Penelope Model

Total cross section

The total cross section of the Compton scattering process is determined from an analytical parameterization. For \(\gamma\) energy \(E\) greater than 5 MeV, the usual Klein-Nishina formula is used for \(\sigma(E)\). For a more accurate parameterization is used, which takes into account atomic binding effects and Doppler broadening [eal96]:

(5)\[\sigma(E) = 2 \pi \int_{-1}^{1} \frac{r_{e}^{2}}{2} \frac{E_{C}^{2}} {E^{2}} \left( \frac{E_{C}}{E} + \frac{E}{E_{C}} - \sin^{2} \theta \right) \times \sum_{shells} f_{i} \Theta(E-U_{i})n_{i}(p_{z}^{max}) \ d(\cos \theta)\]

where:

\[\begin{split}r_e &= \mbox{classical radius of the electron;}\\ m_e &= \mbox{mass of the electron;}\\ \theta &= \mbox{scattering angle;} \\ E_C &= \mbox{Compton energy}\\ &= \frac{E}{1+\frac{E}{m_{e}c^{2}}(1-\cos\theta)} \\ f_i &= \mbox{number of electrons in the *i*-th atomic shell;}\\ U_i &= \mbox{ionisation energy of the *i*-th atomic shell;}\\ \Theta &= \mbox{Heaviside step function;} \\ p_z &= \mbox{projection of the initial momentum of the electron in the direction of the scattering angle} \\ p_{z}^{max} &= \mbox{highest possible value of } p_z \\ &= \frac{E(E-U_{i})(1-\cos\theta)-m_{e}c^{2}U_{i}}{c \sqrt{2E(E-U_{i})(1- \cos\theta)+U_{i}^{2}}}.\end{split}\]

Finally,

\[\begin{split}\begin{array}{rlll} n_{i}(x) = & & & \\ & \frac{1}{2} e^{[ \frac{1}{2}-( \frac{1}{2} - \sqrt{2} J_{i0}x )^{2}]} & \mbox{if} & x < 0 \\ & 1-\frac{1}{2} e^{[\frac{1}{2}-(\frac{1}{2}+\sqrt{2}J_{i0}x)^{2}]} & \mbox{if} & x > 0 \\ % \begin{cases} % \frac{1}{2} e^{[ \frac{1}{2}-( \frac{1}{2} - \sqrt{2} J_{i0}x )^{2}]} & % \textrm{if} \quad x<0\\ % 1-\frac{1}{2} e^{[\frac{1}{2}-(\frac{1}{2}+\sqrt{2}J_{i0}x)^{2}]} & % \textrm{if} \quad x>0\\ % \end{cases} \end{array}\end{split}\]

where \(J_{i0}\) is the value of the \(p_{z}\)-distribution profile \(J_{i}(p_{z})\) for the i-th atomic shell calculated in \(p_{z}=0\). The values of \(J_{i0}\) for the different shells of the different elements are tabulated from the Hartree-Fock atomic orbitals of Ref. [eal75].

The integration of Eq.(5) is performed numerically using the 20-point Gaussian method. For this reason, the initialization of the Penelope Compton process is somewhat slower than the Low Energy Livermore process.

Sampling of the final state

The polar deflection \(\cos\theta\) is sampled from the probability density function

\[P(\cos\theta) = \frac{r_{e}^{2}}{2} \frac{E_{C}^{2}} {E^{2}} \Big( \frac{E_{C}}{E} + \frac{E}{E_{C}} - \sin^{2} \theta \Big) \sum_{shells} f_{i} \Theta(E-U_{i})n_{i}(p_{z}^{max})\]

(see Ref. [eal01] for details on the sampling algorithm). Once the direction of the emerging photon has been set, the active electron shell \(i\) is selected with relative probability equal to \(Z_{i} \Theta (E-U_{i}) n_{i} [p_{z}^{max} (E,\theta)]\).

A random value of \(p_{z}\) is generated from the analytical Compton profile [eal75]. The energy of the emerging photon is

\[E' \ = \ \frac{E \tau}{1-\tau t} \ \Big[ (1-\tau t \cos\theta) + \frac{p_{z}}{|p_{z}|} \sqrt{(1-\tau t \cos\theta)^{2}-(1-t \tau^{2})(1-t)} \Big],\]

where

\[t = \Big( \frac{p_{z}}{m_{e}c} \Big)^{2} \quad \textrm{and} \quad \tau = \frac{E_{C}}{E}.\]

The azimuthal scattering angle \(\phi\) of the photon is sampled uniformly in the interval \((0, 2\pi)\). It is assumed that the Compton electron is emitted with energy \(E_{e} = E-E'-U_{i}\), with polar angle \(\theta_{e}\) and azimuthal angle \(\phi_{e} = \phi + \pi\), relative to the direction of the incident photon. In this case \(\cos\theta_{e}\) is given by

\[\cos\theta_{e} = \frac{E-E'\cos\theta}{\sqrt{E^{2}+E^{'2}- 2EE' \cos\theta}}.\]

Since the active electron shell is known, characteristic x-rays and electrons emitted in the de-excitation of the ionized atom can also be followed. The de-excitation is simulated as described in Atomic relaxation. For further details see [eal01].

Bibliography

eal96

D. Brusa et al. Fast sampling algorithm for the simulation of photon compton scattering. NIM A, 379():167, 1996.

eal01(1,2)

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.

eal75(1,2)

F.Biggs et al. Hartree-fock compton profiles for the elements. At. Data Nucl. Data Tables, 16():201, 1975.