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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.
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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
17
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
CIT
ATI
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.