Frequency Dependence and Dissipation in the Dynamics of Solid Helium

Torsional oscillator experiments on solid ${}^4\mathrm{He}$ show frequency changes which suggest mass decoupling, but the onset is broad and is accompanied by a dissipation peak. We have measured the elastic shear modulus over a broad frequency range, from 0.5 Hz to 8 kHz, and observe similar behavior—stiffening and a dissipation peak. These features are associated with a dynamical crossover in a thermally activated relaxation process in a disordered system rather than a true phase transition. If there is a transition to a dc response, e.g., to a supersolid state, it must occur below 55 mK.

Figure 1

Mechanical response in the 33.3 bar ($20.4{\mathrm{cm}}^3/\mathrm{mole}$) ${}^4\mathrm{He}$ crystal from Ref. 7. (a) Shear modulus (at 200 Hz) and acoustic resonance frequency (left axis) and TO NCRI (right axis). (b) Dissipation corresponding to Fig. 1a. Solid (red) lines are fits described later in the text. The 200 Hz dissipation and fit are vertically offset for clarity.

Figure 2

Frequency dependence of (a) shear modulus and (b) dissipation for an hcp ${}^4\mathrm{He}$ crystal at 38 bar ($20.1{\mathrm{cm}}^3/\mathrm{mole}$). Curves have been vertically shifted for clarity (the modulus scale is for the 2000 Hz data). Dots are transition midpoints plotted in Fig. 3.

Figure 3

Frequency dependence and thermal activation for the crystals of Figs. 1, 2. Open symbols are the temperatures at which 50% of the modulus change has occurred; closed symbols are for the dissipation peaks (as in Fig. 2). Lines correspond to activation energies of 0.73 and 0.77 K.

Figure 4

Fits of thermally activated relaxation process to (a) the shear modulus and (b) the dissipation at 200 Hz for the 33.1 bar crystal of Fig. 1. Fits are explained in the text.