Abstract
We extend the Weak Crystallization theory to the case of metallic alloys. The additional ingredient -- itinerant electrons -- generates nontrivial 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. The condition for stability of iQC that we find, is consistent with the Hume-Rothery rules known empirically for majority of stable iQC; namely, the length of the primary Bragg peak wavevector is approximately equal to the diameter of the Fermi surface.
Highlights
Crystals are best characterized in reciprocal space, where the onset of long-range order is signaled by the appearance of resolution-limited Bragg peaks
Density is highly concentrated near the equilibrium positions of atoms, and the number of relevant Bragg peak harmonics scales in proportion to the ratio of the unit cell size to the atomic size
We find that interionic interactions generated by electrons qualitatively modify the generic weak crystallization theory, stabilizing Face Centered Cubic (FCC), rhombohedral, and, notably, icosahedral quasicrystal states
Summary
Crystals are best characterized in reciprocal space, where the onset of long-range order is signaled by the appearance of resolution-limited Bragg peaks. The optimal e/a ratios have been argued to be associated with a particular geometrical matching condition, when the itinerant (nearlyfree) electron Fermi surface “just crosses” the boundary of the first Brillouin zone[9] Regardless of interpretation, this observation highlights the important role that itinerant electrons play in determining the crystal structure. Despite nominally large conduction electron concentration, their electrical and thermal conductivities are exceptionally low[17,18], consistently with strong scattering around the Fermi surface These observations led to attempts to construct a theory of quasicrystals accounting for the Hume-Rothery rules by perturbatively including electron scattering on quasiperiodic ionic potential[19]. We extend the weak crystallization theory to metallic systems This method is unbiased in the sense that no assumptions regarding the ionic potentials are needed, and the energies of different crystalline and quasicrystalline states can be directly compared. The systematic expansion in terms of electron-ion interactions that we perform here is implicit within these liquid parameters
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