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THESIS

 

 

 

This is to certify that the
dissertation entitled

Calculations of Elastic and Inelastic
Electron Scattering in Light Nuclei with
Shell-Model Wave Functions

presented by

Raad Abdul-Karim Radhi

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Physics

 

 

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CALCULATIONS OF ELASTIC AND INELASTIC
ELECTRON SCATTERING IN LIGHT NUCLEI WITH

SHELL-MODEL WAVE FUNCTIONS

BY
Raad Abdul-Karim Radhi

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1983'

ABSTRACT
CALCULATIONS OF ELASTIC AND INELASTIC
ELECTRON SCATTERING IN LIGHT NUCLEI WITH

SHELL-MODEL WAVE FUNCTIONS

BY
Raad Abdul-Karim Radhi

Shell-model wave functions calculated within the
complete space of Op3/2- Cpl/2 and 0d5/2- 151/2- 0d3/2
configurations are used to calculate elastic and inelastic
form factors of electron scattering from p-shell and
sd-shell nuclei. We analyze the magnetic elastic scattering
data for p—shell and sd-shell nuclei and both the electric
and magnetic inelastic electroexcitation of the even-parity
states of 27Al. Effective operators for the different
multipoles are used to normalize the magnetic elastic form
factors to the experimental data. Different effective
charges are used for £2 and E4 transitions. The longitudinal
form factors and the B(E2) values are well reproduced using
these effective charges. Comparisons are made for the
single-particle wave functions of the harmonic oscillator

and the Woods-Saxon radial wave functions.

To my wife

Shatha

ii

ACKNOLEDGEMENTS

I would like to thank my advisor, Dr. B. H. Wildenthal
for suggesting this problem and for his support and guidance
during my graduate study. I am also grateful to him for
reading and correcting the manuscript.

I would also like to thank Dr. B. A. Brown for his
assistance and guidance. His concern for this work was
essential to its successful completion. I am grateful to the
administration and staff of the cyclotron laboratory for
allowing me to make use of the laboratory facilities,
particularly the computer

I am grateful to the Iraqi government for the financial
support during my graduate study.

I would like to express my deepest thanks to my wife
Shatha and my daughter Sima, for their love and for their
patience and understanding throughout my graduate study.

Lastly, I would like to thank my parents and my brothers
and sisters for their love and encouragement and for their
suffering from my absence during the years of study away

from home.

iii

TABLE OF CONTENTS
page
LIST OF TABLES ........................................ vi
LIST OF FIGURES ....................................... vii
CHAPTER I. INTRODUCTION ............................... 1
CHAPTER II. THE NUCLEAR SHELL MODEL ................... 7
11.1. Introduction ............................ 7
11.2. Matrix elements of one-body operators ... 10
11.3. Radial components of single—nucleon wave
functions ................................ 12
CHAPTER III. ELECTRON SCATTERING ...................... 14
111.1. Introduction ........................... 14
III.2. Single—nucleon form factor for the
longitudinal operator ................... 16
111.3. Single-nucleon form factor for the
transverse magnetic operator ............ 18
111.4. Single-nucleon form factor for the
transverse electric operator ............ 21
111.5. Multi-nucleon form factors ............. 24
111.6. p-shell and sd-shell transition
densities ............................... 27
111.7. Corrections to the electron scattering.. 31
111.8. Conversion of form factors to q-dependent
matrix elements M(q) .................... 38

CHAPTER IV. MAGNETIC ELASTIC ELECTRON SCATTERING ...... 45

iv

IV.1. Introduction ............................ 45
IV.2. Magnetic elastic scattering from 1/2+
nuclei ................................... 46
IV.3. magnetic elastic scattering from other
odd-A sd-shell nuclei .................... 49
IV.4. Magnetic elastic scattering from 17O .... 57
IV.5. Magnetic elastic scattering from 27Al ... 63
IV.6. Magnetic elastic scattering from 39K .... 70
IV.7. Magnetic elastic scattering from p-shell
nuclei ................................... 76
IV.8. Conclusions ............................. 85
CHAPTER V. ELECTROEXCITATION OF EVEN-PARITY STATES IN
27A1 ............ .......... .................. 95
v.1. Introduction ............................. 95
v.2. Elastic scattering for the 5/2+ ground
state ... ..... ............................. 98
v.3. Inelastic scattering for the 0.844 MeV,
1/21+ state .............................. 100
v.4. The 1.014 MeV, 3/21+ state ............. 102
v.5. The 2.211 MeV, 7/21+ state ............. 111
v.6. The 2.735 MeV, 5/22+ state .. ......... .. 116
v.7. The (2.98.3.004) MeV, (3/22+, 9/2lt)
doublet . ....... ........................... 119
v.8. The higher-lying states .................. 124
v.9. Conclusions .............................. 137
APPENDIX . 151
REFERENCES..................... ..... . 154

V

Table

IV.1.

IV.2.

IV.3.

v.2.

LIST OF TABLES
page

Calculated one-body transition density matrix
elements for the ground state of stable sd-shell
nuclei for the wave functions of Ref l7.......... 87
Calculated one-body transition density matrix
elements for the ground state of stable p-shell
nuclei for the wave functions of Ref 18 and Ref
19.......................................... ..... 89
Experimentally determined rms charge radii of
stable p-shell and odd—A sd-shell-nuclei and the
corresponding values calculated in the
harmonic-oscillator model with length parameters
brms and in the Woods-Saxon model................ 91
Measured and calculated magnetic dipole moments
for p-shell and sd-shell nuclei.................. 93
Calculated one-body transition density matrix
elements (OBDM) for [(2Ji= 5, 2Ti= 1, Ni= l) to
(2J1, 2T1= 1, Ni )1 transitions in 27A1 for the
wave functions of Ref 17........ ........ ......... 141
Calculated occupation numbers for sd-shell orbits

in the ground state of 27A1....... ....... . ...... . 150

vi

Figure

III.1.

III.2.

III.3.

III.4.

LIST OF FIGURES
Page

DWBA form factor for the elastic magnetic

scattering from 2 Al (solid line) in comparison
with the PWBA (dashed line). The calculations
incorporate the single-nucleon wave functions of

the HO potential of b=brms- The data are taken

from Ref 40 (circles), Ref 41 (triangles) and

Ref 42 (squares)................................ 36

DWB§7form factor for the 0.844 MeV,1/21+ state

of Al (solid line) in comparison with the PWBA

for the same state (dashed line). The calculations
incorporate the single-nucleon wave functions of

the HO potential of b=brms- The data are taken

from Ref 13...... ......... ...................... 37

Magnetic elastic electron scattering for 27A1
calculated with the HO potential, presented in

the two forms F2(L,q) and M(q) as explained in
section III.8. The magnetic multipoles

contributing to the scattering are M1 (dotted

line), M3 (dashed line) and M5 (dashed-dotted line).
The solid line represents the total incoherent

sum of these three multiploles. The magnetic

dipole moment is displayed in the M(q)
representation. The data are taken from Ref 40
(circles), Ref 41 (triangles) and Ref 42
(squares)....................................... 43

Longitudinal E2 orm factor for the 0.844 MeV,

1/2 state for A1 calculated with the HO
potential, presented in the two forms F2(L,q) and
M(q) as explained in section III.8. The measured
B(E2) value is displayed in the M(q) plot at q=0.
The data are taken from.Ref 13.................. 44

IV.1. Formgfacggrs for 598 magnetic elastic scattering

Si and P calculated with the HO

radial wave functions of b= brms- The cross
signs represent the calculations with the
single-particle model with free-nucleon 9
factors. The configuration-mixing shell model
calculations using free—space 9 factors and
effective M1 g factors are represented by plus
signs and solid lines respectively. The values

vii

IV.2.

IV.3.

IV.4.

IV.

for the M1 effective 9 factors are gsp=5.,
<;;_..,'“=-3.44.99 g,P=1.073 and g,"=-o.044. The

data for F are taken from Ref 47 (circles). The
data for 298i and P are taken from Ref 48
(traingles) and Ref 49 (circles)................. 48

ForT7fac§grs for 299 magnetic elastic scattering
Mg and A1 calculated with the HO
radial wave functions of b= brms- The cross signs
represent the calculations with the
single—particle model and free-space 9 factors.
The configuration-mixing shell model
calculations using free-space 9 factors are shown
by plus signs and those of effective M1 and M3 9
factors are shown by solid lines. The values for
the effective M1 9 factors are those used in
Figure IV.1. The M3 contribution is quenched to
60% of the free-space value. We use the free M5
9 factors for both cases. The decomposition of the
multipoles are Ml (dotted lines), M3 (dashed lines)
and M5 (dashed-dotted lines), calculated with the
empirical 9 factors discussed above. The data for
0 age taken from Ref 50 (circles). The data
for Mg are taken from Ref 37 (squares) and Ref
51 (circles). The data for 2 A1 are taken from
Ref 40 (circles), Ref 41 (triangles) and Ref 42

(squareS)OOOOOOO ..... ......OIOOOOOOOOOOOO00...... 52
Form factgss for the magnetic elastic scattering
of lNe Na and 3S. The conventions of the

presentation are the same as given in the caption
to Figure IV.2. The data for Na are taken from
Ref 52 (CircleS).....I......OOOCOOOOOOOOOOOOOOOOO 54

Form factgrs for the magnetic elastic scattering

of 5C1, C1 and 9K. The conventions of the
presentation are the same as ggven in the caption

to Figure IV.2. The data for K are taken from

Ref 53 (circles) and Ref 54 (squares)............ 56

Form factors foi magnetic elastic electron
scattering for 7O. The single-nucleon radial
wave functions used here are those of the H0
potential of b=brms(Figure IV.5a) and of the

WS potential discused in section II.3

(Figure IV.5a'). Free-space values for the
neutron-g factors are used in both calculations.
The corresponding calculations with quenching the
9 factor of the M3 contributions to 60% are shown
in Figure IV.5b and Figure IV.b'. Figure IV.5c
and Figure IV.5c' are the same as Figure IV.5b
and IV.5b' except that the rms radius of the

viii

IV.6.

IV.7.

IV.8.

IV.9.

valence orbit is reduced by 5%. The same

conventions are used for the different multipoles

as in Figure IV.2. The data are taken from

Ref 50 (circles)................................. 60

ForT7factors for the magnetic elastic scattering

of O. The conventions of the presentation are

the same as given in the caption to Figure IV.5c'
except effective 9 factors for the M1

contributions are used (gsn(eff)= 0.8xgsn ,

g,“(ef£) =9," -182) 62

Form factors £05 magnetic elastic electron
scattering for 7A1. Calculations with the HO
potential of b=brms: assuming only one proton
hole in the d5/2 orbit are shown in Figure IV.7a.
The configuration—mixing contributions of the HO
potential of b=brms and of b reduced 9% from b
are shown in Figure IV.7b and Figure IV.7c
respectively. Free-nucleon values for the 9
factor are used in these calculations. The effect
of reducing b-vaer.of the HO potential by 5% is
shown in Figure IV.7d using different values for
9 factors for M1 contributions to get the exact
ma netic dipole moments (gsn(eff)=.8xgsn,

g (eff)= g nx1.25)‘. Figure IV.7e is the same as
Figure IV.7d except the proton 9 factor of M3
contributions is quenched to 60% of the
free-space value. Figure IV.9f is the same as
Figure IV.7e except the wave functions of the WS
potential whose valence orbits rms radius

reduced 5% are used in place of the HO wave
functions. The same conventions are used for the
different multipoles as in Figure IV.2. The data
are taken from Ref 40 (circles), Ref 41
(triangles) and Ref 42 (squares)................. 65

Form factors £05 magnetic elastic electron
scattering for 7A1 calculated with single-nucleon
radial wave functions of the HO potential of b=brms
using free-space values for the g factors.

Protons contributions only are shown in Figure
IV.8a. The neutrons contributions only are shown

in Figure IV.8b. Figure IV.8c and Figure IV.8d

show the proton gs and 9, contributions
respectively. The same conventions are used for

the different multipoles as in Figure IV.1. The
data are taken from Ref 40 (circles), Ref 41
(triangles) and Ref 42 (squares)................. 69

Form factors fo§9the magnetic elastic electron
scattering for K calculated with single-nucleon

ix

IV.10.

IV.11.

IV.12.

radial wave functions of the HO potential of
b=brms- Free-space values for the 9 factors of
the M1 and M3 contributions are shown in Figure
IV.9a. Figure IV.9b is the same as Figure IV.9a
except the proton 9 factor of the M3
contributions is quenched to 60% of the
free—nucleon value. The orbital angular momentum
contributions alone are shown in Figure IV.9c.
Figure IV.9d shows only the spin contributions.
The same conventions are used for the different
multipoles as in Figure IV.2. The data are taken
from Ref 53 (circles) and Ref 54 (squares)....... 72

Form factors fog the magnetic elastic electron
scattering for 9K calculated by quenching the
9 factor of the M3 contributions to 60% of the
free-space value. Figure IV.10a shows the
calculations with free-nucleon values for the
M1 contributions. Figure IV.10b shows the
calculations with g, (eff)= g,Px1.15 and
gsp(eff)=gsp(free) for the M1 contributions.
Fiure IV.10c are calculated with

gsp(eff)= 0.8xgspand g,p(eff)=0.960g p for the
M1 contributions. Figure IV.10d is t e same as
Figure IV.10c but usig the WS potential whose
valence orbits rms reduced 5%. The same
conventions are used for the different
multipoles as in Figure IV.2. See caption of-
Figure IV.9 for the data........................ 75

Form factors fog the magnitic elastic electron
scattering for Li and N calculated with two
different wave functions, MSU wave functions

(solid lines) and the Cohen-Kurath wave

functions (plus signs) uging the HO potential

of b=brms- The data for Li are taken from

Ref 57 (circles), Ref 28 (squares) and Ref 59
(triangles), and for 1 N are taken from Ref 58
(squares), and Ref 60 (circles)................. 78

Form factora for the magnetic elastic scattering
of Li and Be. The conventions of the
presentation are the same as given in the caption
to Figure IV.11. The decomposition of the
multipoles calculated with the MSU wave functions
are M1 (dotted lines) and M3 (dashed lines).
Free-nucleon values are used here for the 9
factors. The data for Li are taken frgm Ref 61
(circles) and Ref 62 (traingles), for Be are
taken from Ref 63 (circles), Ref 64 and Ref 58
(squares)......... ....... ....................... 80

IV.13.

IV.14.

V.l.

ForT 0factors for the magnetic elastic scattering
ToB and 1 B. The conventions of the

presentation are the same as given in the caption

to Figure IV.12. The data are taken from Ref 58

(squares) and Ref 57 (circles).................. 82

Form 31factors or the magnetic elastic scattering
T3C and The conventions of the
presentation are the same as gigen in the caption
to Figure IV.12. The data for C are taken from
Rgf 63 (circles) and Ref 65 (triangles), and for
N are taken from Ref 66 (squares)............. 84

Experimental spectrum of 27A1 (e,e') in

comparison with the theoretical form factors of

the positive-parity states calculated at the

same angle and incident energy as the measured
spectrum..............................-........... 96

v.2. Measured positive-parity energy levels of 27A1 in

v.3.

comparison with the shell-model calculations...... 97

DWBA elastic scattering form factor calculated

with the single— —nucleon wave functions of the HO
potential of b= brms(solid line) and of the WS
potential (dashed line). The different multipoles

E0, E2 and E4 (dotted lines, plus and ”Y" signs,
respectively) are calculated with the '

single-nucleon wave functions of the HO potential

of b=brm . The data are taken from Ref 41
(circlesi-ZSO Mev, (squares)-500 MeV)............. 99

Longitudiaal E2 form factor M(q) for the 1/21+
state in Al calculated with the single-nucleon
wave functions of the HO potential combined with
the Tassie and valence models (solid and dashed
lines, respectively). The data are taken from Ref

13.0.0000000000 ..... 0.0.0.0.........OOOOOOOOOOI... 101

Longitudisal E2 form factor M(q) for the 1/21+

state in Al calculated with the Tassie model
combined with single nucleon wave functions of the

HO potential of b= brms (solid line) and of the WS
potential (dashed line). The data are taken from

Ref 13............................................ 103

Transverse form factors for the 0.844 MeV, 1/21+
state calculated with the HO radial wave functions
of b=brms- The upper two Figures show the-
contributions from the spin and orbital 9 factors
respectively. The lower two Figures calculated
with free M3 9 factors and quenched M3 9 factors

xi

v.8.

v.9.

to 60% of the free-nucleon values, respectively.

E2 and M3 multipoles are shown by the plus and

cross signs respectively. The data are taken from

Ref 13............................................ 104

Total form factor for the 0. 844 MeV, 1/21+ state
calculated at 8= 90° (solid line). The dashed line
represents the longitudinal form factor, and the
dashed- dotted line represents the transverse form
factor 1.5 FTZ. The data are taken from Ref 13.... 105

Form factors for the 1.014 MeV, 3/21+ state
calculated with the single-nucleon radial wave
functions of the HO potential of b=brms. The
longitudinal, transverse and total form factors
are represented by the dashed, dashed-dotted and
solid lines, respectively. The plus and "Y" signs
in the longitudinal plot represent the
contribution of the E2 and E4 multipoles of the
longitudinal form factors. The decomposition of
the multipoles of the transverse form3 factor are
M1 (dotted line), E2 (plus signs ),

(cross signs ) and E4 ("Y” signsL The total form
factor is calculated at 8=90°. The data are taken
from Ref 13....................................... 108

Longitudinal and transverse M(gi form factors for

the 1. 014 MeV, 3/21 ‘state of Al calculated with

the single-nucleon wave functions of the HO

potential of b=brms- In the longitudinal plot, the
solid and dashed lines represent the calculations
with the Tassie and valence models respectively.

The measured B(E2) value and B(M1) value are shown

at q= 0 in the longitudinal and transverse plots
respectively. The data are taken from Ref 13...... 110

v.10. Form factors for the 2.11 MeV, 7/21+ state. The

conventions of the presentation are the same as

given in the caption to Figure V. 8. The M5

multipole is shown by the triangles. The data are
taken from Ref 13 (circles), Ref 84 (triangles).. 113

Longitudinal and transverse M(q) form factors for

the 2. 211 MeV, 7/21 state. The conventions of the
presentation are the same as given in the caption

to Figure v.9. The data are taken from Ref 13
(circles), Ref 84 (triangles).................... 115

Form factors for the 2.735 MeV, 5/22+ state. The
conventions of the presentation are the same as

given in the caption to Figure v.8. The conventions
for the different multipoles are the same as those in

xii

v.18.

Figure v.3 and Figure v.10. The data are taken
fromRef13............OOOOOOOOOOOO......OOOOOOOO 118

Longitudinal and transvese M(q) form factors for

the 2.735 MeV, 5/22+ state. The conventions of

the presentation are the same as given in the

caption to Figure v.9. The data are taken from

Ref 13 (circles), Ref 84 (triangles)............. 121

Form factors for the (2.98,3.004) Mev,

(3/22+, 9/21+) doublet. The conventions of the
presentation are the same as given in the caption

to Figure v.8. The conventions for the different
multipoles are the same as those in Figure v.10.

The data are taken from Ref 13 (circles), Ref 84
(triangles)...................................... 123

Longitudinal and transverse M(q) for factors for

the (2.98,3.004) MeV ,(3/22+, 9/21+) doublet. The
convections of the presentation are the same as

given in the caption to Figure v.9. The data

are taken from Ref 13 (circles), Ref 84
(triangles)...................................... 126

. ggrm factors for the 3.68 MeV, 1/22+ state of

Al. The upper plot represents the transverse
form factor (dashed—dotted line). The E2 and M3
multipoles are shown by the plus and cross signs
respectively. The lower plot represents the total
form factors calculated at 9= 90°(solid line).
The dashed line represents the longitudinal form
factor, while the dashed-dotted line represents
the transverse form factor including a factor of
1.5. The data are taken from Ref 13.............. 129

Sgrm factors for the 3.957 Mev, 3/23+ state of

A1. The conventions of the presentation are

the same as given in the caption to Figure v.16.

The decomposition of the multipoles of the

transverse form factor are M1 (dotted line), E2

(plus signs), M3 (cross signs), and E4

("Y" signs). The decomposition of the multipoles

of the longitudinal scattering are shown in the

total form factor plot as E2 (plus signs) and E4

("Y" signs)...................................... 131

Egrm factors for the 4.41 Mev, 5/23+ state of
Al. The conventions of the presentation are
the same as given in the caption to Figure v.16.
The decomposition of the multipoles of the
transverse form factor are M1 (dotted line), E2
(plus signs), M3 (cross signs), E4 ("Y"s line)

xiii

and M5 (trianles). The decompositions of the
multipole of the longitudinal scattering are

shown in the total form factor plot as E0

(dotted line), E2 (pluss signs) and E4

("Y" signs)...................................... 134

Egrm factors for the 4.51 Mev, 11/21+ state of
A1. The conventions of the presentation are
the same as given in the caption to Figure v.16.
The decomposition of the multipoles of the
transverse form factor are M3 (cross signs), E4
("Y" line) and M5 (trianles). Only E4 multipole
contributes to the longitudinal scattering....... 136

Egrm factors for the 4.58 Mev, 7/22+ state of
A1. The conventions of the presentation are
the same as given in the caption to Figure v.16.
The decomposition of the multipoles of the
transverse form factor are M1 (dotted line), E2
(plus signs), M3 (cross signs), E4 ("Y" signs)
and M5 (trianles). The decomposition of the
multipoles of the longitudinal scattering are
shown in the total form factor plot as E2
(plus signs) and E4 ("Y" signs)............ ...... 139

xiv

CHAPTER I

INTRODUCTION

Electron scattering has been widely used as a probe of
nuclear structure. Theoretical work on electron scattering
dates from 1929, when Mott (Ref 1) derived the cross section
for the relativistic scattering of Dirac particles by
spinless point nuclei of charge Ze where Z/137 << 1. For the
scattering of high energy electrons from the nucleus, the
de Broglie wavelength becomes equal to or smaller than the
radius of the nucleus, and the interaction of the electron
with the nucleus will be sensitive to the details of the
nuclear charge distribution. The effects of finite nuclear
size on electron scattering were first considered by Guth
(Ref 2) and later independently by Rose (Ref 3).
Corresponding to the Mott formula for the scattering of
electrons from point nuclei, the finite nuclear size can be
taken into account by multiplying the Mott cross section by
a factor which depends on the charge, current and
magnetization distribution of the target nucleus. This
coefficient of the Mott cross section is called the form
factor of the nucleus. Experimentally, the form factor can
be <determined as a function of the momentum transfered (q)

tO"the nucleus, 3 quantity which is determined by the

energies of the incident and scattered electron and the
scattering angle.

The effects of nuclear size were first detected
experimentally by Lyman et a1. (Ref 4), who measured the
scattering cross section of 15.7 Mev electrons by nuclei.
Good agreement was obtained between the experimental data
and calculations which assumed a uniformly distributed
nuclear charge.

The scattering of electrons from a target nucleus can
occur in two ways. In one, the nucleus is left in its ground
state after the scattering and the energy of the electrons
is unchanged. In the other, the scattered electron leaves
the nucleus in different excited state and has a final
energy reduced from the initial just by the amount taken up
by the nucleus in its excited state. These two kinds of
processes are refered to as elastic and inelastic electron
scattering.

Excitation of nuclear levels by electrons was first
discussed theoretically by Mamasachlisov (Ref 5) in 1943.
The first experiment on electron excitation of nuclei to
discrete levels was done in 1940 by Collins and Waldman
(Ref 6). Since that time, electron scattering has become
a major technique for studying the structure of the nucleus
and many experiments have been performed at different
laboratories. The work of Hofstadter et a1. (Ref 7) at the
Stanford university linear accelator in 1953 is considered

as the pioneering experimental study of this subject.

Several review articles have been published discussing the
development of this topic and one can find a detailed
summary of what has been done in this field and lists of
review articles since the early stages of scattering theory
in Ref 8, Ref 9 and Ref 10.

Electron scattering is not the only way to probe
nuclear structure with electromagnetic interactions. An
alternate is photo-excitation. The momentum transfered to
the nucleus in this case equal to the excitation energy (ca)

since the mass of the photon is zero,
qi = q2- wz = 0 ( l)

The three-momentum transfer a in this case cannot be
varied for a given energy level, and the nuclear structure
cannot be studied as a function of momentum transfer. In
the case of charged-particle excitation of nuclear levels,

one can vary q for a fixed w

2 2>0 (2)

u
.0
I
E

and study the form factor of the nucleus as a function of q.
This gives detailed information about the charge and
current distributions in the nucleus, and at q= u), the
results are in principle exactly the same as those of the
photo excitation. Coulomb excitation by heavy charged

particles is one such tool for probing nuclear structure,

but for light nuclei and high energy, the incident particle
may penetrate the Coulomb barrier and the structure effects
of the target cannot be isolated from the strong
interaction, where both of them are not known well. The
electron, on the other hand can penetrate deeply inside the
nucleus with only the electromagnetic force acting between
the electron and the nucleus.

The nucleus interacts with the electromagnetic field via
its charge and current densities. The interaction of the
electron with the charge distributions of the nucleus can be
considered in the first Born approximation as an exchange of
a virtual photon of angular momentum zero along the
direction of q. In this case the electron does not flip
spin, due to the conservation of angular momentum. This kind
of interaction is called Coulomb or longitudinal scattering.
The interaction of the electron with the spin and current
distributions of the nucleus gives rise to the transverse
part of the cross section, where the process can be
considered in the first Born approximation as an exchange of
a virtual photon of angular momentum :1 along the direction
of q. In the limit q —*0 the transition probabilities are
exactly the same as for real photons. The spin of the
electron in this case should flip to conserve angular
momentum. From parity and time reversal invariance one can
see that only electric multipoles can have longitudinal
components, while both electric and magnetic multipoles can

have transverse components. Transverse multipoles must have

angular momentum greater than zero, while longitudinal
multipoles can have angular momentum equal to or greater
than zero.

Longitudinal scattering gives information about the
charge distribution of the nuclear system, while transverse
scattering gives information about the current and
magnetization distributions of the nucleus. The transverse
part can be separated by doing experiments at 180°(Ref 11)
where the transverse form factors dominate the scattering
(equations (14), (15) and (17)). For data at other angles,
the longitudinal and transverse parts can be separated by
making a plot of the cross section against tan2(9/2) at
fixed momentum transfer and energy loss of the electron. The
slope of this plot gives rise to the transverse part, while
the intercept gives rise to the 10ngitudinal part. Such
plots are called Rosenbluth plots.

Our aim in this work is to analyze the electron
scattering data for different nuclei in the p-shell and
sd-shell with a microscopic theory which is not restricted
just to the discription of electron scattering, but has also
been widely used for explaining other static and kinematic
properties of nuclear structure.

We test the validity of the nuclear multi-particle
configuration-mixing shell model in two domains. We analyze
the magnetic elastic electron scattering data for p-shell
and sd-shell nuclei and both the electric and magnetic

inelastic electroexcitation of the even-parity states of

27A1. This nucleus is one of the most interesting systems
in this mass region, because it represents the point at
which nuclear deformations change from prolate ( positive
quadrapole moment for 26M9, or negative spectroscopic
quadrapole moment for the 2+ 26Mg state) to oblate
( negative quadrapole moment for 2881, or positive
spectroscopic quadrapole moment for the 2+ 2881 state)
(Ref 12). Recent measurements (Ref 13) of the many
even-parity states below 7 Mev have been carried out to high
momentum transfers. This allows the comparison of the
shell-model calculations for different states over a large
region of q.

A brief description of the shell-model calculations
is presented in Chapter II. The theoretical formulations of
the longitudinal and transverse form factors are presented
in Chapter III. Results of the elastic magnetic electron
scattering from p-shell and sd-shell nuclei and the
electroexcitation of the even-parity states of 27A1 are

presented and discussed in Chapters IV and V respectively.

CHAPTER II

THE NUCLEAR SHELL MODEL

II.1. Introduction

The configuration-mixing nuclear shell model used here
is a generalization of the classical shell model of Mayer
and Jensen (Ref 14). In the M-J model the nucleons occupy
the lowest available orbits of a spherical potential (which
parallels the nuclear matter distribution) according to
Pauli principle. The properties of the nucleus are
determined by the last unpaired nucleon. Only a few details
of nuclear spectroscopy can be explained by this simple
model. The configuration-mixing shell model (Ref 15) uses a
mixing of different orbits to create the eigenstates. In
this extended model it is still asumed that the nucleus
contains an inert core and active orbits in which the
valence nucleons are distributed according to Pauli
principle. For the sd-shell nuclei, 160 is assumed as an
inert core and no excitations are allowed out of these
filled orbits. The remaining orbits are n1j= Dds/2, 151/2
and 0d3/2 with the valence (A-16) nucleons distributed over
them within the limits of Pauli principle. In the p-shell
4He is assumed as an inert core, and the valence (A-4)

nucleons are distributed over the 0p3/2 -0p1/2 orbits within

the limits of Pauli principle. The problem of shell-model
calculations lies in the specification of the one-body and
residual two-body interactions (Ref 16). The eigenstates of
these interactions are obtained by diagonalization of the
matrices of many-nucleon energy matrix elements. A new
empirical Hamiltonian has been formulated (Ref 17) for the
complete A= 17-39 region. The wave functions obtained from
the diagonalization of this new Hamiltonian are used to
calculate the matrix elements of the sd-shell one—body
operators corresponding to the M1, M3 and M5 moments and
elastic magnetic electron scattering form factors of the
odd-mass nuclei from A= 17-39, and for the inelastic
electromagnetic multipole excitations of 27Al. In the p-shell
we use the eigenstates of the UP " Universal P" interaction
of Chitwood and Wildenthal (Ref 18) and of the Cohen-Kurath
interaction (Ref 19). Single—nucleon wave functions obtained
from either the harmonic oscillator (HO) potential or
Woods-Saxon (WS) potential are combined with these matrix
elements of one-body Operators to create "model-space"
transition densities.

As mentioned before, excitations out of the model space
are excluded from the wave functions we use. However, from
the physical point of view, such excitations must occur at
some level. Nuclear properties such as transition rates
cannot be reproduced properly by the model-space wave

functions if the properties of neutrons and protons are the

same as in free space. The shell-model wave functions have
to be renormalized in order to include such
"core-polarization" effects in describing different nuclear
properties. Renormalizations of the model-space wave
functions can be achieved by introducing effective operators
(Ref 20). For the electric multipole operators additional
charge can be added to the charges of the model space
neutrons and protons to form effective charges. Effective
charges for the protons and neutrons can be justified from
the first-order perturbation theory in terms of 1p-1h
transitions (Ref 21 Section 16.7 and Ref 20) to the giant
electric resonances. With effective charges, shell-model
wave functions can explain the observed values of electric
transition rates. Adding these ingredients to the
model-space transition densities give total transition
densities that can be used in describing different electron
scattering processes. Similar renormalizations for the
magnetic multipole operators involve the introduction of
effective 9 factors.

The details of the calculations of the matrix elements
of one-body opertors are presented in Section 11.2. Two
different models for the single-nucleon wave functions are

described in Section II.3.

10

II.2. Matrix elements of one-body operators

The one—body Operator matrix elements are obtained by
taking the matrix elements of a one—body tensor operator of
rank L between the eigenstates linT > of the interaction
used. These one—body tensor operators of rank L are
expressed in second quantization notation in terms of the
reduced matrix elements between the single-nucleon states
(Ref 21, p. 318)

(L,p/n)
[ a+(j) G>§(j')]

0(L,p/n)=: SNME(O,L,j,j',p/n) W

51'

 

( 3)

The entire set of quantum numbers (n,1,j) are abbreviated by
j. The operators a+(j)(§(j')) are the creation
(annihilation) operators of a neutron or proton in the
single state j (j').

The reduced matrix elements of the tensor operator OL

are obtained according to

<f|| O(L,p/n)|li>=:£OBDM(i,f,L,j,j',p/n) SNME(O,L,j,j',p{n))
4
51'

where the one-body density matrix (OBDM) is given by

1 . .. ., “up/n) .
<f||[a (J)®a(3 )] l|1>

OBDM(i,f,L,j,j',p/n)= ( 5)
F2L+l)

and

 

11

SNME(O,L,j,j',p/n)= <j||O(L,p/n)||j'> ( 6)

We abbreviate the initial/final states (Ai/f, Zi/f, vi/f,
Ji/f, Ti/f) by i/f.

Since the isospin associated with the shell-model wave
functions is a good quantum number, it is convenient to
calculate the OBDM in terms of the isospin—reduced matrix

elements (see the Appendix)

OBDM(p/n)=(-l) xVE' OBDM (AT=0)/2
-Tz 0 T2

Tf- Tz Tf 1 Ti
(+/-) (-1) xVE' OBDM (AT=l)/2
-Tz 0 T2
( 7)

where OBDM(AT) is given by

<f| l|[a+(j) at 5(j')](L'AT)I l|i>
OBDM(i,f,L,j,j',AT)= ( 8)

‘V(2AT+1)V(2L+1)

The triple bars mean that the matrix elements are reduced in

 

both spin and isospin spaces. The values OBDM(AT) are given
in Tables IV.1 and IV.2 for the ground states of stable
sd-shell and p-shell nuclei, respectively, and in Table v.1
for all the excited states of 27A1 considered in this

work. The occupation probabilities for 27A1 are given in

Table v.2.

12

II.3. Radial components of single-nucleon wave functions
The radial components of the single-nucleon wave
functions used here are obtained from two different
potentials, the harmonic oscillator (HO) potential,
characterized by the size parameter b (b2= 41.65/h40, and
the Woods-Saxon (WS) potential. The size parameters of the
HO potentials are set to the values brms which reproduce
the root-mean-square (rms) charge radii of the sd-shell
nuclei (Ref 22) and the p-shell nuclei. The radial
components of the single-nucleon wave functions of the WS

potential are obtained by solving the equations (Ref 23)

h2 d2 h2!((+1)
[ - + + U(r) I R(j,r)= G R(j,r)
2H dr2 2 Hr2
' ( 9)

 

 

where ll= m(p/n)(A-l)/A is the reduced mass. The potential

U(r) contains central, spin-orbit and Coulomb parts (Ref 24)
U(r)= V(r) + Vso(r) <'?.'3> + Gpn VCou1(r) (10)

where 6pn is equal to 1 for protons and 0 for neutrons.

These three components of the potential are

V(r)=V(p/n) [1 + exp(r—R(p/n)/a(p/n)]_l (11)

VCou1(r)=l

 

l3

Ze2
{3 - (r/Rc)2], r s RC

 

212C

Ze2

 

r

1 d

Vso(r)= Vso-——- ——— [1 + exp(r-Rso)/aso]

r dr

(l2)

(13)

where V(p/n), R(p/n) and a(p/n) are the well depth, radius

and diffuseness respectively. Their values and the values

for RC=VE/3 x the experimental rms charge radius,

V50: 12 Mev, R50: 1.1113”3 and 350: 0.65 fm are taken from

Ref 25.

CHAPTER III

ELECTRON SCATTERING

III.1. Introduction

The differential cross section for the scattering of an
electron of initial energy E1 through an angle 9, from a
nucleus of mass M and charge 2 and angular momentum J1, is
given in the one-photon exchange approximation by

(Ref 9, Ref 10, Ref 26)

 

 

d0" do
=( )Mott n E F2(L,q), (14)
dS2 as: L
d0"

 

where ( )Mott is the Mott scattering cross section of a

relativistic electron from a spinless point charge,

 

 

dcr Zacose/Z 2
( )Mott= [ 2 J p (15)
dSZ 2Eisin 8/2
a is the fine structure constant and n is the nuclear
recoil factor,
n = [ 1 + (2E1/M) sin2( /2)]-1 (15)

14

15

The form factor F2 is the sum of "longitudinal" FL2 and

"transverse" FTZ, terms:

4 2

 

q q
F2(L,q)=——E- FL2 +[ “ + tan2(8/2) FT?
q 2q
(17)
where the four—momentum transfer q“ is given by
qi = q2 _ (Bi - Ef)2, (18)
where
2= . - 2 . _ 2
q 4E1Ef S1n (8/2) + ( E1 Ef) (19)

and E1 and Ef denote, respectively the initial and final
total energies of the incident and scattered electron. In
the above equations we use h= c= 1.

The single-nucleon form factors for the longitudinal
electric and the transverse magnetic and electric scattering
are presented in Sections 111.2, 111.3 and 111.4
respectively. The multi-particle form factors are discussed
in Section 111.5. Calculations of the p-shell and sd-shell
transition densities are presented in Section 111.6.
Corrections to the electron scattering form factors are
given in Section 111.7. A derivation of a conversion factor
by which a simultaneous display can be obtained from both
the form factor and matrix elements at zero momentum

transfer is presented in Section 111.8.

16

111.2. Single-nucleon form factor for the longitudinal

operator

The interaction of the electron with the charge
distribution of the nucleus gives rise to the longitudinal

or Coulomb scattering. The Coulomb multipole operator is

defined (Ref 9) by

LLMC(CI)= fd? jL(qr) YLM‘ 91-) P(?) (20)

where P(?) is the charge density operator, which is
considered in the single-particle model as a sum of the

charges of all the nucleons,
pp/n(?)= 2M? - Pk) (21)

A(p/n)= Z/N, the number of protons/neutrons in the nucleus.

In the single-particle model, equation (20) reduces to

Z LLMC(p/n,q,?k)=: jL(qu) YLM( or]: (22)
k k

The reduced single-particle form factor of the Coulomb

operator is given by

fC.p/n(L.q)= <j||LLC(p/n,q,F)IIj'>

17

= /~dr r2 hLC(j,j',r) jL(qr) (23)

C

The radial function hL is given by

hLC(j,j',r)= (jllYLllj') R(j,r) R(j',r), (24)

with
(jllYLllj')= PL(E,2,I') CL(j,j') (25)

where the brackets (II) mean that the integration is taken
over the angles only. The coefficient PL(E,I,£') is the
electric parity-selection-rule operator which guarantees the
correct parity for the Coulomb operator (Appendix A.3e5 of

Ref 21)

l I f'
PL(E,F,I')=_ [1 + (-) + +L] ' (26)
2

 

- (2j+l)(2L+l)(2j'+1) 1/2
ct<j.j'>= <-1)J+1/2 [ 1

4n
3' L J"

x (27)
1/2 0 -l/2

and R(j,r) is the radial component of the single-nucleon wave

functions.

18

111.3. Single-nucleon form factor for the transverse

magnetic Operator

The transverse form factors arise from the interaction
Of the scattering electron with the current and
magnetization distributions Of the nucleus. The transverse
form factor is composed Of electric and magnetic terms. The

multipole magnetic Operator is given (Ref 9) by

TLMmagiq)=]aI-’ RLLM<q5> . 363) (23)

The Operator ML'LM(q,?) is defined by
ML'LM(q,?)= jL'(qr) ?L'LM( Gr) (29)

where EL'L1( 9r) is the vector spherical harmonic

 

{Emu 91-h: <L'M'lqlLM> Ymm 9,) eq (30)
M',q
l
éil=; (éx j; i éy) (31)
2
é0= éz (32)

The factor 3(?) is the sum Of the convection current, 3C, and

_+
the magnetization current, Jm, Of the nucleus, given by

= 3,4?) + {7’ x i1 (F) (33)

where the subscripts c and m stand for the convection and
magnetization parts Of the current respectively, and 3(?)
is the magnetization density Operator.

In the single-particle model, the convection current

and magnetization densities are given (Ref 26) by

 

A(p/n)
_. _, eh 1 _. _. —> 1
JCIP/n(r)= . g! p/n Zléh‘ — rk) Vk ‘ (34)
21mp symm.
k=l
A(p/n)
_) Eh -> -) ->
up/n(?)=—— (1/2)gsp/In Z 6(r - rk) 6;. (35)
2mp c

k=l

where mp is the proton mass and 3k are the Pauli matrices.
Using equations (34) and (35), the multipole magnetic

Operator in the single-particle model reduces to (Ref 9)

 

 

—) .1 l —>
:TLMmag(p/n,q,rk)= q(ieh/2mpc) -2 g! p/n MLLM(q,rk) ._ Vk
q
k k
L+1 1 1 L’ 1 a
+ (1/2 gsP/n): ( ) MLL-1M(q,rk)- < ) MLL+1M(q,rk) .3
2L+1 2L+l k
k (36).

where g! and gs are the orbital and spin 9 factors Of the

20

nucleons.

The single-nucleon form factor Of the magnetic Operator
can be reduced tO a radial function using relationships
between spherical Bessel functions (Ref 10 Appendix A, for

example) and integration by parts,

fT,p/nm39(L.q) (jllTLmag(p/n,q,?)l|j'>

ieh/Zmpc

xu/dr r2[glp/n hLmag(c,j,j',r) jL(qr)

+ esp/n hLmag(m.j,j'.r) 1L(qr)

(37)
where the radial functions Of the convection part (c) and
magnetization part (m) are given (Ref 27 and Ref 28) by

hLmag(C.j.j',r)= CL(jrj') PL(M.I.I') AL(j.j') [L(L+l)]_l/2
x (l/r) R(j,r)R(j',r) (38)

l

 

hLmag(mfj'j'pr)= —[L(L+l)]-l/2 CL(jrj') PL(Mrlr I.)
2
d
x B(j,j') [ R(j,r)R(j',r1
dr

+ B(j,j')-L(L+l)] (l/r) R(j,r)R(j',r) (39)

21

where R(j,r) are the single-particle radial wave functions,

and AL and B are numerical coefficients given by

AL(j,j')= [ 1 + B(j,j')/L][l-B(j,j')/(L+l)] (40)
B(j,j')= 2+ D(j.e)+ D(j'.r') (41)

The numerical coefficient CL is given in equation (27) above,

and the coefficients PL and D are given by

l I ['
PL(M,I,I')=—— 1 + (-1) + HM] (42)
2

D(i,k)= i(i+1)- k(k+1)- 3/4 (43)

111.4. Single-nucleon form factor for the transverse

electric Operator

The transverse electric Operator is given (Ref 9) by

1 —) -> -> —) -> ->
TLMel(q)=-—-— at [w MLLM(q,r) . J(r) (44)
q

Using the definition Of the vector spherical harmonics
§LLMK 9r) with the vector identities (Appendix A.5 Of

Ref 21)

22

—> —> —> -> —> -) a -)
Vx (L M1‘M(r))=iq2 t- MLM(r) + iv(——— r Mum) ) (45)

 

8r
3'. (a A )= a 5;. A + E'a. A , (46)
the transverse electric Operator becomes
TLMel(Q)= TLMell(q) + TLMe12(q) (47)
where -
i _. . _, _. ..
Twell<q)= [L<L+1)1'1/2far {-t-d— r Mum-)1} v. JC (48)
q Br
and
i —» —> »
TLM912(q)=—— [L<t+1)1'1/2far { q2 r Mum). Jc
q
+ q2 FMLMQ). (v x (1)} (49)
and where
MLMG) = jL(qr) rm 9,.) (50)

The nuclear current is conserved, and the continuity
equation can be used to determine the single-nucleon matrix

elements Of the transverse electric Operator (Ref 26)

 

 

all“ 89 —> -+ —> ->
= +V.(JC+VxP)=O (51)
3x“ 8t
_. .. ie ..
V. Jc =- ;— ( ‘i- 6f) 9(1‘) (52)

where 61 and 6f are the single-particle energies Of the
initial and final states respectively and P(?) is the
charge transition density.

In the single-particle model, where the charge, current
and magnetization densities are defined in equations (21),
(34) and (35), respectively, the reduced single-nucleon
matrix elements Of the transverse electric Operator can be

written as

 

fT'p/nel(qu)= fT'p/nell(L,q) + fT'p/nEI2(L,q),
where
l l 2 11
fT'p/ne 1(L,Q)= “ (if r hLe (C,j,j',1‘) jL(qr)
q
(53)
and
2 en 2 12
fT'p/ne1 (L,q)= q jfdr r [gyp/n hLe (c,j,j',r) jL(qr)
2m c
P

+ asp/n ht912<m.j.j'.r) jt<qr)

(54)

e11

and where the radial functions hL are given by

24

 

 

 

( ‘°- 6 )
hLe11(c.j.j'.r)= 1 f [L(L+1)]‘1/2
hc
a
x [2 hLC(j,j',r) + r hLC(j,j',r)],
dr
(55)
a
hLelz(c.j.j'.r)= [L(L+1)1‘1/2 r R(j,r)——-R(j',r)
dr
d
- [———R(j,r)] R(j'.r)] (56)
dr
and
eh l
hLelz(m.j,j'.r)= PL(E,I,E')[ D(j.() - D(j',c') 1
2mpc 2

x[ L(L+1)J‘1/2 CL(j,j') R(j,r) R(j'.r) (57)

The radial function hLC is given by equation (24) above,
and the coefficients PL, D(j,£) and CL(j,j') are

defined in equations (42), (43) and (27) respectively.

111.5. Multi-nucleon form factors

The multi-nucleon form factor for Coulomb scattering

is

FC'p/n(L,q)= dr r2 HL'p/nc(i,f,r) jL(qr), (58)
with the nuclear matrix elements HLC given by
HL'p/nc(i,f,r)= :OBDM(i,f,L,j,j',p/n) hLC(j-,j',r), (59)

is"

25

where the one-body density matrix (OBDM) is given by
equation ( 7) and the single-particle matrix elements hLC
are given by equation (24).

The multi-nucleon form factor for the transverse

magnetic scattering is

ieh

 

FT’p/nma9(L,q)= [ ng/n [2dr r2 HL'p/nmag(c,j,j',r) jL(qr)

2mpc
+ gsp/n d/Pr r2 HL,p/nmag(m.j.j',r) jL(qr) (60)
where the multi-particle matrix elements HLmag are given by

HL’p/nmag(c/m,i,f,r)= Z OBDM(i,f,L,j,j',p/n) hLmag(c/m,j,j ' ,r)
3'3" (61)

The one-body density matrix (OBDM) is given by equation ( 7)
and the single-particle matrix elements hLmag are given by
equations (38) and (39) for the convection and magnetization
currents, respectively. The sums in this case extend over
all the valence orbits.

The multi-nucleon form factor for the transverse

electric scattering are

FT'p/n81(L,q)= FT'p/n611(L,q) + FT'p/ne12(L,q), (62)

where

26

 

l
FT'p/nell(L,q)= -———:/dr r2 HL,p/n911(c,i,f,r) jL(qr) (63)
q
and
eh
FT'p/nelz(L,q)=2 q gIP/n der r2 HL'p/n(c,i,f,r) jL(qr)
m c
P

+ gsp/n J[dr r2 HLe12(m,i,f,r) jL(qr)

(64)

e11

and where the nuclear matrix elements HL are given by

(E' - Ef )
HL911(c,i,f,r)= 1 [L(L+l)]"1/2
hc

 

d
HLC(i,f,r)]

 

x [2 HLC(i,f,r) + r
dr
(65)

and E1 and Ef are the initial and final energies Of the
nuclear states respectively. The multi—nucleon matrix
elements HLC are given in equation (59). The matrix elements

HLelZ are given by

HL912(c/m,i,f,r)= :OBDM(i,f,j,j',p/n) hL912(c/m,j,j',r) (66)
11'

e12

where the radial functions hL are given by equations (56)

and (57) for the convection and magnetization currents

27

respectively. The sums extend over all the valence orbits.

111.6. p-shell and sd-shell transition densities

The multi-particle form factor Of the Coulomb operator
depends on the transition densities HLC(i,f,r). The
transition density can be divided into two parts, one
depending on the model-space transition density and the
other depends on the core-polarization transition density.

The model spaces for the p-shell and sd-shell are
defined by the complete set Of states contructed from the
orbits 0p3/2 -0p1/2 and the orbits 0d5/2 -1s1/2— 0d3/2
respectively. The model-space one-body Operator matrix
elements are Obtained by taking the matrix elements Of a
one-body tensor Operator between the eigenstates Of the

interaction used

GL’p/n(i,f,r)= ZOBDM(i,f,L,j,j',p/n) hLC(j,j',r) (67)
3'3"

where the OBDM are given by equation ( 7) and the hLC are

given by equation (24). The sums extend over all the

valence orbits for L>0. For L=0, the sums includes all the

orbits in the core.

The model-space transition densities G deal with
nucleons in the active orbits only and exclude any effects
from the core. Many nuclear properties cannot be
quantitatively described by using just the model—space

transition density, and the effects from outside the model

28

space must be taken into consideration. The contribution Of
the core tO the transition rates can be explained
(Ref 29 p. 334) as a deformation Of the core when the active
nucleons make a transition from one state to another. The
protons in the closed shell are polarized and their orbits
are slightly distorted which contributes to the total
transition rates by an amount that corresponds to the
Observed values. Mass and state-independent effective
charges have been introduced empirically (Ref 30 and
Ref 25) for the model-space protons and neutrons which are
able tO reproduce the Observed B(E2) values in the sd-shell.
Two models for the effective charge are considered here
for the core-polarization transition densities. One can
assume the transition density tO be proportional tO the
model-space transition density. We will call this the

"valence" (V) model

cL,p/nV(r)= GL,p/n(r) (as)

The Other model is based on a multipole-multipole
interaction which connects the ground state tO the
L—multipole nhu) giant resonance. The shape Of the
core-polarization transition density in this case is given
by the Tassie (T) model (Ref 31)

L-l d
CLIP/nT(r)= r ;— GL=0'p/n(r) (69)

29

where GL=o(r) is the ground state density given in equation
(67).

The total transition densities are Obtained by
combining the model-space transition densities with each Of
the twO models for the radial distribution for the core
polarization. The total transition density corresponding to

first-order perturbation theory is given by
HL’p/nc(i,f,r)= GL’p/n(i,f,r)+ NC'p/n CL'p/n(i,f,r) (70)

where G(r) is the model-space transition density given in
equation (67) above, and C(r) is the core-polarization
transition density given by equations (68) and (69) for
the valence and the Tassie models, respectively, and NC is
a normalization constant to be determined from the matrix

elements Of the L-multipole gamma-ray-transition Operator

ML,
ML=~/.rL HLC (r) r2 dr (71)

The rL radial integrals Of model-space and core-polarization

transition densities are given by

GL=./flrL GL(r) r2 dr (72)

cL=f rL CL(r) r?- dr (73)

30

The gamma-ray-transition matrix elements defined at
q= Ef-Ei, expressed in terms Of the effective charges, are

given (Ref 22) by

ML,p= GL,p(l+6pp)+ GL,n5pn (74)

where 5cv is the polarization charge that arises from the
interaction Of the valence nucleons (v) with the core
nucleons (c). Their relations to the conventional effective

charges ep and en are given by

Opp: 6nn= ep‘l (76)

5pn= 6np= en (77)

It has been found (Ref 30 and Ref 22), that for the
complete sd-shell model space, average values Of the
effective charges for E2 and E4 are close tO ep+en= 1.7e
and ep+en= 2.0e respectively. These values are used
throughout the calculations considered in this work,
together with the isovector effective—charge defined by
ep - en= 1e (Ref 25).

The normalization constants Of equation (70) are
Obtained by comparing the integrals Of both sides Of

equations (74) and (75),

31
Nc'p= (PppGp + PpnGn)/Cp (78)
Nc,n= (5nnGn + 6anp)/Cn (79)

The electromagnetic transition strength B(L,Ji,Jf)

is given by
1
B(L,J1,Jf)= --—-— lML,p|2 (80)
2Ji+l

111.7. Corrections to the electron scattering

The shell-model wave functions used in describing
transition densities give rise tO additional non-physical
excited states called spurious states, due to the fact that
the interaction potential represents an average potential
with respect tO a fixed origin. The Hamiltonian in this
case is in general not translational invariant, and the
motion Of the center Of mass is responsible for these
spurious states. These states can be isolated from the exact
Observed states Of the intrinsic motion Of the nucleons in
the case Of the HO potential (Ref 10), where the Hamiltonian
can be separated into two parts, one representing the motion
Of the center Of mass, and the Other representing the
intrinsic motion Of the nucleus. This is done by including a
factor Gcm given by the H0 in the nuclear form factor.

The center Of mass correction factor Gcm is (Ref 32)

Gcm(q)= exp(q2 b2 /4A) (81)

32

where b is the oscillator length parameter. We use b= be

obtained from a global formula for the oscillator length

h we: 41.46/bo2 = 45 A'l/3 - 25 1‘2/3 , (82)

where A is the total number Of nucleons in the nucleus.

The form factor discussed in the previous sections
assumed the nucleons as point particles. A correction due to
the finite nucleon size should be considered. For
longitudinal scattering, the form factor becomes

(Ref 33)

FC(L,q)= NF Fc'p(L,q) Gfs'p(q)+ Fc'n(L,q) Gfs'n(Q)

where Fc'p/n(L,q) is given by equation (58), and the
free—nucleon form factors Gfs'p and Gfs'n are taken

from Ref 34, including the small Darwin-Foldy relativistic
correction in Gfs (Ref 35). The normalization factor NF

is given by

 

1 4v 1/2
NF =—[ (84)
Z 2J1 + 1
For inelastic scattering we use the approximation
FC, FC,n .
P = L“ (85)

33

where the number Of protons and neutrons in the nucleus are
denoted by z and N respectively. The Coulomb form factor

reduces to

FC(L,q)= NF Fc,p(t,q) [cfs,p(q)+ N/Z cfs,n(q)] Gcm(q)
L>0 (86)

For the transverse form factor, the correction for the
finite nucleon size Gfs described in Ref 36 is used

for both protons and neutrons. The transverse form factors

are

FTmag'el(qu)= NF [FTerag'el(qu) + FTrnmag'el(L'q))

X Gcm(q) Gfs'p(q)
(87)

where FT,p/nmag'el(L,q) is given by equation (60) and (62),
respectively and Gcm(q) is the center Of mass correction

given in equation (81).

The total form factor is Obtained from the sum over all

form-factor multipoles F2(L,q) given in equation (17),
F2(q)=:z F2(L,q) (as)
L

where L is determined from the parity selection rule

Mel = (-1)L (89)

Ammag= (-1)L+1 (90)

34

From the parity and time reversal invariance one can see
that only electric multipoles can have longitudinal
components, while both electric and magnetic multipoles can
have transverse components. Transverse multipoles must have
angular momentum greater than zerO, while longitudinal
multipoles can have angular momentum equal tO or greater
than zero. For elastic scattering, only even multipoles
contribute to the longitudinal scattering, while only Odd
multipoles contribute tO the magnetic scattering. There are
no contributions from the transverse electric elastic
scattering.

The form factors discussed in the previous sections are
formulated in terms Of the first Born approximation, in
which the initial and final states Of the electron are pure
plane waves. This type Of approximation is called the plane
wave Born approximation (PWBA). For nuclei in which zcr<< 1,
the PWBA is expected tO describe the electron scattering
data very well, except in the region Of the diffraction
minima where the PWBA goes tO zero. An improvement tO the
first Born approximation can be Obtained by including the
effects Of the distortion Of the plane wave by the Coulomb
field. This higher-order effect is.incorporated into the
distorted wave Born approximation (DWBA). TO first order,
the effect Of the Coulomb field is tO increase the momentum
transfered tO the nucleus and an effective momentum transfer
can be used tO include these effects. The effective momentum

transfer qeff is related to q (Ref 10) by

35

 

[ V(r) J
Qeff= q l - -—-—-- (91)
Ei(MeV)
fC Z e2
qeff= q l + (92)
Ei(MeV) RC (fm)

where RC =V573-<r2>1/2 , e2=<rhc= 1.44 MeV fm and a'is the
fine structure constant. The value Of fC is determined
from the Coulomb potential energy
-2 e2
V(r)=-—— [ 3Rc2 - r2 1 (93)
2 RC3
where fc= 3/2 if the scattering occurs at the center Of the
nucleus, and 1.0 if the scattering occurs at the surface. An
excellent overlap between DWBA and PWBA is Obtained (Ref 37)
if the experimental data are plotted as a function Of qeff
for the elastic magnetic scattering with fc= 1.2 and the
theoretical form factors calculated in PWBA are plotted
against q. In Figure 111.1 we show the difference between
the DWBA, calculated with the code Duels (Ref 38), and the
PWBA for the elastic magnetic scattering Of 27A1. The DWBA
and the experimental data are plotted versus qeff with
fc= 1.2, while the PWBA are plotted vs. q. NO significant
differences appear between the DWBA and the PWBA
calculations. All the magnetic elastic scattering form
factors presented in this work are calculated in PWBA and

plotted vs. q with the data plotted vs. qeff with

36

 

MSU‘83‘336
i ”r T T 1 ‘1 I T v 1* l 1 I I
' 27 ‘
Al

 

 

 

 

7 1 l J I l
“5.0 I 2 3
(WW)

Figure 111.17 DWBA form factor for the elastic magnetic
sdattering from A1 (solid line) in comparison with the PWBA
(dashed line). The calculations incorporate the
single-nucleon wave functions Of the HO potential Of b=brms-
The data are taken from Ref 40 (circles), Ref 41 (triangles)

and Ref 42 (squares).

37

 

 

 

 

 

 

MSU-83337
L,O.84 Mev, I/2+ 27A)
I0'3- _
' O . -(
(0'4
Rio-5
3
“(’1
(0'6
IO’7
lcy8.. |l _
'1
'0'9 4 l 1 i 1 1
o I 1| 2 3
q(fm )

FiguSe 111.2. DWBA form factor for the 0.844 MeV,l/2+
state Of A1 (solid line) in comparison with the PWBA for
*the same state (dashed line). The calculations incorporate
the single—nucleon wave functions Of the HO potential Of
b=brms. The data are taken from Ref 13.

38

fc=l.2.

The value Of fc=3/2 has been widely used for the elastic
and inelastic electric scattering. In Figure 111.2 we show
the difference between the DWBA calculated with the code
Duels (Ref 38) for the 1/2+ state (0.844 MeV) Of 27A1 and
the PWBA. The DWBA and the experimental data are plotted
versus qeff with fc= 3/2, while the PWBA are plotted vs.

q. The difference between these twO calculations are
significant only in the region Of the first diffraction
minimum, where the PWBA goes tO zero, and no significant
differences appear at other values Of the momentum transfer.
All the excited states Of 27A1 are calculated also in PWBA
and plotted vs. q with the data plotted vs. qeff with fc= 3/2.
The Coulomb elastic scattering Of 27Al is calculated in

DWBA Of the MIT elastic phase-shift code (Ref 39).

111.8. Conversion Of form factors to q-dependent matrix
elements M(q)

We present here a representation for form factors in
which we can display simultaneously both the form factor and
matrix elements at zero momentum transfer. Formally, we wish
to utilyze coversion functions D(L,q) such that
M(q)= F(L,q)/D(L,q).

1n the limit Of small momentum transfer, the PWBA
longitudinal form factor is related tO the gamma-transition

matrix element ML'p defined in equation (74),

39

. 1 4n qL
11m FC(L,q)= -—— ML'p (94)
q-+0 Z 2J1 +1 (2L+l)!!

 

 

TO remove some Of the trivial q dependence at larger q
values, the form factor can be divided by the exponential
dependence exp(-b2 q2/4) which is contained in the HO
radial wave functions. The q- and L-dependent conversion

factor for the Coulomb scattering is written as

l 41T 2
DC(L,q)= qL exp(-bo2 q /4) (2L+1)!!
z 2J1 +1

Dcm(Qrb0 ) Dfs(Q) (95)

 

 

where Dcm and Dfs are the center Of mass and finite proton

size corrections, given by

Dcm(q,b0)= exp(bo2 q2/4A), (96)

and
nfs(q)= exp(-0.43 q2/4), (97)

where be is the harmonic oscillator length parameter
Obtained from the oscillator length given in equation (82).
For the transverse magnetic scattering, the form factor

in the small momentum transfer limit can be written as

1 4"
lim FTmag(q)= ———- -——————- C(L) M(L) qL (98)
q-+0 Z 2J1 +1

40

where M(L) is the magnetic multipole moment and C(L) is

given by

L -1
(2L+1)!! X - (99)

 

C(L)= [ 2mc/h
L+l

where
4W J1 1. J1

x= (100)
2L+l J1 0 -J1

 

for the elastic magnetic scattering and X=1 for the
inelastic magnetic scattering.

In the region Of small momentum transfer, the lowest M
multipole dominates the scattering, and a q- and L- dependent
conversion factor is chosen such that division Of the
inelastic magnetic form factor by this conversion factor
with L set equal to the lowest multipole gives the matrix

elements at q-+0.

Hence ,
l 4Tr
DTmag(L,q)= -———- ---qL exp(-b02 q2/4) C(L)
Z 2J1 +1

Dcm(q,b0) Dfs(q ) (101)

where Dcm and Dfs are the center Of mass and finite proton
size corrections given in equations (96) and (97),
respectively.

For the elastic magnetic scattering, we have chosen a

41

function which is proportional to q at low-q values and
proportional to quax at high-q values. Division Of the
form factors by this function yields results which clearly
display the individual contributions Of the different
multipoles and which have a slower variation with change in

q at region Of high momentum transfer. For this function we

have chosen the expression

Lmax+1 Lmax +1 -1

l -(q/a ) l -(q/a )
H(Lmax)q) -—-e + -—- (1- e )
q q max
(102)

where a is a numerical constant chosen tO be 1 and Lmax is
the highest multipole in the shell, Lmax‘ l, 3 and 5 in the
Os, 0p and ls-Od shells respectively. The complete q- and L-

dependent conversion factor is written as

 

l 41T
DTmag(Lmaxrq)= _\/ exp<-bo2 012/4) H<Lmax.q) C(L=1)
Z 2J1 +1

 

x Dcm(q,b0) Dfs(Q) (103)

where Dcm and Dfs are the center of mass and finite proton
size corrections given in equations (96) and (97).

The new representation for the form factor is

|F(q)|
M(q)= -——————-, (104)

D(L,q)

42

where D(L,q) is given in equations (95), (101) and (103)
with L=2 for the Coulomb, L=l for transverse magnetic and
L= Lmax for magnetic elastic scattering. For the magnetic
elastic scattering, the form factor M(q) is plotted vs. q,
and for the longitudinal scattering and transverse inelastic
scattering, the form factor M(q) is plotted vs. q2.

The two representations Of the form factors F2(q)
and M(q) are shown in Figure 111.3 and 111.4 for the elastic
magnetic scattering and longitudinal E2 transition Of

1/2+ state in 27Al, respectively.

43

 

 

 

 

 

 

 

 

 

 

 

MSU-83-3347
L, 0.84 MeV, l/2+
27A)
i
1
Q a
16° <
-9
I _1 4 J #L A
0o 1 2 3
Mint" )
22.5_ L, 0.84 MeV, l/2+ 4
20.0- .
l7.5r~
(5.0)- .(
Ehzs.
Z
IOO
1
ZS
500
2.5
0 14114 JLIJJJA_LLI
O l 2 3 4 5 6 7 8 9

q2(fm-2)

Figure III.3. Magnetic elastic electron scattering for
27A1 caiculated with the HO potential, presented in the twO
forms F (L,q) and M(q) as explained in section 111.8. The
magnetic multipoles contributing tO the scattering are Ml
(dotted line), M3 (dashed line) and M5 (dashed—dotted line).
The solid line represents the total incoherent sum Of these
three multiploles. The magnetic dipole moment is displayed
in the M(q) representation. The data are taken from
Ref 40 (circles), Ref 41 (triangles) and Ref 42 (squares).-

44

 

  
    
  

 

m-83-0l4
5 27A'
'0'“? "at E
z a
...: i :
C7
E05:— 3
: :
D

1 111111

  

 

Q
q
‘

J.

4L

.4».

4)-

d)-

 

 

 

 

Figure III.4. Longitudigal E2 form factor for the
0.844 Mev, 1/2+ state for Al calculated with the HO
potential, presented in the two forms F (L,q) and M(q) as
explained in section 111.8. The measured B(E2) value is-
displayed in the M(q) plot at q=0. The data are taken from

Ref 13.

CHAPTER IV

MAGNETIC ELASTIC ELECTRON SCATTERING

IV.1. Introduction

The configuration-mixing shell model used here allows
the mixing Of different orbits tO give the full basis
eigenstate. In this case, all the model—space valence
nucleons share in the scattering process, rather than just
the one unpaired nucleon as in the simple shell model. In
the calculation Of the one-body density matrix (OBDM), we
assume that the core is inert and only the motion Of the
valence nucleons need to be considered. However, it has been
shown that higher-order effects such as core polarization
and meson-exchange currents are very important and must be
taken into consideration (Ref 43, Ref 44, Ref 45 and Ref 46).
Assuming an effective two-body interaction between the core
and the valence nucleons, one can carry out microscopic
calculations tO include these effects. Such microscopic
calculations lead tO the introduction Of effective
single-nucleon matrix elements which are different from the
free-space values. Renormalization Of the free-space values
Of the single-nucleon matrix elements Of the different
Operators might take care Of the core-polarization effects.

The renormalization Of the single—nucleon matrix elements

45

46

might be approximated by introducing L-dependent effective 9
factors. The effects Of the meson-exchange current might be
explained by using radial parameters for the valence
nucleons different from those required tO match the rms
radius.

We compare the shell-model calculations with
experimental data for Odd-A ls-Od-shell and 0p-she11 nuclei.
GOOd data are available for the nuclei 170, 27Al and 39K.
Comprehensive comparisons are made for these nuclei using
different effective 9 factors for the different multipoles.
We plot the new representation Of the form factor M(q) vs. q
for all the cases considered in this study, which permits
display Of the magnetic dipole moment at q=0. We will use

the term " form factor" for the new representation M(q).

IV.2. Magnetic elastic scattering from 1/2+ nuclei

The stable Odd-A nuclei Of spin 1/2+ in the sd-shell
nuclei are 19F, 29Si and 31?. Only the M1 multipole
contributes to the magnetic scattering for these systems.

In Figure IV.1 we show the magnetic elastic form
factors for these nuclei using the HO radial wave functions
Of parameter b=brms~ The crosses, plusses, and solid lines
represent the calculations with the single-particle model,
the configuration-mixing shell model with free-nucleon 9
factors and the configuration-mixing shell model with

effective 9 factors, respectively. We use effective M1 9

47

Figure Ivl9' Fqgm factoge for the magnetic elastic
scattering Of Si and P calculated with the HO
radial wave functions Of b= brms- The cross signs
represent the calculations with the single-particle model
with free-nucleon 9 factors. The configuration-mixing
shell model calculations using free-space 9 factors and
effective M1 9 factors are represented by plus signs
and solid lines respectively. Theg values for the M1
effective 9 factors are gsp = 59,9 n: -3. 442, g! 9:1. 078
and g, n=-0. 044. The data2 or F gie taken from Ref 47
(circles). The data for Si and P are taken from
Ref 48 (triangles) and Ref 49 (circles).

Figure IV.1

2.5

2.0

48

MSU-83-3l3

 

 

 

 

 

 

 

 

49

factors resulting from a Chi-square fitting to the
magnetic dipole moments of all the stable odd-A sd-shell-
nuclei. These values are [gsp= 5., gs“ = -3.442, glp= 1.028,
gln= -0.044].

The single-particle picture of 19F is one proton in the
51/2 orbit. From Figure IV.1, we notice that with the
single-particle picture (cross signs) neither the magnetic
dipole moment nor the scattering data are reproduced
satisfactorily. The free-nucleon g factors-configuration
mixing shell model does explain the scattering data much
better than the single-particle model. However, the measured
magnetic dipole moment is not reproduced by the free-nucleon
9 factors. The effective 9 factors-configuration mixing
shell model explains the experimental data very well (solid
lines), and the measured magnetic dipole moment is
reproduced properly for 19F, but it slightly underestimated

for 2981 and 31P.

1V.3. Magnetic elastic scattering from other Odd-A sd-shell
nuclei

We calculate the magnetic elastic form factors for all
other stable sd-shell nuclei. We compare the results of the
single-nucleon shell model (cross signs) with the

configuration-mixing shell model. The calculations for 17O

I

25Mg and 27A1 are presented in Figure IV.2, those for 21Ne,

23Na and 33S are presented in in Figure 1V.3 and those for

50

35€1,37C1 and 37K are presented in Figure IV.4. The
configuration—mixing shell model form factors are calculated
with two different values Of the 9 factors. The lines
denoted by plusses represent the calculations with the
free-nucleon 9 factors for all the multipoles that
contribute to the scattering, while the solid lines
represent the calculations with effective M1 9 factors
obtained from the Chi-square fitting to the magnetic dipole
moments Of all the nuclei considered in this study. The
values for the effective 9 factors [ gsp= 5., gsn= -3.442,
gpp= 1.078 and g,n = -0.044] are used for the M1 multipole.
Also, the M3 contribution is quenched to 60% of the free-space
value. We use the free M5 9 factors for both cases. The
multipole decompositions M1 (dotted line), M3 (dashed lines)
and M5 (dashed-dotted lines) are those of the empirical 9
factors discussed above.

With the empirical g-factors, the agreement with the
measured magnetic dipole moments becomes worse for some
nuclei and improved for others, like 21Ne, 23Na, 25Mg, 35C1,
37C1 and 39K. A common feature noticed in the regions of
high momentum transfer (q > 2 fm’l) is that the data are
higher than the theory for those states which are dominated
by M3 and M5 multipoles. Also, the free-nucleon M3
contribution is an overestimate in the region where it is

important.

In the case Of 25Mg, good agreement with the measured

51

Figure IV12. Fggm factoEs for the magnetic elastic
scattering of 0, Mg and A1 calculated with the HO
radial wave functions of b= brms- The cross signs represent
the calculations with the single-particle model and
free-space 9 factors. The configuration—mixing shell model
calculations using free-space 9 factors are shown by
plus signs and those of effective M1 and M3 9 factors are
shown by solid lines. The values for the effective M1 9
factors are those used in Figure IV.1. The M3 contribution
is quenched to 60% Of the free-space value. We use the free
M5 9 factors for both cases. The decomposition of the
multipoles are M1 (dotted lines), M3 (dashed lines) and M5
(dashed-dotted lines), calculated with the empirical 9
factors discussed above. The dagg for 1 O are taken from
Ref 50 (circles). The data for Mg are taken from Ref 37
(squares) and Ref 51 (circles). The data for 2 A1 are taken
from Ref 40 (circles), Ref 41 (triangles) and Ref 42 (squares).

52

“SO 83 360

j ‘V V f v

 

as» 'Q) .

 

 

 

 

 

 

 

Figure IV.2

53

Figure 1V2}. F059 factor§3for the magnetic elastic
scattering of Ne, Na and S. The conventions of the
presentation are the same as given in the caption Of Figure
IV.2. The data for 3Na are taken from Ref 52 (circles).

 

. . . a imam
2.5- mm .
2.0- .
(.56 1

 

 

 

2.5- 23N°

x l/2
2.0» .

 

 

 

2.0 - . 4
(.5 ~ ‘1
(.o ‘

 

    

 

 

0.5
--Jn., ..
‘gj‘- “ 1 ..°';n‘j AA‘.....IO..I.IO
00 " I 3
a(fm ')

Figure 1V.3

55

Figure Ivgg’ F059 factor§9for the magnetic elastic
scattering Of C1, C1 and K. The conventions of the
presentation are the same as given in the caption of Figure
IV.2. The data for 9K are taken from Ref 53 (circles)
and Ref 54 (squares).

“SJ-8325)

 

 

I'D
).
A

 

r

—-----d

00.00.......‘
.0

A

 

2.5 -

 

+
+0'
+ 2 .
0.5 + I - . “1
I ......
I , '.
I .
‘ .0 00'.
0 ."’°"
c k t : “"‘A :

0
1

'

 

 

Figure IV.4

 

-------‘

00....“
o

 

57

dipole moment is obtained, but the quality of the form
factor data in the region of small- and medium-q values
limits the usefullness of the comparison with theory.

IV.4. Magnetic elastic scattering from 17O

The complete sd-shell space shell-model picture of 17O
is identical to the single-nucleon shell-model picture,
since this nucleus corresponds to only one neutron
outside the 16O inert core. All of the nuclear properties
are determined by this unpaired neutron in the model. The
calculated form factor with this model is shown in
Figure IV.5a and IV.5a' using single-nucleon radial wave
functions of the HO potential and the WS potential,
respectively. The free-space neutron 9 factors are used in
these calculations. From Figures IV.5, it can be concluded
that the M3 contribution is too large and that the
calculated form factor is too small in the region of q > 2
fm'1 .

As mentioned before, the core-polarization effects will
alter the results both of simple-single particle and full
sd-shell model calculations. We will assume that these
effects can be introduced in the form of L-dependent
effective 9 factors. The form factor calculated with
effective 9 factors for the M3 multipole of the neutron
equal to 60% of their free-space values are shown in Figure

IV.5b and IV.5b'. Quenching the M3 multipole to 60% of the

58

free-space value gives better agreement with the
experimental data in the region where M3 is important but
does not improve agreement between experiment and theory in
the region of high momentum transfer (q > 2.0 fm-l). The
discrepancy between theory and experimental data at high
momentum transfers has been discussed for medium-heavy
nuclei (Ref 55). Meson-exchange corrections (MEC)

effects are found to be important in this momentum transfers
region. Simple quenching of the M5 form factors
(dashed-dotted lines) will not help in resolving this
discrepancy.

In Figures IV.5c and IV.5c' we plot the form factors
calculated with the M3 multipole quenched to 60% of the
free-nucleon values and the rms radii of the valence orbit
reduced by 5% from those required to fit the rms charge
radii in the respective model for the single-nucleon wave
functions. With these reductions, enhancements of the form
factor are obtained at high-q values. Reduction of the rms
radius of the valence orbit also has small effects in the
region of small—q values. Best overall agreement between
theory and experiment is obtained with the 0.6 quenching of
the 9 factors for the M3 multipole and the smaller radial
size parameters.

It was found in studies of beta-decay (Ref 56) that the
spin 9 factor of the Gammotheller matrix elements should be

quenched 80% from the free-nucleon value have explored the

59

Figure IV.5. Form factor for magnetic elastic
electron scattering for O. The single-nucleon radial
wave functions used here are those of the HO potential of
b= brms (Figure IV.5a) and of the WS potential discused in
Section II.3 (Figure IV.5a'). Free-space values for the
neutron-g factors are used in both calculations. The
corresponding calculations with quenching the 9 factor of
the M3 contributions to 60% are shown in Figure IV.5b and
Figure IV.b'. Figure IV.5c and Figure IV.5c' are the same
as Figure IV.5b and IV.5b' except that the rms radius of
the valence orbit is reduced by 5%. The same conventions
are used for the different multipoles as in Figure IV.2.
The data are taken from Ref 50 (circles).

60

fiMSUX—OS-O45

 

"2'- .../s

 

I70. H O ..
rrm: V
free ‘i

#01.
° '70. ws

rmas
free

+

4

J

1

 

M(q)

 

 

 

 

 

 

Figure IV.5

 

61

Figure Ivié. Form factors for the magnetic elastic
scattering of O. The conentions of the
presentation are the same as given in the caption of Figure
IV.5c' except effective 9 factors for the M1 contributions
are used (gsn(eff)= 0.8xgsn, gln(eff) = g!n -.182).

62

 

MSUX-83-022
l.8r- (new ”0' WS 1
. ; 0
l6- 1 ”'25 5/° 4
° n
|4+- + 2 €35 )<<:t£3
' ° 921 - O.l8
LZP n3 .
:: t 95 xO,6
3:0- '. .
.2. 2* +
‘ 0.8- . o /.
’3 \
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I I ~\\ _ _'__ .
0.4- l / \ -
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0.2 / , . , . .\. V /d
c- 4 7° '

 

 

 

Figure IV.6

/ -'
(D£fl_¢:::::§ 1 L. 1, \\ T~
l 2

63

consequences of imposing this same quenching

upon the magnetic dipole gs . With this effective gs

for the M1 multipole, the measured magnetic dipole moment
cannot be reproduced without also reducing 9,. In Figure
Iv.6, we show the form factor calculated with the same WS
radial wave functions that used in Figure IV.5c'. Effective
9 factors for the M1 multipole [gsn(eff)= 0.8xgsn, g[n(eff)=
gin-.182] are used, which exactly reproduce the dipole
magnetic moment. It is seen that even with the magnetic
dipole moment reproduced exactly, the agreement with the
form factor data at small momentum transfer data is not as
good as was obtained with the free-space M1 9 factors,
(Figure IV.5c').

No major differences appear between the HO and the WS
potentials in describing the data except in the region of
high-q values, where the WS potential gives better agreement
with the experimental data than the H0 potential (Figure

IV.5c' and IV.5c).

IV.5 Magnetic elastic scattering from 27A1

In the single-particle model, 28Si is considered as a
closed shell, and 27A1 as one proton hole in the d5/2 orbit.
The form factor calculated with this model is shown in
Figure IV.7a using the HO radial wave functions with the
value of the "b" parameter fixed to reproduce the rms charge

radius. This model overestimates the magnetic dipole moment

64

Figure IV.7. Form fastors for magnetic elastic
electron scattering for Al. Calculations with the HO
potential of b= brms: assuming only one proton hole in the
d5/2 orbit are shown in Figure IV.7a. The
configuration-mixing contributions of the HO potential of
b=brms and of b reduced 9% from brms are shown in Figure
IV.7b and Figure IV.7c respectively. Free-nucleon values
for the 9 factor are used in these calculations. The
effect of reducing b-value of the HO potential by 5% is
shown in Figure IV.7d using different values for 9 factors
for M1 contributions to get the exact ma netic dipole
moments (gsp(eff)= 0.8xgsp, g,p(eff)= g, xl.25). Figure IV.7e
is the same as Figure IV.7d except the proton 9 factor of
M3 contributions is quenched to 60% of the free-space value.
Figure IV.9f is the same as Figure IV.7e except the wave
functions of the WS potential whose valence orbits rms radius
reduced 5% are used in place of the HO wave functions. The
same conventions are used for the different multipoles as in
Figure IV.2. The data are taken from Ref 40 (circles),
Ref 41 (triangles) and Ref 42 (squares).

65

 

wo#%
8......9
. x .
saggy.”
. :3
95

         

   

 

:1va U
N

“""‘.H‘ “ .‘I‘

'
Ill- 0
I
O I.

8......
8.. ...Mo
3 ...”o
*0 u :5.
=2
OI

d

‘0

0‘. \

'0” '\ \
O

‘\

\\

    

~.>H ouaomm

 

 

O N . O
oloerhqluu¢1ao. . \.\‘1\

 

 

 

 

’ ' h
“ " ‘i“ ‘ ‘ ‘
0'. ll 0 o o o o
O O O " O
o ’
o o o I

 

 

 

 

 

accusing.

66

by a factor of 1.2. The form factor data are also not
explained satisfactorily. In Figure IV.7b, we present the
form factor calculated with the multi-particle shell model
using the HO radial wave functions whose b parameter is
fixed to reproduce the rms charge radius. The
configuration-mixing shell model reduces the M1 form factor
in the region of small-q values by a factor of 1.4 from that
of the single-particle picture. It similarly reduces the M3
form factor by a factor of 2. It yields good agreement with
the measured magnetic dipole moment. Overall, the
configuration-mixing shell model gives much better agreement
with the experimental data than the single-particle model
discussed above. However, in the region of low momentum
transfer, the theoretical form factor falls below the values
of the data. It would seem that the M1 multipole needs to be
renormalized to get agreement with those data but this would
tend to destroy the agreement with the measured magnetic
dipole moment. The agreement with the low-q data can be
improved by reducing the radial size of the valence nucleons
from that required by the rms radius, as shown in Figure
IV.7c.

In Figure IV.7d, we use effective M1 9 factors
((gsp(eff)= 0.8xgsp, g£n(eff)=g£n+0.247) to reproduce the
measured magnetic dipole moment. We use HO radial wave
functions of radial size 5% smaller than that required by

the rms radius. An enhancement of the M1 form factor at the

67

region of low-q values is thereby obtained, such that the
experimental data at this region of q are well decribed.
Also, an enhancement of the M5 multipole is obtained at
high-q values. The corresponding calculation in which the M3
form factor is quenched to 60% of the free-space value is
shown in Figure IV.7e. In Figure IV.7f we show the
calculation in which the quenching factors are the same as
in Figure IV.7e, but in which single-nucleon wave functions
of the WS potential whose valence nucleons radial size
parameter reduced 5% from that required by the rms radius
are used in place of the HO wave functions. From Figures IVe
and IVf one can see that an overall agreement can be
obtained between theory and the experimental data and that
there are no significant differences between the results of
HO and the WS potentials throughout the momentum-transfer
regions studied.

Even though both protons and neutrons are active
outside the closed 160 core in the multi-particle
configuration-mixing model of 27Al, the magnetic scattering
is dominated by protons (Figure IV.8a). The contribution of
the neutrons is small (Figure IV.8b). The valence protons
contribute to the magnetic scattering both through their
spin and orbital angular momentum. In the region of high
momentum transfer, only the spin part is important, the
orbital part having very small effects, as shown in Figure

IV.8c and IV.8d respectively.

68

Figure IV.8. Form factors for magnetic elastic
electron scattering for Al calculated with
single-nucleon radial wave functions of HO potential of
b= brms using free—space values for the 9 factors. Protons
contributions only are shown in Figure IV.8a. The
neutrons contributions only are shown in Figure IV.8b.
Figure IV.8c and Figure IV.8d show the proton gs and 9,
contributions respectively. The same conventions are used for
the different multipoles as in Figure IV.1. The data are
taken from Ref 40 (circles), Ref 41 (triangles) and
Ref 42 (squares).

69

MSUX¢83~OZO

 

27m. HO
l‘exp. " ;. kl l-‘exp. , d
' neutrons

protons

4.0

   

    

3.0
° W)

2.0

LOI-

 

 

M (q)
k

J£\\\\

\

5 I

4)- I,

in;

F.

 

 

 

 

 

Figure IV.8

70

IV.6. Magnetic elastic scattering from 39K

In Figures IV.9 and IV.10 we show form factors for
magnetic elastic electron scattering in 39K. One proton hole
in the d5/2 orbit is responsible for the scattering. The
single-nucleon shell model and the configuration-mixing
shell model are identical in this case. The form factor
calculated with HO radial wave functions of b=brms and the
free-proton value for the 9 factor is shown in Figure IV.9a.
The calculated magnetic dipole moment disagrees with the
measured value. In the region of medium-q values, where the
M3 multipole is important, the predicted form factor is too
large. This is the same behaviour noticed in 17O at this
region of q. This suggests that the need to quench the M3
strength is a common feature for the electron scattering
process in this mass region.

Quenching the proton 9 factor of the M3 multipole to
60% of its free space value improves the agreement between
theory and experiment at low and medium momentum transfers
values (Figure IV.9b). At the region of high-q values (q > 2
fm'l), the agreement is still poor and the data are
increasing as function of q, while the theory varies
smoothly and approximately steadily as function of q.

In Figure IV.10a we show the calculation which uses the
same quenching factor as those used in Figure IV.9b, and a
radial size parameter reduced 5% from that required to fit

the rms radius. The agreement at high-q values is improved

71

Figure IV.9. Form fggtors for the magnetic elastic
electron scattering for K calculated with single-nucleon
radial wave functions of the HO potential of b=brms~
Free-space values for the 9 factors of the M1 and M3
contributions are shown in Figure IV.9a. Figure IV.9b is
the same as Figure IV.9a except the proton 9 factor of the
M3 contributions is quenched to 60% of the free-nucleon
value. The orbital angular momentum contributions alone
are shown in Figure IV.9c. Figure IV.9d shows only
the spin contributions. The same conventions are used for the
different multipoles as in Figure IV.2. The data are taken
from Ref 53 (circles) and Ref 54 (squares).

72

MSU-83-329

 

 

 

gp'X'

 

 

 

 

 

 

Figure IV.9

 

73

by this reduction.

To reproduce the measured magnetic dipole moment, we-
choose an effective Ml orbital 9 factor for the proton of
g[p(eff)= gpp+0.15. These calculations are shown in Figure
IV.10b. The M1 form factor is increased throughout the
momentum transfer regions except at the high-q values. The
fact that the high-q values do not change is because the
orbital contribution is important only at low—q values as
shown in Figure IV.9c. Figure IV.9d shows the spin
contribution to the elastic magnetic scattering. The form
factors decrease as a function of q at small-q values, while
the high-q values are dominated by the spin part.

Giving the proton effective M1 9 factors (gsp(eff)=
0.8xgsp) and g,p(eff)=0.962xglp) reproduces the measured
magnetic dipole moment. These calculations are shown in
Figure IV.10c and IV.10d with the HO and WS radial wave
functions respectively. The radial size parameter of the
valence proton is reduced 5% from that required by the rms
radius. Much better agreement is obtained with these
empirical values. The WS radial wave functions reproduce the
high-q values better than the H0.

The octupole moment can be calculated using the formula

3
= - _—(gs _ 49!) (11>, (105)
70
with <r2 >1/2= 3.606 fm calculated with the WS radial wave

functions. The binding energy of the valence orbit is fixed

74

Figure IV.10. Form Sgctors for the magnetic elastic
electron scattering for K calculated by quenching the 9
factor of the M3 contributions to 60% of the free-space
value. Figure IV.10a shows the calculations with
free-nucleon values for the M1 contributions. Figure IV.10b
shows the calculations with g p(eff)= glpxl.15 and
gsp(eff)=gsp(free) for the M1 contributions. Fiure IV.10c
are calculated with gsp(eff)= 0.8xgsp and g,P(ef£)= 0.9629,?
for the M1 contributions. Figure IV.10d is the same as
Figure IV.10c but usig the WS potential whose valence
orbits rms reduced 5%. The same conventions are used for
the different multipoles as in Figure IV.2. See caption of
Figure IV.9 for the data.

75

MSU-83-328

 

 
   
   

39K, HO

 

Y

HO
I'm-“5-5% -.
95' X US

M3 3‘ 0.6

(b)

 

 

     

 

92' x 0.962 J
M3xos

 

Figure

IV.10

 

 

76

to be 8.329 MeV. According to this formula, the
single-particle moment is found to be -0.88 PM fmz. The
value with quenching M3 to 60% of the free—nucleon value
(gs(eff)=0.6 gs and gl(eff)=0.6 g, ) is found to be
-0.48 ”N fmz. The rms radius of the valence orbit in this
case is reduced 5% from that required by the rms radius.
Experimental data for the octupole moment of 39K has been
quoted by Lapikas (Ref 53) by extracting the data to q=0,
which cannot be considered as an accurate value. Suzuki
(Ref 46) has calculated the octupole moment by including
the effects of first order core polarization, pair
currents, one-pion exchange currents and isobar currents due
to TT- and P—meson exchanges corrections. He obtains the

value -1.454 “N fmz.

IV.7. Magnetic elastic scattering from p—shell nuclei

In the p-shell region we study the nuclei 6Li, 7Li,
98e, 108, 11B, 13C, 14N, and 15N . The calculations for
6Li and 14N are presented in Figure IV.ll, those for 7Li
and 9Be are presented in Figure IV.12, those for 108 and 118
are presented in Figure IV.13 and those for 13C, and 15N are
presented in Figures IV.14. The lines denoted by crosses
represent the calculations with the single-particle model.
The solid lines and the lines denoted by plusses represent
the calculations with a new empirical Hamiltonian (Ref 18)

and the Cohen-Kurath interaction (Ref 19), respectively, with

77

Figure IV.11. Form factors £95 the magnetic elastic
electron scattering for Li and N calculated with two
different wave functions, MSU wave functions (solid lines)
and the Cohen-Kurath wave functions (plus gigns) using
the HO potentials of b=b ms- The data for Li are taken from
Ref 57(ciicles), Ref 58 (squares) and Ref 59 (triangles),
and for N are taken from Ref 58 (squares)), and

Ref 60 (circles).

78

MSU-83-257

 

2.5-

2.o::s::ov‘""""""x,K

 

M(q)

L5"

 

Li

I4N

 

 

 

 

Figure IV.11

 

79

Figure 1V712. Forg factors for the magnetic elastic
scattering of Li and Be. The conventions of the
presentation are the same as given in the caption of Figure
IV.ll. The decomposition of the multipoles
calculated with the MSU wave functions are M1 (dotted lines)
and M3 (dashed lines). Free-nucleon values are used here for
the 9 factors. The data for Li are takgn from Ref 61 '
(circles) and Ref 62 (traingles), for Be are taken from
Ref 63 (circles), Ref 64 and Ref 58 (squares).

80

MSU-83-256

 

2.5-

 

 

M(q)

ELSr

 

/ . ..00000..............::
0
/ .° °. .
T I V '

 

 

Figure IV.12

 

81

Figure IV1&3. ForTlfactors for the magnetic elastic
scattering of B and B. The conventions of the
presentation are the same as given in the caption of Figure
IV.12. The data are taken from Ref 58 (squares) and

Ref 57 (circles).

MSU-aa-zss

 

 

mm)

2 .0 P “‘1XXX
:M‘ 5"

 

 

 

 

Figure IV.13

 

 

83

Figure IV1§4. ForTSfactors for the magnetic elastic
scattering of C and N. The conventions of the
presentation are the §ame as given in the caption of Figure
IV.12. The data for 1 C are Egken from Ref 63 (circles) and
Ref 65 (triangles), and for N are taken from Ref 66 (squares).

84

 

' , ' '7 MSP-83-254
2.5- DC q
2.0- fl
l.5- + ‘

 

 

mm

2.5%

 

Figure IV.14

 

 

 

 

85

single-nucleon wave functions of HO potentials of b= brms and
free-nucleon 9 factors. Good agreement is obtained for those
nuclei where the M1 is the only multipole which contributes
to the scattering. In the case of 9Be, the M3 multipole is
overestimated and needs to be quenched to describe the data
in this region. The behaviour of the data at large-q values
is the same as in sd-shell nuclei; normaly they increase as
a function of q. To get a better reproduction of these data
with theory, the rms radius of the valence nucleons must be
reduced. The reduction of the rms radius might help also in
7Li, 10B and 11B, where the behaviour is almost the same as
in 27Al, normaly the calculated form factors at low-q values

are lower than the data.

IV.8. Conclusions

From these comparisons, we conclude that the
configuration-mixing shell model succeeds in describing the
magnetic elastic scattering data if we allow for small
modifications to the free-space forms of the magnetic
operators. We use two different types of single-particle
radial wave functions, obtained from the HO and WS
potentials. No major differences between these two
potentials appear in describing the data except in the
region of high momentum transfers. There the WS potential
gives better agreement with the experimental data than the

HO potential in 170 (Figure IV.5c and IV.5c') and 39K

86

(Figure IV.10c and Figure IV.10d). For those nuclei in which
higher multipoles contribute to the scattering, the
high-momentum-transfer data are not be described by these
two potentials when the valence nucleons have a radial size
determined by the rms radius. Reduction of the radial
parameters by 5% seems to resolve the discrepancy at high
momentum transfers for these nuclei. Also in some cases the
lower multipoles are improved by this reduction, as in 27A1,
where the calculated peak at lower momentum transfer falls
lower than the data when the radial parameter of the valence
nucleons determined by the rms radius. For those nuclei in
which M1 is the only multipole which contributes to the
scattering, the radial size parameter determined by the rms
radius seems to describe the data at high—q values without
any reduction in the radius. The M3 contributions are
overestimated in almost all instances. This suggests that
the core-polarization effects cannot be ignored and can be
taken into account by giving the nucleon an M3 effective

g-factor less than its free value.

87'

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+~\m.a~

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O 004

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D FHJ
+N\m.m~

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O 004

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o ”4
+~\m.m~

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+N\m._~

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88

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00.0.0
0000.0:
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0000.0:

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ON

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0

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89

Table IV.2. Calculated one-body transition density
matrix elements for the ground state of stable p-shell
nuclei for the wave functions of Ref 18 and Ref 19.

OBDM(i,f,L,j,j',aaw

A;J" ref 2j-2j' = 3-3 3-1 1-3 1-1
250
6:1+
L=l a 0 0.50116 -0.43568 0.43568 -0.17060
b 0 0.51828 -0.31022 0.31022 -o.22473
7:3/2'
L=l a 0 0.98732 -0.02161 0.02161 0.04009
2 -o.29741 0.33561 —0.33561 0.26912
b 0 0.98594 -0.01682 0.01682 0.04447
2 —0.37212 0.30047 -o.30047 0.23792
L=3 a 0 0.63188 0.00000 0.00000 0.00000
2 -0.65016 0.00000 0.00000 0.00000
b 0 0.72854 0.00000 0.00000 0.00000
2 -0.62924 0.00000 0.00000 0.00000
9:3/2‘
L=1 a 0 0.97682 0.11030 -0.11030 0.07329
2 0.63727 -o.03971 0.03971 -0.09598
b 0 0.99265 0.07927 —o.07927 0.02325
2 0.67171 -0.06522 0.06522 -0.08804
L=3 a 0 0.78341 0.00000 0.00000 0.00000
2 0.73450 0.00000 0.00000 0.00000
b 0 0.84834 0.00000 0.00000 0.00000
2 0.75134 0.00000 0.00000 0.00000
10:3+
L=1 a 0 1.66784 0.14446 -0.l4446 0.01732
b 0 1.68331 0.18029 -0.18029 -0.03158

L=3 a 0 0.31373 0.00000 0.00000 0.00000

 

90

 

lRef'lS

Table IV.2. (cont'd.) -
b 0 0.39980 0.00000 0.00000 0.00000
11:3/2‘
L=1' a 0 0.99489 0.21692 -0.21692 0.01615
2 -0.68575 -0.11247 0.11247 0.25977
b 0 0.99985 0.22396 -0.22396 0.00047
2 -0.70255 -0.11637 0.11637 0.23566
L=3 a 0 0.73701 0.00000 0.00000 0.00000
2 -0.78990 0.00000 0.00000 0.00000
b 0 0.78576 0.00000 0.00000 0.00000
2 -0.75915 0.00000 0.00000 0.00000
13:1/2'
L=1 a 0 0.03275 0.03952 -0.03952 0.89645
2 -0.11993 -0.03815 0.03815 0.81506
b 0 0.01737 0.02962 -0.02962 0.94506
2 -0.09112 -0.03091 0.03091 0.81325
14;1+
Ifi=l a 0 0.02502 0.09508 -0.09508 1.33510
b 0 0.05497 0.15510 -0.15510 1.24037
15; 1/2'
==l a 0 0.00000 0.00000 0.00000 1.00000
2 0.00000 0.00000 0.00000 -1.00000
b 0 0.00000 0.00000 0.00000 1.00000
2 0.00000 0.00000 0.00000 -1.00000
6) Ref 18

91

Table IV.3. Experimentally determined rms charge radii
of stable p-shell and odd-A sd-shell-nuclei and the
corresponding values calculated in the harmonic-oscillator

model with length parameters brms and in the Woods-Saxon

 

 

model.

9 rmsTfmT’ brms(fm) Exp.Ref rms(fm) rms(fm7
NUCLEUS exp HO ws
5L1 2.510(100) 1.880
7L1 2.350(100) 1.740
93a ‘ 2.519(12) 1.763

108 2.400(26) 1.611

118 2.400(26) 1.611

13c 2.472(15) 1.628

14N 2.529(25) 1.645

15m 2.580(26) 1.678

170 2.712(5) 1.763 2.716 2.692

195 2.898 1.833 2.903 2.855

21Ne ‘ (2.984) 1.845 2.989 2.961

23Na 2.896(9) 1.810 2.992 3.034

25Mg 3.003(11) 1.793 3.010 3.097

27A1 -3.058(5) 1.804 3.064 3.158

2951 3.122(15) 1.825 3.134 3.216

319 3.187(3) 1.848 3.197 3.271

33s (3.264) 1.881 3.274 3.321

35c1 3.351(16) 1.921 3.360 3.363

37C1 3.351(17) 1.921 3.359 3.349

39K 3.437(2) 1.95 3.442 3.436

 

a) Ref 67

Table IV.3. (cont'd.)
b) Ref 68

c) Ref 69
d) Ref 70
e) Ref 71
f) Ref 72
9) Ref 73
h) Ref 37
i) Ref 74

j) Ref 75

Table IV.4. Measured and calculated magnetic dipole
moments for p-shell and sd-shell nuclei.

 

 

Experimentc'd

memos a“ (sd)a (sd)b (q->0)

( PN) ( 8N) ( 9N )
6Li 1+ 0.878 0.824 0.822
7Li 3/2’ 3.169 3.234 3.256
98e 3/2‘ -1.115 -1.289 -1.177
108 3+ 1.819 1.811 1.801
118 3/2' 2.509 2.532 2.688
13c 1/2' 0.763 0.701 0.702
14N 1+ 0.339 0.326 0.403
15N 1/2‘ -0.264 -0.264 -0.283
170 5/2+ -1.911 -1.893
19F 1/2+ 2.911 2.628
21Ne 3/2+ -0.824 -0.662
23Na 3/2+ 2.219 2.218
25Mg 5/2+ -0.908 -0.855
27A1 5/2+ 3.584 3.642
2951 1/2+ -0.501 -0.555
31p 1/2+ 1.023 1.132
33s 3/2+ 0.651 0.644
35C1 3/2+ 0.663 0.822
35C1 3/2+ 0.433 0.684
37K 3/2+ 0.124 0.391

 

a) Calculated magnetic moments based on the complete p-shell

and sd-shell space wave functions of Ref 17 Ref 18

94

Table IV.4. (cont'd.)
and the freernucleon 9 factors.

b) Calculated magnetic moments based on the coplete p- -shell
space wave functions of Ref 19 and the
"free- nucleon" 9 factors.

c) Ref 76 ,Ref 77 and Ref 78 for p-shell nuclei.

d) Ref 79, Ref 80, Ref 81, Ref 82 and Ref 12 fortsd-shell
nuclei.

CHAPTER V

ELECTROEXCITATION OF EVEN PARITY STATES OF 27Al

v.1. Introduction

The nucleus 27A1 is one of the most interesting nuclei
in the sd-shell because it represents a point at which the
nuclear deformation changes from prolate (for 26Mg) to
oblate (for 28Si) (Ref 12). The electroexcitation of
27A1 has been analyzed previously (Ref 83 and Ref 13)
in terms of the full 0d5/2- 151/2- 0d3/2 space with the
restriction of J"z 5/2+ levels, and with a truncated space in
which at least 6 particles were restricted to the 0d5/2
shell. In this study we use the full 0d5/2-lsl/2-0d3/2 space
for all states to obtain the eigenfunctions of a new
empirical Hamiltonian (Ref 17). In Figure v.1, we show the
measured 27A1 (e,e') spectrum taken from Ref 13 in
comparison with the calculated spectrum. The experimental
and theoretical energy levels are shown in Figure V.2. In
this chapter, we compare the form factors for all the
measured positive-parity states that have definite
assignments of spin and parity.

Complete separations between the longitudinal and
transverse form_factors have been done only for those states

of excitation energies Exs 3.0 Mev. The calculated

95

96

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t r T 1 U I I
3 4 5

Excitation Energy (MeV)

MSU-83-3l0
W 27m (e, 8’) Theory

IOP ‘
7" e- -
E 7/2*
8 6)- 7,24 " “
9!
ET: 4_ 912* .
T“ 5”? &Q* "a?
“' 512* IV? 9’2’

+ m 9’24) l3/2
2 L- + 3,2 W '1
v2 A l 112* 9/2+
_...4'
A x5 -A we
0 J l A , 1
'0 r I I 1 l ' l V I I I ' l 1 1 ' J
4 27A) (e, 8’) Exp-

A 8- -
z b 7/2‘ 312+ 912* u -
§ 5/2‘ 22)“ 29814-3004
0 6- 95° “1“ -
.4 r 2 u) .
\ v2+ :5/2+ + +
g. _ H 4 t 1‘ 1 I 1 H -

2b . .l 1 a:

o I j I I l T V

o u 2

03—1
fl-

Figure v.1. Experimental spectrum of 27A1 (e,e') in
comparison with the theoretical form factors of the
positive-parity states calculated at the same angle and
incident energy as the measured spectrum.

97

 

 

 

 

 

 

MSU-83-335

5.0” ~
7/2* 7 +
M ”/2

4.5? / '1
/2"
+

4.0 _ 3/2" 41/ 3’2 .
(12* J 1/2"

35* ..

 

 

 

 

3
U
2
V 2.5 r- -(
“1" 712+
2.0 - 4
LS P n
3/2"
~ +
'0’ l2 v2+ 4

 

l/Z" / ‘

0.5? 1

0 0+ 5/2+ 5/2+ ‘

EXPERIMENT THEORY

 

 

 

f 27Figure v.2. Measured positive-parity energy levels
a l in comparison with the shell-model calculations.

98

longitudinal form factors are compared with the experimental
data for these states only. For higher—lying states,
calculated transverse and total form factors are compared

with the data.

V.2. Elastic scattering for the 5/2+ ground state

The measured elastic form factor for the 5/2+ ground
state is shown in Figure v.3 in comparison with the DWBA
calculations which incorporate single—nucleon wave functions
of the HO potential with b=brms (solid line) and WS
potential (dashed line). The different multipoles that
contribute to the elastic scattering are E0 (dotted line),
E2 (plus signs) and E4 ("Y" signs). These multipoles are
calculated with the HO potential of b=brms- The elastic form
factor is dominated by ED up to momentum transfer 2.3 fm'1
where E2 becomes more important. The E4 contribution is very
small and has a negligible contribution to the scattering.
In the region of small momentum transfers, the HO radial
wave functions describe the scattering data very well up to
the first diffraction minimum. Beyound that region, the HO
results fall below the WS results. The latter fit the data
very well in this region. This behaviour is similar to that

noticed for other cases studied in the sd-shell region

(Ref 33).

99

 

(0' -
Io’ *-
10'3 -
IO'4r'

(0'5

I0‘7 1

 

   
 

1

27A: (e.e)27AI
elastic

MSU-83-288
I

 

 

 

(0"

Figure v.3. DWBA elastic scattering form factor

calculated with the single-nucleon wave functions of the
HO potential of b=brms(solid line) and of the WS
potential (dashed line). The different multipoles E0, E2
and E4 (dotted lines, plus and ”Y" signs, respectively)

are calculated with the singlernucleon wave functions

of the HO potential of b=brms. The data are taken from

Ref 42 (circles)-250 Mev,

(squares)-500 Mev.

100

v.3. Inelastic scattering to the 0.844 MeV, 1/21+ state

The longitudinal E2 form factor M(q) for the inelastic
scattering of the 0.844 MeV, 1/21+ state is shown in Figure
v.4, where we compare two models for the core-polarization
transition density, the valence model (dashed line) and the
Tassie model (solid line). The low-q values favour the
valence model.

Figure v.5 shows the form factor M(q) calculated with
the HO potential of length parameter b=brms (solid line) and
the WS potential (dashed line) using the Tassie model for
the core-polarization transition density. The measured B(E2)
value is reproduced reasonably well but the scattering data
are higher than the theory in the region of momentum
transfer l< q2 < 4 fm"2. No differences between
these two potentials appear in describing the data except at
the photon point, where the measured B(E2) value is
reproduced better with the HO radial wave functions.

The transverse form factor is shown in Figure v.6
(dashed-dotted line) calculated with the HO radial wave
functions of b=brms- In this Figure we show the different
contributions from the spin and orbital parts. The different
multipoles that contribute to the scattering are E2 (plus
signs) and M3 (cross signs). The transverse form factor is
dominated by the spin contribution. The orbital contribution
is important only in the region of the diffraction minimum.

The contributions of both multipoles are important in the

101

MSU-83-289

U I T f V I I I I T I U I I I I I

 

225- L, 0.84 Mev, V2+ -

20.0- i
/

l7.5*- / .

(5.0)— / .

 

 

 

 

Figure v.4 Longitudinal E2 form factor M(q) for the
1/21+ state in 2 Al calculated with the single-nucleon
wave functions of the HO potential combined with the
Tassie and valence models (solid and dashed lines,
respectively) The data are taken from Ref 13.

102

region of low-q values up to q==1.7 fm‘l. Beyond that, the
transverse form factor is dominated by the magnetic
scattering. From the discussions of the magnetic elastic
scattering in the previous chapter, we conclude that the M3
form factor needs to be quenched to get a reasonable
explanation of the experimental data. The result of
quenching the M3 form factor to 60% of the free-nucleon
value is shown also in this Figure. Quenching the M3 form
factor improves the agreement with the low—q data. At q >
1.5 fat”1 the form factor is underestimated by a factor of
2.5. From here on in our discussions all the calculations.
are presented with the M3 form factor quenched to 60% of its
free-space value.

In Figure v.7 we show the total form factor for the
0.844 Mev,l/21+ state calculated at 0: 90° by the solid
line. The longitudinal contribution is shown by the dashed
line. The dashed-dotted line shows the transverse form
factor, including the kinematic factor (l/2+tan2£3/2). In the
region of low and medium momentum transfer the scattering is

mostly longitudinal, while at the large momentum transfers,

both longitudinal and transverse are important.

v.4. The 1.014 MeV,3/21+ state
Form factors for the 1.014 MeV, 3/21+ state are shown
in Figure v.8, calculated with the radial wave functions of

the HO potential of b=brms- The longitudinal E2+ E4 form

103

 

 

 

 

. . . MSU-83-290
22.5- L, 0.84 MeV, l/2"' i
20.0)- ..
(7.5-1 -
l5.0- -
3125- -
:5
l0.0r-
7.5- o . / ’ \ ..
O.’ ‘\
5.0- , ~
‘\ . //
2.5- o ’ -
\ /
o o 1 4| 1 1 1 1 1 1 I 1 1 1 1 1 1 1
2 3 4» £5 6 7' (3 9
q2(fm'2)

Figure v.527Longitudina1 E2 form factor M(q) for the
1/21+ state in A1 calculated with the Tassie model
combined with single-nucleon wave functions of the HO
potential of b=brms(solid line) and of the WS potential
(dashed line). The data are taken from Ref 13.

104

 

MSU-83-29l
.o-3 - T. 0.84 Mev, I/2+ 95 free ._ 0., =0 1
g! = 0 9, free

'0" " -(- .4

3204 O. . -

LL .0'5 . .1

7 {##pfififlmfimfig“~ ,

IO. ..
”a

10‘

 

Id”

 

Id‘- .. ‘

 

 

 

 

 

Figure v.6. Transverse form factors for the 0.844 MeV,
1/21+ state calculated with the HO radial wave functions of
b=brms- The upper two Figures show the contributions from
the spin and orbital 9 factors respectively. The lower two
Figures calculated with free M3 9 factors and quenched M3
9 factors to 60% of the free—nucleon values, respectively.
E2 and M3 multipoles are shown by the plus and cross
signs respectively. The data are taken from Ref 13.

105

 

 

 

 

MSU-83-292
90°, 0.84 Mev, V2+ '
(0'3- _
" C . e
0
IO"‘- .0 -
b 1. .
3.0-5h /".\. .0 ... '-
N b / ' \ d
“- I05)— . \ \ 7"\- \ -
p / . l ‘\ . .
. " \ \\
I \ ‘

(0'7 / |
)- 0 | l \ \d
(0'3 ./ II -
i I I, 7

. "

I '9 1 1 1 J 1 1
O 0 l 2 J3
q(fm")

Figure v.7. Total form factor for the 0.844 Mev, 1/21+
state calculated at 9: 90 (solid line). The dashed line
represents the longitudinal form factor, and the
dashed-dotted line represents the transverse form factor
1.5 FTZ. The data are taken from Ref 13.

106

factor (dashed line) is dominated by the E2 contribution
(plus signs). The E4 multipole (”Y" signs) makes a
negligible contribution to the longitudinal scattering. The
data is reasonably well explained, although it is slightly
underestimated in the region of l < q < 1.5 fm’l.

The total transverse form factor is shown by the
dashed-dotted line. The multipole decompositions that
contribute to the transverse scattering are Ml (dotted
line), E2 (plus signs), M3 (cross signs) and E4 ("Y'I signs).
Good agreement is obtained at low- and high—q values,
while in the region of medium-q values, the form factor is
underestimated by a factor of 2.

The total form factor of the 1.014 MeV, 3/21+ state is
shown in this Figure by the solid line, calculated at 9=90°.
The scattering is mostly longitudinal (dashed line) except
at the region of the diffraction minimum where the

transverse contribution 1.5 FT2 is also important

(dashed-dotted line)

2 using

In Figure v.9 we plot the form factor M(q) vs. q
the two models for the core polarization, the valence model
(dashed line) and the Tassie model (solid line). The high-q
values are very well explained by the Tassie model, while
no big diference appears between these two models at the
low-q values. The measured B(E2) value is well reproduced as

shown-in this Figure as q-+0. From previus study (Ref 33)

it was found that the longitudinal scattering data are well

107

Figure v.8. Form factors for the 1.014 MeV, 3/21+
state calculated with the single-nucleon radial wave

functions of the HO potential of b=brms- The longitudinal,

transverse and total form factors are represented by the
dashed, dashed-dottted and solid lines, respectively. The
plus and "Y" signs in the longitudinal plot represent the
contribution of the E2 and E4 multipoles of the
longitudinal form factors. The decomposition of the
multipols of the transverse form factor are Ml (dotted
line), E2 (plus signs), M3 (cross signs) and E4 (”Y"
signs). The total form factor is calculated at 6= 900 The
data are taken from Ref 13.

 

 

 

—; —___ _- .__-_
-.....‘AMH -4

108

MSU-83493

I I j 1 I

L, 1.01 Mev, .3/2“

 

a
9
N

O

a)

-‘

 

*5.
M ‘I
I o-T '- «'7 "7" d
v Vyv “
v
no“ v V 4
t v 7' ..
v V,
v 1 ,
T

09- .
(O‘- ‘

 

 

 

 

 

 

Figure V.8

109

Figure V. 9. Longitudinal and transverie M(q) form
factors for the 1. 014 MeV, 3/21+ state of A1 calculated
with the single- -nucleon wave functions of the HO
potential of b=brms. In the longitudinal plot, the solid
and dashed lines represent the calculations with the
Tassie and valence models respectively. The measured B(E2)
value and B(Ml) value are shown at q= 0 in the longitudinal
and transverse plots respectively. The data are taken from
Ref 13.

 

 

M_(_q)
go

Figure v.9

22.5

110

 

20.0 -
l7.5 .

l5.0 -

I0.0 r

7.5)-

2.5-

 

1 I 1 I l I 1 1 r 1 T

L, (.01 MeV, 3/2"

I

/

1,1 T 1 j

 

 

4.5

4.0-

3.5 P

 

T, (.0) MeV, 3/2“

 

 

 

 

lll

described by the Tassie model in this mass region. We will
use the HO radial wave functions of b=brms and the Tassie
model for the core-polarization transition density

throughout the calculations of the excited states of 27A1.
The predicted magnetic dipole transition matrix element is
slightly smaller than the measured value, as shown in the

M(q) plot for the transverse scattering.

v.5. The 2.211 MeV, 7/21+ state

Form factors for the 2.211 Mev, 7/21+ state are shown
in Figure v.10 calculated with the radial wave functions of
the HO potential of b=brms- The longitudinal E2+ E4 form
factor (dashed line) is dominated by the E2 contribution
(plus signs), as the E4 multipole ("Y" signs) makes a
negligible contribution to the longitudinal scattering. The
experimental data are very well described throughout the
different momentum transfer regions.

The total transverse form factor is shown by the
dashed-dotted line. The decomposition of the multipoles that
contribute to the transverse scattering are M1 (dotted
line), E2 (plus signs), M3 (cross signs), E4 ("Y" signs) and
M5 (triangles). The shape of the form factor is very well
reproduced, but the magnitudes are slightly underestimated.

The total form factor for the 2.211 MeV, 7/21+
state is shown in this Figure by the solid line, calculated

at 9:900. Once again, the scattering is mostly longitudinal

112

Figure v.10. Form factors for the 2.211 MeV, 7/21+
state. The convections of the presentation are the same as
given in the caption to Figure V.8. The M5 multipole is
shown by the triangles. The data are taken from Ref 13
(circles), Ref 84 (triangles).

 

113

1194-83—2”
~ L, 2.2IMeV, 7/2+ 1

0-3 ,. J/‘w‘\ :
\

 

' A

 

“ Y' +
U511 Y7? + YYY ‘
Y Y "
.... Y '7?
V v
U7 1 '7 Y" J
’ v
v v'
.0-8 __ v '7 _,
> Y 'Y 4
Yv
Y L A Y
b '1
)0-3 T -
)- d
10"- J
)- -(

 

 

 

 

 

fifth")

Figure v.10

114

Figure v.11. Longitudinal and transverse M(q) form
factors for the 2.211 MeV, 7/21+ state. The convections
of the presentation are the same as given in the caption
of Figure v.9. The data are taken from Ref 13 (circles),

Ref 84 (triangles).

115

MSU-83-296

VTTfivvvfivvfiT—I—vvvw
.

 

 

 

 

 

 

4.5

I
:4
N
L‘.’
g;
0
.<
\l
\
N
+

4.0 - 7

 

 

 

 

Figure v.11

116

(dashed line), while the transverse part, 1.5 FTZ
(dashed-dotted line), makes a negligible contribution. The
experimental data are very well reproduced at all q values.
Good agreement is obtained with the measured B(E2) value, as
shown in Figure v.11. The predicted B(Ml) is higher than the

measured one by a factor of 2, as shown in the transverse

M(q) plot.

v.6. The 2.735 MeV, 5/22+ state.

Form factors for the 2.735 5/22+ state are shown in
Figure v.12, calculated with the radial wave functions of
the HO potential of b=brms~ In the region of low-q values,
the longitudinal E0+ E2+ E4 form factor (dashed line) is
dominated by the E2 contribution (plus signs), while in the
region of medium-q values it is dominated by the E4
contribution ("Y" signs). In the region of high-q values the
E2 and E4 multipoles contribute equally to the longitudinal
scattering. The E0 contribution (dotted line) is very small
compared to the E2 and E4 multipoles. The shape and
magnitude are very well reproduced except at the region of q
between l—1.5 fm'1 where the form factor is slightly
underestimated.

The total transverse form factor is shown by the
dashed-dotted line. Good agreement is obtained between
theory and experiment. The decomposition of the multipoles

that contribute to the transverse scattering are Ml (dotted

117

Figure v.12. Form factors for the 2.735 MeV, 5/22+ state.
The convections of the presentation are the same as given
in the caption to Figure V.8. The convections for the
different multipoles are the same as those in Figure V.3
and Figure v.10. The data are taken from Ref 13.

 

118

 

 

 

MbU-Ub-JII
f U I j V
L, 2.735 MeV, 5/2+
no" - J
\..
- 4 ‘4'
IO ‘ " 4' ...: \.\. J
* m;
4'
-5 I V" * ’2»
A '0 " l+ 'Y ..... f”. "v\ ‘
3 f y' °.. + ....o " \
N v ' 7 \
u -6 I v + + \
IO )- I 1 4 \ '7
I ¥ \
* Yv +* 0 . Q. \
u. " \
Id? - v '. ‘7
0"“
'0.8 v d
v .
IO.9 § ‘L A w‘ v
T
Ids» ‘
(0’4 t 1

 

 

Figure v.12

 

 

 

119

line), E2 (plus signs), M3 (cross signs), E4 ("Y" signs) and
M5 (triangles).

The total form factor F2(q) is shown by the solid line,
calculated at(3= 90°where the scattering is mostly
longitudinal (dashed line). The transverse form factor, 1.5FT2
(dashed-dotted line), makes a negligible contribution to the
total form factor except in the region of q > 2 fm'l.
The measured B(E2) value is reasonably well reproduced, as
shown in Figure V.13. The measured B(Ml) value is also well

reproduced, as shown in the transverse M(q) plot.

v.7. The (2.98,3.004) MeV, (3/22+, 9/21+) doublet
The theoretical form factor of this unresolved doublet
is obtained by adding the calculated form factors of the 3/22+
and 9/21+ states (Figure v.14). The total longitudinal E2+ E4
form factor is shown by the dashed line. The largest
contribution is due to the E2 form factor (plus signs). The
E2 form factor is dominated by that of the 9/21+ state, while
the 3/22+ state makes a negligible contribution to the E2
form factor. The E4 form factors of both states ("Y" signs)
contribute approximately equally to the longitudinal
scattering and they make negligible contributions except in
the region of the diffraction minimum. The experimental data
are slightly underestimated in the region of q > 1 fm-l.

The total transverse form factor is shown by the

dashed-dotted line. The decomposition of the multipoles that

120

Figure V.13. Longitudinal and transverse M(q) form
factors for the 2.735 MeV, 5/22+ state. The convections
of the presentation are the same as given in the caption
to Figure v.9. The data are taken from Ref 13 (circles),
Ref 84 (triangles).

 

 

121

 

MSU—834514
" L, 2.735Mev, 5/2“ ‘
20 - .

 

 

 

 

 

 

r- II
o : éJS‘A‘BJé‘iJé‘s
q2(fm'2)
1 I I I I I I I Tr I t
4-5' T, 2.735Mev, 5/2” '
4.0" -(
3.55 .
3.0- d

 

 

 

 

Figure v.13

122

Figure v.14. Form factors for the (2.98,3.004) MeV,
(3/22+,9/21+) doublet. The convections of the presentation
are the same as given in the caption to Figure V.8. The
convections for the different multipoles are the same as
those in Figure v.10. The data are taken from Ref 13

(circles), Ref 84 (triangles).

 

 

 

 

 

Figure v.14

123

 

 

MSU-83-327
T I j T 7
L, (3.0, 2.98) MeV (3/2 +, 9/2”)]
9* .. . 4
‘4‘ .
*1- !
4‘ )
10‘» +3 4
”...”4‘ .
’ 1" flrg“! ~. 0
Y + Y ‘

IO-5 ? YY VW\. '7
I Y * ++ YY 73-5 .
I V' * '7 A ,
-6 ‘ Y 4 7' t“ F
‘0 b 1 4. Y7 + J
Y, ‘1 i

YY 17y a

l0.7 1 4 Y",
-e '
l0 . 4.
l
9 Y :
U + —% 4L 1 #v— a].
T 9
IO.3 *- '1'
Id‘t *

 

 

 

 

 

16" ~ ‘J
'9 1 1 l L 1
'oc) I I 2 3
a(fm' )

124

contribute to the transverse scattering are M1 (dotted
line), E2 (plus signs), M3 (cross signs), E4 ("Y" signs) and
M5 (triangles). The transverse form factor is dominated

by the M1, E2 and E4 multipoles. The,shape of the form
factor is very well reproduced, but the magnitude is
overestimated in the region of q < 1.5 fm'1 by a factor of
1.5.

The total form factor F2(q) is shown by the solid line,
calculated at 9= 901 The total form factor is dominated by
the longitudinal contribution of the 9/21... state.

Reasonable agreement is obtained with the measured
B(E2) value of the 9/21+ state, as shown in Figure v.15 for
the longitudinal M(q) plot. The measured B(Ml) value of the
3/22+ state is overestimated, as shown in the transverse

u(q) plot.

V.8. The higher-lying states

We present here the shell—model predictions of the form
factors for the states above 3.0 MeV which have
experimentally known spin and parity assignments. The
experimental longitudinal form factors have not been
separated from the total form factor measured at 90°. The
calculations presented here are for the transverse form
factors measured at angles 1600 and 180°, and for the total
form factors measured at 90°. The decomposition of the

different multipoles of the longitudinal scattering are

125

Figure v.15. Longitudinal and transverse M(q) form
factors for the (2.98,3.004) MeV ,(3/22+,9/21+) doublet. The
convection of the presentation are the same as given in
the caption of Figure v.9. The data are taken from Ref 13
(circles), Ref 84 (triangles).

 

126

 

MSU-83-330

L,(3.0, 2.98) Mews/2*. 9/2“)

 

 

 

 

 

7‘ I 1 1

T,(3.0, 2.98) MeV( 3/2‘, 9/2”)

 

 

 

 

Figure v.15

127

shown in the total form factor plots.

The form factors for the 3.68 MeV, 1/22+ state are
shown in Figure v.16. The transverse form factor is shown by
the dashed-dotted line. It is dominated by the E2
multipole (plus signs), while the M3 multipole (cross signs)
has a very small contribution at q > 2 fm'l. Good agreement
is obtained with the few available data points. The total
form factor calculated at€)= 900 is shown by the solid line.
In the the region of q < 1.5 fm'l, the scattering is mostly
longitudinal (dashed line) and no significant contribution
appears from the transverse 1.5 FT2 part (dashed-dotted line)
at this region of q. In the region of high-q values > 2 fm’l,
the scattering is mostly transverse. Reasonable agreement is
obtained in shape and magnitude throughout the different
momentum transfers regions.

The form factors for the 3.957 MeV, 3/23+ state are
shown in Figure V.17. The transverse form factor is shown by
the dashed-dotted line and the different multipoles that
contribute to the scattering are M1 (dotted line), E2 (plus
signs), M3 (cross signs) and E4 ("Y” signs). The transverse
scattering is dominated by the M1 contribution (dotted
line). The total form factor calculated at 6= 90°is shown by
the solid line. The scattering is dominated by the
transverse 1.5 FT2 part (dashed-dotted line) at all q values
except in the region of q between 1-2 fm‘l, where both

longitudinal and transverse parts are important. The

128

FiguEe v.16. Form factors for the 3.68 Mev, 1/22+
state of A1. The upper plot represents the transverse
form factor (dashed-dotted line). The E2 and M3 multipoles
are shown by the plus and cross signs respectively. The
lower plot represents the total form factors calculated

at 9= 90° (solid line). The dashed line represents the
longitudinal form factor, while the dashed- dotted line
represents the transverse form factor including a factor
of 1. 5. The data are taken from Ref 13.

129

MSU-83-333

 

(0'3
(0‘4

“(0'5

Fzm

10'“

I0'8

109

Io-B ._

104

1TI

I

I

T, 3.68Mev, l/2+

‘-

I

db

1

 

 

Figure v.16

 

 

 

130

Figuge v.17. Form factors for the 3.957 MeV, 3/23+
state of A1. The convections of the presentation are the
same as given in the caption to Figure v.16. The
decomposition of the multipoles of the transverse form
factor are Ml (dotted line), E2 (plus signs), M3

(cross signs), and E4 ("Y" signs). The decomposition of
the multipoles of the longitudinal scattering are shown in
the total form factor plot as E2 (plus signs) and E4

("Y" signs).

131

 

usu-es-ssn
T, 3.96 MeV, 3/2+

@fl' j

ldfif -

 

 

 

 

 

Figure v.17

132

longitudinal form factor is dominated by the E4 contribution
("Y" signs) in the region of q > 1 fm'l, while E2
‘ contribution (plus signs) is more important at q < l fm-l.
{An overall agreement is obtained with the experimental data.
In Figure v.18, we show the form factors for the 4.41
Mev, 5/23+ state. The transverse form factor is shown by the
dashed-dotted line and in the region of low-q values the
transverse scattering is dominated by the M1 contribution
(dotted line). As q increases, E2 (plus signs) and M3 (cross
signs) contributions become more important. The
contributions from E4 ("Y" signs) and M5 (triangles)
multipoles are small and they are important only in the
region where E2 and M3 have their second minima. The
transverse data are very well reproduced. The total form
factor calculated at 9= 900 is shown by the solid line. In
the region of small momentum transfers the scattering is
dominated by the transverse part (dashed-dotted line). At
q > 1 fm'l, the longitudinal part (dashed line) becomes more
important up to q==2.3 fm'l, where the transverse part
becomes again important. The longitudinal form factor
contributes to the scattering through the E0 multipole
(dotted line), E2 multipole (plus signs) and E4 multipole
("Y" signs). The E4 multipole dominates the longitudinal form
factor up to q::2 fm-l. In the region of q > 2 fm'l, the

contributions from E0, E2 and E4 multipoles are all

important. The experimental data are very well reproduced

133

Figufie v.18. Form factors for the 4.41 MeV, 5/23+
state of A1. The convections of the presentation are the
same as given in the caption to Figure v.16. The
decomposition of the multipoles of the transverse form
factor are M1 (dotted line), E2 (plusses line), M3

(cross signs), E4 ("Y" signs) and M5 (trianles line). The
decomposition of the multipoles of the longitudinal
scattering are shown in the total form factor plot as E0
(dotted line), E2 (plus signs) and E4 (”Y" signs).

134

 

   

   

 

MSU-83-332
I I I I 1
T, 4.4 I MeV, 5/2 +
l0aL -
To" — a
—5
3'0 . ..*_.-._.‘9“ "
NV / Fm '\.‘ ‘
u - . 1%.
IO 5 )-/ x .+* '. 22‘s, \ ( J
x++ o 0*. . . ~ ' s
x? .0 + o '
(0'7 "+ . . *
V ‘ 'W
x" A‘ YYY 7’
l0. L X“ f v x
Y
x7 1‘ Y + X ’ Yy 4‘
v - )1
IO’9 : ‘ *4 .1 * 4~ : Y
_ 90°
-4
IO P -(

 

 

 

Figure v.18

 

135

FiguSe v.19. Form factors for the 4.51 MeV, 11/21+
state of A1. The convections of the presentation are the
same as give in the caption to Figure v.16. The
decomposition of the multipoles of the transverse form
factor are M3 (cross signs), E4 ("Y" signs) and M5
(trianles). Only E4 multipole contributes to the
longitudinal scattering.

136

 

MSU-83612

r I I I W I d
.04).. T, 4.5) MeV, “/2+ .
(0'4e 7

 

 

 

 

 

Figure v.19

137

throughout the different q values.

In Figure v.19, we show the form factors for the 4.51
MeV, 11/21+ state. The transverse form factor is shown by
the dashed-dotted line and it is dominated by the E4
multipole ("Y" signs). The M5 multipole (triangles) makes a
negligible contribution. The total form factor calculated at
8= 900 is shown by the solid line, where the scattering is
totally longitudinal (dashed line). Very good agreement is
obtained for all q values.

The form factors for the 4.58 MeV, 7/22+ state are
shown in Figure V.20. The transverse form factor is shown by
the dashed-dotted line and the different multipoles that
contribute to the scattering are Ml (dotted line), E2 (plus
signs), M3 (cross signs), E4 (“Y" signs) and M5 (triangles).
In the region of q < 0.5 fm'l, the M1 multipole dominates
the scattering. As q increases the E2 multipole becomes more
important up to q=:2 fm’l. At q > 2 fm'l, the transverse
scattering is dominated by the E4 multipole. The total form
factor calculated atf3= 90° is shown by the solid line. The
scattering is mostly longitudinal (dashed line) and is
dominated by the E4 multipole ("Y" signs) in the region of q

> 1 fm-l. The experimental data are very well reproduced.

v.9. Conclusions

The complete-space shell-model calculations succeed in

describing all the positive-parity states of 27Al considered

138

FiguSe v.20. Form factors for the 4.58 MeV, 7/22+

state of A1. The convections of the presentation are the
same as given in the caption to Figure v.16. The
decomposition of the multipoles that contribute to the
transverse form factor are Ml (dotted line), E2

(plus signs), M3 (cross signs), E4 ("Y" signs) and M5
(trianles). The decomposition of the multipoles that
contribute to the longitudinal scattering are shown in the
total form factor plot as E2 (plus signs) and E4 ("Y" signs).

139

MSU-83-3l6
I

 

(0'3

lo“4

... (0‘5 -

F2(q

(0’

I

1

T, 4.58 MeV

I

I

, 7/2+

-

 

(0'9

(0'3

5,
.b
Ii

F2(q)

5
l

 

Figure v.20

I

 

 

jIIHVT.D'-m w

140

in this study. The longitudinal form factors for the
low-lying states of excitation energies below 3.0 MeV are
very well described. The B(E2) values are reasonably
reproduced with the isoscalar effective charge of 1.7e. The
transverse form factors of these states are very well
reproduced in shape. However, the magnitudes need to be
adjusted for some multipoles to get better explanation of
the data for all momentum transfers regions. Quenching the
M3 multipole to 60% of the free-nucleon value helps in some
cases in describing the low-q values where the M3 multipole
is important. The measured B(Ml) are not well reproduced for
some cases with the free-nucleon 9 factors of the M1
multipole. This suggests that the M1 multipole needs to be
renormalized by using effective 9 factors. The total form
factors of these low-lying states are dominated by the
longitudinal scattering.

The higher-lying states, of excitation energies above
3.0 Mev, are also well described. Some of these states are
dominated by the longitudinal scattering as in the 4.51 Mev,
11/21+ and 4.58 Mev, 7/22+ states. The 3.957 MeV, 3/23+ state
is dominated by the transverse scattering. The other two
states are dominated alternatively by the longitudinal and

transverse scattering at different q regions.

141

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.000.0

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«.00.01

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.«0..0

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N

0«0.0

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150

Table V.2. Calculated occggation numbers for sd-shell
orbits in the ground state of A1.

p/n 2j= 5 l 3
p 4.l08 0.422 0.470
n 4.787 0.602 0.611

APPENDI X

To obtain the one-body density matrix (OBDM) defined in
equation ( 5) in isospin formalism, we can define the

operator 5 which is a tensor operator in isospin space,

a(t,t3)= 1-11t’t35(t,-t3) (106)

then the operator [a+(t,t3) 0 a(t,t3)] can be written

[a+(t,t3) a a(t,t3)]= (-1)t‘t3[a+(t,t3) Q 5(t,-t3)]

= (-1)t‘t3 Z<t t3 t '-t3 IAT ATZ >
ATATZ
[ a+ up a 1(ATrATz’ (107)

where t= 1/2, t3= l/2 for proton and -l/2 for neutron,

and ATZ= 0 and AT= 0,1.

151

152
[a+(1/2,t3 )qp a(l/2,t3 )]= 1-111/2‘t3{ <1/2 t3 1/2 -t3| o 0>
[a+ ® 51(AT,0)
+ <1/2 t3 1/2 -t3 1 1 0>

[6* G1 5 10131)} (108)
[a+(p/n) a a(p/n)]=‘ri_—_/2 [ 61+ ,3, a ](AT=0)

+/-\[1'/2 [a+ 0 a 1(AT=1) (109)

Using Wigner-Eckart theorem, the matrix element of the

operator a+ 0 5 is

<Tf Tz |[a+ 0 5 1(AT=0)IT1 'I‘z > (-1)Tf'T

x «pf l|[a+ 051(AT=0)IIT1> 1110)

OH

<Tf T2 Ma” «1 1(AT=1)ITi 'rz >= (-1)Tf‘T

-Tz 0 T2

x <Tf'||[a+ o 51(AT=1)||T1> (111)

The OBDM becomes

OBDM(p/n))= <fl[a+ (t,t3 )®a(t‘,t3 )1 li>

153

Tf - Tz Tf 0 Ti
=(-1) xVZ OBDM (AT=0)/2

ff - T Tf l T'

(+/—) (-1) z 1 xVE OBDM (AT=l)/2
‘T2 0 Tz

(112)

where OBDM(AT) is given by

<f| |[a+(t) @ ann‘ATm»
OBDM(i,f,L,AT)= (113)
(2AT+1)

 

Equation ( 8) fOllows from reduction in both spin and

isospin spaces.

100

11.
12.

l3.
14.

15.

16.

‘17.
18.

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