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 3J(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 <e e x p ~ e c a l c ) 2° 3E Ca l C= A J (J + l )

1 0 -I

a 2

10 - 2

/O

® N e

o Na

a Mg

1 0 - 3 J L 1 1 J L18

J L20 22 24 26 28ATOMIC MASS NUMBER

3 0

Fig. 4

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<j iy

where the sums are over the extra core nucleons, V(r.) is the central

interaction between the ith particle and the inert core, and v(r_) is the

residual interaction between the ith and jth particle outside the core. This

calculation removes the degeneracy of states formed by n particles in a

configuration (j)n but retains the prediction that the state of lowest energy

is that of the uncoupled odd nucleon agreeing with the result of the extreme

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 <_ A < 28 sd shell nuclei was to allow nucleons outside an inert core of

16O to occupy only 1 d ^ ^ and 2 s^y2 orbits. The two body effective inter­

actions between the extra-core nucleons was expanded in 16 two-body matrix

elements. The matrix elements plus the d ^ ^ and single particle

energies were determined by using them as parameters in fitting 80

nuclear energy levels in the 20 < A < 28 region. Again generally good

agreement for energy levels was obtained. Wave functions have been

obtained and transition probabilities have been calculated for A = 21 and

A = 23 nuclear systems. A brief discussion of our experimental results

in terms of these new shell model calculations is presented in Section IX.

It should not be surprising that both the shell and collective models

predict levels in approximate agreement with experiment, despite the

seemingly contradictions between the long and short range components

of the nuclear forces that are implicitly assumed in the collective and

shell models, respectively.

Within the mathematical framework of a model (e. g. shell or col­

lective), a complete set of orthogonal eigenfunctions are defined, in which

any well behaved function may be expanded. This well behaved function

m a y be the eigenfunction of a nuclear state and may, therefore, be repre­

sented as a s u m over shell or collective type wave functions. In practice

the space that the basis functions span is necessarily truncated in order to

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-

*<*=-5/2+ GND STATE R O T A T IO N A L B A N D S OF £ =11 N U C L E I

(13/2+)

•45-55-*-ll/2+(7/2) .13

2.869- -33-67-*----9/2+ 2.833--- 36— 64---9/2+(5/2)•09 -.12

1.750-5-95- .16

.350-

'7/2+ 1.723-7-93--. 4

.050 - 2 -■5/2+

•3/2+

.332-

■7/2+(3/2)

-05 0 ——■ 5/2+

■3/2+

N e 2 '10 II N o 21II to

6.236 (IOOhlO-l3/2+(9/2)

5.535-----24-76-^-1|/2+(7/2).18

2.705 — 63-37-2--- 9/2*-.102.079*10-90 ■7/2+

-3/24

N o 23II 12

5.451

Fig. 8

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<J ~§)(JH)(J+3/2)

The constant C incorporates a summation including the effect of all

K = 1/2 bands in the nucleus in question independent of their specific origin.

By exact diagonalization of the Coriolis interaction between a K = 3/2 and

a single K = 1/2 band, the leading term of an expansion of the solution is

shown in Appendix III to have the same angular m o m entum dependence.

In both calculations the third-order Coriolis decoupling term arises from

Coriolis interactions between K = 1/2 and K = 3/2 rotational bands and,

therefore, K is no longer a constant of the motion.

—23. Expansion of Hamiltonian in powers of R

In contrast to Coriolis mixing between two different bands,

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 <j-2Hj+£)(j+3/2)

The third-order Coriolis term arises here as a self-coupling between the

axially symmetric parts of the wave function preserving K as a constant

of the motion in contrast to the previous calculation. The generality

2 2 introduced by expanding in R naturally included higher powers of J

which no longer have to be added in an ad-hoc manner as was done earlier

to obtain the rotation-vibration interaction term. In fact an additional

3 3term in J (J+l) now appears which was earlier omitted. Although the

angular m o m entum dependence of the third-order Coriolis terms are

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 l a t i o n o b t a i n i n g t o t a l n u c l e a r a l i g n m e n t

i n t h e r e s i d u a l n u c l e u s ( A p p e n d i x V I ) . T o t a l n u c l e a r a l i g n m e n t i s

a d v a n t a g e o u s i n t h a t i t y i e l d s t h e m o s t a n i s o t r o p i c g a m m a r a d i a t i o n

p o s s i b l e p e r m i t t i n g e x t r a c t i o n o f m a x i m u m i n f o r m a t i o n f r o m t h e c o r r e l a ­

t i o n d a t a .

S p i n z e r o a l p h a p a r t i c l e s a r e w i d e l y u s e d a s r e a c t i o n p r o b e s a n d

w e r e i n i t i a l l y e m p l o y e d i n t h i s w o r k , b u t t o o b t a i n t o t a l n u c l e a r a l i g n m e n t ,

t h e y m u s t s t r i k e s p i n z e r o o r s p i n o n e h a l f t a r g e t n u c l e i . I n t h e s t u d y o f

2 3 2 1 2 3N a a n d N a t h i s c r i t e r i a i s n o t s a t i s f i e d . N a h a s a g r o u n d s t a t e

21s p i n o f 3 / 2 p e r m i t t i n g p o p u l a t i o n o f t w o s u b s t a t e s a n d N a d o e s n o t

o c c u r n a t u r a l l y . T h e n e x t b e s t p o s s i b i l i t y i s a s p i n o n e h a l f a n d s p i n

z e r o p a r t i c l e i n t h e e n t r a n c e o r e x i t c h a n n e l o n a s p i n z e r o t a r g e t n u c l e u s .

2 6 2 3 2 4 2 1T h i s i s a c h i e v e d h e r e i n b y u s i n g t h e M g ( p , a ) N a a n d M g ( p , a ) N a ,

r e a c t i o n s . T h e s e r e a c t i o n s h a v e t h e a d v a n t a g e o f t o t a l a l i g n m e n t i n t h e

r e s i d u a l n u c l e u s , b u t a r e b e s e t w i t h s o m e e x p e r i m e n t a l i n c o n v e n i e n c e s

t h a t a r e d i s c u s s e d b e l o w .

T h e e n d o t h e r m i c n a t u r e o f t h e ( p . o ) r e a c t i o n o n M g r e q u i r e s h i g h

b o m b a r d m e n t e n e r g i e s i n o r d e r t o l e a v e t h e r e s i d u a l n u c l e u s i n a h i g h

s t a t e o f e x c i t a t i o n ( 6 . 5 M e V ) w i t h a l p h a p a r t i c l e s e m e r g i n g a t b a c k a n g l e s

A . A c c e le ra to r

49

w i t h s u f f i c i e n t e n e r g y t o b e c o n v e n i e n t l y m e a s u r e d .

I t i s a l s o k n o w n ( A l 66 ) t h a t ( p , a ) r e a c t i o n s a t b a c k w a r d a n g l e s

a r e p r e d o m i n a n t l y c o m p o u n d n u c l e a r w i t h a m a r k e d f l u c t u a t i n g e n e r g y

d e p e n d e n c e . T h i s i s a l s o s e e n a t h i g h e r b o m b a r d m e n t e n e r g i e s f r o m t h e

r e s u l t s o f o u r w o r k . T h e r a p i d l y v a r y i n g e n e r g y d e p e n d e n c e o f t h e c r o s s

s e c t i o n s r e q u i r e e n e r g e t i c a l l y s t a b l e b e a m s t o p r e v e n t v a r i a t i o n s i n t h e

y i e l d s t o t h e i n d i v i d u a l s t a t e s o v e r l o n g p e r i o d s o f t i m e , b u t o n t h e o t h e r

h a n d m u s t b e c o n t i n u o u s l y v a r i a b l e i n o r d e r t o c o n v e n i e n t l y c h o o s e a

b o m b a r d m e n t e n e r g y o p t i m i z i n g y i e l d s t o r e l a t e d s t a t e s .

T h e r e d u c e d c o u n t i n g r a t e s i m p o s e d b y t h e c o l i n e a r g e o m e t r y i s a

r e s u l t o f t h e r e l a t i v e l y l a r g e b o m b a r d m e n t e n e r g y f o r c i n g t h e m a j o r

p o r t i o n o f t h e r e a c t i o n y i e l d t o f o r w a r d a n g l e s a n d t h u s t h e w e a k e r y i e l d s

a t b a c k w a r d a n g l e s n e c e s s i t a t e s t h e u s e o f r e l a t i v e l y l a r g e D C b e a m

c u r r e n t s . T h e i m p o r t a n t f a c t o r i s t h a t t h e b e a m c u r r e n t i s D C a n d , t h e r e ­

f o r e , r a d i a t i o n c o u n t e r s a r e n o t s u d d e n l y d r i v e n i n t o n o n l i n e a r e l e c t r o n i c

s a t u r a t i o n b y p e r i o d i c b u r s t s o f b e a m c u r r e n t . T h e o b v i o u s c o i n c i d e n c e

r e q u i r e m e n t s a l s o p u t s a h i g h p r e m i u m o n D C b e a m q u a l i t i e s .

T h e s e r e q u i r e m e n t s a r e e s s e n t i a l a n d a l l a r e u n i q u e l y s a t i s f i e d b y

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 L a b o r a t o r y 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 a n d b e a m t r a n s p o r t s y s t e m . T h e a c c e l e r a t o r p r o v i d e d

r e l a t i v e l y i n t e n s e b e a m s o f p r o t o n s a t s u f f i c i e n t l y h i g h e n e r g i e s w i t h b e a m

q u a l i t y a n d c o n t r o l t h a t w a s n e c e s s a r y t o o v e r c o m e t h e f l u c t u a t i n g e n e r g y

50

A s c h e m a t i c d i a g r a m o f t h e g r o u n d f l o o r l a y o u t o f t h e l a b o r a t o r y

i n c l u d i n g t h e i o n s o u r c e , M P t a n d e m a c c e l e r a t o r , b e a m t r a n s p o r t , g a m m a

c a v e , c o n t r o l r o o m , e t c . i s s h o w n i n F i g . 1 1 .

B . B e a m T r a n s p o r t

T h e b e a m t r a n s p o r t s y s t e m , a s d i s c u s s e d i n t h e i n t r o d u c t i o n , w a s

d e s i g n e d t o f o c u s a h i g h e n e r g y b e a m o f p a r t i c l e s o n t o a t a r g e t w i t h o u t

t h e u s e o f b e a m c o l l i m a t i o n i n t h e v i c i n i t y o f r a d i a t i o n d e t e c t i o n d e v i c e s .

T h i s w a s a c c o m p l i s h e d b y c o n t r o l l i n g t h e s i z e o f t h e b e a m w i t h s l i t s a t a

c r o s s o v e r p o i n t ( a b o u t 4 0 f e e t f r o m t h e t a r g e t ) b e t w e e n t w o q u a d r u p o l e

f o c u s s i n g m a g n e t s s h o w n i n F i g . 1 1 . A t y p i c a l b e a m p r o f i l e t r a n s p o r t e d

b y t h e o p t i c a l s y s t e m i s s h o w n i n F i g . 1 2 . T h e l a s t q u a d r u p o l e w a s u s e d

t o f o c u s t h e b e a m s p o t d e f i n e d b y t h e s e s l i t s ( n o r m a l l y . 020 i n . x . 020 i n . )

t h r o u g h t h e a n n u l a r d e t e c t o r o n t o a t a r g e t . A b o u t t h r e e f e e t u p s t r e a m

f r o m t h e t a r g e t , f o u r i n d e p e n d e n t l y a d j u s t a b l e m i c r o m e t e r h e a d s w e r e

u s e d t o p o s i t i o n f o u r e l e c t r i c a l l y i s o l a t e d t a n t a l u m p l a t e s d e f i n i n g a r e c t a n g ­

u l a r a p e r t u r e , < 0 . 2 5 i n t o t h e s i d e , w h i c h w a s u s e d i n i n i t i a l s t e e r i n g o f

t h e b e a m t h r o u g h t h e a n n u l u s o f t h e d e t e c t o r . T h e r e l a t i v e p o s i t i o n o f

t h e s e s l i t s , t o g e t h e r w i t h o t h e r b e a m p l u m b i n g e q u i p m e n t , i s s h o w n i n

F i g . 1 3 . A n a d d i t i o n a l r e f i n e m e n t o f t h e b e a m t r a n s p o r t s y s t e m , w h i c h

h a s n o t b e e n u s e d t o f u l l e s t p o t e n t i a l , w a s t h e p l a c e m e n t o f f o u r 2 . 2 i n c h

d i a m e t e r , e l e c t r i c a l l y - i n s u l a t e d , t a n t a l u m a p e r t u r e s a t t h e e n t r a n c e a n d

dependent c ro s s sectio ns and the low reactio n y ie ld s.

'I

VERT.

DISPLA

CEME

NT (

IN.)

HORIZ

. DISP

LACE

MENT

(IN.

)

BEAM OPTICS - 0 ° LINE

DISTANCE FROM IMAGE SLIT (IN.)

Fig. 12

-TARGET POSITIONING ROD

MOVABLE SLEEVE FOR LOCATION

GATE VALVE -TELESCOPE

OPTICALFLAT

Fig. 13

51

e x i t o f e a c h o f t h e t w o q u a d r u p o l e f o c u s s i n g m a g n e t s f o r e a n d a f t o f t h e .

b e a m d e f i n i n g s l i t s i n t a r g e t r o o m 2 . T h e p u r p o s e o f t h e s e a p e r t u r e s w a s

t o p h y s i c a l l y c o n f i n e t h e b e a m t o r e g i o n s o f h o m o g e n e o u s m a g n e t i c f i e l d

i n t h e q u a d r u p o l e l e n s e s a n d , a l l o w i n g f o r m a x i m u m b e a m d i v e r g e n c e ,

f u l l y o p t i m i z i n g t h e m a g n e t i c f o c u s s i n g p o w e r o f t h e l e n s e s .

C . G a m m a C a v e a n d G o n i o m e t e r

T h e g a m m a g o n i o m e t e r u s e d i n t h e s e e x p e r i m e n t s i s s h o w n i n t h e

g a m m a c a v e i n F i g . 1 4 . T h e g a m m a c a v e i s a 2 0 f t . s q u a r e t a r g e t r o o m

w i t h p o t a s s i u m a n d t h o r i u m f r e e i l m e n i t e 3 f t . t h i c k c o n c r e t e w a l l s l o a d e d

w i t h b o r o n . T h e g o n i o m e t e r s h o w n i n t h e f i g u r e c o n s i s t s o f a m o v e a b l e

b e d a n d b o o m , c a p a b l e o f s u p p o r t i n g u p t o 4 0 0 0 l b s . a n d 1 0 0 0 l b s . o f

m a t e r i a l , r e s p e c t i v e l y , w h i c h i n a d d i t i o n i s c a p a b l e o f p o s i t i o n i n g a

d e t e c t o r t o a n a c c u r a c y o f 0 . 1 d e g r e e s , b y m a n u a l o r b y d i r e c t c o m p u t e r

c o n t r o l a t a n y o r i e n t a t i o n o n s p h e r i c a l s u r f a c e s o f v a r i a b l e r a d i u s

c e n t e r i n g o n t h e t a r g e t .

I n t h e M e t h o d I I c o r r e l a t i o n w o r k d e s c r i b e d h e r e i n t h e m o v e a b l e

g a m m a r a y d e t e c t o r w a s a t t a c h e d t o t h e b o o m a n d t h e m o t i o n c o n f i n e d t o

a h o r i z o n t a l p l a n e .

D . R a d i a t i o n D e t e c t o r s

T h e g a m m a r a y d e t e c t o r w a s a n R C A 8 0 5 5 p h o t o m u l t i p l i e r t u b e

i n t e g r a l l y m a t c h e d t o a H a r s h a w 5 i n c h x 5 i n c h N a ( T ^ ) I c r y s t a l . T h e

52

e n t i r e a s s e m b l y w a s m a g n e t i c a l l y s h i e l d e d a n d e n c l o s e d i n a c y l i n d r i c a l

a l u m i n u m c a n . T h i s c a n w a s c l a m p e d t o t h e b o o m o f t h e g o n i o m e t e r a n d

t h e a x i s o f t h e N a l c r y s t a l w a s o p t i c a l l y c e n t e r e d w i t h r e s p e c t t o t h e

t a r g e t c e n t e r . T h e d i s t a n c e b e t w e e n t h e t a r g e t a n d f r o n t f a c e o f t h e

c r y s t a l w a s f i x e d a t 2 0 c m . T h e a l i g n m e n t o f t h e c o r r e l a t i o n a p p a r a t u s

w a s c h e c k e d w i t h s t a n d a r d r a d i o a c t i v e s o u r c e s a n d f r o m k n o w n i s o t r o p i c

r e a c t i o n g a m m a r a y d i s t r i b u t i o n s .

T h e o u t g o i n g a l p h a p a r t i c l e s w e r e d e t e c t e d i n a n o m i n a l l y

7 5 m i c r o n , p a r t i a l l y d e p l e t e d , t h e r m o e l e c t r i c a l l y - c o o l e d , O R T E C

a n n u l a r s e m i c o n d u c t o 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 l ~ ^ r i n . f r o m t h e t a r g e t

s u b t e n d i n g a n g l e s o f 1 7 1 ° t o 1 7 5 ° ( 5 . 7 x 1 0 2 s t e r a d i a n s ) a s s h o w n i n F i g .

1 5 . T h e a n n u l a r c o u n t e r w a s o p t i c a l l y a l i g n e d ( F i 6 9 a ) .

T h e p a r t i c l e d e t e c t o r d i s t a n c e a n d s o l i d a n g l e w e r e s e l e c t e d a s a

c o m p r o m i s e b e t w e e n r e a s o n a b l e c o u n t i n g r a t e s , k i n e m a t i c b r o a d e n i n g , a n d

t h e a n g u l a r c o r r e l a t i o n f i n i t e s i z e e f f e c t .

E . A s s o c i a t e d C o m p o n e n t s

1 . S c a t t e r i n g c h a m b e r

T h e d e t e c t o r g e o m e t r y a n d t o p v i e w o f t h e i n t e r i o r o f t h e

s c a t t e r i n g c h a m b e r a r e s h o w n i n F i g . 1 6 . T h e d i a m e t e r o f t h e c y l i n d r i c a l

a l u m i n u m s c a t t e r i n g c h a m b e r i s 5 ^ i n c h e s w i t h a 3 / 1 6 i n c h w a l l

t h i c k n e s s . T h e r i m o f t h e b a s e o f t h e c h a m b e r i s f a s t e n e d w i t h a W N S L

l a b o r a t o r y - s t a n d a r d 5 i n c h V - b a n d c o u p l e r t o a p l a t e w h i c h i s i n t u r n

DETECTOR GEOMETRY

5 in. x 5 in.

Fig. 16

ANNULAR DETECTOR ASSEMBLY

Fig. 15

53

T h e b e a m e n t e r s t h e c h a m b e r a t t h e l e f t , p a s s e s t h r o u g h a n

a p p r o p r i a t e l y t a n t a l u m s h i e l d e d a n n u l a r d e t e c t o r , s t r i k e s t h e t a r g e t a t t h e

c e n t e r , e x i t s a t t h e r i g h t a n d i s s t o p p e d i n a s h i e l d e d b e a m - s t o p 10 f e e t

d o w n s t r e a m . T h e v a r i o u s c o m p o n e n t s t h a t s h i e l d a n d p o s i t i o n t h e a n n u l a r

d e t e c t o r o n t o t h e a l u m i n u m b l o c k a r e i n d i c a t e d a s s h a d e d a r e a s f o r t a n t a l u m ,

c r o s s h a t c h e d a r e a s f o r t e f l o n , a n d t h e u n s h a d e d a r e a s a r e e i t h e r b r a s s

o r a l u m i n u m . T h e x ' s i n d i c a t e t h e d i r e c t i o n o f a m a g n e t i c f i e l d a p p l i e d

f o r s w e e p i n g s e c o n d a r y e l e c t r o n s f r o m t h e d e t e c t o r .

Provision is also made for positioning a large coaxial Ge(Li) detector one inch from the ta rget for coincidence work.

2 . D e t e c t o r s h i e l d

T o r e d u c e l o w e n e r g y p a r t i c l e b a c k g r o u n d i n t h e a n n u l a r

d e t e c t o r a n d t o p r e v e n t s t r a y o r d i r e c t b e a m f r o m s t r i k i n g t h e d e t e c t o r ,

a o n e p i e c e c y l i n d r i c a l t a n t a l u m s h i e l d , t a p e r e d i n s i d e a n d o u t i n t o a

s h o r t t h i n w a l l t u b e ( 0 . 0 0 8 i n c h e s ) , a s s h o w n i n F i g . 1 6 , w a s i n s e r t e d

t h r o u g h t h e a n n u l u s o f t h e d e t e c t o r . T h e t a n t a l u m s h i e l d w a s e l e c t r i ­

c a l l y i s o l a t e d a l l o w i n g t h e m e a s u r e m e n t o f a n y i n c i d e n t r e s i d u a l b e a m

c u r r e n t . I f t h e r e s i d u a l b e a m c u r r e n t g r e a t l y e x c e e d e d . 0 5 % o f t h e

t r a n s m i t t e d b e a m , t h e r e s u l t i n g b a c k g r o u n d r a d i a t i o n ( n e u t r o n a n d g a m m a

r a d i a t i o n ) e m a n a t i n g f r o m t h e s h i e l d s a t u r a t e d t h e p a r t i c l e a n d g a m m a r a y

d e t e c t i n g d e v i c e o b s c u r i n g t h e r e a c t i o n o f i n t e r e s t f r o m t h e t a r g e t . T h i s

se cu re ly mounted in the center of the goniom eter.

54

e f f e c t g r a d u a l l y w o r s e n e d a s t h e b e a m e n e r g y a p p r o a c h e d a n d e x c e e d e d

t h e c o u l o m b b a r r i e r o f t a n t a l u m ( ~ 1 2 . 0 M e V p r o t o n ) , g r a d u a l l y l e v e l e d

o f f a t 1 4 . 0 M e V , a n d w a s n o t m u c h w o r s e a t 2 0 M e V .

3 . D e t e c t o r c o o l i n g

T h e a n n u l a r d e t e c t o r w a s t h e r m a l l y i n s u l a t e d f r o m i t s

s u p p o r t i n g m o u n t b y t e f l o n s p a c e r s i n o r d e r t o p e r m i t u s e o f a 5 w a t t

t h e r m o e l e c t r i c c o o l e r t o r e d u c e t h e l e a k a g e c u r r e n t i n t h e d e t e c t o r .

T o p a r t i a l l y d e p l e t e t h e d e t e c t o r t o 7 0 m i c r o n s , a s w a s d o n e i n

t h e ( p . o ; ) e x p e r i m e n t s , a r e d u c t i o n i n b i a s f r o m 5 0 v o l t s , c o r r e s p o n d i n g

t o d e p l e t i o n d e p t h o f 1 0 0 m i c r o n s , t o 1 5 v o l t s i s r e q u i r e d . T h e r e d u c t i o n

i n b i a s d e c r e a s e s t h e r i s e t i m e c h a r a c t e r i s t i c s o f t h e d e t e c t o r , b u t t h e

r i s e t i m e i m p r o v e m e n t a c h i e v e d b y c o o l i n g t h e d e t e c t o r a d e q u a t e l y c o m p e n ­

s a t e d . I n a d d i t i o n , c o o l i n g r e d u c e s n o i s e i n t h e d e t e c t o r i m p r o v i n g t h e

e n e r g y r e s o l u t i o n a s w e l l a s r e s u l t i n g i n a m o r e s t a b l e o p e r a t i o n o f t h e

d e t e c t o r ( i . e . s m a l l d r i f t s i n v o l t a g e a c r o s s t h e d e p l e t e d r e g i o n c a u s e d b y

l e a k a g e c u r r e n t f l u c t u a t i o n s e s p e c i a l l y w i t h h i g h c o u n t r a t e s ( 5 K C ) o v e r

l o n g p e r i o d s o f t i m e ( 2 4 h r s . ) ) .

4 . E l e c t r o n s h i e l d i n g

C o p i o u s y i e l d s o f l o w e n e r g y s e c o n d a r y e l e c t r o n s b a c k -

s t r e a m i n g f r o m t h e t a r g e t i n t o t h e a n n u l a r c o u n t e r d o m i n a t e d t h e l o w

e n e r g y p a r t o f t h e s p e c t r u m , b u t m o r e s e r i o u s l y w a s t h e a m b i e n t n o i s e

g e n e r a t e d , d e t e r i o r a t i n g t h e e n e r g y r e s o l u t i o n o f t h e d e t e c t o r . A s t a c k

55

o f b a r m a g n e t s o r i e n t e d s o t h a t a m a g n e t i c f i e l d o f a p p r o x i m a t e l y £00

g a u s s w a s m a i n t a i n e d p e r p e n d i c u l a r t o t h e b e a m a x i s , p r e v e n t e d s u c h

l o w e n e r g y e l e c t r o n s k n o c k e d o u t o f t h e t a r g e t f r o m r e a c h i n g t h e d e t e c t o r .

5 . F a r a d a y c u p

T h e F a r a d a y c u p c o n s i s t e d o f a s e c t i o n o f b e a m t u b e c o n ­

n e c t i n g t h e e x i t o f t h e s c a t t e r i n g c h a m b e r t o t h e b e a m s t o p . I t w a s a b o u t

8 f e e t i n l e n g t h a n d e l e c t r i c a l l y i s o l a t e d f r o m t h e s c a t t e r i n g c h a m b e r b y a

0 . 2 5 i n c h t h i c k t e f l o n s p a c e r . T h e F a r a d a y c u p w a s c o n n e c t e d t o a l o w

i n p u t i m p e d a n c e i n t e g r a t o r a n d t h e n o r m a l i z a t i o n d e t e r m i n e d f r o m t h e

i n t e g r a t e d b e a m c u r r e n t w a s f o u n d t o a g r e e , w i t h i n 1 % , w i t h t h a t d e t e r ­

m i n e d f r o m t h e p a r t i c l e d a t a . T h e p o r t i o n o f b e a m t u b e c o n n e c t e d t o t h e

e x i t o f t h e s c a t t e r i n g c h a m b e r w a s l i n e d w i t h t a n t a l u m f o i l a n d c o n i c a l l y

s h a p e d a l l o w i n g t h e N a l d e t e c t o r t o b e p o s i t i o n e d 2 0 ° f r o m t h e b e a m a x i s

a t a d i s t a n c e o f 2 0 c m f r o m t h e t a r g e t . T h i s l i m i t i s i m p o s e d b y t h e

p h y s i c a l s i z e o f t h e b e a m t u b e w h o s e m i n i m u m d i a m e t e r w a s d e t e r m i n e d

f r o m c o n s i d e r a t i o n o f b e a m d i v e r g e n c e r e f l e c t i n g m u l t i p l e s c a t t e r i n g i n

t h e t a r g e t .

F r o m m e a s u r e m e n t s w i t h s t a n d a r d r a d i o a c t i v e s o u r c e s t h e g q m m a

r a y a t t e n u a t i o n , a t 20° , d u e t o t h e i n t e r v e n i n g c o n n e c t i e n s a n d t a n t u l u m

f o i l w a s f o u n d t o b e l e s s t h a n 2% f o r a 5 1 1 k e V g a m m a r a y a n d , t h e r e f o r e ,

w a s c o n s i d e r e d a n e g l i g i b l e c o r r e c t i o n t o t h e a n g u l a r d i s t r i b u t i o n d a t a .

56

I n t h e ( p , a ) e x p e r i m e n t s t h e t a n t a l u m b e a m s t o p b e c a m e

a n i n t e n s e s o u r c e o f n e u t r o n s a n d g a m m a r a y b a c k g r o u n d w h e n t h e b e a m

e n e r g y e x c e e d e d t h e c o u l o m b b a r r i e r f o r p r o t o n s o n t a n t a l u m ( 1 2 . 0 M e " / ) .

Much of the background rad iation resu lted from the neutrons em itted int h e ( p , n ) r e a c t i o n o n t a n t a l u m ( Q = - . 8 M e V ) . B y i n t e r p o s i n g l a y e r s o f

p a r a f i n , l e a d , a n d i r o n b e t w e e n t h e b e a m s t o p a n d t h e N a l c o u n t e r , t h e

b a c k g r o u n d r a d i a t i o n w a s r e d u c e d a p p r o x i m a t e l y 5 0 % . R e p l a c i n g t h e

t a n t a l u m b e a m s t o p w i t h a l a r g e 4 " x 1 / 4 " d i s k o f n a t u r a l c a r b o n a n d

s l i g h t l y m o r e e f f e c t i v e g a m m a r a y s h i e l d i n g , t h e g a m m a r a y b a c k g r o u n d

i n t h e N a l c o u n t e r w a s r e d u c e d b y a n o t h e r 5 0 % ( L i 6 9 b ) . T h e a d d i t i o n a l

i m p r o v e m e n t i s m a i n l y d u e t o t h e f e w e r n u m b e r o f n e u t r o n s g e n e r a t e d f r o m

12t h e b e a m s t o p w h i c h i s a r e s u l t o f t h e l a r g e n e g a t i v e Q v a l u e o f t h e C

( p . n J B 11 r e a c t i o n ( Q = - 1 8 . 2 M e V ) . A s l o n g a s t h e p r o t o n b o m b a r d m e n t

e n e r g y i s l o w e r t h a n t h e t h r e s h o l d f o r n e u t r o n e m i s s i o n t h e d o m i n a n t

s o u r c e o f n e u t r o n s i n t h e b e a m s t o p i s t h e ( p , n ) r e a c t i o n o n t h e 1 %

1 3n a t u r a l l y o c c u r r i n g C .

2 3 2 4 2 6F . P r e p a r a t i o n o f N a , M g , A n d M g T a r g e t s

T h e s o d i u m a n d m a g n e s i u m t a r g e t s w e r e p r e p a r e d ( L i 6 9 c ) b y

c o n v e n t i o n a l t e c h n i q u e s o f v a c u u m d e p o s i t i o n o f m e t a l l i c i s o t o p i c a l l y p u r e

2 3 2 4 2 6N a a n d i s o t o p i c a l l y e n r i c h e d M g ( > 9 9 . 9 % ) a n d M g ( > 9 9 . 5 % ) p r e p a r e d

b y O a k R i d g e N a t i o n a l L a b o r a t o r y . T h e m a t e r i a l w a s e v a p o r a t e d f r o m a

6. Beam stop

57

v e r t i c a l t a n t a l u m c r u c i b l e o n t o t h i n 1 0 -2 0 u g m / c m c a r b o n f o i l s m o u n t e d

o v e r a 3 / 8 i n c h d i a m e t e r h o l e i n a t h i n a l u m i n u m f r a m e . T a r g e t s w e r e

2a p p r o x i m a t e ^ 100 fj, g m / c m t h i c k a n d w e r e o f o p t i m u m t h i c k n e s s e s i n

c o n s i d e r a t i o n o f c o u n t i n g r a t e s a n d e n e r g y r e s o l u t i o n i n t h e a l p h a p a r t i c l e

c h a n n e l . T o m i n i m i z e o x y g e n c o n t a m i n a n t s w h i c h p r o d u c e d p a r t i c l e

g r o u p s o v e r l a p p i n g w i t h t h e p e a k s o f i n t e r e s t , o x i d a t i o n o f t h e t a r g e t s w a s

m i n i m i z e d b y s t o r i n g t h e t a r g e t s i n v a c u u m a n d t r a n s f e r r i n g t h e m t o t h e

s c a t t e r i n g c h a m b e r i n a n e v a c u a t e d p o r t a b l e t a r g e t s t o r a g e c h a m b e r s h o w n

i n F i g . 1 7 . A v a c u u m < 1 ^ c o u l d b e m a i n t a i n e d i n t h e s t o r a g e c h a m b e r

o v e r a p e r i o d o f a b o u t 1 5 m i n u t e s , w h i c h w a s a m p l e t i m e t o c o m p l e t e t h e

t a r g e t t r a n s f e r .

G . V a c u u m

. T h e b e a m t u b e a n d s c a t t e r i n g c h a m b e r w e r e c o n t i n u o u s l y e v a c u a t e d

b y t h r e e 1 5 0 l i t e r p e r s e c U l t e c k i o n p u m p s s p a c e d a l o n g t h e b e a m l i n e .

-6 - 7T h e i o n p u m p s m a i n t a i n e d a v a c u u m i n t h e l i n e b e t w e e n 1 0 a n d 1 0

T o r r .

N e a r t h e s c a t t e r i n g c h a m b e r ( a p p r o x i m a t e l y 1 2 i n c h e s f r o m t h e

t a r g e t ) a n i n l i n e a n n u l a r l i q u i d n i t r o g e n c o l d t r a p w a s u s e d t o p r e v e n t a n y

m o l e c u l a r c o n t a m i n a n t s f r o m d r i f t i n g d o w n s t r e a m i n t o t h e s c a t t e r i n g

c h a m b e r o n t o t h e t a r g e t o r a n n u l a r d e t e c t o r . I n a d d i t i o n , t h e c o l d t r a p

h e l p e d t o i m p r o v e t h e p u m p d o w n t i m e o f t h e s c a t t e r i n g c h a m b e r b y c o n ­

d e n s i n g r e s i d u a l w a t e r v a p o r .

VACUUM TA RG ET STORAGE C H A M B E R

THERMOCOUPLE VACUUM / GAUGE—s.

Fig. 17

58

A c o l d t r a p w a s a l s o u s e d b e t w e e n t h e p u m p i n g p o r t o f t h e c h a m b e r

a n d t h e e x t e r n a l m e c h a n i c a l r o u g h i n g p u m p w h e n e v e r t h e s y s t e m w a s

e v a c u a t e d . I n a d d i t i o n t h e e n t i r e b e a m l i n e a n d s c a t t e r i n g c h a m b e r w e r e

i n i t i a l l y b a k e d o u t w i t h i n f r a r e d l a m p s a t t a i n i n g a p p r o x i m a t e t e m p e r a t u r e s

o f 3 0 0 ° C .

H . E l e c t r o n i c s

S t a n d a r d s o l i d s t a t e n u c l e a r e l e c t r o n i c i n s t r u m e n t a t i o n w a s u s e d

t o d e r i v e p a r t i c l e - g a m m a l i n e a r a n d t i m e i n f o r m a t i o n . A t i m e c o i n c i ­

d e n c e b e t w e e n t h e p a r t i c l e a n d g a m m a r a y w a s d e m a n d e d a n d u s e d t o

s e l e c t l i n e a r p a r t i c l e a n d g a m m a r a y i n f o r m a t i o n t o b e a n a l y z e d . A

m o d i f i e d s t a n d a r d f a s t - s l o w c o i n c i d e n c e a r r a n g e m e n t w a s e m p l o y e d a n d t h e

l o g i c i s s h o w n s c h e m a t i c a l l y i n F i g . 1 8 .

M o d i f i c a t i o n s t o t h e s t a n d a r d s y s t e m t h a t i m p r o v e d t h e t i m i n g

p e r f o r m a n c e w e r e t h e u s e o f t w o O R T E C 2 6 0 t i m e p i c k - o f f u n i t s w h o s e

d i s c r i m i n a t i o n t h r e s h o l d s w e r e s e t o n p u l s e s t a k e n f r o m t h e 10 t h d y n o d e

o f t h e p h o t o m u l t i p l i e r t u b e . O n e d i s c r i m i n a t o r w a s s e t a t a l o w t h r e s h o l d

( 5 0 k e V ) t o m i n i m i z e t i m e w a l k a n d t h e o t h e r s e t a t a h i g h t h r e s h o l d

( 2 5 0 k e V ) t o m i n i m i z e m u l t i p l e t r i g g e r i n g o f t h e t i m e p i c k - o f f u n i t i t s e l f .

B y a p p r o p r i a t e s h a p i n g a n d d e l a y i n g t h e t w o p u l s e s , o v e r l a p b e t w e e n t h e t w o

r e s u l t e d i n a n o u t p u t t i m i n g s i g n a l t h a t p o s s e s s e d t h e r i s e t i m e o f t h e

l o w e r t h r e s h o l d t i m i n g s i g n a l b u t w i t h r e d u c e d m u l t i p l e t r i g g e r i n g , w h i c h

h a d b e e n , i n e f f e c t , g a t e d o u t b y d e m a n d i n g t h e c o i n c i d e n c e . T h e t i m i n g

P A R T I C L E - G A M M A C O I N C I D E N C E

B L O C K D I A G R A M

Fig. 18

59

s i g n a l f r o m t h e a n n u l a r p a r t i c l e d e t e c t o r , d e r i v e d f r o m a s i n g l e t i m e

p i c k - o f f u n i t , s t a r t e d a t i m e - t o - a m p l i t u d e c o n v e r t e r ( T A C ) a n d s i m u l ­

t a n e o u s l y w a s p u t i n f a s t c o i n c i d e n c e w i t h t h e g a m m a r a y t i m i n g s i g n a l .

A f a s t c o i n c i d e n t o u t p u t w a s g e n e r a t e d w h e n t h e t w o i n p u t s i g n a l s o v e r ­

l a p p e d i n t i m e . T h e c o i n c i d e n t o u t p u t g a t e d t h e f a s t l i n e a r N a l s i g n a l

a n d s i m u l t a n e o u s l y t r i g g e r e d t h e s t o p i n p u t o f t h e T A C . A w i n d o w ,

a p p r o x i m a t e l y 3 5 n a n o s e c o n d s w i d e , ( F W @ l / l O M A X . ) , w a s s e t o n

t h e p e a k o f t h e t i m e s p e c t r u m a n d o n e o f e q u a l w i d t h w a s s e t o n t h e f l a t

p o r t i o n , w e l l o f f t h e p e a k ; t h e s e w i n d o w s g e n e r a t e d , r e s p e c t i v e l y , t h e t r u e

a n d a c c i d e n t a l t i m i n g s i g n a l s . T h e s l o w l i n e a r N a l a n d p a r t i c l e s i g n a l s

a n d t h e T A C o u t p u t w h i c h w e r e g a t e d b y t h e t r u e a n d a c c i d e n t a l t i m i n g

s i g n a l s , w e r e b u f f e r e d a n d r o u t e d i n t o s e p a r a t e 1 0 2 4 I B M m o d u l a r a n a l o g

t o d i g i t a l c o n v e r t e r s ( A D C ) m o u n t e d i n t h e N u c l e a r D a t a A c q u i s i t i o n

I n t e r f a c e S y s t e m ( R u 6 7 ) .

I . H a r d w a r e

T h e N u c l e a r D a t a A c q u i s i t i o n I n t e r f a c e S y s t e m i n t e r f a c e s s t a n d a r d

n u c l e a r i n s t r u m e n t a t i o n w i t h t h e I B M 3 6 0 / 4 4 c o m p u t e r a n d a s s o c i a t e d

h a r d w a r e f a c i l i t i e s s h o w n s c h e m a t i c a l l y i n F i g . 1 9 . ( S a 6 8 ) . A n a l o g a n d l o g i c a l

p u l s e s f r o m s t a n d a r d m o d u l a r e l e c t r o n i c s a r e a c c e p t e d b y a n d s t o r e d i n

t h e i n d i v i d u a l c o m p o n e n t i n t h e i n t e r f a c e u n t i l a n a p p r o p r i a t e e v e n t ,

d e t e r m i n e d b y t h e d i o d e p i n p l u g b o a r d c o n f i g u r a t i o n i n i t i a t e s t h e t r a n s f e r

D A T A A C Q U IS IT IO N H A R D W A R E

Fig. 19

60

o f t h e s t o r e d d a t a t o t h e c o m p u t e r . T h e i n d i v i d u a l c o m p o n e n t s a r e

m o d u l a r 1 0 2 4 A D C ' s , s c a l e r / t i m e r s , a n d m o n i t o r r e g i s t e r s . T h e d i o d e

p i n p l u g b o a r d i s a t w o d i m e n s i o n a l a r r a y o f 1 6 p o s s i b l e e v e n t s a n d 3 0

c o m p o n e n t s . T o s e l e c t w h i c h c o m p o n e n t s a r e r e a d b y t h e c o m p u t e r ,

d i o d e p i n s a r e i n s e r t e d i n t o a g r i d i n t e r s e c t i n g t h e c o r r e s p o n d i n g e v e n t

a n d c o m p o n e n t . S p e c i f i c a t i o n o f c o m p o n e n t s a n d d e t a i l e d o p e r a t i o n a l

p r o c e d u r e s m a y b e f o u n d i n r e f e r e n c e R u 6 7 .

J . S o f t w a r e

A b l o c k d i a g r a m o f t h e d a t a a c q u i s i t i o n p r o g r a m i s s h o w n i n F i g . 2 0 ( L i

6 9 ^ - X I h e f i r s t s t a g e i n t h e p r o g r a m i s t h e i n i t i a l i z a t i o n o f a l l a n a l y z e r s ,

s c a l e r s , e t c . T h e p r o g r a m e n t e r s a w a i t l o o p u n t i l i t r e c e i v e s a s t a r t

e v e n t , w h i c h i s a c t i v a t e d b y m a n u a l l y d e p r e s s i n g t h e e v e n t o n t h e i n t e r ­

f a c e o r a u t o m a t i c a l l y b y a n e l e c t r i c a l s i g n a l a p p l i e d t o t h e c o r r e s p o n d i n g

B N C c o n n e c t o r . C o l l e c t i o n o f d a t a b e g i n s i m m e d i a t e l y a n d p r o c e e d s u n t i l

a s t o p e v e n t i s r e c e i v e d b y t h e i n t e r f a c e , a f t e r w h i c h , o r e v e n d u r i n g ,

d a t a c o l l e c t i o n , t h e f u n c t i o n k e y b o a r d m a y b e u s e d t o o u t p u t o r m a n i p u l a t e

s t o r e d d a t a . A n a l y z e r s a n d s c a l e r s m a y b e c l e a r e d a n d t h e p r o c e s s

r e p e a t e d u n t i l t h e e x p e r i m e n t i s t e r m i n a t e d . A t a b u l a t i o n a n d a b b r e v i a t e d

d e s c r i p t i o n o f t h e a v a i l a b l e k e y b o a r d f u n c t i o n s a r e s h o w n i n F i g . 2 1 .

D A T A A C Q U IS IT IO N P R O G R A M B L O C K D IA G R A M

'1INITILIZ E

DEFINEANALYZERS. RETURNSCALERS. MPAETC.

PA

START

-O*EVENTMODE RETURN cEVENT 9: MPA COLLECTSTART DATACOUNTING 1EVENT MODE EVENTS: ANALYZERS. SCALERS, ETC. EVENT 10: STOP COUNTING STOP

OUTPUT

Fig. 20

K E Y B O A R D F U N C T IO N S U B R O U T IN E S

K e y N a m e F u n c t i o n

D l l C L E R E C l e a r s c o n t e n t s o f a n a n a l y z e rD 1 2 C L R A L L C l e a r s c o n t e n t s o f a l l a n a l y z e r sE 12 C L R 2 D C l e a r s c o n t e n t s o f t h e t w o d i m e n s i o n a l a n a l y z e rF I 3 C L S C L R C l e a r s a l l s c a l e r sB 9 C T R S I Z R e a d s t h e d i m e n s i o n s o f t h e t w o d i m e n s i o n a l a n a l y z e rA l D I S P L D i s p l a y s a n a n a l y z e rB 8 G A T E L T D i s p l a y s w i n d o w o f a g a t e d a n a l y z e rA 1 4 G O P r e p a r e s c o m p u t e r f o r a s t a r t e v e n tB 1 5 H O L D I T I n t e r r u p t s c o m p u t e r p r o c e s s i n g b y a F o r t r a n p a u s e

s t a t e m e n tB 1 K B P L 1 D P l o t s a l l o r p a r t o f a n a n a l y z e r o n t h e p r i n t e rH I K B T T e s t k e y b o a r d p a r a m e t e r k e y sA 6 L I N L O G D i s p l a y s t w o d i m e n s i o n a l a n a l y z e r i n c o n t o u r f o r mA 7 L O W E R A d j u s t l o w e r l e v e l o f c o n t o u r d i s p l a y w i n d o wB 3 L P S P E C S e t s t h e w i n d o w o f a g a t e d a n a l y z e r b y m e a n s o f t h e

l i g h t p e nB 1 4 N U M S C L S e t s t h e n u m b e r o f s c a l e r sB 2 P L T A L L P l o t s a l l a n a l y z e r s o n t h e p r i n t e rG 1 5 P R T S P C P r i n t s t h e c o n t e n t s o f a n a n a l y z e rB 1 3 R E S P E C A l t e r s t h e s p e c i f i c a t i o n s o f a n a n a l y z e rB I O R E M O V E R e m o v e s p a r t o f t h e d i s p l a y u s i n g t h e l i g h t p e nG i l R U N S K P I n r e p l a y m o d e , s k i p s d a t a t a p e t o a s p e c i f i e d r u n n u m b e rA 2 S E T G N A d j u s t v e r t i c a l g a i n o f d i s p l a yG 6 S K I P I n r e p l a y m o d e , s k i p s d a t a t a p e t o n e x t S T O P e v e n tA 5 S L I C E R S u m s a n d d i s p l a y s a s l i c e o f t h e t w o d i m e n s i o n a l

a n a l y z e rB 4 T I M E S e t s w i n d o w o n t h e t i m e A D C f o r c o i n c i d e n c e e v e n t sA 3 T R K I n t e g r a t e s c o u n t s b e t w e e n c h a n n e l s s p e c i f i e d b y l i g h t

p e nA l l T U R N O N R e s t a r t s d i s p l a y a f t e r a n e v e n t l i g h t p e n i n t e r r u p tB 7 T Y P A N L T y p e s t h e s p e c i f i c a t i o n s o f a n a n a l y z e rC 1 4 T Y P P R T T y p e t o p r i n t e rC 1 5 T Y P S C L T y p e s c o n t e n t s o f s c a l e r sA 9 T Y P S P C T y p e s u p p e r a n d l o w e r l e v e l s o f c o n t o u r d i s p l a y w i n d o wA 8 W I N D O W A d j u s t t h e u p p e r l e v e l o f t h e c o u n t o u r d i s p l a y w i n d o wA 4 X A X I S D i s p l a y s m a r k e r s a l o n g l o w e r l i m i t o f t h e d i s p l a y s c r e e nH O T E R M T e r m i n a t e s c o m p u t e r p r o c e s s i n g

F ig . 21

61

A l t h o u g h t h e s p e c i f i c e q u i p m e n t e m p l o y e d a n d e x p e r i m e n t a l

d e t a i l s m a y d i f f e r , t h e l o g i c b e h i n d t h e e x p e r i m e n t a l t e c h n i q u e s , p r o ­

c e d u r e s , a n d e l e c t r o n i c s , g e n e r a l l y s p e a k i n g , w a s v e r y s i m i l a r i n t h e

2 3 2 3 2 6 2 3 2 4 2 3s t u d y o f t h e N a ( a . a ' y ) N a , M g ( p , a y ) N a , M g ( t . a y j N a , a n d

2 4 2 1M g ( P . r t y ) N a , r e a c t i o n s . T h i s b e i n g t h e c a s e i t w o u l d o b v i o u s l y b e

r e p e t i t i o u s t o e x p l a i n s e p a r a t e l y p r o c e d u r a l d e t a i l s u s e d i n t h e s t u d y o f

e a c h r e a c t i o n . I n t h i s s e c t i o n a n d i n t h e n e x t t h e d i s c u s s i o n i s d i r e c t e d

2 3 2 6 2 3t o w a r d t h e s t u d y o f N a t h r o u g h t h e M g ( p , o * y ) N a r e a c t i o n . W h e r e

a p p r o p r i a t e , p r o b l e m s e n c o u n t e r e d i n t h e s t u d y o f t h e o t h e r r e a c t i o n s i s

i n j e c t e d .

2 3T o c l e a r l y d i s c e r n s t a t e s u p t o 6 . 5 M e V e x c i t a t i o n i n N a , i t w a s

2 6 2 3n e c e s s a r y t o u s e a b o m b a r d m e n t e n e r g y i n t h e M g ( p , ( y ) N a r e a c t i o n

( Q = - 1 . 8 2 6 M e V ) i n t h e v i c i n i t y o f 1 5 . 0 M e V . E x c i t a t i o n f u n c t i o n s w e r e

t a k e n a t i n c i d e n t p r o t o n e n e r g i e s f r o m 1 2 . 0 t o 1 6 . 0 M e V t o d e t e r m i n e w h i c h

b o m b a r d m e n t e n e r g y m a x i m i z e d t h e y i e l d t o t h e m a j o r i t y o f s t a t e s o f

g r e a t e s t i n t e r e s t . T h e s e w e r e p o s s i b l e h i g h s p i n g r o u n d s t a t e r o t a t i o n a l

b a n d m e m b e r s i n t h e v i c i n i t y o f 6 . 2 M e V a n d t h e 5 . 5 4 M e V s t a t e c o r r e s -

+ +p o n d i n g t o 1 3 / 2 a n d 1 1 / 2 a s s i g n m e n t , r e s p e c t i v e l y . O t h e r s t a t e s o f

i n t e r e s t w e r e n e g a t i v e p a r i t y s t a t e s , o t h e r l o w l y i n g b a n d m e m b e r s , a n d

a h o s t o f h i g h e r l y i n g s t a t e s r e g a r d i n g w h i c h n o i n f o r m a t i o n o t h e r t h a n

e x c i t a t i o n e n e r g y w a s k n o w n . A b o m b a r d m e n t e n e r g y o f 1 4 . 2 5 M e V w a s

21c h o s e n . S i m i l a r c r i t e r i a w e r e u s e d i n t h e s t u d y o f N a , b u t t h e l a r g e

I I I . D A T A A C Q U IS IT IO N

62

2 4 2 1n e g a t i v e Q v a l u e s ( - 6 . 8 7 M e V ) i n t h e M g ( p , o ; ) N a r e a c t i o n n e c e s s i t a t e d

a l a r g e r p r o t o n b o m b a r d m e n t e n e r g y o f 1 7 . 5 M e V .

A 4 0 0 c h a n n e l V i c t o r e e n a n a l y z e r , e x t e r n a l t o t h e c o m p u t e r ,

s t o r e d d i r e c t N a l g a m m a r a y s p e c t r a t o c h e c k p e r i o d i c a l l y f o r g a i n

s h i f t s i n t h e p h o t o - m u l t i p l i e r t u b e . A l s o a 1 0 2 4 c h a n n e l N u c l e a D a t a

a n a l y z e r w a s u s e d t o c o n t i n u o u s l y s t o r e d i r e c t p a r t i c l e s p e c t r a f o r e a c h

a n g u l a r c o r r e l a t i o n r u n . T h i s w a s u s e d a s a c h e c k f o r c o n t a m i n a n t

b u i l d u p , e s p e c i a l l y n e a r t h e a l p h a g r o u p s o n w h i c h a g a t e w a s s e t t o

g e n e r a t e t h e m o n i t o r c o u n t s f o r d a t a a c q u i s i t i o n i n t h e c o m p u t e r .

I n t h e e x p e r i m e n t d e s c r i b e d h e r e i n t h r e e p a r a m e t e r a r r a y s o f t r u e

a n d a c c i d e n t a l c o i n c i d e n t d a t a t o g e t h e r w i t h s c a l e r a n d i d e n t i f i c a t i o n i n f o r m a ­

t i o n w e r e s e l e c t i v e l y s t o r e d i n m e m o r y w i t h a d a t a a c q u i s i t i o n p r o g r a m

f o r p u r p o s e s o f m o n i t o r i n g r e p r e s e n t a t i v e p o r t i o n s o f t h e d a t a d u r i n g d a t a

a c q u i s i t i o n . A t t h e s a m e t i m e a l l i n f o r m a t i o n a r r i v i n g a t t h e i n t e r f a c e

a n d s e l e c t e d b y t h e e v e n t s a n d d i o d e p i n p l u g b o a r d ( i n d e p e n d e n t o f t h e d a t a

a c q u i s i t i o n p r o g r a m ) w a s l o g g e d , e v e n t b y e v e n t , o n m a g n e t i c t a p e b y t h e

c o m p u t e r a n d a s s o c i a t e d h a r d w a r e f a c i l i t i e s . T h e d a t a l o g g e d o n t h e t a p e

w a s s a v e d a n d s u b s e q u e n t l y a n a l y z e d i n d e t a i l w i t h s i m i l a r p r o g r a m s .

R e p r e s e n t a t i v e p o r t i o n s o f d a t a w e r e m o n i t o r e d , d u r i n g a c q u i s i t i o n ,

b y s e t t i n g d i g i t a l g a t e s o n m a n y o f t h e i d e n t i f i a b l e a n d w e l l s p a c e d a l p h a -

2 3p a r t i c l e g r o u p s c o r r e s p o n d i n g t o l o w l y i n g e x c i t e d s t a t e s i n N a . E a c h

g a t e d e f i n e d a 2 5 6 c h a n n e l a n a l y z e r a n d e a c h c o u l d b e d i s p l a y e d e n t i r e l y

b y s e l e c t i n g t h e a p p r o p r i a t e s u b r o u t i n e . I n t h i s m a n n e r c o n t i n u o u s

63

a c c u m u l a t i o n o f g a m m a r a y s s u m m e d o v e r t h e a l p h a g r o u p c o u l d b e

m o n i t o r e d f o r e a c h a n g l e . A d i r e c t m e a s u r e o f t h e g a m m a r a y p h o t o ­

p e a k o f i n t e r e s t c o u l d b e o b t a i n e d i n o r d e r t o d e t e r m i n e w h e t h e r o r n o t

s u f f i c i e n t s t a t i s t i c s h a d b e e n a c c u m u l a t e d , r e s u l t i n g i n a s i m p l e o n l i n e

a n a l y s i s o f t h e a n g u l a r c o r r e l a t i o n s . A n g u l a r c o r r e l a t i o n s f r o m k n o w n

a n d s u s p e c t e d s p i n l / 2 s t a t e s w e r e a n a l y z e d a n d c h e c k e d f o r i s o t r o p y t o

i n s u r e a g a i n s t o b v i o u s m a l f u n c t i o n s . M o n i t o r i n g o f t h e t i m e s p e c t r u m

w a s h e l p f u l i n d e t e r m i n i n g t h e r a t i o o f t r u e t o a c c i d e n t a l c o i n c i d e n c e s a n d ,

m o r e i m p o r t a n t l y , t h e s h a p e o f t h e t i m e p e a k w a s u s e d a s a s e n s i t i v e

c h e c k t o d e t e c t a n y i n i t i a l d e t e r i o r a t i o n o f t h e t i m i n g r e s o l u t i o n . T h i s

a l l o w e d t h e e x p e r i m e n t e r t o m a k e e f f i c i e n t o n l i n e a d j u s t m e n t s t o r e ­

o p t i m i z e t h e t i m i n g i n f o r m a t i o n w i t h o u t s i g n i f i c a n t l o s s o f d a t a .

S h o r t t e r m m o n i t o r i n g o f d a t a c o l l e c t e d b y t h e c o m p u t e r w a s

a c c o m p l i s h e d b y a c c u m u l a t i n g t o t a l p a r t i c l e s i n g l e s , t o t a l g a m m a s i n g l e s ,

t o t a l c o i n c i d e n c e s , t i m e , b e a m c u r r e n t , c h a r g e , e t c . i n a p r e d e f i n e d

n u m b e r o f s c a l e r s f o r a f i x e d t i m e i n t e r v a l a n d t h e n p e r i o d i c a l l y o u t p u t

t h e t i m e a v e r a g e d v a l u e s i n t a b u l a t e d f o r m w i t h t h e o n l i n e p r i n t e r . T h i s

w a s p a r t i c u l a r l y u s e f u l i n c o n t r o l l i n g t h e p a r t i c l e a n d g a m m a r a y d e t e c t o r

c o u n t i n g r a t e s a n d i n g i v i n g a c o n v e n i e n t r u n n i n g a c c o u n t o f t o t a l a c c u m u ­

l a t e d c o i n c i d e n t c o u n t s . I t w a s a l s o h e l p f u l i n c h e c k i n g f o r c o n t a m i n a n t

b u i l d u p a n d t a r g e t d e t e r i o r a t i o n .

2 6 2 3T h e a n g u l a r c o r r e l a t i o n d a t a f o r t h e M g ( p )Q!) N a r e a c t i o n w a s

64

a c c u m u l a t e d w i t h t h e N a l d e t e c t o r a t a n g l e s o f 9 0 , 4 5 , 2 0 , 6 0 , 3 0 , a n d

t h e n r e p e a t e d a t a n g l e s 9 0 , 4 5 , 9 0 , 4 5 , 2 2 . 3 , a n d 4 5 d e g r e e s i n t h e t i m e

o r d e r e d s e q u e n c e t h e r e o f .

65

D u r i n g d a t a a c q u i s i t i o n a m a g n e t i c t a p e w a s g e n e r a t e d c o n t a i n i n g

a l l i n f o r m a t i o n a r r i v i n g a t t h e i n t e r f a c e w h i c h h a s b e e n s e l e c t e d b y t h e

c o n n e c t e d e v e n t s a n d d i o d e p i n p l u g b o a r d . E x p e r i m e n t a l d a t a w e r e

r e p l a y e d b y e f f e c t i v e l y r e p l a c i n g t h e i n t e r f a c e w i t h t h e m a g n e t i c t a p e

( d a t a t a p e ) a s i n p u t t o t h e d a t a a c q u i s i t i o n p r o g r a m . B y s l i g h t a l t e r a t i o n

o f t h e c o n t r o l c a r d s a n d b y a p p r o p r i a t e l y m o d i f y i n g t h e d a t a a c q u i s i t i o n

p r o g r a m t o p e r f o r m t h e d e s i r e d o p e r a t i o n s , t h e d a t a t a p e w a s r e p l a y e d ,

w i t h c o n s e q u e n t r e d u c t i o n o f t h e d a t a t o c o n v e n i e n t f o r m f o r a n a l y s i s .

2 6 2 3I n t h e M g ( p , o ; ) N a r e a c t i o n t h r e e p a r a m e t e r c o i n c i d e n t d a t a a n d

s c a l e r i n f o r m a t i o n w e r e a d d e d t o g e t h e r p r o d u c i n g a s e t o f d a t a f o r f i v e

d i s t i n c t a n g l e s , w h i c h w e r e 2 1 ° , 3 0 ° , 4 5 ° , 6 0 ° , a n d 9 0 ° . T h e d a t a a t

a n g l e s 2 0 ° a n d 2 2 . 3 ° w e r e a d d e d t o g e t h e r t o s i m p l i f y a n a l y s i s a n d t h e

a n g l e w a s r e d e f i n e d a s 2 1 ° . T h i s c o r r e s p o n d s a t m o s t t o a 2 % c o r r e c t i o n ,

b u t s i n c e s t a t i s t i c a l a n d o t h e r u n c e r t a i n t i e s w e r e a c t u a l l y l a r g e r , n o c o r ­

r e c t i o n w a s m a d e a n d i s c e r t a i n l y i n c o n s e q u e n t i a l .

A s u m m a r y o f t h e d a t a m o n i t o r i n g p a r a m e t e r s i s t a b u l a t e d i n

F i g . 2 2 a n d 2 3 . B e g i n n i n g w i t h t h e f i r s t c o l u m n , t h e a v e r a g e b e a m t r a n s ­

m i t t e d t o t h e t a r g e t d u r i n g t h e e x p e r i m e n t w a s 3 0 . 0 n a n o a m p e r e s ( n a ) w i t h

< 0 . 0 5 % o f t h i s i n c i d e n t u p o n t h e t a n t a l u m t u b e s h i e l d i n g a s s e m b l y . T h i s

c o r r e s p o n d e d t o a 3 . 0 k i l o c y c l e ( K C ) c o u n t i n g r a t e o f p a r t i c l e s i n t h e

a n n u l a r d e t e c t o r a n d a 3 6 . 7 K C c o u n t i n g r a t e i n t h e N a l d e t e c t o r w i t h

s h i e l d i n g m a t e r i a l i n t e r p o s e d b e t w e e n t h e t a n t a l u m b e a m s t o p a n d t h e N a l

IV . D A T A R E D U C T IO N

TABLE OF NORMALIZATIONS

ANGLE BEAMCURRENT I NTEG TIME

(DEG) (no) (ftC ) (HRS)2 1 26.9 1 197.5 12.430 30.5 595.1 5.445 25.5 1391. 1 15.260 31.9 599.7 5.290 25.0 1197.4 13.3

NTEGNORM

ANGLE(DEG)

PARTICLEMONITOR

(xSO3 )

MONITOR DEAD TIME

U I O 3 )

CORRECTEDMONITORU IO ^ )

MONITONORM

21 332.9 .5 332.4 .5030 165. 1 1.7 163.4 1.0245 388.9 1.0 387.9 .4360 167.7 .6 167.1 1.0090 336.7 1.7 335.0 .50

Fig. 22

TABLE OF SINGLES AND COINCIDENT COUNTING RATES

ANGLE(DEG)

PARTICLESINGLES(xIO6 )

PARTICLECURRENT

(KC)

GAMMASINGLES

(xIO6)

GAMMACURRENT

(KC)

GAMMASPARTICLES

2 1 131. 2 3.0 1623.8 36.4 12.430 60.2 3.1 729.1 37.4 12. 145 151.7 2.8 1900.3 34.8 12.560 61.4 3.3 742.4 39.4 12. 190 131.4 2.7 5705.9 35.6 13.0

ANGLE

(DEG)

REAL PLUS RANDOM COINC(x IO 3 )

RANDOMCOINC(x SO3 )

REALCOSNC(xIO3 )

REALCOINC

CURRENT(CPS)

REALSRANDOMS

21 368.0 108.1 259.9 5.8 2 .4 030 1 95.8 62.8 133.0 6.8 2. 1245 4 5 4 .4 134.4 3 20 .0 5.8 2 .3 860 1 84.3 48.6 135.7 7.2 2 .7 990 375 .7 98.0 277.7 5.8 2 .8 3

F ig . 23

66

d e t e c t o r . W i t h o u t t h e t a r g e t i n p o s i t i o n a b o u t 5 0 % o f t h e r a d i a t i o n i n t h e

N a l d e t e c t o r c a m e f r o m t h e t a n t a l u m b e a m s t o p . I t w a s l a t e r d i s c o v e r e d

t h a t a c a r b o n b e a m s t o p d e c r e a s e d t h i s b a c k g r o u n d y i e l d b y a f a c t o r o f t w o

o r b e t t e r . I t w a s e m p i r i c a l l y d e t e r m i n e d , u n d e r t e s t c o n d i t i o n s , t h a t t h e

N a l d e t e c t o r c o u l d r e s p o n d a t r a t e s u p t o 5 0 K C b e f o r e c o u n t r a t e n o n -

l i n e a r i t i e s ( d i o d e f a t i g u e ) s i g n i f i c a n t l y d e t e r i o r a t e d i t s e n e r g y a n d t i m e

r e s o l u t i o n .

T h e n e x t c o l u m n l i s t t h e i n t e g r a t e d e l e c t r i c c h a r g e m e a s u r e d i n

m i c r o c o u l o m b s ( M C ) . H a d t i m e p e r m i t t e d m o r e d a t a w o u l d , o f c o u r s e ,

h a v e b e e n a c c u m u l a t e d a t 3 0 ° a n d 6 0 ° t o a c h i e v e t h e s a m e s t a t i s t i c a l

a c c u r a c y a s a t t h e o t h e r a n g l e s . F o r t u n a t e l y t h i s s t a t i s t i c a l u n c e r t a i n t y

p o s e d n o a n a l y s i s p r o b l e m s .

T h e n e x t c o l u m n l i s t s t h e a m o u n t o f t i m e a c t u a l l y s p e n t a c c u m u l a ­

t i n g d a t a .

T h e p a r t i c l e m o n i t o r , d e a d t i m e , a n d p a r t i c l e m o n i t o r r e a d i n g s

c o r r e c t e d f o r d e a d t i m e a r e s h o w n i n t h e n e x t t h r e e c o l u m n s . T h e m o n i t o r

d a t a w e r e t h e n u m b e r o f c o u n t s i n t e g r a t e d i n a s c a l e r , w h e r e t h e i n p u t w a s

2 3a g a t e s e t o n t h e g r o u n d a n d f i r s t e x c i t e d s t a t e s o f N a i n t h e d i r e c t s p e c t r a .

T h e l a s t t w o c o l u m n s e n u m e r a t e t h e n o r m a l i z a t i o n s s e p a r a t e l y d e t e r m i n e d

b y t h e m o n i t o r a n d b y t h e i n t e g r a t e d b e a m c h a r g e .

T h e s e n o r m a l i z a t i o n s a g r e e t o b e t t e r t h a n 1% . T h e a n g u l a r c o r r e l a ­

t i o n d a t a w e r e a c t u a l l y n o r m a l i z e d t o t h e m o n i t o r d a t a .

I n t h e l a s t t w o c o l u m n s o f t h e l a s t r o w a r e l i s t e d t h e r e a l c o i n c i -

67

d e n t c o u n t i n g r a t e s a n d t h e r a t i o o f t r u e t o a c c i d e n t a l c o i n c i d e n t c o u n t s

f o r e a c h a n g l e . T h e v e r y l o w a v e r a g e d c o i n c i d e n t c o u n t i n g r a t e o f 6 . 3

c o u n t s p e r s e c ( c p s ) a n d t h e l o w t r u e t o a c c i d e n t a l r a t i o e m p h a s i z e s t h e

n e c e s s i t y f o r s t a b l e e l e c t r o n i c s , p r e c i s e a c c e l e r a t o r c o n t r o l , a n d a

c o m p u t e r t o f a c i l i t a t e t h e h a n d l i n g o f l a r g e t h r e e p a r a m e t e r a r r a y s o f

t r u e a n d a c c i d e n t a l c o i n c i d e n t d a t a .

T h e p r o c e d u r e f o r r e d u c i n g t h e d a t a a t a g i v e n a n g l e d u r i n g r e p l a y

c o n s i s t e d f i r s t o f s e l e c t i n g t h e a l p h a g r o u p i n t h e t o t a l c o i n c i d e n t a l p h a

p a r t i c l e s s p e c t r u m t a k e n o n l i n e , t h a t w o u l d b e s u m m e d o v e r d u r i n g r e p l a y

2 3t o g e n e r a t e t h e g a m m a r a y s p e c t r a ( i . e . s e l e c t i n g t h e N a s t a t e s w h o s e

d e - e x c i t a t i o n w o u l d b e e x a m i n e d ) . E a c h a l p h a g r o u p w a s d d i n e d b y t w o

c h a n n e l n u m b e r s .

T h e d a t a a c q u i s i t i o n p r o g r a m d u r i n g r e p l a y i n c l u d e d t h e s u b t r a c ­

t i o n o f t h e a c c i d e n t a l c o i n c i d e n t d a t a f r o m t h e t r u e p l u s a c c i d e n t a l d a t a .

B y a n a l y z i n g t h e g a m m a r a y s c o r r e s p o n d i n g t o t h e g r o u n d s t a t e a l p h a

p a r t i c l e g r o u p , i t w a s o b s e r v e d t h a t t h e n u m b e r o f c o u n t s p e r c h a n n e l ,

w h i c h s h o u l d b e i d e n t i c a l l y z e r o i n s u c h a n a l y s i s , w a s , i n d e e d , r e l a t i v e l y

s m a l l a n d s t a t i s t i c a l l y f l u c t u a t e d a b o u t z e r o s e r v i n g a s a v a l u a b l e c h e c k .

D i f f e r e n t r u n s a t a g i v e n a n g l e w e r e p r o c e s s e d i n s e q u e n c e w i t h a l p h a

p a r t i c l e g a t e s r e d e f i n e d t o a c c o u n t f o r s m a l l g a i n s h i f t s a n d t h e n e w d a t a

w h e r e n e c e s s a r y w e r e a d d e d o n t o t h a t a l r e a d y a c c u m u l a t e d . T h i s w a s

c o n t i n u e d u n t i l d a t a f r o m a p a r t i c u l a r a n g l e w a s e x h a u s t e d .

At this point the summed spectrum was plotted with the line-p rin te r

68

w i t h r u n n i n g s u m s i n c l u d e d f o r u s e i n l a t e r a n a l y s i s . T h e p r o c e s s w a s

r e p e a t e d s e q u e n t i a l l y f o r e a c h a n g l e a t w h i c h d a t a h a d b e e n a c c u m u l a t e d .

S p e c t r a l d a t a w e r e a l s o s u m m e d o v e r a l l a n g l e s . T h i s p r o v i d e d

a s p e c t r u m o f g a m m a r a y s f r o m w h i c h b r a n c h i n g r a t i o s c o u l d b e e a s i l y

d e t e r m i n e d w i t h o u t c o r r e l a t i o n c o r r e c t i o n s .

T h e i n p u t i a t a w a s r e p l a y e d a t h i r d t i m e t o s u m t h e d a t a a r r a y

i t s e l f o v e r a l l a n g l e s . T h i s p r o v i d e d a d a t a a r r a y o f h i g h s t a t i s t i c a l

a c c u r a c y s u c h t h a t p l a n e s o f g a m m a r a y s p e c t r a c o u l d b e e x a m i n e d

i n d e p e n d e n t l y r a t h e r t h a n a s b e f o r e w h e n s u m m a t i o n s o v e r a l l t h e p l a n e s

o f a n a l p h a p a r t i c l e g r o u p w a s r e q u i r e d . T h i s w a s u s e f u l i n d e t e r m i n i n g

t h e d o m i n a n t m o d e s o f d e c a y f r o m s t a t e s w h o s e a l p h a p a r t i c l e g r o u p s w e r e

u n r e s o l v e d .

I n a d d i t i o n , t i m e s p e c t r a , t o t a l c o i n c i d e n t a l p h a p a r t i c l e s p e c t r a ,

a n d t o t a l c o i n c i d e n t g a m m a s p e c t r a w e r e o b t a i n e d f o r e a c h a n g l e . I n s o m e

c a s e s t h e t r u e , a c c i d e n t a l , a n d t r u e p l u s a c c i d e n t a l c o i n c i d e n c e g a m m a r a y

s p e c t r a w e r e p l o t t e d s e p a r a t e l y f o r c o m p a r i s o n i n d e t e r m i n i n g t h e s o u r c e s

o f d o m i n a n t c o n t a m i n a n t s a n d a s a c h e c k t o i n s u r e t h a t t h e r e w e r e n o

r e l a t i v e g a i n s h i f t s b e t w e e n g a m m a s p e c t r a f r o m t h e t r u e a n d t r u e p l u s

a c c i d e n t a l c o i n c i d e n c e s .

69

V . D A T A A N A L Y S I S

A n e x h a u s t i v e t r e a t i s e o n t h e t h e o r y o f a n g u l a r c o r r e l a t i o n o f

r a d i a t i o n s h a s b e e n p r e s e n t e d b y D e v o n s a n d G o l d f a r b ( D e 5 7 ) . T h e

a p p l i c a t i o n o f a n g u l a r c o r r e l a t i o n s t o n u c l e a r r e a c t i o n s i n a c o l i n e a r

g e o m e t r y w a s f i r s t s u g g e s t e d b y L i t h e r l a n d a n d F e r g u s o n ( L i 6 1 ) ; t h i s h a s

s i n c e b e c o m e k n o w n a s M e t h o d I I . M e t h o d I i s s i m i l a r i n t h a t i t a v o i d s

t h e n e c e s s i t y o f d e t a i l e d k n o w l e d g e o f t h e r e a c t i o n m e c h a n i s m , h o w e v e r ,

i t s n o n - c o l i n e a r g e o m e t r y d o e s n o t h a v e t h e i n h e r e n t a n a l y s i s s i m p l i c i t y

o f M e t h o d I I u s e d h e r e i n .

R e c e n t l y , t h e s p e c i f i c a n a l y s i s u s e d i n t h e w o r k h e r e i n w a s d i s ­

c u s s e d i n d e t a i l b y P o l e t t i a n d W a r b u r t o n ( P o 6 5 ) ; i t h a s b e e n w i d e l y

u t i l i z e d a n d i s a n o w s t a n d a r d n u c l e a r s p e c t r o s c o p i c t e c h n i q u e . I n v i e w

o f t h i s e x p o s u r e a n d i t s b r o a d a c c e p t a n c e , o n l y a b r i e f d e s c r i p t i o n o f t h e

a n g u l a r d i s t r i b u t i o n f o r m u l a i s g i v e n i n t h e p r e s e n t t e x t . H o w e v e r , a

d e t a i l e d d e r i v a t i o n o f t h e r e l e v a n t f o r m u l a e p a r a l l e l i n g t h a t o f R o s e a n d

B r i n k ( R o 6 7 ) i s p r e s e n t e d i n A p p e n d i x V ; t h e s e a r e s h o w n t o r e d u c e t o

t h e e x p r e s s i o n s o f P o l e t t i a n d W a r b u r t o n ( P o 6 5 ) u s e d i n t h e a n a l y s i s o f

o u r d a t a .

T h e a n g u l a r d i s t r i b u t i o n e x p r e s s i o n u s e d i n t h i s w o r k i s e x p r e s s e d

i n i t s s i m p l e s t f o r m a s ( P o 6 5 )

A . Method I I A n gu lar C o rre la tio n F o rm a lism

70

^m in (2 L ,2 L ',2 a )

w(8) = ^ /^(a)Fk(ab)QkPk(cose) k =0

------------------------------------ a

6

w h e r e 0 i s t h e a n g l e b e t w e e n t h e d i r e c t i o n o f e m i s s i o n o f t h e g a m m a r a y s

a n d t h e a x i s o f q u a n t i z a t i o n w h i c h h a s b e e n c h o s e n a s t h e b e a m a x i s ; a

a n d b a r e t h e s p i n s o f t h e u p p e r a n d l o w e r m e m b e r s o f t h e g a m m a c a s c a d e ,

r e p s e c t i v e l y . P k ( c o s 0) a r e t h e L e g e n d r e p o l y n o m i a l s , w h e r e k t a k e s o n

e v e n v a l u e s f r o m 0 t o M I N ( 2 L , 2 L ' , 2 a ) . L i s t h e l o w e s t a l l o w e d m u l t i ­

p o l a r i t y ( L = a - b ) e x c e p t w h e r e a = b , t h e n L =» 1 a n d L * = L + 1 ,

T h e Q k a r e c o r r e l a t i o n a t t e n t u a t i o n c o e f f i c i e n t s f o r t h e g a m m a

r a y d e t e c t o r r e f l e c t i n g i t s f i n i t e s o l i d a n g l e a n d p ^ ( a ) a r e t h e s t a t i s t i c a l

t e n s o r s w h i c h d e s c r i b e t h e a l i g n m e n t o f t h e i n i t i a l s t a t e a n d a r e g i v e n b y

/^(a) p^(a »0!)p (o;) •aH e r e , a , t h e m a g n e t i c q u a n t u m n u m b e r c o r r e s p o n d i n g t o a , t a k e s o n

v a l u e s 0 < a < a a n d

- (2 - M» ,

w h e r e t h e ( a a a - a | k 0 ) a r e C l e b s c h - G o r d a n c o e f f i c i e n t s d e f i n e d i n

71

C o n d o n a n d S h o r t l e y ( C o 6 3 ) . S i n c e w e a r e d e a l i n g w i t h u n p o l a r i z e d

b e a m s , t h e p o p u l a t i o n p a r a m e t e r s s a t i s f y t h e r e l a t i o n P ( a ) = P ( - a ) a n d

a r e n o r m a l i z e d a c c o r d i n g t o t h e s u m m a t i o n r u l e £ P ( a ) = 1 .O'

T h e F ^ ( a b ) c o e f f i c i e n t d e p e n i s s p e c i f i c a l l y o n 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 s o f t h e g a m m a r a y c a s c a d e a n d 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 n t h i s w o r k i t i s d e f i n e d a s

F k ( a b ) = [ F k ( L L b a ) + 2 6 F k ( L L ' b a ) + a ^ L ' L h a ) ] / f 2

F k ( L L h a ) = ( - l ) b _ a " 1 [ ( 2 L + l ) ( 2 L ' + l ) ( 2 a + l ) ] 2 ( L | L ' - l | k 0 ) W ( a a L L ' ; k b )

c _ <b llL m+1 I la><bllLm lla>

w h e r e a i s d e f i n e d a s t h e m u l t i p o l e m i x i n g r a t i o a n d W ( a a L L ' ; k b ) i s a R a c a h

c o e f f i c i e n t ( R o 66 ) . i s t h e l o w e s t a l l o w e d v a l u e o f t h e m u l t i p o l a r i t y .

T h e a n g u l a r d i s t r i b u t i o n o f t h e s e c o n d g a m m a r a y i n t h e c a s c a d e

i s g i v e n b y

W ( 0 ) P k ( a ) ° k ( a b ) F k ( a b ) Q k P k

w h e r e

U . , ( a b ) =U k ( L a b ) + 6* U k ( L ' a b )

k v ' 21 + f i l

TT *T , . W(abab;Lk) U, (Lab) = „ , - T 'k W(abab;L0)

72

a

®L

'b

c

T h e c o m p u t e r p r o g r a m A N N ( L i 6 9 e ) w a s u s e d t o d e t e r m i n e t h e

b e s t f i t b e t w e e n , W ( 0) , t h e c a l c u l a t e d a n g u l a r d i s t r i b u t i o n a n d , Y ( 0 ) , t h e

m e a s u r e d d i s t r i b u t i o n f o r a g i v e n p a i r o f 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 s a a n d b , t h e m i x i n g r a t i o § w a s s t e p p e d i n d i s c r e t e v a l u e s a n d

f o r e a c h v a l u e a l i n e a r l e a s t s q u a r e s a n a l y s i s w a s p e r f o r m e d o n t h e

p o p u l a t i o n p a r a m e t e r s . T h e b e s t f i t i s d e f i n e d t o c o r r e s p o n d t o t h e

2l o w e s t v a l u e o f v d e f i n e d b y

v e r s u s 5 f o r c o m b i n a t i o n o f s p i n s a a n d b w i l l s h o w m i n i m a , u s u a l l y , f o r

t w o v a l u e s r e f l e c t i n g t h e q u a d r a t i c d e p e n d e n c e o f W ( 0 ) o n t h e m i x i n g r a t i o .

2

w h e r e E ( 0 ^) i s t h e e r r o r a s s i g n e d t o t h e g a m m a r a y y i e l d Y ( 0 p a t a n g l e

20 j, a n d N i s d e f i n e d a s t h e n u m b e r o f d e g r e e s o f f r e e d o m . A p l o t o f x

73

T h e r e l a t i v e d e e p n e s s o f t h e m i n i m a i s a m e a s u r e o f t h e r e l a t i v e p r o b a b i l i t y

t h a t t h e p a r a m e t e r s e t a , b , a n d 6 i s t h e c o r r e c t o n e . T h e s h a r p n e s s o f

t h e s l o p e o r w i d t h o f t h e m i n i m a i s a n a p p r o x i m a t e m e a s u r e o f t h e r e l a ­

t i v e p e r c e n t a g e e r r o r i n t h e i n d i v i d u a l d a t a p o i n t s .

T h e e r r o r a s s i g n e d t o t h e g a m m a r a y y i e l d w a s i n m o s t c a s e s t h e

s a t i s t i c a l e r r o r d e t e r m i n e d b y t a k i n g / Y ( 0p a n d a p p r o p r i a t e l y c o r r e c t i n g

f o r b a c k g r o u n d s u b t r a c t i o n a n d n o r m a l i z a t i o n .

T h e n u m b e r o f d e g r e e s o f f r e e d o m N i s d e f i n e d h e r e a s

N = a ~ f i

w h e r e a i s t h e n u m b e r o f d i f f e r e n t a n g l e s a t w h i c h d a t a h a d b e e n t a k e n a n d

2 6 2 3j3 i s t h e n u m b e r o f l e a s t s q u a r e s f i t t i n g p a r a m e t e r s . I n t h e M g ( p j f y ) N a

r e a c t i o n d a t a w a s t a k e n a t f i v e a n g l e s ( q ; = 5 ) a n d w a s f i t w i t h o n e l i n e a r

p a r a m e t e r ( £ = 1 ) . T h e r e s u l t a n t f o u r d e g r e e s o f f r e e d o m c o r r e s p o n d

2t o a n o r m a l i z e d ^ o f 0 . 8 2 , 1 . 2 , 1 . 9 5 , 3 . 3 , a n d 4 . 6 f o r c o n f i d e n c e l e v e l s

o f 5 0 , 3 3 , 1 0 , 1 , a n d . 1 % , r e s p e c t i v e l y ( Y o 6 2 ) .

B . S p i n A s s i g n m e n t s a n d R e j e c t i o n C r i t e r i a

A c o n f i d e n c e l i m i t o f 0 . 1 % w a s s e l e c t e d s u c h t h a t a n g u l a r m o m e n t u m

2a s s i g n m e n t s c o r r e s p o n d i n g t o a y m i n i m u m a b o v e t h i s l i m i t w e r e

r i g o r o u s l y r e j e c t e d . I n m o s t c a s e s u n i q u e a s s i g n m e n t s c a n n o t b e m a d e

o n t h e b a s i s o f t h e a n g u l a r c o r r e l a t i o n d a t a a l o n e ; m o r e t h a n o n e s p i n

p o s s i b i l i t y i s u s u a l l y a l l o w e d . H o w e v e r , i f t h e a n g u l a r c o r r e l a t i o n

74

r e s u l t s a r e c o m b i n e d w i t h o t h e r s o u r c e s o f s p e c t r o s c o p i c i n f o r m a t i o n ,

t y p i c a l l y 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 a n d l i f e t i m e i n f o r m a t i o n , t h i s

i n f o r m a t i o n m a y b e u s e d t o f u r t h e r r e s t r i c t s p i n a s s i g n m e n t s .

I n p a r t i c u l a r , t h e l i f e t i m e i n f o r m a t i o n t o g e t h e r w i t h t h e e l e c t r o ­

m a g n e t i c d e - e x c i t a t i o n b r a n c h i n g a n d m u l t i p o l e m i x i n g r a t i o s m a y b e

u s e d t o e s t i m a t e t h e t r a n s i t i o n s t r e n g t h . I f t h e e l e c t r i c s t r e n g t h i s

2e n h a n c e d m o r e t h a n Z t i m e s t h e W e i s s k o p f u n i t , t h e c o r r e s p o n d i n g s p i n

a s s i g n m e n t i s r e j e c t e d . R e j e c t i n g s p i n s o n t h i s b a s i s i s l e g i t i m a t e a n d

s t a n d a r d p r a c t i c e , b u t o n e m u s t e x e r c i s e c a u t i o n s i n c e t h e c o n f i d e n c e

l e v e l f o r s u c h r e j e c t i o n i s n o t w e l l d e f i n e d .

B y t h e s a m e t o k e n , w h e n c o m b i n i n g s p i n s f r o m 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 w i t h s p i n s d e d u c e d f r o m R v a l u e a s s i g n m e n t 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 , c a r e f u l c o n s i d e r a t i o n t o t h e v a l i d i t y o f t h e l v a l u e

a s s i g n m e n t m u s t b e g i v e n s i n c e s u c h a s s i g n m e n t s a r e n o t b a s e d o n aIr i g o r o u s s t a t i s t i c a l a n a l y s i s . I n s i t u a t i o n w h e r e t h e s e a p p r o a c h e s w e r e

u s e d , t h e r e a s o n i n g i s d i s c u s s e d i n d e t a i l .

2I n a f e w i n s t a n c e s i n t h e a n g u l a r c o r r e l a t i o n d a t a , t h e l o w e s t y

m i n i m a c o r r e s p o n d i n g t o a n a c c e p t e d s o l u t i o n , d i d n o t r e a c h t h e 5 0 %

c o n f i d e n c e l e v e l . A s s u m i n g t h a t t h e g a m m a r a y t r a n s i t i o n u n d e r a n a l y s i s

i s u n m i x e d , t h i s i s a n i n d i c a t i o n o f t h e p r e s e n c e o f u n a c c o u n t e d f o r

s y s t e m a t i c o r b a c k g r o u n d s u b t r a c t i o n e r r o r s . W h e r e s u c h e r r o r s w e r e

2k n o w n t o b e r e s p o n s i b l e f o r r e l a t i v e l y h i g h ^ m i n i m a , a d j u s t m e n t s w e r e

75

m a d e b y i n c r e a s i n g t h e u n c e r t a i n t y i n t h e a n g u l a r d i s t r i b u t i o n d a t a

p o i n t s t o r e n o r m a l i z e t h e a c c e p t e d s o l u t i o n t o 5 0 % .

C . M i x i n g R a t i o s

OE a c h m i n i m u m i n t h e ^ t h a t o c c u r s b e l o w t h e 0 . 1 % c o n f i d e n c e

l i m i t i s a p o t e n t i a l s o l u t i o n . C o r r e s p o n d i n g t o e a c h m i n i m u m i s a v a l u e

o f t h e m i x i n g r a t i o . T h e e r r o r o n t h e m e a s u r e m e n t i s d e t e r m i n e d b y t h e

2i n t e r s e c t i o n o f t h e l o c u s o f t h e ^ v a l u e a n d t h e 3 3 % c o n f i d e n c e l i m i t

c o r r e s p o n d i n g t o o n e s t a n d a r d d e v i a t i o n . W h e n t h e m i n i m u m o c c u r r e d

a b o v e t h e 5 0 % l i m i t , b u t b e l o w a c o n f i d e n c e l i m i t w h e r e i t c o u l d b e

2r e j e c t e d , t h e ^ p l o t w a s r e n o r m a l i z e d t o p o s i t i o n t h e m i n i m u m a t t h e

5 0 % l i m i t f o r t h e s o l e p u r p o s e o f d e t e r m i n i n g t h e e r r o r o n t h e m i x i n g

r a t i o t o o n e s t a n d a r d d e v i a t i o n . I n c a s e s w h e r e t h e f i n i t e s i z e e f f e c t ( F S E )

o f t h e a n n u l a r c o u n t e r w a s n o t c o m p l e t e l y n e g l i g i b l e t h e e f f e c t i s i n d i c a t e d

2o n t h e x p l o t , a n d a n a v e r a g e v a l u e o f 5 w i t h l a r g e r u n c e r t a i n t y i s q u o t e d

f o r t h e f i n a l m e a s u r e m e n t .

D . F i n i t e S o l i d A n g l e E f f e c t ( F S E )

I n c l u d e d i n t h e d a t a a n a l y s i s p r o g r a m A N N i s a p r o v i s i o n

c o r r e c t i n g f o r t h e f i n i t e s i z e o f t h e a n n u l a r d e t e c t o r ( F S E ) . I n f i r s t o r d e r

t h i s e f f e c t s t h e c o r r e l a t i o n b y p e r m i t t i n g p o p u l a t i o n o f t h e n e x t h i g h e r

s u b s t a t e , P ( 3 / 2 ) i n t h e c a s e o f o u r w o r k . T h e c o u n t e r s u b t e n d s a n g l e s o f

1 7 1 ° t o 1 7 5 ° w h i c h c o r r e s p o n d s t o a n e f f e c t ( P o 6 5 ) o f P ( 3 / 2 ) - 0 . 0 5 P ( l / 2 ) .

76

C l e a r l y s t a t e s o f l o w s p i n a r e m o r e e f f e c t e d t h a n s t a t e s o f h i g h s p i n ,

s i n c e t h e r e l a t i v e p e r c e n t a g e o f a l i g n m e n t i s r e d u c e d i n g r e a t e r p r o p o r t i o n

f o r t h e l o w s p i n s t a t e . T h e n e t e f f e c t o n t h e a n g u l a r c o r r e l a t i o n r e s u l t s

2i s t o d i s p l a c e t h e ^ m i n i m u m t o a s l i g h t l y l a r g e r o r s m a l l e r v a l u e o f t h e

m i x i n g r a t i o , b u t u s u a l l y h a s n o b e a r i n g o n w h e t h e r o r n o t a g i v e n s p i n

i s r e j e c t e d .

A n o t h e r e f f e c t w h i c h c a u s e s s i m i l a r d i s p l a c e m e n t s i n t h e m i x i n g

r a t i o s i s a s l i g h t t i l t o r s l o p e o f t h e m e a s u r e d a n g u l a r d i s t r i b u t i o n . S u c h

e f f e c t s c a n a r i s e f r o m s y s t e m a t i c e r r o r s o r m o r e s p e c i f i c a l l y w h e n s u b -

s t a n t i a l b a c k g r o u n d i s s u b t r a c t e d f r o m t h e p e a k o f i n t e r e s t e s p e c i a l l y

i f t h e r e l a t i v e p r o p o r t i o n s v a r y w i t h a n g l e . U s u a l l y , g a m m a r a y b a c k ­

g r o u n d i s m o s t p r o n o u n c e d a t f o r w a r d a n g l e s a n d g r a d u a l l y d i m i n i s h e s i n

p r o c e e d i n g t o l a r g e r a n g l e s .

77

2 3 2 3A . N a ( o j o / y ) ^ 3.

T h e h i g h a n g u l a r m o m e n t u m t r a n s f e r r e a c t i o n ( & , & ' ) w a s s e l e c t e d

i n t h e h o p e t h a t i t w o u l d p r e f e r e n t i a l l y p o p u l a t e h i g h s p i n s t a t e s w i t h

s u f f i c i e n t y i e l d t o p e r m i t 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 . T h e z e r o s p i n

n a t u r e o f t h e a l p h a p a r t i c l e i n b o t h e n t r a n c e a n d e x i t c h a n n e l a l s o g r e a t l y

f a c i l i t a t e s t h e M e t h o d I I a n a l y s e s a n d r e d u c e s a l l t h e i n h e r e n t a m b i g u i t i e s

m a r k e d l y . E x c i t a t i o n f u n c t i o n s w e r e s t u d i e d w i t h a l p h a p a r t i c l e b o m b a r d ­

m e n t e n e r g i e s , E , i n t h e r a n g e 1 2 M e V < E < 2 8 M e V . T h e y i e l d t o o n e a a.p o s s i b l e h i g h s p i n s t a t e , t h e l l / 2 + g r o u n d s t a t e b a n d m e m b e r a t 5 . 5 4 M e V ,

m a x i m i z e d a t 1 6 . 8 5 0 M e V b o m b a r d m e n t e n e r g y . T h e l e v e l w a s s t r o n g l y

p o p u l a t e d i n c o m p a r i s o n t o t h e o t h e r s t a t e s a n d f o u n d t o d e c a y t o t h e 9 / 2 +

a n d 7 / 2 + m e m b e r s o f t h e g r o u n d s t a t e r o t a t i o n a l b a n d w i t h a b r a n c h e s o f

a b o u t 2 5 a n d 7 5 p e r c e n t , r e s p e c t i v e l y ( F i g . 2 4 ) . U n f o r t u n a t e l y , t h e

M e t h o d I I c o r r e l a t i o n g e o m e t r y w i t h a 3 / 2 g r o u n d s t a t e s p i n o f t h e t a r g e t

n u c l e u s a l l o w s t w o m a g n e t i c s u b s t a t e t o b e p o p u l a t e d a n d i n t h i s c a s e t h e

a l i g n m e n t a c h i e v e d w a s n o t s u f f i c i e n t t o r e s t r i c t t h e r a n g e o f a c c e p t a b l e

5 1 1s p i n s b e y o n d — < J < — ( F i g . 2 5 a n d 2 6 ) .u

2 3I n a d d i t i o n t o t h i s 5 . 5 4 M e V s t a t e , a l e v e l i n N a i n t h e e x c i t a t i o n

r a n g e 6 . 4 < E x < 6 . 0 M e V w a s a l s o s t r o n g l y p o p u l a t e d a n d t h e g a m m a r a y s

o r i g i n a t i n g f r o m t h i s r e g i o n a r e s h o w n i n F i g . 2 7 . A s t r o n g g a m m a r a y

a t 3 . 5 0 M e V i s o b s e r v e d a n d f r o m e n e r g y s y s t e m a t i c s i t c o r r e s p o n d s t o a

t r a n s i t i o n f r o m a l e v e l a t 6 . 2 0 M e V t o t h e 9 / 2 g r o u n d s t a t e b a n d m e m b e r -

V I. E X P E R IM E N T A L R E S U L T S

COUN

TS

PER

CHAN

NEL

I5K.44

IOK

5 K

5 .5 4 L E V E L SUMMED COINCIDENT PHOTONS

N a 2 3 ( a ,a 'y ' ) Na23 Ea = 16 .450 MeV

■72—28—5.54

2.082.71

1.642 . 0 8 4 ^ . 4 4 2.83

.632.71^2.08

2.26 5.54-^2.71 2.7|-£.44

20 40 60 80 100 120 140 160CHANNEL NUMBER F ig . 24

*303coOl

— ro oj* x ^N U M B E R O F C O U N T S

too

2.08

2

7/2

i

CO

UN

TS

PER

CH

AN

NE

J.

CHANNEL NUMBERF i g . 2 7

78

a t 2 . 7 0 M e V . E x c e p t f o r t h e 1 . 8 1 M e V t r a n s i t i o n , t h e o t h e r g a m m a r a y s

o b s e r v e d m a y b e i d e n t i f i e d a s c a s c a d e s f r o m t h e 2 . 7 0 s t a t e . N o o t h e r

d o m i n a n t m o d e s o f d e c a y f o r t h e 6 . 2 M e V s t a t e w e r e o b s e r v e d . T h e 1. 8 1

2 6M e V g a m m a r a y d e - e x c i t i n g t h e f i r s t e x c i t e d s t a t e o f M g i s i n t r u e

c o i n c i d e n c e w i t h l o w e n e r g y p r o t o n s d e t e c t e d i n t h e a n n u l a r c o u n t e r f r o m

2 3 2 6t h e N a ( & , p ) M g r e a c t i o n a n d i s s e e n a s a c o n t a m i n a n t i n t h e g a m m a

2 3r a y s p e c t r a o f t h e h i g h e r l y i n g s t a t e s i n N a .

I t i s i n t e r e s t i n g t o n o t e t h a t b o t h a b o v e l e v e l s a p p e a r i n t h e e x c i ­

t a t i o n s p e c t r u m a b o u t w h e r e t h e l l / 2 + a n d 1 3 / 2 + l e v e l s a r e p r e d i c t e d b y

e m p i r i c a l l y e x t r a p o l a t i n g t h e c u r v e f i t t i n g t h e l o w e r l y i n g b a n d m e m b e r s t o

h i g h e r e x c i t a t i o n e n e r g i e s ( F i g . 3 ) .

B e c a u s e o f t h e c l o s e p r o x i m i t y o f t h e Z - 1 3 / 2 + t o t h e = l l / 2 +

3

l e v e l , t h e E e n e r g y d e p e n d e n c e o f t h e t r a n s i t i o n l i n k i n g t h e s e s t a t e s

w o u l d r e s u l t i n r e l a t i v e i n h i b i t i o n c o m p l e t e l y c o n s i s t e n t w i t h e x p e r i m e n t a l

o b s e r v a t i o n .

A g a i n p o p u l a t i o n o f t w o m a g n e t i c s u b s t a t e s r e s u l t s i n a w e a k l y

a n i s o t r o p i c a n g u l a r c o r r e l a t i o n p r o h i b i t i n g a u n i q u e s p i n a s s i g n m e n t , a s

2 3w a s t y p i c a l o f o t h e r g a m m a r a y t r a n s i t i o n s i n N a e x c i t e d b y t h e

2 3 2 3N a ( a . / y ' y J N a r e a c t i o n . H o w e v e r , t h e l a r g e a n g u l a r m o m e n t u m

t r a n s f e r r i n g ( f » , o* 0 r e a c t i o n d i d s e r v e i t s p u r p o s e i n l o c a t i n g p o s s i b l e

h i g h s p i n s t a t e s t h a t c a n n o w b e s c r u t i n i z e d w i t h a p e r h a p s w e a k e r

y i e l d i n g r e a c t i o n , b u t w i t h t o t a l a l i g n m e n t o f t h e r e s i d u a l n u c l e a r s t a t e s .

79

2 6 2 6 B - M g ( p , a y ) M g

A s w a s d i s c u s s e d e a r l i e r , t o o b t a i n t o t a l a l i g n m e n t , t h a t i s

1 2 6 2 6 p o p u l a t i o n o f m = ± - s u b s t a t e s o n l y , t h e Mg (P,ay)Mg r e a c t i o n w a s

u t i l i z e d a t a p r o t o n b o m b a r d m e n t e n e r g y o f 1 4 . 2 5 M e V s e l e c t e d a s a

c o m p r o m i s e i n t h e y i e l d s t o p o s s i b l e h i g h s p i n s t a t e s a n d m a n y o f t h e

i n t e r e s t i n g l o w e r l y i n g l e v e l s . T h e ( p . a ) r e a c t i o n w a s n o t a s s e l e c t i v e

a s t h e fee . o ' ) r e a c t i o n i n p o p u l a t i n g t h e h i g h s p i n s t a t e , b u t n e v e r t h e l e s s ,

s u f f i c i e n t c o r r e l a t i o n d a t a w a s o b t a i n e d t o i m p o s e l i m i t s o n s p i n

a s s i g n m e n t s t o t h e 6 . 2 a n d 5 . 5 4 M e V l e v e l a s w e l l a s m a n y o t h e r

p r e v i o u s l y u n s t u d i e d s t a t e s .

A d i r e c t a l p h a p a r t i c l e s p e c t r u m i s s h o w n i n F i g . 2 8 a n d a t y p i c a l

t o t a l c o i n c i d e n t a l p h a p a r t i c l e s p e c t r u m c o r r e c t e d f o r a c c i d e n t a l c o i n c i ­

d e n c e s a n d l a b e l l e d w i t h e x c i t a t i o n e n e r g i e s i s s h o w n i n F i g . 2 9 w i t h

m a r k i n g s i n d i c a t i n g t h e g r o u p s o v e r w h i c h v a r i o u s g a m m a r a y s p e c t r a

w e r e s u m m e d i n t h e t w o - d i m e n s i o n a l a r r a y s . C l e a r l y t h e e n e r g y r e s o l u -

2 3t i o n ( 6 5 k e V ) w a s n o t s u f f i c i e n t t o r e s o l v e a l l t h e s t a t e s i n N a . I t w a s

r e c o g n i z e d t h a t s o m e o f t h e u n r e s o l v e d e x c i t e d s t a t e s m i g h t n o t b e

p o p u l a t e d , w h i c h i f k n o w n i n a d v a n c e w o u l d g r e a t l y s i m p l i f y a n a l y s i s o f

t h e g a m m a r a y s p e c t r a . T o t h i s e n d a h i g h r e s o l u t i o n ( 2 5 k e V ) d i r e c t

a l p h a p a r t i c l e s p e c t r u m s h o w n i n F i g s . 3 0 a n d 3 1 w a s o b t a i n e d u s i n g a

2 6t h i n M g t a r g e t w i t h a h i g h e r r e s o l u t i o n , s m a l l s o l i d a n g l e p a r t i c l e

d e t e c t o r a t 1 7 0 ° a n d w i t h t h e s a m e a l p h a p a r t i c l e b o m b a r d m e n t e n e r g y a s

COUN

TS

PER

CH

AN

NE

L

C H A N N E L N U M B E R F i g . 28

CHAN

NEL

NUM

BER

Mg26 (p,ay)Na‘T O T A L C O I N C I D E N T A L P H A S P E C T R U M A F T E R R A N D O M S S U B T R A C T I O N E p = 1 4 . 2 5 0 MeV

, 2 3

50 -

100 -

150

200 -

2 50

300

6.621 6 583 6.347 .311 6 236 -6.194 .6.115 -6.046 -5 965 -5930 -5.779 -5 759 '5.741-5.535-5.380

-4.778-4.431

■—3.915—3.850—3.678

— 2.984— 2.705— 2.640-2.391-2.080

350

4 0 0

450.439

-0.0Na2 3

_ J ________________ |_________________I________________ I___50 100 150 2 00

COUNTS P E R C H A N N E L F ig .

COUN

TS PER

CH

ANNE

L

CHANNEL NUMBERF ig . 30

COUN

TS PER

CH

ANNE

L

CHANNEL NUMBERF ig . 31

80

u t i l i z e d i n t h e c o r r e l a t i o n e x p e r i m e n t . A l l e x c i t e d s t a t e s i n t h e r a n g e .

E < 5 . 5 4 M e V w e r e c l e a r l y r e s o l v e d . A s a n e x a m p l e o f t h e u t i l i t y o f

t h i s a p p r o a c h , o n e p a r t i c u l a r s t a t e l o c a t e d a t 3 . 9 1 M e V w a s n e g l i g i b l y

p o p u l a t e d a n d a s a r e s u l t n o i n f o r m a t i o n o n t h e s t a t e w a s o b t a i n e d ; h o w ­

e v e r , t h e g a m m a r a y c o r r e l a t i o n d a t a f r o m t h e 3 . 8 5 M e V s t a t e w a s

r e a d i l y a n a l y z e d h a v i n g e s t a b l i s h e d t h a t t h e y i e l d t o t h e 3 . 8 5 M e V s t a t e

w a s a p p r o x i m a t e l y a n o r d e r o f m a g n i t u d e g r e a t e r t h a n t h a t t o t h e 3 . 9 1 M e V

s t a t e .

F o r s t a t e s h i g h e r t h a n t h a t a t 5 . 5 4 M e V b e t t e r r e s o l u t i o n w a s

n e e d e d a n d w a s a c h i e v e d b y u s i n g t h e l a b o r a t o r y ' s M u l t i g a p M a g n e t i c S p e c t o -

g r a p h ( K o 6 9 ) . A g a i n a 1 4 . 2 5 M e V p r o t o n b e a m w a s u t i l i z e d w i t h a n a l p h a

p a r t i c l e d e t e c t i o n a n g l e o f 1 7 2 ° . T h e a l p h a p a r t i c l e s p e c t r u m f o r s t a t e s

w i t h e x c i t a t i o n e n e r g i e s 6 . 5 8 < E x < 5 . 5 4 i s s h o w n i n F i g . 3 2 w i t h 9 k e V

e n e r g y r e s o l u t i o n . A l l s t a t e s a r e c l e a r l y r e s o l v e d a n d t h e c o i n c i d e n t

a l p h a p a r t i c l e s p e c t r a i s s h o w n s u p e r i m p o s e d o n t h e d i r e c t s p e c t r a . T h e

h e a v y a r r o w s i n d i c a t e t h e p a r t o f t h e a l p h a g r o u p t h a t w a s u s e d t o o b t a i n

t h e 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 d a t a . F r o m t h e h i g h r e s o l u t i o n a l p h a

p a r t i c l e d a t a i t m a y b e d e t e r m i n e d a p p r o x i m a t e l y w h i c h s t a t e s o f e x c i t a ­

t i o n a n d t h e i r r e l a t i v e m a g n i t u d e w e r e i n c l u d e d i n t h e w i n d o w o v e r w h i c h

g a m m a r a y p l a n e s i n t h e d a t a a r r a y w e r e s u m m e d . I t s h o u l d b e n o t e d

t h a t t h e s t a t e s a t 5 . 7 5 9 , 5 . 7 7 9 , 6 . 3 5 6 , a n d 6 . 6 2 1 M e V w e r e v e r y w e a k l y

p o p u l a t e d a n d f o r t h e p u r p o s e o f a n a l y z i n g t h e g a m m a r a y c o r r e l a t i o n d a t a

c o u l d b e c o m p l e t e l y n e g l e c t e d . F o r s o m e o f t h e o t h e r c l o s e l y s p a c e d

RELA

TIVE

INTE

NSIT

YDIRECT AND COINCIDENT ALPHA PARTICLE SPECTRA

Mg26(p,a)Na23 Ep = l4.250MeV

0 = 172.5°5 0 0 -

4 0 0 -

TOTAL COINCIDENT

A L P H A PARTICLES

F R O M A N N U L A R D E T E C T O R

DIRECT A L P H A PARTICLES

F R O M M A G N E T I C

S P E C T R O G R A P H

3 0 0 -

200 -

100 -

8 5 9 0 9 5 100R E L A T I V E E N E R G Y S C A L E

105

F ig . 32

81

l e v e l s t h e s i t u a t i o n w a s n o t a s c l e a r ; t h e s e w i l l b e d i s c u s s e d b e l o w i n

m o r e d e t a i l .

A t o t a l c o i n c i d e n t g a m m a r a y s p e c t r u m , u n c o r r e c t e d f o r a c c i d e n t a l

c o i n c i d e n c e s , a n d t h e c o r r e s p o n d i n g a c c i d e n t a l c o i n c i d e n c e s p e c t r u m r . r e

s h o w n f o r c o m p a r i s o n i n F i g . 3 3 . I n t h e a c c i d e n t a l s p e c t r u m t h e d o m i n a n t

g a m m a r a y c o n t a m i n a n t s a r e l a b e l l e d a n d t h e p r e s e n c e o f c o n t a m i n a n t s

2 3o v e r l a p p i n g t h e w e a k e r N a l i n e s i n d i c a t e s t h e i m p o r t a n c e o f c o r r e c t i n g

f o r a c c i d e n t a l s . T h e d i f f e r e n c e b e t w e e n t h e t w o c u r v e s r e p r e s e n t s t h e

t r u e c o i n c i d e n c e s p e c t r u m a n d i s s h o w n i n F i g . 3 4 .

T h e s u m m e d c o i n c i d e n t g a m m a r a d i a t i o n s p e c t r a c o r r e s p o n d i n g t o

s t a t e s o r g r o u p s o f s t a t e s w h i c h c o u l d b e r e s o l v e d f r o m t h e t o t a l c o i n c i ­

d e n t a l p h a p a r t i c l e s s p e c t r a a r e s h o w n i n F i g s .35 t o 4 9 . E a c h g a m m a

r a d i a t i o n s p e c t r u m w a s o b t a i n e d b y s u m m i n g t h e g a m m a r a y d a t a o v e r t h e

a n g l e s 2 1 , 3 0 , 4 5 , 6 0 , a n d 9 0 ° . S e l e c t e d s p e c t r a w i l l b e d i s c u s s e d w h e n

r e l e v a n t i n t h e f i n a l a n a l y s i s f o r i n d i v i d u a l s t a t e s .

T o a s s i s t i n i d e n t i f i c a t i o n o f t h e g a m m a r a y t r a n s i t i o n s , p a r t i c u ­

l a r l y f r o m u n r e s o l v e d s t a t e s , a 2 8 c m 2 G e ( L i ) d e t e c t o r a t a n a n g l e o f 9 0 °

w i t h r e s p e c t t o t h e b e a m w a s u s e d i n t h e s a m e c o r r e l a t i o n g e o m e t r y . T h e

d e t e c t e d g a m m a r a y s a n d a l p h a p a r t i c l e s w e r e s t o r e d i n a 4 0 9 6 x 1 0 2 4 a r r a y

a n d s u b s e q u e n t l y a n a l y z e d w i t h r e d u c e d d i s p e r s i o n .

D a t a a c q u i s i t i o n w a s h e r e t r e a t e d i n t h e s a m e m a n n e r a s p r e v i o u s l y

d i s c r i b e d w i t h t h e e x c e p t i o n o f a 4 0 9 6 c h a n n e l A D C i n r e p l a c e m e n t o f t h e

1 0 2 4 c h a n n e l A D C f o r t h e g a m m a r a y s . S e l e c t e d p o r t i o n o f t h e d a t a a r e

COUN

TS

PER

CHAN

NEL

2 5 K -

2 0 K -

I 5 K -

I O K -

i g ^ ® ( p ,G y ) N a ^TOTAL COINCIDENT GAMMA SPECTRA

E p = 1 4 . 2 5 0 MeV

© R E A L S + R A N D O M S O R A N D O M S

8 0 1 0 0 120

CHANNEL NUMBER140 160 1 8 0

F ig . 33

CO

UN

TS

PER

CH

AN

NE

L

2 5 K -

2 0 K -

I5K -

I O K -

5 K -

TO TA L COINCIDENT PHOTONS @ 4 5 °Mg 2 6 ( p , a y) Na

4 . 2 5 0R E A L S

2 3

Ep = 1 4 .2 5 0 MeV

3 .3 05 . 3 8 - ^ 2 . 0 8

3 33 4 3 45.97~®"2.64 4 7Q_i, 4 4

3.92 ' ° /I / 4.946.31^2.39 / I3.50

\ t e B5T°i

_L

5 . 3 8 - ^ 4 4 5 .305.741+>44

5.741-( 5 ' ■o 6.58,

I ■ I r r

6.14+?4

20 40 60 80 100 120 140 160CHANNEL NU M B ER

180 200 220 240

Fig. 34

82

A d i s c u s s i o n o f t h e a n g u l a r c o r r e l a t i o n a n a l y s e s o f t h e d a t a f o r t h e

2 3i n d i v i d u a l s t a t e s i n N a i s p r e s e n t e d i n t h e n e x t s e c t i o n .

2 4 2 3C . M g ( t , a y ) N a

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 w e r e a l s o c a r r i e d o u t t h r o u g h t h e

2 4 2 3M g ( t , c r y ) N a r e a c t i o n a t a t r i t o n b o m b a r d m e n t e n e r g y o f 3 . 3 M e V

p r o v i d e d b y t h e 3 M V V a n d e G r a a f f a t 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 .

H e r e , t h e r e g i o n o f e x c i t a t i o n 2 . 6 4 < E x < 4 . 7 8 M e V w a s f o c u s s e d o n ,

w i t h p a r t i c u l a r i n t e r e s t c n l e v d s a t 2 . 9 8 , 3 . 8 5 , 4 . 4 3 , a n d 4 . 7 8 M e V

2 3e x c i t a t i o n i n N a . S t a n d a r d e l e c t r o n i c a n d a n a l y s i s p r o c e d u r e s s i m i l a r

t o t h o s e p r e v i o u s l y d i s c u s s e d w e r e u s e d t h r o u g h o u t . S e l e c t e d r e s u l t s

o f t h i s e x p e r i m e n t a r e d i s c u s s e d a n d p r e s e n t e d t o g e t h e r w i t h t h o s e

2 6 2 3 d e t e r m i n e d f r o m t h e M g ( p , o ; y ) N a w o r k .

2 4 2 1D . M g f p , o ; y ) N a

21L o w l y i n g e x c i t e d s t a t e s i n N a w e r e s t u d i e d i n M e t h o d I I

2 4 2 1c o r r e l a t i o n g e o m e t r y t h r o u g h t h e M g ( p , a y ) N a r e a c t i o n a t a p r o t o n

b o m b a r d m e n t e n e r g y o f 1 7 . 5 M e V p r o v i d e d b y t h e M P T a n d e m V a n d e

G r a a f f 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 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 .

B e a m t r a n s p o r t , d e t e c t o r g e o m e t r y , e l e c t r o n i c i n s t r u m e n t a t i o n , a n d d a t a

a n a l y s i s p r o c e d u r e s w e r e a l l s i m i l a r i n d e t a i l t o t h a t d i s c u s s e d i n t h e

2 3s t u d i e s o n N a .

A s u m m a r y o f t h e e x p e r i m e n t a l r e s u l t s i s p r e s e n t e d i n s e c t i o n

shown in F ig s . 50 to 55.

83

V I I , t o g e t h e r w i t h r e s u l t s o n t h e s a m e n u c l e u s f r o m o t h e r a u t h o r s .

D e t a i l s o f e x p e r i m e n t a l r e s u l t s a n d t h e r e l e v a n t i n f o r m a t i o n a r e p r e s e n t e d

i n a p r e p r i n t o f a p u b l i s h e d p a p e r i n A p p e n d i x V I I .

4K

3K

2K

I K

0.44 LEVEL SUMMED COINCIDENT PHOTONS

,26 „ _ i m „ 2 3(p,cc / J N a ' Ep = 14.250 MeV

,44-«-0?

LOWERLEVELDISCRI­MINATOR

00

30

30

>0

BACKGROUND STATISTICALLY XIO AVERAGES TO ZERO

20 4 0 6 0CHANNEL NUMBER

8 0 Fig. 35

2.39 LEVELSUMMED COINCIDENT PHOTONS

.26 /_ „ „ i m « 2 3.44

I

1J

Mg (p,a / )N a ‘ Ep = 14.250 MeV

-61-39-2.39

-.44 •01.95 I 2.39-*-0

2.39-=».44

40 60 80CHANNEL NUMBER

2 . 0 8 L E V E L S U M M E D COINCIDENT P HO TO NS

M g^ 6 ( p , a / ) N a 23 E p = 1 4 . 2 5 0 MeV

2 0 4 0 6 0 8 0C H A N N E L N U M B E R fib. so

7K-

6K-

.44

In

2.64, 2.70 SUMMED COINCIDENT PHOTONS

Mg26(p,a /)Na23Ep = 1 4 .2 5 0 MeV

-63—37— 2.70 2.64 2.08

—1C)0 —

2.64 -^0

20 40 60 80 100CHANNEL NUMBER Fic -38

COUNTS

PER

CHANNEL

COUNTS PER

CHANNEL

2.98 LEVEL SUMMED COINCIDENT PHOTONS

Mg26 (p.a y) No23 Ep* 14.250 MeV

- 5 5 - 4 5 — 2 98

40 60 80 100 120CHANNEL NUM8ER ”

3 85 LEVEL SUMMED COINCIDENT PHOTONS

Mg26(p,a/)Na23 Ep= 14 250 MeV

100

60 80 100 120 140 160CHANNEL NUMBER

3 6 7 9 LEVEL SUMMED COINCIDENT PHOTONS

M g26 ( p . a y ) N a23 Ep = 14 2 5 0 MeV

CHANNEL NUMBER

20 40 6 0 80 100 120CHANNEL NUMBER

SUMMED COINCIDENT PHOTONS Mg26(p ,a y )N a 23 E p = 14 .250 MeV-6-5*1-25-15-4 78

20 40 60 80 100 120 140 160 180CHANNEL NUMBER «

2.5K-.44I

UJ<Io

2K-

5 .5 4 L E V E L SUMMED COINCIDENT PHOTONS

M g26 (p .a y) Na23 Ep = 1 4 .250 MeV

-24—76- 5 542 70 2 08 Lu 2K44 zz0 <

2 84ol.5Kcc1 UJ5 54- 2.70I a iKin1c* 3.46 i-A i zI 1 5.54-»2.08

J ! io.5Ku

40 60 80 100 120 140CHANNEL NUMBER

5 .7 4 0 LE VEL SUMMED COINCIDENT PHOTONS

Mg2 6 ( p .a y ) N a 23 E p = 1 4 .2 5 0 MeV

-63-37— 5 740 5.30 5.740*-.440

5740,-0

20 4 0 60 80 100 120 140 160 180 200 220CHANNEL NUMBERrig. 45 rig. 46

COUN

TS

PER

CHAN

NEL

5 .9 2 6 , 5 .9 6 7 ,6 .0 4 3 L E V E L S SUMMED CO INCIDENT PHOTONS

M g2 6 ( p ,a y ) N a23

E d = l4 .2 5 0 M e VK —26----8-17-49-6.043

-I 1----1----1__ I__ I__ I I I__ I__ I I I I I I__ I__ I 1 I20 40 60 80 100 120 140 160 180 200 220

CHANNEL NUMBERFig. 47

CHANNEL NUMBER

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0C H A N N E L N U M B E R Fig. 49

420 440

460 480

500 320

540 360

580 600

620 640

660 680

700 720

740 760

780 CHANNEL NUMBER

CHANNEL NUMBERFll!. 53

FIs. 54

COUNTS PER CHANNEL COUNTS PER CHANNEL

COUNTS PER CHANNEL u> o COUNTS PER CHANNEL

CO

UN

TS

PER

CH

AN

NE

L

C H A N N E L N U M B E R

F ig . 55

84

2 6 2 3I n t h i s - s e c t i o n a l l o u r r e s u l t s f r o m s t u d i e s o n t h e M g ( p , o ; y ) N a

2 3 2 3 2 4r e a c t i o n a n d s e l e c t e d r e s u l t s f r o m t h e N a ( a , o : y ) N a a n d M g ( t . a ; )

2 3N e r e a c t i o n s a r e d i s c u s s e d i n d e t a i l . A d i s c u s s i o n o f t h e r e s u l t s o n

21N a i s p r e s e n t e d , a s m e n t i o n e d i n t h e p r e v i o u s s e c t i o n , i n a p r e p r i n t

o f a p u b l i s h e d p a p e r i n s e r t e d i n A p p e n d i x V I I .

A s u m m a r y o f t h e L e g e n d r e e x p a n s i o n c o e f f i c i e n t s a n d 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 s d e t e r m i n e d f r o m t h e ( P , q ) d a t a a r e

s h o w n i n F i g & 5 6 a n d 5 7 . A s u m m a r y o f s e l e c t e d m u l t i p o l e m i x i n g r a t i o s

d e t e r m i n e d f r o m t h e ( a , a ' ) a n d ( t , o ; ) d a t a i s s h o w n i n F i g . 5 8 .

2 3R e s u l t s f o r N a f r o m t h e s t u d y o f a l l t h r e e r e a c t i o n s a r e d i s c u s -

2 3s e d i n d e t a i l i n t h e f o l l o w i n g t e x t f o r i n d i v i d u a l s t a t e s i n N a i n a s c e n d i n g

o r d e r o f e x c i t a t i o n e n e r g y u p t o t h e 6 . 5 8 4 M e V s t a t e i n c l u s i v e . T h e

2g r a p h s o f x v e r s u s m i x i n g r a t i o a n d c a l c u l a t e d a n d m e a s u r e d a n g u l a r

d i s t r i b u t i o n s f o r t h e i n d i v i d u a l t r a n s i t i o n s a r e p r e s e n t e d i n t o t a l f r o m F i g .

5 9 t o F i g . 9 2 a t t h e e n d o f t h i s s e c t i o n .

2 3T h e g r o u n d s t a t e o f N a

+ 2 3A s p i n / p a r i t y a s s i g n m e n t o f 3 / 2 f o r t h e g r o u n d s t a t e o f N a h a s

b e e n k n o w n f o r s o m e t i m e ( E n 6 7 ) . T h e a c c e p t e d v a l u e o f t h e m a g n e t i c

d i p o l e m o m e n t ( ^ , ) i s 2 . 2 1 7 6 n u c l e a r m a g n e t o n s ( n m ) a n d t h e s t a t i c q u a d ­

r u p o l e m o m e n t ( Q ) i s + 0 . 1 1 b a r n s ( F u 6 9 ) ; t h i s c o r r e s p o n d s t o a n i n t r i n s i c

V II. D ISC U SSIO N O F R E S U L T S

LEGENDRE COEFICIENTS* DETERMINED FROM W = an ( l + ^ . p + ° 4 p j0 a0 2 a0 4

\H THE Mg2 6 (p ,a x )N a 23 REACTION

T R A N S I T I O N a 2 / G o Q 4 / a 0.4 4 0 0 - 0 . 2 0 8 ± 0 .0 1 9 + 0 . 0 6 2 ± 0 .0 3 0

2 .0 7 7 0 + 0 .2 7 4 ± 0 .0 7 8 —0.5 1 2 ± 0.1 1 62 .0 7 7 . 4 4 0 + 0 .0 9 7 ± 0 .0 2 1 + 0 .0 1 9 ± 0 . 0 3 4

(2.077)-®=» 4 4 0 -®300 - 0 . 2 2 1 ± 0 .0 2 5 + 0 .0 5 4 ± 0 . 0 3 92 .3 9 1 o ’ + 0 .0 52 ± 0 . 0 8 4 + 0 .0 1 5 ± 0 .1 2 82 .3 S I -®- . 4 4 0 UNDETERMINED UNDETERMINED2 .6 4 0 0 + 0 .0 2 8 ± 0 .0 1 6 + 0 .0 0 6 ± 0 . 0 2 42 .7 0 3 -o * .4 4 0 + 0 . 5 3 2 ± 0 . 108 — 0 . 3 2 0± 0.1 6 62 .7 0 3 -«»■ 2 . 0 7 7 - 0.1 3 5 ± 0 .0 4 1 + 0 .1 0 1 ± 0 .0 6 32 .9 8 1 0 + 0 .3 54 ± 0 .0 2 2 — 0 .0 5 9± 0 . 0 3 52.9 8 1 -b®* . 4 4 0 + 0 .0 5 7 ± 0 .0 4 2 — 0 .0 1 1 ± 0 . 0 6 53 .6 7 9 . 4 4 0 - 0 . 0 6 9 ± 0 .0 2 2 — 0 .0 0 6 ± 0 . 0 3 4

(3 .6 7 9 ) 4 4 0 - sb®»0 -0 .1 78 ± 0 .0 1 4 — 0 .0 3 3 ± 0 .0 2 13 .6 7 9 2 . 6 4 0 - 0 . 7 6 6 ± 0 . 0 5 4 — 0 .0 6 2 ± 0 .0 8 03.851 -«**• . 4 4 0 - 0 .0 5 0 ± 0 .0 4 2 — 0.0 1 1 ± 0 .0 6 53.851 2 . 0 7 7 - 0 .1 2 6 ± 0 . 0 4 2 + 0 .0 8 1 ± 0 . 0 6 5

(3.851) ■ » 2 .0 7 7 -® * - .4 4 0 + 0 .061 ± 0 .0 4 1 + 0 .0 6 7 ± 0 .0 6 2(4 .431 -c®* 0 ) - 0 . 0 3 1 ± 0 . 0 2 9 — 0.2 1 6 ± 0 . 0 4 64 . 7 7 5 -®®— . 4 4 0 + 0 .0 1 7 ± 0 . 0 3 1 + 0 .0 2 9 ± 0 . 0 4 94 . 7 7 5 2 . 0 7 7 + 0 .4 3 5 ± 0 .0 9 2 —0 .2 3 5 ±0.1 3 95 . 3 8 0 - « - 0 - 0 .5 39 ± 0.1 2 1 + 0 .23 0 ± 0.1 7 75 . 3 8 0 -®*** . 4 4 0 + 0. 1 4 3 ± 0 .0 3 2 — 0 .0 4 8 ± 0 .0 5 05 .3 8 0 -c®* 2 . 0 7 7 - 0. 1 2 0 ± 0 . 0 5 6 — 0 .0 7 9 ± 0 .0 8 8

( 5 . 3 8 0 ) - ^ 2 .0 7 7 —®®**.4 4 0 - 0. 1 2 6 ± 0 .0 4 2 + 0 .0 8 1 ± 0 .0 6 55 . 5 3 5 - » 2 . 0 7 7 + 0 . 4 4 6 ± 0. 130 — 0 .4 4 7 ± 0 . 2 0 55 .5 3 5 -{3s” 2 .7 03 + 0 .0 6 1 ± 0 . 0 3 7 — 0 .0 3 9 ± 0 .0 5 75 . 7 4 0 -°B=* 0 + 0 .0 19± 0 . 4 5 5 + 0.1 0 8 ± 0 . 0 6 95 . 7 4 0 -®“* . 4 4 0 + 0.2 3 0 ± 0 . 0 6 9 — 0 .2 0 6 ± 0 .1 0 56 .2 3 8 -e*- 2 . 7 0 3 + 0. 1 9 6 ± 0 . 0 7 8 — 0 .3 8 3 ±0.1 2 66.31 1 2 .3 9 1 - 0 . 0 2 4 ± 0 .0 5 2 + 0 .0 4 8 ± 0 . 0 7 66 . 5 8 4 -®s= . 4 4 0 + 0 .3 0 4 ± 0 . 0 6 8 — 0 .3 4 3 ± 0 .0 9 36 . 5 8 4 -&£* 2 . 0 7 7 - 0 .5 8 7 ± 0 .0 8 8 — 0 . 2 9 0 ±0. 1 3 06 . 5 8 4 2 . 0 7 7 - ® - .4 4 0 - 0 .0 5 2 ± 0 . 0 5 2 0.0 10 ± 0 . 0 8 2

* C O R R E C T E D FOR F I N I T E S I Z E O F G A M M A R A Y D E T E C T O RFig. 56

Fig. 57

O) fl) (J) UlU l 0 1 CD 00

P Oo-4 4ko

-g4ko

U l+1

-A4ko

uibCMU l

U l

U lCMU l

U lCMCDo

UlCMCDo

UlcmCDO

4* -5 + cm cm cm CMCMOJrorotofororororoio”-g -g 4k b o o cd bo " t t b b b ‘-g ’-g b cm cm "o O "4 *- J -g cm — o> u i o i g g ® © o o . & « ) ( 9 g g ^u i u i o ro — — — c o c o cm cm o g - g o

I i I I I I I j I i I I H I J 11111H i 11111ro■goCM

roo~g•g

roO-g-g

4*4ko

poo-g-g

o o o4*4kOroO■g-g

ro 0 f ° . ° ° o o4k <y> 4> 4k 0 4* 4> 4>g 4 4 4 *

- g o o o4 4 4 4

o o o o

3)>zCDOz

couicocncMcouiCMUicMUi-gco — ui jco — •guicM-guicMuicM-guicM-gmcM — uiui-gmcMUicMCMCMCMcMcoco_____ o -o uiN ^ ^ S ^ v> .S ^ N S S S N S S S S S n S S S N S 's S S S S \ S S N ' \ N S S N V \ \ \ \ S S \ S N S n!s srororororororororororororororororororororororororororoiorororororororororororofororororororororororororof t t i n ? t i t i f i v t t i t v ? 1 1 1 'i I t ? 11 ?•g^imuicoooiuiuiuicocovoco-g-ggi'g-g-g-gcDuiuioJCM-g^i-guiuiuicMuicM-g^i^iuiui —uicncM iuicMuicMOicMCMv>vv> N < '< ^ N '< N N N N N N N N V S N S N N S \\S \S 'v\ s \ \ N \ S N N N N N S N N \ N \ N S N \rorororororororororororororororororororororororororororororororororororororororororororororororororororo

CDpozoosroz>HOz

I I + I + I + + I + Io p p r o o — o o o o o — 4k — b — b -g cm ro K> — CM^oooimooo — cdo) 1+ 1+ 1+ 1+ 1+ 1+ 1+ H-1+1+1+o o O O p p o o o o o b - o 4 > ‘ocncorobbog 4 4 0 U i O O O U O l C Dm • «

CM + +

09* 1+A P O u i t t b iCDbi

+ + I I I + + Io o o o — o o o p— bi — K>bi4*obi —— m-gttOcDttott1+ 1+ |+ A |+ 1+ 1+ 1+ |+o p o 09 o o o o o0 —0 A 4s> fO ro O O -g O cm ro co iv) o co a)

+ iP P — roO U l

1+ H*P Po omro0+roo

i +

oo

CM

+ + +P P P r o r o o

-g cm u i i + i + 1 +

P P PO O O U l 0 ) U l

+

ro

i +

Oiou i

i + 0 — 0 b r o b iA OOD05,1+ i+a P P

C M 4 * 0 *Y*roro

ooi+00 1

+ i +P P Pbl — CMcoco-gi + 1 + 1 +

P P P( M O O a> CM CM

+ C 0 2

• O

5 3

i + m03)o EU 1 Z

m CM oO

i +

Oo

c cz zo or n m

H Hmm3J #cj2 2

Z Zmmo o

+ +P P bob o roi + 1 +

P P — o4k Ul+

•go

i +roio

i +

P Pio ro cn — i+1+P PO — -g 4>

+ 'iro ro■ g t t i + 1+O P

b> CD

+ + +

P P P cm r o b ro ro roi + 1 + 1 +

P P P ■— ott O tt+ + +— — 4>bo — b r : o i ^ i + 1 + 1 +— o po K > b

o

+ + iP P P b o ­ut — oi+1+1+P P P0 0 0

01 ro to «I4k

1 +

O

I C T

O Z

oC C I + I 2 2 0 0 0 O O L

ro H - m0 3)L S

C M i

m

o

m m m ro —om 4k co m m •+ •+ •+3) 3) O p OS 2 o — o

Z Z N --------mmo o

Z

c

T )

0 — r 09 m

1

x

ztt33 >

MI

XI

NG

R

AT

IO

S

FO

R

TR

AN

SI

TI

ON

S

IN N

a“

FR

OM

T

HE

Mg

2S

(p

,a

x

) N

o23

RE

AC

TI

ON

M I X I N G R A T I O S F O R T R A N S I T I O N S I N N s 23 FROM T H E

Na^Ca.a'y) N a 23 R E A C T I O N

T R A N S I T I O N S P I N C O M B I N A T I O N M U L T I P O L E M I X I N G R A T I O ( 8 )

5 . 5 3 8 —< ^ 2 . 0 7 7

5 . 5 3 8 - ^ 2 . 0 7 7

11 / 2 — ^ 7 / 2 9 / 2 — 7 / 2 7 / 2 — e - 7 / 2 5 / 2 — 7 / 2 3 / 2 —^ 7 / 2 11/2 — 9/2 9 / 2 — 9 / 2 7 / 2 — 9 / 2 5 / 2 — 9/2

+ 0 . I 0 ± 0 . 0 5- 0 . 3 5 ± 0 . 0 8 , + ? . 0 ± 3 . 0- l . 5 5 ± 0 . 5 0+ 0 . 2 2 < § < + 2 . 48 < - 0 . 3 , + 0 . 0 ! < 8 < + l . 4 , 8 > + 3 . 0- 0 . 2 3 ± 0 . 0 4-1.85 ± 0 . 3 5 , + 0 . 4 2 ± 0 . 0 8 + 0 . 2 3 ± 0 . 0 7 , + 3 . 0 ± 0 . 6 0 + 0 . 1 6 ± 0 . 2 7 , +1 . 8 ± 0 . 6

M I X I N G R A T I O S F O R T R A N S I T I O N S I N N a 2 3 F R O M T H E

M g 2 4 ( t , a y ) N e 2 3 R E A C T I O N

T R A N S I T I O N S P I N C O M B I N A T I O N M U L T I P O L E M I X I N G R A T I O ( 8 )

2 . 9 8 ------03 . 8 5 —— 2 . 0 8

4 . 7 8 — 2 . 0 8

3 / 2 ------3 / 23 / 2 — a*. 7 / 2 5 / 2 — 7 / 2 7 / 2 — 7 / 2 9 / 2 — s - 7 / 2 3 / 2 — — 7 / 2 5 / 2 — 7 / 2 7 / 2 — ■ - 7 / 2

- 0 . 1 5 ± 0 . 0 5 , - 2 . 4 0 + 0 . 6 0 - 0 . 18 ± 0 . 0 6 , + 2 . 8 ± 0 . 0 5 + 0 . 0 1 ± 0 . 0 4 , l l . 0 < 8 < 3 8 . 0 + 0 . 6 1 ± 0 . 0 5 - 0 . 0 9 ± 0 . 0 3 + 0 . 5 9 ± 0 . 1 5 + 0 . 5 4 ± 0 . 1 9 + 0 . 0 1 ± 0 . 1 5

F ig . 58

85

q u a d r u p o l e m o m e n t ( Q q ) o f . 5 5 a n d a n u c l e a r d e f o r m a t i o n * ( f t ) e q u a l t o

0 . 4 5 . N o e x p e r i m e n t a l i n f o r m a t i o n o n t h e g r o u n d s t a t e w a s d e d u c e d f r o m

t h e p r e s e n t w o r k .

2?T h e 0 . 4 4 0 M e V s t a t e o f N a

T h e a n g u l a r c o r r e l a t i o n r e s u l t s a d m i t t e d a J = 3 / 2 a n d J = 5 / 2

a s s i g n m e n t a n d r i g o r o u s l y r e j e c t e d o t h e r v a l u e s a b o v e a 0 . 1% c o n f i d e n c e

l i m i t ( F i g . 5 9 ) . T h e a c c e p t e d v a l u e o f t h e l i f e t i m e i s 1 . 6 0 ± 0 . 0 8 p s e c s

( E n 6 7 ) a n d t h e p a r t i a l E 2 l i f t e i m e d e t e r m i n e d f r o m c o u l o m b e x c i t a t i o n

( T e 5 6 , S t 6 0 ) i s 0 . 0 4 3 p s e c s . T h i s c o r r e s p o n d s t o a n M 1 / E 2 m u l t i p o l e

m i x i n g r a t i o o f ± 0 . 0 6 ± 0 . 01 i n d e p e n d e n t l y d e t e r m i n e d f r o m t h e s p i n o f

t h e s t a t e . A s p i n a s s i g n m e n t o f 3 / 2 c o r r e s p o n d s t o a m i x i n g r a t i o

+ 0 . 5 9 ± 0 . 1 5 , w h i c h i s m a n y s t a n d a r d d e v i a t i o n s a w a y f r o m t h e m e a s u r e d

v a l u e , a n d J = 3 / 2 i s , t h e r e f o r e , s a f e l y r e j e c t e d . I n a d d i t i o n a J = 3 / 2

a s s i g n m e n t w o u l d c o r r e s p o n d t o a n e n h a n c e d E 2 t r a n s i t i o n s t r e n g t h o f

1 0 0 0 T w ( T “ 1 W e i s s k o p f u n i t ) w e l l a b o v e t h e l i m i t o f Z ? I ^ v , t h e m a x i m u m

e x p e c t e d e n h a n c e m e n t ( W i 6 0 ) i m p o s e d b y t h e n u c l e a r s i z e . T h e r e m a i n ­

i n g a s s i g n m e n t J = 5 / 2 i s i n d e e d t h e c o r r e c t o n e c o r r e s p o n d i n g t o a

m i x i n g r a t i o § = - 0 . 0 9 ± . 0 2 , w h i c h i s i n a g r e e m e n t w i t h t h e c o u l o m b

e x c i t a t i o n a n d l i f e t i m e r e s u l t s a n d t h e c o r r e l a t i o n r e s u l t o f P o l e t t i e t a l .

J . , 2 J - 1 . ^ _ 3 _ „ n 2 , „ _ „ _ , „ A 1 / 3J5'

I V 1 — I < /* . Q = Q o ( — ; ) < — ) ; Q o = — Z R , 0 ( H O . 1 6 0 ; R q = 1 . 3A

86

( P o 66) . A s u m m a r y o f o t h e r w o r k o n t h e 0 . 4 4 0 M e V s t a t e i s g i v e n i n .

E n d t a n d V a n d e r L e u n ( E n 6 7 ) . T h e a g r e e m e n t o f o u r s p i n a s s i g n m e n t

a n d m i x i n g r a t i o w i t h t h a t o f o t h e u s s e r v e d a s a g r a t i f y i n g c h e c k o n o u r

d a t a r e d u c t i o n a n d a n a l y s i s p r o c e d u r e s .

2 3T h e 2 . 0 8 M e V s t a t e o f N a

F r o m t h e a n a l y s i s o f t h e 2 . 0 8 -» 0 . 4 4 M e V t r a n s i t i o n t h e s p i n o f

t h e 2 . 0 8 M e V l e v e l w a s l i m i t e d t o 3 / 2 , 5 / 2 , a n d 7 / 2 ( F i g . 6 0 ) . B y

f i t t i n g t h e c o m b i n e d d i s t r i b u t i o n 2 . 0 8 — 0 . 4 4 — 0 M e V t h e 3 / 2 a n d 5 / 2

s p i n a s s i g n m e n t s d i d n o t f i t a s w e l l , b u t c o u l d n o t b e e l i m i n a t e d a t t h e 0 . 1%

c o n f i d e n c e l e v e l ( F i g . 6 1 ) . H o w e v e r , t h e w e a k e r c r o s s o v e r t r a n s i t i o n

2 . 0 8 — 0 M e V a n g u l a r d i s t r i b u t i o n w a s f i t t e d w i t h a r e l a t i v e l y l a r g e

a ^ / a Q L e g e n d r e p o l y n o m i a l c o e f f i c i e n t i m m e d i a t e l y e l i m i n a t i n g t h e

2J = 3 / 2 p o s s i b i l i t y . F r o m t h e f u l l ^ a n a l y s i s , s h o w n i n F i g . 6 2 , J = 5 / 2

i s a l s o e l i m i n a t e d a t t h e 0 . 1 % c o n f i d e n c e l i m i t . T h i s r e s u l t i n c l u d i n g

t h e 2 . 0 8 — 0 . 4 4 m i x i n g r a t i o a s s i g n s J = 7 / 2 t o t h e s t a t e a n d i s i n a c c o r d

w i t h t h e r e s u l t s o f o t h e r e x p e r i m e n t e r s ( P o 6 6 , M a 6 8 , S o 68a ) .

2 3T h e 2 . 3 9 M e V s t a t e o f N a

T h i s l e v e l w a s v e r y w e a k l y p o p u l a t e d a n d o n l y a l i m i t e d a n a l y s i s

w a s p o s s i b l e . T h e r a d i a t i o n f r o m t h i s s t a t e t o t h e g r o u n d s t a t e w a s i s o ­

t r o p i c w i t h i n s t a t i s t i c a l u n c e r t a i n t y a l l o w i n g p o s s i b l e s p i n a s s i g n m e n t s o f

l / 2 , 3 / 2 , a n d 5 / 2 , s i n c e t h e g r o u n d s t a t e h a s a J = 3 / 2 . I t h a s b e e n

s h o w n c o n c l u s i v e l y f r o m t h e I - 0 s i n g l e p a r t i c l e t r a n s f e r i n t h e r e a c t i o n s

1

87

2 2 3 2 3 2 4 3 2 3N e ( H e , d ) N a p u 6 7 ) a n d M g ( d , H e ) N a p u 6 9 ) t h a t t h e 2 . 3 9 M e V

+s t a t e h a s s p i n / p a r i t y o f l / 2 . O t h e r w o r k p r e s e n t i n t h e l i t e r a t u r e i s i n

a g r e e m e n t w i t h t h i s a s s i g n m e n t p i 6 7 , D u 68 ) .

2 3T h e 2 . 6 4 a n d 2 . 7 0 M e V s t a t e s i n N a

T h e 2 . 6 4 M e V l e v e l w a s u n r e s o l v e d f r o m t h a t a t 2 . 7 0 M e V i n t h e

t o t a l c o i n c i d e n t a l p h a p a r t i c l e s p e c t r u m . H o w e v e r , f r o m o u r G e ( L i ) d a t a

a n d t h e w o r k s o f o t h e r s ( E n 6 7 ) i t w a s s h o w n t h a t t h e 2 . 6 4 M e V s t a t e

d e - e x c i t e d t o t a l l y a n d d i r e c t l y t o t h e g r o u n d s t a t e a n d t h a t a t 2 . 7 0 M e V

d e - e x c i t e d 3 7 % a n d 6 3 % t o t h e 2 . 0 8 a n d 0 . 4 4 M e V s t a t e s , r e s p e c t i v e l y

( F i g . 3 8 ) . T h e r e f o r e , t h e 2 . 6 4 — 0 M e V c o r r e l a t i o n w a s e a s i l y e x t r a c t e d .

T h e i s o t r o p i c g a m m a r a d i a t i o n e m i t t e d i n t h e 2 . 6 4 — 0 M e V t r a n s i t i o n

i m p o s e s t h e l i m i t s J < 5 / 2 f o r t h e 2 . 6 4 M e V l e v e l c o r r o b o r a t i n g t h e w o r k

2 4 3o f P o l e t t i e t a l . ( P o 66 ) . C o m p l e m e n t a r y s t u d i e s f r o m t h e M g ( d , H e )

2 3N a r e a c t i o n ( D u 6 9 ) h a v e d e f i n i t e l y a s s i g n e d a s p i n / p a r i t y o f ( l / 2 ,

3 / 2 ) t o t h e 2 . 6 4 b y I - 1 c h a r a c t e r i z a t i o n o f t h e a n g u l a r d i s t r i b u t i o n .

A l s o , t h e s a m e w o r k f i r m l y e s t a b l i s h e d s p i n / p a r i t y o f ( l /2 , 3 / 2 ) f o r t h e

23 . 68 M e V s t a t e . F r o m t h e ^ a n a l y s i s o f t h e g a m m a r a d i a t i o n e m i t t e d i n

t h e 3 . 68 — 2 . 6 4 M e V t r a n s i t i o n ( F i g . 7 1 ) s p i n a s s i g n m e n t s o f l / 2 a n d

3 / 2 a r e h e r e u n i q u e l y d e t e r m i n e d f o r t h e 2 . 6 4 a n d 3 . 68 M e V l e v e l s ,

r e s p e c t i v e l y .

A n g u l a r c o r r e l a t i o n s f o r t h e 2 . 7 0 ( J ) — 2 . 0 8 ( 7 / 2 ) M e V a n d t h e

+2 . 7 0 ( J ) - * 0 . 4 4 ( 5 / 2 M e V t r a n s i t i o n s w e r e e x t r a c t e d f r o m t h e g a m m a r a y

88

s p e c t r a ( F i g . 3 8 ) . O n t h e b a s i s o f o u r a n g u l a r c o r r e l a t i o n r e s u l t s f o r -

b o t h t r a n s i t i o n s a n d l i f e t i m e c o n s i d e r a t i o n s , s p i n s w e r e l i m i t e d t o 5 /2 a n d

9 / 2 f o r t h e 2 . 7 0 s t a t e ( F i ^ . 6 5 a n d 66 ) . E x p e r i m e n t e r s s t u d y i n g g a m m a -

2 3 2 3g a m m a a n g u l a r c o r r e l a t i o n s f r o m t h e N a ( p . p ' y y ) N a r e a c t i o n h a v e

a s s i g n e d J = 9 / 2 t o t h e 2 . 7 0 M e V s t a t e b y e l i m i n a t i n g t h e J = 5 / 2 p o s s i b i l i t y

a t t h e 1 % c o n f i d e n c e l i m i t ( M a 0 8 , S o 68a ) .

9 9T h e 2 . 9 8 M e V s t a t e o f N a

T h e 2 . 9 8 M e V s t a t e w a s f o u n d t o d e c a y t o t h e g r o u n d s t a t e a n d t o

t h e 0 . 4 4 M e V s t a t e w i t h 5 5 a n d 4 5 p e r c e n t , b r a n c h i n g r e s p e c t i v e l y ( F i g .

3 9 ) i n a g r e e m e n t w i t h o t h e r p u b l i s h e d r e s u l t s ( e . g . E n 6 7 ) . O t h e r w o r k

+ + + h a d r e s t r i c t e d t h e s p i n a n d p a r i t y o f t h i s s t a t e t o 3 / 2 a n d 5 / 2 , w i t h 3 / 2

s l i g h t l y f a v o r e d o v e r 5 / 2 . ( P o 6 6) . T h e p a r i t y w a s d e t e r m i n e d f r o m

£ = 2 c h a r a c t e r i z a t i o n o f t h e l e v e l 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

( D u 6 7 , D u 6 9 ) .

2 6 2 3O u r M g ( p , a y ) N a w o r k d e f i n i t e l y a s s i g n s J = 3 / 2 r e j e c t i n g t h e

J = 5 / 2 v a l u e a t 0 . 5 % c o n f i d e n c e l i m i t ( F i g . 6 7 ) . A d i s c r e p a n c y b e t w e e n

o u r m e a s u r e d v a l u e o f t h e m i x i n g r a t i o 5 = + 0 . 01 ± 0 . 02 a n d t h a t o f

P o l e t t i e t a l . § = - 0 . 1 1 ± 0 . 0 2 r e m a i n s u n r e s o l v e d . B o t h t h e s e v a l u e s d i s ­

a g r e e w i t h t h a t o f K h a n e t a l . - 1 . 4 6 < 5 < - 0 . 3 4 .

F r o m a n i n d e p e n d e n t s t u d y o f t h e 2 . 9 8 M e V l e v e l v i a t h e

2 4 2 3M g ( t , o r y ) N a r e a c t i o n a n u n a m b i g u o u s a s s i g n m e n t o f 3 / 2 i s a l s o

d e d u c e d r e j e c t i n g o t h e r v a l u e s a b o v e t h e 0 . 1 % c o n f i d e n c e l i m i t ( F i g . 6 9 ) .

89

I n t h i s w o r k § = - 0 . 1 5 ± 0 . 0 5 w a s f o u n d i n c l o s e r a g r e e m e n t w i t h P o l e t t i ' s

v a l u e b u t s t i l l i n d i s a g r e e m e n t w i t h o u r ( P , q : ) w o r k . T h e a c t u a l v a l u e o f

t h e m i x i n g r a t i o n r e m a i n s u n d e t e r m i n e d , b u t f r o m t h e s e e x p e r i m e n t s i t

m o s t n o t a b l y l i e s b e t w e e n t h e l i m i t s - 0 . 20 < ft < 0 .

T h e J = 5 / 2 + a s s i g n m e n t m a y a l s o b e r e j e c t e d o n t h e b a s i s o f l i f e ­

t i m e i n f o r m a t i o n ( P o 6 9 a ) .

2 3T h e 3 . 68 M e V s t a t e o f N a

A s w a s d i s c u s s e d i n d i s c u s s i o n o f t h e a n a l y s i s o f t h e 2 . 6 4 M e V

2 4 3 2 3l e v e l , a n I = 1 a s s i g n m e n t w a s m a d e t o t h i s l e v e l i n t h e M g ( d , H e ) N a

2 2 3 2 3r e a c t i o n ( D u 6 9 ) a n d i n t h e N e ( H e , d ) N a r e a c t i o n ( D u 6 7 ) w o r k ,

e s t a b l i s h i n g n e g a t i v e p a r i t y a n d a 1 / 2 o r 3 / 2 s p i n a s s i g n m e n t . F r o m o u r

c o r r e l a t i o n w o r k t h e 3 . 68 — 0 . 4 4 M e V i s o t r o p i c r a d i a t i o n y i e l d e d n o n e w

i n f o r m a t i o n , b u t t h e w e a k e r 3 . 6 8 — 2 . 6 4 M e V b r a n c h w a s s t r o n g l y a n i s o ­

t r o p i c c o m p l e t e l y e l i m i n a t i n g t h e l / 2 p o s s i b i l i t y ( F i g . 7 1 ) . C o n c u r r e n t l y

2 4 2 3a J = 3 / 2 a s s i g n m e n t h a s b e e n e s t a b l i s h e d f r o m t h e M g ( t , a y ) N a

r e a c t i o n i n c o m p l e t e a g r e e m e n t w i t h o u t w o r k ( P o 66 ) .

2 3T h e 3 . 8 5 M e V s t a t e o f N a

F r o m t h e N a l ( F i g . 4 1 ) a n d G e ( L i ) g a m m a r a y s p e c t r a ( F i g . 5 0 )

2 6 2 3o b t a i n e d i n t h e M g ( p , o t y ) N a w o r k , t h i s l e v e l w a s f o u n d t o d e c a y 2 9 ± 6 ,

1 8 ± 5 , 4 7 ± 6 , a n d 6 ± 2 % t o t h e g r o u n d , 0 . 4 4 , 2 . 0 8 , a n d 2 . 6 4 M e V s t a t e s .

T h e b r a n c h t o t h e g r o u n d s t a t e w a s i s o t r o p i c r e s t r i c t i n g t h e s p i n o f t h e

90

s t a t e t o J < 5 / 2 ( F i g . 7 5 ) . T o e x t r a c t t h e 3 . 8 5 — 2 . 0 8 M e V "

g a m m a r a y f r o m t h e 2 . 0 8 — 0 . 4 4 M e V t r a n s i t i o n a G a u s s i a n f i t t i n g p r o g r a m

" P S M " ( 0 1 6 9 ) w a s u s e d . T h e r e s u l t i n g a n i s t r o p i c r a d i a t i o n d e f i n i t e l y

r e j e c t e d t h e J = 1 / 2 p o s s i b i l i t y , w h i l e r e t a i n i n g t h e J = 3 / 2 o r 5 / 2 a s s i g n ­

m e n t ( F i g . 7 3 ) . F r o m t h e m e a s u r e d v a l u e o f t h e l i f e t i m e , 0 . 1 6 8 ± 0 . 0 4

p s e c s ( D u 6 9 4 a n d t h e l a r g e m u l t i p o l e m i x i n g r a t i o , 0 . 2 6 ± . 0 7 , w h e t h e r

t h e m i x t u r e i s E 2 / M 3 o r M 2 / E 3 , t h e J = 3 / 2 a s s i g n m e n t i s s a f e l y

r e j e c t e d o n t h e b a s i s t h a t i t w o u l d c o r r e s p o n d t o a m a t r i x e l e m e n t

e n h a n c e m e n t o f 1 0 o v e r t h e W e i s s k o p f v a l u e . T h e J = 5 / 2 a s s i g n m e n t

i s c o n s i s t e n t w i t h t h e e a r l i e r w o r k o f D u b o i s i n w h i c h a t e n t a t i v e 4 = 3

2 2 3 2 3w a s a s s i g n e d t o t h e l e v e l f r o m N e ( H e , d ) N a p u 6 7 ) , , T h e 4 = 3

a s s i g n m e n t i m p l i e d n e g a t i v e p a r i t y , b u t s i n c e t h e 4 = 3 i s n o t d e f i n i t e ,

n e i t h e r i s t h e p a r i t y a s s i g n m e n t . T h e d e - e x c i t a t i o n o f t h e l e v e l i s c o n ­

s i s t e n t w i t h a n e g a t i v e p a r i t y a s s i g n m e n t a n d w i l l b e d i s c u s s e d i n m o r e

d e t a i l i n t h e n e x t s e c t i o n o n i n t e r p r e t a t i o n o f r e s u l t s .

2 3T h e 3 . 9 1 M e V s t a t e o f N a

T h i s l e v e l w a s t o o w e a k l y p o p u l a t e d t o p e r m i t d e v e l o p m e n t o f a n y

r e l i a b l e s p e c t r o s c o p i c i n f o r m a t i o n . I t h a s b e e n a s s i g n e d 4 = 2 f r o m t h e

2 2 3 2 3N e ( H e , d ) N a r e a c t i o n w o r k a n d i t s d e - e x c i t a t i o n b r a n c h i n g r a t i o h a s

b e e n m e a s u r e d ( P o 6 9 ) t o b e 8 1 , 6 , 1 9 , a n d 2 % t o t h e g r o u n d 0 . 4 4 , 2 . 0 8 ,

a n d 2 . 9 8 M e V s t a t e s , r e s p e c t i v e l y .

91

T h i s l e v e l w a s f o u n d t o d e c a y 9 3 ± 5 , 3 ± 2 , a n d < 4 p e r c e n t t o

t h e g r o u n d , 2 . 3 9 , a n d 0 . 4 4 M e V s t a t e s i n a g r e e m e n t w i t h o t h e r w o r k

( E n G 7 ) , e x c e p t f o r t h e l i m i t o n t h e b r a n c h t o t h e 0 . 4 4 M e V s t a t e ( F i g . 4 2 ) .

I t i s k n o w n 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 ( D u 6 7 , D u 6 9 ) t h a t t h i s

77l e v e l i s c h a r a c t e r i z e d b y a n £ = 0 a s s i g n m e n t c o r r e s p o n d i n g t o a J o f

l / 2 + . T h e r a d i a t i o n e m i t t e d b y t h i s l e v e l s h o u l d t h e r e f o r e b e i s o t r o p i c .

T h e L e g e n d r e e x p a n s i o n c o e f f i c i e n t s d e t e r m i n e d b y a l e a s t s q u a r e f i t t o

t h e a n g u l a r d i s t r i b u t i o n a r e a „ / a = - 0 . 0 3 1 ± 0 . 0 2 9 a n d a 7 a = - 0 . 2 1 6 ± 0 . 0 4 6 ,2 o 4 o

f i v e s t a n d a r d d e v i a t i o n s a w a y f r o m i s o t r o p y . T h e a n i s o t r o p y c o u l d n o t

b e f i t t e d b y t r y i n g o t h e r s p i n s f o r t h e 4 . 4 3 l e v e l .

T h i s a n o m a l y s h o u l d n o t b e t a k e n t o o s e r i o u s l y , s i n c e t h e l e v e l

w a s w e a k l y p o p u l a t e d a n d o n l y a b o u t f i v e t i m e s a b o v e b a c k g r o u n d i n t h e

c o i n c i d e n t s p e c t r u m . T h i s b a c k g r o u n d c o u l d r e s u l t f r o m a l o w e n e r g y ,

p a r t i a l l y s t o p p e d , i n e l a s t i c a l l y s c a t t e r e d p r o t o n s f e e d i n g t h e 4 . 4 3 M e V

12s t a t e i n C , w h i c h w o u l d b e i n r e a l c o i n c i d e n c e w i t h t h e o v e r l a p p i n g 4 . 4 3

12M e V g a m m a r a d i a t i o n e m i t t e d b y t h e C l e v e l . S u c h c o n t a m i n a n t r a d i a ­

t i o n c o u l d a c c o u n t f o r t h i s d i s c r e p a n c y , b u t h a s n o t b e e n c o m p l e t e l y c o n ­

f i r m e d .

I n a d d i t i o n a s m a l l p a r t i a l l y r e s o l v e d c o n t a m i n a n t p e a k , a b o u t o n e

f o u r t h t h e s i z e o f t h e 4 . 4 3 g r o u p , w a s o b s e r v e d i n t h e l i r e c t s p e c t r a

( F i g . 3 1 ) . F i ’o m t h e r e l a t i v e k i n e m a t i c s h i f t o f t h e t w o l e v e l s , t h e o r i g i n

2?The 4. 43 M eV state of Na

92

o f t h e s m a l l e r f a s t e r s h i f t i n g g r o u p i s a l i g h t e r c o n t a m i n a n t n u c l e u s a n d

2 3n o t a p r e v i o u s l y u n d i s c o v e r e d l e v e l i n N a . T h e s p e c i f i c s o u r c e o f t h e

p e a k i s u n d e t e r m i n e d .

9 9T h e 4 . 7 8 M e V s t a t e o f N a

F r o m t h e s u m m e d c o i n c i d e n t p h o t o n s p e c t r u m ( F i g . 4 3 ) t h e 4 . 7 8

M e V l e v e l w a s o b s e r v e d t o d e c a y 6 ± 2 , 5 4 ± 2 , 2 5 ± 2 , a n d 1 5 ± 2 % t o t h e

g r o u n d , 0 . 4 4 , 2 . 0 8 , a n d 2 . 7 1 M e V s t a t e s r e s p e c t i v e l y . T h e a n g u l a r

d i s t r i b u t i o n o f t h e 4 . 7 8 ( J ) — 0 . 4 4 ( 5 / 2 + ) M e V t r a n s i t i o n ( F i g . 7 9 ) w a s

i s o t r o p i c , t h u s l i m i t i n g J < 7 / 2 . T h e 4 . 7 8 ( J ) — 2 . 0 8 ( 7 / 2 + ) M e V t r a n s i ­

t i o n ( F i g . 7 8 a n d 8 0 ) a n g u l a r d i s t r i b u t i o n w a s a n i s o t r o p i c e l i m i n a t i n g

J = 1 / 2 . T h e r e m a i n i n g p o s s i b i l i t i e s 3 / 2 < J < 7 / 2 m a y b e r e d u c e d f u r t h e r

b y c o n s i d e r i n g l i f e t i m e i n f o r m a t i o n a n d m u l t i p o l e m i x i n g r a t i o s . T h e

l i f e t i m e o f t h e l e v e l i s l e s s t h a n 0 . 0 4 p i c o s e c o n d s ( D u 6 9 ) a n d w i t h a

m e a s u r e d m u l t i p o l e m i x i n g r a t i o o f 0 . 5 9 f o r t h e 4 . 7 8 ( 3 / 2 ) — 2 . 0 8 ( 7 / 2 )

M e V t r a n s i t i o n , a n a s s i g n m e n t o f J = 3 / 2 w o u l d c o r r e s p o n d t o a n e n h a n c e d

4t r a n s i t i o n s t r e n g t h g r e a t e r t h a n 10 F f o r t h e h i g h e r o r d e r m u l t i p o l ew

( M 3 o r E 3 ) . O n t h i s b a s i s t h e J = 3 / 2 p o s s i b i l i t y m a y b e r e j e c t e d s a f e l y .

T h e r e m a i n i n g p o s s i b i l i t i e s , J = 5 / 2 a n d 7 / 2 , a r e e q u a l l y p r o b a b l e .

2 2 2 3H o w e v e r , g a m m a - r a y s t u d i e s f r o m t h e N e ( p , y ) N a r e a c t i o n b y B r a b e n

e t a l . ( B r 6 2 ) s u g g e s t e d a J = 7 / 2 o r J = 3 / 2 a s s i g n m e n t b y e l i m i n a t i n g

o t h e r v a l u e s a t a 2 0 % c o n f i d e n c e l i m i t . S i n c e t h e J = 5 / 2 a n d J = 3 / 2

a s s i g n m e n t s a r e m u t u a l l y e x c l u s i v e , a l t h o u g h n o t o n a b a s i s a s r i g o r o u s

93

a s d i s c u s s e d p r e v i o u s l y i n s e c t i o n V - 1 3 w e d o f a v o r t h e J = 7 / 2 a s s i g n ­

m e n t o v e r J = 5 / 2 .

23.T h e 5 . 3 8 M e V s t a t e o f N a

T h e 5 . 3 8 M e V l e v e l w a s d e t e r m i n e d t o d e c a y 1 2 ± 2 , 6 3 ± 2 , a n d

2 5 ± 2 t o t h e g T o u n d , 0 . 4 4 , a n d 2 . 0 8 M e V s t a t e s , r e s p e c t i v e l y ( F i g . 4 4 ) .

A n g u l a r c o r r e l a t i o n s w e r e e x t r a c t e d f o r a l l t h r e e t r a n s i t i o n s ( F i j ^ . 8 1 ,

8 2 , a n d 8 3 ) . A l t h o u g h t h e g r o u n d s t a t e t r a n s i t i o n w a s w e a k e s t , i t w a s

m o s t a n i s o t r o p i c a n d i t s a n a l y s i s e l i m i n a t e d a l l s p i n s a t t h e . 1 % l i m i t

e x c e p t J = 3 / 2 a n d 5 / 2 . T h e p o s i t i v e p a r i t y o f t h e s t a t e h a s b e e n

2 2 3 2 3m e a s u r e d b y a n I = 2 s i n g l e p a r t i c l e t r a n s f e r i n t h e N e ( H e , d ) N a

r e a c t i o n ( D u 6 7 ) .

2 3T h e 5 . 5 4 M e V s t a t e o f N a

E x c e p t f o r t h e w o r k p r e s e n t e d h e r e i n , n o o t h e r i n f o r m a t i o n e x c e p t

e x c i t a t i o n e n e r g y h a s b e e n r e p o r t e d o n t h i s s t a t e . I t w a s p o p u l a t e d i n b o t h

23 2 3 2 6 2 3t h e N a ( a , a y ) ^ a a n d t h e M g ( p . a y J N a r e a c t i o n s . T h e b r a n c h i n g

o f t h e l e v e l d e - e x c i t a t i o n d e t e r m i n e d b y a v e r a g i n g t h e r e s u l t s f r o m b o t h

r e a c t i o n s i s 2 4 ± 5 % a n d 7 6 ± 6% t o t h e 2 . 0 8 ( 7 / 2 + ) a n d 2 . 7 0 ( 9 / 2 + ) M e V

2 3l e v e l s , r e s p e c t i v e l y . S p i n p o s s i b i l i t i e s d e t e r m i n e d f r o m t h e N a (a . o / y )

2 3 2 6 2 3N a r e a c t i o n w e r e 5 / 2 < J < 1 1 / 2 ( F i g . 2 5 a n d 2 6 ) . I n t h e M g ( p , o r y ) N a

2r e a c t i o n , t h e ^ c o r r e s p o n d i n g t o a l l s p i n s e x c e p t J = l l / 2 a n d J = 7 / 2

m i n i m i z e d a b o v e t h e 1 0 % c o n f i d e n c e l e v e l ( F i g . 8 5 ) o r a r e f i v e t i m e s

94

l e s s l i k e l y . I n F i g . 86 t h e m e a s u r e d v a l u e s o f t h e L e g e n d r e c o e f f i c i e n t s

a 2 / a 0 a n d a 4 / a Q a r e p l o t t e d t o g e t h e r w i t h t h e t h e o r e t i c a l v a l u e s , a s

f u n c t i o n s o f t h e m i x i n g r a t i o f o r v a r i o u s s p i n s e q u e n c e s . I t f o l l o w s t h a t

b e s t a g r e e m e n t w i t h t h e m e a s u r e d v a l u e o f a a n d a , i n d i c a t e d b y t h eQ Zc r o s s , i s o b t a i n e d i n t h e r e g i o n s d e f i n e d b y t h e l o c i o f t h e J = 1 1 / 2 o r

3 = 1 / 2 a s s i g n m e n t s .

2 3T h e 5 . 7 4 0 M e V s t a t e o f N a

T h e 5 . 7 4 0 M e V s t a t e i s o n e o f a t r i p l e t o f l e v e l s u n r e s o l v e d i n t h e

t o t a l c o i n c i d e n t s p e c t r u m . F r o m t h e 8 k e V r e s o l u t i o n m a g n e t i c s p e c t r o ­

g r a p h d i r e c t a l p h a p a r t i c l e s p e c t r u m ( F i g . 3 2 ) i t w a s f o u n d t h a t t h e o t h e r

t w o m e m b e r s w e r e n e g l i g i b l y p o p u l a t e d i n c o m p a r i s o n t o t h e 5 . 7 4 0 M e V

s t a t e a t t h e i n c i d e n t p r o t o n e n e r g y u s e d h e r e . T h e r e f o r e , w e c a n , p r e ­

s u m a b l e , s a f e l y a n a l y z e t h e o b s e r v e d a l p h a g r o u p a s p o p u l a t i n g a s i n g l e

s t a t e . T h e d e - e x c i t a t i o n b r a n c h i n g ( F i g . 4 6 ) i s 6 3 ± 2 % a n d 3 7 ± 2 % t o

t h e g r o u n d a n d 0 . 4 4 M e V s t a t e s , r e s p e c t i v e l y . S i m i l a r b r a n c h i n g h a s

2 3b e e n o b s e r v e d b y P o l e t t i i n h i s s t u d y o f t h e s a m e l e v e l f r o m t h e N a ( p . p ' y )

2 3 +N a r e a c t i o n ( P o 68) . T h e 5 . 7 4 0 ( J ) — -0 ( 3 / 2 ) M e V t r a n s i t i o n i s i s o ­

t r o p i c e l i m i n a t i n g J > 5 / 2 . E q u i v a l e n t s p i n l i m i t a t i o n s a n d p o s i t i v e p a r i t y

h a v e b e e n a s s i g n e d b y D u b o i 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 w o r k ( D u 6 7 ) .

2 3T h e 5 . 9 2 6 , 5 . 9 6 7 , a n d 6 . 0 4 3 M e V s t a t e s o f N a

T n e s e t h r e e l e v e l s w e r e u n r e s o l v e d i n t h e g a m m a r a y N a l a n g u l a r

c o r r e l a t i o n w o r k . F r o m t h e d i r e c t a l p h a p a r t i c l e s p e c t r u m ( F i g . 3 2 ) t h e

95

f r o m t h e w e a k e r 5 . 9 2 6 l e v e l w e r e n o t d i s c e r n i b l e . T h e 5 . 9 6 7 l e v e l w a s

o b s e r v e d t o d e c a y p r e d o m i n a n t l y t o t h e 2 . 6 4 M e V l e v e l i n t h e N a l ( F i g . 4 7 )

a n d G e ( L i ) ( F i g . 5 3 ) g a m m a r a y s p e c t r a . N o c o r r e l a t i o n s w e r e e x t r a c t e d

2 4 3 2 3f o r t h i s l e v e l . F r o m M g ( d , H e ) N a d a t a ( D u 6 9 ) a n I = 1 s i n g l e p a r t i c l e

t r a n s f e r h a s b e e n i d e n t i f i e d w i t h t h i s l e v e l e s t a b l i s h i n g s p i n p a r i t y o f ( l /2 ,

3 / 2 “ ) .

T e n t a t i v e b r a n c h i n g r a t i o m e a s u r e m e n t s o f t h e 6 . 0 4 3 M e V s t a t e a r e

2 6 ± 4 , (8 ± 3 ) , ( 1 7 ± 1 0 ) a n d 4 9 ± 8% t o t h e 0 . 4 4 , 2 . 7 0 , 3 . 6 8 , a n d 3 . 8 5

M e V s t a t e s , r e s p e c t i v e l y . T h e a n g u l a r d i s t r i b u t i o n o f t h e 6 . 0 4 3 — 3 . 8 5

2M e V t r a n s i t i o n i s m i x e d w i t h t h e 2 . 7 0 c a s c a d e r a d i a t i o n a n d a ^ a n a l y s i s

i s p r e s e n t e d i n F i g . 8 9 , u n d e r t h e a s s u m p t i o n t h a t t h e d o m i n a n t c o m p o n a n t

i s t h e 6 . 0 4 3 — 3 . 8 5 M e V t r a n s i t i o n . N o d e f i n i t i v e r e s u l t s m a y b e e x t r a c t e d

f r o m t h i s f i t .

2 3T h e 6 . 2 0 0 , 6 : 2 3 8 , a n d 6 . 3 1 1 M e V s t a t e s o f N a

T h e 6 . 2 0 0 , 6 . 2 3 8 , a n d 6 . 3 1 1 M e V l e v e l s w e r e u n r e s o l v e d i n t h e

a l p h a p a r t i c l e c o i n c i d e n t s p e c t r a ( F i g . 2 9 ) . F r o m t h e 8 k e V r e s o l u t i o n

m a g n e t i c s p e c t r o g r a p h d i r e c t s p e c t r u m , t h e y i e l d s t o t h e 6 . 2 3 8 a n d 6 . 2 0 0

M e V s t a t e s w e r e a p p r o x i m a t e l y e q u a l a n d a b o u t o n e t h i r d o f t h a t t o t h e

6 . 3 1 1 M e V s t a t e ( F i g . 3 2 ) .

A 6 . 3 1 1 ( J ) — 2 . 3 9 ( 1 / 2 ) M e V t r a n s i t i o n w a s i d e n t i f i e d i n t h e

s u m m e d c o i n c i d e n t p h o t o n s p e c t r u m ( F i g . 4 8 ) a n d i n t h e 9 0 ° G e ( L i ) c o i n -

re la tiv e population of the three le v e ls may be obtained. The gam m a ra y s

96

c i d e n t d a t a ( F i g . 5 3 ) . T h e a n g u l a r d i s t r i b u t i o n w a s i s o t r o p i c a n d t h e

2x a n a l y s i s ( F i g . 9 1 ) y i e l d e d J = l / 2 a n d J = 3 / 2 p o s s i b i l i t i e s f o r t h e s p i n

o f t h e 6 . 3 1 1 M e V l e v e l . H o w e v e r , t h e l e v e l h a s s i n c e b e e n i d e n t i f i e d a s

2 2 3 2 3a n I = 0 t r a n s f e r i n t h e N e ( H e , d ) N a r e a c t i o n ( D u 6 7 ) e s t a b l i s h i n g

t h e s p i n / p a r i t y a s l / 2 + i n a g r e e m e n t w i t h o u r c o r r e l a t i o n w o r k . N o o t h e r

g a m m a r a y b r a n c h e s f r o m t h e 6 . 3 1 1 l e v e l w e r e c l e a r l y d i s c e r n i b l e .

A s t r o n g g a m m a b r a n c h w a s o b s e r v e d i n t h e N a l s u m m e d c o i n c i d e n t

p h o t o n s p e c t r u m ( F i g . 4 8 ) f r o m o n e o r b o t h o f t h e 6 . 2 0 0 a n d 6 . 2 3 8 M e V

l e v e l s t o t h e 9 / 2 + l e v e l a t 2 . 7 0 M e V a n d t o t h e 5 / 2 + l e v e l a t 0 . 4 4 M e V .

I n t h e G e ( L i ) s p e c t r a t h e t r a n s i t i o n 6 . 2 3 8 ( J ) — 2 . 7 0 ( 9 / 2 + ) M e V w a s

o b s e r v e d a n l a p o s s i b l e 6 . 2 0 0 ( J ) — 2 . 7 0 ( 9 / 2 + ) M e V t r a n s i t i o n ( F i g . 5 2

a n d 5 3 ) . A s s u m i n g t h a t t h e l a t t e r t r a n s i t i o n e x i s t s , t h e d e - e x c i t a t i o n o f

t h e 6 . 2 0 0 M e V s t a t e i s e s t i m a t e d t o b e ( 5 0 ± 2 5 ) % t o b o t h t h e 2 . 7 0 a n d 0 . 4 4

M e V s t a t e s . T h e 6 . 2 3 8 M e V s t a t e i s o b s e r v e d t o d e c a y t o t h e 2 . 7 0 M e V

s t a t e o n l y . S i n c e b o t h t h e 6 . 2 0 0 a n d 6 . 2 3 8 M e V s t a t e s a r e e q u a l l y

p o p u l a t e d ( F i g . 3 2 ) a n d f r o m t h e a b o v e b r a n c h i n g r a t i o e s t i m a t e s , t h e

g a m m a r a y a t 3 . 5 0 M e V i n t h e N a l s p e c t r a i s m i x e d w i t h 7 0 % o f t h e

6 . 2 3 8 — 2 . 7 0 M e V t r a n s i t i o n a n d a t m o s t 3 0 % o f t h e 6 . 2 0 0 — 2 . 7 0 M e V

t r a n s i t i o n . U s i n g t h e l i n e s h a p e o f t h e 4 . 4 3 M e V g a m m a r a y , t h e i s o ­

t r o p i c 3 . 9 1 M e V r a d i a t i o n e m i t t e d i n t h e 6 . 3 1 1 — 2 . 3 9 M e V t r a n s i t i o n

w a s s u b t r a c t e d . B y f i t t i n g t h e a n g u l a r d i s t r i b u t i o n o f t h e s u m m e d t r a n s i ­

t i o n t o a n e x p a n s i o n i n L e g e n d r e p o l y n o m i a l s , a 2 a Q a n c * a 4 / a0 c o e f f i c i e n t s

w e r e f o u n d t o b e + 0 . 1 9 6 ± 0 . 0 7 8 a n d - 0 . 3 8 3 ± 0 . 1 2 6 , r e s p e c t i v e l y . T h e '

97

n e g a t i v e a4 / a0 c o e f f i c i e n t i m m e d i a t e l y r e s t r i c t s t h e s p i n o f o n e o r b o t h o f '

ot h e e m i t t i n g s t a t e s t o J = 9 / 2 o r 1 3 / 2 . T h e ^ , M e t h o d I I a n a l y s i s o f t h e

t r a n s i t i o n i s s h o w n i n F i g . 9 0 r i g o r o u s l y l i m i t i n g t h e s p i n t o J = 9 / 2 o r

1 3 / 2 a t t h e . 1 % l i m i t .

S i n c e t h e 6 . 2 3 8 ( J ) — 2 . 7 0 ( 9 / 2 ) M e V t r a n s i t i o n i s a t l e a s t 7 0 % o f

t h e g a m m a r a y u n d e r a n a l y s i s , i t i s p r o b a b l e t h a t i t d o m i n a t e s t h e a n i s o -

t r o p i c i t y o f t h e a n g u l a r d i s t r i b u t i o n r a t h e r t h a n t h e w e a k e r 6 .2 0 0 — 2 . 7 0

M e V t r a n s i t i o n . T h i s b e i n g t h e c a s e , t h e s p i n a s s i g n m e n t s J = 1 3 / 2 a n d

9 / 2 a r e t h e p o s s i b i l i t i e s f o r t h e 6 . 2 3 8 M e V r a t h e r t h a n t h e 6 . 2 0 0 M e V

s t a t e . T h e s l i g h t d e p a r t u r e o f t h e 1 3 / 2 — 9 / 2 m i x i n g r a t i o f r o m z e r o i s

p r o b a b l y t h e r e f l e c t i o n o f a n u n a c c o u n t e d f o r t i l t i n t h e a n g u l a r d i s t r i b u ­

t i o n s t e m m i n g e i t h e r f r o m b a c k g r o u n d s u b t r a c t i o n o r f r o m t h e u n d e r l y i n g

6 . 2 0 0 — 2 . 7 0 M e V t r a n s i t i o n . A 6 . 2 3 8 ( 1 3 / 2 , 9 / 2 ) — 5 . 5 4 ( 1 1 / 2 ) M e V

t r a n s i t i o n m i g h t b e e x p e c t e d , b u t i s e n e r g e t i c a l l y i n h i b i t e d i n c o m p a r i s o n

t o t h e 6 . 2 3 8 — 2 . 7 0 M e V t r a n s i t i o n . A n u p p e r l i m i t o f 1 0 % i s e s t i m a t e d

f o r t h e w e a k e r u n r e s o l v e d 6 . 2 3 8 — 5 . 5 4 M e V t r a n s i t i o n .

2 3T h e 6 . 5 8 4 M e V s t a t e o f N a

T h e 6 . 5 8 4 M e V s t a t e w a s s t r o n g l y p o p u l a t e d a n d i t s w e a k e r

p o p u l a t e d u n r e s o l v e d n e i g h b o r ( F i g . 3 2 ) m a y b e n e g l e c t e d . T h e d e c a y o f

t h e 6 . 5 8 4 M e V s t a t e i s 4 5 ± 4 , 3 6 ± 3 , a n d 1 8 ± 3 % t o t h e 0 . 4 4 , 2 . 0 8 , a n d

2 . 7 0 M e V s t a t e s , r e s p e c t i v e l y ( F i g . 4 9 ) . T h e a n a l y s i s o f t h e a n g u l a r

d i s t r i b u t i o n f o r t h e 6 . 5 8 4 ( J ) — 2 . 0 8 ( 7 / 2 ) M e V t r a n s i t i o n i s s h o w n i n F i g .

98

9 2 l i m i t i n g 3 / 2 < J < 9 / 2 . T h e r e s u l t s o f t h e 6 . 5 8 4 ( J ) — . 4 4 ( 5 / 2 ) M e V

t r a n s i t i o n s h o w n i n F i g . 9 3 a r e m o r e r e s t r i c t i n g , l i m i t i n g J t o 5 / 2 o r

9 / 2 .

NUMBER OF COUNTS X NUMBER OF COUNTS

.439(J) —►O (3/2)

2Q8(J)— 44(5/2)

M

OCD5ro

ro

AAuiVro

NUMBER OF COUNTS5C

2 39<J)—-0(3/2)

MIXING RATIO (8)

NUMBER OF COUNTS NUMBER OF COUNTS

NUMBER OF COUNTS

No23 No*®

MIXING RATIO (S)FJB. 72

NUMBER OF COUNTS

NUMBER OF COUNTSN Ol A Ul (T) •vlOo

NUMBER OF COUNTS

2xzo73>-jOOJ

oo oo—r~o*oo

NUMBER OF COUNTS

O s

oCD

U/DOOZ

—(DS8C

se°n

N U M B E R .O F CO UNTSro a <j>o o oo o oNUMBER OF COUNTS

zoAv.ro

4.78 (J) — .44 (5/2)

478(J)—• 208(7/2)

MIXING RATIO (8)

NUMBER OF COUNTSu» o. ooI I

A 4 / A o

X NUMBER OF COUNTS

<2/6>

uz—

iDfrtf

i (z/

tio—

(net

s

NUMBER OF COUNTS

SXzo3)>HO5?

No23 No23

5 74(J)——.44 (5/2)

5 740(J) —0(3/2)

MIXING RATIO

(8)

No236.584 (J) —.44(5/2)

2XzoX>Ho

6 3H(J)—2.39(1/2)

6 584 (J) 2 08 (7/2)

99

V I I I . S U M M A R Y O F R E S U L T S O N O D D C O U N T £ = 1 1 N U C L E I

a xr 23A . N a

A s u m m a r y o f e x c i t a t i o n e n e r g i e s , s p i n / p a r i t y a s s i g n m e n t s ,

2 3b r a n c h i n g r a t i o s , a n d l i f e t i m e s a r e s h o w n f o r N a i n t h e e n e r g y l e v e l

d i a g r a i n s h o w n i n F i g . 9 4 .

T h e l e v e l s o f e x c i t a t i o n u p t o 5 . 5 3 8 M e V , i n c l u s i v e l y , a r e l a b e l l e d

w i t h e x c i t a t i o n e n e r g i e s d e t e r m i n e d f r o m o u r G e ( L i ) c o i n c i d e n t d a t a

w i t h a n e x p e r i m e n t a l e r r o r o f a p p r o x i m a t e l y 3 . 0 k e V . L e v e l s o f h i g h e r

e x c i t a t i o n a r e l a b e l l e d w i t h e n e r g i e s d e t e r m i n e d f r o m t h e m a g n e t i c

s p e c t r o g r a p h d a t a s h o w n e a r l i e r i n F i g . 3 2 . E x c i t a t i o n e n e r g i e s d e t e r ­

m i n e d h e r e a r e i n a g r e e m e n t w i t h t h o s e d e t e r m i n e d f r o m o t h e r w o r k

( E n 6 7 , P o 6 9 a ) . N o e v i d e n c e w h a t s o e v e r w a s f o u n d f o r t h e e x i s t e n c e o f

t h e t w o l o w l y i n g l e v e l s a t 2 . 4 1 a n d 2 . 8 7 M e V a s s u g g e s t e d b y L a n c m a n

2 3f r o m t h e b e t a d e c a y o f N e ( L a 6 5 ) .

S p i n / p a r i t y a s s i g n m e n t s w e r e m a d e o n t h e b a s i s o f o u r a n g u l a r

c o r r e l a t i o n r e s u l t s c o m b i n e d w i t h l i f e t i m e i n f o r m a t i o n a n d d e f i n i t i v e I

v a l u e a s s i g n m e n t s . A l e v e l s h o w n i n t h e d i a g r a m t h a t i s c h a r a c t e r i z e d

b y a s i n g l 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 w i t h o u t p a r e n t h e s e s i s

c o n s i d e r e d a u n i q u e a s s i g n m e n t a s d i s c u s s e d p r e v i o u s l y ; u s e o f

p a r e n t h e s e s i m p l i e s t h a t t h e a s s i g n m e n t i s n o t c o m p l e t e l y c e r t a i n , b u t

t h e m o s t p r o b a b l e v a l u e . F o r t h o s e l e v e l s n o t u n i q u e l y l a b e l l e d , q u a n t u m

n u m b e r s w i t h o u t p a r e n t h e s e s i n d i c a t e s u c h q u a n t i t i e s a r e t h e m o s t

p r o b a b l e a s s i g n m e n t s o n t h e b a s i s o f s y s t e m a t i c s , m o d e l d e p e n d e n t a r g u -

100

m e n t s , e t c .

T h e t a b u l a t e d b r a n c h i n g r a t i o s i n t h e e n e r g y l e v e l d i a g r a m h a v e

a l l b e e n d e t e r m i n e d f r o m o u r w o r k e x c e p t f o r t h o s e f o r t h e 3 . 9 1 2 M e V

l e v e l . O u r r e s u l t s a r e a l s o i n a g r e e m e n t w i t h t h o s e p r e v i o u s l y r e p o r t e d

e x c e p t i n t h e c a s e o f t h e 3 . 8 5 1 M e V l e v e l . T h e r e s u l t s e n c l o s e d b y

p a r e n t h e s e s a r e p r e l i m i n a r y a n d t h e s p e c i f i c n u m e r i c a l v a l u e m a y b e

q u e s t i o n a b l e .

T h e l i f e t i m e s , i n u n i t s o f p i c o s e c o n d s , a r e l i s t e d i n t h e f a r l e f t

c o l u m n a n d a r e a v e r a g e d v a l u e s t a k e n f r o m t h e w o r k o f M a i e r e t a l . ( M a 6 8 ) ,

P o l e t t i e t a l . ( P o 6 9 ) , a n d D u r r e l l e t a l . ( D u 6 9 a . ) . S e e R e f . P r 6 9 b .

21B . N a

I n F i g s . 9 5 , 9 6 , a n d 9 7 a r e e n e r g y l e v e l d i a g r a m s s u m m a r i z i n g t h e

2 1 2 1 2 3e x p e r i m e n t a l l y k n o w n i n f o r m a t i o n o n N a , N e , a n d M g . T h e s a m e

2 3c o n v e n t i o n s a r e u s e d i n t h e s e d i a g r a m s a s w a s u s e d i n t h e c a s e o f N a .

21I n N a l o w l y i n g e x c i t a t i o n e n e r g i e s w e r e t a k e n f r o m t h e G e ( L i )

20 21w o r k o f - R o l f s e t a l . ( R o 6 9 ) a n d h i g h e r l y i n g v a l u e s f r o m N e ( p , y ) N a

w o r k o f B l o c b e t a l . ( B 1 6 9 ) a n d t h e r e m a i n i n g v a l u e s f r o m t h e c o m p l i a t i o n

o f E n d t a n d V a n d e r L e u n ( E n 6 7 ) . B r a n c h i n g r a t i o s a n d s p i n / p a r i t y a s s i g n ­

m e n t h a v e a l s o b e e n d e t e r m i n e d b y t h e s e a u t h o r s . T h e m o s t p r e c i s e w o r k

o n t h e l e v e l s b e l o w 3 . 0 M e V h a s b e e n d o n e b y P r o n k o e t a l . ( P r 6 9 c ) .

i n c l u d i n g r i g o r o u s s p i n a s s i g n m e n t s , m u l t i p o l e m i x i n g r a t i o s , a n d b r a n c h ­

i n g r a t i o s ( s e e A p p e n d i x V I I ) . T h e r e s u l t s o f t h i s w o r k a r e s h o w n i n t h e

e n e r g y l e v e l d i a g r a m f o r t h e c o r r e s p o n d i n g l e v e l s b e l o w 3 . 0 M e V . T h e l i f e ­

t i m e o f t h e f i r s t e x c i t e d s t a t e i s g i v e n b y E n d t a n d V a n d e r L e u n ( E n 6 7 ) a n d

t h a t o f t h e s e c o n d e x c i t e d s t a t e h a s b e e n r e c e n t l y m e a s u r e d b y A n t t i l a e t a l . ( A n 6 9 ) .

„ 2 1 , 2 3C . N e a n d M g

21I n N e t h e e x c i t a t i o n e n e r g i e s b e l o w 5 . 0 M e V a r e l a b e l l e d a c c o r d ­

i n g t o t h e G e ( L i ) w o r k o f P r o n k o e t a l . ' ( P r 6 9 ) a n d a b o v e a c c o r d i n g t o t h e

m a g n e t i c s p e c t r o g r a p h w o r k o f F r e e m a n ( F r 6 0 ) .

S p i n a n d p a r i t y a s s i g n m e n t s a r e f r o m t h e w o r k o f m a n y a u t h o r s

( P r 6 9 , H o 6 9 , P r 6 7 , E n 6 7 ) a n d b r a n c h i n g r a t i o s a n d l i f e t i m e m e a s u r e ­

m e n t s a r e f r o m t h e w o r k o f P r o n k o e t a l . ( P r 6 9 ) .

L e v e l s o f e x c i t a t i o n , b r a n c h i n g r a t i o s , a n d s p i n / p a r i t y a s s i g n m e n t

2 3i n M g w e r e t a k e n f r o m t h e c o m p i l a t i o n o f E n d t e t a l . ( E n 6 7 ) , D u b o i s

e t a l . ( D u 6 7 ) , a n d K o z u b ( K o 68 ) .

W i t h t h i s a b u n d a n t a m o u n t o f e x p e r i m e n t a l i n f o r m a t i o n n o w a v a i l -

2 1 2 3a b l e o n t h e £ = 1 1 n u c e l i , i n p a r t i c u l a r N e a n d N a , w e c a n n o w

e x a m i n e i n d e t a i l m o d e l i n t e r p r e t a t i o n s f o r t h e s e n u c l e i , f o c u s s i n g o u r

a t t e n t i o n o n t h e g r o u n d s t a t e r o t a t i o n a l b a n d s .

101

uuui —• - 13/2* f»(9/2)I v U l h J luJJ4 j ' t O l H ”1 1T.O-- u/2,5/2/V.Jv 1 I - - 1/2 ,3/25.740 --63t2“37"t2~" ~ “76- C 11/2<+|(9/2,7/2,5/2)

2-25“ 5/2 ,(3/2‘r)

1(.0003)4.430 --93±5—s1 < .... .. (5/2‘f)1 1 2 0/d

| :2—37(.25) 2.640 ----1C y 3/ L.

1 1 It c

f| eft) 4 4 0

.K z 3Fig. 94

(0.045)

(14.0)

6.51 (3/2,5/2)+6.246.005.835.825.695.464 99 —4.84 —4.47 —4.42 —4.29 —4.15 --3.8643.6803.544

2.833 2 8102.432

1.723

37---63--36 — 64-4 —78- -16-----96---

:6-64

---- ---- -------------------------

--- 7 :2-93±2---

.3320

(5/2,7/2)' 3/2*1/2*(3/2,5/2)“3/2+(3/2",5/2') 5/2 +3/2"5/2’3/2“5 / 2*

9/2t*\(5/2t+>) 1/2*,3/2"1/2"*

7/2(+,,(3/2l+l)

5/2*3/2*

mK oFig. 95

6 748 6 6476.606 6.555 6.4 506 267 6 I 766036 5.9 975822 5.7 7 7 5 694 5 632 5550 5434 5 338K045) 4 7 30 (< 056) 4687(.078) 4.43 I(.078)(<.067)(.12)

(.15)

(22.)

-<50—>50-

3 8863.664

(I) 2866(<.045) 2.7 9 6 (110.) 2.790

1.747

--- <3---81ID 2 L c. D J

..j "• o

D---9

5 ■

5---3

3/2+,5/2+l3/2<+>,(9/2)

3/2+,5/2+ 1/2",3/2” 3/2+,5/2+ 3/2+,5/2+3/2“3/2+, 5/2*3/2+,5/2+ll/2+,(7/2)3/2.5/25/2+,(3/2+)3/2’

9/2'+>1/2 +I/2-.3/2'

7/2(+)

.3500

5/2* 3/2 +

N e21 10 C IIFig. 96

6.540 ---------------------------------------------------------6.508 ---------------------------------------------------------6.444 ---------------------------------------------------------6 379 ---------------------------------------------------------6.238 ---------------------------------------------------------6.194 ---------------------------------------------------------6 128 ---------------------------------------------------------5 984 --------------------------------------------------------- I/2-.3/2-5 931 ---------------------------------------------------------5706 ---------------------------------------------------------5.686 ---------------------------------------------------------5.651 ---------- :----------------------------------------------5450 ---------------------------------------------------------5.284 --------------------------------------------------------- 3/2+,5/2+

4.682 ---------------------------------------------------------4.360 -----100------------------------------------------------- 1/2+

Fig. 97

102

W e h a v e p r e s e n t e d i n t h e l a s t s e c t i o n a s u m m a r y o f t h e k n o w n s t a t i c

a n d d y n a m i c e l e c t r o m a g n e t i c p r o p e r t i e s o f t h e o d d c o u n t £ = 11 n u c l e i .

H e r e i n t h i s s e c t i o n w e a t t e m p t t o a c c o u n t f o r t h e a c t u a l n u m e r i c a l v a l u e s

a n d s y s t e m a t i c s o f s u c h p r o p e r t i e s i n t e r m s o f p a r a m e t e r s c h a r a c t e r i s t i c

o f t h e c o l l e c t i v e r o t a t i o n a l m o d e l s .

I n p a r t i c u l a r w e f o c u s o u r a t t e n t i o n o n t h o s e p a r a m e t e r s w h i c h a r e

m o s t d e f i n i t i v e a n d c a n b e m o s t r e l i a b l y a n d e a s i l y o b t a i n e d f r o m t h e s i m p l e s t

a s p e c t s o f c o l l e c t i v e m o t i o n ; t h e s e p a r a m e t e r s a r e t h e m o m e n t o f i n e r t i a

I , t h e n u c l e a r d e f o r m a t i o n t h e r e l a t e d i n t r i n s i c q u a d r u p o l e m o m e n t Q q ,

a n d t h e c o r e a n d s i n g l e p a r t i c l e g y r o m a g n e t i c r a t i o g ^ a n d g , r e s p e c t i v e l y .

W e u s e f i r s t t h e r o t o r m o d e l w i t h o u t C o r i o l i s m i x i n g t o f i t

e x c i t a t i o n e n e r g i e s , e l e c t r o m a g n e t i c d e - e x c i t a t i o n b r a n c h i n g r a t i o s , a n d

r e l a t i v e m u l t i p o l e m i x i n g r a t i o s t o d e t e r m i n e t o w h a t e x t e n t t h i s s i m p l e

r o t a t i o n a l m o d e l w i l l r e p r o d u c e t h e e x p e r i m e n t a l d a t a . T h e s e c o m p a r i s o n s

a l l o w u s t o d e t e r m i n e t h e m o m e n t o f i n e r t i a p a r a m e t e r r e f l e c t i n g t h e

r i g i d i t y o f t h e n u c l e u s , a n d t h e r e m a i n i n g a f o r e m e n t i o n e d r o t a t i o n a l

p a r a m e t e r s d e t e r m i n i n g t o w h a t e x t e n t t h e r o t a t i o n a l b a n d c a n b e u n i q u e l y

d e f i n e d ; i n p a r t i c u l a r w e d e t e r m i n e t h e c o n s t a n c y o f t h e r a t i o Q / (g " S ^ ) -

S e c o n d l y , c o r r o b o r a t o r y e v i d e n c e i s o b t a i n e d b y s e p a r a t e l y d e t e r ­

m i n i n g t h e i n t r i n s i c q u a d r u p o l e m o m e n t Q q a n d g y r o m a g n e t i c r a t i o s

g p ^ g j ^ f r o m v a r i o u s i n t r a b a n d t r a n s i t i o n s b y c o m p a r i n g t h e c a l c u l a t e d

e l e c t r i c q u a d r u p o l e a n d m a g n e t i c d i p o l e m a t r i x e l e m e n t s w i t h e x p e r i m e n t a l

IX . M O D E L IN T E R P R E T A T IO N S O F O DD C O U N T £ = 11 N U C L E I

103

v a l u e s . T h e e x p e r i m e n t a l v a l u e s o b v i o u s l y d e p e n d d i r e c t l y o n l i f e t i m e

m e a s u r e m e n t s a n d m u l t i p o l e m i x i n g r a t i o s , w h i c h u s u a l l y a r e b e s e t w i t h

l a r g e u n c e r t a i n t i e s p r e v e n t i n g c o m p l e t e l y u n a m b i g u o u s d e d u c t i o n s .

T h i r d l y , h i g h e r o r d e r r o t a t i o n a l a n d C o r i o l i s p e r t u r b a t i o n s a ^ e

c o n s i d e r e d a n d t h e i r i n f l u e n c e o n t h e g r o u n d s t a t e r o t a t i o n a l b a n d e x c i ­

t a t i o n s p e c t r a i s c a l c u l a t e d . T h e s e r e s u l t s g a v e u n s a t i s f a c t o r y f i t s t o

t h e d a t a .

A m o r e s u c c e s s f u l a p p r o a c h w a s t o f i t a l l p o s i t i v e p a r i t y l e v e l s

w i t h t h e f u l l C o r i o l i s c o u p l i n g m o d e l o f M a l i k a n d S c h o l z h o p e f u l l y d e t e r ­

m i n i n g s d s h e l l w a v e f u n c t i o n a d m i x t u r e s i n t h e g r o u n d s t a t e r o t a t i o n a l

2 1 2 1 2 3b a n d s i n N e , N a , a n d N a . A d m i x t u r e s a r e s m a l l b u t m e a s u r a b l e .

E l e c t r o m a g n e t i c p r o p e r t i e s a r e a g a i n c a l c u l a t e d w i t h t h e w a v e f u n c t i o n

a d m i x t u r e s i n c l u d e d a n d c o m p a r e d t o t h e p r e v i o u s u n m i x e d c a l c u l a t i o n s t o

d e t e r m i n e w h e t h e r o r n o t t h e s e a d m i x t u r e s i m p r o v e a g r e e m e n t b e t w e e n

t h e o r y a n d e x p e r i m e n t .

T o f u r t h e r c o r r o b o r a t e t h a t t h e s e s t a t e s o f t h e g r o u n d s t a t e

r o t a t i o n a l b a n d p o s s e s s w a v e f u n c t i o n a d m i x t u r e s f r o m o t h e r c o n f i g u r a t i o n s

d e s p i t e t h e e x c e l l e n t a g r e e m e n t b e t w e e n e x p e r i m e n t a n d t h e c a l c u l a t e d M l

a n d E 2 b r a n c h i n g r a t i o s f r o m p u r e c o n f i g u r a t i o n s , t h e a n o m a l o u s e l e c t r i c

d i p o l e d e - e x c i t a t i o n o f t h e J 77 = 3 / 2 " m e m b e r o f t h e K 77 = l / 2 r o t a t i o n a l

b a n d w a s q u a n t i t a t i v e l y e x p l a i n e d o n t h e b a s i s o f t h e s e a d m i x t u r e s . T h i s

i s a r a t h e r s e n s i t i v e t e s t o f w a v e f u n c t i o n a d m i x t u r e s , s i n c e t h e E l

t r a n s i t i o n s t o t h e d o m i n a n t K 77^ 3 / 2 + c o m p o n e n t i s i n h i b i t e d b y t h e A l a g a

104

N o w t h a t w e h a v e e s t a b l i s h e d t o s o m e d e g r e e t h e p u r i t y o f t h e

g r o u n d s t a t e r o t a t i o n a l b a n d a n d t h e a p p r o x i m a t e n u m e r i c a l v a l u e s a n d

s y s t e m a t i c s o f t h e r o t a t i o n a l f i t t i n g p a r a m e t e r s w i t h i n t h e f r a m e w o r k o f t h e

c o l l e c t i v e m o d e l , i t i s o f i n t e r e s t t o c h e c k t h e r e l i a b i l i t y i n d e t e r m i n i n g

t h e d e f o r m a t i o n p a r a m e t e r £ f r o m t h e e l e c t r o m a g n e t i c p r o p e r t i e s a n d t h e

s e n s i t i v i t y o f s u c h p r o p e r t i e s t o W e a c c o m p l i s h t h i s b y c a l c u l a t i n g t h e

M l a n d E 2 r e d u c e d m a t r i x e l e m e n t s w i t h a s t r o n g c o u p l i n g N i l s s o n m o d e l

u n m i x e d w a v e f u n c t i o n a s a f u n c t i o n o f t h e d e f o r m a t i o n , p a r a m e t e r .

B e c a u s e m o d e l s o t h e r t h a n t h e c o l l e c t i v e h a v e n o t b e e n e x t e n s i v e l y

a p p l i e d t o n u c l e i i n t h i s r e g i o n , c o m p a r i s o n s o f t h e o r y t o t h e e x p e r i m e n t a l

d a t a a r e l i m i t e d a n d a t h o r o u g h s t u d y o f t h e s e o t h e r m o d e l s a s t h e y

p e r t a i n t o t h i s m a s s r e g i o n a s o u t l i n e d i n t h e i n t r o d u c t i o n i s b e y o n d o u r

s c o p e a t t h i s t i m e . H o w e v e r , a l i m i t e d c o m p a r i s o n o f e x c i t a t i o n s p e c t r a

a n d t h e i r e l e c t r o m a g n e t i c d e - e x c i t a t i o n p r o p e r t i e s b e t w e e n t h e o r y a n d

2 1 2 3e x p e r i m e n t f o r N e a n d N a i n t e r m s o f t h e s h e l l a n d S U m o d e l s i sOp r e s e n t e d .

T h e r e m a i n i n g s h e l l d i s c r e p a n c i e s b e t w e e n o u r c o l l e c t i v e t r e a t ­

m e n t a n d e x p e r i m e n t , a n d m o r e i m p o r t a n t l y t h e m a g n i t u d e a n d s y s t e m a t i c s

o f t h e r o t a t i o n a l p a r a m e t e r s a s w e l l a s t h e d e g r e e o f c o n f i g u r a t i o n a l

m i x i n g i n t h e r o t a t i o n a l b a n d s o f t h e £ = 11 n u c l e i a r e a r e s u l t o f t h e

m i c r o s c o p i c c o n s t i t u t i o n a n d p r o p e r t i e s o f t h e n u c l e u s .

asym ptotic se lectio n ru le s .

105

A „ C o l l e c t i v e M o d e l I n t e r p r e t a t i o n o f t h e K 77 = 3 / 2 + G r o u n d S t a t e R o t a t i o n a l

B a n d

1. R o t o r m o d e l p r e d i c t i o n s w i t h N i l s s o n i n t r i n s i c w a v e f u n c t i o n s

I n t h e s p i r i t o f d e t e r r i i n i n g t h e l i m i t a t i o n s o f t h e s i m p l e

c o l l e c t i v e m o d e l a n d t h e d e g r e e t o w h i c h n u c l e a r b e h a v i o r i s r e p r o d u c i b l e

i n t e r m s o f i t s s i m p l e s t c o n c e p t s , n u c l e a r s t r u c t u r e p r o p e r t i e s o f t h e

g r o u n d s t a t e r o t a t i o n a l b a n d m e m b e r s o f t h e o d d c o u n t £ = 11 n u c l e i h a v e

b e e n i n t e r p r e t e d i n t e r m s o f t h e r i g i d r o t o r m o d e l a s a n i n t r o d u c t i o n t o a

m u c h m o r e e x t e n s i v e s y s t e m a t i c s t u d y o f t h e s e n u c l e i . I n S e c t i o n I i t w a s

n o t e d t h a t t h e s e n u c l e i p o s s e s s v e r y s i m i l a r , a n d m a r k e d , r o t a t i o n a l

s t r u c t u r e , a n d t h u s s e r v e a s a n i d e a l s t a r t i n g p o i n t f o r s u c h a n i n v e s t i g a t i o n .

A c o m p a r i s o n o f t h e m e a s u r e d e x c i t a t i o n e n e r g i e s s p i n / p a r i t y

a s s i g n m e n t s , d e - e x c i t a t i o n b r a n c h i n g r a t i o s , a n d m u l t i p o l e m i x i n g r a t i o s

2 1 2 1 2 3 2 3f o r t h e k n o w n b a n d m e m b e r s o f N e , N a , N a , a n d M g w a s s h o w n

i n F i g . 8. I t i s c l e a r f r o m t h e r o t a t i o n a l s e q u e n c e o f e n e r g y l e v e l s , s p i n s ,

a n d i n t r a b a n d e l e c t r o m a g n e t i c d e - e x c i t a t i o n p r o p e r t i e s t h a t r o t a t i o n a l

b e h a v i o r d o m i n a t e s i n t h e s e n u c l e i . W h i l e t h e o v e r a l l c o l l e c t i v e b e h a v i o r '

e x h i b i t e d b y t h e s e n u c l e i i s s t r i k i n g l y s i m i l a r , c l o s e r e x a m i n a t i o n r e v e a l s

s y s t e m a t i c a n d i n t e r e s t i n g d i s s i m i l a r i t i e s o n w h i c h w e n o w f o c u s a t t e n t i o n .

T h e m o s t o b v i o u s o f t h e s e d i s s i m i l a r i t i e s a r e d e p a r t u r e s i n t h e e x c i t a t i o n

2 3e n e r g i e s f r o m t h e J ( J + 1 ) r i g i d r o t o r r u l e f o r t h e m i r r o r p a i r N a a n d

106

2 3 2 1 2 1M g i n c o m p a r i s o n t o t h e b e t t e r a g r e e m e n t i l l u s t r a t e d b y N e a n d N a

i n F i g . 3 . N o r m a l l y , s u c h d e p a r t u r e s i n e x c i t a t i o n e n e r g i e s f r o m t h o s e •

p r e d i c t e d b y a g i v e n m o d e l a r e a c c o m p a n i e d b y e v e n l a r g e r d e p a r t u r e s i n

t h e p r e d i c t i o n s o f e l e c t r o m a g n e t i c d e - e x c i t a t i o n p r o p e r t i e s , s i n c e s u c h

p r o p e r t i e s c h a r a c t e r i s t i c a l l y h a v e a m o r e s e n s i t i v e w a v e f u n c t i o n d e p e n ­

d e n c e . A n e x a m i n a t i o n o f s u c h p r e d i c t i o n s u s i n g a r i g i d r o t o r m o d e l i s

g i v e n b e l o w .

a . E x c i t a t i o n e n e r g i e s , b r a n c h i n g , a n d m i x i n g r a t i o s

T h e e x c i t a t i o n e n e r g y E o f l e v e l s c o n s t i t u t i n g a r o t a -J

t i o n a l b a n d w i t h K / 1 / 2 i s g i v e n b y

ej = I r ■w h e r e I i s t h e m o m e n t o f i n e r t i a , J t h e t o t a l a n g u l a r m o m e n t u m , a n d K t h e

p r o j e c t i o n o f J o n t h e s y m m e t r y a x i s .

F o r e l e c t r o m a g n e t i c t r a n s i t i o n s w i t h i n a r o t a t i o n a l b a n d t h e t o t a l

m a g n e t i c d i p o l e a n d e l e c t r i c q u a d r u p o l e d e - e x c i t a t i o n p r o b a b i l i t i e s f o r a

g i v e n l e v e l a r e g i v e n b y ( P r 6 2 )

4 E 3 1 2 2 , ^ 2 ( J - K H J + K )T ( M 1 > = ? < 7 5 : > < p “ o < % - % > ^ j j i j f c i ) •

w h e r e , t h e n u c l e a r B o h r m a g n e t o n , i s d e f i n e d a s

, = e f t —”o 2mc ’a n d

T(E2> = ^ < ^ > 5 ( e2<5o <J2K° lJ ’K>2

107

( J 2 K O I J ' K )2 = H Q 2 L 3 i g z j a £ t j 9 t i - j i( J 2 K O j J K ) ( J _ 1 ) ( J ) ( 2 J + 1 ) ( J + 1 ) >

and the C le b sch -G o rd a n coefficients a re given a lg e b ra ic a lly as

( J 2 K O LT' K )2 = ( J + K - l ) _( J 2 K O J J K ) - ( 2 J _ 2 ) ( 2 J _ 1 ) ( J ) ( 2 J + 1 ) . - J - 2 .

T h e m u l t i p o l e m i x i n g r a t i o 6 f o r a n i n t r a b a n d t r a n s i t i o n f r o m a

s t a t e w i t h s p i n J t o a s t a t e w i t h s p i n J - l , d e f i n e d b y t h e e q u a t i o n

2 = T ( E 2 )6 T ( M 1 ) ’

i s g i v e n i n t h e s i m p l e r o t o r m o d e l b y ( P r 6 2 )

6 2 = 3 / 2 0 ( E / „ c ) 2 ( Q A )2 - j — ,Q r t ( J - 1 )

w h e r e E i s t h e e n e r g y o f t h e t r a n s i t i o n i n M e V , f t c = 1 . 9 7 x 1 0 11 M e V c m ,

- 2 7 - 1 0f t = 1 . 0 5 4 x 1 0 e r g s e c , a n d e = 4 . 8 0 3 x 1 0 e s u . Q q i s t h e i n t r i n s i c

q u a d r u p o l e m o m e n t , \ i s t h e C o m p t o n w a v e l e n g t h o f t h e p r o t o nc

- 1 4(X = 2 . 1 0 3 x 1 0 c m ) , a n d g a n d g a r e t h e g y r o m a g n e t i c r a t i o s f o r t h eC £") iis i n g l e p a r t i c l e a n d t h e c o r e , r e s p e c t i v e l y . T h e g v a l u e c o r r e s p o n d i n g t o

t h e o d d n u c l e o n i s g i v e n b y

( g s “ g <^ \ . 2 2

Iw h e r e £ > = K f o r a x i a l s y m m e t r i c n u c l e i a n d t h e g y r o m a g n e t i c r a t i o f o r t h e

c o r e i s g i v e n a p p r o x i m a t e l y b y

g j ^ Z / A .

g Q g l 2 , 0 L ( a 4 K - l / 2 " a 4 K + l / 2 } ’

108

t a b l u l a t e d b y N i l s s o n ( N i 5 5 ) a n d g a n d g a r e t h e i n t r i n s i c a n d o r b i t a l gs

v a l u e s r e s p e c t i v e l y f o r t h e o d d n u c l e o n .

W i t h i n t h e f r a m e w o r k o f t h e r i g i d r o t o r m o d e l t h e i n t r i n s i c q u a d ­

r u p o l e m o m e n t a n d a r e c o n s t a n t s w i t h i n a g i v e n r o t a t i o n a l b a n d .

F r o m t h e p r e v i o u s e q u a t i o n s d e f i n i n g T ( M 1 ) a n d T ( E 2 ) , t h e m u l t i p o l e

m i x i n g r a t i o f o r s t o p o v e r i n t r a b a n d t r a n s i t i o n s m a y b e w r i t t e n , a p a r t f r o m

a s i g n a m b i g u i t y , a s

5 = A i ___6 K ,j 2- d 1/2 '

w h e r e t h e c o n s t a n t A i s d e f i n e d a sK.

The a a re the s in g le p a rtic le wave function expansion co effic ien ts

A K = 3 / 2 0 ( Q o / X c ) ( l / f t c ) l / ( g K - g R ) .

B y e x a m i n i n g t h e r a t i o o f m u l t i p o l e m i x i n g r a t i o s , t h e d e p e n d e n c e

f o r a g i v e n b a n d o n t h e c o n s t a n t A ^ i s r e m o v e d . T h e r a t i o m a y t h e n b e

w r i t t e n i n t e r m s o f e n e r g y a n d a n g u l a r m o m e n t u m a s

6 ( J - * J - 1 ) E J ~ E J - 1 (J-l)2-lJ 2 - l

1/2

A t a b l e c o m p a r i n g t h e e x p e r i m e n t a l v a l u e s a n d c a l c u l a t e d v a l u e s

b a s e d o n t h e a b o v e f o r m u l a f o r £ = 11 n u c l e i i s s h o w n i n F i g . 9 8 . I n t h e

. + , +t a b l e t h e c a l c u l a t e d m i x i n g r a t i o s a r e n o r m a l i z e d t o t h e 5 / 2 3 / 2

t r a n s i t i o n r a t h e r t h a n t o t h e p r e c e d i n g s t o p o v e r t r a n s i t i o n . I n a l l f o u r

2 3n u c l e i t h e a g r e e m e n t i s r e m a r k a b l y g o o d , p a r t i c u l a r l y i n N a w h e r e t h e

B A N D E 2 / M I S T O P O V E R M I X I N G R A T I O S

N O R M A L I Z E D T O 5 / 2 + -------- 3 / 2 + T R A N S I T I O N

T R A N S I T I O N ^ E X P E R R O R $ C A L C E R R O R

N e 2 1

5 / 2 + —® *’ 3 / 2 + 0 . 0 5 ± 0 . 0 2 0 . 0 5 ± 0 . 0 2

7 / 2 ^ - s3 * - 5 / 2 + 0 . 1 6 ± 0 . 0 3 0 . 14 ± 0 . 0 5

9 / 2 + - ® = - 7 / 2 + 0 . 0 9 ± 0 . 0 5 0 . 0 8 ± 0 . 0 3

l l / 2 + ^ - S / 2 * 0 . 1 3 ± 0 . 0 6 0 . 0 9 ± 0 . 0 4

l 3 / 2 +-«**» 11 / 2 -5* 0 . 1 0 ± 0 . 0 5 0 . 1 0 ± 0 . 0 4

N a 2 1

5 / 2 + —® = » 3 / 2 + - 0 . 0 5 ± 0 . 0 2 - 0 . 0 5 ± 0 . 0 2

7 / 2 + - < s s - 5 / 2 + - 0 . 14 ± 0 . 0 3 - 0 . 1 4 ± 0 . 0 6

9 / 2 + - ^ 7 / 2 + - 0 . 1 2 ± 0 . 0 3 - 0 . 0 9 ± 0 . 0 3

N a ^

5 / 2 +- b» 3 / 2 * - 0 . 0 9 ± 0 . 0 2 - 0 . 0 9 ± 0 . 0 2

7/2*-&*- 5 /2 * - 0 . 2 0 ± 0 . 0 2 - 0 . 2 3 ± 0 . 0 5

9 / 2 + - * 3» 7 / 2 + - 0 . 1 0 ± 0 . 0 3 - 0 . 0 7 ± 0 . 0 2

I I / 2 4’—o»* 9 / 2 * - 0 . 1 8 ± 0 . 0 3 - 0 . 2 5 ± 0 . 0 6

l 3 / 2 +- ^ * l l / 2 + — — - 0 . 0 6 ± 0 . 0 1

M g 2 3

5 / 2 +—®=* 3 / 2 * 0 . 0 8 ± 0 . 0 2 0 . 0 8 ± 0 . 0 2

7 / 2 + —*=> 5 / 2 + 0 . 1 8 ± 0 . 0 3 0 . 1 9 ± 0 . 0 5

9 / 2 + —«s- 7 / 2 + — — 0 . 0 6 ± 0 . 0 2

I I / 2 + - ® . 9 / 2 + — — 0 . 2 1 ± 0 . 0 5

F ig .' 98

109

e x c i t a t i o n e n e r g i e s d e p a r t e d f r o m r i g i d r o t o r p r e d i c t i o n s . T h e a g r e e m e n t

i s a m e a s u r e o f t h e c o n s t a n c y o f t h e i n t r i n s i c b a n d h e a d p a r a m e t e r s Q q

a n d g K - g R - I t a p p e a r s f r o m t h i s a g r e e m e n t t h a t t h e s e p a r a m e t e r s a r e

i n d e e d a p p r o x i m a t e l y c o n s t a n t a n d t h e r o t o r m o d e l f o r m a l i s m d o e s i n d e e d

r e p r o d u c e r e l a t i v e s t o p o v e r m i x i n g r a t i o s f o r t h e s e n u c l e i .

b . I n t r i n s i c q u a d r u p o l e m o m e n t s a n d g y r m a g n e t i c r a t i o s

A d d i t i o n a l p r o p e r t i e s m a y b e a d d u c e d w h i c h a l s o r e f l e c t

t h e u n i q u n e s s o f t h e i n t r i n s i c q u a d r u p o l e m o m e n t . T h e r a t i o o f s t o p o v e r

a n d c r o s s o v e r t r a n s i t i o n r a t e s m a y b e c a l c u l a t e d a n d c o m p a r e d w i t h

e x p e r i m e n t . W e c a l c u l a t e s p e c i f i c a l l y t h e b r a n c h i n g r a t i o B o f t h e c r o s s -co v e r t r a n s i t i o n d e f i n e d h e r e i n a s

100B =o 1+ T l / T 2w h e r e

T / T 2Kf. djJ.7lJ------------- .1 2 ^ 2 5 ( J + l ) ( J - l - K ) ( J - l + K )6

QoS i n c e w e a r e t e s t i n g t h e d e g r e e t o w h i c h ------------ ; r e m a i n s a c o n s t a n t ,%“%r a t h e r t h a n i t s a b s o l u t e v a l u e , w e a r e i n t e r e s t e d i n c a l c u l a t i n g r e l a t i v e

q u a n t i t i e s r a t h e r t h a n a b s o l u t e o n e s . T o r e d u c e t h e u n c e r t a i n t y i n t h e

e x p r e s s i o n f o r T / T , e x p e r i m e n t a l v a l u e s o f 6 w e r e s u b s t i t u t e d r a t h e rJ. z

t h a n c a l c u l a t e d o n e s . T h i s , i n n o w a y , r e d u c e s t h e g e n e r a l i t y o r r i g o r o f

o u r a p p r o a c h , b u t o n t h e c o n t r a r y , e n h a n c e s t h e r e l i a b i l i t y o f s u c h

110

c o m p a r i s o n , s i n c e l ' e w e r u n c e r t a i n t i e s a r e p r e s e n t i n t h e f o r m u l a .

A t a b l e c o m p a r i n g r e s u l t s f o r t h e £ = 1 1 n u c l e i i s s h o w n i n F i g .

9 9 . I n c a s e s w h e r e t h e e x p e r i m e n t a l v a l u e o f t h e m i x i n g r a t i o i s n o t k n o w n ,

e x t r a p o l a t e d v a l u e s f r o m t h e p r e v i o u s m i x i n g r a t i o c a l c u l a t i o n w e r e u s e d

i n t h e f o r m u l a f o r T ^ / T ^ . A g a i n r e m a r k a b l e a g r e e m e n t i s o b t a i n e d f o r

m o s t t r a n s i t i o n s . T h e l a r g e s t d i s c r e p a n c y o c c u r s f o r t h e c r o s s o v e r

2 3t r a n s i t i o n f r o m t h e 2 . 7 0 M e V s t a t e i n N a . H e r e , h o w e v e r , i t m u s t b e

e m p h a s i z e d t h a t t h e e x p e r i m e n t a l v a l u e o f t h e m i x i n g r a t i o i s m o s t u n c e r ­

t a i n . A s l i g h t l y s m a l l e r v a l u e (5 — . 0 7 ) w o u l d g i v e t h e c o r r e c t c r o s s ­

o v e r b r a n c h i n g .Qo

I n e x a m i n i n g t h e m i x i n g r a t i o p a r a m e t e r s i t w a s s h o w n t h a t -------------g K ~ g R

r e m a i n e d a c o n s t a n t w i t h i n t h e b a n d . H e r e , w e h a v e s h o w n t h a t Q a l s oor e m a i n s a c o n s t a n t a n d i n c o n s e q u e n c e c a n a l s o c o n c l u d e t h a t g j ^ - g ^ I s a

c o n s t a n t w i t h i n t h e g r o u n d s t a t e r o t a t i o n a l b a n d , n o t o n l y f o r o n e , b u t f o r

a l l o f t h e £ = 1 1 r o t a t i o n a l n u c l e i c o m p a r e d . I t m u s t b e e m p h a s i z e d t h a t

t h e s e s t a t e m e n t s a r e v a l i d o n l y f o r t h e b a n d m e m b e r s t h u s f a r s t u d i e d .

A l t h o u g h t h e s e d a t a a l r e a d y i m p l y r e m a r k a b l y r i g i d s t r u c t u r e s i t w o u l d b e

a n t i c i p a t e d t h a t w i t h i n c r e a s i n g t o t a l a n g u l a r m o m e n t u m c e n t r i f u g a l a n d

C o r i o l i s a n t i - p a i r i n g e f f e c t s w o u l d e v e n t u a l l y a c t t o m o d i f y t h e e q u i l i b r i u m

d i s t o r t i o n s , h e n c e q u a d r u p o l e m o m e n t s .

T o f u r t h e r c h e c k t h e c o n s t a n c y o f t h e i n t r i n s i c q u a d r u p o l e m o m e n t

a n d g j ^ - g j ^ i t h e e x p e r i m e n t a l l i f e t i m e s o f t h e b a n d m e m b e r s , t o g e t h e r w i t h

t h e e x p e r i m e n t a l v a l u e o f t h e s t o p o v e r m i x i n g r a t i o , m a y b e u s e d t o d e t e r -

| TRANSITION B R e x p E R R O R B R c a l c E R R O R

[ N e 2 *7/2+-=- 3/2* 5 ± 2 5 ± 2

9/2+-ra» 5/2* 33 ±5 41 ±27

11/2*-®-7/2* 45 ±5 39 ±22

13/2*-®- 9/2* <50 — 41 ±19 '

N o 21I 7/2*-®- 3/2* 7 ±2 4 ±2

9/2*-®» 5/2+ 36 ±6 4 4 ±24

N a 2 37 /2 + - ^ 3/2+ 9 ±2 8 ± 1

9/2+-«— 5/2+ 63 ±2 90 ± 5

II/24'—«*» 7/2+ 24 ±6 18 ±5

I3/2+-* 9/2* 100 ±10 97 ± 1

M g ^7/2+-® -3/2* 15 ± 4 7 ± 2

I l l

m i n e t h e a b s o l u t e v a l u e s o f Q q a n d g ^ - g ^ * O n l y a l i m i t e d c o m p a r i s o n c a n

2 1 2 3b e m a d e , s i n c e l i f e t i m e s a r e o n l y k n o w n f o r l e v e l s i n N e a n d N a . I n

a d d i t i o n t h e l a r g e f r a c t i o n a l u n c e r t a i n t i e s i n t h e l i f e t i m e a n d m i x i n g r a t i o

m e a s u r e m e n t s r e d u c e t h e r e l i a b i l i t y a n d m e a n i n g f u l n e s s o f s u c h c o m p a r i ­

s o n s , b u t n e v e r t h e l e s s , i t d o e s p r o v i d e y e t a n o t h e r i n t e r n a l c o n s i s t e n c y

c h e c k o n a r a t h e r s u r p r i s i n g r e s u l t .

G i v e n t h e e x p e r i m e n t a l v a l u e o f t h e l i f e t i m e j , t h e m i x i n g r a t i o f i ,

t h e s t o p o v e r a n d c r o s s o v e r b r a n c h i n g r a t i o B a n d B , r e s p e c t i v e l y , a n ds c

t h e c o r r e s p o n d i n g e l e c t r i c q u a d r u p o l e t r a n s i t i o n p r o b a b i l i t i e s T ( E 2 ) a n ds

T ( E 2 ) , r e s p e c t i v e l y , t h e i n t r i n s i c q u a d r u p o l e m o m e n t m a y b e c a l c u l a t e d cf r o m t h e f o r m u l a e

( ¥ 1 5 2 2T ( E 2 ) = 1 . 2 3 . (K) ) E ( M & V / Q ( b a r n s ) ( J 2 K O | J ' K ) ,

Ts(E2) = ^ i r ^ r ’ =B

T c ( E 2 ) = * y , J « = J - 2 ,

a n d

O 6 + 1

w h e r e f _ » e > a n d e a r e u n c e r t a i n t i e s i n t h e q u a d r u p o l e m o m e n t ,Q 0 6 t

m i x i n g r a t i o , a n d l i f e t i m e , r e s p e c t i v e l y .

A t a b l e o f i n t r i n s i c q u a d r u p o l e m o m e n t Q w i t h e r r o r e a c h c a l -o %

2 1 2 3c u l a t e d f r o m t h e s e f o r m u l a e f o r N e a n d N a i s g i v e n i n T a b l e 1 .

112

I n t r i n s i c Q u a d r u p o l e M o m e n t s

T a b le 1

T r a n s i t i o n M 21N e2 3

N a

G r o u n d ( 3 / 2 + ) 0 . 4 6 ± 0 . 0 5 0 . 5 5 ± 0 . 0 5

* 5 / 2 + - » 3 / 2 + 0 . 4 3 ± 0 . 0 4 0 . 5 2 ± 0 . 0 3

7 / 2 + -* 5 / 2 + 0 . 3 5 ± 0 . 0 8 0 . 2 3 ± 0 . 0 5

7 / 2 + - 3 / 2 + 0 . 3 5 ± 0 . 0 6 0 . 2 6 ± 0 . 0 3

9 / 2 + - 1 / 2 0 . 4 1 ± 0 . 1 0 1 . 2 0 ± 0 . 4 8

9 / 2 + - 5 / 2 + 0 . 3 5 ± 0 . 0 6 0 . 5 0 ± 0 . 1 5

l l / 2 + - 9 / 2 + 0 . 3 2 ± 0 . 1 4 ----------

1 1 / 2 = * 7 / 2 0 . 3 8 ± 0 . 0 8 ----------

* Q q d e t e r m i n e d h e r e f r o m C o u l o m b e x c i t a t i o n m e a s u r e m e n t o f B ( E 2 ) ( S c 6 9 ) .

21I n N e t h e c a l c u l a t e d Q i s c o n s t a n t w i t h i n t h e u n c e r t a i n t i e s o f t h eom u l t i p o l e m i x i n g r a t i o s a n d l i f e t i m e s u s e d i n t h i s c a l c u l a t i o n . T h i s i s

c o n s i s t e n t w i t h o u r c o n c l u s i o n s d e d u c e d f r o m t h e m i x i n g r a t i o a n d b r a n c h i n g

r a t i o c o m p a r i s o n s .

2 3H o w e v e r , i n N a Q q v a r i e s m a r k e d l y f o r t h e f e w l e v e l s t h a t a r e

c o m p a r e d i n d i s a g r e e m e n t w i t h t h e a b o v e e v i d e n c e f o r c o n s t a n c y o f Q q .

T h e a p p a r e n t v a r i a n c e a m o n g t h e i n t r i n s i c q u a d r u p o l e m o m e n t s , a s h e r e

d i s p l a y e d , i s b e l i e v e t o r e f l e c t p r i m a r i l y i n a c c u r a c i e s i n t h e l i f e t i m e

m e a s u r e m e n t s a n d i n t h e m e a s u r e m e n t s o f t h e m u l t i p o l e m i x i n g r a t i o s i n

2 3N a a n d n o t i n a n y b r e a k d o w n i n t h e r i g i d r o t o r m o d e l .

T h e m o s t o b v i o u s d i s c r e p a n c i e s o c c u r f o r Q q d e t e r m i n e d f r o m t h e

J 11 = 7 / 2 * 2 . 0 8 M e V l e v e l a n d t h e 9 / 2 + => 7 / 2 + t r a n s i t i o n f r o m t h e 2 . 7 0 M e V

113

l e v e l . I n t h e l a t t e r c a s e t h e m i x i n g r a t i o v a r i e s b e t w e e n - 0 . 0 2 <6 < - 0 . 1 2 ,

d e p e n d i n g o n w h o m a d e t h e m e a s u r e m e n t . T h i s u n c e r t a i n t y i f r e f l e c t e d i n

t h e d i s a g r e e m e n t o f Q q w i t h t h a t d e t e r m i n e d f r o m t h e c r o s s o v e r t r a n s i ­

t i o n , w h i c h d o e s n o t i n v o l v e t h e m i x i n g r a t i o a n d a g r e e s w i t h g r o u n d a n d

f i r s t e x c i t e d s t a t e v a l u e s .

T h e t w o s i m i l a r , b u t l o w e r v a l u e s o f Q , d e t e r m i n e d f r o m t h e s t o p -o7T + 2 3o v e r a n d c r o s s o v e r t r a n s i t i o n s f r o m t h e J = 7 / 2 s t a t e i n N a , a r e i n

d i s a g r e e m e n t w i t h o t h e r v a l u e s i n t h e b a n d b y a f a c t o r o f t w o . S i n c e t h i s

d i s a g r e e m e n t i s i n d e p e n d e n t o f w h e t h e r t h e m i x i n g r a t i o i s u s e d a n d b o t h

v a l u e s a r e o f t h e s a m e a m o u n t , t h e l i f e t i m e m e a s u r e m e n t t h a t i s i n

c o m m o n t o b o t h t r a n s i t i o n s i s s u s p e c t e d t o b e i n c o r r e c t . H o w e v e r ,

i n c r e a s i n g b y a f a c t o r o f t w o c o r r e s p o n d s t o r e d u c i n g t h e m e a s u r e d

l i f e t i m e f o r t h e 2 . 0 8 M e V l e v e l b y a f a c t o r o f f o u r . T h i s w o u l d b e o u t s i d e

t h e l i f e t i m e u n c e r t a i n t y s p e c i f i e d i n t h e m e a s u r e m e n t o f D u r e l l e t a l .

( r = 0 . 2 2 ± 0 . 0 5 p s e c s ) . P o l e t t i ( P o 6 9 ) a n d P r o n k o ( P r 6 ^ l i m i t t h e l i f e ­

t i m e t o < 0 . 2 3 a n d < 0 . 3 0 p s e c s , r e s p e c t i v e l y . E i t h e r t h e l i f e t i m e m e a s u r e ­

m e n t i s i n c o r r e c t o r w e h a v e i n d e e d a n a n a m o l o u s b r e a k d o w n o f t h e r i g i d

r o t o r m o d e l i n c o m p a r i s o n t o t h e o t h e r s u c c e s s f u l p r e d i c t i o n s b y t h e m o d e l

i n c l u d i n g t h e b r a n c h i n g o f t h e 2 . 0 8 M e V s t a t e . W e t e n d t o b e l i e v e t h e f o r m e r

i n v i e w o f t h e a l r e a d y a b u n d a n t s u c c e s s f u l p r e d i c t i o n s .

F o r t h e p u r p o s e o f t h e r e m a i n i n g d i s c u s s i o n i t i s a s s u m e d t h a t w e

h a v e i n d e e d p r e s e n t e d t h e c o r r e c t e x p l a n a t i o n a n d t h a t t h e g r o u n d s t a t e

114

r o t a t i o n a l b a n d s o f t h e £ = 11 n u c l e i p o s s e s s a n u n u s u a l l y c o n s t a n t

i n t r i n s i c q u a d r u p o l e m o m e n t a n d g y r o m a g n e t i c r a t i o s ( i . e . g R - g R )- O n

t h i s a s s u m p t i o n w e p r o c e e d t o e x a m i n e t h e d e p a r t u r e o f t h e e x c i t a t i o n

2 3e n e r g i e s f r o m t h e J ( J + 1 ) r u l e p r e d i c t e d b y t h e r i g i d r o t o r m o d e l i n N a ,

w h i l e s u c h a m o d e l h a s f a i r l y w e l l a c c o u n t e d f o r t h e d y n a m i c p r o p e r t i e s

• XT 23i n N a .

2 . H i g h e r o r d e r C o r i o l i s a n d r o t a t i o n a l p e r t u r b a t i o n s

V a r i o u s r o t a t i o n a l p e r t u r b a t i o n s a r e d i s c u s s e d a n d t h e i r

i n f l u e n c e o n t h e e x c i t a t i o n e n e r g i e s a r e c a l c u l a t e d i n A p p e n d i c e s I , I I ,

H I , a n d I V . W e h a v e m a d e a n a t t e m p t t o f i t t h e o b s e r v e d g r o u n d s t a t e b a n d

m e m b e r s w i t h t e r m s b a s e d o n C o r i o l i s p e r t u r b a t i o n s . A r e q u i r e m e n t o f

o n e a p p r o a c h w a s t h e s e l e c t i o n o f p e r t u r b a t i o n s w h i c h w e r e d i a g o n a l i n t h e

s t r o n g c o u p l i n g r o t o r m o d e l c o n f i g u r a t i o n s p a c e . S i n c e s u c h a w a v e

f u n c t i o n w a s s u c c e s s f u l i n p r e d i c t i n g t h e d y n a m i c p r o p e r t i e s , i t w a s h o p e d

t h a t t h e e x c i t a t i o n e n e r g i e s c o u l d b e a c c o u n t e d f o r w i t h a t e r m t h a t d i d n o t

a d m i x d i f f e r e n t c o m p o n e n t s o f K i n t o t h e w a v e f u n c t i o n . A g e n e r a l e x p r e s -

. 2s i o n o f t h e e n e r g y d e d u c e d f r o m e x p a n d i n g t h e H a m i l t o n i a n p o w e r s o f R

i n c l u d i n g d i a g o n a l c o n t r i b u t i o n s t o t h e r o t o r m o d e l w a v e f u n c t i o n u p t o

t e r m s i n R , i n c l u s i v e l y , i s

o 9 o o T-f. 3 / o 3E = A q + A 1 J ( J + 1 ) + A 2 J ( J + l ) + A g [ J ( J + l ) - 8 ( - l ) ) a 3 / 2 ] *

w h e r e t h e c o n s t a n t A , A , A a n d A a r e d e f i n e d i n A p p e n d i x I V .O 1 Z <5A n a l t e r n a t i v e p r o c e d u r e t h a t r e s u l t s i n a s i m i l a r e x p a n s i o n w i t h -

116

Section I-C3. The model has been used to fit the positive parity levels in21 21 23Ne ,N a , and Na up to approxim ately 6. 0 MeV excitation with p a r ti­

cular attention focussed on the fit to the ground state band m em bers. A com parison of the C oriolis couplipg model fits to the experim ental levels is shown in F igs. 100, 101, and 102. The best fit was obtained with the fitting p a ram ete rs labelling the various figures. These p aram eters a retypical of sd shell nuclei (Hi 69) in th is m ass region.

21 21In Ne and Na the fit was considered best when the wavefunctions of the m em bers of the ground sta te rotational band, determ inedfrom the fit, best reproduced the B(E2) transition probabilities and thein trin s ic quadrupole moment of the ground state. Except for the l /2 + holeband (orbit 6) the levels calculated a re in approxim ate agreem ent with

21 21experim ent. Since a J 77 = l /2 + hole band o r state in Ne and Na has not yet been found below 5. 5 MeV, the calculated position of the hole band a t 3. 0 MeV is considered to be in disagreem ent with the model. To ra ise the hole band to a region of higher excitation, a la rg er deform ation p a ra ­m e te r, corresponding to m ore deform ed core is req u ired , but will destroy the agreem ent between the calculated and experim ental B(E2) values.

I t is indeed recognized that the hole band could have a m arkedly different deform ation and should not be entire ly unexpected. In addition a m ore appropriate band head calculation including m icroscopic fea tu res of the nucleus might im prove upon the agreem ent already obtained with the

EXC

ITA

TIO

N

ENER

GY

(Me

V)

CORIOLIS COUPLING MODEL APPLIED TO Ne21

9 -

8 -

7

6

4

2

I

3 / 2 + , 5 / 2 + - ( 9 / 2 ) , 1 3 / 2 + — *

3 / 2 + 5 / 2 + - 3 / 2 + , 5 / 2 + - 3 / 2 + , 5 / 2 + -

3 / 2 + , 5 / 2 + — . ( 7 / 2 ) , 11 / 2 +

3 / 2 , 5 / 2 — - 5 / 2 + , ( 3 / 2 + ) '

7 / 2 +

5/2+■3/2+

7 / 2 +

Ne21

E X P

1 3 / 2 +9 / 2 +

7 / 2 + 9 / 2 +

11/2+ 3 / 2 + 7 / 2 +

5 / 2 +

3 / 2 + ‘5 / 2 +

1/2 + # 55 / 2 +

1/2+ # 6# 9

# 7

7 / 2 + ■NILSSONORBITNO.

5 / 2 +

3 /2 +

3/2 +5 / 2 + # 87 / 2 +

9 / 2 +

O il1/2+3 / 2 +

A = 0.136 MeV C = -0.35 fi w0 D = 0.00 /3= 0.48 Q = 0.65 B = 0 .00

F ig . 100

EXCI

TATI

ON

ENER

GY

(MeV

)CORIOLIS COUPLING MODEL APPLIED TO No21

7

6

5

8N a21EXP

( 3 / 2 , 5 / 2 ) + —

3

2

I/2+-

3 / 2 + '5 / 2 + '

5 / 2 + -

9 / 2 + , ( 5 / 2 + L l / 2 ± , 3 / 2 " - "

I / 2 + —

7 / 2 + ( 3 / 2 + )

5/2+-3/2+-

• 1 3 / 2 +

11/2+

9 / 2 +

3/2+7 / 2 + 5/21 # 8

7 / 2 +

9 / 2 +

9 / 2 +

7 / 2 + ■ 9 / 2 +

3 / 2 + .

1/2 + 3 / 2 +

7 / 2 +

5 / 2 +

3 / 2 + '5 / 2 +

1/2+ # 55 / 2 +

# 77 / 2 +

5 / 2 +

3/2+

# 9>/2+ # 6

NILSSONORBIT

NO.

Hr

A =0.136 MeV C = -0.35 ti w0 D =0.00 f3= 0.48 Q = 0.65 0 = 0.00

Fig. 101

10

9

8

7

6

5

4

3

2

I

CORIOLIS COUPLING MODEL APPLIED TO N o233/2+

9 / 2 , 5 / 2 \ I/2+- ( 9 /2 ) , 13 /2 + -

3 / 2 + 5 / 2 + - (7 /2 ) ,11/2+— 3 / 2 + , 5 / 2 + " '

7 /2 , ( 5 / 2 ) - I/2+-

( 5 /2 + ) -

3 / 2 + — 9 / 2 + — 1/2+— 7 / 2 + —

Na23

EXP

5 / 2 + — «

3 / 2 + — «

13/2+

11/2+

9 / 2 + .

# 7

9 /2 +

11/2+

9 / 2 '

9 / 2 + #11

5 /2 +

3 / 2 +

1/2 +

# 8

7 /2 +

3 / 2 +

5 / 2 +

7 /2 +

5 / 2 +

1/2+5/2+ # 6

3 / 2 +

1/2 +7 /2 + # 9

5 / 2 +

3/2+ ---------

'NILSSONORBIT

NO.

A = 0.223 MeV C =-0.30 ticu0 D = -0.02 ticj0 /3 = 0.70 Q =0.75 6 =0.25

Fig. 102

117

C o r i o l i s m o d e l . H a r t r e e - F o c k c a l c u l a t i o n s e f f e c t i v e l y i n c o r p o r a t i n g

n u c l e o n i c c o r r e l a t i o n s h a v e p r e d i c t e d t h e c o r r e s p o n d i n g h o l e e x c i t a t i o n

2 1 2 3i n t h e v i c i n i t y o f 7 . a n d 6. M e V i n N e a n d N a , r e s p e c t i v e l y ( K e 6 4 ) .

H e r e i n , t h e a g r e e m e n t w i t h t h e B ( E 2 ) i s o f u t m o s t c o n c e r n a n d t h e

h o l e b a n d w i l l b e i g n o r e d ; i t s l o c a t i o n i s c o n s i d e r e d m e r e l y a l i m i t a t i o n o f

t h e C o r i o l i s c o u p l i n g m o d e l a s a p p l i e d h e r e .

A t a b u l a t i o n o f b a n d m i x i n g e x p a n s i o n c o e f f i c i e n t , C T. , p , d e f i n e d

i n t h e C o r i o l i s c o u p l i n g m o d e l a s

| J M > = ^ ^ C k , „ | J K M >

K vw h e r e ( J K M > i s t h e s t r o n g c o u p l i n g m o d e l w a v e f u n c t i o n d e f i n e d i n S e c t i o n

I - C l a n d t h e s u m o v e r v d i s t i n g u i s h e s d i f f e r e n t s i n g l e p a r t i c l e o r b i t s w i t h

t h e s a m e K , f o r t h e g r o u n d s t a t e b a n d m e m b e r s a s s h o w n i n T a b l e 2 f o r

2 3 2 1 2 1N a , N e , a n d N a . I n t h e t a b l e p c o r r e s p o n d s t o t h e N i l s s o n o r b i t

n u m b e r r a t h e r t h a n t h e m e a n i n g g i v e n a b o v e . N o t e t h e r e l a t i v e l y s m a l l

a d m i x t u r e s o f o r b i t 6 w i t h t h e g r o u n d s t a t e b a n d i n c o m p a r i s o n t o o r b i t s

21 219 a n d 5 f o r N e a n d N a r e f l e c t i n g t h e u n i m p o r t a n c e o f t h e p o s i t i o n o f

t h e b a n d h e a d o f o r b i t 6 .

118

Table 2

Table of Coriolis Expansion Coefficients

E x p C a l c J 77 C 3 / 2 , 7 C l / 2 , 9 C l / 2 , 6 C 5 / 2 , 5 C l / 2 , 1 1 C 3 / 2 . 8

0 - f t

+cqCO . 9 9 1

XT 23N a

- . 0 7 9 - . 0 9 8 0 . 021 - .0 1 2

. 4 4 . 6 2 7 5 / 2 + . 9 4 9 . 1 3 6 . 2 3 1 . 1 6 2 . 0 4 9 - . 0 0 4

2 . 0 8 1 . 8 3 6 7 / 2 + . 9 4 5 - . 1 4 0 - . 1 6 7 - . 2 4 0 - . 0 3 6 - . 002

2 . 7 1 2 . 8 6 4 9 / 2 + . 8 7 1 . 1 8 8 . 3 7 0 . 2 5 0 ± . 0 7 4 . 010

5 . 5 4 5 . 1 1 5 1 1 / 2+ . 9 0 7 - . 1 7 2 - . 2 0 1 - . 3 2 6 - . 0 4 2 . 0 1 8

6 .2 6 . 4 6 5 1 3 / 2 + . 8 0 8 .2 1 1 . 4 6 4 . 2 7 9 . 0 9 0 .0 2 2

o . 0.

CO

XT 2 1 XT 21N e , N a

. 9 9 1 - . 1 1 2 - . 0 6 8 0 .0 0 - . 0 1 3 - . 0 0 9

. 3 5 . 5 7 5 / 2 + . 9 2 4 . 2 9 5 . 1 1 9 .2 1 0 . 0 2 8 - . 0 0 4

1 . 7 5 1 . 8 0 7 / 2 + . 9 2 7 - . 1 7 9 - . 1 1 4 - . 3 1 0 022 .0 0 0

2 . 8 7 2 . 7 4 9 / 2 + . 8 2 6 . 4 5 9 . 1 5 5 . 2 8 5 . 0 3 9 . 0 0 4

4 . 4 3 5 . 0 7 l l / 2+ . 8 8 4 - . 2 0 8 - . 1 3 7 - . 3 9 5 0 2 5 . 0 0 9

6 . 4 5 6 . 2 9 1 3 / 2 + . 7 5 6 . 5 6 0 . 1 6 6 . 2 9 2 . 0 4 5 . 0 0 9

2 3I n N a t h e b e s t f i t w a s d e t e r m i n e d s t r i c t l y f r o m e n e r g y l e v e l c o n ­

s i d e r a t i o n s a l o n e . H e r e t h e J 7^ l / 2 + h o l e e x c i t a t i o n i d e n t i f i e d a t 4 . 4 3 M e V

i s f i t b e s t w i t h a d e f o r m a t i o n p a r a m e t e r o f = 0 . 7 . T o o b t a i n a r e a s o n a b l e

f i t t o t h e l e v e l s o f t h e g r o u n d s t a t e r o t a t i o n a l b a n d a l a r g e / S i s r e q u i r e d .

F r o m F i g . 1 0 2 o b s e r v e t h a t a d m i x t u r e s i n t o t h e g r o u n d s t a t e b a n d w a v e

f u n c t i o n s h a v e s h i f t e d t h e l e v e l s i n t h e a p p r o p r i a t e d i r e c t i o n t o g i v e b e t t e r

a g r e e m e n t w i t h e x p e r i m e n t ( L i 6 9 g ) .

A n e x a m i n a t i o n o f t h e a d m i x e d w a v e f u n c t i o n s i n T a b l e 3 s h o w s

119

t h a t t h e d o m i n a n t K a d m i x t u r e s i n t h e g r o u n d s t a t e b a n d o r i g i n a t e f r o m

o r b i t s 6 , 5 , a n d 9 . T h e l o c a t i o n o f t h e p e r t u r b e d b a n d h e a d s c o r r e s p o n d

t o l / 2 + l e v e l s a t 2 . 3 9 M e V ( o r b i t 9 ) a n d 4 . 4 3 M e V ( o r b i t 6) . T h e l o c a t i o n

o f t h e 5 / 2 b a n d h e a d ( o r b i t 5 ) i s t e n t a t i v e l y a s s i g n e d t o t h e l e v e l a t 1 . 3 8

2 3M e V i n N e , b u t i t n o t d e f i n i t e . W e p o i n t o u t t h a t t h e p o s i t i o n o f o t h e r

p o s i t i v e p a r i t y l e v e l s a r e a l s o i n r e l a t i v e l y c l o s e a g r e e m e n t w i t h e x p e r i ­

m e n t , a l t h o u g h t h e y a r e n o t o f p r i m a r y c o n c e r n i n t h i s d i s c u s s i o n .

A c l o s e r e x a m i n a t i o n o f t h e t a b l e o f C o r i o l i s e x p a n s i o n c o e f f i c i e n t s

f o r N a a n d N e r e v e a l s s o m e i n t e i ' e s t i n g s y s t e m a t i c s . F i r s t l y t h e

2 3d o m i n a n t c o m p o n e n t m i x e d i n t o t h e g r o u n d s t a t e b a n d i n N a i s t h e

3 ^ = 1 / 2 + h o l e e x c i t a t i o n c o r r e s p o n d i n g t o o r b i t 6 i n c o n t r a s t t o o r b i t 9 i n

21N e . S e c o n d l y t h e e x p a n s i o n c o e f f i c i e n t s i n t h e g r o u n d s t a t e b a n d i t s e l f

2 1 2 3i n N e a r e s m a l l e r t h a n t h o s e i n N a c o r r e s p o n d i n g t o a s l i g h t l y l a r g e r

21d e g r e e o f C o r i o l i s m i x i n g i n N e . T h i s c o n c l u s i o n i s n o t c o m p l e t e l y

c e r t a i n , s i n c e t h e l o w e r e d J 77 = l / 2+ h o l e s t a t e m a y e r r o n e o u s l y c o n t a i n

l a r g e w a v e f u n c t i o n a d m i x t u r e s i n c r e a s i n g t h e g r o u n d s t a t e a d m i x t u r e s .

2 1 2 3A t f i r s t m o r e C o r i o l i s m i x i n g i n N e t h a n i n N a m a y s e e m

2 1 2 3c o n t r a d i c t o r y t o t h e o b s e r v a t i o n t h a t N e o b e y s J ( J + 1 ) b e t t e r t h a n N e .

H o w e v e r , i t c o u l d b e t h a t t h e a m o u n t o f m i x i n g p r o p o r t i o n a t e l y i n f l u e n c e s

2 1 2 3e a c h l e v e l m a i n t a i n i n g t h e J ( J + 1 ) s e q u e n c e i n N e , w h e r e a s i n N a t h e

i n v e r t e d l / 2 + h o l e e x c i t a t i o n h i n d e r s C o r i o l i s m i x i n g w i t h a l t e r n a t e l e v e l s

s i n c e t h e y a r e e n e r g e t i c a l l y f u r t h e r a w a y . T h i s c o u l d p r o d u c e t h e d e s i r e d

2 3 2 3e f f e c t i n N a a n d i s c o n s i s t e n t w i t h t h e i n t e r p r e t a t i o n t h a t N a i s a n e v e n

120

m o r e r i g i d n u c l e u s t h a n N e .

1 . A b s o l u t e r e d u c e d m a t r i x e l e m e n t c o m p a r i s o n

I n a n a t t e m p t t o o b t a i n m o r e i n f o r m a t i o n o n t h e w a v e

f u n c t i o n s a n d a d m i x t u r e s , t h e e l e c t r o m a g n e t i c n u c l e a r p r o p e r t i e s w e r e

c a l c u l a t e d u s i n g b o t h t h e C o r i o l i s c o u p l i n g m o d e l a n d t h e N i l s s o n m o d e l

w i t h o u t m i x i n g . H e r e w e c o n c e n t r a t e o n i n t r a b a n d m a g n e t i c d i p o l e ( M l )

a n d e l e c t r i c q u a d r u p o l e ( E 2 ) t r a n s i t i o n s . A b s o l u t e r e d u c e d m a g n e t i c

d i p o l e a n d e l e c t r i c q u a d r u p o l e t r a n s i t i o n p r o b a b i l i t y m a t r i x e l e m e n t s w e r e

c a l c u l a t e d u s i n g t h e F o r t r a n c o m p u t e r p r o g r a m M O T R A N

d e v e l o p e d a t Y a l e b y W . S c h o l z ( S c 66) . W e t a b u l a t e i n F i g . 10 3 t h e s e

2 1 2 3q u a n t i t i e s f o r N e a n d N a c o m p a r i n g t h e o r y w i t h e x p e r i m e n t f o r t h e

C o r i o l i s m i x e d a n d u n m i x e d w a v e f u n c t i o n s . T h e p a r a m e t e r s C , D , a n d g

w e r e c h o s e n i n t h e m a n n e r p r e v i o u s l y d e s c r i b e d i n f i t t i n g t h e o b s e r v e d

e n e r g y l e v e l s . I n t h e u n m i x e d c a l c u l a t i o n s , t h e B ( M 1 ) a n d B ( E 2 ) t r a n s i ­

t i o n p r o b a b i l i t i e s w e r e a l s o c a l c u l a t e d w i t h t h e p r o g r a m M O T R A N b y

s e t t i n g t h e e x p a n s i o n c o e f f i c i e n t s c o r r e s p o n d i n g t o a d m i x t u r e s e q u a l t o

z e r o .

21I n N e t h e o v e r a l l a g r e e m e n t b e t w e e n e x p e r i m e n t a n d t h e o r y f o r

b o t h s e t s o f c a l c u l a t i o n s f o r t h e B ( M 1 ) a n d B ( E 2 ) t r a n s i t i o n p r o b a b i l i t i e s

u p t o t h e l l / 2 + b a n d m e m b e r i n c l u s i v e i s e x c e l l e n t ( F i g . 1 0 3 ) . T h e m i x e d

a n d u n m i x e d c a l c u l a t i o n s p r e d i c t s i m i l a r r e s u l t s a n d a r e p r a c t i c a l l y u n d i s -

t i n g u i s h a b l e o n t h e b a s i s o f t h i s t e s t . T h e r e s u l t s p r e d i c t e d f o r t h e B ( M 1 )

21

b. Electromagnetic properties

TABLE OF B(E2) AND B(MI) FOR Na23 AND Ne21

Na 2 3STOPOVER TRANSITIONS

B ( E 2 ) e 2 b2 x I O ' 2

EXP WITHOUT WITHB(M I ) f j L N 2

EXP WITHOUT WITH0 .442.082.71-5.546.24

00 .44 2.08 2.7 I 5.54

5 /27/29/211/213/2

3/25/27/29/211/2

0.890.1072 .03

* . 1.71 1.07

0.701 0 .489 0 .360

1.51 1.01

0.494 0.419 0.194

0.4130.0510.548

0.2960.3970 .4450.4700 .4 8 5

0.214 0.185 0.399 0.171 0.499

CROSSOVER TRANSITIONS EXP WITHOUT WITH2.08-e 0 7/2 - « - 3 / 22.71 —- 0.44 9/2 -e*. 5/25 .54-e=» 2.08 11/2 -<**-7/26 .2 4 -sB&2 .7 1 13/2 -s*» 9/2

0 .0950.540

0.7201.071.271.40

0.7301.041.301.34

C =— 0.30iia;o D = — 0.02 fia>o /3 = 0.70

N eSTOPOVER TRANSITIONS

B(E2)e2b2x I0’ 2EXP WITHOUT WITH

B(M I) fJLN

EXP WITHOUT WITH0.351.752.874.436.45

00.351.752.874.43

5/27/29/211/213/2

3/25/27/29 /21/2

0.6300.2380.2490.102

0.508 0.318 0.208 0.145 0.107

0.4430.2950.1800.1300.086

0.053 0. 128 0.271 0.104

0.2200.2950.3300.3500.363

0.154 0.12 0.333 0.11 0.425

CROSSOVER TRANSITIONS EXP WITHOUT WITH1.752.874.436.45

00.351.752.87

7/29/211/213/2

3/25/27/29/2

0.1640.2630.338

0.2120.3180.3770.414

0.1950.3070.360.414

C = -0 .3 5 riw0 D = 0 .00 (3 ~ 0 .48

* F r o m C o u l o m b e x c i t a t i o n m e a s u r e m e n t s F i g . 1 0 3

121

t r a n s i t i o n s a r e s l i g h t l y d i f f e r e n t . T h i s i s , o f c o u r s e , e x p e c t e d s i n c e t h e

B ( M l ) ' s a r e m o r e s e n s i t i v e l y d e p e n d e n t c n t h e a d m i x t u r e s o f t h e s p e c i f i c

s i n g l e p a r t i c l e o r b i t . T h e s e r e s u l t s a r e c o n s i s t e n t w i t h t h e e a r l i e r

21a p p l i e d t e s t t o t h e e l e c t r o m a g n e t i c p r o p e r t i e s o f N e .

2 3I n N a a g r e e m e n t b e t w e e n e x p e r i m e n t a n d t h e o r y i s p o o r , e v e n

f o r t h e m i x e d c a l c u l a t i o n ( F i g . 1 0 3 ) . S i n c e t h e r e d u c e d e l e c t r o m a g n e t i c

m a t r i x e l e m e n t s d e p e n d d i r e c t l y o n p a r a m e t e r s t h a t a r e f u n c t i o n s o f t h e

d e f o r m a t i o n , i t i s c o n c e i v a b l e t h a t t o f i t t h e h o l e e x c i t a t i o n , w e f o r c e d a

l a r g e r v a l u e o f t h e d e f o r m a t i o n t h a n w h a t i s r e q u i r e d f o r t h e g r o u n d s t a t e

21r o t a t i o n a l b a n d s i m i l a r t o t h e s i t u a t i o n i n N e , b u t p e r h a p s n o t a s s e v e r e .

I n T a b l e 3 t h e e l e c t r o m a g n e t i c p r o p e r t i e s o f t h e g r o u n d s t a t e b a n d

c a l c u l a t e d w i t h t h e C o r i o l i s c o u p l i n g m o d e l f o r t h r e e v a l u e s o f t h e d e f o r m a ­

t i o n 0 = 0 . 3 , 0 . 5 , a n d 0 . 7 , a r e t a b u l a t e d .

I t w a s c o n c l u d e d f r o m a p r e v i o u s d i s c u s s i o n t h a t t h e l i f e t i m e o f t h e

232 . 0 8 M e V l e v e l i n N a c o r r e s p o n d i n g t o t h e J v = 1 / 2 b a n d m e m b e r m a y

b e i n e r r o r . H e r e w e s e e a g a i n v e r y p o o r a g r e e m e n t b e t w e e n e x p e r i m e n t

a n d t h e o r y f o r t h e B ( E 2 ) a n d B ( M 1 ) t r a n s i t i o n s f r o m t h i s s t a t e c o n s i s t e n t

w i t h o u r p r e v i o u s d e d u c t i o n t h a t a f a s t e r l i f e t i m e w o u l d i m p r o v e o u r

c o m p a r i s o n s . N o t e t h a t t h e c a l c u l a t e d m i x i n g r a t i o , y , w h i c h d o e s n o t

d e p e n d o n t h e l i f e t i m e , a r e n o t f a r f r o m t h e e x p e r i m e n t a l v a l u e .

C l o s e r e x a m i n a t i o n o f T a b l e 3 s h o w s t h a t b e t t e r a g r e e m e n t b e t w e e n

e x p e r i m e n t a n d t h e o r y i s o b t a i n e d f o r t h e 5 / 2 + a n d 9 / 2 b a n d m e m b e r s

a t a d e f o r m a t i o n s l i g h t l y g r e a t e r t h a n 0 . 5 i n a g r e e m e n t w i t h 0 d e t e r m i n e d

122

C a l c u l a t i o n o f B ( E 2 ) A n d B ( M 1 ) F r o m C o r i o l i s C o u p l i n g M o d e l F o r N a

Table 323

T r a n s i t i o n E x p | 3 = 0 . 3 = 0 . 5 = 0 . 7

7 / 2 + - 3 / 2 + 0 . 0 9 5 0 . 1 2 7 0 . 3 6 0 0 . 7 3 02 2 -2

9 / 2 + - 5 / 2 + 0 . 5 4 0 0 . 0 0 3 6 0 . 4 9 9 1 . 0 4B ( E 2 ) e b x I O

5 / 2 > 3 / 2 + 2 . 4 5 0 . 1 7 - 2 0 . 6 6 7 1 . 5 1

7 / 2 + -* 5 / 2 + 0 . 1 0 7 0 . 1 4 9 0 . 4 7 6 1 .0 1 B ( E 2 ) e 2b 2x l 0 _2

9 / 2 + - 7 / 2 + 2 . 0 3 0 . 0 0 7 3 0 . 1 9 2 0 . 4 9 4

5 / 2 + - 3 / 2 + 0 . 4 1 3 0 .1 1 0 0 . 1 7 6 0 . 2 1 4, + . +

7 / 2 - * 5 / 2 0 . 0 5 1 0 . 0 2 4 0 . 1 0 4 0 . 1 8 5 B ( M 1 ) p^.. 4 * . +

9 / 2 - 7 / 2 0 . 5 4 8 0 . 0 0 0 6 4 9 0 . 4 1 2 0 . 3 9 9

5 / 2 + - 3 / 2 + - 0 . 0 9 0 . 0 4 5 0 . 0 6 9 0 . 0 9 5

7 / 2 + - 5 / 2 + - 0 .2 0 0 . 3 3 0 . 2 8 0 . 3 1

9 / 2 + - 7 / 2 + - 0 .1 0 0 . 2 3 0 .0 2 2 0 . 0 5 6 6 = E 2 / M 1

l l / 2 + = * 9 / 2 + - 0 . 1 8 0 . 0 6 2 0 . 3 2 0 . 3 7

1 3 / 2 + - l l / 2 + ------- 0 . 0 5 7 0 . 020 0 . 0 3 5

f r o m t h e g r o u n d s t a t e q u a d r u p o l e m o m e n t . A l t h o u g h t h e h o l e e x c i t a t i o n

i s n o t p r o p e r l y f i t i n t h e e x c i t a t i o n s p e c t r a , b e s t a g r e e m e n t b e t w e e n

e x p e r i m e n t a n d t h e o r y i s o b t a i n e d f o r g =“ 0 . 5 , c o r r e s p o n d i n g t o a s i m i l a r

21d e f o r m a t i o n a n d s i t u a t i o n i n N e . F o r m o r e c o n c l u s i v e d e d u c t i o n s ,

i m p r o v e d a n d a d d i t i o n a l l i f e t i m e m e a s u r e m e n t s f o r t h e h i g h e r l y i n g b a n d

m e m b e r s a r e r e q u i r e d .

2 . R e l a t i v e c o m p a r i s o n s

W e n o t e d e a r l i e r t h a t b y c o m p a r i n g r e l a t i v e r a t h e r

123

t h a n a b s o l u t e e l e c t r o m a g n e t i c p r o p e r t i e s w i t h i n t h e g r o u n d s t a t e r o t a t i o n a l

2 3b a n d i n N a , b e t t e r a g r e e m e n t w a s o b t a i n e d w i t h e x p e r i m e n t u s i n g t h e

s i m p l e N i l s s o n m o d e l w i t h o u t C o r i o l i s m i x i n g . H e r e w e c o m p a r e t h e s e

r e s u l t s w i t h t h o s e c a l c u l a t e d b y t h e C o r i o l i s c o u p l i n g m o d e l .

I n F i g . 1 0 4 b r a n c h i n g r a t i o s a n d m i x i n g r a t i o s a r e t a b u l a t e d f o r

a l l f o u r £ = 11 n u c l e i c o m p a r i n g e x p e r i m e n t w i t h t h e o r y w i t h a n d w i t h o u t

C o r i o l i s m i x i n g . I n e a c h c a s e t h e e x p e r i m e n t a l v a l u e o f § h a s b e e n u s e d

i n t h e c a l c u l a t i o n o f t h e b r a n c h i n g r a t i o f o r r e a s o n s p r e v i o u s l y d i s c u s s e d .

E x a m i n a t i o n o f t h e t a b l e i n d i c a t e s r e m a r k a b l e o v e r a l l a g r e e m e n t f o r a l l

f o u r n u c l e i f o r b o t h b r a n c h i n g a n d m i x i n g r a t i o s c a l c u l a t e d u s i n g u n m i x e d

a n d m i x e d w a v e f u n c t i o n s . T h e o n l y m a j o r e x c e p t i o n i s t h e b r a n c h i n g

77 + 2 3r a t i o f o r t h e J = 9 / 2 l e v e l a t 2 . 7 0 M e V i n N a , w h e r e t h e m i x e d c a l c u l a ­

t i o n i s i n m u c h b e t t e r a g r e e m e n t w i t h e x p e r i m e n t , b u t y o u r e c a l l t h a t h e r e

i s a l s o w h e r e fi i s m o s t u n c e r t a i n .

E l e c t r o m a g n e t i c p r o p e r t i e s a r e n o r m a l l y s e n s i t i v e l y d e p e n d e n t

o n w a v e f u n c t i o n a d m i x t u r e s . I n t h e c a s e o f e l e c t r i c q u a d r u p o l e t r a n s i -

77 "I*t i o n s , t h e t r a n s i t i o n p r o b a b i l i t y c o r r e s p o n d i n g t o t h e K = l / 2 c o m p o n e n t

iff . +o f t h e w a v e f u n c t i o n a d m i x e d i n t o t h e K = 3 / 2 b a n d i s m u c h s m a l l e r t h a n

t h e c o n t r i b u t o r s f r o m t h e = 3 / 2 + b a n d . T h i s i s a r e s u l t o f t h e s m a l l e r

e x p a n s i o n c o e f f i c i e n t a n d C l e b s c h - G o r d a n c o e f f i c i e n t c o n n e c t i n g t h e E 2

t r a n s i t i o n i n t h e K u = l / 2 + b a n d . I n m o s t c a s e s t h e s m a l l e r e x p a n s i o n

c o e f f i c i e n t d o m i n a t e s a n d o v e r w h e l m s a n y m a t r i x e l e m e n t e n h a n c e m e n t

f o r i n t r a b a n d t r a n s i t i o n s . T h e r e f o r e , E 2 a n d M l t r a n s i t i o n s a r e r e l a -

No23 /3=0'7 BRANCHING RATIO E2/MI MIXING RATIO

STOPOVER TRANSITIONS EXP WITHOUT WITH EXP WITHOUT WITH

0.44-®=* 0 5 / 2 — > 100 100 100 -0.09 -0.085 -0.097

2.08 - es-0 .4 4 7/2^*-5/2 91 9? 98 -0.20 -0.22 -0.32

2.71 -®-2.08 9/2 -e**-7/2 37 10 44 -0. 10 -0.063 -0.059

5.54-®-2.71 M/2-®-9/2 78 82 65 -0. 18 -0.23 -0.37

6 .2 4 -^ 5 .5 4 13/2^*11/2 <10 3 2 — -0 .049 -0.034

Ne21 0 =0-48 BRANCHING RATIO E2/MI MIXING RATIO

STOPOVER TRANSITIONS EXP WITHOUT WITH EXP WITHOUT WITH

0.35 ~ea& 0 5/2 -**-3/2 100 100 100 0.05 0.044 0.05

l.75-®-0.35 7/2 -=-5/2 95 95 94 0.16 0.12 0.18

2.87-®-1.75 9/2 -^ 7 / 2 67 59 63 0.09 0.072 0.07

4.43-®- 2.87 11/2 - “ 9/2 55 6 1 55 0.13 0.083 0.1 1

6.45-=-4.43 I3/2-S— II/2 >50 59 63 0.10 0.090 0.06

N q 2I 0=0.48 BRANCHING RATIO E2/MI MIXING RATIO

STOPOVER TRANSITIONS EXP WITHOUT WITH EXP WITHOUT WITH

0.33-®— 0 5/2-=— 3/2 100 100 100 -0.05 -0.044 -0 .06

1.72-®**0.33 7/2-®** 5/2 93 96 92 -0.14 -0.13 -0.19

2.83-®-1.72 9/2-®-7/2 64 56 65 -0. 12 -0 .077 -0.05

Mg23 £=0-7STOPOVER TRANSITIONS

BRANCHING RATIO E2/MI MIXING RATIO

EXP WITHOUT WITH EXP WITHOUT WITH

0.45-e—*0 5/2 -«— 3/2 100 100 100 0.08 0.10 0.097

2 .0 5 ^ -0 .4 5 7/2 -®— 5/2 85 93 98 0. 18 0.242 0.32

2.71-®-2.05 9/2-«— 7/2 35 — 44 — 0.076 0.059

Fig. 104

124

t i v e l y i n s e n s i t i v e t o s m a l l w a v e f u n c t i o n a d m i x t u r e s . T h i s i s b o r n e o u t

i n t h e c o m p a r i s o n o f t h e r e l a t i v e e l e c t r o m a g n e t i c p r o p e r t i e s o f t h e g r o u n d

s t a t e r o t a t i o n a l b a n d f o r t h e £ = 1 1 n u c l e i . A s w i l l b e s e e n i n t h e d i s c u s ­

s i o n o n K n = l / 2 n e g a t i v e p a r i t y b a n d e l e c t r i c d i p o l e t r a n s i t i o n s b e t w e e n

b a n d s , e s p e c i a l l y t h o s e t h a t i n v o l v e h i n d e r e d t r a n s i t i o n s b e t w e e n t h e

d o m i n a n t c o m p o n e n t s , a r e e x t r e m e l y s e n s i t i v e t o w a v e f u n c t i o n s a d m i x e d

f r o m o t h e r b a n d s .

B . K 77 = 1 / 2 R o t a t i o n a l B a n d

A f e w n e g a t i v e p a r i t y s t a t e s a m o n g t h e l o w l y i n g p o s i t i v e p a r i t y

2 3s t a t e s i n N a h a v e b e e n u n a m b i g u o u s l y i d e n t i f i e d . S o m e o f t h e c o r r e s ­

p o n d i n g s t a t e s h a v e b e e n f o u n d i n o t h e r £ = 1 1 n u c l e i . A c o m p o s i t e d i a g r a m

i l l u s t r a t i n g t h e i r p r o p e r t i e s i s s h o w n i n F i g . 1 0 5 . . T h e s e s t a t e s a r e

b e l i e v e d t o b e m e m b e r s o f a K 77= l / 2 r o t a t i o n a l b a n d b a s e d o n o r b i t 4 o f

t h e N i l s s o n m o d e l ; t h a t i s , t h e y a r e h o l e e x c i t a t i o n s g e n e r a t e d b y p r o m o t ­

i n g a n u c l e o n f r o m t h e f u l l y o c c u p i e d o r b i t 4 t o t h e p a r t i a l l y o c c u p i e d

o r b i t 7 .

T h e l a c k o f n e i g h b o r i n g n e g a t i v e p a r i t y l e v e l s f r o m o t h e r b a n d s

r e d u c e s t h e p o s s i b i l i t y o f b a n d m i x i n g l e a v i n g t h e r o t a t i o n a l b a n d i n

q u e s t i o n t o e x h i b i t r a t h e r p u r e r o t a t i o n a l b e h a v i o r . H o w e v e r , t h e r e l a ­

t i v e l y c l o s e s p a c i n g o f t h e l e v e l s a n d t h e p o s s i b i l i t y o f c o m p e t i n g E l

t r a n s i t i o n t o t h e l o w e r l y i n g p o s i t i v e p a r i t y s t a t e s r e d u c e s t h e f r e q u e n c y

o f M l a n d E 2 i n t r a b a n d e l e c t r o m a g n e t i c d e - e x c i t a t i o n o f t h e b a n d m e m b e r s .

7TK =1/2 BANDS FOR £=ll NUCLEI

3 .8 9 -3 .6 6

-2 2 -7 8I

2 .7 9 6

f 5 / 2\ 3 / 2 3 . 8 6 - 3 3 - 6 7 ■5/2T’ 3 . 8 5 - 2 9 - 1 8 - 46 O—4 —36 — 3 / 2- 3.68

— i.2 . 7 9 0 -----15 -85

10Ne21

1/22.81'

■1/2" 3 / 2 ”

7 9 — 5 - 1 6 - 3 / 2 " 3 . 6 8 — 3 - 8

U/ 2 , 3 / 2 ) - 2 . 6 4 -

l/2+00

7 /2

5 / 23 / 2

(_) 3 .9 7 — 4 0 — 5 0 ----- 1 0 -. 7 6 " ° 5 / 2 3 . 8 0 —- 6 —9 0

2 - 1 4 - 3 / 2

9 / 2 + 2 . 7 7 — 100 ■1/2l/2+7/2+ __

5 / 2

3/2"

„N a ro .Na2 312

. 4 . - 3 / 2 (1/2)

* - 3 / 2 (5 /2 ) — 1/2“ 3 /2 '

7 /2

5 / 2

3 / 223

l2MgnB-00-875

Fig. 105

125

I n s u f f i c i e n t i n f o r m a t i o n o n m i x i n g r a t i o s a n d h i g h e r b a n d m e m b e r s p r e v e n t

a n a n a l y s i s s i m i l a r t o t h a t g i v e n f o r t h e g r o u n d s t a t e r o t a t i o n a l b a n d .

1 . A s y m p t o t i c s e l e c t i o n r u l e s

H o w e v e r , w e m a y s t i J . a p p l y t h e a s y m p t o t i c s e l e c t i o n r u l e s

o f A l a g a ( A l 5 5 , W a 5 9 ) t o t h e E l t r a n s i t i o n s b e t w e e n t h e m e m b e r s o f t h e

1 / 2 r o t a t i o n a l b a n d a n d t h e . l o w e r l y i n g l e v e l i n t h e 3 / 2 + g r o u n d s t a t e

r o t a t i o n a l b a n d a n d c o m p a r e t h e p r e d i c t i o n s o f t h e s e s e l e c t i o n r u l e s w i t h

e x p e r i m e n t b a s e d o n t h e a s s u m p t i o n t h a t t h e t r a n s i t i o n s a r e b e t w e e n

N i l s s o n o r b i t s 4 a n d 7 . A g r e e m e n t w o u l d s u b s t a n t i a t e t h e c l a i m t h a t

i n d e e d t h e l / 2 b a n d i s a h o l e e x c i t a t i o n b a s e d o n o r b i t 4 .

F i r s t w e p r e s e n t a c o m p a r i s o n b e t w e e n t h e e x p e r i m e n t a l l y k n o w n

p o s i t i o n o f t h e k n o w n a n d s u s p e c t e d b a n d m e m b e r w i t h t h o s e c a l c u l a t e d

f r o m t h e r i g i d r o t o r m o d e l u s i n g N i l s s o n i n t r i n s i c w a v e f u n c t i o n f r o m

o r b i t 4 . T h e r e s u l t s o f t h e c a l c u l a t i o n a r e s h o w n i n F i g . 1 0 6 f o r d i f f e r e n t

1 /2 d e c o u p l i n g c o n s t a n t s c o r r e s p o n d i n g t o d i f f e r e n t a s s u m e d v a l u e s o f t h e

d e f o r m a t i o n 0 R e a s o n a b l e a g r e e m e n t i s a c h i e v e d w i t h a d e f o r m a t i o n i n t h e

n e i g h b o r h o o d o f 0 . 1 5 . T h i s a t f i r s t a p p e a r s t o b e a s m a l l d e f o r m a t i o n —

f a r f r o m b e i n g a n a s y m p t o t i c v a l u e . H o w e v e r , a t t h i s d e f o r m a t i o n t h e

p r o b a b i l i t y d e n s i t y o f t h e w a v e f u n c t i o n i s a l r e a d y 9 0 p e r c e n t o f i t s

a s y m p t o t i c v a l u e .

I n t h e a s y m p t o t i c l i m i t t h e w a v e f u n c t i o n , c h a r a c t e r i z e d b y q u a n t u m

n u m b e r s I N ^ A > > i s a n e i g e n f u n c t i o n o f t h e s y m m e t r i c a n i s o t r o p i c z

EXCIT

ATION

EN

ERGY

(M

eV)

KT*l/2"BAND OF Na23 COMPARED WITH ORBIT# 4 OF NILSSON MODEL------------------ 9 / 2 “

_ _ ------------------ 9 / 2- 7 / 2 . 9 / 2 _ 7 /2 ------------------- 7 /2

6 .0 5 -------------------------7/2_)( 5 / 2 , 3 / 2 )

3.85-------------------- 5/2"’ 5/2 5/2~3.68 3/2" 3 / z .o / c --------------- 3 /2 - 3/2-

- 9 / 2

-7/2

- 5 / 2 - |

- 3 / 2 “

2.64--------------------1/2 1/2 1/2“ 1/2“ 1/2"23 DEFORMATION £=0 £=.11 £=.21 £=.32

Na EXP DECOUPLING Rn_____________________ PARAMETER a -.83 Q-.63 Q-.50

F i g . 106

-00

-87

8

126

oscillator Hamiltonian

w h e r e t h e t e r m C f , * 3 + D £ . - t h a s b e e n n e g l e c t e d . T h i s c o r r e s p o n d s t o t h e

a s s u m p t i o n t h a t t h e d e f o r m a t i o n f o r c e s a r e m u c h g r e a t e r t h a n t h e e f f e c ­

t i v e s p i n - o r b i t f o r c e s a n d i n c o m p a r i s o n t h e l a t t e r c a n b e n e g l e c t e d . T h e

c o r r e s p o n d i n g l y d e r i v e d s e l e c t i o n r u l e s a r e o n l y a p p r o x i m a t e a n d a r e

e x p e c t e d t o " h i n d e r " r a t h e r t h a n " f o r b i d " t r a n s i t i o n s .

T h e e l e c t r i c d i p o l e t r a n s i t i o n s i n q u e s t i o n c o n n e c t N i l s s o n o r b i t s

4 a n d 7 , w h o s e s i n g l e p a r t i c l e e i g e n f u n c t i o n s i n t e r m s o f a s y m p t o t i c

q u a n t u m n u m b e r s a r e c h a r a c t e r i z e d a s | l 0 1 > a n d 12 1 1 > , r e s p e c t i v e l y .

B e t w e e n s u c h o r b i t s = - 1 , & N = - 1 , £ n = “1» an(I AA= 0 . A c c o r d -z

i n g t o t h e A l a g a a s y m p t o t i c s e l e c t i o n r u l e s E l t r a n s i t i o n s b e t w e e n o r b i t s

s a t i s f y i n g t h e s e s e l e c t i o n r u l e s a r e " h i n d e r e d " w i t h r e s p e c t t o t h e

5W e i s s k o p f e s t i m a t e b y a f a c t o r o f 1 0 . A t a b u l a t i o n o f h i n d e r e d a n d

u n h i n d e r e d e l e c t r o m a g n e t i c t r a n s i t i o n s b e t w e e n N i l s s o n o r b i t s o f i n t e r e s t

i n s d s h e l l n u c l e i i s s h o w n i n T a b l e 4 a n d m o r e s p e c i f i c a l l y a t a b u l a t i o n

o f k n o w n E 2 t r a n s i t i o n s t r e n g t h s b e t w e e n o r b i t s 4 a n d 7 m e a s u r e d i n

W e i s s k o p f u n i t s d e f i n e d a s

w h e r e T e ( E l ) i s t h e e x p e r i m e n t a l w i d t h a n d r j [ E l ) i s t h e W e i s s k o p f

e s t i m a t e i s s h o w n i n T a b l e 5 .

127

Table 4

Asymptotic Selection Rules Between sd Shell Nilsson Orbits

O r b i t ( K f f ) - ( K f f ) f E l M 2 M l E 2

4 - 7 1 / 2 ” - 3 / 2 + H i n d U n h i n d

4 — 6 1 / 2 " - l / 2+ H i n d H i n d

4 - 9 1 / 2 " - l / 2+ U n h i n d U n h i n d

6 - 7 l / 2 + - 3 / 2 + U n h i n d U n h i n d

6 - 9 l / 2+ — l / 2+ H i n d H i n d

7 - 6. + . +

3 / 2 - 1 / 2 U n h i n d U n h i n d

7 - 9 3 / 2 - l / 2 + U n h i n d U n h i n d

9 - 7 1 / 2+ - 3 / 2 + U n h i n d H i n d

9 - 6 l / 2+ - l / 2+ H i n d H i n d

1 1 - 7 l / 2 + - 3 / 2 + U n h i n d U n h i n d

1 1 - 6 l / 2+ - l / 2+ H i n d H i n d

1 1 - 9 l / 2+ - l / 2+ H i n d H i n d

5 - 7. + . +

5 / 2 - 3 / 2 U n h i n d U n h i n d

T a b l e 52 3

H i n d e r e d E l T r a n s i t i o n s i n N a

T r a n s i t i o n S p i n | M ( E 1 ) | 2

2 . 6 4 - 0 1 / 2 " - 3 / 2 + 4 . 7 x 1 0 ' 4

3 . 6 8 - . 4 4 3 / 2 " - 5 / 2 + 6 .0 x 1 0 ‘ 3 4 . 0 X 1 0

3 . 8 5 - 2 . 0 8 T1.oaLO 6 . 0 x 1 0 " 4

1 .2 x 1 0 “

3 . 8 5 - . 4 4 5 / 2 " - 5 / 2 + 3 . 1 x 1 0 ” g4 . 2 x 1 0 "

3 . 8 5 - 0 5 / 2 " - 3 / 2 + 3 . 5 x 1 0 5

128

O t h e r p o s s i b l e h o l e e x c i t a t i o n s i n t h e p s h e l l a r e r e j e c t e d o n t h e b a s i s

o f b i n d i n g e n e r g i e s a n d t h e f a c t t h a t s u c h E l t r a n s i t i o n s w o u l d b e u n h i n ­

d e r e d , i n v a s t d i s a g r e e m e n t w i t h o u v e x p e r i m e n t a l r e s u l t s .

2 . C a l c u l a t i o n o f E l t r a n s i t i o n s

S i n c e t h e K n = 3 / 2 + g r o u n d s t a t e b a n d i s n o t c o m p l e t e l y p u r e ,

s m a l l a d m i x t u r e s f r o m o t h e r b a n d s t h a t w o u l d c o r r e s p o n d t o a n u n h i n ­

d e r e d c o u l d y i e l d t r a n s i t i o n s i g n i f i c a n t c o n t r i b u t i o n s t o t h e E l t r a n s i t i o n

p r o b a b i l i t y . A c o m p a r i s o n b e t w e e n t h e e x p e r i m e n t a l e l e c t r i c d i p o l e

b r a n c h i n g r a t i o s a n d c a l c u l a t e d o n e s f o r a n u n m i x e d a n d m i x e d g r o u n d

s t a t e r o t a t i o n a l b a n d t o g e t h e r w i t h a b r i e f s u m m a r y o f t h e c a l c u l a t i o n i s

s h o w n i n F i g . 10 7 . T h e w a v e f u n c t i o n e x p a n s i o n c o e f f i c i e n t s u s e d i n t h e

c a l c u l a t i o n w e r e t a k e n f r o m D u b o i s ( D u 6 7 ) . T h e b a n d m i x i n g c a l c u l a t i o n ( L i 6 9 f )

g i v e s r e s u l t s m u c h c l o s e r t o e x p e r i m e n t a n d i s a s u b s t a n t i a l i m p r o v e m e n t

o v e r t h e u n m i x e d c a s e f o r t h e t r a n s i t i o n s o u t o f t h e 3 . 6 8 s t a t e . T h e l a c k

o f a g r e e m e n t f o r t h e 3 . 8 5 s t a t e c o u l d b e a r e s u l t o f a n i n a c c u r a t e s e t o f

e x p a n s i o n c o e f f i c i e n t s o r m a y b e a r e f l e c t i o n o f t h e f a c t t h a t t h e 3 . 8 5 l e v e l

i s n o t a n e g a t i v e p a r i t y s t a t e a s w a s o r i g i n a l l y a s s u m e d .

T h e s e n s i t i v i t y o f e l e c t r i c d i p o l e t r a n s i t i o n s b e t w e e n o r b i t s 7 a n d

r r „ t t = l / 2+ b a n d a d m i x e d i n t o t h e g r o u n d s t a t e4 c n w a v e f u n c t i o n s f r o m K

b a n d c a n b e u s e d t o d e t e r m i n e t h e a m o u n t o f m i x i n g b y t r e a t i n g t h e e x p a n ­

s i o n c o e f f i c i e n t s a s p a r a m e t e r s i n f i t t i n g t h e e l e c t r i c d i p o l e t r a n s i t i o n s

Indeed, El transitions are hindered by an average factor close to 10 .

ocn

R E L A T I V E E l G A M M A W I D T H S

i r I+h I = { i } l i - ij

^ init =</'i/2"(I ) UNMIXED 1/2“ BAND

r (ed

FINAL=2 a K,'/'K' K MIXED 3/2+ BAND

ABSOLUTE El GAMMA WIDTHWHERERELATIVE El

GAMMA WIDTH

2

r= (22 2.)E3(MeV)|lAK-(I,T)| mev K'= 3/2,1/2,1/2'

A K' = <Ca K * I ^ 1 / 2 " ^ ^rrR = TOTAL(EI)

LEVEL 0 .44 2.0 8A3/2 =a3/2 (~-07 i/ C| a3/2 .986 .872 .914

A |/2 =Q |/2 ( 0 0 5 C2 +.0224 C3 ) a 1/2 -.106 .278 -.148

a I/2‘ =a|/2'(-6625C2-.l300C3 ) al/2 .126 .343 .206

Tr =RELATIVE El GAMMA WIDTHSNILSSON BAND

TRANSITION EXP MODEL MIXINGCALC CALC

2 . 6 4 —b» 0 100 1 0 0 1003 . 6 8 —b O 4 5 0 1 63 . 6 8 —®~.44 9 6 5 0 8 53 . 8 5 —»^0 2 5 15 3 03 .8 5 — 4 4 14 7 4 6 83 . 8 5 —®*2.08 6 1 1 1 2

Fig. 107

129

b e t w e e n b a n d s . T h e e x p a n s i o n c o e f f i c i e n t s d e t e r m i n e d t h i s w a y c a n t h e n

b e c o m p a r e d t o t h o s e c a l c u l a t e d b y t h e C o r i o l i s C o u p l i n g M o d e l . S i n c e

w e h a v e a l r e a d y o b t a i n e d r e a s o n a b l e a g r e e m e n t f o r t h e d e - e x c i t a t i o n o f

3 . 68 M e V s t a t e u s i n g t h e C o r i o l i s C o u p l i n g M o d e l , t h i s p r o c e d u r e w a s n o t

e x p l o i t e d a n d t h o u g h t n o t t o y i e l d a n y n e w i n f o r m a t i o n i f a p p l i e d t o t h i s

l e v e l .

A l t h o u g h t h i s i s n o t t h e c a s e f o r t h e 3 . 8 5 M e V s t a t e , e x p a n s i o n

c o e f f i c i e n t s b y f i t t i n g t h e e l e c t r i c d i p o l e d e c a y h a v e n o t b e e n d e t e r m i n e d .

3 . B a n d p u r i t y

U n f o r t u n a t e l y , t h e p u r i t y o f t h e n e g a t i v e p a r i t y b a n d i s

d i f f i c u l t t o a s c e r t a i n w i t h c e r t a i n t y , s i n c e m o s t t r a n s i t i o n s a r e E l i n t e r ­

b a n d t r a n s i t i o n s w h i c h i n v o k e s s t a t e s f r o m o t h e r b a n d s . A s w e h a v e j u s t

s h o w n s u c h t r a n s i t i o n , e s p e c i a l l y h i n d e r e d o n e s , a r e s e n s i t i v e l y d e p e n ­

d e n t o n w a v e f u n c t i o n a d m i x t u r e s .

B y c o n f i r m i n g t h e n e g a t i v e p a r i t y a s s i g n m e n t t o t h e J = 5 / 2 , 3 . 8 5

M e V l e v e l a n d t h e i d e n t i f i c a t i o n o f h i g h e r b a n d m e m b e r s w h o s e p o s i t i o n s

a r e p r e d i c t e d b y t h e s y s t e m a t i c s o f t h e l o w e r t h r e e m e m b e r s w o u l d

s t r o n g l y s u p p o r t o u r r o t a t i o n a l m o d e l i n t e r p r e t a t i o n .

A p o s s i b l e c a n d i d a t e f o r t h e J u = 7 / 2 m e m b e r i s t h e l e v e l a t 6 . 0 4

M e V . I t s e n h a n c e d e l e c t r o m a g n e t i c d e - e x c i t a t i o n t o s u p p o s e d J n = 5 / 2 ^ \

3 . 8 5 M e V l e v e l a n d i t s p o s i t i o n i n e x c i t a t i o n s u p p o r t t h i s i n t e r p r e t a t i o n .

A d d i t i o n a l l e v e l s i n t h i s v i c i n i t y t h a t a r e p o s s i b l e h i g h s p i n s t a t e s ( e . g .

130

9 / 2 ) a s c a n b e s e e n b y g a m m a r a y d e - e x c i t a t i o n t o t h e J 77^ 9 / 2 ~ l e v e l a t

2 . 7 0 M e V i n F i g . 1 0 8 , a r e t h e 6 . 5 8 4 a n d 6 . 2 0 0 M e V l e v e l s . T h e a n g u l a r

c o r r e l a t i o n r e s u l t s a s s i g n e d J = 5 / 2 o r J = 9 / 2 t o t h e 6 . 5 8 4 M e V l e v e l .

A d d i t i o n a l m e a s u r e m e n t s o n t h e s e l e v e l s a r e r e q u i r e d t o s p e c i f i c a l l y

i d e n t i f y t h e i r r o t a t i o n a l n a t u r e .

C . S e n s i t i v i t y o f E l e c t r o m a g n e t i c P r o p e r t i e s t o t h e N u c l e a r D e f o r m a t i o n

T h e d e f o r m a t i o n p a r a m e t e r g i s f u n d a m e n t a l t o t h e s p e c i f i c a t i o n

o f t h e c o l l e c t i v e r e p r e s e n t a t i o n o f n u c l e i . I n p r a c t i c e g i s d e t e r m i n e d b y

g i v i n g i t a v a l u e w h i c h b r i n g s t h e m o d e l p r e d i c t i o n s i n t o b e s t a g r e e m e n t

w i t h t h e e m p i r i c a l o b s e r v a t i o n s o f t h e s t a t e a n d d y n a m i c n u c l e a r p r o p e r t i e s .

I t s h o u l d b e e m p h a s i z e d t h a t f r o m t h e o u t s e t w e a r e a s s u m i n g t h a t a l l

h e x a d e c o p d e , a n d h i g h e r , d e f o r m a t i o n s m a y s a f e l y b e i g n o r e d , c o n s e q u e n t l y ,

r e s t r i c t i n g c o n s i d e r a t i o n t o q u a d r u p o l e d e f o r m a t i o n s o n l y . F u r t h e r m o r e ,

t h e i n t r i n s i c r i g i d i t y d e m o n s t r a t e d p r e v i o u s l y s u g g e s t t h a t b o t h b e t a a n d

g a m m a v i b r a t i o n s o f t h e n u c l e a r s u r f a c e w o u l d b e v e r y h i g h i n t h e e x c i t a ­

t i o n s p e c t r u m a n d t h e r e f o r e c a n a l s o b e i g n o r e d . W e e m p h a s i z e t o o t h a t a

l a r g e m o m e n t o f i n e r t i a a p p r o a c h i n g t h e r i g i d b o d y v a l u e i s n o t i n i t s e l f

a g u a r a n t e e o f a r i g i d n u c l e a r s t r u c t u r e .

1 . C o r i o l i s c o u p l i n g m o d e l p r e d i c t i o n s

I n t h e C o r i o l i s c o u p l i n g m o d e l g i s a s s u m e d a p r i o r i t o b e

t h e s a m e f o r e a c h r o t a t i o n a l b a n d g e n e r a t e d o n t h e s i n g l e p a r t i c l e o r b i t s

i n t h e s d s h e l l . I n c o m p a r i n g t h e m o d e l p r e d i c t i o n s w i t h e x p e r i m e n t , h o w ­

e v e r , t h e a c t u a l g i n t h e c a l c u l a t i o n r e p r e s e n t s a t b e s t s o m e g a v e r a g e d

CO

INC

IDEN

T AL

PHA

PA

RT

ICL

ES

2.71 Mev CASCADE PHOTONS

A L P H A P A R T I C L E C H A N N E L N U M B E RF i g . 1 0 8

CO

INC

IDEN

T 2.

71

CASC

ADE

RA

DIA

TIO

N

131

o v e r t h e d e f o r m a t i o n s f o r e a c h i n d i v u d a l b a n d a n d t h e s e c l e a r l y n e e d n o t

b e t h e s a m e . T h e h o l e b a n d s i n t h e s e n u c l e i , i d e n t i f i e d t h r o u g h a p p r o p r i a t e

p i c k u p r e a c t i o n s , r e q u i r e c h a r a c t e r i s t i c a l l y l a r g e r d e f o r m a t i o n s f o r t h e i r

m o d e l d e s c r i p t i o n t h a n d o t h e p a r t i c l e b a n d s , e q u i v a l e n t l y i d e n t i f i e d i n

s t r i p p i n g e x p e r i m e n t s . T h e a s s u m p t i o n o f a c o m m o n d e f o r m a t i o n f o r a l l

r o t a t i o n a l b a n d s i s a s e v e r e l i m i t a t i o n o f t h e C o r i o l i s m o d e l , h o w e v e r , i t

n e e d b e e m p h a s i z e d t h a t t h e n u c l e i s t u d i e d h e r e i n w i t h t h e i r e m p i r i c a l l y

d e m o n s t r a t e d r i g i d i t y ' m a y w e l l p r o v i d e t h e c a s e s w h e r e t h i s C o r i o l i s m o d e l

w o u l d b e e x p e c t e d t o h a v e m a x i m u m v a l i d i t y .

2 . I n e l a s t i c s c a t t e r i n g o f p a r t i c l e s

I n t h e s e e x p e r i m e n t s t h e m o d e l c r o s s s e c t i o n i s c o m p a r e d

2t o t h e e x p e r i m e n t a l o n e ; t h e r a t i o i s p r o p o r t i o n a l t o g R ( O w 6 4 ) . A l t h o u g h

t h e d e f o r m a t i o n j3 d e t e r m i n e d h e r e i s c h a r a c t e r i s t i c o f a g i v e n l e v e l r a t h e r

t h a n a s d e s c r i b e d a b o v e , i t s v a l u e i s n o t d e t e r m i n e d d i r e c t l y a n d i s

u n c e r t a i n b y ' t h e c o r r e s p o n d i n g a m o u n t i n R , t h e e f f e c t i v e n u c l e a r r a d i u s .

3 . D i r e c t m e a s u r e m e n t f r o m e l e c t r o m a g n e t i c p r o p e r t i e s

D i r e c t m e a s u r e m e n t o f e l e c t r i c q u a d r u p o l e t r a n s i t i o n s i s

o n e o f t h e m o s t s e n s i t i v e a n d r e l i a b l e d e t e r m i n a t i o n s o f t h e d e f o r m a t i o n g

a n d t h e r e l a t e d p a r a m e t e r , t h e i n t r i n s i c q u a d r u p o l e m o m e n t Q q .

A l t h o u g h m a g n e t i c d i p o l e t r a n s i t i o n p r o b a b i l i t i e s d e p e n d o n 0 ,

r e s u l t s a r e n o t a s r e l i a b l e o r a m e n a b l e t o a l u c i d i n t e r p r e t a t i o n . T h i s

f o l l o w s b e c a u s e o f t h e d o m i n a t i n g s i n g l e p a r t i c l e o r b i t a l t e r m i n t h e w a v e

132

f u n c t i o n s d e s c r i b i n g t h e s t a t e s i n t h e s e n u c l e i . I n t h e E 2 m a t r i x e l e m e n t s

t h e c o r e c o n t r i b u t i o n d o m i n a t e s a n d t h e r e i s e s s e n t i a l l y n o s e n s i t i v i t y t o

s m a l l w a v e f u n c t i o n a d m i x t u r e o f o t h e r s i n g l e p a r t i c l e t e r m s . O n t h e

o t h e r h a n d , t h e M l m a t r i x e l e m e n t r e c e i v e s n o c o n t r i b u t i o n f r o m t h e c o r e

p e r s e a n d a r i s e s e n t i r e l y f r o m t h e s i n g l e p a r t i c l e c o m p o n e n t s . F o r t h i s

r e a s o n i t s m a g n i t u d e i s e x t r e m e l y s e n s i t i v e t o b o t h a m p l i t u d e a n d p h a s e

o f r e l a t i v e l y s m a l l s i n g l e p a r t i c l e a d m i x t u r e s i n t h e w a v e f u n c t i o n s .

F r o m t h i s i t f o l l o w s t h a t t h e C o r i o l i s m o d e l w o u l d b e e x p e c t e d t o

b e t t e r r e p r e s e n t t h e M l t r a n s i t i o n m o m e n t s , a s i n d e e d i t d o e s . H o w e v e r ,

e v e n t h i s m o d e l a p p a r e n t t y m i s s e s c e r t a i n t e r m s o f i m p o r t a n c e r e f l e c t e d

i n t h e r e m a i n i n g d i s c r e p a n c y b e t w e e n m o d e l a n d e x p e r i m e n t a l v a l u e s .

T o t h e e x t e n t t h a t t h e C o r i o l i s m o d e l d o e s g i v e a r e a s o n a b l e r e p r e ­

s e n t a t i o n o f t h e M l m o m e n t s , t h e m o d e l s h o u l d a l s o g i v e p r e d i c t i o n s o f

m u l t i p o l e m i x i n g r a t i o s i n E 2 / M 1 t r a n s i t i o n s .

S i n c e w e a r e i n t e r e s t e d i n t h e s e n s i t i v i t y a n d d e p e n d e n c e o f s u c h

p r o p e r t i e s o n t h e d e f o r m a t i o n r a t h e r t h a n a d e t a i l e d m o d e l f i t , w e c h o o s e

f o r c o m p u t a t i o n a l c o n v e n i e n c e a n d f o r a m o r e d i r e c t i n t e r p r e t a t i o n t h e

s i m p l e r o t o r m o d e l w i t h N i l s s o n i n t r i n s i c w a v e f u n c t i o n s . W e c a l c u l a t e

B ( E 2 ) a n d B ( M 1 ) r e d u c e d m a t r i x e l e m e n t s a n d m u l t i p o l e m i x i n g r a t i o s a s

a f u n c t i o n o f d e f o r m a t i o n 0 f o r o p t i m u m v a l u e s o f t h e C a n d D p a r a m e t e r s

d e t e r m i n e d p r e v i o u s l y f r o m t h e C o r i o l i s c o u p l i n g m o d e l . B y p l o t t i n g t h e s e

2 1 2 3p r e d i c t i o n s a s f u n c t i o n s o f 0 f o r N e a n d N a , a s s h o w n i n F i g s . 1 1 0 t o

133

t h e i r e r r o r r e p r e s e n t e d b y t h e l e n g t h o f t h e a r r o w s , n u m e r i c a l v a l u e s o f

t h e e f f e c t i v e 0 , w i t h a p p r o p r i a t e u n c e r t a i n t y a r e d e t e r m i n e d f o r t h e

i n d i v i d u a l t r a n s i t i o n s i n t h e g r o u n d t a t e r o t a t i o n a l b a n d . T h e s e c u r v e s

p r o v i d e s y s t e m a t i c a n d a b u n d a n t c r o s s c h e c k s o n t h e m a g n i t u d e o f g a n d

o n t h e s e n s i t i v i t y o f i t s d e t e r m i n a t i o n . A g r a p h o f g a s a f u n c t i o n o f t h e

2 3 2 1r e l a t e d p a r a m e t e r Q q f o r N a a n d N e i s s h o w n i n F i g . 1 0 9 .

21F o r N e , a v e r a g e v a l u e s o f t h e d e f o r m a t i o n g w e r e d e t e r m i n e d

f r o m t h e c r o s s o v e r B ( E 2 ) , s t o p o v e r B ( E 2 ) , a n d m i x i n g r a t i o a s 0 . 4 4 ,

2 30 . 4 6 , a n d 0 . 6 4 , r e s p e c t i v e l y . F o r N a , 0 i s n o t a s w e l l d e f i n e d , p r i m a r i l y

b e c a u s e o f t h e i n a c c u r a c y i n t h e e x p e r i m e n t a l l i f e t i m e s , b u t f r o m t h e

m i x i n g r a t i o c o m p a r i s o n i n F i g . 1 1 7 , a v a l u e o f g = 0 . 6 3 w a s d e t e r m i n e d . I n

2 1 2 3T a b l e 6 t h e m e a n e f f e c t i v e d e f o r m a t i o n g i s t a b u l a t e d f o r N e a n d N a

a s o b t a i n e d t h r o u g h v a r i o u s m e a s u r e m e n t s .

117, in c lu s iv e , and sup erim p o sin g the experim ental m easurem ents with

T a b l e 6 . D e f o r m a t i o n g

s t a t i c q u a d r u p o l e B ( E 2 ) t o f i r s t B ( E 2 ) f r o m M i x i n g r a t i o N u c l e u s m o m e n t o f G N D e x c i t e d s t a t e l i f e t i m e s a n d f r o m r o t o r

s t a t e f r o m C o u l o m b r o t o r m o d e l m o d e le x .

N e 2 1 0 . 5 1 0 . 4 8 0 . 4 5 0 . 6 4

N a 23 0 . 5 2 0 . 5 0 0 . 5 0 0 . 6 3

E x a m i n a t i o n o f t h e e n t r i e s i n T a b l e 6 r e a d i l y s h o w s t h a t i n d e e d t h e

d e f o r m a t i o n i s w e l l d e f i n e d f o r t h e g r o u n d s t a t e b a n d o f t h e n u c l e i t e s t e d

INTRINSIC QUADRUPOLE MOMENT Qn (eb)

*5OQoCD

OmT]o

H

>ioo■ft

DEFORMATION t/3)Fig. 110

DEFORMATION </3)

Fig. 112

B(E

2) ez

bz ilO

"

DEFORMATION {£)Fig. I l l

DEFORMATION i$)

Fig. 113

10.0

1.0

o.ot

‘ I ' I ' 1 ' 1 ' 1 '1 ! 1 .. ■»* 11 1 | IN<

NILSSON MODEL (WITHOUT CORIOLIS MIXING)

l3/2-**9/2 11/2-*-7/2

1

ii

9/2-*-

7/2

5/2

3/2

■;..... Ii i

11

-- / Z Z■

W //

/ / '/ ///A / "/ // i/

\ /

--- I ll /-

0 3 0.4 0 5 0.6 0.7DEFORMATION (0)

Fig. U4

0.9

COo<<rozX2MUJ

DEFORMATION (£)KlR. 116

0.5

0.1

0.05

0.01

B(E2) «2

b2 *I0-

DEFORMATION </9)

Fig. 115

■ i ■ i ' 1 1 7'*“ 12 3

1NILSSON MODEL (WITHOUT CORIOLIS MIXING 1

-0 30 tiw0

•"wo

-EXPERIMENTAL 11/2-te7/2-**

9/25/2

8 =0.81 E(NeV) / B(E21 V B[MI)e2t2

5/2-3/2

r - 9/2-7/2;.13/2-**11/2:

j / '

0AVE. 0.63

/

. . i ----1----- ■0.1 0.2 0.3 0.4 0.3 0.6 0.7 0.8 0.9 1.0

DEFORMATION (0)F ig . 117

134

a l t h o u g h N a d o e s a p p e a r t o h a v e a s l i g h t l y l a r g e r d e f o r m a t i o n .

T h e d e f o r m a t i o n g w a s n o t w e l l d e f i n e d b y t h e c o m p a r i s o n o f m o d e l

p r e d i c t i o n s w i t h t h e m e a s u r e d m a g n e t i c d i p o l e t r a n s i t i o n s t r e n g t h s . T h e

s o u r c e o f t h e g e n e r a l d i s a g r e e m e n t , h e r e f o u n d , i s n o t o b v i o u s , b u t a t

21l e a s t i n t h e c a s e o f N e , b e t t e r a g r e e m e n t w a s o b t a i n e d w h e n C o r i o l i s

m i x i n g w a s i n c l u d e d . B y e x a m i n i n g t h e m a t r i x e l e m e n t s o f t h e i n d i v i d u a l

c o n t r i b u t i o n s f r o m d i f f e r e n t r o t a t i o n a l b a n d s , i t b e c o m e s e v i d e n t t h a t t h e

B ( M 1 ) r e d u c e d m a t r i x e l e m e n t s a r e m o r e s e n s i t i v e t o w a v e f u n c t i o n a d m i x ­

t u r e s t h a n a r e t h e B ( E 2 ) ; t h i s i s a l m o s t c e r t a i n l y t h e d o m i n a n t s o u r c e o f

t h e d i s a g r e e m e n t r e f l e c t i n g s m a l l e r a d m i x t u r e s n o t y e t p r o p e r l y i n c o r p o r a ­

t e d w i t h i n t h e m o d e l f r a m e w o r k .

O n e o f t h e m o s t s u r p r i s i n g a s p e c t s o f a l l t h e s e c o m p a r i s o n s i s t h e

s y s t e m a t i c a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l m u l t i p o l e m i x i n g r a t i o a n d

t h e r o t o r m o d e l p r e d i c t i o n s a t d e f o r m a t i o n s t h a t a r e 1 5 % l a r g e r t h a n t h o s e

21d e t e r m i n e d f r o m t h e B ( E 2 ) m a t r i x e l e m e n t s f o r N e . A s s u m i n g t h a t t h e

f o r m u l a s r e l a t i n g g t o t h e c a l c u l a t e d e l e c t r o m a g n e t i c p r o p e r t i e s a r e c o r r e c t

a n d t h e a p p r o x i m a t i o n s a r e n o t b r e a k i n g d o w n i n t h e l i m i t o f l a r g e g , t h e

m o d e l i s s y s t e m a t i c a l l y o v e r e s t i m a t i n g t h e B ( E 2 ) e l e m e n t s o r u n d e r ­

e s t i m a t i n g t h e B ( M 1 ) m a t r i x e l e m e n t s ; t h i s , h o w e v e r , i s n o t c o n s i s t e n t

w i t h t h e e x p e r i m e n t a l B ( M l ) ’ s w h i c h a r e o v e r e s t i m a t e d b y t h e m o d e l r a t h e r

t h a n t h e c o n v e r s e . A n o t h e r p o s s i b i l i t y , o f c o u r s e , i s t h a t t h e e x p e r i m e n t a l

m i x i n g r a t i o s a r e s y s t e m a t i c a l l y t o o l a r g e , b u t t h i s i s i m p r o b a b l e . T h e

135

s e n s i t i v i t y o f 5 t o i s s h o w n b y t h e l e n g t h o f t h e a r r o w s i n F i g s . 1 1 3 a n d

1 1 7 . S u c h l a r g e e r r o r s s c a n n i n g £ f r o m 0 . 3 t o 0 . 8 m a k e i t d i f f i c u l t t o p u t

m u c h m e a n i n g i n t h i s d i s c r e p a n c y .

S i n c e w e a r e c o n c e r n e d w i ' t h n u c l e i t h a t g e n e r a l l y h a v e l a r g e d e f o r m a ­

t i o n s 0 5 , c a r e f u l c o n s i d e r a t i o n m u s t b e g i v e n t o t h e p o s s i b i l i t y o f b r e a k ­

d o w n o f v a l i d i t y o f t h e u s u a l f o r m a l i s m w h e r e i n e x p a n s i o n i n £ a r e t y p i c a l l y

c a r r i e d o n l y t o f i r s t o r d e r . A t g ~ 0 . 5 , I . c o n t a i n s c o r r e c t i o n > 1 0 % f r o mr i g

s u c h h i g h e r o r d e r t e r m s n o r m a l l y n e g l e c t e d a s d o e s t h e i n t r i n s i c q u a d r u p o l e

m o m e n t Q .oD i s a g r e e m e n t s i n e v a l u a t i n g Q q f o r n u c l e i i n t h i s m a s s r e g i o n f r o m

C o u l o m b e x c i t a t i o n a n d f r o m t h e r e - o r i e n t a t i o n e f f e c t h a v e b e e n n o t i c e d

b y S c h w a l m ( S c 6 9 ) . T h e l a t t e r m e a s u r e m e n t s g i v e 3 0 % s y s t e m a t i c a l l y

h i g h e r v a l u e s o f Q q . T h e s o u r c e o f t h e s e i n c o n s i s t e n c i e s a r e n o t y e t k n o w n .

I t s h o u l d b e k e p t i n m i n d t h a t £ i s a p a r a m e t e r m e a s u r i n g t h e

s t r e n g t h o f t h e Y 0 ( Qi ( r$ t e r m i n t h e H a m i l t o n i a n a n d i t i s o n l y i n t h e l i m i t

o f o u r r i g i d r o t o r m o d e l t h a t i t c a n b e " c l a s s i c a l l y " i n t e r p r e t e d a s t h e

p h y s i c a l d e f o r m a t i o n o f t h e n u c l e u s .

O t h e r m o d e l s g e n e r a t e d e f o r m e d s i n g l e p a r t i c l e o r b i t a l s ( K e 6 4 ) ,

w h i c h a r e v e r y s i m i l a r t o t h o s e g e n e r a t e d b y a d e f o r m e d h a r m o n i c o s c i l ­

l a t o r , a s a f u n c t i o n o f a s t r e n g t h p a r a m e t e r p r o p o r t i o n a l t o t h e i n t e n s i t y

o f a t w o b o d y r e s i d u a l i n t e r a c t i o n . H e r e t h e m e a n i n g o f t h e p a r a m e t e r

c h a r a c t e r i z i n g t h e s i n g l e p a r t i c l e o r b i t s i s f r o m t h e m o d e l v i e w p o i n t q u i t e

136

d i f f e r e n t f r o m t h e d e f o r m a t i o n p a r a m e t e r u s e d i n t h e N i l s s o n m o d e l a n d

i n p a r t i c u l a r t h e a p p e a r a n c e s o f a m o m e n t o f i n e r t i a c o m p a r a b l e t o t h e

c a l c u l a t e d r i g i d b o d y v a l u e i s n o t , i n i t s e l f , a g u a r a n t e e t h a t t h e i n t r i n s i c

s t a t e i s w e l l d e f i n e d o r t h a t t h e n u c l e u s i t s e l f i s r i g i d .

D . O t h e r N u c l e a r M o d e l P r e d i c t i o n s

O n l y r e c e n t l y h a v e a n y c a l c u l a t i o n s , o t h e r t h a n c o l l e c t i v e , b e e n

a p p l i e d i n t h i s m a s s r e g i o n . A s a r e s u l t o n l y a l i m i t e d a n d c r u d e c o m p a r i ­

s o n c a n b e m a d e b e t w e e n t h e m .

1 . E x c i t a t i o n e n e r g i e s

T h e e x c i t a t i o n e n e r g i e s h a v e b e e n c a l c u l a t e d i n t h e m o d e l

f r a m e w o r k o f C o r i o l i s c o u p l i n g ( t h i s w o r k ) , S h e l l ( W i 6 9 ) , H a r t r e e - F o c k

21p r o j e c t i o n ( J o 6 9 ) , a n d S U t r u n c a t i o n ( A k 6 9 ) f o r N e . T h e C o r i o l i s a n do2 3 2 1 2 3

S h e l l m o d e l s h a v e a l s o b e e n a p p l i e d t o N a . R e s u l t s f o r N e a n d N a

a r e s h o w n i n F i g s . 1 1 8 a n d 1 1 9 , r e s p e c t i v e l y . T h e s h e l l m o d e l c a l c u l a ­

t i o n o f B o u t e n e t a l . ( B o 6 7 ) , d e s c r i b e d i n S e c t i o n I - C 2 , f o r a l l f o u r c o l ­

l e c t i v e £ = 1 1 n u c l e i i s s h o w n i n F i g . 1 2 0 a s a f u n c t i o n o f t h e r e l a t i v e

s t r e n g t h o f a s p i n o r b i t f o r c e . G e n e r a l l y o v e r a l l a g r e e m e n t i s b e s t b e t w e e n

2 3 2 1t h e C o r i o l i s m o d e l a n d e x p e r i m e n t f o r b o t h n u c l e i N a a n d N e w i t h t h e

e x c e p t i o n t h a t t h e e x t e n d e d s h e l l m o d e l c a l c u l a t i o n o f W i l d e n t h a l e t a l ( W i 6 9 )

21p r o v i d e s a c o m p a r a b l e r e p r o d u c t i o n o f t h e N e d a t a t o t h a t o f t h e C o r i o l i s

m o d e l .

EX

CIT

AT

ION

EN

ERG

Y

(MeV

)

8EXP

CORIOLIS COUPLING

(THIS WORK)-9/2+

3/2+,5/2+— (9 /2 ) ,l3 /2 t-

- 3/2+,5 /2 + ^ ;

3/2+,5/2+> (7/2), II/2+-3/2+,5/2+— 5/2+,(3/2+)'

9/2+-I/2+'

A4 / 2 + .'✓3/2 + =—9/2+ — 13/2+ -9 /2 + *7/2+l/2+’< 3 / 2 +7 /2+

-5 /2 +-5 /2+-3 /2 +“5 /2 +-l/2+-9 /2 +-I / 2+

7/2+— -7/2H

5/2+3 /2 +—

Fig.

21

SHELL (Wi 69)

HARTREE- FOCK bU3

PROJECTION TRUNCATION(Jo 69) (Ak 69)

— 13/2+—13/2+

^ 3 /2 + ------------------- -3 /2 + _-11/2+ — — 1 / 2 + --------------------- 7/2+'3 /2 + 1/2 —5/2+■5/2+ 5 /2 + ------------------— 1/2+

3/2+ — 11/2+-5/2+ 5/2+-9/2+- 1 / 2 + --------------------- 9 /2 + < 5 / | + ~

— 9/2+ 4-l/2+

-7/2+ ’ -l/2+

-5/2 + -5/2+____________ -5 /2+

- 3 / 2 + — -----------------*— — 3 / 2 + ■------------------------------------ -3 / 2 +

118

EXC

ITA

TIO

N

ENER

GY

(MeV

)9 —

8 EXP

9/2 ,5/2 — 1/2+~~ (9/2),l3/2+"3/2+,5/2+—II/2+-3/2+,5/2+"

7/2 ,(5/2)- 1/2 —

4 — (5/2+)-

3 h - 3/2+- 9/2+- I/2+- 7/2+-

5/2+-

3/2+-

N a

CORIOLIS COUPLING

(THIS WORK)

23

SHELL (Wi 6 9 )

— 9 / 2 + /7 /2+ - — 1 3 / 2 +

" ^ 3 / 2 +

11 /2+' ^ - 5 / 2 + 7 / 2 +^5/2+— 1/2+ — 5 / 2 +

.13/2 + A7/2 + -^3/2 +

=i-5/2 + 1 / 2 +11/2 +9/2 +

-7/2 + -3/2 + '5/2+

-9/2'3/2+-1/2 + -7/2+

-5/2 + '9/2+-1/2+

7/2"1

-5/2+

-3/2+— 5/2+— 3/2+

F ig . 119

LS C

OU

PLIN

GN e - N a -

(81)22 (81)22(8 I>22

(81)22 (62)22(8 I)22 (62)22(8 I)22

S P I N - O R B I T S T R E N G T HX

Fig. 120

jj C

OU

PLI

NG

137

a . S t a t i c c o m p a r i s o n s -

T h e m a g n e t i c d i p o l e m o m e n t a n d t h e e l e c t r i c q u a d r u p o l e

m o m e n t a r e t y p i c a l s t a t i c e l e c t r o m r g n e t i c q u a n t i t i e s u s u a l l y c a l c u l a t e d

b y a m o d e l a n d c o m p a r e d t o e x p e r i m e n t . I n T a b l e 7 s u c h q u a n t i t i e s f o r t h e

2 1 2 3g r o u n d s t a t e s o f N e a n d N a a r e s h o w n c o m p a r i n g t h e m o d e l p r e d i c t i o n

w i t h e x p e r i m e n t . T h r o u g h o u t t h e t a b l e g e n e r a l o v e r a l l a g r e e m e n t i s m a i n ­

t a i n e d w i t h n o o b v i o u s a n o m a l i e s . A m o r e s e n s i t i v e t e s t t o m o d e l p r e d i c ­

t i o n s i s t o c o m p a r e t h e d y n a m i c p r o p e r t i e s .

T a b l e 7

E l e c t r i c Q u a d r u p o l e M o m e n t ( Q i n b a r n s )

N u c l e u s E x p R o t o r C o r i o l i s S h e l l H a r t r e e S UF o c k

N e 2 1 + 0 . 0 9 + 0 . 0 7 7 + 0 . 0 7 5 + 0 . 0 9 7 + 0 . 1 3

2 3N a 0 . 1 0 + 0 . 1 0 + 0 . 1 0 + 0 . 1 2 + 0 . 1 4

2. Electromagnetic properties

M a g n e t i c D i p o l e M o m e n t i n n u c l e a r m a g n e t o n s )

N e 21 - 0 .6 6 - 0 . 4 0 - 0 . 5 5 - 0 . 5 8 - 1 . 0 2

N a + 2 . 2 2 + 1 . 9 7 2 . 2 2 + 2 . 4 4 + 2 . 3 2

b . D y n a m i c c o m p a r i s o n s

The electric quadrupole and magnetic dipole reduced

138

m a t r i x e l e m e n t s f o r t r a n s i t i o n i n t h e g r o u n d s t a t e r o t a t i o n a l b a n d o f N e

2 3a n d N a a s p r e d i c t e d o n t h e b a s i s o f t h e r o t o r , C o r i o l i s , a n d s h e l l m o d e l s

a r e c o m p a r e d w i t h e x p e r i m e n t i n T a b l e 8 . S u c h q u a n t i t i e s w e r e c a l c u l a t e d

i n t h e r o t o r a n d C o r i o l i s m o d e l i n t h i s w o r k u s i n g p a r a m e t e r s C , D , a n d $

s h o w n i n t h e t a b l e f o r t h e c o r r e s p o n d i n g n u c l e i . E l e c t r o m a g n e t i c p r o ­

p e r t i e s h a v e o n l y r e c e n t l y b e e n c a l c u l a t e d o n t h e s e n u c l e i a s e x t e n s i v e a s

s h o w n h e r e b y W i l d e n t h a l e t a l . ( W i 6 8 , W i 6 9 ) a n d J o h n s t o n e e t a l ( J o 6 9 ) .

A n e x a m i n a t i o n o f t h e e n t r i e s i n t h e t a b l e i m p l y t h a t t h e s h e l l m o d e l

p r e d i c t i o n s o f t h e B ( E 2 ) a n d B ( M 1 ) m a t r i x e l e m e n t s f o r b o t h n u c l e i ( W i 6 9 )

a r e o n t h e a v e r a g e s y s t e m a t i c a l l y t o o l a r g e i n c o m p a r i s o n t o e x p e r i m e n t .

21I n N e t h e r o t o r a n d C o r i o l i s m o d e l s p r e d i c t q u i t e w e l l w i t h e q u a L

s u c c e s s t h e B ( E 2 ) t r a n s i t i o n s , b u t i n t h e c a s e o f t h e B ( M 1 ) t r a n s i t i o n s t h e

C o r i o l i s m o d e l p r e d i c t s s y s t e m a t i c a l l y s m a l l e r v a l u e s i n b e t t e r a g r e e m e n t

2 3w i t h e x p e r i m e n t t h a n t h e r o t o r m o d e l . A s i m i l a r s i t u a t i o n e x i s t s i n N a ,

a s w o u l d b e e x p e c t e d o n t h e b a s i s o f a c o l l e c t i v e m o d e l .

21

139

Table 8

Dynamic Electromagnetic Properties

N e 21 B ( E 2 ) x l O 2 e 2b 2 B ( M 1 ) y 0T r a n s i ­ E x p R o t o r C o r i o l i s ' S h e l l E x p R o t o i ’ C o r i o l i s S h e l lt i o n

5 / 2 - 3 / 2 0 . 6 3 0 0 . 5 5 0 . 4 4 3 0 . 7 9 8 0 . 0 5 3 0 .2 2 0 . 1 5 4 0 . 0 7 9

7 / 2 — 5 / 2 0 . 2 3 8 0 . 3 5 0 . 2 9 5 0 . 6 2 0 0 . 1 2 8 0 . 3 0 0 .1 2 0 0 . 1 4

9 / 2 - 7 / 2 0 . 2 4 9 0 . 2 3 0 . 1 8 0 0 . 3 4 4 0 . 2 7 1 0 . 3 3 0 . 3 3 3 0 . 4 5

1 1 / 2 - 9 / 2 0 . 1 0 2 0 . 1 6 0 . 1 3 0 0 . 2 6 8 0 . 1 0 4 0 . 3 5 0 .1 1 0 0 . 4 0

1 3 / 2 = * 1 1 / 2 0 .1 2 0 . 0 8 6 — 0 . 3 7 0 . 4 2 5

7 / 2 - 3 / 2 0 . 1 6 4 0 . 2 3 0 . 1 9 5 0 . 3 4 20 = 0 . 4 8

9 / 2 - 5 / 2 0 . 2 6 3 0 . 3 5 0 . 3 0 7 0 . 4 3 8 C - 0 . 3 5 f t (j$1 1 / 2 - 7 / 2 0 . 3 3 8 0 . 4 1 0 . 3 6 0 0 . 4 9 8

D = 0 .U

0 h a 01 3 / 2 = i 9 / 2 “ "*■ 0 . 4 5 0 . 4 1 4 0 . 3 5 9

9°,N a B ( E 2 ) x l O 2

2 , 2 e b B ( M 1 )

5 / 2 - 3 / 2 0 . 8 9 0 . 8 4 0 . 6 6 7 0 . 9 2 0 0 . 4 1 3 0 . 2 8 0 . 1 7 6 0 .1 2

7 / 2 - 5 / 2 0 . 1 0 7 0 . 5 2 0 . 4 7 6 0 . 7 5 2 0 . 0 5 1 0 . 3 8 0 . 1 0 4 0 . 0 7 5

9 / 2 - 7 / 2 2 . 0 3 0 . 3 5 0 . 1 9 2 0 . 4 0 6 0 . 5 4 8 0 . 4 2 0 . 4 1 2 0 . 3 6

1 1 / 2 - 9 / 2 0 . 2 4 0 . 1 4 0 — 0 . 4 5 0 . 0 9 7 —

1 3 / 2 - 1 1 / 2 0 . 1 8 0 . 0 6 4 2 — 0 . 4 6 0 . 5 2 9 —

7 / 2 - 3 / 2 0 . 0 9 5 0 . 3 5 0 . 3 6 0 0 . 3 5 6

9 / 2 - 5 / 2 0 . 5 4 0 0 . 5 3 0 . 4 9 9 0 . 4 3 2g = + 0 . 5 0

1 1 / 2 - 7 / 2 0 . 6 3 0 . 6 3 1 ---- C = - 0 . 3 0 f t

1 3 / 2 - 9 / 2 0 . 6 9 0 . 6 3 7 ---- D = - 0 . 02 f t(jo^

140

X . S U M M A R Y A N D C O N C L U S I O N S

T h e o d d c o u n t £ = 11 n u c l e a r s y s t e m h a s b e e n e x a m i n e d s y s t e m ­

a t i c a l l y a n d i n d e t a i l h e r e i n b y p e r f o r m i n g M e t h o d I I a n g u l a r c o r r e l a t i o n

2 3 2 1s t u d i e s e x t r a c t i n g n e w i n f o r m a t i o ' n o n N a a n d N a a n d b y c o m p a r i n g

m o d e l p r e d i c t i o n s b a s e d o n t h e s i m p l e r o t o r , C o r i o l i s , a n d s h e l l m o d e l s

w i t h o u r e x p e r i m e n t a l r e s u l t s a n d o t h e r f o r t h e £ = 11 c o l l e c t i v e n u c l e i .

W e h a v e f o u n d t h a t t h e r o t o r a n d C o r i o l i s m o d e l s g i v e r e a s o n a b l e

r e p r o d u c t i o n o f d a t a a n d r e p r e s e n t i m p r o v e m e n t s o v e r t h e s h e l l m o d e l .

T h e r o t o r a n d C o r i o l i s m o d e l s p r e d i c t w i t h a p p r o x i m a t e l y e q u a l s u c c e s s

B ( E 2 ) t r a n s i t i o n s w h i l e t h e C o r i o l i s m o d e l i s a n i m p r o v e m e n t o v e r t h e

r o t o r B ( M 1 ) p r e d i c t i o n s , w h i c h a r e m o r e s e n s i t i v e t o w a v e f u n c t i o n

a d m i x t u r e s i g n o r e d h e r e i n i n t h e r o t o r m o d e l .

I t h a s b e e n d e t e r m i n e d t h a t t h e o d d c o u n t £ - 1 1 n u c l e i p o s s e s s

a l m o s t r i g i d b o d y v a l u e s o f t h e m o m e n t o f i n e r t i a w i t h a w e l l d e v e l o p e d

g r o u n d s t a t e r o t a t i o n a l b a n d . F r o m a c l a s s i c a l p o i n t o f v i e w t h e r i g i d

b o d y m o m e n t o f i n e r t i a i s d e f i n e d i n t e r m s o f t h e n u c l e a r d e f o r m a t i o n

a n d t h e r e f o r e t h i s c o n c e p t i s o n l y a s g o o d a s t h e m e a s u r e m e n t o f t h e

d e f o r m a t i o n i t s e l f . H e r e w e h a v e d e t e r m i n e d t h a t t h e r e l a t e d q u a n t i t y ,

t h e i n t r i n s i c q u a d r u p o l e m o m e n t , i s w e l l d e f i n e d a n d c o n s t a n t i n a

r o t a t i o n a l b a n d w i t h i n t h e u n c e r t a i n t y o f t h e e x p e r i m e n t a l m e a s u r e m e n t s .

T h e a d d i t i o n a l p a r a m e t e r , t h e d i f f e r e n c e o f t h e g y r o m a t i c r a t i o s , g ^ - g p ,

h a s a l s o b e e n f o u n d t o b e c o n s t a n t w i t h i n t h e r o t a t i o n a l b a n d . W e t h e r e f o r e

141

c o n c l u d e f r o m t h i s e v i d e n c e t h a t t h e s e l i g h t n u c l e i a r e i n d e e d w e l l d e f o r m e d ,

a c t u a l l y r i g i d , a n d s a t i s f y a s w e l l a s a n y t h e b a s i c p r e m i s e s o n w h i c h t h e

s i m p l e r o t o r m o d e l i s f o u n d e d .

H o w e v e r , e v e n h e r e w h e r e m o s t a H e v i d e n c e i n d i c a t e s t h a t t h e

2 3n u c l e i a r e m o s t r i g i d , i n p a r t i c u l a r N a , t h e e x c i t a t i o n e n e r g i e s d e v i a t e

f r o m t h e J ( J + 1 ) r u l e , i n d i c a t i n g t h e e x i s t e n c e o f u n a c c o u n t e d f o r r o t a t i o n a l

p e r t u r b a t i o n s . A n a t t e m p t t o e x p l a i n t h e s e d i s c r e p a n c i e s b y i n c l u d i n g

s i m p l e h i g h e r o r d e r C o r i o l i s b a s e d p e r t u r b a t i o n s f a i l e d , b u t w i t h t h e p o s ­

s i b i l i t y o f f u l l c o n f i g u r a t i o n a l b a n d m i x i n g i n t h e s d s h e l l , t h e C o r i o l i s

m o d e l p r e d i c t i o n s o f t h e e x c i t a t i o n s p e c t r a o f t h e g r o u n d s t a t e r o t a t i o n a l

b a n d w a s a n i m p r o v e m e n t o v e r t h e r o t o r m o d e l p r e d i c t i o n s .

C o n n e c t e d w i t h t h i s s e e m i n g l y C o r i o l i s b a s e d p e r t u r b a t i o n i s t h e

a n o m a l y t h a t i n c o n t r a s t t o t h e u s u a l s i t u a t i o n t h e d y n a m i c e l e c t r o m a g n e t i c

d e - e x c i t a t i o n p r o p e r t i e s w i t h i n t h e g r o u n d s t a t e r o t a t i o n a l b a n d a r e i n

b e t t e r a g r e e m e n t w i t h r o t o r m o d e l p r e d i c t i o n s t h a n t h e c o r r e s p o n d i n g s t a t i c

p r o p e r t i e s ( e x c i t a t i o n e n e r g i e s ) . -

O n t h e b a s i s o f o u r e x t e n s i v e a n d s y s t e m a t i c c o m p a r i s o n o f t h e

e x p e r i m e n t a l s t a t i c a n d d y n a m i c n u c l e a r p r o p e r t i e s w i t h r o t o r m o d e l p r e ­

d i c t i o n s , a n d f r o m t h e c o r r e s p o n d i n g e m p i r i c a l l y d e t e r m i n e d s e t o f c o l -

2 1 2 3l e c t i v e f i t t i n g p a r a m e t e r s f o r N e a n d N a i n p a r t i c u l a r , w e c o n c l u d e

t h a t t h e s e n u c l e i a n d t h e i r m i r r o r p a i r c o n s t i t u t i n g t h e o d d c o u n t £ = 11

n u c l e a r m u l t i p l e t m a y i n d e e d b e t h e m o s t r i g i d n u c l e i i n n a t u r e .

142

A k 6 9

A l 5 7

A n 6 9

A r 6 2

B a 6 7

B i 66

B l 6 9

B o 5 2

B o 5 3

B o 5 5

B o 6 7

B o 6 9

B r 5 7

B r 6 0

B r 6 2

B r 6 2 a

B r 6 2 b

REFERENCES

Y . A k i y a m a , A . A r i m a , a n d T . S e b e , p r e p r i n t ( t o b e p u b l i s h e d ) .

G . A l a g a , N u c l . P h y s . 4 , 6 2 5 ( 1 9 5 7 ) .G . A l a g a , K . A l d e r , A . B o h r , a n d B . R . M o t t e l s o n , D a n . M a t . F y s . M e d d 2 £ , N o . 9 ( 1 9 5 5 ) .

A . A n t i l l a a n d M . B i s t e r , P h y s . L e t t e r s 2 9 B , 6 4 5 ( 1 9 6 9 ) .

S . E . A r n e l l a n d E . W e r n b o m , A r k i v F o r F y s i k 2 3 , 3 0 1 ( 1 9 6 2 ) .

F . E l - B a t a n o n i a n d A . A . K r e s n i n , N u c l . P h y s . A 1 0 2 , 4 7 3 ( 1 9 6 7 ) .

E . V . B i s h o p , T h e s i s , Y a l e U n i v e r s i t y , 1 9 6 6 .

R . B l o c h , T . K n e l l w o l f , a n d R . E . P i x l e y , N u c l . P h y s . A 1 2 3 ,1 2 9 ( 1 9 6 9 ) .

A . B o h r , M a t . F y s . M e d d . D a n . V i d . S e l s k . 2 6 , N o . 1 4 ( 1 9 5 2 ) .

A . B o h r a n d B . R . M o t t e l s o n , M a t . F y s . M e d d . D a n . V i d . S e l s k . 2 7 , N o . 1 6 ( 1 9 5 3 ) .

A . B o h r a n d B . M o t t e l s o n , D a n . M a t . F y s . M e d d . 3 0 , N o . 1 ( 1 9 5 5 ) p . 1 7 .

M . C . B o u t e n , J . P . E l l i o t t , a n d J . A . P u l l e n , N u c l . P h y s . A 9 7 ,1 1 3 ( 1 9 6 7 ) .

A . B o h r a n d B . R . M o t t e l s o n i n N u c l e a r S t r u c t u r e V o l . I I W . A . B e n j a m i n , I n c . , N e w Y o r k , 1 9 6 9 .

D . A . B r o m l e y , H . E . G o v e , a n d A . E . L i t h e r l a n d , C a n . J . P h y s . 3 5 , 1 0 5 7 ( 1 9 5 7 ) .

D . M . B r i n k , P r o g , i n N u c l . P h y s . 9 9 ( 1 9 6 0 ) .

D . W . B r a b e n , L . L . G r e e n , a n d J . C . W i l m o t t , N u c l . P h y s . 3 £ , 5 8 4 ( 1 9 6 2 ) .

D . M . B r i n k a n d G . R . S a t c h l e r , A n g u l a r M o m e n t u m , O x f o r d U n i v e r s i t y P r e s s , 1 9 6 2 , p . 3 2 .

S e e R e f . B r 6 2 a , p . 1 1 7 .

143

C h 6 3

C l 6 1

C o 6 3

C r 6 9

D a 6 9

D e 5 7

D u 6 7

D u 6 7 a

D u 68

D u 6 9

D u 6 9 a

E l 5 5

E l 5 8

E l 6 2

E l 6 7

E n 6 7

F e 5 5

F e 6 5

F I 5 4

A . B . C l e g g a n d K . J . F o l e y , P h i l , M a g . 7 , 2 4 7 ( 1 9 6 1 ) .

E . V . C o n d o n a n d G . H . S h o r t l e y i n T h e T h e o r y o f A t o m i c S p e c t r a , C a m b r i d g e U n i v e r s i t y P r e s s , 1 9 6 3 .

G . C r a i g , T h e s i s , Y ~ l e U n i v e r s i t y , 1 9 6 9 .

J . P . D a v i d s o n i n C o l l e c t i v e M o d e l s o f t h e N u c l e u s , A c a d e m i c P r e s s , N e w Y o r k a n d L o n d o n , 1 9 6 8 , p . 1 8 1 .

S . D e v o n s a n d L . J . B . G o l d f a r b i n H a n b u c h d e r P h y s i k , V o l . 4 2 , e d . b y S . F l U g g e , S p r i n g e r , B e r l i n , 1 9 5 7 , p . 3 6 2 .

J . D u b o i s , N u c l . P h y s . A 9 9 , 4 6 5 ( 1 9 6 7 ) .J . D u b o i s , N u c l . P h y s . A 1 0 4 , 6 5 7 ( 1 9 6 7 ) .

J . D u b o i s a n d L . G . E a r w a k e r , P h y s . R e v . 1 6 0 , 9 2 5 ( 1 9 6 7 ) .

J . D u b o i s , N u c l . P h y s . A 1 1 6 , 4 8 9 ( 1 9 6 8 ) .

H . D u h m , p r i v a t e c o m m u n i c a t i o n ( t o b e p u b l i s h e d ) .

J . D u r e l l , P . R . A l d e r s o n , D . C . B a i l e y , L . L . G r e e n , M . W . G r e e n e , A . N . J a m e s , a n d J . F . S h a r p e y - S c h a f e r , P h y s . L e t t e r s 2 9 B , 1 0 0 ( 1 9 6 9 ) .

J . P . E l l i o t t a n d B . H . F l o w e r s , P r o c . R o y . S o c . 2 2 9 , 5 3 9 ( 1 9 5 5 ) .

J . P . E l l i o t t , P r o c . R o y . S o c . A 2 4 5 , 1 2 8 ( 1 9 5 8 ) .J . P . E l l i o t t , P r o c . R o y . S o c . A 2 4 5 , 5 6 2 ( 1 9 5 8 ) .

J . P . E l l i o t t a n d M . H a r v e y , P r o c . R o y . S o c . A 2 7 2 , 5 5 7 ( 1 9 6 2 ) .

J . P . E l l i o t t a n d C . E . W i l s d o n , P r o c . R o y . S o c . A 3 0 2 , 5 0 9 ( 1 9 6 7 ) .

P . M . E n d t a n d C . V a n d e r L e u n , N u c l . P h y s . A 1 0 5 , 1 ( 1 9 6 7 ) .

M . F e r e n t z a n d N . R o s e n z w e i g , A r g o n n e N a t i o n a l L a b o r a t o r y R e p o r t N o . A N L - 5 3 2 4 ( 1 9 5 5 ) .

A . J . F e r g u s o n i n A n g u l a r C o r r e l a t i o n M e t h o d s i n G a m m a R a y S p e c t r o s c o p y , N o r t h - I - I o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 6 5 , p . 8 1 .

B . H . F l o w e r s , P h i l . M a g . 4 5 , 3 2 9 ( 1 9 5 4 ) .

B. E. Chi and J. P. Davidson, Phys, Rev. 131, 366 (1963).

144

G 1 6 4

G r 6 3

G r 6 4

I - I a 68

H a 68a

H a 6 9

H a 6 9 a

H i 6 9

H o 6 5

H o 6 9

I n 5 4

J o 6 9

K e 6 3

K e 6 4

K o 68

K o 6 9

Fr 60

Fu 69

La 65

G . H . F u l l e r a n d V . W . C o h e n , N u c l e a r M o m e n t s , A p p e n d i x 1 t o N u c l e a r D a t a S h e e t s , 1 9 6 5 .

W . G l t f c k l e , Z e i t s c h r i f t f u r P h y s i k 1 7 8 , 5 3 ( 1 9 6 4 ) .

J . S . G r e e n b e r g , D . A . B r o m l e y , G . C . S e a m a n , a n d E . V . B i s h o p , P r o c e e d i n g s o f t h e T h i r d C o n f e r e n c e o n R e a c t i o n s B e t w e e n C o m p l e x N u c l e i ( U n i v e r s i t y o f C a l i f o r n i a P r e s s , 1 9 6 3 ) p . 2 9 5 .

Y . T . G r i n a n d I . M . P a r l i c h e n k o v , P h y s . L e t t e r s 9 , 2 4 9 ( 1 9 6 4 ) .

E . C . H a l b e r t , J . B . M c G r o r y , a n d B . H . W i l d e n t h a l , P h y s . R e v . L e t t e r s 2 0 , 1 1 1 2 ( 1 9 6 8 ) .

M . H a r v e y i n A d v a n c e s i n N u c l e a r P h y s i c s , e d . t y . M . B a r a n g e r a n d E . V o g t , P l e n u m P r e s s , N e w Y o r k , 1 9 6 8 , p . 6 7 .

I . H a m a m a t o a n d T . U d a g a w a , p r e p r i n t ( t o b e p u b l i s h e d ) .

0 . H S u s s e r , B . W . H o o t o n , D . P e l t e , a n d T . K . A l e x a n d e r , P h y s .R e v . L e t t e r s 2 2 , 3 5 9 ( 1 9 6 9 ) .

R . G . H i r k o , T h e s i s , Y a l e U n i v e r s i t y , 1 9 6 9 .

A . J . H o w a r d , J . P . A l l e n , a n d D . A . B r o m l e y , P h y s . R e v . 1 3 9 ,B 1 1 3 5 ( 1 9 6 5 ) .

A . J . H o w a r d , J . G . P r o n k o , a n d C . A . W h i t t e n , J r . , ( t o b e p u b l i s h e d ) .

D . R . I n g l i s , P h y s . R e v . 9 6 , 1 0 5 9 ( 1 9 5 4 ) .

1 . P . J o h n s t o n e a n d H . G . B e n s o n , N u c l . P h y s . A 1 3 4 , 68 ( 1 9 6 9 ) .

I . K e l s o n , P h y s . R e v . 1 3 2 , 2 1 8 9 ( 1 9 6 3 ) .

I . K e l s o n a n d C . A . L e v i n s o n , P h y s . R e v . 1 3 4 , B 2 6 9 ( 1 9 6 4 ) .

R . L . K o z u b , P h y s . R e v . 1 7 2 , 1 0 7 8 ( 1 9 6 8 ) .

D . K o v a r , W . C a l l e n d e r , L . M c V a y , W . M e t z , C . M c G u i r e , a n dC . K . B o c k e l m a n ( t o b e p u b l i s h e d ) .

I I . L a n c m a n , A . J a s i n s k i , J . K o w n a c k i , a n d J . L u d z i e j e w s k i , N u c l . P h y s . J 3 9 , 3 8 4 ( 1 9 6 5 ) .

J. Freeman, Phys. Rev. 120, 1436 (1960).

145

L i 6 1

L i 6 9

L i 6 9 a

L i 6 9 b

L i 6 9 c

L i 6 9 d

L i 6 9 e

L i 6 9 f

L i 6 9 g

M a 5 0

M a 66

M a 6 7

M a 68

M a 68a

M e 6 4

M i 6 4

M i 66

Li 58 A . E . L i t h e r l a n d , H . M c M a n u s , E „ B . P a u l , D . A . B r o m l e y , a n dH . E . G o v e , C a n . J . P h y s . 3 6 , 3 7 8 ( 1 9 5 8 ) .

A . E . L i t h e r l a n d a n d A . J . F e r g u s o n , C a n , J . P h y s . 3 9 , 7 8 8 ( 1 9 6 1 ) .

R . A . L i n d g r e n , R . G . H i r k o , J . G . P r o n k o , A . J . H o w a r d , M . W .S a c h s , a n d D . A . B r o m l e y , B u l l . A m . P h y s . S o c . J L 3 , 1 3 7 1 ( 1 9 6 8 ) .

R . A . L i n d g r e n , C o o p C o m p u t a t i o n B o o k N o . 3 6 , 1 9 6 7 , p . 3 5 .

R . A . L i n d g r e n , C o o p C o m p u t a t i o n B o o k N o . 5 9 , 1 9 6 7 , p . 1 1 1 .

R . A . L i n d g r e n , C o o p C o m p u t a t i o n B o o k N o . 5 , 1 9 6 7 .

M P A L - a m u l t i p a r a m e t e r a n a l y z e r p r o g r a m o r i g i n a l l y w r i t t e n b y M . W . S a c h s a n d R . G . H i r k o f o r u s e w i t h t h e I B M 3 6 0 / 4 4 d a t a a c q u i s i t i o n s y s t e m a n d s u b s e q u e n t l y r e w r i t t e n a n d g e n e r a l i z e d b y R . A . L i n d g r e n a n d R . I l a d s e l l ( u n p u b l i s h e d ) .

A N N - a c o m p u t e r c o d e u s e d f o r L i t h e r l a n d a n d F e r g u s o n M e t h o d I I a n a l y s i s t h a t w a s o b t a i n e d f r o m J . A . B e c k e r , L o c k h e e d P a l o A l t o L a b o r a t o r i e s ; i t w a s s u b s e q u e n t l y m o d i f i e d b y R . H i r k o a n d R . L i n d g r e n f o r u s e a t t h i s l a b o r a t o r y .

R . A . L i n d g r e n , J . G . P r o n k o , J . W . O l n e s s , R . G . H i r k o , a n d D . A .B r o m l e y , B u l l . A m . P h y s . S o c . L 4 , 5 3 1 ( 1 9 6 9 ) .

R . A . L i n d g r e n , J . G . P r o n k o , a n d D . A . B r o m l e y , C o n t r i b u t i o n s , I n t e r n a t i o n a l C o n f e r e n c e o n P r o p e r t i e s o f N u c l e a r S t a t e s , M o n t r e a l C a n a d a , 1 9 6 9 , p . 9 3 .

M . G . M a 3' e r , P h y s . R e v . 7 8 , 1 6 ( 1 9 5 0 .M . G . M a y e r , P h y s . R e v . 7 8 , 2 2 ( 1 9 5 0 ) .

F . B . M a l i k a n d W . S c h o l z , P h y s . R e v . 1 5 0 , 9 1 9 ( 1 9 6 6 ) .

F . B . M a l i k a n d W . S c h o l z , N u c l e a r S t r u c t u r e , e d . b y A . H o s s a i n ,N o r t h - H o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 6 7 , p . 3 4 - 5 7 .

M . A . J . M a r i s c o t t i , G e r t r u d e S c h a r f f - G o l d h a b e r , a n d B r i a n B u c k ,P h y s . R e v . 1 7 8 , 1 8 6 4 ( 1 9 6 9 ) .

I I . J . M a i e r , T h e s i s , U n i v e r s i t y o f F r e i b u r g , 1 9 6 8 .H . J . M a i e r , J . G . P r o n k o , a n d C . R o l f s , B u l l . A m . P h y s . S o c . 1 3 , 6 5 2 ( 1 9 6 S )I I . J . M a i e r , J . G . P r o n k o , a n d C . R o l f s ( t o b e p u b l i s h e d ) .

F . R . M e t z g e r , P h y s . R e v . 1 3 6 , B 3 7 4 ( 1 9 6 4 ) .

V . M . M i k h a i l o v , I z v . A n . S S S R , S e r . f i z . 2 8 , 3 0 8 ( 1 9 6 4 ) .

V . M . M i k h a i l o v , I z v . A k a d , N a u k . U S S R , S e r . F i z . j ! 0 , 1 3 3 4 ( 1 9 6 6 ) .

146

M o 6 7

N a 6 5

N a 6 9

N e 5 9

N e 6 0

N i 5 5

0 1 6 9

O w 6 4

P a 5 8

P e 66

P o 6 5

P o 66

P o 68

P o 6 9

P o 6 9 a

P r 6 2

P r 6 2 a

P r 6 7

Mo 60 B . M o t t e l s o n , P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e o n N u c l e a r S t r u c t u r e , K i n g s t o n , O n t a r i o , 1 9 6 0 , e d . b y D . A . B r o m l e y a n d F . W . V o g t , ( U n i v e r s i t y o f T o r o n t o , 1 9 6 0 ) p . 5 2 5 .

B . R . M o t t e l s o n i n P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e o n N u c l e a r S t r u c t u r e , e d . b y J . S a n a d a , J . P h y s . S o c . , J a p a n ,V o l . 2 4 , 1 9 6 8 , p . 8 7 .

O . N a t h a n a n d S . G . N i l s s o n , A l p h a , B e t a , a n d G a m m a R a y S p e c ­t r o s c o p y , e d . b y K . S i e g b a h n , V o l . 1 , N o r t h - I - I a o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 6 4 , p . 6 7 4 .

K . N a k a i , F . S . S t e p h e n s , a n d R . M . D i a m o n d i n C o n t r i b u t i o n s , I n t e r n a t i o n a l C o n f e r e n c e o n P r o p e r t i e s o f N u c l e a r S t a t e s , M o n t r e a l C a n a d a , 1 9 6 9 , p . 9 3 .

T . D . N e w t o n , C a n . J . P h y s . 3 7 , 4 4 4 ( 1 9 5 9 ) .

T . D . N e w t o n , C a n . J . P h y s . 3 8 , 7 0 0 ( 1 9 6 0 ) .

S . G . N i l s s o n , M a t . F y s . M e d d . D a n . V i d . S e l s k . _ 2 9 , N o . 1 6 ( 1 9 5 5 ) .

P S M - a g a m m a r a y f i t t i n g p r o g r a m o b t a i n e d f r o m J o h n O l n e s s a t 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 . I t w a s m o d i f i e d a n d a d a p t e d t o t h e W N S L I B M 3 6 0 / 4 4 c o m p u t e r b y R . H a d s e l l .

L . W . O w e n a n d G . R . S a t c h l e r , N u c l . P h y s . 5 1 , 1 5 5 ( 1 9 6 4 ) .

E . B . P a u l , N u c l . P h y s . 8 , 6 1 ( 1 9 5 8 ) .

D . P e l t e , P h y s . L e t t e r s 2 2 , 4 4 8 ( 1 9 6 6 ) .

A . R . P o l e t t i a n d E . K . W a r b u r t o n , P h y s . R e v . 1 3 7 , B 5 9 5 ( 1 9 6 5 ) .

A . R . P o l e t t i a n d D . F . H . S t a r t , P h y s . R e v . 1 4 7 , 8 0 0 ( 1 9 6 6 ) .

A . R . P o l e t t i , J . A . B e c k e r , R . E . M c D o n a l d , a n d A . D . W . J o n e s ,B u l l . A m . P h y s . S o c . j L 3 , 6 5 2 ( 1 9 6 8 ) .

A . R . P o l e t t i , J . A . B e c k e r , a n d R . E . M c D o n a l d ( t o b e p u b l i s h e d ) .

A . R . P o l e t t i , A . D . W . J o n e s , J . A . B e c k e r , R . E . M c D o n a l d , a n d R . W . N i g h t i n g a l e ( t o b e p u b l i s h e d ) .

M . A . P r e s t o n , i n P h y s i c s o f t h e N u c l e u s , A d d i s o n - W e s l e y P u b l i s h i n g C o . , 1 9 6 2 , p . 1 4 5 .

S e e R e f . P r 6 2 , p . 3 4 0 , 3 4 3 .

J . G . P r o n k o , C . R o l f s , a n d H . J . M a i e r , N u c l . P h y s . A 9 4 , 5 6 1 ( 1 9 6 7 ) .

147

P r 6 9

P r 6 9 a

P r 6 9 b

P r 6 9 c

R a 5 7

R a 66

R i 6 7

R o 66

R o 6 7

R o 6 7 a

R o 6 9

R o 6 9 a

R u 6 7

S a 68

S c 66

S c 68

S c 6 9

S c 6 9 a

J . G . P r o n k o , C . R o l f s , H . J . M a i e r ( t o b e p u b l i s h e d ) .

J . G . P r o n k o , R . A . L i n d g r e n , a n d D . A , B r o m l e y ( t o b e p u b l i s h e d ) .

J . G . P r o n k o , R . A . L i n d g r e n , a n d D . A . B r o m l e y , B u l l . A m . P h y s .S o c . 1 4 , 1 2 3 ( 1 9 6 9 ) .

J . G . P r o n k o , R . A . L i n d g r e n , a n d D . A . B r o m l e y , B u l l . A m . P h y s .S o c . _ 1 4 j 5 3 1 ( 1 9 6 9 ) .

G . R a k a v y , N u c l . P h y s . 4 , 3 7 5 ( 1 9 5 7 ) .

V . K . R a s m u s s e n a n d N . A . K h a n , P h y s . R e v . 1 5 2 , 1 0 2 7 ( 1 9 6 6 ) .

A . R i c h t e r a n d W . v o n W i t s c h , N u c l . P h y s . A l 0 0 , 6 8 3 ( 1 9 6 7 ) .

M . E . R o s e i n E l e m e n t a r y T h e o r y o f A n g u l a r M o m e n t u m , J o h n W i l e y a n d S o n s , I n c . N e w Y o r k , 1 9 6 6 .

H . J . R o s e a n d D . M . B r i n k , R e v . M o d . P h y s . _ 3 9 , 3 0 6 ( 1 9 6 7 ) .

D . J . R o w e , " P h e n o m e n o l o g i c a l C o l l e c t i v e M o d e l s " , i n F u n d a ­m e n t a l s i n N u c l e a r T h e o r y , e d . b y A . D e S h a l i t a n d C . V i l l i , I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y , V i e n n a , 1 9 6 7 , p . 9 3 .

C . R o l f s , W . T r o s t , E . K u h l m a n n , R . K r a m e s , a n d F . R i e s s ,N u c l . P h y s . A 1 2 9 , 2 5 1 ( 1 9 6 9 ) .

C . R o l f s , p r i v a t e c o m m u n i c a t i o n ( t o b e p u b l i s h e d ) .

J . D . R u s s e l i n O p e r a t i n g P r o c e d u r e s : T h e I B M / Y a l e N u c l e a r D a t a A c q u i s i t i o n I n t e r f a c e S y s t e m , T e c h n i c a l N o t e 2 1 . 5 7 5 - 2 1 ,A d v a n c e d T e c h n o l o g y I S D D L a b o r a t o r y , K i n s s t o n , N e w Y o r k , 1 9 6 7 .

M . W . S a c h s , D . A . B r o m l e y , J . B r i n b a u m , a n d H . L . G e l e r n t e r ,W r i g h t N u c l e a r S t r u c t u r e L a b o r a t o r y , Y a l e U n i v e r s i t y , I n t e r n a l R e p o r t N o . 3 2 ( u n p u b l i s h e d ) .

W . S c h o l z a n d F . B . M a l i k , P h y s . R e v . 1 4 7 , 8 3 6 ( 1 9 6 6 ) .W . S c h o l z a n d F . B . M a l i k , P h y s . R e v . 1 5 3 , 1 0 7 1 ( 1 9 6 7 ) .

L . I . S c h i f f i n Q u a n t u m M e c h a n i c s , T h i r d E d i t i o n , M c G r a w - H i l l B o o k C o m p a n y , 1 9 6 8 , p . 2 4 4 .

D . S c h w a l m a n d B . P o v h , P h y s . L e t t e r s 2 9 B , 1 0 3 ( 1 9 6 9 ) .

D . S c h w a l m a n d B . P o v h , i n S u p p l e m e n t t o C o n t r i b u t i o n s , I n t e r n a t i o n a l C o n f e r e n c e o n P r o p e r t i e s o f N u c l e a r S t a t e s , M o n t r e a l , C a n a d a , 1 9 6 9 , p . 1 5 .

S o 68 P . C . S o o d , C a n . J . P h y s . 4 6 , 1 4 1 9 ( 1 9 6 8 ) .

148

S t 6 0

T e 5 6

W a 5 9

W a 6 9

W i 6 0

W i 68

W i 6 9

W o 6 7

So 68a

Yo 62

B . D . S o w e r b y , D . M . S h e p p a r d , a n d W „ C . O l s e n , N u c l . P h y s .A 1 2 1 , 1 8 1 ( 1 9 6 8 ) .

P . H . S t e l s o n a n d F . K . M c G o w a n , N u c l . P h y s . JL6 , 9 2 ( 1 9 6 0 ) .

G . M . T e m m e r a n d N . P . H e y d e n b u r g , P h y s . R e v . 1 0 4 , 9 8 9 ( 1 9 5 6 ) .

A . H . W a p s t r a , G . J . N ' j g h , a n d R . V a n L i e s h o u t , N u c l e a r S p e c ­t r o s c o p y T a b l e s , N o r t h - H o l l a n d P u b l i s h i n g C o . , A m s t e r d a m , 1 9 5 9 , p . 1 1 9 .

P . W a s i e l e w s k i , T h e s i s , Y a l e U n i v e r s i t y , 1 9 6 9 .

D „ H . W i l k i n s o n i n N u c l e a r S p e c t r o s c o p y , P a r t 2 , e d . b y F . A j z e n b e r g - S e l o v e , A c a d e m i c P r e s s , 1 9 6 0 , p . 8 5 2 .

B . H . W i l d e n t h a l , J . B . M c G r o r y , E . C . H a l b e r t , a n d P . W . M . G l a u d e m a n s , P h y s . L e t t e r s 2 6 B , 6 9 2 ( 1 9 6 8 ) .

B . H . W i l d e n t h a l , e t a l . , p r i v a t e c o m m u n i c a t i o n ( t o b e p u b l i s h e d ) .

J . E . W o l l r a b i n R o t a t i o n a l S p e c t r a a n d M o l e c u l a r S t r u c t u r e , e d b yE . M . L o e b l , A c a d e m i c P r e s s , N e w Y o r k , 1 9 6 7 .A l s o s e e G . H e r z b e r g i n M o l e c u l a r S p e c t r a a n d M o l e c u l a r S t r u c t u r e I . D i a t o m i c M o l e c u l e s , P r e n t i c e - H a l l , I n c . , N e w Y o r k 1 9 3 9 .

H . D . Y o u n g i n S t a t i s t i c a l T r e a t m e n t o f E x p e r i m e n t a l D a t a , M c G r a w - H i l l B o o k C o m p a n y , I n c . , 1 9 6 2 , p . 1 6 3 .

149

H e r e w e w i s h t o s h o w t h a t t h e c h a n g e i n e n e r g y d u e t o t h e C o r i o l i s

i n t e r a c t i o n c a l c u l a t e d i n s e c o n d o r d e r p e r t u r b a t i o n t h e o r y b y u s i n g s t r o n g

c o u p l i n g m o d e l w a v e f u n c t i o n s i s e q u i v a l e n t t o a c h a n g e i n t h e m o m e n t o f

i n e r t i a a n d d o e s n o t a d d a n y n e w a n g u l a r m o m e n t u m d e p e n d e n c e .

T h e s e c o n d o r d e r c o r r e c t i o n t o t h e e n e r g y E ° i s g i v e n i n s t a n d a r d

n o n d e g e n e r a t e p e r t u r b a t i o n t h e o r y b y ( S c 68 )

Ew . v ’ ^ , (1)i i m »

w h e r e E ^ a n d E ^ a r e u n p e r t u r b e d e n e r g y s t a t e s c o r r e s p o n d i n g t o e i g e n ­

f u n c t i o n s < m | a n d | n > , c h a r a c t e r i z e d b y q u a n t u m n u m b e r s m a n d n ,

r e s p e c t i v e l y .

I n t h e s t r o n g c o u p l i n g m o d e l t h e r e l e v a n t q u a n t u m n u m b e r s a r e J a n d

K , w h i c h r e p r e s e n t t h e e i g e n f u n c t i o n s g i v e n b y ( s e e S e c t i o n I - C l )

APPENDIX I

|JK> \ K *0+ ("1)J i D-k x-si) • <2)T h e i n t r i n s i c f u n c t i o n s X o a r e g i v e n b y ( D a 6 9 )

X —. = / C X - ^ ( 3 )

jw h e r e C . _ a r e e x p a n s i o n c o e f f i c i e n t s ( e g N i l s s o n c o e f f i c i e n t s ) a n d x . n

j i / J i /—♦

a r e s i n g l e p a r t i c l e i n t r i n s i c w a v e f u n c t i o n s w i t h a n g u l a r m o m e n t u m j . I n

o u r c a s e o f a x i a l l y s j u n m e t r i c n u c l e i K = & .

T h e p e i ' t u r b i n g C o r i o l i s H a m i l t o n i a n i s

I P = - 2 A ( J + j _ + J - j + ) ( 4 )

w h e r e t h e l a d d e r o p e r a t o r s h a v e b e e n d e f i n e d i n S e c t i o n I - C l .

150

(2)To evaluate E first consider the matrix element < m IH1 |n> .

S u b s t i t u t e E q . ( 2 ) a n d ( 4 ) i n t o E q . (1 ) s e p a r a t i n g i t i n t o t w o t e r m s a s

< m | I T | n > = - 2 A [ < J K | J + j | J K * > + < J K | J - j | J K * > ] ( 5 )

S u b s t i t u t e E q . ( 2 ) i n E q . ( 5 ) a n d c o n s i d e r i n g t h e f i r s t t e r m

< J K | J + j | J K » > w e o b t a i n

<JK|j+j_ |JK.>= 2J±|<D J % + x { j + j _ |Dd ,x +(-l)J-JD_J X_ >. (6)

1 677

T h e o n l y n o n v a n i s h i n g t e r m i s

2 ' ~ K A0 ' ~ J - l~ K , A P1677

S e p a r a t i n g t h e c o l l e c t i v e a n d i n t r i n s i c p a r t o f t h e m a t r i x e l e m e n t w e

o b t a i n

< J K | j + j . | j K ' > = e ^ < D d | j + l D d 1x X n | j _ l x p , > ( 8 )1 6 77

L i k e w i s e ,

' 1677

a n d s i n c e J - j i s i n t e g r a l , t h e f a c t o r ( - l ) 2 3 ^ i s u n i t y . U s i n g t h e r e l a t i o n s

<DK j JX > = f s r P±K+1K«K)a n d

<xj n t l M xjQ>= (j±Q+1)(jTp) (10)a n d s i n c e t h e s u m i n E q . ( 1 ) e x t e n d s o v e r K * , w e g e t c o n t r i b u t i o n s f r o m

t e r m s K ' = 1 / 2 a n d K ' = 5 / 2 .

T h e K * = 1 / 2 t e r m s a r e

< J 3 / 2 | j + j _ | J 1 / 2 ' > = ~ ( J + 3 / 2 ) ( J - l / 2 ) < X 3 / 2 l j _ I X t y 2 '

151

< J - 3 / 2 | J - j + | J - l / 2 > = 1 / 2 - ^ / ( J + 3 / 2 ) ( J - l / 2 ) < x _ 3 /2 | j + | X _ l / 2 > ( H )

w h e r e r t d i s t i n g u i s h e s d i f f e r e n t s i n g l e p a r t i c l e s t a t e s w i t h K = 1 / 2 .

S u b s t i t u t i n g E q s . ( 1 1 ) i n E q s . ( 8) a n d ( 9 ) , w e o b t a i n t h e c o n t r i b u t i o n s t o

t h e m a t r i x e l e m e n t < m | H ' | n > f o r K - l / 2 b a n d s , w h i c h w e d e s i g n a t e

[ < m | I P | n > ] l / 2 = - A ^ J + 3 / 2 ) ( J - l / 2 ) ( < X 3 / 2 I L | X i / 2 > + < X _ 3 / 2 1 I X ? i / 2 >}

T o s h o w t h e e q u i v a l e n c e o f t h e l a s t t w o t e r m s , d e f i n e u s i n g E q . ( 3 ) ,

C . ,

(12)

X 3 / 2 I j j 3 / 2 x j 3 / 2 j

a. = \ r ry X l / 2 L j l / 2 X j l / 2

j

X - 3 / 2 ^ ( " 1 ) j C j 3 / 2 x j - 3 / 2

J

O! = \ I iJ+1/ 2 c aX - l / 2 ’ j l / 2 Xj - 1 / 2 ( 1 3 )

j

w h e r e w e h a v e u s e d C . = ( - 1 )1+*^2 C . ,3-n J+OS u b s t i t u t i n g E q s . ( 1 3 ) i n t o E q . ( 1 2 ) , t h e f i r s t t e r m b e c o m e s

< X 3 / 2 I M x “ / 2 C j » 3 / 2 C j l / 2 < x j ’ 3 / 2 ^ - l X j l / 2 > ( U )j j '

a n d a f t e r s u b s t i t u t i n g E q . ( 1 1 ) , w e o b t a i n

<X3/2 e - l x V =I I CJ'3/2C?l/2 l/» ::i^ > < ^ <Xj.3/2 lxj3/2> <15>j ' j

a n d f u r t h e r r e d u c t i o n l e a d s t o

< x 3 / 2 U - l x “ / 2 > = l c j 3 / 2 c n / z i m / 2 ) 0 " 1 / 2 ) • j

L i k e w i s e ,

<X.3/2li+ lx"1/2> = I Cj3/2CJ“l/2 V O*372*0- 172’ 'j

T h e r e f o r e ,

<x3/2 1 - 1 1?2> ~ <X-3/2 lX-l/2 >T h e r e f o r e ,

< x 3 / 2 ^ - l x 4 / 2 > _ < ^ _ 3 / 2 I L 1x - l / 2 >

s o t h a t E q . ( 1 2 ) b e c o m e s

[ < m | H ' | n > ] l / 2 = - 2 A " ^ / ( J + 3 / 2 ) ( J - l / 2 ) < X3 / 2 U - I X i / 2 >

T h e K ' = 5 / 2 t e r m s a r e e v a l u a t e d

< J 3 / 2 j J - j + | J 5 / 2 > = \ - / ( J - 3 / 2 ) ( J + 5 / 2 ) < X 3 /2 l L | ^ / 2 >

< J 3 / 2 J J + j _ [ T 5 / 2 > = | ^ f l T - 3 / 2 ) ( J + 5 / 2 ) < X _3 / 2 I * - 1 > ^ / 2 >

a n d s i m i l a r l y w e o b t a i n

[ < m | H | n > ] 5 /2 = - 2 A “ | / ' ( J + 5 / 2 ) ( J - 3 / 2 ) < X _3 / 2 1 ^ | / _ 5 / 2 >

w h e r e

<X-3/2W /-5/2> = I Cj3/2Cf5/2VG+5/2)(i- 3/2' '

153

S u b s t i t u t i n g E q s . ( 2 2 ) a n d ( 2 0 ) i n t o E q . ( 1 ) a n d p e r f o r m i n g t h e

s u m m a t i o n , w e o b t a i n

21 ( J + 3 / 2 ) ( J - l / 2 )1' - l x l / 2 > , , x < X - 3 / 2 1 y _ 5 / 2 ^

“ 3 / 2 1 /2 s ^ 3 / 2 E 5 / 2 > ( 2 4 )

w h e r e a , 0 a r e s u m m e d o v e r a n d r e f e r t o d i f f e r e n t s i n g l e p a r t i c l e s t a t e s

w i t h K = 1 / 2 a n d 5 / 2 , r e s p e c t i v e l y . H o w e v e r , t h i s c a n b e w r i t t e n a s

E < 2 > = A + A J ( J + 1 ) ( 2 5 )

w h e r e

Ai = - 4 a o [<3/4>a3 + ^- a4J ( 2 6 )

A2 “ 4 ( 21J [ A3 + A4] ’

A 3 =< x 3 / 2 ^ ~ l X l / 2 >

^ ( E 3 ' / 2 " E l / 2 )

(27)

(28)

a n d

= <X-3/2'j-'x^5/2>4 /-• /t-> T1 \

6 ' 3 / 2 5 / 2

(29)

H e n c e , t h e s e c o n d o r d e r c o r r e c t i o n t o J ( J + 1 ) m a y b e a d d e d a n d a m o u n t s

t o a n e f f e c t i v e c h a n g e i n t h e m o m e n t o f i n e r t i a . W e h a v e , t h e r e f o r e

154

2E = ( E q + A x ) + ( | - + J ( J + 1 ) ( 3 0 )

o

155

APPENDIX II

H e r e w e w i s h t o c a l c u l a t e t h e c h a n g e i n e n e r g y d u e t o t h e C o r i o l i s

i n t e r a c t i o n i n t h i r d o r d e r p e r t u r b a t i o n t h e o r y u s i n g t h e u n p e r t u r b e d s t r o n g

c o u p l i n g w a v e f u n c t i o n a s i n A p p e n d i x I . T h e t h i r d o r d e r c o r r e c t i o n t o t h e

e n e r g y E q i s g i v e n i n s t a n d a r d p e r t u r b a t i o n t h e o r y b y ( S c 68)

E (3) _ < m | H ’ | k >E - E .

m k

V < k | H t [ n > < n | l - I ' [ m > < k [ H 1 | m >

nE - E m n

E - E , m k

w h e r e a g a i n t h e e i g e n f u n c t i o n s a n d e i g e n v a l u e a r e e q u i v a l e n t t o t h o s e

d e f i n e d i n A p p e n d i x I . L i k e w i s e , H ' i s t h e C o r i o l i s H a m i l t o n i a n a n d s i n c e

w e a r e n o t c o n s i d e r i n g K = l / 2 b a n d s t h e r e a r e n o d i a g o n a l c o n t r i b u t i o n s

( i . e . < m I H * I m > = 0 ) r e d u c i n g E q . ( 1 ) t o

, ( 3 ) < m [ H 1 [ k > \ < k [ H 1 | n x n | H ' | m >L jn

E - E . m k E - E m n

U s i n g t h e s a m e n o t a t i o n a s i n A p p e n d i x I a n d s u b s t i t u t i n g

j n > = | J K >

a n d

H « = - 2A ( J + j _ + J - j + ) ,

(1)

(2)

(3)

t h e m a t r i x e l e m e n t < n | H ' | m > i s w r i t t e n a s

< n | H ' | m > = - 2 A < J K ' j J + j _ j J K > - 2 A < J K ' j J - j + | J K >

S u b s t i t u t i n g t h e s t r o n g c o u p l i n g m o d e l w a v e f u n c t i o n ,

(4)

| J K > =■16 it

D K ( - i ) D _ K x _ P

156

< J K ' | J + i - l « > = e i ! f - < D l K 1 P + | D 4 K > < X _ 0 , P - l x . f ? ( - l ) 2 < J ' 1) ( 5 )16ira n d s i n c e J - j i s i n t e g r a l t h e f a c t o r ( - l ) 2 ^ ^ i s u n i t y . U s i n g t h e r e l a t i o n s

2< D K ± l l J J D K 5 = ( J ± K + 1 ) ( J + K )

< X j j ? ± x l i 4 I X i r , > = ( j ± 0 + 1 ) ( j + 0 ) (6 )

a n d r e c a l l i n g t h a t K = 3 / 2 w e o b t a i n , a f t e r s u m m i n g o v e r K ' ,

2 J + 1< J l / 2 | J + j _ | J 3 / 2 > = ^ 2 ------------ - { ( j - l / 2 ) ( J + 3 / 2 ) < X „ i / 2 I1- | x _ 3 / 2 > • ( 7 )

L i k e w i s e ,

, - ^ J + l -----------------< J l /2 | J - j + [ J 3 / 2 > = ^ ---------- ^ | ( J - l / 2 ) ( J + 3 / 2 ) < x l / 2 | j + | x 3 / 2 >

a n d s i m i l a r t o t h e p r e v i o u s l y d e r i v e d r e l a t i o n i n A p p e n d i x I , w e h a v e

<Xi/2 Ik lx3/2> ” <x-l/2 l1-lx -3/2> *S u b s t i t u t i n g , w e o b t a i n

< n | H ’ | m > = - 2 A ( - 1 ) 2 J + ^ ( J - l ) ( J + 3 / 2 ) < x T i / 2 I3- l x _ 3 / 2 > ^

w h e r e t h e l a b e l a h a s b e e n i n s e r t e d t o d i s t i n g u i s h o t h e r i n t r i n s i c s t a t e s

w i t h t h e s a m e K .

L i k e w i s e ,

< k | H ’ | n > = - 2A ( - 1 )2 J+ 1 ( J + l / 2 ) < / l / 2 | j _ l x ° l i / 2 > ( U )

a n d

into Eq. (4) we obtain as the only nonvanishing matrix element

(8)

0)

157

< m | H ’ | k > = - 2 A - J ( J + 3 / 2 ) ( J - l / 2 ) < X 3 /2 | 5 J x i / 2 > • <1 2 >

S u b s t i t u t i n g E q s . ( 1 0 ) , ( 1 1 ) , a n d ( 1 2 ) i n t o E q . ( 2 ) , w e o b t a i n

E < 3 > = ( - 2A ) 3 ( - l ) 3 J + 3 /2 ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) ' S - ~ 3- 2- ^ ~ ' X l/ 2

B E3/2 ' El/2

x , <X 1/2 -1/2><X -1/2 1 j- -3/2>‘—< -p Tptt ’ ( 1 3 )

3 / 2 " 1 / 2

S e p a r a t i n g t h e i n t r i n s i c d e p e n d e n c e f r o m t h e J d e p e n d e n c e , E q . ( 1 3 ) i s

w r i t t e n a s

2 3E ( 3 ) = - 8 ( - l )J + 3 / 2 ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) C ( 1 4 )

w h e r e C i s d e f i n e d i l l u s t r a t i n g t h e ot - 8 t e r m e x p l i c i t l y a s

a 1 /2L-J /T7 r?<y- \a E3/2“El/2^

+ <X3/2 |i-k 1/2><X 1/2 l^-lx_i/2><:x _x/2 ^-lx_3/2>L i L t r - a W F v ( I 5 )« « 3 / 2 1 / 2 3 / 2 “ 1 / 2

a t jS

a n d

a. = ^ ( - i ) 3+l/2 ( j+ l/ 2 )| C .l / 2 |2 (16)l/ 2

3

158

w h i c h i s t h e u s u a l d e c o u p l i n g p a r a m e t e r f o r a ^ ^ b a n d .

T h e i n t r i n s i c m a t r i x e l e m e n t s ( e . g . < x 2/ 2 l ^ _ | X i / 2 > m a y 1,6

f u r t h e r r e d u c e d a n d w r i t t e n a s

<x»/a L lx “1/2> = I c j3/2 .j

<X_1/2li+lx_3/2> “ < X3/2 U-IXi/2 > ’

a n d

< X l / 2 ^ a- l / 2 > - l Ci U 2 Cl / 2 fl+l/2) ' j

159

I t i s s h o w n t h a t t h e a n g u l a r m o m e n t u m d e p e n d e n c e o f t h e t h i r d

o r d e r C o r i o l i s d e c o u p l i n g t e r m d e r i v e d i n A p p e n d i c e s I a n d I I m a y b e

o b t a i n e d b y e x p a n d i n g t h e e x a c t s o l u t i o n f o r C o r i o l i s m i x i n g b e t w e e n a

K = 3 / 2 a n d K = l / 2 b a n d . T h e e x a c t s o l u t i o n i s g i v e n b y ( D a 6 9 )

APPENDIX III

E ( J ) = 1 / 2 [ E ( J , 1 / 2 ) + E ( J , 3 / 2 ) ]

( J - l / 2 ) ( J + 3 / 2 ) | 2 A |2 .± 1 / 2 [ E ( J , l / 2 ) - E ( J , 3 / 2 ) ] [ l + ~ } l / ( 1 )

( E ( J , l / 2 ) - E ( J , 3 / 2 ) )

w h e r e 2Ak = - ? f <X3/2 |3-IX i / 2> . <2)

o

2E ( J , 3 / 2 ) = E ° + * j r J ( J + 1 ) , ( 3 )o

a n d 2E ( J , l / 2 ) = E ° + ^ [ J ( J + 1 ) + a ( - l ) J + 1 / 2 ( J + l / 2 ) ] . ( 4 )o

l /2U s i n g t h e a p p r o x i m a t i o n ( 1 + x ) = - 1 + l / 2 x a n d s u b s t i t u t i n g E q . ( 3 ) a n d

E q . ( 4 ) i n t o E q . ( 1 ) , w e o b t a i n

2 2E ( J ) = 1 / 2 ( E ° + E ° + ( J ) ( J + 1 ) + ^ a ( - l ) J + l / 2 ( J + l / 2 )o o

* ^ / 2 - > ^ < V + l / 2 <J + 1 / 2 ) [ 1 + ( J * ' / 2 ) ( j + 3 / 2 ) I 2 A > 1 } ( 5 )

&E2

160

w h e r e

AE = E3/2- El/2+i t a(-1»J+1/2<J+1/2» 'C o l l e c t i n g a n d r e a r r a n g i n g t e r m s i n d e s c e n d i n g o r d e r , w e o b t a i n t h e

d e s i r e d r e s u l t

E ( J ) = E q + A J ( J + 1 ) + C ( - l ) J + 1 / / 2 ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 )

w h e r e

(6)

P)

Eo " 1/2 K / 2 + E3/2- 3/4 B» ’A = f f / 2 I o ( 1 + B ) ,

<X3/2 li-IXi/2: I0 AE

0 = 4 a i t o<x3/ 2 l i - lx i /2>

AE

(8)

(9)

(10)

(11)

and

= 2 < - 1)i+1/2(i+l/2>lCjl/2 |2 • (12)

161

APPENDIX IV

A s d i s c u s s e d i n S e c t i o n I - F 3 , t h e H a m i l t o n i a n m a y b e e x p a n d e d

i n a p o w e r s e r i e s o f t h e f o r m

H = (1)y=oH e r e w e e v a l u a t e a l l t e r m s d i a g o n a l i n t h e s t r o n g c o u p l i n g m o d e l w a v e

f u n c t i o n

(2)f o r a K = 3 / 2 b a n d u p t o y = 3 i n c l u s i v e i n E q . ( 1 ) .

S u b s t i t u t i n g R = J - j a n d e x p a n d i n g t h e s q u a r e s , w e o b t a i n

S 2 = J 2 - 2 1 - 1 + j2 , (3)

R 4 = J4-4J2(J-'J) + 2J2j2+4(J-'J)2 -4j2 (jT)+j4 , (4)

a n d

3 6 = J 6 - 6J 4 ( < ? • ! ) + 3 J 4 j 2 + 12J 2 ( J ’" j )2 + - 1 2 J 2 j 2 (J* .“j ) - 8 ( J * j ) 3

+ 1 2 ( T j ) 2 j 2 + 3 J 2 j 4 —6 ( J - j ) j 4 + j 6 . ( 5 )

T h e o n l y n o n v a n i s h i n g t e r m s u p t o t h i r d o r d e r ( y = 3 ) t h a t a r e

d i a g o n a l i n I J K > , t h e s t r o n g c o u p l i n g m o d e l w a v e f u n c t i o n , a r e

A < J K | J 2 | J K > = A 1 ( J ( J + 1 ) ) , (6)

162

A <JK|J4|JK>=A (J2(J+1)2) , (7)

A < J K | J 6 | J K > = A ( J 3 ( J + 1 ) 3 ) , ( 8)

a n d t h e t e r m a n a l o g o u s t o C o r i o l i s d e c o u p l i n g i n a K = 1 /2 b a n d ,

L3 <- 8A < J K | ( J - j )3 | J K > , w h i c h w e n o w e v a l u a t e b e l o w ,

W e r e w r i t e

(?-j)3 = (J? + J+3-+ J-J+)3 (9)0

a n d s i n c e J i s o b v i o u s l y d i a g o n a l w e n e e d o n l y e x p l i c i t l y c a l c u l a t e t h e z3

n o n v a n i s h i n g p a r t ( J + j _ + J - j + ) . T h e c r o s s t e r m s a r e n o n - d i a g o n a l a n d

t h e r e f o r e a r e o m i t t e d .

T h e d i a g o n a l c o n t r i b u t i o n s a r e g i v e n i n f i r s t o r d e r p e r t u r b a t i o n t h e o r y

b y m a t r i x e l e m e n t s o f t h e f o r m

E ( 1 ) = < m | H » | m > . ( 1 0 )m 1 1S u b s t i t u t i n g | m > = | J K > a n d H * = J + j _ + J — w e o b t a i n

< J K | ( J + j _ + J - j + ) 3 | J K > ^

e ^ < DKXJ1+(-i)J' iD! KX-nl<J+i - + J-W3| < x n+<-1)J' i^ Kx.0> ■ <u>16'rrT o e v a l u a t e t h e m a t r i x e l e m e n t f i r s t c o n s i d e r

( J + 3_ + J - j + ) | d + 3 / 2 X 3 / 2 + ( ~ 1 ) "D - 3 / 2 X - 3 / 2 >

- y j ( J - l / 2 ) ( J + 3 / 2 ) | D 3 / 2 > ! j + x 3 / 2 > + ( J - l / 2 ) ( J + 3 / 2 ) | D _ i / 2 > | 3 - X . i / 2 >(12)

w h e r e n o n d i a g o n a l m a s s t e r m s h a v e b e e n o m i t t e d s i n c e t h e y w o u l d

e v e n t u a l l y v a n i s h .

R e p e a t e d a p p l i c a t i o n o f (J+j_ + J-j+) y i e l d s f o r a K = 3 / 2 b a n d

< J K | ( J + j _ + J - j + ) 3 | J K > =

( - l ) 3 < J - i ) ( J - X / 2 ) ( J « / 2 ) ( J + 3 / 2 ) S £ a < D J_3 /2 | D 3 3 / 2 > < x _3 /2 | i ? | x + 3 / 2 >I 677

2 J + 1 J J 3+ ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) — < D + 3 / 2 | D + 3 / 2 > < X f 3 / 2 | i . | X . 3 / 2 > .

1 6 TTJ 2U s i n g < D ID > = § K ' K ’ t h e a b o v e e x p r e s s i o n r e d u c e s t o

< J K | ( J + j _ + J - j + ) 3 | J K > =

( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) f , . 3 , . , . 3 , . , 1x3 ( J - j ) _3 / 2 ^ l x + 3 / 2 > + ^ + 3 / 2 13 - ™ .

E x p a n d i n g t h e i n t r i n s i c w a v e f u n c t i o n s

X - 3 / 2 C j - 3 / 2 ^ - 3 / 2J

a n d

^ 3 / 2 = ^ C j + 3 / 2 > < + 3 / 2j

w e o b t a i n

< X - 3 / 2 l L l x + 3 / 2 > = + 3 / 2 ) ^ C . _ 3 / 2 C .3 /2

j

a n d

< X + 3 / 2 |i~ lx~3/2> = 0 - l / 2 ) a + l / 2 ) 0 « / 2 ) ,

164

b u t s i n c e t h e C . a r e d e f i n e d a s r e a l , w e h a v e 30

< x - 3 / 2 I k l X 3 / 2 > _ K X + 3 / 2 ^ - l )< - 3 / 2 > '

T h e r e f o r e t h e d e s i r e d r e s u l t i s

< J K | ( J + j _ + J - j + ) 3 | J K > = ( - l ) J + 3 / 2 ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) a ^ ( 2 0 )

a „ = Y ( - l ) " a + 3 / 2 ) ( j - l / 2 ) ( j + l / 2 ) ( j + 3 / 2 ) | C . f ( 2 1 )

w h e r e

l 3 / 23

i n c o m p l e t e a n a l o g y w i t h t h e a j y 2 d e c o u p l i n g p a r a m e t e r . I n c l u d i n g a l l t h e

6d i a g o n a l c o n t r i b u t i o n s u p t o S t h e e q u a t i o n f o r t h e e n e r g y b e c o m e s

2E = zTo { Ao+Ai<JHJ+1> + A2(J)2(J+1)2+A3 [(J)3(J+1)3

- 8 ( - l ) J + 3 / 2 ( J - l / 2 ) ( J + l / 2 ) ( J + 3 / 2 ) a 3 /2 ] } ( 2 2 )

i n w h i c h a l l t e r m s i n d e p e n d e n t o f J h a v e b e e n i n c o r p o r a t e d i n t o t h e c o n s t a n t s

V A 1 ’ A 2' a n d A 3-

165

APPENDIX V

A d e r i v a t i o n o f t h e a n g u l a r d i s t r i b u t i o n f o r m u l a a s u s e d i n t h e

M e t h o d I I a n g u l a r c o r r e l a t i o n f o r m a l i s m d i s c u s s e d i n S e c t i o n V - A i s

p r e s e n t e d b e l o w .

T h e H a m i l t o n i a n o f a s y s t e m o f n u c l e o n s i n t e r a c t i n g w i t h a n

e l e c t r o m a g n e t i c f i e l d d e s c r i b e a b l e b y a v e c t o r p o t e n t i a l A i s

V i s a n u n p e r t u r b e d p o t e n t i a l w h o s e e i g e n s t a t e s a r e t h e n u c l e a r e n e r g y

—*l e v e l s a n d M , P , e ^ , a n d y a r e t h e m a s s , m o m e n t u m , c h a r g e , a n d

m a g n e t i c m o m e n t , r e s p e c t i v e l y , o f t h e j t h n u c l e o n .

E x p a n d i n g t h e s q u a r e w e g e t

a c t i o n s i n s e c o n d o r d e r p e r t u r b a t i o n . T h e H a m i l t o n i a n i s n o w r e w r i t t e n

i n t h e f o r m

(1)

w h e r e

H ( r . ) = C U R L A ( r . )y Y (2)

( 3 )

T h e t e r m P . - X ( r . ) c o m m u t e s i n t h e c h o s e n g u a g e

D I V 1 = 0 ( 4 )

s o t h a t•4 + maA - P + P - A = 2 A - P (5)

T h e t e r m i n A i s n e g l e c t e d s i n c e i t c o r r e s p o n d s t o s m a l l e r i n t e r -2

166

w h e r e

H = Hq + H' (6)

p 2H = V + ) ( 7 )o l _ j 2 M . v '

j 3a n d

(8)

B y s u b s t i t u t i n g = g ^ S . w h e r e g i s t h e s p i n g v a l u e o f t h e j

n u c l e o n w i t h i n t r i n s i c s p i n S . a n d m u l t i p l y i n g A ( r . ) . P . b y g , w h e r eJ J j <0

g . = 0 f o r a n e u t r o n a n d g . . = 1 f o r p r o t o n , w e g e t

V eiftH * = " ) a T T : f 2 g , - ^ . * A ( r . ) + g . S . * H ( r . ) ] , ( 9 )i _ , M . C 1 6 t j ) v y s ] ] y 1 '1 3

w h e r e t h e o p e r a t o r s P . a n d S . a r e d e f i n e d a sJ J

V ‘ 1 ?and § = ! ? , (io)w h e r e a i s t h e P a u l i s p i n o p e r a t o r .

T h e e l e c t r o m a g n e t i c f i e l d m a y b e d e s c r i b e d b y a c l a s s i c a l v e c t o r

f i e l d X o r b y a q u a n t i z e d f i e l d w i t h X a s a v e c t o r q u a n t u m m e c h a n i c a l o p e r a t o r .

•s fiI n t h e c l a s s i c a l c a s e A i s g i v e n b y t h e p l a n e w a v e

A ( i l c - r . - i ^ t ) ( - i l o > r , + i ^ t )[Te J + t * e 3 ) . (ID

I n t h e q u a n t i z e d f i e l d c a s e X i s g i v e n b y

V 1 /2 - ( i i t . r - i o j t ) ( - i k * r + i ^ t )JTcr-.t) => ( t e 1 a, + > e J a t ] (12)j Z j t V n k n ‘• n k nlm L K

167

w h e r e i n b o t h c a s e s e i s a p o l a r i z a t i o n v e c t o r a n d k t h e w a v e n u m b e r .

I n t h e l a s t e q u a t i o n a . a n d a * a r e t h e u s u a l c r e a t i o n a n d a n n i h i l a t i o nk n k n

o p e r a t o r s f o r a p h o t o n w i t h w a v e n u m b e r k a n d s t a t e q u a n t u m n u m b e r n .

B o t h d e s c r i p t i o n s l e a d t o t h e s a m e a n g u l a r d i s t r i b u t i o n f o r m u l a .

A l t h o u g h t h e q u a n t i z e d f i e l d a p p r o a c h i s i n t e l l e c t u a l l y m o r e s a t i s f y i n g i n

i t s g e n e r a l i t y , t h e c l a s s i c a l d e s c r i p t i o n o f t h e f i e l d A w i l l b e u s e d f o r t h e

s a k e o f s i m p l i c i t y i n d i r e c t l y a r r i v i n g a t t h e f o r m u l a o f i n t e r e s t .

S u b s t i t u t i n g t h e c l a s s i c a l e x p r e s s i o n f o r a p l a n e w a v e o f a r b i t r a r y

p o l a r i z a t i o n g i v e n b y E q . ( 1 1 ) i n t o E q . ( 8) , w e o b t a i n

I B = y [ H x e " l w t + H 2 e l a ! t ] , ( 1 3 )

w h e r e — —* —^ e . f t l k . r . i k . r .

Hl = - Z i t T o [2gt i V ? e » x ? > e J ] <14)j 3

a n d

h2 = h; •B y d e f i n i n g e = e = - ^ ( e , + i q e , ) , w h e r e q = 1 c o r r e s p o n d s t o r i g h tq h x yh a n d e d a n d q = - 1 t o l e f t h a n d e d c i r c u l a r p o l a r i z a t i o n w i t h r e s p e c t t o eZi1

w h e r e e . i s a u n i t v e c t o r h a v i n g t h e d i r e c t i o n k a n d e , a n d e , a r e t w o z ’ x ' y '

m u t u a l l y p e r p e n d i c u l a r u n i t v e c t o r s , w e m a y r e w r i t e H ' a s— — .7* -*e . f t l k - r . l k . r .

H -. = " / ^ 7 7 * ^ g - S . ' V x e e 3 . ( 1 5 )1 l _ j 2 M C j q s j j qj 3 — i k z

A p l a n e , t r a n s v e r s e , c i r c u l a r p o l a r i z e d w a v e e e m a y b eqe x p a n d e d i n t e r m s o f t h e e l e c t r i c a n d m a g n e t i c m u l t i p o l e c o m p o n e n t s o f

t h e t r a n s v e r s e p a r t o f t h e e l e c t r o m a g n e t i c f i e l d a s

168

- ilcz 1 \ - > m ~*e

v “ - y • <16)L

where _^e V x tyLMLM =--7 ~ ---- • ” =<-1>

k L ( L + 1 )

- m _ ^ L M _ LL M . ff ( 1)

L(L+1)

‘(’LM = ‘L(2L+1) L (kr) C LM«’-<I>> ’

1/2and

C L M4ff

2L+1

where 3 (la?) and Y (@ 5 (p) are the usual spherical bessel and spherical

harmonic functions, respectively.

We can now pass to the more general expansion of a plane wave off

the z axis by making a rotation of axes

e e1 "r = - -r- y (q X ” 1 + X® ) D L (R) (18)q / 2 A L M L M ; M q ' ' '

R = (a; 0,y) is the set of Euler angles that describe the rotation of the z

axis into the direction of k, the z' axis.

By maldng the standard long wave length approximation

jT (kr) - ----JL V ' (21+1)!!

(W “ X L r L c L M (£|''I,> •

X L = (ik)L/(2L-1)!! (19)

169

the electric and magnetic field components become

iX.t e JL M k

.-v [L+T LA n r v (rj c L M ) ,

and

AiXT m L_

L M “ V <rj C L M )x?j • (20)

L(L+1)

Substituting Eq. (5) into Eq. (4), using the long wave length approximation

and recalling standard vector identities, we obtain

H.

where

L M

a L ( 2 L - 1 ) ! !

L+l2L

1/2

(21)

m _ . ea L 1 L ’

Q L M k V (rj C L M )‘Pj »

Q , L M L + l E (rj C L M )o ’j

M L M L 2gl f i L+l V (rj C L M ^ j

M L M lySs j V (rj C L M ),Rj

and

J

1 ' -240 = ^ ^ = 5. 05 x 10 erg/gauss (proton). (22)

170

T l m s o that may be compactly written in a form in which all operators

have the same transformation properties.

H l = - I q,' T L l / D M q ‘R > • <2 3 >

L.M .rr .q

The summation runs over all L,M,tt, q which can contribute to the

transition. A summation over the nucleons is understood. The operator

77T L M 7S *or electric and magnetic components as

0 e eT =f T = n (Q + Q* )L M L M L L M ^ L M

Now it is convenient to define an "Interaction Multipole Operator"77

T* = T?1 = (MT,T + M>L M L M L v L M L M 7

xr = 0/1 for electric/magnetic transition. (24)

The transition probability or angular distribution corresponding to

an emission of a gamma ray from an initial state \ to a final state u is

given by

w = f c ; (S c 6 8 > • (25)

For computational convenience we define a transition amplitude

A M l M / > = < d - ) l / 2 < X l H l < ^ 1 / 2 < J 1M 1 I H i IJ 2 M 2 > (2e»

or after substituting Eq. (23) into Eq. (26)

AM,M / > = ^ 1 / 2 1 q , ,<Jl Ml l TM l J 2M2 > DMq<R> <27)LMxr

171

W < J M l - J 2M ; q E ) = | A ^ (E) | 2 (28)1 2

In the application we are considering, we are not interested in

observing a particular substate M of the final system, therefore,

and

W U ^ - y q l O ^ l A ^ M <k)| , (29)

M 2 1 2

nor are we interested in the polarization,

W(J M r j2 ;E)=£[|A7 (E)|2 + I*m M ] <2°>

m 1 2 1 22

By choosing the quantization axis along a symmetry axis of the

radiating system (e. g. beam axis), is a constant of the motion. Then

the total radiative probability for gamma transitions is obtained by weighing

each W(J M -*J f) by a population parameter, P(M ), defined as theX JL A 1

relative population of substate M . Multiplying Eq. (30) by P(M^) and

summing over M^ we have

W ( J j - J 2 ,S) = 2 , J 2 ,E) (31)

M 1

which we now rewrite, after substituting Eq. (30) into Eq. (31) as

W(fl)=^Wq(g) (32)

q=±l

where

W q<6 ) “'j |AqM M (k) | 2 . (33)-. 1 2

' 1 2

172

The direction of the emitted gamma ray with respect to the quantization

axis is now designated by the angle 0 rather than k*and the obvious quantum

numbers J and J have been omitted.J. d

To evaluate W q(0 ) in Eq. (33), expand the expression for the

transition probability amplitude |A^ (k) |2 as follows1 2

lAMM2fi|2 = ^ ’ I 1L M tt L ' M y

< J 1M l l T [ M t r2 M 2 > < J lM l l < . M 'lJ 2 M 2 > * ' (34>

Using the well known reduction formula for the product of rotation matrices

(Br G2a)

X <il i2ml m2lKH)0l )2,1l “2lKN)DHN (35)h h v . k

D D m in i m 2 n 2 KHN

and recalling that

= (-l)M,_q D L' (36)M'q ' -M'-q

a n d . K

D O 0 = PK (00S 81

Eq. (34) reduces to

(37)

L M tt L ' M y KN

(LL'M-M* |KN) (LL'q-q |KO)

<rJ M IT77 IJ M ■> <J M IT17 |J M >* . (38)1 1 * L M r 2 2 1 1 1 L’M' 2 2 .

173

Since the system possesses cylindrical symmetry, the eigenstates

are aligned (i„ e. M = M T) and, therefore N = 0. Substituting Eq. (37) into

Eq. (38) with N = 0, we obtain

Zy < q7T'qTr(-l)M ’"q(LL'M-M|KO)(LL’q-q|KO)PK cos0)

1 2 LL' vv' I'M

< J l M l l T m l J 2 M 2 > < T M l l T [ ' M . I J 2 M 2 > • (39)

Summing over the final M substates, we obtaind

^ Ia m m ™ 1 2

2

2l7fiZL \ ^ (LLtq-q|K °) ^ (-l)M_q(LL’M - M |KO) P K (cos e j q ^ ’

LL* 7TTT' K M 2,M

X < J 1M 1 1T 1 m 2 M 2 > < J 1M 1 1 T M * $ 2 * 2 * * <40)

Using the Wigner-Eckart theorem

< J 1M llT m l J 2 M 2 > = ( - 1 ) 2 L ( J 2 L M 2 M I T M l) < J l H T [ l p 2 > ' <41)

and substituting into Eq. (40), we obtain

M 2

Z 1 / (LL'q-q |KO)I-l)Rh2E*2L' ^ (-1)“ (LL'M-M |KO)Jc_

2?i h L /LL* 7777’ K M 2M

x ( ^ L M g M l J ^ M J g L ’M g M l J ^ )

x P ^ o o s r t q ^ k J j I I + T ^ l l ^ x J j l T ^ l l J ^ * . ( « )

174

To reduce Eq. (42) we use the well known formula for contracting three

Clebsch-Gordan coefficients into one Clebsch-Gordan and one Racah

coefficient (Br (2b)

X' M^(-l) (ll’m-m|ko)(j2 l m 2m|j1m 1)(j2 l«m2m|j 1m 1) =

MJ -M

. (2J+1)(-1) 1 1(J1J1M 1- M 1 |KO)W(J1J1LL';KJ2)(-l)L'"I/fJl“J2''K

Substituting Eq. (43) into Eq. (42) and Eq. (42) anto Eq. (33), Eq. (33)

becomes

v J i"M i

w q < 9 ) = 2V n L P ( M i )(_1) 2 J i+ 1 ( V A ^ i l * 0 *

M.1

q+ '-I+J - J - K

and

^ ( J J M - M |KO)

>K (J1) = j_j P(M1) (J J M -M |00)O - c M ^

(43)

2^ (LL’q-q |KO) (-1) “ ~1 “2 “ 2Jj+l

K L L ’tttt'

x q77*77' P K (cos 0) W(JlJ1LL»;KJ2)<J1 | |T^ | |J2> < J 1 I IT ij'I lJ2>* * ^

Note that^ 1 + M 1

(J J M -M |°°) = ^ ------- (45)2 J X+ 1

(46)

define the statistical tensor coefficient

Using this definition of p^(J^) to rewrite W^(0 ), we obtain

175

q+L’-L+J -J -K1 2 n r ir+TT'W(1(e)= 2^ } , p ^ J ^ ^ L ’q-qlKOJfyl) 1 2 2 ^ + 1 q

K L L ’tttt’

x PK (cos e)W(JiJiL L ’;KJ2)<Ji | |T^| l ^ x J j |T '| |J2>*. (47)

Define R q (LL'JJ0) as K 1 Z

q+J -J +L' -L-KR q (LL'J^) =(-1) 2J1+1 2L+1 2L«+1

x (LL'q-q |KO) W (J^LL' ;KJ2) (48)

and substitute Eq. (48) in Eq. (47) and get

k \ 2q tt+ tt’ q < I IT l I ^ I i 1w a(0 ) = - T — ) P ! _ ( J J ( - l ) V * (L L ' J i J 2 ) ------------------------q 277ft Lj K 1 2 2Dfl 2LH1 (49)

K L L W

Recalling Eq. (32) and summing over q we obtain

W(0) = W'(0 )+ W _1(q) (50)

where W(0 ) is defined as the probability per unit time per unit solid angle

for the emission of a gamma ray at angle 0 measured with respect to the

quantization axis.

Using the relation

R i?=(-1)I* L'-K 4 (51)

Eq. (50) becomes

176

W ( 8 ) = Z 1 <> W L '-K ) R K ( L L . J l J 2 ) P K (ocse)

KLL'tttt'

< j i I I t I I I j 2 > < j i I I < . I I V *x ------------------

2 L+l 2L'+1

The electromagnetic multipole mixing ratios are now defined in terms of

the reduced matrix elements in Eq. (52) as

(52)

<Ji I iT I° 1 ' 2 L °+ 1

6 l = ~o------ <5 3 >

where L° and 77° correspond to the lowest order multipolarity in the

transition.

The normalization of W(Q) is now chosen such that the coefficient

of p (cos 0) is unity. Using this condition and substituting Eq. (53) into o

Eq. (52), we obtain

6 ff6 irf

W ( 0 ) = ^ p ^ H R ^ L L ’ J ^ ) L 2 ] f 1 + ( - l ) ^ L , + i r + i r , " K ] P K ( Co s 0 ) .

LL'tttt'K 2tT<6 L} (54)L 7t

If the initial and final states have definite parity as is the case in

our application and because the electromagnetic interaction conserves

parity, L+L’+fl+n’’ must be an even integer. Therefore, the term,

(l+(-l)E+L +7T+n ), vanishes unless k is even. Since the parity is well

defined for the initial and final states and the change in parity is related

to L through the equations tt = (-1)E and it = (-1) corresponding to

electric and magnetic transitions, respectively, it is redundant to specify

177

both L and v and consequently the superscript (nj on 6 is omitted.

We may now write Eq. (54) as

6 l AV ' A AW ( 0) = I Pk<Jl> R K <LL1 J1J2> P K (C0S 9 > (55)

LL«K even

The coefficient R „ is related to the coefficient of Ferentz andiv K.

Rosenzweig (Fe 55) by the relation

R ^ L L ' J ^ ) - (-1)L "L’+K F K (LL’J2J1) . (56)

Substituting Eq. (56) into Eq. (55), Eq. (55) becomes

T T | J. O t Or f

W(e)=Z Pk(Jl)(_1) " F K (L L>S2J 1) s 6 2 " P K (C0S e) • (57)LL« L L

Since higher order multipoles are normally negligible in comparison to

the lowest order, it is standard procedure to omit these terms in the formula

for the angular distribution of gamma rays. Including only the two lowest

orders, Eq. (57) becomes<min (2L,2L',2J^)

W « ) = I Ok<Jl>k=0 ,2,4...

1 + 62

x P T (cos 0) (58)

178

„ V U 2,l>t S F K<LLt¥ l t t5 V 1'Ltl'Vll -----------------------;------------------------------ . (59)

1 + 5 ^

Substituting Eq. (59) into Eq. (58), the expression for W(0 ) reduces

to<min(2L,2L'2J )

W ( 0 ) = ^ p ( J 1 ) F k ( J 1 , J 2 ) Q k P k ( c o s 9 ) . ( 6 0 )

K even k

where the coefficient FK (J-jJ2) is deliaed as

Eq. (60) is the exact form of the angular distribution formula used

in our work described earlier in section V-A. The usual attenuation coef­

ficient Q is inserted into the formula to take account of the finite size of

L-L!the gamma ray detector (Fe 65). The coefficient of 6 in Eq. (58), (-1)

is assumed positive as was done in the formula described by Poletti (Po 65).

Since positive and negative values of 6 are allowed, no serious difficulties

arise.

179

We prove here that in a colinear reaction geometry the maximum

substate populated in the residual nucleus is equal to the algebraic sum

of the spins of the target nucleus, incident projectile, and outgoing

particle.

Let

= angular momentum of incident projectile,

? = angular momentum of target nucleus, z

1 IJg = angular momentum of residual nucleus,

and = angular momentum of outgoing particle.

From conservation of angular momentum we have

' (1)

Solving for if and substitutingO

? i - V 8 i ■

J2 = V §2 •

V V §3 •

and y4 = E 4 + § 4 • <2>

APPENDIX VI

we obtain

% = + % “ §4 + AL ’ (3)

180

where

==* —♦A L - Lx + Lg - (Lg + L4) . (4)

In the laboratory frame of reference L = 0 since the target is atd

rest. The above equation reduces to

A L = Lx - (L3 + L4) . (5)

W e now' define

L i = r i x P i •

L3 = r3 x P 3

and L . = r . x P . (6)4 4 4

The constraint of colinear geometry now defines the direction of

the linear momentum P , T , and P . The beam direction (i. e. projectileX d u

P^) is defined along the axis of quantization. Therefore,

H A . —AP1 = klp1l (?)

and since the outgoing particle (i. e. P ) is detected at 0° or 180° with

respect to the beam

■4P = ± k | P , | (8)

and, consequently, to conserve momentum, the residual nucleus (i.e. Pg)

must have

p3 = - p4 . O)

181

Substituting Eqs. (7), (8), and (9) into Eq. (6) and Eq. (6) into

Eq. (5), we obtain

A L = [ P3i 1 ? i ± | P 3 | r 3 T | P 4 | r 4 ] x £ . (10)

Any vector (i. e. AL) defined by the cross product of any two other

—+ —>vectors (e. g. r , k) is perpendicular to the plane defined by the two

vectors. Hence, from Eq. (10)

*4 —*AL-L k

and, therefore,

(AL) = 0. (11)z

The maximum substate of the residual nucleus, Mg, that can be

populated is equal to the algebraic sum of the magnetic substates or z

components of the angular momentum of the individual terms in Eq. (3),

which we now write as

M 3 =(Sl»Z +(S2>z + 'S4>z+ <A L »z • (12)

But from Eq. (11) (AL) = 0 and, therefore, the maximum substate populatedz

is equal to the algebraic sum of the intrinsic spins of the target, incident

projectile, and outgoing particle given by

M 3 = S1 + S2 + S4' (13)

APPENDIX VII

21 24 21STRUCTURE OF Na F R O M THE Mg(p,a) Na REACTIONt

J. G. Pronko, R.A. Lindgrenf and D.A. Bromley'

Wright Nuclear Structure Laboratory Yale University

New Haven, Connecticut 06520

t Work partially supported under U.S. Atomic Energy' Commission Contract

AT(30-1)3223

| Present address: Physics Department, University of Maryland, College Park,

Maryland.

2

21 24 21Abstract: The lower excited states of Na were studied using the Mg(p,0!) Na

reaction at a bombarding energy of = 17. 5 MeV. From analysis of

particle-jy ray angular correlations the spins of the 332, 1723, and 2834 keV

states were found to be_J = 5/2 or (3/2) 7/2 or (3/2), and 9/2 or (5/2),

respectively. Branching and multipole mixing ratios were obtained as well

as evidence for a doublet of states with excitation energies 2833. 8 ± 4. O'and

2803.5 ± 5. 0 keV. This removes a puzzle of long standing concerning an

21 21 apparent missing state in Na as compared to the mirror nucleus Ne.

The experimental information obtained in the present study is compared

21with that for the mirror nucleus, Ne, and discussed in terms of a

Coriolis coupling model. These data, and those to be reported in subsequent

papers for adjacent nuclei, suggest strongly that the intrinsic structure of

these nuclei is perhaps the most rigid of any region of the periodic table.

The model comparisons indicate that the particle bands have significantly

- smaller deformation than do the hole bands; this is consistent with the long

established rapid increase in the magnitude of the static prolate deformation

in moving from A = 20 to A = 24.

3

E

24 ?1N U C L E A R REACTIONS Mg(p,a) Na, E =17.5 MeV; measured— p

21Ex , ay(®), 1 »a(0_). Na levels, deduced_J, branching and

multipole mixing ratios. Enriched target.

1. IntroductionL- X - . . . . - \ -■ V.-* --------

There presently exists much experimental evidence on the nuclear

structure of £ =ll nuclei, winch indicates that pronounced collective effects

1 2should be expected in this mass region. In particular, the evidence ’ ) for the

21 23nuclei Ne and ' Na demonstrates well developed ground state rotational bands

17 +up to and including _J- = 13/2 members. Of particular interest, however, is

21the fact that whereas the Ne band member excitations closely follow a pure

23rotor J(J+1) spectrum, those of Na show a striking oscillatory perturbation

suggestive of higher order Coriolis effects. In view of this striking anomaly,

and since considerably less spectroscopic information is available concerning

21 23the mirror nuclei, Na and Mg, the present experiment on the former

nucleus was initiated as a means of further extending the understanding of

this system. Attention was also centered on the region of excitation of ~3 MeV

21 21in Na inasmuch as three states were known in Ne but only two had been found

21in this region of excitation in Na.

To the present this nucleus has been studied mainly via the (p,p'), (p,y)

2 0and (d, n) reactions on Ne. None of these reactions allow a critical examina­

tion of the electromagnetic decay of individual states lying near or below the

2 0 21 proton breakup threshold to Ne, which is at ~2.45 M e V excitation for Na.

24 21With this in mind the Mg(p,ay) Na reaction (Q^ = - 6. 85 MeV) was chosen

for the present investigation. A beam energy of E = 17.5 MeV (provided byP

the Yale M P Tandem accelerator) was chosen after a search had been made to

find an optimum energy at which all the low lying states were populated with an

acceptable strength. It should be noted that the excitation functions here show

rather striking structure and that precise/energy control is required to select

and maintain a given fluctuation maximum.

21 3The spin and parity of the ground state of Na were known ) to be

7T +_J'“ = 3/2 . The spin and parity of the first excited state has been shown from

20 TTthe J, = 2 pattern in the Ne(d,n) reaction, to be = 3/2 or 5/2 with the

5 9latter spin favored byy -ray angular correlation measurements ’ ). Although

the possibility of having J- = 4 associated with the formation of the 1723 keV

2 0 21 3 7state in the Ne(d, n) Na reaction has been entertained ’ ), there appears to

be no substantial support for this, and the consequent spin and parity assign-

5 *4“ment. Thus, the recent restriction ) of_J~s 7/2 is felt to be more appropriate

5for this state. Rolfs et_ ah ) report the existence of a state at 2833 keV which

has _J — 9/2. They found no evidence for another state in this region but suggest

the possibility of one which would correspond to an energy of 2810 keV, the3

excitation energy normally quoted ) for the state located in this region. The

recent work of Bloch jt al.4) has established the spins and parities of the 3544,

IT + - -,3680, and 3864 keV states as_J“ = 5/2 , 3/2 , and 5/2 , respectively. These

latter authors also established the electromagnetic de-excitation modes of

these states. A further summary of available experimental data, as well as

information derived from the present experiment, is presented in the synopsis

of results of sect. 4.

The main mode of investigation/mvolved the study of particle-gamma

ray angular correlations in a co-linear geometry. This resulted in the

determination of spin and multipole mixing ratio information for the low lying

excited states and their subsequent electromagnetic decay; this study is

presented in sect. 3. A secondary investigation involving study of the alpha

particle groups themselves is reported in sect. 2, while sect. 4 contains the

21discussion of the available experimental Na information in terms of the

collective model.

2. Alpha particle group studies

21As previously noted, the excited states in Na above an excitation energy

of ~2. 45 MeV are unstable against proton emission. Since proton decay widths

can be much larger for these states than the corresponding widths for y-decay,

a study additional to the y-ray angular correlation measurements was advantageous.

• The subsequent effort in this direction was the examination of comparatively high

■ resolution (29 keV FWH M ) alpha particle spectra which were of importance in

21locating the Na excited states. The data for these studies were acquired using

a 76.2 cm ORTEC scattering chamber in which was mounted a target consisting

2 24 2ofa20/igm/cm layer of enriched Mg evaporated onto an 8 pgm/cm natural

carbon foil. The particle detector, a 100 pm thick surface barrier counter, was

highly collimated and under very low bias in order to reduce the background

corresponding to protons which passed completely through the detector. Alpha

particle groups corresponding to states up to an excitation energy of 5. 5 MeV

21in Na were recorded. (See Fig. 1) Angular distributions of some of these

particle groups were obtained but no detailed analysis of these curves in terms

of a (p,a) transfer reaction mechanism has yet been completed; such analyses

will be reported subsequently.

At all angles the alpha particle group to the state at 2. 83 McV appeared

consistently broader than did nearby single groups, suggestive of the possible

existence of a doublet of states. This particle group, obtained at an angle of 55°,

6

7

is illustrated in Fig. 2. The peak was fitted with two Gaussian curves each

having the same width as/es?ab]ished experimentally for a single group. The

solid lines represent the standard shapes for the single peaks obtained in the

fitting while the dashed curve is the combined envelope. A calibration of the

4 5particle spectrum was obtained using the available excitation energies ’ ) of

other excited states as measured recently in terms of their de-excitation

radiations in Ge(Li) detectors; theresulting centroids of the two experimentally

unresolved peaks were found to be at energies of 2833. 8 ± 4. 0 and 2803. 5 ± 5. 0

keV. The former energy agrees quite well with the excitation energy, 2833 ± 7

5keV, obtained for a state in this region by Rolfs et_aL ); this latter experiment

would not have been sensitive to'the presence of a second state unless that

state de-excited mainly via y-ray emission to the first excited state. The above

states will be discussed in more detail in sect. 5. The centroids of other

observable peaks not used in the calibration, were measured as corresponding

to 4308 ± 4, 4425 ± 7, and 4988 ± 8 keV excitation. Further studies involving

21the excitation energies of the Na excited states will be made using the Yale

multigap spectrograph.

3. Particle-y ray angular correlations

The angular correlations of y-rays in coincidence with alpha particle

groups detected at ~177° were obtained in a co-lincar geometry utilizing the

now familiar Method II of Litherland and Ferguson10); this method removes all

requirements for knowledge of the populating reaction mechanism. The y-radiation

was detected in a 12. 7 q x 12. 7 cm Nal(TL) counter which was arranged in a fast-

slow coincidence configuration (2 t = -1 0 0 nscc) with a 100 thick annular surface

8

chamber and beam dump used at this laboratory for the present study, as well

as otherj/-ray spectroscopy studies involving high-energy beams, will be found

elsewhere ). The target consisted of ~150 ugm/cm 2 of isotopically pure 24M g

2deposited on a 20 /igm/cm carbon foil and the beam current v'as restricted,

by counting rate restrictions, to the general region of 25 na.

A typical ungated alpha particle spectrum is illustrated in Fig. 3b where

21groups up to and including those corresponding to ~4. 5 MeV excitation in Na

are evident. An alpha particle spectrum in coincidence (random coincidences

have been subtracted) with all y-rays is shown in Fig. 3a. The particle groups

corresponding to the first four excited states are clearly observable and y-ray

angular correlations were obtained for these states from coincidence y-ray

spectra such as shown in Fig. 4. Although random coincidence counts were

. subtracted from the spectra used in extracting the angular correlations, this

operation has not yet been performed on this spectra of Fig. 4 in order to

illustrate the quality of y-ray spectra to be expected under the present experi­

mental conditions. The angular correlations were obtained over the angles

20°,to 90° and were extracted from the photo-peaks of the coincident y-ray

spectra. The channel spins of the present reaction are such that only y-rays

from magnetic substates m = ± 1 / 2 are in principle selected although a slight

contribution from m = ± 3/2 originates from the finite solid angle of the particle

detector. Using the equations of Litherland and Ferguson19), it was estimated

that for angular momentum I ^ 4 the effect resulting from the solid angle of thejy

particle counter allowed a contribution, in terms of relative population para­

meters, of P(3/2)/P(l/2) 0. 0G. A correction for this as well as one for the

barrier detector. A detailed description of the beam line, goniometer, target

solid angle of the y-ray counter were incorporated into the subsequent analyses

of the angular correlations. In the anatyses, the experimental angular correla­

tions were compared to the theoretically predicted correlations for a given spin

2sequence, in a least squares fit and standard y test. It is assumed in these

analyses that, for the fast coincidence resolving time used, octupole transitions

are not observable. Figs. 5 to 7 illustrate the angular correlations as well as

2the accompanying x analysis; individual states will be dealt with in more detail

in the next section. Listing of the branching ratios and mixing ratios and

Legendre pofynomial coefficients obtained from this experiment are given in

Tables 1,2, and 3, respectively.

4. Synthesis of results

4.1 THE 332 keV STATE

2The y-ray angular correlation as well as the corresponding x analysis

77 -ffor the decay of this state to the J~ = 3/2 ground state is illustrated in Fig. 5.

Acceptable fits to the data were obtained for assumed initial spins ofjl = 3/2

and 5/2; the accompanying mixing ratio solutions are listed in Table 3. Utilizing

1 2these mixing ratios and the known lifetime ' ) of r = 14 ± 3 psec, the transition

strengths for this decay were calculated and are presented in Table 4. Unrea­

sonably large quadrupole enhancements are obtained for a spin of_J = 3/2 while

for J = 5/2 the indicated M2 enhancement of 154 ± 90 W. U. makes a negative

parity assignment improbable. The E2 enhancement of 6 ± 4 W. U. is typical of

a possible collective transition and is in accord with the positive parity assign-

2 0 21 (j- qment obtained from Ihe-t- = 2, Ne(d,n) Na stripping reaction pattern ).

9

10

The spin assignment of J = 5/2 for this state is in agreement with y-ray

9angular correlation work of van dor Leun and Mouton ) where a spin of J_= 5/2

was found to be 20 times more probable than J = 3/2.

4.2 THE 1723 keV STATE

The angular correlations of the y-ray cascade of this state through the

77 += 5/2 first excited state are shown in Fig. 6. Acceptable fits for these data

were obtained for initial spins of_J= 3/2 and 7/2; the mixing ratio obtained for

the respective minima are listed in Table 3. Since the lifetime of this state

is unknown,no further restrictive arguments regarding a spin assignment can

be made on the basis of transition strengths. The angular correlation of the

ground state transition yielded an assignment of_J — 7/2 but was of no help in

further restricting the spin of this state.

4. 2 THE 2432 keV STATE

77 +The 2432 keV state has been given a_J— = 1/2 assignment from the

2 0Jkp = 0 pattern associated with the neutron- distribution in the Ne(d, n)

6“ 8reaction ). The measured angular correlation of the decay of this state to

the ground state must be isotropic and was consequently used as an aid in the

normalization of the data for the other correlation points. The y-ray decay5

mode of this state has been measured to be >95% to the ground state ).

4. 4 T H E 2834 A N D 2804 keV STATE

The gate on the 2. 83 M e V particle group (see Figs. 3 and 4) resulted in

a spectrum containing y-ray lines corresponding to energies of 332, 1110, 1391,

and 2502 keV. The results of Sec. 2 indicates that this group consists of at least

11

two states at excitation energies of 2834 and 2804 keV. The y-ray lines seen

in coincidence with these two state are attributed to the higher lying state and

5are the same lines seen by Rolfs et^aL ) in their triple coincidence experiment

which first pointed out the existence of a state at 2834 keV. The absence of other5

y-ray decay modes for this doublet both in this work and that of Rolfs et ah )

20would suggest that the 2804 keV state has a large width of proton decay to Ne.

This is further substantiated by evidence that the 3680 keV state cascades to a

4 13state in the region of 2. 8 MeV with no secondary y-ray being observed ’ ).

2The angular correlations and accompanying x analysis for the 1110

and 1391 lceV lines may found in Fig. 7. Acceptable fits were obtained for

initial spins of J = 9/2 and 5/2 withjj = 7/2 being ecducbdat the 2% limit on the

, grounds that it is about 25 times less probable than the former two spins. The

• 2834 -• 332 keV transition yielded an assignment of _J_ — 9/2 and was of no help

in effecting a further restriction on the spin.

5. Discussion

5.1 THE G R O U N D STATE ROTATIONAL B A N D

21Although a more rigorous de-limiting of spins for states in Na is not

21presently possible, it is suggested from a comparison of this nucleus with Ne

'(see Fig. 8) that the 0, 338, 1723, and 2834 keV states could be associated with a

ground state rotational band. This band ostensibly follows a J(J+1) rule but an

1 8attempt to reproduce the nuclear properties with a simple Nilsson model fails );

this is particularly true of the Ml transitions which are predicted to be as much

as four times more enhanced than what is observed. This is the result which

12

wave function components on which the B(M 1 ) value depends sensitively'; the B(E2)

value, in contrast, is almost entirely dominated by the core effect and is relatively

insensitive to the presence of small single particle wave function components.

It is concluded that band mixing must be involved in the model interpretation in

order to account for all of the static and dy'namic properties of these two nuclei.

In'the simplest such mixed model the mixing is accomplished by involving the

Coriolis perturbation in first order; in view of the fact that there are several bands

in this mass region available for Coriolis mixing it appears essential to consider

more realistic inclusion of the Coriolis effects. One model which satisfies this

requirement and which may serve well as a reference framefor understanding the

grQss properties of this collective structure is the Coriolis coupling model as

1 6interpreted by Malik and Scholz ). Although the application of the model to

certain mass regions has been open to some question in view of the basic assump­

tion of a rigid core for all particle bands, it is this very feature which would lead

one to expect a fruitful application of this model to the apparent rigid-nuclei found

in the £ = 11 system. Since the exact identity of most states above 3 MeV has not

yet been experimentally established, the application of this model herein was

directed towards reproducing only' the properties of the ground state rotational

band for the two nuclei presently considered. The six 2s-ld positive parity bands,

ld5/2(K = l/2, 3/2, 5/2), 2 s 1 / 2 (K = 1 / 2 ) , and ld3/2(K = l/2,3/2), were allowed

to mix through Coriolis coupling under fixed parameterization. A comparison

between the measured and calculated static and dynamic nuclear properties for

various sets of parameters produced a final choice of B = 0. 48, C = - 0. 35, D= 0,

would be anticipated qualitatively' as a result of neglect of small single particle

13

2 , . andj? /20= 136 keV. The ensu ing theoretical B(E2) and B(M1) values for the

various intraband transitions are presented in Tables 5 and 6 along with their

experimentally obtained equivalents; the measured and calculated ground state

static quadrupole and magnetic moments are listed in Table 7. The unavailability

of all of the experimental absolute strengths for transitions within the ground state

21rotational band in Na, necessitates an examination of theoretical and experimental

relative strengths in the form of branching and mixing ratios if one is to judge the effec­

tiveness of band mixing in this case; such a comparison can be made through the

21 21illustrations of Figs. 8 and 9 for both Na and Ne. The agreement between mea­

sured and calculated nuclear properties is, in general, reasonably good with the

IT +exception of the prediction of a ldj_^2 (K-'=l/2 ) hole band head at ~ 3 MeV excitation.

21 21 TTThere is no evidence in Ne or Na for a J = 1/2 state below an excitation of

5 M e V beyond the now well established^11= l/2+ particle state at 2796 and 2432 keV

excitation, respectively. One way in which a higher predicted excitation energy

may be realized for this band is bj£ an increase in the deformation. It is not

unreasonable to assume, with the degree of rigidity found in this mass region, that

2 2 2 2upon removal of a neutron or proton from a basic Ne or M g core, respectively,

the deformation associated with the original core is maintained in contradistinction

2 0to the Ne core assumed for the particle states. Thus the increase in deformation

of the core from a part icle to a hole state would be consistent with the equivalent

increase from Ne^or Mg. In view of the fact that the Malik and Scholz version

of the Coriolis coupling model has the intrinsic limiting feature of using a single

deformation and moment of inertia for all bands and of calculating all band head

14

energies instead of allowing them to vary as parameters, the few noted discrep­

ancies between calculated and observed properties need not be considered as a

serious restriction to the use of this model as a framework for interpreting the

21 21general nuclear structure of such rigidly deformed nuclei as Na and Ne.

It is interesting to note that an attestation of the degree of rigidness in the

21case of Ne is manifested not only by an adherence to the J(J+1) rule, but also

by virture of the fact that the intrinsic quadrupole moment for states of increasing

J in the ground state rotational band are, within experimental errors, constant1).

Further considerations and generalizations in this direction will be presented in

foithcoming papers which are concerned with the continuing investigation of the

£ = 11 system. The entirety of currently available data does, however, suggest

* that the nuclei in this system may well be among, if nofc.'the most rigid in nature.

5.2 THE 2804 keV STATE

17 4Howard et aL ) and Bloch et_ aL ) have recently shown the existence of

21 21negative parity states in Ne and Na, respectively. The de-excitation of the

23non-normal parity states in these nuclei as well as a third £ = 11 nucleus, Na,

are illustrated schematically in Fig. 10. The similarity in the decay of the

' TT —«P*= 3/2 states at approximately 3. 7 MeV excitation in all three of these nuclei

21 21 23’ suggests that the 2791, 2804, and 2604 keV states in Ne, Na, and Na,

77 -respectively, one structurally equivalent states with a spin and parity o£J~= 1/2“

7or 3/2“. The stripping pattern observed by Ajzenberg-Sclove et al. ) to the 2S04

2 0 21keV state in the Ne(d, n) Na reaction can be reasonably fit with = 0 or 1

and is not contrary to the above interpretation. The lowest negative parity state

in all three of these nuclei could possibly be interpreted as the band head of a

15

TT -K " = 1 / 2 rotational based on a hole in the lp , orbit. A treatise on this inter- — 1 / 2

pretation bj* the authors will be presented in a forthcoming article.

References 16

1. J. G. Pronko, C. Rolfs and II. J. Maier, Phys. Rev. 186 (19G9) XXXX

2. R. A. Lindgren, J. G. Pronko and D. A. Bromley, Proc. Int. Conf. on

Properties of Nuclear States (Aug. 1969), Montreal, Canada.

3. P. M. Endt and C. van der Leun, Nuclear Physics A105 (1967) 1

.4. R. Bloch, T. Knellwolf and R. E. Pbzley, Nuclear Physics. A123 (1969) 129

5. C. Rolfs, W. Trost, E. Kuhlmann, R. Kramer and F. Riess, Nuclear

Physics A129 (1969) 231

6. W. R. Gibbs and W. Grifebler, Nuclear Physics 62 (1965) 548

7. F. Ajzenberg-Selove, L. Cranberg and F. S. Dietrich, Phys. Rev.

124 (1961]_1548

8. M. B. Burband, G. G. Frank, N. E. Davison, G. C. Neilson, S.S. M. Wong

and W. J. McDonald, Nuclear Physics A119 (1968) 184

9. C. van der Leun and W. L. Mouton, Physica J30 (1964) 333

1 0 . A. E. Litherland and A. J. Ferguson, Can. J. Phys. 39 (1961) 788

11. R. G. Hirko, R.A. Lindgren, A. J. Howard, J. G. Pronlco, M. W. Sachs

and D.A. Bromley, to be published

1 2 . A. Bamberger, K. P. Lieb, B. Povh and D. Schwalm, Nuclear Physics\

Alll (1968) 12

13. F.X. Haas, J. K. Bair and C. H. Johnson, Bull. Am. Phys. Soc. 13

(1968) 1371 and private communication

D. Schwalm and B. Povh, Proc. Int. Conf. on Properties of Nuclear

States (Aug. 19G9), Montreal, Canada

E.Kuhlman, R. Kramer, F. Riess, and C. Rolfs, to be published

B. Malik and W. Scholz, Phys. Rev. 150 (I960) 919

A. J. Howard, J. G. Pronko, and C. A. Whitten, Jr., Phys. Rev. 186

(1969) X X X X

A. J. Howard, J. P. Allen and D. A. Bromley, Phys. Rev. 139 (1965) Bll

Fig. 1 A particle spectrum obtained at an angle of 55° using a

2 2420 ugm/cm M g target.

Fig. 2 The particle group, correqoonding to ~2. 83 MeV excitation in

21 oNa, obtained at 0 = 55 . The solid lines represent the

a

standard Gaussian shapes for two single peaks obtained from a

least squares fit and the dashed curve is the combined envelope.

Fig. 3 a) The coincidence total alpha-particle spectrum obtained during

the y-ray angular correlation experiments with an annular counter

positioned at 180° to the beam direction. Random coincidences

have been subtracted, b) The direct alpha-particle spectrum

obtained at 180° to the beam direction with an annular counter.

Fig. 4 The y-ray spectra obtained in coincidence with alpha-particle

groups leading to a) the 332 keV state, b) the 1723 keV state,

c) the 2432 keV state, and d) the 2834 keV state. The alpha-

particles were detected in an annular detector positioned at 180°

to the beam direction. Random coincidences have not been

subtracted from these spectra.

Fig. 5 The y-ray angular correlations for the 332— 0 transition and the

2accompanying x analyses. The solid line through the data points

represent the best fit for a spin_J = 5/2 and has been corrected

for the solid angle effects of the y-ray detector.

18

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

The y-ray angular correlations for the 1723-332-0 cascade and

2the accompanying x analyses. The solid line through the data

points represent the best fit for a spin_J =7/2 and 3/2 and has

been corrected for the solid angle effects of the y-ray detector.

The y-ray angular correlations for the 2834— 1723— 0 cascade

2and the accompanying x analyses. The solid line through the

data points represent the best fit for the spin J_= 9/2 and 5/2 and

has been corrected for the solid angle effect of the y-ray detector.

21A plot of J(J+1) versus the excitation energy of states in Na

21(solid line) and Ne (dashed line) belonging to the ground state

rotational band. The spectroscopic information for the states

21 1 15 21in Ne is taken from Ref. ’ ) while that for Na is taken from

the present work. The phases of the mixing ratios are such to

be consistent with the notation of Ref. 10). ,

The theoretically predicted spectroscopic parameters for the

21 21nuclei Na and Ne using the Coriolis coupling model of Malik

1 6and Scholz ). The parameters used in the calculation were

,' ,0 = 0. 48, C = - 0. 35, D = 0, and ft2/20 = 136 keV.

The de-excitation of the J*~ = 3/2 state at ~3. 7 MeV excitation

in three of the £ = 11 nuclei.

19

Table 1. A summary of results obtained for the branching ratios of states in Na.21

Initial states (MeV)

Final States (MeV)

Present results Previous results ; reference adopted

0. 3 3 2 0. 1 0 0 - 1 0 0

1. 7 2 3 0 7 ± 2 < 6; 5 7 ± 2

0 . 3 3 2 9 3 ± 2 > 9 4 ; 5 9 3 ± 2

2 . 4 3 2 0 > 9 6 >95; 4 > 96

0 . 3 3 2 < 4 ' - < 4

1 . 7 2 3 < 4 - < 4

2. 8 3 4 0 < 2 - < 2

0 . 3 3 2 3 6 ± 6 4 0 ±15; 5 3 7 ± 5

1 . 7 2 3 6 4 ± 6 6 0 ± 1 5 ; 5 6 3 ± 5

2 . 4 3 2 < 6 < 6

Table 2. The results of least-squares Legendre-polynomial fits to the alpha-gamma angular correlation

3.measurements obtained in this experiment ).

State(MeV)

•1Transition

(MeV)a /a

2 0 V ao 2X

0.332 0. 332— 0 -0.292 ± 0. 002 -0. 032 ± 0. 003 0. 03

1.723 1. 723— 0 +0.46 ± 0 . 2 1 -0. 31 ±0. 26 0. 20

1.723— 0.332 -0. 07 ± 0. 01 +0. 05 ± 0. 02 0.18

0.332-0 -0.23 ± 0. 02 -0. 05 ± 0. 02 0. 65

2.432 2. 432-0 isotropic

2. 834 2. 834-1.723 -0. 09 ± 0. 04 +0 . 1 1 ± 0. 06 1 . 1 2

2. 834— 0.332 +0. 85 ± 0. 21 -0.16 ± 0.24 2 . 1

1.723— 0.332 -0. 08 ± 0. 02 +0. 02 ± 0. 03 0. 60

0. 332— 0 -0.18 ± 0. 02 -0. 01 ± 0. 03 0. 27

) The values of ajc/aQ have been corrected for finite solid angle effects of the Nal(Tl) detector.3.

Table 3. Multipole mixing ratios for various y-ray transitions in Na.21

E.(MeV) • E (MeV) J.1

Multipole mixing ratios3-) Present results

Previous results'3); reference

adopted ■ average0)

0.332 0 5/2 3/2 -0. 05 ± 0. 02 -0. 04 ± 0. 04; 5

-0. 05 ± 0. 05; 9

-0. 05 ± 0. 02

0.332 0 (3/2) • 3/2 +0.49 ± 0.13 or +4. 0 ±1. 0 — —

1.723 0 7/2 3/2 assumed zero — assumed zero

1.723 0 (3/2) 3/2 -3. 2 ± 2.2 or -0. 04 ± 0.12 — —

1.723 9.332 7/2 5/2 -0.14 ±0. 03 -0.16 to -1. 75; 5 -0.14 ± 0. 03

1.723 0.332 (3/2) 5/2 +0. 03 ± 0. 02or +3. 7 ±1.0

+0. 70 to +3. 74; 5 —

2. 834 0.332 9/2 5/2 assumed zero — assumed zero

2. 834 0.332 (5/2) 5/2 -0. 63 ± 0. 8 — —

2.834 1.723 9/2 7/2 -0.12 ±0.03 — -0.12 ± 0. 03

2.834 1.723 (5/2) 7/2 +0. 04 ± 0. 04 or +5.7 ± 2. 0 — —

a £,+1) The mixing ratio is defined as 6 (— -— ) where L is the lowest allowed multipole. The phase convention used in the

1 0present work is that of Ref. ).

k) Where the phase convention for the mixing ratio differed in the original reference, the appropriate sign change was

made to agree with the convention used in the present work.

) The adopted average is given only for the most probable spin assignments for a given transition.

23

Tabic 4, Transition strengths (Weisskopf units) for the 0.332 to g. s. transition using3

the measured mixing ratios and known lifetime ).

J.1

6 |M(E1) | 2 |M(M1 ) | 2 |M(E2) |2 |M(M2) j2

5/2 3/2 -0. 05 ± 0. 02 (1. 8 ± 0. 05)10-3 (4.3 ± 1. 0)10- 2 6 ± 4 154 ± 90

-3 -2 43/2 3/2 > 0. 46 < 1. -5 x 10 < 3. 6 x 1 0 > 400 > 1 0

a 12) The lifetime used was that of Ref. ) which is T = 14 + 3 psec.

Table 5 . A comparison of the experimentally and theoretically obtained B(E2) and21

B(Ml) values for the ground state rotational band in Ne. The theoretical

values were obtained using band mixing while the experimental values are

taken from the tabulation of Ref. "S.■1

J.— J- 1 f

Meas.B(E2)2 4 (e fm )

Calc.B(E2)2 4 (e fm )

Meas.B(M 1)

2 - 2 (M 10 )

Calc.B(M 1)

2 - 2 (n io )n

(13/2)— 11/2 > 0. 3 8 >3 42

(13/2)— 9/2 a) 41 — —

(1 1/2)— 9/2 13 ± 10 13 14 ± 5 11

(11/2)— 7/2 43 ±19 36 — —

9/2 — 7/2 21 ± 14 18 22 ± 6 33

9/2 — 5/2 22 ± 7 30 — —

7/2 — 5/2 24 ± 10 29 13 ± 4 12

7/2 — 3/2 16 ± 6 19 — —

5/2 — 3/2 63 ±13 ) 44 6 ± 2 15

) Not determined.

k) This value of B(E2), which is the result of a recent Coulomb excitation experi-14 1

ment ), is used in preference to the one derived from the known lifetime ),

because its magnitude is independent of the multipole mixing ratio.

25

Tabic G. A comparison of the exerpimental and theoretical values of

B(E2), B(M1 ) and lifetime for the = 5/2+, 332 keV stateit +

and its transition to the J = 3/2 , ground state.

•

Meas. Calc.

B(E2) (e2fm4) 30 ± 14 a) 57

B(M1) (p^lO"2) 11 ± 2 a) 17

r (psec) 14 ± 3 8. 6

) These values were calculated using the indicated measured lifetime )

and the mixing ratio obtained in the present experiment.

26

Table 7. A comparison of the experimental and calculated values for

the ground state quadrupole and magnetic moments. The

theoretical values were obtained using band mixing while the3

experimental values were taken from the tabulation of Ref. ).

21 MNa 21 XTNeMoment Meas Calc Meas Calc

Quadrupole (b) a) +0.089 +0. 091 +0.075

Magnetic (p ) +2.386 +2.137 -0. 66 -0.55

a) Not experimentally determined.

CO

UN

TS

PER

CH

AN

NE

L

F i g . 1

COUN

TS

PER

CH

AN

NE

L

C H A N N E L

F ig . 2

CO

UN

TS

PER

CH

AN

NE

L

1.5 K

I .O K

0.5K

IOOK

50K

a)2 4 M g (p ,a ) 21 Na

E p = 17.5 MeV

C O I N C I D E N C E a P A R T I C L E S P E C T R U M

3 8 6 43 6 8 03 5 4 4

2 8 3 4 2 8 1 0

2 4 3 2

7 2 3

3 3 2

0

2 4 3 2

2 8 3 4 3 8 6 4 i

3 6 8 0 "

3 5 4 4 2 3 6 7 I 3 Nj

G.S."

— 3 3 2 - ,

1 7 2 3 il 3 N G.S.

D I R E C T a P A R T I C L E

S P E C T R U M

b - y 1 5 0 100 150CHANNEL NUMBER

Fig. 3

C H A N N E L N U M B E R

COS2 00 0.25 0.5 0.75 1.0

- 9 0 ° - 4 5 ° 0 ° 4 5 ° 9 0 °

A R C T A N 8 F i g _ 5

CO

UN

TS

PER

AN

GL

E

c o s ZQ

A R C T A N 8 j Fig. G

COUN

TS

PER

AN

GLE

c o s 2 60 0.25 0.5 0.75 1.0

ARCTAN 8 ,

Fig. 7

Ex(M

eV)

■I

(13/2*1

2834 ■6416 3616

•(9/2*) 2867

1723

-------------8 — 0.1210.03

9312 712 J_

332

(7/2+) 1747

J U + I)

8 —0:0510.02 0 *■

21II N ° 10

21 \|pIOINe II

T ( p s e c )

<0.04

0.081003

0 1010 02

0151004

2213

Fig. 8

A-00-II69

4

E x (keV) \TTT ( p s e c )

2834 •

1723-

65 35

8=0.06

92 0 8—o— 1—

8 =0.19

3 3 2—9"8 =t0.05

■19/2*)-----------

•(7/2+ ) 0.14 —■

*5/2+

"3/2+ —

8.8'

II N a i o Theo

E x ( k e V

6447 —

i7T

63 —o—

33

8=0.06

\

\N4431-

55 45—o ■ <$-

— 2867-

8=0.11

68 32' ■ <>■

8 = 0.07

94 0 6 1747— s — i — v

8 = 0.18

" 3 5 0 -8='0.05

— 0 — 1----- 1

_ i

Nel O 1 II

-U3/2+)

(II/2+)

■9/2-*

7/2+

•5/2+

•3/2+

T ( p s e c )

0.01

0.07

0.08

0.16

F ig . D

E* (keV) TT Ex(keV) i TT

3 / 2 + '

1 t- r5 16 7 9

foNen

3 / 2 +

3 6 8 0 --------------------3 / 2 “

2 1 Na

T T T T2 19 7 7 2

2 8 3 42 8 0 4 9 / 2 « -

2 4 3 2 -----------

1 7 2 3

l/2“,3 / 2 “ —

1 /2+ — '■

7/2 (+).

3 3 8

0

5/2+•

3/2*

23n i o Na

11 1 2

i x(keV).

3 6 7 8

2 7 0 42 6 4 0

2 3 9 0

2 0 7 7

441

0

Tig. 1 0