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

Gergely Stomfai1, Gergely Kovács2, Benedek Nagy3

  • 1Peterhouse, University of Cambridge, Cambridge, United Kingdom.

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Summary

This study analyzes minimal paths on face-centered cubic (f.c.c.) grids, modeling crystal structures. Researchers provide explicit, computationally efficient O(1) methods for finding these shortest paths.

Keywords:
Gomory methodchamfer distancesf.c.c. latticeshortest pathsimplex method

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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.