Short-Collision Time Approximation

Calculating the scattering law \(S(\alpha,\beta)\) for large momentum changes (i.e. large \(\alpha\) values) can become costly. To avoid prohibitively costly calculations, LEAPR employs the short-collision time approximation (SCT) to describe scattering in a solid for these conditions of large momentum transfer. The SCT approximation is defined as

\[S(\alpha,-\beta)=\frac{1}{\sqrt{4\pi\alpha w_s \overline{T}/T}}~\mathrm{exp}\left[-\frac{(\alpha w_s-\beta)^2}{\alpha w_s\overline{T}/T}\right]\]

and

\[S(\alpha,\beta)=\mathrm{e}^{-\beta}~S(\alpha,\beta)\]

where the effective temperature \(\overline{T}\) is defined as

\[\overline{T}=\frac{T}{2w_s}\int_{-\infty}^\infty\beta^2P(\beta)\mathrm{e}^{-\beta}~d\beta\]

and \(w_s\) is the weight for the continuous, solid-type spectrum.