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 $\ensuremath{\Theta}$ and much more than the impurity exchange temperature $\frac{\ensuremath{\hbar}J}{{k}_{B}}$, it is shown that the diffusivity is given by $D=(\frac{J{a}^{4}}{{\ensuremath{\sigma}}^{*}x})$. The effective cross section ${\ensuremath{\sigma}}^{*}$ 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 $(\frac{\ensuremath{\hbar}J}{{k}_{B}\ensuremath{\Theta}}){(\frac{T}{\ensuremath{\Theta}})}^{7}\ensuremath{\ll}x\ensuremath{\ll}1$. Observation of the concentration dependence $D\ensuremath{\propto}\frac{1}{x}$ would constitute strong evidence of quantum mobility, which has been of considerable theoretical interest in recent years.

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