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 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 g_e (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 μ_B is the Bohr magneton.

The gyromagnetc ratio for the self-rotation of the electron is twice 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 an one-electron cyclotron:

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

The electron gyromagnetic ratio is given by NIST as

$\left| \gamma_\mathrm{e} \right| = 1.760\,859\,708(39) \times 10^{11}\, \mathrm{\ \frac{rad}{s\cdot T} }$

$\left| \frac{\gamma_\mathrm{e}}{2\pi} \right| = 28\,024.952\,66(62) \mathrm{\ \frac{MHz}{ T}}.$

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.

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,

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

Here, 1/2 σ^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, according to, 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.

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. 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. 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	267.513	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

See also

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

References

Note 1 note

• ^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.