Quantum diffusion of positive muons in copper

Abstract

The diffusion of positive muons (${\mu }^{+}$≊0.11×proton mass) in copper was studied experimentally by the zero-field muon-spin-relaxation method in a temperature (T) range from 70 mK to 190 K, using a pulsed muon beam suitable for the present method. The measurements were performed in three copper samples of different purities to see the effect of impurities on the nature of the ${\mu }^{+}$ diffusion. We find that the diffusion (hopping) rate ν in ultrapure copper decreases rapidly with decreasing temperature, reaches a minimum at T=30–70 K, then begins to increase. The T dependence in the low-temperature region follows $T^{\mathrm{-}\alpha }$(α≃0.67±0.03) at 0.5–10 K and levels off below 0.5 K. The behavior is strongly modified by impurity consisting of ≊100 ppm iron below 10 K. The T dependence in the pure copper is accounted for by new theories developed independently by Kondo and by Yamada, predicting a hopping rate ν∝$T^{2K\mathrm{-}1}$(0≤K≤1/2 for a single charged particle), where the factor $T^{2K}$ comes from the renormalization of the muon tunneling matrix due to the nonadiabaticity of the conduction electron interacting with the moving muon. From the present results the constant K is determined to be K=0.16±0.01.