The Compton scattering is an inelastic gamma scattering on atom with the ejection of an electron. In the standard sub-package two model G4KleinNishinaCompton and G4KleinNishinaModel are available. The first model is the fastest, in the second model atomic shell effects are taken into account.

Cross Section

When simulating the Compton scattering of a photon from an atomic electron, an empirical cross section formula is used, which reproduces the cross section data down to 10 keV:

\[\sigma(Z,E_{\gamma}) = \left [ P_{1}(Z) \ \frac{\log(1+2X)}{X} + \frac{P_{2}(Z)+P_{3}(Z) X + P_{4}(Z) X^{2}}{1+aX+bX^{2}+cX^{3}} \right ] .\]

where

\[\begin{split}Z & = \mbox{atomic number of the medium} \\ E_{\gamma} & = \mbox{energy of the photon} \\ X & = E_{\gamma}/mc^2 \\ m & = \mbox{electron mass} \\ P_{i}(Z) & = Z (d_{i} + e_{i}Z + f_{i}Z^{2}) .\end{split}\]

The values of the parameters can be found within the method which computes the cross section per atom. A fit of the parameters was made to over 511 data points [JHHOverbo80, SI70] chosen from the intervals

\[1 \leq Z \leq 100\]
\[E_{\gamma} \in [10 \mbox{ keV} , 100 \mbox{ GeV}] .\]

The accuracy of the fit was estimated to be

\[\begin{split}\frac{\Delta\sigma}{\sigma} = \left\{ \begin{array}{lrl} \approx 10\% & {\rm for } & E_{\gamma} \simeq 10 \mbox{ keV} -20 \mbox{ keV} \\ \leq 5-6\% & {\rm for } & E_{\gamma} > 20 \mbox{ keV} \end{array} \right.\end{split}\]

To avoid sampling problems in the Compton process the cross section is set to zero at low-energy limit of cross section table, which is 100 eV in majority of EM Physics Lists.

Sampling the Final State

The Klein-Nishina differential cross section per atom is [KN29]:

\[\frac{d\sigma}{d\epsilon} =\pi r_e^2 \ \frac{m_e c^2}{E_0} \ Z \left[\frac{1}{\epsilon}+\epsilon\right] \left[1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2}\right]\]

where

\[\begin{split}r_e & = \mbox{classical electron radius} \\ m_e c^2 & = \mbox{electron mass} \\ E_0 & = \mbox{energy of the incident photon} \\ E_1 & = \mbox{energy of the scattered photon} \\ \epsilon & = E_1/E_0\end{split}\]

Assuming an elastic collision, the scattering angle \(\theta\) is defined by the Compton formula:

\[E_1 = E_0 \ \frac{m_{\rm e}c^2}{ m_{\rm e}c^2 + E_0(1-\cos\theta )} .\]

Sampling the Photon Energy

The value of \(\epsilon\) corresponding to the minimum photon energy (backward scattering) is given by

\[\epsilon_0 = \frac{m_{\rm e}c^2}{m_{\rm e}c^2+2E_0} ,\]

hence \(\epsilon \in [\epsilon_0, 1]\). Using the combined composition and rejection Monte Carlo methods described in [MC70, NHR85, BM60] one may set

\[\Phi(\epsilon) \simeq \left[ \frac{1}{\epsilon}+\epsilon \right] \left[ 1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2} \right] = f(\epsilon) \cdot g(\epsilon) = \left[ \alpha_1 f_1(\epsilon) + \alpha_2 f_2(\epsilon) \right] \cdot g(\epsilon) ,\]

where

\[\begin{split}\begin{array}{lcl} \alpha_1 = \ln (1/\epsilon_0) & ; & f_1(\epsilon) = 1/(\alpha_1\epsilon) \\ \alpha_2 = (1-\epsilon_0^2)/2 & ; & f_2(\epsilon) = \epsilon/\alpha_2 . \end{array}\end{split}\]

\(f_1\) and \(f_2\) are probability density functions defined on the interval \(\lbrack\epsilon_0, 1\rbrack\), and

\[g(\epsilon) = \left[ 1 - \frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right]\]

is the rejection function \(\forall \epsilon \in [\epsilon_0, 1] \Longrightarrow 0 < g(\epsilon) \leq 1\). Given a set of 3 random numbers \(r, r', r''\) uniformly distributed on the interval [0,1], the sampling procedure for \(\epsilon\) is the following:

  1. decide whether to sample from \(f_1(\epsilon)\) or \(f_2(\epsilon)\): if \(r < \alpha_1/(\alpha_1+\alpha_2)\) select \(f_1(\epsilon)\), otherwise select \(f_2(\epsilon)\)

  2. sample \(\epsilon\) from the distributions corresponding to \(f_1\) or \(f_2\):

    • for \(f_1 : \epsilon = \epsilon_0^{r'} \qquad (\equiv \exp(-r' \alpha_1))\)

    • for \(f_2 : \epsilon^2 = \epsilon_0^2 + (1-\epsilon_0^2)r'\)

  3. calculate \(\sin^2\theta = t(2-t)\) where \(t \equiv (1-\cos\theta) = m_e c^2 (1-\epsilon)/(E_0 \epsilon)\)

  4. test the rejection function: if \(g(\epsilon) \geq r''\) accept \(\epsilon\), otherwise go to step 1.

Compute the Final State Kinematics

After the successful sampling of \(\epsilon\), the polar angles of the scattered photon with respect to the direction of the parent photon are generated. The azimuthal angle, \(\phi\), is generated isotropically and \(\theta\) is as defined in the previous section. The momentum vector of the scattered photon, \(\overrightarrow{P_{\gamma1}}\), is then transformed into the World coordinate system. The kinetic energy and momentum of the recoil electron are then

\[\begin{split}T_{el} & = E_0 - E_1 \\ \overrightarrow{P_{el}} & = \overrightarrow{P_{\gamma0}} - \overrightarrow{P_{\gamma1}} .\end{split}\]

Doppler broadening of final electron momentum due to electron motion is implemented only in G4KleinNishinaModel. For that empirical electron density profile function is used.

Atomic shell effects

The differential cross-section described above is valid only for those collisions in which the energy of the recoil electron is large compared to its binding energy (which is ignored). In the alternative model (G4KleinNishinaModel) atomic shell effects are taken into account. For that a sampling of a shell is performed with the weight proportional to number of shell electrons. Electron energy distribution function is approximated via simplified form

\[F(T) = \exp{(-T/E_{b})}/E_{b},\]

where \(E_{b}\) is shell bound energy, \(T\) is the kinetic energy of the electron.

The value \(T\) is sampled and scattering is sampled in the rest frame of the electron according the algorithm described in the previous sub-chapter. After sampling an inverse Lorentz transformation to the laboratory frame is performed. Potential energy \((E_{b} + T)\) is subtracted from the scattered electron kinetic energy. If final electron energy becomes negative then sampling is repeated. Atomic relaxation are sampled if deexcitation module is enabled. Enabling of atomic relaxation for Compton scattering is performed in the same way as for photoelectric effect Relaxation.

Bibliography

BM60

J.C. Butcher and H. Messel. Nucl. Phys., 20(15):, 1960.

JHHOverbo80

J.H. Hubbell, H.A. Gimm and I. Øverbø. Pair, Triplet, and Total Atomic Cross Sections (and Mass Attenuation Coefficients) for 1 MeV-100 GeV Photons in Elements Z=1 to 100. Journal of Physical and Chemical Reference Data, 9:1023–1148, October 1980. doi:10.1063/1.555629.

KN29

O. Klein and Y. Nishina. Z. Physik, 52:853, 1929.

MC70

H. Messel and D. Crawford. Electron-Photon shower distribution. Pergamon Press, 1970.

NHR85

W.R. Nelson, H. Hirayama, and D.W.O. Rogers. EGS4 code system. SLAC, Dec 1985. SLAC-265, UC-32.

SI70

H. Storm and H.I. Israel. Nucl. Data Tables, A7:565, 1970.