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We have studied natural purity $^{4}$He single crystals and polycrystalsbetween 10 and 600 mK using a torsional oscillator with a 2 cm$^{3}$ rigid cellmade of sapphire with a smooth geometry. As the temperature was lowered, weobserved sample dependent but reproducible resonant frequency shifts that couldbe attributed to a supersolid fraction of order 0.1%. However, these shiftswere observed with single crystals, not with polycrystals. Our results indicatethat, in our case, the rotational anomaly of solid helium is more likely due toa change in stiffness than to supersolidity. This interpretation wouldpresumably require gliding of dislocations in more directions than previouslythought.

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