The ground state energy and the lowest excitations of a two dimensional Wigner crystal in a perpendicular magnetic field with one and two electrons per cell is investigated. In case of two electrons per lattice site, the interaction of the electrons {em within} each cell is taken into account exactly (including exchange and correlation effects), and the interaction {em between} the cells is in second order (dipole) van der Waals approximation. No further approximations are made, in particular Landau level mixing and {em in}complete spin polarization are accounted for. Therefore, our calculation comprises a, roughly speaking, complementary description of the bubble phase (in the special case of one and two electrons per bubble), which was proposed by Koulakov, Fogler and Shklovskii on the basis of a Hartree Fock calculation. The phase diagram shows that in GaAs the paired phase is energetically more favorable than the single electron phase for, roughly speaking, filling factor $f$ larger than 0.3 and density parameter $r_s$ smaller than 19 effective Bohr radii (for a more precise statement see Fig.s 4 and 5). If we start within the paired phase and increase magnetic field or decrease density, the pairs first undergo some singlet- triplet transitions before they break.
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