Quantum Theory of Diffusion with Application to Solid Helium

A quantum theory of diffusion is presented and applied to the diffusion of isotopic impurities in solid helium. For temperatures much less than the Debye temperature $\Theta$ and much more than the impurity exchange temperature $\frac{\hslash J}{k_B}$, it is shown that the diffusivity is given by $D=\left(\frac{J{a}^4}{{\sigma }^{*}x}\right)$. The effective cross section ${\sigma }^{*}$ for the scattering of two mobile impurity atoms is of the order of a square lattice spacing $a^2$, and the mole fraction $x$ of the impurity atoms is assumed to obey $\left(\frac{\hslash J}{k_B\Theta }\right){\left(\frac{T}{\Theta }\right)}^7\ll x\ll 1$. Observation of the concentration dependence $D\propto \frac{1}{x}$ would constitute strong evidence of quantum mobility, which has been of considerable theoretical interest in recent years.