Positron - Electron Annihilation

Introduction

The process G4eplusAnnihilation simulates the in-flight annihilation of a positron with an atomic electron. As is usually done in shower programs [NHR85], it is assumed here that the atomic electron is initially free and at rest. Also, annihilation processes producing one, or three or more, photons are ignored because these processes are negligible compared to the annihilation into two photons [NHR85][MC70].

Cross Section

The annihilation in flight of a positron and electron is described by the cross section formula of Heitler [Hei54][NHR85]:

\[\sigma(Z,E) = \frac{Z \pi r_e^2}{\gamma +1} \left[ \frac{\gamma^2 + 4 \gamma +1}{\gamma^2 -1} \ln \left( \gamma +\sqrt{ \gamma^2 -1} \right) - \frac{\gamma +3}{\sqrt{\gamma^2 -1}} \right]\]

where

\[\begin{split}E & = \mbox{total energy of the incident positron} \\ \gamma & = E/m c^2 \\ r_e & = \mbox{classical electron radius}\end{split}\]

Sampling the final state

The final state of the \(e+e-\) annihilation process

\[e^+ \; e^- \to \gamma_a \; \gamma_b\]

is simulated by first determining the kinematic limits of the photon energy and then sampling the photon energy within those limits using the differential cross section. Conservation of energy-momentum is then used to determine the directions of the final state photons.

If the incident \(e^+\) has a kinetic energy \(T\), then the total energy is \(E_e = T + mc^2\) and the momentum is \(Pc = \sqrt{T(T+2mc^2)}\). The total available energy is \(E_{tot} = E_e + mc^2 = E_a + E_b\) and momentum conservation requires \(\vec{P} = \vec{P}_{\gamma_a} + \vec{P}_{\gamma_b}\). The fraction of the total energy transferred to one photon (say \(\gamma_a\)) is

\[\epsilon = \frac{E_a}{E_{tot}} \equiv \frac{E_a}{T+2mc^2} .\]

The energy transferred to \(\gamma_a\) is largest when \(\gamma_a\) is emitted in the direction of the incident \(e^+\). In that case \(E_{a, max} = (E_{tot}+Pc)/2\) . The energy transferred to \(\gamma_a\) is smallest when \(\gamma_a\) is emitted in the opposite direction of the incident \(e^+\). Then \(E_{a, min} = (E_{tot}-Pc)/2\) . Hence,

\[\begin{split}\epsilon_{max} &= \frac{E_{a, max}}{E_{tot}} = \frac{1}{2} \left\lbrack 1+ \sqrt{\frac{\gamma - 1}{\gamma+1}} \right\rbrack \\ \epsilon_{min} &= \frac{E_{a, min}}{E_{tot}} = \frac{1}{2} \left\lbrack 1- \sqrt{\frac{\gamma - 1}{\gamma+1}} \right\rbrack\end{split}\]

where \( \gamma = (T + mc^2)/mc^2\) . Therefore the range of \(\epsilon\) is \( \lbrack \epsilon_{min} ; \epsilon_{max} \rbrack \) \( ( \equiv \lbrack \epsilon_{min} \; ; \; 1-\epsilon_{min} \rbrack) \).

Sampling the Gamma Energy

A short overview of the sampling method is given in Monte Carlo Methods. The differential cross section of the two-photon positron-electron annihilation can be written as [Hei54][NHR85]:

\[\frac{d \sigma (Z, \epsilon)} {d \epsilon} = \frac{Z \pi r_e^2}{\gamma - 1} \ \frac{1}{\epsilon} \ \left[ 1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon} \right]\]

where \(Z\) is the atomic number of the material, \(r_e\) the classical electron radius, and \(\epsilon \in [ \epsilon_{min} \; ; \; \epsilon_{max} ]\) . The differential cross section can be decomposed as

\[\frac{d \sigma (Z, \epsilon)} {d \epsilon} = \frac{Z \pi r_e^2}{\gamma - 1} \alpha f(\epsilon) g(\epsilon)\]

where

\[\begin{split}\alpha &= \ln (\epsilon_{max}/\epsilon_{min}) \\ f(\epsilon) &= \frac{1}{\alpha \epsilon} \\ g(\epsilon) &= \left[ 1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon} \right] \equiv 1-\epsilon+\frac{2 \gamma \epsilon -1}{\epsilon (\gamma +1)^2}\end{split}\]

Given two random numbers \(r, r' \in [0,1]\), the photon energies are chosen as follows:

  1. sample \(\epsilon\) from \(f(\epsilon)\): \(\epsilon =\epsilon_{min} \left( \frac{\epsilon_{max}}{\epsilon_{min}} \right)^r\)

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

Then the photon energies are \(E_a = \epsilon E_{tot} \qquad E_b = (1-\epsilon) E_{tot}\).

Computing the Final State Kinematics

If \(\theta\) is the angle between the incident \(e^+\) and \(\gamma_a\), then from energy-momentum conservation,

\[\cos \theta = \frac{1}{Pc} \left[ T+mc^2 \frac{2\epsilon -1}{\epsilon} \right] = \frac{\epsilon(\gamma +1) - 1}{\epsilon \sqrt{\gamma^2 -1}} .\]

The azimuthal angle, \(\phi\), is generated isotropically and the photon momentum vectors, \(\vec{P_{\gamma_a}}\) and \(\vec{P_{\gamma_b}}\), are computed from energy-momentum conservation and transformed into the lab coordinate system.

Annihilation at Rest

The method AtRestDoIt treats the special case when a positron comes to rest before annihilating. It generates two photons, each with energy \(k=mc^2\) and an isotropic angular distribution.

Penelope Model for positron-electron annihilation

Total Cross Section

The total cross section (per target electron) for the annihilation of a positron of energy \(E\) into two photons is evaluated from the analytical formula [Hei54][NHR85]

\[\sigma(E) = \frac{\pi r_{e}^{2}}{(\gamma+1)(\gamma^{2}-1)} \times \Big{\{} (\gamma^{2}+4\gamma+1) \ln \Big[ \gamma + \sqrt{\gamma^{2}-1} \Big] -(3+\gamma)\sqrt{\gamma^{2}-1} \Big{\}}.\]

where \(r_{e}\) = classical radius of the electron, and \(\gamma\) = Lorentz factor of the positron.

Sampling of the Final State

The target electrons are assumed to be free and at rest: binding effects, that enable one-photon annihilation [Hei54], are neglected. When the annihilation occurs in flight, the two photons may have different energies, say \(E_{-}\) and \(E_{+}\) (the photon with lower energy is denoted by the superscript “\(-\)”), whose sum is \(E+2m_{e}c^{2}\). Each annihilation event is completely characterized by the quantity

\[\zeta = \frac{E_{-}}{E+2m_{e}c^{2}},\]

which is in the interval \(\zeta_{min} \le \zeta \le \frac{1}{2}\), with

\[\zeta_{min} = \frac{1}{\gamma + 1 + \sqrt{\gamma^{2}-1}}.\]

The parameter \(\zeta\) is sampled from the differential distribution

\[P(\zeta) = \frac{\pi r_{e}^{2}}{(\gamma+1)(\gamma^{2}-1)} [S(\zeta)+S(1-\zeta)],\]

where \(\gamma\) is the Lorentz factor and

\[S(\zeta) = -(\gamma+1)^{2}+(\gamma^{2}+4\gamma+1) \frac{1}{\zeta}-\frac{1}{\zeta^{2}}.\]

From conservation of energy and momentum, it follows that the two photons are emitted in directions with polar angles

\[\cos \theta_{-} = \frac{1}{\sqrt{\gamma^{2}-1}} \Big( \gamma+1-\frac{1}{\zeta} \Big)\]

and

\[\cos \theta_{+} = \frac{1}{\sqrt{\gamma^{2}-1}} \Big( \gamma+1-\frac{1}{1-\zeta} \Big)\]

that are completely determined by \(\zeta\); in particular, when \(\zeta=\zeta_{min}\), \(\cos\theta_{-}=-1\). The azimuthal angles are \(\phi_{-}\) and \(\phi_{+} = \phi_{-} + \pi\); owing to the axial symmetry of the process, the angle \(\phi_{-}\) is uniformly distributed in \((0,2\pi)\).

Bibliography

Hei54(1,2,3,4)

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

MC70

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

NHR85(1,2,3,4,5)

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