On the diffusion approximation of the Boltzmann equation for slow electrons

In this article we investigate the validity of the lowest-order diffusion or P₁ approximation of the Boltzmann equation. This is done by considering a narrow band of electrons in energy space. The analysis shows that if the spatial derivative can be neglected, i.e., for a distributed source of electrons, the P₁ approximation is valid all the cases considered (for e.g., N₂). When the spatial derivative cannot be neglected, which is the case for a spatially localized source of electrons, then the region of validity of the P₁ approximation is much more restrictive. High-pressure discharges are examples of a distributed source. Electrons generated for swarm experiments are examples of a spatially localized source of electrons.