Lattice Instabilities Along the Transformation from Hexagonal to Cuboidal Structures in Hard- and Soft-Sphere Models
The diffusionless Burgers-Bain phase transition from a hcp arrangement to a cuboidal lattice (fcc and bcc) is analysed in great detail for Lennard-Jones solids. From the lattice vectors of an underlying bi-lattice smoothly connecting these phases, we are able to express the corresponding lattice sums for inverse power potentials in terms of fast converging Bessel function expansions resulting in an efficient evaluation to computer accuracy for cohesive energies. From the kissing hard-sphere limit we derive exact analytical expressions for the lattice parameters varying along the minimum energy path of the phase transition. This simple model suggests that for the Burgers-Bain transformation of a LJ solid requires a minimum of four lattice parameters, $(a,α,β,γ=c/a)$, describing the change in the base lattice lengths $a$ and $c$, the shear force acting on the hexagonal base plane ($α$), the sliding force of the middle layer of the original hexagonal packing arrangement($β$), and the cuboidal transformation ($γ=c/a$). This choice results in a two-step process: hcp$\to$fcc$\to$bcc. However, a further extension of the parameter space including an additional slide parameter for the middle layer, one suddenly observes a distinct symmetry-breaking effect along the hcp$\rightarrow$fcc transition path with a bifurcation point appearing joining the original Burgers with the Bain path of the bcc$\rightarrow$fcc cuboidal transition. Furthermore, for soft LJ potentials the bcc phase appears as a local minimum along the Burgers hcp$\rightarrow$fcc path with two transition states to either the hcp or fcc phase. The underlying topology of the Burgers-Bain phase transition also incorporates the rhombohedral distortion of the bcc phase, which is analyzed in detail.