μ = γ I (for nucleus)
μ = γ S (for electron)
The value of the gyromagnetic ratio (γ) varies by atomic species. The units of γ are typically given in the form of [frequency] ÷ [magnetic field strength], such as (radians/sec)/gauss or MHz/tesla. The reason for these units will become apparent in subsequent Q&As where the famous Larmor equation relating resonance frequency to field strength will be discussed.
Most atomic nuclei have positive gyromagnetic ratios, but a few nuclei and the electron have negative values (γ < 0). When γ > 0 the spin and magnetic moment point in the same direction; for negative γ they are still collinear but lie in opposite directions. Two particles with positive and negative γ's precess in opposite directions.
Advanced Discussion (show/hide)»
For simplicity with the general readership, note that in the description above I have treated I, S, and μ as simple vectors, whereas they are actually quantum mechanical operators.
The collinearity and proportionality between spin angular momentum (I or S) and magnetic moment (μ) is a consequence of a fundamental symmetry relation in quantum mechanics (the Wigner-Eckart Theorem). As a direct result of this theorem, the expectation value of any vector operator of a system is equal to the projection onto its total angular momentum. For a more complete discussion/derivation see the following reference:
Berkeley Dept of Physics, Physics 221A. Lecture Notes: Irreducible Tensor Operators and the Wigner-Eckart Theorem (pdf), 2010.
"Gyromagnetic Ratio", Wikipedia. The Free Encyclopedia.
What is a magnetic moment? Is it the same as a magnetic dipole?
What is spin?
Who was Larmor and how did he discover his famous frequency?