Gyromagnetic ratio

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In physics, the magnetogyric ratio (more commonly called the gyromagnetic ratio) of a particle or system is the ratio of its magnetic dipole moment to its angular momentum. Its SI units are hertz per tesla (Hz/T), and it is often denoted by the symbol γ.

1 Magnetogyric ratio for a classical rotating body
2 Magnetogyric ratio for an isolated electron
3 Magnetogyric ratio for a nucleus
4 See also
5 References

[edit] Magnetogyric 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 on account of its rotation. It can be shown that as long as its charge and mass are distributed identically (e.g., both distributed uniformly), its magnetogyric 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

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

as desired.

[edit] Magnetogyric 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 is in fact a fundamentally different, quantum-mechanical phenomenon with no true analogue in classical physics. Consequently, there is no reason to expect the above classical relation to hold. In fact it does not, 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):

$\gamma_\mathrm{e} = \frac{-e}{2m_e}g_e = -g_e \mu_B/\hbar,$

where μ_B is the Bohr magneton. The electron g-factor g_e is a bit more than two, and has been measured to twelve decimal places (Gabrielse, 2006).

The electron magnetogyric ratio is given by NIST ^[1] as

$\gamma_\mathrm{e} = 1.760\,859\,74(15) \times 10^{11}\,s^{-1}T^{-1}.$

[edit] Magnetogyric ratio for a nucleus

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:

$\gamma = \frac{e}{2m_p}g = g \mu_p/\hbar,$

where μ_p is the nuclear magneton, and g is the g-factor of the nucleon or nucleus in question.

The magnetogyric ratio of a nucleus is particularly important because of the role it plays in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). These procedures rely on the fact that nuclear spins precess in a magnetic field at a rate called the Larmor frequency, which is simply the product of the magnetogyric ratio with the magnetic field strength.

Approximate values for some common nuclei are given in the Table below (Bernstein 2004).

Nucleus	γ / 2π (MHz/T)
¹H	42.576
⁷Li	16.546
¹³C	10.705
¹⁴N	3.0766
¹⁵N	-4.3156
¹⁷O	-5.7716
²³Na	11.262
³¹P	17.235

[edit] See also

[edit] References

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.

S.J. Brodsky, V.A. Franke, J.R. Hiller, G. McCartor, S.A. Paston, and E.V. Prokhvatilov, A nonperturbative calculation of the electron's magnetic moment, Nuclear Physics B 703 (2004) 333.

Matt A. Bernstein, Kevin F. King, Xiaohong Joe Zhou, In: Handbook of MRI Pulse Sequences, Publisher: Academic Press (September 7, 2004), Pages 960-1