# Gyromagnetic ratio

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The gyromagnetic ratio (sometimes magnetogyric ratio), γ, is the constant of proportionality between the magnetic moment (μ) and the angular momentum(J) of an object:

${\displaystyle {\boldsymbol {\mu }}=\pm \gamma \mathbf {J} \ ,}$

where the sign is chosen to make γ a positive number.

The units of the gyromagnetic ratio are SI units are radian per second per tesla (s−1·T−1) or, equivalently, coulomb per kilogram (C·kg−1). When the object is placed in a magnetic flux density B, because of its magnetic moment it experiences a torque and precesses about the field at the Larmor frequency, which is given (in radians/s) by the product of the field strength and the gyromagnetic ratio.[1]

A closely related quantity is the g-factor, which relates the magnetic moment in units of magnetons to spin: in terms of the gyromagnetic ratio, g = ±γℏ /μ with ℏ the reduced Planck constant and μ the appropriate magneton (the Bohr magneton for electrons and the nuclear magneton for nucleii). The sign of the g-factor is negative when the magnetic moment is oriented opposite to the angular momentum (it is negative for electrons and neutrons) and positive when the two are aligned the same way (it is positive for protons). More detail is below.

## Examples

The electron gyromagnetic ratio is:[2]

${\displaystyle \gamma _{\rm {e}}=2|\mu _{\rm {e}}|/\hbar =\mathrm {1.760\ 859\ 770\ \times \ 10^{11}\ s^{-1}T^{-1}} \ ,}$

where μe is the magnetic moment of the electron (−928.476 377 × 10−26 J T−1), and ℏ is Planck's constant divided by 2π and ℏ/2 is the spin angular momentum.

Similarly, the proton gyromagnetic ratio is:[3]

${\displaystyle \gamma _{\rm {p}}=2\mu _{\rm {p}}/\hbar =\mathrm {2.675\ 222\ 099\ \times \ 10^{8}\ s^{-1}T^{-1}} \ ,}$

where μp is the magnetic moment of the proton (1.410 606 662 × 10−26 J T−1).

The neutron gyromagnetic ratio is:[3]

${\displaystyle \gamma _{\rm {n}}=2|\mu _{\rm {n}}|/\hbar =\mathrm {1.832\ 471\ 85\times 10^{8}\ s^{-1}T^{-1}} \ ,}$

where μn is the magnetic moment of the neutron (−0.966 236 41 × 10−26 J T−1).

Other ratios can be found on the NIST web site.[4]

## Theory and experiment; g-factors

Comparison between theory and experiment for particles usually is made using the g-factor rather than the gyromagnetic ratio because it is a dimensionless number.

### Electron

The relativistic quantum mechanical theory provided by the Dirac equation predicted the electron to have a magnetic moment of exactly one Bohr magneton, where the Bohr magneton is:[5]

${\displaystyle \mu _{B}={\frac {e\hbar }{2m_{e}}}=\mathrm {927.400915\times 10^{-26}\ J/T} \ ,}$

with e the elementary charge. If magnetic moment is expressed in units of Bohr magnetons, the gyromagnetic ratio becomes the g-factor and the magnetic moment becomes:

${\displaystyle \mu _{e}=-\gamma _{e}{\frac {\hbar }{2}}={\frac {\mu _{e}}{\mu _{B}}}{\mu _{B}}=g_{e}{\frac {\mu _{B}}{2}}\ ,}$

so the gyromagnetic ratio and the g-factor are related as:

${\displaystyle g_{e}=-\gamma _{e}{\frac {\hbar }{\mu _{B}}}\ .}$

The value of the g-factor for the electron is:[6]

${\displaystyle g_{e}=2{\frac {\mu _{e}}{\mu _{B}}}=\mathrm {-2.0023193043622} \ ,}$

The Dirac prediction μe = μB results in a g-factor of exactly ge = −2. Subsequently (in 1947) experiments on the Zeeman splitting of the gallium atom in magnetic field showed that was not exactly the case, and later this departure was calculated using quantum electrodynamics.[7]

### Proton

Similarly, the nuclear magneton is defined as:[8]

${\displaystyle \mu _{\rm {N}}={\frac {e\hbar }{2m_{\rm {p}}}}=\mu _{B}{\frac {m_{e}}{m_{p}}}=\mathrm {5.050\ 783\ 24\ \times \ 10^{-27}\ J\ T^{-1}} \ ,}$

with mp the mass of the proton, and the proton g-factor is:[9]

${\displaystyle g_{\rm {p}}=2{\frac {\mu _{\rm {p}}}{\mu _{\rm {N}}}}=\mathrm {5.585694713} \ ,}$

corresponding to a proton magnetic moment of about μp = 2.793 nuclear magnetons.

This surprising value suggests the proton is not a simple particle, but a complex structure, for example, an assembly of quarks. So far, a theoretical calculation of the magnetic moment of the proton in terms of quarks exchanging gluons is a work in progress, with the present estimate as 2.73 nuclear magnetons.[10]

### Neutron

The neutron g-factor is:[11]

${\displaystyle g_{\rm {n}}=2\mu _{\rm {n}}/(e\hbar /2m_{\rm {p}})=\mathrm {-3.82608545} \ ,}$

corresponding to a neutron magnetic moment of about μn = −1.913 nuclear magnetons. The theoretical calculation of the magnetic moment of the neutron in terms of quarks exchanging gluons is −1.82 nuclear magnetons.[10]

### Deuteron

The deuteron is a bound system consisting of a neutron and proton. Because both constituents are spin 1/2 particles, the bound state must have both spins parallel.[12] The magnetic moment for the deuteron in nuclear magnetons is:[13]

${\displaystyle \mu _{\rm {d}}/\mu _{\rm {N}}=\mu _{\rm {d}}/(e\hbar /2m_{\rm {p}})=\mathrm {0.8574382308} \ ,}$

the same as its g-factor because it is a spin 1 particle. The neutron and proton magnetic moments oppose each other, with a vector sum of about 0.880 nuclear magnetons.

## Measurement

The basis for measuring gyromagnetic ratios is the relation between the spin precession frequency ω of the object (the Larmor frequency) and the magnetic flux density B:

${\displaystyle \omega =\gamma |\mathbf {B} |\ .}$

The resonance frequency is affected by the surrounding medium and, for example, for protons in water the resulting values are called "shielded" values, referring to the shielding by the electrons in the water molecule, and are denoted with a prime: γ'p.[14] This dependence of the resonance frequency upon the environment of a nucleus is called a "chemical shift" and used to explore the matrix surrounding the nucleus in the field of nuclear magnetic resonance (NMR).[15]

## Notes

1. For a number of entertaining exercises on this subject, see Charles Pidgeon (1996). “Larmor frequency versus field strength”, Advanced Tutorials for the Biomedical Sciences: Animations, Simulations, and Calculations Using Mathematica. Wiley-VCH, pp. 165 ff. ISBN 0471186465.
2. Electron gyromagnetic ratio. The NIST reference on constants, units, and uncertainty. Retrieved on 2011-03-28.
3. Proton gyromagnetic ratio. The NIST reference on constants, units, and uncertainty. Retrieved on 2011-03-28. Cite error: Invalid <ref> tag; name "NIST1" defined multiple times with different content
4. A general search menu for the NIST database is found at CODATA recommended values for the fundamental constants. National Institute of Standards and Technology. Retrieved on 2011-03-28. For example, the magnetic moment.
5. Bohr magneton. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
6. Electron g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
7. An historical summary can be found in Toichiro Kinoshita (2010). “§3.2.2 Early tests of QED”, B. Lee Roberts, William J. Marciano, eds: Lepton dipole moments. World Scientific, pp. 73 ff. ISBN 9814271837.  An introduction to the behavior of the electron in a magnetic flux is found in Yehuda Benzion Band (2006). “§5.1.1 Electron spin coupling”, Light and matter: electromagnetism, optics, spectroscopy and lasers. Wiley, pp. 297 ff. ISBN 0471899313.
8. Nuclear magneton. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
9. Proton g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
10. See, for example, Brian Martin (2009). “Table 3.5”, Nuclear and Particle Physics: An Introduction, 2nd ed. Wiley, p. 104. ISBN 0470742747.  and Steven D. Bass (2008). “Chapter 1: Introduction”, The spin structure of the proton. World Scientific, pp. 1 ff. ISBN 9812709460.
11. Neutron g factor. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
12. For a discussion of the deuteron, see for example S. B. Patel (1991). “§8.2 Ground state of the deuteron”, Nuclear physics: an introduction. New Age International, pp. 276 ff. ISBN 8122401252.
13. Deuteron magnetic moment to nuclear magneton ratio. Fundamental physical constants. NIST. Retrieved on 2011-03-28.
14. For example, see BW Petley (2001). “Chapter 2: The fundamental constants and metrology; §3 The fundamental constants and electrical measurements”, T. J. Quinn, S. Leschiutta, Patrizia Tavella, eds: Metrologia e costanti fondamentali recenti sviluppi: Volume 146 of Proceedings of the International School of Physics "Enrico Fermi". IOS Press, pp. 128 ff. ISBN 1586031678.
15. Robin A. De Graaf (2007). “Chemical shift”, In vivo NMR spectroscopy: principles and techniques, 2nd ed. John Wiley and Sons, p. 18. ISBN 0470026707.