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Thomson problem

The Thomson problem is that of determining the minimum (ground state) energy configuration of N classical electrons on the surface of the 2-sphere $S 2$ . The electrons repel each other with a force given by Coulomb's law. This problem is named for J. J. Thomson, who posed it in 1904 as part of the development of the plum pudding model of the atom. In this model, the electrons formed spherical shells. This one hundred year old puzzle (Thomson wrote "I have not as yet succeeded in getting a general solution") has many important physical realizations including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps. The Thomson problem is solved exactly for a few particles on the sphere and the energy per particle is known for a large number of particles on the surface of the sphere. Numerical results provide the best known solutions for a wide range of particle numbers. The configurations that are found have a great variety of geometrical structure.

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One can also ask for ground states of particles interacting with arbitrary potentials: this is the generalized Thomson problem. The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which 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 the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

The Thomson problem is of outstanding mathematical interest not only for the asymptotics (large N behavior) of the growth of minimum energy configurations but also for the characteristics of the minimum energy configurations themselves.

Minimum Energy Configuration for Coulomb Potential

N	$E 1$	Symmetry	$\left\| \sum \mathbf{r}_{i} \right\|$	$v 3$	$v 4$	$v 5$	$v 6$	$v 7$	$v 8$	$e$	$f 3$	$f 4$	$θ 1$	Equivalent Polyhedron
2	0.500000000	$D_{\infty h}$	0	-	-	-	-	-	-	-	-	-	180.000°
3	1.732050808	$D 3 h$	0	-	-	-	-	-	-	-	-	-	120.000°
4	3.674234614	$T d$	0	4	0	0	0	0	0	6	4	0	109.471°	tetrahedron
5	6.474691495	$D 3 h$	0	2	3	0	0	0	0	9	6	0	90.000°	triangular dipyramid
6	9.985281374	$O h$	0	0	6	0	0	0	0	12	8	0	90.000°	octahedron
7	14.452997414	$D 5 h$	0	0	5	2	0	0	0	15	10	0	72.000°	pentagonal dipyramid
8	19.675287861	$D 4 d$	0	0	8	0	0	0	0	16	8	2	71.694°	square antiprism
9	25.759986531	$D 3 h$	0	0	3	6	0	0	0	21	14	0	61.190°	triaugmented triangular prism
10	32.716949460	$D 4 d$	0	0	2	8	0	0	0	24	16	0	64.996°	gyroelongated square dipyramid
11	40.596450510	$C 2 v$	0.013219635	0	2	8	1	0	0	27	18	0	58.540°
12	49.165253058	$I h$	0	0	0	12	0	0	0	30	20	0	63.435°	icosahedron
13	58.853230612	$C 2 v$	0.008820367	0	1	10	2	0	0	33	22	0	52.317°
14	69.306363297	$D 6 d$	0	0	0	12	2	0	0	36	24	0	52.866°	gyroelongated hexagonal dipyramid
15	80.670244114	$D 3$	0	0	0	12	3	0	0	39	26	0	49.225°
16	92.911655300	$T d$	0	0	0	12	4	0	0	42	28	0	48.936°
17	106.050404829	$D 5 h$	0	0	0	12	5	0	0	45	30	0	50.108°
18	120.084467447	$D 4 d$	0	0	2	8	8	0	0	48	32	0	47.534°
19	135.089467557	$C 2 v$	0.000135163	0	0	14	5	0	0	50	32	1	44.910°
20	150.881568334	$D 3 h$	0	0	0	12	8	0	0	54	36	0	46.093°
21	167.641622399	$C 2 v$	0.001406124	0	1	10	10	0	0	57	38	0	44.321°
22	185.287536149	$T d$	0	0	0	12	10	0	0	60	40	0	43.302°
23	203.930190663	$D 3$	0	0	0	12	11	0	0	63	42	0	41.481°
24	223.347074052	$O$	0	0	0	24	0	0	0	60	32	6	42.065°	snub cube
25	243.812760299	$C s$	0.001021305	0	0	14	11	0	0	68	44	1	39.610°
26	265.133326317	$C 2$	0.001919065	0	0	12	14	0	0	72	48	0	38.842°
27	287.302615033	$D 5 h$	0	0	0	12	15	0	0	75	50	0	39.940°
28	310.491542358	$T$	0	0	0	12	16	0	0	78	52	0	37.824°
29	334.634439920	$D 3$	0	0	0	12	17	0	0	81	54	0	36.391°
30	359.603945904	$D 2$	0	0	0	12	18	0	0	84	56	0	36.942°
31	385.530838063	$C 3 v$	0.003204712	0	0	12	19	0	0	87	58	0	36.373°
32	412.261274651	$I h$	0	0	0	12	20	0	0	90	60	0	37.377°
33	440.204057448	$C s$	0.004356481	0	0	15	17	1	0	92	60	1	33.700°
34	468.904853281	$D 2$	0	0	0	12	22	0	0	96	64	0	33.273°
35	498.569872491	$C 2$	0.000419208	0	0	12	23	0	0	99	66	0	33.100°
36	529.122408375	$D 2$	0	0	0	12	24	0	0	102	68	0	33.229°
37	560.618887731	$D 5 h$	0	0	0	12	25	0	0	105	70	0	32.332°
38	593.038503566	$D 6 d$	0	0	0	12	26	0	0	108	72	0	33.236°
39	626.389009017	$D 3 h$	0	0	0	12	27	0	0	111	74	0	32.053°
40	660.675278835	$T d$	0	0	0	12	28	0	0	114	76	0	31.916°
41	695.916744342	$D 3 h$	0	0	0	12	29	0	0	117	78	0	31.528°
42	732.078107544	$D 5 h$	0	0	0	12	30	0	0	120	80	0	31.245°
43	769.190846459	$C 2 v$	0.000399668	0	0	12	31	0	0	123	82	0	30.867°
44	807.174263085	$O h$	0	0	0	24	20	0	0	120	72	6	31.258°
45	846.188401061	$D 3$	0	0	0	12	33	0	0	129	86	0	30.207°
46	886.167113639	$T$	0	0	0	12	34	0	0	132	88	0	29.790°
47	927.059270680	$C s$	0.002482914	0	0	14	33	0	0	134	88	1	28.787°
48	968.713455344	$O$	0	0	0	24	24	0	0	132	80	6	29.690°
49	1011.557182654	$C 3$	0.001529341	0	0	12	37	0	0	141	94	0	28.387°
50	1055.182314726	$D 6 d$	0	0	0	12	38	0	0	144	96	0	29.231°
51	1099.819290319	$D 3$	0	0	0	12	39	0	0	147	98	0	28.165°
52	1145.418964319	$C 3$	0.000457327	0	0	12	40	0	0	150	100	0	27.670°
53	1191.922290416	$C 2 v$	0.000278469	0	0	18	35	0	0	150	96	3	27.137°
54	1239.361474729	$C 2$	0.000137870	0	0	12	42	0	0	156	104	0	27.030°
55	1287.772720783	$C 2$	0.000391696	0	0	12	43	0	0	159	106	0	26.615°
56	1337.094945276	$D 2$	0	0	0	12	44	0	0	162	108	0	26.683°
57	1387.383229253	$D 3$	0	0	0	12	45	0	0	165	110	0	26.702°
58	1438.618250640	$D 2$	0	0	0	12	46	0	0	168	112	0	26.155°
59	1490.773335279	$C 2$	0.000154286	0	0	14	43	2	0	171	114	0	26.170°
60	1543.830400976	$D 3$	0	0	0	12	48	0	0	174	116	0	25.958°
61	1597.941830199	$C 1$	0.001091717	0	0	12	49	0	0	177	118	0	25.392°
62	1652.909409898	$D 5$	0	0	0	12	50	0	0	180	120	0	25.880°
63	1708.879681503	$D 3$	0	0	0	12	51	0	0	183	122	0	25.257°
64	1765.802577927	$D 2$	0	0	0	12	52	0	0	186	124	0	24.920°
65	1823.667960264	$C 2$	0.000399515	0	0	12	53	0	0	189	126	0	24.527°
66	1882.441525304	$C 2$	0.000776245	0	0	12	54	0	0	192	128	0	24.765°
67	1942.122700406	$D 5$	0	0	0	12	55	0	0	195	130	0	24.727°
68	2002.874701749	$D 2$	0	0	0	12	56	0	0	198	132	0	24.433°
69	2064.533483235	$D 3$	0	0	0	12	57	0	0	201	134	0	24.137°
70	2127.100901551	$D 2 d$	0	0	0	12	50	0	0	200	128	4	24.291°
71	2190.649906425	$C 2$	0.001256769	0	0	14	55	2	0	207	138	0	23.803°
72	2255.001190975	$I$	0	0	0	12	60	0	0	210	140	0	24.492°
73	2320.633883745	$C 2$	0.001572959	0	0	12	61	0	0	213	142	0	22.810°
74	2387.072981838	$C 2$	0.000641539	0	0	12	62	0	0	216	144	0	22.966°
75	2454.369689040	$D 3$	0	0	0	12	63	0	0	219	146	0	22.736°
76	2522.674871841	$C 2$	0.000943474	0	0	12	64	0	0	222	148	0	22.886°
77	2522.674871841	$D 5$	0	0	0	12	65	0	0	225	150	0	23.286°
78	2662.046474566	$T h$	0	0	0	12	66	0	0	228	152	0	23.426°
79	2733.248357479	$C s$	0.000702921	0	0	12	63	1	0	230	152	1	22.636°
80	2805.355875981	$D 4 d$	0	0	0	16	64	0	0	232	152	2	22.778°
81	2878.522829664	$C 2$	0.000194289	0	0	12	69	0	0	237	158	0	21.892°
82	2952.569675286	$D 2$	0	0	0	12	70	0	0	240	160	0	22.206°
83	3027.528488921	$C 2$	0.000339815	0	0	14	67	2	0	243	162	0	21.646°
84	3103.465124431	$C 2$	0.000401973	0	0	12	72	0	0	246	164	0	21.513°
85	3180.361442939	$C 2$	0.000416581	0	0	12	73	0	0	249	166	0	21.498°
86	3258.211605713	$C 2$	0.001378932	0	0	12	74	0	0	252	168	0	21.522°
87	3337.000750014	$C 2$	0.000754863	0	0	12	75	0	0	255	170	0	21.456°
88	3416.720196758	$D 2$	0	0	0	12	76	0	0	258	172	0	21.486°
89	3497.439018625	$C 2$	0.000070891	0	0	12	77	0	0	261	174	0	21.182°
90	3579.091222723	$D 3$	0	0	0	12	78	0	0	264	176	0	21.230°
91	3661.713699320	$C 2$	0.000033221	0	0	12	79	0	0	267	178	0	21.105°
92	3745.291636241	$D 2$	0	0	0	12	80	0	0	270	180	0	21.026°
93	3829.844338421	$C 2$	0.000213246	0	0	12	81	0	0	273	182	0	20.751°
94	3915.309269620	$D 2$	0	0	0	12	82	0	0	276	184	0	20.952°
95	4001.771675565	$C 2$	0.000116638	0	0	12	83	0	0	279	186	0	20.711°
96	4001.771675565	$C 2$	0.000036310	0	0	12	84	0	0	282	188	0	20.687°
97	4177.533599622	$C 2$	0.000096437	0	0	12	85	0	0	285	190	0	20.450°
98	4266.822464156	$C 2$	0.000112916	0	0	12	86	0	0	288	192	0	20.422°
99	4357.139163132	$C 2$	0.000156508	0	0	12	87	0	0	291	194	0	20.284°
100	4448.350634331	$T$	0	0	0	12	88	0	0	294	196	0	20.297°
101	4540.590051694	$D 3$	0	0	0	12	89	0	0	297	198	0	20.011°
102	4633.736565899	$D 3$	0	0	0	12	90	0	0	300	200	0	20.040°
103	4727.836616833	$C 2$	0.000201245	0	0	12	91	0	0	303	202	0	19.907°
104	4822.876522746	$D 6$	0	0	0	12	92	0	0	306	204	0	19.957°
105	4919.000637616	$D 3$	0	0	0	12	93	0	0	309	206	0	19.842°
106	5015.984595705	$D 2$	0	0	0	12	94	0	0	312	208	0	19.658°
107	5113.953547724	$C 2$	0.000064137	0	0	12	95	0	0	315	210	0	19.327°
108	5212.813507831	$C 2$	0.000432525	0	0	12	96	0	0	318	212	0	19.327°
109	5312.735079920	$C 2$	0.000647299	0	0	14	93	2	0	321	214	0	19.103°
110	5413.549294192	$D 6$	0	0	0	12	98	0	0	324	216	0	19.476°
111	5515.293214587	$D 3$	0	0	0	12	99	0	0	327	218	0	19.255°
112	5618.044882327	$D 5$	0	0	0	12	100	0	0	330	220	0	19.351°
113	5721.824978027	$D 3$	0	0	0	12	101	0	0	333	222	0	18.978°
114	5826.521572163	$C 2$	0.000149772	0	0	12	102	0	0	336	224	0	18.836°
115	5932.181285777	$C 3$	0.000049972	0	0	12	103	0	0	339	226	0	18.458°
116	6038.815593579	$C 2$	0.000259726	0	0	12	104	0	0	342	228	0	18.386°
117	6146.342446579	$C 2$	0.000127609	0	0	12	105	0	0	345	230	0	18.566°
118	6254.877027790	$C 2$	0.000332475	0	0	12	106	0	0	348	232	0	18.455°
119	6364.347317479	$C 2$	0.000685590	0	0	12	107	0	0	351	234	0	18.336°
120	6474.756324980	$C s$	0.001373062	0	0	12	108	0	0	354	236	0	18.418°
121	6586.121949584	$C 3$	0.000838863	0	0	12	109	0	0	357	238	0	18.199°
122	6698.374499261	$I h$	0	0	0	12	110	0	0	360	240	0	18.612°
123	6811.827228174	$C 2 v$	0.001939754	0	0	14	107	2	0	363	242	0	17.840°
124	6926.169974193	$D 2$	0	0	0	12	112	0	0	366	244	0	18.111°
125	7041.473264023	$C 2$	0.000088274	0	0	12	113	0	0	369	246	0	17.867°
126	7157.669224867	$D 4$	0	0	2	16	100	8	0	372	248	0	17.920°
127	7274.819504675	$D 5$	0	0	0	12	115	0	0	375	250	0	17.877°
128	7393.007443068	$C 2$	0.000054132	0	0	12	116	0	0	378	252	0	17.814°
129	7512.107319268	$C 2$	0.000030099	0	0	12	117	0	0	381	254	0	17.743°
130	7632.167378912	$C 2$	0.000025622	0	0	12	118	0	0	384	256	0	17.683°
131	7753.205166941	$C 2$	0.000305133	0	0	12	119	0	0	387	258	0	17.511°
132	7875.045342797	$I$	0	0	0	12	120	0	0	390	260	0	17.958°
133	7998.179212898	$C 3$	0.000591438	0	0	12	121	0	0	393	262	0	17.133°
134	8122.089721194	$C 2$	0.000470268	0	0	12	122	0	0	396	264	0	17.214°
135	8246.909486992	$D 3$	0	0	0	12	123	0	0	399	266	0	17.431°
136	8372.743302539	$T$	0	0	0	12	124	0	0	402	268	0	17.485°
137	8499.534494782	$D 5$	0	0	0	12	125	0	0	405	270	0	17.560°
138	8627.406389880	$C 2$	0.000473576	0	0	12	126	0	0	408	272	0	16.924°
139	8756.227056057	$C 2$	0.000404228	0	0	12	127	0	0	411	274	0	16.673°
140	8885.980609041	$C 1$	0.000630351	0	0	13	126	1	0	414	276	0	16.773°
141	9016.615349190	$C 2 v$	0.000376365	0	0	14	126	0	1	417	278	0	16.962°
142	9148.271579993	$C 2$	0.000550138	0	0	12	130	0	0	420	280	0	16.840°
143	9280.839851192	$C 2$	0.000255449	0	0	12	131	0	0	423	282	0	16.782°
144	9414.371794460	$D 2$	0	0	0	12	132	0	0	426	284	0	16.953°
145	9548.928837232	$C s$	0.000094938	0	0	12	133	0	0	429	286	0	16.841°
146	9684.381825575	$D 2$	0	0	0	12	134	0	0	432	288	0	16.905°
147	9820.932378373	$C 2$	0.000636651	0	0	12	135	0	0	435	290	0	16.458°
148	9958.406004270	$C 2$	0.000203701	0	0	12	136	0	0	438	292	0	16.627°
149	10096.859907397	$C 1$	0.000638186	0	0	14	133	2	0	441	294	0	16.344°
150	10236.196436701	$T$	0	0	0	12	138	0	0	444	296	0	16.405°
151	10376.571469275	$C 2$	0.000153836	0	0	12	139	0	0	447	298	0	16.163°
152	10517.867592878	$D 2$	0	0	0	12	140	0	0	450	300	0	16.117°
153	10660.082748237	$D 3$	0	0	0	12	141	0	0	453	302	0	16.390°
154	10803.372421141	$C 2$	0.000735800	0	0	12	142	0	0	456	304	0	16.078°
155	10947.574692279	$C 2$	0.000603670	0	0	12	143	0	0	459	306	0	15.990°
156	11092.798311456	$C 2$	0.000508534	0	0	12	144	0	0	462	308	0	15.822°
157	11238.903041156	$C 2$	0.000357679	0	0	12	145	0	0	465	310	0	15.948°
158	11385.990186197	$C 2$	0.000921918	0	0	12	146	0	0	468	312	0	15.987°
159	11534.023960956	$C 2$	0.000381457	0	0	12	147	0	0	471	314	0	15.960°
160	11683.054805549	$D 2$	0	0	0	12	148	0	0	474	316	0	15.961°
166	12597.649071323	$D 2 d$	0	0	0	16	146	4	0	492	328	0	15.607°
170	13226.681078541	$D 2 d$	0	0	0	12	158	0	0	504	336	0	15.569°
177	14364.837545298	$D 5$	0	0	0	12	165	0	0	525	350	0	15.269°
182	15206.730610906	$D 5$	0	0	0	12	170	0	0	540	360	0	15.155°
184	15550.421450311	$T$	0	0	0	12	172	0	0	546	364	0	14.932°
187	16072.975186320	$D 5$	0	0	0	12	175	0	0	555	370	0	14.848°
192	16963.338386460	$I$	0	0	0	12	180	0	0	570	380	0	14.819°
197	17878.340162571	$D 5$	0	0	0	12	185	0	0	585	390	0	14.147°
202	18817.204718262	$D 5$	0	0	0	12	190	0	0	600	400	0	14.189°
204	19199.540775603	$T h$	0	0	0	12	192	0	0	606	404	0	14.291°
212	20768.053085964	$I$	0	0	0	12	200	0	0	630	420	0	14.118°
214	21169.910410375	$T$	0	0	0	12	202	0	0	636	424	0	13.771°
217	21779.856080418	$D 5$	0	0	0	12	205	0	0	645	430	0	13.902°
232	24961.252318934	$T$	0	0	0	12	220	0	0	690	460	0	13.260°
256	30506.687515847	$T$	0	0	0	12	244	0	0	762	508	0	12.572°
257	30749.941417346	$D 5$	0	0	0	12	245	0	0	765	510	0	12.672°
272	34515.193292681	$I h$	0	0	0	12	260	0	0	810	540	0	12.335°
282	37147.294418462	$I$	0	0	0	12	270	0	0	840	560	0	12.166°
306	43862.569780797	$T h$	0	0	0	12	294	0	0	912	608	0	11.628°
312	45629.362723799	$I$	0	0	0	12	300	0	0	930	620	0	11.581°

References

J. J. Thomson, "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", Philosophical Magazine Series 6, Volume 7, Number 39, pp. 237--265, March 1904.

Mark J. Bowick, Cris Cecka and Alan A. Middleton: http://physics.syr.edu/condensedmatter/thomson/thomsonapplet.htm

David J. Wales and Sidika Ulker: http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html

Category: Electron

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