Gyromagnetic ratio

The gyromagnetic ratio (sometimes magnetogyric ratio), &gamma;, is the constant of proportionality between the magnetic moment (&mu;) and the angular momentum(J) of an object:

\boldsymbol {\mu} = \gamma \mathbf{J} \ .$$ Its 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.

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 &mu;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 2&pi; 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 &mu;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 mangeton, where the Bohr magneton is:


 * $$\mu_B = \frac{e \hbar }{2 m_e } = \mathrm{927.400 915 \times 10^{-26}\ J/T}\, $$

with e the elementary charge. If magnetic moment is expressed in units of Bohr magnetons, taking the spin 1/2 of the electron into account, this prediction results in a gyromagnetic ratio of exactly &gamma;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.