Stejskal-Tanner equation ($S/S_0=\exp [-D(\gamma g\delta {)}^2(\Delta -\delta /3)]$)

The Stejskal-Tanner equation relates diffusion-sensitive echo attenuation to a rectangular gradient-pair experiment: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.