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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 . 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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