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$^4$He Crystal Quality and Rotational Response in a Transparent Torsional Oscillator

We have studied natural purity $^{4}$He single crystals and polycrystals between 10 and 600 mK using a torsional oscillator with a 2 cm$^{3}$ rigid cell made of sapphire with a smooth geometry. As the temperature was lowered, we observed sample dependent but reproducible resonant frequency shifts that could be attributed to a supersolid fraction of order 0.1%. However, these shifts were observed with single crystals, not with polycrystals. Our results indicate that, in our case, the rotational anomaly of solid helium is more likely due to a change in stiffness than to supersolidity. This interpretation would presumably require gliding of dislocations in more directions than previously thought.