Dependence of Hysteresis Effect in Torsionally Oscillated Solid $^4$He on $^3$He Impurity Concentration

We studied the effects of $^3$He-impurity on the hysteresis phenomenon in solid $^4$He grown in an annular container as measured by a compound torsional oscillator (TO). The sample is cooled while oscillating at an initial high ($\sim$ 500 $\mu$m/s) velocity from a high temperature to a regulated target temperature where the Non-Classical Rotational Inertia fraction(NCRIf) is measured. The velocity is decreased to below a critical velocity of about 15 $\mu$m/s and then increased back to the original velocity where the NCRIf is measured again. The difference in the two NCRIf represents hysteresis. The difference is found to vanish (i.e. reversible) when the temperature is above a temperature $T_h(x_3)$ which depends on the $^3$He impurity concentration $x_3$. $T_h$ was measured for $x_3$ = 0.3, 6, 25 ppm. Surprisingly, $T_h$ coincides with the phase separation temperature evaluated according to Edwards and Balibar theory [1]. [1] D. O. Edwards and S. Balibar, Phys. Rev. B 39, 4083 (1989)