Gyromagnetic ratio

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:

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

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

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

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

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

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

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 g_e (or just g when there is no risk of confusion):

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

where μ_B is the Bohr magneton.

The gyromagnetic ratio for the self-spinning electron is two times bigger than the value for an orbiting electron.

In the framework of relativistic quantum mechanics,

${\displaystyle g_{\mathrm {e} }=2(1+{\frac {\alpha }{2\pi }}+\cdots ),}$

where ${\displaystyle \alpha }$ is the fine-structure constant. Here the small corrections to the relativistic result g = 2 come from the quantum field theory calculations of the anomalous magnetic dipole moment. The electron g-factor is known to twelve decimal places by measuring the electron magnetic moment in a one-electron cyclotron:[3]

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

The electron gyromagnetic ratio is given by NIST[4][5][6] as

${\displaystyle \left|\gamma _{\mathrm {e} }\right|=1.760\,859\,644(11)\times 10^{11}\,\mathrm {\ {\frac {rad}{s\cdot T}}} }$

${\displaystyle \left|{\frac {\gamma _{\mathrm {e} }}{2\pi }}\right|=28\,024.951\,64(17)\mathrm {\ {\frac {MHz}{T}}} .}$

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

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.[7]

Physical spin-1/2 particles which can not be described by the linear gauged Dirac equation satisfy the gauged Klein–Gordon equation extended by the ge/4σ^μνF_μν term according to,[8]

${\displaystyle \left((\partial ^{\mu }+ieA^{\mu })(\partial _{\mu }+ieA_{\mu })+g{\frac {e}{4}}\sigma ^{\mu \nu }F_{\mu \nu }+m^{2}\right)\psi =0,\quad g\not =2.}$

Here, 1/2σ^μν and F^μν stand for the Lorentz group generators in the Dirac space, and the electromagnetic tensor respectively, while A^μ is the electromagnetic four-potential. An example for such a particle,[8] is the spin-1/2 companion to spin-3/2 in the D^(1/2,1) ⊕ D^(1,1/2) representation space of the Lorentz group. This particle has been shown to be characterized by g = −2/3 and consequently to behave as a truly quadratic fermion.

The sign of the gyromagnetic ratio, γ, determines the sense of precession. 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:

${\displaystyle \gamma _{\rm {n}}={\frac {e}{2m_{p}}}g_{\rm {n}}=g_{\rm {n}}\mu _{\mathrm {N} }/\hbar ,}$

where ${\displaystyle \mu _{\mathrm {N} }}$ is the nuclear magneton, and ${\displaystyle g_{\rm {n}}}$ is the g-factor of the nucleon or nucleus in question. The ratio of ${\displaystyle {\frac {\gamma _{n}}{2\pi g_{\rm {n}}}}}$, equal to ${\displaystyle \mu _{\mathrm {N} }/h}$, is 7.622593285(47) MHz/T.[9]

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.[10][11] Approximate values for some common nuclei are given in the table below.[12][13]

Nucleus	${\displaystyle \gamma _{n}}$ (106 rad⋅s−1⋅T−1)	${\displaystyle \gamma _{n}/(2\pi )}$ (MHz⋅T−1)
1H	267.52218744(11)[14]	42.577478518(18)[15]
2H	41.065	6.536
3H	285.3508	45.415[16]
3He	−203.789	−32.434
7Li	103.962	16.546
13C	67.2828	10.7084
14N	19.331	3.077
15N	−27.116	−4.316
17O	−36.264	−5.772
19F	251.815	40.078
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

• Charge-to-mass ratio
• Chemical shift
• Dirac equation
• Landé g-factor
• Larmor equation
• Proton gyromagnetic ratio

• ^Note 1 : Marc Knecht, The Anomalous Magnetic Moments of the Electron and the Muon, Poincaré Seminar (Paris, Oct. 12, 2002), published in : Duplantier, Bertrand; Rivasseau, Vincent (Eds.); Poincaré Seminar 2002, Progress in Mathematical Physics 30, Birkhäuser (2003), ISBN 3-7643-0579-7.