= Pulsed field gradient NMR
= PFG NMR
{c}
{synonym}
<Pulsed field gradient NMR>, or <PFG NMR>, uses timed <magnetic field> gradients to encode position in the <phase> of a <nuclear magnetic resonance> <signal>. An appropriately refocused pair of encoding episodes cancels the position-dependent <phase> for stationary spins. Motion between the episodes can prevent cancellation, making the echo sensitive to displacement.
For an original narrow-pulse derivation in one spatial direction, let each effective encoding pulse have gradient magnitude $g$ and short duration $\delta$, and write $q=\gamma g\delta$, where $\gamma$ is the angular <gyromagnetic ratio>. Neglect displacement during each pulse. For positions $x_1,x_2$ separated by an observation interval $\Delta$, a choice of encoding sign gives $\phi=q(x_2-x_1)$. The normalized signal is the ensemble average of $e^{i\phi}$.
For free <diffusion> with <diffusion coefficient> $D$, displacement has a zero-mean <normal distribution> of variance $2D\Delta$. Its <characteristic function> gives
$$
\frac{S(q)}{S(0)}=\left\langle e^{iq(x_2-x_1)}\right\rangle=e^{-Dq^2\Delta}.
$$
For invented values $q=10^5\,\mathrm{m^{-1}}$, $D=10^{-9}\,\mathrm{m^2\,s^{-1}}$ and $\Delta=0.1\,\mathrm s$, the attenuation is $e^{-1}\approx0.368$. Finite pulse duration, restricted motion and other sequence details require their corresponding model rather than automatic use of this narrow-pulse result.
The 1965 experiment of <E. O. Stejskal> and <J. E. Tanner> combined a derivation for time-dependent gradients with experimental tests. Reducing gradients during radio-frequency pulses and echo detection relaxed practical limitations of a continuously strong gradient; the authors measured slow <diffusion> in dry <glycerol>.<Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient by E. O. Stejskal and J. E. Tanner>[Page288, Abstract; Theory and experimental tests]{ref}
<PFG NMR> can also use a <stimulated echo> for storage between encoding periods. Distinguishing motion-induced attenuation from other signal losses is part of the measurement, not a consequence of the method's name alone.
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