Bohr–van Leeuwen theorem
The Bohr–van Leeuwen theorem is a theorem in the field of statistical mechanics. The theorem states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero. This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for diamagnetism, paramagnetism or ferromagnetism.
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 significance of this discovery is that classical physics does not allow for such things as paramagnetism, diamagnetism and ferromagnetism and thus quantum physics and relativity are needed to explain the magnetic events. This result, "perhaps the most deflationary publication of all time," may have contributed to Bohr's development of a quasi-classical theory of the hydrogen atom in 1913.
An intuitive proof
The Bohr–van Leeuwen theorem applies to an isolated system that cannot rotate (an isolated star could start rotating if exposed to a field). 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.
The probability that the system will be in a given state of motion is predicted by Maxwell–Boltzmann statistics to be proportional to , where is the energy of the system, is the Boltzmann constant, and is the absolute temperature. This energy is equal to the kinetic energy for a particle with mass and speed and the potential energy.
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
We will consider, for simplicity, a system with electrons. 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. Each electron has a negative charge and mass . If its position is and velocity is , it produces a current and a magnetic moment
The above equation shows that the magnetic moment is a linear function of the position coordinates, so the total magnetic moment in a given direction must be a linear function of the form
where the dot represents a time derivative and are vector coefficients depending on the position coordinates .
Maxwell–Boltzmann statistics gives the probability that the nth particle has momentum and coordinate as
The thermal average of any function of these generalized coordinates is then
In the presence of a magnetic field,
so the moment is a linear function of the momenta .
The thermally averaged moment,
is the sum of terms proportional to integrals of the form
where represents one of the moment coordinates. The integrand is an odd function of , so it vanishes. Therefore, .
Applications of the Bohr–van Leeuwen theorem
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)
- Aharoni 1996
- Bohr 1972
- van Leeuwen 1921
- van Vleck 1932
- van Vleck 1992
- Feynman, Leighton & Sands 2006
- Roth 1967
- Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. ISBN 0-19-851791-2.
- Bohr, Niehls (1972) [originally published as "Studier over Metallernes Elektrontheori", Københavns Universitet (1911)]. "The Doctor's Dissertation (Text and Translation)". In Rosenfeld, L. Early Works (1905-1911). Niels Bohr Collected Works 1. Elsevier. pp. 163, 165–393. doi:10.1016/S1876-0503(08)70015-X. ISBN 978-0-7204-1801-9
- Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006). The Feynman Lectures on Physics 2. ISBN 0-8053-9045-6.
- Roth, Reece (1967). "Plasma Stability and the Bohr-Van Leeuwen Theorem". NASA. Retrieved 2008-10-27.
- van Leeuwen, Hendrika Johanna (1921). "Problèmes de la théorie électronique du magnétisme". Journal de Physique et le Radium 2 (12): 361–377.
- van Vleck, J. H. (1932). The theory of electric and magnetic susceptibilities. Clarendon Press. ISBN 0-19-851243-0.
- van Vleck, J. H. (1992). "Quantum mechanics: The key to understanding magnetism (Nobel lecture, 8 December 1977)". In Lundqvist, Stig. Nobel Lectures in Physics 1971-1980. World Scientific. ISBN 981-02-0726-3