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.

The diffusion weighting b-value quantifies the encoding applied by a magnetic field gradient waveform in a diffusion measurement. For a refocused experiment in one spatial direction, defineHere $\gamma$ is the angular gyromagnetic ratio, in radians per second per tesla, and $G_{\mathrm{eff}}$ includes the sign reversal caused by the refocusing pulse. The echo time is $T$, with $q(T)=0$ for refocusing of stationary spins. Thus $q$ has units ${\mathrm{m}}^{-1}$ and $b$ has units $\mathrm{s}{\mathrm{m}}^{-2}$. The squared encoding history matters even though its final value is zero.^[ref]

For unrestricted, isotropic Gaussian diffusion with constant diffusion coefficient $D$, the otherwise matched signal ratio is $S/S_0=e^{-bD}$. Here $S_0$ uses the same pulse timing and detection conditions with the applied diffusion-encoding gradients removed. The diffusion weighting b-value describes the experiment's weighting; $D$ describes the assumed sample model.

For an original unit check, $1000\mathrm{s}\mathrm{m}{\mathrm{m}}^{-2}=10^9\mathrm{s}{\mathrm{m}}^{-2}$. With $D=10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the ideal attenuation is $e^{-1}\approx 0.368$. Quoting a numerical $b$ without its units can therefore cause a millionfold scaling error.

The Stejskal-Tanner equation gives $b=(\gamma g\delta {)}^2(\Delta -\delta /3)$ for its equal rectangular pulses. More general waveforms require their own integral. For direction-dependent diffusion, a scalar $b$ alone cannot specify the complete directional encoding and response; a tensor description is needed.

The Stejskal-Tanner equation relates diffusion-sensitive echo attenuation to a rectangular gradient-pair experiment:It assumes free Gaussian diffusion with a constant scalar diffusion coefficient $D$, ideal refocusing and two equal nonoverlapping rectangular gradient pulses of magnitude $g$ and duration $\delta$, whose starting times are separated by $\Delta$. The gyromagnetic ratio $\gamma$ is angular. The baseline $S_0$ is an otherwise matched echo without the applied diffusion-encoding gradients. These definitions are part of the formula, rather than optional choices of notation.^[ref]

The finite-duration term follows directly from the encoding history. Write $q_0=\gamma g\delta$. The effective encoding rises linearly from zero to $q_0$, stays constant for $\Delta -\delta$, then falls linearly to zero. Each ramp contributes $q_0^2\delta /3$ to the diffusion weighting b-value, soThe two physical gradients can have the same polarity: the intervening $180^{\circ }$ pulse reverses the effective phase-encoding sign.

For the illustrated $\delta =2\mathrm{ms}$ and $\Delta =30\mathrm{ms}$, the normalized integral $b/q_0^2$ is $29\frac{1}{3}\mathrm{ms}$, rather than $30\mathrm{ms}$. This isolates the pulse-duration correction without assigning an unmeasured gradient amplitude.

The 1965 work combined theory with apparatus tests, including slow self-diffusion in dry glycerol.^[ref] Restricted diffusion, flow, background gradients and measurement noise require their own interpretation; fitting a single attenuation to this equation does not establish all of its assumptions.