Low-temperature heat capacities and Raman spectra of negative thermal expansion compounds ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8$

Heat capacities of ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8$ were precisely measured between 1.8 and 330 K. Heat-capacity curves of ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8$ are very similar to each other. The heat capacity of ${\mathrm{HfW}}_2{\mathrm{O}}_8$ at low temperature is larger than that of ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ due to atomic mass effect, but both heat capacities cross around 220 K. Raman spectra of ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8$ were recorded at room temperature. Frequency distributions of lattice vibrations were estimated through an analysis of the heat capacities for ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8.$ It is found that the difference in the frequency distributions between ${\mathrm{ZrW}}_2{\mathrm{O}}_8$ and ${\mathrm{HfW}}_2{\mathrm{O}}_8$ arises from the different atomic mass and bond strength, and causes the different temperature dependence of the heat capacities. The properties of the optical-phonon modes with a large negative mode-Grüneisen parameter are discussed.