Global minimum for Thomson's problem of charges on a sphere

Area of Science:

• Physics
• Computational Science
• Materials Science

Background:

• Thomson's problem involves minimizing electrostatic energy of N unit charges on a sphere.
• Finding global energy minima configurations is crucial for understanding charge distributions.

Purpose of the Study:

• To identify likely global energy minima configurations for Thomson's problem at N=306 and N=542.
• To investigate the stability of tetrahedral and dihedral configurations for larger N.
• To contribute to the theoretical understanding of Thomson's problem by analyzing icosadeltahedral configurations.

Main Methods:

• Numerical simulations were employed to determine energy configurations.
• Analysis focused on specific symmetric configurations like tetrahedral (T(h)) and dihedral (D5).
• Comparison with existing theoretical models, including Dodgson and Moore's work, was performed.

Main Results:

• Tetrahedral (T(h)) configuration identified as a likely global minimum for N=306.
• Dihedral (D5) configuration identified as a likely global minimum for N=542.
• Analogues of these configurations for N > 306 and N > 542 were found not to be global minima, supporting defect theory.

Conclusions:

• The study provides the largest known N values (outside icosadeltahedral series) with identified global minima for Thomson's problem.
• Results support the theory that dislocation defects lower lattice strain and energy in symmetric configurations as N increases.
• A comprehensive analysis of icosadeltahedral configurations for N<1000 is presented, aiding refinement of Thomson's problem theories.