Gyromagnetic ratio

The gyromagnetic ratio (sometimes magnetogyric ratio), γ, is the constant of proportionality between the magnetic moment (μ) and the angular momentum(J) of an object:

${\boldsymbol {\mu }}=\pm \gamma \mathbf {J} \ ,$

where the sign is chosen to make γ a positive number.

The units of the gyromagnetic ratio are SI units are radian per second per tesla (s⁻¹·T ^-1) or, equivalently, coulomb per kilogram (C·kg⁻¹). When the object is placed in a magnetic flux density B, because of its magnetic moment it experiences a torque and precesses about the field.

A closely related quantity is the g-factor, which expresses the magnetic moment in units of magnetons: in terms of the gyromagnetic ratio, g = ±γℏ /μ with ℏ the reduced Planck constant and μ the appropriate magneton (the Bohr magneton for electrons and the nuclear magneton for nucleii). The sign of the g-factor is is negative when the magnetic moment is oriented opposite to the angular momentum (it is negative for electrons and neutrons) and positive when the two are aligned the same way (it is positive for protons). More detail is below.

Examples

The electron gyromagnetic ratio is:

$\gamma _{\rm {e}}=2|\mu _{\rm {e}}|/\hbar =\mathrm {1.760\ 859\ 770\ \times \ 10^{11}\ s^{-1}T^{-1}} \ ,$

where μ_e is the magnetic moment of the electron (-928.476 377 x 10^-26 J T^-1), and ℏ is Planck's constant divided by 2π and ℏ/2 is the spin angular momentum.

Similarly, the proton gyromagnetic ratio is:

$\gamma _{\rm {p}}=2\mu _{\rm {p}}/\hbar =\mathrm {2.675\ 222\ 099\ \times \ 10^{8}\ s^{-1}T^{-1}} \ ,$

where μ_p is the magnetic moment of the proton (1.410 606 662 x 10^-26 J T^-1). Other ratios can be found on the NIST web site.

Theory and experiment

The relativistic quantum mechanical theory provided by the Dirac equation predicted the electron to have a magnetic moment of exactly one Bohr magneton, where the Bohr magneton is:

$\mu _{B}={\frac {e\hbar }{2m_{e}}}=\mathrm {927.400915\times 10^{-26}\ J/T} \ ,$

with e the elementary charge. If magnetic moment is expressed in units of Bohr magnetons, the gyromagnetic ratio becomes the g-factor and the magnetic moment becomes:

$\mu _{e}=-\gamma _{e}{\frac {\hbar }{2}}=-{\frac {\mu _{e}}{\mu _{B}}}{\mu _{B}}=g_{e}{\frac {\mu _{B}}{2}}\ ,$

so the gyromagnetic ratio and the g-factor are related as:

$g_{e}=-\gamma _{e}{\frac {\hbar }{\mu _{B}}}\ .$

The value of the g-factor for the electron is:

$g_{e}=-2{\frac {\mu _{e}}{\mu _{B}}}=\mathrm {-2.0023193043622} \ ,$

The Dirac prediction μ_e = μ_B results in a g-factor of exactly g_e = −2. Subsequently (in 1947) experiments on the Zeeman splitting of the gallium atom in magnetic field showed that was not exactly the case, and later this departure was calculated using quantum electrodynamics.

Similarly, the nuclear magneton is defined:

$\mu _{\rm {N}}={\frac {e\hbar }{2m_{\rm {p}}}}=\mu _{e}{\frac {m_{e}}{m_{p}}}=\mathrm {5.050\ 783\ 24\ \times \ 10^{-27}\ J\ T^{-1}} \ ,$

with m_p the mass of the proton, and the proton g-factor is:

$g_{\rm {p}}=2{\frac {\mu _{\rm {p}}}{\mu _{\rm {N}}}}=\mathrm {5.585694713} \ ,$

corresponding to a proton magnetic moment of about 2.79 nuclear magnetons.

This surprising value suggests the proton is not a simple particle, but a complex structure, for example, an assembly of quarks. So far, a theoretical calculation of the magnetic moment of the proton in terms of quarks exchanging gluons is a work in progress, with the present estimate as 2.73 nuclear magnetons.