Thomson problem

In the Thomson problem , n electrons should be distributed on the surface of a unit sphere in such a way that the total electrostatic potential, which is established by the Coulomb force , assumes its minimum. The physicist Joseph John Thomson formulated this problem in 1904 after developing his atomic model .

Mathematically, it's one of the Smale problems .

Mathematical description

The electrostatic potential that arises between two electrons can be described using Coulomb's law . ${\ displaystyle U (N)}$

{\ displaystyle U_ {ij} (N) = k_ {e} {e_ {i} e_ {j} \ over r_ {ij}}}

.

Where and are the charges of the electrons, is the Coulomb constant (given by ; is the electric field constant ) and is the distance between the two electrons. To simplify the problem, and can be set. ${\ displaystyle e_ {i}}$ ${\ displaystyle e_ {j}}$ ${\ displaystyle k_ {e}}$ ${\ displaystyle {\ frac {1} {4 \ pi \ varepsilon _ {0}}}}$ ${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle r_ {ij} = | \ mathbf {r} _ {i} - \ mathbf {r} _ {j} |}$ ${\ displaystyle e_ {i} = e_ {j} = 1}$ ${\ displaystyle k_ {e} = 1}$

In the case of a configuration of electrons, the potential arises ${\ displaystyle N}$

{\ displaystyle U (N) = \ sum _ {i <j} {\ frac {1} {r_ {ij}}}}

.

a. The goal is now to find the form in which this total potential assumes a minimum. Finding a solution is usually done using numerical methods .

Well-known solutions

${\ displaystyle N = 1}$ : The solution is trivial for only one single electron, because regardless of where the electron is on the surface of the sphere, the same potential is always established.

${\ displaystyle N = 2}$ : With two electrons the potential minimum is then present when they are diametrically opposite (e.g. north and south pole).

${\ displaystyle N = 3}$ : With three electrons the configuration forms an equilateral triangle on a great circle of the sphere.

${\ displaystyle N = 4}$ : The four electrons form a tetrahedron .

${\ displaystyle N = 5}$ : In 2010, computer-aided evidence was provided for five electrons, according to which they form a triangular bipyramid .

${\ displaystyle N = 6}$ : The six electrons form an octahedron .

${\ displaystyle N = 12}$ : This configuration forms a regular icosahedron .

literature

Carlos Beltrán: The State of the Art in Smale's 7th Problem. In: Felipe Cucker , Teresa Krick, Allan Pinkus, Agnes Szanto (eds.): Foundations of Computational Mathematics. Budapest 2011 (= London Mathematical Society. Lecture Note Series. 403). Cambridge University Press, Cambridge et al. 2013, ISBN 978-1-107-60407-0 , pp. 1-15.
LL Whyte: Unique arrangements of points on a sphere. In: American Mathematical Monthly . Vol. 59, No. 9, 1952, pp. 606-611, JSTOR 2306764 .
Edward B. Saff, Arno BJ Kuijlaars : Distributing many points on a sphere. In: The Mathematical Intelligencer . Vol. 19, No. 1, 1997, pp. 5-11, doi : 10.1007 / BF03024331

Web links

Interactive Java applet on the Thomson problem

Individual evidence

↑ Joseph 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. In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6, Vol. 7, No. 39, 1904, ZDB -ID 5450-1 , pp. 237-265, doi : 10.1080 / 14786440409463107 .
↑ Ludwig Föppl : Stable arrangements of electrons in the atom. In: Journal for pure and applied mathematics . Vol. 141, 1912, pp. 251-302, ( digitized version ).
^ Richard Evan Schwartz: The 5 Electron Case of Thomson's Problem . In: Mathematical Physics . January 21, 2010, arxiv : 1001.3702 .
^ VA Yudin: The minimum of potential energy of a system of point charges . In: Discrete Mathematics and Applications . tape 3 , no. 1 , 2009, p. 75-82 , doi : 10.1515 / dma.1993.3.1.75 .
↑ Nikolay N. Andreev: An extremal property of the icosahedron. In: East Journal on Approximations. Vol. 2, No. 4, 1996, ISSN 1310-6236 , pp. 459-462, MR 97m: 52022, Zbl 0877.51021.

[1] Joseph 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. In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6, Vol. 7, No. 39, 1904, ZDB -ID 5450-1 , pp. 237-265, doi : 10.1080 / 14786440409463107 .

[2] Ludwig Föppl : Stable arrangements of electrons in the atom. In: Journal for pure and applied mathematics . Vol. 141, 1912, pp. 251-302, ( digitized version ).

[3] Richard Evan Schwartz: The 5 Electron Case of Thomson's Problem . In: Mathematical Physics . January 21, 2010, arxiv : 1001.3702 .

[4] VA Yudin: The minimum of potential energy of a system of point charges . In: Discrete Mathematics and Applications . tape 3 , no. 1 , 2009, p. 75-82 , doi : 10.1515 / dma.1993.3.1.75 .

[5] Nikolay N. Andreev: An extremal property of the icosahedron. In: East Journal on Approximations. Vol. 2, No. 4, 1996, ISSN 1310-6236 , pp. 459-462, MR 97m: 52022, Zbl 0877.51021.