phonons.

Discussion
==========

In materials where NTE has been studied through lattice dynamics calculations of dispersion curves and associated mode Grüneisen parameters (for example, Cu$_2$ [@Cu2O_Gupta; @Leila_Cu2O], ScF$_3$ [@ScF3_phonons], Zn(CN)$_2$ [@Hong_Zn(CN)2], MOF-5 [@Leila_MOF5]), it has usually been found that there is a well-defined set of low-frequency modes with negative mode Grüneisen parameters. In this study we have found that Y$_2$W$_3$O$_{12}$ is unusual in that a high degree of eigenvector mixing occurs, leading to the existence of broad bands of phonons with either negative or positive mode Grüneisen parameters.

More specifically, this study has found that NTE in Y$_2$W$_3$O$_{12}$ is primarily driven by phonons in the 0–5 THz frequency range, with the strongest contribution coming from those phonons at the lowest frequencies. There is also some weaker contribution to NTE from another band of phonons in the 9–10 THz frequency range. All other phonons drive PTE.

We had previously shown that any material containing octahedra within a corner-linked network with no non-bridging bonds cannot support RUMs [@Hammonds_octahedra] (the ScF$_3$ structure being the one exception, as discussed in reference [@Dove_RUMs]). Thus it follows that NTE in Y$_2$W$_3$O$_{12}$ does not arise from the ‘traditional’ RUM mechanism [@NTE_theory_Welche] based on rigid YO$_6$ octahedra and 