  We consider a long quantum wire at low electron densities. In this strong
interaction regime a Wigner crystal may form, in which electrons comprise an
antiferromagnetic Heisenberg spin chain. The coupling constant J is
exponentially small, as it originates from tunneling of two neighboring
electrons through the segregating potential barrier. We study this exponential
dependence, properly accounting for the many-body effects and the finite width
of the wire.
