Coherent diffusion of isotopic impurities in solid helium

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.