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It is invariant under the symmetry, ~ri ! ~ri +~a, ~Ra ! ~Ra +~a. However, the energy
can usually be minimized by forming a crystal. At low enough temperature, this will
win out over any possible entropy gain in a competing state, so crystallization will
occur. Why is the crystalline state energetically favorable? This depends on the type
of crystal. Dierent types of crystals minimize dierent terms in the Hamiltonian.
In molecular and ionic crystals, the potential energy is minimized. In a molecular
crystal, such as solid nitrogen, there is a van der Waals attraction between molecules
caused by polarization of one by the other. The van der Waals attraction is balanced
by short-range repulsion, thereby leading to a crystalline ground state. In an ionic
crystal, such as NaCl, the electrostatic energy of the ions is minimized (one must
be careful to take into account charge neutrality, without which the electrostatic