Applicability of exponentially attractive and repulsive interactomic potential functions in the description of cubic crystals

Abstract

The well‐known Morse function φ(r) = D {exp [−2α(r−r₀)] −2 exp [−α(r−r₀)]} can be considered to be a particular case of the general family of potential functions with exponential attraction and exponential repulsion, viz., φ_m(r) = [D/(m−1)] {exp[−mα(r−r₀)] −m × exp[−α(r−r₀)]}. This study examines the suitability of applying the functions φ_m (r) to the description of mechanical properties of cubic crystals. For bcc lattices, the ratio C₁₁/C₁₂ for the theoretical model of the crystal made up of atoms with φ_m (r) interatomic interactions is shown to be too small to provide a realistic description of bcc metals; also, the bcc crystals are found to be inherently unstable for larger values of αa₀, where a₀ is the lattice parameter of the crystal. For fcc lattices, however, the theoretical model of the crystal is found to be mechanically stable for arbitrary (i.e., m>1) exponentially attractive and repulsive interactions and the elastic moduli C₁₁ and C₁₂ of the theoretical fcc crystals are found to be capable of conforming to the respective experimental values of fcc metals. The results are in agreement with Born's analyses of lattice stability. Numerical values of D, α, and r₀ are calculated for several different values of the parameter m for fcc Ni using experimental values of C₁₁, C₁₂, and lattice parameter. The element Ni is selected because, among the fcc metals, Ni most closely obeys the Cauchy condition C₁₂ = C₄₄. The functions φ_m are then used to calculate theoretical pressure versus volume (P‐V) behavior (at pressures large enough to produce considerable anharmonicity) and the results are compared with experiment; good agreement is found between the calculated and experimental P‐V relations.