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This is to certify that the
dissertation entitled

THE STUDY OF (P,D) REACTIONS ON 13C, 118, AND 1088

IN INVERSE KINEMATICS

presented by
Xiaodong Liu

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics and Astronomy

’LhimAkiflbflg Ikyuapy

Mala Professor 5 SignatureU

5/6/05

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THE STUDY OF (p, d) REACTIONS ON 13C, ”B, AND
“’Be IN INVERSE KINEMATICS

By

Xiaodong Liu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2005

ABSTRACT

THE STUDY OF (p, d) REACTIONS ON 13C, “B, AND
“’Be IN INVERSE KINEMATICS

By

Xiaodon g Liu

. . . ll . .
This work studied the one neutron transfer reactions on 13C, B, and 10Be In Inverse
kinematics using detectors with high angular and high energy resolution. The (p,d)

reactions were performed using secondary beams of 13C, 11B, and 10Be on polyethylene
targets (CH2)n. The experiment was performed at the National Cyclotron

Superconducting Laboratory (NSCL) at Michigan State University. Neutron

spectroscopic factors have been extracted for the transfers from the ground states of 13C,
II 10 12 10 9 13
B, and Be to the ground states of C, B, Be and from the ground state of C to the

first excited state of 12C. The theories of DWBA and ADBA were reviewed and the

approximations of zero-range, finite range, and non-locality were examined. Sensitivities
of the optical-model potentials in the extraction of the spectroscopic factor were
analyzed. The results of this work indicated that a transfer reaction in inverse kinematics
provides a unique tool for the study of the structures of the radioactive nuclei and that
reliable spectroscopic factors must be extracted with a systematic and consistent

approach using global optical-model potentials.

To
my wife - Yu Liang
and

my daughter - Liying Liu

iii

ACKNOWLEDGMENTS

First of all, I would express my deepest gratitude to my advisor, Dr. Betty M. Tsang. I
could not have completed this thesis without her tremendous support and guidance. She
taught me how to solve somehow impossible problems by detailed inspections and
unremitting effort. The experience of making progress through gloomy zenith has been
wonderful and invaluable. I also learned much from her enthusiasm for life as well as for
the nuclei.

I own my sincere thanks to C. K. Gelbke, P. Hosmer, T. Liu, W. G. Lynch, R.
Shomin, W. Tan, G. Verde, A. Wagner, M. Wallace, H. Xi and H. Xu. I am also indebted
to L. Beaulieu, B. Davin, Y. Larochelle, T. Lefort, R. T. de Souza, R. Yanez and V. E.
Viola from Indiana University, as well as R. J. Charity and L. G. Sobotka from
Washington University. I would also express my gratitude to the REU students, K.
Chalut, S. Farges, M. D. Lennek, M. Ramos, and S. Seun for their massive work.

I would like to thank Dr. F. Nunes, Dr. M. Famiano, and Dr. F. Delaunay for their
great help on theoretical calculations and discussions. I am deeply indebted to Dr. J. A.
Tostevin who offered the TWOFNR program from Surrey University and gave us
invaluable help on theoretical analyses.

Great thanks to Dr. Daniel Bazin who taught me a lot about the S800 spectrometer
and beam transmission. I would also thank Dr. Marc-Jan van Goethem and Michal
Mocho for their help during my analyses.

Special thanks go to Dr. William G. Lynch who always gave me excellent ideas and
philosophical understanding of physics throughout my thesis experiment and data

analyses.

iv

I would like to acknowledge the excellent support at the National Superconducting
Cyclotron Laboratory at Michigan State University. I appreciate the help from J. Yurkon
of the detector laboratory, L. Morris of the design group, J. Vincent of electronics group,
R. Fox and B. Pollack of the computer group.

I would also like to thank Dr. S. D. Mahanti, Dr. B. Sherrill, and Dr. C. P. Yuan for
reading my dissertation and also serving on my guidance committee.

Last but not least, I would like to thank my wife and my daughter for their love and

belief in me.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................ viii

LIST OF FIGURES ............................................................................... x
CHAPTER 1

INTRODUCTION ................................................................................. l

1.1 Motivation ................................................................................. 1

1.2 Inverse Kinematics ....................................................................... 3
CHAPTER 2

THEORETICAL DESCRIPTION ............................................................... 9

2.1 Overview ................................................................................. 9

2.2 Theoretical Spectroscopic Factor ..................................................... 10

2.3 Reaction Theory ......................................................................... 14

2.3.1 Distorted-Wave Born Approximation (DWBA) ............................. 14

2.3.1.1 Optical-Model Potential ............................................. 18

2.3.1.1.1 Overview .................................................. 18

23.1.1.2 Global Optical-Model Potentials ....................... 19

2.3.1.1.3 Proton Global Optical-Model Potentials . . . . 2O

2.3.1.1.4 Deuteron Global Optical-Model Potentials .......... 28

2.3. 1.1.5 Nucleon-Nucleus Optical-Model Potential .......... 35

23.1.1.6 JLM Optical-Model Potential .......................... 43

2.3.1.2 Zero-Range Approximation .......................................... 53

2.3.1.3 Finite-Range Approximation ....................................... 54

2.3.1.4 Non-Locality Correction ............................................ 58

2.3.1.5 Neutron Form Factor ................................................ 64

2.3.2 Adiabatic Deuteron Breakup Approximation (ADBA) ................... 70

2.4 Momentum Matching .................................................................. 75

2.5 Summary ................................................................................. 76
CHAPTER 3

EXPERIMENTAL SETUP AND CALIBRATION ......................................... 79

3.1 Overview ................................................................................ 79

3.2 MWDC Detectors ....................................................................... 84

3.2.1 Principle of MWDC Detector ................................................ 84

3.2.2 Position Calibration ............................................................ 86

3.3 LASSA Detector Array ................................................................ 94

3.3.1 Overview ........................................................................ 94

3.3.2 Geometric Setup ................................................................. 95

3.3.3 Silicon Strip Detector Array .................................................. 95

3.3.3.1 Overview ............................................................ 95

3.3.3.2 Energy Calibration .................................................. 97

vi

3.3.3.3 Particle Identification ................................................ 98

3.3.3.4 Position Calibration ................................................ 99

3.3.4 CSI(T1) Crystals ................................................................ 101

3.3.4.1 Detectors ............................................................ 101

3.3.4.2 Energy Calibration ................................................ 102

3.3.4.3 Particle Identification ............................................. 104

3.4 8800 Spectrometer ................................................................... 114

3.4.1 Overview ...................................................................... 114

3.4.2 Cathode Readout Drift Counters ........................................... 115

3.4.3 Ion Chamber .................................................................. 116

3.4.4 Plastic Scintillators .......................................................... 116

3.4.5 Summary ...................................................................... 117

3.5 Data Acquisition Electronics ....................................................... 124
CHAPTER 4

EXTRACTION OF AN GULAR DIFFERENTIAL CROSS SECTIONS ............... 126

4.1 Overview ............................................................................... 126

4.2 Deuteron Spectra ...................................................................... 126

4.3 Extraction of Angular Differential Cross Sections ............................... 128
CHAPTER 5

EXTRACTION OF SPECTROSCOPIC FACTORS ....................................... 141

5.1 Overview ................................................................................ 141

5.2 12C(d,p)l3C (g.s.) Reaction ........................................................... 142

5.3 l3C(p,ti)'2C (g.s.) and p(‘3C,d)‘-2C (g.s.) Reactions ................................ 154

5.4 13C(p,ti)‘2C (2") and p(l3C,d)12C (2*) Reactions .................................. 158

5.5 l"I3(ti,p)l ‘B (g.s.), “B(p,d)‘°B (g.s.), and p(‘ ‘B,d)‘°E (g.s.) Reactions 162

5.6 9Be(d,p)10Be (g.s.) and p(loBe,d)9Be (g.s.) Reactions ............................ 168
CHAPTER 6

SUMMARY ...................................................................................... 173

BIBLIOGRAPHY ............................................................................... 178

vii

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3.1

3.2.

3.3

3.4

4.1

4.2

5.1

5.2

5.3

LIST OF TABLES

An overview of the input parameters and options for TWOFNR ................ 13
The parameters of global nucleon potentials ......................................... 23

Deuteron global parameters. N=neutron number, Ezdeuteron laboratory

energy in MeV. For Daehnick potential, ,6 = 4&6) 2 , 11,-: (54421132 ,

where M ,- = magic numbers (8,20,28,50,82,126) ................................... 30

Parameters for the global nucleon-nucleus optical potential of CH89 [Var91]... 38
Parameters for J LM potentials ......................................................... 48

The neutron potentials and the binding energies of the neutron form factors

in the reactions in this experiment ..................................................... 65
Summary of the input parameters used in TWOFNR ............................... 78
The properties of the secondary beams and the targets ............................. 81

Geometric setup of the telescopes and the configuration of the silicon

strip detectors ............................................................................ 105
The calibrated parameters in Equation 3.3.1 .......................................... 105
Characteristics of the S800 spectrometer ............................................ 120

Contributions to the energy resolution of the single neutron pickup
reactions in inverse kinematics ......................................................... 135

Experimental angular differential cross sections and statistical errors .......... 136

List of references and Spectroscopic factors for the 12C(d,p)13C (g.s.)
reactions ................................................................................... 147

List of references and spectroscopic factors for the ‘3C(p,d)12C (g.s.)
and p(13C,d)12C (g.s.) reactions ........................................................ 157

Extracted spectroscopic factors of the lp3/2 neutron from the
l3C(p,ti)‘2C (2”) and p('3C,d)'2C (2*) reactions .................................... 160

viii

5.4

5.5

Extracted spectroscopic factors of the 1pm neutron from the reactions

of ”B(p,d) l"I3 (g.s.), ‘°B(d,p)“B (g.s.) and p(“B,d)‘°B (gs)

Extracted spectroscopic factors of the 1pm neutron from the reactions

of p(loBe,d)9Be (g.s.), and 9Be(d,p)]0Be (g.s.)

ix

166

.171

1.1

1.2

1.3

2.1

2.2

2.3

2.4

2.5

2.6

LIST OF FIGURES

Velocity diagrams for normal kinematics (a) and inverse kinematics (b) as in

(p,d) reactions. V00" is the velocity of the center of mass in the laboratory frame;

leab and Vd cm are the deuteron velocities in the laboratory frame and the center
of mass, respectively; Blob and 66", are the emitted angles in the laboratory
frame and the center of mass, respectively ............................................. 6

The deuteron emitted angles in center of mass vs. emitted angles in the laboratory

frame. The solid line presents the inverse kinematic reaction of p(l3C,d)lzC (g.s.)
at bombing energy of 47.9 MeV per nucleon; the dashed line stands for the

reaction of l3C(p,d)12C (g.s.) at proton energy of 48.3 MeV ........................ 7

Kinematic broadening vs. angles in the laboratory frame. The solid line presents
the inverse kinematic reaction of p(l3C,d)lzC at bombing energy of 47.9 MeV per

nucleon; the dashed line stands for the reaction of l3C(p,d)12C at proton energy of

48.3 MeV . ................................................................................. 8

The target nucleus A is composed of the core nucleus B and one neutron n.
The proton picks up a neutron to form the deuteron. O is the center of mass

of nucleus B and 7,, points to the neutron; O’ is the center of mass of nucleus
A; F and 7,, are the proton coordinates relative to the neutron and the center

of mass of nucleus A, respectively. R is the coordinate of the deuteron
center relative to nucleus B .............................................................. 17

Proton global optical-model potentials of 13C at incident energy of 12.5 MeV... 24

Calculations by different global optical-model potentials for the reaction of

13C(p,p)13C (g.s.) at incident energy of 12.5 MeV compared with the data
[Wel78] ..................................................................................... 25

Proton global optical-model potentials of 13C at incident energy of 30.95
MeV ........................................................................................ 26

Calculations by different global optical-model potentials for the reaction of
l3C(p,p)l3C (g.s.) at incident energy of 30.95 MeV compared with the data
[Bar88] .................................................................................... 27

Deuteron global optical-model potentials of 12C at incident energy of 11.8
MeV ......................................................................................... 31

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

Calculations by different global optical-model potentials for the reaction of

12C(d, d)12C (g.s.) at incident energy of 11.8 MeV compared with the data
[Fit67] ....................................................................................... 32

Deuteron global optical-model potentials of 12C at incident energy of 34.4
MeV ........................................................................................ 33

Calculations by different global optical-model potentials for the reaction of

”C(d, d)‘2C (g.s.) at incident energy of 34.4 MeV compared with the data
[New67] ................................................................................... 34

Comparison of the CH89 proton potentials of 13C with the proton potentials
of Menet and Perey & Perey at incident energy of 12.5 MeV ...................... 39

Calculations for proton elastic scattering on 13C at incident energy of
12.5 MeV using the potentials of Menet, Perey & Percy, and CH89
compared with the data [Wel78] ...................................................... 40

Comparison of the CH89 proton potentials on 13C with the proton potentials
of Menet and Perey & Perey at incident energy of 30.95 MeV .................... 41

Calculations for proton elastic scattering on 13C at incident energy of
30.95 MeV using the potentials of Menet, Perey & Perey, and CH89
compared with the data [Bar88] ....................................................... 42

Comparison of the JLM proton potentials of 13C with the proton potentials
of Menet and CH89 at incident energy of 12.5 MeV ............................... 49

Calculations for the l3C(p,p)‘3C (g.s.) reaction by JLM, Menet and CH89
potentials at incident energy of 12.5 MeV compared with the data [Wel78] ....... 50

Comparison of the JLM proton potentials of 13C with the proton potentials
of Menet and CH89 at incident energy of 30.95 MeV ............................... 51

Calculations for the l3C(p,p)l3C (g.s.) reaction by JLM, Menet and CH89
potentials at incident energy of 30.95 MeV compared with the data [Bar88] 52

Calculations for reaction of l3C(p,d)12C (g.s.) using finite-range

approximation (solid line) and zero-range approximation (dashed line) at

incident energies of 15 MeV and 48.3 MeV. The cross sections at proton

energy of 15 MeV have been multiplied by 10 so that the calculations at

the two energies can be seen more clearly ............................................. 56

xi

2.19

2.20

2.21

2.22

2.23

2.24

2.25

2.26

2.27

2.28

2.29

3.1

Finite-range DWBA calculations for reaction of l3C(p,d)12C (g.s.) by

TWOFNR (solid line) and DWUCKS (dashed line) at incident energies of

15 MeV and 48.3 MeV. The cross sections at proton energy of 15 MeV

have been multiplied by 10 ............................................................. 57

Comparison of the local (dashed line) and non-local (solid line) proton
potentials (CH89) of 13C at proton energy of 48.3 MeV ........................... 61

Calculated differential cross section for l3C(p,d)12C at proton energy of
48.3 MeV by non-local proton potential (solid line) increased 12% at the
forward angles compared to that by local proton potential (dashed line) 62

Calculated differential cross sections for l3C(p,d)]2C at incident energy of 15
MeV using non-local proton potential (solid line) and local proton potential
(dashed line) ............................................................................... 63

Neutron form factors for the reactions of (a) p(13C,d)12C (g.s.),
(b) p('3C,d)‘2C (2"), (c) p(”B,d)'°B (g.s.), and (d) p(‘°Be,d)9Be (g.s.) 66

Calculations on the variation of neutron radius parameter r0, where the neutron

diffuseness a0 is fixed to 0.65 fm and the spin-orbit strength Vso is 0.0 MeV.
The cross sections at proton energy of 15 MeV are multiplied by 10 ............ 67

Calculations on the variation of neutron diffuseness do, where the neutron

radius parameter r0 is fixed to 1.25 fin and the spin-orbit strength Vso is 0.0
MeV. The cross sections at proton energy of 15 MeV are multiplied by 10 ..... 68

Calculations on the variation of neutron spin-orbit strength V50, where the neutron

radius To and diffuseness a0 are fixed to 1.25 fin and 0.65 fm separately. The cross
sections at proton energy of 15 MeV are multiplied by 10 .......................... 69

Comparison of Daehnick global deuteron potential (dashed line) with the
adiabatic deuteron potential (solid line) constructed by CH89 potentials,

for 12C at Ed: 49.2 MeV ................................................................ 72

Comparison of ADBA (solid line) and DWBA (dashed line) calculations for
reaction of l3C(p,d)12C at proton energy of 48.3 MeV. The ADBA increases

the cross section at the forward angles and faster fall off than DWBA .......... 73
Comparison of ADBA (solid line) and DWBA (dashed line) calculations

for reaction of l3C(p,d)12C at proton energy of 15 MeV ........................... 74
Schematic diagram of A1200 ........................................................... 82

xii

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.14

3.15

Schematic diagram of the facilities in 3800 vault .................................. 83

Schematic drawing of wire plane of MWDC detector. All the anode wires are

connected to a micro-strip delay line, which has two timing outputs T1 and T2.
Every other field wire is connected together to form two groups and gives the

left—right ambiguity signals E1 and E2 respectively 88

An enlarged drawing of the ions drifting between the wires. The positive

ions drift toward the field wires and the negative ions drift toward the

anode wire. The term dt is the drift time of the negative ions from the

hit point to the anode wire ................................................................ 89

Spectrum of the wire sequential number k. Each individual sharp peak
indicates the wire with its sequential number ......................................... 90

Spectrum of drift time (it, where the time is scaled in units of microseconds ..... 91

Spectrum of E2 versus E1. There are two groups of particles. The left-right
ambiguity can be clarified by which side yields larger signal than the other 92

(a) Patterns on the mask used to calibrate the MWDC detector. The distance
between adjacent small holes is 2.54 mm; the distance between adjacent big
holes is 10.2 mm; (b) The two-dimensional position Spectrum of the mask.
The corresponding areas are surrounded by the dotted rectangle in (a)

and (b) ...................................................................................... 93
Structure of LASSA telescope . ...................................................... 106
Schematic of the geometric setup .................................................... 107

One double-sided silicon strip detector with the flat printed circuit board
cables ..................................................................................... 108

Calibration curve for silicon strip detector, by which the channel readout of
the silicon strip detector is converted to particle energy in units of MeV.
This curve stands for the No. 6 strip of EF detector in telescope 3 108

Particle identification using the energy spectrum of BBB vs. E51: for
telescope 7 ............................................................................... 109

Geometry calibration system composed of one optical telescope and a mirror
mounted on a turntable with two orthogonal axes that rotate in horizontal and
vertical planes. The center of the mirror is the position of the target in the
experiment and the optical telescope is mounted in the beam line ............... 110

The shape of CsI(Tl) crystal ........................................................... 110

xiii

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

4.1

4.2

4.3

4.4

Schematic of the CsI calibration. The deuteron emitted angle is determined

by the pixel on DE and EF(EB) silicon strip detectors. The deuteron emitted
energy is obtained by kinematic calculation. The deuteron deposited its

energy into the CsI crystal after going through target, window foil, DE

and EF Silicon strip detectors .. ......................................................... 111

Calibration for CsI detector, by which the channel readout of the CsI
Detector is converted to particle energy in units of MeV. This figure
shows the calibration for the No. 3 crystal in telescope 3 .......................... 112

Particle identification in the energy Spectrum of 4E DE + E EF vs. Emm,
for telescope 3 ........................................................................... 113

Schematic of the focal plane detector of 8800 spectrometer. It consists
of two CRDC detectors, one ion chamber, and four plastic scintillators ........ 119

Schematic of the CRDC detector. A particle ionizes the gas as it passes

through the detector. The electrons drift to the anode wire where they

are collected. The induced image charges on the cathode pads provide

horizontal position information. The drift time of the electrons to the

anode wire provide vertical position information ................................... 121

(a) Patterns on the mask. (b) Position spectrum of the mask placed in

front of the first CRDC detector ...................................................... 122
Spectrum of the energy deposited in E1 scintillator versus the

time-of-flight for the p(1 lB,d)lOB reaction ........................................... 123
Schematic of the electronics .......................................................... 125

Deuteron energy spectrum of the p(13C,d)12C reaction at the laboratory
angle of 19° measured by telescope 7 ................................................ 132

Deuteron energy spectrum of p(1 iB,d)10B reaction at the laboratory
angle of 13° measured by telescope 7 ................................................. 133

Deuteron energy spectrum of p(loBe,d)9Be reaction at the laboratory
angle of 15° measured by telescope 7 ................................................ 134

The measured angular differential cross section of p(13C,d)12C (g.s.)
(open circle) reaction compared to the published data of Ref. [Cam87]
(closed circle) and Ref. [Sco70] (diamonds) .......................................... 137

xiv

4.5

4.6

4.7

5.1

5.2

5.3

5.4

5.5

5.6

5.7

The measured angular differential cross section of p(13C,d)12C (2+)

(open circle) reaction compared to the published data of Ref. [Cam87]
(closed circle) and Ref. [Sco70] (diamonds) ........................................ 138

The measured angular differential cross section of p(1 lB,d)loB (g.s.)
reaction ................................................................................... 139

The measured angular differential cross section of p(loBe,d)9Be (g.s.)
reaction .................................................................................. 140

Spectroscopic factors for 12C(d,p)l3C (g.s.) and l3C(p,d)]2C (g.s.) reactions
extracted from the literatures (see Table 5.1 and Table 5.2) ..................... 148

Angular distributions for 12C(d,p)l3C (g.s.) reactions for beam energy from 7 to
56 MeV: solid lines present ADBA (JLM); dotted lines present ADBA (CH89);
dashed lines present DWBA. Each distribution is displaced by factor of 10 from
adjacent distributions. The overall normalization factor is 1 for the 19.6 MeV
data. References are listed in Table 5.1 .............................................. 149

Extracted spectroscopic factors in the present work for 12C(d,p)l3C (g.s.),

l3C(p,d)12C (g.s.), and p(l3C,d)'2C (g.s.) reactions. The dashed lines represent the
shell model prediction of Cohen and Kurath [Coh67] of 0.62. See text for detail
explanation ........................................................................ 150

Comparison of the existing measurements of 12C(d,p)13C (g.s.) reaction for
deuteron energy at 4.5 MeV, a [Gur69], b [Gal66], and c [Bon56] .............. 151

Comparison of the existing measurements of 12C(d,p)l3C (g.s.) reaction for
deuteron energy at 11.8 MeV [Sch64], 12.0 MeV a [Lan88], 12.0 MeV b
[Sch67], and 12.4 MeV [Ham6l] ..................................................... 152

Comparison of the existing measurements of 12C(d,p)]3C (g.s.) reaction
for deuteron energy at 14.7MeV [Ham61], 14.8MeV [MchS], and 15 MeV
[Dar73] ................................................................................... 153

Angular distributions for l3C(p,d)12C (g.s.) and p(l3C,d)12C (g.s.) reactions for

beam energies from 35 to 65 MeV: solid lines present ADBA (JLM); dotted lines
present ADBA (CH89); dashed lines present DWBA. The calculations have been
normalized by the spectroscopic factors. Each distribution is displaced by factor
of 10 from adjacent distributions. The overall normalization factor is 1 for the
65.0 MeV data ........................................................................... 156

XV

5.8

5.9

5.10

5.11

5.12

5.13

5.14

6.1

Calculations for reactions of l3C(p,d)12C* (4.439MeV) and p(13C,d)]2C*

(4.439MeV) [Toy95, Cam87, Tak68]; solid lines present ADBA (JLM) potentials;
dotted lines present ADBA (CH89). The calculations have been normalized by the
spectroscopic factors. Each distribution is displaced by factor of 10 from adjacent
distributions. The overall normalization factor is 1 for the 55.0 MeV data ...... 159

Extracted spectroscopic factors of the reactions 13C(p,d)12C* (4.439MeV)

(circle) and p(13C,d)lzC* (4.439MeV) (square). The dashed lines represent
the shell model prediction of 1.12 by Cohen and Kurath [Coh67] ............... 161

Calculations of 10B(d,p)”B (g.s.) reaction at 10.1MeV [Hin62], 12.0 MeV
[Sch67], 13.5 MeV [Bar65], and 28 MeV [Slo62]: solid lines present ADBA
(JLM); dotted lines present ADBA (CH89). The calculations have been
normalized by the Spectroscopic factors. Each distribution is displaced

by factor of 10 from adjacent distributions. The overall normalization
factor is 1 for the 28.0 MeV data ......................................................... 164

Calculations of llB(p,d)10B (g.s.) (closed points) p(l lB,d)mB (g.s.) (open

points): solid lines present ADBA (JLM); dotted lines present ADBA (CH89).
The calculations have been normalized by the spectroscopic factors. Each
distribution is displaced by factor of 10 from adjacent distributions. The

overall normalization factor is 1 for the present data ............................... 165

Extracted spectroscopic factors for the reaction of p(l lB,d)'0B (g.s.) (open
square), llB(p,d)mB (g.s.) (open circle), and l0B(d,p)l lB (g.s.) (closed circle).
The dashed lines represent the shell model prediction of 1.09 by Cohen and
Kurath [Coh67] .......................................................................... 167

ADBA calculations based on CH89 potentials for the reactions of
9Be(d,p)lOBe (g.s.) (closed circle) [Ze101, Gen00, Va387, Sch64, Dar76,
And74, 81062] and p(lOBe,d)9Be (g.s.) (open circle). The calculations have
been normalized by the spectroscopic factors. Each distribution in closed
circle is displaced by factor of 10 from adjacent distributions. The overall

normalization factor is 1 for the data at 11.0 MeV. The present data is
reduced by a factor of 100 ............................................................. 170

Extracted spectroscopic factors for the reactions of p(10Be,d)9Be (g.s.) (open

square), and 9Be(d,p)'oBe (g.s.) (closed circle). The dashed lines represent
the shell model prediction of 2.35 by Cohen and Kurath [Coh67] .............. 172

Comparison of the extracted spectroscopic factors with the predictions of
the modern shell model [BroO4] for 79 nuclei ranging from Li to Cr [Tsa05].
Good agreement with most isotopes except Ne, F, and Ti isotopes .............. 177

xvi

CHAPTER 1

INTRODUCTION

1 . 1 Motivation

The Study of nuclei far away from stability has been the focus of nuclear study in
recent years. We want to know the extent to which the nuclear shell model theory is valid
for nuclei beyond the stability limits. Such understanding is especially important since
unstable nuclei are essential components in the nuclear synthesis process.

Since the discovery of the shell model, which explains many structural properties of
the nuclei, transfer reactions have been used to study the configuration of the valence
nucleons. Spectroscopic factors (SF) are important quantities that tell us the structure of
the single nucleon orbit. In this work, we define the Spectroscopic factor as the ratio of
the experimental cross section from the transfer reaction to the theoretical calculation
based on a reaction model that assumes the orbital fully occupied by the transferred
nucleon. Since unstable nuclei cannot be made into targets, the transfer reactions must be
performed in inverse kinematics using rare isotope beams.

Currently there are unanswered questions in the extraction of spectroscopic factors. In

the reaction theory, which uses the Distorted-Wave Born Approximation, DWBA, a fast
one-step direct process of less than 10'22 sec is assumed. Elastic scatterings are used to

describe both the entrance and the exit channels. It is usually believed that an accurate

optical-model potential, which is derived from the best fitting of the elastic scattering

data, would give the correct incoming and outgoing wave functions and hence the correct
extraction of the spectroscopic factor. Unfortunately, such practice has failed to provide a
consistent extraction of the spectrosc0pic factors in part due to the ambiguity in the
parameters needed to describe the optical-model potential. In contrast, there are also
arguments that superior results would be obtained if global optical-model potentials that
describe a range of nuclei and incident energies are used instead. There are statements in
literature that such an average optical-model potential tends to give more reasonable
Spectroscopic factors than individual potential [Sch67]. However, such statements have
not been well quantified. One purpose of this thesis is to compare the different strategies
and find a reliable method to extract consistent spectroscopic factors.

Another goal of this work is to study the structure of deformed unstable nuclei such
as 10Be via the (p,d) reaction. It was envisioned that this would become the starting point
of a series of studies of the N=6 isotones in inverse kinematics. The valence neutron of
10 . . . 9 10 .

Be had been prevrously studied vra the Be(d,p) Be reaction. The extracted

spectroscopic factors, however varied from 0.97 to 2.07, in some cases differing from the

theoretical value of 2.35 based on the shell model. We want to know if there is new

physics in 10Be that makes it different from the shell model expectation. Furthermore,
understanding the structure of 10Be may help us to understand the structure of more
neutron-rich isotopes of Beryllium such as 11Be.

This is the first time that the secondary radioactive beam of l0Be was used to perform

the (p,d) reaction in inverse kinematics. For these experiments, we used a high-angular

and high—energy resolution detector, Large Area Silicon Strip Array (LASSA), to detect

the deuterons emitted in the reactions. In addition, we also measured (p,d) reactions on
another N=6 isotone llB as well as '3C. This letter reaction was used to obtain energy

calibrations. High quality data were obtained in this reaction and used in our systematic
studies to find a strategy to extract the spectroscopic factor.

Various properties of the inverse kinematic reaction are analyzed in the next section.
The theoretical background is presented in chapter 2. This latter chapter includes the
description of the theoretical spectroscopic factor (Section 2.2) and of the reaction
theories (Section 2.3). Specifically, the theories of distorted-wave Born approximation
(DWBA) and adiabatic deuteron breakup approximation (ADBA) are introduced in
Section 2.3.1 and Section 2.3.2 respectively. Detailed descriptions of the experimental
setup are provided in chapter 3, which includes the descriptions of the various detectors
and electronics. Chapter 4 describes the analyses of the deuteron spectra and the
extraction of the angular differential cross sections. Theoretical calculations and the
extraction of the spectroscopic factors are explained in chapter 5. Chapter 6 gives the

summary of this thesis.

1.2 Inverse Kinematics

Nuclear reactions involving nucleon transfer between stable beams and target nuclei
have been a very useful source of nuclear structure information, and many theoretical
tools have been developed to extract spectroscopic information. However, for the Study
of radioactive nuclei far from the stable region, which has become the new focus of

studies in nuclear astronomy and nuclear structure beyond the shell model in recent years,

inverse kinematics becomes necessary since the radioactive nuclear targets, especially
those with a very short half-life, are usually not available. Thus, transfer reactions
induced by radioactive beams on proton and deuteron targets have great potential for
probing single-particle structures in new regions [For99, Win01, Reh98, Oga99].

One advantage of the inverse kinematic reaction is that it is relatively easy to cover
the forward scattering angle in the center of mass. In the normal kinematic reaction,
where the light projectile bombards the heavy target, the small scattering angle in the
center of mass can only be covered at the most forward angle in the laboratory frame.
Figure 1.1(a) shows the velocity diagram for the (p,d) reaction in normal kinematics,

cm lab

where V0 is the velocity of the center of mass in the laboratory frame; Vd and

Vdcm are the deuteron velocity in the laboratory frame and in the center of mass; 61a},
and 66m are deuteron emitted angles in the laboratory and in the center of mass. In the
inverse kinematic reaction, as shown in Figure 1.1(b), the deuteron scatters backward in
the center of mass. Smaller (96m can be obtained at relatively large 610,). Figure 1.2
shows the relations between the deuteron emitted angles in the laboratory frame and the

emitted angles in the center of mass for the reactions of p(13C,d)12C g.s. (solid line) and

13C(p,d)12C g.s. (dashed line) at the equivalent bombing energy respectively. The

detector in the inverse kinematic reaction covers smaller angles in the center of mass than
that in a normal kinematic reaction at the same laboratory angle.
One disadvantage of inverse kinematics is the kinematic broadening. Figure 1.3

shows the kinematic broadening vs. the emitting angle in the laboratory frame. The solid

line presents the inverse kinematic reaction of p(]3C,d)12C at bombing energy of 47.9

MeV per nucleon; the dashed line stands for the 13C(p,d)]2C reaction at a proton energy

of 48.3 MeV. Except for the very forward angles, the kinematic broadening is much more
severe for the inverse kinematic reaction than the normal kinematic reaction. For
example, the kinematic broadening increases dramatically from 650 keV at 30 degree to
1.27 MeV at 35 degree Therefore in this reaction, deuterons should not be detected
beyond 35 degree in the laboratory frame. For the forward angles, detectors with high
angular resolution as well as high energy resolution are required. Silicon strip detectors
are widely employed to achieve high angular resolution and high energy resolution. The

techniques of using these detectors will be discussed in Section 3.3.

 

 

I W
\
I i ‘ \
’ s
I x
’ \
r
’ lab ‘3
.’ Vd
6 \
: lab V cm \
I r I g
Beam t cm I
\‘ V0 8 cm ’r
\ I
\ I
\ l
x l/
\\ I,
‘ r
s ‘ I
\ ‘ ~ 4.. ’ a ’
( a )
2” “\
I ‘ \
’ \
I \
I \
’ \
’ \
lab ’ cm \
Vd , Vd r‘
I I
L ' i |
r I I I
Beam 9 t‘ V cm 9 i
lab r 0 cm I
.. I
\ I
\ l
x l/
x \ I
x I ’
\ ‘ ’
\ I

(b)

Figure 1.1: Velocity diagrams for normal kinematics (a) and inverse kinematics (b) as in

(p,d) reactions. VOC'" is the velocity of the center of mass in the laboratory frame; leab and

Vd cm are the deuteron velocities in the laboratory frame and the center of mass, respectively;

61a}, and 6m are the emitted angles in the laboratory frame and the center of mass,
respectively.

 

60

9am (deg)
\
\

20— /

 

 

 

0 10 20 30 40
elab (deg)

Figure 1.2: The deuteron emitted angles in the center of mass vs. emitted angles in the
laboratory frame. The solid line presents the inverse kinematic reaction of p(l3C,d)12C
(g.s.) at bombing energy of 47.9 MeV per nucleon; the dashed line stands for the reaction
of l3C(p,d)12C (g.s.) at proton energy of 48.3 MeV.

 

Kinematic broadening
(MeV/deg )

 

 

 

 

Figure 1.3: Kinematic broadening vs. angles in the laboratory frame. The solid line
presents the inverse kinematic reaction of p(l3C,d)12C at bombing energy of 47.9 MeV

per nucleon; the dashed line stands for the reaction of l3C(p,d)12C at proton energy of
48.3 MeV.

CHAPTER 2
THEORETICAL DESCRIPTION

2.1 Overview

The main goal of this thesis is the extraction of the neutron spectroscopic factors from
measurements. The experimental spectroscopic factor is defined as the ratio of the
experimental differential cross section to the calculated differential cross section based on
a reaction model that assumes the relevant orbit is fully occupied. The extraction of the
experimental differential cross sections measured in this thesis will be described in
Chapter 4. This chapter mainly describes how the theoretical differential cross sections
are calculated. The theoretical nucleon spectroscopic factor will be introduced in Section
2.2. The most widely used models, the distorted—wave Born approximation (DWBA) and
the adiabatic deuteron breakup approximation (ADBA), will be discussed in Section 2.3.
The effects of different input parameters including the choices of optical-model potentials
in DWBA will be covered in the subsections of Section 2.3. At the end of this chapter, a
list of standard input parameters for the DWBA and ADBA calculations will be

proposed.

2.2 Theoretical Spectroscopic Factor

In the theory of the shell model, the valence nucleon in the nucleus is described as a
single-particle state of a particular orbit. Each orbit is assigned the number n, l, and j
corresponding to the node number, the orbital momentum, and the total spin momentum
of the nucleon. The assumption that the nucleon occupies a pure single-particle state is an
idealization, which is true only in few cases in real nuclei. Due to the interactions among
other nucleons, each nucleon may occupy several single-particle states. The occupation
of a nucleon in a pure single-particle state is called the spectroscopic factor, which
contains the information of the nuclear structure and how well the shell model theory
describes the real nuclei. Thus the nucleon Spectroscopic factor is among the most
fundamental tests of shell model theory [Ban85].

In a nucleus composed of A nucleons, the spectroscopic factor can be deduced from

the expansion of the wave function I// (A) in terms of a summation over the complete set
of single-particle states ¢,,,j(f,,) and the states III (B) of the residual core nucleus B is

composed of A-l nucleons [Gle04]:

wfij‘tam = Bzflflnfl (A. B')A[¢nlj(7n)lfl13'(3)] 31" (2.2.1)

where A is an antisymmetrization operator, ,BnIJ-(A,B') are coefficients of fractional

parentage and their values depend on the detailed structure of the nuclear wave function.

10

The square bracket denotes vector coupling:

[Q10 (mil/13' (3) MA = Z CJB'jJA ¢m

- M'
M'mMA nlj(r") WJB'(B) (2'22)

Mm

The spectroscopic factor for a specific single particle state (n1 j) is:
5",,- = fl,,,,-2(A,B') (2.2.3)

For the pickup (p,d) reaction, the spectroscopic factor is related to the experimental

angular differential cross section and the theoretical calculation by:

 

 

 

“(6') = "U (“a”) (2.2.4)
d9 d9 theory
do(6) . . . .
where rs calculated assuming the neutron m the exact state (121]). The
theory

theoretical calculations are performed either in the distorted-wave Born approximation
(DWBA) theory or in the adiabatic deuteron breakup approximation (ADBA) theory. The
introduction of the theories will be in Section 2.3. In this work, all the theoretical
calculations for (p,d) and (d,p) reactions are performed using the code TWOFNR, which
was initially developed by M. Igarashi in 1977 [Iga77]. This code is relatively easy to use
since it supplies multiple options with default values for every step and component in the
calculations. All the inputs, including the parameters and option choices, are converted
automatically into a standard input file for TWOFNR by a partner program FRONT. The
input parameters and options are listed in Table 2.1. For example, the integration ranges
and the number of partial waves can be specified by the user or the default values can be
adopted. The user can choose the global optical-model potentials for proton and deuteron
or specify the parameters for individual potential. When the ADBA theory is employed,

the Johnson-Soper adiabatic potential for deuteron can be constructed using three

11

different nucleon-nucleus potentials. For the application of JLM potential, users can input
their own parameters for the target density and potential scaling factors following the
prompts of the program. There is a switch either to zero-range approximation or to finite-
range approximation. If the finite-range approximation is chosen, the finite-range factor
can be the default value or be specified. The same strategy is also applied to the options
of neutron binding potential, the vertex constant, and non—locality correction.

We choose to use TWOFNR because there are many default options available and it
is easier to perform many calculations in a systematic study. Another popular finite-range
DWBA code is DWUCKS [Kunz], which performs finite-range calculations with
deuteron wave function instead of the finite-range approximation in TWOFNR. We
compared them in Section 2.3.1.3 and found that the results from these two programs are
very close to each other for the same input parameters (see Figure 2.19). We believe

essentially the same results will be calculated if different codes than TWOFNR are used.

12

Table 2.1: An overview of the input parameters and options for TWOFNR

 

Integration ranges

Specified or default value (30 fm in 300 steps)

 

Number of partial waves

Specified or default value (70)

 

Proton potential

Choose built-in options of global optical-model potentials:
Bechetti-Greenlees;
Chapel-Hill 89 (CH89);
Perey & Perey;
Menet;
JLM;

Or Specified parameters for V,, r,, av, WV, W5, rw, aw, V50,
r50, ago, and RC ’

 

Deuteron potential

Choose built-in options of global optical-model potentials:
Lohr-Haeberli;
Perey & Perey;
Daehnick;
Johnson-Soper adiabatic (ADBA);

Or specified parameters for V,, r,, av, WV, W5, rw, aw, V50,
r30, ago, and RC

 

Johnson-Soper
adiabatic potential

Choose built-in options of global optical-model potentials:
Bechetti-Greenlees;
Chapel-Hill 89 (CH89);
JLM;

 

Choose built-in options:

 

Target density Negele form;
for JLM potential Specify rrns radius;

Modified Harmonic oscillator form;
JLM potential scaling A Specified or default values (Av=1.0, Aw=0.8)

 

Neutron binding potential

Specified
or default values (r0=1.25 fm, ao=0.65 fm, VSO=6 MeV)

 

Zero-range
approximation

Use or not

 

Finite-range
approximation

Use or not
If use, finite range factor can be specified or choose default
value of 0.7457 fm

 

Vertex constant D02

Specified or default value of 15006.25 MeV2 - fm 3

 

 

Non-locality correction

 

Use or not.
If use, non-locality range can be specified or choose
default value (0.85 fm for proton; 0.54 frn for deuteron).

 

l3

 

2.3 Reaction Theory

2.3.1 Distorted-Wave Born Approximation (DWBA)

Transfer reactions have been an important tool in the study of nuclear structure. The
results obtained from the studies of the pickup A(p,d)B and stripping B(d,p)A reactions,
involving single neutron transfer, help to validate the nuclear shell model by identifying
the single-particle states. To a large extent, the (p,d) reaction can be understood as one in
which the neutron is removed from a single particle state of the target nucleus A. In the
(d,p) reaction the neutron in the deuteron is deposited to a sin gle-particle state of the final
nucleus A. Thus, the theoretical description of the (d,p) reaction is similar to that of the
(p,d) reaction.

In the pickup reactions, A(p,d)B, where A=B+n, a neutron in a single-particle state in
A, is picked up by the incident proton to form the deuteron. The process is illustrated in
Figure 2.1. The transition amplitude for this reaction under the distorted-wave Born

approximation (DWBA) theory is written as [Gle04]:

T(#pJAMA_)#dJBMB):ZCJBjJ/I C(1/2)(1/2)(1)Cl(1/2)j
1'

MijMA Hunt/Jr! mlpmnj
1 (2.3.1)
l n];

14

where

Bml(kp,kd)=i-l(21+l) —((-l/2)J‘Z) *(kd RWMIU'; )

.(v +Va—U mam-1‘ ’(k .r )dFd?
W! p p p p p n (232)

51/2 A _ _
Snlj :JIW: *(A)A[WJB(B)¢an(;}190 531).“ dBdrn

where A and B refer to the nucleon coordinates and Spins of nucleus A and B; 7,, and

’17

are the coordinates of neutron and proton; 7,, and R are the relative and center-of-
mass coordinates of the deuteron; 1P(+)(i€p,Fp) is the distorted-wave function

describing the elastic scattering of the incoming proton by the proton optical-model

potential U I); the distorted-wave function zd(-)*(l;d,R) describes the elastic scattering

of the emitted deuteron by the deuteron optical-model potential U d? i/I(A) is the wave
function of the target nucleus A; (11(3) is the wave function of the core nucleus B; and

¢d (f) is the internal wave function of the deuteron. The term VP" + VpB — U p is called

the residual interaction, where V is the interaction between the proton and neutron,

pit
and VpB is the interaction between the proton and the remaining B nucleus. The term

¢nlj(fn ,6") is the neutron wave function in Specific single-particle state (n1 3' ), which

is also called the neutron form factor:

61"" (7n 5n)= [(0,210 )Xt/2(0,.)l"' (2.3.3)

nlj

raj/(r,)=u,,,(r,,>Y;”’(r,,) (2.3.4)

15

Assuming VpB and U p approximately cancel each other in Equation 2.3.2 [Aus70,
Sat71], the 8;", (12p , 12d ) becomes

3;"‘(i'r‘p kd)=i_l(21+l)_(1/2)JZ()*(kd ,R)¢""(r)

‘ (+) __ (2.3.5)
Vpn'¢d(r)'Zp (kp’rp)d?nd;p
The cross section is
do mmd p de
—(fl JAMA *fldJBMB)= (2.3.6)

where rm; and mp are the reduced masses.

In order to obtain the distorted-wave functions of 2’de and 1p”) , we need the

optical-model potentials for the deuteron and proton. In the next section, the choices and

detailed descriptions about the optical-model potentials will be discussed.

16

 

Figure 2.1. The target nucleus A is composed of the core nucleus B and one neutron n.
The proton picks up a neutron to form the deuteron. O is the center of mass of nucleus B

and 7,, points to the neutron; O’ is the center of mass of nucleus A; F and 7,, are the

proton coordinates relative to the neutron and the center of mass of nucleus A,
respectively. R is the coordinate of the deuteron center relative to nucleus B.

17

2.3. 1. 1 Optical-Model Potentials

2.3.1.1.1 Overview

The Schrodinger equation of the collision system of a + b can be written as:
(H - E)‘I’ = 0 (2.3.7)
where H includes the intrinsic energy HO, the kinetic energy T, and the potential U
between a and b:
H =H0+T+U (2.3.8)
The Schrodinger equation is separable into the nuclear intrinsic coordinates and relative
coordinates so that the solution w can be written as a product of the nuclear intrinsic
wave function WaWb and a relative wave function ¢(f), which satisfies the optical-
model Schrodinger equation:
(T+U —E)¢(F) =0 (2.3.9)
Since U depends only on the relative coordinates of the two nuclei, it produces no change
in the nuclei and describes only the elastic scattering. As the nucleon force is Short-
ranged, and since the density p of nucleons in the nucleus is fairly constant in the interior
and falls smoothly to zero at the nuclear surface, it is reasonable to assume that U has a

radial shape that is similar to the density. Usually the optical potential is expressed in the

Woods-Saxon form [WooS4]:

l8

U(r) = —va(r,Rv,av)—iWVf(r,RW,aW)

+4iWsaW—al-f(r,Rw,aw)
d’ 1 d (2.3.10)
+ 2.0(v50 +iwm)(—r——d—r f(r,R,0,a,0)Z-a)

+ Vc

where the Woods-Saxon shape function f (r, Rk ,ak) is :

1 l

r,R, = , R = -A‘ 2.3.11
fl kak) l+exp[(r—Rk)/ak] k r" 3 ( )

 

Here rk is the radius parameter and ak is the diffuseness parameter; V, and W, are the

depths of the real and imaginary potentials, respectively; Ws is the depth of surface term
of the imaginary potential. Vso and Wso are the depths of the real and imaginary parts of
the spin-orbit potentials; Z is the orbital angular momentum of the relative motion of the
scattered particle; and 6' is the spin operator. VC is the Coulomb interaction, which is
taken for a uniformly charged sphere of radius RC with different expressions inside and

outside the radius RC .

 

2
Z Z e
a b , r>Rc
r
V = 2.3.12
C i ZaZbez r2 ( )
3— ), rSRc
2R6 R02

2.3. 1. 1.2 Global Optical-Model Potentials

In principle, all the parameters of the optical-model potential can be obtained by

fitting them to the experimental data of the elastic scattering. For the best fit to individual

19

nucleus at specific energy, all parameters could be optimized. However, the parameters
of the optical potential usually vary smoothly with energy and are similar for neighboring

nuclei. Thus global optical potentials could be obtained by fitting a group of nuclei with a

total of N points in a certain energy range by minimizing 12 :

(2.3.13)

 

2
12 : N[ath(6i)-Uexp(6i)]

l
N E AUCXP(6i)

where 0' and 0' are the calculated and ex erimental values of the cross sections at an
th exp P

angle of 6,- and Aoexpis taken to be the experimental error; N is the number of data

points.

Over the years, many global optical potentials have been develOped for both protons
and deuterons. In the following sections, we discuss mainly those potentials which have
been provided as options for TWOFNR [Iga77] that we have used to calculate the

theoretical differential cross sections.

2.3.1.1.3 Proton Global Optical-Model Potentials

In this section, we introduce three sets of proton global optical-model potentials
developed by Bechetti-Greenlees [Bec69], Menet [Men71], and Percy & Perey [Per76].
The Becchetti-Greenlees [Bec69] global proton potential has been developed for A>40
nuclei and proton energies up to 50 MeV. Menet [Men7l] developed a global proton
potential for 12<A<208 nuclei and proton energies from 30 MeV to 60 MeV. The global

proton potentials developed by Perey & Perey [Per76] are good for 30<A<100, and

20

proton energies up to 20 MeV. The parameters of the above three global optical
potentials are listed in Table 2.2.

Figure 2.2 Shows the real (upper panel of figure) and imaginary (lower panel) global
proton potentials on 13C at proton energy of 12.5 MeV. These potentials have very
similar shapes and depths in the surface regions of the real parts, which dominate the
scattering at the forward angles. However, there is a lot of difference among the

imaginary parts, which scatter more strongly at large angles. Figure 2.3 shows the

calculations based on the above global potentials and the comparison to the proton elastic
scattering data on l3C at an incident energy of 12.5 MeV [Wel78]. The calculation by
Menet potential gives a higher cross section, and the potentials of Perey & Perey and

Bechetti-Greenlees give similar cross sections up to 56 degree in the center of mass.

Figure 2.4 shows the real (upper panel of the figure) and imaginary (lower panel)
global proton potentials on 13C at proton energy of 30.95 MeV. These potentials also
have similar shapes and depths at the surface regions of the real parts. The imaginary
parts are different. Figure 2.5 shows calculations based on the above global potentials
and the comparison to the proton elastic scattering data on 13C at an incident energy of
30.95 MeV [Bar88]. The calculations agree with each other rather well at the forward
angles (less than 10° in the center of mass) and start to differ from each other for angles
larger than 15° in the center of mass. We see that, at higher energy, the potentials of
Menet and Bechetti-Greenlees give similar results up to 80° and the potential of Percy &

Perey gives lower cross section compared to others.

Based on the above comparisons, we see that the proton global optical-model

potentials give consistent calculations at forward angles, less than 10°. They give

21

relatively good fitting to the data in the region from 10° to 40°. Since the extraction of a
spectroscopic factor is mostly determined by the reaction in forward angles, proton global
potentials may provide a reasonable approach for describing the elastic scattering
channel. In Section 2.3.1.1.5 and 2.3.1.1.6, we will introduce, respectively, potentials of
CH89 and JLM, which improve the accuracy of the proton potential based on the folding

model and nuclear matter approaches.

22

Table 2.2: The parameters of global proton potentials

 

Bechetti-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Potentials Greenlees Menet Perey & Perey
( ro ton) (proton) (proton)
Parameters [Eec69] [Men71] [Per76]
A A>40 12<A<208 30<A<100
E E<50 MeV 30<E<60 MeV E< 20 MeV
54.0-0.32E 49.9 —0.22E 533-0555
v, + 24.0—N—_—Z- + 26.4-1!———§- + 27 Liz—
A A A
+0.4Z.A‘”3 +0.4z-A““3 +0.4Z-A‘“3
rv 1.17 1.16 1.25
av 0.75 0.75 0.65
M 4.2-0.05E
ax
N — Z
W" [(0.22E-2.7),0] +15.5 T 13.5
7w 1.32 1.37 1.25
N _ Z 0.74-0.008E
aw 0.51+0.7 —— N — Z 0.47
A +1 .0———
A
Max[11.8-0.25E
W - .
3 +120 , 0] 1.2+0.09E 0 0
A
Vso 6.2 6.04 7.5
r30 1.01 1.064 1.25
am 0.75 0.78 0.47
RC 1.25 1.25 1.25

 

 

 

 

23

 

 

 

 

 

 

 

 

 

 

 

 

0
§ -—20 —
G.) l-
5
> _4() — /.,~’/ —: Menet
: xc/ —————— : Perey8cPerey
— ' l ------------ : Bechetti—Greenlees
—60 . I 1 L
g : I _
Cl) , -.
E —10 f; ........ ,x
i: —20 E
g :
_3OP..II.......IILL
0 2 4 6 8

Figure 2.2: Proton global optical-model potentials of 13C at incident energy of 12.5 MeV.

24

 

105 ' ' ' 4 i ' v r r I r r . 4
13 m
C(p,p) C (g.s.) Ep=12.5MeV

 

 

 

 

104— .
————:Menet
A 3 ------ : Perey8cPerey
L; 10 ’ """""" :BechettI—Greenlees 4
to ..
\
..Q
g 102—
C2
33 10
b
"O
1.0- 2
10_1‘***#‘AL.1...L
0 50 100 150

9am.(deg)

Figure 2.3: Calculations by different global optical-model potentials for the reaction of
13C(p,p)l3C (g.s.) at incident energy of 12.5 MeV compared with the data [Wel78].

25

 

 

 

 

 

 

 

 

O_ /..-
t //
—10}
’9 i
CD —20_—
5 .
”a -803- ..
> : / (fr —: Menet
—40 E: T ///' — — —: Perey8cPerey
/ ------- : Bechetti—Greenlees
' 1 . . a .L
9
<1)
5
3 —20,—
3 i
_3OF....1E...I....I....
O 2 4 6 8

r (fm)

Figure 2.4: Proton global optical-model potentials of 13C at incident energy of 30.95
MeV.

26

5
10....T..rtl

 

 

 

 

 

13C(p,p)13C (g.s.) Ep=30.95MeV
104— r
—— : Menet

A 3 —————— : Perey&Perey
$53. 10 ~ ---------- :BechettI-Greenlees
\
"Q 2
g 10 —
C2
3 10
b
"C

1.0 r

10‘1 r . r r I r . L L l . r I r \‘r

0 50 100 150

9am.(deg)

Figure 2.5: Calculations by different global optical-model potentials for the reaction of
l3C(p,p)l3C (g.s.) at incident energy of 30.95 MeV compared with the data [Bar88].

27

2.3. 1.1.4 Deuteron Global Optical—Model Potentials

There are three widely-used deuteron global optical-model potentials: Lohr-Haeberli
[Loh74], Perey & Perey [Per76], and Daehnick [Dae80]. These three potentials are
available as options in the code of TWOFNR [Iga77].

The Lohr-Haeberli deuteron global potential is for nuclei with A>40 and for deuteron
energies from 8 MeV to 13 MeV; the Perey & Percy deuteron global potential is for
nuclei with 2212 and deuteron energies from 12 MeV to 25 MeV; the Daehnick deuteron
potential spans the energy range from 11.8 MeV to 90 MeV and includes nuclei ranging
in mass from 27Al to 238Th. The parameters of the above three global optical-model
potentials are listed in Table 2.3.

As a comparison, Figure 2.6 Shows the three global deuteron potentials of 12C at
incident deuteron energy of 11.8 MeV. Figure 2.7 shows the elastic scattering
calculations based on the above global potentials, together with the experimental data at
incident energy of 11.8 MeV [Fit67]. Unlike Figure 2.3 and Figure 2.5, Figure 2.6 plots
the ratios of scattering differential cross section divided by the Rutherford differential
cross section. The calculations agree with each other at the forward angles (less than 25°),
but there exist slight deviations from the data.

Figure 2.8 shows the three global deuteron potentials on 12C at incident deuteron
energy of 34.4 MeV. Figure 2.9 shows the elastic scattering calculations with the
experimental data at 34.4 MeV [New67]. The calculations agree with each other within
the standard error of 5.3% at the forward angles (less than 20°), and they fit the elastic

scattering well up to 36° in the center of mass.

28

Based on the above comparisons, we see that the present deuteron global optical-
model potentials describe the deuteron elastic scattering better at higher energy than at
lower energy and at smaller scattering angles better than at larger scattering angles. At
higher energy, The Daehnick potential gives better fitting than others, so we choose
Daehnick deuteron potential in our DWBA analyses. For the ADBA, we use the adiabatic

deuteron potential that will be introduced in Section 2.3.2.

29

Table 2.3: Deuteron global parameters.

energy in MeV. For Daehnick potential, ,6 = —(I§0) 2, fl, =(M—i2fl) 2, where M i:

magic numbers (8,20,28,50,82,126).

=neutron number, E=deuteron laboratory

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Potentials Lohr-Haeberli Perey & Perey Daehnick
(deuteron) (deuteron) (deuteron)
Parameters [Loh74] [Per76] [Dae80]
A A>40 2212 27<A<238
E 8<E<13MeV 12MeV<E<25MeV 1 l.8<E<90MeV
91.13 81.0-0.22E 88.5-0.26E
V" +2.2z-A'“3 +2.02-A‘“3 +0.882-A““3
r, 1.05 1.15 1.17
a, 0.86 0.81 0.709+0.0017E
WV - 0.0 (12.2+0.026E)(l-eB)
rw 1.43 1.34 1.325
0.53 + 0.07711 ’ 3
aw 0.50+0.013 A 2 I 3 0.68 _ 0.0427.) —n 1'
r
W, 218A"” 3 14.4+0.24E (12.2+0.026E) eB
V50 7.0 - 7.33-0.029E
r50 0.75 - 1.07
as, 0.5 - 0.66
Re 1.3 1.15 1.3

 

30

 

 

 

 

 

 

 

 

 

0 .
_20 _
T; .
(D —40 T
E .
7: —60i— , .
:> I // —— : Lohr—Haeberli
—80 777/ ------ : Perey8cPerey
:/ ............. I Daehnick
—100 _
% 20 :
5 :
3 —-40 i
3 i
—60 h A I E . l . l A
O 2 4 6 8

r (fm)

Figure 2.6: Deuteron global optical-model potentials of 12C at incident deuteron energy
of 11.8 MeV.

31

(dU/dOI/UknkflDnnh

 

10

'I

 

f

f

 

r T I T I 1 I I T Y I I I Y ‘1' I I I I T 4

12C(d,d)12C (g.s.) Ed=11.8MeV

 

 

 

: Lohr—Haeberli
3 """ : Perey8cPerey
"""""" : Daehnick
10—2 .1.n..1....1.1111.
O 20 4O 6O 80
9am.(degl

Figure 2.7: Calculations by different global optical-model potentials for the reaction of
”C(d, d)12C (g.s.) at incident energy of 11.8 MeV compared with the data [Fit67].

32

 

 

 

 

 

 

 

O .
E‘
Q)
5
E
:> -—-— : Lohr—Haeberli
- - — : Perey8cPerey
. -------- : Daehnick
— 100 _
IT: 20 Z
5 :
3 —40 I
3 i
_60' A 1 . 1 . In L
0 2 4 6 8

T (im)

Figure 2.8: Deuteron global optical-model potentials of '2C at incident deuteron energy
of 34.4 MeV.

33

(dU/dQ)/(dU/dQ)Ruth

|._s
O

H
O

 

 

I...r[....rrTrrf.

12C(d,d)12C (g.s.) Ed=84.4MeV

1 ti.
l! : Lohr-Haeberli

lj' " — —: Perey8cPerey

' """" : Daehnick
l L L .l l l 11 1 L L l A g A l l

 

 

 

20 40 60 80
9am. (deg)

Figure 2.9: Calculations by different global optical-model potentials for the reaction of
12C(d, d)12C (g.s.) at incident energy of 34.4 MeV compared with the data [New67].

34

2.3. 1.1.5 Nucleon-Nucleus Optical-Model Potential

The global optical-model potentials discussed above are derived from the fitting to
the elastic scattering data in particular mass and energy regions. One consequence is that
the derived global optical-model potentials cannot cover all the nuclei over a wide energy
region. Thus, derivation of an optical-model potential using a much more extensive
database of elastic scattering than previously used is desirable. A parameterization of the
nucleon-nucleus optical-model potential based on data for A from 40 to 209, proton
energies from 16 to 65 MeV and neutron energies from 10 to 26 MeV, was developed by
R. L. Vamer et. a1. [Var91]. This parameterization, which is called Chapel-Hill 89
(CH89), is based on the current understanding of the basis of the optical potential, such as
the folding model and nuclear matter approaches instead of the determination of optical-
model potentials phenomenologically. The extensive database includes nearly 300
angular distributions (9000 data points) of proton and neutron differential cross sections
and analyzing powers, which is significantly more accurate and complete than previous
analyses [Per76, Men71, Bec69].

This parameterization adapts the basic Woods-Saxon form of Equation 2.3.10 but
some parameters have Slight modifications. One special feature of the parameterization of
CH89 is that, based on the parameterization of nuclear charge radii [Mye73], offset

values are added to the conventional radius parameters:

RV = rvAl/3 + rvw), Rw = rwA1/3 + rwm)
R50 = rsoAl/3 + rig), RC = rcAl/3 + r20) (2.3.14)

35

0 0 0 . . . . .
where rvw) , rso( ) , rw( ) , and rc( ) are offset radrus of the real, Imaginary, sprn-orbrt, and

Coulomb potentials. The other special feature of CH89 is that the depths of the potential

have more complex dependence on the energy and proton-neutron number.

 

 

 

 

 

 

 

v, =vO :v, +(E—EC)Ve (2.3.15)
6Z6 2 f
, or proton
EC =4 5Rc (2.3.16)
l O, for neutron
W,
W, — W ‘0 E E (2.3.17)
l+exp[ veO_( — C)]
erW
N - Z
W30 i Wst T
w, = E E w (2.3.18)
l+exp[( _ C)_ 5‘30] 1
W56”)

where ‘+’ is used for protons and ‘—’ for neutrons. The parameters used in potential CH89

are listed in Table 2.4.
Figure 2.10 shows the shapes of CH89 proton potentials on 13C at incident energy of

12.5 MeV. The global potentials of Menet and Perey & Perey are plotted together for the

convenience of comparison. Figure 2.11 shows the calculations for proton elastic
scattering on 13C at incident energy of 12.5 MeV using the potentials of Menet, Perey &
Perey, and CH89. The potential CH89 gives better fitting to the data.

Figure 2.12 shows the shapes of proton potentials on 13C for CH89, Menet, and Perey
& Perey at incident energy of 30.95 MeV. The surface regions of the real parts are close

to each other. Figure 2.13 shows the calculations for proton elastic scattering on 13'C at

36

incident energy of 30.95 MeV employing the potentials of Menet, Percy & Perey, and
CH89. It is obvious that the potential CH89 gives better fitting to the data. Based on the
above comparisons, we adopt the potential of CH89 in our calculation in a wide energy

region.

37

Table 2.4: Parameters for the global nucleon-nucleus optical-model potential of CH89

 

 

 

 

 

 

 

 

 

 

 

 

[Var91] ‘

Parameters Value Parameters Value
V0 52.9 MeV aso 0.63 fm
vt 13.1 MeV wv0 7.8 MeV
v, -0299 wv,O 35 MeV
rv 1.250 fm erw 16 MeV

r,‘°) -0225 fm wso 10.0 MeV
av 0.690 fm w,, 18 MeV
rc 1.24 fm wseO 36 MeV
r30) 0.12 fm wsew 37 MeV
vso 5.9 MeV-rm2 rw 1.33 fm
no 1.34 fm rW‘O’ -0.42 fm
rsom) .1.2 fm aw 0.69 fm

 

 

 

 

 

38

 

 

 

 

 

 

 

 

 

 

 

0
—10_—
”>7 i
(D —207
2 .
v _
”t? —307
v _ ///
:> // —: Menet
—40 h /-’,’ ------ : Pere &Pere
/.. Y Y
/2/’ _._. ..... . CH89
Ara/1:04.; l .
A K“\\
T) 10'
2 ; """""""
v _
A ’
t... —20;‘
v _
3 .
_30’....1....1....1...L
O 2 4 6 8

Figure 2.10: Comparison of the CH89 proton potentials of 13C with the proton potentials
of Menet and Perey & Perey at incident energy of 12.5 MeV.

39

104
A 3
5.} 10 '
(I)
\
PO 2
g 10
c:
E 10
o
"O
1.0
10‘1

 

 

I I Y T I l

13C(p,p)13C (g.s.) Ep=12.5MeV

—— : Menet

______ : Perey8cPerey
_.-.-._.- :CH89

 

l I I I A l

 

 

50 100
98m.(deg)

Figure 2.11: Calculations for proton elastic scattering on 13C at incident energy of 12.5
MeV using the potentials of Menet, Perey & Perey, and CH89 compared with the data

[Wel78].

40

 

 

 

 

 

 

 

0_ /,/»~
' /./
—10} “
g i
CD ~20?
E .
2: —3oi-
;> I /// ——:Menet
—40 f" T / — — —: Perey8cPerey
?/’// — —-n CH89
_.50’..._\ . 1 . 1 I 1
A LLL‘ \\ /
> : 7" \ _,/ / /'
Cl) _ — \ /
2 10: \w/
v i
E —207
3 I
O 2 4 6 8

r(fin)

Figure 2.12: Comparison of the CH89 proton potentials of 13C with the proton potentials
of Menet and Perey & Perey at incident energy of 30.95 MeV.

41

 

5
10.4vrrtvvat

 

 

 

 

E 13C(p,p)13C (g.s.) Ep=30.95MeV
104? _
- —— :Menet q
’ j ------ : Perey&Perey 7
E 103?— 7°} :CH89 —.
\ . .
PO 2
E 10 E‘
C C
E 10 r
b E
“O
1.0 E—
10_1* . . L . 1 . . . . 1 . . . . N:
O 50 100 150

0am. (deg)

Figure 2.13: Calculations for proton elastic scattering on 13C at incident energy of 30.95

MeV using the potentials of Menet, Percy & Percy, and CH89 compared with the data
[Bar88].

42

2.3. 1. 1.6 J LM Optical-Model Potential

Instead of fitting elastic scattering data phenomenologically, the optical-model
potential could be determined from nuclear matter theory, which may supply more
microscopic understanding of the nuclear interior and overcome the uncertainties of the
geometry parameters in the global optical—model potentials described previously. One
such optical-model potential developed via realistic nucleon-nucleon interaction and
nuclear matter density is the JLM (the initials of the threeauthors: Jeukenne, Lejeune,
and Mahaux) potential [Jeu77].

The JLM potential started from the Brueckner-Hartree-Fock approximation and
Reid’s hard core nucleon-nucleon interaction, which was folded with the nuclear matter
density. The complex optical-model potential in infinite nuclear matter is parameterized
for nuclei with mass numbers 12 S A S 208 and for energies E up to 160 MeV. For the
nucleus whose nuclear matter densities are available experimentally, the JLM potentials
may model the shape of the optical potentials more accurately than the phenomenological
ones.

The real and imaginary JLM potentials are expressed as:

4 4.2
— _ —3 V50) . ’"l 3.
VE(r)_/t,-(bJ?r_) —p—(—r)—Ip(r)exp -——bZ—— d r

_ _3WE(r) IF-f'l2 3

WE(r)=/1w-(b\/;t_) ——jp(r')ex — d r' (2.3.19)

p(r) b2

where b = 1.2 fm, corresponding to the range of effective interaction [Gil71]; the scaling

factors 2., and xlw is 1.0 and 0.8 for the real and imaginary potentials, respectively

43

[Pet85]; the VE(r) and WE(r) are the real and imaginary nucleon potentials derived in the

local density approximation (LDA). LDA implies that the value of the potential at each
point of the nucleus is the same as in a uniform medium with the same local density. In

the case of a neutron with energy E, the LDA potential in uniform nuclear matter with

density p and neutron excess 8 is given by:
V,,(p.E) =Vo(p.E)+5 -Vt(p.E)
W,,(p,E) =W0(p,E)+6-W1(p,E) (2.3.20)
where the neutron excess is measured by the asymmetry parameter 5

6_pn—pp

_ (2.3.21)
pn + pp

There are different models to parameterize the density distributions of protons and
neutrons. One that is used in this work is the modified harmonic-oscillator model [Dej74]

as defined by:

ptr) = 700[1+0’('2)2] exp[—[-;-]2] (2.3.22)

The parameters a and or can be read from Ref. [Dej74]. The density p0 is [Neg70]:

 

p0“): 3 3'; 2 2 , k=NorZ (2.3.23)
472Cp (1+7: ap /Cp )

where ap= 0.54 fin, and

Cp = (0.978+0.0206A“3)A“3 (fm) (2.3.24)

The quantity V0 (p, E) is parameterized to

44

M

Di

“he

3 . .
V0(p,E)= z aijp'El“ (2.3.25)
i,j=l

The coefficients aij are listed in Table 2.5. The parametric form of the imaginary potential

W0 (p.15) is

4 . .
2 are/2'5!“
i,j=l

W0(,0, E) = (2.3.26)

 

D

1+-———2
(E—é'F)

Where D = 600 MeV 2, 3F (,0) = p(—510.8 +3222p—6250p2), the coefficients d),- are

listed in Table 2.5. The function V,(,0,E) and W1 (p, E) have the forms:

 

171 ,E
Vl(p9E) :—(”%—2R6N(p,E)
W1(p.E) = _ ’" ImN(p,E) (2.3.27)
m(,0,E)
where iii adapts the form:
WEE) 3 i ‘—1
—=1— 2 cl-jp E] (2.3.28)
m i,j=1

The coe fficients Cij are listed in Table 2.5. The IT: is calculated by

 

, *
m=mf a3»)
m
where th e effective mass m * is defined as
a:
M). = 1——d—V0(p, E) (2.3.30)
m (IE

45

N(p,E) is the auxiliary function in Brueckner-Hartree-Fock approximation [Jeu77]. The

real part of N is parameterized by:

3 . .
ReN: z bijp'EJ-l (MeV) (2.3.31)
i,j=1

with coefficients by- listed in Table 2.5. The imaginary part of N is parameterized by

4 . .
2 ftjplEj l
i,j=l

 

Im N(,0, E) = (2.3.32)

F
E—8F

1+

 

where F=l.0 MeV. The coefficients fij are listed in Table 2.5.

In the case of a proton with energy E in the additional presence of a Coulomb field

Vc, the corresponding real and imaginary potentials are given by
Vp(.0,E) = Vo(p.E) +Ac(p,E)-§ V1(;0.E-Vc)
Wp(p,E) = W0(p,E) +WC(p,E) —5 Wl(p,E —VC) (2.3.33)
where
AC(P,E) = Vo(,0,E—Vc)—Vo(,0,E)
WC(,0,E)=W0(p,E—VC)—W0(p,E) (2.3.34)
Figure 2.14 shows the JLM proton potentials on 13C at incident energy of 12.5 MeV.

For comparison, the global potentials of Menet and CH89 are plotted in the same figure.
The surface regions of the real potentials are similar to each other but the interior part of

the real J LM potential is deeper.

46

Figure 2.15 shows the elastic scattering calculations based on the above potentials.
All of these calculations are quite similar at the forward angles (less than 10°). The
calculated angular distribution by the JLM potential gives good fitting up to 125°.

Figure 2.16 shows the JLM proton potentials on 13C at the incident energy of 30.95

MeV together with the global potentials of Menet and CH89. Again, the surface regions
of the real potentials are similar to each other but the interior part of the real JLM
potential is deeper. In addition, contrary to the other potentials, the imaginary JLM
potential in the nuclear interior is positive.

Figure 2.17 shows the elastic scattering calculations based on these potentials. The
calculated angular distribution by the JLM potential looks similar to that by the CH89
potential. All of these potentials give quite similar results at the forward angles.

Based on the above comparison, we can see that both the JLM and CH89 potentials
are better than other global optical-model potentials; the JLM is even better than CH89.
The disadvantage of the JLM potential is that it requires the information of nuclear
density. When the nuclear density is available, we perform calculations with both the

JLM and CH89 potentials; if the nuclear density is not available, only the CH89 is used.

47

Table 2.5: Parameters for JLM potentials

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b0

 

(a) aij for Vo(P,E) (b) dij for Wo(p,E)
i
. j=1 i=2 j=3 , j=1 j=2 j=3 j=4
J J
i = 1 -974 11.26 00425 i = l -l483 37.18 -0.3549 0.001119
i = 2 7097 -125.7 0.5853 1 = 2 29880 -93l.8 9.591 -0.0316
= 3 -19530 418 -2.054 1 = 3 -212800 7209 -77.52 0.2611
1 = 4 512500 -l7960 198 06753
(C) Cij fOI' 171(p, E) (d) fij fOI' MN
i
j=1 j=2 j=3 j=1 j=2 j=3 j=4

L—nv

 

 

4.557 -0.00529l 0.6108E-5 546.1 -ll.2 0.1065 -3.54113-4

_

 

do

 

-2.051 -0.4906 0.001812

h
ll

~847l 230.0 -2.439 0.008544

 

WN—

 

 

 

-65.09 3.095 -0.01190 51720 -1520 17.17 -0.06211

 

 

 

 

 

II
#0319—

-1 14000 3543 -41.69 0.1537

—

 

 

 

 

 

 

 

(e) bij for ReN

 

b0

j=1 j=2 j=3

 

360.1 -5.224 0.02051

 

-2691 51.3 -0.247

 

MINI-

 

 

 

 

 

7733 ~17l.7 0.8846

 

48

 

 

 

 

 

 

 

 

0.
_10_.
i
A i
> —207
Q) t
E :
—30r
A _
La .
v
> —40—
-—50 7:}
l'
_GOl...i....1.,_H._.12.LL
A »::.‘__‘::~..\~‘ 1%—
> : 1“‘\\\“ -__,<,"’"
Q) t ‘~__ "
2 ~10:
V P
A t
é; —20:
_3OWIILL
0 2 4 6 8

Figure 2.14: Comparison of the JLM proton potentials of 13C with the proton potentials
of Menet and CH89 at incident energy of 12.5 MeV.

49

5
10 . . , ' I Y , v r I

 

 

 

 

 

13C(p,p)13c (g.s.) Ep=12.5MeV
104—
A 3
L; 10 '
(I)
\\
PO 2
g 10 —
C2
31 10
b
'0
Lo—— _
10—1 i . . . l a i . . I i . l .
o 50 100 150

90m.(deg)

Figure 2.15: Calculations for the l3C(p,p)13C (g.s.) reaction by JLM, Menet and CH89
potentials at incident energy of 12.5 MeV compared with the data [Wel78].

50

 

 

 

 

 

 

 

 

OF
3 I /
E _40 —/
A h /
L. : /
> 60L // —:Menet
: —-——:JLM
; _._.-:CH89
m . . i l . 1 L i L i r J r l
—80_\\
A - \
i ' ‘\
2 o i \ ,_.
1? :
E _20 r 11.1, L r 1 [1 I
O 2 4 6 8

r (fm)

Figure 2.16: Comparison of the JLM proton potentials of 13C with the proton potentials
of Menet and CH89 at incident energy of 30.95 MeV.

51

t; 10

m i

\

PD 2

g 10

c:

E 10

b

"C .
1.0E
10‘1

 

 

l?"rl

13C(p,p)13C (g.s.) Ep=30.95MeV

— : Menet
------ : JLM
: CH89

 

 

 

Figure 2.17: Calculations for the l3C(p,p)]3C (g.s.) reaction by JLM, Menet and CH89

potentials at incident energy of 30.95 MeV compared with the data [Bar88].

52

2.3.1.2 Zero-Range Approximation

The DWBA expression for the transition amplitude in Equation 2.3.2 involves a 6-

fold integration over 7,, and Fp after the integration of the nuclear coordinates B. The 6-

fold integration has been discussed by [Au564] and Sawaguri [Saw67]. To simplify the
integration, it is usually assumed that the transition amplitude receives contributions only
from the region where the coordinates of the proton and neutron coincide so that we have
the zero-range approximation:

0(7) 2 v,,,, (rm (f) = Do 66) (2.3.35)

The value of Do can be obtained by integrating this equation over f :
_ 2 — _
1)0 _ j r Vpn(r)¢d(r)dr (2.3.36)
and the vertex constant 002 is [Lee64, Knu75] :

002 =15006.25 MeV2 . fm3 , (2.3.37)

When the zero-range approximation is made, the coordinates are transformed to:

B 7 (2.3.38)

__.)
3+1"

 

rn—>R, rp

The term Brl(iP,Ed)becomes
mz- - _.—1 —(1/2) (—)*- - ml — (+)- B — —
Bl (kp,kd)—z (21+1) DOI/rd (kd,R)¢nl(R)zp (kp,B+lR)dR

(2.3.39)

53

2.3.1.3 Finite-Range Approximation

In general, deuteron has finite range of radius and the interaction between proton and
neutron exists in a finite range. The zero-range approximation over-emphasizes
contributions coming from the nuclear interior. A means has been found to approximate
the finite-range effect so that it reduces to the form of the zero-range approximation

multiplied by a radial dependent factor A(R) [But64].

ml- - _.-1 —(1/2) (—)*- - ml — (+)‘ B - -
B, (kp,kd)—z (21+1) Dojzd (kd.R>A(R)¢n,(R)zp(k,,.B+1R)dR

(2.3.40)
The factor A(R) is
A(R) = l—(a/fl)2(l/Ed)[Ud(R)-V,,(R)—Up(R)-Ed]
(2.3.41)
= h 2 1/2, : mdMB
a (#Ed/ ) ‘u md + MB

where B is the finite range parameter with the value of 0.7457 [Knu75], Ed is the
deuteron binding energy, U d and U p are the deuteron and proton optical potentials, and

V" is the neutron potential that binds the neutron to the core nucleus B.

Figure 2.18 shows the comparison of the DWBA calculations in this finite-range
approximation (solid line) and zero-range approximation (dashed line) for the reaction of
l3C(p,d)12C at proton energies of 15 MeV and 48.3 MeV. The calculations use CH89 as
the proton potential and Daehnick global potential for the deuteron. Thus finite—range

approximation increases the cross section by 4.8% at the peak region for the proton

energy of 15 MeV. For incident proton energy at 48.3 MeV, the enhancement is 8.9% at

54

forward angles. Thus the effects, although not negligible, are not very large for the
reactions that we studied.

It should be noted that results from this finite-range approximation are very close to
the exact finite-range calculations with numerical solution of the deuteron wave function.
The latter kind of calculation is available by another widely used finite-range DWBA

code: DWUCKS [Kunz]. Figure 2.19 shows the comparison between the calculations
from two codes for the l3C(p,d)12C reaction. They are very close to each other especially

at the forward angles. At proton energy of 15 MeV, the curve from TWOFNR is just
1.5% lower than that from DWUCKS at the region of first peak. At proton energy of 48.3
MeV, the curve from TWOFNR is 2.0% higher than that from DWUCKS at the forward

angles. However, TWOFNR is more user friendly with many options to choose from.

55

 

 

 

 

 

 

' ..- — \ 13 12
f \ C(p,d) c (g.s.) «
A
e ‘ \ /,2\
m 10. : \ / —.
r-\ \ _, / :
E = \ 15 MeV \9;
Cl ’ \\ '
"C5 \\\ 48 3 MeV
\ - \ \°
b 10“—
U Finite range
— - — Zero range
10‘2 1 1 1 1 l 1 1 1 L l L 4 m 1 1 1 1
O 20 4O 60 80

Gem (deg)

Figure 2.18: Calculations for reaction of l3C(p,d)12C (g.s.) using finite-range
approximation (solid line) and zero-range approximation (dashed line) at incident
energies of 15 MeV and 48.3 MeV. The cross sections at proton energy of 15 MeV have
been multiplied by 10 so that the calculations at the two energies can be seen more
clearly.

56

 

 

dU/dQ (mb/sr.)

 
 

TWOFNR (Finite—range)
“““““ DWUCK5

 

 

10—2 . . . 1 1 . . . . 1 1 1 . . l 1 . . 1 1 .
O 20 4O 6O 80

(deg)

 

6

CH]

Figure 2.19: Finite-range DWBA calculations for reaction of l3C(p,d)12C (g.s.) by
TWOFNR (solid line) and DWUCKS (dashed line) at incident energies of 15 MeV and
48.3 MeV. The cross sections at proton energy of 15 MeV have been multiplied by 10.

57

2.3. 1.4 Non-Locality Correction

The optical-model potential is usually taken to have simple local form, which means
that, at the point r, the particle feels the potential only at that point. The Schrodinger
equation reads

h 2
—-2——V2 +UL(f)—E 1/1(f)=0 (2.3.42)
,u

The real situation is more complicated and the optical potential should be non-local,

which means that the wave function at point 7 is affected within the range of non-local

potential. U L (FM/(f ) in Equation 2.3.42 should be replaced by
IU(?,?')1//(F')d?' (2.3.43)

where U (?,F')is the non-local potential. Non-locality can be expected wherever the

potential is energy dependent that comes from the exchange terms required by the
asymmetry of the overall wavefunction.
This effect has been studied by Percy and Buck [Per62]. They separated the non-

local kernel U (7, f') into a potential fonn U times a Gaussian non-locality function.

7+?

 

U(F,f')=U[

 

 

)H (F — 7') (2.3.44)

where H (F — 'r") was chosen to be a Gaussian function:

 

H(? — 7') = ex"(— ((7 - PM” )2) (2.3.45)

3
[#2131113]

58

where ,BNL is the range of non-locality. The value of flNL that Percy and Buck found to

yield the best fit to the data of neutron scattering on Pb over an energy range from 4.1

MeV to 24.0 MeV is 0.85 fm [Per62]. For the deuteron, flNL is 0.54 fm [Per74].

Figure 2.20 shows the comparison between the local and non-local proton potential

from CH89. The local potentials are generally weaker than the non-local potentials,
|VL| <|V~L| and IWLI <|WNL| , especially within the interior of the nuclei. Thus, non-

locality reduces contributions to transfer reaction from the interior of the nucleus. The
main change produced in the cross section is the reduction of the large angle scattering

while increasing the forward or peak cross section [Phi68]. Figure 2.21 shows the
calculations for the reaction of l3C(p,d)12C at incident energy of 48.3 MeV, where proton

potential chooses the global potential of CH89 and the deuteron potential adapts the
global potential of Daehnick [Dae80]. Finite-range approximation is employed in these
calculations. The dashed line shows the result from the local proton potential; the solid
line shows the result when non-locality correction is applied to the proton potential. We
can see that the cross section by the non-local proton potential increases 12% at forward
angles, where the spectroscopic factors are extracted. Similar effect is obtained when the
non-locality correction is applied to the deuteron potential. The total effects are
cumulative when the non-locality correction is applied to the proton and the deuteron
simultaneously. Thus it is important to include non—locality corrections in transfer
reaction calculations.

The non-locality correction at lower energy is also examined. Figure 2.22 shows the
calculations for the same reaction at incident proton energy of 15 MeV. The calculations

with and without non-locality correction are very close at the region of first peak.

59

Therefore, the extracted SF are not strongly affected by the non-locality correction at low

energy.

60

 

 

v(1~) (MeV)

 

 

 

 

 

 

T;

(l)

5

E

B

_4OF111111111L111L11111
O 2 4 6 8

r(fm)

Figure 2.20: Comparison of the local (dashed line) and non-local (solid line) proton
potentials (CH89) of 13C at incident proton energy of 48.3 MeV.

61

 

2
10;'rrfi1""1""1""1'3

13C(p,d)12C (g.s.) Ep=4eeMeVi

   

 

 

 

 

—— Nonlocal
— — — Local
10-2 1 1 1 1 l 1 1 1 L 11 1 1 1 I 1 1 1 1
O 20 4O 60 80
Gem (deg)

Figure 2.21: Calculated differential cross section for l3C(p,d)12C at incident proton
energy of 48.3 MeV by non-local proton potential (solid line) increased 12% at the
forward angles compared to that by local proton potential (dashed line).

62

‘13 101
\
E
C 1.0;
"C .
\
b
"C

 

 

 

—— Norrlocal

— -— — Local

1L11l

 

l l l l l l l 1 1 L I l. l l l J

 

20 4O 6O 80.
Gem (deg)

Figure 2.22: Calculated differential cross sections for 13C(p,d)12C at incident energy of
15 MeV using non-local proton potential (solid line) and local proton potential (dashed

line).

63

2.3.1.5 Neutron Form Factor

In most analyses of single neutron transfer reactions, it is assumed that the neutron is
picked up or deposited into a shell model single—particle state. In the standard energy

separation procedure, the corresponding single-particle wave function ¢nlj (7,1), called
the neutron form factor, is usually taken to be an eigenfunction of a Woods-Saxon

potential whose geometry is fixed (ro=1.25 fm, ao=0.65 fm, R = r0 -Al/ 3 fm) and depth

is adjusted so that the eigenvalue is equal to the experimental neutron separation energy
[Pin65, Ber65].

Table 2.6 lists the information of the neutron form factors for the four reactions we

studied in this thesis. Figure 2.23(a) shows ¢an (r) in the 1 pl ,2 orbit for the reaction of

p(l3C,d)lzC (g.s.). The depth Vn is adjusted to be —39.779 MeV corresponding to the

neutron binding energy of —4.946 MeV. The neutron form factor for the reaction of

p(l3C,d)12C (2+ , 4.439 MeV) is shown in Figure 2.23(b). In this case, the neutron orbit is
1173/2 and the neutron separation energy is —9.385 MeV. Similarly, Figure 2.6 (c) and ((1)

show the neutron form factors for the reaction of p(”B,d)lOB (g.s.) and p(10Be,d)9Be

(g.s.), separately. The well-depths and the binding energies are listed in Table 2.6
It is important to examine the sensitivity of the calculations to the parameters r0 and

210 of neutron potential. Figure 2.24 shows the dependence on the neutron radius
parameter for the reaction of l3C(p,d)12C (g.s.) at proton energies of 15 MeV and 48.3

MeV. CH89 and Daehnick potentials are used for proton and deuteron respectively.

(Finite-range approximation and non-locality correction are employed in all following

64

calculations). The neutron radius parameter r0 is changed from 1.2 fm to 1.3 fm while the

spin-orbit strength is fixed to zero and the neutron diffuseness is fixed to 0.65 fm. The
change of 0.1 fm (corresponding to 8% change) changes the cross sections at forward
angles by 16% at higher incident energy and 11% at lower incident energy. Figure 2.25

shows the dependence on the neutron diffuseness parameter, where the neutron
diffuseness changes from 0.6 fin to 0.7 fm with the r0 fixed to 1.25 fin and Vso fixed to

zero. The increase of 0.1 fm (corresponding to 16% change) in the neutron diffuseness
increases the cross sections at forward angles 20% at higher energy and 17% at lower
incident energy.

Figure 2.26 shows the dependence on the spin-orbit strength. When the spin-orbit
strength of the neutron potential changes from 0.0 MeV to 6.0 MeV, the cross sections at
forward angles decrease by 6-8%. Since this effect is small, the spin-orbit strength is set

to zero in all of the following analyses.

Table 2.6 The neutron potentials and the binding energies of the neutron form factors in
the reactions of this experiment

 

 

 

 

 

 

 

 

 

 

 

 

Reaction N222?“ Emdgllc’lgeflnfrgy (181g, ) (f1: ) (331)
p(‘3c,d)‘2c g.s. 1191/2 -4.946 -39.779 1.25 0.65
p(‘3c,d)‘2c 2+ 1123/2 -9.385 48.257 1.25 0.65
p(”B,d)'°B g.s. 1103/2 41.455 -56.853 1.25 0.65

p(lOBe,d)9Be g.s. 1103/2 -6.811 -50905 1.25 0.65

 

65

 

110 E 1

-2
10 5

 

 

10"3

Form factor

I jTTIVTI

 

 

 

 

 

 

10 5 10

R (fm)

Figure 2.23: Neutron form factors for the reactions of (a) p(l3C,d)12C (g.s.), (b)
p(‘3c,d)‘zc (2*), (c) p(”B,d)‘°B (g.s.), and (d) p(‘°Be,d)9Be (g.s.).

66

 

do/dQ (Inb/sr.)

 

 

 

 

6cm (deg)

Figure 2.24: Calculations on the variation of neutron radius parameter to, where the

neutron diffuseness a0 is fixed to 0.65 fm and the spin-orbit strength Vso is 0.0 MeV. The
cross sections at proton energy of 15 MeV are multiplied by 10.

67

3
10 ""I'fi7rl

 

 

 

 

 

 

102 —
”'5
5—1 —\
g 10. Ni
I \
E \
V 10 __ “a
C: E
"o 1
E ‘
0‘1— -
'U 1 ao=0.6 frn i
- — -— a0=0.7 fm l
10‘2 1 . . - ' .
0 20

Gem (deg)

 

Figure 2.25: Calculations on the variation of neutron diffuseness a0, where the neutron
radius parameter r0 is fixed to 1.25 fin and the spin-orbit strength Vso is 0.0 MeV. The

cross sections at proton energy of 15 MeV are multiplied by 10.

68

 

 

 

 

 

 

 

,5
L;
U)
\
E
c: _ .
"o
\
9 10"—- ‘~
U vso=oo MeV ‘
_ — — — VSO=B.O MeV
10‘2 1 1 1 1 I 1 1 1 m l 1 1 1 1 1 1 1 1 1
o 20 4o 60 80
Gem (deg)

Figure 2.26: Calculations on the variation of neutron spin-orbit strength V50, where the

neutron radius re and diffuseness a0 are fixed to 1.25 fm and 0.65 fm separately. The
cross sections at proton energy of 15 MeV are multiplied by 10.

69

2.3.2 Adiabatic Deuteron Breakup Approximation (ADBA)

Deuteron is composed of two loosely bound nucleons, a proton and a neutron. Since
the separation energy between the proton and neutron is 2.224 MeV, deuteron breaks up
easily in the field of core nucleus. Thus the extraction of the spectroscopic factors from
(p,d) and (d,p) reactions using the DWBA calculations is usually not very reliable
especially at high incident energy because of inadequate treatment of the breakup effect
of deuteron [Pea66, But67].

Johnson and Soper [Joh70] extended the DWBA theory involving deuterons to
adiabatic deuteron breakup approximation (ADBA), which requires only a specification
of the nucleon-target interactions. In this approximation, the effective two-nucleon-
nucleus interaction is assumed to be the sum of the nucleon optical-model potentials
evaluated at one-half the incident deuteron kinetic energy. The deuteron adiabatic

potential is defined as:
4 1 _ 1_ - 11 1 _
Ud(R)=——j Un(R+—r)+Up(R——r) Vpn(r)¢d(r)d? (2.3.46)
D0 2 2

where U" and Up are the neutron and proton optical potentials at one half the deuteron

bombarding energy, R is the coordinate of the deuteron center of mass and f is the

relative coordinate between proton and neutron, VP" ('r‘) is the interaction between proton

and neutron, ¢d (7) is the deuteron wave function, and D0 is defined in Equation 2.3.36.

The exact (d,p) and (p,d) transfer reaction amplitudes require knowledge of the
adiabatic three-body wave function only at small neutron-proton separations. There, the

adiabatic distorting potential governing the center of mass motion of the deuteron is well

70

described by the sum of the neutron- and proton- target optical potentials. It is important
to stress that this adiabatic distorting potential generates the three-body wave function in
that limited region of configuration space needed to evaluate the transfer amplitude, and

it does not describe deuteron elastic scattering at the beam energy.
Figure 2.27 shows the comparison of two deuteron potentials of 12C at Ed = 49.2

MeV. The dashed line presents the Daehnick global deuteron potential and the solid line
represents the adiabatic deuteron potential constructed by the CH89 nucleon potentials.
The adiabatic potential based on CH89 is deeper in the interior and is shallower at the
surface. The effect on the stripping or pickup cross section is to cause a faster fall off
with angle and to create stronger oscillations at higher incident energy. Figure 2.28 shows

the calculations of ADBA (solid line) and DWBA (dashed line) for reaction of
l3C(p,d)12C. At the forward peak, the ADBA calculation is 29% larger than that of the
DWBA calculation.

The situation is different at lower incident energy. Figure 2.29 shows the calculations
for the same reaction at proton energy of 15 MeV. The ADBA (solid line) has similar
peak value as the DWBA (dashed line) and thus has little effect on the extraction of
spectroscopic factors. However, to be consistent in the use of deuteron potentials

throughout the range of energy, we choose ADBA calculations.

71

 

 

V(r) (MeV)

 

 

 

 

 

 

’>‘
Cl)
5 .
E
a )-
_50111111111111111
0 2 4 6 8

r (fm)

Figure 2.27: Comparison of Daehnick global deuteron potential (dashed line) with the

adiabatic deuteron potential (solid line) constructed by CH89 potentials, for 12C at Ed 2
49.2 MeV. ‘

72

 

    
 

 

 

 

A P 18C(p,d)12C gs. Ep=48.3MeV :
‘13 101 —:
\
"E
'U - \\\ :
\ Z \
g \
1.0—1:— —
I —— ADBA
— — — DWBA
10‘2 1 1 1 1 l 1 1 1 n l_1
O 20 4O

 

6 (deg)

CH1

Figure 2.28: Comparison of ADBA (solid line) and DWBA (dashed line) calculations for

reaction of l3C(p,d)12C at proton energy of 48.3 MeV. The ADBA increases the cross
section at the forward angles and faster fall off than DWBA.

73

 

do/dQ (mb/sr.)

 

 

 

 

0 20 4O 60 80

Gem (deg)

Figure 2.29: Comparison of ADBA (solid line) and DWBA (dashed line) calculations for
reaction of l3C(p,d)12C at proton energy of 15 MeV.

74

2.4 Momentum Matching

Assuming PI) is the incident proton momentum and Pd is the momentum of the

deuteron, the transferred momentum P, of the neutron is given by conservation of

momentum [Fes92]:

P, = Pd — Pp (2.4.1)
From this equation one can immediately determine the magnitude of P,:
Pz-P2+P2—2PP 6
, — p d p d cos (2.4.2)

Where 6’ is the angle between the direction of the final deuteron and the direction of the

incident proton. The angular momentum transferred, hl, , must be less than P, R, where R

is the projectile-target separation at which the reaction occurs. Hence
1
h 2(1, +5) 2 3 [1,2197- (2.4.3)
So that

(19,119)2 +(kpR)2 —(1, 41%)2

cos 6 S (244}

 

2(kd R)(kpR)

where hk as usual equals p. This equation expresses the approximate relation between
the angular position of the first peak in the angular differential cross section and the
transferred orbital momentum so that the angular position of the first peak in the

measured angular differential cross section will tell us the value of the transferred orbital

momentum.

75

The transferred momentum is bounded by the momentum of the transferred neutron

by [Fes92]:

2m|€|
h 2

A
-——k —k <
A+l p d

 

(2.4.5)

 

 

where 6‘ is the binding energy of the neutron. Good momentum matching that satisfies
the Equation 2.4.5 gives slow radial oscillations and large overlaps in the nuclear surface

region [Au387]. Based on the neutron binding energy listed in Table 2.6, the upward

limits of the proton incident energies in l3C(p,d)12C reaction to the ground state and the
first excited state are 32 MeV and 66 MeV respectively; for reactions of llB(p,d)mB

(g.s.) and 10Be(p,d)9Be (g.s.), the upward limits of the proton incident energies are 82

MeV and 46 MeV respectively.

2.5 Summary

As one of the fundamental tests of the shell-model theory, the spectroscopic factor
measures the occupancy of a nucleon in a pure single-particle state. It can be derived
from the ratio of the measured cross section to the calculated cross section assuming pure
single-particle state. The theoretical cross sections are calculated via DWBA and ADBA
models.

The proton and deuteron global optical-model potentials have been discussed. The
nucleon-nucleus potentials of CH89 and JLM usually give better descriptions to the
proton elastic scatterings than other global potentials. An adiabatic deuteron potential can

be constructed based on the nucleon-nucleus potentials such as CH89 and JLM. ADBA

76

calculations give better predictions than the regular DWBA calculations especially at
higher incident energy where deuteron break-up effect is significant.

The wave function of the transferred neutron (neutron form factor) is obtained by
adjusting the depth of the neutron potential to match the neutron separation energy to the
experimental value. The radius parameter and the diffuseness of the neutron potential are
usually fixed to 1.25 fin and 0.65 fm respectively. The sensitivities of the calculations to
the geometry parameters of neutron potential have been examined.

We chose target densities to have the form of modified Harmonic oscillator [Dej74]
for the JLM potentials. The scaling factor )1 of the JLM potentials were chosen to be 1.0
and 0.8 for the real and imaginary parts respectively [Pet85].

The momentum matching is discussed in Section 2.4. All the standard input

parameters used in TWOFNR for our calculations are listed in Table 2.7. We adopted the
value of 15006.25 MeV2 - fm3 [Knu75] for the vertex constant 002. Finite-range

approximation is employed in the calculations with the Hulthen finite-range factor of
0.7457 fm [Knu75]. Non-locality correction is also employed with the non-locality range

IBNL to be 0.85 fm [Per62] and 0.54 fm [Per74] for the proton and deuteron potential

respectively.

77

Table 2.7: Summary of the input parameters used in TWOFNR

 

DWBA

Adiabatic CH

Adiabatic JLM

 

Proton potential

Chapel-Hill [Var91]

Chapel-Hill [Var91]

JLM [Jeu77]

 

Deuteron potential

Daehnick [Dae80]

Adi abatic [J oh70]

Adiabatic [J oh70]

 

 

from CH ‘ from JLM
Modified Harmonic
Target densities oscillator density
[Dej74]
Woods-Saxon, Woods-Saxon, Woods—Saxon,
r0=l.25 fm, ro=l.25 fm, ro=1.25 fm,
n-binding pctential ao=0.65 fm, a0=0.65 fm, ao=0.65 fm,
depth adjusted, depth adjusted, depth adjusted,

no spin-orbit

no spin-orbit

no spin-orbit

 

Hulthen finite range

 

 

 

 

factor (frn) 0.7457 0.7457 0.7457
[Knu75]
Vertex constant Do2
( Met/2 . fm3) 15006.25 15006.25 15006.25
[Knu75]
JLM potential Av=1.0 and 71w=0.8
scalin g A N/A N/A [Pet85]
p: 0.85 fm; p: 0.85 fm; p: 0.85 fm;
Non-Locality range d: 0.54 frn; d: 0.54 fm; d: 0.54 frn;
n: N/A; n: N/A; n: N/A;

 

 

 

 

78

 

CHAPTER 3

EXPERIMENTAL SETUP AND DETECTOR
CALIBRATION

3.1 Overview

This experiment was performed at the National Superconducting Cyclotron

Laboratory (NSCL) at Michigan State University. One experimental objective is to study

the nuclei with neutron number, N=6, such as 1'B and 10Be which can be produced by
bombarding the production target of 98c with a primary beam of 13C (E=80.4 ~A MeV)
produced from the K1200 cyclotron. In addition to HB and 10Be beams, a secondary

beam 13C was also produced and the reaction p(l3C,d)12C was used for energy calibration
of the CsI detectors as explained in Section 3.3. The schematic diagram of the beam
fragment separator A1200 is shown in Figure 3.1. Fragmentations from the collision of
the primary beam with the production target of 9Be are bent by the two dipoles. A

momentum slit at dispersive image #1 selects the desired particles according to their mass
and momentum. A wedge at dispersive image #2 could be used to further disperse the
particles according to their energy loss. There is another momentum slit at the final
achromatic image that select the desired secondary beam. The thickness of the production
target (98e) and the beam intensity are listed in Table 3.1.

The experiment was carried out in the S800 vault at NSCL. Figure 3.2 shows the

schematic of the facilities in the 8800 vault. The secondary beam particles produced after

79

the A1200 were transported to the 8800 line. They were bent through the analysis line
consisting of Sextupoles, an Intermediate Image, Dipoles, and a Quadrupole Triplet

before the target chamber. The (p,d) reactions took place inside the target chamber which

contains the reaction targets of polyethylene (CH2)n foils. The thickness of the (CH2)n

targets are listed in Table 3.1. The total detection system includes the Multi Wires Drift
Counter (MWDC) detectors, the Large Area Silicon Strip Array (LASSA) [Wag01,
DavOl], and the S800 spectrometer. The following sections will describe each of these

detectors separately.

80

 

 

 

 

 

 

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83

3.1 MWDC Detectors

3.2.1 Principle of MWDC Detector

The Multi Wire Drift Counters (MWDC) are used to measure the beam positions and

angles at the intermediate image chamber of the 8800 vault. Each MWDC detector has an
active area of 11.2xl 1.2 cm2 covered by PPTA (p-Phenylene Terephthalamide) [Fuj89]
foil of 50 pm thick at the front and back windows. It is filled with P30 gas (70% Ar and
30% CH4) at a pressure of 500 torr. Beam particles transversing through the production
targets and the MWDC detectors lose energy due to interactions of the beam with the

target and other materials used to construct the counters. Taking into account the energy

loss which could be substantial with thick Be targets, the beam energies before the
(CH2)n targets are calculated and listed in Table 3.1.

Each MWDC detector has two orthogonal wire planes for the X direction and the Y
direction, respectively. The front and back of these two wire planes are covered by 12 um
thick mylar foil at front and back. Another mylar foil with the same thickness is placed
between the two wire planes. A schematic drawing of one wire plane is shown in Figure
3.3. In each plane, there are 14 anode wires and 15 field wires. The working voltage of
the anode wires is 580 Volts. The separation of the adjacent anode wires is 8.0 mm, and

all the anode wires are connected to a micro-strip delay line. This delay line has two

timing outputs T1 and T2. The field wires are separated into two groups. Every other field

84

wire is connected together and gives the left-right position signals E1 and E2,

respectively.
An enlarged drawing for the ions drifting between the wires is shown in Figure 3.4.

The positive ions drift towards the field wires and the negative electrons drift towards the
anode wire. The time signals T1 and T2 are expressed as:

T1 = dtl + (k — 1) - Z0 + dz (3.2.1)

T2 = dtz +(l4—k)-ZO +dt (3.2.2)
where 20 is the delay time between adjacent anode wires, dtl and dtz are the delay times
from the left and right wire ends to the preamplifier respectively, dt is the drift time of the
negative electrons from the hit point to the anode wire, and k is the wire sequential
number. Subtracting T2 from T1 yields :

T1 —T2 = dtl —d12 +(2k —-15)-ZO (3.2.3)

Then the wire sequential number k is

(T1 —T2 —dtl +d12) +1—5—
220 2

k:

 

(3.2.4)

A spectrum of the wire sequential number k is shown in Figure 3.5. By adding T1 and T2,
we get
T, + T2 = dtl + dt2 +1320 + 2dr (3.2.5)

Then the drift time dt is

: T1+T2 -dtl -d12 _1320
2

 

dt (3.2.6)

A spectrum of the drift time dt is shown in Figure 3.6, where the time is scaled in units of

microseconds ( ,u sec). The sharp peak near the zero drift time and the tail at the drift

85

time around 80 us correspond to the nonlinear electron drifting. Analysis indicated that
5% of the total particles are affected by this nonlinear behavior.
The left-right ambiguity is resolved by comparing the amplitudes of the two signals

E1 and E2 from the field wires. When the negative ions move close to the anode wires,

they produce a significant multiplicative effect and produce lots of positive ions around
the anode wire. Some of the positive ions will drift to the field wires. Since more positive
ions are produced on the side that the negative ions drift from, the particles hitting at one
side of anode wire produce larger signals on field wire in this side than the other. Figure
3.7 depicts a typical spectrum of the signals from one side of field wires versus the
signals from the other side.

Since alternate field wires are connected together, the particle position is calculated as

following in units of millimeters (mm).

k dt 1, E1<E2
=8.0k+ —1 -C~)-——D, 9: .2.7
p ( ) 20.0 {—1. 1551>1~32 (3 )

where k is the wire sequential number, dt is the drift time, D is the alignment center in

value of 60.0 mm, and ('9 is the left-right ambiguity function of E. and E2.

3.2.2 Position Calibration

To ensure that all the equations used in the position deterrrrination of MWDC
detectors are correct, a mask made of a 3.1 mm-thick brass plate is used to calibrate the
position. Figure 3.8(a) shows the pattern on the mask. The distance between adjacent big

holes is 10.2 mm; the distance between adjacent small holes is 2.54 mm. There is one L-

86

shaped slit with the width of 2.0 mm. The area in the middle surrounded by a dotted

rectangle is the area that detected the passing beam particles. Figure 3.8(b) is the
reconstructed two-dimensional position spectrum of the mask using the 11B beam, where

the beam was focused on the middle position between the first and second MWDC

detectors. A position resolution of 0.4 mm was achieved.

87

 

 

 

 

 

 

\ \ i J

 

 

 

 

 

 

 

 

 

T2

r-’ Tl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.3: Schematic drawing of wire plane of MWDC detector. All the anode wires are
connected to a micro-strip delay line, which has two timing outputs T1 and T2. Every
other field wire is connected together to form two groups and gives the left-right
ambiguity signals E1 and E2, respectively.

88

El E2

(JD—r
dt

 

 

 

 

A
0
V

Figure 3.4: An enlarged drawing of the ions drifting between the wires. The positive ions
drift toward the field wires and the negative ions drift toward the anode wire. The term dt
is the drift time of the negative ions from the hit point to the anode wire.

89

 

5000 *- ~ -

Counts

 

 

2500 r -

 

 

 

 

1 1 1 AL 11‘
7 8 9 10

 

91)
OB

Wire Number

Figure 3.5: Spectrum of the wire sequential number k. Each individual sharp peak
indicates the wire with its sequential number.

90

 

1500

1000 - -

Counts

500 —

 

 

 

 

 

O l l l l 1 J
0 20 40 60 80 100

Drift Time (,as)

Figure 3.6: Spectrum of drift time dt, where the time is sealed in units of microseconds.

91

 

250—

200-

150 —

E2 (Channel)

100-

50-

 

 

 

r n r I r
0 50 100 150 200 250

E1 (Channel)

Figure 3.7: Spectrum of E2 versus E1. There are two groups of particles. The left-right
ambiguity can be clarified by which side yields a larger signal than the other.

92

 

 

 

 

 

 

 

 

 

O O O O O
/ )

0 000000000 0
ooooooooo
000000000

0 OoooOoooO 0
000000000
000000000
000000000

V
O O O O O

 

 

 

A
a:
v

 

.0;

Q»... :
$312
0

vi
a 9 f '5‘“. .'..

II".

131940;.
«is:
o o -0 03“

 

 

"iii
19:!

I . .

 

A
5'
v

Figure 3.8: (a) Patterns on the mask used to calibrate the MWDC detector. The distance
between adjacent small holes is 2.54 mm; the distance between adjacent big holes is 10.2
mm; (b) The two-dimensional position spectrum of the mask. The corresponding areas
are surrounded by the dotted rectangle in (a) and (b).

93

3.3 LASSA Detector Array

3.3.1 Overview

The Large Area Strip Silicon Array (LASSA) was designed and constructed to
provide excellent energy, angular, and isotope resolution for the detection of charged
particles. It has been used successfully to study the isospin degree of freedom in heavy
ion multifragmentation experiments at NSCL [Tan02, LinS]. It consists of 9 individual
telescopes, which can be arranged into various configurations. Each LASSA telescope

consists of two silicon strip detectors and a cluster of four CsI(Tl) scintillators. The side
view of the telescope assembly is shown in Figure 3.9. A 5 mg/cm2 SnPb foil covers the

top window and provides a dark environment to the Si detectors. In addition, the foil also
protects the Si detectors from electrons produced in nuclear collisions. The two silicon
strip detectors are mounted inside the top frame below the SnPb foil. Right behind the
two silicon detectors are four CsI crystals with light guides and photodiodes mounted on
the back. The preamplifiers, and their motherboards, for the CsI detectors are placed
directly behind the photodiodes. A cooling bar is mounted at the back of each telescope

to keep the telescope at constant temperature.

94

3.3.2 Geometric Setup

In this experiment, LASSA is used to detect the deuteron particles emitted from the
(p,d) reactions. The 9 telescopes were separated into three groups and mounted on three
independent frames. A schematic drawing of the detector setup is shown in Figure 3.10.

The following coordination in the laboratory frame has been adopted: the beam
direction is defined as the z axis and the upward direction is the x axis; the polar angle 0
defined the angle of the particle direction with respect to the beam axis; the angle Otis the
angle between the particle projection on the z-y plane and the z axis; the angle B is the
angle of the particle direction with respect to its projection on the z-y plane. For
reference, each telescope is assigned a number as labeled in Figure 3.10. The geometry
of the centoid of each telescope is specified in Table 3.2. The detectors cover the angular
range of 3.6°<0<36.9° in the laboratory frame, which covers mainly the first peaks of the
transfer reactions. Beyond this region, the kinematic broadenings increase dramatically in

inverse kinematics.

3.3.3 Silicon Strip Detector Array

3.3.3.1 Overview

Silicon strip detectors are widely used in nuclear experiments because of their
excellent energy resolution and linear response for charged particles. Both layers of

silicon strip detectors used in LASSA are ion-implanted, passivated devices, Si(IP),

95

obtained from Micron Semiconductor [Micr]. For all the strip detectors used in our

experiment, the width of each strip is 3 mm, and there is a 0.1 mm wide gap between

adjacent strips. The active area on each surface is about 50x50 mmz.

The front silicon detector, which is labeled as DE, is about 65 um thick. It has 16
strips on the front surface. There are two different types of silicon strip detectors for the
second Si detector. One is double-sided and about 500 um thick, and the other type is
single-sided and about 1000 pm thick. In the double-sided detector, the strips on the front
are perpendicular to the strips on the back. For convenience, we refer the front strips and
the back strips of the double-sided detector as EF and EB respectively. For the single-
sided second detector, only the label EF is used. The close-packed design of the
telescopes right next to each other with a minimum dead area required the development
of a highly flexible flat printed circuit board cable connecting the silicon strip detectors
with the pre-amplifier housings. Figure 3.11 shows the picture of a double-sided silicon
strip detector with the flat printed circuit board cables. The combinations of the silicon
strip detectors are listed in Table 3.2. The reason for choosing 1.0 mm silicon detector for
telescope 2, 4, 5, and 9 is that the deuterons emitted at smaller angles have lower energies
and will be stopped inside the 1.0 mm silicon detectors. The deuterons emitted at larger
angles with higher energies (E>10.8 MeV) would punch through the 500 um silicon
detectors and be stopped in the CsI(Tl) crystals.

One advantage of silicon-strip detectors is their position resolution. For the double-
sided detectors, we can use the orthogonal strips on EF and EB to obtain each particle’s
(x, y) pixelwise position. When the 1.0 mm detector is used, its strips are oriented

orthogonally to the strips in the DE silicon detector to provide a two-dimensional

96

position. As the strips are 3.1 mm apart, the 50mmx50mm lateral dimensions of each
telescope are divided into 256 (16x16) square pixels with a resolution of 3x3 mmz. At a

distance of 205.3 mm, the angular resolution of the pixel is 0.43 deg. Telescopes 4, 5, and
7 were placed at a greater distance of 392.6 mm and their angular resolution is 0.22 deg.

All the position and angular information is summarized in Table 3.2.

3.3.3.2 Energy Calibration

One advantage of silicon detectors is their linear and largely particle independent
energy response. In this experiment, the relevant deuteron energies range from 9.0 MeV
to 20 MeV. The silicon energy response in this range is very linear. However, due to the
nonlinearity of electronic system including preamplifiers, shapers, and ADCs, energy
calibration must be performed.

A BNC (Berkeley Nuclear Co.) precision pulser generator was used to calibrate the
silicon detectors. The pulser has a group of attenuation switches to change the amplitude
of the output signal. Three attenuation settings were chosen corresponding to three

different dynamic ranges. An absolute calibration was obtained from the measurements
of 228Th or source for these three settings. The linear relation between the pulser dial
value and its equivalent energy was obtained:

E=a~W+b (3.3.1)

where E is the equivalent energy of the pulser signal in the units of MeV, and W is the

dial value in the Volts. The values of a and b for the three different settings, as listed in

97

Table 3.3, show that the output of the pulser is not strictly proportional to the dial voltage
and the offsets are not zero.

Right after the experiment was finished with all the electronics setup intact, the
calibrated pulser signals were sent as inputs into the preamps of each strip. Then one-by-
one, the pulser-calibration was carried out for all 352 silicon channels. The signals were
read by the DAQ program and analyzed with the analysis program SMAUG. A linear
fitting was performed to the channel reading C and the energy E converted from the
pulser dial value W by Equation 3.3.1. The relation between C and E was obtained for
each strip:

E=gi ‘C‘l‘hi (3.3.2)

where i stands for each strip. Equation 3.3.2 was used to convert the channel readout
from each strip into particle energy in units of MeV. There is a total of 352 calibration
curves for the Si-strips. As an example, Figure 3.12 shows the calibration curve for the

No.6 strip of the EF detector in telescope 3.

3.3.3.3 Particle Identification

In a heavy ion collision, many kinds of particles like protons, deuterons, tritons, and
other fragments from the projectiles are detected in LASSA detectors. Since we are only
interested in the deuterons, a particle identification (PHD) must be performed to
distinguish the other particles. The PID can be performed by a combination of AE and E

detectors.

98

For particles passing through a detector, the energy loss is approximated by the Bethe

formula [Bar96]:

kAZ2
~ E

AE

 

dx (3.3.3)

where dx is the detector thickness, k is a proportional constant, A is the mass number,
and Z is the atomic number of the particles. For a fixed dx, a plot of AE versus E yield a
family of contours with AE cc l/E . Each line corresponds to an integer value of Z and A.

Figure 3.13 shows the AE-E spectrum of telescope 7. The x axis is the particle energy
deposited in the EF strip detector; the y axis is the energy loss deposited in the DE strip
detector. By vetoing the particles stopped in CsI(Tl) crystals, we were able to separate the

deuterons and protons as well as tritons.

3.3.3.4 Position Calibration

The position determination of the emitted deuterons is critical in this experiment. To
determine the position of each pixel of the telescope accurately, we need to perform the

position calibration. The angular position (61,6) of the center of each telescope was

determined optically by using a system composed of a optical telescope and a mirror as
shown in Figure 3.14. The mirror was mounted on a turntable which can be rotated in
both horizontal and vertical planes and the rotation angles can be read from the turntable.
The center of the mirror was placed at the center of the target in the experiment. The
optical telescope was mounted in the beam line. The mirror was rotated until the center of

each detector is visible and aligned with the optical telescope, then the angular positions

99

(a, ,6) of the detector center are:

a = 2 - a0

13 = 2 - flo (3.3.4)
where do and ,60 are the angles of the mirror rotated in the horizontal and vertical

planes. The rotation angles can be read to the accuracy of 0.01 degree so the accuracy of
the measurement is 0.02 degree. The angular position of the center of each detector is
listed in Table 3.2. The coordinates 7(x. y. Z) of each pixel of the LASSA telescopes are
obtained by:

f(x. y, z) = R1(a)-R,1(fl)- fo(xo.yo.zo) (3.3.5)
where 20 is the distance between the target and the center of the detector; x0 and yo are

the vertical and horizontal distances between the pixel and the center of the detector,

respectively; R,C (a) and Rv(,6) are the rotation matrices along the x axis and y axis.

They are defined as:

0 0 O
Rx(a) = 0 cosa sina
0 — sin (1 cos a

cosfl 0 - sin ,6
Ry (,5) = 0 O 0 (3.3.6)
sin ,6 0 cosfl

We performed the calibration before and after the experiment. The positions did not

change during the experiment.

100

3.3.4 CsI(Tl) Crystals

3.3.4.1 Detectors

The CsI(Tl) crystals are produced by Scionix [Scio]. A non-unifonnity in light output
of CsI(Tl) crystals better than 1% was obtained via crystal selection and a quality control

procedure [Mic99, Tan02]. Each crystal is trapezoidal in shape as shown in Figure 3.15,

and the length of the crystal is 6.0 cm. The front and back areas are 2.64x2.64 cm2 and

3.38x3.38 cmz, respectively. To allow compact packing, the sides between adjacent

crystals are at right angles to each other while the sides next to the frame are cut at an
angle of 7.09 degrees. Each crystal is wrapped with two layers of cellulose nitrate on the

outer surfaces (next to the frame) and one layer on the inner surfaces. One layer of
aluminized mylar foil (0.15 mg/cm2 mylar + 0.02 mg/cm2 Al) is inserted between
adjacent crystals to ensure optical isolation.

Each crystal is optically coupled to a clear 1.0x3.5x3.5 cm3 acrylic light guide with
optical epoxy BC600 [Bicr]. This light guide is in turn optically connected to a 2.0x2.0
cm2 Hamamatsu S3204 photodiode [Hama] with clear silicone rubber compound

RTV615 [Gene]. To prevent light leak and cross-talks between adjacent crystals, the
outer sides of the light guide and the photodiode are painted with a reflective white paint
BC620 [Bicr].

To reduce the noise level, the charge-sensitive preamplifiers are mounted right behind

the crystals to reduce the length of the input cables and minimize the capacity input to the

101

preamplifiers. There are two motherboards in one telescope. Each one hosts two
preamplifiers.

Fig 3.9 shows the internal mounting structure, the outside of the detector box is Open
and the two silicon strip detectors are placed on the side. An aluminum mylar foil covers
the top of the wrapped crystals to maximize light reflection and improve the energy
resolution. One motherboard of the preamplifiers can be seen under the crystals.

Three major precautions are taken to reduce the cross-talks between the preamplifiers
of the CsI detectors. The first is to place a grounded copper shielding between the two
motherboards to minimize broadcasting; the second is to put a 110 Q resistor on the test
input line connecting the two preamplifiers on the same motherboard to terminate each
amplifier; the third is to use shielded coaxial cables instead of twisted—pair cables to
reduce the broadcasting between cables. With this set up, the cross-talks are reduced to

the level of 0.1% [Mar98].

3.3.4.2 Energy Calibration

The fluorescence emitted by the CsI(Tl) scintillator has two major components of a
fast (500 ns) and a slow (7 us) decay time constants. The relationship of light output and
energy is mass and charge dependent. Therefore the CsI calibration cannot be performed
by different kind of particle, neither by pulsers. In addition, the light output of a CsI
crystal also depends on the T1 doping of CsI crystals. Since every CsI crystal may have
different doping during manufactory, it is necessary to perform calibration for each CsI

crystal individually.

102

For heavy ions at low energy, the light output of a CsI crystal shows non-linear
response to the deposited energy [Lar94, BirSl]. However, for the isotopes of hydrogen,
Tan [Tan02] and Handzy [Han95] found that linear functions result in good fitting.
However, previous calibrations did not extend deuteron calibration below 20 MeV, so the
deuteron response function for the CsI crystals was not known in our energy region of
interest.

Calibration of the CsI(Tl) crystals was achieved by the reaction of p(l3C,d)12C. 13C is
the primary beam with high beam density. The emitted deuterons corresponding to the
ground state and the first excited state of 12C can be identified clearly in the energy
spectrum. As the deuteron scattering angle is known from position calibration of the
pixels, the deuteron energy is obtained by kinematic calculation. As shown in Figure
3.16, the emitted deuteron goes through target, SnPb foil on the window of telescope, DE
silicon strip detector, and EF(EB) silicon strip detector before being stopped in C51
detector. The deuteron energy deposited into the CsI detector is:

ECsl = Ed — AEtar ’ AESnPb ‘ AEDE — AEEF (3-3-7)
where Ed is the emitted energy of the deuteron from the target determined from

kinematics, AEm, and AEsnpb are the deuteron energy losses in target and SnPb foil

respectively. These energy losses are obtained using the program ENLOSS [Enlo]

according to the material components and thickness. AE DE and AEEF are the energies
deposited into DE and EF(EB) silicon strip detectors. Then the channel readout CCsl
from €81 detector can be calibrated to ECsI by a linear fitting:

ECsl =41 'CCsl +101 (333)

103

where 1 stands for each CsI crystal. This equation was used to convert the readout of the
CsI detectors into particle energy in units of MeV. Figure 3.17 shows the calibration
curve for the No. 3 crystal in telescope 3. Clearly, the linear fitting works very well for
the deuteron calibration from 1 MeV to 14 MeV. This result is consistent with the
observations of Tan [Tan02] and Handzy [Han95]. Our fitting results in a precision of the

calibration better than 2%.

3.3.4.3 Particle Identification

For the particles that stopped in CsI(Tl) crystals, the particle identification can be
performed by the combination of silicon strip detectors and CsI(Tl) detector. Figure 3.18
shows the AE-E spectrum of telescope 3. The x axis is the particle total energy, including
the energies deposited in DE, EF(EB) silicon strip detectors and CsI detector; the y axis is
the sum of 4 times the energy loss in DE detector and the energy loss in EF(EB) detector.

The deuterons and protons are well separated.

104

Table 3.2: Geometric setup of the telescopes and the configurations of the silicon strip

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

detectors.
Thickness Thickness
of of EF/EB
Telescope 9 or B Dist. Angular DE silicon
No. (deg.) (deg.) (deg.) (mm) resolution silicon strip
strip detector
detector (um)
(pm)
1 21.5 -14.2 -l6.3 205.3 i0.43° 67 480
2 14.2 -14.2 0 205.3 i0.43° 68 978
3 21.5 -l4.2 16.3 205.3 i0.43° 64 500
4 7.0 0 -7.0 392.6 i0.22° 64 913
5 7.0 0 7.0 392.6 i0.22° 65 982
6 27.9 23.0 -16.3 205.3 10.43° 67 481
7 16.7 0 16.7 392.6 i0.22° 66 476
8 27.9 23.0 16.3 205.3 i0.43° 70 482
9 23.0 23.0 0 205.3 i0.43° 66 993
Table 3.3: The calibrated parameters in Equation 3.3.1
Attenuating Maximum energy a b
setting range

x 2 30 MeV 5.0592 01081

x 5 16 MeV 2.0347 -0.0933

x 20 3 MeV 0.5115 -0.l223

 

 

105

 

 

 

 

 

Figure 3.9: Structure of LASSA telescope.

106

    
 

A X
Entrance
of S800

 

Figure 3.10: Schematic of the geometric setup.

107

 

Figure 3.11: One double-sided silicon strip detector with the flat printed circuit board

cables.

 

 

 

 

10 m ' ' ' ' T

E = 0.0021C-0.0265

S‘

0

a

DJ 5 - -l

0 . . . . 4L . 4%
0 2000 4000

Channel

Figure 3.12: Calibration curve for silicon strip detector, by which the channel readout of
the silicon strip detector is converted to particle energy in units of MeV. This curve

stands for the No. 6 strip of EF detector in telescope 3.

108

EDE (arbitrary unit)

100

 

 

 

l

|

t

 

 

1 0 0
E5]: (arbitrary rmit)

150

200

Figure 3.13: Particle identification using the energy spectrum of BBB vs. EEF for telescope 7.

109

Detector
Telescope

Mirror

Eve Telesmpe \ \
<1 ‘

[I]
i

/—

 

 

ll

 

'I‘m‘ntable

 

 

 

 

Figure 3.14: Geometry calibration system composed of one optical telescope and a mirror
mounted on a turntable with two orthogonal axes that rotate in horizontal and vertical
planes. The center of the mirror is the position of the target in the experiment and the
optical telescope is mounted in the beam line.

 

 

 

 

a= 2.64 cm

 

 

 

 

Figure 3.15: The shape of CsI(Tl) crystal.

110

13C beam

 

 

CH2 target

Figure 3.16: Schematic of the CsI calibration. The deuteron emitted angle is determined
by the pixel on DE and EF(EB) silicon strip detectors. The deuteron emitted energy is
obtained by kinematic calculation. The deuteron deposited its energy into the CsI crystal
after going through target, window foil, DE and EF(EB) silicon strip detectors.

lll

 

15

E = 0.0188C - 0.1777

E (MeV)

 

 

 

 

0 l 1 fi
0 200 400 600 800

Channel

Figure 3.17: Calibration for CsI detector, by which the channel readout of the CsI
detector is converted to particle energy in units of MeV. This figure shows the calibration
for the No. 3 crystal in telescope 3.

112

 

 

 

 

5‘5 200 .—
E.
E. 150 -
ha
m
[:1
+ 100 — a,
'5‘ '9'. a
I: ' “nigh
_ _ .39.“.
50 “- -J _ ' H . ..
p 3 ' 331.913: 9‘. - .- H
0 I r l " l"
0 50 100 150 200,
Brant (arbitraryrmit)

Figure 3.18: Particle identification in the energy spectrum of 4E DE + E EF vs. Emm, for
telescope 3.

113

3.4 8800 Spectrometer

3.4.1 Overview

The working principle of magnetic spectrometer is following: A particle with charge
q and mass m, traveling at speed v, passing through a uniform magnetic field with

strength B, will travel in a circular path with radius p given by

m3: 8,0 (3.4.1)
q

Relativistically, the mass is m, where m is the rest mass and y is the Lorenz
transformation factor. Thus, for a given magnetic field setting, particles with identical
momentum to charge ratios are deflected the same amount by the magnet.

A schematic of the 8800 spectrometer is shown in Figure 3.2. It stands behind the
target chamber and consists of one quadruple doublet, two dipoles, and one focal plane
detector. The advantages of S800 spectrometer are the high energy resolution and large
solid angle acceptance [Zha97, Yur99, Cag99]. Some of notable characteristics of the
S800 spectrometer are listed in Table 3.4.

Figure 3.19 shows the schematic of the focal plane detector of 8800. It consists of
two Cathode Readout Drift Chambers (CRDC), one ion chamber, and four plastic
scintillators. The CRDC detectors measure the two transverse positions and angles of the
particles; the ion chamber measures energy loss in the gas; the plastic scintillators
measure the particles energies.

The particle flight time is measured relative to the cyclotron radiofrequency (RF)

pulses. Different species of particles emitted from the reactions have different velocity,

114

and hence different flight time to the focal plane. This flight time measurements can then
be used in conjunction with the energy loss measurements or total energy measurement to

identify the particle species that arrive at the focal plane.

3.4.2 Cathode Readout Drift Counters

The CRDC detectors have an active area of 30 cm x 59 cm and an active depth of 1.5
cm. They are filled to a pressure of 140 Torr with 80% CF4 and 20% C4H10. Figure 3.20

shows a schematic illustrating the principles of their operation. Ions traveling through the
gas create ionizations. A constant vertical electric field in the detector move the electrons
toward an anode wire, where charge amplification takes place in the high electric field
close to the wire. The anode wires are placed below a ground Frisch grid and held at a
constant voltage, typically 1400 Volts. The electrons are collected on the anode wire.
Cathode pads are Tocated in front and back of the anode wires. The charges collected on
the anode wire induce positive charges on the cathode pads. There are 224 pads in each
CRDC detector. The centroid of the Gaussian fit to the charge distribution is used as the
horizontal position in the detector.

The vertical position is determined by the drift time of the electrons to the anode wire.
The typical drift time of the electrons to the anode wire is 0-20 us, depending on their
vertical position. Measuring the time between the scintillator signal and the anode wire
signal provides a direct vertical position measurement of the particle track.

Masks with well-defined holes and slit patterns as shown in Figure 3.21(a) are placed

in front of the CRDC detectors to calibrate the detector positions. Figure 3.21(b) is the

115

position spectra taken with 10Be beam with the mask placed in front of the first CRDC

detector. The position resolution of 0.2 mm is achieved.

3.4.3 Ion Chamber

Immediately following the CRDCs, the beam particles pass through an ionization
chamber. The ion chamber (IC) used in the S800 is a standard Frisch grid ion chamber
[Yur99]. It is designed to measure the energy loss as the beam particles ionize the gas in

the detector by sampling the signal generated along sixteen anode strips. The gas used is
P10, which is composed of 90% argon (Ar) and 10% CH4 (methane). The energy loss in

the ion chamber combined with the time-of—flight or the energy deposited in scintillator

detectors can provide particle identification.

3.4.4 Plastic Scintillators

There are four plastic scintillators in 8800 spectrometer. They are made of BC-408
scintillant plastics manufactured by Bicron [Yur99]. In the order from first to last, with
respect to the beam direction, the scintillators are labeled as E1, E2, E3, and E4 in Figure
3.19 with the thickness of 3 mm, 5 cm, 10 cm, and 20 cm respectively. Light guides are
mounted on each end to enhance the collection of the light in the photomultiplier tubes
(PMT). The light travels through the plastic as well as the light guide and is collected in
the PMT’s on the top and bottom ends of the scintillator. The energy deposited in the

scintillator is calculated by:

116

 

E, = J 15,-”? x EiDOW" (3.4.2)

In the meanwhile, we get the number of particles that enter into S800. We performed
normalization run at the beginning of each kind particle beam, when the targets are
moved out of the position and the beam particles enter 8800 directly. We measure the
beam transfer efficiency by the ratio of particle number in 8800 to the particle number
going through BLT2 scintillator. The normalization procedure will be introduced in
Section 4.3.

Since both energy degraded beam particles and the residual particles from the (p,d)
reaction enter 8800 spectrometer simultaneously, we need to separate them out by the
combination of deposited energy in E1 vs. time-of—flight. As an example, Figure 3.22
shows the spectrum of the deposited energy in E1 versus the particle time-of-flight for the

reaction of p(“B,d)loB. The residual particle of '0B is separated from the incident beam

ll

of B.

3.4.5 Summary

Originally, we plan to use S800 spectrometer to detect the recoiled residual nuclei in
coincidence with the deuterons detected by LASSA detector to perform complete
kinematic measurement. Based on the above analyses, 8800 supplies excellent particle
position determination via CRDC detectors and good particle identification via the
combinations of energy loss in ion chamber, energy deposited in scintillators, and time-
of—flight of particles. Unfortunately, there were errors in writing the data from 8800

spectrometer onto the tape in this experiment so that some 8800 data were lost. In the

117

present work, the S800 spectrometer was used only for the normalization, when the data

from 8800 are complete.

118

 

El. E2. E3. E4 Scintilators

\l
4,41.

1 . ' III
4. . , lon Chamber
l/Q’RM‘ #2

   
   

      

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.19: Schematic of the focal plane detector of S800 spectrometer. It consists of
two CRDC detectors, one ion chamber, and four plastic scintillators.

119

Table 3.4: Characteristics of the S800 spectrometer

 

 

 

 

 

 

 

 

 

 

 

 

AE _
Energy resolution 7?— =10 4
AP -
Momentum resolution 7; = 5 x 10 5
Energy range 11.6 %
Momentum range 5.8 %
Solid angle 20 msr
Angular resolution 52 mrd
Horizontal detector
. 0.3 mm
resolution
Vertical detector
. 0.3 mm
resolution
Maximum rigidity 4.0 T-m
Maximum dipole field 1.42 Tesla

 

120

 

.1. ‘ lllul

1' - ls-~i~i-s-sl~\1
1§R§N§§§§N§S§§§N

1....l~.x‘..\‘x....\.1|\\u.\l..\l.\ .1.‘ .x, ..s .x .x. ..c. .\ 1.. ..4 .,.
....\. ~§§\ _ x... 1.. . \ . \x . .. ..1 x .\ 1...

x. 1..
x \ x \ ..x
\1 n! I . . 1 . . .1

x. 1 a .. u \ ~ \ 1 \ a
—. \ )0 x xx . x . . . 4 as. . .
\ K .2. . x. .. .- ~ . .\ v ..
~ 4
a! \ _. .. x . \ \. .
x .. .. . .. . x
. .\ ll/ 4. . .~ . .. .. x x. .\ .. H
\ _ . H 1.. . .
. w. x . .
‘ \x I . 1.
\ R . .. .
. . 4 .x
\ \ .‘
ax

 

       

      

 

 

       
    

    

 

 

   

 

    

 

 

         

 

1..
..r'sll.\
..‘.\.l!.

'.

x
i

 

Anode wire

Figure 3.20: Schematic of the CRDC detector. A particle ionizes the gas as it passes

through the detector. The electrons drift to the anode wire where they are collected. The

induced image charges on the cathode pads provide horizontal position information. The

drift time of the electrons to the anode wire provide vertical position information.

121

0.10

 

 

 

 

 

 

 

 

 

 

 

 

A bi'l'l'llTr IUIIIIIIIIITI IIIIIIIUIIITUI UTTITITII'IT' WI .1
12 E ° 3
2 0.05 —- ° . . —
.5 I o o c j
1: I 3 3 Z
:3 0.00 -_— o o o o o o o o o o .1:
3. : ° 3
{>4 “0.05 [_- ° 0 o T.
_0_ I 0 :r_Lu11_L_r..r_r_r_Lr. W .r.r..r_LL_r_r_L.Lr_r_r_r_rur_r_r_1_‘.
-0.3 -0.2 -O. I 0.0 0.1 0.2 0.3
X position (meters)
(8)
0.05
0 0 t . t ‘ . E. . ‘
' . l l l
-0.05 r 1 1 1
-0.15 —0.l -0.05 0.0 0105 0.1 0.15 0.2

(b)

Figure 3.21: (a) Patterns on the mask. (b) Position spectrum of the mask placed in front of

the first CRDC detector.

122

 

 

 

 

 

 

CE __

5

E”

.E 140 —

.fl

3

E _

B

H 100 ——

60 | l l l l I
120 140 160 200
Time of Flight (arbitrary unit)

Figure 3.22: Spectrum of the energy deposited in E1 scintillator versus the time-of—flight
ll 10 .
for the p( B,d) B reaction.

123

3.5 Data Acquisition Electronics

Figure 3.23 is a schematic of the electronics used in this experiment. The signals from
the up and down PMT’s of the first scintillator El are sent to constant fraction
discriminator (CFD) module. The outputs of the CFDs are AND-ed to give the S800
premaster signal. The S800 premaster signal and the CFD outputs of the anode wires of
CRDC detector are used as the start and stop for the drift time in the CRDCs. The TAC
output is input to module of analog-to-digital converter (ADC). The cathode pads are read
by the fast encoding and reading ADCs (FERA). The gate for the FERAs is given by the
AND of the 8800 premaster and the anode pulse.

The signals from the silicon and CsI(Tl) detectors are digitized in Phillips Scientific
peak-sensing ADCs (7164H). The signals from the EF silicon strip detectors are sent to
Shaper-Discriminator-TFC dual modules. For this module, the shaper outputs are sent to
ADCs; the TFC outputs are sent to Lecroy 4300B fast encoding and reading ADCs
(FERA) to give the time signals; the trig outputs from all 9 telescopes are OR—ed to give
LASSA Premaster signal.

The LASSA Premaster will be AND-ed with 8800 premaster to give coincidence
Premaster signal. The LASSA Premaster will also be delayed and downscaled to give

LASSA trigger signal.
The Master signal is logically AND-ed with the Busy signal from the computer,

coincidence Premaster, S800 premaster, and LASSA trigger. The Master signal is the
start signal to the computer and stop signal for the TFC. The gates for the modules of

ADCs and FERAs are also supplied by the Master signal.

124

 

 

 

 

 

 

 

 

 

Down scale {Delay

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ELUP L
Coincidence
E1_Up ~ CPD ssoo LASSA Premaster
Pr emaster 0 Pre mas ter (AND) :9:
(AND) 3
ELDown CFD “‘ E
LASSA
Trigger
ELDN Busy
start 3801
TAC
ADC
/ stop TAC
CRDC Anode —{>— CFD H I
‘ AND gate
FERA Pads
Cathode Pad FD" Shaper in out
Master : ‘ FIFO DGG
Start To LASSA TF C
l\\
CsI Detectors l/ Shaper ADC
1\
DE Detectors L/ Shaper ADC
l\ {—
EB Detectors V Shaper ADC
Shaper j
BF Detectors —{>_ Disc ADC
TFC 5 3
Disc FERA
Master Stop Trig "I
:: LASSA _ Down scale __ LASSA ‘
—p OR Premaster {Delay Trigger
—I~
—' T c ' T
o om. o
Premaster Master

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.23: Schematic of the electronics.

125

CHAPTER 4

EXTRACTION OF AN GULAR DIFFERENTIAL CROSS
SECTIONS

4.1 Overview

This chapter will discuss the extraction of the deuteron spectra and analyze the
contributions to the energy resolution in Section 4.2. In section 4.3, the procedure to
extract the differential cross sections will be introduced and the measured data are

presented.

4.2 Deuteron Spectra

Applying the PID gates obtained in Section 3.3, we can pick out the deuterons and
obtain their energies in laboratory frame via Equation 3.3.2 and Equation 3.3.8 for silicon
strip detectors and CsI(Tl) crystal detectors respectively. The deuteron energy in the
center of mass is obtained by converting the measured deuteron energy in the laboratory
frame to the center of mass frame.

EC," = %mV2 + -:—mV02 - mV vO cos a (4.2.1)

where m is the deuteron mass, V is the deuteron velocity, V0 is the velocity of the center

of mass, and 6 is the emitted angle of deuteron. Figure 4.1 shows the deuteron energy

spectrum of the p(l3C,d)12C reaction at the laboratory angle of 19°. The peaks of ground

126

state (0+) and first excited state 4.439 MeV (2+) can be distinguished clearly. The peaks
at 7.654 MeV (0+) and 12.71 MeV (1+) do not have enough statistics but still can be

identified. The peaks at 15.11 MeV (1*) and 15.44 MeV (2") cannot be resolved

completely.

To estimate the full width at half maximum (FWHM) of the peaks in the laboratory
frame AEL, we need to take into account that the emitted deuterons emitted from the

reactions have to go through the remainder of the target and the SnPb foil before reaching

the Si detectors.

 

AEL = ‘55,," 2 + AESnPb 2 + A592 (4.2.2)
where AEtar is the rms width of the deuteron energy loss distribution in the target, AESnpb
is the deuteron energy straggling in the SnPb foil, and the AEB is the kinematic

broadening due to the angular resolution of the strip detectors. AE.ar is larger than the

width given by energy loss straggling because of the variation of energy loss in the target
depend on how much of the target is traversed before the reaction occurs. The beam
broadening and the beam straggling in the target are not included in Equation 4.2.2
because they contribute little to the deuteron resolution. The FWHM in the center of mass
derived from Equation 4.2.1 is

AEcm =AEL—mV0cost9-AVL +mVLV0 sin 6-A6 (4.2.3)
where A0 is the angular accuracy of pixelation (i0.l°), AVL is the FWHM of deuteron

velocity in laboratory frame. Table 4.1 lists the contributions to deuteron resolution. The

energy straggling are calculated by the program SRIM [Srim]. The energy broadening

127

AEe due to the angular resolution and the energy loss in the target contribute most to the
final energy resolution. The experimental FWHM for the peaks of 0.0 MeV and 4.439
MeV are 800 keV. The estimated energy resolutions are pretty close to the measured

OIICS.

Figure 4.2 shows the deuteron spectrum of the p(“B,d)loB reaction at the laboratory

angle of 13°. The energy resolutions are about 640 keV for the ground state and the first
excited state at 1.74 MeV. The other states cannot be evaluated because of low statistics.
From Table 4.1, we see again that the kinematic broadening due to the angular resolution
and the energy loss in target contribute most to the energy resolution. A smaller

kinematic broadening and a thinner target will result in a better energy resolution.
Figure 4.3 shows the deuteron spectrum of the p(lOBe,d)9Be reaction at the laboratory
angle of 15°. The energy resolution is 800 keV for the ground state. The excited states

cannot be distinguished because of low statistics. Contributions to the energy resolution

are also listed in Table 4.1.

4.3 Extraction of Angular Differential Cross Sections

Before the extraction of the angular differential cross sections, we need to know the

beam transfer efficiency f and the total beam particles Npar that hit the target. The beam

transfer efficiency is measured in a normalization run at the beginning of each
experiment, where the target is moved out of the beam line. The beam particle delivered
before the target is measured by the BLT2 scintillator and the beam particle through the
target is measured by the El scintillator of S800 spectrometer. The beam transfer

efficiency is:

128

.1 Na
Nam 'RLT

 

f (4.3.1)

where RLT is the life time of the data acquisition system. The total beam particles that hit

the target Npar are calculated by the summation over all the runs:

Npar :ZNBLTZI 'RLTl 'f (4.3.2)

I

where N 3112'. , and R [1" are the number of particle detected by the BLT2 scintillator and

the life time of the data acquisition system for each experimental run, i.

The angular differential cross section in the laboratory frame is obtained for each

(do) i Nd
— = (4.3.3)
d9 6L dQL'Ntar 'Npar

telescope individually:

 

where i denotes individual telescope, BL is the angle in the laboratory frame, Nd is the
number of deuterons detected in the interval of il.0° relative to BL, dQL is the solid

angle in the laboratory frame, and NW is the target thickness in number of hydrogen

atoms per centimeter square. The statistical error for each telescope is calculated by
A [do I _ do I 1
do 6L do 6L ,/ Nd (4'34)
The average angular differential cross section and statistical error are obtained by
(L0) 1’
2 d9 6L
. 2
A(LU) 1
(dd) _ dQ 6 L
6L _

214311.104

L" .1 1 ..
A(dfljflz Z[A[do) i]_2 (436)

 

(4.3.5)

 

 

 

129

The angular differential cross section and statistical error in the center of mass are

[12) _.(iz)
d9. 6,", do 6L

 

 

 

A (fl—9) = y - A (£9) (4.3.7)
d9 60,, d9 19L
1+ flcos 6’6,"
m2
y :

2 2
m m
1+ ——1—2— + 2—l—cos66m
"12 m2

where y is the ratio of dQL / dflcm , m1 and m2 are the mass of projectile and target

nuclei.

For the p(l3C,d)]2C reaction, the angular differential cross sections to ground state
and first excited state at 4.439MeV have been extracted. For the reaction of p(“B,d)mB

and p(loBe,d)9Be, only the angular differential cross sections to the ground state have

been extracted because of the low statistics of the excited states. The data and the
statistical errors are listed in Table 4.2.

The Open red symbols in Figure 4.4 show our measured angular differential cross
sections of p(13C,d)12C (g.s.) reaction. It is compared to the published data of Ref.
[Cam87] at proton energy of 41.3 MeV (solid red circles). There are additional data in
Ref. [Sco70] at proton energy of 50 MeV. However, the latter set of data was published
in arbitrary unit. We match Scott’s data [Sco70] at 121° to the data of this measurement

and get the normalization factor of 2.45. The three sets of data show fairly good

agreement especially when the difference in beam energies is taken into consideration.

130

Figure 4.5 shows the comparison of the differential cross section of p(l3C,d)12C to the

first excited state from our measurement (open circle), data of ref. [Cam87] (closed red
circles), and data from ref [Sco70] (diamonds) with the same normalization factor of
2.45. Again, our data agree with the past measurements fairly well suggesting that the
experimental procedures we used for measuring angular distributions for inverse

kinematic reactions with high resolution strip detectors work rather well.

Figure 4.6 shows the angular differential cross sections of p(”B,d)lOB (g.s.).

Unfortunately, the 11B data were taken with relatively short time so that the total statistics

we have collected are low. Only telescopes l, 2, 3, and 7 yield significant counts to the

measurements. The measured data have large error bars.

Figure 4.7 shows the angular differential cross section of p(loBe,d)9Be to the ground
state. However, we have problem with the absolute normalization. In this particular
reaction, the 8800 trigger some time did not fire. When that happens, the LASSA trigger

fired alone but with a downscale factor of 5. We have to add the events by S800 trigger

together with 5 times of the events by LASSA trigger. This problem only happened in the
beam of 10Be. We still do not understand the reason of this problem thus there are

unresolved questions about the absolute value of the cross section.

131

 

 

 

 

 

 

 

 

 

 

30 l * r I r'f)‘
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*" Z¥
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24 3

16 2 40

E d(MeV)

Figure 4.1: Deuteron energy spectrum of the p(l3C,d)12C reaction at the laboratory angle
of 19° measured by telescope 7.

132

 

 

 

 

 

 

 

30 ' l r l r l
" >
0
2
O.
> o
m _ _
a 20 1 9 l
a 9 as —
o is 2.4
‘13 sl"
L9. to)
co "3
10- ‘ m _
Oat—L11 Ilfl 1 1 1iflfl r0
16 20 24 28 32

Ed(MeV)

Figure 4.2: Deuteron energy spectrum of p(1 lB,d)lOB reaction at the laboratory angle of
13° measured by telescope 7.

133

 

—

3011111

B 20 — —
C. .
:3
o
C.) >
g >>
go 22
10 7 1‘. LOCI) —
_ (0 OD-

 

 

F‘—

l
_ 11 :AM
00101111111111.411111111 11111111111 0111111

16 24 4O
Ed(MeV)2

 

Figure 4.3: Deuteron energy spectrum of p(lOBe,d)9Be reaction at the laboratory angle of
15° measured by telescope 7.

134

 

 

 

 

 

 

 

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135

Table 4.2 Experimental angular differential cross sections and statistical errors

(a)

 

p(l3 C.d)12c. Ex: 0.0 MeV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(10' da'

_ A _
6cm d9, d9

(deg) (mb/ sr) (mb/sr)

4.0 1 1.54 1 .07
5.2 11.48 1.19
6-4 1 1.88 1.59
7.7 1 1.09 1.80
8.9 10.07 0.58
10.2 9.60 0,49
1 1.6 8.70 0.61
13.0 7.85 0.45
14.5 6.68 0.41
16.1 5.39 0.37

 

 

(c)

 

p(' ‘B.d>‘°B. Ex: 0.0 MeV

 

 

 

 

 

 

 

 

do' do

_ A _
6cm d9 dg

(deg) (Mb/ Sf) (mb/ sr)

6.8 6.12 1,49
3-4 6.73 1.28
10.0 7.59 1,27
1 1.7 8.44 1.16
13.5 7.45 1.2
15.4 6.84 1.25

 

 

 

 

136

(b)

 

P<l3C.d)12C. Ex: 4.439 MeV

 

 

 

 

 

 

 

 

 

 

 

 

d 0' do'

6 :15 A55
cm
(deg) (mb/sr) (mb/sr)

7.5 7.19 1.04
8-9 8.74 1.51
10.4 9.54 0.45
1 1.9 9.20 040
13.6 8.33 0.45
15.3 7,35 031
17.1 6.08 0.27
19.1 4.87 0.26

 

(d)

 

p( 1036109813, Ex: 0.0 MeV

 

 

 

 

 

 

 

 

 

 

 

(10' do

6 79 A25
cm
(deg) (mb/ sr) (mb/sr)

3.2 27.40 5.27
4.4 27.06 4.20
7.1 27.95 6.04
9.8 26.18 1.96
1 1.3 22.91 351
12.8 19.43 3.30
14.4 16.68 1.70
16.1 12.89 1.18
17.9 10.98 1.24

 

 

 

 

 

 

1000 . . r . rrfi

 

 

r

o p(13C,d)12C g.s. Ep=48.3 MeV
o 13C(p,d)12C g.s. Ep=41.3 MeV

T

 

 

”T
$031 1 9 13C(p,d)12C g.s. Ep=50.0 MeV 1
\
..Q 10.0_ 7;
E :
c; 5:
EU
\ o
b
"C 1.01— “ —.
- O
9 18;. 1
O
9 o
1.9
0.1 ' ‘ ‘ ‘ ' ‘ ' l ' ' L '
O 20 4O 60
Gem (deg)

Figure 4.4: The measured angular differential cross section of p(l3C,d)12C (g.s.) (open
circle) reaction compared to the published data of Ref. [Cam87] (closed circle) and Ref.

[Sco70] (diamonds).

137

 

 

 

 

100.01 ' r ' ' I 1 v 1 1 l
P O p(130.d)‘zc (2+) 5113:4183 MeV
A 9 13C(p.d)‘zc (2") Ep=41.8 MeV
gm"; 5 13C(p.d)‘2C (2+) Ep=50.0 MeV
E 10 0
° :— _5330 j
E I 99‘ Ge :
v O
0
C:
3 1°
5 -° .1 3
"U 1.0 t— G 1
0.1 ‘ ‘ 5 . I . . . . I . 1 . .
0 20 40 60
6 (deg)

cm

Figure 4.5: The measured angular differential cross section of p(l3C,d)lzC (2+) (open
circle) reaction compared to the published data of Ref. [Cam87] (closed circle) and Ref.
[Sco70] (diamonds).

138

 

100_ ' ' 1 r1 v . . 'l

p(“B,d)1°B g.s. Eg=44.1 MeV

 

 

 

”Q
(I)
\
..Q
E
G 1o~ T _
Pd : TOT j
E gig-53%
'6 l
1 1 m. . l . . . .l
0 10 20 30
Gem (deg)

Figure 4.6: The measured angular differential cross section of p(1 lB,d)IOB (g.s.) reaction.

139

 

100

 

 

 

6 T6 6 0
iii 33.5 5 55:5
\\\ J—O
"Q 0
E 9'
v
107
C: .
73 r
\\3 -
C)
73
p(loBe,d)9Be g.s. E13=49.8MeV
1 . . . . l 1 1 L 1 L . 1 . .
O 10 20 30

Gem (deg)

Figure 4.7: The measured angular differential cross section of p(lOBe,d)9Be (g.s.)
reaction.

140

CHAPTER 5

EXTRACTION OF SPECTROSCOPIC FACTORS

5.1 Overview

Single-nucleon transfer reactions, such as the (d,p) and (p,d) processes, have been
used extensively to study the spectroscopy of nuclei [But51, Mac60, Aus70, Mac74,
Sat83, Cat02], in particular for the extraction of single-nucleon spectroscopic factors
(SF). The SF is obtained by taking the ratio of the measured cross section to that
calculated using a reaction model [Mac60, Mac74]. Theoretically, the spectroscopic
factor can be considered as the overlap integral between the initial and final state of the
target nuclei and yield information on the occupancy of a given single-particle orbit
[Mac60, Mac74, Sat83].

The properties of rare isotopes far away from the valley of stability are very important
in the study of astrophysics and in the understanding of how heavy elements are created
in the universe. Spectroscopic information about the orbits of the valence nucleons in
these unstable nuclei have led to novel and surprising properties for the corresponding
unstable nuclear states [Han95, MadOl]. Due to their short life time, transfer reactions in
inverse kinematics with rare nuclei as projectiles are the optimal way to study these
nuclei [Cat02] and they are becoming a viable tool to explore single-particle states in
neutron- and proton-rich nuclei [For99, Win01, HanOl, Cat02, Han03]. However, since
rare isotope beam intensities remain very much less than that of stable beams, and the

history and experience of the spectroscopy of rare isotopes is much shorter, it is critical to

141

understand the limitations of transfer reaction theory by selectively re-examining the
consistency of more precise measurements made with intense light-ion beams.

It is well documented that large uncertainties have been associated with the extraction
of spectroscopic factors. In a systematic compilation of spectroscopic factors for sd-shell
nuclei, Endt [End77] noted that very different values of the spectroscopic factors arise
from different analyses and/or experiments. By examining a large amount of data, and
using consistency checks when available, Endt extracted an error of about 25% for
individual spectroscopic factor for the sd-shell. The uncertainty increases to 50% if only
(p,d) and (d,p) transfer reactions are used. However, this analysis, performed in 1977 and
limited to heavy nuclei with A between 21 to 44, did not provide the systematic
uncertainties associated with the method and relied heavily on the analyses of different
authors. To assess the systematic uncertainties associated with the extraction of

spectroscopic factors using a standardized procedure, we re-analyze the reaction of
12 I3 . . . 13 12
C(d,p) C(g.s.) and its inverse reaction, C(p,d) C(g.s.), measured over a range of

incident energies from 12 MeV to 60 MeV from the published literatures [Lin4].

5.2 12C(d,p)”C (g.s.) Reaction

There are published angular distributions for the 12C(d,p)l3C reaction at incident

deuteron energies from 0.4 to 56 MeV. Until now, spectroscopic factors have been
extracted from only a subset of these experiments. The associated analyses relied mainly

on distorted-waves Born approximation (DWBA) calculations, but with no consistent

choice of input parameters. The published 12C(d,p)”C (g.s.) spectroscopic factors are

142

listed in Table 5.1 and shown by closed points in Figure 5.1 as a function of the incident

deuteron energy. For comparison, the published spectroscopic factors for the reaction of
13C(p,d)l‘o'C (g.s.), to be discussed in Section 5.3, are shown by open points as a function

of the equivalent incident deuteron energy. The values fluctuate from 0.3 to 1.4 with no
evident correlation with incident energies. In some experiments, multiple values were
deduced from different optical model parameter sets or different theories. In these cases
the higher values are shown as squares in Figure 5.1. The dashed line shows the
theoretical prediction (0.62) of the Cohen and Kurath shell model calculation [Coh67].
Large fluctuations in the data highlight the problem of extracting of a meaningful
empirical spectroscopic factor that can be compared to theoretical value.

Our calculations use a modified version of the code TWOFNR [Iga77]. All

calculations include the local energy approximation (LEA) for finite range effects
[But64] using the zero-range strength (Do) and range ([3) parameter of the Reid Soft core
331—3D1 neutron-proton interaction [Knu75]. For simplicity, no spin-orbit coupling is
included as explained in Section 2.3.1.5. Non-locality corrections [Per62] are included in
the proton and deuteron channels. All the parameters are listed in Table 2.7.

We first perform calculations where both the exit channel proton and the entrance
channel Johnson-Soper (JS) adiabatic potentials using the JLM nucleon-target optical
potentials [Pet85]. These are calculated by folding the density-dependent JLM nucleon-
nucleon effective interaction [Jeu77], assumed to have a Gaussian form factor of range 1
fm [Me183], with the assumed target matter density in the mid-point local-density

approximation [Me183]. The matter density distribution for both 12C and 13C are

evaluated assuming the modified Harmonic oscillator density parameters (a=1.247 fm,

143

a=l.649 fm for 12C; (1:1.403 fm, a=1.635 fm for 13C) compiled in Ref. [Dej74]. The
corresponding root-mean-square (rrns) charge radii are 2.46 fin and 2.44 fm for 12C and
13C respectively. The resulting real and imaginary parts of the nucleon potentials are

scaled by the factors Av=l.0 and Aw=0.8, respectively, obtained from a systematic study

of light nuclei [Pet85].
The spectroscopic factor is extracted by fitting the theory to the data at the first peak
in the angular distribution, since the backward angle data are more sensitive to the effects

of inelastic couplings and other hi gher—order effects. To be consistent, the spectroscopic

factors are extracted by minimizing x2, including only angular points that are (i) within

30% of the maximum yield at the predicted angle and (ii) at 9cm< 30°.
In the present analyses we consider measured angular distributions for the reactions
of 12C(d,p)mC (g.s.) over a range of incident energies from 7 MeV to 56 MeV. The

calculated angular distributions normalized by the extracted spectroscopic factors are
shown as solid lines in Figure 5.2. Each is displaced by a factor of 10 for ease of
presentation, the displacement factor being unity for the angular distribution at 19.6
MeV. The associated spectroscopic factors are listed in Table 5.1 and shown at the
bottom of Figure 5.3 include re-analysis of the data shown in Figure 5.1 (closed circles)

[Mor60, Zai6l, Dan63, Sch67, Fet7l, Dar73, Ohn86, Lan88] and of additional data sets

(closed squares) [Mcg55, Mor60, Rob6l, Ham61, Sch64]. Available data at Ed=28 and

56 MeV are not included since their angular distributions do not include the first peak.
The rise of the spectroscopic factors with decreasing energy below 12 MeV shown in

Figure 5.3 has been observed before [Sch67] and has been attributed to the effect of

144

resonant structures in the elastic scattering of deuterons [Ohl63] and in the 12C(d,p)l3C
reactions [Eva63]. Excluding measurements affected by compound nucleus formation

and resonances, the extracted spectroscopic factors for Ed=12-60 MeV provide an

average spectroscopic factor of O.58i0.09. In contrast, the published values in Figure 5.1
vary from 0.3 to 1.5. Our consistent, theoretically motivated analyses thus reduce the
fluctuations substantially.

To find out whether the uncertainty comes from the accuracy in the measurements or
from the calculations, we examine the consistency in different measurements at the same

incident energy. Figure 5.4, 5.5, and 5.6 show the comparisons of the measurements of
l2C(d,p)l3C reaction for deuteron energies at around 4.5 MeV, 12 MeV, and 15 MeV

respectively. In Figure 5.4, the difference among the data is more than 30% in the region
from 20° to 50° in the center of mass with no quoted error in the literatures. In Figure 5.5,
the quoted error from Ref. [Sch67] and Ref. [Ham6l] are 15% and 25% respectively. The
difference among the data is close to 20% for the first peak from 10° to 30° in the center
of mass. In Figure 5.6, the quoted error from Ref. [Ham61] and Ref. [Mcg55] are both
20%. Ref. [Dar73] did not give error of the data. The difference in the region around 20
deg in the center of mass is more than 20%. From the above analyses, the measured
angular distributions do not agree to better than 20%. Thus the variations in the
spectroscopic factors we obtained partly come from the uncertainties in the experimental
measurements.

To assess the stability of the above adiabatic three-body model calculations, we have
repeated the analyses by replacing the JLM nucleon optical potentials everywhere with

the Chapel-Hill (CH89) [Var91] global potential set. The calculated angular distributions

145

normalized by the spectroscopic factors are shown by the dotted curves in Figure 5.2. The
spectroscopic factors are shown in the center of Figure 5.3. Overall, the values are quite
similar, but consistently higher. The average spectroscopic factors is 0.7li0.10. It should
be noted that light nuclei are not included in the extraction of the CH89 potential
evaluation. Thus it is remarkable that the CH89 potential seems to work well.

For a final comparison, we also analyzed the full data set within the DWBA
formalism, neglecting the role of deuteron break-up channels. To remove energy-
dependent optical potential ambiguity, we used the CH89" and Daehnick [Dae80] global
potentials for the proton and deuteron channels, respectively. The calculated angular
distributions normalized by the spectroscopic factors are shown as dashed curves in
Figure 5.2 and the deduced spectroscopic factors are plotted at the top of Figure 5.3.
Again, the extracted values are more consistent than the published values; see Figure 5.1.
The average value is 0.79i0.19. Comparisons with the JS adiabatic calculations suggest
that neglect of the break-up channel within the DWBA is a significant contributing factor

at high energies.

146

Table 5.1: List of references and spectroscopic factors for the 12C(d,p)wC (g.s.) reaction

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R°°°°°n £2311“) (1.181;) (115.1131) (C329) (135/11:31:.) Ref‘
12C(d,p)l3C(g.s.) 4 0.99 0.62 0.61 0.65 [Gal66]
12C(c1,p)'3(:(g.s.) 4.5 0.72 0.69 0.69 0.6 [Gal66]
12C(d,p)l3C(g.s.) 4.5 0.52 0.53 0.4 [Bon56]

0.55
12C(d,p)l3C(g.s.) 4.5 0.6 0.42 0.43 0.49 [Gur69]
0.8
[2C(d,p)l3C(g.s.) 7.15 0.53 0.89 0.93 0.94 [Zai6l]
12C(c1,p)'3c (g.s.) 8.9 0.8 0.9 0.91 [Rob61]
12C(d,p)l3C(g.s.) 9.0 0.84 N/A N/A N/A [8111163]
12C(d,p)l3C(g.s.) 10.2 0.68 0.79 0.81 [Ham6l]
12C(d,p)13C(g.s.) 11.8 0.61 0.74 0.77 [Sch64]
12C(d,p)l3C(g.s.) 12 1.15 0.50 0.63 0.68 [Lan88]
”6 [Sch67]
12C(d,p)l3C(g.s.) 12 3'3: 0.75 0.85 0.86 [D0b70]
0:85 [01175]
12C(d,p)l3C(g.s.) 12.4 0.63 0.74 0.78 [Ham6l]
12C(c1,p)‘3C(g.s.) 14.7 0.61 0.74 0.79 [Ham6l]
12C(d,p)13C(g.s.) 14.8 0.64 0.75 0.78 [Mcg55]
12C(d,p)]3C(g.s.) 15 {‘13 0.53 0.67 0.74 [Dar73]
12C(d,p)13C (g.s.) 16.6 0.85 0.48 0.59 0.66 [Mor60]
'2C(d.p)‘3C(g.s.) 19.6 0.52 0.65 0.76 [Mor60]
12C(d,p)13C(g.s.) 25.9 0.7 0.59 0.69 0.79 [Dan63]
12C(c1,p)‘3C(g.s.) 28 0.82 1.06 1.49 [81662]
12C(d,p)l3C(g.s.) 30 0.77 0.52 0.65 0.79 [Ohn86]
12C(d,p)l3C(g.s.) 51 0.95 0.66 0.82 1.06 [[53772]
0.63
12C(d,p)l3C(g.s.) 56 0.75 1.05 1.26 1.44 [Hat84]
1.26
(average) 0.58 0.71 0.79
50.09 50.10 10.19

 

 

 

 

 

 

 

 

 

1.5 r l T r I
I
(I)
E I
8 o . o 1
‘H 1 0 —- o :1 ..
.2 . 8 °
Q—c
o ' o I . . C] O
O o
m o o
E : ° ° ° "
8 0.5 — ° —
94 D-
U) D,
13 12 O
:1 O .
l o lzgég’dgmggs
1p gs
OO . . . i 1 L i . 1 l L L 1 J L
O 20 4O 60
Ed (MeV)

Figure 5.1: Spectroscopic factors for 12C(d,p)lB'C (g.s.) and l3C(p,d)lzc (g.s.) reactions
extracted from the literatures (see Table 5.1 and Table 5.2) .

148

 

dU/dQ (mb/sr)

 

 

 

 

 

 

98m. (deg)

Figure 5.2: Angular distributions for 12C(d,p)BC (g.s.) reactions for beam energies from
7 to 56 MeV: solid lines present ADBA (JLM); dotted lines present ADBA (CH89);
dashed lines present DWBA. Each distribution is displaced by factor of 10 from adjacent
distributions. The overall normalization factor is 1 for the 19.6 MeV data. References are
listed in Table 5.1.

149

 

1.5.1rr.r.lfir..l....l..4firf.

 

 

 

E
T T
. i ‘L f T .
3 1.0—37 T DWBA 1 E—
0 "Li—Fl T T —v— '1‘ ‘
+> 2‘ g? I E E
O 'I— I 4—
“ :-——-*———_1__—————————————:I_ ———————— 4
H 1.0—IT -1- T O '1‘ “
.2 ”53:9. 4 9 i E 5 ‘
g, 4.2 I T 3 T CH 5 L
8 .———2—‘2—;-£———-—-—_% ——————————————— .
8 10:733— —
o e — 4 H —————— a —_,— — ~— 4:; ______ 2 _ .. .1:—
CD 0 5— I 2'. I 2 9 -
Cg, ' -- -— JLM
OO . l L . 4 . l . . . i l . . i i 1 1 . . . l . . . .
10 20 30 4O 5O 60
Ed (MeV)

Figure 5.3: Extracted spectroscopic factors in the present work for 12C(d,p)13C (g.s.),

13C(p,d)nC (g.s.), and p(]3C,d)12C (g.s.) reactions. The dashed lines represent the shell
model prediction of Cohen and Kurath [Coh67] of 0.62. See text for detail explanation.

150

 

 

 

 

4 . . 1 . . . . I . . . . I . . . . I
12C(d,p)13C Ed=4.5 MeV
? 20 b O: a —
m 1. X: b
3 X c
E x .
g :1 O 1: x
b 10 _ X O X —
Pd 1- D 1
D o x x x
E]
01210 CDC) 0 O
O . . . . a . . . 1 1 . . L . 1 . . . . 1 .
O 20 4O 6O 80

Gem (deg)

Figure 5.4: Comparison of the existing measurements of 12C(d,p)wC (g.s.) reaction for
deuteron energy at 4.5 MeV, a [Gur69], b [Gal66], and c [Bon56].

151

 

 

 

 

 

 

 

 

3O . . r . T T l
12C(d,p)13C Ed~12.0 MeV 4
1: 1‘2-
_ O: 11.8 MeV
E 20 _ l: + [3+ X: 12.0 MeVa z
E j T" +: 12.0 MeVb
v 141 ,_ + 1:1: 12.4 MeV
c: ._ ET
1:: ’ +
\ “ 1
b -- 1
Pa 10 — my _
>11 13.;
T“ X15
3%: >111) X5 E] [:1
O . . . i l l
O 20 4O 60
Gem (deg)

Figure 5.5: Comparison of the existing measurements of 12C(d,p)BC (g.s.) reaction for
deuteron energies at 11.8 MeV [Sch64], 12.0 MeV a [Lan88], 12.0 MeV b [Sch67], and

12.4 MeV [Ham6l].

152

 

 

 

 

 

 

 

 

 

 

 

1 . . . . 1 . . -
12C(d,p)13C Ed~15.0 MeV
__ T __ O: 14.7 MeV
E >< __ X: 14.8 MeV
E X O X E]: 15 MeV
V 6 9
C: «4 "d” E3 13
3 5 1
b 10 L— T T
U 0
El
T .l
0
i1:]
[:10 CED [:10 C
O L 1 L L r J r
O 20 4O 60
9m (deg)

Figure 5.6: Comparison of the existing measurements of 12C(d,p)nC (g.s.) reaction for
deuteron energies at 14.7 MeV [Ham61], 14.8 MeV [Mcg55], and 15 MeV [Dar73].

153

5.3 l3C(p,d)lzC (g.s.) and p(13C,d)12C (g.s.) Reactions

Systematic analyses are performed to the measurements of 13C(p,d)lzC reaction to the

ground state [Toy95, Cam87, Tak68, H0880] listed in Table 5.2. The proton energies
range from 35 MeV to 65 MeV. Same parameters listed in Table 2.7 and same procedure
as described in section 5.2 are employed.

The data and calculations multiplied by the corresponding spectroscopic factors listed
in Table 5.2 are shown in Figure 5.7. The angular distributions at 35 MeV, 41.3 MeV,
and 65.0 MeV do not have data at forward angles. These data without the first peak may
not give reliable SF. The data at 55.0 MeV have data at forward angles but the shape is
different from that of the calculations. The extracted SF at 65.0 MeV is almost twice the
expected value. Thus the data at 65.0 MeV may not be correct. The extracted SF from the
data at 35.0 MeV, 41.3 MeV, and 55.0 MeV are plotted in Figure 5.3 as open circle
points. As the existing measurements do not give reliable SF, a new measurement to

cover the first peak in inverse kinematics is desirable.
The data and calculations for the inverse kinematic reaction of p(l3C,d)12C (g.s.)

performed in the present work are plotted as the third set of data (open points) and lines
in Figure 5.7 The extracted spectroscopic factors, as shown in open squares in Figure 5.3,
are 0.74, 0.91, and 1.18 for ADBA (JLM), ADBA (CH89), and DWBA calculations,
respectively. One possible reason that the (p,d) reactions give higher spectroscopic
factors than the (d,p) reactions in Section 5.2 is that the (p,d) reactions are performed at
higher energies. The overall averaged spectroscopic factors from all the (d,p) and (p,d)

reactions are 0.62i0.09, 0.761011, and 0.89:0.20 for ADBA (JLM), ADBA (CH), and

154

DWBA calculations, respectively. The ADBA calculations based on JLM potentials give
the best result compared to theory. Again, the SF values from the DWBA calculations are
higher. As the importance of the deuteron break-up effects has been demonstrated, we

will not discuss the DWBA calculations in the remaining part of this chapter.

155

dU/dQ (mb/sr)

 

 

 
 

F 1 r V T I I I r I

0 13C(p,d)lzc (g.s.)
o p(13C,d)12C (g.s.)

—

 

35.0 MeV

 

 

 

Figure 5.7: Angular distributions for l3C(p,d)lzC (g.s.) and p(13C,d)12C (g.s.) reactions
for beam energies from 35 to 65 MeV: solid lines present ADBA (JLM); dotted lines
present ADBA (CH89); dashed lines present DWBA. The calculations have been
normalized by the spectroscopic factors. Each distribution is displaced by factor of 10
from adjacent distributions. The overall normalization factor is 1 for the 65.0 MeV data.

156

Table 5.2: List of references and spectroscopic factors for the l3C(p,d)lzC (g.s.) and

P(13C,d)12C (g.s.) reactions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reaction SF SF SF SF Ref
(MeV) (Liter) (JLM) (CH89) (DWBA) '
0.7
”C(p,d)”mge.) 35 0.8 0.66 0.85 1.16 [Toy95]
1.0
”a d)12C( s) 413 0'91 078 098 131 [Cam87]
l3C(p,d)”c (g.s.) 55 0.82 0.66 0.82 1.05 [Tak68]
0.26
13C(p,d)12C(g.s.) 65 0.31 1.22 1.57 1.33 [H0580]
0.43
( 3 0.70 0.88 1.17
°V°rag° i007 i009 :013
p(13C,d)12C(g.s.) 483* 0.74 0.91 1.18

 

* Equivalent proton energy

157

 

5-4 l3C(p,d)12C(2+) and p(l3C,d)12C (2+) Reactions

Theoretically the transferred neutron in the reaction of l3C(p,d)lzC to the first excited

state 2+ at 4.439MeV of 12C is predicted to be in pure lp3/2 orbit [Cam87]. The

theoretical spectroscopic factor from Cohen and Kurath [Coh67] is 1.12. This clearly

identified state provides another opportunity to test our strategy to extract the SF.
Systematic analyses are performed to the existing measurements of l3C(p,d)lzC

reaction to the first excited state [Toy95, Cam87, Tak68]. The proton energies range from

35 MeV to 65 MeV. The same procedure as described in section 5.2 is employed.
The data and calculations for the present measurement of p(l3C,d)12C* (4.439MeV)

are plotted as the third set (open points) in Figure 5.8 together with the data (closed
points) from the literatures [Toy95, Cam87, Tak68]. Among the published data, only the
data at 55.0MeV have reasonable coverage at forward angles. The data at 35.0 MeV and
41.3 MeV are not reliable since they did not include the first peak. However, the
extracted spectroscopic factors by fitting the slope of these two data sets may provide
consistent checks. The extracted spectroscopic factors are listed in Table 5.3 and plotted
in Figure 5.9. The averaged spectroscopic factors from ADBA calculations based on JLM
potentials and CH89 potentials are 0.92:0.09 and 1.08:0.13, respectively. The extracted
SF for present measurement are 1.03 and 1.2 from ADBA(JLM) and ADBA(CH89),
respectively. These values are listed in Table 5.3. The SF values from CH89 potentials

are usually higher than that from J LM potentials.

158

 

W,,....
- 13c(p,d)1°c* (4.439 MeV)
o p(130,d)120* (4.439 MeV)

 
   
    

35.0 MeV

dU/dQ (mb/sr)

 

 

l

0 20 40 60
em (deg)

0.1

Figure 5.8: Calculations for reactions of 13C(p,d)12C* (4.439MeV) and p(l3C,d)12C*
(4.439MeV) [Toy95, Cam87, Tak68]; solid lines present ADBA (JLM); dotted lines
present ADBA (CH89). The calculations have been normalized by the spectroscopic

factors. Each distribution is displaced by factor of 10 from adjacent distributions. The
overall normalization factor is 1 for the 55.0 MeV data.

159

Table 5.3: Extracted spectroscopic factors of the lp3/2 neutron from the l3C(p,d)lzC (2+)

and p(l3C,d)]2C (2+) reactions.

 

 

 

 

 

 

 

 

 

 

 

Proton SF SF
Reaction energy ADBA ADBA Ref.
(MeV) (JLM) (CH89)
l3C(p,d)12C* (2*) 35 0.92 1.08 [Toy95]
I3C(p,d)'2C* (2*) 41.3 1.01 1.2 [Cam87]
'3C(p,d)l2C* (2*) 55 0.84 0.95 [Tak68]
(average) (0.92i0.09) ( 1 08:0. 13)
p('3c.d)'2c* (2*) 433* 1.03 1.2

 

* Equivalent proton energy

160

 

 

 

 

 

2.0? - a r r - r

1.5} -—
z: T - CH

T o C]

s ------- s ————— 1 —————— - ------ T - —
O 1.0j 4— o ‘
.53 ; 4‘
O “ 4
"-t 1.5 1
3* : JLM l
0 l. — e — e e e e H — -.s —————— e. ————————— .
8 l.0~ - s -
L. f ET] AL 0
+2 l e 4
O r 13 12 :I: l
a 0‘5? O C(p,d) C (4.439MeV) _
m g o p(13c,d)lzc* (4.439MeV)

0.0b . l l l l l l l l l A . l .

30 4O 5O 60

Ed (MeV)

Figure 5.9: Extracted spectroscopic factors of the reactions ”C(p,d)IZC“ (4.439MeV)

(circle) and p(l3C,d)12C* (4.439MeV) (square). The dashed lines represent the shell
model prediction of 1.12 by Cohen and Kurath [Coh67].

161

5-4 103&1,pr (g.s.), 11B(p,d)1°B (g.s.), and
P(llB,d)l0B (g.s.) Reactions

Systematic analyses are performed for the reaction of 11B(p,d)loB (g.s.) [Leg63]
[Ku168] [81062] and its inverse reaction of ‘°B(cl,p)“B (g.s.) [Hin62] [Sch67] [Bar65].

The transferred neutron is in lp3/2 orbit and its form factor is plotted in Figure 2.23. The

theoretical spectroscopic factor from the shell model is 1.09 [Coh67]. Applying the
consistent procedure as described above, the ADBA calculations are performed based on

CH89 and JLM potentials respectively. The modified harmonic oscillator densities
(a=0.837 fm, a=l.7l fm for "’B; a=0.811 fm, a=l.69 fm for ”13) compiled in Ref.

[Dej74] are used in the JLM potentials.

The data and calculations are shown in Figure 5.10 for the (d,p) reactions and in
Figure 5.11 for the (p,d) reactions, respectively. The ADBA calculations based on CH89
and JLM potentials give similar results. The spectroscopic factors have been extracted by
fitting the first peaks as described in section 5.2. The results are listed in Table 5.4 and
plotted in Figure 5.12. The spectroscopic factors from the published data present a trend

with larger value at 30 MeV and lower value at lower (10 MeV) and higher (50 MeV)
energies. Particularly, the measurement of llB(p,d)mB at incident energy of 19 MeV (not

plotted) gave a much higher SF compared to other experiments. In general, the absolute
cross sections increase with incident energy. However, instead of lower cross section, the
measured cross sections at 19 MeV are nearly twice as large as the cross sections

measured at 33.6 MeV and 44.1 MeV. Thus, we believe this data set has normalization

162

problems and disregard it. The average value of the extracted SF except the data at 19

MeV is 1.37i0.34 and 1.34:0.31 by using potentials of JLM and CH89 respectively.
The theoretical angular distribution calculations for the p(“B,d)lOB reaction are

shown in Figure 5.11 as open symbols. The spectroscopic factors calculated from ADBA
based on JLM and CH89 potentials are 1.05 and 0.97 respectively. Due to low statistics,
the data of this measurement have larger error bar. The statistical uncertainty of the
extracted SF is 17.2% for both the JLM potential and CH89 potential. However we
cannot determine systematic errors due to our concern about the absolute normalization
of those cross sections. Some data from the 8800 scintillator are missed. Although the
data for present measurements seem reasonable, we are not sure the normalization is

absolutely correct.

163

 

do/dQ (mb/sr)

 

 

 

E 10.1MeV f
103:— 0 —:
: 12Mev ° 3
o -' d
2 .'
10 :— . —.
” ' o. i
.. 13.5MeV , o '_ .
.l . q
10; 28Mev o . ‘5
: ' _.-- "0 .70 :
_._. g
. 10 11 .
B(d.p) B (g.s.)
01 r . I l . r . l . 4
O 20 4O 60

(deg)

Figure 5.10: Calculations of loB(d,p)“B (g.s.) reaction at 10.1 MeV [Hin62], 12.0 MeV
[Sch67], 13.5 MeV [Bar65], and 28 MeV [81062]: solid lines present ADBA (JLM);
dotted lines present ADBA (CH89). The calculations have been normalized by the
spectroscopic factors. Each distribution is displaced by factor of 10 from adjacent
distributions. The overall normalization factor is l for the 28.0 MeV data.

9

cm

164

 

19.0 MeV

3 — s‘, \ i —
L. . .
U)
E °
E 102 g‘ . x 38.6 -5
V : - i
g no
\ ' O O r
b
”d 10 ":

    

1.0 E ...... .
o 11B(p.<il)1°B (g.s.)
o p(“B,d)1°B (g.s.)

0.1L14.lirlrlrrir
O 20 4O 60

Gem (deg)

 

 

 

Figure 5.11: Calculations of llB(p,d)wB (g.s.) (closed points) p(”B,d)loB (g.s.) (open
points): solid lines present ADBA (JLM); dotted lines present ADBA (CH89). The
calculations have been normalized by the spectroscopic factors. Each distribution is
displaced by factor of 10 from adjacent distributions. The overall normalization factor is
l for the present data.

165

Table 5.4: Extracted spectroscopic factors of the 1pm neutron from the reactions of
“B(p,d)'°B (g.s.), '°B(d,p)' 'B (g.s.), and p(“B,d)'°B (g.s.)

 

 

 

 

 

 

 

 

 

Beam SF SF
Reaction energy ADBA ADBA Ref.
(MeV) (JLM) (CH89)
”B(p,d)‘°13 (g.s.) 19 3.25 3.22 [Leg63]
11 10
B (W) B 33.6 1.35 1.24 [Kul68]
(g.s.)
‘°B(d,p)' ‘13 (g.s.) 10.1 0.94 0.94 [Hin62]
loB(d,p)l '13 (g.s.) 12 1.20 1.22 [Sch67]
'°B(d.p)' 'B (g.s.) 13.5 1.56 1.61 [Bar65]
'°B(d,p)' ‘8 (g.s.) 28 1.82 1.68 [51662]
(average) (I .37i0.34) (l.34i0.3l)
p(' ‘B,d)'°8 (g.s.) 44.1* 1.05 0.97

 

 

 

 

 

 

 

* Equivalent proton energy

166

 

 

 

 

:13. BE 3
o E + 3‘ 3
'8 ~__ g _,_ CH89 _«
«5 WE ........................... § ................
W 1:— . E] .1
0 E 3
”—1 _ _.l
8 : 3
g 02‘ 3 “3
8 — 9. 5JLM ~
-+-> , .......... .0 ........................... 7“ ......... D ..... .
0 1.— o ‘
CD :
Q-I L. _
U? e
O ’ 1 1 l
O 20 4O 60
Ed (MeV)

Figure 5.12: Extracted spectroscopic factors for the reaction of p(”B,d)10B (g.s.) (open
square), llB(p,d)lOB (g.s.) (open circle), and loB(d,p)“B (g.s.) (closed circle). The
dashed lines represent the shell model prediction of 1.09 by Cohen and Kurath [Coh67].

167

5.6 9Be(d,p)lOBe (g.s.) and p(10Be,d)9Be (g.s.) Reactions

Systematic analyses are performed on the angular distributions measured from the

9Be(d,p)mBe reactions that were published in the literatures [ZelOl, GenOO, Va587,

Sch64, Dar76, And74, $1062]. The transferred neutron is in the 1pm orbit and its form

factor is shown in Figure 2.23. The SF value obtained from the theoretical prediction of
shell model is 2.35 [Coh67]. The ADBA calculations are based on CH potentials as

described in section 5.2. The JLM potential is not used since the nucleon radius
information of 10Be is not available. The experimental data and calculations are shown in

Figure 5.13. The extracted spectroscopic factors are listed in Table 5.5 and plotted in
Figure 5.14. The systematic studies do not give a consistent value of spectroscopic factor.
The data from 6.0 MeV to 11.0 MeV came from one reference of [GenOO], which gave
the spectroscopic factor value around 1.0 for deuteron energies from 7.0 MeV to 11.0
MeV. The other data yield spectroscopic factor values from 0.97 to 2.59. There is big
difference in the measured cross sections at 15 MeV and at 15.3 MeV, which give
spectroscopic factor of 1.83 and 1.19 respectively at nearly the same energies. The
average value (to give the same weight for different systems, only one set from Ref.
[GenOO] at 11.0 MeV is included) is 1.401041 with rather large uncertainty.

The data and the calculations for the present measurement in the inverse kinematics
of p(loBe,d)9Be are shown as open points in Figure 5.13. Our ADBA (CH89) calculation

gives spectroscopic factor of 2.99, which is 27% higher than the theoretical value and
nearly a factor of two higher than the values obtained from seven (d,p) reactions

measured by different groups. Since we have problems in the absolute normalization as

168

discussed in Section 4.3, it is not clear if the discrepancies arise from problems in our

measurements.

169

 

10 IYIUTIITII’Wfirr 'VIUIT'YIIWYUU YI'YIUIY'IYYTY

' 9Be(d,p)mBe (g.s.)
106 — 6.0MeV - op(1°Be,d)9Be (gs) — 11.8MeV *‘

  
 

 

9.0MeV

  
 
 

dU/dQ (mb/sr‘)

 

 

 

 

 

10—2 W11....l....l..11
O 20 40 0 20 40

Gem (deg)

Figure 5.13: ADBA calculations based on CH89 potentials for the reactions of
9Be(d,p)'°Be (g.s.) (closed circle) [ZelOl, GenOO, Vas87, Sch64, Dar76, And74, 31662]
and p(10Be,d)9Be (g.s.) (open circle). The calculations have been normalized by the
spectroscopic factors. Each distribution in closed circle is displaced by factor of 10 from
adjacent distributions. The overall normalization factor is l for the data at 11.0 MeV. The
present data is reduced by a factor of 100.

170

Table 5.5: Extracted spectroscopic factors of the 1pm neutron from the reactions of
p(lOBe,d)9Be (g.s.), and 9Be(d,p)mBe (g.s.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Incident SF
Reaction Energy ADBA Ref.
(MeV) (CH89)
9Be(d,p)loBe (g.s.) 6 2.05 [GenOO]
gBe(d,p)lOBe (g.s.) 6.5 1.43 [GenOO]
9Be(d,p)wBe (g.s.) 7 1.3 [GenOO]
9Be(d,p)lOBe (g.s.) 7.5 1.04 [GenOO]
gBe(d,p)10Be (g.s.) 8 1.12 [GenOO]
gBe(d.p)loBe (g.s.) 8.5 1.01 [GenOO]
9Be(d,p)IOBe (g.s.) 9 0.97 [GenOO]
9Be(d,p)mBe (g.s.) 9.5 1.01 [GenOO]
gBe(d,p)loBe (g.s.) 10 1.07 [GenOO]
9Be(d,p)mBe (g.s.) 10.5 1.08 [GenOO]
9Be(d,p)‘°Be (g.s.) 11 1.03 [GenOO]
9Be(d.p)lOBe (g.s.) 11.8 1.44 [Sch64]
9Be(d,p)‘°Be (g.s.) 12.5 1.29 [Vas87]
9Be(d,p)lOBe (g.s.) 15 1.83 [Dar76]
9Be(d,p)loBe (g.s.) 15.3 1.19 [ZelOl]
gBe(d.p)loBe (g.s.) 17.3 0.97 [And74]
9Be(d,p)loBe (g.s.) 28 2.07 [81062]
(average) (1.40:0.4 l)
p( mBe,d)9Be (g.s.) 49.8* 2.99

 

* Equivalent proton energy

171

 

 

 

 

. I - T I -
2
8 4— CH89 —
o
3 T
.2 " 0 ~
0. . L 1
8 r ....................................................... .
<03) 2 L 0 fi_ E E
5 + _2
8 1' 3.0: 3‘
m C

L

O i . . . . 1 . . . . l . . . .
O 20 4O 60
Ed (MeV)

Figure 5.14: Extracted spectroscopic factors for the reactions of p(mBe,d)9Be (g.s.) (open

square), and 9Be(d,p)loBe (g.s.) (closed circle). The dashed lines represent the shell
model prediction of 2.35 by Cohen and Kurath [Coh67].

172

CHAPTER 6

SUMMARY

This experiment is originally designed to study the structure of the valence neutron of

10Be and 1'B by extracting the spectroscopic factors. The angular differential cross
. . 10 ll 13 . .
sections of (p,d) reactions on Be, B, and C are measured, wherein the reaction on

13C is performed as a calibration system and later used as a systematic study to devise a
strategy to extract spectroscopic factors using the (p,d) and (d,p) reactions. Since target of

. . 10 . . . . . .
the radioactive nucleus Be is not available, the reaction is performed in inverse

. . . ll 13 . . . .
kinematics. The reactions on B and C are also performed in inversed kinematics to

keep all the three experiments similar to reduce systematic errors and to learn about the

new technique of using reverse kinematics of radioactive beams. All three secondary
beams are produced by bombing a thick 9B6 target with the 13C primary beam.

This experiment provides a learning experience of how to study (p,d) transfer
reactions using secondary beams. The characteristics of the reaction in inverse kinematics
are analyzed in this work. The advantage of the reaction in inverse kinematics is that the
emitted light particles can be easily detected at forward angles (Section 1.2). The
disadvantage of the reaction in inverse kinematics is that the energy broadening requires
high angular resolution of the detectors (Section 1.2). The contributions to the energy

resolution are analyzed in Section 4.2, which states that thin target and high angular

173

resolution of the detectors are the essential keys to achieve high energy resolution for the

reactions in inverse kinematics.

The angular differential cross sections are measured for the reaction of p(l3C,d)12C to

the ground state (0+) and the first excited state (2+). The distributions are in good

agreement with the published data in the literatures at adjacent energies (Section 5.3 and
5.4). The extracted SFs are in good agreement with the theory expectation. This means
that the experimental techniques are good and the strategy to extract the SF works fine.

The angular differential cross sections are measured for the reaction of p(“B,d)loB to

the ground state. The extracted SF is 1.05. Past measurements give higher SF values
even though this experiment in inverse kinematics presents the best agreement with the
shell model prediction.

The angular differential cross sections are measured for the reaction of p(mBe,d)9Be

to the ground state. The extracted SF is 2.99, which is 27% higher than the theoretical
value of 2.35. This value is not confirmed as we had problems in the beam normalization.
Since the published data give lower values of spectroscopic factor, it is desirable to re-

measure the differential cross section. The measurements of the differential cross sections

to the excited states of 10B and 9Be are not performed because of the low particle counts.

Therefore higher intensity of '1B and loBe beams, which are available from the new

Coupled Cyclotron Facility, are desirable in future measurements.
The energy resolution in this work is around 600 keV to 800 keV, which may not be
high enough for the separation of some other excited states. Higher angular resolution

achieved by placing the detector further away or by using smaller spacing of the strips is

174

desirable. This is currently under development in the construction of the HiRA (High
Resolution Array) at NSCL.

In the course of this study, we have developed the strategy to extract the
spectroscopic factor by using a standard set of input parameters listed in Table 2.7 for the
adiabatic deuteron breakup approximation (ADBA) calculations. We find that the
Optical-Model Potential (0MP) obtained from fitting individual data of elastic scatterings
do not give consistent and reliable spectroscopic factors due to the ambiguity of the
0MP; however, global optical-model potentials for proton and deuteron give consistent
good “relative” spectroscopic factors. The 0MP based on the folding model and the
effective nucleon-nucleus interactions such as CH89 and JLM potentials seem to give
better agreement with data.

Based on this work, specifically the analysis procedure provided, recent extraction of
ground state neutron spectroscopic factors of 79 nuclei for elements ranging from Li to
Cr [Tsa05]. These values are in consistent agreements with shell model predictions
[BroO4]. Figure 6.1 shows the comparison of the extracted spectroscopic factors with the
predictions of the modern shell model. Good agreements are achieved except for Ne, F,
and Ti isotopes. Such agreement raises the possibility that the extracted spectroscopic
factors are not only relative but absolute values. Furthermore, the agreement between the
extracted values and the shell model predictions suggest that long-range n-n and n-core
interactions can be described by modern day shell model. The disagreement between the
spectroscopic factors extracted from transfer reactions and knockout reactions using the
electron probe could be explained by the short-range nucleon-nucleon interactions since

the electron probes the interior of the wave function where n-n interaction is more

175

important than that at the surface of the wave function where the transfer reactions are
more sensitive. The present work has stimulated a lot of interest in the use of transfer
reactions to extract spectroscopic factors, not only for rare nuclei but for stable nuclei as

well.

176

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177

[And74]

[Aus64]

[Aus70]

[Au387]

[Ban85]

[Bar88]

[Bar65]

[Bar96]
[Bec69]
[Ber65]
[Bicr]

[Bir51]

[Bon56]

[BroO4]

[ButS 1]
[But64]

[But67]

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