byRichard A. Lindgren
B.A. , University of Rhode Island, 1962
M.A. , Wesleyan University, 1964
A STUDY OF LIGHT RIGID ROTOR NUCLEI
A Dissertation Presented to the Faculty of the
Graduate School of Yale University in
Candidacy for the Degree of
Doctor of Philosophy
1969
To my Family
The author would like to thank his graduate research adviser,
Professor D. A. Bromley, for the opportunity to carry out this work, for
his inspirational advice, for his encouraging suggestions, and especially
for his constructive criticism in the preparation of this manuscript. These
moments will long be remembered.
I also would like to thank Drs. J. G. Pronko and A. J. Howard for
continued interest and thought-provoking discussions on this work and
many related matters. The author also wishes to acknowledge Dr. M. W. Sachs
for his consultation in computer programming and m y student colleague,
Dr. R. G. Hirko for his assistance and participation in numerous aspects
of this work. The stimulating discussions with and assistance from other
graduate students will always be remembered.
The author thanks the entire technical staff of the A. W. Wright
Nuclear Structure Laboratory, whose individual assistance and coordinated
efforts are greatly appreciated.
I sincerely thank m y family, in particular, m y wife, Ruth, and
children, who have endured without complaint m y negligence as a husband
and father.
The United States Atomic Energy Commission is gratefully
acknowledged for its financial support of this entire research.
ACKNOWLEDGEMENTS
ABSTRACT
In an attempt to empirically determine the degree to which the concepts of the simple rigid rotor model approach to light collective nuclei is valid, a study of four similar; strongly deformed, prolate nuclei in the odd count £ = 11 nuclear multiplet has been undertaken. Two members of this multiplet, Na21 and Na22, have been studied utilizing the M g 24(p,Q!'y'Na21, M g 28(p,o!'y)Na28, M g 24(t,o('y)Na22, and Na22(ry, a* y)Na2 reactions in the standard Method n angular correlation geometry of Litherland and Ferguson.
N e w angular m o m e n t u m quantum numbers and electromagnetic de-excitation properties have been determined for levels and transitions in these nuclei. These data together with works of others have been systematically examined by comparing model predictions based on the rotor, Coriolis, and shell models with experiment for the Ne21, Na21, Na22, and M g 22 odd count £ = 11 nuclei.
W e have determined from these comparisons that the rotor and Coriolis model predictions of excitation spectra, electric quadrupole and magnetic dipole reduced transition probabilities in the K ff= 3/2+ ground state rotational band in the £ = 11 nuclei are in better agreement with experiment than the shell model results. Further, the Coriolis results including other single particle configurations reproduce the magnetic dipole transitions matrix elements better than the pure rotor model calculation. Both models, rotor and Coriolis, are equally as effective in reproducing the electric quadrupole transitions as would be expected in a well deformed nuclear system. ^
Despite the remarkably accurate predictions of the electromagnetic transition properties by the simple rotor model in this multiplet, validating the use of the model, there still remains an unexplained oscillatory pertur- bative deviation of the excitation energies from the J(J+1) rule in the mirror pair Na22 and M g 22. Although only partially confirmed, a Coriolis based explanation appears appropiate to correct this anomaly as is seen by the improvement of the Coriolis model over the rotor model in reproducing the excitation spectra of the ground state rotational band.
The large, almost limiting, rigid body values of the moment of inertia (>90%) and the strong prolate nuclear deformations ( 8 ~+0.5), completely supporting the premises on which the rigid rotor model is based, characterize the odd count £ = 11 group of collective nuclei without exception as perhaps the most rigid rotors in nature.
TABLE OF CONTENTS
Abstract
A cknowledgements
Introduction.................................................. 1
A. Motivation............................................ 1
B. General Considerations............................... 2
1. Model overlap..................................... 2
2. Evidence for collectivity............................ 4
3. Limitations of collectivity.......................... 5
C. Nuclear Models....................................... 9
1. The strong coupling collective m o d e l ............... 9
a. The Nilsson m o d e l.............................. 13
b. The Hartree-Fock method...................... 14
c. The Coriolis coupling model.................... 15
2. The shell m o d e l ................................... 17
a. The extreme single particle shell m o d e l ....... 17
b. The single particle shell m o d e l ................. 18
c. The individual particle shell m o d e l ............. 19
3. The SU m o d e l ................................... 2123
D. Literature Survey on Model Interpretations of N a ....... 2321 23
E. Rotational Structure of Na and Na .................... 26
1. Rotor behavior..................................... 26
2. Nuclear rigidity................................... 29
F. Rotational Perturbations................................. 31
1. Nonconstant moment of inertia.................... 32
2. Higher order Coriolis perturbations............... 34-2
3. Expansion of Hamiltonian in powers of R ......... 36
4. Wave function admixtures............................ 39
G. Model Comparison Using Electromagnetic Properties . . . 41
H. Experimental M e t h o d ..................................... 42
1. G a m m a ray angular distribution from aligned nuclei. 43
2. Reactions......................................... 44
3. Background radiation difficulties.....................45
I. S u m m a r y .................... 46
n. Apparatus.................................................... 48
A. Accelerator.............................................. 48
B. B e a m Transport......................................... 50
C. G a m m a Cave and Goniometer............................ 51
D. Radiation Detectors......................................... 51
E. Associated Components................................... 52
1. Scattering c h a m b e r ............................... 52
2. Detector shield.......................................53
3. Detector cooling................................. 54
4. Electron shielding............................... 54
5. Faraday c u p ....................................... 55
6. B e a m stop......................................... 5623 24 26
F. Preparation of N a , M g , and M g Targets........... 56
G. V a c u u m .................................................. 57
H. Electronics................................. 58
I. Hardware . . . ........................................... 59
J. Software................................................ 8°HI. Data Acquisition........................................... 81
IV. Data R e d u c t i o n ............................................ 85
V. Data Analysis . . .......................................... 89
A. Method n Angular Correlation F o r m a l i s m ...................69
B. Spin Assignments and Rejection Criteria................. 73C. Mixing Ratios............................................ 79D. Finite Solid Angle Effect (FSE)............................ 75
VI. Experimental Results...........................................7723 23 77
A. N a faa1 y)Na ......................................... 7726. ,XT 23 7q
B. M g (p,o! y)Na .........................................
vn.vm.
ix .
„ 2 4 23C. M g (t,ay)Na ......................................... 8224 21
D. M g (p,a y)Na ......................................... 82
Discussion of Results....................................... 84
Summary of Results on Odd Count £ = 11 Nuclei.............. 999 Q
A. Na .................................................... 99
B. N a 21.................................................... 1009 1 9 9
C. Ne and M g ......................................... 101
Model Interpretations of Odd Count £ = 11 N u c l e i .......... 102
A. Collective model interpretations of K ff=3/2+ ground state
rotational b a n d ........................................... 105
1. Rotor model predictions with Nilsson intrinsic wave
functions......................................... 106
a. Excitation energies, branching and mixing ratio. 106
b. Intrinsic quadrupole moments and gyromagnetic
ratios......................................... 109
2. High order Coriolis and rotational perturbations. . 114
3. Coriolis coupling m o d e l .......................... 115
a. Excitation energies.......................... 115
b. Electromagnetic properties................... 120
1. Absolute reduced matrix element comparisons 120
2. Relative comparisons...................... 122
B. Rotational B a n d ................................ 124
1. Asymptotic selection r u l e s ........................ 125
2. Calculation of El transitions.................... 128
3. Band purity.............. 129
C. Sensitivity of Electromagnetic Properties to the Nuclear
Deformation.............................................. 130
1. Coriolis coupling model predictions............... 130
2. Inelastic scattering of particles.................. 131
3. Direct measurement from electromagnetic properties 131
D. Other Nuclear Model Predictions......................... 136
1. Excitation energies............................... 136
2. Electromagnetic properties........................ 137
a. Static comparisons.............................. 137
b. Dynamic comparisons.......................... 137
X. S u m m a r y and Conclusions.......................................140
References............................. 142
Appendix I .................................................... 149
Appendix II.................................................. 155
Appendix III.................................................. 159
Appendix I V .................................................. 161
Appendix V . .............................................. 165
Appendix V I ................................................. 179
Appendix V I I ................................................ 182
The complexity of the interactions between nucleons of a nuclear
system precludes a rigorous mathematical treatment of nuclei in terms
of fundamental internucleon forces, especially in view of our insufficient
knowledge of the force producing meson-exchange fields. Even if these
force fields were precisely known, however, we would not be able to solve
the nuclear problem, since state of the art mathematical techniques are
not capable of handling the many body problem in any exact fashion; Not
only are we uncertain of the nuclear forces and limited in our mathe
matical framework, but application of nonrelativistic Schrodinger quantum
mechanics to interactions confined to subnuclear dimensions is not
entirely correct, in the light of marginally important relativistic correc
tions necessarily imposed by Heisenberg’s Uncertainty Principle. However,
such corrections are small and assumed to have a negligible consequence
on the main body of nuclear structure.
Within the Schrodinger formalism the unknown nuclear forces and
unsolvable many body problem are avoided by constructing solvable models
whose salient features approximate the nuclei of interest and the physical
properties of which can be calculated explicitly for comparison with
experiment. The success of this approach is measured by the inherent
plausibility of the selected model and by the degree of agreement attained
I. INTRODUCTIONA. Motivation
2
between theory and experiment, not only for any given nucleus, but also,
and more importantly, for groups of systematically selected nuclei
spanning related areas in the periodic table. F r o m such models certain
basic underlying features emerge, which must necessarily be incorporated
into any fundamental understanding of nuclear structure and behavior. It
seems probable that major improvement in our understanding of the
nucleus will occur in such an empirical and phenomenological manner.
B. General Considerations
1. Model overlap
Since the successful application of the strong-coupling
collective model to light nuclear systems (Br 57, Li 58), nuclei in the
mass region 19 ̂ A ̂ 2 5 have been of growing experimental and theoretical
interest. Additional and more extensive studies, including the present
work, have definitely established nuclear collectivity in this mass region
(Ho 65, P o 66, D u 67, Pr 67, Pr 69a). These nuclei possess strong, rigid
prolate deformations with rotational structure of varying degrees of
purity and relatively large moments of inertia, greater than 90% of rigid
body values in some cases.
Early shell model calculations in the sd shell were first applied
to the mass 18 and 19 nuclear systems (El 55) and, recently, more realistic
shell model calculations have been applied to nuclei throughout the mass
region defined above (Bo 67, H a 68, Wi68). Although the shell model features
are not as prominent as the collective ones, the excitation spectra can be
3
well reproduced, if appropriate amounts of configuration mixing in the sd
shell is included; wave functions from such calculations, however, have
not been thoroughly tested. Comparison of the results of applying these
various models to similar nuclei has already led to a much improved
understanding of nuclear behavior and to isolation of the more fundamental
aspects of the nuclear problems involved.
As an example, Elliott (El 58), in reproducing collective behavior
19in F by expanding over appropriately selected shell model wave
functions, developed the SU Coupling Model, in which wave functionsOwere classified according to the symmetries of the special unitary group
in three dimensions (SU ). This model has been applied with moderateOsuccess in the first half of the sd shell (El 62, El 67, Ha 68a) and has
resolved what had for some time appeared to be rather fundamental dif
ferences between the apparently equally successful shell and hydrodynami-
cally based collective models.
The mass region defined herein is basically, as yet, the only
region containing a sufficient number of nuclei which exhibit pronounced
collective behavior to permit a detailed systematic study in terms of the
collective model, yet with few enough extra-core nucleons involved to
permit treatment within the framework of the shell model. This situation
is unique in defining a testing ground overlapping the region of applicability
of three major nuclear models, collective, shell and SU^, which can now
be directly compared; such comparisons had not been exploited in full
4
detail in the past for lack of adequate precise nuclear spectroscopic
information on the nuclei involved. By systematic comparison of the nuclei
spanning the aforementioned mass region in terms of dynamic as well as
static nuclear nuclear properties, we hope to gain a more fundamental
understanding of the behavior and symmetries of nuclear systems and of
the interrelationship of these nuclear models.
2. Evidence for collectivity
Within the collective framework, the strong-coupling model
based on Nilsson intrinsic states (Ni 55) has been applied to odd A nuclei
in the mass region 19 ̂ A ^25 with overwhelming success in comparison
to the often mediocre results obtained with the shell and SU^ models. Not
only has it been successful, but also the simplicity with which calculations
can be performed and compared with experiment has made it the most
popular mode of data interpretation in this mass region.
The model consists of a single extra core nucleon coupled to a
rigid, well-deformed core. The interaction between the orbiting particle
and the core is represented by a deformed simple harmonic oscillator
potential which is discussed in more detail in section I-C.
The early, macroscopic collective models typically parametrized
experimental data in terms of physically meaningful quantities such as
the moment of inertia, nuclear deformation, etc. A systematic study of
the evolution of these parameters throughout a major shell (sd) can yield
valuable nuclear structure information regarding the rigidity and shape of
5
the nucleus and, indirectly, the importance of the various components of
the nuclear force as the shell fills (Hi 69). A plot of the moment of
inertia obtained as a parameter in fitting the ground state rotational bands
of available nuclei according to the crude rotor equation, E = A J(J+D,
2where A = ft /2I, is shown in Fig. 1. It should be noted that I for a rigid
2 2 spheroid is given (Bo 55) by I = 2/5 M R (1 + 0. 31 8+ 0. 44g + . . .) and
that it is, therefore, a relative measure both of the deformation gand of
the rigidity of the nucleus; it is clear from this figure that the apparent
deformation maximizes in the region defined by the mass 21, 22, and 23
nuclei.
More conclusive evidence for rotational behavior in this region is
signified by enhanced intraband E2 matrix elements, which are shown in
Fig. 2, plotted in Weisskopf units versus the atomic weight of the nucleus
involved. Rotational enhancements are again most marked for these same
nuclei. W e have, herein focussed our attention on these nuclei comprising
the region of m a x i m u m deformation and rigidity, within the sd shell; this
has been done in the hope of testing the extent to which the apparent collec
tive characteristics m a y be extrapolated before the simple concepts on
which they are based require modification.
3. Limitations of collectivity
Of particular interest in this work is the class of odd A
collective nuclei, whose last odd nucleon is the eleventh located in orbit
7 (K77 = 3/2+) of the Nilsson model. This defines the nuclear multiplet of
10
9
8
7
6
o 5I / f l
4
3
2
I
18 2 0 2 2 2 4 2 6 2 8 3 0ATOMIC MASS NUMBER
M O M E N T OF IN E R T IA V E R S U S A T O M IC M A S S
* 2A = n / 2 I A Is a parameter determinedby a least squares fit to the ground state rotational band
I i I l I I I I I 1-------1-------1------ L
Fig. 1
|M|
FOR
E2 Q0
(bar
ns2)
ATOMIC WEIGHTF ig. 2
6
Ne , Na , Na , and M g categorically referred to as the odd count
£=11 nuclei. On the basis of the most simple (single particle) Nilsson
model interpretation these nuclei should exhibit indistinguishable nuclear
structure. In search for rotor behavior among these nuclei we have found
an unexpected marked difference in the sequence of excitation energies of
21 21the ground state rotational bands for the mirror pairs (Ne , Na ) and
23 23(Na , M g ) with the latter pair manifesting distinct departures from the
almost pure rotor behavior exhibited by the former pair (Fig. 3). In
contrast the g a m m a ray de-excitation properties are in relatively good
agreement with rigid rotor behavior for both mirror pairs of nuclei. This
is rather surprising in that dynamic properties are usually more sensitive
to wave function admixtures, and small departures in predicting excitation
energies usually result in larger departures in predicting transition
probabilities. A detailed account of this anomaly is deferred until more
general considerations are discussed.
Two other known nuclei that fall in the £ = 11 nuclidic category, but
are best described in terms of the shell model, as verified by the predicted
ground state spins of 5/2 in contrast to the collective prediction of 3/2 for
19 25the aforementioned nuclei, are O and Na . Although these nuclei are
not specifically considered in this work, they are important in that they
border the concerned region of collectivity reflecting the dependence of
the nuclear deformation on isospin and subshell closure. As an illustra
tion of this dependence, the mean square deviation of the excitation energies
E(M
eV)
E(M
eV)
E X C IT A T IO N E N E R G Y V E R S U S J(J + I)
7.06.05 .04 .03.02 .0 1 0
6.450-
4.431-
2.867-
1.747-
.3 5 0 - *- 0N e 2 i
J (J+1)
7.06.05 .0 4 03.02.0
1.0
2 .705- h 2 .079-
.4 3 90
No 2 3 J(J + I)F ig. 3
7
from the calculated values determined by a least squares fit to the ground
state rotational band members by the equation E = A(J)(J+1) for Ne, Na,C c t iC
and M g isotopes, is plotted versus atomic mass number in Fig. 4. For each
isotopic group there exist optimum rigid rotor behavior characterized by
the minima of the parabolic curve drawn through a given isotopic sequence.
This apparent simple parabolic dependence of rigid rotor behavior
on neutron occupation number is rather remarkable considering the c o m
plexity of the deformation-reducing short range pairing forces, competing
long range forces tending to align nucleonic orbitals maintaining deformation,
and subshell closure effects. A discussion of these effects can be found
in references Br 60, M o 60, Ro 67a, and Bo 69.
In this region, where nuclei m a y well be among the most rigid in
the periodic table, an addition or subtraction of a proton or neutron
markedly effects the rotational character as is seen by the sharpness of
the slopes and narrowness of the curves shown in Fig. 4. In light of this
the differences between the relative location of the band members with
respect to rotor model predictions shown in Fig. 3, which are unexplained
on the basis of a macroscopic model, is not then surprising.
These differences are a result of the fact that nuclei are obviously
not perfectly symmetric rigid bodies with well defined nuclear surfaces.
Even in regions where the model works best there are unexplained systematics
and discrepancies between theory and experiment. A particularly interesting
1 0 ° O’ 2 V E R S U S A T O M IC M A S S
o ' 2 * 4 * I
8
example of this has very recently appeared in the measurements of the
20 22intrinsic quadrupole moments of certain even A nuclei such as Ne , Ne ,
24 +and Ne in this mass region, when in their first excited 2 states (Ha 68a,
Na69, Sc 69, Sc 69a). These measurements have yielded the very
surprising results that these states have quadrupole moments 30% greater
than those which would be expected for rigid rotors having the experi
mentally determined deformation of the 0+ ground states in each case.
Not only are these results in themselves not yet understood, but also the
contrast between these nuclei and the adjacent odd mass isotopes, as
studied herein, is most striking.
These seemingly inexplicable disagreements are, of course,
manifestations of many unaccounted for degrees of freedom intrinsic to
the microcomposition of the nucleus. In search for possible explanations
of these departures from pure collectivity, we have concentrated on
examining, systematically, the static and dynamic properties of this
group of nuclei in terms of the present strong coupling Nilsson model and
in terms of possible extensions and modifications thereof incorporating new
and previously unaccounted for degrees of freedom. In this way we have
attempted not only to explain present disagreements, but also to determine
the limitations of the simple collective approach.
To put this problem in perspective, a brief review of nuclear
models is presented emphasizing aspects of each according to their
9
importance in the work discussed herein.
C. Nuclear Models
1. The strong coupling collective model
The strong coupling collective model of Bohr and Mottelson
(Bo 52, Bo 53) as applied to odd A nuclei consist of a rotating deformed
core with angular momen t u m R coupled to a single extra-core nucleon
orbiting about the core with angular m o m e n t u m j (Fig. 5). The total
angular m o m e n t u m J of the core plus particle is given by
J = R + j .
The Hamiltonian in the strong-coupling model framework m a y then
be written as
h = a r 2 + h 'sp
—►2where A R is the rigid rotor contribution from the core, A is the moment
2 /of inertia parameter h /2I, and H^ is a single particle Hamiltonian
representing the interaction of the single odd nucleon with the core. Since
2R is not a constant of the motion, it is more useful to substitute
R = J - r,
expand the square (J- j )2 and dot product (J • j ) obtaining,
H = A J 2 - 2AJ3i3 - 2A(J1j1 + J2y +Aj*2+ ff'p .
Since the core is assumed, in this simple model, to be axially symmetric,
the projection of R on the body-fixed symmetry axis is zero (Rg=0).
Using the notation K=J0 and (1= j and substituting R =0 in the equationu O O
STRONG COUPLING MODEL PICTURE OF ODD A NUCLEI
z
F ig . 5
we find K = Q. That is, the projection of the total angular m o m e n t u m on
the symmetry axis is equal to the corresponding projection of the angular
m o m entum of the orbiting single particle outside the core and is a constant
of the motion in the absence of the familiar Coriolis coupling of intrinsic
and rotational motion in such a system.
By defining ladder operators in the usual fashion,
J ± = J l± i J 2
for the components of J and j, substituting into the expression J j + J j ,JL J. d dand rearranging terms, the Hamiltonian in the strong coupling model m a y
be written in the form
H = H + H + H , rot cor sp
where
H 4 = A(J2 - 2K2) , rot
Hcor = 2A(J+i- + J> >
and H = A j 2 + H' .sp sp
2The core Hamiltonian A R has been subdivided into a pure rotational
part H and a part H coupling the rotation of the core with the single rot cor
extra-core nucleon in analogy with the form of the classical Coriolis
rotational coupling term co* j . The last component, H , is a generalsp
single-particle Hamiltonian whose specific form is dependent on the choice
of interaction between the odd particle and the core.
11
By neglecting the Coriolis coupling terms we may write the
Hamiltonian as
H = ft2/2I ( J ^ K 2) + H .sp
Without specifying the exact form of and restricting ourselves to
deformed axially symmetric nuclei, the Hamiltonian may be conveniently
diagonalized in a basis defined by eigenfunctions of the form (Da 69)
where are single particle intrinsic eigenfunctions of H ,D^_ (0 .)Sp lv i.1 V 1
are rotation matrices of Euler angles (0 .), and C _ a r e expansion
coefficients. The wave function | J K M > is characterized by total momentum
J with projections on the body fixed and spaced fixed axes of K and M,
respectively.
This form of the wave function is characteristic of the strong coupling
model and does not depend on the specific choice of H gp« The two part
wave function depending on K and -K is a result of the axial symmetry of
the core and it is just this symmetry that is responsible for diagonal
contributions to the Hamiltonian from the Coriolis coupling terms in the
case of K=l/2 bands and in higher bands when correspondingly higher order
powers of the Coriolis perturbation are included. F r o m this simplified
form of the Hamiltonian it follows that for relatively large moments of
inertia the model predicts a series of closely spaced rotational levels
built on more widely spaced single particle levels. The faster orbiting
12
single particle "follows" the slower rotating core with no significant
perturbation of the orbit of the single particle (adiabatic approximation).
In heavy odd A nuclei, where I is large enough to validate
this assumption, the Coriolis coupling can be by and large safely
neglected. In contrast to the heavy nuclei, the smaller moments of
inertia of light nuclei, although almost rigid body values, cause complete
overlap of single particle and rotational levels in violation of the adiabatic
approximation. In spite of this violation, rotor-like spectra surprisingly
22have been still identified. In some cases such as Na the absence of
nearby rotational bands satisfying the K band mixing selection rule £K=±1
21preserves the rotational structure, while in other nuclei such as Ne ,
23Na , etc. no equivalently simple explanation has been presented to account
for the preservation of the ground state rotational bands in the presence
of possible full Coriolis coupling.
Before specifying H gp> follows from the form of the wave function
that the calculation of any dynamic properties linking levels in a given
rotational band depends solely on the properties of the rotation matrices,
since the intrinsic parts of the wave function remain unchanged. This is
an extreme simplification and comparison of predicted and experimental
E2 transition strengths have provided characteristic signatures for
rotational behavior as have the large static ground state quadrupole
moments observed in these cores.
The band heads are not confined to single particle origin. In even-
13
even nuclei H is replaced by a vibrational Hamiltonian (Bo 53) where the
nucleus is assumed to undergo vibrations similar to those of a liquid drop.
The Coriolis coupling term is replaced by an analogous term coupling the
rotations and vibrations of the nuclear surface. Applying the adiabatic
approximation to this model and likening the vibrational motion to the
single particle motion, rotational levels are built on vibrational band
heads denoted as beta (axial vibrations) and g a m m a (nonaxial vibrations)
bands. Rotational bands of this nature are c o m m o n in rare earth and
actinide nuclei, but have not as yet been found, unambiguously, in light
sd shell nuclei.
a. The Nilsson model
To obtain further detail from the strong coupling
model the form of H gp must be specified. In light odd A nuclei the most
simple and successful approach has been the Nilsson Hamiltonian (Ni 55)
in the form
h = h + cT- r + dT- rN o
where
Ho = l b + t maJo r2 f1- 2 ̂Y2 0 ^ ) -
H q is a single-particle deformed harmonic oscillator potential with
deformation 0 , characteristic mass m, and frequency coQ . S and I are the
intrinsic spin and orbital angular m o m e n t u m of the particle, and C and D
2are parameters measuring the strength of the spin-orbit and L terms,
respectively. The form of these latter terms is given by the comparison
14
has shown that the coefficients there derived are not physically realistic;
hence C and D have been introduced as fitting parameters in the model.
The result of diagonalizing the Nilsson Hamiltonian is a sequence
of single particle deformed orbitals which are functions of the nuclear
deformation ]3. The constants C and D were originally chosen to give the
appropriate shell model level splittings in the limit of zero deformation.
A typical Nilsson energy level diagram is shown in Fig. 6 illustrating the
23filling of the single particle orbits of Na for a given deformation
3 B 5(T) = / — ). The total Hamiltonian that is diagonalized in the strong
coupling Nilsson model is
H = A(J 2) + H n
where the Coriolis coupling term has been completely neglected. Application
to light nuclei in the sd shell, particularly in the first half of the shell,
has been surprisingly successful (Li 58, Ho 67, Po 66, Pr 67), especially
in light of the non-fullfilment of the adiabatic approximation. Where
necessary the Coriolis interaction has been introduced as a perturbation
operator using the eigenfunctions of the above Hamiltonian as a basis
set.
b. The Hartree-Fock method
In an attempt to determine a more general set of
single particle deformed orbitals Levinson and Kelson (Ke 63, Ke 64)
employed a Hartree-Fock variational method leading to a Hamiltonian that
of the j term appearing in the above R expansion, how ever, experience
V
Fig. 6
15
included a harmonic oscillator part plus spin orbit and -T terms and a
two-body Rosenfeld interaction having a Yukawa radial dependence. The
strength of the two body interaction was used as a parameter and serves
in a capacity similar to the nuclear deformation in the Nilsson model. The
calculated positions of the single particle orbits are similar to those of
Nilsson and the results only differ substantially in the placement of hole
excitations. In the Hartree-Fock calculations the hole excitations are
consistently located higher in excitation than in the Nilsson case. The
large gap between filled and unfilled orbitals reflects the inclusion of
exchange forces in the Hartree-Fock Hamiltonian, which are absent in the
Nilsson case. This is a vitally important aspect of the nuclear many body
problem analogous to that characteristic of the superfluid and super
conducting states in the theory of condensed matter.
c. The Coriolis coupling model
The nuclei under discussion are filling sd subshells
16outside an assumed inert core of O . Rotational bands are based on
single particle or hole excitations where the former are generated by
promoting the last odd nucleon to higher lying orbit s previously illustrated
in the Nilsson energy level diagram in Fig. 6. Hole excitations differ
in that the nucleon is promoted from a fully occupied lower lying orbit to
a previously partially occupied higher lying one. Within the sd shell there
are six possible positive parity orbits into which a given nucleon can be
excited including, in the case of the odd count £ = 1 1 nuclei, a hole
16
1T +excitation based on a K =1/2 band derived from the d . subshell.
5/2
Low lying negative parity states have also been, heretofore,, identified
as hole excitations originating from the lower fully occupied p shell.
In the simple rotor or Nilsson model, interactions between the single
particle or hole levels are ignored. This approximation is justified if
the interacting band heads are much further apart than are the rotational
23levels within any given band. For most levels in Na this criteria is not
satisfied and proper account of these interactions is accomplished by
including the Coriolis term in the total Hamiltonian.
The most consistent and complete treatment of Coriolis band
mixing in the sd shell has been developed in the Coriolis Coupling model
of Malik and Scholz (Ma 67). Here the total Hamiltonian to be diagonalized,
in the sd shell subspace, is written as
H = H a+ H + H rot cor sp
where H H , and H have been previously defined as the rotor,rot cor sp
Coriolis Coupling, and Nilsson terms, respectively.
The single particle band heads are calculated from the equation
(Ne 60)
E = £ +£ r > r+ D t.-fc -where E is the Nilsson energy of the individual nucleons and the sum»£> Vis over all nucleons in the nucleus. M is the true nucleon mass and p is
an effective mass defined by Newton (Ne 59). The parameters A, |3, C,
and D are varied until the best fit between the calculated energy levels
and the experimental ones is obtained.
The range of the parameter C is restricted to values that lie
17 39between the d 5/2- d 3/2 level splitting of O and of Ca in the limit
of zero deformation. Using this model, generally good fits have been
obtained systematically for sd shell nuclei with reasonable sets of para
meters (Ma 67, Hi 69).
2. The shell model
In contrast to the simplicity with which the collective model
m a y be applied to light nuclei in the sd shell, the complexity of realistic
shell model calculations requires the use of high-speed, large-memory
computers to perform large matrix diagonalization even for a system of
a few active nucleons. These calculations have been prohibitive in the
past and only because of the recent availability of such computers has it
become possible to treat nuclei in this framework.
a. The extreme single particle shell model
Simple shell model treatments are completely
inadequate as m a y easily be demonstrated by attempting to predict the
+ 23anamolous 3/2 ground state spin of Na on a shell m o d e l basis.
In the extreme single particle shell model of odd A nuclei, the nucleus is
approximated by a single particle moving in a potential well given by
18
where V(r) is a central potential, the behavior being intermediate between
a square and a simple harmonic oscillator well, and -t and s are the orbital
and spin angular momenta of the single particle. This model, enunciated
by Mayer (Ma 50), which correctly predicted the ground state spin of
23almost all stable nuclei, incorrectly predicted a 5/2 assignment for Na , the
55only other major discrepancy at the time being M n . In view of this,
application of a limited shell model to these nuclei should be with parti
cular reservation.
b. The single particle shell model
A slightly more sophisticated approach involves
incorporation of a residual interaction between the nucleons outside a
closed shell, but not so strong that it cannot be treated by first order
perturbation theory. This approximation is called the single particle
model (Pr 62) where the total potential of the extra core nucleon can be
schematically written as
H = £ V(r.) + L At.* s. + T v(r..)sp i i' i i i i
1 9
c. The individual particle shell model
This rule no longer applies and the ground state spin
23of Na is correctly predicted when including strong residual interactions
(FI 54, El 55) between the extra core nucleons especially in cases where
configuration mixing includes the s 1 y 2 and d3/2 orBits as w e R as tlie % / 2
(Bo 67, Wi 6 8). In this approach the Hamiltonian differs from that of the
single particle model in that the strength of the residual interactions is
stronger, requiring a total diagonalization of the Hamiltonian, since
neither LS nor jj coupling are diagonal representations of the perturbations.
W e refer to this approach as the individual particle model or intermediate
coupling model. Reasonable agreement for the energy levels up to 4. 0 M e V
23in Na , as well as in other sd shell nuclei, were obtained in the calcula
tions of Bouten et al. (Bo 67) performed within the framework of this
model. The technique consisted of calculating excitation spectra in the
two extremes of LS and jj coupling and using first order perturbation
theory to calculate small departures from each extreme as a function of
the strength of ts coupling. Assuming that the eigenvalues are smooth,
monotonic functions between the two extremes, energy eigenvalues for
arbitrary degrees of intermediate coupling were then obtained by inter
polation. The approximation appears to be valid for eigenvalue inter
polation, but fails in calculating wave functions which are necessary for
calculation of any of the electromagnetic dynamic properties.
single particle model.
20
Another approach, taken by Wildenthal et al. (Wi 68), applied to
20
21
perform the calculation. Therefore, the actual nuclear wave function is
approximated by the expansion functions, but in certain cases the
approximation is quite good.
19In the specific case of F , both the shell and collective models
gave a reasonable fit to the energy levels and, indeed, it was observed
that rotational behavior could be derived by expanding over a limited
number of shell model configurations (El 58). This initiated the
application of group theoretic techniques to the classification of nuclear
energy levels. This has the advantage of exploiting the nuclear symmetry
through the transformation properties of orbital angular m o m entum eigen
functions.
3. The SU modelOIf a given Hamiltonian is invariant under the symmetry
operations of a group, then there corresponds to each eigenstate of the
Hamiltonian an irreducible representation of the group, by which the eigen
state m a y be labelled. The degeneracy of the eigenstate is given by the
dimensionality of the group.
Elliott (El 58) showed that the symmetry group of the three dimen
sional harmonic oscillator Hamiltonians is the special unitary group in
three dimensions (SU~). Therefore, the eigenstates of the harmonicOoscillator Hamiltonian can be labelled according to the irreducible repre
sentations of the SU group. Assuming the radial dependence of theOnuclear Hamiltonian is dominated by harmonic oscillator like terms leading
22
to a long range effective force of the r. r P 2(cos0„) type and that spin-
orbit coupling or spin-dependent forces are negligible, then SU will beuan approximate symmetry group of the nuclear Hamiltonian. Because of
the principle of indistinguishability of identical nucleons comprising a
nuclear system, the Hamiltonian must also be invariant under the permu
tation group. Since SU and the permutation group operate in differentOspaces, eigenstates of the Hamiltonian can be simultaneously classified
according to the irreducible representations of each.
A class of states possessing particular permutation symmetry,
determined in part by requiring the total wave function to be antisymmetric,
m a y be labelled by a partition [f] and then, for a given partition, the
individual eigenstates of the class are labelled by the SU quantum numbersO(X,pi) in addition to the usual L,S,T,etc. quantum numbers. The lowest
lying states are labelled by irreducible representation (A, p ) that correspond
to states of m a x i m u m orbital symmetry with X " ^ > p corresponding to prolate
shapes and \ « p corresponding to oblate shapes.
23In Na the lowest lying positive parity states are classified
according to the partition [43] with the (8,3) leading representation of S U g(Ha 68a).
For non-zero ground state intrinsic spin nuclei the intrinsic state or
leading representation contains more than one eigenstate of the same J.
The " J projection scheme" is used to classify states of the same J
according to their K labels. In the example given K = 1/2, 3/2, 5/2, and
7/2 and for each K, J = K, K + 1, . . . K + A. Indeed, K = 1/2, 3/2, and
2 2
23
23 TT +5/2 rotational bands have been identified in Na . In the K = 3/2
23 +ground state band of Na , members up to the = 13/2 level inclusive
have been identified, but not as yet has the predicted cutoff of J n = 19/2+
19 i t +been reached. In F members of the ground state K = l/2 band are
__ .j.known up to the J = 13/2 SU predicted cutoff limit and at this time noOvalues are known which exceed the SU limit. It would clearly be of greatOinterest to firmly establish the validity or breakdown of these SU_ predic-Otions in the form of higher spin band members. As yet the only discrep-
g
ancy is the rather special one in Be .
23D. Literature Survey on Model Interpretations of Na
23The earliest attempts at calculating the low lying spectra of Na
were done with the collective model. The calculations of Litherland (Li 58),
Rakavy (Ra 57), Paul and Montague (Pa 58), and of Clegg and Foly (Cl 61)
were performed with Coriolis coupling included in the strong coupling
Nilsson model. Moments of inertia and band head excitations were used
as parameters in obtaining fits for the first three excited states. The
lack of definitive experimental spectroscopic information discouraged any
detailed comparisons. The asymmetric-core collective model of Chi and
Davidson (Ch 63) and the Hartree-Fock approach of Kelson and Levinson
23(Ke 64) also fit the first few states of Na and predicted approximate
locations for some of the higher lying levels. With the exception of
Litherland and of Rakavy, who admixed only two bands, the above authors
24
mixed the = 3/2+ , l/2+ , and 5/2+ band and neglected the other three
band heads in the sd shell on the basis that they were too high in excitation
to contribute to the low lying spectra. This is partially true, but it has
since become known that the ground state band mixes strongly with a l/2+
23hole band located at about 4.4 M e V excitation in Na . The calculation of
Glockle (G1 64) included the l/2+ hole excitation (Nilsson orbit 6) in
•J*addition to the 3/2 , 1/2 , and 5/2 bands and obtained a reasonable
comparison with the experimental data known at that time. But the hole
excitation band head was incorrectly positioned at 2, 64 MeV, for which
state the parity has since been shown to be negative.
The work of Howard et al. (Ho 65) was the first to compare dynamic
21 21properties for the 3/2, 5/2, and 7/2 band members of Ne , Na , and
23Na . Considering that Howard completely neglected band mixing, good
agreement was obtained for the few comparisons made.
23Aware of the need for additional experimental information on Na ,
Poletti and Start (Po 66) measured mixing and branching ratios and
rigorously limited spin assignments for levels up to 2. 98 M e V excitation
23in Na . Poor experimental statistics and generally weak correlations
precluded any new or unique spin assignments. Experimental studies have
also been reported on the J=-| states at (2.39, 2. 64) and 4. 43 M e V (Pe 66)
and (Me 64), respectively , and on the 2.98 M e V state (Ra 66). Earlier,
branching ratios and approximate spin assignments were made to some
levels up to 4. 78 M e V through study of resonance proton capture on
25
calculations with Coriolis coupling between the four lowest lying bands,
(using as parameters the band head energy, different moment of inertia
for each band, deformation, and a generalized spin-orbit coupling constant,
23for levels below 5. 0 M e V in Na ) were performed by El-Batanoni and
Kresnon (Ba 67). Also, calculations by Malik and Scholz (Ma 67) mixing
in all siz bands in the sd shell with a single moment of inertia and the
23deformation as parameters, were done for levels up to 8. 0 M e V in Na .
In both calculations the overall fit was good, but again both groups of
authors were led astray through fitting low lying assumed positive parity
states that have since been shown to have negative parity. A n important
difference between the above two calculations is that Malik and Scholz
calculated band head energies and used the same moment of inertia for
each band, which reduced considerably the number of parameters used
in fitting the data. It is also interesting to note that the latter calculation
predicts the ll/2+ and 13/2+ ground state band members at excitations
between 5.0 and 7.0 MeV, which is the approximate position predicted by
the rotor model as well.
Additional information determined from g a m m a - g a m m a angular
correlations was reported by Maier (Ma68a); particularly defini-
“I*tive 7/2 and 9/2 assignments identified the second and third ground state
23band members of Na . Branching ratio and lifetime information was also
22Ne (Ar 62, Br 62 ).With the additional experim ental inform ation, new co llective model
26
In addition to the ground state rotational band and the higher lying
positive parity states, another series of levels of current interest are the
negative parity states, believed to be hole excitations generated by
promoting a particle from orbit 4 to orbit 7. Since the beginning of this
work, a few negative parity states have definitely been established.
Reflecting the absence of other low lying negative parity bands, this
77 _K =1/2 band should exhibit pure rotational behavior, the degree of which
we have studied herein.
During the course of this experiment other theoretical and experi
mental information was reported, but will be discussed later together with
the results of our current work. A s u m m a r y of the experimental informa-
23 21tion on Na and Na at the outset of the present measurements is shown
in the energy level diagram in Fig. 7.
E. Rotational Structure of Odd Count £ = 1 1 Nuclei
1. Rotor behavior
In odd-even nuclei the single particle structure is determined
as is evident from the Nilsson energy level diagram in Fig. 6, by promoting
the last odd nucleon into various unoccupied orbits creating single particle
excitations. Hole excitations m a y be generated by promoting a particle
from a lower occupied orbit, e. g. (#4), to a higher partially occupied one,
e. g. (#7). Both types of excitations have been found experimentally in the
£ = 11 nuclei. In these nuclei, where the number of odd count nucleons
obtained on som e of the other lev e ls in Na .
6.311 ----------------------------------------------- 1 / 2 *
Fig. 7
27
is the same, and under the assumption that the nuclear structure is -
determined by the last odd nucleon, the sequence of orbits available for
occupation by the excited single nucleon are identical and, therefore, the
spectra of nuclei in this scheme should be very similar. As an example,
excitation spectra of four of the £ = 11 nuclei are shown in Fig. 8 illustra
ting the 3/2* ground state rotational bands. Other similarities exist in
these nuclei but are omitted for purposes of clarification. In each nucleus,
it should be noted that the ground state has the same spin and parity
followed by a series of rotational levels. This is by no means a trivial
example; as was noted earlier, in the case of two of the neutron rich
19 25£ = 11 nuclei, O and Na , ground state spins are not even predicted by
the strong coupling collective model and the excitation spectra of these
nuclei possess no obvious rotational structure.
It is of interest to examine the excitation level sequence more closely.
21The excitation of the members of the ground state rotational bands of Ne
21and Na are approximately linearly dependent upon J(J+1), up to the recently
, + 21 , + 21 established 11/2 m e m b e r in Ne (Ro 69) and the 9/2 members in Na
21(Pr 69). Higher levels in Na have not been heretofore identified and a
. + 21possible 13/2 state in Ne is under study (Ro 69a) with excitation energy
consistent with the J(J+1) rule. The members of the corresponding
23 +bands in Na w e r e previously known up to the 9/2 m e m b e r
+ +and new measurements, presented herein, identify the 11/2 and 13/2
members substantiating the systematics suggested by the low lying band
EXCITATION
ENERGY
(MeV)
V
6.450-
4.431-
*
28
members. A plot of excitation energy versus J(J+1) was shown earlier in
21 23Fig. 3 for Ne and Na illustrating basic structural differences in the
rotational bands.
23In Na the levels show oscillatory systematic departures from a
pure rotor spectrum implying the existence of rotational perturbations,
21in contrast to the almost pure rotor behavior in Ne . These differences
provide evidence for rotational anomalies unaccounted for in previous
collective treatments of these nuclei. The little experimental information
23 21available on the corresponding mirror nuclei M g and Na confirms the
above systematics and, therefore, the differences are not an accidental
23peculiarity of Na itself. Additional evidence supporting this type of
159rotational behavior has been found in Tb (Gr 67). Here the level
TT +sequence in the K =3/2 ground state band oscillates in much the same
23manner as in Na , but the departures from pure rotor behavior are much
smaller.
A possible collective mechanism capable of producing the observed
23level ordering in Na is Coriolis band mixing. Strong Coriolis coupling
1T + +between a K =3/2 ground state band and a higher lying 1/2 band,
having a large decoupling parameter producing level inversion in the 1/2
band itself, could account for the observed effect.
rr +A higher order effect that could be unusually large in a K =3/2
band is third^order Coriolis decoupling in the K = 3/2 band itself, similar
to the decoupling in the K = 1/2 band. Such mechanisms were suggested
2 9
to account for the results in Tb and are discussed in more detail in
Section I-F. Both mechanisms are possible explanations but neither
23 21within the model framework favor Na over Ne .
F r o m a microscopic point of view, and perhaps more realistically,
21 23the difference between N e and N a is that the latter has an additional
7T *1*proton and a neutron in the K = 3/2 orbit. It might be expected that the
23two neutrons in the K = 3/2 orbit in Na are inertly paired and are
effectively incorporated in the core leaving the odd proton to generate
single particle excitations. If residual interactions between the extra-
20core particles are to be considered, all three nucleons outside of Ne
23(in the case of N a one proton and two neutrons) would have to be treated,
which is beyond the scope of contemporary strong coupling models. Up to
20three neutrons outside a close core of Ne has been considered and such
approaches have been applied to the neon isotopes with favorable success
(Cr 69). Also odd-odd nuclei (i. e. one proton and one neutron outside an
inert core) have been treated including complete Coriolis coupling with
proper account of isobaric spin (Wa 69). In any case, multiple nucleonic
excitations in an unfilled subshell outside a closed core are not considered
in the spirit of the approach taken herein.
2. Nuclear Rigidity
Continuing the spirit of the simple rotor model, states of
high spin should be generated by successive rotations of the nuclear core.
Locations of the states are determined from the equation
30
E = ft2/2l J(J+l)
in the absence of any departure from rigidity. Searching for states of
high angular m o m e n t u m in light nuclei, where the SU and shell modelsOpred'ct finite limits on the magnitude of the angular m o m e n t u m quantum
number terminating a rotational band will have interesting consequences.
In particular, a value exceeding the cutoff would certainly question the
detailed validity of the shell or, more particularly, the SUQ model to
23nuclei in this region. A plot of excitation energies versus J(J+1) for Na ,
up to an extrapolated spin of 17/2 , is shown in Fig. 9. F r o m an empirical fit to the data l A r. 97> found; these almost rigid body values, typical
of nuclei near mass number 23 are in marked contrast to typical rare
earth values of l / Y g “ 9- 3- Systematic application of the Coriolis
coupling model to sd shell nuclei predicted similar results as shown in
Fig. 10 (Hi 69). Positive deformations have been measured and determined
from best fits of the data for the nuclei considered herein and, together
with rigid body moments of inertia, these imply rigid, well-deformed
prolate nuclei, capable of maintaining rigid deformations up to large values
of angular m o m e n t u m without significant centrifugal stretching.
The high spin states, as apparent in the J(J+1) plot, lie at high
excitation energies exceeding thresholds for particle emission. However,
cascade de-excitation by electromagnetic g a m m a decay might well remain
as favored over particle emission in view of the large angular m o m e n t u m
which would necessarily be carried by the emitted particle. These states
EXCI
TATI
ON
ENER
GY
EXCITATION ENERGY VERSUS J(J+I)
M A S S N U M B E R
F ig. 10
31
radiation to and from the lower band members. F r o m the existing
systematics of the rotational levels and the apparent rigid deformations
21 23of Ne and Na , it is probable that little centrifugal stretching occurs
and that a relatively constant moment of inertia is maintained. Therefore,
levels of excitation may be expected to follow the J(J+1) rule up to large
angular m o m e n t u m quantum numbers exceeding the largest presently
measured value of 13/2 ft in light, odd-A, sd shell nuclei.
F. Rotational Perturbations
The static properties, and enhancements of certain dynamic
properties, of the low-lying states of many nuclear species, particularly
the rare earth nuclei, have been reasonably well accounted for by the simple
rotor model. Although not quite as successfully, the model has been used
to interpret properties of the low lying levels of light sd shell nuclei,
particularly in the first half of the shell.
In both regions of the periodic table particular nuclei clearly depart
from rotor behavior without obvious physical reason. These discrepancies
m a y be explained,on occasion, by adding higher order corrections to the
rotor Hamiltonian or by treating the existing terms in the Hamiltonian to
higher order perturbation theory, in effect approximately diagonalizing
the Hamiltonian. Perturbation treatments in lowest order usually suffice,
especially in light of the increased computational difficulty in exact
diagonalization of the total Hamiltonian. Because the exact diagonalization
may therefore be identified from the ch aracteristics of the cascading
*6
32
is done in a truncated space of basis wave functions, this approach is, in
any case, only reliable for the first few lowest states in the energy
spectrum.
In cases where a larger range of states in a rotational band,
including ones of high excitation energy, are of interest, a perturbation
treatment is frequently more useful. Here, by exploiting the symmetry
of the perturbing Hamiltonian, and inspecting the matrix elements in the
perturbation expansion, it is frequently possible to identify which terms
give significant corrections to the energy eigenvalues. The calculational
advantages in applying perturbation theory to the rotor model will be
demonstrated herein. W e consider the effect of various rotational pertur
bations on the energy eigenvalues determined from an unperturbed rotor
model using strong coupling model wave functions.
1. Nonconstant moment of inertia
It was noted earlier that if the rotational motion was not
sufficiently slower than the vibrational motion the adiabatic approximation
was invalid and that rotation-vibration interaction terms must then be
incorporated into the total Hamiltonian. The first order correction to the
Hamiltonian for this interaction is a term of the form (Wo 67, Na 65)
Hr o t-v ib = -B'J»2'J+1>2where the numerical sign of B has been explicitly written. Such a term
is well known from the simplest molecular physics studies involving an
angular m o m e n t u m expansion of the rotational energies (Wo 67). The
33
Hamiltonian used to fit the early data for M g (Li 58) included this term,
for example, and its inclusion together with higher order members of the
angular m o m e n t u m expansion has been most clearly demonstrated in the
case of heavier nuclei.
The coefficient B has been calculated by Hamamato (Ha 69) using
the cranking model formula (In 54) with a Nilsson model Hamiltonian
including a pairing interaction. The magnitude and sign of B, thus
obtained, agreed fairly well with data taken for heavy nuclei. By casting
the above equation in a slightly different form it is possible to interpret
the correction factor as an apparent increase in the moment of inertia.
W e m a y write
H = ft2/2I * J(J+l)
whereI ' = I / ( 1 - B / a (J)(J+i))
As J increases the effective moment of inertia increases, which
physically corresponds to a stretching of the nucleus as it undergoes
successively faster rotation. Recently, this idea has been more fully
exploited, resuting in two very interesting, simple, semiclassical
approaches that have been remarkably successful in accounting systematically
for collective properties of many heavy, even-A nuclei.
One is a centrifugal stretching of a classical rotator (So 68) and
the other is the variable moment of inertia model (VMI) of Mariscotti
et al. (Ma 68). These models warrant attention in view of their mathema
tical simplicity and especially in light of their success over a broad range
25
34
of nuclei. No doubt that the moment of inertia is not a constant of the
motion and that any reasonable collective model should include this effect
in some form or another. However, to employ the centrifugal stretching
models, a broad range of nuclei with similar rotational properties must
be kr.own in order to obtain reliable fitting parameters averaged over many
nuclei. In the mass 21 and 23 region the onset and deterioration of
rotational behavior is so brief and abrupt that no such averaged fitting
parameters can be determined.
In addition both models are designed to apply to even A nuclei
rather than to odd A nuclei, which are of particular interest in this work.
2. Higher order Coriolis perturbations
In light, odd-A nuclei, departure from rotor behavior is
usually attributed to Coriolis band mixing, where the band heads are
intrinsic, single-particle states. In heavy deformed nuclei the band heads
include beta and g a m m a vibrations on the nuclear surface as well as
single particle excitations. No analogous vibrations have been found as
yet in light odd A nuclei. We, therefore, confine our attention to rotational
perturbations generated by the interactions between the orbiting odd
particle and the core, of which the most c o m m o n form is the Coriolis
interaction.
A systematic departure from pure rotor behavior similar to that
23 +illustrated earlier in Na , was found in the K 7T= 3/2 ground state band
159of Tb (Gr 63, Bi 66). Although the deviations were much smaller, they
35
persisted up to the highest then known band m e m b e r (J = 23/2). It was
discovered that the energy levels of the 3/2 band could be fitted with an
expansion of the form
where A,B, and C are parameters determined by fitting the data (Gr 63).
The exact physical origin of. the last term, which we shall refer to as
third order Coriolis decoupling, is unclear in that the angular m o m e n t u m
dependence of such a term can be calculated by starting with different
forms of the Hamiltonian. To obtain a better understanding of its possible
origins, we consider it in more detail.
Treating the Coriolis interaction
as a perturbation and using the unperturbed strong coupling model wave
functions
the corrections to the energy eigenvalues may be conveniently calculated
corrections are calculated in the appendices given herein.
Since the basis functions form a complete orthogonal set and the
E(J) = A(J)(J+1) + B J2(J+1)2 + C(-l)J+^(J4)(J+|)(J+3/2)
H ' = -2A (J+j_ + J-j+)
up to third order in a perturbation expansion of the usual form (Sc 68)
m m m m m
where is the unperturbed energy of the mth state and the additional
36
Coriolis perturbation has no diagonal contribution in a K = 3/2 band,
there is no contribution in first order (i. e. = 0).
It is shown in Appendix I that corrections in second order can be
written in the form
E (2) = A i+A 2 (J)(J+3)
and are equivalent to a renormalization of the moment of inertia and band
head energy. Such corrections are obviously already included in the simple
rotor model when the coefficient of J (J+l) is treated as a parameter in
fitting the data.
The first nontrivial correction appears in third order and the cal
culation in Appendix II yields a term of the form
e(3) = ("1)J+3/2
37
another approach leading to a term of the same angular momentu m
dependence, but without inclusion of band mixing, is to effectively expand
—*-2the Hamiltonian in a power-series in R . Recall that the rigid rotor part
of the Hamiltonian was written as
W e begin by phenomenologically assuming that l / l m a y be written as a
-*2slowly varying function f(aR ) where a is a small constant (Mi 64).
Justification for such a function is by no means rigorously based. Since
the Hamiltonian must be a scalar and the effects of centrifugal core
distortion are independent of the axis of rotation and in analogy with the
treatment of molecular rotation (Wo 67), the simplest nontrivial non
vector ial fuction that can be expanded in a power series is a function of
2the form f(aR ). Substituting
~=f(aR2)
2 2 in the above Hamiltonian and expanding f(aR ) in a power series in aR
f(aR2) = l / l (B + B (aR2) + B (aR2)2 + . . .) ,O O i / 2the rotor Hamiltonian m a y then be written in the form
H = *- E A (R2)w ,2Iq i/=0 uwhere I is the moment of inertia at rest and A are constants determined o vby the internal motion of the nucleus.
It should be noted that this is not the most general angular m o m e n t u m
38
expansion (Mi 64), but it suffices to illustrate how higher order diagonal
contributions m a y be included in the rotor model. However, assuming
that centrifugal distortions preserve axial symmetry and R remainsOzero, the above form is indeed correct within the framework of the rotor
model. A n expansion in powers of J ± systematically including higher order
Coriolis perturbations is given by Bohr and Mottelson (Bo 69).
Recalling that S = J-j and using the strong coupling rotor model
wave functions, the calculation of all diagonal contributions or first order
2 3corrections to the energy, up to the (R ) term, is shown in Appendix IV.
The result is an expression for the energy written as3
E = AJ(J+l)+B J2 (J+l)2+C [ J3(J+l)3-8(J-^(J+|)(J+|)(-l)J+2"a3/2]
where *s definec* as K = 3/2 third order decoupling parameter
r i J-3/2ln ,2w _iwiJ
j
in complete analogy with the more familiar K = 1/2 decoupling parameter.
a3/2= £ (“1) lC j3/2 I
39
identical, the Hamiltonian expansion in power of R reflects vastly
different physical behavior than does the strong coupling rotor model with
Coriolis interaction. In the former the nucleus is imagined to undergo
complicated rotations and vibrations, neither motion being spelled out
explicitly, while in the latter the nuclear motion is predominantly
rotational with strong Coriolis coupling between the odd particle and the
core. Distinguishing between these two kinds of motion is virtually
impossible from energy level consideration alone, primarily because of
the similarity of their angular m o m e n t u m dependence.
4. Wave function admixtures
Calculations have only been presented to show the effects
of perturbations on the rotational levels themselves, but generally speaking,
the corresponding wave functions will have additional components resulting
from these rotational interactions. A n exception to this generality is the
angular m o m e n t u m expansion calculation where only diagonal contributions
to the energy were considered. Here, the strong coupling rotor model
wave function remains unchanged. In the explicit band mixing calcula
tions, K no longer remained a good quantum number introducing con
figurational mixing into the wave functions. Although the eigenvalue
dependence on the angular m o m e n t u m is the same up to third order, the
eigenfunctions corresponding to the Hamiltonians are different. These
unmixed and admixed wave functions m ay all be used to calculate transi-
2
40
tion strengths and by comparing the results of the calculation with
experiment the form of the Hamiltonian that best describes the nuclear
motion may be determined.
In transitions that are predominantly forbidden or hindered by the
Alaga asymptotic selection rules (Al 57, W a 59), the transition probabilities
obviously may become more sensitive to the wave function admixtures.
In certain instances the smaller admixed portion of the wave function may
provide the dominant contribution to the transition. Examples are
electric dipole interband transitions between the excited K 77 = 1/2 and the
— + 23K = 3/2 ground state rotational band in Na . Here it is essential to
include the wave function admixtures to obtain any agreement with experi
ment as will be demonstrated herein.
A more general approach to the problem has been discussed by
Mottelson (Bo 69, M o 67, Mi 66, Gr 64) where the El transition matrix element
is written effectively as an expansion in powers of angular m o m entum
similar to the above mentioned energy expansion. In particular the El
transition probability was written as
B(E1) = M 1 + [I.(I.+1) -If(If+l)] M 2
where M is the usual electric dipole contribution and the second term isi ■*-the lowest order correction from the angular m omentum expansion. This
approach was found to provide agreement with the available experimental
data (Mo 67). The disadvantage of this particular form of expression is
41
that it is independent of the particular interaction and can not be used to"
select the perturbation mechanism. The power expansion does provide us
with the lowest order I dependence and at least we know the analytic form
that should be approximated by any explicit calculation.
G. Model Comparison Using Electromagnetic Properties
The quality of a model calculation for sd shell nuclei has been
judged in most cases by fitting excitation energies and angular momentum
quantum numbers, and in certain cases spectroscopic factors. This crude
test of a model is sufficient when differentiating between the asymptotic
extremes of the shell and collective models. In most cases nuclei show
intermediate behavior and on the basis of the above comparisons the
models may be virtually indistinguishable. A more rigorous test of the
model wave function, must then be obtained. Such a test involves
comparison of the matrix elements of the dynamic observables in addition
to the static ones. More specifically, herein we examine the electro
magnetic transition rates, g a m m a de-excitation branching ratios, multi
pole mixing ratios, etc. In particular we concentrate on the electro
magnetic properties of the nucleus in these comparisons. These are
experimentally and theoretically the most reliable in view of our relatively
fundamental and extensive knowledge of the electromagnetic interactions
in nuclei. The fact that this interaction is weak in comparison to the strong
interactions of the nuclear constituents permits the use of lowest order
perturbation theory (e. g. Fermi’s Golden Rule II) to calculate transition
42
probabilities. Even in the case where higher order effects are of
interest, it is an ardous but straight forward computational task to
include them.
It was the inherent inability of the shell model to correctly predict
the large quadrupole moments of nuclei and the enhanced E2 transition
probabilities that motivated the development of the collective model (Bo 52,
Bo 53), These enhanced properties have since become the signature of
nuclear collective behavior, and are most pronounced in the present work.
H. Experimental Method
The experimental techniques for measuring angular m o m entum and
nuclear dynamic electromagnetic properties are standard and the objec
tivity of data interpretation combines to lead to a very reliable source of
spectroscopic information. In addition a rigorous statistical analysis of
errors can be performed on such measurements and can be used in dis
criminating against unsatisfactory fitting parameters or models on the
basis of well established and precisely defined confidence levels.
Incorporating these desired features is the standard Method II
particle-gamma angular correlation technique first suggested by
Litherland and Ferguson (Li 61).- In the experiment reported herein, the
method was used in the form described by Poletti and Warburton (Po 65).
A derivation of the angular distribution formula (Ro 67) used herein is
given in Appendix V.
43
1 . G a m m a r a y a n g u l a r d i s t r i b u t i o n f r o m a l i g n e d n u c l e i :
O u r e x p e r i m e n t s b a s i c a l l y c o n s i s t o f m e a s u r i n g a n a n g u l a r
d i s t r i b u t i o n o f g a m m a r a y s e m i t t e d f r o m p r e f e r e n t i a l l y a l i g n e d n u c l e a r
s t a t e . T h e r e s i d u a l n u c l e u s i n t h e s e l e c t e d n u c l e a r r e a c t i o n i s a l i g n e d
w i t h r e s p e c t t o t h e p r o j e c t i l e b e a m a x i s ; t h i s a l i g n m e n t i s a c h i e v e d b y
c o n s t r a i n i n g t h e e f f e c t i v e p o p u l a t i o n o f m a g n e t i c s u b s t a t e s t h r o u g h t h e
d e t e c t i o n o f g a m m a r a y s i n c o i n c i d e n c e w i t h t h e l i g h t o u t g o i n g r e a c t i o n
p r o d u c t i n a n a n n u l a r c o u n t e r a x i a l l y p o s i t i o n e d v e r y c l o s e t o 1 8 0 d e g r e e s .
T h e a x i a l g e o m e t r y d e f i n e d b y t h i s c o u n t e r p e r m i t s p o p u l a t i o n o f t h o s e
m a g n e t i c s u b s t a t e s w h o s e m a g n e t i c q u a n t u m n u m b e r i s l e s s t h a n o r e q u a l
t o t h e s u m o f t h e s p i n s o f t h e t a r g e t n u c l e u s , t h e i n c o m i n g , a n d t h e o u t
g o i n g p a r t i c l e s ; a s i m p l e p r o o f o f t h i s c o n d i t i o n i s g i v e n i n A p p e n d i x V I .
T h i s t e c h n i q u e m i n i m i z e s t h e n u m b e r o f f i t t i n g p a r a m e t e r s u s e d i n t h e
a n a l y s i s o f t h e d a t a a n d i s e s s e n t i a l i n m a k i n g u n i q u e s p i n a s s i g n m e n t s .
M o s t i m p o r t a n t , h o w e v e r , i s t h e b a s i c f a c t t h a t i t m a k e s t o t a l l y u n n e c e s
s a r y a n y k n o w l e d g e o f t h e m e c h a n i c s o r i n t e r m e d i a t e s t a t e s o f t h e p o p u l a
t i n g r e a c t i o n a s i s e s s e n t i a l i n , f o r e x a m p l e , t h e m o r e f a m i l i a r a n a l y s i s
o f a n g u l a r c o r r e l a t i o n s f o l l o w i n g r e s o n a n c e r e a c t i o n s .
I f t h e a n g u l a r m o m e n t u m q u a n t u m n u m b e r o f t h e l e v e l o f i n t e r e s t
i n t h i s a p p r o a c h i s n o t u n i q u e l y a s s i g n e d , i t i s a t l e a s t r i g o r o u s l y l i m i t e d
t o a f e w p o s s i b i l i t i e s . A t t h e s a m e t i m e t h e e l e c t r o m a g n e t i c m u l t i p o l e
m i x i n g r a t i o i s d e t e r m i n e d a s a f i t t i n g p a r a m e t e r f r o m t h e m e a s u r e d
44
a n g u l a r d i s t r i b u t i o n o f g a m m a r a y s a n d b y s u m m i n g t h e g a m m a r a y y i e l d
f r o m a g i v e n s t a t e o v e r a l l a n g l e s , t h e b r a n c h i n g r a t i o o f t h e s t a t e m a y
b e d e t e r m i n e d .
C o m p l i m e n t a r y t o s u c h a n g u l a r c o r r e l a t i o n d a t a a r e l i f e t i m e i n f o r
m a t i o n ( i . e . a b s o l u t e t r a n a t i c n m a t r i x e l e m e n t s ) a n d R v a l u e a s s i g n m e n t s
f r o m s i n g l e p a r t i c l e t r a n s f e r d a t a y i e l d i n g u n i q u e p a r i t y a s s i g n m e n t s a n d
a g a i n a f e w s p i n p o s s i b i l i t i e s . B y c o m b i n i n g t h e s p i n p o s s i b i l i t i e s f r o m
b o t h s e t s o f d a t a , p a r t i c u l a r l y w h e n , w i t h o n e e x c e p t i o n , t h e s e s e t s a r e
m u t u a l l y e x c l u s i v e , a r i g o r o u s a s s i g n m e n t m a y b e m a d e .
N u c l e a r l i f e t i m e i n f o r m a t i o n m a y b e u s e d t o d e t e r m i n e t h e a b s o l u t e
t r a n s i t i o n m a t r i x e l e m e n t s o f a g a m m a r a y t r a n s i t i o n w h e n c o m b i n e d w i t h
t h e m i x i n g r a t i o o f t h e t r a n s i t i o n . I n p a r t i c u l a r , i f t h e e l e c t r i c s t r e n g t h
2e x c e e d s t h e W e i s s k o p f e s t i m a t e b y Z , t h e c o r r e s p o n d i n g s p i n m a y
r e a s o n a b l y b e r e j e c t e d . W e m a y t h e n e x t r a c t t h e r e d u c e d t r a n s i t i o n
p r o b a b i l i t y f o r t h e a c c e p t e d s p i n a n d w e h a v e a n a d d i t i o n a l e l e c t r o m a g n e t i c
q u a n t i t y t o c o m p a r e w i t h a n u c l e a r m o d e l .
2 . R e a c t i o n s
I n t h e w o r k d i s c u s s e d h e r e i n , M e t h o d I I c o r r e l a t i o n s t u d i e s
2 3 2 1 2 3 2 3 2 6w e r e c a r r i e d o u t o n N a a n d N a t h r o u g h t h e N a ( a , o : V ) ^ a » ( P . c v y )
2 3 2 4 2 3 2 4 2 1N a , M g ( t . a y J N a , a n d M g ( p , o » y ) N a r e a c t i o n s , r e s p e c t i v e l y . T h e
M P t a n d e m V a n d e G r a a f f a c c e l e r a t o r i n t h e W r i g h t N u c l e a r S t r u c t u r e
45
L a b o r a t o r y a t Y a l e U n i v e r s i t y p r o v i d e d b o t h t h e a l p h a - p a r t i c l e a n d t h e
p r o t o n b e a m s w h i l e t h e 3 M V V a n d e G r a a f f a c c e l e r a t o r a t t h e B r o o k h a v e n
N a t i o n a l L a b o r a t o r y p r o v i d e d t h e t r i t o n b e a m .
3 . B a c k g r o u n d r a d i a t i o n d i f f i c u l t i e s
A t Y a l e , w h e r e t h e p r o t o n i n d u c e d r e a c t i o n s w e r e c a r r i e d
o u t a n d f r o m w h i c h t h e b u l k o f t h e e x p e r i m e n t a l d a t a w a s o b t a i n e d , h i g h
e n e r g y p r o t o n b e a m s o f 1 4 . 2 5 M e V a n d 1 7 . 5 0 M e V w e r e r e q u i r e d i n o r d e r
t o c l e a r l y d i s c e r n t h e h i g h e r s t a t e s o f e x c i t a t i o n . T h e s e h i g h b o m b a r d
m e n t e n e r g i e s , n o t n o r m a l l y u s e d i n M e t h o d I I c o r r e l a t i o n s t u d i e s , p r e
s e n t e d s o m e s e v e r e e x p e r i m e n t a l d i f f i c u l t i e s t h a t h a d t o b e s u r m o u n t e d
b e f o r e t h e e x p e r i m e n t s w e r e s u c c e s s f u l l y c o n d u c t e d . T h e m o s t c h a l l e n g
i n g w a s t h e r e d u c t i o n o f t h e i n t e n s e n e u t r o n a n d g a m m a r a y b a c k g r o u n d
r a d i a t i o n g e n e r a t e d b y a d d i t i o n a l o p e n r e a c t i o n c h a n n e l s a t t h e h i g h e r
p r o t o n b o m b a r d m e n t e n e r g i e s . T o m i n i m i z e t h e r a d i a t i o n d i f f i c u l t i e s
t h e b e a m t r a n s p o r t s y s t e m w a s d e s i g n e d t o f o c u s t h e b e a m t h r o u g h a n
a n n u l a r c o u n t e r w i t h o u t s t r i k i n g t h e s h i e l d i n g m a t e r i a l o r , f o r t h a t
m a t t e r , a n y m a t e r i a l i n t h e v i c i n i t y o f t h e r a d i a t i o n d e t e c t o r s ( p a r t i c l e o r
g a m m a r a y ) o r t a r g e t . M o r e s p e c i f i c b e a m t r a n s m i s s i o n a n d f o c u s s i n g
c o n d i t i o n s a n d o t h e r p r o b l e m s a r e d i s c u s s e d i n t h e a p p r o p r i a t e s e c t i o n s
o f s u c c e e d i n g c h a p t e r s .
46
T h e w o r k p r e s e n t e d h e r e i n f o c u s s e s o n t h e m a s s 2 1 a n d 2 3 r e g i o n
w h i c h h a s l o n g b e e n r e c o g n i z e d a s o n e d e m o n s t r a t i n g m a r k e d c o l l e c t i v e
2 1 2 3b e h a v i o r . A d e t a i l e d s t u d y o f t w o s e l e c t e d n u c l e i , N a a n d N a h a s
b e e n u n d e r t a k e n u t i l i z i n g f c e . a ' y ) , ( P . a y ) . a n d ( t , a y ) r e a c t i o n s o n i
a p p r o p r i a t e t a r g e t s . N o w s t a n d a r d c o l i n e a r c o r r e l a t i o n g e o m e t r i e s h a v e
b e e n u s e d t o s t u d y g a m m a r a d i a t i o n f r o m a l i g n e d r e s i d u a l s t a t e s a n d a n
o n - l i n e c o m p u t e r s y s t e m h a s b e e n u t i l i z e d i n d a t a a c q u i s i t i o n a n d r e d u c t i o n .
P a r t i c u l a r i n t e r e s t h a s b e e n f o c u s s e d o n t h e K ^ = 3 / 2 + g r o u n d
21s t a t e r o t a t i o n a l b a n d s . I n N a , c l o s e a g r e e m e n t o f t h e o b s e r v e d l e v e l
e x c i t a t i o n w i t h t h o s e e x p e c t e d i n a p u r e r o t o r s p e c t r u m h a s b e e n f o u n d .
2 3W h e r e a s i n t h e s u p p o s e d l y d i r e c t l y c o m p a r a b l e s i t u a t i o n i n N a , a l s o a
£ = 1 1 n u c l e u s , a n d t h e r e f o r e e q u i v a l e n t o n t h e b a s i s o f a N i l s s o n m o d e l ,
s t r i k i n g o s c i l l a t o r y d e v i a t i o n s f r o m r o t o r p r e d i c t i o n s a r e o b s e r v e d .
T h i s s u g g e s t s a C o r i o l i s b a s e d e x p l a n a t i o n b u t s u c h i s n o t y e t a v a i l a b l e i n
s a t i s f a c t o r y f a s h i o n .
C o m p l i c a t i n g t h e s i t u a t i o n i s t h e f a c t t h a t i n b o t h n u c l e i , t h e m o m e n t s
o f i n e r t i a a r e i n e x c e s s o f 9 0 % o f t h e r i g i d b o d y v a l u e s a n d t h e i n t r i n s i c
e l e c t r i c q u a d r u p o l e m o m e n t s o f t h e m e m b e r s o f t h e r o t a t i o n a l b a n d s a p p e a r
t o r e m a i n r e l a t i v e l y c o n s t a n t u p t o t h e h i g h e s t e x c i t a t i o n s s t u d i e d ( J = 1 3 / 2 ) .
T h e s e d a t a s u g g e s t t h a t t h e s e n u c l e i m a y w e l l b e t h e m o s t r i g i d i n t h e
I . Sum m ary
47
p e r i o d i c t a b l e , b u t a l s o t h e r e a r e a s p e c t s o f c o l l e c t i v i t y e v e n i n t h e s e
r e l a t i v e l y s i m p l e n u c l e i , w h i c h a r e n o t a d e q u a t e l y u n d e r s t o o d .
!
48
I I . A P P A R A T U S
T h e u s e o f s p i n z e r o a n d s p i n o n e h a l f p a r t i c l e s a s n u c l e a r
r e a c t i o n p r o b e s i n M e t h o d I I c o r r e l a t i o n s t u d i e s o n s p i n 0 t a r g e t s
r e s t r i c t s m a g n e t i c s u b s t a t e p o p u
