First Quantum-Mechanical Correction to the Classical Viscosity Cross Section of Hard Spheres

Abstract

The classical and first quantum correction terms in a high-energy expansion of the viscosity cross section ${Q}^{(2)}$ for a Boltzmann gas of hard spheres is derived. The first correction is found to be proportional to ${(\frac{1}{k\ensuremath{\sigma}})}^{\frac{4}{3}}$, which is a term nonanalytic in $\ensuremath{\hbar}$ (i.e., ${\ensuremath{\hbar}}^{\frac{4}{3}}$), and results from scattering near the edge of the sphere. A bound is established showing the remainder of the asymptotic series to be of $O[\frac{\mathrm{ln}(k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}]$. This asymptotic formula is compared with calculations based on the exact phase-shift expressions and its range of validity is established. The next correction terms are deduced to be proportional to $\frac{(\mathrm{ln}k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}$ and $\frac{1}{{(k\ensuremath{\sigma})}^{2}}$ which involve ${\ensuremath{\hbar}}^{2}\mathrm{ln}\ensuremath{\hbar}$ and ${\ensuremath{\hbar}}^{2}$, respectively.