Superfluid field response to edge dislocation motion

We study the dynamic response of a superfluid field to a moving edge dislocation line to which the field is minimally coupled. We use a dissipative Gross-Pitaevskii equation, and determine the initial conditions by solving the equilibrium version of the model. We consider the subsequent time evolution of the field for both glide and climb dislocation motion and analyze the results for a range of values of the constant speed $V_D$ of the moving dislocation. We find that the type of motion of the dislocation line is very important in determining the time evolution of the superfluid field distribution associated with it. Climb motion of the dislocation line induces increasing asymmetry, as function of time, in the field profile, with part of the probability being, as it were, left behind. On the other hand, glide motion has no effect on the symmetry properties of the superfluid field distribution. Damping of the superfluid field due to excitations associated with the moving dislocation line occurs in both cases.

Figure 1

Top panel: An edge dislocation is illustrated executing climb motion. The arrow indicates the direction of climb. The bottom panel illustrates the glide motion of an edge dislocation.

Figure 2

The absolute value of the equilibrium $|\overline{\psi }(\overline{x},\overline{y})|$ of the superfluid field. A broad maximum is seen in the attractive region of the dislocation strain potential. The two plots correspond to different visual orientations (see text). Results were obtained by solving Eq. (4).

Figure 3

Plot of the absolute value of the wave function $|\overline{\psi }(\overline{x}=0,\overline{y};\overline{t})|$ at different times for a climbing edge dislocation line. The top panel corresponds to $V_D=5\times {10}^{-4}$ and the bottom panel to $V_D=1.5\times {10}^{-3}$. The times are indicated in the legend.

Figure 4

The asymmetry parameter $(B-B_0)/B_0$ (see text) during climb motion is shown as a function of time $\overline{t}$ in the main plot for three different values of $V_D$ as indicated in the legend. The inset illustrates the procedure employed to extract $B$ as defined in Eq. (10). The value for $B_0=0.538$.

Figure 5

In the top panel, the absolute value of the wave function $|\overline{\psi }(\overline{x},\overline{y}={\overline{y}}_{\text{max}};\overline{t})|$ is shown for $\overline{t}=0$, 2196, and 6590 during glide. $V_D=5\times {10}^{-4}$ is used. In the bottom panel, the equilibrium $|\overline{\psi }(\overline{x}={\overline{x}}_{\text{max}},\overline{y};\overline{t}=0)|$ is offset by $\overline{x}=3.308$ along the positive $x$ direction in order to compare it to $|\overline{\psi }(\overline{x},\overline{y}={\overline{y}}_{\text{max}};\overline{t}=6590)|$.

Figure 6

Plot of $|\overline{\psi }(\overline{x}={\overline{x}}_{\text{max}},\overline{y};\overline{t})|$ at $\overline{t}=0$ and $\overline{t}=6590$ during glide for $V_D=5\times {10}^{-4}$. No change in the shape $|\overline{\psi }|$ along $\overline{x}={\overline{x}}_{\text{max}}$ is observed.

Figure 7

Plots of the total normalization $\mathcal{N}$ vs $\overline{t}$. In the top panel, three different values of $\gamma =0, {10}^{-3}$, and ${10}^{-1}$ are used for both climb and glide motion at a fixed velocity of $V_D=5\times {10}^{-4}$. In the bottom panel the change in $\mathcal{N}$ at varying velocities $V_D=0, 5\times {10}^{-4}$, and $1.5\times {10}^{-3}$ for climb at two different values of $\gamma ={10}^{-3}$ and ${10}^{-2}$ is shown.