# Bohr magneton (μB)
The **Bohr magneton** (symbol μB) is the natural physical unit for the magnetic moment of an electron, defined as \( \mu_{\mathrm{B}} = \dfrac{e\hbar}{2m_{\mathrm{e}}} \), where \( e \) is the elementary charge, \( \hbar \) the reduced [Planck constant](https://www.edgepedia.org/planck-constant), and \( m_{\mathrm{e}} \) the electron mass. Its 2022 CODATA recommended value is 9.274 010 0657(29) × 10⁻²⁴ J T⁻¹, with a relative standard uncertainty of 3.1 × 10⁻¹⁰[1](https://physics.nist.gov/cgi-bin/cuu/Value?mub%7Csearch_for=bohr+magneton). It sets the scale of atomic magnetism: orbital and spin magnetic moments of electrons, and the magnetic moments of atoms and ions, are quoted in multiples of it.
| Key fact | Detail |
|---|---|
| Definition | \( \mu_{\mathrm{B}} = e\hbar/2m_{\mathrm{e}} \), the natural unit of electron magnetic moment[1](https://physics.nist.gov/cgi-bin/cuu/Value?mub%7Csearch_for=bohr+magneton) • [2](https://openstax.org/books/university-physics-volume-3/pages/8-2-orbital-magnetic-dipole-moment-of-the-electron) |
| Value (2022 CODATA) | 9.274 010 0657(29) × 10⁻²⁴ J T⁻¹; relative uncertainty 3.1 × 10⁻¹⁰[1](https://physics.nist.gov/cgi-bin/cuu/Value?mub%7Csearch_for=bohr+magneton) |
| Energy-unit form | 5.788 381 7982(18) × 10⁻⁵ eV T⁻¹[3](https://www.physics.nist.gov/cgi-bin/cuu/Value?mubev=) |
| Frequency form | \( \mu_{\mathrm{B}}/h \) = 1.399 624 491 71(44) × 10¹⁰ Hz T⁻¹[4](https://physics.nist.gov/cuu/pdf/JPCRD2022CODATA.pdf) |
| Orbital moment | \( \mu_z = -\mu_{\mathrm{B}} m \), with the magnetic quantum number \( m = -l, \ldots, 0, \ldots, l \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-2-orbital-magnetic-dipole-moment-of-the-electron) |
| Spin moment | \( \mu_z = \pm\mu_{\mathrm{B}} \) for the two spin states \( m_s = \pm 1/2 \)[5](https://openstax.org/books/university-physics-volume-3/pages/8-3-electron-spin) |
| Electron moment | \( \mu_{\mathrm{e}}/\mu_{\mathrm{B}} = -1.001\ 159\ 652\ 181\ 28(18) \)[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC9888147/) |
| Nuclear counterpart | Nuclear magneton \( \mu_{\mathrm{N}} = e\hbar/2m_{\mathrm{p}} \) = 5.050 783 7393(16) × 10⁻²⁷ J T⁻¹, about 1/1836 of μB[4](https://physics.nist.gov/cuu/pdf/JPCRD2022CODATA.pdf) |
## Origin in orbital motion
A charged particle moving in a circular orbit carries a magnetic moment proportional to its angular momentum, with the ratio \( q/2m \) independent of the orbit's radius or speed[7](https://www.feynmanlectures.caltech.edu/II_34.html). For an orbiting electron, quantum mechanics restricts the orbital angular momentum to discrete values \( L = \sqrt{l(l+1)}\,\hbar \), so the magnitude of the orbital magnetic moment becomes \( \mu = \mu_{\mathrm{B}}\sqrt{l(l+1)} \), and its z-component is \( \mu_z = -\mu_{\mathrm{B}} m \). The negative sign reflects the electron's negative charge: the magnetic moment points antiparallel to the orbital angular momentum[2](https://openstax.org/books/university-physics-volume-3/pages/8-2-orbital-magnetic-dipole-moment-of-the-electron).
In an external magnetic field \( B \) along the z-direction, the potential energy of the orbital moment is \( U = m\,\mu_{\mathrm{B}}B \). Because \( m \) takes the \( 2l+1 \) values from \( -l \) to \( l \), a single energy level splits into that many components; for an \( l = 1 \) state the energies are \( -\mu_{\mathrm{B}}B \), 0, and \( +\mu_{\mathrm{B}}B \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-2-orbital-magnetic-dipole-moment-of-the-electron). More generally, an atomic state of angular momentum \( j \) splits into \( 2j+1 \) magnetic sublevels with energies \( U_{\mathrm{mag}} = g\,\mu_{\mathrm{B}}B\,(J_z/\hbar) \), where \( g \) is the dimensionless Landé g-factor of the state[7](https://www.feynmanlectures.caltech.edu/II_34.html).
## Spin and the electron magnetic moment
The electron's intrinsic spin contributes a magnetic moment twice as large per unit angular momentum as the orbital contribution: in vector form \( \vec{\mu}_s = (e/m_{\mathrm{e}})\vec{S} \), a factor of 2 above the orbital ratio that special relativity accounts for. Since the spin projection quantum number has only the two values \( m_s = \pm 1/2 \), the z-component of the spin magnetic moment takes exactly two values, \( \mu_z = \pm\mu_{\mathrm{B}} \)[5](https://openstax.org/books/university-physics-volume-3/pages/8-3-electron-spin).
The measured electron moment is close to, but not exactly, one Bohr magneton. CODATA gives \( \mu_{\mathrm{e}}/\mu_{\mathrm{B}} = -1.001\ 159\ 652\ 181\ 28(18) \), corresponding to an electron g-factor of \( -2.002\ 319\ 304\ 362\ 56(35) \)[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC9888147/). The small excess over the Dirac value of 2 is the anomalous magnetic moment, explained by quantum electrodynamics. A one-electron quantum cyclotron measurement reported \( g/2 = 1.001\ 159\ 652\ 180\ 73(28) \), an uncertainty of 0.28 parts per trillion, and used the result together with QED theory to determine the fine-structure constant[8](https://arxiv.org/abs/0801.1134).
In a hydrogen atom the spin moment interacts with the magnetic field produced by the orbital motion, a phenomenon called spin-orbit coupling, which splits spectral lines and explains the fine structure of the hydrogen spectrum[5](https://openstax.org/books/university-physics-volume-3/pages/8-3-electron-spin).
## The nuclear magneton and related units
The corresponding unit built on the proton mass, \( \mu_{\mathrm{N}} = e\hbar/2m_{\mathrm{p}} \), is the nuclear magneton. Because the proton is about 1836 times heavier than the electron, \( \mu_{\mathrm{N}} \) is correspondingly smaller: 5.050 783 7393(16) × 10⁻²⁷ J T⁻¹, or 3.152 451 254 17(98) × 10⁻⁸ eV T⁻¹[4](https://physics.nist.gov/cuu/pdf/JPCRD2022CODATA.pdf) • [9](https://physics.nist.gov/cgi-bin/cuu/Value?munev%7Csearch_for=t). Nuclear magnetic moments are quoted in nuclear magnetons; the proton's, for example, is 2.792 847 344 63(82) \( \mu_{\mathrm{N}} \)[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC9890581/). NIST's constant tables also list the Bohr magneton in derived forms, including Hz/T, K/T, and inverse meters per tesla, for use in resonance and spectroscopy calculations[11](https://physics.nist.gov/cgi-bin/cuu/Results?search_for=bohr).
## How the accepted value has changed
CODATA, the Committee on Data for Science and Technology, issues internationally recommended values of the fundamental constants roughly every four years, determined by a least-squares adjustment of all available theoretical and experimental information[12](https://www.nist.gov/programs-projects/codata-values-fundamental-physical-constants). The Bohr magneton's uncertainty has tightened by orders of magnitude over the decades: the 1986 adjustment gave 9.274 015 4(31) × 10⁻²⁴ J T⁻¹[13](https://physics.nist.gov/cuu/pdf/codata86.pdf), the 2018 adjustment gave 9.274 010 0783(28) × 10⁻²⁴ J T⁻¹ with a relative uncertainty of 3.0 × 10⁻¹⁰[10](https://pmc.ncbi.nlm.nih.gov/articles/PMC9890581/), and the 2022 adjustment gives 9.274 010 0657(29) × 10⁻²⁴ J T⁻¹[1](https://physics.nist.gov/cgi-bin/cuu/Value?mub%7Csearch_for=bohr+magneton). The value itself has shifted only in its last digits; the improvement lies in the uncertainty, which now sits at a few parts in 10¹⁰.
## References
1. [CODATA Value: Bohr magneton, NIST](https://physics.nist.gov/cgi-bin/cuu/Value?mub%7Csearch_for=bohr+magneton)
2. [8.2 Orbital Magnetic Dipole Moment of the Electron, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/8-2-orbital-magnetic-dipole-moment-of-the-electron)
3. [CODATA Value: Bohr magneton in eV/T, NIST](https://www.physics.nist.gov/cgi-bin/cuu/Value?mubev=)
4. [CODATA recommended values of the fundamental physical constants: 2022, J. Phys. Chem. Ref. Data 54, 033105 (2025)](https://physics.nist.gov/cuu/pdf/JPCRD2022CODATA.pdf)
5. [8.3 Electron Spin, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/8-3-electron-spin)
6. [CODATA recommended values of the fundamental physical constants: 2018](https://pmc.ncbi.nlm.nih.gov/articles/PMC9888147/)
7. [The Feynman Lectures on Physics Vol. II Ch. 34: The Magnetism of Matter](https://www.feynmanlectures.caltech.edu/II_34.html)
8. [New Measurement of the Electron Magnetic Moment and the Fine Structure Constant, arXiv:0801.1134](https://arxiv.org/abs/0801.1134)
9. [CODATA Value: nuclear magneton in eV/T, NIST](https://physics.nist.gov/cgi-bin/cuu/Value?munev%7Csearch_for=t)
10. [CODATA recommended values of the fundamental physical constants: 2018](https://pmc.ncbi.nlm.nih.gov/articles/PMC9890581/)
11. [CODATA Values of the Fundamental Constants, search results for "bohr", NIST](https://physics.nist.gov/cgi-bin/cuu/Results?search_for=bohr)
12. [CODATA values of the fundamental physical constants, NIST](https://www.nist.gov/programs-projects/codata-values-fundamental-physical-constants)
13. [CODATA recommended values, 1986 adjustment, NIST](https://physics.nist.gov/cuu/pdf/codata86.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Magnetostatics › Magnetic dipoles*
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