Wave modes in shear-deformed two-dimensional plasma crystals

A theory of wave modes in shear-deformed two-dimensional plasma crystals is presented. Modification of the dispersion relations upon the pure and simple shear, and the resulting effect on the onset of the mode-coupling instability, are studied. In particular, it is explained why the velocity fluctuation spectra measured in experiments with sheared crystals exhibit asymmetric “hot spots”: It is shown that the coupling of the in-plane compressional and the out-of-plane modes, leading to the formation of an unstable hybrid mode and generation of the hot spots, is enhanced in a certain direction determined by deformation.

Figure 1

Sheared hexagonal lattice cells (left) and the corresponding reciprocal-lattice cells in $\mathbf{k}$ space (right). The upper row (a,b) and lower row (c,d) show the cases of pure and simple shear, respectively. Upon pure shear, the crystalline lattice is contracted (by the factor $\mu =0.924$) along the $x$ axis (a), so for the reciprocal lattice the contraction is in the $k_y$ direction (b). Upon simple shear, the strain (of $\tau =0.08$) is in the $y$ direction (c), so in the reciprocal space the strain is along the $k_x$ axis (d). The basis vectors ${\mathbf{b}}_{1,2}$ of the reciprocal lattice are shown; the gray regions within the reciprocal cells are the first Brillouin zones.

Figure 2

Example of a sheared 2D plasma crystal observed in experiments. (a) The hexagonal lattice is subject to pure shear, resulting in the contraction with $\mu =0.924$ along one of the principal lattice axes [as in Fig. 1]. The thin solid lines drawn along the lattice axes are a guide for the eye. (b) The velocity fluctuation spectrum integrated over frequency (in the range $18.5<f<21.5$ Hz) demonstrates the asymmetric mode coupling—generation of the hot spots in the direction of contraction (instead of the regular patterns with a period of ${60}^{\circ }$). The hot spots emerge near the border of the first Brillouin zone, obtained from the lattice structure factor (indicated by the dashed line).

Figure 3

Modification of the wave modes in a hexagonal lattice under pure shear. The upper row (a,b) is for the contraction with $\mu =0.924$ (corresponding to the experiment in Fig. 2), while the lower row (c,d) demonstrates the effect of extension with $\mu =1/0.924=1.08$. The solid lines show the dispersion relations in the sheared crystal, and the dashed lines are for the regular (unsheared) hexagonal lattice. The upper and lower acoustic branches are the in-plane compressional and shear modes, respectively, and the optical branch is the out-of-plane mode. The left and right panels depict the dispersion relations for $\theta =0^{\circ }$ and ${90}^{\circ }$, respectively [see Fig. 1].

Figure 4

Onset of the asymmetric wake-induced mode coupling under pure shear $\left(\mu =0.924\right)$. (a) The dispersion relations for $\theta =0^{\circ }$ (solid lines) and $\theta ={60}^{\circ }$ (dashed lines) are different due to the shear-induced asymmetry. Therefore, the in-plane compressional (acoustic) and out-of-plane (optical) modes first intersect in the direction of contraction, forming an unstable hybrid mode with a nonzero imaginary part [the dotted line shows the growth rate $\mathrm{Im}\omega \left(k\right)$ multiplied by 10]. (b) The contour plot of the growth rate for the hybrid mode, emerging near the border of the first Brillouin zone (indicated by the dashed line) and representing the hot spots observed in the experiment [Fig. 2].

Figure 5

Anisotropy of the long-wavelength wave modes upon the pure shear. Shown are the squared velocities $C^2$, determining the dispersion relations of the acoustic shear (a) and compressional (b) wave modes as well as of the optical out-of-plane mode (c) in the limit $k\to 0$, plotted vs the angle $\theta$. The solid and dashed lines illustrate the contraction $\left(\mu =0.924\right)$ and extension $\left(\mu =1/0.924=1.08\right)$, respectively, and the thin horizontal lines are for a regular lattice $\left(\mu =1\right)$. The lower panel (d) presents the dependence of the anisotropy amplitude $\mathrm{\Delta }{C}^2$ on $\mu$.

Figure 6

Wake-induced coupling of the in-plane shear mode, modified by simple shear with the strain $\tau =0.08$. The branches for $\theta =0^{\circ }$ (a) and $\theta ={90}^{\circ }$ (b) are plotted; the solid lines show the dispersion relations in the sheared crystal, and the dashed lines are for the regular (unsheared) lattice. The out-of-plane (optical) mode crosses both the compressional (upper acoustic) and shear (lower acoustic) modes, and the dotted lines show the imaginary part of the resulting hybrid modes, $\mathrm{Im}\omega \left(k\right)$, multiplied by 10.

Figure 7

Same as in Fig. 5, plotted for the simple shear vs the strain $\tau$. The solid and dashed lines in (a)–(c) illustrate the positive $\left(\tau =0.08\right)$ and negative $\left(\tau =-0.08\right)$ strain, respectively.