Discrete Oscillator (Einstein Crystal)

Polyatomic molecules normally contain a number of vibrational modes that can be approximated as discrete oscillators. These would appear in the phonon distribution as a Dirac-\(\delta\) functions with some corresponding weight, \(\omega_{i}\delta(\beta_i)\).

If there exist any peaks in the scattering law that the user wants to approximate as a \(\delta\) function, the corresponding scattering law contribution \(S_d(\alpha,\beta)\) can be computed as

\[\begin{split}\begin{align} S_{i}(\alpha,\beta)&=\mathrm{e}^{-\alpha\lambda_i}\sum_{n=-\infty}^\infty\delta(\beta-n\beta_i)~I_n\left[\frac{\alpha\omega_i}{\beta_i\sinh(\beta_i/2)}\right]~\mathrm{e}^{-n\beta_i\,/2}\\ &=\sum_{n=-\infty}^\infty A_{in}(\alpha)~\delta(\beta-n\beta_i) \end{align}\end{split}\]

where the discrete oscillator Debye-Waller coefficient is defined as

\[\lambda_i=\frac{\omega_i\,\coth(\beta_i/2)}{\beta_i}\]