Effect of rotation on the elastic moduli of solid ${}^4\mathrm{He}$

We report measurements of elastic moduli of hcp solid ${}^4\mathrm{He}$ down to 15 mK when the samples are rotated unidirectionally. Recent investigations have revealed that the elastic behavior of solid ${}^4\mathrm{He}$ is dominated by gliding of dislocations and pinning of them by ${}^3\mathrm{He}$ impurities, which move in the solidlike Bloch waves (impuritons). Motivated by the recent controversy of torsional oscillator studies, we have performed direct measurements of shear and Young's moduli of annular solid ${}^4\mathrm{He}$ using pairs of quarter-circle-shape piezoelectric transducers (PZTs) while the whole apparatus is rotated with angular velocity $\mathrm{\Omega }$ up to 4 rad/s. We have found that shear modulus $\mu$ is suppressed by rotation below 80 mK, when shear strain applied by PZT exceeds a critical value, above which $\mu$ decreases because the shear strain unbinds dislocations from ${}^3\mathrm{He}$ impurities. The rotation-induced decrement of $\mu$ at $\mathrm{\Omega }=4$ rad/s is about 14.7(12.3)% of the total change of temperature dependent $\mu$ for solid samples of pressure 3.6(5.4) MPa. The decrements indicate that the probability of pinning of ${}^3\mathrm{He}$ on dislocation segment $G$ decreases by several orders of magnitude. We propose that the motion of ${}^3\mathrm{He}$ impuritons under rotation becomes strongly anisotropic by the Coriolis force, resulting a decrease in $G$ for dislocation lines aligning parallel to the rotation axis.

Figure 1

Two cases of shear motions of hcp ${}^4\mathrm{He}$ that contribute to change in $c_{44}$. (a) The hcp basal plane glides to [1010], [1100], or [0110] directions. (b) (1010), (1100), or (0110) planes glide to the direction of $c$ ([0001]) axis. (c) Schematic drawing of dislocation network (cubic shape is assumed for simplicity). The vibration of dislocation lines perpendicular to glide direction and in parallel to the gliding plane (shown by dashed lines) only contributes to shear stress. The direction of sample rotation $\mathrm{\Omega }$ is also shown by an arrow for later discussion.

Figure 2

The experimental cell. (a) Schematic view. Capillary is used for supplying line of ${}^4\mathrm{He}$. (b) Inner structure. The gaps between the PZTs are 0.5 mm. (c) Movements of two PZT's. Top : neutral position. The middle and the bottom express movements of shear and compressive PZTs, respectively.

Figure 3

Shear modulus of two solid samples as a function of temperature. Data are shifted so as to coincide at 300 mK for each strain. Scales of shear modulus for two samples are different. (a) $P=3.6$ MPa. Data are shown for three different strains and $\mathrm{\Omega }=0,1,2,3,4$ rad/s. (b) $P=5.4$ MPa, for two strains and $\mathrm{\Omega }=0$ and 4 rad/s.

Figure 4

(a) A sequence of measurement of $\mathrm{\Omega }$-dependent shear modulus $\mu$ at $\epsilon =3.96\times {10}^{-8}$. Brown lines show temperature, while other colors show $\mu$ at different $\mathrm{\Omega }$'s. Black points indicate data at $\mathrm{\Omega }=0$ rad/s between 1 and 2 rad/s, 2 and 3 rad/s, after 4 rad/s measurements. Shear modulus has a minimum around 300 mK and has a plateau as the lowest temperature approaches. (b) $\mathrm{\Omega }$ dependence of the “minimum” and the “plateau” values of $\mu$ [${\mu }_{\mathrm{ave}}\left(300\mathrm{mK}\right)$ and ${\mu }_{\mathrm{ave}}\left(20\mathrm{mK}\right)$]. The lowermost (red) and middle (black) dashed lines are results of the linear least-square fitting. The uppermost (red) line is a parallel shift of the minimum line (indicated by red arrows), thus shows a “supposed” change in $\mu$ at the lowest $T$ if there were no suppression by rotation. Suppression of $\mu$ by rotation is described by blue arrow (see text).

Figure 5

Young's modulus $E$ measured in cooling scans. (a) $E$ measured at $\epsilon =7.35\times {10}^{-9}$ and $3.68\times {10}^{-8}$ without rotation. (b) $E$ measured at $\epsilon =3.68\times {10}^{-8}$ with and without rotation. Data at $\mathrm{\Omega }=4$ rad/s are shifted to coincide at 300 mK.

Figure 6

(a) Schematic illustration of distribution in $L_{\mathrm{N}}$, which corresponds to the abscissa of graph (b). $L_{\mathrm{i}}$ and $R_{\mathrm{c}}$ are also shown. (b) Number density of dislocation lengths $L_{\mathrm{N}}$ and $L_{\mathrm{i}}$, denoted as $N\left(L_{\mathrm{N}}\right)$ and $N\left(L_{\mathrm{i}}\right)$, as a function of $L_{\mathrm{N}}$ and $L_{\mathrm{i}}$ (for temperatures at 50, 100, and 300 mK), respectively. These are anticipated by Eqs. (14) and (15). Measured shear modulus $\mu$ for the 3.6- and 5.4-MPa samples is shown as a function of $L_{\mathrm{c}}$, which is given by Eq. (13). We focus on the location of $\mu$ at $L_{\mathrm{c}}=5\times {10}^{-5}\mathrm{m}$ ($\epsilon =3.96\times {10}^{-8}$). The location is indicated by dashed line. The dislocation segment between nodes with lengths $L_{\mathrm{N}}>L_{\mathrm{c}}$ is depinned from ${}^3\mathrm{He}$ by applied shear stress. This is indicated as orange colored area of $N\left(L_{\mathrm{N}}\right)$. $R_{\mathrm{c}}$ shown by colored (green and turquoise) area and double-headed arrows, which indicate uncertainty (see text), is much smaller than $L_{\mathrm{c}}$. Therefore, the pinning of ${}^3\mathrm{He}$ to dislocation segments at lengths $L_{\mathrm{N}}\le L_{\mathrm{c}}$ will be suppressed, resulting in the decrement in $\mu$.

Figure 7

Schematic view of dislocation network and circular impuriton motion viewed in the direction of rotation axis $z\parallel \mathrm{\Omega }$. As discussed in Sec. 2, edge dislocation lines parallel to $z$ mostly contribute to $\mu$ when shear is applied to $\theta$ direction. For simplicity, four dislocation lines are shown as $\perp$ keeping distance $L_{\mathrm{N}}$. When $R_{\mathrm{c}}\ll L_{\mathrm{N}}$, impuritons do not encounter the dislocations, and sticking such impuritons to dislocation lines in $r$ and $\theta$ directions (not shown) does not contribute significantly to the change in $c_{44}$. Impuritons rotating with $R_{\mathrm{c}}\sim L_{\mathrm{N}}$ will be captured by the $z$ dislocations, so the rotation effect will disappear as molar volume of solid ${}^4\mathrm{He}$ increases.

Figure 8

The average shear modulus $\mu$ with standard deviation as error bar for each $\mathrm{\Omega }$ taken from Fig. 3 in the main paper. Lines are guide to eye.