English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We formulate a theory of thermal expansion based on the self-consistentphonon (SCP) theory, which nonperturbatively considers the anharmonic effect.We show that the Gr\"unseisen formula holds within the SCP theory by replacingthe phonon frequency by the SCP frequency. By comparing it with thequasiharmonic approximation (QHA), we derive explicit formulas of thecorrection to the QHA result. We show that the phonon anharmonicity gives asmall correction of $O(\langle{\hat{U}_4\rangle}/\langle{\hat{U}_2\rangle})$ tothe thermal expansion coefficient $\alpha$, where $\hat{U}_2$ and $\hat{U}_4$are the harmonic and the quartic terms of the potential energy surface. On theother hand, we show that the phonon anharmonicity gives two correction terms tothe temperature ($T$)-dependent phonon frequency shift which are comparable tothe original QHA term. In strongly anharmonic materials such as NaCl and MgO,these two correction terms tend to cancel out each other, which explains whythe QHA sometimes gives reasonable values for the $T$-dependent phononfrequency shift while it fails for thermal expansion.

Anharmonic Gr\"uneisen theory based on self-consistent phonon theory: Impact of phonon-phonon interaction neglected in the quasiharmonic theory