A calculation of the elastic constants of yttrium and the rare-earth metals

A modified cellular method developed by Raimes was extended to scandium, yttrium, and the rare earth metals. The assumption that the valence electrons are free and share the same ground state wave functions at zero wave number, was capable of giving fairly good agreement between the calculated and experimental values of the atomic radii, compressibilities, and total energies of the trivalent rare-earth metals as well as for scandium and yttrium. In addition the calculated variation of atomic radius and compressibility of the hexagonal rare-earth metals with atomic number was in qualitative agreement with experiment. Calculations based on the assumption that europium and ytterbium are divalent in the solid state were capable of giving reasonable agreement with the observed atomic radii and compressibilities of these elements. Calculations for cerium did not give satisfactory agreement with the assumption of either a trivalent or quadrivalent atomic core. This failure probably results from the fact that the assumption of equivalent behavior of the valence electrons at zero wave number is quite poor for this element. The compressibilities of promethium and scandium were predicted. A calculation of the elastic shear constants of hexagonal close-packed yttrium at 0°K was made based on the assumption of nearly-free electron behavior for the valence electrons. The method developed by Reitz and Smith for hexagonal close-packed metals was applied. In order to obtain reasonable agreement with the experimental values it was assumed that electron overlap had occurred across the {110,1} and {000, 2} faces of the Brillouin zone and that there are an appreciable nwnber of holes in the zone. The results are in agreement with the measured resistivity of single crystals of yttrium