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Thomson problem

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The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904[1] after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.

Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

Mathematical statement

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The electrostatic interaction energy occurring between each pair of electrons of equal charges (, with the elementary charge of an electron) is given by Coulomb's law,

where is the electric constant and is the distance between each pair of electrons located at points on the sphere defined by vectors and , respectively.

Simplified units of and (the Coulomb constant) are used without loss of generality. Then,

The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interaction energies

The global minimization of over all possible configurations of N distinct points is typically found by numerical minimization algorithms.

Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".[2] The main difference is that in Smale's problem the function to minimise is not the electrostatic potential but a logarithmic potential given by A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N.

Example

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The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, , or

Known exact solutions

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Schematic geometric solutions of the mathematical Thomson Problem for up to N = 5 electrons.

Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases.

  • For N = 1, the solution is trivial. The single electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge.
  • For N = 2, the optimal configuration consists of electrons at antipodal points. This represents the first one-dimensional solution.
  • For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.[3] The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. Also, this represents the first two-dimensional solution.
  • For N = 4, electrons reside at the vertices of a regular tetrahedron. Of interest, this represents the first three-dimensional solution.
  • For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.[4] Of interest, it is impossible for any N solution with five or more electrons to exhibit global equidistance among all pairs of electrons.
  • For N = 6, electrons reside at vertices of a regular octahedron.[5] The configuration may be imagined as four electrons residing at the corners of a square about the equator and the remaining two residing at the poles.
  • For N = 12, electrons reside at the vertices of a regular icosahedron.[6]

Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Numerical solutions for N = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, the cube and dodecahedron respectively.[7]

Generalizations

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One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional

Traditionally, one considers also known as Riesz -kernels. For integrable Riesz kernels see the 1972 work of Landkof.[8] For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.[9] Notable cases include:[10]

  • α = ∞, the Tammes problem (packing);
  • α = 1, the Thomson problem;
  • α = 0, to maximize the product of distances, latterly known as Whyte's problem;
  • α = −1 : maximum average distance problem.

One may also consider configurations of N points on a sphere of higher dimension. See spherical design.

Solution algorithms

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Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:[10]

  • constrained global optimization (Altschuler et al. 1994),
  • steepest descent (Claxton and Benson 1966, Erber and Hockney 1991),
  • random walk (Weinrach et al. 1990),
  • genetic algorithm (Morris et al. 1996)

While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest.

Continuous spherical shell charge

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The extreme upper energy limit of the Thomson Problem is given by for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Significantly lower energy of a given N-electron solution of the Thomson Problem with one charge at its origin is readily obtained by , where are solutions of the Thomson Problem.

The energy of a continuous spherical shell of charge distributed across its surface is given by

and is, in general, greater than the energy of every Thomson problem solution. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. For example, a spherical shell of represents the uniform distribution of a single electron's charge, , across the entire shell.

Randomly distributed point charges

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The expected global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by

and is, in general, greater than the energy of every Thomson problem solution.

Here, N is a discrete variable that counts the number of electrons in the system. As well, .

Charge-centered distribution

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For every Nth solution of the Thomson problem there is an th configuration that includes an electron at the origin of the sphere whose energy is simply the addition of N to the energy of the Nth solution. That is,[11]

Thus, if is known exactly, then is known exactly.

In general, is greater than , but is remarkably closer to each th Thomson solution than and . Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.

Relations to other scientific problems

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The Thomson problem is a natural consequence of J. J. Thomson's plum pudding model in the absence of its uniform positive background charge.[12]

"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."

—Sir J. J. Thomson[13]

Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements.[14]

The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.

The generalized Thomson problem arises, for example, in determining arrangements of protein subunits that comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory. An example with long-range logarithmic interactions is provided by Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.

Configurations of smallest known energy

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In the following table[citation needed] is the number of points (charges) in a configuration, is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, is the number of vertices where the given number of edges meet, is the total number of edges, is the number of triangular faces, is the number of quadrilateral faces, and is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal. Thus, except in the cases N = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column.[15]

N Symmetry Equivalent polyhedron
2 0.500000000 0 1 180.000° digon
3 1.732050808 0 3 2 120.000° triangle
4 3.674234614 0 4 0 0 0 0 0 6 4 0 109.471° tetrahedron
5 6.474691495 0 2 3 0 0 0 0 9 6 0 90.000° triangular dipyramid
6 9.985281374 0 0 6 0 0 0 0 12 8 0 90.000° octahedron
7 14.452977414 0 0 5 2 0 0 0 15 10 0 72.000° pentagonal dipyramid
8 19.675287861 0 0 8 0 0 0 0 16 8 2 71.694° square antiprism
9 25.759986531 0 0 3 6 0 0 0 21 14 0 69.190° triaugmented triangular prism
10 32.716949460 0 0 2 8 0 0 0 24 16 0 64.996° gyroelongated square dipyramid
11 40.596450510 0.013219635 0 2 8 1 0 0 27 18 0 58.540° edge-contracted icosahedron
12 49.165253058 0 0 0 12 0 0 0 30 20 0 63.435° icosahedron
(geodesic sphere {3,5+}1,0)
13 58.853230612 0.008820367 0 1 10 2 0 0 33 22 0 52.317°
14 69.306363297 0 0 0 12 2 0 0 36 24 0 52.866° gyroelongated hexagonal dipyramid
15 80.670244114 0 0 0 12 3 0 0 39 26 0 49.225°
16 92.911655302 0 0 0 12 4 0 0 42 28 0 48.936° tetrahedrally diminished dodecahedron
17 106.050404829 0 0 0 12 5 0 0 45 30 0 50.108° double-gyroelongated pentagonal dipyramid
18 120.084467447 0 0 2 8 8 0 0 48 32 0 47.534°
19 135.089467557 0.000135163 0 0 14 5 0 0 50 32 1 44.910°
20 150.881568334 0 0 0 12 8 0 0 54 36 0 46.093°
21 167.641622399 0.001406124 0 1 10 10 0 0 57 38 0 44.321°
22 185.287536149 0 0 0 12 10 0 0 60 40 0 43.302°
23 203.930190663 0 0 0 12 11 0 0 63 42 0 41.481°
24 223.347074052 0 0 0 24 0 0 0 60 32 6 42.065° snub cube
25 243.812760299 0.001021305 0 0 14 11 0 0 68 44 1 39.610°
26 265.133326317 0.001919065 0 0 12 14 0 0 72 48 0 38.842°
27 287.302615033 0 0 0 12 15 0 0 75 50 0 39.940°
28 310.491542358 0 0 0 12 16 0 0 78 52 0 37.824°
29 334.634439920 0 0 0 12 17 0 0 81 54 0 36.391°
30 359.603945904 0 0 0 12 18 0 0 84 56 0 36.942°
31 385.530838063 0.003204712 0 0 12 19 0 0 87 58 0 36.373°
32 412.261274651 0 0 0 12 20 0 0 90 60 0 37.377° pentakis dodecahedron
(geodesic sphere {3,5+}1,1)
33 440.204057448 0.004356481 0 0 15 17 1 0 92 60 1 33.700°
34 468.904853281 0 0 0 12 22 0 0 96 64 0 33.273°
35 498.569872491 0.000419208 0 0 12 23 0 0 99 66 0 33.100°
36 529.122408375 0 0 0 12 24 0 0 102 68 0 33.229°
37 560.618887731 0 0 0 12 25 0 0 105 70 0 32.332°
38 593.038503566 0 0 0 12 26 0 0 108 72 0 33.236°
39 626.389009017 0 0 0 12 27 0 0 111 74 0 32.053°
40 660.675278835 0 0 0 12 28 0 0 114 76 0 31.916°
41 695.916744342 0 0 0 12 29 0 0 117 78 0 31.528°
42 732.078107544 0 0 0 12 30 0 0 120 80 0 31.245°
43 769.190846459 0.000399668 0 0 12 31 0 0 123 82 0 30.867°
44 807.174263085 0 0 0 24 20 0 0 120 72 6 31.258°
45 846.188401061 0 0 0 12 33 0 0 129 86 0 30.207°
46 886.167113639 0 0 0 12 34 0 0 132 88 0 29.790°
47 927.059270680 0.002482914 0 0 14 33 0 0 134 88 1 28.787°
48 968.713455344 0 0 0 24 24 0 0 132 80 6 29.690°
49 1011.557182654 0.001529341 0 0 12 37 0 0 141 94 0 28.387°
50 1055.182314726 0 0 0 12 38 0 0 144 96 0 29.231°
51 1099.819290319 0 0 0 12 39 0 0 147 98 0 28.165°
52 1145.418964319 0.000457327 0 0 12 40 0 0 150 100 0 27.670°
53 1191.922290416 0.000278469 0 0 18 35 0 0 150 96 3 27.137°
54 1239.361474729 0.000137870 0 0 12 42 0 0 156 104 0 27.030°
55 1287.772720783 0.000391696 0 0 12 43 0 0 159 106 0 26.615°
56 1337.094945276 0 0 0 12 44 0 0 162 108 0 26.683°
57 1387.383229253 0 0 0 12 45 0 0 165 110 0 26.702°
58 1438.618250640 0 0 0 12 46 0 0 168 112 0 26.155°
59 1490.773335279 0.000154286 0 0 14 43 2 0 171 114 0 26.170°
60 1543.830400976 0 0 0 12 48 0 0 174 116 0 25.958°
61 1597.941830199 0.001091717 0 0 12 49 0 0 177 118 0 25.392°
62 1652.909409898 0 0 0 12 50 0 0 180 120 0 25.880°
63 1708.879681503 0 0 0 12 51 0 0 183 122 0 25.257°
64 1765.802577927 0 0 0 12 52 0 0 186 124 0 24.920°
65 1823.667960264 0.000399515 0 0 12 53 0 0 189 126 0 24.527°
66 1882.441525304 0.000776245 0 0 12 54 0 0 192 128 0 24.765°
67 1942.122700406 0 0 0 12 55 0 0 195 130 0 24.727°
68 2002.874701749 0 0 0 12 56 0 0 198 132 0 24.433°
69 2064.533483235 0 0 0 12 57 0 0 201 134 0 24.137°
70 2127.100901551 0 0 0 12 50 0 0 200 128 4 24.291°
71 2190.649906425 0.001256769 0 0 14 55 2 0 207 138 0 23.803°
72 2255.001190975 0 0 0 12 60 0 0 210 140 0 24.492° geodesic sphere {3,5+}2,1
73 2320.633883745 0.001572959 0 0 12 61 0 0 213 142 0 22.810°
74 2387.072981838 0.000641539 0 0 12 62 0 0 216 144 0 22.966°
75 2454.369689040 0 0 0 12 63 0 0 219 146 0 22.736°
76 2522.674871841 0.000943474 0 0 12 64 0 0 222 148 0 22.886°
77 2591.850152354 0 0 0 12 65 0 0 225 150 0 23.286°
78 2662.046474566 0 0 0 12 66 0 0 228 152 0 23.426°
79 2733.248357479 0.000702921 0 0 12 63 1 0 230 152 1 22.636°
80 2805.355875981 0 0 0 16 64 0 0 232 152 2 22.778°
81 2878.522829664 0.000194289 0 0 12 69 0 0 237 158 0 21.892°
82 2952.569675286 0 0 0 12 70 0 0 240 160 0 22.206°
83 3027.528488921 0.000339815 0 0 14 67 2 0 243 162 0 21.646°
84 3103.465124431 0.000401973 0 0 12 72 0 0 246 164 0 21.513°
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86 3258.211605713 0.001378932 0 0 12 74 0 0 252 168 0 21.522°
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88 3416.720196758 0 0 0 12 76 0 0 258 172 0 21.486°
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90 3579.091222723 0 0 0 12 78 0 0 264 176 0 21.230°
91 3661.713699320 0.000033221 0 0 12 79 0 0 267 178 0 21.105°
92 3745.291636241 0 0 0 12 80 0 0 270 180 0 21.026°
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97 4177.533599622 0.000096437 0 0 12 85 0 0 285 190 0 20.450°
98 4266.822464156 0.000112916 0 0 12 86 0 0 288 192 0 20.422°
99 4357.139163132 0.000156508 0 0 12 87 0 0 291 194 0 20.284°
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102 4633.736565899 0 0 0 12 90 0 0 300 200 0 20.040°
103 4727.836616833 0.000201245 0 0 12 91 0 0 303 202 0 19.907°
104 4822.876522746 0 0 0 12 92 0 0 306 204 0 19.957°
105 4919.000637616 0 0 0 12 93 0 0 309 206 0 19.842°
106 5015.984595705 0 0 0 12 94 0 0 312 208 0 19.658°
107 5113.953547724 0.000064137 0 0 12 95 0 0 315 210 0 19.327°
108 5212.813507831 0.000432525 0 0 12 96 0 0 318 212 0 19.327°
109 5312.735079920 0.000647299 0 0 14 93 2 0 321 214 0 19.103°
110 5413.549294192 0 0 0 12 98 0 0 324 216 0 19.476°
111 5515.293214587 0 0 0 12 99 0 0 327 218 0 19.255°
112 5618.044882327 0 0 0 12 100 0 0 330 220 0 19.351°
113 5721.824978027 0 0 0 12 101 0 0 333 222 0 18.978°
114 5826.521572163 0.000149772 0 0 12 102 0 0 336 224 0 18.836°
115 5932.181285777 0.000049972 0 0 12 103 0 0 339 226 0 18.458°
116 6038.815593579 0.000259726 0 0 12 104 0 0 342 228 0 18.386°
117 6146.342446579 0.000127609 0 0 12 105 0 0 345 230 0 18.566°
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119 6364.347317479 0.000685590 0 0 12 107 0 0 351 234 0 18.336°
120 6474.756324980 0.001373062 0 0 12 108 0 0 354 236 0 18.418°
121 6586.121949584 0.000838863 0 0 12 109 0 0 357 238 0 18.199°
122 6698.374499261 0 0 0 12 110 0 0 360 240 0 18.612° geodesic sphere {3,5+}2,2
123 6811.827228174 0.001939754 0 0 14 107 2 0 363 242 0 17.840°
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126 7157.669224867 0 0 2 16 100 8 0 372 248 0 17.920°
127 7274.819504675 0 0 0 12 115 0 0 375 250 0 17.877°
128 7393.007443068 0.000054132 0 0 12 116 0 0 378 252 0 17.814°
129 7512.107319268 0.000030099 0 0 12 117 0 0 381 254 0 17.743°
130 7632.167378912 0.000025622 0 0 12 118 0 0 384 256 0 17.683°
131 7753.205166941 0.000305133 0 0 12 119 0 0 387 258 0 17.511°
132 7875.045342797 0 0 0 12 120 0 0 390 260 0 17.958° geodesic sphere {3,5+}3,1
133 7998.179212898 0.000591438 0 0 12 121 0 0 393 262 0 17.133°
134 8122.089721194 0.000470268 0 0 12 122 0 0 396 264 0 17.214°
135 8246.909486992 0 0 0 12 123 0 0 399 266 0 17.431°
136 8372.743302539 0 0 0 12 124 0 0 402 268 0 17.485°
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138 8627.406389880 0.000473576 0 0 12 126 0 0 408 272 0 16.924°
139 8756.227056057 0.000404228 0 0 12 127 0 0 411 274 0 16.673°
140 8885.980609041 0.000630351 0 0 13 126 1 0 414 276 0 16.773°
141 9016.615349190 0.000376365 0 0 14 126 0 1 417 278 0 16.962°
142 9148.271579993 0.000550138 0 0 12 130 0 0 420 280 0 16.840°
143 9280.839851192 0.000255449 0 0 12 131 0 0 423 282 0 16.782°
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145 9548.928837232 0.000094938 0 0 12 133 0 0 429 286 0 16.841°
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147 9820.932378373 0.000636651 0 0 12 135 0 0 435 290 0 16.458°
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149 10096.859907397 0.000638186 0 0 14 133 2 0 441 294 0 16.344°
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159 11534.023960956 0.000381457 0 0 12 147 0 0 471 314 0 15.960°
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179 14698.754594220 0.000125113 0 0 13 165 1 0 531 354 0 14.968°
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192 16963.338386460 0 0 0 12 180 0 0 570 380 0 14.819° geodesic sphere {3,5+}3,2
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198 18064.262177195 0.000011149 0 0 12 186 0 0 588 392 0 14.237°
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202 18817.204718262 0 0 0 12 190 0 0 600 400 0 14.189°
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212 20768.053085964 0 0 0 12 200 0 0 630 420 0 14.118° geodesic sphere {3,5+}4,1
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282 37147.294418462 0 0 0 12 270 0 0 840 560 0 12.166° geodesic sphere {3,5+}4,2
292 39877.008012909 0 0 0 12 280 0 0 870 580 0 11.857°
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358 60341.830924588 0 0 0 12 346 0 0 1068 712 0 10.647°
372 65230.027122557 0 0 0 12 360 0 0 1110 740 0 10.531° geodesic sphere {3,5+}4,3
382 68839.426839215 0 0 0 12 370 0 0 1140 760 0 10.379°
390 71797.035335953 0 0 0 12 378 0 0 1164 776 0 10.222°
392 72546.258370889 0 0 0 12 380 0 0 1170 780 0 10.278°
400 75582.448512213 0 0 0 12 388 0 0 1194 796 0 10.068°
402 76351.192432673 0 0 0 12 390 0 0 1200 800 0 10.099°
432 88353.709681956 0 0 0 24 396 12 0 1290 860 0 9.556°
448 95115.546986209 0 0 0 24 412 12 0 1338 892 0 9.322°
460 100351.763108673 0 0 0 24 424 12 0 1374 916 0 9.297°
468 103920.871715127 0 0 0 24 432 12 0 1398 932 0 9.120°
470 104822.886324279 0 0 0 24 434 12 0 1404 936 0 9.059°

According to a conjecture, if is the polyhedron formed by the convex hull of the solution configuation to the Thomson Problem for electrons and is the number of quadrilateral faces of , then has edges.[16][clarification needed]

References

[edit]
  1. ^ Thomson, Joseph John (March 1904). "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure" (PDF). Philosophical Magazine. Series 6. 7 (39): 237–265. doi:10.1080/14786440409463107. Archived from the original (PDF) on 13 December 2013.
  2. ^ Smale, S. (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291. S2CID 1331144.
  3. ^ Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. 141 (141): 251–301. doi:10.1515/crll.1912.141.251. S2CID 120309200..
  4. ^ Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv:1001.3702 [math.MG].
  5. ^ Yudin, V.A. (1992). "The minimum of potential energy of a system of point charges". Discretnaya Matematika. 4 (2): 115–121 (in Russian).; Yudin, V. A. (1993). "The minimum of potential energy of a system of point charges". Discrete Math. Appl. 3 (1): 75–81. doi:10.1515/dma.1993.3.1.75. S2CID 117117450.
  6. ^ Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462. MR1426716, Zbl 0877.51021
  7. ^ Atiyah, Michael; Sutcliffe, Paul (2003). "Polyhedra in physics, chemistry and geometry". arXiv:math-ph/0303071.
  8. ^ Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
  9. ^ Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
  10. ^ a b Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle" (PDF). IMFM/TCS. Archived from the original (PDF) on 25 June 2018.
  11. ^ LaFave Jr, Tim (February 2014). "Discrete transformations in the Thomson Problem". Journal of Electrostatics. 72 (1): 39–43. arXiv:1403.2592. doi:10.1016/j.elstat.2013.11.007. S2CID 119309183.
  12. ^ Levin, Y.; Arenzon, J. J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63 (3): 415. arXiv:cond-mat/0302524. Bibcode:2003EL.....63..415L. doi:10.1209/epl/i2003-00546-1. S2CID 18929981.
  13. ^ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
  14. ^ LaFave Jr, Tim (2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics. 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001. S2CID 118480104.
  15. ^ Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
  16. ^ "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.

Notes

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