Magnetic ordering of the three-dimensional Wigner crystal

Using Path Integral Monte Carlo, we have calculated exchange frequencies as electrons undergo ring exchanges of 2, 3, and 4 electrons in a "clean" 3D Wigner crystal (bcc lattice) as a function of density. We find pair exchange dominates and estimate the critical temperature for the transition to antiferromagnetic ordering to be roughly 1 × 10^-8 Ry at melting. In contrast to the situation in 2D, the 3D Wigner crystal is different from the solid bcc ³He in that the pair exchange dominates because of the softer interparticle potential. We discuss implications for the magnetic phase diagram of the electron gas.