The anomalous stiffness of $^4$He crystals

In 2007, Day and Beamish have discovered that the supersolid transition of $^4$He crystals is accompanied by an increase in their shear modulus $\mu$. They proposed that this stiffening is due to the pinning of dislocations by $^3$He impurities. However, we have measured $\mu$ in single crystals which are free of any impurity. In such ultrapure crystals, we have found a stiffening which calls for a mechanism independent of $^3$He content. This new mechanism might be Kuklov's roughening transition of dislocations or some interaction between Anderson's vortex liquid and the lattice. In order to discriminate between these two scenarios, we have built a cell which allows acoustic measurements in ideal $^4$He crystals where there are no impurities and no dislocations. We will analyze the stiffness of single crystals in the presence of variable disorder.