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The Distribution of Charge and Current in an Atom consisting of many Electrons obeying Dirac's equations

Published online by Cambridge University Press:  24 October 2008

D. R. Hartree
D. R. Hartree
Affiliation:
Christ's College.
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Abstract

The approximation is made that in a many-electron atom each electron is in a stationary state in the field of the nucleus and the remaining electrons, and further that this field is central.

To this approximation it is known that, if the electrons obey Schrödinger's equation, then a complete n, l group of electrons [i. e. 2 (2l + 1) electrons with the same n, l] can be divided into two half-groups for each of which the distribution of charge is spherically symmetrical. To the same approximation it is shown that, if the electrons obey Dirac's equation, a complete Stoner sub-group [i. e. 2j + 1 electrons with the same n, l, j (j = l ± £)] can be divided into two halves, for each of which the distribution of charge is spherically symmetrical.

The distribution of current and the resultant magnetic moment for a solution of Dirac's equation in a central field are obtained, and it is shown that for a complete Stoner sub-group the net current and magnetic moment are zero. The Landé g-formula for one electron is derived from the formula for the magnetic moment by neglecting ‘relativity’ terms.

The formula for the magnetic moment suggests the question whether it is ever possible to specify the direction of the spin axis of the electron (as distinct from the magnetic moment of the whole atom). This is investigated, and it seems justifiable to specify the direction of the spin axis for states for which the magnetic quantum number m has its extreme values ±j, but not otherwise.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 2 , April 1929 , pp. 225 - 236
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* See Unsöld, A., Ann. d. Phys. LXXXII, p. 355, §§ 5, 6 (1927);CrossRefGoogle ScholarSommerfeld, A., Phys. Zeit. XXVIII, p. 238 (1927).Google Scholar

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† I am indebted to MrMott, N. F. for this result.Google Scholar

* The negative sign appears to arise from the fact that χ, ω are defined to be the polar angles of the magnetic moment, which is in the opposite direction to the angular momentum on account of the negative charge on the electron.Google Scholar