Download - Triangle Universities Nuclear Laboratory

c 2008 by Qiang Ye
Copyright All rights reserved
Abstract
The search for the existence of a nonzero neutron electric dipole moment (nEDM)
has the potential to reveal new sources of T and CP violation beyond the Standard
Model and may have a significant impact on our understanding of the universe. A
new experiment aiming at two orders of magnitude improvement (∼ 10−28 e·cm) over
the current experimental upper limit has been proposed in the United States. In the
experiment, the measurement cell will be made of dTPB-dPS coated acrylic and filled
with superfluid 4 He at ∼300-500 mK. The measurement of the neutron precession
frequency will rely on the spin-dependence of the cross section of the nuclear reaction
~ → p + t + 764 keV. Polarized
between polarized neutrons and 3 He atoms: ~n + 3 He
3
He will also be used as a comagnetometer based on the nuclear magnetic resonance
technique. The 3 He polarization needs to have sufficiently long relaxation time so
that little polarization is lost during the measurement period in order to achieve the
proposed sensitivity. Understanding the relaxation mechanism of 3 He polarization in
the measurement cell under the nEDM experimental conditions and maintaining 3 He
polarization is crucial for the experiment.
With the presence of superfluid 4 He, 3 He relaxation time measurements in a
dTPB-dPS coated cylindrical acrylic cell at the temperature of 1.9 K and ∼400 mK
have been performed at the Triangle University Nuclear Laboratory (TUNL) on the
campus of Duke University. The extracted depolarization probabilities of polarized
3
He on the cell surface are on the order of (1 − 2) × 10−7 at 1.9 K and ∼ 4.7 × 10−7
at ∼400 mK. The extrapolated relaxation time of polarized 3 He in the nEDM cell
geometry is ∼ 4870 seconds at ∼400 mK, which is sufficiently long for the nEDM
experiment and further improvements are anticipated.
iv
Dedicated to my parents and Shouyue Yu
v
Contents
Abstract
iv
List of Tables
x
List of Figures
xi
Acknowledgements
xviii
1 Introduction
1.1
1.2
1.3
1
CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
CP Violation In Kaon Decay . . . . . . . . . . . . . . . . . . .
3
1.1.2
CP Violation in the Standard Model . . . . . . . . . . . . . .
5
Neutron Electric Dipole Moment . . . . . . . . . . . . . . . . . . . .
8
1.2.1
What Is The Neutron EDM & Why Is It Important
. . . . .
8
1.2.2
Neutron EDM Experiments & Techniques . . . . . . . . . . .
10
Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . .
13
2 Neutron EDM Experiment Overview
2.1
15
Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1.1
Measurement Principle . . . . . . . . . . . . . . . . . . . . . .
15
2.1.2
Ultra Cold Neutron Production . . . . . . . . . . . . . . . . .
17
2.1.3
Neutron Frequency Measurement . . . . . . . . . . . . . . . .
20
2.1.4
3
He Comagnetometer . . . . . . . . . . . . . . . . . . . . . . .
22
2.2
Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3
Experimental Apparatus & Procedures . . . . . . . . . . . . . . . . .
25
2.4
3
28
He’s Role In The Experiment . . . . . . . . . . . . . . . . . . . . . .
vi
3
3
He Relaxation Studies at Low Temperatures Experimental Apparatus
30
3.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.1
Spin Exchange Optical Pumping
. . . . . . . . . . . . . . . .
32
3.2.2
3
He Relaxation Mechanism . . . . . . . . . . . . . . . . . . . .
36
3.2.3
3
He Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . .
39
1.9 K Test Experimental Apparatus . . . . . . . . . . . . . . . . . . .
47
3.3.1
Double-Cell System and Gas Handling System . . . . . . . . .
47
3.3.2
Cooling System and Temperature Monitoring . . . . . . . . .
49
3.3.3
Magnet System and NMR-AFP system . . . . . . . . . . . . .
51
3.3.4
Laser and Optics . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.5
Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
58
400 mK Test Experimental Apparatus . . . . . . . . . . . . . . . . .
60
3.4.1
Double Cell System and Gas Handling System . . . . . . . . .
60
3.4.2
Cooling System and Temperature Monitoring . . . . . . . . .
63
3.4.3
Magnet System . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4.4
NMR System . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.4.5
Narrowed Laser and Optics . . . . . . . . . . . . . . . . . . .
73
3.4.6
Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
75
3.3
3.4
4 Results and Data Analysis
77
4.1
1.9 K Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2
1.9 K Test Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.2.1
82
Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
4.2.2
Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.3
400 mK Test Result . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.4
400 mK Test Data Analysis . . . . . . . . . . . . . . . . . . . . . . .
97
4.5
Discussion Of The 1.9 K & 400 mK Tests . . . . . . . . . . . . . . . . 100
5 Conclusion and Future Outlook
102
A Geometric Phase Study at 300 K
104
A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.2 Experimental Technique and Apparatus
. . . . . . . . . . . . . . . . 107
A.3 Results of Geometric Phase Study at 300 K . . . . . . . . . . . . . . 108
A.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B
3
He Injection Test
112
B.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.2.1 Polarize 3 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.2.2
3
He Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.2.3
4
He Film Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.3.1 Atomic Beam Source . . . . . . . . . . . . . . . . . . . . . . . 119
B.3.2 Cooling System . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.3.3 Pyrex Glass Reservoir . . . . . . . . . . . . . . . . . . . . . . 125
B.3.4 Gas Handling System . . . . . . . . . . . . . . . . . . . . . . . 126
B.3.5 Magnet System . . . . . . . . . . . . . . . . . . . . . . . . . . 128
B.3.6 Pulse NMR System . . . . . . . . . . . . . . . . . . . . . . . . 129
viii
B.3.7 Film Burner and the Cs Ring . . . . . . . . . . . . . . . . . . 130
B.4 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 130
C Measuring the RF B-field
132
D dTPB-dPS Material Manufacturing Procedure
133
E dTPB-dPS Coating Procedure
137
F Reciprocity Theorem
139
Bibliography
140
Biography
147
ix
List of Tables
1.1
Physical quantities’ behavior under P and T transformation . . . . .
3
1.2
The kaon systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Experimental upper EDM values of different particles. . . . . . . . . .
9
1.4
Neutron energy distribution. . . . . . . . . . . . . . . . . . . . . . . .
11
2.1
The steps in nEDM measurement cycle.
. . . . . . . . . . . . . . . .
28
3.1
3
He adsorption energy on different materials. . . . . . . . . . . . . . .
38
3.2
Plarized 3 He NMR-AFP measurements’ parameters. . . . . . . . . . .
56
3.3
Polarized 3 He NMR-AFP measurements’ parameters. . . . . . . . . .
73
4.1
3
He relaxation time measurements at ∼1.9 K. The error bars are the
quadrature sum of the statistical and systematic uncertainties. . . . .
80
4.2
The amounts of 3 He in the vapor and in the liquid. . . . . . . . . . .
83
4.3
3
96
He relaxation time measurements at ∼400 mK. . . . . . . . . . . . .
x
List of Figures
1.1
The decay of KL and KS . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The nEDM experimental upper limit as a function of time. The experimental techniques are highlighted in the legend. The preliminary
result from the new nEDM experiment is expected around 2014. Predicted ranges for the nEDM values from various theoretical approaches
are shown to the right of the figure. . . . . . . . . . . . . . . . . . .
10
2.1
The schematics of the nEDM experimental setup. . . . . . . . . . . .
17
2.2
Free neutron dispersion curve and superfluid 4 He elementary dispersion
curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
The schematic overview of the full detector apparatus for the neutron
EDM apparatus. This view most clearly demonstrates the relationship
between the upper and lower cryostats. The upper cryostat contains
the refrigeration and 3 He systems. The lower cryostat contains the
entrance port for the neutrons, the magnets/magnetic shielding and
the measurement cells. . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Experimental cryostat. The neutron beam enters from the right. Two
neutron cells are between the three electrodes (ground, high voltage,
ground). The second picture shows the central region of the detector
including the measurement cells, HV electrodes, light guides, etc. . .
27
Spallation Neutron Source in ORNL. Beamline 13 is reserved for the
nEDM experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1
Optical pumping of Rb outer shell electrons. . . . . . . . . . . . . . .
34
3.2
Spin exchange between Rb outer shell electrons and 3 He nuclei. . . .
35
3.3
The effective magnetic field in the rotating reference frame. . . . . . .
42
3.4
Holding field curve of NMR-AFP measurement for polarized 3 He. The
holding field is ramped from below the resonance field to above it and
then back down. The RF field is on during the ramping. . . . . . . .
43
2.3
2.4
2.5
xi
3.5
3.6
The NMR-AFP signal read
lock-in
p amplifier. The average
p from the
2
2 and
is the AFP
x2down + ydown
of the fitted amplitudes x2up + yup
signal of the sweep. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.7
A schematic of the 1.9 K test experimental setup (courtesy of D. Dutta). 47
3.8
1.9 K test experimental setup. . . . . . . . . . . . . . . . . . . . . . .
48
3.9
1.9 K test double cell system. . . . . . . . . . . . . . . . . . . . . . .
49
3.10 Top part of the double cell system. . . . . . . . . . . . . . . . . . . .
50
3.11 Bottom acrylic cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.12 Picture of the gas handling system with pneumatic valves, gas tanks
and turbo pump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.13 The dewar used for the 1.9 K test. The double cell system is mounted
onto the top of it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.14 The 4 He vapor pressure versus the temperature graph. The normal
phase-superfluid phase transition temperature is 2.17K. . . . . . . . .
53
3.15 NMR coils in the pumping cell position. The purple ring is the color
of the laser from the camera. . . . . . . . . . . . . . . . . . . . . . . .
54
3.16 NMR coils in the dewar. . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.17 NMR-AFP circuit diagram. Red “GPIB” indicates that the equipment
is connected to PC via GPIB cable. . . . . . . . . . . . . . . . . . . .
55
3.18 The schematic of laser optics setup. . . . . . . . . . . . . . . . . . . .
56
3.19 1.9 K test optical pumping system setup. . . . . . . . . . . . . . . . .
57
3.20 The absorption line of the laser after its passing through the optical
pumping chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
xii
3.21 Before chasing the Rb into the pumping cell, the ampule is sealed
inside the side arm and broken under vacuum. . . . . . . . . . . . . .
59
3.22 A schematic of the 400 mK test experimental setup. . . . . . . . . . .
61
3.23 400 mK test experimental setup. . . . . . . . . . . . . . . . . . . . . .
62
3.24 Detachable cell which can be polarized and transported onto the 8-coil
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.25 The 3 He in the detachable cell is polarized in physics building, brought
over in a portable magnetic field, then put on top of the DR in French
Family Science Center room 1127. . . . . . . . . . . . . . . . . . . . .
64
3.26 The dTPB-dPS coated acrylic cell in the vacuum chamber of the DR.
65
3.27 Dilution refrigerator unit. It consists of the 1 K pot, the Still, the
50 mK plate, the mixing chamber and a series of heat exchangers. . .
66
3.28 Dilution refrigerator mixing chamber and copper buffer volume. . . .
67
3.29 Gold plated 99.999% pure copper wires and the NMR RF coil. . . . .
68
3.30 Grooves on the acrylic cell to house the cooling wires separately. A
small copper piece is attached to the 1 in. copper transition piece to
house one cooling wire and a temperature sensor. . . . . . . . . . . .
69
3.31 A 0.25 mm diameter capillary tube is positioned above the copper
buffer volume to help reduce the superfluid 4 He film flow. . . . . . . .
70
3.32 8 coil magnetic field simulation. The left graph is using the same
current in all 8 coils. The right graph is using larger current in the
outer two coils than that in the inner 6 coils. Iouter2 ∼ 1.37Iinner6 . . .
70
3.33 The three curves correspond to three different currents in the outer
two coils. The black curve gives the smallest magnetic field gradients
in the z direction close to the edge of the 8-coil system. . . . . . . . .
71
3.34 The polarization station for polarizing the 3 He in a detachable cell.
Typical holding field is ∼20 G. . . . . . . . . . . . . . . . . . . . . . .
72
xiii
3.35 Portable magnetic field powered by car batteries to hold the polarization of the detachable cell. . . . . . . . . . . . . . . . . . . . . . . . .
72
3.36 The NMR-AFP RF coils around the acrylic cell. The pickup coil is
behind the cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.37 The optical setup for the polarizing station in Figure 3.34. Configuration is the same as Figure 3.18. . . . . . . . . . . . . . . . . . . . . .
75
The relaxation time of polarized 3 He as a function of the amount of
He in the measurement cell at a temperature of ∼1.9 K. . . . . . . .
81
The polarized 3 He relaxation time as a function of the amount of 3 He.
The amount of 4 He is held constant at 0.404 mole. . . . . . . . . . . .
81
Illustration of model I. Depolarization probability in the vapor is Pv .
Surface relaxation time constant in the liquid is Ts . . . . . . . . . . .
84
The fitting of Equation 4.15 to the data points. Pv is the depolarization
probability of 3 He in the vapor. Ts is the relaxation time constant on
the wall below the liquid surface. . . . . . . . . . . . . . . . . . . . .
86
Diffusion model II of the 1.9 K test. The depolarization probabilities
on the walls in the vapor and liquid are two parameters that can be
varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
(a) and (b) are NMR measurements of the 3 He signal (green triangles) at 1.9 K as a function of time with the amount of 4 He equal to
0.135 mole (0.34 cm). Red squares are the simulated total signal in
the pickup coil consisting of the contributions from the vapor (blue
triangles) and liquid (pink circles). (c) is reduced χ2 obtained from
the best fit as a function of Pv (red circles, top axis) and Pl (black
squares, bottom axis) showing how different values of Pv and Pl can
fit the data due to the fact that with low liquid level, the 3 He atoms
in the vapor is close to the pickup coil. . . . . . . . . . . . . . . . . .
90
4.1
4
4.2
4.3
4.4
4.5
4.6
4.7
The amount of 4 He in the acrylic cell is 1.08 mole (2.71 cm). For
Pv = 1 × 10−9 and Pv = 1.21 × 10−7 , equally good fits can be obtained. 91
xiv
The amount of 4 He in the acrylic cell is 0.673 mole (1.69 cm). The experimental data (green triangles) consisting of the contributions from
the vapor (blue triangles) and liquid (pink circles) are fitted to the
simulation results (red squares). . . . . . . . . . . . . . . . . . . . . .
92
Reduced χ2 for the fits versus Pl and Pv values for 4 He amounts of
0.404 mole (black squares), 0.538 mole (red triangles) and 0.673 mole
(blue circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.10 3 He relaxation time measurements at room temperature in two acrylic
cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.11 The relaxation time of polarized 3 He as a function of the amount of
4
He in the measurement cell at a temperature of ∼400 mK. The error
bars are the quadrature sum of the statistical (determined from the exponential fit) and systematic uncertainties (determined from the AFP
spin flip inefficiency). . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.12 The inverse of relaxation time of polarized 3 He as a function of the S/V
values. For black points, S is the surface area covered by superfluid
4
He (cell bottom and cell side) and V is the volume of bulk 4 He. The
red point also include the top surface of the acrylic cell in the S. . . .
98
4.13 The inverse of relaxation time of polarized 3 He as a function of the
S/V values. S is the entire inner surface area of the acrylic cell and
V is the volume of bulk 4 He. The red line is a fit of the data points.
1
= (0.00041 ± 0.00000756) VS . The blue circle is the extrapolated
T1
1/T1 for nEDM cell geometry. . . . . . . . . . . . . . . . . . . . . . .
99
4.8
4.9
4.14 Two AFM images of the acrylic pieces using the old and new coating
procedures. The surface using the new coating method is much smoother.101
A.1 A cross section view of the cylindrical trap bounded by a circular
~ are
sidewall. A particle is undergoing specular reflections. B0z and E
perpendicular to the paper. The frequency shift depends only on the
component of the trajectory in the plane perpendicular to the axis (vr ).105
A.2 The 3 He correlation function measurement experimental setup. . . . . 108
A.3 The relaxation rates (corrected with the AFP loss) of the sealed cell
versus the square of different external magnetic field gradients at a
holding field of 24 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xv
A.4 ω 2 Sr (ω) values at different frequencies. . . . . . . . . . . . . . . . . . 109
B.1 A block diagram of the 3 He subsystem in the nEDM experiment, including the injection volume, 3 He purifier, evaporator, etc. . . . . . . 113
B.2
3
He injected from the ABS and collected by the collection volume
sitting in a cos θ magnet. . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.3 Quadrupole configuration of permanent magnets in the ABS system.
116
B.4 The cross section view of the 3 He injection test experimental setup.
The lower part of the picture is an expanded view of the measurement
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.5 A 3-D view of the 3 He injection test experimental setup. . . . . . . . 120
B.6 The atomic beam source (ABS) for the nEDM experimental. . . . . . 121
B.7 One of the eight quadrupole magnets along the ABS axis. . . . . . . . 121
B.8 The home-made refrigerator inside the ABS to cool the 3 He atoms
down to ∼1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.9 Cross sectional view of the cryostat with the DR inserted. . . . . . . 123
B.10 Outer vacuum vessel, heat shields and the liquid helium vessel, from
left to right, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.11 The ABS lower radiation limiter made by baffles mounted along the
beam line in the transfer tube. . . . . . . . . . . . . . . . . . . . . . . 124
B.12 Autodesk Inventor drawing of the thermal link from the MC to the
film burner and the injection tube. . . . . . . . . . . . . . . . . . . . 125
B.13 The pyrex glass reservoir (collection volume) with the side arm for Cs
coating. The bottom small cell is where NMR-FID measurements are
carried out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.14 The schematic and the picture of the gas handling system. . . . . . . 127
B.15 Tri-coil system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xvi
D.1 Distill the 99% styrene in the complicated glassware with an Argon
atmosphere ≤150 mBar at a temperature ∼95 degrees Celsius. . . . . 133
D.2 There are small boiling stones at the bottom of the vessel to prevent
the liquid from boiling too much. . . . . . . . . . . . . . . . . . . . . 134
D.3 Polymerization of the distilled styrene. Argon atmosphere, temperature of ∼200 degree Celsius. . . . . . . . . . . . . . . . . . . . . . . . 135
D.4 Drip the d-polystyrene and d-toluene solution into d-methanol in order
to remove the monomer. . . . . . . . . . . . . . . . . . . . . . . . . . 136
E.1 A teflon piece made to occupy most of the space in the acrylic cell to
minimize the use of dTPB-dPS material. . . . . . . . . . . . . . . . . 137
E.2 “Swinging method” to coat the acrylic pieces. . . . . . . . . . . . . . 138
xvii
Acknowledgements
First of all, I would like to thank my advisor, Professor Haiyan Gao for giving me the
opportunity to work on this challenging project (nEDM experiment) at Duke University. I
am grateful for her support and guidance. She shared with me a lot of her expertise and
research insight to make my work at Duke come to a success. I would also like to express
my gratitude to Professor Robert Golub and his wife, Dr. Ekaterina Korobkina from NC
State University. I learned from them a lot of expertise in low temperature physics, neutron
physics and wavelength shifting materials.
In the past six years of study and research, I not only acquired a lot of experimental
and theoretical knowledge, but also learned that scientific research is not an easy endeavor,
about 90% of one’s time is spent in preparing to make measurements and conducting trials
that ultimately fail. Only 10% or less of the time you will be able to obtain some useful
results. This means that you will feel depressed most of the time during this process and
definitely need perseverance and diligence to carry on. My parents have been teaching me
about these, giving me endless love, care and support from the moment I was born twenty
eight years ago. They educated me about life and how to be a good boy and a good man.
Without their love and support, I would never have crossed the ocean to pursue greater
knowledge and my future.
During my earlier years at Duke, Dipangkar Dutta, Nick Boccabello, Scott Singer,
Lars Hannelius, and the technical staff at TUNL helped me start Haiyan’s lab from an
empty space. I learned a lot of engineering and experimental techniques from them. Other
people in Haiyan’s group, including Wei Chen, Kevin Kramer, Rongchun Lu, Xin Qian,
Wangzhi Zheng, Xiaofeng Zhu, Xing Zong and several undergraduate students, have helped
me and given suggestions for my projects. I would also like to thank J. Rishel from the
University Research Glassware Co. for making all the glasswares for the 1.9 K and 400
mK measurements. During the neutron electric dipole moment experiment collaboration
meetings, many professors from other universities shared with me useful and insightful
xviii
ideas. I feel lucky to have joined this collaboration from the beginning of my thesis project.
I would like to express my gratitude to my advisory committee members, including
Professor John Thomas, Professor Calvin Howell, Professor Thomas Mehen and Professor
Joshua Socolar. They are working in diverse fields in physics and offer different points of
views to my projects. I also thank the staff in the physics department, especially Donna
Ruger, for their continuous assistance to make my life easier and happier in the department.
I also want to thank my classmates at Duke, including Carolyn Berger, Zheng Gao, Jie
Hu, Matt Kiser, Le Luo, Peidong Yu, Jianrong Deng, Botao Jia as well as other students
from NC State University and UNC - Chapel Hill, including Joe Newton, Timothy Daniels,
Charles Arnold, Chris Swank and many others. I have great memories of the old days with
them when we discussed homework together, played basketball, practiced spoken English...
They made my life at Duke a joyful one during the hard times.
Finally, I am extremely grateful to my wife, Shouyue Yu. Meeting her at Duke and
marrying her at the Sarah P. Duke Gardens made my dream come true. She has always
been by my side, encouraging me and supporting me. To my parents and her, I dedicate
this thesis.
xix
Chapter 1
Introduction
In the year 2002, a proposal [1] was submitted to the Department of Energy of the United
States to carry out a new search for the neutron electric dipole moment (nEDM). It was
the same year I joined the Physics Department of Duke University and found this project
fascinating. Many research and development (R&D) experiments need to be completed
before the construction of the real experiment can proceed. The measurement of relaxation
times of polarized 3 He under the nEDM experimental condition was one of these R&D
experiments due to the fact that sufficiently long relaxation time of 3 He polarization (∼ 104
s) during the measurement period is required to help the experiment achieve the expected
sensitivity, which is on the order of 10−28 e·cm. And no one has ever studied the relaxation
behavior of polarized 3 He in a dTPB-dPS (deuterated tetraphenyl butadiene - deuterated
polystyrene) coated acrylic cell at low temperatures. Such a surface was chosen for its
low neutron absorption cross section and for its good wavelength shifting efficiency (see
Section 2.1.3). At Duke University, supervised by Professor Haiyan Gao, I started to work
step by step towards this final goal: to understand the relaxation mechanism of polarized
3 He
under nEDM experimental conditions and find ways to improve the relaxation time
if needed. At first, 3 He relaxation time measurements at ∼1.9 K (below the liquid 4 He
λ point) in a cylindrical dTPB-dPS coated acrylic cell in the presence of superfluid 4 He
were carried out with a holding magnetic field of 21 G. With many people’s help, this
experiment was completed in early 2006 and a paper [2] was published in Physical Review
A in 2008. Then I went on with measurements at ∼400 mK using a dilution refrigerator
(on loan to Professor R. Golub from HMI in Germany) at the Triangle University Nuclear
Laboratory (TUNL) with a holding magnetic field of ∼7 G and the dTPB-dPS coated
acrylic cell full of superfluid 4 He. This experiment was one more step further towards the
1
nEDM experimental conditions and the extrapolated 3 He relaxation time based on surface
to volume ratio was already sufficient for the nEDM experiment. Further improvement
on the relaxation time is anticipated using the newly developed coating technique. The
measurements of 3 He relaxation times at ∼1.9 K and ∼400 mK make up the major part of
my thesis.
During the same time, I was also actively involved in the 3 He injection test, which was
a first experiment using the cryogenic assembly and the 3 He atomic beam source (ABS)
specifically designed for the nEDM experiment to measure the 3 He polarization and relaxation time after the polarized 3 He is injected from the ABS into the glass collection reservoir
filled with superfluid 4 He. This experiment is now being constructed and will be carried
out in late 2008 at Los Alamos National Laboratory (LANL). Besides this experiment, I
completed the room temperature measurements of the geometric phase study, which is an
important part of determining the systematic errors from the geometric phase effect. The
~ field with mag“Geometric Phase Effect” [3, 4] arises from the interaction of the ~v × E
netic field gradients resulting in a frequency shift proportional to the external electric field,
mimicking an EDM signal. The injection test and the geometric phase study are included
in the appendix.
1.1
CP Violation
The symmetries of three discrete transformations P, T and C play an essential role in the
understanding of our universe. Parity transformation, P, changes the sign of the space
coordinates x, y and z. Time reversal transformation, T, changes the sign of the time
coordinate, t. Charge conjugation, C, transforms a particle into its anti-particle. Table 1.1
lists the behavior of some common physical quantities under P and T transformation in
classical physics.
Parity was considered to be a good symmetry until 1956, Lee & Yang [5] discussed
the possibility of parity non-conservation in weak interactions and suggested some possible
2
Table 1.1: Physical quantities’ behavior under P and T transformation
Name
Symbol
P
T
Time
t
+
-
Position
~r
-
+
Energy
E
+
+
Momentum
p~
-
-
Spin
~s
+
-
Electric field
~
E
-
+
Magnetic field
~
B
+
-
experiments of parity conservation. In 1957, parity violation was experimentally discovered
by Wu et al. [6] in the β decay of Co60 . In the experiment, the Co60 nuclei spins were
aligned in the direction of the external magnetic field and the intensities of the emitted β
particles were measured in both directions with respect to the spin. If parity is conserved
in the β decay process, the intensity of the electrons should be the same in either direction
but a non-symmetric intensity distribution with respect to the nuclei spin direction was
found, which indicated that the parity was not conserved in the β decay process. In the
same year, Garwin et al. [7] and Friedman et al. [8] also discovered parity violations in
meson decays.
1.1.1
CP Violation In Kaon Decay
After parity violation had been found, the combined operation C and P was believed to
be conserved for all physical phenomena until in 1964, Christenson, Cronin, Fitch and
Turlay [9] discovered CP violation in neutral kaon decays. Table 1.2 lists the quark compositions, strangenesses and isospins of the kaon systems [10]. The two CP eigenstates are
3
Table 1.2: The kaon systems.
K0
K+
K̄ 0
K−
Quark Model
s̄d
s̄u
sd¯
sū
Strangeness
+1
+1
-1
-1
Isospin
− 21
1
2
1
2
− 12
defined as:
K1 =
K2 =
1
√ (K 0 + K̄ 0 )
2
1
√ (K 0 − K̄ 0 )
2
(1.1)
(1.2)
in which K1 decays to two pions (π 0 π 0 or π + π − ) and has CP= +11 while K2 decays to
three pions (π + π − π 0 or π 0 π 0 π 0 ) and has CP= −1. The two weak eigenstates (the states
that decay via the weak force) have different lifetimes and are called KL and KS , with the
lifetimes of τL = 5.17 × 10−8 s and τS = 8.93 × 10−11 s [10], respectively. Before the CP
violation was discovered, KL should only decay to three pions (due to small phase space
for three pions) with a CP= −1 and KS should only decay to two pions with a CP= +1.
So KS =K1 and KL =K2 . However in 1964, an experiment by Christenson, Cronin, Fitch
and Turlay [9] first discovered that the long-lived neutral kaon state KL could also decay
to two pions with a CP= +1 with a branching ratio of ∼ 2 × 10−3 (Figure 1.1), where CP
was violated to a small degree. From then on, KS and KL are distinguished from K1 and
K2 since they are not CP eigenstates, and each of them is an admixture of the two CP
eigenstates.
1
KS = p
(K1 − K2 )
1 + ||2
1
KL = p
(K2 + K1 )
1 + ||2
with ∼ 2 × 10−3 .
1
The parities of π 0 , π + and π − are all −1 [10].
4
(1.3)
(1.4)
Figure 1.1: The decay of KL and KS .
The scenario described above is called the “Indirect” CP violation due to the mixing
of the CP eigenstates. Another “Direct” CP violation also occurs in the 2-pion decay
amplitudes of the isospin I = 0 or I = 2 states. Details of the direct CP violation can
be found in [11]. The same CP violation mechanisms can also occur in B 0 − B̄ 0 and
D0 − D̄0 [12] systems. The Belle [13] experiment at the High Energy Accelerator Research
Organization (known as KEK) in Japan and the BaBar [14] experiment at SLAC are
dedicated to precision determinations of the Standard Model parameters in the CKM matrix
(see the next section) and have presented data for CP violation in the neutral B meson
system.
1.1.2
CP Violation in the Standard Model
The Standard Model (SM) [15] describes the fundamental particles and how they interact
with each other. The SM includes the strong (with gluons g as the mediating particles),
electromagnetic (with photons γ as the mediating particles) and weak (with W ± and Z 0
bosons as the mediating particles) interactions between quarks and leptons, which are the
basic particles of the SM. The three generations of quarks are up (u) and down (d), charm
(c) and strange (s), top (t) and bottom (b). The three generations of leptons are electron
(e) and electron neutrino (νe ), muon (µ) and muon neutrino (νµ ), tau (τ ) and tau neutrino
5
(ντ ). Each of these particles has its own anti-particle.
In the SM, there are two known possible sources [16] of CP violation. The first SM CP
violation mechanism involves the weak interaction, the only interaction in which a quark
can change its flavor and is only allowed to change by a unit amount of electron charge.
Since there are three generations of quarks, the charge-raising weak current can be written
as [17]:



 d 
 Vud Vus Vub
5)



γ
(1
−
γ
µ
J µ = ( ū c̄ t̄ )
U
, U =
s 


 Vcd Vcs Vcb
2
 

b
Vtd Vts Vtb






(1.5)
where γµ = {γ 0 , −γ 1 , −γ 2 , −γ 3 } are the gamma matrices and γ 5 = iγ 0 γ 1 γ 2 γ 3 . The 3 × 3
matrix U is called the Cabibbo-Kobayashi-Maskawa or CKM matrix [17], which couples
the u, c, t quarks states (with charge + 23 e) with the orthogonal combinations of the mass
eigenstates d, s, b (with charge − 31 e) and has nine complex coupling constants. For this
complex and unitary matrix, there are four independent parameters, including three mixing
angles and one complex phase, δ [18]. It is shown in [17] that this complex phase violates
CP symmetry by imposing CP transformation on the Hamiltonian (H =
4G µ †
√
J Jµ ),
2
where
G is the weak coupling constant. HCP 6= H † if δ is non-zero.
Another possible source of CP violation in the SM is the θ term in the Lagrangian
of Quantum ChromoDynamics (QCD), a theory of the strong interactions. The QCD
Lagrangian [19] is:
X
1
LQCD = − Gαµν Gαµν −
ψ¯n [6 ∂ − ig 6 Aα tα + mn ]ψn
4
n
(1.6)
where Aαµ is the color gauge vector potential, Gαµν the color gauge-covariant gluonic field
strength tensor, g the strong coupling constant and tα the complete set of generators of color
SU (3) (see [19] for details). It was realized that this Lagrangian can be generalized [20, 21,
22, 23] by including an Lθ term:
Lθ = θ
g 2 αµν e α
G Gµν
32π 2
(1.7)
e αµν ≡ 1 µνρσ Gαρσ , µνρσ the total antisymmetric tensor and µνρσ = −µνρσ . The
where G
2
θ parameter is one of the inputs in the Standard Model and it is shown in [24] that the
6
Lθ term violates both P and T symmetries (C-symmetry is conserved, therefore CP is
violated) if θ is non-zero. At the moment there is no prediction for the θ parameter from
QCD. One can determine it experimentally using predictions from models which relate
physical quantities such as the neutron EDM to this unknown parameter. While such
determination is model dependent, it provides insight into the origin of the strong CP
problem. Calculations [25, 26, 27] have shown that the neutron EDM ∼ O(10−16 θ) e·cm
and the current experimental upper limit of the neutron EDM is 2.9 × 10−26 e·cm [28]. To
meet this experimental bound, θ needs to be < 10−10 . The “Strong CP problem” refers to
this observation that θ is so close to zero instead of being of order one as one may expect.
There is a CPT theorem [29, 30] which states that CPT is always conserved in any local
quantum field theory with Lorentz invariance, Hermitian Hamiltonian, and spin-statistics
(Bose-Einstein and Fermi-Dirac statistics). The Standard Model satisfies the assumptions
of the CPT theorem. If an interaction is not invariant under one of the C, P, T operations,
it must be accompanied by the violation of the other two combined together. Thus CP
violation means T violation assuming CPT symmetry.
While the CP violation in the Standard Model suffices to explain what has been observed in the kaon and B meson systems, it is insufficient to explain the Baryon Asymmetry
of the Universe (BAU) [31]. If the Big Bang produced equal amounts of matter and antimatter at the beginning of the universe, one would expect the universe today to consist
only of photons due to the pair annihilation. Obviously this is not the case. Some physical
processes must have happened to change the ratio of matter and anti-matter. The baryon
asymmetry parameter is defined as rbau =
nB −nB̄
nB +nB̄
where nB and nB̄ are the number den-
sities of baryons and anti-baryons. Since today’s universe is made up of mostly matter,
rbau |today ∼
= 1. If there is no BAU, rbau should be zero. At the early stage of the universe, rbau |early can be estimated in this way. Since pairs of baryons and anti-baryons have
annihilated after the Big Bang and only the difference is left behind, the (nB − nB̄ )|early
is approximately the baryon number density nB |today times a scaling factor S taking into
account the expansion of the universe. And because B + B̄ → 2γ, the (nB + nB̄ )|early is
7
approximately the number density of photons today nγ |today (can be estimated from the
Cosmic Microwave Background Radiation (CMBR)) times the scaling factor S. Therefore rbau |early ∼
S·nB
S·nγ |today
=
nB
nγ |today
∼ (6.1 ± 0.3) × 10−10 [32, 33]. In 1967, Andrei
Sakharov [34] proposed three necessary conditions that would eventually lead to a baryon
asymmetry. These “Sakharov conditions” were baryon number B violation, C-symmetry
and CP-symmetry violation and non-equilibrium processes. There is no experimental evidence yet that the first condition is met. The second condition is discussed already and
the third condition means that the particles and anti-particles have not had the time to
reach thermal equilibrium since the pair annihilation processes happens at a lower rate
due to the rapid expansion of the universe. The CP violation in the Standard Model is
orders of magnitude smaller than that needed to explain the BAU [31]. There must be New
Physics beyond the Standard Model to account for the extra CP violation. The precision
measurement of a neutron electric dipole moment presents a great opportunity to search
for violations of T and CP symmetries (see next section) and has the potential to identify
new sources of CP violation and Physics beyond the Standard Model and contributes to
explain the BAU.
1.2
1.2.1
Neutron Electric Dipole Moment
What Is The Neutron EDM & Why Is It Important
In the rest frame of a non-degenerate quantum system with a non-zero spin, the direction
of the spin ŝ is the only vector to characterize the system. The “non-degenerate system”
means there is no degeneracy besides that due to the 2s + 1 possible orientations of the
spin [35]. No other vectors can be selected since the system will then be degenerate, contrary
to the property of the system. So the electric dipole moment d~ can only be along the spin
ŝ direction [36], d~ = dŝ. This applies to electrons, neutrons, protons, muons and other
particles.
~ its Hamiltonian
Take a neutron for example, if it is placed in an external electric field E,
8
~ where dn is the neutron electric
due to the interaction with the electric field is H = dn ŝ · E,
dipole moment. According to the transformations in Table 1.1, the Hamiltonian changes
sign under both P and T transformation if dn is non-zero. Assuming CPT is a good
symmetry, CP is also violated. Since the CP violation discussed in the Standard Model
is not sufficient to explain the observed baryon asymmetry, precision measurements of the
nEDM present an opportunity to search for a direct violation of the T symmetry and help
identify new sources of T and CP violation to contribute to explain the BAU. Table 1.3 [37]
is a list of the upper limits on the electric dipole moments of different particles.
Table 1.3: Experimental upper EDM values of different particles.
Particle
Experimental EDM Value / Limit (e·cm)
Electron, e
(0.18 ± 0.16 ± 0.10) × 10−26
Neutron, n
< 2.9 × 10−26 [90% C. L.] [28]
Proton, p
(−3.7 ± 6.3) × 10−23
Lambda Hyperon, Λ
< 1.5 × 10−16 [95% C. L.]
Tau Neutrino, ντ
< 5.2 × 10−17 [95% C. L.]
Muon, µ
(3.7 ± 3.4) × 10−19
Tau, τ
< 3.1 × 10−16 [95% C. L.]
Because the gluonic field operator G in Equation 1.7 relates to the gluon exchange in
strong interactions and does not involve the quark flavor change (which only happens in
the weak interactions), the neutron EDM dn is more sensitive to the θ parameter in Strong
CP mechanism than it is to δCKM in the weak interactions [38]. Thus the measurement of
dn also helps determine an important parameter of the SM.
At present, the neutron EDM’s experimental upper limit is 2.9 × 10−26 e·cm (90%
C.L.) [28], and the Standard Model calculations have predicted dn to be ∼ 10−32 − 10−31
e·cm [39, 40](∼ 10−30 e·cm [41]), which is ∼5-6 orders of magnitude smaller. Many other
theories predict the nEDM values to lie between 10−31 e·cm and 10−26 e·cm. The supersym-
9
metric (SUSY) extensions of the SM [42, 43, 44, 45, 46], left-right symmetric models [47, 48],
a class of non-minimal models in the Higgs sector [49, 50, 51, 52, 53] allow for the CP violation mechanisms not in the Standard Model and have their own predictions of the neutron
electric dipole moment (Figure 1.2 [54]). New measurements with improved sensitivity (∼2
orders of magnitude) of dn will help narrow the possible theories of New Physics and make
critical tests of the validity of the Standard Model.
Figure 1.2: The nEDM experimental upper limit as a function of time. The experimental techniques are highlighted in the legend. The preliminary result from the new
nEDM experiment is expected around 2014. Predicted ranges for the nEDM values
from various theoretical approaches are shown to the right of the figure.
1.2.2
Neutron EDM Experiments & Techniques
In 1932, Chadwick [55, 56] discovered the neutron and since then measurements of the
neutron properties have been pursued with great interest. In 1950, Purcell and Ramsey [57]
10
pointed out that it was possible to test the symmetry of parity by measuring the neutron
EDM. The first experiment [58, 59] was using magnetic resonance technique at Oak Ridge
National laboratory (ORNL) and set the earliest nEDM upper limit of 5×10−20 e·cm. It was
then recognized that a non-zero nEDM would also violate time reversal symmetry [60, 61]
and Ramsey [62] emphasized the need to check T symmetry experimentally. In 1964,
CP violation [9], which is directly linked to T violation assuming CPT symmetry, was
discovered in the neutral kaon system. In the next several decades, a series of measurements
with greater precision have been performed, and the current best nEDM upper limit of
dn < 2.9 × 10−26 e·cm (90% C. L.) was obtained at the Institute Laue-Langevin (ILL)
reactor at Grenoble [28].
The nEDM experiments over the years utilized three techniques to probe the neutron
EDM: neutron scattering, nuclear magnetic resonance (NMR) technique using thermal or
cold neutron beams (1957-1977), and NMR using bottled Ultra Cold Neutrons(UCNs) (after 1980). Figure 1.2 shows the nEDM experimental upper limits using different techniques
versus the year of publication. For convenience, Table 1.4 shows the neutron energy distribution with different names.
Table 1.4: Neutron energy distribution.
Name (neutron)
Energy (eV)
Speed (m/s)
fast
>1
> 14000
epithermal
0.025 - 1
2200 - 14000
thermal
∼ 0.025
∼ 2200
cold
5 × 10−5 - 0.025
98 - 2200
very cold
3 × 10−7 − 5 × 10−5
7.6 - 98
ultra cold
< 3 × 10−7
< 7.6
In the neutron scattering experiment [63, 57], neutrons were scattered off solid and
molten lead which was chosen due to its large Z and its small absorption cross section for
11
slow neutrons. If the observed n-e interaction strength was due to the non-zero nEDM, an
upper limit of dn < 3 × 10−18 e·cm was extracted. In the NMR measurements using the
thermal or cold neutron beams, the precession frequencies of the neutrons are measured
accurately in the parallel magnetic and electric fields using the separated oscillatory fields
technique developed by Ramsey [64, 65]. The neutron EDM is proportional to the difference
between the frequencies with the electric field parallel and anti-parallel to the magnetic field.
The most sensitive result came from [66], in which dn < 3 × 10−24 e·cm. Since the speed of
~ = 1 ~v × E
~ viewed
the neutron was relatively fast (∼150 m/s), the additional magnetic field B
c
from the neutron’s rest frame could not be neglected if there was a misalignment between
~
the magnetic and electric fields. So the dominant systematic error came from the ~v × E
effect (also called the motional field effect) and the fluctuation of the magnetic field, which
determined the main neutron precession frequency (Section 2.1.1). In the following NMR
experiments with bottled ultra cold neutrons (UCNs), both of these systematic errors can
be highly suppressed due to the UCNs’ much slower speed (< 7.6 m/s) and randomization
of the neutrons’ velocity directions in the storage cell. Moreover, using bottled UCNs will
greatly increase the effective interaction time (102 − 103 seconds) between the neutrons and
~ E
~ fields, which will significantly improve the experimental sensitivity. Therefore
the B,
among all these nEDM experiments, NMR using bottled UCNs provides the most sensitive
measurement to date.
As the nEDM sensitivity is increased by using UCNs, the magnetic field noise and
systematic effects will eventually set the limit of the experimental sensitivity. Ramsey [67]
first published an analysis using an in situ magnetometer (referred to as a comagnetometer)
which can deliver the information of the magnetic field directly experienced by the UCNs.
The latest ILL UCN measurements [68] used polarized
199 Hg
as a comagnetometer to
reduce the systematic error. Presently, the upgraded ILL experiment is in phase 1, hoping
to obtain a sensitivity of 10−27 e·cm. Phase 2 plans to upgrade the experiment [69] and
move it to Paul Scherrer Institut (PSI) in Switzerland where a more intense UCN source
can be obtained. A sensitivity of 10−28 e·cm is the goal of this experiment. Besides this
12
experiment, a multi-cell experiment [70] from Petersburg Nuclear Physics Institute (PNPI)
is also under construction and will be moved to PSI for higher UCN density. Similar
sensitivity at the 10−28 e·cm level is expected.
In the United States, a new approach [1] using polarized 3 He as the comagnetometer
and superthermal method to produce UCNs has been proposed to measure the neutron
EDM to greater precision. In this new neutron EDM experiment which is planned to be
carried out at the Spallation Neutron Source (SNS) in ORNL, the neutron storage cell will
be made of deuterated TetraPhenyl Butadiene-doped deuterated PolyStyrene (dTPB-dPS,
a wavelength shifting material) coated acrylic and filled with superfluid 4 He. The experiment will use the nuclear magnetic resonance technique to measure the neutron precession
frequency by comparing with that of the polarized 3 He using the spin dependence of the
~ → p + t + 764 keV. Polarized 3 He will be used as the
nuclear absorption process: ~n + 3 He
comagnetometer to monitor the magnetic field in situ during the experiment and will take
part in the spin dependent reactions. Therefore understanding the relaxation mechanism
of polarized 3 He in the storage cell under the experimental conditions and maintaining 3 He
polarization is crucial to the success of the nEDM experiment.
In this thesis, I present the first 3 He depolarization study in a dTPB-dPS coated cylindrical acrylic cell at a temperature of 1.9 K with the presence of superfluid 4 He at a magnetic
holding field of 21 G. I then present the measurements at ∼400 mK (the proposed nEDM
experimental temperature) using a dilution refrigerator at TUNL at a magnetic holding
field of ∼7 G. The results presented in this thesis are essential to the success of the future
nEDM experiment.
1.3
Dissertation Organization
After the introduction of the motivation and rationale of the neutron EDM experiment,
Chapter 2 describes the new method [1] to measure the neutron EDM with greater precision
and emphasizes the importance of long relaxation time of polarized 3 He in the nEDM
13
experimental conditions. Since this experiment uses polarized 3 He as the comagnetometer,
and to produce the scintillation light by reaction products from neutron capture on polarized
3 He,
the material used to construct the measurement cell needs to be experimentally verified
to have sufficiently long polarized 3 He relaxation time for the nEDM experiment.
Chapter 3 and chapter 4 present the 3 He relaxation time measurements on dTPB-dPS
coated acrylic surfaces at two temperatures: 1.9 K and 400 mK. The results are presented
and discussed. This is the major part of this dissertation.
The new neutron EDM experiment is still in the conceptual design phase and many R&D
experiments need to be carried out in the near future. Chapter 5 presents the conclusion
of the dissertation and describes the future studies.
As the nEDM experiments’ sensitivities are getting higher, the systematic error coming
~ field and the external B-field gradient can no longer
from the interaction between the ~v × E
be ignored. This systematic uncertainty is called the “Geometric Phase Effect” and its
principles are explained in appendix A. I present the first room temperature measurement
of the geometric phase effect, which measures the correlation function that determines
the absolute frequency shift of the neutron precession frequency under given experimental
conditions. Preliminary results and analysis are also presented.
Before the ∼100% polarized 3 He atoms coming out of the atomic beam source are introduced into the measurement cell, they need to be collected in a Cs coated glass reservoir for
a short period of time and then transferred over. Therefore 3 He atoms need to stay polarized in the collection volume for as long as possible. Appendix B describes the techniques
used in the 3 He injection test and the apparatus being built to verify the high polarization
and long relaxation time of 3 He atoms in the collection reservoir. This experiment will be
carried out in late 2008 at Los Alamos National Laboratory.
The appendix also gives out some useful information on measuring the RF B-field,
dTPB-dPS wavelength shifting material making and coating procedures, and the reciprocity
theorem used to analyze the 1.9 K test data.
14
Chapter 2
Neutron EDM Experiment Overview
This chapter explains the importance of a long relaxation time (∼ 104 s) of 3 He polarization
in this innovatively designed neutron EDM experiment [1]. And this is why I am tying to
understand the relaxation behavior of polarized 3 He under the nEDM experimental conditions. To elaborate this in a natural way, details of this nEDM experiment are presented
first.
The current upper limit of the neutron EDM is 2.9 × 10−26 e·cm (90% C.L.) [28]. This
new experiment has the potential to measure the neutron EDM magnitude or to at least
reduce the upper limit by ∼2 orders of magnitude. It provides a great opportunity to search
for direct violation of time reversal symmetry, challenge the models of New Physics beyond
the Standard Model and search for New Physics in the CP violation sector to help explain
the Baryon Asymmetry of the Universe.
This experiment requires a research team with a broad range of technical expertise
and experience including atomic physics, nuclear physics, low temperature physics, surface
physics, and ultra cold neutron physics. I am glad to join this important and ambitious
project at the beginning of it.
2.1
2.1.1
Experimental Technique
Measurement Principle
If a magnetic moment µ
~ is placed in an external magnetic field, this magnetic moment will
~ field direction,
tend to align with the field. If it is directed at an angle with respect to the B
~ which will cause it to
the field will exert a torque onto the magnetic moment ~τ = µ
~ × B,
precess in a plane perpendicular to the magnetic field. This precession is called “Larmor
15
Precession” [71]. For a particle of spin 1/2 in a magnetic field, the quantum energy difference
between the two spin states is 2µB and the corresponding Larmor precession frequency is
ν = 2µB/h. The ratio between the angular precession frequency and the magnetic field is
called the “gyromagnetic ratio” [71] γ = 2πν/B. If the particle has a non-zero EDM which
is along the spin axis d~ = dŝ, the corresponding Larmor precession frequency due to the
EDM in an external electric field is ν = 2dE/h.
~ with a parallel electric field E,
~
If a neutron is placed in an external magnetic field B
assuming the neutron’s magnetic (electric) dipole moment is µn ŝ (dn ŝ), where ŝ is the spin
direction, the Hamiltonian of the neutron from its interaction with the external electric and
magnetic fields is
~ + dn ŝ · E)
~
H = −(µn ŝ · B
(2.1)
The reason of the minus sign is due to µn < 0. So the Larmor precession frequency is
νn = −(2µn B + 2dn E)/h
(2.2)
If the electric field direction is reversed, the Larmor precession frequency will be νn =
−(2µn B − 2dn E)/h. The difference is ∆ν = 4dn E/h. From this equation, it can be seen
clearly that the Larmor precession frequency shift of the neutron is proportional to the
neutron’s electric dipole moment if this quantity is non-zero. If we put in the neutron’s
magnetic moment of µn = −1.91µN [10], where µN = 5.05 × 10−27 J·T−1 is the nuclear
magneton, a magnetic field of B=10 mG, an electric field of E=50 kV/cm, the main Larmor
frequency will be νn =29.2 Hz and the frequency shift will be ∆ν = 0.19 µHz = 0.66×10−7 νn
if we assign a nominal neutron EDM value of dn = 4 × 10−27 e·cm. Measuring this small
frequency shift is extremely difficult. Both the statistical and systematic errors need to
be controlled to a very low level while minimizing the magnetic field and maximizing the
electric field.
To reach this level of accuracy, the nEDM collaboration [1] has proposed the technique
to use polarized ultra cold neutrons, polarized 3 He and superfluid 4 He. Figure 2.1 shows
the schematics of the measurement cell. A storage cell made of acrylic is located in a weak
16
Figure 2.1: The schematics of the nEDM experimental setup.
magnetic field of ∼10 mG with a parallel strong electric field of ∼50 kV/cm. The cell is
filled with superfluid 4 He at a temperature of ∼300-500 mK. Polarized neutrons and 3 He
atoms will be introduced in the cell for measuring the neutron precession frequency to a
great precision.
2.1.2
Ultra Cold Neutron Production
To measure the neutron’s EDM, neutrons need to be obtained in the first place. As discussed
in Chapter 1, ultra cold neutrons have been used in the EDM experiments since 1980
because they can highly suppress the motional field effect and maximize the interaction
time with the surrounding fields. The designed nEDM experiment will be carried out
using the fundamental physics neutron beam line of the Spallation Neutron Source (SNS)
at the Oak Ridge National Laboratory (ORNL). A large amount of neutrons is created by
bombarding a liquid mercury target with a high energy proton beam [72]. The neutrons are
then slowed down to cold neutrons using the H2 and H2 O moderators. The cold neutrons
at a wavelength of 0.89 nm are produced using monochromators made of alkali-intercalated
graphite and this technique has been developed for measurements of the neutron lifetime [73]
in the National Institute of Standards and Technology (NIST). In order to obtain the
neutrons with polarization greater than 95%, a neutron guide with an internal polarizing
17
“supermirror” is designed by the nEDM collaboration [74]. The basic idea of a supermirror
is to combine layers of magnetic alloys and layers of non-magnetic materials to achieve
different coefficients of reflection for different neutron spin states so that the neutrons with
the desired spins can be filtered out to a certain direction. Neutrons with polarization
higher than 95% can be produced using this technique and the details can be found in [74].
After the polarized cold neutron beam has been obtained, it will easily penetrate the
walls of the measurement cell made of acrylic filled with superfluid 4 He at a temperature
of ∼300-500 mK. Cold neutrons will be down-scattered into ultra cold neutrons via the socalled “Superthermal” process [75, 76]. The basic idea is from the free neutron dispersion
curve and the superfluid 4 He elementary excitation dispersion curve (the Landau-Feynman
dispersion curve) in Figure 2.2 [77]. The free neutron’s dispersion curve is ~k 2 /2m, which is
Figure 2.2: Free neutron dispersion curve and superfluid 4 He elementary dispersion
curve.
a parabola. The Landau-Feynman dispersion curve [77] is close to a linear function (phonon
relation) ω = vk when k approaches zero and v is the velocity of longitudinal sound waves.
The two curves intersect at 2π/k ∗ = 8.9 Å point (En ∼0.001 eV and vn ∼ 440 m/s) and
the zero point. Since the energy and momentum are both conserved during the scattering
process, only the neutrons with energy E ∗ = (~k ∗ )2 /2m ∼12 K can be down-scattered into
the UCNs (E < 0.13 µeV, v < 5 m/s) by emitting a phonon. On the other hand the UCNs
18
can also absorb energy E ∗ and be up-scattered to cold neutrons but this up-scattering
process occurs much less frequently due to the Boltzmann factor of e−E
∗ /T
in the density
of the 12 K phonons at at low temperatures. The reason why superthermal process is used
is because it produces more UCNs than any other known method.
Superfluid 4 He is not only used for the production of UCNs in the nEDM experiment,
it also serves as a good insulator for the high electric field thanks to its good electrical
insulating properties [78]. The high voltage part of the proposed nEDM experiment will
not be discussed in this thesis.
Deuterated TetraPhenyl Butadiene-doped deuterated PolyStyrene (dTPB-dPS) material is chosen for the coating on the acrylic measurement cell is due to small neutron absorption cross section (this is why some nuclear reactors use deuterated water D2 O, also called
heavy water, to slow down the neutrons) and its good wavelength shifting efficiency [79],
which will be discussed later. The deuterated polystyrene’s wall potential is 0.134 µeV [1]
and the neutrons with lower energy will not be able to cross over this potential and be
trapped in the measurement cell until they are lost by β decay (n → p + e− + ν̄e ) reaction,
3 He
absorption and wall absorption. The total neutron loss rate can therefore be written
as:
1
1
1
1
=
+
+
τ
τβ
τ3He τwall
(2.3)
where τ is the time constant for each loss mechanism. Superfluid 4 He in the measurement
cell needs to be isotopically purified (remove the unpolarized 3 He in 4 He) to reduce the
3 He
neutron absorption rate. The low temperature of ∼300-500 mK also helps suppress
the neutron absorption mechanisms on the walls so that the UCN storage time can be of
the same order as the neutron lifetime. The goal for the mean life time of a neutron in the
trap filled with superfluid 4 He is ∼500 seconds and the nominal production rate of trapped
UCN is ∼0.3 UCN/(cm3 ·sec). So after ∼ 500 seconds of UCN production, the neutron
density will reach ρn ∼ 150 UCN/cm3 in the superfluid 4 He. The UCN density scales with
the flux of the cold neutrons, and it will certainly reach higher values with a more intense
cold neutron source.
19
2.1.3
Neutron Frequency Measurement
After ultra cold neutrons are produced by the superthermal method, resulting in ∼150
polarized UNCs/cm3 , ∼100% polarized 3 He atoms are then introduced into the measurement cell. One thing must be made clear at this point is that the EDM of 3 He atom is
highly suppressed, or at least much smaller than that of the neutron due to the shielding
from the two bound electrons. The reason is that for a neutral atom in an electric field, it
is under equilibrium condition. The atom’s internal structure must rearrange so that the
electrostatic forces exerted on each constituent is zero to meet the equilibrium condition.
If seen from the outside, there is an internal electric field generated to cancel the external
electric field [80] so that the EDMs of the constituent particles are basically shielded from
the outside world. This is called “Schiff Shielding” and its quantum mechanical proof can
be found in [81]. The polarized 3 He is introduced for two purposes. The first purpose, also
the main task of polarized 3 He, is to help measure the neutron’s precession frequency to a
great precision. The second purpose is in the next section. Initially the spins of polarized
3 He
and polarized neutrons are aligned parallel to each other in the direction of the mag-
netic field. Radio frequency (RF) coils are used to generate independent RF pulses at the
Larmor precession frequencies of the 3 He and neutron to rotate the 3 He and neutron spin
directions into the plane perpendicular to the magnetic field. Then the neutron and 3 He
spins will start to precess, initially with their spins parallel. Since the density of the polarized 3 He atoms will be ∼ 1012 /cm3 , ∼10 orders of magnitude higher than that of UCNs, the
precession signal of polarized 3 He can be easily detected directly using the Superconducting
QUantum Interference Device (SQUID) coils mounted adjacent to the measurement cells.
The SQUID coils have sufficient sensitivity to measure the 3 He precession frequency ν3He
precisely. Details of the SQUID system can be found in [82].
Because the 3 He’s magnetic dipole moment is µ3He = −2.128µN [10], which is about
10% higher than that of the neutron, the precession frequency of 3 He will be ∼10% faster
than that of the neutron. Between the polarized 3 He atoms and neutrons, the nuclear
20
absorption cross section for the reaction
~ → p + t + 764 keV
~n + 3 He
(2.4)
is highly spin-dependent because the intermediate spin-0 excited state cannot be formed
when the spins of neutron and 3 He are parallel [83]. The cross sections are σparallel ∼ 59 b
and σopposite ∼ 11 kb at thermal neutron energy (25.3 meV) [84, 85, 86], and both of them
scale inversely with the neutron velocity. This means that the probability that the reaction
will happen when the 3 He and neutron’s spins are anti-parallel is ∼200 times higher than
that when two species’ spins are parallel. As they precess in the plane perpendicular to
the magnetic field, the cross section for this nuclear reaction is time-dependent because
the 3 He atom’s magnetic moment precesses faster than that of the neutron. The recoiling
charged particles from the reaction, including proton and triton, will generate scintillation
light in the superfluid 4 He [87]. This scintillation light has a broad spectrum centered at 80
nm (Extreme Ultraviolet (EUV)) and conventional PhotoMultiplier Tubes (PMTs) cannot
detect it. This is where the dTPB-dPS coating material comes into play. The scintillation
light can be easily transmitted to the wall of the cell where the dTPB-dPS material can
absorb it and re-emit it at a wavelength centered at ∼430 nm [79]. This wavelength-shifted
deep blue light can then be collected with light guides made of acrylic and transmitted to
the PMTs outside of the magnetic field region. The details of the light guides and PMTs
can be found in [1].
Since the cross section for the spin-dependent nuclear reaction (Equation 2.4) is timedependent, the scintillation light detected by the PMTs will be a function of time also.
If the neutrons and 3 He atoms are both polarized, the spin dependent UCN loss rate is
written as:
1
1
(1 − P~3He · P~n ) =
(1 − P3He Pn cos θn3 )
τ3He
τ3He
where
1
τ3He
(2.5)
is half the neutron-3 He absorption rate when the spins of 3 He and neutron are
anti-parallel. θn3 is the angle between the neutron and 3 He spin directions. P3He and Pn
are the 3 He’s and neutron’s polarizations. So after the spins of neutron and 3 He start to
21
precess freely, the net scintillation rate will be [1]:
1
Φ(t) = Φbg + N e−t τucn [
1
1
(1 − P3He (t)Pn (t) · cos(2π(ν3He − νn )t + φ))]
+
τβ
τ3He
(2.6)
where Φbg is the background scintillation rate (assumed to be constant). N is the number
of UCNs trapped in the cell.
1
τucn
=
1
τβ
1
1
+ τwall
+ τ3He
is the total loss rate of UCNs coming
from the β decay rate, cell wall loss rate and average 3 He absorption rate. P3He (t) =
P3He (0)e−t/T3 and Pn (t) = Pn (0)e−t/Tn are the time evolution of the 3 He and neutron’s
polarizations. T3 and Tn are the relaxation time of polarized 3 He and neutron in the
measurement cell, respectively. ν3He − νn is the difference between the neutron and 3 He
precession frequencies, which is like a modulation, and φ is an arbitrary phase. Here ν3He is
a measured quantity (see the next section) and νn is to be determined. The resulting PMT
signal gives a direct measurement of the neutron precession frequency νn when combined
with the knowledge of the 3 He precession frequency (ν3He ) information from the SQUID
measurement. The same measurement can be repeated when the electric field direction
is reversed and another νn can be determined. Then the neutron EDM can be calculated
using the formula dn = h∆νn /4E.
2.1.4
3
He Comagnetometer
The method described above sounds easy to do but in reality, the measurement’s sensitivity
depends on how well the statistical and systematic errors are minimized. Error analysis
will be discussed in the next section. One of these systematic errors comes from the nonuniformity and fluctuations of the magnetic field because the magnetic field determines the
main Larmor precession frequency. Since the goal of the nEDM experiment is to achieve two
orders of magnitude improvement in sensitivity over the current experimental upper limit,
it requires additional magnetometry to monitor the magnetic field noise. The magnetic
field needs to be spatially uniform (1 part in 2000 at a magnetic field of ∼10 mG) and
temporally stable. This is very challenging to achieve and must be confirmed by direct
measurement.
22
The second purpose of introducing polarized 3 He is to serve as a comagnetometer.
Ramsey [67] first published the analysis of using an in situ magnetometer. Such comagnetometer must meet two requirements. One is that the comagnetometer does not have its
own EDM. The second requirement is that the comagnetometer must be able to uniformly
sample the magnetic field experienced by the UCNs. The polarized 3 He’s EDM is small
due to Schiff Shielding and its diffusion coefficient is large below 1 K [88] (∼730 cm2 /s
at a temperature of 400 mK). Thus the two requirements are met. Because the neutron
density is ∼10 orders of magnitude smaller than that of 3 He, the amount of 3 He reacting
with neutrons to generate scintillation light is negligible compared to the total amount of
3 He.
The precession signal from the neutrons will be too small to detect by the SQUID
system due to the neutron’s low concentration. The sinusoidal signal from polarized 3 He
will be picked up by the SQUID system which will determine the 3 He precession frequency
(ν3He ) directly. The in situ magnetic field averaged over the same volume as experienced
by the trapped UCNs during the measurement period is:
B=−
h · ν3He
2µ3He
(2.7)
Analyzing the scintillation light signal while taking into account the in situ magnetic field
information will greatly reduce the nEDM experiment’s systematic error.
2.2
Error Analysis
The statistical error of the nEDM experiment can be estimated in this way. For a known
electric field, the uncertainty of dn is
δdn = h
δ∆ν
4E
(2.8)
To reduce this error, one of the challenges is to generate as large an electric field as possible
with the presence of a weak magnetic field. From the uncertainty principle, ∆E∆t ≥ ~, we
have
hδ∆ν · ∆t ≥ ~
23
(2.9)
Moreover, because the statistical error is inversely proportional to the square root of the
sample size [89], which is the UCN number,
δ∆ν ≥
1
√
2πTm N
(2.10)
Here Tm is the precession frequency measurement time and N is the effective number
of neutrons contributing in the measurement. So by multiplying h/4E on both sides of
Equation 2.10, the statistical error to the frequency uncertainty for m measurements is
δdn ≥
~/4
~/4
√
= √
e · cm
E Tm N t
ETm N m
(2.11)
where t = mTm is the total measurement time and this equation gives the lower bound
(order of magnitude) of the statistical error. With the proposed E = 50 kV/cm, Tm = 500
sec, N = 1.0 × 106 neutrons/cycle and m = 5700 repeated cycles, δdn ≥ 10−28 e·cm. In
comparison to the most recent result at ILL [28], |dn | < 2.9 × 10−26 e·cm (90% C.L.), with
the electric field of 50 kV/cm (5 times that of ILL), UCN density of 150/cm3 (150 times
that of ILL) and UCN storage time of 500 seconds (4 times that of ILL), the figure of merit
of this nEDM experiment is increased by a factor of ∼120, which means a sensitivity of
10−28 e·cm is feasible. Higher sensitivity can be reached if the UCN density can be further
increased.
There are also many systematic errors involved in this nEDM experiment. The major
~ effect is the first one. It is also called the
systematic errors have been identified. The ~v × E
~ m is:
“motional field effect”. In neutron’s rest frame, the additional magnetic field B
~ m = 1 ~v × E
~
B
c
(2.12)
~ field is parallel to the B
~ field,
where ~v is the neutron velocity in the lab frame. Since the E
~ m will be in the yz plane due
which is in the x direction in Figure 2.1, this motional field B
to the horizontal motion of the neutrons. If the electric field is not completely aligned with
~ m will have a non-zero component along the B
~ direction.
the magnetic field, then the B
When the electric field is reversed, this component will also reverse direction and simulate
a non-zero neutron EDM. For instance, if the angle between the electric and magnetic fields
24
is 1.5×10−3 radians, a false EDM as big as ∼ 10−23 e·cm will be generated if the neutron is
traveling at a speed of 100 m/s. This effect is already highly suppressed by using the UCNs
because of the obvious reason and the suppression can be further enhanced by using bottled
UCNs, in which the neutrons’ directions of motion can be randomized. Still, a stringent
tolerance needs to be imposed on the alignment of the electric and magnetic fields.
The fluctuation of the magnetic field is another source of systematic errors. Since
neutrons are moving in different paths in the measurement cell, any spatial non-uniformity
and temporal instability will enlarge the uncertainty in the neutron frequency measurement.
Fortunately the introduction of polarized 3 He as the comagnetometer can help monitoring
the in situ magnetic field and the effect can be compensated later in the data analysis.
~ field and this requires
It is also realized that a quadratic term still exists in the ~v × E
that the reversal of the electric field be extremely accurate. This part will not be discussed
in this thesis. And more importantly, a “Geometric Phase Effect” [90] is also realized to
be non-negligible and needs to be controlled. I have completed a very first study of the
geometric phase effect at room temperature. The principle and the preliminary result are
included in the appendix A.
2.3
Experimental Apparatus & Procedures
The conceptual design for the proposed apparatus is shown in Figure 2.3 [74]. The apparatus is divided into two parts. The lower cryostat is where the measurement will take
place and the upper cryostat is where the polarized 3 He is injected from the Atomic Beam
Source (see Appendix B) as well as where the dilution refrigerator is located. Two dilution
refrigerators will be used to cool down the entire central liquid 4 He volume as well as the
measurement region to ∼300-500 mK. Four layers of magnetic shielding will be able to
isolate the measurement region from the outside world.
The lower cryostat contains two measurement cells (blue squares) sandwiched between a
red high voltage (HV) electrode and two green ground electrodes as shown in Figure 2.4 [74].
25
Figure 2.3: The schematic overview of the full detector apparatus for the neutron
EDM apparatus. This view most clearly demonstrates the relationship between the
upper and lower cryostats. The upper cryostat contains the refrigeration and 3 He
systems. The lower cryostat contains the entrance port for the neutrons, the magnets/magnetic shielding and the measurement cells.
In this way, two orientations of the electric field for a fixed B field (along the electric field
direction) will be measured simultaneously to reduce the systematic errors. The region in
the cryostat but outside the measurement cells will be filled with normal superfluid 4 He (not
isotopically purified) to make use of its good electrical insulating properties. Any UCNs
produced there will be absorbed in coatings on the vessel wall to prevent wall activation.
Table 2.1 lists the procedures in one measurement cycle with duration time.
From the description above, this experiment has many scientific ideas built into its
design and many physics principles stand behind them. That’s how difficult and interesting
it is. This experiment is currently still in the conceptual design phase. More research &
development experiments are still being carried out. The overall goal of the experiment is
to search for an nEDM with a statistical and systematic uncertainty of 2.2 × 10−28 e·cm
26
Figure 2.4: Experimental cryostat. The neutron beam enters from the right. Two
neutron cells are between the three electrodes (ground, high voltage, ground). The
second picture shows the central region of the detector including the measurement
cells, HV electrodes, light guides, etc.
(90% C.L.), and this experiment will be carried out in beamline 13 (Figure 2.5 [91]) in
Spallation Neutron Source (SNS) in ORNL around year 2014.
27
Table 2.1: The steps in nEDM measurement cycle.
Step
Description
Duration (s)
1
Produce polarized UCNs
1000
2
Diffuse polarized 3 He atoms into measurement cell
100
3
Apply π/2 pulse to rotate both 3 He and UCN spins to
10
be perpendicular to B field
4
Precession frequencies & scintillation light measurement
500
5
Remove 3 He atoms from the cryostat using purifier
100
6
Return to step 1
2.4
3
He’s Role In The Experiment
We rely heavily on the spin dependence of the nuclear absorption cross section for the reaction (Eqn.2.4) to determine the neutron precession frequency relative to that of 3 He. Since
the number density of polarized 3 He (∼ 1012 /cm3 ) is far greater than that of polarized
neutrons (∼ 102 /cm3 ), the amount of 3 He involved in the spin-dependent neutron absorption reaction is negligible compared to the total number of 3 He atoms. However, if the 3 He
atoms depolarize too fast during the measurement period (∼500 s), it will affect the final
measurement sensitivity. So the polarized 3 He needs to have a long relaxation time (∼ 104
s [1]) under the nEDM experimental conditions.
Moreover, it is crucial to have a complete understanding of the holding field experienced
by UCNs, and it is not possible to determine the magnetic field by measuring the signal
from polarized neutrons due to its extremely low concentration. Measuring the precession
signal of polarized 3 He can be accomplished using a SQUID system. In this way, a direct
measurement of the magnetic field (Equation 2.7) in situ becomes possible, and it will assist
us in analyzing the systematic error from the temporal fluctuation of the holding field.
All of the above need 3 He polarization to have a long relaxation time under the neutron
EDM experimental conditions. In the next two chapters, I discuss the 3 He relaxation time
28
Figure 2.5: Spallation Neutron Source in ORNL. Beamline 13 is reserved for the
nEDM experiment.
measurements performed at ∼1.9 K and ∼400 mK in dTPB-dPS coated acrylic cells, and
then I present the results and analysis.
29
Chapter 3
3
He Relaxation Studies at Low
Temperatures Experimental Apparatus
3.1
Overview
Our new neutron electric dipole moment experiment is planned to be carried out at Spallation Neutron Source in Oak Ridge National Laboratory using ultra cold neutrons trapped
in an acrylic cell coated with dTPB-dPS material and filled with superfluid 4 He at a temperature of ∼300-500 mK. The experimental method used is based on the nuclear magnetic
resonance technique and the spin dependence of the nuclear absorption cross section of the
following process:
~ → p + t + 764 keV
~n + 3 He
(3.1)
In addition, the magnetic field experienced by UCNs will be monitored in situ using a 3 He
comagnetometer
B=−
h · ν3He
2µ3He
(3.2)
where ν3He and µ3He are the polarized 3 He precession frequency and magnetic moment.
The application of these techniques requires that the relaxation time of polarized 3 He be
sufficiently long (∼ 104 s [1]) to achieve the needed statistical and systematic accuracy.
Therefore understanding the relaxation mechanism of polarized 3 He in the storage cell under the experimental conditions and maintaining 3 He polarization is crucial to the success
of the experiment. A number of other experiments [92, 93, 94, 95, 96] have studied 3 He
longitudinal relaxation times in mixtures of 3 He-4 He at temperatures similar to ours. However, no experimental data on relaxation time of polarized 3 He under such conditions (in
a dTPB-dPS coated acrylic cell at low temperatures) has ever been reported before. My
Ph.D. work is the first systematic study of the relaxation behavior of polarized 3 He close
30
to nEDM experimental conditions. This is one of the most important R&D experiments
and its result will greatly affect the feasibility of the nEDM experiment.
In order to study the 3 He depolarization behavior, measurements are taken step by step
to approach the nEDM experimental conditions. In this thesis, two sets of measurements
are presented. The first one is the 3 He relaxation time measurement in a dTPB-dPS coated
cylindrical acrylic cell at a temperature of 1.9 K (below the 4 He λ point) in the presence
of superfluid 4 He at a magnetic holding field of 21 G. This is the first step to approach
the environment at low temperatures with the special coating. With many people’s help
(see Acknowledgements), the 1.9 K test was completed in the physics building of Duke
University using a conventional liquid helium dewar in early 2006. This experiment is
relatively easier because of the simple cooling system and the short turn-around time.
The second step is a similar set of measurements at ∼400 mK (nEDM experimental
temperature) with a holding field of ∼7 G. The measurement cell’s dimension and the
coating are the same as those in the 1.9 K test. However, the amount of effort to reach a
temperature of ∼400 mK using a dilution refrigerator1 is significantly greater than that in
the 1.9 K test. It took Professor R. Golub from NC State University and I several months to
get the DR working successfully. The 400 mK test was completed in the middle of 2008 at
Triangle University Nuclear Laboratory (TUNL) in Duke University French Family Science
Center (FFSC).
In both measurements, polarized 3 He atoms are introduced into a dTPB-dPS coated
acrylic cell at low temperatures in the presence of superfluid 4 He with a small magnetic
holding field. Spin exchange optical pumping method is used to polarize the 3 He atoms.
The nuclear magnetic resonance technique is applied to measure the 3 He polarization as a
function of time to extract the relaxation time.
In this chapter, the experimental techniques and the apparatus for these two sets of
measurements are described. The data analysis and results will be presented in Chapter 4.
1
The dilution refrigerator is on loan to R. Golub from HMI in Germany.
31
3.2
Experimental Technique
Both of these two sets of experiments (1.9 K and 400 mK tests) use the spin exchange
optical pumping (SEOP) method to polarize 3 He nuclei in a glass cell [97]. After that,
polarized 3 He will be transferred into the acrylic measurement cell at a low temperature,
where the polarization is measured as a function of time using the adiabatic fast passage
(AFP) technique of Nuclear Magnetic Resonance (NMR) [98]. In this section, I will describe
the SEOP technique, 3 He relaxation mechanisms and the NMR-AFP technique.
3.2.1
Spin Exchange Optical Pumping
To study the the polarized 3 He’s depolarization behavior in a certain condition, we need
to polarize the 3 He nuclei first. Since the spin of the 3 He nucleus is I=1/2, in an external
magnetic field, it can have two quantum states. The polarization of 3 He nucleus is defined
as:
P =
N+ − N−
N0
(3.3)
where N+ and N− are the number of nuclei in the spin + 21 state and − 12 state and N0 =
N+ + N− is the total number of 3 He nuclei in the sample. In thermal equilibrium in an
external magnetic field, the polarization of 3 He is given by
P3Hethermal = tanh(
µ3He B
)
kB T
(3.4)
where µ3He = −2.128µN = −1.075 × 10−30 J/G is the 3 He’s magnetic moment. kB is the
Boltzmann constant and T is the temperature. At room temperature, the polarization is
only ∼ 7.8 × 10−9 with a magnetic field of 30 G, which is very small. Other techniques
need to be used in order to obtain higher polarization.
People mostly use two optical pumping techniques to polarize 3 He nuclei. One technique
uses the direct optical pumping of the metastable 23 S1 state of 3 He [99], which is called
metastability exchange optical pumping (MEOP). The basic idea is to use a weak RF
discharge in a low pressure 3 He gas to create a metastable 23 S1 state. A right-handed
32
circularly polarized laser light (selection rule ∆m = 1) with a wavelength of 1.08 µm excites
the nuclei in this state to the 3 P0 state, from which they can decay back to the metastable
state with probabilities according to Clebsch-Gordan coefficients [100]. This process will
polarize the 3 He nuclei in the metastable state. Then the polarization is transferred to
the ground state 3 He nuclei by collisions. The details of this process can be found in [99].
MEOP can only be used effectively when the density of 3 He is relatively low (a few torr),
however in our study, we are mainly interested in the 3 He depolarization mechanism in
a dTPB-dPS coated acrylic surface at low temperatures, and the 3 He precession signal is
much easier to observe if the concentration of polarized 3 He in superfluid 4 He is relatively
high. Even though higher nuclear polarization can be obtained using MEOP, we use the
second technique, spin exchange optical pumping (SEOP) method, to polarize 3 He in that
a high density (∼1 amagat2 ) of polarized 3 He can be easily obtained with a rubidium (Rb)
spin-exchange source.
In this technique 3 He is polarized in two steps. The first step is to produce plenty of
polarized Rb outer shell electrons that can later collide and transfer their polarization to the
3 He
nuclei. Rb has one electron in the 5s shell, which dominates the Rb atomic magnetic
moment. The 3 He’s atomic magnetic moment is dominated by the 3 He nucleus because its
1s shell is fully occupied by two electrons. The Rb’s valence electron can be polarized by
optical pumping method [97, 101]. In the optical pumping process, the cell is heated up to
∼ 190◦ C so that a lot of Rb atoms are vaporized. The Rb outer shell electrons are then
polarized by a laser tuned to the 5S1/2 → 5P1/2 transition, known as the D1 transition.
Right-handed circularly polarized photons at a wavelength of 794.8 nm [102, 103] can excite
electrons of the 5S1/2 (mJ = −1/2) state to the 5P1/2 (mJ = 1/2) state due to ∆m = 1
selection rule, while the left-handed circularly polarized light can do the opposite (From
5S1/2 (mJ = 1/2) state to the 5P1/2 (mJ = −1/2) state). Figure 3.1 shows the optical
pumping process. After the electrons are excited from the 5S1/2 (mJ = −1/2) state to
5P1/2 (mJ = 1/2) state, they can spontaneously decay back to the ground state but cannot
2
1 amagat=2.687 × 1019 /cm3 , a unit of density equals 1 atmosphere of ideal gas at 0◦ C.
33
Figure 3.1: Optical pumping of Rb outer shell electrons.
be excited again from the mJ = +1/2 state. The decay rates back to 5S±1/2 states are given
by the Clebsch-Gordon coefficients [100],
2
3
to mJ = −1/2 state and
1
3
to mJ = +1/2 state.
Thus in the absence of any relaxation processes, the 5S1/2 (mJ = −1/2) ground state will
become depopulated, and gradually all electrons will accumulate in the mJ = +1/2 state.
But when excited electrons decay to the ground state, photons at the same D1 transition
wavelength will be emitted. These photons are not polarized and can depolarize the already
polarized electrons if absorbed. This depolarization effect can be reduced by introducing
nitrogen into the system as a buffer gas. The collisions between the Rb atoms and N2
molecules will allow the electrons to decay without emitting a photon since the nitrogen
absorbs energy into its rotational and vibrational motion during a collision, usually referred
to as non-radiative quenching [101]. Moreover, buffer gas collisions randomize the P states
(collisional mixing), and the excited electrons decay back with equal probabilities to both
ground state levels. The amount of N2 is chosen to be orders of magnitude less than the
3 He
density and orders of magnitude more than the Rb density. Only about 5% of excited
electrons decay by emitting a photon after introducing N2 . The typical amount of N2
introduced into the 3 He cell is ∼50-100 torr. The average Rb vapor polarization can be
34
expressed as:
< PRb >=
R
R + ΓSD
(3.5)
where ΓSD is the spin-destruction rate, and R is the polarizing rate from the laser. The
more laser power at the absorption frequency with less spin-destruction rate, the higher the
Rb polarization can be obtained. In most cases, PRb ∼100% in the region where the laser
beam is incident.
The second step is the collisional transfer of polarization between optically pumped Rb
electrons and 3 He nuclei, which is the key process in SEOP (Figure 3.2 [104]). The polarized
Figure 3.2: Spin exchange between Rb outer shell electrons and 3 He nuclei.
Rb electrons will collide with 3 He nuclei and exchange their spins through a hyperfine-like
interaction. The equilibrium 3 He polarization can be written as [105]:
P3He =< PRb > ·
γSE
γSE + Γ
(3.6)
where γSE is the spin exchange rate between Rb and 3 He and Γ is 3 He polarization destruction rate. Only approximately 3% of polarized Rb atoms lose their polarization during
the 10−12 s binary collision time with 3 He. This makes it an inherently inefficient process.
Usually it takes ∼15-20 hours for our spherical 3 He cell (2 in. diameter, filled with ∼1.5
atmosphere of 3 He and ∼100 torr of N2 at room temperature) to reach its maximum
polarization. Recent development in the SEOP can significantly reduce the polarization
35
pump-up time by using a “hybrid” cell, which not only has vaporized Rb, but has vaporized potassium as well. The basic idea is to polarize Rb atoms first, then Rb atoms
transfer the polarization to the K atoms. After that the K atoms will subsequently spin
exchange with the 3 He atoms much more efficiently than Rb atoms [106]. The pump-up
time can be reduced to within 10 hours in a recently developed high-pressure polarized 3 He
gas target [107] at Duke University.
3.2.2
3
He Relaxation Mechanism
There are many effects which can depolarize 3 He. The 3 He depolarization rate can come
from the three most important sources of depolarization:
Γ = Γdipole + Γwall + Γ∇B
(3.7)
Here Γ is the total 3 He polarization destruction rate. ΓDipole is the depolarization from the
3 He-3 He
dipole-dipole collisions, Γwall is the depolarization from wall effects, and Γ∇B is
from magnetic field gradients.
The dipolar spin relaxation comes from the dipole-dipole interaction between the 3 He
atoms’ magnetic moments. The leading order of the dipole-dipole potential can be written
as [105]:
V =(
µ3He 2 1 ~ ~
3(I~1 · ~r)(I~2 · ~r)
) 3 (I1 · I2 −
)
I
r
r2
(3.8)
where I~1 and I~2 are the nuclear spins of two 3 He atoms separated by a distance ~r and
I = |I~1 | = |I~2 |. At room temperature, the dipole-dipole spin relaxation rate can be written
as [105]:
Γdipole =
1
Tdipole
=
[3 He]
hrs−1
744
(3.9)
where [3 He] is the 3 He density in amagats. The details of the calculation can be found
in [105]. Essentially the empirical interatomic potentials are used to numerically calculate
the relaxation rate between the 3 He-3 He magnetic dipoles. The glass cells used in the 1.9 K
and 400 mK tests are usually filled with 0.5∼1.5 atmosphere of 3 He, and the corresponding
dipolar spin relaxation rate is very small.
36
If there is a magnetic field gradient over the volume of a 3 He cell, the 3 He atoms moving
inside the cell will experience varying magnetic fields which tends to change the direction of
the polarization. The 3 He relaxation rate due to magnetic field gradient can be calculated
using the formula [108]:
Γ∇B =
1
T∇B
=
< v̄ 2 > |∇Bx |2 + |∇By |2
Tc
2
3
Bz
1 + ω02 Tc2
(3.10)
Here Bz is the magnitude of the main holding magnetic field in the z direction, and ω0 is the
Larmor precession frequency ω0 = γBz , where γ = 2π× 3.24 kHz/G is the gyromagnetic
q
ratio for 3 He. v̄ is the average speed of 3 He defined as v̄ = 8kT
πm . ∇Bx and ∇By are the
gradients of the two transverse field components Bx , By and have 6 components altogether.
Tc is the mean time between atomic collisions Tc =
λmf p
v̄ ,
where λmf p is the mean free path.
The mean free path λmf p can be calculated using the equation from [109], page 6-34:
λmf p = √
kT
2πP d2
(3.11)
where P is the pressure, T the absolute temperature, k the Boltzmann constant, d the diameter of the 3 He atoms. In the limit of ω0 Tc << 1, which is usually the case, Equation 3.10
can be rewritten as [110]:
Γ∇B =
1
T∇B
where the diffusion coefficient D3He =
= D3He
v̄λ
3
|∇Bx |2 + |∇By |2
B02
(3.12)
can be used. The magnetic field gradient effect
can be reduced by using a better designed coil configuration or a higher magnetic holding
field to suppress the environmental field gradients which should be relatively constant.
Among all three major sources of depolarization, the wall effect is the most difficult part
to quantify due to multiple reasons. The relaxation can be due to paramagnetic impurities
like Rb2 O in the walls or the out-gassing of paramagnetic materials like O2 or NO from the
cell walls when heated. It can also be caused by 3 He sticking to the surface of the cell for
a long time (probably trapped in micro-fissures, if present) and relaxing by local magnetic
field gradient [111], by exchanging with other nuclear spins [112], or by the combination of
those reasons. However, recent evidence from Schmiedeskamp et al. [113] shows that distant
37
dipolar coupling to paramagnetic impurities in the glass (iron ions) cannot be the dominant
relaxation mechanism. Instead the dangling-bond type defects in the glass interact much
stronger via the isotropic Fermi contact interaction (details can be found in [113]). This was
proved by measuring the 3 He relaxation time in the vessel made of iron rich and iron poor
glasses (30 ppm - 4800 ppm) and similar relaxation times were observed [113]. Even if you
fabricate two cells in exactly the same way, the wall effect may still be different. People have
found that coating glass cell walls with alkali metal can significantly reduce the relaxation
rate of polarized 3 He due to the much lower adsorption energy [114]. Table 3.1 lists some
of the 3 He adsorption energies on different materials. Some oxidation of a Cs coating can
also assist to reduce the relaxation rate [114].
Table 3.1: 3 He adsorption energy on different materials.
Material
3
He Adsorption Energy, Ead
Glass
∼0.01 eV [115]
Cs
∼0.2 meV
Rb
∼0.24 meV
Mg
∼2.5 meV
Al
∼5 meV
A depolarization probability can be derived to characterize how “friendly” or “unfriendly” a surface is to polarized 3 He. Consider a sample of 100% polarized 3 He atoms in
a container of volume V and surface area S. The average number of 3 He atoms arriving
at the wall per unit time per unit area is 14 n · v̄ from statistical mechanics, where n =
N
V
is the number density of 3 He atoms. A polarized 3 He atom will have a probability Pd to
depolarize after each collision with the cell surface. So the total number of 3 He atoms that
will lose polarization per unit time is
−
dN
1N
1
S
=
· v̄ · S · Pd = N · v̄ · Pd
dt
4V
4
V
38
(3.13)
Solving this equation gives us
1
S
t
N
= N0 e− 4 v̄·Pd · V t = N0 e− T
(3.14)
1
T
=
1
S
S
v̄ · Pd ·
=ρ
4
V
V
(3.15)
Here ρ = 14 v̄ · Pd is the depolarization coefficient and
S
V
is the surface to volume ratio of
a particular cell. In analyzing the data from 1.9 K and 400 mK tests, the depolarization
probability Pd is extracted to characterize the depolarization behavior of the surfaces.
3.2.3
3
He Polarimetry
The magnetic moment of an atom is the product of the gyromagnetic ratio and its nuclear
~ where for 3 He γ3He = 2π×3.24 kHz/G and for proton γproton = 2π×4.26
spin µ
~ = γ I,
kHz/G [116]. When a magnetic moment, a 3 He atom for instance, is put in an external
magnetic field and has an angle with respect to the magnetic field direction, there will be
components of the nuclear spin magnetization parallel and perpendicular to the external
magnetic field. Different physical processes are responsible for the relaxation of these two
components.
The spin-lattice relaxation time (also called longitudinal relaxation time or T1 ) is a
time constant to characterize the average time that the spin’s parallel component with the
magnetic field can stay in the current state. If a collection of unpolarized 3 He atoms are put
in an external magnetic field, they will eventually reach the thermal polarization defined in
Equation 3.4. The time it takes to recover 1− 1e ∼ 63% of the thermal polarization is defined
as T1 . The time evolution of the polarization is therefore Pl (t) = Pl (thml)(1−e−t/T1 ), where
Pl (thml) is the thermal polarization. On the other hand, if the 3 He atoms are already
polarized to Pl (i) >> Pl (thml), the time it takes to reach
1
e
∼37% of Pl (i) − Pl (thml) is
also defined as T1 . Since Pl (thml) is usually much smaller than 1, the time evolution of
the polarization is Pl (t) = Pl (i)e−t/T1 . The environment of the spins in the experiment
is called the “lattice” and hence the name spin-lattice relaxation time. The interaction
between the spin and the lattice results in heat and thermal energy to relax the spin to low
39
energy states.
The spin-spin relaxation time (also called transverse relaxation time or T2 ) is a time
constant to characterize the average time that the spin’s component perpendicular to the
magnetic field can stay in the current state. Since there is no thermal polarization of
this transverse component, the time evolution of the transverse polarization is Pt (t) =
Pt (i)e−t/T2 . After one time constant T2 , the transverse magnetization will drop to ∼37%
of its original magnitude. The cause of T2 relaxation is the existence of the non-stationary
magnetic field. Even though the stability of the external magnetic field can be controlled to
a high precision, each 3 He magnetic moment will not only experience this external magnetic
field, but also the magnetic fields generated by the neighbouring atoms’ magnetic moments.
As the 3 He atoms move around, there will be random fluctuations of the local magnetic
field seen by different 3 He atoms and therefore change the Larmor precession frequencies
of different spins. Some spins will precess faster and others slower so they are gradually
out of phase. The overall transverse magnetic moment will then decrease in the transverse
plane as time goes on. As a result, the initial phase coherence of the nuclear spins is lost,
until finally no net transverse magnetization is left. Because T2 relaxation involves only the
phases of nuclear spins, it is often called “spin-spin” relaxation.
The spins’ phase decoherence due to the magnetic field inhomogeneity is not a pure
“spin-spin” relaxation because it is not random but dependent on the location of the magnetic moment in the field. The corresponding effective transverse relaxation time constant
is T2∗ , which is defined as:
1
1
1
=
+
T2∗
T2 Tinhom
(3.16)
where T2 is the intrinsic (spin-spin) transverse relaxation time and Tinhom represents the
time constant due to the inhomogeneity of the magnetic field. In the high-pressure limit,
McGregor [117] give the result of
1
Tinhom
1
8γ 2 R4 |∇Bz |2
=
+
2T1
175D
(3.17)
where R is a spherical cell’s radius, D the diffusion coefficient and |∇Bz | is the gradient
40
of the Bz component. T1 here is the longitudinal relaxation time due to the magnetic field
gradient, which is the same as T∇B in Equation 3.10 and 3.12.
NMR - Adiabatic Fast Passage
To monitor the 3 He polarization signal, Adiabatic Fast Passage (AFP) technique from
nuclear magnetic resonance (NMR) [98] is used. The strength of the NMR signal is proportional to the amount of polarization, and relative polarization can be obtained using
the NMR-AFP method. Water calibration and other techniques are needed to extract the
absolute polarization.
A magnetic moment experiences a torque when placed in an external magnetic field
B0 [118]:
~
dM
~ × B~0
|lab = γ M
dt
(3.18)
If we describe the system in a rotating frame that rotates around the holding magnetic field
~ in a rotating reference frame can
direction at a frequency ω. The motion of any vector M
be described as:
~
~
dM
dM
~
|lab =
|rot + ω
~ ×M
dt
dt
(3.19)
Equate it with equation 3.18, we have
~
dM
~
~ × (B~0 + ω
|rot = γ M
)
dt
γ
(3.20)
~ in the rotating reference frame. The effective holding field
which describes the motion of M
in this rotating frame is therefore B~0 + ω~γ .
Consider a magnetized sample in a static magnetic field B0 ẑ. An RF field B~1 =
2B1 cos(ωt)x̂ is applied, where ω is the rotating frame frequency and its magnitude B1 <<
B0 . The RF field can be written as
B~1 = (B1 cos(ωt)x̂ + B1 sin(ωt)ŷ) + (B1 cos(ωt)x̂ − B1 sin(ωt)ŷ)
(3.21)
which represents two magnetic fields rotating at the frequencies ω
~ = ωẑ and ω
~ = −ωẑ. In
the reference frame rotating at the frequency −ωẑ, the RF field becomes a static field B1 x̂
41
(the other component will be rotating at a frequency 2ω, which is too far off resonance
and can be ignored) and the holding field is B~0 +
ω
~
γ
= (B0 − ωγ )ẑ. So the effective field
experienced by the sample becomes (Figure 3.3):
~ ef f = (B0 − ω )ẑ + B1 x̂
B
γ
(3.22)
During AFP, the RF coil frequency ω is held as a constant and the holding field B0 changes
Figure 3.3: The effective magnetic field in the rotating reference frame.
from below ω/γ to above it while the RF field is applied. When the holding field is ramped
~ ef f deviates from the ẑ axis to x̂ axis, and the magnetization of
towards resonance (ω/γ), B
~ ef f . When at resonance, the 3 He atoms’ magnetic moments induce an
the sample follows B
EMF signal in the pickup coil, which is perpendicular to both the RF field and the holding
field, and this signal is proportional to the 3 He polarization. As the holding field is ramped
~ ef f and 3 He polarization will be pointing in the opposite
beyond the resonance value ω/γ, B
direction, which is called spin flip. The measurement is done by ramping the magnetic field
from below the resonance to above it and then back. Figure 3.4 shows the holding magnetic
field ramping curve of one NMR-AFP measurement. This procedure results in two spin
flips, inducing the EMF signal in the pickup coil twice.
The measured signal from the pickup coil is proportional to the instant transverse
component of the 3 He magnetization. So the AFP signal curve can be fitted into this
42
Figure 3.4: Holding field curve of NMR-AFP measurement for polarized 3 He. The
holding field is ramped from below the resonance field to above it and then back
down. The RF field is on during the ramping.
expression [119]:
< M > B1
nmr
S3He
(t) = p
+ m · B0 (t) + C
(B0 (t) − ω/γ)2 + B12
(3.23)
where < M > is the fitted signal amplitude which is proportional to the 3 He polarization.
m · B0 (t) + C is the fitted signal background. Figure 3.5 shows the NMR-AFP signal from
q
2 and
one sweep. After fitting the curve to Equation 3.23, the average value of x2up + yup
q
2
is the AFP signal of the sweep.
x2down + ydown
During the AFP sweep, the sweep rate must be slow enough [120]:
1 dB0 << γB1
B1 dt (3.24)
so that the magnetization of 3 He follows the effective magnetic field (this is why it is called
“adiabatic”), while it must be fast enough for [120]:
1 dB0 1 1
>>
,
B1 dt
T1 T2
(3.25)
so that spin relaxation at resonance is minimal (this is why it is called “fast”). Here T1
and T2 are the longitudinal and transverse relaxation times.
43
Figure 3.5: The NMR-AFP signal read from the lock-in amplifier. The average of
p
p
2
2 and
the fitted amplitudes x2up + yup
x2down + ydown
is the AFP signal of the sweep.
NMR - Free Induction Decay
The effective transverse relaxation time T2∗ (Equation 3.16) is much more rapid than T2
relaxation because spin coherence is lost quickly due to the movement of spins in a static
field with gradients. This decay is exponential and is called free induction decay (FID).
Initially most of the spins are aligned with the external magnetic field. By applying a short
pulse at the Larmor frequency, the magnetization can be tipped away from that direction.
After the pulse, the transverse component of the magnetization freely precesses around
the magnetic field direction at the Larmor frequency while the amplitude is exponentially
decaying with time. The tip angle can be controlled by the pulse’s length and strength
θ = γB1 t, where θ is the tipping angle, B1 is the amplitude of the RF field and t is the
length of the pulse. A θ = π/2 pulse will tip the magnetization to the transverse plane with
no Bz -component left. If the tipping angle is 0.2 rad for instance, the final longitudinal
polarization will be cos 0.2 = 98.2% of the initial polarization. A single coil with its axis
perpendicular to the magnetic field can be used to both generate the RF pulse and pick up
the FID signal after the pulse.
44
Water Calibration
Free induction decay and adiabatic fast passage themselves can only measure the 3 He
relative polarization. They can measure the absolute polarization once calibrated. Water
calibration is one of the methods to calibrate the 3 He polarization.
A glass cell with the same geometry as the optical pumping cell is filled with deionized
water and sealed. The glass water cell is mounted in the same position as the optical
pumping cell and the thermal polarization of the protons in water is given by [98]:
Pp = tanh(
µp B
µp ωp
)
) = tanh(
kB T
kB T γp
(3.26)
where µp = 2.793µN = 1.41 × 10−30 J/G is the proton magnetic moment. kB is the Boltzmann constant and γp is the gyromagnetic ratio of proton. At room temperature and with
a small magnetic field (∼ 20 G), we have:
µp · ωp
Pp ∼
∼ 7 × 10−9
=
kB T γp
(3.27)
The ratio of the 3 He and proton signals can be written as [98]:
R=
Q3 · ω3 · P3 · n3 · V3 · µ3
Qp · ωp · Pp · np · Vp · µp
(3.28)
where the Q-values Q3 = Qp since the same pickup coils are used. ω3 = ωp , V3 = Vp mean
the resonance frequencies and the volumes of the water cell and 3 He cell are the same. We
carried out NMR-AFP measurements by using the same resonance frequency for both the
3 He
and the water. Therefore the resonance took place at different magnetic holding fields
for 3 He (∼24.1 G) and water (∼18.3 G).
P3,p , n3,p , µ3,p are the polarization, number density and the magnetic moment of 3 He
and proton. Replace Pp in Equation 3.28 with Equation 3.26 and simplify the equation, we
have
R=
P3 · n3 · kB · Tp · µ3 · γp
ωp · np · µ2p
(3.29)
Since the number density n3 = P s3 /(kB T3 ) and T3 = Tp =300 K, where P s3 is the pressure
of the 3 He in the cell,
R=
P3 · µ3 · γp · P s3
ωp · np · µ2p
45
(3.30)
We have the polarization of 3 He
P3 =
R · ωp · np · µ2p
µ3 · γp · P s3
(3.31)
where np = 6.7 × 1028 /cm3 is the proton density in water, µ3 the 3 He magnetic moment.
Since the NMR-AFP signal of the water is small, one must average out hundreds of AFP
sweeps in order to see the water signal if the surrounding RF noise is not controlled to a low
enough level. Because the relaxation times of protons in the water is ∼3-4 seconds, which
is much shorter than the 3 He’s T1 , it is important to wait for ∼4 seconds after the magnetic
field’s ramping up before the ramping down for water polarization to be re-established.
Otherwise the signal in the down sweep can hardly be seen. Figure 3.6 shows the holding
field ramping curve for the AFP measurements of water. The resonance frequency is the
same as that in 3 He case (Figure 3.4).
Figure 3.6:
By using the water calibration, information about absolution polarization of 3 He can
be recorded. Another method of measuring the 3 He absolute polarization is through the
Electron Paramagnetic Resonance technique [121] which will not be discussed here.
46
3.3
1.9 K Test Experimental Apparatus
The schematic of the entire apparatus is shown in Figure 3.7 [2]. Figure 3.8 is what the real
system looks like.
The 67 in. diameter red Helmholtz coils provide a uniform magnetic
Figure 3.7: A schematic of the 1.9 K test experimental setup (courtesy of D. Dutta).
field over the region of the double-cell system so that the 3 He can be polarized by the laser
in the upper glass cell and later lowered down into the acrylic cell at low temperatures.
NMR-AFP is used to measure the relaxation time of polarized 3 He both in the upper cell
position and in the dewar. Details of the apparatus are explained next.
3.3.1
Double-Cell System and Gas Handling System
A two-chamber apparatus for polarizing the 3 He nuclei and for measuring their relaxation
time at cryogenic temperatures is constructed from aluminosilicate glass (GE180) and a
47
Figure 3.8: 1.9 K test experimental setup.
cylindrical acrylic cell. The two chambers are connected via a 3-mm-diameter, 21-in.-long
pyrex capillary tube, and are separated by a glass valve (Figure 3.9). The top chamber
is a spherical cell with a diameter of 2.0 in. (Figure 3.10), while the bottom chamber
is a 2.0 in. long cylindrical acrylic cell with an outer diameter of 2.0 in. and an inner
diameter of 1.45 in. It is attached to the glass via a a 0.5-in.-long glass to copper seal
with a diameter of 3 mm (Figure 3.11). The copper seal is attached to the acrylic cylinder
using the low-temperature epoxy Emerson and Cuming Stycast 1266. Since the thermal
expansion coefficient of glass and acrylic is very different (at 20◦ C, α = 4 × 10−6 /K for
pyrex glass and α = 76 × 10−6 /K for acrylic), they cannot be connected together directly
since cracks will appear once cooled down. Thus this short copper piece (α = 17 × 10−6 /K)
is used to join these two parts together. Both the top chamber and the bottom acrylic
cell can be independently evacuated and filled with either 3 He and nitrogen gas in the top
chamber or 4 He gas in the bottom chamber, and they can be isolated from the gas handling
system (Figure 3.12) via a pair of glass valves. The N2 gas is introduced for efficient optical
48
Figure 3.9: 1.9 K test double cell system.
pumping which will freeze on the wall of the capillary tube at low temperatures. The inner
surfaces of the acrylic cell is coated with dTPB-dPS material made by R. Golub. The
making and coating procedures can be found in Appendix D and E.
3.3.2
Cooling System and Temperature Monitoring
The double-cell system is mounted onto a dewar (Figure 3.13) from Precision Cryogenic
System, Inc.. The reason why a big dewar is needed is due to the size of the NMR RF coils
(12 in. in diameter). The dewar does not have a liquid N2 jacket but does have two vacuum
jackets instead. The temperature of the bottom cell in the dewar can be lowered to ∼1.8
K by pumping on the helium vapor above the liquid 4 He. A liquid 4 He level sensor (read
out by LM-500 Liquid Cryogen Level Monitor) from Cryomagnetics Inc. is used to monitor
the liquid 4 He level. Figure 3.14 [122] is a diagram showing the 4 He vapor pressure versus
the temperature. The lambda point of 4 He is 2.17 K, and an oil pump (Leybold Sogevac
49
Figure 3.10: Top part of the double cell system.
Figure 3.11: Bottom acrylic cell.
SV25) with a high pumping speed is used to achieve the helium vapor pressure of a few
torr. The temperature is monitored by a calibrated Cernox thin film resistance cryogenic
50
Figure 3.12: Picture of the gas handling system with pneumatic valves, gas tanks
and turbo pump.
temperature sensor from Lakeshore company. The sensor is attached onto the side of the
acrylic cell and read out using a temperature transmitter 234D from Lakeshore.
3.3.3
Magnet System and NMR-AFP system
A set of red Helmholtz coils provides the holding field for the 1.9 K test apparatus. The
coil’s radius is 67 in. in order to provide a uniform magnetic holding field (the typical
holding field is 21G) over the optical pumping cell region as well as the low temperature
cell which is immersed in the cryogenic dewar. The Helmholtz coils are powered by a PSC-4
power supply (from Walker LDJ Scientific Inc.), which is controlled by an Agilent 33120A
function generator voltage output. The Agilent function generator is controlled through a
GPIB interface to a Windows-based PC running LabVIEW program.
Since our double-cell system is ∼ ±27cm with respect to the center of the magnetic
51
Figure 3.13: The dewar used for the 1.9 K test. The double cell system is mounted
onto the top of it.
field in the x-y plane (z direction is the holding field direction), the magnetic field gradients
may be large so that the 3 He relaxation time will be affected. The gradient is measured
in the pumping chamber position and the averaged gradient in the transverse plane is less
than 10 mG/cm, which will give us a relaxation time of ∼125 hours due to the gradient
effect. In the acrylic cell position in the dewar, it is hard to measure the actual magnetic
field gradient using a Gauss Meter. So to estimate the magnetic field gradient effect at 1.9
K at the acrylic cell position, a sealed 1-in.-diameter spherical cell made of aluminosilicate
glass (GE180) from T. Gentile at National Institute of Standards and Technology (NIST)
is used. This sealed cell is filled with 100 torr of 3 He, 50 torr of N2 , and 535 torr of 4 He at
room temperature. At the center of a magnetic field where the magnetic field gradient is
negligible, the cell’s longitudinal relaxation time Tcenter is measured to be on the order of 102
52
Figure 3.14: The 4 He vapor pressure versus the temperature graph. The normal
phase-superfluid phase transition temperature is 2.17K.
hours. Then this cell is put in the dewar and the relaxation time Tdewar is measured. The
1
1
difference of the two relaxation rates ( Tdewar
− Tcenter
) is thus purely from the contribution
of the magnetic field gradient and can be put into the left hand side of Equation 3.10. the
|∇Bx |2 + |∇By |2 is determined to be ∼0.009 (G/cm)2 . This result is then put back into
the same equation and applied to the acrylic cell case, which is at 1.9 K and filled with
0.00139 mole of 3 He. The calculated relaxation time due to the magnetic field gradient is
5.26 × 105 seconds.
Two NMR-AFP systems are built in order to measure the 3 He relaxation times in both
the top (pumping) cell (Figure 3.15) and the bottom cell (Figure 3.16).
Each NMR
system consists of a pair of RF coils (powered by AG 1021 RF amplifier from T&C Power
Conversion Inc.) which is 12 in. in diameter, and one or two pickup coils attached to
the pyrex cell and the dTPB-dPS coated acrylic cell as shown in Fig 3.7. The RF and
pickup coils for the low temperature (bottom) cell are placed inside the dewar and hence
are immersed in superfluid helium during a measurement cycle. The 3 He relaxation time in
the pumping cell is measured at room temperature to establish a baseline for comparison
53
Figure 3.15: NMR coils in the pumping cell position. The purple ring is the color
of the laser from the camera.
Figure 3.16: NMR coils in the dewar.
with the relaxation time measured in the bottom cell at low temperatures.
It is important to know the holding magnetic field versus the function generator output
voltage curve both in the pumping cell chamber and the acrylic cell position so that the
54
magnetic field can be recorded during the NMR-AFP measurement. The relationship between the voltage output and the magnetic field is linear B = αV +β where α and β are the
calibrated slope and offset, respectively. The model 7010 single-channel GAUSS/TESLA
meter from F. W. Bell is used for this calibration. Figure 3.17 shows the NMR-AFP circuit
diagram. Table 3.2 lists some NMR-AFP parameters of the measurements.
Figure 3.17: NMR-AFP circuit diagram. Red “GPIB” indicates that the equipment
is connected to PC via GPIB cable.
3.3.4
Laser and Optics
The diode laser and the optical system that produces circularly polarized photons is illustrated in Figure 3.18. A convex lens is placed in front of the laser fiber with the distance
equal to its focal length so that the diverging laser beam becomes parallel after passing
through it. Then the beam is split into two linearly polarized beams (s wave and p wave)
by a cubic beam splitter. The beam passing through the beam splitter is the p wave with
~ component polarized in the horizontal plane. It is reflected by a mirror and then
its E
55
Table 3.2: Plarized 3 He NMR-AFP measurements’ parameters.
Parameter
Value
Holding B-field
21 Gauss
RF field magnitude
∼0.1 Gauss
RF field frequency
78 kHz
B-field ramping range
21→27→21 Gauss
Pre-amp magnification
200
Time interval between AFP sweeps
50-220 seconds
Figure 3.18: The schematic of laser optics setup.
circularly polarized by a quarter wave (λ/4) plate. The beam reflected by the beam splitter
~ component polarized in the vertical plane. After passing the λ/4
is the s wave with its E
56
plate, being reflected by a mirror and passing through the λ/4 plate again, the s wave
becomes a p wave and it can pass through the beam splitter. Then a λ/4 plate makes it
circularly polarized. However the beam splitter is not 100% efficient and up to 10% the s
wave is reflected back to the laser fiber. This amount of light can do damage to the fiber
and even the laser diode. So one needs to tilt the beam splitter a little to prevent this from
happening.
Figure 3.19 shows the optics arranged to polarize 3 He using one FAP LX 60 W diode
laser from Coherent Inc. The output laser wavelength needs to be fine tuned to D1 transition
by adjusting the laser diodes’ temperatures.
Figure 3.19: 1.9 K test optical pumping system setup.
The total power incident on the 3 He cell is measured to be ∼55 W after transmissions
though the cube and reflections from the mirrors. After the laser beam is incident on
the cell which is heated up to ∼190 degrees Celsius, the photons at the D1 transition are
absorbed. Those photons away from D1 transition will pass through. Figure 3.20 shows
the readout of the spectrometer placed after the cell. A big valley is in the figure due to
Rb absorption.
57
Figure 3.20: The absorption line of the laser after its passing through the optical
pumping chamber.
3.3.5
Experimental Procedure
The Rb metal is originally sealed in a breakable ampule from Alfa Aesar, and it requires
a special method to chase the Rb into the pumping chamber. Figure 3.21 shows what the
pumping chamber looks like before chasing the Rb. The following procedures are followed
to chase the Rb from the sealed ampule into the pumping chamber without introducing
impurities.
• After Rb ampule is put into the side arm, a small pyrex coated cylindrical magnet is
put on top of the ampule’s sealing tip. The top of the side arm is then sealed with a
torch.
• Bake the glass piece from the pumping chamber to the side arm at 350◦ C under
vacuum for at least 3 days until the vacuum reaches 10−8 torr level.
• Use a big magnet to attract the small magnet on top of the ampule’s tip from the
outside and break the seal under vacuum.
• Chase the Rb using a torch to the middle sink on the horizontal pyrex tube. Remove
58
the side arm from the pull-off using a torch under vacuum. Bake the rest of the
system under vacuum for another 2 days.
• Chase the Rb into the pumping chamber and remove the horizontal pyrex tube from
the pull-off. Now the top chamber is ready to be filled with 3 He and N2 .
Figure 3.21: Before chasing the Rb into the pumping cell, the ampule is sealed
inside the side arm and broken under vacuum.
Once the top cell is ready, a known amount of 3 He is introduced into the cell for each
measurement (the amount can be varied as desired). N2 (∼50-100 torr filled at room
temperature) is also added as a buffer gas. The top cell is enclosed in an oven (Figure 3.15)
and heated to 190◦ C (temperature controlled), and the 60-watt circularly polarized laser
light at 794.8 nm is incident onto the cell to polarize the 3 He atoms through SEOP process
overnight. While the 3 He atoms in the top cell are being polarized, liquid 4 He is filled into
the dewar and the temperature of the bottom (acrylic) cell is lowered below the liquid 4 He
lambda point by pumping on the 4 He vapor using the SV25 pump. Once the acrylic cell has
59
reached the desired temperature ∼1.9 K, a known amount of 4 He gas is slowly introduced
into the acrylic cell to condense into liquid, which usually takes several hours. The laser
is then turned off, and the top cell is cooled down to room temperature, after which the
middle glass valve separating the two chambers is opened to allow the polarized 3 He atoms
to diffuse into the bottom acrylic cell. The N2 gas condenses on the way down and does
not enter the bottom cell. The valve is closed after 30 seconds and a series of NMR-AFP
measurements are performed with a time interval between 50 to 220 seconds. After the
measurement, the double-cell system is warmed up slowly while pumping so that 4 He in
the acrylic cell can be pumped out when vaporized. The top cell is then filled with fresh
3 He
and the measurement cycle is repeated.
3.4
400 mK Test Experimental Apparatus
The schematic of the entire apparatus is shown in Figure 3.22. Figure 3.23 is what the
real system looks like. A double-cell system is mounted through the dilution refrigerator
(DR) sitting in an eight-coil system providing a vertically uniform magnetic field. The DR
control is located behind the eight-coil system and a gas handling system is sitting on the
ground to the left.
3.4.1
Double Cell System and Gas Handling System
Similar to the double-cell system in the 1.9 K test (Figure 3.9), polarized 3 He atoms are
introduced into the bottom dTPB-dPS acrylic cell from a detachable 2 in. diameter glass
cell sitting ∼86 in. above it outside of the Dilution Refrigerator (Figure 3.22). The top
cell is made of pyrex glass and also has a valve and threaded stem. Its end can be sealed
using a 1/4 in. O-ring onto a 1/4 in. outer diameter glass tube, which makes it detachable
(Figure 3.24). The valve is ∼6 in. away from the cell body by a capillary since the rubber
O-ring (made of Viton) is not polarized 3 He friendly. The detachable cell is usually filled
with ∼1.5 atmosphere 3 He and ∼100 torr N2 gas at room temperature, and spin-exchange
60
Figure 3.22: A schematic of the 400 mK test experimental setup.
optical pumping technique is used to polarize the 3 He. To simplify the complicated system,
it is decided to polarize the 3 He on the polarizing station, which is located in another
building, then transfer it to the cryogenic system using a portable magnetic field system
(Figure 3.25). The bottom acrylic cell has the same dimension as the cell used in the 1.9 K
test (1.45 in. inner diameter and 2.0 in. long cylinder), except the outside has a lot of
grooves cut to house the cooling wires from the mixing chamber of the DR (Figure 3.26).
The two cells are connected via a long pyrex capillary tube (6 mm OD, 2 mm ID, ∼86 in.
long) and separated by a glass valve. A glass-to-copper seal (1/8 in. OD, 1 in. long) is used
as the transition between the pyrex capillary and the acrylic cell to serve as the transition
between the two materials with different coefficients of thermal expansion. Stycast 1266
is applied to make the vacuum seal between copper and acrylic. Even though copper is
not the best surface for polarized 3 He, 3 He goes through it within a short time thus the
61
Figure 3.23: 400 mK test experimental setup.
depolarization effect is negligible.
The acrylic cell needs to be filled with superfluid 4 He before introducing 3 He. A 7.5 liter
aluminum volume is mounted onto the gas handling system shown on the lower-right corner
of Figure 3.22 with a baratron (pressure gauge) attached to it. So the exact amount of 4 He
gas inside the volume is known. Once the acrylic cell has reached the right temperature,
62
Figure 3.24: Detachable cell which can be polarized and transported onto the 8-coil
system.
a metering valve connecting the gas handling system and the DR 4 He capillary is opened.
4 He
is flowed slowly through a liquid N2 trap first and then into the low temperature region,
condensing into liquid slowly. The reason a metering valve is needed is because the 4 He
gas will be bringing a lot of heat into the cooling system (1 K pot, Still, 50 mK plate and
MC, see next section), and the dilution refrigerator will not be able to handle that amount
of heat if the 4 He gas flow is too fast.
3.4.2
Cooling System and Temperature Monitoring
The cylindrical dewar shown in Figure 3.23 is composed of four layers: the outer vacuum
chamber (OVC), the liquid N2 layer, the liquid 4 He main bath and the inner vacuum chamber (IVC). A dilution refrigerator (DR) is mounted through it for obtaining temperatures
below 0.5 K at the mixing chamber position. The model “Minikelvin 126-TOF” dilution
refrigerator made by Leiden Cryogenics [123] is used to cool the acrylic cell to ∼400 mK in
63
Figure 3.25: The 3 He in the detachable cell is polarized in physics building, brought
over in a portable magnetic field, then put on top of the DR in French Family Science
Center room 1127.
this experiment. The working principle of the dilution refrigerator can be found in [124].
The basic idea of the DR is to circulate the 3 He atoms in the 3 He-4 He mixture so that when
3 He
atoms absorb the energy from the mixing chamber to cross the 3 He-4 He boundary, they
effectively cool the mixing chamber. At low temperatures, the cooling power of the dilution
refrigerator is small because the amount of heat removed is also small. The DR unit is
composed of 1 K pot, Still, 50 mK plate, mixing chamber, and a series of heat exchangers
shown in Figure 3.27 [123].
A gold plated copper buffer volume (Figure 3.28) is attached to the bottom of the DR
mixing chamber, the coldest part of the DR at normal operating conditions. The pre-
64
Figure 3.26: The dTPB-dPS coated acrylic cell in the vacuum chamber of the DR.
cooled 4 He liquid used to fill up the acrylic cell goes into the buffer volume first, then drips
into the acrylic cell slowly. Since the 4 He density changes from ∼0.125 g/cm3 to ∼0.145
g/cm3 [124] when temperature drops below the lambda point, the liquid volume expands
when the system is warming up. This buffer volume is there for this extra 16% volume of
4 He
to expand into. Attached to the bottom of the buffer volume is an oxygen free copper
cap with twenty gold plated 99.999% pure copper wires (Figure 3.29) brazed in. Nineteen
wires (each 20 cm long) are extended onto the outside of the acrylic cell, where grooves
are made to house all of them (Figure 3.30) to increase the thermal contact between the
copper wires and the acrylic. Vacuum grease is also applied between the acrylic and the
wires afterwards. A Ruthenium oxide (RuO2 ) sensor is thermally connected to the top of
the acrylic cell to monitor the cell’s temperature. The Model 340 cryogenic temperature
controller from Lake Shore Cryotronics, Inc. is used as the readout device. The reason why
the acrylic cell cannot be closer to the mixing chamber is due to the size of the NMR-AFP
RF coils. The acrylic cell needs to be in the center of the Helmholtz RF coils, and the
65
Figure 3.27: Dilution refrigerator unit. It consists of the 1 K pot, the Still, the
50 mK plate, the mixing chamber and a series of heat exchangers.
copper buffer volume needs to be as far from the RF coils as possible to minimize the
eddy current heating during the measurement. Figure 3.30 also shows a small copper piece
attached to the 1 in. copper transition piece (between the acrylic cell and pyrex capillary)
66
Figure 3.28: Dilution refrigerator mixing chamber and copper buffer volume.
to house one extra cooling wire from the buffer volume and another RuO2 temperature
sensor.
Moreover, a smaller pyrex tube (0.25 mm diameter, 1 mm long) is positioned in the
6 mm × 2 mm pyrex tube right above the copper buffer volume to reduce the 4 He film
flow since the film flow effect is proportional to the inner circumference of the tube [77,
124]. Since superfluid 4 He has a high thermal conductivity and tends to flow to high
temperature regions [124], the 4 He film will thermally connect the high temperature and
the low temperature regions, thus bringing in a lot of heat load to prevent from cooling to
67
Figure 3.29: Gold plated 99.999% pure copper wires and the NMR RF coil.
lower temperatures. This is why the film flow needs to be restricted. By incorporating all
the modifications described above, the acrylic cell is successfully cooled to ∼416 mK filled
with superfluid 4 He.
3.4.3
Magnet System
In the 1.9 K test, liquid helium vapor is continuously pumped to reduce the sample temperature. However, it is not possible to reach below 1 K using this method. Instead, a
dilution refrigerator is used to cool the acrylic cell to 400 mK. Since it sits in a long and
heavy cylindrical dewar, the 1.9 K test magnet system will not fit. An eight-coil (33 in.
diameter each, 16.5 in. separation) cylindrical magnet system is assembled to produce a
vertical uniform magnetic field of ∼7 G along the dilution refrigerator (Figure 3.23). The
coils are numbered 1 to 8 starting from bottom and up. Figure 3.32 shows that a smaller
magnetic field gradient can be obtained by applying a larger current in the outer two coils
68
Figure 3.30: Grooves on the acrylic cell to house the cooling wires separately. A
small copper piece is attached to the 1 in. copper transition piece to house one cooling
wire and a temperature sensor.
(coil #1 and coil #8).
In order to compensate for the edge effect, coil #1 and #8 are
controlled by one power supply (Kepco ATE 36-30M), and the other six coils are controlled
by a separate power supply of the same model. Each power supply is controlled by one
function generator’s output offset, and the 1.9 K test NMR-AFP LabVIEW program is
rewritten to control these two function generators separately to implement the ramping
of the holding field. After optimizing the capabilities of the power supplies, it is decided
that (Figure 3.33) 1.61A flow through the inner six coils and 2.20A current flow through
the outer two coils. During the NMR-AFP field sweeping process, the ratio of the currents
in the outer two and inner six coils still remains the same(∼1.37). This configuration will
produce ∼7 G magnetic field at the acrylic cell position with the magnetic field gradient in
z-direction smaller than 15 mG/cm.
The magnetic field used for polarizing 3 He (Figure 3.34) is made up of two 58 in.
69
Figure 3.31: A 0.25 mm diameter capillary tube is positioned above the copper
buffer volume to help reduce the superfluid 4 He film flow.
Figure 3.32: 8 coil magnetic field simulation. The left graph is using the same
current in all 8 coils. The right graph is using larger current in the outer two coils
than that in the inner 6 coils. Iouter2 ∼ 1.37Iinner6 .
diameter, 2.5 in. thick coils, supported by PVC pipes. The portable magnetic holding field
is made of yellow styrofoam (Figure 3.35) with a diameter of 24 in. and a height of 12 in..
70
Figure 3.33: The three curves correspond to three different currents in the outer two
coils. The black curve gives the smallest magnetic field gradients in the z direction
close to the edge of the 8-coil system.
Each coil has 250 turns of gauge 22 (0.644 mm diameter) copper wires. A 2” diameter hole
is made at the center of the coils to hold the detachable cell at the center of the portable
magnetic field. The total resistance of the coils is ∼71 ohms. Three car batteries are used
to power this Helmholtz coil and everything is put onto a cart for easy transport. The
coil is separated from all magnetic materials on the cart by at least one meter to minimize
the magnetic field gradients. With a current of ∼0.5 amp in the coils, the magnetic field
strength at the center is ∼7 G and very stable as long as the batteries are in good working
condition. The directions of the 8-coil magnetic field, the portable magnetic field and the
optical pumping magnetic field should be the same in order for the polarized 3 He in the
71
Figure 3.34: The polarization station for polarizing the 3 He in a detachable cell.
Typical holding field is ∼20 G.
detachable cell to maintain polarization during the transfer process.
Figure 3.35: Portable magnetic field powered by car batteries to hold the polarization of the detachable cell.
72
3.4.4
NMR System
NMR-AFP technique is used to measure the 3 He polarization signal. As shown in Figure 3.36, two 9 in. diameter NMR coils are mounted in the inner vacuum chamber (IVC)
of the dewar, below the copper buffer volume to avoid the eddy current heating in the
copper while the RF field is turned on. The acrylic cell is positioned in the center of the
RF coils with the cooling wires and the glass to copper seal as the support structures. A
pickup coil (0.75 in. diameter, 250 turns) is glued onto the outside of the acrylic cell and
finely adjusted so that the RF background noise is minimized with the IVC closed. The
NMR-AFP circuit diagram is almost the same as the diagram shown in Figure 3.17 except
that two Agilent 33120A function generators are now being used to control the outer 2
coils and inner 6 coils separately with the currents’ ratio of ∼1.37. A series of NMR-AFP
measurements are performed to measure the 3 He longitudinal relaxation time.
Table 3.3 lists the NMR-AFP parameters of the measurements in the 400 mK test.
Table 3.3: Polarized 3 He NMR-AFP measurements’ parameters.
3.4.5
Parameter
Value
Holding B-field
7 Gauss
RF field magnitude
∼0.08 Gauss
RF field frequency
33 kHz
B-field ramping range
7→13→7 Gauss
Pre-amp magnification
100
Time interval between AFP sweeps
70-225 seconds
Narrowed Laser and Optics
Figure 3.37 shows the optical setup for the 3 He polarizing station. The configuration is
also the same as Figure 3.18. The 60 W laser used in the 1.9 K test was borrowed by
73
Figure 3.36: The NMR-AFP RF coils around the acrylic cell. The pickup coil is
behind the cell.
another experiment going on at the same time, so a 40-watt narrowed laser diode driven by
a Model 5600 laser diode driver and cooled by a Model 3150 temperature controller from
Newport was used instead to polarize the detachable 3 He cell. Even though the narrowed
laser’s overall power is smaller than the Coherent diode laser (60 W), with a line width of
∼0.2-0.3 nm (FWHM), the power is more focused on the Rb D1 transition. One special
polarization preserving mirror3 is used to reflect the horizontal laser beam by 90 degrees
so that the incident laser is parallel to the vertical magnetic field direction. The type of
3
On loan from Jian-Ping Chen at Jefferson Lab.
74
Figure 3.37: The optical setup for the polarizing station in Figure 3.34. Configuration is the same as Figure 3.18.
material, the thickness, the number of layers of the dielectric coating on the mirror are
carefully chosen so that the coating can introduce additional phase to the laser beam’s
electric field vector to compensate for the phase change after reflection. In this way the
circular polarization of the light is preserved.
3.4.6
Experimental Procedure
First, the acrylic cell is made and coated with the dTPB-dPS material. Stycast 1266 is used
to connect the glass-copper seal to the top of the acrylic cell. A series of strict leak tests
(most time-consuming) are done after the cell is in position and the dewar’s four layers are
put on one by one. Cooling the DR will take at least one week’s time.
The detachable cell is filled up with ∼1.5 atmosphere of 3 He and ∼100 torr of N2 and
being polarized in the laser lab overnight. Once the dTPB-dPS coated acrylic cell is cooled
by the dilution refrigerator mixing chamber to ∼400 mK, the calibrated volume mounted
75
on the gas handling system is filled up with gaseous 4 He. The metering valve is then opened
slowly to let the 4 He gas go through the liquid N2 trap and then cooled by the DR unit in
the dewar. The gas condenses into superfluid 4 He and fills up the acrylic cell slowly. This
process usually takes hours.
After the acrylic cell is filled with the desired amount of superfluid 4 He, the 3 He detachable cell which is being polarized at ∼190 degrees Celsius is cooled down to room
temperature and brought over to the top of the DR using the portable magnetic field dedicated for transporting the polarized 3 He detachable cell. A small region is opened to the air
while connecting the 3 He detachable cell to the top of the double-cell system. Pumping this
region to vacuum is necessary before opening the glass valve separating the two cells. After
the pressure of that region goes below 1 × 10−6 torr (which takes about 20-30 minutes),
the valve will then be opened to allow the polarized 3 He atoms to diffuse to the bottom
acrylic cell at the low temperature. A set of RF and pickup coils for the low temperature
NMR system in the inner vacuum chamber are then used to perform a series of NMR-AFP
measurements to measure the 3 He longitudinal relaxation time (T1 ).
76
Chapter 4
Results and Data Analysis
Chapter 3 explained the details of the experimental setup of the 1.9 K test and the 400
mK test. In this chapter, I will present the results and data analysis of the depolarization
behavior of spin polarized 3 He in a mixture of 3 He-4 He at temperatures below the 4 He λ
point in an acrylic cell coated with dTPB-dPS material.
As discussed in Chapter 3, the depolarization probability is the probability of losing
the polarization of a 3 He atom when it hits a particular surface. From Equation 3.15, the
depolarization probability Pd is defined as:
1
1
S
4
= v̄ · Pd ·
⇒ Pd =
T
4
V
T v̄ · S/V
where
S
V
(4.1)
is the surface to volume ratio.
The 1.9 K test results are first analyzed in a relatively simple way. The diffusion
equation
∂M
∂t
= D∇2 M (~r, t) is solved in cylindrical coordinates with the diffusion coefficient
of polarized 3 He in superfluid 4 He D = 2.4 × 10−4 cm2 /s [125, 126]. The 3 He depolarization
probability in the 4 He vapor and the 3 He relaxation time constant on the wall under the
superfluid 4 He are obtained. The next step taken in analyzing the data is to use a software
package COMSOL Multiphysics which uses finite element method (FEM) to numerically
solve the diffusion equation. The corresponding boundary conditions are imposed and
depolarization probabilities of polarized 3 He in the vapor and superfluid are determined by
fitting the simulation results to the experimental data points.
The diffusion equation does not have to be solved to analyze the 400 mK test results
because the 3 He diffusion coefficient is D ∼ 730 cm2 /s [127] in superfluid 4 He at ∼400
mK, which is 6 orders of magnitude larger than that at 1.9 K. Thus the system reaches
equilibrium within a very short time. A sufficiently long relaxation time of polarized 3 He
under nEDM experimental condition is extrapolated from the result and steps are taken to
77
further improve it.
4.1
1.9 K Test Results
The NMR signal during the AFP sweep (Figure 3.5) are fitted to Equation 3.23:
< M > B1
nmr
+ m · B0 (t) + C
S3He
(t) ∝ q
~ 0 (t) − ω/γ)2 + B 2
(B
(4.2)
1
The fitted value < M > from the sweep up and sweep down are averaged to be the NMR
signal from this sweep.
3 He
relaxation times are then extracted by fitting this data as
a function of time to an exponential decay form with corrections for AFP spin flipping
inefficiency.
AN = A0 (1 − x)N e−tN /T
(4.3)
P
af terAF P
. Pbef oreAF P
where x is the signal loss from each AFP spin flip, defined as x = 1 − Pbef
oreAF P
and Paf terAF P are the NMR signals before and after one AFP sweep. T is the relaxation
time and N is the number of AFP sweeps. The AFP spin flip loss x can be determined
by doing a consecutive 4-5 AFP sweeps, which in total takes about 1 minute. During this
time, the polarization loss is mainly due to the AFP measurements because the relaxation
time is on the order of hours. By taking the average of the signal differences between these
sweeps, the AFP loss is determined to be ∼ (1 ± 1)% at 1.9 K.
The relaxation times from the top glass cell (pumping chamber) made of aluminosilicate glass (GE180) are measured at room temperature before the low temperature measurements. T1 varies from 5980 and 6700 seconds. As can be seen from Figure 3.10, it is
an open-cell system which has more chances for the impurities to sneak in and does not
perform as good as a sealed cell (one of the sealed cells reaches a T1 ∼700 hours). It is
not a big issue because the 3 He will be let into the low temperature region immediately
after the pumping chamber is cooled to room temperature and little polarization is lost.
By comparing with the water signal at the pumping chamber position using Equation 3.31,
the absolute polarization of the 3 He is ∼ 10%, which is sufficient for our purpose.
78
The 21-in.-long pyrex capillary tube connecting the two cells (the pumping chamber and
the acrylic cell) has an ID of 3 mm. The amount of 3 He in the capillary tube is calculated
to be ∼ 5 × 10−6 mole assuming a linear temperature gradient from the acrylic cell to the
top. Compared with the total amount of 0.00139 mole of 3 He, this number is negligible.
Measurements are then carried out with a dTPB-dPS coated acrylic cell at 1.9 K.
The statistical error of the relaxation time is determined by minimizing the reduce χ2 of
fitting the AFP signal to Equation 4.3. The systematic errors, including the fluctuation
of the temperature, the amount of 3 He gas in the glass capillary, external RF noise, lockin amplifier and pre-amplifier’s accuracies and noises, NMR signal loss in the wire, are
mostly built into the AFP loss parameter x in the same equation, which is the dominant
source of uncertainty of the relaxation time because the index N in Equation 4.3 becomes
big if many AFP sweeps are carried out. It is found that the relaxation time of 3 He is
consistently shorter than 10 seconds with no 4 He inside the cell at a temperature of around
1.9 K. Basically no 3 He signal can be detected after it is lowered into the acrylic cell,
which means the bare dTPB-dPS coated acrylic surface is not 3 He “friendly”. After we
have started introducing more and more 4 He into the cell, a strong correlation between the
3 He
relaxation time and the amount of 4 He introduced is observed. Table 4.1 shows the
main results of measured 3 He relaxation times with a 21 G holding field from a dTPB-dPS
coated acrylic cell at ∼1.9 K. In Figure 4.1, the amount of 4 He is varied from 0.067 to
1.076 mole during these measurements while the amount of 3 He introduced into the acrylic
cell is fixed at 0.0014 mole. Figure 4.2 shows the measured 3 He relaxation time versus
the amount of 3 He in the cell for a fixed amount (0.404 mole) of 4 He. The results show
that the relaxation time (∼1700-1800 s) is almost independent of the amount of 3 He in the
range of the measurements (0.00056 mole to 0.0086 mole). The longest 3 He relaxation time
obtained at ∼ 1.9 K from the dTPB-dPS coated acrylic cell is 3152 ± 86 (statistical) ±
473 (systematic) seconds when the cell is filled with 2.71 cm high superfluid 4 He.
The main contributions to the depolarization of 3 He atoms are the dipole-dipole relaxation, the magnetic field gradient effect, and the surface effect at the wall explained in
79
Table 4.1:
3
He relaxation time measurements at ∼1.9 K. The error bars are the
quadrature sum of the statistical and systematic uncertainties.
4
He amount(mole)
4
He height(cm)
3
He amount(mole)
3
He relaxation time(s)
0.0675
0.17
0.00139
630±36
0.135
0.34
0.00139
855±73
0.269
0.68
0.00139
1200±72
0.404
1.02
0.00056
1768±146
0.404
1.02
0.00139
1637±132
0.404
1.02
0.00413
2130±508
0.404
1.02
0.00139
1708±150
0.538
1.35
0.00139
2580±590
0.673
1.69
0.00139
2301±249
0.808
2.03
0.00139
2715±398
1.076
2.71
0.00139
3152±480
Section 3.2.2. In the 1.9 K test, the total amount of 3 He in the cell is 0.00139 mole and
the cell’s volume is 54 cm3 . The 3 He number density is therefore 1.55 × 1019 /cm3 , which is
0.0577 amagat. From Figure 3 in [128], the dipole-dipole relaxation time for a 10 amagat
polarized 3 He is ∼36000 s. Since the dipole-dipole relaxation time is inversely proportional
to the 3 He number density, the calculated dipole-dipole relaxation time is 6.24 × 105 s in
our case. The relaxation rate is the inverse of this number. As for the magnetic field gradient effect, Section 3.3.3 shows that the calculated relaxation time due to the gradient is
5.26 × 105 seconds, which is two orders of magnitude larger than the longest measured 3 He
relaxation time in the acrylic cell (and so is the dipole-dipole relaxation time). Therefore,
we conclude that the surface effect on the acrylic walls is the dominant contribution to the
80
Figure 4.1: The relaxation time of polarized 3 He as a function of the amount of 4 He
in the measurement cell at a temperature of ∼1.9 K.
Figure 4.2: The polarized 3 He relaxation time as a function of the amount of 3 He.
The amount of 4 He is held constant at 0.404 mole.
3 He
relaxation time in the 1.9 K test.
The initial improvement observed in the 3 He relaxation time shown in Figure 4.1 can be
attributed to the formation of a superfluid 4 He film [95] on the dTPB-dPS coated acrylic
wall, which tends to expel the 3 He atoms away from the surface. However as the liquid 4 He
level becomes higher, the 3 He diffusion time in superfluid 4 He cannot be ignored because it
81
takes time for 3 He to diffuse to the walls where the depolarization takes place. Hence the
measured 3 He relaxation time is due to the convolution of the 3 He longitudinal relaxation
time and the diffusion time. In the next section, the relaxation time data is analyzed
using two models, from a relatively simple one to a more complicated diffusion model,
to understand the depolarization behavior of the polarized 3 He as the amount of 4 He is
increased.
4.2
4.2.1
1.9 K Test Data Analysis
Model I
As we introduce more and more 4 He into the acrylic cell, the 4 He atoms liquefy and collect
at the bottom of the cell. The liquid level ranges from 0.17 to 2.71 cm shown in Table 4.1.
To extract the depolarization parameters of the wall effect, one must know exactly the
amounts of 3 He in the vapor phase and in the liquid phase. At 1.9 K, the fraction of 3 He
atoms that are in the liquid phase increases rapidly as the amount of 4 He is increased. The
vapor pressure of 3 He-4 He mixtures as a function of molar concentration of 3 He in liquid
state (XL ) is a well measured quantity [129, 130]. Using Equation (2) and Table III in [129],
the vapor pressure can be extracted using the the densities in our experiment. Then the
ratio of 3 He atoms that are in the liquid phase compared to those in the vapor phase are
numerically calculated to range from 0.6 to 22.9 as the liquid 4 He level increases from 0.17
to 2.71 cm. Table 4.2 shows the calculated numbers. The total amount of 3 He is 0.00139
mole. Very close results can be calculated using the equation that the 3 He densities satisfy
at equilibrium [131]:
nv3
m 3
−EB
= ( ∗ ) 2 exp(
)
l
m
kT
n3
(4.4)
where m∗3He = 2.4m3He [132] is the effective mass of 3 He dissolved in superfluid 4 He and
EB = 2.8k is the solvation energy for 3 He in liquid 4 He, where k is the Boltzmann constant.
Using these results, the functional relationship nl3 (h) between the number of moles of 3 He
82
Table 4.2: The amounts of 3 He in the vapor and in the liquid.
4
He height(cm)
3
He in vapor nv3 (mole)
3
He in liquid nl3 (mole)
nl3 /nv3
0.17
0.000860
0.000530
0.617
0.34
0.000587
0.000803
1.369
0.68
0.000345
0.00104
3.027
1.02
0.000233
0.00116
4.974
1.35
0.000169
0.00122
7.244
1.69
0.000126
0.00126
9.995
2.03
0.0000968
0.00129
13.355
2.71
0.0000582
0.00133
22.903
atoms in the liquid phase (nl3 ) and the height of liquid 4 He in the cell (h) can be extracted.
And the corresponding number of 3 He atoms in the vapor is nv3 (h) = 0.00139 − nl3 (h).
From [131], the 3 He atoms’ exchange rate at the superfluid 4 He surface is calculated
to be on the order of 1023 /s, which is much larger than the total amount of 3 He. So the
3 He
amounts in the vapor and liquid can reach the right number shown in the Table 4.2 in
a short time. Since the approximate time required for diffusion over a given distance h is
∼h2 /2D [133] (D = 2.4 × 10−4 cm2 /s [125, 126] is the diffusion coefficient), the estimated
diffusion time of 3 He from the top of the liquid surface to the bottom ranges between ∼ 60
and 15300 seconds as the height of the liquid increases. Therefore, the system is not in
equilibrium for most of the measurements.
The net relaxation rate can be written as:
1
1
1
=
+
T1 (h)
Tvap (h) Tliq (h)
(4.5)
which is simply the sum of the relaxation contribution from the vapor and the liquid. In
the vapor phase the 3 He atoms depolarize by diffusing through the thin superfluid film and
depolarize at the top and the side walls that are above the liquid 4 He. The relaxation rate
83
for the 3 He atoms in the vapor from the wall can be written in terms of the depolarization
probability Pv :
1
Tvap
where N =0.00139 mole, the v̄3 =
=
q
1
Sv nv3 (h)
·
· Pv · v̄3 ·
4
Vv
N
8kB T
πm∗3
(4.6)
is the average velocity of the 3 He atoms. Sv is
the surface area above the liquid and Vv is the volume of the vapor. For a cylindrical cell
of length H and radius R, when filled with liquid 4 He up to height h from the bottom of
the cell,
Sv
Vv
=
2
R
+
1
H−h
(Figure 4.3).
Figure 4.3: Illustration of model I. Depolarization probability in the vapor is Pv .
Surface relaxation time constant in the liquid is Ts .
Define the 3 He-4 He interface to be z = 0, the bottom wall is where z = h. R=1.84 cm is
the radius of the cell, H=5.08 cm is the total height of the cell. If the 3 He magnetization in
the liquid 4 He is M, the diffusion equation in cylindrical coordinates is (no φ dependence):
∂M
1 ∂ ∂M
∂2M
= D(
(r
)+
)
∂t
r ∂r ∂r
∂z 2
(4.7)
By separating the variables, its solution is:
M (r, z, t) =
∞ X
∞
X
Am J0 (βm r) cos(kn z)e−λ
n=1 m=1
84
2 Dt
(4.8)
2 . Using the initial condition M (r, z = h, t = 0) = 0, which means the 3 He
Here λ2 = kn2 +βm
magnetization on the bottom of the cell in the beginning is zero, we have kn = (n − 12 ) πh .
And M (r = R, z, t = 0) = 0, which means the 3 He magnetization on the side walls under
the liquid is zero, we have βm = χm /R, where χm is the mth root of the Bessel function
J0 (x).
Using another initial condition M (r, z = 0, t = 0) = n0 , which means at the liquid
surface, the initial magnetization is M0 :
M0 =
∞
X
Am J0 (βm r)
(4.9)
m=1
We have:
Am =
2M0
2
R J12 (βm R)
R
Z
rJ0 (βm r)dr
(4.10)
0
So the total solution is,
Z R
∞ ∞
2
2 )t
2M0 X X J0 (βm r)
1 π
−D( 2.42 +((n− 12 ) π
)
h
R
M (r, z, t) =
cos((n − ) z)e
rJ0 (βm r)dr
R2
2 h
J 2 (βm R)
0
n=1 m=1 1
(4.11)
In the first order approximation, n = 1, m = 1, R = 1.84, χ1 = 2.40:
Z 1.84
2
2M0 J0 (1.30r)
π
−D(1.70+ π 2 )t
4h
z)e
cos(
rJ0 (1.30r)dr
R2 J12 (2.4)
2h
0
π2
π
= 1.61 · M0 · J0 (1.30r) · cos( z) · e−D(1.70+ 4h2 )t
2h
M (r, z, t) =
Here M0 is the initial 3 He magnetization and D(1.70 +
3 He
π2
)
4h2
(4.12)
(4.13)
can be thought of as the
relaxation rate in liquid 4 He, which is basically the inverse of the average diffusion
time to the walls if we assume that the walls are completely absorptive (a 3 He atom loses
polarization when a wall is hit). But in practice the walls are not completely absorptive,
which requires that we take into account the surface relaxation time on the wall, Ts [134].
So the total relaxation rate due to diffusion and the subsequent relaxation on the wall in
superfluid 4 He is:
1
1
D(1.70+
π2
)
4h2
85
+ Ts
(4.14)
So the total relaxation rate is:
1
1
Sv nv3 (h)
= · Pv · v̄3 ·
·
+
T1
4
Vv
N
1
1
D(1.70+
π2
)
4h2
+ Ts
·
nl3 (h)
N
(4.15)
The above equation is fitted to the existing data points and the best fit we obtained
has Pv = (6.45 ± 0.49) × 10−7 and Ts = 993 ± 306 seconds, respectively. The fitting of the
curve is plotted in Figure 4.4, in which the fitted value Ts has a big error. The shape of
Figure 4.4: The fitting of Equation 4.15 to the data points. Pv is the depolarization
probability of 3 He in the vapor. Ts is the relaxation time constant on the wall below
the liquid surface.
the fitted curve is mainly determined by Pv and is not sensitive to the surface relaxation
time Ts under the liquid due to the relatively long diffusion times.
This simple model has shortcomings. Whether Equation 4.14 is correct or not is yet
to be confirmed. Also the model just presents the overall behavior of the polarized 3 He in
the superfluid, not the exact signal in the pickup coil read by the lock-in amplifier. Hence
a time-dependent diffusion model is necessary to simulate the 3 He signal in the pickup coil
in our experiment.
86
4.2.2
Model II
The major part of the content and the results from Model II have been published in [2].
Here a more detailed analysis is presented.
Model Description
One can set up all kinds of boundary conditions and initial conditions in the software
package COMSOL Multiphysics and it can use the finite element method to numerically
solve the differential equations. This diffusion model assumes that all 3 He atoms are in the
vapor state immediately after the 3 He atoms enter the acrylic cell. So the initial 3 He density
is assumed to be uniform in the vapor and zero in liquid 4 He. 3 He atoms then start diffusing
both in the vapor and liquid, in which the diffusion coefficients are different. At 1.9 K,
Dl = 2.4 × 10−4 cm2 /s [125, 126] is the diffusion coefficient of 3 He in liquid 4 He. The 3 He
diffusion coefficient in 4 He vapor is calculated using Dv = 1.463 × 10−3 · T 1.65 · P −1 = 0.018
cm2 /s [135]. The boundary condition at the liquid surface is written using the flux exchange
between the vapor and liquid. The flux going from the vapor to the liquid is:
1
|~jvl | = cv vv
4
(4.16)
EB
vv m
|~jlv | = ( ∗ )3/2 e− kT cl
4 m
(4.17)
and in the opposite direction [131]:
cv and cl are the concentration of the polarized 3 He atoms in the vapor and liquid, req
8kT
4
spectively. The average speed of 3 He in the vapor is vv =
πm3 = 1.15 × 10 cm/s.
Other parameters are the same as those in Equation 4.4. The pickup coil is mounted at
the bottom of the acrylic cell and measures the change of the magnetic flux caused by the
spin-flip of the 3 He magnetic dipoles in the cell (both in the vapor and in the liquid) during
an NMR-AFP sweep. The pickup coil flux is usually calculated by integrating the flux
generated by all the 3 He magnetic moments within the area circled by the pickup coil. But
this method is slow and involves loops of integration. In order to calculate this flux faster
87
and easier, we use the reciprocity theorem (see Appendix F). Using this method, the flux
going through the pickup coil can be calculated as proportional to the integration of the
magnetic fields produced by a current in the pickup coil at the location of the 3 He dipole.
Figure 4.5: Diffusion model II of the 1.9 K test. The depolarization probabilities
on the walls in the vapor and liquid are two parameters that can be varied.
The wall depolarization effect in our measurements is best characterized by the depolarization probability (DP) per wall collision. In this analysis we allow this probability to
be different on the walls covered with bulk liquid 4 He, Pl , and the walls covered with superfluid film only, Pv (Figure 4.5). The wall boundary condition is that the depolarization
rate on the wall is the product of the number of atoms reaching the wall per unit time and
the corresponding DP. In the 4 He vapor, the boundary condition on the cell top and the
cell side is:
n̂ · Dv ∇cv = −|~jvl |Pv
(4.18)
where n̂ · D∇c is the normal diffusive flux. The boundary condition on the vapor-liquid
surface from above the surface is:
n̂ · Dv ∇cv = |~jlv | − |~jvl |
88
(4.19)
In the liquid 4 He, the boundary condition on the cell bottom and the cell side is:
n̂ · Dl ∇cl = −|~jlv |Pl
(4.20)
And the boundary condition on the vapor-liquid surface from beneath the surface is:
n̂ · Dl ∇cl = |~jvl | − |~jlv |
(4.21)
So the depolarization behavior of the model can be changed by varying the Pv and Pl
parameters.
After all those parameters described above are inputed into the simulation program,
the diffusion equation
∂cv,l
+ ∇ · (−Dv,l ∇cv,l ) = 0
∂t
(4.22)
is solved numerically for a set of Pv and Pl values both in the vapor and in the liquid. Here
cv,l is the concentration of polarized 3 He either in the vapor or liquid. The program also
calculates the magnetic field in the cell produced by a current in the pickup coil, performs
an integration of the product of the magnetic field and the polarized 3 He atoms’ magnetic
R
~ · µ3He
moments over the entire cell volume cell cv,l B
~ . The integrated result is proportional
to the NMR signal produced in the pickup coil by the polarized 3 He atoms in the entire
acrylic cell according to the reciprocity theorem (Appendix F). Each set of Pv and Pl
will generate a time-dependent pickup coil signal curve. These simulated signals are then
fitted to the NMR-AFP measured data points (starting from injection of the polarized 3 He
into the 4 He liquid) using the least square fits (minimizing the reduced χ2 ) by varying the
proportionality constant. For each data set of a certain 4 He height, thousands of simulations
are carried out in order to find the best set of Pv and Pl values.
Data Analysis
Figure 4.6 (a) shows a typical simulation result. The green triangles are the NMR data
points for that measurement. The red squares are the simulated 3 He precession signal in
the pickup coil. The purple circles are the contribution from the 3 He in the liquid and the
blue triangles are the contribution from the 3 He in the vapor.
89
Because 3 He atoms dissolve into the liquid 4 He rapidly without losing polarization, the
signal increases from zero in the beginning of the measurement and then decays after it
saturates. Figure 4.6 (a) and (b) shows the trends of the total 3 He signal, 3 He signal in
vapor and 3 He signal in the liquid. Pv value will influence the short time buildup of the
signal in the pickup coil, and Pl will determine the long time behavior. This means that
in the graph, a larger Pv and smaller Pl will move the peak of the signal (red squares) to
the left. In the measurements made with small amounts of 4 He, the long time behavior is
Figure 4.6: (a) and (b) are NMR measurements of the 3 He signal (green triangles)
at 1.9 K as a function of time with the amount of 4 He equal to 0.135 mole (0.34
cm). Red squares are the simulated total signal in the pickup coil consisting of the
contributions from the vapor (blue triangles) and liquid (pink circles). (c) is reduced
χ2 obtained from the best fit as a function of Pv (red circles, top axis) and Pl (black
squares, bottom axis) showing how different values of Pv and Pl can fit the data due
to the fact that with low liquid level, the 3 He atoms in the vapor is close to the pickup
coil.
influenced by both Pv and Pl , making it difficult to extract unique values of the parameters
from these data because big Pv and small Pl or small Pv and big Pl can lead to the fittings
of the curve with the almost the same reduced χ2 . As an example, Figure 4.6 (a) and
(b) show the varying contribution of the 3 He in the liquid (pink circles) and vapor (blue
triangles) to two equally good fits (red squares and green triangles) for Pv,l varying by
90
about a factor of 10. Figure 4.6 (c) shows the plot of reduced χ2 obtained from the best
fits as a function of Pl (bottom axis) and Pv (top axis).
Figure 4.7: The amount of 4 He in the acrylic cell is 1.08 mole (2.71 cm). For
Pv = 1 × 10−9 and Pv = 1.21 × 10−7 , equally good fits can be obtained.
Because the pickup coil is located on the bottom of the measurement cell, it is more
sensitive to 3 He dissolved in the liquid. It becomes less sensitive to the 3 He in the vapor as
the amount of 4 He is increased. For the measurements with high 4 He levels, the results are
not sensitive enough to Pv to allow the extraction of a value for this quantity. Figure 4.7
shows two equally good fitting curves with Pv values two orders of magnitude in difference.
The amount of liquid 4 He in the cell is 1.08 mole.
Since the NMR-AFP technique only measures the relative 3 He polarization, the fitting
is made more difficult due to the lack of absolute polarization information in the acrylic cell.
The normalization of the curve needs to be treated as a free parameter. This is another
reason that a range of parameters can give good fits in the low filling cases (when the 4 He
amount is small). In addition, our operational procedures were such that in most cases
data taking started after the peak had been passed. Only the measurement with 0.673
mole (1.69 cm) of 4 He shows the peak of the signal (Figure 4.8), and we are able to extract
−7
a reasonable value of Pl from the fit shown in Figure 4.9 (a), Pl = (3.9+2.0
−0.7 ) × 10 . The
error bar is determined by the standard method of varying the Pl parameter so that the
reduced χ2 is increased by 1 [89].
91
Figure 4.8: The amount of 4 He in the acrylic cell is 0.673 mole (1.69 cm). The
experimental data (green triangles) consisting of the contributions from the vapor
(blue triangles) and liquid (pink circles) are fitted to the simulation results (red
squares).
For measurements with larger amounts of 4 He, diffusion to the bottom wall plays a
more significant role (it takes ∼15300 seconds for the 3 He to diffuse to the bottom wall in
the 2.71 cm case) than the relaxation on the cell surface. For instance, in Figure 4.7, the Pl
values change from 17.4×10−6 to 32.3×10−6 , the reduced χ2 values change little. Therefore
it is hard to extract meaningful values of Pl from the data from the measurements with
high 4 He levels. Figure 4.9 (a) also shows the reduced χ2 plots for the measurements with
0.404 mole (1.02 cm) and 0.538 mole (1.35 cm) of 4 He. From these plots, we can extract
Pl = (1.7 ± 0.2) × 10−7 and Pl = (1.6 ± 0.4) × 10−7 respectively. The minima in reduced
χ2 are much broader when plotted versus Pv (Figure 4.9 (b)). For the measurements with
−7
0.404 mole (1.02 cm) 4 He, Pv ∼ (10+2
−1 ) × 10 . For the measurements with 0.538 mole
−7
4
(1.35 cm) 4 He, Pv ∼ (8+4
−3 ) × 10 . For the measurements with 0.673 mole (1.69 cm) He,
−7
Pv ∼ (4+2
−1 ) × 10 . In all, the most important results obtained in the 1.9 K test is the wall
depolarization probability for polarized 3 He in superfluid 4 He (Pl ) which is on the order of
(1 − 2) × 10−7 . The Pv values are on the order of 8 × 10−7 and the error bars are big.
92
Figure 4.9: Reduced χ2 for the fits versus Pl and Pv values for 4 He amounts of 0.404
mole (black squares), 0.538 mole (red triangles) and 0.673 mole (blue circles).
The neutron EDM experimental cell has a dimension of 7.6 cm × 10.2 cm × 50.0 cm and
its surface to volume ratio (S/V ) is ∼ 0.5 cm−1 . The average speed of 3 He atoms in liquid
q
8kB T
4 He at ∼400 mK (the approximate nEDM experimental temperature) is v̄ =
3
πm∗ =
3
3.4×103 cm/s. Using a Pl value of 1.7×10−7 and Equation 4.1, the extrapolated relaxation
time of polarized 3 He at nEDM experimental temperature is ∼ 1.4 × 104 s, which is much
longer than the measurement time ∼500 s. However the diffusion coefficient changes a lot
with temperature and the depolarization probability may change when the temperature is
below 1 K. It is thus necessary to extend the measurements to ∼400 mK.
No relaxation measurements of polarized 3 He on dTPB-dPS coated acrylic surface has
ever been carried out before but we can compare the result to some previous measurements
on pyrex surface to some extent. Lusher et al. [111, 95] carried out a series of measurements
with open pyrex glass chambers as well as sealed pyrex glass cells. Their results showed
that the formation of a superfluid 4 He film on a hydrogen coated glass surface reduces
the depolarization of 3 He from the surface. Since we are also using an open cell system,
the results from similar conditions can be compared (no hydrogen coating on our surface
though). For an open cell they observed a relaxation time of ∼500 seconds with a magnetic
93
holding field of 0.23 Tesla at a temperature of 1.9 K. The 3 He bulk number density for these
measurements was 5.2 × 10−6 mole/cc (cell volume 4.2 cc) and the 3 He : 4 He atomic ratio
was 1:16 (ours is 1:769). As shown in Figure 4.1 we have observed relaxation times in excess
of 3000 seconds at 1.9 K for a holding field of 21 G. The surface to volume ratio of our cell
is 50% of the cells used in measurements of [95, 111], and our measured relaxation time
is a convolution of 3 He T1 and the 3 He diffusion time. Their corresponding depolarization
probability is determined to be ∼ 1.9 × 10−7 , which is similar to our Pl value, though ours
is obtained from a dTPB-dPS coated acrylic surface under the superfluid 4 He liquid.
4.3
400 mK Test Result
In the 1.9 K test, the dewar can only maintain a stable temperature of ∼1.9 K for about
8 hours (4 He vapor is being pumped out) and the time is not enough to fill the acrylic
cell completely with superfluid 4 He and perform the following NMR measurement. Since
the neutron EDM experiment needs the cell to be completely filled with superfluid 4 He at
∼300-500 mK, it is necessary to extend the similar 3 He relaxation time measurements to
meet this requirement so that the results can be directly compared to the real condition.
Chapter 3.4 gives a detailed description of the 400 mK test experimental setup. The dilution
refrigerator can keep the measurement cell at ∼400 mK for as long as we need.
Before each cooling cycle, room temperature relaxation time measurements of the cylindrical acrylic cells are carried out. Since both the detachable cell and the measurement cell’s
temperatures are at 300 K, only about half of the 3 He atoms in the pumping chamber enter
the acrylic cell after the valve is opened. Figure 4.10 shows the 3 He relaxation time measurements at room temperature in two acrylic cells in 2007. The first acrylic cell used in
April, 2007 had a big leak at low temperature even though its 3 He T1 at room temperature
was longer. It was not possible to cool the cell down due to the leak. The cell was replaced
by another acrylic cell which had a shorter T1 (∼675 s) with an AFP loss ∼0.5%. After
around one week of cooling with the dilution refrigerator, this cell was cooled down to 0.53
94
Figure 4.10: 3 He relaxation time measurements at room temperature in two acrylic
cells.
K with 1.23 mole of superfluid 4 He. The 3 He relaxation time was measured to be ∼1263
seconds with an AFP loss of ∼0.5% at a holding field of ∼7 G.
After this measurement, many improvements were done to the cooling system, including:
• Adding a 0.25mm hole in the 6x2 mm glass capillary through which the 3 He is
introduced into the acrylic cell. This restriction is used to limit the superfluid 4 He
film flow to thermally connect the MC to the 50 mK plate, the Still and the 1 K pot
(Figure 3.31).
• Gold plating copper buffer volume’s bottom part and the copper wires (99.999%
pure) are designed to make better thermal conduction (Figure 3.29).
• Applying vacuum grease between the copper wires and the grooves on the acrylic cell
to make better thermal contact between the copper wires and the acrylic cell.
• Replacing the bulky acrylic clamp with tie-wraps in order to have less material to
cool down.
95
• Using one gold-plated wire directly from the MC buffer volume on to the glasscopper-acrylic transition and a temperature sensor is mounted on this copper wire
(Figure 3.30).
All these improvements were implemented to increase the cooling power from the MC
to the acrylic cell so that it can be cooled down to lower temperatures. However, a leak
was found between the copper-acrylic joint on top of the acrylic cell, and it was fixed by
adding more stycast 1266 epoxy and applying vacuum grease outside the stycast.
At room temperature, the 3 He relaxation times from the detachable glass cell on top of
DR is ∼40 hours at ∼7 G. With no 4 He in the acrylic cell, the T1 is ∼560 seconds with a
magnetic holding field of ∼7 G at 300 K. After the acrylic cell is cooled down (∼400 mK),
all of the 3 He atoms in the detachable cell enter the acrylic cell. With 4 He being introduced
into the acrylic cell at ∼400 mK, 3 He relaxation time also has a strong correlation with the
amount of 4 He atoms like in the 1.9 K measurements. Table 4.3 and Figure 4.11 show the
3 He
relaxation times with a 7 G holding field from a dTPB-dPS coated acrylic cell at ∼400
mK. The AFP spin flip inefficiency is determined to be (0.3±0.25)%. The amount of 4 He
Table 4.3: 3 He relaxation time measurements at ∼400 mK.
Run#
Cell Temp.(K)
3009
0.39
3015
4
3
He Amount(mole)
T1 (s)
0.4
0.0042
331 ± 51
0.39
1.28
0.0028
1101 ± 29
3019
0.42
2.08 (cell full)
0.0028
1606 ± 52
3032
0.416
2.2 (cell full)
0.0028
1666 ± 56
He Amount(mole)
is varied from 0.4 to 2.2 moles. The acrylic cell can hold up to 2 moles of superfluid 4 He,
and the extra 0.2 mole is added to ensure that the cell is full of superfluid 4 He (the extra
4 He
will be in the copper buffer volume, see Figure 3.28). The temperature range for these
measurements is from 0.39 K to 0.42 K. The longest 3 He relaxation time obtained from the
dTPB-dPS coated acrylic cell is 1666 ± 1.6 (statistical) ± 56 (systematic) seconds.
96
Figure 4.11: The relaxation time of polarized 3 He as a function of the amount of
4
He in the measurement cell at a temperature of ∼400 mK. The error bars are the
quadrature sum of the statistical (determined from the exponential fit) and systematic
uncertainties (determined from the AFP spin flip inefficiency).
4.4
400 mK Test Data Analysis
To estimate the dipole-dipole effect at 400 mK, from Figure 3 in [128], the dipole-dipole
relaxation time for a 10 amagat polarized 3 He is ∼12 hours at 0.4 K. The density of 3 He
in the acrylic cell is about 0.173 amagat, so the calculated dipole-dipole relaxation time is
∼ 2.5 × 105 s at ∼400 mK, which means the dipolar effect is negligible. The magnetic field
gradient in the vertical direction is measured using a Gauss Meter and it is smaller than 15
mG/cm. This will corresponding to a relaxation time of ∼ 2 × 106 s due to the gradient,
whose effect is even smaller than the dipole-dipole interaction. Based on the calculations
above, the wall depolarization effect is still the dominant source of relaxation in the 400
mK test.
At temperatures below 1 K, the diffusion coefficient D scales as T −7 [127]. D ∼ 7.3×102
cm2 /s at a temperature of ∼400 mK, which is much larger than that in the 1.9 K test. This
97
shows that the system goes to equilibrium very fast after the polarized 3 He is introduced
into the acrylic cell. A diffusion model in this case is not necessary and Equation 3.15 can
be used to obtain the depolarization probability of the cell wall directly.
The 1/T1 versus the S/V values are plotted in Figure 4.12. The surface area used in
the analysis is the surface wet by the superfluid 4 He. The red line which fits the three black
points has a large negative intercept with the y axis, which is unphysical. This indicates
that the model is not appropriate to describe the data.
Figure 4.12: The inverse of relaxation time of polarized 3 He as a function of the
S/V values. For black points, S is the surface area covered by superfluid 4 He (cell
bottom and cell side) and V is the volume of bulk 4 He. The red point also include
the top surface of the acrylic cell in the S.
From Figure 4.11, the 3 He relaxation time seems to be linear with the superfluid 4 He
amount (the volume of the 4 He). So an 1/T1 versus the S/V values graph will be likely
to go through the zero point if S (surface area) is kept as a constant. M. Hayden from
the Physics Department of Simon Fraser University in Canada points out that the surface
area should be the entire inner surface of the cell and the data can be reanalyzed shown in
Figure 4.13. His idea [136] is that the experimental cell is composed of three reservoirs: the
4 He
bulk liquid, the vapor and the film. At temperatures well below 1 K, most of the 3 He
98
Figure 4.13: The inverse of relaxation time of polarized 3 He as a function of the
S/V values. S is the entire inner surface area of the acrylic cell and V is the volume
of bulk 4 He. The red line is a fit of the data points.
1
T1
= (0.00041 ± 0.00000756) VS .
The blue circle is the extrapolated 1/T1 for nEDM cell geometry.
atoms stay within the bulk liquid. In equilibrium, the 3 He density satisfies the Equation 4.4
and
nv
nl
= 2.45 × 10−4 , which indicates that almost all of the 3 He atoms are in the bulk
liquid. The time scale for the exchange of 3 He between the liquid and the vapor is given
by [131]
τ=
4Vl m∗ 3
EB
(
) 2 exp(
)
Sv̄αvl m
kT
(4.23)
where S = πR2 (the free liquid surface area) and α ∼ 0.95 is the averaged probability that
a 3 He atom in the vapor hitting the vapor-liquid surface will go into the liquid 4 He. The
calculated time scale τ is short compared to the relaxation times. Thus the 3 He exchange
between the three reservoirs is rapid. The relevant surface area that should be used in the
analysis is thus the entire inner surface of the acrylic cell, rather than simply the area wet
by the bulk liquid. Figure 4.13 is a plot of 1/T1 verses the surface to volume ratio using the
inner surface of the entire cell for the four data points. From Equation 4.1, the slope of the
line is 14 v̄ · Pd . The depolarization probability comes out to be Pd ∼ (4.79 ± 0.09) × 10−7
using Hayden’s model.
99
When the cell is completely filled with superfluid 4 He (Run # 3032) at 400 mK, the
T1 extracted is 1666±56 seconds. The corresponding 3 He depolarization probability Pd ∼
(4.72 ± 0.16) × 10−7 . This is a model independent extraction of this quantity and it is on
the same order of magnitude of that extracted from the 1.9 K measurements. Moreover
it is in excellent agreement with the number derived from Hayden’s model, which gives us
confidence that the model does explain the overall 3 He behavior in the measurement cell
at 400 mK.
4.5
Discussion Of The 1.9 K & 400 mK Tests
Since the 3 He diffusion coefficient in liquid 4 He at 1.9 K is not big, the diffusion and
relaxation of polarized 3 He atoms happen at the same time in a dTPB-dPS coated acrylic
cell at 1.9 K. The diffusion model constructed to simulate the pickup coil signal shows
that it’s possible to achieve values of wall depolarization probability (Pl ) on the order of
(1−2)×10−7 for polarized 3 He in the superfluid 4 He at 1.9 K on a dTPB-dPS coated acrylic
surface. The two models in the 1.9 K data analysis give out similar numbers for the 3 He
wall depolarization probability in the 4 He vapor, which is on the order of (3 − 12) × 10−7 .
The neutron EDM experimental cell will be full of superfluid 4 He and its surface to
volume ratio (S/V ) is ∼ 0.5 cm−1 . So the extrapolated relaxation time of polarized 3 He
in the nEDM cell geometry is ∼ 4870 seconds at ∼400 mK (the blue circle in Figure 4.13).
The 3 He relaxation time needs to be much larger than the measurement time, which will be
close to the neutron lifetime in the acrylic cell (∼500 s). This extrapolated relaxation time
for the nEDM experiment is already several times longer than the measurement time. It is
a good number for the nEDM experiment and we anticipate improvements in the relaxation
time by improving the coating technique discussed below.
It is shown that it is possible to achieve values of wall depolarization probability of
(4.72 ± 0.16) × 10−7 for polarized 3 He in the superfluid 4 He at ∼400 mK. New dTPB-dPS
coating procedures are being developed (Appendix E). Figure 4.14 shows the Atomic Force
100
Figure 4.14: Two AFM images of the acrylic pieces using the old and new coating
procedures. The surface using the new coating method is much smoother.
Microscope (AFM) images of the acrylic cells using the old and new coating procedures. The
peak-to-valley height of the acrylic surface using the old coating method is ∼700 nm while
the number is reduced to only 16 nm using the new coating procedure.
3 He
relaxation
time measurements using rectangular acrylic cells with this new coating method will be
performed. It remains to be seen how sensitive depolarization probabilities are to surface
preparations. Measurements have shown that surfaces with smoother AFM images provide
better target performance both in terms of smaller depolarization probability from the wall
as well as smaller deuterium/hydrogen recombination probability from the wall [137]. This
needs to be demonstrated in the 3 He case. If the same improvement as described in [137]
can be achieved, the relaxation rate can be more than twice smaller, which means the
relaxation time can reach 104 s as proposed in the nEDM experiment pre-proposal [1].
101
Chapter 5
Conclusion and Future Outlook
Searches for the neutron electric dipole moment have been going on for nearly 60 years.
A non-zero nEDM is a direct violation of time reversal symmetry and has the potential of
revealing new sources of CP violation to explain the baryon asymmetry of the universe. In
this dissertation, the motivation for measuring the neutron EDM and an innovative nEDM
experiment has been presented. This nEDM experiment has the potential to measure the
nEDM or to lower the current experimental upper limit by two orders of magnitude, which
will put to the test possible theories of New Physics beyond the Standard Model.
The experiment requires the use of polarized 3 He to be the comagnetometer and to
generate scintillation light from reacting with polarized neutrons. Thus the 3 He polarization needs to have a sufficiently long relaxation time. My work is mainly focused on the
depolarization study of polarized 3 He under the nEDM experimental conditions. Chapter
3 and 4 describe such studies at two temperatures of 1.9 K and 400 mK in a cylindrical
dTPB-dPS coated acrylic cell. 3 He depolarization probabilities of (1 − 2) × 10−7 at 1.9 K
and (4.72 ± 0.16) × 10−7 at 400 mK are extracted. From these results, the extrapolated
relaxation time under the nEDM experimental conditions is ∼ 4870 seconds. This result
means that during the ∼500 s measurement period in the nEDM experiment, the 3 He will
only lose ∼10% (or below) polarization, which is sufficient for the experiment and further
improvement is anticipated using new coating techniques.
So far, all the measurements have been done using cylindrical cells. Appendix E describes the procedures for coating the cylindrical acrylic cell and the flat acrylic pieces with
dTPB-dPS material. Since the nEDM measurement cell is rectangular, a similar rectangular cell with smaller dimensions will be made and coated with dTPB-dPS material using the
newly developed “Swinging method”. Longitudinal relaxation time measurements will be
102
carried out using this cell under the nEDM experimental temperatures in the near future.
If the same improvement as described in [137] can be achieved, the relaxation time can
reach 104 s at the nEDM experimental temperature as proposed in the nEDM experiment
pre-proposal [1].
In the nEDM experiment, polarized 3 He will be entering from the atomic beam source
into the two measurement cells through the collection volume, the transfer tube and the
valves. Studying the relaxation mechanism on glass and acrylic is not enough. More
relaxation time measurements need to be carried out on different material surfaces so that
the right material can be selected to minimize the polarization loss during the polarized
3 He
transfer process.
In summary, this neutron EDM experiment is a challenging project and a lot of R&D
experiments are being carried out. The nEDM collaboration is on track in receiving the
approval of the experiment by the U.S. Department of Energy. The construction of the experiment will begin after the experiment is approved and preliminary results are anticipated
around 2014.
103
Appendix A
Geometric Phase Study at 300 K
A.1
Overview
The proposed nEDM sensitivity is ∼ 10−28 e·cm (Chapter 2). To achieve this goal, it is
important to identify and control the systematic errors that can induce a false electric dipole
~ field
moment signal. One of these systematic errors comes from the interaction of the ~v × E
with the external magnetic field gradients (also called the “Geometric Phase Effect” [3,
4]). It is basically the Berry’s phase [138] in the experiment, which is a pure geometric
effect [3] and hence the name. This interaction produces a frequency shift proportional
to the external electric field, just like an EDM signal. This effect has been pointed out
in the measurement of electron electric dipole moment (eEDM) experiment using atomic
beams [139, 3] where the error has already been estimated to be non-negligible. Although
bottled ultra cold neutrons will be used, this effect becomes important as the sensitivity
of the nEDM searches increases. A general analytical approach based on the relationship
between the systematic frequency shift and the velocity autocorrelation function of the
stored particles has been developed [4] to describe the geometric phase effect observed in a
recent nEDM experiment [68] at ILL.
The principle of the geometric phase effect is desribed below, which can be found
in [90]. In a cylindrical storage cell, assume there is a radial magnetic field due to a
magnetic field gradient in the z direction (the electric field is also along z direction). Since
~ =
∇·B
∂Bx
∂x
+
∂By
∂y
z
+ ∂B
∂z = 0 and cylindrical symmetry,
∂Bx
∂x
=
∂By
∂y
z
= | 12 ∂B
∂z | = a. Consider
that the particles’ orbits are roughly circular due to specular reflection around the bottle
shown in Figure A.1 [4]. The wall collision rate is 1/τc (τc is the time between collisions) and
the orbital frequency is ωr = 2α/τc , where α is the incident angle relative to the surface.
104
Figure A.1: A cross section view of the cylindrical trap bounded by a circular
~ are perpendicular
sidewall. A particle is undergoing specular reflections. B0z and E
to the paper. The frequency shift depends only on the component of the trajectory
in the plane perpendicular to the axis (vr ).
~ × ~v
If we transform into a rotating frame at frequency ωr , the radial field (including the E
field) is
BR =
R ∂Bz
ωr RE
± BE = aR ±
2 ∂z
c
(A.1)
where aR is the magnetic field due to the gradient in the transverse plane at radius R and
~ × ~v field and the ± sign refers to the rotation direction. In
±BE is the radially directed E
this rotating frame,
B 2 = (B0 − ωr /γ)2 + (BR )2
(A.2)
In the limit of BR << B0 , the magnetic field is transformed back into the laboratory frame,
B = B0 +
1 (aR − ωr RE/c)2
aR2 ωr E/c
= B0 −
2
B0 − ωr /γ
B0 − ωr /γ
(A.3)
while keeping only the linear term in BE . After averaging over the rotation directions (the
~ × ~v field generate a systematic frequency shift of
± sign of ωr ), the gradient field and the E
δω = γδB = −
γ 2 av 2 E
c(ω02 − ωr2 )
So δω is proportional to E, mimicking an EDM signal.
105
(A.4)
Further studies in [90] used the density matrix approach to find the general solution to
the frequency shift due to the fluctuating magnetic fields in the x − y plane (Equation 40
in [90]):
∞
Z
δω = −γab
−∞
where b =
γE
c .
ψ(ω)
dω
(ω02 − ω 2 )
(A.5)
ψ(ω) is the Fourier transform of the velocity correlation function. ω0
is the resonance frequency. Since the geometric phase effect is highly dependent on the
operating conditions of the experiment, a method was proposed by Barabanov, Golub
and Lamoreaux [140, 141] to directly measure the correlation function that determines the
frequency shift under the exact conditions of a given experiment. The correlation function is
directly related to the relaxation rate of polarized 3 He at a certain frequency. Our current
3 He
relaxation time experimental setup (Figure A.2) is ideal for this purpose. A sealed
cell’s longitudinal relaxation times are measured at different holding fields to extract the
spectrum of this correlation function described below.
The frequency shift is given by (Equation 26 in [90])
Z
δω = γab lim
t→∞ 0
t
R~r~v (τ ) cos ω0 τ dτ
(A.6)
Here R~r~v (τ ) is the position-velocity correlation function defined in [90], Equation 27.
R~r~v (τ ) =< ~r(t) · ~v (t − τ ) − ~r(t − τ ) · ~v (t) >
(A.7)
[141] proposed a method to directly measure the correlation function R(τ ). If the applied
uniform magnetic field gradient dBz /dz is large enough,
∂Bx,y
1 ∂Bz
=−
= −a
∂x, y
2 ∂z
(A.8)
1
γ 2 a2
=
Sr (ω0 )
T1
2
(A.9)
Following [117],
Here T1 is the 3 He longitudinal relaxation time at different holding fields (corresponding to
different ω0 ) and
Z
∞
Sr (ω0 ) =
R~r~r (τ ) cos ωτ dτ
−∞
106
(A.10)
2
Using R~v~v = − d dτR2~r~r [142], where R~r~r and R~v~v are the position and the velocity correlation
functions, respectively. They are defined similarly in Equation A.7.
Z ∞
2
R~v~v (τ ) cos ωτ dτ
ω Sr (ω0 ) =
(A.11)
−∞
So we have
R~v~v
1
=
2π
Z
∞
ω 2 Sr (ω) cos ωτ dω
(A.12)
−∞
And compare with the velocity correlation function in [90], Equation 38,
1 2
ω Sr (ω)
2π
ψ(ω) =
(A.13)
Put this back into Equation A.5,
γab
δω = −
2π
Z
∞
−∞
ω 2 Sr (ω)
dω
(ω02 − ω 2 )
(A.14)
So by determining the function Sr (ω), we can predict the absolute frequency shift at any
holding field using the formula above. This requires measuring the 3 He relaxation times
(T1 ) at different holding fields with a known external magnetic field gradient at the nEDM
experimental temperatures.
A.2
3 He
Experimental Technique and Apparatus
longitudinal relaxation time measurements at different holding fields are carried out
using the NMR-AFP technique described in Section 3.2.3 with the presence of external
magnetic field gradients. The AFP inefficiency is found to be large due to the magnetic
field gradient and thus needs to be taken into account to extract an accurate T1 .
Figure A.2 shows the correlation function measurement experimental setup. A sealed 1
in. diameter pyrex 3 He cell is filled with 550 torr 3 He and 50 torr N2 at room temperature
and spin-polarized using SEOP method (Section 3.2.1) at the center of the holding field
(67 in. diameter) with negligible external magnetic field gradients. After the cell reaches
maximum polarization after ∼20 hours optical pumping, it is cooled down to room temperature and a known magnetic field gradient is turned on. The magnetic field gradient is
107
generated by two 24 in. diameter coils connected in an anti-Helmholtz coil configuration
and measured using a Gauss meter. A series NMR-AFP measurements are then performed
to determine the T1 corrected for the AFP inefficiency.
Figure A.2: The 3 He correlation function measurement experimental setup.
A.3
Results of Geometric Phase Study at 300 K
The sealed cell’s relaxation time with no external magnetic field gradient is ∼600-700 hours,
close to the dipolar relaxation limit [105]. With the external magnetic field gradient turned
on, the relaxation rates (1/T1 ) versus (dBz /dz)2 are plotted in Figure A.3 with the holding
field of 24 G (based on Equation A.9).
Using Equation A.9, Sr (ω) values can be calculated from the slope of the blue line in
Figure A.3 at different holding fields. Figure A.4 shows the measured ω 2 Sr (ω) values at
different magnetic fields (different ω) at room temperature. The reason why ω 2 Sr (ω) is
plotted is because it is directly related to the Fourier transform of the velocity correlation
108
Figure A.3: The relaxation rates (corrected with the AFP loss) of the sealed cell
versus the square of different external magnetic field gradients at a holding field of
24 G.
function (Equation A.13). From the measurements, ω 2 Sr (ω) = 12.96 ± 1.14 is almost a
constant throughout the frequency range.
Figure A.4: ω 2 Sr (ω) values at different frequencies.
One way to judge whether the result is reasonable or not is to compare the mean free
path of a 3 He atom calculated from the ω 2 Sr (ω) value with the mean free path obtained
109
from other methods. From Equation 7 in [141], the equation for the velocity correlation
function ψ(τ ) is:
d2 ψ(τ )
1 dψ(τ )
+
+ < ω02 > ψ(τ ) = 0
dτ 2
τc dτ
(A.15)
where τc is the average time between collisions. The solution is:
ψ(τ ) = c1 e−η1 τ + c2 e−η2 τ
(A.16)
where
1
η1 =
+
2τc
s
1
− < ω02 >,
4τc2
1
η2 =
−
2τc
s
1
− < ω02 >
4τc2
(A.17)
Plus the boundary condition of ψ(τ )
Z
τ
ψ(t)dt =
h(τ ) =
0
R(τ )
→ 0,
2
when τ → ∞
(A.18)
Thus the velocity correlation functional form is:
ψ(τ ) =
η1 v 2
η2
(e−η1 τ − e−η2 τ )
η1 − η2
η1
(A.19)
After Fourier transformation,
η2
ω 2 Sr (ω)
1 v2
η2
=
( 2 1 2 − 2 2 2)
2π
π η1 − η2 η1 + ω
η2 + ω
In the overdamped limit
1
2τc
(A.20)
>> ω0 , which is the same as the short mean free path limit,
v
1
η1 ∼
= , η2 ∼
==
= τc < ω02 >, η1 >> ω >> η2
τc
λ
(A.21)
we have
ω 2 Sr (ω)
2π
=
1 v2
π η1
ω 2 Sr (ω) = 2v 2 τc = 2vλ
(A.22)
(A.23)
Using the mean velocity of the 3 He atoms at room temperature, the mean free path is
calculated to be (4.5 ± 0.4) × 10−5 cm.
From the diffusion theory calculation in [90], equation (68) gives
ψ(ω) =
1X
4
Dω 2
( 2
)( 2
2 )2 )
π n x1,n − 1 ω + (Dk1,n
110
(A.24)
Here we are in the high frequency limit, so
ψ(ω) =
1 X 4D
π n x21,n − 1
(A.25)
Using the zeros of J10 (x): x1,1 = 1.84, x1,2 = 5.33, x1,3 = 8.54, x1,4 = 11.7,
ψ(ω) = 0.607D =
1 2
ω Sr (ω)
2π
(A.26)
The diffusion coefficient D = v 2 τc /2 = vλ/2 since we are dealing with a two-dimensional
problem. Put this into Equation A.26, the mean free path is calculated to be λ = (4.7 ±
0.4) × 10−5 cm.
Using the mean free path equation and the diameter of helium of 2.2 × 10−8 cm in the
handbook [109] (calculated from the viscosity), the calculated value of the mean free path
is λ ∼ 2.6 × 10−5 cm.
A.4
Discussion
The measured mean free paths from the overdamped limit ((4.5 ± 0.4) × 10−5 cm) and the
diffusion theory ((4.7 ± 0.4) × 10−5 cm) agree with each other, while the calculated mean
free path is λ ∼ 2.6 × 10−5 cm from the handbook [109]. These numbers are on the same
order of magnitude and agree with each other reasonably well. We can say that the theory
works at room temperature to some extent.
So far, all measurements have been carried out at room temperature with a relatively
large holding magnetic field. The final geometric phase effect experiment needs to be carried
out under nEDM experimental conditions and geometry, which means small holding fields
and low polarized 3 He concentration in a rectangular cell. To accomplish this task, the 400
mK experimental setup can be modified to incorporate the magnetic field gradient in all
directions and use a rectangular acrylic cell. NMR-FID method will be incorporated into
the existing 400 mK test setup. The SQUID system has a much higher sensitivity but it
requires extensive vibration isolation, RF shielding and magnetic shielding techniques. If
it can be incorporated, measurements at even lower magnetic fields will be possible.
111
Appendix B
3
He Injection Test
B.1
Overview
The purpose of the 1.9 K and 400 mK tests is to measure the 3 He relaxation time under
nEDM experimental conditions and the spin exchange optical pumping method is used
to polarize 3 He. SEOP is able to produce large amounts of polarized 3 He atoms but the
polarization has not been able to reach higher than 70% experimentally [143]. In the real
nEDM experimental cell, the 3 He density will only be ρ3He = 0.8 × 1012 /cm3 and the
polarization needs to be ∼100%. SEOP cannot achieve this goal. Another method which
can achieve a polarization near 100% is to pass an atomic beam of 3 He in a magnetic
field gradient, the so-called atomic beam source (ABS). Even though the ABS production
rate for polarized 3 He is orders of magnitude smaller compared to the conventional optical
pumping method, it is sufficient for the nEDM experiment. Before the polarized 3 He atoms
coming out of the atomic beam source are introduced into the measurement cell, they need
to be collected in a reservoir (made of glass and coated with Cs) for a short period of time
then transferred over. This chapter describes the techniques and apparatus being built for
the upcoming injection test. This experiment will be carried out in late 2008 at Los Alamos
National Laboratory.
Figure B.1 shows a block diagram of the entire 3 He polarizing, transporting, and 4 He
purification systems of the nEDM experiment. Superfluid 4 He is used for creating UCNs
and for detecting reaction products from interactions between 3 He nuclei and UCNs. The
number of unpolarized 3 He must be reduced to the level of 1010 atoms/cm3 in order to
prolong the UCN storage time (ultra cold neutrons will be absorbed by unpolarized 3 He
in superfluid 4 He). Therefore the 4 He needs to be isotopically purified. The nEDM ex-
112
Figure B.1: A block diagram of the 3 He subsystem in the nEDM experiment, including the injection volume, 3 He purifier, evaporator, etc.
perimental technique also employs polarized 3 He (∼100%) dissolved in superfluid 4 He as
a comagnetometer. The preparation and transport of this mixture of polarized 3 He and
purified liquid 4 He to the measurement cells is the 3 He subsystem.
Figure 2.3 shows the overall nEDM experimental apparatus. Figure B.2 [144] is a
closer look at the upper cryostat. ∼100% polarized 3 He atoms will be injected from the
3 He
Atomic Beam Source (ABS) and collected by a volume filled with superfluid 4 He at
∼300-400 mK in an external cos θ magentic field of ∼10 mG. The polarized 3 He will then
be transferred to the measurement cells located in the lower part of the cryostat where
measurements take place. One has to make sure that the 3 He atoms in the collection
volume lose little polarization during the injection and transport periods. The ABS has
been demonstrated to be able to deliver a beam of 99.6% polarized 3 He with a flux of
∼ 1.7 × 1014 atoms/s [145]. The purpose of the injection test is to demonstrate that
polarized 3 He from the ABS can be injected and stored in a glass reservoir filled with
superfluid 4 He at low temperatures with acceptable polarization loss. Furthermore, a long
3 He
longitudinal relaxation time T1 needs to be demonstrated for the collection volume.
113
Figure B.2:
3
He injected from the ABS and collected by the collection volume
sitting in a cos θ magnet.
Due to the small concentration of polarized 3 He (∼ 1014 atoms/cm3 ) and the NMR
signal being proportional to the Larmor frequency, a superconducting tri-coil magnetic
holding field system, instead of the 10 mG cos θ coil, will be used as the holding field
for the injection test. It can reach a magnetic field of ∼1.2 kG for a high precession
frequency. Superconducting quantum interference device (SQUID) coils are proposed to
pick up the signal from the precessing 3 He magnetic dipoles during the experiment. It
would be beneficial that the SQUID system be incorporated into the injection test so that
more subsystems of the nEDM experiment can be tested at the same time. However, the
SQUID system requires stringent magnetic and RF shielding, which will make the alreadycomplicated system much more complicated. It is decided to test the SQUID in a different
setup. Pulse NMR (Section 3.2.3) is used instead to measure the 3 He polarization signal,
which can be calibrated later.
Like in the 400 mK test, the superfluid 4 He film will climb up from the cold region to
the warmer part and bring in extra heat load to the part that needs to be cooled down
due to the superfluid 4 He’s good thermal conductivity. In the injection test, it is necessary
114
to keep a high vacuum in the injection tube. Otherwise 3 He atoms coming from the ABS
will be deviated from their ballistic trajectories by collisions with 4 He atoms and will result
in polarization losses. Superfluid 4 He film will very likely creep up from the collection
volume to high temperature region, vaporize, and destroy the vacuum. Two methods will
be discussed below to suppress the 4 He film flow: the Cs ring and the passive film burner.
Detailed experimental techniques, apparatus and procedures are explained in this chapter. Presently this test is still in the construction phase and data taking will start once the
construction is complete later this year.
B.2
B.2.1
Experimental Technique
Polarize 3 He
~ r) is written as
The energy of a magnetic dipole µ
~ in an external static magnetic field B(~
~ r)
U (~r) = −~
µ · B(~
(B.1)
and the force imposed on the dipole is given by
~ r)
F~ (~r) = µ(ŝ · ∇)B(~
(B.2)
where ŝ is the spin direction. For 3 He, µ = −~γ3 /2 where γ3 = 2.04 · 108 /T is the 3 He
gyromagnetic ratio. From this equation, the magnetic forces exerted on particles with
different spins have different directions. For a magnetic quadrupole configuration shown
in Figure B.3 [1], the magnetic field at the center is the weakest. At places further away
from the center, the magnetic field becomes stronger. In this configuration, particles of one
spin state are focused along the interaction region and those of the other spin state become
defocused. If a beam of unpolarized atoms is traveling along the center, only atoms with
one spin state will be selected and make it to the end of the magnet. In this way, the atoms
in the beam are “polarized” by selecting the ones with only one spin state. However, unlike
other atomic beam sources, the 3 He nuclear magnetic moment is very small (Section 3.2.3).
115
The interaction time between the atoms and the magnetic field needs to be increased and
the kinetic energy of the atoms must be decreased to achieve high polarization.
Figure B.3: Quadrupole configuration of permanent magnets in the ABS system.
In a 3 He atom’s rest frame, the magnetic field changes in both magnitude and direction
as the atom travels through the polarizer. The change cannot be too fast otherwise the
atomic beam will lose polarization. To maintain the atoms’ polarization, the adiabatic
condition (Section 3.2.3) needs to be satisfied so that the atoms’ spins will be able to
adiabatically follow the direction of the magnetic field.
|Ḃ|
<< |γ3 B|
|B|
(B.3)
where Ḃ = dB/dt and γ3 B is the Larmor frequency.
Since the magnitude of the magnetic field is theoretically zero at the center of the polarizer, polarized atoms travelling through this region of a zero field may become unpolarized
and reduce the net polarization of the beam. It is necessary to add a weak axial magnetic
field Bz to maintain the polarization. Only atoms with one spin state are experiencing
restoring forces and will make it through the polarizer.
116
B.2.2
3
He Polarimetry
In the injection test, the concentration of polarized 3 He is very small (∼ 1014 atoms/cm3 ).
The signal from NMR-AFP discussed in Section 3.2.3 will be too small to detect. The
SQUID system is much more sensitive than conventional NMR but it requires stringent
magnetic and RF shielding, which is very difficult to incorporate into the injection test
apparatus. Pulse NMR (NMR-FID, Section 3.2.3) has the sensitivity between the NMRAFP and the SQUID and it will be used to calibrate the 3 He polarization.
The measurement volume will be filled with superfluid 4 He at ∼0.3-0.4 K. Unpolarized
3 He
will be introduced using the gas handling system (not from the ABS) so that the
concentration of the 3 He (ρ3cal ) in the liquid 4 He is high. The volume will be inside a ∼1.2
kG magnetic field so that the 3 He’s thermal polarization is given by
P3cal = tanh(
µ3He B
)
kB T
(B.4)
where µ3He is the 3 He magnetic moment, and kB is the Boltzmann constant. The density
of polarized 3 He is then written as P3cal · ρ3cal . The NMR-FID signal measured is defined
as A3cal .
If the ABS is used, the 3 He density coming out of the ABS and collected in the volume is
ρ3ABS and the NMR-FID signal measured is A3ABS . The 3 He polarization in the collection
volume can then be written as
P3ABS =
ρ3cal · P3cal A3ABS
ρ3ABS
A3cal
(B.5)
This is one of the important parameters that needs to be determined in the injection test.
B.2.3
4
He Film Flow
Superfluid 4 He film tends to flow to warmer temperature regions and eventually evaporates.
Metallic Cs has the property that it is not wet by liquid 4 He below a temperature of ∼2
K [146, 147]. So a ring of bulk Cs can prevent the superfluid 4 He film from flowing over it.
117
The 3 He concentration in superfluid 4 He is extremely low in our case, so it will not induce
the wetting of Cs [148].
Another way to prevent the 4 He film from flowing to warmer regions is to use a film
burner. The basic working principle is that the helium film is flowing from the low temperature region (∼300 mK) to an evaporation plate. A condensing plate will be very close to
the evaporation plate. When the film burner is working, the evaporation plate is heated up
to maintain a temperature over ∼420 mK and the condensing plate is kept below 310 mK.
The 4 He atoms evaporated from the evaporation plate will have a probability of ∼100% to
strike the condensing plate, stick to it and condense back into the liquid 4 He. 4 He atoms
will be going through this cycle when the whole system is operating normally so that the
4 He
film flow is stopped. The film burner will generate heat and the dilution refrigerator
needs to have enough cooling power to maintain the temperatures of the evaporation and
condensing plates as well as the condensation energy of the helium film to keep the whole
system in a dynamically stable state.
B.3
Experimental Apparatus
Figure B.4 and Figure B.5 [144] are the cross section view and the 3-D view of the 3 He
injection test experimental setup, respectively.
The 3 He injection test apparatus includes a collection volume (3 He reservoir) made of
pyrex with an injection tube tilted by 45 degrees and a measurement cell, a 3 He/4 He gas
handling system, a superfluid 4 He film burner, the ABS’s lower beam line, a pulse nuclear
magnetic resonance (pNMR) system for measuring 3 He polarization, a superconducting
solenoid magnet for spin transport, and a superconducting tri-coil magnet system for providing a uniform magnetic field in the collection and the measurement cell region. They
are described in details below.
118
Figure B.4: The cross section view of the 3 He injection test experimental setup.
The lower part of the picture is an expanded view of the measurement region.
B.3.1
Atomic Beam Source
Figure B.6 [149] shows the atomic beam source constructed for the nEDM experiment.
The unpolarized 3 He atoms are cooled down to around 1 K before passing through the
ABS nozzle so that the speed of the atoms is less than 100 m/s. The ABS provides a well
119
Figure B.5: A 3-D view of the 3 He injection test experimental setup.
collimated ∼99.6% polarized 3 He beam with an intensity of ∼ 1014 atoms/s and an average
velocity of ∼ 100 m/s. The beam has an angular divergence of ∼0.008 radian (half-angle).
At the ABS exit, the velocities and spins of 3 He atoms are along the ABS downstream axis.
Figure B.7 [149] shows a single permanent quadrupole magnet which is one of the eight
quadrupole mangets along the ABS axis. Figure B.8 [149] is a home-made refrigerator
inside the ABS to cool the 3 He atoms down to ∼1 K. In this way the 3 He atoms’ speed is
slow (∼100 m/s) so that the interaction time with the magnetic field is increased. The ABS
support structure has been modified to allow tilting. And it has already been demonstrated
to work when tilted for 45 degrees as required by the injection test. The 3 He atoms entering
the injection port from the ABS will impinge directly on the superfluid 4 He liquid surface
as shown in Figure B.4.
120
Figure B.6: The atomic beam source (ABS) for the nEDM experimental.
Figure B.7: One of the eight quadrupole magnets along the ABS axis.
B.3.2
Cooling System
The entire injection test setup is enclosed in a liquid helium cryostat designed for the
nEDM project’s R&D experiments. The cryostat is intended to be used with the dilution
refrigerator (Leiden Cryogenics Model DRS3000), whose working principle has been detailed
121
Figure B.8: The home-made refrigerator inside the ABS to cool the 3 He atoms
down to ∼1 K.
in Section 3.4.2, except that this DR has more cooling power than the one used in our 400
mK test. The cryostat consists of an outer vacuum vessel, two heat shields (one at 50 K
and the other at 4 K), and a liquid helium vessel (main bath). The outer vacuum vessel
provides the isolation vacuum to the heat shields, the liquid helium vessel, and the DR.
The top part of the DR (4 K plate and above) will be immersed in liquid helium in the
main bath and the bottom part of the DR, namely the 1K pot, the Still, the 50 mK plate
and the mixing chamber will be in the vacuum. The heat shields will be cooled by helium
122
boil-off gas from the liquid helium vessel flowing through a copper pipe mounted on the
heat shields. Figure B.9 [150] shows the schematic of the cryostat with the DR inserted.
Figure B.9: Cross sectional view of the cryostat with the DR inserted.
Figure B.10: Outer vacuum vessel, heat shields and the liquid helium vessel, from
left to right, respectively.
Figure B.10 shows the outer vacuum vessel, heat shields and the liquid helium vessel.
While the atomic beam source is operating at a temperature of ∼1 K, the ABS exit is at
123
room temperature. Therefore the heat load due to thermal radiation needs to be suppressed
by several “blackened” thermal baffles mounted along the beam line (Figure B.11 [144]) in
the transfer tube.
Figure B.11: The ABS lower radiation limiter made by baffles mounted along the
beam line in the transfer tube.
The 4 He molecules introduced into the injection cell are cooled down on their way to
the glass reservoir by going through a heat exchanger which is connected to various stages
of the DR (Figure B.12). The 3 He circulation speed in the dilution refrigerator determines
the cooling power at a certain temperature. The DR from Leiden Cryogenics is modified so
that the cooling power is increased at ∼300-500 mK but the price paid is that the minimum
temperature the DR can reach (usually < 50 mK) is higher, which is not an issue for the
injection test.
A ductile thermal link between the DR and the pyrex cell is implemented using Oxygen
Free High Conductivity (OFHC, thermal conductivity λ ∼ 100 W/m ) copper foils (Figure B.12 [144]). The outer surface of the injection tube is covered with 20 layers of copper
foils. GE Varnish will be applied to glue the pyrex and the copper foils to ensure good
surface contact.
124
Figure B.12: Autodesk Inventor drawing of the thermal link from the MC to the
film burner and the injection tube.
B.3.3
Pyrex Glass Reservoir
After passing a vacuum transfer tube of ∼1 m in length, 3 He atoms enter the collection
volume, a pyrex cell filled with superfluid 4 He at ∼300-400 mK shown in Figure B.4. Except
for the ABS, all other parts of the apparatus are inside the cryostat with 2 layers of thermal
radiation shields at temperatures of 4 K and 50 K, respectively (Section B.3.2).
The pyrex glass reservoir is composed of an injection tube and a measurement cell. The
injection tube is 45-degree tilted and provides a liquid 4 He surface area large enough (∼4.0
cm diameter) to accommodate the 3 He beam profile. The measurement cell is a cylinder (2.0
cm in length and 2.0 cm inner diameter) designed for pNMR polarization measurement. The
reservoir is covered with Oxygen Free High Conductivity (OFHC) copper foils thermally
linked to the dilution refrigerator’s mixing chamber so that cooling power can be transferred
efficiently.
The injection cell inner surface will be coated with Cs due to its ability to inhibit nuclear
spin relaxation on the pyrex cell walls (Section 3.2.2). Cs metal will be chased around in
125
the pyrex reservoir using a torch, getting in contact with all inner surfaces of the cell and
leaving a Cs coated surface. Experiments have shown that Cs-treated pyrex and bulk Cs
are weak binding surfaces with low surface relaxation rates for pure 3 He [151]. Figure B.13
shows the pyrex glass reservoir (collection volume) with the side arm for Cs coating. The
procedures of the coating process are the same as described in Section 3.3.5.
Figure B.13: The pyrex glass reservoir (collection volume) with the side arm for Cs
coating. The bottom small cell is where NMR-FID measurements are carried out.
B.3.4
Gas Handling System
A gas handling system is specially designed and made for filling a known amount of ultra
pure 3 He/4 He gas into the collection reservoir using a calibrated volume through a capillary
tube. It is used for NMR calibration purpose (Section B.2.2) and controlling the height of
the liquid 4 He in the collection volume. A metering valve is used to control the flow rate
126
Figure B.14: The schematic and the picture of the gas handling system.
of the gaseous 3 He/4 He when condensing the gas into liquid. The flow rate cannot be high
so that the DR can stay at normal operation state and successfully condense the gas into
liquid. The gas handling system panel will be mounted to the bottom platform of the ABS
supporting rack (Figure B.5).
127
B.3.5
Magnet System
The magnetic field should continuously be applied to maintain the polarization of 3 He
during the injection period along the injection path. The magnet system includes a superconducting transport solenoid magnet (manufactured by American Magnetics, Inc.) and
a superconducting tri-coil magnet (manufactured by Cryomagnetics, INC.). The combination of these two coils provides a holding field for 3 He spin transport and a strong, uniform
field for the subsequent polarization measurement. Both coils use superconducting NbTi
wires to reduce the heat load to the cryostat. The solenoid coil is cooled by mounting it
onto the 4 K shield inside the 45-degree snout of the cryostat. The tri-coil will be sitting
in a can filled with liquid 4 He.
The superconducting solenoid coil of 40 cm in length and 11.6 cm in diameter is located
outside the injection tube (Figure B.4), providing ∼20 G axial magnetic holding field. Due
to geometric limits, a 20 G holding field around the pyrex collection reservoir is limited to
the vertical down direction, which is provided by a superconducting tri-coil system. Along
the injection trajectory, the spins of 3 He atoms need to be rotated by 45 degrees. The
magnet system to control the spin rotation while maintaining the polarization is facilitated
by the tri-coil together with the transport solenoid coil. As long as the NMR-Adiabatic Fast
Passage condition
|Ḃ|
|B|
<< |γ3 B| is met, the 3 He atoms will follow the effective magnetic
field and stay polarized. Based on the field information generated by TOSCA [152], a
Monte-Carlo simulation of the spin rotation along the trajectory shows the polarization
loss is less than 1% during the spin transport [153].
After the 3 He atoms are injected into the collection reservoir, which is sitting in the
center of the tri-coil system, the tri-coil holding field will be ramped from 20 G to 1.2
kG in order to enhance the pNMR signal. Figure B.15 shows the superconducting tri-coil.
The distance of the outside coils to the central coil is 0.76 times the radius of the coils,
and the ratio of the current flowing in the central coil and that in the outer coil is exactly
0.531 [154]. This configuration will provide a larger uniform region of magnetic field than
that of the Helmholtz coil. The magnetic field gradient in the measurement cell region
128
Figure B.15: Tri-coil system.
will be ∼ 100 ppm/cm and the 3 He longitudinal relaxation time will be dominated by wall
effect.
B.3.6
Pulse NMR System
The measurement cell dimensions together with magnetic field gradients are optimized to
achieve a reasonably long transverse relaxation time T2 (>1 ms). A dedicated cryogenic
pulse NMR system is being constructed. The resonant tank circuit composed of a probe coil
and a tunable capacitor, a duplexer (the circuit equivalent to a quarter wavelength cable),
an RF power amplifier, and a low noise pre-amplifier is being tested. The pNMR probe
includes a solenoid coil and two mechanically tunable capacitors (trim piston) enclosed by
a copper Faraday cage. The resonance frequency (3.89 MHz) and the impedance matching
(50 ohms) of the probe are tuned by two long G10 screw drivers mounted on a 3 He injection
test flange on top of the dual use cryostat. Since the 3 He NMR signal will be very small,
the signal to noise ratio needs to be improved a lot in order to successfully detect the 3 He
129
precession signal.
B.3.7
Film Burner and the Cs Ring
Because deviation of 3 He atoms from their ballistic trajectories results in the loss of polarization, collisions of 3 He atoms with the 4 He atoms in the vapor should be avoided as much
as possible by reducing the vapor pressure in the injection tube. To suppress the evaporation of the superfluid film, Cs rings (∼45 mm in diameter) and a passive film burner are
being designed to block the superfluid 4 He film flow, reduce the evaporation rate, and reach
good vacuum.
Figure B.12 shows the approximate locations of the film burner and three Cs rings.
The superfluid film burner, mounted upstream of the 45-degree port of the pyrex reservoir,
recondenses evaporated helium vapor to maintain a high vacuum in the 3 He injection region.
High vacuum (better than 10−7 torr) along the ABS injection beam line is essential to
guarantee that the mean free path of 3 He is long. The design of the film burner has not
been finalized, but the surface area of the condensing plate will be large to re-condense all
the vaporized 4 He atoms from the evaporation plate.
Three Cs rings will be made along the injection tube by chasing Cs vapor with a torch
and condensing the vapor with dry ice attached to the outside of the ring area. Cs ring
manufacture and coating of inner surface with Cs will be done at the same time. If the
Cs ring works effectively to block the superfluid film flow, it will not only reduce the heat
load from the film burner but also limit the surface area seen by the polarized 3 He atoms,
therefore increasing the 3 He longitudinal relaxation time.
B.4
Experimental Procedures
The injection test consists of two periods. The first period is the collection period, when
the 3 He atoms from the ABS are collected and dissolved in superfluid 4 He in the pyrex
reservoir. Both the transport solenoid magnet and the tri-coil magnet are energized at 20
130
G, providing curved magnetic field to rotate the 3 He spin by 45 degrees. This period will
last for ∼100 seconds and eventually ∼ 1016 3 He atoms will be collected in ∼50 cm3 of
superfluid 4 He. The second period is the measurement period. The ABS output will be
closed, the transport solenoid coil will be turned off, and pulse NMR measurement will start
to measure the 3 He polarization and the relaxation time. During the measurement period,
the polarization measurement works with a low density of polarized 3 He atoms ∼ 1014 /cm3 .
Since the signal size is proportional to the Larmor precession frequency, the current in the
tri-coil system will be ramped up to 1.2 kG with a 3 He resonance frequency of ∼3.89 MHz.
The parameters of the pNMR system are being fine tuned to maximize the signal to noise
ratio and minimize the recovery time.
The injection test is one of the many R&D efforts to ensure that polarized 3 He can
successfully maintain the polarization in the collection volume (made of pyrex coated with
Cs) before being transferred to the measurement cell (made of acrylic coated with dTPBdPS material). It is planned to be carried out at Los Alamos National Laboratory in the
Fall of 2008 and will be the first experiment on the cryogenic assembly specifically designed
and manufactured for the nEDM project.
131
Appendix C
Measuring the RF B-field
The oscillating RF B-field cannot be measured by a Gauss meter. A small N-turn probe
coil with area S can help measure the RF field.
The RF B-field is 2B1 cos(ωt) in Section 3.2.3. If a probe coil is put into the center of
the Helmoltz coil and an oscilloscope is used to read its output signal,
pp
dφ
=−
2
dt
N d(B · S)
dt
dB
= −N S
dt
= −
= N S2B1 ωsin(ωt)
(C.1)
(C.2)
(C.3)
where pp is the peak-peak signal from the oscilloscope. Then we have
pp
max
2
= N S2B1 ω
(C.4)
= N SB1 4πf
(C.5)
so the magnitude of the RF B field is
pp
max
B1 =
N S8πf
where f is the oscillating frequency.
132
(C.6)
Appendix D
dTPB-dPS Material Manufacturing
Procedure
Extreme ultraviolet (EUV) light (∼80 nm) is produced by the recoil of the charged proton
Figure D.1: Distill the 99% styrene in the complicated glassware with an Argon
atmosphere ≤150 mBar at a temperature ∼95 degrees Celsius.
and triton from the capture of neutrons by polarized 3 He in superfluid 4 He. This light
is wavelength shifted to visible light so that it can be detected by photomultiplier tubes
(PMTs). On the inner surface of the acrylic cell, the walls are coated with deuterated
polystyrene (dPS) doped with the deuterated organic fluor 1,1,4,4-tetraphenyl buta-1,3-
133
diene (dTPB). The dTPB absorbs the EUV photons and emits blue light with a spectrum
peaked at 430 nm and a width of approximately 50 nm [155]. The main part of making
the dTPB-dPS procedure is to make the deuterated polystyrene. The procedures (from
Professor R. Golub) to make this dTPB-dPS wavelength shifting material are as follows:
1. Distill the deuterated styrene (Figure D.1). A small boiling stone (Figure D.2) is
necessary to prevent the liquid from boiling too much. The distillation process needs
to be completed in ∼100 mBar Argon atmosphere at a temperature of ∼ 80 − 100◦ C.
The distilled deuterated styrene accumulates in the small pipets (∼14 g d-styrene
comes out).
Figure D.2: There are small boiling stones at the bottom of the vessel to prevent
the liquid from boiling too much.
2. Polymerization of the distilled styrene (Figure D.3). This process needs to be done
134
under Argon atmosphere at a temperature of ∼200 degrees Celsius for one night.
The next day the oil bath needs to be cooled down really slow with a cooling rate of
∼5◦ C/hour. The deuterated polystyrene will harden on the glass and breaking the
glass is the only way to separate them.
Figure D.3: Polymerization of the distilled styrene. Argon atmosphere, temperature
of ∼200 degree Celsius.
3. Dissolving deuterated polystyrene into deuterated Toluene. The same setup as in
Figure D.3 is used. ∼100-130 cm3 d-toluene is put into the flask along with the
mixture of ∼14 g d-polystyrene and some small glass pieces. A big teflon stirrer is
put in the d-toluene liquid heated up to 40-50 degrees Celsius with Argon atmosphere
to help the dissolving process.
4. Dripping into deuterated methanol. After the d-polystyrene is completely dissolved
into the d-toluene, 500 cm3 deuterated methanol is prepared in a big beaker, and
a pipet is used to drip the solution into the d-methanol slowly (Figure D.4). The
135
d-methanol helps dissolve the leftover monomer in the d-polystyrene and only pure
polymerized styrene is precipitated out of the solution and becomes white rubber-like
material.
Figure D.4: Drip the d-polystyrene and d-toluene solution into d-methanol in order
to remove the monomer.
5. Put the pure d-polystyrene into a shallow culture container and bake it under vacuum at ∼45◦ C for a few hours to get rid of the d-methanol and d-toluene. The
d-polystyrene hardens again.
6. Break the d-PS into small pieces and use the same setup in Figure D.3 to dissolve
the d-PS into d-toluene at 40-50◦ C with Argon atmosphere overnight.
7. Weigh the right amount of (dTPB:dPS∼2:3) dTPB material and use the step 6 to
dissolve it into the dPS solution. The final dTPB-dPS solution is then completed.
136
Appendix E
dTPB-dPS Coating Procedure
The best way to make a thin coating onto acrylic material is to let the liquid drip down on
the surface once and dry out by itself. This will leave a thin and transparent coating on the
acrylic surface. The acrylic cell used in the 400 mK test is cylindrical, which is composed
of three parts, the top part, the middle cylinder and the bottom part.
Since it is hard and also time-consuming to make dTPB-dPS material, a teflon piece
(Figure E.1) is made to occupy most of the inner space of the cell and a clearance of ∼1.5
Figure E.1: A teflon piece made to occupy most of the space in the acrylic cell to
minimize the use of dTPB-dPS material.
mm is left between the teflon piece and the cell’s inner surface. In this way, only minimum
amount of dTPB-dPS material is used for the coating process.
Before coating, the top and middle parts are glued together using stycast 1266. A soft
pipe is connected to the center hole on the top part and used to suck the liquid up. The
dTPB-dPS liquid will occupy the clearance and leave a coating on the inner surface when
the liquid falls back down. The bottom part of the acrylic cell is coated by putting liquid
onto it and then tilting it to let the liquid drip out. After the coating, the top-middle and
137
bottom parts of the cell are dried out and glued together using stycast 1266.
The old method described above only applies to cylindrical cells. However the nEDM
experimental cell is rectangular, which has six sides and needs to be coated separately and
glued together. A new “Swinging method” is developed to coat the acrylic pieces. As
Figure E.2: “Swinging method” to coat the acrylic pieces.
described in Figure E.2, the bottom of a shallow culture container is filled with dTPB-dPS
liquid. The acrylic piece is attached to a rod, immersed in the liquid and swung to one side
to let the liquid drip down. Some test pieces are already made and atomic force microscope
(AFM) shows very smooth surface after the coating. Figure 4.14 shows the AFM images
of the acrylic cells using the old and new coating procedures.
138
Appendix F
Reciprocity Theorem
Consider two coils C1 and C2 . There is a current i in C1 . The flux through C2 is given
by [156]
Z
Φ2 =
ds~2 · A~1 (r~2 )
Z
µ0 i
ds~1
4π C1 r12
Z Z
ds~1
µ0 i
ds~2 ·
= Φ1
4π C2 C1
r12
(F.1)
C2
A~1 (r~2 ) =
Φ2 =
(F.2)
(F.3)
which is the same as the flux through C1 produced by current i flowing in C2 . Now shrink
the coil C2 to a very small area. The flux through it is
~ r~2 ) · S~2
Φ2 = B(
(F.4)
in which S~2 represents the shrunken area. This is the flux through C1 when a current i
flows in the shrunken coil, which then would have a magnetic moment of iS~2 . Thus the
flux through C1 due to a magnetic moment µ~2 at r~2 is
Φ1 =
~ r~2 )
B(
· µ~2
i
(F.5)
~ r~2 )/i over the volume of the measurement cell will give
So the integration of n3 (r~2 )P~3 (r~2 )· B(
the expected NMR signal, where n3 (r~2 ) and P~3 (r~2 ) are the concentration and polarization
of the 3 He atoms at r~2 position.
139
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Biography
Qiang Ye was born in Nanjing, China on March 30th, 1980. He grew up in Nanjing
and graduated from Nanjing Foreign Language School, where he attended the Fourteenth
National Contest in Physics for High School Students of Jiangsu Province and won the first
prize. He obtained his B. S. degree in the Department of Physics in 2002 from Nanjing
University. During his undergraduate study, he was awarded People’s Scholarship several
times. In the fall of 2002, he was enrolled as a Ph.D. student in the physics department
at Duke University. Later he joined Professor Haiyan Gao’s Medium Energy Group in
2003 and started the polarized 3 He relaxation study for neutron electric dipole moment
experiment. He was awarded the M. A. degree in physics in 2005, Henry W. Newson
Fellowship at Triangle University Nuclear Laboratory in 2007 followed by the Ph.D. degree
in physics in 2008. During his graduate study, he helped with the paper “A High-pressure
Polarized 3 He Gas Target for the High Intensity Gamma Source (HIγS) Facility at Duke
Free Electron Laser Laboratory” (K. Kramer et al., Nuclear Instruments and Methods
in Physics Research Section A, 582, 318-325, 2007) and published “Relaxation Of Spin
Polarized 3 He In Mixtures Of 3 He And 4 He Below The 4 He Lambda Point” (Q. Ye et al.,
Phys. Rev. A, 77, 053408, 2008). Currently he is involved in more R&D projects for the
nEDM experiment.
147