Theoretical properties of cubic crystals at arbitrary pressure—III. Stability

A complete account is presented of the application of the principles of bifurcation analysis for general materials to the particular case of cubic crystals subjected to hydrostatic loading. The treatment of crystal stability is classical in that (i) the loading environment is fully specified, to sufficient order and in both its active and passive modes, and (ii) the potential energy of the system as a whole is examined in all the nearby, possibly inhomogeneous, configurations allowed by the kinematic constraints. Computations are made of the pressures and the bulk and shear moduli of the entire Morse-model family of fcc, bcc, and sc monatornic crystals under pure hydrostatic compression and tension. The stable range of each lattice as well as the potential bifurcations at the range limits are presented and discussed in terms of the role of the particular lattice structure and the effective range of the interatomic potential function (as specified by the parameter log β). The fee lattices are stable in compression and in tension up to an all-round stretch λ = Λ_k, at which point the bulk modulus vanishes; Λ_k, is a monotonically decreasing function of log β. The bcc lattices are stable for λ < Λ_CR, where the bulk modulus or the shear modulus μ vanishes (depending upon the value of log β) at λ = Λ_CR. For very large values of log β a second range of bcc stability is located in a region of hydrostatic expansion. The sc crystals are stable only in a range of hydrostatic tension and only for relatively short-range interatomic interactions (large log β); the present work appears to be the first in which a theoretical range of stability of sc crystals has been revealed. The question of the possibility of assessing lattice stability under load with the aid of higher order moduli at zero load is given consideration quantitatively for the fee lattices and the bcc lattices that are stable at zero load. Finally, the present approach to crystal stability is distinguished from some simplistic notional criteria based upon local convexity of strain energy and, for the Morse-model cubic crystals, quantitative comparisons are made with the present classical treatment of stability in a hydrostatic environment.