The theory of the Pauling–Simon law on lattice constants of close-packed ordered alloys

In close-packed ordered alloys of composition AB_n the lattice constant a and all interatomic distances are determined to a good approximation by the quantity (R_A + nR_B). This observation, the Pauling-Simon law [Pauling (1957). Acta Cryst. 10, 374-375; Simon (1983). Angew. Chem. 22, 95-113], is analogous to Vegard's law [Pearson (1972). The Chemistry and Physics of Metals and Alloys. New York: Wiley] for random alloys. No exact proof is possible but here a theoretical discussion is given using the spirit of Froyen & Herring's 'proof' of Vegard's law [Froyen & Herring (1981). J. Appl. Phys. 52, 7165-7167]. The effect of a change in radius ratio can be discussed precisely with the virial Σ_i F_iD_i either in terms of the exact crystal structure surrounding each type of atom or by approximating it as a close-packed medium treated as a uniform isotropic continuum. The essential point is that the A and B sites have to be elastically equivalent in the sense that a radius change δR on either site has to produce the same overall volume change. From this it is found that the interatomic force constant is proportional to D^-m, where D is the spacing between atoms and m lies between 3 and 4, more or less consistent with Badger's law [Badger (1934). J. Chem. Phys. 2, 128-131].