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Ff Investigation of Transient E Ects in a Time-Dependent Magnetic Kicker Sietse Buijsman

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Ff Investigation of Transient E Ects in a Time-Dependent Magnetic Kicker Sietse Buijsman faculty of mathematics and natural sciences van swinderen institute for particle physics and gravity Investigation of Transient Eff ects in a Time-Dependent Magnetic Kicker Sietse Buijsman University of Groningen J-PARC Muon g-2 EDM Experiment Master Thesis Investigation of Transient Effects in a Time-Dependent Magnetic Kicker Supervisors Author Dr. ir. C.J.G. Onderwater S.F. Buijsman Prof. dr. K.H.K.J. Jungmann 26 June, 2019 Abstract The most recent developments in the experimental determination of the muon anomalous magnetic dipole moment (g − 2) have indicated a difference of 3.6 standard deviations between the theory and experiment. The muon g − 2 EDM experiment situated at J-PARC is designed to simultaneously measure g − 2 and the electric dipole moment (EDM) at low energies by measuring the spin precession vector. This is done by measuring trajectories of decay positrons, which are, by violation of parity conservation, preferably emitted in the direction of the spin of the muon in the muon rest frame. The J-PARC experiment aims to improve upon the previous measurement to a precision of 0.1 ppm (from 0.54 ppm, by BNL), in parallel with the experiment situated at Fermilab, which aims for similar precision goals. The muons are injected in a storage ring of a 33 cm radius in a 3 T magnetic field. A weak magnetic focusing field keeps the muons in the storage region. A time-dependent magnetic kicker brings the muons from the injection volume to the storage volume. Various systematic effects caused by the magnetic kicker have been investigated. The electric field induced by the time-varying kicker field has a negligible effect on the dynamics of the injected muons. The electric field decayed below the desired 10 mV/cm after about 150 ns after the kicker was switched off. The volume of acceptance was determined to reduce calculation time. The muon spin was successfully tracked using a numerical method that preserves the length of the spin vector. Oscillations in the pitch angle have been observed. The frequencies present in this oscillation 5 6 correspond well with theory. Their values are found to be !1 = 4:5 · 10 · 2π rad/s and !2 = 1:4 · 10 · 2π rad/s. Effects caused by retardation, transient currents and broadening of the current waveform due to dispersion have been investigated. Muon injection did not succeed for effective current speeds below ∼ 7=10 c. It has been shown that successful injection is prevented for a kicker coil of multiple turns. Retardation effects are negligible compared to transient and dispersion effects. Broadening of the current waveform prevents injection for a broadening greater than ∼ 1/12 ns for each ns of propagation time. To still successfully trap muons while taking these effects into account, the kicker parameters need to be adjusted accordingly. When these effects are taken into account, muon injection is expected to be successful. Contents 1 Context and Motivation 3 2 Introduction 4 2.1 Physics . .4 2.1.1 Anomalous Magnetic Dipole Moment . .4 2.1.2 Electric Dipole Moment . .5 2.2 Recent Developments . .6 2.2.1 Theoretical . .6 2.2.2 Experimental . .6 2.2.3 Future Experiments . .7 2.3 J-PARC . .7 2.3.1 The Muon Beam Line . .7 2.3.2 Muon Injection and Storage . .8 2.3.3 Measurement of Decay Positrons . .8 2.4 Goal of this Thesis . 10 3 Numerical Calculation of Magnetic Fields 11 3.1 Numerical Calculation of Magnetic Fields . 11 3.1.1 Loop . 11 3.1.2 Coil . 12 3.2 The Magnetic Field Class . 12 3.3 Fitting the Fields . 13 3.3.1 Fitting the Main Field . 14 3.3.2 Fitting the Weak Focusing Field . 16 3.3.3 The Kicker Coils . 18 3.4 Precision and Accuracy . 19 3.4.1 Main Field . 19 3.4.2 Focus Field . 20 3.5 Conclusions . 20 4 Muon Injection 21 4.1 Dynamics . 21 4.1.1 The Equation of Motion . 21 4.1.2 The Runge-Kutta Fourth Order Algorithm . 21 4.2 Trapping of a Muon . 22 4.2.1 Weak Focusing . 22 4.2.2 Kicker . 22 4.2.3 Injection . 24 4.3 The Effect of Electric Fields on Dynamics . 24 4.3.1 Adjusting the Equation of Motion . 24 4.3.2 The RK4 Algorithm for Electric Fields . 25 4.3.3 Electric Field of a Thin Circular Wire . 26 4.3.4 Results for Electric Field Simulations . 27 4.3.5 Electric Field Inside the Storage Volume . 27 4.4 Conclusions . 28 1 5 Phase Space 29 5.1 Concepts and Dimensionality . 29 5.1.1 Emittance and Acceptance . 29 5.1.2 Determining the Acceptance . 29 5.2 Integrating Over Phase Space . 31 5.3 Conclusions . 31 6 Spin Tracking 32 6.1 Spin Precession . 32 6.1.1 The T-BMT Equation . 32 6.1.2 The RK4 for Spin Tracking . 33 6.2 Results of Spin Tracking . 34 6.2.1 Injection . 34 6.2.2 Trapping . 34 6.3 Conclusions . 36 6.3.1 Injection . 37 6.3.2 Trapping . 37 7 Transient Currents and Retardation 38 7.1 Retardation . 38 7.2 Transient Current . 40 7.2.1 Finite Propagation Speed . 40 7.2.2 Dispersive Effects . 40 7.3 Results . 41 7.3.1 Single Turn Kicker . 41 7.3.2 Multiple Turn Kicker . 42 7.3.3 Dispersion . 43 7.4 Conclusions . 43 8 Overall Conclusions and Outlook 45 8.1 Magnetic Field Calculation . 45 8.2 Muon Injection . 45 8.3 Phase Space Acceptance . 45 8.4 Spin Tracking . 46 8.5 Transients and Retardation . 46 8.6 Outlook . 46 Acknowledgements 47 Appendices 50 A Runge-Kutta Spin Vectors 51 B Magnetic Field Class 52 C Muon Tracking 59 2 Chapter 1 Context and Motivation For a long time, science has been the practice of trying to understand, explain and predict natural phenom- ena. Ever since.
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