Coherent diffusion of isotopic impurities in solid helium

Abstract

A theory of coherent diffusion (impuriton propagation) of dilute isotopic impurities in solid He is described. An expression for the mean square distance 〈R 2 (t)〉 traveled by the impurity is derived. For short times pure band propagation occurs and so 〈R 2 (t)〉 grows as t 2. For times longer than a characteristic impurity-phonon scattering time this time dependence becomes linear, and a diffusion constant can be defined. This characteristic scattering time is estimated to be ~7 × 10−5 T −9 sec (T in K) for a 3He impurity in 4He at V m = 21 cm3/mole. The corresponding diffusion constant is consistent with that measured in the NMR experiment of Allen and Richards if the 3He-4He tunneling frequency J 34 is taken as 7 × 105 sec−1.