Stejskal–tanner equation derived in full†

The pulsed field gradient spin-echo nuclear magnetic resonance method is used to measure the translational diffusion of solvents and solutes in homogeneous and heterogeneous systems. It is widely used in the characterization of molecules and materials in general, via phenomena such as 'diffusion interference'. It is also the basis of tissue discrimination in magnetic resonance imaging, via diffusion tensor imaging. The mathematical equation used to analyze data from a simple noninteracting solute (using a pair of magnetic field gradient pulses that are applied in the experiment) was first derived by Stejskal and Tanner. However, in the article and in subsequent presentations the basic derivation, which we call the the "theoretical physics" of the theory, is not presented in extenso. Conversely, many papers in which the exploration of the effects of magnetic field gradient pulses of shapes other than simple rectangles generally begin with the time-dependent integral that emerges from the original theoretical physics. To fill this "pedagogical gap" we use here a rigorous step-by-step approach to the theoretical physics of the Stejskal-Tanner equation and indicate how it was based on earlier theories. We also take the opportunity to indicate a contemporary approach to deriving new relationships between user-defined magnetic field gradient pulse shapes and the diffusion coefficient; and we show how these can be rapidly and accurately derived using symbolic computation.