Compare Phase Enthalpies at Common Pressure

Plot both branches and inspect their structures

A real comparison starts by opening the competing phase structures, checking their state identities, and plotting each energy-volume or enthalpy-pressure branch over the same range. At a chosen pressure, inspect the minimizing volume on both branches and the pressure-volume contribution before locating a crossing. Compare the pressure range and phase candidates with the relevant literature through visual tools and literature sources.

Optional common-pressure exercise: both phase branches are invented conceptual data. Use them only after inspecting two accepted real branches; the script tests common-pressure bookkeeping and cannot support a material phase boundary or stability claim.

Use this fixture after two phase branches have been fitted over a common supported pressure interval. It checks the common-pressure operation with invented analytic alpha and beta branches; it is not a material calculation.

From the repository root, print the complete sampled enthalpy report:

python3 examples/practical-guides/eos_phase_enthalpy.py

The output contains the fixture crossing and sampled enthalpy records. Inspect both branches, not only the reported crossing.

Minimize each phase independently

At every external pressure $p$, evaluate

$$H_i(p)=\min_V\left[E_i(V)+pV\right].$$

Alpha and beta normally minimize at different volumes. They must use the same pressure, energy convention, and cell or formula-unit normalization. The fixture converts GPa to eV per cubic angstrom with ASE 3.29.0 before adding $pV$.

Confirm in each sample that the pressure is common, the minimizing volume is phase-specific, and $pV$ has energy units. Comparing both phases at one arbitrary common volume is not this calculation.

Locate and challenge the crossing

The invented beta branch begins $0.08$ eV per abstract cell above alpha at zero pressure but has a smaller equilibrium volume. In the fixture it is lower in enthalpy by 6 GPa, and bisection finds equality near 2.6209 GPa.

Treat that value as a regression target. For real phase curves, repeat the minimization across accepted EOS forms, fit windows, numerical settings, and candidate branches. A crossing is usable only when both minimizing volumes lie inside supported data ranges and each phase retains its intended structural and electronic identity.

Claim boundary

A crossing satisfies

$$H_\alpha(p_t)=H_\beta(p_t)$$

for the represented branches. Report the accepted pressure range, both minimizing volumes, phase-set scope, and sensitivity. If a third phase lies lower, or fit and numerical variation are comparable to the claimed resolution, the two-phase transition is unresolved or pre-empted.

The fixture contains no DFT run, pathway, barrier, nucleation model, phonons, elastic tensor, finite-temperature contribution, or experimental pressure calibration. It verifies common-pressure minimization, unit conversion, bracketing, and sign change only.

Official and primary sources

• IUPAC definition of enthalpy
• Vinet and co-workers, compressibility of solids
• Mouhat and Coudert, elastic stability conditions
• Phonopy quasiharmonic documentation