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Signa Ture Redacted' Signature Redacted Ultracold Neutron Storage Simulation Using the MASSACUTS ILNSTITUTE Kassiopeia Software Package OF TECHNOLOGY by JUL 2 4 2019 Zachary Bogorad LIBRARIES Submitted to the Department of Physics ARCHIVES in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 Massachusetts Institute of Technology 2019. All rights reserved. Signa ture redacted' A uthor ............................ D 4 tment of Physics May 10, 2019 Signature redacted_ C ertified by ....................... ose . Formaggio rofessor of Physics Thesis Supervisor Signature redacted Accepted by ................ Nergis Mavalvala Associate Department Head, Department of Physics 2 Ultracold Neutron Storage Simulation Using the Kassiopeia Software Package by Zachary Bogorad Submitted to the Department of Physics on May 10, 2019, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract The Kassiopeia software package was originally developed to simulate electromagnetic fields and charged particle trajectories for neutrino mass measurement experiments. Recent additions to Kassiopeia also allow it to simulate neutral particle trajectories in magnetic fields based on their magnetic moments. Two different methods were implemented: an exact method that can work for arbitrary fields and an adiabatic method that is limited to slowly-varying fields but is much faster for large precession frequencies. Additional interactions to simulate reflection of ultracold neutrons from material walls and to allow spin-flip pulses were also added. These tools were used to simulate neutron precession in the Paul Scherrer Institute's neutron electric dipole moment experiment and predict the values of the longitudinal and transverse relax- ation times as well as the trapping lifetime. All three parameters are found to closely match the experimentally determined values when simulated with both the exact and adiabatic methods, confirming that Kassiopeia is able to accurately simulate neutral particles. This opens the door for future uses of Kassiopeia to prototype the next generation of atomic traps and ultracold neutron experiments. Thesis Supervisor: Joseph A. Formaggio Title: Professor of Physics 3 4 Acknowledgments I would like to thank Joseph Formaggio for being an invaluable mentor to me for the last several years. I would also like to thank Mathieu Guigue for some helpful discussion, as well as to acknowledge the KATRIN and Project 8 collaborations for their support of this project. This work was supported by the MIT Undergraduate Research Opportunities Program, by the Department of Energy Office of Nuclear Physics under award number DE-SCO011091, and by SERI-FCS under award number 2015.0594. Other allied works were supported by Sigma Xi under grant number G2017100190747806. 5 6 Contents 1 Introduction 15 2 The PSI nEDM Experiment 17 2.1 The Neutron Precession Chamber .................... 19 3 Equations of Motion for Spin 21 3.1 Exact Equations of Motion ...... ........ ......... 21 3.2 Adiabatic Equations of Motion . ........ ........ ..... 22 4 Surface Interactions of Ultracold Neutrons 25 5 The Structure of Kassiopeia 29 5.1 Initialization ......... ............... ........ 29 5.2 Sim ulation ..... ........ ......... ........ ... 31 6 Implementation of the Simulation 33 6.1 Additional Kassiopeia Features ............. ........ 33 6.2 A nalysis ........................ .......... 35 7 Results & Discussion 37 7.1 The Longitudinal Relaxation Time T1 ................. 37 7.2 The Transverse Relaxation Time T2 .................. 38 7.3 The Effective Trap Lifetime T ...................... 38 7.4 Summary of Results and Discussion . ............ ...... 39 7 8 Other Applications 41 A XML Bindings for New Kassiopeia Features 43 A .1 G enerators .... ...... .. ...... ...... .... 43 A.2 Trajectories ........................................ 45 A.3 Interactions ........................................ 46 A .4 O utput .... ...... ...... ...... ....... .. 46 B Figures 49 C Tables 57 8 I "W'11RMIR I List of Figures B-1 A plot of the z component of the magnetic field at the center (length- wise) of the PSI precession chamber in one of the field configurations. The larger dots are the field values provided to us (measured in [39J) while the smaller dots are our interpolated values. The 0(1 nT) vari- ations in the field are typical of the entire trap, both configurations, and all three components. ..... ....... ....... ..... 49 B-2 A VTK image of three neutrons tracked for 1 second inside of the PSI precession chamber. Colors correspond to kinetic energies, with higher energies at the blue end of the spectrum. Note the approximately specular reflections and the slight curvature in the tracks induced by gravity. ...... ............ ............ ..... 50 B-3 The vectors used to define the adiabatic coordinates m and # from the spin vector s in (3.5). .............. ............. 50 B-4 Plots of the exact (left, see (4.2)) and approximate (right, see (4.4b)) outgoing angle distributions for a UCN incident on a surface with wk = 2.4 (typical for our simulations) and an angle to the surface normal of 7r/7. Note how the two distributions are reasonably similar, though the approximation is rough since wk ~ 1. This was acceptable for us since there was no reason for our simulation results to depend strongly on the exact angular distribution function. ........... .... 51 9 B-5 A comparison of neutron spin angles around the local magnetic field (in the coordinate system defined by (3.5)), including an adiabatic trajectory with an integration time step of 100 ps and exact trajectories with time steps of 5, 10, and 20 ps. These were run in a version of the neutron precession chamber described in this work, but with the magnetic field scaled up by a factor of 1000, resulting in a precession frequency of approximately 30 MHz. As this comparison shows, the adiabatic trajectory agrees well with the exact trajectory at a much shorter step size, but the exact trajectory rapidly loses accuracy as the step size approaches the precession period of 33 ps. Note that the plot necessarily suffers from aliasing, but sampling was done simultaneously so that points at the same time can be compared. ........... 52 B-6 The distribution of UCN energies after 300 seconds, using adiabatic simulations of neutrons in one field configuration. Note that the av- erage energy is now far below the initial 150 neV, and in fact a sharp cutoff around the real part of the optical potential (approximately 150 neV ) is apparent. ........... .................. 53 B-7 A comparison of the fractional remaining longitudinal polarization measured at PSI (Figure 5.18 in [23], p.114) with the results of our adiabatic and exact simulations, including 1l- confidence intervals for the simulations and 1l- error bars for the experimental data. The x- axis labels are hidden as the PSI data sets are not yet published. Here, a 0.95 relative polarization corresponds to an average spin along the z-axis equal to 0.95 times the initial average spin along the z-axis. Each data set averages over the two field configurations and over ini- tially aligned and anti-aligned neutrons. We see good agreement over the entire period; the deviation of the exact data at long times is not statistically significant. ........... ............ ... 54 10 B-8 A comparison of the fractional remaining transverse polarization mea- sured at PSI (Figure 2.38 in [42], p.73) with the results of our adiabatic and exact simulations, including la confidence intervals for the simu- lations and lr error bars for the experimental data. The x-axis labels are hidden as the PSI data sets are not yet published. Here, a 0.95 relative polarization corresponds to an average spin along the axis of average precession equal to 0.95 times the initial average spin along the initial polarization axis. Each data set averages over the two field configurations and over two initial spin directions. We see good agree- ment over the entire period; the deviation of the exact data at long times is not statistically significant. ....... ........ .... 55 B-9 A comparison of the fraction of neutrons remaining in the trap mea- sured at PSI (Figure 2.37 in 142], p.72) with the results of our adiabatic and exact simulations, including la confidence intervals for the simula- tions and la error bars for the experimental data. The x-axis labels are hidden as the PSI data sets are not yet published. Averages for each data set are taken as in Figure B-7 and Figure B-8. Good agreement is seen along the entire PSI data set. Note that the exact data uses a shifted y-axis for visual clarity. The experimental values are arbitrarily normalized as the initial neutron count is not known. ......... 56 11 12 List of Tables C. 1 The results of X2 testing of our results against the PSI experimental data, including left-sided p-values. Note that ndf=11 for the T1 and T2 measurements and ndf=9 for the lifetime measurements. These values suggest that we are over-estimating our errors, likely due to our results being strongly correlated. The T1 adiabatic lifetime data's dis- agreement with the experimental data may indicate that our assumed value of r/ was not correct. ...... .................. 57 13 14 Chapter 1 Introduction At sufficiently low energies, neutrons can be reflected from material walls by the coherent strong interaction [41, 35]. Such neutrons can be stored in material traps, and are termed "ultracold neutrons" (UCNs). The ability of UCNs to be stored for prolonged periods makes them useful for mea- suring the neutron lifetime [32]. UCNs can be stored in a trap for various times, and then the remaining number can be counted in order to extract a lifetime. However, the precision of this method is limited by uncertainties in other neutron losses-for example, neutrons leaking out of the trap-so a thorough understanding of neutron dy- namics inside the trap is crucial for these measurements. One tool for understanding these dynamics is numerical simulation, which will be the focus of this work.
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