Critical velocities and the effect of steady and oscillating rotations on solid He4

We apply our recently developed model of a Bose condensate of quantum kink wave in solid He4 to understand recent torsional oscillator experimental results of the citical velocities and the effect of the steady and oscillating rotations at around 0.1 degree K. When the D.C. rotation is present we find a decrease of the Q factor given by $Q^{-1} \propto f_{sf}\times \Omega_{D}/\omega_{TO}$where$f_{sf}$isthesuperfluidfraction;$\Omega_{D}$,theD\mathrm{.}C\mathrm{.}angularrotationvelocity,$\omega_{TO}$, the torsional oscillator oscillating frequency. We estimate the AC critical velocity ${\mathrm{\Omega }}_A^{crit}$ as that required to generate a kink wave of wavevector $2\pi \mathrm{/}{L}_d$ where $L_d$ is the distance between nodes of the dislocation network. We generalize this to include a steady rotation and find a D. C. critical velocity ${\mathrm{\Omega }}_D^{crit}\propto ({\mathrm{\Omega }}_A^{crit}{)}^{1\mathrm{/}2}$. Estimates for both the steady and the oscillating critical velocities are in order of magnitude agreement with experimental results. We have also examined an alternative mechanism of kink tunnelling through a node in the dislocation networm and find that there is also a dependence on the torsional oscillator frequency: ${\mathrm{\Omega }}_D^{crit}=[{\mathrm{\Omega }}_A^{crit}{\omega }_{TO}2\pi {]}^{1\mathrm{/}2}\mathrm{.}$ The DC critical velocity ${\mathrm{\Omega }}_D^{crit}$ is ten times higher than the experimental value.