A DEFORMED NUCLEAR SHELL MODEL WITH APPLICATION TO NEON, MAGNESIUM,

AND SILICON ISOTOPES

George Craig • 1970

Dissertation Presented to the Faculty of the Graduate School of Yale University

in Candidacy for the Degree of Doctor of Philosophy

Abstract

A program of deformed shell model (DSM) calculations for spheroidal nuclei with several active valence nucleons is developed. The DSM is formulated in terms of the strong- coupling limit of the unified model, as was suggested originally by A. Bohr and B.R. Mottelson. The DSM is then applied to several nuclei in the first half of the s-d shell.

Many nuclei in this region of the periodic table are thought to possess spheroidal equilibrium shapes and, as a consequence, are frequently described in terms of simple strong-coupling models. In general, these models fail to account for all of the low-lying levels observed in these nuclei. Thus, it is not known whether these extra states are rotational in design or are due to other modes of exci­tation important in s-d shell nuclei.

Hartree-Fock calculations of deformed orbitals in this mass region strongly suggest that some s-d shell nuclei may have several easily excited valence nucleons outside a stable permanently deformed core. This leads one to expect, within the framework of the DSM, that some of the extra degrees of freedom evident in s-d shell nuclei may represent rotational levels built on different intrinsic excitations of several valence nucleons.

Detailed DSM calculations are made for Ne^>22,23 an(j Mg25j26,27 as 1, 2, and 3 active valence neutrons outside a Ne^0 and Mg^4 core respectively. A simple pairing residual interaction is employed between the valence nucleons.

Spin and parity predictions are made for these nuclei.The importance of the Coriolis perturbation on the energy level spacings and spectroscopic factors is demonstrated.In particular, we note it shifts the J1T=6+ state in Ne^2 by 6 MeV, bringing agreement with experiment. In other instances, very small admixtures in the wave functions can enhance the spectroscopic factor by a factor of two.

The silicon isotopes are also examined in the DSM for­malism. The DSM describes the neon isotopes very well, the magnesium isotopes reasonably well, and the silicon isotopes not at all. This trend is interpreted as evidence for im­portant vibrational correlations near the middle of the s-d shell.

i i

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Acknowledgement

I wish to thank Professor D. Allan Bromley for his critical reading of the first draft of my thesis. I shall always remember his exuberant exhortations to forge on.

It is a pleasure to acknowledge several valuable discussions with Dr. A.J. Howard about the experimental realities vis-a-^vis the neon isotopes. I am grateful to Dr. Robert Ascuitto for further theoretical suggestions regarding this work and for his long trip to offer them.This problem was originally suggested to me by Dr. I. Kelson.

I am -grateful to my wife, Nahide, for helping me see this thesis through to its completion. I also wish to thank Mrs. Brenda Preston for her efforts in typing this thesis.

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Table o f Contents

PageAbstract».................................................. 11Acknowledgements.......................................... ivList of Illustrations and Graphs........................ xList of Tables........................................... xii

Chapter I - IntroductionA. Nuclear Many-Body Realizations................. 1B.. Nuclear Models................................... 2C. The Unified Model............................... 2D. Motivation for this Work........................ 3E. Scope of the Present Investigations............ 4

Chapter II - .The Nuclear Shell ModelA. Shell Models..................................... 8B. The Nuclear Many-Body Problem.................. 9C. Quintessence of the Nuclear Shell Model

1. The Approximation.............................“ .132. Evaluation of Hab............................ 133. Discussion of Equations (2.9)............... 164. Shell Model Interpretation of Hab ............ 18

D. Origin and Implementation of Shell Model Phenomenology1. General Remarks.............................. 202. Phenomenological Pitfalls......... 213. Internucleon Interaction..................... 22

v

4. Theory of Effective Interactions......5. Shell Model Parameterizations.......... 30

E. Core Excitations............................. 36

Chapter III - The Unified ModelA. Core Excitations and the Nuclear Shell

Model......................................... 39B. Bohr's Liquid Drop Characterization of

Core Excitations1. <X2y Quantization........................ 422. $2y Quantization........................ ^3. Relationship Between Both

Quantization Schemes.................... 50*}. Rotational Nuclei....................... 5^

C. The Average Potential Generatedby the Core.................................. 57

D. Core Excitations and the EffectiveInternucleon Interaction.................... 58

E. A Unified Model Hamiltonian1. Interplay Between Core and Valence

Nucleon Degrees of Freedom............. 592. Weak-Coupling Limit......... 603. Strong-Coupling Limit

(A Deformed Shell Model)............... 6l

Chapter IV - Froward Unified Model NucleiA. The Nuclear s-d Shell....................... 6^B. Neon 20....................................... 6*1C. Neon 22

1. Disparity With Neon 20................. 66

v i

C. Candidate Nuclei for Several ActiveValence Nucleons............................ 70

Chapter V - The Deformed Shell ModelA. Introduction................................. 72B. Strong-Coupling Basis Functions 'l'(JMKa)

and Ko-Configurations1. Strong-Coupling Formalism............... 7*12. Generalization to Several Valence

Nucleons................................. 783. Discussion of Eq. (5.10)................ 80

C. Reduction of the Strong-Coupling Hamiltonian With Respect to the K Quantum Number1. General Considerations.................. 822. RPC Matrix Elements..................... 833. H Matrix Elements....................... 864. Discussion of Eq. (5.2*0............... 885. PPC Matrix Elements..................... 89

D. Reduction of Structure Factors1. General Considerations.................. "912. (t,p) and (d,p) L-Transfers and a

Strong Coupling Selection Rule......... 9*13. Spectroscopic Factor for (d,p)

Stripping................................ 95*1. Discussion of Eq. (5.*10)................ 1035. Working Assumption...................... 10*1

E. Summary...............’....................... 10*1

vil

2 . J^=2+ In tr u d in g L e v e l ............................................. 67

A. Introduction.................................. 106B. Single Particle Energies and Wave

Functions1. General Considerations.................. 1072. The Nilsson Model........................ 1093. The Hartree-Fock Model.................. 1144. Comparison of Both Models............... 1155. Relevant Single Particle Parameters.... 125

C. Moment of Inertia Parameter A ............... 128D. Residual Interaction .................... 132

Chapter VII - Deformed Shell Model Results and PredictionsA. Introduction................................. 137B. Neon Isotopes

1. Preliminary Remarks..................... • 1402. Neon 21.................................. 1413. Neon 22.................................. 1484. Neon 23.-................................. l6l

C. Magnesium Isotopes-1. Preliminary Remarks..................... 1772. Magnesium 25............................. 1783. Magnesium 26.............................. 1864. Magnesium 27............................. 196

D. Silicon Isotopes............................. 207

Chapter VI - D i s c u s s i o n o f Parameters

v i i i

A . Summary .................................B. The Ne2^(t,p) Ne2'1' Reaction................C. Theoretical Refinements for Neon Isotopes..D. Validity of Strong-Coupling Unified Model

in s-d Shell........................... .....E. Denouement...................................

References

AppendicesA.B.C.D.

Chapter V I I I - Summary and C onclusions

Reduction of H , .......................abDerivation of Eq. (2.15)..............The Hartree-Fock Potential of the Core Strong-Coupling Matrix Elements......

L i s t o f I l l u s t r a t i o n s and Graphs

Figure

(2 .1 )

(3.1)

(3*2)

(3.3)

(3.'0

(4.1)

(4.2)

(6 .1 )

(6 .2 )

(7.1)

(7.2)

(7.3)

(7.4)

Title Page

N-Particle Basis Space 11

Rotating-Vibrating Spheroid 45

Excitation Spectrum of a Liquid Drop (I) 51

Excitation Spectrum of a Liquid Drop (II) 52

Hydrodynamic Threshold for 3 and y 55Vibrations (After A. Bohr)

20Positive Parity Excitation Spectra of Ne 6522and Neighboring Ne

Davydov Diagram for an Axially Asymmetric 68Rotator

Nilsson Diagram for the s-d Shell 111

Hartree-Fock Single Particle Energies for 121XT 2 0Ne

21Deformed Shell Model Spectrum for Ne as 1 22Da Ne Rotator Core Plus a Single Valence

Nucleon22Deformed Shell Model Spectrum for Ne as 1*19

ona Ne Rotator Core Plus Two Interacting Valence Nucleons

P PNe Without a Residual Interaction Between 153the Valence Nucleons

Comparison of Spectra from Different Theore- 158 tical Models for Ne22

x

Figure(7 . 5 )

(7 . 6)

(7 . 7)

(7 . 8 )

(7 . 9 )

(7 . 10)

(7 . 11)

(7 .12)

(7 . 13)

( 7 . 1 4 )

Title23Deformed Shell Model Spectrum for Ne

22as a Ne Rotator Core Plus a Single Valence Nucleon

23Deformed Shell Model Spectrum for Ne 20as a Ne Rotator Core Plus Three

Interacting Valence Nucleons

Yrast Lines for Ne2 and Ne22

Yrast Lines for Ne21 and Ne2^25Deformed Shell Model Spectrum for Mg

ohas a Mg Rotator Core Plus a Single Valence Nucleon

2 6.Deformed Shell Model Spectrum for MgOilas a Mg Rotator Core Plus Two Inter­

acting Valence Nucleons

Comparison of Spectra from Different Theoretical Models for Mg

27Deformed Shell Model Spectrum for Mg Pfias a Mg Rotator Core Plus a Single

Valence Nucleon27Deformed Shell Model Spectrum for Mg oilas a Mg Rotator Core Plus Three

Interacting Valence Nucleons

Comparison of Spectra from Different27Theoretical Models for Mg

Page

163

166

175

176

180

188

192

198

201

206

x i

List of Tables

Table(5.1)

(5.2)

(6 .1 )

(6 .2 )

(6.3)

(7.1)

(7.2)

(7.3)

(7.*0

TitlePossible Final State Spins and L-Values for (d,p) and (t,p) Transitions To Ne22 and Ne23

Possible Final State Spins and L-Values for (d,p) and (t,p) Transitions To Mg2 and Mg27

"Nilsson Coefficients" for the In­trinsic States <J>j Ne2<

Hartree-Fock Single Particle Energies 20for Ne

Dominant Experimental

Dominant Experimental

Dominant Experimental

Dominant Experimental

xii

Moment of Inertia Parameter

Ne23-=Ne2^+l DSM Parameters, Configuration Amplitudes, and References

Ne22=Ne20+2 DSM Parameters, Configuration Amplitudes, and References

Ne23=Ne22+l DSM Parameters, Configuration Amplitudes, and References

23 20Ne -Ne +3 DSM Parameters, Configuration Amplitudes, and

Page96

97

118

123

131

142

150

164

167

(7.5)Table

(7.6)

(7.7)

(7.8)

(7.9)

Mg2^=Mg2^+l DSM Parameters, Dominant 182Configuration Amplitudes, and ExperimentalReferences

Mg2^=Mg2^+2 DSM Parameters, Dominant 189Configuration Amplitudes, and ExperimentalReferences

Mg27=Mg2<3+l DSM Parameters, Dominant 199Configuration Amplitudes, and ExperimentalReferences

27 ?4Mg -Mg +3 DSM Parameters, Dominant 202Configuration Amplitudes, and ExperimentalReferences

T i t l e Page

"Nilsson Coefficients" c^j for the 210Hartree-Fock Intrinsic StatesXfiv 'Ecv j * j n - f o r Ne2° ’ Mg2i<’ and Si28 (Ri 68)

x i i i

1

A. Nuclear Many-Body RealizationsAs experimental techniques for probing and partitioning

the nucleus become more refined and as our theoretical understanding becomes more comprehensive it has become clear that nuclei throughout the periodic table exhibit a rich structure arising from the interplay of particle and collec­tive degrees of freedom. For example, in light nuclei there appear collective 4 particle-4 hole excitations as well as quasi-molecular excitations (Br 60, Br 64a,b). And, at the other end of the periodic table, the theory of fission, the prediction of super-heavy elements and conjectures about shape isomerisms have drawn on both independent particle and collective features (Br 69b, Me 66, Gr 69).

Each of these ideas or discoveries represents an imagina­tive penetration into the behavior of the nuclear many-body system since at present there is no single theory encompassing all nuclear phenomena. Presumably such properties are con­tained in the many-body Schroedinger equation, the Pauli Principle, and additional assumptions about the nuclear inter­action. Calculations of nuclear properties from these first principles are in progress and some successes may be near at hand (Ba 69). But unfortunately these results seem to emerge slowly, piecemeal, and only through tremendous effort.

Chapter I

I n t r o d u c t io n

B. Nuclear ModelsIn the Interim, and recognizing that the solutions of

the actual nuclear Hamiltonian equation are accessible to us as the nuclear quantum states even though the equation Itself may still elude us, major progress has been possible through the introduction of the familiar and often apparently contradictory nuclear models. Each of these is in effect a caricature of the actual situation designed to emphasize those aspects of it which dominate the phenomena under dis­cussion to the exclusion of all or many of the other aspects. To the extent that the model can reproduce all available in­formation on these phenomena— and. is amenable to mathematical extrapolation and prediction— it is a most useful adjunct to nuclear study.

Inasmuch as all these models are from their very nature interrelated aspects of a general understanding of the nuclear many-body problem the elucidation of these interrelationships and the range of validity of the various approaches provides an important and powerful channel for study of the many-body problem.

C. The Unified ModelWe wish to generalize herein one model which has been

remarkably successful in describing collective and single particle phenomena. Depending upon which features of the model are emphasized, it is called the Liquid Drop Model, the Bohr-Mottelson-Nilsson Model, the Coriolis Model, and

2

3

also the Unified Model. In the spirit that this model attempts to vied macroscopic and microscopic, we shall refer to it as the unified model.

The unified model is a shell model. By this we imply that it attempts to describe the low-lying levels of a nucleus in terms of a few valence nucleons moving in the potential generated by a core of nucleons. It differs from the conventional shell model in that the core is not inert. The core appears explicitly as a quantized liquid drop of nuclear matter with simple modes of excitation. In the uni­fied model, we study these modes and their coupling to the valence spectrum.

D. Motivation for This WorkOne region of the periodic table rich in collective

phenomena yet still amenable to individual particle calcu­lations of manageable dimensions is the first half of the s-d shell. Many even-even nuclei in this region clearly exhibit a J7T= 0+ ,2+ ,iJ+!> . . ground state spin sequence reaching through their spectrum. The members of the sequence arelinked by enhanced E2 transitions and approach a J(J+1)

20 2kenergy rule. The most striking examples are Ne , Mg , and2 8Si . As a consequence, the unified model has been applied

enthusiastically to these nuclei and their neighbors. Typically, in this region of the periodic table, the model is that of a simple rotator for even-even nuclei or a rotator

core with a single valence nucleon for odd-even nuclei.And, generally, in this form, the model fails to accountfor all the low-lying levels observed in a given nucleus.

20 21In point of fact, the simple rotator model describes Ne ’ 'very well but provides less than an adequate representation

22 23for the neighboring nuclei Ne * . Therefore it is notknown whether the extra states in these latter nuclei are of a rotational design or are other modes of excitation im­portant for s-d shell nuclei.

E. Scope of the Present InvestigationsIn an effort to examine this" question and augment the

experimental investigation of the "extra" states, we extend the present development of the unified model in order to treat quantitatively, and microscopically, the problem of several valence nucleons coupled to a rotator core. The Hamiltonian for the system is given by

H = A$2 + H + V .. (1.1)sp ri

Here, ft is the angular momentum of the core and A its moment of inertia. The single particle Hamiltonian Hsp describes the motion of the valence nucleons in the average field generated by the core nucleons. And, V is a "residual" interaction acting between the valence nucleons.

4’

Surprisingly few authors have considered this problem.The form or even the importance of V for a strongly deformed nucleus is an open question because unified model calculations are invariably limited to zero or one valence nucleon. For this reason and also because our treatment of the remaining parameters is different than the traditional approach based on the Nilsson Model for H , we begin this research in Chapter II with a review of shell model phenomenology for the single particle Hamiltonian and residual interaction.

Our objective in Chapter II is twofold. First, we wish to establish the relevant parameters for our unified model calculation. From considerations' presented, we later conclude that the Nilsson parameterization obscures non-local anti- symmetrization effects in H which are important In s-d shell nuclei. Second, and equally important, we ultimately intend to compare whenever possible applications of the nuclear shell model and the unified model to s-d shell nuclei. The s-d shell is presently developing into a testing ground for the importance of core excitations.

In Chapter III, we discuss the need for and the limita­tions of the core Hamiltonian adopted in Eq. (1.1). In the Bohr model, this form of the core Hamiltonian actually corres­ponds to the strong-coupling limit for a valence particle polarizing a liquid drop core. The restrictions placed by the strong-coupling assumption on choosing a good core are evaluated.

5

6

Just recently, a significant step towards deriving the strong-coupled Hamiltonian from a microscopic point of view has been made thereby justifying our connection of the basic features brought out in Chapters II and III (Vi 70).

The strong-coupling assumption implies that the coreof a strongly deformed even-even nucleus can be treated asa quantum mechanical rotator. In Chapter IV we discuss the

22likelihood that the "extra" states observed in Ne imply anaxially asymmetric oore for this nucleus or that they mayrepresent 8 or y vibrations as others have conjectured. We

22conclude that the extra degrees of freedom evident in Ne are more reasonably due to two active valence nucleons out-

4 .3 M 20side a Ne core.Having carefully examined the meaning of each term

leading up to Eq. (1.1) and encouraged by our preliminary 22analysis of Ne , we proceed to develop in Chapter V the

formalism necessary to perform some interesting calculations involving several valence nucleons coupled to a rotator core. In addition to excitation spectra, this includes spectroscopic factors because they provide a more stringent test of the model wave functions.

In Chapter VI, we compare the Hartree-Fock and Nilsson input parameters for this formalism in the light of the phenomenology discussed In Chapters II and III. We conclude that the deformation should not be an available parameter in the unified model description of s-d shell nuclei if this

Using the Hartree-Fock input parameters, we interpretin Chapter VII Ne2"*">22,2 and Mg2'*’2^ ’2' as 1, 2, and 3

20 24active neutron systems with Ne and Mg cores respectively. These results are compared with the available experimental information about these nuclei. Whenever possible we also attempt an intercomparison with other theoretical interpre­tations of these nuclei. Beyond this, detailed predictions about the multi-particle composition of many levels are made and additional experiments are proposed for testing these conclusions .

It Is clear from the results of Chapter VII that the strong-coupling unified model with several valence nucleons works best for the neon isotopes, fair for the magnesium isotopes, and fails for the silicon isotopes. We discuss the significance of this observation in the final chapter, Chapter VIII.

7

description Is to be based on first principles.

8

A. Shell ModelsThe unified model is a shell model. As such, we wish

to derive as well as possible its basic form from the first principles of nuclear structure. Our objective in this and subsequent Chapters is to develop a deformed shell model interpretation of the strong-coupling limit of the unified model (cf. Eq. (1.1)). These considerations will facilitate comparing the unified model with other shell models by establishing the basic features they do or do not have in common.'

Specifically, in this Chapter, we motivate the Bloch- Horowitz statement of the nuclear shell model and discuss its phenomenological implementation. In doing so, we develop the shell model concept of the nucleus as a "core" with active "valence" nucleons. We show that in the shell model approximation, (1) the valence nucleoris move in the Hartree-Fock potential generated by the core nucleons and (2) the valence nucleons interact via an effective many- body interaction instead of the two-body internucleon po­tential gleaned from nucleon-nucleon scattering data. Further­more, we show that in diagonalizing the effective inter­action, one needs to use basis functions antisymmetrized only in the coordinates of the valence nucleons.

Chapter I I

The Nuclear S h e l l Model

In the conventional (spherical) shell model, the core is considered to be inert. Accordingly, the shell model approximation implies that the wave function for the N-body nucleus is a product of a constant core func­tion and a Slater determinant for the valence nucleons.In Section (II. E) and in Chapter III, we consider the necessity for introducing collective excitations of the "core" nucleons. In the unified model, the core wave function is not constant but describes, phenomenologically, a quantized drop of nuclear matter with simple modes of excitation. One cannot fail to appreciate the outstanding successes of this model in view of the bold assumptions leading to its formulation.

B. The Nuclear Many-Body ProblemWe wish to solve the N-body Schroedinger equation

H'F = E¥ (2.1a)

N . NH = E T, + j I V. . (2.1b)

i=l 1 i , j =1 1Ji / j

with Vi^=V( |r»i—r*j | ) and T1=p2/2mi . For our purposes we neglect the neutron-proton mass difference. Being a system of fermions, the eigenstates ¥(1,2,...N) are antisymmetric under the exchange of the coordinates of any two particles.

9

10

Equations (2.1) have resisted an exact solution for thirty years. So, one Is led quickly through various approximations to a model Hamiltonian which is solved to some degree or conviction of exactitude. These approxi­mations reduce the number of degrees of freedom to be solved for from 3N to a few by partitioning the system into core and a few valence nucleons.

In the simplest situation, the core is considered to be inert. Its presence produces an average central poten­tial in which the valence particles move. In this poten­tial, the core nucleons occupy the lowest orbitals. The low energy properties of the N-body system are characterized as excitations of interacting valence nucleons.

In general, one expands the eigenstate T (1 ,2 ,3...N) in a complete basis of NxN Slater determinants which are made up of single particle orbitals {<f>}. Calling the lowest orbitals below some cutoff the core orbitals {c^} and the remainder above the cutoff valence orbitals {v^}, see Fig. (2.1a), the N-particle basis is seen to consist of two kinds of states: Those in which the core is excited and those inwhich it is not. A core excited state is obtained by pro­moting one or more nucleons from the core orbitals to the valence orbitals. In particle-hole terminology, this great variety of states is classified by the number of nucleons in valence orbitals and the number of nucleons removed from the core. The former are called "particles” and the latter

11

_S£_

M

{Ci}

-Je

| 2 p - O h > 1 3 p - I h >

(a) ( b ) (c)

Fig. (2.1) N-Particle Basis Space. (a) ($,} is a complete set of single particle orbitals. The orbitals above and below some cutoff are called valence orbitals {v-j_} and core orbitals (c^} respectively. The N-particle basis space consists of all possi­ble N-fold antisymmetrized products of the generated by distributing the N nucleons in tne

in accordance with the Pauli Principle. The resulting set of N-particle basis functions can be partially classified according to the number of nucleons occupying valence orbitals ("particles") and the number of nucleons removed from core orbitals ("holes"). (b) and (c) are examples of 2-particle 0-hole and 3-particle 1-hole states.

"holes", see Figs. 2b and 2c.Nothing has yet been stated concerning the number of

nucleons to be included In the core. In fact the number of orbitals <{> which should be included in {c^} is im­material in principle and important in practice. Suppose for example that we split the N-particles system into c core particles and m valence particles. Then, were we able to diagonalize the Hamiltonian in the basis of N-particle states, the eigenstates of the system would be written

¥ = ZAa |mp-0h>a + EB^ | (m+l)p-lh>b + ... + ERr | Np-ch>i<.

(2 .2 )

The individual sums'above indicate that there are an infinitenumber of states of a given p-h classification. Clearly,how we choose the core affects the rate of convergence of theexpansion in p-h configurations. Part of the game of the

18core is deciding, for example, whether 0 is most easily1

characterized as 2 particles outside an 0 core or as 220holes inside a Ne core. Either description, carried

through, should yield the same results. But It goes without saying that in many situations one approach is usually more to the point than the other. (Interestingly, in this example both of these descriptions are competitive (Fe 57))-

12

13

C. Quintessence of the Nuclear Shell Model1. The ApproximationWe now make precise our previous description of the N-

body system as valence nucleons moving in an average poten­tial created by the core nucleons. We return to the simplest case and assume that the core is inert and closed. The wave function Y in the basis of valence configurations is then taken to be

This expansion neglects the contribution of core excited states to the iow energy properties. Of course the accuracy of this approximation is inversely proportional to the size of the core for if m+N we have the complete N-body problem.

2. Evaluation of H .abOur objective is to examine the Hamiltonian In the space

of |mp-0h> states for the valence nucleons:'

Since the |mp-0h> state is an antisymmetric function of all N nucleons, the reduction of H ^ Is most easily accomplished using the methods of second quantization. In this formula­tion, the many-body problem is written

Y = ZAa |mp-0h>a . ( 2 . 3 )

( 2 . 4 )

14

H4' = EY (2.5a)

H " ijTiJaiaJ + 2 ^ k£Vijk£aj+aiakar (2.5b)

The creation and annihilation operators satisfy the anti­commutation relations

4 . 4 . 4 .

{ a . , a , } = a ,a , + a .a , = 6 . , (2.6a)i j i j J i ij

(a^,aj) = (a^,aj) = 0 • (2 .6b)

and matrix elements of the kinetic energy and two body inter­action are given by

T1j = /<J>*(r) |^j(r)d3r (2.7a)

and

Vijk£ = //4>J(r1)(j)*(r2)V( |r1-r2 | )<J>k (r1)<j)Jl(r2)d^r;Ld^r2 . (2 .7b)i \^3„ ^3,

And finally, the various components of the particle hole expansion of Y , Eq. (2.2), are given by

15

|mp-m'h> n a.+ |0 > (2 .8)1=1 1

where the N a^'s create m nucleons in valence orbitals {v^} and (N-m) nucleons in core orbitals Thedifference between the number of core orbitals and the number of nucleons in them is the number of holes m r in the core. It follows immediately from the commutation relations that the particle-hole basis |mp-m,h> is anti­symmetric in all N nucleons.

The nature of the single particle states, a^|0>=<}>^ remains unspecified at this time. They could be plane wave, harmonic oscillator, Hartree-Fock, or other complete set of states depending on the choice of the unperturbed or zeroth order Hamiltonian 1-1 . Actually, as we shall see,Hq is not completely at our disposal. In a properly anti­symmetrized shell model, Hq is necessarily non-local. We consider later the consequence of this fact.

It is now an elementary exercise using this formalism to evaluate the matrix element Hab in the N-particle basis of |mp-0h> valence configurations as was originally proposed. Using the index y when summing over core orbitals {c^} andthe indices y, v, p, a when summing over valence orbitals{v^}, we have from Appendix A,

Hab = a<mP-°h lH lmp-0h>b = EcSab + a<mp-0h | Hy | mp-0h>b

i*

(2.9a)

16

where

(2 .9b)

(2 .9c)

and

( 2 . 9 d )

We have attached an index c to the one body U potential to stress that it is defined in terms of the core orbitals.

3. Discussion of Equations (2.9)Equations (2.9) reveal the basic structure of the matrix

element H and provide the theoretical foundation for the nuclear shell model. We can discover their physical content more easily in the coordinate representation.

Suppose we rewrite the N-body Hamiltonian in coordinate space in the following way:

The subscripts c and v refer to coordinates of core and valence nucleons, respectively. Although we cannot identify particular nucleons as core nucleons and others as valence nucleons because antisymmetrization insures that all nucleons

in ¥(1,2,...N) are indistinguishable, we, nevertheless, neglect for the moment the antisymmetrization between nucleons in core and valence orbitals and write

¥(1,2,...N) = ¥(1 2 ...)¥ (1 ,2 ....) = |(mp)(0h)> .Lt CL V V Cl

(2 .11 )

The parentheses in the bra |(mp)(0h)> indicate antisymmetri­zation of the m valence nucleons among themselves and the (N-m) core nucleons among themselves. Clearly then the matrix element H&b in this approximation,

a<(mp)(Oh)|H|(mp)(0h)>b = ^c6ab + &<(mp)(Oh)|HV |(mp)(0h)>b,

(2 .1 2 )

will miss an exchange contribution from the core and valence nucleons which is contained in the exact calculation of Hab, Eqs. (2.9). Despite this, it is evident from Eqs. (2.10) through (2.12) that we can construct a new Hamiltonian

H = I(T + Uc ) + k v , (2.13)v v v 2 vv'

which acts only on the m valence nucleons. The presence ofthe remaining (N-m) core nucleons adds a constant coreenergy E and generates an average potential c

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18

Uu (ry ) = <Y(lc2c ...)|£V(r,r)|T(l 2 ,...)> (2.14)c

which binds the valence nucleons to the core. This one body potential for the valence nucleons is written as

z uJLaV, Uc = ZV (2.15)pv y v yv yyvy

in second quantized notation (see Appendix B). Comparing this with Eq. (2.9d) show that Uc is the direct portion of Uc .The remaining portion of Uc is the anticipated exchange term which arises in the correct treatment of antisymmetrization.

4. Shell Model Interpretation of Hab From the sum of the proceeding considerations we can

now draw several conclusions as to the meaning of the matrix element H ^ given by Eqs. (2.9). First, in the |mp-0h> valence space, the many-body Hamiltonian can be written as the sum of a constant Hamiltonian H equal to the energy of the (N-m) core nucleons and a Hamiltonian Hv which operates only on the m valence nucleons. Second, H and H are

L V

coupled to one another via the core potential U°:

H = H + H (Uc). (2.16)C V

Third, the second quantized expression of Hy given by Eq. (2 .9c) has the coordinate representation

19

H = vmE

V = 1(T UC)v

+ I2m2

v,v' = v/v'

Vvv (2.17)

And lastly, the net effect of antisymmetrizing the N particles in an |mp-0h> space is to make the average potential experienced by the m valence nucleons a non-local operator given by Eq.(2 ,9d).

Equation (2.17) encourages quite naturally the notion of a nuclear shell model wherein a many-body system of strongly interacting nucleons is characterized by a few active nucleons In the average potential produced by the core nucleons. Of course, valence nucleons are fictitious particles inasmuch as all the nucleons are indistinguishable by virtue of the anti- symmetrization.

The physical meaning in this picture is developed even further by an appropriate choice of basis functions From Appendix C it is clear that the average potential Uc given by Eq. (2.9d) is the non-local Hartree-Fock potential for (N-m) nucleons in their ground state configuration. This suggests that a natural basis for the shell model is that defined by the single particle Hamiltonian H which is the Hartree-Fock Hamiltonian for the core nucleons (cf. Eq. (2.25)):

Ho<i>i = (T + Uc(<Di ) )^ = e1<}.i . (2.18)

Then the Hamiltonian Hv describing the valence nucleons assumes

20

m 1

a <mp-0h |HV lmp-0h>b = z b ^ a b + a <mp_0h 12'CVvv' lmp_0h>b

which says that the valence nucleons are distributed in the unoccupied Hartree-Fock orbits of the core.

There is no explicit reference to the core in this expression. As_ a consequence we need to diagonallze the two-body Interaction with states explicitly antisymmetrized in the m valence nucleons only, ■

m +jmp-0h> = II a. |c> = |mp>, (2 .20)i=l 1

thereby effectively reducing the number of degrees of freedom to be solved for from 3N to 3m.

D. Origin and Implementation of Shell Model Phenomenology1. General RemarksAlthough E q. (2.17) with its shell model interpretation

is a very appealing formulation of the N-body problem, attempts at solving it rigorously depart abruptly from this simplicity. Since non-locality is an inherent feature of the average potential there is always the formidable problem of determining in a self-consistent fashion the single

the particularly simple form

(2.19)

particle energies and wave functions of Hq . Thereare also substantial problems involving the nature of the two-body interaction V and the effects of truncating the infinite |mp-m'h> space.

For these reasons, several phenomenological shell models have been proposed in order to illuminate the shell structure of Eq. (2.17). These models generally choose a simple local single particle Hamiltonian Hq containing a harmonic oscillator or Woods-Saxon potential to define the basis The single particle energies {e^} appear to bemore sensitive to non-local effects and therefore are taken from experiment. Finally, the influence of distant states and uncertainties in V are absorbed into an effective inter­action V for the valence nucleons.

2. Phenomenological PitfallsThere is now no doubt that a wide variety of nuclear

phenomena can be explained in terms of interacting valence nucleons. Nevertheless, we recognize that apart from these phenomenological confirmations, the theoretical foundation of the nuclear shell model is still incomplete. Consequently, it is not always evident what precisely the phenomenological successes are confirming especially since this approach is not without pitfalls, the pseudonium calculations being startling examples of this danger (Co 66). Before the vali­dity or limitations of the original non-relativistic

21

22

Hamiltonian formulation of the many-body problem, Eqs.(2.1), can be fully ascertained, many formidable theore­tical questions about the nuclear shell model, its genera­lization and its implementation must be answered.

Although we are interested in one phenomenological model, the unified model, we venture briefly into the con­text of these fundamentals to discover the source for the variety of the phenomenological shell models and to establish the position of the unified model in the spectrum of appro­ximations to the nuclear shell model.

3. Internucleon InteractionThe major obstacles precluding an exact shell model

calculation are threefold. In the first place, there is no closed analytical expression for the internucleon poten­tial V(|r^-r^j). After more than 20 years of theoretical investigations, we have only fragmentary knowledge of its analytical structure (Ro 49, Be 55, Ba 69, Si 69):

r"+00" r= | ri-rj | <0 . 5f

V±j - ( ? 0.5f<r<2f

Tt *t 0[ gJ! *Oj+S4J{ ( r / a ) 2+ 3 ( r / a ) + 3 } ] e “r//a- 3 J x*5

(2 .21 )

The long range portion of V is called the one pion exchange potential ("OPEP"). It consists in part of a Yukawa radial

23

potential and a non-central tensor force operator S^j

suggestive of a dipole-dipole interaction between the

spin 1/2 nucleons. The intermediate range part of V is

believed to be dominated by multiple pion exchange and/or

the exchange of heavier mesons. These contributions to V

have not been calculated sufficiently accurately (Si 69).From the shell model point of view, it is not this un­

certainty but the hard core of . which presents the greatest

technical problem: Matrix elements of , in particular

the average potential Uc , diverge for familiar basis func­

tions {<{> } inasmuch as such functions do not vanish below

0.5f. V, on the other hand, has -a large or infinite repul­

sive core in this region. Methods of a limited nature have

been devised to accommodate the hard core (Sc 6l).

Equations (2.21) should not be construed as implying

that there is no accurate representation of the Internucleon

potential. To the contrary, V is well-defined from an

experimental point of view in terms of its phase shifts and

bound state data (Si 69). Unfortunately, as far as shell

model calculations are concerned, one generally assumes an

analytic or parameter representation of V other than that

given directly by experiment. In at least one Instance,

however, an effort is being made to express shell model

matrix elements of the two-body interaction directly in terms

of the phase shifts in a way which avoids hard core diffi­

culties (El 68). We add parenthetically that the philosophy

24

behind this approach is that the hard core has no effect

on low-energy nuclear properties.

4. Theory of Effective Interactions

The second obstacle to an exact shell model calculation

is the manner by which the calculation proceeds. Although

the physical eigenvector Y extends throughout the infinite

|mp-m’h> space as indicated by Eq. (2.2), we calculate only

a finite number of its component amplitudes. This comes

about because we replace the infinite dimensional Hamiltonian

matrix for the complete space by a finite dimensional one

in a process of double truncation'. First we truncate an

infinite number of core excited components in order to set up

the shell model Hamiltonian given by Eq. (2.17). Next we

must truncate the infinite |mp-0h> space to a large, albeit

finite, basis in order to diagonalize the shell model

Hamiltonian. Normally one includes enough states in the

shell model basis so that the resulting eigenvalues and eigen­

vectors are unaffected by the addition of a few more basis

states.

This operational definition of convergence is commonly

assumed in shell model calculations. It is, however,

completely inadequate for the shell model Hamiltonian given

by Eq. (2.17). The difference between the typical shell model

calculation with an "effective" interaction V and our shell

model with a "bare" interaction V are important perturbative

corrections to the double truncation procedure. These

corrections reflect the fact that t h e .convergence of a

shell model wave function in a finite space is not a

reliable measure of the convergence of physical quantities

calculated from the model wave function.

It is not difficult to see why this is true. We know

that the eigenvectors of our shell model Hamiltonian, Eq.

(2.17), are orthonormal. At the same time we also know

that the eigenvectors ¥, Eq. (2.2), of the exact Hamilton­

ian are orthonormal. This means that their projections

into the finite model space of |mp-0h> states are not

orthonormal. Consequently our shell model wave functions

over estimate the |mp-0h> components since the sum of the

corresponding intensities in ¥ is less than unity. The

normalization strength of those states outside the finite

model space has effectively been redistributed inside the

model space. The resulting shell model wave function ¥o II I

may or may not be an accurate representation of the physical

wave function ¥. This is governed by the size of the

|mp-0h> model space and the degree to which core excited

configurations can be neglected in the shell model. More

specifically, such particle-hole states should not appear

among or nearby the low-lying energy levels of the nucleus.

When testing Y , another consequence of the double

truncation becomes apparent. Since we do not measure ¥

directly but rather its expectation values <¥ 10|¥> andO

transition strengths |<¥^|o|¥^>| , the accuracy of the

25

26

physical quantity calculated from ¥gm depends on how the

operator 0 weighs the various components of ¥g m . To see

this, we consider expectation values of the Hamiltonian

and unit operators. Their expansion in the complete

|mp-m’h> space is given by

E = < ¥ |H|¥> = E c.c.H (2.22a)i,j = l 1 3 1J

and

1 = < ¥ |1 |¥> = I c c,6,,. (2.22b)i,j=l 1 3 3

These two series neatly exemplify the difference between the

rate of convergence of ¥ and that of quantities calculated

from ¥. The matrix element brings to bear infinitely

many more terms in the eigenenergy than 6.. does in the± Jnormalization. Thus not only is the energy more sensitive

to errors in the mixing coefficients c^ brought about by

truncation but it also depends on contributions from states

outside the model space. Even though the mixing coefficients

of such "distant"- states may be extremely small, their

accumulation, weighed by H, may not be negligible. One

anticipates from the point of view of diagonalizing the

Hamiltonian matrix that the lower lying states would be pushed

even lower by the higher lying states were they not truncated.

These general observations about truncation of the

particle - hole basis, convergence of the model wave func­

tion, and the nature of operators go a long way towards

explaining our recourse to a variety of phenomenological

shell models. The proper treatment of truncation effects

leads to an exact formulation of the nuclear shell model

and the formidable theory of effective interactions (Ma 69).

The theory of effective interactions is actually a

very general theory applicable to any eigenvalue problem.

For our purposes, it leads to the conclusion that all dyna­

mical effects associated with core excitations and |mp-0h>

excitations outside the model space can be incorporated into

an effective shell model Hamiltonian for the valence nucleons

and a prescription for constructing the physical wave func­

tion (Br 66, Ma 69). The basic equation for the effective

Hamiltonian

27

is due to Bloch and Horowitz (B1 58). The effective inter­

action V is defined by a series expansion in powers of the

nucleon-nucleon potential. The leading terms in this ex­

pansion is the bare interaction V of Eq. (2.17). Trunca­

tion effects are offset by the higher order terms which

induce transitions to intermediate states outside the model

space and back again. There are no known closed expressions

mZ (T

v=l+ U C ) + ~ v v 2

m Z

v,v* =1

v^v'V (2.23)

for U G and V (Br 66).This situation has inspired innumerable shell model

calculations some searching for adequate parameteriza-

tions of the effective interactions and others attempting

detailed calculations of them. The most extensive cal-

l8 l8culations of this latter kind have been on 0 and F ,

treating them as an core plus two nucleons confined

to the s-d shell. The effective interaction for this case

involves core excited intermediate states and two kinds of

|2p-0h> intermediate states, namely those with one or both

particles excited above the s-d shell. Calculations show

that perhaps the most important group of intermediate states

are those for which both valence particles are excited out

of the s-d shell. In this approximation the effective

interaction is the-Breuckner reaction matrix or G-matrix.

The model wave functions do not of course contain these.

|2p-0h> components explicitly. Instead, their consideration

is clearly revealed in the depression of the lowest shell

model eigenenergies with respect to results employing only

the bare interaction V given by Eq. (2.21). This is in

accord with earlier comments about higher-lying states18 l8repelling lov/er-lying states. For 0 and F , these energy

level shifts amount to 1 or 2 MeV depending on the particular

state involved (Ma 69). Agreement with experiment is even

better when contributions to the effective interaction from

core excited intermediate states are included. Probably the

most widely known product of these calculations are the

28

29

Kuo-Brown G-matrix elements for the effective interaction

for two particles in the s-d shell (Ku 66). "

These matrix elements are frequently used for shell

model calculations involving three or more valence par­

ticles in the s-d shell. A leading example of this type

of calculation which we refer to later is the work of E.C.

Halbert et al. (Ha 68a). Such attempts to test shell model

concepts in broader regions of the periodic table move in­

evitably into the game of'finding adequate parameterizations

of the effective interaction. This is warranted because

the effective interaction is as complicated as possible.

The Kuo-Brown approximation of the effective interaction

applies rigorously only to two particle states In the s-d

shell. However, the effective interaction, being a sum of

products of V s , is, in general, a many-body operator. This

is written symbolically as

V = V 2 + V 3 + + ... (2.24)

Consequently, in a many particle shell model calculation V

connects states differing by two or more particles. The

importance of effective three-body forces is of great interest

now (We 69, Qu 69).

4.There are some numerical errors In the Kuo-Brown matrix elements. Furthermore, it Is not clear If ordinary per­turbation theory is reliable for the calculation of the G-matrix elements. For the latest theoretical comment see (Ba 70).

30

Microscopic calculations of V for systems possessing

more than two valence nucleons are only beginning (Ma 69). Nevertheless, many qualitative features of these more

complicated nuclear systems are already known. If not

given directly by the experimental observations, they are

drawn from phenomenological shell models.

5. Shell Model Parameterizations

The starting point for these models is the Hamiltonian

for the valence nucleons:

The one-body portion of Hv defines the single particle

energies and wave functions {Hq : ,<f) }. As mentioned earlier,

H describes the motion of the valence particles in the oHartree-Fock potential U° of the core. In Appendix C it

is shown that the single particle Schroedinger equation

corresponding to Hq is given by the integrodlfferential

equation

The shell model Hamiltonian matrix in this basis is given

by

(2.23)

T4>^(r) + /U°(r ,r' )<j)i (r * )d^r1 = e ^ ^ r ) . (2.25)

(2.26)

This equation is similar to Eq. (2.19) the difference being

the effective interaction V replacing the bare interaction

V.

In phenomenological shell model calculations one0

attempts to parameterize the potentials U and V in a

reasonable manner (In 53, El 55, Ku 56, El 58, Bo 67, Ha 68a).

The valence particles are distributed in the unoccupied

orbits of the core potential U c . The resulting finite set

of antisymmetric m particle basis functions constitute the

space in which the Hamiltonian matrix is diagonalized. The

primary objective is to learn how various interactions V

split the degeneracies of U c . In actual practice only effec­

tive two-body forces are considered. This modesty pre­

vails only because at present there is no theoretical or

experimental concensus about the nature or importance of

effective three-body forces (Ha 68a).

Frequently simple analytical expressions are taken for

Uc and . The single particle energies and wave functions

are invariably calculated from Eq. (2.25) with a local appro­

ximation for the non-local Hartree-Fock potential:

U c (r,rT) = 6(r-r')[U(r) + £jt‘s]. (2.27)

Depending on whether or not surface effects are important,f

U(r) is a harmonic oscillator or Woods-Saxon potential.

•j*'A very interesting discussion of the harmonic approximation to the Hartree-Fock potential can be found in (Mo 69).

31

32

The spin-orblt strength £ is adjusted to reproduce the

observed level splittings in nuclei with a single valence

nucleon. The spin-orbit term probably arises from the

tensor part of the internucleon potential (cf. Eq. (2.21))

(Pr 62). Some non-local effects can be simulated by using

an effective mass m* of about 1/2 the nucleon mass (Ma 67c,

Pr 62).

So called "intermediate coupling" shell model calcula­

tions, specific examples of which we quote in Chapter VII,

attempt to determine, among other things, the relative

strength of the central potential and the spin-orbit inter­

action in Eq. (2.27) (Bo 67). The limits U->-0 and £-+•()

correspond to pure jj coupling and pure LS coupling respectively.

An appropriate choice for the effective two-body inter­

action V 2 is the reaction or G-matrix for the presumed trunca­

tion. More typical, however, is the choice

V 2 (|?i-?j |) = V STV(rij/a) (2.28)

which is modeled after the Rosenfeld part of the internucleon

interaction (Ro 49). The strength of the interaction in the

four possible spin-isospin states of two nucleons is VgT with

S,T=0,1.

A more general and in some ways a less physical parameteri­

zation of the shell model Is the method of Slater integrals

as developed by I. Talmi (deSh 63). In this approach only

two general assumptions are made about U° and First,

U c is assumed to be a central field. Then the single

particle wave functions factorize into radial and tensor

components:

33

♦i = ? ' W r ) Y t a < n >x 1/ 2 I„ • <2 -2 9>s

Second, the effective two-body interaction V 2 can be ex­

panded in terms of tensors,

V i V f t b = v sT 2Y(ri - r j ) p i ( c o s e i j ) ( 2 - 30)

V STj 2JI+1 V ^ r i ’r j Y * m ^ i ^ Y Jlm^j * xm

By virtue of these factorizations, |mp-0h> matrix elements of

V 2 are a function of tensor and radial products. The tensor

products are reduced by techniques of Racah algebra. The

radial products are Slater integrals of the type

P = //Rn 1 4 (rl )Rn 2i2 (r2 )ve (rl-r 2 )Rn 3ll3 (rl )Fin 1|ll1,(r2 )drldr2-

(2.31)

Since radial shapes for Uc and V 2 are never assumed, the2

values of these integrals are determined by a x fit °T the

34

eigenvalues of the shell model Hamiltonian to the physical

spectrum. The best fit is expressed as the set of best

values for the matrix elements of .

These two ways of parameterizing the nuclear shell

model, viz., E q s . (2.27), (2.28), and (2.31), differ In the

number of assumptions they make for Uc and • In a Talmi-

type calculation one essentially varies the matrix elements

of ^ 2 to obtain the best possible fit to the experimental

spectra. It is expected that significant discrepancies in

the fit will demonstrate the existence of effective three-

body forces (We 69). Furthermore, the "best" two-body matrix

elements can be compared with those from the assumption of

specific potentials and those from detailed calculations of

the effective interaction V as they become available for more

nuclei. (See however (Ma 69) for a criticism of this latter

comparison. )

A disadvantage of the Talmi parameterization is that the

calculations yield spectra but not model wave functions. Since

dynamical properties requiring wave functions cannot be calcu­

lated it is impossible to get a full picture of what the nuclear

shell model can accomplish. This problem arises because the

radial part of the basis function cannot be extracted from

the Slater integral without assuming some radial dependence

for V £ . Doing so would lead of course to the second type of

parameterization of the nuclear shell model. Therein one

varies assorted physical parameters such as the form, strength,

and range of U c and V t h e spin-orbit strength £ and the

effective mass m * . Needless to say conclusions based on

this phenomenology must be tentative (cf. Pitfalls in Sec­

tion (II. D.2)). The Interdependence of these quantities

is rarely if ever examined. The main value in this approach

lies in the qualitative features of nuclear physics which

are brought into light.

In terms as these the shell model becomes an expedient

and gainful tool. The subsequent discovery of "shell model

nuclei" throughout the periodic table has firmly established

the ubiquity of nuclear shell phenomena. In many instances

phenomenological models have successfully predicted missing

energy levels, the reactions by which they may be found, and

the basic mechanisms leading to enhanced and inhibited transi­

tions and reactions.

From these successes there has emerged a phenomenological

pairing plus multipole characterization of the effective two-

body interaction:

35

V2 = V + v ^ - D j + v 2 Q.-Qj + V j t y O j +

(2 .3 2 )

The dipole, quadrupole, octupole, and other multipole operators

are those in the tensor expansion of Eq. (2.30). The v^ are

strengths of the various multipole interactions. Presumably

36

they vary slowly and smoothly from nucleus to nucleus. In

a major shell, say |mp-0h> states In the s-d shell, matrix

elements of the odd multipole tensors vanish by parity

considerations. This interaction is then known as the

"pairing plus quadrupole" interaction. The zero range 6-

potential and the long range quadrupole potential produce

different and competing correlations between the valence

nucleons. The merits of these individual potentials are well-

known (Br 64b). And, their interplay has been thoroughly

Investigated in the "P + Q" model (Be 59 > Be 69).

The relative importance of these two potentials for a

given nucleus depends on the number of nucleons outside the

closed inert core. As more and more nucleons are added to

the valence space, the long range quadrupole force produces

a stronger correlation between all the valence particles than

does the zero range 6-force which favors a pairwise coupling

of nucleons. The nuclear "SU^ Model" wherein the quadrupole

interaction is diagonal has been extremely successful in

demonstrating that such a long range interaction can generate

the rotational-like spectra evident throughout the lower-

half of the s-d shell (Ha 68b).

E. Core Excitations

From what has been said about the theory of effective

interactions, it is all too clear that a general program of

exact shell model calculations which can match or improve upon

37

phenomenological shell model calculations lies far in the

future. The matrix formulation of the nuclear shell model

leads unavoidably to the truncation of an infinite number of

|mp-0h> components and all core excited components. Though

these components may comprise only a very small percentage

of the physical state 4*, their proper reckoning leads to

important "renormalizations" of quantities calculated from

the model wave functions. Because no closed form of the effec­

tive interaction V is known, one must rely on phenomenological

representations of V or at best corrected Kuo-Brown G-matrix

elements for s-d shell calculations.

The present state of the art notwithstanding, the theory

of effective interactions provides an exact statement of the

nuclear shell model for interacting valence nucleons. Accord­

ing to the theory, .one constructs model wave functions and

effective operators to describe the eigenstates of a nucleus.

The model wave functions are the projections of the physical

wave functions into a finite space of |mp-0h> basis functions.

Theoretically, the theory of effective interactions says that

the complete eigenfunction is calculable from its nonvanishing

projection in the model space. As applied to the nuclear shell

model, most detailed calculations to date assume that most of

4' lies within the |mp-0h> model space (Ma 69). On the other

hand if 4' is composed mostly of states outside the model space,

knowing its |mp-0h> components is knowing almost nothing about

its basic structure. From these components and the effective

interaction one must build up the rest of ¥ to see what

it is like.

The construction of such a state is of immediate in­

terest if it lies among the usual shell model states of the

spectrum. The overriding question in this situation is:

Just what nature of excitation is competing with excitations

of independent valence particles? It has long been suspected

that core excited states appear among the low-lying excita­

tions of the nuclear system. It has only been relatively

recently that nucleon transfer reaction experiments have given

some clues as to the detailed structures of the wave functions.

Perhaps is the outstanding example of where the lowest

excited state is known to be a |4p-4h> excitation of the core.

Unfortunately it is the combined circumstances of a low-

lying state of markedly different character than the |mp-0h>

basis states of the shell model which make the effective inter­

action more pathological than perverse. Prom a small |mp-0h>

projection one must construct via V the essential character

of the wave function. Herein lies the third obstacle to an

exact shell model calculation. The presence of low-lying

collective particle-hole excitations of the core introduces

serious complications to the nuclear shell model (Ma 69).We consider in the main body' of this thesis a phenomeno­

logical approach to the study of the coupling of collective

excitations of the core to valence nucleons.

38

39

A. Core Excitations and the Nuclear Shell Model

In the previous Chapter, vie developed the shell model

concept of the N-body nucleus as a "core" with a few active

"valence" nucleons. In particular, we showed that in the

shell model approximation the wave function for the nucleus

can be written as a product of a core function Yc and an

antisymmetric function for the valence nucleons Y . In the

spherical shell model, the core is treated as if it were

inert and the low energy properties of the N-body system are

characterized as excitations of interacting valence nucleons.

It Is now widely known that there are low-lying excita­

tions in many nuclei which cannot be described in terms of

several active nucleons outside a closed core. Indeed, it is

well-known from the theory of superconductivity and the

concomitant energy gap between the ground and excited states

that an important degree of freedom of the many-body system

is the collective excitation of all the particles in the system

(Bo 58, Pi 62).

Detailed microscopic calculations of the eigenstates of

the core became feasible in the late 1950's with the advent

of the random phase approximation (RPA) for the many-body

Schroedinger equation (Br 64b). In contrast to the shell

model approximation of the Schroedinger equation in terms of

|mp-0h> states, the RPA calculations showed that there can be

Chapter IIIThe Unified Model

40

low-lying excitations of the core which are built largely

from coherent superpositions of many |2p-2h> configurations,

each contributing a small amplitude.

Herein we are interested in the interplay between core

and particle excitations. Inasmuch as the microscopic pro­

blem of the coupling of valence particles to the aforementioned

core excitations is only now being investigated (Br 64b), we

consider a physically graphic and comprehensive description

of the Interplay between core and particle degrees of freedom

first proposed by-A. Bohr in the early 1950's (Bo 52, Bo 53).

Bohr envisioned the core as a liquid drop of nuclear matter

and core excitations as quantized surface oscillations of this

drop. The average potential of the core is assumed to oscillate

in unison with the core thereby coupling the valence particles

to the behavior of the core.

Since its introduction, the ramifications of the liquid

drop description have been exploited relentlessly and with

great expectations bebause the successes enjoyed by this model

appear to be without bound. This situation is all the more

phenomenal in light of crude macroscopic caricature of the core.

Of course, these same successes suggest some perplexing ques­

tions. First, can the parameters of the unified model be

derived from first principles? Second, why does the model

work so well in some nuclei yet prove quite inadequate for

neighboring nuclei? And third, how does one distinguish be­

tween a unified model and shell model description in nuclei

where both models seem applicable?

V/e forego, to a large extent, the first question

(cf. Section (III. E. 3)) in favor of examining the overall

serviceability of the unified model and its relation to the

shell model, when these comparisons can be made.

Returning to the shell model matrix element H ^ with an

effective interaction V for the m valence nucleons, we intro­

duce an index a and operator a corresponding to the eigen­

states of the core. By analogy with equations (2.16) and

(2.23), the Hamiltonian describing core excitations as well

as excitations of the valence nucleons is taken to be

HT = ET (3.1a)

where

( 3 .1 b )

and

T = Tc (a)Tv (cO . (3.1c)

These are the basic equations for the unified model of

collective and particle motion. The individual quantities

in these equations are generalizations of the corresponding

42

shell model quantities. The m valence particles move in

the average potential of the core as before. Now however,

the core assumes a vital role: (1) there is a spectrum of

core states described by ¥c (a), (2) the average potential

for the valence nucleons depends on the state of excitation

of the core, and (3) the effective interaction is one

appropriate to an enlarged model space of core and particle

excitations. We expect from our previous experience with

antisymmetrization that the average potential is inherently

non-local and perhaps to a fair approximation much like the

Hartree-Fock potential of the inert core.

One reason that the unified model works for a given

nucleus but not its neighbors may be that even after all these

years the full potential of the unified model has not been

utilized. Most calculations to date concentrate exclusively

on the excitation modes of the core and on the coupling of

these modes to a single valence particle. We now examine the

basic tenets of the unified model for the purpose of ulti­

mately generalizing these results to include several valence

nucleons.

B. Bohr’s Liquid Drop Characterization of Core Excitations

1. 0.2^ Quantization (5-Dimensional Harmonic Oscillator)

The fundamentals of the theory of quantized nuclear

surface oscillations are contained in the classic papers of A.

Bohr (Bo 52, Bo 53). The surface of the oscillating liquid

43

drop Is defined by an expansion in spherical harmonics:

(3.2)

The o^pCt) describe the time development of the multipole

moments of an arbitrarily distorted surface. Treating the

as generalized coordinates of the surface, the Hamiltonian

of the liquid drop is Taylor expanded about its undisturbed

spherical equilibrium. For small oscillations the core

Hamiltonian can then be written as

to lowest order in the generalized coordinates (Al 69). Small

variations in the nuclear surface are thus described by a set

of uncoupled harmonic oscillators with excitation energies

nuclear density, the balance of surface tension against

Coulomb repulsion, and other hydrodynamic properties of the

liquid drop. One generally takes the excitation energy as

a parameter of the model.

A closer inspection of E q . (3.2) reveals that the A=0

and A=1 excitation modes correspond to nuclear density and

(3.4)

The mass parameter B and force constant C depend on theA A

never been observed and the latter are spurious. Hence

unified model calculations first consider A=2 deformations

of the core. In this case the Hamiltonian for the core is

a 5-dimens.ional harmonic oscillator in the a n . The2yharmonic oscillator is quantized by the usual prescription

of defining conjugate coordinates to the a2p and applying

the Boson commutation relation

center of mass oscillations respectively. The former have

The vibrational quanta each carry two units of angular momentum

and are called surfons or phonons. The excited states of the

core are given by the wave function |n J M> which corresponds

to n phonons coupled to angular momentum J and projection M.

suggested in Fig. (3.1), that the a 2y ^ depict a surface

wave moving across the liquid drop. From this vantage point,

the liquid drop resembles a vibrating spheroid rotating in

space. The Hamiltonian for the core, transformed to the

principle axes of the spheroid, is

(3.5a)

i"2v’a2v ' i h ' (3.5b)

2. 32p Quantization (Rotating-Vibrating Spheroid)

There is an alternative and more descriptive set of

generalized coordinates than the five a„ . It may be, as

45

z

Fig. (3*1) Rotating-Vibrating Spheroid. The 8^ are three Euler angles specifying the orientation of the principle axes 2' of the spheroid with respect to the space axes z.

46

H (a) = c c

without loss of generality. The 3 and y variables are simple

angles which specify the orientation of the principle axes z'

with respect to the space axes z. The overall deformation of

the core from sphericity is measured by 3 and deviations of

the resulting spheroid from axial symmetry by y. The quanti­

ties Ik and are the components of the moment of inertia

t habout the k principle axis and the angular velocity of the

core along that axis. The moments of inertia of the core are

those associated with the nuclear matter comprising the surface

w a v e ,

and not that of a solid spheroid. The indices k=l,2, and 3

correspond to the x ’, y ’-, and z' principle axes respectively.

Assuming that the velocity field of the nuclear field

forming the liquid drop is irrotational, the total angular

momentum of the core is given by

transformations of the a_ given in terms of the three EulerC- p

k = 1,2,3 (3-7)

(3.8)

and the Hamiltonian for the core is now

This form of the Bohr Hamiltonian affords a more in­

tuitive picture of core excitations than that permitted by

the 5-dimensional harmonic oscillator variables . As Eq.2)i(3.9) demonstrates, core excitations can also be described

in terms of rotations and g and y vibrations of a spheroid.

For a constant non-zero deformation g and zero asymmetry y,

the core Hamiltonian simplifies to

3 Jk <3 .10)

and describes a quantum mechanical rotator. Apart from the

hydrodynamic estimates .for the moments of inertia, the axially

symmetric form of this Hamiltonian has much phenomenological

merit, as we shall see.2

In the limit of axial symmetry, commutes with J , J z ,

and J ,, the generator of infinitesimal rotations about the zsymmetry axis. Furthermore, the irrotational assumption per­

mits quantizing the rotator in accordance with the commutation

relation

[Jx ,,Jy f ] = -iJz , (3.11)

which is appropriate for angular momentum components projected

into a rotating coordinate system (Da 65b). It is worth noting

that this commutator is not the usual one for angular momen­

tum operators. The appearance of the minus sign reflects

the fact that the principle axes of the spheroid are rotating

in space. As we shall see in Chapter V, this interchanges

the action of the J+ and J_ operators in the rotating coordinate

system.

From these considerations it- follows that the basis

functions for the quantum mechanical rotator are Wigner D-

functions D ^ ( 0 ^ ) which are generalizations of the familiar

angular momentum functions T^(0^). The D-functions are eigen-p

functions of J , J , and J , with eigenvalues J(J+1), M and6 Z

K, respectively.

Incorporating rotational invariance about the z ’ axis

and reflection invariance through the x'-y' plane, the

properly symmetrized and normalized wave function for an

axially symmetric rotator is (Pr 62)

c(3.12a)

and the corresponding eigenvalues of are

[J(J + 1) - K2 ] + ~2 2

= 21 K ( 3 .1 2 b )z

where I=Ix ,=Iy l .

49

Empirically it is *-ound that the moment of inertia I

needed in Eq. (3.12b) is greater than that predicted by the

hydrodynamic model (cf. Eq. (3.7)) and less than that of a

solid spheroid. On the other hand, I r is very small inzagreement with the hydrodynamic estimate. Referring to Eq.

(3 .12b), this implies ti.lt the low-lying levels of an axially

symmetric nucleus have no component of collective angular

momentum along the z' axis and it then follows from Eq. (3.12a)

that only even values of angular momentum are allowed. Thus

the spectrum of an axially symmetric rotator is simply

The general solution of Eq. (3-9) is not the simple rotator

of above because the rotational and vibrational motions are

inextricably coupled through the moment of inertia terms.

Equation (3.9) must be quantized by constructing the differ­

ential operators associated with 8 and y. The resulting wave

equation is complicated but does allow for a separation of

variables given by

ft2EJ = 5T J(J + 1} J = 0,2,4 K=0. (3.13)

(3.14)

where 82^ stands for 8 , y, 6^.

50

3. Relationship Between Both Quantization Schemes

Within the irrotational assumption, Eq. (3.8), the 32

variables for the vibrating-rctating spheroid are equivalent

to the c*2^ variables for the 5-dimensional harmonic oscillator.

Consequently, in either representation the energy spectrum is

that of a harmonic oscillator of A=2 phonons with energy

■ftw2 (see Fig. (3.2)). The equidistant levels of the harmonic

oscillator are wholly unlike the J(J+1) spacing for the

symmetrical top. The connection between these two represen­

tations is thoroughly masked by the rotation-vibration inter­

action. Bohr proved that the J=0 states in Fig. (3.2) corres­

pond to 3-vibrations. The other angular momentum states are

not as easily interpreted because they are extensive admixtures

of rotating and 3 and y vibrations.

The 32^ representation would be of little interest were

it not for the empirical observation that the low-lying levels

in many even-even nuclei follow a J(J+1) rule. The rotation-

vibration interaction appears to be very weak and the spectrum

of the core splits into well-defined rotational and vibrational

levels. This is illustrated In Fig. (3.3). On each vibrational

state is built a sequence of low energy rotational excitations.

In general, the 3 excitations preserve the axial symmetry of

the spheroid and have quantum number K=0. Rotation and reflec­

tion invariance of the spheroid restrict the rotational levels

built on a 3-band to even angular momentum values: J=0,2,4,...

On the other hand, the y excitations can have K=2,4,6... with

51

Fig.

r

NO. OF PHONONS

4 O t22,42,5,6,8

3 --------- J------ 0,2,3,4,6noj2

2 . .--------------------f -------------- ° ’2 ’ 4I -------------------------- 2

0

(3.2) Excitation Spectrum of a Liquid Drop (I). Small quadrupole (X—2) vibrations about a spherical equilibrium exhibit a 5-dimensional harmonic oscillator spectrum. Each quadrupole phonon carries two units of angular momentum. All states have positive parity.

52

8' 6'

PyS “ In y - 0

K = 0

2 ro +

n/9= 0 n ^ = I K = 2

i +'

4"

Fig. (3«3) Excitation Spectrum of a Liquid Drop (II). • In the limit of a weak rotational-vibrational interaction, the liquid drop resembles a rotating spheroid. The 5-dimensional harmonic oscillator spectrum of the core splits into well defined rotational levels built on 3 and y vibrational states (Ba 60).

J=K, K+l, K+2... (Da 65a). These spin sequences clearly

stand out in Fig. (3-3) for the weak rotation-vibration

interaction (called the strong coupling limit) and are

embedded in Fig. (3.2) for the contrary case.

Bohr also showed that the coupling of a single valence

particle to the liquid drop tends to polarize it. The sur­

face of this drop then acquires a stable deformation in which

case an oscillating spheroid description becomes appropriate.

The 3 and y vibrations are approximately harmonic and occur

with excitation energies (Ch 54, Ba 60),

fio) 0 = h u = ftw- ' (3.15)p y £

The cumulative effect of many valence nucleons outside a liquid

drop core is frequently that of producing an effective,

permanently-deformed core comprised of all the nucleons

("strong coupling"). The low-lying spectrum of the effective

core is very accurately described by a quantum mechanical

rotator.

The exact positions of the 3 and y band heads in Fig. (3-3)

are difficult to establish because of the Increasing density

of states at higher excitation energies from all modes of

excitation. For this reason, the band head energies n^hu^

and n^hiOy are additional parameters tentatively assigned to

likely states. The effect of all higher-lying states on the

lower-lying rotational states is to compress them and thereby

53

modify the simple J(J+1) rule. The singular effect of the

rotation-vibration interaction on the ground state rotational

band is given by the perturbation result that the ground

state band is depressed according to

Ej = A J (J + 1) - BJ2 (J + l)2 (3.16)

with B>0 (Da 65a).

4. Rotational Nuclei

Nuclei with pronounced rotational structure have atomic

mass numbers 20-28, 150-190, and 220 and above. An interest­

ing comparison of nuclei in these three rotational regions of

the periodic table is presented in Fig. (3.4). All the energy

levels of Ne2< , Gd^^^,.and Pu2^ up to the lowest level which

is not a member of the ground state rotational band are plotted.

In each case the intruding level is a 0+ state. Beneath the

spectrum for each nucleus are the values in MeV of the

reciprocal moment of inertia A and the rotation-vibration

parameter B as determined from the energies of the 2+ and 4+

states.

The excited 0+ states just mentioned lie in the energy

range where 6 and y vibrational excitations can be expected.

This is indicated in Fig. (3.4) by the hw-curve for Eqs.

(3.4) and (3.15). This curve gives the vibrational excitation

energy of a uniformly charged irrotational liquid drop of

54

EX

CIT

AT

ION

EN

ERG

Y (M

eV

)

55Fig. (3.^0 Hydrodynamic Threshold for 3 and y Vibrations

(After A. Bohr). The lowest level intruding into the ground state band of many rotational nuclei has spin and parity J-,T=0+ and lies in the energy range where a J ^ C r K=0 vibrational state is expected on the basis of hydrodynamic argu­ments. For each nucleus, the moment of inertia A and rotational-vibrational parameter B is given for the presumed rotational-vibrational perturbation of the ground state rotational band,

8

4

0

10 Ne

- 0 + 20

8 +6 + ,4+

+

64 Gd

-- 0

156

- o +.8+ - / 6 + - / 4 +

9 4 Pu

■o+ 238 ATOMIC

MASS NO.

r A = 0 .3 0

8 = 0 .40x10

A = 0 . 14x10“ '- 2 8 = 0 .29x10 -A

A = 0 . 7 3 x I0 "*1-

B = 0 .3 6 x 1 0 "°

( A and B in MeV)

56

constant density (Bo 52, Bo 53). The decreasing energy

needed to excite a vibrational mode as one moves higher in

the periodic table is due to the domination of the repulsive

Coulomb forces over the nuclear surface tension thereby

making the liquid drop softer for low energy vibrations.

The values of A and B for the heavy element in Fig. (3-4)

reproduce to better than a few percent all levels of the

ground state rotational band below the Intruding level. In

contrast, the rotational structure of the lighter nucleus20 20 Ne is not as well developed. The Ne spectrum is unlike

that found in the heavier nuclei primarily in that the ground

state band extends over a broader energy range and the intrud­

ing level appears much lower in the rotational spectrum. The+ 20 excited 0 level at 6.72 MeV in Ne is the lowest of a number

of intruders In the region of possible band 3 and y states.20The values of A and B for Ne predict the 1^=6 level at

5.02 MeV. By way of contrast, in the rigid rotation limit

B is set to zero and A is determined from the 0+-2+ spacing.

This yields A=.27 and B=0 MeV which is not very different from

the values in Fig. (3.4). Nonetheless, the J 7T=6+ level is now

pushed way up to 11.41 MeV. In actuality, the J ir=6+ level lies

within these two limits. Clearly then, the ground state

rotational band is compressed by various excitation modes

appearing above 6 MeV, but not quite as naively as predicted

by Eq. (3 .16) .

With these considerations in mind, it seems reasonable to

characterize the rotational levels below the intruding level

57

as arising'from the rigid rotation of a symmetrical top.

For these levels, the rotation-vibration correction to Ej

is small and can be neglected in a first approximation.

C. The Average Potential Generated by the Core

Thus far a simple description of core excitations has

been outlined. The theory of quantized surface oscillationsA

of a liquid drop leads one to a core Hamiltonian Hc (a) with

much phenomenological merit. There is overwhelming evidence

that core excitations can be assigned quantum numbers corres­

ponding to a2p phonons for vibrations of a spherical core and

B2p phonons and rotons for vibrations and rotations of a

permanently deformed core.

The next problem in the unified model for collective andc Aparticle motions is - determining the non-local potential U (a)

of Eq. (3.1b) which is generated by the core and in which the

valence particles move. Unfortunately, treating the core as a

nuclear fluid precludes treating properly the antisymmetrization

between excited core nucleons and valence nucleons. Neverthe­

less, the average potential generated by the core ought not be

radically different from the non-local Hartree-Fock potential

of the core in its ground state.

As the core rotates and vibrates, the oscillating field

it generates affects the motion of the valence particles so

that they no longer move undisturbed in a static shell model

potential. Assuming that the surfaces of constant density of

58

the oscillating liquid drop are equipotentials, the average

single particle potential can be written

U°(o,r) = Uc(o)(r) - r I a, 7, (SI) + ... (3.17)dr Ap Xu Xu

to lowest order in the deformation parameters ci (Pr 62,

G1 69). The zeroth order term is the shell model potential

generated by the inert core. The higher order terms bring

to bear the oscillatory action of the core on the motion of

the valence nucleons. Most considerations of Eq. (3.17) are

confined to linear quadrupole deformations .

D. Core Excitations and the Effective Internucleon Interaction

Recalling the discussion of Chapter II, nucleons outside

the core interact among themselves by means of an effective

interaction V. The many-body effective interaction appeared

as a consequence of truncating the infinite dimensional N-

particle basis space. It is known only as a series expansion

in powers of the free (or bare) two nucleon interaction V.

Furthermore, its detailed character depends on the particular

model space in which the calculations are made. In view of

these forbidding constraints, V is generally granted a

phenomenological existence. This situation can be expected to

prevail in the unified model too.

Virtually all unified model descriptions of the interplay

between collective and particle degrees of freedom are limited

59

to one or no valence nucleons thereby obviating from the

beginning the need for an efffective nucleon-nucleon

interaction. Some of the few exceptions which do involve

several valence nucleons outside a liquid drop core are

discussed in the following section. Otherwise there has

been too little experience to establish the phenomenological

effective interaction most appropriate for unified model cal-c a

culations. Presumably, as with the average potential U (a),~ A

V(a) is not entirely different from that employed for valence

nucleons outside an inert core, namely Eq. (2.33).

E. A Unified Model Hamiltonian

1. The interplay Between Core and Valence Nucleon Degrees of Freedom

All points considered, within the theory of quantized

surface oscillations the coordinates for a phenomenological

model of collective and particle motion separate according to

H - H^°^ = H (a, ) + H (x) + H (a, ,x) + H ,(x,x') c c Ay v cv Ay5 • vv* *

(3 .1 8 )

The a. are the collective coordinates of the core and x the Ayspace-spin coordinates of the valence nucleons. The core

Hamiltonian H (a, ) is given by E q s . (3-3) or (3.9) for the c a y

a 2y or ^2y rePresentati°ns respectively. Following Eq. (3-17),

the Hamiltonian for the valence nucleons and their interaction

60

with the core phonons is given by

(3.19a)

and

(3.19b)

The phenomenological effective interaction Hv v , is ill-defined

and presumably a small perturbation whose necessity and merits

must be judged by the agreement it produces with experiment.

the average non-local potential it generates.

2. Weak Coupling Limit

There are two approaches generally pursued in studying the

interplay between collective and particle motion. They are

called the "weak and strong coupling limits" depending on

whether the valence particles experience a spherical or deformed

field on the average. In weak coupling, the particle-core

interaction given by Eq. (3.19b) is treated as a weak per­

turbation to the motion of valence particles in a spherical

potential. The particle-core interaction is diagonalized in

a basis of phonon states |n J M> vector coupled to valence

particles in shell model orbitals <j>jm of Eq. (3.19a). In

actual practice, the radial strength of the coupling is taken

Finally, is the Hamiltonian for the Inert core and uc ^°^

as an adjustable parameter k. The problem of coupling

several valence particles to the 5-dimensional harmonic oscillator has been examined by Alaga et al. (Al 69).They allow the valence particles to interact among them­

selves, via a phenomenological pairing interaction.

3. Strong Coupling Limit (the Deformed Shell Model)

Alternatively, the strong coupling approach recognizes

that the low-lying levels in many nuclei can be described by

assumption of a quantum mechanical rotator for the core. As

pointed out earlier, in these nuclei the rotational and vi­

brational modes of excitation are for the most part indepen­

dent and uncoupled degrees of freedom. Then, in the strong

coupling limit, the Hamiltonian for collective and particle

motion simplifies to

H _ H<o) = ZAkR 2 + Hv (S,Y,x) + Hv v ,. (3.20)

For a fixed 3 and y distortion, it is convenient to combine

H and H into a single particle Hamiltonian H (3,y,x)\ C V V

governing the motion of the valence nucleons in a deformed

field. Strong coupling is distinct from weak coupling on

the essential point that the core and average potential are

permanently deformed. Hence, it follows that the angular

momentum j of a valence nucleon in a non-spherical potential

is no longer a constant of the motion. The wave function for

6.1

62

a valence nucleon in a deformed potential is a superposition

of shell model wave functions , . Having absorbed theJ in e

Hcv term into H , it is now evident that part of the coupling

of the valence nucleons to the coi’e results from their sharing

of the total angular momentum of the system

J = it + Ej (3.21)v

which is a good quantum number. This so called "Coriolis"

coupling is developed in full detail in the next chapter.

F. Villars has recently derived the strong-coupling

Hamiltonian, including Coriolis coupling, from a microscopic

point of view (Vi 70). He concludes that the phenomenological

unified model is substantially correct for nuclei with large

deformations and that the particle or intrinsic Hamiltonian

Hv in Eq. (3.20) is amenable to Kartree-Fock methods. The

best known phenomenology for the intrinsic Hamiltonian is due

to S. Nilsson (Ni 55). He makes a local approximation

analogous to Eq. (2.27) for the shell model potential and then

parameterizes the deformed potential well as an anisotropic

harmonic oscillator.

A comparison of the Hartree-Fock and Nilsson treatments

of the intrinsic Hamiltonians is deferred until Chapter VI

when the parameters of the strong-coupling unified model are

discussed.

63

We now explore within the framework of this model the

problem of several valence nucleons coupled to a rotator

core. There are many calculations In the literature which

consider a single valence nucleon but surprisingly few which

consider several (Fo 53, Ga 62a,b, As 68, Wa 70). Only in

the most recent calculations was an effective Interaction

included between the valence nucleons (Ne 62, As 68, Wa 70).

We restrict ourselves to the s-d shell where rotational

nuclei exist and where an intensive program of shell model

calculations is underway (Ha 68a, Ak 69). The s-d shell is

proving to be a testing ground for the importance of core

excitations -.

64

A. The Nuclear s-d Shell

The first half of the s-d shell has long been presumed

a region of the periodic table where nuclei possess stable

deformations from a spherical shape. Naturally, as might

be expected, few nuclei fit so conveniently Into the

simple theoretical picture provided by a rotator core for

even-even nuclei or a i?otator core with a single valence

nucleon for odd-even nuclei. Nonetheless, the experimental

picture of this region of the s-d shell consistently demon­

strates a substantial underlying collectivity for there are

unmistakable rotational-like spin sequences obeying an

approximate J(J+1) energy rule. Moreover, another signature

of collectivity is the marked enhancement of E2 y-ray transi­

tion strengths linking the members of each of these spin

sequences. These enhanced strengths reflect a certain co­

herence between many nucleons which cannot be duplicated by

a few particles jumping from one orbit to another.

B. Neon 20

An excellent example illustrating differing degrees of

complexity in two neighboring collective nuclei can be seen

in Fig. (4.1). The J=0, 2, 4, 6, 8,... spin sequence and

20spacing suggest quite convincingly that Ne can be described

Chapter IVFroward Unified Model Nuclei

EX

CIT

AT

ION

EN

ERG

Y (M

eV

)

65

10 b MANYL E V E L S

9

8

7

6

5

4

3

2

8'

2+-

M A N YL E V E L S

2 +

cre 20 N e22

2 0F i g . (4.1) Positive Parity Excitation Spectra of N e * and

Neighboring N e ^ .

66

in part as an axially symmetric rotator. There are of course

perturbations to these rotational levels which originate in

axial asymmetry and coupling to all other modes of excita-20 20 tion in Ne . Above 6 MeV, the density of states in Ne

increases rapidly. In the context of the hydrodynamic

model for the liquid drop core, some of these high-lying

states are rotational levels built on the lowest excited

3 and y vibrational states. Recalling the previous Chapter,

corrections to the J(J+l)-rule from the rotation-vibration

20interaction were estimated for Ne and found to be small.

Furthermore, it is also reasonable to expect that admixtures

20into the Ne ■ ground state rotational band from other kinds

of excitations are small. Indeed, since states of a given

angular momentum and parity can mix only with states of the20same angular momentum and parity and since, in N e " , there

is more than a 6 MeV gap between the lowest 0^ state and the

lowest excited 0* state and similarly for the 2* and 2*, 4*

and 4* states, and so on, one is led to believe that the20ground state rotational band in Ne should be very pure.

C. Neon 22

1. Disparity with Neon 20

A J=0, 2, 4, 6,... rotational spin sequence can also be

identified in Ne22 in Fig. (4.1). The apparent similarity

20 22 with Ne ends here inasmuch as the Ne spectrum exhibits

several new features. For one, the ground state band Is

significantly compressed with respect to its Ne counter­

part. Second, a large number of levels appear below the

anticipated energy range for 3 and y vibrations.

Specifically, 3 and y states are not expected below 5 MeV

excitation energy (cf. Eq. (3.15) and Fig. (3*4)). And

third, the proximity of perturbing states is halved from

6 to 3 M e V . These three observations are interrelated and

together suggest the presence of large admixtures in the 22Ne ground state rotational band. In short, the increased

number of higher-lying states weigh heavily on the ground

state band.

2. J 1T~='2+ Intruding Level

22Another noteworthy point about Ne which might, at

22first sight, make Ne a conventional unified model nucleus

after all is the fact that the lowest intruding level inTTthe ground state band has spin and parity J =2 . This ob-

20servation suggests that the relationship between Ne and

22Ne could be one of a smooth transition from an axially

symmetric to an axially asymmetric rotator. In Fig. (4.2),

we give the Davydov diagram for the excitation spectrum of

an axially asymmetric rotator (Pr 62). The Hamiltonian for

the system is given by Eq. (3.10), namely,

67

20

H = A 1R 2 + A.R2 + A 3R 2 . (4.1)

The hydrodynamic expressions for the Ak (3,y) are given by

68

Fig. (4.2) Davydov Diagram for an Axially Asymmetric Rotator. The limit of zero asymmetry y corresponds to the excitation spectrum of an axially symmetric rotator. As the asymmetry increases, additional low-lying rotational excitations become feasible Inasmuch as the moment of inertia I3 about the z 1 axis is no longer vanishingly small (v. Eq. (3-7)) •

fi2/

4B

/3

69

y (degrees)

F i g . ( 4 . 2 )

E q s . (3.7) and (3-10). In the limit of zero asymmetry y,

the Davydov diagram corresponds to an axially symmetric

rotator with a J(J+1), J=0,2,4... spectrum. As the asymmetry

increases, A^ becomes finite with the consequence that addi­

tional low-lying rotational excitations become feasible.

The lowest of these levels in even-even nuclei is always

a J ir=:2+ state.

Unfortunately an asymmetric rotator characterization of

20 22Ne and Ne does not provide a complete account of the ex­

perimental picture. On the one hand, there are at least 11? P 7T "f*positive parity states in Ne upto and including the J =6

state. However, the asymmetrical model predicts only 6

levels at most. On the other hand, it can be understood

from Fig. (4.2) that the ground state spin sequence is not

seriously perturbed cnergywise by any amount of asymmetry.

22This implies that the striking compression of the Ne ground

state band cannot be due to asymmetry alone.

C. Candidate Nuclei for Several Active Valence Nucleons22Thus it seems that the rotational foundation of Ne is

22uncertain at best. Significantly, Ne is not alone in this

respect. A picture similar to Fig. (4.1) also holds for the

pairs M g 2 - Mg2^ and SI^® - S i ^ .

In all of these cases, the addition of two neutrons to

a good rotator core results in a structure substantially

more complex than that given by any form of rotator. Perhaps,

then, some of the extra states observed in these nuclei

70

71

represent extra degrees of freedom associated with excitations

of the last two neutrons. Not surprisingly, there are con­

jectures (cf. Chapter VII) that some collective nuclei in

the first half of the s-d shell exhibit rotational spin

sequences built on different intrinsic excitations of several

valence nucleons.

By way of exploring these conjectures about the inter­

play between collective and valence particle degrees of

freedom, we attempt to interpret, N e 2^ ,22,2^ } Mg2"’52^ ’2^,29 30 31and Si * as 1, 2, and 3 active neutron systems with

20 2-4 23Ne , Mg , and Si rotator cores respectively.

72

A. Introduction

It is clear from the preceding chapter that our under­

standing of presumably rotational nuclei in the first half

of the s-d shell is rather incomplete. To be sure, at some

energy, some of the extra states observed in these nuclei

must come from the microscopic degrees of freedom associated

with the excitation of a few nucleons. For these reasons,

we now Investigate the problem of coupling several valence

nucleons to a rotator core.

Towards this end, we adopt the axially symmetric strong-

coupling Hamiltonian

H = AR2 + Eh. + Ev,, (5.1)1 i< j

for investigating the Interplay between collective and valence

particle degrees of freedom in s-d shell nuclei. The quanti- *■>

ties A and R are the reciprocal moment of inertia and the

angular momentum of the core. The single particle Hamiltonian

h describes the motion of the valence particles in the non­

local permanently deformed field generated by the core nucleons.

And lastly, a small residual interaction v is effective be­

tween the valence nucleons.

In this Chapter we consider the eigenvalue problem

defined by Eq. (5.1). First of all, this entails generalizing

Chapter VThe Deformed Shell Model

the strong-coupling wave function of Bohr to several valence

nucleons. Since the Hamiltonian is not diagonal in the

strong-coupling representation, we then evaluate the matrix

elements of the Hamiltonian in the space of these basis

functions. The eigenrnodes of the system of core and valence

nucleons are then determined by diagonalizing the Hamiltonian

matrix.

As experience with nuclear models has shown, reproducing

the experimental excitation spectrum is not a particularly

sensitive test of.the model assumptions. The theoretical

excitation spectrum represents but one possible average of

the components of the model wave function. A gratifying

theoretical spectrum may actually becloud the fact that other

quantities calculated from the wave function bear little

resemblence to the given nucleus. Sometimes this reflects

uncertainties about the effective operators used for these

quantities (cf. Chapter II). Othertimes this failure implies

that the underlying structure of the physical state is con­

trary to that predicted by the model.

Therefore, in the fourth section of this Chapter, we

calculate the spectroscopic factor for this model. The

spectroscopic factor for (d,p) transitions provides a test

of specific components of the model wave functions. Briefly,

in the stripping approximation, one assumes that the target

nucleons captures a neutron without perturbing its ground

state configuration. Thus the captured neutron enters the

73

empty orbitals of the target thereby forming the residual

nucleus in various excited particle states. The strength

with which these transitions take place can aid in locating

the multi-particle states among the low-lying excitations

in the physical spectrum.

Of particular interest in this respect is the puzzle

regarding the origin of the low-lying J=K=2 state intruding

into the rotational band of s-d shell nuclei. As pointed

out previously, it seems unlikely that it corresponds to a

permanent axially asymmetric state with <y>/0 or the beginning

of a y-vibrational band with <y >=0 and <y >/0. Spectroscopic

calculations may help answer whether or not this state results

from an excitation of loosely bound valence neutrons.

B. Strong-Coupling Basis Functions T(JMKa) and Ka-configurations

1. Strong-Coupling Formalism

It is convenient to rewrite the Hamiltonian in terms of

the total angular momentum of the system which is

J = R + j (5.2)

-* m -Vwith j= I j. for m valence neutrons. This yields

i=l x

-*■ -*■ -+PH = A(J - 2J * j + j ) + Zh1 + I v ^ . (5.3)

7^

75

In the limit of a rotator core and zero number of

valence nucleons, only the first term in this Hamiltonian

is relevant. As previously discussed, this corresponds to

a pure J(J+1) rotational band with J=0, 2, 4,... and K = 0 .

The strong-coupling wave function in this example is an

eigenfunction of the rotator Hamiltonian and is given by

Eq. (3.12a).

Where this simple model provides an adequate description

of some even-even nuclei, the natural generalization is to

that of a rotator core plus a single valence nucleon for

neighboring nuclei. From E q . (2.18) we have the fundamental

shell model result that the valence nucleons move in the

Hartree-Fock orbitals generated by the core nucleons:

h *av = W s i v ( ! M )

However, in contrast to the standard shell model ansatz of

spherical orbitals 4>jm > we now permit a more general vari­

ational calculation and actually look for solutions of Eq.

(5.4) which correspond to an average spheroidal field. The

functions x^v then represent deformed orbitals in the prin­

ciple axis frame z' of the core. For the assumed spheroidal

symmetry, the particle angular momentum j is no longer a good

quantum number although its projection j_,=ft on the symmetrytt

axis still is. The single particle wave functions and energies

are therefore labeled by ft. An additional quantum number

76

index v is needed to distinguish between orbitals with the

same projection ft. In the limit of zero deformation, the

deformed orbitals x^v converge to the usual shell model

wave functions <j>. with good angular momentum j for

spherical symmetry. Otherwise they are linear combinations

of shell model states

where j stands for N j JL Finally we have that each orbital

Invariance of the spheroidal potential through the x'-y'

plane.

The strong-coupled wave function for the core plus

particle is essentially .a product of a Wigner D-function

Several remarks are in order as regards the constitution

of the strong-coupling wave function given by the above ex­

pression. The overall symmetry of ¥-manifested in the super­

position of terms with +K and -K projections-is dictated by

the requirement of invariance under reflections in the x'-y*

plane. In addition, ¥ reflects the empirical constraint

(5.5)

v is twofold degenerate with j ,=ift corresponding reflectionz

rotations and a single particle function x^v for

intrinsic excitations of the system (Pr 62):

(5.6a)

77

mentioned earlier that the collective angular momentum R ,zabout the symmetry axis is zero for low-lying states of

the system. Hence, we have Imposed the corresponding

subsidiary condition that K=Q. A point of constant con­

fusion about Y is the phase which premultiplies the -K

projection terms. Frequently one sees it written as ( - l / -

This form of the phase dates back to the earliest works of

A. Bohr where the particle coupled to the core was treated

as if having good angular momentum j (Bo 52, Bo 53). In a

non-spherical potential, j is not a good quantum number.

Then, (-1)J “J' X_Kv has the meaning (-1)J Z(-1)“^ c ^ j 4>j _kj

(NI 55). For the purpose of our eventual generalization of

Eq. (5.6a) to the case of several valence nucleons outside

a rotator core, we prefer the notation that

x±Kv ~ ^vj^jiK (5.6b)

and use the theorem of Preston that invariance with respect

to inversion of the z-'— axis implies (Pr 62)

Here IT is the parity of x* Therefore, with the above defini- XJ — 1/2tion of x we need the phase (-IT H as opposed to the

j •

phase (-1) . This is also a convention adopted by other

authors (Ma 66). Our final remark about the constitution

of 4' is that j strictly speaking, the D-funetions are not

now eigenfunctions of the core because they refer to the

total angular momentum J and not the core angular momentum

R.

The resulting spectrum for an odd-even nucleus is richer

in structure than is that of its even-even parent. More than

just a ground state band, there are also J=K, K+l,... rota­

tional bands built on the intrinsic excitations x^v tbe

valence nucleon. However, the spacing of these levels is

not given so simply by a J(J+l)-rule because the appearance

of the rotator-particle coupling term in the Hamiltonian

RPC = -2AJ-J (5.7)

serves to mix states with AK=1. As we shall see, RPC can

render the J(J+l)-rule inutile in an even-even nucleus and

at the same time improve the theoretical calculations

immeasurably.

2. Generalization to Several Valence Nucleons

The strong-coupling formalism for the rotator core plus

valence nucleon can easily be generalized to Include several

valence nucleons. This eventually was briefly mentioned by

Bohr and Mottelson in their pioneering development of the

unified model (Bo 53).

78

79

From Chapter II, it follows that an explicit treatment

of antisymmetrizatlon is essential only in the valence space

vth deformed orbital of h with j-projection ft on the z*

axis, then the intrinsic state for m valence neutrons is

given by

where K=Eft^ is the projection of the total particle angular

momentum on the z* axis and where a = ( > ^2V2 ’* * *'^mvm^

is an m-tuple quantum number epitomizing the single particle

composition of the m-body state X^a *

The creation operators for deformed orbitals are related

to these for spherical orbitals by

in accordance with Eq. (5.5).

Incorporating the necessary rotation and reflection

symmetries (Pr 62), the strong-coupling basis functions for

a rotator core plus m valence neutrons can be written

of the model. Thus, if we let b create a neutron in the

x Ko = £b f t i V i b ft2V2 • • • b ft v ^ K o l 0>m m ( 5 . 8 )

J - m / 2

V m - K x - K ctJ

(5.10)

80

where the represents normalized D-functions:

■I-

2J+1 nJ 8II2 MK ’ (5.11a)

( 0i )d.ei6J J « 6M M ' 6 K K ' *

(5 .11b)

and where the Intrinsic state X_Ka is related to xKa by

ft^-ft.^ in Eq. (5.8). The 6-function normalization <$KQpW

vanishes in all instances except when the intrinsic state

xKa consi-sts of an even number of neutrons placed in the

deformed orbitals pairwise. In other words, if a=(...ft^v^,

. ..), then K=0 pairwise (i.e., K=0PW) and dK0PVJ= “

3. Discussion of Eq. (5.10)

The wave function Y(JMKo) represents a rotational band

J=K, K+l,... built on the Ka-configuration of the valence

neutrons. That Y encompasses all previously discussed features

of the strong-coupling limit can be easily demonstrated.

First, with m=l for one valence particle, we must have ^^opw=^*

In this case, Y reduces immediately to Eq. (5.6a) which is

the strong-coupling wave function for an even-even rotator

core plus a single valence nucleon.

Next, for the case of two valence neutrons outside the

core, we have m=2. Assuming that these two neutrons occupy

the same intrinsic single particle state with opposite pro­

jections ft, we have 61<rnD =1 as well as the parity of theKU £ Vj

product state n =+1. Furthermore, whenever ^k o p V7=3'>X

Xj£a and X_jra are not orthogonal. Instead a simple mani­

pulation of the creation operators in Eq. (5-8) yields the

relationship

v = (-l')rri//2Y (5 12)xKa=0PW ' ; x- K a = 0 P W ;

As a consequence, ¥ this time reduces to

nJHKo=0PW) = 4-d0 XKo=0pw . ( 5 . 13)

By virtue of the bracketed term, we conclude that a K =0PW

wave function exists only' for even values of the angular

momentum. Moreover, if the two valence neutrons are placed

in the lowest extra-core orbitals pairwise, ¥(Ko=0PW) is

precisely the analogue of the wave function E q . (3.12a) for

the ground state band of the even-even rotator: J=0,2,4,,..,

K=0. Our generality in ¥ allows the valence neutrons to be

excited out of their ground state configuration.

The wave functions ¥(JMKa) are not eigenfunctions of the

strong-coupling Hamiltonian. States of different K and a•4-

are mixed by the rotator-particle coupling term (RPC=-2AJ•j ),

the particle-particle coupling term (PPC=A(Ej^) ), and the

residual interaction • We use ¥(JMKa) to define a

basis space in which to set up and diagonalize the Hamiltonian

matrix. The excitation modes of the fully coupled system are

81

82

then given by the eigenvalues and eigenvectors of the H-

matrix.

The basis consists of all x^a states produced by dis­

tributing the m extra-core particles In the deformed orbi­

tals of the valence space. The particles can be placed in

each orbital v with projection -J2. Since ¥(JM-Ka) =

(-l)J_m/2n ¥(JMKa), an independent set of basis functions is Xgiven by all Ka-configurations with K ^ O . Henceforth, we

impress the convention that the symbol K always signifies

a non-negative integer or half-integer.

C. Reduction of the Strong-Coupling Hamiltonian with Respect to the K Quantum Number

1. General Considerations

For the eigenvalue problem and the spectroscopic cal­

culation which we consider later it is necessary to construct

matrix elements of ¥(JMKa). Because ¥ has two additive parts,

matrix elements <¥'|r|¥> of an operator r expand into a sum

of four terms of similar form

(-1)P <4>' X 1 |r|cf>x>

for the combinations -K and - K ' . These four terms are ex­

plicitly written out in Appendix D together with the phases

(-1) which connect them. Hereafter, we set n =+1 since weA

are ultimately Interested in multi-particle excitations in

the s-d shell.

For r=H, symmetry v;ith respect to inversion of the

z' axis allows us to write

83

HK' o ' ,K a <4'(JMK’ a ’) |H | Y(JMKa) >

1/2] (5.1*0

K'OPW KOPW

Further reduction of the Hamiltonian matrix element is tedious.

Nevertheless, it is worthwhile to exhibit the K-selection

rules which are operative.

Considering the first term in the parenthesis and sub­

stituting in E q . (5.1) for the Hamiltonian, we have

To proceed, we resolve the scalar product 2J*j in the

body fixed axes:

(5.15)

+ Zeft,<xK ,a t lxKa> + <XK'a' IEvi j *xKa> * I

2. RPC Matrix Elements

84

2 J ‘ j - J +11_ + + 2 J Z.J'Z» • ( 5 . 1 6 )

As mentioned earlier, the components of the total angular

momentum in the body frame satisfy commutation relations

appropriate for a rotating coordinate system. These commu­

tation relations, Eq. (3.11), interchange the familiar action

of raising and lowering operators. In the rotating frame,

J+ lowers and J_ raises the projection of the total angular

momentum on the z* axis (Da 65b):

The particle operators in Eq. (5.15) act on the particle

state X with the expected raising and lowering effect since

all particle quantities are defined with respect to the body

fixed frame. For a single particle state, we have

(5.17a)

(5.17b)

For brevity, we have introduced the notation

(JK)* = /(J±K)(JTK+1) = (J-K)T . (5.18)

(5.19a)

( 5 . 1 9 b )

85

and, in general,

J+ = 50 (JR)+aJS)±lajSl’ (5.20a)J M

V = (5.20b)

From E q s . (5.11b), (5.15), (5.16), and (5.17a) it follows that

RPC connects Ka~configurations having AK=0,-1:

^ M K ^ K ' a ’ I 2J ’ l<j)MKXKa> = 6K'K-1^J K <XK »a 1 3 - I xKa>

(5 .21 )

+ ^K'K+l^"71 <xK'o' U + lxKa> + 26K'KK <xK ,a»lxKa> *

An immediate corollary of this result is the RPC matrix

element appearing in the K-*-K portion of the Hamiltonian

matrix element given by Eq. (5.14). Since K*,K>0 by con­

vention, we find

<<J)MK'XK ra' I 2^ ’ <,)M-Kx-Ka>

^<SK'16K0 + 6K'l/26Kl/2 + 6K'06K 1 ^ J K <x K' a ' U + I x_Ka> *

(5.22)

86

This expression is immediately recognized as containing

the generalization of the decoupling parameter for a

rotator core with a single valence nucleon (Pr 62) to one

with several valence nucleons:^"

a <xl/2a’ lx-l/2a> * (5.23)

3. H Matrix Elements

With these key results, we are now in a position to

write down the Hamiltonian matrix element for a rotator core

with sgveral valence nucleons in a form completely reduced

with respect to the K quantum number. It remains only to

collect all terms in E q s . (5.14), (5.15), (5.21), and (5.22)

For K'<K (see below), the final result is given by

HK'a'Ka g6K ' K 6a •■</A J ^J+1 2AK +

1/2 , 1/2+st i + "g ] I-1 + § -I

x K *OPW x KOPW

* {<sK 'KA ^<xK ,a r I xKc> + / 6K » 05K0<xK ' a ’ I J lx-Kc>-'

+ 6K ,K l-<XK ,a' I Evij I xKa> + / 6K ‘ 06K0<xK ' a ' I E VIj I xKa>

brief comment about phases is appropriate at this juncture, Frequently one sees the decoupling parameter for a single valence nucleon defined by a=~<K=l/2|j,|K=-l/2>. In E q .(5 .23)5 the minus sign has been absorbed into our definition of X_j^a (cf. Eqs. (5.6b) and (5.6c). Recognizing this, our expression for the decoupling parameter Is precisely that of others for the limit of a single valence nucleon (Ni 55,Pr 62, Na 65).

37

6 6 < yK'O K1 AK'cr' IJ+ I ^-Kcr^

6K'l/26Kl/2 (5.2*0

It is worth noting that Eq. (5.2*1) includes a little

economy on the number of terms it is necessary to evaluate.

We have taken advantage of the fact that the Hamiltonian

matrix is a symmetric matrix. If the rows and columns of

the Hamiltonian matrix are arranged in an ascending sequence

of K-values, that is, then the matrix elements of

the upper triangle and diagonal of H are given by H > .

Equation (5.24) is in a convenient form for encoding for a

symmetric matrix diagonalization subroutine.

The g-factor appearing tv/ice in Eq. (5.24) provides a

formal statement of the fact that Kc=OPW wave functions exist

only for even values of the total angular momentum J, a prime

example of this being the ground state band of an even-even

nucleus (cf. Eq. (5.13)). In order to treat all Ka-configura-

tions on an equal footing, we have

where, for a lack of a better notation, <5JEy EN is taken to

rK ’a ' = OPW & Ka = OPW

g =< (5.25)

1 Otherwise

88

mean that the associated matrix elements are nonexistent

unless J is even.

4. Discussion of Eq. (5.24)

We can establish contact between these results and the

simpler versions of the unified model by reviewing the

diagonal elements of Eq. (5.24). This amounts to a first

order perturbative calculation of the spectrum of the

rotator core and valence nucleons:

E(JKo) = g{A[ J (J + 1) - 2K2 - <$K1/2(-l)J"m / 2 (J + 1/2 )a]

+ + A<J2> + <v>). (5.26)

The decoupling parameter a is given by Eq. (5.23) and the ,

expectation values’<j"2> and <v> are read from Eq. (5.24).

This expression is much less formidable to interpret.

It represents a straightforward extension of the familiar

results for a rotator core for an even-even nucleus and a

rotator core with a single valence nucleon for an odd-even

nucleus. In the first case, K is an integer so there is no

diagonal RPC contribution (cf. Chapter VIII). For the ground

state configuration of the neutrons, we have Ka=0PW and

J=0, 2, 4,... by virtue of the g-factor. In the second case

of an odd-even nucleus, g is unity and the RPC term contri­

butes to Ko-configurations with K=l/2.

89

In either case, rotational bands J=K, K+l,... are

built on each possible configuration of the valence nucleons.

The relative ordering of the band heads of different Ko-

configurations depends on two components. The first of these

is analogous to the shell model result for |mp-0h> excita­

tions. Just as In Eq. (2.26), it consists of the sum of the

single particle energies of the states occupied by the

valence nucleons plus the residual interaction energy of the

valence nucleons in these states. The second contribution

to the band head energy originates in the coupling of the

core and particle angular momenta.;.. Besides the decoupling

energy for K=l/2 configurations, there is in all cases an

additional one and two body interaction

which we refer to as particle-particle coupling (PPC).

5. PPC Matrix Elements

In most unified model calculations involving a rotator

core with a single valence nucleon, the one body operator

j is never evaluated in detail but rather is absorbed into

the single particle energies which are ultimately fit to the

spectrum in one way or another. Extending this philosophy

to systems with several nucleons, we would absorb the two

body portion of the j operator into the phenomenology for

the residual interaction.

This manner of dismissing PPC is unsatisfactory on two

accounts. In the first place, the residual Interaction would

not necessarily be a small perturbation. And, in the second

place, the single particle energies and residual interaction

would depend strongly on the Ka-configuratlon involved. Both

these points are born out by explicit evaluation of PPC. On

the one hand, we find that the magnitude of the one body por­

tion of PPC is a varying and sometimes sizable fraction of the

relative spacing between the single particle energies. While

on the other hand, we find that the magnitude of the two body

portion of PPC can in some instances be more than the decoupling-

parameter squared. In detail, this is established by expanding

the dot product in terms of the j+ operators and comparing the

outcome with Eq. (5-23).

For these reasons, we calculate the PPC terms in Eq. (5.24)

exactly. In general, PPC connects Ka-configurations with

AK=0.

It is Interesting to note in this light that recent micro­

scopic treatments of rotational motion are also beginning to

stress the role of PPC terms in the determination of collective

quantities like the moment of inertia of the core (Vi 70).

We pick up the ramifications of this and related points in the

Chapter VI where we discuss the parameters of the unified

model.

90

-*2

1. General Considerations

As we noted earlier, in evaluating the success of a

particular model, it is essential to test the model wave

functions as entities in their own right apart from the

eigenvalue problem because In that context the model

Hamiltonian and associated parameters are always tailored

to produce a gratifying spectrum.

Since the novel feature of our model is the appearance

of multi-particle excitations among the low-lying levels of

rotational nuclei, we would really like to investigate the

single particle character of the theoretical and physical

eigenstates. Towards this end, we turn to one and two

nucleon transfer reactions as the ideal mechanism for bring­

ing into focus the particle composition of the model wave

functions.

To see this, we consider the (d,p) and (t,p) reactions

which are most naturally disposed to our model. In these

reactions, neutrons impinge upon and are captured by the tar­

get nucleus thereby forming a..neutron enriched residual

nucleus in various excited states. There are two basic types

of states which can be excited in this process. Either the

bombarding neutrons collide with many nucleons in the target

and loose their energy to the target as a whole or they

scatter into the unoccupied orbitals of the core without break­

ing up the ground state configuration of the core. According

91

D. Reduction of Structure Factors

92

to these two idealizations of the reaction mechanism, the

final states are built up of many particle-hole or core

excited components or, alternatively, they are built up of

valence particle configurations. The excitation of the

final state in these two limits is thought of as proceeding

through a many or single step process,.respectively.

Clearly then, if the one and two neutron (d,p) and (t,p)

stripping reactions proceed in a single step they should po­

pulate many of the Ka-configurations of our model and in

doing so they should reveal the extent to which we may view

the nuclei of interest as rotator cores with one, two, and

three valence nucleons.

When a level of the residual nucleus is excited in a

transfer reaction, one can tell from the signature of the

cross section whether that particular state was reached

through a single or many step process. If the reaction pro­

ceeds in a single step, it is then possible to analyze in

detail the components of the final state which the reaction

fed. In this case, we expand the final state over all possi­

ble ways that the transformed cluster of nucleons can popu­

late the valence orbitals and vector couple to the ground

state of the target. In general, the wave function of the

final state can be written

y = e & a[y *yJLS]j + nR yJLS Y T YJ1j JR R

(5 .2 8)

93

where Y is the wave function of the target nucleus,T

^yJLS describes the motion of the captured nucleons in the

valence orbitals, and Y' gives that portion of the finalR

state which contains excited state components of the target

which cannot be reached in a single step reaction. The

momenta and Internal structure of the cluster are given by

L, S, J and y. The index y indicates, in other words, which

orbitals the transferred nucleons occupy. The total angular

momentum of the target ground state and the cluster J are

coupled to the final state value JR . The operator A Implies

that all valence nucleons outside the chosen core (not nece­

ssarily the target nucleus) are antisymmetrized.

The spin states which can be excited by the transfer of

a cluster of nucleons to the target are given by

where J is the vector sum of the spin of the cluster and its

orbital angular momentum relative to the target. For the

reactions of interest,

In the triton, the spin function for the two neutrons must

be antisymmetric; hence S=0. In both case, the parity of.

J R = J T + J (5.29)

rL + 1/2 (d,p)

(5-30)

L (t ,p)

94

the final state is given by n:-(-l)B (G1 63)In general, the cross section for a transfer reaction

is a superposition of all possible multiple step events.

If the single step process dominates to the exclusion of

all others then the cross section is a function only of the

final state components 3 TT0 which describe the system as aYU Lo

cluster orbiting about the target nucleus (G1 63):

, 2J +1 b 2 2dft = N(2J t + 1 )LSj M 2S+t I 2ByJLSB y Lm I * (5.31)

Here is the amplitude for the. absorption of the cluster

and N is a density of final states factor for the transmitted2

particles. For the (d,p) and (t,p) reactions, bg equals

l/26gi/2 and respectively.

Detailed calculations of the amplitude show that it has

a pronounced dependence on the value of the orbital angular

momentum L transferred by the cluster. Furthermore, and

significantly, low L transfers are favored. In many instance

a final state is populated by a definite L transfer which can

readily be determined from the shape of the cross section.

2. (t,p) and (d,p) L-Transfers and a Strong-CouplingSelection Rule

In our model, there are only a few values of L which can

be expected. Since stripping to the s-d shell can excite

positive parity states only, L is restricted to the values

0, 2 and 0, 2, 4 for the (u,p) and (t,p) reactions respec­

tively. In general, several L transitions may occur to a

given final state. The possible transitions to final states

In the nuclei of interest are listed in Table (5.1) and (5.2).

Our prediction of specific Ka-configurations for these levels

determines which deformed orbitals are occupied and may

further restrict the allowable L transfers. This is because

there is a selection rule arising from the assumption of a

unified model-characterization'of these-nuclei . The magnitude-

of the transferred angular momentum must be least its pro­

jection on the symmetry axis:

J * | f t | = |Kt - Kr |. (5.32)

Here ft is the sum of the ft of the deformed orbitals occupied

by the stripped particles. This rule is exploited later.

In some situations it eliminates low L transfers to Ka-

conf igurations with large K values. (The origin of this

selection rule is evident in our derivation of Eq. (5.44a)).

3. Spectroscopic Factor for (d,p) Stripping

Because of the coherent sum in E q . (5-31) over the

occupied orbitals, the (t,p) reaction can lead to valuable

Information about the relative phases of the Ka-configurations

in our model wave functions. However, the calculation of the

cross .section for this reaction is beyond the scope of our

95

96

Table (5.1)

Possible Final State Spins J . and L-Values

for (d,p) and (t,p) Transitions

To Ne22 and Ne23

M 22Ne Ne23

J/ L L„ j „ Lf n 2n f n 2n

0 2 0 1/2 0 2

1

C\Jo

- 3/2 2 0,2

2 0,2 2 5/2 2 2,4

3 2 - 7/2 - 2,4

4 2 4 9/2 - 4

11/2 — 4

97

Table (5.2)

Pos sible Final State Spins and L-•Values

for (d,p) and (t ,p) Transitions

To Mg26 27and Mg

Mg 26 Mg27

Jf Ln L2n J fLn L„2n

0 2 0 1/2 0 2

1 2 - 3/2 2 2,4

2 0,2 2 5/2 2 0,2,4

3 0,2 - 7/2 - 2,4

4 2 4 9/2 - 2,4

5 2 — 11/2 — 4

1 3 / 2 - 4

98

immediate objectives. Since very little (t,p) work has

been reported on our nuclei, vie hold our discussion of the

(t,p) reaction to predictions given by the above selection

rules.

The (d,p) reaction, on the other hand, is a popular

experimental tool for assigning spins and parities. In this

case, a single neutron is transformed to the target. Inas­

much as there is no cluster structure to worry about, vie

loose the coherence effect mentioned above. The advantage

gained is one of greater simplicity. Now the essence of

the reaction cross section

, . 2J +1dft = N 2J,r+l £SL aL (5.33)

is just its dependence on the orbital angular momentum trans­

fer. And, as previously mentioned, one L transfer frequently2

dominates all the others. Here 3 ^ = 2 3 ^ and is the intrin-J

sic single particle cross section (Sa 58). As vie shall see,

the spectroscopic factor expresses the degree to which

the residual nucleus can be regarded as a single nucleon in

a particular orbit outside the target nucleus.

We now derive in the framework of our model the spec­

troscopic amplitude Bj^ for (d,p) stripping. From Eq. (5.28)

we have

In evaluating this expression, we adapt the method of G.R.

Satchler (Sa 58) to a rotator core with several valence

nucleons.

Being eigenstates of the strong-coupling Hamiltonian,

the initial and final state wave functions are superpositions

Y(JMA) = Z T. „ Y(JMKa) (5-35)Kc X ’Ka

of strong coupling wave functions corresponding to different

Ka-configurations. The quantum number X distinguishes the

different eigenvectors of the Hamiltonian matrix. It then

follows from Eqs. (5*34), (5.35), and Section (V.C.l) that

the heart of the spectroscopic amplitude lies in the matrix

element

< MRKRXKRaR (m+1} I aJLM I (},M^KTxKTaT (m} > * (5.36)

Since the intrinsic states Xj(CT describe the motion of

nucleons in the principle axis frame z * , we transform the

transfer operator a^ from the laboratory frame to the body

frame and express it in terms of transfers to deformed or­

bitals. This yields

100

a Ti m - r D L a i ? „ - X d J„ c ® bl, (5.37)JLM " MS] JLfi " Mfi vj fivft ftv

with the aid of Eq. (5.9). Substituting this into Eq. (5.36)

and integrating over the Euler angles with the outcome

/deiIV R ( 18 J DM « ( 0i )DV t ( 6i > = 1 JRM R><JTJKTn 1 W

(5.38)

we ■ find

< / * x Ia+ l / T X ’ >=RKR RaR J L M 1VMT KT xKT aT

2j r + i <JTJMTM I JRMR ><JTJ KT ^ JRKR >cvj<XKRaR lbftv I XK^,c^1> C5 - 39)

with the following provisions regarding 9, v and phases.

The second vector coupling coefficient implies

fl=KR-K,p. The object < X ’(m+l)|b | X(m) > is essentially a

6-function with a phase as can be verified with the second

quantized formalism of E q . (5.8). It formally states that

in a single step (d,p) reaction, the final intrinsic state

cannot differ by more than one occupied orbital from the

initial intrinsic state. This one extra orbital is the one

populated by the stripped neutron and is denoted by v. In

101

a symbolic sense, v=cR -a,p. The phase associated with the

6-function depends on the relative ordering of the creation

operators in the initial and final Intrinsic state. The

determinant whose elements are 6-functions of the individual

ftpVp quantum numbers appearing in the (m+1)-particle wave

Substituting E q . (5-39) into the Appendix D expression

for matrix elements in a strong coupling representation, we

find, after collecting terms, that the spectroscopic ampli­

tude for (d,p) stripping into rotational levels built on

pure Ka-configurations can be written

Hamiltonian matrix element.

In a moment, we demonstrate hew this expression for

8jP reduces in special cases to the familiar results of

1*complete reduction of<x'lb |x> yields an (m+1) by (m+1)

+functions x* and b X* Tbe values of this determinant is 0

or -1 according to the details of the overlap.

1/2]

k d o p wA

KT 0PW

(5.40)

which, structurally, is identical to Eq. (5.14) for the

102

Satchler. Before doing so, we note that, in actuality, the

model wave functions are mixtures of Kcr-configurations .

From E q s . (5.34) and (5.35), it follows that the amplitude

for stripping into eigenstates of the Hamiltonian is

6J L (JR XR ;JT XT ) I I R X K o ^ A X o . p ^ L ^ r V r ^ t V t ^R R TaT 1 ‘ R R 1 1 1

(5.41)

Thus, In dealing with several valence particles, unlike

Satchler, it is essential to start from Eq. (5.40) to assure

a proper accounting of the relative phases arising from the

overlaps between various KRoR and configurations.

In this vein, we add that according to Eq. (5.33) for

the (d,p) cross section, the relevant physical quantity to

be calculated is the spectroscopic factor for the (d,p)

transition from the target ground state, J , gs, to final

state, J,-,!,;,, of the residual nucleus:A A

2SL (JTgs+JR XR ) = z|BJ L (JR XR ;JTgs)| . (5.42)

J

It is clear from E q . (5-41) that the spectroscopic factor SR

involves a coherent admixture of the components of the model

wave function. As such, it should provide a demanding test

of the configuration mixing in our model.

To derive Satchler's results, we'examine the two

determinants <x|<[b ^IXj_> appearing in Eq. (5.40). As

discussed earlier, these objects are 6-functions (with a

phase) which vanish if the initial state differs from the

final state by more than one occupied orbital. There are

three cases of interest: (1) Kd O^/OPW and KmOm^OPW (2)n n 1 1

V t -OPW and KR aR=( P W * the tarSet and residual

nuclei were given by pure Ka-configurations, these last two

cases would correspond to stripping on an even-even target

and stripping into the ground state band of an even-even

residual nucleus.

If neither KRcR nor K^a^ is OPW, then only one of the

determinants, <XKr 0r I b ^ I or <XKr 0r lb*J x _V t >

may be non-vanishing. If either KRaR or K^a^ is OPW, then

the two determinants are not distinct, according to Eq. (5.12)

Hence, to within a phase, we can write Bj L for the three

cases as does Satchler:

103

4. Discussion of E q . (5.40)

p 2JT+13Jl/JRKR 0R ;JTKTaT') ~ SKRaR ;KT aT 2JR+ ^

< J T j i K T n I J R K R > C v J < X K r 0 r l b L I X ± V t > < 5 • 4 3 a )

where ft=K^+KT and

104r

/2 [ ^ - 2 ----- ] KTaT = OPW

RUR ’ TUT(5.43b)

1 Otherwise

with recourse to Eq. (5.6c) as necessary. The g-factor re­

flects both the 5 Qp ,r normalization for Ka=0PW states and

the fact that these states exist only for even angular

5. V/orking Assumption

In our calculation of the spectroscopic factor for (d,p)

stripping on an even-even target, we make the working assump­

tion that the target ground state is a single Ka=0PW con­

figuration. In this case we can use the simpler result for

3j L given by Eqs. (5.43) with (-l)p=l and JT =0.

E . Summary

In this Chapter, in retrospect, we have developed the

formalism necessary for unified model calculations for a

rotator core with several valence nucleons. Specifically,

we have derived the Hamiltonian matrix for the eigenvalue

4*As a passing remark vie mention that in deriving a sum rule from Eqs. (5.43) for stripping into the ground state band of an even-even nucleus Satchler overlooks the fact that Jp must be even and as a consequence overestimates the sum rule strength by a factor of two.

momenta.t

problem and the spectroscopic factor for testing the model

wave-functions. We have shown how these results reduce to

the more familiar results for a rotator core and a rotator

core plus a single valence nucleon.

Before we can exercise this model on real nuclei, we

must extract from the non-local single particle problem,

defined by Eq. (5.4), the valence space orbitals and energies

for valence particles moving in an average deformed field.

In the next Chapter, we direct our attention to this problem.

105

1 0 6

A . Introduction

In the previous Chapter, we set up the Hamiltonian matrix

for the unified model description of nuclei in terms of a

rotator core with several valence nucleons. In summary, the

Schroedinger equation for the system is given by

HY = EY (6.1a)

where

H = A(tf-J)2 + Eh, + Ev, . . (6.1b)i i< j

We have assumed since Chapter II that the space of valence

orbitals is defined by the single particle Hamiltonian h

with

hxftv = eftv xftv# (6.2)

hence, before we can diagonalize the Hamiltonian matrix for

the coupled system, we need as input the single particle

energies and wave functions of h. Furthermore, we must

decide on the value of the reciprocal moment of inertia A

of the core and the nature of the effective interaction

Vpj acting between the valence nucleons.

Chapter VIDiscussion of Parameters

107

B. Single Particle Energies and Wavefunctions

1. General Considerations

In Chapter II, we discussed the basic program for

nuclear structure calculations which is implied by the

many-body Schroedinger equation. In one respect, a re­

markably simple picture of a very complicated system

emerged: we may replace the many body system by a fictitious

model nucleus conceptually divided into core and valence

nucleons. From the primeval matrix element of the many body

Hamiltonian, H ^ given by Eq. (2.4), we extracted an effec­

tive Hamiltonian, Hv given by Eqs. (2.17) and (2.23), which

operates only on the m valence nucleons outside the core.

From these considerations emerged the fundamental shell model

principle: The valence nucleons move in the average field

generated by the core nucleons.

The not so simple aspect of this simple picture is that

the single particle Hamiltonian h describing the motion of

the valence nucleons is the non-local Hartree-Fock Hamiltonian

of the core, Eq. (2.18). As such, E q . (6.2) for the single

particle energies and wave functions defines an eigenvalue

problem of substantial computational complexity. It entails

solving the integrodifferential equation, E q . (2.25), of the

Hartree-Fock problem for the core.

Vie re we dealing with the usual shell model for spherical

nuclei, we could reasonably avoid this labor. We would take

the single particle energies from that nucleus which has a

1 0 8

single valence nucleon outside the spherical core of interest—

assuming that the lowest levels of this nucleus have pure

single particle strengths. The wave functions corresponding

to the tightly bound states, are, to a reasonable approxi­

mation, states of the appropriate angular momentum in a

harmonic oscillator or Woods-Saxon potential (Ba 69). Alter­

natively, if we did not wish to specify the potential we could

use the method of I. Talmi for parameterizing the energy levels

in terms of various radial moments of the effective interaction

(cf. Eq. (2.31)).

Unfortunately, where deformed nuclei are concerned, we

cannot so evade the Hartree-Fock problem for the single par­

ticle energies and wave functions. One reason springs from the

fact that the average field of the core is non-spherical. The

single particle wave functions Xq v are, as a consequence,

admixtures of shell model states of good angular momentum:

Xn = j <j> j n • ( 6 . 3 )Aftv j vjYjft

The expansion coefficients c ^ depend in detail on the nature

of the deformation produced by the internucleon forces. Not

only this, in addition, the single particle strengths are,

in general, smeared out over the physical spectrum by the

rotator-particle (RPC) perturbation. Therefore, since we can

neither read the from neighboring odd-even nuclei nor use

any elementary wave functions for x^v we must turn to a more

detailed analysis of the single particle problem for the

motion of particles in a deformed field.

The quest for quantitative information concerning deformed

orbitals has progressed through three distinct stages. At

the start, with the introduction of the unified model by A.

Bohr in 1952, deformed orbitals were approximated by states

of good j (Bo 52). A few years later, in 1955, S.G. Nilsson

proposed a local approximation to the Hartree-Fock Hamiltonian

(Ni 55). The Nilsson model for h is much like that used for

the spherical shell model, given by E q . (2.27), but of course

a deformation parameter is introduced. The deformed orbitals

of this model have proved to be a vast Improvement over pure

j-states. However, it was not until 1963 that the theoretically

crucial breakthrough was achieved. Finally, the microscopic

origin of deformed orbitals was revealed in the restricted

Hartree-Fock solution of Eq. (6.2) by C.A. Levinson and I.

Kelson (Ke 63a, Ke 63b).

Under normal circumstances, we would, without further

comment, take the Hartree-Fock solutions as input to our

model calculations. However since virtually all other unified

model calculations adopt .the Nilsson approximation we feel it

worthwhile to comment further on these two phenomenologies.

2. The Nilsson Model

The Nilsson model Hamiltonian

109

h = | m-^2r2 (l-28Y2O(0,<j>)) + cl-s + dI ’2 (6.4)

1 1 0

defines the motion of nucleons in a spheroidal harmonic

oscillator potential. The equipotentials of the potential

are congruent to the quadrupole surface deformation which

is given by

R( 0, «*.) = R0( l + BY20(e,<fr)). ( 6 . 5 )

The magnitudes of the deformation is given by 3. Prolate

deformations, such as that depicted in Fig. (3*1), correspond

to 3>0 and oblate deformations to 3<0. The spin-orbit and

well flattening parameters, C and D, are adjusted to reproduce

the observed shell model spacings for 3=0 nuclei. Some of the

effects of non-locality are simulated by an effective mass

parameter m * . The frequency w is related to the RMS radius1/0

of the nucleus. This is usually stated as fta)=4l/A MeV.

The Nilsson single particle eigenvalues and eigen­

functions Xq v are constructed by diagonallzing h of Eq. (6.4)

in the shell model basis space The deforming action2

of the r Y20 perturbation mixes states with different angular

momenta and AN=2 where N is the principle quantum number.

Usually the mixing is confined to a single major shell.

The behavior of the resulting deformed orbitals is well

illustrated by a Nilsson diagram. Figure (6.1) demonstrates

the splitting of the shell model energy levels in the s-d

shell as the deformation increases. The deformations are

frequently quoted in terms of the spheroidicity parameter

Ill

Fig. (6.1) Nilsson Diagram for the s-d Shell. Thisfigure schematically illustrates the behavior of the eigenvalues of the Nilsson Hamiltonian (v. Eq. (6.4)) as a function of the quadrupole (A=2) deformation of a spheroidial harmonic oscillator potential. Each orbit is doubly degenerate corresponding to Jz'=±K. Nilsson's designation of the orbits is indicated on the left side of the diagram; our notation for these orbits on the right.

1 1 2

- 77-

u lT NOTATION * H E R E IN

Fig. (6.1)

6=/ij5/16tt 8=.953. The wave functions Xfiv= X^v(6) are also

functions of the deformation. The Nilsson coefficients

c ^ ( 6 ) are easily calculable. They are also readily accessi­

ble in published tables (Ch 66, Ne 60).

Each Nilsson orbital is doubly degenerate corresponding

to projections j_,=-ft. The exclusion principle thus allowsztwo neutrons and two protons to be placed in each orbit.

Theoretically, the value of the deformation of an odd-even

nucleus is determined by minimizing, with respect to 3, the

total energy of all the nucleons. The ground state spin of

the odd-even nucleus should also be predicted by this proce­

dure. It is given by the value J=|ft| of the last orbit which

is occupied. For an even-even nucleus, the ground state spin

is J=0.

Again and again the Nilsson model for h has convincingly

explained all manner of single particle systematics in deformed

nuclei. Because of its repeated successes and underlying

simplicity, the Nilsson model has become the standard tool for

investigating single particle properties in the framework of

the unified model.

Still, it is worth stressing that the fullest realization

of the eigenvalue problem for h involves achieving a measure

of self-consistently between the single particle potential

and orbitals as is implied by the Hartree-Fock nature of the

original problem. The Hartree-Fock solution of Eq. (6.2)

yields single particle energies e^v and wave functions xfiv

113

much like selected examples from the Nilsson model. Despite

this overlap, the reason Hartree-Fock quantities have not

been employed in unified model calculations certainly must

reflect the fact that they have been enshrouded in a more

elaborate mathematical and computational scheme. Unfortun­

ately, this has long deterred a desireable evaluation of the

Nilsson phenomenology. We propose below a more fundamental

phenomenology for deformed orbitals which is based on the

Hartree-Fock solutions of E q . (6.2).

3. The Hartree-Fock Model

The Hartree-Fock problem is defined by E q s . (6.2) and

(2.25). It may be cast into a matrix diagonalization for the

energies and coefficients c^j of x^v (R i 68). In general,

this wave function should be expanded over a complete set of

basis functions. The practical expendient is to expand x^v

only over shell model states in a major shell - the same

method of operation as is adopted in the Nilsson model. The

lower shells are then treated as comprising an inert core.

This approximation constitutes the so called "restricted

Hartree-Fock Model". It differs from the general solution In

that no radial variations arising from contributions from

other shells are included In the solution for x^ • TheAftvpresence of the Inert core is acknowledged phenomenologically

by taking its single particle energies from experiment. This

is identical to the shell model recipe discussed in Chapter II.

114

This phenomenology is also no different from that employed

in the Nilsson model for h; in that approach, the parameters

C and D are fixed to reproduce the spacing between the shell

model states at zero deformation.

The final step in the Hartree-Fock reduction of Eq.

(6.2) is the introduction of an effective interaction between

the extra-core nucleons. Typically, and primarily for reasons

of mathematical tractability rather than any compelling

physical argument, the effective interaction assumed is that

given by a Rosenfeld mixture with a simple radial dependence

and an overall strength V q :

V2 = j V 0 (t 1 *t 2 )(0.3 + 0.7o1 -a2 )f(r1J/a). (6.6)

The radial part of the interaction is commonly of a Gaussian

or Yukawa form with a in the range 1.4 to 1.5f*

The remarkable conclusion of the Hartree-Fock analysis

is that deformed orbitals x^v appear as a direct consequence

of the Interaction between the extra-core nucleons and pre­

sumably all other nucleons, were the calculation not restricted

to a major shell.

4. Comparison of Both Models

Putting aside, momentarily, important quantitative

differences between the Hartree-Fock and Nilsson solutions,

we first remark on a notable qualitative distinction between

115

the Hartree-Fock and Nilsson phenomenologies. Treating

V Q as a parameter, one can make a Nilsson diagram out of

the Hartree-Fock solutions (Ke 63a). It would resemble

Fig. (6.1) with V Q fulfilling the role of the deformation

parameter 6 in the conventional Nilsson diagram. In the

Hartree-Fock treatment, however, V q is not a free parameter

as is 6 in the Nilsson model. The strength V q of the effec­

tive interaction is fixed for the entire shell by recourse to

binding energy arguments (Ri 68). The implications of this

development are not widely acknowledge in unified model

calculations. It leads us to conclude that the deformation

cannot be an available parameter In the unified model des­

cription of a given nucleus if the description is to be based

on first principles. This is not to say that the deformation

is constant throughout the shell because V Q is constant

throughout the shell. On the contrary, the deformation

associated with the Hartree-Fock wave functions depends not

so much on V q as it does on the total number of extra-core

nucleons. Thus each nucleus having a different number of

nucleons has in turn a different equilibrium deformation. As

a qualification to these remarks, we add that the Hartree-Fock

single particle energies and wave functions are derived self-

consistently only for the ground state of the system. The

detailed properties of the Hartree-Fock solution depend upon

which orbitals are occupied. Hence it is reasonable to expect

that a slightly different deformation might be appropriate for

116

It is instructive to compare the intrinsic structure

of the deformed orbitals calculated in the Nilsson and

Hartree-Fock models. In Table (6.1) we present five differ­

ent sets of c^j coefficients and single particle energies

for the six deformed orbitals in the s-d shell. The first20three columns correspond to Hartree-Fock solutions for Ne

with three different effective interactions; the first two

are of Yukawa and of Gaussian radial dependencies and the

third is based on the reaction matrix (cf. Chapter II). The

last two columns in Table (6.1) contain the Nilsson quantities

for deformations 6=0.1 and 6=0.2.

To facilitate comparing any two of these sets of de­

formed orbitals, we calculate the six possible overlap

integrals.between the six orbitals in both sets. The smallest

of the resulting numbers vie call the index of comparability

(IC) for the two sets of deformed orbitals. A high IC implies

that the intrinsic structure of two sets is similar. As an

example, the IC of the Nilsson orbitals for 6=0.1 and 6=0.2

is 92%. Specifically, 0.92 is the value of the overlap

integral for the two K=l/2' orbitals. Taking this as a guide,

we adopt 92% as a reference value indicating that two sets

of orbitals correspond to different deformations or are in­

trinsically different in some other respect. In general,

the IC is representative of the overlap integral for one of

the three orbitals K=l/2, 1/2’, and 1/2". These orbitals

117

excited configurations.

1 1 8

Table (6.1) "Nilsson Coefficients" c^i for the Intrinsic States Xftv=£c$j<J>jq of Ne<0. The first three columns correspond to different choices of the effective interaction for the Hartree- Fock solution of Ne20 (cf. Caption of Fig.(6.2)). The absolute value of the Hartree- Fock single particle energies is given on the same line as the orbit designation. The 4th and 5th columns give the Nilsson wave functions for deformations 6=+0.1 and 6=+0.2 (Ch 66).

119

Reaction

Table (6.1)

Yukawa Gaussian Matrix' 6=. 1 6

K=3/2' -1.00 -.25 + 3.81

d 3/2 0.993 .993 0.985 0-992 0d5/2 0 .122 .117 0.169 0.125 0K=l/2" -3.19 -2.35 + 1.29

d 3/2 0.901 -0.888 0.831 0.895 0sl/2 0 .400 0.435 -0.537 -0.440 -0d5/2 0.167 -0.148 . 0.100 -0.061 -0

K=5/2 -5.70 -5 .14 -2.55

d5/2 1.000 1.000 1.000 1.000 1

K=l/2' -6.25 -5.19 -1.48

d3/2 0.187 -0.252 0.391 0.424 0sl/2 -0.706 -0.730 0.696 0.806 0d5/2 0.684 -0.636 0.593 0.413 0

K=3/2 -7.25 -6.58 -4.83

d3/2 -0.122 0.117 -0.169 -0.125 -0d5/2 0.993 -0.993 0.985 0.992 0

K=l/2 -16.79 -14 .58 -12.26

d3/2 -0.391 0.385 0.393 -0.132 -0sl/2 0.585 0.527 0.457 -0.395 -0d5/2 0.711 -0.758 -0.797 0.908 0

Note : When comparing the wave functions, the eigenvectorsof different authors may differ by an overall phas eand/or the phase of the sl/2 component to complywith their definition of the Laguarre polynomials.

= .2

.979

.202

.694

.687

.209

.000

.677

.529

.510

. 2 0 2

.979

.240

.496

.834

respond most sensitively to the shape of the average field.

(We recognize throughout this discussion that overlap

integrals are deceptive objects and can mask important

variations in individual components of the wave functions.)

Returning now to Table (6.1) we present several im­

pressions. Even though a different effective interaction is

used in each of the Hartree-Fock calculations, the Hartree-

Fock wave functions are virtually imperturbable. Indeed,

in the sense defined above, the IC for different pairs of

Hartree-Fock solutions are 97%, 98%, and 99%. At the same

time we note that the Hartree-Fock and Nilsson deformed or­

bitals while similar in special instances in general possess

different intrinsic structures. The IC between the Hartree-

Fock solutions and Nilsson functions for 6=0.1 and 0.2 all

hover about 93% and 85% respectively.

In Fig. (6.2) we plot the Hartree-Fock single particle

energies associated with the deformed orbitals in Table (6.1).

Here we see a greater sensitivity to the choice of the

effective interaction; it affects the position of the single

particle spectrum as a whole. For our purposes, we focus on

the relative energy spacing of the deformed orbitals. In Table

(6.2) we give the Hartree-Fock and Nilsson single particle

energies relative to the K=3/2 level, the first unoccupied PO

level in Ne . It is clear between Fig. (6.2) and Table (6.2)

that a huge 7-9 MeV gap prevails between the K=l/2 and K=3/220orbitals in the Hartree-Fock solutions of N e ' . The Hartree-

1 2 0

1 2 1

20Fig. (6.2) Hartree-Fock Single Particle Energies for Ne The three sets of single particle energies correspond to three different choices of the effective interaction: (a) Rosenfeld mixturewith Yukawa radial dependency (Ke 63a), (b) Rosenfeld mixture with Gaussian radial depen­dency (Ri 68), and (c) Reaction matrix (Pa 67).A gap between the occupied and unoccupied or­bitals In Ne20 occurs in each case. The relative ordering of the single particle energies depends on the choice of the effective inter­action .

1 2 2

HARTREE-FOCK SINGLE PARTICLE ENERGIES

FOR N e 20

MeV

5

- 5

-10

-15

-20

3 / 2 '

1/2

3 / 2

1/2

-5/2 -I/2'/-1/2'-" 3/2• 3 / 2

1/2

3 /2

1/ 2 "

1/ 2 '1/2" / ' ------------------- 5 / 2

> 5 / 2 ^ / 3 / 2

1/2

(a) (b) (c)

Fig. (6.2)

123

Hartree-Fock Single Particle Energies for Ne

For Yukawa (Ke 63a), Gaussian (Ri 68), and

Reaction Matrix (Pa 67) Effective Interactions,

and Nilsson Single Particle Energies, Based on

Table (6.2)

20

ReactionYukawa Gaussian Matrix 6=.1 <S = .2

e3/2, 6.25 6.33 8.64 4.80 6.00

e1/2„ 4.06 4.23 6.12 3-75 4.65

e5/2 1,55 1,i44 2,28 1,20 2,70

e1/2, 1.00 1.39 3.35 1.80 1.95

e3/2 0 0 0 0 0

£l/2 -9-54 -8.00 . -7.43 -1.05 -2.40

124

Fock gap is a consequence of the non-locality of the average

potential generated by the core nucleons (Ri 68). The

absence of a corresponding gap in the Nilsson energies re­

flects the fact that the Nilsson model approximates the

average potential by a local potential (cf. E q s . (2.27) and

(6.4)).

It is the existence of this gap which lends credence to 20our choice of Ne as a good core for our unified model cal-

20culations. As the gap implies and as the spectrum of Ne20confirms, see Fig. (4.1), Ne is very stable against low-

lying particle excitations. Furthermore, the close spacings

of the unoccupied levels above the gap indicate that valence20nucleons outside the Ne core are easily excitable. It is

22our contention that many of the low-lying levels in Ne ,

for example, are based on just such excitations.

The Hartree-Fock gap also appears between the occupiedp h p O

and unoccupied orbitals of Mg and Si (Ke 63a, Ke 63b,

Ri 68). These gaps are not as large as that encountered in 20Ne but are still sizable (5-7 MeV) compared to the relative

spacing between the valence orbitals. For this reason, we

also assume that these nuclei make stable cores to which may

be appended a few active valence nucleons.

It should be mentioned that the Hartree-Fock gap in24Mg is most pronounced for the axially asymmetric solution

(Ba 65). Inasmuch as experimental evidence is not at vari­

ance with the symmetrical interpretation (Ro 67), we shall

use the corresponding Hartree-Fock solution. Furthermore,

for our purposes, the differences between the of the

symmetric and asymmetric wave functions is not an important

effect.p O

The situation regarding Si is more problematic. In

Appendix C, we show that the Hartree-Fock solution of Eq.

(6.2) minimizes the total energy of the many-body system if

one chooses a Slater determinant for the trial wave function

of the variational calculation. In terms of the total energy

of the system, the prolate, spherical, and oblate Hartree-

28Fock solutions for Si are nearly degenerate. Consequently,

2 8it is not clear what the equilibrium shape of Si is.

Experimental evidence indicates an oblate deformation (Na 70);

however, as we shall see In Chapter VII the oblate strong-29coupling description of Si is less than an adequate proposi­

tion. Our interpretation of this result in Chapter VIII is

relevant to the Hartree-Fock degeneracy mentioned above.

5. Relevant Single Particle Parameters

Another aspect of the Hartree-Fock theory is evident

in Table (6.2). As we have already pointed out, the Hartree-

Fock wave functions x^v are almost completely stable with

respect to the effective interaction used. On the other

hand, the associated single particle energies are more

responsive to the choice of the interaction. Of particular

interest in this respect Is the relative ordering of the

125

126

K=5/2 and K=l/2* levels (see Fig. (6.2)). Contrasting

this development is the Nilsson model wherein the x^v

and are strongly correlated. There too the relative

order of the z^/2 and el/2 levels maY change-but if

it does there is an accompanying change in the deformation

and the wave functions. From our Hartree-Fock review the

strongest conclusion we can draw regarding the single par-

20t i d e energies in Ne is their ordering

el/2 < e3/2 < e5/2 ~ el/2' < el/2" < e3/2' (6,7)

and their spanning of a 6-8 MeV energy range.

It seems that the volatility of the Hartree-Fock energies

and the constancy of the wave functions x^v are persistent

features of the general Hartree-Fock method. These charac­

teristics prevail even when one examines a fuller spectrum of

Hartree-Fock style calculations, for example those based on

(1) Hartree-Fock-Bogoliubov theory-which is Hartree-Fock theory

with an effective interaction with pairing forces (Sa 69a,

Sa 69b), (2) Hartree-Fock theory with Coulomb and center of

mass corrections (Gu 68), (3) Hartree-Fock theory with a

Woods-Saxon basis expansion for the x^v (Bo 69), and an endless

host of other variations on this theory.

One new proposal, however, does have a particular re­

levance to our unified model calculations. As mentioned in

the introduction, F. Villars has recently derived the strong-

coupled form of the unified model Hamiltonian from a micro­

scopic starting point (Vi 70). Villars proposes that the

Hartree-Fock variational calculation also include the PPC

terms, (Zj)2 of Eq. (5.27). Then,

hVillars h + A(^ i )2 • (6-8)

From our general considerations, we are led to expect that

this will only renormalize the single particle energies. The

wave functions should remain unperturbed - certainly as

measured by the 92% IC criterion established earlier.

This completes our brief survey of Hartree-Fock syste-

matics in s-d shell nuclei. In view of our observations, we

feel that the relevant set of parameters for unified model

calculations are the single particle energies We intend

to vary these about their Hartree-Fock values. The appropriate

set of wave functions for our calculations are those given by

any Hartree-Fock solution of the rotator core under considera­

tion. In a certain sense, we have returned to the standard

spherical shell model phenomenology of predefined orbitals

and adjustable single particle energies.

Before leaving the problem of the deformed orbitals, we

wish to reemphasize that the deformation is not a parameter

in our calculations. For a rotator core with several valence

nucleons, the implied deformation is that associated with the

Hartree-Fock field of the core nucleons. Though we never

127

need to say precisely what this deformation is, it does

provide an intuitive characterization of the deformed or­

bitals. Earlier we showed that the Hartree-Fock and Nilsson

wave functions are not always comparable deformationwise.

What is needed is a deformation index for the Hartree-Fock

potential. Such an index has been proposed very recently

(St 69); but, it has not yet been evaluated numerically.

Thus we turn again to the Nilsson model as a rough guideline

for interpreting Hartree-Fock solutions. Guided by the IC

overlap criterion discussed earlier, we say that the Hartree-

20Fock wave functions for Ne correspond to a 6=0.1 deformation.+

Small changes in the Hartree-Fock parameters (other than V Q )

can bring this to being although we recognize that there is

no compelling reason to do this. In the same way, we say that

24the Hartree-Fock wave functions for Mg correspond to a

6=0.2 deformation. Finally, the prolate, spherical, and28oblate wave functions for Si correspond to deformations

6=+0.2, 6=0.0 and 6=-0.2 respectively.

C. Moment of Inertia Parameter A

So far, we have considered the average field of the core

only from the point of view of the single particle spectrum

it generates for the valence nucleons. In the strong-coupling

limit of the unified model the average field is permanently

deformed and thus has its own rotational degree of excitation.

This rotational mode is characterized by the moment of inertia

+ 20 Calculation of the intrinsic quadrupole moment of Ne usingthe aforementioned Hartree-Fock wave functions implies thatthe "physical" deformation of Ne^O (i.e. 6=AR/R) is +0.3.(Ri 68).

parameter A=h /2I which appears in the unified model

Hamiltonian, Eq. (6.1b).

The microscopic origin of the moment of inertia I

seems more elusive today than it ever has been. Several

theories for the calculation of the moment of inertia have

been advanced over the past two decades, beginning with the

cranking model of D.R. Inglis (In 53). Various refinements

of, and variations on, this first theory have appeared. Most

of the microscopic calculations of I involve a variational

procedure which is intimately related to the Hartree-Fock

problem for the single particle energies and wave functions.

For this reason we have tried to understand how the moment

of inertia depends on the Hartree-Fock parameters discussed

in the preceding section.

At first, it seemed that the lowest order cranking model

result was appropriate for this purpose (Ke 63b):

I = 2h2 Z l< Q lJxlp > 1 . (6.9)o,y ea "

But, very recently, it has been shown that this expression-

and its Thouless-Valatin modiflcation-are of dubious validity

(Ke 67, Gu 68). In Eq. (6.9), the sum extends over the

occupied and unoccupied orbitals of the core. The single

particle energies of these orbitals are denoted by eQ and

ep , respectively.

1292

130

Since the moment of inertias predicted by the other

theories are too small as compared with the experimental

value given by the AJ(J+l)-rule (Gu 68), we must take the

experimental value of A from the appropriate core nucleus

for our unified model calculations. We assume that the

quantitative nature of the deformed orbitals which we use

is consistent with the eventual theoretical calculation the

moment of inertia. Noteworthy in this regard are the PPC

contributions to the moment of inertia proposed by F. Villars

in his new theory of the unified model Hamiltonian (Vi 70).

It remains to be seen how these terms affect the numerical

value of A. As it now stands, the calculation of the moment

of inertia is an open question.

Phenomenologically speaking, the moment of inertia

parameter is simply related to the excitation spectrum of

the rotator core. This is a welcome respite to the somewhat

uncertain issue of choosing the single particle parameters

from a multitude of levels in the excitation spectrum.

Estimates of A are readily obtained from the A J (J+l)-rule,

modified perhaps, for the rotational-vibrational interaction

(cf. Eq. (3.16)). To a first approximation, A is given by

the 0+-2+ level spacing of the ground state band of the core.

Including rotational vibrational corrections, A can be es­

timated by fitting the 0+-2+-4+ spacings. As Table (6.3)

shows, these two estimates of A for each core nucleus are

virtually the same. The value of A so determined is some-

131

Table (6.3)

Moment of Inertia Parameter

Ej = A'J(J+1)

Ej = A J (J + l ) - B J 2 (J+1)2

20 24Ne Mg

A 1 .27 .23

A .30 .24

B .42xl0-2 .15xl02

.30

.33

.47xl0“2

what less for Mg2 than for Ne2^ and Si2^.

1 3 2

D. Residual Interaction

In this section, we conclude the discussion of the

parameters for our calculations by turning to one of the

very interesting features of the unified model. We refer

to the residual interaction acting between the valence

nucleons outside the rotator core. This area of the unified

model is virtually undeveloped because most applications of

the model are limited to the extreme single particle or zero

particle cases. The relevant physics has then been the ex­

citation modes of the core alone or the coupling of a single

valence nucleon to the core.

A few authors have ventured to couple several valence

nucleons to a liquid drop core. In at least one instance, a

pairing residual interaction was employed between the valence

nucleus outside a spherical vibrator (Al 69). In the strong-

coupling limit, most recent investigations have focused on

odd-odd nuclei and the residual interaction between the extra­

core proton and neutron. The residual interactions used have

had a Gaussian radial form with a Serber exchange mixture

(Ne 62, As 68), or are given by Kuo-Brown reaction matrix

elements (cf. Chapter II) or a Yukawa radial form with a

Rosenfeld exchange mixture (Wa 70). There have also been

earlier qualitative studies of heavy deformed nuclei with two

or more valence nucleons. One of these omitted any residual

133

interaction (Fo 53) and the other prescribed two quasi­

particle excitations from the BCS theory of pairing (Ga 62b).

Apart from these few individual researches, there has

been no systematic study of the residual interaction appro­

priate for unified model calculations. In view of our

discussion of the theory of effective interactions in Chapter

II, it is impossible at this stage of development to judge

the theoretical merits of any of the residual interactions

which have been used for deformed nuclei. When thinking of

truncation effects and configuration dependencies, it is not

surprising that for some deformed nuclei a residual interaction

may be necessary to generate the proper level orderings while

for others it seems to have little effect (As 68).

As we shall see, it is absolutely essential that a

residual interaction be included in our model. The combina­

tion of the residual interaction v^j between the valence

nucleons and the Coriolis coupling J-j of these nucleons to

the rotator core is largely responsible for the level

structure of many s-d shell nuclei.

Operationally, we define the residual interaction as

that two-body interaction acting between the valence nucleons

which is not included in the average field through which they

move. As noted in Chapter II, a popular phenomenological

residual interaction for shell model calculations In a major

shell is the "pairing plus quadrupole" force. Since the

average Hartree-Fock field is of a quadrupole nature (Ri 68),

we assume that the dominant residual interaction between

the valence nucleons is of the pairing type.

Since the residual interaction is presumably a small

perturbation to the basic Hamiltonian, we employ for trial

purposes a diagonal pairing interaction (DPI) suggested by

I. Kelson at the beginning of this research.

Before writing the DPI, we mention the qualitative

effects we expect from a pairing interaction (PI). It is

observed throughout the periodic table that the spacing

between the J=0 ground state and first excited state of even-

even nuclei is larger than the average spacing between other

low-lying levels. This is interpreted as evidence for a

pairing interaction (Bo 58). Specifically, PI splits the

degeneracy in a many-particle configuration according to the

seniority quantum number. In the spherical shell model,

the seniority quantum number s measures the number of particles

in these states which do not couple to form zero spin pairs.

The ground state of the system has s=0 or s=l for even-even

and odd-even nuclei respectively.

The unified model statement of PI is that the particles

couple to form zero spin-projection pairs. The two-body

matrix elements of PI with deformed orbitals are given by

(Na 65, Ro 65)

134

<n1v 1n2v2 |vp I |n3v 3n 1)v 1(> = <6 -iOa)

135

The single parameter associated with this interaction is

its strength G,

90G = =— MeV (6.10b)

where A is the atomic mass number of the system.

In effect, PI measures the K=0PW parentage of a Re­

configuration. In a deformed even-even nucleus with many

possible valence configurations, PI selectively depresses

Ka=0PW configurations below their unperturbed energies.

Furthermore, the rotational levels based on these configura­

tions are strongly admixed. Extending the seniority concept

to zero spin-projection pairs, we can say that in odd-even

nuclei the s=l levels are depressed with respect to the 8=3

and higher seniority levels.

Our experience indicates that the PI given by Eqs. (6.10)

would not satisfactorily reproduce the observed level spacings

22in Ne . The spectrum was too compressed for values of G of

1.0-2.0 MeV.

Since our primary objective is a reasonable identifica­

tion of the Ka-configurations important in s-d shell nuclei,

we simplified PI to its diagonal form with the intention of

separating the states of different seniorities as suggested

by experiment. Therefore, for rotator cores with two and

three valence nucleons we can write DIP compactly as

One value of P is determined for each of the isotope sets

N e , Mg, and Si. More revealing than the specific values of

P we use is the evolution of the theoretical spectrum as a

function of P.

These and other numerical conclusions for the unified

model of a rotator core with several valence nucleons are

presented in the next Chapter.

137

A. Introduction

In this Chapter, we present the results of treating

He21’22,23, M g 25,26,27, and si29’30,31 as rotator cores with

1, 2, and 3 active valence nucleons. These calculations are

summarized in the form of excitation spectra and of spectro­

scopic factors for (d,p) stripping to low-lying levels in

these nuclei.

21 25Before proceeding, we mention that the nuclei Ne , Mg

29and Si are well-known to the unified model. In fact, their

interpretation in terms of a rotator core plus a single valence

nucleon first demonstrated the applicability of the unified

model to nuclei in the first half of the s-d shell (Fr 60,Li 58, Br 57). However since this early analysis was based

on the phenomenology relevant to orbitals calculated from the

Nilsson Hamiltonian (cf. Chapter VI), we have added these

nuclei to our calculations to test our assumption of deformed20orbitals defined by the Hartree-Fock solution of the Ne ,

O jj o Q OMg , and Si cores. The coefficients c . defining these

'■'vorbitals xfiv a^e tabulated in Table (7.9). The RPC pertur­

bation is included in these core +1 calculations (as it is in

all our calculations). In the earlier calculations this was

not always the case.

As we shall see, the structure of all the nuclei treated

herein is vitally influenced by the rotator-particle coupling

Chapter VIIDeformed Shell Model Results and Predictions

(RPC=-2AJ,j) and the residual interaction P acting between

the valence nucleons. This collective particle interplay

should not be neglected when investigating rotational nuclei

in this region of the periodic table.

A second reason for including the standard core +1

nuclei is to establish acceptable values of the single par­

ticle energies which are needed for the core +2 and core

+3 calculations. This procedure is necessary because,

recalling from the previous Chapter, the Hartree-Fock single

particle energies are much more sensitive to the parameters

of the two-body interaction than are the associated wave

functions. Consequently, we must determine the from

experiment.

Our primary objective is a description of the even-even

nuclei Ne2 2 , M g 2^, S i ^ and their odd-even neighbors Ne2^,

27 31Mg and Si . These nuclei have never been analyzed in the

context of the strong-coupling unified model as anything

other than a rotator core or a rotator core with a single

valence nucleon respectively. As pointed out in Chapter IV,

the spectra of these even-even nuclei are substantially more

complex than that given by a simple J=0, 2, 4... rotational

spin sequence (see Fig. (4.1)). Herein vie interpret them as

core +2 systems. The situation regarding the odd-even nuclei

is more complicated. To help establish our description of

23 27 31Ne , Mg , and Si as core +3 systems, vie also present

calculations based on a core +1 interpretation for comparison.

1 3 8

The results of these calculations using parameters

discussed in the preceding Chapter are summarized in a

series of figures and associated tables. Each figure

refers to one nucleus and consists of its experimentally

determined spectrum and two related theoretical spectra.

The spectrum "EXP" represents a summary of the known energy

levels, spins, spectroscopic factors, Ln and L2n values for

(d,p) and (t,p) transitions. In all energy level diagrams,

we use the following format: For each level, L-values are

listed at the right and are follov/ed by the spectroscopic

factor, and lastly by the spin. In some cases, two values

of the spectroscopic factor are given. If these values are

separated by a dash, they Indicate the range of values quoted

by different authors. If they are separated by a comma, the

lower value corresponds to the higher spin assignment and

vice versa. Lastly, we omit from the experimental spectra

all known negative parity levels as being outside our present

considerations.

Our theoretical interpretation of the nucleus as a

rotator core plus 1, 2, or 3 valence nucleons is given by

the spectrum denoted by "THY". The eigenvalues in THY are

determined by diagonalizing the unified model Hamiltonian

given by E q s . (5.1), (5.24), and (6.11). The spectroscopic

factor for (d,p) stripping on an even-even target are cal­

culated from Eqs. (5.42) and (5.43). Finally we include the

spectrum "FOPT" giving the calculation of our theory in first

139

order perturbation theory, according to Eq. (5.26). Since

each state in FOPT corresponds to a pure Ka-configuration,

the difference between THY and FOPT is a consequence of the

RPC H- PPG + P perturbations, Eqs. (5.7), (5.27), and (6.11)

respectively.

Each level in FOPT is labeled K^, according to

which Ka-configuration it belongs. The correspondence be­

tween the K^, 1<2. . . and the different Ka-conf igurations is

given in the table immediately following the figure. This

table also contains the parameters of the calculation, A,

e^v , and P, the leading Ka-components of the eigenvectors

for the lowest levels in THY and references for the experi­

mental data used.

B. Neon Isotopes

1. Preliminary Remarks

For the neon isotopes, the extra-core particles are dis­

tributed in the 3/2, 5/2, 1/2’, 1/2", and 3/2* deformed or­

bitals in the s-d shell. Our notation for these orbitals is

indicated in Fig. (6.1). According to the Pauli Principle no

more than two neutrons can occupy any one orbital. This

implies a basis of 5 possible states for Ne20+1, 25 for Ne20+2,PO PP60 for Ne +3, and 4 for Ne +1. Vie diagonalize the unified

model Hamiltonian given by Eqs. (5.1) and (5.24) in the full

space for each case.

By- way cf illustration, the lowest configuration for each^

of the three neon nuclei is denoted by Ko-3/2(3/2),

1*10

Ka=OPW(3/2,-3/2), and Kc=5/2(3/2,-3/2,5/2). All other

configurations are written in a similar fashion. In the2 8 22core +1 treatment of Ne , Ne~ is assumed to be the

closed core and the lowest configuration is then given by

Kff=5/2(5/2).

2. Neon 21

Vie turn now to specific cases. As we mentioned earlier,21Ne has been interpreted in terms of the unified model by

many authors (Fr 60, Bh 62, Ch 63, Dr 63, Ke 63b, Ho 65,P 1

La 67, Li 70). Our results for Ne are similar to these

and are given in Figure (7.1) and Table (7.1). In the light

of the earlier calculations, we recall that our analysis of 21Ne serves two purposes. First, we are testing the success

of quantitative calculations based on the use of deformed20orbitals defined by the Hartree-Fock solution of Ne (cf.

Table (7.9)). Second, we determine the values of the single

particle energies to be used in our core +2 and core +3

22 28calculations of Ne and Ne . Vie also give the theoretical

21spectroscopic factors for Ne . The only other known cal­

culation of the spectroscopic factors which includes Coriolis

mixing is based on the Nilsson phenomenology (La 67).21Referring to Fig. (7.1), we see that Ne can be exceed­

ingly well accounted for as a rotator core to which is coupled

a single valence neutron. Remarkably, the first six experi­

mental and theoretical levels agree to within 0.1 MeV. This

141

142

Figs. (7.1)-(7 .3) , (7.5), (7.6) Deformed Shell Model Excitation Spectra for Neon Isotopes - Namely Ne2!, Ne22, and Ne23 as a Ne20 rotator core plus 1, 2, and 3 valence nucleons respectively. "FOPT" denotes first order perturbation theory; "THY" the complete diagonalization; and "EXP" the experimental values.

Tables (7-1)— (7.4) Deformed Shell Model Parameters and Dominant Configuration Amplitudes for Neon Isotopes. The following notation is used

. for the wave functions: K=3/2, 5/2, 1/2',1/2", and 3/2' deformed orbitals are denoted by 30, 50, 10, 11, and 31 respectively.

The complete explanation of these figures and tables is given on pp. 139-140 and Section (VII. B) .

EX

CIT

AT

ION

E

NE

RG

Y

(MeV

)

343

N e21

N e2 0

8 EXP T H Y FOPT

1/20.12

0.73

- 9/2

- 1/2

“ 3/2

/

7/2 /

0.19

/ 0.39I

9/2 K 3 9/2 K 2

1/2 K 43/2 K 4

22

(HoToTiT"0.18,0.27

3/2,5/2 3/2,5/2

0.14,0.21

0.05

3/2,5/2 0.083/2,5/2 11/2,7/2

(3/2,5/2)0.03~cCo5~* 5/2,(3/2) O03_

0.80^9/2• 1/2 0.60

11/2

3/2

5/2

5/2

9/21/2

/

/

/

0.13

0.33

0.53

9/2 K l 7/2 K 3

7 3/2 l<2

5/2 K 2

5/2 K 3

7/2 K |

1/2 K2

22

0.62

WEAK

7/2

5/2

3/2

0.68

0.00

7/2

//

//

5/2 /

3/2 0.00 K,

Fig. (7.1)

Table (7.1) Ne21=Ne20+l DSM Parameters and Dominant Configuration Amplitudes

A 0 . . 2 5

e 3 / 2 0 ,. 0 0

e 5 / 2 = 4 , . 0 0

e l / 2 t = 4 , . 0 0

e i / 2 " = 7 . 5 0

C 3 / 2 ' = 9 . . 0 0

DOMINANT.CONFiG_ AMPLITUDES FOR . THY_______________ PAR.T1 c *-E CONFIG FOR FOPT STATES

1= 3/2 ,_3/2 (_30 . - 0 - 0 ) _ . 0 . 1 5 . K=__ 1 / 2 < 10 . -0 - 0 ) . .0.01 K= 3 / 2 ( 3 1 . - 0 . -0 1___________ a 3/2 _ K=_ . 3 / 2 C 30 _-0 r 01 Kiu.-------------- Is.. _ 5 /2 _ _ _P.88_K=_ _ 3 / 2 ( . 30 - 0 ~0>_ - 0 . 3 7 , K= 5 / 2 { 50 - 0 -0 ) 0 . 3 0 K = 1 / 2 1 10 - 0 - 0 ) a 5/2 . 3 / 2 ( 30 . - 0 - 0 ) - KI-------------- J=_ r£..?f>_S=_ _.3 /2 <_ 30_ r0 . . r0 ) ._ .. 0 . 47_ K=__ 5 / 2( 50 -O.-OJ- __-0. 21 K.= . . l / 2 < . 10 - 0 - 0 1 __________ n 1/2. _K=»__ 1 / 2 ( _ io _ -o . .r O ) __E_2

1 = U.?__ C« 99 K = l / 2 ( 10 - 0 - 0 ) - 0 . 1 2 K= 1 / 2 ( 11 - 0 - 0 ) 0 . 0 0 K = 3 / 2 ( 30 - 0 - 0 ) 7 /? K = _ 3 / 2 ( . 5 / 2 (

3 0 _ - 0 - 0 ) 50 - 0 -01

^ 1__________ T=_ _ ?72__ . 0. 8 0 _ K = .. 3 /2 ( _ 30 -0 . -0 ) . . . - 0 . 4 5 .. 5 / 2 ( 50 - 0 - 0 ) . 0 . 4 0 K = l / 2< 10 - 0 - o i ........ .. ;. a 5/2 K-5 -------1__________J-=_ _572 .-P..8lJK=_ _5/2(..5D_.-D_..rO)__ - 0 . 5 6 _ .K.=__1 / 2( .10.. - 0 . - 0 ) - _. -0. 15. K=_. 3 / 2 ( . 3 0 . - 0 .TO)__________ 3 .5/2 -1 . /2I . . ip_r9-rP> R_2.

T = 5/2__ „ 0 . 7 6 _K = _J /2 ( 10 - 0 -01 . - 0 . 4 5 K=» 3 / 2 ( 30 - 0 -01 - 0 . 4 5 K= 5 / 2 ( 50 - 0 - 0 ) _ - /? K= . .1 /2 1 5 / 2 t

1O

O1

’ 10

o1

' 1o

o

Kn ___0T _ _ 3 /2 _ _ ._ C. 9 3 K=_ . 1 / 2 ( _ 1 0 - 0 - 0 ) _ - 0 . 1 5 K= 3 / 2 1 30 - 0 - 0 ) . - 0 . 0 9 K = 3 / 2 ( 31 . - 0 - 0 ) .........._.. 3 7/ 2 K =__________ I3_ 11/?__ rO. 81. K = . _ 3 / 2 ( .30 .-0 - 0 ) . . 0 . 5 3 *=__ _5/2( 5 0 . - 0 - 0 ) . - 0 . 2 3 K = 1 / 2 ( 10 . - 0 -01 = 9/2 _ K= 3 / 2 ( 3 0 - 0 -01 K1

i= Ill - C . 87 K = _5/ 2 ( .. 60 - 0 _ - 0 ) . _ - 0 . 4 1 . K= 3 / 2 C 30 - 0 - 0 ) - 0 . 2 7 K = 1 / 2( 10 - 0 - 0 ) = 3/2 K* 1 / 2 < 11 - 0 - 0 } Kh

i = .7-0* 9 2. _K.=__ 1 /2 (_.ii_.-.o..rQ) 0 . 4 0 K = 3 / 2 ( 31 - 0 - 0 ) 0 . 0 1 K= 3 / 2 ( 30 - 0 - 0 ) a 1/2 K = l / 2 { 11 - 0 - 0 ) K 4_________ j=_ -XJlJ.._ 0-.9.9_K.=.. _. l /2(__H..-0. . .-O)__ _.0.12„ K.=__ .1 / 2 ( _ 10_- 0 . - 0 ) _ ,_.0.00 K = 3/2<_ 30. - 0 rO!___________ .5/2 __ _K=_._ 1 / 2 ( - IP . 7 ° . r 0) k 2

T = 9/2 0 . 7 9 K = 1/2 < 10 - 0 - 0 ) 0 . 6 1 K=__ 5 / 2 ( 50 - 0 - 0 ) o . 10; K=__ l / 2 ( _ JJLrP. - 0 ) = 5/2 K= 512 ( 50 - 0 - 0 ) k 3

Experimental References for Ne

Table (7.1) (Continued)21

K.H. Bray et.al., Nature 215. (1967) 501.

A.L. Catz and S. Amiel, Phys. Lett. 20 (1966) 291.

B. Chambon et.al., Nucl. Phys. A136 (1969) 311.

A.J. Howard, J.G. Pronko, and C.A. Whitten, Jr., Phys. Rev.184 (1969) 1094.

A.J. Howard, J.G. Pronko, and C.A. Whitten, Jr., Yale Preprint 3223-192, to be published in Nucl. Phys.

L. Jonsson et.al., Arkiv ftir Fysik, 35_ (1968) 403-

R. K8mpf, Phys. Lett. 21 (1966) 671.M. Lambert et.al., Nucl. Phys. A112 (1968) l6l.

D. Pelte, B. Povh, and W. Scholz, NP 55 (1964) 322.

D. Pelte, B. Povh, and B. Schtirlorn, NP 73. (1965) 481.

D. Pelte and B. Povh, NP 73. (1965) 492.

P.J.M. Smulders and T.K. Alexander, PL 21 (1966) 664.

accuracy Is very gratifying but really reflects small

adjustments of the parameters, the most important of which

are the moment of inertia A and the single particle energies

and ei/2'* (">ur values Por these quantities (cf. Table

(7-1)) are similar to those of Freeman (Fr 60) and of Kelson

and Levinson (Ke 63b). Much more significant than small

variations in these parameters is the perturbation produced

by the Coriolis interaction. Comparing the THY and FOPT

spectra, it follows that RPC shifts the lowest 5/2, 7/2, and

9/2 levels relative to the 3/2 ground state by a substantial

0.8, 1.4, and 2.4 MeV respectively. Considering the magnitudes

of these shifts and their dependence on the spin, it is all

the more striking that the 5/2, 7/2, 9/2 spin sequence fits

the experimental spectrum to 0.1 M e V . Continuing in this

vein, we add that the Coriolis interaction depresses the 11/2

spin level by 2.7 M e V . However, compared with the above

accuracy, it is significant that the 11/2 state is nearly

1 MeV higher than the only known experimental candidate

located at 4.43 MeV. VJe remark on a possible reason for this

discrepancy in our concluding Chapter.

Another notable aspect of the Coriolis interaction which

22 2°will be found in Ne ' and Ne J is the considerable admixing

it produces between the Ka-configurations of the basis space21

as can be seen in Table (7.1). This mixing in Ne is

responsible for the twofold enhancement of the spectroscopic

factor for the first 5/2 state. This, too, agrees very well

with the experimental situation.

146

1*17

As for other levels, our calculations predict that the

level at 3.74 lieV has spin 5/2. The levels at 4.53 and 4.69

MeV probably have spins 5/2 and 3/2 respectively. These two

spin assignments have a particular bearing on the nature of

the deformed orbitals used in this calculation. We discuss

this point shortly. Finally we note that the experimental

level at 3.89 MeV probably has negative parity as suggested

by systematics of hole excitations in the lp shell (Ho 69,

Li 70). In summary, the success of our unified model des-21 22 cription of Ne encourages us to attempt the same for Ne

and Ne2 3 .

However, before vie do so, we wish to reiterate that

these results have been obtained with deformed orbitals given

20by the Hartree-Fock solution of the Ne core. As discussed

in Chapter VI, the deformation is not a parameter in our

calculations. In most of the aforementioned calculations of 21Ne which'are based on the Nilsson phenomenology, the de­

formation is generally taken to be 6=+.2 or ri=+4, where

n=205. This implies a decoupling parameter (cf. E q . (5.23))

of a=+0.5 which implies in turn a normal spin sequence

J=l/2, 3/2, 5/2... for the K=l/2' band. The corresponding

decoupling parameter calculated from the Hartree-Fock co­

efficients c^j is a=+1.6. This value yields a J=l/2, 5/2,

3/2... spin sequence for the K=l/2' band. No positive

deformation 6 of the Nilsson orbital K=l/2' can generate a

decoupling parameter this large and produce an inversion of

the J = 3/2 and 5/2 levels (Ch 66). If the normal 3/2, 5/2

ordering applied, then the Coriolis interaction would pre­

serve this ordering and further separate these states as

can be seen in going from FOPT to THY in Fig. (7.1). Hence,

we interpret the near degeneracy of the 4.53 and 4.69 MeV■{• -J-

levels together with one 3/2 and one 5/2 spin assignment -H” "Vas opposed to two 3/2 or two 5/2 assignments - as further

evidence for the use of Hartree-Fock deformed orbitals.

As mentioned at the beginning of this Chapter, our

21interpretation of Ne in terms of the unified model is not

in itself new. However, our methodology is different than

the traditional application of the unified model with the

Nilsson Hamiltonian for the deformed orbitals. Our approach

is intimately related to the work of I. Kelson and C.A.

Levinson and the Hartree-Fock representation of these orbitals

(Ke 63b).

3. Neon 22

Despite the long standing belief - as convincingly

21demonstrated by Ne - that the unified model is a viable22model for nuclei in the first half of the s-d shell, Ne

has long resisted a comprehensive unified model interpreta­

tion. As a simple rotator, it fails to possess the observed

density of states.2 ? 20Our interpretation of Ne as a Ne rotator core with

two active valence neutrons is given in Fig. (7*2) and Table

148

E X C ITA T IO N ENERGY (M eV )o

rro oj cn <T> CD

ro roo+ro

O+ roro ro

H-C3

- A

ro

O

ro/

ro\

\\

\\

\

ro

4b/

ro cr rocu oj roro 4b \r y

/ / 1■

O C D

\ I \l \l

/

\\ \I \\ \ \ l

\ \ 1 I \ \

ro 4b-

U lrooJOOJ I 4b Icn I O) I

II

ro A Aoj ro cdo 4b o ui —

■ ro O — 4b o ro oj ro

oj

ro

mX"0

—IX-<

T|o~u

ororo

33

CDroo

+ro

j—' VO

X Xro

X X X X X X X OJ 4b O l <J> — ->J OJ 4b O

Xcn

Table (7.2) Ne^2=Ne20+2 DSM Parameters and Dominant Configuration Amplitudes

A 0 ,.30

e3/2 = 0 ,.00

e5/2 = 4,.00

e l / 2 ' =4,.25

£ 1 / 2 "= 7..50

e3/2,=Q> 1.00

P -2 ,.00

D O M INANT. C O N FIG . A M Pl I TU.OES .F n R _ . THY ___________ PART IC_t..E_CaNF I G F C R FOPT STATES

________ir _ _ 2 ______ 0 . 92. K= ..Q( 3 0 -3 0 )

0 .9 6 .

J=_

J =

K= , . 0 ( 5 0 - 5 0 ) . - 0 . 1 3 K = . . . 0 ( 1 0 - 1 0 1 - _ C K = 0 ( 3 0 - 3 0 )Vi t iX

K= . 1 ( - 3 0 5 0 ) ___0 . 20 K - .. 1 ( 3 0 - 1 0 ) ................ ......................... ................... ... 2 ___K = 0 ( . 3 0 - 3 0 ) .y

. . i V k

K = _. . 1 ( - 3 0 5 0 ) ___ 0 . 3 3 . K=. . _ 1 ( . 3 0 - 1 0 ) ___________________________________ = _ _.9_ ____K = _ _4(, 3 0 5 0 )K -iC

K= . J l ( . 3 0 - 1 0 ) . - 0 . 9 A K= 2 ( 3 0 1 0 ) ■ 3 1 K= 1( - 3 0 5 0 )

K = . .3 ( . 5 0 1 0 ) 0 . 1 2 K = . 2 ( 3 0 1 0 ) ___________ ____ ____________ . 2 . . 2 ( 3 0 1 0 ) . .

K=_ . l . ( . 3 0 - 1 0 ) _ - 0 . 0 V K = _..0< . 3 0 - 3 1 _________________________ r . _ 0 _ _____K=_ _ 0 !

0in1 .

oin

" 5 •

K = 0 ( 5 0 - 5 0 )

onT0o

i K = ... 1 ( - 3 0 5 0 ) = 1 K = 1 ( 3 0 - 1 0 )

K = . 1 (.. 3 0 - 1 0 ) _ - 0 . 3 5 K= . 3 ( 5 0 1 0 ) .. ..................... ............................. ........... 3 4 K = . o< 3 0 - 3 0 ) .. , . 3 _

K= . _ 0 ( . 1 0 - 1 0 ) ___ 0 . 4 0 K = ... 2 ( 5 0 - 1 0 ) ___________________________________ = _ _ c _ ___ K =_ _P( _ 1 0 - 1 0 ) _ " 7

K= 1 ( - 3 0 5 0 ) 0 . 3 9 K= 1 ( 3 0 - 1 0 ) _ 2 K = l ( - 3 0 5 0 ) K3

K= 0 { 1 0 - 1 0 ) 0 . 1 3 K 3 0 ( 3 0 1 0 ) K 4.

K=.. . . 2 ( . 3 0 . 1 0 ) ... - 0 . 4 6 K = , _1 ( - 3 0 5 0 ) __________ __________________ _____ f_ .. 2_ ___ _ l ( . . 3 0 - 1 0 ) . - J k . .

K = _0_(_.5_ 0 . - 5 0 ) __r 0_ . 2 3. K= _ o < 3 0 - 3 0 ) 3 2 K= 0 ( 5 0 - 5 0 ) K —

I—1 vj; o

151

E xperimental References for N e _

S. Buhl, D. Pelte, and B. Povh, NP A9l (1967) 319.

B. Chambon et.al., Nucl. Phys. A136 (1969) 311.

W.G. Davies, C. Broude, and J.S. Forster, Bull. Am. Phys. Soc. (April 1970) EE3.

A.J. Howard, J.G. Pronko, and R.G-. Ilirko, Yale Preprint 3223-185, to be published in Nucl. Phys.

W. Kutschera, D. Pelte and G. Schrieder, Nucl. Phys. Alll(1968) 529.

D. Pelte, B. Povh,-and V/. Scholtz, Nucl. Phys. 52_ (1964) 333

H. Ropke and D. Pelte, Z. Physik (Feb. 12, 1968) p. 179, 210

W. Scholz et.al., Phys. Rev. Lett. 22_ (1969) 949-

W. Scholz et.al., Proc. Montreal Conf. on Nuclear Structure(1969) 311.

B.H. Wildenthal and E. Newman, Phys. Rev. 175 (1968) 1431.

Table (7.2) (Continued)22

(7.2). The two neutrons can now be excited from the

K=0PW(3/2,-3/2) configuration of the ground state of the

simple rotator to other unoccupied Nilsson orbitals. It

is immediately evident from the figure that the resulting

density of states is dramatically improved. In fact, a one-

to-one correspondence can be made for 11 states up to and

including the J Tr=6+ state at 6.35 KeV. In making this

correspondence, we predict that the 5.34 MeV level has spin

and parity 1+ , the 5.52 MeV level 4+ , and the 5*92 MeV level

152

The parameters used for this calculation are essentially21those established in the earlier Ne analysis. Small

variations in the moment of inertia A and the single particle

energies are discussed below. A new feature entering

21this calculation as opposed to the Me calculation is the

residual interaction acting between the two valence neutrons.

We have assumed a pairing type interaction given by E q . (6.11).

Its strength P is the crucial parameter of this calculation.

That a residual interaction is absolutely essential can be22seen in Fig. (7.3) where we find the spectrum of Ne with

P=0. On the other hand once vie have Pru-2 MeV the basicP Pstructure of Ne in Fig. (7.3) is not significantly changed

by 1/2 MeV variations about this value. This value of P is

not wholly unlike the strength G of the usual pairing inter­

action (cf. Eq. (6.10b)) for this region of the periodic

table. However, as we mentioned in Chapter VI, calculations

or

no

E X C IT A T IO N EN E R G Y (M e V )

OJ 4b Ol OT

01

ro ro crro ojoj ro ”4*

ro ocn Jl A— rooJO oj

"010)

mxT3

<ZL.CD

ror 0

-oLO

IO ro ro / \

4b — ro OJ Aro 0 3 0 4 o at —1oj

•011I

ro x-<

s£~CD

fOO

*rro

ro X Lro 4 b - ro oj / / /\ ro 4b a i— at

Uii HO ±o ^

r-»VAOJ

with the usual pairing interaction left the J=0,2,4,6...

spin sequence too compressed.

The 11 state correspondence between theory and experi­

ment in Fig. (7*2) is a consequence of the RPC + PPC + P

perturbations. We can isolate the effect of each of these

perturbations in Fig. (7.2). The major contribution in first

order perturbation theory (FOPT) is that from the pairing

interaction which lowers s=0 seniority states with respect

to s=2 states. In moving from FOPT to exact diagonalization

(THY) the Coriolis interaction has the greatest effect in

shaping the spectrum. Indeed, RPC shifts the 2^, 4^, and 6^

states with respect to the ground state by 0.9, 3.2, and 6.3

M e V . These are large energy shifts.

Before commenting further on these energy shifts, we

also note the consequences of the PPC perturbation. Like

RPC, PPC results from the substitution R 2=(J-j)2 eliminating

explicit reference to the core angular momentum in the unified

model Hamiltonian. PPC is a two-body force, quadratic in j,

which many connect intrinsic states with AK=0. Hence it

follows that in our calculations, PPC is entirely responsible

22for the mixing In the J=0 states in Ne . It is also the

dominant perturbation repelling the two lowest 1+ states in

Fig. (7.2) as the corresponding wave functions in Table (7.2)

indicate. In these cases, PPC shifts the unperturbed level

less than 1 M e V . In contrast to RPC, PPC is not an increasing

function of the spin J.

154

Generally, for J£2 the various Ka-configurations are

strongly mixed by RPC + PPC + P. The leading components

of the wave functions are given in Table (7.2); they in-

22dicate that most levels in Ne are not of a simple charac­

ter, say that given by exciting a neutron from the ground

state to a particular deformed orbital. A typical example

of mixing is given by the "ground state rotational band".

These levels, beginning with the J=2 member, contain a sub­

stantial percentage of Ka=l(-3/2,5/2) and Ka=l(3/2,-1/21)

configurations admixed into the unperturbed ground state con­

figuration Ka=0PW(3/2,-3/2) by the Coriolis interaction. The

single exception to extensive mixing is the 42 state which is

92% K =4(3/2,5/2) and may correspond to the 5*52 MeV J 1T=(3+ ,4 + )

level in the experimental spectrum. If it is, this may be the

first K=4 state identified in the s-d shell.

Although the Coriolis interaction thoroughly mixes the

22Ka-configurations in most of the Ne wave functions, its

affect on the eigenvalues is most pronounced for the ground

state rotational band. We previously noted that the 6- level

is depressed 6.3 MeV and matches the experimental value. The

shift in the 6- state is surprisingly large. It is in fact

much larger than that obtainable by any reasonable variation

of the parameters of the calculation. That this is so lends

22increasing credence to our core +2 model for Ne

Further evidence for the Coriolis compression of the

ground .state rotational band is the recent discovery of the

155

0 member at 11.15 MeV (Da 70). Our calculation places this

state at 11.9 M e V ! The Coriolis shift of the 8"1 state is

9.7 MeV.

Greatly encouraged by this extension of our model to high

spin states and high excitation energies, we venture to pre-•f

diet that the 10 member of this band should appear at aboutO p

18 MeV excitation in Ne' . Our calculated value of 19.8 MeV

is expected to be an upper limit on the excitation energy.

The Coriolis shift for the 10+ state is 13.2 MeV.22Another point of fundamental concern regarding Ne is

the nature of the excitation mechanism underlying the secondTT *$*J =2 state at 4.46 MeV. Shell model calculations based on

the SU(3) classification of states predict a low-lying K=2

state (Ha 68b). Even in the shell model framework, this

level is frequently referred to as the beginning of a y-band

with K=2 (Ak 69). Of course in terms of the Bohr liquid drop22model such a description evokes a vision either that Ne may

undergo low-lying axially asymmetric vibrations, in other22words, y-vibrations - see Fig. (3.3) - or that Ne has a

permanent axially asymmetric deformation - see Fig. (4.2).

In either case, a series of rotational levels J=2, 3, 4,...

is built on the y-band head. In Chapters III and IV, we

pointed out that both these interpretations seemed unlikely

in view of the high excitation energy for y-vibrations and

low density of states from axially asymmetric rotations.

156

In contrast to this situation, our core +2 model for

He ' has the correct density of states and excitation energies7T +and has a J =2 candidate which is in excellent agreement

with the experimental level in question at 4.46 MeV. Signi­

ficantly, the 22 state in our model is not a y-band head

because vie presume an underlying axial symmetry. Hence we

conclude that the essential excitation mechanism is that

provided by the two extra-core neutrons. Again, as with the

ground state rotational band, there is no convenient charac­

terization of the 22 state. It can be seen in Fig. (7.2)

that this state is displaced more than all other J=2 states

by the RPC and PPC perturbations. Also noteworthy, in view

of the prevailing interpretations, is the fact that the K=2

component in this state is only 19% of the 22 wave function.

The wave function also includes sizable superpositions of K=0

and K=1 configurations. The precise composition can be found

in Table (7.2) .

It is interesting at this point to see how other modelsp P

for N e ' compare with ours. In Fig. (7*4) we find the presentp p

theoretical perspective of Ne . E.C. Halbert et al. (Ha 68a)22perform a shell model calculation viherein they treat Ne as

1 fian inert 0 core with six valence particles confined to the

s-d shell. They employ Kuo-Brown reaction matrix elements for

the residual interaction. Another shell model calculation of

Interest Is that of Y. Akiyama et al. (Ak 69). This calcu­

lation differs from the former in two ways. It uses

157

22

EXP UNIFIED MODEL (THIS WORK)

A<2,3,4,5,6

"^0— 2_ /3- 3 , 4=R2

(1,2)

0 0

ro 4

— ro

oj ro

oi o

-^o

oi

1 ^ 2 2

SHELL MODEL SHELL MODEL PROJECTION (Ha 68a) (Su(S)TRUNCATiON) (Sc 6 3 )

(A k 6 9 )

^2—AS

102

6-5'0

2

4

42

4 -4-2

VJ1c?

---------------------- o 0 o

F i g . ( 7 . 4 )

phenomenological residual interactions, namely Gaussian

and Yukawa interactions with exchange mixtures given by

l8 l8fits to 0 and F or the particular nucleus in question.

These authors also truncate their basis space to the leading

SU(3) representations for six particles in the s-d shell.

Finally, we mention a model calculation which begins

as does ours. L. Satpathy and S.C.K. Nair (Sa 68) also22assumes a core +2 model for Ne . They allow the two neutrons

20to move in the Hartree-Fock deformed orbitals outside the Ne

core and to interact via a Yukawa force with a Rosenfeld

mixture. The similarity with our model ends here because

Instead of diagcnalizing the interaction in a space of strong-

coupled wave functions ~ with the accompanying Coriolis

interaction - they diagonalize the two-body interaction in

an "intrinsic space" defined by xKa with K=0. They consider

22the lowest eigenstate to be the intrinsic state of Ne

Projecting from this intrinsic state, they find the ground

state band J=0, 2, 4, 6 for Ne2 2 .

Overlooking the fact that the Satpathy and Nair model

applies only to the intrinsic state for the ground state band,

it is not clear that our ground state bands J=0,2,4,6 are in

any way similar. If we assume that the strong-coupled wave

function is a low order approximation to the corresponding

state of good J projected from the intrinsic state (Ri 68),

then our interpretation and their differs significantly in

that we- find large K=1 contributions to the J>2 members of

159

160

the ground state band. These components appear in our

wave functions as a consequence of the Ccriolis coupling

of Ka-configurations not included in their "intrinsic

space".

As for the two shell model calculations we note that

the 4^ and 22 states are nearly degenerate. Furthermore,

in the SU(3) version there are no low-lying 1 states to

be found. It seems that our unified model calculation and

the shell model calculation with the effective interaction

are most similar. It would be interesting to know where

this shell model predicts the 8+ and 10+ levels to be. In

addition it is desirable that additional quantities such as

transition rates be calculated to extend this intercomparison.

22In concluding our unified model analysis of Ne , we

make some last remarks about the values of the parameters which

have been used. In view of the exploratory nature of this

calculation we have not tried to determine a best fit of the

22parameters A, and P to the Ne spectrum. We regard

the greatest uncertainty about these parameters to be the

relative ordering of the z^/p and Ei/2 ' sin£le particle

energies as discussed in the previous Chapter. One can see

directly from the experimental spectra that the 5/2 level is21 23higher than the 1/2* level in N e " and lower than it in Ne .

To see how the relative ordering of £5/2 and el/2’ affe°ted 22the Ne spectrum, calculations were performed with e^/p< ei / p 1

21and e5/2>el/2'5 ^eeP anS near the Ne estimates at all

times. This interchange produces almost no change in the

ground state band and its wave functions and little dis-

cernable affect in the rest of the spectrum.

4. Neon 23

2 PJust as a simple even-even rotator description of Ne2?is untenable, so we anticipate that a Ne core with a single

valence neutron will be less than an adequate representation 23

of Ne . This point of view is born out by those few unified2 3model interpretations of Ne which have tried to attribute

its structure to rotational bands built on a single valence

nucleon (Fr 60, Ho 67, La 67, Na 69). Unfortunately the

23experimental data available on Ne is paucious and very few

spin assignments have been determined unambiguously. As a

consequence the identification of band heads and rotational

sequences in this nucleus has been largely speculative. Most

23of this speculation is based on the assumption that Ne J with

its 13th odd particle being the active particle should have25 25the same rotational structure as Mg and Al which are also

13th odd particle nuclei (Li 58). Nevertheless with each

attempt to expldit this analogy, it is becoming increasingly

23clear that He J is unlike the other core +1 nuclei. A.J.

Howard et' al. (Ho 67) were among the first to point this out.

They noted that the electromagnetic decay properties of the

1.83 MeV J Tr=3/2+ state in Ne2^ are at variance with the syste-

25 25matics observed in the Mg and Al ' systems.

161

In Fig. (7.5) we give a core +1 description of Ne J

which is similar to the earlier calculation with Coriolis

mixing made by J, Freeman (Fr 60). This figure serves only

to confirm the difficulties encountered in a core +1 inter­

pretation. In this figure we clearly see the necessity for

calculating more than energy levels.' Within the uncertain­

ties of the experimental spin assignments it appears that we

can make a one-to-one correspondence with the lowest five

levels. However, two serious discrepancy cast considerable

doubt into this interpretation. In the first place, the

theoretical spectroscopic factor predicted for the 1.83 MeVTT H"J =3/2 level implies strong (d,p) stripping to this level

whereas experimentally this state exhibits no stripping be­

havior in the Ne22 (d,p) Ne2^ reaction (La 67, Na 69, Ho 70a).

In the second place, this model suggests no likely candidates23for both the 2.31 and 2.52 MeV levels in Ne . Moreover

raising the band head reveals a great disparity in the

theoretical and experimental density of states.

The observations about the electromagnetic and stripping

properties of the 1.83 MeV state have encouraged conjectures

that this level may be the band head of a K=3/2 hole (Nilsson

orbit No. 7) relative to the Ne22 ground state (La 67, Na 69). Presumably, the neutron pair in the K=3/2 orbit is broken and

one of the neutrons is excited to the level of the 13th odd

particle. The ensuing state appears low in the spectrum be­

cause there is no net loss in the pairing energy of the system.

162

2 2

EX

CIT

ATI

ON

EN

ER

GY

(M

eV

)163

N e 2 3 = M e 2 2 + I

6 r-

0

(2)

0

EXP T H Y

1/2

2 ■Q:.1-6.’ ? -2.7■■ 3 /2 ,5 /2

2 2 ^ 3 3 . 3 /2 }5/2

0.17

0.02

0.10

5 /2 ,7 /2 ,9 /20.02

2 0 ;0 5 ,0 J 0 _ 3 /2 j5 /2 0.

WEAK 3 / 2 -----------------(7/2)

0.65

0.700.70

1/2

2 — --- — 5 /2 0.34

- 3 /2

_ /5 /2~^9/2- 7 /2

- 3 /2

1/25/2

3/27/2

1/2

5 /2

FOPT

0.00

0.01

0.49 /

0.03

0.39

0.53

0.33

5/2 K4

5 / 2 K 3

7 / 2 K 3

3/2 j$49 /2 K |

3 /2 K2

0J3------_ 5 /2 K20H9---------- 1/2 Ks

— 3/2 K 3

— 7/2 K|

1/2 K2

5 /2 K,

Fig. (7.5)

28 22Table (7-3) Ne =Ne +1 DSM Parameters and Dominant Configuration Amplitudes

A 0,.25

e5/2 = 0,.00

el/2'= 1,.90

£l/2"= 2,.35

G 3/2,=h ,.50

D C y i N A N T C O N F I G A M P LI TUCES FDR T H Y ____________ _ _ _ . .? j?R I I 5 >- E_S9.Nf LG F0R F 0 P T STAT ES

i = 5 / 2 1 . 0 0 K= 5 / 2 (

01oU"S - 0 ) 0 . 0 3 K= 3 / 2( 3 1 - 0 - 0 )

»—< O0 O1

■ 1 K = l / 2 ( 11 - 0 - 0 ) - 5 / 2 K= 5 / 2 ( 50 - 0 - 0 ) X 1

i = 1 / 2 0 . 9 3 K= 1 / 2 < 1 0 - 0 - 0 ) - 0 . 3 8 K= 1 / 2 { 11 - 0 - 0 ) 0 ^ 0 0 K = 3 / 2 ( 31 - 0 - 0 ) re 1 / 2 K = l / 2 ( 1 0 - 0 - 0 ) X 2

i = 7 / 2 1 . 0 0 ;<= 5 / 2 ( 5 0 - 0 - 0 ) 0 . 0 5 3 / 2 ( 3 1 - 0 - 0 ) - 0 . 0 3 K = l / 2 ( 11 - 0 - 0 ) 3 7 / 2 K = 5 / 2 ( 5 0 - 0 - 0 ) A

i = 3 / 2 _ - 0 . 9 5 K = ! / ? < 11 - 0 - 0 ) 0 . 30 K = 3 / 2 < 3 1 - 0 - 0 ) 0 . 0 2 K= l / 2 < 10 - 0 - 0 ) 3 / 2 K= 1 / 2 ( 11 - 0 - 0 ) K -,

—i = _ 5 / 2 C. 9 7 K = 1 / 2 ( 10 - 0 - 0 ) - 0 . 2 9 K= 1 / 2 ( 11 - 0 - 0 ) - 0 . 0 6 K= 3 / 2 ( 31 - 0 - 0 ) s 1 / 2 K = 1 / 2 ( 11 - 0 - 0 ) K ?

_ I / 2 _ _ _ C . 9 . 3 . K=_. . .1 / 2 ( _ .1.1 - o . - 0 ) 0 . 3 8 K=_ _ .1 / 2 1. . 1 0 . r 0 . - 0 ) 0 . 0 0 K =__ 3 / 2 ( 3 1_ r 0 . . - 0 ) _____________ J3 - 5 / 2 _ _ K = _ _ 1 / 2 ( 1 0 _ r 0 _ r 0 ) 2 *

i = 3 / 2 C ,9 . 6_ K = 1 / 2 ( _ 10 - 0 - 0 ) - 0 . 2 8 K= _ 3 / 2 ( _ 3 1 - 0 - 0 ) - 0 . 0 7 K= l / 2 { 11 - 0 - 0 ) 3 3 / 2 K = l / 2 ( 1 0 - 0 - 0 )n 2

___________i = _ _ J 7 2 - _r . C- . 90_ K=__ 1 / 2 t 1 1 . . - 0 . - 0 ) . 0 . 9 3 K - . _ 3 / 2 ( 3 1 - 0 - 0 ) - 0 . 1 1 K= . l / 2 ( 1 0 - 0 - 0 ) .............._____ . S / 2 . K=J 5 / 2 ( 5 0 - 0 - 0 1 . „ k jl

_____T =_ _ . 9 / 2 _ _ _ l . C 0 „ X r . . . . J / 2 ( _ 50. r 0 - 0 ) . . . . . 0 . 0 7 K = . .. 3 / 2 ( 3 1 . - 0 - 0 ) __- 0 . 0 2 K = l / 2 ( 11 - 0 . - 0 ) ____________ a _ . 3 / 2 _ K = _ _ 3 / 2 ( 3 1 - 0 - 0 1

1 = 5/_2 0 . 7 1 K 3 l / 2 { 11 - 0 - 0 ) - 0 . 6 9 K= 3 / 2 ( 3 1 - 0 - 0 ) 0 . 1 3 K = 1 / 2( 1 0 - 0 - 0 ) a 7 / 2 K = 1 / 2 t 11 - 0 - 0 ) K n ---------j -

__?_3A li

; =

j -

_ 3 Y 2 _

9 / 7

_ r 0 . 91_K.=__

C. 9 3 K -

3 / 2 (_

1 / 2 (

3.1..r.O .

10 - 0

r Q ) . _

- 0 )

- 0 . 2 9

- 0 . 1 7

K=

K=

1 / 2 (

l / 2 <

.11 . .“ P.

1 1 - 0

, r 0 )

- 0 )

_ . _ - : 0 . . 2 8

- 0 . 1 1

K= _

K =

1 / 2 C

3 / 2 (

. 10.

31

- 0 . - 0 ) _____________

- 0 - 0 )

a _ 5 / 2 _ _ K=_

K =

. ! / ? <

3 / 2 (

_ i i „ r 0 _ r 0 )

31 —0 - 0 )

1 = 5 / 2 - 0 . 7 2 K= 3 / 2 ( 3 1 - 0 - 0 ) - 0 . 6 6 K = . _ l / 2 < 11 - 0 - 0 ) - 0 . 2 1 K = l / 2 ( 10 - 0 - 0 ) 5 / 2 K= 1 / 2 ( 1 0 - 0 - 0 )

J

In view of our previous conclusion that Ne behaves

like a rotator with two easily excitable neutrons, It is

not difficult to anticipate that such a "hole" excitation

could play an important role in the rotational structure 23

of Ne -. Moreover, the addition of a third neutron to the 20Ne core leads us to consider the impact on this structure

of a wide variety of other s=l and s=3 seniority excitations.23In our core +3 representation of Ne , the unperturbed ground

state is given by Ka=5/2(3/2,-3/2,5/2). The aforementioned

"hole" configuration is the three particle state denoted by

Ka=3/2(3/2,5/2,-5/2). Altogether there are 60 possible Re­

configurations in our valence space. As before, these are

mixed by the RPC + PPC + P perturbations.

23Our core +3 Interpretation of Ne J is Illustrated in

Fig. (7.6). The single particle energies employed in this21calculation are essentially those established in the Ne and

22 2 3Ne analysis. In Ne , however, there Is no ambiguity about

the relative order of £5/2 and el/2 f* M°st assuredly, theTTJ =.1/2 state at 1.02 MeV defines the necessary energy splitting,

namely £ j / 2 1 -e5 /2~® ' On the other hand, determination

of the strength P of the residual interaction is problematical.

2 2In the previous Ne calculation, it was absolutely essential

to postulate a small residual interaction between the two22valence nucleons. Indeed, the rotational structure of Ne

depends critically on a non-vanishing strength P. In view of23these facts, it is surprising to find that in Ne neither the

165

22

CIT

ATI

ON

EN

ERG

Y (M

eV

)

I6 r-

n EXP

M e 2 3 = f\ !e2 0 + 3

TH Y

166

FOPT

2

1/2

0.16,0.27

0 .22 ,0 .33

3/2 ,5 /2

3 /2 ',5/2

0.11

0.59

0.05

- 1/2

^ 3 / 2 " 1 1/2

- 3 /2- 7/2- 5 /2

9 /2

0.000 J § “

0 3 9

0.00

0.00

0.03

< ' / 2 |<7I /2 k 5

'9 /2 j<6'3/2 K 5

9 /2 K|

3 /2 K4

5/2 K:

3/2 K2

LU (2)

0

-------- 5 /2 ,7 /2 ,9 /2 ^0.05,0.10 3 /2 ,5 /2 0.01

WEAK

0.70

3 /2 (7 /2 )x

\

0.02

0.09\

1/2 0.55

- 9 / 2 y '3 /2 3 /2 0.13

/

5 /2 //

/

7/21/2

/

0.53

, 5 /2 [<2-7/2 K

/ /~0.00 ' 3 / 2 K3

1/2 K 2

00.2.2 5 /2 0.29 5 /2 0.33 5/2 K

Fig. (7.6)

2 o 20Table (7.4) Ne °=Ne +3 DSM Parameters and Dominant Configuration Amplitudes

A zz o U) o

£ 3/2 = 0.00

e5/2= 3.50

el/2f= 4.25

el/2"= 7.50

£3/2' 9.00

P 55-■2 .00

_____________________________ DOMI NANT. C O NF I G . A M P L I T U D E S FOR THY _____________________________ PART I C LE CONF I G_F OR_ FO PT ST AT E S_

J = 5 / 2 ____ r .0 ..° .3 _ K = _ 5 / 2 ( 3 0 - 3 0 5 0 ) _- 0 . 3 3 .K*__ 3 / 2 ( . 3 0 5 0 - 5 0 ) ___________ 0 9 K = _ _ 3 / - 2 < - 3 0 5 0 . 1 0 ) ____________ 1=__5 / 2 _ K=__ 5 / 2 ( . 3 0 - 3 0 5 0 ) __ ^ 1 _

i . = _ _ l / 2 _____ ,C. ? 7 . . K = _ _ 1 / 2 ( 3 0 - 3 0 1 0 ) . . . - 0 . 2 4 K= , l / 2 < 5 0 - 5 0 1 0 ) . . . - 0 , 0 8 K= .. l / 2< 3 0 - 3 0 11 ) ___________ I = . 1 / 2 . . K= 1 / 2 ( 3 0 - 3 0 1 0 ) _ ^ 2 -

1 = __7 / 2 _____0 . 5 , 3 . , K = __ 5 / 2 (. 3 0 - 3 0 5 0 ) . . . . 0 . 3 0 _ K = _ _ 3 / 2 ( . 3 0 5 0 - 5 0 ) _ „ - 0 . 2 2 K * . . _ 7 / 2 ( 3 0 5 0 - 1 0 ) ______ '_____ I =»___ 2 / 2 _ _ K = _ _ 3 / 2 ( . 3 0 5 0 r 5 0 ) ___ ^ 3..

T= 5 / ? 0 . 9 0 K = l / 2 ( . 3 0 - 3 0 . . . 1 0 . ) - 0 . 24__K=______ 3 / 2 ( _ . 3 0 . . 1 0 r l 0 ) _______ 0 . 2 2 _ K = _ 3 / 2 { - 3 0 . 5 0 _ 10 )____________ I j f ____ 7 / 2 K=__5 / 2 ( . 3 0 - 3 0 5 0 ) ^ 1

l 5 _ _ 3 / 2 0 . 81 K = _. 1 / 2 ( 3 0 - 3 0 1 0 ) . . - 0 . 3 9 K= _ . 3 / 2 ( 3 0 5 0 - 5 0 ) . . - 0 . 3 4 K= l / 2 ( - 3 0 5 0 - 1 0 ) ________ I=*. . 5 / 2 . . K= ... 1 / 2 ( 3 0 - 3 0 1 0 )

J = _ _ 3 / 2 _ J _ C . 87. K = _ _ 3 / 2 { . 3 0 5 0 r 5 0 ) 0 . 3 0 . K=__. 1 / 2 < . . 3 0 - 3 0 1 0 ) . . - 0 . 2 4 K=>_. 1 / 2 1 - 3 0 5 0 - 1 0 ) ____________ I = . _ 2 / 2 _ _ K= _ _ 1 / 2 (. . 3 0 r 3 0 1 0 ) _____

T = Q / 7 - 0 . 7 3 K = 5 / 2 ( 3 0 - 3 0 ___5 0 ) ___- 0 . 3 5 . . K * _3 / 2 ( _ 3 0 _ 5 0 - 5 0 >_____r 0 . 3 1 . _ K = ___ 1 / 2 ( _ 3 0 - 3 0 _ 1 0 ) ____________ I f 5 / 2 K= 3 / 2 ( 3 0 5 0 - 5 0 ) K 3

K •J = _ _ 5 ?/ 2___ 0 . : 7 5 . K.= - . , l / 2 ( . 3 0 - 3 0 1 0 ) _ _ - 0 . 3 4 ,K=. _ 5 / 2 ( 3 0 - 3 0 5 0 ) r 0 . 2 9 K= .. 3 / 2 ( 3 0 1 0 - 1 0 ) _____________1= . 3 / 2 K = . . 3 / 2 ( _ 3 0 1 0 - 1 0 ) _ _ _ A .

, J = _ _ J / 2 _ . . . 0 . £6 K= 3 / 2 ( 3 0 5 0 - 5 0 1 . . . . - 0 . 2 8 K =* . _ _ 5 / 2 ( 3 0 - 3 0 . 5 0 ) ’ . - 0 . 2 7 K = . 3 / 2 ( 3 0 1 0 - 1 0 ) ___________I . S / 2 . . . K = _ 5 / 2 ( . 3 0 - 3 0 5 0 ) . . j A

r = 7 / 7 ___ - 0 • 71 _ K = ___ l / 2 ( . . 3 0 - 3 0 . _ 1 0 ) ____ 0 . 4 3 . K= _ _ 3 / 2 3 0 . 1 0 - 1 0 ) 0 . 4 2 . K = _ _ l / 2 ( - 3 0 5 0 - 1 0 ) ___________1=_____2 / 2 __K= l / 2 < _ . 3 0 - 3 0 1 1 ) ____

, J = 3 / 2 ____5 . 8 £ _ k = _ _ 3 / 2 I _ . 3 . 9 _ 1 0 -1 .0 1 ___ 0 . 3 5 . K f ^ , _ 3 / 2 ( . - 3 0 , 5 . 0 _ 1 0 . ) _____0 . 1 7 . k r _ _ l / 2 ( . . 3 0 r 3 0 : 1 0 ) _____________ 1 2 — 5./ 2 — i S f _ _ ? / ? . ( _ 3 0 _ 5 0 1 0 ) _ _ _ ^ 6 _]/-. J J 1 / 2 _ . „ Co 7.6 K = _ 5 / 2 < 3 0 - 3 0 . 5 0 ) . _ „ 0 . 4 0 K = _ _ 3 / 2 < „ 3 0 5 0 - 5 0 ) __._rP . 3 5. K = . . 7 / 2 { 3 0 . 5 0 - 1 0 ) ....................1 _ 1 / 2 K - __ 1 / 2 ( _ _ 3 0 - 3 0 _ l 1 ) 5 __

T= 3 / 2___ rC » .£ 5 „K = ___ l_ / .2 .(_ 3 .0 -3 .0 . .1.11____0 . . 2 9 . K f ___ 3/_2(__3_0.-3 0 _ 3 1 >_____ Q . 2 0 , _K=_____1 / 2 ( _ . 5 0 - . 5 0 _ _ l l ) _ ___________1 =____ 1 / 2 __K» l / 2 ( - 3 0 5 0 - 1 0 )

168

Experimental References for Ne

Table (7.4) (Continued)23

B. Chambon et.al., Nucl. Phys. A136 (1969) 311.

Dumazet et.al., Comp. Rend. 264B (1967) 1514.

D.B. Fossan et.al., Phys. Rev. 141 (1966) 1018.

A.J. Howard, J.G. Pronko, and C.A. Whitten, Jr., Yale Preprin3223-192, to be published in Nucl. Phys.

M. Lambert et.al., Proc. Japanese Conf. on Nuclear Structure(1967) 112.

H. Nann et.al., Z. Physik 218 (1969) 190.

A.J. Howard et.al., Phys. Rev. 154 (1967) IO67.

energy levels nor the spectroscopic factors are very sensi­

tive to large variations in P.

As a result, a constant feature of our model is the

prediction of a fixed sequence of seven low-lying levels

where only six have yet be.en found. In addition to this,

we note the persistence of a small 1 MeV gap centered about

3 MeV excitation energy. It may only be coincidental that

a similar gap appears in the experimental spectrum. Very

little Is known about the spins and parities of states above

this gap. The most conclusive statement which can be made

TTis that both theory and experiment predict a J =1/2 state

near 5 MeV excitation energy. Some of the experimental

states in Fig. (7-6) above 4 MeV undoubtedly have negative

parity assignments and thus lie outside the scope of our 2 3model for Ne . These negative parity states may correspond

to hole excitations in the lp shell (Li 70) and particle ex­

citations to the 2p-lf shell (Ho 67, Ho 70a).

The energy shifts produced by the Coriolis mixing in

23this core +3 calculation for Ne are comparable to those

21found in Ne . As Fig. (7.6) shows, the ground state band —

at least as regards the 7/2 member — is too compressed.

Interestingly however, there exists a possible candidate for

the 9/2 member at the correct excitation energy. Another

apparent anomaly in our calculation is the following. If the

5/22 level corresponds to one of the observed states at 2.31

or 2.52 MeV then it invariably falls too low for all reasonable

choices of the reciprocal moment of inertia and single

particle energies.

The original conjecture that there is a low-lying

K=3/2 "hole” in Ne2^ is substantially confirmed by this

calculation. According to Table (7.4), the 3/22 level is

lS% pure Kcr=3/2(3/2,5/2,-5/2)— the "hole" configuration in

question. In reviewing the wave functions in Table (7.4),

we find that this "hole" configuration strongly admixes

into the ground state band. Moreover, another type of K=3/2

"hole" state is seen to contribute to the low-lying rota-23

tional structure of Ne . Specifically, we refer to the

Ka=3/2(3/2,l/2',-1/2’) component in the 5/22 state. These

two K=3/2 "hole" states are largely responsible for the

depression of the ground state rotational band and the 5/22

state as mentioned above. We view this development as evi­

dence for a more subtle choice of the residual Interaction

between the three valence nucleons.

There is some uncertainty regarding the appropriate ex­

perimental candidates for the 3/2-^, 3/22 , and 5/22 levels.

Mindful of our criterion for rejecting the core +1 model

23for Ne , we note that the spectroscopic factors for the

states under consideration are not inconsistent with the

available experimental information. Inasmuch as the theore-23tical spectroscopic factors for many states in Ne are very

small and the observed levels have been excited almost ex-22 2*3clusively by the Ne (d,p) Ne D reaction it is a definite

1 7 0

possibility that not all of the low-lying levels in Ne23

are known (Ho 70b).

An ideal probe into the core +3 structure of Ne23

should be the Ne2^(t,p) Ne23 reaction. If the (t,p) re­

action proceeds via a single step mechanism, the two

neutrons will be transferred to the unoccupied orbitals 21of Ne without breaking up the target ground state. As

we have previously seen, the ground state of Ne21 is J7r=3/2+

and corresponds to a very pure Ka=3/2(3/2) configuration.

Hence we are led to expect that the two neutrons which are

transferred in the (t,p) reaction will preferentially popu-23late state in Ne J with Ka-components of the form Ka=K(3/2,ft^,ft2)

where K=|3/2-fi^-fi2 I and therefore provide a more rigorous

test of the coupling scheme and mixing in our model.

Of particular interest in this respect are (t,p) transi­

tions to K=3/2 "hole" states in our model. These consist of

a class of Ka-configurations given by Ka=3/2(3/2,ft,-ft). In

general, simple angular momentum considerations limit (t,p)

transitions to Ju=3/2+ states in Ne23 to L2n=0,2 transfers

(see Table (5.1)). Recalling the K-selection rule for stripping

given by Eq. (5.32) we find that either L2n=0 or 2 is allowed.

On the other hand, transitions to the Ju=3/2+ member of a

K=l/2 configuration can proceed only through E2n=2 transfers

according to the K-selection rule. Therefore within the limits*IT 2 ^of the mixing, we expect the J =3/2 states in Ne ^ which are

largely- based on K=3/2 "hole" configurations to have a pro-

171

minent L2n=0 stripping signature and as a consequence be

readily Identifiable.

At present, this is no known report in the literature.2 1 p q

of the Ne (t,p) Ne 5 reaction.

As a postscript to the present theoretical and experi­

mental status of Ne2^, we note with curiosity the omission23of shell model interpretations of Ne J in several compre­

hensive surveys of s-d shell nuclei (Bo 6 7, Ha 68a, Ak 6 9).

In presenting our unified model interpretation of the21 22 23three neon isotopes Ne ' * , we have concentrated on the

salient mechanisms in the model which are relevant to each

isotope. These results can be synthesized into a simple

inter-comparison of the isotopes which emphasizes their

common origin. The reciprocal moment of inertia which we

have used in these calculations is assumed to be that of 20the Ne core. This is estimated to be A=0.27 MeV from the

0+-2+ spacing in Ne2^. This value is about twice as large2i 2?

as that commonly thought to apply to Ne and Ne (Go 68,

Hi 6 9, Li 70). We, however, have consistently used A=0.25 -

0.30 MeV for our core +1, core +2, and core +3 calculations21 29 po

of Ne , Ne , and Ne 3 in agreement with the assumption of

a Ne29 core.

In Fig. (7.7) we plot the excitation energy of the mem-9n

bers of the ground state rotational band of Ne as a function

of J(J+1) where J is the total angular momentum. If Ne2^

were an ideal rotator, the resulting curve would be a straight

line with slope equal to the reciprocal moment of inertia

1 7 2

of the Ne2^ core. Although the 0+-2+ spacing In Ne2

implies that A=0.27 MeV,.the extended curve defines an

effective reciprocal moment of inertia given by Aeffl=0.l8

MeV which is more In agreement with the aforementioned21 ? ? estimations for Ne and Ne .

It is convenient to refer to the Ej vs J(J+1) curve

as the yrast line (Gr 67). The yrast line is the locus

of points which points represent the lowest lying states

of a given angular momentum. Technically, the yrast line

is plotted as a function of J instead of J(J+1). Neverthe­

less, its meaning remains the same. For the neon isotopes,

the yrast line consists of the locus of points determined

by the ground state rotational band.

In Figs. (7.7) and (7.8) we plot the yrast line for Ne2'L 22and Ne . Remarkably, the theoretical calculations predict

effective reciprocal moments of inertia in substantial agree­

ment with those suggested by the yrast lines for these nuclei.

We find for Ne21 A ff=0.l4 MeV and for Ne22 A ff=0.15 MeV.23From Fig. (7.8) the theoretical estimate for Ne is Ae^ .=

0.17 MeV. The agreement between these values and that for20the Ne core, namely Aeff=0.l8 MeV, is astonishing.

Since the theoretical calculations use A=0.25 - 0.30 MeV,

it is apparent from these remarks that not only does the

Coriolis mixing generate the kinks in the yrast line as is

well-known but it is also responsible for rotating the yrast

line clockwise thereby obscuring hhe true moment of inertia

173

174

Figs. (7.7) and (7.8) Yrast Lines for Neon Isotopes.Open and closed circles represent deformed shell model calculations and experimental values respectively. Dashed line yields effective slope of yrast line.

EXC

ITA

TIO

N

ENER

GY

(MeV

) EX

CIT

ATI

ON

EN

ERG

Y (M

eV)

175

N e 20

J (J + l)

Ne2 2

j (j + i )

F i g . ( 7 . 7 )

Fig. (7.8)

EXCITATION ENERGY (MeV)

O — ro w m

d+

p)r

EXCITATION ENERGY (MeV)

O ~ ro w 4> cn

of the core. Finally, it is appropriate to reiterate a

point we mentioned much earlier and that is that the

Coriolis mixing also prevents us from determining the

single particle energies directly from the spectra of

odd-even nuclei.

C. Magnesium Isotopes

1. Preliminary Remarks

We now attempt to describe the magnesium isotopes Mg2**,26 27 p li

Mg , and Mg as a Mg core with 1,2, and 3 valence nucleons.

Our procedure is analogous to that established for the neonPfiisotopes. The motivation for this endeavor is that Mg is

substantially more complex than a simple even-even rotator 27and Mg appears to have a low-lying hole state not account­

able for in a simple core +1 model. This is precisely the22 23situation we encountered in Ne and Ne . Thus it is

natural to ask to what extent the extra degrees of freedom 26 27evident in Mg and Mg can be attributed to several active

24valence nucleons outside a Mg core?

We distribute the valence nucleons in the Hartree-Fock24 24 20orbitals of the Mg core. In the case of Mg , unlike Ne ,

we encounter an axially symmetric and axially asymmetric

representation of the Hartree-Fock deformed orbitals (Ba 6 5 ,

Pa 68). Current experimental evidence is consistent with an24axially symmetric interpretation of Mg (Ro 6 7 ). The

appropriate Nilsson coefficients c ^ for this Hartree-Fock

177

1 7 8

solution of the deformed orbitals are listed in Table (7.9).24They correspond to a prolate deformation of the Mg core.

In this case,.the unoccupied orbitals in the s-d shell are

K=5/2, 1/2', 1/2", and 3/2'. Accordingly, the basis space

for Mg is 4-dimensional and that for Mg Is 16-dimensional.

Again we contrast the core +1 and core +3 interpretations for 27Mg . The corresponding dimensions are 3x3 and 28x28

respectively.

2. Magnesium 25 25Mg was first described in terms of the unified model

by A.E. Litherland et al. in 1958 (Li 58). Using first order

perturbation theory as given by Eq. (5.26), they demonstrated

that the strong-coupling wave function is a good approximation

for this nucleus. However, detailed calculations of the level

ordering, electromagnetic properties and spectroscopic factors

from the intrinsic wave functions of the Nilsson model produced

only fair agreement with experiment. More recently, F.B. Malik,

and W. Scholz (Ma 6 7 ) reproduced an excellent fit to the low-

lying energy level spectrum within the Nilsson phenomenology

when Coriolis mixing was included between all bands with4 *

AK=1 instead of just the diagonal contribution from K=l/2

bands as in first order perturbation theory. Unfortunately,

they did not test the validity of the admixing In the wave

functions.

^and K=K’=l/2 but a/o’ (cf. Eq. (5.23)).

the spectroscopic factors for the Coriolis mixed wave

functions. Of course our primary reasons for reviewing 25Mg are to test the assumption of the Hartree-Fock deformed

24orbitals of the Mg core and to estimate the band headOJT 0 7

energies for our Mg and Mg calculations.25Our Mg ^ calculation is summarized in Fig. (7.9). A

number of comments regarding these results are in order.

Using three parameters, namely A, > and ei/2" as dis_

cussed in Chapter VI, we are able to fit the first seven

theoretical levels to within 0.1 MeV of their experimental

counterparts. Moreover, the predicted spin sequence of the

lowest ten levels agrees with experiment provided we assign

a J =3/2+ spin to the 2.80 MeV level. As expected, the ground

state rotational band is very pure because there is no first

order Coriolis coupling to the K=5/2 band in our space. The

Coriolis mixing between the other bands is stronger and pro­

duces the desirable shifts in the 5/22 and 7/22 levels. It

also enhances the spectroscopic factors for the 1/2^ and 3/22

levels.21More so than in our Ne test of Hartree-Fock deformed

25orbitals, discrepancies are evident in our Mg test case

which may be attributable to the inflexible assumption of

axially symmetric Hartree-Fock intrinsic states. The large

gap between the l/22 and 3/22 levels compared to that ob­

served between their experimental counterparts at 2.56 and

2.80 MeV reflects a small decoupling parameter predicted for

179

25I n o u r u n i f i e d m ode l c a l c u l a t i o n o f Mg , we e v a lu a te

180

Figs. (7.9), (7.10), (7.12), (7.13) Deformed Shell Model Excitation Spectra for Magnesium Isotopes - Namely Mg25s Mg26, and Mg27 as a Mg2 rotator core plus 1, 2, and 3 valence nucleons respectively. "FOPT" denotes first order perturbation theory; "THY" the complete diagonalization; and "EXP" the experimental values.

Tables (7-5)— (7.8) Deformed Shell Model Parameters andDominant Configuration Amplitudes for Magnesium Isotopes. The following notation is used for the wave functions: K=5/2, 1/2', 1/2", and 3/2’deformed orbitals are denoted by 50, 10, 11, and 31 respectively.

The complete explanation of these figures and tables is given on pp. 139-140 and Section (VII. C).

EX

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ON

EN

ERG

Y (M

eV)

181

M g 2 5 = M g 2 4 + I

£ n EXP t h y f o p t

6 r

1/2MANY

LEVELS0 .27

9 / 2

3 /2 M I _______ , 3 / 2 K40 ^ 0 --------- . . c7/2 K2

- 5/2 K30.01

0.01 ( 9 / 2 ) g / 2 K|— ------------ 3 / 2 , 5 / 2 / ------------------- 9 / 2

0 0 6 3 /2( 9 / 2 ) '

----------- 3 / 2 *<3

2 0 -2 2 : -Q-4 0 ^3 / 2 , 5 / 2 7 /2 0 .10 c /o Ko0 0 .09 -0 .15 w a > 0 3 7 ---------- 1/2 ^ ^ ^

2 5 / 2 5 /2

7 / 2 ---------( 7 / 2 ) - " o . 2 6

2 0 . 2 0 - 0 . 5 0 3 /2 2 ^ 2 . 3 / 2

O 33 P*J.2---------- |/2 K2Q 0 . 2 5 - 0 . 5 0 ) / 2 --------------------0 133---------- | / 2 i / g 2

2

7 /2 K l

3 / 2 k 2

0 . 3 3 - 0 . 4 2 g / 2 ________ 0 . 3 5 5 / 2 ___________ 9,3 3______ 5 / 2 x l

F i g . ( 7 . 9 )

pc phT a b le ( 7 . 5 ) Mg ^=Mg +\ DSM P a ra m e te rs and D om inan t C o n f i g u r a t i o n A m p l i t u d e s

A 0.25 MeV

e5/2 = 0.00

el/2,= 0.75

el/2"= 3.00

nCM\onu 5.00

_c f MJAANJ_i:cNn5_*mJJu_cEi_Fjp_R_^j±y___________________________________________________________________________ pab i m i . f rjNfjfL f £B_ f s p j_ s I a I e s _________KT = 5/.? 1 .0 0 K* 5 /2 t 5C -Q - 0 ) ____0 .0 6 K= 3/ 2 (...2.L-.0 -01------ = 0 .3 2_K=----1/ 2 t.JLQ—-Q --C J__________ 1= 5 /2 K= 5 /? ( 50 -0 - 0 ) 1

l=__3J2.__3j3J.Jk:— JJ2J.33.-3.-31— s3^23-&?— JJ2.i-13--9-r21— Qj2.0.&*__3./2A-31--0--OJ----------------— 122 ____ UY2j;_10_-D_rOi ^ 2 _____V-

3 J3J_7o__J7iU_JJ--3_-3J__.r3.»3JL_*J?__37.2J_3-l--.0_.r.0J---------------- J =__ 332_______ 17.2i_JL0_-0_rQl____ 2______KT = 1 H 0 .9 9 K - 5 / 2 t 3C -C -C ) 0 .11 K= 3/2 ( 31 -_Q_-.QJ------ =J1..0S_K.?----1J 2.C_10_rJ>_^0 J__________ L=___ 172__K= 5 /? t 50 -0 - 0 ) 1

l=__3J2-----------------------------------------------------------------------------------------------------_b..JS_X?__3/2l_31_-JO_r.OJ---------------_J =__UY^__JK3__U72J_ll_-0_rQl___ ^ 3 _____V

L =__J7iL l_f^3J_ *J:__J73J_JJ-r£_-.C J___Q^2S-X^— U 21.13.-0--91-------Q*Q0-te— 2J2S- 3J_jr.D_.r.0J----------------J =— 5J2— 1221.1D.rO.=Ql____2_____T = 7 /2 O.fiO K - I /2 ( 10 -0 -0 ) ____0 .4 8 K= 1/ 2 (_ .U . .-Q—-.Q)____ -0.-3 3_K=__ 3 / 2 (_3 L_rO_r.OJ__________ 1= 3 /2 K= ! / ? ( 11 -0 -0 ) K 3

...l3..3J2.s3^52.33— JJ21.13.-2.-3J____3.-39.J< = ..S/xl.JO. -3__-3J-----D..CjiK?— 3/2 < _ 3 1 _ _-.0_r 3 ) ----------------J _SI2._ _ 3 7 2 J _ 3 0 _ -0 _ r 0 J____ ? 1 _____K J .522. IU?3_ J<_=__522S.39..3.-3J__ 9.-33. 53‘..332S.3X.-D.r91— A? 1/2_<_ U0_ _-.0_ - OJ___________ J “__372__B =__.172J_ .U _-3_r01_____3_‘____KT. 5 /2 0.8C K - 1 / 2 ( 11 -0 - 0 ) - 0 .5 4 K= 3 / 2 ( 31 -0 -01 _£L^Z5_K-_l./.2X_JLO_rQ_-OJ__________ L= 7 /2 K= l/2 < 10 - 0 -0 ) 2K J_= 322. 0^53.7_=__ 2221_33.s3. ~3J__ 3.-U..K5_ U2J.33. ~D_r31-------3.»1.4_Jlr— 17.2J_Jl_-.0_-3J ---------------- Ji?__37il__B =__372J_31_-0.:rQ l____4 ______

lj__5J2— 3.-J:3-3J— JJ3J-J3— -3-s3J— -3--J>t>_X=— J./2J-JLL-J)--3J— -.0^23-&?— 3.S2,l-2±.-£--.QJ----------------J_=__522-_ 7_=__2J21.21. -3_ r 01________1» 1 1 /2 - 0 .6 2 K= 5 /2 t SC -C - 0 ) 0 .5 7 K= l / 2 ( 10_r0 -01------ CL..23_&H 17.2 ( . 1 . - 0J----------------Ls UL2 102 I /? t 11 - 0 -3 1_____________

182

183

Table (7.5) (Continued)

25Experimental References for Mg

Bibijana Cujec, Phys. Rev. 136 (1964) B1305.

P.M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1.

S.M. Lee et.al., Nucl. Phys. A122 (1968) 97.

184

the Ka=l/2(1/2") band by the Hartree-Fock model. However

we readily add that recourse to axially asymmetric Hartree-

Fock states may alleviate this problem but leads to too

large of a spacing between the 1/2^ and 3/2.^ levels (Ba 66).

Further related to the symmetric solution for the K=l/2"

band, we remark that the spectroscopic factors for the l/22 ,

3/22 , and 3/2^ levels are in less than fair agreement with

experiment.

This may not be entirely the fault of Hartree-Fock or­

bitals. Regarding the J=3/2 levels, a definite element of

the problem is that the K=3/2* band head (Nilsson orbit No. 8)

has never been identified in Mg2"3 (Li 58, Cu 64). A possible

explanation for its apparent absence is suggested by the

Coriolis mixing. According to Eqs. (5.42) and (5.43), the

maximum spectroscopic, factor the J=3/2 member of the K=3/2'

band could have is S=0.50 which is attained for the Nilsson

coefficient c(v=3/2' ; j=ft=3/2)-*T .0. The maximum value cal­

culated from the Hartree-Fock single particle wave functions

is S=0.47 as can be seen by referring to FOPT in Fig. (7.9).

These spectroscopic strengths alone suggest that the K=3/2*

band head should stand out sharply in a (d,p) stripping re­

action. However, as Table (7.5) shows, even though the 3/2^

state is 94% pure Ka=3/2(3/2') (Nilsson orbit No. 8) the

Coriolis mixing depletes nearly 42% of the spectroscopic

strength of this state. This is a trend in the right direc­

tion and suggests why the K=3/2' state has never been populated

in a (d,p) reaction.

185

Of course for this to hold rigorously, the 3/2^ state

should have a zero spectroscopic factor. Calculations were

performed with the K=3/2' band head at a lower position than

that shown in Fig. (7.9). This resulted in a transfer of

spectroscopic strength from the 3/2^ state to the 3/22 state

and to a lesser extent to the 3/2^ state. This improves

the theoretical description of the 3/22 and 3/2^ states

somewhat but cannot be regarded as yielding conclusive evi­

dence as to where the K=3/2' band head might be found in the 25Mg spectrum. Unfortunately, the K=3/2* band cannot be

identified by (d,p) stripping to its J=5/2 member because the

spectroscopic factor for this level is negligible under all

circumstances relevant herein.

It is generally believed on the strength of the similarity

of Mg2-3 and Al2"3 as 13th odd particle nuclei that the K=3/2'

band head in Mg2-3 is located near 4 MeV excitation energy

(Li 58, Cu 64).

Apropos to the location of the missing K=3/2' band headpc 2*7

in Mg is the position of the same for Mg and Mg . As

we shall see, the K=3/2' band head stands out unequivocally 27in Mg and the theoretical spectroscopic factor predicted

for this state — using the same K=3/2' band head energy and25Hartree-Fock intrinsic states as for Mg — is in excellent

agreement with experiment. This result may be mostly fortui-

25tous in view of our qualifications of the Mg wave functions.

Nevertheless, this provides an interesting and enlightening

contrast to the problems encountered in our detailed analysis

of Mg25.

Despite several quantitative shortcomings of the axially

symmetric (and axially asymmetric (Ba 66)) Hartree-Fock24representations of the deformed orbitals of Mg , there can

25be little doubt that the low-lying levels of Mg can be

grouped into interacting rotational bands are adequately

describable by the unified model. It is therefore of great

interest to display the qualitative structure this model0*7

implies for Mg and Mg in terms of a core +2 and core +3

interpretation.

3. Magnesium 26

In recent years there has been considerable speculation

about the nature of the excitation mechanisms manifested in2 6the low energy spectrum of Mg . This speculation has grown

26because the disparity between Mg and its rotational neighbor

Mg2** is truly striking. In the first place, Mg2^ has approxi­

mately three times the density of states below 7 MeV excita-

24tion energy as has Mg . Furthermore, it is not even clear

if Mg has a ground state rotational band embedded in these<tr -f- ^ *|*

states. In the lowest states, given by J =0 , 2 , 2 , 0 , 3 ,

we find one of the most unusual spin sequences observed in a

nucleus in a presumably rotational region of the periodic

table.

It- is precisely these anomalies which have encouraged the

characterization of this nucleus with all manner of collective

186

a K=0 ground state rotational band comprising the and 2^

levels, (2) a K=2y-band including the 2^ and 3^ states

(Cu 64), and (3) a K=0 6-vibrational band beginning with the

C>2 state (Ro 6 7 ) although others have surmised that this

state may be a K=0 band head for a series of rotational

levels based on an excited configuration of the last two24neutrons outside of a Mg core (Cu 64, Hi 6 5).

The conjectures about the 6 and y bands have been ex­

amined in detail in terms of the rotational-vibrational,

asymmetric rotator, and vibrational models (Ro 6 7 ). The

conclusion of these calculations is that collective effects

26do not play an apparent role In the structure of Mg

This is not surprising in view of our earlier conclusion

(cf. Chapters II and III) that 6 and y excitations are not

even to be expected among the lowest lying states (i.e. below

5 MeV) in nuclei in the first half of the s-d shell. Our

22subsequent description of Ne as a core +2 system further

supports this belief. Hence it remains to consider the

possibility that Mg exhibits interacting rotational bands

built on excited configurations of two neutrons outside a 24Mg core.

2 6The results of our core +2 interpretation of Mg are

shown in Fig. (7.10). These results can best be regarded as

schematic. Overall, our model cannot reproduce the observed

22density- of states as it did so successfully in Ne . As

187

e x c i t a t i o n s . The lo w e s t l e v e l s have been g ro uped i n t o ( 1 )

EXCI

TATI

ON

ENERGY

(MeV)

188

0+2

2 6 _ n _ 2 4 + g

(2) ------------------ (3) d , Kq--------------------o k 7

0 0 : 0 x2 0 2 4 3

= — < 2 K52 —_____________ A3) 2 4 4

------------------ - ( 1 , 2 ) ------------------ 44 2 4

2 (1,2) K0 ( 2 K _____________ 0 \ Kf

, ----------------- 4 - " ' i 4

!2n £ n EXP THY FOPT

(2) Ov______________/ 3 . // 4

C (l, 2 ) /^//i\ / — — 5(4)

(2) 0 3 2 3 K4

0 2 0 I 2 K3

0 K2

2 -----------------------------------2 Kl

0 *- 0 2 0 0 0 K l

F i g i ( 7 . 1 0 )

o o hT a b le (7.6) Mg =Mg +2 DSM P a ra m e te rs and D om inan t C o n f i g u r a t i o n A m p l i t u d e s

E5/2

= 0 . 2 5 MeV

= 0 . 0 0

£ ]_/2 *= ^

el/2"= "

E 3/2'= ^ ^P =-1.50

________________________ _C C V_I N A NJ_ _C C A_F_I G_ AMf LJJ UC E_S_ f 0 R_ J± Y_______________________________________________f i i? JJ .£U _ f.CMJ CL f CfL f £ PJ_ 51 Al £ 5________1= o 1 .0 0 Kc C( 5 0 -5 0)____________ O .U i.-3 .U ___ J)..£)0„K=___0.{_ 10-.I0.)__________________________ 1= Q k. nt 4n-4n> K 1___________

_I_=___2______ J_,C>CL J<_=_ _ £J__5Cb_5C )_ __ CLC. 5_ *_=_ _1J_ 5 0_-31J _ _ _-.C. .0 3_ J<?_ . 2 1 .5 0 -1 0 .)_____________________________ i= _ _ 2 _______________________________________J _=_--P____ r£L.S3_ J<_«_ - _ JCLJOJ_ _ rC L iM J< _ OJ_ J J - l i J____0_,J£LJS5__.C-<-3J^3.1J_______________________ _____ i= _ _ £ _____________________________^ 2 ________1= 2 - 0 .5 6 k- 2 ( 5 0 -1 0 )____SU12_JLr_U_5.Q.rllJ_____ 0 ,0 5 ___1JLLLCL3U_____________________________ 1= 2 K= 2( 5 0 -1 0 )___ K3___________

J_=__3_____"CL5 6_ M=_ _ JJ MCL JC J_ _ M ., JL=U _ iLL 30 - J1J _ _ _ 0. .0 5_ * =_ _ 21 _ JCL 3 JU___________________________ :_J=__3____ ^ _ _ 3 J _ 3 3 _ iO J ____ ? 4 ___________J_=___2______________________________________________________ 0_.2.1_jtL=__.lCrl0_32J_____________________________ J=__2_^__J<=__D l_a0ri0J___ ? 2 ___________1= 3 0 .6 6 K= 2 (- 50-1C ) =O.40_J<x 3_(_5.C_1 I ) d?_.23_J<f___l.(_OC- 3 L)_____________!________________ 1= 4______K= 0( s r - s m K1___________

J_=__5_____ DJi^_K_=__3J_>5_3i)J__-i:.^_J<_=__i)J-^_-3.C.)__-i..33_J<_=__A(_35_3.1J____________________________ _ l5 _ _ 3 ____ P=__2J_.50ri.OJ___ ? 3 ___________J_=___A______O^J_5_ J< _ OJ_ -50_-J0J_ _ oO., J?2_ M=_ _ 3J_ 3.CL J0U_ _ _0,2.5_ _ .4J _ 5 0 . 31 J ___________________________________ JS=__3J_30_JDJ___ii)____________

KT= 4 0 .7 3 K = 2( 5 0 -1 0 ) - 0 .4 7 K= 3( 5Q 11) - 0 .2 6 K= C( 1 0 -1 0)______________________________ 1= 2 K= 7 1 50-1 1 » 5__________

J_=_„2____ ___________________________ 0.,21_ JL=_ _ JJ _ 3D-31 J_ _ _ 0_,Jd_ J<_=_ _2J_JQ _32J____________ J________________J=__3____ Jfc?__3J_30_.UJ_____ §___________J _-XL, JJL JL=_ _ OJ _ JO.-JOJ___0-, Ai_J<J= U -JO LJJJ ML, 23. K_=_ _ J i r J3_ 3 J J _______________________ L____ J=__.Q____ .&3__Oi_.10Ml.l____^ 7 ___________

1= 0 - 0 .5 5 K» C( 1 0 -1 1 ) C. 14 K= 0( 31 - 3 1) C.O2 K = 0( 1 1 -1 1 )______________________________ 1= 1 K= 1( 5G -21) K8___________(->oovo

Experimental References for Mg

T.R. Canada, R.D. Bent, and J.A. Haskett, Phys. Rev. 187(1969) 1369.

Bibijana Cujec, Phys. Rev. 136 (1964) B1305.

T. Daniels, J.M. Calvert, and A. Adams, Nucl. Phys. A110(1968) 339.

P.M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1.

0. Hausser, T.K. Alexander, and C. Broude, Can. J. Phys. 46 (1968) 1035.

S. Hinds, H. Marchant, and R. Middleton, Nucl. Phys. 67 (1965) 257.

S.W. Robinson and R.D. Bent, Phys. Rev. 168 (1968) 1266.

190

T a b le ( 7 . 6 ) (C o n t in u e d )2 6

Fig. (7.11) shows, this problem is also symptomatic of the

few shell model calculations which have been attempted for

this nucleus. Of particular interest, we note the presenceTT +of a J =0 state near 5 MeV which lies outside the scope

of the spherical and deformed shell models. It seems likely

on the basis of our estimate of the hydrodynamic threshold

for 8 and y vibrations (see Fig. (3.4)) that this state could

reasonably be a K=0 8-vibrational band head. In addition to

this State, there are many other states near and above 5 MeV

excitation energy which are not accountable for by these shell

models.

Consequently we make only a few cursory remarks about26our core +2 model for Mg . In fact, we limit most of our

remarks to those levels below 4.5 MeV where the mixing from

the higher-lying modes of excitation can be minimized as much26as possible . Of course our wave functions for Mg are sus­

pect because they are missing contributions from the 8 and y

and other degrees of freedom so evident in this nucleus.

In Fig. (7.10) we see that the Coriolis mixing has little

apparent effect on the energy levels of the ground state

rotational band — in marked contrast to the situation en­

countered in Ne22. Looking into Table (7*6) at the relevant

wave functions, we find this to be true only as far as the 0^

and 2^ levels are concerned. They are virtually 100% pure

Ka=0PW(5/2,-5/2) which is the unperturbed ground state con­

figurations formed by adding two neutrons to the K=5/2 orbital

191

EXC

ITA

TIO

N

ENER

GY

(MeV

)EXP

6 -

0

\

0

2 6

UNIFIED MODEL SHELL MODEL SHELL MODEL(THIS WORK) (Su(3)TRUNCATION) (INTERMEDIATE

(St 6 8 ) COUPLING)

(3) .....I

(1,2) 3(4) 3

- 0 ' 0- 2 *4

(3) 2 _ p“"(1,2) 4

4 ' I‘ (1,2) " I

^ 0 4- ( 2 , 3 ) 4

2 4/ 3

____________ 4--------------------- 3

3 0

( Bo 67)

Fig. C7.ll)

192

outside the Mg core. On the other hand, the J=4 member

of the ground state rotational band is, in general, a

thorough admixture of the unperturbed ground state configura­

tion and other K=l, 2, 3, and 4 configurations. Further

comments about this state are given below.

The interpretation of the other levels below 4.5 MeV

is straightforward because they are dominated by a single

Ka-configuration. The 0^ state corresponds to the excitation

of the two valence neutrons from the ground state configuration

to the configuration given by Ko=0PW(l/21,-1/21). The 2^, 3^

and 32 levels involves the excitation of a single valence

neutron to intrinsic states given by Ka=2(5/2,-l/2*) or

Ka=3(5/2,l/2’).

TTExcept for the J =4 states, all of the aforementioned ■

theoretical states remain very pure even under large varia­

tions of the reciprocal moment of inertia parameter, the

single particle energies, and the residual interaction strength.

As for the J 1T=4+ states, their Ka composition is extremely

sensitive to the values of the parameters employed. It is

difficult to reconcile which of the 4i and 42 states should

be identified as belonging to the ground state band. The

fractured strength of the Ka=0PW(5/2,-5/2) configuration is

easily shifted from one of these states to the other. Not­

withstanding this delicate mixing problem, it is reassuring

that experimental evidence favors assigning the J7r=4 +

4.32 MeV state in Mg to the ground state rotational band

193

24

by virtue of its exclusive decay to the f =2+ state at

1.81 MeV (Ha 68c).

Referring again to Fig. (7.11), we note that the SU(3)26shell model calculation for Mg predicts the correct spin

sequence for the first six levels. Alternatively, the core

+2 model provides eight candidates for the eight experimental

levels observed below 4.5 MeV. In particular our model

suggests that the middle member of the triplet at 4.3 MeVTT *fshould have spin J =2 . The most disturbing aspect of our

calculation is the fact that the 22 state lies above the 02

state. Presumably, the inclusion of the higher excitation

modes and the choice of a better residual interaction would

push the 22 lower and remedy this situation.

It is instructive to extend this intercomparison between

the SU(3) shell model and the core +2 model further. It is

well-known that the SU(3) model predicts a low-lying K=2 ? 6band in Mg and other s-d shell nuclei (Ha 68b). Not only

does our core +2 calculation confirm this, it also.provides

a simple description of the excitation mechanism. Specifically,

the 22 state corresponds to the excitation of one of the

valence nucleons from the K=5/2 level to the K=l/2* level in

a Nilsson diagram. By way of contrast, the SU(3) model with

its irreducible representations and projections therefrom is

not amendable to such physical visualizations.26The intuitive appeal of a core +2 interpretation of Mg

is well, illustrated by its description of the Mg2^ (d,p) Mg2^

TTreaction. From our earlier results, we know that the J =5/2

194

Assuming then that the (d,p) reaction simply transfers a 25neutron to Mg without breaking up its ground state con-

? 5figuration, the states excited in Mg will contain compon­

ents of configurations of the form Ko=K(5/2,ft) where

K = | 5 / 2 i f l | . Furthermore, as Table ( 5 . 2 ) shows, angular25momentum considerations for (d,p) stripping on Mg imply

that only J 1T=2+ and 3+ states in Mg28 may be populated by

Ln=0 transfers.

We find that except for the 02 state, the theoretical

levels below 4.5 MeV are consistent with the above (d,p)

selection rules. The 22 and 3- levels have dominant compon­

ents of the form Ka=2(5/2,-1/2*) and 3(5/2,l/2'). In

addition, the transfer of a particle to the K=l/2' orbital

can proceed via Ln= 0 .. On the other hand, the 2^ state con­

tains no major components of the form Ka=K(5/2,ft) and

accordingly — in agreement with experiment — cannot be

populated in one step (d,p) stripping. In contrast to these

examples, we note that an ^n=2 transfer to the 02 state is at

variance with the dominant Ka=0PW(l/2',-1/2*) character of

this state. This discrepancy may reflect the presence of some

Kcr=0PW(5/2,-5/2) component in the C>2 wave function which would

be present had we used a pairing interaction of the type given

by Eqs. (6.10).

Finally, we note in passing the prediction of a 99%

pure Kc=0 (1/2 1 ,-1/2") J7r=0+ state above 6 MeV. Experimentally,

195

25g ro u n d s t a t e o f Mg i s a v e r y p u re K c r= 5 /2 (5 /2 ) c o n f i g u r a t i o n .

196

there i s a J =0 candidate in t h i s v i c i n i t y .

4. Magnesium 27

The last of the magnesium isotopes which we wish to27consider in the framework of the unified model is Mg

This nucleus should provide an interesting exercise for our

model because it is evident from the current literature that

27the rotational structure of Mg is entirely conjectural

(Cu 64, Me 69). Indeed, in terms of the unified model, there

are conflicting attempts to ascribe its properties to the

motion of an odd neutron in Nilsson orbitals associated withPfi

a prolate (La 66, Sy 6 9) and oblate (G1 6 5 ) Mg core.

Certainly”, a large measure of this confusion stems from

the fact that in either representation, the ground state spin

of Mg27 of Jir=l/2+ is consistent with the odd neutron being

in the K = l / 2 ' o r b i t a l ( N i l s s o n o r b i t a l No. 9) over a la rg e

range o f d e f o r m a t io n s , - 0 . 2 < 6 < + 0 . 2 , as F i g . ( 6 . 1 ) s u g g e s t s .

Further difficulties with the rotational characterization of 27Mg may originate in its being situated in the mass region

A=26-29 where some nuclei have prolate equilibrium shapes and

others oblate shapes (Br 57, Li 5 8 , Ne 60, Ke 63a). In such

a transitional region, the nuclear surface may be soft against

vibrational excitations. Indeed, vibrational correlations

are found to play an important role in microscopic calculations

of the structure of nuclei in this mass region (Ca 70).

Forewarned about the doubtful validity of the strong-coupling

TT 4*

we attempt nonetheless to describe some of the properties 27of Mg in this formalism.

Whether choosing a prolate or oblate deformation for Mg27

it is difficult to reconcile which levels belong to the K=l/2’

ground state and other rotational bands. We have carried out

detailed core +1 calculations using the parameters of the

oblate solution suggested by R.N. Glover (G1 65). We could

reproduce his spectroscopic factors but the overall fit to

the energy level spectrum was very poor. D.H. Sykes et.al.

(Sy 6 9 ) were also unable to fit the energy levels by assuming

a negative deformation. As a consequence, we find an oblate27interpretation of Mg unsatisfactory.

27At the same time, a prolate core +1 description of Mg

is less than satisfactory. As Fig. (7.12) shows, the most

glaring flaw is the prediction of only one low-lying J 7r=5/2+

state where two are in fact observed.

J.M. Lacambra et.al. (La 66) have suggested that one of

the Jir=5/2+ states i§ formed by core excitation. Specifically

they conjecture that one of these states represents a K=5/2

"hole" excitation resulting from the promotion of a neutron

from the K=5/2 orbital in the Mg2^ core to the K=l/2* orbital.

According to the Nilsson diagram given by Fig. (6.1), this

excitation could be expected at reasonably low energies — forTT +positive deformations. The other J =5/2 state is assumed

of course to belong to the K=l/2' ground state rotational band

197

wave f u n c t i o n i n t h i s r e g i o n o f th e p e r i o d i c t a b l e (R i 6 8 ) ,

EX

CIT

ATI

ON

EN

ERG

Y (M

eV

)198

2 7 _ R/j „ 2 6

7 r-^ 2 n EXP T H Y

4

MANYLEVELS

1/2

WEAK0.00

0.01WEAK

0.03

5 /2 (3 /2 ) 3/2 ,5 /2 ,9 /23 /2 ,5 /2

0.01

0.33

WEAK2 2-

> 1/2 . 3 / 2 <0 .40

0.03-0.13 ^ 3 /2

0.16-0.60 WEAK

WEAK

="l / 2 —

7 /2 ^

0.42

00

WEAK

0 . 10- 0 .2 05/25 / 2 ------------------0-14

1- 2 2 M 2 z ° £ ° 3 / 2 ^\ \ \ 0 5 9

0 0 .40 -0 .80 | / 2 ___________ 0-28

FOPT

/

9 /2

5 /2

3 /2

3 /2

0.010.01

1/2 / /

/7 /2 7

0.53

0.10

//

//

5 /2 /

0.26

3 /2 '

1/2-------- 0.16

^ 5 / 2 ^2 ~"5/2 K3

7 /2 K |

/ 0.19 ^3/2 }<2/ ^ " 3 / 2 K3

1/2 k 2

5 / 2 K|

3 / 2 K|

1/2 K[

F i g . ( 7 . 1 2 )

T a b le < 7 .7 ) Mg27=Mg2^ + l DSM P a ra m e te rs and D om in an t C o n f i g u r a t i o n A m p l i t u d e s

el/2'

■ 1 /2 "

'3/21

0.30 MeV

0 . 0 0

3 -8 0

4.50

_______ 1_________________HJ'J J_ _C CNf_I G_ AAPiJ J UD E_S_ FOB TJHY____________________________________ JPAAJ JPl-f_ SP NF J5_ f f f PJ_ STATES______

______ 1= 1/2 e.qo K= 1/2 { 10 -C -Q)__=£L.17_ ?__1/ 2 L_1 l_-_Q_r.O)___0 . O.G_K=>__3/21_3l_r0_-0)______ 1= 1/7 K* 1 /?( :o -n -o i K 1K

______________________________________________________________-OJ__-JD..23_X?__J/2i_31_rO-i-JO)________ J_?__372__Jf =___L/2J__ljD_-Q_rQ 1_____1_

_________ I_=__5J2.____J?J.52-J<_=__Jyyj_JJ)_--i)_-i)J._-D-.3.0_A=__.l/2(_JJL-J3_-DJ__ DJ.Z6..K?__jyi£_’JLjrl3.r-.0J_____ ______ J =__ 5J2__JJ=____XJ2S _J.D_.rQ_.rQi______ JL__

______ 1= 7/2 0.E3 K - 1/ 2 ( 1C -C -C) 0.40 K= 1/21 11 -Q -01 -n ,39_K2_3V 21 31 -Q -0 J________ 1=. 1/7 K = l/?i ii -n -m K2_____ J_=__jy7___iL,5.9_y_=__jyyj_J.:L_-.£L_-.CJ___Q_.i-7_.K_=__J.A2J_i0_rQ_rD.J_-_Q..QQ-K?__3/2<_31_rQ-rQ)_________J=„3y2U_J!^„372J_3i_rQ_rQl___^3

V_____ J_=__yyiLi_3J.J3_J<J'__3yiLi_33-_-3_-3J__-3 33_yj’-_jyyi_3J__-J3_^XD-- Q^3i>_X=__ i J21 _JLD_ -J0_ rQ.)_________i.?— 3/2__Xj?__ JLy2J_JJ_rQ_rQl_____2__

______ T= 3 / 2 0.79 K = 1/21 11 -C -C) O.fel K= 3/2( 31.-0_-0)_-0.07 . K= l / 2 ( 10 -0 -0 )________ 1= 7/7 K= 1/7f If) -n -n ) K 1

______ I_=__jyiL__i?^- K3__jy i_Ji_-J3_-i)J__jr-0..«>.7_J<.f-_J/.2J_J.l.-q.r3J__-.0.02_Kr.._ 1/21_.1.0_r.0_rQ)________ J =__372_JS=___3/2J _ 3i_rQ_ r Q)___H i _

______J_=___sy2_ __ TU2JL/_=__J72J_JD_rQ__-QJ__ -X>_.36__K? 1/2(_ Ji-_-D_rQjJ _z\0..3.2_ K =_ _ 3 / 2 < _ 3i_rQ_-.0)_________ 1= 5/2 k=i i/?t n -n - m E_2.__.______ 1= 11/2 0.77 K ° 1/2 ( 10 -0 -C 1___0.45. K = . 1/2.(. 11 _-Q. -0)__-Q.44..KS__3/2( 31 -0 -0) 1= 4/2 K= !/?( 10 -0 -n >________

______ L=__-5yy__-^i>J_y3__jyyj__LL_-i)_-3J__-^i>iLy_=__3yyJ_3-l--D_r.0J„-Xi..43_^=__jy.21_J3_r3_-3J________ J =__J72_JSs__3y2J_31_rQ_rQl________

I_=__jyy._'_J?-.J5__K_=__jyjJ_JJ-_-i)_-j:j___i)-.33_jK5__3/y.(_3C_r.0_ri)J__-.0 3.1_ r__.l/iU_IJ_rJ3-rJ3J________ Js__jy2__JKs__iy2J_lJ_-D_rQ 1________

______1= 7/2 0 . 6 6 K = 1/21 11 -0 -0) 0.47 K° 3/2( 31 -0 -0) . -0.20 K» l/2( 10 -0_-01________ 1= A/2 K= */?( -o -n)________I—1VOVO

Which of the J 7r=5/2+ states is a member of the ground state

rotational band and which is the core excited state is

currently disputed (La 66, Me 6 9).27In our unified model treatment of Mg as three in-

24teracting valence neutrons outside a prolate Mg core, we

address ourselves that the questions about the rotational27structure of Mg and the possibility of a low-lying K=5/2

"hole" excitation. We begin by distributing the three

neutrons in the unoccupied orbitals K=5/2, 1/2', 1/2", and

3/2'. The aforementioned K=l/2' ground state and the K=5/2

"hole" state are given by the configurations Ka=l/2(5/2,-5/2,

1/2') and Ka=5/2(5/2,l/2’,-1/2*) respectively. There are a

total of 28 Ka-configurations defining the basis space in

which we diagonalize the unified model Hamiltonian. As inper

the Mg and Mg calculations, the deformed orbitals we use

are those given by the prolate Hartree-Fock solution of the 24

Mg core (see Table (7.9)).

The spectrum and spectroscopic factors for our core +3

interpretation of Mg27 are given in Fig. (7.13). The core +3

description is an immediate improvement over the core +1

description by virtue of the prediction of two low-lying

JU=5/2+ states in agreement with experiment. Our calculation

confirms the conjecture of Lacambra et al. (La 66) that one

of these states is a K=5/2 "hole" state. Indeed, as Table

(7.8) shows, the 5/22 state is 93% pure Ka=5/2(5/2,l/2',-1/2*).

From a comparison of the theoretical and experimental

spectroscopic factors of the two 5/2 states, we favor assigning

200

EXCI

TATI

ON

ENERGY

(MeV)

201

M g 2 7 = M g 2 4 + 3

7 £-2n 4 EXP THY FOPT

1/2

MANY 0-00LEVELS

2 £ 2 ---------- 1/2 K61/2 0 .00 |/2 K5

WEAK n j

5 /2 K3

2 WEAK 5 / 2 0 /2 ) N ------------------ 9 /2 / 2^1 5/2 K4 3 /2 ,5 /2 ,9 /2 0.00 ^5/2

2----2 ^ 3 --------- 3 /2 ,5 /2 / ----------------- s J 7fWEAK °-°2 £ 4 9 .3 /2 K3----------------- >1/2 ^7/20 .40 3 / 2 - ^ 7/2 K |

2-^0 .0 3 -0 .13 > 3 /2 ^ 0.34 , /0 - /=■:- -■ ■■■: ~ 1 / 2 —- 3 /2

0.16-0.60 WEAK ~ " ~ ~ 0 4 4 ---------- 1/2 / ° ' 47 < 3 /2 j<4__________ 7/2 7 0153 " 1/2 K 3WEAK \ 7/2 /

\\ /

/ •N— ;------------- 7/2 2 £ 0 5 /2 K2

0 ^ ---------5 / 2 ------------------M 2 _____ 5/7 5/2 K|5 /2„ 0 .1 0 -0 .2 0 ' 0 2 5 /2 -^ /

0.14/ /

5 / 2 x

0.57 3 /2 '

0 L o P-4 ° - 0 .:g.Q. | / 2 -------------------------- | /2 ------------------------ 1 /2 K l

F i g . ( 7 . 1 3 )

T a b le ( 7 . 8 ) . Mg27=Mg2 **+3 DSM P a ra m e te rs and D om in a n t C o n f i g u r a t i o n A m p l i t u d e s

A 0.25 MeV

e5/2 = 0 . 0 0

el/2,= 1.20

el/2,,= 5.00

e3/2'= 5.00'

P -1.50

____________________________ CCK_IJNANJ_ C L N f_ IG _ AN_PLJJUCJ_S_ FO R __ JFY__________________________________________ P A R T J £JL J _ CO N F_I F C R_ f P f T_ S I A J | S _________ _

1= 1 / 2 0 . 9 9 K = l / 2 ( . _ 5 0 - i5 C . _ 1 0 i___ r P . . J iP _ k = 1 / 2 < - 5 C r 5 0 _ l L ) _____ = .0 . .02_J< =___ 1 / 2 1 _ 5 0 r . l l ^ 3 1 )_____________ 1 = _____ 1 / 2 K = l / 2 ( 5 0 - 5 0 1 0 ) K 1

I _ = _ _ jy ^ _ _ _ 0 _ .9 4 _ _ K _ = _ _ jy 2 J _ J i_ - A C _ a 5 J _ _ r 0 . . 2 5 _ K = _ _ .3 y 2 J _ ^ .C . r ^ O _ 3 .1 J — J D . » J 2 3 _ J < 5 _ _ i / 2 . t _ 5 0 r 5 0 _ I lJ _______ ._____ J =___2 J Z ___JS=____L / 2 i _ 5 0 r 5 Q _ l Q l ____ £ l . ______ ,

________ L = _ - - 5 7 2 0 _ , 9 i L y j ‘_ _ j y iU _ J 3 _ - 3 3 - 3 0 J _ _ - 0 - * 2 J _ y _ = _ _ 3 . / 2 J _ . 5 0 r 3 0 _ 3 2 J _ _ - 0 . . 2 i> _ K ? l / 2 i _ 5 . 0 r 3 0 _ l 1 J _ _ _ _ _ _____ J = 5 2 2 1!-= 1 2 2 1 - 5 Q r 5 Q ^ l Q l _______? 1 _____ _

1= 5 / 7 - 1 . 9 7 K = 5 / 2 1 5 0 1 C - 1 0 ) - Q . 1 3 K = _ 3 / 2 ( 5 C - 1 0 - 1 1 1 ______ 0_.JL2_J<^___ 3 ./ .2 .L - 5 .C _ 1 0 . -3 1 ) 1 = 5 / 2 K = 5 / 7 ( 5 0 i n - 1 0 ) K 2

_ I = „ j y _ 2 ____0_._35_ _K_=_ _ j y . 2 J _ J0_ -_50_ JLOJ - 5 . - 3 9 - 7 =_ _ . 3 / 2 .( _ .5 0_ -5 0 _ 3 .1 J _ _ _ 0 . . 3 .6_ J<=_ _ 1 J 2 1 . 5 0 r 5 D . 1 _L)______________J = 1 1 2 J 5 = _ _ J . y 2 i _ 3 0 r 5 D _ l l l _____ _ 3 _ ______ ,

l S — 3 J 2 - ' s ^ 3 2 - £ S — 5 J 2 1 - 5 Q . 1 3 r J S i l — r ! > - , 2 S ) - £ S — 2 J 2 1 - 5 0 r . \ D - 2 1 1 £ U 2 0 . .& ?_•_J J Z l . 5 D _ l . 0 _ i n _____________ I s . . 2 1 2 . . U s . . 2 1 2 1 . 5 0 - 5 0 . 3 1 1 ________ _______ ,

1= 1 / 2 - 0 . 5 9 K - l / 2 ( 5 C - 5 0 1 1 ) - 0 . 1 3 K = 1 / 2 . ( _ 5 J S l5 0 ._ lC J _______Q ^ I Q - K i : ___ l / . 2 i _ 5 0 r J . 0 . 7 J l J _____________ 1 = 7 / 2 K = 1 / 2 1 S 0 - 5 n m i 1

K0J = _ _ 3 J 2 — 0 ^ 3 3 . 3!-5— 3 J 2 1 - 5 £ s 5 S - 3 1 J — — O s 2 S - X . 5 — l / Z l - 5 0 s 5 Q - 1 0 J i — = 0 - . 2 1 . K s — 1 / 2 1 . 5 . 0 : 5 . 0 . 1 1 1 ------------------------ I s . . 3 1 2 — & = — 5 1 2 1 . 5 0 . 1 0 - 1 0 1 _____ £ ■ .______ t

J _ = _ _ 9 y y _ _ s 0 . - J J . 5 s . . 5 1 2 1 - 5 5 . 1 0 - 1 0 J . . s 0 s 2 ( > . l f . S . . 1 1 Z S . 5 0 s 5 0 . 1 0 1 . . - 0 . 2 5 . ) l s . . J . / 2 1 . 5 0 - 1 0 . 2 1 1 _____________ J =_ _ 2 1 2 . _ .172 .1 _ 3 0 r 3 0 _ 1 1 1 _______ A _____

1 = - 3 / 2 - C . . 5 4 k = 1 / 2 ( 5 C - 5 C . 1 1 ) ____d L ..2 _ 6 _ I< j! l L U - 5 £ s 5 0 L 3 . U _____ JL«JLA_K.= 1 /2 T _ 5 0 - 5 Q _ J L O J _________ !_____ 1 = 5 / 2 K ° 3 / ? ( s o - s o n > K l)

I _ = _ _ 3 y j_ _ _ 0 _ .J J _ K _ = _ _ jy _ 2 . ( _ J O _ - 3 C _ 3 J J _ _ r 0 . .3 _ B _ J < = _ _ J /2 J _ 3 .C r 3 0 _ J J J ____ O . .O . 5 _ K ? _ _ J /2 1 _ 3 .C r 3 0 _ J O J __________ _ _ J ^ _ 3 y 2 _ _ r = _ _ J / 2 J _ . 5 £ ) - . 5 Q _ l l l ____ ? 3 _ ______

_________ L = _ _ 3 y y ._ _ J ? . * J C _ K j _ _ J . J 2 J . 5 O s 5 0 . 1 O J . . r O . ’2 5 . H s . . 5 1 . 2 1 . 5 O . 2 0 s l O J . . s O s 2 1 . 1 1 s . . 3 1 . 2 1 . 5 O r 5 O . 2 1 1 ______________I s J 7 2 31s U 2 1 - 5 £ r l Q = 3 1 1 ____ ^ 5 _______

1= 1 / 2 0 . 9 9 K « 1 / 2 ( S 0 - 1 C - 3 1 ) 0 . 1 0 K = 1 7 2 .L 2 L C .rJ5 0 _ J J J ,Q_.p. 8_J<,g___ 1 X 2 1 5Qr i 1 - 3 1 ) • 1 = 1 / 2 K - 1/21 10- 10 I I ) K 6 ____r ooro

27Experimental References for Mg

T a b le ( 7 . 8 ) (C o n t in u e d )

203

Bibijana Cujec, Phys. Rev. 136 (1964) B1305.

P.M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1.

R.N. Glover, Phys. Lett. 16 (1965) 147.

J.M. Lacambra, D.R. Tilley, and N.R. Roberson, Phys. Lett. 20 (1966) 649.

L.C. McIntyre, Jr., P.L. Carson, and D.L. Barker, Phys. Rev. 184 (1969) 1105.

D.H. Sykes et .al.,. Nucl. Phys. A135 (1969) 335.

204

the "hole" configuration to the 1.94 MeV state. This Is

in agreement with McIntyre et al. (Me 69) who assume from

the B(E2) values deduced from lifetime measurements that

the J ir=5/2+ state at 1.69 MeV belongs to the ground state

rotational band. On the other hand, Lacambra et al. (La

66) and Sykes et al. (Sy 69) make just the opposite asso­

ciations based on a description of the electromagnetic

decay properties of these states using first order pertur­

bation theory (FOPT).

Our calculations suggest that FOPT may be inadequate

and that a complete treatment of configuration mixing is

27essential for the description of Mg For example, the

ground state is also very pure, being 98% of its unperturbed

configuration Ka=l/2(5/2,-5/2,l/2'). However, it takes only

a 2% admixture of the Ko=l/2(5/2,-5/2,l/2") configuration in

the ground state to produce nearly a twofold enhancement of

the spectroscopic factor. A similar enhancement is also

shown for the 3/2-^ state.

In general, the mixing compresses the ground state band

too severely. On this point, it is interesting to note that

the relative spacing and the spectroscopic factors for the25 27rotational members of the K=l/2' band in Mg and Mg are

27very similar. That the band is too compressed in Mg must

be regarded as evidence for choosing a better residual inter­

action between the three valence nucleons.

In spite of our oversimplification in this respect, we

remark that the predicted spin sequence of the first five

205

levels agrees with experiment. Our calculations suggest

that the weakly excited level at 3.42 MeV may have a

J ir=7/2+ spin assignment. Following this, there are un­

equivocal candidates in the experimental spectrum for the

l/22 and 3/22 levels. These can be identified by their

large spectroscopic factors. Interestingly, the 3/22 level

is essentially K=3/2' band head (Nilsson orbit No. 8) which25has never been identified in Mg . We also anticipate that

the 4.39 MeV state in Mg27 has a J7r=9/2+ spin assignment.

Comparing the core +1 and core +3 interpretations of

Mg27, Figs. (7.12) and (7*13) respectively, it appears that

the addition of the K=5/2 "hole" state to the calculation has

little apparent effect other than adding another rotational

band to the spectrum. Inspection of Table (7.8) shows how­

ever that fragments of the rotational levels built on the

K=5/2 "hole" configuration can be found throughout the spec­

trum .27In concluding our discussion of Mg , we note that the

core +3 model cannot account for the observed density of

states above 3.5 MeV. This is reminiscent of the shortcoming26we encountered in our core +2 description of Mg . Again we

are confronted with evidence of other modes of excitation,

perhaps 8 and y vibrational degrees of freedom, which must

be incorporated into a more complete treatment of these two

nuclei. Our current theoretical understanding of these nuclei,

see Figs. (7.11) and (7.14), is growing. Still we can only

EXCI

TATI

ON

ENERGY

(MeV

)

E X PUNIFIED M O D E L

M g 24+3

• 5 /2 ,(3 /2 ) ------------------- 9/2■ 3/2,5/2,9/2 5/2• 3 /2 ,5 /2 ' 9/2✓ > l /2 - 3 / 2_^3/2 = I /2

7/2

5/25/2

3 /2

3 /21/2

7/2

7 /2

5/2

5 /2

3 /2

0 1/2 1/2

2 7

UNIFIED M O D E L M g 26 + I

S H E L L M O D E L

(Sy69/Ha68a)

5/2

3 /2

S H E L L M O D E L (INTERMEDIATE COUPLING)(Bo 67)

3/2

1/2

7 /2

5/2

5 /27/29 /2

5/2

5/2 3 /25 /2

3 /2

3 /2

1/2 1/2 1/2

F i g . ( 7 . 1 4 )

206

speculate about the nature of the other inodes of excitation

which contribute to the structure of these nuclei.

D. Silicon Isotopes

Inasmuch as we are predisposed to the strong-coupling

limit for nuclear structure, it is tempting to propose an

interrelationship between the silicon isotopes si2^ ,2^ ,3*3,33‘

analogous to that envisioned for the neon and magnesium

isotopes. Indeed, there is much experimental and some

theoretical evidence to encourage this point of view. Of

course for the purpose of the ensuing synthesis, we neglect

contributions from & and y degrees of freedom anticipated in

the prolate-to-oblate transitional region A=26-29.

In this approximation then the energy level spectrum of

2 8Si is that of a simple even-even rotator. Furthermore28Hartree-Fock calculations of the deformed orbitals of Si

indicate that the equilibrium shape of this nucleus is that

of an oblate spheroid (Ke 63a). A recent measurement of the

28quadrupole moment of the first excited state in Si confirms

the oblate interpretation for a presumed ground state rota­

tional band (Na 70). Preceding this direct evidence regarding28 29 the shape of Si , many of the properties of Si were found

to be consistent with a core +1 interpretation where the odd

neutron moves in the K=l/2', 3/21 and 1/2" orbitals associated

with a negative deformation (Br 57)* As for the other nuclei,

Si3k like Ne22 and Mg2^, exhibits a greater density of states

207

than can be accounted for by a simple rotator characteri-

31zation. And, Si may be an oblate spheroid with evidence

for low-lying core excitations or "hole" excitations of

the type described by the unified model formalism developed

herein (We 68).

As in the preceding neon and magnesium analyses, we

begin to develop our expectations starting from the simplest

case available, namely a core +1 description of Si2^. The

considerable influence of Coriolis mixing on the spectrum,29spectroscopic factors, and electromagnetic properties of Si

has previously been demonstrated (Ma 60, Ma 6 7 , Hi 6 9 ). Since

these calculations were based on the phenomenology of Nilsson,

our issue is to test the intrinsic structure of the deformedp O

orbitals x^v defined by the Hartree-Fock solution of Si

(see Table (7.9)).

In marked contrast to the success of this modus operand! 21 25for Ne and Mg , we find that a core +1 calculation using

Hartree-Fock orbitals cannot reproduce the low-lying spectrum29or spectroscopic factors of Si .

The crux of the difficulty encountered in employing

Hartree-Fock intrinsic states xfiv in the strong-coupling

approximation (cf. Eqs. (5.8) and (5.10)) lies with the magni­

tude of the decoupling parameter predicted for the K=l/2'

band (Nilsson orbit No. 9). The Hartree-Fock value of a=+1.89

is larger than the largest value a=+1.40 (for 6=-0.1) given

by the Nilsson model (Ch 66). As a consequence the inversion

208

209

is unreasonably large and no choice of the single particle

energies £^/2' and el/2" cou-1-<* produce the observed spin2Qsequence of the lowest levels in Si .

?QTheoretically, a few displaced levels in Si could be

qualified if predictions of other properties formed a con­

sistent picture. Unfortunately, we found the spectroscopic

factors to be extremely sensitive to the Nilsson coefficients

of the intrinsic states Xfiv> the single particle energy

parameters efiv, and the Coriolis mixing.

Since no overall concensus with experiment could be 29achieved for Si % a program of core +2 and core +3 calcula-

30 31tions for Si and Si employing Hartree-Fock orbitals for 2 8Si was considered ill-advised. Instead of adopting the

Nilsson phenomenology in order to implement these calculations,

we believe it necessary to review in our concluding Chapter

our development of the strong-coupling unified model and its

expected validity in the middle of the s-d shell.

of the J=3/2 and 5/2 members of the ground state band K=l/2*

210

Table (7.9)

"Nilsson Coefficients" c^j for the Hartree-Fock

ft , 20 „_24Intrinsic States X n . r ^ ' l ’in for Ne

Si28 (Ri 68).ftv vj Yjft Mg and

K=3/2'

K=l/2"

K=5/2

K=3/2

K=l/2

Ne 20Mg

24 Si 28

(Prolate)S I 28

(Oblate)

d3/2d5/2

0.99320.1167

0.97060.2406

0.95310.3027

-0.69350.7204

d3/2sl/2d5/2

-0 .8 8 8 10.4352-0.1481

-0.62350 . 7 2 9 00.2825

-0.60990 . 6 6 3 00.4340

-0 . 8 8 3 0-0.44940.1358

d5/2

K=I/2 *

d3/2sl/2d5/2

1.0000

-0 .2 51 6-0 .7 29 8-0 .6 35 7

1.0000

-0 .7 30 2-0.4139-0.5436

1.0000

-0.7833-0.4212-0.4572

1.0000

0.3130-0.47020.8044

d3/2d5/2

0.1167-0.9932

0.2406-0.9706

0.3027-0.9531

0.72040.6935

d3/2sl/2d5/2

0.38470.5273-0.7576

0.27940.5452-0.7904

0.12030 .6 18 8-0.7763

0.2977-0.7596-0.5783

Chapter VIII

Summary and Conclusions

211

A . Summary

The results of the preceding Chapter represent the

embryonic stage of a program of phenomenological deformed

shell model calculations for s-d shell nuclei. The ex­

tension of the strong-coupling formalism of the unified

model to several valence nucleons has provided a systematic

and quantitative description of the low-lying states of

Ne21, Ne22, Ne23, Mg25, Mg26, and Mg27.

At the same time, these results illustrate the challenges

and obstacles that a general program of this nature will

encounter. Indeed, it is clear from the preceding analyses

that the model of a rotator core plus several interacting

valence nucleons is a promising representation of the degrees

of freedom evident in the neon isotopes; it provides a partial

characterization of the magnesium isotopes; and, it fails for

the silicon isotopes.

In attempting to assess these results and observations,

we are led to several conclusions regarding the utility and

validity of the strong-coupling limit of the unified model,

its generalization to several valence nucleons, and its

applicability to s-d shell nuclei in general. We start from

a utilitarian point of view and afterwards consider the

validity of the strong-coupling approximation as one moves

from the beginning to the middle of the s-d shell.

L T ^B. The Ne2^(t,p) Ne2^ Reaction

As previously mentioned, our model appears to have its

greatest success with the neon isotopes. The predicted

energy levels, spins and parities, and spectroscopic factors21 22 23for Ne , Ne , and Ne J are, on the whole, in excellent

agreement with the available experimental information. Further

information regarding the spins and parities of the low-lying 23levels of Ne J is especially desirable to complete this pic-

21ture. We wait with interest for the results of the Ne 2 ?(t,p) Ne J reaction. As discussed in the preceding Chapter,

the (t,p) reaction is the ideal reaction for locating the

K=3/2 "hole" states in Ne28. Moreover, as Table (5.1) shows,

this reaction should also be useful for locating the 7/2, 9/2,

23and 11/2 spin states in Ne . Assuming a direct reaction

mechanism is operative, the 9/2 and 11/2 states should exhibit

clean L2n=4 signatures. This should establish whether or notO '* +

the 2.52 MeV level in Ne J has a 9/2 spin assignment and

perhaps if it or the 2.31 MeV levels is actually a doublet.

This in turn would help clarify our prediction of seven low-23lying levels in Ne J where only six are known as of the present

time.

C. Theoretical Refinements for Neon Isotopes

Altogether, our experiences with the neon isotopes

encourage us to suggest additional theoretical excursions

212

213

within the framework of the deformed shell model. One

necessary development in this direction is the calculation

of the electromagnetic properties of these nuclei. More

generally, we propose using the neon isotopes to test

refinements of the deformed shell model just as the oxygen

isotopes are used to test refinements of the spherical shell

model. Specifically, we have in mind the implementation of

a theory of effective interactions for deformed nuclei pro­

posed by several authors (Br 59, Un 6 3). This theory is

analogous to the Talmi parameterization of the residual

interaction for spherical nuclei which was discussed in Chapter

II. As we pointed out in the previous Chapter, Ne23 is in

need of a residual interaction that is more sophisticated

than the naive pairing interaction vie have adopted herein.

In addition to .alternative choices of a residual inter-

20action, a more general representation of the Ne core is

desirable. For example, the expansion

E = f (R2 ) = AR2 + BR** + ... ' (8.1)

20has been suggested in recognition of the fact that the Ne

core does not correspond to that of a perfect quantum mechani­

cal rotator (Ke 68, Mo 70, K1 70)'. In actuality, the higher

spin states are depressed below their perfect rotator values

20as the yrast line for Ne indicates (see Fig. (7 -7)) •

If we adopt the representation of the Ne core given

by Eq. (8.1), we can expect two new types of effects to

appear in the theoretical calculations for the remaining

neon isotopes. The more subtle effects will be those

associated with the higher order Coriolis terms arising

from the substitution of (J-^)2 for $ 2 in the core Hamil­

tonian.

The more obvious effects will be the expected depression

of higher spin states. We have already noted evidence for

this. In particular, this should remove the 1 MeV displace-+ 21ment of the 11/2 level in Ne and yield a better prediction

for the higher spin "members’' of the ground state rotational

band. With regard to the ll/2+ state, we remarked earlier that

its 1 MeV displacement is a notable discrepancy in view of the

fact that the lower members of the ground state rotational

band fit to within 0.1 MeV of their experimental values (see

F i g . (7.1)). In addition, this effect should also yield a

more accurate prediction of the excitation energy of the 10+

22state and perhaps higher spin states in Ne , none of which

have yet been found. From both a theoretical and experimental

point of view, the prediction and discovery of J^12 "members"

22of the ground state rotational band in Ne would be an

extremely exciting development because they would signify

a greater degree of collective core excitation than can be

achieved in conventional shell model calculations which assume

six particles outside an inert core. We return to this

point shortly.

214

20

The refinements suggested above for the neon iso­

topes can be extended to the magnesium isotopes too. As21anticipated in Ne , they should permit more accurate

predictions regarding the high spin members of the ground

states rotational band of Mg2^ (see Fig. (7*9)). Un­

fortunately, these modifications will not help the deformed2 6shell model reproduce the density of states observed in Mg

27and Mg . As a specific example, these modifications in­

volve no mechanism for generating a J=0+ state at 5 MeV

excitation energy in Mg

D. Validity of the Strong Coupling Unified Model in the s-d Shell

Such omissions force us to question the general validity

and applicability of the deformed shell model to s-d shell

nuclei. Indeed, in the previous Chapter we witnessed the

gradual quantitative and then qualitative demise in the model's

ability to predict excitation energies and spectroscopic21 25 29factors in moving from Ne to Mg ^ to Si . Our interpre­

tation of these circumstances is intimately related to the

Hartree-Fock results we have presumed herein. In order to

develop this argument, it is desirable to recapitulate the

basic principles behind the deformed shell model which we set

forth in Chapters II and III.

In Chapter II, we began with the N-body Schroedinger

equation,

215

216

HY = EY (8.2a)

where

H = Z T, ,a,+a, + | lj

(8.2b)

and the Pauli Principle which states that Y must be anti­

symmetric in the coordinates of all N fermions. The nuclear

shell model interpretation of these equations rests on the

following two assumptions. First, there exists a particle-

hole expansion of Y(cf. Eq. (2.2)). And second, a good

approximation to Y is given by a finite number of |mp-0h>

states. V/ithin the validity of these assumptions, the solu­

tion of the N-body Schroedinger equation is given by the

eigenvectors of the Hamiltonian matrix

It is worthwhile stressing that each |mp-0h> state is anti­

symmetric in the space-spin-isospin coordinates of all N

nucleons. Needless to say, this implies that all the neutrons

(protons) are indistinguishable from one another.

Consequently, upon evaluation of this matrix element,

it is rather remarkable to find that the original system of

N interacting nucleons can be replaced by a simpler model of

(N-m) inert "core nucleons" and m active "valence nucleons"

and that we need only to antisymmetrize the "valence nucleons".

Hab = a <mP-°h lH lmP-°h>b * (8^3)

Herein we have the quintessence of the nuclear shell

model. The Hamiltonian for the original system has effec­

tively been split into two non-interacting spaces,

H = H + h + v. (8.4)v

Hc is the energy of the inert core. T can then be inter­

preted as a product of a constant wave function for the inert

core nucleons and an antisymmetrized product wave function

for the m valence nucleons. The single particle wave functions

and energies of the valence nucleons are defined by the non­

local Hartree-Fock potential, h, generated by the core nucleons

(cf. Eqs. (2.9d) and (2.25)). In general, the effective inter­

action v^j between the valence nucleons is as complicated as

possible and depends on the details of the calculation (cf.

Eq. (2.24)).

Several general programs of shell model calculations for

s-d shell nuclei have been underway for a number of years

(Bo 6 7 , Ha 68a, Ak 6 9 ). From these surveys] it is now

generally recognized that there are low-lying excitations in

s-d shell nuclei which continuously defy a shell model des­

cription in terms of several valence nucleons outside an

inert core. Examples can be found in the simple limit

of two nucleons outside (Ma 6 9). Within the vagaries of

parametrizations and effective interactions for the shell

model, these results are thought to imply that some degree

of core excitation is in evidence.

217

By way of testing this, a recent shell model calcu­

lation of F 3-9 and Ne2^ has been made assuming an inert C^2

core (Me 70). Remarkably, it seems that the amount of core

excitation is still underestimated. The implications of this

statement are not clear, however. Even with a smaller core,

or alternatively, with the inclusion of multiple particle-

hole excitations in the conventional shell model cal-20culations still indicate that the "ground state band" in Ne

TT 4*cuts off at J =8 (Go 70) whereas the strong-coupling limit

of the unified model predicts no such cut-off. In the

context of the shell model, this question can be resolved only

by choosing an even smaller core and facing a concomitant com­

binational problem of staggering dimensions for the basis

space.

In the Interim, these results certainly suggest that

many core nucleons may be excited in a coherent fashion and

thereby contribute to the low-lying spectra of s-d shell

nuclei to a greater extent than heretofore expected from con­

ventional shell model calculations with an inert core.

For this reason, we turned in Chapter III, to A. Bohr's

heuristic description of core excitations (Bo 52, Bo 53). The

core now enters as a dynamical entity in its own right. Core

excitations are described in terms of the quantized surface

oscillations of a liquid drop of nuclear matter. In the

"weak coupling" limit, the liquid drop has a harmonic spectrum

corresponding to small vibrations about its spherical equili-

218

brium shape. In the "strong coupling" limit, surface waves

viewed from the principle axis frame suggest that the liquid

drop is a permanently deformed spheroid rotating in space.

In this case, the excitation spectrum of the core is that

of a quantum mechanical rotator (cf. Eqs. (3.10) and (3.12)

and Fig. (4.2)).

These "limits" provide a convenient phenomenology for

describing the low-energy properties of many nuclei in terms

of the independent motions of valence nucleons and the

collective behavior of core nucleons. In the unified model

of A. Bohr, one studies the rotational and vibrational degrees

of freedom of the core, their interaction, and their coupling

to the degrees of freedom of the valence nucleons.

The unified model Hamiltonian appropriate for nuclear

structure studies in the first half of the s-d shell is that

given by the strong-coupling limit:

H - H^o) = A$2 + h + v. (8.5)

The wave function T is now approximated by a superposition of

products of a Wigner D-function times an antisymmetrized pro­

duct wave function for the m valence nucleons. In the above

expression, is the trivial zeroth order term arising from

the expansion of the core about its equilibrium shape.

It would seem that the liquid drop characterization of

the core precludes a proper treatment of antisymmetrization

of all N nucleons. Actually, the microscopic derivation of

219

the strong-coupling limit of the unified model Hamiltonian

by F. Villars (Vi 70) confirms our defintion of Y above and

our interpretation of h as the Hartree-Fock Hamiltonian of

the core.

This being so, we are now in a position to understand

the breakdown of the strong-coupling limit of the unified21 25model that we encountered in moving from Ne to Mg to

29Si . The Hartree-Fock problem,

h X = £X, <8 -6 )

for the single particle energies e^v and wave functions

Xfiv can be interpreted as a variational calculation of the

total energy of the N-body system (see Appendix C). In the

restricted Hartree-Fock solution (cf. Chapter VI), the energy

surface for the system is a complicated function of the ex­

pansion coefficients c ^ . The solution of Eq. (8.6) is that

self-consistent set of single particle orbitals corresponding

to the absolute minimum of the energy surface.

A comparison of the resulting coefficients c ^ with

those of the Nilsson model generally suggests a prolate,

spherical, or oblate equilibrium shape for the nucleus.

Unfortunately, the energy surface may have several relative

minima of about the same energy (Ri 68). In the case where

there is no well separated absolute minimum, it is difficult

to decide upon a definite equilibrium shape. From a Hartree-

220

Fock point of view, the equilibrium shape of Mg is notP O

well defined and that of Si not really defined at all.24In Mg , the prolate symmetric and axially asymmetric

solutions are nearly degenerate energywise; and, In Si2^,

the oblate, spherical, and prolate solutions are virtually

degenerate (Ke 63a, Ba 6 5 , Ri 68, Pa 68).

Hence it is not too surprising that the strong-coupling

limit of the unified model, which presumes a well-defined

permanent deformation, falters on the magnesium isotopes and

breaks down for the silicon isotopes. The extension of the

deformed shell model towards the middle of the s-d shell

must include vibrational degrees of freedom (Gu 6 7 , Ca 70).

E. Denouement

In conclusion, we find that the strong-coupling limit

of the unified model has its greatest validity at the be­

ginning of the s-d shell. It seems reasonable from the

number of states predicted a!nd from their spins, spacings

and spectroscopic factors that two and three valence nucleons

may be providing the extra degrees of freedom observed as22 23low-lying levels in Ne and Ne Hence, we feel that a

general program of deformed shell model calculations can be

profitably extended to other neighboring nuclei. Indeed,

22the low-lying levels of Na have already been successfully

20described as an odd neutron and odd proton outside a Ne

core (Wa 70).

Perhaps the greatest merit in such a program lies in

its simple description of the low-energy properties of those

221

24

222

nuclei in terms of a few active valence particles in de­

formed orbitals. In some instances, the collective-particle

interplay can be visualized as rotational levels built on

relatively pure intrinsic excitations of the valence nucleons.

In these instances, further experimental investigations are

readily envisioned.

Generally, there is extensive mixing between a few Ka-

conf igurations . The physical picture is then a little more

complicated. Nevertheless, this is to be contrasted with

the shell model alternative of perhaps a hundred component

representation of the same states in terms of many particlesTP i

outside an inert C or 0 core and/or the superposition of

a few abstruse irreducible representations of an SU(3)

classification of the shell model basis states.

223

Ak 69

Al 69

As 68

Ba 60

Ba 65

Ba 66

Ba 69

Ba 70

Be 55

Be 59

Be 69

Bh 62

BI 58

Bo 52

Bo 53

Bo 58

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Appendix A

Reduction of H , ab

We wish to evaluate the Hamiltonian matrix element

Hab = a <mP“0h lH lmP-°h>b (A.1)

230

where

H = ijTijaj- a «J + 2 ijia V;LJkA ^ ^ & k a * ( A ‘ 2 )

and

N|mp-0h> = n a?|0>* (A.3)

i=l 1

The summations in Eq. (A.2) extend over all the orbitals

in the basis space. • |mp-0h> is an antisymmetric wave func-a.

tion (Slater determinant) of N fermions, m of which are in

valence orbitals and (N-m) of which are in core orbitals.

The core orbitals are completely filled in accordance with

the Pauli Principle. The subscripts a and b denote differ­

ent valence configurations of the m valence nucleons.

In reducing Ha b , it is convenient to split the many-

body Hamiltonian given by Eq. (A.2) into summations over

core and valence orbitals. Letting the indices a,3,Y,<S

represent core orbitals and the indices p,v,p,a represent

valence orbitals, H can be written as the sum of four terms

for the kinetic energy operator and sixteen terms for the

231

two-body interaction operator:

T = I T . a j a . + s T a +a +£ T a +a +1 T a +a (A.it)a6 “ 6 “ 8 au 0,1 “ 11 mo wa M ° uv uv M vand

V = \ E V -ala^a a.+ i E V a a^a^a a a8y6 aeY<S 3 a Y 6 aByy a3YP 3 a 7 p

+ ... + 5 - 2 V , a ^ a ^ a a .2yvpo pvpa v T p a

(A.5)

We later enumerate all sixteen terms of V and simplify it

to nine distinct ones.

Fortunately, few of these terms for T and V contribute

to Hafc. Mathematically speaking, the ultimate reason for

this is because

a^|mp-0h> = <mp-0h|aa = 0 (A.6 )

by virtue of the anticommutation relations given by Eq. (2.6)

and the fact that the same a^ also appears in the |mp-0h>

state since all the core orbitals are occupied by definition.

We now calculate Hab explicitly. Reordering the operators

in the first term of T, viz. a aQ = 6 0-a0a , we find thata p exp p a

Tab = a<mp-0h|T|mp-0h>t) - ET 6 + <mp-0h|£ T ^ J a |mp-0h>b .a yv K

(A.7)

The 2nd'and 3rd terms in the expression for T vanish because

the core is full, as discussed above.

To expedite the enumeration of the sixteen terms of V,

we introduce the following notation:

V = = (A.8)

The sixteen terms are then given by

V = £ [(a3Y<S) + (a3YV>) + (a3PY) + ( a 3 y v )

+ (ap3Y) + (ay3v) + (ayv3) + (ayvp) (A.9)

+ (ya3Y) + ( y a 3 v ) + ( y a v 3 ) + ( ya vp )

+ ( y v a 3 ) + ( yv ap ) + ( yv p a) + ( y v p a ) ] .

The first two terms and last term in Eq. (A.9) are the

corresponding terms written out explicitly in Eq. (A.5).

We can write Eq. (A.9) in a more compact form. To do

so, we note that V can be written as

232

V = = T (1J « )a <A -10)

where Va is the antisymmetrized matrix element

V ijk£ V ijk£“Vij£k* (A.11)

We have, also introduced an appropriate notation with the

properties

233

(ijk£,)a = (ijk£) + (ij£k) = (iJk£) + ( jik£) = 2(ijkfc). (A.12)

From these results, it readily follows that Eq. (A.9) for V

consists of nine basic terms:

V = ^-(a3Y<5)a + •(a3YM ) a + |-(aii3Y)a

+^-(a3yv)a + (ap3v)a + ^(yva3)a (A.13)

+^-(ayvp)a + 7j-(yvap)a + ^(yvpa)a .

We now show that |mp-0h> matrix elements of most of these

terms vanish. Reading from left to right, we note that terms

2, 3, 7 and 8 consist of indices for one occupied and three

unoccupied orbitals or vice versa. The heart of term 2 is

given by

<mp-0h| (a3YU )a |mp-0h>. ■+ <mp-0h | ata^a^ a |mp-0h>K .a D a p Ct j |J D

From this it is clear that term 2 entails the annihilation

of one valence and one core nucleon. This is followed by the

creation of two core nucleons; however, since only one core

orbital is vacant, this matrix element vanishes. This is

essentially the same argument we encountered in the evaluation

234

of Tab earlier. It also explains the demise of terms 3,

7, and 8.

Terms 4 and 6 also vanish because they involve the

creation of nucleons in occupied core orbitals.

Written out, the surviving matrix elements are given

uy

V . = <mp-0h | i Z Va .a!a+a a. + Z Va a +a +a 0a ab a 1 4a3Y<s a3y6 3 a y 6 ay6v ay3v y a 3 v

+ i- Z Va a^a^a a |mp-0h>. . (A.l4)% v p a pvpa v p p a b

The last term above, recalling Eqs. (A.8) and (A.10),

Is recognized as the final term in Eq. (2.9c).4* 4.

Reordering a a D = <S 0-a0a , the middle term is recognized ot p ctp p qas Uav in Eq. (2.9c).

After arranging the first term above in normal order"f* 1 c(i.e. a is on the right), we find it reduces to 2^ ^ $ ^

Eqs. (2.9a) and (2.9b).

We can combine these expressions for Uc in the following

form

N-m

^ AyA'y AyyA

which Is Eq. (2.9d). Uc is the average one-body potential

generated by the core. Later we show it is the Hartree-Fock

field generated by the core.

235

This concludes the derivation of Eqs. (2.9a)-(2.9d)

for H& b . The shell model interpretation of these results

is developed in Chapter 2.

As a postscript, we add a few useful expressions for

fermion operators:

(A.l6a)

(A .16b)

[ab,c] = a[b,c] + [a,c]b ( A .l 6 c )

*f* *f* "t*[,a, & o , a a 1. ] _ a-.a 1.6la2 ,a3a4 1 4 23 x 3 24 h d. 13 d 4 14*

(A.l6d)

This last commutator is useful for evaluating the first term

in E q . (A.14).

236

Appendix B

Derivation of Eq. (2.15)

We wish to write the equation

U° < V = ^ c K v ' V (B-1>c

in second quantized notation. This expression appears in

Chapter 2 in a heuristic argument pertaining to the nature

of the average field generated by the core nucleons and

experienced by the valence nucleons. For this argument we

assumed that core and valence nucleons are distinguishable

and as a consequence the |mp-0h> state can be written as a

product state

|mp-0h> = Y(1C2C ...)¥(lv2v ...) (B.2)

where the core wave function ¥ and the valence wave functionc

¥v are antisymmetric in (N-m) and m nucleons respectively.

In the space of the core nucleons EV(|rc-rv |) Is a one-c

body operator. Hence we can write

^ c ' ^ c v ' V " ( B -3)c a3

4. 4.Rewriting a a„ = 6 „-a„a and noting that all core states are

& a 3 a3 3 aoccupied, this equation becomes

237

<¥„ |ZV lY > = Z<<f> (r ) |V U (r )>. (B.4)c 1 c v 1 c a c c v 1 a c \c a

We have called this result U (ry )*

In the space of valence particles, U°(rv ) is a one-

body operator of the form

Z Uc a+a . yv y v

We then have

Uc = <4 (r ) IZ<d> (r ) |V 14 (r ) > U (r )> (B.5)yv y v g a c 1 c v |Va c 1Tv v v

which, by convention regarding the implied ordering of integra­

tion coordinates, can be written

U° = Z<yaIV Iva> = ZV (B.6)yv yavaa a

and which is the desired result.

Appendix C

The Hartree-Fock Potential of the Core

238

We now show that Uc given by Eq. (2.9d) or Eq. (A.15)

is the Hartree-Fock potential generated by (N-m) core nucleons

in their ground state configuration. Specifically, we have

N-m

the shell model matrix element a<mp-0h|H|mp-0h>b . U° is the

average non-local potential in which the valence nucleons

move.

In a Hartree-Fock variational calculation one chooses a

Slater determinant trial wave function,

and determines the single particle energies and wave func-

of the many-body system. Presumably, $ is a good approxi­

mation to the ground state wave function of the many-body

system (Br 6 9 ). In particle-hole notation, $ is denoted by

the |0p-0h> state. In the context of the strong-coupling

limit of the unified model, it is the so called "intrinsic

state" pf the core (Ri 68, Vi 70).

(C.l)

Uc arises in Chapter 2 and Appendix A in the evaluation of

# = na^|0> ( C . 2)

+tions <j> = a^|0> which minimizes the total energy <$|H|$>

239

The variational procedure (Th 6l) is defined by

0 = 6<$|H|<I>> = <6$ | H | $> (C.3)

where first order changes in $ are given by

4>(n) = 4>+na+a $y a ( C . 4)

and where H is the many-body Hamiltonian given by Eq. (A.2).

Equations (C.3) and (C.4) imply that

This in turn implies that matrix elements of H vanish between

the |0p-0h> state and all states of the form |lp-lh> which

are obtained by promoting one particle in |0p-0h> from an

occupied to an unoccupied orbital. In other words, the

Hartree-Fock Solution is that single particle basis space

wherein $ is stable against single particle excitations. -

To reduce Eq. (C.5), we use Eq. (A.6) to write

<$|a+a Hi$> = 0. ' a y ' (C.5)

0 = <$|a^a H|4>> = <$|[a"^a ,H]I$> • a y 1 ' a y * 1 (C .6)

and expand the commutator according to Eqs. (A.16). After

some algebra, we find

- ET.a.a* + i Z Va .. a*a. a „a + 1 yi i a 2ijk yijk j k £ a

where V a is given by Eq. (A. 11).

The ^-expectation value of most of these terms vanishes

because (1) the core is full and can accept no additional

particles and (2) the valence orbitals contain no particle

to be annihilated. Furthermore, <<£ |a^a^ | <!>> vanishes unless

i=j=Y where y represents a core orbital. Consequently, the

variational condition given by Eqs. (C .3) or (C.5) simplifies

to

0 = T + E (V -V ) (C.8)ya yyotY PYY<*

since = V ji£k as E q * (2,7b) shows.

It follows by inspection that the single particle wave

functions which satisfy Eq. (C.8) are those defined by the

Hartree-Fock-Schroedinger Equation (Th 6l)

241

Indeed, premultlplying Eq. (C.9) by <|>|(r) and Integrating

yields

N-m

T« + ^ 1<viYjy-v iYYj) -

Hence, from Eqs. (C.8)-(C.10) we identify Uc given by

Eqs. (2.9d) or (A.15) as the Hartree-Fock potential of the

(N-m) core nucleons in their ground state configuration.

As a postscript to this Appendix, we remark that Chapter

2, Appendix A, and Appendix C represent but three brief ex­

cursions into the many-body problem and the theoretical

foundations of the nuclear shell model. They serve to guide

our implementation of the deformed shell model. Fuller ex­

positions on the microscopic origin of the nuclear shell

model can be found In the following list of references:

(Th 61, Pr 62, Br 64, Da 6 7 , Ma 67b, Ma 67c, Ba 6 9 , Br 69a,

Ma 6 9 ).

242

Appendix D

Strong-Coupling Matrix Elements

The strong-coupling wave function for m valence nucleons

is given by Eq. (5.10), namely,

The meaning of each term in Y(JMKo) is discussed in Sections

(V. B. 1) and (V. B. 2).

By convention, we have K $0.

The number of valence nucleons described by Y is impli­

citly contained in the definition of a, cf. Eq. (5.8). The

number of valence nucleons m appears explicitly in the phase

(-I)'7-111/2 which premultiplies the -K portion of Y.

We now abbreviate Y(JMKo) by Y^. Inasmuch as Y^ is

the sum of two terms, corresponding to +K and -K, strong-

coupling matrix elements of an arbitrary operator r are given

by

r(JMKa) =

(D.l)

K'OPW 1 + 6K0PW

1 / 2 -j 1 / 2] ] ( D . 2 )

(Continued)

243

(Continuation)

( <<1)K i Xk i l I’ l <{)KXK > + ( “ 1 ) J J ^ 1X - K ’ r ^ - K X - K >

+ (-l)J"m/2

C<<®>K , X K I I r I <>- K X - K > + ( - 1 ) J , " J ” ( r n , " m ) / 2 < < f>_K . X _ K i l r l<frKXK > ] } •

In general, we must evaluate in detail the matrix element

where K' and K may be positive and/or negative quantities.

In Chapter V, we evaluated the specific cases of r=H,+

the Hamiltonian operator, and r=a the spectroscopic factor

operator for single particle transfer reactions.