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

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 lattice vectors of a cuboidal unit cell. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy where the interaction is described by a Lennard-Jones (LJ) interaction potential or a sticky hard-sphere (SHS) model with a $r^{-n}$ long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging Bessel function series. Applying the whole range of lattice parameters for the SHS and LJ potentials we prove that the bcc phase is unstable (or, at best, metastable) toward distortion into the fcc phase in the low temperature and pressure limit. 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.

Figure 1

The four lattices acc, bcc, mcc, and fcc along the cuboidal transition path. The corresponding primitive cell basis vectors according to Eq. (1) are shown for the bcc lattice. For the fcc lattice the lighter colored atoms moving towards the central atom (0,0,0) become nearest neighbors with the overall cuboidal fcc structure displayed.

Figure 2

Difference in cohesive energies $\mathrm{\Delta }{E}^{*}(b,A)=\frac{1}{2}[L(b,A=1)-L(b,A)]$ between the cuboidal lattices and fcc for various exponents $b$ and lattice parameter $A$ of the SHS model. The contour interval chosen is 0.1. The vertical black line at $b=5.493634$ shows the point of least instability for the bcc lattice.

Figure 3

Minimum distance $R_{\min }^{*}(a,b,A)$ for various $(a,b)$ LJ potentials and for the ELJ potential for argon (taken from Ref. [9]) dependent on the lattice parameter $A$.

Figure 4

Cohesive energy differences $\mathrm{\Delta }{E}^{*}(R_{\min }^{*},a,b,A)=E^{*}(R_{\min }^{*},a,b,A)-E^{*}(R_{\min }^{*},a,b;A=1)$ for the $(a,b)$ LJ potential dependent on the lattice parameter $A$ and for the two ELJ potentials of argon and chromium (see Appendix pp4).

Figure 5

Energy difference $\mathrm{\Delta }{E}_{\mathrm{bcc},\mathrm{fcc}}^{*}(a,b)=E^{*}(R_{\min ,\mathrm{bcc}}^{*},a,b,\frac{1}{2})-E^{*}(R_{\min ,\mathrm{fcc}}^{*},a,b,1)$ between the bcc and the fcc lattice for the $(a,b)$ LJ potential. The contour interval chosen is 0.0625. The almost linear (yellow) curve in the lower left corner of the plot describes the phase transition line to a metastable bcc state.

Figure 6

Potential energy curves $V^{*}\left(r^{*}\right)$ (in dimensionless units) for a (12-6) LJ potential, and for ${\mathrm{Ar}}_2,{\mathrm{Li}}_2$, and ${\mathrm{Cr}}_2$ (see Appendix pp3).

Figure 7

Lattice sums $L(a,\frac{1}{2})$ and ${\partial }_A^2{L(a,A)|}_{A=1/2}$ (bcc lattice) as a function of the exponent $a$. Note that ${\partial }_A{L(a,A)|}_{A=1/2}=0$ for all $a$ values.

Figure 8

Lattice sum ratio $\frac{{\partial }_A^2{L(a,A)|}_{A=1/2}}{aL(a,\frac{1}{2})}$ against the exponent $a$ at lattice parameter $A=\frac{1}{2}$.