The 50 K anomaly in the shear modulus of $\beta $-PdH$_{0.71}$

When palladium hydride, PdH$_{x}$, is rapidly cooled to liquid helium temperature and then slowly reheated, both the heat capacity and electrical resistivity show a peak in the range 50~$<$~$T$~$<$~80~K, depending on the composition $x$. This ``50~K anomaly'' has been previously explained in terms of formation of long-range ordered hydrogen superlattice structures. However, several aspects of the 50~K anomaly are inconsistent with an ordering phase transition, namely, the temperature of the anomaly depends on the rate of cooling, and the magnitude of the anomaly is larger for a fast cooling rate than for a slow cooling rate. We have studied the 50~K anomaly by measuring the elastic constants of single-crystal PdH$_{0.71}$ in the temperature range 1.4~$<$~$T$~$<$~300~K during both fast cooling and slow warming. During warming, we observed a peak in the shear modulus $C^{\prime }$~=~($C_{11}$~-~$C_{12}$/2) at 55~K, which we attribute to the 50~K anomaly. In contrast, we observed no peak in the temperature dependence of the shear modulus $C_{44}$ or of the bulk modulus $B$. We propose that the 50~K anomaly arises not from the formation of long-range ordered hydrogen superlattice structures, but from freezing of the hydrogen short-range order as the hydride is cooled.