University of Zurich¨ Script - Lecture PHY 432 Physics with Muons: From Atomic Physics to Solid State Physics Alex Amato March 2020 Contents Forwords 7 Literature 9 1. Introduction: the Muon 11 1.1. The muon: an elementary particle ...................... 11 1.2. Muon discovery................................ 12 1.3. The pion: the parent particle ......................... 17 1.3.1. Pion properties............................ 17 1.3.2. Pion production reactions ...................... 17 1.3.3. The pion decay............................ 19 1.4. Muon properties................................ 21 1.5. The muon decay................................ 23 1.5.1. Generalities.............................. 23 1.5.2. Differential positron emission .................... 23 1.5.3. Decay of a muon ensemble ...................... 26 1.6. Muon magnetic moment and spin precession . 27 1.6.1. Muon magnetic moment ....................... 27 1.6.2. Muon spin precession ........................ 28 1.6.2.1. Classical view ....................... 28 1.6.2.2. Quantum mechanics view . 29 1.7. Atmospheric muons .............................. 30 1.8. “Man-made” muons .............................. 36 1.8.1. Pion production: 3 different possible accelerators . 36 1.8.2. Pion production: for example at the Paul Scherrer Institute . 38 1.8.3. Muon beams for condensed matter . 42 1.8.3.1. “High-energy” muons ................... 42 1.8.3.2. “Surface” muons ...................... 46 1.8.3.3. Few words about a typical beamline and beam optics . 48 Dipole magnets ........................ 49 Quadrupole magnets ..................... 50 Separator (Wien filter) and spin rotator . 55 2. Implanting Muons in Matter 59 2.1. Energy loss of particles in matter ....................... 59 2.1.1. Energy loss by ionization: classical approach . 61 2.1.2. Energy loss: Bethe formula ..................... 64 2.2. Range and thermalization time ........................ 70 2.2.1. Range of muons ........................... 70 2 2.2.2. Thermalization time ......................... 73 2.3. “Free” muon vs muonium........................... 74 3. µSR Technique 77 3.1. The µSR signal ................................ 78 3.2. Key features of the µSR technique ...................... 83 3.2.1. Local probe - volume sensitivity ................... 83 3.2.2. Larmor frequency - magnetic field sensitivity . 86 3.2.3. Typical time window ......................... 87 3.2.4. Other important features ....................... 87 3.3. Experimental setup .............................. 89 3.3.1. At a “continuous-wave” (cw) beam . 89 3.3.2. At a pulsed beam ........................... 94 3.3.3. Muon-on-request setup ........................ 95 3.4. The different measurement geometries .................... 96 3.4.1. ZF & LF geometry .......................... 96 3.4.2. TF geometry ............................. 99 4. Depolarization Functions 101 4.1. Depolarization function for static internal fields (ZF geometry) . 101 4.1.1. The simple case: single value of field . 103 4.1.1.1. Single-crystal case . 103 Some examples . 105 4.1.1.2. Polycrystal case . 107 Some examples . 110 4.1.2. Randomly oriented fields . 112 4.1.2.1. Gaussian distribution . 112 Some examples . 115 4.1.2.2. Lorentzian distribution . 117 4.1.2.3. “In between” . 119 4.1.3. Generalization ............................ 120 4.2. Depolarization function for applied external fields (TF geometry) . 121 4.3. Dynamical effects ............................... 124 4.3.1. Stochastic processes . 124 4.3.2. The strong collision approximation . 124 4.3.2.1. The muon depolarization . 127 Approximations for some limiting cases . 128 Examples . 130 Fluctuation time or correlation time? . 133 4.3.3. Testing for the dynamical character with a longitudinal external field (LF) .................................. 136 4.3.3.1. LF external field and static field distribution . 138 4.3.3.2. LF external field and dynamical effects . 143 4.3.4. Dynamical effects in TF external field . 151 5. Studying magnetism with the µSR technique 153 5.1. Local magnetic fields in magnetic materials . 153 5.1.1. The interaction muon - electron . 153 3 5.1.2. Dipolar and contact contributions .................. 154 5.1.2.1. Orbital field ........................ 155 5.1.2.2. Dipolar field........................ 155 5.1.2.3. Contact field........................ 157 5.1.3. Demagnetizing field and Lorentz sphere . 158 5.2. Examples of magnetic states studied by µSR . 161 5.3. Special cases magnetic states . 172 5.3.1. Incommensurate vs commensurate magnetic structure . 172 5.3.1.1. The simple case . 172 5.3.1.2. The slightly more difficult case . 176 5.3.2. Study of spin glasses . 179 5.4. Determining magnetic volume fractions . 183 5.5. Studying the magnetic response in the paramagnetic or diamagnetic states: the Knight shift ................................ 186 5.5.1. Knight shift (contact term): Studying the paramagnetism of the con- duction electrons . 187 5.5.1.1. Pauli susceptibility . 188 5.5.2. Knight shift in materials with local moments . 192 5.5.2.1. The dipole field contribution . 192 5.5.2.2. The enhanced contact field contribution . 193 5.5.2.3. The total Knight shift . 194 5.5.3. Determining the muon-stopping site . 195 5.6. Depolarization created by nuclear moments . 200 5.6.1. Classical calculation . 200 5.6.1.1. The TF case . 200 5.6.1.2. The ZF case . 203 5.6.1.3. Comment . 205 6. µSR in the Superconducting State 207 6.1. Introduction .................................. 207 6.2. Two characteristic lengths in superconductors . 209 6.2.1. The magnetic penetration depth . 210 6.2.1.1. The London equations . 210 6.2.1.2. Field and current decay in the Meissner state . 211 6.2.2. The coherence length . 213 6.2.2.1. The Ginzburg Landau theory . 213 6.3. Two types of superconductor . 216 6.3.1. Condensation energy and energy balance . 216 6.3.2. Type I and type II superconductors . 217 6.4. Abrikosov state of a type II superconductor . 220 6.4.1. Field in the Abrikosov state . 221 6.4.1.1. Field due to one vortex . 223 6.4.1.2. Field distribution of an extreme type II superconductor . 224 6.4.1.3. Corrections due to the coherence length and the magnetic field ............................ 227 4 6.5. Obtaining the characteristic lengths from µSR . 232 6.5.1. Obtaining the penetration depth . 232 6.5.1.1. Multi-Gaussian approach . 236 6.5.2. Obtaining the coherence length . 238 6.6. Testing the superconducting gap symmetry . 239 6.7. Determining the anisotropy of the magnetic penetration depth . 243 6.8. Multiple superconducting gaps . 244 6.9. Uemura relation, Uemura plot: Correlation between Tc and σ . 245 6.10. Dynamics of the FLL ............................. 248 6.10.1. Melting through temperature . 248 6.10.1.1. Stabilization with defects . 248 6.10.2. Moving the FLL with an applied current . 248 7. Low energy muons: a tool to study thin films and heterostructures 249 7.1. Introduction .................................. 249 7.2. Generation of low energy muons . 250 7.2.1. Moderation in thin layers of cryosolids . 250 7.2.2. Laser resonant ionization of muonium . 253 7.3. The Low-Energy Muon (LEM) instrument at PSI . 255 7.4. Stopping profiles of Low-Energy Muons in thin films . 259 7.5. Examples of LEM studies . 264 7.5.1. Magnetic field penetration at the surface of superconductors . 264 7.5.1.1. Strongly type-II superconductors . 264 7.5.1.2. Weak type-II superconductors and type-I superconductors 266 7.5.2. Giant proximity effect in cuprate heterostructures . 268 7.5.3. Probing the spin injection in an organic spin valve . 270 8. Muonium 273 8.1. Introduction .................................. 273 8.2. Muonium ground state and hyperfine interaction . 274 8.2.1. Ionisation energy . 274 8.2.2. Hyperfine interaction . 275 8.2.3. Adding an external field . 277 8.3. Time evolution of the muon polarization in the muonium state . 281 8.3.1. Introduction .............................. 281 8.3.2. Longitudinal (and zero) field case . 283 8.3.3. Transverse field case . 285 8.4. Few examples of muonium studies . 289 8.5. Anomalous muonium and weakly bound muonium . 291 A. Annex 297 A.1. Magnetic Moment ............................... 297 A.1.1. Introduction .............................. 297 A.1.2. Relation to the angular momentum . 298 A.1.2.1. Orbital angular momentum . 298 A.1.2.2. Spin angular momentum . 299 A.1.2.3. Spin angular momentum of the muon . 299 5 A.2. Spin Angular Momentum........................... 301 A.2.1. Spin Operators............................ 301 A.2.2. Spin Space .............................. 301 2 A.2.3. Eigenstates of S z and S . 303 A.2.4. Pauli Representation......................... 304 A.2.5. Relating Spinor to Spin Direction .................. 307 A.3. The canonical momentum (or generalized momentum) . 308 A.3.1. Legendre transformation....................... 309 A.3.2. Rewriting the Hamiltonian...................... 310 A.4. The demagnetizing field ........................... 311 A.5. Useful formula from Quantum Mechanics.................. 313 A.5.1. Time evolution of an operator . 313 A.5.2. 2-D Matrix .............................. 314 A.6. Useful vector relations ............................ 315 A.6.1. Laplacian ............................... 315 A.6.1.1. Laplacian operator . 315 A.6.1.2. Vector Laplacian . 315 A.6.2. General identities . 315 A.6.3. Gradient, divergence and curl . 316 A.6.4. Some examples ............................ 316 Bibliography 317 Bibliography 317 6 Forwords The first version of this lecture notes was composed in December 2017 and Januar 2018. Though