  The decoherence of a fast quantum particle in a gas is studied by applying
the Kramers-Moyal expansion to the quantum master equation for the reduced
density matrix of the particle. This expansion leads to a general form of the
Caldeira-Leggett master equation accounting for the angular variation of the
differential cross section. The equation describes the decoherence in both the
longitudinal and transverse directions with respect to the particle motion. It
is shown that, when the differential cross section is concentrated in the
forward direction, transverse decoherence dominates. The coherence region off
the diagonal of the density matrix is characterized by coherence lengths, which
can be deduced, for Gaussian states, from the momentum covariance matrix
according to a Heisenberg-type uncertainty relation. Finally, the
longitudinal-to-transverse ratio of the coherence lengths is estimated for an
alpha particle of a few MeVs. This ratio indicates that the coherence region
looks like an ellipsoid elongated in the direction of motion.
