How MRI Imaging Works (Larmor)

Equivalent forms

\omega_0 = \gamma B_0

f = (\gamma/2\pi)\,B_0

The precession frequency depends only on the field, not on how hard the proton is tipped — so a position-dependent field writes spatial information directly into a frequency the body itself emits.

Symbol	Name	SI unit	Dimension	Range
$f$	Larmor frequencyoutput Resonant precession frequency	MHz	T⁻¹	0 – 300
$B₀$	Main field Static scanner field	T	M·T⁻²·I⁻¹	0.5 – 7
$γ$	Gyromagnetic ratio γ/2π for the nucleus (proton 42.58 MHz/T)	MHz/T	M⁻¹·T·I	10 – 60

Unit systems

Symbol	SI	CGS	Imperial
$f$	MHz	MHz	MHz
$B₀$	T	G	T
$γ$	MHz/T	MHz/T	MHz/T

Where it holds

Valid for nuclei with non-zero spin in a uniform

B_{0}

within the linear (non-relativistic) regime. Chemical shift

(\sim ppm)

and relaxation (T1, T2) add structure exploited in spectroscopy; very high fields need careful RF (SAR) limits.

Dimensional analysis

$(\gamma /2\pi )\cdot B_{0} = (MHz/T)\cdot T = MHz \checkmark$

Discovery

Isidor Rabi / Paul Lauterbur / Peter Mansfield · 1973

Rabi measured nuclear precession in 1938 (NMR). Lauterbur and Mansfield later showed that adding magnetic-field gradients encodes spatial position into the Larmor frequency, turning NMR into imaging — the MRI scanner, recognized with the 2003 Nobel Prize in Medicine.

Try this

A giant magnet maps the soft tissue inside you without a single X-ray. What makes your own protons sing back a signal?

In a 1.5 T MRI scanner, at what radio frequency do hydrogen protons precess? (proton γ/2π = 42.58 MHz/T)

Research status: stable

Real-world applications

Clinical MRI soft-tissue imaging
NMR spectroscopy for chemical structure
Functional MRI (fMRI) brain mapping
Magnetic resonance angiography

Common misconceptions

MRI uses ionizing radiation like X-rays — it uses harmless radio waves and magnetism
The precession frequency depends on how hard you tip the spin — it depends only on $B_{0}$
Stronger magnets just make bigger machines — they raise f and signal-to-noise, improving resolution

Experimental verification

Rabi's molecular-beam resonance (1938) and every NMR spectrometer confirm

f = \gamma B_{0}/2\pi to

extraordinary precision; the proton at 1.5 T resonates at 63.87 MHz exactly as predicted, and clinical scanners tune their RF to that line daily.

Derivation

A magnetic moment µ in field B feels torque

\tau

= µ

\times B = dL/dt

with µ

= \gamma L

.

This gives

dL/dt = \gamma L\times B

, the equation of a vector precessing about B at angular rate

\omega _{0} = \gamma B_{0}

— independent of the tip angle.

Dividing by

2\pi

gives

f = \gamma B_{0}/2\pi

.

Limiting cases

B_{0} \to 0

⟶

f \to 0

No field, no preferred axis, no precession — and no MRI signal.

Add gradient G⟶

f(x) = (\gamma /2\pi )(B_{0}+Gx)

Position-dependent field makes frequency encode location — the key to imaging.

Different nucleus⟶ f scales with

\gamma

^{13}C

or ^{31}P

resonate at their

own \gamma

— the basis of spectroscopic imaging.

What if…

What if you upgrade from 1.5 T to 3 T?

Larmor frequency doubles $to \sim 128\,\mathrm{MHz}$ and signal-to-noise roughly doubles — sharper images, but stricter RF heating limits.

What if you image

^{31}P

instead

of ^{1}H

?

$Its \gamma is \sim 2.5\times$ smaller, so it resonates at a lower frequency — used in metabolic phosphorus spectroscopy.

What if you add a spatial field gradient?

Frequency becomes a map of position; Fourier-analyzing the received signal reconstructs the image.

1

Protons at 1.5 T

Given ·

γ:: 42.58
B₀:: 1.5

Find · f

Steps

Use the proton constant $\gamma /2\pi = 42.58\,\mathrm{MHz/T}$
$f = 42.58 \times 1.5 = 63.87\,\mathrm{MHz}$
That's why 1.5 T scanners transmit/receive near 64 MHz

Result ·

f = 42.58\,\mathrm{MHz/T} \times 1.5\,\mathrm{T} = 63.87\,\mathrm{MHz}

Larmor Formula→Cyclotron Frequency→Magnetic Dipole Moment & Torque→