# Gyromagnetic ratio

In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic dipole moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla (rad·s−1·T−1) or, equivalently, the coulomb per kilogram (C·kg−1).

The term "gyromagnetic ratio" is sometimes used[1] as a synonym for a different but closely related quantity, the g-factor. The g-factor, unlike the gyromagnetic ratio, is dimensionless. For more on the g-factor, see below, or see the article g-factor.

## Gyromagnetic ratio and Larmor precession

Main article: Larmor precession

Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field B (measured in teslas) that is not aligned with its magnetic moment, will precess at a frequency f (measured in hertz), that is proportional to the external field:

$f=\frac{\gamma}{2\pi}B.$

For this reason, values of γ/(2π), in units of hertz per tesla (Hz/T), are often quoted instead of γ.

This relationship also explains an apparent contradiction between the two equivalent terms, gyromagnetic ratio versus magnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to a gyric (rotational, from Greek: γύρος, "turn") property (i.e. angular momentum), it is also, at the same time, a ratio between the angular precession frequency (another gyric property) ω = 2πf and the magnetic field.

## Gyromagnetic ratio for a classical rotating body

Consider a charged body rotating about an axis of symmetry. According to the laws of classical physics, it has both a magnetic dipole moment and an angular momentum due to its rotation. It can be shown that as long as its charge and mass are distributed identically (e.g., both distributed uniformly), its gyromagnetic ratio is

$\gamma = \frac{q}{2m}$

where q is its charge and m is its mass. The derivation of this relation is as follows:

It suffices to demonstrate this for an infinitesimally narrow circular ring within the body, as the general result follows from an integration. Suppose the ring has radius r, area A = πr2, mass m, charge q, and angular momentum L = mvr. Then the magnitude of the magnetic dipole moment is

$\mu = IA = \frac{qv}{2\pi r}\times\pi r^2 = \frac{q}{2m}\times mvr = \frac {q}{2m} L .$

## Gyromagnetic ratio for an isolated electron

An isolated electron has an angular momentum and a magnetic moment resulting from its spin. While an electron's spin is sometimes visualized as a literal rotation about an axis, it cannot be attributed to mass distributed identically to the charge. The above classical relation does not hold, giving the wrong result by a dimensionless factor called the electron g-factor, denoted ge (or just g when there is no risk of confusion):

$| \gamma_\mathrm{e} | = \frac{|-e|}{2m_\mathrm{e}}g_\mathrm{e} = g_\mathrm{e} \mu_\mathrm{B}/\hbar,$

where μB is the Bohr magneton. As mentioned above, in classical physics one would expect the g-factor to be g = 1. However in the framework of relativistic quantum mechanics,

$g_e = 2(1+\frac{\alpha}{2\pi}+\cdots),$

where $\alpha$ is the fine-structure constant. Here the small corrections to the relativistic result g = 2 come from the quantum field theory. Experimentally, the electron g-factor has been measured to twelve decimal places:[2]

$~g_\mathrm{e} = 2.0023193043617(15).$

The electron gyromagnetic ratio is given by NIST[3][4] as

$\left| \gamma_\mathrm{e} \right| = 1.760\,859\,708(39) \times 10^{11}\, \mathrm{\ \frac{rad}{s\cdot T} }$
$\left| \frac{\gamma_\mathrm{e}}{2\pi} \right| = 28\,024.952\,66(62) \mathrm{\ \frac{MHz}{ T}}.$

The g-factor and γ are in excellent agreement with theory; see Precision tests of QED for details.

## Gyromagnetic factor as a consequence of relativity

Since a gyromagnetic factor equal to 2 follows from the Dirac's equation it is a frequent misconception to think that a g-factor 2 is a consequence of relativity; it is not. The factor 2 can be obtained from the linearization of both the Schrödinger equation and the relativistic Klein–Gordon equation (which leads to Dirac's). In both cases a 4-spinor is obtained and for both linearizations the g-factor is found to be equal to 2; Therefore, the factor 2 is a consequence of the wave equation dependency on the first (and not the second) derivatives with respect to space and time.[5]

## Gyromagnetic ratio for a nucleus

The sign of the gyromagnetic ratio, γ, determines the sense of precession. Nuclei such as 1H and 13C are said to have clockwise precession whereas 15N has counterclockwise precession.[6][7] While the magnetic moments shown here are oriented the same for both cases of γ, the spin angular momentum are in opposite directions. Spin and magnetic moment are in the same direction for γ>0.

Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above. The ratio is conventionally written in terms of the proton mass and charge, even for neutrons and for other nuclei, for the sake of simplicity and consistency. The formula is:

$\gamma_n = \frac{e}{2m_p}g_n = g_n \mu_\mathrm{N}/\hbar,$

where $\mu_\mathrm{N}$ is the nuclear magneton, and $g_n$ is the g-factor of the nucleon or nucleus in question.

The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). These procedures rely on the fact that bulk magnetization due to nuclear spins precess in a magnetic field at a rate called the Larmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength. With this phenomenon, the sign of γ determines the sense (clockwise vs counterclockwise) of precession.

Most common nuclei such as 1H and 13C have positive gyromagnetic ratios.[6][7] Approximate values for some common nuclei are given in the table below.[8][9]

Nucleus $\gamma_n$ (106 rad s−1 T −1) $\gamma_n/(2\pi)$ (MHz T −1)
1H 267.513 42.576
2H 41.065 6.536
3He -203.789 -32.434
7Li 103.962 16.546
13C 67.262 10.705
14N 19.331 3.077
15N −27.116 −4.316
17O -36.264 -5.772
19F 251.662 40.052
23Na 70.761 11.262
27Al 69.763 11.103
29Si −53.190 −8.465
31P 108.291 17.235
57Fe 8.681 1.382
63Cu 71.118 11.319
67Zn 16.767 2.669
129Xe -73.997 -11.777

## References

### General note

1. ^ For example, see: D.C. Giancoli, Physics for Scientists and Engineers, 3rd ed., page 1017. Or see: P.A. Tipler and R.A. Llewellyn, Modern Physics, 4th ed., page 309.
2. ^ B Odom, D Hanneke, B D'Urso and G Gabrielse (2006). "New measurement of the electron magnetic moment using a one-electron quantum cyclotron". Physical Review Letters 97 (3): 030801. Bibcode:2006PhRvL..97c0801O. doi:10.1103/PhysRevLett.97.030801. PMID 16907490.
3. ^ NIST: Electron gyromagnetic ratio. Note that NIST puts a positive sign on the quantity; however, to be consistent with the formulas in this article, a negative sign is put on γ here. Indeed, many references say that γ<0 for an electron; for example, Weil and Bolton, Electron Paramagnetic Resonance (Wiley 2007), page 578. Also note that the units of radians are added for clarity.
4. ^ NIST: Electron gyromagnetic ratio over 2 pi
5. ^ Greiner, Walter. Quantum Mechanics: An Introduction. Springer Verlag. ISBN 9783540674580.
6. ^ a b M H Levitt (2008). Spin Dynamics. John Wiley & Sons Ltd. ISBN 0470511176.
7. ^ a b Arthur G Palmer (2007). Protein NMR Spectroscopy. Elsevier Academic Press. ISBN 012164491X.
8. ^ M A Bernstein, K F King and X J Zhou (2004). Handbook of MRI Pulse Sequences. San Diego: Elsevier Academic Press. p. 960. ISBN 0-12-092861-2.
9. ^ R C Weast, M J Astle, ed. (1982). Handbook of Chemistry and Physics. Boca Raton: CRC Press. p. E66. ISBN 0-8493-0463-6.