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SURVEY OF NEUTRON SPECTROSCOPIC FACTORS AND
ASYMMETRY DEPENDENCE OF NEUTRON CORRELATIONS IN

TRANSFER REACTIONS
By

Hiu Ching Lee

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

DOCTOR OF PHILOSOPHY

Physics and Astronomy

2010

 

Kw" '

ABSTRACT

SURVEY OF NEUTRON SPECTROSCOPIC FACTORS AND ASYMMETRY
DEPENDENCE OF NEUTRON CORRELATIONS IN TRANSFER REACTIONS

By
Hiu Ching Lee

Transfer reactions have been used as an experimental tool to obtain abundant
spectroscopic information for stable nuclei in the past half-century. With the advance of
high intensity radioactive beams, transfer reactions can be used in the same spirit to
explore the structures of exotic nuclei, where dramatic changes in nuclear structure have
been observed near the drip lines.

This thesis work is partly motivated by the revived interest in transfer reactions
using rare isotope beams. The first part of this dissertation describes the survey of
spectroscopic factors extracted from an extensive amount of data on single neutron (p,d)
and (d,p) transfer reactions on stable isotopes from Lithium to Nickel. Utilizing the
global optical model potential, Chapel-Hill 89, and the conventional single-particle
geometry in the adiabatic distorted wave approximation (ADWA) model, the measured

ground-state and most excited-state neutron spectroscopic factors for nuclei with 3 _<_ Z S

28 agree with the predictions from large-basis-shell-model to better than 30%. The
established systematics provides an essential framework to analyze the structural
information obtained from transfer reactions for both stable and exotic nuclei. The
suppression of spectroscopic factors from shell model values due to nucleon-nucleon
correlations is explored by using different optical model parameters and single nucleon

bound state geometry. For most nuclei, the neutron ground-state spectroscopic factors are

reduced by about 30% compared to large-basis-shell-model predictions if the ADWA
reaction model uses transferred-neutron bound state and microscopic Jeukenne, Lejeune
and Mahaux (JLM) nucleon-target optical potential geometries constrained by Hartree-
Fock calculations. The magnitude of reduction is similar to that obtained in (e,e’p)
measurements near the closed shells.

The neutron-proton asymmetry dependence of the reduction in the spectroscopic
factor is related to the neutron-neutron and neutron-proton correlations. The experimental
efforts in this dissertation are dedicated to the study of the asymmetry dependence of

neutron correlations through spectroscopic factor measurements using (p,d) neutron
. . . 34 . 46 . . .
transfer reactions wrth proton-rich Ar and neutron-rich Ar radioactive beams in

inverse kinematics. The kinematically complete experiments use a high resolution silicon
strip array to detect the deuterons in coincidence with the recoil residues detected in a

mass spectrometer. The experimental results show little reduction of the ground-state

neutron spectroscopic factor of the proton-rich nucleus 34Ar compared to that of neutron-

rich 46Ar. The results suggest that correlations, which generally reduce such

spectroscopic factors, do not depend strongly on the neutron-proton asymmetry of the
nucleus in this isotopic region. The results are consistent with the systematics established
from the extensive studies of spectroscopic factors from transfer reactions. They are,
however, contradictory to the strong dependence obtained in knockout reactions. Our
new results pose intriguing questions about the reaction mechanisms of transfer and
knockout reactions as well as the nature of neutron correlations in nuclei with extreme

isospin asymmetry.

to my parents (Hiu Y ing Chan and Kwong Ming Lee)
' and my sister (Hiu Man Betty Lee)

iv

Acknowledgements

First and foremost, I would like to express my deepest appreciation to my Ph.D
supervisor Professor Betty Tsang for her invaluable guidance and continuing support
throughout my graduate studies. Her powerful insight and leadership as well as
persistence to accomplish goals in the intellectual ways are essential to the completion of
this dissertation. The wisdom, enthusiasm and determination I learned from her have
profound influence on my future career and personal life. I would also like to express my
heartiest gratitude to her for giving me encouragement and numerous opportunities

during these years.

Much gratitude goes to Professor William G. Lynch for countless fruitful
discussions and patience. His intellects and critical observation as well as deep
understanding of Physics are invaluable to me. I am sincerely thankful to all the members
in HiRA group, including the former HiRA post-docs Drs. Vladimir Henzl and Daniela
Henzlova and fellow graduate students Dan Coupland, Micha Kilbum, Andy Rogers,
Alisher Sanetullaev and Mike Youngs. Their experimental skills and working attitudes
are remarkable. The experiment could not be accomplished without their hard work. I am
especially indebted to Andrew Rogers. His great help fiom setting up the experiment to
addressing endless problems I experienced in the beginning of data analysis are highly

appreciated.

I would like to acknowledge the contributions from all the collaborators. The
productive collaborations led to the success of this experiment. I would like to thank to

Dr. Daniel Bazin at NSCL, Dr. Zhiyu Sun at Institute of Modern Physics in China, Prof.

V

Robert Charity and Prof. Lee Sobotka at Washington University, Dr. Sylvie Hudan at
Indiana University, Prof. Michael Famiano at Western Michigan University, Dr. Mark
Wallace at LANL, Dr. Dan Shapira at ORNL, Dr. Bill Peters and Patrick O’Malley at
Rutgers University, Kyle Schmitt and Dr. Andy Chae at University of Tennessee and Hoi

Kit Cheung at Chinese University of Hong Kong.

Many thanks go to the excellent support from the incredible NSCL staff members.
I appreciate the help from John Yurkon of the detector laboratory, Len Morris, Craig
Snow and Renan Fontus of the design group, Jim Vincent of electronics group, Ron Fox
of the computer group, Andrew Thulin of vacuum group, Dr. David Sanderson for laser
alignment measurements and all the machine shop staff. In addition, A1900 people
including Drs. Mauricio Portillo, Marc Hausmann, Tom Ginter have always been

extremely helpfirl. Their great help made the experiment possible.

The extensive studies of spectroscopic factors (SF) could not be achieved without
the following people. Dr. Xiaodong Liu (a former NSCL graduate student of Prof. Betty
Tsang) developed a systematic framework to extract consistent experimental SFs, which
is the foundation of my dissertation. Shi Chun Su and Jiayan Dai at the Chinese
University of Hong Kong came to NSCL in summer and reanalyzed massive amounts of
existing experimental data to obtain SFs for a wide range of nuclei. Special thanks to Shi
Chun Su for developing a powerful and user-friendly database to organize all the data and
calculations. Scott Warrant maintained and improved such a fantastic database. Prof.
Mihai Horoi at Central Michigan University calculated the theoretical SFs with full fp

shell-model space and Dr. Hang Liu formerly at NSCL used the high performance

vi

computing (HPC) systems in MSU for large-basis shell model calculations. Angelo
Signoracci performed the SF calculations for 46Ar nucleus. I greatly appreciate their
work. In additional, I would also like to thank Prof. B. Alex Brown and Prof. Jeff

Tostevin for the use of the programs Oxbash and TWOFNR.

I would like to express my sincere thanks to Professor Pawel Danielewicz for
giving me the opportunities to get involved in his theoretical project from which I have
benefited a lot. His deep intuition, great ability to visualize interactions in nature and
brilliant ideas are admirable. I deeply appreciate his opening the window for me to

another piece of elegant Physics world.

I am grateful to Prof. B. Alex Brown, Prof. Edward Brown, Prof. William G.
Lynch and Prof. Chien-Peng Yuan for reading my dissertation and serving on my
guidance committee. Their generous advice and suggestions on the thesis are greatly

appreciated.

I would also acknowledge other members and visitors of the HiRA group
including Dr. Zbigniew Chajecki, Dr. Franck Delaunay, Dr. Tilak Kumar Ghosh, Rachel
Hodges, Dr. Sergei Lobastov, Fei Lu, Michal Mocho, DanYang Pang, Patrick
Quarterman and Jack Winkelbauer for their support and help as well as friendship. Many
thanks to Rachel Hodges and Dr. Tilak Kumar Ghosh for reading my thesis and giving

suggestions.

I would like to personally thank the Physics Department of the Chinese University
of Hong Kong, where I did my Bachelor degree, and Prof. Ming Chung Chu for offering

me the opportunity to participate the Summer Undergraduate Research Experience
vii

(SURE) program and work with Prof. Betty Tsang (who became my Ph.D advisor) at
NSCL during the summer in my senior year. It was all because of the fantastic research
experience at NSCL and excellent supervision of Prof. Betty Tsang that summer, I started

to be fascinated with nuclear physics and decided to pursue a doctorate degree in this

field.

I would also like to thank Dr. Chueh-an Hsieh who has been my wonderful
roommate for more than 4 years during our graduate studies. I deeply appreciate her

understanding, sharing and invaluable friendship.

Last, but not least, my deepest gratitude goes to my parents Hiu Ying Chan and
Kwong Ming Lee for always being there to give me unconditional support and believing
in me. Many thanks to my sister Hiu Man (Betty) Lee for being such a wonderful sister
and friend always deeply caring about and understanding me. The love I have received
fi'om my family is the main source of power in my pursuit of graduate study and my

entire life.

viii

TABLE OF CONTENTS

Chapter 1 Introduction
1.1 Nucleon correlation in nuclear structure and spectroscopic factor 1
1.2 Spectroscopic factors in (p,d) and (d,p) one-nucleon transfer reactions... ............ 8
1.3 Motivation of present experiment ......................................................... 14
1.4 Organization ofDissertation 17

Chapter 2 Spectroscopic factors from transfer reactions

2.1 Reaction theory“ .19
2.1.1 Adiabatic Distorted-Wave Born Approximation (ADWA) .................... 20

2. 1.2 Optical model potential .......................................................................... 23
2.2 Methodology of extracting experimental spectroscopic factor... ........................ 27
2.2.1 Input parameters in reaction model.. .......................................................... 28
2.2.2 Compilation and digitization of angular distribution data ................... 30
2.2.3 Extraction of spectroscopic factor from transfer reaction ..................... 31
2.2.4 Evaluation of the angular distribution measurements .......................... 33
2.2.5 Transfer reaction at high and low energy ....................................... 38
2.2.6 Uncertainty of the extracted spectroscopic factor ............................ 40

2.3 Theoretical spectroscopic factor ....................................................... 44

2.4 Survey of spectroscopic factors (SF) using CH89 global optical model

potential .................................................................................... 46
2.4.1 Comparison with shell model predictions ..................................... 47
2.4.1.1 Ground-state SFs for Z=3-24 nuclei ................................... 47

2.4.1.2 Excited-state SFs for Z=8-28 nuclei ................................... 51

2.4.1.2.] sd-shell nuclei .................................................. 52

2.4.1.2.2 Ca, Ti and Crsotopes .......................................... 54

2.4.1.2.3 Ni isotopes ..................................................... 56

2.4.2 Comparison with Endt’s “best values” ......................................... 62
2.4.3 Comparison with ENDSF values ................................................. 65

2.5 Reduced SF values using J LM optical model potential ............................. 66
2.6 Asymmetry dependence of reduction factors for stable nuclei in .................. 72
transfer reactions ......................................................................... 75

2.7 Applications of the established systematics in spectroscopic factors

Chapter 3 Experimental Setup and Detectors

3.1 Experimental designs ..................................................................... 90
3.1.1 Method of measurement ........................................................... 90
3.1.2 Radioactive beams and reaction targets ..................................... ' ..... 93
3.1.3 Geometry of HiRA detector Array ................................................ 96
3.1.4 Total energy and kinematics measurements ................................... 102

3.2 HiRA Detector Array .................................................................... 103

ix

3.2.1 Silicon strip detectors and CsI(T1) crystals .................................... 105
3.2.2 Readout Electronics - Application Specific Integrated Circuit (ASIC) 108

3.3 S800 Spectrometer ....................................................................... 110
3.4 Micro Channel Plate detectors ......................................................... 113
3.5 Position measurements of experimental setup ........................................ 117
3.5.1 Laser Based Alignment System (LBAS) ....................................... 118
3.5.2 Setup of position measurement .................................................. 121
3.5.3 Offset correction .................................................................. 124
3.6 Analysis of LBAS measurements ...................................................... 127
3.6.1 Transformations of laser coordinate systems .................................. 128
3.6.2 Reaction target position .......................................................... 131
3.6.3 HiRA pixel position ............................................................... 132
3.6.4 MCP foil position .................................................................. 135
3.7 Electronic and Data acquisition system ............................................... 137

Chapter 4 Data Processing and Analysis

4.1 Overview of the analysis ............................................................... 144
4.2 High Resolution Array (HiRA) ................................... - ..................... 1 47
4.2.1 Silicon Energy calibration ....................................................... 147
4.2.1.1 Readout order problem ................................................. 147

4.2.1.2 Readout order correction ................................................ 154

4.2.1.3 Energy calibration ....................................................... 159

4.2.1.4 Uniformity of Silicon thickness ........................................ 163

4.2.1.5 Determination of Dead-layer thickness .............................. 164

4.2.2 CsI Energy calibration ........................................................... 166
4.2.3 Pixelation technique .............................................................. 170
4.2.4 Particle identification ............................................................. 173
4.2.5 Deuteron total energy ............................................................. 175
4.3 S800 Spectrometer ...................................................................... 178
4.3.1 Beam particle identification .................................................... 178
4.3.2 CRDC position calibration ...................................................... 181
4.3.3 S800 Particle identification ...................................................... 183
4.3.4 Beam and reaction residue energy ............................................. 192
4.3.5 Trajectory reconstruction ........................................................ 195

4.4 Micro Channel Plate detectors analysis .............................................. 196

Chapter 5 Experimental results

5.1 Reaction kinematics ...................................................................... 199
5.2 Q-value calculations ...................................................................... 202
5.3 Background subtraction ................................................................. 206
5.4 Extraction of cross sections and spectroscopic factors .............................. 208
5.4.1 Absolute normalization and error analysis ...................................... 208
5.4.2 Differential Cross Sections and spectroscopic factors ....................... 217
5.5 Theoretical spectroscopic factors and reduction factors ........................... 233
5.6 Asymmetry dependence of reduction factors ......................................... 236

Chapter 6 Summary and conclusion ................................................... 243

Bibliography ................................................................................. 249

xi

LIST OF TABLES

2.1 Summary of parameters for CH89 global optical model potential ..................... 25
2.2 List of nuclei with SFs obtained from both (p,d) and (d,p) reactions .................. 42
2.3 Table of the extracted ground-state SFs for Z=3-24 isotopes ........................... 76
2.4 List of neutron excited-state SFs for the SD shell nuclei ................................ 78
2.5 List of neutron excited-state SFs for the Ca, Ti and Cr isotopes ....................... 80

2.6 List of experimental neutron SFs and ENDSF’s SF values for the Ni isotopes ......81
2.7 List of experimental neutron SFs and shell model predictions for the Ni isotopes . 86

2.8 List of SF3 and reduction factors obtained in (d,p), (p,d) transfer, (e,e’p) and
knockout reactions ................................................................................. 87

2.9 Spin assignments and SFs of three excited states in 27Mg .............................. 89

3.1 Summary of the distances and the angles deviated from the telescope center (Si face)
relative to the target center ....................................................................... 135

4.1 Averaged corrections for the first two readout order on the motherboard for EF and

EB within each telescope ........................................................................ 158
4.2 Averaged corrections for the readout order on the motherboard for
EF within each telescope ........................................................................ 158
4.3 Thickness of DE detectors given from manufacture and deduced from experimental
data ................................................................................................. 163
4.4 Thickness of E detectors given from manufacture and deduced from experimental
data of NSCL Expt 02023 ....................................................................... 169
4.5 Summary of $800 CRDC calibrations ................................................... 183
4.6 Summary of the beam energy from A1900 and the real beam energy ............... 193
33,35,45

5.1 Energy levels (< 6MeV) with spin and parity assignments for Ar ...201

5.2 Contributions to the energy resolutions (FWHM) simulated by Geant4 code and

comparisons to the FWHM of the peak corresponding to ground-state transitions in the
Q-value spectra for the reactions in inverse kinematics ...................................... 206

xii

5.3 Differential cross sections in center-of-mass frame for p(34’36’46Ar,d) p0pu1ating to

ground-state and excited states of 33,35,45Ar ................................................. 219
5.4 Theoretical spectroscopic factors of 34Ar for energy levels less than 10MeV .....234
5.5 Theoretical spectroscopic factors of 36Ar for energy levels less than 10MeV .....234
5.6 Theoretical spectroscopic factors of 46Ar for energy levels less than 5MeV .......235
5.7 Extracted SF values with the associated uncertainties for different reactions ...... 235

5.8 Extracted ground-state SF 3 for stable Ar and Ca isotopes ............................. 240

xiii

LIST OF FIGURES
Images in this dissertation are presented in color
1.1 Chart of the nuclei .............................................................................. 2
1.2 Nuclear shell structure ........................................................................ 4

1.3 Fluctuations in the published ground-state spectroscopic factors of 41Ca extracted
4o 41 .
from Ca(d,p) Ca reactions ...................................................................... 12

1.4 Reduction factor (Rs) as a function of the difference of neutron and proton
separation energies ................................................................................... 16

2.1 Comparison of tabulated data and digitized data the same measurement of
“*N(d,p)'5N ........................................................................................... 31

2.2 Angular distributions of the deuteron obtained in the 44Ca(p,d) 43Ca reaction at
incident proton energy of 40 MeV ............................................................... 33

2.3 Comparisons of the angular distributions of the proton measured in the llB(d,p)12B
reactions in three different experiments ........................................................ 34

2.4 Comparisons of the angular distributions of the proton measured in the 12C(d,p)l3C
reactions in four different experiments ......................................................... 36

2.5 Comparison of the published SFs for 41Ca isotope ....................................... 37

2.6 Angular distributions for 40Ca(d,p)41Ca reactions for beam energy from 4.69 to 56
MeV ................................................................................................ 39

2.7 Comparisons of spectroscopic factors obtained from (p,d) and (d,p) reactions ......42
2.8 Spectroscopic factors obtained for the Ca isotopes using consistent approach ..44
2.9 Experimental and theoretical ground-state neutron SFs for Ca isotopes ............... 48

2.10 Comparison of experimental SFs to predictions from the independent
particle model ....................................................................................... 49

2.11 Comparison of experimental SFs to predictions from the large-basis shell-model . 50

2.12 Comparison of experimental excited states SFs for sd-shell nuclei to predictions
from large-basis shell-model calculations using the USDA and USDB interactions ......54

xiv

2.13 Comparison of experimental SF 8 to predictions from large-basis shell- model for the
Ca, Ti and Cr isotopes .............................................................................. 56

2.14 Comparison of the ground—state experimental SFs to the shell model calculations
using different interactions ........................................................................ 5 9

2.15 Comparison of the experimental SFs to the shell-model calculations with the
GXPFlA interaction in the pf model space and the XT interaction in gfp model space . 60

2.16 Plot of energy levels below 2 MeV for Ni isotopes with SF values ................... 62

2.17 Comparisons of ground-state SFs obtained using systematic approach CH89 and
from the compiled values of Endt ................................................................. 64

2.18 Comparisons of excited-state SFs obtained using systematic approach CH89 and
from the compiled values of Endt ................................................................ 64

2.19 Comparisons of SFs obtained using systematic approach CH89 and from the
compiled values of ENDSF ....................................................................... 66

2.20 Ratios of the experimentally deduced SFs to those of the independent particle shell-
model for Ca isotopic chain ....................................................................... 70

2.21 Asymmetry dependence of reduction factors observed in (d,p), (p,d) transfer,
(e,e’p) and knockout reactions .................................................................... 74

3.1 A 3-D rendering of the $800 spectrometer aligned with the scattering chamber and

analysis beam line .................................................................................. 92
3.2 Schematic layout of the designed experimental setup ................................... 92
3.3 Momentum matching for neutron for the (34Ar,d) 33Ar and p(46Ar,d) 45Ar

Reactions ............................................................................................. 95
3.4 Schematic of the Coupled Cyclotron Facility and A1900 at the NSCL ............... 96

3.5 Calculated deuteron angular distributions of p(34’36’46Ar,d) for ground-state

transitions in center-of-mass frame (lefi panels) and in laboratory fi'ame (right panels) at
beam energy of 33MeV/ nucleon ................................................................. 98

3.6 Velocity diagrams for (p,d) reaction in inverse kinematics ............................. 99
3. 7 Deuteron emitted angles 1n center of mass angle versus the emitted angle in

laboratory frame for p(3 6 ’46Ar,d) of ground-state transitions at beam energy of
33MeV/nucleon ...................................................................................... 99

XV

3.8 Photograph of the HiRA detector array and the downstream MCP .................. 100

3.9 Efficiency of HiRA setup where 16 telescopes were placed at 35 cm from the

reaction target ...................................................................................... 100
3.10 HiRA configuration from mechanical design in the Cartesian coordinate systems
and Spherical coordinate systems ............................................................... 101
3.11 Kinematic curves of deuterons from p(34’36’46Ar,d) reactions at beam energy of
33MeV/ nucleon ................................................................................... 103
3.12 Different HiRA setup used in four different experiments ........................... 104
3.13 Schematic of a single HiRA telescope containing two layers of silicon detectors
with thickness of 65 um and 1.5mm backed up by a cluster of CsI crystals .............. 105
3.14 Image of HiRA single-sided silicon strip-detector .................................... 106
3.15 Image of HiRA double-sided silicon strip-detector ................................... 106

3.16 Side and front view of a CsI crystal, light guide and photo-diode assembly..... . ..108

3.17 Block diagram of the HiRA ASIC ...................................................... 109
3.18 HiRA chipboard and motherboard ...................................................... 110
3.19 Schematic of S800 spectrometer ......................................................... 111
3.20 The schematic of the focal plane detector system in S800 ........................... 112
3.21 Schematic of CRDC detector ........................................................... .113
3.22 Schematic design of a micro-channel plate ............................................ 114

3.23 The detection system with two micro-channel plates and the view of chevron style

MCPs ............................................................................................... 115
3.24 Schematic setup of MCP detection system with MCP, two permanent magnets, foil
and anode layer .................................................................................... 116
3.25 Photograph of Laser Based Alignment System ............. '. ........................... 119
3.26 Data points from a typical laser scan across an edge ................................. 120
3.27 Schematic setup of position measurement ............................................. 122

xvi

3.28 Photograph of setup with laser being placed at position 0 ............................ 122
3.29 Photograph of reference posts in laser measurements ................................ 123
3.30 Layout of the beamline in the laboratory ................................................ 124
3.31 Schematics of laser Based Alignment System with offset problems indicated .. 125
3.32 Schematic diagram to present the strategy of laser scanning on the post 129

3.33 Reaction target ladder, schematic diagram of the target mask and the strategy of
laser scanning ..................................................................................... 132

3.34 Comparison between the telescope angles on theta and phi in laboratory frame
obtained from the LBAS measurements and mechanical design ........................... 134

3.35: Comparison between the telescope positions obtained from the LBAS

measurements and mechanical design ........................................................... 134
3.36: A photograph of a MCP mask for position calibration ................................. 136
3.37: The configuration of six large holes in the MCP mask ................................ 137
3.38: Schematic of data acquisition system ............................................... 141-143
4.1 Flow chart of the data analysis ............................................................ 146
4.2 Spectrum of IU pulser ramp in one electronic channel ................................. 149

4.3 Cross-talk problem on the neighboring strips (strip 9 and 11) when only one strip
(strip 10) is pulsed by 1U pulser ................................................................ 150

4.4 Diagram of the decay chain of 228Th
...................................................... 152

4.5 Alpha spectrum in raw channel of one EF strip without readout order correction .153

4.6 Alpha spectrum in raw channel of one EF strip after readout order correction applied
on both motherboard and chipboard readout ................................................ 153

4.7 Spectrum of alpha source gated on OrdDet-=1 different values of OrdTow 1 56

4.8 Spectrum of IIJ pulser ramp gated on different OrdDet corrected with readout order
of motherboard ................................................................................... .157

xvii

4.9 Zoom in of Figure 4.3 on the regions where the same peaks with different OrdDet

appear in the spectrum ............................................................................ 157
4.10 Photograph of the flame with pin source and the slot between the DE and E
detectors in the front part of HiRA telescope ................................................. 160
4.11 Pin source spectrum in raw channel of one pixel with MB and CB readout
sequences problem corrected .................................................................... 160
. 4.12 Geometry of setup for pin source calibration ....................................... 165
4.13 A 2D spectrum of EF in MeV vs. CsI in channel and the linear fit between the
ADC channel and the particle energy in C81 .................................................. 167
4.14 The zero offset of each CsI crystal converted into energy ........................... 168
4.15 Energy difference between a pair of matched EF and EB strip verus energy in the
corresponding EB strip ........................................................................... 172
4.16 PID spectrum of EF vs DE energy in unit of MeV .................................. .174
4.17 PII) spectrum of CSI vs BF energy in unit of MeV .................................... 175

4.18 Linear fits of the calculated and calibrated total energy in laboratory frame for one
in p(34Ar,d) reaction of ground-state transition and the corresponding deuteron total

energy in center-of-mass frame ................................................................. 177
4.19 Calculated kinematics broadening of p(34’36’46Ar,d) for ground-state transitions in
laboratory frame ................................................................................... 177
4.20 The beam PID spectrum for p(36Ar,d) reaction ........................................ 179
4.21 The beam PID spectrum for p(34Ar,d) reaction ....................................... 180
4.22 The beam PID spectrum for p(46Ar,d) reaction ....................................... 180
4.23 Pattern of the mask used in the CRDC position calibration ........................ 182
4.24 Two-dimensional position spectra of CRDC] and CRDC2 ........................ 182
4.25 ssoo PID spectrum for p(36Ar,d) reaction ............................................ 184

4.26 8800 PID spectrum of p(34Ar,d)33Ar with and without trajectory corrections in
ToF and energy loss in the ion chamber ....................................................... 185

xviii

4.27 Spectrum of ToF-RF signal of 33Ar fi'agment gated on 34Ar beam versus the
dispersive angle in S800 focal plane. And the centroids of ToF-RF in the bin of AFP .187

4.28 Spectra of the position dependence of time-of—flight for 33Ar gated on 34Ar for
CRDC] and CRDC2 .............................................................................. 188

4.29 Centroids of ToF.RF in the bin of CRDC positions from corresponding spectrum in
Figure 4.17 ......................................................................................... 189 ‘

4.30 Spectra of the position dependence of energy loss in S800 Ion Chamber for 33Ar
gated on “M for CRDC] and CRDC2 ........................................................ 190

4.31 Centroids of IC.de in the bin of CRDC positions from corresponding spectrum
shown in Figure 4.19 .............................................................................. 191

4.32 S800 PID spectrum for p(34Ar,d) and p(46Ar,d)
reaction ............................. 192

4.33 The dispersive position (F P.x) of the incoming beams of 36Ar, 34Ar and
46
Ar. . 193

4.34 The calculated energy of reaction residues of 35Ar, 33Ar and 45A1“ at ground state in

laboratory frame ................................................................................... 194
4.35 The energy spectrum of 33Ar produced from reaction of p(34Ar ,d)33Ar ground-

state transition gated on scattering angle between 0.17°-0.23° .............................. 195
4.36 Multiplicity of MCPO and MCPl ....................................................... 197
5.1 Deuteron kinematics for p(34Ar,d)33Ar in laboratory frame .......................... 200
5.2 Deuteron kinematics for p(36Ar,d)35Ar in laboratory frame .......................... 200
5.3 Deuteron kinematics for p(46Ar,d)45Ar in laboratory frame .......................... 201

5.4 p(34Ar,d)33Ar Q-value spectrum with laboratory angular coverage from 5° to
23° .................................................................................................... 204

5.5 p(36Ar,d)35Ar Q-value spectrum with laboratory angular coverage from 5° to
26° .................................................................................................... 205

5.6 p(46Ar,d)45Ar Q-value spectrum with laboratory angular coverage from 5° to
19° .................................................................................................... 205

xix

5.7 Carbon target reaction data with 46Ar beam ............................................. 208

5.8 Live time of data acquisition determined by the ratios of live to raw rate of MCPO,
MCPl and the clock in the p(34’36’46Ar,d) reactions

.......................................... 211
5.9 MCPl efficiency of the p(34’36’46Ar,d) reactions ........................................ 212
5.10 The RF and XF P signals for the same data run of p(34Ar,d) reaction ............. 213

5.11 A1900 XF P efficiency of the p(34’36’46Ar,d) reactions
................................ 213

5.12 Proportion of 34Ar in the incoming secondary beams for the p(34Ar,d)
reactions...215

5.13 Comparisons of absolute differential cross sections at certain angles for p(34’ 36’

46Ar,d) reactions respectively over the entire experiment ................................... 217

5.14 Differential cross sections of p(34Ar,d)33Ar in center-of—mass frame for ground—
state transition ..................................................................................... 222

5.15 Differential cross sections of p(34Ar,d)33Ar in center-of-mass frame for transitions
to 1.359 and 1.798 MeV states .................................................................. 223

5.16 Differential cross sections of p(34Ar,d)33Ar in center-of—mass frame for transitions
to 3.456 and 3.819 MeV states .................................................................. 223

5.17 Differential cross sections of p(36Ar,d)35Ar in center-of-mass frame for ground-
state transition ..................................................................................... 225

5.18 Differential cross sections p(36Ar,d)35Ar in center-of-mass frame for transition to
first excited state .................................................................................. 225

5.19 Differential cross sections of p(46Ar,d)45Ar in center-of-mass frame for group of
transitions to ground state and first excited state ............................................. 227

5.20 Q-value spectrum of p(46Ar,d)45Ar at 0 lab=12.5 :t 05° (0 COM~7.5°)
........... 227

5.21 Differential cross sections of p(46Ar,d)45Ar in center-of-mass frame for group of
ransitions to ground state and first excited state at forward angles ........................ 228

5.22 Q-value spectrum of p(46Ar,d)45Ar at lab=28-29° (9 COM~21°) .................. 229

XX

5.23 Differential cross sections of p(46Ar,d)45Ar in center-of-mass frame for ground-
state transition ................................................................................... 229

5.24 Deduced differential cross sections of p(46Ar,d)45Ar in center-of-mass fiame for
first excited-state transition ...................................................................... 230

5.25 Differential cross sections of p(46Ar,d)45Ar in center-of-mass frame for group of
transitions to 1.734-1.911 MeV ................................................................. 231

5.26 Differential cross sections of p(AéAr,d)45Ar in center-of-mass frame for group of
transitions at about 3.950 MeV ................................................................. 232

5.27 Reduction factors SF(expt)/SF(LB-SM) of the proton-rich 34Ar and neutron
rich46Ar, together with the well bound 3641M isotopes .................................... 238

5.28 Reduction factors of Ca isotopes obtained from the dispersive optical-model
(DOM) calculation (solid line) and from neutron (d,p) and (p,d) transfer reactions
analyzed using J LM+HF approach ............................................................. 240

5.29 Reduction factors (Rs) as a function of the difference between neutron and proton

separation energies AS. Solid circles and open triangles represent the Rs deduced from
present work and knockout reactions [Gade] respectively .................................. 242

xxi

Chapter 1

Introduction

1.1 N ucleon correlation in shell structure and spectroscopic factor

One of the basic aims of fundamental science is to understand the properties of matters
that make up the universe. All terrestrial substances are composed of atoms. At the core
of an atom, surrounded by electrons, lies the nucleus. Protons and neutrons are the basic
building blocks of the nucleus. Properties of all isotopes are determined by the number of
protons and neutrons. The number of protons in a nucleus (Z) establishes the chemical
identity of the element. A variety of isotopes of a specific element are formed from
different numbers of neutrons (N). The possible N/Z ratio of an isotope depends on the
nature of the nuclear force which binds the protons and neutrons in the nucleus. About

300 stable isotopes, with large binding energies, exist in nature.

Figure 1.1 displays the chart of the nuclei with proton number and neutron
number [ISF]. The region with black squares represents the stable isotopes, called the
valley of stability. As the N/Z ratio decreases (proton-rich) or increases (neutron-rich),
compared to the stable isotopes the nuclei are unstable and decay towards the valley of

stability. These observed short-lived exotic isotopes are shown in green in Figure 1.1.

One of the ultimate goals in nuclear physics is to determine the limits within
which nucleus can exist or the exact positions of the proton and neutron drip lines in the
nuclear chart. Since the nucleosynthesis processes, for instance r- and rp-process, take

place in regions of extreme N/Z ratios as indicated in Figure 1.1 [Wal8l,Cow91,Sch98],
1

understanding the properties of exotic nuclei near the drip lines is of critical importance

and would offer new glimpses into the origin of the elements in the universe.

 

 

120 _ I I r I I I I I ‘ I

1m ' I -r
E 80 ' ' // 133-]
E
3 60
c _ . r -process
9 [1) process ---_.:..-
2 I
n- 40 h -l

20 _ I Observed nuclei
I Stable nuclei
0 I 1 1 1 1 4 r

 

 

0 20 40 60 80 100 120 140 160 180
Neutron number
Figure 1.1: Chart of the nuclei with proton number and neutron number. Stable
isotopes are shown in black while the observed unstable nuclei are shown in

green. Dashed lines indicate the conventional magic numbers. Nuclei relevant in
the astrophysical 171- and r- processes are also indicated (adopted from [ISF]).

In 1930s, the gross feature of nuclear binding energies for stable isotopes was
successfully described by the Bethe-Weizsacker Formula [Boh75]. This simple
macroscopic approach based on the liquid drop model includes the volume, surface,
Coulomb, pairing and proton-neutron symmetry effects in a nucleus. A systematic
comparison of experimental nuclear binding energies to predictions reveals periodic
deviations where unusual stabilities arise from nuclei with numbers of neutrons or

protons at 8, 20, 28, 50, 82 and 126 [Mye66,Boh75]. Such observation can be

2

satisfactorily explained by the shell model of atomic nuclei proposed by Mayer and
Jensen [May60], for which they shared the 1963 Nobel Prize for Physics. The shell model
has been immensely robust and successful in describing both systematic and specific
features of nuclear structures. The simplest shell model, ofien called the Independent
Particle Model (1PM), requires each nucleon to occupy a single-particle eigenstate in the
nuclear spherical mean-field potential from the core. The mean-field is usually described
by a harmonic oscillator or Woods-Saxon potential [May60,Cas90,Pre93]. This model
neglects any correlations in the motions of particles outside the core or residual
interactions between the valence single particles and the core. Protons and neutrons fill
the eigenstates independently. Each nucleon state is characterized by its single-particle
energy and quantum numbers (n, U), where n is associated with the number of nodes of
the radial wave function, and l and j are the orbital and total angular momenta. The
maximum occupancy in each neutron (proton) eigenstate is (2j +1) neutron (proton)
[May60,Cas90,Pre93]. With a strong spin-orbit force included in the nuclear spherical
mean-field potential from the core, the shell model produces significant energy gaps at
the magic numbers of 8, 20, 28, 50, 82 and 126 (Figure 1.2)
[Br005,May60,Cas90,Pre93]. Stable nuclei with proton or neutron numbers
corresponding to a magic number exhibit stability and natural abundance. This simple
shell model also gives the correct ordering of the single-particle orbits for stable isotopes
as shown in Figure 1.2. In addition, it reasonably describes the nuclear static and dynamic
properties of low-lying states in spherical stable nuclei and serves as a foundation of the

more sophisticated modern shell model development [BroOl ].

10

E
d
o

sings-particle energy (MeV)
ti:
o

do
0

-40

 

IllIIIIIITIIIIIIPT11fiTIIIIllllllllllllllIIlllerIIllflllrl

 

Neutron Single-Particle Energies

[56] 168 N-6
—-————-

 

[42] 112 N=5

[30] 7011-4
—_W

[20] 40 N-3

_‘

[12] 20 N=2

——fi

[6] e~=1

%

[2] 214-0

Harmonic
33.1121

~
\
\

‘ \ [26]1380i
———<

[6]1122p

‘ ..—
\

\

\

\ [22192011
———-<

[217023

a;
\
\

\
\

‘ [1915309

\
\ s

, 4 161401;:

 

~
~
\
*

 

\

\ J1413401 A

 

\

\ \
\ \ [212013

 

‘ \ [10]180d

\

xterm

 

[2] 20a

 

2116833
10 1662d
- - - -

\

I : [14110611‘f

\

~’~’

I

I\l|{

Woods-Saxon

Potential

1 1

J11

2184 263
]1800j15
81164197
2 156 331

6 1542t15
i 5 =/_ [1211430111
[10]136199
[211262p1
1124115
/ 41182pa
Jail” 133::
,_—_:_—[_[a] 90117
1213201111
/ _[ 70201
4 68163
6 641d5
:ij 53097
50
[10150099
401 p1
——’l 38133
[6] 34015
28
___[a] 29017
20
[2] 20101
[4113mm
[61140115
8
2 a 1
:jd 6&3
2
[2] 2031
Woods- Saxon
Orbit
otent

lLlllllllllllllllllllll[LII]liJJJlJJlllJlllllllllJJllllllL

 

 

Figure 1.2: Nuclear shell structure [Br005].

4

Although the mean-field model provides a basic framework to understand nuclear
structure, a complete description of nuclei requires consideration of the nucleon-nucleon
(N-N) correlations which modify the nuclear wave functions. The short-range component
of the correlation in nuclei originates from the strong repulsion of nucleons at very short
distances. Such repulsive force extends the momentum components of the nucleon wave
functions to very high energy single-particle orbits and modify the compressibility of
nuclei and nuclear matter [Akm98,Dic08]. The long-range component of the correlation
arises from the attractive interactions between valence nucleons. This induces collective
modes and pairing correlations within nuclei and leads to the coupling 0f nucleon orbits
to low-lying collective excitations and giant resonances [Gol99]. The correlated interplay
between the single-particle and collective dynamics spreads the single particle strength

over a large range in excitation energy [Dic04,Dic08].

One signature of correlations is the fractional occupation of single-particle
orbitals. The total occupancy of a single-particle state is not directly observable, but it can
be quantified by the spectroscopic factor (SF). The SF probes the overlap between many-
body wave functions of the initial and final states and measures the degree of single-
nucleon overlap for a state in the specific transition [Aus70]. It characterizes the structure
of nuclear states and connects experimental results to theoretical nuclear structure
calculations. The measurement of (e,e ’p) reactions on nuclei near closed proton shells is
one well-known example of using SFs to investigate the effects of correlations. Results
show that the absolute experimental proton SF values are reduced systematically by 30-

40% compared to the mean-field theory of the Independent Particle Model [Pan97,

KraOl]. This observation suggests that significant contributions of N-N interactions are
absent in the Independent Particle Model descriptions [Pan97,Dic04,Bar09].

The large-basis shell-model approach has been developed to include some of the
N-N correlations in the nuclear structure studies [BroOl]. To account for the

shortcomings in an approximation with a one-body mean-field potential, a two-body

nucleon-nucleon interaction VNN is employed in the Hamiltonian:

-—_-2
H = 203;) + Z vnnc‘r‘t— 7;) (1.1)

Because of the difficulties in obtaining the exact solution in such a many-body aspect, a

single-particle spherical potential is introduced to approximate the Hamiltonian into two
terms as an independent particle moving in an effective average potential ”(0) and

residual interactions H (1).

H = 2: [333+ aria] +1; VNNG'; — a) —Z 0079] = Hm) + HG) (1.2)
<1

Consequently nucleons in a nucleus can be grouped into the core part and the
valence part, where the core part is completely occupied. Starting from the core and
observed single-particle state, the residual two-body interaction in the Hamiltonian is
adjusted to describe other properties of nuclei in a given valence space. By
approximating the configuration mixing with a basis of nucleon quasi-particle orbits, the
effective-interaction theories of current shell models can describe much of the long-range
component of the interactions [BroOl,Dic04]. Limited by the dimension space, the

valence nucleon Hilbert space has to be truncated in the calculations. This confines the

6

description of long-range correlations in the vicinity only near the Fermi surfaces.
Currently available residual interactions provide satisfactory descriptions for the light to
medium mass nuclei in sp-, sd- and pfg— shell regions [Cau05]. The correlations that arise
from short-range and long-range with coupling to vibrational excitations are beyond the
effective interactions employed in the current effective-interaction shell models. Progress

with other theoretical approaches to describe firll nucleon correlations is being made

[Dic04,SteO6,BarO9-3].

The results from (e,e ’p) clearly indicate the absence of nucleon correlations in the
mean field model descriptions for nuclei near the closed shell. Understanding the
correlations and the single-particle structure in nuclei with large proton-neutron
asymmetry becomes compelling. With the advent of radioactive beam facilities, nuclei
far from the stability line with large neutron or proton excess can be investigated. It has
been observed that the nucleon correlations play a prominent role in the breakdown of the
conventional magic numbers with the appearance of new “magic” numbers and new
ordering of single-particle states and exotic modes of excitations for exotic nuclei

[Sor08]. For instance, new shell gaps and magic numbers appear at N=14, 16 instead of
N=8, 20 for neutron-rich nuclei in the sd-shell, and the swapping of 31/2 and aim single-

particle states at N=16 for Oxygen isotopes [Sor08]. To study the evolutions in the
nuclear structure from the valley of stability toward the drip lines, it is essential to
investigate the roles of the proton-neutron asymmetry dependence of the mean field and
the long- and short-range correlations. Many experimental approaches provide insight
about such evolutions. For instance, one can measure the variations of effective single

particle energies which illustrate the monopole interaction effect that acts between

proton-neutron spin-orbit partners [Gau06, Sig07]. Another approach is to investigate the
asymmetry dependence of single-particle occupancies [Gad08]. Both approaches require

accurate determination of spectroscopic factors.

1.2 Spectroscopic factors from transfer reactions

Direct reaction is a powerful probe of single-particle structures. With minimum
rearrangement of the nucleons during collisions, direct reaction involves a few degrees of
freedom. In the past half century, reaction theories have been developed to allow
determination of spectroscopic factors which quantify the nucleon occupancies and
degrees of correlations in the single-particle wave functions. In addition to transfer
reactions, there are two other experimental techniques to extract spectroscopic factors.
Firstly, the electron-induced proton knockout reaction (e,e ’p) which was developed to
measure the spectroscopic factors of proton single-particle hole states in well-bound
nuclei in the 1980’s [KraOl]. The results from (e,e'p) reactions indicate 30-40% depletion
of occupancy attributed to the nucleon correlations. Even though it is believed that
reaction theory is more exact and that electrons probe the interior of the wave functions,
neutron removal and particle-state cannot be studied by electron scattering. In addition,
this technique can be applied to study only stable isotopes due to lack of rare isotope
targets. To extend the systematic studies to exotic isotopes, considerable progresses in
both theories and experimental techniques in single-nucleon knockout reactions using fast

beams have been made in the last decade [Han03, Tos04]. The low beam intensity
requirement of only an order of 102 particles-per-second at incident energy above 50

MeV/u offer a powerful approach to systematically study the hole-state spectroscopic

factors for exotic nuclei [Han03, T0504, Gad08]. Knockout reactions suggest a systematic

trend of spectroscopic factors and nucleon correlations.

The main source of spectroscopic factor information for stable nuclei in the past
half century is from transfer reactions. Compared to (e,e ’p) and one-nucleon knockout
reactions, transfer reactions probe both particle- and hole-states exclusively for stable as
well as exotic nuclei. Beam energies used in transfer reactions can vary from a few MeV
to a hundred MeV per nucleon depending on the experimental purposes. With the recent
revived interest of transfer reactions due to the availability of radioactive beams, it is
essential to review and understand the existing theoretical reaction models of transfer
reactions, and develop a systematic framework to extend the study of the nuclear

structures from stable to exotic nuclei.

One-nucleon transfer reactions such as (p,d), (d,p), (d,n), (3He,d) and (61,3He) are
frequently used to probe correlations and properties of single-particle wave functions and
the degree to which the nuclear state is a pure single-particle state. The angular
distributions of the emitted d or p are characteristic of the angular momentum of the
transferred nucleon. For two-nucleon transfers such as (p, t) and (t,p), the transferred
angular momentum is shared between two transferred nucleons and therefore the
correlations of nuclear structure govern the strength of the reaction. This important
feature provides a means to probe nuclear correlations between two nucleons which is not
accessible to one-nucleon transfer reactions [AusO7,Tan08]. The two-nucleon transfer
reaction can also be used to study unbound states in exotic nuclei which are restricted in
one-nucleon transfer techniques [SuzO9]. However the applicability of the two-nucleon

transfer reactions is limited by the complications in the reaction theories and low reaction

9

rates. Similarly, multi-nucleon transfer reactions may be a useful tool for studies of
correlations [AusO7]. Unfortunately, there is not yet a well developed theory to describe
the data. This thesis is dedicated to single-nucleon transfer reactions which have more

well developed theories and extensive existing data.

For one-nucleon transfer reactions, the description of the reaction mechanism
generally utilizes the distorted wave Born approximation (DWBA) [Sat80,G1e04, Aus70].
The differential cross section is proportional to the square of transition amplitude, which

can be described by two main factors as shown in Equation 1.3. S“ is the spectroscopic
factor which is determined by the wave functions of nuclear states involved but
independent of the kinematics; Blm’ contains the reaction dynamics information through

the overlap of the wave functions of the relative motions and the wave function of the
transfer nucleon [Aus70, Sat80, Gle04].

do
Tact Irafliz at 225013;”? (1.3)

1’”!

Since 5],- reflects the nature of wave functions, it provides a means for comparison
between experiments and theories. The experimental spectroscopic factor in a transfer
reaction is obtained by taking the ratios of the experimental cross sections to the
predicted cross sections fiom the reaction model [Aus70,Sat80,Gle04]. Extraction of
meaningful and reliable spectroscopic factors is essential through accurate descriptions of
reaction theory. As discussed in Chapter 2, non-locality and finite-range DWBA
calculations are developed for realistic descriptions of kinematics. The calculations also

require optical model potentials to construct the distorted waves of incoming and

10

outgoing reaction channels and the inputs of the bound-state radial wave functions of the

transferred particle.

However, a survey of the published spectroscopic factors for certain nuclei shows
that they are not consistent. It is not unusual to find published spectroscopic factors for a

particular nucleus that fluctuate by a factor of two as demonstrated in Figure 1.3 for the
ground-state spectroscopic factors of 41Ca extracted fi'om 4OCa(d,p)“Ca reactions.

Similarly, one can often find that published spectroscopic factor values for the same
reaction by different authors agree within uncertainties, even though the data used to
extract them are not in agreement [Lee07]. Many of the difficulties in the past extractions
of spectroscopic factors have been associated with ambiguities in the optical model
pararneterizations which are used in the DWBA model to construct the distorted waves of
incoming and outgoing reaction channels. Geometries of the bound-state radial wave
functions of the transferred particle are strongly correlated to the magnitudes of the
spectroscopic factors [Aus70,Sat80,Gle04]. In addition, the deuteron breakup effects
have been found to be important in the calculated angular distributions and deduced
spectroscopic factors [Joh70]. The inconsistencies in spectroscopic factor extractions
hinder the use of transfer reactions from studying evolution of nuclear structures, such as
the shell evolution along the isotopic or isotonic chains. Therefore it is crucial to develop
a systematic approach to extract consistent and reliable relative spectroscopic factors

from transfer reactions.

11

1.5 . r \

 

 

 

 

 

V
40Ca(d,p)41Ca
7:?
g e
.59. o .
F' 1.0
“S . 0 e
3 e e 9 0
127-1 . .
U) 0 . . O
O . O
. r R
0.5 .
10 20 \56
Ed(MeV)

Figure 1.3: Fluctuations in the published ground-state spectroscopic factors of
4lCa extracted from 40Ca(d,p) lCa reactions.

A consistent three-body analysis of neutron transfer reaction data involving
minimal assumptions has been developed for (p,d) and (d,p) transfer reactions [Liu04,
Liu05]. The methodology uses Adiabatic Distorted-wave Born approximation (ADWA),
an extension of DWBA theory with the Johnson-Soper (JS) adiabatic approximation to
the neutron, proton, and target three-body system [Joh70] taking into account the effect
of deuteron break up. To avoid the ambiguity in optical potentials obtained from
individual best fits to elastic scattering data, global nucleon optical model potentials or
microscopic potentials are applied consistently at all incident energies and target nuclei.
Additionally, a consistent set of input parameters for the reaction mechanism and single-

particle bound state is adopted in the cross-section calculations [Liu04, Liu05, Lee07].

12

One may argue that it would be better to fit the experimental elastic scattering
data at the same incident energy by adjusting the parameters in optical model potentials
for nucleon transfer reaction. This, however, does not seem to result in an overall
improved description, mainly because data at a single incident energy do not lead to an
unambiguous determination of the nuclear optical potential. Instead, it is more
appropriate to use elastic scattering data measured over many energies to obtain a global
optical potential that can be used without the need for an additional measurement of
elastic scattering for a particular transfer reaction [Per76]. Such practice becomes
desirable when a large range of beam energies and nuclei is needed in systematic studies
or when radioactive beams are involved where beam time for elastic scattering is often
not available.

The systematic ADWA analysis approach has been proven to sufficiently remove

the long-standing ambiguities in the cross-section calculations and give consistent
spectroscopic factors for 12C(d,p)13C and l3C(p,d)12C reactions [Liu04,Liu05]. To

examine the possibility of using such a consistent approach for a global study of nuclear
structure using spectroscopic factors, the first part of this dissertation is dedicated to the
systematic comparisons of an extensive amount of single neutron (p,d) and (d,p) transfer
reaction cross-section data on stable isotopes from Lithium to Nickel, obtained in the past
40 years, to the angular distributions calculated in the ADWA model
[TsaOS,Lee06,Lee07,Tsa09,Lee09]. This large-scale survey utilizes the existing data to
extract meaningful spectroscopic factors and establishes reliable systematics in the
ground-state and excited-state spectroscopic factors. This provides an essential

framework for studying nuclear structures using transfer reactions.

13

1.3 Motivation of present experiment

Based on the consistent ADWA framework, the comprehensive analyses of transfer
reaction data described in Chapter 2 suggest that the reduction factors Rs, the ratio of the
experimental spectroscopic factor (SF) to prediction, are independent of the neutron
binding energy for stable nuclei within statistical errors [Tsa05, Lee06]. The reduction
factor quantifies the percentage of correlation absent in the shell model descriptions. To

understand the regions of extreme N/Z, the experimental efforts of this dissertation focus

. . . . 34

on the measurements of (p,d) neutron transfer reactrons mvolvrng proton-rich Ar and
. 46 . . . .

neutron-rich Ar beams. These measurements were done 1n 1nverse krnematrcs to extract

the experimental neutron SF 5 for 34Ar and 46Ar. The difference between neutron and
proton separation energy (AS =Sn-Sp for neutron SF; AS=Sp-Sn for proton SF), which
characterizes the asymmetry dependence of the relative shift for the neutron and proton
Fermi surface, is 12.41 MeV and -10.03 MeV for 34Ar and 46Ar respectively. This
significantly expands the separation energy difference of isotopes investigated. In
previous global studies the maximum AS is 7.31 MeV with a few measurements at AS
smaller than -10 MeV. The 34Ar results would highlight the correlation’s effects in the

proton-rich regime [Leel 0].

Large suppression (up to 75%) in SF values compared to shell model predictions
for strongly bound valence nucleons has been observed in one-nucleon knockout
reactions [Gad08] as represented as solid triangles in Figure 1.4. In particular, the neutron

reduction factor R5 of 34Ar is smaller by a factor of 2 compared to that of 46Ar [GadO4-2,

Gad05]. This suggests that the valence nucleon of the deficient nucleon species is much

14

more strongly correlated compared to those of the excess nucleon species [Gad08]. The
trend of strong suppression experienced by the deeply bound nucleon is not found in the
spectroscopic factors obtained in transfer reactions [Tsa05,Lee06] or (e,e ’p)
measurements [KraOl] for nuclei close to stability, where the (e,e ’p) results are shown as
open circles in Figure 1.4. The large asymmetry dependence of correlations in knockout
reactions, however, has not been explained by theories or other global analyses [Bar09-
l,Bar09-2,Bar09-3,Cha06,Cha07]. For example, the dispersive optical model (DOM)

analysis of elastic-scattering and bound-level data on Ca isotopes (4049Ca) suggests weak

asymmetry dependence of proton correlations, where the change in proton SF magnitude

fiom 4"Cu to 48Ca is about 10% [Cha06,Cha07].

The present measurements of the (p,d) reaction on proton—rich 3"'Ar and neutron-

rich 4(V’Ar provide the opportunity to compare the asymmetry dependence of correlation

obtained from transfer and knockout reactions [LeelO]. The conclusion fi'om the present
transfer reaction experiments investigates the compatibility of different experimental
techniques and reaction theories for a unified description of the nature of correlations and

the underlying physics.

15

 

 

 

 

 

lczb1""Ter'Trrr'I"r'I‘
A
2
”1’
CD
1—1
v
ELI
U)
\
A
4.)
Qt
>4
0)
v
% t 32 ‘
0.2r O (e,e'p) Ar _‘
IVA n(p) knockout .
0.01....1....1....l...1L
-20 -10 0 10 20

AS(MeV)

Figure 1.4: Reduction factor (Rs) as a function of the difference of neutron and
proton separation energies (AS). The open circles and closed triangles denote the
Rs obtained from (e,e ’p) and knockout reaction measurements respectively. The
solid line is the averaged Rs from (e,e’p) reaction data, while the dashed line is
the best fit of Rs fi'om the knockout reaction data.

16

 

1.4 Organization of Dissertation

This dissertation is organized in the following way. Chapter 2 introduces the
methodology to extract spectroscopic factors systematically using published angular
distributions of (p,d) and (d,p) transfer reactions. This chapter also presents a survey of
deduced ground- and excited-state spectroscopic factors of Z=3-28 nuclei. The
systematics of spectroscopic factors is used to provide checks of the residual interactions
and Hilbert spaces used in the current nuclear structure calculations. Chapter 3 focuses on

34,36,46

the experiments of the p( Ar,d) transfer reactions performed at NSCL. This is

followed by the descriptions of the calibrations and data analysis in Chapter 4. Chapter 5

presents the cross section and spectroscopic factor extractions for these measurements as
well as the associated uncertainties. In this chapter, the 34Ar and 46Ar data are compared

to large-basis shell-model predictions and the discussion of the underlying physics is

given in great detail. The summary and conclusions are given in Chapter 6.

17

Chapter 2

Spectroscopic factors from transfer reactions

Spectroscopic factors (SF) quantify the nature and occupancy of the single-particle orbits
in a nucleus. Measurements of spectroscopic factors therefore provide quantitative
information about the single particle structure of nuclei in the shell model. In our present
work, we extract the neutron spectroscopic factors of nucleus B in the reactions of
B(p,d)A and A(d,p)B, where nucleus B is considered to be composed of the core A plus
the valence neutron n. The overlap integral between the wave function of one state in
nucleus A and another in B provides the theoretical neutron spectroscopic factor of
nucleus B, which are usually calculated using shell models. The ratio of the measured
cross sections divided by the cross sections calculated with a reaction model defines the

experimental SF for transfer between these states.

Since 1960, (p,d), (d,p) and other single nucleon transfer reactions have been
extensively used to extract spectroscopic information for single nucleon orbits. However,
the published SF values in the literatures for a particular nucleus often varied widely,
more than a factor of 2 in the case of 40Ca(d,p)‘flCa reaction as shown in Figure 1.3, for
instance, reflecting uncertainties and ambiguities introduced by the choice of optical
model parameters and bound state potential parameters employed in the reaction models
[Liu04, Lee07]. To allow comparisons of the experimental spectroscopic factors with
theoretical predictions over a broad range of nuclei, we have adopted a systematic and

consistent approach involving minimal assumptions and have reanalyzed the existing

18

transfer reactions data. A priori, transfer reactions do not yield absolute spectroscopic
factors as the analysis depends on other input parameters such as the geometry of the
neutron bound state wave function as well as the optical potentials used in the reaction
model. However, if the analysis utilizes a consistent set of parameters, the relative
spectroscopic factors can be determined reliably. It should be emphasized that relative
normalized SF values are sufficient to study the evolution of nuclear structure; while the
absolute normalization in SF magnitude is only required to determine how much
correlation remove the single-particle strengths from the Hilbert space in which the shell

model states are degenerated.

This chapter explains the basic principles of both transfer reaction model and shell
structure model. The methodology of extracting consistent experimental SFs with
minimum assumptions is also demonstrated here. The comparisons of the experimental
SFs to the shell-model predictions as well as to the other SF compilations are presented
and the underlying physics is discussed. The results shown in this section have been

published in several papers [Tsa05, Lee06, Lee07, Tsa09, Lee09].

2.1 Reaction theory

The mechanism of direct transfer reactions (A +a-)B+b) can be described by the
distorted-wave Born approximation (DWBA). In this theory, the reaction is regarded as a
perturbation in the elastic scattering that induces a transition to occur between two
channels. The relative motions are governed by an optical potential. The resulting waves,
distorted from the inelastic scattering plane wave, are used to obtain the approximate

transition amplitude which is a nuclear overlap fimction. The model calculates the

19

differential cross sections. It is important to note that the DWBA theory is based on the
assumptions that the reaction is a one-step direct process and the interaction which drives
the reaction is weak enough for the reaction to be treated in the first-order perturbation
theory. Adiabatic approximation can be incorporated in the DWBA framework to take
into account the influence of deuteron break-up. Such approach is designated as adiabatic
distorted-wave Born Approximation (ADWA). This subsection presents the DWBA
formalism and the use of adiabatic approximation in ADWA. Many global optical
potentials have been developed over the past years. Among them, two well determined
and widely used optical-model potentials derived using global and microscopic

approaches respectively are introduced in this chapter.
2.1.1 Adiabatic Distorted-Wave Born Approximation (ADWA)

The basis of the distorted-wave Born approximation for direct transfer reactions
(A +a-)B+b) is the distorted-wave transition amplitude which is expressed as Equation

2.1[Aus70,Sat80,Jac70,KraOl].
T533" = (1962.5) wan/u Iva— ”a1 wwaxft’me‘»

= 1de Id‘r‘xxg" (Em—p?) (Bbiva- UalAa) x93" (Er?) (2.1)

where a (B) is the entrance (exit) channel with projectile (ejectile) a(b) and target (final)
nucleus A(B) and f; (r733 is the relative coordinates between projectile a and A nucleus
(between b and B). Va is the interaction between the projectile, a, and the target nucleus.
1]) A(III 3) is the internal wave function of the projectile (target) nucleus. And

x2) kmfi) is the solution of the Schrodinger equation for the incoming particle with

20

the distorting potential Ua in channel a. The exact wave function of the system is
described by a product of the internal wave functions of the outgoing particle 111b, the

residual nucleus th and a function x§)@,fi§) associated with the elastic scattering of

the outgoing particle from the final nucleus B. It is a common practice to assume that the
term of Va - Ua can be replaced by the interaction between the transferred nucleon and

the projectile nucleus Va”.

The nuclear matrix element (BbIVa- UaIAa), with integration performed over all
coordinates independent of i; and r757 , can be expanded into the nuclear overlap integral

as shown in Equation 2.2 , which carries single-particle state information (n,lj,m), and
into the overlap function between the projectile and ejectile as presented in Equation 2.3.
Eventually the transition amplitude can be expressed in Equation 2.4 and the differential

cross section is proportional to the square of the transition amplitude [KraOl ].
i —h . ....
I “3103(0) $218312) = XanmUBJ M5 mUA MA)‘/Snlj ¢nljm(R ) (2-2)

f (7' ) = MILK, ?)|Van(?‘)l ¢a(Ca)) (23)

1‘59”,“

Lam/3:711 die I diet)?" 07M?) MAE ) f(1" 1x8“ 1‘01th (2.4)

where (I 3 j M 3 ml] A M A) is a Clebsch-Gordon coefficient, ,3 n1, is the spectroscopic

amplitude and <1) III/m0?) is the normalized single-article wave function and f is the

internal coordinate.

21

For more realistic descriptions of the reaction mechanism, the optical potential
should be non-local, which means that the wave function at a point is affected within the
range of non-local potential. This directly affects the wave functions of the projectile and
ejectile in the region where the transfer takes place. The nonlocality corrections have
been obtained by fitting the experimental data [Per62]. In addition, interactions among
protons and neutrons exist in a finite range. Without such consideration in the
calculations, the contributions coming from the nuclear interior are over-emphasized. The
effect of finite-range of the interaction for deuteron can be taken into account

approximately using local energy approximation (LEA) [But64].

With the small separation energy of 2.224 MeV, deuteron breaks up easily in the
field of core nucleus especially at high incident energy. This would change the total
reaction cross sections and consequently the amplitudes of spectroscopic factors. Johnson
and Soper extended the DWBA to include the effects of the breakup of the deuteron in
the field of the target and of the transfer of the neutron into (or out of) the breakup
continuum [Joh70]. The use of adiabatic approximation to the neutron, proton, and target
three-body system is designated as Adiabatic Distorted-Wave Born Approximation
(ADWA). In the Johnson-Soper (JS) adiabatic approximation, the effective nucleon-
nucleus interaction is formulated as the sum of the nucleon optical-model potentials

evaluated at half of the deuteron incident energy as shown in the following equation 2.5:

udrk‘) = Diojrun (ii + i?) + 11,,(13 — gr) } vpnrr‘) cor?) d? (2.5)

22

 

where Up and Ud are the optical-model potentials for proton and deuteron respectively
and VP" is the interaction between proton and neutron, while (P0 represents the internal

wave function of deuteron. Do is strength of the Reid sofi-core 3S1-3D1 neutron-proton

interaction.

2.1.2 Global optical model potential

The optical-model potential is the main model-dependent ingredient in the transfer
reaction model. The local optical-model potential for nucleon-nucleus scattering can be

written in the Woods-Saxon form [Aus70]:

d
”(7') = "Vrf(T:RVoaV) " ier(r,Rw,aW) ‘1' 4iwsaw —f(T:Rmaw)
dr

 

1 d
+ 2 (Vso+ iWso) (;E;f(r.Rso. 050% ° 0) + Vc (2-6)
where
2
Za—ZLe—’ r 2 RC
f(r R a) = 1 V = r
ZRC EC? ' C

Here, R is the nuclear radius and a is the diffuseness; V and Wrepresent the depths of the
real and imaginary potential with the subscript r, s, so and c corresponding to volume
term, surface term, spin-orbit and coulomb respectively. L is orbital angular momentum
of relative motion of the scattered particle and a is the spin operator. Re is the radius of

nucleon charge distribution, which is taken to be a constant density for r < Re.

23

In the past, the optical model potential parameters for specific nucleus-nucleus
reactions were obtained from fitting elastic scattering data for the relevant entrance and
exit channels. However, such approach becomes difficult when radioactive beams are
involved where beam time for elastic scattering is often not available or when a large range of
beam energies and nuclei is needed in systematic studies. In addition, the optical potential
parameters cannot be accurately determined from the data at a single incident energy. To
be more accurate and to apply the analysis to a wide range of target-nucleus and
bombarding energies, it is desirable to use global optical model potentials which is
determined by fitting a large amount of data over a broad range of nuclei and energies, or
a microscopic global optical potential which is derived from the relevant nuclear densities
using effective Skyrme interactions. In the following section, we will briefly discuss two

widely used optical-model potentials in transfer reactions.
Chapel-Hill s9 (CH89)

Chapel-Hill 89 (CH89) is a parameterization of the nucleon-nucleus optical model
potential for nuclear mass range 405A _<_209 and nucleon laboratory energy range
103E565 MeV [Var91]. It is obtained by fitting simultaneously a large database of nearly
300 proton and neutron differential cross sections and analyzing powers. CH89
parameterization is determined based on the current understanding of the basis of optical
potential, such as folding model and nuclear matter approaches. Equations 2.7 -2.10
summarize the parameterization and features of CH89 optical model potential and Table
2.1 lists the parameters of CH89. Of many global optical model potentials, it has been

shown that CH89 potential best describes the (d,p) experimental data [Liu05, Nun08]

24

1 1
RV = TVA? + r60) , Rw = TWA? + r63)

1 1
R50 = T50143- + T}? , RC = 71-143 '1' rem) (2.7)
N.
V0“) = V0 i Vt ('72) + (E — Ec)Vc (2-8)
W

WV")=1+exp 419%.; “)
W80 iWst (EA-2'2)

 

W80) = E—E -Wse0 (23)
1+ exp(§—v§L—sew )
2
Ec = -6-5-Z—:c— for proton; EC = 0 for neutron (2.10)

Table 2.1: Summary of parameters for CH89 global optical model potential

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameters
v0 52.9 MeV vso 5.9 MeV.fm2 wst 18 MeV
vt 13.1 MeV rso 1.34 fm wseo 36 MeV
v, 0.299 MeV rs,” -1.2 frn wsew 37 MeV
rv 1.250 frn 3,, 0.63 frn rw 1.33 frn
rvw) -0.225 frn wv0 7.8 MeV rw(°) -o.42 fm
av 0.690 fin wveo 35 MeV aw 0.69 fm
r, 1.24 frn wvew 16 MeV
r5") 0.12 fin wso 10.0 MeV

 

25

JLM Optical-model potential

To include microscopic description of the nuclear interior, a set of optical-model
potential parameterization is derived from the nuclear matter effective nucleon-nucleon
interaction of Jeukenne, Lejeune, and Mahaux (JLM) [Jeu77] based on the Brueckner-
Hartree-Fock approximation and Reid’s hard core interaction. This approach can be
applied consistently for nuclei with mass numbers 12 S A S 208 and for energies up to

160 MeV. Equation 2.11 gives the real and imaginary JLM potential respectively:

V50) = 4v (by/'1? )'3 2;? I 90’) exp (- LEE] )d3r’
W5(r) = Aw (bx/E )‘3 VII-1%) f p(r’) exp 6%] )d3r’ (2.11)

The range of effective interaction b is set to be 1.2 fm. We adopt the conventional

scale factors of AV = 1.0 and AW = 0.8 which are consistent with an analysis of data on

several systems [Pet85]. V5 and WE are the real and imaginary nucleon potential derived
in the local density approximation (LDA), where the LDA potential with energy E in

uniform nuclear matter with density p and asymmetry 8 is expressed in Equation 2.12.

1,110,157: V0 (p,E) + 6V1 03,5) » Wn(p,E) = W0 (P,E) + 6W1 (p,E) (2'12)

where

p, -p,,

6':
Pn'l'Pp

 

The resulting energy- and density-dependent effective interactions are folded with

the target one-body densities, using the mid-point local density prescription [Pet85].

26

These required densities are taken from Hartree Fock (HF) calculations based on a
Skyrme parameterization that offers quantitative agreement with experimental nuclear
size parameters. Specifically, we use the recent SkX parameter set [Bro98], determined
from a large set of data on spherical nuclei, including nuclei far from stability. The
parameter set accounts for the binding energy differences of mirror nuclei [BroOO],
interaction cross sections [Br001-2], and nuclear charge distributions [R1003]. The
aforementioned agreement of the systematics of the Skyrme SkX HF predictions with
nuclear size parameters suggests that this theory will also give a good description of
individual single-particle states. The computed neutron and proton HF densities were
used individually in evaluating the isovector contribution to the JLM optical potentials

[Bro99].

2.2 Methodology of extracting experiment spectroscopic factor

The purpose of this section is to explain the criteria that we used in the data evaluation
and the quality control that we applied to the existing cross sections data in transfer
reactions. Furthermore, we explain the procedure we used to extract a consistent set of
spectroscopic factors from (p,d) and (d,p) transfer reactions. This section is organized as
follows. We begin in sub-section 2.2.1 with a brief description of the input parameters
used in ADWA three-body adiabatic reaction model. This is important because
spectroscopic factors are usually extracted by dividing the measured differential cross
sections by theoretical cross sections predicted by a reaction model. We then explain in

sub-section 2.2.2 how the data have been compiled and the uncertainties introduced in the

27

process. We explain in sub-section 2.2.3 the procedure for extracting the SFs. Problems
with consistencies between measurements and analysis with too low and too high
incident energies are discussed in sub-sections 2.2.4 and 2.2.5 respectively. Sub-section
2.2.6 deals with the internal consistency of the approach. As the pickup (p,d) reaction is
the inverse of the stripping (d,p) reaction, from detail balance, the ground state SFs
obtained separately by the (p,d) and (d,p) reactions should be the same within
experimental uncertainties. We use this fact to assess the consistency of our method and

to assign uncertainties to the extracted spectroscopic factors.

2.2.1 Input parameters in reaction model

The transfer cross sections are calculated within the Johnson-Soper adiabatic
approximation [Joh70], which approximates the full many-body system by a three-body
system consisting of a neutron, a proton, and an inert core. The core would be the target
in a (d,p) reaction or the final nucleus in a (p,d) reaction. The phenomenological proton-
and neutron-optical model potentials (CH89) [Var91] are folded to construct the deuteron
optical potential that is used in ADWA model. By using the folded potential instead of a
phenomenological deuteron optical potential, one includes the main corrections to the

transfer cross section from the breakup of the deuteron in the field of the target.

Even though the breakup effect is mainly important for energies above 15 MeV
per nucleon, to be consistent, we constructed the deuteron potential using the Soper-
Johnson approach at all incident energies. Similar results are obtained if DWBA is used
at low energy. The potential of the transferred neutron to the inert core was chosen to be
Woods-Saxon in shape, where we used the widely adopted values of the fixed radius
parameter of 1.25 fin and diffuseness parameter of 0.65 fm [Liu04, Liu05]. The depths of

28

 

the central potential wells are adjusted to reproduce the experimental binding energies.
Consistent with the findings of Ref. [Ver94], we find that the surface properties of the
neutron bound state wave function are dominated by the central potential. Thus, we have
neglected for simplicity the spin-orbit interaction in constructing the valence neutron
wave function. (We have studied this effect. In the light nuclei, 4 5 Z S 24 studied here,
the effect is of the order of 10% or less. Such effect may become important for heavier

nuclei.) All calculations adopt the local energy approximation (LEA) for finite range

effects [But64] using the zero-range strength (Doz=15006.25 MeV2 fm3) and range

=O.7457 fm) parameters of the Reid sofi-core 3S1-3Dl neutron-proton interaction

[Knu75]. Nonlocality corrections with range parameters of 0.85 fin and 0.54 fm are
included in the proton and deuteron channels, respectively [Per62]. The same set of input
parameters is used for all the reactions analyzed here. We label our SF values as
SF(ADWA) in our figures, to distinguish them from other SF values obtained when
different input parameters or potentials are used. The transfer reaction calculations were
carried out using the University of Surrey version of the code TWOFNR [Iga] which
respects the detailed balance between (p,d) and (d,p) reactions that connect the same
states. The code TWOFNR is chosen mainly for convenience as it contains all the input
options discussed below. With the same input parameters, we have compared the
calculations from two other widely used reaction model codes, DWUCKS and FRESCO,
and find that they provide predictions that are basically the same as those provided by

TWOFNR [Kee04, De105, Liu05].

29

2.2.2 Compilation and digitization of angular distribution data

Nearly all the existing angular distributions we used have been digitized from the
published figures [Lee07]. The few exceptions are some tabulated data found in the
Nuclear Science References (NSR) database of the National Nuclear Data Center
(NNDC) [NSR]. The sources of these data came from the Former Soviet Union or Japan
whose journals are not widely available in the United States. These non-US and non-
European data complement our search in the Physical Review, Physical Review Letters,
Nuclear Physics and occasionally in Physics Letters and Journal of Physics G. While we
have made an effort to find all the relevant experiments that published the absolute
differential cross sections, we could have missed some reactions especially if the incident

energy is below 10 MeV or above 70 MeV.

By checking some of the data carefully and sometimes repeating the digitization
several times, we estimate the uncertainties introduced by the digitization process to be

less than 0.5 deg in determining the angles and less than 10% in extracting the differential
cross sections. For illustration, we use the data for the reaction MN(d,p)15N at Ed=12

MeV [Sch67, Hef77]. This set of data was first published in tabulated form in ref.
[Sch67]. The tabulated data are plotted as closed points in Figure 2.1. Later the authors in
ref. [Hef77] plotted the data in a figure, which we digitized. We compare our digitized
data (open points) with the tabulated data (closed points) in Figure 2.1. We see a
difference of less than 10% between the two sets of data. Of course, the digitization
errors also depend on the actual size of the graphs available in the original literature. As
described later, generally, errors introduced by digitization are relatively small compared

to the uncertainties in the absolute cross section measurements.

30

 

."’
O

     

dU/dfl (mb/sr)

  

14N(d’p)15N
Ed=12MeV

0 10 20 30 4O 50 60
96.... (deg)

.0
g-s

      

 

 

 

Figure 2.1: Comparison of tabulated data (closed points) [Sch67] and digitized
data (open points) [Hef77] from the same measurement of the angular
distributions of the protons obtained in the 14N(d,p)15N reaction at incident
deuteron energy of 12 MeV. The curve is the predicted angular distributions from
the code TWOFNR as described in the text, multiplied by 1.12 which is the
spectroscopic factor.

2.2.3 Extraction of spectroscopic factors

The experimental SF value for transfer reaction is defined as the measured cross-section

divided by the cross section calculated with a reaction model. For most of the nuclei we
calculated, we use the E values determined from the angular distributions and the jJr

values of the valence neutron ground states found in the isotope tables [NNDC]. In
general, the experimental angular distributions at larger angles are more sensitive to
details of the optical potential, the effects of inelastic couplings and other higher order
effects that are not well reproduced by most reaction models. Furthermore, discrepancies
between the shapes from calculations and experiment are much worse at the cross section
minimum, which could give these points an unduly large weight in a least squares
minimization procedure. Thus, we follow the procedures used by many groups in the past

31

40 years that the spectroscopic factor is extracted by fitting the reaction model
predictions to the angular distribution data at the first peak, with emphasis on the
maximum. The accuracy in absolute cross section measurements near the peak is most
important. When possible, we take the mean of as many points near the maximum as we

can to extract the spectroscopic factors. We will use the angular distributions of
l4N(a',p)15N shown in Figure 2.1 to illustrate the procedure we adopt to extract the

spectroscopic factors.

In Figure 2.1, the first three data points with 0cm<25° have been used to determine

the ratios of the measured and calculated differential cross sections. The mean of these
three ratios is adopted as the spectroscopic factor. For example, for the two sets of data
plotted in Figure 2.1, the spectroscopic factors are 1.1 and 1.2 for tabulated data [Sch67]
and digitized data [Hef7 7] respectively. The difference in the spectroscopic factors
represents the uncertainties introduced by digitization. The theoretical angular
distributions, obtained from TWOFNR have been multiplied by the spectroscopic factor,

1.1, and plotted as the solid curve in the figure.

In cases when a “first peak” is not obvious or that the angular distributions of the
forward angles are nearly flat, e. g. in the reaction of 44Ca(p,d)43Ca at Ep=40 MeV
[Mar72] as shown in Figure 2.2, we find that fitting the shoulder gives more consistent
results. In general, the agreement of the measured shape of the angular distributions in the
vicinity of the first peak or the shoulder to the shape predicted by the transfer model gives
some indication as to the quality of the spectroscopic information that can be extracted by

comparing the model to data. When there are more than one set of data that can be used

32

to determine a given spectroscopic factor, we use the number of measured data points
from a given measurement that lie in the peak or shoulder region where data and theory
are in good agreement to assign a relative weight for the SF extracted from that
measurement. Various SF ’s extracted from different measurements are combined in a

weighted average to compute the mean spectroscopic factors presented here.

 

 

 

 

 

10:0 I I I I
’1:
VJ
\
.o
E
61.0; T.
I'U D
E
"‘3 “Ca(p,d)43Ca
Ep=4OMeV
0.1 l J. ..... 1
o 10 20 30 4o 50 60
9cm (deg)

Figure 2.2: The angular distributions of the deuteron obtained in the

44Ca(p,d)43Ca reaction at incident proton energy of 40 MeV [Mar72]. The curve

is the predicted angular distributions fi'om the code TWOFNR as described in the
text, multiplied by the spectroscopic factor.

2.2.4 Evaluation of the angular distribution measurements

Even though most published papers state the uncertainties of their cross section
measurements to be 10-20%, the actual disagreements between experiments are ofien
larger than the quoted uncertainties. An example is illustrated in the reactions 1]B(d.p)1213
reactions. From the literature, we find two measurements: one measurement at deuteron
incident energy of 11.8 MeV [LinI] and another measurement at 12 MeV [Sch67]-

Since the incident deuteron energy is nearly the same, one would expect the angular

33

distributions from the two data sets plotted in Figure 2.3 to be the same within
experimental error. Ref. [Sch67] (open circles) stated that the accuracy of the absolute
cross section measurements is 15% while ref. [Liu01] (closed circles) quoted an error of
6%, which is smaller than the symbols in Figure 2.3. Not only do the cross sections differ
sometimes by a factor of two, the shapes of the distributions (especially the first peak) are
not even the same. In this case, the shape of the angular distributions in ref. [Liu01]
agrees with the calculation (solid curve) better than that measured in ref. [Sch67].
Fortunately for this reaction, we are able to find another measurement in the NNDC
database [Fic74] (open diamonds). Near the peak at forward angles, this latter angular
distribution agrees with ref. [Liu01] and so we disregard the measurements of ref
[Sch67]. Data in ref. [Liu01] were measured nearly 40 years later than data in ref.
[Sch67] and one might be tempted to attribute the difference to the availability of better

beam quality and detection systems for the measurements.

 

 

10.0
’1?

tn
\ .
lo 1
5

CLO 7 1
1'0 >
E

'U “Ca(p,d)‘3Ca

Ep=40MeV

 

 

 

0.1 r A r J L 1
0 10 20 3O 40 50 60
9m. (deg)

Figure 2.3: Comparisons of the angular distributions llB(d,p)lzB reactions. Open
circles, closed circles, open diamonds represent data from refs. [Sch67], [Liu01]

and [Fick74] respectively. The curve is the predicted angular distributions from
the code TWOFNR normalized by the spectroscopic factor.

34

However, when another reaction,12C(d,p)13C at Ed=l 1.8 MeV from ref. [Liu01]

(closed circles) is compared to three other published angular distributions in Figure 2.4 at
Ed=11.8 MeV (closed diamonds) [Sch64], 12 MeV (open circles) [Sch67], 12 MeV (open
diamonds) [Lan88], the cross sections in the first peak measured in ref. [Liu01] is
consistently low. No uncertainties in the measurements are given in ref. [Sch64] and ref.
[Lang] but it is clear that data in ref. [Liu01] do not agree with the other measurements,
especially in the most forward angle region. Thus we disregard the SF values derived
from ref. [Liu01] in our compilation of 12C(d,p)13C reactions. The authors of ref [Liu01]
cannot explain the discrepancies described here [Liu-priv]. In general, data taken by the
same group with the same setup have similar systematic errors that lead to rejection of
the entire data set. When there are independent measurements available for comparison,
however, cross comparisons to other data can allow one to be more selective. The

existence of confirming data, allowed us to keep 11B(d,p)12B data and discard the

12C(d,p)l3C data even though both sets of data come from ref [Liu01].

35

 

N
O

do/dfl (mb/sr)

 

 

 

 

Figure 2.4: Comparisons of the angular distributions of the proton measured in the

12C(d,p)]3C reactions in four different experiments. Open circles and closed
circles, open and closed diamonds represent data from refs. [Sch67], [Liu01],
[Lan88] and [Sch64] respectively. The curve is the predicted angular distributions
from the code TWOFNR multiplied by the spectroscopic factor.

Cross comparisons of angular distributions sometimes help to establish common

systematic problems when one set of measurements was performed by the same group
with the same set up. An example is illustrated in the 40Ca(d,p)‘flCa reactions in ref.
[Lee64] where the ground state angular distributions of 41Ca at E; =7, 8, 9, 10, 11 and 12
MeV have been measured. Figure 2.5 shows the extracted spectroscopic factors (labeled
as SF(ADWA)) as a function of incident deuteron energy for all the 40Ca(d,p)“Ca
reactions. For clarity in presentation, no error bars are plotted. Except for the point at
Ed=7 and 12 MeV, the extracted spectroscopic factors from ref. [Lee64] (open circles)

are consistently larger than the spectroscopic factors extracted from other experiments
that probed the same reaction at the same energy. Detailed comparisons of the angular

distribution data show essentially the same effect, that the differential cross sections

36

measured in ref. [Lee64] are systematically higher than the other measurements [Sch64,
Kat65, Hjo65, Nie66, And68, F0869, Set70, Koc7l, Bro74, Han75, Boy76, A1178]
measured by different groups. Clearly, there were problems in the determination of the
absolute cross sections in ref. [Lee64]. As it is not possible to find the cause of this
discrepancy after so many years, we disregard the spectroscopic factor values determined
in ref. [Lee64] in our review of the data. The disagreements between data sets generally
exceed the quoted uncertainties of the data. Indeed, we have found that the most
important aspect of quality control of the data is to have as many independent
measurements as possible. Comparisons of different measurements help to identify

problematic measurements.

 

 

 

 

 

1.5 ' I ' r #1—‘\"‘
i e 0
e
. l
A . O 0
g 0 O . 8. e .
g 1.0 ~ 0 :trfi
Eh
(I) .
e
l t
0.5 - . - - 4 ‘\
10 20 \56
Ed(MeV)

Figure 2.5: (Color online) Comparison of spectroscopic factors, SF(ADWA),
obtained from Ref. [Lee64] (open circles) and from other measurements (closed
circles). The increase of spectroscopic factors observed at Ed<10 MeV has been
observed before [Liu05, Sch64] and has been attributed to the resonance

structures in the elastic scattering of the deuterons [Ohl63]. The solid line is the
mean SF(ADWA) between 10 and 56 MeV.

37

2.2.5 Transfer reactions at high and low energy

When Q-value, the momentum transfer or angular momentum transfer are not well-
matched or when there are significant contributions from the compound nucleus, the
shape of the experimental angular distributions may be poorly described by theory. We
find better agreement for ground state transfers at incident energies of around 10-20 MeV

and poorer agreement at very low or high (> 50 MeV) beam energies. Figure 2.6 shows

the angular distributions of protons emitted from the 40Ca(d,p)“Ca (g.s) reaction from

Ed=4.7 to 56 MeV. Only one angular distribution is shown at each incident energy. The

agreement between data and prediction for the first peak improves with increasing
energy. At very low incident energy, the shapes of the measurements and the calculated
transfer cross sections do not agree. This phenomenon is also seen in other reactions. The
spectroscopic factors as a function of incident energy are shown in Figure 2.5. The
increase of spectroscopic factors at Ed<10 MeV has been observed before [Liu05, Sch64]
and has been attributed to the resonance structures in the elastic scattering of the

deuterons [Ohl63]. As explained in the last section, the open points based on the data

from ref. [Lee64] are discarded. Between 10 to 56 MeV, we find that the mean
spectroscopic factor, 1.01 :1: 0.06 shown by the solid line in Figure 2.5, describes the data

at all energies within experimental errors.

38

 

105 ...l...., '1 ...l. .

...s
O
4;

do/dfl (mb/sr)

 

 

 

 

 

1 0 °
0 40 0
0 cm. (deg)
Figure 2.6: Angular distributions for 40Ca(d,p)MCa reactions for beam energy

from 4.69 to 56 MeV. Each distribution is displaced by factors of 10 from

adjacent distributions. The overall normalization factor is 10 for the 7.2 MeV
data. References are listed in ref.[Lee07].

In reactions which have large negative Q values such as 12C(p,d)”C (Q = —l6.5
MeV), the center of mass energy available in the exit channel is very small even at ~20
MeV proton incident energy [Win01]. The validity of the calculated angular distribution
is questionable at these energies and we discard these data. For other reactions measured
at low incident energy (<10 MeV), the data could be dominated by compound nucleus
emissions, or resonances in the low energy elastic scattering [Ohl63]. When possible, we
exclude spectroscopic factors obtained with incident beam energy less than 10 MeV

when computing the mean values of the spectroscopic factors.

39

Even though we exclude data with incident energy lower than 10 MeV fiom the
calculation of the mean SF, these low energy data are still valuable. In cases where very

few (sometimes only one) measurements with incident energy greater than 10 MeV are
available, they provide checks for consistency of the measurements. In the 43Ca(d,p)‘MCa
reaction, only data at 8.5 MeV [Bje67] are available. Similarly, we only have data at 7.5

MeV for the 50V(d,p)51V reaction [deL67] and at 7.83 MeV for the 23Na(d,p) 24Na

reaction [Dau63] . We adopt these results despite their low incident energies.

At high energies, momentum transfer and angular momentum transfer are
mismatched so conditions may not be optimized to extract reliable spectroscopic factors.
Furthermore, the global nucleon-nucleus potentials (CH89) [Var91] are fitted only to 65
MeV for protons and to 26 MeV for neutrons. Thus, we do not include data fiom
reactions at incident energy greater than 65 MeV in this work. In examining data over a
wide range of d or p incident energies, we find that the optimum beam energies for

studying transfer reactions lie between 10-20 MeV.

2.2.6 Uncertainty assignment to the extracted spectroscopic factors

Since the ground-state transition of neutron pickup (p,d) and neutron stripping (d,p)
reactions are inverse reactions, they should yield the same values for the spectroscopic
factors. We used nuclei which have been studied reasonably well by both neutron pickup
and stripping reactions to the ground state. The averaged SF values are listed in the
second and fourth column of Table 2.2. The numbers of measurements contributing to the

averages are listed next to the mean values in the third and fifth column.

40

There are strong correlations between the ground state neutron spectroscopic
factors determined from the (p,d) and (d,p) reactions as shown in Figure 2.7. The solid
line corresponds to perfect agreement. These are independent values determined using the
procedure outlined above. The scatter of the data points about the solid line can be used
to determine an overall uncertainty for such analyses. As explained in Section 2.3.4, the
quoted experimental uncertainties are not always reliable. In the absence of a completely
independent criterion that can be applied to each data set without comparison to others,
we assume the uncertainty of each measurement to be the same, even though data from
certain experiments may actually be more accurate than others. If we require the chi-
square per degree of freedom of the scatter data shown in Figure 2.7 to be unity, we can
extract a random uncertainty of 20% for a given measurement. The extracted uncertainty

of 20% is consistent with comparisons with analysis on systems that have large number
of measurements such as 12C(d,p)]3C, l60(p,ct')150, 160(d,p)l70, 4oCa(d,p)‘flCa and
other reactions. In Table 2.2 and Figure 2.7, we have excluded measurements for 7Li, 34S

and 10Be nuclei due to large errors associated with either the (p,d) or (d,p) measurements.

If we include these three measurements, the estimated uncertainty in a given

measurement increases to 28%

41

Table 2.2: List of nuclei with spectroscopic factors obtained from both (p,d) and (d,p)
reactions. di and Ndp denote the number of (p,d) and (d,p) independent measurements
included in the analysis.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B Baum di A(d,p)B Nd, B 30th NM A(d,p)B Nd,
”Be 0.57 1 0.49 2 3°51 0.87 1 0.79 2
”B 1.29 1 1.55 3 “Ca 2.12 2 1.82 3
13c 0.81 4 0.73 12 “Ca 0.63 1 0.63 2
14c 1.50 3 1.82 2 “ea 3.93 3 5.14 1
”N 1.65 2 1.39 4 “Ti 0.11 3 0.14 1
”o 0.81 3 0.99 10 49“ 0.26 2 0.23 1
18o 1.60 2 1.80 1 son 5.50 2 7.12 2
”Ne 0.03 1 0.04 2 51v 1.10 2 1.58 1
261.4,; 2.80 3 2.01 1 53c, 0.37 1 0.39 8

I I I I T [:51
5.0 - 1.1
AZ-O ' T .:T 01 ..
91‘ boggy].
,0 .
v 1_0 - to -
In. 2 .3—1
(I) I——*--i
0.5 - e-i .
1— -1 *
.1
0.2 T 1 ~
.1
0.1 l I P l I
0.1 0.2 0.5 1.0 2.0 5.0
SF(p.d

Figure 2.7: Comparisons of spectroscopic factors obtained from (p,d) and (d,p)
reactions as listed in Table 2.2. The line indicates perfect agreement between the
two values.

42

As mentioned in sub-section 2.2.3, the final SF values are obtained from the
weighted average of independent measurements from both (p,d) and (d,p) reactions. In
these weighted averages, the low energy (<10 MeV) data and the inconsistent data
(nominally marked with asterisks) that differ significantly from other sets of data are
excluded. For values determined from only one measurement without the consistency
checks provided by other independent measurements, an associated uncertainty of 28% is
assigned. For values determined by more than one measurement (N), we take into
account the distribution of the SF’s around the mean. Figure 2.8 illustrates this procedure.
The open stars in Figure 2.8 represent the spectroscopic factors extracted from the good

measurements of the calcium isotopes. However, the spread of the data are more than
20% for the 44Ca and 48Ca nuclei even though three “good” measurements are found for
each of these nuclei. For these nuclei, it is more realistic to assign the uncertainty using

the standard deviations of the mean of the data points. Each of the associated

uncertainties is determined by adopting either the standard deviation of the mean or an

uncertainty of 20%/W, depending on which of the two uncertainties is larger. For
comparison, the mean SF values with the associated uncertainties are plotted as the solid

stars with error bars in Figure 2.8.

43

 

 

 

 

10.0 1 r 7 1 . l ' 1g; 1 j
50.? *1 WL .
gr .
A20’ at:
‘3‘
910 0*
E305 1% 3‘
0) ° a;
air
0.2
0'139 41 ‘ 43 ‘ 45 I 47 J 49

ACa

Figure 2.8: Spectroscopic factors obtained for the Ca isotopes. The open stars
represent individual measurements. The accompanying solid stars are the
weighted averaged values with the associated uncertainties determined from the

standard deviations or 20%/WV, of the mean SF values whichever is larger.

2.3 Theoretical spectroscopic factor

Mathematically, spectroscopic factor is defined by a matrix element between the initial
state in the entrance channel and the final state in the exit channels [BroOS]. For an
B(p,d)A reaction, this matrix element evaluates the degree to which the wave function of

the initial nucleus B can be described by the final nucleus A plus a neutron in a specific
single particle orbit. The wave function for a specific state 111,-” in initial nucleus can be

expanded in terms of a summation over the complete set of states 112’; in final nucleus:

up? = 2,916»? (2.13)

The (I); is the overlap function between the wave function of one state in nucleus A (104)

and another in B(¢3). The normalization of overlap function defines the theoretical

spectroscopic factor between these states:
SF = 10%;)? 617' , where 011?) = < w}! up}? > (2.14)

The simplest way to compute the wave function is to use independent particle
model (IPM). IPM refers to the description of a nucleus in terms of particles in the orbits
of a spherically symmetric potential. The particles in the last orbit are called valence
nucleons. These nucleons are non-interactive except that maximum number of valence
nucleons are paired up. Thus nuclei with even number of valence nucleons always have
spin zero. Nucleons in the cores collectively produce the mean-field potential. This mean-
field potential results in the well-known magic numbers (e.g. 2, 8, 20, 28 etc) for stable
nuclei and the extreme single-particle values for the spectroscopic factors. This model

works well for some nuclei, especially with valence nucleons near a closed shell.

In general, spectroscopic factors for nuclei with even number of valence neutrons
generally exceed those of the neighboring nuclei with odd number of valence neutrons.
This results from the pairing interaction, which couples pairs of neutrons to spin zero
similar to the Cooper pairs in a superconductor. For nuclei in the vicinity of a closed
shell, this trend can be well replicated by calculations that consider only pairing
modifications to the independent particle model (IPM). Assuming maximal pairing
(minimum Seniority), one can obtain a simple relationship between the spectroscopic

factor and the number of valence nucleons (n) with total angular momentum j [Aus70] as

45

shown in Equation 2.15. The success of this Eq. 2.15 is illustrated by the ground state SF

of Ca isotopes discussed in details in section 2.4.1.

SF=n ,forn=even; SF=1—£.'T11-,forn=odd (2.15)

In reality, however, most states are an admixture of the particle-hole
configurations and therefore the residual interaction between nucleons is needed in the
shell model calculation. The residual interaction introduces some long-range range
correlations in the nuclear wave function beyond the mean field and can result in the
fragmentation in the single-particle strength. Modern large-basis shell-models (LB-SM)
were developed to take into account the shell-model configuration mixing and construct
the wave function with different effective interactions in a specific Hilbert space. The
effective interactions are usually obtained by fitting the two-body matrix elements and
single-particle energies to the experimental energy levels, empirical nuclear masses,
charge radii and low-lying excited state spectra for a wide range of nuclei within the
region being studied [BroOl]. In our work, we used Oxbash [Br004], the LB-SM code, to

calculate the wave functions and theoretical spectroscopic factors.

2.4 Survey of spectroscopic factors (SF) using CH89 global optical

model potential

About 640 neutron spectroscopic factors for ground-state and excited-state nuclei have
been extracted by systematically analyzing more than 2500 measured (d,p) and (p,d)

angular distributions. The systematic analysis as discussed in section 2.2 adopts global

46

 

optical-model potential CH89 and radius of transferred neutron orbital rms of ro=l .25 frn.

We compare the extracted spectroscopic factors to values predicted by large-basis shell-
model (LB-SM). The survey of spectroscopic factors obtained over a wide range of
nuclei provides important benchmarks and consistent analysis framework for neutron-
transfer reaction studies. In addition, the spectroscopic factors provide an independent
evaluation to the effective interactions used in shell models. In section 2.4.1, the
extracted spectroscopic factor values are listed and compared to the LB-SM for ground-
state and excited-state in sd— and fp- shell nuclei respectively. In section 2.4.2 and section
2.4.3, the extracted spectroscopic factors values will be compared to other compilations

from Endt and ENDSF [End77, ENDSF].

2.4.1 Comparison with shell model predictions

2.4.1.1 Ground-state SF for Z=3-24 nuclei

The ground state neutron spectroscopic factors for 80 nuclei ranging in Z fiom 3 to 24
have been extracted by analyzing the past measurements of the angular distributions from

(d,p) and (p,d) reactions [Tsa05, Lee07]. Table 2.3 lists the theoretical and experimental

SF values determined for 80 nuclei, from 6Li to 55Cr, studied in this work.

The thin bars in Figure 2.9 show the predictions of Equation 2.15 as a firnction of
the mass number A for the transfer of an fm neutron to or from Ca isotopes; the extracted
neutron SF’s are represented by star symbols. The excellent agreement reflects the fact
that configuration mixing of fm neutrons outside the double magic 40Ca core is well

described by a pairing interaction, with little discemable contribution from core

47

~. ’9 fi ..

’1

polarization or higher lying orbits. Most nuclei display more significant configuration
mixing than these Ca isotopes. Figure 2.10 compares experimentally extracted SF ’s for
nuclei with predictions from the independent particle model plus pairing. Open symbols
represent odd Z elements and the closed symbols represent even Z elements. The solid
line indicates perfect agreement. Most extracted values are less than the predictions of the
IPM-plus-maximal-pairing as represented by Equation 2.15. This is expected because the
IPM neglects many important correlations between the nucleons in orbits near the Fermi
energy as well as nucleon-core and short-range corrections. In the events that other

correlations or interactions are negligible, the LB-SM predictions will be the same as the

 

 

 

 

 

predictions from IPM.
I T I I 1 fl
.1
5 0 r r
A _ i-
<t:
g 2.0 - a .
<3
V
LII! 1.0 r ‘h 1
U) __ .
{r I
0.5 r -- ‘
i :1?“ f .
* Data
- shun 1160.11 ‘1'
0.2 ‘ ‘ ‘ ‘ ‘ 7
41 43 A 45 47 49
Ca

Figure 2.9: Ground state neutron spectroscopic factors for Calcium isotopes with

valence neutrons in the fig orbit, star symbols represent SF values extracted from
present analysis. Thin bars are IPM values and thick bars represent predictions
from LB-SM using the program Oxbash.

48

 

' .981 AP As T .
A PVCI VAr I
<2: mt *Ca I ;
32.0.05c ori ‘ I. 4
Q ICr
<3 i
V
£141.07 '1
U) i T T T l
0.5 ‘ “
. T f T .
T
. 1'. I I
0.2 g . A A - - l l +i A L
0.2 0.5 1.0 2.0 5.0
SF(IPM)

 

If"rr

ioBe 1:113 IC ON 00

 

 

 

49

Figure 2.10: Comparison of experimental spectroscopic factors to predictions
from the IPM. Open and closed symbols denote elements with odd and even Z
respectively. The three different colors of green, blue and red represent Z=3-8, 9—
18 and 19-22 isotopes respectively. The solid lines indicate perfect agreement.

If one diagonalizes the residual interaction within a LB-SM [BroOl], that involves
the mixing of several different orbitals in the shell model basis that are close to the Fermi
energy, one can obtain a better description of nuclei. Using Oxbash code with the PPN,
SPSDPF, SDPN, SD, and FPPN model space and the corresponding CKPPN, WBP,
WPN, W, and FPBPPN interactions [BroO 1 , Br004] as input, the ground state neutron
SF’s for 74 nuclei have been calculated with uncertainties of about 10-20% [BroOl].

Their predicted values are listed in Table 2.3 and plotted in Figure 2.11. (In Figure 2.11,
we exclude the deformed 24Mg, Li, F and Ne isotopes, some of which have small
calculated or measured SF values, which, in general, tend not to be accurate.) In contrast

to the IPM-plus-pairing calculations, the agreement between data and LB-SM predictions

are within 20% for most cases, as indicated by the two dashed lines. This shows that the

correlations between nucleons in orbits near the Fermi energy included in the LB—SM

41-48

calculation are the most important one to consider. For the Ca isotopes, there is close

agreement between LB-SM predictions and IPM values as shown in Figures 2.9.

 

 

 

 

:OBe 1:113 IC ON 00 XI.“
50L°Na ”180A! " .
' , .81 AP ‘8 T y .
A b VCI VAT x”:
4 fix *Ca I "
32.01-080 0T1 . 11:1 -
a .Cr T I, Xi
V . ,
12141.0 r ’ ‘
(I) i T”. 2 .
0 5 " ‘
. . . - 3t
0.2 - I- A I a - - - I l - . 1 - x a
0.2 0.5 1.0 2.0 5.0
SF(LB-SM)

Figure 2.11: Comparison of experimental spectroscopic factors to predictions
from the LB-SM. Open and closed symbols denote elements with odd and even Z
respectively. The three different colors of green, blue and red represent Z=3-8, 9-
18 and 19-22 isotopes respectively. The solid lines indicate perfect agreement.
For reference, the two dashed lines indicate i20% of the solid line.

Due to the absorption of flux into other channels in the nuclear interior, the
adiabatic three-body model samples the neutron bound state wave function mainly at the
nuclear surface. Transfer reactions constrain the exterior but not the interior contributions
to the overlap integral that defines SF. In this analysis, we assume a smooth variation of

the potential radius (R =roAI’3) for the bound neutron global potential. Consistent with

findings in [Ver94], we find the surface properties of the neutron bound-state wave

50

function to be dominated by the central potential; for simplicity, we have neglected the
spin-orbit interaction in constructing the neutron wave function. The good overall
agreement between calculated and measured SF ’5 indicates that these assumptions are
reasonable, and that the relative magnitudes of SF ’s from nucleus to nucleus appear to be
well described. However, the absolute values of the surface contributions to the SF are

influenced by these geometrical assumptions.
2.4.1.2 Excited-state SF for Z=3-28 nuclei

In astrophysics calculations, states involving resonances near the nucleon thresholds are
also relevant [DesOO]. For instance, the astrophysical rate calculations of the captures or
emissions of single-nucleons in stellar burning processes often involve short-lived
nuclear states with small spectroscopic factors which are used to determine the decay
widths of resonance states [Sch05]. With current experimental technique, it is difficult or
even impossible to measure all the spectroscopic factors which are necessary inputs to
such astrophysical network calculations, particularly those for the short-lived nuclear
states. In these cases, shell-model calculations provide the principal means to estimate

these spectroscopic factors.

In general the shell model describes the properties of ground state nuclei very
well, but the success of the shell model is less certain in describing such excited states
[Sch05]. It is therefore important to address the accuracy of such predictions using
different Hilbert spaces [Sch05]. Such an assessment can be obtained by comparing the
calculated spectroscopic factors to those extracted using well-calibrated experimental

probes, such as (p,d) and (d,p) transfer reactions. We therefore adopt the analysis

51

procedure described in section 2.21 to extract the neutron spectroscopic factors of the

excited states of the following sd shell nuclei: ‘70, 180, 21Ne, 24Na, 25Mg, 26Mg, 27Mg,

41,43,45,47,49 47,49,51
Ca,

29Si, 3081, 31Si, 33S, 35S. We also extend our analysis to Ti and

51,53,55 57,59,61,62,63,65

Cr isotopes as well as Ni isotopes with neutrons in the pf shell.

Comparisons have been performed by matching between the theoretical and experimental
levels based on the exact agreement of the quantum numbers (l,j) and spin-parity J’r of the

transferred neutron and the approximate agreement of the energy of the states. In general,
the agreement between energy levels calculated with LB-SM and the data is within 300
keV. Comparison of experimental and theoretical spectroscopic factors provides an
independent method to evaluate the interactions used in shell-model calculations, most of
which have been obtained from fitting the binding energy and excitation energies of a

range of nuclei.
2.4.1.2.1 sd-shell nuclei

In ref. [Br006], two new interactions, USDA and USDB, have been obtained to describe
sd shell nuclei with an inert '60 core. In extracting the USDA and USDB interactions,

constraints based on the binding energy and energy levels were used and the rms
deviation of the predicted energy levels ranges from 130-170 keV. To further test the
validity of these interactions, we compare the experimental and calculated spectroscopic
factors, which were not used to determine the parameters of the USDA or USDB ,
interactions. This comparison includes all (d,p) transfer reaction data on these nuclei for
which spectroscopic factors can be calculated in the corresponding Hilbert spaces in

large-basis shell-model (LB-SM). In this section, we show a quantitative overall

52

evaluation of the success of LB-SM calculations in describing the SFs. There are only a
handful of states where the agreement between the experimentally extracted and
predicted SFs is unusually poor. These states are identified and discussed. The

experimental and predicted SF values are listed in Table 2.4.

Figure 2.12 shows the comparisons of the experimental SFs to shell-model
predictions. The average SF values obtained with USDA and USDB (x co-ordinates) are
plotted vs. the experimental extracted SF values obtained in this study labeled as
SF(ADWA) (y co-ordinates). The horizontal error bars indicate the range of the USDA
and USDB results. For most of the cases, the two values are nearly identical. The solid
diagonal line indicates perfect agreement between theoretical predictions and
experimental data. Nearly all the data cluster around the solid line. For excited states SF,
the measured angular distributions are often of poorer quality than those for the ground
state. Even though we adopted the deduced experimental uncertainties of 20-30% in the
figure as discussed in section 2.2.6, the larger scatter of the excited state data could imply
larger experimental uncertainties closer to 40%. For reference, the dashed lines in all the
figures indicate 40% deviation from the solid line. There are three states (3.908 MeV
(5/2“) state in 25Mg, 7.692 MeV (3/2+) and 8.290 MeV (572*) states in 298i), with very
small calculated spectroscopic factors (<0.005), outside the range of the established
systematics. Small calculated SFs originate from large cancellations of contributions
from different components of the wave functions, which are hard to control even in the
best shell-model calculation. Indeed, the calculated values using USDA and USDB
interactions differ from each other by more than a factor of two and under-predict the

experimental values by more than a factor 10. Clearly, these cases would be important to

53

examine further, both experimentally and theoretically as the capability of predicting very
small (<0.005) spectroscopic factors of sd shell nuclei is important for astrophysical

applications.

 

1.000 -
0.600

v1 v-

0.100 E'
0.050 }

 

SF(ADWA)
l
1
q.

0.010 ,-
0.006 ’-

i

 

 

 

”iii-3“ ‘ “ii-2 ‘ “ii-t * “ an
SF(LB-su)

0.001

 

Figure 2.12: Comparison of experimental excited states spectroscopic factors,
SF(ADWA), to predictions from large-basis shell-model calculations, SF(LBSM),
using the USDA and USDB interactions. The ends of the horizontal error bars
indicate the range of values predicted by USDA and USDB interactions. Symbols
indicate the averaged values. The solid line represents perfect agreement between

data and theory. The dashed lines correspond to i40% deviations (expected
experimental uncertainties) fi'om the solid line.

2.4.1.2.2 Ca,Ti and Cr isotopes

Beyond the sd shell nuclei, regions of interest will be around the N=20, 28 and 2:20, 28
magic shell closures. The ground states of Ca isotopes are good single-particle states with
doubly magic cores. Spectroscopic factor predictions by both the Independent Particle

Model and by the fp shell LB-SM are nearly the same and agree with the experimental

54

 

values to within 20% as shown in Figure.2.13. Figure 2.13 compares excited states

41,43,45,47,4 47,4 ,51 . 51,53,55
9Ca, 9 T1 and

neutron spectroscopic factors for Cr isotopes. Most of

the values lie within the experimental uncertainties of 40% indicated by the deviations of
the dashed lines from the solid line. On the other hand, the calculated and measured SF s
near the boundaries of the fi) shell-model space can disagree by factors of hundreds. The
largest discrepancies when using the modern GXPFlA interaction [Hon04, HonOS] occur

for the 2.462 MeV (p3/2) and 6.870 MeV (f5/2) states in 41Ca, the 2.944 MeV (p3/2) state

in 43Ca and the 4.312 MeV (p1 ,2) and 4.468 MeV (pl/2) states in 45Ca, most of which

have theoretical predicted spectroscopic factors near unity. Due to proximity of these
nuclei to the sd shell, their excited state wave functions have strong contributions from
particle-hole excitations that have hole in the sd shell. These lie outside the fp model
space [Hon02]. It is rather difficult at the present time to include hole excitations of the
sd shell core due to the huge model space that would require reliable effective

interactions for the larger Sd®fi) shell Hilbert space. In contrast, the excited states of mid-

shell nuclei, such as Ti and Cr, where the contribution of the hole excitations of the sd
shell core is relatively smaller, do not have this problem and are consequently better

described by the shell model. The experimental and predicted SF values are listed in

Table 2.5.

55

 

41 4a 4a '
1.000 .- 0 Cu O Ce 0 Ca x ."1
E A"’Ca v“Ca 0"1'1 .- ' 3
i l“’l‘1 Ou'l‘i A“Cr 3 ' ,th ‘
: V”Cr *“Cr 5.’ " “L
1- ’ J, 0
20.100 - r1. 1
’ ' I, T i
a i i a' J" E :
v I 4
% o
0.010r § § ‘:
l :
L § :
i,’ x
r x
0.001 “ ‘ ‘ “‘ *‘ “‘ ‘ ‘ ‘*““‘

 

 

 

0.001 0.010 0.100 1 .000
SF(LB-su)

Figure 2.13: Comparison of experimental spectroscopic factors, SF(ADWA), to
predictions from large-basis shell-model for the Ca, Ti and Cr isotopes, SF(LB-
SM). Complete basis with the interaction GXPFlA is used in the theoretical
calculations. The solid line indicates perfect agreement between data and
predictions. Dashed lines represent :t40% deviations from the solid line.

2.4.1.2.3 Ni isotopes

The shell structure of the unstable doubly magic nucleus 56Ni (N=Z=28) has attracted
much attention recently [Yur04, YurOS, Ber09, Min06, Kra94]. In most shell model
calculations, the N=28 core in 48Ca is assumed to be a well-established closed shell.

However, Relativistic Hartree+Bogoliubov calculations predict a strong suppression of

the N=28 shell gap for neutron rich nuclei [Lal99]. While experimental investigations of

. ,4
the 2+ energres of 36’” 0

Si provide evidence for the weakening of the N=28 shell gap in
nuclei with large neutron excess [Cam06], the evidence is inconclusive for the case of

47Ar [Gau06, Sig07]. Recent measurements of the nuclear magnetic moment of the

56

ground state of 57Cu, which could be viewed as a valence proton outside a closed 56Ni
core, suggests significant breaking of the fig shell [Min06]. To further explore the
property of the single particle states outside 56Ni, we extracted the neutron spectroscopic
factors for 57Ni to 65Ni isotopes and the values are listed in Table 2.6. The extracted
spectroscopic factors are important bench marks in evaluating different pf—shell model
interactions that are used to predict the structure of Ni or Cu nuclei, particularly the 78Ni
which is a major waiting point in the path of the r-process [Kra93]. Isotopes around 56Ni

are of special importance for the stellar evolution and electron capture in supernovae.

Shell model calculations for Ni isotopes have been available since 1960’s. In the
early calculations, such as ref. [Aue67], 56Ni is assumed to be an inert core and the

influence of core excitation was taken into account in the effective residual interaction
between the valence nucleons in the pf-shell. With advances in computational capability,
many new effective interactions, which are the key elements for successful predictions,

have been proposed. The GXPFlA interaction, a modified version of GXPFl with five
matrix elements, involving mostly the p1/2 orbitals, has been obtained by adjusting the

parameters used in the interaction to the experimental data [Hon02]. Another interaction

KB3 [Pov01, CauOS] has also been used to predict properties in the pf shell nuclei. Both

of these calculations require full pf model space with 40Ca as inert core. Recently, a new
T=l effective interaction for the fs/z, p3/2, pl /2, g9/2 model space has been obtained for the
6N1-78N1 region by fitting the experimental data of N1 isotopes from A=57 to A=78 and

N=50 isotones for 89Cu to 100Sn [LisO4]. This interaction provides an improved

57

 

Hamiltonian for 2:28 with a large model space and new Hamiltonian for N=50. It has
been mainly used to describe heavier Ni isotopes using a 56Ni core. Following the

convention established in ref [Ver07], this new interaction is called JJ4PNA in the

present work.

To demonstrate the sensitivity of the spectroscopic factors to interactions used in

the shell model calculations. We first obtained the ground state neutron spectroscopic
factors with the Auerbach interactions [Aue67] and JJ4PNA interactions with 56Ni inert
core [Lis04] using Oxbash code [BroO4]. For calculations with GXPFlA and K83
interactions with 40Ca inert core, we use the m-scheme code Antoine [Cau99]. The latter

calculations are CPU intensive. The comparison of the ground-state spectroscopic factors
between experiments and calculations are shown in Figure 2.14 The solid lines are the
least square fits of the linear correlations between data and predictions. The slopes of the
lines are labeled inside each panel. The predicted spectroscopic factors using the KB3,
JJ4PNA, and Auerbach interactions are about 25% larger than the experimental values.
The results using the full pf model space and the GXPFlA interaction, shown in the
upper left panel of Figure 2.14, give better agreement with the data as indicated by the
slope (0.93i0.06) of the solid line in the upper left panel. This is consistent with the
observation that with the improved modification in the monopole and pairing matrix

elements of the Hamiltonian, the GXPFlA interaction is better than KB3 for the lighter
isotopes around 56Ni [Hon05, Hor06]. This overall agreement with the results fi'om

GXPFIA interaction is consistent with the trends established in nuclei with Z=3-24 as

shown in section 2.4.1.1.

58

3 Ni Isoto-es
41Ca Core
03310.06 T

     

.. A=57 058 56Ni Cor_
'- 59 I 60 T :
61 V62 '3
63 V65

-

' ’ 0.72:0.04 .:
c, L JJ4PNA 3
561“ core
0.75:0.05 I

 
     

q
-

 

AAA AALA AAAA AAAA AAAA LA

1 2 3

Figure 2.14: Left panels: Comparison of the ground-state experimental
SF(ADWA) values and the shell model calculations using GXPFlA (top) and
KB3 (bottom) interactions in full in gfi) model space . Right panels: Same, but
with the shell model calculations using JJ4PNA (top) and N. Auerbach (bottom)

based on 56Ni inert core. The solid lines are the linear fits with y-intercept fixed at
zero. The slopes of the lines are listed in the individual panels.

Figure 2.15 shows comparisons of predictions using the GXPF 1A interaction (left
panel) and the XT interaction (right panel) to experimental values of the excited states in
Ni isotopes. The current analysis yields spectroscopic factors that cluster around the
large-basis shell-model predictions. Due to difficulties in identifying states at higher
excitation energy, only SF values for a few states less than 3 MeV are obtained from
calculations using the GXPFIA interaction. More states from calculations using the XT

interaction are compared to data as shown in the right panel. The data and the predictions

59

agree to within a factor of two. The scatter in the calculated values is large compared to
experimental uncertainties, which are estimated to be around 40% as indicated by the

dashed lines above and below the solid line. For small calculated SF values of less than

0.01, two of the data points (the 4.709 MeV, 9/2+ state in 59Ni and the 2.124 MeV, 1/2'

state in 6lNi) deviate from the systematics in the right panel. Unfortunately, there are not

enough data to draw a firm conclusion on the reliability of small calculated SF values
from calculations with the XT interaction in the Ni isotopes. Calculations for Ni isotopes

with GXPF 1A in complete fp model basis were performed by M. Horoi.

 

 

 

 

 

 

OA=57D59 " . Ni Isotopes , .
“’00 F261 vez ,' 1: ,I
; ,’ I, :i T I, I 1
0.500 _ 63 was , _ I .
I 1' f ,
’1 I Q I ’I
A I .I, v A l .5
4 'L’ ‘7’ TJ/ I" .5
E o. 100 :- I” ’I .1 E» I I” ..
: I, I” i» T “D, TI’ :
0.050 " , I, T D ‘ J: T
E b X I" Q h ‘L T I «L ‘
m ’ I , ’1’ "'
ll” g ’1’ Y
0.010 r I; ,a" 1: l." I." '3
E.’ / GXPFIA 1: l ’ 1
0.005 ,4 ,‘Complete Basis 1’ KT 1
..’...-i - nun“: - . ---...1 -.Lu . - “uni - - -.--..i
0.01 0.10 1.00 0.01 0.10 1.00

SF(LB—SM)

Figure 2.15: Comparison of the experimental SF values, SF (ADWA) and
the shell-model calculations with the GXPFlA interaction in the pf model
space (left panel) and the XT interaction in gfp model space (right panel).
The solid line indicates perfect agreement between data and predictions.
Dashed lines represent :l:40% deviations fi'om the solid line.

60

The predictions from the JJ4PNA interaction with 56Ni core [LisO4], using Oxbash

code [BroO4], as well as predictions fi'om the GXPFlA interaction with 40Ca core using

the Antoine code [Cau99] are listed in Table 2.7. All the states in fp-shell have the same
parity assignments (“-“ for odd and “+” for even nuclei). In both calculations, the number
of states, which have corresponding matched states in the shell model calculations,
decreases with energy of the levels. For these reasons, a large number of experimental
states have no counterparts in the shell model predictions and Table 2.7 lists 43 levels as

compared to 177 experimental levels listed in Table 2.6.

More insights regarding the residual interactions may be obtained by combining
the spectroscopic factors with energy level information. Each panel in Figure 2.16
compares the experimental energy levels and SF values to corresponding values obtained
from shell model calculations for one isotope using the GXPF 1A and JJ4PNA
interaction. The lengths of the horizontal bars represent the values of the spectroscopic
factors. As described earlier, very few states above 2 MeV have been obtained in the full
pf model space using GXPFIA interaction. Figure 2.16 only show states with energy

levels up to 2 MeV Of 57,59.6l,62.63.65

Ni nuclei. In the upper left panel of Figure 2.16, only
three states have been measured for 57Ni. The description of the data by both calculations

is quite reasonable. In 6lNi, the ordering of the states is not reproduced by calculations

using either one of the two interactions. In general, shell model calculations tend to

predict larger spectroscopic factors for the low-lying states, thus assigning larger single

particle characteristics to these states. Due to the limitation of the model space, no g9/2

61

states (dashed lines in 63Ni and 65Ni nuclei) are predicted by calculations using the

 

 

 

 

 

 

 

 

 

GXPF 1 A interactions
H 57 . 1/2 3/2 . H 59 . ' .
SF=0.2 N1 5/2 9/2 0.2 " N1 =- -
I — r - - . q
— r .
17 _ :7 - — ‘.
A :: = — :
5 0 - — — __ -- — — — -
H , 61 . - L H 62 .
a . 02 N1 - 0.25 '. N1 . .
g ' z ' ' ‘ - ..
no 1.” 'T .
Lt . ,_ .
Q) _ . .
Ci — ' .
m 0 - = -— — -
H 63 . l H 65 . -
02 N P--- 0.1 N1
1- - ‘
o — E : : — —.:—- —— —— .1
DATA GXPF 1A JJ4PNA DATA GXPFlA JJ4PNA
Fig. 2.16: Lefi panels: Plot of energy levels below 2 MeV for 57‘61’63Ni nuclei

with the length of the horizontal bars representing the values of the spectroscopic
factors. Right panels: Same as lefi panels, but for 59'62’65Ni nuclei. The scale of
the SF factor is given in the upper left comer of each panel.

2.4.2 Comparison with Endt’s “best values”
In 1977, Endt compiled a list of the “best” spectroscopic factor values for A=21-44
isotopes. Most of them are sd-shell nuclei [End77]. For the neutron spectroscopic

factors, Endt compiled the published spectroscopic factors from (d,t), (Rd), (3119,”) and

62

 

(d,p) reactions. An uncertainty of 25% is assigned to the values. (When only the (p,d) and
(d,p) reactions are studied, Endt assigned 50% uncertainties.) Endt’s best values for
ground-state SFs are listed in Table 2.3. Figure 2.17 compares the ground-state
spectroscopic factors determined by Endt and the ground-state SF(ADWA) for Z=3-24.
There are strong correlations between the two procedures even though the values scatter
around the dashed line, which indicates perfect agreement. From the consistency check
with (p,d) and (d,p) reactions, we expect that our values should have smaller random
uncertainties because a systematic approach is used to extract the SF values directly fiom
the measured angular distributions while Endt’s compilation depended on the analysis by
different authors and relied on the communication with the authors concerning the
normalizations of the spectroscopic factors. We also have the advantage that many more
measurements are included in our work than those that were available for Endt’s
compilations. The comparison of excited-state SFs between Endt’s values with 25% error
bars and SF(ADWA) for sd—shell nuclei is plotted in Figure 2.18. Endt’s excited-state SFs
appear to be larger than the SF(ADWA). More than a factor of 10 difference between
Endt’s values and some small SF(ADWA) less than 0.1 are found Endt’s values for

excited—state SFs are listed in Table 2.4.

63

1.0

SF(ADWA)

 

 

 

 

 

1.0
SF (Endt)

10.0

Figure 2.17: Comparisons of the ground-state spectroscopic factors
obtained from this work SF(ADWA) and the compiled values of Endt
[End77]. All the values are listed in Table 2.3. The line indicates perfect
agreement between our values and Endt’s compilation.

1.00
it?
B 0.10
.3,
re.
m

0.01

 

US

I
r

 

r
l
I

l

- VMg A Ne

OSi

I F-
40% T4?
T
I”' 4
7% ti. Hii

..-

l

 

I

 

 

0.01

0. 10
SF(Endt)

1.00

Figure 2.18: Comparisons of the excited-state spectroscopic factors
obtained from this work SF(ADWA) and the compiled values of Endt
[End77]. The line indicates perfect agreement betWeen our values and
Endt’s compilation. All the values are listed in Table 2.4.

64

2.4.3 Comparison with ENDSF

From the published angular distributions, which are of reasonable quality, we extracted
177 spectroscopic factors, SF (ADWA), for the Ni isotopes as discussed in sub-section
2.2.4.2.3 [Lee09]. These values are listed in Table 2.6. When available, spectroscopic
factors from the Evaluated Nuclear Structure Data File (ENSDF) compiled by the
National Nuclear Data Center (NNDC) [ENDSF] are also listed in Table 2.6 in the last
column. Unlike Endt’s best values which have been evaluated by Endt, SF(ENSDF)
values are taken directly from the published values, which came from different
experiments and might be analyzed differently using different optical potentials and
different reaction models. As a result, these SF values may not be consistent with each
other or with the results from the present work. Figure. 2.19 compares the spectroscopic
factors obtained in this work, SF(ADWA), (y-ordinate) to those listed in ENDSF (x-
abscissa). The solid line indicates perfect agreement. Most of the ENDSF values are

about 30% larger than the values obtained in the present work. (The spectroscopic factors
for the data set 61Ni(d,p)62Ni are not included in the comparison because of the

discrepancies between the ENSDF and the values published in ref [ENDSF]. The second
set of ENDSF values obtained from reference [F ul63] for the same reaction does not have

published angular distributions in the paper.

65

 

 

 

 

. OA=58 059 '60 ,. ",4
. 061 A63 V6 -
100 a . 5 .1' a
l N1 Isotopes ’ :
A 1’ I” 'L
<: —1 - ' ,. _
2: 1° *3?
% ’ TI’I’ J- ‘
10—2 r I. g ‘:
i xi, {TI 3
. 1’,’ l U 4
b ’ §I J- ‘
10"3 '4" - - ‘ *‘
10"3 10'”2 10"1 10°
SF(ENDSF)

Figure 2.19: Comparison of the extracted SF(ADWA) values in the
present work and the compiled values in ENDSF [ENDSF]. The solid line
indicates perfect agreement and dashed lines represent i30% of solid line.

2.5 Reduced SF values with JLM optical model potential

A recent analysis of (e, e ’p) reactions suggests that proton SF values deduced from nuclei
near closed shells, including 40Ca, are suppressed by about 30—40% compared to the
independent particle model (IPM) expectations [KraOl]. This suppression has been
attributed, in different proportions, to short- and medium-range (tensor) nucleon-nucleon
correlation effects and to longer range correlations arising from couplings, by nucleons

near the Fermi surface, to collective degrees of freedom.

Historically, conventional distorted wave Born approximation (DWBA) transfer

reaction analysis has shown little need for such a systematic suppression of single-

66

particle strength [End03, TsaOS]. Marked differences in proton SF 8 extracted from the (e,
e ’p) and (61,3He) proton transfer reaction analyses were reconciled by a reanalysis of the
((1,3He) data [KraOl], requiring significant changes to the (DWBA) ((1,3He) calculations
used there. These included fine-tuning of the root mean squared (rms) radii of the active

proton single-particle orbitals to be consistent with the (e, e 'p) data analysis. The optical
model potentials for deuteron and 3He were obtained from the elastic scattering

measurements [Kra0 1 , Ban85].

In this section we assess related effects in (d,p) and (p,d) ground-state neutron
transfer reactions by constraining the geometry of the nucleon optical interactions with
the target and the transferred neutron orbital rms radii using modern Hartree-Fock (HF)
calculations [Bro98]. In this way we not only reduce long-standing parameter ambiguities
but also introduce into the transfer reaction description the systematic behavior
manifested by the HF calculations across extended mass regions [Br000, BroOl-2,
Ric03]. This agreement between the systematics of the HF predictions and a range of
nuclear size parameters suggests that the theory should also provide a good description of

the spatial extent of individual single-particle states.

We thus propose the following consistent three-body analysis with HF theoretical
input for ground-state-to-ground-state neutron transfer reactions. The analysis would be
the same as the framework we discussed in sub-section 2.2.1, except different global
optical model potential and transferred neutron orbital rms radii are used. The key inputs
of the approach are that we calculate the transfer reaction amplitudes using the Johnson-
Soper adiabatic approximation [Joh70] and adopt the JLM nuclear matter effective

nucleon-nucleon interaction [Jeu77] rather than CH89 phenomenological nucleon-target

67

interactions [Var91]. The remaining critical ingredient is the geometry of the potential

used to generate the neutron overlap function. Basically, we fix the diffuseness parameter

at a0 = 0.65 fm, to which the calculations are only weakly sensitive. We then adjust the
radius parameter r0 for each reaction to reproduce the rms radius of the relevant
transferred neutron orbital, as given by the HF calculation. More precisely, r0 is adjusted
so that the mean squared radius of the transferred neutron orbital is <r2> = [A/(A -

1)]<r2>m=, where <r2>HF is the HF calculation value. This adjustment is carried out using

the HF separation energy. The small mass correction factor corrects the fixed potential
center assumption used in the HF approach. For the binding potential choices, the depths
of the central potential wells are adjusted to reproduce the experimental separation

energies to ensure the correct asymptotic form of the overlap functions. A spin-orbit
potential of strength 6 MeV, with the same (central) geometry parameters, r0 and a0, was

included.

Before we examine a broader range of targets, we first focus on available data for

the 4045'4749Ca isotopes. Since 4oCa is doubly magic, with closed proton and neutron

sd shells, the additional neutrons in 41448Ca fill the fm orbit. These valence neutron wave

functions in the Ca isotopes are expected to have good single-particle property. Indeed,
the predicted SF values from large-basis shell-model (LB-SM) calculations, which
include configuration mixing, and the IPM, which neglects such effects, are essentially
equal in the calcium isotopes. Figure 2.20 shows the ratios of the extracted SFs to these
SF(IPM) as a function of mass number A. The solid stars in Figure 2.20 represent the

ratios SF(JLM+HF)/ SF(IPM) for the calcium isotopes. These SF(JLM+HF) are the new

68

results of our constrained analysis, using the HF ~inspired neutron binding potential
geometries and the J LM nucleon optical potentials obtained using the HF densities of the
targets. We observe an overall reduction in the SF (JLM+HF) compared to the IPM values
of about 25%—30%. The additional data point for A = 40 (open circle) is the proton SF
value, 0.645(50), as deduced from the (e,e’p) analysis of Ref. [KraOl]. Within the

assigned experimental uncertainties, the neutron SF(JLM+HF) and the proton SF (e, e’p)
for 40Ca agree. For comparison, we also plot the SF values from the conventional
adiabatic three-body model reaction analysis using the CH89 global potential and the
standard binding potential geometry of r0=l.25 frn. The open stars in Figure 2.20 show
the corresponding SF ratios, SF(ADWA)/SF(IPM). The ratios for these latter calculations

are close to unity within experimental uncertainties, although three odd-A isotopes, 43Ca,
45 49 . 49

Ca, and Ca, are somewhat suppressed. The suppressron for Ca may be traced to a
sharp increase in the rms radius of the 2pm orbit in 49Ca (4.59 fm), compared to that of
the lfm orbit (3.99 fm) in neighboring 48Ca, when using the standard geometry.

. . . . 4
However, this explanat1on cannot address the reduction in the SFs for the 43Ca and 5Ca

nuclei. The deduced SF(JLM+HF) are consistently reduced compared to the SF(CH89).
The reduction of the SF(JLM+HF) values comes from both the changes of optical
potential and the use of more realistic (larger) neutron bound-state wave functions. On
average, the use of the JLM potential folded with nucleon densities calculated with
Skyrme SkX HF instead of the CH89 global potential results in reduction of the SF
values by 15%. Similar effects were observed in ref. [Liu05]. The rms radii of the neutron

bound-state wave functions based on the Skyrme SkX HF predictions are also, on

69

average, about 2% larger than the rrns radii of 1.25 frn. This results in further reduction of
the SF values by about 15% [Lee06]. The observed suppression is thus a manifestation of
both effects. As was stated earlier, we believe that these changes, constrained by the same
(HF) theoretical systematics, will better determine the all-important overlap of the

distorted waves and bound-state wave functions at the nuclear surface.

 

 

 

 

 

 

 

l ' l ' l ‘ l T l
< u ——->
1am 7" 2p
1.25 - «- W
a? l. l
m J L T i
1:. 1 it
Q l - 1
A 0 75 - r i L .
'3. $ “ Ir 1r 1 l
..L i l
6 0 50- L I -
v ' .L
a
o 25 * JLM-t 1111‘
' it CHBQ
O (e’f'p) 1 l 1 l
0.00 ‘ 4 t 4 ‘
411 43 46 47 49
ACa

Figure 2.20: Ratios of the experimentally deduced spectroscopic factors to those
of the independent particle shell-model SF(IPM) for the calcium isotopic chain.
The open symbols, from SF(CH89), result from the use of conventional, three-
body adiabatic model calculations using the CH89 global nucleon optical
potentials and a fixed neutron bound-state geometry of 1.25 frn. The solid
symbols, from SF(JLM+HF), are the results of constrained three body model
calculations, where both the nucleon optical potentials (the JLM microscopic
optical model) and the neutron bound-state potential geometry are determined by
the Skyrme (SkX) HF calculations of Ref. [Bro98].

The Ca isotope SF(CH89) values are included in the large-scale survey of 80
nuclei, studied via (p,d) and (d,p) transfer reactions discussed in sub-section 2.4.1.1. In
the survey, it was shown that within experimental and theoretical uncertainties, most

extracted SF(CH89) values, like those for the Ca isotopes, agreed with the predictions of

70

the LB-SM. To examine whether or not the reductions in the deduced SF(JLM+HF) are
limited to the calcium isotopes, we have applied the same HF -constrained analysis to a
selection of the 80 nuclei. As the HF is less appropriate for the description of single-
particle configurations of very light systems, we limit the analysis to A > 11.
Additionally, beyond the calcium isotopes, the IPM does not take proper account of
configuration mixing effects, so we now compare the extracted SF(JLM+HF) to large—
basis shell-model SF(LB-SM) predictions, which are calculated with the code OXBASH
[BroO4]. These ratios are listed in Table 2.8 and shown in Figure 2.21 as a function of the
difference between the neutron and proton separation energies (AS). The physics
implication of using AS is discussed in Section 2.8. For clarity, only those points with an
overall uncertainty of less than 25% are included. Data with uncertainties much larger
than 20% (the random error assigned to each measurement) have quality-control
problems in the evaluation. In such cases, there is either no second measurement to
corroborate the validity of a data set or the standard deviations of the measurements used
to extract the SF values are larger than 25%. For the (statistically most significant) cases
presented, we note once again an overall SF(JLM+HF) reduction of order 30% compared
to the SF(CH89), but with significant residual fluctuations between the values for
different nuclei.

In principle, like other optical potentials based on the microscopic folding model,
JLM approach should be more appropriate than the global optical potential such as CH89
to extract structural information, since such formulation of the optical potential is more
fundamental and possesses no free parameters. More importantly, the JLM microscopic

model adopts the nuclear density and an effective nucleon-nucleon interaction. However,

71

we found that JLM parameterizations display discontinuity in the calculated cross
sections and give energy dependence in the SF determinations for some reactions in the
energy region of ~10-50 MeV. Bauge et al located the discontinuity at 10 MeV in the
imaginary part of J LM parameterizations, which consequently yields a discontinuity of
the volume integral [Bau96]. Using the refitted JLM parameterizations from Bauge et al.,
some significant discontinuities in the cross section calculations can be removed,
however, we still obtain instabilities in the energy dependence which is not completely
understood. Due to these limitations and the difficulties in calculating the nuclear
densities for nuclei at excited states, we have not extended similar transfer reaction

studies using JLM parameterization to extract excited-state SFs.

2.6 Asymmetry dependence of reduction factors for stable nuclei in

transfer reactions

Figure 2.21 plots the reduction factor, which is the ratio of experimental SF to LB-SM
predictions, as a fimction of the difference between the neutron and proton separation
energies AS (AS = Sn - Sp for neutron SF and AS = Sp - Sn for proton SF), where AS
characterizes the asymmetry dependence of relative shifts for neutron and proton Fermi
surfaces. For the nuclei investigated using transfer reaction based on the systematic
approach discussed above, there is no evident asymmetry dependence of the observed
reduction factors on AS. Limiting the observations to the calcium isotopes (the solid stars

in Figure 2.20), which span neutron-proton separation energy differences fi'om —11.3

MeV (”Ca) to 7.3 MeV (40Ca), one draws the same conclusion. The open circles in

Figure 2.21 are the corresponding ratios of the proton ground state SF for 7Li, 12C, 16O,
72

30 .31 40 48
S

l, p, Ca, Ca, 5'v, 902:, and 208Pb (as listed in Table 2.8), studied with the (e, e’p)

reaction [Kra01, Wes92]. Similarly, the solid triangles show the ratios of the deduced SF
to SF(LB-SM) values from both exclusive and inclusive studies of intermediate-energy
nucleon knockout reactions. Neutron (proton) knockout values are shown as inverted

(upright) triangles (and listed in Table 2.8). These include, at the extremes of the |Sn-Sp|

scale, 15C [Ter04], 22o [Gad04-1], 34Ar [Gad04-2], and 32Ar [Gad04-l], whereas the

values for 8B, 9C [End03], 46Ar [Gad05], 12C, and 16O [BroO2] overlap the AS values of

both the transfer and the (e, e’p) analyses. In the case of the inclusive knockout reaction
analysis of Ref. [BroO2], effective neutron and proton removal energies, obtained by
weighing the physical separation energies to each final state by the corresponding cross-
sections, were used. The suppression with respect to the SF(LB-SM) is similar for the
three different reactions within the AS region in which they overlap. A dependence of the
suppression on AS is indicated by the nucleon-knockout data that extend the data set into
regions of significant neutron and proton asymmetry. H0wever, SFs from past transfer
data focus on stable nuclei and the large systematic uncertainties from different
experiments reduce the sensitivity in determining the dependence. Thus, to confirm this
behavior with a transfer reaction probe, measurements over a larger range of asymmetry

corresponding to a larger AS are needed.

73

1.2 l

 

 

 

 

E 1.0 -r I i —

r? I Iii—l ? I l l

a 0.8 - §ul1lfli me -

v T I D via? ‘ { ,pftii i

B06 I It‘lsgiég

\ . —- I Ij"? ~— 4

A A I

"'5. f E if? g V 6 T

g 0.4 — I —

EL! ~1-

m 0.2 J: Eptd')43(d.p) 1.

lVA n(p)pll(nockout I I 1

0'0—20 —10 o 10 20

AS(MeV)

Figure 2.21: Ratios of experimentally deduced SF to those of the large-basis shell-
model calculations SF(LB-SM) for nuclei with A = 12—49 as a function of the
difference of neutron and proton separation energies, AS (see text). The solid
circles and stars are the present results from transfer reactions. The solid stars
represent the Ca isotopes, as in Fig. 1. The open circles are ground-state proton
SF from (e, e ’p) analysis and the triangles are the results from proton knockout
reactions (with inverted triangles for neutron knockout). The data points are listed
in Table 2.8 and are referenced in the text.

74

2.7 Applications of the established systematics in spectroscopic factors

As shown in Section 2.4, we have evaluated the accuracy for different interactions
and established the systematics of SFs in different shell regions. The established
systematics suggests the theoretical SFs calculated in large-basis shell-model using good
interactions agree to the experimental SF deduced using CH89 approach in ADWA
model to within 20-30%. The definition of experimental SF is the ratio of the
experimental to calculated differential cross sections. Based on the systematics, we can
predict reliable experimental cross sections using the predicted SF value and calculated
cross sections in CH89 analysis. Such cross-section estimation is needed when designing

(p,d) and (d,p) transfer reaction experiments.

The established systematics can be also used in confirming the spin assignment.
In sd-shell nuclei with USDA/USDB interactions in LB-SM, for instance, we have used

the established systematics between the experimental and theoretical spectroscopic
factors to assign the spins of three selected states in 27Mg that have no definitive spin
assignments from measurements. The spins for these states listed in Table 2.9 can either
be 3/2+ or 5/2+ (second column) according to NUDAT [NUDAT]. Since the measured

angular distributions are sensitive to the angular momentum [but not very sensitive to the
spin .1, the extracted SF values (third column) are similar for different values of J.
However, the shell-model spectroscopic factors (sixth column) for the 3/2+ and 5/2+
states within 100 keV of the 5.627 MeV state differ by more than a factor of 25. The
systematics of Figure 2.13 indicates that the spins of the 5.627 MeV, 3.491 MeV and

4.150 MeV states are consistent with a J=3/2+, J=3/2+ and J=5/2+, respectively.

75

Table 2.3: List of isotopes with the extracted spectroscopic factors, SF (ADWA) and other
information such as the mass number (A), charge number (Z) and neutron number (N) for
the nuclei. j“, T and Sn are the spin and parity, isospin and neutron separation energy of
the nuclei. For completeness, we list the root mean square radii of the neutron wave
functions. Endt compiled values are also listed when available. The model space and
interactions used in Oxbash [Br004] are listed together with the predicted SF values
labeled as LBSM.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B A Z N j" T Sn rms Endt SF (ADWA) LBSM Model Space Interaction
‘Li 6 3 3 1/2‘ 0 5.66 2.91 1.12 10.32 0.68 PPN CKPPN
7“ 7 3 4 1/2' 1/2 7.25 2.81 1.85 10.37 0.63 PPN CKPPN
8Li 8 3 5 1/2‘ 1 2.03 3.66 06210.18 1.09 PPN CKPPN
9“ 9 3 6 172' 3/2 4.06 3.23 09810.28 0.81 PPN CKPPN
93c 9 4 5 3/2‘ 1/2 1.67 3.86 0.451003 0.57 PPN CKPPN
7‘3, 10 4 6 3/2' I 6.81 2.96 1.581015 2.36 PPN CKPPN
“Be 11 4 7 “2* 3/2 0.50 7.11 05110.06 0.74 SPSDPF WBP
703 10 5 5 3/2‘ 0 8.44 2.85 05010.07 0.60 PPN CKPPN
“3 11 5 6 3/2 1/2 11.45 2.73 1.481019 1.09 PPN CKPPN
"B 12 5 7 1/2' 1 3.37 3.46 0.451006 0.83 PPN CKPPN
”c 12 6 6 3/2‘ 0 18.72 2.53 3.121036 2.85 PPN CKPPN
13c 13 6 7 1/2‘ 1/2 4.95 3.26 07510.10 0.61 PPN CKPPN
"c 14 6 8 1/2' 1 8.18 3.00 1.631033 1.73 PPN CKPPN
15c 15 6 9 10* 3/2 1.22 5.51 1.121032 0.98 SPSDPF WBP

“N 14 7 7 1/2‘ 0 10.55 2.87 07710.12 0.69 PPN CKPPN
”N 15 7 8 1/2' 1/2 10.83 2.89 1.481024 1.46 PPN CKPPN
“N 16 7 9 372* 1 2.49 4.26 0.421012 0.96 SPSDPF WBP

“o 16 8 8 1/2' 0 15.66 2.74 2.461026 2.00 PPN CKPPN
"o 17 8 9 5/2+ 1/2 4.14 3.48 0.941013 1.00 so uso
mo 18 8 10 5/2+ 1 8.04 3.24 1.661019 1.58 so USD

"0 19 8 11 5/2+ 3/2 3.95 3.57 0.431006 0.69 so uso

”F 19 9 10 1/2“ 1/2 10.43 2.66 1.601023 0.56 so USD
my 20 9 11 3/2+ l 6.60 3.39 ~0.01 0.02 so USD

“Ne 21 10 11 3/2 1/2 6.76 3.41 0.01 0.031001 0.03 so uso

"Ne 22 10 12 3/2’“ 1 10.36 3.27 0.19 0.241003 0.13 so USD

”Ne 23 10 13 5/2+ 3/2 5.20 3.58 0.24 0.241003 0.32 so Uso

“N. 24 11 13 172* 1 8.87 3.49 0.30 05910.17 0.39 so USD

“Mg 24 12 12 3/2+ 0 16.53 3.13 04110.06 0.41 so uso
75M: 25 12 13 5/2+ 1/2 7.33 3.50 0.37 0.291003 0.34 so USD

“Mg 26 12 14 5/2” 1 11.09 3.35 1.80 2.431050 2.51 so USD

 

 

 

 

 

 

 

 

 

 

 

 

 

 

76

Table 2.3 (cont’d)

 

B A Z N j" T s“ rms Endt SF (ADWA) LBSM Model Space Interaction

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mg 27 12 15 1/2 3/2 6.44 3.90 0.58 04510.13 0.46 so uso
"A. 27 13 14 50+ 1/2 1306 3.31 1.10 1.401020 1.10 so USD
”A; 28 13 15 “2+ 1 7.73 3.78 0.50 0.661010 0.60 so USD
”s1 28 14 14 5/2+ 0 17.18 3.22 4.401124 3.62 so USD
”s1 291415 “2* 1/2 8.47 3.73 0.55 04210.13 0.45 so USD
3°51 30 14 16 “2+ 1 10.61 2.87 0.89 0.841010 0.82 so USD
3'81 31 14 17 3/2+ 3/2 6.59 3.70 0.75 05410.07 0.58 so uso
32]) 32 15 17 “2+ 1 7.94 3.64 0.80 0.581010 0.60 so USD
325 32 16 16 “2+ 0 15.04 3.40 1.511043 0.96 so USD
338 331617 30* 1/2 8.64 3.63 0.70 0.701020 0.61 so USD
3‘3 34 16 18 3/2+ 1 11.42 3.53 1.90 1.431035 1.83 so uso
s 35 16 19 372* 3/2 6.99 3.77 0.38 0.301009 0.36 so USD
37s 37 16 21 7/2' 5/2 4.30 4.02 08810.12 0.92 SDPF SDPFNOW
35c. 35 17 18 3/2+ 1/2 12.64 3.51 0.351010 0.32 so USD
“c1 36 17 19 1/2“ 1 8.58 3.70 1.20 06810.19 0.77 so uso
37c] 37 17 20 1/2+ 3/2 10.31 3.64 0.95 09710.43 1.15 so USD
c1 38 17 21 1/2' 2 6.11 3.94 0.78 1.811051 0.95 SDPF SDPFNOW
36A, 36 18 18 30+ 0 15.26 3.45 3.341089 2.06 so USD
37A, 3718 19 3/2+ 1/2 8.79 3.71 0.49 03610.05 0.36 so USD
3‘11, 38 18 20 30+ 1 11.84 3.60 2.50 2.471070 3.04 so USD

 

”A, 39 18 21 7/2' 3/2 6.60 3.94 0.64 0.811011 0.83 SDPF SDPFNOW
A, 40 18 22 7/2' 2 9.87 3.83 1.20 1.081031 1.91 SDPF SDPFNOW
A, 41 18 23 7/2' 5/2 6.10 4.01 0.47 0.551008 0.65 SDPF SDPFNOW
”K 39 19 20 3/2+ 1/2 13.08 3.58 2.00 2.121060 1.72 so USD

K 4019 21 5/2‘ 1 7.80 3.90 0.94 1.711048 0.98 SDPF SDPFNOW
K 41 19 22 5/2' 3/2 10.10 3.84 0.56 0.911026 1.06 SDPF SDPFNOW
4219 23 1/2' 2 7.53 3.96 0.34 08110.11 0.88 SDPF SDPFNOW
4°C. 40 20 20 3/2 0 15.64 3.81 4.351062 4.00 so USD

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“‘ca 41 20 21 7/2' 1/2 8.36 3.90 0.85 1.011006 1.00 FPPN FPBPPN
42c. 42 20 22 7/2' 1 11.48 3.82 1.60 1.931017 1.81 FPPN FPBPPN
43a,. 43 20 23 7/2‘ 3/2 7.93 3.97 0.58 0.631007 0.75 FPPN FPBPPN
“c. 44 20 24 7/2' 2 11.13 3.87 3.10 39311.08 3.64 FPPN FPBPPN
45c. 45 20 25 7/2' 5/2 7.41 4.03 03710.05 0.50 FPPN FPBPPN
47c. 47 20 27 7/2' 7/2 7.28 4.08 0.261004 0.26 FPPN FPBPPN
“c. 48 20 28 7/2' 4 9.95 3.99 73511.42 7.38 FPPN FPBPPN
49c. 49 20 29 3/2' 9/2 5.15 4.59 06910.07 0.92 FPPN FPBPPN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

77

Table 2.3 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B A Z N j" T sll rms Endt SF (ADWA) LBSM Model Space Interaction
“Sc 45 21 24 3/2' 3/2 11.32 3.89 0.34 0.301008 0.35 FPPN FPBPPN
“Sc 46 2125 1/2' 2 8.76 4.00 0.511014 0.37 FPPN FPBPPN
“n 46 22 24 7/2' 1 13.19 3.85 2.421034 2.58 FPPN FPBPPN
47“ 47 22 25 5/2‘ 3/2 8.88 4.01 0.031001
7311 48 22 26 5/2' 2 11.63 3.94 01110.01
7911 49 22 27 7/2' 5/2 8.14 4.08 02510.03
7611 50 22 28 7/2‘ 3 10.94 4.00 63611.10
5‘11 51 22 29 3/2' 7/2 6.37 4.46 1.25.1035
51v 51 23 28 5/2' 5/2 11.05 4.01 1.281032
15c, 50 24 26 5/2' 1 13.00 3.94 0.111002
51c, 51 24 27 7/2' 3/2 9.26 4.08 03010.08
520 52 24 28 7/2' 2 12.04 4.00 6.241088
53Cr 53 24 29 3/2' 5/2 7.94 4.34 03910.03
54c, 54 24 30 3/2' 3 9.72 4.22 0711020
"550 55 24 31 3/2‘ 7/2 6.24 4.53 0.631013

 

Table 2.4. List of neutron excited-state spectroscopic factors for the sd—shell nuclei. We
adopt the energy levels compiled in the data base NUDAT by the National Nuclear Data
Center [NNDC]. SF(ADWA) is the deduced experimental SF value with the uncertainty
from the present work. SF(USDA) and SF(USDB) are the large-basis shell-model
calculations using USDA and USDB interactions in SD shell respectively. <SF(LB-SM)>
is the average of the SF(USDA) and SF(USDB). Endt compiled values are also listed
when available.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sr <sr>
Nucleus Ex I j ( ADW A) Error SF(USDA) SF(USDB) (LB-SM Endt

17o 0.871 0 1/2 + 1.020 10.194 1.000 1.000 1.000
"Ne 0.351 2 5/2 + 0.564 10.107 0.624 0.616 0.620 0.71
3.736 2 5/2 + 0.028 10.005 0.022 0.012 0.017 0.04

4.526 2 5/2 + 0.136 10.041 0.117 0.130 0.124

5.550 2 3/2 + 0.118 10.029 0.130 0.262 0.196

6.608 2 3/2 + 0.192 10.047 0.219 0.199 0.209

"Na 2.563 2 5/2 + 0.142 10.043 0.061 0.064 0.063
”Mg 0.585 0 1/2 + 0.392 10.055 0.507 0.483 0.495 0.51
0.975 2 3/2 + 0.215 10.040 0.232 0.224 0.228 0.35
1.965 2 5/2 + 0.079 10.019 0.084 0.089 0.086 0.11

 

78

 

Table 2.4 (con’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex 1 1 (2115321) Error SF(USDA) SF(USDB) (1:15;) Endt
2.563 0 1/2 + 0.169 10.036 0.102 0.114 0.108 0.13
2.801 2 3/2 + 0.242 10.046 0.290 0.285 0.288 0.36
3.908 2 5/2 + 0.018 10.006 0.002 0.001 0.001 0.51
4.359 2 3/2 + 0.009 10.003 0.010 0.012 0.011 0.19
5.475 0 1/2 + 0.121 10.036 0.104 0.114 0.109
"ML 3.589 2 5/2 + 0.201 10.049 0.232 0.247 0.239
4.972 2 5/2 + 0.043 10.013 0.039 0.022 0.031
"Mg 0.985 2 3/2 + 0.405 10.122 0.396 0.389 0.392 0.60
1.699 2 5/2 + 0.089 10.027 0.122 0.123 0.123 0.13
3.476 0 1/2 + 0.213 10.064 0.163 0.191 0.177 0.29
3.491 2 3/2 + 0.049 10.015 0.085 0.108 0.097 0.40
3.787 2 3/2 + 0.095 10.028 0.118 0.093 0.105 0.56
4.150 2 5/2 + 0.025 10.007 0.027 0.031 0.029
5.029 0 1/2 + 0.024 10.007 0.018 0.014 0.016
5.172 2 5/2 + 0.003 10.001 0.004 0.006 0.005
5.172 2 3/2 + 0.004 10.001 0.004 0.005 0.005
5.627 2 3/2 + 0.129 10.039 0.141 0.147 0.144
5.764 2 3/2 + 0.011 10.003 0.033 0.030 0.032
5.764 2 5/2 + 0.007 10.002. 0.005 0.000 0.005
”81 1.273 2 3/2 + 0.453 10.119 0.679 0.651 0.665 1.00
2.028 2 5/2 + 0.136 10.041 0.142 0.155 0.148 0.19
3.067 2 5/2 + 0.054 10.013 0.057 0.058 0.058 0.10
4.840 0 1/2 + 0.024 10.005 0.034 0.047 0.041 0.06
5.949 2 3/2 + 0.029 10.009 0.012 0.018 0.015 0.10
7.692 2 3/2 + 0.025 10.008 0.000 0.001 0.001
8.290 2 5/2 + 0.071 10.021 0.000 0.002 0.001
W1 5.232 2 5/2 + 0.012 10.004 0.022 0.025 0.023 0.01
“Si 0.752 0 1/2 + 0.221 10.031 0.224 0.251 0.237 0.46
2.317 2 3/2 + 0.033 10.008 0.028 0.035 0.032 0.06
2.788 2 5/2 + 0.034 10.010 0.050 0.058 0.054
4.259 2 3/2 + 0.031 10.009 0.048 0.049 0.049 0.15
4.720 0 1/2 + 0.088 10.026 0.073 0.078 0.075
6.252 2 3/2 + 0.005 10.001 0.011 0.013 0.012
33s 0.841 0 1/2 + 0.210 10.063 0.221 0.233 0.227 0.40
2.313 2 3/2 + 0.047 10.014 0.043 0.050 0.046 0.07
2.868 2 5/2 + 0.049 10.015 0.052 0.069 0.060 0.05
1“s 1.573 2 1/2 + 0.103 10.025 0.101 0.102 0.102 0.21
2.717 2 5/2 + 0.019 10.006 0.009 0.017 0.013 0.43

 

79

 

Table 2.5. List of neutron excited-state spectroscopic factors for the Ca, Ti and Cr
isotopes. We adopt the energy levels compiled in the data base NUDAT by the National
Nuclear Data Center [NNDC]. SF(ADWA) is the deduced experimental SF value with
the uncertainty from the present work. SF (LB-SM) is the large-basis shell-model
calculations using GXPF 1A in pf-shell.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex I j SIilADWA) Error SF(LB-SM)
“Ca 2.462 1 3/2 0.157 10.014 1.000
4.753 1 1/2 1.111 10.157 1.000
6.870 3 5/2 0.01 1 10.004 1.000
“Ca 2.938 1 3/2 0.044 10.015 0.976
4.239 1 1/2 0.314 10.125 0.310
“Ca 3.442 1 1/2 0.159 10.079 0.185
4.312 1 1/2 0.004 10.001 0.237
4.468 1 1/2 0.012 10.005 0.547
4.885 3 5/2 0.007 10.003 0.008
5.373 1 3/2 0.003 10.001 0.002
5.440 1 3/2 0.005 10.002 0.005
5.764 3 5/2 0.025 10.010 0.041
"Ca 2.014 1 3/2 0.640 10.090 0.949
2.875 1 1/2 0.199 10.079 0.108
4.058 1 1/2 0.374 10.150 0.850
4.785 3 5/2 0.090 10.045 0.076
“Ca 2.023 1 1/2 0.766 10.108 0.966
3.991 3 5/2 0.478 10.085 0.942
"'11 1.550 1 3/2 0.393 10.083 0.185
1.794 1 1/2 0.122 10.049 0.129
2.163 3 5/2 0.009 10.004 0.010
2.548 1 3/2 0.046 10.018 0.053
2.600 3 7/2 0.253 10.101 0.190
2.839 3 5/2 0.089 10.036 0.1 18
”Ti 1.380 1 3/2 0.510 10.204 0.291
5‘11 1.167 1 1/2 0.945 10.378 0.571
5‘Cr 0.749 1 3/2 0.243 10.097 0.365
0.777 1 1/2 0.253 10.101 0.237
1.353 3 5/2 0.171 10.068 0.228
1.557 3 7/2 0.069 10.028 0.032
3.056 1 1/2 0.043 10.017 0.039
4.040 1 1/2 0.166 10.067 0.185

 

 

 

 

 

 

 

 

 

 

80

 

Table 2.5 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex I j u SF(ADWA) Error SF(LB-SM)
53Cr 0.564 1 1/2 - 0.330 10.110 0.382
1.006 3 5/2 - 0.237 10.031 0.324
1.290 3 7/2 - 0.049 10.025 0.033
2.321 1 3/2 — 0.182 10.057 0.212
2.723 1 1/2 - 0.132 10.053 0.052
3.617 1 1/2 - 0.254 10.036 0.326
55Cr 0.518 3 5/2 - 0.164 10.066 0.409
1.215 3 5/2 - 0.038 10.019 0.015
2.008 3 5/2 - 0.086 10.034 0.043

 

 

 

 

 

 

 

 

 

Table 2.6. List of neutron spectroscopic factors for the Ni isotopes. We adopt the energy
levels compiled in the data base NUDAT by the National Nuclear Data Center [NNDC].
SF(ADWA) are SF values from the present work and SF (ENDSF) values are obtained
from ENDSF data base [ENDSF]. Spin value, I, enclosed in ”( )” represents state with
uncertain I value and the symbol “*” represents doublet state. State with undetermined
parity (1!) is labeled “N”.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex (MeV) I J 11: SF(ADWA) Error SF(ENDSF)
57m 0.000 1 3/2 - 0.954 10.286
0.769 3 5/2 - 1.400 10.420
1.113 1 1/2 - 1.000 10.300
Wm 0.000 1 0 + 0.890 10.087 1.250
mm 0.000 1 3/2 - 0.444 10.045 0.816
0.339 3 5/2 - 0.472 10.059 0.677
0.465 1 1/2 - 0.424 10.060 0.620
0.878 1 3/2 - 0.046 10.006 0.072
1.301 1 1/2 - 0.166 10.031 0.286
1.680 3 5/2 - 0.062 10.016 0.093
1.735 1 3/2 - 0.004 10.001 0.009
1.948 3 7/2 - 0.013 10.007 0.037
2.415 1 3/2 - 0.013 10.006 0.008
2.627 3 7/2 - 0.016 10.008 0.039
2.640 1 (1/2) - 0.022 10.007
2.640 1 (3/2) - 0.01 1 10.003
2.681 3 (5/2) - 0.019 10.010 0.022
3.026 1 1/2 - 0.023 10.007
3.026 1 1/2* - 0.009 10.002
3.026 3 (5/2*) N 0.016 10.003
3.061 4 9/2 + 0.479 10.096

 

 

 

 

 

 

 

 

 

81

Table 2.6 (con’t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex (MeV) l J n SF(ADWA) Error SF(ENDSF)
3.429 0 (1/2) N 0.010 10.003
3.452 1 3/2 - 0.022 10.003 0.034
3.546 2 (5/2) N 0.019 10.006
3.652 3 (5/2) N 0.018 10.009 0.021
3.858 1 3/2 - 0.019 10.013 0.025
4.036 1 (3/2) - 0.031 10.016 0.012
4.506 2 5/2 + 0.175 10.053 0.234
4.542 2 5/2 - 0.161 10.023
4.709 4 9/2 + 0.049 10.024 0.098
4.822 2 (5/2) N 0.040 10.020
4.939 1 (1/2) N 0.054 10.027
5.069 1 1/2 - 0.009 10.003 0.017
5.149 0 1/2 + 0.065 10.019 0.093
5.213 2 5/2 + 0.018 10.005 0.026
5.258 2 (5/2) N 0.017 10.009
5.429 4 (9/2) + 0.080 10.015
5.458 2 (5/2) + 0.151 10.075
5.528 0 1/2 + 0.120 10.060
5.569 0 (1/2) + 0.021 10.006 0.024
5.692 0 1/2 + 0.077 10.023 0.126
5.894 2 (5/2) + 0.014 10.004
6.142 1 1/2 - 0.027 10.005
6.142 1 3/2 - 0.014 10.003
6.206 2 (5/2) + 0.023 10.01 1 0.01 1
6.284 2 (5/2) N 0.053 10.026
6.380 0 1/2 + 0.039 10.012 0.078
6.648 2 3/2 + 0.036 10.01 1
6.648 2 5/2 + 0.024 10.007
7.073 0 1/2* + 0.027 10.008 0.029
7.073 2 5/2* + 0.007 10.002 0.012
7.204 0 1/2* + 0.017 10.005 0.019
7.204 2 5/2* + 0.005 10.001 0.012
7.302 3 7/2 - 0.011 10.003 0.017
7.353 2 5/2 + 0.040 10.020 0.007
7.604 2 3/2 + 0.004 10.001
7.604 2 5/2 + 0.013 10.004
76m 0.000 1 0 + 1.915 10.383 1.640
“N1 0.000 1 3/2 - 0.263 10.026 0.346
0.067 3 5/2 - 0.368 10.1 10 0.507
0.283 1 1/2 - 0.363 10.051 0.615
0.656 1 1/2 - 0.015 10.005 0.027
1.100 1 3/2 - 0.012 10.004 0.027

 

 

 

 

 

 

 

 

 

 

 

82

 

Table 2.6 (con’t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex (MeV) I J 11: SF(ADWA) Error SF(ENDSF)
1.132 3 5/2 - 0.036 $0.011 0.067
1.185 1 3/2 - 0.049 $0.015 0.064
1.729 1 3/2 - 0.006 $0.002 0.01 1
2.122 4 9/2 + 0.499 $0.071
2.124 1 1/2 - 0.242 $0.034
2.640 1 1/2 - 0.028 $0.009
2.640 1 3/2 - 0.014 $0.004
2.697 2 5/2 + 0.062 $0.019 0.087
2.765 1 3/2 - 0.008 $0.003 0.014
2.863 1 1/2 - 0.010 $0.003
2.863 1 3/2 - 0.005 $0.002
3.062 0 1/2 + 0.023 $0.007
3.273 1 (3/2) - 0.002 $0.001 0.003
3.382 1 1/2 - 0.007 $0.002
3.382 1 3/2 - 0.003 $0.001
3.506 2 3/2 + 0.158 $0.047
3.506 2 5/2 + 0.105 $0.031
3.686 1 1/2 - 0.018 $0.005
3.686 1 3/2 - 0.009 $0.003
4.568 2 (3/2) + 0.006 $0.002
4.568 2 (5/2) + 0.004 $0.001
4.600 2 5/2 - 0.004 $0.001 0.005
5.112 1 1/2 - 0.035 $0.010
5.112 1 3/2 - 0.018 $0.005
5.185 0 1/2 + 0.027 $0.008 0.051
5.309 0 1/2 + 0.012 $0.004 0.027
5.723 2 (3/2) N 0.055 $0.016
5.723 2 (5/2) N 0.036 $0.01 1
5.987 0 1/2 + 0.021 $0.006
6.016 2 (3/2) + 0.006 $0.002
6.016 2 (5/2) + 0.004 $0.001
6.346 2 3/2 + 0.019 $0.006
6.346 2 5/2 + 0.013 $0.004
6.371 2 3/2 + 0.008 $0.002
6.371 2 5/2 + 0.006 $0.002
6.609 2 3/2 + 0.005 $0.002
6.609 2 5/2 + 0.004 $0.001

“N1 0.000 1 0 + 1.619 10.324
1.173 1 2 + 0.218 $0.065
2.049 1 0 + 0.280 $0.084
2.336 3 4 + 0.274 $0.082
2.891 1 0 + 0.505 $0.152
3.059 3 2 + 0.233 $0.070
3.158 1 2 + 0.052 $0.016

 

 

 

 

 

 

 

 

83

 

Table 2.6 (con’t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex (MeV) I J n SF(ADWA) Error SF(ENDSF)

3.262 3 (2) + 1.119 10.336
3.370 1 1 + 0.295 $0.089
3.370 1 2 + 0.177 10.053
3.519 1 2 + 0.248 10.074
3.757 4 3 — 0.361 10.108
3.849 1 0 + 1.028 10.309
3.849 1 l + 0.343 10.103
3.849 1 2 + 0.206 10.062
4.393 3 (2) N 0.144 10.043
4.503 4 (3) - 0.264 10.079
4.720 4 (3) - 0.791 10.237
4.863 4 5 - 1.079 10.324
4.863 4 6 - 0.913 10.274
5.331 2 (3) - 0.163 10.049
5.545 4 3 - 0.653 10.196
5.545 4 4 — 0.508 10.152
5.545 4 5 - 0.416 10.125
5.545 4 6 - 0.352 10.106
5.628 2 3 - 0.024 10.007
6.103 2 1 - 0.451 10.135
6.103 2 2 - 0.270 10.081
6.103 2 3 - 0.193 10.058
6.103 2 4 - 0.150 10.045
6.540 2 1 - 0.350 10.105
6.540 2 2 — 0.210 $0.063

“N1 0.000 1 1/2 — 0.176 10.025 0.370
0.087 3 5/2 - 0.234 10.070 0.563
0.156 1 3/2 - 0.177 10.053 0.275
0.518 1 3/2 - 0.042 10.008 0.080
1.001 1 1/2 - 0.184 10.037 0.330
1.292 4 (9/2) + 0.565 $0.169 0.750
1.324 1 3/2 - 0.028 $0.008 0.063
2.297 2 5/2 + 0.189 10.027 0.142
2.697 1 1/2 - 0.023 10.003 0.045
2.953 0 1/2 + 0.128 $0.038 0.190
3.104 2 3/2 + 0.016 10.005
3.104 2 5/2 + 0.011 10.003
3.283 2 (5/2) N 0.041 10.012 0.053
3.292 2 5/2 + 0.037 10.01 1
3.740 2 (3/2) N 0.030 10.009 0.040
3.951 2 5/2 + 0.074 10.022 0.100
4.387 2 5/2 + 0.038 10.011 0.062
4.622 2 3/2 + 0.053 10.016
4.622 2 5/2 + 0.036 10.005

 

84

 

Table 2.6 (con’t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nucleus Ex (MeV) I J n SF(ADWA) Error SF(ENDSfl

5.060 2 (3/2) + 0.009 10.003
5.060 2 (5/2) + 0.006 10.002

65m 0.000 3 5/2 - 0.218 10.031 0.338
0.063 1 1/2 - 0.399 $0.056 0.620
0.310 1 3/2 - 0.022 10.003 0.035
0.693 1 3/2 — 0.093 $0.028 0.235
1.017 4 9/2 + 0.738 10.221 0.085
1.418 1 1/2 - 0.038 10.011 0.257
1.920 2 5/2 + 0.173 10.052
2.163 1 (1/2) N 0.031 10.009
2.325 3 (5/2*) N 0.030 10.009
2.325 4 (9/2*) N 0.050 10.015
2.336 3 (5/2) N 0.085 10.025
2.336 3 (7/2) N 0.063 10.019 0.003
2.712 2 3/2 + 0.003 10.001
3.044 1 (1/2) N 0.022 10.007
3.044 1 (3/2) N 0.01 1 10.003
3.41 1 2 (3/2) + 0.130 10.039
3.41 1 2 (5/2) + 0.087 10.026
3.463 2 (3/2) N 0.008 10.002
3.463 2 (5/2) N 0.005 10.002 0.082
3.563 2 5/2 + 0.065 10.013 0.042
3.743 2 5/2 + 0.031 10.009 0.068
3.907 2 5/2 + 0.058 $0.018
4.391 2 3/2 + 0.057 10.017
4.391 2 5/2 + 0.038 10.011

 

 

 

 

 

 

 

 

85

 

Table 2.7: Comparison of experimental and large-basis shell-model energy levels and

spectroscopic factors for Ni isotopes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex (MeV) 81‘
Nucleus 1 J P NUDAT GXPFIAJJ4PNA ADWA Error GXPFlA JJ4PNA
57m 1 3/2 - 0 0 0 0.954 10.286 0.783 1.000
3 5/2 - 0.769 0.825 0.714 1.400 10.42 0.76 1.000
1 1/2 - 1.1 13 1.184 1.302 1.000 10.3 0.698 1.000
mm 1 0 + 0 0 0 0.890 $0.087 1.105 1.118
75m 1 3/2 - 0 0 0 0.444 10.045 0.477 0.574
3 5/2 - 0.339 0.364 0.472 10.059 0.597
1 1/2 — 0.465 0.595 0.424 10.06 0.504
1 1/2 - 1.301 1.371 1.103 0.166 10.031 0.175 0.685
3 5/2 - 1.68 1.439 0.062 $0.016 0.032
1 3/2 - 1.735 1.906 0.004 10.001 0.008
4 9/2 + 3.061 3.454 0.479 $0.096 0.938
4 9/2 + 4.709 4.540 0.049 10.024 0.007
4 9/2 + 5.429 5.418 0.080 10.015 0.028
“m 1 0 + 0 0 0 1.915 10.383 1.746 2.496
“N1 1 3/2 - 0 0 0.547 0.263 $0.026 0.244 0.278
3 5/2 - 0.067 -0.006 0.364 0.368 10.1 1 0.527 0.727
1 1/2 - 0.283 -0.008 0 0.363 10.051 0.609 0.683
1 1/2 - 0.656 1.457 0.015 10.005 0.188
3 5/2 - 1.132 1.277 0.036 10.011 0.072
1 3/2 - 1.729 1.835 0.006 10.002 0.010
4 9/2 + 2.122 2.516 0.499 10.071 0.917
1 1/2 - 2.124 2.280 0.242 10.034 0.007
1 3/2 - 3.686 3.669 0.009 10.003 0.001
62m 1 0 + 0 0 0 1.619 10.324 1.635 2.522
1 2 + 1.173 1.148 0.218 $0.065 0.284
1 0 + 2.049 2.188 2.263 0.280 10.084 0.075 0.259
3 4 + 2.336 2.256 2.317 0.274 10.082 0.247 0.275
1 0 + 2.891 2.740 0.505 10.152 0.153
63M 1 1/2 - 0 0 0 0.176 10.025 0.412 0.634
3 5/2 - 0.087 0.158 0.171 0.234 10.07 0.476 0.576
1 3/2 - 0.156 0.373 0.319 0.177 10.053 0.083 0.138
1 3/2 - 0.518 0.77 0.643 0.042 10.008 0.163 0.107
1 1/2 - 1.001 1.216 1.282 0.184 10.037 0.118 0.079
4 9/2 + 1.292 1.546 0.565 $0.169 0.811
1 3/2 - 1.324 1.363 1.491 0.028 10.008 0.014 0.012
1 1/2 - 2.697 2.79 0.023 10.003 0.014

 

 

86

 

Table 2.7 (cont’d)

 

 

 

 

 

 

 

 

 

Ex (MeV) sr

Nucleus 1 J r NUDAT GXPFIA JJ4PNA ADWA Error cxrrm JJ4PNA
1 1/2 - 0.063 0.025 0 0.399 10.056 0.526 0.594
13/2 - 0.310 0.453 0.022 10.003 0.109
13/2 - 0.693 0.864 0.093 $0.028 0.056
49/2+ 1.017 1.082 0.738 10.221 0.797
1 1/2 - 1.418 1.100 1.425 0.038 10.011 0.040 0.024
49/2+ 2.325 2.474 0.05 10.015 0.083

 

 

 

 

 

 

 

 

 

 

 

 

Table 2.8. List of isotopes plotted in Fig. 2.21. J1r is the angular momentum and parity of
the transferred nucleon. For the (p,d), (d,p) [Tsa05, Lee07], and (e,e’p) [Kra01,Wes92]
reactions, only ground-state SFs are extracted. The theoretical SF values are obtained
from the LB-SM code OXBASH [Br002,Lee07,Br004]. For the neutron and proton
knockout reactions [TerO4,Gad04-l,Gad04-2,Gad05mEbd03], the deduced quantities are
the cross-section reduction factors Rs , which are equivalent to SF(expt)/SF(LB-SM).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1M] ii (d,p) J Sn-Sp SF(expt) SF B—S SF ex t /SF B-S

B 1/2- 4072 0.4 1 0.06 0.83 0.48 1 0.07
”c 3/2- 2.75 2.16 1 0.25 2.85 0.76 1 0.09
13c 1/2- -12.58 0.54 1 0.07 0.61 0.88 1 0.12
”c 1/2- -12.65 1.07 1 0.22 1.73 0.62 1 0.12
”N 1/2— 3.00 0.48 1 0.08 0.69 0.69 1 0.11
”N 1/2- 0.62 0.93 1 0.15 1.46 0.64 1 0.1
1‘0 1/2- 3.53 1.48 1 0.16 2.00 0.74 1 0.08
1"’0 5/2+ -9.64 0.75 1 0.1 1.00 0.75 1 0.1
‘80 5/2+ -7.9 1.46 1 0.17 1.58 0.92 1 0.11
190 5/2+ -1312 0.35 1 0.05 0.69 0.51 1 0.07
stg 5/2+ 4.73 0.21 1 0.02 0.34 0.61 1 0.07
2“Mg 5/2+ -306 1.83 1 0.38 2.51 0.73 1 0.15
"A; 5/2+ 4.79 0.93 1 0.13 1.10 0.84 1 0.12
”A; 1/2+ -1.82 0.57 1 0.08 0.60 0.95 1 0.14
3°31 1/2+ .2.9 0.55 1 0.07 0.82 0.67 1 0.08
3‘51 3/2+ -7.78 0.42 1 0.07 0.58 0.72 1 0.11
32]) 1/2+ -071 0.39 1 0.07 0.60 0.65 1 0.11
34s 3/2+ 0.54 1.11 1 0.27 1.83 0.61 1 0.15
37A, 3/2+ 0.08 0.27 1 0.04 0.36 0.74 1 0.1
4°C, 3/2+ 7.31 3.2 1 0.46 4.00 0.80 1 0.11
“c. 7/2- -0.53 0.73 1 0.04 1.00 0.73 1 0.04
“ca 7/2- 1.2 1.31 1 0.12 1.81 0.72 1 0.06

 

 

 

 

 

 

 

87

Table 2.8 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, (p,dg, (d,p) J Sn-Sp SF(expt) SF(LB-SIVD SF(expg/SfiLB-SM)
c, 7/2- -275 0.44 1 0.05 0.75 0.59 1 0.07
45c. 7/2- -4.88 0.26 1 0.04 0.50 0.52 1 0.07
47c. 7/2- -6.93 0.19 1 0.03 0.26 0.74 1 0.1
48Ca 7/2— -5.86 5.41 1 1.05 7.38 0.73 1 0.14
“c. 3/2- -1 1.3 0.74 1 0.08 0.92 0.81 1 0.08
“n 7/2- 2.85 1.61 1 0.23 2.58 0.62 1 0.09
(e,e'Ji) .1 Sp—Sn SF(expt) SF(LB-SM) SF(expt)/SF(LB-SM)
71,. 3/2- 2.73 0.42 1 0.04 0.67 0.63 1 0.06
”c 3/2- -275 1.72 1 0.11 2.85 0.60 1 0.04
1‘0 1/2— -353 1.27 1 0.13 2.00 0.64 1 0.07
303: 5/2+ 2.9 2.21 1 0.2 3.8 0.58 1 0.05
31p 0+ 501 0.4 1 0.03 0.58 0.68 1 0.04
“c. 3/2+ -73 2.58 1 0.19 4.00 0.65 1 0.05
4801 1/2+ 5.86 1.07 1 0.07 1.98 0.54 1 0.04
51v 7/2- -299 0.37 1 0.03 0.75 0.49 1 0.04
902, 1/2- -3.62 0.72 1 0.07 1.28 0.56 1 0.05
mm, 1/2+ 0.63 0.98 1 0.09 2.00 0.49 1 0.05
n-knockout J Sn-Sp SF(expt) SF(LB-SM) SF(expt)/SF(LB-SM)
12C incl 3.07 0.49 1 0.02
15c 1/2+ -19.86 0.98 0.96 1 0.04
160 incl 7.64 0.56 1 0.03
To 5/2+ -16.39 5.22 0.7 1 0.06
32Ar 5/2+ 19.2 4.12 0.25 1 0.03
34Ar incl 13.94 0.41 :1: 0.07
“A, 7/2- -1003 5.41 0.85 1 0.12
p—knoekout .1 Sp-Sn SF(expt) SF(LB-SM) SF(expt)/SF(LB-SM)
83 incl -12.82 0.86 1 0.07
1? 3/2- -12.96 0.94 0.82 1 0.06
r2c incl -243 0.53 1 0.02
160 incl 0.68 1 0.68 1 0.04

 

88

 

Table 2.9: Spin assignments of three excited states, 3.491, 4.150 and 5.627 MeV of 27Mg.
The NUDAT [NNDC] values listed in column 2 are not confirmed by experiments. The
shell-model information is listed in Column 4, 5 and 6. Our recommended spin values are
listed in the last column.

 

 

 

 

 

 

 

 

 

1" " SF(ADWA) E(LB-SM) " SF SF(ADWA)/SF(LB—SM) " (““5
(MeV) (NUDAT) (LB-SM) (LB-SM) won-IQ
0.049 1 0.097 1
3.49 (3/2+) 0.01 5 3.562 3/2+ 0.012 0.511017 3/2+
0.032 1
(5/2+) 0.010
0.038 1
4.15 (3/2+) 0.01 1
0.025 1 0.029 1
(5/2+) 0.007 4.097 5/2+ 0002 0.861025 5/2+
0.129 1 0.144 1
5.627 (3/2+) 0.039 5.561 3/2+ 0. 003 0.891027 3/2+
0.085 1 0.0054 1
(5/2+) 0.026 5.690 5/2+ 0' 1614.8

 

 

 

 

 

 

 

 

 

 

89

Chapter 3

Experimental Setup and Detectors

To understand the nature of neutron correlations in the regions of asymmetric nuclei,

spectroscopic factor measurements using (p,d) neutron transfer reactions have been

performed using proton-rich 34Ar and neutron-rich 46Ar beams in inverse kinematics. The

experiments of p(34Ar,d)33Ar and p(4 6A1’,d)45Ar with beam energy of about 33

MeV/nucleon were performed at the National Superconducting Cyclotron Laboratory at
Michigan State University. In this chapter, the method of measurement and experimental
design are discussed. It is followed by the description of the detectors. In addition, the
method and results of position measurement for the experimental setup are presented.

The electronic and data acquisition system for the present experiments are also included.

3.1 Experimental design

3.1.1 Method of measurement

The goal of the present experiment is to determine the neutron spectroscopic factors of
4 . . . . . . .
3 Ar and 46Ar us1ng (p,d) smgle-nucleon transfer reactions 1n 1nverse k1nemat1cs. The

differential cross sections of emitted deuterons are the experimental observables. We
used the high resolution silicon array, HiRA [WalO7], to measure the energies and angles
of the emitted deuterons. To ensure the particles observed were actually the deuterons of

interest, forward going recoil residues were detected in coincidence in the S800 mass

90

spectrometer [Yur99, BazO3]. With these kinematically complete measurements, the
coincident deuteron energies and angles could be used to identify the states p0pu1ated in

the final nuclei.

Studies of unstable p-rich 34Ar and n-rich 46Ar measurements were carried out in
inverse kinematics by impinging the radioactive 34Ar and 46Ar beams on polyethylene
(CH2)n targets. Since 36Ar(p,d) reaction has been studied before in normal

kinematics[Koz68], the 36Ar induced reaction was used as the calibration beam for our

detectors. Due to the considerable size of the secondary beam spot, two multi-channel
plate detection systems (MCP) [ShaOO-l, Sha00-2] with thin Carbon foils were employed
to track the positions of the incoming beams and correct the deuteron angles for the beam

positions on the target [RoglO-Z]. Figure 3.11 summarizes the designed experimental
setup. The HiRA array with 16 telescopes was placed at 35 cm from (CH2)n reaction

target. The design of this configuration is explained in Section 3.1.3. Two MCP foils
were positioned individually at 10 cm and 60 cm upstream of the reaction target. To
avoid blocking the beam with the MCP detectors, the foils are orientated at an angle of
60° relative to the beam line. The MCP also monitored the absolute beam intensities
throughout the experiment for overall normalization of deuteron cross sections. The
HiRA, MCP and reaction targets were accommodated in the $800 scattering chamber in

front of the S800 spectrometer as shown in Figure 3.2.

91

Designed Experimental Setup

4'1“ A, 1
74M...

..71 .1 “

ce relative a ct:
MCPO - 600 mm Target - 0 °
MCP] - 100 mm MCPO - 60 °
HiRA - 350mm MCP! - 60°

 

Figure 3.1: Simplified schematic diagram of the designed experimental setup
inside the 8800 chamber. The 4 posts were not placed inside the chamber during
experiment. They were used only in the LBAS measurement as discussed in
Section 3.5.

 

$800 Spectrometer

   
 
 
 

Scattering
Chamber

Figure 3.2: 3-D rendering of the $800 spectrometer aligned with the scattering
chamber and analysis beam line.

 

 

 

92

Precise reconstruction of the reaction dynamics requires accurate knowledge of
the beam trajectory and alignment of detectors with respect to the reaction target. Sub-
millirneter position measurement without mechanically touching the experimental setup
(especially for the fragile MCP Carbon foils and reaction targets) is essential. The
working principle and performance of the position measurements using a laser-based

alignment system (LBAS) are discussed in Sections 3.5 and 3.6.
3.1.2 Radioactive beams and reaction targets

Our survey of (p,d) and (d,p) transfer reactions suggests that incident energies ranging
from 10 to 20 MeV per nucleon permit accurate comparison of theoretical and
experimental angular distributions, as discussed in Section 2.2.5. The radioactive beam is
produced at 120 MeV/nucleon or higher, and its intensity generally decreases as the beam
energy is decreased by degrading with absorbers in the A1900 fragment separator.
Therefore there is a compromise between the intensity of the radioactive beam and the

optimum choice of bombarding energy. With all the optimal and practical issues
considered, we chose the energy of 33 MeV/nucleon for the 34Ar and 46Ar secondary

beams in the present measurements.

It is desirable to check whether the beam energy satisfies the angular momentum
matching condition presented in Equation 3.1, even though it is not a necessary condition

[Joh89].

LZQ'Rlein‘ outl'R (3-1)

93

where L is the momentum of the transferred nucleon, K5,, and K0", are the wave number

in the entrance and exit channel respectively and R= roAm.

If such condition is well-matched, the transferred angular momentum is given by
the condition to Q-R-l _<_ Ls Q-R+l. In such cases, the nucleon-transfer probability to that
particular state will be relatively large and dependent on the accurately calculated parts
of the DWBA integrals. As a result, the simple one-step DWBA description to the data is

more valid [Sch09]. Figure 3.3 shows the momentum matching calculations for valence

neutron in the p(34Ar,d)33Ar and p(46Ar,d)45Ar reactions. A beam energy of 33
MeV/nucleon gives Q-R of 0.1 and 1.5 for p(34Ar,d)33Ar and p(46Ar,d)45Ar reactions
respectively. The neutron transition to the ground-state of 33Ar is 2s1/2; while to the
ground- and first-excited states of 45Ar are lfi/z and 2p3/2. Even though the I-transfer to

the ground-state of 45Ar is slightly outside this criterion, the 33 MeV/nucleon beam

energy is not unreasonable for the reactions being studied.

94

 

; Momentum matching for
. transferred neutron .

3 _m' p(“Ar,d)‘5Ar

'
a
a
a
a
a
o
I
4'
I
o
O
t
O
’
‘o
I
C
a
’d
a

Q‘R

 

 

 

 

20 4O 80

60
Em (MeV/ A)

Figure 3.3: Momentum matching for neutron for the [=0 ground-state transition of
p( 4Ar,d) 33Ar and [=3 ground-state transition of p(4 6Ar,d) 45Ar reactions.

The primary beams 36Ar and 48Ca were produced by the Coupled-Cyclotron

Facility (CCF) [NSCL94, Mor94, She02] which consists of two superconducting

cyclotrons, the K500 and the K1200. Using the in-flight projectile fiagmentation

technique, secondary beams of 34Ar and 46Ar at 33 MeV/nucleon were produced by
fragmentation of 150 MeV/nucleon 36Ar primary beam on a 1480 mg/cm2 9Be production

target and 140 MeV/nucleon 48Ca beam on a 1763 mg/cm2 9Be production target

respectively. After fragmentation, the radioactive beams were separated by the A1900

large-acceptance fragment separator [Mor03]. A schematic of A1900 fragment separator
is shown in Figure 3.4. A 375 mg/cm2 thick achromatic aluminum wedge degrader and
momentum slits at the dispersive image of the separator were employed to purify the

beams. The secondary 34Ar and 46Ar beams were identified unambiguously by the time

95

of flight from cyclotron to the A1900 focal plane, using the radio-fi'equency (RF) of
cyclotron and a scintillator at the extended focal plane of the A1900. In addition, 36Ar

primary beam was degraded to 33 MeV/nucleon for calibration purposes. (CH2)n targets

34,36

with thickness of 7.10 mg/cm2 for the p( Ar,d) reactions and 2.29 mg/cm2 for the

6 . . . . . .
p(4 And) reactlon were chosen, which compromrse between maxrmrzrng the cross-

section yields and minimizing the energy loss and the angular and energy straggling of

deuterons in the target.

RT-ECR Q, Q 20 n

if: 10m
K500 —————
couplingline ........ .1 :

 

Figure 3.4: Schematic of the Coupled Cyclotron Facility at the NSCL. The K500
and K1200 are connected by the coupling line. The 1900 fragment separator is
enclosed in the dashed-line area.

3.1.3 Geometry of HiRA detector Array

In this experiment, HiRA detects the deuteron particles emitted from the p(34’36’46Ar,d)

reactions. Deuteron angular distributions in the center-of-mass frame calculated using

TWOFNR code for the ground-state transfers for the three different reaction systems are
96

displayed in the left panels of Figure 3.5. The right panels show the corresponding
angular distributions in laboratory frame. In inverse kinematics, all deuterons are emitted
to the forward angles and any single angle in laboratory corresponds to two different

deuteron energies and COM angles. This can be understood by the velocity diagram

displayed in Figure 3.6, where VOCOM is the velocity of the center of mass in the
lab COM . .
laboratory frame; Vd and Vd are the deuteron velocrty 1n the laboratory and center

of mass flame. The corresponding laboratory and center-of-mass angles are 0131, and OCOM

respectively. The circle represents the velocity of the deuteron in the center-of-mass
frame. The relations between the deuteron emitting angles in the center of mass and

laboratory frame for all three reactions are presented in Figure 3.7.

Figure 3.8 shows a photograph of the HiRA detector array and the downstream

MCP. A total of 16 HiRA telescopes were arranged in five towers positioned at 35cm
from the reaction target and subtended polar angles of 4°s 0m, 545°. The gap at angles

less than 4° allows the beam and heavy reaction fragments go to the 8800 spectrometer.
Due to the forward focusing of the deuteron particles, this setup covers most of the
important solid angle in the center-of-mass frame for all three reactions. The geometrical
efficiency of the HiRA array setup, calculated based on Monte-Carlo principle, is shown
in Figure 3.9, where 30-40% coverage is achieved in general. In addition to a large
geometry efficiency in the angular regions covering the first peak, the design of the HiRA
configuration also optimizes the energy and angular resolutions. At 35 cm setup, the

pixelation of the HiRA telescope allow angle determination to a precision of 1 016°.

97

The HiRA coverage is plotted in the Cartesian and Spherical coordinate systems
as shown in Figure 3.10. The beam direction is defined as the z axis and vertical direction
is defined as the x axis. The polar angle 6 defines the angle of the particle direction with
respect to the beam axis and the azimuthal angle 03 defines the angle between the particle
projection on the x-y plane and y axis. For reference, each telescope is assigned a number

as labeled in Figure 3.10.

 

100.0
10.0

 

100.0 . . . .

 

n...
O
O

 

pa
F’ $3
00-5

 

F’
a;

 

 

do/dQ (mb/sr)
3

 

 

 

 

 

 

 

1.0 ’
0.1 ’ (”Ar.d)”Ar‘§
’--4-l--1-l-1.-l.--.l‘
10 20 30 40
Blah (deg)
Figure 3.5: The calculated deuteron angular distributions of p(34’36’46Ar,d) for

ground—state transitions in center-of-mass flame (lefi panels) and in laboratory
flame (right panels) at beam energy of 33 MeV/nucleon.

98

Beam

 

Figure 3.6: Velocity diagrams for (p,d) reaction in inverse kinematics.

 

v r . v 'r y r f r T v

; Kinematics broadening

N
or

F

E - p(“AndWAr ,5

N
o

i ------ p(“Ar.d)‘”Ar
p(“Ar,d)“Ar

r- 1*
O 0'

Broadening (MeV/ deg)
O
in

 

40

 

 

P
o

 

Figure 3.7: The deuteron emitted angles in center of mass angle versus the emitted
angle in laboratory frame for p(34’36’46Ar,d) of ground-state transitions at beam

energy of 33 MeV/nucleon.

99

 

Figure 3.8: Photograph of the HiRA detector array and the downstream MCP.

 

HiRA Efficiency

0.5

   

0.4

0.3

Efficiency

0.2

0.1

 

00 5 1 0 1 5 20 25 30 35 40 45 50
91-h (deg)

Figure 3.9: Efficiency of HiRA setup in 2-degree bin where 16 telescopes were
placed at 35 cm flom the reaction target.

I-IiRA Position (Beam View)

 

     

 

 

 

 

 

   
 

 

 

 

 

ZOTT‘W" *rffi *Tfi'fitfi'fi’rr'fir
10 ~ u m 9
E L 1
3 0 r 1
12
.. _ I 3
u ~.
_20’ L1 - 1 _ 1 - - 1 1 _ 1 -
— 30 —-20 10 o 10 20 30
y (cm)
HiRA Angular Coverage
' ' I f ' f f T ' ' ' I T ' ' ' Wfi ' ' '
300 1 ~
A
m 1
m 1
L. .
8°
13 200 r r
v db
n .
2
e- .
100 ~ -
o - -
0 5o

elab (degree)

Figure 3.10: The HiRA configuration flom mechanical design in the Cartesian
coordinate systems (top) and Spherical coordinate systems (bottom) with a
number assigned for each telescope. The beam position is indicated as a red cross.

101

3.1.4 Total energy and kinematics measurement

In the present (p,d) measurements, deuteron energies were recorded from all the silicon
detectors, with the total energy in the telescopes calculated flom the sum of the DE, E
and CsI energies. The calculated kinematic curves for deuterons corresponding to
ground-state transitions in different reactions are plotted in Figure 3.11. The deuteron
punch-through energy of the silicon detectors is about 21.5 MeV. If deuterons are emitted
at the forward angles, they would have energy less than 21.5 MeV and therefore stop in
silicon E detectors. Particle identifications (PID) can be constructed by using DE and E.
The Csl crystals covering the forward angles are mainly used to veto energetic particles
that punch through the silicon detector and reduce the background in the DE-E PID
spectra. In the lower right panel in Figure 3.11, the red/blue shaded part in the detector

array represents the region where deuterons flom the (p,d) reactions stop in the thick
silicon detectors for 34Ar (red) and 46Ar (blue and red) reactions. Deuterons emitted at

larger angles have energies larger than 21.5 MeV and Stop in the CsI crystals. E-Csl
spectra can be used for PID. No DE was mounted to telescope 5 and 6 which allow

monitors of alpha calibration for E detectors thoughout the whole experiment.

102

50 .
40
30
20 :
10 L —g.s. ,
' -- Ex=1.38 MeV ; 5
50 .:; -:,;-;;1:,, ..
40 ' p(3°Ar.d)
30
20
10 7 —g.s. _’ . ‘
3.. . . 3:11-1:11 11°11- .. ..afl'gg.

O ..
0 10 20 30 400 10 20 30 40
91:11: deg

 

wvvlu

v'vvvv'vvv Iv

p “Ar.d) 1

l [1 1
vv'Irvv'Ivvv
a

—g.a. -
_:_ -- Ex=o.54 uev

   

 

-_
I

Energy in lab.(MeV)

    

.l

(Sap) . “Bib

 

 

 

 

Figure 3.11: (Top panels and bottom lefi panel) Kinematic curves of deuterons
flom p(34‘36’46Ar,d) reactions at beam energy of 33 MeV/nucleon. (Bottom Left)
HiRA configuration in the spherical coordinate system with the shaded part
representing the region where deuterons stop in the thick silicon detectors for 34Ar
(red) and 46Ar (red and blue) reactions

3.2 HiRA Detector Array

The High Resolution Array (HiRA) is a state-of-the-art detection array capable of
detecting charged particles produced in nuclear reactions. It allows precise measurement
of energy, charges and masses of various charged particle species with high angular
resolution. Currently, HiRA consists of twenty identical detector telescopes. Depending

on the experimental requirements, HiRA can be configured into different geometries with
103

the flexibility in using different number of detectors as demonstrated in photographs of
actual experimental set ups in Figure 3.12. Each HiRA telescope contains 2 layers of
position-segmented silicon detectors backed by four separate CsI(Tl) crystals mounted in
quadrants. An expanded view of a single telescope is shown in Figure 3.13. Of critical
importance in HiRA is the ability to identify the mass and element number of various
particles to an excellent degree of resolution. For particle identification, HiRA relies on
the technique of AE vs E identification where the energy loss AE is roughly proportional
to AZZ/E. A plot of AE vs E uniquely identifies A and Z of the charged particle that stop

in the detectors. Further details of the HiRA detectors can be found in Ref. [Wa107].

 

Figure 3.12: Photographs of different HiRA setup used in four experiments,
where the bottom-right shows the configuration used in the present measurement.

 

Figure 3.13: Schematic of a single HiRA telescope containing two layers of
silicon detectors with thickness of 65 um and 1.5mm backed up by a cluster of
four CsI crystals.

3.2.1 Silicon strip detectors and Csl(Tl) detectors

Silicon detectors have been widely used in nuclear experiments because of their excellent
energy resolution and linear energy response over a large dynamic range for charged
particles. As displayed in Figure 3.13, a HiRA telescope consists of a single-sided thin
(65 um) and double-sided (1500 um) silicon detectors. Each has active surface area of
6.25 cm x 6.25 cm segmented into 32 position-sensitive strips with pitch of 1.95 mm. The
32x32=1024—pixel constructed by the flout-vertical and back-horizontal strips in the
double-sided E silicon detector allows excellent angular measurement of the detected
particles. If the array is set up at 35cm from the reaction target, the angular resolution
achieved is i 0.l6°. There is a gap of 25 um between the active strips. Its effect is

included in the efficiency determination of HiRA.

The thin detector is denoted as “DE” as it is used to measure the energy loss by

the particle passing through the thin detector. The thick detector is denoted as “E” with
105

the front and back sides labeled as “EF” and “EB” respectively. To allow the close-
packed design of the array, flexible polyimide cables with small printed circuit boards
connected to the 34—pin Amp female header are used to read the signals flom the
detectors. Figure 3.14 and 3.15 show photographs of the flont side of a single-sided and a

double-sided Si strip-detector respectively.

 

Figure 3.14: HiRA single-sided silicon strip-detector in a circular plastic container.

 

Figure 3.15: HiRA double-sided silicon strip-detector in a circular plastic container.
106

Scintillators fabricated from thallium doped CsI (Cesium-Iodide) crystals have
been extensively used for detecting charged particles because they are cost effective and
easily machined into different shapes. In addition, CsI crystals have reasonable energy
resolutions and light output uniformity. Each HiRA telescope contains four Csl(Tl)
scintillation crystals arranged in quadrants behind the silicon detectors. The trapezoidal

crystals are 3.5 x 3.5 cm2 on the flont and 3.9 x 3.9 cm2 in the rear. For compact packing,

the sides between adjacent crystals are cut straight back while the sides next to the flame
are cut at an angle of 5.3°. With a thickness of 4 cm, all CsI(Tl) crystals can stop proton,
deuteron and alpha with energy up to 115.8 MeV, 154.8 MeV and 462.3 MeV

respectively. A light guide is attached to the back of each CsI. Glued on the back of the
light guide is a photodiode with active area of 1.8 x 1.8 cm2 which is painted with

reflective white paint BC600 to prevent 1ight leak and cross-talks. Figure 3.16 displays
the side and flont view of a CsI crystal, 1ight guide and photo-diode assembly. Each
_ crystal is individually wrapped in cellulose nitrate membrane filter paper to maximize the
light collection efficiency. To ensure optical isolation and maximize light reflection, the
front surface of the crystal array and the sides between crystals are covered with 1.9 um

thick aluminized Mylar foil.

107

Light Guide Photo Diode

 

 

 

 

CsI(Tl) Crystal

Figure 3.16: Side and flont view of a CsI crystal, light guide and photo-diode assembly.

3.2.1.2 Readout Electronics -- Application Specific Integrated Circuit

(ASIC)

The HiRA array with 20 telescopes of DE and E consists of 1920 individual silicon strips.
Application Specific Integrated Circuits (ASIC) were developed for signal processing to

greatly reduce the cost and space to implement the readout electronics [Wa107].

The HiRA ASIC contains charge sensitive amplifiers, pseudo constant fraction
discriminators, shaping amplifiers, time to analog convertors and a sample and hold
circuit along with the digital logic necessary for communication and multiplexing the
signals out. Figure 3.17 shows the block diagram of the HiRA ASIC. Each ASIC chip
processes signals for 16 individual channels. Since HiRA silicon detectors have 32 strips
on each surface, two ASIC chips with other high density electronics mounted on a

chipboard are used for each surface. Input and output signals of several chipboards are

108

merged by a circuit board called the “motherboard”. Each motherboard can accept 15
chipboards. Since three are used for each telescope, this allows up to 5 detectors
connected within one motherboard. Photographs of a motherboard and chipboard are

shown in Figure 3.18. Details of the HiRA electronics can be found in ref [Eng07].

In this experiment, 16 telescopes were used. All the chipboards for EF and EB
were inserted to four motherboards placed inside the scattering chamber with each
motherboard supporting readout for four telescopes. The chipboards for DE were
processed by another two motherboards placed outside the chamber. DE signals were

pre-amplified externally because of higher noise due to low capacitance of the thin Si

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

detectors.
Shaper -o« HP-Filter Peak Find Peak Hold
—1 OR r- H1 .
Pseudo CFD H“ Reg
E 1 WC ak F d P ak H 1d
. —- P in e 0
External Stop 2:: e

 

 

 

 

 

 

 

0 Inspection Points

Figure 3.17: Block diagram of the HiRA ASIC.

109

 

Figure 3.18: (Lefl) Photograph of a chipboard with two ASIC’s attached. A US.
quarter is used for size reference. (Right) Photograph of a motherboard with 6
chipboards inserted.

3.3 8800 Spectrometer

The 8800 Spectrograph is a high resolution and high acceptance spectrometer
[Yur99, BazO3]. It covers a solid angle of 20 msr and momentum acceptance of 5%. The
setup is composed of two parts: the analysis line and the spectrograph itself. Figure 3.19
shows the schematic of $800 spectrometer. The analysis line can be operated in two
optics modes: focus mode and dispersion matching mode. In this experiment, focus mode
was used where the secondary beam is focused on the reaction target and dispersed on the
$800 focal plane. This mode allows large momentum acceptance of approximately : 2%,
but it limits the momentum resolution at the focal plane of the spectrograph to about 1
part in 1000. For best resolution, tracking of the incident beam is necessary to recover

momentum resolution of the reaction products. Behind the 8800 scattering chamber are
110

two superconducting dipoles for selecting the reaction flagments of interest by setting the
corresponding magnetic rigidity. The selected fragments will reach the focal plane
detector array which measures their positions, energies and timing. Three quantities
obtained in the $800, including magnetic rigidity time-of-flight and energy loss, suffice

to identify the residues of the reactions.

    

From A1900

Figure 3.19: Schematic of S800 spectrometer.

The focal plane detector array of the S800 spectrograph consists of two Cathode
Readout Drifi Chambers (CRDC) followed by an ion chamber and a stack of three plastic
scintillators [Yur99, BazO3]. The schematic of the focal plane detector system is shown
in Figure 3.20. Two CRDCs are gas-filled high-voltage single-wire drift detectors with
active areas of 30 cm x 59 cm separated by approximately one meter. Figure 3.21 shows
the schematic of CRDC detector. Particles traveling though the detectors ionize the gas.

The electrons drift to the anode wire by the constant vertical electric field in the detectors

111

and the drift time of the electrons gives the vertical position information (non-dispersive).
A total of 224 cathode pads with pitch of 2.54 mm located in the flont and back of the
anode wires collect the positive image charges induced by the anode currents. The
horizontal position (dispersive) is obtained flom the centroid of the Gaussian fit to the
charge distribution. The dispersive and non-dispersive angles can be calculated flom a
pair of associated position measurements in the CRDC detectors. The segmented
ionization chamber (1C) following the CRDCs is used for energy loss measurement.
Additional energy loss and total energy measurement as well as timing and trigger
information are provided by the three plastic scintillators labeled as E1, E2 and E3 with
thickness of 5, 10 and 20 cm respectively. Signals collected by the photomultiplier tubes

on both top and bottom ends of the scintillator were used.

 

El, E2, E3 Scintillators

 

 

 

 

 

 

Figure 3.20: The schematic of the focal plane detector system in $800.

112

  
  

      
 

A

b” _V G

 

//,/

   
         

////////.

Figure 3.21: Schematic of CRDC detector.

3.4 Micro Channel Plate detectors

In order to resolve the energy states of the residuals at high resolution level,
accurate determination of deuteron emission angle is required. Therefore beam tracking
that determines the beam positions to 1-1.5 mm, much smaller than the beam spot size
which is of the order of 10 mm (FWHM) and the beam incident angles on the reaction
target is necessary. Micro Channel Plate detector (MCP) is employed for beam tracking
in this experiment. The micro-channel plate is a compact electron multiplier array made
with many tiny glass tubes closely packed together. The plate itself has an active
diameter of 40 mm and thickness of 0.46 mm. The diameter of each tube is about 10 um

and the center to center distance between adjacent tubes is about 12 um. Figure 3.22
113

shows the schematic design of a Micro Channel Plate. The working principle involves the
amplification and detection of the secondary electron emission flom the reaction of the
beam on the MCP foil. An electron strikes the inner surface which is coated with material
having a low work function for electron emission. The impact then starts a cascade of
electrons that propagates through the tube as shown in Figure 3.23. An avalanche of
electrons will eventually be generated and emerges on the other end of the tube. In our
MCP detection system, there are two micro—channel plates in which the second plate with
angled channels oriented in the opposite angle as shown in Figure 3.23 to significantly
enhance the secondary electron emissions and amplify the signals. The secondary
electrons emitted flom the MCP are collected by a thin anode resistive layer placed
behind the second micro-channel plate. The position of the electrons on the anode is
determined flom the signal amplitude of the four corners of the anode resistive layer

[Sha00-l, Sha00-2].

 

Schematlc Construction and Operatlng Prlnclple of MCP
CHANNEL

 

 

 

 

 

Figure 3.22: Schematic design of a micro—channel plate (adopted flom [Wa105]).

114

Primary
electron

 

 

 

 

 

 

Anode

 

Two MCPs

Figure 3.23: The detection system with two micro-channel plates and the view of
chevron style MCPs (adopted flom [Wa105])..

The schematic setup of MCP detection system is shown in Figure 3.24. Secondary
electrons are produced when a beam ion strikes the foil. The electrons are accelerated by
an electric field generated by applying 1000 V bias between the foil and front surface of
the MCP. The strong voltage also ensures the drift time of electrons is sufficiently short
for excellent timing resolution. The overall fast timing response allows the MCP to
function at beam rates of up to 1x106 pps. To ensure good spatial resolution, a strong
magnetic field created by two permanent magnets (14 kG on the surface), manufactured
by Magnet Sales & Manufacturing Inc, part number: 35NERR192, confined the electrons
to narrow helical orbit while traveling towards the MCP. In our MCP detection system,
the magnetic field measured at the MCP and the MCP target foil positions are 5000 G

and 3000 G respectively.

115

 

 

Figure 3.24: The schematic setup of MCP detection system with MCP, two
permanent magnets, foil and anode layer (adopted flom [Wa105])..

In the experiment, two MCP detection systems were used and placed at 50 cm
apart, with MCPO and MCP] denote the upstream and downstream MCPs respectively.
The reaction target was located at 10 cm downstream of the MCPl as displayed in Figure
3.2. From the position information of the beam given by MCPO and MCPl, beam
positions and angles on the reaction target can be deduced accurately. The Ar beams used
in this experiment have masses of 34, 36 and 46 with charges of 18+, their trajectories are
influenced by the strong magnet fields in the MCP setup. When assuming the beam come
horizontally, the angle being deflected under the B-field can be simply estimated using
80=B-d/Bp, where B is the magnetic B-field at MCP foil (~3000G), d is the beam
traveling distance in B-field (~ 8cm) and Bp is the magnetic rigidity of the beam. The

estimated deflection angles are therefore 066°, 071°, 056° for 34Ar, 36Ar and 46Ar

respectively. The fields in the two MCP are in the opposite direction, so the beam angle is

not changed. Instead, if there is a net effect as a vertical displacement of the beam, this

116

basically means that the displaced dispersive angle calculated by the MCP is off by 80.
This needs to be taken in to account in the calibration. As explained later that we are

unable to use the MCP for actual beam checking, details for the optimal operation of the
MCP is described in the 56Ni(p,d) experiment NSCL Expt 06035 [Expt06035] which was

performed after the runs with Ar isotope beams.

3.5 Position Measurements with Laser Based Alignment System (LBAS)

Direct transfer reactions carried out in inverse kinematics using short-lived heavy

beams bombarding light targets, suffer from strong kinematic broadening. For example,

the kinematic broadening (dE/d0) of p(46Ar,d) at 33 MeV/nucleon in this experiment is

up to 2 MeV per degree for 91ab >36°. To resolve the energy states of the residual nuclei

at high resolution level, it is critical to accurately reconstruct the reaction dynamics where
sub-millimeter determination in positions of beam tracking systems for beam particles
and array of highly segmented detectors for reaction flagments is needed. Such accuracy
is desired as the actual position of the detectors and the designed position can deviate by
more than 1 mm due to torque and the weight of the detectors even with the best care in

assembling the designed mechanical mounts of the detectors.

Such measurement should be done without touching the detectors to avoid
altering the configuration setup and damaging the flagile components (such as the
detectors, reaction target and MCP foils). The device should be portable and small

enough to operate in the S800 scattering chamber. Based on these requirements, we used

117

a high precision alignment tool, the Laser Based Alignment System (LBAS), to perform

the measurement [Rog 1 0-1 ].
3.5.1 The Laser Based Alignment System (LBAS)

The LBAS is composed of 2 basic devices as shown in Figure 3.25: Acuity laser [Acuity]
and two OWIS rotary stages [OWIS]. The acuity laser (Schmitt Measuring Systems, Inc,
USA; Model no: Acuity AR600) is contained in a box of 90 mm long, 26 mm wide and
154 mm tall. Laser beam is projected flom the housing and is reflected flom the target
surface to the laser collection system. The laser collection system contains lens for
focusing the image of the spot on a CCD digital camera. The laser sensor employs
triangulation measurement principles to determine distance with high sensitivity to very
low amounts of reflected light. LBAS measures distance in the range of 25.4 to 40.6 cm

with resolution of 45.6 um.

The sensor housing box is mounted on two rotary stages with OWIS stepping
motors [OWIS] and controllers (Wheedco, Model no: IMJ-lOSD-l-D). The LBAS
stepping motors are controlled by two digital signal processing based motion controllers
independently. Two Rotary stages provide angular motion to the laser sensor for the
capability of measuring 360 degrees for both theta and phi with resolution better than
0.006 degrees. The overall position measurement can achieve resolution of 52.8-61.9 um.
A Java language based program was developed to control the laser, scan with specified
step sizes and output the distance, theta and phi in spherical coordinates to a file. Detailed

descriptions of LBAS components and operation can be found in ref. [RoglO-l].

118

Window for

laser beam 0 rotary stage

   

(p rotary stag

Figure 3.25: Photograph of Laser Based Alignment System (LBAS) with an
Acuity laser accommodated inside a box and two OWIS rotary stages.

To determine the position of an edge, laser beam is used to scan along the surface
planes and across the edge in between. The abrupt change in the trend of scanned points
locates the edge point. Figure 3.26 presents the measured angles in 0-direction verse
distances (p) obtained in a typical laser scan across a hole on a plate where (p-rotation
stage remains stationary. The scanned object is displayed in the right panel of Figure
3.26. A circular hole of 2 mm diameter is located at the center of the rectangular plate,
which is used as target mask as described in Section 3.6.2. The hole was covered by a
plastic foil on the back side of the plate, which allows laser-light reflection in the hole

opening. The red dotted line indicates the path of the laser scan.

The slight fluctuation of scanned points can be attributed to the surface
imperfection and roughness of the object; while the outlining data points can be attributed
to the variation of the laser beam transportation and deflection. The data with larger

distance corresponds to the measurements in the hole region and reflect the plate

119

thickness of 1.56 mm by design. The abrupt change in the scanned points locates two
edge positions of the hole as indicated in the right panel of Figure 3.26. The separation
between two edges of about 1.78 mm suggests the laser does not travel through the center
of the hole. With another laser scan in the horizontal direction, a total of 4 edge positions
of the hole are obtained and the hole center can be interpolated. The measured data points
need to be corrected for the offsets due to a non-superposition of the rotation axis of

motor stage and the laser source as described in Section 3.5.

 

 

  
   

 

 

 

 

 

 

 

365 365
p=362.57mm
A2}, - 0-9037'
3"»; ' 02;”? hirer Mask
364 364‘ _ ,
2 ’ \2
A <33.
E 1- -. _
5383 i"“i‘1'-,%,,:~'i 23,3. :2: sea— 35;“
.' ' I.-. : '3'; .'_'.' '1 ’ p= . mm
0. . , -. ‘i‘: :‘3 . 'gtigl. gflxfifigg . e: ”-66.
- ' :. .1; c:-{_ 3 - o=-o.os-
.2 " ‘1 - -' -
3o .'.‘ 362‘
IIII Illllllllllllll‘llllllll

 

 

89.5 90 90.5 91 91.5 -0.1-0.05 0 0.05 0.1 0.15 0.2
0“, (dog) 9“, (deg)

Figure 3.26: Data of angles in 0-direction verse distances (p) obtained in a laser
scan across a hole on a plate where (p-rotation stage remains stationary. The
scanned object with values of measured positions of two edges is shown.

120

3.5.2 Setup of position measurement

To measure the exact position of the HiRA Array, targets and the two MCP foils, we
have to cover an overall distance of 95 cm. Considering the size of the LBAS and the
limitation in its measurement range, we had to measure the HiRA array and the target
separately with LBAS mounted at different positions. In addition, since it is important to
know the positions of the MCP foils relative to the target center, measurements of the
MCPO and MCPl masks were also performed. The schematic setup of position
measurement is shown in Figure 3.27. Along the beam lime, the laser was first placed at
position 0 between two MCPs systems (see figure) to measure the reaction target and
MCPO mask. To measure the MCP] mask, the laser was then moved to the position 1,
between the HiRA array and the MCPl detection system. Finally, the MCPl detection
system and the reaction target ladder were removed, and the laser was moved to position
2 in flont of the HiRA array to measure each telescope. A photograph of the setup with

laser being placed at position 0 is shown in Figure 3.28.

In order to transform the positions of the HiRA telescopes, target center and
MCP masks measured by laser at different locations into one absolute coordinate system,
we need to know the laser positions (position 1, 2 and 3) relative to each others.
Reference objects were used for such purpose. The positions of the reference objects are
shown as grey squares in Figure 3.27. Because of its working principles, LBAS works

reasonably in determining the sharp edges. The reference object was therefore designed
and machined to be a 1.9 x 1.9 cm2 square reference post, 35 cm high. Two posts are

shown in Figure 3.29. By scanning the edges of the reference posts at different laser

121

positions, the same comers of the post with respect to different laser positions could be

determined and used to match all three laser coordinate systems.

 

Pesto Postl

[11:1

Laser target Laser

       

Laser Position 0 | P | l Position 1
I i I
l I I
(Measure MCP 0 and Target Mask)! (Maggy. E E (Measure
y MCPO HIM) Mcpl MCP 1)
1 (out of screen) Post 2 Post 3 . "IRA

Schematic drawing -

° ° ° Corners determined from laser at
Diagram not drawn to scale -

Warts

 

 

Figure 3.27: Schematic setup of position measurement. The dashed lines indicate
the positions of target and foils of MCPO and MCPl detection systems. The
drawing is not drawn to scale.

LBAS laser Detector (HiRA)

 

 

MCP Mask Reference objects (posts)

Figure 3.28: Photograph of setup with laser being placed at position 0.
122

 

Figure 3.29: A photograph of reference posts used for transforming different laser
positions.

To do beam tracking using the $800 inverse mapping, it is important to know
the position of our setup (LBAS coordinates) relative to the entire laboratory (global
coordinates) and relate our setup to the magnetic elements of the 8800 spectrometer. A
high-precision laser tracker has been used to measure the locations of various monuments
installed around the laboratory and the fiducials on the beamline magnets and defines the
location and orientation of the beamline system in the laboratory [San06]. In this
experiment, the laser tracker measured the position of the center of balls sitting on
reference points along the HiRA alignment posts with respect to the global coordinate
system. The comers of the HiRA alignment posts can be deduced flom the measured ball
positions according to the known dimensions of the posts. By transforming the comers
measured in LBAS coordinate to those obtained by laser tracker in the NSCL global
coordinate system, we can match our experimental setup to the beamline, all the way

back to the K1200 cyclotron. Figure 3.30 shows the layout of the beamline in the

123

laboratory. Our experiment was performed in S3 vault. The distance between the origin

of the NSCL global coordinate system and S3 vault is about 76 meters.

ECRK500 N1N2N3N4 N51“

 

 

 

~76 meters

 

Figure 3.30: Layout of the beamline in the laboratory. The 8800 spectrograph is
located inside S3 vault.

3.5.3 Offset correction for LBAS

The most significant problem of the LBAS is the non—superposition of the rotation axis of
motor stage and the laser source. Ideally, the coordinate axes that define the laser box and
OWIS motor stages would be perfectly aligned under rotations. However, due to the
inherent mechanical design of how the laser box is rotated, all 3 body axes of the laser
source is displaced during rotations as demonstrated in Figure 3.31. The offsets in

rotation axes have to be corrected for accurate position measurements.

124

virtual origin

   

4, physical laser origin

Figure 3.31: Schematics of laser Based Alignment System (LBAS) where the
intersection of two dash lines and blue line defines the virtual origin, while the
intersection of two solid lines and blue line defines the physical laser origin.

Figure 3.31 defines the axes for 0-rotation perpendicular to that for the (p-rotation
and the offsets which need to be determined. There exists a unique plane which is normal

to the (p-rotation axis and containing the 0-rotation axis. The intersection of this plane
with the (p-rotation axis defines the origin which is shown as the intersection of two dash

lines and blue line in Figure 3.31. If the rotation is perfect, this origin remains stationary.
Because the laser is not located in this origin, actual laser position after rotations is
changed. Thus we would name the origin before rotation as “virtual origin” and after
rotation as “physical laser origin” as labeled in Figure 3.31. The offset correction in the
axes can be considered as two components: Laser offset correction and theta-stage offset
correction. The former one is the correction with respect to the theta-stage rotation axis
for aligning laser box with theta-stage axis. The latter one corresponds to the correction

with respect to the phi-stage rotation axis for aligning the theta-stage assembly with the

125

phi-stage. As a result, the offset correction can be written analytically in the following

Equation 3.4:
x' psinacos<p
(y') = (psinBsimp)
z' pcosB
cosrp sinrp 0 cos(0 — 90°) 0 -sin(9 — 90°) Ax
+ (—sinrp cosrp 0) t 0 1 0 )( 0 )
0 0 1 in(9 — 90°) 0 cos(0 - 90°) Az
cosrp sinrp 0 Axtp
+ (—sinrp cosrp 0) AMP (3-4)
0 0 1 o

The sum in the RHS of Equation 3.4 consists of three parts: The first part is the
vector point flom the physical laser origin to the measured object. The second part is the

theta-stage offset correction, denoted by the vector (Ax, 0, A2), which aligns the physical

laser origin to 9 axis. The third part is the phi-stage offset correction, denoted by the
vector (qup qu, ,0), which aligns the 0 axis to the (p axis. The second and third terms of

the sum together give the vector pointing flom the virtual origin to the physical laser

origin.
In the Cartesian coordinate system, the direction of (p = 00 and 9 = 90° defines the

x direction. The (p axis (i.e. 0 = 0°) defines the z axis. y direction is defined by the z x x.

The coordinate system is hence right-handed. The convention used in LBAS is that x is
the beam direction, 2 direction is vertical-up, and y is defined by z X x. It is important to
note that the LABS convention is different flom the one used in standard laboratory

convention where z is the beam direction for instance. The reason of using the current
126

convention in LBAS is because it closely coincides with the basic movement and

orientation of LBAS.

Displacements of all components (Ax, Az, Ax (p, Ayv) were measured on the granite

surface plate with tall vertical caliper and steel calibration plates. It is found that the
largest correction is the 7.5431 mm displacement of the laser box with respect to the

theta-rotation stage (A2). The other corrections are relatively smaller with Ax=-O.1300 mm,

Ax ¢=-0. 1400m and Ay¢=-0.4l98mm.

3.6 LABS Position measurements

After extracting all the edge points flom the laser seaming corrected for laser offsets as
discussed in section 3.5, we analyzed the data and determined the position of objects in a
quantitative and systematic approach. The whole method relies on the least square plane
fitting constrained by dimensions of the objects. There are three types of objects
measured: reference posts, HiRA telescopes and masks for both reaction target and MCP
foils. With the systematic and general analytical method, 0.3 millimeters (which
corresponds to 005°) precision, well below the intrinsic position resolution of the
detectors, is achieved in present experiment. Such accuracy is sufficient for the transfer
experiment. In the following sections, data analysis and results on the LBAS

measurement for the entire experimental system will be discussed in detail.

127

3.6.1 Transformations of laser coordinate systems

Due to the limitations of the laser measurement ranges, laser scanning data for the whole
system in the scattering chamber were acquired in three different positions as shown in
Figure 3.27. Transformations among different laser positions are determined by scanning
the reference posts (gray squares) as shown in Figure 3.27. For each reference post,
LBAS measured five edges with three scans made on each edge as demonstrated in
Figure 3.32. Three planes can be consequently constructed by least-square plane fitting
which requires the planes to be orthogonal. The intersection of the 3 planes defines the
comer position of the post. Lasers at different locations measured different corners within
one post which can be matched using the known mechanical dimension of the post.
Transformation between the positions of the same comers with respect to different lasers

could be performed by Equation 3.5:
Xa = R(6)xb +dl (3.5)
where rta and xb are the same point in space relative to different laser positions.

R=Rx(0x)Ry(t9y)Rz(62) is the rotation matrix which is characterized by 3 angles 0,, By and

62; and d is the translation vector characterized by 3 vector components dx, dy and dz.

There are altogether six parameters that need to be determined.

128

I’lanc l.\

Plane 3

 

2L1 _
1 1 1 341/
o 1 2 912
_ /
6 9-2
Plane2’/
/13
7— 104
/14
a— 11"

 

 

 

 

Figure 3.32: Schematic diagram to present the strategy of laser scanning on the post.

We selected the co-ordinates of laser position 2 as the LBAS reference where

HiRA telescopes were measured. For the transformation flom laser position 0 to laser

position 2, we found that the rotation (6x, 6y, 192)=(-0.l6°, -0.30°, 0°) and the translation

(dx, dy, dz) = (-401.59mm, 0.58mm, -1.l7mm). The -l.17 mm difference in z direction

between laser 0 and laser 2 reflects the non-flatness of the aluminum table where the
HiRA and MCP detection systems are placed. The transformation flom laser 1 to laser 2
could not be determined flom fitting due to insufficient data points taken with LBAS.
Therefore, we only calculated the translation vector by subtracting two laser
measurements on the same point which gives (dx, dy, dz)=(123.15mm, -0.84mm,

0.15mm).

Regarding the transformation of the LABS to the NSCL global coordinates
system [San06], we employed the same algorithm as Equation 3.5. Balls placed on the

129

stand mounted on the sides of the posts were measured by the laser tracker and the
centers of the balls were consequently deduced in global coordinates system. The mount
is mechanically attached to the post by dowel pins and screws and can be moved to
different heights as shown in Figure 3.29. Using the dimension of the post, we could

determine the ball center flom the deduced corner positions in LABS coordinate system.

We transformed the global coordinates to laser 2 position and found the rotation (65,, 19y,
62)=(0.01°, -0.06°, 547°) and the translation (61,, dy, dz)=(-75191.7mm, -l4193.2mm,
5512.7mm). It is known that the S800 beam line is at an angle of 5.46° relative to the
cyclotron beam axis in the global coordinates system, the 192 value of 547° in our

deduced transformation matrix shows good agreement. This verifies proper alignment of
the detectors in the chamber and the $800 magnets and indicates accurate position

determinations with LBAS.
3.6.2 Reaction target position

A large target with dimensions of 4.1 cm x 5.7 cm was used in this experiment to
accommodate the large beam spot (~2 cm in diameter for 46Ar beams). To determine the

position of the target center, a mask with 5 well-defined holes as shown in Figure 3.33
occupied one of the target positions. During the experiment, the target mask was
remotely inserted by the target drive to the position where the center hole is located at the

center of the target. Figure 3.33 shows the target ladder with a beam scintillator viewer
placed on the top following by four (CH2)n reaction targets of different thickness, a

highly uniform thickness of carbon target and target mask. Two laser scans were

130

performed on each hole on the mask giving a total of 20 edge points as shown in Figure
3.33. We obtained the first-order estimate of the centers by averaging the scanned edge
points of each hole. Determination of center positions was done by requiring that all the
centers lie on the plane forming a 1 cm x 1 cm square. The center of the central hole

corrected for the target mask thickness represents the reaction target center.

The results indicate the target mask is titled by 0.7 degree with respect to the
beam axis. We have problems in determining the radius of the hole precisely because the
holes appeared to be elliptical instead of circular in the analysis. It could be contributed
by the long distance flom the laser to the target mask which is about ~36 cm apart (the
optimal measurement range of LBAS is 25 .4 - 40.6 cm) and the asymmetric shape of the

finite-size laser beam spot.

131

 

Scan 3 Scan 4 Scan 5

Sm“ 0 _<:>Hole l CjHole 2

 

(\Hole 3
Scanl k/

\HOIC 4

y

— Laser Scam
x Edge points measured by laser
x Center to be determined

 

\I‘lole 5
J

 

Scan 2

 

 

 

K‘\
F‘\

 

 

 

 

Figure 3.33: (left) Target ladder with a beam scintillator viewer placed on the top
following by four (CHz)n reaction targets of different thickness, a highly uniform carbon
target (appeared as a black square and target mask. (Right) Schematic diagram of the
strategy of laser scanning on the target mask.

3.6.3 HiRA pixel position

A layer of 265 ug/cm2 aluminized mylar foil is attached to the flame that covers the
window of each HiRA telescope. This foil is used to provide a dark environment to the Si
detectors and protect them flom electrons produced in nuclear collisions. More
importantly, the Al layer on the mylar in contact with the flame provides a faraday cage
to the telescope. For each HiRA telescope, the flame was measured with three scans
performed on each side. Four comer positions of the flame were determined by fitting the

plane with the constraints of 4 comers forming a square lying on the same plane. Since

132

the distance between the flame and the silicon detectors is known, the positions of the
pixels were then deduced flom the 4 comers of the flame with information of the strip

dimension. The center of reaction target is defined as the origin in our reaction system.

Figure 3.34 and 3.35 compare the positions between the LBAS measurement and
mechanical design. Most of them are in good agreement, while some of the pixels deviate
up to 09°. Table 3.1 summarizes the distance flom the center of each telescope (Si face)
to the target. Our LBAS measurements give 35.97 i 0.12 cm while the design is 35.76
cm. It is also shown in Table 3.1 that the normal of some telescopes did not point to the
target center. The maximum deviation, the angle between of normal of telescope face and
the line joining the center of telescope to center of target, is 258° (telescope 5). These

systematic and consistent discrepancies reflect the distortion in the mounting structure.

133

HiRA Angular Coverage

 

300 *

¢1ab (degree)
N
8

 

  

 

 

‘00 ’ . LBAS
,1 0 Design
0 , m A—
__.._1. .1 e 1. 1. 14 . 1 l ‘14.. 11L L ..A .1. L -111 .1 11k
0 10 20 30 40 50

9.8,, (degree)

Figure 3.34: Comparison between the telescope angles on theta and phi in
laboratory frame obtained from the LBAS measurements and mechanical design.

HiRA Position (Beam View)

 

 

x (cm)

DE mlE‘]
IE1 “[133 ‘

LBAS Design
-30 4130 — 10 0 10 20 30

y (cm)

 

 

~20

 

Figure 3.35: Comparison between the telescope positions obtained flom the
LBAS measurements and mechanical design.

134

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tower Telescope Titled angle (d3) distance (mm)
0 0 1.03 360.73
0 1 1.44 360.52
0 2 1.39 360.16
0 3 1.43 359.02

3-1 4 0.24 360.57
3-1 19 0.42 359.37
3-0 5 2.58 360.43
3-0 6 2.10 359.57
1 10 1.17 360.30

1 11 1.43 359.53

1 12 1.30 359.29

1 13 1.34 358.52
2 14 2.09 359.87
2 15 1.92 359.11
2 16 1.76 358.63
2 17 1.75 358.86
Aveggi 359.66

Design 357.63

 

 

 

 

Table 3.1: Summary of the distances of the telescope center and target center, and
the angles between the normal of the telescope (Si face) and the normal of
reaction target.

3.6.4 MCP foil position

To obtain the information about the MCP foil positions, laser scanning on the
MCP mask was performed. A MCP mask for position calibration is show in Figure 3.36.
A brass MCP mask with an array of holes was inserted to the foil position for each MCP
detection system. The holes are 5 mm apart. The smaller holes have a diameter of 0.75
mm while an “L” pattern, used for orientation of the mask pattern, is composed of six

larger holes, each with a diameter of 1.5 mm.

135

Figure 3.37 displays the configuration of six large holes on each MCP mask.
Following the same strategy used in the target mask measurement, four edge points were
obtained for each hole and used to constrain fitting to determine the centers of holes. The
results show that the angle between MCP foil and the beam is 58.46° for MCPO and
59.79° for MCPl. It should be 60° by design. The centers of the MCPO and MCPl foil
deviate fi'om the target center by approximately 1.5 mm and 0.7 mm in vertical and 0.91
mm and 0.54 mm in horizontal direction respectively. To achieve the best angular and
energy resolution at large laboratory angles, these offsets are taken into account in the

analysis.

It is interesting to find that the radii of the holes in the MCP mask are better
determined compared to the target mask. This may be attributed to the fact that the
measuring distance for MCP mask better fulfilled the measurement range of LBAS
(Laser was put at about 24-26 cm from the MCP masks in present measurement) and the
MCP mask is thinner (0.13 mm) than the target mask (1.57 mm) which allows better

determination of the positions of edges around the hole.

 

Figure 3.36: A photograph of a MCP mask for position calibration.

136

 

2 1
H01“ 3 Hole 6 5 4 3
4 h—H
5 5 mm 2
1:5 mm
5 l
MCP 0 MCP l

 

 

 

Figure 3.37: The configuration of six large holes in the MCP mask.

3.7 Electronic and Data acquisition system

Figure 3.38 (a-c) are the schematic diagrams of the electronic circuit of this experiment.
The data acquisition system can be divided into three parts, the S800 (Fig. 3.38a), the
HiRA (Fig. 3.38b) and the MCP's (Fig. 3.38c). The kinematically complete
measurements were achieved using the high resolution silicon array (HiRA), to detect the
deuterons in coincidence with the recoil residues detected in the S800 mass spectrometer.
Therefore, the “AND” signal from HiRA and 8800 serves as the main trigger in the 8800
trigger box. The MCP signals came as slave signals which were recorded for the

coincidence events or downscaled MCP events.

Concerning the 8800, the signals from the up and down PMT’s of the first
scintillator E1 are amplified and split. One of them is read by its own fast encoding ADC
(F ERA), and another is sent to a constant fraction discriminator (CFD) module to provide
a logic pulse. The outputs of the CFDs are logically AND-ed to give the S800 premaster

signal. The S800 premaster signal is used as the start for the drift time in the CRDCs,
137

while the CFD output of the anode wires in the CRDC detector is used as the stop for the
flight time information taken relative to the RF pulses of the cyclotron. The Time to
Analog Converter (TAC) output is digitized by the analog-to-digital converter (ADC).
Each cathode pad (a total of 224 in each CRDC) is read by the STAR Front End
Electronic (FEE) system. The STAR F EE read is enabled by the S800 premaster and
stopped by the anode pulse. Signals such as E1 timing or XF P timing are digitized by the
S800 TDC (Phillips 7186H). The input mapping of the 8800 TDC is listed in Ref.[8800].
In addition, the S800 premaster timing signal is delayed and sent to the MCP time-to-

digital convertor (TDC) (CAEN V775).

In the HiRA data electronics system, the signal from DE, EF or BB is sent to the
amplifier, the pseudo-constant fraction discriminator CFD) and the shaper implemented
in the ASIC. If the signal passes through the CFD, then the time, energy and event
address are sent to the HiRA motherboard buffer. In addition, the CF D time signal is sent
to a logical OR to trigger the XLM universal logic module via the ECL-NIM module.
The “OR” signals from different motherboards are fanned out and logically OR-ed to
give the HiRA premaster signal. The HiRA premaster is then AND-ed with the S800
signal to give coincidence premaster signal. In addition, the OR-ed signal from all towers

for E detectors is delayed and sent to the MCP TDC.

The control of the readout and storing of address from each motherboard along
with the Flash analog-to-digital converter (FADC) clock are handled by an XLM. Once
the XLM is triggered, the event address information is sent directly to the XLM and the

corresponding time and energy information are sent to the FADC. A clock signal issued

138

by XLM is input to FADC to initiate FADC to digitize the time and energy signals and
store them in its internal buffer. When the requested data is no longer acknowledged by
the motherboard, a logic “complete” signal is sent out by XLM and used as the start for
the trigger latch. On the other hand, the master is then logically AND-ed with the NOT
BUSY signal from the computer and coincidence premaster. The master signal serves as

the start signal to the computer and stop signal for the NIM Latch.

The NIM output signals from three latch modules are AND-ed to preserve the
slowest signal and give the HiRA singles trigger signal which is sent to the 8800 trigger
box. The HiRA singles (labeled as “Secondary” in the $800 trigger box) was selected

when HiRA was used for data taking such as alpha source data runs.

The signal from €31, amplified by the preamplifier board located within the
telescope, is sent to the external CAMAC Pico System shaper modules. Output fiom the
variable gain shaping amplifiers is routed to the ADC (CAEN V785). The gating and
readout of the peak sensing ADC is triggered by the master signal. The ADC module is
operated in zero compression mode ensuring that only those channels with signal heights
above the thresholds are read in the individual ADC channels. There are four other CsI
crystals used for monitoring the reaction rates. The OR signals from the crystals is further
OR-ed with the MCP OR signal to provide another trigger labeled as “External 1” in the

8800 trigger box.

In the MCP detection system, signal of each comer is amplified by the fast
amplifiers (ORTEC 820). The signal is split and digitized by the charge-to-digital
converter (QDCs) (CAEN V792N) with and without firrther being amplified (High and

139

low gain). The timing signal from the MCP is sent to the fast amplifiers (ORTEC VT120)
and constant-fraction discriminator, then the logic signal is split into two. One will be
delayed and eventually be digitized by TDC. Another signal is OR-ed with the same
timing signal from another MCP. The OR signal is downscaled by the rate divider and
further split into two. One is delayed and serves as the common stop to the TDC. Another
is delayed and OR-ed with the master trigger to generate the gate signals for the QDCs.
The latter one is also OR—ed with the OR signal from four CsI monitor crystals and used

as one of the trigger conditions (External 1).

The data acquisition was performed by running the standard NSCL data
acquisition system NSCLDAQ [Daq] in the Linux operating system. The data were
collected and distributed by the SpectroDaq server. In addition to writing the data on
disks, the NSCLDAQ allows user to run online analysis using NSCL SpecTcl software

[Spectcl].

140

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Chapter 4

Data Processing and Analysis

4.1 Overview of the analysis

Figure 4.1 shows the flow chart of the data analysis for converting the raw data into the
final physics results. In general, the analysis splits into three streams comprising HiRA,
8800 and MCP. The raw data stored in the event files are decoded and sorted into three
different ROOT [ROOT] files, each containing the data relevant to its own pathway. This
strategy reduces the sizes of data files and time needed in the analysis of each device. The

analysis data from each device is combined to provide the final measurements.

In HiRA analysis, we first analyzed data of precision pulser and corrected the
readout order problems in the chip electronics. Then the energy calibrations for silicon
and CsI detectors were performed. The HiRA analysis incorporates the pixelation
procedure which associates specific EF and EB strips hit by the same particle, and further
links the identified particle in the EF-EB pixel to its corresponding DE strip and CsI
crystal. Eventually, the HiRA particle identification (PID) gates were constructed. On the
other hand, the 8800 analysis aims at constructing identification gates of both beam
particles and reaction residues. Identification of the residues requires CRDC position
determination and the trajectory corrections to the time-of-flight and energy in the 8800

focal plane. Regarding MCP analysis, the main objective is to construct beam trajectory

144

and correct deuteron scattering angles using the actual beam positions on the reaction

target.

By merging the calibrated data files generated in each analysis pathway,
deuterons produced from the reactions of interest with the scattering angles corrected for
beam positions could be identified using HiRA, MCP and S800 gates. The corresponding
reaction Q-value was calculated on an event-by-event basis. Finally, the full set of
information consisting of the kinematic energies and emission angles for the deuterons of
interest, Q—values as well as the counts of deuterons within defined angular bins was
recorded in the final physics ROOT file. In the subsequent sections of this chapter, the
analysis and associated problems will be discussed in detail. In order to obtain the
reaction cross sections, background from random coincidence should be subtracted. In
addition, normalization is needed to determine the cross sections in absolute magnitudes.
The detection efficiency would have to be calculated and corrected due to the limited
coverage of HiRA array. The analysis results and the extraction of cross sections will be

discussed in Chapter 5.

145

 

[Raw Data Tapes (.evt?J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ERA Raw 1 r 5800 Raw “ MCP Raw).
(ROOT) J L (ROOT) ' (ROOT) J
l ' Beam ID Pulser Gain
[Si Pulser Cal ‘ m 1 Match
(DE‘EREB) J [ ssoo - CRDC] Fed a1
1' . Calibrations est
Readout-Order W. CsI Calibration j. l [3,;an
Correction ; (Gain-Matched) j

 

 

 

J 9 Corrections on i
i i. check TOF and energy [ MCP ]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[ DE Calibration j 1 Calibration
LCsI pulser Cal? LS800 PID ]
EF/EB calibrations ]
. _ (S800 Cam}. File) (\‘ICP Callb. \
L Pixelation? TOP. energy, File: Position &
position & angle angle on MCPs
HiRA Calib. File: Energ) :nCRDBC ande 8‘ target, timing,
& angle , HiRA PID Real ' 3:200 3:; man
multiplicity jk mu rp crty /

 

 

 

\ //

Physics File (Energy and
angle of deuteron in Lab. & . _. [ BaCkgrounq

 

 

 

 

 

 

 

COM. reaction Q-value, subtraction
deuteron counts J
Absolute diflereutial . .
cross-sections in Lab. ‘— Eéilttieomemcal Js—[ Normalization ]
a COM y

 

 

 

Figure 4.1: Flow chart Of the data analysis

146

4.2 High Resolution Array (HiRA)

The high resolution array (HiRA) was employed to measure the energies and positions Of
the emitted deuterons in this experiment. Each HiRA telescope contains silicon strip
detectors and CsI crystals. This section focuses on the calibration procedures which
include the corrections on readout order, actual silicon detector thickness and total
calibrated energies. The thickness uniformity and dead-layer thickness of silicon
detectors are examined. The difficulties in CsI calibrations are discussed. For proper
particle identification, pixelation method which utilizes the position information to

associate energies measured in DE, E and CsI for a single particle is also presented.

4.2.1 Silicon energy calibrations

4.2.1.1 Readout Order Problems

This method to correct the readout order is mainly based on the findings of V. Henzl in
the data analysis of NSCL Expt03045 [Expt03045]. The main defect of the current ASIC
electronics is the readout order problems on the chipboards and motherboards. It is
discovered that the raw channel number read out from the chipboard (CB) and
motherboard (MB) is significantly shifted by various amounts depending on the readout
order. The response also depends on the particular channel in the chipboard and its
position within the motherboard. NO universal corrections can be applied because the

shifts would vary for different settings of the gains and dynamic ranges etc.

The problem is illustrated by the double peak structures that were Observed in the

pulser ramp on the EF strips with the precision IU pulser [TU]. IU pulser is an automated
147

pulsing system which is stable to within ~0.1 %. It injects a precise voltage into the test
input of each channel of the silicon detector electronics while the detectors were still
attached. The pulser ramp (a series Of signals with a constant increment of pulsed
voltage) was carried out for all strips individually. Figure 4.2 shows the spectrum of IU
pulser ramp in one electronic channel. The double peak structures, which are clearly
identified around channel 3100 and 4200, were found to be the consequence of cross-talk
induced on the neighboring detector strips. Figure 4.3 shows the cross-talk problem on
the neighboring strips when pulsing only telelO EF strip 10 by the IU pulser from

channel 4000 to 16000.

The top two panels of Figure 4.3 show that cross-talk was induced on strips 9 and
11 respectively when the pulse on strip 10 was sufficiently high. And the bottom panel
shows the correlation of two cross-talk signals on strip 9 and 11 when their common
neighbor (strip 10) was fired. Such cross-talk induced by high energy pulses is also
Observed in the experimental data. The cross-talk induction depends on the energy of the
mother pulse and threshold on the corresponding electronic channel of the recipient strip.
In addition, by performing the IU pulser ramp with and without detector attached, the
cross-talk signals are shown to come from the detector itself, not on the cables,

chipboards or IU pulser.

148

 

ITOIO‘IO EF 8MB 10 - IU Pulsar]

 

 

 

 

 

 

 

 

 

 

 

 

200
100
100
140
£120
3100
00
00
40
20
o L 2.411.. 1.. 1.. .11.
2500 3000 3500 4000 4500
Channel

Figure 4.2: Spectrum of IU pulser ramp in one electronic channel.

Such cross-talk causes a serious problem in our readout electronic system. On the
chipboard, chip 0 is the first chip readout. With the ASIC chip mapping file used in this
experiment, the chip with Odd channels is always read prior to the chip with even
channels for EF and the order is reversed for BB. If the main signal on the even strip is
too small to induce cross-talk, then it will be read first. But if it is large, cross-talks can
be induced up to 2 strips left and right from the pulsed strip, resulting in the pulsed strip
being read as the second to the forth in the readout sequences. On the motherboard, the
chipboards are readout from the bottom slot to the top. Since the chipboard for BB is
conventionally inserted in a lower slot than the corresponding EF, BB is read first and
therefore read-order correction affects EF most. If the signal is big enough to induce
severe cross-talk in the adjacent strips on the EB side, readout correction for EF pulsed

strip needs to be determined up to significantly high order.

149

 

Cross-talk in strip 9 induced from strlp1o ”it”, H" 1*

ill II'HH'”

 

 

817po
N
8
lllllllllilllllll
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.. “5°:— ”H
.2; m:— m: HI-mmm-HHIHH
5 5 HH-Hlliiiml

 

 

 

 

strlp 9

 

 

 

 

000
strip 11

Figure 4.3: Cross-talk problem on the neighboring strips (strip 9 and 11) when
only one strin (strip) is pulsed by [U pulser.

150

228Th source calibration data are used to examine the effect of readout order

problem on energy resolution. The decay of 228Th is characterized by eight well-known

alpha energies and the alpha peaks are separated by at least 100keV. The diagram of the

228Th decay chain is displayed in Figure 4.4. There are five strong alpha peaks with

energies of 8.787, 6.778, 6.288, 5.685 and 5.423 MeV. Figure 4.5 is a spectrum of alpha

228

particles emitted from the decay of Th. Without any readout-order corrections, there

are severe peak shifts in raw channel for one EF strip read out directly from the
electronics. Figure 4.6 shows the same alpha spectrum after readout order corrections are
applied. It is apparent that readout order correction is critical and essential for correct

calibration and energy resolution of silicon E detector.

For DE detectors with preamps outside the chamber, it is found from the alpha
source data that the statistics in the second (or higher) readout sequence is significantly
lower than that of the first order. While low statistics is not sufficient to determine the
readout corrections for DE, it indeed suggests that the readout order issue in DE can be

neglected.

151

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152

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tsle1o EF strip 10 - wlo Readout Ordar Corrsctlonl
m I-
we
4005-
- I
2 '2
0 3m +
u _-
not
100
° 2500 3000 0500 4000 4500
Channel

Figure 4.5: Alpha spectrum in raw channel of one EF strip without readout
order correction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tsls10 EF strip 10 - wl Readout Ordsr Corrsction
1200
I 5.685 6.288
‘°°°'.‘ 6.778
;
cool—
a C 5.423 8.785 MeV
E L
3 m:-
o _
400i—
200 _
. .U. .LJL-....JL...
2500 3000 3500 4000 4500
Channel

Figure 4.6: Alpha spectrum in raw channel of one EF strip afier readout
order correction applied on both motherboard and chipboard readout

153

4.2.1.2 Readout Order Corrections

As shown in Section 4.2.1.1, it is crucial to determine the readout order corrections and
incorporate the corrections before carrying out any energy calibrations and data analysis
in silicon detectors. To determine the correction values, the readout order of each signal
within the motherboard (MB) and chipboard (CB) were identified when unpacking the
data from an event file using two parameters. One of them is a 32-bit array which is used
to represent channels being fired in a CB with the corresponding bits flipped to 1 from 0.
Another parameter counts the number of channels read in a tower. From these two
parameters, we can deduce the readout sequences of a hit in its corresponding MB and

CB, which are labeled as ”OrdTow” and “OrdDet” respectively in the analysis code.

Since the MB and CB readout order problems do not seem to interfere with one
another, the readout order corrections for the motherboards and chipboards could be
distinguished and determined independently. Procedures used for readout order

correction as described in the following are established by V. Henlz [Hen08]. To

228

investigate the MB readout correction alone, we can look at Th alpha spectrum for

each strip gated on the first event of each CB (OrdDet=l) with different order on MB
(OrdTow=1,2 and so on). Figure 4.7 shows the alpha spectrum with OrdTow=l and 2
gated on OrdDet=l. There are small shifts in the peaks between OrdTow=l and 2. The
shift in the channel number for a particular MB readout order relative to the first readout
order gives the MB readout correction for that particular readout sequence. Since there is
no evidence that the readout order correction on the motherboard is energy dependent, the

correction for each strip is the average of shifts in four alpha peaks with energies of 5,69,

154

6.29, 6.78 and 8.78 MeV at the particular MB readout sequence. The MB readout
corrections in unit of channel averaged within each telescope are listed in Table 4.1. It
should be noticed that the average values in the table present the general idea about
readout order effect for each telescope. The readout corrections are slightly different
among the strips within a telescope and each strip is corrected individually with its own
correction values. The non-integral shift in channel is due to the randomization in the

analysis corrected for data digitization.

For the CB readout correction, we first applied the MB readout order corrections
to the IU pulser data and then determined the peak shifts in IU pulser ramp for individual
strip gated on different readout order on each CB (OrdDet=l ,2 and so on). The spectrum
of IU pulser ramp for one strip gated on different OrdDet is shown in Figure 4.8. Peak
shifis could be found by comparing the channels of the same peak with different OrdDet.
Figure 4.9 shows a zoom-in of Figure 4.8 between channel 13200 to 15500 where the
same peaks with different OrdDet coexist. This enables determination of the second and
third order correction in CB. Depending on the response of each channel, the third and
fourth order corrections could be determined. TU pulser ramp was performed for the
chips on BF only, because we used energy from EF for analysis due to its better energy
resolution than EB. Table 4.2 lists the averaged CB readout corrections in unit of channel
for EF within each telescope. Again, the averaged corrections only serve as a summary
of overall effect. Slightly different correction values were applied to each strip

individually within a telescope.

155

Cou nts

 

 

Tele10 EF strlp 10 - Alpha gated on dlflerent OrdToj
35°,

 

 

200 1
1.. 'i

100

 

50

 

 

 

 

 

 

 

 

 

 

f

“30020002000300032003400m0030004000
Channel

Figure 4.7: Spectrum of alpha source gated on OrdDet=l with OrdTow=l
(red) and OrdTow=2 (blue) respectively.

156

Counts

250

Counts

 

Haw EF strlp 10 - IU pulser gated on different OrdDet!
500

 

 

§§§§§§

llllIllllllllllllllllllllllll‘lllillllllllllllllTi

 

 

 

O

4000 6000 8000 10000 12000 14mm 16000
Channel

Figure 4.8: Spectrum of IU pulser ramp gated on different OrdDet
corrected with readout order of motherboard. The OrdDet=1,2,3 are
highlighted in red, blue and green respectively.

 

 

OrdDet=2 (blue) and 3 (green) ]

 

 

13400 1300 13800 14000

 

14200 14400 14800 14800 15000 15200 15400
Channel

Figure 4.9: Zoom-in of Figure 4.8 on the regions where the same peaks
with different OrdDet appear in the spectrum.

157

 

Averaged readout order correction for MB on EF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TowerO Towerl Tower2 Tower3
Tele 1 st 2nd Tele 1 st 2nd Tele 1 st 2nd Tele 1 st 2nd
0 -6.72 -12.33 10 -6.97 -10.51 14 -0.09 2.01 4 -7.86 -10.53
1 -4.07 -5.98 l l -7.54 -12.96 15 -0.27 1.85 5 —6.72 -8.32
2 -4.59 -7.90 12 -6.89 -10.45 17 0.64 2.17 6 -7.06 -10.68
3 -5.68 -9.39 13 -7.19 -11.08 16 -4.00 -5.42 19 -7.39 -11.87
Averaged readout order correction for MB on EB
TowerO Towerl Tower2 Tower3
Tele 1 st 2nd Tele 1 st 2nd Tele 1 st 2nd Tele 1 st 2nd
0 18.14 29.51 10 37.60 65.47 14 32.48 61.32 4 14.61 25.27
1 33.91 54.71 11 30.56 46.70 15 26.20 42.97 5 15.59 25.95
2 35.82 54.50 12 32.92 60.86 17 33.90 62.95 6 16.73 26.33
3 31.59 45.86 13 37.06 68.59 16 31.11 55.74 19 17.55 27.09

 

Table 4.1 Averaged corrections for the first two readout order on the motherboard for EF
and EB within each telescope.

 

Avera ed readout order correction for CB on EF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TowerO Towerl
Tele 1 st 2nd 3rd Tele 1 st 2nd 3rd
0 56.03 90.49 98.71 10 55.32 85.80 94.43
1 82.88 94.74 1 1 56.92 89.78 96.53
2 52.94 89.32 12 54.50 87.67 103.18
3 54.36 85.88 13 . 60.17 100.37 115.52
Tower2 Tower3
Tele 1 st 2nd 3rd 4th Tele 1 st 2nd 3rd 4th
14 50.72 82.09 109.06 4 49.79 77.14 84.02
15 52.53 84.01 114.05 5 53.97 86.23 102.78
17 34.87 57.60 80.71 94.12 6 44.67 70.03 86.12 95.14
16 34.39 54.67 76.42 19 52.03 83.74 93.96

 

Table 4.2 Averaged corrections for the readout order on the motherboard for
EF within each telescope.

158

 

 

4.2.1.3 Energy Calibration

Afier readout order corrections, energy calibrations for DE and E silicon detectors can be

carried out using alpha particles emitted from 228Th source. The source was put in the

target position. The 228Th alpha decay peaks of each strip are shown in Figure 4.6. We

use the five strongest alpha peaks which correspond to energies of 8.787, 6.778, 6.288,
5.685 and 5.423 MeV and establish a linear function to convert the channel readout from
each strip into particle energy in unit of MeV. The energy resolution is typically
represented by the FWHM of the 6.778 MeV peak. More accurate calibrations can be

achieved if the dead-layer, mylar foil covering each telescope and the window of the

228

alpha source are taken into account. The mylar foil and the Au window of the Th

source in this experiment are 265 ug/cm2 and 50 ug/cm2 thick respectively.

Determination of dead-layer thickness is discussed in Section 4.2.1.5.

In this experiment, DE was placed in front of E silicon detector and alpha
particles could not penetrate the DE. To calibrate E detectors, the HiRA telescopes had to
be disassembled from the array in order to dismount the DEs and then assembled back to
the system. To check if there is any change in the E calibrations due to the handling of
silicon detectors, pin source was used to calibrate E detector without removing DEs in
each telescope. A pin source is a 0.5” dowel pin activated by electroplating the tip with

daughter nuclei fi'om 228“. The pin was attached to the frame and inserted in the slot

me is the

between DE and E detector for each telescope as shown in Figure 4.10.
primary deposition on the pin which emits strong alpha particles at 8.785 MeV, 6.050

159

MeV and 6.090 MeV as presented in Figure 4.4. The pin source is not covered and sharp

alpha energy spectrum can be obtained as shown in Figure 4.11.

 

Figure 4.10: Photograph of the frame with a pin source attached. A slot between
the DE and E detectors in the front part of HiRA telescope shown on the right is
designed for the frame to be inserted. Extra pins are shown on the bottom lefi. A
ruler is displayed for size reference.

 

I was ens-ears- Pln Source I

8.705 HOV

Counts
s§§§§§§§

 

  

 

°2sooaoooasoowoo4soo
channel

Figure 4.11: Pin source spectrum in raw channel of one pixel (EFl6-EBIS in
telel9) with MB and CB readout sequences problem corrected.

160

In general, all DE and EF achieve good energy resolutions of 45-55keV and 60-75
keV respectively. Regarding strips on EB, about 80% of them have resolutions of 60-85
keV while the rest is about 90-120 keV. Therefore energies in EF strips were used to
represent the energy deposited in E silicon detectors in the analysis. Since the energy of
alpha particles used in the calibration ranges only from 5.42 to 8.79 MeV while the
dynamic ranges of DE and EF were set up to be about 13 and 38 MeV, respectively, in
this experiment, IU and BNC precision linear pulse systems were used to pulse each strip
to check the linearity of energy response over the ADC range. Overall, all strips have
good linearity suggesting the calibrations obtained from alpha source spectra can be

applied for the full ADC ranges.

To check if there is any gain shifi in silicon detectors during the experiment, we
chose three groups of alpha source data taken at different periods of time. We first used
the alpha runs at the end of the experiment to obtain calibration parameters for silicon
detectors and applied the same parameters to calibrate the alpha runs performed at the
beginning and in the middle of the experiment. The average energy shift of the alpha
peaks indicates the stability of the silicon detector. It is found that the performance of DE
was very stable over the entire experiment. We examined the stability of E detectors
using telescope 5 and 6 since these are the two telescopes without DE. There is no shifi
observed in the alpha runs in the middle of the experiment compared to those taken at the
end. However about 70-85 keV energy shifts are found in EF at the beginning of the
experiment, while the shifis in BB are relatively smaller. It is not conclusive that the

calibration parameters for E detectors obtained from the alpha data taken five days before

the first beam (36Ar) came in even though the detectors were not changed during that
161

 

period of time are applicable. Also, the performance of telescopes 5 and 6 could not
represent all the E detectors in the system. Nevertheless, since absolute energy calibration
is not required in this experiment, such problems would be corrected when calculating the

total energy of detected particles as discussed in section 4.2.5.

It is reported from the manufacture that the thickness of silicon detector is known

to 10% accuracy. The energy calibrations in silicon detectors can be used to determine
thickness of DES. In the data run, we use 4He particles which stopped and deposited 6
MeV in E detector and found the corresponding energy deposited in DE. Using LISE++
code, we can deduce DE thickness from the 4He energy deposit in DE with the dead-layer

thickness taken into account. The reasons of using EF energy at 6 MeV are because the
DE-E energy relation is steep in this region which allows better energy determination.
Furthermore, the corresponding DE energy would be approximately at 5.5-6.5 MeV
depending on the actual DE thickness resulting in both energies in DE and E lie in the
calibration regions. Table 4.3 compares the thickness fi'om the manufacturer and our
analysis. In general, the agreement is good and the deviations from the manufactuer’s

values are at most about 12%.

162

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tele Manu.§um) Deduced (mm) % difference
0 63 62 -1.59
1 65 58 -10.77
2 62 62 0.00
3 66 62 -6.06
4 62 63 1.61
10 66 63 -4.5 5
1 1 66 62 -6.06
12 65 62 -4.62
13 66 58 -12.12
14 62 62 0.00
15 61 62 1.64
16 64 63 -1.56
17 63 62 -1.59
19 63 63 0.00

 

 

 

 

 

Table 4.3 Thickness of DE detectors given from manufacture and deduced from
experimental data.

4.2.1.4 Uniformity of Silicon thickness

Non-uniformity of silicon detector thickness would result in position-dependent variation
in the energy loss of particle across the silicon surface and poor energy resolution of the
detector. Therefore, the effect of non-uniformity of silicon thickness needs to be
investigated and incorporated in the calibrations if necessary. To study the non-
uniformity of silicon, we used the alpha particles which punched through the thin silicon
(65 um) but stopped in the thick silicon detectors. We further gated these alpha particles
on two regions of DE energy around 5.4 and 7.4 MeV with the corresponding E energy
around 35 to 40 MeV and 25 to 30 MeV respectively. Then we used the E energy of each
event and calculate the corresponding DE energy based on the DE thickness provided by

the manufacture. The ratio of the measured to the calculated DE energy for the alpha data

163

gated on DE energy of around 5.4 and 7.4 MeV for one DE detector was studied in
another HiRA experiment -- NSCL Expt03045 [Expt03045]. Energy loss of a particle
passing through the silicon detector is proportional to the detector thickness. The
difference between the calculated DE energy to the measured DE energy for those alpha
particles on event-by—event basis reflects the non-uniformity of silicon thickness. It is
found that the effect of thickness variation on DE energy is less than 2%, which
corresponds to less than 5% variation in thickness. This DE detector (HiRA part number:
2266-4) shows the most significant variation compared to the other DE detectors. Since
the E detector is more than 20 times thicker than the DE silicon detector, the effect of
non-uniformity on E detector is negligible. No correction for the non-uniformity of

thickness of E detector was applied in the data analysis.

4.2.1.5 Determination of Dead-layer thickness

Dead-layer is the metallic electrodes and non-depleted region on the surface of the silicon
detector. Energy loss of the particles in these regions could not be measured. The
thickness of the dead-layer has to be considered for an accurate energy calibration. To
determine the dead-layer thickness of E detector, we use the calibration data of E
detectors with the pin source as described in Section 4.2.1.3. Schematically, the pin
source was located 2.72 m away from the front of the E detector as shown in Figure
4.12. The energy loss of the detector can be described by Equation 4.1, where E is the
measured energy deposited in the silicon detector, E’ is the total energy of alpha particle

emitted from the source, DL is the dead-layer thickness, 6 is the oblique angle of alpha

164

particle, and dE/dx is the stopping power of alpha. The stopping power of alpha particle

in silicon at certain energy can be found in the ref. [Sky67].

l.95mm+
0.025mm

'A """""""""" g Q.-
. -~-- ,. r-

2.72mm q‘ . 3"“ 30L
i O O.

oooo'g"
...xf

Figure 4.12: Geometry of pin source calibration.

E-E’ ‘15 DL 41
- dxCosH (')

 

The pin source is capable of hitting all pixels on the E detector. Due to the close
proximity to the silicon surface, the alpha particles that strike the pixels near the edge of
the surface have much larger oblique angles compared to those near the center. We
assume the pin is a point source. Since the thickness of dead-layer that the particle
passing through depends on the oblique angle, the dead-layer near the edge will
sufficiently shift the energy of the alpha particle. The dead-layer thickness could be
determined by varying the DL thickness until the difference between the measured
energy from data and calculated energy from Equation 4.1 is minimum. Our result
indicates the averaged thickness of the dead-layer for E detector is 1.0 :l: 0.5 um silicon
equivalent. In the analysis, we assumed the same averaged dead-layer thickness for both

DE and E detectors.

165

4.2.2 CsI Energy calibration

The light output of CsI(Tl) crystals are mass and charge dependent and non-linear. It also
depends on the T1 doping of the CsI crystal. Therefore, calibration has to be performed
for each CsI individually. Two-dimensional spectrum of calibrated EF energy vs. raw
signal from CsI is generated to calibrate each CsI as shown in Figure 4.13. For a fixed
ADC channel in Csl, the corresponding energy in the Si detector could be found in each
identified isotope. Using the energy loss from calculations using LISE++ [LISE], we can
obtain a set of points with their individual energies deposited in Si and the corresponding
Csl, and eventually establish a 5th order polynomial fimction between energy in Si
detector and CsI crystal for each particle species. Since the light output is roughly
proportional to the energy deposited in the C31, a linear relation between the ADC
channel and the particle energy could be determined as demonstrated in Figure 4.13. In

this experiment, we used deuterons for C511 calibrations because it is the particles of

interest.

166

 

 

25
20

EF Energy (MeV)

  

 

 

500 1000 1500 2000 250 3000 3500 4000
Channel

 

a
GI

Value = =-1.23 + 0.0201‘channel

  

88

.5
G

Cal Energy (MeV)
d N
O O

llllllllllllllllllllllll]llllll

 

 

 

400 000 800 1000 1200 1400 1000 1000
Channel

Figure 4.13: (Top panel) 2D spectrum of EF in MeV vs. CsI in channel. (Bottom
Panel) Linear fit between the ADC channel and the particle energy in Csl.

The BNC pulser with linearity specifications was also used to extract absolute
zero offset based on the linear response of CsI crystal. The zero offset of each CsI crystal
was converted into energy using the calibration parameters. The offsets are compared to
the offsets obtained from BNC pulsers. Figure 4.14 shows the corresponding energy of
zero offset in units of MeV, where each crystal ID represents one Csll crystal with a total
of 63. One CsI was not working in the experiment. Ifthe zero offset given by the pulser is
correct, the deviation in calibrated energy of zero offset from 0 MeV indicates that there
may be about 0.5 MeV inaccuracies in the CsI calibration. In principle, the zero offset

and CsI punch-through points are sufficient to constraint the calibrations parameters

167

assuming linear response of the light output from CsI(Tl) crystals. Unfortunately, there
are no punch-through points in this experiment because of the setting of CsI dynamics

range and the low energy of the incoming beam.

 

 

 

of pulser zero offset
S‘ ' .
g o . fl“ 0.
V . e. .. . .. . e 0 .e .. e
> g . . e ' . e0 e. .e e
9 e . . ~ . e
0 'o
:
Ill

 

trillrrfrlrrnllrrnrlrrmrlr
30 50 60

22m c stall
Figure 4.14: The zero offset of each CsI crystal converted into energy using the

calibration parameters from Si-CsI energy relations, where each crystal ID
represents one CsI crystal with a total of 63.

 

Using AE-E relation for calibration would require knowing the exact thickness of
E detectors. There is 10% uncertainty in the thickness given from the manufacture.
However, the thickness of some E detectors have been determined using the deuteron
elastic data on Au target in another HiRA experiment — NSCL Expt02023 [Rog09]. In the
elastic data, the energies of the deuteron peaks were known from the beam magnetic
rigidity Bp. Based on the calibrated energy from the C51, the energy deposited in the E
detectors could be determined and used to deduce the E thickness using LISE++. Table

4.4 lists the thickness of E detectors from the manufacture and our results. It is found in

168

general that the E detector is about 45 um thinner than the thickness given by the

manufacture.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

From Expt 02023 - elastic scatterirfi

Telescm Mann. (um) Deduced (um) Difference (um) % difference
0 1496 1442.09 53.91 -3.74
1 1503 1452.71 50.29 -3.46
2 1490 1453.91 36.10 -2.48
3 1488 1411.99 76.01 -5.38
4 1489 1444.93 44.07 -3.05
5 1499 1447.25 51.75 -3.58
6 1471 1435.47 35.53 -2.48
10 1511 1462.66 48.34 -3.31
11 1491 1457.57 33.43 -2.29
12 1452 1410.90 41.11 -2.91
13* 1473 1430.64 42.36 -2.96
14* 1533 1490.64 42.36 -2.84
15* 1506 1463.64 42.36 -2.89
16 1513 1475.43 37.57 -2.55
17 1496 1333.35 162.65 -12.20
19 1541 1512.16 28.84 -1.91

Avg: 51.98
Best Avg. 42.36
*Deduce from best avg

 

Table 4.4 Thickness of E detectors given from manufacture and deduced from

 

experimental data of NSCL Expt 02023 [Rog09]. The telescope index with “*” means the
specific telescope was not used in Expt 02023, and the thickness was deduced from the
average value.

The deduced E thickness could be used to establish an accurate AE-E relation for
each individual telescope. For each deuteron produced fi'om the reactions of interests, we
calculated the corresponding CsI energy based on its energy deposited in E detectors
using LISE++ code and used the calculated CsI energy for firrther data analysis. There
are three E detectors (telescope l3, l4 and 15) employed in this experiment which were

not being used in the previous experiment with elastic scattering data in NSCL Expt
169

02023 [Rog09]. Since the results from the elastic scattering data indicate a systematical
difference of about 42 um on average between the data-deduced and manufacture value,

we assume those three detectors follow the same trend.
4.2.3 Pixelation technique

The highly-segmented 32-strip double-sided silicon detector allows the construction of
1024-pixels within one telescope, giving precise position determination of the detected
charge-particles. To fully utilize this advantage, associating the proper EF and EB strips
hit by the specific particle is important and necessary. Such identification is clear if only
one single particle hits the telescope during one event. For the multiple-hit case, however,
there can be ambiguities in assigning the position of each particle. In addition, particles
hitting the inter-strip may result in charge signal splitting between two neighboring strips.
Cross-talk and noise issues also need to be considered. Therefore, the following
procedure [Hen08] is used to properly assign particles to their respective pixels in all

different combinations of situations.

When more than one strip on FF or EB are fired within a telescope in one event,
we first examine two adjacent channels of each registered strip and disgard the signal
with energy smaller than the threshold defined as 1.25+1% of the energy of the registered
channel. Such small signals are regarded as noise or cross-talk. After that, the remaining
signals in the particular telescope are sorted according to their signal amplitudes
(calibrated energies) in descending order for EF and EB separately. On each face, hits in
separated and non-neighboring strips are considered as “singlet”; while hits in

neighboring strips are labeled as “doublet” unless it is found to be accidental neighboring

170

single hit. If hits are identified as “singlet” on both sides, then we can pair them up
according to their energies. When the energy difference between a “singlet” on one side
and the sum of two “doublet” signals on another side is smaller than 1+5% of the energy
of singlet energy, then we combine the “doublet” hits together and pair it to the
corresponding singlet. When two singlet signals are on one side and one singlet is on
another side, we consider it to be described in one of the following circumstances. The
double-hit situation is when two particles in separate strips on one side hit only one strip
on another side. Another corresponds to the hit on one of the strips giving no output
signals due to either the problem of electronics or strips with broken wire bonds. These
situations can be identified and proper matching can be done according to the signal
energies. Same principles are applied for the more complicated situation, for instance two
doublets and three singlet signals. After the pairing process, the discrepancy in calibrated
energy between EF and EB for almost all particles is less than +/- 0.5 MeV as shown in
Figure 4.15. This is one of the constraints used in the data analysis. In addition to the
particle position and energy, the real multiplicity in each telescope can also be

determined after the pixelation procedure.

171

 

IEergy (11
0.5 . var

flerence between matched EF and EB]

        
   

  

'6 adult: 416:1!

EB Energy (MeV)
Figure 4.15: Energy difference between a pair of matched EF and EB strip
versus energy in the corresponding EB strip.

To obtain the correct total energy of the particle, DE and CsI also need to be
associated properly to the EF-EB pair for specific particle. Since DE strips are parallel to
EF strips and the separation between DE and EF is only 0.8 cm, we assumed the particle
on the EF strip has passed through the specific DE strip with allowance of +/- l strip off.
It is validated by GEANT4 simulations [geant4] which show that more than 99% of
particles can be described by this assumption. 86% of them hit exactly the same
corresponding DE and EF strips. Therefore, we pair up the signal which has the largest
amplitude within three adjacent DE strips to the hit in the corresponding EF strip. For
Csl, if more than one charged particles punch through two layers of silicon detectors and
stop in the same CsI crystal, signals induced by multiple particles superimposed in C31
could not be separated. In principle, such events do not give correct particle ID and

would be disregarded in the analysis. Furthermore, to ensure that the particles in EF-EB

172

would hit the corresponding Csl quadrant, we discard the middle strips (strip 15 & 16) in

both EF and EB as they cover the gaps between four separated CsI crystals.

When particles hit bad strips on DE, EF and EB in a telescope, the reconstruction
fails because the position and energy information could not be retrieved. All the bad
strips are taken into account in the geometric efficiency of HiRA array as discussed in

Section 5.4.

4.2.4 Particle Identification of HiRA

After associating energies measured in DE, E and CsI to a single particle, proper particle
identification (PID) can be performed. HiRA relies on the technique of AE vs. E for
particle identification. The principle of this can be approximated by Bethe-Bloch energy

loss formula:

[M 22

1158 E

 

dx (4.2)

where dx is the detector thickness, k is proportionality, A is the mass of the particle and Z

is the atomic number of the particle. From the formula, energy loss AB is proportional to
AZ2/E. A plot of AE vs E could be used to uniquely identify the A and Z of the charged

particle. For particles stopping in the E detectors, particle identification can be achieved
by using DE vs. BF as shown in Figure 4.16. For particles having higher energies that
penetrate the E detector and deposit energies in Csl, a plot of EF vs. Csl as displayed in
Figure 4.17 is used for PID. The punch-through energies of deuteron in E detector and

Csl crystal are about 22 MeV and 155 MeV respectively. In either case, one can clearly

173

see a family of bands corresponding to p, d, t of hydrogen isotopes as well as bands of

3He and 4He of helium isotopes in the PID plots.

 

HiRA PID using DE-EF (all telescopes) I

   

   

.'A.'. .'
_.-

    

DE Energy (MeV)

 

"witnfile.'fg~9~c ~ A I I I I'- - "
15 20 25
EF Energy (MeV)

Figure 4.16: PID spectrum of EF vs. DE energy in unit of MeV.

174

 

 

  
  
     

 

 

HiRA PID using EF-Csl (all telescopefll
"-1 - ..... 3'1“};"35 ‘ .
3 1 120
E l
> 100
O
3 so
15
LL 60
1.1.! _~ _ - . -
1.»:M..,1~“—' (.1 . . ; I
_. .0 rev-«w ‘ 1 2°
0 I I -- 2.“. J: . -. .. .--l- -., o
10 20 30 40 50 60 70 80 90 100

Csl Energy (MeV)
Figure 4.17: PID spectrum of Csl vs. BF energy in unit of MeV.

4.2.5 Deuteron total energy

After applying the PID gates from HiRA, beam and S800, deuterons produced from the
reactions of interests could be identified. The total energy of deuteron is the sum of
energy in both silicon detectors and Csl detector as well as the energy loss in the target,
Mylar foil and dead-layer. The energy loss tables for deuteron in different materials were
generated using LISE Excel [LISE] for each individual telescope based on the thickness

of DE and E detectors.

As discussed in section 4.2.2, the CsI energy was calculated from the
corresponding energies in E detectors. Uncertainties in the deduced silicon thickness and
the silicon calibrations would therefore result in the inaccuracies in the total energy of

deuteron and the reaction Q-value calculations. To improve the final energy of deuteron

17S

and the energy matching between different telescopes, we gated on the deuterons in the
reactions of ground-state transitions and extracted the total energy of deuteron at three
different laboratory angles within one telescope. The total energy is corrected by linear
fitting to the corresponding calculated kinematics values. The left panel of Figure 4.18

demonstrates the linear relation for Telescope 10 with reaction of p(34Ar,d) of ground-

state transition.

In general, kinematic broadening increases dramatically at the large angles. To
check the final deuteron total energy corrected by the kinematics in laboratory frame, the
deuteron energy in the center-of-mass is extracted and compared to the calculations. The
right panel of Figure 4.18 plots the deuteron center-of-mass total energy from p(34Ar,d)
reaction for telescope 10 which was located at angle of about 23° in the region of large
kinematics broadening. The center-of-mass deuteron energy in this telescope is 15.50

MeV which is consistent with the calculation of 15.53 MeV.

176

Calculated (MeV)

 

 

 

 

 

 

 
 
   
 

 

  
  
 

27' - a
Deuteron Total Energy] - Tel-1° (all Galena-1.)]

100* em 934

a. p("Ar,d)’°Ar“ l l m 01:12
25 00F”
2‘ cool:
23 i
‘0)—
” T le10 E
C .-

2, y = 1.120 x - 2.263 20: 91“” “PM”.

P..l...l...l..l l l..l..l

 

 

 

 

230 21 22 Z! 24 25 20 27 °el 1o 12 14 «writ-d 22' 24 25
0811578100 (NOV) Deuteron Energy to COM (MeV)

Figure 4.18: (Left) Linear fits of the calculated and calibrated total energy in

laboratory frame for telescope 10 in p(34Ar,d) reaction of ground-state transition.
(Right) Total energy of deuteron in TelelO in the center-of-mass frame.

 

v r v v ‘7 v v ‘v r I v v v I v v v-[v
n
I

: Kinematic
. broadening

F—p("‘Ar.d)33Ar :

.N
or

9’
o

l
r """ p(aeAr,d)35Ar

g—e
O

Broadening (MeV/ deg)
C rd
in or

 

 

.___
-2¢-"‘-r----r-- r----

20 A A30
61111) (deg)

 

.°
0

40

Figure 4.19: The calculated kinematic broadening of p(34’36’46Ar,d) for
ground-state transitions in laboratory frame.

177

4.3 S800 Spectrometer

In this experiment, the time-of-fiight and energy loss were measured by the S800
spectrometer to identify various beams and fragments. To achieve the best separations
between different species in the particle identification, time-of-flight for the fragment of
interest was corrected by its trajectory using the position and angle detected in the 8800
CRDC detectors on an event-by-event basis. In addition, the actual average beam energy
for precise kinematics analysis can be determined from the dispersive position of the
unreacted beam in the S800 focal plane. This section presents all the methods and results

in details.

4.3.1 Beam particle identification

34,36,46

Ar isotope beams of Ar were used in this experiment. The E1 scintillator signal in

$800 spectrometer provided the event trigger in data acquisition and the time-of-flight
(ToF) start. Two different time signals are measured with respect to the 8800 El trigger.
One is the start time from the A1900 extended focal plane (XFP) which is a plastic
scintillator with thickness of 100 mm and an area of 150 x 100 mm2 placed at the end of
the A1900 fragment separator. Another start time is the radio frequency (RF) pulse from
the cyclotron. Incoming beams are identified by plotting the TOF-RF verse TOF-XFP.
Figure 4.20 — 4.22 display the beam PID of 36Ar, 34Ar and 46Ar respectively. Several
bunches of beams appear in the spectra. They are the RF bunches due to the multi-turn
extractions in the K1200 cyclotron. They were combined in the analysis using the known
radio frequency. Incident beams can be identified unambiguously using the ToF

178

technique. It is found that the secondary 34Ar beam is approximately 94% pure with main

contamination of 33Cl. 46Ar beam is nearly 100% pure.

 

  

 

 

 

 

 

130 .
: Incoming Beam for p(“Ar,d)""Ar . °”’ I13“):
1150— j
‘8 — .:""".-
5140— as Ar ‘ 10’
l — n
1:: - -"
$3120?!“ \9”;;)
l— _ . yt/ ' ‘
h f _ 10
100 . ' .'
[I
lr 1
i- f_'
3930 -20 -1o 30 4o 50' ' ‘ ‘60 1

0 1o 20
ToF.XFP (ns)

Figure 4.20: The beam PID spectrum for p(36Ar,d) reaction.

179

 

11v~ '5.

 

    
 

 

 

 

 

“v _ ‘1'
: Incoming Beam for p(“Ar,d)”Ar _- _“IZI; ‘;:" ' i i
200: 341”“ _.-- 1
t 33 - ~ I —: 10’
reo— GIN ‘ "3 ‘ I
a 3 sniff; ‘
5160— . £511." ‘
l _ ‘3’, :. .'
G. : “A? 2"?" i
‘3 14°: \f'i? ‘ ‘ ‘°
" : “as?“ 3
120 — ...x :2.“
: f.“ 1 3.16“
1m.— 1;» 7 . I" 1
.-
8350 40 -2o 0 20 40 so so

ToF.RF (ne)

ToF.XFP (ns)

Figure 4.21: The beam PID spectrum for p(34Ar,d) reaction.

 

 
 

 

 

 

1a“. 5.
- Incomlng Beam for p(‘°Ar,d)‘5Ar _ ,- - 1
1150— : . - 1 _

'_ 4e --<-=' - -. 10’
‘4°_ Ar «3:.- :

: \- -/<* - r :
120*— _ . ‘13:! _ ‘. .

'— 1’ e
100; ‘ .

_ _ ‘J’ ‘ '
so? I" J}- _
%%_ I- I L_L r r _L r r r l .1 r I r r I I I i "

40 -2o 0 20 4o

 

 

ToF.XFP (ns)

Figure 4.22: The beam PID spectrum for p(46Ar,d) reaction.

180

4.3.2 CRDC position calibration

CRDC mask calibrations have to be performed to relate the channel numbers and true
physical distance for the position-sensitive CRDC. A mask with well-determined pattern
of slits and holes are inserted remotely in front of each CRDC. The pattern of the mask
used in the position calibration is displayed in Figure 4.23. First order polynomial fitting
is used for the calibration which is accurate enough for our experiment. Figure 4.24 plots
the two-dimensional position spectra of CRDCl and CRDC2. The centroids of the image
of well-defined holes were extracted and compared to the pattern of the mask. In the non-
dispersive direction (y-direction), first—order polynomial fitting is used to relate the raw
channel to the distance in mm. On the other hand, only the offset value needs to be
determined in the dispersive direction (x-direction) because the slope of the first-order
polynomial is fixed by the geometry of the cathode pads which are separated by 2.54
mm/pad and 224 pads are evenly spaced across the detector. There is a beam axis hole on

the mask which provides the optical center of the detector. Table 4.5 summaries the
results of CRDC calibrations for 34Ar and 46Ar reactions. Unfortunately the position
calibration data for 36Ar were not saved onto the disk during the experiment, we assumed
the pressure in the gas system was stable so that the non-dispersive position calibration
from 34Ar is applicable to the 36Ar reactions. Mask data run on regular basis would be

useful to monitor the stability over the entire experiment.

181

. t.

r

 

 

 

 

 

 

 

 

 

’E r...
0 TV'YUVV'VYII' IYV'TV‘IIIVVI' III'VIVVerF' ‘
,_. . :
g . . . .:
v 2 - : 1
= e z e :eee e e -:
.8 - :
0— 4
O . ° .1

d

> rrlrrrllrrrrl rrrlrrnurrnrl rrlrnrrlrrrrm

-0.3 -0.2 -0.1 0.0 0.1 0.2 03

X position (meters)

Figure 4.23: Pattern of the mask used in the CRDC2 position calibration,
when the mask is inverted for CRDCl.

 

 

 

 

rfleelr spectrum of 03001 lMaek m of CRDC2!

 

 

 

 

 

 

 

 

 

 

 

 

 

C
’ ' i r ‘ -: '
1eoo~ * reoo- .3 , .,
b C' i . 1
l. ' :_ .. 4 ‘ ‘
._ ‘ .. Q - _
1m_ ,1 . Q 1 M . . . 21m?- ' l ‘
’ . ‘ ' C L _-
— . ' ‘ 2 _ 1 1 1 O z.” | l
1aoo~ . , oreoof ( .
h l 1 _ l
b ‘ . I l _ )
1ooor é -: ' 1zoo~ j’
1‘ - : F '.
:LLLJ. 1. . _LLJ" ‘ h-rr'rlrjmrmlrrrrlrrrrlrrr
b 50 1m 150 200 B 50 100Wd150 200

Figure 4.24: Two-dimensional position spectra of CRDC] (left) and CRDC2
(right). The pattern in CRDCl is upside down compared to that in CRDC2.

182

 

 

34

46

I! L‘-

 

 

 

 

 

 

 

 

 

 

 

 

 

Ar Ar
CRDC] Y slope [mm/T AC ch] -0.09 l 3 -0.0988
Y offset [mm] 127.4091 125.6478
X ofi'set [mm] -281.432 -283.21
CRDC2 Y slope [mm/1‘ AC ch] 0.09140 0.09969
Y offset [mm] -132.828 -132.023
X ofi‘set [mm] -279.4 -280.924

 

 

Table 4.5: Summary of 8800 CRDC calibrations.

4.3.3 8800 Particle Identification

Particle identification in S800 spectrometer was achieved by using the AE—ToF

technique. Figure 4.25 shows the 8800 PID spectrum of p(36Ar,d)35Ar reaction. Energy

deposited in the ion chamber AB is roughly proportional to the square of nuclear charge
(Z2) according to Equation. 4.2, the Bethe-Bloch formula [Bethe]. Time of flight (ToF)
refers to the time between the radio frequency (RF) pulse and S800 El time signal.
Because of the magnetic rigidity Bp in the spectrometer, the TOP can be expressed as a
function of the A/Z ratio as shown in Equation 4.3. As a result, isotopes of particular
element form the tilted bands; while fragments with a constant neutron excess N-Z form

vertical bands. The straight-vertical band corresponding to the nuclei with N =Z and the

neighboring one corresponding to the N-Z=1 or N-Z= -l.

ToF - L LyUA

 

where y =
1-8
183

A
_C—fl= ‘377; 3P =Bl’u57

(4.3)

is the relativistic factor and u is the atomic mass unit.

(if .1.

saoo PID . 33Ar d 5Ar
600

    
 
 

E

anneD
N M
N
8

IC.de‘(Ch
§

as
8

3
8

1200

80 0 120

0 100 11
ToF.RF (ns)

Figure 4.25: S800 PID spectrum for p(36Ar,d) reaction where the fragment 35Ar is
circled.

The large momentum acceptance of the 8800 spectrograph leads to a relatively
large spread in the TOP, which adversely affects the PID resolution. In particular for the
secondary beam when the beam spot size is relatively large (diameter ~20 mm for 34Ar
beam), the fragments of interests detected in the focal plane have broad position and
angular distributions in the dispersive direction, resulting in bad 8800 PID as shown in
the left spectrum of Figure 4.26. The timing resolution can be improved by correcting the
ToF for the trajectory dependence. In addition, the trajectory correction is necessary to
achieve good energy loss resolution, because the energy loss in the ion chamber depends
on the travelling distance of the particle. After the trajectory correction in both ToF and
energy loss, a well-resolved PID can be obtained as shown in the right panel in Figure

4.26.
184

 

    
     

        

 

 

   

p(°‘Ar,d)”Ar wlo Trajectory Correctlone 2300— -, ,_ _m "W Ml?"
23007‘,’ a“ . i ' " ' ' ‘
— ,6! _ cl
2200:: - 3314‘s ' 22°"
A c ._
._ _ o _
gm» :21”;-
5 _ E
g T‘-
_ 02000
93"” v r
' o
‘8 3' F! L
61m:" 01900
woof. mo;
r. . . _, .- .. ..
,LLIJ'WJ, ‘ " ‘ .3 .1-1' .:'-‘ 0- g r‘ '. "x'
17m 100 110 120 130 140 150 1700 100 110 120 130 140 150
ToF.RFlnel _ ToF.RF(ne)

Figure 4.26: (Left) S800 PID spectrum of p(34Ar,d)33Ar without trajectory
corrections in ToF and energy loss in the ion chamber. (Right) Same as left panel,
but with trajectory corrections where the fragment 33Ar is circled.

The trajectory dependences of ToF and energy loss are expressed explicitly in
Equations 4.4 and 4.5. As seen in Equation 4.4, part of the ToF resolution is correlated
with the angle in the dispersive direction of the S800 focal plane (S800.AFP), because
particles with larger incident angles into the CRDCs compared to the central path would
have deviated from the central trajectories. This means that those particles take longer
time to travel from the reaction target to the focal plane. Therefore, parameter a1 is
needed to minimize the angle-dependence. Since AFP is calculated from the particle
positions on two CRDCs, trajectory corrections can be improved by removing the
dependence on the x and y components of CRDC (CRDC.x and CRDC.y). On the other

hand, the energy loss in the ion chamber is positively correlated to the travelling distance

185

 

of particles, which can be deduced from position information of CRDC. Trajectory

correction to the energy loss is made based on Equation 4.5.
ToF.RF(Corr.) = ToF. RF(orig.) — E1. time + a1 - S800.AFP + a2 - CRDC1.x

+a3 - CRDCZ.x (4.1) + a4 - CRDCI. y + a5 - CRDCZ.x (4.4)

IC.de(Corr.) =1C.de(orig.)+ b1 - CRDC1.x + b2 - CRDCZ.x + b3 - CRDCl.y
+b4 - CRDCZ. y (4.5)

The procedure of determining the trajectory correction parameters (a,- and b i) is
demonstrated below using p(34Ar,d)33Ar as an example. To achieve good resolution in
S800 PID for the region of interest, we gated on the fragment from the reaction being
studied and adjusted the parameters a,- until the angle and position dependences are

minimized as displayed in Figures 4.27-4.29. Since AFP and X-Y corrections are

correlated, several iterations between them are needed. The left panel of Figure 4.27

shows the ToF-RF of 33Ar fiagment corrected for the trajectory (ToF.RF(Corr.)) gated on

34Ar beam versus the dispersive angle (AFP) in S800 focal plane; while points in the
right panel denote the corresponding centroids of ToF-RF. The red line is the linear fit of
the data points. The horizontal straight line indicates the disappearance of angle-
correlation. Figure 4.28 shows the spectra of position dependence for 33Ar gated on 34Ar
beam in each CRDC after corrections are applied. The respective centroids are extracted
and plotted in Figure 4.29. Same procedure was used to adjust the correction parameters

186

 

 

(b1) until we could remove the trajectory dependence on energy loss in ion chamber as
demonstrated in Figure 4.30 and 4.31. The corrections parameters (at and bi) were then
applied based on Equations 4.4 and 4.5 for S800 detected particles on the event-by-event

basis. The right spectrum of Figure 4.26 is the S800 PID of p(34Ar,d) 33Ar reactions

compensated for the correlation with the trajectory for fiagments of interest.

 

 
  

 

 

 

 

 

3.3ng
"‘ ' . 11
II- .-
q a. r
:§ 310%
1mb
d “ElrrLrUrJrl
«0.05 0 0“ «0.05 o 0.00
3000AFP(red) seooAFPmd)

Figure 4.27: (Left) Spectrum of the corrected ToF—RF signal of 33Ar fragment

gated on Ar beam versus the dispersive angle in S800 focal plane after the
trajectory corrections are applied. (Right) Centroids of the corrected ToF-RF
shown in the left panel in the bin of AFP. The red line is the linear fit of the data

points

187

 

 

 

 

 

 

 

 

 

 

   

 

1- . 1 -.
5““ 33Ar gated on “Ar 5' '
gm _ - ., -. 5'120 _
“E -‘
ll.
.2 1.5
0 5011113 " 020 l '71
011001 .11 (mm) CRDCt.y (mm)
1 1
A‘ '
I
5130
75120
811
fi'.’
nun
s ..
.—
o 7: 1oo . 'm *h ' -2o . 'o
CRDC2.x (mm) 080024 (mm)

Figure 4.28: Spectra of the position dependence of the corrected time-of-flight

ToF.RF for 33Ar gated on 3'4Ar beam for CRDC1.x (upper left), CRDC1.y (upper
right), CRDC2.x (bottom left), CRDC2.y (bottom right) after the trajectory
corrections are applied.

188

 

 

33Ar gated on “Ar

  
   
 

 

 

'8‘
v 11 .WW
4 x.
r;
“- 105
O
)—
100
05
CRDC1.x (mm)
120‘
11
’2’
m—M ’
I 1‘ ”’W ' “Vb:
“E
1.1.
O 105
)—

 

100 150 zoo
CRDC2.x (mm)

Figure 4. 29: The centroids of the corrected ToF. RF in the bin of CRDC positions
from corresponding spectrum shown 1n Figure 4. 28. The red line rs the linear fit

of data points.

 

189

 

 

 

 

120‘
11
3 .1
:11fiWWtflww
m_
“-105
.9.
1
.222 **.2.*.**2.**
CRDC1.y(mm)
120‘
11
2
:11 #MMW,VJ¢
1:.
140-105
'—
1
“*‘fi‘.1r“‘é“‘aar“

 

 

CRDC2. y (mm)

 

l
I
l
l
5-

 

 

l

33Ar gated on 3"Ar

“ - ' 1 . p - s

01’»? am no)».
2 . ' ‘2 , e ‘1 .

    

 

2100

N
g

 

g

 

lc.de(Corr§) (Channel) .
lc de(Corr ) (Channel)

33

 

 

d ‘ d d i
‘ r

 

VJ'

 

‘ '{cnoc1.mm) CRDC” (m)

 

 

lc.de(Corr.) (Channel)

 

 

 

 

 

. W
ca'ocz‘x (mm) CRDC2.y (mm)

Figure 4.30: Spectra of the position dependence of the corrected energy loss in

S800 Ion Chamber ic.de for 33Ar gated on 34Ar beam for CRDCl.x (upper left),
CRDCl.y (upper right), CRDC2.x (bottom left), CRDC2.y (bottom right) after
the trajectory corrections are applied.

190

 

 

 

 

 

 

 

 

 

 

 

 

 

E2250:- 3"'Ar gated on 34Ar E2250:-
5 2150 .- 0 2150.— ‘ ‘ W
V Wu- V i .- {Agwfl
32100 - MM fzioohr" W N
3 - '6 E
$3,205. 0 2050E
‘- zooo — E2000.—
.2 "so - .2 1950:—

1000 lo “0° .40 do 0 20

CRDC1.x (mm) CRDC1.y (mm)

€22.50; 7, 2250—
s E s
.2 2200;- 2 mm
82150;- 82150}! k '
Emm?’“v" ' """"' '~ Mwfiavw'lM} €21w}‘+‘ !M:’F ‘ v.
8 zosog— 8 mod-
:gzooog— $2000—
.2109;— .Q 1950—

‘°°°- 0 so 100 150 zoo sun “W -2o 0

CRDC2.x (mm) CRDC2.y (mm)

Figure 4.31: The centroids of the corrected ic.de in the bin of CRDC positions
from corresponding spectrum shown in Figure 4.30. The red line is the linear fit
of data points.

Figures 4.32 show the trajectory corrected S800 PID spectrum gated on emitted
deuterons and incoming beams for p(34Ar,d) and p(46Ar,d) reactions. The reaction
residues produced from each transfer reaction could be identified unambiguously as
circled in the spectra and the corresponding S800 residue-gates could be generated for
further analysis. For p(46Ar,d) 45Ar, the centroid of 45Ar gate in the PID spectrum had to
be shified by 27 channels in IC.dE starting from the middle of the experiment. The shift

corresponds to approximately additional 0.11 MeV/nucleon energy loss. Since the

191

magnetic setting remained the same over the entire experiment, it is suspected that the

shift in energy might be attributed to the change in gas pressure of the ion chamber

 

 

     

 

 

 

 

 

 

m g(°‘Ar,d)fAr | $300 PIDtorg(“Ar,d)‘_‘Ar I
:33; i t ‘ ‘ “ mo; ' " ‘ ‘
2200:: ’
$2100: $2100
C '1“: C
I: -. I:
2 :' 2 -
0 : I: O
'u
(51900— ' 3 -
_ 3 .1900
I - 1m
1100 1‘ I 140 150 7o

10' 12b
ToF.RF (ns)

Figure 4.32: (Lefi) S800 PID spectrum for p(34Ar,d) reaction where the fragment 33Ar is
circled. (Right) Same, 8800 PID spectrum for p(46Ar,d) reaction where the fi'agment 4sAr
is circled

4.3.4 Beam and reaction residue energy

From past experiments, the beam energy given by the A1900 magnetic setting have been
found to be different from the real beam energy by about one percent. To determine the
real energy of the beam, we allowed the beam to travel directly to the $800 focal plane
without passing through the reaction target. In the unreacted beam runs, the magnetic
rigidity of S800 was set according to the A1900 magnetic setting for the incoming beam.
If the real beam energy is the same as expected from the A1900 Bp setting, the incoming

192

 

beam would be focused in the middle of the dispersive axis (x-axis) of the $800 focal "V
plane. The radius of S800 is 2.8028 meters and the momentum dispersion is 95.6 mm/%.
Therefore the 95.6 mm deviation from center for the beam dispersive position on 8800
focal plane would imply 1% momentum dispersive and consequently 1% difference in Bp
value compared to the setting. Figure 4.33 plots the dispersive position (FP.x) of the

incoming beams (36Ar, 34Ar and 46Ar) with the centriods of beam obtained from

Gaussian-fit. The real beam energy can be determined accordingly from the position
deviations. Table 4.6 summaries the beam energy from A1900 and the beam energy

determined from unreacted runs.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

38A:
Comtant mo
Mom 47.57
8 35mm 6.002
C
.3
1
...1....1....1....1.. .1 “1....“ . ‘
450-100 0 50 100 150 212 - 450-100 -50 0 50 100150 zoo - 450400-50 0 50 100150 2W
FP.x (mm) FP.x (mm) ”’4 (mm)
Figure 4.33: The dispersive position (F P.x) of the incoming beams of 36Ar, 34Ar
and 46Ar. The centriods of beam are obtained using Gaussian-fit and shown in the
spectra.
Al900/ Real beam % diff.
S800Brho Energy expected Real Brho of energy in
Beam [Tm] [MeV/nucleon] FP.x [mm] Beam [Tm] [MeV/nucleon] energy_
36Ar 1.6691 33.07 47.57 1.6608 32.75 -0.98
“Ar 1.5741 32.96 52.32 1.5654 32.6 -1.1
46Ar 2.1308 33.00 47.41 2.1202 32.68 -0.98

 

 

 

 

 

 

 

 

 

Table 4.6: Summary of the beam energy determined from the Brho ofA1900 and S800.

193

In Figure 4.34, the calculated residue energies of 35Ar, 33Ar and 45Ar at ground

state are plotted versus laboratory scattering angles. The emitted angles span from 0 to
l.75° and selected Bp setting ensures the desired fragments are measured in the S800

without being blocked by the beam blocker. To estimate the energy resolution given by

the S800 spectrometer, we studied p(34Ar ,d)33Ar ground-state transition and plotted the

resulting 33Ar fragments with scattering angle between 0.17°-0.23° as displayed in Figure

4.35. This range of the residue angles correspond to the deuteron emitted angles of 6°-8°.
The reason of choosing the angular range of O.l7°-0.23° is because there is no angle-
dependence in the residue energy as seen in the calculations. By performing Gaussian
fitting to the peak in Figure 4.35, the energy of 33Ar corrected for the energy loss in the
target was found to be 32.04 MeV/nucleon with resolution of 490 keV/nucleon FWHM.
The measured energy is consistent to the calculations of 32.06 MeV/nucleon, however

the energy resolution is not better than 1.5%.

CD
uh

 

O)
N

(D
O

 

Energy in lab.(MeV/A)
N
O

 

 

AA-.I- 1-- I --4

0.5 i 1.0 A ‘15-
Blah (deg)

 

.0
0

Figure 4.34: The calculated energy of reaction residues of 35Ar, 33Ar and 45Ar at
ground state in laboratory frame.

194

i" d

 

 

fill; (017° < 0",, < 023°) “

 

 

 

 

 

 

fliimnliini

 

 

PILLLJILJLJJJLJ lLJLl
98 29 32 33

30 31
Energy (MeV/A)

Figure 4.35: The energy spectrum of 33Ar produced from reaction of p(34Ar

,d)33Ar ground-state transition gated on scattering angle between 0.17°-O.23°.
Energy is corrected for the energy loss in the target.

4.3.5 Trajectory reconstruction

Inverse map was employed to reconstruct the particle trajectories so that the scattering
angles of the particles on the reaction target could be deduced analytically from measured
dispersive and non-dispersive angles in the CRDC on an event-by-event basis. The
Inverse map was calculated using the ion optics code COSY Infinity with correction of
optical aberrations to the fifth order according to the magnet and Bp settings in the
spectrometer [BazO3]. In addition, the reconstruction procedure assumes perfect
alignment of magnetic elements. The distance between the reaction target and pivot point
of the S800 optics is also required for the calculation of the map. In this experiment, the
reaction target was placed at 59.4 cm upstream of the pivot point. The beam trajectories

deduced from the MCP detection systems were combined with the deuteron angles from

195

 

HiRA to calculate the scattering angles of recoil residues, which can be compared to the
scattering angles reconstructed from the positions in CRDC. However, the MCP
performance in present experiment was not satisfactory as discussed in Section 4.4.
Detailed analysis of the MCPs and the comparisons between the angles obtained from
MCP and S800 can be found Ref. [Expt06035]. Expt06035 [Expt06035] to study the
spectroscopic factors from p(56Ni,d) reaction, was carried out with the same experimental

set up right after this experiment.

4.4 Micro Channel Plate detectors

In this section, the performance of MCP and associated problems are discussed. Due to
the considerable size of the beam spot and the actual beam trajectory, corrections for the
beam positions and angles on reaction target is necessary in order to achieve good
angular resolutions. Ideally, we could use the position information given by MCPO and

MCPl to deduce the position and angle of beam trajectory on the target.

We first ran the experiment with the stable beam 36Ar. It is found that substantial

amounts of position signals in the central region of the resistive anode plate in the MCP
detector are too weak to be registered by all four comers of the anode plate. In MCP
detection system, signal amplitude is proportional to the amount of secondary electrons
produced along the photo-multiplier tube. The significant decrease of signal amplitude
indicates severe deterioration of the central part of the micro-channel plates. Since the
central region suffers from the highest count rates, we speculate that the excessive

196

exposure from the secondary electrons induced from the focused beam damages the
inside of the channel walls. Such observation raises concern about the practical lifetime
of MCP-PMT under high counting rates. The problems are partly solved by splitting the
corner signals into two groups and amplified one of them before sending the signals to
the charge-to-digital convertor (QDC) for digitization. Unfortunately, the high-gain

system was implemented only for MPCl in the reactions using 3”Ar and 46Ar beams due

to the limitation of time and insufficient electronic modules. Figure 4.36 shows the corn
multiplicity of MCPO and MCPl. Corner multiplicity indicates how many comers receive
signal above the preset threshold for an event. Position could not be determined for an
event when multiplicity is less than 4. Without operating in high-gain mode for MCPO, it
is obvious that the number of events with multiplicity 3 is comparable to those with

multiplicity 4, resulting in significant loss of position information for a lot of events.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x10’ X103
m;- MCPO-Multlpllclty 1 14cm - Munlpncuv
no; 1
C 1
g”: g‘
3 1 81
o o
.1 .0
oo
20!- 4°
20
l l l or ‘r 1] l
1 a 4 5 2 3 4 5
Comer Multpllclty Corner multipllclty

Figure 4.36: Multiplicity of MCPO and MCPl.

197

To avoid losing significant amount of usefiil data, we assumed zero-degree beam
angle and the position of beam particle on MCPl to be the same as that on the reaction
target which is only 10 cm downstream of MCPl. However, no improvement in the
angular resolutions is observed. This can be attributed to neglecting the effects from the
beam angles and the actual beam position of the reaction target, as well as the
unsatisfactory position calibrations in MCPI. A mask as discussed in Section 3.6.4 was
used for MCP position calibration. For MCPI, the mask coverage is relatively limited
due to the relatively small beam spot size. Analysis of the defoused beam which has
much larger beam spot was tried with no success due to the unsatisfactory beam intensity.
In addition, there was efficiency drop of the MCP] performance for 46Ar during the
experiment. The MCP] corner amplifications only give 91% and 94% multiplicity-4
signal for 34Ar and 46Ar respectively. Multiplicity equal to 4 is required to determine the
position. With the unsatisfactory performance of the MCP we decided not to include the
MCP data in the present analysis. Nevertheless, whether to include the MCP data or not
does not change the conclusions from the experimental results. The MCP performance
was improved significantly for another transfer reaction experiment [Expt06035] which
allows precise determination of the beam angles, beam position and emitting deuteron

angles resulting in improved position and energy resolution.

198

 

Chapter 5

Experimental results

Afier proper energy and position calibrations for all detectors, important physical

observables including reaction kinematics, Q-values and differential cross sections were
extracted. This chapter presents the experimental results for p(34’36’46Ar,d) 33’35’45Ar

reactions and describes the method for absolute normalization and its associated
uncertainties. In addition, structural information such as the orbital angular momentum
and spectroscopic factor of the populated states, deduced by comparing the experimental
differential cross sections to the reaction model calculations, is also presented.
Theoretical spectroscopic factors from shell model predictions are discussed. The
extracted asymmetry dependence of the reduction factors in transfer reactions is
compared to the measurements from knockout reactions. The result is also compared to

trends obtained from the results from dispersive optical-model analysis on Ca isotopes.

5.1 Reaction kinematics

The kinematics (energy and angles) of deuterons from p(34Ar,d)33Ar, p(36Ar,d)35Ar and

p(46Ar,d)45Ar reactions were obtained by gating on deuterons, beams and recoil residues.

Figures 5.1-5.3 present the deuteron kinematics (energy vs. angle) for different reaction
systems in laboratory frame. In each spectrum, the bands correspond to deuteron in
coincidence with various states of reaction residues and the curves are the calculated

kinematics according to the excitation energies compiled in the NNDC database

199

 

.
l

I

0
q

. ‘1'

W“

[NNDC]. The energy levels of 33Ar, 35Ar and 45Ar below 6 MeV, with spin and parity
assignments if available, are listed in Table 5.1. Those highly populated states with
corresponding kinematics calculations shown in Figures 5. l —5 .3 are highlighted in bold. It
is apparent from the kinematics spectra that the agreement between the data and
calculations is satisfactory and the ground-state kinematics bands for 36Ar and 34Ar are
well-separated from the excited states. Such agreement allows clear identification of the

strongly populated states.

W88 1

1 111

 

Energy of deuteron (MeV)
21‘

 

 

_a
.9
v H.
at

'10 15 t “20'-
91.1.0109)
Figure 5.1: Deuteron kinematics for p(34Ar,d)33Ar in laboratory frame.

A-

8, '8

'0
0|

.2
0|
1

 

Energy of deuteron (MeV)
M
O

_a
O
.TY

 

 

25 30 35

10

GI

15 l 20
9..., (699)
Figure 5.2: Deuteron kinematics for p(36Ar,d)35Ar in laboratory frame.

200

 

N
1

SP

.5
O

.5
Q

_n
5

vaTrv1I111l111l11111.

.a
N

-A

Energy of deuteron (MeV)
O

 

Figure 5.3: Deuteron kinematics for p(1 6Ar,d)45Ar in laboratory frame.

Table 5.1: Energy levels (< 6 MeV), spin and parity assignments for each reaction
[NNDC] where the energy levels with corresponding kinematics plotted in Figure 5. l -5 .3

are presented in bold.

Ar

 
 
 

”Andfurjf "

s 1.; mgfigya 3."

 

201

 

5.2 Q-value calculations

For best resolution, the reaction Q-value calculation is mainly based on the excitation
energy of residue determined from the measured kinetic energy and angle of the emitted
deuteron. From conservation of energy and momentum of the binary reaction of A +p 9B
+d, we can express the kinematic energy of deuteron in center-of-mass (COM) frame

with relativistic approach:

COM = (ECOM - ma)2 - (m3?

rd 25mm (5.1)

where m is the rest mass of deuteron; m3* is the rest mass of residue plus the excitation

energy; ECOM is the center-of-mass energy which can be calculated from the beam

energy and the rest mass of the beam particle and proton.

By Lorentz transformation of F our-Vector momentum Pd = (Ewfi), the kinetic
energy of emitted deuteron in COM frame ( tdCOM ) can be determined from its measured

kinetic energy and angle in the laboratory frame. Here the beam angle is assumed to be

sufficiently small and is not considered in the calculations.

After determining mB" using Equation 5.1, the reaction Q-value of p(A,d)B* is

obtained on an event-by-event basis:
Q=mA+ma—md—mf3 (5.2)

where mA, mp, md are the rest mass of the beam particle, proton and deuteron

respectively; m3* is the rest mass of the residue plus its excitation energy.

202

 

 

urn-n—l- - 1“"

Figures 5.4-5.6 show the reaction Q-value spectra for p(34Ar,d)33Ar,

p(36Ar,d)35Ar and p(iéAr,d)45Ar respectively. For the 34Ar Q-value spectrum (Figure

5.4), the peak corresponding to the ground-state transition is well-distinguished with
FWHM of approximately 500 keV; while the peak at the second lowest excitation is

comprised of the first two low-lying states (1.358 and 1.798 MeV). In the Q-value
spectrum of 36Ar (Figure 5.5), the ground-state and the first excited state (1 . 184 MeV) are

clearly identified, where the FWHM of the ground-state peak is 470 keV. Since the
resolution of CsI crystals is much worse than the resolution of the Si detectors, the Q-

value spectra presented here exclude events of deuterons punching though the thick Si

detector. Therefore, the laboratory angular coverage in the Q-value spectrum for 34Ar and
36Ar is up to 23° and 26° respectively. In addition to excluding deuterons punching

through silicon detectors, the 46Ar Q-value spectrum shown in Figure 5.6 covers

laboratory angles only up to 19°. Since the beam positions and angles on reaction target
are not well-determined from MCP data as discussed in Section 5.3, the ground-state and
first excited state (0.542 MeV) at larger angles could not be clearly resolved. The FWHM

of the ground-state peak of p(4 6Ar,d)45Ar is about 416 keV. The excitation energies of a

number of low-lying states were also determined as indicated in the spectrum.

The FWHM of the peaks in the Q-value spectra were verified by GEANT4
simulations [AgoO3]. The simulations take into account the resolution of HiRA detectors,
angular and energy straggling, kinematics broadening as well as energy resolution of the
beam and the fmite-size of the beam spot. The resolution of the detection system due to

pixelation and detector performance is about 350 keV. Other contributions to energy

203

resolutions from energy straggling and position resolution on target for each reaction at
specific angles are summarized in Table 5.2. The simulated resolutions in center-of-mass

34,36,46

frame for p( Ar,d) correspond to all the laboratory angles where deuterons do not

punch-through the thick silicon detectors. In general, the resolutions obtained fi'om the
experiments are consistent with the predicted resolution fi'om simulations. The main
contributions to the resolutions were effects due to the target thickness and the kinematic
dispersion from the large beam spots. The small discrepancy in energy resolutions
between data and simulations for 36Ar case may be attributed to the problems in energy

calibrations.

 

p(“Ar,d)’°Ar

g.e.
\

1.358, 1.798 HOV

I \
o 200|L 3.45s, 3.319 Mall

1.

1 \.

cunts I 75kev

 

 

 

F 1 A _. 1 L 1 L 1
920 -13 -1o -14
O-value (MeV)

Figure 5.4: p(34Ar,d)33Ar Q-value spectrum with laboratory angular coverage
from 5° to 23°.

204

 

1|

 

 

125

° WWW“ 1.1.11.1

\

2.”, 3.103 mv

2 \
15.572116v

\

Counts I 75keV

 

 

 

 

 

 

 

 

1 I 1 n 1 m 1 1
920 -1a -1e -14 -12
O-value (MeV)
Figure 5.5: p(36Ar,d)35Ar Q-value spectrum with laboratory angular coverage
fi'om 5° to 26°.
' 1.734, 1770,
’ p(“Ar,d)“Ar 1270,1911 “0V
>2ool- /
3 ~ g...
m
" /
\ 0.542 MeV
g \.
0
L 1 1 1 Ln
-7 -o -5

 

G-value (MeV)

Figure 5.6: p(46Ar,d)45Ar Q-value spectrum with laboratory angular coverage
from 5° to 19°.

205

Table 5.2: Contributions to the energy resolutions (F WHM) simulated by Geant4 code
and comparisons to the FWHM of the peak corresponding to ground-state transitions in
the Q-value spectra for the reactions in inverse kinematics.

 

 

 

 

 

 

. Geant4 Simulations - energy resolutions
Reactions (g.s) in keV Data
Lab.
9L b Target 83;!“ Straggling Position at 32%;?!" Q-value

‘1 thickness in Tar et resolution keV

(FWHM) g gLab w/o Csl) ( )

find) 24° 75 um 3 mm 447 425 650 420 470
p(34Ar,d) 21° 75 um 10 mm 470 750 893 495 500
p(46Ar,d) 30° 25 um 8 mm 397 717 839 377 416

 

 

 

 

 

 

 

 

 

 

5.3 Background subtraction

The data runs in this experiment were taken with HiRA -8800 coincidence triggers. The
deuterons of interest were obtained from the d-33Ar, d-35Ar and d-45Ar coincidence
events with the corresponding beam gates. Therefore the main background would mainly

be random coincidence. The reaction target was polyethylene (CH2)n. To assess

contributions from the reactions with C nuclei, a 17 mg/cm2 thick uniform carbon target

was employed to evaluate the background. Analyzing the carbon target data in the same
way as the normal data runs would directly give the number of deuterons from C

background coincidence. Judging from the background in the energy vs. angle spectra in
Figures 5.1-5.3, p(46Ar,d) 45Ar reaction has the worse background, probably due to the
thin target that was used. The carbon target run was measured using 46Ar beams with

HiRA-S800 coincidence trigger. The left panel of Figure 5.7 shows the spectra of the

S800 PID gated on 46Ar incoming beam.

206

Since the number of carbon particles in the target and the duration of runs are
different between the carbon target runs and normal data run, normalization in the

reaction occurrence is necessary for correct estimate of the background counts. The

number of carbon particles on the 17 mg/cm2 pure carbon target and the 25 um (CH2),,
target are 8.52x1020/cm2 and 9.88x1019/cm2 with the corresponding total beam particles
of 2.83x109 and 4.15x10lo respectively. Therefore the normalization factor is 1.7. The
right panel of Figure 5.7 shows the deuteron kinematics spectrum gated on the 46Ar beam

particles and 45Ar residues. Multiplying the number of counts in this spectrum by a factor
of 1.7 would give the actual background level. From Figure 5.3, it is clear that the
random coincidence events could be neglected in 46Ar reaction system. The same
conclusion should also be valid for 34Ar and 36Ar as the background problems are much

less severe because thicker targets were used. Therefore no background subtraction is

included in the analysis discussed here.

207

 

t'l". g'v'q'V'h’-

 

 

.. EM PID gated on 4°Ar] Ljnematicsgated on 47°Ar 83 V2731]

 

 

2‘ _ ~~~~A *5
:Carbontarget _- . ' - :'.~ '
2300. ‘ ~- >01 '
A : g
0 L r: to
£2200: g
' 1
g . O' 14 1
92100 8
. : E 12
1: - o
02W_ 31
— : .6 ‘
' 1: 8
“0° 11:
o

 

 

 

 

 

05 ”TOSFBEZO551O115120 0 10 e15(&)25 so 35

Figure 5.7: Carbon target reaction runs with 46Ar beam. (Left) S800 PID spectrum
gated on the incoming 46Ar beam. (Right) Deuteron kinematics spectrum gated on
46Ar beam particle and 45Ar residues.

5.4 Extraction of Cross sections and spectroscopic factors

Normalization analysis is critical in the extractions of absolute cross sections. This

section describes the procedures used to obtain absolute normalization and differential

cross sections as well as the methods to estimate the measurement uncertainties arising

from data acquisition live time and various detector efficiencies. In addition, comparisons

of cross section data to reaction model predictions and the extraction of spectroscopic

factors are discussed.

5.4.1 Absolute normalization and error analysis

The absolute differential cross sections as a function of angle could be expressed by the

following equation 5.3:

208

 

lab
do Nd101)
= 5.3
(an) 6i Afl(9i) NmNbeam ( )

where Nbeam is the total beam particles that hit the target for all the runs; NW is the
number of hydrogen nuclei per centimeter square in the reaction target; Nd is the actual
number of deuterons from the desired reaction emitted in the i m angular bin of 1 1° for a

given scattering angle in laboratory frame; AQ(0,) is the laboratory solid angle subtended

by the scattering angle.

By using Jacobian transformation from the laboratory to center-of-mass
coordinates for the ith angular bin, the differential cross sections in center-of-mass frame

can be obtained using Equations 5.4-5.5:

 

(E = ME) = “_pmse‘m' (5 4)
‘19 COM d“ lab ' (1+p2—2pC050f0”)3/2 '
COM
= mA m E
P 4.1,... (W) ‘55)

where y is the Jacobian transformation, EEOM is the center-of-mass kinetic energy of

deuteron, Q is the reaction Q-value, mg, mg, mp and md are the rest mass of the beam

particle, residue, proton and deuteron respectively and Him” is the center-of-mass angle

corresponding to the I'm angular bin.

The actual number of deuterons, N4, is determined from the measured recoil-

coincident deuterons with consideration of the live time of data acquisition (EDAQ),

209

 

A1900 XFP efficiency (expp), S800 acceptance and transmission efficiency (63300), and

the HiRA geometric efficiency (6(91)H1RA) corrected for N31) e"(01) using Equation5.6.

 

Det
. _ "n (92')/ “919111101

As seen in Equation 5.3, the absolute normalization of the cross sections requires
direct continuous measurement of beam intensities Nbeam which was achieved by the

Multi-Channel Plates (MCP) detection system placed 10 cm in front of the reaction
target. Since the number of beam particle was determined from MCP, the MCP

efficiency is required in order to obtain the actual beam particles on target.

The statistical uncertainty in cross sections is 1/ NE“ . Weighing several pieces

of target foils suggests less than 1% uncertainty in the target thickness which was
neglected in the analysis. Other uncertainties stemming from the determinations of beam
intensities, detector efficiencies and live time of DAQ system are considered as
systematic uncertainties. In this subsection, the detection efficiencies and the overall

normalization uncertainties are discussed.

Detection efficiencies

A. Live time of data acquisition

The live time of the data acquisition is the period when the data acquisition system is

accepting the arriving events. The counting rates of events as a function of time were

210

 

‘13

recorded by electronic sealer modules. By comparing the live to raw (total) rates of
trigger events, the DAQ live time can be evaluated. Figure 5.8 shows the ratios of the live

to raw rates from MCPO, MCPl and the clock in the 19(34’36’46

And) reactions. The drop in
the live time in p(34Ar,d) 33Ar measurement is correlated to the beam rate. The MCPl

live time, which is similar to the MCPO and the clock live time, was used as DAQ live

time in this experiment.

 

 

f
1:
1E

 

 

 

 

 

 

c

.1

$0.0

$0.7

.1” “Ar 0

mono

0.5 , 1 men
150 200 250 000 350 400 450

Run number

Figure 5.8: Live time of data acquisition determined by the ratios of live to raw
rate of MCPO, MCPl and the clock in the p(34’36’46Ar,d) reactions

B. MCP efficiency

The efficiency of MCP is determined by taking the ratio of the number of registered MCP
time events gated on S800 El signal to the E1 time events. Since MCP] time signal was
used for MCP-OR input in the DAQ system, MCPl efficiency is assumed to be 100%
relative to MCPO. Figure 5.9 plots the MCP] efficiency for the reaction runs. The central

part of the MCP] is intensely exposed from the secondary electrons, because the beam

211

 

1;! 9*

was focused at the reaction target which is 10 cm downstream from MCPl. Therefore the
MCP efficiency generally deteriorates with time. To recover the efficiency, bias voltage

was raised from 2290V to 2450V at run 398.

 

| MCPt Efficiency |

11—————*———‘:

 

 

:‘ : K g; e
C E 5'. 3; 9
0.91— ' ' 5 1, E
: 1» § ‘3; s .
. '1' i ‘
§ 0.01: '9 t. '°.1
é ” ° ’9‘
111 0'7.— é a
: a'Ar §°‘Ar “Ar
05E E i
l I l l L 1 l LLL l m

 

7 l L 1 1 A 1 r l 1 2 1 A r 1 1 n L 1
”5150 200 250 300 350 400 450
Run number

Figure 5.9: MCPl efficiency of the p(34’36’46Ar,d) reactions.
C. A1900 extended focal plane efficiency

The A1900 extended focal plane (XF P) efficiency was monitored during the experiment.
The performance of the scintillator is affected by radiation damage. For slight
deterioration, the XF P scintillator bias was simply raised or the XFP scintillator would be
moved so that the incoming beam would hit a fresh spot on the scintillator. If the
efficiency dropped significantly, the XF P scintillator is replaced. Since beam gates
imposed in the analysis were constructed With the TOP -XF P, XFP efficiency needs to be
included for absolute cross section determination. The RF and XFP signals for the same
data run are shown in the left and right spectrum in Figure 5.10. The total number of RF
and XFP event recorded are listed in the spectra. The XF P efficiency for each run is then

determined from the ratio of the total event number of XFP to RF based on the fact that

212

 

there is no efficiency problem in RF. Figure 5.11 summarizes the XF P efficiency for each
data run. Same XFP efficiency is obtained by taking the ratio of the total event number of

XFP to E1 of 8800 spectrometer.

 

 

 

 

 

 

 

 

 

 

ToF.RF
“I”
111111
E
400:
g 2 a
8’0? 5
209.L
1002" hug-1.40490
-Cniilirrlirrlrrilr
‘b 200 400 000 000

ToF.RF ToF.XFP

Figure 5.10: The RF (left) and XFP (right) signals for the same data run of p(34Ar,d)
reaction. The total number of RF and XFP event recorded are listed in the figure.

 

 

[ XFP Efficiency I
fig 0 .°
0.90 . M9: 0 ° .
E 9 . : z a & ..~ 0 O
0.901; . °

 

   

 

”‘150 200.250 300 350 400 450
Runnumber

Figure 5.11: A1900 XFP efficiency of the p(34’36’46Ar,d) reactions.

213

 

 

D. 8800 acceptance and transmission efficiency

In the present experiment, a tungsten beam blocker was inserted on the low rigidity side
of the S800 magnet in order to block the high-momentum tail of the unreacted beam to
protect the focal plane detectors. To ensure that the residues of interest were not blocked
by the blocker, we investigated the fragments of interest in CRDC dispersive positions
with and without being gated on deuterons detected in HiRA. We found that all the
fragments corresponding to the reactions of interest are within S800 acceptance. Thus no
correction is needed in the normalization factor. Unfortunately, there is no reliable
method to obtain the S800 transmission efficiency in this experiment. We assumed the

efficiency to be 100%.
E. HiRA geometric efficiency

In the experiment, 16 HiRA telescopes were placed at 35 cm downstream from the center
of reaction target. The actual positions of pixels obtained from LBAS measurement as
discussed in Section 3.6.3 were used to calculate the actual HiRA geometric efficiency.
One CsI crystal in telescope 15 and a total of 38 silicon strips (17 DE, 13 EF and 8 EB)
did not work. In addition, strip 15 and 16 are not used in the data analysis because of the

gaps between Csl crystals. All these non-functioning crystal and strips have to be taken
into account to determine the correct HiRA geometric efficiency. For 46Ar reaction, some

of the deuterons completely stopped in EF silicon detector in telescope 5 and 6 without
being identified because no DE detector was mounted. Geometric efficiency for the

angular regions covered by these two telescopes was corrected.

214

 

F. Beam Purity

Degraded 3{’Ar beam and secondary 46Ar beam were obtained in this experiment.

Secondary beam 34Ar with main contaminant 33Cl was identified by time of flight from

cyclotron to the A1900 focal plane, using the radio-frequency (RF) of cyclotron and a
scintillator at the extended focal plane at the end of the A1900 spectrometer. Since

momentum acceptance at image 2 of A1900 was adjusted from 0.5% to 1% to increase

the beam intensity on target, the purity of 34Ar changed accordingly. Figure 5.12 displays

the purity of 34Ar beam for each data run.

 

 

 

 

 

 

Run number

Figure 5.12: Proportion of 34Ar in the incoming secondary beams for the
p(34Ar,d) reactions.

215

.f ’-

in. . ...e.‘ .1. a...“

A]

 

Overall normalization uncertainties

In our analysis, the cross sections presented in Table 5.3 were extracted from all the data
available. To estimate the overall normalization uncertainties arising fiom detection
efficiencies and systematics uncertainties, we divided the whole set of data for each (p,d)
reaction into several groups and extracted the cross sections from each data groups
individually. By examining the consistency of the absolute cross sections extracted from

different data groups, we can estimate the overall normalization uncertainty.

We grouped approximately every 100 counts in 2-degree bin size and analyzed

the cross section using the data in each group individually. The extracted cross sections

from each group at deuteron angles 01ab=9° for p(34Ar,d), 01ab=23° for p(36Ar,d) and

0131,: 15° for p(46Ar,d) are presented in Figure 5.13. The solid lines indicate the cross

sections at the same corresponding angles extracted using all available data in the
experiments. Fluctuations between different times of data-taking are found, but the

overall normalization uncertainties are better than 10% indicated by the corresponding
dashed lines. The large deviations in the second and third data group of p(4 6And) can be

explained by the instability of MCP.

216

 

 

[Comparison of Cross Sections]

 

 

 

 

 

 

 

 

25_
__-...-... ------------ ....-..-..---.-... ---------------------------------------
20':- V I . i V .
A
3 -~-----------. -------------------------------------------------------------
g ...,
V -
g 10;“ ---------------------------------------------- 1' ----- -.- ---------------- 1
E ....1 ..... E ...... f. ..... i ...... i ..... . ................... f ............
~ 34
5»- ' Ar
7 39
_, ___________________ 1 _________________________________________ ° Ar
............ run...“.--.5.-.._.‘...-..g...-..-....-.......
- 4s
' r J . 1 . Ar
0 2 4 o o 10
Group

Figure 5.13 Comparisons of absolute differential cross sections at certain angles

for p(34’ 36’ 46Ar,d) reactions respectively over the entire experiment. The symbols
represent the cross sections deduced from about every 100 counts. The solid lines
indicate the cross sections extracted using all the available data, while the dashed
lines are i10% of the solid line.

5.4.2 Differential cross sections and spectroscopic factors

After properly determine the normalization for each reaction, the experimental
differential cross sections in center-of-mass and the associated statistical errors for

p(34’36’46Ar,d) reactions at 33 MeV/nucleon leading to the ground state, first excited state

and groups of excited states are extracted and listed in Table 5.3. The data are also plotted
in solid circles with purely statistical uncertainties as shown in Figures 5.14-5.19, 5.21,
5.23-5.26. For the ground states, differential cross sections determined from the present
measurements are compared to calculations in the framework of the ADWA using two
different prescriptions: (I) CH89 global optical-model potentials and transferred neutron

orbital with Woods-Saxon potential of fixed radius 1.25 fm (discussed in section 2.2.1);

217

 

 

(II) JLM potential geometries and single-particle parameters constrained by Hartree-Fock
calculations (discussed in section 2.5). For clarity and simplicity, we labeled these two
approaches as “CH89” and “JLM+HF”. It should be noted that JLM+ HF descriptions are
not available for excited states here because they require HF calculations of the nucleon
densities and radii of transferred nucleons for excited-states. Calculations plotted in
Figures 5.14-5.19, 5.21, 5.23-5.26 are normalized to the data around the first maximum.
Blue and purple lines represent the CH89 and JLM+HF calculations respectively. The
normalization factors are the SF values with the associated uncertainties. They are listed

in Table 5.4.

218

 

Table 5.3: Differential cross sections in center-of-mass .fiame for p(34’36’46Ar,d)

o ulatin to ound-state and excited states of 33’35’45Ar. The uncertainties resented
P P g 81’ P
here are purely statistical.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

113414803348 p(34Ar,d)33Ar, p(34Ar,d)33Ar.

Ex=0 MeV Ex=1.359+1.798 MeV Ex=3.456+3.819MeV
9c0M Ada/d0 912014 dU/dfl Ada/d0 900M daldfl Ada/d1)
(deg) d0'/d.(1(mb/sr) (mb/sr) (deg) (mb/g) (mb/sg E.) (mb/sr) (mb/sr)

6.9 11.75 0.52 7.8 0.90 0.13 8.94 0.52 0.09
9.0 8.97 0.30 10.1 0.90 0.08 l 1.64 0.54 0.06
l 1.1 5.47 0.18 12.5 0.93 0.07 14.47 0.68 0.05
13.4 3.00 0.11 15.1 1.07 0.06 16.33 0.70 0.09
15.7 1.45 0.08 17.8 1.29 0.07 20.71 0.96 0.07
18.3 1.11 0.07 20.7 1.40 , 0.06 24.27 0.83 0.04
21.0 1.53 0.07 28.7 1.15 0.07 28.29 0.88 0.04
23.7 2.09 0.09 32.0 0.99 0.04 33.03 0.61 0.03
28.6 2.10 0.09 37.5 0.70 0.02 39.03 0.44 0.02
31.7 1.54 0.05
37.0 0.60 0.02

p(36Ar1d)35Ar1 056Ar1d)35Ar.

Ex=0 MeV Ex=l.184 MeV
(Icon Ada/d0 9cm do/dfl Ada/d0

(dej) d0'/d.Q (mb/sr) (mb/sr) (deg) (mb/sr) (mb/sr)

6.2 1.98 0.31 6.6 14.04 0.76
8.0 2.06 0.21 8.6 12.01 0.47
9.9 1.55 0.14 10.6 6.58 0.27
11.8 1.92 0.13 12.7 3.80 0.17
13.9 2.04 0.14 15.0 1.87 0.12
16.1 2.42 0.14 17.4 1.24 0.09
18.5 2.69 0.14 20.0 1.69 0.10
21.1 2.75 0.13 22.8 2.41 0.11
23.9 2.68 0.11 ‘ 25.2 2.80 0.16
26.4 2.18 0.13 29.8 2.19 0.08
31.2 0.82 0.04 34.5 1.04 0.04

 

 

 

 

 

 

 

 

 

 

219

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.3 (cont'd)
p(“Ar,d)"5Ar,
Ex=0+0.542 MeV
9cm Adar/d0.
deg.) da'ldfl (mb/srL (mb/srL
4.1 19.34 1.31
5.3 19.97 0.89
6.5 20.30 0.69
7.8 19.26 0.58
9.1 19.41 0.59
10.4 18.72 0.60
11.9 16.89 0.64
13.4 16.29 0.69
15.0 14.12 0.52
16.8 14.15 0.44
18.7 13.25 0.35
20.9 12.95 0.33
23.3 11.24 0.28
26.3 8.75 0.21
p(“ArAf‘Ar,
Ex=l.734-1.911 MeV
OCOM dcr/dfl Ada/d0
(deg) (mb/sr) (mb/sr)
4.5 19.45 1.19
5.8 18.74 0.78
7.2 15.03 0.54
8.6 11.07 0.39
10.0 8.92 0.36
11.6 6.14 0.31
13.2 5.24 0.32
14.9 5.33 0.35
16.7 6.10 0.31

 

 

 

 

 

 

 

p(46Ar,d)‘5Ar, Ex=0 MeV

 

 

 

 

 

 

 

 

 

0mm da'ldfl Ado/d0

(deg) (mb/srL me/sr)
4.1 1 1.47 0.98
5.3 12.09 0.67
6.5 1 1 .59 0.51
7.8 10.75 0.41
20.9 12.95 0.33
23.3 1 1.24 0.28
26.3 8.75 0.21

 

 

 

 

 

p(46Ar,d)45Ar, Ex ~ 3.950 MeV

 

 

 

 

 

 

9cm do/dfl
(deg) (mb/srL Ada/d0 (mb/sr)
5.02 4.25 0.50
6.51 3.58 0.31
8.03 2.67 0.20
9.61 2.13 0.15

 

 

 

 

220

 

F.197'.‘ "Hm "I
1

 

213‘Ar,g)33Ar

For p(34Ar,d)33Ar, the shapes of the predicted angular distributions from CH89 (solid

curve) and JLM+HF (dashed curve) are in good agreement to the data for the ground-
state transition shown in Figure 5.14. The angular distribution clearly characterizes the
[=0 angular momentum transfer in the reaction. SF(CH89) = 1.10 and SF(JLM+HF) =

0.85 listed in Table 5.7 were extracted by using the first three data points.

The spectroscopic factors for excited states use only the CH89 algorithms.

JLM+HF calculations are not available to describe the excited states. The first two low-
lying states, 1d3/2 at 1.358 MeV and My; at 1.795 MeV, cannot be resolved.

Unfortunately these two states have the same orbital angular momentum [=2 and
distinguishing the individual contributions of these two states using angular distributions
cannot be achieved. The total cross sections are presented in Figure 5.15. CH89
calculations reproduce the experimental data well. Since the calculations of angular
distributions for transfer reactions are not sensitive to the spin value (J). Fitting the data
around the peak gives normalization of 2.34. Large-basis shell-model calculations
suggest a 3/2+ and 5/2+ states at 1.46 and 1.96 MeV respectively. Their corresponding

SF values are 0.517 and 1 respectively.

There is another distinct group at excitation energies of 3.456 and 3.819 MeV as
shown in the Q-value spectrum in Figure 5.4. The total differential cross sections are
extracted and plotted in Figure 5.16. According to NNDC database, the tentative spin
assignments of 3.456 MeV and 3.819 MeV states are (7/2+) and (5/2+) respectively.
Large-basis shell-model calculations suggest a 5/2+ state at 4.08 MeV with SF value of

221

 

:r’
._ .;__ -..—_th

 

 

1.407 as shown in Table 5.4. Since the model space used in the calculations is restricted
in sd-shell, no prediction of 7/2' state is available. The 1 =2 in the CH89 calculations,
which is the dash blue line in Figure 5.16, reasonably reproduce the data and give
SF(CH89) = 1.98. Attempt to associate the peak to the J“=7/2' state has been made. The

black line in the figure is the total contribution of j=7/2 and j=5/2 transition, where the
blue dotted and dashed lines represent the individual calculated angular distributions

multiplied by the respective SF values. Using the combined fit for all the points except
the most forward angle point, the SF(CH89) for the J"=5/2Jr and Jn=7/2' are 1.49 and
0.43 respectively. Based on the present data, however, no firm conclusion can be drawn
on whether the single-particle strength of J"=7/2' is significant around this region of
energy levels. However, the extracted SF values in good agreement with the LB-SM

predictions strengthens the tentative assignment of J'[ =5/2+ for the 3.819 MeV states

 

H
O

da/dQ (mb/sr)

H

 

 

\
l . . L1

 

 

 

‘ ‘ ‘30. 40
9am. (deg)

Figure 5.14: Differential cross sections for p(34Ar,d)33Ar in the center-of-mass
frame for the ground-state transition. The red solid circles are the data from the
present experiment. The solid and dashed curves are the calculations using CH89

and JLM sets of parameters respectively. The calculations are normalized to the
data.

222

 

 

 

 

 

 

10.0, v f ' V I V ' i fii ' ' ' ' l ' ' ' ' I 1
i 34 33
A ‘ P( AI‘,d) Arex. 1
(’73 Ex=1.359+1.798 MeV 1
“\\
Q -
E __ 0r" ““““ ax.
v 1 O F~~T . . . .‘ '1
c: ‘‘‘‘‘ ~.- 3
“d ~. .
E
vs CH89
0.1 A . A A l A A . .44 . 4*; l 4 A . A l
O 10 20 30 40
9cm. (deg

Figure 5.15: Differential cross sections for p(34Ar,d)33Ar in the center-of-mass

frame for transitions to the 1.359 and 1.798 MeV excited states of 33Ar. The red
solid circles are the data from the present experiment. The solid curve is the
calculation using the CH89 parameters normalized to the data.

I
|
r

 

 
 

 

 

 

5.0==vvuvf=~.--....--.
' CH89 p(34Ar,d)33Arex.
A _
$5). 2.0 _<15/2 Ex—3.456+3.819 Mev_
\ ..... 7/2 p
“g 1.0 _ _ d5/2'1'f'7/2 _
v ¥ .:
C 0.5” /
'C3 L--“
\ ......... \
t) .L ................................... \\
PC 0.2 ............................................ 1
01
0 10 20 30 40

Figure 5.16: Differential cross sections for p(34Ar,d)33Ar in center-of-mass frame
for transitions to the 3.456 and 3.819 MeV excited states. The red solid circles are
the data from the present experiment. The black line is the total contribution of
j=7/2 and j=5/2 transitions, where the blue dashed and blue dotted lines represent
the individual calculated angular distributions multiplied by the respective SF
values. Curves are calculations using the CH89 parameters normalized to the data.

 

223

[36Ar D35Ar

The angular distributions of p(36Ar,d)35Ar obtained from present measurements at 33

MeV/nucleon for transitions to ground-state (Figure 5.17) and first-excited state (Figure
5.18) are compared to the previous work of Kozub et a1 [K0268] on the same reaction
performed at 33.6 MeV/nucleon in normal kinematics. Our data and the results of Kozub
et al are denoted by red closed circles and open squares respectively. The error bars
shown in Figure 5.18 of Ref. [K0268] are smaller than the size of the symbols presented
here. The differential cross sections from Kozub et a1 and the present measurement are in

reasonable agreement.

As shown in Figures 5.17-5.18, the calculated distributions using CH89 and

J LM+HF approaches give good descriptions for the p(36Ar,d)35Ar reactions, leading to

the ground state transitions of 35Ar. The CH89 method reproduces the cross sections of

ground-state transition better at the very forward angles. For ground state transition, the
six points in the first peak were used to obtain spectroscopic factors and the extracted
values are SF(CH89) = 2.29 and SF (JLM+HF) = 1.60. For the first excited state, the first
three points were used to obtain SF(CH89) = 1.23. No JLM+HF calculation was done for

excited states.

224

 

 

 

   

   

 

 

 

36 35 . 1”"
1:10.05— p( Ar,d) Argso 3
U) 5.0 L 9
\ - 4
PD ****
E
(3 L0.
“d :
\\.05.
.8 ’ othis work CH89 ,
’ URef.(Koz68) — JLM+HF l
0.1 - - A A l A + . . l x - - . l . . - - l
0 1o 20 30 40
9cm. (deg

Figure 5.17: Differential cross sections for p(36Ar,d)35Ar in the center-of-mass
frame for the ground-state transition. The red solid circles represent the data fi'om
present experiment in inverse kinematics and the black open squares represent
results of the (p,d) reaction at 33.6 MeV/nucleon in normal kinematics fi'om
Kozub et a1 [K0268]. The dashed and solid curves are the calculations using
CH89 and JLM sets of parameters respectively. The calculations are normalized

to the data.

 

 

 

 

: 36 35
A t ~~~~ .‘ P( And) Are.x.
$4 . _
.0 ’ x ‘
E 3. .0
v 1 0 ‘ /“‘bc1
t, . . ,
% 10 - “ ' ‘5 -
\\ 5"-CH89 \ j q 5
b i . ‘1 i \“ :
'U » 0thls work'.‘ ;
U Ref.(K0263)
0.1 . . A x I . . - ‘u l - . A #1 A A . .
0 10 20 3O 40
9m (deg)

Figure 5.18: Differential cross sections for p(36Ar,d)35Ar in the center-of-mass
frame for the transition to first excited state (1.18 MeV). The red solid circles
represent the data from the present experiment in inverse kinematics and the black

open squares represent results for the (p,d) reaction at 33.6 MeV/nucleon in
normal kinematics from Kozub et al [K0268]. The dashed curve is the calculation

using CH89 set of parameters. The calculations are normalized to the data.

225

 

gflArfl‘sAr

For p(46Ar,d)45Ar reaction, the ground state and first excited state of 45Ar are separated

by 0.542 MeV. Due to the severe kinematics broadening and inefficiency of the MCP
detection system in the experiment, we did not completely resolve the ground-state and
first-excited state transitions at all angles. Figure 5.19 shows the experimental data
contributed by both the ground and first excited state transitions. The respective [=3 and
l=1characteristics for the ground and first excited state transition allows us to determine
their individual contributions to the differential cross sections. At forward angles where
kinematic broadening has the least effects, we could distinguish the peaks of the ground
and first excited states at the small angles up to 13°. Figure 5.20 plots the Q-value spectra
at laboratory angle of 125° :t 0.5°, which corresponds to center-of-mass angle of ~7.5°.
For angles less than 13°, we can extract differential cross sections for ground and first
excite states using double-Gaussian fitting. These cross sections are presented in Figure
5.21. The SF values of 5.28 and 0.51 for ground-state and first-excited state obtained by
normalizing the calculations to the data points at forward angles would be used to guide
the two-1 value fitting for composite angular distributions. The dashed and dotted lines in
Figure 5.19 represent the angular distributions of [=3 ground-state and l=l first excited-
state transitions and the solid line is the total angular distributions. All the data points are
used for two-I value fitting constrained by the respective SF extracted at small angles. In
general the shape of the experimental data is reasonably reproduced and the resulting

SF(CH89) for ground-state and first-excited state are 5.08 and 0.51 respectively.

226

 

 

“firm -
‘1

 

U1
C

 

p(46Ar.d)45Ar..x f
Ex=0+0.542 MeV

    

N
O

......................
...............

H
O

do/dO (mb/sr)
U"

 

 

 

CH89
2 __ g.s. q
----- Ex=o.542MeV
1 1 -Ex=0+0.542 Mev1
O 5 10 15 20 25 30
9cm. (deg)

Figure 5.19: Differential cross sections of p(4 6Ar,d)45Ar in center-of-mass frame
for transitions to ground state and first excited state. The red solid circles are the
data from present experiment. The dashed and dotted curves are the calculations
using CH89 for ground-state (1f7/2-) and first excited state (2p3/2-) multiplied by

the corresponding SF respectively. The solid curve is the total contributions of
these two transitions.

 

 
  

 

60_ p(“Ar,d)“Ar
b 9 =12-13°
50g ""
>0 :
x .
4o
13 2
£30:
a :
o .
020_
10
L,

 

30 -a -7 -e 4
O-valuo (MeV)
Figure 5.20: Q-value spectrum of p(46Ar,d)45Ar at 0,,b=12.5 a: 05° (OCOM~7.5).
The red curve is the Double-Gaussian fits to extract cross-sections corresponding
to the ground- and first excited-state transitions.

227

 

 

 

 

50 ""j 'TI'TY'1"fiTI""Tf' '3 20.0 T'* ‘ 1T"l

 

 

 

 

 

 

 

. - 4e 45
:_ 46 45 j P( Ar,d) ArEx=O.542MeV
q: 30* P( Ar,d) ATg.s. 3 7: 10.0 -
m 20; ; m , i'i‘
\ r , \ _____
.0 ’ i .0 5.0-
3.3, E .
g _ g 2.0.
E 5 - CH89 E
“U 3: — JLM+HF : "d 1.0.} CH89
2' ............ 1.1.11 ......... l 0.5 .................
o 5 10 15 20 25 30 o o 15 20 25 30
9cm (deg) 6m (deg

Figure 5.21: Differential cross sections of p(4 6Ar,d)45Ar in center-of-mass frame

for group of transitions to ground state (left) and first excited state (right) at
forward angles.

The different 1 values for the ground state (l=3) and first excited states (l=1) result

in very different angular distributions as shown in Figure 5.19. Specifically between

9C0M =20-28°, the angular distributions for the excited state exhibit a deep minimum,

suggesting that the data points at 9C0M =20.9°, 23.3° and 263° (corresponding 9131, =29°,
31° and 33°) reflect mainly the cross-sections from the ground state. To verify this, the Q-
value spectrum of p(46Ar,d)45Ar at 913b=28-29° (OCOM~21°) is investigated as shown in
Figure 5.22. The Gaussian fit for the peak at the lowest excitation energy gives a centroid
of -5.75 MeV with FWHM ~550 keV. The adopted Q-value for ground-state transition is
-5.80 MeV. The GEANT4 simulations suggest the resolution of the ground-state peak in
this angular range to be ~520 keV. The reasonable agreement between the data and
simulations suggests the cross sections within the angular range of 900M =20.9°-26.3°
shown in Figure 5 .19 can be regarded as pure contribution from ground-state transitions.

As displayed in Figure 5.23, we therefore fitted these three data points plus the other four

228

 

 

points obtained at forward angles (shown in the left panel in Figure 5.21) using CH89
calculations and deduced SF(CH89) = 5 .29 which is consistent with the ground-state SF
of 5.08 deduced from the constrained two-l value fitting to within 4%. Similar analysis

with JLM +HF approach is performed for this data set and gives SF(JLM+HF) of 3.93.

 

 

 

9m -9

-a -7 -6 -5 " L4
Q-value(MeV)

Figure 5.22: Q-value spectrum of p(4 6Ar,d)45Ar at 913b=28-29° (9C0M~21°)-

U1
C
4
1
q
4
1
1
q

 

NOD
OO

 

 

 

da/dQ (mb/sr)

 

 

 

73

5 'CH89 ':

" —JLM+HF 7

3r ‘

2'.-..I-...l.-..i--..i....1....‘

0 5 10 15 20 25 30
9c.m.(deg)

Figure 5 .23: Differential cross sections of p(46Ar,d)45Ar in center-of-mass
frame for ground-state transition.

229

 

 

The above discussion shows that the CH89 calculations reasonably describe the
experimental ground-state angular distributions. The cross sections from the first
excited-state transition can be deduced by subtracting the ground-state contributions from
the data presented in Figure 5.19, where the ground-state contribution is the calculated
angular distributions multiplied by the deduced SF value. The points in Figure 5.24

represent the deduced cross sections and associated statistical uncertainties for the first
excited state. The [=1 calculations nicely reproduce the maximum regions at 900M<10°-
Fitting the calculations to those five points around the peak gives SF(CH89) = 0.53.
However the predicted angular distribution could not reproduce the shape of the first

peak, where the predictions drop less rapidly compared to the data. The discrepancies in

these angles are also reflected in the two-l fitting in Figure. 5.19.

 

20.0 ' '1' l' ‘ |'-v-'v-v-'-..
46 45
p( Ar ’d) ArEx=0.542MeV
1: 10.0 - ‘0 .
”J .42" "“x.
\ .--- .‘
“a 5.0 " . -
V \\\
g 2.0 - 2 2“ -
\ \
b
"d 10 - CH89 ~. ..
§ .
0.5H""“‘"‘--1-A--l-“-..L”...

 

 

 

0 15 20 25 .30
9cm. (deg)

Figure 5.24: Deduced differential cross sections of p(46 Ar,d)45Ar in
center-of-mass frame for the first excited-state transition.

230

 

 

A peak with significant statistics around Q value of -7.5 MeV is clearly seen in
Figure 5.6. The Q-value spectrum includes data with 91ab up t019°. This peak results from
the contributions of the transitions leading to four different states with excitation energies
1.734, 1.770, 1.876 and 1.911 MeV (see Table 5.1). The angular distribution for this
group of states was extracted and shown in Figure 5.25. Apart from the spin assignment

for the 1.876 state which are 1/2’, and 3/2' listed in NNDC, the structural information for

these states is unknown as presented in Table 5.1. From the present data, it is found that
the angular momentum transfers of [=0, 2 (s, d) orbits need to be included in order to
reproduce the angular distribution in Figure 5.25. The extracted SF values are 0.41 and
0.49 for the s- and d-transfer respectively. The solid curve is the total contributions with

each distribution multiplied by the corresponding SF value.

 

 

 

 

 

100.0E
7:500
U)
\
“210.0,
v 5.0:
c: .
'0
\1.OE'___38 '3
.8 0.5}
----- 2d .
—total
0.1...-l‘...l...-l. ..l--.-lL...
0 510152025 30
0m. (deg

Figure 5.25: Differential cross sections of p(46Ar,d)45Ar in center-of-mass frame
for group of transitions to 1.734-l.911 MeV. The red solid circles are the data
from present experiment. The dashed and dotted curves are the calculations using
CH89 for 3s and 2d transitions respectively. The solid curve is the total
contributions.

231

 

Another group of states is found at approximate excitation energy of 3.95 MeV,
where no information about orbital configuration is available according to NNDC. The
corresponding forward angle differential cross sections can be clearly identified and are
plotted in Figure 5.26. The shape of the present angular distributions are consistent with
the [=0 calculations using CH89, suggesting the existence of [=0 strength for the state at
3.95 MeV. The SF(CH89) deduced is 0.15. Predictions for [=0 is not available in the pf-

shell model calculations for 46Ar.

 

 

 

 

46 45
10.0 .- P( AI‘,d) Arex.-.
A 3.“ ‘
$5). ‘ Ex=~3.950 MeV
E *‘a
E 1 '0 -----
v ‘1‘ X,
c 1 -0 r 1
“U . “‘ ' \ 1
\ , ‘1
b '
“U 1 CH89...
0.1 A A A . 1 L A A A l A L4. A l k . A A l
0 10 20 30 40

9m. (deg

Figure 5.26: Differential cross sections of p(4 6Ar,d)45Ar in center-of-mass
frame for group of transitions at about 3.950 MeV.

232

 

 

5.5 Theoretical spectroscopic factors and reduction factors

Experimental spectroscopic factor is extracted by taking the ratio of the measured cross-
sections to the cross sections calculated with a reaction model. As discussed in section
5 .4, the statistical uncertainty was determined better than 10%. The main uncertainty in
the measured data is the absolute normalization of the cross-sections which is determined
to about 10%. Therefore we assign 10% uncertainty to the experimental SFs summarized
in .Table 5.7. No uncertainty arising from the choice of optical mode parameters

associated to the deduced SF values is assigned.

The Large-basis shell-model (LB-SM) calculations were performed using the

code Oxbash [Br004] in sd-shell model space with USDB effective interaction [Br006]

for 33’35Ar, where the calculated SFs for excitation energy less than 10 MeV are listed in

Tables 5.4 and 5.5. For 46Ar, interaction of Nummela in sd-pf model space et a1. is used

[Sig07]. The results for energy level less than 5 MeV are presented in Table 5.6.

The LB-SM predictions and the reduction factors which are the ratios of the

experimental spectroscopic strengths with respect to LB—SM predictions are listed in
Table 5.7. The values of the ground-state SF(CH89) for neutron-rich 46Ar and stable 36Ar
isotopes and the first excited-state of 36Ar obtained in the present experiments are in good

agreement with the LB-SM predictions, which follow the systematics and benchmarks

established by our extensive studies on a wide range of stable nuclei as discussed in
Chapter 2. The experimental ground-state SF(CH89) for proton-rich 34Ar is about 15-

20% smaller than predictions. If the approach with JLM potential and HF constraints is

233

 

adopted to analyze the ground states, an approximately 30% reductions in the
experimental SF(JLM) are obtained for 34Ar, 36Ar and 46Ar relative to LB-SM prediction.
This is also consistent with our previous results of ~30% systematic reduction as
discussed in section 2.5. Such consistent suppression in SFs for 34Ar and 46Ar in the

transfer reaction analysis verifies the applicability of a systematic framework with

consistent input parameters to extract reliable relative SF s.

Table 5.4: Theoretical spectroscopic factors of 34Ar for energy levels less than 10 MeV
using USDB interaction in sd-shell model space.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex J" SF(LB- Ex J“ SF(LB- Ex J“ SF(LB-
(MeV) SM) (MeV) SM) (MeV) SM)
0.00 1/2+ 1.311 5.70 1/2+ 0.017 7.61 3/2+ 0.008
1.46 3/2+ 0.517 6.26 1/2+ 0.214 7.64 5/2+ 0.214
1.96 5/2+ 1.000 6.54 3/2+ 0.002 7.93 1/2+ 0.008
2.63 3/2+ 0.000 6.59 5/2+ 0.046 7.94 3/2+ 0.004
3.44 3/2+ 0.021 6.71 3/2+ 0.007 8.22 1/2+ 0.003
3.65 5/2+ 0.270 6.91 3/2+ 0.041 8.37 5/2+ 0.020
4.08 5/2+ 1.407 6.99 1/2+ 0.008 8.86 5/2+ 0.058
4.35 1/2+ 0.069 7.00 5/2+ 0.416 8.96 1/2+ 0.004
4.99 3/2+ 0.001 7.19 5/2+ 0.030 9.21 1/2+ 0.005
5.13 5/2+ 1.715 7.28 3/2+ 0.003 9.55 1/2+ 0.005

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.5: Theoretical spectroscopic factors of 36Ar for energy levels less than 10MeV
using USDB interaction in sd-shell model space.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex SF (LB— Ex SF(LB- Ex
(MeV) J1: SM) (MeV) J1: SM) (MeV) J7: SF(LB-SM)

0.00 3/2+ 2.104 5.71 5/2+ 1.358 7.85 3/2+ 0.011
1.23 1/2+ 1.368 5.80 3/2+ 0.000 8.05 5/2+ 0.009
1.68 5/2+ 0.001 5.83 5/2+ 0.050 8.17 3/2+ 0.000
2.63 3/2+ 0.291 6.4] 5/2+ 1.587 8.25 5/2+ 0.046
3.10 5/2+ 1.108 6.86 3/2+ 0.001 8.68 3/2+ 0.000
3.95 3/2+ 0.004 6.97 5/2+ 0.152 8.82 5/2+ 0.519
3.98 1/2+ 0.005 7.1 1 1/2+ 0.052 9.26 1/2+ 0.000
4.61 5/2+ 0.027 7.31 1/2+ 0.175 9.43 l/2+ 0.000
4.77 3/2+ 0.002 7.37 3/2+ 0.015 9.72 l/2+ 0.004
4.93 l/2+ 0.054 7.83 1/2+ 0.058

 

 

 

 

 

 

 

 

 

234

 

 

Table 5.6: Theoretical spectroscopic factors of 46Ar for energy levels less than 5 MeV

using interaction of Nummela et al in sd-pf model space [Sig07].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex JR SF(LB- Ex J“ SF(LB- Ex J“ SF(LB-

(MeV) SM) (MeV) SMl (MeV) SM)

0.00 7/2- 5.161 3.27 7/2- 0.071 4.43 3/2- 0.000
0.42 3/2- 0.776 3.50 3/2- 0.001 4.46 7/2- 0.060
1.24 3/2- 0.005 3.51 7/2- 0.033 4.55 3/2- 0.001
1.33 5/2- 0.057 3.75 1/2- 0.01 1 4.62 5/2- 0.002
1.79 1/2- 0.001 3.76 5/2- 0.000 4.68 5/2- 0.044
2.14 1/2- 0.037 3.79 5/2- 0.001 4.76 7/2- 0.010
2.26 7/2- 0.097 4.03 5/2- 0.000 4.81 7/2- 0.001
2.37 7/2- 1.073 4.05 7/2- 0.058 4.85 3/2- 0.000
2.42 5/2- 0.006 4.15 5/2- 0.006 4.91 5/2- 0.000
2.71 3/2- 0.007 4.18 3/2- 0.000 4.98 3/2- 0.000
2.87 5/2- 0.004 4.20 7/2- 0.136 4.98 7/2- 0.001
2.87 3/2- 0.013 4.20 5/2- 0.004 5.00 5/2- 0.001
2.95 5/2- 0.001 4.28 3/2- 0.002
2.96 7/2- 0.065 4.34 1/2- 0.001
Table 5.7: Extracted SF values.
r an -s ate n tr n ectr o i fact 1'
Sn-S SF(LB-

Isotopes I 1" mid?) SM) SF(CH89) Rs(CH89) SF(JLM) Rs(JLM)
”A, 3”; 12.41 1.31 1.10 1 0.11 0.84 a: 0.08 0.85 4 0.09 0.65 1 0.07
“A, d”; 6.75 2.10 2.29 1 0.23 1.09 1 0.11 1.60 1 0.16 0.76 :h 0.08
46Ar fm' -1003 5.16 5.29 :t 0.53 1.02 1: 0.10 3.93 i 0.39 0.76 1 0.08

 

 

 

 

 

 

 

 

 

 

First excited-state neutron spectroscopic factor

NNDC

1.184
0.542

LB-SM

235

Present work

1.23 i0.12

 

0.49 i 0.05

0.90 :t 0.09
0.63 d: 0.06

 

Table 5.7 (cont'd)

Normalization value of oalculation to data for a group of oxoitod stateo

NNDC LB—SM Present work

Isotopes Ex(MeV) 1;" Egg $13.93- SF(CH89) Rs(CI-189)
34Ar 1.359+1.798 d(,,,., 512-) 1.46(3/2+) 0.52 23410.23

1.9 (5/2+) 1.00

 

 

 

 

 

34

 

 

 

 

 

 

 

 

 

 

 

 

3.456 lfm- -- -- 0.40 4 0.04
“A: 3.819 2dm+ 4.08 1.407 1.49 1 0.15 1.06 1 0.12
“A, 1.734-1911 2d(y2‘) -- -- 0.49 4 0.05
“A, 1734-1911 331,24. -- -- 0.41 1 0.04
46m ~3.950 35“.; -- -- 0.15 :1: 0.02

 

5.6 Asymmetry dependence of reduction factors

The error bars listed in Table 5.7 reflect the uncertainties in the absolute cross-section

determination of 10%. The values of Rs deduced using CH89 or J LM+HF for symmetric

36 . 46 . . . . . . . .
Ar and neutron-rich Ar are Similar. This is conSistent With the preVious systematic

studies with stable nuclei [Tsa05, Lee07]. The extracted value of Rs for proton-rich 3"1Ar

is about 15-20% smaller.

To show the asymmetry dependence for the chain of Ar isotopes, reduction
factors of proton-rich 34Ar and neutron-rich 46Ar and stable 36Ar nuclei obtained from the

present experiments are plotted in closed symbols and the reduction factors of the stable

37'41Ar isotopes deduced from previous measurements [U607] are plotted in open

symbols as a function of AS (defined as the difference in the neutron and proton
separation energy [Lee06]) in Figure 5.27. The difference of neutron and proton

separation energies, AS, of 46Ar and 34Ar are -10.03 and 12.41 MeV, significantly

236

 

increasing the span of the separation difference from isotopes 36'MAr near stability which

have AS from -6.70 MeV to 6.57 MeV. The Rs deduced fiom CH89 and JLM+HF are

plotted in top and bottom panels of Figure 5.27, respectively.

There are two sets of measurements for 40Ar, but the extracted SFs are different

by about a factor of 2. Such inconsistency may be attributed to the inconsistency of

absolute normalization in cross sections between two experiments [Joh68,Ton77]. Based
on the consistency check as explained in Ref. [Lee06], the SF for 40Ar shown in Figure
5 .27 is extracted using data in Ref. [Joh68]. Furthermore, there is only one measurement
available for 38Ar and no other independent measurement exists for consistency check

[00075]. The theoretical and the experimental SFs from transfer reactions for stable Ar

isotopes are listed in Table 5.8, where LB-SM calculations use USDB interactions in sd

model space for 37'38Ar isotope and SDPFNOW interaction in sdpf model space for 39'
41Ar. Within experimental uncertainties, little suppression in the reduction factor is found

for proton-rich 34Ar compared to the well-bound nuclei or neutron-rich 46Ar.

237

Eh}: ..W m” I
1

T3“ in

 

 

 

 

 

 

 

 

 

 

A1.25 E- 39 37Ar _?
2 46 “Ar Ar 36Ar 1
”3100 - l. l. ' -
m f l? i 1 38Ar “Ar j
C1 1 l rfr %
LL. 0.75 f q
(I) .
> i.
g 0.50 " 40Ar —1
11‘. : .
80.25 7 0 this work CH89 j
'53 751’ other work n-transfer
0.00 L*“~-‘~AA1-----.4..1...
-10 -5 0 5 10
AS(MeV)
E 1.25 - _
”1’ .
a 1.00 "' ”Ar 37Ar _
v 46Ar 41 30A
Eh Ari l r 1
£0.75 _- E l l i 34A}. _1
A ' i l lat:Ar }
[:4 I
m 0.50 T l 1
+ 140 1
2 Ar ‘
$0.251. this work JLM—HF 1
v ’ —- .
% :* other work n transfer 1
0.00“"““‘ -l-----|i.
-10 —5 O 5 10
AS(MeV)

Figure 5.27: (top) Reduction factors deduced using CH89 approach
SF(CH89)/SF(LB-SM) of the proton-rich 34Ar and neutron rich46Ar, together

with the well bound 3641A: isotopes. (Bottom) Same as top panel, except the
calculations use J LM+HF method.

238

 

 

- Ink-EM

“889...“..3‘
‘.

 

Recent dispersive-optical model (DOM) analysis of elastic—scattering and bound-

4048

level data on Ca isotopes ( Ca) suggests that proton correlations exhibit weak

dependence on the asymmetry where the proton SF magnitude from 40Ca to 48Ca changes
by about 10% [Cha06, Cha07]. Based on the principle of isospin asymmetry, about 2%
reduction in neutron SF value for 34Ar relative to 36AI is predicted [Cha-priv]. The weak

dependence of reduction factors on the asymmetry of the three Ar isotopes in this

experiment is similar to the trends obtained from the recent Dispersive-Optical-Model

analysis of elastic-scattering and bound-level data for 40.49Ca isotopes [Cha06, Cha07].

However, there are not enough existing data to constrain the asymmetry
dependence for neutrons in the DOM analysis. Using no asymmetry term for neutrons,
the asymmetry dependence of neutron occupation probabilities in Ca isotopes from the
DOM result is plotted as solid line in Figure 5.28. Such results describe the trend of
neutron reduction factors for Ca isotopes in transfer reactions [Cha07]. Here the SF
values in transfer reactions were deduced using JLM+HF approaches [Lee06] and SF
(LB-SM) values of Ca isotopes were calculated in fiJ-shell model space with GXPF 1A
interactions [GXPFIA]. The shell-model and the experimental SF values are listed in

Table 5.8.

239

 

 

 

Reduction Factor (Rs)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

 

 

ngl

    

ll.
{31

if? JLM+HF

.l.--.

"l

v ' Viv

fiT 1 T v

40-45.47-49Ca

 

 

n—transfer

’ — DOM predictions
‘ l . . . . I . .

1
A l n A A

 

 

—10

-5

O 5

AS(MeV)

10

Figure 5.28: Reduction factors of Ca isotopes obtained from the dispersive

optical-model (DOM) calculation (solid line) and from neutron (d,p) and (p,d)
transfer reactions analyzed using J LM+HF approach.

Table 5.8: Extracted ground-state SFs for stable Ar and Ca isotopes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“NOW 11" (813;?) sggbn- SF(CH89) Rs(CH89) ”gs“ mglfiM“
”A: dm‘ 0.07 0.36 0.36 :1; 0.05 1.00 :L- 0.14 ‘ 0.27 i: 0.04 0.75 :t 0.11
38A: 0,; 1.60 3.04 2.47 :l: 0.70 0.81 :1: 0.23 1.78 :l: 0.50 0.59 :l: 0.17
39A: fm' 413 0.83 0.81 2: 0.11 0.98 at 0.13 0.65 :l: 0.09 0.78 i 0.11
40Ar rm' -2.66 1.91 1.08 :l: 0.31 0.57 :t 0.16 0.84 :l: 0.24 0.44 :l: 0.12
“Ar f,,,' -6.70 0.65 0.55 i 0.08 0.85 at 0.12 0.46 :L- 0.07 0.71 :1: 0.10

-_‘I°Ca (102+ 7.31 4.00 4.35 :t 0.62 1.09 :l: 0.16 3.20 :t 0.46 0.80 :l: 0.12
“Ca f,,,' -0.53 1.00 1.01 :t 0.06 1.01 :l: 0.06 0.73 i 0.04 0.73 4: 0.04
”Ca fm‘ 1.20 1.95 1.93 :t 0.17 0.99 i 0.09 1.31 a: 0.12 0.67 i 0.06
”Ca fm' -2.74 0.75 0.63 :1: 0.07 0.84 :l: 0.09 0.44 :l: 0.05 0.59 :l: 0.07
44Ca fm' -1.03 3.86 3.93 :l: 1.08 1.02 i 0.28 2.67 :l: 0.31 0.69 a: 0.08
”Ca fm‘ -4.88 0.50 0.37 :t 0.05 0.74 :t 0.10 0.26 :l: 0.04 0.52 i 0.08
47Ca rm“ 6% 0.25 0.26 :1 0.04 1.04 :1: 0.16 0.19 d: 0.03 0.76 a: 0.12
“Ca fm' -5.86 7.70 7.35 :t 1.42 0.95 :l: 0.18 5.41 d: 1.05 0.70 :l: 0.14

flea pm' —1 1.31 0.95 0.69 :t 0.07 0.73 :t 0.07 0.74 :1: 0.08 0.78 i 0.08

 

 

240

 

 

The weak dependence of reduction factors on the asymmetry observed from the present
transfer reaction experiments are, however, in contradiction to the strong systematic
suppression in SFs obtained using the intermediate-energy nucleon knockout reactions
[Gad08]. Figure 5.29 compares the asymmetry dependence AS of reduction factors (Rs)
for the transfer (closed and open circles) and knockout (open triangles) reactions. For
transfer reaction data, the open circles represent Rs(CH89)= SF(CH89)/SF(LB-SM)) and

the closed circles represent (Rs(JLM+HF)= SF(JLM+HF)/SF(LB-SM). For knockout

32,34,46

reaction data, the dashed line is the best fit of Rs of Ar from knockout reactions.

The 34Ar AS values from the present work and ref [Gad04-2,Gad08] are different.
In this work, spectroscopic factor of the ground-state is determined. In knockout
reactions, ground-state Rs are obtained for 46Ar [GadOS] and 32Ar [Gad04-l]; while
inclusive cross sections including contributions from the excited states to the ground
states were measured for 34Ar [Gad04-2]. Thus the 34Ar AS value in Ref. [GadO4-2] is
weighted by the nucleon separation energy of the excited states. In principle, the
knockout value would give the upper limit of the 34Ar ground-state Rs value as

contributions from excited states should increase the reduction factor values.

As shown by the open triangles in Figure 5.29, the neutron Rs extracted from

knockout reactions for 34Ar is approximately a factor of two smaller than that for 46Ar

[Gad08]. Even larger reductions have been observed for neutron knockout from 32Ar
[Gad08], a nucleus for which transfer data is not available. Within experimental
uncertainties of i10% in the present transfer reaction measurements, the reductions in the

spectroscopic factors, deduced using both CH89 and JLM+HF analysis, for 34Ar relative

241

 

to 46Ar of 0-3 5% are possible but much larger reductions are excluded. This suggests that

there is a systematic difference between the conclusions drawn from these two probes for
the spectroscopic factors of strongly bound particles. Thus a reexamination of the
reaction theory description of transfer reactions or knockout reactions including the input

parameters used in these analyses may be needed.

 

 

 

 

1.25’I ' ' ' ' l '—' f v j r v f v I . 1 f , ‘
1.00? g % _
: 4'GAI. ‘\‘\ 36Ar-l- § ‘
0.75— I _
I “ .\ :34 j
m” J. Ar .
0.50 ' \\\\ T -
L ‘¥\
0'25.- O n-transfer (CH89) 32 Y1
o n—transfer (JLM+HF) Ar .
. Vn—knockout ‘
o,oo'---41---.n.....-....
—20 —10 0 10 20

AS(MeV)

Figure 5.29: Reduction factors Rs=SF(expt)/SF(LB-SM) as a function of the
difference between neutron and proton separation energies, AS. The solid and
open circles represent Rs deduced in JLM+HF and CH89 approach using the
present transfer reaction data respectively. The open triangles denote the Rs from

knockout reactions [Gad08]. The dashed line is the best fit of Rs of 32’34’46Ar from
knockout reactions. The use of different AS values from the present work and
knockout reactions in Ref. [Gad08] is explained in text.

242

 

 

 

Chapter 6

Summary and conclusions

In this thesis, we investigate the asymmetry dependence of neutron correlations in nuclei
over a large span of Ar isotopes. Such correlations govern the nuclear properties of the
drip line nuclei. The experimental spectroscopic factor (SF), defined as the ratio of the
measured cross sections divided by the cross sections calculated with a reaction model
for transfer of a nucleon between the initial and final states, plays a special role in linking
the microscopic theory of nuclear structure to the reaction model. The calculated cross
sections depend on the wave functions of the nuclei involved and the transferred nucleon.
Transfer reactions comprise the preponderance of determining experimental
spectroscopic factors in the literature and provide an important technique to study the
properties of rare isotopes. In addition to performing a survey of the experimental neutron
SF for nuclei Z S 28 and comparing the results to the shell model, this thesis also
describes the first (p,d) transfer reaction measurement with Ar isotopes performed at the

NSCL using the High Resolution Array (HiRA) and the S800 spectrometer.

In the first part of the thesis, we survey the neutron spectroscopic factors extracted
from the available (p,d) and (d,p) angular distribution data on nuclei with Z S 28.
Utilizing global optical-model potentials and consistent sets of parameters systematically,
the long-standing inconsistency and ambiguities in the nucleon-nucleus potentials as well
as single-particle parameters can be resolved. As a result, relative spectroscopic factors
can be determined over a wide range of nuclei. We re-analyze the past measurements of

243

 

 

(p,d) and (d, p) transfer reaction data using nucleon-nucleus global optical-model

potential, Chapel-Hill 89 (CH89) and the neutron potential of Wood-Saxon shape with
conventional radius of ro=1.25 fin and diffuseness a0=0.65 frn in the fi’amework of

Adiabatic Distorted Wave approximation (ADWA). The extracted ground-state neutron
spectroscopic factors for 88 nuclei ranging in Z=3-28 are in agreement with the large-
basis shell-model (LB-SM) calculations to within 20%. The results illustrate the success
of the approach for providing systematic and consistent spectroscopic factor values with

minimum assumptions.

The excited-state spectroscopic factors are of particular interest because they are
needed in astrophysical network calculations. Based on the same methodology, we have
extracted 565 neutron spectroscopic factors of sd- and fi2-shell nuclei. We are able to
compare 125 of the extracted spectroscopic factors to values predicted by large-basis-
shell—model (LB-SM) calculations and evaluate the accuracies of spectroscopic factors
predicted by different interactions. For Z S 24 nuclei far away from closed shells with SF
values larger than 0.005, the agreement between the extracted SF and shell-model
predictions are better than 30%. Immediately outside the 40Ca core, especially for light
Ca isotopes near the closed sd shell, the shell model predicts pure single-particle states
with much larger spectroscopic factors than experimental values. This is a further
indication that a large sd-fp model space may be needed due to the effects of core
excitations and particle-hole collective states. For Ni isotopes, the agreement is poor and

the result suggests that the effective interactions in this region need improvements.

244

 

 

The survey of neutron spectroscopic factors provides important benchmarks for
the neutron-transfer reaction studies and a stringent means to test the residual interactions
and Hilbert spaces used in the current nuclear structure calculations. Beyond the sd-shell,
better effective interactions and larger valence spaces should be accommodated in the
current large-basis shell-model calculations. Both directions require more efficient shell-
model codes/algorithms and larger computational facilities, which is becoming available
in the forthcoming years. The good agreement for small spectroscopic factors in the sd-
shell nuclei suggests that experiments can be reliably performed to extract spectroscopic
factor values down to 0.005. The ability to measure and calculate small spectroscopic

factor values is important for network calculations of astrophysical interest.

Since the shell model includes the effects of correlations within the model basis,
but not those outside the model basis, the shell model spectroscopic factors represent the
upper limit. Reduction of the spectroscopic factors is one signature of the correlations
experienced by the valence nucleon. This is evidenced in the smaller SF values predicted
by the large-basis shell-model compared to those predicted by the Independent Particle
Model. Constant quenching (~30-40%) of the proton spectroscopic factor values are
observed in the studies of (e,e ’p) reactions for nuclei near the closed shell compared to
the independent particle model (IPM) expectations. For nuclei near the closed shell, the
LB-SM calculations give similar predictions as the IPM model. The suppression of the
proton spectroscopic factors extracted in the (e,e ’p) study has been attributed to
insufficient treatment of nucleon-nucleon correlations including both short- and long-
range correlations in the current shell models. More intriguing is the large (up to 75%)

245

 

 

suppression in the spectroscopic factor values for the strongly bound nucleon of rare
isotopes compared to shell model predictions obtained using the intermediate—energy
nucleon knockout reactions. Currently, no theory is able to account for the large
reduction of the spectroscopic strength.

We study the suppression of the spectroscopic factors in transfer reactions over a
range of nuclei using the microscopic potentials with constraints based on physics input.
We use the Jeukenne, Lejeune and Mahaux (JLM) nucleon optical interactions and
constrain the density of the potential as well as the transferred neutron orbital rrns radii
with the Hartree-Fock (HF) calculations. With the JLM+HF method, an overall ~30%
suppression in SF values for most nuclei is obtained. Similar quenching of the proton SF
values have been observed in the studies of (e,e ’p) reactions. However, unlike the
knockout results, the reduction factors Rs (the ratio of experimental spectroscopic factors
to LB-SM predictions) obtained from the comprehensive transfer reactions data have no
dependence on the neutron binding energy within experimental error. Similarly, reduction

factors obtained fi'om the studies of (e,e ’p) reactions are constant.

The survey data include very few extremely neutron-rich or neutron-deficient
nuclei. Furthermore, systematic uncertainties inherent in comparing spectroscopic factors
from different experiments performed over four decades are large. To reduce the

uncertainties and to extend the region of study to more extreme N/Z nuclei, we extract the

experimental neutron spectroscopic factors for 34Ar and 46Ar using (p,d) neutron transfer

reactions with proton-rich 34Ar and neutron-rich 46Ar beams in inverse kinematics. The
. 34 33 6 45 .

experiments of p( And) Ar and p(4 Ar,d) Ar were performed at the National

246

 

.1“

 

Superconducting Cyclotron Laboratory at Michigan State University at beam energy of
33 MeV per nucleon. The kinematically complete measurements were achieved by using
the high resolution silicon array, HiRA, to detect deuterons in coincidence with the recoil
residues detected in the S800 mass spectrometer. The success of the current experiment
demonstrates the techniques of transfer reactions in inverse kinematics using radioactive
beams to study single-nucleon properties of exotic nuclei. The experience learned from
this experiment is essential for future HiRA transfer reaction measurements, especially

for those which require excellent energy and angular resolution.

The deuteron differential cross sections obtained in the experiments are compared

to cross sections calculated using both JLM+HF and CH89 methods. Consistent with

previous systematic studies with stable nuclei, the values of Rs for symmetric 36Ar and

neutron-rich 46Ar are similar. The extracted value of Rs for proton-rich 34Ar is about 15-
20% smaller. With the experimental uncertainties of i10% in the measured differential

cross-sections, reductions in the spectroscopic factors for proton-rich 34Ar relative to

neutron-rich 46Ar of 0-35% are possible but much larger reductions are excluded. The

weak dependence of reduction factors on the asymmetry of the Ar isotopes (including

other Ar isotopes obtained in the survey) is similar to the trends obtained from the recent

4049
Ca

Dispersive-Optical-Model analysis of elastic-scattering and bound-level data for
isotopes. In the latter study, calculations using the dispersive-optical model suggest that
the proton spectroscopic factors from 40Ca to 48Ca have a weak (10%) dependence on the

asymmetry. The experimental trend is consistent with weak asymmetry dependence in the

neutron correlations for N>Z Ca isotopes.
247

 

VS

 

In single nucleon knockout reactions in intermediate energy, the deficient nucleon

species have stronger reductions in spectroscopic factors than the weakly bound excess

species, in particular the neutron reduction factor, Rs, of 34Ar is smaller by a factor of two
compared to 46Ar. Unlike the trends observed for knockout reactions, comparison of the

extracted spectroscopic factors for proton-rich 34Ar and neutron-rich 46Ar using transfer

reactions suggests a weak dependence of correlations on neutron-proton asymmetry in
this isotope region. The origin of such discrepancy between the spectroscopic factors
extracted from transfer and knockout reactions is not clear. The new results pose an
intriguing question about the reaction mechanisms of transfer and knockout reactions as
well as the nature of neutron correlations in nuclei with extreme isospin asymmetry.
Further theoretical study would be needed to resolve the inconsistency in the reaction
mechanisms and better describe the asymmetry dependence of nucleon correlations when

approaching to the drip lines.

248

 

Wei“

 

 

Bibliography

[Acuit] http://www.acuitvlascr.com/AR600/index.shtml

[AgoO3] S. Agostinelli et al., Nucl. Instrum. Meth. A, 506, 250 (2003).
http://www.geant4.org/gcant4/

[Akm98] A. Akmal et cl., Phys. Rev. C58, 1804 (1998).

[Alf78] W. P. Alford et al., Nucl. Phys. A302, 12 (1978).

[And68] S. A. Andersen and Ole Hansen, Nucl. Phys. A120, 421 (1968).
[Aue67] N. Auerbach, Phys. Rev 163, 1203 (1967).

[Aus70] N. Austem, Direct Nuclear Reaction Theories, John Wiley & Sons, New York,
1970.

[BarO9-l] C. Barbieri, Phys. Rev. Lett 103, 202502 (2009).

[Bar09-2] C. Barbieri et al., [nucl-th] arXiv:0901.1920v1.

 

[Bar09-3] c. Barbieri et al., Nucl. Phys. A834 (2010) 788.

[Bau96] E. Bauge et. al., Proceedings of the Specialists' Meeting on the Nucleon Nucleus
Optical Model up to 200 MeV, 1996. www.nea.fr/html/science/om200/bauge.pdf

[BazO3] D. Bazin et al., Nucl. Instrum. Methods Phys. Res. B 204, 629 (2003).
[Ber09] J. Berryman et al., Phys. Rev. C 79, 064305 (2009).

[Bje67] J. H. Bjerregaard and Ole Hansen, Phys. Rev. 155, 1229 (1967).
[Bob06] R.J. Charity et al., Phys. Rev. Lett. 97, 162503 (2006).
[Bob07l R.J. Charity et al., Phys. Rev. C76, 044314 (2007).

[Boh75] A. Bohr and B. R. Mottleson, Nuclear Structure (Benjamin, New York,
1975),Vol. I

[Boy76] R. N. Boyd et al., Phys. Rev. C 14, 946 (1976).
[BroOO] B. A. Brown, W. A. Richter, and R. Lindsay, Phys. Lett. B483, 49 (2000).

[BroOl] B. A. Brown, Frog. Part. Nucl. Phys. 47, 517 (2001).

249

[BroOl-l] B.A. Brown, Progress in Particle and Nuclear Physics, 47, 517 (2001).
[BroOl-2] B. A. Brown, S. Type], and A. Richter, Phys. Rev. C 65, 014612 (2001).

[Br002] B. A. Brown, P. G. Hansen, B. M. Sherrill, and J. A. Tostevin, Phys. Rev. C 65,
061601(R) (2002).

[BroO4] B. A. Brown et al., Computer program,
http://www.nscl.msu.edu/~brown/resources/oxbash-augsut-2004mdf.

[BroOS] B. A. Brown, http://www.nscl.msu.edu/~brown/Jina-workshop/BAB-lecture-
notespdf, 2005.

[BroO6] B. A. Brown and WA. Richter, Phys. Rev. C 74, 034315 (2006).

[Bro74] G. Brown, A. Denning and J. G. B. Haigh, Nucl. Phys. A225, 267 (1974).
[Bro98] B. A. Brown, Phys. Rev. C 58, 220 (1998).

[Bro99] B.A. Brown, J. Phys. G: Nucl. Part. Phys. 25, 623 (1999).

[But64] P. J. A. Buttle and L. J. B. Goldfarb, Proc. Phys. Soc. London 83, 701 (1964).
[Cam06] C. Campbell et al., Phys. Rev. Lett. 97, 112501 (2006).

[Cas90] R.F. Casten, Nuclear structure from a simple perspective (Oxford University
Press Inc., New York, 1990).

[Cau05] E. Caurier et al., Rev. Mod. Phys. 77, 427 (2005).

[Cau05] E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A.P. Zuker, Rev. Mod.
Phys. 77, 427 (2005).

[Cau99] E. Caurier, shell model code ANTOINE, IRES, Strasbourg 1989-2004, E.
Caurier and F. Nowacki, Acta Physica Polonica 30, 705 (1999).

[Cha06] R.J. Charity et al., Phys. Rev. Lett. 97, 162503 (2006).

[Cha07] R.J. Charity et al., Phys. Rev. C76, 044314 (2007).

[Cow9l] J.J. Cowan, F .-K. Thielemann and J .W. Truran, Phys. Rep.208, 267 (1991).
[Daq] http://sourceforge.net/projects/nscldaq/

[Dau63] C.Daum, Nucl. Phys. 45, 273 (1963).

250

nnsu: m2" gal: ‘ = ‘

 

[De105] F. Delaunay, F. M. Nunes, W.G. Lynch, M.B. Tsang, Phys. Rev. C 72, 014610
(2005)

[deL67] M. E. de L6pez et al., Nucl. Phys. A94, 673 (1967).
[DesOO] P. Descouvemont, The Astrophysical Journal, 543,425 (2000).
[DicO4] W.H. Dickoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004).

[Dic08] W. H. Dickhoff and D. V. Neck, Many-body theory exposed, World Scientific,
Singapore (2008).

[End03] J. Enders et al., Phys. Rev. C 67, 064301 (2003).

[End77] P. M. Endt, Atonic Data and Nuclear Data Tables 19, 23 (1977).
[ENDSF] http://www.nndc.bnl.gov/ensdf/

[Eng07] G. L. Engel et al., Nucl. Instrum. Meth. 573, 418 (2007).
[Expt03045] M. Kilburn, http://groups.nscl.msu.edu/hira/03045/index.htm

[Expt06035] A. Sanetullaev, http://groupsnscl.msu.edu/hira/06035/index.htm

[Fic74] D.Fick, J,NUK, 19,693 (1974),
http_://www.nndc.bnl. gov/ ex for/ servlet/X4sGetSubent?subID=l 503 26002&master=ND

[Fos69] C. C. Foster, W. E. Maddox, and D. W. Miller, Phys. Rev. 181, 1529 (1969).
[Ful63] R.H.Fulmer and A.L.McCarthy, Phys. Rev. 131, 2133 (1963).

[Gad04-l] A. Gade et al., Phys. Rev. Lett. 93, 042501 (2004).

[Gad04-2] A. Gade et al., Phys. Rev. C 69, 034311 (2004).

[Gad05] A. Gade et al., Phys. Rev. C 71, 051301(R) (2005).

[Gad08] A. Gade et al., Phys. Rev. C77, 044306 (2008) and reference therein.
[Gau06] L. Gaudefroy et al., Phys. Rev. Lett. 97, 092501 (2006).

[Gle04] Norman K Glendenning, Direct Nuclear Reactions, World Scientific Publishing,
2004.

[Gol99] G. Golo, P.F. Bortignon, and RA. Broglia, Nucl. Phys. A649, 335 (1999).

[Goo75] D. R. Goosman, P. D. Parker, and A. J. Howard, Nucl. Phys. A250, 309 (1975).

251

 

[Han03] P.G. Hansen and J .A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003).
[Han75] O. Hansen et al., Nucl. Phys. A243, 100 (1975).

[Hef77] E. F. Hefier, E. T. Boschitz, V. Heidt and Ch. Weddigen, Nucl. Phys. A275, 212
(1977)

[Hen08] V. Henlz, private communication.
[Hjo65] Sven A. Hjorth, J. X. Saladin and G. R. Satchler, Phys. Rev. 138, B1425 (1965).

[Hon02] M. Honma, T. Otsuka, B.A. Brown and T. Mizusaki, Phys. Rev. C 65,
061301(R) (2002).

[Hon04] M. Honma, T. Otsuka, B.A. Brown and T. Mizusaki, Phys. Rev. C 69, 034335
(2004)

[Hon05] M. Honma, T. Otsuka, B.A. Brown and T. Mizusaki, Eur. Phys. J. A 25, Suppl.
1, 499 (2005).

[Hor06] M. Horoi et al., Phys. Rev. C 73, 061305(R) (2006).

[Iga] M. Igarashi et al., Computer Program TWOFNR (Surrey University version).

[ISF] https://intra.nscl.msu.edu/illustrations/gallery.php?id=34

[IU] Electronic Instrument Services, Department of Chemistry, Indiana University, Bloomington.
[Jac70] D.F. Jackson et al., Nuclear Reactions, Methuen, London, 1970.

[Jeu77] J.-P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev. C 15, 10 (1977); 16, 80
(1977)

[Joh70] R.C. Johnson and P.J.R. Soper, Phys. Rev. C 1, 976 (1970).

[Joh68] R. R. Johnson and R. J.Griffiths, Nucl. Phys. A108, 113 ( 1968).

[Joh89] R.C. Johnson, E.J. Stephenson and J .A. Tostevin, Nucl. Phys. A505. 26 (1989).
[Jon68] R.R. Johnson et al., Nucl. Phys. A108, 113 (1968).

[Kat65] Syohei Kato et al., Nucl. Phys. 64, 241 (1965).

[Kee04] N. Keely et al., Phys. Pev. C69, 064604 (2004).

[Knu75] L. D. Knutson, J. A. Thomson, and H. 0. Meyer, Nucl. ths. A241. 36 (1975).

252

 

[Koc7l] D. C. Kocher and W. Haeberli, Nucl. Phys. A172, 652 (1971).
[K0268] R.L. Kozub, Phys. Rev. 172, 1078 (1968).

[KraOl] G. J. Kramer, H. P. Blok, and L. Lapik'as, Nucl. Phys. A679, 267 (2001), and
references therein.

[KraOl] G.J. Kramer et. al., Nucl. Phys. A. 679, 267 (2001) and references therein.
[Kra93] K.L. Kratz et al., Ap. J. 403, 216 (1993).

[Kra94] G.Kraus et al., Phys. Rev. Lett. 73, 1773 (1994).

[Lal99] G. A. Lalazissis et al., Phys. Rev. C 60, 014310 (1999).

[Lan88] J. Lang eta1., Nucl. Phys. A477, 77 (1988).

[Lee06] Jenny Lee, J .A. Tostevin, B.A. Brown, F. Delaunay, W.G. Lynch, M.J. Saelim,
and MB. Tsang, Phys. Rev. C 73, 044608 (2006).

[Lee07] Jenny Lee, M. B. Tsang and W. G. Lynch, Phys. Rev. C 75, 064320 (2007) and
references therein.

[Lee09] Jenny Lee, M.B. Tsang, W.G. Lynch, M. Horoi, S.C. Su, Phys. Rev. C 79,
054611 (2009).

[LeelO] J. Lee et al., Phys. Rev. Lett. 104, 112701 (2010).
[Lee64] L. L. Lee eta1., Phys. Rev. 136, B971 (1964).
[Lis04] A. Lisetskiy eta1., Phys. Rev. C 70, 044314 (2004).
[LISE] http://groupsnscl.msu.edu/lise/lise.html

[Liu01] Z. H. Liu et al., Phys. Rev. C 64, 034312 (2001).
[Liu04] X. D. Liu et. al., Phys. Rev. C 69, 064313 (2004).

[LinS] X.D. Liu, PhD thesis, Michigan State University, (2005).
http://grormsnscl.msu.edu/nscl library/Thesis/Liu,%20Xiaodongpdf

[Liu-priv] Z.H. Liu, private communications.

[Mar72] P. Martin, M. Buenerd, Y. Dupont and M. Chabre, Nucl. Phys. A185, 465
(1972)

253

[May60] M.G. Mayer and J. H. D. Jensen, Elementary Theory of Nuclear Shell Structure
(John Wiley and Sons, London, 1960).

[Min06] K. Minamisono et al, Phys. Rev. Lett. 96, 102501 (2006).
[Mor03] D.J. Morrisey et al., Nucl. Instrum. Methods B 204, 90 (2003).

[Mor97] D. J. Morrissey. The Coupled Cyclotron Project at the NSCL. Nuclear
Physics A, 616:45c—55c, 1997.

[Mye66] D. Myers et al., Nucl. Phys. 81,1 (1966).
[Nie66] H. Niewodniczanski et al., Phys. Rev. 146, 799 (1966).

[NNDC] Richard B. Firestone, Virginia S. Shirley, Table of Isotopes, Wiley-Interscience;
2 Vols. 8th edition (December 4, 1998), http://wwwnndcbnlgov/

 

[NSCL94] The K500+K1200, A coupled cyclotron facility at the National
Superconducting Cyclotron Laboratory. NSCL Report MSUCL-939, July 1994.

[NSR] http://www.nndc.bnl.gov/nsr

[Nun08] F. Nunes, Private communications.

[Ohl63] G.G. Ohlsen and RE. Shamu, Nucl. Phys. 45, 523 (1963).
[OWIS] http://wwwowis-staufcn.com

[Pan97] V. R. Pandharipande et al., Rev. Mod. Phys. 69, 981 (1997).
[Per62] F. Perey and B. Buck, Nucl. Phys. 32. 353 (1962).

[Per76] C.M. Perey et al., At. Data Nucl. Data Tables 17, l ( 1976).

[Pet85] J. S. Petler, M. S. Islam, R. W. Finlay, and F. S. Dietrich, Phys. Rev. C 32, 673
(1985)

[PovOl] A. Poves et al., Nucl. Phys. A694, 157 (2001).

[Pre93] M.A. Preston and R.K. Bhaduri, Structure of The Nucleus (Westview Press,
1993)

[Ric03] W. A. Richter and B. A. Brown, Phys. Rev. C 67, 034317 (2003).
[Rog09] A. Rogers, PhD thesis, Michigan State University, (2009).

254

 

 

[RoglO-l] A. Rogers, in preparations

[Rog10-2] A. Rogers, in preparations

[ROOT] http://root.cern.ch/drupal/
[8800] httpzflzroupsnscl.msu.edu/s800/Technical/Elcctronics/Electronics framcsethtm

[San06] D.P. Sanderson,
http://wwwslac.stanford.edu/econf/C060925l 1/papcrs/WEPOZ3 .PDF, (2006)

[Sat80] G.R. Satchler, Introduction to Nuclear Reactions, Macmillan, London, 1980.
[Sch05] H. Schatz et al., Phys. Rev. C 72, 065804 (2005).

[Sch09] J. Schiffer, lecture notes of the 8th Summer School on Exotic Beam Physics,
Berkerly, 2009.

[Sch64] U.Schmidt-rohr, R.Stock and P. Turek, Nucl. Phys. 53, 77 (1964).
[Sch67] J. P. Schiffer et al., Phys. Rev. 164, 1274 (1967).

[Sch98] H. Schatz et al., Phys. Rep. 294, 167 (1998).

[Set70] Kamal K. Seth, J. Picard, G.R. Satchler, Nucl. Phys. A140, 577 (1970).
[Sha00-l] D. Shapira et al., Nucl. Inst. and Meth. A. 449, 396 (2000).
[Sha00-2] D. Shapira et al., Nucl. Inst. and Meth. A. 454, 409 (2000).

[She02] B. M. Sherrill. Scientific opportunities with the nscl coupled cyclotron facility.
Prog. Theor. Physics, 146260—69, 2002.

[Sig07] A. Signoracci et al., Phys. Rev. Lett. 99, 099201 (2007).
[Sky67] D. J. Skyrme, Nucl. Instrum. Meth. 57, 61 (1967)

[Sor08] O.Sorlin and M.-G. Porquet, Prog. Part. Nucl. Phys. 61, 602 (2008) and reference
therein.

[Spectcl] http://docs.nscl.msu.edu/daq/spectcl/

[Ste06] I. Stetcu et al., Phys. Rev. C 73, 037307 (2006).

255

 

[SuzO9] D. Suzuki etal. Phys. Rev. Lett. 97,092501 (2006).
[Tan08] I. Tanihata et al., Phys. Rev. Lett. 100, 192502 (2008).

[Ter04] J. R. Terry et al., Phys. Rev. C 69, 054306 (2004).
[Ton77] J. F. Tonn et al., Phys. Rev. c 16, 1357 (1977).

[T0504] J. Tostevin et al., Nuc. Phy. A746, 166 (2004).
[Tsa05] M. B. Tsang, Jenny Lee and W. G. Lynch, Phys. Rev. Lett. 95, 222501(2005).

[Tsa09] M.B. Tsang, Jenny Lee, S.C. Su, J .Y. Dai, M. Horoi, H. Liu, W.G. Lynch, S.
Warren, Phys. Rev. Lett. 102, 062501 (2009).

[Var91] R.L.Varner et al., Phys. Rep. 201, 57 (1991).

[Ver07] D. Vemey et al., Phys. Rev. C 76, 054312 (2007).

[Ver94] J. Vemotte et al., Nucl. Phys. A571, 1 (1994).

[Wa105] M.S. Wallace, PhD thesis, Michigan State University, (2005).

[Wa109] M.S. Wallace et al., Nucl. Instrum. Methods Phys. Res. A583, 302 (2007).
[Wal81] R. K. Wallace and S. E. Woosley, Astrophys. J. Suppl. 45, 389 (1981).
[Wes92] J. Wesseling et al., Nucl. Phys. A547, 519 (1992).

[Win01] J. S. Winfield et al., Nucl. Phys. A683, 48 (2001).

[Yur04] K.L. Yurkewicz et al., Phys. Rev. C 70, 054319 (2004).

[Yur06] K.L. Yurkewicz et al,. Phys. Rev. C 74, 024304 (2006).

[Yur99] J. Yurkon et al., Nucl. Instrum. Methods Phys. Res. A 422, 291 (1999).

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