# Bohr magneton

The value of Bohr magneton
system of units value unit
SI[1] 9.274009994(57)×10−24 J·T−1
CGS[2] 9.274009994(57)×10−21 erg·G−1
eV[3] 5.7883818012(26)×10−5 eV·T−1
atomic units 1/2 /me

In atomic physics, the Bohr magneton (symbol μB) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum.[4][5] The Bohr magneton is defined in SI units by

${\displaystyle \mu _{\mathrm {B} }={\frac {e\hbar }{2m_{\mathrm {e} }}}}$
and in Gaussian CGS units by
${\displaystyle \mu _{\mathrm {B} }={\frac {e\hbar }{2m_{\mathrm {e} }c}}}$

where

## History

The idea of elementary magnets is due to Walther Ritz (1907) and Pierre Weiss. Already before the Rutherford model of atomic structure, several theorists commented that the magneton should involve Planck's constant h.[6] By postulating that the ratio of electron kinetic energy to orbital frequency should be equal to h, Richard Gans computed a value that was twice as large as the Bohr magneton in September 1911.[7] At the First Solvay Conference in November that year, Paul Langevin obtained a ${\displaystyle e\hbar /(12m_{\mathrm {e} })}$ .[8] Langevin assumed that the attractive force was inversely proportional to distance to the power ${\displaystyle n+1,}$  and specifically ${\displaystyle n=1.}$ [9]

The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1911.[10][11] The value is sometimes referred to as the "Bohr–Procopiu magneton" in Romanian scientific literature.[12] The Weiss magneton was experimentally derived in 1911 as a unit of magnetic moment equal to 1.53×10−24 joules per tesla, which is about 20% of the Bohr magneton.

In the summer of 1913, the values for the natural units of atomic angular momentum and magnetic moment were obtained by the Danish physicist Niels Bohr as a consequence of his atom model.[7][13] In 1920, Wolfgang Pauli gave the Bohr magneton its name in an article where he contrasted it with the magneton of the experimentalists which he called the Weiss magneton.[6]

## Theory

A magnetic moment of a charged particle can be generated by two ways. First, a moving electric charge forms a current, hence the orbital motion of an electron around a nucleus generates a magnetic moment by Ampère's circuital law. Second, the inherent rotation, or spin, of the electron has a spin magnetic moment.

In Bohr's atomic model, a natural unit for the orbital angular momentum of an electron was denoted ħ. The Bohr magneton is the magnitude of the magnetic dipole moment of an electron orbiting an atom with such angular momentum. According to the Bohr model, this is the ground state, i.e. the state of lowest possible energy.[14]

The spin angular momentum of an electron is 1/2ħ, but the intrinsic electron magnetic moment caused by its spin is also approximately one Bohr magneton since the electron spin g-factor, a factor relating spin angular momentum to corresponding magnetic moment of a particle, is approximately two.[15]

## References

1. ^ "CODATA value: Bohr magneton". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2012-07-09.
2. ^ O'Handley, Robert C. (2000). Modern magnetic materials: principles and applications. John Wiley & Sons. p. 83. ISBN 0-471-15566-7. (value was slightly modified to reflect 2014 CODATA change)
3. ^ "CODATA value: Bohr magneton in eV/T". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2012-07-09.
4. ^ Schiff, L. I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. p. 440.
5. ^ Shankar, R. (1980). Principles of Quantum Mechanics. Plenum Press. pp. 398–400. ISBN 0306403978.
6. ^ a b Keith, Stephen T.; Quédec, Pierre (1992). "Magnetism and Magnetic Materials: The Magneton". Out of the Crystal Maze. pp. 384–394. ISBN 978-0-19-505329-6.
7. ^ a b Heilbron, John; Kuhn, Thomas (1969). "The genesis of the Bohr atom". Hist. Stud. Phys. Sci. 1: vi–290. doi:10.2307/27757291. JSTOR 27757291.
8. ^ Langevin, Paul (1911). La théorie cinétique du magnétisme et les magnétons [Kinetic theory of magnetism and magnetons]. La théorie du rayonnement et les quanta: Rapports et discussions de la réunion tenue à Bruxelles, du 30 octobre au 3 novembre 1911, sous les auspices de M. E. Solvay. p. 404.
9. ^ Note that the formula
${\displaystyle I_{o}={\frac {m}{Me}}{\frac {h}{8\pi }}{\frac {n}{n+2}}}$

on page 404 should say
${\displaystyle I_{o}={\frac {Me}{m}}{\frac {h}{8\pi }}{\frac {n}{n+2}}.}$

10. ^ Procopiu, Ștefan (1911–1913). "Sur les éléments d'énergie" [On the elements of energy]. Annales scientifiques de l'Université de Jassy. 7: 280.
11. ^ Procopiu, Ștefan (1913). "Determining the Molecular Magnetic Moment by M. Planck's Quantum Theory". Bulletin de la Section Scientifique de l'Académie Roumaine. 1: 151.
12. ^ "Ștefan Procopiu (1890–1972)". Ștefan Procopiu Science and Technology Museum. Archived from the original on 2010-11-18. Retrieved 2010-11-03.
13. ^ Pais, Abraham (1991). Niels Bohr's Times, in physics, philosophy, and politics. Clarendon Press. ISBN 0-19-852048-4.
14. ^ Alonso, Marcelo; Finn, Edward (1992). Physics. Addison-Wesley. ISBN 978-0-201-56518-8.
15. ^ Mahajan, Anant S.; Rangwala, Abbas A. (1989). Electricity and Magnetism. McGraw-Hill. p. 419. ISBN 978-0-07-460225-6.