Larmor Precession: Individual Spins in Thermal Equilibrium

ω₀ = γB₀ (Larmor frequency = gyromagnetic ratio × magnetic field)

Key Concept: Individual nuclear spins exist in quantum states (parallel or anti-parallel to B₀) but quantum mechanics requires them to precess around B₀ at the Larmor frequency. At thermal equilibrium, spins are randomly distributed around the precession cone, resulting in zero net transverse magnetization.

Magnetic Field (B₀): 1.5 T

Temperature: 300 K

Animation Speed: 1.0x

Number of Spins: 8

Larmor Frequency: 63.9 MHz Population difference: 0.0098% Net M_z: +0.005%

Individual Spins (3D View)

Green: spin-up (parallel to B₀)
Red: spin-down (anti-parallel to B₀)

Transverse Plane (x-y)

Random phase distribution → zero net signal
Purple vector: net transverse magnetization

Energy Level Populations

Boltzmann distribution determines populations
ΔE = γℏB₀

Spin Array View

Individual spins in spatial array
Color = phase, arrow = transverse component

Quantum Mechanical Picture:

Spin states: |↑⟩ (m = +½) parallel to B₀, |↓⟩ (m = -½) anti-parallel to B₀
Energy difference: ΔE = γℏB₀ = ℏω₀
Population ratio: N↓/N↑ = exp(-ΔE/kT) ≈ 1 - γℏB₀/kT
Precession: Both states precess at ω₀, but with opposite z-components
Phase coherence: At equilibrium, phases are random → no net transverse magnetization
RF excitation: Creates phase coherence and population inversion