# 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

e is the elementary charge,
ħ is the reduced Planck constant,
me is the electron rest mass and
c is the speed of light.

The electron magnetic moment, which is the electron's intrinsic spin magnetic moment, is approximately one Bohr magneton.[6]

In general, and without reference to the electron's properties, the Bohr magneton may also be defined as

${\displaystyle \mu _{\mathrm {B} }={\frac {q\hbar }{2m_{q}c}}}$

in Gaussian CGS units, where q is any charge and mq is its mass.[7]

## 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.[8] 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.[9] At the First Solvay Conference in November that year, Paul Langevin obtained a submultiple.[10] The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1911.[11][12] The value is sometimes referred to as the "Bohr–Procopiu magneton" in Romanian scientific literature.[13] 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.

The Bohr magneton is the magnitude of the magnetic dipole moment of an orbiting electron with an orbital angular momentum of ħ. According to the Bohr model, this is the ground state, i.e. the state of lowest possible energy.[14] In the summer of 1913, this value was naturally obtained by the Danish physicist Niels Bohr as a consequence of his atom model.[9][15] 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.[8]

Although the spin angular momentum of an electron is 1/2ħ, the intrinsic magnetic moment of the electron caused by its spin is still approximately one Bohr magneton. The electron spin g-factor is approximately two.

## 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. McGraw-Hill. p. 440.
5. ^ Shankar, R. (1980). Principles of Quantum Mechanics. Plenum Press. pp. 398–400. ISBN 0306403978.
6. ^ Mahajan, Anant S.; Rangwala, Abbas A. (1989). Electricity and Magnetism. McGraw-Hill. p. 419. ISBN 978-0-07-460225-6.
7. ^ Zettili, Nouredine (2009). Quantum Mechanics: Concepts and Applications (2nd ed.). Chichester: Wiley. p. 365. ISBN 978-0-470-02678-6.
8. ^ 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.
9. ^ a b Heilbron, John; Kuhn, Thomas (1969). "The genesis of the Bohr atom". Hist. Stud. Phys. Sci. 1: 232.
10. ^ 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. 403.
11. ^ Procopiu, Ștefan (1911–1913). "Sur les éléments d'énergie" [On the elements of energy]. Ann. Sci. Univ. Jassy. 7: 280.
12. ^ Procopiu, Ștefan (1913). "Determining the Molecular Magnetic Moment by M. Planck's Quantum Theory". Bull. Sci. Acad. Roum. Sci. 1: 151.
13. ^ "Ștefan Procopiu (1890–1972)". Ștefan Procopiu Science and Technology Museum. Retrieved 2010-11-03.
14. ^ Alonso, Marcelo; Finn, Edward (1992). Physics. Addison-Wesley. ISBN 978-0-201-56518-8.
15. ^ Pais, Abraham (1991). Niels Bohr's Times, in physics, philosophy, and politics. Clarendon Press. ISBN 0-19-852048-4.