  Binary-pairing Monte-Carlo methods are widely used in particle-in-cell codes
to capture effects of small angle Coulomb collisions. These methods preserve
momentum and energy exactly when the simulation particles have equal weights.
However, when the interacting particles are of varying weight, these physical
conservation laws are only preserved on average. Here, we 1) extend these
methods to weighted particles such that the scattering physics is correct on
average, and 2) describe a new method for adjusting the particle velocities
post scatter to restore exact conservation of momentum and energy. The efficacy
of the model is illustrated with various test problems.
