# Larmor Equation - Classical Derivation

**Part 1—general Bloch equation.**

We need three basic equations to get us going. The direct relationship between the magnetic moment (μ) and the spin angular momentum vector (S) is, from experiment

μ = γS (Eq. 1)

where the gyromagnetic ratio γ = 2.675*10^{8} rad s^{-1} T^{-1}.

(Note ~~γ~~ = γ/2π = 42.58 MHz T^{-1}. T is the *Tesla* unit of magnetic field.)

When this magnetic moment is in the presence of an external magnetic field B it experiences a torque and precesses around the field. The formula for the net torque (N) on any current distribution is

N = μ x B (cross product, Eq. 2)

Nonzero total torque on a system implies that the system's total angular momentum (arising from spin only, in this discussion) must *change* according to

dS/dt = N (Eq. 3)

From Eq. 1 and 2 we see that Eq. 3 reduces to the general Bloch equation:

dμ/dt = γμ x B (Eq. 4)

Since (by cross-product definition) dμ/dt is perpendicular to both μ and B, then in the event that μ and B are not aligned (e.g. after energy input into the system which drives the magnetised spin system into a state of resonance), μ must move in a circular path. This is precession.

Now see **Part 2—consider geometry**.

**Part 2—consider geometry.**

Precession of magnetic moment μ around an external magnetic field B

From the geometry of the situation (the arc length

|dμ| = radius.|d*φ*|), we may write

|dμ| = *μ*sin*θ*|d*φ*| (Eq. 5)

From the general Bloch equation (Eq. 4 from Part 1: dμ/dt = γμ x B) we may therefore also state

|dμ| = γ|μ x B|dt = γ*μB*sin*θ*dt (Eq. 6)

Eq. 5 and 6 combine to produce γ*B*dt = d*φ* with *B* ≡ |B|. Since the rate of change of *φ* is the angular precessional frequency

(*ω* = -d*φ*/dt (the rotation is left-handed in the direction of B)), we can use Eq. 5 and 6 to yield

ω = -γ*B* (Eq. 7)

If the external magnetic field is constant (*B*_{0}) then *φ* = -ω_{0}t + *φ*_{0}, where *φ*_{0} is the initial angle, and since ω = -d*φ*/dt = -(-ω_{0}) we may write the Larmor equation:

**ω _{0} = γB_{0}**

If the scalar Larmor frequency *f*_{0} is calculated instead of the angular Larmor frequency ω_{0}, and ~~γ~~ is used (~~γ~~ = γ/2π): *f*_{0} = ~~γ~~*B*_{0}. This is more convenient, and Larmor frequencies can be calculated in MHz.

Further reading on this topic:

Books: MRI: Physical Principles & Sequence Design p29, MRI From Picture to Proton p136