A solid expands when it is heated. What actually decides how much it expands is its stiffness — the bulk modulus BT. The relation that comes out of the quasiharmonic approximation (QHA) αV = γCV / (BTV) says exactly that in a single line. In ①, a cubic crystal is built as a 3D supercell and its atoms are made to vibrate as the temperature is raised — the amplitude follows the formula for the atomic displacement parameter ADP exactly, so the vibration does not stop even at 0 K. In ②, αV and BT are plotted against each other for 38 cubic crystals. In ③, temperature is the horizontal axis, and you follow how the expansion, the stiffness, and the specific heat change. When the Grüneisen parameter γ is negative, the crystal shrinks even though it is heated (negative thermal expansion, Cu2O).
Every curve is the direct output of first-principles calculations (QHA).
V(T), αV(T), BT(T), and
CV(T) are read as they are from the phonopy-qha calculation results
behind Y. Mochizuki et al., J. Phys. Chem. C 128, 525 (2024),
over the range 0–595 K
(cubic crystals only, 38 materials). The lattice constant is also built from this volume as a = (natV)1/3,
so the a shown on the screen is a calculated value, not an experimental one
— the median deviation is 0.6 %, and only Fe (−4.8 %) and Ce (−10.9 %) deviate substantially,
and the reason for each is given in the note for that material.
Only the range above 595 K, and Cu2O, come from a model
— the details are in “How the temperature dependence was obtained” in ③.
ΘD is the value obtained by fitting the Debye model to the CV(T)
produced by the same calculation. When the ADP estimated with it is compared with the measured Debye–Waller
B parameters (room temperature) for 20 materials, the model/measured ratio is 1.02 (median) and 1.01 (mean)
(diamond 0.153 vs. 0.11, Cu 0.530 vs. 0.55 Ų).
The underlying idea is described in the derivation of the Mie–Grüneisen equation of state (in Japanese).
Data sources: all the temperature dependence comes from the QHA calculations of the paper above,
the 300 K values of αV, γ, and BT from Tables S2/S3 of the same paper,
the room-temperature experimental lattice constants (for comparison) and the melting points, decomposition temperatures, and transformation points
from the CRC Handbook of Chemistry and Physics and Wyckoff’s Crystal Structures,
the radii used for drawing from metallic radii, covalent radii, and Shannon ionic radii,
and the measured Debye–Waller B values are
standard values of the kind tabulated in the International Tables for Crystallography.
The ΘD here is “the Debye temperature that best reproduces the calculated specific heat,”
which is a different quantity from the literature value determined from low-temperature specific heat
(for diamond, 1835 K versus 2230 K in the literature. Fe and Cu2O are the only two that are not fitted. The Fe calculation does not include magnetism, so its soft modes have imaginary frequencies and CV reaches only 1.86 kB at 300 K, which a Debye model cannot match; the literature value of 470 K is used instead. Cu2O has no QHA data, so its thermal vibration uses the literature value of 188 K and its CV uses 612 K).