Doppler broadening as a lower limit to the angular resolution of next generation Compton telescopes

The angular resolution of a telescope which detects gamma-rays via the Compton effect is fundamentally limited below a few hundred keV by the fact that the target electrons have an indeterminable momentum inside their atom which introduces an uncertainty in the recoil energy of the Compton electron and the scattered photon. This additional component in the energy and momentum equation results in a Doppler broadening of the angular resolution compared to the standard Compton equation for a target at rest. The deterioration in resolution is most pronounced for low photon energy, high scatter angle, and high Z of the scatter material. This physical limit to the angular resolution of a Compton telescope is present even if all other parameters (e.g. energy and position) are measured with high accuracy. For different Compton scatter materials such as silicon, germanium and xenon, which are used in current telescope designs, the best possible angular resolution as a function of photon energy and scatter angle is calculated. Averaged over all scatter angles and energies, the Doppler-limited angular resolution of silicon is a factor of -1.6 better than that of germanium and a factor of -1.9 better than that of xenon. Looking at the Doppler limit of materials from Z=1 to 90 the best angular resolution can be reached for alkaline and alkaline earth metals, the worst for elements with filled p-orbitals (noble gases) and dorbitals (e.g. Pd and An). Of all semiconductors which might be used in a next generation Compton telescope, silicon seems to be the best choice.