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Coils, fields and xenon : towards measuring xenon spin precession in a magnetic field for the UCN collaboration Wienands, Joshua Nikolai 2016

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Coils, Fields and XenonTowards Measuring Xenon Spin Precession in a Magnetic Field for the UCN CollaborationbyJoshua Nikolai WienandsBachelor of Science, Astronomy, The University of British Columbia, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)August 2016c© Joshua Nikolai Wienands, 2016AbstractIn this thesis I present my work on building a set of magnetic coils for the purpose of performing nuclear magneticresonance (NMR) on Boltzmann polarized protons in water, and on hyperpolarized 129Xe. The coils were designedto be used as a method for testing the degree of polarization achieved in 129Xe, and for testing the capability of anin-house developed continuous wave (CW) ultraviolet (UV) laser to drive a 2-photon transition in 129Xe. This laserwill be used to measure the precession frequency of 129Xe in a magnetic field, in order to precisely measure themagnitude of that field.This work is being done for the ultra-cold neutron (UCN) collaboration’s flagship experiment: to measure theneutron electric dipole moment (EDM). Previous neutron EDM experiments have only found an upper limit, and havebeen limited in precision largely because of systematic errors in the magnetic field strength measurement. Theseexperiments, such as the one performed at Institut Laue-Langevin (ILL), which has given us the current lowest limit,used 199Hg as a co-magnetometer. The UCN EDM experiment will add 129Xe in addition to the 199Hg, to make adual co-magnetometer. By using multiple species of atoms in the measurement, systematic effects can be greatlyreduced.I have characterized the coils that I built by performing NMR on protons in water. I measured the inhomogeneityin the B0 field, across the sample container, to be 18.9±0.9 µT. It turns out that the homogeneity of the B0 fieldcan be improved significantly, and it will likely be necessary to do so in order to perform similar experiments onhyperpolarized 129Xe. I also found the T1 time of water in this setup to be 2.7±0.2 s.iiPrefaceThis thesis covers some of the research I did for the ultra-cold neutron (UCN) collaboration’s flagship experiment,which attempts to find a non-zero value for the neutron electric dipole moment (EDM). This work was done underthe supervision of Dr. David Jones and Dr. Kirk Madison at the University of British Columbia (UBC). In thisthesis, there are some brief descriptions of two laser systems that are being built by Emily Altiere, in Chapter 1. Avery similar system is described in detail in her thesis[1]. With permission from Emily, I have used and modified afigure created by her for my Fig. 1.6. There are also some descriptions of a 129Xe polarizer that is being built byEric Miller, in Chapters 1 and 2.The work described in this thesis focuses on the construction of magnetic coils that will be used to test theeffectiveness of the ultraviolet (UV) lasers, and the degree of 129Xe achieved. The simulations described in Chapter3 were coded and run by myself, with advice from Dr. Chris Bidinosti and Dr. Jeff Martin from the University ofWinnipeg as to what coil geometries to pursue. The measurements in Chapter 4 were performed by myself.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Neutron Electric Dipole Moment (EDM) Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Neutron EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Ramsey’s Method of Separated Oscillating Fields . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 Past Neutron EDM Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.5 Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.6 Co-Magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.7 Future Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Xe-129 Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 UCN collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 High Power CW UV Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Testing the UV Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 NMR and Free Induction Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Particles in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iv2.2.2 Projective Measurements of Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . 162.3 Spin Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Boltzmann Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Hyperpolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 T2 and T∗2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.3 Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.1 Detection Via Pickup Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.2 Optical Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Adiabatic Fast Passage (AFP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.2 Speed of the B0 ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6.3 Detection and Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Free Induction Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7.2 Detection and Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7.3 FID in the Context of UCN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Coils and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Helmholtz Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Saddle Coil Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Analytic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Image Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6 B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 Pickup Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.8 Shielding and Improving Homogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.8.1 AC Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.8.2 DC Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.8.3 Shimming the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Measurements and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1 Final Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1.1 AC Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1.2 B0 Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1.3 B1 Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.4 Pickup Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54v4.1.5 Lock-in Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Adiabatic Fast Passage (AFP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.1 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.2 AFP results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Free Induction Decay (FID) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.2 FID results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.3 Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.1 AC Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.2 B0 Coil Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.3 B1 Pulse for FID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.4 FID Repetition Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.1 Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2 Saddle Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.2.1 Saddle Coil Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.2.2 Magnetic Field From a Curved Section of Wire . . . . . . . . . . . . . . . . . . . . . . . . 87A.2.3 Magnetic Field From a Straight Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.2.4 A Function to Read the Simulation Output File . . . . . . . . . . . . . . . . . . . . . . . . 90viList of TablesTable 1.1 angular momentum quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Table 1.2 Here are the state equations for all of the possible F and mf states of interest in 129Xe. . . . . . . 10Table 3.1 Test for precision for the calculation of the magnetic field of a pair of saddle coils. . . . . . . . . 40Table 3.2 Convergence test for the image field produced by a steel optical table. . . . . . . . . . . . . . . . 43Table 3.3 Gyromagnetic ratios and recession frequencies for some sources in a 1.5 mT magnetic field. . . . 45viiList of FiguresFigure 1.1 neutron and its quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Figure 1.2 Ramsey’s method of separated oscillating fields . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 Ramsey fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.4 An oscillating field under the rotating wave approximation . . . . . . . . . . . . . . . . . . . . 6Figure 1.5 The Xe-129 energy level diagram for the ground state and some useful excited states. . . . . . . 9Figure 1.6 UV laser schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.7 A schematic of the test coils, the UV enhancement cavity, and the Xe-129 cell. . . . . . . . . . 13Figure 2.1 Precession of spin for a particle in a magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.2 A particle with spin state given by angle θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.3 A schematic of the Rubidium energy levels used for optical pumping. . . . . . . . . . . . . . . 19Figure 2.4 Optical pumping schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.5 The change in signal when shining linearly vs. circularly polarized light into a Rubidium cellwhen doing optical pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.6 T1 relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.7 A spin echo experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.8 An example of a demodulated signal from a spin echo experiment. . . . . . . . . . . . . . . . . 24Figure 2.9 The dark state in 129Xe when driving the 252.4 nm two photo transition with circularly polar-ized light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.10 Adiabatic fast passage in a frame rotating at the B1 frequency. . . . . . . . . . . . . . . . . . . 28Figure 3.1 A schematic of the coils and Xe cell above the steel optical table. . . . . . . . . . . . . . . . . 32Figure 3.2 A pair of current loops in the Helmholtz and anti-Helmholtz configuration . . . . . . . . . . . . 32Figure 3.3 Convergence test for the Helmholtz pair simulation. . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.4 Simulated and analytic results for the magnetic field in the z direction, along the z axis, for aHelmholtz pair of coils, with total current of 404 A, and a coil radius and separation of 300 mm. 35Figure 3.5 Saddle coil configuration with typical dimensions shown. . . . . . . . . . . . . . . . . . . . . . 36Figure 3.6 Calculating the magnetic field from straight wire . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.7 Convergence tests for the simulated field from a pair of saddle coils. . . . . . . . . . . . . . . . 38Figure 3.8 A comparison of analytic and simulated results for the magnetic field created by saddle coils inthe z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41viiiFigure 3.9 image current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.10 A calculation of the field along the z axis due to the image current from a saddle coil. . . . . . . 43Figure 3.11 Magnetic field from saddle coils with a nearby steel table. . . . . . . . . . . . . . . . . . . . . 44Figure 4.1 The full assembly of coils and the Xe cell, and cavity optics. The cell has a pickup coil wrappedaround it. The water bottle used to make the measurements in this chapter is about the samewidth, but is shorter than the Xe cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.2 A photo of the full assembly with an old pickup coil meant for a larger water bottle. . . . . . . . 52Figure 4.3 B0 field homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.4 A closeup photo of the pickup coil and the mechanical cross-talk decoupling mechanism. Alsoshown is the B1 coil, and the water bottle used. . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.5 A sample screenshot of the oscilliscope when performing AFP. . . . . . . . . . . . . . . . . . . 58Figure 4.6 A schematic of the AFP circuit, including the circuits for the B0, B1 and pickup coils. . . . . . 59Figure 4.7 A fit of the data obtained by varying the B1 field strength to the Landau-Zener model. . . . . . 61Figure 4.8 The phase of the signal from AFP depends on the direction of initial polarization of the sample. 61Figure 4.9 This is a plot of the heights of the second AFP peak (the first data point is the initial peak) whenthe time between peaks is scanned. An exponential decay is fit to the data, approaching thenegative of the initial peak height. T1 is found to be 2.7+/-0.2 seconds. . . . . . . . . . . . . . 62Figure 4.10 The effect of adding a Q-killing resistor to the pickup coil circuit on cross-talk from the B1 coil. 64Figure 4.11 FID circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.12 Initial FID attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.13 The data from the second attempt at FID. Shown are two measurements with the nuclei preces-sion close to resonance with the B1 field, and also the measurement with them far off resonance.The signals from the two on resonance FID measurements should be clear when subtracting theoff resonance baseline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.14 This is the FID signal at B0 = 6.95 A, with an exponential fit through the data. . . . . . . . . . 69Figure 4.15 This is the FID signal at B0 = 6.98 A, with an exponential fit through the data. . . . . . . . . . 69Figure 4.16 The circuit that is used in our spin echo experiment. It is similar to the circuit for FID, withadditional function generators for the pi pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.17 The signal from a spin echo experiment. There is evidence of a spin echo, but the original FIDsignal is missing. This experiment should be repeated for confirmation. . . . . . . . . . . . . . 72ixGlossaryAC alternating currentAFP adiabatic fast passageBBO barium borateCW continuous waveDC direct currentEDM electric dipole momentEMF electromotive forceFID free induction decayIC integrated circuitILL Institut Laue-LangevinIR infraredLBO lithium triborateMDM magnetic dipole momentMRI magnetic resonance imagingNMR nuclear magnetic resonanceOPSL optically pumped semiconductor laserPCB printed circuit boardPPM parts per millionRF radio frequencySEOP spin exchange optical pumpingSNR signal to noise ratioxTTL transistor-transistor logicUBC the University of British ColumbiaUCN ultra-cold neutronUV ultravioletxiAcknowledgmentsI would like to thank Dr. David Jones and Dr. Kirk Madison, my supervisors, for their guidance and help during myMaster’s at the University of British Columbia (UBC). Thanks also to Arthur Mills, Emily Altiere and Eric Miller.Their presence in the lab, their advice, and the conversations I had with them were invaluable for the completionof this thesis. Jeff Maki, a fellow graduate student at UBC, gave me hours of his free time to help me understandthe quantum mechanics involved in the optical transitions and nuclear magnetic resonance (NMR) that are describedin this thesis. Thanks to Bernhard Zender, from the Physics Engineering projects lab, both for his help in usingthe water jet cutter, the laser cutter and the 3d printer effectively, and also for his enthusiasm in helping me, EmilyAltiere and Eric Miller design and construct the apparatuses we needed to conduct our experiments.Special thanks to Dr. Chris Bidinosti and Dr. Jeff Martin from the University of Winnipeg are also in order, fortheir help in my understanding of NMR, and their advice on the various possible coil geometries.Thanks to my family, Uli, Sylvia and Robin, and to my Mother-in-law, Jennifer, and sister-in-law, Rebecca,for all of their support during my time at UBC. Thanks, finally, to my wife Kimberly, for supporting my decisionto go away for two and a half years to pursue this degree. I could not have done this without your patience andunderstanding.xiiChapter 1Introduction1.1 Neutron Electric Dipole Moment (EDM) Experiments1.1.1 Neutron EDMThe neutron is an electrically neutral particle, but it is made up of three charged quarks (an up quark with charge+2/3, and two down quarks, each with a charge of -1/3), so there is the possibility of a non-zero electric dipolemoment (EDM). The neutron is also known to have a magnetic dipole moment (MDM)[2], so a non-zero EDMwould result in a violation of both parity (P) and time reversal (T ) symmetry. This can be seen by analyzing theHamiltonian of a neutron in a magnetic and electric field.H = µnB · S|S| −dnE ·S|S| , (1.1)where µn is the neutron MDM, B the magnetic field it sits in, S/|S| is the direction of the spin vector, dn the neutronEDM, and E the electric field. Under P reversal, P(B · S) = B · S but P(E · S) = −E · S and under T reversal,T (B ·S) = −B ·S but T (E ·S) = E ·S [3]. In both cases, the Hamiltonian is not invariant if the neutron has botha non-zero MDM (µ), and a non-zero EDM (d). Since the neutron is already known to have a non-zero MDM, themeasurement of the neutron EDM is a useful test for fundamental symmetries in the universe. The T violation is ofFigure 1.1: A neutron is made up of 3 charged quarks, an up quark with charge +2/3, and two down quarks,each with a charge of -1/3. Despite being electrically neutral, there is a possibility for the neutron to havea non-zero electric dipole moment1particular interest because under the CPT theorem, it also implies a CP violation. In the standard model, this CPviolation comes from the weak interaction, and it predicts a neutron EDM on the order of 10−32e cm[4]. Modelsbeyond the standard model, such as supersymmetry theories, have additional sources of CP violation, and predict alarger neutron EDM. Thus, constraining this value is one test for the validity of these theories[3].In principle, the method for measuring the neutron EDM is quite simple. A nucleus with nonzero spin, in anelectric and/or magnetic field will precess over time. That is, the direction of the angular momentum of that nucleuswill change. The frequency of this precession is determined by the strength of the electric and magnetic fields aswell as the electric and magnetic dipole moments of the nucleus. To determine the effect from the electric fieldalone (and thus the EDM), the precession frequency can be measured with parallel electric and magnetic fields andthen anti-parallel fields. If the magnetic field stays constant, and the electric field is flipped, the change in precessionfrequency can be used to calculate the EDM.The challenge in performing this experiment comes from the scale of the neutron MDM and EDM. The neu-tron MDM has been measured to be µn = −1.91304272(45)µN, where µN = 1.05155× 10−14e cm is the nuclearmagneton[5]. With the best constraint on the neutron EDM being |dn| < 2.9×10−26e cm[6], this is a difference ofat least 12 orders of magnitude. This massive difference makes the EDM measurement very challenging. It can bemitigated somewhat by making the magnetic field as small as possible and the electric field as large as possible,but there are practical constraints that limit this. For example, electric breakdown limits the possible strength of theelectric field, and the ability to shield the experiment from external magnetic fields limits how small the experimen-tal magnetic field can be. The difference in magnitude between the two terms is still large enough that uncertaintiesin the magnitude of the magnetic field term dominate over the entire value of the electric field term. For this reason,one major goal with current neutron EDM experiments is to reduce uncertainties in the measurement of the magneticfield strength. I will discuss the method the ultra-cold neutron (UCN) project is planning on using to do so in Section1.2.Chapters 1 and 2 will go over some of the theory of how the magnetic field measurement is performed, aswell as some details about the theory of several nuclear magnetic resonance (NMR) techniques and how sources arepolarized. Section 2.7.3 details which techniques are used specifically in the experiments I performed for this thesis.The time constrained reader may wish to refer to that section to help inform themselves about which theory sectionsto concentrate on.1.1.2 Ramsey’s Method of Separated Oscillating FieldsThe neutron EDM is measured using Ramsey’s method of separated oscillating fields. This method involves using asource of spin polarized neutrons, placed initially in a strong, static magnetic field, B0. The method is shown in Fig.1.2. The coordinate system is defined such that the z axis is along the B0 field. The spins of the polarized neutronsare initially spin up, as shown in step a). An RF B1 field, at the frequency of the neutrons’ precession, is then appliedbriefly in a pi/2 pulse to shift the spin vector of the neutrons into the xy plane, step b). The pulse duration can begiven by the following equation:tpi/2 =pi2γB1, (1.2)2Figure 1.2: This is a representation of Ramsey’s Method of separated oscillating fields. The top row showsa Bloch sphere representation of the particle’s spin, and the second the state of the B1 field. This fieldprovides the pi/2 spin flips, and for the method to work, the two flips need to be exactly in phase. TheB1 field is on during the darkened portions of the sine wave, and the grey shows how the phase wouldevolve if the field were on. A) Initially the particles are all spin up. B) An RF pi/2 pulse, at the frequencythat the particles precess, is applied, flipping the spin into the xy plane. C) The particles’ spin precessesfreely in the static field. D) A second pi/2 pulse is applied, in phase with the first. E) If the pi/2 pulsesare exactly on resonance, the spin is now in the down state.where γ is the nucleus’s gyromagnetic ratio, and B1 the strength of the RF field. This is simply the Larmor precessionequation, ω = γB, solved for the time it takes to complete 1/4 of a period. Ideally the B1 field would rotate withthe precessing neutrons, but such a field is very difficult to create. Instead a field oscillating on some axis in thexy plane is used, with a correction to the field’s strength or duration, according to the rotating wave approximation,explained in detail in Section 1.1.3.After the initial pi/2 pulse, the neutrons’ spin vectors are aligned on the xy plane, and will be precessing co-herently around the B0 field, seen in step c) of Fig. 1.2. This coherence will be maintained as long as the fieldis homogeneous over the population of the source neutrons. The effects of inhomogeneities will be discussed inSection 2.4.2. The neutrons are allowed to precess freely for a duration tfp, and are then subject to another pi/2pulse, step d). At this point, if the pi/2 pulses were exactly on resonance with the neutron spin precession due to theB0 field, and were applied for the proper duration, then the entire population will be put into the opposite spin statefrom the original, step e). This is because the two Ramsey pulses are in phase with each other, and at resonance,will also be in phase with the neutron spin. The final spin state is then measured along the z axis. For an out ofresonance pair of Ramsey pulses, the final spin state will be some superposition of the original and opposite spinstate.In order to precisely measure the resonant frequency, this experiment is done many times in succession, withthe frequency of the B1 field adjusted slightly for each data point. The result of these measurements is a patternof fringes, called Ramsey Fringes. A theoretical representation of these fringes is shown in Fig. 1.3. These occurbecause there is a strong revival of the original spin state when the neutrons are exactly opposite in phase compared3to the second Ramsey pulse at the end of the free precession period. The second pulse then flips the spin back to theoriginal state, rather than driving it to the opposite state. There are also periodic dips when they are in phase at theend of the free precession period. If the Ramsey pulse frequency is slightly off-resonant, a pi/2 pulse will actuallyshift the spin by less than pi/2. The final state is then a superposition, with a small amplitude in the original spinstate. These dips get more and more shallow as one moves off resonance. At the resonant frequency, one measuresthe strongest signal in spin in the opposite direction. This is the precession frequency of the neutrons in the electricand magnetic field.Figure 1.3: The Ramsey fringe pattern resulting from doing repeated measurements using Ramsey’s methodof separated oscillating fields, scanning the B1 frequency across the resonance. In theory, the central dipwill go to zero, and the full population of particles will always return to the original spin state if they areexactly out of phase at the end of the free precession time.41.1.3 Rotating Wave ApproximationSince the spin vector of the neutrons is precessing around the B0 field as this pi/2 B1 pulse is being applied, the B1field cannot simply be a constant field. In the ideal case it would be rotating in the xy plane, at the same frequencyas the Larmor precession of the source atoms around the B0 field. However, it is simple to show, using the rotatingwave approximation, that a field oscillating along the x or y axis can also rotate the neutrons’ spin vectors in thesame way as a rotating field would, up to a shift in the effective field strength, called the Bloch-Siegert Shift.Consider a B1 field oscillating along the x-axis, which can be defined like so:B1 = Bcos(ωt)iˆ. (1.3)This field is shown in Fig. 1.4, panel a). Mathematically, it can also be represented as two counter-rotating wavesinstead:B1 =B2(cos(ωt)iˆ− sin(ωt) jˆ)+ B2(cos(ωt)iˆ+ sin(ωt) jˆ), (1.4)shown in panel b). The total field experienced by the neutrons also includes the B0 field, so the total field can bewritten asBtotal =B2(cos(ωt)iˆ− sin(ωt) jˆ)+ B2(cos(ωt)iˆ+ sin(ωt) jˆ)+B0kˆ. (1.5)Now, convert this to a frame rotating at frequency ω ′ in the xy plane. In this frame, the frequencies of the counter-rotating B1 fields are changed, increasing one and decreasing the other. There is also a correction that has to bemade to the total field when considering the effect of the total field on the neutrons,−ω ′/γ kˆ. This correction comesfrom how the time derivative of the angular momentum behaves in the rotating frame, see [7] for the mathematicaldetails.Brot =B2(cos((ω−ω ′)t)iˆ− sin((ω−ω ′)t) jˆ)+ B2(cos((ω+ω ′)t)iˆ+ sin((ω+ω ′)t) jˆ)+(B0−ω ′/γ)kˆ. (1.6)If the rotating frame is chosen to follow the precession of the neutrons, then ω = ω ′. In this frame, one of therotating B1 fields is stationary, while the other rotates at 2ω . Also, B0 = ω ′/γ , completely cancelling the field alongthe kˆ direction. The total effective field experienced by the neutrons isBeff =B2iˆ+B2(cos(2ωt)iˆ+ sin(2ωt) jˆ). (1.7)These two fields are shown in the rotating frame, in Fig. 1.4, in panel c). In this rotating frame, the neutrons willprecess around Beff. This precession is much slower than 2ω , so the quickly rotating part of Beff has little effect.The precession around Beff is almost entirely from the static part of the field.The rapidly rotating field does have a small contribution to the neutron precession, called a Bloch-Siegert Shift.This shift actually manifests itself as a change in precession frequency around the B0 field in the lab frame. Thecorrection is[8]5ωBS =(γB1)22ω. (1.8)For the experiments performed in this thesis, this corresponds to a shift of less than one Hz, compared to theprecession frequency of tens of kHz, and so, is completely neglected. In general, this term becomes less and lessrelevant as the magnitude of B1 is reduced compared to B0. A full derivation of this shift can be found in a 1955paper by Ramsey[9], or for the even more general case of an elliptical B1 field, in the original 1940 paper on thesubject by Bloch and Siegert[10].Figure 1.4: Shown is an oscillating field under the rotating wave approximation. The actual field in the labframe is shown in a). It is decomposed into the clockwise and counter-clockwise parts in b). The twocounter-rotating fields are shown in c), in a frame that rotates with one of the fields. That field is thenstationary, while the other rotates at twice the frequency as it does in the lab frame.1.1.4 Past Neutron EDM ExperimentsEarly neutron EDM experiments used fast-moving beams of neutrons, so the experiments had to be performed invery short time scales. The very first experiment was performed by J. H. Smith, E. M. Purcell and N. F. Ramsey, andused a beam of neutrons at a temperature of about 500 K. They found a neutron EDM of -0.1±2.4×10−20 e cm[11].At the time, a non-zero neutron EDM was not expected, since there was no reason to believe that time reversalinvariance was violated in the universe. However, CP invariance was found to be violated in 1964, so making moreprecise measurements of the neutron EDM became very interesting[12].There were a number of further experiments done using a hot beam of neutrons until techniques were discoveredto cool neutrons further. Ultra-cold neutrons (UCNs) were used starting in the early 80s[13]. The first result froman experiment using UCNs was performed at the Leningrad Nuclear Physics Institute, obtaining a result of |dn| <1.6×10−24 e cm[14]. The neutrons are cooled to about 6 m/s, cold enough to undergo total internal reflection withthe chamber walls. The paper predicts that the neutron EDM measurement can be brought down to about 10−27 ecm with this technique, by increasing the neutron storage time (the storage time in their experiment was about 5 s).6There is a limit to this time, however, because neutrons are known to have a mean lifetime of only 880.7±1.3±1.2s[15].The current upper limit on the neutron EDM is |dn| < 2.9× 10−26, from a measurement performed at Insti-tut Laue-Langevin (ILL)[6]. Modern neutron EDM experiments measure the exact magnetic field strength in theexperiment by performing spectroscopy on a co-habitating atomic species, called a co-magnetometer.1.1.5 Ultracold NeutronsIt was speculated, in the 40s and 50s, that neutrons which were cold enough, would undergo total internal reflec-tion at any angle of incidence, off of certain types of surfaces, and so could be stored in bottles for their entirelifetime[16]. This speculation came from the scattering formula deduced and experimentally confirmed, by EnricoFermi:sin(θ)≤√V/E (1.9)where θ is the angle of incidence at which the neutrons undergo total internal reflection, E is the neutron’s energy,and V ≈ 10−7 eV for many relevant materials, is now known as the material’s ”Fermi potential.”[16] At energiesbelowV , all angles θ satisfy the inequality. This also corresponds to the neutrons’ de Broglie wavelengths becominglarge compared to the inter-atomic spacing of the material of the walls.The first neutron EDM experiment to use UCNs was performed in 1980, in the Leningrad Nuclear Physics In-stitute. There, a Beryllium converter, kept cold by flowing 20 K Helium through it, was used to cool the neutronsto about a velocity of 6.8 m/s. Despite a neutron flux density of about an order of magnitude lower than previousexperiments done at ILL with a warm neutron beam, at LNPI they were able to significantly reduce the upper limiton the neutron EDM measurement, to |dn|< 1.6×10−24[14].1.1.6 Co-MagnetometerNoise and drifts in the magnetic field strength will be seen in the measurement of the neutron precession frequency,so to correct for these, it is necessary to carefully monitor the magnetic field during the EDM experiment. In earlyexperiments, this was done by placing magnetometers around the experimental chamber. Precision was improved byintroducing a magnetometer that co-habitates with the neutrons, a “co-magnetometer”. This has generally been doneusing 199Hg, since it interacts little with neutrons and has a well known EDM and MDM. Such a co-magnetometer(with modifications that will be described in the next section, 1.1.7) will be used in the UCN EDM experiment.The Ramsey method described in Section 1.1.2 will be used to measure the EDM of the neutron, but a simi-lar method will also be used to precisely measure the magnetic field the neutrons occupy. The atoms in the co-magnometer will be polarized along the same axis as the neutrons, and a pi/2 flip will also be applied to them.After this flip, the atoms are allowed to precess freely, and unlike for the neutrons, their precession frequency canbe monitored constantly rather than only at the end of their free precession time, by optical detection, described inSection 1.2. By using atoms with a well-known EDM and MDM, the magnetic field strength can be calculated fromthe measured precession frequency of these atoms. This field measurement is then used to apply corrections to theneutron precession frequency that was measured.71.1.7 Future ExperimentsOne major constraint on the precision of previous neutron EDM experiments comes from systematic effects fromusing 199Hg as a co-magnetometer, such as the ~v× ~E effect. This effect comes about when the 199Hg is movingin a strong electric field. In its own frame of reference, it also experiences an associated magnetic field. Undercompletely random motion, this effect should average out, but if there is even a slight overall rotation of the 199Hg,then the precession frequency will systematically be measured as too high or too low. Since the direction of theelectric field changes during the experiment, the shift from the~v×~E effect changes as well. This looks exactly likethe effect expected from a non-zero neutron EDM, so it is absolutely critical that this effect is well understood andreduced as much as possible in the experiment.There is also inevitably a small gradient in the uniform magnetic field, and thus some field inhomogeneity. So,for the UCN experiment, any 199Hg nuclei that spend more time along the edges of the experimental cell will expe-rience a different field, and different precession frequency than those that do not, introducing additional uncertaintyin the field measurement.Both of these effects can be mitigated by introducing a second type of atom to the magnetometer. This secondatom would be affected differently by these systematic effects, since it would have a different gyromagnetic ratio,as well as a different average velocity at room temperature (and so, experience a different strength of~v×~E effect).This gives us additional data points that can be used to reduce uncertainty in the magnetic field measurement. Theatom chosen needs to interact as weakly as possible with the neutrons themselves, however. 129Xe is an atom thatfits all of these criteria.1.2 XenonXenon is a noble gas, atomic number 54 on the periodic table. One reason 129Xe is ideal as a co-magnetometer inneutron EDM experiments is that it interacts very little with neutrons. This means that a higher density of 129Xe canbe used, which improves the signal to noise ratio (SNR) for the magnetic field strength measurement. Unfortunately,other aspects of the experiment still constrain the pressure of 129Xe that can be used. The major one being thatat the strength of electric fields produced in the experimental chamber, too high of a pressure results in electricalbreakdown.1.2.1 Xe-129 Energy LevelsThe precession frequency of the 129Xe is measured by driving a σ+ transition with an ultraviolet (UV) laser. Twophotons of 252.4 nm will drive the transition[17], and due to the two units of angular momentum added by the2-photon absorption, the spin +1/2 ground state has no available excited state. The relevant transition, as well as theavailable decay channels after absorption are shown in Fig. 1.5. As a 129Xe atom precesses, it moves in and out ofthe this “dark” state, so the absorption rate varies sinusoidally. Since it is a two photon transition, this absorptionrate is very small, so rather than trying to measure it directly, it is much easier to measure the resulting infrared (IR)fluorescence after absorption. Eventually, the 129Xe decays back down to the ground state.A note on the energy level designations and terminology. The angular momenta of various parts of the atom (thenucleus, the electron cloud, the spin, etc.) are each designated by their own quantum number, listed and describedin table 1.1. In this format of writing down the states, we are interested the F, I, and J quantum numbers. The8Figure 1.5: An energy level diagram of the relevant 129Xe transition. The diagram on the left is shown withoutabsorption or decay channels for clarity. Light with wavelength of 252.4 nm drives the transition shownon the right, in a 2-photon process. When measuring the 129Xe precession frequency, σ+ polarized lightis shined on the atoms. Under 2-photon absorption, this adds two units of angular momentum to theatom, meaning that the spin +1/2 ground state has no available excited state. This is called a “dark” state.As the atom’s precession brings it in and out of this dark state, a sinusoidally varying absorption rate isobserved.Letter descriptionS spin angular momentumL orbital angular momentumJ = L + S total electron angular momentumI total nucleus angular momentumF = I + J total atomic angular momentumTable 1.1: The various angular momentum quantum numbers and what they represent.ground state is designated as 5p6(1S0). The term before the parentheses, 5p6, is the term that describes all of the“core” electrons, with all but the outermost layer of electrons truncated. For the ground state, this means that all ofthe orbitals, up to and including the 5p state, are filled. In this case, the total angular momentum of the electronsadds up to 0. Since the electrons must each occupy a different state by the Pauli exclusion principle, the spin andorbital angular momenta of electrons in a filled orbital must add up to 0, and in a ground state noble gas, there areno partially filled orbitals. Because the electrons have no total angular momentum, there is no hyperfine splittingfor ground state 129Xe.The nucleus for 129Xe has angular momentum I = 1/2, so for the ground state, F = 1/2 and mf =−1/2 or +1/2.In the excited state, 5p5(2P3/2)6p, one 5p electron has been excited to the 6p state. The core has one less electron(so, 5p5) and the excited electron is now considered a valence electron, and appears in the term at the end. In thiscase, the total electron angular momentum adds up to J = 2. The nucleus’ angular momentum has not changed, so9I = 1/2 still. Combining J and I, F can take on the values F = 3/2 or 5/2, the antisymmetric and symmetric states,respectively. These F states are only split by about 2 GHz, due to hyperfine splitting. In general, pulsed lasersare too broad, spectrally, to resolve these states, which is unfortunate since they are better suited for detecting a2-photon transition, having much higher peak intensities than continuous wave (CW) lasers. Table 1.2 shows thestate equations for each of the F states and their projections, mf. The coefficients for each state in the superpositionsare found by using Clebsch-Gordan coefficients, which are used when adding up angular momenta in quantummechanical systems.Electrons at the same principle quantum number, n have their degeneracy lifted by various means. Due to theelectron’s spin and charge, it has an MDM. It is moving in an electric field generated by the nucleus and otherelectrons around it, and so, in its own frame, experiences a magnetic field. The energy of the electron is shifted bythe interactions between its MDM and this magnetic field. This is known as spin-orbit coupling, and is responsible, inpart, for fine splitting, the splitting of the energy of different orbital types within a principle energy level. Hyperfinesplitting is due to a similar effect for the proton, where its magnetic dipole moment interacts with the magnetic fieldcreated by the moving electron. This effect is several orders of magnitude weaker, and is responsible for splittingthe F states.Energy levels of the individual mf states are also split via Zeeman splitting in a magnetic field. The shift infrequency for a given mf state is:∆ν =−γmfB02pi(1.10)This shift, and splitting between mf states, is very weak for the fields used in this experiment, however, and can beignored. For the mf = 5/2 state, ∆ν = 4.22 kHz, compared to the hyperfine splitting of the F states, about 2 GHz,and the line-width of the CW UV laser being developed, which is hundreds of MHz.F state mf state mj and mi statesF = 52mf =±52 |52 ,±52〉= |±2〉|± 12〉mf =±32 |52 ,±32〉=√15 |±2〉|∓ 12〉+√45 |±1〉|± 12〉mf =±12 |52 ,±12〉=√25 |±1〉|∓ 12〉+√35 |0〉|± 12〉F = 32mf =±32 |32 ,±32〉=√45 |±2〉|∓ 12〉−√15 |±1〉|± 12〉mf =±12 |32 ,±12〉=√35 |±1〉|∓ 12〉−√25 |0〉|± 12〉Table 1.2: Here are the state equations for all of the possible F and mf states of interest in 129Xe. The F =5/2 states are all symmetric and F = 3/2 states are anti-symmetric. For each F and mf state, the possiblecombinations of mj and mi are shown.1.3 UCN collaborationThe UCN collaboration’s flagship experiment is to measure the neutron EDM at TRIUMF. One of the major improve-ments over previous neutron EDM experiments will be to reduce uncertainty due to systematic errors in the magneticfield strength measurement. This will be done by introducing 129Xe as a co-magnetometer in addition to the 199Hg10used in previous experiments. The precession frequency is measured by driving a two-photon UV transition, wherethe 129Xe precesses in and out of a dark state.1.3.1 High Power CW UV LaserFigure 1.6: A schematic of the UV laser that will be used for 129Xe spectroscopy. This diagram shows workdone by a fellow graduate student, Emily Altiere. The 1009 nm optically pumped semiconductor laser(OPSL) is frequency doubled twice in two optical cavities, using non-linear crystals. The total conversionefficiency is about 10%. Light from this laser will be sent to an enhancement cavity surrounding the129Xe cell. This figure was modified and printed with permission from Emily Altiere. [1]This laser is an IR OPSL that we generate the 4th harmonic from. Figure 1.6 is a schematic of the design. Thelaser itself operates at about 1009 nm, and has a free running line-width of about 100 MHz. Light from the laseris directed into an enhancement cavity, which has a lithium triborate (LBO) crystal in the optical path. This crystalconverts a portion of the light into its second harmonic, which is green light at about 505 nm. The reason for usingan enhancement cavity is that the conversion efficiency of the crystal depends on the intensity of the light going11through it. Outside of the cavity, a single pass of 3W of IR would result in microwatts of green light. With thecavity, we are able to approach 50% conversion efficiency. The output from that cavity is then directed into anotherenhancement cavity, which generates the second harmonic of the green light, or fourth harmonic of the IR, at 252.4nm. The type of crystal used in this cavity, barium borate (BBO), is quite sensitive to damage due, in part, to howeasily it absorbs moisture from the air, so the cavity is sealed and dry air is flowed through it during operation.One of the major challenges in developing this laser is mode matching the light into the cavities. Light couplinginto the cavities must match the size and shape of the beam that can circulate in the cavity. A system of lensesis used to ensure this. Mode matching into the LBO cavity is relatively simple, since the beam is roughly circularalready. It is far more challenging to mode match into the BBO cavity. The green light generated in the LBO cavityis not circular at all, and due to astigmatism from curved mirrors and from the crystal, the vertical and transversemodes of the BBO cavity require a slightly ovular input beam. Cylindrical lenses are used to shape each axis of theinput light’s profile properly.The output we get from these enhancement cavities is about 300 mW of 252.4 nm UV light. This light is thencollimated so that it is circular in profile and mode matched into the experimental cavity.1.3.2 Testing the UV LaserTo test the UV laser’s ability to drive this transition and detect the precession frequency of 129Xe in a magnetic field,I have built a set of magnetic coils that are used to perform NMR. Building and characterizing these coils, and usingthem to perform NMR on protons in water is the work that I will present in the rest of this thesis. The coils are set upto be like a simple version of the co-magnetometer that will be used in the actual neutron EDM measurement. Thereis a cell, containing only 129Xe, placed in the middle of a magnetic field, and after a pi/2 pulse, the 129Xe nuclei willprecess around that magnetic field. Detection can be done with a pickup coil, or with the UV laser, for comparison,and to make sure that there is a signal present when the laser is being tested. Figure 1.7 shows the laser and the testcoils. In Chapter 2 I will go over some NMR theory, and in Chapter 3 I will discuss how to create the magnetic fieldsthat are required for these experiments. The results of my work are shown in Chapter 4.12Figure 1.7: Here is a schematic of the test coils, the UV enhancement cavity, and the 129Xe cell. The actualmeasurement is the intensity of the IR light emitted as the 129Xe decays back down to the ground state,and we can choose to measure either the 895.5 nm or 823.4 nm emission. The back mirror of the UVenhancement cavity is transparent to IR, but the emission is in all directions, so the detector could beplaced elsewhere if needed.13Chapter 2NMR and Free Induction Decay2.1 IntroductionNuclear magnetic resonance (NMR) is a technique that many people are familiar with through magnetic resonanceimaging (MRI), in which NMR is used to non-invasively image organs and other parts of the body. There arenumerous other applications for NMR, such as determining the purity of a sample. This can be done by comparingthe expected signal, given that the sample were 100% pure, to the actual NMR signal obtained, for example. For theultra-cold neutron (UCN) neutron electric dipole moment (EDM) experiment, we are interested in using NMR to veryprecisely determine the strength of a magnetic field.In general, NMR experiments take advantage of the fact that the spin vector of a nucleus precesses around amagnetic field, usually called B0, if the spins are not aligned with that field. The angular frequency of this precessionis given by the gyromagnetic ratio, γ , of the nucleus, which is simply the ratio of the frequency over the magneticfield strength. That is,ωprecession = γB0. (2.1)Precession is described in detail in Section 2.2.1Conventionally, the z axis is taken to be along the B0 field, a convention I will keep in this thesis. The x and yaxes are chosen such that they make a right handed coordinate system. The nuclei are initially polarized along theB0 direction, either due to Boltzmann polarization, or by hyperpolarizing them by external means. These conceptswill be described in Section 2.3. The nuclei’s spins need to be rotated off of the B0 axis in the experiment, which isusually done using a radio frequency (RF), field. Depending on the experiment, this field may only be on for a shortduration, with the goal being a specific rotation of the spin. These are usually called pi/2 or pi pulses, depending onthe desired rotation. These pulses are described in Section 2.7, on free induction decay (FID). Other experimentsmay leave the RF field on for the entire measurement run.There are two types of NMR that are of interest to us. The first is a technique called adiabatic fast passage (AFP).Using this technique, it is relatively easy to obtain a signal, making it a useful tool when trying to find evidence ofhyperpolarization in a sample, as well as to quantify improvements of that hyperpolarization. The other techniqueis free induction decay (FID). This technique can be used to precisely determine strength and homogeneity of a14magnetic field, and also most closely resembles the technique that will be used in the UCN neutron EDM experiment,making it the ideal method for testing the capability of our ultraviolet (UV) laser. Detection in an FID experiment istraditionally done with a pickup coil. This is the method that I use to determine what signal we should expect fromthe optical detection method that will be used later to test the UV laser. AFP and FID are described in Sections 2.6and 2.7, respectively.2.2 Particles in a Magnetic Field2.2.1 PrecessionPrecession is a rotation of the angular momentum vector around some axis over time. A simple example is aspinning top in a gravitational field. If you start it spinning, and the spin axis isn’t aligned perfectly with thedirection of gravity, the top wobbles around this axis. A similar effect happens to magnetic dipoles with angularmomentum in magnetic fields.Consider a particle with spin angular momentum ~S in a magnetic field. It has a gyromagnetic ratio, γ , a dipolemoment ~µ = γ~S, and experiences a torque, ~τ = ~µ ×~B. This torque is perpendicular to both the direction of themagnetic dipole (which is axis of angular momentum for a non-zero spin nucleus), and the magnetic field. Theresult is that the particle’s spin vector ~S rotates around the magnetic field.Torque is defined as the change in angular momentum over time, or~τ =d~Sdt. (2.2)But, as before, we also have~µ = γ~S, (2.3)which can be combined to getd~Sdt= γ(~S×~B). (2.4)If ~B is along the z axis, as B0 is usually defined in NMR experiments, the form that the solution to equation 2.4will take is~S(t) = (Sxy cos(γBt+φ),Sxy sin(γBt+φ),Sz). (2.5)The tip of ~S traces out a circle at height Sz on the Bloch Sphere, over time. Equation 2.1 is easily taken from 2.5; theprecession frequency is ω = γB, or f = γB/2pi . The direction of the precession can be determined by the right handrule (or by working out the cross product). It is important to note, however, that if γ is negative, such as for 129Xe,the precession will be in the opposite direction. Figure 2.1 shows precession of the spin of a particle with positive γin a magnetic field pointing along the z axis.15Figure 2.1: The spin vector of a particle precesses in a magnetic field, around the field axis. The right handrule is useful to determine the precession direction, given by ~S×~B. It is important to note, however, thatparticles with negative gyromagnetic ratio, γ , will precess in the opposite direction. The figure showsprecession for a particle with positive γ .2.2.2 Projective Measurements of Spin Angular MomentumPart of any NMR experiment is a measurement of the nuclei’s spins along a particular axis. In some methods, thismeasurement results in a projection of the spin state onto that axis. In the case of a spin-1/2 particle, this projectionthen determines the probability of each possible state, +1/2 or -1/2, being measured.I will follow the derivation in Sakurai’s “Modern Quantum Mechanics” [18], with details changed to best fit mywork. I will take the x axis to be the axis that the spin is measured along, and the z axis to be the axis the staticmagnetic field, B0 is along. This is the axis the spins will be precessing around. The y axis direction is such that thecoordinate system is right handed.First, take the case where a single particle’s spin is precessing in the xy plane at angular frequency ω . In amagnetic field, with no electric field, the Hamiltonian of this system isH =−µ ·B. (2.6)The energy eigenstates of a spin 1/2 system are thenE± =∓h¯µB. (2.7)The precession frequency is ω = µB. When H is time independent, like in the case of a static magnetic field, thetime evolution operator is given byU (t, t0) = exp(−iHh¯), (2.8)and as long as this operator acts on an energy eigenstate, the Hamiltonian in the exponential can be replaced by theenergy of the state being acted on. In this case, there are two eigenstates, spin up and spin down, with energies given16by equation 2.7. Note that the energies of these states depend on B; in the absence of a magnetic field, the two spinstates are degenerate.Equations 2.6, 2.7, and 2.8 can be combined to rewrite the time evolution operator, in a static magnetic field, interms of the frequency of precession:U (t,0) = exp(−iωSzth¯)(2.9)In general, the initial state is superposition of spin up and spin down.|Ψ, t = 0〉= c+|Sz+〉+ c−|Sz−〉. (2.10)If, at some point the particle is measured to be in the |Sx+〉 state, there is an equal chance of subsequently measuringspin up or spin down along the z axis, so the initial state in that case is|Sx+, t = 0〉= 1√2|Sz+〉+ 1√2|Sz−〉 (2.11)Now, add the time dependence to see how this state evolves in time as the particle’s spin precesses around themagnetic field along z.|Ψ(t)〉= 1√2exp(−iωt2)|Sz+〉+ 1√2exp(iωt2)|Sz−〉 (2.12)The probability, then, to measure spin up along the x axis, Sx+, over time isP(|Sx+〉, t)= |〈Sx+ |Ψ(t)〉|2 =∣∣∣∣∣[c+〈Sz+ |+ c−〈Sz−|]·[1√2exp(−iωt2)|Sz+〉+ 1√2exp(iωt2)|Sz−〉]∣∣∣∣∣2(2.13)When multiplying this through, it is useful to remember the following properties:〈Sz+ |Sz+〉= 〈Sz−|Sz−〉= 1 (2.14)〈Sz+ |Sz−〉= 〈Sz−|Sz+〉= 0 (2.15)The result is the following probability:P(|Sx+〉, t)= ∣∣∣∣12exp(−iωt2)+12exp(iωt2)∣∣∣∣2 = cos2(ωt2)=12+12cos(ωt) (2.16)So, the probability of measuring positive spin along the x axis from a particle that is precessing in the xy plane overtime is sinusoidal with frequency equal to the precession frequency. This result should not be surprising, as one cansimply think of how the projection of the spin onto the x axis changes as it precesses around the z axis on the Blochsphere.It is also useful to look at a more general case. In actual experiments, there is always the possibility that the spinflip is not exactly the correct magnitude or duration. That is, the flip may cause the spin of the particle to over- or17Figure 2.2: A particle with a spin state that is a superposition of spin up and spin down along z, with the am-plitudes of each state given by the angle θ . The arrow is only there to more easily define θ . In a quantummechanical spin system, the “coordinates” on the Bloch Sphere cannot all be known simultaneously, sothe state is more accurately represented as a ring at some height.undershoot the xy plane on the Bloch Sphere. Furthermore, as will be discussed in Section 2.4.1, the particles’ spinswill relax toward thermal equilibrium in the magnetic field over time, which is a rotation of the spin vector towardthe z axis on the Bloch Sphere.Take angle θ to be the polar angle in spherical coordinates (shown in Fig. 2.2), that is, 0◦ is up along the z axis,90◦ in the xy plane and 180◦ down along the z axis. The initial state is described quantum mechanically by choosingc+ and c− appropriately. Using the Bloch Sphere makes this easy, by looking at the projection of the spin vectoronto the xy plane, c+ = cos(θ/2) and c− = sin(θ/2). Plug that into equations 2.10 and 2.13, to get:P(|Sx+〉, t;θ)= (12 + cos(θ/2)sin(θ/2))cos2(ωt2)+(12− cos(θ/2)sin(θ/2))sin2(ωt2)(2.17)This is the probability of measuring spin + in x for a particle whose initial z spin state is arbitrary, defined by θ ,and then precesses in a magnetic field along the z axis. Notice, the probability becomes 50% when θ = 0◦ or 180◦,as expected, and equation 2.16 is recovered when θ = 90◦. Measuring the spin along the y axis instead simplycorresponds to a phase shift in the probability to measure spin up or spin down.2.3 Spin Polarization2.3.1 Boltzmann PolarizationA sample that is polarized according to thermal equilibrium is said to be “Boltzmann polarized.” In the absence ofan electric or magnetic field, the spin states in a given orbital have the same energy (although it turns out that even18the electric and magnetic fields from the proton(s) in the nucleus are enough to measurably break this degeneracy).A magnetic field will break the degeneracy between spin states along the direction of the field. Since there is adifference in energy, atoms will not necessarily occupy those states in the same proportion. Instead, the distributionof states in the sample will depend on its temperature and the energy difference between states. In a spin 1/2 systemthis results in an imbalance between the two spin states. That is, the system will polarize to some degree.Take a proton in a magnetic field, B, for example. This is a spin 1/2 particle, so there are two spin states, whichare not degenerate in this field. The energy difference between the two states is ∆E = 2µpB. When the system is inthermal equilibrium, the protons will follow the Boltzmann distribution:Figure 2.3: These are the energy levels of interest for optically pumping Rubidium. Initially, the ground spinstates are populated according to the Boltzmann distribution. Atoms in the dark state cannot absorb thecircularly polarized 795 nm light [19], and stay in that spin state. Atoms in the bright state will absorba photon and eventually decay randomly back to either the bright or dark ground state. Any atom thatdecays into the dark state (the +1/2 state) will stay there, so over time the dark state becomes morepopulated and the sample becomes hyperpolarized.f (E) = 1− 1Aexp( EkT) (2.18)where f is the probability of a given particle being in the state with energy E, k is the Boltzmann constant, T thetemperature, and A is a normalization constant. The degree of polarization, N+/N−, can be determined from thedifference in probabilities for each state:19N+N−= exp(−∆EkT). (2.19)For a room temperature sample of protons, this corresponds to a polarization of about 4.4 parts per million (PPM)[20].If such a sample is subject to a pi/2 pulse, this polarization remains, and there will be a measurable signal fromthe spin precession, unlike in the case of a completely unpolarized sample. Even though the polarization is small,a Boltzmann polarized sample of liquid water can be used to generate a measurable NMR signal from protons (byapplying the proper fields to the hydrogen atoms in the water molecules).2.3.2 HyperpolarizationA sample that has been polarized beyond its Boltzmann polarization is said to be “hyperpolarized.” A hyperpolarizedsample can generate much larger signals than a Boltzmann polarized sample, or generate a signal of similar strengthwith far fewer particles. This makes it possible to perform NMR on a thin gas. There are several ways to achievehyperpolarization, but a commonly used one is optical pumping.To perform optical pumping, circularly polarized light is shined on the atoms, such that for a given spin state,there is no available excited state due to the angular momentum that would be added by absorbing that photo. Atomsin the other spin state, the “bright” state, are able to absorb a photon, however. After absorption, the excited atomswill eventually decay back to the ground state, and end up in a random spin state again. Those that end up back inthe bright state will absorb another photon, but those in the dark state will remain there. Over time, the dark stategets filled up, and the entire population is transferred to that state.Figure 2.4: A schematic for doing optical pumping of Rubidium. The half wave plate and polarizer are usedto ensure that the light is polarized (light reflected from the polarizer is dumped or can be used as away to measure the power of the laser if the degree of polarization from the laser is known and does notchange over time). A quarter wave plate then circularly polarizes the light. A pair of lenses acts as atelescope to blow up the beam to about the size of the Rubidium cell and collimate it. After the cell,light is focused onto a spectrum analyzer. In general, some method of attenuating the light is necessaryto avoid damaging the analyzer, which is not shown in this diagram. Methods include reflecting the lightoff of a piece of glass, or using a neutral density filter. Figure 2.5 shows the change in signal expectedwhen shining linearly polarized light versus circularly polarized light into the cell.Rubidium is often hyperpolarized using this technique. Light from a 795 nm laser [19] is circularly polarizedusing a quarter wave plate. The relevant states are shown in Fig. 2.3. This beam is expanded to fill the entirety ofthe cell containing Rb. With the circularly polarized light, the cell quickly becomes transparent as the dark state isfilled. If the quarter wave plate is rotated so that it passes the linearly polarized light through, unchanged, the cellis no longer transparent to the light. By focusing the light onto a spectrum analyzer, it is possible to check for this20Figure 2.5: Shown are the expected signals from a spectrally broad laser being shined on a Rubidium cellunder linear polarization (left plot), and circular polarization (right plot), after sufficient time for opticalpumping to occur. When shining linearly polarized light, such as the schematic shown in Fig. 2.4 withthe quarter wave plate removed or set such that it does not alter the linear polarization, the Rubidium isable to absorb all of the light within the transition’s bandwidth. However, when the light is circularlypolarized, one of the spin states in the ground state does not have a corresponding excited state withthe correct angular momentum to transition to. Atoms in this “dark state” do not absorb photons. Afterseveral decay periods, all of the atoms are driven into this dark state as they absorb a photon and thendecay randomly back into the bright state (where they can absorb another photon) or the dark state (wherethey remain and are transparent to the light). Once all of the atoms are in the dark state, the entire cell istransparent and the dip in the spectrum disappears.change in absorption. Figure 2.4 shows a typical optical pumping setup, and Fig. 2.5 shows the difference in thespectrum between sending linearly polarized light vs. circularly polarized light through the Rubidium cell.To polarize the 129Xe in our experiments, we use a process called spin exchange to transfer spin polarizationfrom a sample of optically pumped Rubidium to the 129Xe through van der Waals interactions. Nitrogen is used asa mediator in these interactions. This technique overall is called spin exchange optical pumping (SEOP) [21]. Thesetup involves flowing the 129Xe mixture through the Rubidium cell, and directly from there to the experimentalchamber.2.4 Spin RelaxationA sample that is in a static magnetic field aligned along the z axis, but polarized along some axis in the xy plane willlose that polarization over time. There are several of these relaxation mechanisms, and they are generally describedby their time scales, called T1, T2, and T∗ T1The T1 lifetime is defined by how quickly the magnetization of the atoms in the direction of the B0 field reachesthermodynamic equilibrium, or Boltzmann polarization. This necessarily results in a de-magnetization of the atomsin the xy plane. This effect is shown in Fig. 2.6. If the sample is initially polarized along x or y, the magnetizationalong the z axis over time is21Figure 2.6: Shown is the effect of T1 relaxation. The spin moves toward thermodynamic equlibrium along thez axis (specifically, along the direction of B0). This causes the projection in the xy plane to decrease inamplitude, and a loss of signal strength in NMR experiments.Mz(t) =Mz,eq(1− e−t/T1). (2.20)In this case, the T1 time is the time it takes for the sample to recover 1−1/e or about 63% of its magnetization atthermal equilibrium.Historically, this is also known as the spin-lattice relaxation time, since this relaxation depends on interactionswith the sample’s surroundings. This time becomes very short when the rotation rate of the molecules or atoms issimilar to that of the Larmor precession frequency. In general, stronger magnetic fields are associated with longerT1 times, since the rotation rate is generally smaller than the precession frequency, even with the B0 field at around atenth of a mT in strength. Stronger magnetic fields bring the precession frequency even further from this resonance.2.4.2 T2 and T∗2The T2 lifetimes are determined by local field inhomogeneities, which cause atoms to precess at different frequen-cies, resulting in dephasing over time. This effect is split into two types, the T2 lifetime, caused by time-varyinginhomogeneities, and T∗2, caused by inhomogeneities that are constant or slowly varying over the lifetime of theexperiment. It is not strictly correct to call T∗2 a “relaxation” process, since it is not random, and in fact, as will bedescribed in this section, the dephasing caused by these effects can be reversed, and a signal can be recovered in aspin echo experiment.The T2 lifetime describes how the component of the magnetization perpendicular to B0 relaxes:M⊥(t) =M⊥(0)exp(−tT2− γ∆B0), (2.21)Where ∆B0 is the difference between the maximum and minimum field strength in the region of interest. This is22also called spin-spin relaxation, since one possible source of a rapidly time-varying inhomogeneity occurs when twoatoms move past each other. The field one atom generates can perturb the total field, and the precession frequency,of the other. Whenever two atoms interact in this way, they are dephased from the rest of the sample. The T2 lifetimealso includes other rapidly varying inhomogeneities such as atoms moving across spatially small regions of constantinhomogeneity, or random fluctuations in the field to current noise, and so on. Any process that causes atoms toexperience a non-uniform field that changes on the time scale of the experiment is included in the T2 lifetime.The T∗2 lifetime is caused by a gradient in the B0 field. Two atoms that are spatially separated experience adifferent, but constant in time, magnetic field, so they have different precession frequencies. The T∗2 lifetime isdefined as1T∗2=1T2+ γ∆B0. (2.22)Because this inhomogeneity is invariant in time, it does not cause a random dephasing process and it is theoreticallypossible to recover a signal from a sample that has dephased this way. Note that in general as you increase thestrength of B0, its gradient will also increase, making the T∗2 time faster. In fact, it is the number of periods ofprecession it takes for the sample to dephase that will stay constant (ignoring non-linear effects), rather than thetime.2.4.3 Spin EchoFigure 2.7: A spin echo experiment. The atoms begin with spin aligned with the B0 field. A) They are given api/2 pulse. B) They precess freely in the static field. Atoms in a stronger part of the field precess faster,colored red here. Those in a weaker field precess more slowly, colored blue. Since the field strengthvaries smoothly, there will be a distribution of phases, shown as a gradient here. C) After some time,a pi pulse flips the spins around the x axis. The more slowly precessing atoms are now ahead in phasecompared to the more rapidly precessing atoms. D) After a time equal to the time between pulses, theatoms are all in phase again.T∗2 and T2 can be determined by performing a spin echo experiment. In such an experiment, a sample that ispolarized along the B0 direction is first subject to a pi/2 pulse. The spins precess around the B0 field, but due to23Figure 2.8: Here is what a demodulated signal from a spin echo experiment should look like. Multiple echoesare shown, but only the first is annotated. The numbers correspond to the stages of the spin flip experimentdescribed in Fig. 2.7. The signal is recovered after each pi pulse, but is weaker each time. The peakenvelope can be fit exponentially, with a decay constant due to the T1 and T2 times. T∗2 is given by howquickly the signal is lost after each peak.the field’s gradient, the individual nuclei precess at slightly different frequencies. This results in dephasing andeventually the signal is lost. This decay in signal will be exponential, with T∗2 as the time constant. A pi pulse thenrotates all of the spins about some axis in the xy plane. This effectively swaps the phases of all of the atoms. Theslower precessing atoms are now ahead in phase compared to the faster precessing atoms, and vice versa. After anamount of time equal to the time between the pi/2 and pi pulses, the atoms are briefly back in phase, and a signalis seen again. This is called a spin echo. The echo is always weaker than the initial signal because the polarization(and thus, magnetization) is never completely recovered. All of the factors that contribute to weakening the echo,such as noise in the field, nuclei interacting with each other, and so on, determine the T2 lifetime.This experiment can be repeated, changing the delay between pulses, and by tracing the envelope of peakheights, one should see an exponential decay. The time constant of this decay is T2. A spin echo experiment isdetailed and diagrammed in Figs. 2.7 and Detection2.5.1 Detection Via Pickup CoilA common method for measuring the precession frequency is to measure the electromotive force (EMF) induced byFaraday induction from the changing magnetic field generated by the spin of the particles. Faraday induction on asingle loop of wire is given by:ε =−dφdt(2.23)where ε is the EMF and φ = BA is the magnetic flux through the coil; the field strength times the area enclosed bythe loop. For N loops, the signal is simply multiplied by N, so24εtot =−N dφdt (2.24)The negative sign is due to Lenz’s law. The EMF generated drives a current that creates a magnetic field whichopposes the change in the field due to the precession of the nucleus.In a stronger magnetic field, the nucleus precesses more rapidly, which means that the magnetic field generatedby the its spin changes more rapidly, and generates a larger EMF in the pickup coils. However, as discussed inSection 2.4 and Chapter 3, a stronger field also results in a signal that decays more rapidly, unless the B0 field ismade more homogeneous.The signal detected in the pickup coil will be sinusoidal, as dφ/dt will change based on the angle made betweenthe pickup coil axis and the polarization axis. The frequency of the signal will be exactly the precession frequencyof the nuclei, and since that should be well known, one can use a lock-in amplifier to amplify the signal and filterout noise at other frequencies.2.5.2 Optical DetectionFigure 2.9: 129Xe has a dark state when driving the 252.4 nm two photon transition from the ground state usingcircularly polarized light. When using σ+ light, the +1/2 state cannot absorb the UV light, since there isno excited state with the proper amount of angular momentum. The atoms in this state are transparent tothis light.Detection in the UCN experiment will be done optically, instead of using a pickup coil. Circularly polarizedlight will be shined on the 129Xe. For one spin state in the 129Xe ground state there is no excited state available dueto the additional angular momentum from the absorbed light, creating what is called a dark state, shown in Fig. 2.9.Atoms in the dark state will not absorb this light, so the absorption is dependent on the spin state of the 129Xe. As25the atom precesses, the probability of absorption varies sinusoidally, at the precession frequency of the 129Xe.This type of measurement results in a projection of the atom’s spin along the axis of absorption, so the behaviourof the signal should look like it is described in Section 2.2.2. An atom that absorbs a photon also ends up dephased,since the decay back to the ground state is completely independent of the phase of precession in the atoms aroundit. This can be an issue when there is a high rate of absorption, but in the case of 129Xe, we are driving a two photontransition, so the absorption rate is very low, and dephasing from this is negligible.The amplitude of the signal from this method depends entirely on the absorption rate of the UV light, and isindependent of the rate of precession. This is useful for the UCN EDM measurement, since the magnetic field willbe made as weak as possible. Unfortunately this absorption rate is very low, so it is much easier to detect theinfrared (IR) emission as the 129Xe decays back to the ground state rather than try to detect the actual absorption.The relevant 129Xe energy levels, and wavelengths of the light used and measured are shown in Fig. 2.9. There aretwo IR decay channels from the excited state, and we will decide on which one to measure based on calculations ofthe branching ratio between the two channels.2.6 Adiabatic Fast Passage (AFP)2.6.1 OverviewIn an AFP 1 experiment, nuclei are first polarized along the B0 field direction. This field is also called the static field,although it is actually ramped slowly. The RF field, B1, is left on for the duration of the experiment. For most ofthe B0 ramp, the B1 field is out of resonance with the nuclei’s spin precession. In this case, it has little effect onthe spin of the nuclei. However, as the ramp scans through the field strength at which the B1 field is resonant withthe nuclear spin precession, this spin gets flipped. The B0 field then ramps back out of resonance and the spin axisremains stationary again until the next ramp.To understand how the spin flip works in detail, it is useful to look at a rotating reference frame, that rotatesat the B1 frequency. Figure 2.10 shows the spin flip process. In this rotating frame, the nuclei and fields behavesomewhat differently. B1 is a static field, and the precession frequency of the nuclei around B0 is ω ′ = ω −ωB1 .This also means that the B0 field strength is effectively reduced, so B′0 =B0−ω/γ .Starting at the peak of a ramp of B0, the nuclei are precessing slightly faster than the B1 frequency. That is, inthis rotating frame, ω ′ is small and positive. The strength of B′0 is then also small and positive, and the total effectivefield felt by the nuclei is Beff =B′0+B1. At the peak of a ramp, Beff is dominated by B′0, and the field points mostlyalong the z axis, which is the axis of the B0 field and of the nuclei’s initial polarization. As the B0 field ramps down,however, B′0 approaches 0, and Beff tilts towards the xy plane, as can be seen in the first two plots in Fig. 2.10. If theramp is slow enough to be adiabatic, the spin of the nuclei will follow Beff as it tilts.The ramp down continues, and B′0 becomes negative, so Beff tilts below the xy plane, until B′0 once againdominates over B1, this time in the negative direction. The polarization has been transferred to the spin down state.This process is shown in the last plot in Fig. 2.10. This figure also shows how the spin vector rotates on the Blochsphere during the spin flip.1The name “adiabatic fast passage” may sound odd, but it is called fast to differentiate it from adiabatic slow passage, where the ramp isslow compared to the relaxation times. A detailed description is found in Bloch’s 1946 paper on nuclear induction[22].26Due to this new polarization direction, the nuclei are not in thermal equilibrium anymore, however, so over time,determined by the T1 lifetime, they will relax back to a thermal polarization. The B0 ramp needs to be quick enoughthat this relaxation takes a long time compared to the spin flip.2.6.2 Speed of the B0 rampThere are some constraints on how quickly or slowly B0 can be ramped in a successful AFP experiment. Forobvious reasons, the B0 field must ramp through the spin flip faster than the time constant for any relaxation process(relaxation effects are discussed in a previous section, 2.4). That is,B1T1,B1T∗2<<dB0dt. (2.25)B0 must also ramp slowly enough, though, for the spin to follow adiabatically. The spins of the nuclei will “lock”to the local direction of the magnetic field as long as that field does not change too rapidly. The condition foradiabaticity in this case is[23]:dB0dt<< γB21. (2.26)In general, these constraints on the ramp speed are actually rather forgiving, since the relaxation times and theadiabatic condition tend to be quite far apart.2.6.3 Detection and SignalDetection is usually done via pickup coil, which is the method we use in our setup for AFP. The signal is a voltage,generated by Faraday induction from the precessing spins, in the pickup coil. Since the B0 field is ramped repeat-edly, a signal is produced every time the precession frequency scans through the B1 frequency. For hyperpolarizedsamples, the signal decays over time as the sample returns to the Boltzmann polarization, but for Boltzmann polar-ized samples, this repetition is seen indefinitely with no loss. This is a tremendous advantage, since such samplescan be used to average a signal over many ramps, to greatly improve signal to noise ratio (SNR). This is invaluableespecially when working with a newly constructed setup, where the noise characteristics are not yet well known.A well tuned and noise controlled AFP experiment should produce an easily recognizable signal without averaging,however.The signal itself is a sin wave, which increases in amplitude as the ramp approaches resonance. Each ramp upor down produces a peak in the envelope of the wave. The height of this peak, as well as the phase of the sin waveitself within this envelope, depends on how quickly the ramp is repeated compared to the T1 lifetime. Because ofthis, it is possible to use AFP to measure the T1 time of the sample used, by varying how quickly the ramp throughresonance is repeated.27Figure 2.10: Adiabatic fast passage in a frame rotating at the B1 frequency. B1 is static in this frame, andthe total effective magnetic field, Beff is the sum of the fields B0, B1 and ω/γ . Beff tilts from pointingup along the z axis to down along the z axis as B0 ramps through the B1 resonance. At the top, thedirections and strengths of the fields are shown at various points on the B0 ramp. During the ramp, themagnitude of B0 crosses over the magnitude of ω/γ . At the bottom a Bloch Sphere representation ofthe direction of polarization through this process is shown.282.7 Free Induction Decay2.7.1 OverviewFree induction decay is, in principle, a very simple NMR experiment. Nuclei are spin polarized and placed in amagnetic field, B0, and are allowed to precess freely after a pi/2 pulse is applied. Over time, the nuclei de-polarizedue to various relaxation processes that were described in Section 2.4.A pi/2 pulse is an RF pulse that is applied for exactly the duration necessary to rotate the spin vector by pi/2radians. It is driven by the B1 field. Since the pulse needs to be applied for a precise period of time, effects fromthe rotating wave approximation need to be taken into account. The approximation is explained in detail in Section1.1.3, but I will repeat the basic principle here. The nuclei are precessing, and their spin vectors trace out a circle onthe xy plane. The pi/2 pulse would ideally be driven by a field that follows the spin vector precisely, but such a fieldis difficult to create. A field that oscillates along some axis in the xy plane is equivalent to two counter-rotating fieldson that plane. The field following the precession drives the pi/2 flip, and the field rotating in the opposite directiondoes so quickly enough to have a negligible effect on the spin flip (this small effect is called the Bloch-Siegert shiftand is small enough to ignore for FID). The main thing to note is that such a field has only half the effect on thenuclei’s spins than a field that only rotates along with the spin vectors, so twice the duration than one might expect isneeded to drive the pi/2 pulse. To determine the pulse duration needed, one can simply calculate how long a fourthof a period of precession is for a given B1 field strength:ω = γB1 (2.27)T =2piγB1(2.28)The duration of the pulse is a fourth of this period, but with half of the strength of B1, soTpi/2, RWA =piγB1. (2.29)with B1 being the actual amplitude of the RF pulse.After the pi/2 pulse, the polarization is measured as it diminishes over time due to relaxation effects. FID isan excellent way to measure the homogeneity of a magnetic field, as long as the T1 and T2 times are already wellknown, or are at least known to be much longer than T∗2, since inhomogeneities have a clear and easily measuredeffect on the decay rate of the FID signal.2.7.2 Detection and SignalIn essence, FID is a method for determining how quickly a non-equilibrium spin magnetization returns to equilibriumin a sample of atoms, and so, a potentially very accurate way to measure the B0 field’s homogeneity. Unlike AFP,which is quite resilient to relaxation processes, the signal in an FID experiment decays on the T∗2 time scale. With aBoltzmann polarized sample, FID can be repeated if the sample has been given enough time (several T1 periods) toreturn to its Boltzmann polarization. If the experiment is repeated before this T1 relaxation, the amplitude of the FIDsignal will be diminished. With a hyperpolarized sample, the sample needs to be replaced to repeat the experiment.29This makes it very difficult to average many FID runs, potentially limiting the SNR that can be achieved.Detection can be done with a pickup coil, just like AFP, where Faraday induction generates an EMF in the coilas the spins precess. The signal is sinusoidal, with frequency equal to the precession frequency of the sample in theB0 field, with an exponentially decaying envelope. Unless the pi/2 flip is repeated after the sample relaxes back toBoltzmann polarization, that is the entirety of the experiment. A spin echo can also be performed, by applying a pipulse after T∗2 relaxation is over, but not T1. The details are explained in Section 2.4.2. The result is a brief revivalof polarization, but weaker than the initial signal.2.7.3 FID in the Context of UCNFID is the technique that will be used to measure the magnetic field in the neutron EDM experiment. Detectionwill be done optically, with two atomic sources used simultaneously to independently measure the field and reducesystematic effects that occur when only one source is present. It is somewhat of a misnomer to call this method freeinduction decay, since there is no induction being measured, however, the technique differs only by the detectionmethod. A continuous wave (CW) UV laser has been developed by another graduate student in our group, EmilyAltiere, which will be used for detecting the precession frequency of 129Xe.To demonstrate that this laser is capable of measuring the precession frequency of the 129Xe, I am constructinga set of coils to generate the necessary fields for FID. The first test is to perform AFP on Boltzmann polarized water(protons), with a pickup coil. This has several benefits. One is that detection with a pickup coil depends on fewervariables than with the laser. We do not need to ensure a precise wavelength, and we do not need to worry aboutalignment between many optical components, cavities and detectors. Using water as a source also eliminates theuncertainty in how well polarized 129Xe we have, as well as allowing us to see a baseline to compare the eventual129Xe signal to. The signal from water can also be repeated and averaged over many runs, since its polarization isbased on thermal equilibrium. After AFP, FID is performed on the water sample as well, in order to more preciselycharacterize the magnetic field generated by the coils. These two experiments are the scope of this thesis.In the future, the coils can be adjusted and optimized based on the results of these experiments, and then thesame experiments can be performed on hyperpolarized 129Xe, to determine the degree of polarization achieved, andto determine what signal to expect from the optical detection. The 129Xe itself is externally polarized via SEOP andthen transported to the measurement cell.On the optical side of the experiment, the first step is to see successful 129Xe spectroscopy. Simply, we willshine UV light on the 129Xe atoms and look for the IR emission. This will be done in a UV enhancement cavity.After that, we will look for a precession signal.Once it has been determined that the laser is capable of measuring the magnetic field, it will first be used tomeasure the 129Xe EDM. This value is not yet known to enough precision to achieve the precision goal of the UCNneutron EDM measurement.30Chapter 3Coils and Fields3.1 IntroductionThe experiments described in this thesis require several different magnetic fields, with different constraints oneach of them. There is a strong, static magnetic field, B0, and a weaker, radio frequency (RF) field that oscillatesat the source particle’s precession frequency in the B0 field, the B1 field, which must be perpendicular to B0.There also needs to be a way to measure the precessing nuclei’s spins, one method being to use a pickup coil andmeasure the Faraday induction as the magnetic field generated by the polarized particles changes direction duringprecession. Such a coil needs to be perpendicular to B0 as well, and as will be described in Section 3.7, should alsobe perpendicular to B1.These experiments rely on a large number of in phase nuclei to make measurements. The precession frequencyof the nuclei depends on the B0 field, and since nuclei with even slightly different precession frequencies within asource will rapidly go out of phase, the magnitude of B0 needs to be very uniform. The exact requirement dependson experimental factors, such as the actual precession frequency of the nuclei, and the in phase time needed to makea reasonable measurement. The bulk of this chapter will discuss the generation of a homogeneous B0 field underthe conditions imposed by the optical measurement with our ultraviolet (UV) laser.The fields themselves are generated by an electrical current, and there are several potentially useful coil geome-tries for creating a uniform field. In Sections 3.2 and 3.3 I will go over the two geometries I considered for the B0coils, the Helmholtz configuration, and the saddle configuration. To determine their usefulness in these experiments,I coded some simulations, and took advantage of some known analytic solutions to compare my results along theaxes where the analytic solution is easily solvable. The goal is to make as homogeneous as possible a B0 field,across the experimental region, which is a glass cell, 200 mm long and with a diameter of 25.4 mm, filled with129Xe. One challenge is that the B0 field needs to be directly above a steel optical table. As will be discussed inSection 3.4, the steel distorts the magnetic field, potentially increasing the field’s inhomogeneity.Figure 3.1 shows the coils that are used in this experiment (in this case with saddle coils for the B0 field), andthe cell, above the steel optical table. B0 points straight up out of the table, and defines the z axis in this thesis.The x axis is down the long axis of the 129Xe cell, and the y axis is oriented such that the coordinate system is righthanded. The center of the cell defines the origin.All of the Matlab code used in the simulations described in this chapter is reproduced in the Appendix.31Figure 3.1: This is a schematic of the coils needed to perform NMR experiments, shown with Saddle shapedcoils, as well as the 129Xe cell that will be used. The entire setup is suspended above a steel optical table.The cell is a 25.4 mm diameter cylinder, 200 mm long, and placed in the center of the setup. A pickupcoil is wrapped around it (red). The B0 (orange) and B1 (blue) coils are also shown.3.2 Helmholtz ConfigurationThe Helmholtz configuration is named after the German physicist Hermann von Helmholtz. It is a very simplegeometry for creating a uniform magnetic field in a small space between the coils. This configuration, shown in Fig.3.2, consists of two loops of wire of equal radius. There is an equal current, running in the same direction, througheach loop, and they are separated by their radius. The geometry can be adjusted by changing their separation ifFigure 3.2: (a) A pair of current loops in the Helmholtz configuration. The distance between them is equal tothe radius of each loop, and the field generated points straight up. The field is relatively homogeneousnear the center of the coils. (b) The anti-Helmholtz geometry, where the current in each loop runs inopposite directions. The magnetic field generated from these is a gradient, which is linear near the centerof the coils. The field strength at the center is 0.32certain effects are desired. For example, by moving them farther from each other, there is a local minimum inthe field at the center. The current in one loop can also be reversed, to produce an anti-Helmholtz geometry. Theresulting field is a linear gradient.Calculating the field along the z axis for the Helmholtz configuration is simple, since all of the field componentsother than the z component cancel out. A loop of wire in the xy plane generates a field on the z axis that is givenby[24]:Bz(z) =µ0I2R2(R2+ z2)3/2. (3.1)Modifying the equation to get the field of a Helmholtz pair is simply a matter of moving the coordinates a bit so thatthe origin is at the center of the pair, and adding the contributions of both coils.Bz(z) =µoI2R2(R2+(z+R/2)2)3/2+µoI2R2(R2+(z−R/2)2)3/2 . (3.2)The field elsewhere does not have a simple analytic solution, and to calculate it, I used a finite differencessimulation instead. In such a simulation, a differential equation is split into many discrete parts, and the contributionof each part is added together. In this case, the equation is the Biot-Savart Law:d~B=µ0Id~l×~r4pi|r|3 , (3.3)where, µ0 = 4pi × 10−7 is the permeability of free space, I is the current going through the piece of wire and~r isthe vector pointing from the piece of wire, d~l, to the point you are calculating the field for. The integral form of thisequation leads to equation 3.1.To calculate the field at a given point with a computer, d~l cannot be infinitely small, like when using the analyticsolution. First, a size for d~l is chosen, and the Biot-Savart law is used to calculate the field generated by the pieceof wire with length d~l. This is then iterated over the entire loop, and all of the contributions from the pieces of wireare added up. However, these pieces of wire are necessarily approximated as straight, so the simulation actuallygenerates the result for a field from a polygon. This approximation gets more accurate for smaller d~l, but at thecost of computing time since there are more iterations to go through and sum up. Thus, there is a trade-off betweenaccuracy and speed in the simulation. Fortunately, the simulation can be compared to the analytic result along the zaxis, so there is a means to determine what size d~l is necessary for the desired precision.It is also useful to test that the simulation really does get more accurate for smaller d~l. The simulation is runrepeatedly over the region where the analytic solution is known, for smaller and smaller d~l. Since a small region iscalculated, it is possible to go to higher precision than would be practical for the entire simulation. The results areanalyzed to look for convergence (the simulation should converge on a value rather than, for example, oscillating)and to see what size d~l is necessary to achieve the desired precision. This is known as a convergence test.The result of testing my simulation for a Helmholtz pair is show in Fig. 3.3. The simulation was tested both atthe center (top plot) and at the edge of the 129Xe cell (bottom plot). The value at the center does not require muchresolution at all in the simulation, and the field at the edge of the cell converges very rapidly. It is accurate to 5decimal places, even at only 16 pieces per loop. Since this simulation does not take long to run, I divided up theloops into 50 pieces.33Figure 3.3: This is the convergence test for the Helmholtz pair simulation. The y axis shows the value of themagnetic field at the center and edge of the 129Xe cell calculated at the resolution given by the x axis. Theresolution is the number of straight line pieces the coils are divided into when performing the simulation.At the center of the cell, the simulation is accurate even at very low resolutions. The field calculated atthe edge of the cell benefits from a higher resolution, but converges rapidly.Figure 3.4 shows the results of the analytic and the simulated results, for the magnitude of the B0 field along thez axis. These plots are for coils with radius R = 300 mm, 100 windings and a current of 4.04 A, generating a fieldat the center of about 1.2 mT. There is good agreement for both methods, suggesting that the simulation should beaccurate over the entire experimental region.3.3 Saddle Coil ConfigurationAnother possible configuration to create a homogeneous magnetic field is called the saddle coil. This geometry alsoconsists of two coils, but rather than than two loops, each coil consists of two straight pieces, or “rungs,” and twocurved sections, “arcs,” connecting the rungs. Figure 3.5 shows the geometry needed to create a vertical magneticfield.3.3.1 SimulationThe field generated by the rungs has a simple analytic expression, which can be derived directly from the integralform of the Biot-Savart law, and using some quantities defined in Fig. 3.6.∫ BAµ0I4pi~r×d~l|r|3 (3.4)It is simplest in this case to calculate the magnetic field’s magnitude, and determine the direction by other methodssuch as the right hand rule, rather than by carrying through the vector properties of the cross product, so:34Figure 3.4: Simulated and analytic results for the magnetic field in the z direction, along the z axis, for aHelmholtz pair of coils, with total current of 404 A, and a coil radius and separation of 300 mm.|~r×d~l|= rdlsin(θ +90) = rdlcos(θ). (3.5)Both r and dl can also be written in terms of θ :r =scos(θ)(3.6)rdθ = dlcos(θ) (3.7)As shown in Fig. 3.6, s is the distance from the wire to the measurement point and θ the angle made by s and r.Equations 3.4, 3.6 and 3.7 can be put together, and after canceling some terms, and changing the limits of integrationto the initial and final angles:35B =∫ θ2θ1µ0Icos(θ)4pisdθ =µ0I4pis[sin(θ2)− sin(θ1)] . (3.8)Equation 3.8 only gives the magnitude of the magnetic field, but the direction is then easily determined by the righthand rule. For example, in Fig. 3.6, the field comes out of the page at the point shown.The contributions to the field by the arcs of the saddle coil must be calculated with a finite difference simulationagain, like the calculation for the Helmholtz configuration. Figure 3.7 shows the convergence test on the number ofpieces to divide the coil into. In this test, the exact result was used for the rungs, which was added to the simulatedresult for the arcs. The x axis of these plots is the number of pieces the arcs were divided into for the simulation.The plots show the result for the field at the center, as well as for the inhomogeneity along the x axis. In mysimulations for the saddle coils I divided up the arcs into 200 pieces. This slightly overestimates the field strengthand inhomogeneity, but the computation time becomes prohibitive for higher resolutions.The final results of these simulations are shown at the end of the next section, in Fig. 3.8, along with the analyticresults to compare them to.3.3.2 Analytic FormulaeThere are also analytic formulae that can be used to check the simulation results. They are actually valid for allspace, but are computationally very expensive except near the axes. Because of this, it is actually more efficient touse a simulation to determine the field, and compare it to the analytic results along the axes.The field generated by a saddle shaped coil can be found to be[25]:Figure 3.5: The saddle coil geometry. Shown are common ratios for parameters: the half-length l is twice theradius a of the arcs, which have a half-span ϕ of 60◦. For consistency with the Helmholtz configuration,B at the center points along the z axis, and the coil axis is in the x direction. The current flows in the samedirection in each coil.36BρBφBx= µ02pi ∞∑m=−∞eimφ∫ ∞−∞dk eikx×k2ima2Pm(a)Fmx (k)kρ a2Qm(a)Fmx (k)k2m a2Qm(a)Fmx (k) (3.9)wherePm(a) =I′m(kρ)K′m(ka) when ρ < aI′m(ka)K′m(kρ) when ρ > a (3.10)Qm(a) =Im(kρ)K′m(ka) when ρ < aI′m(ka)Km(kρ) when ρ > a (3.11)andFmx (k) =−i4Ipikasin(kl)sin(mϕ)δm,odd. (3.12)Here, ρ , φ and x are cylindrical coordinates (I use x rather than z to maintain consistency with the z direction beingthe direction of the generated magnetic field), µ0 = 4pi × 10−7 is the permeability of free space, a is the radius, ϕhalf of the span, and l half of the length of the coils, I and K are the modified Bessel functions and I′ and K′ aretheir derivatives.The region of interest is the field inside the coils, so ρ < a. The result is given in Bidinosti 2005 [25], but I willFigure 3.6: Calculating the magnetic field generated by a straight wire. (a) Shows the variables used in theintegral in equation 3.4. (b) Shows the relevant variables that are used to directly calculate the field forthe whole length of wire, equation 3.8. Note that θ1 is negative when taken directly from s as shown.37(a) Convergence test for the magnitude of the field at the centerof a saddle coil configuration. The field strength converges to a bitbelow 1.488 mT.(b) Convergence test for the inhomogeneity of the field along thex axis. The inhomogeneity converges to about 15 parts per million.Figure 3.7briefly show the derivation for Bρ . The first step is to combine the equations 3.9, 3.10, and 3.12 (equation 3.11 isused for Bφ and Bx, but not Bρ ). Constants can be pulled out to the front and combined, leaving:Bρ =−2µ0aIpi2∞∑m=−∞[cos(mφ)+ isin(mφ)]δm,odd∫ ∞−∞dk [cos(kx)+ isin(kx)]kmI′m(kρ)K′m(ka)sin(kl)sin(mϕ), (3.13)where I have also used eix = cos(x)+ isin(x). This can be simplified by considering the symmetry in m→−m andk→−k. These identities for the modified Bessel functions and their derivatives are also useful[26]:I′m(x) =12[Im−1(x)+ Im+1(x)] , (3.14)K′m(x) =−12[Km−1(x)+Km+1(x)] , (3.15)I−m(x) = Im(x), (3.16)K−m(x) = Km(x). (3.17)Putting them together, I′ and K′ are symmetric functions, as is cos. Sin and k/m, however are antisymmetric for eachtransformation, so out of the four terms that come out of multiplying out the exponential, only the cos(mφ)cos(kx)term is symmetric overall. The rest are all antisymmetric and cancel out when the whole sum or integral is computed,as they both run from −∞ to ∞. For the surviving term, the limits of integration and the sum can both be changed tobe completely positive. The term then needs to be multiplied by four, since each change in the limit of integration38or sum removes half of the contribution to the total field. The result is:Bρ =−8µ0Iapi2∞∑m=1,3,5,...cos(mφ)∫ ∞0dkkmcos(kz)I′m(kρ)K′m(ka)sin(kl)sin(mϕ). (3.18)The equations for Bφ and Bx can be obtained in a similar fashion and after some rearranging of terms forcompactness, are[25]:BρBφBx= 8µ0Iapi2 ∞∑m=1,3,5,...sin(mϕ)∫ ∞0dk sin(kl)K′m(ka)×−km cos(mφ)cos(kx)I′m(kρ)1ρ sin(mφ)cos(kx)Im(kρ)kmcos(mφ)sin(kx)Im(kρ) . (3.19)This equation can be further manipulated to put it into a more useful form to compare simulations to. The zcomponent (which is vertical out of the table, transverse to the saddle coil, see Fig. 3.1) of the field can be calculatedalong the x, y and z axes. The x axis is fairly simple, since for ρ = 0, only the m= 1 term contributes to the field inthe z direction. This is because as ρ → 0,Im(kρ)→(kρ2)m/m! (3.20)andI′m(kρ)→12(kρ2)m−1/(m−1)!. (3.21)For m > 1 these both go to 0, but for m = 1, I′1(kρ)→ 1/2. So, to get the field in the z direction, along the x axis,the Bρ equation for ρ = 0 and φ = 0, and equation 3.15 can be used, to obtain:Bz(x) =2µ0api2sin(ϕ)∫ ∞0dk ksin(kl)cos(kx) [K2(ka)+K0(ka)] . (3.22)The z axis goes up out of the table, so φ = 0. The field in the z direction along this axis is just Bρ with ρ = z,again making use of equations 3.14 and 3.15:Bz(z) =2µ0Iapi2∞∑m=1,3,5,...sin(mϕ)∫ ∞0dkkmsin(kl)cos(kx) [Km−1(ka)+Km+1(ka)] [Im−1(kz)+ Im+1(kz)] . (3.23)And finally, the y axis is parallel to the table, at φ = pi/2, and with the field in the z direction being Bφ , andρ = y.Bz(y) =4µ0Iapi2∞∑m=1,3,5,...sin(mϕ)sin(mpi2)∫ ∞0dksin(kl)y[Km−1(ka)+Km+1(ka)] Im(ky). (3.24)Equation 3.22 can be calculated for good precision in the magnetic field calculation, even when truncating theintegration for computation time. Equations 3.23 and 3.24 involve an infinite sum, so these need to be approximatedby also truncating the sum in addition to truncating the integration. Fortunately, I found them to converge veryrapidly, and including just the first three terms (m = 1,3,5) is sufficient for accuracy to about 10 parts in 109.39Number of terms in sum limit of integration value for B0 (T) normalized B0(m = 1, 3, 5...) at cell edge2 100 0.001487225401 1.0000151505892 1000 0.001487225399 1.0000151492452 10000 0.001487225399 1.0000151492453 100 0.001487202871 1.0000000013453 1000 0.001487202869 13 10000 0.001487202869 14 100 0.001487202961 1.0000000618614 1000 0.001487202960 1.0000000611894 10000 0.001487202960 1.000000061189Table 3.1: Test for precision for the calculation of the magnetic field of a pair of saddle coils. I chose to include3 terms in the sum and use k = 1000 as the limit of integration. This should be accurate to 10 parts in 109.Truncating the integration to k = 1000 for calculating the field along each axis is acceptable for similar precision.This is easily enough precision to be useful as a check for the accuracy of the Biot-Savart simulation. The test forprecision of the analytic calculation is shown in table 3.1. Higher values of m are required for an accurate calculationas one goes further from the origin, so this precision test was done at the edge of the cell containing the polarized129Xe, or about 12.7 mm from the center of the coils, rather than at the center.Shown in Fig. 3.8 is the results of running these analytic formulae in Maple, as well as the results from thesimulation described in Section 3.3.1. For the y and z axes, the calculation is run over the cell’s radius in eachdirection, which is half an inch, or about 12.7 mm. For the x axis, which is the long axis of the coil and cell, it is runover the length of the cell, which is about 100 mm in each direction.For both the Helmholtz configuration, and the saddle configuration, the cell region has very small inhomo-geneity, small enough for either to be useful without modification to perform adiabatic fast passage (AFP) and freeinduction decay (FID). However the magnetic field can be distorted by external fields, or by nearby magnetic met-als. This becomes a serious issue due to the constraints imposed on our setup by the requirements to do the opticaldetection, specifically the requirement that these coils are placed just above our steel optical table. This is describedin Section 3.4. The saddle geometry has the benefit that it can be extended along the x direction, meaning that theregion of good homogeneity can easily be made larger along that axis without increasing the size of the coils inother directions. This is useful since the 129Xe cell being used is quite long and thin.3.4 Image FieldsDue to the nature of the 129Xe spectroscopy experiment, these coils need to be placed just above the same opticaltable that the UV laser is built on. This is mainly due to the enhancement cavity for the UV light that will surroundthe 129Xe cell. The optical components in this cavity must be mechanically very stable, so that the feedback andlocking electronics can reduce fluctuations in the cavity length to sub-wavelength amplitudes. There is a significantdrawback for this in terms of the homogeneity of the magnetic fields, though. The optical table is made of steel,which will deform nearby fields significantly due to its relatively high µr.Fortunately, these effects can be approximated and simulated, using the method of images. The idea is similarto using image charges in electrostatics, where an imaginary “image charge” can be placed on the other side of a40(a)(b)(c)Figure 3.8: A comparison of the analytic and simulated results for the magnetic field in the z direction, createdby saddle coils. Plot (a) shows the field along the x axis, plot (b) the y axis and plot (c) the z axis. Theresults agree very closely.41Figure 3.9: The image current of a piece of straight wire above a high permeability material such as steel. Thecurrent of the image (i) goes the same direction as the actual current (I).planar conductor from a test charge, to create a plane of zero potential. In the magnetostatic case, one can use an“image current” instead. Assuming an infinite, flat plane boundary, the image current is simple to calculate[27]:i=µr−1µr+1I, (3.25)where µr = µ/µ0 is the relative permeability of the material creating the boundary, I is the actual current, and i isthe image current. The image current sits on the opposite side of the boundary, the same distance away as the actualcurrent. There are two limiting cases in which calculating the image current is trivial. One case is when µr = 0, suchas in the case of a superconducting boundary, in which i=−I. The other case is when µr 1, so i≈ I. Steels tendto have µr on the order of 1000 or more, so I use i = I in my image field calculations. A straight piece of currentabove a high-µ material and its image are shown in Fig. 3.9.Once the image current has been calculated, the field generated by this current can be calculated using the sametechniques as for the field from the actual current. The only change is that the position of the image current isdifferent; it is below the table rather than around the origin. In the case for geometries where the coils don’t havecurrent running in the same direction (the anti-Helmholtz configuration, for example), care must be taken to ensurethat the image currents are chosen to be going in the correct direction for the top and bottom coils. The image is amirror of the actual coils.Equation 3.9 can also be used to find the field outside of the coils in a saddle coil geometry, analytically, whichis useful to calculate the image field. The relevant formulae can be obtained in the same way as equation 3.19, butchoosing the appropriate Pm and Qm for ρ > a. The result is very similar to the field inside the coils:BρBφBx= 8µ0Iapi2 ∞∑m=1,3,5,...sin(mϕ)∫ ∞0dk sin(kl)I′m(ka)×−km cos(mφ)cos(kx)K′m(kρ)1ρ sin(mφ)cos(kx)Km(kρ)kmcos(mφ)sin(kx)Km(kρ) , (3.26)and the corresponding field in the z direction along the z axis isBz(z) =2µ0aIpi2∞∑m=1,3,5,...sin(mϕ)∫ ∞0dk sin(kl)[I′m(ka)K′m(kz)]. (3.27)I ran this calculation in Maple along the z direction to have something to compare the simulation to. Thecalculation off axis was prohibitive in terms of the time required (and in this case, the x and y axes are not “on axis”42Figure 3.10: A calculation of the field along the z axis due to the image current from a saddle coil.Number of terms in sum value for B0 (T) normalized B0(m = 1, 3, 5...) at center2 0.00013643552773 1.00622020163 0.00013551496688 0.99943101014 0.00013559262322 1.00000373025 0.00013559262322 1.00000373026 0.00013559207121 0.99999965917 0.00013559211775 1.00000000248 0.00013559211775 1.00000000249 0.00013559211743 1Table 3.2: Precision test for the image field produced at the center of a pair of saddle coils by a steel opticaltable, with the integration limits being k = 0 to k = 1000. Seven terms in the sum is sufficient for 10 partsin 109 precision.for the image field), but just calculating the exact field on the z axis is still a good check. Figure 3.10 shows theresult of this calculation across the diameter of the cell.Even on axis, the integral and the sum in equation 3.26 need to be truncated to calculate within a reasonabletime frame. Table 3.2 shows the results of adjusting the limits of the sum. To get to 10 parts in 109 precision, 7terms are needed. Figure 3.11 shows the analytic and simulated results for the B0 field in the z direction for a saddlecoil with radius a = 180 mm, half-length l = 500 mm and half-span ϕ = 60◦, sitting 310 mm above a steel table. Theeffects of the table are very obvious (compare Fig. 3.11 to Fig. 3.8); there is a strong gradient along the z directionthat will need to be compensated for. Methods for doing so are described in Section 3.8.43Figure 3.11: This is the magnetic field in the z direction, along the z axis, generated by saddle coils withdimensions radius a = 180 mm, half-length l = 500 mm and half-span ϕ = 60◦ with a high-µr steel table310 mm away in the -z direction. The plots are calculated using the analytic formula (in red), and usingthe simulation (in blue). There is strong agreement.3.5 External FieldsThe final contribution to the total magnetic field in the experimental region is from all of the other magnetic fieldspresent due to external sources. In most environments, this will be dominated by the earth’s own local magneticfield. The strength of this field varies somewhat over the surface, but is generally about 0.25 to 0.65 Gauss, or 25-65µT. In particular, around Vancouver, Canada where these experiments took place, it is about 54 µT[28].Other contributions can include currents from nearby electronics, and nearby permanent magnets. These tend tocontribute very little to the static field, though. The currents used to generate the fields I am using are on the orderof hundreds of amps, and are close to the experimental region. There is no reason to have permanent magnets inthe experimental region in this experiment, so those are not a concern either, although it is important to make surethat there aren’t any placed nearby accidentally. For example, some optical mounts and bases have magnets built in,and these are frequently placed right next to these coils to perform measurements on the UV laser. These must beremoved before performing NMR experiments with the coils.Finally, there is the possibility of nearby magnetic materials, which will distort the field, like the optical tabledoes. This is best mitigated by simply going over all of the components with a weak magnet and determining whichones are attracted to it. If possible, all such components are replaced with non-magnetic ones, such as those madeof aluminum or brass. In our lab the biggest offenders are our precision translation stages.3.6 B1So far, this chapter has focused on the generation of a uniform, static B0 field. NMR experiments also require an RFfield to perform the necessary spin flips, however. This B1 field oscillates at the resonant frequency of the nuclei in44Source γ (MHz/T) Frequency at 1.5 mT (kHz)129Xe -11.777 17.666H2O (proton) 42.577 63.866Table 3.3: Gyromagnetic ratios γ and precession frequencies for some sources in a 1.5 mT magnetic field.the B0 field, and so is at a different frequency for different sources. The purpose of this field is to rotate the spin axisof the source nuclei. In general, NMR experiments begin with a pi/2 pulse to put the spins into the xy plane, fromthe initial polarization along z.The reason this field needs to be oscillating, rather than direct current (DC) like the B0 field is that the nuclei’sspins will be precessing around B0 during the flip. An additional DC field would simply rotate the axis of precession.Instead, the oscillating B1 field follows the source nuclei around the z axis, to provide a constant torque towards thexy axis. When doing FID, B1 is turned off after the spin has been rotated by pi/2. The details of this spin flip and therotating wave approximation have been explained in Section 1.1.3, but I will give a brief summary here.The nuclei are precessing, and their spin vectors trace out a circle on the xy plane. The pi/2 pulse would ideallybe driven by a field that follows the spin vector precisely, but such a field is difficult to create. A field that oscillatesalong some axis in the xy plane is equivalent to two counter-rotating fields on that plane. The field following theprecession drives the pi/2 flip, and the field rotating in the opposite direction does so quickly enough to have anegligible effect on the spin flip (this small effect is called the Bloch-Siegert shift and is small enough to ignore forFID). It is important to note, however, that half of the B1 field’s strength is lost to the counter-rotating portion, so theduration of the pulse needs to be double than it would be if B1 were to rotate in the plane with the precessing nuclei.Fortunately, the homogeneity of the B1 field is far less critical than the B0 homogeneity. The effect of aninhomogeneous B1 field on an FID experiment is minor. Source nuclei at different locations in the RF field willexperience a slightly different field strength. The result is a spin flip that does not rotate the nuclei’s spins by exactlypi/2. This nucleus will then precess around the z axis, with the spin vector slightly above or slightly below the xyplane on the Bloch sphere. This means that when the time variation of the projection of its spin along the x or yaxis is measured, contrast is lost. The particle will never be fully aligned with the measurement axis, so there willalways be a probability of finding it in the opposite spin state. However, unlike inhomogeneities in the B0 field, thiseffect is not cumulative over time. In fact, the loss of contrast is also not a particularly strong effect to begin with,since the projection onto the x axis will not change significantly.Table 3.3 has the gyromagnetic ratios, γ , and precession frequencies for 129Xe and for H2O (proton) in a 1.5 mTmagnetic field. The frequencies are of the order of 10s of kHz.3.7 Pickup CoilsThe usual method for detecting an NMR signal is through Faraday induction in a pickup coil, as the magnetic fluxthrough the pickup coil, from the precessing nuclei changes. This method is described in Section 2.5.1. Thissignal is very small, however, making the dampening of noise in the pickup coils and amplification of the signalvery important. The signal can be increased by making the pickup coil circuit into a resonant circuit, by adding acapacitor, which is chosen such that the circuit is resonant at the frequency of interest. This turns the pickup coil45Figure 3.12: Here is a typical pickup coil circuit. The coil itself is at the left, and the leads go to a capacitorthat tunes the circuit. The voltage across the coil is measured by a lock-in amplifier (LIA), through apre-amp. The capacitor is usually placed in a small box to protect it from being short circuited by metalobjects in the environment. A bucking coil can also be placed in this box to help cancel any cross-talkleft from imperfect mechanical decoupling between the pickup and B1 coils.circuit into an LRC circuit, whose impedance will depend on the frequency of any alternating current (AC) signaldriven in the pickup coil. There is a resonance ω0 atω0 =1√LC(3.28)where L is the inductance in the circuit and C the capacitance. The resonant frequency does not depend on theresistance in the circuit. Tuning the circuit in this way will make any signal at the resonant frequency build up overseveral cycles, the same way an optical cavity builds up power circulating within it. Unwanted signals or noise atother frequencies will not build up this way, so this improves signal to noise ratio (SNR). A lock-in amplifier isalso usually used to amplify the signal, which amplifies only in a narrow frequency band, and actually damps anynoise at frequencies outside of this band. Such amplifiers are typically used when the signal is weak, but of a knownfrequency. Figure 3.12 shows a typical circuit for detecting an NMR signal.One significant challenge comes from the fact that the B1 field itself oscillates at the same frequency of thesignal that is being measured. This can potentially generate an unwanted signal that is orders of magnitude strongerthan the signal from the precessing particles, called cross-talk. The best way to combat this issue is to make theB1 and pickup coils perpendicular to one another. This decouples the pickup coil from the B1 field, since thereis no field from the B1 field along the pickup coil axis. Perfect alignment is never quite possible, and a leftoversignal from slight misalignment must still be significant. One way this can be compensated for is by introducinga “bucking” coil to the pickup circuit that drives an electromotive force (EMF) that opposes the leftover cross-talk,which is described in more detail in Section Shielding and Improving HomogeneitiesExternally generated RF fields at the same frequency of the precessing nuclei can be a significant source of noise inNMR experiments. If these are constant, they can theoretically be cancelled with the bucking coil, but changes in46direction or amplitude of the externally generated fields cannot be easily dealt with this way. Shielding of some sortis required to improve the SNR. It is also possible to shape the DC (B0) field using a shield made out of a high-µmaterial, with the right geometry, potentially improving the field’s homogeneity.Other techniques can also be used to improve the homogeneity of the B0 field. I will describe two in this section:using shim coils and adjusting the current ratio in the main coils. In brief, shim coils are weaker sources of magneticfield that are placed strategically around the main coils to compensate for the inhomogeneities present. Adjustingthe current ratio is simply a matter of having a different current go through each coil in a pair. In both cases, themagnetic field generated by the B0 coils and their image fields are modified, and other external static fields can alsobe compensated for.3.8.1 AC ShieldingOne method for shielding the pickup coil from external AC fields is the use of a second coil, with fewer turns but alarger radius, or area. This can be set up in such a way that the ratioNinnerAinner = NouterAouter (3.29)holds, but with the coils wired in opposing directions. For an externally generated field, the magnetic field gradientwill be small across the coils, and so the flux through both of these coils will be very similar. Since they are counter-wound, the resulting EMF will mostly cancel. This is not the case for a magnetic field generated by a small sourceinside the coils, such as the spin signal we are trying to measure. This field will decay on a length scale givenby the source dimensions, and so the induced EMF will be weaker for the outer coil. The result is that the signalis only diminished slightly, while noise from externally generated fields is almost completely removed, improvingthe SNR. Increasing the radius of the outer coil compared to the inner coil (and reducing the number of windingsappropriately) results in a less diminished signal, but also is less effective in canceling externally generated fields.It is more effective, however, if the external fields can be shielded away without reducing the signal at all. Thiscan be done with an RF shield. A shell of conductive material is placed around the coils so that external RF fieldsgenerate a current on this shell, but do not penetrate it. This works as long as the cage is sufficiently thick. Therequired thickness depends on the skin depth for the particular frequency of field and the material used, and alsodepends on the shield geometry itself. For a cylindrical shield, at a frequency and material such that the skin depthis δ , and for radius R, the required thickness d0 is[29]d0 = δ 2/R. (3.30)At tens of kHz, this is significantly thinner than just the skin depth alone. This is due to Faraday induction. Theshield is, effectively, a current loop, and the oscillating RF field causes a changing magnetic flux, inducing an EMFaround the shield. This current generates a field that opposes the RF field, diminishing it inside the shield.Using a shield like this has the advantage that the signal from the precessing spins is not reduced, but it can useup a lot of room in the experiment, and can make it difficult to access parts of the experiment to make adjustments.473.8.2 DC ShieldingA shield made out of magnetic material, whose permeability, µ , is much different than the permeability of free space,µ0, will also distort DC fields, in addition to blocking AC fields. A useful way to analyze this distortion is to expandthe field in terms of a uniform field and multipoles. The effect of the shield on each multipole can be described interms of its “reaction factor,” or, the proportional change in field strength. For a saddle coil configuration, if thematerial is high µ , such as steel or mu-metal, the reaction factor will be greater than 1, since the image currentsgenerated are in the same direction as the actual currents. For a low µ material, such as a superconductor, thereaction factor will be less than 1, due to the negative image currents[25]. This means that the overall strength ofthe field is intensified with a shield made of high µ material, and diminished if made of low µ material. With theright geometry, the homogeneity of static fields inside the shield can also be improved[30].In theory, for an infinitely long, cylindrical high-µ shield, if the coil’s radius is 0.7784× the shield’s inner radius,the reaction factor for the uniform part of the field is maximized compared to other orders. The shield distorts thefield inside, just like the optical table distorts nearby fields, but it does so in a way that is beneficial, rather thanharmful to the homogeneity of the static B0 field. Since the shield also reduces the strength of the field due to the B0coils outside of it, it also mitigates or eliminates the effects from nearby magnetic metals, such as the steel opticaltable.The analytic results for the magnetic field generated by saddle coils can be modified to include a cylindricalshield [31].Rm(k) =− I′m(ka)Km(kb)Im(kb)(3.31)if the coil radius is smaller than the inner diameter of the shield. To calculate the total resulting field, in equation3.19, K′m(ka) is replaced with (K′m(ka)+Rm(k)).Unfortunately, for a finite length shield, with or without end caps, it is not clear exactly what the effect on theB0 field would be. To avoid potentially worsening the homogeneity, it was decided not to make a high-µ shield forour setup. Instead, we used aluminum, which will block AC fields but have little effect on DC fields, having µr ≈ Shimming the FieldShimming a magnetic field refers to the addition of weaker currents placed around the primary source of the mag-netic field, or an adjustment of the current ratio between the field’s primary source coils. Either method can beused to cancel a gradient in the magnetic field. This gradient can come from externally generated fields, such as theearth’s magnetic field, or from field distortions of the B0 field.A simple way to shim a linear gradient in a field is to use an anti-Helmholtz pair of coils. This geometry wasdescribed in Section 3.2, and generates a linear gradient of its own. By creating a gradient that opposes the existinggradient that needs to be compensated for, a much more uniform field is generated. This method of shimming makesit easy to adjust the field gradient while having a minimal effect on the field’s strength at the center, which is veryuseful for NMR experiments.In the case of a saddle coil, adjusting the current ratio also generates a linear gradient. The adjustment can bemade several ways. One is to build it into the coils by wrapping fewer windings on one of the coils, but there is nosimple way to make further adjustments in that case. Another method is to simply run a different amount of current48through each coil. This can be achieved with a variable current shunt, or by simply driving each coil with separatepower supplies. This method is simpler than building a separate set of anti-Helmholtz coils, but the field’s strengthchanges along with the homogeneity. Initially, I will use a current ratio to shim the B0 field, but it may be necessaryto add anti-Helmholtz coils in the future.49Chapter 4Measurements and ResultsIn this chapter I will describe the final design of the magnetic coils that I constructed, and I will go over the resultsfrom doing nuclear magnetic resonance (NMR) experiments on protons in water, which was done to characterize thefields generated by the coils, as well as to provide a baseline for later comparisons to similar experiments on 129Xe.The data from these experiments were taken under time constraint, and so were analyzed later. From this analysis, ithas become clear that there are several straightforward improvements that can be made to obtain stronger or longerlasting signals in the future.4.1 Final Coil DesignThe final design is shown in Fig. 4.1, which includes the AC shield, the B0 and B1 coils, and the 129Xe cell. Thesupports, mounts and optics needed for the UV enhancement cavity are also shown. The UV laser itself is behindthe shield, towards the left side of the assembly, in this drawing’s perspective. The NMR experiments in this chapterare performed on protons in water molecules, so the 129Xe cell is replaced with a water bottle, with a pickup coilwrapped around it. This bottle is about the same width, but shorter than the 129Xe cell. A photo of the bottom halfof the AC shield and B0 coil, the B1 coil, and an older version of the pickup coil, meant for a wider water bottle thanthe one used is shown in Fig. AC ShieldSince the signal generated in the pickup coil is passed through a lock-in amplifier, noise at frequencies other thanthe precession frequency is eliminated, but noise near that frequency is amplified along with the signal. I designedan AC shield made out of aluminum to shield the pickup coil from externally generated noise sources.This shield is 2 m long, and has a radius of 220 mm. The aluminum itself is about an eighth of an inch thick,and was rolled into a cylinder by the the University of British Columbia (UBC) machine shop. Our shop is onlyable to roll pieces that are up to about 3 feet wide, due to the size of the rollers, so the shield was made in 4 parts,which were riveted together. Since we are doing NMR at low magnetic field strengths, the frequencies that we areinterested in are quite low, only tens of kilohertz, which correspond to wavelengths in the tens of kilometers. Thismeans that holes in the shield should not diminish its effectiveness. Some holes in the shield are needed to passwiring to the coils and UV cavity electronics, and to let the UV laser light in. The shield is also cut in half down the50Figure 4.1: The full assembly of coils and the 129Xe cell, and cavity optics. The cell has a pickup coil wrappedaround it. The water bottle used to make the measurements in this chapter is about the same width, but isshorter than the 129Xe cell. The AC shield is also shown, as well as the mounts used to hold that shield inplace. The B0 coils are mounted to a wooden frame that attaches to the shield, and the B1 coil is mountedon an acrylic frame. Holes are cut in the shield to accommodate the aluminum pillars that hold up theslab for mounting the B1 coil, cell, and cavity optics. The UV laser’s output is behind the shield towardsthe left side of this assembly. As shown, the z axis points up in this figure, and the x axis down the shield.The y axis is also in the transverse plane, such that the coordinate system is right handed. The origin istaken to be the center of the 129Xe cell.long axis, so that the top half can be easily removed for access to the UV cavity, the B1 coil, and the pickup coil.The top half mounts to the bottom half with brass screws and a strip of aluminum.Mechanical stability is extremely important for the UV cavity to function, but the cavity needs to be at the centerof the magnetic fields, so it needs to be raised above the optical table. A heavy, nonmagnetic platform, with threadedbrass inserts is raised up on aluminum columns and glued into place with epoxy to use as a platform for the cavityas well as the B1 and pickup coils. The lower half of the shield has holes drilled to accommodate these columns,and is mounted on wooden mounts before the platform is epoxied. The columns are secured to the optical table withaluminum clamps.4.1.2 B0 CoilThe B0 coil design was motivated by the need for a very homogeneous magnetic field, and like described in Chapter3, a saddle coil geometry is ideal for creating a uniform field across the size and shape of the 129Xe cell. The exactcoil parameters also need to take into account the distorting effects from the high-µr steel optical table. The final B0coil design has 20 windings of 14 gauge wire wrapped in a saddle shape with rungs 1 m long, and arcs with a radius51Figure 4.2: A photo of the shield, the bottom of the B0 coil, the B1 coil, and an older version of the pickupcoil, meant for a wider water bottle than the one used. On the left side there is a mirror that is used toguide the UV laser light to the UV enhancement cavity, which is removed in this photo. The top half ofthe shield also holds the top half of the B0 coil.of 180 mm and a span of 120◦ that generate a field along the z axis. The B0 coils were wrapped around a purposebuilt wooden frame, then lifted off and mounted to the inside of the AC shield. Part of the bottom half can be seenin Fig. 4.2.The strength of the field that the coils can generate is limited by the heating of the wire as the current throughthem is increased. In principle, the current can be increased until either the insulation breaks down and the wireshorts, or until the copper itself melts, but there are other potential problems as the wire temperature increases. Theresistance in the wire is dependent on its temperature, so when using a voltage source, such as the power supplywe used, the current drifts as the wire approaches the equilibrium temperature. With the AC shield, this heat getstrapped, warming the pickup and B1 coils as well. It is also sometimes necessary to reach into the shield, so thereare safety concerns with the wires getting too hot. This coil starts becoming painful to the touch at about 10 A.For these experiments, this limit is not of consequence, since the power supply being used to drive the B0 coil52Figure 4.3: This is a plot of the B0 field homogeneity measurement along the z axis. The field strength isnormalized to 0 at the top of the water bottle. The vertical extent of the water bottle is shown by theblack lines, at about -17 mm and +17 mm. The height is zeroed at the cell center. The blue plot is whatthe experiments in this chapter were performed under, with about 0.42 A taken from the top coil. Thevariation of the magnetic field strength along the z axis is about 0.1 Gauss (10µT), or about 1.5% of themagnitude of the magnetic field. Removing the potentiometer, or moving it to the lower coil improvesthe field homogeneity.cannot drive more than 8A. It is usually set to about 7 A for the experiments in this chapter, since the adiabatic fastpassage (AFP) experiments require ramping this current up and down around its set point. It takes about an hour forthe temperature inside the shield to stabilize after turning on the B0 coil.The field generated by the B0 coils in the cell region was measured with a flux gate. The direct current (DC)field generated by these coils, when placed above the steel optical table, with no compensation for the gradient itcreates in the field, was measured to be 0.943 G/A (94.3 µT/A) at their center. The field gradient along the z axiswas also measured, which can be compensated for by adjusting the current ratio between the top and bottom coils.This adjustment is currently done by connecting a potentiometer in parallel with one of the coils, to divert somecurrent away from that coil. Since the flux gate was borrowed from another group, I ended up having to make thesemeasurements after the experiments in this thesis had already been performed, so the current ratio was set based onthe simulations described in Chapter 3. However, I mistakenly connected the potentiometer to the top coil ratherthan the bottom coil, and actually made the gradient worse. Figure 4.3 shows the field gradient in the z directionfor various current ratios. The black vertical lines are at the water bottle’s walls, so the field inhomogeneity for theexperiments in this chapter should be about 0.1 G, or 10 µT.53The B0 coil circuit consists of a power supply, the coils, and a current stealer circuit attached in parallel to oneof the coils. The power supply also has a control input, where a ramped signal is sent for AFP, which is controlledby a function generator.4.1.3 B1 CoilThe B1 coil generates the radio frequency (RF) field that flips the nuclei’s spins. This field is generated by anothersaddle coil, and points perpendicular to the B0 field, along the y axis. The homogeneity of the B1 field is not nearlyas critical as the B0 field, so it can be made much smaller, physically. This B1 coil is made of 10 windings of 26gauge wire in a saddle shape with rungs 400 mm long, and arcs with a 75 mm radius, and a span of 135◦, wrappedaround an acrylic frame. The coil can be seen in the photograph, Fig. 4.2.For a DC field, the B1 coils produce about 1 G/A, measured with a gaussmeter. In principle, for AC fields, thismight be attenuated somewhat by the presence of the aluminum shielding. This attenuation should be weak, since theshield radius is considerably larger than the coil radius. Measuring the AC field is not possible with the measurementdevices we had in the lab, unfortunately. This matters the most when doing free induction decay (FID), since thefield strength determines how long the pi/2 pulse needs to be on. It will also affect the adiabaticity condition forAFP. The best test available to us is to simply do FID repeatedly with different currents or pulse durations, to findthe maximum signal amplitude from FID.The B1 coil circuit consists of a function generator which drives the AC current, the coil itself, and a 1 Ω powerresistor that is used to monitor the current in the coil.4.1.4 Pickup CoilThe pickup coil measures the changing magnetic flux from precessing spins. By Faraday induction, an electromotiveforce (EMF) is induced, the current from which generates a magnetic field that tries to oppose an external changein magnetic field. This coil can be easily replaced with a different one in the setup, and is usually matched to thesample container. For example, for the proton NMR experiments done in this section, the coil is wrapped directlyaround the water bottle used to hold the sample. This maximizes the signal from precession, while minimizing noisefrom other oscillating fields. The EMF generated in the pickup coil is very small (on the order of a µV or less inthese experiments), so the current in the wires is tiny, and they can be quite thin without worrying about thermaldamage. The pickup coil used was wound from 26 gauge wire, and has 190 windings, in a solenoid configuration.The pickup coil is inductive, of course, so rather than connecting it directly to the amplification circuit, a resonantcircuit can be made by adding a capacitor. This is done in an external box (the “tuning box”), so that it is easy tochange the tuning capacitor if the pickup coil is changed. The capacitor needs to be chosen appropriately to createa resonant circuit at the correct frequency. The resonant frequency of the circuit is given byω =1√LC, (4.1)where L is the inductance of the pickup coil and C the capacitance of the tuning capacitor.The pickup coil used will depend on the sample, so its inductance L needs to be measured, and then a tuning boxwith the correct capacitor needs to be made for each pickup coil. The desired precession frequency also needs tobe known, so that the circuit’s resonant frequency, ω can be made to match. After a capacitor has been chosen, the54Figure 4.4: This is a closeup photo of the pickup coil, wrapped around the water bottle used for these experi-ments. Rotating the plastic rod at the bottom of the photo changes the alignment between the pickup andB1 coils, to mechanically decouple them and reduce cross-talk. The B1 coil can also be seen, mounted toan acrylic frame. In the back left, there is a hole in the shield to allow the UV laser light in. It is wrappedin black tape to protect wires from sharp edges, since the insulation on the wires used to make the coilsis very thin.actual resonant frequency needs to be measured since tolerances on components, and other inductive and capacitivecircuit elements can shift this frequency. The easiest way to measure the resonance of the pickup coil circuit issimply to measure the cross-talk from the B1 coil with the lock-in amplifier. Since this amplifier also drives the B1coil, changing the frequency on the amplifier changes both the signal in the pickup coil, and the frequency that ismeasured by the amplifier. The resonant frequency is the frequency where the cross-talk is maximized. For theseexperiments, the pickup coil’s inductance was measured to be 454 µH. We wanted a precession frequency of around27 kHz for the protons in our NMR experiment, since that corresponds to about 7 A on the B0 coils, so a capacitorof 76 nF would tune the circuit. The measured resonance with such a capacitor was at 27.07 kHz, so this is thefrequency that is used for all of the experiments in this chapter.For the actual NMR measurements, cross-talk is not desirable, so the pickup coil’s angle to the B1 coil can be55adjusted. The corner of one of the mounts can be moved around to change the horizontal angle of the containercompared to the B1 coil by rotating a threaded rod. This is usually good enough to bring the cross-talk down toabout 15 mV before the lock-in amplifier (but after the pre-amp), which has been sufficient for these experiments.Figure 4.4 shows a closeup view of the pickup coil, water bottle, B1 coil and the decoupling mechanism.The pickup coil circuit consists of the pickup coil and a tuning capacitor. The output is sent to a 100x pre-ampand then into the lock-in amplifier.4.1.5 Lock-in AmplifierAll of the NMR measurements taken in this chapter were done through a lock-in amplifier, which is commonly usedwhen a signal is of a constant, known frequency. They are somewhat different than traditional amplifiers in thatthey amplify a signal from a specific frequency, while damping any signal or noise at other frequencies. This isdone by multiplying the signal by the reference sine wave, and integrating over some time. If the signal being sentto the lock-in amplifier is Asin(ωt), and the amplifier’s reference frequency is ωlia, then the output depends on theintegration time, and the phase between the signal and reference, φlia:Vout(t ′) =t ′∫0Vin(t)sin(ωt)sin(ωliat+φlia)dt. (4.2)The output’s amplitude diminishes as ω gets further from ωlia, and does so more rapidly when t ′ is longer. Inthe limit where one integrates over all time, so t ′ → ∞, then the only contribution to the signal is from sourcesat exactly the reference frequency. For finite integration times, this multiplication creates a bandpass filter, withthe width increasing for shorter integrations. That means that there is potentially more noise left in the amplifiedsignal when set to a shorter time constant. However, if the signal changes quickly in amplitude, like FID does whenthe field inhomogeneity is too high, a short time constant is required so that this signal does not get averaged out.This averaging can also make measurements slightly inaccurate. For example, the height of a peak during AFP willbecome smaller due to this averaging when going to long integration times. The positions of features will also bedelayed a bit, which needs to be taken into account when comparing to other simultaneous measurements. Theseinaccuracies increase as the integration time increases. To compensate for the lost signal to noise ratio (SNR) bygoing to short integration times, many data runs can be taken and averaged together.The amplified signal’s amplitude also depends on the phase between the raw signal and the reference sine waveused by the lock-in amplifier, φlia. When they are out of phase, the strength of the output signal will be diminished,going to 0 when the two sine waves are 90◦ out of phase. The lock-in amplifier I used has two outputs, withthe reference frequency for each being out of phase with each other by 90◦, called “x” and “y.” The y output isdetermined by equation 4.2, but with φlia replaced with φlia+ 90◦. With the outputs phased in this way, the trueamplitude of the amplified signal can be obtained by adding them in quadrature.The lock-in amplifier also has a variable setting for the amount of amplification. This is labeled as “sensitivity,”since it sets the input range. The output is always -10 V to +10 V, so if the sensitivity is set to 5 mV, for example,there is 2000x amplification. There is also a pre-amp which has a frequency independent amplification of 100x. Theoutput from the lock-in amplifier is sent to an oscilloscope to read and save the measurement data.564.2 Adiabatic Fast Passage (AFP)The first experiments performed with these coils was AFP on protons in water molecules. Since the experiment isrun on a Boltzmann polarized1 sample, there is a known and repeatable polarization. This is convenient because thisway a lack of signal cannot be due to insufficient polarization, and indeed, it was quite simple to find an AFP signalby scanning the B0 field, and so the precession frequency, over a large range. Having such a repeatable polarizationalso makes it easy to average many runs to improve SNR. This allows us to dramatically reduce the integration timeon the lock-in amplifier to get a more accurate signal. By varying the parameters used in the AFP experiments, it ispossible to begin characterizing the coils and the fields they produce. This characterization is continued with FID inSection 4.3.AFP was described in detail in Section 2.6, but I will go over it again briefly here. The oscillating B1 field is lefton constantly, and the B0 field is ramped slowly. As the B0 field is ramped, the precession frequency of the nucleichanges. As the precession frequency ramps through the B1 frequency, the spins of the nuclei are adiabaticallytransferred to the opposite z state as they were originally. As they are transferred, they spiral through the transverseplane and induce an EMF in the pickup coils. Figure 4.5 shows an example of what signals are monitored on theoscilloscope during an AFP measurement. The resonant frequency of the pickup coil circuit is 27.07 kHz, and forprotons to precess at that frequency, a B0 field of 0.636 mT (6.36 G) is required. A current of just under 7 Agenerates this field strength, using these coils.The power supply being used has a tuning input, so to ramp the field, a ramped input from a function generatorcan be used. The lock-in amplifier and the B1 frequency need to be exactly the same, which is accomplished mosteasily by using the lock-in amplifier’s reference sin wave itself as a source for the B1 current. One additionalfunction generator needs to be used to drive the bucking coil that cancels residual cross-talk, the details of which aredescribed in Section 4.2.1. The entire circuitry and the equipment used is shown in Fig. 4.6. For the pickup coil, thetuning capacitor is placed in a tuning box to make it easy to swap out for a different one. Conveniently, the buckingcoil can also be placed in this box. The pre-amp amplifies the signal by 100x and outputs the amplified difference involtage between the two leads of the pickup coil circuit. This signal is passed to the lock-in amplifier. The currentthrough the B1 coil is measured by measuring the voltage across a 1 Ω power resistor on the grounded side of thecircuit.4.2.1 Cross-talkCross-talk is a significant spurious signal that prevents us from being able to go to lower sensitivity on the lock-inamplifier. In this case, it refers to any signal that is generated in the pickup coil by the oscillation B1 field. So, tomaximize SNR, it is important to minimize cross-talk. Other than the mechanical decoupling described in Section4.1.4, a “bucking” coil is used to drive a current in the pickup coil circuit that is exactly out of phase with thecross-talk when doing AFP. The resulting destructive interference can reduce the cross-talk by several orders ofmagnitude. Empirically, we did notice that the noise in the signal increases if the bucking coil needs to do morework, however, so the mechanical decoupling still needs to be as good as possible. A good goal for the mechanicaldecoupling is to get under 15 mV after the pre-amp.After mechanically decoupling the coils as well as possible, the voltage of the remaining cross-talk is noted1see Section 2.3.1 for a description of Boltzmann polarization57Figure 4.5: Here is a sample screenshot of the oscilloscope while performing AFP. The green plot measuressignal sent to the power supply for the B0 ramp. The noisy magenta plot is the EMF generated in thepickup coil, de-modulated and amplified by the lock-in amplifier. The light pink plot above that is anaverage of 115 sweeps of the magenta plot. The blue plot is the current through the B1 coil, and oscillatesat 27.07 kHz. At longer time constants, the SNR straight out of the lock-in amplifier is similar to theaveraged signal here. The magenta plot was taken with a time constant of 1 ms, and at a time constant of100 ms, the data look similar to the light pink plot.and then the B1 coil is turned off. The bucking coil is then turned on and its amplitude is adjusted until the signalthrough the lock-in amplifier matches that of the cross-talk. Then, the B1 coil is turned back on, and the phase onthe bucking coil is adjusted until the cross-talk is minimized. The amplitude and phase of the current through thebucking coil often needs to be adjusted again after the B0 coil has reached thermal equilibrium.4.2.2 AFP resultsMost of these AFP experiments were performed with the lock-in amplifier at a 5mV sensitivity (which correspondsto a 2000x amplification, plus an additional 100x from the pre-amp), 1 ms time constant, and 24 dB slope2. Iperformed AFP on protons in water while varying a number of parameters. The goal was to determine the T1 timeconstant and to confirm the adiabaticity conditions for water for these coils, and so also confirm the B1 field strength.From the width of the AFP signals, it would also be theoretically possible to determine the B0 field gradient, and2The slope here refers to how steep the edges of the bandpass filter are in frequency space. A 24 dB slope is the steepest setting on thislock-in amplifier.58Figure 4.6: Here is a schematic of how the function generators and circuits are hooked up for AFP. On the leftis the pickup coil circuit. FG1, the first function generator, drives the bucking coil, which is placed insidethe tuning box (shown as the dashed box around the tuning capacitor and bucking coil). The coil needsto maintain phase with the B1 coil, but since they are driven by separate devices, this is not guaranteed.To ensure phase, FG1 is actually run in burst mode, triggered by the lock-in amplifier. The voltage acrossthe pickup coil is measured by the LIA, after a 100x pre-amp. The amplifier demodulates the signal, andalso drives the B1 coil at the same frequency as this demodulation. A 1 Ohm power resistor is placed inthe B1 circuit, and is used to measure the current through this circuit. A second function generator, FG2,drives the ramp on the power supply, PS, which drives the necessary current through the B0 coil. Thiscircuit is completely separate from the pickup and B1 circuits.from there the T∗2 time, although FID is better suited for that measurement, since there are a lot of factors that canaffect the width of the AFP signal that would all need to be carefully accounted for.When performing AFP, it is important to make sure that the ramp rate is not too fast, or the spins will not followthe field adiabatically, reducing the amplitude of the AFP peak. The condition for adiabaticity isdB0dt<< γB21. (4.3)With the B0 ramp rate at 4.1 µT/s, B1 should be much greater than 0.3 µT. This can be confirmed by repeatingAFP measurements for varying strengths of B1. As the B1 field becomes weaker, the chance that a given nucleustransfers to the new spin state decreases.59The probability that a nucleus’ spin is not transferred to the new spin state is[32]:Plost = e− pi2 Ω2∂δ/∂ t , (4.4)where Ω is the precession frequency of the nucleus, in rad/s, around the B1 field (not the B0 field), and ∂δ/∂ t is theramp rate of the precession frequency around the B0 field, in rad/s2.By varying the strength of B1, Ω is varied:Ω(B) = γB1(I) = 2pi ∗42.576∗106 ∗B1(I), (4.5)where B1(I) = c ∗ I, some constant times the current. Based on measurements performed with a gaussmeter, c =97±4 µT/A.The B0 ramp rate for this experiment was kept constant at 4.1 µT/s, so∂δ∂ t= 4.1∗10−6 ∗2pi ∗42.576∗106 = 1096.8 rad/s2 (4.6)is the ramp rate in the proper units for equation 4.4.The data that is obtained by varying the B1 field strength does not give the probability of losing polarization,though, it is the polarization that remains. So, the equation that I fit to isPremaining = A∗ (1− e− pi2(2pi∗42.576∗106∗c∗I)21096.8 ). (4.7)Here, I multiply the whole equation by a scaling factor, A, since the data are not normalized. The fit parameter is c,the field strength per amp through the B1 coils.The result from the fit is shown in Fig. 4.7, and was found to be c = 10.9± 0.3 µT/A, almost a full orderof magnitude lower than what was measured with the gaussmeter at DC, c = 97± 4 µT/A. It is possible that theattenuation from the shield is greater than expected, but more likely that one of the measurements, either the DCfield per amp, or the data and its fit, are incorrect. Since the shield’s diameter is about 3x that of the B1 coils, theexpected attenuation is approximately 1/32, so the reduction should be by about 11%, rather than the 89% differencefound here, between the DC measurement and the result from the fit. [30] Looking back at the adiabaticity conditionas well, equation 4.3, for this ramp rate, B1 needs to be much greater than 0.3 µT. According to the fit, with c= 10.9µT/A, this condition would correspond to a B1 current of much greater than 27 mA. The gaussmeter measurement,with c = 97 µT/A, would put the condition on the current at much greater than 3 mA for adiabaticity. From Fig.4.7, it can be seen that at a B1 current of 27 mA, AFP appears to already be adiabatic since the peak heights havereached their maximum. So, it seems likely that the gaussmeter measurement is more accurate. This inconsistencyhas potential implications for FID experiments as well, since the strength of the B1 field needs to be known so thatthe duration can be set properly. A method for directly measuring the field strength per amp at 27.07 kHz may berequired.The other parameter I varied was the time between AFP peaks. After each AFP peak, the sample will return toBoltzmann polarization on the T1 time scale, but if another AFP ramp happens before this relaxation is complete, aspin flip still occurs. However, the signal from this flip will be somewhat out of phase with the lock-in amplifier’sreference sin wave. Figure 4.8 shows an example of two AFP spin flips that begin with a sample polarized in the60Figure 4.7: This is the data obtained by varying the B1 field strength. The data in blue is the height of the AFPpeak at that field strength, and the red line is a fit to the Landau-Zener model. This model states that theprobability of a spin not transferring to new spin state is given by P = epi2Ω2∂δ/∂ t , where∂δ/∂ t is the rateat which the precession frequency of the protons is ramped, and Ω = 2piγcI, with γ = 42.576 MHz/T, Ibeing the current through the coils, and c being the fit parameter; the strength of the B1 field per amp ofcurrent through it. The fit found c= 10.9±0.3 µT/A, almost a full order of magnitude lower than whatwas found by measuring the DC field produced by a known current with a gaussmeter.Figure 4.8: The phase of the signal from AFP depends on the direction of initial polarization of the sample.Shown here is a portion of the spin flip during AFP, in the rotating frame for simplicity. In panel a), thespin is initially up along the z axis, and halfway through the spin flip, points in the positive y direction.In panel b), the spin is initially down along the z axis, and points in the negative y direction during thespin flip. The two examples are 180◦ out of phase with each other, and so is the signal they generate inthe pickup coils.61up state in z, panel a), and a sample polarized in the down state, panel b). In this example, the signals generated byeach of the spin flips are out of phase with each other by 180 ◦. For this experiment, the lock-in amplifier’s phase isset to maximize the signal amplitude out of the “x” output for one of the AFP peaks. As long as the time before thenext AFP peak is not large compared to the T1 time of the sample, the AFP signal from the next ramp through theresonance will be out of phase with the first, which will result in a smaller peak, or even a negative peak in the “x”output.Figure 4.9: This is a plot of the heights of the second AFP peaks (the first data point is the initial peak), whenthe time between peaks is scanned. An exponential decay is fit to these points, where the polarization isA(1−2e−t/T1). T1 is found to be 2.7±0.2 seconds.The experiment is repeated, for varying times between AFP peaks, and the height of the second peak is recordedeach time. These peak heights can be fit to an exponential decay. The fit equation is found by examining exactlyhow the polarization behaves after the initial AFP spin flip.After the first AFP ramp, the nuclei end up polarized in the opposite spin state from Boltzmann equilibrium. Thez polarization (or, magnetization) then changes like:Mz(t) =Mz, eq(1−2e−t/T1), (4.8)where Mz, eq is the Boltzmann polarization. In this case, Mz(0) is the polarization at the end of the first spin flip.Over time, the polarization returns to equilibrium (Boltzmann polarization), until the second AFP spin flip is applied,at time t = t2. The phase of the signal generated during this second spin flip compared to the phase of the first flip isφ = acos(Mz(t2)Mz, eq). (4.9)62The relative amplitude of the signal in the “x” output of the lock-in amplifier will depend on this phase:V2V1= cosφ , (4.10)where V1 is the amplitude of the signal from the first AFP peak, and V2 the amplitude from the second. The threeequations 4.8, 4.9, and 4.10 can be combined to arrive at the equation the data should be fit to:V2 =V1(1−2e−t/T1). (4.11)This equation is fit to the peak heights (including also the initial peak height, where t=0, as the first data point),which is shown in Fig. 4.9. The result of the fit is a T1 time of 2.7±0.2 seconds.4.3 Free Induction Decay (FID)FID was described in detail in Section 2.7, but I will briefly go over it again here. The spins of a sample that isinitially polarized along the B0 field direction are tipped into the transverse plane by a pi/2 pulse from the B1 coils.The sample is then allowed to precess freely around the B0 field. The transverse polarization will disappear overtime due to relaxation effects and the B0 field gradient, so the EMF induced in the pickup coil decays exponentially.For a number of reasons, finding an FID signal is usually more challenging than performing AFP. In principle,the initial signal strength should be the same as the amplitude of the AFP peaks, since in both cases the spins are attheir maximum transverse polarization. However, if the FID experiment’s B1 pulse is not well tuned, it will not tipthe spins by exactly pi/2. In that case, the transverse polarization will not be as strong as it could be, diminishingthe initial signal. Measuring the B1 field at its operating frequency is not possible with the equipment in our lab,so it must be estimated. I measured the field strength per amp for a DC field, and found it to be 0.97±0.04 G/A.For the first attempts at FID, I assumed that there would be a negligible difference to the field strength at 27.07 kHz.The presence of the aluminum shield could introduce a frequency dependent attenuation, however. To determinethe optimum B1 pulse duration, FID can be repeated while varying the duration or strength of the B1 pulse, lookingfor the maximum FID signal strength. Cross-talk poses a larger problem than for AFP, since even with a buckingcoil, the ring down in the resonant pickup coil circuit of the B1 coil’s cross-talk after the B1 pulse turns off is anexponentially decaying signal, just like the FID signal we are looking for. It is mitigated by using a Q-killing resistor,instead, described in detail in Section 4.3.1.The B0 field needs to be precisely calibrated such that the nuclei precess at the same frequency as the B1frequency. One of the easiest ways to do so is to perform AFP first, and adjust the center of the B0 ramp such thatthe AFP peaks are precisely centered. When the ramp is turned off, the field should be sitting exactly on resonance.Care must be taken, however, since the lock-in amplifier’s time constant results in a slight time delay of the AFPpeaks, due to the averaging it does. This manifests itself as an error in the exact current that resonance occurs, andis seen as an asymmetric positioning of the peaks on the up and down ramps. The time constant should be reduceduntil this shift is negligible compared to the width of the AFP peaks. Generally, a time constant of a few ms hasworked for this purpose in these experiments.Cross-talk is not completely eliminated, so to make sure that any signal I find is actually from FID, I first tookdata with the nuclei precession on resonance with the B1 field, and then again with them far off resonance. These63Figure 4.10: Shown is the effect of adding a Q-killing resistor to the pickup coil on how much cross-talk isseen from the B1 coil. The top plot is the signal in the pickup coil without the analog switch (or withthe switch turned off). For the bottom plot, the B1 coil is at the same strength, and the mechanicaldecoupling has not changed, but the analog switch is turned on. This way, a large amount of the currentgenerated in the pickup coil is dumped through the switch, damping the cross-talk significantly. Themost significant effect is that the ring-down is reduced to almost nothing, so data can be taken as soonas possible after the B1 pulse is turned off. The ring-down in each case is circled in the sets are then subtracted from each other, and I look for evidence of FID in the data after the B1 pulse has turnedoff.4.3.1 Cross-talkCross-talk poses some issues for FID experiments, even though the B1 field is off during the actual signal collecting.Since the pickup coil is connected to a resonant circuit, any resonant signal induced in it will build up over severalcycles, and also ring down when the source is turned off. Cross-talk generates a resonant signal, and the ring downlooks qualitatively identical to an FID signal; it is an exponentially decaying signal that starts just when the pi/2pulse is turned off.Cross-talk during the B1 pulse could be mitigated with the bucking coil, just like during AFP, but that requiresadditional function generators, and setting the phase correctly for the bucking coil would be more difficult thanfor AFP. There is a more elegant solution. For FID, the ideal configuration would be to have a high-Q resonantcircuit during the data gathering part of the experiment, but a damped, low-Q circuit during the pi/2 pulse. Thiscan be achieved by adding a switching resistor to the tuning box. I built the circuit on an external printed circuit64board (PCB), which connects to the tuning box via a short BNC cable. This way, the box can be easily used for bothFID, with the switching resistor attached and AFP without it. A 5 V signal is sent to the switch during the pi/2 pulse,which turns on the resistor, damping the circuit significantly. The resistor turns off after the pulse is over, restoringthe high Q-factor, so the signal from the precessing nuclei is maximized. The resistor is a CD4066BE integratedcircuit (IC), with a resistance of about 75 Ω when on (and effectively infinite resistance when off), and is controlledby a function generator, set to “pulsed” output, to +5 V when B1 is on, and 0 V otherwise.Figure 4.10 shows the results of testing this method. These plots were taken without the use of the lock-inamplifier (although the pre-amp, which amplifies the signal by a factor of 100, was used), so they are not demodu-lated, leaving the underlying sine wave in the signal intact. The pickup and B1 coils were intentionally left slightlycoupled, so that the effect of the resistor could be seen more clearly. The damping is significant, enough to reducethe cross-talk almost down to the noise floor under these conditions.Figure 4.11 is a schematic of how the coils and electronics are connected, including the Q-killing resistor in thepickup coil circuit. With the switch open (0 V), the resistor has no effect on the circuit, so it maintains a high Qfactor. With it closed, I measured about 75 Ω of resistance. Some of the power generated in the pickup coil due tothe cross-talk is dissipated in this resistor. The lower the resistance, the more effective this method is. Finding an ICwith lower resistance would be potentially beneficial.4.3.2 FID resultsWhen performing FID on protons in water, the experiment can be done repeatedly on the same sample, since theprotons return to their Boltzmann polarization on their T1 time scale. For the FID experiments in this section, Irepeated the B1 pulse every 1.5 seconds, and averaged many runs together, to increase SNR.For the first FID attempt, the lock-in amplifier was set to a sensitivity of 5 mV, a time constant of 1 ms and aslope of 24 dB. In this initial run, a very short T∗2 time was expected. Based on the simulations described in Chapter3, and the center field strength measured from the actual coils, it was expected that the inhomogeneity would bearound 2-3 µT, corresponding to a T∗2 time of around 10-15 ms. Figure 4.12 shows the results of this experiment,including a baseline that was subtracted out to get the FID signal itself, in green on this plot.The result was encouraging since the T∗2 time looked about right. However, since the potentiometer was mis-takenly connected to the wrong coil, the T∗2 time should actually be considerable shorter. Like described in Section4.1.2, the inhomogeneity is actually closer to about 10 µT, corresponding to a T∗2 time of around 2 ms.Figure 4.13 shows the results of repeating the experiment with a shorter time constant on the lock-in amplifier.Due to increased noise, more averages needed to be taken to get adequate SNR. In this case, about 3000 FID runs wereaveraged together, over a period of about an hour and a half, with a 1.5 second delay between pi/2 pulses. Figures4.14 and 4.15 show the signals from the two FID measurements, with the data taken during cross-talk truncated, andan exponential fit through the remaining data. The T∗2 time was measured by these fits to be 1.287±0.015 ms, and1.203±0.016 ms by the two runs, close to the expected 2 ms. Based on these results, the inhomogeneity across thewater bottle is 18.9±0.9 µT.Despite the averaging, there is still considerable noise, which is clearly sinusoidal, at about 700 Hz, which isparticularly noticeable in the baseline measurement, shown in red in Fig. 4.13. Because of this noise, the baselinewas not subtracted from the data for the fit. The lock-in amplifier demodulates the signal it receives, so this noiseshould actually be found at either 27.77 kHz, or 26.37 kHz (700 Hz above or below the amplifier’s setting of 27.0765Figure 4.11: Here is a schematic of how the circuits and function generators are hooked up for doing FID.Channel 1 on the dual output function generator is used to drive the pi/2 pulse on the B1 coil. Tomake sure that phase is maintained between the pickup coil circuit’s signal and the lock-in amplifiersdemodulation, channel 2 of the same function generator is a long burst at the same frequency that is fedinto the LIA’s reference input. When doing FID on protons in water, the experiment can be repeated.In this case, both channels are run in burst mode, with the reference channel’s burst lasting almost theentire burst period, skipping only one cycle. The function generator’s burst also triggers a single outputfunction generator, FG1. This is set to send a 5V pulse to the resistor in the pickup coil circuit for theduration of the pi/2 pulse. The pickup coil’s signal is sent to the LIA input through a 100x pre-amp.kHz). To investigate this, I measured the pickup coil’s output through the lock-in amplifier, while scanning thereference frequency and with no known sources driving a signal in the pickup coil. This way, any EMF generated inthe pickup coil is from a source of noise. I found that there was clearly stronger noise at 27.77 kHz than other nearbyfrequencies. The frequency or phase of this noise does seem to drift slowly over time, which is why averaging thesignal reduces it. It can be filtered out by increasing the time constant, as can be seen from the first FID attempt,with a time constant of 1 ms, but that is not an option unless the B0 field gradient is improved significantly so thatthe actual FID signal lasts longer. The source of this noise is still unknown.66Figure 4.12: The results from the initial FID attempt. This experiment was performed with a time constant ofthe lock-in amplifier set to 1 ms . Data were gathered twice, once with the sample on resonance (blueplot), and once with the sample off resonance by adjusting the B0 field strength (red plot). This way thecross-talk from the B1 coil can be subtracted out. The result of this subtraction is the green plot here.There is some evidence of an FID signal.67Figure 4.13: Another FID attempt, this time with a time constant of 100 µs. This plot was generated by takingthe x and y channel inputs, and adding them in quadrature for each run. Shown are two data sets withthe nuclei near resonance with the B1 coil, at 6.95 A and 6.98 A through the B0 coil, as well as a data setwith the nuclei far off resonance, as a baseline (the current was about 6 A). In the baseline measurement,the nuclei would be unaffected by the pi/2 pulse, and there would be no precession, and no EMF in thepickup coil. The FID runs look very similar, and there is a noticeable signal above the baseline just afterthe B1 coil is turned off. The subtracted signals are shown in Figs. 4.14 and 4.15, with an exponentialfit of each.68Figure 4.14: This is the FID signal at B0 = 6.95 A (blue) and an exponential fit through the data (red). Thedata before the 5 ms point is removed, since it is contaminated by the remaining cross-talk from the B1pulse. T∗2 is measured to be 1.287±0.015 ms.Figure 4.15: This is the FID signal at B0 = 6.98 A (blue) and an exponential fit through the data (red). Thedata before the 5 ms point is removed, since it is contaminated by the remaining cross-talk from the B1pulse. T∗2 is measured to be 1.203±0.016 ms.694.3.3 Spin EchoThe T2 time can be found by doing a spin echo experiment, like described in Section 2.4.3. Figure 4.16 shows howthe function generators and other electronics are set up to generate the needed pulses. This can be simplified a greatdeal by using programmable function generators or switching circuits, but we did this experiment with little time,and with equipment we could find in our lab. We combined the outputs from two function generators to generatethe two pulses for the B1 coil, and the outputs from two other function generators to provide the transistor-transistorlogic (TTL) signal for the Q-killing switch. These function generators are all triggered on each other so that thepulses and TTL signals are active at the same times. The delay between the pi/2 and the pi pulses is adjustable onone of the function generators.This experiment is very similar to FID, but with an added pi pulse that comes after the sample has depolarizeddue to an inhomogeneity in the B0 field. After the pulse, the spins will briefly be in phase again, generating a peakin the signal after the pi pulse. This ”revival peak” will come at a time after the pi pulse exactly equal to the timebetween the pi/2 pulse and the pi pulse.Figure 4.17 shows the initial attempt I made at finding a spin echo. This experiment was a partial success; thereappears to be a small signal where it is expected to be, but the original FID signal is missing, potentially drowned outby leftover ringdown from cross-talk with the B1 coil. It seems very likely that the peak is indeed a spin echo sinceit would be difficult to explain the existence of a signal there in the data otherwise. Unfortunately, without beingable to measure the height of the original FID peak, this data is not useful for determining the T2 time of the sample.Once better data can be taken, the experiment can be repeated, adjusting the delay time between the pulses eachtime, and recording the height of the revival peak. The heights of these peaks can be fit to an exponential decay,with the time constant of the decay being the T2 time.70Figure 4.16: The spin echo experiment’s schematic is quite similar to the FID’s. There are two additionalfunction generators used to generate the second pulse. FG2’s output is added to FG1’s, to supply thetwo 5V pulses to the Q-killing resistor. FG3 adds a delayed pi pulse to the pi/2 pulse already generatedby CH1 of the dual output FG.71Figure 4.17: The signal from a spin echo experiment. The green is the subtracted signal, and there is evidenceof a spin echo seen after the second pulse, the pi pulse. However, the original FID signal is not seen.This experiment should be repeated for confirmation, and if it is successful, repeated for various timedelays so that the spin echo peak height can be fit to an exponential decay to determine the T2 lifetime.72Chapter 5Conclusions and Future Work5.1 ConclusionI have designed and built a set of magnetic coils to use in nuclear magnetic resonance (NMR) experiments, and usedthem to perform adiabatic fast passage (AFP) and free induction decay (FID) on protons in water. I have shown thatthey should be useful in their current configuration for measuring the degree of polarization of 129Xe.I have characterized the coils, measuring the field they generate, per Amp, as well as the B0 field’s homogeneity,especially along the z axis. The B0 coils generate a field of 94.3 µT/A at their center, when placed above the steeloptical table, and with no compensation for the gradient it produces. The B0 field’s inhomogeneity along the verticalaxis of the water bottle, for the experiments in Chapter 4 was measured with a flux gate to be about 10 µT The T∗2time of water was measured twice, with the results T∗2 = 1.287± 0.015 ms and T∗2 = 1.203± 0.016 ms. The total B0field gradient across the entire water bottle, calculated from these times is 18.9 ± 0.9 µT. This means that a largeportion of the inhomogeneity is still from the gradient along the z direction due to the steel table’s distorting effectson the B0 field. There was an attempt made to do a spin echo experiment, but it was inconclusive.There is disagreement in the B1 coil’s field measurement at 27.07 kHz. When measuring the field strength atdirect current (DC) and then using theory to predict the damping effect of the alternating current (AC) shield for thefield at 27.07 kHz, the strength should be about (97 ± 4)∗(1− rcoils/rshield), or about 87.3 ± 3.6 µT/A. However,fitting the AFP peak heights to a Landau-Zener model gives the result of 10.9± 0.3 µT/A. Looking at the adiabaticitycondition and the peak heights, it seems more likely that the DC field measurement is more accurate.5.2 Possible ImprovementsThe next milestone for this project is to successfully perform 129Xe FID. To do so, it would be very helpful, andpossibly necessary, to improve the signal to noise ratio (SNR) to the point that we do not need to average 100sor 1000s of runs to find an FID signal. Improvements to the AC shield may help in this regard, depending on theactual source of the noise, especially the strong noise at about 27.77 kHz can be damped. A higher SNR can alsobe achieved by improving the B0 field homogeneity, to maximize the T∗2 time. If the FID signal lasts longer, we canuse longer integration times on the lock-in amplifier, improving the signal amplitude, and narrowing the frequencyband of the amplifier.73Because all of the data for the results described in Chapter 4 were taken before analyzing them, there are anumber of improvements that can be made to the setup which are now clear to us, that were not implemented forthis thesis.5.2.1 AC ShieldThe effectiveness of the AC shield was never tested rigorously, but there is still a significant noise source at about27.77 kHz that is seen clearly in the FID data. This noise is either generated somehow inside the shield, is relatedto the electronics in the pickup coil circuit, or is externally generated noise that is not sufficiently damped by theshield. The easiest way to test the shield, if the noise source is external to it, is to remove the upper B0 coil from theshield, and do NMR without the top half of the shield present. If the noise is no worse than with the shield attached,then the shield is likely not being very effective.5.2.2 B0 Coil HomogeneitySince the B0 field measurements were done after the NMR experiments, it was not discovered, until too late, thatthe compensating potentiometer was connected to the wrong coil. The field homogeneity can be easily improved byattaching it to the correct coil. However, fine-tuning this potentiometer is challenging, since the best way to see ifthe homogeneity overall has improved is to do FID and find the T∗2 time of the sample in that field. By compensatingfor the field gradient this way, the center field strength is also changed, meaning the resonance frequency needs tobe found again each time FID is performed after adjusting the gradient. A more time efficient method is to use ananti-Helmholtz pair of coils as the compensation method. The field of such a pair of coils is 0 at the center, so whenadjusting their strength, the gradient is changed, but not the center field strength. This method was initially rejecteddue to concerns about having the space for such coils, but it would be worth the time and effort to look for ways tomake it work.5.2.3 B1 Pulse for FIDIn analyzing the AFP data taken while varying the B1 strength, there is strong disagreement between the expected B1field per amp of current, and that found by using the Landau-Zener model. This means that when performing FID,the pi/2 pulse may not have actually tipped the spins by pi/2. The best way to truly measure the B1 field strengthper Amp might be to adjust the duration or amplitude of the pi/2 pulse until the FID signal strength is maximized.5.2.4 FID Repetition RateWhen I performed FID, I repeated the experiment every 1.5 s and averaged the results to improve SNR. However, ifthe experiment is repeated too quickly, the spins have not had time to return to Boltzmann polarization. The resultis that the amplitude of the FID signal is diminished. The T1 time of water is about 1.7 s under most conditions[33],but the measurement I made suggests a longer time, of almost 3 s. The repetition rate for FID should be decreasedsignificantly. Despite the longer repetition time, the increased SNR will make it so fewer runs need to be averaged,which will hopefully result in a quicker, more accurate FID measurement. This makes improving the B0 homogeneityand optimizing the pi/2 pulse quicker, since both improvements require repeated FID experiments.745.3 Next StepsThe next milestone for this project is to successfully perform 129Xe FID. To do so, it would be very helpful, andpossibly necessary, to improve the SNR to the point that we do not need to average 100s or 1000s of runs to findan FID signal. The improvements described in Section 5.2 are all intended to work towards this goal. The B0 coilhomogeneity, and the B1 field strength per amp of current are the most important improvements, since they will alsoimprove the signal seen when testing the ultraviolet (UV) laser using this setup.By performing AFP on 129Xe, its polarization can be estimated by comparing the amplitude of the signal tothat of water, which has a known polarization. There are a number of factors to keep in mind when doing so,however. The precession frequency of 129Xe is smaller than that of a proton, so each 129Xe nucleus will contributeless to the electromotive force (EMF) generated in a pickup coil than each proton. Since the precession frequenciesare different, a separate tuning box needs to be made for each of the samples. These boxes will inevitably havea different Q-factor. The Q-factor for each tuned circuit needs to be measured so this can be corrected for. Thecomparison is made the most difficult by the fact that the pickup coil and container are also different between thewater and the 129Xe sample. The signal amplitude depends on factors such as how many windings are in the pickupcoil, its diameter, and how much of the space inside the coil is filled with the sample. Despite these challenges, acareful comparison the signal from a water sample is the best way to estimate the 129Xe polarization.After finding evidence of sufficiently hyperpolarized 129Xe, and having successfully performed FID on it, theUV laser can be tested for its ability to measure the B0 magnetic field strength. The laser will be circularly polarizedbefore being shined on the precessing 129Xe. Due to the additional angular momentum imparted on the 129Xe atomswhen absorbing a photon of this light, only atoms with a particular spin state will be able to absorb the light and betransferred to an excited state. This absorption will be sinusoidal due to the spin precession. Emission from theseatoms as they decay back to the ground state is detected, and the frequency of that emission is used to calculate themagnetic field strength from the 129Xe gyromagnetic ratio.75Bibliography[1] Emily Altiere. The adventures of nikita and casper. Master’s thesis, University of British Columbia, 2014. →pages iii, 11[2] J. Beringer (Particle Data Group). Neutron properties. PR, D86(010001), 2012. → pages 1[3] Maxim Pospelov and Adam Ritz. Electric dipole moments as probes of new physics. Annals of Physics,2005. → pages 1, 2[4] S. Dar. The neutron edm in the sm: A review. arXiv:hep-ph/0008248v2. → pages 2[5] P.J. Mohr, B.N. Taylor, and D.B. Newell. Codata recommended values of the fundamental physical constants:2010. NIST, March 2012. → pages 2[6] C.A. Baker et al. An improved experimental limit on the electric-dipole moment of the neutron. Phys. Rev.Lett., 97(13):131801, Sep 2006. → pages 2, 7[7] I.I. Raby, N. F. Ramsey, and J. Schwinger. 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Medical Physics, 1984. → pages 7477Appendix AMatlab CodeThis appendix lists holds the simulations that I coded to obtain the data for Chapter 3.A.1 Helmholtz CoilsThis simulation calculates the magnetic field created by a Helmholtz pair of coils. It is all contained in one file.1 % Created by Joshua Wienands23 % This i s a Program to ca l cu l a t e the magnetic f i e l d o f a pa i r o f c o i l s i n4 % Helmholtz con f i gu ra t i on , or a t some other separa t ion d is tance .5 % I t i s assumed tha t the atoms are i n a c y l i n d r i c a l c e l l .67 % The f i e l d i s ca l cu la ted using the f i n i t e d i f f e r ences technique . The8 % Biot−Savart Law i s a d i f f e r e n t i a l equat ion t ha t t e l l s us the s t reng th o f9 % magnetic f i e l d produced by an i n f i n i t e l y sho r t p iece of s t r a i g h t wi re .10 % The exact so l u t i o n would be to i n t eg r a t e t h i s over the en t i r e wi re ( which11 % i s a c o i l i n t h i s case ) . The approx imat ion comes from ins tead tak ing a12 % f i n i t e leng th and summing ra the r than i n t e g r a t i n g . This i s equ iva len t to13 % the c o i l s being made up of shor t , s t r a i g h t pieces o f wire , r a t he r than14 % being ac tua l c i r c l e s .1516 % You can def ine the f o l l ow i ng constants :1718 % FOR THE MAGNETIC FIELD :19 % co i l s i ze20 % co i l separa t ion21 % cur ren t22 % number o f windings23 % length and rad ius o f c e l l ( which i s c y l i n d r i c a l , cannot eas i l y be changed )24 %25 % reso l u t i o n i n r and d is tance to go i n r26 % reso l u t i o n i n z and d is tance to go i n z27 % reso l u t i o n i n dL f o r the Biot−Savart c a l c u l a t i o n282930 % NOTE:31 % For vec to rs con ta in ing data such as pos i t i on , ve l o c i t y , magnetic f i e l d ,32 % the f i r s t en t ry i s the x d i r e c t i on , second the y d i r e c t i on , and t h i r d the z33 % d i r e c t i o n .347835 % The ax is going through the center o f the two c o i l s i s the z ax is .36 % The program assumes a c y l i n d r i c a l c e l l , w i th i t s long ax is po i n t i ng i n37 % the r ( or equ i va len t l y , x or y ) d i r e c t i o n3839 % Values def ined i n a l l caps are e i t h e r phys i ca l constants or constant40 % parameters t ha t can be chosen . The i r value should not be a l t e r ed by the code41 % l a t e r i n the program .4243 % 10A, 100 windings r e su l t s i n 30G f i e l d a t center f o r he lmhol tz con f i g .4445 clear a l l ;46 format long ;4748 %Constants :49 MM = 10ˆ−3;50 K = 10ˆ−7; % k = 0 /4∗ p i = 10ˆ−7 N/Aˆ25152 % simu la t i on r e so l u t i o n53 Z STEPS = 100; % The number o f data po in t s i n z d i r e c t i o n . Inc reas ing t h i s increases s imu la t i on t ime l i n e a r l y54 MAX D = 20∗MM; % Max d is tance from the center o f the c o i l pa i r along z to ca l cu l a t e f i e l d55 % Make sure to inc lude the s ize o f the ce l l ’ s rad ius i n t h i s56 R STEPS = 1; % number o f data po in t s i n R d i r e c t i on , Inc reas ing t h i s increases the s imu la t i on t ime most ly l i n e a r l y57 MAX R = 120∗MM; % Max r a d i a l d is tance from center o f the c o i l pa i r to ca l cu l a t e f i e l d58 % Make sure to inc lude the en t i r e center to corner Ce l l l eng th w i t h i n t h i s59 L STEPS = 50; % reso l u t i o n o f the leng th element dL of the wi re i n c o i l s60 % 100 or so steps here looks to be enough to a t l eas t 1% accuracy61 % inc reas ing t h i s increases s imu la t i on t ime l i n e a r l y6263 %Coi l Parameters :64 SEPARATION = 300∗MM; % separa t ion between the c o i l s i n meters , along z ax is65 RADIUS = 300∗MM; % Radius o f the c o i l s i n meters66 CURRENT = 4.04 ; % cu r ren t i n Amps67 NUM WINDINGS = 100; % number o f windings6869 % Cel l parameters70 CELL RADIUS = 12.7∗MM; % rad ius o f the c e l l i n meters71 CELL LENGTH = 200∗MM; % leng th o f the c e l l i n meters7273 B0 = zeros ( [ 2∗R STEPS+1 , 2∗Z STEPS+1 , 4 ] ) ; % i n i t i a l i z e the B0 mat r i x74 % I t i s s u f f i c i e n t to ca l cu l a t e 1 quadrant75 % the res t i s symmetric7677 z s t ep s i ze = MAX D/Z STEPS ; % ca l cu l a t e the z step s ize i n m78 r s t e p s i z e = MAX R/R STEPS; % ca l cu l a t e the r step s ize i n m79 l s t e p s i z e = 2∗pi∗RADIUS/ L STEPS ; % ca l cu l a t e the leng th o f dL i n m808182 % Calcu la te the f i e l d from the two c o i l s using f i n i t e d i f f e r ences technique83 % and the Biot−Savart Law . This step can probably s t i l l be opt imized84 % s i g n i f i c a n t l y by e l im i na t i n g nested loops i n favo r o f v e c t o r i z a t i o n . Most85 % of the commented out code here i s from attempts to do so .8687 t he ta = 0:2∗ pi / L STEPS: (2∗ pi−l s t e p s i z e ) ;88 dL = [−1.0∗RADIUS∗2∗pi∗sin ( the ta ) / L STEPS ;RADIUS∗2∗pi∗cos ( the ta ) / L STEPS ; zeros (1 ,L STEPS ) ] ;89 for i = 0 :Z STEPS90 i %#ok<NOPTS> % p r i n t the i t e r a t i o n the program i s on91 z = z s tep s i ze∗ i ; % s t a r t the z steps a t z = 07992 for i i = 0 :R STEPS93 r = r s t e p s i z e∗ i i ; % s t a r t the r steps a t r = 09495 for i i i = 1 :L STEPS9697 % con t r i b u t i o n from lower c o i l98 % vecto r po i n t i ng from wire to measurement po in t99 x = [ r , 0 , z + SEPARATION/2]− [RADIUS∗cos ( the ta ( i i i ) ) ,RADIUS∗sin ( the ta ( i i i ) ) , 0 ] ;100 xmag = sqrt ( ( x ( 1 ) ˆ 2 + x ( 2 ) ˆ 2 + x ( 3 ) ˆ 2 ) ) ; % magnitude of t h i s vec to r101 dB0 = K∗CURRENT∗NUM WINDINGS∗cross ( dL ( : , i i i ) , x ) / ( xmag ˆ 3 ) ;102103 % con t r i b u t i o n from upper c o i l104 % vecto r po i n t i ng from wire to measurement po in t105 x = [ r , 0 , z − SEPARATION/2]− [RADIUS∗cos ( the ta ( i i i ) ) ,RADIUS∗sin ( the ta ( i i i ) ) , 0 ] ;106 xmag = sqrt ( ( x ( 1 ) ˆ 2 + x ( 2 ) ˆ 2 + x ( 3 ) ˆ 2 ) ) ; % magnitude of t h i s vec to r107 % This i s the change in B f i e l d i n Tesla , add i t to the change from other c o i l108 dB0 = dB0 + K∗CURRENT∗NUM WINDINGS∗cross ( dL ( : , i i i ) , x ) / ( xmag ˆ 3 ) ;109110111112 % add up the f i e l d c on t r i b u t i o n .113 % cu r r e n t l y j u s t the magnitude , d i r e c t i o n i s commented out114 % above , and only f o r one quadrant .115 % Ca lcu la t i on only needs to be done f o r one quadrant , but need116 % to populate the whole matr ix , so add app rop r i a t e l y to each of117 % the 4 quadrants by cen te r ing around R STEPS and Z STEPS118 % Also , be ca r e f u l to avoid over lap and avoid t r y i n g to w r i t e119 % to the 0 th index of a mat r i x120121122 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) + i , 1) = B0 ( (R STEPS + 1) + i i , . . .123 (Z STEPS + 1) + i , 1) + dB0 ( 1 ) ;124 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) + i , 2) = B0 ( (R STEPS + 1) + i i , . . .125 (Z STEPS + 1) + i , 2) + dB0 ( 2 ) ;126 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) + i , 3) = B0 ( (R STEPS + 1) + i i , . . .127 (Z STEPS + 1) + i , 3) + dB0 ( 3 ) ;% Lower r i g h t quadrant128129 i f ( i ˜= 0)130 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 1) = . . .131 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 1) + dB0 ( 1 ) ;132 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 2) = . . .133 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 2) + dB0 ( 2 ) ;134 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 3) = . . .135 B0 ( (R STEPS + 1) + i i , (Z STEPS + 1) − i , 3) + dB0 ( 3 ) ;% Lower l e f t136 i f ( i i ˜= 0)137 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 1) = . . .138 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 1) + dB0 ( 1 ) ;139 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 2) = . . .140 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 2) + dB0 ( 2 ) ;141 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 3) = . . .142 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) − i , 3) + dB0 ( 3 ) ;% Upper l e f t143 end144 end145146 i f ( i i ˜= 0)147 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 1) = . . .148 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 1) + dB0 ( 1 ) ;80149 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 2) = . . .150 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 2) + dB0 ( 2 ) ;151 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 3) = . . .152 B0 ( (R STEPS + 1) − i i , (Z STEPS + 1) + i , 3) + dB0 ( 3 ) ;% Upper r i g h t153 end154155 end156157 B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)+ i , 4 ) = sqrt (B0 ( (R STEPS+1)+ i i , . . .158 (Z STEPS+1)+ i , 1 ) ˆ 2+B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)+ i , 2 ) ˆ 2+ . . .159 B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)+ i , 3 ) ˆ 2 ) ;160 B0 ( (R STEPS+1)− i i , ( Z STEPS+1)+ i , 4 ) = sqrt (B0 ( (R STEPS+1)− i i , . . .161 (Z STEPS+1)+ i , 1 ) ˆ 2+B0 ( (R STEPS+1)− i i , ( Z STEPS+1)+ i , 2 ) ˆ 2+ . . .162 B0 ( (R STEPS+1)− i i , ( Z STEPS+1)+ i , 3 ) ˆ 2 ) ;163 B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)− i , 4 ) = sqrt (B0 ( (R STEPS+1)+ i i , . . .164 (Z STEPS+1)− i , 1 ) ˆ 2+B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)− i , 2 ) ˆ 2+ . . .165 B0 ( (R STEPS+1)+ i i , ( Z STEPS+1)− i , 3 ) ˆ 2 ) ;166 B0 ( (R STEPS+1)− i i , ( Z STEPS+1)− i , 4 ) = sqrt (B0 ( (R STEPS+1)− i i , . . .167 (Z STEPS+1)− i , 1 ) ˆ 2+B0 ( (R STEPS+1)− i i , ( Z STEPS+1)− i , 2 ) ˆ 2+ . . .168 B0 ( (R STEPS+1)− i i , ( Z STEPS+1)− i , 3 ) ˆ 2 ) ;169170171 end172 end173174175 % Plo t a s l i c e i n the xz plane , w i th the c e l l edges drawn in176177 % draw the c e l l edges178 for i = 1 : f loor (1+CELL RADIUS / z s t ep s i ze )179 r i ndex = f loor (1+ sqrt (CELL RADIUSˆ2 − ( ( i −1)∗ z s t ep s i ze ) ˆ 2 ) / r s t e p s i z e ) ;180 B0(R STEPS + r index , Z STEPS + i , 4 ) = 0 ;181 B0(R STEPS − ( r index −1) ,Z STEPS + i , 4 ) = 0 ;182 B0(R STEPS + r index , Z STEPS − ( i −1) ,4) = 0 ;183 B0(R STEPS − ( r index −1) ,Z STEPS − ( i −1) ,4) = 0 ;184 end185186 % p l o t the magnitude as a heat map.187 % th i s i s a s l i c e i n the xz plane , going through the o r i g i n188 imagesc (B0 ( : , : , 4 ) ) ;189 colorbar ;190 caxis ( [ 0 B0(R STEPS,Z STEPS , 4 )∗1 . 5 ] ) % cons t ra in the co lo r ax is because f i e l d gets huge near c o i l s191 % cons t r a i n t i s based on the mag. o f192 % f i e l d a t the center .193194 % re tu rn ; % end here unless you want the next p l o t a lso195196 % Can also p l o t a s l i c e on the xy plane :197198 B0 xy = zeros ( [ 2∗R STEPS, 2∗R STEPS, 2∗Z STEPS ] ) ; % I n i t i a l i z e B0 mat r i x i n xy plane199200 % Due to symmetry , the magnitude of the f i e l d i s i d e n t i c a l f o r a l l t he ta a t201 % a given r and z . B0 i s a mat r i x w i th the magnitude of the f i e l d as a202 % func t i on o f r and z . To p l o t t h i s i n a heat map, we need to create a203 % mat r i x o f the f i e l d magnitude as a func t i on o f x and y f o r a given z204 % do t h i s by loop ing over a l l x and y pos i t i ons , and choose the magnetic205 % f i e l d s t reng th by look ing up i t s value from r = sq r t ( x ˆ2 + y ˆ 2 ) , t ak ing a81206 % weighted average i n cases where the index wouldn ’ t be an i n t ege r .207 % since we loop over a l l x and y up to R MAX, there are many spots where we208 % have not ca l cu la ted the magnetic f i e l d . Just leave those as 0. I d e a l l y209 % these would be l e f t b lank i n the p lo t , I ’m not sure how to accomplish210 % tha t .211212 for i = 1 :Z STEPS % can create a s l i c e f o r every z213 for i i = 1 :R STEPS214 for i i i = 1 :R STEPS % these two loops loop over x and y215 r = sqrt ( i i i ˆ2 + i i ˆ 2 ) ; %f i n d index f o r the value o f r f o r t ha t x and y216 i f ( r +1)<=R STEPS % don ’ t have any i n f o f o r x ˆ2 + y ˆ2 > R MAXˆ2 , so leave them as 0217 %look f o r c l oses t r ava i l ab le , rounded down , get f i e l d there218 B0 up = B0(R STEPS + f loor ( r +1) , Z STEPS + i , 4 ) ;219 B0 down = B0(R STEPS + f loor ( r ) , Z STEPS + i , 4 ) ; %same th ing , rounded up220 % take a weighted average of the f i e l d s , t h i s i s the f i e l d221 % at t ha t x and y to good approx imat ion222 % Bet te r approx imat ion can be made by cons ider ing the223 % actua l form of the f i e l d vs . r , r a t he r than j u s t assuming224 % i t ’ s l i n e a r225 B i n t e r = (1−mod( r , 1 ) )∗B0 down + mod( r , 1)∗B0 up ;226227 % Ca lcu la t i on only needs to be done f o r one quadrant , but need228 % to populate the whole matr ix , so add app rop r i a t e l y to each of229 % the 4 quadrants by cen te r ing around R STEPS230 % Also , be ca r e f u l to avoid over lap and avoid t r y i n g to w r i t e231 % to the 0 th index of a mat r i x232 B0 xy (R STEPS+( i i i −1) , R STEPS+( i i −1) , Z STEPS + ( i −1)) = B i n t e r ; % Lower r i g h t quadrant233 B0 xy (R STEPS−( i i i −1) , R STEPS+( i i −1) , Z STEPS + ( i −1)) = B i n t e r ; % Upper r i g h t234 B0 xy (R STEPS+( i i i −1) , R STEPS−( i i −1) , Z STEPS + ( i −1)) = B i n t e r ; % Lower l e f t235 B0 xy (R STEPS−( i i i −1) , R STEPS−( i i −1) , Z STEPS + ( i −1)) = B i n t e r ; % Upper l e f t236237 B0 xy (R STEPS+( i i i −1) , R STEPS+( i i −1) , Z STEPS − ( i −1)) = B i n t e r ; % Lower r i g h t quadrant238 B0 xy (R STEPS−( i i i −1) , R STEPS+( i i −1) , Z STEPS − ( i −1)) = B i n t e r ; % Upper r i g h t239 B0 xy (R STEPS+( i i i −1) , R STEPS−( i i −1) , Z STEPS − ( i −1)) = B i n t e r ; % Lower l e f t240 B0 xy (R STEPS−( i i i −1) , R STEPS−( i i −1) , Z STEPS − ( i −1)) = B i n t e r ; % Upper l e f t241242 % Draw the c e l l edges by se t t i n g the f i e l d to be 0 there .243 % long edge f i r s t244245 i f i i == round (CELL RADIUS / r s t e p s i z e )246 i f i i i < round (CELL LENGTH/ (2∗ r s t e p s i z e ) )247 B0 xy (R STEPS+( i i i −1) , R STEPS+( i i −1) , Z STEPS) = 0; % Lower r i g h t quadrant248 B0 xy (R STEPS−( i i i −1) , R STEPS+( i i −1) , Z STEPS) = 0; % Upper r i g h t249 B0 xy (R STEPS+( i i i −1) , R STEPS−( i i −1) , Z STEPS) = 0; % Lower l e f t250 B0 xy (R STEPS−( i i i −1) , R STEPS−( i i −1) , Z STEPS) = 0; % Upper l e f t251252253 end254 end255256 % shor t edge next257 i f i i i == round (CELL LENGTH/ (2∗ r s t e p s i z e ) )258 i f i i < round (CELL RADIUS / r s t e p s i z e )259260 B0 xy (R STEPS+( i i i −1) , R STEPS+( i i −1) , Z STEPS) = 0; % Lower r i g h t quadrant261 B0 xy (R STEPS−( i i i −1) , R STEPS+( i i −1) , Z STEPS) = 0; % Upper r i g h t262 B0 xy (R STEPS+( i i i −1) , R STEPS−( i i −1) , Z STEPS) = 0; % Lower l e f t82263 B0 xy (R STEPS−( i i i −1) , R STEPS−( i i −1) , Z STEPS) = 0; % Upper l e f t264 end265 end266267268 end269 end270 end271 end272273274275 % p l o t the magnitude as a heat map276 % th i s i s f o r a s l i c e i n the xy plane277278 imagesc ( ( 10 ˆ4 )∗ B0 xy ( : , : , 1 ) ) ; % the l a s t index determines z pos i t i o n . 1 i s the center279 colorbar ;280281 % no need to re−scale co lo r ax is as long as we are not near the c o i l s282 % can s t i l l be use fu l to increase f i e l d magnitude r e so l u t i o n around the c e l l i t s e l f283284 caxis ( [ ( 1 0 ˆ 4 )∗ B0 xy (R STEPS,R STEPS)∗0.99 (10ˆ4)∗B0 xy (R STEPS,R STEPS ) ] ) ;A.2 Saddle CoilsThis simulation calculates the magnetic field created by a pair of coils in the saddle geometry. The simulation callson two functions which calculate the contribution from a small piece of a curved part of the wire, or the length ofone of the rungs, respectively. There is also a function I wrote to read the file generated by this simulation.A.2.1 Saddle Coil Simulation1 MM = 10ˆ(−3); % conver t from mm to m by ∗MM23 TABLE POS = 310∗MM; % dis tance from the center o f the c e l l to the tab l e4 TABLE ORIENT = 1; % How the tab l e and c o i l s are o r i en ted .5 % 1: c o i l s produce f i e l d perpend icu la r to t ab l e6 % 2: c o i l s produce f i e l d p a r a l l e l to t ab l e7 % other value : don ’ t ca l cu l a t e image f i e l d8 USE FUNC = 1;9 C RATIO = 1;1011 SHIFT = 0∗MM;12 DEFORM = 0∗MM;1314 ALPHA = 135∗pi / 180 ; %span of each c o i l i n rad ians15 RADIUS = 75∗MM; %rad ius o f the c o i l s i n meters16 LENGTH = 400∗MM; %leng th o f the c o i l s i n meters17 K = 10ˆ(−7); % k = 0 /418 CURRENT = 1; %cu r ren t i n amps19 WINDINGS = 10; %number o f windings o f the c o i l2021 num dL curves = 200;2223 xy s teps = 1; % number o f po in t s i n the x and y d i r e c t i o n s8324 B STEPS Z = 200; %number o f po in t s the z d i r e c t i o n2526 CELL RADIUS = 12.7∗MM; %rad ius o f the Xenon c e l l i n meters27 CELL LENGTH = 200∗MM; %leng th o f the Xenon c e l l i n meters2829 dL curves mag = ALPHA∗RADIUS/ num dL curves ;3031 d the ta = ALPHA/ num dL curves ; % How much the ta changes over dL3233 b s tep s i ze = CELL LENGTH/B STEPS Z ; % step s ize i n the z d i r e c t i o n3435 xy s tep s i ze = CELL RADIUS∗2/ xy s teps ; % step s ize i n the x or y d i r e c t i o n3637 B f i e l d = zeros ( xy steps , xy steps , B STEPS Z , 4 ) ; % i n i t i a l i z e the magnetic f i e l d38 dBrods = [ 0 ; 0 ; 0 ] ;39 dBcurved = [ 0 ; 0 ; 0 ] ;40 dBr image = [ 0 ; 0 ; 0 ] ;41 dBc image = [ 0 ; 0 ; 0 ] ;42 deform array = 2∗DEFORM∗rand (40 ,1 ) − DEFORM;43 deform angle = 2∗pi∗rand ( 40 , 1 ) ;44 d ar ray = zeros ( num dL curves , 1 ) ;45 d angle = zeros ( num dL curves , 1 ) ;464748 for i = 1 : num dL curves49 index = ce i l ( i ∗10/ num dL curves ) ;50 d ar ray ( i ) = deform array ( index ) ;51 d angle ( i ) = deform array ( index ) ;52 end5354 deformat ion = [ d ar ray d angle ] ;555657 for bx = 1: xy s teps58 for by = 1: xy s teps59 bx60 by61 for bz = 1:B STEPS Z6263 i f B STEPS Z == 1 && xy steps == 164 b pos = [065 066 0 ] ;6768 e l se i f B STEPS Z == 16970 b pos = [ ( xy s tep s i ze∗bx − CELL RADIUS)71 ( x y s tep s i ze∗by − CELL RADIUS)72 0 ] ;7374 e l se i f xy s teps == 17576 b pos = [077 078 ( b s tep s i ze∗bz − 0.5∗CELL LENGTH ) ] ;7980 else8481 b pos = [ ( xy s tep s i ze∗bx − CELL RADIUS)82 ( x y s tep s i ze∗by − CELL RADIUS)83 ( b s tep s i ze∗bz − 0.5∗CELL LENGTH ) ] ; %pos i t i o n the f i e l d i s being measured at84 end858687 dBrods = r o d s B i o t Sa va r t f i e l d (ALPHA,LENGTH,RADIUS, b pos ,CURRENT∗WINDINGS, C RATIO , . . .88 0 , SHIFT ) ;89 dBcurved = cu r ved B i o tSava r t f i e l d (ALPHA,LENGTH,RADIUS, dL curves mag , b pos , . . .90 CURRENT∗WINDINGS, d theta , C RATIO , 0 , SHIFT , deform array ) ;9192 i f TABLE ORIENT == 193 d i s t = [−2∗TABLE POS; 0 ; 0 ] ;94 image pos = b pos − d i s t ;95 dBr image = r od s B i o t Sa va r t f i e l d (ALPHA,LENGTH,RADIUS, image pos ,CURRENT∗WINDINGS, . . .96 C RATIO , 1 , SHIFT ) ;97 dBc image = cu r ved B i o tSava r t f i e l d (ALPHA,LENGTH,RADIUS, dL curves mag , image pos , . . .98 CURRENT∗WINDINGS, d theta , C RATIO , 1 , SHIFT , deform array ) ;99100 else101 dBr image = [ 0 ; 0 ; 0 ] ;102 dBc image = [ 0 ; 0 ; 0 ] ;103 end104105106 B f i e l d ( bx , by , bz , 1) = B f i e l d ( bx , by , bz , 1 ) + dBrods (1 ) + dBcurved (1 ) + . . .107 dBr image (1 ) + dBc image ( 1 ) ;108 B f i e l d ( bx , by , bz , 2) = B f i e l d ( bx , by , bz , 2 ) + dBrods (2 ) + dBcurved (2 ) + . . .109 dBr image (2 ) + dBc image ( 2 ) ;110 B f i e l d ( bx , by , bz , 3) = B f i e l d ( bx , by , bz , 3 ) + dBrods (3 ) + dBcurved (3 ) + . . .111 dBr image (3 ) + dBc image ( 3 ) ;112113 B f i e l d ( bx , by , bz , 4) = sqrt ( B f i e l d ( bx , by , bz , 1 ) ˆ 2 + . . .114 B f i e l d ( bx , by , bz , 2 ) ˆ 2 + . . .115 B f i e l d ( bx , by , bz , 3 ) ˆ 2 ) ;116 end117118 end119 end120121122123 %prepare every th ing to record to t e x t f i l e124 bsize = size ( B f i e l d ) ; %s ize o f the magnetic f i e l d mat r i x125 xs ize = bsize ( 1 ) ; %s p l i t i n t o x , y and z s izes126 ys ize = bsize ( 2 ) ;127 zs ize = bsize ( 3 ) ;128 %open a f i l e to w r i t e to , add some desc r i p t i o n to the f i l e name129 %number o f dL steps and c o i l s i ze a t l eas t130131 % bu i l d the f i l e name132 alphadeg = ALPHA∗180/ pi ;133 i f CELL LENGTH > 0.3 | | CELL RADIUS > 0.02134 reg ion = ’ b ig ’ ;135 else136 reg ion = ’ c e l l ’ ;137 end85138139 i f B STEPS Z > xy s teps140 axes = ’ z ’ ;141 e l se i f B STEPS Z < xy s teps142 axes = ’ xy ’ ;143 else144 axes = ’ xyz ’ ;145 end146147 i f USE FUNC == 1148 func = ’ wfunc ’ ;149 else150 func = ’ ’ ;151 end152153 i f TABLE ORIENT == 1154 img = ’ ximage ’ ;155 e l se i f TABLE ORIENT == 2156 img = ’ yimage ’ ;157 else158 img = ’ ’ ;159 end160161162 leng = LENGTH/MM;163 r ad i = RADIUS/MM;164165 f i lename = spr in t f ( ’%ddLsteps %ddeg %dmmx%dmm coi l %dA %s reg ion %s%s%s%1.3 f I r a t i o %1.3 f y s h i f t ’ , . . .166 i n t 16 ( num dL curves ) , i n t 16 ( alphadeg ) , i n t 16 ( leng ) , i n t 16 ( r ad i ) , i n t 16 (CURRENT∗WINDINGS ) , . . .167 region , axes , func , img , C RATIO , SHIFT ) ;168 i = 2 ;169170 %i f the f i l e a l ready ex i s t s , add a number to the end of i t .171 i f ex is t ( spr in t f ( ’%s . t x t ’ , f i lename ) , ’ f i l e ’ )172 while exist ( spr in t f ( ’%s%d . t x t ’ , f i lename , i ) , ’ f i l e ’ )173 i = i +1;174 end175 f i lename = spr in t f ( ’%s%d ’ , f i lename , i ) ;176 end177 f i lename = spr in t f ( ’%s . t x t ’ , f i lename ) ;178 f i l e i d = fopen ( f i lename , ’w ’ ) ;179180 %p r i n t a header ; the length , rad ius o f co i l , cur ren t , span of the c o i l i n181 %degrees , as we l l as step s izes and number o f steps i n each d i r e c t i o n182 %Matlab gets upset when reading i f there are any non−numeric characters ,183 %also there should be 4 values per l i n e184 f p r i n t f ( f i l e i d , ’%f\ t%f\ t%f\ t%f\ t0\ t0\ t0\n ’ , LENGTH, RADIUS, CURRENT∗WINDINGS, 180∗ALPHA/ pi ) ;185 f p r i n t f ( f i l e i d , ’%d\ t%d\ t%d\ t%d\ t%d\ t%f\ t0\n ’ , xy steps , xy s tep s i ze , xy steps , xy s tep s i ze , . . .186 B STEPS Z , b s tep s i ze ) ;187188 % wr i t e the magnitude of the magnetic f i e l d to a t e x t f i l e189 for i = 1 : xs ize190 for j = 1 : ys ize191 for k = 1: zs ize192 xcoord = i ∗ xy s tep s i ze ;193 ycoord = j ∗ xy s tep s i ze ;194 zcoord = k∗b s tep s i ze ;86195 %p r i n t i n s c i e n t i f i c no ta t i on (%e )196 f p r i n t f ( f i l e i d , ’%e\ t%e\ t%e\ t %1.10e\ t %1.10e\ t %1.10e\ t %1.10e\n ’ , xcoord , ycoord , zcoord , . . .197 B f i e l d ( i , j , k , 4 ) , B f i e l d ( i , j , k , 1 ) , B f i e l d ( i , j , k , 2 ) , B f i e l d ( i , j , k , 3 ) ) ;198 end199 end200 end201202203 fclose ( ’ a l l ’ ) ; %close the f i l eA.2.2 Magnetic Field From a Curved Section of Wire1 function dBc = cu r ved B i o tSava r t f i e l d ( alpha , len , radius , dLmag , bpos , cur ren t , . . .2 d theta , c u r r e n t r a t i o , image , s h i f t )34 % Calcu la te the magnetic f i e l d due to the 4 curved po r t i ons o f56 i f image == 17 c o i l s h i f t = [0 s h i f t ] ;8 else910 c o i l s h i f t = [ s h i f t 0 ] ;11 end12 K = 1e−7;13 zr = 0.5∗ l en ; % z pos stays constant14 dBc = [ 0 ; 0 ; 0 ] ;15 i = 1 ;1617 for a = 0.5∗ d the ta : d the ta : alpha1819 % Find the x and y components f o r a po in t on a curved pa r t20 % of the co i l , get the d is tance to the measurement po in t ,21 % use Biot−Savart law to get the con t r i b u t i o n to the f i e l d2223 the ta1 = alpha /2 − a ; %angle f o r the f i r s t curved pa r t24 xr = rad ius∗cos ( the ta1 ) ;25 yr = rad ius∗sin ( the ta1 ) ; %x and y components f o r the po in t262728 r1 = bpos − [ x r ; y r ; z r ] + [ c o i l s h i f t ( 1 ) ; c o i l s h i f t ( 2 ) ; 0 ] ; %get d is tance29 r1mag = sqrt ( r1 ( 1 ) ˆ 2 + r1 ( 2 ) ˆ 2 + r1 ( 3 ) ˆ 2 ) ; %dis tance squared3031 %get dL d i r e c t i on , magnitude was obta ined before32 dLx = dLmag∗sin ( the ta1 ) ;33 dLy = −1.0∗dLmag∗cos ( the ta1 ) ;34 dLr1 = [ dLx ; dLy ; 0 ] ;3536 %con t r i b u t i o n from tha t curved piece373839 %do the same th i ng again f o r the next piece40 % same co i l , but o ther side , so f l i p z41 r2 = bpos − [ x r ; y r ; −1.0∗zr ] + [ c o i l s h i f t ( 1 ) ; c o i l s h i f t ( 2 ) ; 0 ] ;42 r2mag = sqrt ( r2 ( 1 ) ˆ 2 + r2 ( 2 ) ˆ 2 + r2 ( 3 ) ˆ 2 ) ;43 dLr2 = −1.0∗dLr1 ;4487454647 %again , f o r curved piece 348 %f l i p x , y pos i t i on , +z again though49 r3 = bpos − [−1.0∗ xr ; −1.0∗yr ; z r ] + [ c o i l s h i f t ( 2 ) ; c o i l s h i f t ( 1 ) ; 0 ] ;50 r3mag = sqrt ( r3 ( 1 ) ˆ 2 + r3 ( 2 ) ˆ 2 + r3 ( 3 ) ˆ 2 ) ;51 dLr3 = dLr1 ;52535455 %f i n a l l y , the 4 th curved piece56 %f l i p x , y , z ( or add ra the r than sub t rac t )57 r4 = bpos + [ x r ; y r ; z r ] + [ c o i l s h i f t ( 2 ) ; c o i l s h i f t ( 1 ) ; 0 ] ;58 r4mag = sqrt ( r4 ( 1 ) ˆ 2 + r4 ( 2 ) ˆ 2 + r4 ( 3 ) ˆ 2 ) ;59 dLr4 = −1.0∗dLr1 ;6061 i f image == 162 dB1 = K∗cu r ren t ∗(cross ( dLr1 , r1 ) / r1mag ˆ 3 ) ;63 dB2 = K∗cu r ren t ∗(cross ( dLr2 , r2 ) / r2mag ˆ 3 ) ;64 dB3 = K∗cu r ren t∗ c u r r e n t r a t i o ∗(cross ( dLr3 , r3 ) / r3mag ˆ 3 ) ;65 dB4 = K∗cu r ren t∗ c u r r e n t r a t i o ∗(cross ( dLr4 , r4 ) / r4mag ˆ 3 ) ;666768 else69 dB1 = K∗cu r ren t∗ c u r r e n t r a t i o ∗(cross ( dLr1 , r1 ) / r1mag ˆ 3 ) ;70 dB2 = K∗cu r ren t∗ c u r r e n t r a t i o ∗(cross ( dLr2 , r2 ) / r2mag ˆ 3 ) ;71 dB3 = K∗cu r ren t ∗(cross ( dLr3 , r3 ) / r3mag ˆ 3 ) ;72 dB4 = K∗cu r ren t ∗(cross ( dLr4 , r4 ) / r4mag ˆ 3 ) ;73 end7475 dBc = dBc + dB1 + dB2 + dB3 + dB4 ;76 i = i +1;7778 end7980 endA.2.3 Magnetic Field From a Straight Rod1 function dBr = r o d s B i o t Sa va r t f i e l d ( alpha , len , radius , b pos , cur ren t , c u r r e n t r a t i o , image , s h i f t )234 % Calcu la te the exact f i e l d a t b pos due to the 4 rod po r t i ons o f the5 % saddle c o i l pa i r . No f i n i t e d i f f e r ences needed , the general formula6 % fo r the magnitude i s :7 %8 % B = (K∗ I / s )∗ [ s i n ( the ta2 ) − s in ( the ta1 ) ]9 %10 % K i s a constant , I i s the cur ren t , s i s the d is tance between b pos11 % and the rod , the ta1 i s the angle between s and the vec to r po i n t i ng12 % from the beginning o f the rod to b pos , and theta2 i s the angle13 % between s and the vec to r po i n t i ng from the end of the rod to b pos .1415 K = 1e−7; % constant1617 % x and y pos i t i o ns o f the rods . X ax is goes through the co i l s , y ax is8818 % goes between them . Z ax is goes down the c o i l ax is . Rods are centered19 % around the o r i g i n .2021 i f image == 122 x = [ rad ius∗cos ( alpha / 2 )23 rad ius∗cos ( pi − alpha / 2 )24 rad ius∗cos ( pi + alpha / 2 )25 rad ius∗cos(−1.0∗alpha / 2 ) ] ; % x pos i t i ons o f the 4 rods2627 y = [ rad ius∗sin ( alpha / 2 )28 rad ius∗sin ( pi − alpha / 2 ) + s h i f t29 rad ius∗sin ( pi + alpha / 2 ) + s h i f t30 rad ius∗sin (−1.0∗alpha / 2 ) ] ; % y pos i t i ons o f the 4 rods3132 else3334 x = [ rad ius∗cos ( alpha / 2 )35 rad ius∗cos ( pi − alpha / 2 )36 rad ius∗cos ( pi + alpha / 2 )37 rad ius∗cos(−1.0∗alpha / 2 ) ] ; % x pos i t i ons o f the 4 rods3839 y = [ rad ius∗sin ( alpha / 2 ) + s h i f t40 rad ius∗sin ( pi − alpha / 2 )41 rad ius∗sin ( pi + alpha / 2 )42 rad ius∗sin (−1.0∗alpha / 2 ) + s h i f t ] ; % y pos i t i ons o f the 4 rods43 end4445 % dis tance between the rod and the po in t . I f the po in t i s past the edge46 % of the rod along the z axis , then t h i s i s the d is tance to where the47 % rod would be i f i t were longer .48 d i s t = [ sqrt ( ( b pos (1)−x ( 1 ) ) ˆ 2 + ( b pos (2)−y ( 1 ) ) ˆ 2 )49 sqrt ( ( b pos (1)−x ( 2 ) ) ˆ 2 + ( b pos (2)−y ( 2 ) ) ˆ 2 )50 sqrt ( ( b pos (1)−x ( 3 ) ) ˆ 2 + ( b pos (2)−y ( 3 ) ) ˆ 2 )51 sqrt ( ( b pos (1)−x ( 4 ) ) ˆ 2 + ( b pos (2)−y ( 4 ) ) ˆ 2 ) ] ;5253 % get the angle made by the x ax is and the vec to r po i n t i ng from the rod54 % to b pos .55 ph i = asin ( ( b pos (1 ) − x ) . / d i s t ) ;5657 % angle between s and the l i n e connect ing the beginning o f the rod to58 % b pos59 the ta1 = [ atan ((−0.5∗ len − b pos ( 3 ) ) / d i s t ( 1 ) )60 atan ((−0.5∗ len − b pos ( 3 ) ) / d i s t ( 2 ) )61 atan ((−0.5∗ len − b pos ( 3 ) ) / d i s t ( 3 ) )62 atan ((−0.5∗ len − b pos ( 3 ) ) / d i s t ( 4 ) ) ] ;6364 % angle between s and the l i n e connect ing the end of the rod to b pos65 the ta2 = [ atan ( ( 0 . 5∗ l en − b pos ( 3 ) ) / d i s t ( 1 ) )66 atan ( ( 0 . 5∗ l en − b pos ( 3 ) ) / d i s t ( 2 ) )67 atan ( ( 0 . 5∗ l en − b pos ( 3 ) ) / d i s t ( 3 ) )68 atan ( ( 0 . 5∗ l en − b pos ( 3 ) ) / d i s t ( 4 ) ) ] ;697071 i f image == 172 % f i e l d c on t r i b u t i o n to the x d i r e c t i o n i s the magnitude∗cos ( ph i ) . Phi73 % was ca l cu la ted above .74 dBrx = (K∗cu r ren t / d i s t ( 1 ) )∗ ( sin ( the ta2 ( 1 ) ) − sin ( the ta1 ( 1 ) ) )∗ cos ( ph i ( 1 ) ) + . . .8975 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 2 ) )∗ ( sin ( the ta2 ( 2 ) ) − sin ( the ta1 ( 2 ) ) )∗ cos ( ph i ( 2 ) ) + . . .76 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 3 ) )∗ ( sin ( the ta2 ( 3 ) ) − sin ( the ta1 ( 3 ) ) )∗ cos ( ph i ( 3 ) ) + . . .77 (K∗cu r ren t / d i s t ( 4 ) )∗ ( sin ( the ta2 ( 4 ) ) − sin ( the ta1 ( 4 ) ) )∗ cos ( ph i ( 4 ) ) ;7879 % f i e l d c on t r i b u t i o n to the x d i r e c t i o n i s the magnitude∗s in ( ph i ) . Phi80 % was ca l cu la ted above .81 dBry = (K∗cu r ren t / d i s t ( 1 ) )∗ ( sin ( the ta2 ( 1 ) ) − sin ( the ta1 ( 1 ) ) )∗ sin ( ph i ( 1 ) ) + . . .82 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 2 ) )∗ ( sin ( the ta2 ( 2 ) ) − sin ( the ta1 ( 2 ) ) )∗ sin ( ph i ( 2 ) ) + . . .83 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 3 ) )∗ ( sin ( the ta2 ( 3 ) ) − sin ( the ta1 ( 3 ) ) )∗ sin ( ph i ( 3 ) ) + . . .84 (K∗cu r ren t / d i s t ( 4 ) )∗ ( sin ( the ta2 ( 4 ) ) − sin ( the ta1 ( 4 ) ) )∗ sin ( ph i ( 4 ) ) ;8586 else87 % f i e l d c on t r i b u t i o n to the x d i r e c t i o n i s the magnitude∗cos ( ph i ) . Phi88 % was ca l cu la ted above .89 dBrx = (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 1 ) )∗ ( sin ( the ta2 ( 1 ) ) − sin ( the ta1 ( 1 ) ) )∗ cos ( ph i ( 1 ) ) + . . .90 (K∗cu r ren t / d i s t ( 2 ) )∗ ( sin ( the ta2 ( 2 ) ) − sin ( the ta1 ( 2 ) ) )∗ cos ( ph i ( 2 ) ) + . . .91 (K∗cu r ren t / d i s t ( 3 ) )∗ ( sin ( the ta2 ( 3 ) ) − sin ( the ta1 ( 3 ) ) )∗ cos ( ph i ( 3 ) ) + . . .92 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 4 ) )∗ ( sin ( the ta2 ( 4 ) ) − sin ( the ta1 ( 4 ) ) )∗ cos ( ph i ( 4 ) ) ;9394 % f i e l d c on t r i b u t i o n to the x d i r e c t i o n i s the magnitude∗s in ( ph i ) . Phi95 % was ca l cu la ted above .96 dBry = (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 1 ) )∗ ( sin ( the ta2 ( 1 ) ) − sin ( the ta1 ( 1 ) ) )∗ sin ( ph i ( 1 ) ) + . . .97 (K∗cu r ren t / d i s t ( 2 ) )∗ ( sin ( the ta2 ( 2 ) ) − sin ( the ta1 ( 2 ) ) )∗ sin ( ph i ( 2 ) ) + . . .98 (K∗cu r ren t / d i s t ( 3 ) )∗ ( sin ( the ta2 ( 3 ) ) − sin ( the ta1 ( 3 ) ) )∗ sin ( ph i ( 3 ) ) + . . .99 (K∗cu r ren t∗ c u r r e n t r a t i o / d i s t ( 4 ) )∗ ( sin ( the ta2 ( 4 ) ) − sin ( the ta1 ( 4 ) ) )∗ sin ( ph i ( 4 ) ) ;100 end101102 % make a vec to r from the components . There i s no f i e l d i n the z103 % d i r e c t i o n a t any po in t due to the rods .104 dBr = [ dBrx ; dBry ; 0 ] ;105 endA.2.4 A Function to Read the Simulation Output File1 % header r e f e r s to how many l i n e s make up the header i n the t e x t f i l e2 % before the B1 data s t a r t s .34 function [ B1 B1x B1y B1z ] = read B1 data3 ( i n p u t F i l e )56 formatSpec = ’%e %e %e %e %e %e %e ’ ;7 sizeA = [7 I n f ] ; % 7 columns of data89 f i l e i d = fopen ( i n p u t F i l e )1011 A = fscanf ( f i l e i d , formatSpec , sizeA ) ; % read the data i n1213 fclose ( ’ a l l ’ ) ;1415 A = transpose (A ) ; % transpose the data to make ana lys i s eas ie r1617 length = A(1 ,1 ) % f i r s t row conta ins c o i l data18 rad ius = A(1 ,2 )19 cu r ren t = A(1 ,3 )20 alpha = A(1 ,4 )2122 xsteps = A(2 ,1 ) % second row conta ins s imu la t i on data9023 x s t ep s i ze = A(2 ,2 )24 ysteps = A(2 ,3 )25 y s t ep s i ze = A(2 ,4 )26 zsteps = A(2 ,5 )27 z s t ep s i ze = A(2 ,6 )28 i = 3 ;2930 B1 = zeros ( xsteps , ysteps , zsteps ) ; % the res t i s the magnetic f i e l d31 B1x = zeros ( xsteps , ysteps , zsteps ) ; % at each po in t32 B1y = zeros ( xsteps , ysteps , zsteps ) ; % i n i t i a l i z e the output data33 B1z = zeros ( xsteps , ysteps , zsteps ) ;343536 % div i de the data i n t o the magnitude of the magnetic f i e l d37 % and the f i e l d i n each of the ca rd i na l d i r e c t i o n s38 for x = 1: xsteps39 for y = 1: ysteps40 for z = 1: zsteps41 B1( x , y , z ) = A( i , 4 ) ;42 B1x ( x , y , z ) = A( i , 5 ) ;43 B1y ( x , y , z ) = A( i , 6 ) ;44 B1z ( x , y , z ) = A( i , 7 ) ;4546 i = i +1;47 end48 end49 end505152 end91


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