Using in-depth structural and spectroscopic analysis, we unravel the nature of phenomena specific to the Fe3+/Fe4+ redox couple in P2-Na0.67−z[NiyMn0.5+yFe0.5−2y]O2.

Despite the widespread use of silicon in modern technology, its peculiar thermal expansion is not well understood. Adapting harmonic phonons to the specific volume at temperature, the quasiharmonic approximation, has become accepted for simulating the thermal expansion, but has given ambiguous interpretations for microscopic mechanisms. To test atomistic mechanisms, we performed inelastic neutron scattering experiments from 100 K to 1,500 K on a single crystal of silicon to measure the changes in phonon frequencies. Our state-of-the-art ab initio calculations, which fully account for phonon anharmonicity and nuclear quantum effects, reproduced the measured shifts of individual phonons with temperature, whereas quasiharmonic shifts were mostly of the wrong sign. Surprisingly, the accepted quasiharmonic model was found to predict the thermal expansion owing to a large cancellation of contributions from individual phonons.

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Inelastic neutron scattering was performed on silicon powder to measure the phonon density of states (DOS) from 100 to 1500 K. The mean fractional energy shifts with temperature of the modes were ∆εi/εi∆T = −0.07 giving a mean isobaric Grüneisen parameter of +6.95±0.67, significantly different from the isothermal parameter of +0.98. These large effects are beyond the predictions from quasiharmonic models using density functional theory (DFT) or experimental data, demonstrating large effects from phonon anharmonicity. At 1500 K the anharmonicity contributes 0.15 kB/atom to the vibrational entropy, compared to the 0.03 kB/atom from quasiharmonicity. Excellent agreement was found between the entropy from phonon DOS measurements and the reference JANAF thermodynamic entropy from calorimetric measurements.2

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