Larmor precession ω = γB₀, RF pulses, T1/T2 relaxation & the
physics of MRI
Larmor freq. ω₀
—
MRI frequency
— MHz
Flip angle θ
0°
Mz (longitudinal)
1.00
Mxy (transverse)
0.00
Gyromagn. ratio γ
—
Physics & equations
Nuclear spin precession: nuclei with non-zero spin
(¹H, ¹³C, etc.) behave like tiny magnets. In external field B₀, the
magnetic moment precesses around B₀ at the
Larmor frequency: ω₀ = γB₀, where γ is the gyromagnetic
ratio (e.g. γ/2π = 42.58 MHz/T for ¹H).
RF pulse: a radiofrequency pulse at the Larmor
frequency rotates the net magnetisation M by a flip angle θ =
γ·B₁·τ_pulse. A 90° pulse tips M fully into the transverse plane for
maximum signal; 180° inverts it.
Bloch equations: after the pulse, M relaxes back
via T1 (longitudinal/spin-lattice) and T2 (transverse/spin-spin):
dMz/dt = −(Mz − M₀)/T1 | dMxy/dt = −Mxy/T2
MRI uses gradient fields to spatially encode Larmor
frequencies; tissues differ in T1/T2 → contrast. Clinical scanners:
1.5 T (63.87 MHz) and 3 T (127.74 MHz) for ¹H.
NMR spectroscopy: chemical shifts in Larmor
frequency reveal molecular structure (shielding/deshielding by
electron density).
About Spin Precession
A quantum spin-½ particle in a static magnetic field B0 precesses about the field axis at the Larmor frequency ωL = γB0, where γ is the gyromagnetic ratio (for a proton, γ/2π = 42.577 MHz T⁻¹). This precession is not a classical orbit but a quantum-mechanical evolution of the spin state described by the Bloch equations, which also include longitudinal relaxation (time constant T1) as spin populations return to thermal equilibrium, and transverse relaxation (T2) as phase coherence between spins dephases. These principles underpin nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI), one of the most valuable diagnostic tools in modern medicine.
Visualise the spin vector as a point on the Bloch sphere. Apply a resonant RF pulse at exactly ωL to tip the spin away from B0, then watch T1 and T2 relaxation return it to equilibrium. Adjust field strength, gyromagnetic ratio, and pulse flip angle to explore the full parameter space.
Frequently Asked Questions
What is Larmor precession and what determines its frequency?
Larmor precession is the rotation of a magnetic moment about an applied magnetic field, analogous to a gyroscope precessing in gravity. The precession frequency fL = γB0/2π depends on both the field strength B0 and the gyromagnetic ratio γ of the nucleus. For protons in a 3 T MRI scanner, fL = 42.577 × 3 ≈ 127.7 MHz — in the radio-frequency band, which is why RF coils are used to detect the signal.
What is the Bloch sphere and how does it represent a spin state?
The Bloch sphere is a unit sphere where every point on the surface represents a pure quantum state of a spin-½ system. The north and south poles correspond to spin-up (|↑⟩) and spin-down (|↓⟩) eigenstates of B0; points on the equator are equal superpositions with different relative phases. The Bloch equations describe how the tip of the magnetisation vector M moves on this sphere under the combined influence of B0, RF pulses, and relaxation.
How does a resonant RF pulse work in MRI?
A rotating magnetic field B1 applied at exactly the Larmor frequency ωL exerts a steady torque on the spin in the rotating frame. The spin nutates (tips) away from the z-axis at rate ω1 = γB1. A "90° pulse" of duration t = π/(2γB1) tips the spin to the equator, maximising the detectable transverse magnetisation. A 180° pulse inverts the spin — used in spin-echo sequences to refocus dephasing.
What is the difference between T1 and T2 relaxation?
T1 (spin-lattice or longitudinal relaxation) is the time constant for the z-component of magnetisation to return to its thermal equilibrium value M0 after a perturbation; it involves energy exchange between spins and the surrounding lattice. T2 (spin-spin or transverse relaxation) governs the decay of in-plane coherence without energy exchange — caused by fluctuating local fields from neighbouring spins. Always T2 ≤ T1; in biological tissue at 3 T, grey matter has T1 ≈ 1,300 ms and T2 ≈ 80 ms.
Why must the RF pulse be exactly at the Larmor frequency?
Only an RF field rotating at ωL appears stationary in the rotating reference frame, so it applies a steady tipping torque. At any other frequency, the effective field in the rotating frame has a z-component that prevents efficient nutation — the spin nutates off-resonance and the flip angle is reduced. This resonance condition is why NMR can selectively excite specific nuclei (¹H vs ¹³C) just by tuning the RF frequency.
What is free induction decay (FID)?
After a 90° pulse tips spins to the transverse plane, they precess and induce a sinusoidal voltage in the detection coil. Different spins in a sample precess at slightly different rates due to chemical-shift differences and field inhomogeneities, causing the transverse magnetisation to dephase. The resulting exponentially decaying oscillation picked up by the coil is the FID; its Fourier transform gives the NMR spectrum with peaks at different chemical shifts.
How does MRI contrast work?
MRI contrast between tissues arises primarily from differences in T1, T2, and proton density. By choosing the repetition time TR and echo time TE, radiographers weight the image to emphasise one parameter: short TR/short TE gives T1-weighted images (fat bright, fluid dark); long TR/long TE gives T2-weighted images (fluid bright, useful for detecting oedema). Gadolinium-based contrast agents shorten local T1, enhancing blood vessels and tumours.
What is chemical shift in NMR spectroscopy?
The exact Larmor frequency of a nucleus depends on its local electron environment. Electrons partially shield the nucleus from B0, so chemically distinct hydrogen atoms in a molecule resonate at slightly different frequencies, typically within a range of 10–15 ppm. These chemical shifts, measured in ppm relative to a reference compound (TMS), form the backbone of ¹H NMR spectra used to determine molecular structure — a daily tool in organic chemistry and drug development.
Can spin precession be used outside medical imaging?
Yes — widely. Atomic magnetometers exploit spin precession of alkali-metal vapours to measure magnetic fields with sensitivity below 1 fT Hz⁻½ (femtotesla), enabling brain imaging without superconducting magnets (magnetoencephalography, MEG). Muon spin rotation (µSR) uses implanted muons to probe magnetic order in materials. Nitrogen-vacancy (NV) centres in diamond precess in local fields, enabling nanoscale magnetometry at room temperature for imaging single biological cells.
What is quantum computing's link to spin precession?
Many quantum bit (qubit) implementations are physical spin-½ systems: electron spins in quantum dots, nuclear spins in silicon, and superconducting circuits that behave like artificial spins. Single-qubit gates correspond to RF pulses that rotate the Bloch vector by controlled angles. The same T1 and T2 relaxation times that limit MRI signal determine qubit coherence time — and extending T2 from microseconds to milliseconds is a central challenge in building fault-tolerant quantum computers.
What is spin-orbit coupling and how does it affect precession?
Spin-orbit coupling is an interaction between a particle's spin and its orbital angular momentum, arising relativistically from the particle's motion through an electric field. It adds an effective internal magnetic field that causes spin to precess even without an external B0. In semiconductors like GaAs this leads to Rashba and Dresselhaus splitting, enabling spin manipulation by electric rather than magnetic fields — the basis of proposed spin field-effect transistors (spin-FETs) in spintronics.