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Frequency shifts and dissipations of a compound torsional oscillator inducedby 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 modefrequencies ($f_1$ = 493 and $f_2$ = 1164 Hz) at temperatures ($T$) between0.02 and 1.1 K. The fractional frequency shifts of the $f_1$ mode were muchsmaller than those of the $f_2$ mode. The observed frequency shifts continuedto decrease as $T$ was increased above 0.3 K, and the conventionalnon-classical rotation inertia fraction was not well defined in all sampleswith $x_3 \geq$ 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 peakdissipation 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 modefrequencies and inverse peak temperatures. The average activation energy amongall samples was 430 mK, and $\tau_0$ ranged from 2$\times 10^{-7}$ s to5$\times 10^{-5}$ s in samples with $x_3$ = 0.3 to 25 ppm. The characteristictime increased in proportion to $x_3^{2/3}$. Observed temperature dependence ofdissipation were consistent with those expected from a simple Debye relaxationmodel \emph{if} the dissipation peak magnitude was separately adjusted for eachmode. Observed frequency shifts were greater than those expected from themodel. The discrepancies between the observed and the model frequency shiftsincreased at the higher frequency mode.

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