2.1.1 Rayleigh scattering

Rayleigh scattering, also often called coherent scattering, consists of the elastic change of direction of an incoming photon by atomic electrons. The Rayleigh scattering model from PENELOPE [12], which is based on atomic form factors [13] and anomalous scattering factors [14], is used.

The Rayleigh cross-sections, which described the elastic scattering of photons ($p=p'=\gamma$) are given by [13–15]

\[\sigma_{s}^{i}(E,\mu) = \pi r_{e}^2 \left(1+\mu^2\right) \left[\left(F_{i}(E,\mu)+f_{i}'(E)\vphantom{\frac{1}{2}}\right)^{2} + \left(f_{i}''(E)\vphantom{\frac{1}{2}}\right)^{2}\right] \,,\]

where $F_{i}(E,\mu)$ is the atomic form factor for the $i^{\text{th}}$-element, where the factors $f_{i}'(E)$ and $f_{i}''(E)$ respectively are the real and imaginary parts of the anomalous scattering factors for the $i^{\text{th}}$-element, which are all tabulated by the Japanese evaluated nuclear data library (JENDL-5) [16], which are based on the EPDL library [17]. The double differential cross-sections are given by

\[\sigma_{s}^{i}(E\rightarrow E',\mu) = \sigma_{s}^{i}(E,\mu) \delta(E'-E) \,.\]

2.1.1.1 Scattering cross-sections of the incoming photon

The Rayleigh Legendre moments of the differential scattering cross-sections of the incoming photon are simply given by

\[\Sigma_{s,\ell}^{\gamma\rightarrow\gamma}(E) = \sum_{i=1}^{N_{e}} \mathcal{N}_{n,i} f_{i} \int_{-1}^{1} P_{\ell}(\mu)\sigma_{s}^{i}(E,\mu) \,,\]

which are integrated using numerical quadrature.

2.1.1.2 Total cross-sections

The Rayleigh total cross-sections are given by

\[\Sigma_{s,\ell}^{\gamma}(E) = \Sigma_{s,0}^{\gamma\rightarrow\gamma}(E) \,.\]