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What is today known as the Bohr-van Leeuwen theorem was discovered by Niels Bohr in 1911 in his doctoral dissertation and was later rediscovered by Hendrika Johanna van Leeuwen in her doctoral thesis in 1919. In 1932, van Vleck formalized and expanded upon Bohr's initial theorem in a book he wrote on electric and magnetic susceptibilities.
The Bohr-van Leeuwen theorem applies to an isolated system that cannot rotate. If the isolated system is allowed to rotate in response to an externally applied magnetic field, then this theorem does not apply. If, in addition, there is only one state of thermal equilibrium in a given temperature and field, and the system is allowed time to return to equilibrium after a field is applied, then there will be no magnetization.
where is the electric field and is the magnetic flux density. The rate of work done is and does not depend on . Therefore, the energy does not depend on the magnetic field, so the distribution of motions does not depend on the magnetic field.
In zero field, there will be no net motion of charged particles because the system is not able to rotate. There will therefore be an average magnetic moment of zero. Since the distribution of motions does not depend on the magnetic field, the moment in thermal equilibrium remains zero in any magnetic field.
A more formal proof
So as to lower the complexity of the proof, a system with electrons will be used.
This is appropriate, since most of the magnetism in a solid is carried by electrons, and the proof is easily generalized to more than one type of charged particle.
The Bohr-van Leeuwen theorem is useful in several applications including plasma physics: "All these references base their discussion of the Bohr-van Leeuwen theorem on Niels Bohr's physical model, in which perfectly reflecting walls are necessary to provide the currents that cancel the net contribution from the interior of an element of plasma, and result in zero net diamagnetism for the plasma element."
Diamagnetism of a purely classical nature occurs in plasmas but is a consequence of thermal disequilibrium, such as a gradient in plasma density. Electromechanics and electrical engineering also see practical benefit from the Bohr-van Leeuwen theorem.
^John Hasbrouck van Vleck stated the Bohr-van Leeuwen theorem as "At any finite temperature, and in all finite applied electrical or magnetical fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes identically." (van Vleck, 1932)
^Bohr, Niehls (1972) [originally published as "Studier over Metallernes Elektrontheori", Københavns Universitet (1911)]. "The Doctor's Dissertation (Text and Translation)". In Rosenfeld, L.; Nielsen, J. Rud (eds.). Early Works (1905-1911). Niels Bohr Collected Works. 1. Elsevier. pp. 163, 165-393. doi:10.1016/S1876-0503(08)70015-X. ISBN978-0-7204-1801-9.