scholarly journals Two and three electrons on a sphere: A generalized Thomson problem

2018 ◽  
Vol 97 (23) ◽  
Author(s):  
Liu Yang ◽  
Zhenwei Yao
Keyword(s):  
Thomson Problem

scholarly journals Nearly uniform sampling of crystal orientations

2018 ◽  
Vol 51 (4) ◽  
pp. 1162-1173 ◽  
Author(s):  
Romain Quey ◽  
Aurélien Villani ◽  
Claire Maurice
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Uniform Distributions ◽  
Orientation Space ◽  
Unit Quaternions ◽  
Spatial Uniformity ◽  
Unit Hypersphere ◽  
Open Source Software Package ◽  
Diffraction Patterns ◽  
Thomson Problem

A method is presented for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry. The method is based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere – a configuration of high spatial uniformity. Orientations are represented as unit quaternions, which lie on a unit hypersphere in four-dimensional space. Expressions of the electrostatic potential energy and Coulomb's forces are derived by working in the tangent space of orientation space. Using the forces, orientations are evolved in a conventional gradient-descent optimization until equilibrium. The method is highly versatile as it can generate uniform distributions for any number of orientations and any symmetry, and even allows one to prescribe some orientations. For large numbers of orientations, the forces can be computed using only the close neighbourhoods of orientations. Even uniform distributions of as many as 106 orientations, such as those required for dictionary-based indexing of diffraction patterns, can be generated in reasonable computation times. The presented algorithms are implemented and distributed in the free (open-source) software package Neper.

scholarly journals Patchy particles by self-assembly of star copolymers on a spherical substrate: Thomson solutions in a geometric problem with a color constraint

Soft Matter ◽  
2019 ◽  
Vol 15 (46) ◽  
pp. 9394-9404
Author(s):  
Tobias M. Hain ◽  
Gerd E. Schröder-Turk ◽  
Jacob J. K. Kirkensgaard
Keyword(s):  
Self Assembly ◽  
Geometric Problem ◽  
Patchy Particles ◽  
Star Copolymers ◽  
Thomson Problem

Star copolymers on a sphere self-assemble into patchy particles with structure and coordination corresponding to those found in the famous Thomson problem.

Hugoniot-type conditions and weak solutions to the phase-field system

1999 ◽  
Vol 10 (1) ◽  
pp. 55-77 ◽  
Author(s):  
V. G. DANILOV ◽  
G. A. OMEL'YANOV ◽  
E. V. RADKEVICH
Keyword(s):  
Phase Field ◽  
Weak Solutions ◽  
Wave Train ◽  
Limit Problem ◽  
Mushy Region ◽  
Field Equations ◽  
Phase Field Equations ◽  
Phase Field System ◽  
Thomson Problem ◽  
Free Interfaces

We consider a new concept of weak solutions to the phase-field equations with a small parameter ε characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ε→0. For the case of a large number M(ε) (M(ε)→∞ as ε→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as ε→0.

scholarly journals Stability of charge inversion, Thomson problem, and application to electrophoresis

2003 ◽  
Vol 67 (3) ◽  
Author(s):  
Michael Patra ◽  
Marco Patriarca ◽  
Mikko Karttunen
Keyword(s):  
Charge Inversion ◽  
Thomson Problem

scholarly journals Two-dimensional packing of soft particles and the soft generalized Thomson problem

Soft Matter ◽  
2011 ◽  
Vol 7 (16) ◽  
pp. 7552 ◽  
Author(s):  
William L. Miller ◽  
Angelo Cacciuto
Keyword(s):  
Two Dimensional ◽  
Soft Particles ◽  
Thomson Problem
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