Mechanical and thermodynamic basics

Tasks 9.6.1-9.6.3. References: pymatgen analysis.elasticity and analysis.eos; Birch (1947); Vinet, Ferrante, Smith and Rose (1987); Murnaghan (1944); Voigt (1889), Reuss (1929) and Hill (1952).

Elastic tensor

The stiffness is the derivative of the stress with respect to an engineering strain. For each Voigt component j, two structures are built with +epsilon and -epsilon:

C_ij = (sigma_i(+epsilon) - sigma_i(-epsilon)) / (2 epsilon)

The stress comes from the engine's analytic stress (GPa); UFF is the native default, and any Calculator that reports stress can be used. The matrix is symmetrized as (C_ij + C_ji)/2 to remove finite-difference noise. The compliance S = C^-1 gives the Reuss moduli,

B_R = 1 / (S11 + S22 + S33 + 2 (S12 + S13 + S23))
G_R = 15 / (4 (S11 + S22 + S33) - 4 (S12 + S13 + S23)
            + 3 (S44 + S55 + S66))

and the Voigt moduli use the stiffness directly. The Hill values are the arithmetic means; Young's modulus and Poisson's ratio follow from E = 9 B G / (3 B + G) and nu = (3 B - 2 G) / (2 (3 B + G)). The stability flag is a Cholesky factorization of the stiffness: positive definiteness is the Born criterion in this approximation.

Equation of state

Three models are fitted with Levenberg-Marquardt and numerical Jacobians:

• Birch-Murnaghan (third order): E = E0 + 9 V0 B0 / 16 [(eta^2 - 1)^3 B0' + (eta^2 - 1)^2 (6 - 4 eta^2)] with eta = (V0/V)^(1/3).
• Vinet: the universal equation with x = (V/V0)^(1/3).
• Murnaghan: the linear pressure derivative form.

B0 is reported in GPa (1 eV/Angstrom^3 = 160.21766208 GPa), B0' is dimensionless and the parameter errors come from sigma^2 (J^T J)^-1 with sigma^2 = RSS/(n-4). The initial guess uses the minimum-energy point, a three-point curvature for B0 and B0' = 4.

Formation energies and the convex hull

The formation energy per atom of a phase is

E_f = (E_total - sum_i n_i mu_i) / N

where the chemical potentials mu_i come from the pure-element entries or from an explicit map. Every composition is converted to element fractions, so all phases live in one simplex.

For each phase the hull energy is the solution of the linear program

minimize    sum_j w_j E_f,j
subject to  sum_j w_j x_j = x_target
            sum_j w_j = 1,  w_j >= 0

solved with a two-phase simplex. The decomposition is the set of weights above 1e-9; the distance above the hull is E_f - hull and a phase is stable when it is at most 1e-6 eV/atom above it. The formulation is independent of the number of elements.