Thermal expansion, ultrasonic velocity, and attenuation measurements in Ti${\mathrm{S}}_2$, Ti${\mathrm{Se}}_2$, and Ti${\mathrm{S}}_{0.5}$${\mathrm{Se}}_{1.5}$

Measurements have been made of the velocity and attenuation of ultrasonic plate mode waves in Ti${\mathrm{S}}_2$, Ti${\mathrm{Se}}_2$, and Ti${\mathrm{S}}_{0.5}$ ${\mathrm{Se}}_{1.5}$ between 4 and 300 K. Ti${\mathrm{Se}}_2$ exhibits large anomalies associated with the occurrence of a charge-density-wave transition near $T_c=200\mathrm{}$ K. Measurements of the expansion coefficient and of the $\alpha$-axis uniaxial stress dependence of $T_c$ have also been made, and these measurements, together with other data in the literature, are discussed in terms of the Ehrenfest relationships. A large discrepancy is observed between results obtained from thermal expansion and those obtained from elastic properties. The temperature dependence of the velocity in Ti${\mathrm{Se}}_2$ below $T_c$ is accounted for by an anharmonic contribution and an electron-phonon contribution, which reflects the opening of an energy gap on the Fermi surface by the charge-density wave. Above $T_c$, the velocity changes are found to be affected by fluctuations over a wide temperature range and it is suggested that these fluctuations are associated with the phonon mode softening near the $L$ point of the Brillouin zone as $T_c$ is approached. Reduced-dimensionality effects are suggested.