Absence of a low-temperature anomaly in the Debye-Waller factor of solid ${}^4\mathrm{He}$ down to $140\mathrm{mK}$

The mean-square atomic displacement in hcp-phase solid ${}^4\mathrm{He}$ has been measured in crystals with a molar volume of $21.3{\mathrm{cm}}^3$. It is temperature independent from $1\mathrm{K}\text{to}140\mathrm{mK}$, with no evidence for an anomaly in the vicinity of the proposed supersolid transition. The mean-square displacement is different for in-plane motions $(0.122\pm 0.001{\mathrm{\AA }}^2)$ and out-of-plane motions $(0.150\pm 0.001{\mathrm{\AA }}^2)$.

Figure 1

The mean-square displacement $⟨u^2⟩$ of ${}^4\mathrm{He}$ hcp crystal, as measured for the three selected peaks as a function of temperature.

Figure 2

The crystallite-dependent quantity $\ln N$, as measured from the ${}^4\mathrm{He}$ hcp crystal for the three selected peaks as a function of temperature.

Figure 3

The temperature dependence of the $c$ and $a$ lattice parameters from two of the crystallites studied in the text.

Figure 4

(Color online) The temperature dependence of the $a$ lattice parameter as measured here (closed points) and as calculated from Fig. 2 of Ref. 22 (open points), assuming that the temperature-independent part of $a$ is the same for both data sets. The lines are based on the different models for vacancy concentration: the solid line is the exponential from the thermal activation model, with an activation energy of $8.6\mathrm{K}$; the broken lines are fits to the model proposed in Ref. 21—a $T^4$ dependence. The dotted line is a fit to the data from Ref. 22 only and the dashed line includes the data measured in this paper.