Representation of Incoherent Elastic through Debye-Waller Factors

In Continuous, Solid-type Spectrum, we found that the phonon expansion for solid-type spectra resulted in the following sum:

\[S_s(\alpha,\beta) \approx \mathrm{e}^{-\alpha\lambda_s}~\sum_{n=1}^N\frac{1}{n!}\Big[\alpha\lambda_s\Big]^n\mathcal{T}_n(\beta)\]

where \(n\) corresponds to the number of phonons created/destroyed in a collision. This summation used intentionally left out the \(n=0\) term (which, since no phonons are created/destroyed, corresponds to elastic scattering). The zero-phonon term is intead considered here, so that if translational motion is desired, it can be properly treated.

No Translational Component Included

If the user does not request that a translational component be considered in while preparing the scattering data, then the zero-phonon term may simply be added without additional steps.

\[S_{inc.el}(\alpha,\beta)=\mathrm{e}^{-\alpha\lambda}~\delta(\beta).\]

The corresponding differential scattering cross section is

\[\sigma(E,\mu)=\frac{\sigma_b}{2}~\mathrm{e}^{-2WE(1-\mu)}\]

and the integrated cross section is

\[\sigma(E) = \frac{\sigma_b}{2}~\frac{1-\mathrm{e}^{-4WE}}{2WE}.\]

Where the Debye-Waller factor \(W\) is computed using the Debye-Waller coefficient \(\lambda\),

\[W=\frac{\lambda}{Ak_bT},\]

where \(A\) is the ratio of the scatterer mass to the neutron mass and \(\lambda\) can be readily computed from the phonon density of states, as was seen in Continuous, Solid-type Spectrum.

LEAPR writes the bound scattering cross section \(\sigma_b\) and the Debye-Waller factor \(W\) as a function of temperature into a section of ENDF-6 output with MF=7 and MT=2.

Translational Component Included

If a translational term is considered, then the translational and continuous scattering law contributions are combined as follows:

\[S(\alpha,\beta) = S_t(\alpha,\beta)~\mathrm{e}^{-\alpha\lambda_s} + \int_{-\infty}^\infty S_t(\alpha,\beta')~S_s(\alpha,\beta-\beta')~d\beta'\]

Note that while doing this convolution, the values of the translational piece \(S_t(\alpha,\beta')\) and the solid-type piece \(S_s(\alpha,\beta-\beta')\) are precomputed and interpolated on.