Contribution of optical phonons to the elastic moduli ofPdHxandPdDx

Sound-velocity and ultrasonic-attenuation measurements were carried out on dilute $\alpha$-phase and concentrated ${\alpha }^{\prime }$-phase $\mathrm{Pd}{\mathrm{H}}_x$ and $\mathrm{Pd}{\mathrm{D}}_x$ alloys between 10 and 300 K. Both the bulk modulus $B$ and the angular shear modulus $C_{44}$ of ${\alpha }^{\prime }-\mathrm{Pd}{\mathrm{H}}_x\left(\mathrm{Pd}{\mathrm{D}}_x\right)$ are lower than those of pure Pd, as expected from the lowering of the acoustic-phonon branches at the Brillouin-zone boundary observed in neutron scattering experiments. The tetragonal shear modulus $C^{\prime }$, on the other hand, increases with hydrogen or deuterium concentration. As a consequence, at 0 K the Debye temperatures of Pd${\mathrm{D}}_{0.652}$ and Pd${\mathrm{H}}_{0.66}$ are almost equal to that of Pd (${\Theta }^{\mathrm{D}}=276\mathrm{}$ K). A marked isotope effect is found in the temperature variation of all elastic moduli, the temperature coefficients $\frac{d{C}_{\mathrm{ij}}}{\mathrm{dT}}$ being significantly more negative for the deuterides. This isotope effect, which is due to the different energies of the optical phonons in $\mathrm{Pd}{\mathrm{H}}_x$ and $\mathrm{Pd}{\mathrm{D}}_x$, is well described by a quasiharmonic model in which the transverse- and longitudinal-optical phonons are treated as Einstein oscillators with different Grüneisen parameters ${\gamma }_t$ and ${\gamma }_l$. The Einstein temperatures are ${\Theta }_t^{\mathrm{H}}=650\mathrm{}$ K, ${\Theta }_l^{\mathrm{H}}=910\mathrm{}$ K in Pd${\mathrm{H}}_{0.66}$, and ${\Theta }_t^{\mathrm{D}}=450\mathrm{}$ K, ${\Theta }_l^{\mathrm{D}}=640\mathrm{}$ K in Pd${\mathrm{D}}_{0.652}$. Our analysis implies that $|{\gamma }_l-{\gamma }_t|$ is large compared to the average optical-phonon Grüneisen parameter $\overline{\gamma }=\frac{1}{3}({\gamma }_l+2\mathrm{}{\gamma }_t)$ determined from thermal-expansion measurements. For the longitudinal mode $C_L$ the ultrasonic attenuation exhibits a maximum around 220 K which is interpreted as being due to a reorientation of pairs or clusters of vacancies in the hydrogen sublattice under uniaxial strain. For the shear modes the increase in attenuation is approximately 1 order of magnitude higher than in $C_L$. This strong temperature dependence of the attenuation has, however, no measurable influence on the $C_{\mathrm{ij}}$.