The properties of dislocations in $^4$He crystals*

We have measured (1,2) the response of oriented $^4$He crystals to an ac-driving strain as a function of temperature, strain amplitude, frequency, and $^3$He content. The very large softening of these crystals around 0.2~K is due to the free motion of dislocations parallel to the basal planes in the absence of dissipation from collisions with thermal phonons or from the binding of $^3$He impurities. We have built a complete model for the mechanical properties of $^4$He crystals, which is in full quantitative agreement with all experimental results so that most of the properties of these dislocations are now well established. These properties are incompatible with the two scenarios that had been proposed for supersolidity in $^4$He. Dislocations have a density between 10$^4$ and 10$^6$ cm$^{-2}$. They move like free strings down to 20~mK, meaning that the kink energy is negligible. They have a large distribution in length and a small connectedness. $^3$He impurities bind to dislocations with an energy distributed around 0.67~K and move with them below 45~$\mu$m/s.\\ 1- A. Haziot et al., Phys. Rev. Lett. 110, 035301 (2013), Phys. Rev. B 87, 060509(R) (2013), and Phys. Rev. B 88, 014106 (2013).\\ 2- A. D. Fefferman et al., submitted to Phys. Rev. B, Nov. 2013.\\