Inelastic (Incoherent Approximation)¶

As introduced in Scattering Law, the thermal neutron scattering cross section is

\[\sigma(E\rightarrow E',\mu) = \frac{\sigma_b}{k_bT}\sqrt{\frac{E'}{E}}~S(\alpha,\beta).\]

The two steps to preparing cross sections are:

Calculate the scattering law \(S(\alpha,\beta)\). This is primarily done by the LEAPR module.
Use the scattering law to calculate the cross section, and prepare the final data on ENDF files. This is performed by the THERMR module.

For more information on these steps, please see [LINK TO LEAPR AND THERMR DOCUMENTATION HERE]

Currently, the inelastic thermal neutron scattering uses the following approximations:

Warning

What approximations are made? In describing the scattering behavior with a continuous, solid-type spectrum, a number of approximations are made, which are summarized below.

Approximation	Description	Comments
Gaussian approximation	In the definition of \(S(\alpha,\beta)\), \(\mathrm{exp}\Big(-\alpha\gamma(t) +\alpha^2\gamma_2(t)+\dots\Big)\) is approximated as \(\mathrm{exp} \big(-\alpha\gamma(t)\big)\).	See Parks [REFERENCE] for more discussion. Not typically considered a significant source of error
Incoherent approximation	While this is not an approximation made in the above equations, it is important to note that LEAPR uses the incoherent equations to desribe both coherent and incoherent scattering.	This is typically valid practice for materials with either low coherent cross sections or randomly-oriented crystallites.
Validity of phonon DOS	The continuous calculation requires an an input phonon spectrum, which are specific to material composition, crystalline structure, and temperature.	Before using a published phonon spectrum, ensure that it corresponds to the correct material, structure, and temp.
Randomly oriented spins	In order to separate scattering into coherent and incoherent components, spin-correlation effects are ignored.	Spin-correlation is important while considering materials like liquid hydrogen/deuterium, which are considered separately later.
Edge effects are ignored	Edge effects are not considered, meaning that the scatterer is considered to be infinitely large.	Ignoring edge effects is a standard approximations that is is not typically considered to cause significant error.
Atoms of only one kind are present	Atoms of only one kind are considered. Treatment for mixed moderators will be discussed later.	In a material like \(\mbox{H}_2\mbox{O}\), scattering from hydrogen is significantly more important than that from oxygen, so that only hydrogen exists is a valid simplification. This is not the case for materials like beryllium oxide, however.
Simplified crystal structure	Atoms are assumed to be bound by isotropic, harmonic forces in a crystal with cubic symmetry and one atom per unit cell.	This approach is applied to many materials that do not display these characteristics, and is believed to be an adequate approximation.

Topic	Resources
Phonon frequency spectrum theory	Please see Vibrational (Phonon) Density of States
Phonon frequency spectrum availability	1. http://materialsproject.com/ 2. ENDF-B/VIII release available at https://www.nndc.bnl.gov/endf/b8.0/ has a thermal neutron scattering sublibrary, with ENDF outputs and the LEAPR inputs with which the ENDF files were generated. In those LEAPR inputs are phonon distributions.