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measurements, this technique is non-resonant and so we could measure the shear modulus over a wide frequency range, from a few Hz (a limit set by preamplifier noise) to a maximum of 2500 Hz (due to interference from the first acoustic resonance in our cell, around 8 kHz). The technique is very sensitive, allowing us to make modulus measurements at strains as low as ϵ ≈ 10 $^{$_{−}$9}$, correspond- ing to stresses σ ≈ 0.02 Pa. The drive voltage can be increased substantially without heating the sample, so measurements can be made at strains up to ϵ ≈ 10 $^{$_{−}$5}$, al- lowing us to study the amplitude dependence, including hysteretic effects, at low temperatures. 

Figure 1 shows the temperature dependence of the nor- malized shear modulus µ / µ$_{o}$ for an hcp $^{4}$He sample at a pressure of 38 bar and a frequency of 2000 Hz. The crys- tal was grown from standard isotopic purity $^{4}$He (con- taining about 0.3 ppm $^{3}$He) using the blocked capillary method. A piezoelectric shear stack[21] was used to gen- erate strains in a narrow gap (D = 500 µ m) between it and a detecting transducer. Other experimental de- tails are the same as in refs. 16 to 18. The curves in Fig. 1 correspond to different transducer drive voltages, i.e. to different strains, and were measured during cool- ing from a temperature of 0.7 K. The shear modulus in- creases at low temperature, with Δ µ / µ$_{o}$ ≈ 17% at the lowest amplitude. Similar stiffening was seen in all hcp $^{4}$He crystals[16, 17], although the onset temperature was lower in crystals of higher isotopic purity and the magni- tude of the modulus change varied by a factor of about 2 from sample to sample (the TO NCRI varies much more, by a factor of 1000, although its temperature dependence is always essentially the same[18]). The amplitude de- pendence of the shear modulus stiffening is essentially the same as that of the TO NCRI. At the lowest drive voltages and temperatures, both are independent of am- plitude but they decrease at high amplitudes. The onset of stiffening shifts to lower temperatures at high ampli- tudes, as does the onset of TO decoupling. However, in the case of elastic measurements, it is more natural to think of this behavior in terms of a critical stress (pro- portional to the strain, i.e. to transducer displacement), rather than a critical velocity. 

The amplitude dependence of the shear modulus is shown in more detail in Fig. 2. Open circles show the modulus at 48 mK, taken from the temperature sweeps (i.e. the points marked by open circles in Fig. 1). The solid circles are the modulus measured when the drive voltage was reduced at fixed temperature (48 mK), after cooling from high temperature at the highest drive am- plitude (3 V$_{pp}$). Figure 2 also shows the corresponding amplitude dependence at temperatures well above the shear modulus anomaly (open squares are data at 700 mK from the temperature sweeps of Fig. 1; solid squares are from amplitude sweeps at 800 mK). The data taken with the two protocols agree very well. The critical drive voltage (where the modulus becomes amplitude depen- 

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dent) is around 30 mV$_{pp}$ (corresponding to σ ≈ 4x10 $^{$_{−}$8}$, σ ≈ 0.8 Pa) and at the highest drive levels (3 V$_{pp}$ corre- sponding to σ ≈ 80 Pa) the stiffening is almost completely suppressed. The amplitude dependence of the modulus closely resembles that of the NCRI in TO experiments, behavior which is attributed to a superflow critical ve- locity (typically around 10 µ m/s). However, even in TO measurements there are inertial stresses in the helium, produced by its acceleration, which could lead to the ob- served amplitude dependence. 

A number of experiments[6, 19, 22, 23] have shown that 

