Non-linear Elastic Response in Solid Helium: critical velocity or strain? James Day, Oleksandr Syshchenko, and John Beamish Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2G7 (Dated: October 15, 2018) Torsional oscillator experiments show evidence of mass decoupling in solid $^{4}$He. This decoupling is amplitude dependent, suggesting a critical velocity for supersolidity. We observe similar behavior in the elastic shear modulus. By measuring the shear modulus over a wide frequency range, we can distinguish between an amplitude dependence which depends on velocity and one which depends on some other parameter like displacement. In contrast to the torsional oscillator behavior, the modulus depends on the magnitude of stress, not velocity. We interpret our results in terms of the motion of dislocations which are weakly pinned by $^{3}$He impurities but which break away when large stresses are applied. PACS numbers: 67.80.bd, 67.80.de, 67.80.dj Helium is a uniquely quantum solid and recent tor- sional oscillator (TO) measurements[1–8] provide evi- dence for a supersolid phase in hcp $^{4}$He. At tempera- tures below about 200 mK the TO frequency increases, suggesting that some of the $^{4}$He decouples from the os- cillator. This evidence of non-classical rotational iner- tia (NCRI) inspired searches for other unusual thermal or mechanical behavior in solid helium. Heat capacity measurements[9, 10] do show a small peak near the on- set temperature of decoupling, supporting the idea of a phase transition in solid $^{4}$He. Mass flow is an obvious possible signature of supersolidity but experiments in this temperature range[11–13] show no pressure-induced flow through the solid (although recent measurements[14, 15] at higher temperatures showed intriguing behavior). the blocked annulus experiment[1, 19] in which NCRI is greatly reduced by inserting a barrier into the flow path, thus indicating that long-range coherent flow is involved. The other is the reduction of the NCRI fraction when the oscillation amplitude exceeds a critical value[1]. In anal- ogy to superfluidity in liquid helium, this is interpreted in terms of a critical velocity v$_{c}$ (of order 10 µ m/s), above which flow becomes dissipative. However, torsional oscil- lators are resonant devices and measurements made at a single frequency cannot distinguish an amplitude depen- dence which sets in at a critical velocity from one which begins at a critical displacement or a critical accelera- tion. A recent experiment[6] used a compound torsional oscillator which operated in two modes, allowing mea- surements to be made on the same solid $^{4}$He sample at two different frequencies (496 and 1173 Hz). The ampli- tude dependence scaled somewhat better with velocity than with acceleration or displacement, supporting the superflow interpretation of TO experiments. However, recent measurements[20] in which one mode was driven at large amplitude while monitoring the low-amplitude response of the other mode gave unexpected results. The suppression of NCRI, as seen in the low amplitude mode, appeared to depend on the acceleration generated by the high amplitude mode, rather than on the velocity. To settle the question of whether the amplitude dependence seen in torsional oscillators reflects a critical velocity for superflow, measurements over a wider frequency range are needed. We recently made low frequency measurements[16, 17] of the shear modulus of hcp $^{4}$He and found a large stiff- ening, with the same temperature dependence as the fre- quency changes seen in TO experiments. This modulus increase also had the same dependence on measurement amplitude and on $^{3}$He concentration and, like the TO decoupling, its onset was accompanied by a dissipation peak. It is clear that the shear stiffening and the TO de- coupling are closely related. Subsequent experiments[18] with $^{3}$He showed similar elastic stiffening in the hcp phase below 0.4 K, but not in the bcc phase (the bcc phase exists only over a temperature range in $^{4}$He). TO mea- surements with hcp $^{3}$He did not, however, show any sign of a transition in the temperature range where the stiff- ening occurred, nor was a transition seen with bcc $^{3}$He. The stiffening appears to depend on crystal structure (ap- pearing in the hcp but not the bcc phase) while the TO frequency and dissipation changes occur only for the bose solid, $^{4}$He. We have made direct measurements[16] of the ampli- tude dependence of the shear modulus, µ , of hcp $^{4}$He in a narrow gap of thickness D. An AC voltage, V, with angular frequency ω = 2 π f, is applied to a shear trans- ducer (with piezoelectric coefficient d 15 ) to generate a displacement δ x=d 15 V at its surface. This produces a quasi-static shear strain in the helium, ϵ = δ x/D. The resulting stress, σ , generates a charge, q, and thus a current, I = ω q, in a second transducer on the oppo- site side of the gap. The shear modulus µ = σ / ϵ is then proportional to I/fV. In contrast to torsional oscillator Although it is clear that solid $^{4}$He shows unusual be- havior below 200 mK, the interpretation in terms of su- persolidity rests almost entirely upon torsional oscillator experiments. In addition to frequency changes which im- ply mass decoupling, two other features of the TO ex- periments are invoked as evidence of superflow. One is arXiv:1001.0189v2 [cond-mat.other] 29 Jan 2010