Weak crystallization theory of metallic alloys

Crystallization is one of the most familiar, but hardest to analyze, phase transitions. The principal reason is that crystallization typically occurs via a strongly first-order phase transition, and thus rigorous treatment would require comparing energies of an infinite number of possible crystalline states with the energy of liquid. A great simplification occurs when crystallization transition happens to be weakly first order. In this case, weak crystallization theory, based on unbiased Ginzburg-Landau expansion, can be applied. Even beyond its strict range of validity, it has been a useful qualitative tool for understanding crystallization. In its standard form, however, weak crystallization theory cannot explain the existence of a majority of observed crystalline and quasicrystalline states. Here we extend the weak crystallization theory to the case of metallic alloys. We identify a singular effect of itinerant electrons on the form of weak crystallization free energy. It is geometric in nature, generating strong dependence of free energy on the angles between ordering wave vectors of ionic density. That leads to stabilization of fcc, rhombohedral, and icosahedral quasicrystalline (iQC) phases, which are absent in the generic theory with only local interactions. As an application, we find the condition for stability of iQC that is consistent with the Hume-Rothery rules known empirically for the majority of stable iQC; namely, the length of the primary Bragg-peak wave vector is approximately equal to the diameter of the Fermi sphere.