Instability of the body-centered cubic lattice within the sticky hard sphere and Lennard-Jones model obtained from exact lattice summations

Abstract

A smooth path of rearrangement from the body-centered cubic (bcc) to the face-centered cubic (fcc) lattice is obtained by introducing a single parameter to cuboidal lattice vectors. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy. This is described by a Lennard-Jones (LJ) interaction potential and the sticky hard sphere (SHS) model with an r −n long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging series of Bessel functions. Applying the whole range of lattice parameters for the SHS and LJ potentials demonstrates that the bcc phase is unstable (or at best metastable) toward distortion into the fcc phase. Even if more accurate potentials are used, such as the extended LJ potential for argon or chromium, the bcc phase remains unstable. This strongly indicates that the appearance of a low temperature bcc phase for several elements in the periodic table is due to higher than two-body forces in atomic interactions.

Cuboidal bcc to fcc Transformation of Lennard-Jones Phases under High Pressure Derived from Exact Lattice Summations

“…For the exact evaluation of the pressure dependent free energy and transformation path for the bcc ↔fcc transition, we introduce the following lattice vectors describing the primitive cell of a body-centered tetragonal (bct) lattice 35 i k j j j j y…”

“…lattice as an average between the bcc (A = 1/2) and the fcc lattice (A = 1), 35,42 or directly from the conventional bct cell (space group I4/mmm) with lattice constants a 1 = a 2 and a 3 (…”

“…The cuboidal lattice is shown in Figure 2 for the special case of a bcc cell. The lattice vectors defined in eq 2 lead to the following Gram matrix G(A) 35,42 i k j j j j j j j j j y…”

“…, ∈ + are real numbers, and we require a > b > 3 to avoid divergencies for the lattice summations. 35 The analytical expression for the cohesive energy in terms of lattice sums L(a, A) and nearest neighbor distance R* = R/r e for a lattice defined by the lattice vectors in eq 2 then becomes 34…”

“…The lattice sums L(a, A) introduced by Lennard-Jones and co-workers can be evaluated to computer precision by various expansion techniques leading to fast converging series. 34,35 The volume can be defined through the Gram matrix (eq 4) 35 V…”