There are three methods for initialization of position: random, Gaussian, or face-centered cubic (fcc) lattice structure.
When the ``random'' position initialization method is chosen, particles' x-, y-, and z-positions are randomly distributed within a cube (centered on the origin) whose dimensions are set by the user. Caution: this initialization method does not check to see if repelling particles are unrealistically close to each other, so it is possible that enormous repulsive forces will arise from initialization in an unlikely configuration of high potential energy. However, the ``random'' form of initialization is usually satisfactory, given a moderately-sized region that does not require unrealistically high particle density. If necessary, damping (Sec. ) can be applied to extract excess energy and stabilize the system.
The ``Gaussian'' position initialization method uses the Box-Muller algorithm to assign particle positions such that the x-, y-, and z-coordinate distributions correspond to Gaussian statistics. The Box-Muller algorithm generates normal (Gaussian) distributions for uniform (equal probability over a finite range) distributions according to the transformations:
where the distribution of each y is described by:
and , are uniform deviates, i.e.:
Note: As with the ``random'' initialization, the ``Gaussian'' initialization method does not check to see that position assignments describe a realistic (in terms of particle proximity) configuration.
When the ``lattice (fcc)'' option is selected, the ions will be initialized in a face-centered cubic lattice structure. The geometry of a face-centered cubic lattice is described by a cubic lattice with particles at all vertices and in the center of each cube face. To generate such a lattice, one must first construct a unit cell of four atoms, with one atom at the origin and one atom centered on each of the faces describing the positive quadrant (x>0, y>0, z>0) of space. The lattice is easy to create once the unit cell is constructed: simply displace the unit cell by a vector, , whose components are described by integral multiples of the lattice distance, a. The lattice distance, a, is the distance between two consecutive (non-diagonal) vertices of the lattice.
TrapApp provides two options for assigning the lattice distance, a: