Bohr magneton

The value of the Bohr magneton
System of units	Value	Unit
SI‍^[1]	9.2740100657(29)×10⁻²⁴	J·T⁻¹
Gaussian^[2]	9.2740100783(28)×10⁻²¹	erg·G⁻¹
eV/T^[3]	5.7883818060(17)×10⁻⁵	eV·T⁻¹
atomic units	⁠1/2⁠	⁠eħ/m_e⁠

In atomic physics, the Bohr magneton (symbol $μ B$ ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.^[4]^[5] In SI units, the Bohr magneton is defined as $\mu _{\mathrm {B} }={\frac {e\hbar }{2m_{\mathrm {e} }}}$ and in the Gaussian CGS units as $\mu _{\mathrm {B} }={\frac {e\hbar }{2m_{\mathrm {e} }c}},$ where

$e$ is the elementary charge,
$ħ$ is the reduced Planck constant,
$m e$ is the electron mass,
$c$ is the speed of light.

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 the Planck 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 $e\hbar /(2m_{\mathrm {e} })$ .^[8] Langevin assumed that the attractive force was inversely proportional to distance to the power $n+1,$ and specifically $n=1.$ ^[9]

The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1913.^[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 an electron in an atom is composed of two components. First, 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 the Bohr model of the atom, for an electron that is in the orbit of lowest energy, its orbital angular momentum has magnitude equal to the reduced Planck constant, denoted ħ. The Bohr magneton is the magnitude of the magnetic dipole moment of an electron orbiting an atom with this angular momentum.^[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, which results in the electron spin g-factor, a factor relating spin angular momentum to corresponding magnetic moment of a particle, having a value of approximately 2.^[15]

References

^ "2022 CODATA Value: Bohr magneton". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
^ O'Handley, Robert C. (2000). Modern magnetic materials: principles and applications. John Wiley & Sons. p. 83. ISBN 0-471-15566-7. (value updated to correspond to CODATA 2018)
^ "CODATA value: Bohr magneton in eV/T". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2022-08-28.
^ Schiff, L. I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. p. 440. ISBN 9780070856431.
^ Shankar, R. (1980). Principles of Quantum Mechanics. Plenum Press. pp. 398–400. ISBN 0306403978.
^ ^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.
^ ^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.
^ 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.
^ Note that the formula $I_{o}={\frac {m}{Me}}{\frac {h}{8\pi }}{\frac {n}{n+2}}$ on page 404 should say $I_{o}={\frac {Me}{m}}{\frac {h}{8\pi }}{\frac {n}{n+2}}.$
^ Procopiu, Ștefan (1911–1913). "Sur les éléments d'énergie" [On the elements of energy]. Annales scientifiques de l'Université de Jassy. 7: 280.
^ 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.
^ "Ștefan Procopiu (1890–1972)". Ștefan Procopiu Science and Technology Museum. Archived from the original on 2010-11-18. Retrieved 2010-11-03.
^ Pais, Abraham (1991). Niels Bohr's Times, in physics, philosophy, and politics. Clarendon Press. ISBN 0-19-852048-4.
^ Alonso, Marcelo; Finn, Edward (1992). Physics. Addison-Wesley. ISBN 978-0-201-56518-8.
^ Mahajan, Anant S.; Rangwala, Abbas A. (1989). Electricity and Magnetism. McGraw-Hill. p. 419. ISBN 978-0-07-460225-6.

[physconst-muB-1] "2022 CODATA Value: Bohr magneton". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.

[O'Handley_tati-2] O'Handley, Robert C. (2000). Modern magnetic materials: principles and applications. John Wiley & Sons. p. 83. ISBN 0-471-15566-7. (value updated to correspond to CODATA 2018)

[3] "CODATA value: Bohr magneton in eV/T". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2022-08-28.

[4] Schiff, L. I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. p. 440. ISBN 9780070856431.

[5] Shankar, R. (1980). Principles of Quantum Mechanics. Plenum Press. pp. 398–400. ISBN 0306403978.

[Keith-6] 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.

[Heilbron-7] 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 $I_{o}={\frac {m}{Me}}{\frac {h}{8\pi }}{\frac {n}{n+2}}$ on page 404 should say $I_{o}={\frac {Me}{m}}{\frac {h}{8\pi }}{\frac {n}{n+2}}.$

[proc1-10] Procopiu, Ștefan (1911–1913). "Sur les éléments d'énergie" [On the elements of energy]. Annales scientifiques de l'Université de Jassy. 7: 280.

[proc2-11] 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.

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History

Theory

See also

References