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*10⁸ 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.

From the geometry of the situation (the arc length
|dμ| = radius.|dφ|), we may write

|dμ| = μsinθ|dφ| (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 = γμBsinθdt (Eq. 6)

Eq. 5 and 6 combine to produce γBdt = dφ with B ≡ |B|. Since the rate of change of φ is the angular precessional frequency
(ω = -dφ/dt (the rotation is left-handed in the direction of B)), we can use Eq. 5 and 6 to yield

ω = -γB (Eq. 7)

If the external magnetic field is constant (B₀) then φ = -ω₀t + φ₀, where φ₀ is the initial angle, and since ω = -dφ/dt = -(-ω₀) we may write the Larmor equation:

ω₀ = γB₀

If the scalar Larmor frequency f₀ is calculated instead of the angular Larmor frequency ω₀, and γ is used (γ = γ/2π): f₀ = γB₀. This is more convenient, and Larmor frequencies can be calculated in MHz.