Compton Scattering by Linearly Polarized Gamma Rays - Livermore Model

The Cross Section

The quantum mechanical Klein-Nishina differential cross section for polarized photons is [Hei54]:

\[\frac{d\sigma}{d\Omega} = \frac{1}{2}r_0^2 \frac{h\nu^2}{h\nu_o^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}- \sin^2\Theta \right]\]

where \(\Theta\) is the angle between the two polarization vectors. In terms of the polar and azimuthal angles \((\theta, \phi)\) this cross section can be written as

\[\frac{d\sigma}{d\Omega} = \frac{1}{2}r_0^2 \frac{h\nu^2}{h\nu_o^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}-2 \cos^2\phi \sin^2\theta \right] .\]

Angular Distribution

The integration of this cross section over the azimuthal angle produces the standard cross section. The angular and energy distribution are then obtained in the same way as for the standard process. Using these values for the polar angle and the energy, the azimuthal angle is sampled from the following distribution [Dep03]:

\[P(\phi)= 1 - 2 \frac{a}{b} \cos^2\phi\]

where \(a = \sin^2\theta\) and \(b = \epsilon + 1/\epsilon\). \(\epsilon\) is the ratio between the scattered photon energy and the incident photon energy.

Polarization Vector

The components of the vector polarization of the scattered photon are calculated from [Dep03]:

\[\begin{split}\vec{\epsilon'_\bot} &= \frac{1}{N} \left( \hat{j} \cos\theta - \hat{k} \sin\theta \sin\phi \right) \sin\beta \\ \vec{\epsilon'_\|} &= \left[ N \hat{i}- \frac{1}{N} \hat{j} \sin^2\theta \sin\phi \cos\phi - \frac{1}{N} \hat{k} \sin\theta \cos\theta \cos\phi \right] \cos\beta\end{split}\]

where

\[N = \sqrt{1-\sin^2\theta \cos^2\phi} .\]

\(\cos\beta\) is calculated from \(\cos\theta = N \cos\beta\), while \(\cos\theta\) is sampled from the Klein-Nishina distribution.

The binding effects and the Compton profile are neglected. The kinetic energy and momentum of the recoil electron are then

\[\begin{split}T_{el} &= E - E' \\ \vec{P_{el}} &= \vec{P_\gamma} - \vec{P_\gamma '} .\end{split}\]

The momentum vector of the scattered photon \(\vec{P_\gamma}\) and its polarization vector are transformed into the World coordinate system. The polarization and the direction of the scattered gamma in the final state are calculated in the reference frame in which the incoming photon is along the \(z\)-axis and has its polarization vector along the \(x\)-axis. The transformation to the World coordinate system performs a linear combination of the initial direction, the initial polarization and the cross product between them, using the projections of the calculated quantities along these axes.

Unpolarized Photons

A special treatment is devoted to unpolarized photons. In this case a random polarization in the plane perpendicular to the incident photon is selected.

Bibliography

Dep03(1,2)

G.O. Depaola. New monte carlo method for compton and rayleigh scattering by polarized gamma rays. Nucl. Instrum. and Methods A, 512():619, 2003.

Hei54

W. Heitler. The Quantum Theory of Radiation. Oxford Clarendon Press, edition, 1954.