Scattering Law¶
The thermal scattering cross section is defined as
\[\sigma(E\rightarrow E',\mu) = \frac{\sigma_b}{k_bT}\sqrt{\frac{E'}{E}}~S_{n.sym}(\alpha,\beta)\]
where \(S_{n.sym}(\alpha,\beta)\) is the non-symmetric form of the thermal scattering law, \(\sigma_b\) is the bound scattering cross section, and \(k_bT\) is the temperature in eV.
Here, \(\alpha\) is unitless momentum exchange and \(\beta\) is unitless energy exchange, as seen below:
where initial and final neutron energies are \(E\) and \(E'\) respectively, \(A\) is the ratio of the scatterer mass to the neutron mass, and \(\mu\) is the cosine of the scattering angle in the laboratory. Note that \(\beta\) is positive for neutron energy gain and negative for neutron energy loss.
The purpose of LEAPR is to prepare the scattering law (along with Bragg edges, Debye-Waller factors, etc.) for further use by supplementary codes like the THERMR module.
Symmetric vs. Non-Symmetric¶
In the definition of the inelastic thermal neutron scattering cross section above, we made use of the non-symmetric scattering law \(S_{n.sym}(\alpha,\beta)\). For systems in thermal equilibrium, there is a relationship between upscatter and downscatter called “detail balance” that is a consequence of microscopic reversibility. It requies that
If we define a function \(S_{sym}(\alpha,\beta)\) to be
then the detail balance requirement states that
which indicates that \(S_{sym}(\alpha,\beta)\) is a symmetric function in \(\beta\). Thus, \(S_{sym}(\alpha,\beta)\) is known as the symmetric scattering law.
Note
The symmetry \(S_{sym}(\alpha,\beta)\) is not present for liquid hydrogen or liquid deuterium.
LEAPR performs all its calculations on \(S_{n.sym}(\alpha,-\beta)\) because its contents have less extreme values than \(S_{sym}(\alpha,\beta)\) or \(S_{n.sym}(\alpha,\beta)\). Typically, however, the scattering law is converted to \(S_{sym}(\alpha,\beta)\) prior to being written to the output tape.