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Multiple Mode Torsional Oscillator Studies and Evidence for Supersolidity in Bulk 4He

The discovery by Kim and Chan (KC) of an anomalous decrease in the period of torsional oscillators (TO) containing samples of solid $^4$He at temperatures below 0.2 K was initially interpreted as a superfluid-like decoupling of a fraction of the solid moment of inertia from the TO. These experiments appeared to confirm the thirty-year-old theoretical prediction by Chester, Andreev, and Leggett of the existence of a low-temperature Bose-condensed supersolid phase in solid $^4$He. The initial results of KC lead to a flurry of experimental and theoretical activity and their results for bulk $^4$He samples were soon confirmed in a number of laboratories. The early excitement in this field, however, has been tempered by the realization that an anomaly in the shear modulus of solid helium $^4$He may explain most if not all of the period shifts observed in the early TO experiments. In principle, it is possible to distinguish experimentally between shear modulus induced period shifts and period shifts resulting from other physical mechanisms through the use of multiple frequency torsional oscillators. In this paper we shall discuss our recent results on bulk solid $^4$He with double and triple mode TOs, using both open cylindrical cells and annular cells. In these experiments, we have observed a small frequency independent contribution to the period shift signal such as might be expected in the presence of a supersolid phase. Since the interplay between the elastic properties of the solid and the mechanics of TOs can be subtle and depends on the specific design of the TO, we shall discuss in detail the mechanics of the TOs employed in our measurements applying both analytic and finite element methods (FEM) for the analysis.