= Stejskal-Tanner equation
{c}
{title2=$S/S_0=\exp[-D(\gamma g\delta)^2(\Delta-\delta/3)]$}
The <Stejskal-Tanner equation> relates diffusion-sensitive echo attenuation to a rectangular gradient-pair experiment:
$$
\frac{S}{S_0}=e^{-bD},\qquad b=(\gamma g\delta)^2\left(\Delta-\frac{\delta}{3}\right).
$$
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.<Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient by E. O. Stejskal and J. E. Tanner>[Pages289–290: Theory and rectangular-gradient case]{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>, so
$$
b=\frac{q_0^2\delta}{3}+q_0^2(\Delta-\delta)+\frac{q_0^2\delta}{3}=q_0^2(\Delta-\delta/3).
$$
The two physical gradients can have the same polarity: the intervening $180^{\circ}$ pulse reverses the effective phase-encoding sign.
\Image[/spin-echo-gradient-timing.png]
{title=Reused original illustration of a refocused gradient pair and its accumulated encoding}
{id=stejskal-tanner-equation-gradient-timing}
{description=The chosen illustrative timings are not a reconstruction of a historical run. Both physical gradient pulses have the same sign; refocusing reverses their effective encoding sign. Squaring the lower trace and integrating gives the finite-duration correction.}
For the illustrated $\delta=2\,\mathrm{ms}$ and $\Delta=30\,\mathrm{ms}$, the normalized integral $b/q_0^2$ is $29\tfrac{1}{3}\,\mathrm{ms}$, 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>.<Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient by E. O. Stejskal and J. E. Tanner>[Pages288 and291–292: motivation, apparatus and experiments]{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.
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