# Gyromagnetic Ratio

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Related to Magnetogyric ratio: Larmor frequency

## Gyromagnetic ratio

The ratio of angular momentum to magnetic moment for atomic systems. This ratio is usually expressed in terms of the magnetomechanical factor *g*^{′}, as in Eq. (1).

*e/c*and

*m*are the charge and mass of the electron. The factor

*g*

^{′}is sometimes loosely called the gyromagnetic ratio.

The magnetomechanical ratio is the inverse of the gyromagnetic ratio. It is usually denoted by *γ* and is equal to *g*^{′}*e*/2*mc*. The magnetomechanical ratio of a substance identifies the origin of the magnetic moment. For example, for electron spin the angular momentum is ½*ℏ*, where *ℏ* is Planck's constant divided by 2π. The magnetic moment is the Bohr magneton *eℏ*/2*mc*. Thus, the magnetomechanical ratio

*γ*=

*g*

^{′}

*e*/2

*mc*, for electron spin

*g*

^{′}= 2. For orbital angular momentum,

*γ*=

*e/mc*and

*g*

^{′}= 1. The experimental values of

*g*

^{′}for most ferromagnetic materials are in the neighborhood of 2, showing that the major contribution to the magnetization comes from the electron spin. In superconductors, on the other hand, the fact that

*g*

^{′}= 1 shows that the diamagnetic currents which cause the Meissner effect are caused by electrons.

*See*Meissner effect, Superconductivity

## Gyromagnetic Ratio

the ratio of the magnetic moment of elementary particles and systems of elementary particles, such as atoms, molecules, and atomic nuclei, to their angular momentum (mechanical moment). The gyromagnetic ratio has a definite value for each elementary particle with nonzero mechanical momentum, or spin. The values may be determined for various states of an atomic system from the equation *γ = gγ _{o}*, where γ

_{o}is the unit of the gyromagnetic ratio and is the Lande factor. In this case, the unit of the gyromagnetic ratio is its value for the orbital motion of the electron in an atom,

*—e/2m*, where

_{e}C*e*is the magnitude of the elementary electric charge,

*m*is the mass of an electron, and

_{e}*c*is the speed of light. The gyromagnetic ratio for nuclei is the analogous quantity for the proton in a nucleus:

*e/m*, where

_{p}C*m*is the mass of a proton.

_{p}The value of the gyromagnetic ratio determines the effect of magnetic fields on a system that has a magnetic moment. According to classical theory, the magnetic moment in an external magnetic field of intensity H undergoes precession, which is uniform rotation about the direction of H, with the preservation of the angle of inclination, at an angular velocity ω = — *γH*. In a special case, in which the magnetic moment is due to the orbital motion of the electrons, Larmor precession results. The characteristic value of magnetic splitting of the energy levels in a magnetic field is determined by the gyromagnetic ratio according to quantum theory. In this case the value is *γhH* = *gγ _{o}hH*, where

*h*is Planck’s constant.

M. A. EL’IASHEVICH