MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT TOTAL CROSS SECTION DIFFERENCE FOR THE

MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT TOTAL CROSS SECTION DIFFERENCE FOR THE
MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT
TOTAL CROSS SECTION DIFFERENCE T FOR THE
SCATTERING OF POLARIZED NEUTRONS
FROM POLARIZED PROTONS
by
Wesley Scott Wilburn
Department of Physics
Duke University
Date:
Approved:
N. Russell Roberson, Supervisor
Lawrence E. Evans
Daniel J. Gauthier
Christopher R. Gould
Werner Tornow
Dissertation submitted in partial fulllment of
the requirements for the degree of Doctor
of Philosophy in the Department of
Physics in the Graduate School
of Duke University
1993
ABSTRACT
(Physics-Nuclear)
MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT
TOTAL CROSS SECTION DIFFERENCE T FOR THE
SCATTERING OF POLARIZED NEUTRONS
FROM POLARIZED PROTONS
by
Wesley Scott Wilburn
Department of Physics
Duke University
Date:
Approved:
N. Russell Roberson, Supervisor
Lawrence E. Evans
Daniel J. Gauthier
Christopher R. Gould
Werner Tornow
An abstract of a dissertation submitted in partial
fulllment of the requirements for the degree
of Doctor of Philosophy in the Department
of Physics in the Graduate School
of Duke University
1993
MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT
TOTAL CROSS SECTION DIFFERENCE T FOR THE
SCATTERING OF POLARIZED NEUTRONS
FROM POLARIZED PROTONS
by
Wesley Scott Wilburn
Although the existence of a tensor component in the nucleon-nucleon interaction has
been known since the discovery of the electric quadrupole moment of the deuteron, its
strength has remained poorly determined, especially at low energies. The tensor force
is also important as it is responsible for much of the binding energy of the triton, and it
contributes signicantly to the binding of other few-nucleon systems. At low energies,
the strength of the tensor interaction in neutron-proton scattering is characterized by
the 1 phase-shift parameter which gives the mixing between the 3S1 and 3D1 states.
Most observables which are sensitive to 1, such as the spin-correlation coecients
Ayy () and Azz (), are also sensitive to other phase-shift parameters, particularly the
phase shifts associated with the 1P1 and 3P1 states. The longitudinal and transverse
spin-dependent dierences in total cross section, L and T , however, are very
sensitive to 1 and insensitive to most other phase-shift parameters. In addition, L
and T are predicted to cross through zero, allowing very accurate measurements
to be made at these energies as they are unaected by most systematic errors.
Measurements of T have been made at six energies in the region of the zerocrossing using a polarized proton target and a polarized neutron beam. The polarized
proton target is of the brute-force type and consists of TiH2 cooled to below 17 mK
by a 3He-4He dilution refrigerator in a 7 T magnetic eld. The polarized neutron
beam is produced via the 3H(~p,~n)3He reaction using a polarized proton beam obtained
from the Triangle Universities Nuclear Laboratory atomic beam polarized ion source
and accelerated by a tandem Van de Graa accelerator. The beam polarization is
monitored by a carbon-foil polarimeter. The temperature of the dilution refrigerator
is measured by 3He melting curve thermometers. In addition, polarization transfer
i
coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies
using a 4He neutron polarimeter.
A zero-crossing energy of Ezc = 5:08 0:10 MeV has been observed for T . A
simple phase-shift analysis using this value results in 1 = 0:30 0.17 . This result
is somewhat lower than predicted by the full Bonn potential, which has the smallest
tensor force of all realistic nucleon-nucleon potential models.
0
ii
ACKNOWLEDGMENTS
The success of this project is due to the hard work of many people. I would like to
rst thank my thesis advisor, Dr. Russell Roberson for teaching me the art of experiment and for his advice, encouragement, and patience. I am grateful to Dr. Chris
Gould who has been a teacher and advisor, beginning when I was an undergraduate
and continuing to this day. I have greatly enjoyed working with Dr. David Haase
who has taught me cryogenic techniques and more. To Dr. Werner Tornow, who
rst suggested this measurement, I am greatly indebted for his tireless eorts and
considerable expertise.
I would like to thank Chris Keith and Paul Human for helping to set up and
run the experiments and for doing whatever was necessary to get the job done. Tim
Murphy and Brian Raichle deserve thanks for the many midnight shifts they worked.
Dr. Jim Koster's help in the early days of the experiment was invaluable.
An experiment of this complexity would not be possible without a superb technical
sta such as the one at TUNL. The machinists in the instrument shop, Al Lovette,
Gene Harris, and Bob Hogan, could make anything I could draw, and almost always
had a better way of doing it. Sidney Edwards and Pat Mulkey in the electronics shop
deserve thanks for either building, xing, or modifying much of the electronics in
this experiment, often on short notice. Paul Carter, John Dunham, and Ken Sweeton
have my gratitude for xing what was broken and for keeping the lab running, despite
vacuum failures, chiller woes, and water main breaks.
I have beneted from many discussions on the theoretical aspects of this work
with Dr. Alec Schramm. I appreciate the timely help of Dr. H. O. Klages at KFA
in Karslruhe, Germany who provided the titanium hydride for the target. Without
this material, the experiment would have been delayed for several months. Chris
Westerfeldt has been a source of help and advice both before and during this project.
Thanks are due also to Dr. Tom Clegg and Dr. Eric Crosson for their work with the
TUNL Polarized Ion Source.
I would like to thank my parents, Ed and Mary Wilburn, for encouraging my
interest in science and for making my education possible. Finally, I would like to
thank my son Grey for his smiles and laughs. I only hope he will be as happy in his
pursuits as I have been in mine. This dissertation is dedicated to my wife Dianne
whose constant love and encouragement has meant so much to me during this journey.
iii
Contents
Abstract
i
Acknowledgments
iii
List of Figures
vi
List of Tables
ix
1 Introduction
1
2 Theoretical Overview
3
2.1
2.2
2.3
2.4
Nuclear Tensor Interaction : : : : : : : : : : : :
Phase-Shift Parameters : : : : : : : : : : : : : :
Spin-Dependent Total Cross Section Dierences
Potential Models : : : : : : : : : : : : : : : : :
3 Experimental Apparatus
3.1 Charged-Particle Beam : : : : : : :
3.1.1 Polarized Ion Source : : : :
3.1.2 Acceleration and Transport
3.1.3 Spin Transport : : : : : : :
3.2 Charged-Particle Polarimeter : : :
3.3 Neutron Beam : : : : : : : : : : : :
3.3.1 Neutron Beam Production :
3.3.2 Neutron Beam Collimation :
3.3.3 Neutron Detection : : : : :
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3.4 Neutron Polarimeter : : : : : : : : : : : : :
3.5 Polarized Proton Target : : : : : : : : : : :
3.5.1 Dilution Refrigerator : : : : : : : : :
3.5.2 Titanium Hydride Target : : : : : :
3.5.3 Thermometry : : : : : : : : : : : : :
3.6 Data Acquisition Electronics : : : : : : : : :
3.6.1 T Measurements : : : : : : : : : :
3.6.2 Neutron Polarization Measurements :
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4 Experimental Procedure
4.1 T Measurements : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.2 Kyy Measurements for the 3H(~p,~n)3He Reaction : : : : : : : : : : : :
0
5 Data Analysis
5.1 T Measurements : : : : : : : : : : : : : : : : : : : : : : : :
5.1.1 Calculation of the Neutron-Transmission Asymmetries
5.1.2 Calculation of the Average Neutron Beam Polarization
5.1.3 Calculation of T : : : : : : : : : : : : : : : : : : : :
5.1.4 Determination of the Zero-Crossing Energy of T : :
5.2 Kyy Measurements for the 3H(~p,~n)3He Reaction : : : : : : : :
5.2.1 Determination of Kyy : : : : : : : : : : : : : : : : : : :
5.2.2 Interpolation of Kyy Data : : : : : : : : : : : : : : : :
0
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102
106
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108
110
6 Comparison of Data with Potential Models
114
7 Conclusions and Summary
118
A Calculation of Neutron Depolarization Due to Magnetic Fields
120
B The Spin-Dependent Total Cross Section Dierence 151
tensor
References
154
Biography
161
v
List of Figures
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4
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3.1 Energy Level Diagram for the Hydrogen Atom : : : : : : : : : : : : :
3.2 Low Energy Beam Transport : : : : : : : : : : : : : : : : : : : : : : :
3.3 High Energy Beam Transport : : : : : : : : : : : : : : : : : : : : : :
33
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36
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
Feynman Diagram for One-Pion Exchange : : : : : : : : : : : :
Potential-Model Predictions of the 1 Mixing Parameter : : : : :
Triton Binding Energy versus Deuteron D -State Probability : :
Potential-Model Predictions for L at Low Energies : : : : : :
Potential-Model Predictions for T at Low Energies : : : : : :
Sensitivity of L to a 1 Variation in 1S0 Using Arndt SP89
Sensitivity of T to a 1 Variation in 1S0 Using Arndt SP89
Sensitivity of L to a 1 Variation in 3P0 Using Arndt SP89
Sensitivity of T to a 1 Variation in 3P0 Using Arndt SP89
Sensitivity of L to a 1 Variation in 1P1 Using Arndt SP89
Sensitivity of T to a 1 Variation in 1P1 Using Arndt SP89
Sensitivity of L to a 1 Variation in 3S1 Using Arndt SP89
Sensitivity of T to a 1 Variation in 3S1 Using Arndt SP89
Sensitivity of L to a 1 Variation in 3P1 Using Arndt SP89
Sensitivity of T to a 1 Variation in 3P1 Using Arndt SP89
Sensitivity of L to a 1 Variation in 3D1 Using Arndt SP89
Sensitivity of T to a 1 Variation in 3D1 Using Arndt SP89
Sensitivity of L to a 1 Variation in 1 Using Arndt SP89 :
Sensitivity of T to a 1 Variation in 1 Using Arndt SP89 :
Zero-Crossing Predictions for L : : : : : : : : : : : : : : : : :
Zero-Crossing Predictions for T : : : : : : : : : : : : : : : : :
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3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
3.33
3.34
3.35
3.36
Beam Current Integration Electronics : : : : : : : : : : : : : : : : : :
Madison Convention Coordinate System : : : : : : : : : : : : : : : :
Spin Precession for a Proton Beam, Transverse Horizontal on Target :
Wien Filter Calibration : : : : : : : : : : : : : : : : : : : : : : : : : :
Proton Beam Polarimeter : : : : : : : : : : : : : : : : : : : : : : : :
Proton Beam Polarimeter Electronics : : : : : : : : : : : : : : : : : :
Tritiated-Titanium Foil Holder : : : : : : : : : : : : : : : : : : : : : :
ENDF/B-VI Total Neutron Cross Section of Carbon : : : : : : : : :
ENDF/B-VI Total Neutron Cross Section of Oxygen : : : : : : : : :
Measured Total Neutron Cross Section of Carbon : : : : : : : : : : :
Measured Total Neutron Cross Section of Oxygen : : : : : : : : : : :
Neutron Collimation and Shielding : : : : : : : : : : : : : : : : : : :
Neutron Pre-Collimator : : : : : : : : : : : : : : : : : : : : : : : : :
Neutron Detector Shield : : : : : : : : : : : : : : : : : : : : : : : : :
Neutron Detector Assembly : : : : : : : : : : : : : : : : : : : : : : :
Neutron Detector Electronics : : : : : : : : : : : : : : : : : : : : : :
Neutron Polarimeter : : : : : : : : : : : : : : : : : : : : : : : : : : :
High-Pressure Helium Gas Cell : : : : : : : : : : : : : : : : : : : : :
Helium Cell Filling System : : : : : : : : : : : : : : : : : : : : : : : :
Electronics for the Forward Neutron Detector Pair : : : : : : : : : : :
Electronics for the Backward Neutron Detector Pair : : : : : : : : : :
Coincidence Electronics for the Neutron Polarimeter : : : : : : : : : :
Dilution Refrigerator : : : : : : : : : : : : : : : : : : : : : : : : : : :
Dilution Refrigerator Cryostat : : : : : : : : : : : : : : : : : : : : : :
Titanium Hydride Target : : : : : : : : : : : : : : : : : : : : : : : : :
3
He Melting Curve Thermometer : : : : : : : : : : : : : : : : : : : :
Data Acquisition Electronics for T Measurements : : : : : : : : : :
Data Acquisition Timing Diagram : : : : : : : : : : : : : : : : : : : :
Beam Prole Monitor Feedback Steering Electronics : : : : : : : : : :
Beam Prole Monitor Collector Spectrum : : : : : : : : : : : : : : :
Beam Prole Monitor Fiducial Spectrum : : : : : : : : : : : : : : : :
Neutron Polarimeter Data Acquisition Electronics (Part 1) : : : : : :
Neutron Polarimeter Data Acquisition Electronics (Part 2) : : : : : :
vii
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5.1 Dead-Time Pulser Spectrum : : : : : : : : : : : : : : : : : : : : : : : 94
5.2 T Data with Fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107
5.3 3H(~p,~n)3He Polarization-Transfer Coecient Data : : : : : : : : : : : 112
6.1 Comparison of T Measurements with Potential-Model Predictions : 115
6.2 Comparison of the Measured Ezc with Potential-Model Predictions : : 116
6.3 Comparison of 1 from T Data with Potential-Model Parameters : 117
viii
List of Tables
2.1 Predicted Deuteron Properties for Various Potential Models : : : : :
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Wien Filter Settings : : : : : : : : : : : : : : : : : : : : :
Analyzing Powers for 12C(p,p0)12C at lab = 40 : : : : : :
Calculated Average Neutron Energies and Energy Widths :
Measured Average Neutron Energies : : : : : : : : : : : :
Asymmetry Dilution Factor : : : : : : : : : : : : : : : : :
Calculated Neutron Depolarization : : : : : : : : : : : : :
Eective Analyzing Powers for the Neutron Polarimeter : :
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53
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66
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
Average Neutron-Transmission Asymmetries : : : : : : : : : : : : : :
Analyzing Power Fits for 12C(p,p0)12C at lab = 40 : : : : : : : : : :
Average Proton and Neutron Beam Polarizations : : : : : : : : : : :
Asymmetries Normalized to the Proton Polarimeter Asymmetry : : :
Asymmetries Normalized to the Proton Beam Polarization : : : : : :
Asymmetries Normalized to Beam and Target Polarizations : : : : : :
Calculated Values of T at a Neutron Energy of 1.94 MeV : : : : :
Measured Values of T : : : : : : : : : : : : : : : : : : : : : : : : :
Neutron Counts in the Neutron Polarimeter Detectors : : : : : : : : :
Average Neutron Beam Polarizations : : : : : : : : : : : : : : : : : :
Average Proton Beam Polarizations : : : : : : : : : : : : : : : : : : :
Measured Values of the 3H(~p,~n)3He Polarization-Transfer Coecient :
3 H(~
p,~n)3He Polarization-Transfer Coecient Data : : : : : : : : : : :
Interpolated Values of Kyy : : : : : : : : : : : : : : : : : : : : : : : :
98
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100
103
104
105
105
105
108
109
109
110
111
113
0
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12
Chapter 1
Introduction
Although the existence of a tensor component in the nucleon-nucleon interaction has
been known since the discovery of the electric quadrupole moment of the deuteron,
its strength is still poorly determined, especially at low energies. The tensor force is
also important as it is responsible for much of the binding energy of the triton, and it
contributes signicantly to the binding of other few-nucleon systems. At low energies,
the strength of the tensor interaction in neutron-proton scattering is characterized by
the 1 phase-shift parameter which gives the mixing between the 3S1 and 3D1 states.
The notation used to describe the scattering states is 2S+1LJ , where S is the total spin,
L refers to the orbital angular momentum, `, (S , ` = 0, P , ` = 1, D , ` = 2,
etc.), and J = S + ` is the total angular momentum. Most observables which are
sensitive to 1, such as the spin-correlation coecients Ayy () and Azz (), are also
sensitive to other phase-shift parameters, particularly the phase shifts associated with
the 1P1 and 3P1 states. The longitudinal and transverse spin-dependent dierences in
total cross section, L and T , however, are very sensitive to 1 and insensitive to
most other phase-shift parameters. In addition, L and T are predicted to cross
through zero, allowing very accurate measurements to be made at energies where they
are unaected by most systematic errors.
Measurements of T have been made at six energies in the region of the zerocrossing using a polarized proton target and a polarized neutron beam. The polarized
proton target is of the brute-force type and consists of TiH2 cooled to below 17 mK
in a 7 T magnetic eld. The target is cooled by a 3He -4He dilution refrigerator and
the magnetic eld is produced by a superconducting split-coil solenoid. The polar1
2
Chapter 1. Introduction
ized neutron beam is produced via the 3H(~p,~n)3He reaction using a polarized proton
beam. The proton beam polarization is monitored by a carbon-foil polarimeter. The
temperature of the dilution refrigerator is measured by 3He melting curve thermometers. Transmitted neutrons are detected at 0 using two organic liquid scintillator
cells mounted to photomultiplier tubes.
Because of their importance in the analysis of the T data, polarization-transfer
coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies
using a neutron polarimeter. A high-pressure 4He gas cell is used as an analyzer
and the scattered neutrons are detected in four side detectors. The gas cell is an
active target, scintillating from the recoiling alpha particles. Coincidences between
the scattered neutrons and the recoiling alphas greatly reduce the background counts.
A zero-crossing energy, Ezc , has been extracted from the T data and used in a
simple phase-shift analysis to determine the value of 1 at that energy. The data and
the resulting values of Ezc and 1 are compared to potential-model predictions.
0
Chapter 2
Theoretical Overview
2.1 Nuclear Tensor Interaction
At present, the only satisfactory theories of the nuclear force are meson-exchange
models. These models treat nucleons as fundamental particles which interact through
the exchange of mesons with Yukawa-type potentials. Although nucleons are known
to actually be composite, no realistic quark model of the nuclear interaction has as
yet been constructed due to the computational diculties of such an approach. At
the other extreme, phenomenological models exist, but they are unable to provide
insight into the underlying mechanisms. Thus it is reasonable to look for the origins
of a nuclear tensor interaction in meson exchange.
The Yukawa potential arises from considering the wave equation for massive particles Yuk35].
!
2c2
2
m
2
r ; h 2 = c12 @@t2
(2:1)
Here, m is the mass of the exchanged particle. The solutions to this equation fall o
exponentially with distance, with a decay constant proportional to the mass. The
strength of the interaction is given by a coupling constant g.
;mcr=h
= 4g e r
(2:2)
Since pions have the smallest mass of the mesons involved in the nuclear force, they
will have the largest contribution at long range. Because pions have negative intrinsic
3
4
Chapter 2. Theoretical Overview
E ,´ p´
u´
u´
E,´ –p´
m ps
gps
E ,p
u
gps
u
E,–p
Figure 2.1: Feynman Diagram for One-Pion Exchange
parity, a pseudoscalar coupling is chosen to conserve parity. The Lagrangian for onepion exchange (represented by the Feynman diagram in Figure 2.1) is then of the
form
Lps = gps i
5ps:
(2:3)
The matrix element for this interaction can be written
2
0
u02i
5u2)
Mfi = gps((pu1;i
5pu0)12)(
(2:4)
+ m2ps :
p and p0 are the initial and nal momenta, and u and u0 are the initial and nal wave
functions. In the non-relativistic limit, the wave functions have the form
1
0
s
1
(2:5)
u1 = E2+MM B
@ 1p CA 1
E +M
where M is the nucleon mass, and p are the spin and momentum operators, and is the spin wavefunction. Evaluating the matrix element gives the result
2
Mfi = ;4Mgps2 (1q2q+)(m22 q) (2:6)
ps
2.2. Phase-Shift Parameters
5
with q = p0 ; p. This interaction can be expressed as a momentum-space potential
;
gps2 (1 k)(2 k)
(2:7)
Vps (k) = 4M 2 k2 + m2 :
ps
By performing a three-dimensional Fourier transform, the coordinate-space potential
may be obtained Mac86].
(
"
#) ;mpsr
gps2 m2ps
3
3
Vps (x) = 4 12M 2 1 2 + S12(^x) 1 + m r + (m r)2 e r
(2:8)
ps
ps
S12 is the tensor operator
S12(^x) = 3(1 x^ )(2 x^ ) ; 1 2:
(2:9)
The rst term of the potential is a spin-dependent central interaction while the second
is a tensor interaction which mixes states with dierent orbital angular momentum.
This type of interaction also arises from considering the tensor coupling of vector
mesons such as the rho.
(2:10)
Lt = 2gMv @ ps
This coupling results in a coordinate-space potential
(
"
#) ;mv r
2
2
g
m
3
3
v
v
Vt(x) = 4 12M 2 21 2 ; S12(^x) 1 + m r + (m r)2 e r
(2:11)
v
v
containing a tensor term of the same form but opposite in sign compared to the
pseudoscalar coupling. The tensor force is thus reduced at shorter distances where
the heavier rho contributes to the nuclear interaction.
2.2 Phase-Shift Parameters
The elastic scattering of one spinless particle from another can be described asymptotically as the superposition of an incident plane wave with a scattered spherical
wave Sch68, DeB67]. If only central forces are involved, the orbital angular momentum, `, is conserved and the incident plane wave can be written as a partial-wave
expansion.
i = Aeikz
= A
1
X
`=0
i`(2` + 1)j` (kr)P` (cos )
(2.12)
6
Chapter 2. Theoretical Overview
For large values of r,
1
sin(
kr
;
2 ` )
j` kr
;i(kr;` )
; e+i(kr;` ) :
ie
2kr
2
2
(2.13)
The incident plane wave can then be described (in the asymptotic limit) as the sum
of incoming and outgoing spherical waves and is given by
1
i
h
X
(2:14)
i = 2Akr i`+1(2` + 1) e;i(kr;` 2 ) ; e+i(kr;` 2 ) P` (cos ):
`=0
Since the scattered wave consists only of outgoing spherical waves, it can be included
by modifying the second term of Equation 2.14 by a factor `. The total wave function
is then
1
h
i
X
= 2Akr i`+1(2` + 1) e;i(kr;` 2 ) ; ` e+i(kr;` 2 ) P` (cos ):
(2:15)
`=0
The scattered wave is then obtained by subtracting the incident wave from the total.
s = ; i
1
X
= 2Akr i`+1(2` + 1)(1 ; `)ei(kr;` 2 )P` (cos )
`=0
1
ikr X
e
i
= A 2k r (2` + 1)(1 ; ei` )P` (cos )
`=0
(2.16)
(2.17)
Here ei` = ` and ` are the phase shifts. The total wave function can be written
asymptotically in terms of the scattering amplitude, f (), as
"
ikz #
e
ikz
= A e + f ()
(2:18)
r :
Comparing this result to Equation 2.15 gives
1
X
i
f () = 2k (2` + 1)(1 ; ei` )P` (cos ):
(2:19)
`=0
2.2. Phase-Shift Parameters
7
The dierential cross section is then
() = jf ()j2
X
2
1
1
i
(2.20)
= k2 (2` + 1)e ` sin ` P` (cos )
:
`=0
Integrating over gives the total cross section in terms of the phase shifts.
1
X
4
= k2 (2` + 1) sin2 `
(2:21)
`=0
There exists one scattering eigenstate corresponding to each value of `. The contribution of each eigenstate is determined by its corresponding phase shift, `. The phase
shifts are related to the scattering matrix, S, through the expression
8
>
< e2i` `f = `i = `
Sfi = >
(2:22)
: 0
`f 6= `i
when purely central forces act. The subscripts i and f refer to the initial and nal
states. O-diagonal terms will occur in S if non-central interactions exist.
In a phase-shift description of nucleon-nucleon scattering, the spin 1/2 nature of
the nucleons must be considered. The principal quantum number is now the total
angular momentum, J . Whereas before there was one state for each value of `, there
are now four states for each value of the total angular momentum J , for J 1. There
is a spin-singlet state with orbital angular momentum ` = J and three spin-triplet
states with ` = J ; 1, ` = J , and ` = J + 1. Total spin S is conserved under the
assumption of charge independence, while ` is conserved for pure central forces. Since
the tensor force is manifestly non-central, mixing can occur for states with the same
J , S , and parity, but with dierent ` the triplet states with ` = J ; 1 and ` = J + 1.
For the simplest case in which mixing can occur, J = 1,1 the four scattering states
are:
1P
1
3S
1
3
P1
3D
1
1
There is no mixing for J = 0 because there are no states with ` = J 1.
8
Chapter 2. Theoretical Overview
and mixing of states can occur between 3S1 and 3D1. In the case of proton-proton
and neutron-neutron scattering, only states with total isospin T = 1 exist, whereas
in the case of neutron-proton scattering states with both T = 0 and T = 1 occur
DeB67]. In addition, the Pauli principle requires
(;1)S+T +` = ;1:
(2:23)
Thus, for identical particles, only singlet states with even ` and triplet states with
odd ` are allowed. Mixing can then occur only for states with even J in the pp and
nn cases.
When a tensor interaction exists, the states with ` = J ; 1 and ` = J + 1 are
no longer energy eigenfunctions. Instead, the new states are J and J which are a
mixture of the two states of denite `. The part of the scattering matrix corresponding
to these states has the form Bla52a, Bla52b]
S = U;1e2i U
where
0
U = [email protected] cos J
; sin J
0
= [email protected] J 0
0 J
1
sin J C
A
cos J
1
CA :
(2:24)
(2.25)
(2.26)
J and J are the phase shifts and J is the mixing parameter, in the notation of
Blatt and Biedenharn. By convention, these states are referred to as 3S1 and 3D1,
even though they are not states of pure `, except at E = 0. At low energies, 1
is the phase-shift parameter which most directly characterizes the strength of the
tensor interaction. Because of the dierence in ` of the two states coupled by the
tensor interaction, no mixing can occur at En = 0. At the lowest energies (below
approximately 2 MeV), 1 is constrained by this kinematic consideration and by the
properties of the deuteron.
2.3. Spin-Dependent Total Cross Section Dierences
9
2.3 Spin-Dependent Total Cross Section Dierences
Phase-shift parameters are not directly measurable. Instead, experiments designed
to measure the strength of the tensor force at low energies must measure observables
sensitive to the 1 mixing parameter. Ideally, these observables will not be sensitive
to any other phase-shift parameters. In order to be sensitive to a spin-spin interaction
such as the tensor force, two polarizations must be measured by the observable. Such
observables include the spin-correlation parameters Ayy () and Azz (). Unfortunately,
these measurements are also very dependent upon the 1P1 phase shift except at =
90 which is experimentally very dicult.
Another set of observables are the spin-dependent dierences in total cross section,
, Bug80, Tor88] where
= (0 I ;s) ; (0 I s):
(2:27)
I is the spin of the proton target, and s is the spin of the neutron projectile. In
the simplest case, I and s are along the same axis and can either be longitudinal
or transverse with respect to the direction of propagation. More general cases are
discussed in Appendix B. The longitudinal and transverse cases are dened by
L = (!
(2.28)
) ; (!
! )
T = ("#) ; (""):
(2.29)
Where the top or rst arrow refers to the target spin and the bottom or second refers
to the projectile spin. The total cross section can be decomposed into two parts
= 0 + ss
(2:30)
where 0 is even under a reversal of projectile spin and ss is odd. The total cross
section dierences depend only on the second term.
= ( 0 + ss) ; ( 0 ; ss )
= 2 ss
(2.31)
The spin-spin cross section can then be divided into scalar and tensor parts
(2:32)
ss = s ( 1 2 ) + tS12 10
Chapter 2. Theoretical Overview
where the tensor operator, S12, is dened in Equation 2.9. This gives
L
T
= 2( s + 2 t)
= 2( s ; t):
(2.33)
(2.34)
L and T are then two dierent linear combinations of the scalar and tensor
parts of the spin-spin cross section. These observables can be written in terms of the
phase-shift parameters in the Stapp convention as Tor91]
2
3
;
2
+
cos
2
+
cos
2
+
3
cos
2
1
S
0
3
P
0
1
P
1
66
77
66 ; cos 23S1 ; 3 cos 3P 1 + cos 23D1
77
6
7
p
L = 2 666 ; 4 2 sin(3S1 + 3D1) sin 21 + 5 cos 21D2 777 (2.35)
k 6
77
66 ; cos 23P 2 ; 5 cos 23D2 + cos 23F 2
75
4
p
; 4 6 sin(3P 2 + 3F 2) sin 22 + 2
3
cos
2
;
cos
2
+
3
cos
2
1
S
0
3
P
0
1
P
1
66
77
66 ; cos 23S1 ; 2 cos 23D1
77
6
77
p
77 :
(2.36)
T = k2 666 + 2 2 sin(3S1 + 3D1) sin 21
66
77
64 + 5 cos 21D2 ; 2 cos 23P 2 ; 3 cos 23F 2 75
p
+ 2 6 sin(3P 2 + 3F 2) sin 22 + In these equations, 1S0, for example, refers to the phase shift for the 1S0 state.
2.4 Potential Models
Potential models of the nucleon-nucleon interactions vary considerably in the strength
of the tensor force. In terms of the phase shifts, this variation can be seen in the 1
mixing parameter. Figure 2.2 shows 1 as a function of energy for several potential
models as well as the phase-shift analysis, SM92, of Ardnt Mac87, Bra88, Mac89,
Lac80, Nag78, Arn92]. This variation in tensor strength can also be observed in
the predictions of the properties of the deuteron. Table 2.1 lists the deuteron D state probability, PD , magnetic dipole moment, d , and electric quadrupole moment,
Qd, for various potential models Mac87, Bra88, Mac89, Nag78, Lac80, Rei68, Kel39,
2.4. Potential Models
11
1 (degrees)
5:0
Arndt SM92
Full Bonn
Bonn A
Bonn C
Paris
4:0
3:0
2:0
1:0
0:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
Figure 2.2: Potential-Model Predictions of the 1 Mixing Parameter
30:0
12
Chapter 2. Theoretical Overview
Potential
Qd (fm2)
d (N ) PD (%)
Full Bonn
0.2807
0.8555
4.25
Bonn A
0.274
0.8548
4.4
Bonn B
0.278
0.8514
5.0
Bonn C
0.281
0.8478
5.6
Nijmegen
0.2715
5.4
Paris
0.279
0.853
5.8
Reid Soft Core
0.2796
6.5
Experiment
0.2859
0.857406
0.0003
0.000001
Table 2.1: Predicted Deuteron Properties for Various Potential Models
Eri84]. The binding energy of the triton is also greatly aected by the choice of tensor
strength. Figure 2.3 shows the strong correlation that exists between the deuteron
D -state probability and the triton binding energy Bra88]. Models with a stronger
tensor force, as indicated by a larger PD , underestimate the triton binding energy
more than models with a weaker tensor force.
Predictions of L and T can be calculated from the potential-model phase
shifts. Figures 2.4 and 2.5 show these results for several models and for the Arndt
phase-shift analysis in the energy range of interest. In addition, sensitivity calculations can be performed by varying the phase-shift parameters of one model. For
this purpose, the Arndt phase-shift analysis, SP89, is chosen Arn89]. The phase
shifts are varied by 1 and the resulting shifts in L and T are plotted in Figures 2.6{2.19 for J 1. Higher partial waves do not contribute signicantly to these
observables. These gures show that L and T are more sensitive to 1 than to
the other phase-shift parameters. L and T can be expressed to a good approximation in terms of only 1S0, 3S1, and 1. The sensitivity to 1 is a maximum at
about 10 MeV: approximately 51 mb/degree for L and 26 mb/degree for T . Of
particular interest from an experimental viewpoint are the zero-crossing energies of
L and T , which also display large sensitivities to 1. Measurements made at the
zero-crossing energies are insensitive to most systematic uncertainties. Predictions
for the zero-crossings of L and T are shown in Figures 2.20 and 2.21.
2.4. Potential Models
13
EB (MeV)
;7:0
Experiment
Argonne V14
Bonn A
Bonn B
Bonn C
Paris
Reid Soft-Core
de Tourreil et al
;7:2
;7:4
?
4
;7:6
b
3
3
+
2
4
?
b
;7:8
;8:0
2
;8:2
+
;8:4
;8:6
;8:8
;9:0
7:0
6:5
6:0
5:5
PD (%)
5:0
4:5
4:0
Figure 2.3: Triton Binding Energy versus Deuteron D -State Probability for Various
Potential Models
14
Chapter 2. Theoretical Overview
L (mb)
1000:0
Arndt SM92
Full Bonn
Bonn A/B/C
Paris/Nijmegen
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
2:0
4 :0
6:0
8:0
En (MeV)
10:0
12:0
14:0
Figure 2.4: Potential-Model Predictions for L at Low Energies
2.4. Potential Models
15
T (mb)
1000:0
Arndt SM92
Full Bonn
Bonn A/B/C
Paris/Nijmegen
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
2 :0
4:0
6:0
8:0
En (MeV)
10:0
12:0
14:0
Figure 2.5: Potential-Model Predictions for T at Low Energies
16
Chapter 2. Theoretical Overview
L (mb)
1000:0
S0
S ;1
S +1
1
1
0
1
0
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.6: Sensitivity of L to a 1 Variation in 1S0 Using Arndt SP89
2.4. Potential Models
17
T (mb)
1000:0
S0
S ;1
S +1
1
1
0
1
0
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.7: Sensitivity of T to a 1 Variation in 1S0 Using Arndt SP89
18
Chapter 2. Theoretical Overview
L (mb)
1000:0
P0
P ;1
P +1
3
3
0
3
0
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.8: Sensitivity of L to a 1 Variation in 3P0 Using Arndt SP89
2.4. Potential Models
19
T (mb)
1000:0
P0
P ;1
P +1
3
3
0
3
0
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.9: Sensitivity of T to a 1 Variation in 3P0 Using Arndt SP89
20
Chapter 2. Theoretical Overview
L (mb)
1000:0
P1
P ;1
P +1
1
1
1
1
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.10: Sensitivity of L to a 1 Variation in 1P1 Using Arndt SP89
2.4. Potential Models
21
T (mb)
1000:0
P1
P ;1
P +1
1
1
1
1
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.11: Sensitivity of T to a 1 Variation in 1P1 Using Arndt SP89
22
Chapter 2. Theoretical Overview
L (mb)
1000:0
S1
S ;1
S +1
3
3
1
3
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.12: Sensitivity of L to a 1 Variation in 3S1 Using Arndt SP89
2.4. Potential Models
23
T (mb)
1000:0
S1
S ;1
S +1
3
3
1
3
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.13: Sensitivity of T to a 1 Variation in 3S1 Using Arndt SP89
24
Chapter 2. Theoretical Overview
L (mb)
1000:0
P1
P ;1
P +1
3
3
1
3
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.14: Sensitivity of L to a 1 Variation in 3P1 Using Arndt SP89
2.4. Potential Models
25
T (mb)
1000:0
P1
P ;1
P +1
3
3
1
3
1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.15: Sensitivity of T to a 1 Variation in 3P1 Using Arndt SP89
26
Chapter 2. Theoretical Overview
L (mb)
1000:0
D1
D1 ; 1
D1 + 1
3
3
3
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.16: Sensitivity of L to a 1 Variation in 3D1 Using Arndt SP89
2.4. Potential Models
27
T (mb)
1000:0
D1
D1 ; 1
D1 + 1
3
3
3
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.17: Sensitivity of T to a 1 Variation in 3D1 Using Arndt SP89
28
Chapter 2. Theoretical Overview
L (mb)
1000:0
1
1 ; 1
1 + 1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.18: Sensitivity of L to a 1 Variation in 1 Using Arndt SP89
2.4. Potential Models
29
T (mb)
1000:0
1
1 ; 1
1 + 1
800:0
600:0
400:0
200:0
0:0
;200:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 2.19: Sensitivity of T to a 1 Variation in 1 Using Arndt SP89
30
Chapter 2. Theoretical Overview
L (mb)
100:0
Arndt SM92
Full Bonn
Bonn A
Bonn C
Nijmegen
Paris
50:0
0:0
;50:0
;100:0
5:6
5:8
6 :0
6 :2
6:4
6:6
En (MeV)
6:8
7:0
Figure 2.20: Zero-Crossing Predictions for 7:2
L
7:4
2.4. Potential Models
31
T (mb)
100:0
Arndt SM92
Full Bonn
Bonn A
Bonn C
Paris/Nijmegen
50:0
0:0
;50:0
;100:0
4:0
4:2
4:4
4:6
4:8
5:0
5:2
En (MeV)
5:4
Figure 2.21: Zero-Crossing Predictions for 5:6
T
5:8
6:0
Chapter 3
Experimental Apparatus
3.1 Charged-Particle Beam
Since it is not possible to accelerate neutral particles directly, polarized neutron beams
must be produced as secondary beams from charged-particle reactions. In this experiment, the charged-particle beam consists of polarized protons. The beam is created
by an ion source capable of producing nuclear spin-polarized beams of negative hydrogen ions. The beam must be transported to the accelerator where its energy is
increased and then to the neutron production target where a polarized neutron beam
is created via the 3H(~p,~n)3He reaction. In addition, the spin direction of the beam
must be set to its proper orientation.
3.1.1 Polarized Ion Source
The polarized proton beam used in this experiment is produced in the TUNL Atomic
Beam Polarized Ion Source Cle89]. The ABPIS is capable of producing high-intensity
beams of vector polarized protons, or vector or tensor polarized deuterons. It is an
atomic beam source and it uses an electron cyclotron resonance (ECR) ionizer. Positive beams are extracted directly from the ionizer, while negative beams for acceleration by the tandem Van de Graa are produced by passing the positive beam through
a cesium charge-exchange oven. The source can produce negative hydrogen beams
with microampere intensities and polarizations of approximately 60%. The entire
source rests on a high-voltage platform biased to ;72 kV to provide the beam with
32
3.1. Charged-Particle Beam
33
1
m I=+ 1/2
2
Energy
–1/2
WF2
SF2
–1/2
3
+ 1/2
4
Magnetic Field
Figure 3.1: Energy Level Diagram for the Hydrogen Atom
sucient energy to be transported to the accelerator.
The polarized ion source produces nuclear polarization in a neutral atomic beam
through a combination of two sextupole magnets and two adiabatic fast-passage transition units. The atomic beam is created by a dissociator which uses an RF discharge
to break the molecular bonds of hydrogen. The atomic beam sprays out of a copper
nozzle and passes through two sextupole magnets. The nozzle is cryogenically cooled
and continuously coated with nitrogen to reduce molecular recombination. Atomic
beam type polarized ion sources make use of the separation in energy of the magnetic substates of the hydrogen atom in a magnetic eld (Figure 3.1). The magnetic
eld of a sextupole increases as a linear function of radial distance, thus focusing the
hydrogen states labeled 1 and 2, and defocusing states 3 and 4. This eect is used
to produce atomic polarization. Nuclear polarization is then obtained by using one
34
Chapter 3. Experimental Apparatus
of the two transition units. The rst transition unit operates with a strong magnetic
eld and is called SF2, while the second operates with a weak magnetic eld and is
called WF2. SF2 consists of a DC magnetic eld of approximately 100 G and an
oscillating eld with a frequency of 1450 MHz. This unit produces a transition from
state 2 to state 4, giving a positive nuclear polarization along the z^-direction, Pz ,
dened by
+ ; N;
(3:1)
Pz = N
N +N
+
;
where N+ and N; are the numbers of atoms with nuclear spin projections +1=2 and
;1=2. WF2 consists of a DC magnetic eld of approximately 10 G and an RF eld
with a frequency of approximately 15 MHz. This unit causes a transition from state
1 to state 3, giving a negative Pz . The transition units can be rapidly toggled on
and o by modulating the RF supplies, allowing the beam polarization to be quickly
reversed. The spin state produced by SF2 will be referred to as spin up, while that
produced by WF2 will be called spin down.
After exiting from the sextupoles and transition units, the atomic beam is ionized
by the ECR ionizer. This type of ionizer uses a plasma driven by microwave radiation
to strip the electrons from the neutral hydrogen atoms. The plasma consists mostly
of nitrogen introduced into the ionizer through a mass-"ow controller. Electrons from
this buer gas are excited to high energy by the microwaves to perform the task of
ionization. The plasma is conned in the radial direction by permanent magnets in
a sextupole conguration and in the axial direction by two solenoids. Electrostatic
lenses extract and focus the beam from the ionizer. Following the ionizer, a cesium
oven converts the H+ beam to H; through double charge exchange with an eciency
of approximately 10%. The beam is then accelerated to 26 keV after which it passes
through the Wien lter where the spin direction is rotated (Section 3.1.3). Upon
leaving the Wien lter, the beam receives an additional 46 keV acceleration.
3.1.2 Acceleration and Transport
The beam produced by the polarized ion source, having an energy of approximately
72 keV,1 is analyzed by a magnet with a 30 de"ection. This angle aligns the direction
Since charge exchange occurs in the cesium oven, the net acceleration increases the beam energy
by 2eVCs , where VCs is the voltage applied to the Cs oven, typically a few hundred volts.
1
3.1. Charged-Particle Beam
35
Magnetic Quadrupole
Polarized
Ion Source
Electrostatic Quadrupole
Magnetic Steerer
Wien Filter
Einzel
Inflection
Magnet
Tandem
Accelerator
Figure 3.2: Low Energy Beam Transport
of the beam with the axis of the accelerator. Magnetic steerers are manually tuned
to control the beam position, while a combination of three electrostatic quadrupole
triplets, two einzels lenses, and one magnetic quadrupole doublet focuses the beam
into the low energy end of the accelerator (Figure 3.2). Typically, beam currents of
about 4 A are obtained on the low energy Faraday cup.
The accelerator is a tandem Van de Graa, High Voltage Engineering Model FN.
A tandem accelerator operates by accelerating a negative beam (in this case H;)
toward a positively charged terminal, changing the charge of the beam, and then
accelerating the positive beam (H+) away from the terminal VdG60], giving the
beam an energy of 2eVTerminal . The charge state of the beam is changed by passing it
through a thin carbon foil which strips away the electrons. In this experiment, a foil
with a thickness of approximately 5 g/cm2 is used. The terminal of the accelerator is
biased by two National Electrostatics Corporation (NEC) Pelletron charging chains.
The chains consist of aluminum pellets joined by nylon links. The chains are mounted
on pulleys driven by motors so that they run continuously between the terminal and
each end of the accelerator. Charge is induced on the pellets by high-voltage DC
supplies at the ground potential ends and is removed in the terminal. A corona
current feedback system regulates the voltage of the terminal to keep the energy
of the beam constant. The accelerator is contained in a pressure vessel lled with a
mixture of carbon dioxide, nitrogen, and sulfur hexa"uoride. The beam is transmitted
36
Chapter 3. Experimental Apparatus
Tandem
Accelerator
Steerer Feedback
Analyzing Magnet
Corona Feedback
Proton Polarimeter
Magnetic Quadrupole
Magnetic Steerer
Beam Current Slits
Cryostat Leg
Bending Magnet
Beam Profile Monitor
Neutron
Production Target
Figure 3.3: High Energy Beam Transport
through a series of four evacuated acceleration tubes, consisting of alternating glass
and stainless steel sections. The eciency for transporting the beam through the
accelerator typically ranges from 55% at the lowest energies used in this experiment
to 70% at the highest.
Upon leaving the accelerator, the proton beam must be transported to the neutron
production target (Figure 3.3). The beam is analyzed by passing through the switching magnet with a de"ection of 59 . The eld of the magnet is regulated by feedback
from a nuclear magnetic resonance probe. Currents from horizontal slits placed at
the exit of the magnet feedback to the corona circuit of the accelerator, keeping the
3.1. Charged-Particle Beam
37
energy of the beam constant. A nal bending magnet (cryostat leg bending magnet)
directs the beam toward the neutron production target. The beam is focused by three
magnetic quadrupole doublets. A carbon-foil polarimeter allows the measurement of
the beam polarization (Section 3.2). Most of the beam steering between the exit of
the accelerator and the cryostat leg bending magnet is accomplished by three sets of
slit feedback loops. These feedback loops keep the beam centered on the slits in the
horizontal plane by steering the beam such that the dierence in current from the
left slit and the right slit is zero. Similarly, the beam is centered in the vertical plane
using currents from the up and down slits Gou84]. The rst feedback loop keeps the
beam centered at the entrance to the analyzing magnet. The second centers the beam
at the entrance to the polarimeter, and the third centers the beam at the entrance to
the cryostat leg bending magnet. The size of the slit openings can be adjusted to t
the size of the beam.
After the cryostat leg bending magnet, the beam is transported approximately
2 m to the neutron production target. A NEC beam prole monitor allows the shape
of the beam to be viewed on an oscilloscope. In addition, signals from the beam
prole monitor are read into the data acquisition system and used to steer the beam
so that its centroid is centered on the beam pipe (Section 3.6). A nal slit feedback
loop holds the beam centered on the neutron production target. This set of slits
is a xed quad collimator and is described in Section 3.3.1. Since the proton beam
must approach the cryostat quite closely, it must be shielded from the eld of the
superconducting magnet. For this reason, the last 1.2 m of the beam pipe is made of
soft iron with 5.6 mm walls. Inside of the pipe are two concentric cylinders of a high
permeability alloy,2 1 mm thick. The beam current striking the neutron production
target is integrated using a Brookhaven Nuclear Instruments 1000 whose output is
a series of pulses with a frequency proportional to the current. These pulses are
converted to a NIM level signal by a Ortec 416A gate and delay generator and then
the width is set with a LeCroy 222 gate and delay before being sent to the data
acquisition electronics (Figure 3.4). The analog output of the current integrator goes
to a discriminator with upper and lower levels. The discriminator sends an inhibit
signal to the computer to stop data acquisition when the beam current is outside of
the window dened by the upper and lower levels. The beam current on target is
2
CO-NETIC AA, Magnetic Shield Corporation, Perfection Mica Company.
38
Beam
Stop
Chapter 3. Experimental Apparatus
1 µs
Gate
&
Delay
Current
Integrator
Gate
&
Delay
75 ns
BCI
Analog
Discriminator
INHIBIT
Figure 3.4: Beam Current Integration Electronics
y
ˆ
x
ˆ
ŝ
ẑ
k̂
Figure 3.5: Madison Convention Coordinate System
typically 0.5{1.5 A.
3.1.3 Spin Transport
Just as important as transporting the polarized proton beam to target is having it
arrive with the proper spin orientation. The Madison convention Sat71] denes a
coordinate system for describing the polarization of the beam (Figure 3.5). In order
3.1. Charged-Particle Beam
39
to produce the transversely polarized neutron beam required for the T measurements, the proton beam must arrive at the neutron production target with its spin
oriented transverse to the beam direction (^z) and in the horizontal plane (^x direction).
Although the proton beam emerges from the polarized ion source with a longitudinal
spin orientation, its direction can be changed by the Wien lter spin precessor. Propagation of the beam through dipole bending magnets can aect the spin direction
and must be taken in to account in setting the spin precessor. For beams polarized
in the y^ direction (transverse vertical), spin precession does not occur in the bending
magnets and the only consideration is to properly set the initial spin direction at the
polarized ion source. For beams polarized in the x^ direction (transverse horizontal),
however, spin precession does occur in the bending magnets and must be compensated for by setting the spin direction at the source. This is the case for the T
measurements as the spin of the beam must be in the same direction as that of the
polarized target, which is determined by the superconducting magnet.
When a beam of spin 1=2 particles is bent by a magnet, the spin direction is
not necessarily changed by the same angle as the propagation direction. A spin
component normal to the magnetic eld will precess. If the angle of de"ection is ,
and the angle of spin precession is , it can be shown that the two angles are related
by the g-factor of the particle Lew87].
= g
(3:2)
For the special case g = 1, the spin precession angle is the same as the de"ection
angle, maintaining the relationship between the beam propagation direction and the
spin direction. For g =
6 1, the angle between the propagation and spin directions, ,
changes. In the case of a proton beam with spin oriented in the x-z plane, increases
according to
+
+
final
= initial
+ (gp ; 1)
(3:3)
where gp = 2:7927. For a negative hydrogen ion beam, however, the beam de"ection
is in the opposite direction.
;
;
final
= initial
; (gp + 1)
(3:4)
Since the spin precession depends only on the angle of de"ection, the spin orientation at any point along the beam path can be calculated (Figure 3.6). Because the
40
Chapter 3. Experimental Apparatus
ŝ
IP
IS
ŝ
165°
30°
52°
Accelerator
59°
54°
20°
ŝ
ŝ
90°
Target
Figure 3.6: Spin Precession for a Proton Beam, Transverse Horizontal on Target
3.2. Charged-Particle Polarimeter
41
cryostat leg bending magnet is located between the polarimeter and the target, there
is a dierence in spin angle between the two. The de"ection of the magnet has been
determined by surveying to be 19.82 0.04 . This gives a dierence in spin angle of
35.53 0.07 . All beam polarization measurements made with the carbon-foil polarimeter (Section 3.2) when the spin is transverse horizontal on target must then be
corrected by cos(35.53 ).
Since the beam produced by the polarized ion source is polarized along the beam
direction, a spin precessor is needed to produce transverse polarization. The Wien
lter3 performs this task with crossed electric and magnetic elds. The magnetic
eld precesses the spin, while the electric eld compensates for the de"ection to keep
the beam straight. Electrostatic quadrupole doublets reduce the size of the beam
so that all particles traverse nearly the same magnetic eld. The Wien lter can
be mechanically rotated about the beam direction, allowing any spin orientation to
be achieved. The Wien lter has been calibrated by measuring the polarization in
the x^ direction at the polarimeter as a function of magnetic eld (Figure 3.7). The
non-linearity observed at the higher elds is due to a problem with the gaussmeter
used to measure the magnetic eld.4 For this reason, only eld settings below 1200 G
are used for spin precession. The portion of the curve from 0 to ;1200 G has been
t to a sine function
2B
max
Px = Px sin B + 0
(3:5)
0
where B0 = 1653:12 G and 0 = 2:3224. From this data, Wien lter magnetic eld
and mechanical angle settings, B and , can be determined for any spin orientation.5
Table 3.1 lists the settings which are important for this experiment.
3.2 Charged-Particle Polarimeter
Since T is proportional to the neutron polarization, the beam polarization must
be determined for each measurement. Although it is dicult to measure the neutron
beam polarization directly, it is relatively easy to monitor the polarization of the
ANAC model 2170 Spin Precessor.
The gaussmeter has since been replaced. The new unit does not exhibit this behavior.
It should be noted that a magnetic eld is in the y^-direction ( = 0 ) rotates the spin orientation
in the x-z plane.
3
4
5
42
Px
Chapter 3. Experimental Apparatus
1:0
0:8
0:6
b
b
b
b
b
b
b
b
0:4
b
0:2
b
b
b
b
b
b
b
0:0
b
b
;0:2
b
b
;0:4
b
b
;0:6
b
b
b
;0:8
;1:0
0:0
;500:0
;1000:0
;1500:0
;2000:0
;2500:0
B (G)
Figure 3.7: Wien Filter Calibration for the 59 Beam Line as Measured by the
Proton Polarimeter
3.2. Charged-Particle Polarimeter
43
B (G) Spin Orientation
0 ;861:17 ;x^ (Target)
0 ;1024:32 ;x^ (Polarimeter)
0
;34:61 +^x (Target)
0 ;197:76 +^x (Polarimeter)
+90 ;413:28 ;y^
+90 +413:28 +^y
0 +378:67 ;z^ (Target)
0 +215:52 ;z^ (Polarimeter)
0 ;447:89 +^z (Target)
0 ;611:04 +^z (Polarimeter)
Table 3.1: Wien Filter Settings
charged-particle beam and then relate this value to the neutron beam using known
polarization-transfer coecients, Kyy . For this reason, a carbon-foil polarimeter is
used to monitor the polarization of the proton beam during T measurements. This
polarimeter has been calibrated by a neutron polarimeter in separate measurements
(Section 4.2).
The polarimeter consists of a carbon foil and two solid state charged-particle
detectors contained within a small scattering chamber (Figure 3.8). The chamber is
made of a cylindrical body with two arms as particle "ight paths in order to minimize
the internal volume and thus reduce the need for vacuum pumping. The chamber
is constructed of aluminum with the arms welded to the body. Dependex "anges
are used at the entrance and exit of the chamber and at the ends of the arms. The
detectors are mounted on blank dependex "anges at the ends of the arms with three
sets of tantalum collimators dening the scattered beams with an angular acceptance
of 3:46 . The carbon foil is 22 mm in diameter and 5 g/cm2 thick and is mounted
on a plunger so that it can be removed from the beam when not in use. Slits in front of
the polarimeter dene the beam to be 12.7 mm 12.7 mm. The slit currents are fed
back to a steerer which keeps the beam centered on the entrance of the polarimeter.
The detectors and all but the rst set of collimators are electrically isolated to reduce
noise pickup. The detectors are silicon charged-particle detectors with 300 mm2 active
0
44
Chapter 3. Experimental Apparatus
Detector
40°
Beam
Foil
Tantalum Collimator
Figure 3.8: Proton Beam Polarimeter
areas and depletion depths of 1000 m. The angle of the arms is xed, for simplicity,
at 40 . This angle is chosen to provide a relatively large analyzing power over a
range of energies for protons elastically scattered from carbon.
The signals from the detectors pass through Ortec 142 preampliers at the polarimeter and are then further amplied by Ortec 572 ampliers in the control room.
The unipolar outputs of the ampliers are fed to Ortec 551 timing single-channel analyzers (TSCA). This setup is shown schematically in Figure 3.9. The TSCA windows
are set to select one peak in the linear energy spectrum from the ampliers and the
outputs are sent to the data acquisition electronics. The window is set by observing
the delayed output of the amplier on an oscilloscope while triggering on the TSCA
output.
When a proton beam is used, only three peaks are observed. In order of increasing
energy of the detected particle they are 1H(p,p)1H, 12C(p,p1)12C, and 12C(p,p0)12C.
In the case of elastic scattering from hydrogen, it is the recoiling proton which is
detected. Note that because of its excitation energy of 4.44 MeV, the rst excited
state of carbon will not be observed at lower beam energies. The TSCA windows are
3.2. Charged-Particle Polarimeter
Top
Solid State
Detector
Bottom
Solid State
Detector
Preamp
45
Amp
Preamp
TSCA
Amp
TSCA
Gate
&
Delay
Gate
&
Delay
75 ns
TPOL
75 ns
BPOL
Figure 3.9: Proton Beam Polarimeter Electronics
set around the 12C(p,p0)12C peak and the counts from both detectors are read by the
data acquisition electronics into scalers. Asymmetries between the number of counts
from each detector, "ppol, are then calculated for each spin state.
L ; NR
(3:6)
"ppol = N
NL + NR
Here, NL and NR refer to the number of counts in the left and right detectors by convention, although the detectors are physically up and down for most measurements.
The proton beam polarization, Pp, is given by
(3:7)
Pp = A"ppol
y sin where Ay is the analyzing power at 40 for 12C(p,p0)12C (Table 3.2) and is the
azimuthal angle of the spin axis in the polarimeter.6 The analyzing powers are obtained by tting published analyzing power data as a function of angle Mos65, Ter68]
(Section 5.1.2). The neutron beam polarization is then obtained by knowing the polarization transfer coecient Kyy for the 3H(~p,~n)3He source reaction.
0
Pn = Kyy Pp
0
(3:8)
Beam polarizations have been observed to remain very constant with time Kos90],
allowing the polarization measurements to be made every 2{4 hours. Removing the
Since the polarimeter is placed before the last bending magnet, the angle of the spin is not the
same in the polarimeter and on target. For protons the dierence is 35.53 0.07 .
6
46
Chapter 3. Experimental Apparatus
Ep
4.66
5.04
5.41
5.78
5.89
6.18
6.77
Ay
;0:496
;0:724
;0:535
;0:775
;0:812
;0:851
;0:520
Ay
0.022
0.009
0.024
0.015
0.022
0.009
0.009
Ep
7.21
7.55
7.99
8.66
8.90
9.15
9.60
Ay
;0:278
;0:339
;0:131
;0:015
0.125
;0:010
;0:348
Ay
0.015
0.016
0.017
0.014
0.041
0.015
0.054
Table 3.2: Analyzing Powers for 12C(p,p0)12C at lab = 40
carbon foil between polarization measurements increases the beam current on target
by approximately 40{100%, depending on the beam energy.
3.3 Neutron Beam
The polarized neutron beam is produced as a secondary beam from the polarized
proton beam. The beam is produced by an essentially unshielded source and is
collimated after passing through the polarized proton target. The neutron detectors
are placed at 0 inside a large shield. It is important to know the characteristics of the
beam including its polarization, average energy, and energy spread. In addition, it is
important to determine the amount of unpolarized background in the beam and the
amount of depolarization caused by the superconducting magnet used in the polarized
target.
3.3.1 Neutron Beam Production
The polarized neutron beam is produced by the 3H(~p,~n)3He reaction Don71, Sim73].
This reaction is chosen because the threshold of 0.764 MeV makes the production of
low energy neutron beams with an acceptable energy spread possible. By comparison,
the 2H(~d,~n)3He reaction Lis75] has a positive Q value of 3.3 MeV making it dicult
to produce neutron beams at 0 with energies less than 6 MeV. In addition, the
3H(~
p,~n)3He reaction has a large polarization transfer coecient at 0 (Section 5.2).
The tritium is in the form of tritiated-titanium on a 0.51 mm thick 58Ni backing. A
3.3. Neutron Beam
47
Ceramic
Insulator
Suppressor Ring
Proton Beam
Tritiated
Foil
Havar
Foil
He
Dependex
Flange
Neutron Beam
He Fill Line
Quad
Collimator
Figure 3.10: Tritiated-Titanium Foil Holder
solid compound is used instead of gaseous tritium for safety considerations. The 58Ni
backing stops the proton beam without producing neutrons due to the high threshold
of 9.44 MeV for the 58Ni(p,n)58Cu reaction. To contain any tritium which might be
released from the foil, as well as "akes of Ti 3H2, a 1 bar helium cell with a 2.5 m
Havar entrance foil is placed in front of the tritium foil deR89]. A four-segment
tantalum collimator denes the proton beam to be 4.76 mm 6.35 mm (vertical
horizontal) and provides feedback to a magnetic beam steerer. The foil holder,
shown in Figure 3.10, is electrically isolated to allow beam current integration and a
suppressor voltage of ;300 V is applied to prevent leakage currents from secondary
electrons. Since the number of neutrons produced is proportional to the number of
protons incident on the tritiated-titanium foil, it is not necessary to measure the
neutron "ux directly. Because tritiated-titanium foils cannot hold as much tritium
as a gas cell, the neutron count rates are low, typically of order 100 s;1 after being
attenuated by the target and with a detector eciency of approximately 25%.
The proton beam loses energy in passing through the havar foil, the helium gas,
and the tritiated-titanium, making the energy of the neutron beam dicult to determine exactly. In addition, these losses determine the energy spread of the neutron
beam. For these reasons, it is important to both estimate the energy losses for the proton beam and to directly measure the energy of the neutron beam. The energy losses
are calculated using the FORTRAN code BABEL Bow82]. The average energy of
the beam is taken to be that of the neutrons produced half way through the tritiated-
48
Chapter 3. Experimental Apparatus
Ep (MeV) En (MeV) En (keV)
3.02
2.00
114.5
4.66
3.71
80.6
5.41
4.48
71.7
5.89
4.97
67.1
6.18
5.27
64.6
6.77
5.87
60.2
7.21
6.31
57.3
Table 3.3: Calculated Average Neutron Energies and Energy Widths
titanium. The energy width is taken to be the dierence in energy between neutrons
produced at the start and the end of the tritiated-titanium added in quadrature to
the straggling of the proton beam as it passes through the havar and the helium. The
straggling is estimated to be one tenth the total energy loss in these materials. The
results are shown for the energies of interest in Table 3.3. To determine the neutron
energy more accurately, direct measurements are needed. For this purpose, yield
curves have been measured for carbon and oxygen targets for neutron energies in the
vicinity of known resonances using an unpolarized beam Nat91, Nat90]. Figure 3.11
shows the total cross section for carbon in this region, while Figure 3.12 shows the
total cross section for oxygen. The carbon target is a block of graphite approximately
130 mm thick and the oxygen target is a plastic cylinder of water approximately
127 mm thick. The targets are placed between the cryostat and the collimator and
the number of neutrons at 0 counted for a xed amount of proton beam charge. The
measurement is repeated with the target removed in the case of carbon, and with the
target replaced by an empty plastic cylinder in the case of oxygen. The total cross
section is then given by
Nin (3:9)
Total = ;x ln
Nout
where Nin and Nout are the number of neutrons counted with the target in and out and
x is the target thickness. The results of this measurement are shown in Figure 3.13
for carbon and in Figure 3.14 for oxygen. Since only the positions of the resonances
are important, it is not necessary to know x accurately. Comparing the positions of
the resonances with the known values gives a calibration for the relationship between
3.3. Neutron Beam
49
(mb)
10:0
8:0
6:0
4:0
2:0
0:0
0:0
2:0
4:0
6:0
8:0
10:0
En (MeV)
Figure 3.11: ENDF/B-VI Evaluation for the Total Neutron Cross Section of Carbon
50
Chapter 3. Experimental Apparatus
(mb)
10:0
8:0
6:0
4:0
2:0
0:0
0:0
2:0
4:0
6:0
8:0
10:0
En (MeV)
Figure 3.12: ENDF/B-VI Evaluation for the Total Neutron Cross Section of Oxygen
3.3. Neutron Beam
51
; ln(Nin =Nout )
2:5
b
b
b
2:0
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1:5
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1:0
0:5
0:0
2:0
3:0
4:0
5:0
Ep (MeV)
6:0
7:0
Figure 3.13: Measured Total Neutron Cross Section of Carbon
8:0
52
Chapter 3. Experimental Apparatus
; ln(Nin =Nout )
3:0
2:5
b
b
b
b
b
b
2:0
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1:5
1:0
0:5
0:0
5:0
5:2
5:4
5:6
5:8
6:0
6:2
Ep (MeV)
6:4
6:6
6:8
Figure 3.14: Measured Total Neutron Cross Section of Oxygen
7:0
3.3. Neutron Beam
53
Ep (MeV) En (MeV)
3.02
1.94
4.66
3.65
5.41
4.42
5.89
4.91
6.18
5.21
6.77
5.81
7.21
6.25
Table 3.4: Measured Average Neutron Energies
En (MeV) 1.94
1.000
3.65
1.077
4.42
1.450
4.91
1.500
5.21
1.530
5.81
1.330
6.25
1.430
Table 3.5: Asymmetry Dilution Factor Due to the Unpolarized Neutron Background
the incoming proton energy and the outgoing neutron energy (Table 3.4).
Time-of-"ight studies using an unpolarized beam have shown that although no
neutrons are produced by the 58Ni beam stop, background neutrons are produced by
the slits, the havar foil, and titanium in the Ti 3H2, as well as by 13C deposited on the
havar. No polarization has been observed of these neutrons in measurements using
the polarized beam and the neutron polarimeter. The net eect of this background
is to dilute the measured asymmetries by a factor . Values of as determined from
the time-of-"ight spectra are listed in Table 3.5. The values of do not increase
monotonically with energy as they depend upon the detector thresholds.
Since the neutron beam must pass through the eld of the superconducting magnet, there is the possibility of spin precession. Fortunately, the neutron spins are
54
Chapter 3. Experimental Apparatus
En (MeV) P=P0
1.94
0.978
3.65
0.984
4.42
0.986
4.91
0.987
5.21
0.988
5.81
0.989
6.25
0.989
Table 3.6: Calculated Neutron Depolarization Due to the Superconducting Magnet
parallel to the direction of the magnetic eld and ideally are not eected. There are,
however, regions where neutrons not exactly on the beam axis, due to the nite extent
of the beam, experience small transverse eld components. This eect is greatest as
the beam passes between the two coils of the split-coil magnet. The precession caused
by these elds acts in opposite directions for neutrons on either side of the beam axis
so that the average spin direction is preserved and the net eect is to depolarize the
beam by some amount. In order to calculate this eect the FORTRAN code NEUTRONS is used (Appendix A). This program performs a Monte Carlo simulation of
the beam passing through the magnetic eld. The trajectories chosen are a random
sampling of those allowed by the collimator. The spin precession of each neutron is
tracked and averages calculated over all trajectories. The results for 10,000 neutrons,
tabulated in Table 3.6, are used in determining the neutron polarization during T
measurements.
3.3.2 Neutron Beam Collimation
In order to obtain a large solid angle, the neutron source must be placed as close as
possible to the polarized target as its size limits that of the beam. For this reason,
the collimator is placed after the target instead of in front of it as in a shielded source
arrangement. Having a completely unshielded neutron source would, however, illuminate the superconducting magnet and other parts of the cryostat, greatly increasing
the background. To alleviate this problem, a small pre-collimator is used. In addition, the neutron detectors are placed within a large shield (Figure 3.15) to further
3.3. Neutron Beam
0 m
55
1 m
2 m
3 m
4 m
Cryostat
Detectors
Neutron Beam
Copper
Polyethylene
Pre-Collimator Collimator
Polyethylene
Detector Shield
Figure 3.15: Neutron Collimation and Shielding
Figure 3.16: Neutron Pre-Collimator
suppress the background.
The pre-collimator, shown in Figure 3.16, is constructed of a copper cone with
a rectangular double-truncated bore. The entrance opens at a total included angle
of 30 in each plane, while the exit opens at an angle of 1.0 in the horizontal plane
and 3.0 in the vertical plane. The size of the opening at the narrowest point is
5.33 mm 8.76 mm. The total included angle of the copper cone is 70 and is
197 mm in diameter at its base. The thickness of the cone in the beam direction
is 133 mm, giving a neutron attenuation factor of 44 at 10 MeV Che87]. The precollimator is mounted on three stainless-steel rods 25.4 mm in diameter. Each rod
56
Chapter 3. Experimental Apparatus
has a turnbuckle in the center and a dial indicator, allowing vertical adjustment to
25 m. The rods are connected to a rotating stage which sits on an x-y translation
table. This arrangement allows precise control of the axial, horizontal, and vertical
position of the pre-collimator as well as the horizontal and vertical angles. The
positioning mechanism is constructed such that the pre-collimator tilts and rotates
about its narrowest opening. This denes a beam spot at the center of the polarized
target having dimensions 9.35 mm 25.65 mm with a solid angle of 0.53 msr.7
The collimator is a rectangular block of polyethylene with a tapered rectangular
bore. It is 500 mm high, 400 mm wide, and 660 mm long. As in the case of the
pre-collimator, the bore opens with a total included angle of 1.0 in the horizontal plane and 3.0 in the vertical plane. The opening at the entrance is 12.70 mm
38.10 mm and at the exit is 24.23 mm 72.69 mm. The collimator is mounted
on precision linear bearings which ride on cylindrical rails. This allows the collimator
to be rolled away from the cryostat, allowing access for maintenance. The frame
supporting this mechanism is made of stainless-steel angle stock welded together.
Non-magnetic materials are used whenever possible in order to minimize the force on
the superconducting magnet. The collimator is designed to provide an attenuation
factor of at least 400 for neutron energies up to 14 MeV. The entrance and exit of the
collimator are used as reference points in aligning all other devices along the neutron
beam path. The alignment of the collimator with the target has been veried with
the target at 4 K using X-ray lm. The lm is exposed by bombarding a deuterium
gas cell with 5 MeV deuterons. Gammas produced in the gas and in the tantalum
beam stop are believed to be responsible for exposing the lm. Two lms are used,
one between the cryostat and the collimator and one after the collimator with an
exposure time of 30 minutes. Rare-earth radiators are placed in front of the lm to
improve the sensitivity.
The detector shield (Figure 3.17) is a rectangular block of polyethylene measuring
806 mm 806 mm 1016 mm. It has a straight rectangular bore, 57 mm 190 mm.
The two neutron detectors are inserted from the top and bottom into a cylindrical
hole. They are attached to aluminum plates on the top and bottom of the shield and
contained within an iron pipe to provide magnetic shielding. This arrangement allows
Since the pre-collimator is not very thick at the narrowest point, the actual solid angle is closer
to the value of 0.9 msr dened by the collimator.
7
3.3. Neutron Beam
57
Magnetic Shield
Beam
Detectors
Figure 3.17: Neutron Detector Shield
the detectors to be easily removed for sighting through the collimation assembly. The
shield is constructed of stacked 25.4 mm thick sheets of polyethylene with aluminum
plates on the top and bottom and with vertical rods compressing the entire stack.
The bore of the shield extends all the way through, allowing the neutron beam to
exit without scattering back into the detectors. The iron pipe also has windows to
reduce the attenuation of the beam. Additional shielding is placed around the beam
entrance and around the detector access holes on the top and bottom.
3.3.3 Neutron Detection
In order to measure T , the transmitted neutron "ux at 0 must be detected. Unlike
charged particles, neutrons are not detected directly, but through secondary reactions.
Recoil protons from a hydrogen-rich target provide one such method, and in the
case of organic liquid scintillators, the scintillation "uid is also the hydrogen target.
The neutron detectors in this experiment consist of such scintillators attached to
photomultiplier tubes which detect the "ashes of light from the neutron events. These
detectors are also sensitive to gammas, and a means of discriminating the two particles
is necessary. Pulse-shape discrimination provides this information and is implemented
using commercially available electronic modules. Two detectors are used to cover the
entire solid angle of the neutron beam, using readily available equipment.
58
Chapter 3. Experimental Apparatus
Organic liquid scintillators operate by detecting the recoil protons from 1H(n,n)1H
reactions. The protons then interact with the scintillator, either by excitation or
ionization, which emits light during de-excitation or recombination Swa60]. This
light strikes the photocathode of the photomultiplier tube, producing electrons. The
number of electrons is increased through secondary emission by the dynodes of the
electron multiplier. The nal electron current is collected by the anode. As the
light has dierent decay times for neutrons and gammas, the current appears as a
pulse with dierent rise and fall times. This property is exploited by the pulse-shape
discrimination electronics.
The scintillator cells are aluminum cylinders 127 mm in diameter and 127 mm
in length with one face made of glass to allow the light to exit. The insides of the
cells are coated with a white re"ector paint to reduce light loss. The scintillator
"uid is Bicron BC-501 and an expansion tube is provided to accommodate thermal
expansion of the liquid without bubbles. The photomultiplier tubes are Hamamatsu
type R1250, which have 127 mm diameter photocathodes. As photomultiplier tubes
are greatly aected by magnetic elds, shields made of a high permeability material
are placed around them. The scintillator-photomultiplier assembly (Figure 3.18) is
mounted to a "ange which bolts to the detector shield.
The electronics for the neutron detectors, shown in Figure 3.19, take the anode signals and produce a NIM level output pulse for each neutron detected. Signals used for
determining dead-time corrections are also produced. The Link 5020 modules perform
the functions of amplication, pulse-shape discrimination, and single-channel energy
analysis Ada78, Lin]. This module distinguishes between neutrons and gammas by
measuring the signal decay-times. The neutron output of the Link 5020 is converted
to a fast NIM signal using a LeCroy 222 gate and delay generator before being sent
to the data acquisition electronics. A second LeCroy 222 is congured as a 100 kHz
pulser. Its output is ANDed with the live-time signal from the Link 5020. This signal
and the ungated pulser signal are sent to the data acquisition system for calculating
a dead-time correction, .
= NNpulser
(3:10)
gated
Although this calculation only takes into account the 400 ns per event dead-time
of the Link 5020, this is a valid assumption since the dead-time of the detector is
approximately 25 ns per event.
3.3. Neutron Beam
59
Mounting Flange
Aluminum Plate
Stainless Steel
Support Rod
Polyethylene
Detector Shield
Phototube
Clamp
Scintillator
Support Ring
Beam
Figure 3.18: Neutron Detector Assembly
60
Chapter 3. Experimental Apparatus
Top
Anode
Neutron
Detector
PSD
n out
Gate
&
Delay
Live Time
75 ns
TOP
75 ns
AND
TGAT
100 kHz
Pulser
75 ns
AND
BGAT
Live Time
Bottom
Anode
Neutron
Detector
PSD
n out
Gate
&
Delay
75 ns
BOT
Figure 3.19: Neutron Detector Electronics
3.4 Neutron Polarimeter
Although the polarization of the neutron beam is usually determined by measuring
the polarization of the charged-particle beam, it is important to be able to measure the neutron polarization directly. A neutron polarimeter provides a means of
calibrating such secondary polarimeters and measuring polarization transfer coecients. The polarimeter consists of a 4He target and neutron detector pairs placed
symmetrically about the beam direction (Figure 3.20). Since the analyzing power
of neutrons scattering elastically from helium can be calculated with an accuracy of
approximately 0:01, the neutron beam polarization can be extracted from the measured left-right asymmetry. The polarimeter is placed between the neutron collimator
and the detector shield when in use and is oriented to measure polarization in the x^
direction (horizontal plane). The distance from the neutron source to the center of
the polarimeter target is 1.79 m.
The target consists of helium gas at a pressure of 100 bar in a steel cylinder
with 1 mm walls Tor74]. The active volume of the cell has a diameter of 44.6 mm
and a height of 158.2 mm. The gas has approximately 5% xenon added in order
3.4. Neutron Polarimeter
61
Neutron Detectors
Collimator
Beam
Target
Cell
Figure 3.20: Neutron Polarimeter
to make it scintillate with the recoiling alpha particles. This arrangement makes
it an active target and allows the measurement of timing coincidences and alpha
energy, reducing the background from the neutron detectors. The gas cell, shown
in Figure 3.21, has windows at both ends to allow the light from scintillations to be
detected in photomultiplier tubes. The inside of the gas cell is coated with 0.8 mm
of magnesium oxide as a re"ector. Evaporated on top of the re"ector is 120 g/cm2
of p -quaterphenyl which acts as a wavelength shifter, improving the response of the
photomultiplier tubes. The photomultiplier tubes are Hamamatsu type H1161 having
a diameter of 51 mm. The cell is lled using the system shown in Figure 3.22.
The xenon is loaded rst by liquefying the gas in the condensation chamber and
then allowing it to evaporate and expand into the target cell until a pressure of
approximately 5 bar is attained. This method is necessary as the xenon is stored at
too low of a pressure to ll the cell directly. Then, 99.999% helium is bled into the
cell in stages until the nal pressure of 100 bar is reached. The gas lling system is
constructed completely of stainless-steel with viton o-rings used as valve stem seals
and is designed to withstand pressures of 330 bar. A sintered stainless-steel lter
62
Chapter 3. Experimental Apparatus
Photomultiplier
Tube
Glass Window
He – Xe
Gas
Fill Line
Figure 3.21: High-Pressure Helium Gas Cell
3.4. Neutron Polarimeter
63
Pressure
Gauge
Target Cell
Rupture
Disc
Xe
Turbo-Molecular Pump
He
Backing Pump
Trap
Condensation
Chamber
Figure 3.22: Helium Cell Filling System
cooled to liquid nitrogen temperatures is used to trap contaminants during the helium
ll. The lter must be warmed above the melting point of xenon when lling with
this gas. As small amounts of contaminants, such as nitrogen and oxygen, greatly
reduce the performance of the cell by absorbing the light of scintillation, the cell and
gas lling system are pumped by a turbo-molecular pump before use.
The neutron detectors are placed symmetrically about 0 at angles where the
product of the square of the analyzing power and the cross section is a maximum for
4He(n,n)4He. There are two such angle pairs, one forward and one backward, and two
detector pairs are used. At the lowest energy measured, however, one maximum is too
low in magnitude to be useful and only one detector pair is used. The detectors are
organic liquid scintillators coupled to photomultiplier tubes through light guides. The
light guides match the geometry of the 45 mm 158 mm 76 mm thick rectangular
scintillator cells to the 51 mm diameter phototubes. The photomultiplier tubes are
Hamamatsu type H1161. The neutron detectors are mounted to a detector ring having
1 graduations. The entire apparatus has been aligned optically with the neutron
beam collimator. The forward detector pair uses an electronic setup (Figure 3.23)
similar to that used for the main neutron detectors (Section 3.3.3). Since there are
64
Chapter 3. Experimental Apparatus
40 ns
Timing
L1
Anode
Neutron
Detector
PSD
n out
Gate
&
Delay
Live Time
L1TIM
75 ns
L1PSD
75 ns
AND
L1GAT
100 kHz
Pulser
75 ns
AND
R1GAT
Live Time
R1
Anode
Neutron
Detector
PSD
n out
Gate
&
Delay
75 ns
R1PSD
40 ns
Timing
R1TIM
Figure 3.23: Electronics for the Forward Neutron Detector Pair
not enough Link 5020 PSD modules for all four detectors, a dierent setup must be
used for the backward angle pair (Figure 3.24). In this setup, a Canberra 2160A PSD
module is used to determine the zero-crossing of the anode signal after the peak of
the pulse Can83]. By measuring the time between the start of the pulse, determined
by a constant-fraction discriminator (CFD), and this zero-crossing, neutrons can be
distinguished from gammas.
Monte Carlo techniques have been used to calculate the eective analyzing power
of the neutron polarimeter Tor75]. These calculations make corrections for the nite
geometry of the target cell and neutron detectors, and for double-scattering events.
Double scattering which includes neutrons scattering from materials in the target
cell other than helium must be considered. The double-scattering events which are
considered are He-He, He-Xe, Xe-He, He-Fe, and Fe-He. Since only events which
3.4. Neutron Polarimeter
L2
Anode
Neutron
Detector
65
Stop
PSD
Strobe Start
TAC
250 ns
CFD
75 ns
AND
L2
20 ns
2 ns
L2TIM
Amp
&
TSCA
L2TAC
R2
Anode
Neutron
Detector
Stop
PSD
Strobe Start
TAC
250 ns
CFD
75 ns
R2
AND
20 ns
2 ns
Amp
&
TSCA
R2TIM
R2TAC
Figure 3.24: Electronics for the Backward Neutron Detector Pair
66
Chapter 3. Experimental Apparatus
En (MeV) d (mm)
Ay
1.94
107
408
0:764 0:010
5.21
50
408 ;0:629 0:010
121
338
0:919 0:015
5.81
51
408 ;0:632 0:010
121
308
0:916 0:015
Table 3.7: Eective Analyzing Powers for the Neutron Polarimeter
include the detection of an alpha are recorded, single scattering events from elements
other than helium need not be considered. Cross sections and analyzing powers are
obtained from phase-shift data sets. The phase shifts for helium come from experimental data Sta72] as do the phase shifts for xenon and iron at the higher energies.
At the lowest energy measured, 1.94 MeV, experimental cross section data for xenon
are used with analyzing powers assumed to be zero. The calculated eective analyzing
powers, Ay , are shown in Table 3.7 for the neutron energies at which measurements
have been made. The length of the "ight paths, d, from the center of the target cell
to the center of the detector is also given.
For an event to be considered valid, three criteria must be satised. First, there
must be a coincidence between the top and bottom photomultiplier tubes of the
target cell, reducing the noise from the tubes. Second, the signal from the neutron
detector must meet pulse-shape discrimination requirements, eliminating signals from
gammas. Third, there must be a coincidence between the target cell signal and one of
the neutron detector signals, in order to reject neutron background events. Figure 3.25
shows the electronic setup which performs these functions. The anode signals from the
target cell photomultiplier tubes are used to form the coincidences, while the dynode
signals provide the alpha recoil energy information. Summing the dynode signals
improves the energy resolution of the detector as only part of the light of scintillation
is deposited in each photomultiplier tube. The gains of the two tubes are matched
using a 137Cs source. Since the alpha recoil energy is quite dierent for the forward
and backward angles, the summed dynode signal is split into two signals which are
amplied and delayed by dierent amounts before being recombined. This process
creates two pulses for each alpha, one large in amplitude and one small. The large
3.4. Neutron Polarimeter
67
High Gain
Linear
Fan
In/Out
250 ns
Linear
Fan
In/Out
270 ns Linear
Gate
Amp
E
Low Gain
Top
Photo
Tube
Bottom
Photo
Tube
Dynode
Anode
Anode
CFD
200 ns
AND
CFD
Dynode
20 ns
100 ns
Start
AND
L1TIM
20 ns
Stop
L1TOF
TAC
20 ns
L1
Start
AND
R1TIM
20 ns
Stop
R1TOF
TAC
20 ns
R1
Start
AND
L2TIM
Stop
TAC
L2TOF
SCA
L2TAC
20 ns
R2TIM
Start
AND
Stop
TAC
R2TOF
SCA
R2TAC
OR
OR
300 ns
Figure 3.25: Coincidence Electronics for the Neutron Polarimeter
68
Chapter 3. Experimental Apparatus
amplitude pulse is gated through to the data acquisition system only when an event
occurs in one of the backward angle detectors. Conversely, the small amplitude pulse
is gated through only when an event occurs in one of the forward angle detectors.
This arrangement allows a single ADC to digitize the alpha recoil energy for both
detector pairs. Neutron time-of-"ight from the target cell to the neutron detectors is
determined using Ortec 467 time-to-amplitude converters (TAC). The start signal for
each TAC comes from the timing signal of the corresponding neutron detector after
the coincidence with the target cell anode coincidence signal. The coincidences are
formed such that the timing signal from the neutron detector determines the timing
of the TAC start signal. The stop signal comes from a delayed version of the target
cell anode signal. This delay is adjusted to give a suitable range for the neutron timeof-"ight signals (typically 100{200 ns). Because of the coincidence requirements, the
count rates are low, typically a few per second.
3.5 Polarized Proton Target
The polarized proton target uses the brute-force technique SP86] of cooling the target
material to low temperatures in a large magnetic eld. This method has the advantage
of producing very thick targets with fewer polarized impurities than other methods
such as laser pumping. The brute-force method makes use of the energy dierence
between the dierent nuclear magnetic substates. The population pm of a substate
with spin projection m measured in the direction of the magnetic eld B is given by
emgB=kT
pm = mX
(3:11)
=+I
mgB=kT
e
m=;I
where g is the nuclear g-factor and k is the Boltzmann constant. For protons, I = 1=2
and the target polarization PT is given by
p;1=2 :
(3:12)
PT = pp+1=2 ;
+1=2 + p;1=2
For a given temperature T and magnetic eld B , the polarization is then
gB PT = tanh kT :
(3:13)
3.5. Polarized Proton Target
69
The magnetic eld is produced using a 7 T superconducting magnet.8 The magnet
is a split-coil solenoid with Nb-Ti windings. The magnetic eld in the sample region
is uniform, having a homogeneity of 0.1% over 1 cm. The magnet is charged to its
rated current of 86.2 A and placed in persistent mode, causing the eld to be essentially constant over the course of the measurement. The inductance of the magnet is
approximately 30 H, storing over 100 kJ of energy at full eld. For protons in a 7 T
eld, the polarization is given by
PT = tanh(T0=T )
(3:14)
with T0 =7.15 mK. The sample must then be cooled to a temperature of order T0 to
produce signicant polarization. A 3He -4He dilution refrigerator is used to achieve
these temperatures.
3.5.1 Dilution Refrigerator
Since measurements of T require that the proton target remain polarized for several days at a time, a means of continuous cooling at temperatures of 10{20 mK
is required. The only continuous method for cooling below 300 mK is the 3He -4He
dilution refrigerator Lou74]. The principle of operation of the dilution refrigerator
Lon51, Lon62] relies on a phase separation which occurs in 3He -4He mixtures at temperatures below several hundred millikelvin. One phase (dilute phase) is a mixture of
approximately 6% 3He and 94% 4He, while the other (concentrated phase) is essentially pure 3He. Below 500 mK, the 4He, obeying Bose-Einstein quantum statistics
because of its 0 spin, is super"uid in its ground state and has no entropy. The 3He,
however, is spin 1/2 and obeys Fermi-Dirac statistics. Thus it has a non-zero heat
capacity and entropy even as the temperature approaches absolute zero. A diusion
of 3He from the concentrated phase to the dilute phase is analogous to evaporation
and produces cooling. A dilution refrigerator (Figure 3.26) maintains an osmotic
pressure gradient across the phase boundary by continuously pumping 3He from the
dilute phase and restoring it to the concentrated phase. The phase separation occurs
in the mixing chamber. 3He which is to be returned to the mixing chamber is rst
condensed by the cold plate and is then cooled as much as possible by heat exchangers
8
American Magnetics, Incorporated, Oak Ridge, Tennessee.
70
Chapter 3. Experimental Apparatus
He
Out
He In
Cold Plate
He Out
Condenser
Still Heat Exchanger
Still
Continuous Heat Exchanger
Discrete Heat Exchangers
Concentrated Phase
Mixing Chamber
Dilute Phase
Figure 3.26: Dilution Refrigerator
3.5. Polarized Proton Target
71
in thermal contact with the dilute phase. 3He is removed from the dilute phase in the
still. A heater in the still increases the recirculation rate by raising the vapor pressure
of the 3He. The 4He, having a much lower vapor pressure remains behind. The cold
plate is cooled to approximately 1 K by pumping on liquid 4He obtained from the
surrounding liquid helium bath. In principle, the ultimate temperature of a dilution
refrigerator is limited only by the eciency of the heat exchangers and temperatures
of a few millikelvin have been reached by this method.
The dilution refrigerator used in this experiment is an S. H. E. Model 430. The
refrigerator has an ultimate temperature of 4.5 mK and a cooling power of 1.5 W
at 10 mK. The 3He is recirculated at a rate of 500 mol/s by an Edwards 9B3 diffusion pump backed by a hermetically sealed mechanical pump. A copper cold nger
is connected to the mixing chamber for attaching the target. The dilution refrigerator and the superconducting magnet are contained within a liquid helium cryostat
(Figure 3.27). The cryostat holds the liquid helium necessary for operating the refrigerator and cooling the magnet, and provides thermal insulation. The cryostat requires
100 liters of liquid helium for the initial cooldown, and approximately 30 liters each
day thereafter. Hollow brass cylinders are inserted in the magnet bore as spacers to
remove most of the liquid helium from the neutron beam, reducing the attenuation
of the beam.
3.5.2 Titanium Hydride Target
The criteria for choosing a suitable material for use as a polarized proton target are
very stringent. The hydrogen in the material must be polarizable by a magnetic eld
and it must have a large heat conductivity at millikelvin temperatures. All other
nuclei contained in the material must not be polarized by the magnetic eld. Molecular hydrogen, H2, is not suitable since the two nuclei in the molecule pair o with
opposite spins in the para-hydrogen conguration. The heat capacity associated with
converting the para-hydrogen to ortho-hydrogen is too large to be overcome in practice. Most hydrocarbons can be ruled out because of their low heat conductivities
and because of the presence of 13C, which becomes polarized. Many other hydrogen compounds do not have a suciently high hydrogen content or have unwanted
contaminants. One material which is suitable for use as a polarized proton target is
72
Chapter 3. Experimental Apparatus
0 m
Liquid Nitrogen Shield
Liquid Helium
Vacuum Can
Still
1 m
0.5 K Shield
Mixing Chamber
Superconducting
Magnet
Beam
Spacer
Target
2 m
Figure 3.27: Dilution Refrigerator Cryostat
3.5. Polarized Proton Target
73
titanium hydride, TiH2, as demonstrated by Aures et al. Aur84].
Titanium hydride is made by heating very pure titanium in a hydrogen atmosphere. The TiH2 is then in the form of a powder which must be compressed to form
a solid sample. The proton density of such a target is 9:0 1022 cm;3 as compared to
5:3 1022 cm;1 for solid H2. It has been shown that heat conductivities of approximately 5 10;5 W/cmK at temperatures of 10{20 mK can be attained with this type
of sample Hee85]. This value is an order of magnitude below that of pure titanium
Chi76]. Recent work indicates that the conductivity depends greatly on the purity
of the material Li92]. The rst attempts to produce a target for this experiment
used commercially available TiH2 with a purity of 99%. The powder was pressed into
a silver box under a pressure of 0.4 GPa. Silver is chosen because of its excellent
heat conductivity and because it does not become polarized. The polarization was
determined to be only 15% by measuring the asymmetry in the neutron transmission
(Section 4.1). This value corresponds to a temperature of approximately 50 mK.
Since polarizing a sample and measuring its polarization is an expensive and time
consuming process, a simpler method of determining the quality of the TiH2 target
is needed. The Wiedemann-Franz law
= LT
(3:15)
states that below 1 K the thermal conductivity, , of a metal is proportional to its
electrical conductivity, , where L = 25 nW#/K2 is the Lorentz constant and T is the
temperature. Because of this relation, a measurement of the electrical conductivity as
a function of temperature gives an indication of the thermal properties of the material.
Electrical resistance measurements of the rst TUNL TiH2 target at 300 K and at
80 K indicate that the resistance decreases as a function of increasing temperature.
This behavior indicates the material is behaving as an electrical insulator and suggests
that its thermal conductivity will be low as well. Similar measurements for the target
used by Aures et al at KFA Karlsruhe, Germany Hee85] indicate that their sample
behaves electrically as a metal, implying a relatively high heat conductivity. The
dierence is thought to be due to impurities in the commercial TiH2.
The nal TiH2 target is made from the same material as the Karlsruhe target.9
The powder is pressed into a rectangular copper box with the front and back faces
We gratefully acknowledge the assistance of H. O. Klages at KFA Karlsruhe, Germany in providing the material for this target.
9
74
Chapter 3. Experimental Apparatus
Silver Alloy Solder
Copper
Beam
Titanium Hydride
Figure 3.28: Titanium Hydride Target
open, since the copper becomes polarized. Copper is chosen to more closely approximate the Karlsruhe design. The box, made from 2 mm copper plate, is placed within
a hardened steel die. The powder is pressed into the box under a pressure of 1.5 GPa.
A 0.5 mm silver foil was placed at the bottom of the die to allow the connection of
thermometers to the TiH2, but good thermal contact was not established. As a result,
the target polarization must be determined by nuclear measurements (Section 4.1).
The rear face, which suered from mechanical defects, has been machined to give a
smooth surface. A bored copper rod is soldered to the top of the box for attachment
to the dilution refrigerator cold nger. The TiH2 target (Figure 3.28) has a density
of 3.75 g/cm3 and a thickness of 22.4 mm giving a proton thickness of 0.203 b;1. The
area of the TiH2 face is 14.0 mm 34.1 mm. The total amount of hydrogen contained in the target is 1.6 mol. Electrical resistivity measurements indicate a metallic
behavior similar to the Karlsruhe target. Proton polarizations obtained from this
target are in the range 40{50%, corresponding to temperatures of 13{17 mK.
3.5. Polarized Proton Target
75
3.5.3 Thermometry
Although attempts to accurately measure the temperature of the TiH2 directly have
been unsuccessful, it is still important to monitor various temperatures inside the
cryostat as an indication of how well the dilution refrigerator is working. The simplest thermometers are resistors whose resistance changes with temperature. Approximately twelve carbon or thick lm resistors are attached to various parts of the
refrigerator, including the cold nger, mixing chamber, still, and the discrete heat
exchangers. In addition, two calibrated germanium resistors are used, one attached
to the cold nger, and one to the mixing chamber. The resistance measurements are
made using an AC four-lead resistance bridge. Although the resistance thermometers are the easiest to use, they are not useful below approximately 50 mK as the
resistance saturates.
In the temperature range 5{40 mK, nuclear orientation thermometry may be used.
54
Mn and 60Co are the most common isotopes used for this purpose Mar83]. A cobalt
nuclear orientation thermometer consists of a small amount of 60Co embedded in a single crystal of natural cobalt. Below about 100 mK, the cobalt nuclei become aligned
along the crystal direction, due to the large eective magnetic eld (several tesla) of
the ferromagnetic material. The 60Co decays with a half-life of 5.3 years, emitting
gammas with energies of 1333 keV and 1173 keV. For an unpolarized crystal, the
radiation is emitted isotropically. As the crystal becomes more polarized, the radiation becomes anisotropic with a minimum "ux along the crystal axis. Such a crystal
is mounted inside the cryostat in thermal contact with the cold nger. The cobalt
is located outside of the bore of the superconducting magnet to minimize the eect
of the magnetic eld on its polarization. An intrinsic germanium gamma detector
is placed outside of the cryostat along the crystal axis. The gamma count rate is
measured and compared to the rate at temperatures above 40 mK to determine the
temperature. Measurements to an accuracy of 1 mK can be obtained by counting
for approximately 300 s. Since the presence of a neutron beam makes accurate measurements impossible, this method is not always practical. In addition, since warm
counts are required, the germanium detector cannot be moved once the refrigerator
is cold.
A third type of thermometer which is used to measure the temperature of the
dilution refrigerator is the 3He melting curve thermometer (MCT) Kei90, Kei92].
76
Chapter 3. Experimental Apparatus
Capacitor
Plates
Epoxy
He Chamber
Fill Line
BeCu
Mounting Screw
OFHC Copper
Ag Powder
11.4 mm
Figure 3.29: 3He Melting Curve Thermometer
The MCT relies on the precise knowledge of the pressure versus temperature of 3He
along the melting curve at low temperatures Gre86]. The MCT contains a mixture
of solid and liquid 3He in what is essentially a constant volume (Figure 3.29). The cell
is lled at 1 K through a capillary tube leading to a room temperature gas handling
system. As the temperature decreases, a plug of solid 3He forms in the tube, sealing
the cell at a constant average density. The top of the 3He chamber is a 250 m
thick diaphragm to which one plate of a capacitor is attached. Changes in pressure
are observed as changes in the capacitance, which can be measured to an accuracy
of 1 10;5 pF out of a total capacitance on the order of 10 pF. This corresponds
to a temperature sensitivity of approximately 1.5 K at 10 mK. By calibrating the
capacitance versus pressure characteristics of the cell at 1 K with a precision Bourdon
tube gauge, the absolute accuracy of the temperature measurement is estimated to
be 0.36 mK. Two such devices are installed in the cryostat, one attached to a silver
foil which touches the TiH2 target and one attached to the cold nger.
3.6. Data Acquisition Electronics
77
3.6 Data Acquisition Electronics
Data from the measurements are collected by CAMAC modules controlled by a MBD11 Multiple Branch Driver Rob81]. The MBD is connected to a Digital Electronics
Corporation VAXStation 3200 which supervises the data acquisition, stores the data,
and performs online analysis. In addition, this computer performs part of the feedback
steering of the proton beam. The data acquisition software runs under the TUNL
XSYS data acquisition and analysis system Gou81].
3.6.1
T
Measurements
For the T measurements it is necessary to measure count-rate asymmetries to an
accuracy of order 1 10;4 . Instrumental asymmetries due to the data acquisition
system must therefore be of order 1 10;5 or less. To achieve this level of stability,
several methods are employed. First, the spin of the neutron beam is rapidly reversed
at a rate of 10 Hz to reduce the eects of detector gain drifts and other instrumental
shifts. Precision timing is used to insure that the same amount of time is spent
counting in each spin state. Finally, the data are collected and stored in 800 ms time
slices, allowing asymmetries to be calculated for very short periods of time before
being averaged together. All the data are collected by counting pulses in scalers since
all particle discrimination takes place in hardware (Section 3.3.3). In addition to the
neutron detector and polarimeter counts, pulses from the beam current integration
and dead-time measurements are recorded.
The fast spin-reversal is controlled by a NIM Spin State Control (SSC) module
constructed in the TUNL electronics shop. The module produces an eight-step spin
sequence: + ; ; + ; + +; which cancels the eects of detector drifts to second
order in time Kos90]. All data fed to the data acquisition system pass through
a Phillips 706 sixteen-channel discriminator. Gate and delay generators are used
to create a veto signal which blocks the data during a spin reversal (Figure 3.30).
One pair of gate and delay generators determine the exact timing, vetoing the data
starting 2 ms before a spin reversal until 5 ms past. This amount of time is needed
to insure that the polarization of the source has stabilized Kos90]. A second pair
of gate and delay generators blank the output of the rst when an inhibit signal is
received by the SSC. This arrangement allows the timing circuits of the rst set to
78
Chapter 3. Experimental Apparatus
3 µs
Gate
&
Delay
Gate
Linear
Gate
Stretcher
ADC
Gate
CAMAC
Preset
Scaler
250 ns
250 ns
Gate
&
Delay
ADC
Interface
SCAN ROUT
INHIBIT
SF
+
Start
CAMAC
Spin State
Output
–
Reset Controller
Register
WF
IPIS
Spin
Control
Gated Clock
Del
Gate
&
Delay
10 Hz
Clock
Del
Gate
&
Delay
Blank
Gate
&
Delay
Del
Gate
&
Delay
7 ms
100 ms
Veto
75 ns
125 ns
TOP
BOT
TGAT
BGAT
PULS
CAMAC
Scalers
BCI
TPOL
BPOL
Figure 3.30: Data Acquisition Electronics for T Measurements
3.6. Data Acquisition Electronics
79
run continuously, avoiding drifts which occur when they are turned on and o. Data
collection is inhibited when the beam current drops below a preset level and while
writing the data to disk at the end of a run. The SSC determines the spin state of
the ion source by sending TTL level signals to the ABPIS through ber optic cables.
The signals modulate the RF supplies of the two transition units. The SSC provides
duplicates of these signals, labeled \+" and \;", for routing information. The \+"
signal is read into an analog-to-digital converter (ADC) to signal the computer that
a spin reversal has occurred. The computer reads all the CAMAC scalers and stores
them according to spin state, a high value in the ADC indicating spin up , and a low
spin down . At the end of an eight-step sequence, the CAMAC preset is decremented.
A timing diagram is shown in Figure 3.31. The data are stored in the computer as
spectra of counts versus time. Each time channel corresponds to one 800 ms eight-step
spin sequence. There are two spectra for each scaler, corresponding to the two spin
states. A run is comprised of 1023 such spin sequences. At the end of a run, the data
are written to disk and asymmetries are calculated for display on the control-room
monitor. The end of a run is signaled by the CAMAC preset scaler reaching zero.
The beam prole monitor (BPM) operates by rotating a helical wire through the
proton beam. The geometry is such that the wire travels across the beam rst horizontally and then vertically Nat84]. The current collected by the wire is preamplied
before being sent to the control room. In addition, a magnetic pickup on the rotating
shaft picks up signals from three magnets mounted on the chamber. The rst magnet produces the strongest signal and is used as a trigger to indicate the beginning
of a scan. The other magnets generate ducial marks which indicate the centers of
the x and y passes, respectively. The signals from the collector and the ducials
pass through isolation ampliers since they are referenced to dirty ground. In addition, the ducial signal is split with one branch going through a broad-band isolation
transformer to preserve timing information. The trigger signal (the largest of the
ducial marks) res an EG&G T140/N zero-crossing detector. The output of the
zero-crossing detector starts a linear ramp. The collector signal, the ducial signal,
and the ramp are sampled by linear gate stretchers before being sent to ADC's. The
gate signals are generated by a variable frequency clock which runs during the ramp.
This setup is shown in Figure 3.32.
The beam prole information (Figures 3.33 and 3.34) is stored in two sets of
80
Chapter 3. Experimental Apparatus
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 s
Clock
+
–
Veto
Blank
Clock
5 ms
2 ms
Veto
3 ms
Blank
0.0
0.1 s
Figure 3.31: Data Acquisition Timing Diagram
3.6. Data Acquisition Electronics
81
Beam
Collector Iso
Profile
Amp
Monitor
Linear
Gate
Stretcher
Fiducials
ADC
Gate
ADC
Interface
Isolation
Amplifier
Linear
Gate
Stretcher
Iso
Amp
Zero
Crossing
Detector
ADC
Gate
Gate
Or
ADC
Interface
10 ns
75 ns
Gate
&
Delay
Gate
Linear
Gate
Stretcher
Ramp
Generator
Period
ADC
Gate
Gate
Or
Gate
Gated
Clock
ADC
Interface
1 µs
Bit 14
2.5µs
SCAN ROUT
Del
Gate
&
Delay
+x
CAMAC
Output
Register
–x
+y
–y
1 µs
Gate
&
Delay
Beam
Steerer
Control
Figure 3.32: Beam Prole Monitor Feedback Steering Electronics
82
Chapter 3. Experimental Apparatus
Counts
1000000
800000
600000
400000
200000
0
0
100
200
300
400
500
Channel
Figure 3.33: Beam Prole Monitor Collector Spectrum
spectra. The rst set has the ramp as the x-axis and the collector as the y-axis. The
second again has the ramp as the x-axis, and the ducials as the y-axis. Each set has
one spectrum for spin up , one for spin down , and one which is the sum of the two. A
FORTRAN code running as a subprocess calculates the centroids of the two collector
peaks and the centers of the x and y ducial marks. Bits are sent at 1 s intervals to
an automatic steerer according to the relative position of the centroids and ducial
marks. The output signals are capacitively coupled to preserve the ground isolation.
3.6. Data Acquisition Electronics
83
Counts
10000000
8000000
6000000
4000000
2000000
0
0
100
200
300
400
Channel
Figure 3.34: Beam Prole Monitor Fiducial Spectrum
500
84
Chapter 3. Experimental Apparatus
3.6.2 Neutron Polarization Measurements
For the neutron polarization measurement, it is necessary to measure asymmetries
to an accuracy of order 5 10;3 . Although this requirement is somewhat relaxed
compared to the T measurements, the fast spin reversal techniques described above
are used. There are three major dierences, however. First, the signals obtained from
the polarimeter are in the form of energy and time-of-"ight information and must be
read by ADC's. In contrast, the detector signals for the T measurements are simply
read by scalers. Second, the routing of the scaler signals is performed by fanning out
the signals to two sets of scalers and selectively enabling just one set according to the
spin state (Figures 3.35 and 3.36) instead of using software to perform the routing.
Third, the ADC and scaler data is accumulated for an entire run instead of being
stored in 800 ms time slices.
Two ADC's are used for the neutron polarimeter. The rst records the alpha
recoil energy signal from the center detector. Gates for this ADC can come from
coincidences with any of the four neutron detectors (Section 3.4). The gates pass
through the Phillips 706 discriminator module so that they are vetoed during spin
reversal as are the scaler signals. Next, coincidences are formed between gates from
the forward detectors and their corresponding PSD signals. The PSD coincidences
for the backward detectors are generated at an earlier stage. Routing information is
created by the ADC interface according to the neutron detector and the spin state.
Delays are applied to the linear signal and the gates to allow a proper time relation
to be established. The second ADC records the neutron time-of-"ight signals from
the four neutron detectors. The linear signals are are fanned together using sum and
invert ampliers. The gate for this ADC comes from the interface for the rst ADC
so that the ADC's are triggered together and both alpha energy and neutron time-of"ight are recorded for each event. The events are stored in two-dimensional spectra
with neutron time-of-"ight as the x-axis and alpha recoil energy as the y-axis. There
are two spectra for each of the four neutron detectors, corresponding to the two spin
states.
3.6. Data Acquisition Electronics
85
3 µs
Gate
&
Delay
Gate
Linear
Gate
Stretcher
ADC
Gate
CAMAC
Preset
Scaler
250 ns
250 ns
Gate
&
Delay
ADC
Interface
ROUT +
INHIBIT
ROUT –
SCAN ROUT
+
Start
CAMAC
Spin State
Output
–
Reset Controller
Register
SF
WF
IPIS
Spin
Control
Gated Clock
Del
Gate
&
Delay
10 Hz
Clock
Del
Gate
&
Delay
Blank
Gate
&
Delay
Del
Gate
&
Delay
7 ms
100 ms
Veto
75 ns
125 ns
TGAT
SCAL1
BGAT
SCAL2
PULS
SCAL3
BCI
SCAL4
TPOL
SCAL5
BPOL
SCAL6
L1PSD
GATE1
R1PSD
GATE2
L2TIM
GATE3
R2TIM
GATE4
Figure 3.35: Neutron Polarimeter Data Acquisition Electronics (Part 1)
86
Chapter 3. Experimental Apparatus
L1TOF
R1TOF
Del
Amp
L2TOF
ADC
Gate
R2TOF
ADC
Interface
Del
Amp
E
125 ns
Gate
&
Delay
GATE4
GATE3
GATE2
250 ns
Gate
&
Delay
R1TIM
1 µs
AND
Gate
&
Delay
AND
Gate
&
Delay
125 ns
250 ns
GATE1
Gate
&
Delay
Gate
&
Delay
L1TIM
ADC
Gate
ADC
Interface
Or
Bit 15
125 ns
ROUT +
ROUT –
125 ns
Enable
SCAL1
SCAL2
SCAL3
CAMAC
Scalers
SCAL4
SCAL5
SCAL6
Enable
CAMAC
Scalers
Figure 3.36: Neutron Polarimeter Data Acquisition Electronics (Part 2)
Chapter 4
Experimental Procedure
4.1
T
Measurements
As the total cross section is related to the forward scattering amplitude by the optical
theorem, T can be measured by observing the change in transmitted neutron "ux
through the polarized target upon reversing the beam polarization. The transmitted
"ux, N , is given by
N = N0e;x
(4:1)
where N0 is the incident "ux, x is the target thickness, and is the cross section.
For the case of a polarized beam and a polarized target, the cross section can be
expressed as
1 (4:2)
= 0 ; Pn PT 2
where 0 is the unpolarized cross section, PT is the proton target polarization, and Pn
are the neutron beam polarizations for the two spin-states. The beam polarizations
can be expressed in a more convenient form as
Pn = Pn Pn
where
+
;
Pn = Pn +2 Pn +
;
Pn = Pn ;2 Pn :
87
(4:3)
(4.4)
88
Chapter 4. Experimental Procedure
The neutron transmissions for the transverse geometry are then
N = N0e;x(0 ; 2 Pn PT T )e 12 Pn xPT T :
1
An asymmetry can be formed using the two beam transmissions
N+ ; N;
" = N
+ + N;
(4:5)
(4.6)
+ 12 Pn PT xT ; e; 12 Pn PT xT
e
(4.7)
= + 1 Pn PT xT ; 1 PnPT xT
+e 2
e 2
(4.8)
= tanh 21 Pn PT x T
which is independent of the incident "ux and the unpolarized cross section. Since the
argument of the hyperbolic tangent is small ( 0:025), the approximation tanh(x) x
can be made
" = 21 Pn PT x T :
(4:9)
Then in order to determine T , the neutron transmission asymmetry, ", must be
measured as well as the two polarizations, Pn and PT , and the target thickness, x. The
neutron beam polarization, Pn, can be inferred from the proton beam polarization
measured by the carbon-foil polarimeter. This measurement is relatively fast, with
a sucient statistical accuracy being reached in a few minutes, allowing it to be
repeated every few hours. The product of the proton target polarization and target
thickness, PT x, is determined by measuring the neutron transmission asymmetry at
an energy below 2 MeV where the uncertainty in T is small. This measurement,
however, is slow, taking on the order of 12 hours to reach a sucient statistical
accuracy. Because of the length of time required, this measurement is usually only
performed once, at the beginning of an experiment. In addition, the instrumental
asymmetry of the system due to errors such as beam misalignment must be measured.
This requires taking neutron transmission asymmetry data with the target warmed
to a high enough temperature (> 800 mK) to depolarize the protons. The true
asymmetry is then taken to be the dierence between the cold and warm values.
Because the cold and warm measurements contribute equally to the nal uncertainty,
the most ecient use of time is to measure them to equal statistical precision.
4.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
0
89
The cooling of the target begins 3{5 days before the start of measurements, which
typically last for 7 days. This allows the dilution refrigerator to run for at least
48 hours before the target is needed, giving it time to reach thermal equilibrium. Because of the long time required to polarize the target, the cold measurements are made
rst. The beam polarization is measured at an energy where the 12C(p,p0)12C analyzing power is known, and the energy is then lowered to En = 1:94 MeV to determine
PT x. TSCA windows are set for the carbon-foil polarimeter and PSD parameters
are adjusted for the neutron detectors. The gas cell in front of the tritiated-titanium
foil is evacuated and then lled with helium. Helium is added to the cell approximately once per day to replace the gas which diuses through the beam entrance foil.
Neutron transmission asymmetries are measured for approximately 12 hours at this
energy. Beam polarization measurements are made every few hours to monitor the
state of the polarized ion source.
Once the target polarization has been determined, measurements are made at
higher neutron energies where T is not known. With each energy change, the
polarimeter TSCA windows and neutron detector PSD settings must be reset. Data
is collected for 18{48 hours at each energy, with beam polarization measurements
made every few hours. After making measurements at all energies of interest, the
target is warmed to approximately 1 K. The asymmetry measurements are then
repeated at each energy, including 1.94 MeV.
4.2
Measurements for the 3H(~p,~n)3He Reaction
Kyy
0
The polarization-transfer coecient, Kyy , can be written as the ratio of outgoing
neutron polarization to incoming proton polarization
0
Kyy = PPn :
p
0
(4:10)
The measurement of the proton beam polarization is discussed in Section 3.2 (see
Equation 3.7). The neutron polarization is measured by scattering the neutrons
from helium. The scattered neutron "uxes observed by a pair of detectors placed
90
Chapter 4. Experimental Procedure
symmetrically about the beam direction can be written as Hai72]
NL = N0(1 + PnAy )L
NR = N0(1 ; PnAy )R
(4.11)
where refers to the spin state, N0 are the neutron "uxes into the detector for an
unpolarized beam, Ay is the analyzing power, and contains the detector eciency
and solid angle. Since the "ux from the polarized ion source is dierent for the two
spin states, N0+ and N0; must be specied separately. N0 and can be eliminated
by forming the following ratio of count rates.
NL+ NR; = N0+L(1 + Pn+ Ay )N0;R(1 ; Pn; Ay )
(4.12)
NL; NR+
N0;L(1 + Pn; Ay )N0+R(1 ; Pn+ Ay )
+ Pn+ Ay )(1 ; Pn; Ay )
(4.13)
= (1
(1 + Pn; Ay )(1 ; Pn+ Ay )
Substituting for Pn using Equation 4.3 gives
(1 + Pn Ay )(1 ; Pn Ay ) = 1 + (Pn Pn )Ay ]1 ; (Pn Pn )Ay ]
(4.14)
= (1 Pn Ay )2 ; Pn2A2y (4.15)
(1 Pn Ay )2
(4.16)
where the approximation Pn2A2y 1 has been made. Since Pn+ and Pn; are opposite
in sign and equal in magnitude to within 20%, this is a valid approximation. The
neutron polarimeter asymmetry, "npol, is then dened as
r
NL+ NR ; 1
N N
(4.17)
"npol = r NL NR+ L+ R + 1
NL NR+
;
;
;
;
= Pn Ay :
(4.18)
The neutron detectors are placed at an an angle corresponding to a maximum
in the product of the square of the 4He analyzing power and the cross section, A2y .
4.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
0
91
At most energies, two detector pairs are used to take advantage of both the forward
and backward maxima. The ght path from the helium cell to the neutron detectors
is set as a compromise between time resolution and counting rate. The coincidence
timing between the center and side detectors is set using a 22Na source. The neutron
polarization is measured continuously for a period of approximately 24 hours. Such
a long time is necessary due to the low counting rate. The proton polarization is
measured approximately every 2 hours. With each change in energy, the detector
angles are changed and the amplier gains and coincidence timing is checked. In
addition, PSD must be set for each detector at each energy.
Chapter 5
Data Analysis
5.1
T
Measurements
Following Equation 4.8, as
T
can be written in terms of the experimental parameters
(5:1)
T = xP2"P T n
where " is the observed asymmetry in neutron transmission, Pn is the neutron beam
polarization, PT is the proton target polarization, and x is the target thickness. As
discussed in Section 2.2, T is constrained at low energies by kinematic considerations and by the properties of the deuteron. At energies below 2 MeV, the values
of T obtained from potential models, phase-shift analysis, and eective-range parameters are in good agreement. Thus it is possible to write
" " (E ) #
T (E ) = "Pn(E ) T (E0)
(5:2)
Pn 0
where E0 2 MeV is the calibration energy at which T can be calculated, and E is
a higher energy at which T is to be determined. The asymmetry at E0 eectively
measures the product of target thickness and polarization, xPT . Normalizing to
the asymmetry at E0 eliminates all energy-independent quantities from the analysis.
Additionally, measurements made with the target unpolarized allow the subtraction
of instrumental asymmetries due to eects such as target misalignment. The data
has been taken in four sets, labelled A, B, C, and D, each having a dierent target
polarization. In two cases, a measurement has been repeated within the same set.
92
5.1. T
Measurements
93
These measurements are distinguished by the subscripts 1 and 2. In the analysis of
the data, statistical and systematic errors will be treated separately.
5.1.1 Calculation of the Neutron-Transmission Asymmetries
The data is collected in time-ordered spectra with each channel giving the number of
counts collected during one eight-step spin sequence lasting 800 ms. Since the data is
sorted by spin state, there are two spectra for each scaler value corresponding to data
taken with spin up and with spin down . Because the amount of time spent collecting
data in each spin state is determined by hardware, interrupting the data collection
will make these times unequal during the current spin sequence. This problem occurs
when the beam current falls outside of the discriminator window and an inhibit signal
is sent to the data acquisition system. These events must be detected by the o-line
analysis and the data from these spin sequences rejected. This task is accomplished
by analyzing the dead-time pulser spectra for irregularities.
The dead-time pulser produces regular pulses with a frequency of approximately
100 kHz, equivalent to 40,000 counts per channel in each spin state. The pulser is
stable to approximately 5 10;4 over the time spanned by an entire spectrum and
any deviations larger than this value indicate that the time spent collecting data was
incorrect. When data collection is interrupted, one or both of the pulser spectra will
show a spike or dip in an otherwise "at spectrum (Figure 5.1). These irregularities
are used to identify the channels to be rejected by the FORTRAN program CHAY.
This program makes three passes through each set of spectra in evaluating the data.
The rst pass generates a histogram of the number of channels versus counts per
channel with a bin width of 250 counts. The rst-pass average counts per channel is
then taken to be that of the bin with the most number of channels. This method of
estimating the average removes most of the spikes and dips which would otherwise
bias the result, as they are not statistically distributed. The second pass calculates
the average pulser counts per channel, excluding all channels which are more than 200
counts away from the rst-pass average. This value is essentially the average which
would be measured if no interruptions in the data collection occur. In addition, a
standard deviation is calculated. The third and nal pass extracts the data, excluding
94
Chapter 5. Data Analysis
Counts
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0
200
400
600
Channel
Figure 5.1: Dead-Time Pulser Spectrum
800
1000
5.1. T
Measurements
95
all channels where the pulser counts in either spectrum are more than four standard
deviations away from the average. Since the dead-time pulser is independent of the
detectors, this process in no way aects the statistical distribution of the neutron
data. Typically, less than 1% of the data must be rejected.
The data which is not rejected is used to calculate asymmetries in the transmitted
neutron "ux on a channel-by-channel basis. First, neutron yields, N~ , are calculated
by normalizing the detector counts, N , to the integrated beam current, I , and multiplying by the dead-time correction, .
(5:3)
N~ = NI The asymmetry for either of the two neutron detectors is then
~
~
" = N~+ ; N~;
(5.4)
N+ + N;
(5.5)
= ;
+
where the subscripts \+" and \;" refer to the spin state and
+ +
(5.6)
= N
N;; = II+ :
(5.7)
;
Since the detector counts obey a Poisson distribution, the standard uncertainty is
simply given by the square-root of the number of counts.
p
N = N
(5:8)
The counts from the beam current integration, however, are not normally distributed.
Instead, the uncertainty arises from the nite resolution due to digitization. An
uncertainty of 1/2 count occurs when changing spin state. Because of the order of
the eight-step sequence (+ ; ; + ; + +;), three such changes occur for each spin
state.1 Adding the uncertainties in quadrature gives the uncertainty in the integrated
Since the time spent vetoing data during spin reversal is short compared to the time between
BCI pulses, ;; and ++ are treated as if they are one continuous measurement in this analysis.
1
96
current for one spin state.
Chapter 5. Data Analysis
p
I = 23
From these results and can be found.
q
= 1=N+ + 1=N;
(5:9)
(5.10)
p q
3
= 2 1=I+2 + 1=I;2
The uncertainty in the asymmetry for one detector is then
s
2
" = ( + )2 1=N+ + 1=N; + 34 (1=I+2 + 1=I;2 ):
(5.11)
(5:12)
The asymmetries for the two detectors can be combined in two ways. Averaging the
two asymmetries gives the same result as a single detector located at 0 . It is this value
which will be used to obtain T . Taking the dierence of the two asymmetries gives
essentially a measure of the analyzing power for the 3H(~p,~n)3He source reaction and is
proportional to the neutron beam polarization. The sum and dierence asymmetries
are dened as
(5.13)
"S = "T +2 "B "D = "T ;2 "B :
(5.14)
The uncertainties for these combined asymmetries, however, cannot be obtained simply by adding the uncertainties for the individual asymmetries in quadrature. Since
"T and "B are calculated using the same beam current values, I+ and I;, a correlation
exists. The uncertainties in "S and "D are given properly as
s
(5.15)
"S = 14 ("T 2 + "B 2) + ( + 2)2T(B + )2 2
T
B
s
"D = 14 ("T 2 + "B 2) ; ( + 2)2T(B + )2 2:
(5.16)
T
B
The correlation term increases the uncertainty in "S and decreases it in "D.
5.1. T
Measurements
97
Once the asymmetries have been calculated for each 800 ms channel and for all
runs, they are averaged and standard deviations are calculated.
n " !
X
i
"2
(5.17)
" =
2 "
i
i=1
v
u
1
u
" = u
(5.18)
n
u
X
t 1="2i
i=1
v
u
n
u
X
u
("i ; ")2="2
u
u
i=1
(5.19)
" = t
n(n ; 1)
Here, n is the total number of 800 ms measurements and " is the reduced standard
deviation. The results for the neutron detector asymmetries for all measurements are
tabulated in Table 5.1. In all subsequent calculations, the standard deviations will
be used as the uncertainties in these asymmetries.
5.1.2 Calculation of the Average Neutron Beam Polarization
The neutron polarization is determined by measuring the polarization of the proton
beam with a carbon analyzer as described in Section 3.2. These measurements are
typically made every 2{4 hours, with the exception of data at the calibration energy
E0 = 1:94 MeV. At this energy the analyzing power is small (approximately ;0:15),
making beam polarization measurements susceptible to systematic eects. For this
reason, polarization measurements are made at a higher energy either before or after
the data is taken. In some cases, the polarization is measured both before and after
and the results averaged. Polarimeter asymmetries are measured for spin up and
spin down and the dierence taken to cancel systematic eects.
(5:20)
"ppol = "ppol+ ;2 "ppol;
Runs taken between two polarization measurements are assigned a polarimeter asymmetry equal to the average of the two. Runs at the beginning or end of a T measurement are assigned an asymmetry by a linear extrapolation from the two closest
98
Chapter 5. Data Analysis
En (MeV) Set
n
1.94
A
47,174
B
48,651
C
49,428
D1 39,448
D2
7,127
3.65
C
57,865
D
20,263
4.42
A
63,316
4.91
A
66,302
5.21
D1 44,931
D2 185,087
5.81
A
47,236
6.25
B
42,061
C 142,101
En (MeV) Set
n
1.94
A
37,624
C
54,789
D
47,699
3.65
C
67,179
D
23,126
4.42
A
60,624
4.91
A
64,152
5.21
D 221,298
5.81
A
68,713
6.25
B 173,181
C 128,196
"S
179.60
;115:00
;177:10
;107:20
;114:40
;33:81
;25:99
7.45
1.54
0.72
;0:06
;4:83
6.24
5.06
Target Polarized
"S ("S )
4.36 (4:52)
5.06 (5:27)
6.26 (6:89)
4.18 (4:42)
9.97 (11:03)
3.22 (3:31)
6.29 (6:46)
2.74 (2:75)
2.83 (2:85)
5.08 (5:14)
1.39 (1:41)
3.00 (3:03)
5.54 (5:59)
2.33 (2:36)
"D
;70:44
60.72
68.83
56.15
52.53
;26:09
;23:64
;44:47
;26:10
17.75
20.96
;10:59
15.19
16.24
"D
4.08
4.74
6.00
3.52
8.44
3.03
5.38
2.52
2.64
4.37
1.31
2.84
5.16
2.24
("D )
(4.21)
(4:90)
(6:73)
(3:62)
(8:50)
(3:13)
(5:39)
(2:52)
(2:64)
(4:40)
(1:31)
(2:85)
(5:25)
(2:26)
Unit
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
"S
;7:21
20.21
4.54
4.76
3.60
2.93
;3:03
;3:32
;1:08
;3:11
;4:88
Target Unpolarized
"S ("S )
4.94 (5:12)
5.92 (6:70)
4.06 (4:80)
2.92 (2:98)
4.51 (4:71)
2.51 (2:50)
2.53 (2:56)
1.38 (1:39)
2.44 (2:47)
2.57 (2:59)
2.35 (2:38)
"D
;68:81
60.86
60.62
;16:61
;29:15
;51:89
;24:47
23:87
;4:51
23.55
22.04
"D
4.60
5.65
3.50
2.76
4.07
2.37
2.39
1.30
2.32
2.40
2.27
("D )
(4:77)
(6:40)
(3:55)
(2:85)
(4:04)
(2:37)
(2:39)
(1:29)
(2:31)
(2:44)
(2:30)
Unit
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
Table 5.1: Average Neutron-Transmission Asymmetries
5.1. T
Measurements
Ep (MeV)
4.66
5.04
5.41
5.78
5.89
6.18
6.77
7.21
7.55
7.99
8.66
8.90
9.15
9.66
n `max 2
13
7 1.09
13
10 1.56
9
8 2.34
9
5 0.99
13
8 1.12
11
7 9.53
13
10 3.34
13
9 3.28
13
9 1.16
13
9 0.72
13
6 1.48
16
8 1.53
20
6 1.29
12
8 0.77
99
d
d
(mb)
158.0
162.0
132.8
182.5
178.6
195.0
218.0
177.0
125.0
148.0
133.0
102.3
129.9
85.0
Aexp
y
;0:50
;0:72
;0:521
;0:779
;0:849
;0:82
;0:53
;0:30
;0:36
;0:14
+0:00
+0:124
;0:011
;0:342
Aexp
y
0.03
0.01
0.026
0.041
0.034
0.01
0.01
0.02
0.02
0.02
0.02
0.041
0.044
0.058
Afit
y
;0:496
;0:724
;0:535
;0:775
;0:812
;0:851
;0:520
;0:278
;0:339
;0:131
;0:015
+0:125
;0:010
;0:348
Afit
y
0.022
0.009
0.024
0.015
0.022
0.009
0.009
0.015
0.016
0.017
0.014
0.041
0.015
0.054
Table 5.2: Analyzing Power Fits for 12C(p,p0)12C at lab = 40
polarimeter measurements. The average polarimeter asymmetry for an entire T
measurement is then calculated, weighting each asymmetry by the number of runs
which it spans.
Analyzing powers for the polarimeter are determined from published data Mos65,
Ter68] for the elastic scattering of protons from carbon. The products of dierential
cross section, dd , and analyzing power, Ay , are t as a function of center-of-mass
angle using rst-order associated Legendre polynomials. The number of terms used
in the t, `max, is chosen at each energy to minimize the reduced chi-square, 2 . The
exp
results of the t, Afit
y , as well as the experimental value, Ay , at the angle of interest
(lab = 40 ) are shown in Table 5.2, where n is the number of experimental data
used in each t. Proton beam polarizations are calculated using Equation 3.7 and
neutron beam polarizations using Equation 3.8 and the depolarization factor listed
in Table 3.6.
Kyy Pp
Pn = P=P
(5:21)
0
0
Table 5.3 lists the average beam polarizations, Pp and Pn, for each measurement.
The uncertainties, Pp and Pn , come from uncertainties in the values of the carbon
analyzing power and in the polarization-transfer coecient and will be treated as
100
Chapter 5. Data Analysis
En (MeV) Ep (MeV)
1.94
5.89
7.21
4.66
4.66
6.18
3.65
4.66
4.42
4.91
5.21
5.41
5.89
6.18
5.81
6.25
6.77
7.21
Set
A
B
C
D1
D2
C
D
A
A
D1
D2
A
B
C
Target Polarized
"ppol
Pp
;0:499 0.755
0.151 ;0:667
0.279 ;0:691
0.245 ;0:607
0.314 ;0:453
0.270 ;0:669
0.213 ;0:528
;0:276 0.634
;0:496 0.751
0.290 ;0:419
0.412 ;0:595
;0:285 0.673
0.153 ;0:676
0.162 ;0:716
Target Unpolarized
Pp
En (MeV) Ep (MeV) Set "ppol
1.94
5.41
A ;0:291 0.668
7.21
C
0.150 ;0:663
4.66
D
0.273 ;0:675
3.65
4.66
C
0.268 ;0:664
D
0.273 ;0:676
4.42
5.41
A ;0:294 0.675
4.91
5.89
A ;0:440 0.666
5.21
6.18
D
0.430 ;0:621
5.81
6.77
A ;0:256 0.605
6.25
7.21
B
0.149 ;0:659
C
0.154 ;0:681
Pp
0.020
0.036
0.031
0.027
0.005
0.030
0.023
0.028
0.020
0.004
0.006
0.012
0.036
0.039
Pn
0.496
;0:438
;0:454
;0:399
;0:298
;0:471
;0:372
0.439
0.532
;0:310
;0:440
0.547
;0:559
;0:592
Pn
0.030
0.034
0.032
0.028
0.017
0.033
0.025
0.028
0.023
0.011
0.015
0.020
0.036
0.038
Pp
0.030
0.036
0.030
0.029
0.030
0.030
0.018
0.007
0.010
0.036
0.037
Pn
0.439
;0:436
;0:444
;0:468
;0:476
0.468
0.472
;0:460
0.491
;0:545
;0:563
Pn
0.031
0.034
0.031
0.032
0.033
0.030
0.021
0.016
0.017
0.035
0.036
Table 5.3: Average Proton and Neutron Beam Polarizations
5.1. T
Measurements
101
systematic uncertainties. The analysis of the polarization-transfer coecient, Kyy , is
discussed in Section 5.2.
Knowledge of the beam polarizations allows neutron asymmetry measurements
made at the same energy and target polarization to be combined. In order to simplify the calculation of uncertainties, whenever possible constants containing systematic uncertainties are not included in the normalization. Instead, they are factored in
at a later stage. Thus, the neutron asymmetries are normalized to the proton polarimeter asymmetries whenever the analyzing power is the same for all measurements
at that energy. This is possible at all energies except E0 where the polarization has
been determined at a variety of beam energies. In this case the neutron asymmetries
are normalized to the proton beam polarization. Since the polarization-transfer coefcients are the same for all measurements at a given energy, Kyy is not included until
later in the analysis. Once normalized, the values are weighted by their statistical
variance and averaged.
"X
#
n
y
i
2
y =
(
(5.22)
stat y)
2
i=1 (stat yi )
v
u
1
u
(5.23)
stat y = u
n
u
t X1=(stat yi)2
0
0
i=1
#
sys yi
sys y =
(stat y)2
(5.24)
2
(
y
)
i=1 stat i
In these expressions yi refers to either "="ppol or "=Pp . Note that unpolarized target data taken in dierent sets can be combined as the target polarization is the
same in all cases, namely zero. This is not true, however, for the polarized target
data, and asymmetries from dierent data sets cannot be combined at this point.
Finally, instrumental asymmetries are removed by subtracting the warm data (target
unpolarized) from the cold data (target polarized).
"X
n
"="ppol = ("="ppol)c ; ("="ppol)w
(5.25)
"=Pp = ("=Pp)c ; ("=Pp )w
(5.26)
The normalized asymmetries for the cases cold, warm, and cold minus warm as well
102
Chapter 5. Data Analysis
as the appropriate averages are shown in Tables 5.4, and 5.5. Here, stat and sys
refer to the statistical and systematic uncertainties, respectively.
5.1.3 Calculation of
T
The next step in the analysis of the data is to combine measurements made at the same
energy, but in dierent data sets which have dierent target polarizations. Since the
target polarization information is contained in the calibration data, the other energies
are normalized to these values, forming the ratio .
"
"ppol (E )
(5:27)
= " (E )
Pp 0
The averaging is performed as before and the results shown in Table 5.6.2 T values
can now be determined by including the constants from Equation 5.2 and multiplying
by the calculated value of T at E0.
8
9
3
"
#" y
#2
<
=
Ky (E0) 4 PP0 (E0) 5
(
E
)
T = :Ay (E ) (E )
(5:28)
(
E
)
T 0 Kyy (E ) PP0 (E )
0
0
0
The 12C(p,p0)12C analyzing power for the calibration measurement does not appear
in this expression as it was included at an earlier stage of the analysis. The corrections
for unpolarized neutron background, , are obtained from Table 3.5 and values for
the polarization-transfer coecient, Kyy from Table 5.14. T (E0) is obtained by
taking the average of predictions from potential models, phase-shift analysis, and
eective-range parameter calculations (Table 5.7).3 The full Bonn is a relativistic
meson exchange model. Bonn A, B, and C are one-boson exchange approximations
to the full Bonn. The three dier only in the strength of the tensor interaction. A
systematic uncertainty is assigned equal to the standard deviation of the predicted
values. Table 5.8 lists the values of T obtained from the analysis of the data. The
total uncertainty, , is given by
q
(5:29)
= (stat)2 + (sys )2:
0
There exists a correlation in the values being averaged due to using the same warm asymmetries
in both cases. The eect, however, is small in practice and will be ignored in this analysis.
3Predictions from the Paris and Nijmegen Potentials are not included as they use 1S phase shifts
0
obtained from p-p scattering. 1 S0 for n-p scattering is known to be dierent by several degrees.
2
5.1. T
Measurements
103
Target Polarized
En (MeV)
Set
("="ppol )c
stat
3.65
C
;125:22
12.26
D
;122:02
30.33
4.42
A
;26:99
9.96
4.91
A
;3:11
5.75
5.21
D1
2.48
17.72
D2
;0:15
3.42
Average
;0:06
3.36
5.81
A
16.95
10.63
6.25
B
40.78
36.54
C
31.24
0.00
sys
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Unit
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
Target Unpolarized
En (MeV)
Set
("="ppol )w
stat
3.65
C
17.76
11.11
D
13.19
17.25
Average
16.42
9.34
4.42
A
;9:97
8.50
4.91
A
6.89
5.82
5.21
D
;7:72
3.23
5.81
A
4.22
9.65
6.25
B
;20:87
17.38
C
;31:69
15.46
Average
;26:91
11.55
sys
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Unit
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
Target Polarized Minus Target Unpolarized
En (MeV)
Set
"="ppol
stat
sys
3.65
C
;141:64
16.55
0.00
D
;138:44
31.74
0.00
4.42
A
;17:02
13.09
0.00
4.91
A
;10:00
8.18
0.00
5.21
D
7.66
4.66
0.00
5.81
A
12.73
14.36
0.00
6.25
B
67.69
38.32
0.00
C
58.15
18.59
0.00
Unit
10;4
10;4
10;4
10;4
10;4
10;4
10;4
10;4
Table 5.4: Neutron-Transmission Asymmetries Normalized to the Proton Polarimeter Asymmetry
104
Chapter 5. Data Analysis
Target Polarized
En (MeV)
Set
("=Pp )c
stat
1.94
A
237.88
5.99
1.94
B
172.41
7.90
1.94
C
256.30
9.97
1.94
D1
176.61
7.28
D2
252.54
24.35
Average 182.84
6.98
sys
6.30
9.31
11.50
7.86
2.79
7.44
Unit
10;4
10;4
10;4
10;4
10;4
10;4
Target Unpolarized
En (MeV)
Set
("=Pp )w
stat
1.94
A
;10:79
7.67
C
;30:48
10.11
D
;6:72
7.11
Average ;13:20
4.63
sys
0.47
1.66
0.30
0.65
Unit
10;4
10;4
10;4
10;4
Target Polarized Minus Target Unpolarized
En (MeV)
Set
"=Pp
stat
sys
1.94
A
251.08
9.73
6.33
1.94
B
185.61
9.16
9.33
1.94
C
269.50
10.99
11.52
1.94
D
196.04
8.38
7.47
Unit
10;4
10;4
10;4
10;4
Table 5.5: Neutron-Transmission Asymmetries Normalized to the Proton Beam
Polarization
5.1. T
Measurements
105
En (MeV) Set
stat sys Unit
3.65
C
;52:56
6.50
2.25 10;2
D
;70:62
16.48
2.69 10;2
Average ;54:99
6.05
2.31 10;2
4.42
A
;6:78
5.22
0.17 10;2
4.91
A
;3:98
3.26
0.10 10;2
5.21
D
3.91
2.39
0.15 10;2
5.81
A
5.07
5.72
0.13 10;2
6.25
B
36.47
20.73
1.83 10;2
C
21.58
6.96
0.92 10;2
Average 23.09
6.60
1.01 10;2
Table 5.6: Neutron-Transmission Asymmetries Normalized to Beam and Target
Polarizations
T (E0) (mb)
Full Bonn
899.8
Bonn A
958.5
Bonn B
959.2
Bonn C
964.0
Arndt SM92 Analysis
879.2
Eective Range Parameters 972.7
Average
938.9 39.1
Table 5.7: Calculated Values of T at a Neutron Energy of 1.94 MeV
Model
En (MeV) T (mb)
stat
sys
3.65
260.2
28.6
29.1
4.42
45.9
35.3
4.8
4.91
42.1
34.5
3.9
5.21
;39:3
24.1
3.9
5.81
;26:5
29.9
3.0
6.25
;67:8
19.4
7.6
Table 5.8: Measured Values of T
40.8
35.6
34.7
24.4
30.1
20.8
106
Chapter 5. Data Analysis
5.1.4 Determination of the Zero-Crossing Energy of
T
In order to determine the zero-crossing energy, a continuous function must be t to
the T data. For this purpose, the prediction of the Bonn potential is used with
the transformation
E ! a + bE:
(5:30)
The coecients a and b are allowed to vary to nd the smallest total chi-square.
" i
Bonn(a + bE ) #2
X
(
E
)
;
T
T
2 =
(5:31)
i )(E )
(
T
i
The uncertainties used are the total uncertainties of Table 5.8. The best t is obtained
with a = ;0:800 MeV, b = 1:127, resulting in a total chi-square of 2 = 3:98 and
a reduced chi-square of 2 = 0:995. The data and t are shown in Figure 5.2. The
zero-crossing energy obtained from the t is
Ezc = 5:08 0:10 MeV:
The errors on the zero-crossing energy are obtained by shifting a by an amount
sucient to increase 2 by 1.0. The zero-crossing energy determined by the shifted t
gives the one-sigma bound on Ezc . A simple phase-shift analysis has been performed,
allowing only the 1 mixing parameter to vary4 in order to reproduce Ezc . The result
obtained from the analysis is
1 = 0:30 0.17 :
These results will be discussed further in Chapter 6.
5.2
Measurements for the 3H(~p,~n)3He Reaction
Kyy
0
Since the determination of T depends upon the polarization-transfer coecients,
Kyy , for the 3H(~p,~n)3He neutron source reaction, it is important to know these parameters well. In particular, it is essential to know the polarization transfer at the
calibration energy, E0. Existing data only extends down to a neutron energy of
0
4
All phase-shift parameters except 1 are obtained from the full Bonn potential.
5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
107
0
T (mb)
600:0
500:0
400:0
300:0
200:0
100:0
0:0
;100:0
;200:0
3:0
3:5
4:0
4:5
5:0
En (MeV)
5:5
Figure 5.2: T Data with Fit
6:0
6:5
7:0
108
Chapter 5. Data Analysis
Forward Angle
Backward Angle
En (MeV) NL+ NL; NR+ NR; NL+ NL; NR+ NR;
1.94
| | | | 1316 2192 1757 1117
5.21
1133 676 632 1006 204 422 353 197
5.81
198 361 303 157
81 29 41 107
Table 5.9: Neutron Counts in the Neutron Polarimeter Detectors
2.12 MeV, requiring an extrapolation to E0 = 1:94 MeV. In addition, it is useful to
conrm the earlier data in the region of the T zero-crossing. For these reasons, Kyy
has been measured at neutron energies of 1.94, 5.21, and 5.81 MeV. The new data
is combined with the existing data to predict the polarization transfer at all energies
at which T measurements are made.
0
5.2.1 Determination of Kyy
0
The neutron polarimeter asymmetry is calculated for each pair of detectors by forming
the asymmetry for the two detectors as dened in Section 4.2.
rN N
L+ R
NL NR+ ; 1
"npol = r N N
(5.32)
L+ R + 1
NL NR+
q
NL+ NR;(NL; + NR+) + NL; NR+(NL+ + NR;)
"npol =
(5.33)
p
2
p
NL+NR; + NL; NR+
;
;
;
;
NL and NR refer to the number of counts in the left and right detectors respectively.
The subscripts \+" and \;" refer to the spin-state of the beam. This method of averaging reduces the sensitivity of the asymmetry to beam misalignments and detector
eciency dierences. In addition, since two detectors are used, it is not necessary
to know the incident neutron "ux. Table 5.9 lists the total number of counts in
each detector and spin-state combination summed over all runs. The average neutron
polarizations can be calculated from the average asymmetries using
(5:34)
Pn = A "(npol
y 4He)
5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
0
En (MeV) "npol
"npol Pn
stat Pn
1.94
107 ;0:2362
0.0122 ;0:309
0.016
5.21
50
0.2405
0.0166 ;0:382
0.026
121 ;0:3163
0.0277 ;0:344
0.030
Average
;0:366
0.020
5.81
51 ;0:3045
0.0300 0.482
0.048
121
0.4594
0.0560 0.502
0.061
Average
0.490
0.038
Table 5.10: Average Neutron Beam Polarizations
109
sys Pn
0.004
0.006
0.006
0.006
0.008
0.008
0.008
En (MeV) Ep (MeV) "ppol
Pp
Pp
1.94
6.18
0.401 ;0:471
0.005
5.21
6.18
0.395 ;0:464
0.005
5.81
6.77
;0:316 0.608
0.011
Table 5.11: Average Proton Beam Polarizations
where the 4He(n,n)4He eective analyzing powers, Ay (4He), are obtained from Table 3.7. The uncertainties in the analyzing powers are treated as systematic uncertainties. The results for each angle pair at each energy as well as the weighted
averages over both angles are listed in Table 5.10.
The average proton polarimeter asymmetries are obtained by taking the dierence
between the spin up and spin down asymmetries as in Equation 5.20 and averaging
over all runs. The average proton polarization is given by
(5:35)
Pp = A "(pol
y 12C)
where the 12C(p,p0)12C analyzing powers, Ay (12C), are obtained from Table 5.2.
Again, the uncertainties in the analyzing powers are treated as systematic uncertainties. As in the T measurements, the proton beam polarization for the En =
1:94 MeV measurement is obtained at a higher energy. Table 5.11 lists the average
proton polarizations at each energy.
Once the neutron and proton beam polarizations are known, the polarization-
110
Chapter 5. Data Analysis
En (MeV)
1.94
5.21
5.81
Kyy
0.656
0.789
0.806
stat Kyy
0.034
0.043
0.063
0
sys Kyy
0.011
0.015
0.020
0
0
Kyy
0.036
0.046
0.066
0
Table 5.12: Measured Values of the 3H(~p,~n)3He Polarization-Transfer Coecient
transfer coecient is simply the ratio of the two.
Kyy = PPn
p
(5:36)
0
Table 5.12 lists the values of Kyy obtained, as well as statistical and systematic uncertainties. The total uncertainty, Kyy , is given by
q
(5:37)
Kyy = (stat Kyy )2 + (sys Kyy )2:
0
0
0
0
0
5.2.2 Interpolation of Kyy Data
0
The present Kyy measurements are combined with data from an earlier experiment
at Los Alamos National Laboratory (LANL) Don71, Hai72] in order estimate the
values at the energies of interest. These data are listed in Table 5.13 and plotted in
Figure 5.3. The value of Kyy is estimated at other energies by linear interpolation
between the two nearest measurements. The uncertainty is estimated by a linear
interpolation of the variances. Estimates of Kyy are made by this method at the
neutron energies used in the T measurements. These values are listed in Table 5.14.
0
0
0
5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
111
0
En (MeV)
1.94
2.12
3.12
4.15
4.66
5.16
5.21
5.67
5.81
6.17
6.67
7.17
7.68
8.18
9.18
10.19
11.18
12.18
13.18
14.19
15.19
Kyy
0.656
0.632
0.661
0.719
0.681
0.724
0.789
0.788
0.806
0.828
0.808
0.789
0.769
0.785
0.670
0.681
0.567
0.551
0.444
0.509
0.393
0
Kyy
0.036
0.048
0.033
0.052
0.031
0.036
0.046
0.037
0.066
0.045
0.040
0.047
0.048
0.043
0.050
0.048
0.043
0.056
0.046
0.039
0.036
0
Source
TUNL
LANL
LANL
LANL
LANL
LANL
TUNL
LANL
TUNL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
LANL
Table 5.13: 3H(~p,~n)3He Polarization-Transfer Coecient Data Measured at LANL
and TUNL
112
Kyy
Chapter 5. Data Analysis
0
1:0
TUNL LANL 2
0:9
0:8
0:7
0:6
2
2
2
2 2 2 2 2
2 2
2
2 2
2 2
0:5
2
0:4
2
2
0:3
0:2
0:1
0:0
0:0
2:0
4 :0
6:0
8:0
10:0
En (MeV)
12:0
14:0
16:0
Figure 5.3: 3H(~p,~n)3He Polarization-Transfer Coecient Data Measured at LANL
and TUNL
5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction
113
0
En (MeV)
1.94
3.65
4.42
4.91
5.21
5.81
6.25
Kyy
0.656
0.691
0.699
0.703
0.789
0.806
0.825
0
Kyy
0.036
0.044
0.042
0.034
0.046
0.066
0.044
0
Table 5.14: Interpolated Values of the 3H(~p,~n)3He Polarization-Transfer Coecient
Chapter 6
Comparison of Data with
Potential Models
The measured values of T in Table 5.8 can now be compared with potential-model
predictions. Figure 6.1 shows the measured values of T along with the t obtained
in Section 5.1.4, the phase-shift analysis SM92 of Arndt, and the predictions of several
potential models. The best agreement is obtained with the Bonn potentials, which
have a relatively weak tensor force. The region of the zero-crossing is expanded
in Figure 6.2 to compare the t to the potential models. In this plot, the error
bars indicate the uncertainty in the zero-crossing energy. The value of the 1 phaseshift parameter obtained by performing a simple phase-shift analysis at the zerocrossing energy (Section 5.1.4) is plotted in Figure 6.3. Also included in this plot are
the values from potential models and the Arndt SM92 phase-shift analysis. Results
obtained from recent experiments sensitive to 1 are also shown Ock91b, Ock91a,
Sch88]. The Bonn results are from measurements of the neutron-proton polarization
transfer coecient, Kyy .1 The Erlangen/T)ubingen result is from a measurement of
the neutron-proton spin-correlation parameter, Ayy (), at cm = 90
0
A value of 1 was not reported for the Bonn measurement at 17.4 MeV. Instead, a phase-shift
analysis similar to the one for the TUNL data was performed by W. Tornow to obtain the plotted
value Tor92].
1
114
115
T (mb)
600:0
TUNL (tpn) TUNL (ddn) ?
Fit
Arndt SM92
Full Bonn
Bonn A/B/C
Paris/Nijmegen
500:0
400:0
300:0
200:0
100:0
0:0
;100:0
;200:0
3:0
3:5
4:0
4:5
5:0
En (MeV)
5:5
6:0
6:5
7:0
Figure 6.1: Comparison of T Measurements with Potential-Model Predictions
116
Chapter 6. Comparison of Data with Potential Models
T (mb)
100:0
Ezc +
Fit
Arndt SM92
Full Bonn
Bonn A
Bonn C
Paris/Nijmegen
50:0
+ + +
0:0
;50:0
;100:0
4:0
4:2
4:4
4:6
4:8
5:0
5:2
En (MeV)
5:4
5:6
5:8
6:0
Figure 6.2: Comparison of the Measured Zero-Crossing of T with Potential-Model
Predictions
117
1 (degrees)
5:0
TUNL Erlangen/Tubingen 2
Bonn 4
Arndt SM92
4:0
Full Bonn
Bonn A
Bonn C
Paris
3:0
4
2:0
1:0
0:0
2
4
;1:0
;2:0
0:0
5:0
10:0
15:0
En (MeV)
20:0
25:0
30:0
Figure 6.3: Comparison of 1 Obtained from T Measurements with PotentialModel Parameters
Chapter 7
Conclusions and Summary
Measurements of the spin-dependent dierence in total cross section, T for the
scattering of polarized neutrons from polarized protons have been made at six energies
from 3.65 to 6.25 MeV. T crosses through zero in this energy range, and a zerocrossing energy has been extracted from the data:
Ezc = 5:08 0:10 MeV.
Since a zero-crossing measurement is unaected by most systematic errors, the uncertainty is due almost entirely to counting statistics. A phase-shift analysis has been
performed at the zero-crossing energy in which the mixing parameter 1 is allowed
to vary, while the other parameters are xed to the full Bonn potential values. The
result of this analysis is
1 = 0:30 0.17 :
A complete phase-shift analysis allowing all phase-shift parameters to vary and including all relevant data will be performed at a later date in collaboration with R. A.
Arndt at Virginia Polytechnic Institute and State University. However, no signicant
changes are expected from such an analysis. The present result suggests that the
tensor force is relatively weak in this energy range. Comparison of the data with
potential models shows agreement with potentials such as Bonn A which has a weak
tensor force and comes the closest to predicting the correct triton binding energy. The
result is also consistent with recent theoretical work suggesting that three-body forces
and relativistic eects are not signicant in the binding of the triton Pic92, Sam92],
118
119
leaving a weak tensor force as the most likely solution of the triton binding energy
problem. It is important to extend the present measurements to higher energies in
order to verify these conclusions.
Because of their importance in the analysis of the T data, polarization-transfer
coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies
using a neutron polarimeter. The results are in agreement with earlier measurements
made at Los Alamos National Laboratory.
0
Appendix A
Calculation of Neutron
Depolarization Due to Magnetic
Fields
A beam of neutrons passing through a magnetic eld has the spins of the neutrons
precessed about the eld component transverse to the spin direction. Since the beam
has a non-zero spatial extent, the trajectories of the particles are not identical. Thus,
particles with dierent trajectories traverse dierent magnetic elds and experience
diering spin precessions. This process results in an eective depolarization of the
beam which must be calculated by Monte Carlo techniques. The FORTRAN code
NEUTRONS has been developed to calculate the spin precessions of a beam of neutrons passing through an arbitrary magnetic eld. The program is based on the
computer code SPINX Bow91].
The program represents the wavefunctions of the neutrons as two-component Dirac
spinors (the lower two components are neglected).
0 1
=B
(A:1)
@ 1 CA
2
The precession is then represented by a 2 2 rotation matrix, T, which gives the
relationship between the initial and nal wavefunctions.
f = Ti
120
(A:2)
121
The precession is calculated as a product of small precessions which can be treated
as innitesimal.
T = TN TN ;1!TN ;2 : : : T3!T2T1
Tn = I cos 2 + i sin 2 B^ (A.3)
(A.4)
where I is the 2 2 identity matrix, is the angle of precession, B^ is a unit vector
in the direction of the magnetic eld, and is the Pauli spin vector
= xx^ + y y^ + z z^ :
The Pauli spin matrices are dened as
x
y
z
0
0
= B
@
1
0
0
= B
@
i
0
1
= B
@
0
1
1C
A
0
1
;i C A
0
1
0C
A:
;1
(A:5)
(A.6)
(A.7)
(A.8)
The spin precession angle for an innitesimal rotation, , is given by
= #pdB (A:9)
E
where d is the distance travelled, B is the magnitude of the magnetic eld, E is
the energy of the neutron and # = ;1:325 10;5 (MeV) 21 /(Gcm) is a constant.
The step size is made small enough such that the dierence between calculating the
rotation matrix in one step, Tn , and in two half-steps, TLn and TUn , is less than a
specied amount, , for each component.
2
Tij ; X
TikU TkjL (i = 1 2 j = 1 2)
(A:10)
k=1
122
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
The initial wavefunction is calculated from the angles i and i from the input
le which give the initial spin direction according to the Madison convention (Section 3.1.3).
!
!
!
!
(A.11)
1i = cos 2 cos 2 ; i cos 2 sin 2
!
!
!
!
2i = sin 2 cos 2 + i sin 2 sin 2
(A.12)
Trajectories are chosen by picking a point at random within the initial rectangular
beam spot, and picking a direction from a cosine distribution, with the constraint
that the trajectory ends inside the nal beam spot. The dimensions of the rectangular
initial and nal beam spots are specied in the input le.
Spin projections are calculated for the nal wavefunction along each cartesian
axis, and along an axis specied in the input le. The spin projection along an axis
specied by a unit vector, n^ , is given by
hP (^n)i = hf j P (^n) j f i (A.13)
^
I
+
(
n
)
P (^n) = 2 :
(A.14)
The spin projections are averaged over all trajectories and standard deviations are
calculated. The angles f and f which give the nal spin direction are also calculated.
The magnetic eld used by NEUTRONS is calculated in a separate program on a
three-dimensional grid. The components Bx, By , and Bz are calculated at each point.
For this experiment, the magnetic eld of the superconducting solenoid has been
calculated by summing over all windings. The eld of one winding can be calculated
from the vector potential of a current loop in cylindrical coordinates Jac75]
"
#
2
(2
;
k
)
K
(
k
)
;
2
E
(
k
)
4
Ia
(A.15)
A = q
k2
c (a + )2 + z2
k2 = (a + 4a
)2 + z2 where a is the radius of the winding, I is the current, c is the speed of light in vacuum,
and K (k) and E (k) are elliptic integrals. The solution is given as a series and is in
123
cgs units.
"
#2
1
X
4
Iaz
n
(2
n
+
1)
(2
n
;
1)!!
2n
B = c(a + )2 + z2]3=2
k
(A.16)
n
2 n!
n=0 n + 1
" (2n ; 1)!! #2
1
2
X
n
2
Ia
2n
k
(A.17)
Bz = c(a + )2 + z2]3=2 (2n + 1) 1 ; n + 1 a
n
2 n!
n=0
124
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
PROGRAM NEUTRONS
C
C
Written 1/8/91 by W. Scott Wilburn, TUNL
C
Based on the FORTRAN program SPINX by J. D. Bowman, LANL
C
C
***********************************************************
C
This program uses three files which must be assigned to
C
FORTRAN units before execution.
C
to FOR001. The format of this file can be obtained from
C
reading the comments about the input parameters given
C
below.
C
created by this program.
C
assigned to FOR003.
C
declared as BFIELD below must match that of the array
C
stored in this file.
C
***********************************************************
The input file is assigned
The output file is assigned to FOR002 and is
The B field array file is
Note that the size of the array
C
C
This program performs a Monte Carlo simulation of the
C
precession of the spins of a neutron beam as it passes
C
through a magnetic field.
C
SPINPROP.FOR in that it obtains the B field at each step
C
from an array previously generated by BFIELD.FOR instead of
C
calculating the values as they are needed.
C
trajectories are chosen to begin within a specified
C
rectangular area in the x-y plane and an initial z position
C
and to end within another rectangular area in the x-y plane
C
at a final z position.
C
equal probability within the specified plane, and the
C
direction of travel is chosen with a cosine probability
C
distribution such that a trajectory parallel to the z axis
C
is most probable.
C
function is calculated from the initial values of the
C
angles theta and phi given.
It differs from the program
Random
The initial position is chosen with
The initial-two component spinor wave
Each neutron is transported
125
C
along its chosen trajectory in small enough steps that the
C
spin precession can be treated as an infinitesimal
C
rotation.
C
the product of all the rotation
C
is computed along the way.
C
rotation matrix representing the entire precession has been
C
generated.
C
criteria which compares the result of taking a whole step
C
with that of taking two half steps.
C
the rotation matrix from the two step sizes differ by more
C
than a specified amount, the step size is cut in half and
C
we try again.
C
spinor wavefunction is calculated by operating on the
C
initial wavefunction with the rotation operator.
C
projection expectation values are calculated along the x,
C
y, and z axes and along the target spin direction and
C
written to output.
C
squares of sums for all neutrons are kept up with so that
C
the statistical mean and standard deviation can be
C
calculated.
C
calculate the mean theta and phi spin angles with standard
C
deviations.
A rotation matrix is generated for each step and
matrices (for one neutron)
At the end of the trajectory, a
The step size is determined with a convergence
If the components of
At the end of each trajectory, the final
Spin
Also, the sums of spin projections and
Finally, the mean spin projections are used to
C
C
Input Variables (read from FOR001):
C
C
E
Neutron Energy (MeV) (common block DATA)
C
ETA
Magnet Angle (deg converted to rad) (common
C
block DATA)
C
N
Number of neutrons
C
PHII
Initial azimuthal spin angle (deg conv. to rad)
C
THETAI
Initial polar spin angle (deg conv. to rad)
C
XMAXF
Maximum final x position (cm)
C
XMAXI
Maximum initial x position (cm)
126
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
C
YMAXF
Maximum final y position (cm)
C
YMAXI
Maximum initial y position (cm)
C
ZF
Final z position (cm)
C
ZI
Initial z position (cm)
C
C
Input Variables (read from FOR003):
C
C
BFIELD
Array of B field vectors (G) (common block
C
FIELD)
C
NBMAX
Dimensions of BFIELD
C
RMAX
Maximum x, y, and z covered by BFIELD (cm)
C
RMIN
Minimum x, y, and z covered by BFIELD (cm)
C
(common block FIELD)
C
C
Output Variables (written to FOR002):
C
C
PAVE
Mean values of spin projections
C
PDEV
Standard deviations of spin projections
C
PHIF
Mean azimuthal angle of final spin (deg)
C
PHIFDEV
Standard deviation of final phi (deg)
C
RF
Final position (cm)
C
RI
Initial position (cm)
C
THETAF
Mean polar angle of final spin (deg)
C
THETAFDEV
Standard deviation of final theta (deg)
C
P
Spin projections on cartesian axes and
C
target spin direction
C
C
Internal Variables:
C
C
CVG
C
C
C
Indicates whether convergence has occurred in
calculating rotation matrix
D
Step size in fraction of total distance to be
travelled
127
C
DELTA_T
C
Difference in component of rotation matrices
calculated using step D and D/2
C
DR
Magnitude of step (cm)
C
DX
Interval between points in BFIELD (cm)
C
(common block Field)
C
F
Fraction of total distance travelled
C
GAMMA
Relativistic gamma (common block DATA)
C
J
Counting variable
C
K
Counting variable
C
L
Counting variable
C
M
Counting variable
C
PSIF
Final spinor wavefunction
C
PSII
Initial spinor wavefunction
C
PSQSUM
Sum of squares of spin projections
C
PSUM
Sum of spin projections
C
R
Position vector (cm)
C
R_B
Position vector of B field for full step (cm)
C
RD
Position difference vector, RF-RI (cm)
C
RL_B
Position vector of B field for lower half
C
C
step (cm)
RU_B
C
Position vector of B field for upper half
step (cm)
C
S
Spin direction vector
C
SIN_ALPHA
Sine of horizontal trajectory angle
C
SIN_AMAX
Sine of maximum alpha
C
SIN_AMIN
Sine of minimum alpha
C
SIN_BETA
Sine of vertical trajectory angle
C
SIN_BMAX
Sine of maximum beta
C
SIN_BMIN
Sine of minimum beta
C
T
Rotation matrix for a full step
C
TL
Rotation matrix for lower half step
C
TT
Matrix product TU*TL
C
TU
Rotation matrix for upper half step
128
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
C
U
Rotation matrix for entire trajectory
C
W
Unit vector along target spin
C
C
Constants:
C
C
D0
Initial step-size in fraction of total distance
C
to be moved
C
E0
Neutron rest energy (common block CONSTANTS)
C
EPS
Convergence criteria on rotation matrices for
C
determining step-size
C
I
Imaginary unit (common block CONSTANTS)
C
PI
Pi (common block CONSTANTS)
C
SEED
Seed for random number generation for choosing
C
trajectories
C
SI
2x2 identity matrix (common block CONSTANTS)
C
SIGMA
2x2x3 Pauli spin vector (common block CONSTANTS)
C
SX
\
C
SY
} Pauli spin matrices (common block CONSTANTS)
C
SZ
/
C
X
Unit vector along x (common block CONSTANTS)
C
Y
Unit vector along y (common block CONSTANTS)
C
Z
Unit vector along z (common block CONSTANTS)
C
C
*****************************
C
*** Variable Declarations ***
C
*****************************
C
IMPLICIT NONE
C
LOGICAL*1
CVG
INTEGER*2
J,K,L,M,N
C
C
129
INTEGER*4
NBMAX(3),SEED
REAL*4
D,DX(3),D0,DELTA_T,DR,E,E0,EPS,ETA,F,GAMMA
C
>
,PAVE(4),PDEV(4),PHIF,PHIFDEV,R(3),R_B(3),RD(3)
>
,RF(3),RI(3),RL_B(3),RMIN(3),RU_B(3),RMAX(3)
>
,SIN_ALPHA,SIN_AMAX,SIN_AMIN,SIN_BETA,SIN_BMAX
>
,SIN_BMIN,THETAF,THETAFDEV,W(3),X(3),XMAXF
>
,XMAXI,Y(3),YMAXF,YMAXI,Z(3),ZF,ZI
C
REAL*8
>
BFIELD(11,27,407,3),P(4),PHII,PI,PSQSUM(4)
,PSUM(4),S(3),THETAI
C
COMPLEX*8
I,SI(2,2),SIGMA(2,2,3),SX(2,2),SY(2,2),SZ(2,2)
COMPLEX*16
PSIF(2),PSII(2),T(2,2),TL(2,2),TU(2,2),TT(2,2)
C
>
,U(2,2)
C
EQUIVALENCE (SIGMA(1,1,1),SX),(SIGMA(1,1,2),SY)
>
,(SIGMA(1,1,3),SZ)
C
COMMON /CONSTANTS/
E0,I,PI,SI,SIGMA,X,Y,Z
COMMON /DATA/
E,ETA,GAMMA
COMMON /FIELD/
BFIELD,DX,RMIN
C
C
C
DATA D0
/1.0/
DATA E0
/939.573/
DATA EPS
/1.0E-5/
DATA I
/(0.0,1.0)/
DATA SEED
/67459873/
DATA SI
/(1.0,0.0),(0.0,0.0),(0.0,0.0),(1.0,0.0)/
DATA SX
/(0.0,0.0),(1.0,0.0),(1.0,0.0),(0.0,0.0)/
130
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
DATA SY
/(0.0,0.0),(0.0,-1.0),(0.0,1.0),(0.0,0.0)/
DATA SZ
/(1.0,0.0),(0.0,0.0),(0.0,0.0),(-1.0,0.0)/
DATA X
/1.0,0.0,0.0/
DATA Y
/0.0,1.0,0.0/
DATA Z
/0.0,0.0,1.0/
C
C
********************************
C
*** Preliminary Calculations ***
C
********************************
C
C
Calculate value of pi
C
PI=DACOS(-1.0D0)
C
C
Calculate relativistic gamma
C
GAMMA=1+E/E0
C
C
******************
C
*** Read Input ***
C
******************
C
C
Get input parameters:
C
C
ZI
Initial z position of neutrons (cm) (R*4)
C
ZF
Final z position of neutrons (cm) (R*4)
C
XMAXI
Maximum initial x position (cm) (R*4)
C
YMAXI
Maximum initial y position (cm) (R*4)
C
XMAXF
Maximum final x position (cm) (R*4)
C
YMAXF
Maximum final y position (cm) (R*4)
C
N
Number of neutrons to be tracked (I*2)
C
E
Energy of neutrons (MeV) (R*4)
C
THETAI
Initial spin angle theta of neutron (deg) (R*4)
131
C
PHII
Initial spin angle phi of neutron (deg) (R*4)
C
ETA
Angle of magnet (deg) (0 is longitudinal) (R*4)
C
C
Open input file
C
OPEN (UNIT=1,STATUS='OLD',ACCESS='SEQUENTIAL',READONLY)
C
C
Read data
C
READ (1,*) ZI,ZF
READ (1,*) XMAXI,YMAXI
READ (1,*) XMAXF,YMAXF
READ (1,*) N,E
READ (1,*) THETAI,PHII
READ (1,*) ETA
C
C
Close input file
C
CLOSE (UNIT=1)
C
C
Open output file
C
OPEN (UNIT=2,STATUS='NEW',ACCESS='SEQUENTIAL')
C
C
Write input data and header for trajectory information to
C
output file
C
WRITE (2,100) ZI
WRITE (2,110) ZF
WRITE (2,120) XMAXI
WRITE (2,130) YMAXI
WRITE (2,140) XMAXF
WRITE (2,150) YMAXF
132
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
WRITE (2,160) N
WRITE (2,170) E
WRITE (2,180) THETAI
WRITE (2,190) PHII
WRITE (2,200) ETA
WRITE (2,210)
C
100
FORMAT (' Initial z position (cm)
',F7.2)
110
FORMAT (' Final z position (cm)
',F7.2)
120
FORMAT (' Maximum initial x position (cm)
',F7.2)
130
FORMAT (' Maximum initial y position (cm)
',F7.2)
140
FORMAT (' Maximum final x position (cm)
',F7.2)
150
FORMAT (' Maximum final y position (cm)
',F7.2)
160
FORMAT (' Number of neutrons
',I7)
170
FORMAT (' Neutron energy (MeV)
180
FORMAT (' Initial spin angle theta (degrees)
190
FORMAT (' Initial spin angle phi (degrees)
',F7.2)
200
FORMAT (' Magnet angle eta (degrees)
',F7.2)
210
FORMAT (/,' NUMBER
>
,'Px
Xi
Py
Yi
Pz
',F5.2)
Xf
',F6.2)
Yf
Pt')
C
C
Open file containing B field array
C
OPEN (UNIT=3,FORM='UNFORMATTED',ACCESS='SEQUENTIAL'
>
,STATUS='OLD')
C
C
Read header information
C
READ (3) NBMAX
READ (3) RMIN
READ (3) RMAX
READ (3)
C
'
133
C
Read B field array
C
READ (3) BFIELD
C
C
Close file
C
CLOSE (UNIT=3)
C
C
********************************
C
*** Preliminary Calculations ***
C
********************************
C
C
Convert angles from degrees to radians
C
THETAI=THETAI*PI/180
PHII=PHII*PI/180
ETA=ETA*PI/180
C
C
Calculate unit vector along target spin
C
W(1)=SIN(ETA)
W(2)=0.0
W(3)=COS(ETA)
C
C
Calculate interval between points in BFIELD
C
DO K=1,3
DX(K)=(RMAX(K)-RMIN(K))/(NBMAX(K)-1)
END DO
C
C
Calculate initial spin wavefunction, PSII, from initial
C
theta and phi
C
134
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
PSII(1)=DCOS(THETAI/2)*DCOS(PHII/2)-I*DCOS(THETAI/2)
>
*DSIN(PHII/2)
PSII(2)=DSIN(THETAI/2)*DCOS(PHII/2)+I*DSIN(THETAI/2)
>
*DSIN(PHII/2)
C
C
C
C
Initialize variables.
PSUM are the spin projection
C
expectation values for each axis summed over all
C
trajectories calculated.
PSQSUM are the sums of the
C
squares of these values.
They are used at the end to
C
calculate averages and standard deviations.
C
DO K=1,3
PSUM(K)=0.0
PSQSUM(K)=0.0
END DO
C
C
**************************
C
*** Propagate Neutrons ***
C
**************************
C
C
Calculate N random trajectories, propagate a neutron spin
C
wavefunction along each trajectory, and calculate
C
expectation values of the spin projections on the x, y, and
C
z axes.
C
DO J=1,N
C
C
*************************
C
*** Choose Trajectory ***
C
*************************
C
135
C
Choose a random trajectory for a neutron starting at
C
z=ZI and ending at z=ZF.
C
randomly chosen in the range -XMAXI<=x<=XMAXI and
C
-YMAXI<=y<=YMAXI respectively.
C
chosen by picking random trajectories with a cosine
C
probability distribution such that the pass through the
C
region -XMAXF<=x<=XMAXF, and -YMAXF<=y<=YMAXF.
C
RF are the initial and final vectors so chosen and RD is
C
RF-RI.
C
in the x and y planes respectively.
C
SIN_AMIN are the sines of maximum and minimum values of
C
alpha which allow the trajectory to pass through the
C
specified region.
C
for beta.
C
randomly chosen values of alpha and beta for this
C
trajectory.
Starting x and y values are
Final x and y values are
RI and
Alpha and beta are the angles of the trajectory
SIN_AMAX and
SIN_BMAX and SIN_BMIN work the same
SIN_ALPHA and SIN_BETA are the sines of the
C
RI(1)=XMAXI*(2*RAN(SEED)-1)
RI(2)=YMAXI*(2*RAN(SEED)-1)
RI(3)=ZI
C
SIN_AMAX=(XMAXF-RI(1))/(ZF-ZI)
SIN_AMIN=(-XMAXF-RI(1))/(ZF-ZI)
SIN_BMAX=(YMAXF-RI(2))/(ZF-ZI)
SIN_BMIN=(-YMAXF-RI(2))/(ZF-ZI)
C
SIN_ALPHA=(SIN_AMAX-SIN_AMIN)*RAN(SEED)+SIN_AMIN
SIN_BETA=(SIN_BMAX-SIN_BMIN)*RAN(SEED)+SIN_BMIN
C
RF(1)=(ZF-ZI)*SIN_ALPHA+RI(1)
RF(2)=(ZF-ZI)*SIN_BETA+RI(2)
RF(3)=ZF
C
136
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
DO K=1,3
RD(K)=RF(K)-RI(K)
END DO
C
C
*************************
C
*** Transport Neutron ***
C
*************************
C
C
Transport a neutron spin along the trajectory from RI to
C
RF starting with initial spin wave function PSII. PSIF
C
is the calculated final spin wave function.
C
C
Set initial values of position (R), fraction of
C
trajectory travelled (F), and incremental distance(D).
C
DO K=1,3
R(K)=RI(K)
END DO
C
D=D0
F=0.0
C
C
Initialize U, a 2x2 matrix corresponding to the
C
cumulative rotation of the neutron spin
C
DO K=1,2
DO L=1,2
U(K,L)=SI(K,L)
END DO
END DO
C
DO WHILE (F.LT.1.0)
C
137
C
Check to see if we are at the last step and it will
C
take us past the end of the trajectory.
C
adjust the last step to end at the right point.
If so,
C
IF ((F+D).GT.1.0) THEN
D=1-F
END IF
C
C
Calculate the rotation matrix for a whole step (T)
C
and for the interval taken as two half steps (TL and
C
TU).
C
more than EPS, it is not a good approximation to
C
approximate the step with an infinitesimal rotation,
C
and we cut the step size in half and try again.
If T differs from the product of TL and TU by
C
DO K=1,3
R_B(K)=R(K)+(D/2)*RD(K)
RL_B(K)=R(K)+(D/4)*RD(K)
RU_B(K)=R(K)+(3*D/4)*RD(K)
END DO
C
DR=D*SQRT(RD(1)**2+RD(2)**2+RD(3)**2)
C
CALL ROT(R_B,DR,T)
CALL ROT(RL_B,DR/2,TL)
CALL ROT(RU_B,DR/2,TU)
C
C
Calculate TT and the difference between it and T,
C
DELTA_T
C
CALL MULT(TL,TU,TT)
C
CVG=.TRUE.
138
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
C
DO K=1,2
DO L=1,2
DELTA_T=ABS(T(K,L)-TT(K,L))
IF (DELTA_T.GT.EPS) THEN
CVG=.FALSE.
END IF
END DO
END DO
C
C
If the convergence criteria is not met, cut the step
C
size in half.
C
of the distance travelled, F, by the step taken, D,
C
multiply the cumulative rotation matrix, U, by the
C
rotation performed in this step, T, and set the step
C
size back to its original value.
If it is met, increment the fraction
C
IF (CVG.EQ..FALSE.) THEN
D=D/2
ELSE
F=F+D
DO K=1,3
R(K)=R(K)+D*RD(K)
END DO
CALL MULT(U,T,U)
D=D0
END IF
C
END DO
C
C
*******************************************
C
*** Calculate Final Spinor Wavefunction ***
C
*******************************************
139
C
DO K=1,2
PSIF(K)=(0.0,0.0)
END DO
C
DO K=1,2
DO L=1,2
PSIF(K)=PSIF(K)+U(K,L)*PSII(L)
END DO
END DO
C
C
*****************************
C
*** Calculate Projections ***
C
*****************************
C
C
Calculate spin projection expectation values for at the
C
end of the trajectory and write to output file
C
CALL PROJ(PSIF,X,P(1))
CALL PROJ(PSIF,Y,P(2))
CALL PROJ(PSIF,Z,P(3))
CALL PROJ(PSIF,W,P(4))
C
WRITE (2,300) J,RI(1),RI(2),RF(1),RF(2),P
C
300
FORMAT (2x,I5,4(3x,F5.2),4(2x,F6.4))
C
C
*****************************
C
*** Accumulate Statistics ***
C
*****************************
C
C
C
Add spin projections to statistical accumulations
140
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
DO K=1,4
PSUM(K)=PSUM(K)+P(K)
PSQSUM(K)=PSQSUM(K)+P(K)**2
END DO
C
END DO
C
C
*************************************
C
*** Calculate Statistical Results ***
C
*************************************
C
C
Calculate averages and and standard deviations for the spin
C
projections.
C
DO K=1,4
PAVE(K)=PSUM(K)/FLOAT(N)
IF (N.NE.1) THEN
PDEV(K)=SQRT(ABS((PSQSUM(K)-FLOAT(N)*PAVE(K)**2)
>
/FLOAT(N-1)))
ELSE
PDEV(K)=0.0
END IF
END DO
C
C
Calculate the average theta and phi for final spin
C
wavefunction, with standard deviations
C
C
Get unit vector for spin direction.
C
DO K=1,3
S(K)=2*PAVE(K)-1
END DO
C
141
THETAF=ACOS(S(3))
IF (ABS(S(3)).EQ.1.0) THEN
THETAFDEV=0.0
ELSE
THETAFDEV=(2/SQRT(1-S(3)**2))*PDEV(3)
END IF
C
IF ((S(1).EQ.0.0).AND.(S(2).EQ.0.0)) THEN
PHIF=0.0
PHIFDEV=0.0
ELSE
IF (S(2).GE.0.0) THEN
PHIF=ACOS(S(1)/SQRT(S(1)**2+S(2)**2))
ELSE
PHIF=2*PI-ACOS(S(1)/SQRT(S(1)**2+S(2)**2))
END IF
PHIFDEV=(2/(S(1)**2+S(2)**2))*SQRT(S(2)**2*PDEV(1)**2
>
+S(1)**2*PDEV(2)**2)
END IF
C
C
Convert angles from radians to degrees
C
THETAF=THETAF*180/PI
THETAFDEV=THETAFDEV*180/PI
PHIF=PHIF*180/PI
PHIFDEV=PHIFDEV*180/PI
C
C
********************
C
*** Write Output ***
C
********************
C
WRITE (2,400) PAVE(1),PDEV(1)
WRITE (2,410) PAVE(2),PDEV(2)
142
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
WRITE (2,420) PAVE(3),PDEV(3)
WRITE (2,430) PAVE(4),PDEV(4)
WRITE (2,440) THETAF,THETAFDEV
WRITE (2,450) PHIF,PHIFDEV
C
400
FORMAT (//,' Averages',/,'
PX = ',F6.4,' +/- ',F6.4)
410
FORMAT ('
PY = ',F6.4,' +/- ',F6.4)
420
FORMAT ('
PZ = ',F6.4,' +/- ',F6.4)
430
FORMAT ('
PT = ',F6.4,' +/- ',F6.4,/)
440
FORMAT (' THETA = ',F6.2,' +/- ',F6.2)
450
FORMAT ('
PHI = ',F6.2,' +/- ',F6.2,/)
C
END
C
C
C
SUBROUTINE MULT(U1,U2,U3)
C
C
Multiplies two spin rotation operators to obtain a third,
C
U3=U2*U1
C
C
Input Variables:
C
C
U1
Rotation operator acting first
C
U2
Rotation operator acting second
C
C
Output Variable:
C
C
U3
Matrix product U2*U1
C
C
Internal Variables:
C
C
K
Counting variable
143
C
L
Counting variable
C
M
Counting variable
C
UT
Matrix product U2*U1 (temporary)
C
IMPLICIT NONE
C
INTEGER*2
K,L,M
COMPLEX*16
U1(2,2),U2(2,2),U3(2,2),UT(2,2)
C
C
C
Initialize UT, which will temporarily be the product.
C
allows U3 to be the same variable as U1 or U2, if desired.
C
DO K=1,2
DO L=1,2
UT(K,L)=(0.0,0.0)
END DO
END DO
C
C
Calculate UT=U2*U1
C
DO K=1,2
DO L=1,2
DO M=1,2
UT(K,L)=UT(K,L)+U2(K,M)*U1(M,L)
END DO
END DO
END DO
C
C
Set U3=UT
C
DO K=1,2
DO L=1,2
This
144
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
U3(K,L)=UT(K,L)
END DO
END DO
C
RETURN
END
C
C
C
SUBROUTINE ROT(R1,DR1,T1)
C
C
Constructs the approximate rotation operator for an
C
incremental step by assuming it is infinitesimal
C
C
Input Variables:
C
C
DR1
Length of step (cm)
C
R1
Position at which field is to be calculated (cm)
C
(center of step)
C
C
Output Variable:
C
C
T1
Rotation matrix for step
C
C
Internal Variables:
C
C
B
Magnetic field vector at R1 (G)
C
BMAG
Magnitude of B (G)
C
BUN
Unit normal vector along B
C
FB
Fraction R is away from nearest grid point in
C
each dimension
C
K
Counting variable
C
L
Counting variable
145
C
M
Counting variable
C
P
Counting variable
C
THETA
Angle of rotation (rad)
C
C
Constants:
C
C
BFIELD
Array of B field vectors (G) (common block
C
C
FIELD)
DX
Interval between points in BFIELD (cm) (common
C
block FIELD)
C
E
Energy of neutron (MeV) (common block DATA)
C
E0
Neutron rest energy (MeV) (common block
C
CONSTANTS)
C
ETA
Angle of magnet (rad) (common block DATA)
C
GAMMA
Relativistic gamma (common block DATA)
C
I
Imaginary unit (common block CONSTANTS)
C
NB
Array coordinates of R1 in BFIELD
C
OMEGA
Precession of 1 MeV neutron after travelling 1
C
cm in 1 G field
C
PI
Pi (common block CONSTANTS)
C
RMIN
Minimum x, y, and z covered by BFIELD (cm)
C
(common block FIELD)
C
SIGMA
Pauli spin vector (common block CONSTANTS)
C
SI
2x2 identity matrix (common block CONSTANTS)
C
X
Unit vector along x (common block CONSTANTS)
C
Y
Unit vector along y (common block CONSTANTS)
C
Z
Unit vector along z (common block CONSTANTS)
C
IMPLICIT NONE
C
INTEGER*2
K,L,M,NB(3),P
REAL*4
DR1,DX(3),E,E0,ETA,FB(3),GAMMA,OMEGA,R1(3)
C
146
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
>
,RMIN(3),X(3),Y(3),Z(3)
C
REAL*8
B(3),BFIELD(11,27,407,3),BMAG,BUN(3),PI,THETA
COMPLEX*8
I,SI(2,2),SIGMA(2,2,3)
COMPLEX*16
T1(2,2)
C
C
C
COMMON /CONSTANTS/
E0,I,PI,SI,SIGMA,X,Y,Z
COMMON /DATA/
E,ETA,GAMMA
COMMON /FIELD/
BFIELD,DX,RMIN
C
DATA OMEGA
/-1.325E-5/
C
C
Calculate magnetic field vector by interpolating between
C
points in BFIELD array
C
C
Calculate coordinates of R in BFIELD array.
NB(K) are the
C
integer parts while FB(K) is are the fractional parts minus
C
1/2.
C
C
Initialize B.
C
DO K=1,3
NB(K)=IINT((R1(K)-RMIN(K))/DX(K))+1
FB(K)=(R1(K)-RMIN(K))/DX(K)+1.0/2-NB(K)
B(K)=0.0
END DO
C
C
Linearly interpolate between nearest 8 points in BFIELD.
C
DO K=0,1
DO L=0,1
147
DO M=0,1
DO P=1,3
B(P)=B(P)+(1.0/2-(-1)**K*FB(1))*(1.0/2-(-1)**L
>
*FB(2))*(1.0/2-(-1)**M*FB(3))
>
*BFIELD(NB(1)+K,NB(2)+L,NB(3)+M,P)
END DO
END DO
END DO
END DO
C
C
Calculate magnitude of B
C
BMAG=DSQRT(B(1)**2+B(2)**2+B(3)**2)
C
C
Calculate unit vector in direction of B
C
DO K=1,3
IF (BMAG.EQ.0.0) THEN
BUN(K)=0.0
ELSE
BUN(K)=B(K)/BMAG
END IF
END DO
C
C
Calculate precession of spin during step
C
THETA=(OMEGA/(GAMMA*SQRT(E)))*DR1*BMAG
C
C
Initialize T1, rotation matrix
C
DO K=1,2
DO L=1,2
T1(K,L)=(0.0,0.0)
148
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
END DO
END DO
C
C
Calculate T1=Icos(THETA/2)+isin(THETA/2)(SIGMA*BUN)
C
DO K=1,2
DO L=1,2
T1(K,L)=T1(K,L)+DCOS(THETA/2)*SI(K,L)
DO M=1,3
T1(K,L)=T1(K,L)+I*DSIN(THETA/2)*BUN(M)
>
*SIGMA(K,L,M)
END DO
END DO
END DO
C
RETURN
END
C
C
C
SUBROUTINE PROJ (PSI1,N1,P1)
C
C
Calculates the projection of the spin wavefunction PSI1 on
C
the normal vector N1.
P1 is the result
C
C
Input Variables:
C
C
PSI1
Spinor wavefunction
C
N1
Unit vector
C
C
Output Variable:
C
C
P1
Spin projection of PSI1 on N1
149
C
C
Internal Variables:
C
C
J
Counting variable
C
K
Counting Variable
C
L
Counting variable
C
PHI
Result of operating on PSI1 with projection
C
operator
C
C
Constants:
C
C
E0
Neutron rest energy (MeV) (common block
C
CONSTANTS)
C
I
Imaginary unit (common block CONSTANTS)
C
PI
Pi (common block CONSTANTS)
C
SIGMA
Pauli spin vector (common block CONSTANTS)
C
SI
2x2 identity matrix (common block CONSTANTS)
C
X
Unit vector along x (common block CONSTANTS)
C
Y
Unit vector along y (common block CONSTANTS)
C
Z
Unit vector along z (common block CONSTANTS)
C
IMPLICIT NONE
C
INTEGER*2
J,K,L
REAL*4
E0,N1(3),X(3),Y(3),Z(3)
REAL*8
P1,PI
COMPLEX*8
I,SI(2,2),SIGMA(2,2,3)
COMPLEX*16
PHI(2),PSI1(2)
C
C
C
C
C
150
Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields
COMMON /CONSTANTS/
E0,I,PI,SI,SIGMA,X,Y,Z
C
C
Initialize PHI, result of operating on PSI1 with projection
C
operator
C
PHI(1)=(0.0,0.0)
PHI(2)=(0.0,0.0)
C
C
Calculate PHI=1/2I+(SIGMA*N1)]PSI1
C
DO J=1,2
DO K=1,2
PHI(J)=PHI(J)+SI(J,K)*PSI1(K)/2
DO L=1,3
PHI(J)=PHI(J)+SIGMA(J,K,L)*N1(L)*PSI1(K)/2
END DO
END DO
END DO
C
C
Initialize P1, the expectation value of the projection of
C
PSI1 along N1
C
P1=0.0
C
C
Calculate P1=<PSI1|PHI>
C
DO J=1,2
P1=P1+REAL(CONJG(PSI1(J))*PHI(J))
END DO
C
RETURN
END
Appendix B
The Spin-Dependent Total Cross
Section Di
erence tensor
Following Equation 2.31, a general expression for the spin-dependent total cross section dierence can be written
(1 2) = 2(1 2) s + S12 t]
(B:1)
where 1 and 2 are the angles of the proton target spin and neutron beam spin with
respect to the beam direction, x^ . As before, s and t are the scalar and tensor parts
of the spin-spin cross section, ss. The equation can be expressed in terms of 1 and
2 to give1
= 2(sin 1 sin 2 + cos 1 cos 2) s ; (sin 1 sin 2 ; 2 cos 1 cos 2) t]:
(B:2)
Thus, it is possible to measure dierent linear combinations of s and t by changing
the spin directions of the target and beam.
The measurement of two dierent linear combinations is sucient to completely
determine the two spin-spin cross sections. Although the measurement of both L
and T fullls this requirement, they are not the most desirable for two reasons.
First, changing between the longitudinal and transverse geometries requires that the
magnet be rotated. This cannot be accomplished while the target is polarized and
precludes measuring both T and L in the same experiment. Second, since the
1
An identical result is obtained using the statistical polarization tensor formalism Kei93].
151
152
Appendix B. The Spin-Dependent Total Cross Section Dierence tensor
goal of these measurements is to determine the strength of the tensor interaction, the
most useful measurement is one in which the linear combinations which are diagonal
in s and t. Instead, L and T are mixtures of these cross sections. There is,
however, a pair of geometries which avoids both of these problems. These special
cases of will be referred to as scalar and tensor . tensor is the case in which
1 2 = 0. The result is then given by
tensor = 3 sin 21 t
(B:3)
where the upper sign applies in the case 1 ; 2 = + 2 , and the lower sign in the case
1 ; 2 = ; 2 . A maximum is obtained for 1 = 4 , 34 , giving
max = 3 :
tensor
(B:4)
t
scalar is the case in which S12 = 0. As a result,
scalar = p 3 sin 221 s (B:5)
1 + 3 cos 1
with the constraint tan 1 tan 2 = 2. The upper sign refers to the case ; 2 2 + 2 .
Fixing the target spin angle at 45 (1 = + 4 ) then gives the result
tensor = 3 t 2 = ;45 (B.6)
= p6 s 2 63 :
(B.7)
10
Thus, the experiment can be switched between tensor and scalar by simply resetting the spin direction of the neutron beam. The target is left unchanged.
It is often useful to have expressions for these cases in terms of the simpler longitudinal and transverse cross section dierences. Through Equations 2.33 and 2.34,
the general case can be expressed in terms of L and T .
= L cos 1 cos 2 + T sin 1 sin 2
(B:8)
The special cases tensor and scalar are then given by
tensor = sin2 21 ( L ; T )
(B.9)
scalar
scalar
1
sin 21
= p 2
( + 2 T ):
1 + 3 cos2 1 L
(B.10)
153
Using Equations 2.35 and 2.36, tensor and scalar can be written in terms of
phase-shift parameters.
2
3
66 2 ; 2 cosp23P 0 + 3 cos 3P 1 ; 3 cos 23D1 77
6 + 6 2 sin(3S1 + 3D1) sin 21
77
sin
2
1 6
6
7
tensor =
2 k2 666 ; cos 23P 2 + 5 cos 23D2 ; 4 cos 23F 2 777 (B.11)
4
5
p
+ 6 6 sin(3P 2 + 3F 2) sin 22 + 2
3
66 ;2 + 3 cos 21S0 ; cos 23P 0
77
66 + 9 cos 21P 1 ; 3 cos 23S1
77
1
6
77
sin
2
66
1
scalar = p 2
77 (B.12)
;
3
cos
;
3
cos
2
3
P
1
3
D
1
1 + 3 cos2 1 k2 66
77
66 + 15 cos 21D2 ; 5 cos 23P 2
75
4
; 5 cos 23D2 ; 5 cos 23F 2 + References
Ada78] J. M. Adams and G. White. A Versatile Pulse Shape Discriminator for
Charged Particle Separation and Its Application to Fast Neutron Time-ofFlight Spectroscopy. Nuclear Instruments and Methods, 156(1978) 459{476.
Arn89] R. A. Arndt. Phase-Shift Analysis SP89, 1989. Private Communication.
Arn92] R. A. Arndt. Phase-Shift Analysis SM92, 1992. Private Communication.
Aur84] R. Aures, W. Heeringa, H. O. Klages, R. Maschuw, F. K. Schmidt, and
B. Zeitnitz. A Brute-Force Polarised Proton Target as an Application of a
Versatile Brute-Force Polarisation Facility. Nuclear Instruments and Methods, 224(1984) 347{354.
Bla52a] J. M. Blatt and L. C. Biedenharn. The Angular Distribution of Scattering
and Reaction Cross Sections. Reviews of Modern Physics, 24(1952) 258{
272.
Bla52b] J. M. Blatt and L. C. Biedenharn. Neutron-Proton Scattering with SpinOrbit Coupling. I. General Expressions. Physical Review, 86(1952) 399{
404.
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Biography
Wesley Scott Wilburn
Personal
Born in Pensacola, Florida, March 9, 1964
Married Dianne Williams, August 11, 1989
Education
B.S. Physics, North Carolina State University, Raleigh, North Carolina, 1987
B.S. Electrical Engineering, North Carolina State University, Raleigh, North
Carolina, 1987
A.M. Physics, Duke University, Durham, North Carolina, 1990
Academic Positions
President, NCSU Society of Physics Students, 1986{1987
Townes-Perkins Fellow, Duke University, 1987{1990
Teaching Assistant, Duke University, 1987{1989
Research Assistant, Duke University, 1989{1993
Memberships
American Physical Society
Sigma Pi Sigma
161
162
Biography
Publications
W. Tornow, O. K. Baker, C. R. Gould, D. G. Haase, N. R. Roberson, and W. S.
Wilburn. Experimental Determination of the NN Tensor Force. In J. Sowinski
and S. E. Vigdor, editors, Physics with Polarized Beams on Polarized Targets.
World Scientic, Singapore, 1990.
J. E. Koster, E. D. Davis, C. R. Gould, D. G. Haase, N. R. Roberson, L. W.
Seagondollar, S. Wilburn, and X. Zhu. Direct Reaction Test of T Violation in
2 MeV Neutron Scattering from Aligned 165 Ho. Physics Letters B, 267(1991)
23{26.
C. D. Keith, C. R. Gould, D. G. Haase, N. R. Roberson, W. Tornow, and W. S.
Wilburn. 3He Melting Curve Thermometry in a Nuclear Polarization Experiment. Hyperne Interactions, 75(1992) 525{532.
Contributed Abstracts
W. S. Wilburn, N. R. Roberson, K. A. Sweeton, E. W. Hoen, and T. B. Clegg.
Computer Control System for a Polarized Ion Source. Bulletin of the American
Physical Society, 33(1988) 2201.
W. S. Wilburn, N. R. Roberson, W. Tornow, C. R. Gould, and D. G. Haase. Experimental Determination of the Zero-Crossing of the Spin-Dependent NeutronProton Total Cross-Section Dierences T and L and its Bearing on the
NN Tensor Force. In Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, Paris, 1990, page 27A.
W. S. Wilburn, P. R. Human, J. E. Koster, N. R. Roberson, W. Tornow, C. R.
Gould, D. G. Haase, and C. D. Keith. Measurement of T in Polarized Neutron/Polarized Proton Scattering. Bulletin of the American Physical Society,
37(1992) 902.
Biography
163
W. S. Wilburn, P. R. Human, J. E. Koster, N. R. Roberson, W. Tornow, C. R.
Gould, D. G. Haase, and C. D. Keith. The NN Tensor Force from ~n-~p Scattering. In I. R. Afnan and R. T. Cahill, editors, Proceedings of the XIII International Conference on Few-Body Problems in Physics, Adelaide, 1992, page
124.
Co-Authored Abstracts
T. Clegg, K. Felder, W. Hooke, J. Hunn, H. Lewis, A. Lovette, H. Middleton,
H. Pf)utzner, R. Roberson, H. Robinson, K. Sweeton, and S. Wilburn. Design of an Atomic Beam Polarized Ion Source with an ECR Ionizer. Bulletin
of the American Physical Society, 33(1988) 905.
K. Felder, H. G. Robinson, H. W. Lewis, W. S. Wilburn, and T. B. Clegg. Development of New RF Cavities for Use in a Polarized Ion Source. Bulletin of the
American Physical Society, 33(1990) 2200.
J. E. Koster, C. R. Gould, D. G. Haase, W. Seagondollar, N. R. Roberson, W. S.
Wilburn, and X. Zhu. Holmium Deformation Eect Measured with a Rotating
Aligned Target. Bulletin of the American Physical Society, 35(1990) 926.
C. D. Keith, D. G. Haase, C. R. Gould, W. S. Wilburn, N. R. Roberson, and
W. Tornow. Application of 3 He Melting Curve Thermometry in a Nuclear Orientation Cryostat. Bulletin of the American Physical Society, 35(1990) 2361.
T. B. Clegg, D. J. Abbott, T. C. Black, E. R. Crosson, R. K. Das, K. A. Fletcher,
C. R. Howell, H. J. Karwowski, J. E. Koster, S. Lemieux, E. J. Ludwig, M. AlOhali, N. R. Roberson, K. A. Sweeton, W. Tornow, W. S. Wilburn, and J. Z.
Williams. Report on Operation of a New Intense Polarized Ion Source and ECR
Ionizer. In Proceedings of the 7th International Conference on Polarization
Phenomena in Nuclear Physics, Paris, 1990, page 3E.
164
Biography
W. Tornow, N. R. Roberson, W. S. Wilburn, C. R. Gould, and D. G. Haase. Experimental Determination of the Nucleon-Nucleon Tensor Force via T and L
Measurements. In Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, Paris, 1990, page 29A.
C. D. Keith, C. R. Gould, D. G. Haase, P. R. Human, N. R. Roberson, W. Tornow,
and W. S. Wilburn. Level Structure of 4 He and the Spin Dependence of the n3 He Total Cross Section. Bulletin of the American Physical Society, 37(1992)
1257.
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