Penelope Model

Total cross section

The total cross section of the Rayleigh scattering process is determined from an analytical parameterization. The atomic cross section for coherent scattering is given approximately by [Bor69]

(2)\[\sigma(E) = \pi r_{e}^{2} \int_{-1}^{1} \frac{1+\cos^{2}\theta}{2} [F(q,Z)]^{2} \ d \cos\theta,\]

where \(F(q,Z)\) is the atomic form factor, \(Z\) is the atomic number and \(q\) is the magnitude of the momentum transfer, i.e.

\[q \ = \ 2 \ \frac{E}{c} \ \sin \Big( \frac{\theta}{2} \Big).\]

In the numerical calculation the following analytical approximations are used for the form factor:

\[\begin{split}\begin{array}{rlll} F(q,Z) = f(x,Z) = & & & \\ & Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5}x^{4})^{2}} & \mbox{or} & \\ & \max[f(x,Z),F_{K}(x,Z)] & \mbox{if} \ Z>10 \ \mbox{and} \ f(x,Z) < 2 & \\ % \begin{cases} % f(x,Z) = Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5} % x^{4})^{2}} & \\ % \max[f(x,Z),F_{K}(x,Z)] & \textrm{if} \ Z>10 \ \textrm{and} \ % f(x,Z)<2\\ % \end{cases} \end{array}\end{split}\]

where

\[F_{K}(x,Z) = \frac{\sin(2b \arctan Q)}{bQ(1+Q^{2})^{b}},\]

with

\[x = 20.6074 \frac{q}{m_{e}c}, \quad Q = \frac{q}{2m_{e}ca}, \quad b = \sqrt{1-a^{2}}, \quad a = \alpha \Big( Z-\frac{5}{16} \Big ),\]

where \(\alpha\) is the fine-structure constant. The function \(F_{K}(x,Z)\) is the contribution to the atomic form factor due to the two K-shell electrons (see [eal94]). The parameters of expression \(f(x,Z)\) have been determined in Ref. [eal94] for \(Z=1\) to 92 by numerically fitting the atomic form factors tabulated in Ref. [eal75]. The integration of Eq.(2) is performed numerically using the 20-point Gaussian method. For this reason the initialization of the Penelope Rayleigh process is somewhat slower than the Low Energy Livermore process.

Sampling of the final state

The angular deflection \(\cos\theta\) of the scattered photon is sampled from the probability distribution function

\[P(\cos\theta) = \frac{1+\cos^{2}\theta}{2} [F(q,Z)]^{2}.\]

For details on the sampling algorithm (which is quite heavy from the computational point of view) see Ref. [eal01]. The azimuthal scattering angle \(\phi\) of the photon is sampled uniformly in the interval \((0, 2\pi)\).

Bibliography

Bor69

M. Born. Atomic physics. Ed. Blackie and Sons, edition, 1969.

eal01

F. Salvat et al. Penelope - a code system for monte carlo simulation of electron and photon transport. Technical Report, Workshop Proceedings Issy-les-Moulineaux, France; AEN-NEA, 5-7 November 2001.

eal94(1,2)

J.Baró et al. Analytical cross sections for monte carlo simulation of photon transport. Radiat. Phys. Chem., 44():531, 1994.

eal75

J.H. Hubbel et al. Atomic form factors, incoherent scattering functions and photon scattering cross sections. Phys. Chem. Ref. Data, 4():471, 1975. Erratum: ibid. 6,615 (1977).