KAHIBARO
12.5 Positron Physics
Table of Contents
Energy loss
When a positron moves through matter in Geant4 it is treated as a charged particle very similar to the electron, but with opposite charge and an additional annihilation process at the end of its life. In a physics list you do not usually add separate “special” processes for positrons by hand, because the reference physics lists already include the standard positron processes. However, it is useful to understand what is happening, especially when you analyze dose or energy spectra that involve positrons.
From the transport point of view the main continuous processes that control positron energy loss are ionization and multiple scattering. The ionization process models the average energy that the positron deposits as it slows down through many small collisions with atomic electrons. This is often described by a stopping power, which is the mean energy loss per unit path length. In Geant4 this is provided by the electromagnetic model and you usually just select an appropriate standard or low energy electromagnetic physics builder in your physics list. Multiple scattering describes the many small angular deflections of the positron due to Coulomb scattering on nuclei and electrons. It does not change the kinetic energy directly, but it modifies the path that the positron takes through the material.
In addition to ionization, positrons in Geant4 can lose energy through Bremsstrahlung. In Bremsstrahlung, a high energy positron is deflected in the electric field of a nucleus and emits a photon. The probability and spectrum of these photons depend on the material and on the positron energy. In practice, this process becomes important at higher energies. In a typical physics list this is handled by the same Bremsstrahlung process that is used for electrons, but configured for the positron particle type.
The detailed list of processes attached to the positron is managed by the electromagnetic physics constructor that you choose, for example in reference lists like FTFP_BERT or QGSP_BIC. The constructor will attach at least ionization, multiple scattering, and Bremsstrahlung, plus annihilation. You can inspect this in your code by enabling verbose output or by printing the physics table. For a beginner application you normally just trust the reference list, but if you ever create your own physics list you will have to register these processes for the $e^+$ if you want realistic energy loss.
Positrons in Geant4 lose energy mainly through ionization and Bremsstrahlung, with multiple scattering changing their direction. These processes are provided by the electromagnetic physics part of the physics list and must be present for realistic positron transport.
From the analysis point of view, energy loss of positrons will appear as continuous energy deposition along their track. If you record step by step energy deposition with a sensitive detector or SteppingAction, you will see many small $dE$ values as the positron slows down. This is different from the discrete deposition from annihilation photons, which appears at a specific point when the positron finally annihilates. The energy spectrum of deposited energy in a detector that contains positrons is therefore a combination of continuous slowing down contributions and the annihilation related signals that we discuss next.
Positron annihilation
Once a positron has slowed down to very low kinetic energy, it no longer behaves like a continuously losing charged particle. Instead it will eventually annihilate with an electron in the material. In Geant4 this is handled by a dedicated annihilation process attached to the $e^+$ in the electromagnetic physics.
In the simplest and most common case, a free or weakly bound electron and a positron annihilate at rest. The total initial momentum in this case is approximately zero so to conserve momentum the annihilation produces two photons with equal energy that fly in nearly opposite directions. Since the rest mass of the electron and the positron is the same, the available energy is
$$
E_{\text{tot}} = 2 m_e c^2 \approx 2 \times 0.511 \,\text{MeV} = 1.022 \,\text{MeV}.
$$
For annihilation at rest, this is shared equally between two photons of $511 \,\text{keV}$ each. Geant4 generates these photons and then transports them according to the gamma interaction physics that you have selected. This is therefore the origin of the characteristic 511 keV line that appears in many detectors and that is crucial for PET simulations.
For annihilation at rest in Geant4, a positron and an electron produce two photons of 511 keV each, emitted in opposite directions, to conserve energy and momentum.
If the positron still has significant kinetic energy at the moment of annihilation, the situation is slightly different. The center of mass of the electron positron system is then moving, so the annihilation photons are not exactly back to back and do not have exactly equal energies in the laboratory frame. Geant4 includes this effect. For many applications at medical or moderate energies, annihilation at rest dominates, but for higher energy beams or in thin materials in high energy physics simulations, in flight annihilation can become relevant.
From the point of view of building and using physics lists, positron annihilation is automatically included as part of the standard electromagnetic physics constructors. If you create a custom physics list, you must add the annihilation process to the positron, otherwise your positrons will slow down and then simply disappear without producing the 511 keV photons. This would break energy conservation in your simulation and would remove an important observable signal in detectors that are sensitive to annihilation photons.
In analysis, annihilation affects both energy and timing distributions. The annihilation photons may travel some distance before interacting in your detector, depending on the surrounding geometry and materials. In PET, for example, you rely on coincident detection of two annihilation photons. Geant4 models the production and transport of these photons, and the coincidence logic is usually implemented in your user code that processes the hits. If you are looking at depth dose or shielding problems, the annihilation photons can contribute to dose downstream of the positron stopping region in a way that differs from the continuous ionization energy loss.
Finally, annihilation is also important if you are validating your simulation against experimental spectra. The presence, position, and intensity of the 511 keV line is a sensitive check that positron physics in your physics list is enabled and working as expected. If the line is missing or has the wrong shape, it is a sign that you should check your chosen physics list, materials, or detector modeling.
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