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John E. Tanner was a physical chemistry researcher whose work with Edward O. Stejskal established pulsed field gradient NMR as a method for measuring self-diffusion. He studied physical chemistry at Oberlin, Indiana and Wisconsin, and later worked at Idaho National Laboratory. His early NMR work received renewed recognition with an honorary fellowship in 2016.^[ref]

The appeal of pulsed field gradient NMR was practical as well as theoretical: strong magnetic field gradients could encode motion without remaining strong during excitation or detection. In a 2017 interview with Derek Jones, Tanner recalled choosing Stejskal's laboratory because it had a magnet and because he wanted to measure slow diffusion. He also described rebuilding the probe to deal with eddy currents generated by switching the magnetic field gradients.^[ref]

Tanner made restricted diffusion experimentally concrete. He described assembling 99 layers of mica with aluminium foil spacers, carefully removing wrinkles and burrs to obtain regular gaps. Such a controlled geometry lets an echo probe displacement between known barriers, rather than treating every loss of NMR signal as free diffusion with a smaller diffusion coefficient.^[ref]

His 1970 work used a stimulated echo to extend the observation interval when spin-lattice relaxation is slower than transverse relaxation. The interval can store the relevant pattern as longitudinal magnetization, rather than keeping it transverse throughout.^[ref] For an illustrative ideal comparison, take $T_1=1\mathrm{s}$, $T_2=0.1\mathrm{s}$ and storage time $T=0.5\mathrm{s}$. The storage-dependent nuclear spin relaxation factors are $e^{-T/T_1}\simeq 0.607$ and $e^{-T/T_2}\simeq 0.00674$. Even allowing the ideal three-pulse pathway's initial factor of $1/2$, the former factor is about 45 times larger. This compares only that storage contribution: pulse losses, the other transverse intervals and diffusion attenuation still matter.^[ref]

Restricted diffusion is diffusion whose displacement statistics are constrained by boundaries that particles cannot cross. A reflecting boundary condition models an impermeable wall: particles continue moving, but there is no normal diffusive flux through it. Finite permeability would require a different boundary model.

A physical demonstration uses a liquid confined between solid lamellae. In his first-person account, John E. Tanner described constructing 99 mica layers separated by aluminium-foil strips, removing wrinkles and burrs to obtain regular spacing; Edward O. Stejskal inspected the stack with a stereomicroscope. Pulsed magnetic-field gradients encode displacements in the nuclear magnetic resonance signal, allowing motion normal to the barriers to be compared with less confined directions.^[ref] This is a material confinement experiment, not a change in the liquid's chemistry.

For an original one-dimensional model, let particles diffuse in $0\le x\le L$ with constant diffusion coefficient $D>0$:The wall conditions are homogeneous Neumann boundary conditions for the diffusion equation. Start in equilibrium, so $X(0)$ has the uniform distribution on $[0,L]$. Its variance isAt long time separation, the positions decorrelate. Consequently the mean squared displacement approachesAt short times relative to $L^2/D$, most displacements have not encountered a wall, and the leading scale is the free-diffusion result $2D\tau$. For a hypothetical gap $L=10\mu \mathrm{m}$, the long-time plateau is $100/6\mu {\mathrm{m}}^2$ and its root is about $4.08\mu \mathrm{m}$.

The initial condition matters: fixing $X(0)=L/2$ instead gives the long-time value $L^2/12$. Likewise, echo attenuation in a confined sample must not automatically be converted into a free-diffusion coefficient. Confinement geometry, timing and the selected boundary model all affect the inference.

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.