Quantum diffusion of positive muons in copper

Abstract

The diffusion of positive muons (${\ensuremath{\mu}}^{+}$\ensuremath{\approxeq}0.11\ifmmode\times\else\texttimes\fi{}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 ${\ensuremath{\mu}}^{+}$ diffusion. We find that the diffusion (hopping) rate \ensuremath{\nu} 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 ${T}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$(\ensuremath{\alpha}\ensuremath{\simeq}0.67\ifmmode\pm\else\textpm\fi{}0.03) at 0.5--10 K and levels off below 0.5 K. The behavior is strongly modified by impurity consisting of \ensuremath{\approxeq}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 \ensuremath{\nu}\ensuremath{\propto}${T}^{2K\mathrm{\ensuremath{-}}1}$(0\ensuremath{\le}K\ensuremath{\le}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\ifmmode\pm\else\textpm\fi{}0.01.