Ground state of a large number of particles on a frozen topography

Problems consisting in finding the ground state of particles interacting with a given potential constrained to move on a particular geometry are surprisingly difficult. Explicit solutions have been found for small numbers of particles by the use of numerical methods in some particular cases such as particles on a sphere and to a much lesser extent on a torus. In this paper we propose a general solution to the problem in the opposite limit of a very large number of particles $M$ by expressing the energy as an expansion in $M$ whose coefficients can be minimized by a geometrical ansatz. The solution is remarkably universal with respect to the geometry and the interaction potential. Explicit solutions for the sphere and the torus are provided. The paper concludes with several predictions that could be verified by further theoretical or numerical work.

Figure 1

Typical distortions due to geometric constraints on a perfect square lattice. If an additional row of atoms somewhere in the middle of the crystal were added, variations of the lattice constant would become very small.

Figure 2

(Color online) Triangle considered in the argument.

Figure 3

(Color online) $l\left(r\right)-r$ on a sphere for both $q=1$,0; for a comparison, the results for the plane are also shown.

Figure 4

(Color online) Characteristic of grain boundaries of a spherical cap with different aperture angles with and without a disclination at the origin. Squares are sevenfold vertices and spheres are fivefold vertices. This plot corresponds to $L∕a=50$.

Figure 5

(Color online) $l\left(r\right)-r$ on a torus at $r=1.2$ for both $q=1$,0. If the central defect is oriented as shown in Fig. 8 below only three functions are needed for $q=1$ (fivefold). In this case $R_1∕R_2=1.2$ and $\beta =0$.

Figure 6

(Color online) Energy of the spherical cap $(L=50)$ with and without defects. The contribution proportional to the number of dislocations is not shown.

Figure 7

(Color online) Triangular wedges used to solve the $(n,n)$ and the $(n,0)$ configurations. Dashed lines follow the directions of the grain boundaries. $\text{Filled circles represent}+\text{disclinations}$, also marked with D.

Figure 8

(Color online) Representation of the torus, with the definition of the subtended angle. The two circles are geodesics of maximum and minimum curvature. A fivefold defect on the positive curvature of the torus has been shown.

Figure 9

(Color online) $l\left(r\right)-r$ on a torus at $r=1.2$ for both $q=1$. The different functions are shown. $R_1∕R_2=1.2$ and $\beta =0$.