Coherent Inelastic Approximations

Skold

While preparing the scattering law (for inelastic thermal scattering processes), it is standard to use the incoherent approximation. As introduced in Incoherent Approximation, the incoherent approximation neglects the distinct contribution of the pair distribution function, which neglects interference between different atoms.

Intermolecular interference occurs when there is both a significant bound coherent scattering cross section (property of the atoms) as well as some correlation between the positions of nearby molecules (property of the lattice). If these requirements are met, coherent scattering may become non-negligible, at which point its effect can be accounted for by using the Skold approximation.

Here, the scattering law is separated into a coherent and an incoherent contribution, which are weighted using a coherent fraction \(c\). The incoherent contribution to the scattering law can be obtained using the phonon density of states (which is explained more further in [LINK TO LEAPR DOCUMENTATION HERE]), but the coherent contribution must be approximated.

\[S(\alpha,\beta)=\big(1-c\big)S_{inc}(\alpha,\beta)+c~S_{coh}(\alpha,\beta)\]

The Skold approximation approximates the coherent scattering law by using the static structure factor \(S(\kappa)\) to modify the incoherent scattering law.

\[S_{coh}(\alpha,\beta)=S_{inc}\left(\frac{\alpha}{S(\kappa)},\beta\right)\times S(\kappa)\]

The static structure factor \(S(\kappa)\) is a user-provided input that describes correlation in molecular positions, where \(\kappa\) is wave number, defined as

\[\kappa = \frac{\sqrt{2Mk_bT\alpha}}{\hbar}\]

where \(M\) is the mass of the scatterer. Using these relations, the coherent-corrected scattering can be obtained by solving the above three equations for all \(\alpha,\beta\) values.