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Cubic Elastic Constants

Computes \(C_{11}\), \(C_{12}\), and \(C_{44}\) for a cubic (or orthogonal) cell from three targeted energy-volume/energy-strain deformation modes, rather than a general stress-strain fit.

Overview

CubicElasticConstantsAnalyzer is a lighter-weight, cubic-only alternative to the general ElasticConstantsAnalyzer. Instead of fitting a full symmetry-dependent stress-strain system, it applies three specific distortion families to the input cell: uniform (volumetric), orthorhombic (tetragonal shear), and monoclinic (pure shear), and extracts the bulk modulus and the two independent shear-related moduli directly from the curvature of energy vs. distortion. This only requires the energy property (not stress), at the cost of only being valid for cubic or orthogonal cells.

Caution

This analyzer only works with cubic or orthogonal cells.

Theory

Bulk modulus from uniform compression/expansion

The uniform distortion scales the cell isotropically by \((1+\delta)\) along all three axes for a range of \(\delta\) (-delta_max to delta_max, step step_size). The energy-volume curve is fit to an equation of state (eos_name, default Birch-Murnaghan; see Equation of State for the functional form) to obtain the bulk modulus \(B\) directly in GPa.

Shear moduli from volume-conserving distortions

The orthorhombic distortion (diag\((1+\delta, 1-\delta, 1/(1-\delta^2))\)) and the monoclinic distortion (a shear of magnitude \(\delta\) between the \(x\)/\(y\) axes, with \(z\) rescaled to preserve volume) are both volume-conserving to second order. For either family, the energy is quadratic in \(\delta\) near \(\delta=0\):

\[ E(\delta) \approx E(0) + a\,\delta^{2} \]

A degree-2 polynomial fit to \(E(\delta)\) over the sampled \(\delta\) range gives the leading coefficient \(a\), from which the corresponding modulus follows as

\[ M = \frac{a}{2\,V_0} \]

(in eV/ų, converted to GPa) where \(V_0\) is the undistorted cell volume. Applied to the orthorhombic distortion this gives the tetragonal shear modulus \(G' = \tfrac{1}{2}(C_{11}-C_{12})\); applied to the monoclinic distortion it gives the shear modulus \(C_{44}\) directly.

Assembling C11, C12, C44

\[ C_{11} = B + \tfrac{4}{3}G', \qquad C_{12} = B - \tfrac{2}{3}G', \qquad C_{44} = C_{44} \]

From these three constants, the same Voigt-Reuss-Hill machinery described in Elastic Constants is used to derive the bulk/shear/Young's moduli, Poisson's ratio, Pugh's ratio, and Chen-Vickers hardness.

References