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Scatterings of electrons at quasiparticles or photons are very important formany topics in solid state physics, e.g., spintronics, magnonics or photonics,and therefore a correct numerical treatment of these scatterings is veryimportant. For a quantum-mechanical description of these scatterings Fermi'sgolden rule is used in order to calculate the transition rate from an initialstate to a final state in a first-order time-dependent perturbation theory. Onecan calculate the total transition rate from all initial states to all finalstates with Boltzmann rate equations involving Brillouin zone integrations. Thenumerical treatment of these integrations on a finite grid is often done via areplacement of the Dirac delta distribution by a Gaussian. The Dirac deltadistribution appears in Fermi's golden rule where it describes the energyconservation among the interacting particles. Since the Dirac deltadistribution is a not a function it is not clear from a mathematical point ofview that this procedure is justified. We show with physical and mathematicalarguments that this numerical procedure is in general correct, and we commenton critical points.

Physical and mathematical justification of the numerical Brillouin zone integration of the Boltzmann rate equation by Gaussian smearing