The Stejskal–Tanner equation generalized for any gradient shape—an overview of most pulse sequences measuring free diffusion

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Abstract
The widely known Stejskal–Tanner (ST) equation is central to any diffusion NMR or so‐called pulsed gradient spin‐echo experiment, describing the signal attenuation due to loss of coherence caused by diffusion as a function of the experimental parameters. What is less widely known is that the equation itself is not invariable when applying different pulse sequences or different gradient pulse shapes and should thus be modified accordingly. This concept is not new, but nevertheless experimentalists applying diffusion NMR in their research often overlook or are completely oblivious to this fact. In this article, the derivation of the ST equation through the Bloch–Torrey equations is discussed in detail, followed by a discussion of the most basic NMR experiments that measure free diffusion. This derivation is performed here in a novel way, leading to expressions for the ST equation that do not assume any gradient shape beforehand, leaving only a few parameters that are determined solely by the gradient pulse shape. This new approach to presenting the ST equation increases awareness of its dependence on gradient shape. Moreover, it relieves designers of future diffusion NMR pulse sequences of struggling with the cumbersome task of deriving and reporting the equation for each gradient shape separately. A quick reference table of the ST equation for the basic diffusion NMR pulse sequences for any gradient shape is given, as well as an overview for most other diffusion NMR pulse sequences described in the literature. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40A: 39–65, 2012.

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