Effects of $^3$He Impurity on Solid $^4$He Studied by Compound Torsional Oscillator

Frequency shifts and dissipations of a compound torsional oscillator induced by solid $^4$He samples containing $^3$He impurity concentrations ($x_3$ = 0.3, 3, 6, 12 and 25 in units of 10$^{-6}$) have been measured at two resonant mode frequencies ($f_1$ = 493 and $f_2$ = 1164 Hz) at temperatures ($T$) between 0.02 and 1.1 K. The fractional frequency shifts of the $f_1$ mode were much smaller than those of the $f_2$ mode. The observed frequency shifts continued to decrease as $T$ was increased above 0.3 K, and the conventional non-classical rotation inertia fraction was not well defined in all samples with $x_3 \geq$ 3 ppm. Temperatures where peaks in dissipation of the $f_2$ mode occurred were higher than those of the $f_1$ mode in all samples. The peak dissipation magnitudes of the $f_1$ mode was greater than those of the $f_2$ mode in all samples. The activation energy and the characteristic time ($\tau_0$) were extracted for each sample from an Arrhenius plot between mode frequencies and inverse peak temperatures. The average activation energy among all samples was 430 mK, and $\tau_0$ ranged from 2$\times 10^{-7}$ s to 5$\times 10^{-5}$ s in samples with $x_3$ = 0.3 to 25 ppm. The characteristic time increased in proportion to $x_3^{2/3}$. Observed temperature dependence of dissipation were consistent with those expected from a simple Debye relaxation model \emph{if} the dissipation peak magnitude was separately adjusted for each mode. Observed frequency shifts were greater than those expected from the model. The discrepancies between the observed and the model frequency shifts increased at the higher frequency mode.