# Larmor Equation

For a particular magnetic field strength, the (resonant) frequency of precession can be calculated via the Larmor equation. The Larmor equation may be derived using classical physics or quantum mechanics. It links these two descriptions of nuclear magnetic resonance, and is fundamental to MRI physics. The Larmor equation is

**ω _{0} = γ B_{0}**

where ω_{0} is the angular frequency of precession of protons in an external magnetic field, γ is a proportionality constant called the gyromagnetic ratio, and *B*_{0} is the strength of the external magnetic field. γ = 2.67*10^{8} radians s^{-1} T^{-1}. In some texts the right-hand side of the equation is negative; this simply depends on the direction in which a positive rotation is defined.

We may also write *f*_{0} = ~~γ~~ *B*_{0}, where ~~γ~~ = 42.56 MHz T^{-1} (~~γ~~ = γ/2π).

**Example:** the Larmor frequency *f*_{0} for a 1.5 T magnet is 1.5(T)*42.56(MHz T^{-1}) = 63.8 MHz.

Further reading on this topic:

Books: MRI From Picture to Proton p107, MRI The Basics p26

Online: Hyperphysics