EMPIRICAL RENORMALIZATION 0F SHE-LL- MODEL
HAMILTONIANS AND MAGNETIC DIPOLE MOMENTS 0F
sd-SHELL NUCLEI

Dissertation for the Degree of PIT. D.
MICHIGAN STATE UNIVERSHY
WILTON CHUNG
1975

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EMPIRICAL RBNORMALIZATION 0F SHELL-MODEL
HAMILTONIANS and MAGNETIC DIPOLE
MOMENTS OF sd-SHELL NUCLEI
presented by

Wilton Chunq

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ABSTRACT

EMPIRICAL RENORMALIZATION OF SHELL-MODEL
HAMILTONIANS and MAGNETIC DIPOLE
MOMENTS OF Sd-SHELL NUCLEI

BY

Wilton Chung

A refinement of the technique of using energy-level
data to renormalize shell-model Hamiltonians is described.
The one- and two-body matrix elements of the Hamiltonian
are treated as parameters and determined by an iterative
least-squares fit to experimental energy-level data. To
overcome the problems associated with the large number of
correlated parameters involved, the least-squares fit is
reformulated in terms of orthogonal linear combinations of
the one- and two-body matrix elements. -Empirical Hamil-
tonians for full 0d5/2-151/2-0d3/2 model space shell-model
calculations are determined by the described technique
using energy-level data at either the lower or upper end of
the sd-shell. For the lower end of the sd-shell, the
Hamiltonian is renormalized with respect to 197 measured
level energies in the A=17-24 region. For the upper end,
the data set is comprised of 134 measured level energies

in the A=32-39 region. In either case, the initial

Hamiltonians are of the realistic variety of Kuo. A single

Wilton Chung

mass independent (l+2)-body Hamiltonian is found to be
inadequate to simultaneously fit the data sets at.both ends
of the sd-shell. Results of fits to both the upper and
lower ends of the sd-shell show that in each case only a
few orthogonal parameters are very well determined, and
less than half of the orthogonal parameters are at all
well determined by the data sets. The dominant result of
the empirical renormalization obtained for the Kuo matrix
elements is a reduction in attractiveness of the dS/Z-Sl/Z'
dS/Z-d3/2' and s]_/2--d3/2 diagonal two-body matrix
elements. Ground-state binding energies and low-lying
spectra of a number of sd-shell nuclei are calculated with
the renormalized Hamiltonians. The agreement with experi-
ment is very good, except for some missing levels in a few
active particles or active holes systems which are presum-
ably intruder states. Band shifting in which entire
excited bands are predicted overbound with respect to the
ground state, the main defect of present interactions, is
corrected and the improvement is found to extend beyond
the region of nuclei from which the data sets were taken
in the least-squares fits. Ground-state wave functions of
nuclei in the middle of the sd-shell also look more
"normal" than the results of realistic interactions as is
shown by the wave functions generated for 2BSi. It is
hoped that the two sets of renormalized Hamiltonians will
complement each other to give a good description of all

nuclei in the sd-shell region.

,Wilton Chung

Using wave functions generated from the renormal-
ized Hamiltonians, magnetic dipole moments of some ground
and excited states of sd-shell nuclei are calculated.
Results are given both for using the bare-nucleon values of
the single-particle reduced u matrix elements and values
obtained by a fit to available precise measured magnetic
moments. Agreement with experiment is good with either
operator for A=l7-26. However, for A=28—39, results agree
less well with experiment using the bare-nucleon Operator
than using the fitted Operator. Effective orbital
g-factors and intrinsic moments are also obtained from

the fitted operators.

EMPIRICAL RENORMALIZATION OF SHELL-MODEL
HAMILTONIANS and MAGNETIC DIPOLE

MOMENTS OF sd-SHELL NUCLEI

BY

Wilton Chung

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics

1976

ACKNOWLEDGMENTS

I wish to express my deepest appreciation to my
thesis adviser, Professor Hobson Wildenthal, whose influ-
ence goes beyond this work, for introducing me to the many
areas of current interest in nuclear physics, especially
the shell model and this study, and for his guidance and
continuing help and encouragement throughout the course of
this work and the writing of the thesis.

I am grateful to Dr. Jerzy Borysowicz for the
enlightening discussions on least-squares fitting and
error analysis.

To the members of the Cyclotron Laboratory computer
staff, especially to Mr. Richard Au and Mr. Robert Howard,
go my thanks for their excellent maintenance of the
computer and computer assistance, without whom this study
would have been impossible. The patience of the entire
professional staff of the Cyclotron Laboratory is also
appreciated.

Special thanks go to Dr. Rex Whitehead and Dr. Sandy
Watt for their generosities in making the Glasgow Shell-

Model Code readily available.

ii

'Parts of this work were done with the computer
facilities at Oak Ridge National Laboratory, Brookhaven
National Laboratory, Glasgow University in Scotland, and
Rutherford Laboratory in England. I wish to express my
appreciation, especially to Dr. Joe McGrory and Dr. Duane
Larson at Oak Ridge National Laboratory, and Dr. Sandy
Watt at Glasgow University for their help and assistance.

I also acknowledge the financial support of the
National Science Foundation throughout my graduate career

at M.S.U.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES. . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . vii
I. EMPIRICAL RENORMALIZATION OF SHELL-MODEL
HAMILTONIANS (FOR Sd-SHELL NUCLEI) . . . 1
1.1. Introduction. . . . . . . . . l
1.2. Method. . . . . . . . . . . 5
1.3. Details of the Calculation . . . . l7
I.3.A. Initial Attempt, "Intruder
States" and Coulomb
Energies . . . . . . 17
1.3.8. "Particle" Hamiltonian
(A=l7-24 fit) . . . . 26
I.3.C. "Hole" Hamiltonian
(A=32-39 fit) . . . . 35
I.3.D. Computer Codes. . . . . 42
1.4. Results . . . . . . . . . . 43
I.4.A. Orthogonal Parameter Fit . 43
I.4.B. Comparison of Two-body
Matrix Elements. . . . 47
I.4.C. Ground-State Binding
Energies and Spins. . . 56
I.4.D. Ground-State Wave Function
of 2851 . . . . . . 70
I.4.B. Energy Spectra. . . . . 73
1.5. Summary and Conclusion . . . . . 103
II. MAGNETIC DIPOLE MOMENTS OF sd-SHELL
NUCIEI. O O O O O O O O O I O O 106
11.1. Introduction. . . . . . . . . 106
II.2. Details of Calculation . . . . . 108
11.3. Results . . . . . . . . . . 114
II.4. Summary . . . . . . . . . . 124

iv

III

LIE

Page
III. SUGGESTIONS FOR FURTHER STUDY. . . . . . 126

LIST OF REFERENCES 0 O O O O O O O O O C 129

 

IO

LIST OF TABLES

Table Page

1. Binding and excitation energies of states
comprising the data set used to determine
the "Particle" Hamiltonian (MeV). . . . . 27

2. The two-body matrix elements <jajb|V|jde>JT
of "Particle," KUOl4 and PW Hamiltonians
(MeV). . . . . . . . . . . . . . 32

3. Binding and excitation energies of states
comprising the data set used to determine
the "Hole" Hamiltonian (MeV) . . . . . . 37

4. The two-body matrix elements <jajbIVchjd>JT
of "Hole" and K12.5P Hamiltonians
(MeV). . . . . . . . . . . . . . 40

5. Strengths of orbit-orbit interactions (MeV). . 51

6. Matrix elements <d5/2jIVId5/2j>J (Tzl) of the
effective neutron-neturon intefaction 4
(MeV). . . . . . . . . . . . . . 53

7. Ground state nuclear binding energies (MeV)
relative to 160 calculated with the
"Particle" and "Hole" empirical
Hamiltonians . . . . . . . . . . . 57

8. Mass excesses of neutron-rich nuclei (MeV) . . 65

9. Magnetic moments of some ground and excited
states of sd—shell nuclei . . . . . . . 111

10. Comparison between bare-nucleon and fitted

single-particle reduced u matrix elements
(n.m.) . . . . . . . . . . . . . 118

vi

LIST OF FIGURES

Figure Page

1. Eigenvalues d of the D-1 error matrix and'
the deviatigns between corresponding
starting and fitted orthogonal parameters. . 14

2. Percentage change in x2 for a 200 keV
change in each orthogonal parameter. . . . 46

3. Diagonal two-body matrix elements of the
"Particle," "Hole," KUOl4 and K12.SP
Hamiltonians . . . . . . . . . . . 48

4. Off-diagonal two-body matrix elements of
the "Particle," "Hole," K0014 and
K12.5P Hamiltonians . . . . . . . . . 49

5. Deviations between measured and calculated
ground-state nuclear binding energies
for the "Particle" Hamiltonian . . . . . 61

6. Deviations between measured and calculated
ground-state nuclear binding energies
for the "Hole" Hamiltonian. . . . . . . 62

7. Configuration probabilities in the ground-
state of 238i calculated with "Particle,"

"Hole," and KUOl4 Hamiltonians . . . . . 71

8. Energy spectra of A=l7 (T=1/2), A=18 (T=0,l),
and A=19 (T=1/2, 3/2) . . . . . . . . 75
9. Energy spectra of A=20 (T=0,1,2) . . . . . 77
10. Energy Spectra of A=21 (T=l/2, 3/2, 5/2). . . 79
11. Energy spectra of A=22 (T=0,l,2) . . . . . 80
12. Energy spectra of A=23 (T=1/2) . . . . . . 82
13. Energy Spectra of A=23 (T=3/2, 5/2) . . . . 84

vii

Figure Page
14. Energy Spectra of A=24 (T=0). . . . . 85
15. Energy spectra of A=24 (T=l,2) . . . . 87
16. Energy spectra of A=25 (T=l/2) . . . . 88
17. Energy spectra of A=26 (T=0). . . . . 89
18. Energy spectra of A=32 (T=0). . . . . 90
19. Energy spectra of A=32 (T=l,2) . . . . 92
20. Energy spectra of A=33 (T=l/2) . . . . 93
21. Energy spectra of A=33 (T=3/2, 5/2) . 94
22. Energy spectra of A=34 (T=0,1,2) . . . 95
23. Energy spectra of A=35 (T=l/2, 3/2, 5/2). 97
24. Energy spectra of A=36 (T=0,l,2) . . . 99
25. Energy spectra of A=37 (T=1/2, 3/2),

A=38 (T=0,l), and A=39 (T=l/2) . . . . . 100
26. Magnetic dipole moments of some ground and

excited states for A=l7-25. . . . . . . 115
27. Magnetic dipole moments of some ground and

excited states for A=29-39. . . . . . . 116
28. Effective orbital g-factors and intrinsic

moments from the fitted single-particle

reduced u matrix elements (n.m.). . . . . 120

viii

I. EMPIRICAL RENORMALIZATION OF SHELL-MODEL

HAMILTONIANS (FOR sd-SHELL NUCLEI)

1.1. Introduction

Shell-model calculations have proved to be success—
ful in describing not only energy levels of nuclei, but
other prOpertieS such as spectroscopic factors, electro—
magnetic transitions and moments as well.1-12 However, a
serious limitation Of the method is the rapid increase in
the dimensions of the model space as the number of parti-
cles considered active is increased. Present calculations
have to be done in well-chosen truncated spaces. The major
problem is then that of finding an apprOpriate effective
Hamiltonian for the model space.

For mass A=18-38 "s-d shell" nuclei, many aspects
Of nuclear prOperties can be well reproduced by treating
as active while the

the three orbits 0d 0d

5/2' 151/2' 3/2
031/2, 0p3/2, Opl/2 orbits are filled, forming an inactive

16O core.l-3' 5-8

The model space is spanned by all
Pauli-allowed states formed from distributing A—16 active
nucleons in the three active orbits. The effective

Hamiltonian is assumed to consist of only one-and two-body

parts. The one-body terms represent the interaction

energies of the active nucleons with the core, while the
two-body terms represent the residual effective inter-
actions among the active nucleons. For each A,J,T combina-
tion, the many-body Hamiltonian is constructed and
diagonalized in the model basis space. The eigenvalues are
interpreted as energy levels and compared with corresponding
experimentally Observed levels. The associated eigen-
vectors are used to calculate other experimentally
measured nuclear properties.

The advent Of sophisticated computer codes like

13

the J-T coupling code of French et a1. (Oak Ridge-

Rochester) and the M-scheme code of Whitehead14 (Glasgow)
have made shell-model calculations relatively straight-
forward in a computational sense. Nuclei of the sd-shell

have since been extensively studiedl"8

with various
effective interactions obtained by different techniques.
First, there are realistic effective interactions derived
from nucleon-nucleon scattering data, such as those of Kuo
and Brown15 and Kuo.16 The successes and failures of this
type of interactions have been discussed extensively for a
few particles (A=18-22),l and a few holes (A=34-38)2 in the
sd-shell; and more recently for more than 6 active nucleons

(A=23-31) in the Sd-shell.5-8

Secondly, the residual
two-nucleon interaction may be assumed to have a simple
general functional dependence. The variables in the

function are then adjusted to best reproduce the experi-

mental level energies. For example, the depths Of the

II

various spin-isospin components of a potential with a
gaussian radial dependence,17 or the strengths of the two
isospin components of surface—delta or modified surface-
delta interactions18 can be varied to best reproduce the
experimental level energies. Finally, the one- and two-
body matrix elements of the Hamiltonian can be treated as
basic parameters of the model,19 independent of concern
about any underlying potential, and determined empirically
from available experimental level energies.

The technique Of direct empirical determination of
the two-body matrix elements had early successes with
nuclear levels approximately described by models of one or
two "j" orbits?'lo'20'21'22 The problem inherent in this
technique is the rapid increase in the number Of two-body
matrix elements (2bme parameters) with larger model Spaces.
For example, the Hamiltonian for the (f7/2)n model space is
specified by only eight 2bme, while the lsl/2—0d3/2 model
space requires fifteen. For the full sd-shell model space,
sixty-three 2bme are needed to specify the Hamiltonian.
Attemptsl'3 have previously been made to empirically
improve some features of the realistic interactions Of
Kuo.16 However, these attempts circumvented the problem of
"too many" parameters by adjusting only selected 2bme.
Specifically, the Preedom-Wildenthal (PW) interaction3 was
fitted to 72 experimental level energies in the A=18-22

region by adjusting only the 2bme which do not involve the

 

'1?

dI

An

to

5/2'd3/2

and d3/2-d3/2 interactions. The success Of the PW inter-

d3/2 orbit together with only the centroids of the d

action in the A>22 region, as shown by recent full sd—shell
calculationss.8 by the GlaSgow group, is encouraging. The
problem remains of how to manage the larger number of
parameters in an Optimum way. A more systematic method of
extracting as much information as possible from the level-
energy data while at the same time varying the right number
of two-body matrix elements is certainly desirable.

In the following, the formulation Of theoretical
binding energies in terms of one- and two-body matrix
elements is briefly outlined and, from this, expressions
for the least-squares fit to experimental energies are
derived. A method is described in which the least-squares
fit problem is reformulated in terms of uncorrelated linear
combinations Of the one- and two-body matrix elements or
parameters. The uncorrelated parameters are reordered
according to increasing uncertainty, and the well deter-
mined separated from the poorly determined. The well deter-
mined uncorrelated parameters are varied while the poorly
determined uncorrelated parameters are kept fixed. A new
set of one- and two-body matrix elements is derived by
applying the inverse transformation to the uncorrelated
parameters.

Applications of the technique are then described.
An attempt was initially made to Obtain an empirical Hamil-

tonian for the whole sd-shell region by least-squares

fitting to level energies in A=18-24 and A=32-38 simultane-
ously. This attempt was not successful. The problem was
then divided into two separate least-squares fits, in the
A=18-24 and A=32-38 regions, respectively. Results of the
empirical Hamiltonians Obtained are presented and compared
to the realistic Hamiltonians Of Kuol6 used as the starting
sets. Ground-state binding energies of sd-shell nuclei,
and spectra of A=18-24, 25Mg, 26Al and A=32-38 are also

presented. Mass excesses of neutron-rich nuclei are com-

pared with predictions using other mass formulae.

1.2. Method

The Hamiltonian is assumed to consist of one- and

two-body matrix elements only:23

a+a+a a (l)

+
H ’ zeiaiai + zVkimn k l m n

where ei are the single particle energies, are the

vklmn
. + .

two-body matrix elements, and ai and ai are the Single

particle annihilation and creation Operators, respectively.

For a more compact definition, we use:

x = ei or Vklmn

and

equation (1) becomes:

H = E x 6 , (2)
l

where p is the total number Of one— and two-body matrix
elements.
k .th . . .
Let ¢i denote the 1 pure configuration baSis
state where k stands for a set of quantum numbers, e.g,
A-l6 number Of particles, angular momentum J, isospin T,
and parity n. One- and two-body operator matrix elements

can be defined as:

_ k k

Note that the Operator matrix elements are independent of
the interaction and only dependent on the model space. A
matrix element of the many-body Hamiltonian can then be

expressed as:

(I:
I

_k'k
<¢i|Hl¢j>
Pk k
- §<¢i|61|¢j>x1

pk
— Zpijlxl . (4)
1
Let Wk be an eigenstate with corresponding eigen-
k

k . . .
value A W can be expressed as a linear combination of

the basis states OE;

(5)

where a? are the amplitudes Of the wavefunction, and D is
the dimension Of the state k in the model space. Equations

(4) and (5) give:

<wklnlwk>

>2
I

k k k k
Ijaiaj‘¢i'fi'¢j>

N
' M
'9
O
t
x

31 1 (6)

»
II

I—I M'O
'm

x1 (7)

where 81 = Z a jl (8)
Equation (7) expresses the eigenvalue 1k as a linear
expression of the one- and two-body matrix elements. The
Bi's are just combinations of the operator matrix elements
and amplitudes of the wavefunction. It should be noted
that the 8's, unlike the p's, depend on the Hamiltonian
through the amplitudes a's.

Changes in the one- and two-body matrix elements
which improve the agreement of the 1k with experimental

level-energies can be Obtained by minimizing the quantity:

n
x =Z(E -)\) (9)
k

k . .
where Eexp's are experimental level energies corresponding

to the lk's for the shell-model eigenstates Wk's, and n is
the total number of such level energy data. ngp is the
binding energy relative to the model core, with Coulomb

energies extracted. The p number of parameters in the

Hamiltonian gives p equations of

If.

0 m=l,2,3,...,p (10)

C)
X
5

Equations (7), (9) and (10) give:

 

P
n p k
k k 32 B x
X (E -28 x ) , l' 1'
k exp 1 l l l _ 0
3x
m

m=l,2,3,...,p (11)

It has been pointed out that the 8's depend indirectly on
the interaction through the a's. However, the a's change
slowly with the interaction. For small changes in the
interaction, the 8‘s can be treated as constants; more
precisely, with an assumption Of approximate linearity of
equation (7), the minimization Of equation (9) can be done
iteratively until the interaction converges. Equation (11)
then becomes .

n k k

P k

n
Defining Y = 28 am (13)
k

exp = gEk k

and em k exme (14)
Equation (12) becomes:
gY x = Eexp m=l 2 3 (15)
lmll m I I I'OOIP

The least-squares fit reduces to that of solving p equa-
tions for p unknowns. Equation (15) in more compact

matrix notation becomes
GX = E (16)

where G is a pxp matrix whose matrix elements are the

Y 1'3; x is a vector composed of the (unknown) parameters,

m
and E is a vector derived from the experimental level-
energies. It should be pointed out here that the single-
particle energies and/or some of the two-body matrix
elements may be fixed in the least-squares fit; in this
case p would denote the number of free parameters, not
necessarily the total number of onee and two-body matrix
elements. For the rest Of this section, however, p is
taken to be equal to the number of two-body matrix elements.
The procedure commonly followed in the past in

l

solving equation (16) has been to solve for G- by

numerical methods. This would then be followed by a matrix

10

multiplication G-IE to Obtain the parameters X. However,
the problems of the large number of parameters (or equiva-
lently the correlations among the parameters), and an
insufficient data set, make it difficult to do the minimi-
zation effectively. In previous cases, some additional
assumptions were usually made. For example, certain para-
meters were assumed to be poorly determined by the data
set and were fixed at prior theoretical values. In deter-
mining the PW interaction, for example, blocks of para-
meters were assumed to vary by only one single additive
constant rather than independently. The centroids of the
dS/Z-d3/2 and d3/2-d3/2 interactions were adjusted in this
way.

MacFarlane has previously compared24 empirically
determined 2bme with realistic effective interactions. He
noted that the eigenvectors of the error matrix give
uncorrelated linear combinations of the two-body matrix
elements, and that the corresponding eigenvalues give the
uncertainties in these linear combinations. From the error
matrix for the p-shell calculation of Cohen and Kurath,9
MacFarlane found that of the 11 independent two-body
parameters, only seven were well determined. Similar
results were found in other shell-model least-squares fits,
with an increasing prOportion of poorly determined linear
combinations as the number of two-body matrix elements

increased.

11

In View of the difficulty in thus determining a
large number of two-body matrix elements, the question may
be asked whether the x2 can be minimized in terms Of
uncorrelated linear combinations Of the two-body matrix
elements. The mutual independence Of the parameters should
then make the problem more manageable. In others words,
can the G in equation (16) be diagonalized as in the case
of the error matrix investigated by MacFarlane. The eigen-
vectors thus derived would give uncorrelated linear
combinations Of the two-body matrix elements.

It follows from equation (13) that Ym so

1 = Yim'
that G is a symmetric matrix. It can be diagonalized with
the same numerical method used to diagonalize the symmetric
Hamiltonian in shell-model calculations. The least-
squares fit can then be reformulated in terms of uncorre—
lated linear combinations of the two-body matrix elements
or for a shorter name "orthogonal parameters."

Let A be the transformation matrix formed from the
eigenvectors of G. Matrix A being orthogonal, A.1 is just

AT. Let D be

D = AGAT (17)

It is interesting to note that the uncorrelated linear
combinations Obtained here are exactly the same as those
obtained by MacFarlane from the error matrix. This is

easily seen by getting the inverse of D, in which case

12

where G-1 is just the error matrix. The eigenvalues of
the matrix D are just the inverses of the eigenvalues of
the error matrix. The same transformation is applied to

the right hand side Of equation (16)

~

C = A6 (18)
A new set of orthogonal parameters is then Obtained by
Y = C/D (19)

where the matrix division denotes dividing each component
of C by the corresponding diagonal matrix element Of D.
Applying the inverse transformation immediately gives a new

set of fitted two-body matrix elements, i.e.,
X=AY (20)

The above procedure merely replaces the inversion
of G by the diagonalization of G, it does not solve all
problems of the least-squares fit. We next note that since
the eigenvalues of the error matrix are the squares of the
uncertainties of the corresponding orthOgonal parameters,
the eigenvalues of the D matrix are just the inverses Of
the squared uncertainties. The orthOgonal parameters can
then be ordered according to the increasing or decreasing

certainty with which the data set determines them, and the

13
well determined orthogonal parameters separated from the
poorly determined orthogonal parameters.

In Figure l are plotted the eigenvalues dm of the
D- error matrix, and the deviations between corresponding
starting and fitted orthogonal parameters. The fitted
orthogonal parameters are derived from the least-squares
fit, i.e., Y in equation (19). The starting orthogonal
parameters are derived by applying the same transformation
on the starting Hamiltonian for the iteration.

It is clear that the deviations for the very well
determined orthogonal parameters are systematically much
smaller than the others. The deviations for the poorly
determined orthogonal parameters are large and randomly
distributed, a large part of which must be contributed from
round-off errors. For these parameters with large uncer-
tainties, the starting orthogonal parameters are as "good"
as the fitted orthogonal parameters in terms of fitting
the data set. To avoid the round-Off errors, and in the
spirit of the linear approximation of equation (17),

i.e., keeping the change in the interaction small in each
iterative least-squares fit, the procedure in obtaining
a new interaction is modified. First, the transformation

A is applied on the starting Hamiltonian, i.e.,

Y8 = A xS (21)

l4

 

 

 

 

 

 

 

l -x -IO’3
x
x
Xxx
x x
x x
S x" - xx 0.
E xx x 53
EIO'I- xxx xx “IO-43.
E xx x ‘33
U . x
xx
x
x
'2 I l l l l liii '5
IO 5 IO l5 /\r 50 55 60 41,0
4,
' nn
40IL
.T‘ .0
>
0930 -
2
v o
820 ""
N
< 0
IO '0 o 0 ° 0
o
0 0t 0 e nae/\rul-m-OWM
5 IO IS 50 55 60
nn

Figure l. Eigenvalues dm of the D.1 error matrix and the
deviations between corresponding starting and
fitted orthogonal parameters.

15

where XS specifies the starting Hamiltonian and Y8 is the

vector of starting orthogonal parameters. A new vector Y'

is defined by:
s
' =
Y Y(6:dm) + Y (6<dm) (22)

The 6 is an arbitrarily set uncertainty level, and the dm's
are the diagonal matrix elements of the D"1 matrix. The
vector Y' is made up of components from Y if the corres-
ponding dm is less than or equal to the uncertainty level 6:
otherwise, components from YS are used. The dm's are the
squared uncertainties Of the corresponding orthogonal
parameters, so that the 6 in effect sets a limit on the
uncertainties Of the orthogonal parameters. A new inter-
action for the iteration is then derived by applying the

inverse transformation on Y',

x' = AT y' (23)

As a result Of the modification of equation (22) some con-
straints have been added to the least-squares fit.
Ideally, the fitted interaction is held to the starting
interaction except for the parts that are well determined
by the data set.

The above procedure describes one iteration of the
least-squares fit. The process is repeated until the
interaction converges, i.e., until the differences between
the (n+1)th set and nth set of matrix elements are

negligibly small. A reasonable realistic interaction,

16

e.g., Kuo's matrix elements16 derived from the Hamada-
Johnston potential,25 or a schematic interaction, e.g.,

the MSDI,18 is usually used as the initial Hamiltonian for
the process. The many-body Hamiltonian matrices are
created and diagonalized in the shell-model space to Obtain
the level energies and corresponding eigenvectors. The
B's, and so the y's, are calculated from the eigenvectors.

(l)

A new interaction X is then derived from the least-

squares fit. A new set of level energies, eigenvectors

and hence the 8's and y's are calculated from the inter-

(1)

action X . From the ensuing new least-squares fit a new

(2)

and further improved interaction X
X(3)

is derived. Again,

(2)

a third set of interaction is calculated from X ,

X(4) from X(3)

, and so on, until the interaction converges.
The diagonalization of the G matrix, or the
reformulation Of the least-squares problem in terms of
orthogonal parameters, attacks the main problems of the
minimization of x2, namely those of the large number of
correlated matrix elements and the insufficiency of the
data set. The matrix elements of G are calculated from
the eigenvectors of the Hamiltonians, so that the data set
indirectly determines the orthogonal linear combinations of
the two-body matrix elements, and separates the well
determined from the poorly determined parameters. The
least-squares problem boils down to having a reasonable
initial Hamiltonian for the shell-model space and Of fixing

the uncertainty level.

17

There exist now many realistic interactions calcu-
lated from potentials derived from nucleon-nucleon

scattering data, such as the Hamada-Johnston potential25

or the Reid potential.26 Schematic interactions, such as

the MSDI,18 have also been shown to reproduce nuclear
observables quite well. The uncertainty level 6 allows

the fixing of the parameters to be varied. Only very well
determined orthogonal parameters may be varied, thus only
making slight improvements in the interactions. In the
other extreme, even poorly determined orthogonal parameters
may be varied with the possible inclusion Of round-off
errors. A compromise between the two is possible by set-
ting the 6 at an intermediate level. The parameters are
found to converge in at most two iterations for a fixed
uncertainty level. Thus, varying the 6 does not affect the
total number Of iterations needed for the whole calculation.
In fact, it is useful in that the change in the inter-

actions can be kept small in each iteration, in line with

the linear approximation of equation (7).
I.3. Details of the Calculation

I.3.A. Initial Attempt, "Intruder States" and
Coulomb Energies

The A=18=22 and 34-38 regions have previously been

1’2 with a

investigated extensively in shell-model studies
full sd-shell model space, i.e., with (A-16) active
particles distributed in the OdS/Z' 131/2, 0d3/2 orbits.

Comparisons were made between experiments and results of

18

realistic, schematic, and limited-fit interactions for
which only a few matrix elements were varied to fit experi-
mental energy data. In general, the results were incon-
clusive as to which interaction was better. Preedom and
Wildenthal3 further expanded the shell-model least-squares
fit in the A=18-22 region by allowing more degrees of
freedom in the two-body matrix elements. All two-body
matrix elements which did not involve the 613/2 orbit, plus
the centroids of the dS/Z-dB/Z and d3/2-d3/2 interactions
were varied. As the active particles are mainly filling
the d5/2 orbit in the A=18-22 region, the matrix elements
involving the d3/2 orbit are hence not well fixed by the
data set. Quantitatively, the PW interaction results were
found to agree better with experiment relative to results
from previous interactions in essentially all cases in the

5-8 in the full

A=18-22 region. More recent calculations
sd-shell model space by the Glasgow group showed that the
improvement extends beyond the A=18-22 region.

Under the stimulation of the success of the PW
interaction, the initial purpose Of the present study was
to find a single empirically determined interaction for
the whole sd-shell region. The Glasgow shell-model
developments have made it possible to include more measured
level energies in the least—squares fit. Specifically, the
two sd-shell regions previously studied can now be expanded

to include measured level energies of A=23,24 and A=32,33

in the fit.

19

More measured level energies will presumably deter-
mine more matrix elements of the Hamiltonian. Using the
above described least-squares method, a single masse
independent (l+2)-body Hamiltonian was fitted simultaneously
to measured level energies in both the expanded A=l7-24 and
A=32-39 regions, with the level binding energies corrected
relative to 160 and with Coulomb energies extracted. It
was found that the least-squares X2 was fairly constant as
the number of data points increased in the A=l7-24 region.
However, adding more measured level energies from the
A=32-39 region into the fit increased the X2 by almost a
factor Of 2. Besides the poorer fit, nuclear observables
calculated from wave functions derived from the fitted
interaction thus obtained were in poorer agreement with
experiment than was the case when the process was confined
to one or the other subsets Of the data. A single set of
mass-independent (l+2)-body Hamiltonian was hence found to
be inadequate for the description of the entire sd-shell
model space. Accordingly, the least—squares fit was divided
into two parts; the A=l7-24 region and the A=32-39 region.

Before discussing the two least—squares fits
separately, it is appropriate here to comment on how
"intruder states" and the extraction of Coulomb energies
are treated in the present study. By "intruder states,"
we mean experimentally Observed states which cannot be even
qualitatively described by a (l+2)-body Hamiltonian in a

A-16

(0s)4(0p)12(ls,0d) shell-model space. Trivially, these

20

include the negative-parity states, for which the model
cannot yield theoretical partners. More important, there
are certain positive—parity states which appear to be domi-
nated by configurations from outside of the sd-shell model
space. Examples are core-excited states where 2,4 or more
particles are excited out of the 160 core, or states where
2,4 or more active particles are excited out of the sd-
shell and into the (Of,1p) orbits. For states in the
A=18-20 and A=36-38 regions, i.e., close to the boundaries
of the model space, those levels suspected to be intruder
states or to involve large admixtures of intruder states
are not included in the least-squares fits. For example,
the second Observed 1+, first observed 2+ and second
Observed 3+ levels in 18F are not included in the fits.
The only exceptions are the second and third observed 0+
in 18O, and the second and third observed 2+ in 38Ar. In
these cases, an approximate centroid for the two states is
used. However, for states in the A=21-24 and A=32-35
regions, i.e., farther away from the model space boundaries,
intruder state problems are generally ignored. The low-
lying Observed positive-parity states are taken to have a
one-to-one correspondence with the calculated results, the
inherent assumptions being that these states have only
small and constant admixtures of intruder states. A
fundamental assumption is that this sort of effect can be

treated by a renormalization of the Hamiltonian through the

least-squares fit.

95

mi

Pr

21

The Hamiltonian used in the least-squares fit
includes no Coulomb terms. Hence the measured level
binding energies have to be corrected by removal of the
Coulomb component. However, only estimates can be made Of
the Coulomb part Of the binding energies. In most previous
work, the Coulomb part of the binding energies was para-

meterized in some form, for example23

_ 1 _ z
EC0u1(A,Z)—CZ + §az(z l)+b[§-] ,

where Z is the number of active protons, [g] is the largest
integer :_% and C, a, b are parameters which depend on the
active orbits in which the protons are distributed. The
energy differences between mirror nuclei having either

Tz=iT=:l/2 or Tz=iT=il is then given by

A1(A) c + Za + %{l-(-l)z}b ,

and

A2(A) 2C + (ZZ + l)a + b ,

respectively.

The parameters C, a, b are determined by a least-squares

fit to measured energy differences between mirror nuclei.22
The Coulomb part of the binding energies can then be
estimated as a function of A and Z.

Other estimates Of the empirical binding energies
minus the Coulomb part are possible with the use of
T> = To+l and T>> = To+2 analOgue states in To nuclei. At

present, all T=l analogue states in T=0 Odd-Odd nuclei, and

22

all T=1 and T=2 analogue states in T=0 even-even nuclei,

corresponding to the ground states Of neighboring nuclei,

27,29

are known in the sd-shell region. For example, the

corrected ground-state binding energy EC r Of the A=18,

or
T=1 system can be Obtained by taking the difference between

measured binding energies Of 16O and 18O. From reference

28:
' - _ _ 18 _ 16
= -l39.813 + 127.624
= -12.189 (MeV).
The Ecorr Of the A=18, T=0 system is estimated from the
excitation energy of the T=1 analogue state in 18F, corres-
ponding to the ground state Of 18O, i.e., from reference
29:
E (A=18 T=0) = E (A=18 T=1) - Ex E (18F T=1)
corr ' corr ' ° ° I

-12.189 - 1.042

-13.231 (MeV).

Assuming the difference in measured binding energies of

18O and 18Ne to be due to the Coulomb contribution, the

Ecorr of A=20, T=0 system is Obtained from the difference
20

in measured binding energies of 18Ne and Ne, i.e.,

23

Ecorr(A=20’ T=0) = B.E.(zoNe) - B.E.(18Ne)
+ Ecorr(A=18' T=1)
= -160.651 + 132.146 - 12.189
= -40.694 (MeV).
The process is repeated with 20Ne to Obtain the Ecorr Of
22Na, 24Mg and so on. The Ecorr of the Odd-A nuclei and

other even-A nuclei with higher isospin are obtained from
the differences in the measured binding energies of the

corresponding isotopes.

20

However, the above estimate for EC of Ne is

orr
not unique; there are other possibilities. Use can be made

20

of the T=1 analogue state in Ne corresponding to the

ground state Of 20F, then:

_ _ _ _ _ 20 _ 18
Ecorr(A-20, T-l) - Ecorr(A—18, T—O) + B.E.( F) B.E.( F)
= -l3.231 - 154.407 + 137.375
= -30.263 (MeV),
and
E (A=20 T=0) = E (A=20 T=1) - Ex E (2°Ne T=1)
corr ' corr ’ ' ' ’
= -30.26 - 10.26
= -40.52 (MeV).
Or use can also be made of the T=2 analogue state in 20Ne
20

corresponding to the ground state of O, in which case:

24

- _ _ 20 _ 16
Ecorr(A-20, T-Z) - B.E.( O) B.E.( O)
= -151.374 + 127.624
= -23.750 (MeV).
and
E (A=20 T=0) = E (A=20 T=2) - Ex E (2°Ne T=2)
corr ' corr ’ ° ’ '
= -23.750 - 16.728
= -40.478 (MeV),
and
E (A=20 T=1) = E (A=20 T=0) + Ex E (2°Ne T=1)
corr ’ corr ’ ° ° '

= -40.48 -+ 10.26

-30.22 (MeV).

The above described procedures are schematically

shown in the following diagram:

 

 

 

160<T=O> ___ III +200(T=2)
180(T=i) -———$——» 18Ne<T=1I I III
18F(T=0) ———EE——»-20F(T=1) II ; 20Ne(T=0)

where the procedures are numbered I, II and III. It is

noted that the three procedures give different estimates

of Ecorr of 20Ne. The discrepancy between the two extremes,

i.e., I and III, is more than 200 keV. It does raise the

25

question of which is the best estimate. It Should be
remarked here that the above procedures do not exhaust all
possibilities of estimates of Coulomb energies. Other
analogue states, such as T=2 states in T=1 nuclei, or
T=3/2 states in T=1/2 nuclei and so on, if experimentally
known, offer many more alternatives to the above procedures.
Qualitatively, part of the discrepancies may be accounted
for as due to the effects Of different numbers of active
neutrons in the different nuclei used in the procedures.
However, a full understanding of the differences is a
problem to which a solution is not attempted in the present
study.

Arbitrarily, procedure III was adopted for the

estimate of corrected ground-state binding energies. The
40

only exception is that of deriving Ecorr of Ca from
36Ar. Use of procedure III would entail use of the
36

unmeasured binding energy of Ca. Procedure I is used
instead, with an additional 200 keV correction for the
discrepancy between procedure I and III which is found to
be quite general in the sd-shell region. The corrected
ground-state binding energies of Odd-A nuclei, and even-A
nuclei with higher isospin were Obtained from the differ-
ences in measured binding energies of the corresponding

isotOpes.

oh
le
th
st

bi

ar
in

12;
as

l3(
Cor
the
(:05
SM

the

11m

26
1.3.3. "Particle" Hamiltonian (A=17-24 fit)

The word "Particle" will henceforth refer to the
least-squares fit in the A=l7-24 region simply because these
shell-model calculations were done in the particle
formalism. The fit in the A=32-39 region will corres-
pondingly be referred to with the word "Hole"; as will be
discussed later, the shell-model calculations there were
done in the hole formalism (relative to 40Ca).

The corrected binding energies of 197 experimentally
Observed states included in the data set for the "Particle"
least-squares fit are tabulated in Table 1, together with
their excitation energies. The considerations for intruder
states and the procedure for the correction Of measured
binding energies relative to 16O for Coulomb contributions
are as described above. The present least-squares fit
including level energies of A=23 and 24 involves diagonal-
izing Hamiltonians of large orders. These can be as large
as 2000 in the j-j coupling scheme (Oak Ridge Code) or
13000 in the m-scheme (Glasgow Code). Large amounts of
computer time and data storage are required not only for
the diagonalization, but as well for the calculation of the
coefficients 8 of equation (7). As a time and storage
saving step, the data set was slightly reduced except for
the last iteration.

The above time and storage considerations, the

linear approximation in the formulation requiring only a

2'7

 

 

 

TABLE l.--Binding and excitation energies of states comprising the data set
used to determine the "Particle" Hamiltonian (MeV).
a a
A N 2T v Ex EB A 2.] 2T v- Ex EB
17 01 ' 01 1 0.87 - 3.27 00 02 1 - 3.53 -26.69
03 01 1 5.79 + 1.60b 02 02 1 1.06 -29.16
05 01 1 0.00 - 9.19 2 3.99 -26.73
3 9.08 -26.19 .
18 02 00 1 0.00 -13.23 09 02 1 0.00 —3o.22
2 3.72 — 9.51 2 2.09 -28.18
09 00 l 3-89 - 9-39 06 02 1 0.66 -29.56
06 00 1 0.99 -12.29 2 2.20 -23,028
2 9.12 - 9.11 a 2.97 -27.258
10 00 1 1.12 -12.11 08 02 1 0.82 -29.90h
- - 2 3.68 -26.598
00 02 1 0.00 -12.19 10 02 1 1.82 .28.9oh
2 8.80 - 7.39c 12 02 1 9.51 -25.71§
09 02 1 1.98 -10.21 19 02 1 9.59 -25.631
2 3.92 - 8.27
06 02 1 5.37 - 6.82 00 09 1 0.00 -23.75
08 02 1 3.55 - 8.69 2 9.95 .19.30
09 09 1 1.67 -22.08
19 01 01 1 0.00 -23.68 2 9.07 -19.68
2 5.39 48.39d 08 09 1 3.57 -20.18
03 01 1 1.56 -22.12
05 01 1 0.22 —23.96 21 01 01 1 2.80 -99.99,
2 9.56 -19.12 2 5.78 -91.96J
07 01 1 9.38 -19.30 03 01 1 0.00 -97.29
2 5.96 -18.22 2 9.69 -92.55k
09 01 1 2.79 -20.89 3 5.39 -91.902
11 01 1 6.50 -l7.18 9 5.55 -91.69k
2 7.99 -15.79 05 01 1 0.35 -96.69
13 01 1 9.65 —19.03 2 3.79 ~93.50
3 9.53 -92.71‘
01 03 1 1.97 -19.67 07 01 1 1.75 -95.99
03 03 1 0.10 -16.09 2 5.93 —91.811
2 3.07 -13.076 09 01 1 2.67 -99.37
3 5.96 -lO.68f 11 01 1 9.93 -92.81
05 03 l 0.00 -16-19 13 01 1 6.95 ~90.79m
2 3.15 -12.99
3 9.71 -11.93 01 03 1 0.28 -38.ll
07 03 1 2.78 -13.36e 03 03 1 1.73 -36.66n
09 03 1 2.37 ~13.77e 2 3.51 -39.88
05 03 1 0.00 -38.39
20 00 00 1 0.00 -90.98 2 3.93 -39.96
2 6.72 -33.76 09 '03 1 1.76 ~36.63n
09 00 1 1.63 -38.85
2 7.96 -33.02 22 02 00 1 0.58 -57.65
08 00 1 9.25 -36.23 2 1.99 -56.29
12 00 1 8.78 —31.70 3 3.99 -59.29
16 00 1 11.95 -28.53 09 00 1 3.06 -55.17

2E3

 

 

 

A 23 2? v Ex 88a A 20 27 V Bx EBa
0“ 00 2 “-36 -53.87 01 03 1 1.02 -61.78
06 00 l . 0.00 -58.23 2 3,39 -53,95Y
2 1.98 -56.2S° 03 03 i 1.83 —60.97
3 2.97 -55.26 2 3.99 -59.362
08 00 1 0.89 -57.39 3 3.99 -58.8iz
2 9.77 ~53.96 05 03 1 0.00 -62.80
10 00 1 1.53 -56.70 2 2.31 -60.992
2 9.71 -53.52 3 3.83 -58.97aa
3 5.83 ~52.90 07 03 1 1.70 -61.10
12 00 1 3.71 -59.52° 2 9.93 -58.37aa
2 6.58 -51.65P 09 03 1 2.52 -60.28bb
14 00 l 9.52 -53.71P 11 03 1 n.27 -58.53
2 9.05 -99.18P
16 oo 1 8.60 -99-63P 05 05 1 0.00 -51.10aa
18 oo 1 9.86 -98.37P
2 12.62 -95.61P 29 00 00 1 0.00 -87.11
20 00 1 13.58 -99.65p 2 6.99 -80.67
02 00 1 7.75 -79.36
00 02 1 0.00 -57.61 2 9.83 -77.28
2 6.29 -51.37q 09 00 1 1.37 -85.79
3 7.39 -50.273 2 9.23 -82.88
02 02 l 5.39 -52.27 3 7.35 -79.76
2 6.86 -50.7sr 9 8.65 -78.96
09 02 1 1.28 -56.33 06 00 1 5.23 -81.88
2 9.96 -53.15 08 00 1 9.12 -82.99
3 5.36 -52.25 2 6.01 -81.10
9 5.92 -51.69 3 8.99 -78.67
06 02 1 5.69 -51.97 10 00 1 7.81 -79.30cc
08 02 1 3.36 -59.25 12 oo 1 8.12 . -78.99C°
2 5.52 ~52.09 2 9.53 -77.58CC
12 02 1 6.35 -51.265 16 00 1 11.86 -75.25dd
16 02 1 11.01 -96.60 2 13.21 -73.90cc
t 3 19.19 -72.97dd
08 09 1 0.00 -93.56
02 02 1 0.97 -77.18
23 01 01 1 2.39 -68.37 2 1,35 .75,30
2 9.93 -66.33 09 02 1 0.56 -77.09
3 6.31 -69.95 2 1.39 -76.31ee
03 01 ' 1 0.00 -70.76 3 1.85 -75.80ee
2 2.98 -67.78 06 02 1 1.39 -76.31ee
05 01 1 0.99 -70.32 2 1.89 -7S.7Gee
2 3.91 -66.85 08 02 1 0.00 -77.65
3 5.38 -65.38u 10 02 1 1.51 -76.19ee
07 01 1 2.08 -68.68
2 9.78 ~65.98 00 09 1 0.00 -7l.67
3 5.77 -69.99 2 9.76 -66.91
09 01 1 2.70 -68.06 09 09 1 1.98 -69.69
11 01 1 5.53 ~65.23V 2 3.87 -67.80
13 01 1 6.23 -69.53v 3 5.58 -66.09
15 01 1 9.09 -61.72" 08 09 1 3.96 -67.71
19 01 1 19.29 -56.52x

TABLE l.--Continued.

259

 

 

 

A 20 2? 9 Ex 283 A 20 2T 9 Ex 28a
25 01 05 1 0200 -7s.97ff 33 03 05 1 0.00 -175.19u
26 02 09 1 0.29 -92.o9gg 35 03 01 1 0.00 -215.39mm
09 09 1 0.10 —92.1888 -
2 0.98 -91.8086 01 ~ 05 1 0.00 -200.98"“
06 09 1 0.00 -92.28hh
,, 38 06 00 1 0.00 -251.19mm
28 02 06 1 0.00 -102.6611
,, . 00 02 1 0.00 -251.06mm
29 03 07 1 0.00 -106.99%%111 09 02 1 2.17 ~298.89mm
os 07 1 0.00 -106.9911»33
- 39 03 01 1 0.00 -269.22mm
31 05 05 l 0.00 -198.62kk 01 01 1 2.50 -26i.72mm
05 01 1 6.71 -257.51mm»°°

 

aUnless otherwise noted, the ground-state binding energies are taken from reference

28, corrected for Coulomb energies and relative to

excitation energies are taken from references 27 and 29.

0’

Reference 30, see text for discussion.

C

‘1.

Reference
Q

Reference
f

Reference

8Reference

38.
39.
90.
#1.

hReferences 91

1Reference
jReference
kReference
lReference
mReference
nReference
oReference
pReference
qReference
rReference
sReference
tReference
uReference
vReference

w
Reference

93.
1N.
95.
96.
97.
98.
99.
50.
51.
52.
53.
59.
55.
56.
57.

and 92.

160; the spin assignments and

. . . . . +
Spectroscopic-factors-weignted centr01d of second and third observeu 0 state.

3O

xReferences 58 and 59.

yThe spin-parity assignment is not definite in reference 60. The
22Ne(d,p) 23Ne results were not sufficient to distinguish between
i=0 and i=1 transitions.

2Reference 60.

aaReference 61.
bb

Reference 62.
ccReference 63.
ddReference 69.

eeReference 65
ffReferences 66 and 67.
88Reference 68.

hhReferences 68 and 69.

11Reference 69. .

jjPre-final calculation showed a douplet of J"=3/2+ and 5/2+ for the
lowest two states in 29Na, both were fitted to the observed ground-
state binding energy.

kkReference 70.

2'Reference 71.

mm . 2

Weighted by (0.25) .

rmReference 72.

00Reference 31, see text for discussion on d -hole strength in 39K.

5/2

31

small change in the interaction in each iteration, and the
rapid convergence of the present method of doing the least-
squares fit in terms of orthogonal linear combinations of
the parameters at a fixed uncertainty level resulted in the
ad0ption Of the following fit procedure. Four iterations
in all were done to arrive at the final fitted interaction
or "Particle" Hamiltonian listed in Table 2. The uncer-
tainty level was kept at a small value in the first itera-
tion and increased slowly for the following three iterations
to ensure only a small change in the interactions in each
iteration. The data sets in the first and second iterations
included only the level energies of A=l7-21 and A=22, T=0
systems listed in Table 1. For the third iteration, level
energies of A=22, T=1,2 and A=23 systems were added to the
data set. The data set was expanded further to include all
level energies listed in Table 1 for the fourth and last
iteration.

The initial set Of 63 two-body matrix elements used

17

is the "K+ 0“ interaction, which was extensively investi-

gated in reference 1. It is one of the realistic effective

16

interactions calculated by Kuo from the Hamada-Johnston

25 It will be referred to as the KUOl4 inter-

potential.
action since a harmonic-oscillator parameter of‘hw=14 MeV
was used in generating the interaction. The PW interaction

was also derived from the KUOl4 interaction. The two-body

matrix elements of the original KUOl4 interaction, the PW

32

Table 2.--The two-body matrix elements <jajb|V|jcjd>JT of
"Particle", KU014, and PW Hamiltonians (MeV).a

 

 

 

213 21b 2jc 23d JT "Particle" K0019b ch

5 5 5 5 01 -2.0099 -2.9381 -2.1293
5 5 5 5 10 -0.8660 -1.0289 -o.9937
5 5 5 5 21 -1.0399 -1.0358 -1.2312
5 5 5 5 30 -1.3939 -0.8589 -l.7788
5 5 5 5 91 0.0208 -0.0502 0.1611
5 5 5 5 50 -9.0307 -3.6690 -9.0232
5 5 5 1 21 -0.6176 -0.8592 -0.6599
5 5 5 1 30 -1.3830 -1.5659 -l.1865
5 5 5 3 10 3.3882 3.1651 3.2056
5 5 5 3 21 -0.9781 -0.3969 -0.9020
5 5 5 3 30 1.9909 1.8796 1.8986
5 5 5 3 91 -1.3293 —1.3626 -1.3801
5 5 1 1 01 -1.3225 -0.9677 -1.9058
5 5 1 1 10 -0.6255 -0.5959 -0.9291
5 5 1 3 10 -0.9292 —0.2368 -0.2399
5 5 1 3 21 -0.9602 -0.8369 —0.8971
5 5 3 3 01 —3.8935 -3.7882 -3.8367
5 5 3 3 10 1.7200 1.6209 1.6917
5 5 3 3 21 -1.2395 -0.9039 -0.9199
5 5 3 3 30 0.8725 0.9996 0.5060
5 1 5 1 20 0.0660 -0.6222 0.1766
5 1 5 1 21 -0.8189 -1.2879 -0.8995
5 1 5 1 30 -3.5513 -3.6919 -3.6603
5 1 5 1 31 0.7762 0.1723 0.7838
5 1 5 3 20 —l.0366 -1.9988 -1.9679
5 1 5 3 21 0.2028 -0.2181 -0.2209
5 1 5 3 30 1.2093 1.1561 1.1709
5 1 5 3 31 -0.3350 -0.0892 -0.0903
5 1 1 3 20 -2.9571 -2.5788 -2.6118
5 1 1 3 21 —1.6881 —1.5511 —1.5710
5 1 3 3 21 -0.9668 -0.7936 -0.7531
5 1 3 3 30 0.0502 0.0269 0.0272
5 3 5 3 10 -5.5217 —5.8276 -5.3692
5 3 5 3 11 0.5267 -0.1257 0.9058
5 3 5 3 20 -3.7876 -9.5271 -9.0520
5 3 5 3 21 0.6659 -0.2037 0.3268
5 3 5 3 30 -0.5305 -1.1313 -0.6127
5 3 5 3 31 0.5976 0.1316 0.6669
5 3 5 3 90 -3.9056 -9.3137 -3.8359
5 3 5 3 91 -1.1927 -1.6603 -1.1985

Table 2.--Continued.

33

 

 

 

Zja 2jb 2jc 2jd JT "Particle" KUOllIb PWC

5 3 l l 10 1.7223 1.7125 1.7345
5 3 l 3 10 —1.6277 -1.9132 -1.9378
5 3 1 3 11 —0.1106 -0.0976 -0.0989
5 3 1 3 20 -l.3218 -l.5404 -1.5602
5 3 l 3 21 -0.2836 -0.7697 -0.7796
5 3 3 3 10 0.1337 0.0383 0.0388
5 3 3 3 21 -0.8424 -1.0101 -1.0230
5 3 3 3 30 2.0286 2.1579 2.1856
1 1 l 1 01 -2.3068 -1.9493 -2.2643
1 l l l 10 -3.3275 -3.1839 -3.4227
1 l l 3 10 0.2719 0.3085 0.3125
1 1 3 3 01 -0.8385 -0.7448 -0.7543
1 1 3 3 10 -0.2569 -0.2127 —0.2154
1 3 l 3 10 -3.0871 ~3.2771 -2.7861
1 3 1 3 11 0.2733 0.2167 0.7525
1 3 1 3 20 -l.34l4 -1.6099 -l.0974
l 3 l 3 21 -0.1653 -0.3267 0.2022
1 3 3 3 10 0.7599 0.7995 0.8097
1 3 3 3 21 -0.1856 -0.2071 —0.2097
3 3 3 3 01 —0.8119 -0.8076 -0.2849
3 3 3 3 10 -0.4708 -0.4695 0.0576
3 3 3 3 21 0.1747 0.0770 0.6110
3 3 3 3 30 —2.6098 -2.5872 -2.0873

 

a
Phase conventions are from reference 1.

Reference 1.

CReference 3.

34

interaction, and the fitted "Particle" interaction are

listed in Table 2.

The initial single-particle energies used were

-4.l4, -3.27, +1.60 MeV for the Od5/2'1 31/2, Od3 3/2 orbits

respectively. The energies for the Od5/2 and 131/2 orbits

were taken from the ground and first 1/2+ states of 170.

Instead of using the energy of the first 3/2+ state in

17O, the centroid of five Observed 3/2+ resonances in 17O,

30

which contain nearly 100 percent of the Od /2 strength,

3

was used for the energy of the Od3/2 orbit. The centroid

energy Of 5.74 MeV plus the binding energy of the ground
state of 170, relative to 16O, -4.14 MeV, give +1.60 MeV

for the energy of the 0d orbit.

3/2
All 63 two-body matrix elements were varied in all
four iterations, although the number of orthogonal para-
meters varied was in all cases less than 63. The number
Of orthogonal parameters varied increased with the uncer-
tainty level. The single-particle energies were fixed at
the chosen values for the first three iterations. On the
final iteration, the energy of the Od3/2 orbit was varied

together with the 63 two-body matrix elements as free

parameters. The fragmentation of the 0d
17

3/2 strength in

O raises the question of what is the best energy to use
for the Od3/2 orbit for this type Of shell-model calcula-
tion. It was found that the energy shifted to +0.88 MeV,
close to the +0.94 MeV which would result if the energy of

the first 3/2+ in 17O was used.

35
I.3.C. "Hole" Hamiltonian (A=32-39 fit)

In shell-mOdel theory, there is a complementary
relation between a particle and a hole representation of
an eigenstate, such that shell-model calculations can be
done either in particle or in hole formalism. In certain
cases, one formalism may be preferable to the other. For
example, in the present study of A=l7-39 sd-shell nuclei,
the same shell-model eigenstates can be described either by
distributing (A-16) particles in the OdS/Z' 131/2, Od3/2
orbits, or by distributing (40-A) holes in the same orbits.
For the A=32-39 region, the hole formalism is preferable
than the particle formalism for two reasons. First, the
number of active holes is smaller than the number of active
particles, so the effects Of three- or more-body contribu-
tions to the interaction should be smaller. Second, it is
also more economical Of computer time and data storage when
doing the least-squares fit. The one- and two-body Operator
matrix elements defined in equation (3) are only dependent
on the model space, the number of active particles or
holes, and the angular momentum and isospin of the eigen-
state. A single set Of Operator matrix elements need only
be generated once and can then be used for both the
"Particle" and the "Hole" least-squares fits. The hole
formalism was used for the A=32-39 region and thus the name

"Hole" Hamiltonian.

36

The data set for the "Hole" least-squares fit, com-
posed Of 134 corrected binding energies of observed states,
is listed in Table 3, tOgether with the excitation
energies. The considerations for intruder states are as
described previously. The measured binding energies were
first corrected as in the “Particle" least-squares fit, and
the "Hole" corrected binding energies relative to 40Ca were
Obtained by simply subtracting the "Particle" corrected
binding energy of 40Ca from the "Particle" corrected
binding energies. The orders of the Hamiltonians are the
same as in the "Particle" case. The fit procedure is also
similar. Again, four iterations were done to arrive at the
final fitted interaction or "Hole" Hamiltonian listed in
Table 4. The same variations on the uncertainty level were
also performed. The data set in the first and second
iterations included the level energies of only A=34, T=0
and A=35-39 systems listed in Table 3. Level energies of
A=34, T=1,2 systems were added to the data set in the third
iteration. Finally, all level energies listed in Table 3
were included in the fourth iteration of the fit.

The initial set of 63 two-body matrix elements used
was the K12.5P interaction extensively studied in reference
2. As described there, the interaction was generated in
the same way as the KUOl4 interaction, except that a
harmonic—oscillator parameter Of'fiw=12.5 MeV was used to
take into account an increase in nuclear size for

increasing A. The two-body matrix elements of the original

137

TABLE 3.--Binding and excitation energies of states comprising the data used
to determine the "Hole" Hamiltonian (MeV).

 

 

 

A 23 2? 0 Ex 583 A 23 21 9 Ex 88a
39 01 01 1 2.50 18.13 35 01 01 1 1.21 65.72
03 01 1 0.00 15.63 2 3.96 68.97
3 9.72 69.23
38 02 00 l 0.95 29.11 9 6.63 71.19
2 1.70 30.36 03 01 1 0.00 69.51
09 00 1 3.93 32.09b 2 2.69 67.20
06 00 1 0.00 28.66 05 01 1 1.75 66.26
2 3.00 67.51
00 02 1 0.00 28.79 3 5.12 69.631
02 02 1 5.55 39.39c 9 5.98 69.99
09 02 1 2.17 30.96 5 5.59 70.101
2 9.90 33.19d 6 6.03 70.59
7 6.83 71.39
37 01 01 1 1.90 92.09 07 01 1 2.65 67.16
03 01 1 0.00 90.69 09 01 1 3.99 68.95
05 01 1 2.79 93.93 -
2 ~ 3.17 93.818 01 03 1 1.56 71.72
07 01 1 2.22 92.86 03 03 l 0.00 70.16
01 03 1 1.73 97.36f 01 05 1 0.00 79.371‘
03 03 1 0.00 95.63
2 9.02 99.65 39 02 00 1 0.96 77.61
05 03 1 3.09 98.728 2 0.67 77.82
2 9.80 50.93r 3 2.59 79.79
9 3.13 80.28
36 00 00 1 0.00 99.92 09 oo 1 1.29 78.39
2 9.33 53.75 . 2 1.89 79.09
09 00 1 1.97 51.39 06 00 1 0.15 77.30
- 2 9.99 53.86 2 2.19 79.39
06 00 1 7.19 56.56 3 2.62 79.77
08 00 1 9.91 53.83 08 00 1 2.38 79.53
2 6.36 55.78
00 02 1 0.00 77.15
00 02 1 3.12 59.15 - 2 - 3.92 81.07
02 02 1 1.16 57.19 3 5.23 82.38
2 1.60 .57.63h’i 02 02 1 9.08 81.23
3 2.67 58.70h'1 2 5.39 82.59
9 3.97 59.50 09 02 1 2.13 79.28
09 02 1 0.00 56.03 2 3.31 80.96
2 1.96 57.99 3 9.12 81.27
3 2.99 58.52 9 9.89 82.09
06 02 1 '0.79 56.821 5 5.99 83.19
2 2.86 58.89 06 02 1 9.88 82.03
08 02 1 9.69 81.89
00 09 1 0.00 60.27 2 6.25 83.90'
02 09 1 9.52 69.79 ~
09 09’ 1 3.29 63.56 02 09 1 0.00 87.83

rum 3. --C0ntin08d.

138

 

 

2J

2?

¢

M
X

EBa

2J

2T

6

EB

 

33

32

01

03

05

07

09

01
O3

05

07

03

00

02
09

06
08

00

02

09

06

08

01

01

01

01

01

03

03

03

03‘

05
00
00
00

00
00

02

02

02

02

02

rd OJ-E QIKDP*O)NJk*F* rarOI~ S’ODN>P*ODK3F'OJNDF‘

NJP‘F*ODK>F*P*NJFJ

HUMH$UNH30MHMP

O

ccnzwnpwnozco
OOCDHCDQO‘DQOQOG

PO
:0
(DO

2.59
3.28
1.85
3.99
9.05
9.20
3.63

O
O
.O

05301013'0300

O I
cccmwmqqo
HOPU‘CDCOOQO

U1
'0

3.99
0.00
1.15
2.29
2.75
0.08
1.33
2.22
2.66
1.76
2.18
3.01
3.15

qucwmmwwomm:

89.91
92.62
92.95
88.57
90.88

92.50m

90.53

91.931»m

92.90
92.71
91.59

92.66m
92.62m

99.07
95.50
”96.61
97.35
95.92
97.56
98.12
98.27
97.70

109.71n

97.21
100.99
101.91

99.99
101.99
102.76
102.62
101.67

103.620

107.65?

109.21
105.36
106.95
106.96
109.29
105.59

106.93?
106.87P

105.97
106.39

107.22P

107.36

31

29

00

09

05

03

09

09

05

07

MHNH

5...

109.26
119.25
111.20
113.99

131.23q

172.91r

 

39

3Unless otherwise noted, the ground state binding energies are taken
from Reference 28, corrected for Coulomb energies and relative to

“OCa; the
Reference

bReference

c
Reference

spin assignments and excitation energies are taken from
27.

73.
79.

Spectroscopic-factors—weighted centroid Of second and third Observed

2+ state.
eReference.
fReference
gReference
hReference
iReference
jReference
Reference
Reference

Reference

255107?

Reference
0Reference
pReference
qReference

r
Reference

75.
76.
77.
78.
79.
80.
72.
81.
82.
71.
83.
89.
70.
85.

Table 4.--The two-body matrix elements <jajb|VIj

40

"Hole" and K12.5P Hamiltonians (MeV).a

cjd>JT O

 

 

 

Zja 23b ch 23d JT "Hole" K12.5Pb
5 5 5 5 01 -2.1234 -2.2766
5 5 5 5 10 -0.8983 -0.9790
5 5 5 5 21 -0.5549 -0.8799
5 5 5 5 30 -0.6833 -0.7269
5 5 5 5 41 0.4434 -0.0323
5 5 5 5 50 -2.9351 -3.0479
5 5 5 1 21 -0.5527 -0.7416
5 5 5 1 30 -1.3040 -1.3368
5 5 5 3 10 2.6485 2.8734
5 5 5 3 21 -0.1927 -0.3841
5 5 5 3 30 1.5701 1.6060
5 5 5 3 41 -0.8755 -1.1738
5 5 1 1 01 -0.7769 -0.8938
5 5 1 l 10 -0.5285 -0.5859
5 5 l 3 10 -0.2574 -0.2170
5 5 1 3 21 -0.8774 -0.7169
5 5 3 3 01 -3.4471 -3.3550
5 5 3 3 10 1.4097 1.4574
5 5 3 3 21 -0.7272 -0.8010
5 5 3 3 30 0.4723 0.4262
5 l 5 l 20 -0.4291 -0.5203
5 1 5 1 21 -0.3676 -1.1110
5 l 5 l 30 -3.0649 -3.0699
5 1 5 l 31 1.0105 0.1493
5 1 5 3 20 -0.9922 -1.2315
5 1 5 3 21 0.3111 -0.2158
5 1 5 3 30 0.9444 0.9998
5 1 5 3 31 -0.4935 -0.0488
5 1 l 3 20 -2.0269 -2.1718
5 1 l 3 21 -1.3537 *1.3349
5 1 3 3 21 -0.7392 -0.6419
5 1 3 3 30 0.0660 0.0295
5 3 5 3 10 -5.0568 -5.3266
5 3 5 3 11 0.2352 -0.1367
5 3 5 3 20 -3.6694 -3.8860
5 3 5 3 21 0.4477 -0.l664
5 3 5 3 30 -0.9271 -0.9606
5 3 5 3 31 0.6083 0.1122
5 3 5 3 40 -3.3154 -3.5935
5 3 5 3 41 -0.3100 -1.4490

f

Table 4.--Continued.

41

 

 

 

Zja Zjb ch Zjd JT "Hole" K12.5Pb
5 3 1 1 10 1.4758 1.6018
5 3 1 3 10 -1.6758 -1.6509
5 3 1 3 11 0.0483 —0.0654
5 3 1 3 20 -0.7370 -1.3372
5 3 1 3 21 -0.3077 -0.6623
5 3 3 3 10 0.0956 -0.0712
5 3 3 3 21 -0.3325 -0.8646
5 3 3 3 30 1.7037 1.8108
1 1 1 1 01 -1.3430 -1.8186
1 1 1 1 10 -2.8093 -2.9245
1 1 1 3 10 0.4661 0.2866
1 1 3 3 01 -0.6696 -0.6906
1 1 3 3 10 -0.2055 -0.1306
1 3 1 3 10 -2.9441 -2.7934
1 3 1 3 11 0.4955 0.1668
1 3 1 3 20 -1.0458 -1.3606
1 3 1 3 21 -0.0240 -0.2924
1 3 3 3 10 0.8651 0.7384
1 3 3 3 21 0.3064 -0.1952
3 3 3 3 01 -0.9707 —0.8197
3 3 3 3 10 -0.4862 -0.4922
3 3 3 3 21 0.0800 0.0571
3 3 3 3 30 -2.1908 -2.1795

 

a
Phase conventions are from reference 1.

Reference 2.

42

K12.5P and the fitted "Hole interactions are listed in

Table 4.

The initial single-particle energies used were

+22.34, +18.13, +15.63 MeV, for the 0d 2, 0d3

5/2’ ls1/
orbits, respectively. The energies for the 0d

3/2

3/2 and
151/2 orbits are taken from the ground and first l/2+

states of 39K. For the OdS/Z orbit, a problem arises from

hole strength in 39K. The

5/2
energy used is taken from recent 40Ca(d, 3He) data by

Doll et al.31 It is the centroid of spectroscopic weighted

the fragmentation of the 0d

energies of all states with i=2 transfer in 39K between
5.27 to 9.75 MeV. The sum of the spectrosc0pic factors
(C28), assuming a dS/Z pick-up for all the states used,

is 4.97, still somewhat smaller than the theoretical total
strength of 6.

All 63 two-body matrix elements were varied in the
first three iterations as in the "Particle" case. Because
of the uncertainty in the OdS/Z orbit energy, its energy
was varied together with the two-body matrix elements in
the last iteration. The energy was then found to shift

down slightly, to +21.75 MeV.
I.3.D. Computer Codes

Various computer codes were used in the present

study. The Oak Ridge Code13 and a modified version of the

Glasgow Codel4 were used to generate the one- and two-body

. . . 1
operator matrix elements p's. A modified verSion of SMIT

43

was used to combine the Operator matrix elements with the
eigenvectors to obtain the linear equations (15), and then
perform the least-squares fit in terms of orthogonal linear
combinations of the parameters. A further modified version
of the Oak Ridge Code which uses the Lanczos iterative
diagonalization method was used to calculate many of the
ground state binding energies and the spectra of A=23,24,
25,26,32 and 33 systems to be discussed in the next

section.
I.4. Results
I.4.A. Orthogonal Parameter Fit

Since the Hamiltonian parameters in the trans-
formed representation are linearly independent (orthoqonal),
it is interesting to ascertain how many of them are well

determined by each of the data sets.
The change in the least-squares x2 from a change

in parameter xk of Axk can be estimated as follows. The

definition of x2 is:

_ 2
—E(8Wmm£ (9)

If parameter xk changes by Axk, the new x2 is:

X2(Axk) =2(282mx m + sfikAxk - 8,12 (24)

44

The change in x2 is obtained from equations (9) and (24):

A(x2) = x2(Axk) - x2

A(x2) = 2 (is x + s Ax - E )2 - (26 x - E )2 (25)
2 m 2m m 2k k 2 m £m.m 2

Equation (25) can be simplified to:

A(x2) = §8£k2(Axk)2 + Axk[%z(282mxm - E£)Blk (26)
2 m '

2
The second expression on the right is just Axk 52., which is

equated to zero for the minimization of x .

Hence, equation
(26) is simply:

A(x2)

2 2
Esgk (Axk) , and

2 _ 2
The change in x2 from a change Axk for parameter xk is just
(Axk)2 multiplied by the corresponding diagonal matrix
element of the G matrix.

In terms of the diagonal matrix
elements dm of the error matrix D-l,

(A )
A(x2) — —:-k— (28)
k
or

MXZ)

H

m D
xx”
‘1)

(29)

where 8k is the uncertainty of parameter xk.

noted that the x

It should be
k affects the x2

independently without

45

affecting the other parameters, as the orthogonal para-
meters are linearly independent.

The x? of the "Particle" Hamiltonian is 0.22 MeV
and for the ”Hole" Hamiltonian is 0.28 Mev. It should be
noted that the uncertainty in the Coulomb correction of the
binding energies can be as much as 200 keV or more. Using
the results of the last iterations of the "Particle" and
"Hole" fits, the change in x2 was calculated for a change
in each of the orthOgonal parameters. In Figure 2 is
plotted on a semi-log scale the percentage change in x2 for
a 200 keV change in each of the better determined ortho-
gonal parameters. The steepness of the curves is to be
noted; only a few orthogonal parameters are extremely well
determined by the data in each case. Less than half of the
63-orthogonal parameters affect the x2 by more than 1 per-
cent in either the "Particle" case or the "Hole" case for
a 200 keV change in their value. The shape of the two
curves are similar, though more orthogonal parameters seem
to be determined by the data set in the "Particle" case
than in the "Hole" case. However, the data set for the
"Particle" fit is bigger than for the "Hole" fit.

In the final iteration of the "Particle" fit,
thirty orthogonal parameters were varied as free parameters,
while twenty were varied in the final iteration of the
"Hole” fit. In either case, all the orthogonal parameters
kept constant affect the x2 by less than 1 percent as shown

in Figure 2. Thus, the X2 obtained for the "Particle" and

46

 

 

IO4:'
E. CHANGE IN X2 FOR 200 keV _
P CHANGE IN EACH ORTHOGONAL
F PARAMETER
‘ ,03_ ° . "PARTICLE"
E . "HOLE"
i .
.QIOZEf
N E . O
3: :2 ° °.
.7 .. ,
Z n
V '0 F ..o 00
E .3. o
.. .0... 0000
_ °. °%
I L.— ...o 0°00
5 060
I '- °0
0 00°
- 0°
- g st
0. 0°
'04 1 1 1 I w u 1 1 004
0 IO 20 30 40 20 30 4o

ORTHOGONAL PARAMETERS IN INCREASING'UNCERTAINTY
_.

 

Figure 2. Percentage change in x2 for a 200 keV change in
each orthogonal parameter.

47

"Hole" Hamiltonian should not be affected much by varying

additional of the less-sensitive orthOgonal parameters.
I.4.B. Comparison of Two-Body Matrix Elements

The two-body matrix elements of the "Particle"
Hamiltonian, the original KUOl4 interaction and the PW
interaction are all listed in Table 2. The two-body matrix
elements of the "Hole Hamiltonian and the original K12.5P
interaction are listed in Table 4. Comparison of the
different sets of interactions is now a matter of comparing
the different lists of two-body matrix elements. The task
is difficult since the least-squares fits were done in
terms of orthogonal linear combinations of the two-body
matrix elements and hence every two-body matrix element has
changed. In Figure 3 are plotted the diagonal two-body
matrix elements of the "Particle" Hamiltonian, the "Hole"
Hamiltonian, and the original KUOl4 and K12.5P realistic
Hamiltonians. The off-diagonal two-body matrix elements
of the four Hamiltonians are plotted in Figure 4. The dots
are the original Kuo matrix elements, and the crosses are
the new "Particle" or "Hole" matrix elements. The "Parti-
cle" interactions are plotted to the right of the "Hole"
interactions. In general, the changes have no clear
pattern, though more matrix elements tend to change in the
positive direction (become less attractive).

One can compare the strengths of diagonal orbit-

orbit interactions for the different Hamiltonians defined as:

48

.mcmwcouaflsmm mm.NHM cam «Hoax
=.oaom= =.maowuumm= on» mo mucoEmHm xwuumfi hponIOSu accommflo .m ousmflm

 

 

 

 

mu. . .ma
- t. .
n... r 1......
I o o 1
cl. / NI. +/ o 1c!
1 or. .Y «0’ o/ 1
n1... Luv jet. 1. I. 1”!
f .. ., .
Nu- ,3 +,.. «v.9.
. . / . .
kr + o \o o .o.
l + r
.1. .t% v # . K, M¥kf +d. in
fi .00... too? + 0 NO... P 1 W
o .4 3 4.x o a
.7 o. to no A
. J ENE. ; A 1
.9
_+. + 1_+
. m_.~b1,~FN_1rb1Nmm_;vn~_1nm-mMEmmgumbo
O . _ 1.0 - _ O .3 O 1. _ - O . _ -11 O - .1J.h
mt ammzmmv am___m_v A______v amm=mmv A_m___mv Amm=mnv. 1~+

 

 

49

.mcmwcoudflemm mm.mam 0cm vHODM =.mHom=

=.mH0fluumm= on» mo mucosmaw xfluume hponlozu Hmcommwplmmo

.¢ musmflm

 

 

 

 

 

 

VA +6 1?...

T (v 1
ml. 1m:

r ‘0’ L
N01 o 00 1NI

. . «f +. .. ..
_ .9, r .2... 4. +, 5.. 4... _

§’ 0’ o. 0 ARom? .9 00+ )4 00.? * to.
v I \ v e. O I O V 1
a... it. o o + + 3. AL 3 {7 3? W

O F It r1 +468 4 14.71“... AV 11 .91” v 11.1 01 an AV ll. 0 M

. .... mm: In. 28 98 :8 new 96 .1... ammo can :8 man... .68 1
T. .03.. n... .3 1 ._+

. t. 32.. .. ... .
N+v ’N‘ 0. 1N...

' w l
n. v Os. 1m+

v k 1
cffi iv...

50

JXT(2J+1)(2T+1) <3a3b|V|3a3b>JT , (30)

where ja and jb are single-particle angular momenta and
J,T are the coupled angular momentum and isospin. In
Table 5 are listed the strengths of the orbit-orbit
interactions for the different Hamiltonians. The strength
of the dS/Z-dS/Z interaction does not change in the fitted
"Particle" and PW Hamiltonians from the original KUOl4

Hamiltonian. The strengths of the d and d -d

5/2'51/2 5/2 3/2
interactions are, however, reduced in both cases by

approximately a factor of 2. The strength of the s1/2--sl/2

interaction is again found to be unchanged by both the

u - n ' ' ’
present Particle fit and the PW fit. The sl/2 d3/2

d3/2-d3/2 interaction strengths are, however, very different

and

for the "Particle" and PW Hamiltonians." Both PW strengths
are very much reduced from the KUOl4 Hamiltonians. For the
u . fl . . _ . .
present Particle Hamiltonian, the d3/2 d3/2 interaction
strength is not changed; the Sl/Z-dB/Z interaction strength
is slightly reduced, though less than the factor of 2
found for the dS/Z-Sl/Z and d5/2-d3/2 interaction strengths.
However, the Sl/Z-dB/Z and d3/2_d3/2 interactions are not
well determined from either the PW or "Particle" data set.
Essentially the same picture is obtained in com-
paring the strengths of the orbit-orbit interactions of the
u :1 ° ° _
K12.5P and Hole Hamiltonians. The <13/2 (33/2
1/2 interaction strengths are not changed, while the

and 31/2-
5

-d3/2 and dS/Z—d3/2 interaction strengths are reduced

s1/2

51

Table 5.--Strengths of orbit-orbit interactions (MeV).
Z (2J+1)(2T+l)<jajb|V|jajb>JT

 

 

 

J,T

a u n b C n n
ja jb KU014 Particle PW K12.5P Hole
5/2 5/2 - 75.9 -77.4 -80.0 - 62.5 -42.5
5/2 1/2 - 44.6 -20.5 -21.0 - 37.6 - 7.9
5/2 3/2 ~133.1 -74.5 -83.6 -llS.0 -56.6
1/2 1/2 _ 15.4 -l6.9 -17.l - 14.2 -12.5
1/2 3/2 - 20.8 -16.0 - 4.0 - 18.1 -l0.0
3/2 3/2 - 20.8 -l9.5 - 6.1 - 18.3 -18.5

 

8Reference 1.
bReference 3.

cReference 2.

52

by approximately a factor of 2 in the "Hole" Hamiltonian.
The d5/2-d5/2 and dS/Z-Sl/Z interactions are the ones not
well determined in the "Hole" least-squares fit for the

and d

same sort of reasons that the sl/z-d -d

3/2 3/2 3/2
interactions are not well determined in the "Particle"
least-squares fit. It may be concluded then that the
result of both empirical renormalization of the Kuo's
realistic Hamiltonians is a lessening of the attractiveness
of the dS/Z-Sl/Z’ d5/2-d3/2, and Sl/Z-dB/Z interaction
strengths.

Li et al.32 have recently investigated the
17O(d,p)180 reaction at a deuteron bombarding energy of
18 MeV, and observed 12 states in 180 up to an excitation
energy of 6.34 MeV. From the observed excitation energies
and extracted absolute Spectroscopic factors, they deduced
the diagonal matrix elements of the effective neutron-

neutron interaction for (dB/2)2 + and (d

0+,2+,4 5/2'51/2)2+,
3+, T=1 configurations. Their matrix elements are listed
in Table 6, together with the corresponding matrix elements
of the K0014, PW and "Particle" Hamiltonians. The uncer-
tainties for the "Particle" Hamiltonian matrix elements are
obtained by assuming a 200 keV theoretical error for each
calculated energy.

Li et a1.32 pointed out that their deduced matrix
elements may be too attractive because of the omission of

transitions to higher excited states not seen in the

experiment, and made some theoretical estimates of the

TABLE 6.--Matrix elements <d5/2jIVId5/2j>J.(T=1) of the
effective neutron-neutron interaction (MeV).

53

 

 

 

 

2j Expt.a Est. Errorb KUOl4c "Particle"d PWe
5 -2.77 1.00 -2.44 -2.01:0.44 -2.12
5 -1.06 0.13 -l.04 -1.04ip.12 -l.23
5 -0.35 0.36 -0.05 +0.02:0.04 +0.16
1 -0.79 0.20 -l.29 -0.82:0.l4 -0.85
1 +0.60 +0.17 +0.78:0.09 +0.78
a

Reference 32.
bSee text for discussion on error estimates; reference 32.
cReference l.

dSee text for discussion on uncertainties.
e

Reference 3.

54

error in the deduced matrix elements; these are also listed
in Table 6. Li et al.32 also derived these diagonal matrix
elements from the theoretical energies and spectroscopic
factors for the two lowest 0+ and 2+ levels and the lowest
4+ level of reference 33 by the same technique they used
with their data. These values were then compared to the
actual values of the matrix elements used in the calcula-
tion of reference 33. The discrepancies are then estimates

+ and 4+ levels.

of error introduced by omitting higher 0+, 2
These error estimates were found to be model
dependent, though the model used is the same as the one for

the different Hamiltonians listed in Table 6, i.e., a 160

core with (sd)l and (sd)2 configurations for 170 and 180

respectively. No estimated error was given for the

<5/2 l/2IVI5/2 l/2>3,l matrix elements because the lowest

3+ state in 18O is essentially a pure (dS/Z—Sl/Z) configura-

tion. Different (d5/2-51/2)3+,(T=1) matrix elements

merely predict different energies for the lowest 3+ state.
Comparison of the (d5/2)2 diagonal two-body matrix

elements shows that the KUOl4 realistic matrix elements

all agree with the deduced experimental matrix elements

within the theoretically estimated errors, with the experi—

mental values being more attractive. The PW matrix

elements, however, are in poorer agreement with the experi-

mental values. The present “Particle” matrix elements

using the orthogonal least-squares search, on the other

hand, are again in good agreement with the experimental

55

values. The (d5/2)20+ matrix element is even less attrac-

)2
5/2 2
matrix element is not changed from the KUOl4 value. The

tive, but well within the estimated error. The (d +
(d5/2)24+ matrix element changes sign, but again the
estimated uncertainty of 0.04 MeV is still within the
estimated error. In short, the KUOl4 and "Particle"
(d5/2)2 diagonal two-body matrix elements and experimen-
tally deduced values are in rather good agreement, with

the observed deviations understandable as mostly due to the
omission of higher excited states.

Comparison of the (dS/Z-Sl/Z) diagonal two-body
matrix elements gives quite a different picture. The PW
and "Particle" matrix elements are in good agreement with
the experimental values. The KUOl4 matrix elements, how-
ever, are much more attractive for both the (d5/2-81/2)2+,
(T=1) and (dS/Z-Sl/2)3+,(T=l) matrix elements than the
experimental values. This is consistent with the above
conclusion that the result of empirical renormalization is
the lessening of the attractiveness of dS/Z-sl/Z'
d5/2-d3/2' and 81/2-d3/2 interaction strengths. The
evident question then is why the diagonal dS/Z-dB/Z'
dS/Z-sl/Z and 81/2—d3/2 interaction strengths are over-
attractive in the Kuo realistic interactions while the
diagonal dS/Z-dS/Z’ 81/2-31/2 and d3/2-d3/2 interaction

strengths are not.

56
I.4.C. Ground-State Binding Energies and Spins

The corrected measured ground-state binding
energies relative to 16O are listed in Table 7, together
with the calculated binding energies of the "Particle"
and "Hole" Hamiltonians. The single particle energies used
with the "Particle" Hamiltonian are -4.l4, -3.27, and
+0.88 MeV for the OdS/Z’ Isl/2, Od3/2 orbits, respectively.
For the "Hole" Hamiltonian, they are +21.75, +18.13,
+15.63 MeV for the OdS/Z' 151/2, Od3/2 orbits, respectively.

The deviations between calculated and measured
ground-state binding energies are also plotted in Figure 5
and Figure 6. The energy deviations for the "Particle"
Hamiltonian are plotted in Figure 5 as a function of
mass A. For each A, the energy deviations are plotted in
order of increasing isospin, starting from the lowest
isospin. The ground-state binding energy deviations for
the "Hole" Hamiltonian are plotted similarly in Figure 6.

The ground state binding energies are well repro-
duced in the A=17-24 region with the "Particle" Hamiltonian.
The energy deviations are all smaller than 0.5 MeV, except
for 21O and 22O, which were not included in the least-
squares fit. The observed binding energies of 21O and
220, however, have large uncertainties. Beyond A=24, the
energy deviation increases with A, with a clear isospin

dependence, i.e., less binding for higher isospin. The

effect of adding seven energy levels from A=35, 38 and 39

TABLE 7.--Ground state nuclear binding energies (MeV) relative to 16
calculated with the "Particle" and "Hole" empinical Hamiltonians.

5'7

 

 

 

 

 

THEORY (EXPT.-THEORY)

Nucleus J Expt.a Particle Hole Particle fible
160 0 0.00 --- 30.83 --- -3o.83
170 5/2 - 4.14 - 4.14 21.54 0.00 -25.68
180 0 - 12.19 - 12.21 9.18 0.02 -21.37
190 5/2 - 16.14 - 16.31 0.85 0.17 -16.99
200 0 - 23.75 - 23.98 10.50 0.23 -13.25
210 5/2 - 26.3 :gng - 27.69 17.56 1.4 - 8.7
220 0 - 32.2 :g:§b - 34.78 27.58 2.8 - 4.6
230 1/2 - 37.13 32.48

2“0 0 - 41.54 38.28

250 3/2 - 40.14 38.83

.250 0 - 40.49 41.30

270 3/2 - 37.95 40.84

280 0 - 37.43 42:36

18? 1 - 13.23 - 13.35 9.10 0.12 -22.40
19? 1/2 - 23.68 - 23.87 4.54 0.19 -19.14
2°r 2 - 30.22 - 30.47 15.61 0.25 -14.67
21F 5/2 - 38.39 - 38.51 27.02 0.12 -11.37
22? 4 «3.561003c - 43.61 35.04 0.05 - 8.52
23F 5/2 41.101017d - 51.30 45.17 0.20 - 5.93
2”? 3 9 - 54.74 50.69

25? 5/2 - 59.62 57.05

255 1 - 60.47 59.76

27F 5/2 - 61.50 62.72

28? 3,2 - 60.13 63.46

29? 5/2 - 60.02 65.42

20Ne 0 - 40.48 - 40.60 23.55_ 0.12 - 16.99
2146 3/2 - 47.24 - 47.29 33194 0.05 - 13.30
2246 0 - 57.61 - 57.64 47.46 0.03 - 10.15
23Ne 5/2 - 62.80 - 62.88 55.40 0.08 - 7.45

TABLE 7.--Continued.

 

 

 

 

 

THEORY (EXPT.-THEORY)
Nucleus J VExpt.a Particle Hole Particle Hole
2”Ne 0 - 71.67 - 72.04 - 66.72 0.37 - 5.00
25Ne 1/2 45.971010f - 75.95 - 72.68 0.03 - 3.21
26Ne o - 81.83 - 79.70
2746 3/2 - 82.88 - 82.32
2846 0 - 85.95 - 87.34
29Ne 3/2 - 84.79 - 88.32
3°Ne 0 - 85.78 - 91.53
22Na 3 - 58.23 - 58.23 - 47.48 0.00 -10.75
23Na 3/2 - 70.76 - 70.76 - 62.30 0.00 - 8.46
2”Na 4 - 77.65 - 77.75 0.10
25Na 5/2 - 86.66 - 87.04 0.38
26Na 3 - 92.28:p.02g — 92.50 0.22
27Na 3/2 - 99.0710.06h - 99.43 - 97.40 0.36 - 1.67
28Na 1 -102.66ip.08h -lO2.76 -102.10 0.10 - 0.56
29Na 5/2 406941010h -106.63 -107.50 0.31 0.56
3°Na 1 -109.3010.20h -107.03 -110.01 2.27 0.71
31Na 5/2 -llS.lflip.80h -108.46 -ll3.62 6.68 - 1.52
2uMg 0 - 87.11 - 87.49 - 80.23 0.38 - 7.01
254g 5/2 - 94.44 - 94.83 0.39
26Hg 0 ~105.51 -106.33 -102.27 0.82 - 3.24
27Mg 1/2 -111.97 -112.68 -109.99 0.71 - 1.98
28Mg 0 -120.48 -121.44 -119.40 0.96 - 1.08
29Mg 3/2,1/2i -1'24.12ip.40j -124.75 -123.78 0.63 - 0.34
3°Mg 0 -130.55 -131.08
alMg 3/2 -131.16 -133.65
32Mg 0 ~134.06 -138.71
26Al 5 -105.80 -106.45 0.65
27Al 5/2 -118.52
28Al 3 -126.25
29A1 5/2 -135.68

TABLE 7.--Continued.

59

 

 

 

 

 

THEORY (EXPT.-THEORY)

Nucleus J Expt.a Particle Hole Particle Hole
3°81 3 -141.4310.04 -141.01 0.42
31A1 5/2 -148.62:p.10k -149.46 -149.14 0.84 0.52
32Al 1 -152.63 -153.96

33A1 5/2 -156.56 -159.51

2851 0 -135.70 -138.03 -134.36 2.33 1.34
2981 1/2 -144.18 -146.37 -143.46 2.19 0.72
3°51 0 -154.79 -157.11 -154.59 2.32 0.20
3151 3/2 ~161.37 —163.25 -161.09 1.88 0.28
3251 0 -170.59 —171.97 -170.80 1.38 0.21
3351 3/2 -175.14¢0.05£ -175.24 -175.29 ’0.10 0.15
3”51 0 -181.81 4183.28

3°? 1 -155.46 ~157.52 ~155.09 2.06 0.37
31? 1/2 -167.77 -l70.36 -167.88 2.59 ' 0.11
32F 1 -175.64 ~177.79 ~175.74 2.15 0.10
33F 1/2 -185.78 -187.85 -186.34 2.07 0.56
3“? 1 -192.02:_0.05z -192.75 —192.09 0.73 0.07
35P— 1/2 -200.4810.08m -200.54 -200.96‘ 0.06 0.48
325 0 -182.64 -185.78 -182.85 3.14 0.21
338 3/2 -191.28 -194.43 -191.32 . 3.15 0.04
34s 0 -202.70 -205.54 -203.08 2.84 0.38
355 3/2 -209.69 -212.22 -209.58 2.53 0.11
358 0 -219.58 -220.99 -218.50 1.41 0.08
3“01 3 ' -202.55 -205.87 -202.63n 3.32 0.08
3501 3/2 4215.34 -219.07 -215.69 3.73 ‘ 0.35
3601 2 -223.82 -225.93 -223.84 2.11 0:02
3701 . 3/2 —234.22 -235.65 —234.08 1.43 0.14

6 0 .

TABLE 7.--Continued.

 

 

 

 

 

THEORY ' (EXPT.-THBORY)

Nucleus J Expt.a Particle Hole Particle Hole
36

Ar 0 ~230.43 -232.57 -230.75 2.14 0.32

7
3 Ar 3/2 -239.21 -240.72 -238.98 1.51 -0.23
38Ar 0 -251.06 -251.8c -250.67 0.74 -0.39
38

K 3 -251.19 -253.04 -2Sl.16 1.85 -0.03
39x 3/2 -264.22 -265.39 -264.22 ' 1.17 0.00
40

Ca 0 -279.85 -280.69 --- 0.84 ---

 

aUnless otherwise noted, the ground-state binding energies are taken from
reference 28, corrected for Coulomb energies and relative to 160. The
uncertainties of the uncorrected binding energies are less than 30 keV,
except where explicitly specified.

U‘

Reference 86.

0

Reference 5*.
6Reference 61.

. . 17+ . 11+.
eTne "Particle" Hamiltonian gives a J =3 ground-state with a J =2 first
excited state at 150 keV. The "Hole" Hamiltonian reverses the two states
with a splitting of 80 keV.

fReferences 66 and 67.
gReferences 68 and 69.
hReference 69.

1The "Particle" Hamiltonian gives a Jfl=3/2+ ground state with a J"=l/2+ first
excited state at 420 keV. The "Hole" Hamiltonian reverses the two states
with a splitting of 24 keV.

jReference 87.
kReference 70.
zReference 71.
mReference 72.

nFor the "Hole" Hamiltonian, the first Jfl=l+ state overbinds and is 120 keV
below the J'"=3+ ground-state.

61

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meN VN NNON m.
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822:8: .2986... 91..
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965m 265-285

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111111111111111
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m+.

 

 

 

62

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Hmoaosc oumumlpcsowm woumHsono cam powsmmme swmzuon mcoflumw>mo .o owsmflm

 

 

4.4
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___J_____.52..._._+.1_____
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l N IN:
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:23 82638 4658..
.. . 9.65m 265-285 1..
_____________cc________

 

 

 

63

into the fit with a weighted factor of (0.25)2 was to reduce
the energy deviations in the upper end, reversing the
increase in energy deviation with A. In general, the
"Particle" Hamiltonian tends to overbind.

For the "Hole" Hamiltonian, the A=32-39 region
ground state binding energies are well-reproduced, consis-
tent with the interaction being fitted to observed energies

in these nuclei. Except for 33

P, the energy deviations are
all smaller than 0.5 MeV. For A<32, the "Hole" Hamil-
tonian tends to under-bind. The energy deviation increases
with decreasing A below A=32, again with a clear isospin
dependence which now has the character of more binding
energy for higher isospin. The much reduced attractiveness
of the d5/2-d5/2 and dS/Z-Sl/Z two-body matrix elements can
reasonably be assumed to be responsible for the increasing
under-binding in the lower half of the sd-shell. These
matrix elements are more and more important as the holes
are filling up the OdS/Z-orbit.

The mass relation formulae of Garvey and Kelson,34
and later Garvey et al.35 have been quite successful in
predicting masses of nuclei near to stability. The
accuracy of using such recurrence relations connecting
mass excess values of neighboring nuclei depends upon the
accuracy of the input data. Thibault and Klapisch36 have

recomputed mass excesses for light neutron-rich nuclei

using the Garvey et a1.35 mass relations with more current

64

and complete data on T232 nuclei. These results are listed.
in Table 8 under the column TK. Jelley et al.37 have
further extended such fits to include the recently measured"
massexcesses of Tz=5/2 sd-shell nuclei in the input data.
Their results are also listed in Table 8 under column G.

I Under column M are also listed results Obtained
with a modified shell-model mass equation employed by
Jelley et al..37 The modified shell-model mass equation
differs from that of Garvey et al.35 mainly in the para?
meterization of the residual neutron-proton interaction,
where shell structure is more explicitly taken into account.
The calculated mass excesses of neutron-rich sd-shell nuclei
using the "Particle" and the "Hole" Hamiltonians are also
listed in Table 8 for comparison.

Comparison of columns TK and G shows that inclusion

of the mass excesses of Tz=5/2 sd-shell nuclei in the
input data for the Garvey et al.35 mass relations does
improve the agreement of the results with experiments,

except for 21O, 22O and 31Na. The results of the modified

shell-model mass equation of Jelley et al.37 are very
similar to that of Garvey et a1.35 as can be seen from
columns G and M. The calculated mass excesses of the
"Particle" and "Hole" Hamiltonians combined are in good
agreement with the measured mass excesses, i.e., depending
on mass A and isospin Tz, either or both Hamiltonians

give mass excesses which agree with experiments to within

1 MeV, except again for 21O, 220 and 31Na.

TABLE 8.--Mass-Excesses of Neutron-rich Nuclei (MeV).

 

 

 

 

 

a b Jelley et a1.C Theory
Nucleus Expt. TK
G M Particle Hole
200 3.80 3.74 3.57 17.05
210 9.3:8‘; 8.82 8.74 8.39 7.93 18.06
220 11.5:g::d 9.84 9.42 9.35 8.91 16.11
230 16.44 15.48 15.40 14.64 19.28
240 20.41 19.70 19.44 18.30 21.56
250 28.91 27.77 29.08
260 33.97 35.49 34.68
270 43.26 46.10 43.21
280 49.90 54.69 49.76
22F 2.83:0.038 2.83 2.78 11.35
23F 3.36:0.l7f 3.87 3.40 3.36 3.16 9.29
24F 8.71 8.04 7.79 11.84
25F 12.42 11.75 11.26 10.98 13.55
26F 18.84 18.21' 18.92
27F 23.06 25.25 24.03
2889 31.06 34.69 31.36
29F 36.87 42.87 37.47
24Ne - 5.95 - 5.90 - 6.32 - 1.00
25Ne - 2.18:0.10h - 1.33 - 1.95 - 2.12 - 2.16 1.11
26Ne 0.30 0.17 - 0.27 0.03 2.16
27Ne 5.89 6.52 6.58 7.05 7.61
28Ne 8.82 12.06 10.67
29Ne 15.99 21.29 17.76
30Ne 20.62 28.37 22.62

 

lullllll ll '1‘!

TABLE 8.--Continued.

66

 

 

 

 

 

a b Jelley et a1.c Theory
Nucleus Expt. TK
G M Particle Hole

26Na - 6.90:0.021 - 6.90 - 6.94 - 7.12

27Na - 5.62:0.06j - 6.11 - 5.71 - 5.73 - 5.98 - 3.95
28Na - 1.14:0.08j - 1.81 - 1.02 - 1.24 - 0.58
29Na 2.65:0.10j 0.29 2.32 2.66 2.96 2.09
30Na 8.37:0.20j 6.28 8.50 10.64 7.66
31Na 10.60:O.80j 10.13 12.70 14.38 17.28 12.12
28Mg -15.02 -15.05 -15.98 -13.94
29149k -10.59:0.40£ -11.58 -lO.7O -10.75 -11.22 -10.25
30Mg -10.66 - 9.37 - 9.21 - 8.94 - 9.47
31Mg - 5.45 - 3.73 - 3.17 - 1.48 - 3.97
32Mg - 2.94 3.69 - 0.96
30Al -15.89:0.04 -15.89 -15.47
31Al -15.01:0.10m —15.75 -15.00 -15.05 -15.85 -15.53
3211 -11.88 -11.14 -10.95 -12.28
3341 -10.17 - 9.34 - 8.65 - 6.80 - 9.75
3251 -24.09 -24.13 -25.47 -24.30
3351 -20.57:_0.05n -21.05 -20.71 -20.67 -2o.67 -20.72
348i -20.66 -20.57 -20.32 -19.17 -2O.64
34p -24.55:_0.05n -24.55 —25.28 -24.62
359 -24.04:0.08° -24.79 -24.90 -24.81 -25.00 -25.42
365 -30.67 -30.70 -32.08 -30.59

 

67

a .

Unless otherwise noted, the mass excesses are taken from reference 28.
The uncertainties of the measured mass excesses are less than 30 keV,
except where explicitly specified.

bReference 36.

cReference 37; G indicates using Garvey et al.35 mass relations, and M
indicates modified shell-model mass relations.

dReference 86.
eReference 54.
fReference 61.

n + + .
gThe ground-state has J =3 and 2 for the "Particle" and "Hole"
Hamiltonians, respectively.

hReferences 66 and 67.
1References 68 and 69.
3Reference 69.

n + +
kThe ground-state has J =3/2 and 1/2 for the "Particle" and "Hole"
Hamiltonians, respectively.

to

Reference 87.

8

Reference 70.

Reference 71.

O :3

Reference 72.

68

It was observed that aside from a mass dependence,
the energy deviations between calculated ground-state
binding energies and experiments have an isospin depen-
dence, which corresponds to less binding for the "Particle"
Hamiltonian, and more binding for the "Hole" Hamiltonian, for
increasing isospin. Comparison of calculated mass excesses
of neutron-rich nuclei of the “Particle" and "Hole"
Hamiltonians with those of Thibault and Kalpisch36 seems to
indicate that the isospin dependences of the "Particle"
and "Hole" Hamiltonians complement each other in giving a
good description of all sd-shell nuclei. The agreement
with the mass excesses of Thibault and Klapisch36 tends to
shift from the "Particle" to the "Hole" Hamiltonian as mass
A and isospin T increase. This encourages the hope that
the two Hamiltonians will yield overlapping descriptions in
the middle of the sd-shell.

The calculated ground state spins are adso listed
in Table 7. The agreement with all experimentally known
states is excellent. The "Particle" and "Hole" Hamiltonians

agree for all ground-state spins except for 28F, 29Mg and

34Cl. In 28F, the "Particle" Hamiltonian predicts a 3+
ground state with a first excited 2+ state; the "Hole"
Hamiltonian reverses the two states. In either case, the
splitting between the two states is less than 150 keV. It
may be noted that the "Particle" Hamiltonian predicts 28F
to be particle unstable by more than 1 MeV; while the

"Hole" Hamiltonian predicts a neutron separation energy of

69

740 keV. Thibault and Klapisch36 predict a neutron separa-

tion energy of 60 keV. In 29Mg, the "Particle" Hamiltonian

. + . .
predicts a 3/2 ground state and l/2+ first exc1ted state;
the "Hole" Hamiltonian again reverses the two states. The
splitting of the two states is 420 keV for the "Particle"

Hamiltonian and 24 keV for the "Hole" Hamiltonian. Both

are consistent with a recent observation of Jfl=3/2+, 1/2+

by Goosman et al.87 In 34C1, the first l+ (T=0) state

comes below the lowest 3+ (T=O) state for the "Hole"

Hamiltonian.

The predicted spin of 5/2+ for 210 ground state

agrees with that of the PW interaction.6 The recent
assignment of Jfl=4+ for the 22F ground state by Davids
et al.54 was included in the data set for the "Particle"

least-squares fit; however, the "Hole" Hamiltonian also

reproduces a Jn=4+ 22F ground state. The same statement

23

is also true for the J“=5/2+, F ground state recently

assigned by Goosman and Alburger,61 and the Jn=l/2+, 25Ne

ground state experimentally observed by Goosman et al.66

to have J"=1/2+(3/2+). Both the "Particle" and "Hole"

Hamiltonians predict a Jfl=3/2+ ground state for 27Na

consistently with Jfl=3/2+, 5/2+ proposed by Alburger

et al.88 The ground state spins of other Na isotopes are

calculated to have Jfl=l+, 5/2+, 1+, 5/2+ for 28Na, 29Na,

30 31

Na and Na respectively. The Jfl=l+ 28

Na ground state
spin was recently assigned by Roeckl et a1.85 and included

in the "Particle" fit, however, the "Hole" Hamiltonian also

70

reproduces a Jfl=l+ ground state for 28Na. The 29Na ground

state was fitted to the two lowest theoretical eigenstates
of J"=3/2+ and J"=5/2+ in the "Particle" fit, and to the
lowest Jfl=3/2+ state in the "Hole" fit. In both cases, a
J"=5/2+ ground state results from the final interaction.
The Jfl=l+ 34F ground state recently assigned by Goosman

et a1.71 is reproduced by the "Particle" Hamiltonian, but

was not included in the data set for the least-squares fit.

I.4.D. Ground-State Wave Function of 285i

As shown in the previous subsection, the ground-
state binding energies and spins are well reproduced by
the "Particle" and "Hole" Hamiltonians combined. The
deviations between calculated and observed binding energies
in the regions of nuclei from which the data sets were
taken are less than or equal to 0.5 MeV. The improvement
in the calculated binding energies further extends beyond
the regions of nuclei included in the least-squares fit.
It is hoped that the two Hamiltonians will complement each
other to give a good description of all sd-shell nuclei,
i.e., provide overlapping and similar descriptions of
nuclei in the middle of the sd-shell region. To gain some
idea of how the Hamiltonians compare in the middle of the
sd-shell, we compare the ground-state wave functions of
28Si as generated by the different Hamiltonians.

In Figure 7 are plotted the configuration proba-

bilities of n active particles in the dS/Z-orbit for the

71

flm

mm

.mGMRGCuHflEmm

«Hoax can emaome :.oaowuumme :ufiz cmumasoamo
mo oumgmlpsnowm may cw moauwaflnmnoum soflumwsmfimsoo

.h musmwm

 

 

 

 

 

 

     

./

,o_. m.

 

 

 

.7..-. ---... -- — -— ——
--------.-----——~- --- -
_ - _--__- -

 

 

 

‘-

 

o--._.- ---...
--—--------_-
-. ..e A -

L-

 

 

L
[

 

I

N w

 

 

 

 

I. . /
. e .
Z _ .4 _
r , ., z.
. _ 1.; .
f/. _. ./ , .
./” _.. /1 w .
l/ . .. ./ ..
../ .. I. s
f __ . Vx/L ..
. ... I .-
_.. 2. r.—
.. f/ #
.... // . . _.
1.. 77.4 ..
.... r1
.— v A
.._. y
... //
,... I
.
r

 

7'. , 77
54.’///r—;zi

 

 

 

 

 

 

[ZZZ/737,; ///7///////C‘2}.292/Ai’:}1jiffy/”33:75:; 17>
7/////////éY/ / if/

r/V—V— /. r _ I
1.2.7472. '-

mzszoSSq: Rzmmmta TE;

8:38.30 58 to NEE 028%

ME. 2_

mw_._._..=m<momd ZOF<¢DOEZOU

g g

NOIlONfld BAVM :10 "/6

C)
N)

 

 

72

ground state of 2881, as calculated from the "Particle,"

"Hole," and KUOl4 Hamiltonians. It can be seen that the
"Particle“ and "Hole" Hamiltonians give similar descrip-
tions of the ground-state of 28Si, one which is quite dif-
ferent from that given by the Kuo interaction.

The KUOl4 interaction gives almost no component
of (d5/2)12 configuration in the wave function. The
(dS/z)lo configuration has also a very small component.
The large components are those of the (d5/2)6, (dS/Z)7
and (ds/z)8 configurations. The (average) occupation
numbers are 6.84, 2.68, and 2.48 for the d5/2' Sl/Z' and
<13/2 orbits, respectively. The empirical Hamiltonians give
a description of the 2BSi ground-state more in keeping with

12

simple expectations. The (d and (dS/z)10 configura-

5/2’
tions are much more heavily occupied than in the Kuo wave
function. The (averaged) occupation numbers of 9.13, 1.45,
and 1.41 for the "Particle" and 8.79, 1.64, and 1.58 for the
"Hole" Hamiltonians, for the d5/2' sl/Z’ and c13/2 orbits,
respectively, are in better agreement with relevant experi-
mental data than are the Kuo values.

The difference between the Kuo and empirical
results may be explained by the less attractive dS/Z-sl/Z
and d5/2‘d3/2 interactions in the "Particle" and "Hole"
Hamiltonians. The same argument may also be used to
account for the small differences in the configuration
probabilities for (dS/Z)8 and (dB/2)12 configurations

between the "Particle" and "Hole" Hamiltonians. The

73

dS/Z-dS/Z and dS/Z-Sl/Z interactions are more attractive in

the ”Particle" than in the "Hole" Hamiltonian.
I.4.B. Energy Spectra

The energy spectra calculated with the "Particle"
Hamiltonian for A=17-24, 25(T=l/2), 26(T=0) and with the
"Hole" Hamiltonian for A=32-39 nuclei are shown in
Figures 8-25. The angular momentum labels in the figures
give J for even-A nuclei and 2J for odd-A nuclei. Brackets
indicate the experimental assignment is tentative. An
assignment such as (3) means that the spin is probably 3;
and a (3,5) means that the spin is probably 3 or 5. An

(+)

assignment such as 5

( )

means that the parity is probably
positive; and a 5 means that the parity is unknown. A
bracket around a line means that the observed level is
probably there. A dashed line indicates a negative parity
state, and a solid line with no label indicates a positive
parity state or a state with only the energy known. All
labeled lines with no parity assignment indicate positive-
parity states. A dot indicates that above that energy
there are levels not shown. All known or calculated
levels are included before the first dot. The levels are
plotted in terms of calculated or corrected measured
binding energy relative to 16O or 4oCa. The spectra are
not shifted except for A=26 (T=0) nucleus, where the whole

theoretical spectrum is shifted by 0.5 MeV for easier

comparison. ‘The spectra plotted do not necessarily reflect

74

the energy levels included in the data sets for the least-
squares fits as listed in Tables 1 and 3. The lines
connecting calculated and experimentally observed energy
levels are merely to guide the eyes for easier comparison.
The experimetnal spectra are taken from references 27, 29,
and Tables 1 and 3.

In the following paragraphs we discuss some of the
relationships between the model spectra and the experi-
mental data.

(1) A=l7, T=1/2 (Figure 8):

The d strength is observed to be fragmented in

3/2
17

O. The single-particle energy for d was treated as a

3/2
free parameter in the last iteration of the least-squares
fit and was found to move down to +0.88 MeV, close to the
energy of the first 3/2+ state in 17O.

(2) A=18, T=0 (Figure 8):

The 1*, 2+, and 3+ states at 1.70, 2.52, and 3.36
MeV, respectively, were assumed to be intruder states;
otherwise there is good agreement between the calculated
and observed Spectra in the low-energy region.

(3) A=18, T=1 (Figure 8):

The question of whether the second (3.63 MeV) or
third (5.33 MeV) observed 0+ state in 18O is dominated by
core-excited, particle-hole configurations is an unresolved
issue in shell-model calculations. The theoretical second

+

0 state was fitted to the spectrosc0pic-factor-weighted

centroid of the energies of the two observed excited 0+

75

 

'2‘23': ....... \
#- 3___\__3
or I:
2.-
. =;=I
l-———-—-l 2
4’ 5—»-——5 —4
.. EXP TH
a 4—
2 6 T 3V2
—4 (Pat—3
o -—-| 2 ------
e A = I? ......
8)- 2__ _%
E >- g_. :? =N—4 _
5' '0’ — _2 2———2 _
'3 .- 2i (3g:/:39
d 5——_ 5 -——5
a; '2” 3'::_"§_5 0—4.._.._0 ..—...
L | 3 EXP TH 5= _3
-—-\__. (72:22::: 0
g l4- EXP TH T: (9)—‘——;
E " T80 '—\.__._'
l|—° —-ll
9 '4 A = I8 ,5 3===g
— t 9 “' EXP TH
E ”7==T:::
(n ,8- 3 s 1 7 T=3/2
£15: ' -
0: "' '33:: __.5
< 57:21:07
2420— 3-
A = l9
.2) " 9_\__9
22~ 3E:T~—-3
" 5__
1:2:‘1—5
4..
2 EXP TH'
” T=l/2
26...

 

Figure 8. Energy Spectra of A=17 (T=1/2), A=18 (T=0,l),
and A=l9 (T=1/2, 3/2).

76

states. The third observed 2+ state at 5.25 MeV is gener-
ally accepted as an intruder state. The theoretical second
4+ state is always predicted higher than the one experi-
mentally observed at 7.11 MeV.

(4) A:l9, T=1/2 (Figure 8):

The observed 3/2+ state at 3.91 MeV in 19F is
missing in the calculated spectrum. This is very probably
an intruder state. A recent weak-coupling particle-hole
model calculation by Ellis and Engeland33 which included
(sd)3 configurations also is not able to reproduce this
state.

(5) A=l9, T=3/2 (Figure 8):

The agreement is good between the calculated and
observed spectra, although additional experimental data are
required to establish more firmly the correlations between
theoretical and observed levels. The observed possible

l/2+ state at 3.24 MeV in 19

0 (not labeled in Figure 8)
is missing in the calculated spectrum.
(6) A=20, T=0 (Figure 9):

The ground state rotational band of 0+, 2+, 4+, 6+,
8+ states is well-reproduced. Between 6 and 9 MeV, there
are three observed 0+ and three observed 2+ states, while
the calculated Spectrum has only one 0+ and one 2+ state.
(7) A=20, T=1 (Figure 9):

Below.2.5 MeV, agreement between calculated and

observed spectra is good; however above 2.5 MeV more

experimental data are needed for a definite comparison

77

 

4. 024 2 O
I. _ \
.3 _ _ _ e. T
E
024. 9. O
7.023 5 4
7 6141323 '32
.h
_ 12:12:... 6
:2 \ l \ .. 2
:: 2:1: m T ..
. .4.03?5I43oe AH
6 5403
.153824 2... 6 2 0 2 O
:__:__ ___ _ _
/___\ __.____1 __.__> m _.
66 443460220 . 2 O
8 ?2 O
—| — h p — h p _ h b — _J~ L ~ — h _ $8 — m
m m. a m m w w 4.x M x 3 4
$2.... 06.. 9 ”.553“... Emuu 62525 .5382

Energy Spectra of A=20 (T=0,l,2).

Figure 9.

78

(8) A=20, T=2 (Figure 9):

The five lowest observed states with known spin-
parity assignment are well reproduced.
(9) A= 21, T=1/2 (Figure 10):

The low-lying states are very well reproduced. The
first 15/2+ state is predicted at 9.76 MeV.
(10) A=21, T=3.2 (Figure 10):

Only the Spins of a few states have been uniquely
determined experimentally. However, the level density is
well reproduced, except for two of the four states observed
at about 2 MeV. It is possible that these are negative-
parity states.

(11) A=21, T=5/2 (Figure 10):

The ground state of 21O is predicted to have
Jfl=5/2+, although the binding energy is 1.4 MeV larger than
the experimental value.

(12) A=22, T=0 (Figure 11):

The 1+ and 3+ members of the K=0+, T=0 excited band
in 22Na are overbound in the calculated spectrum, although
the higher spin members of the same band are not. The
extra 2+ state predicted by the PW interaction near 3 MeV
is moved up to around 4 MeV. The observed 4.32 MeV l+
state is missing in the calculated spectrum.

(13) A=22, T=1 (Figure 11):
The first excited 0+ state observed at 6.24 MeV is

predicted to come 1 MeV too low in excitation. Otherwise,

27’

4|?

nuium mumn aemw:eun~510'%>uuw

44

,5,

 

47-

Figure 10.

33‘

37 .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T' HE

 

79

 

 

 

 

 

 

 

 

 

 

 

 

S
T-me
l3
5 -——n
5552 ____.
' —— """" 5 H
-—.___ l3
__ —_5.T
5T:T___ 53 ’
==\ 57
22:2: ____9
3
_ I
—‘ 5
EXP TH
T . 3/2
A==2l
Energy Spectra of A=21 (T=1/2, 3/2, 5/2).

80

III III
IIIIIIIa a

II I H | NIH
Ill.

22

I
1

 

 

1'45 \—_5

 

 

 

: I
I I
. : i
2 § 2 EEE § 5 1?
(.0 G

 

 

of A=22 (T=0,l,2).

11. Energy spectra

Figure

81

the agreement is good for individual excitation energies
and the density of states is also well-reproduced.
(14) A=22, T=2 (Figure 11):

Contrary to the compilation of Endt and Van der
Leun,27 the ground state is predicted to have Jn=4+. This
is in agreement with the experimental results of Davids
et al.54 I
(15) A=23, T=1/2 (Figure 12):

Compared to the PW interaction, the present calcu-
lated spectrum agrees much better with experiment. The
low-lying non-ground-state band member observed levels are
overbound, but the discrepancies are all less than 250 keV.
The 9.04, 9.81, 14.24, and 14.70 MeV excited states were
observed recently in 12C(lzc,n)23Mg by Speer et a1.59
These were proposed to be the 15/2+, 17/2+, 19/2+ and 21/2+
members of the ground state rotational band. The 9.04 and
9.81 MeV excited states were also observed in 12C(12C,p)23Na

58 The 9.04 (IS/2+) and 14.24 (l9/2+) MeV

by Bibber et a1.
excited states were fitted to the theoretical counterparts
in the "Particle" fit. As shown in Figure 12, the calcu-
lated first 15/2+ state agrees well with the prOposed spin
assignment for the observed 9.04 MeV state. However, the
observed 9.81, 14.24 and 14.70 MeV states are closer in
energies to calculated excited states of 15/2+, 1'7/2+ and
19/2+ spin respectively. The 15/2+ assignment for the 9.81

1

MeV state agrees with a recent 2C(12C,p)23Na measurement

57

by Kekelis et al., which eliminated a 17/2+ assignment

82

 

 

 

 

 

 

 

 

 

 

.9; I 23
w.—
5._
52I—
53—
54_ ---'-—2I
l9
2 55* ._._- .7
g 56— Ian 3:9
2 am
58— --——I7
5., '5I5
F _.
5.59 —. ms
nu ——l3
m 60— "—"n
0)
g 6l— (:7) SIS
“’ 0&-————~ “‘L—‘JH5
a sa— -————".3
0
Z- 63— II
£2 '3"
a 64*- g
5 ' I3 ‘ {“39
65— (3.5) H __.___. 3%
g (5) 57”
z 66—- Y r
67— STETEEE\ 5
> 7'
69— 7
7o»-
5 5
7|-— 3 1
EXP TH
72— . T=|/2
A=23

Figure 12. Energy spectra of A=23 (T=1/2).

83

for this state through angular correlations and lifetime
data. The calculated spectrum shows a number of high-spin
states rather close together. Following the J(J+1) rule,
it is not clear which of the 13/2+, 15/2+, or 17/2+ states
are members of the ground-state rotational band. Further
investigation of the structure of the ground-state band is
needed.

(16) A=23, T=3/2 (Figure 13):

The reversal of the order of the observed '7/2+ and
3/2+ states at 1.70 and 1.82 MeV by the PW interaction is
corrected in the present calculated spectrum.

(17) A=23, T=5/2 (Figure 13):

The ground state of 23F was recently assigned a
Jfl=S/2+ by Goosman and Alburger.61 The state was included
in the "Particle" fit and the binding energy is well
reproduced.

(18) A=24, T=0 (Figure 14):

The energies in general are overbound compared with
experiment. The K=2 excited band which was predicted 1.5
MeV below its observed position by the KU014 realistic
interaction and 0.5 MeV by the PW interaction is well
reproduced in the present case. The first excited 0+ state
is still predicted to be underbound. In the 10-12 MeV
region, there are three observed 0+ states, compared to

only two theoretical counterparts.

84

x#

.A~\m

MN

4

anuh

.~\mu8v mmnd mo muuommm hmumcm .MH wusmwm
NB»...
mi 1... dxm m mm.
mm
b m
hiII
mI n 6 m
m. I a. w
m I 6v cm W
MI ........ fine. m
m . 3.3 3
x. . 2.03 W.
m H
mm mm m
_ u 9
n. . km 8
Z . 3
w.
on 1
m. . M
3
on m
I a...
BI... W.
n. . mm M .
No
$8 2

 

 

 

 

 

 

85

.76 .I2
8 8 5860 452062465 0 4 3 42

 

 

 

 

 

«TH

T=O

EXP

 

8 8 7&0 0 Ol3246l2 O4 3 24 O
l\ 64. 2 5
- _ p L! b . . . .— _ _ _ _ . p p
3 5 6 7 8 I 2 3 4 5 7
n 7 M 7 7 7 7 .818 m 8 8 8 8 8 8 %

$2): Om. O... w>:.<..mm >omwzw 02.025 mam-.032

A=24

Energy spectra of A=24 (T=0).

Figure 14.

86

(19) A=24, T=1 (Figure 15):

The experimental spectrum is well reproduced below
3 MeV. More experimental data are needed for comparison
above 3 MeV.

(20) A=24, T=2 (Figure 15):

The calculated spectrum strongly suggest the 4.89
MeV state to have Jfl=3+.

(21) A=25, T=1/2 (Figure 16):

Except for the counterpart of the 1/2+ excited
state observed at 2.56 MeV in 25Mg, all the calculated
low-lying states are overbound compared to experiment. The
calculated Spectrum agrees very well with experiment in
terms of sequence and Spacings, however.

(22) A=26, T=0 (Figure 17):

The calculated binding energies are too large com-
pared to experiment. In Figure 17, the calculated spectrum
has been shifted up by 0.5 MeV for an easier comparison
with the experimental spectrum. The agreement is remarkably
good, suggesting that the observed 2.66 and 3.07 MeV states
have Jfl=2+ and 3+ respectively.

(23) A=32, T=0 (Figure 18):

The calculated binding energies are too large
compared to experiment. However, the shifting of the
excited states down relative to the ground-state by the
K0014 interaction is corrected. The first excited 0+ state
is also correctly calculated; it is underbound by more than

2 MeV by the KU014 interaction relative to the ground-state.

87

 

 

02 3 3O 42 2 O
\ : \ \ ,. a
- n m
n? 0. 42 2 O
7 T 67 I4 2%! 4 23 43 235l32 2| 4
_ _ . _ _ _ _ _ _ _ _ _ _ _
e a a a m n n n u n n n n

.>02. 00. Oh 9:515: muamuzw 0252.0 «(H.032

Energy Spectra of A=24 (T=1,2).

Figure 15.

88

 

 

9l5 59 3573” 7 I
5 B733H399H57BIH593 59 9 hf. :37 3 I 5
.0. H
T
W 5
x 2
3 .-
T A
m _ n _
_ _ _ . P
. _ u _ . X
_ _ _ _ E
m. .m) _ 9 .9. .
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5 no 7. 8 9 no I. 9. no 4 5
no no no 8 8 .9 a. q. o. q. 9

325 02 o._. m>_._.<...wm mmamwzm 02.025 mdmgoaz

Energy spectra of A=25 (T=1/2).

Figure 16.

89

2855 920.20 mzmmemm 3.52m 3 50 2.45

 

 

 

 

 

 

 

 

m m m m m m m m
_ _ 1 _ _ _ _ _
4 I
4 4 6
nmuaHZy 35 43 4.764352? |23 I30? I 3 5
.. _H
* T
i O %
= .-
\ / T A
0.. .P
VA
.t
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I43 a P...
a.
_ _ ._. _ _ _ _ _ _
m m m m m m m m m

305: Ow. 0k m>_._.<4mm mmamwzm 02.025 mam-.032

Figure 17.

Energy spectra of A=26 (T=0).

90

 

 

6
l05- : 54
I -'==“=="24
(2-4) E 3'
g '04— (2—5) _________ ————- 02
8 .........
9, IO3-— --------- .
2____
O
h- 3
I; \ I
I 4 / 4
2
{.3 IOI \ 2
— o
\ .
w I
(D
Z
5 I00—
z
m
3:" 2 x 2
g 99—
Z
98—
o———\
97" EXP To
T=O

 

A=32

Figure 18. Energy spectra of A=32 (T=0).

91.

(24) A=32, T=1 (Figure 19):

The J"=o+ and 2+ states observed at 0.51 and 1.32
MeV, respectively, are overbound in the calculation. Other-
wise the agreement is good for the low-lying states.

(25) A=32, T=2 (Figure 19):

More experimental data are needed for comparison
above 4 MeV. Two excited 0+ states are predicted near
5 MeV, while only one is observed with a possible assign-
ment of Jfl=0+ at 4.98 MeV.

(26) A=33, T=1/2 (Figure 20):

Some of the excited states are calculated to be
overbound compared with experiment; the order of the second
5/2+ and second 3/2+ states is reversed. The agreement is
good in general.

(27) A=33, T=3/2 (Figure 21):

The low-lying states are well reproduced, though
more experimental data are needed for comparison above
4 MeV. The observed 3.28 MeV state is suggested to have
J"=3/2+ by our results.

(28) A=33, T=5/2 (Figure 21):

The binding energy of 3381 is well reproduced. The
observed energy and J1T of 3/2+ was included in the "Hole"
least-squares fit.

(29) A=34, T=0 (Figure 22):
The calculated binding energy of the 3+ ground state

agrees well with experiment. However, the first excited

us
"4
"3

3 II2

I}

I9 III

S

.-

S

u no

a:

(D

9.’

(9

I5 .09

z .

DJ

0

Z

5 I08

3

m

5

w IO?

.J

U

:3

2
I06
I05
IO4

_

 

92

 

 

 

 

 

 

E 3
I =
'3’ _____|42
(0P————— 0
O
2 \
2
6 2 / 2
0
5 5 tAr \ O
. “is
0-3) ‘ 3 T=2
"""" I
4
(023:::::_ 330 A=32
3?) 4
I 3'
2(l) 2
l
2M3 3|
2
3 \ 3
2..____ .
'_‘_‘ I
2
l—_\-——I
EXP TH
T=|

Figure 19. Energy spectra of A=32 (T=1,2).

93

96—

TH

 

 

 

__/

\

\
T=I/2
A=33

EXP

 

 

_ _ _ f _

95—
4
3
2
I
O

39L

88-—-

30.5 coat 0... w>:.<4mm mmaxmzm 6252.0 mdwqosz

Energy spectra of A=33 (T=1/2).

Figure 20.

94

.Am\m .m\mnav mmud m0 whuowmm amumcm ,.Hm musmwm

«x? p
E L mm
_ |||| axm
////IIIIII_ .|I¢m
nIIIII... I mm
.1 mm

Illhm

 

 

 

U
““300”
IDF'O"
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IIImm

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mml ll 00.

mm

II
4

 

 

 

ll .0.

II no.

Maw} 030‘, 0.1. HAHN-138 SEISHBNE SNIONIB HVB'IOflN

NB . k
E axu II «9

 

IJ no.

 

95

 

 

.A~.H.ouav «mud mo muuowmm hmumcm

MN -

[0'0
N? N - O ¢NON

Wm

 

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GN N-N-IO

mhI

OmI

NmI

wmI

 

.NN wusmwm

OJ. BAILV'IBH ASHENB SNIGNIB BVBWOON

(MW) .000,

96

1+ state is overbound and comes out below the ground state.
Some other excited states are also overbound.
(30) A=34, T=1 (Figure 22):

The 0+, T=l ground state of 34S is correctly calcu-
lated to come below the T=0 states in 34C1. The calculated
spectrum agrees well with experiment except that between
S and 6 MeV there are two observed 0+ state while there is
only one calculated 0+ state in this region. The next
calculated 0+ state is at 7.46 MeV.

(31) A=34, T=2 (Figure 22):

The calculated binding energy of the l+ ground
state agrees very well with experiment. A very close first
excited state is predicted to have Jfl=2+.

(32) A=35, T=1/2 (Figure 23):

The calculated spectrum is somewhat expanded com-
pared with experiment; the agreement is otherwise good.
(33) A=35, T=3/2 (Figure 23):

Only the two lowest observed states have uniquely
determined spin and positive parity, so that more experi-
mental data are needed for comparison with the calculated
spectrum.

(34) A=35, T=5/2 (Figure 23):

The calculated binding energy for the J"=l/2+

ground state is too large by 0.40 MeV. A Jn=3/2+ state is

predicted for the first excited state.

(MeV)

IRKLEAR EWIMWB ENEKB' RELNHVE ‘ND ‘PCO

-n~

~66 '-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__——\ s 9
_____ ”‘3'
__ ——g§
::3::: 75
.EEEEE__——————5|
_ fifjf 3
- ————I
&._-‘-‘-'—‘_:§< 7
39
_____\\\ s
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EXP TH
T=V2

Figure 23.

97

 

 

 

 

 

 

 

 

 

 

 

 

 

I
o —_5 3
: 3 '
— 3
3
(3) /
l—‘——_|
3 -—--" 3
EXP TH
T = 3/2

Energy spectra of A=35 (T=1/2, 3/2, 5/2).

98

(35) A=36, T=0 (Figure 24):

More experimental data above 4.5 MeV are needed for
better comparison of calculated and observed spectra. An
observed Jfl=(1,2)+ state (not labeled in Figure 24) at 4.95
MeV in 36Ar is missing in the calculated spectrum.

(36) A=36, T=1 (Figure 24):

The agreement between calculated and observed

spectra is good. The observed 1.60 and 2.86 MeV states in

36C1 have been reassigned to have Jn=l+ and 3+, respec-

tively, following Rice et al.79
(37) A=36, T=2 (Figure 24):
The observed first excited O+ state is predicted
to be 1.14 MeV underbound, and the observed second 2+ state
is also predicted to be 1.63 MeV underbound. These two
experimental states may be dominated by intruder state
configurations.
(38) A=37, T=1/2 (Figure 25):
The five lowest positive parity states are well-
reproduced. The results suggests the observed 3.60 MeV

37Ar to have Jfl=3/2+, consistent with the recent

75

state in
observation by Gadeken et al. of J"=3/2+ or 5/2-.
(39) A=37, T=3/2 (Figure 25):

Above 4 MeV, there are many more positive parity

states observed than are calculated. These are presumed to

be mostly dominated by intruder state configurations.

 

 

 

 

 

 

 

 

 

 

"VF
b
-6&
3
v - D
3 a»
- I?
2 6
m D
Z
I...
<
9’”
E-v :
I“ 3 ' g
'2 l ————I
g 4 / ‘9
E'“' :
I- 2 4
g 40 -------- ‘:€i g
2 ___
-5'. 2
0
-«» \\‘ 0
EXP TH

Figure_24.

(3
('gfizL—v-z: __

99

_
——

 

 

-----:-\_

2m\

(l)——-————

 

'—x

3 /

U-~N —O b

H

 

9

 

 

EXP TH

N

I>
ll

36

U

-N

NO

 

EXP TH

Energy spectra of A=36 (T=0,l,2).

100

 

_“ 5
—7—3
-50... 5 _3
L- 53;:7/_5
b . h__,»——l
- —— EXP H
-4 .. .
5. «mm- __;;s Isa/2
o 7___
E . .-_-_-_-_;___.7
O p ._2—3
-40 EXP TH
E? r W.
*9 I 3 4 A=37
E ’ __3
'35- _....o
5 . —_'2g)——'T_‘—|
g 4 Zreh\_2
* (a -—-I 3”*’
E ‘ "< ..... --2 ._ ——2
.- """ 2 a,
l——~”—”
E-w-
w I- l—\_' /—o
o . sap—m3 EXPT_ITH
5 ~ "° '
a r
‘25” (5) A g 38
5 4 or——
(5) \
g * wr——§
g - (5)____ 5
r. (5)—7
(5) ,
-mr -:.
P LL.
‘. I-— '''' —---l
)-
..|5.. 3EXP TH 3
T=|l2
A==39

Figure 25. Energy spectra of A=37 (T=1/2, 3/2), A=38
(T=0,l), and A=39 (T=1/2).

101

(40) A=38, T=O (Figure 25):
Again intruder states are important in the spectrum.

+ and 1+ are well

The lowest three states of Jfl=3+, 1
reproduced. The first 2+ is overbound by approximately
0.5 MeV, while the third 1+ is underbound by approximately
0.5 MeV.

(41) A=38, T=1 (Figure 25):

The first excited 0+ state is predicted at 5.83
MeV, while there are two excited O+ states observed at
3.38 and 4.71 MeV. There are also more excited 2+ states
observed between 3 and 6 MeV than are calculated. In
analogy to the case of 180, these states are very probably
mixtures of (sd)-2 configurations and intruder states.

(42) A=39, T=1/2 (Figure 25):

The dS/Z-hole strength is observed to be frag-
mented. The dS/z-hole single particle energy was treated
as a free parameter in the last iteration of the least-
squares fit. The energy was found to be 21.75 MeV.

In summary, the observed spectra are well-reproduced
by the "Particle" and "Hole" Hamiltonians in their respec-
tive domains, except for some levels which are missing in
a few nuclei. These are mostly in ffew particle" or "few
hole" systems, and can reasonably be assumed to be intruder
states. In the extensive studies made by the Glasgow
Group,5-8 the main defect of the KU014 realistic inter-

action was the shifting of whole bands of levels relative

to each other. The KU014 interaction predicts

102

qualitatively correct interband spacings, but the relative
positions of the band heads can be wrong by several MeV.
The PW interaction gives a good description of A122 nuclei,
as is to be hOped since it was derived by a least-squares
fit to these nuclei. However, the improvement over the
K0014 interaction diminishes as A increases, as the ten-
dency to shift bands again shows up beyond A=22. The
"Particle" Hamiltonian gives very similar spectra for A122
nuclei as the PW interaction. The A=23, and 24 spectra are
better described by the "Particle" Hamiltonian than the PW
interaction, as these were included in the present least-
squares fit and not in the PW derivation. The band
shifting is corrected even in A=25 and 26. From the calcu-
lated spectra, it appears that the "Particle" Hamiltonian
gives a better description of lower sd-shell nuclei than
the PW interaction, and definitely better than the KU014
interaction. '

It is hard to compare the "Hole" Hamiltonian with
the K12.5P interaction as only the A=35-38 nuclei were
studied previously.2 The KU014 interaction was found to
give a very poor description of the upper sd-shell nuclei,
in many cases even predicting the wrong ground state spin.
The PW interaction is not expected to show much improvement
over the KU014 interaction in the upper sd-shell nuclei,
as the matrix elements involving the d3/2-orbit which were
least determined in the fit become more dominant. Compari-

son of A=32 and 33 spectra did bear out this fact, though

103

the low-lying levels are still predicted in roughly the
correct order.5 The same spectra are very well described

by the "Hole" Hamiltonian.
1.5. Summary and Conclusion

Following the success of the PW interaction,3 an
attempt was made to derive a single empirical Hamiltonian
for the entire sd-shell region nuclei. The matrix elements
were treated as parameters and least-squares fitted to
measured binding energies of nuclei in the A:18-24 and
A=32-38 regions. It was found that a single set of mass-
independent (1+2)-body Hamiltonian was inadequate for the
entire sd-shell, presumably because of differing renormali-
zations at either end of the sd-shell. The data set was
divided into two parts and a Hamiltonian was obtained in
similar fashion from each part separately.

Starting from the KU014 realistic interaction, the
"Particle" Hamiltonian was obtained by an iterative least-
squares fit to measured binding energies in the A=18-24
region. The "Hole" Hamiltonian was fitted to measured
binding energies in the A=32-48 region instead, starting
from the K12.5P interaction. The least-squares fit was
reformulated in terms of orthogonal linear combinations
of the matrix elements as parameters. With from 134 to 197
entries in the data sets, it was found that only a few

orthogonal parameters were very well determined and that

104

less than half of the orthogonal parameters were at all
well determined by the data.

The dominant result of the empirical renormaliza-
tion obtained for the Kuo matrix elements is the reduction

in attractiveness of the dS/Z—sl/Z' dS/Z-dB/Z' and sl/Z-d

diagonal two-body matrix elements. Efforts should now be

3/2

made to understand why the realistic interactions are too
attractive for these "unlike orbit" cases.

The ground state binding energies and spins have
been calculated using both the "Particle" and "Hole"
Hamiltonians. In both cases, the agreement in the region
of nuclei from which the data set were taken is very good,
with the deviations between calculated and observed
binding energies being less than or equal to 0.5 MeV. In
both Hamiltonians, the energy deviations increase with
increasing number of active particles or active holes,
with a further isospin dependence superimposed. Low-lying
spectra for a number of nuclei have also been calculated
with either the "Particle" or the "Hole" Hamiltonian. The
agreement with experiment is good, except for some missing
levels in a few active particles or active holes systems
which are presumably intruder states. The "Particle"
Hamiltonian gives descriptions of A522 nuclei rather
similar to the PW interaction. Beyond A=22, the "Particle"
Hamiltonian is a better interaction than the PW interaction
in that band shifting, the main defect of previous inter-

actions where whole excited bands are predicted overbound

105

with re3pect to the ground State, is further corrected. The
"Particle" and "Hole" Hamiltonians appear to give a good
description of nuclei beyond the region where the Hamil-
tonians were fitted, overlapping in the middle of the sd-
shell. Whether the two sets of Hamiltonians will complement
each other to give a good description of all nuclei in the
sd-shell region remains to be seen. Further investigations
of other nuclear prOperties, such as spectrosc0pic factors,
electromagnetic transitions and moments, B-decays and so on,
specifically for nuclei in the middle of the sd-shell is

needed.

 

 

 

 

 

II. MAGNETIC DIPOLE MOMENTS OF

sd-SHELL NUCLEI
II. 1. Introduction

The deviations of observed magnetic moments of
odd-A nuclei from the single-particle Schmidt values have
been the subject of many theoretical studies. Modification
of the independent-single-particle shell-model wave func-

tions or configuration mixing89-91

have been thought to
explain the majority portion of the deviations, while
corrections of the magnetic dipole Operator for such

92-94

effects as mesonic currents have generally been con-

sidered small, except possibly in high spin states.94
However, previous studies have mainly been on selected
odd-A nuclei with simple configurations, such as one
particle or one hole outside a j-j or 1-s closed shell.
A more quantitative and comprehensive survey, e.g., of all
measured dipole moments of sd-shell nuclei, is definitely
desirable. The calculations described in section I make
such a survey feasible.

The magnetic dipole moment calculations discussed

here are carried out Wlth the untruncated dS/Z-Sl/Z-dB/Z

shell-model wave functions obtained in the work just

106

107

described. The magnetic dipole operator, a one-body
operator, does not change the principal nor the orbital
quantum numbers of the particle it acts on. A "good" wave
function defined in one major shell, e.g., the full Os-ld
model space, should at least be able to account fully to
first order for configuration mixing corrections to the
observed deviations of measured dipole moments from the
Schmidt values. The dipole moments thus provide a sensi-
tive test of the wave functions or the interaction used to
generate the wave functions. Conversely, a "good" set of
wave functions can be used to find effective dipole
operators which may be necessary for other types of cor-
rections which cannot be included in the wave functions.
Magnetic moments of ground and excited states of

sd-shell nuclei (except for 27'28 30

Al, Si) were first
calculated using the wave functions with the "bare"-(or
"free"-)nucleon gyromagnetic ratios for the dipole
operator. Agreement with experiment was found to be good
for A=l7-26, but poor for A=28-39. With the same set of
wave functions, the dipole Operator matrix elements were
then treated as parameters and determined from a least-
squares fit to available precise values of measured dipole
moments. When the magnetic moments were recalculated with
this revised operator, the good agreement with experiment

for A=l7-26 was not changed, while that for A=28-39 was

much improved. Subsequently, effective g-factors and

108

intrinsic moments were derived from the Operator matrix

elements determined from the fit.
II.2. Details of Calculation

In isospin formalism, the magnetic moment of a

state WJT is given by:

 

 

_ 1 <J J 1 o|J J > <T T2 I OIT Tz > JT + JT
U-X L <‘1’ IlluIIII‘P >
I=0 /2J+1 V2T+1
(40)
where <WJT|||EI|||WJT> is a double-reduced matrix element

with respect to space and isospace. The subscript I
equals 0 or 1 for the isoscalar and isovector components
respectively of the dipole operator. The Operators are

defined as follow:

A g1 + 92
EO=Z {<—P——£) 33+ (u +u)'§} (41)
._ p n
1—1 2
A 92 _ 92
131= {(-E——’-‘>7i+(u-u)§}w(i) (42)
-_ p n 2
1-1 2
where T2 = +1 for proton and -l for neutron. The 9:, up

and gn, “n are the orbital g-factors and intrinsic moments
of the proton and neutron respectively. They are, for a

bare- or free-nucleon,

2.79 n.m.

1.0 n.m. , u

5 N p N
'U

0.0 n.m. -l.9l n.m.

12'
ll

109

JT JT
The double-reduced matrix element <W IIIKIIIIW > can be
further expressed in terms of single-particle reduced

matrix elements and transition density matrix elements

JT + JT , . JT
<W IIIUIIIIW >=fiz <1IIIEIIIIJ> pijI , (43)
1]

where i,j denotes single-particle states, and the transi-

JT

tion density matrix elements pi. are defined as:

 

31
J <wJT|||(aT x a.) )IlleT>
p.T. - 1 3 11 (44)
lJI /3T§E¢IT’

. . . . . +
The (a: x aj) 13 just the Single—particle creation (ai)

11
and annihilation (aj) Operators coupled to rank 1 and iso-
spin I. Combining equations (40) and (43), the dipole

moment u can be expressed as a linear expression of the

single-particle reduced matrix elements

u :2 <i|llfillllj> egg-‘1 <45)
:1:le
with coefficients:
CJT = <J J 1 0|J J><T Tz I o | T Tz> pJT (46)

 

 

ijI /(2J+IT(2T+17 131

The transition density matrix elements piT and hence the

11'
igl' contain all the necessary information

from the mixed-configuration wave functions.

coefficients C

For the present calculation, the "Particle"

Hamiltonian was used to generate the wave functions for

110

A=l7-28, while the "Hole" Hamiltonian was used to generate
the wave functions for A=28-39. As mentioned earlier, the
exact overlap of the two Hamiltonians in the middle of the
sd-shell remains to be seen from future calculations.
However, A=28 where the sd-shell is half-filled is a
nautral boundary. And as will be seen, the two sets of
Hamiltonians are very similar with respect to reproducing
the dipole moment of the first 2+ state of 288i. The
transition density matrix elements were calculated from
the set of wave functions using equation (44), and the
dipole moments using equation (45) and (46).

With the linear expression (equation 45) and the
coefficients Cigl fixed from the generated wave functions,
the single-particle reduced matrix elements can be treated
as parameters and determined by a least-squares fit to

measured dipole moments. This was done by simply mini-

mizing the quantity:

2

(6102 = ) (47)

(“theory - uexpt.

with respect to the parameters. The measured dipole
moments were not weighted by any uncertainties as only the
more precise measured moments were included in the fit.

In the full Os-ld model space, 1- and j- selection rules
limit the number of independent single-particle reduced
matrix elements to only eight. The eight parameters were
fitted to thirty-seven measured dipole moments, which are

listed in Table 9, except where otherwise noted. Where the

1211.

TABLE 9.--Magnctic moments of some ground and excited states of sd-shell nuclei.

 

 

 

Nucleus Ex.E. a n uexpt.b “calc(n'm')
(keV) I (s) J T (n.m.) bare fitted
17 o stable 5/2 1/2 -1.89c -1.91 -1.84
17 0 6.6x101 5/2+ 1/2 +4.72 +4.79 +4.78
180 1982 3.8x10'12 2+ 1 -0.59:_o.03d'e -o.85 -0 82
18Ne 1887 4.9xlO-13 2+ 1 3.02 +3.10
180 3553 >3x1o'12 4+ 1 :2.48_+_0.40d'f -1.99 -2.00
1886 3376 4.4x10'12 4+ 1 +6.38 +6.60
18? 0 6.6x1o3 1+ 0 +0.85 +0.73
18? ”937 6.8::10'll 3+ 0 +1.87 +1.84
18? 1122 2.2x1o'7 5+ 0 +2.85:p.03 +2.88 +2.94
190 0 2.7x1o1 5/2+ 3/2 -1.50 -1.49
190 96 2.0x10’9 3/2+ 3/2 -o.69_+0.09d'g -0.91 -o.84
19? 0 stable 1/2+ 1/2 +2.63 +2.90 +2.77
19 e o 1.7x10l 1/2+ 1/2 -1.89 -2.04 -2.03
192 197 1.3::10’7 5/2+ 1/2 +3.60:0.01 +3.65 +3.53
1986 238 2.6x10-8 5/2+ 1/2 -0.74:p.c1 -o.7s -0.58
20 1672 1.3x10"11 2+ 2 :p.78:p.08h -o.67 -0.72
20 0 1.1x1o1 2+ 1 +2.09 +2.06 +1.99
zoNa o 4.1x1o'1 2+ 1 +0.37 +0.48 +0.47
20Ne 1634 1.2x10'12 2+ 0 +1.08:0.08i +1.02 +1.10
20Ne 4247 9.3x10-14 , 4+ 0 +2.04 +2.21
21? o 4.3x1oO 5/2+ 3/2 +3.84 +3.79
Zlue o stable 3/2+ 1/2 -o.66 -0.77 -0.66
21Na 0 2.354101 3/2+ 1/2 +2.39 +2.50 +2.41
ZlNe 350 2.0x1o'11 5/2+ 1/2 —o.61 -0.52
21Na 332 1.4::10'1-1 5/2+ 1/2 +3.38 +3.40
222 o 4.2x10o 4+ 2 +2.59 +2.50
22Ne 1275 4.9x10'12 2+ 1 +0.65_+_0.o3j +0.76 +0.78
22Ne 3356 3.6x10‘13 4+ 1 +1.91 +2.03
22Na 0 2.6 years 3+ 0 +1.75 +1.78 +1.78
22Na 583 3.5x10"7 1+ 0 +0.54:p.01 +0.53 +0.56
23Ne 0 3.8x101 5/2+ 3/2 -1.08:p.01 -1.07 -1.06
23Na 0 stable 3/2+ 1/2 +2.22 +2.10 +2.04
24Na 0 5.4x104 4+ 1 +1.69 +1.59 +1.57

ILJLZ

TABLE 9.--Continued.

 

 

 

Nucleus Ex.E. uexpt_b . uca1c(n.m.)
(keV) Ta(s) J1T T (n.m.) bare fitted
24Na 472 2.0x1o'2 1+ 1 -1.77 -1.68
24Mg 1369 1.7::10'11 2+ 0 +1.02:o.04k +1.03 +1.11
24Mg 4238 1.0x1o'13 2+ 0 +1.03 +1.10
24Mg 4123 5.5x1o'14 4+ 0 +2.06 +2.22
25Na 0 6.0x101 5/2+ 3/2 +3.67ip.032 ‘ +3.39 +3.51
25Mg 0 stable 5/2+ 1/2‘ -0.85 -o.85 -0.79
26149 1809 7.25410"13 2+ 1 +1.64+_o.32d'm +1.61 +1.75
26A1 0 7.2x10S years 5+ 0 +2.82 +2.91
2621 417 1.8x10'9 3+ 0 +1.77 +1.79
2851 1779 6.8x10’13 2+ 0 +1.12ip.18d'm +1.03n +1.10n
28s1 1779 6.8x1o'l3 2+ 0 +1.12_+_o.18d'm +1.03° +1.1o°
2951 o stable 1/2+ 1/2 -0.56 -o.36 -o.55
299 o 4.2x1oo 1/2+ 1/2 ‘ +4.23 +0.97 +1.18
‘"y 0 1.5x102 1+ 0 , +0.57 +0.60
"p 0 stable ' 1/2+ 1/2 +1.13 +0.94 +1.12
329 o 14 days 1+ 1 -o.25 -0.24 -0.27
32$ 2230 2.2::10'13 2+ 0 +0.99 +1.07
335 o stable 3/2+ 1/2 . +0.64 +0.50 +0.55
3301 0 2.554100 3/2+ 1/2 +0.86 +0.95
345 2127 4.1::10'13 2+ 1 +0.85 +1.01
34c1 146 1.9x1o3 3+ 0 +1.33 +1.48
355 0 87 days 3/2“ 3/2 11.oo:p.o4 +0.90 +0.90
3501 o stable 3/2+ 1/2 +0.82 +0.74 +0.87
353: 0 18:100 ” 3/2+ 1/2 +0.63 ' +0.63 +0.64
36Cl 0 3x105 years 2+ 1 +1.29 +1.34 +1.44
36x o 3.4x10'1 2+ 1 +0.55 +0.35 +0.49
36Ar 1970 4.1x10-13 2+ 0 +0.98 +1.06
37c1 0 stable 3/2+ 3/2 +0.68 +0.32 +0.57
372: o 35 days 3/2+ 1/2 . +0.95:_o.2od +1.39 ' +1.34
37x o 1.2x100 3/2+ 1/2 +0.20 -o.13 +0.11
38 13 +

At 2168 5.3x10' 2 1 +0.38 +0.72

TABLE 9.--Continued.

113

 

 

 

 

Nucleus Ex.E. ' a n uexpt.b ucalc(n'm')
(keV) T (s) J T (n.m.) bare fitted
+
38K O 4.6x102 3 0 +1.37 +1.23 +1.43
38K 461 7.4x10'10 1+ 0 +0.43 +0.50
+
39K stable 3/2 1/2 +0.39 +0.13 +0.4l
- +
3905 8.8x10 1 3/2 1/2 +1.02p +1.15 +1.04
+ +
6T is Tl/Z for ground states and Tm for excited states, except for 24Na (1 ) and 34C1 (3 )

excited states where T

1/2

are specified instead.

b 4 I Q I 0
Unless otherwise noted, the measured magnetic moments are taken from the compilation in

Reference

C
Reference

d

Not included in the least—squares fit.

eReference

f
Reference

9Reference

h
Reference

p.

Reference
Reference
Reference

Reference

3 ho a- u

Reference

27.
29.

96, 97.
98.

99.

100.

101.

102.

101, 103.
104.

105.

n . . .
"Particle" Hamiltonian.

0 . .
"Hole" Hamiltonian.

9Reference

106.

 

114

sign of the measured dipole moment is not known, it is taken
to have the same sign as the bare-nucleon calculated

moment.
II.3. Results

In Table 9 are listed the measured dipole moments
and calculated dipole moments, both with the bare-nucleon
value and the fitted single-particle reduced matrix
elements, of some ground and excited states of sd-shell
nuclei. Also listed are the excitation energies, life-
times, spins, parities, and isospins of each state. The
measured and calculated dipole moments for A=17-25and
A=29-39 are also plotted in Figure 26 and Figure 27
respectively. It is clear that good agreement is obtained
with either the bare-nucleon or the fitted Operators for
A=17-25. For A=29-39, agreement is much better with the
fitted operator than the bare-nucleon Operator. The RMS
deviation between measured and calculated dipole moments
remains almost unchanged for the 22 dipole moments for
A=l7-25 included in the least-squares fit, it is 0.11 n.m.
and 0.10 n.m. for the bare-nucleon and fitted Operators,
respectively. On the other hand, the RMS deviation shows
a big improvement for the 15 dipole moments for A=29-39
included in the fit with the fitted Operators, it is
0.07 n.m. compared to 0.20 n.m. with the bare-nucleon

operators. Overall, the RMS deviation changes from

115

.mmlhand How

 

 

 

 

 

mwumum pmufioxm one ossoum meow mo mucmEoE maomfio Oflumcmmz .mm musmflm
mmwweeaaunmmmwwmmmmv
MMIOMMIOOMZIOMMIIOM;
Iv-
Ln.
9% 91mm-
8 0 .ee 0 o. Aim;
J. u .i n n “L. O.U
e .99 SE h_m\
® 0 B0 9 hm
® E Inn
6 e n.m..
.28 D6 .36 + mtmEoz 285 25522 ®lm

116

 

 

 

I.5L—, V
_ v
_ * 8 <3
- O {7
. 0 v
_, V7
LO T <1; o o
C Y '
- C)
_. V 9 ‘
EQS— {7 8
, v .
5' . o O
:L L .
- C) g7
' C t 4* : r + . + +7 +
' O
1 . Magnetic Dipole Moments 9
. + expt. O,v colc. 0
05+ v
" S 1’ § $1 ‘t‘ “8’ $ 3 T $ $
< ,9, 8 +3 :1: 8 :8 8 :8 8 .7. 8

 

 

 

Figure 27. Magnetic dipole moments for some ground
and excited states for A=29-39.

117

0.15 n.m. to 0.09 n.m. for all 37 measured moments included
in the least-squares fit.

The fitted single-particle reduced matrix elements
are listed in Table 10, together with the bare-nucleon
single-particle reduced matrix elements for comparison.
The uncertainties were estimated by assuming a 0.09 n.m.
(obtained RMS deviation) uncertainty or error for each
calculated dipole moment. The dS/Z-dS/Z isoscalar and
isovector matrix elements only change very little, pre-
sumably due to the already good agreement with the bare-
nucleon operators for the lower half of the sd-shell,
where the d —d /2 matrix elements are more important.

5/2 5

In general, except for the s -s isoscalar and

1/2 1/2
dS/Z-dS/Z isovector matrix elements, the single-particle
reduced matrix elements get more positive.

Obviously, the next question is whether the set
of fitted single-particle reduced matrix elements can tell
us anything about effective g-factors and intrinsic
moments. The isoscalar and isovector single-particle
reduced matrix elements are defined explicitly in the

following:

1 2
g + 9 -
<i||li30|||j>= (-P——2 “) <i|||1|||j>

o + O
+ (up + u.) <1||ls|||3> <48)

 

 

 

 

 

ith'lr :t b ’lFIL- -16 ”an I:-
.

118

TABLE 10.--Comparison between bare-nucleon and fitted
single-particle reduced u matrix elements (n.m.).

 

 

 

<£j|||KI|||£'j'>a Bare Fittedb
<dS/zlllfiolllds/2> ' 2.88 2.94:0.06
<dS/2|||fioll|d3/2> - 0.41 -o.ie:0.15
<sl/2|||301||sl/2> 0.74 0.69:0.14
<d3/2|||Eo|||d3/2> 1.13 1.29:0.04
<d5/2|||El|||ds/2> 11.62 ii.49:0.18
<ds/2|||Ell||83/2> - 7.79 -6.52:0.28
<sl/2|||fi1|||sl/2> 6.89 7.25:0.25
<d3/2|||fil|||d3/2> - 1.58. -o.97:0.13

 

 

aI equals 0 or 1 for isoscalar and isovector components,
respectively.

See text for description on uncertainties.

119

l l
I + I g — g 0 I
<1II|U1|||J> = (_E_7_JE) <1|||1T2||l3>

(up-un)<illl§Tzlllj> (49)
where the reduced matrix elements of I, 3, 3T2, 3T2 are
easily evaluated with angular—momentum algebra. The effec-
tive g-factors and intrinsic moments can be treated as
unknowns and determined by two separate least-squares fits
to the fitted isoscalar and isovector single-particle
reduced matrix elements. However, with only two parameters
in each case, a graphical representation of the least-
squares fits, which is feasible, is more helpful and gives
a clearer picture.

In Figure 28(a) and Figure 28(b) are plotted the
straight lines corresponding to the eight fitted isoscalar
and isovector single-particle reduced matrix elements
respectively. The dash lines indicate the uncertainties
as listed in Table 10. The cross in each figure corres-
ponds to the bare—nucleon value g-factors and intrinsic
moments.

It is seen from Figure 28(a) that the area bounded
by the intersections of the four lines is small. In
Figure 28(b), the corresponding area is larger with the
isovector Sl/Z-Sl/Z clearly outside the boundary. If the
least-squares fits were simply done using Equations (48)
and (49) fitted to the four isoscalar and four isovector

fitted single-particle reduced matrix elements; the

llil'll'llllllll

 

120

I /

}/ d5/2’d3/2

\ I / ‘
/ ds/z‘dwz

 

 

(a)

 

 

N4!-
L”

 

 

 

 

 

a; 1
4 SI/z-SI/zl l
\ds/z‘dS/z lyd —d
‘3 V/ 5/2 3/2
(b) 3‘ /’ I
«2 / .f ' d3/2/‘d3/2
.2 \ \\ / / //I I
| ”"$la«fii
”‘,;’3/'_/ \
.f’a"" J /9’. l I
4 l 2 37/4 I“ 6
I

-Figure 28. Effective g-factors and intrinsic
moments from the fitted single-particle
reduced u matrix elements (n.m.).

121

isoscalar part would give a good fit, and the isovector
part a bad one.

The SIOpes of the straight lines are fixed by the
reduced matrix elements of I and 3, and 1T2 and ETZ. The
fitted single-particle reduced matrix elements merely trans-
late the lines. In both Figure 28(a) and Figure 28(b), the
directions of change of the g-factors and intrinsic moments
closely parallel the dS/Z-dS/Z lines. Thus the changes for
the d5/2-d5/2 single particle reduced matrix elements are
small compared to the others. It should be noted that in
Equations (48) and (49), we have neglected the radial part,
i.e., assumed the overlap of radial wave functions in each
case to be one. For the dS/Z-dB/Z' an overlap less than
one would move both lines for the isoscalar and isovector
cases in the direction of making the area of intersections
smaller. The translation should be larger for the iso-
vector than for the isoscalar. The resulting fits should
be better than are shown in Figure 28(a) and 28(b).

The increase in the isovectors Sl/2 intrinsic
moment may be a little surprising. However, it is not
inconsistent with what is found for the magnetic moments
of 3H and 3He. The Schmidt value for the isovector
magnetic moment, i.e., u(3H)/2-u(3He)/2, is too small
compared to the experimental value. The effective
intrinsic moments for a Sl/Z particle (with i=0) can be

obtained directly from the single-particle reduced matrix

elements listed in Table 10 and equations

The results are:

eff eff
up (SI/2) + un
eff eff
up (31/2) un
eff
or up (Sl/Z)
eff (s )
11n 1/2

122

(48) and (49) .

i 0.17 n.m.

: 0.17 n.m.

n.m.

(SI/2) = 0.76
(31/2) = 4.95
= 2.85 i 0.17
= -2.09 i 0.17 n.m.

The effective g-factors and intrinsic moments for

d5/2 and <33/2 orbits can be read off Figure 28(a) and

Figure 28(b).

A more quantitative analysis, i.e., least-

squares fits to the four isoscalar and three isovector

(excluding sl/Z-sl/Z) fitted Single-particle

matrix elements listed in

Table

reduced

10 gives:

9g'eff + gfi'eff = 1.07 n.m.
gg’eff - fi'eff = 1.12 n.m.
pgff + piff = 0.79 n.m.
ugff - ufiff = 4.36 1 0.01 n.m.
or 92"eff = 1.09 n.m.

123

9R"eff = -0.02 n.m.

n

ugff = 2.58 i 0.01 n.m.
ufiff = -1.79 _+_ 0.01 n.m.

The present calculation with the full Os-ld shell-
model space does not include major shell crossing second
and higher order configuration mixing corrections for the
dipole moments via the mixed-configuration wave functions.
The effective g-factors and intrinsic moments are thus due
to the combined effects of these higher-order configuration-
mixing corrections, mesonic exchange currents and other
possible corrections. Mavromatis and Zamick91 have pre-
viously calculated second order configuration-mixing
correction for the dipole moments of mass 17 and 39 with
up to 25w excitations from the ground state. Their
results, using the bare-nucleon g-factors and intrinsic
moments, show the corrections to be non-negligible.
Further studies are needed to untangle the different
effects in the effective g-factors and intrinsic moments.

It may be mentioned in passing that the quenching
of the intrinsic moments is in agreement with the results
of Miyawaza92 and Drell and Walecka93 for the effects of

mesonic exchange currents. The effective orbital g-factors

are close to the empirical estimates of Nagamiya and

 

 

 

 

 

 

 

124

Yamazaki95 over the whole mass region, i.e., 9:,eff =
1.09 .t 0.03 n.m. and gfi'eff = -0.06 i 0.04 n.m.

II.4. Summary

Magnetic dipole moments in the range A=l7-39 have
been investigated using mixed-configuration shell-model
wave functions generated from empirical Hamiltonians in the
full Os-ld model space. Dipole Operators were treated as
parameters and determined from a least-squares fit to
precise measured moments. Good agreement was found for
the whole range with the fitted operators, while the bare-
nucleon Operators could only give good agreement for
A=l7-25. Effective g-factors and intrinsic moments were
derived from the fitted operators. The intrinsic moments
are quenched compared to the bare-nucleon value; while the
change in the orbital g-factors are fig: = +0.09 n.m. and
égn = -0.02 n.m. The isovector intrinsic moment of the
sl/z-orbit on the other hand increases, however, this is
not inconsistent with the observed deviations of the

3H and 3He from the Schmidt value. It

magnetic moments of
can safely be said only that the effective g-factors and
intrinsic moments arise from the combined effects of many
different corrections other than the intra-major-shell
configuration-mixing corrections included in the present

mixed-configuration wave functions. More rigorous and

quantitative treatment of the various other corrections to

125

the magnetic moment, specifically as a renormalization of

the g-factors and intrinsic-moments, would be very helpful.

III. SUGGESTIONS FOR FURTHER STUDY

The principal aim of this study was to obtain an
empirical Hamiltonian for use in shell-model calculation
that would give a good description of all sd-shell nuclei
using a full ls-Od model space. A single set of mass
independent (l+2)-body Hamiltonian was found to be inade-
quate. Instead two Hamiltonians were obtained by iterative
least-squares fits to energy-level data in the lower and
upper end of the sd-shell. Comparison of calculated
ground-state binding energies and spins, and energy spectra
with experiments seems to indicate the two Hamiltonians
combined will complement each other to give a good descrip-
tion of all sd-shell nuclei. The good agreement of
calculated magnetic dipole moments of some ground and
excited states in sd-shell nuclei with experiments is an
initial confirmation of this expectation. More complete
tests are needed, however. Further calculations of energy
spectra of nuclei in the middle of the sd-shell, and
further tests of the generated wave functions with other
nuclear observables such as quadruple moments, electro-

magnetic transitions, B-decays, spectroscopic amplitudes

126

127

and electron scattering form factors will yield a thorough
picture of our present level of understanding.

The unambiguous result of the empirical renormali-
zation of the Kuo's realistic Hamiltonians was the reduction
in attractiveness of the "unlike orbit" diagonal two-body
matrix elements. In the light of the problems that still
exist in the theory of effective interactions for shell-
model-type calculations derived from the free-nucleon
interaction, an understanding of the reduction in attrac-

tiveness may provide a key to the solution.

 

More quantitative treatment of the various effects
on orbital g-factors and intrinsic moments of valence
nucleon due to higher-order configuration mixings, mesonic
exchange currents and others are needed for a better under-
standing of the effective g-factors and intrinsic moments
derived from the fitted single-particle u reduced matrix
elements.

The technique of empirical renormalization of
shell-model Hamiltonian in terms of uncorrelated (ortho-
gonal) linear combinations of one- and two-body matrix
elements can be applied easily to other regions of nuclei
of interest (with manageable ordersof Hamiltonians)
without the previous problems of too many parameters or
insufficiency of data. Examples are shell-model calcula-
tions in a 0f7/2-lp3/2 model space, 0p3/2-0p1/2-0d5/2-1s1/2-

0d3/2 model space, 0f5/2-lp3/2-lp1/2 model space and so on.

128

Better descriptions of the nuclei of interest should be
achieved compared to previous attempts.

However, a shell-model calculation of major
interest with the full Of-lP model space is still prohibi-
tively large even with the present available techniques.
The result of the orthogonal parameter fit on the other
hand is promising in that it shows only a few orthogonal
parameters are important in describing the low-lying
spectra. Understanding of such orthogonal linear combina-
tions may provide simplications for shell-model description

of low-lying Spectra.

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