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Upper limit of supersoldity in solid helium

The resonant period drop observed at low temperatures in torsional oscillators containing solid helium had been interpreted as a signature of supersolid. However, it was found that the shear modulus increase found in solid helium at the same low temperature could also decrease the resonant period of the torsional oscillator. We report the results of a study in two different torsional oscillators that were specially designed to minimize the shear modulus effect and maximize any possible supersolid response. We were able to place an upper limit on the nonclassical rotational inertia or supersolid fraction of $4\times10^{-6}$. Moreover, we have repeated an earlier experiment on hcp $^3$He solid, which shows similar low temperature stiffening as in hcp $^4$He. We found that the small drop of the resonant period measured in the hcp $^3$He sample is comparable in size to that found in the hcp $^4$He samples. These results strongly suggest that the resonant period drop reported in most torsional oscillator studies in the last decade is primarily a consequence of the shear modulus stiffening effect.