  We study scattering of charge and spin excitations in a system of interacting
electrons in one dimension. At low densities electrons form a one-dimensional
Wigner crystal. To first approximation the charge excitations are the phonons
in the Wigner crystal, and the spin excitations are described by the Heisenberg
model with nearest neighbor exchange coupling. This model is integrable and
thus incapable of describing some important phenomena, such as scattering of
excitations off each other and the resulting equilibration of the system. We
obtain the leading corrections to this model, including charge-spin coupling
and the next-nearest neighbor exchange in the spin subsystem. We apply the
results to the problem of equilibration of the one-dimensional Wigner crystal
and find that the leading contribution to the equilibration rate arises from
scattering of spin excitations off each other. We discuss the implications of
our results for the conductance of quantum wires at low electron densities.
