Distances in the face-centered cubic crystalline structure applying operational research

Area of Science:

• Crystallography
• Materials Science
• Computational Physics

Background:

• The face-centered cubic (f.c.c.) grid is a fundamental crystal structure found in numerous minerals and materials.
• Understanding atomic-level movement and interactions within f.c.c. lattices is crucial for predicting material properties and phenomena.
• Existing models may not fully capture the complexities of pathfinding in these multi-lattice structures.

Purpose of the Study:

• To define and analyze the concept of minimal paths between any two points within an f.c.c. grid.
• To develop an efficient computational method for determining these shortest paths.
• To establish a mathematical model for spreading phenomena in f.c.c. structured crystals.

Main Methods:

• Modeling the f.c.c. grid as composed of four interlinked cubic lattices.
• Defining two distinct types of steps (type 1 and type 2) between grid points, each with potentially different weights (lengths).
• Developing an explicit algorithm to calculate minimal paths with O(1) computational complexity.

Main Results:

• Explicit mathematical descriptions of minimal paths within the f.c.c. grid have been derived.
• A highly efficient computational method (O(1)) for finding these minimal paths is presented.
• The study confirms that steps within the same type share a uniform weight, while type 1 and type 2 steps may differ.

Conclusions:

• The developed methods provide a computationally inexpensive way to determine shortest paths in f.c.c. structures.
• This research offers a valuable mathematical framework for modeling diffusion and other spreading processes in crystalline materials.
• The findings contribute to a deeper understanding of transport phenomena in face-centered cubic systems.