Next: Imaging parameters Up: Signal and noise in Previous: Introduction Contents

The MR signal

The MR signal is the electromagnetic force (emf) induced in a coil by a rotating magnetization. The larger this magnetization, the larger the MR signal. The magnetization M₀ per unit volume is given by [6]:

M₀	=	$\displaystyle N_s \sum_{m=-I}^I \rho_m \hbar\gamma m$	(2.1)
	=	$\displaystyle N_s {\gamma^2 \hbar^2 I(I+1) B_0 \over 3 k_b T_s}\quad ,$	(2.2)

with N_s the number of spins at resonance per unit volume.
An elegant way to determine the emf was proposed by Hoult and Richards [8]:

$\begin{displaymath} S(t) = -{\partial \over \partial t} \left(\vec B_1\cdot \vec M_0\right)\quad . \end{displaymath}$

(2.3)

Here, $\vec B_1\equiv\vec B_1(\vec r, t)$ denotes the magnetic induction, produced by a coil carrying unit current at $\vec M_0\equiv M_0(\vec r)$ . For a sample with volume V_s the emf becomes:

$\begin{displaymath} S(t) = -\int\limits_{V_s} {\partial \over \partial t} \left(\vec B_1 \cdot \vec M_0\right) d \vec r\quad . \end{displaymath}$

(2.4)

If $\vec B_1$ is assumed to be homogeneous over the sample, the signal is easily seen to be:

$\begin{displaymath} S(t) = K \omega_0 B_{1, xy} M_0 V_s \cos\omega_0 t \quad , \end{displaymath}$

(2.5)

with $\omega_0$ the angular frequency with which the magnetization $\vec M_0$ precesses in the main magnetic induction $\vec B_0$ . The factor K includes possible inhomogeneities of $\vec B_1$ over the sample.

Imaging parameters

Next: Imaging parameters Up: Signal and noise in Previous: Introduction Contents

Jan Sijbers
1999-01-04