Stejskal–tanner equation derived in full

Abstract

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. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40A: 205–214, 2012.

Structure and Dynamics of Confined C-O-H Fluids Relevant to the Subsurface: Application of Magnetic Resonance, Neutron Scattering, and Molecular Dynamics Simulations

“…where I is the observed signal intensity for the gradient power g, I 0 is signal intensity for zero gradient power g, D is diffusion coefficient, γ is gyromagnetic ratio of the observed nuclei, δ is duration of PFG, and is diffusion time (Kuchel et al, 2012).…”

Diffusion on a molecular scale as observed using PGSE NMR

“…[3], the sign of the accumulated phase is effectively reversed to become 1cB 0 s [Ref. The evolution of the accumulated precessional phase begins at t 5 0 when the p=2 RF pulse is applied.…”

“…MðtÞ ¼ Mð0Þexpð2i/ðtÞÞ [3] with /ðtÞ being the accumulated precessional phase in the interval (0,t). If B 0 is the only applied magnetic field, /ðtÞ ¼ x o t.…”

“…[3], the sign of the accumulated phase is effectively reversed to become 1cB 0 s [Ref. [3), Fig. 2, the red line].…”

“…[5] and [6] corresponds to the effect of the p RF pulse. The above two equations use the approximation that the gradient field seen by the diffusing spin is a constant during the time d, which requires that the spin does not diffuse a significant distance during time d. Kuchel et al (3), develop the detailed calculations that are needed to remove this approximation. If the spin was stationary, the phase angles accumulated during the first and second gradient pulses would cancel.…”