IMG-01

Larmor frequency

ω = γ B0, f = γ B0 / 2π. Precession of a spin in B0.

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MRILarmor

Governing equation

\omega=\gamma B_0,\quad f=\gamma B_0/2\pi

$B_0 \in \left[0.2, 7\right]$

where

B_0: Field (T)
\gamma/2\pi: Gyromagnetic ratio (MHz/T)
f: Larmor frequency (MHz)
\omega: Angular frequency (rad/s)

The figure, part by part

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①

Governing law — red

\omega=\gamma B_0,\quad f=\gamma B_0/2\pi

$B_0 \in \left[0.2, 7\right]$

$B_0 = 0.2$ $f = 8.516$

Along the curve

②

Domain of the variable — blue

B_0_{\min} = 0.2,\quad B_0_{\max} = 7

$B_0 \in \left[0.2, 7\right]$

$B_0 = 0.2$ $B_0 = 0.2$

Along the curve

③

Response curve — green

\omega=\gamma B_0,\quad f=\gamma B_0/2\pi

$B_0 \in \left[0.2, 7\right]$

$B_0 = 0.2$ $f = 8.516$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f}{\mathrm{d}B_0}

$B_0 \in \left[0.2, 7\right]$

$B_0 = 0.243$ $f^{\prime} = 42.58$

Along the curve

Lecture brief

Historical brief

Larmor precession, CT Beer projections, SNR and Nyquist sampling are why MRI and CT images exist as numbers. The lab computes frequency, dose and resolution limits. This sheet (IMG-01 — Larmor frequency) is the form associated with Larmor. Working symbols: $B_0$ , $\gamma/2\pi$ $\rightarrow$ $f$ , $\omega$ . For ¹H, γ/2π ≈ 42.58 MHz/T. At 1.5 T this is about 64 MHz; at 3 T about 128 MHz.

Purpose

Purpose: compute

f

\omega

from

B_0

\gamma/2\pi

in Medical imaging via

\omega=\gamma B_0,\quad f=\gamma B_0/2\pi

ω = γ B0, f = γ B0 / 2π. Precession of a spin in B0. Use it when a real medical imaging question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

B_0 = 1.500\,\mathrm{T}

\gamma/2\pi = 42.580\,\mathrm{MHz/T}

, the governing relation

\omega=\gamma B_0,\quad f=\gamma B_0/2\pi

yields

f = 63.870\,\mathrm{MHz}

\omega = 4.0e+8\,\mathrm{rad/s}

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Calculator

Inputs

Field B_01.5 TGyromagnetic ratio \gamma/2\pi42.58 MHz/T

Outputs

Larmor frequency f63.870 MHz
Angular frequency \omega401307045.6 rad/s

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Narration of this film

A spinning arrow, a field, a circling tip.

For ¹H, γ/2π ≈ 42.58 MHz/T. At 1.5 T this is about 64 MHz; at 3 T about 128 MHz.

Reading speed

Watch on YouTube