  The electronic structure (electron charges and spins) of a perfect crystal
under external magnetic field is analyzed using the spin-polarized
Thomas-Fermi-Dirac-von Weizsacker model. An extension of the classical
Cauchy-Born rule for crystal lattices is established for the electronic
structure under sharp stability conditions on charge density wave and spin
density wave. A Landau-Lifschitz type micromagnetic energy functional is
derived.
