Mutual effect of $^3$He impurities and Peierls potential on shear modulus softening in solid $^4$He

We investigate numerically dislocation crossover from quantum smooth to classically rough state in solid $^4$He in presence of both - Peierls potential and $^3$He impurities as pinning centers providing gaussian trapping potential. Monte Carlo simulations have been performed within the formalism [1]. $^3$He is modeled as classical particles localized on dislocations according to thermal equilibrium with the bulk at some activation energy $E_0$, the $^3$He total fraction $x_3$ as well as the dimensionless dislocation density $x_d<<1$ (in units of interatomic distance). It is shown that the softening of the shear modulus $\mu(T)$ observed in Ref.[2] cannot be explained within the simple $^3$He evaporation model under the assumption of zero Peierls potential: for realistic $E_0$, $x_3$ and $x_d$ the temperature range over which the softening occurs is much narrower, $\approx E_0/\ln(x_d/x_3^2)$ (where $x_3 \sim x_d$), than the one observed in [2]. Inclusion of the Peierls potential smoothens out the crossover and allows good fit of the data [2]. \\[4pt] [1] D. Aleinikava, E. Dedits, A. B. Kuklov, D. Schmeltzer, arXiv:0812.0983 \\[0pt] [2] J. Day and J. Beamish, Nature {\bf 450}, 853(2007).