# 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 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 often used 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.

## 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.

### Heuristic derivation

The derivation of this relation is as follows: First we must prove that the torque resulting from subjecting a magnetic moment $\mathbf {m}$  to a magnetic field $\mathbf {B}$  is ${\boldsymbol {\mathrm {T} }}=\mathbf {m} \times \mathbf {B}$ . The identity of the functional form of the stationary electric and magnetic fields has led to defining the magnitude of the magnetic dipole moment equally well as $m=I\pi r^{2}$ , or in the following way, imitating the moment p of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges $\pm q_{\rm {m}}$  on the two poles and vector distance between the poles $\mathbf {d}$  under the influence of the magnetic field of earth $\mathbf {B}$ . By classical mechanics the torque on this needle is ${\boldsymbol {\mathrm {T} }}=q_{\rm {m}}(\mathbf {d} \times \mathbf {B} ).$  But as previously stated $q_{\rm {m}}\mathbf {d} =I\pi r^{2}{\hat {\mathbf {d} }}=\mathbf {m} ,$  so the desired formula comes up.

The model of the spinning electron we use in the derivation has an evident analogy with a gyroscope. For any rotating body the rate of change of the angular momentum $\mathbf {J}$  equals the applied torque $\mathbf {T}$ :

${\frac {d\mathbf {J} }{dt}}=\mathbf {T} .$

Note as an example the precession of a gyroscope. The earth's gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot. In the place of the gyroscope imagine a sphere spinning around the axis and with its center on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the center of the sphere, upwards $\mathbf {J}$  and downwards $\mathbf {m} .$  Replace the gravity with a magnetic flux density B.

${\frac {d\mathbf {J} }{dt}}$  represents the linear velocity of the pike of the arrow $\mathbf {J}$  along a circle whose radius is $J\sin {\phi }$ , where $\phi$  is the angle between $\mathbf {J}$  and the vertical. Hence the angular velocity of the rotation of the spin is $\quad \omega =2\pi f={\frac {|d\mathbf {J} |}{dt\cdot J\cdot \sin {\phi }}}={\frac {|\mathbf {T} |}{J\cdot \sin {\phi }}}={\frac {|\mathbf {m} \times \mathbf {B} |}{J\cdot \sin {\phi }}}={\frac {mB\sin {\phi }}{J\cdot \sin {\phi }}}={\frac {mB}{J}}=\gamma B.$

Consequently, $f={\frac {\gamma }{2\pi }}B.\quad {\text{q.e.d.}}$

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.

The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency, the resonance frequency of an ionized plasma being under the influence of a static finite magnetic field, when we superimpose a high frequency electromagnetic field.

## 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

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

## 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.

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,

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

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

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

The electron gyromagnetic ratio is given by NIST as

$\left|\gamma _{\mathrm {e} }\right|=1.760\,859\,644(11)\times 10^{11}\,\mathrm {\ {\frac {rad}{s\cdot T}}}$
$\left|{\frac {\gamma _{\mathrm {e} }}{2\pi }}\right|=28\,024.951\,64(17)\mathrm {\ {\frac {MHz}{T}}} .$

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

## 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.

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/4σμνFμν term according to,

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

Here, 1/2σμν and Fμν stand for the Lorentz group generators in the Dirac space, and the electromagnetic tensor respectively, while Aμ is the electromagnetic four-potential. An example for such a particle, 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.

## 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. 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:

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

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

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. Approximate values for some common nuclei are given in the table below.

Nucleus $\gamma _{n}$  (106 rad⋅s−1⋅T−1) $\gamma _{n}/(2\pi )$  (MHz⋅T−1)
1H 267.52218744(11) 42.577478518(18)
2H 41.065 6.536
3H 285.3508 45.415
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 251.662 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