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

Cauchy-Born rule and spin density wave for the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model