 Pulsed field gradient NMR

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 $\varphi =q(x_2-x_1)$. The normalized signal is the ensemble average of $e^{i\varphi }$.

For free diffusion with diffusion coefficient $D$, displacement has a zero-mean normal distribution of variance $2D\Delta$. Its characteristic function givesFor invented values $q=10^5{\mathrm{m}}^{-1}$, $D=10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $\Delta =0.1\mathrm{s}$, the attenuation is $e^{-1}\approx 0.368$. 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.^[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.