Measurements of
and fLT structure
fLT, fTT,
functions in a non-quasi-elastic
2H(', e'p)
reaction
at 210 (MeV/c) missing momentum
by
Jiang Chen
B.S., University of Science and Technology of China (1991)
Submitted to the Physics Department
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1999
© Massachusetts Institute of Technology 1999. All rights reserved.
Author ............ ...........................
......
.....
. ,....
Physics Department
August 26, 1999
.17
Certified by ...
.
-,,....
William Bertozzi
Professor
Thesis Supervisor
Accepted by .........................
Thomas
Professor, Associate Department Head for
MASSACHUSET TS
INSTITUTE
F TECHNOLOGY
LUBRA
reytak
ducation
Measurements of
non-quasi-elastic
2
fLT, fTT,
and fLT structure functions in a
H(-, e'p) reaction at 210 (MeV/c) missing
momentum
by
Jiang Chen
Submitted to the Physics Department
on August 26, 1999, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
We have studied the 2 H (e, e'p) reaction using three out-of-plane spectrometers to
detect protons in coincidence with an electron spectrometer in non-quasi-elastic kinematics at Q2 =D.15 (GeV/c) 2 . These kinematics emphasize the effects of sub-nuclear
degrees of freedom. The experiment was performed with a 40% polarized, 800 MeV
electron beam. The experiment was part of the commissioning of the out of plane
spectrometers (OOPS) at the Bates Linear Accelerator Laboratory. The OOPS modules were positioned in a ,q=(0,90,1 8 0)0 configuration for a given value of Opq. This
allowed the simultaneous measurement of the fLT, fTT, and fLT structure functions at
a central missing momentum of 210 (MeV/c). Experimental results will be presented
and discussed.
Thesis Supervisor: William Bertozzi
Title: Professor
A,
2
Acknowledgments
There are many people I would like to thank, for their contributions to this experiment
and for their contributions to my graduate education.
First, I would like to thank my advisor, Prof. William Bertozzi, for his guidance,
encouragement and support throughout my graduate career. I particularly appreciate
William Bertozzi's efforts of proof reading this thesis. I am also grateful to my thesis
committee members, Prof. Aron Bernstein and Dr. William Donnelly, for reading
this thesis and making valuable suggestions.
I am indebted to the members of the OOPS Collaboration. Without their efforts,
this work would never be finished. I would like to particularly thank Dr. Zilu Zhou,
for his tireless involvement in the experiments and data analysis. Without his help,
this thesis would have been taken significantly longer. My special thanks also go
to Ricardo Alarcon, Adam Sarty, Steve Dolfini, Jeff Shaw, Shalev Gilad, Christian
Kunz, Karen Dow, Dan Tieger and the Bates staff for their contributions to the OOPS
project.
I should thank my fellow OOPS graduate students Shiaobin Soong, Xiaodong
Jiang and Alaine Young. Without their contributions, this experiment would not be
possible.
I would like to thank my friends and colleagues who have provided help and
companionship. I would like to thank Dr. Jianping Chen, Dr. Jianguo Zhao for all
the fun time we spent together, and other friends Dr. Jian Tang, Dr. Li Cai and Dr.
Kevin Lee for making my graduate school experience both educational and enjoyable.
Finally, I would like to thank my wife, Qinqin Wang, for her patience and support
during my time in graduate school.
3
Contents
1
2
16
Introduction
1.1
Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.2
Inclusive Electron Scattering . . . . . . . . . . . . . . . . . . . . . . .
21
1.3
Coincidence Electron Scattering . . . . . . . . . . . . . . . . . . . . .
22
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
. . . . . . . . . . . . . . .
24
1.4
Theoretical Calculation . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.5
Extraction of structure functions
. . . . . . . . . . . . . . . . . . . .
34
1.6
Previous Experimental Data on 2 H structure functions
1.3.1
K inem atics
1.3.2
Plane wave impulse approximation
. . . . . . . .
36
. . . . . . . . . . . . .
. . . . . .
36
. . . . . . . . . . . . . . . . .
. . . . . .
39
. . . . . .
44
. . . . . .
45
1.6.1
Measurements of fL and
1.6.2
Measurements of
fLT
1.6.3
Measurements of
fLT . . . . . . . . . . . . . . . . ..
1.6.4
Measurements of fTT
fT
. . . . . . . . . . . . . . . . .
53
Experimental Setup
2.1
Overview of Experiment
. . . . . . . . . . . . . . . . . . . . . . . . .
53
2.2
Bates Linear Accelerator Center . . . . . . . . . . . . . . . . . . . . .
54
2.3
Electron Beam Monitors . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.4
Polarized Electron Source
. . . . . . . . . . . . . . . . . . . . . . . .
58
2.5
Moller Polarimeter
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.6
Liquid Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.7
The OOPS Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . .
66
The OOPS Focal Plane Instrumentation Overview . . . . . . .
67
2.7.1
4
2.8
2.9
2.7.2
The OOPS Horizontal Drift Chambers
2.7.3
The OOPS Scintillators
. . . .
68
. . . . . . . . . . . .
72
. . . . . . . . . . . . . . .
74
2.8.1
The OHIPS Focal Plane Instrumentation . . .
76
2.8.2
The OHIPS VDCX . . . . . . . . . . . . . . .
76
The OHIPS Spectrometer
Electronics Logic circuit
. . . . . . . . . . . . . . . .
80
2.9.1
The OHIPS Trigger Electronics Circuit . . . .
80
2.9.2
The OOPS trigger and coincidence electronics
83
2.9.3
Veto System . . . . . . . . . . . . . . . . . . .
87
2.10 Data Acquisition
. . . . . . . . . . . . . . . . . . . .
3 The Data Analysis
4
89
91
3.1
An Overview
3.2
Coordinate System . . . . . . . . . .
92
3.3
OOPS Analysis . . . . . . . . . . . .
93
3.3.1
The wire numbers . . . . . . .
93
3.3.2
Left and right decision . . . .
95
3.3.3
Meantime Correction . . . . .
96
3.3.4
Converting drift time to drift distance
97
3.3.5
Determination of Wire Plane Coordinates
98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.4
Optics Studies . . . . . . . . . . . . .
101
3.5
Particle Identification . . . . . . . . .
103
3.5.1
OOPS Particle Identification
104
3.5.2
OHIPS Particle Identification
106
3.6
One-Per-Beam-Burst Correction . . .
107
3.7
Beam Polarization Determination . .
108
Monte Carlo simulations and normalization
114
4.1
Spectrometer models . . . . . . . . . . . . .
. . . . . . . . . . .
115
4.2
OHIPS single arm acceptance . . . . . . . .
. . . . . . . . . . .
116
4.2.1
. . . . . . . . . . .
116
The solid angle of OHIPS
. . . . . .
5
4.2.2
4.3
OOPS Single Arm Acceptance . . . . . . .
4.3.1
OOPS focal plane efficiency profile
4.3.2
Extended Target
117
119
120
123
4.4
Coincidence Acceptance Simulation . . . .
127
4.5
Radiative Correction . . . . . . . . . . . .
128
4.5.1
Internal Bremsstrahlung . . . . . .
128
4.5.2
External Bremsstrahlung . . . . . .
130
4.5.3
Landau Straggling
. . . . . . . . .
131
Coincidence H(e, e'p) measurements . . . .
132
. . . . . . . .. . . . . .
133
2
4.6.1
Kinematics
4.6.2
Cross Sections of H(e, e'p)
. . . . 133
136
H( e#, e'p) Data Analysis
5.1
6
. . . . . . . . . . . . .
. . . . . . . . . .
4.6
5
OHIPS focal plane efficiency profile
Time-of-Flight Spectrum . . . . . . . . .
. . . .
136
5.1.1
Proton Path Length Correction
. . . . . . .
138
5.1.2
Electron Path Length Correction
. . . . . . .
138
. . . . . . .
139
. . . . . . . .
5.2
Missing Mass Calculation
5.3
Phase Space Matching
5.4
Software Cuts . . . . . . . . . . . . . . .
. . . . . . .
145
5.5
Asymmetry Extractions
. . . . . . . . .
. . . . . . .
146
. . . . . . . . . .
5.5.1
ALT' . . . . .
. - . - .. . . .. .
5.5.2
ALT . . . . .
. . . . . . . .
5.5.3
ATT
. . . . . . . 144
..
- 147
S- - .
- 147
.
-.
. . . . . . . . . . . . . . . .
. . . . . . . 148
5.6
Absolute Cross Section . . . . . . . . . .
. . . . . . . 148
5.7
Structure Function Extractions
. . . . .
. . . . . . . 150
5.8
Systematic Errors . . . . . . . . . . . . .
. . . . . . . 150
152
Results and Discussions
6.1
Results of 2 H(e, e'p) . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
6.2
Comparison with Theory . . . . . . . . . . . . . . . . . . . . . . . .
155
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
6.3
C onclusions
6.4
O utlook
A H(e, e'p) kinematics
B
C
2 H(e,
165
e'p) Kinematics
168
-cc, calculation of 2 H(e, e'p)
173
D The OOPS and OHIPS matrix elements
D.I OOPS Matrix Elements
........................
D.2 OHIPS Matrix Elements .............................
E Turtle Models of OOPS and OHIPS
F Alt measurement in
12
C(e, e'p) reaction
7
176
176
178
180
196
List of Figures
1-1
A Typical inclusive electron scattering spectrum of (e - nucleus).
1-2
Kinematic definitions for the A(e, e')B reaction in the one-photon exchange approximation. The
#,,q
20
. .
is the out-of plane angle and the
.,q
is the reaction angle. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-3
Feynman diagram illustrating the one virtual photon exchanged in
(e, e'X ) reaction.
1-4
23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
(a) PWIA e-p scattering, (b) PWIA e-n scattering, (c) final state interaction, (d) pion exchange or called meson current and (e) isobar
configures ........
30
.................................
1-5
Schematic Experiment Setup. . . . . . . . . . . . . . . . . . . . . . .
1-6
Separated fL and fT structure functions from Bates [39] and the NIKHEF
35
[40] experiment of van der Schaar et al.[42]. The NIKHEF data (q =
380 MeV/c ) are averaged over 5 MeV/c bins in pm. The Bates data
(
q = 400 MeV/c) are averaged over in the range of 30 to 70 MeV/c in
1-7
Pm. Only statistical errors are shown. . . . . . . . . . . . . . . . . . .
Ratio of measured fL and fT structure functions to Arenh6vel's calcu-
37
lation and the Saclay experiment of Ducret et al.[41]. Only statistical
errors are shown.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
38
1-8
Asymmetry ALT of the 2 H(e, e')p cross section measured at NIKHEF
[40], Bonn [36], Saclay [41] and Bates [43]. The data are compared
to calculations of Arenh6vel et al.[11] with (solid curves) and without
(dotted curves) relativistic corrections. Also shown are the relativistic
calculations of E. Hummel and J. A. Tjon [17] (long-dashed curve) for
the NIKHEF [40] data and calculations of Nosconi and P. Ricci [13]
with relativistic corrections for both NIKHEF [40] and Saclay [41] data. 40
1-9
The ratio between measured spectral (LT) function on 2H versus the
PWIA plus Paris NN potential calculation. Tjon and Hummel's calculations are within a full relativistic frame. This figure is taken from
the J. E. Ducret's paper [35].
1-10 ALT and
fLT
. . . . . . . . . . . . . . . . . . . . . .
41
measured at NIKHEF [40]. The shaded areas indicate the
size of the systematic errors. The solid curve represents the relativistic
calculation of Tjon et al.[17] the dashed(dotted) curves are calculations
of Mosconi et al.[13] with (without) relativistic corrections. . . . . . .
42
1-11 ALT as a function of missing momentum at Q2 = 1.2 (GeV/c) 2 measured at SLAC [37] compared with various non-relativistic and relativistic calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-12 Cross section -(e, e'p), asymmetry A', and
fLT'
43
verse Opq and pm mea-
sured at Bates [22]. The curves correspond to calculations performed
in the non-relativistic framework of Arenv6vel et al. [32] using the Paris
potential. The errors shown are statistical only. . . . . . . . . . . . .
9
44
1-13 Differential cross section and fTT response function for the 2 H(e, e')p
reaction as a function of 0" measured at NIKHEF [40]. The black dot
data is obtained by assuming the
fLT
term is zero; the white dot data
is obtained by using the Arenh6vel's predication of
fLT
term. Only
statistical errors are shown. The various curves represent calculations
by Arenh6vel et al.[32].
Dotted curve: N; dashed curve: N+i-MEC;
solid curve: N+MEC+IC, calculated within the coupled-channel (CC)
model; dot-dashed curve: N+MEC+IC, calculated in the impulse approximation framework.
. . . . . . . . . . . . . . . . . . . . . . . . .
48
1-14 ALT, ALT, and ATT curves are shown based on Arenh6vel's [62] calcu. . . . . . . . . . . . . . . . . . . . . . . .
lation for this experiment.
1-15
fLT
curves are shown based on Arenh6vel's [62] calculation for this
experim ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-16
fLT'
fTT
51
curves are shown based on Arenh6vel's [62] calculation for this
experim ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1
50
curves are shown based on Arenh6vel's [62] calculation for this
experim ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-17
49
52
A schematic view of the experimental setup showing OHIPS and three
O O PS m odules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2-2
Schematic of the Energy Compression System
. . . . . . . . .
57
2-3
The Diagram of energy levels in GaAs crystal.
. . . . . . . . .
59
2-4
The Bates polarized electron source . . . . . .
. . . . . . . . .
61
2-5
Layout of Mollor apparatus
. . . . . . . . . .
. . . . . . . . .
62
2-6
Schematic of the basel loop target . . . . . . .
. . . . . . . . .
64
2-7
OOPS Spectrometer Layout . . . . . . . . . .
. . . . . . . . .
67
2-8
OOPS Spectrometer Cross Section View . . .
. . . . . . . . .
68
2-9
OOPS detector package
. . . . . . . . . . . .
. . . . . . . . .
69
2-10 Inside of an OOPS chamber . . . . . . . . . .
. . . . . . . . .
70
. . . .
71
2-11 The cross section diagram of HDC with the gas flow.
10
2-12 Left figure is the HDC efficiency verse the operating voltage, right
figure is the HDC efficiency verse the argon percentage at 2550 V . .
72
2-13 Setup for measuring scintillator efficiency . . . . . . . . . . . . . . . .
73
2-14 OHIPS Spectrometer Layout . . . . . . . . . . . . . . . . . . . . . . .
75
2-15 OHIPS Detector Package . . . . . . . . . . . . . . . . . . . . . . . . .
77
2-16 DCOS Readout System . . . . . . . . . . . . . . . . . . . . . . . . . .
79
2-17 OHIPS Trigger Diagram . . . . . . . . . . . . . . . . . . . . . . . . .
82
2-18 The OOPS scintillator trigger logic. . . . . . . . . . . . . . . . . . . .
84
2-19 The coincidence circuit logic diagram.
. . . . . . . . . . . . . . . .
86
2-20 The one-per-beam-burst diagram.....
. . . . . . . . . . . . . . . .
88
2-21 The front end veto logic diagram.....
. . . . . . . . . . . . . . . .
90
3-1
Transport angle definitions and spectrometer coordinate systems . . .
93
3-2
X plane wire location and wire number spectra . . . . . . . . . . . . .
94
3-3
Typical OOPS O-E signal spectra . . . . . . . . . . . . . . . . . . . .
96
3-4
Typical OOPS HDC Drift time spectra . . . . . . . . . . . . . . . . .
98
3-5
Typical drift distance versus drift time . . . . . . . . . . . . . . . . .
99
3-6
(1)Typical X Plane Resolution (2)Typical Y Plane Resolution
3-7
Image of OOPS sieve-slit collimator in the focal plane. Figure courtesy
of Alaine Young, Arizona State University
. . . . 101
. . . . . . . . . . . . . . . 103
3-8
Typical Carbon elastic peak are obtained in OOPS A and OOPS C. .
3-9
Typical OOPS average scintillator pulse height in S2 versus in S3 in
XY axis, and event counts in z axis . . . . . . . . . . . . . . . . . . .
3-10 OHIPS particle identification.
104
105
a) Two-dimensional ADC sum his-
togram of the first lead-glass versus the second lead-glass.
b)Sum
(pbgsum) of all lead-glass ADC. c) Cerenkov ADC sum with pbgsum
< 900. d) Cerenkov ADC sum with pbgsum > 900. Note no Cerenkov
signal for e- events can be found in c). . . . . . . . . . . . . . . . . . 111
3-11 Typical quadruple real scan in Moller polarimeter, the x-axis is the
relative voltage on the quadruple magnets called the shunt voltage
11
. 112
3-12 Typical quadruple peak scan in Moller polarimeter, the x-axis is the
relative voltage on the quadruple magnets called the shunt voltage . . 112
3-13 Measured beam polarization against run to run
. . . . . . . . . . . . 113
117
4-1
OHIPS horizontal opening angle measured . . . . . . . . . . . . . . .
4-2
OHIPS horizontal opening angle in Monte Carlo . . . . . . . . . . . . 118
4-3
OOPS spectrometer angular acceptance. Angles at the target in the
dispersion direction are plotted against the relative momentum 6 for
OOPS and compared with the Monte Carlo simulation. . . . . . . . . 120
4-4
A typical OOPS focal plane efficiency profile comparing with Monte
Carlo simulation.The thin line is the Monte Carlo simulation, and the
thick line is the measured focal plane efficiency profile . . . . . . . . .
4-5
123
The measured OOPS extended target efficiency, compared with a TURTLE simulation (solid line) . . . . . . . . . . . . . . . . . . . . . . . .
124
4-6
The measured OHIPS extended target efficiency . . . . . . . . . . . . 125
4-7
Relative target length assuming the target diameter is 1 unit of length. 126
4-8
Feynman diagrams for radiative process . . . . . . . . . . . . . . . . . 129
5-1
The left figure is the Raw time-of-flight, the right figure is the Corrected time-of-flight. Data are shown for OOPS C. These spectra were
obtained under the condition ( -3 MeV < Emiss < 8 MeV) ( See Section
5 .2).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
. . . . . . . . .
142
5-2
The procedure to subtract background missing mass.
5-3
The true missing mass spectrum for three OOPSs. . . . . . . . . . . . 143
5-4
Plot of W -
Opq
for the forward OOPS (A) , backward OOPS (C) and
out-of-plane OOPS (B) from the experimental data. . . . . . . . . . .
5-5
145
The time dependence of the cross section of three OOPS spectrometers
in this experim ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
149
6-1
The inclusive (e, e') cross section verse o through a program from Prof.
LightBody [63]. The 2N in the figure indicates strength of the two-body
current contributions and A indicates strength of the A excitation state. 154
6-2
ALT,
ALT'
and
ATT
compared with different calculations. The solid
dots are measured values in this experiment. Only the statistical error
is show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7
6-3
fLT
compared with different calculations. Only the statistical error is
shown........
6-4
fLTI
fTT
158
compared with different calculations. Only the statistical error is
shown........
6-5
....................................
....................................
159
compared with different calculations. The solid dot is the experi-
mental result. The circle is the folded theoretical calculation including
the N+RC+MEC+IC. Only the statistical error is shown.
6-6
. . . . . . 160
Three OOPS' cross sections dependence on w are shown. The curve is
the Arenh6vel's full calculation (N+MEC+IC+RC) folded with spectrom eter's acceptance.
6-7
. . . . . . . . . . . . . . . . . . . . . . . . . . 161
Absolute Cross Section compared with different theories is shown. The
solid line is Arenh6vel full calculation (N+RC+MEC+IC), the dash
line is N+RC, the short dash line is PWBA+RC and the short dot line
is N+M EC+IC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F-i
162
A typical phase space in q - w dimension. The data with in black box
are used to extract the Alt asymmetry. . . . . . . . . . . . . . . . . . 197
13
List of Tables
2.1
Experimental Parameters. . . . . . . . . . . . . . . . . . . . . . . . .
54
2.2
MIT-Basel Loop Target Parameters . . . . . . . . . . . . . . . . . . .
65
2.3
Summary of OHIPS properties . . . . . . . . . . . . . . . . . . . . . .
76
3.1
Offsets for the OOPS HDCS. The units are cm in offsets and mr in the
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
3.2
The One-per-beam-burst Correction Factor . . . . . . . . . . . . . . .
108
3.3
Systematic uncertainties in electron beam polarization
. . . . . . . .
110
4.1
Cross sections of the OHIPS
rotations.
12
C(e,
e') measurements. Cross sections
in (pb/sr) for the ground state are listed with statistical errors only. .
119
4.2
Internal Bremsstrahlung Correction . . . . . . . . . . . . . . . . . . . 130
4.3
External Bremsstrahlung Correction
. . . . . . . . . . . . . . . . . .
131
4.4
Landau Straggling Correction . . . . . . . . . . . . . . . . . . . . . .
132
4.5
H(e, e'p) Cross Section Compared with Dipole Fit(pb/sr) . . . . . . .
135
5.1
Contribution to width of timing peak . . . . . . . . . . . . . . . . . .
137
5.2
Proton Path Length Corrections . . . . . . . . . . . . . . . . . . . . .
138
5.3
OHIPS Path Length Corrections . . . . . . . . . . . . . . . . . . . . .
139
5.4
Comparison between the folded theoretical calculations in the matched
phase space using a Monte Carlo simulation with the unfold theoretical
calculations at the central kinematics of this experiment. Arenh6vel's
full calculations (See Chapter 6) are used for the theoretical calculations. 146
5.5
Systematic errors in this experiment . . . . . . . . . . . . . . . . . . .
14
151
6.1
Summary of kinematical quantities of this experiment . . . . . . . . .
6.2
Cross Sections of three OOPSs in the matched phase space (See Secnb
tion 5.6) are in e
. The Folded theoretical calculations are
153
Me V - (sr)2
Arenh6vel's full calculation (N+RC+MEC+IC) averaged over experimental acceptance using AEEXB. The asymmetries are in percent.
The structure functions are in fm. The first error in each measured
quantity is the statistical error, and the second error is the systematic
error (See Section 5.7). The
ALT'
in A and C is 0 due to the fact that
sin 0', 1800=0 in the equation 1.45.
6.3
. . . . . . . . . . . . . . . . . . . 153
Cross sections dependence of w, where w is in MeV and cross sections
are in
(sr. Only the statistical errors are shown. The folded
MeV - (sr)2
theoretical calculations are the Arenh6vel's full calculation averaged
over the matching experimental acceptance using a Monte Carlo simulation .
6.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinematic conditions for the future deuteron measurements.
155
. . . . . 164
A.1 Hydrogen Kinematics Effects . . . . . . . . . . . . . . . . . . . . . . . 167
B.1
2
H (e, e')p Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
C.1 Deuteron Momentum Distribution . . . . . . . . . . . . . . . . . . . . 175
D.1 OOPS 6 Matrix Elements
. . . . . . . . . . . . . . . . . . . . . . . . 177
D.2 OOPS 0 Matrix Elements
. . . . . . . . . . . . . . . . . . . . . . . . 177
D.3 OOPS
#
Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . 177
D.4 OHIPS Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . 179
15
Chapter 1
Introduction
This thesis describes a measurement of the structure functions
a non-quasi elastic
2 H(e,
fLT,
fTT,
and Q'T in
e'p) reaction at 210 (MeV/c) missing momentum.
This
experiment is one part of a deuteron program using the out-of plane spectrometer
system at Bates Linear Accelerator Center. The goal of the entire program is to
understand the two-nucleon system and its electro-magnetic currents.
Interactions between nucleons should be understood in terms of the underlying
theory of strong interaction, which is quantum-chromo dynamics (QCD). In practice,
the complete calculation of nuclear forces in the medium energy region below 1 GeV
through QCD is extremely difficult, even for two-body system. Currently it yields
only some qualitative results at best. Instead the phenomenological descriptions of
the nucleon-nucleon force which have proven to be quite successful in describing most
of the data are used.
The deuteron is an important nucleus to study because it provides the simplest
system for studying the nucleon-nucleon (NN) interaction and sub-nuclear degrees of
freedom without the complications arising from multi-nucleon effects that occur in
larger nuclei. It consists of a loosely bound proton and neutron, with the binding
energy of 2.22 MeV, and no bound excited state. It has a magnetic dipole moment of
0.857
I-N
(nucleon magneton), and an electric quadrupole momentum of 0.286 fim2 .
The small non-zero quadrupole moment of the deuteron implies that the deuteron
has some mixture of S-wave and D-wave in the ground state wave function. The
16
D-wave is due to the tensor force of the NN interaction.
The percentage of the
D-wave is not a physical observable and varies from 4% to 7% according to various
models. One way to measure the D-wave property is to measure the cross section of
the reaction 2 H(e, e'p) at high initial momenta of protons [1]. Another way is through
the measurement of the tensor component t 20 from 2 H(e, e'd) reaction to extract the
combination of electric quadrupole and electric monopole form factor.
Another fundamental issue is the electromagnetic structure of the neutron. The
knowledge on the electric form factor of the neutron G' is still imprecise and controversial. This is partly due to the absence of free neutron targets and to the fact
that G' is very small. However, a polarized 3 He target is a good approximation of a
polarized neutron target. It has been suggested that a 3 He
(e, e'n)p experiment
with
longitudinally polarized electrons and a vector polarized deuteron target can provide
data on Gn [2, 3]. Therefore, understanding the deuteron well is important to the
interpretation of these data.
Another technique is the measurement of the final state proton polarization in
either (e, e'p) or (e e'p) reactions through a focal plane polarimeter (FPP) [8]. The
focal plane polarimeter can measure the recoil proton's polarization components in
the electron scattering plane, either longitudinal, transverse to the recoiled proton
momentum which are called Pt and P, or a recoil polarization compoment normal
to the (e, e') plane which is called Pn. Note that the ratio P/P gives the ration of
the form factors GP/GP independent of the beam helicity if the polarized beam used
and the polarimeter analyzing power. Pn is induced by the final state interactions.
Comparing the three measured polarization observables in both p(, e'p5) and d(, e'p-)
scattering allowed a sensitive, model-independent test of the impulse approximation
for deuteron [4, 5, 6, 7]. The understanding of G' is an important step to measure
the GE physical quantity.
The fact that careful study of the deuteron is fundamental to nuclear physics is
also due to the fact that reliable calculations can be performed in both non-relativistic
and relativistic models for a given NN potential [9, 10, 11, 12, 13, 17]. This feature
makes deuterium the first testing ground for any NN potential model. From the
17
electrodisintegration of the deuteron, information on not only the ground state wave
function [19, 20] but also the electromagnetic currents which connect to the continuum
np system [21] can be obtained. Those currents are directly related to NN potentials based on one-boson-exchange through meson-exchange currents, and they are
also connected to the internal nucleon structure through isobar configurations. A detailed understanding of both these currents and the effects of final-state interactions
is crucial for the extraction of precise information on the neutron electromagnetic
current.
More stringent constraints on the NN potential and reaction models can be provided through measurements of the several 2 H(e, e'p) structure functions [11, 13, 17].
By separating the electron scattering cross section into longitudinal(fL), transverse
(fT),
longitudinal-transverse
(fLT
and
fLT)
and transverse-transverse (fTT) inter-
ference functions, reliable information on the deuteron wave function and reaction
mechanism can be obtained. For example, it is known that the longitudinal-transverse
structure function fLT is particularly sensitive to the relativistic effects [42] in some
region, while
fiT
arises purely through the final-state interactions [22]. The trans-
verse responses (fT and fTT) appear mostly sensitive to meson-exchange currents and
isobar configurations [23], and these sensitivities are much greater at the non-quasielastic kinematics than at the quasi-elastic kinematics.
We performed precise measurements of the deuteron structure functions in both
QE and non-QE regions. The data in non-QE region is the subject of this thesis
while the QE region are the subjects of other thesis in the same program [24, 25].
The measurement was performed at the Bates Linear Accelerator Center in January
and February of 1997. The electron beam energy was 800 MeV, and the target was
a liquid deuterium target.
The scattered electrons and the ejected protons were
detected in coincidence at the energy transfer w = 155 MeV, the momentum transfer
q = 414.0 (MeV/c), and the initial proton momentum inside the deuteron was 210
(MeV/c).
In this experiment we measured three coincidence cross sections simultaneously
at the azimuthal angles (Pp. = 00, 900, 1800. The electron kinematics was fixed in the
18
dip region. The scattered electrons were detected in coincidence with the knockedout protons. The "dip region" is between quasi-elastic scattering kinematics and the
kinematics leading to the A excition. This work is the first measurement of three
structure functions in the dip region and represents an important milestone for the
development of the out-of plane spectrometer (OOPS) system and for the out-of plane
physics program at the Bates Linear Accelerator Center.
1.1
Electron Scattering
Electron scattering is a powerful tool for probing the electromagnetic properties of
nuclei and nucleons. It is based on the following facts:
" Electron scattering process is precisely calculable in the framework of quantum
electro-dynamics (QED) [26]. Therefore, the results from electron scattering
experiments can be accurately linked to the properties of the electromagnetic
currents inside nuclei and the structure of these nuclei.
1
), which
137.036
allows the interaction to be described by using only first order perturbation
" The interaction is relatively weak (the coupling constant a =
theory for light nuclei (one-photon exchange approximation). It also means the
virtual photon can probe the entire nuclei volume, in comparison with hadronic
probes which probe mainly the surface of the nuclei in the medium to low energy
domain.
" The momentum transferred to the target (-4)
is independent of the energy
which is transfered (w). The only constraint is that the virtual photon which
is exchanged must be space-like (W2 ;
w2).
This allows the momentum de-
pendence of specific transition matrices to be mapped out. For a given w, this
allows us to determine the spatial distribution of those matrices. This may be
compared with real photon experiments in which
There are some disadvantages:
19
2
is equal to
W 2.
ELASTIC
Q[ASIE1 ASTIC
NUCLEUS
DELTA
Giant Resonace
2A2N
D.LT
-+
300MNeV
2N1
Figure 1-1: A Typical inclusive electron scattering spectrum of (e - nucleus).
" Because of the weaker interaction comparing with the strong interactions, the
cross section tends to be smaller and the experiment tends to be longer.
" The analysis and interpretation of electron scattering is complicated by the
process of radiation of the electron in the presence of the target nuclei.
In an electron scattering experiment, we aim a beam of electrons at a target and
detect electrons which are scattered from the target in a particular direction and with
a particular energy. Measurements in which only one particle is detected are called
single-arm measurements or inclusive experiments, because one integrates all final
states of the nucleon system for the given (q, w). Those in which one or more particles
are detected in coincidence with the electrons are called coincidence measurements
and they are generally exclusive experiments since only a selected number of possible
final states are measured. The experiment described in this thesis is a coincidence
measurement since both the scattered electron and proton from the same nucleus are
detected.
20
1.2
Inclusive Electron Scattering
A generic spectrum of the cross section as a function of a for a fixed
Q2
2 _ 72
for an inclusive electron scattering is given in Fig.1-1. As showed in the picture, with
increasing energy transfer, the first feature is the elastic peak at w
where the nucleus remains in the ground state.
=
-Q
2
/2114 ,
With increasing energy transfer,
one observes a set of broader bumps that are caused by the excitation of collective
modes, called "the giant resonances.". Above that, one observes a broad quasi-elastic
peak which is located at w = -Q 2/2AI,
where the virtual photon is absorbed by a
single proton, which is emitted from the nucleus. Above the quasi-elastic peak, one
observes the peaks which correspond to the excitation of a nucleon to the A and the
N* excitations. Between the quasi-elastic peak and the delta peak is a region called
the "dip region".
This thesis is about the measurement of three structure functions
at one point in this region. Above these peaks, one observes a large region called
deep inelastic scattering. In this region, the virtual photon interacts with the quarks
inside of the nucleus.
For light nuclei, electron scattering is well approximated by one-phone exchange.
In the one-photon approximation the cross section for the single-arm electron scattering can be decomposed into portions correspondent to longitudinal and transverse
virtual photon polarizations as follows [29];
d3a
dk'dQk'
where
9
47raM Q 4 R1
4RL (IQ
M AIkdi
M
T''2
-Q2
)+(
V
t
2
-))RTlq' 7W1(1.-1)
+ tan2 2--)TI7~)
(1
e is the electron scattering angle and am is the Mott cross section which is:
a 2 cos 2 (Oe/2)
=M
4E2 sin 4 (Oe/2)
(1.2)
where c is the energy of the incident electron, and RL, RT are respectively the longitudinal and transverse structure functions. RL measures the nuclear response to
the Coulomb excitations, while RT measures the response to excitations involving the
21
transverse electromagnetic current density.
Coincidence Electron Scattering
1.3
In a coincidence or exclusive scattering experiment, one or more scattered particles
are detected in coincidence with the scattered electron. This exclusive measurement
allows the selection of a particular final state. This thesis focuses on one area in
the coincidence electron scattering: the
(e,e'p) reaction.
The advantages of exclusive
electron measurements are:
" A particular reaction channel can be selected, like (e, e'p), (e, e'ir+), or (e, e'7r-)
while in the single arm experiment one measures the contributions from all
processes.
" By measuring the missing energy, one can map out the nucleon spectral function S(E, P) which is the proton energy and momentum distribution inside the
nuclei.
Thus the (e, e'p) reaction offers an attractive method for studying nuclear structure
in detail.
1.3.1
Kinematics
We use the following notation [27] for describing the kinematics of the (e, e'p) reaction in Fig.1-2. The incident electron has 4-momentum k, = (k0 , k) in the laboratory
frame, the scattered electron k' = (k/, k').1 We only consider ultra-relativistic electrons, so we neglect the electron mass. The difference between initial and final electron
4-momenta is the 4-momentum transferred to the target nucleus.
qj, = (w, q') = kt, - k'A ,
(1.3)
'We work with the metric g9 where goo = -1 and gij = 61. 6.j is the Kronecker delta symbol,
equals 1 for i = j and 0 otherwise.
22
Y
X
B
Figure 1-2: Kinematic definitions for the A(e, e')B reaction in the one-photon exchange approximation. The #, q is the out-of plane angle and the 0,q is the reaction
angle.
where w = E - E' for the energy transfer and q = k - k' for 3-momentum transfer.
A=
(EA, PA) is the 4-momentum of the target nucleus. Since the target is at rest in
the laboratory frame, PA = 0 and EA
=
MA
is the mass of the target. The scattered
proton has 4-momentum pA =(E,,pJ) and the recoiling nucleus (of mass MB) has
4-momentum PIe, = (Ere,, Prec). The conservation of 4-momentum yields
W+ MA= Ep+ Erec = M + T + MB+ Trec ,
(1.4)
where T, and Tree are the kinetic energies of the ejected proton and recoiling nucleus
respectively, and
q= po +
23
Prec
.(1.5)
We define the missing momentum as follows:
Pm =
q
(1.6)
-Prec,
and the missing energy as follows:
Em=w
TL
TP
-
rec=
*_i+M
(1.7)
-Ma.
In the Plane Wave Impulses Approximation (PWIA), it is assumed that:
* The entire momentum transfer is absorbed by a single proton in the target
nucleus.
" The knocked-out proton exits the nucleus without further interaction with the
recoiling nucleus.
With these assumptions, the missing momentum is simply the initial momentum of
the proton in the nucleus:
PA = 0 = A + Prec,
(1.8)
p = -Prec = Pm
In Fig.1-2, the incident and scattered electron 3-momentum vectors, k and kI', define
the scattering plane. The 3-momentum transfer q and the final proton 3-momentum
)define the reaction plane. The electron scatters at an angle 9e with respect to the
incident electron beam. The angle between the outgoing proton and q is
9
pq,
while
the angle between the electron scattering plane and reaction plane is <pq.
1.3.2
Plane wave impulse approximation
For light or medium nuclei at high q2 , it is a good approximation to simplify the
scattering process by assuming the one-photon exchange reaction. A diagram of the
process is shown in Fig.1-3. The general formalism of the (e, e'p) reaction is reviewed
in detail by A. S. Raskin and T. W. Donnelly [29]. The exclusive scattering cross
24
e'
X
B
B
A
Figure 1-3: Feynman diagram illustrating the one virtual photon exchanged in (e, e'X)
reaction.
section is formed by the contraction of the lepton tensor L,, and the hadron tensor
-
MI 2
=
rW
(1.9)
.
The lepton tensor is known from QED and it is given by:
m ) -ih
2 [kyk' + k' k, - gt(k -
k
.
(1.10)
Similarly, the hadron tensor W,, is constructed from the nuclear electromagnetic
current J,, by
W
yjJ,*J,
.
(1.11)
spin
The nuclear current J can be decomposed into the parallel and perpendicular to the
momentum transfer, :
J = th0n9 +
25
trans
(1.12)
From the current conservation, we have:
0 .
i -r
T e+
(1.13)
Then taking the Fourier transformation, we have:
u) - p(q) - q - (qj = 0 .
Since q* -=
(1.14)
|Jiong,
"
iong
p=
(1.15)
W
If one choose the coordinate with the z-axis along the direction of q,
eo
=
=
where
, 9,
(1.16)
,
-)
1
(1.17)
denote unit vectors in the three Cartesian coordinate directions,
=
ong
tran
Joso ,
(1.18)
J+s+ + Jd,
(1.19)
and
J
=
Jtran
=
1
-(J
(1.20)
i iJY).
After lengthy calculation [29], we get the cross section for (e', e'p) as following:
d6 .
dQedQpdEpdw
SUM[VL
Op,Ep) + VTWT(q W, 9p,Ep) +
WL (,
w, O, Ep) +
VLTWLT(Q w, ,,, Ep) + VTTWTT(J,
VLT'WLT'(q, w, Op,
26
Er)]
,
(1.21)
where the v's are kinematics factors given by:
VT
=
ULT
-
VTT
=
VLT'
=
IQ 2 2
(
+ tan
1 Q2
-(-s-)
2
9 t )+) an (-)2(
(+
1Q22
S 2
(
(1.23)
2
)
(
1 Q2
V2
(
)
-
(1.25)
02
(1.26)
The response function W can be expressed as the combination of the nuclear current
J0, J±
WL
=
WT
=IJ2
W
2
WTT =
WLT
P12 = (P)2 1 j01 2
=
,(1.27)
j+j
1,2
(1.28)
)(Ja)
,
(1.29)
2Re[p*(J+ - J)]
(1.30)
-2-Re[Jo*(J+ - J_)]
WLT'
(1.31)
- 2Re[p*(J++A)]
-2 q Re[Jo*(J+ + J-)].
WL and WT are the longitudinal and transverse parts of the nuclear electromagnetic
current. WLT measures the real part of the interference between the longitudinal and
transverse nuclear electromagnetic current. WTT measures the interference response
between two transverse nuclear electromagnetic current.
WLT'
measures the real
part of imaginary part interference between the longitudinal and transverse nuclear
electromagnetic current.
One widely cited procedure in determining the off-shell cross section is that by
deForest [9], who imposes the current conservation requirement by a suitable choice
of gauge field. He expressed the (e, e'p) as follow:
27
= P 2P
r
ped
d ed
d~ed~pdPedp
(1.32)
/eN
S(Pm, Em) ,
and the off-shell cross section reN is calculated:
JeN
where A
Aott{A 2RL
+
(A/2 + tan 2 (O/2))RT
+
1/2A
A + tan2 (/
, 0 is the scattering angle, and
-
(1-33)
2 )RLT
cos
+1/2ARTTcos 20}
# is the out-of plane angle.
In deForest
paper, there were two different expressions for the structure functions which were
based on two different off-shell extrapolations of the nucleon current. These are named
o-cci and Occ2.
In most cases, both give similar results for cross section calculations.
Since -cc1 is simpler in the form of the vertex coupling,
Uccl
is more widely used to be
compared with experiment results and will be compared with this experiment results
later.
In the
Ucci
model:
2 2E'
+ E]
-
+
2m
2
2
2
[GP] 2
m
RL
=
[F? + TF ][
RT
=
[F
RLT
=
2E' +E |j7'|
sin6pq
-2[F? +TF] 2
RTT
=
[F
4
,
(1.34)
2 ,
(1.35)
-12
F ][-] sin2
P
'2
F]
2T[G
Opq+
i2,
,
,
(1.36)
(1.37)
P
where the response function R's are expressed in term of the nuclear electromagnetic
current via
RL
RT
RLT
=
|p(q)12
=
(q/w) 2 1J(0, q)1 2
-
J(+1,j2 + Ij(-1q12 I
-2Re(J(+,
q*J(-l, q)
28
RTT
=
-2(q/w)Re[J(0,q)*(J(+1,q) - J(-1,q))
=
2Re(J(+1,q)*J(-1,q)) .
(1.38)
-2
In which j =_E' - E, q'
q(,), and T _=4m . and F and F2 are the Dirac [34]
P
and Pauli form factors [34] of the proton. In terms of the Sach form factors [34],
GE
and GAL, they are related by
F 2)
F1 (qA) =
GE (q 2)+
TGm(q 2
I+ T _
Gm(q 2) -
F2 (q ) =
A
where
Q2
1+
(1.39)
GEE(q 2
_
T
A.
(1.40)
__* 0,
Gp -- + 1, Gp
where M, = 2.79 and An = -1.91
-+
I, G"n--+
0, G
-+ p(1.41)
nuclear magnetons for the proton and neutron
respectively.
The energy transfer is now associated with the initial and final energies of the
struck proton rather than the energy of the incident and scattered electron. This has
the effect of incorporating the initial momentum of the proton into the nucleon vertex
function.
In Appendix C, the detailed formula of
cc
and Van Ordan's p(p) of deuteron
spectral function are listed. This experimental results are also compared with o-cc,
calculation in Chapter 6.
1.4
Theoretical Calculation
The most important lowest-order diagrams contribution to the 2 H(e, e'p) cross section
are shown in Fig.1-4. The diagram (a) is the PWIA e-p scattering. The diagram (b)
is the PWIA e-n scattering. The diagram (c) represents the proton rescattering from
the neutron before it exits the nuclei. The diagram (d) represents when two nucleons
29
e
e
d
p
n
P
n
P
n
(a)
(b)
(c)
A, N*
P
P
7T
Tr
n
n
(e)
(d)
Figure 1-4: (a) PWIA e-p scattering, (b) PWIA e-n scattering, (c) final state interaction, (d) pion exchange or called meson current and (e) isobar configures
exchange a virtual meson, the proton also absorbs the virtual photon. An isovector
7r
two-body exchange term is included for the long-range meson-exchange current,
and p and w exchange current terms are included for the short-range meson-exchange
current. The diagram (e) represents the so called isobar configures either proton or
neutron could be excited as virtual A and N* state after it absorbs the virtual photon.
There are two different approaches in the calculations for the 2 H(e, e'p) structure
functions: One approach is within a full relativistic framework. The other approach is
within a non-relativistic framework with relativistic corrections. For example, Tjon's
and his colleagues' [17] calculations are based on the full relativistic framework. They
used a covariant approach which is based on an approximation of the Bethe-Salpeter
equation. However, the Bethe-Salpeter equation is difficult to solve and their calculations on deuteron disintegration are only becoming available recently and in some
limited circumstances. The second approach is to modify the Schr6dinger equation
to include the relativistic effects. For example, the works by Arenhbvel et al. [11],
Mosconi et al. [14] belong to the second approach. The most systematic theoretical
30
calculations on 2 H(e, e'p) have been done by Arenh6vel et al. [30, 11, 12, 15, 32, 21, 16]
Their calculations based on the Schrddinger equation in conjunction with an NN potential such as Bonn [31] or Paris [18] potential. They can also include the effects
due to meson-exchange currents, isobar configurations and final state interaction in
the calculation. The relativistic effect was included by expansion the electromagnetic
p
. The relativistic corrections are
charge and current operators in the order of
discussed in more detail in references [32, 21]. Mosconi's calculations are very similar to Arenh6vel's calculation. The main difference between Mosconi's calculation
and Arenh6vel's calculation is the order of partial wave functions in the treatment
of FSI and the order of the
expansion in the current operator. Not surprisely,
their calculation results generally do not differ much at low energy and momentum
transfer.
Furthermore, there are some exciting developments incorporating the relativistic
correction into the non-relativistic reduction of the electromagnetic current operator
in the calculations of electro-nuclear reactions by S. Jeschonnek and T. W. Donnelly
[33]. They improved the current operator to incorporate relativistic effects without
any approximation in the transferred momentum and energy. Their results show that
the relativistic contribution was large, specially in the longitudinal-transverse (LT)
structure functions. They show that the LT response function has two sources: one
is the product of the first-order spin orbit term and the zero-order magnetic current,
and the other is the product of the zero-order charge operator and the first-order
convection current [33]. The former sources contribute the majority of the LT term.
The spin-orbit operator can be included in the relativistic terms. This improvement
in the current operator will have an impact on the fLT calculations.
Since at the moment, only Arenh6vel et al. has the complete calculation in the
kinematic region of this experiment, the experimental results will primarily compared
with Arenh6vel's calculation. In the following section I describe the formalism which
Arenh6vel [11] used in the calculation of the structure functions.
In the one-photon exchange approximation the general expression for the differ-
31
ential cross section with polarized beam and target is given by
da =
2a
_
q~kok'
4(P -_,
- q)(ktk' + kk - kk'g,,)T"'
(1.42)
where
= E < Pf5|J(0)|P, >*< P5|14 (0)|P, >
T
-"
(1.43)
Here, k and k' denote the four-momenta of incoming and outgoing electrons q = k - k
the four-momentum transfer, P and Pf the four-momenta of initial deuteron and final
n-p systems, respectively, and J. the current operator of the two-nucleon system.
Usually, the current matrix elements are evaluated in the final n-p center of mass
system or anti-lab system. Then, one obtains for the coincidence cross section in the
lab system
d5o
dk'dQlab
d dG b
e
Q
(9'"M
do
P
= dk'dQad
kd(1.44)
e
" q0gla'
P
P
where
d or
luPLL
dk'dQ bdQcm
a
kol
6r kq
C
+ pLTfLT Cos <n
+
PTfT
+
PTTfTT cos 2(b
+ hpLTI fLT' sin 0
(1.45)
-
is given in terms of nuclear structure functions fAg and kinematic functions describing
the polarization matrix of the exchanged virtual photon
PL
=
PLT
=
(1.46)
,
2
(±
qL
1q 2(1 +
pr=
PTT
=
PLTI
=
2
v /--,
32
(1.47)
)
(1.48)
(1.49)
v4,,
1 Oq2
)I
.
(1.50)
The four nuclear structure functions fAg (E,", qcm, Am) are related to the current
matrix elements
fL
=
Xoo
fT
=
2X
fLT
=
4ReXoi
fTT
=
2X 1-1 ,
fLT'
=
4mXOI
,
(1.31)
11 ,
(1.52)
(1.53)
,
(1.54)
,
(1.55)
MMdt
(1.56)
where
Z
Xu,, =
sm,um
Smmd
is given in terms of T-matrix elements:
tsm,um
=
kCM
-27r(
ei"Pcm <
smS~J 1 (q)|md>
The initial state is characterized by the deuteron spin projection
md
(1.57)
and the final
state by the relative n-p momentum km, total spin s and projection m with respect
to kC". The operator J, denotes the charge density (p = 0) and the transverse current
(p =+1).
The Jacobian transforming the proton angular distribution from the cm to the
lab frame is given by
lab
Q Cm
J Q=ab
Mlab
labMlab
2p 4mElab cos
) Elab
(
n2)-
(1.58)
where Miab = Enm + 2M. The structure functions are related to the W's:
fT
=
127r2 caWL/J,
(1.59)
fT
=
127r 2 aWT/J ,
(1.60)
-
127 2aWLTIJ,
(1.61)
=
127 2 aWTTJ,
(1.62)
fLT
fTT
=
33
127rFaWLT'/J
fLT'
(1.63)
,
where a is the fine structure constant and J is the Jacobianl.58.
1.5
Extraction of structure functions
From equation 1.45, we see that if the electron kinematics is fixed, fLT and
can be extracted by measuring the cross sections at different
#pq;
and the
fLT'
fTT
can be
extracted by measuring the cross sections out-of the scattering plane (hence sin(Opq) #
0) and using a polarized electron beam. There are several configurations to measure
the cross sections: 1) By measuring the cross sections at the angles which are integers
(1,3,5,7) multiplying 7r/4 in qpq; or 2) By measuring the cross section at the angles
which are integer(0,1,2,3) multiplying 7r/2 in #pq.
In this experiment, the second
configuration was chosen. Ideally four proton spectrometers should be used in order to
minimize the systematic errors. At the time of this experiment, the support structure
was not ready for the fourth OOPS's and only three OOPS's were used. They were
aligned at #pq
= 00, 900,
1800 angle.
The asymmetry and response functions can be expressed as follows:
ALT'
=
7-~/2
+
ALT
0
-
O+
ATT
=
o
T0
PLT'!LT'
PLfL+PTfT-IPrTfT
_
7r/
+/2
PLTfLT
PLfL+PTfT+PTTfTT
07
Or
+ a,
2
PTTfTT
-20/2
+ (77r+2r/2
(1.64)
(
(1.66)
PLfL+PTfT
and
(1.67)
7r/2
7r/2 2
fLT'
CPLT'
fLT
=
UP
fTT
=
uo + or4
2CPLT
Cptt
34
(1.68)
,
2a/
2
(1.69)
e
detector
Beam
dump
Exit
line
Scattering
7T
plane2
0
Beam
f
4
line
3 -r
2
Figure 1-5: Schematic Experiment Setup.
a kol
.ko4
61r2 kq;
The above methods to extract the response functions and their asymmetries are
where C =
applied to this experiment data. The results and comparision with theory calculations
are presented in Chapter 6.
35
1.6
Previous Experimental Data on 2 H structure
functions
Data on separated response functions are not abundant. In part this is due to the
stringent requirement to control the systematic uncertainties made the measurements very difficult and time consuming. Still, each of those 5 response functions
in 2 H(C, e'p)n were measured at least at one kinematics.
In this section, the results from NIKHEF [42], Saclay [35], Bonn [36], SLAC [37]
and Bates [22] data will be discussed. A complete review of the status of recent
measurements and their comparison to theories can be found in the paper [38].
1.6.1
Measurements of fL and fT
There are three experiments measuring the fL and fT using the Rosenbluth separation method. First we compare the results from Bates [39] with NIKHEF [40] measurements. The measurements were done around
Q2 =
0.2 (GeV/c)2 and at missing
momentum range of 20 to 100 MeV/c. While both data sets agree in the transverse
part, they disagree significantly in the longitudinal part, as shown in Fig.1-6. Comparison between Bates [39] and Saclay [41] measurements and the calculations provided
by Arenh6vel are shown in Fig.1-7. The calculation provided by Arenh6vel includes
FSI + MEC + IC with the Paris [18] NN potential in the non relativistic frame.
The Bates data agree with the trend of the Saclay data. The calculation reproduces
the measured transverse response except at -100 MeV/c, but the calculation overestimates the longitudinal response for the points at pm of -20, 50 and 100 MeV/c.
Hence, the experimental data are not completely conclusive. However, theory cannot
reproduce the experimental longitudinal and transverse response simultaneously.
36
8 NIKHEF
0 Bates
6J
S4I
2
01
20
30
40
50
60
70
88 NIKHEF
6
IBates
4
201
20
30
40
50
Pm (MeV/c)
60
70
Figure 1-6: Separated fL and fT structure functions from Bates [39] and the NIKHEF
[40] experiment of van der Schaar et al.[42]. The NIKHEF data ( q = 380 MeV/c ) are
averaged over 5 MeV/c bins in p,. The Bates data ( q = 400 MeV/c) are averaged
over in the range of 30 to 70 MeV/c in pm. Only statistical errors are shown.
37
110 -
. Saclay
Bates
100
$90
80
780
70
. . .
-100
. .
-50
.
. .
0
50
100
110
ZA00 - - - - -----------------0
-------
#
~90-
Saclay
A Bates
0
80
70
. . ..
-100
-50
50
0
100
Pm (MeV/c)
Figure 1-7: Ratio of measured fL and fT structure functions to Arenhdvel's calculation and the Saclay experiment of Ducret et al.[41]. Only statistical errors are
shown.
38
1.6.2
Measurements of
fLT
There are several measurements of the fLT interference response function or asymmetry ALT from NIKHEF [40], Bonn [36], Saclay [35] as well as Bates. Results are
shown in Fig.1-8. Those measurements were performed in the quasi-elastic region
with
Q2 around
0.2 (GeV/c) 2 . The asymmetries are compared to the calculations of
Arenh6vel et al. with or without the relativistic corrections. In addition, the NIKHEF
data are compared to the calculations from Tjon et al. and both NIKHEF data and
Saclay data are compared to the calculations of Mosconi et al. with relativistic corrections. It is noticed that the calculations which include relativistic effects reproduce
the asymmetry ALT data much better than those which do not. In Saclay's experiments [35], it is noteworthy that the calculations which do not include the relativistic
effects seem to reproduce the response function fLT better as shown in Fig.1-9.
Recently the response function
!LT
and the cross section asymmetry ALT for the
reaction 2 H(e, e'p) have been measured at NIKHEF with
Q2
= 0.2(GeV/c) 2 which
is slightly above the quasi-elastic region. The data are presented in Fig.1-10. The
data here are compared with the calculation by the Tjon et al. [17] and Mosconi et
al. [14]. Data reasonably agree with the relativistic calculation in both asymmetry
and response function. There was one measurement of the asymmetry in the high
Q2 = 1.2 (GeV/c) 2 in SLAC [37] in the quasi-elastics region as shown in Fig.1-11.
The data is reproduced by the calculation of Tjon [17] and also by calculations which
include relativistic effects by Arenh6vel et al.[11]. It is not surprising that up to pm
= 100 MeV/c, these data can described well by PWIA calculation by using acc and
the Paris spectral function for the deuteron [18], above pm = 100 MeV/c, the data
does not agree with any calculations.
It seems that the calculation with the relativistic corrections overestimate the
fLT but correctly predict the asymmetry in one experiment in Fig.1-9, and while the
calculations with the relativistic corrections reproduce other experiments in Fig.1-8
and Fig.1-10. This is a troublesome situation and further studies, both in experiment
and theory, are needed to resolve those discrepancies.
39
0
-0.2
-0.4
2
0 2-0.20 (GeV/c)
.
0
.
Socloy - 1992
Botes - 1996
-0.2
-0.4
02-0.15 (GeV/c) 2
Bonn
-
1993
0
-0.2
-0.4
02=0.18 (GeV/c)2
0
20
40 60
80 100 120 140 160 180
p, [MeV/c]
Figure 1-8: Asymmetry ALT of the 2 H(e, e')p cross section measured at NIKHEF
[40], Bonn [36], Saclay [41] and Bates [43]. The data are compared to calculations
of Arenhdvel et al.[11] with (solid curves) and without (dotted curves) relativistic
corrections. Also shown are the relativistic calculations of E. Hummel and J. A. Tjon
[17] (long-dashed curve) for the NIKHEF [40] data and calculations of Mosconi and
P. Ricci [13] with relativistic corrections for both NIKHEF [40] and Saclay [41] data.
40
n
0
d- 1.2
Qa
en
1
-.....
----.....
--
0.8
.---------
Arenhovel without rc
Arenhovel with rc.
0.6
Tjion and Hummel
Mosconi with rc
.........
..................
0.4
0.2
n
.
0
.
I
25
.
.
I
I
50
I
I
i
I
I
75
I
I
I
I
I
100
I
,
,
,
,
I
125
,
,
,
,
,
150
.
I ,
175
.
I
I
200
Primss
Figure 1-9: The ratio between measured spectral (LT) function on 2H versus the
PWIA plus Paris NN potential calculation. Tjon and Hummel's calculations are
within a full relativistic frame. This figure is taken from the J. E. Ducret's paper
[35].
41
0.0
I
I
I
I
-0.1
H
-J
-0.2
-0.3
--
-0.4 -IT r77
-0.5
'
I
- +1*
.
..
..
..
..
-2.10-4
>6
-4
(D -4.10
-- 6.10 -4
-4
140
L~
-
~
160 180
200
220
- - - 140 160 180 200
220
pm [MeV/c]
pm [MeV/c]
Figure 1-10: ALT and fLT measured at NIKHEF [40]. The shaded areas indicate the
size of the systematic errors. The solid curve represents the relativistic calculation of
Tjon et al.[17] the dashed(dotted) curves are calculations of Mosconi et al.[13] with
(without) relativistic corrections.
42
0.2---
0 .0 -=
------- --
--------
-- ----
-0.2 ---Arenhoevel NR
-0.4-
-
-
Arenhoevel REL
-0.6 - -------- Tjon NR
-----0.8-
Tjon REL
Gc1c
0
50
100
150
20
pm [MeV/c]
Figure 1-11: ALT as a function of missing momentum at Q2 = 1.2 (GeV/c) 2 measured
at SLAC [37] compared with various non-relativistic and relativistic calculations.
43
1.6.3
Measurements of
fLT'
fLT comes from the real part of image part of the longitudinal and transverse current
interference [11]. In order to measure the fLT', one has to use the out-of plane technique. There is only one measurement of the spin dependent longitudinal-transverse
interference response function fLT'. It was done at Bates Laboratory [22].
sults are shown in Fig.1-12.
none zero
fLT'
The re-
Due to a large statistical error, only the indication of
can be obtained. The fLT' is correlated with the final state interac-
tions. Without the final state interaction (FSI), fLT' should be zero. Since there is
only one measurement of the fLT' with a marginal precision, clearly further precise
measurements are necessary.
Pm (MeV/c)
50
100
150
' '
I
Z
I'
200
I|
'' | ' ' I--
Full calculation
UPWBA
102
S101
S100
10-1
0.0
-0.1
-0.2
--------------
---------
-0.3
-0.4
-0.5
0.00
---
- - - - - - - -- - - - - - - -
-0.02
- --0.04
-----0.06
0
10
30
20
0,,
40
(deg)
Figure 1-12: Cross section o-(e, e'p), asymmetry A', and fLT' verse Opq and pm measured at Bates [22]. The curves correspond to calculations performed in the nonrelativistic framework of Arenvdvel et al. [32] using the Paris potential. The errors
shown are statistical only.
44
1.6.4
Measurements of
fTT
The fTT response function is a small but interesting response function. It strongly
links with the isobar contributions and with the meson-exchange current contribution
to the reaction mechanism [32].
NIKHEF [23].
Recently there is one measurement of fTT from
The measurement was performed in the A resonance region.
These
NIKHEF data are presented in Fig.1-13, together with calculation by Arenh6vel et
al. [32].
In order to measure the ftt term, one has to do the measurement out of the reaction
plane. The NIKHEF experiment used a large acceptance detector (HADRON4) to
accept the out-of plane knocked-out proton. HADRON4 had the out-of plane angle
acceptance ±22.30.
The results are shown in Fig.1-13. It shows that isobar configuration (IC) plays an
important role in the A resonance region. The diagram (e) in Fig.1-4 is the Feynman
diagram for this process. In Fig.1-13. The full curve represents the complete calculation explicitly including the A degrees-of-freedom. The details of the dynamical
treatment of the A-isobar can be found in the paper [30]. There are two approaches.
One is calculated in the impulse approximation framework and the other is within
the coupled-channel model. In the coupled-channel model, internal nucleon degreesof-freedom are explicitly admitted in the nuclear Hamiltonian. The resulting wave
function will contain the isobar configurations which correspond to one or more internally excited nucleons. The amplitude for the electromagnetic excitation of the
A-isobar and subsequent one-pion-exchange is then obtained by evaluating the NA
transition current between an NN component of the initial state and NA component
of the final state. In this way, the intermediate propagation of the A isobar is automatically included in the isobar wave function. Within this approach, in principle, a
complete solution of the coupled-channel equations for the various nucleon and isobar
components of the two-body wave function is required. However, since a complete
solution is rather involved, a perturbation treatment called impulse approximation is
frequently used. In this impulse approximation one retains only the couplings of the
45
various channels to the NN one channel, which gives rise to a "modified" diagonal NN
potential, implicitly containing the dispersive contribution of the isobar channels. As
a result, the NN wave function can be calculated for a given NN potential, and the
isobar configurations can be directly determined through their exclusive coupling to
the NN channel [44].
By comparing the experiment data with different theoretical curves, one observe
the Arenh6vel's calculation including IC, MEC, and FSI within the coupled-channel
(CC) model reproduced the cross section quite nicely. However, his calculation including IC,MEC, and FSI within the impulse approximation framework reproduced
the fTT better in strong contrast with the cross section result.
The results suffered the large systematic errors in determining fTT. The systematic errors were estimated to be 20% from the published paper [23]. Aside these 20%
systematic errors, I feel that there maybe some more systematic errors in the experiment. First because the out-of plane angle was small (t22.30), any errors due to the
measurement of the out-of plane angle has an effect on the error in the ftt extraction.
Second, the different efficiency in different locations in the detectors might have some
impact on the resulting accuracy, since the H(e, e'p) measurement was done only in
a smaller acceptance. Third, in the analysis it was assumed that the value of PLTfLT
was about 2.5% which is the value of prTfTT according to various models in this
experiment kinematics [44]. This procedure has some model dependence errors.
Thereforem, a new precise measurement on the fTT is not only desirable but also
crucial to guide the theoretical calculations.
In summary, no theoretical model can describe the all experimental data consistently. The theorists reproduce the transverse response function quite well, while
they overestimate about 20% the strength of the longitudinal response function. For
the longitudinal and transverse interference response function, the asymmetry agrees
with the theory with relativistic corrections, but the response function agrees with
theory without relativistic corrections except for the results from recent Scalcay [35]
data. For the helicity-dependent longitudinal and transverse interference function,
data from the experiment is not conclusive due to large statistical errors. For
46
fTT,
the theory correctly estimate the fTT by using the impulse approximation framework,
but overestimates the total cross section by 20% to 25% [23], if using the impulse approximation framework. Thus a consistent theoretical and experimental comparison
is not established. It is desirable to measure several responses at the same kinematics.
And also, measurements at the same time allow a better control over the kinematic
parameters and systematic uncertainties.
In the dip region, we know the calculation of
fLT
is sensitive to the way one
treats the relativistic effect. The prediction of
fLT'
is sensitive to whether or not
one includes the final state interaction effects.
fTT
is sensitive to inclusion of the
IC currents and meson-exchange currents in calculations. In this experiment, these
three structure functions were measured simultaneously for one kinematics. The result from this experiment will place a stringent constraint on theoretical models. The
Out-of Plane Spectrometer (OOPS) was designed to do the measurement of structure
functions. It can easily go out-of plane up to 900 with the proper satellite support
system. It is light-weight (16 ton) and can be calibrated in order to understand the
detailed properties of each spectrometer. The OOPS cluster is one of best instruments available to measure three structure functions simultaneously. In Fig.1-14, the
theoretical prediction of different calculation models for ALT, ALT, and ATT from
Arenh6vel for this experiment are shown. In Fig.1-15, the theoretical prediction fLT
from Arenhdvel for this experiment are shown. In Fig.1-16, the theoretical prediction
fLT'
from Arenh5vel for this experiment are shown. In Fig.1-17, the theoretical
prediction fTT from Arenhdvel for this experiment are shown.
47
(a)
()
CCJ
1
I-
00-
(b)
41
3
iiI~
2
1
CD 1
85
95
115
105
reaction as a function of
135
145
(degrees)
0C
Figure 1-13: Differential cross section and
125
fTT
response function for the 2 H(e, e')p
measured at NIKHEF [40]. The black dot data is ob-
tained by assuming the fAT term is zero; the white dot data is obtained by using
the Arenh6vel's predication of fAT term. Only statistical errors are shown. The various curves represent calculations by Arenh6vel et al.[32]. Dotted curve: N; dashed
curve: N+MEC; solid curve: N+MEC+IC, calculated within the coupled-channel
(CC) model; dot-dashed curve: N+MEC+IC, calculated in the impulse approximation framework.
48
800 MeV, DIP region, Q2 =0.15 (GeV/C) 2
ALT
0
-0.2
-4
-0.4
-0.6
0.025
0
ALTO
x40%
-0.025
N
- - - - -
-0.05
N+MEC
N+MEC+IC
-0.075
N+MEC+IC+RC
0.1
0.05
ATT
0
-0.05
0
10
20
30
40
50
0'
60
[deg)
Figure 1-14: ALT, ALT, and Arr curves are shown based on Arenh6vel's [62] calculation for this experiment.
49
800 MeV, DIP region, Q2=0. 15 (GeV/c) 2
0
F (T
(fin)
-0.005
-0.01
-0.015
V.1
-0.02
PWBA
N
N+MEC
-0.025
..... .....
N+MEC-i-iC
N+MEC+IC+RC
-0.03
-0.035
-0.04
0
10
20
30
40
50
0'
60
deg]
Figure 1-15: fLT curves are shown based on Arenh6vel's [62] calculation for this
experiment.
50
800 MeV, DIP region, Q2 =0.15 (GeV/c) 2
0.002
FLTV
(fim)
0
-0.002
,,,/
-0.004
7
w/
-0.006
N
N+MEC
.
N+MEC+IC
N+MEC+IC+RC
-0.008
-0.01
0
10
20
30
40
50
0 '
60
[deg]
Figure 1-16: fT' curves are shown based on Arenh6vel's [62] calculation for this
experiment.
51
x 10
-2
800 MeV, DIP region, Q2=0.15 (GeV/c) 2
0.05
0.025
(fi
0
-0.025
-0.05
-0.075
-0.1
-0.125
'/
A
-0.15
N
- -- -
-0.175
N+MEC
N+MEC+t-C
N+MEC+IC+RC
-0.2
-0.2
0
10
20
30
40
60
50
pq
[deg]
Figure 1-17: frr curves are shown based on Arenh6vel's [62] calculation for this
experiment.
52
Chapter 2
Experimental Setup
This experiment was performed in the South Experimental Hall at the Bates Linear Accelerator Center in Middleton Massachusetts during the early spring of 1997.
It used the One-Hundred-Inch-Proton Spectrometer (OHIPS) in the South Experimental Hall for the detection of the outgoing electrons, and three Out-Of Plane
Spectrometers (OOPS) for the detection of the protons. The experiment is a part of
the program to study the deuteron structure functions.
This chapter describes the experimental setup that was used for this experiment,
including the accelerator, the polarized electron source, the Moller Polarimeter, the
liquid target (MIT-Basel loop target), the OOPS and OHIPS spectrometers, the
electronics logic circuit, the data acquisition system.
2.1
Overview of Experiment
The kinematics condition was located in the so called dip region, with a momentum
transfer q of 414 MeV/c and an energy transfer w of 155 MeV. The experimental
parameters are summarized in the following table and the schmatic view of the experimental setup is shown in Fig 2-1:
53
Table 2.1: Experimental Parameters
Beam Energy
Momentum Transfer (q)
Energy Transfer (w)
Missing momentum
Electron Spectrometer:
Angle 0,
Center Momentum
Momentum Acceptance
Horizontal Acceptance
Vertical Acceptance
Solid Angle
Proton Spectrometer:
Lab angle 01,b
Out-of Plane angle 4kc"
Out-of Plane angle 4
Scattering angle gq
Momentum Setting
Momentum Acceptance
Horizontal Acceptance
Vertical Acceptance
Solid Angle
2.2
Forward
29.90
00
00
38.50
509.8MeV/c
±20%
±12mr
±25mr
1.2 msr
800 MeV
414.0 MeV/c
155.0 MeV
210 MeV/
OHIPS
31.00
645 MeV/c
±4%
±20 mr
±55 mr
4.4 msr
Out-of-Plane
53.40
900
23.50
38.50
509.8MeV/c
±20%
±12mr
±25mr
1.2 msr
Backward
76.90
1800
0
38.50
509.8MeV/c
±20%
±l2mr
±25mr
1.2 msr
Bates Linear Accelerator Center
An overview of the Bates Linear Accelerator Center is given in Fig.2.2. The accelerator is designed to produce a pulsed electron beam with a duty factor of approximately
1% and a maximum energy of 1 GeV. This experiment used polarized electrons of
800 MeV at a nominal average current of 5 pA with a typical pulse width of 15 ps
and a pulse repetition rate of 600 Hz.
The beam is injected into the accelerator with an initial energy of 360 KeV.
The accelerator consists of a series of radio frequency cavities to boost the energy of
electron beam up to 500 MeV. For a higher energy, the beam is recirculated by sending
the beam pulse through the accelerator second time before entering the experimental
halls. This experiment used the recirculator and the Energy Compression System
(ECS) shown in Fig.2-2 which reduces the energy spread of the beam to the 0.05%
54
Figure 2-1: A schematic view of the experimental setup showing OHIPS and three
OOPS modules
[45] level. After passing the ECS section, the beam enters the beam switch yard
where the beam is transported to either the South Hall (B-Line), the 14 degree area
or the North Hall (S-Line). This experiment was performed on the B-Line in the
South Hall.
In each second, 20 out of 600 beam bursts were blank, those blank bursts contained
no electron. The blank bursts provided a way to measure the level of background of
the experiment.
The position and size of the beam can be monitored visually by the berylliumoxygen target which can be removed during data taking. The beam position and halo
were recorded for each burst by the beam position monitors (See section 2.3).
55
-n,
(a
N
CD
SAMPLE
EXPERIMENT
/,t
POLARIZED SOURCE
4EA
RESEARCH
64LDING
ADMIN8STRATION
BUI.{NG
VAL _I'-
EPRMENTA
'cl
HALL
4
-
R.F.
(D
GALLERY
01
(D
-
RECNIRCULATORt
SYSTEM
0
CD
(D
SIEIA
EXPERIMENT
ASSEMBLY
%IBRIN4
/NK
BATES LINEAR ACCELERATOR CENTER
0
32 48 14FET
1s
0 5
10
WM
15
IT
OOPS
ENGINEERING
OHIPS
BUILDING
SOUTH
20 METERS
EXPERIMENTAL
--
A
H-HALL RING
LAST
E
P ;
,1;. 7
E
L
*EBT 3
C
E
v.,
EF
ECK)
E
Figure 2-2: Schematic of the Energy Compression System
2.3
Electron Beam Monitors
There are three ferrite-core toroidal transformers mounted on the beam line entering
the South Hall. They are labeled as BT1, BT2 and BT3. These toroids provide
a continuous non-interfering measurement of the beam current. BT1 is positioned
about 10m upstream of the Moller scattering chamber. BT2 and BT3 are about one
meter upstream of the South Hall scattering chamber. Signals from BT1 and BT2 are
digitized by Analog-to-Digital Converters (ADC) to measure the beam charge pulseby-pulse. BT3 signal is sent to a BIC integrator [45] in the counting bay to measure
the total change delivered during a given time and at the same time monitored the
online average beam current.
To accurately determine the beam charge, BT1, BT2 and BT3 are carefully cali57
brated. The calibration procedure is as follows:
* Each beam toroid has a built-in one-turn loop of wire, called a Q-loop. The
current output from a precise charge pulser is fed to the Q-loop to simulate
the pulsed electron beam.
By varying the peak current and pulse width of
the charge pulser, the relation between the toroid ADC channels and the input
charge is determined.
* The charge pulser is calibrated against the BIC integrator. The BIC integrator
is a very precise charge integrator with an accuracy of 0.1%.
Two NIKHEF beam position monitor (BPM) [45] are mounted about 2m and
12m upstream of the target chamber. They provide information of the horizontal (X)
and vertical (Y) positions of the electron beam. The analog signal outputs of the
BPMs are digitized and histogrammed. The centroid of the resulting pulse-height
distribution provide a measure of beam position. Both BPMs are calibrated against
a LUTE during the experiment. The device can monitor the beam position up to 1
mm. It is also possible to monitor the beam incoming angle relative to the beam line.
Two photo-multiplier tubes with no scintillators are located inside the beam vacuum pipe as beam halo monitors. One is located upstream 2m of the Moller scattering
chamber, the other one is upstream 1m of the South Hall scattering chamber. The
outputs of these PMTs are digitized and histogrammed. These histogram is very
helpful to the accelerator operators in tuning the electron beam to reduce its halo
and thus background.
2.4
Polarized Electron Source
The polarized electron source used for this experiment is based on the design from
SLAC [47]. GaAs has two very important properties that make it useful as a potential
polarized electron source.
* Its band structure permits a given spin state to be preferentially pumped into
the conduction band.
58
* Its surface can be treated to develop a negative work function (so called negative
electron affinity).
The band gap between the energy maximum of the valence band and energy minimum of conduction band is Eg=1.5 2 eV. The electron wave function has P symmetry
at the maximum of the valence band and S symmetry at the minimum of the conduction band. The spin-orbit splitting of the valence band of GaAs cause the otherwise
degenerate P state to be split into a four-fold degenerate P3/ 2 state and a two-fold
degenerate P/ 2 state. The P/ 2 is located lower by an amount A=0.34 eV in energy.
Fig.2-3 is the diagram of the energy levels in the GaAs crystal.
free electron
2.50 eV
1.52 eV
hv
0.34 eV
j
3/2
p 1/2
Figure 2-3: The Diagram of energy levels in GaAs crystal.
For circularly polarized light, the selection rules require that Am 3 = +1 for the
positive helicity and Amj = -1
for the negative helicity.
Assuming that a circularly polarized photon of positive helicity is incident on a
GaAs crystal, if the photon energy is the difference between the P 3 / 2 and the S1 /2
energy level, then the transitions of P 3 / 2 states to S1/ 2 states can only be allowed.
There are only two possible transitions:
59
m,= -3/2 in P 3/ 2
m = -1/2
+ m=
--
in P 3 /2 -=
m
-1/2 in Si/ 2
+1/2 inSl/
(2.1)
2
The P3 / 2,-3/ 2 state, 13/2, -3/2>, can be decoupled to the orbital and spin angle
momentum 11, -1> 11/2, -1/2>. The P 3 / 2,- 1/ 2 state, 13/2, -1/2>, can be decoupled
to the orbital and spin angle momentum
1/311,-i> 1/2, 1/2> + V2311, 0> 1/2,
-1/2>. Therefore the probability from P 3/ 2 ,-3/ 2 to SI/ 2 ,-1/ 2 is three times more likely
than the probability from P 3 / 2,-1/ 2 to S1/2,+1/2.
P = 3
3+1
1 = 50%.
(2.2)
Here P is the spin polarization of the emitted electrons for a circularly polarized
photon of positive helicity.
In order to make a polarized electron source, polarized electrons in the conduction
band, which are created with circularly polarized photons, must leave the GaAs crystal. In a normal GsAs crystal, the energy gap from the conduction band to the free
electron state is about 2.50 eV. By treating the surface of GsAs with Cs, the energy
gap from the bottom of the conduction band to the free electron state decreases below
zero, so that a negative electron affinity is developed.
Quantum efficiency is defined as the ratio of the number of released electrons and
the number of photons. The typical quantum efficiencies in this experiment were in
the range 0.5% to 2.0%.
2.5
Moller Polarimeter
The spin polarization of the electrons in the beam was measured by elastic scattering
with polarized atomic electrons. The device that monitors this scattering is called
the Moller Polarimeter as shown in Fig.2-5. The Moller Polarimeter on beam line B
at Bates that was used in this experiment was installed in early 1989 and later was
60
POLARIZED
ELECTRON
GUN(6OkeV)
WIEN FILTER
300 keV
DC ACCELERATOR
COLUMN
FLOOR LEVEL
-
90 DEGREE BEND
ACCELERATOR
-
Figure 2-4: The Bates polarized electron source
improved by Dr. K.Joo. A detailed description can be found in Dr. K.Joo's thesis
[4]. Here, I summarize some important aspects.
The cross section for polarized, elastic electron-electron scattering can be written
as [48]
dQi (1 + j Pi P3
du
'3)
where P.B(PT) are the components of the beam (target) polarization as measured
in the rest frame of the beam (target) electrons. Here, the z-axis is along the beam
momentum. The nine asymmetries Aij can be calculated in QED to lowest order [48]:
- ZZ
Azz
(7+
=
A=
cos 2 0cm) sin 2
-4y
- An =
-
= AxZ
A~x
=
A,
~
y
=
=
61
(2.4)
(2.4)
0"c
(3cos
+ 29cm)2
2i
C
sin 4 92 cm5
cos
(32 +sins
3 6 9cm)2
3
2 sin =cm
'y(3 + cos 2 9cm)2
=
,
26
26
(2.7)
Target
Quadrupole
Magnet
Collimator
Detectors
Internal
shielding
Beam
Figure 2-5: Layout of Mollor apparatus
where
0
cm
is the scattering angle in the center of mass frame and y is the Lorentz
factor between the laboratory and center of mass frames. At
is at its largest, where A,
6
cm
= 900
where A,,
= - 7/9, A,, = -1/9, AYY = 1/9, Az =A., = 0. The
asymmetry in the cross section due to the helicity of beam is:
1
+
pTpB
7
A =-= 9 P, X + -9 YY -y, -9 PT
,
(2.8)
Under the assumption that both the beam and the target are polarized only along
the beam momentum and there is no background, the asymmetry simplifies to
7
Aphy =
pBpT
(2.9)
The electron beam polarization is determined by measuring the asymmetry for both
positive and negative helicities of longitudinally polarized electron. The measured
asymmetry is given by
Ameasure = N+
N_'
(2.10)
where N+ and N_ are the yields for the positive and negative helicities normalized
with total beam charge. In the Chapter 4, the measurements and analysis of the
beam polarization is discussed.
For this experiment, the Moller apparatus was located in a shielded experimental
62
area on beam line B approximately 10 m upstream of the main target. Fig.2-5 shows a
schematic layout. The electron beam was incident from the left on the ferromagnetic
target foil, which was contained in a vacuum target chamber. The chamber was
surrounded by a pair of Helmholtz coils, which provided the polarizing magnetic field.
The electrons scattered from the foil passed through a lead collimator. Subsequently,
a quadruple magnet defected the Moller electron horizontally away from the beam
direction. A pair of Cherenkov detectors (C1/C2) detected the electrons. The position
of detector C1 and C2 could be adjusted within a small angular range to maximize
the detector acceptances for the specific angles of scattering set by the collimator and
target positions. A central opening in the collimator allowed the beam to pass into
the main experimental hall as shown in Fig.2-5.
For this experiment, the apparatus was configured for 800 MeV beam energy,
corresponding to a distance between target foil and collimator of 110.1 cm and Moller
lab scattering angle of 2.05'( 90'm )-
Several targets were installed on a target ladder inside the target chamber of the
Moller apparatus. Two ferromagnetic Fe-Co alloy foils of 13pm and 25pum thickness,
made of Supermendur (49% Fe, 49% Co and 2% Va by mass), were provideed for the
beam polarization measurements. Only the 13pam foil was used in this experiment. A
fluorescent BeO target was used for beam position monitor, and an empty frame was
provided to allow the beam to pass undisturbed into experimental hall where there
was no Moller measurement.
2.6
Liquid Target
The liquid target for this experiment is the MIT-Basel Loop target. This target
system consists of two loops for cryogenic liquid and a stage to mount solid targets.
The two liquid loops are identical in construction and instrumentation. The solid
targets are BeO, Carbon, and a blank. We used the BeO and C targets for beam
energy measurements and focal plane calibrations. The blank target was used for
background measurements.
63
Refrigerant
in: Aout
DTS - Diode Tempature Sensor
RTS - Resistive Tempature Sensor
LL - Liquid Level Sensor
DTs i
. s2
Hydrogen In/Out
I0
* S4
2-------
--
Ts
Hyroe
I/uln
Hyroe
I/u
l
* sS
Ilow>
Figure 2-6: Schematic of the basel loop target
64
Table 2.2: MIT-Basel Loop Target Parameters
Liquid
Cell Diameter (cm)
Cell Wall Thickness
Nominal Pressure (atms)
Nominal Temperature (K)
Nominal Liquid Density (g/cm 2 )
Bottom
LH 2
1.6
4.3(pm)
1.0
20.3 K
0.0793
Top
LD 2
1.6
4.3(pim)
1.0
23.7 K
0.160
A schematic of the target is shown in Fig.2-6. Each loop consisted of a heat
exchanger to cool the liquid, a heater to maintain a constant liquid temperature, two
resistive temperature sensors to monitor the temperature of the liquid, and a fan to
circulate the liquid. The two loops were cooled in series by gaseous helium. The
helium refrigerator was 200 Watt Koch model 1420 [4]. The target instrumentation
information was monitored and recorded by an IBM PC compatible computer in the
South Hall Counting Bay. The target information was collected through a GPIB to a
CAMAC module over a serial port. The information included the date and time, the
top and bottom temperatures of the target cells and the pressure for each cell and it
was written to the event data stream once per minute.
The density of the liquid hydrogen was calculated from the following expression
[22]:
PLH 2 = PC + A1 -ATO.
+ A 2 AT +
AT
+ A 4 - A T 5 / 3 + A5 - A T 2 ,
(2.11)
for T < T, where
Pc = 0.01559moles/cm 3 , Tc = 33.0K, AT = Tc - T, A1 = 7.32. 10-
3
3
.
A2
=
-4.4.
10-3 , A 3 = 6.6
- 10-3, A 4 = -2.9-
10-3, A 5 =
4.0. 10-
,(2.12)
(2.13)
For a small region of temperature around the critical point, the density of liquid
65
deuterium in moles of nuclei per cm3 is:
PLD 2
-
0.210
PLH 2
0.160
(2.14)
for a small region of temperature around the critical point. Since in the experiment,
the temperature sensors were stable within ± 0.2K. This uncertainty translates to an
error in the target density of ± 0.3%.
2.7
The OOPS Spectrometer
We used OOPS to detect protons and measure both the magnitude and direction
of their momentum. The design and construction of the OOPS was a collaborative
effort: A group from the University of Illinois at Urbana-Champaign and scientists
from Bates [49] were primarily responsible for the spectrometer magnets and module
construction, while the MIT group (including the author) built the detector packages
and electronic logic circuits.
For this experiment, we built three identical OOPS
spectrometers and three identical OOPS detector packages. Fig.2-8 is a cross-sectional
illustration of the OOPS showing the positions of the magnets, shielding, baffles, and
the detector package.
Each OOPS is a relatively lightweight [22] (16 tons) dipole-quadrupole spectrometer designed for convenient positioning out of the electron scattering plane. For this
experiment, we used three OOPS spectrometers. Two were in plane, and the other
one was out of plane. Maximum central momentum of OOPS spectrometer is 832
MeV/c. The nominal central momentum for this experiment was 509.8 MeV/c. Momentum acceptance of OOPS is about ±25%. The focal plane is tilted at an angle
about 130 with respect to the central ray. The focal plane detectors measure the position of particle trajectory perpendicular to the central ray. The OOPS momentum
resolution is about 1%. The OOPS angular resolution is about 1 mr in both in-plane
and out-of plane angle (Details in Chapter 4).
66
rear shielding plug
internal top/bottom shielding
internal shielding plug
quadrupole magnet
shielding plug
dipole magnet
detector
package
external side
shielding
front collimator
Figure 2-7: OOPS Spectrometer Layout
2.7.1
The OOPS Focal Plane Instrumentation Overview
The OOPS detector package consists of three LAMPF [39] style horizontal drift chambers (HDC) and three scintillators. Each of the HDC's contain a pair of X/Y wire
planes which measures the position in the x and y direction. The detector package
of OOPS consists of three such chambers even through only two HDCs are needed
to get the angle information. The third chamber allows the continuous monitoring of
the efficiency and the resolution of the wire chambers in the experiment and increases
the overall detection efficiency of the detector package by choosing the combination
of any two of three fired chambers. The HDCs are spaced 12.7 cm apart.
Each end of the scintillators is connected by the a layer of fiber-optics to a photomultiplier tube. The signals from the photo-multiplier tubes provide the OOPS trigger and the start fiducial time for the chamber readout system.
The focal plane instrumentation for OOPS has to be small in order to fit in
the small room inside of OOPS shielding can. The entire package is mounted on
an aluminum frame and can be easily slid in and out of OOPS shielding can. It
67
Magnets
Lead Shielding
Lead Collimators
andi
Bafles
0'~
21.79
60.4 Cni
60.9
140
cm
12.86
121.92 cm
8.9 3 -Ray
cm
)
Focal
Plane
34.50 cin Center to Center
34.37 cm Ray exit to
entrance
Target
Figure 2-8: OOPS Spectrometer Cross Section View
was verified that this motion can be accomplished while accurately reproducing the
alignment of detectors with respect to the spectrometer.
2.7.2
The OOPS Horizontal Drift Chambers
All the horizontal drift chambers (HDCs) used in three detector packages were constructed and tested by the MIT Nuclear Interaction group. Each chamber consists of
two detection planes providing a X and Y measurement of the particle position. The
chambers are small, with active area that is 17 cm in the x direction and 32 cm in the
y direction. They are constructed of eight 4.8 mm thick machined aluminum plates
that are stacked on top of each other and are sealed from outside by O-rings. There
are two 0.25 mil aluminized mylar sheets on both side of the chamber to provide
the isolation between the chamber and the outside atmosphere. There are four more
aluminized mylar planes in side of chambers as the ground plane for the wire planes.
There are 21 signal wires in the x plane and 38 signal wires in the y plane at
a spacing of 8 mm. The diameter of the signal wires is 20 jpm, and they are at a
distance of 4 mm from the two ground planes. The error of the spacing between 2
68
HDC 3
HDC 2
HDC 1
S3
S 2
S 1
Particles
Figure 2-9: OOPS detector package
signal wires is about ±1% out of 8 mm. Each signal wire is connected to a PC-board
delay line. The signal wires are held at a potential of about 2500 to 2600 volts, and
are connected to a high voltage box which provides a positive potential. Between the
signal wires, there are ground wires which are 76 pm in diameter. These are held at
ground potential, and are bussed together alternatively to two lines which go to the
odd/even amplifiers. The odd/even amplifiers are read out by a self-gated ADC. The
entire chamber is filled with a gas mixture of 49.25% argon and 49.25% isobutane and
0.5% alcohol. The alcohol is added to the gas mixture by bubbling the gas through
a bottle filled with alcohol in an refrigerator. The alcohol is used to prevent buildup
of deposits on the wires. It helps prolong the lifetime of the chambers and maintain
their stability. The cross section of HDC is shown in Fig.2-11. The gas flow insides
the HDC is also shown.
When a charged particle passes through a working chamber, it will ionize some
argon atom. Those electrons are attracted to the signal wire by a strong electric field.
They drift along the field line. In the neighborhood of the sense wires, those electrons
69
P"
Odd-even Busses
Signal Wire
7
Guard Wire
7
Delay Line
Figure 2-10: Inside of an OOPS chamber
gain so much energy between the collision with the atoms that they ionize the gas. The
freed electrons repeat the process. The result is an avalanche of electrons. Meanwhile
the isobutane is a long molecule, it has rotation excitation states and vibrational
excitation states to absorb the photons which are produced by the electron avalanche.
The isobutane keeps the avalanche localized and maintains the stable condition of the
chamber.
In the HDCs, the shortest drifting path to a signal wire is almost horizontal
because the incident angle is almost perpendicular to the plane. When those electrons
arrive at the signal wire, the avalanche produces a small a negative electric pulse,
typically 5-10 mV in height and 200 ns in duration. This pulse travels down the
delay line, and through a capacitor, then to a high gain amplifier. The analog signal
is converted to a digital logic signal, and the logic signal is sent to a Time-to-Digital
Converter (TDC) and its value is recorded in the event data stream.
When an avalanche occurs, there are also small induced positive charges on the
neighboring ground wires. These signals are sent to a high gain and fast amplifier,
which makes an (O-E) signal and an (O+E) signal. Only the event with its (O+E)
signal larger than a preset value, its (O-E) signal would process through a gate. The
70
Gas Out
Gas In
Sense Wire
: Ground Wire
: Aluminum mylar sheet
Bolt
Figure 2-11: The cross section diagram of HDC with the gas flow.'
threshold for (O+E) signal in this experiment is 10 mV. The width of (0-E) signal
after the amplifier is 80 ns, the signal is offset by -80 mV and clamped between -20
mV to -140 mV before O-E signal is sent to an ADC. This signal is used to resolve
the left-right ambiguity (Details are presented in Chapter 3).
The performance of the left-right separation is very sensitive to the chamber operating voltage and gas mixture ratio. By experimental study, we found that generally
the chamber efficiency increased with a higher voltage and a higher percentage of
argon gas. However, the total efficiency of chamber is the vicinity of 95% to 97% as
shown in Fig.2-12. We found for the best condition for the chamber considering both
the efficiency of chambers and stability of chambers to be at about 50% argon and at
a voltage of about 2550 volts.
The performance and resolution of these chambers were studied extensively using
cosmic rays, a
90 Sr
source and the electron beam. The intrinsic resolution of the
chambers, when the multiple scattering from windows is unfolded, is found to be
155±9 Mm. When the chambers and the software were properly optimized this results
in a trajectory reconstruction resolution (FWHM) of 165pm (Details are presented
71
100
1C0
80
80
60
60
40
40
20
20
0
0
2000
2200
2400
2600
2800
L . . . .
20
300Q10
Votage V)
30
40
50
60
70
Argon %
Figure 2-12: Left figure is the HDC efficiency verse the operating voltage, right figure
is the HDC efficiency verse the argon percentage at 2550 V
in Chapter 3). This translated into an angular resolution 0.75 mr in the focal plane.
2.7.3
The OOPS Scintillators
The OOPS detector has three plastic scintillators that are mounted downstream the
HDCs as shown in Fig.2-9. The first scintillator has a thickness of 1/16 inch and
each of the next two scintillators have a thickness of 3/16 inch. Each scintillator is
coupled to two photo multiplier tubes through light pipes. Each photo-tube has its
own magnet shielding.
The voltage on the PMTS is set to ensure that all proton events produce a large
enough pulse to trigger the discriminator. Typical signal height in this experiment is
from -100 mV to -150 mV, and typical signal width from PMT is from 15 ns to 25
ns.
Timing of each scintillators is determined by the signal from the left-side of a
scintillator. This is done by delaying the signal from the left-side of scintillator by
extra 10 ns comparing the signal from the right-side of scintillator. The timing of
each OOPS detector package is determined by the signal from the left-side of the
72
Sr Source
Photo
Tube
Scintillator I
Scintillator 2
Electron
Scaler
I
ANDD
Figure 2-13: Setup for measuring scintillator efficiency
second scintillator. In order to reduce the time walk effect due to the variation of
signal size of scintillator signals. The signal from the left-side of the second scintillator
is discriminated at -30 mV, while all other scintillator signals are discriminated at
-50 mV. The time variation due to the variation of size in the scintillator for OOPS
detector is less than 0.1 ns.
The efficiency of the scintillators has been carefully studied. We used a collimated
9 0Sr
source on top of two closely stacked scintillators as shown in Fig.2-13. Each elec-
tron which arrived at the second scintillator must pass through the first scintillator,
any difference in the singles rate of the second scintillator and the coincidence rate is
inefficiency of the first scintillator. The efficiency of the top scintillator is thus:
Scaler #2
73
We mapped each individual scintillator, and we found each scintillator to have an
efficiency higher than 99.5%, except at the very edge (~~ 1/2 cm towards the edge of
scintillator), it drop slowly towards 50%.
2.8
The OHIPS Spectrometer
The OHIPS (One-Hundred-Inch-Proton Spectrometer as shown in Fig.2-14) was originally designed to be a proton detector. In this experiment, OHIPS was rebuilt and
converted into a high efficiency, low background electron spectrometer. OHIPS is a
Q-Q-D (quadrupole-quadrupole-dipole) magnetic spectrometer with a 90' up-bean in
the vertical direction. OHIPS is designed to focus point to point in both the dispersive plane and the transverse plane. The maximum OHIPS central momentum is 1.3
GeV/c, the momentum acceptance is 8.75%. as shown in Table 2.3.
An eight-inch-thick rectangular lead collimator is attached to the front end of
OHIPS. The collimator is placed inside the snout of OHIPS which extends the vacuum closer to the target in order to minimize multiple scattering in the air between
the scattering chamber and OHIPS. The collimator has the horizontal opening of 17.3
cm and the vertical opening 7.5 cm and was located 158.5 cm away from the target.
This design gives a geometric solid angle of 5.16 msr, if the front end of the collimator determined the solid angle. We found that several internal structures of the
spectrometer were cutting off the acceptance in transverse direction and it changed
the solid angle to 4.34 msr (Detail are presented in Chapter 4).
There are two quadrupole focusing modes for OHIPS. One is Normal Mode which
the first quadrupole focuses in the horizontal direction and second quadrupole focuses
in the vertical direction. The other mode is Reverse Mode which the first quadrupole
focuses in the vertical direction while the second quadrupole focuses in the horizontal direction. OHIPS in the Reverse Mode has a three times better scattering
angle resolution than in the Normal Mode which is essential for the measurements of
ALT,
fLT, ATT
and
fTT.
The Reverse Mode was used in this experiment.
74
OHIPS
1.0 meter
I
I
DETECTOR
PACKAGE
P
I
DIPOLE
I II
COLLIMATOR
Q1
TARGE
02
r
II
I
I I
i I II II I I -I --.1
Figure 2-14: OHIPS Spectrometer Layout
75
Table 2.3: Summary of OHIPS properties
2.8.1
Drift Distance
Quadrupole Focusing Mode
Solid Angle
0 Acceptance
0 Acceptance
Maximum Central Momentum
Maximum Momentum Accept-ace
Momentum Resolution (FWHM)
Radius of Curvature
Bending Constant
2.04 meter
Reverse Mode
4.34 msr
± 54.6 mr
± 19.9 mr
1300 MeV/c
t 8.75%
1.2 x 10-3
2.54 meter
77.82 MeV/kG
Bending Angle
900
Path Length to Focal Plane
9.7 meter
The OHIPS Focal Plane Instrumentation
The OHIPS Focal Plane Instrumentation consists two cross-wired vertical drift chambers (VDCX1 and VDCX2), three scintillators (S1, S2 and S3), one gas Cherenkov
and two layers of the lead glass as shown in Fig.2-15. The three scintillators provide
the OHIPS single arm trigger. The two VDCX chamber determine the position and
direction of the particles. The Cherenkov counter is used to distinguish between pions
and electrons. The Cherenkov was calibrated in 1995 and was documented in the theses of Zhifeng An [50] and Dr. Xiaodong Jiang [51]. The two layers of the lead glass
are used to reduce the noise from cosmic rays, although the noise from cosmic rays
is not a issue in this experiment. For some high
Q transfer
single arm experiments,
due to extremely low counting rate, The lead glasses Cherenkov is required.
2.8.2
The OHIPS VDCX
Each of VDCX chambers is made of two cross-wired vertical drift chambers (VDC).
Each VDC has two planes of aluminized mylar at -9.0 kV. Between those two high
voltage planes are 128, 20 pm thick, gold plated tungsten wires at a distance of
4.23 mm from each other as signal wires, and 50 pm thick beryllium-copper wires
successively placed between them as guard wires as shown in Fig.2-16. The chamber
76
iSPECTROMETER
CENTER LINE
00
SHOWER
COUNTER
00 0
CHERENKOV
COUNTER
S2
45'40
S1
VDCX
VDCX1
VACUUM
WINDOW
WINDOWTYPICAL
TRACK
OHIPS DETECTOR
PACKAGE
1 foot
Figure 2-15: OHIPS Detector Package
is filled with equal amounts of argon and isobutane.
A charged particle going through the chamber ionizes gas atoms. The knocked-out
electrons drift along the field lines towards the signal wires at constant speed. When
they come close to a signal wire where the fields are the strong, they gain enough
energy between collision and ionize the gas atom. An avalanche is formed. A signal
is produced in the sense wire due to the motion of electron avalanche.
The measurement of the drift time of the electron use the LeCroy 4290 Drift Chamber Time Digitizing System (DCOS). DCOS is a stand alone system for multi-wire
drift chamber data acquisition. The system, organized as a sub-system of CAMAC,
consists of amplifier discriminator cards, time digitizer modules, readout units and
CAMAC interface buffers.
DCOS allows one TDC per wire of the drift chamber.
Each TDC channel in DCOS has the ability for the self calibration. The resolution of
77
TDC is 1 ns for a full time scale of 512 ns. If no stop signal is received within the full
time scale, all TDC channels are reset. Valid data are stored in the DATABUS [51]
Interface module which can be accessed by the data acquisition computer through
the branch highway [24].
78
(a)
VDCX2
VDCX1
(not to scale)
(not to scale)
-HV
-HV
GND - x -
x
x
12.7 mm
x - x
11.8 mm
x -
GND
-HV
-HV
36.5 mm
GND
,
-HV
GND
-HV
23.6 mm
FIELD WIRE
"x
SIGNAL WIRE
-1 .*
4
-HV
SIGNAL WIRE
4.230 mm
J
4
-HV
[-4.243 mm
WIRES IN THE BOTTOM PLANE ARE ROTATED BY 900
(b)
DRIFT CHAMBER OPERATING SYSTEM
(DCOS)
TO COMPUTER AND
OTHER CAMAC CRATES
VDCX
\ \\
%\ \ \\
~\ ~ \\ \ \ \\ ~\
N-277
AMP./D IS.
DCOS CAMAC
OHIPS CAMAC
DCOS COM. STOP
z
L4291B
TDC
L4298 TDC
CONTROLLER
Figure 2-16: DCOS Readout System
79
L4299
DATABUS
2.9
Electronics Logic circuit
The electronics that formed the hardware logic and digitization were built from standard standard Nuclear Instrumentation Module (NIM) and CAMAC [52] modules
and crates. The MicroVax was interfaced to the electronics via a Microprogrammable
Branch Drive (MBD)
[53],
which does the actual data acquisition between the user
and the data acquisition system for control, on-line data analysis, and data storage.
The electronics logic circuit consists three parts. One part is the OHIPS trigger electronis circuit, one part is the OOPS trigger electronics circuit, and last one is the
coincidence trigger electronics circuit.
2.9.1
The OHIPS Trigger Electronics Circuit
The OHIPS scintillators consist of Bicron BC-408 plastic scintillator material. Both
scintillator S1 and S2 are 1/4 inch thick and each has an active area of 8.0 inches wide
and 25.0 inches long, whereas S3 is 1/2 inch thick, 9.0 inches wide and 29.0 inches
long. RCA-8575 photo-tubes are connected by optical fibers to both sides of S1 and
S2. Only one narraw side of the S3 is viewd by a photomultiple tube . The OHIPS
electronics were set up on the OHIPS platform to avoid sending signal over long cables.
Typical pulse heights of raw photo-tubes signals are above 100 mV. The threshold of
discriminators receiving these signals was set to 50 mV. Logic signals from both ends of
the S1 and S2 discriminators passed through mean-time modules and then coincidence
with the signals from S3 logical signal. The S2 signal was delayed so that it always
determined the timing of OHIPS trigger. The trigger logic diagram of OHIPS is
shown in Fig.2-17. Coincident logic signal from the three scintillators forms a OHIPS
scintillator trigger signal. This signal was called the OHIPS pilot signal. The OHIPS
pilot signal was sent to the OOPS/OHIPS coincidence logic circuits (See Section
2.8.4) along with the OOPS scintillator trigger signal. If the event is not selected by
the OOPS/OHIPS coincidence logic circuits, the OOPS/OHIPS coincidence circuit
would issue a self clear signal to clear all OHIPS ADCs and TDCs. If the OHIPS pilot
signal is selected, all OHIPS ADCs' and TDCs' value is stored in the modules. The
80
time of OHIPS self clear was about 0.8 ps. During this time, all OHIPS scintillator
triggers were inhibited. The inefficiency caused by this inhibit is called the OHIPS
self-inhibit inefficiency. The final results were corrected for this efficiency by using
the OHIPS scaler information.
81
S1L
OHIPS TRIGGER LOGIC
ADC
D
F
s
TDC
M
D
INH
1R
D
F
S
s
D
H
OHIP S PILOT
S
TDC
ADCH
FASI TCLEAR
ADC
ADC
S2L
F
DTDC DC
INH
3-FOLD
COIN.
MD
_D
JH
S2R
F
470 ns,
sM
DC
680 ns
ADC
1.3.us
INH
D
S3 :1j
F
sDC
DCOS COM.
STOP
ADC
PIT READ2
LEGEND
SCALER
-D-
AND
FAN-OUT
:I>
OR
~- DELAY
GATE GENERATOR
DISCRIMINATOR
H+
HELICITY
"+"
SIGNAL
MEAN-TIMER
H~
HELICITY "-"
SIGNAL
Figure 2-17: OHIPS Trigger Diagram
82
2.9.2
The OOPS trigger and coincidence electronics
The OOPS trigger electronic circuits and OOPS/OHIPS coincidence electronic circuits are set up in the underground area just below the target area. The reason for
this location is to reduce distances for the small signals traveling from OOPS detector
packages to the electronics circuits. Another reason is to reduce the radiation damage
on the electronic modules by placing the electronic circuits in a shielded underground
place. Details of the OOPS trigger layout is showed in the Fig.2-18.
In this experiment, three types of events: OHIPS single arm events, OOPS single
arm events and Coincidence events, are recorded in the event data stream. The
single arm event rate for both OHIPS and OOPS selected is scaled down by a factor
of about 1000. Those signal events are used as the run-time calibration monitors of
luminosity. The coincidence events are the events which the time difference between
one of three OOPS scintillator triggers and the OHIPS pilot is within 100 ns. Both
true and accident coincidence events are recorded in this experiment.
The type of event is checked before the readout of the electronic modules begins.
Only the electronic modules which associate with the spectrometers that have an
event are be read out. For this purposes, the LATCH system was setup. There
are two LATCHs. Latch One exams the direct events from the spectromters. Latch
Two exams the output of the OOPS/OHIPS coincidence and pre-scal counters from
the four spectrometer.
is determined.
From Latch One, the spectrometers which have an event
The microcomputer reads out the information from the electronic
modules accordingly. From Latch two, the type of events is recorded in the data
stream for later analysis.
In order to read the electronic modules as fast as possible, several techniques
have been applied. All the unnecessary data words are eliminated. The data in the
electronic modules are read out in a specific order. Since the digitalization of the
ADC is much slower than the that of the TDC, it is natural to read all the TDCs
first then the ADCs. One hardware clear signal is used to clear all the modules after
all the TDCs and ADCs are read out (instead of to use the software clear command
83
Sl
SILDADC
ADAC
ADC
H. D
S2L
TDC
L.D
OOPS Trigger
TDC
OOPS Timing
LD
AND
TDC
TDC
S2R
H. D
Octal Logic Unit Modular
AADC
S3R
3H. D
--
TDC
S3R
ADC
S
H.D: High Threshold Discriminator (50 mV). L.D: Low Threhold Discriminator (30 mV). S: Scaler
Figure 2-18: The OOPS scintillator trigger logic.
84
for each modules). The time for a microcomputer to read out one OOPS electronic
system is about ims. The time for a microcomputer to read out one OHIPS electronic
system is about 2ms.
Timing is the one of the most important part of the electronic circuit. The pulsed
beam with 0.5 mA peak current is used in this experiment. The accident coincidence
rate between OOPS and OHIPS is quite high. For a true coincidence event, the time
between OOPS and OHIPS is fixed in an ideal case. One could use the time between
OOPS and OHIPS to increase the signal to noise ratio. Since the signals from both
OOPS and OHIPS travel through a lot of electronic modules, and along the way there
could be some ambiguities in the signal's timing. In the electronic circuit, it is ensured
that the start fiducial time for an OOPS electronic system is always the signal from
the second left scintillator, and start fiducial time for the OHIPS electronic system
is always the meantime of signals from the both sides of the second scintillator. The
overall resolution of timing property of the electronics circuit is determined about
0.1ns FWHM.
85
UUS A Timing
OOPS
OOPS Timing OR
B Timing
AND
OOPS C Timing
OOPS A Trigger
R
OOPS
s
Prescal
OOPSB
Trgger
_
OR
OR
AND Z
Gate
>OR
Generator
Event
[AND
Trigger
OHIPS Trigger
RND Pulse
3AND
all
OR
Veto
OHIPS Timing
AND
Time
-7OHIPS Start
Figure 2-19: The coincidence circuit logic diagram.
86
2.9.3
Veto System
There are three veto systems which are implemented in this experiment.
" The OHIPS self veto: When a trigger is formed by the OHIPS scintillators, this
veto prevents any OHIPS triggers for 1.5 Ms.
" The one-per-beam-burst vote: This veto ensure that there is at most one event
to recored per beam burst.
" The front-end Veto: This veto ensures that we only record an event when the
data acquisition system is ready.
Each of these three vetoes systems are required for unambiguous data quality.
They are the key components in the electronic logic circuits.
The reason for the OHIPS self veto is that the round trip of the OHIPS pilots to
the OOPS/OHIPS coincidence module is about 500 ns, and all signals in the DCOS
system used in OHIPS can not wait such a long time. The solution of this problem is to
start the processing of OHIPS events immediately after the OHIPS scintillator trigger
is formed. It the meantime, the OHIPS pilot is sent to the OOPS/OHIPS coincidence
circuit. If this is not the signal which made to the final trigger, the OOPS/OHIPS
coincidence electronics circuit would send a signal back to the OHIPS electronics
circuit to stop digitalization and clear the OHIPS TDCs and ADCs. Selection as
the final trigger means that the OHIPS pilot signal is a coincidence event with the
OOPStrigger or a pre-scaled event selected by the pre-scaled counter. The time to
clear the OHIPS TDC and ADC module is around 800 ns. If Another OHIPS pilot
signal are to make the same request during the time when there is an OHIPS event
processing. it would confuss all unread OHIPS TDC and ADC values. Therefore
within that period time, the OHIPS electronic circuit is not allowed to send its pilot
signal to the OOPS/OHIPS coincidence circuits or the any of the internal OHIPS
DCOS electronics. The OHIPS scintillator trigger is vetoed by 1.5 ps second after
any OHIPS trigger. However, for the normalization purposes, it is very important
87
1.5 Micro Second Veto
n
WIPS
r;r
*9
* 9
t---------
OOPS
50 Micro Second Veto
Event 8
Trigger
Beam Burst
15 Micro Second
Figure 2-20: The one-per-beam-burst diagram.
that scalers kept track of the number of raw OHIPS triggers and the number of raw
OHIPS triggers that are sent to the coincidence module
The one-per-beam-burst veto is to ensure there is at most one event recorded
during each beam burst. The time for a microcomputer to read out all TDC and
ADC information is ims to 3ms. A second trigger in the same event would corrupt
the unread TDC and ADC information. However, the number of these second triggers
is recorded in scalers for the normalization purposes.
The front-end veto circuit is to ensure: the entire electronic circuit is vetoed under
the following situations.
" no gun pulse
" computer is busy in reading data out of the CAMAC module
" run is suspended or finished
" chambers are tripped off.
88
The mathematical way to express this logic is
Veto Signal = no gun
U computer busy U run off U chamber off .
(2.16)
In the setup of the front end veto system, there are certain details that need to
be considered.
" Since there is no accurate way to calculate the charge of a partial beam pulse, no
part of the gate of the front end veto can exist during the beam pulse. Therefore
this veto never stops and starts during a beam burst.
" In the event of abnormal experimental conditions, such as when the chamber
tripped or the beam went off, the front end veto should immediately turn on.
The front-end veto signal is used to veto the OOPS and OHIPS scintillator discriminators. We also use this veto signal to veto the charge counter. The procedure
greatly reduces the possibility of incorrect calculation of the correction for computer
busy time in all experimental condition. The electronic logic diagram is shown in
Fig.2-21.
2.10
Data Acquisition
The data acquisition system used in this experiment, named
Q, was
developed at the
Los Alamos Meson Physics Facility for the VMS operating system and is described in
LAMPF document MP-1-3401-3, Introduction to
Q. Q is
a general purpose CAMAC
data acquisition system which, in conjunction with a micro programmable branch
driver (MBD) and a micro VAX computer operating under VMS, provides data acquisition, analysis, and storage.
Q is setup
as an event-driven data acquisition system.
CAMAC modules were used to record various parameters such as times, pulse height,
number of events, etc. The individual CAMAC crates were daisy-chained together
using a Branch Highway which is connected to the MBD. An Event Trigger module is
89
Run Gate
Beam Gate
OR
ANE__
F
Front End
CB
Z
F
IVeto
Reset
Trigger
G
Computer Busy
Figure 2-21: The front end veto logic diagram.
used in the first CAMAC crate to initiate read out of the various events. The CAMAC
modules are initialized, read out and cleared with a user written
Q program:
QAL
[25]. The QAL program controls the modules and defines the data stream structure.
Data is stored in the 8mm tapes for the future analysis.
90
Chapter 3
The Data Analysis
The bulk raw data for this experiment is about tens of gigabytes in total size. The
off-line analysis involves a detailed examination of physical quantities. A considerable
amount of time was spent on improving the analysis methods and many details of
the calibration of the apparatus. This means that new software had to be developed.
In this chapter we describe the detailed analysis including the methods used for
the OOPS chamber decoding, OOPS chamber alignment, mean-time correction, the
OOPS optics, electronic inefficiencies and radiative correction.
Since the OHIPS
properties were explained very detailed in the thesis of Dr. Xiaodong Jiang [51], I
skip the part of OHIPS and concentrate on the OOPS analysis.
v
3.1
An Overview
During the experiment, we taped the data using 8mm tapes with the on-line
Q ana-
lyzer [25]. After the experiment, we first rewrote all the data from 8mm tapes into a
number of CD-ROMs. It is much faster and also safer to use a CD-ROM driver to replay and analyze. We developed data analysis software in c,+to decode the data and
calculate all the physical quantities, and produced the CERN format database files
(HBOOK). We used the PAW (Physicist Analysis Workstation) (http://www.cern.ch)
software package in the CERN library as our main tool to analyze these HBOOK files.
91
The advantage of this approach is the speed. The time it takes from the raw data to
the final results is more than ten times faster than using the Q analyzer previously
used by the Bates Laboratory.
The one of important aspects of the data analysis is the understanding of the
apparatus. This includes the wire chamber decoding for both OOPS and OHIPS, the
optics of the spectrometers, the particle identification, OOPS and OHIPS focal plane
efficiencies, and target thickness. Some of these require a Monte Carlo program to
simulate and compare with the experiment data. In this chapter, we focus on the
wire chamber decoding, the spectrometer optics, the particle identification, the beam
polarization. In chapter 4, we discuss the Monte Carlo method which is used in this
analysis, and some issues about focal plane efficiency profiles for OOPS and OHIPS,
target effective thickness, acceptance, and radiative corrections.
3.2
Coordinate System
A coordinate system is defined as shown in Fig.3-1. For the OOPS spectrometer
focal plane coordinate, we define the z axis to be along the the central line of the
spectrometer and its direction is the same as the particle direction, and the x axis
to be along the opposite direction of the dipole bending direction. For the OHIPS
spectrometer focal plane coordinate, we define the z axis to be along the central line of
the spectrometer and its direction is the same as the particle direction, and x axis to
be along the opposite direction of the dipole bending direction. In target coordinates,
the z axis is along the beam direction, and the x axis is towards the floor.
The definition of
#
is the smallest angle between the particle trajectory and the
xz plane, and 0 is the smallest angle between the particle trajectory and the yz plane.
A useful quantity to define is 6, 6 =
, where P, is the central momentum for
P0
a particular magnetic setting, and P is the particle momentum.
92
Transverse
Plane
Dispersion Plane
y.
Focal PI ane
Coordin ate s
-.....
x
A
y
Target
Coordinates
-.
Centra Ray
A
X
Figure 3-1: Transport angle definitions and spectrometer coordinate systems
3.3
OOPS Analysis
Inside each OOPS spectrometer, there is a detector package which has three scintillators and three horizontal wire chambers. OOPS analysis identifies the incoming
particles as either 7r+ or p, and measures the position where the particles hit each
plane of the three chambers. From this information, the position and angles of a
particle trajectory are determined.
3.3.1
The wire numbers
When a particle passes through a drift cell, it ionizes the chamber gas. The free
electrons drift along the electric field lines towards the sense wires at a drift velocity
of about 50Mm/ns [54]. The time it takes for an electron to reach the sense wire is
proportional to the distance the electron travels. When the electrons reach the sense
wire, an avalanche forms which causes a small electrical pulse of about 5 to 10 mV.
This signal is amplified and then sent to a discriminator to produce a logic signal. It
93
travels through a delay line to a Time-to-Digital converter (TDC) module. The TDC
information is read out for each event by a microcomputer.
If n is the number of the wire that produces the signal, and each delay line element
has a constant delay time of r, the total time it take the pulse to reach the TDC is:
Tieft
=
tdrif + n --
Tright
=
tdrift +
(3.1)
+ T er,
(N - n)
-,r +
(3.2)
Tother
right
where Tieft and Tright are the times for the signal from two sides of the same plane
to reach a TDC, tdrift is the electrons drift time in the wire chamber, N is the total
number of wires in a wire plane. Tf*te' and T,,er are the times for the signal traveling
within the electronic circuits, which are constant.
The wire number is obtained by subtracting equation 3.2 from 3.1:
n =
2250
2000
1 -(Tie
- Tright)
+
-B
1
1
N-
(Tot her -o
t
(3.3)
her)
5000
4000
1750
1500
3000
1250
1000
2000
750
500
1000
250
A
-50 -40 -30 -20 -10
0
10
20
x plane time difference
30
40
0
50
ns
0
L - I
L
2
6
4
n L
8
10
12
14
16
18
20
wire number
Figure 3-2: X plane wire location and wire number spectra
In practice there are non-linear terms due to the dispersive effects in the delay
94
line. A correction is applied by adding a second order term to equation 3.3. The wire
number n can be written as:
(Tdiff) = ao + a1 - Tdiff + a 2 -Td2J
,
(3.4)
where Tdff is Tiet -- Tight. The procedure used to obtain the parameters ao, a, and
a 2 is:
" Use the Carbon target, set the spectrometer magnet to detect the proton in
(e, p) reaction.
" All the wire chamber planes inside the spectrometer are covered.
" Do the least square fit on the n(Tdiff) to the nearest integer n, get the parameters
ao, a, and a 2 .
We use the equation 3.4 to find out which wire is fired in this experiment.
3.3.2
Left and right decision
After we know which wire is fired, we need to know the particle passes through the
left or right side of the sense wire. This is done by analysis of the (0-E) signal. As
stated in Chapter 2, the (0-E) signal is the signal produced by the difference between
two neighboring ground wires.
There are three peaks in the (0-E) signal spectra. The left most peak is the
pedestal of an ADC, which is populated when the particle goes through the chamber
does not produce a strong (O+E) signal. The ratio of the number of events in the
pedestal peak to the total events is the inefficiency of (0-E) signal. We include this
inefficiency into the chamber total inefficiency. The next two peaks are used to decide
which side of the sense wire the particle passes through. If the event (0-E) signal is
in the left peak of these two peak, then it passes through the left side of sense wire,
otherwise it passes through the right side of sense wire.
95
O-E signal
4000
3500
3000
2500
2000 -I
1500
1000
500
0
0
200
400
600
800
1000
800
1000
x plane
4000
3500
3000
2500
2000
1500
1000
500
0
0
200
400
600
y plane
Figure 3-3: Typical OOPS O-E signal spectra
3.3.3
Meantime Correction
After we know which side of the sense wire the particle passes through, the next step
is to obtain the drift distance from where the particle passes through the plane of
the sense wires to the sense wire. Before we do the drift time to the drift distance
conversion, we need to understand the OOPS timing properties.
In the OOPS detector system, the start time for all TDCs in this experiment
was determined by the signals from the left arm of the second scintillator in the
detector package. The reference time varies depending on where the particle hits on
the scintillator. An appropriate correction is needed. Assume that Te ft is the TDC
value of the left arm signal from the second scintillator which will be constant since
96
the same signal starts and stops the TDC,
Tight
is the TDC value of the right arm
signal from the same scintillator, tj is the time for the photon to travel inside the
scintillator before reaching the left side photo-tube, and t 2 is the time for the photon
to reach the right side photo-tube. Due to the constancy of the speed of the photon
inside of scintillator,
t 1 +t
2
Tieft
Tright
constant = Co.
=
=
(3.5)
constant = C 1
t 2 +C
2
-t
1
-C
1
=t
2
-t
1
+C
3
(3.7)
=Co - 2 -t + C3
S-2-t
1
+C
4
(3.6)
(3.8)
.
We want the start time to be Tieft + t1 . That can be accomplished by:
Tieft + t1 = Tet
e
+Tright + C5,
(3.9)
where Co, C1, C2, C3, C4, C5 are constants. It means for all the TDC values in the
OOPS detector, the mean time of (Tieft
+ Tright)/ 2
should be subtracted to obtain
the position independent fiducial time of an event.
3.3.4
Converting drift time to drift distance
The drift time is calculated by taking the sum of Tie ft and Tright for the same plane.
The drift time is found by adding equation 3.2 and 3.1:
TSM = 2 - tdriet+ N sum rtf
+ (T
oter
eft
+_ rtr)
'right
(3.10)
In order to get the drift distance from the drift time, the wire chamber is calibrated
during a quasi-elastic carbon run. Under this kinematics, the cross section is almost
flat across the focal plane. We now can safely assume the particle distribution is
uniform in each wire chamber cell. Fig.3-4 shows a typical drift time histogram. The
97
700
600
500
400
300
200
100
- -'..,.-
0
0
200 400 600 800 1000 1200 1400 1600 1800 2000
time sum
Figure 3-4: Typical OOPS HDC Drift time spectra
drift time histogram is a plot of dN/dt, the number of events per time. The shape
of the spectrum results from the variation of electron drifting velocities inside a wire
chamber cell:
dN
dN dx
dt -
dx
=
const -
.
(3.11)
By integrating Eqn.3.11, we obtain the relation between the drift distance and the
drift time:
t dN
Dt
Dt Dmax
a
=
Ljo dt,
-~a
d,.N
fto
dt
(3.12)
where Dmax is the width of a cell which is 4.064 mm, and Dt is a function between
the drift distance to the drift time as shown in Fig.3-5
3.3.5
Determination of Wire Plane Coordinates
Once the position of the particle track is known for each of the three chambers, the
location and angle of this track is calculated by making a straight line fit to these
three points. The precision of alignment of the chambers is important. Hardware
alignment is carefully done but the ultimate alignment is done in software with data.
98
Distance vs lime
E
E
---
4
--
3.5
-..-- .....-
. ..
...... ..... ... ....
0....................
50
100
--.. ----.. -.. --..
.. --..
.. ..
-...
.................
150
.1
200
250
ns
Figure 3-5: Typical drift distance versus drift time
The position of the chambers is relative to the central ray of the spectrometer.
Toward the end, a series of (e, e') experiment on
12 C
elastic scattering were performed
for each OOPS spectrometer. The beam was 200 MeV, the scattering angle for OOPS
was 89.6*. The OOPS magnetic field is set to detect elastic electrons at 197 MeV/c.
A sieve-slit with 13 holes was set up in front of the OOPS spectrometers. The center
hole on the sieve-slit was on the central ray of the spectrometer. The center of the
image of the electrons passing through the center hole in each wire chamber plane
defined the origin of each wire plane. The physical offset of each plane X and Y was
obtained by this method. The Z offset was obtained by a physical measurement of
the detector. The results are in Table 3.1.
We define the resolution in the X dimension as XD and the resolution in the Y
dimension as YD. They are:
+
(X 1 -_Xfit)2 + (X 2 - Xflt)2
2
2
XD
YD
=
S(YY
1 -
)2+(Y2
-Y
2
(X
(
-
Xfit)2
)2 +(Y 3 -yfi)2
,
,XV
(3.13)
(3.14)
where X 1 , X 2 , X 3 are the measured X positions in three planes, and Xf", Xfi, Xf"t
99
Table 3.1: Offsets for the OOPS HDCS. The units are cm in offsets and mr in the
rotations.
OOPS
A
A
A
A
A
A
B
B
B
B
B
B
C
C
C
C
C
C
Chamber
1
1
2
2
3
3
1
1
2
2
3
3
1
1
2
2
3
3
Plane
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
Offset
-0.32
0.28
-0.30
0.44
-0.36
0.49
-0.40
0.21
-0.53
0.15
-0.74
0.20
-0.39
0.29
-40
0.17
-0.37
0.25
100
Z-Offset
0.0
0.953
12.7
13.653
25.4
26.353
0.0
0.953
12.7
13.653
25.4
26.353
0.0
0.953
12.7
13.653
25.4
26.353
Rotation
0.0
0.0
-2.6
-2.6
0.0
0.0
0.0
0.0
4.75
4.75
0.0
0.0
0.0
0.0
0.69
0.69
0.0
0.0
35000
30000
40000
j/n&2822E+05/ 197
Constant 0.3290E+05
0.2784E-02
Mean
Sigma o0.6842E-02
35000
-
30000
25000
2
25000
20000
20000
15000
15000
10000
--
10000
5000
0
197
X/n&3520E+05/
Constant 0.3378E+05
0.3060E-02
i Mean
0.6435E-02
Sigma
5000
-0.2 -0.15 -0.1 -0.05
0
0.05
0.1
0.15
0
-0.2 -0.15 -0.1 -0.05
0.2
0
0.05
0.1
0.15
0.2
YD
XD
Figure 3-6: (1)Typical X Plane Resolution (2)Typical Y Plane Resolution
are the fitted X positions in three planes; Y1 , Y2 , Y3 are the measured Y positions in
three planes, and Yfit Yfit Yfi
are the fitted Y positions in three planes. In the
Fig.3-6, XD and YD are plotted for a typical experiment run. The mean of the XD
and the mean of the YD spectra are centered at zero with an error less than 30 pm.
The FWHMs of XD and YD are less than 160 pm. These results mean that the angle
and position resolution of the OOPS detector in the focal plane are:
3.4
A(9focal)
=
+0.75 mr,
A(Xfocal)
=
+163
A(#focal)
= ±0.75 mr,
(3.15)
pm, A(Yfocal) = +165 pm.
Optics Studies
The established method in obtaining spectrometer optics parameters has been documented in several theses [49, 55, 39]. Here we just outline some important points.
The target quantities Q(P, Xt, Y, Zt, 9 t, #t) are related to the focal plane quantities
in a Taylor expansion:
Q = {
igf f~,
Qijik f Ykq
101
(3.16)
where Qijk are the optical matrix elements of the spectrometer, or sometimes QijkI
are called reverse matrix elements. Since there are only four independent focal plane
quantities, we can not get six independent quantities in the target. We controlled
the electron beam position so that the beam position in the x and z position at the
target were zero for the optics measurement and the experiment.
Because there exists a middle plane symmetry around the x-z plane the matrix
elements in 6 and Ot will be zero when k is odd, while those in
#t are
zero when k is
even.
To get the best set of the optical matrix elements Qijkl, one needs to collect a full
set of data with knowledge of the target coordinates and the momentum, covering the
full range of the acceptance of a spectrometer. Since one needs to know the incoming
particle energy precisely, "C(e, e') elastic scattering was used to calibrate both OOPS
and OHIPS optics.
As documented in section 3.3.5, a sieve-slit was also used to measure the matrix
elements for both OOPS and OHIPS. This involves placing a collimator with an array
of holes in the snout of spectrometer. Particles passing through the sieve-slit holes
arrive at the spectrometer focal plane at locations which correspond to the angular
positioning of the holes. Since the angular positions of the sieve-slit holes are known
in term of the target coordinates, one can determine the mapping of measured focal
plane variables back to the target variables at the hole position.
Fig.3-7 displays a histogram of
Vfocal
versus Ofocal for OOPS. The image of the
sieve-slit holes is clear. The OOPS optics achieve good resolution (Appendix D) in
both the in-plane (#) and out-of-plane (0) angles.
In Appendix D, we list the matrix elements used for OOPS and OHIPS in this
experiment. The measured carbon elastic peak spectra are shown in Fig.3-8. The
x-axis is 6 in the unit of percentage, and the y-axis is the event counts. The OOPS A
spectrum was obtained under the following condition: The beam energy is 345 MeV,
the 6 = 330,p = 750 and P = 337.28 MeV/c. The OOPS C spectrum was obtained
under the following condition: The beam energy is 345 MeV, the 6 = 330, Op = 50.40
and P = 341.10 MeV/c. The FWHMs of the elastic peaks for both spectrometers
102
2D
Z12
D
40
0
0'
-ao
-800
Figure 3-7: Image of OOPS sieve-slit collimator in the focal plane. Figure courtesy
of Alaine Young, Arizona State University
are around 1.1%.
3.5
Particle Identification
Some reactions produce particles such as 7r+, 7r- and e+ which are indistinguishable
from protons or electrons using only information on their momentum, charge and
relative timing between the OOPS and OHIPS events. To ensure that the events we
study are indeed (e, e'p) events, particle identification is required in both OOPS and
OHIPS.
103
500
FWHMI,stic = 1.1*10-2
400
300
200
K
100
-
20
-17.5
-15
-12.5
-10
-7.5
-5
-2.5
0
2.5
5
-2.5
0
2.5
5
OOPS A delta for "C(e,ez)
3000
2500
FWHMOcti
2000
= 1.09*10-2
1500
-
1000
500
n
-20
-17.5
-15
-12.5
-10
-7.5
-5
OOPS C delta for '2 C(e,ez)
Figure 3-8: Typical Carbon elastic peak are obtained in OOPS A and OOPS C.
3.5.1
OOPS Particle Identification
The OOPS spectrometer detector package is designed to detect hardons. Apart from
protons, 7r+ and deuterons with the same momentum could also enter the OOPS
spectrometer. The Bethe-Bloch equation [52] for the energy loss of charged particles
passing through materials is
dE
pdx
where
f
Z 1 2my 2 3
= 0.307-
A
2
(1n
I
2
-
2)
,
(3.17)
and y are the usual relativistic quantities, I is the atomic ionization po-
tential, and p, Z and A are the target material density, charge and atomic number
104
respectively.
Because of the large difference in mass between 7r+, p and d, their 3 are quite
different for a given momentum. Each deposits a different energy in the scintillators.
From the AE spectra, one can separate 7r+, p and d in the OOPS detector. In Fig.3-9,
the x-axis is the average scintillator pulse height in the second scintillator (S2), the
y-axis is the average scintillator pulse height in the third scintillator (S3), and the
z-axis is the number of events. In this experiment, the final trigger requires the triple
coincidence of three scintillators. Since deuterons with momentum of 510 MeV/c stop
in the second scintillator, there is no deuteron signal in Fig.3-9.
Particle Identification
20000
17500
15000
12500
10000
7500
5000
2500
1200
0 200
40060
1000
800i
800
600
20n400
1009
'1400
0
200
Figure 3-9: Typical OOPS average scintillator pulse height in S2 versus in S3 in XY
axis, and event counts in z axis
105
3.5.2
OHIPS Particle Identification
The setup OHIPS detector package consists a gas Cerenkov detector and two layers
of lead-glass to discriminate between e- and
7r-.
A Cerenkov detector is used to differentiate e- from 7r-. It has a gas path of 1.4
meter, photo-multiplier tubes and mirrors to reflect the Cerenkov light to the photomultiplier tubes. The tank is filled with isobutane gas at an atmospheric pressure.
The index of refraction of isobutane is n=1.0013. The threshold /3 is
S-=
1
n
0.9987.
(3.18)
The energy threshold is 10 MeV for electrons, 2.0 GeV for p- and 10.0 GeV for
7r-. Only electrons produce the Cerenkov radiation in this experiment.The efficiency
of the Cerenkov detector was extensively checked. The detailed method and results
can be found in the thesis of Dr. X. Jiang [51]. We quote the efficiency was 99.5%.
The lead-glass detector measures the amount of Cerenkov light generated by the
a charge particle passing through it. The approximate dimensions of each block are
10 cm wide by 25 cm long by 10 cm deep. 14 of the lead glass blocks were used
in this experiment. The lead glass has a density of 5.18 g/cm 3 , index of refraction
of 1.80, and a radiation length of 1.68 cm. A PMT views each lead glass block.
The Cerenkov threshold momentum for electrons is only 0.34 MeV/c. The Cerenkov
threshold momentum for pions is 161.2 MeV/c. The lead glass blocks are useful for
cosmic background rejection in low count-rate experiments. We used the lead-glass
blocks to help confirm a valid particle event by recording the pulse height of the
hardware sum of the PMT signals, as measured in an ADC.
Fig.3-10 (a) is two-dimensional ADC sum histogram of the first layer lead-glass
versus the second layer lead-glass. Separation between the e- and 7r- are clear. Fig.310 (b) is a sum of all lead-glass ADCs. Fig.3-10 (c) is Cherenkov ADC sum under the
condition that the sum of all lead-glass ADC is less 900 ( 7r-). All the signals are at
channel zero. It means that 7r- at that momentum can not produce Cherenkov light.
106
3.6
One- Per-Beam- Burst Correction
In this experiment, the electronic logic circuit processed at most one event for each
beam burst. A correction for possible multiple events in one beam burst is applied.
There are two methods which could obtain the correction. In the hardware setup, we
had two scalers. One recorded the total events before the one-per-beam-burst veto
was applied, and the other one recorded the total events after the one-per-beam-burst
veto (see Chapter 2, section 2.8.5) was applied. These two numbers in the scalers
were recorded in the data stream for every run. The ratio of these two numbers is
the one-per-beam-burst correction.
Another approach to obtain the one-per-beam-burst correction is based on the
theory of probability. One assume that the events happen uniformly within a beam
burst. As a result, the number of the events occurring within one beam burst is
distributed as a POSSION distribution [52] in time. If P(n) is the probability of
finding n counts in one beam burst, and r is the number of the real average counts
within one beam burst, then:
Pr (n) = rner
(3.19)
n!
If nmeasure is the measured average number of events in one beam burst, then:
nmeasure
=
=
Z 1 - P(n) =
e-r(er
-
E
1
(3.20)
(2
1) = 1 - e~
Then the correction factor between the nmeasure and the real average number of events,
r, is
=-
r
=
ln(1
-
nmeasure)
nmeasure
C -nmeasure .
(3.21)
(3.22)
In this experiment, nmeasure is around 0.15, and the correction is about 1.08. Depending on the beam current, this number may vary by a few percent.
107
Table 3.2: The One-per-beam-burst Correction Factor
Run Number
2004
2010
2012
3141
nmeasure
0.147
0.153
0.201
0.105
±
±
±
±
0.003
0.004
0.005
0.001
Measured Factor
1.077 ± 0.006
1.086 ± 0.008
1.121 ± 0.009
1.062 ± 0.003
Calculated Factor
1.082 ± 0.004
1.085 ± 0.005
1.124 ± 0.007
1.056 ± 0.002
We compare these two methods in each run, Table 3.2 is the typical result (All
errors are statistical errors). The difference between the measured correction factor
and the calculated correction factor is around ±0.5% within the statistical errors. We
use the measured correction factor as the one-per-beam-burst correction factor in the
data analysis in this experiment.
3.7
Beam Polarization Determination
The beam polarization is determined by measuring the asymmetry in the counting
rates in the both Cerenkov detectors in the Moller spectrometer as the beam helicity
is changed,
AN = Y+/Q+ + Y-/Q-
(3.23)
Y+1Q + + Y -1Q-
where Y' and
Q±
are yield and charge for each helicity, and AN is the asymmetry.
If the beam polarization is the along the beam direction, we have (See Section
2.5)
AN
=
7
PBPT COS OT
9
,
(3.24)
where the PB is the beam polarization, PT is the Moller target polarization, and the
9T is the angle of the target foil and the incident beam.
In this experiment, the
target electrons are polarized with a 150 Gauss magnetic field using Helmholtz coils.
The target polarization was found to be 8.02±0.12% in an earlier measurement at
MIT-Bates [4] and 97 was 300 in this experiment.
108
The measured asymmetry,
4
measure was diluted by the background.
AN
Ameasure =1 + B/S
(3.25)
where S is the rate from the Moller scattering and B is the background rate. Then
the beam polarization can be written as
_
Ameasure (1 + B/S)
BTP cos
(.
A quadrupole magnet field scan was performed to find out the signal to noise ratio
and the position of the Moller scattering peak. The procedure was called the "real
scan
The normalized yield curve is fitted to the function
Y(x) = S - exp[ (-l~7A)P
W
P
+ B(x)
,
(3.27)
where x is the quadrupole relative voltage, B(x) is a linear background function, S
is the signal amplitude, and w,c and p are fitting parameters which determine the
shape of signal. The result is shown in Fig.3-11. In Fig.3-11, the peak position (called
CENTROID) is to be at 0.6040 shunt voltage, the signal at 0.6040 shunt voltage is
0.3167, and the background is 0.1680. Thus the signal to noise at the peak position
(S/B) is 1.88.
After the real scan was completed, the peak position and ratio of the signal to
the background (S/B) were determined. A quadrupole magnet field was performed
again near a narrow range which contained the Moller peak. The asymmetry shown
in Fig.3-12 is the asymmetry of the measured yields of two helicity states of the
electron beam in the Cerenkov counters. This asymmetry is called the pulse-pair
asymmetry. The pulse-pair asymmetry then is fitted by a guassian function to find
the maximum asymmetry which is 1.41% in Fig.3-12.
After S/B correction, the
measured asymmetry Ameasure is 2.16±6% in Fig.3-12. From the Eqn. 3.26, the
beam polarization is obtained which is 40.0+1.2%.
109
Table 3.3: Systematic uncertainties in electron beam polarization
Description
APB
Beam position fluctuation
Target thickness uncertainty
Target polarization uncertainty
Target angle uncertainty
Total system error
2.4%
1.4%
1.5%
3.0%
5.0%
Throughout the entire experiment, the beam polarizations were measured at least
once every other day. The measured beam polarization in this experiment is in the
Fig.3-13. The Fig.3-13 represents around 50 days of runtime.
The systematic errors for the polarization determination are listed in the Table 3.3.
After all major sources of systematic uncertainties are combined together, the error is
estimated to be 5.0% in the beam polarization. The final average beam polarization
for this experiment is 38.6 ± 1.8± 5.0 %, where the 1.8% is the statistical error and
5.0 is the systematic error.
110
100 75 i 50
0 25-00
-
...............
6
500
1000
First layer lead-.gj 0 5 AD sum 1500
1000
750
0B
00
400 NvG
0 5
2000
200
s
O(N
e
- "C(e, e-) Q.E. region
EO = 750.6 MeV
- 4. = 79-7
(b)
C
o 500 250 0
2000
1500
1000
500
0
PBGSUM (ADC sum of all lead-glass blocks, in channels)
4000 -W(d
4000(c)
3000
(d)
pbgsuma9OO
-pbgsum<900
4a-
C
C2000
200
1000
0
0
500
1000
1500
0
0
500
1000
1500
Cherenkov ADC sum (channels)
Cherenkov ADC sum (channels)
Figure 3-10: OHIPS particle identification. a) Two-dimensional ADC sum histogram
of the first lead-glass versus the second lead-glass. b)Sum (pbgsum) of all lead-glass
ADC. c) Cerenkov ADC sum with pbgsum < 900. d) Cerenkov ADC sum with
pbgsum > 900. Note no Cerenkov signal for e- events can be found in c).
111
.5
N
E
0
.4
-
.3
-
0.3167(50)
=
0.1680(12)
=
PEAK SIGNAL
NOISE
1.88(4) = S/B
0.6040(6) = CENTROID
'
9
,
0
.2-2
0\
z
A. -1
i
.38
.25
I
i
.64
.51
(V)
Voltage
Shunt
.77
.90
Figure 3-11: Typical quadruple real scan in Moller polarimeter, the x-axis is the
relative voltage on the quadruple magnets called the shunt voltage
2.0
2.16(6)% = ASYMMETRY
(S/B corrected)
40.0(1.2)% = BEAM POL
0.80
E
E
0.4
-0.4
= x 2
-
-
ci)
(I3
d-
-2.0
1
.30
.42
I
.66
.54
Shunt Voltage (V)
.78
.90
Figure 3-12: Typical quadruple peak scan in Moller polarimeter, the x-axis is the
relative voltage on the quadruple magnets called the shunt voltage
112
70
05
0
60
N
50
40
-
I
I
~
I
I!
LI
302010
01
0
I I
III
5
I
I
I
I
I
I
15
10
I
I
I
20
I
I
I
I
25
Run Number
Figure 3-13: Measured beam polarization against run to run
113
I I
Chapter 4
Monte Carlo simulations and
normalization
The purpose of using Monte Carlo simulations in the analysis is to understand the
acceptance properties of apparatus. One is concerned about the focal plane efficiency
profile, the phase space volumes for both single and coincidence kinematics, the acceptance profile of the extended target, and the energies lost in various processes.
The Monte Carlo used in this analysis is based on the AEEXB program originally
developed by Dr. Joe Mandeville [49], and later was improved by Dr. Costa Vellides
[56]. This program has two main parts. One is the event generator which generates the desired physical events, the other is the TURTLE [57] model which traces
the generated particles from the target to the focal plane instruments. The event
generator takes into account the effects of multiple scattering, ionization energy loss,
and electron radiation in the field of the same target nucleus from which it scatters.
The details of AEEXB is in the manual for AEEXB [56]. AEEXB is a program for
simulation of coincidence electron scattering experiments of the type A(e, e'x)B. The
important features of AEEXB include:
" It stimulates magnetic spectrometers within the context of the optics program
TURTLE.
* The events are generated according to any theoretical cross section model on
114
an event-by-event basis.
The event generator is the key aspect of the program. It processes events in the
following way:
1. Two transport rays, one for the scattered electron and one for the coincidence
particle X, are sampled within a specified acceptance about their respective
central rays. In addition, the incident electron helicity (± 1), the Cartesian
coordinates of the reaction point along the target length and beam diameter
are randomly sampled.
2. The five structure functions of whatever a theoretical model is used, tabulated
over a 3-dimensional grid in the independent dynamical variables (ef, 6 e,, Opq),
are interpolated for any particular kinematic point using either a cubic spline
or polynomial interpolation.
3. Energy loss due to ionization, multiple scattering, and electron bremsstrahlung
in the target is included for the incident and scattered electron and particle X.
4. There exists options which do or do not include the radiative processes.
4.1
Spectrometer models
A modified version of TURTLE is used to model both the OOPS and the OHIPS
spectrometers. The main purpose of using a spectrometer Monte Carlo model is to
trace charged particles through elements of the spectrometers on an event-by-event
basis to see if they are stopped by any internal aperture. Since neither OOPS nor
OHIPS has an acceptance solely determined by the front collimators, the determinations of their acceptance have to rely on realistic spectrometer models. Also since
many details of the spectrometers and detector packages have been changed over the
years, dimensions and major elements such as the baffles, vacuum pipes , entrance and
exit windows, and collimators were measured again to rebuild the TURTLE models.
115
In Appendix E, the TURTLE models of OHIPS and OOPS used in this simulation
are listed in detail.
Although each of the OOPS spectrometers may have slightly different properties
in terms of their magnetic settings and detector alignments, differences in optical
properties and acceptances are found to be small between different OOPS modules.
These properties are discussed in the reference [24] and the differences in the optical
properties between OOPSs are listed in Appendix D. For this reason only one generic
OOPS model is used in the simulation and the design value of the relation between
currents and magnetic field of OOPS dipole is used. The magnetic setting of OHIPS
is taken directly from the field measurement.
If a simulated trajectory reaches the final instrument of the detector package, it is
an accepted event in the simulation, otherwise it is a rejected event. To simulate the
finite resolution of the spectrometers, the TRANSPORT coordinates (See Section 2.1)
at the target position are smeared randomly according to a gaussian distribution with
widths determined by the resolutions obtained from the optics studies (See Section
3.4).
4.2
OHIPS single arm acceptance
Single arm (e, e') cross sections for the elastic scattering from
12
C were measured
during the commissioning of the OHIPS spectrometer. A position scan of the carbon
elastic peak across the OHIPS focal plane was performed to study the variation of
the focal plane efficiency at different relative momenta.
The OHIPS single arm
12
C(e,
e') data were collected at the beam energy of 345
MeV with the OHIPS spectrometer set at Oe = 33.4 with respect to the beam line.
The thickness of the carbon target was 24.2 mg/cm2
4.2.1
The solid angle of OHIPS
The first thing found in the data is that the OHIPS solid angle is not defined by the
front end collimator. If the solid angle were define by the front end collimator, it
116
should be 5.16 msr with a horizontal opening angle ± 23.5 mr and a vertical opening
angle ± 54.6 mr. From the measured cross sections based on this value of solid
angle yield 20% lower than the well known cross section values [51].
Furthermore
the AEEXB Monte Carlo simulation showed that the entrance of the dipole magnet
defines the transverse opening angle: the horizontal openning angle is instead of ±
23.5 mr, it becomes ± 19.9 mr. Our data support this finding by comparing Fig.4-1
which is obtained from the quasi-elastic
12 C(e,
e') with Fig.4-2 which is obtained from
the Monte Carlo stimulation.
23.5 mr
19.9
9.9 mr
mr
900
23.5 mr
800
700
600
500
400
300
200
100
0
Ii
-50
-40
-30
-20
-10
liii'
0
111111
10
20
30
40
50
OHIPS Horizental Opening
Figure 4-1: OHIPS horizontal opening angle measured
4.2.2
OHIPS focal plane efficiency profile
The OHIPS focal plane efficiency profile was obtained through the cross section measurement of 12C(e, e') elastic reaction. The code ALLFIT [22] is used to extract the
intergrated area of the elastic peak and take the radiative corrections into account.
The program performs the least x 2 fit to each peak using a predefined function. The
117
600
Monte Carlo
500
400
300
200
100
0
-50
-40
-30
-20
-10
0
10
20
30
40
50
OHIPS Horizontal Openning Angle
Figure 4-2: OHIPS horizontal opening angle in Monte Carlo
function is an asymmetric Gaussian function with 10 parameters. A resolution function of the spectrometer convoluted with a theoretical radiation tail is built into the
fitting routine. AllFIT also takes the inputs of the solid angle, target thickness, beam
charge, and total correction factor to calculate the experimental cross section of the
elastic peak.
The expected 12 C(e, e') elastic cross section of the ground state is obtained from
the code ELASTB [51] and was averaged over the acceptance of OHIPS. The code
ELASTB interpolates carbon form factors between existing data points and predicts
the corresponding
12C(e, e')
elastic cross sections. The absolute OHIPS focal plane
efficiencies can be determined by taking the ratio of the measured 12 C(e, e') cross
sections over the predicted value from ELASTB.
From the Table 4.1, it is clear that the OHIPS focal plane profile is very flat across
118
Table 4.1: Cross sections of the OHIPS 12C(e, e') measurements. Cross sections in
(pb/sr) for the ground state are listed with statistical errors only.
Run #
Q
Etrack
R780
R781
R782
R783
R784
R785
R786
R787
ELASTB
0.398
0.255
0.249
0.248
0.237
0.223
0.231
0.209
0.901
0.888
0.917
0.929
0.931
0.932
0.933
0.933
I
I
6 ground
state
-3.27
-2.31
-1.32
-0.34
0.59
1.66
2.73
3.57
1
_
23.76
0.12
23.48 ± 0.11
23.22 t 0.12
23.63 ± 0.13
23.45 ± 0.10
23.42 ± 0.12
23.15 ± 0.12
22.82
0.11
24.15 [51]
the focal plane (97.3%). The small difference between the measured cross section and
the predicted value could be the error from the ELASTB Program. No correction is
applied for this difference.
4.3
OOPS Single Arm Acceptance
The OOPS single arm acceptance is more complex than OHIPS single arm acceptance.
There are a few reasons for that. One is due to the dipole quadrupole configuration:
the dipole quadrupole configuration introduces a non-uniform shape in the momentum
acceptance, and the other is due to the baffles inside the dipole magnet which are
designed to eliminate the low energy particles [22]. The baffles interfere with the edge
of focal plane images.
119
OOPS-A
50
' 2 5
0
------
--------
fl-g
-
-25
1
-50
hiif
50
O P-
-
-
0
-
-25
WIT
-1
-50
rMtc
50
0
Carlo
mtronte
fr ont coll
-25dioe-
t;I,.
front collimator
--
-
-50
-30
-20
-10
0
10
20
30
Figure 4-3: OOPS spectrometer angular acceptance. Angles at the target in the
dispersion direction are plotted against the relative momentum 3 for QOPS and compared with the Monte Carlo simulation.
4.3.1
OOPS focal plane efficiency profile
Since the QOPS focal planes are not uniformly efficient, the focal plane efficiencies
have been mapped. The relative focal plane efficiencies are determined by the following procedure (The profile of absolute focal plane efficiencies is determined by
measureing a well known reaction cross section such as H(e, e'p). See Section 4.5):
120
"
Measure a reaction with a smoothly varying cross section at different momentum
settings of the spectrometer.
" Decouple the results of the measurements into a relative efficiencies in the focal
plane and cross sections c-(P) which is explained below. The idea is from the
fact that by changing the momentum setting of the OOPS dipole, the same cross
section of the reaction is measured by many different positions in the focal plane,
therefore the profile of the relative focal plane efficiencis is obtained.
The carbon target was used in this experiment. The kinematics is set at quasi-elastic
peak. The OOPS spectrometers are set to detect the electrons. The cross sections of
different momentum setting were measured.
We used the program RELEFF [46] to decouple the focal plane efficiencies from the
(e, e') spectrum. RELEFF approximates the cross section as the sum of polynomials
ft
up to order n:
anfn(pij)
i =
(4.1)
.
n
here o-ri is the cross section and pij is the momentum of the ith channel for the
measurement.
The polynomials
fn
jth
can be either regular polynomials of the form
X" or Legendre polynomials. The Legendre polynomials are used in this case. The
coefficients an are then varied by minimizing x 2 where
(
2=
-
N
- ei)2
.
(4.2)
ii
Cij is the number of measured counts, and wij is the statistical weight in channel i for
the run j, Nj is the normalization factor for the run j and ei is the relative efficiency
of the channel i. An iterative procedure is used to determine Ei and the coefficients
an. The E, are initialized at unity and x 2 is minimized with respected to an:
iDan
= 0, foreveryn.
121
(4.3)
This gives n linear equations which determine a,:
Xn = m Mrn am = 0 ,
(4.4)
where
Xn = E WijCi'EijNj
fn(Pi) ,(4.5)
and
wij (eNj)2 fm(ij)fn(pij) .
Mmn =
(4.6)
ii
The coefficients am are found by inverting the matrix M and calculating M-'X.
With these coefficients new efficiencies are computed:
ei=
Ej Cij(47
.
(.7
E j En Njanfn(Pij)
Note that this procedure preserves the total number of counts. The Ej determined
are put back into equation. This procedure is repeated until x
2
converges.
The relative focal plane efficiencies are measured and compared with the monte
carlo stimulation of spectrometers. Fig.4-4 is a typical OOPS focal plane efficiency
and its Monte Carlo spectrum.
The Monte Carlo simulation agrees with the ex-
perimental result very well. It means that the Monte Carlo model (AEEXB) can
reproduce the properties of spectrometers.
122
OOPS Focal Plane Profile compare with Monte Carlo
1.2
Monte Carlo
1
Data
0.8 -Real
0.6
0.4
0.2
0
-20
-15
-10
-5
5
0
10
15
20
25
odelta
Figure 4-4: A typical OOPS focal plane efficiency profile comparing with Monte
Carlo simulation.The thin line is the Monte Carlo simulation, and the thick line is
the measured focal plane efficiency profile
4.3.2
Extended Target
A cryogenic liquid target in a cylinder was used in the experiment OOPS modules
viewed the target from different angles. As a result, the extended target affects
the acceptances of the individual OOPS sepectrometers and this difference must be
calibrated.
The extended target response of each OOPS module was measured by detecting
123
quasi-elastic electrons from a slant carbon target. Motion of the target ladder up
and down translated into the movement of the interaction point of the electron with
the carbon target along the beam line. The measured relative efficiencies of the focal
plane for OOPS A, B and C, compared with the Monte Carlo simulation, are shown
in Fig.4-5.
Q)
1.2
C
C)J
**
0.9
0.8
-r-L
0.7
* OOPS A
0.6
A
OOPS B
OOPS C
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Y, in cm
Figure 4-5: The measured OOPS extended target efficiency, compared with a TURTLE simulation (solid line)
The extended target response of OHIPS module was measured by detecting quasoelastic electrons from a slant carbon target. Motion of the target ladder up and down
124
translated into the movement of the interaction point of the electron with the carbon
target along the beam line. The measured relative efficiencies of the focal plane for
OHIPS, compared with the Monte Carlo simulation, are showin in Fig.4-6
1
U
C
>1
U
a)
-e
0.9
wD
Liquid Target Diameter
0.8
-K
0.7
0.6
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Yt in cm
Figure 4-6: The measured OHIPS extended target efficiency
In this experiment, we used a small diameter target cell: the diameter of liquid
target was 1.6 cm. Because of the variation of the beam position, the effect length
of the liquid target changes from run to run, affecting the luminosity. Throughout
the experiment, we tried to measure the beam position by using the beam position
monitors. Unfortunately, the device we used for the beam position monitor was not
125
always reliable. After this experiment, the spot on the target wall from the electron
beam was measured. It was centered along the target cell with a diameter of 2 mm.
Since the OHIPS angle acceptance is larger than the target cell, we found the
OHIPS single rate over the charge was an excellent tool to measure the relative target
length. The yields in OHIPS single rates are proportional to the relative target length.
From the rate of the yield in each run to the maximum yield of the total runs, the
correction factor of the target length variation is obtained. In this experiment we
have the following yields in Fig.4-7 from run to run:
Relative Luminosity
1.2
1
---
*0*~+~I
@
O
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
Figure 4-7: Relative target length assuming the target diameter is 1 unit of length.
126
4.4
Coincidence Acceptance Simulation
The acceptance for the single arm experiment is the product of solid angle of the
spectrometer and its momentum acceptance. It is calculated by AEEXB as
AZeAe
-AZsingiearm -
(
NA
where A6e and -Aqe are electron angle sampling ranges specified in the AEEXB parameter file, NT is the total number of events generated and NA is the number of
acceptable events in the spectrometer.
The coincidence acceptance is calculated as
AQcoin
where zAO, and
ZA\/
-
zAw /AOeZA
eAOP
.
-P
.
(4.9)
are proton angular sampling ranges, AWe is the electron momen-
tum sampling range.
AEEXB also simulate extended target effects.
For an extended target, the ac-
ceptance has one more physical dimension, the target length. It is evaluated by the
Monte Carlo simulation as
NA
< Q, >= ALAweAeAeAOPAqP
NT
(4.10)
where AL is the extended target length. The extended target efficiency is folded
into the simulation. All the extracted asymmetries and structure functions from the
experiments are not directly comparable with the theoretical calculations, because the
extracted values from the experiments are averaged over a range of a finite acceptance.
In order to compare the extracted asymmetries and structure functions from theoretical models, AEEXB is used to fold theoretical calculations over the experimental
acceptance.
In addition, the same Monte Carlo simulation is used to average the
radiative correction factors and kinematics factors, which are needed for the response
function extractions.
127
4.5
Radiative Correction
As stated in Chapter 1, radiative corrections are required in electron scattering for
accurate comparison experiment cross sections in (e, e'p) with un-radiated theories.
The radiative processes add tails to peaks and reduce the cross sections observed in
the peaks. Useful articles on this subject can be found in references [58, 28, 59].
Radiative corrections can be separated into three main categories:
o Internal Bremsstrahlung: Emission of real and virtual photons in the field of
the scattering nucleus.
o External Bremsstrahlung: Emission of real photons before or after the main
scattering in the presence of other nuclei.
o Landau Straggling: Energy loss due to collisions with atomic electrons in the
target.
Protons can also radiate. However the intensity is inversely proportional to the
square of the mass of the particle [28]. This effect is very small for this experiment
and is ignored.
4.5.1
Internal Bremsstrahlung
The Feynman diagrams for internal bremsstrahlung are given in Fig.4-8. The corrections for internal bremsstrahlung were first calculated by Schwinger [60] and later
improved by Mo and Tsai [28]. The diagram (a) corresponds to a real photon emission before or after the scattering. Because a real photon is emitted, the kinematic
condition of the reaction is altered. The correction factors for diagram (a) depend on
the interval of the missing energy, AE. A smaller AE corresponds to a larger correction factor. In diagram (b), the radiative process will not change the kinematics
before or after the reaction, but they have an overall renormalization of the vertex
and thus change the cross section.
128
a)
e+
b)
e-
Figure 4-8: Feynman diagrams for radiative process
The equations of the internal bremsstrahlung are:
Cintbrem
d3oorracorr
exp
(4.11)
=1 -6' ,
d3 Oexp
-Clntbrem
(4.12)
de~
where -r,acor is the cross section which has been corrected for the internal bremsstrahlung
correction. The raw experiment cross section is
cexp. CExtbrem
is the internal bremsstrahlung
correction factor. The other parameters are:
'2a
13
7r
12
= -6(AE)
=
(21n
2a 1n /
7r
AE
17
2+ -36 + Iln
4
ef
1_~_
Me
(21n
Me
2
1 r
26
- L 2 (cos 2)]
2
(4.13)
(4.14)
) .
The Spencer function L2 is given by:
L 2 (x)
=
jx 1n(1
Sy
-
dy
(4.15)
Also ej and ef are the incident and scattered electron energies, Me is the electron
mass and a is the electromagnetic fine structure constant. We average the internal
129
Table 4.2: Internal Bremsstrahlung Correction
AE
8.00 MeV
Internal Bremsstrahlung Correction
1.2161
bremsstrahlung correction formula over the experimental acceptances using a Monte
Carlo technique (Chapter 4).
We used AE = 8.00 MeV for the missing energy region of integration, and the
result in internal bremsstrahlung correction,
4.5.2
Cntbrem,
is 1.2161.
External Bremsstrahlung
We used the external bremsstrahlung correction formula which is given by Friedrich
[58]. This is based on an approximation of the probability for the electron to radiate
one photon. This probability can be integrated analytically to give an approximate
one photon correction factor:
1EZ-A.E
CExtbrem(zA E)
1-
=
-
fo
6
E E2
t E [1 - 7 E + Iji]dEf
Xo(Ei - Ef)
Ei
E
(4.16)
(4.17)
,a
where the CExtbrem(AE) is the external bremsstrahlung correction factor. The integration yields:
r
=
X0
[--)+(2.-)ln(
2
AE
)+(2-7)
In order to have a proper limiting behavior as AE -+
AE
2 Ej2
].
(4.18)
0, the (1 - grad) is replaced
by e-rad. By rewriting (2 - 77) with the function (, the correction becomes:
CExtbrem = e -x [
)+2)
AE
+
(4.19)
Here e is the electron energy. The external bremsstrahlung correction is applied twice.
Once before the reaction, where the c is the beam energy, and once after the reaction,
130
Table 4.3: External Bremsstrahlung Correction
AE
8.00 MeV
External Bremsstrahlung Correction
1.0111
where the E is the final electron energy. The function ( and the radiation length Xo
are:
f (Z)
=
(Za) 3 1.202 + (Za)2 [-1.0369 + 1.008(Za) 2 /((Za)
Z+1
1
],
-[12 +
9
11Z+12
716.405
X0
11
=
5.31 ,
12
=
6.114
Again we chose AE
Z(Z(l 1
f(Z)) + 12)
,
(4.20)
(4.21)
(4.23)
.
(4.24)
=
8.00 MeV for the missing energy region of integration, and
the result in external bremsstrahlung correction,
4.5.3
+ 1)]
(4.22)
A
-
2
CExtbrem,
is 1-0111-
Landau Straggling
Charged particles traversing a material can lose energy due to collisions with atomic
electrons; the energy transferred to the atomic electron can result either in excitation
of the atom or ionization. This process is called Landau Straggling [28]. Landau
straggling can result in the loss of an event from the missing energy peak. Like the
correction for external bremsstrahlung, the Landau straggling correction depends on
the thickness of the target. The correction factor is
1
CLandau =
131
1 -o
,
(4.25)
Table 4.4: Landau Straggling Correction
AE
8.00 MeV
Landau Straggling Correction
1.0026
where
6, =
-y
=
(4.26)
A(A +lnA + y)
(.6
0.577... is Euler's constant, and the other parameters are give by
I
e
=
0.154 - .
=
averageionizationpotential
(4.28)
=
13.5- 10-6 . ZMeV
(1 - 32
2.718"__,
(4.29)
mostprobableenergyloss
(4.30)
=
eo=
t(g/cm 2 )
,
2Ale
(4.27)
=((1n-+1-7),
el
A
=
-
(4.31)
The formula for the most probable energy loss yields eo ~ 0.5 MeV for this
experiment. The correction factor, CLandau, is about 1.0026. Thus the effect of Landau
straggling is negligibly small for this experiment.
The total radiative correction for this experiment with an 8.00 MeV interval in
missing energy is:
CRad = CIntbrem * CExtbrem * 0 Landau = 1.233 -
4.6
(4-32)
Coincidence H(e, e'p) measurements
The coincidence H(e, e'p) measurements are very important calibration in our experiments. The kinematics of the reaction is determined by either the electron arm or
132
the proton arm. This property allows us cross-check the beam energy, angle determination, and momentum calibration (In Appendix A, these features are presented in
detail). As a coincidence measurement, it provides a clean way to validate the calculation of coincidence phase space and all the efficiencies. Finally the elastic scattering
reaction H(e, e'p) has been studied intensively in the past [61], thus the cross section
is well known and can be used as the standard of normalization. Comparisons of data
with Monte Carlo simulations on every observables in the H(e, e'p) measurements also
serve as tests of the overall understanding of the combined OHIPS-OOPS system.
4.6.1
Kinematics
Liquid hydrogen was measured immediately after every the deuterium experiment.
The kinematics conditions are chosen as close as possible to the condition of the
deuterium experiments. The kinematics condition is as follows:
EO
=
800.0MeV ,
(4.33)
=58.010
O,
37.270
e=
Pe = 486.35MeV/c ,
P
where EO is the beam energy,
Oe
=
681.16MeV/c ,
is the angle of OHIPS relative to the incident beam,
6, is the angle of OOPS relative to the incident beam, and Pp is the proton momentum
and Pe is the electron momentum.
4.6.2
Cross Sections of H(e, e'p)
The coincidence H(e, e'p) elastic scattering cross sections are determined by
do-
Ncoin - Crad
dQe
K - Vcoin
133
(4.34)
where Ncoin is the number of coincidence events, Crad is the radiation correction factor,
and V"coin is the coincidence solid angle. The factor K is given by:
K =-
Qp
e
t * NA
Eot
* fht '
Eoc
' fhc *El/BB 'si-
(4.35)
where
Q
=
Totalcharge,
e
=
Electroncharge = 1.6 - 10-19
(4.36)
p =
Targetdensity ,
t
TargetEffectiveLength
N4
=
=
6.02 - 1023
OOPStriggerefficiency ,
et
eht
Avogadro'sconstant
=
OHIPStriggerefficiency
OOPSchamberefficiency
c=
Chc
=
OHIPSchamberefficiency
C1/BB
=
One - per - beam - burstvetocorrection
fsi= OHIPSselfinhibitefficiency .
The measured cross sections are compared with the H(e, e') cross section derived
from the Mainz fit of the proton form factors [61]. The comparison is listed in Table
4.5. The Mainz cross section was averaged over the experimental acceptance using
AEEXB. The ratio of the measured cross section versus the expected cross section is
98.6% ± 0.2%. This correction is applied in the final results of this experiment.
134
Table 4.5: H(e, e'p) Cross Section Compared with Dipole Fit(pb/sr)
frad
dui
dQ,
133960
1.290
1.20
0.071
0.6124
0.9813
0.9984
0.9254
1.0
1.0
1.263
0.2968 ± 0.0008
Dipole Fit
0.3010 [61]
Nc
msr
/Qe
t(cm)
p (mol/cm 3 )
6oc
6hc
E1/BB
6
si
Cot
6ht
135
Chapter 5
2 H(e+
e'p) Data Analysis
This chapter describes the procedure used in the 2H(eY e'p) data analysis. It involves:
1. Time-of-Flight Correction;
2. Phase Space Matching Techniques;
3. Various Software Cuts;
4. Asymmetry Extractions;
5. Absolute Cross Section Determination;
6. Response Function Extractions;
7. Estimation of the Systematic Errors;
5.1
Time-of-Flight Spectrum
The time-of-flight (TOF) is the time difference between the OOPS scintillator trigger
and the OHIPS scintillator trigger in a coincidence event. It is related but not identical
to the difference between the electron time of flight from the target to the detectors
and the corresponde proton quantity. If the absolute time difference between the
OOPS scintillator trigger and the OHIPS scintillator trigger is less than 100ns, it is
a coincidence event. The TOF TDC was started by an OOPS trigger and stopped
136
Table 5.1: Contribution to width of timing peak
OHIPS
4.2 ns
0 ns
±0.3 ns
±0.1 ns
Contribution
Paths
Speeds
Time Walk
Electronics Fluctuation
OOPS
1.4 ns
8.8 ns
±0.1 ns
±0.1 ns
by an OHIPS trigger. Ideally, a TOF peak would be a delta function. However, it is
typically broadened by the following effects:
" Particles travel different path lengths through the spectrometer dependence of
OT, OT
"
Particles have different speeds in the spectrometer.
" Different timing from diferent interaction locations in both the OOPS and
OHIPS trigger scintillators.
" Time-walk effect due to the variations of scintillator pulse heights.
" Electronic Fluctuation
It is important to correct for the above effects to increase the signal to noise ratio.
In Table 5.1 we summarized the contributions from different sources to the width of
Time-of-Flight signal.
The relation between a corrected TOF and a raw TOF is:
TOFcorr = TOFraw + Tp
-
Te
-
toopsscintcorr - tohipsscintcorr ,
(5.1)
where Tp is the time correction of the proton's path length and speed, Te is the time
correction of the electron's path length, toopsseintcorr is the time correction from the
OOPS scintillators such as the location dependence and the time-walk effect, and
tohipsscintcorr
is the time correction from the OHIPS scintillators such as the time-walk
effects.
137
Table 5.2: Proton Path Length Corrections
457.0
0.077
0.031
0.730-10-3
0.860-10-'
0.140.10-2
10
< 110 >
< 1k >
< 1162 >
< 1102 >
< 1106 >
< 112 >
5.1.1
0.290-10-2
Proton Path Length Correction
The proton path length correction is calculated given the focal plane coordinates and
the transport matrix elements as follows:
= 10+ <110 > 0+ < 116 > 6+ < 1102 > 02 +
-P
< 1162 > 62+ < 1106 > 06+ < 11|
t
p
=
2
1P
> 02
(5.2)
(5.3)
p
cii%
where the angle is in mr, 6 is in percents , the length is in cm. The coeffiencies are
in Table 5.2.
5.1.2
Electron Path Length Correction
The electron path length correction is calculated given the focal plane coordinates
and the transport matrix elements as follows:
le = < 110>0<116 > 6+ < 11t >
tp
ip
(5.4)
(5.5)
C
where the angle is in mr, 6 is in percents , the length is in cm. The coefficenies are
in the Table 5.3.
Due to a large threshold setting (50mV) in the discriminators for the OHIPS
138
Table 5.3: OHIPS Path Length Corrections
< 116 >
-0.4166
< 1k6 >
1.45
scintillator signals, a non-negligible time-walking effect is found. It is corrected by:
41.9
Ttime-walk
/A
DC21 + 1.0 ,
(5.6)
where the ADC 21 is the ADC value of the left arm of the second scintillator in OHIPS.
The ADC 21 value is around 1000, and 1.0 in Egn. 5.6 is not important and its main
purpose is to prevent the software crash due to possible zeros in the denominator.
This time-walk effect is discovered during the data analysis.
Fig.5-1 is the spectrum of raw time-of-flight of the backward OOPS (OOPS C)
and OHIPS and the spectrum of its corrected time-of-flight. The FWHM of TOF
improves from 3.0 ns in raw time-of-flight to 1.45 ns in corrected time-of-flight. The
spectrum is fit by a function:
2
(t - P3)
2
-)
2(P4)
P1 + P2 -exp(-
(5.7)
where the P1, P2, P3 are fit parameters which are not important, and P4 is the - of
the gaussian fit.
5.2
Missing Mass Calculation
Missing Mass is one of key physics quantities in the data analysis. For deuterium, it
is the binding energy of deuteron which is 2.2 MeV. It defines as:
= w - Tp - T. ,
Emiss
139
(5.8)
800
)e/ndf
P1
P2
P4
P4
700
402.2
/197
64.19
666.5
99.11
1.351
x/ndf
P1
P2
P3
P4
1600
1400
600
290.9
/
217
64.86
1373.
76. 35
.66
1200
500
1000
400
300
600
200
400
100
200
A
0
0
20
40
60
80
100 120 140 160 180 200
0
Raw OOPS C TOF on Missing Mass Cut
20
40
Corrected
60
80
100 120 140 160 180 200
OOPS C TOF on Missing Mass
Cut
Figure 5-1: The left figure is the Raw time-of-flight, the right figure is the Corrected
time-of-flight. Data are shown for OOPS C. These spectra were obtained under the
condition ( -3 MeV < Emis < 8 MeV)( See Section 5.2).
where T, and T, are proton and neutron kinetic energies respectively, and W is the
energy transfered. T, and T can be obtained through:
(5.9)
TT =
p 2 +M
Pn
PPq+ M2 - Mnq
=
P
V=P2-
2
-M
2 -P
qcos~pq+ q2 .
(5.10)
(5.11)
where P., P, are the momentum of the neutron and the proton, q is the transfer
momentum. Since there are accidental events in the raw missing mass spectrum,
several steps need to be applied to obtain the "true" missing mass spectrum (it is
defined later). The following steps outline the procedure.
o Using a corrected time-of-flight spectrum. We performed a gaussian fit of the
time-of-flight peak. In Fig.5-2(a), the region WR in location of the peak with
140
1
the width of 6- are the region of the real events, and the region V4, and W
A2
are the region of the accidentals.
" By applying the timing constrains just defined on the missing mass spectrum,
we obtain the missing mass spectra called the real missing mass spectrum and
background missing mass spectrum respectively as shown in Fig.5-2(b) and
Fig.5-2(c).
" The accidental missing mass spectrum is scaled by the ratio of those two time-offlight widths (
W;
+R W
(W A1 + WA2)
and subtracted from the real missing mass spectrum
to form the true missing mass spectrum as shown in Fig.5-2(d). The three true
missing mass spectra for OOPS A, OOPS B, and OOPS C are shown in Fig.5-3.
141
a
C
2
0
30( 10 -Corrected
TOF
25( 0 20C 0
(a)
WAE
1I
0ft
X-
--
-
60
140
120
160
Time of Flight(ns)
S
S
£
2500-
C
2
a
Missing Mass
Accidentals
140
Missing Mass
Reals
U
120 a
2000
100
(c)
sa 0
(b)
60 0*
1000 -
40 0
50
20 0 -
o
eO
-50 -40 -30 -20 -10
0 10 20 30
Missing Mass (MeV)
a
I 1200
0
"..1.
I....
71 2 73.1 . .. 1. .,,.,,
-20 -10 0 10 20 30 40
-30
-40
-50
MissIng Mm (M*V)
Missing Mass
Trues
1000
a00
600
0
-40 -50 -40 -30 -20 -10
0
10 20 30 40
(d)
Missing Mass (MoV)
Figure 5-2: The procedure to subtract background missing mass.
142
E.
S
U
300
250
200 MeV
150 -4.18
.
100
50
.0
-60
-50
-40
-30
-20
-10
0
10
20
30
40
OOPS A: Missing Mass (MoV)
800
4
3.34 MeV
400
200-
--
5
. . .1
-450
&~1
---
0-50
-40
. .
.
. .
-20
-10
3.48 MeV
_...
-30
l
. .
.
.......... ,
.
0
40
20
30
10
OOPS B: Missing Mass (MaV)
1200--
1000NO0
Goo
400
2000-60
.1 ....
-50
-40
-30
-20
-10
0
I....
I
10
20
30
40
OOPS C: Missing Mass (MaV)
Figure 5-3: The true missing mass spectrum for three OOPSs.
143
5.3
2 H(e.
Phase Space Matching
e')p coincidence cross section is a function of w, q,
0
pq
and
#pq.
Since the
coincidence phase space of each OOPS and OHIPS has a different range of the W, q,
0
pq
and
#pq,
the phase space matching has to be applies in order to obtain the
ALT,
fLT, ATT, and fTT
In order to extract fLT and fTT independent of theory, one must use use the Phase
Space Matching method. That means all kinematics variables, w, q, 6 pq and #pq, have
to be matched for all OOPS spectrometers involved.
Two dimensional W -
Opq
histograms for the forward, backward and out of plane OOPS are shown in the Fig.54. The phase space shapes for the forward, backward and out of plane OOPS are very
different. Only the small amount of data which lie in the matched area can be used
to extract the LT, TT asymmetries, and structure functions independent of module.
The matched phase space is defined as the phase space which all three OOPS and
OHIPS overlapp in W-Opq and q-#pq histgrams. There are two methods to obtain the
matched phase space. One is to use the Monte Carlo matched phase space, the other
is to used the raw data. The raw data from the experiment is preferred. Every Monte
Carlo simulation is only an approximation of reality, and there are errors associated
with the Monte Carlo both in statistical and systematic sense.
Since the measured cross sections are averaged over the acceptances of spectrometers, the measured cross sections are compared with the theoretical calculations folded
with the matched phase space which defined above in a Monte Carlo simulation.
In Table 5.4, the Monte Carlo averaged theoretical cross sections, asymmetries
and structure functions are compared with central kinematics values of theory. From
this table, we see that the differences between the folded Monte Carlo results of
Arenh6vel [62] theoretical calculations and Arenh6vel central kinematics calculation
of this experiment are quite small. Results of this experiment are presented in the
Chapter 6.
144
60 -
OOPS A
50403020OOS
10-
--
00
907
I.
16
I
1
I
20
ilii
I
22
24
26
28
p
30
(deg)
Figure 5-4: Plot of w - 9 pq for the forward OOPS (A) , backward OOPS (C) and
out-of-plane OOPS (B ) from the experimental data.
5.4
Software Cuts
Generic cuts for data analysis are as follows:
* Cuts are made in the OHIPS Cherekov ADC values. These cuts ensure that
only electrons were analyzed in the OHIPS instead of other particles.
o Cuts are made in the acceptance of the electron momentum. Since the Monte
Carlo simulation does not simulate the edge of the spectrometer well, the electron momentum is limited in range by the software momentum cuts on the
electron momentum.
145
Table 5.4: Comparison between the folded theoretical calculations in the matched
phase space using a Monte Carlo simulation with the unfold theoretical calculations at
the central kinematics of this experiment. Arenh6vel's full calculations (See Chapter
6) are used for the theoretical calculations.
ATT:
Cross Section GOPS A
Cross Section OOPS B
Cross Section OOPS C
fi
ft
Folded Theo. Cal.
0.0163
0.1454 nb
0.2734 nb
0.4195 nb
-0.01025 fm
-0.00049 fm
Unfold Theo. Cal.
0.0167
0.1501 nb
0.2849 nb
0.4391 nb
-0.01010 fm
-0.00048 fm
" Cuts are made in both OOPS and OHIPS angle acceptances. These cuts reject
the particles which were out of the spectrometer angle acceptance which were
due to the track errors.
" Cuts are made in the OOPS scintillator ADCs. These cuts ensure that only the
proton events are analyzed.
" Cuts are made in the both OOPS and OHIPS chambers TDCs and ADCs.
These cuts ensure that only traceable particles are analyzed.
" Cuts are made in corrected time-of-flight spectrum. These cuts ensure that the
true coincidences are taken and also the accidental level.
" Cuts are made in the missing mass spectrum. These cuts ensure the deuterium
breakup events are analyzed and the value of the interval on the missing mass
spectrum of these cuts are also used for the radiative correction.
5.5
Asymmetry Extractions
By measuring an asymmetry, one hope to eliminate some systematics uncertainties in
such quantities as effect target length, beam currents, and the absolute normalization
of the spectrometers. However the absolute cross sections are also observables and
important physical quantities.
146
5.5.1
ALT'
The ALT' asymmetry is the most simple one to measure. Only one OOPS is involved
and the
ALT,
asymmetry measures the cross section dependence on the beam helicity.
The formula used to obtain the ALT, asymmetry is
A_
N+ + N_
N+-
(5.12)
ALT' = N,+N
where N+ is the number of counts where the helicity is plus and N_ is the number of
counts where the helicity is minus. The statistical error is
6
ALT' =
(N+ + N_)
+
2
,
(5.13)
The major systematic error comes from the measurement of beam polarization.
5.5.2
ALT
For the ALT asymmetry, there are two OOPS are involved. One is in forward side
of q direction and the other one is in backward side of q. The Phase space volumes
are different for the forward and backward OOPS. Also, all the triggering and wire
chamber efficiencies are different. The true counts are corrected for these efficiencies.
The formula for ALT is:
NA
NA
ALT
NA+
-
N0
NC
(5.14)
Nc
where NA and Nc are:
NA =
'NAf
C
V
~ dfa
, NC=
NtC . gd
V
(5.15)
Where Nf and NFc are the true counts, EA and EC are the efficiencies, flad and fcad
are the radiative correction factors and VA and VC are the phase space volumes for
for both OOPS. The statistical error can be calculated by following:
6 ALT
=
(NA + Nc)
2
4N 2Nc +4N
147
2
NA.
(5.16)
5.5.3
ATT
For the ATT, there are three OOPS spectrometers involved. The formula for the ATT
and its statistical error 6 ATT are:
NA++ Nc - 2 - NB
T =NA
Nc + 2. NB
6ATT =
1
(NA + Nc+ 2 - NB )
2
2
2
4. N2(6 NA +5 NC) + 2- (Ni + N2)6 NB
(5.17)
(5.18)
here the NA, NB and Nc are the same definition as in the ALT.
5.6
Absolute Cross Section
The coincidence 2 H(e, e'p) scattering cross sections are determined by
dodwedQedQp
Neoin *frad
K -Vcoin
(5.19)
where Ncin is the number of coincidence events, and frad is the radiation correction
factor, and Vin is the coincidence phase space volume. The factor K is given by:
K
e
p * t * NA - Eot
' ht
* e6oc
* hc * E1/BB * Esi ,
(5.20)
where
Q
=
Totalcharge ,
e
=
Electroncharge
p
(5.21)
Targetdensity ,
t = TargetEffectiveLength
NA
=
Avogadro'sconstant(6.02 - 1023)
fot = OOPStriggerefficiency
fht =
OHIPStriggerefficiency
148
COC
= OOPSchamberefficiency,
ehc
=
OHIPSchamberefficiency
61/BB
=
One - per - beam - burstvetocorrection ,
68i
=
OHIPS - self - inhibitvetocorrection .
In the Fig.5-5, the cross section of each OOPS spectrometer is plotted against
different group of runs. Each run group represents about 15 hours beam time.
800 MeV, DIP region, q=414 MeV/c, w=155 MeV
>0. 5
U)
U)
OOPS C
C 0.4
0
U
)
OOPS B
U)
0
0.3
0.2
OOPS A
-
0.1
0
'
0
2
'
'
4
8
6
15
10
12
hours run time for each point
Figure 5-5: The time dependence of the cross section of three OOPS spectrometers
in this experiment
149
5.7
Structure Function Extractions
Extracting the structure functions from the measured cross sections is straight forward. The equations are:
o
+
7/2
fLT'
-
r/ 2
(5.22)
2 - C - JPLT'
U 0 + U7r
2 -C. JPLT
o0 + oir - 2 0'7r,/2
4-C - JPTT
fLT
fTT
(5.23)
(5.24)
Where C is the Mott cross section and J is a Jacobian factor. The structure functions
obtained are the structure functions averaged over the acceptance. The statistical
errors for
fLT', fLT, fTT
are:
620±
6
fLT'
+
fLT
fLT
(U 0
62
6
5.8
fTT
7r/2 +
7r/2
fLT'
(O7r/2 +
6
6 2 0-u
+
rr/2 2
PB
PB
(5.26)
+ 6 2 0;
U7r 2
+ 6 2 0,,+ 46 2 Or./
= + r + 29,/2)
0
(5.25)
2
(5.27)
Systematic Errors
The major sources of the systematic uncertainties are:
" The uncertainty in the beam energy: 0.8 MeV.
* The uncertainty in the spectrometer angle: 0.10 for OHIPS and 0.05' for OOPS.
" Coincidence phase space volume: 5% of the total phase space volume.
* Target Thickness: 3% of the total target length.
" Radiative correction: 1% of the total radiative correction.
150
Table 5.5: Systematic errors in this experiment
Sources
A7r,
ALT
ATT
ro
9r/2
0'7
fT/2
fLT
fTT
E,
0.8
2.3
2.4
0,
1.6
1.6
0.6
Op
3.4
0.7
0.7
V
t
fracd
PB
Q
-
-
-
-
-
-
-
-
-
-
3.0
1.0
0.1
6
3.0
3.0
3.0
3.0
3.0
1.0
1.0
1.0
1.0
1.0
-
0.1
0.1
0.1
0.1
0.1
6Or 7 r/2 /0r/
2.3
1.1
1.2
5.0
1.7
2.3
2.4
5.4
0.9
0.5
1.0
2.5
2.4
1.2
1.0
1.2
0.8
2.0
0.3
5.0
5.0
5.0
5.0
5.0
5.0
-
Total Systematic Error%
±3.7
cA7 ,/AW 2,
6
2.8
ALT/ALT
ATT/ATT
=
±2.3
6cro/uo = ±6.4
2
=
±6.1
6o,/a, = ±6.1
= ±8.7
f
±8.7
=
fLT/IfLT
6
fTT/IfTT= ±6.3
" Beam Polarization: 5% in the total beam polarization.
" Charge uncertainty: 0.1% of the total charge.
To estimate the systematic errors for all measured physics quantities caused by
these uncertainties, Arenh6vel's full cross section [62] is used. In the table 5.5, the
total systematic errors for each measured quantities and the contributions from each
source are listed
151
Chapter 6
Results and Discussions
The final results of this experiment are summarized in this chapter. The experiment
results are compared with various Arenh6vel's calculations [62].
This chapter also
includes the discussion of future directions for the out-of-plane spectrometer program.
6.1
Results of 2 H(&, e'p)
The final experimental results are summarized in Table 6.2 and in Table 6.3. The
cross sections, asymmetries and corresponding structure functions are listed. Since the
measured physical quantities were averaged within the spectrometers acceptance, the
measured physical quantities are compared with the Arenh6vel's calculations folded
within the same spectrometers acceptance. To extract Alt, and fitw, the entire coincidence acceptance was taken for each individual OOPS. For the extraction of the LT
and TT asymmetries and structure functions, only the data in the matched phase
space are used (see Section 5.3).
In Table 6.3, the cross section dependence on w for each OOPS is listed along with
the folded cross sections of full Arenh6vel's calculation (N+RC+MEC+RC) [62] by
using a Monte Carlo simulation.
The centeral kinematical quantities of this experiment are listed in table 6.1.
152
Table 6.1: Summary of kinematical quantities of this experiment
OOPS
#
OOPS A
OOPS B
OOPS C
olab
(0)
31.0
w
(MeV)
155.0
0
(0)
38.5
CM C
(0)
0
90
180
ab
(0
23.5
olab
olab
PP
Pmiss
()
(0)
0
23.5
0
(MeV/c)
509.8
(MeV/c)
210
29.86
53.35
76.85
Table 6.2: Cross Sections of three OOPSs in the matched phase space (See Section
5.6) are in
b
. The Folded theoretical calculations are Arenh6vel's full calMeV . (sr)2
culation (N+RC+MEC+IC) averaged over experimental acceptance using AEEXB.
The asymmetries are in percent. The structure functions are in fim. The first error in
each measured quantity is the statistical error, and the second error is the systematic
error (See Section 5.7). The ALT' in A and C is 0 due to the fact that sin 0', 180 =0
in the equation 1.45.
OOPS A:
OOPS B:
OOPS C:
ALT:
ATT:
in A:
ALT' in B:
ALT' in C:
fit
fit'
ALT'
ftt
Experimental Results
Folded Theoretical Calculations
0.1372 ± 0.0075 ± 0.0088
0.2650 ± 0.0110 ± 0.0161
0.4098 ± 0.0066 ± 0.0250
0.0213 ± 0.013
-0.4783
0.0067 ± 0.004
0.0157
0.0004
0.0180
0.0116
-0.0254 ± 0.0127 ± 0.002
0.0040 ± 0.0107 ± 0.0001
-0.01004 ± 0.00037 ± 0.00087
-0.00408 ± 0.00204 ± 0.00036
-0.00045 ± 0.00065 ± 0.00052
0.1454
0.2734
0.4195
-0.4852
0.0163
0.0
-0.0227
0.0
-0.01065
-0.00412
-0.00049
153
The location of this experiment kinematics is shown in Fig.6-1. It is at W = 155
MeV above the quasi-elastic peak.
e =800 MeV
1,= 31*
V)
10
qfs
total (radiated)
2N
2
0
100
200
300
400
500
600
ci [MeV]
Figure 6-1: The inclusive (e, e') cross section verse w through a program from Prof.
LightBody [63]. The 2N in the figure indicates strength of the two-body current
contributions and A indicates strength of the A excitation state.
154
Table 6.3: Cross sections dependence of w, where w is in MeV and cross sections are
nib
in
(s). Only the statistical errors are shown. The folded theoretical calcuAMe V - (sr)2
lations are the Arenh6vel's full calculation averaged over the matching experimental
acceptance using a Monte Carlo simulation.
W
130-140
140-150
150-160
160-170
170-175
6.2
o
0.1351±0.0116
0.1303±0.0085
0.1474±0.0098
0.1304±0.0113
0.1497±0.0145
folded ao
0.1410
0.1413
0.1470
0.1484
0.1510
Qr/2
0.2256±0.0101
0.2504±0.0075
0.2874±0.0080
0.3370±0.0092
0.4507±0.0115
folded o r/2
0.2443
0.2613
0.2923
0.3440
0.4721
0.2084±0.0084
0.2842±0.0067
0.4257±0.0080
0.6585±0.0100
1.0997±0.0133
folded or,
0.2204
0.3012
0.4434
0.6835
1.0556
Comparison with Theory
As discussed in Chapter 1, Arenh6vel and co-workers have performed systematic
theoretical calculations of deuteron electrodistintegration [9, 10, 11, 12, 13, 17]. A
brief overview of Arenh6vel's approach has been presented in Section 1.4.
In Arenh6vel's calculation, he provided us five theoretical calculation sets for these
experiment conditions:
" one set was calculated using the plane wave bonn approximation calculation
(PWBA).
" one set was calculated by including the final state interaction (N).
* one set was calculated by including the final state interaction and the relativistic
corrections (N+RC).
" one set was calculated by including the final state interaction, the relativistic
corrections and meson-exchange currents (N+RC+MEC).
" one set was calculated by including the final state interaction, the relativistic
corrections, meson-exchange currents and isobar configurations (N+RC+MEC+IC).
These five sets of calculation show the effects of the different nuclear interaction
mechanisms in fit,
f/,
and ftt.
155
The deuteron initial state and the interactions of the outgoing n-p system in these
calculations are based on the Paris potential [18].
Various calculations show very
little sensitivity to the choice of a realistic NN potential model [22].
The comparisons of cross section, asymmetries and structure functions are presented in Fig.6-2, Fig.6-3, Fig.6-4, Fig.6-5, Fig.6-6 and Fig.6-7 respectively. In these
figures, the solid dots are measured values in this experiment, and the vertical error
bars are statistical errors only.
156
800 MeV, DIP region, Q2=0.15 (GeV/c) 2
ALT
0
-0.2
-0.4
-0.6
0.025
0
ALTO
x40%.
-0.025
-0.05
-0.075
- - - - -
N+MEC
-
N+MEC+IC+RC
0.1
0.05
ATT
0
-0.05 I
0
10
20
30
40
50
60
9p"' [deg]
Figure 6-2: ALT, ALT' and ATT compared with different calculations. The solid dots
are measured values in this experiment. Only the statistical error is shown.
157
800 MeV, DIP region, Q2=0.15 (GeV/c) 2
0
-.
F LT
(fim)
-0.005
-0.01
-0.015
N
.I
-0.02
Ps
N+MEC
-0.025
N+MEC+IC+RC
-0.03
-0.035
-0.04
0
10
20
30
40
60
50
pq
Figure 6-3:
ATr
[deg]
compared with different calculations. Only the statistical error is
shown.
158
800 MeV, DIP region, Q2=0.15 (GeV/c) 2
0.002
FLTC
(fim)
0
-0.002
--
V
7,
7/
/
-0.004
/
-0.006 1
..
N
N+MEC
N+MEC+IC+RC
-0.008
-0.01
0
10
20
30
40
60
50
pq
Figure 6-4:
shown.
fLT'
[deg]
compared with different calculations. Only the statistical error is
159
x 10
-2
800 MeV, DIP region, Q2=0.15 (GeV/c) 2
0.05
0.025
F
(f n)
0
-0.025
-0.05
-0.075
-0.1
-0.125
/
_7
-0.15
N
N+MEC
-0.175
N+MEC+IC+RC
-0.2
0
10
20
30
40
60
50
[deg]
Figure 6-5: fTT compared with different calculations. The solid dot is the experimental result. The circle is the folded theoretical calculation including the
N+RC+MEC+IC. Only the statistical error is shown.
pq
160
800 MeV, DIP region
1.2
En
#OOPS C
En
1
0.8
0
0.6
U)
U)
V)
0
L-
OOPS B
0.4
OOPS A
0.2
e
I
0
130
140
I
160
150
170
180
w (MeV)
Figure 6-6: Three OOPS' cross sections dependence on w are shown. The curve is
the Arenh6vel's full calculation (N+MEC+IC+RC) folded with spectrometer's acceptance.
161
DIP, Q2 =0.15 (GeV/c) 2 , x=0.52
0.5
0.4
'
b
...
.......
...
0.3
0.2
LO
0.1
0
0
50
150
100
c4)pq
200
[deg]
Figure 6-7: Absolute Cross Section compared with different theories is shown. The
solid line is Arenh6vel full calculation (N+RC+MEC+IC), the dash line is N+RC,
the short dash line is PWBA+RC and the short dot line is N+MEC+IC.
6.3
Conclusions
The dip region structure functions
fLT, fLT'
and
fTT
in the 2 H(e, e'p)n dip region
were measured simultaneously. The results of this experiment show:
" The data (cross sections, asymmetries and structure functions) agree with Arenh6vel
full calculation within statistical and systematic uncertainties.
" Both Alt and fit quantities are better reproduced by including the relativistic
correction, which disagrees with an earlier measurement [35].
162
* fw' is non-zero within a confidence interval of 2-. It implies the final state
interaction is of importance.
" The result of the measured ftt and A4t favours the calculations which include
the isobar configurations and meson-exchange currents.
6.4
Outlook
As stated in Chapter 1, this experiment is only a part of a program to measure
all deuteron structure functions at various kinematics [64] systematically. With the
completion of the fourth OOPS module, and together with a support system which
permits all four OOPS modules to arrayed azimuthally about a symmetry axis in
the scattering plane, more comprehensive measurements can be conducted with fully
controlled systematic uncertainties.
Future improvements at the Bates accelerator
facility, such as increasing the duty factor to 100% with the new constructed pulse
stretcher ring and doubling the beam polarization by using strained gallium-arsenide
crystals, will significantly reduce the statistical errors for the asymmetry and structure
function measurements.
The kinematic conditions of the remaining deuterium program at Bates [66] are
summarized in Table 6.4. The agenda for deuterium physics in Bates Laboratory will
be focused at the high missing momentum or large 0'
region where the deuteron D-
state dominates and where accurate measurements of the 2 H(e, e'p)n response functions will place even more constrains on the different models of the NN interaction
[64].
One of the primary goals of the OOPS program at Bates Laboratory is to study the
N -- + z transition on the proton through the H(e, e'p)7r' reaction [67]. The study of
the N -+
-A reaction in the nucleon provides a deep insight about hadronic physics,
which involves the resonant quadrupole excitation of the A. In the spring 1998, part
of the original N -+
A proposal, H(e, e'p)7r0 and H(e, e'p)7r+, were completed at
Bates. Two OOPS spectrometers were used for the simultaneous measurements of
RLT'
and
RLT
response functions. The data analysis is underway.
163
Kin
E,
MeV
4A
q
q2
Enp
Opc
Pr
MeV
MeV/c
fm~ 2
MeV
o
MeV/c
800
265.0
438
3.14
217
34.8
315
2B
800
118.6
486
5.72
56.0
78.5
300
3B
4B
800
600
155.0
265.0
414
438
3.79
3.14
110
217
61.4
70.6
290
442
W
|
Table 6.4: Kinematic conditions for the future deuteron measurements.
In addition, several new experiments have been approved by the Bates PAC, which
will use the unique OOPS facility. These experiments fall into four physics categories:
" Studies of the quadrupole component in the N
--
A transitions:
1. Exp 87-09 H(e, e'p)7r0 [68].
2. Exp 97-05 H(e, e'p)y [70].
3. Exp 97-04 H(e, e'7r+)n [71]
" Studies of the electromagnetic currents in the deuteron:
1. Exp 89-14 2 H(e, e'p)n [66].
2. Exp 89-10 2 H(e, e'p)n [67].
" Studies of the generalized polarization of the proton through virtual Compton
scattering:
1. Exp 97-03 H(e, e'p)> [72].
o Studies of few-nucleon systems:
1. Exp 97-06 3 Ie(', e'd)p [73].
2. Exp 97-01 4 He(e, e'd)p [74].
With the completion of these experiments, one will have much better understanding of the properties of nucleon and the electromagnetic currents between nucleons
inside the nucleus.
164
Appendix A
H(e, e'p) kinematics
Often we use the H(e, e'p) reaction to determine the beam energy given the angle of
6e and O,, or determine the Fe, or Pj given the beam energy E0 . In our optics study,
one of OOPS optics is obtained through the reaction H(e, e')p. Since the proton
has a large kinematics correction, the proper correction has been to be done before
obtaining the correct optical matrix elements.
1) Assuming the 0e and O, are given, the beam energy is determined as follows.
This method is one way to determine the beam energy although its accuracy is limited.
The electron scatters from the proton at angle 0, with final energy Ef, while the
proton recoils at angles O,, energy Ep, and momentum p. At the extreme relativistic
limit, pe = Ee. Conservation of momentum in the direction of the incident electron:
E!cosOe + pcosOp,
E = F
(A.1)
and in the perpendicular direction:
Ef sinOe = psin,
.
(A.2)
Conservation of energy yields:
E + M,= E + E.
165
(A.3)
From the first two equations, we get:
e 0).
szn(Oe+ P)'
E( = E '
p = Ei . sl
(A.4)
(A.5)
,
sin(Oe+ Op)
From equation (A.3), we get:
E + M - Ef,
E
=
E
= E + M2
+
(A.6)
(A.7)
2Er MI - 2EiEf - 2E M,
E
=
+ p2
(A.8)
,
After some calculation, we get
Ei
=
2M - f(Oe, Op)
(A.9)
where
f(0e, O)
= sin (0e + Op)(sinOp - sin(Oe + Op))
(sinp - sin(Oe + Op)) 2 - sin2 Oe
(
In one of liquid hydrogen run, we set OHIPS at 37.270 and OOPS at 58.010. From
the above equation, we get the beam energy is 799.7 (MeV). The error in OHIPS
angle is 0.10 and the error in OOPS angle is 0.060. These errors translate the error
in the beam energy is 6.5 (MeV), which one order larger than the measurement from
the Energy Compression System which has the error around 0.8 MeV. The hydrogen
target is a useful method to validate the beam energy, but not accuracy enough to
determine the beam energy due to the error in the angle measurements in electron
arm and proton arm.
2) Assuming the beam energy is known, from 0e, one can get Ef or from O,, one
can also get momentum of the ejected proton. These formula is important to get the
166
Table A.1: Hydrogen Kinematics Effects
Offset in O,
Offset in 0e
Proton Mom
Electron Mom
mrad
-20
-16
-12
-8
-4
0
4
8
12
16
20
mrad
27.1729
21.7030
16.2509
10.8165
5.39959
0
-5.38245
-10.7479
-16.0966
-21.4287
-26.7444
MeV/C
503.880
500.375
496.869
493.363
489.856
486.348
482.840
479.331
475.822
472.312
468.802
MeV/C
672.979
674.633
676.278
677.914
679.542
681.161
682.770
684.371
685.963
687.546
689.119
OOPS optics through the hydrogen target. The Ef or momentum p can be obtained
through the following equations:
E.
2
-
sin (0e
+ O) (sinp - sin(e
(sirnt
sn(Oe + Op))
-
2
-
+ Op))
(A 11)
sin 2O,
If Ej and O, are given, then from above equations, the
0
e
can be numerically
obtained.
Ef
sino
sin(pe + Op)
=E
E,
=ZE1
,
(A.12)
,e
(A.13)
sin(Oe + Op)
Where the Ef and p are obtained. In the calibration of spectrometer optics, the
kinematics effect of H(e, e/)p effect can be corrected. In the case for this experiment,
we took the optics data for OOPS A using the liquid hydrogen at the O, = 58.010 with
the beam energy at 800 MeV. The center proton momentum is 486.35 (MeV/C). The
table [A.1] is the correction in above kinematics. In order to get the correct optics,
for the different holes, one need the use the different moments.
167
Appendix B
2 H(e,
e'p) Kinematics
In the 2 H(e, e'p) reaction, one often need to calculate the magnitude of the ejected
proton momentum given the virtual photon (P,q), and the angle between ' and
the outgoing proton momentum pf is
0
pq
in the laboratory frame; and the Jacobian
relating the proton solid angle in laboratory frame, QL to the solid angle 92"
in the
cross section. Both are used intensively in the Monte Carlo program called AXXEB.
1) The magnitude of the ejected proton momentum given the virtual photon (w,
q), and the angle between q and the outgoing proton is 9 pq in the laboratory frame.
We assume that the final proton momentum p' and the initial proton momentum
(inside the deuteron),
j',
are related by
p~f =p+qf,(B.1)
The target deuteron has zero momentum in the laboratory frame, so to conserve
momentum the recoiling neutron must have
-=
-#,
(B.2)
and energy
En=
2+
168
Mn,
(B.3)
The energy of the ejected proton must be:
(B.4)
Ef = w + Md- En
=
Substituting for the A3 in
W + M-
2 + M2,
( B.4):
Ef= w + Md -
(pr- q)
2
(B.5)
+ M, ,
rearranging the terms and squaring them:
2+ M
P - 2pfqcosOp + q2 + M2
where Ef =
E2 + 2wiVld - 2(w + Md)Ef
+
p-f2 + MA2, after some length
(B.6)
calculation, we obtain the following
equation:
(B.7)
Ap + Bpf +C = 0,
where
A
B = -4(q2
C = (q2
W2 -M
=
4q 2 cos 26pq-
4(w + Md) 2
- Md - 2wMAd)qcosO,,
- 2w M)
2
-4(w+
M)
2
M2
(B.8)
(B.9)
(B.10)
The appropriate solution is
-B - v'B 2 - 4 AC
Pf =
2A
169
(B.11)
The missing momentum is
Pmiss
= q- pf
PMiss = \q 2 + I
2
-
(B.12)
qpfcosOpq
2) Jacobian calculation The Jacobian is given by:
(P
aQL
qLML
2pcmEL
3ML
VL
m
EL
)
(B.13)
where
+ Md) 2
E(w
-
(qL)
2
ML= En + 2I
EL =
The angle Oc'
(ML)2 + (qL)2
(B.14)
(B.15)
(B.16)
is:
cospc_
"p
cOS 2 0L . (I- p)
(B.17)
no
\I
- pCOS2gn0L
where
(qcm) 2
=
0n
(qcm)
2
(B.18)
+M2
is the angle between the q and the relative np momentum py -p 7, in the lab frame.
The transformations of q and w from the lab frame to the cm frame are given by:
qCM = 7(qL - Ow L)
wC"
= -(WL
(B.19)
/3qL)
-
(B.20)
L
/ =
(B.21)
+±Md
1
1
Vl-
170
=(B.22)
Here are the table for the this experiment at the central kinematics wj = 155
(MeV), Oe = 31.00.
171
Opq
Degree
50.354
49.354
48.354
47.354
46.354
45.354
44.354
43.354
42.354
41.354
40.354
39.354
38.354
37.354
36.354
35.354
34.354
33.354
32.354
31.354
30.354
29.354
28.354
27.354
26.354
25.354
24.354
23.354 '
22.354
21.354
20.354
19.354
18.354
17.354
16.354
15.354
14.354
13.354
12.354
11.354
O
Degree
79.7782
78.3334
76.8801
75.4185
73.9488
72.4712
70.9857
69.4927
67.9923
66.4846
64.9698
63.4482
61.9199
60.3851
58.8439
57.2966
55.7432
54.1840
52.6192
51.0489
49.4733
47.8926
46.3069
44.7165
43.1214
41.5219
39.9181
38.3102
36.6984
35.0828
33.4637
31.8411
30.2153
28.5863
26.9545
25.3199
23.6828
22.0432
20.4014
18.7575
Table B.1: 2 H(e, e')p Kinematics
Proton Momentum Missing Momentum
MeV/C
416.941
421.066
425.160
429.221
433.247
437.233
441.178
445.080
448.935
452.742
456.497
460.198
463.843
467.429
470.954
474.415
477.811
481.137
484.393
487.577
490.684
493.715
496.666
499.535
502.321
505.021
507.634
510.157
512.588
514.927
517.170
519.317
521.366
523.316
525.164
526.910
528.553
530.090
531.521
532.845
II
172
MeV/C
353.522
348.717
343.850
338.922
333.936
328.893
323.795
318.644
313.443
308.194
302.900
297.562
292.185
286.771
281.323
275.845
270.340
264.813
259.268
253.709
248.141
242.569
236.998
231.436
225.888
220.361
214.862
209.400
203.984
198.624
193.328
188.110
182.980
177.953
173.043
168.265
163.636
159.174
154.899
150.832
Jacobian
1.84089
1.87017
1.89929
1.92823
1.95698
1.98551
2.01380
2.04183
2.06958
2.09703
2.12416
2.15095
2.17737
2.20342
2.22906
2.25427
2.27905
2.30336
2.32720
2.35053
2.37335
2.39562
2.41735
2.4385
2.45905
2.47901
2.49833
2.51702
2.53505
2.55241
2.56908
2.58505
2.60030
2.61483
2.62861
2.64164
2.65391
2.66540
2.67611
2.68602
Appendix C
U~ccl calculation of 2 H(e, e'p)
In the PWIA, the cross section of 2 H(e, e')p can be written as
6 ~d p
dwd Ep,dedp
=
Kd
d~e
p(p) - 6(Em - EB)
(C-1)
The p(p) is the proton distribution inside deuteron; o(e-P) is the cross section of
ep scattering. Because in the deuteron, the ep reaction is off-shell. When we use
the on shell vertex, there has to have a correction. There are well known method to
correct this effect, most widely cited are occ and -cc2. In our experiment region, o-cc1
and Ucc2 give almost the identical results, and also acc, is a little simple in the formula
than
cc2.
Here the detail calculation of acc in the reaction 2 H(e, e')p is presented.
Within the general framework for (e, e')N in the PWIA, off-shell extrapolation
of the electron nucleon cross section amounts to choosing a method to calculate the
off-shell current:
j'" =< piJF'(q)jp>
.
(C.2)
On-shell one has
JA
= iU(p1)(-,"F(qA) - a,,q,,F2(q)U2m
(C.3)
When one goes off-shell one must decide which form factors. operator and spinor
173
to use. Some arguments can be made for keeping the term 7,F 1 as it is: one expects
such term to be present from minimal coupling. How one should should treat the
term c-,,q,F
2
associated with the anomalous magnetic moment, however, is not at
all clear. One must hope that the different off-shell extrapolations do not differ very
much. One possible form is :
=
iU(j)(y(F1(q2) + KF2(q )) +i(p' + p)' 4 F2 (q ))U(P)
2m
.
(C.4)
In the ep cross section:
UeN
(C.5)
p2Z
*
=r2g2
2 2
2
q1- k~1
1
4uWC + (22+ tan2
=
+
(
qq2
+ tan 2 )
2
TW
2 Wcos5
+ q2 cos2
+tan 2)Ws}
using the off-shell nucleon current, we obtain:
1
WT
=
=S
2 E'(F + KF 2 )2 ,
2EE'
' 2 si 2 62
E'
(+
4m2
W,
WhereE= v/p2m2,=
FE'
~EI)
+P
(F,
(C.7)
4M
F)2
2
+
qt 2
4m2"
(C-8)
2(C9
2
C9
E' - E and pq is the angle between p'and q'
The spectral function p(p) comes from Van Ordan. From the -ep and spectral
function, we can calculate the cross section and all the response functions in -cc,
model.
174
Table C.1: Deuteron Momentum Distribution
Proton Momentum in
Deuteron in (GeV)
0.OOOOOE-02
2.OOOOOE-02
4.OOOOOE-02
6.OOOOOE-02
8.OOOOOE-02
1.OOOOOE-01
1.20000E-01
1.40000E-01
1.60000E-01
1.80000E-01
2.OOOOOE-01
2.20000E-01
2.40000E-01
2.60000E-01
2.80000E-01
3.OOOOOE-01
3.20000E-01
3.40000E-01
3.60000E-01
3.80000E-01
4.OOOOOE-01
4.20000E-01
4.40000E-01
4.60000E-01
4.80000E-01
5.OOOOOE-01
5.20000E-01
5.40000E-01
5.60000E-01
5.80000E-01
6.OOOOOE-01
6.20000E-01
6.40000E-01
6.60000E-01
6.80000E-01
7.OOOOOE-01
7.20000E-01
7.40000E-01
7.60000E-01
7.80000E-01
II
Distribution
MVeV- 3
1.50000E+01
9.37075E+00
4.09536E+00
1.62474E+00
6.68644E-01
2.96785E-01
1.41045E-01
7.09261E-02
3.73659E-02
2.04694E-02
1.15919E-02
6.76989E-03
4.07377E-03
2.52812E-03
1.62250E-03
1.07968E-03
7.46893E-04
5.37697E-04
4.02300E-04
3.11723E-04
2.48829E-04
2.03419E-04
1.69356E-04
1.42846E-04
1.21560E-04
1.04064E-04
8.94245E-05
7.69933E-05
6.63395E-05
5.71687E-05
4.92604E-05
4.24216E-05
3.64995E-05
3.13739E-05
2.69466E-05
2.31299E-05
1.98394E-05
1.70036E-05
1.45622E-05
1.24640E-05
Proton Momentum in
Deuteron in (GeV)
1.00000E-02
3.OOOOOE-02
5.OOOOOE-02
7.OOOOOE-02
9.OOOOOE-02
1.10000E-01
1.30000E-01
1.50000E-01
1.70000E-01
1.90000E-01
2.10000E-01
2.30000E-01
2.50000E-01
2.70000E-01
2.90000E-01
3.10000E-01
3.30000E-01
3.50000E-01
3.70000E-01
3.90000E-01
4.10000E-01
4.30000E-01
4.500OE-01
4.70000E-01
4.90000E-01
5.10000E-01
5.30000E-01
5.500OOE-01
5.70000E-01
5.90000E-01
6.10000E-01
6.30000E-01
6.500OOE-01
6.70000E-01
6.90000E-01
7.10000E-01
7.30000E-01
7.500OOE-01
7.70000E-01
7.90000E-01
175
Distribution
MeV-3
1.22267E+01
6.36775E+00
2.58426E+00
1.03190E+00
4.41580E-01
2.02983E-01
9.93876E-02
5.12074E-02
2.75328E-02
1.53435E-02
8.82522E-03
5.23212E-03
3.19633E-03
2.01648E-03
1.31735E-03
8.93547E-04
6..30589E-04
4.62941E-04
3.52686E-04
2.77576E-04
2.24392E-04
1.85249E-04
1.55329E-04
1.31655E-04
1.12404E-04
9.64340E-05
8.29638E-05
7.14650E-05
6.15834E-05
5.30697E-05
4.57178E-05
3.93541E-05
3.38437E-05
2.90785E-05
2.49675E-05
2.14237E-05
1.83687E-05
1.57370E-05
1.34731E-05
1.15296E-05
Appendix D
The OOPS and OHIPS matrix
elements
We measured the OOPS and OHIPS in the fall of 1996 by using various targets. The
majority data are taken from the
12C
target, some are taken from the liquid hydrogen.
The detailed analysis in the Chapter 3 in this thesis, here we listed the results.
D.1
OOPS Matrix Elements
The OOPS target quantities are calculated as
60 +
< 6|x > xf+ < 610 > f+ < 6xO >xfOf+ < 6|02 > 02
< j y2 > OfY2 + 6102
+
9t =6
(D.1)
o + < 010 > Of+ < |x6 > xfOf+ < |x2 > x2
+
<9Ixy > Xfyy+ < Ojx0 2 >
(D.2)
Xf02+ <9Ix 2 0 > X2f
Ot = 0+ < 01y > Yj .
(D.3)
The unit in these equations are cm for the length, mrad for the angle, % for the J
176
Table D.1: OOPS 6 Matrix Elements
OOPS B
0.00
4.39
2.15E-2
-4.42E-3
OOPS C
-0.03
4.46
2.02E-2
5.57E-3
60
< 61X >
< 61x >
< 610 >
OOPS A
0.28
4.64
2.57E-2
-4.25E-3
< 6102 >
7.71E-5
1.15E-4
7.29E-5
>
4.48E-5
-3.50E-5
-2.29E-5
< 61y 2 >
< 6l# 2 >
-1.04E-3
0.00
7.03E-4
0.00
3.48E-3
2.11E-5
< 610y
2
Table D.2: OOPS 0 Matrix Elements
OOPS A
-0.19
-0.27
-4.26
-3.69E-2
0.32
-7.65E-3
OOPS B
3.34
-0.28
-4.08
-2.83E-2
0.0
0.0
OOPS C
0.08
-0.28
-3.58
-2.89E-2
0.36
0.0
>
0.0
0.0
-1.41E-4
< O|X20 >
0.00
0.00
2.56E-3
00
>
< |x >
< 0x0 >
< 01|
< 0X2
>
< Oxy >
<
2
O|x0
Table D.3: OOPS # Matrix Elements
00
<
#|y
>
OOPS A
0.00
1.10
OOPS B
0.00
1.09
177
OOPS C
0.00
1.10
D.2
OHIPS Matrix Elements
First there is an offset correction for the chamber in the OHIPS.
Xc
=
Xch+0. 4
Yc
=
Ych - 0.24 - 0.0022x, + 0.0002x,
(D.4)
,
7r2
ch --
0C=
=C
Where Xch,
Ych,
0
ch
Och +
+ 2.3 - 1.605xc + 0.0017xc ,
4
3.74 + 0.093xc - 0.0007xc
(D.5)
(D.6)
(D.7)
and Och are the raw quantities of chamber coordinates; and x,, yc, Oc
and Oc are the corrected quantities of chamber coordinates.
The target quantities can be expressed as following:
Sr=<xO~z'ye"k4> x'6y"'c",
(D.8)
k,1,m,n
6t
=
S
<6X40' y"n
> Xkoly"c
(D.9)
< yxlyk
> Xk0'y",q,
(D.10)
k,l,m,n
yt
=
S
y,"Onq5
k,l,m,n
ot
=
5
<q
4X9Ily"onq
>
0'l y"c ,
(D.11)
k,l,m,n
The units in above equation are [cm] for length, [mrad] for angle and [%] for 6
178
Table D.4: OHIPS Matrix Elements
Olk Im n I <6|x9'y?#q"
o o0 0
00 0 1
00 0 2
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 0 2
0 1 1 0
0 1 1 1
0 20 1
03 0 1
1 0 0 0
1 00 1
1 0 1 2
1 1 00
2 00 0
3 00 0
5 00 0
8.936E-1
0.000
1.320E-4
0.000
0.000
6.180E-3
0.000
-3.857E-6
0.000
0.000
0.000
0.000
1.658E-1
0.000
0.000
1.769E-4
-2.099E-4
0.000
8.210E-9
>
11
< O|xk'
y '">
0.000
0.000
0.000
0.000
2.636E-2
-1.289
0.000
-1.204E-4
0.000
8.027E-4
0.000
0.000
0.000
0.000
0.000
3.237E-3
0.000
0.000
0.000
179
< yIx 9'y '" >
0.000
-6.916E-2
0.000
-7.987E-1
0.000
0.000
-1.542E-3
0.000
0.000
0.000
4.571E-5
0.000
0.000
-1.680E-3
0.000
0.000
0.000
0.000
0.000
I < O|x'y?@n
0.000
-1.097
0.000
5.273E-1
0.000
-2.424E-2
-4.281E-3
0.000
1.221E-2
0.000
1.167E-4
1.656E-6
0.000
-5.709E-3
5.346E-5
0.000
-7.709E-5
-7.709E-5
0.000
>
Appendix E
Turtle Models of OOPS and
OHIPS
(This is a TURTLE deck for Joe Mandeville's version of the program.)
(DESIGN-MOMENTUM 0.625)
(This is an OOPS module deck)
(in a coincident simulation.)
(Use second order optics and enforce aperture in the magnets.)
SECOND ON
APERTURES ON
(Give the dipole vertical and horizontal width/2.)
(Following slits are more restrictive)
DIPOLE-APERTURE 15.24,4.1275
(See the TURTLE manual for fringe fields;)
FRINGE-FIELD .7,4.4
(Write the target coordinates to the output file.)
DETECTOR
180
(Shift any target positions here.)
(SHIFT 0. 0. 0. 0. 0. 0.)
(Drift to the target chamber window)
DRIFT 0.254
(scattering chamber window)
(
---
mass [MeV], L/L.r)
( Kapton dens = 1.42 )
( CH2
dens = 0.92 -- 0.95 g/cm^3
(air
L_r = 44.8 g/cm^2
)
L_r = 30420 cm
)
(MULTIPLE-SCATTER 938.0 1.04E-4)
(Drift through air )
DRIFT 1.0423
(air--multiple scattering)
(MULTIPLE-SCATTER 938. 3.2873E-3)
(spectrometer entrance window )
(MULTIPLE-SCATTER 938. 1.04E-4)
(The vertical acceptance of the front collimator )
(test run is used below.
This corresponds to an aperture of +-31mr)
(in theta-target.)
RECTANGULAR-SLIT 1 4.1778
(With the collimator insert used for the data cycle )
(of 1991, the acceptance in theta-target was reduced to +-25 mr.)
(RECTANGULAR-SLIT 1
3.24075)
(The horizontal acceptance is +-12 mr.)
RECTANGULAR-SLIT 3
1.5634
DETECTOR
181
(Drift to the effective field boundary of the dipole )
(the front collimator.)
DRIFT .058958
(0 0 P S
D I P O L E)
(Model the OOPS dipole.
The total distance is 1.317366 m,)
(and the field is 6 kG.
We have partitioned the dipole )
(into many parts to include the baffles as slits.)
(beginning of dipole field inside front collimator)
OFFSET-RECTANGULAR-SLIT 1
-4.22
POLE-FACE-ROTATION
12.8616
Dipole
0.059959
4.17
6.0
0.00000
(end of top of front collimator)
OFFSET-RECTANGULAR-SLIT 1
-4.36
+4.40
OFFSET-RECTANGULAR-SLIT 3
-1.72
+1.72
Dipole
0.018385
6.0
0.00000
(end of bottom of front collimator)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-7.30
0.284477
+4.50
6.0
0.00000
(1st top baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-6.10
0.068472
+11.23
6.0
0.00000
(2nd top baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-6.33
0.081898
+10.45
6.0
0.00000
(1st bottom baffle, 3rd top baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-6.61
0.070855
(2nd bottom baffle)
182
+6.46
6.0
0.00000
OFFSET-RECTANGULAR-SLIT 1
Dipole
-11.39
0.038675
+6.67
6.0
0.00000
(4th top baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-6.90
0.020247
+8.11
6.0
0.00000
(3rd bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-8.18
0.050833
+6.89
6.0
0.00000
(4th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-11.28
0.044326
+7.03
6.0
0.00000
(5th top baffle, 5th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-7.31
0.040070
+7.17
6.0
0.00000
(6th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-9.11
0.037596
+7.28
6.0
0.00000
(7th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-11.07
0.036304
+7.40
6.0
0.00000
(8th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-12.90
0.022248
+7.51
6.0
0.00000
(6th top baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-7.66
0.013061
+8.34
6.0
0.00000
(9th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
Dipole
-8.27
0.035989
(10th bottom baffle)
183
+7.63
6.0
0.00000
OFFSET-RECTANGULAR-SLIT 1
-9.94
Dipole
0.037747
+7.73
6.0
0.00000
(11th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
-11.62
Dipole
0.041258
+7.83
6.0
0.00000
(7th top baffle, 12th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
-8.07
Dipole
0.046455
+7.96
6.0
0.00000
(13th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
-10.05
Dipole
0.055635
+8.08
6.0
0.00000
(14th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
-12.31
Dipole
0.070104
+8.27
6.0
0.00000
(15th bottom baffle)
OFFSET-RECTANGULAR-SLIT 1
-12.05
Dipole
0.142960
POLE-FACE-ROTATION
+8.45
6.0
0.00000
8.8616
(end of dipole field --
still inside vacuum box)
OFFSET-RECTANGULAR-SLIT 1
-10.15
+12.98
(The end of the rear flange of the dipole vacuum box)
DRIFT .089060
ELLIPTICAL-SLIT
1
RECTANGULAR-SLIT 3
8.62
3
8.62
3.33248
DRIFT .012692
(A circular lead collimator between the dipole and quad kills)
(bad rays before they enter the quad.)
(The front of the ring collimator)
184
ELLIPTICAL-SLIT
1
8.62
3
8.62
DRIFT .063492
(The end of the ring collimator)
ELLIPTICAL-SLIT
1
8.62
8.62
3
(The dipole-quad distance for the North Hall OOPS is slightly
This is presumably taken up here )
(shorter than the design value.
(in the bellows.)
DRIFT .049446
(DRIFT .039446
--This is the value for the North Hall OOPS)
This is the end of the )
(The beginning of the quad pipe.
(<7" diameter region; the pipe inner diameter is <8".)
ELLIPTICAL-SLIT
1
8.62
8.62
3
DRIFT .048804
(0 0 P S
Q U A D R U P O L E)
(Model the OOPS quadrupole.
The total length is .6925 m.)
(The dipole/quad field ratio is 1.185027.)
(The dipole/quad field ratio for the OOPS design is 1.185972.)
(this slightly changes the design field below to 5.05914.)
QUADRUPOLE .115417 5.063176 9.995
QUADRUPOLE .115417 5.063176 9.995
QUADRUPOLE .115417 5.063176 9.995
QUADRUPOLE .115417 5.063176 9.995
QUADRUPOLE .115417 5.063176 9.995
QUADRUPOLE .115417 5.063176 9.995
185
DRIFT .123635
(The end of the quad vacuum pipe)
ELLIPTICAL-SLIT 1 9.995 3 9.995
DRIFT .076215
(The front edge of the OOPS rear vacuum collimator)
(The lead plate number 6)
RECTANGULAR-SLIT 1
6.8707
RECTANGULAR-SLIT 3
11.430
DRIFT 0.04445
(The lead plate number 5)
RECTANGULAR-SLIT 1 6.4389
RECTANGULAR-SLIT 3 11.9253
DRIFT 0.04445
(The lead plate number 4)
RECTANGULAR-SLIT 1 6.0007
RECTANGULAR-SLIT 3 12.433
DRIFT 0.04445
(The lead plate number 3)
RECTANGULAR-SLIT 1 5.5753
RECTANGULAR-SLIT 3 12.9413
DRIFT 0.04445
(The lead plate number 2)
RECTANGULAR-SLIT 1 5.1943
RECTANGULAR-SLIT 3 13.4493
186
DRIFT 0.04445
(The lead plate number 1)
RECTANGULAR-SLIT 1 3.7503
RECTANGULAR-SLIT 3 12.9286
DRIFT 0.0381
(The back side of plate number 1)
RECTANGULAR-SLIT 1 3.7503
RECTANGULAR-SLIT 3 12.9286
(The rear window flange of the quad vacuum box extension)
DRIFT .052507
RECTANGULAR-SLIT 1
5.08
RECTANGULAR-SLIT 3 17.78
(0 0 P S
D E T E C T 0 R
S Y S T E M)
(Note that the HDCs for the North Hall are off center.
Future OOPS)
(modules will not be this way.)
DRIFT .050033
(The 1st HDC intersects the center of the focal plane)
RECTANGULAR-SLIT 1 6.5
RECTANGULAR-SLIT 3 14.0
(When we are reconstructing data or wish for some other reason )
(to save the standard focal plane variables, we must include a )
(DETECTOR card here since this is normal position in z for the )
(focal plane variables.)
DETECTOR 0.03 3. 0.03 3.
187
DRIFT .127
(The 2nd HDC)
RECTANGULAR-SLIT 1 7.0
RECTANGULAR-SLIT 3 15.5
DRIFT .127
(The 3rd HDC)
RECTANGULAR-SLIT 1 8.0
RECTANGULAR-SLIT 3 17.0
DRIFT .0753
(The 1st scintillator)
RECTANGULAR-SLIT 1 8.890
RECTANGULAR-SLIT 3 19.05
DRIFT .0508
(The 2nd scintillator)
RECTANGULAR-SLIT 1 8.890
RECTANGULAR-SLIT 3 19.05
DRIFT .0762
(The 3rd scintillator)
(The trigger requires that all three scintillators were hit.)
(Generally, we just check to see that the last scintillator was hit.)
RECTANGULAR-SLIT 1 8.890
RECTANGULAR-SLIT 3 19.05
(Put a detector card here to see if particles make it this far.)
(Call this the trigger.)
DETECTOR
188
(This is a TURTLE deck for Joe Mandeville's version of the program.)
(DESIGN-MOMENTUM 0.2691)
(This is an OHIPS module deck )
(in a coincident simulation.)
(It is in the HIGH RESOLUTION or NORMAL MODE,)
( which is defined by -Q +Q)
(The LOW RESOLUTION or HIGH THETA ACCEPTANCE mode,)
( which is defined by +Q
-Q)
(This file is derived from the following sources:)
)
( 1: Thesis of Robert Steven Turkey Feb 1984
( 2: A drawing file of the
design OHIPS detection system)
( 3: Dan Tigers 1.77M OHIPS Turtle file
)
(Use second order optics and enforce APERTURES in the magnets.)
SECOND
ON
APERTURES ON
(Write the target coordinates to the output file.)
DETECTOR
(Shift any target positions here.)
(SHIFT 0. 0. 0. 0. 0. 0.)
(first drift space)
(scattering chamber vacuum)
DRIFT 0.254
(scattering chamber window
(
---
mass
)
[MeV], L/L-r)
189
(MULTIPLE-SCATTER 0. 0.9200 )
(spectrometer entrance window )
(MULTIPLE-SCATTER 938. 1.04E-3)
(vacuum to squads collimator ---
Tieger collimator)
(DRIFT 1.311)
(vacuum to quads collimator ---
Vellidis collimator)
(DRIFT 1.152)
DRIFT 1.3153
(---
Front window --- )
(The vertical
acceptance is 17.46 cm.)
RECTANGULAR-SLIT 1
8.65
(The horizontal acceptance is
RECTANGULAR-SLIT 3
(---
(---
3.75
19.255 cm thickness ---
DRIFT
7.62 cm.)
)
0.19255
Rear window --- )
(The vertical
acceptance is 19.70 cm.)
RECTANGULAR-SLIT 1
10.0
(The horizontal acceptance is
RECTANGULAR-SLIT 3
9.84 cm.)
4.3
DETECTOR
(---0 H I P S
F I R S T
Q U A D R U PO
(drift to the entrance of the first quad)
DRIFT
0.2406
190
L E
--- )
(vacuum pipe)
ELLIPTICAL-SLIT
1.
13.97
3.
13.97
(negative field for "high resolution"
(---- NEG field for "normal"
mode)
mode)
(QUADRUPOLE 0.708 -1.65012 15.24)
(----
POS field for "reverse"
QUADRUPOLE 0.708
(---0 H I P S
mode)
1.86913 15.24
S E C 0 N D
Q U A D R U PO
(drift to the entrance of the second quad)
DRIFT 0.1307
(vacuum pipe)
ELLIPTICAL-SLIT
1.
13.97
3.
13.97
(positive field for "high resolution"
( ----
POS field for "normal"
(QUADRUPOLE 0.708
( ----
mode)
mode)
0.63626 15.24)
NEG field for "reverse"
mode)
QUADRUPOLE 0.708 -1.80225 15.24
(vacuum pipe)
ELLIPTICAL-SLIT
1.
13.97
13.97
3.
DRIFT 0.262175
(transition piece)
RECTANGULAR-SLIT 1 20.32
RECTANGULAR-SLIT 3 9.525
(
---
0 H I P S
D I P 0 L E
---
)
191
L E--)
(drift to the entrance of the dipole)
DRIFT 0.2008
(vacuum pipe)
RECTANGULAR-SLIT 1 21.2725
RECTANGULAR-SLIT 3 9.6043
(Give the dipole vertical and horizontal width/2.)
(The subsequent slits are more restrictive,)
(so these are effectively ignored.)
DIPOLE-APERTURE 20.32,9.6043
(See the TURTLE manual for fringe fields; )
(this is unclamped Rogowski.)
FRINGE-FIELD 0.7,4.4
(dipole field)
POLE-FACE-ROTATION 0.0
DIPOLE
3.
3.5339
0.0
(vacuum pipe)
RECTANGULAR-SLIT 1 20.32
RECTANGULAR-SLIT 3 9.604
(clamped Rogowski)
FRINGE-FIELD 0.4,4.4
DIPOLE
0.9898 3.5339
POLE-FACE-ROTATION
0.0
0.0
(vacuum pipe)
RECTANGULAR-SLIT 1 20.32
RECTANGULAR-SLIT 3 9.6043
(Drift 1.626 m to the center of the focal plane)
(DRIFT 1.626)
DRIFT 0.534
(vacuum pipe)
192
RECTANGULAR-SLIT
1 38.1
RECTANGULAR-SLIT
3 8.6
DRIFT 0.457
(vacuum pipe)
RECTANGULAR-SLIT
1 38.1
RECTANGULAR-SLIT
3 15.3
(multiple scattering on exit)
(MULTIPLE-SCATTER 0. 0.9200 )
( ---
0 H I P S
D E T E C T 0 R
S Y S T E M
---
)
(vdcx)
( Wire chamber --VDC 1-- Low momentum side
DRIFT
)
0.4106
OFFSET-RECTANGULAR-SLIT 1
-22.45
1000
RECTANGULAR-SLIT 3 8.89
( Wire chamber --VDC 2-- Low momentum side
)
DRIFT 0.04625
OFFSET-RECTANGULAR-SLIT 1
-33.5
1000
RECTANGULAR-SLIT 3 15.0
(Center of focal plane:center of VDCX1)
DRIFT 0.17828
DETECTOR .03 3. .03 3.
(measurement errors: dx,dy = .03 cm ; dth,dph = 3 mr)
( Scintillator --S1-- Low momentum side
DRIFT 0.03556
193
)
OFFSET-RECTANGULAR-SLIT 1
-21.556
1000
RECTANGULAR-SLIT 3 15.0
( Drift to the center of VDCX2)
DRIFT 0.121285
( Wire chamber --VDC 1-- High momentum side )
( No cut on y)
DRIFT 0.0677
OFFSET-RECTANGULAR-SLIT 1
-1000
22.45
(Drift to the center of S1)
DRIFT 0.026594
( Wire chamber --VDC 2-- High momentum side )
( No cut on y)
DRIFT 0.2049
OFFSET-RECTANGULAR-SLIT 1
-1000
29.92
( Scintillator --S1-- High momentum side )
DRIFT 0.010668
OFFSET-RECTANGULAR-SLIT 1
( Scintillator --S2--
-1000
)
DRIFT 0.26703
RECTANGULAR-SLIT 1 30.48000
RECTANGULAR-SLIT 3 10.16
( The Cerenkov detector )
194
21.556
DRIFT 0.11270
RECTANGULAR-SLIT 1 61.27750
RECTANGULAR-SLIT 3 23.8125
DRIFT 0.71999
RECTANGULAR-SLIT 1 61.27750
RECTANGULAR-SLIT 3 23.8125
DRIFT 0.08573
RECTANGULAR-SLIT 1 61.27750
RECTANGULAR-SLIT 3 23.8125
( Scintillator --S3--
)
DRIFT 0.13452
RECTANGULAR-SLIT 1 35.56
RECTANGULAR-SLIT 3 11.43
(Put a detector card here to see if particles make it this far.)
(Call this the trigger.)
DETECTOR
( PbG )
DRIFT 0.07810
RECTANGULAR-SLIT 1 36.67252
RECTANGULAR-SLIT 3 12.5
DRIFT 0.20955
RECTANGULAR-SLIT 1 36.67252
RECTANGULAR-SLIT 3 12.5
195
Appendix F
At measurement in
12 C(e,
e'p)
reaction
Before this experiment, we did two experiments of Alt measurement in
12C(e, e'p)
reaction. In one experiment, OOPS A and OOPS C were used to detect the in-plane
proton; in another experiment, OOPS A and OOPS C were used to detect the inplane proton at the same kinematics condition. The kinematics condition for both
experiments are as follows:
Ebeam energy = 660MeV , Oe = 33.71* ,
pi = 50.40
,
9
9 lab =
75.00 ,
(F.2)
590.20 ,
(F.3)
p2 =
Pp = 321.23MeV/c , E=
12.30 , OP'
(F.1)
= 13.890
(F.4)
The data analysis follows the procedure which is described in this thesis. Here is
the typical spectrum of the phase space for two OOPS' in Fig. F-1
The Alt is obtained as follows:
ALT
NA NA +
196
Nc
Nc
(F.5)
Phase Space cut in Min(AC) Carbon
1110
100
oops C
-
forward
90
80
-Ar
70
60
50
70
60
50
80
90
100
110
130
120
140
Pmis(MeV)
Figure F-1: A typical phase space in q - w dimension. The data with in black box
are used to extract the Alt asymmetry.
where NA and Nc are:
N=
NA-
_
-V
,N
N fcf
EC -V
(F.6)
Where N A and NF are the true counts, EA and Ec are the efficiencies, ff. and fcde
are the radiative correction factors and VA and Vc are the phase space volumes for
for both OOPS. The statistical error can be calculated by following:
6 ALTr =
(1
JAT (NA +NC)2
4NA2 2Nc + 4NC22NA.
197
(F.7)
In the OOPS A and OOPS C configuration, we obtain the results as follows:
Alt = -0.201 ± 0.032
,
(F.8)
where 0.032 is the statistical error. In the OOPS A and OOPS B configuration, we
obtain the results as follows:
Alt = -0.187 ± 0.021,
(F.9)
where 0.021 is the statistical error.
We conclude that the same physical quantity (Alt) can be obtained through either
set of OOPS configuration. A similar experiment was performed which is the subject
of the thesis of Dr. Xiaodong Jiang [51].
198
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