with a given spin was tackled. The results were mixed. Of
the four sets of data analyzed, only the 235U data yielded positive results. The level spacing
distribution, the A3(L) statistic, and the distribution of reduced widths all gave realistic
values for this fraction. The problem is assumed to lie in the technical difficulties of taking

high quality uncontaminated data.

Next the main problem of the dissertation was described and formulated. The regu-
larities in nuclear spectra, especially the spin—zero ground states of even—even nuclei, are
usually ascribed to the effect of the nuclear pairing interaction. While no-one doubts this, it
is seen to be an incomplete picture in general. Regularitics, similar to those in nuclear spec-

tra, are seen in the spectra of random two-body Hamiltonians. The largely ignored role of

76

geometric chaoticity is shown to be the driving force behind this. Some simple illustrative
examples of this mechanism are discussed, where the probability of the ground state spin
for a simple system with a random spin-spin interaction is separated into a product of terms,
one containing the random interaction parameter, and the other the geometric information,
particle spin and particle number. This separation of the different aspects of the interaction
is ultimately achieved for a more complicated system described in Chapter 3. A system of
N spin— j fermions under the influence of a random two-body angular momentum conserv-
ing interaction was studied. The spectra exhibited many interesting features, the strongest
of these was a propensity for the ground states to be magnetically aligned (maximum spin)
or anti-aligned (spin-zero). The role of pairing, the V0 parameter, was unambiguously
eliminated as the culprit. A statistical theory based on equilibrium statistical mechanics
was developed, and an expression for the equilibrium energy derived. The important point
is that the equilibrium energy for the angular momentum-J state was expressed as a sum
of products of two terms, one containing all the random interaction information, the other
containing geometric information. The theory was used to successfully describe the salient

features of the ensembles studied.

The conclusion of it all is that the previously ignored role of quantum chaos in shap-
ing nuclear spectra dominates in systems of randomly interacting spectra. At the very
least, quantum chaos cannot be dismissed in studies of the properties of nuclear spectra.
Although this work was motivated by issues in nuclear spectroscopy, the theory was devel-
oped without specific reference to nuclear structure, only statistical mechanics. The results
concern any small Fermi systems and can be applied to atomic systems, quantum dots,
and other mesoscopic systems of condensed matter physics. For the first time in quantum
many-body chaos correlations between blocks with different quantum numbers were shown
to play an important role. This is a fresh angle on the theory of quantum chaos where the

union between the chaotic structure of the eigenfunctions and the geometry of the finite

77

Fermi—system spawn regularity in the spectrum.

78

Chapter 6

Appendices

6.1 The Darwin-Fowler Method

This section details a derivation of Eq. (4.2) for the single-particle occupation numbers
which is the foundation of the statistical theory of the text. The method is very elegant, and

can be applied to get the density of levels labelled with any additive quantum number.

The goal is to get an expression for the number of states with angular momentum pro-
jection M, in a system of A fermions, each with spin J. Of course one can immediately
write

p(A,M) = 2801 — n)8(M —M.-(n)) (6.1)

where M,(ni) is the i’h possible value of M for an N—particle system. This expression is

ex act, but not at all useful as it stands. Let’s take the Laplace transform of (6.1):
L [p(A,M)] : X] (1.4] dMexp(—M7t—Ap) 5(A —n) 5(M— Mi(n))
m- o 0

= Z e~M,-(n)X—Ap

n,i

: ”(1+e—rnl—41)

= Z(7t,p) (6.2)

79

where m takes all possible values of the single—particle projection.

Now this is something one can handle, because taking the inverse Laplace transform
gets us back to p(A,M) but without the nasty B-function. The essential point is that the

Fourier representation of the 8—functions has been used.

P(AM) = L"‘tZ»(x.u)l
1 2 [co ioo
: __ dk/ d Ml-Hip 1 —-m?t~p
(2m) ./..-.. ...-... ”e In,“ +8 )
1 2 ice ioo 1 mk-p
___ __ x/ MX+Ap+£mlnH+e .)
1 2 ice iw F041)
= R / all/_.. due A (6.3)
where
F(7t,,i) z. Mr+Au+Zinu +e‘ WP). (6.4)
m

The saddle-point approximation can be used to evaluate this integral:

F()\,[J) : F(A01#0

v

+ (A — XO)FA(A‘03“0)
+(fl-H0)Fu(7\0..#ol+ (k—kofmwmo)

+(7x — 10W»! —uo)Fr,,(7\o,uo) +

NIHNI"

(#1 _”0)2Fm1(7\01“0) (65)

where the notation Flam/~10) = EQXF (7t, ,u)|;‘O 410 etc has been employed. The point (3.0410)
is the saddle-point, ie. F),(7ko,1uo) = Fp()\.(),fl()) = 0.

80

Straightforward differentiation gives the following set of equations:

m
map) = M—XW

: A—ZZ—f—fi +e1m1t;y)

I712 emlafiu

1+ emkep)2

 

emk+p

 

,. <1 + emw)?
(6.6)

From the definition of the saddle—point we have

M—2 .711.

 

1
A:Z(]+ern)\rorflo).

m

(6.7)

 

which gives (4.2).

The following definitions clean up the notation a little;

f0 2 F(A‘0nu0)
a = ‘2 m2 emit"€710 ,,
2 m (1+emM+Htt)~

 

:1 eM+H0
22< ‘1.+em}\.0 H2110)

m

 

m "tank; .110
Z 6;“ 1+ emM filo)?-

 

(6.8)

Now we are left with

2 'oo 'oo

= _1 ' ' f0+110"-lolz-‘r-btpwof'+'C(?v--lo)(rI-~rlo)

P(AaM) 27t' d?» (111 e ,
l ——-ioo — [00

81

Completing the square in the exponent and remembering that (6.7) give us ko(A,M) and

u0(A,M) we have the final result:

A M WM) 6 9
m ’ )_ 2n \/4a(A,M)b(A,M) -—c(A,M)3' ( ° )

 

 

6.2 Sums Involving Clebsch-Gordan Coefficients

7
In order to evaluate sums like {A 1.2 |C{t,’\2| m’l or 211,1,2 [Cfé‘l mfm‘z’ we’ll need a few

tricks based on the vector model and the following relationships for the Clebsch Gordan

coefficients

C13?” . = (—1)1'2”"’2 2’3 + 1C1." 1"“ . = (—1)j"""‘ 2———j3 + 101% .
le’"l:12:m2 2jl +1 JZ:"’"2113:’"3 2j2 .+_] J39’n3rjla—‘ml,

(6.10)

 

which means that

2
"If

 

)2 [ca

A,M1 rmz

 

2L+1 Z,

_._ Jml
—2j+1 Cj:m-._»,.,21A| ml

A,ml,m2

2L 1
=;2j:1<1,mrlmp(§|1,— I712) j,—mgI) |jml)

 

2L+1 ,,

—— .. ml (6.11)
”11214—1

 

For more complicated sums we need the vector model in which, for the coupling of two

momenta j1 +j2 = L, we have

(LAljlijgleA> = (aLiLk+b8,-k +iC€ijkLk >, (6.12)

82

where a, b and c are coefficients to be found. Evaluating the expectation value of j12

and (jl oL)3 in the state |LA) we get

’3 '7 . “7 L2 .2_.% 2 o
j1“=aL"+3b, (Jl-L)”=(-—+%l———J—”) :aL4+(b+c)L“, (6.13)

where L2 = L(L+ 1), and j:2 = j(j+1). Coefficient c can be found by evaluating

commutator [jg jj] and using a usual vector model for (LAM-ILA), which gives

(6.14)

In our casejl2 =j22 =j2 and from Eqs. (6.13)

_3L3—4j3—3
_ 8L3—6

 

4sz2 — 2j2 —L4 +L2
8L3 — 6 ’

 

b:

L2 + 3j2

C: m (6°15)

where L4 = L2(L+1)2.

It is convenient to collect the following elements together:

83

1 .
(”I“) : _._ 2.171" = '3_j(j+1)7

<m4> : $111+ 1113f +3j— 1) = §<mB><9<nF> — 1),

 

 

 

 

 

 

 

 

 

2 1C??? mi = -,-<2L+ 1)j(j+1),
A,1.2 ‘
(1A 1 1 . .
ZCIZA 2III1II13=I\Z:C —A2—m'1) =16L(L+])(2L+])—§(2L+1)j(j+1),
A,,21A,,21
" 1 .2 .
2C , m,=— —(2L+1)j (j+1)(3]"+3j—1),
A,,21 ~ 15
C1? - m‘}A2 2 (IA4 + bA2 ,
1,2 '
1 .3 . 1 4 3

C112A 2mlm3=fi(2L+1)j(j+l)(3j‘+3j—1)+2;(—?:—2(1)A —2bA‘,

Al,2 A

 

 

2 2 3
C1?! (mlmz +m1mg): —B(2L+1)j(j+1)(3j‘+3j— 1)

 

>

,1.

+Z(3a— éw +3bA2, (6.16)
A

Some of these identities follow directly from A:2 2 (ml +1713)2.

6.3 An expression for (H2)

For the sake of completion the result of the statistical theory for the equilibrium value of

(H 2) is stated here without derivation.

84

(H2) is given by the statistical theory as:

<H2>

N 4 . 4 _
ZVLVL’ — 1(2L+1)(2L’+1)(1—- —7—) + (2L+ 1)511.’
LL, 0) 21+]

9J2 1 7 1 ,2 2.7 (2L+1)(2L’+1)
———— —L-+—L ——~ 1—8
N2j4((6 6 3J N 2j+1

 

— (2L+ ”51.11)

4.2 I J j L 1 2 1o2
31 +4(2L+1)(2L+1)§(2L+1){ L, j L }(—L —-J)

h—I
h—A
J

4<2L+1138,.UZ(2L+1){§ j :}< Lia->11 (6.17)

L

85

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86

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