Figure 1

Dependence of the dispersion relation on the magnetic field strength, which is represented by the parameter ${\omega }_c/{\omega }_0$, where ${\omega }_c$ is the dust-cyclotron frequency and ${\omega }_0$ the dust-plasma frequency defined as ${\omega }_0=\sqrt{Q^2/4\pi {\epsilon }_0{m}_d{a}^3}$, where $a$ is the mean interparticle distance. Top figures (a1), (b1), and (c1) are for ${\omega }_c/{\omega }_0=0.52$ (${\omega }_c/{\omega }_0<1$, a weakly-magnetized plasma crystal); middle figures (a2), (b2), and (c2) for ${\omega }_c/{\omega }_0=2.62$; and bottom figures (a3), (b3), and (c3) for ${\omega }_c/{\omega }_0=6.85$ (${\omega }_c^2/{\omega }_0^2\gg 1$, a strongly-magnetized plasma crystal), where a screening parameter $\kappa =a/{\lambda }_D$ is 1.0. (a1)–(a3): The dispersion relation from theoretical analysis described by Eq. (4) that is normalized by ${\omega }_c$ for $\omega$, and $\pi /a$ for $k$, where ${\omega }_H$ and ${\omega }_L$ label the high-frequency and low-frequency branches, respectively. (b1)–(b3): The dispersion relation obtained from thermal motion of dust particles in a molecular dynamics simulation, where the color represents the ratio of the wave-amplitude components $v_x(k_x,\omega )/v_y(k_x,\omega )$. The purple color represents a dominating transverse mode, while yellow belongs to a dominating longitudinal mode. Red regions show where both wave modes are of the same quantity. (c1)-(c3): The theoretical ratio of the wave modes $P$, where $P_H$ and $P_L$ are labels for the high-frequency and low-frequency branches, respectively. $P=1$ corresponds to a pure longitudinal mode and $P=0$ to a pure transverse mode.Reuse & Permissions

Figure 2

Phase difference (${\varphi }_x-{\varphi }_y$) between $v_x(k_x,\omega )$ (longitudinal mode) and $v_y(k_x,\omega )$ (transverse mode) for ${\omega }_c/{\omega }_0=2.62$, obtained from simulation data. Here, ${\varphi }_x-{\varphi }_y$ is normalized by $\pi /2$, and the negative values (purple and black regions) and the positive values (yellow and red regions) show right-hand and left-hand polarizations, respectively. The solid lines correspond to the dispersion relations that we derived from theory.Reuse & Permissions