Movement of dislocations dressed with ${}^3\mathrm{He}$ impurities in ${}^4\mathrm{He}$ crystals

Solid ${}^4\mathrm{He}$ is a unique example of crystal where dislocations may move at macroscopic speeds with impurities attached to them. In ${}^4\mathrm{He}$ crystals, the only impurities are ${}^3\mathrm{He}$ atoms, whose concentration can be reduced to zero and measured down to the ppt $\left(10{}^{-12}\right)$ level. We present measurements of the mobility of dislocations dressed with ${}^3\mathrm{He}$ impurities as a function of the crystal purity. They show that the damping of dislocation motion is proportional to the concentration of ${}^3\mathrm{He}$ bound to these dislocations. It has allowed us to measure the ${}^3\mathrm{He}$ binding energy $E_B$ to dislocations without any ambiguity. Our results solve the controversy concerning $E_B$: We confirm our previously measured value $0.7 \pm$ 0.1 K, and we demonstrate that it cannot be 0.2 or 0.4 K as estimated by other authors. Finally, we present a simple model for the damping magnitude, where dissipation is due to the emission by moving ${}^3\mathrm{He}$ impurities of transverse waves along the dislocation lines.

Figure 1

(Solid curves) Dissipation measurement on cooling C2 (${}^3\mathrm{He}$ concentration $x_3=2.3\times {10}^{-8}$) at $\omega /2\pi =16037$ Hz and driving strain $\epsilon =2.3\times {10}^{-6}$. The three different colors correspond to measurements on three different days, which demonstrate excellent reproducibility. The dashed line shows the linear variation of $Q^{-1}$ in the low $\omega T^3$ limit, which is predicted by Eq. (6) and allows a determination of the average network length $L_N=158 \mu \mathrm{m}$ and dislocation density $\Lambda =4.1\times {10}^5 \mathrm{cm}{}^{-2}$. (Inset) The measurement cell (see text).

Figure 2

The dissipation and shear modulus of crystal C1 whose ${}^3\mathrm{He}$ concentration was $2.3\times 10{}^{-6}$. Measurements made on cooling at a low rms driving strain of $2.7\times {10}^{-9}$ and low frequency, where the low temperature damping is due to the binding of ${}^3\mathrm{He}$ impurities only.

Figure 3

A semilog plot of the relaxation time vs inverse temperature. Different symbols correspond to six different crystals with various ${}^3\mathrm{He}$ concentrations. The measurements were made at low driving strain $\epsilon =2.7\times {10}^{-9}$ and at frequencies in the range 1–100 Hz. This plot is used to determine the binding energy of ${}^3\mathrm{He}$ atoms to dislocations (see text).

Figure 4

The dependence on ${}^3\mathrm{He}$ concentration $x_3$ of the prefactor of the exponential in the expression of the damping coefficient $B_3$ (see text). $x_3$ is measured within 5% in two different mass spectrometers. The agreement with $n=1$ demonstrates that the damping is proportional to $x_3$ (i.e., $n=1$), not to $x_3^2$ (i.e., $n=2$).