Calculation of the structure factor of metals with the complete multiphonon series

It is shown that to a high accuracy (better than 7%) the $n$-phonon term $S^{\mathrm{}\left(n\right)\mathrm{}}\mathrm{}\left(q\right)(n\ge 2)$, is simply expressed in terms of the exponent of the Debye-Waller factor $2W$ as $S^n\mathrm{}\left(q\right)=\frac{{\left(2\mathrm{}W\right)}^n}{n!}$. This follows from the realization that the multiphonon contribution to the structure factor is mostly incoherent. Based on this result an expression accurate to 0.7% at all temperatures is proposed for the static structure factor $S_E\mathrm{}\left(q\right)\mathrm{}$ used in electronic transport calculations, namely: $S_E\mathrm{}\left(q\right)=S_E^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\left(q\right)\{1+{\left[S_E^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\right(q\left)\right]}^{-1\mathrm{}}(e^{2W}-1-2\mathrm{}W\left)\right\}{e}^{-2\mathrm{}W}$. This form is convenient because the right-hand side can be directly computed from experimental phonon dispersion curves. Futhermore, it represents an order-of-magnitude improvement in accuracy over the usual approximation $S_E\mathrm{}\left(q\right)=S_E^{\mathrm{}\left(1\right)\mathrm{}}\mathrm{}\left(q\right)\mathrm{}$.