Editing Bohr magneton

{{Short description|Unit of magnetic moment}}
{| class="wikitable" align="right" style="text-align:center"
|+ The value of the Bohr magneton
! [[System of units]] !! Value !! Unit
|-
! [[SI]]{{physconst|muB|ref=only}}
| {{physconst|muB|unit=no|ref=no}} || [[Joule|J]]·[[tesla (unit)|T]]<sup>−1</sup>
|-
! [[Gaussian units|Gaussian]]<ref name="O'Handley tati">
{{cite book
 |first=Robert C. |last=O'Handley
 |author-link=Robert O'Handley
 |year=2000
 |title=Modern magnetic materials: principles and applications
 |url=https://archive.org/details/modernmagneticma00ohan |url-access=limited |page=[https://archive.org/details/modernmagneticma00ohan/page/n109 83]
 |publisher=[[John Wiley & Sons]]
 |isbn=0-471-15566-7
}} (value updated to correspond to CODATA 2018)</ref>
| {{val|9.2740100783|(28)|e=-21}} || [[erg]]·[[Gauss (unit)|G]]<sup>−1</sup>
|-
! eV/T<ref>
{{cite web
 |title=CODATA value: Bohr magneton in eV/T
 |url=http://physics.nist.gov/cgi-bin/cuu/Value?mubev
 |work=The NIST Reference on Constants, Units, and Uncertainty
 |publisher=[[NIST]]
 |access-date=2022-08-28
}}</ref>
| {{val|5.7883818060|(17)|e=-5}} || [[electronvolt|eV]]·[[tesla (unit)|T]]<sup>−1</sup>
|-
! [[atomic units]]
| {{sfrac|1|2}} || {{sfrac|''eħ''|''m''<sub>e</sub>}}
|}

In [[atomic physics]], the '''Bohr magneton''' (symbol {{math|''μ''<sub>B</sub>}}) is a [[physical constant]] and the natural unit for expressing the [[magnetic moment]] of an [[electron]] caused by its [[Angular momentum#Angular momentum in quantum mechanics|orbital]] or [[Spin (physics)|spin]] angular momentum.<ref>
{{cite book
 |first=L. I. |last=Schiff |author-link=Leonard I. Schiff
 |year=1968
 |title=Quantum Mechanics
 |page=440
 |publisher=[[McGraw-Hill]]
 |url=https://archive.org/stream/QuantumMechanics_500/Schiff-QuantumMechanics#page/n455|edition=3rd
 |isbn=9780070856431
}}</ref><ref>
{{cite book
 |first=R. |last=Shankar |author-link=Ramamurti Shankar
 |year=1980
 |title=Principles of Quantum Mechanics
 |url=https://archive.org/details/principlesofquan0000shan |url-access=registration |pages=[https://archive.org/details/principlesofquan0000shan/page/398 398–400]
 |publisher=[[Plenum Press]]
 |isbn=0306403978
}}</ref>
In [[SI units]], the Bohr magneton is defined as
<math display="block">\mu_\mathrm{B} = \frac{e \hbar}{2 m_\mathrm{e}}</math>
and in the [[Gaussian units|Gaussian CGS units]] as
<math display="block">\mu_\mathrm{B} = \frac{e \hbar}{2 m_\mathrm{e} c} ,</math>
where
*{{mvar|e}} is the [[elementary charge]],
*{{mvar|ħ}} is the [[Planck constant|reduced Planck constant]],
*{{math|''m''<sub>e</sub>}} is the [[electron mass]],
*{{math|''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''.<ref name="Keith">{{cite book
 |first1=Stephen T. |last1=Keith |first2=Pierre |last2=Quédec
 |year=1992
 |chapter=Magnetism and Magnetic Materials: The Magneton
 |title=Out of the Crystal Maze
 |pages=384–394
 |isbn=978-0-19-505329-6
}}</ref> 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.<ref name="Heilbron">{{cite journal
 |first1=John |last1=Heilbron 
 |first2=Thomas |last2=Kuhn
 |year=1969
 |title=The genesis of the Bohr atom
 |journal=[[Historical Studies in the Physical Sciences|Hist. Stud. Phys. Sci.]]
 |volume=1 |pages=vi–290 
 |doi=10.2307/27757291
 |jstor=27757291 
 |doi-access=free}}</ref> At the [[Solvay Conference#First Conference|First Solvay Conference]] in November that year, [[Paul Langevin]] obtained a value of <math>e\hbar/(2m_\mathrm{e})</math>.<ref>
{{cite conference
 |first=Paul |last=Langevin
 |author-link=Paul Langevin
 |year=1911
 |title=La théorie cinétique du magnétisme et les magnétons
 |trans-title=Kinetic theory of magnetism and magnetons
 |conference=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
 |conference-url=https://archive.org/details/lathoriedurayo00inst/page/404/mode/2up
 |page=404
}}</ref> Langevin assumed that the attractive force was inversely proportional to distance to the power <math>n+1,</math> and specifically <math>n=1.</math><ref>Note that the formula
<math display="block">I_o=\frac m{Me}\frac h{8\pi}\frac n{n+2}</math>
on page 404 should say
<math display="block">I_o=\frac {Me}m\frac h{8\pi}\frac n{n+2}.</math></ref>

The [[:Category:Romanian physicists|Romanian physicist]] [[Ștefan Procopiu]] had obtained the expression for the magnetic moment of the electron in 1913.<ref name = proc1>{{cite journal
 |first=Ștefan |last=Procopiu
 |year=1911–1913
 |title=Sur les éléments d'énergie |trans-title=On the elements of energy
 |journal=[[Annales scientifiques de l'Université de Jassy]]
 |volume=7 |page=280
}}</ref><ref name = proc2>{{cite journal
 |first=Ștefan |last=Procopiu
 |year=1913
 |title=Determining the Molecular Magnetic Moment by M. Planck's Quantum Theory
 |journal=[[Bulletin de la Section Scientifique de l'Académie Roumaine]]
 |volume=1 |pages=151
}}</ref>  The value is sometimes referred to as the "Bohr–Procopiu magneton" in Romanian scientific literature.<ref>{{cite web
 |title=Ștefan Procopiu (1890–1972)
 |url=http://www.etti.tuiasi.ro/sibm/old/Technical%20Museum/html/en/Stefan_Procopiu_en.htm
 |publisher=Ștefan Procopiu Science and Technology Museum
 |access-date=2010-11-03
 |archive-url=https://web.archive.org/web/20101118065825/http://www.etti.tuiasi.ro/sibm/old/Technical%20Museum/html/en/Stefan_Procopiu_en.htm
 |archive-date=2010-11-18
 |url-status=dead
 }}</ref> The [[Weiss magneton]] was experimentally derived in 1911 as a unit of [[magnetic moment]] equal to {{val|1.53|e=-24}} [[joule]]s 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 [[Bohr model|his atom model]].<ref name="Heilbron"/><ref>
{{cite book
 |first=Abraham |last=Pais
 |year=1991
 |title=Niels Bohr's Times, in physics, philosophy, and politics
 |publisher=[[Clarendon Press]]
 |isbn=0-19-852048-4
}}</ref>   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]].<ref name="Keith"/>

==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 [[Angular momentum#Angular momentum in quantum mechanics|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.<ref>{{cite book
 |first1=Marcelo
 |last1=Alonso
 |first2=Edward
 |last2=Finn
 |year=1992
 |title=Physics
 |publisher=[[Addison-Wesley]]
 |isbn=978-0-201-56518-8
 |url-access=registration
 |url=https://archive.org/details/physics00alon
 }}</ref>

The spin angular momentum of an electron is {{sfrac|1|2}}''ħ'', but the intrinsic [[electron magnetic moment]] caused by its spin is also approximately one Bohr magneton, which results in the electron spin [[g-factor (physics)|''g''-factor]], a factor relating spin angular momentum to corresponding magnetic moment of a particle, having a value of approximately 2.<ref>{{cite book
 |first1=Anant S. |last1=Mahajan |first2=Abbas A. |last2=Rangwala
 |year=1989
 |title=Electricity and Magnetism
 |url=https://books.google.com/books?id=_tXrjggX7WwC&q=%22intrinsic+dipole+moment%22+and+electron+%22Bohr+magneton%22&pg=PA419
 |page=419
 |publisher=[[McGraw-Hill]]
 |isbn=978-0-07-460225-6
}}</ref>

==See also==
* [[Anomalous magnetic dipole moment|Anomalous magnetic moment]]
* [[Electron magnetic moment]]
* [[Bohr radius]]
* [[Nuclear magneton]]
* [[Parson magneton]]
* [[Physical constant]]
* [[Zeeman effect]]

==References==
{{Reflist}}

[[Category:Atomic physics]]
[[Category:Niels Bohr]]
[[Category:Physical constants]]
[[Category:Quantum magnetism]]
[[Category:Magnetic moment]]