Unusual compressibility in the negative-thermal-expansion material ZrW$_{2}$O$_{8}$

The negative thermal expansion (NTE) compound ZrW$_{2}$O$_{8}$ has been well-studied because it remains cubic with a nearly constant, isotropic NTE coefficient over a broad temperature range. However, its elastic constants seem just as strange as its volume because NTE makes temperature acts as \textit{positive} pressure, decreasing volume on warming and, unlike most materials, the thermally-compressed solid$_{ }$\textit{softens}. Does ZrW$_{2}$O$_{8}$ also soften when pressure alone is applied? Using pulse-echo ultrasound in a hydrostatic SiC anvil cell, we determine the elastic tensor of monocrystalline ZrW$_{2}$O$_{8 }$near 300 K as a function of pressure. We indeed find an unusual decrease in bulk modulus with pressure. Our results are inconsistent with conventional lattice dynamics, but do show that the thermodynamically-complete constrained-lattice model can relate NTE to elastic softening as increases in either temperature or pressure reduce volume, establishing the predictive power of the model, and making it an important concept in condensed-matter physics.