Lattice Instabilities Along the Transformation from Hexagonal to Cuboidal Structures in Hard- and Soft-Sphere Models

TL;DR

The study addresses the microscopic mechanism of diffusionless hcp↔fcc/bcc transitions (the Burgers-Bain pathway) in hard- and soft-sphere models by constructing a bi-lattice with a monoclinic unit cell and a seven-parameter Burgers-Bain path, parameterized by $α∈[0,1]$. It treats cohesive energies via exact lattice sums recast as fast-converging Bessel expansions, derives analytic hard-sphere limits, and uncovers symmetry breaking that yields a bifurcation point $α_c$ where Burgers and Bain paths merge and potentially reveal a metastable bcc minimum for soft LJ potentials. The work extends to soft potentials and includes DFT-VC-NEB validation for solid Argon, showing dispersion forces strongly influence barrier heights and that the Burgers path provides an upper bound on activation energy for solid-state martensitic-like transitions. Overall, the framework clarifies rhombohedral distortions of the bcc phase and offers a computationally efficient route to estimate transition barriers and study parameter dependencies relevant to real materials.

Abstract

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.

Figures (20)

• Figure 1: The (distorted) cuboidal (blue lines shown on the left) and the hcp structure both with a ABABAB... sequence shown in a hexagonal unit cell with corresponding basis vectors with cell parameters $|\vec{b}_1|=a$, $|\vec{b}_2|=b$, and $|\vec{b}_3|=c$. The ratio $\gamma_{2}=c/a=\sqrt{8/3}$ together with $\angle(\vec{b}_1,\vec{b}_2)=60^\circ$ and $|\vec{b}_1|=|\vec{b}_2|=a$ leads to the optimal hcp lattice with 12 kissing spheres around a central atom. For $c=a$ and $\angle(\vec{b}_1,\vec{b}_2)=90^\circ$ we obtain the bcc lattice with 8 kissing spheres. Alternatively, we have $|\vec{b}_1|=|\vec{b}_2|=a$, $c/a=\sqrt{2}$ and $\angle(\vec{b}_1,\vec{b}_2)=90^\circ$ for the fcc lattice with 12 kissing spheres.
• Figure 2: Relationship between the parameter $\alpha$ and the angle between the two base vectors $\vec{b}_1$ and $\vec{b}_2$.
• Figure 3: The shearing in the A-layer (gray spheres, change of angle between $\vec{b}_1$ and $\vec{b}_2$) of the base hexagonal plane and sliding of the B-layer (red spheres, sitting at $c/2$ above the A-layer) in the Burgers transformation. Left: bcc lattice with $\angle(\vec{b}_1,\vec{b}_2)=90^\circ$, $\gamma_{2}=1$ and fractional coordinates $\vec{u}^\top=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, or fcc lattice with $\gamma_{2}=\sqrt{2}$ and $\vec{u}^\top=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$. Right: hcp structure with $\angle(\vec{b}_1,\vec{b}_2)=60^\circ$, $\gamma_{2}=\sqrt{\frac{8}{3}}$ and $\vec{u}^\top=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$.
• Figure 4: bcc phase shown in a $(3\times 2\times 3)$ super-cell. Left: the conventional bcc bi-lattice shown in red color (left) with the body-centered atom in grey; middle: the bi-lattice shown in blue with unit cell identical to the one defined in Fig.\ref{['arxiv_manuscript:fig:hcp']} with $(\angle(\vec{b}_{1},\vec{b}_{2})=70.5287794^\text{o}=180^\text{o}-\theta_T)$, where $\theta_T$ is the tetrahedral angle; right: the rhombohedral primitive bcc cell of equal lengths shown in green (containing no central atom), with a ratio for the length of the lattice vectors compared to the conventional unit cell of $\frac{\sqrt{3}}{2}$. The angles between the lattice vectors are $180^\text{o}-\theta_T$. Note that the primitive bcc cell has half the volume of the conventional one.
• Figure 5: Burgers transformation using a kissing hard-sphere model potential along different values of $\alpha$ from $-0.2\rightarrow0.0\rightarrow0.5\rightarrow1.0\rightarrow1.2$. The continuous black lines show the bonds from the atom at the origin to its nearest neighbors (kissing spheres). The red atoms lie in the Wyckoff positions of the unit cell and form layers above and below (B layer) the layer of gray atoms (A layer). Including the bonds for the lower B layer we count the number of solid lines and get the kissing numbers as shown in \ref{['arxiv_manuscript:eq:KHSE']}.