Structural properties of anisotropically confined binary Coulomb balls

We investigate the structural properties of a small binary system of dusty plasma subjected to an anisotropic external confinement. We have found that the ground state can form various symmetrical configurations, which generally correspond to variations of a multiple-ring structure. The presence of such structures is given through a detailed phase diagram constructed for the systems with $N=18$ and 24 particles. Furthermore, we show that the configurations of multiple rings can present a persistent behavior for the case in which the number of less-charged particles is equal to a multiple of the number of particles per ring. Finally, we discuss the main characteristics of the transitions of first and second orders since these are related to the main configurational changes of the ground states.

Figure 1

$z$ spatial coordinate of the A (red lines) and B (black lines) types of particles as a function of the charge ratio parameter, $\beta$, for the system with $N=18$, $N_B=14$, and $\alpha =0.4$.

Figure 2

GS configurations for the system with $N=18$, $N_B=14$, $\alpha =0.4$ and several values of the charge ratio parameter, $\beta$, as indicated within figure. The red and black balls represent, respectively, particles of types A and B. Bonds between balls are used only to enhance visualization. The $z$ axis points to the top of the page and is perpendicular to its base.

Figure 3

First derivative of the energy, $E$, with respect to the charge ratio parameter, $\beta$, for the system with $N=18$, $N_B=14$, and $\alpha =0.4$. The left inset shows the discontinuity of the derivative $dE/d\beta$ that occurs at $\beta =0.886$. The inset on the right highlights the unstable behavior of the same derivative during the vitric phase. In this last inset, the black arrow indicates the value of $\beta$ that marks the end of the vitric phase.

Figure 4

The largest critical charge ratio parameter, ${\beta }_c$, occurring outside the vitric phase, as a function of $N_B$, for the system with $N=18$ and $\alpha =0.2$ (black circles), $\alpha =0.4$ (blue squares), and $\alpha =0.7$ (red triangles).

Figure 5

Representation of the lowest energy configurations for the system with $N=18$, $\alpha =0.4$, and different values $N_B$ as indicated within the panels. Configurations belonging to the same column are of equal value of $N_B$, and those located in the first (second) row has an $\alpha$ value larger (smaller) than the value of ${\beta }_c$. Particles of types A and B are represented by the spheres in red and black, respectively, while the connections drawn between them are only used to enhance visualization.

Figure 6

Examples of the five possible kinds of symmetric ground states. (a) Multiple rings of type B, (b) multiple rings of type A, (c) multiple rings of the type AB, (d) multiple rings with external defect, and (e) multiple rings with internal defects. Configurations within the same column are of the same kind. Particles of types A and B are, respectively, represented by the spheres in red and black, while the connections drawn between the spheres are only used to enhance visualization.

Figure 7

Phase diagram for the system with $N=18$ particles containing the five types of symmetric configurations found for the ground state, i.e., multiple rings of type A (red full lines), multiple rings of type B (black full lines), multiple rings of the type AB (solid blue lines), multiple rings with internal defects (dashed blue lines), and multiple rings with external defects (dashed black lines). The vitric phase is represented by black dotted lines. The nomenclature adjacent lines only refers to the ordering of the particles making up the structures without defects. From top to bottom panels refer to $\alpha =0.2$, $0.3$, $0.4$, $0.5$, and $0.7$.

Figure 8

(a) Eigenfrequencies, [(b) and (c)] first derivative of the energy with respect to $\beta$, and (d) second derivative of energy with respect to $\beta$ for the system with $N=18$, $N_B=9$, and $\alpha =0.7$. All the above quantities are given as a function of the parameter $\beta$.

Figure 9

Representations of the lowest energy configurations for the systems with (a) $\beta =0.497$, (b) $\beta =0.350$, and (c) $\beta =0.200$. All configurations are for the same set of parameters: $N=18$, $N_B=9$, and $\alpha =0.7$. Particles of types A and B are represented by the spheres in colors red and black, respectively, while the connections drawn between them are only used to enhance visualization.

Figure 10

Representations of the lowest energy configurations for the systems with (a) $\beta =0.400$ and (b) $\beta =0.250$. All configurations are for the same set of parameters: $N=18$, $N_B=10$, and $\alpha =0.4$. This figure follows the same pattern of symbols and colors used in Fig. 8.

Figure 11

Graphics of the eigenfrequency (a) and the first derivative of the energy with respect to $\beta$ (b) for the system with $N=18$, $N_B=10$, and $\alpha =0.4$. In all the above cases such amounts are given as a function of the parameter $\beta$.

Figure 12

The largest critical charge ratio parameter, ${\beta }_c$, occurring outside the vitric phase, as a function of $N_B$, for the system with $N=24$ and $\alpha =0.15$ (black circles), $\alpha =0.40$, (blue squares), and $\alpha =0.90$ (red triangles).

Figure 13

Representation of the lowest energy configurations for the system with $N=24$ and different values of $\alpha$ and $N_B$ as indicated within the panels. Configurations belonging to the same column are of equal values of $\alpha$ and $N_B$, and those located in the first (second) row has an $\beta$ value larger (smaller) than the value of ${\beta }_c$. Particles of types A and B are represented by the spheres in colors red and black, respectively, while the connections drawn between them are only used to enhance visualization.

Figure 14

Phase diagram for the system with $N=24$ particles containing the five types of symmetric configurations found for the ground state, i.e., multiple rings of type A (red full lines), multiple rings of type B (black full lines), multiple rings of type AB (solid blue lines), multiple rings with internal defects (dashed blue lines), and multiple rings with external defects (dashed black lines). The vitric phase is represented by black dotted lines. The nomenclature adjacent lines only refers to the ordering of the particles making up the structures without defects. From top to bottom panels refer to $\alpha =0.2$, $0.4$, and $0.7$.

Figure 15

Representations of the lowest energy configurations for the systems with (a) $\beta =0.40$, (b) $\beta =0.30$, and (c) $\beta =0.15$. All configurations are for the same set of parameters: $N=24$, $N_B=10$, and $\alpha =0.4$. Particles of types A and B are represented by the spheres in colors red and black, respectively, while the connections drawn between them are only used to enhance visualization.

Figure 16

(a) Eigenfrequencies, [(b) and (c)] first derivative of the energy with respect to $\beta$, and (d) second derivative of energy with respect to $\beta$ for the system with $N=24$, $N_B=10$, and $\alpha =0.4$. All the above quantities are given as a function of the parameter $\beta$.