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Cross-polarization magic-angle spinning, abbreviated CPMAS, combines two operations in solid-state nuclear magnetic resonance. Cross-polarization (nuclear magnetic resonance) transfers nuclear spin polarization from a more strongly detected spin population, commonly protons, to a less sensitive one such as carbon-13. Magic angle spinning rotates the sample to average appropriate orientation-dependent interactions. Signal enhancement and spectral narrowing address different experimental obstacles.

The angle is fixed by a geometrical condition:This nulls the relevant second-rank angular factor under the averaging assumptions; it does not remove every interaction or every source of linewidth. Chemical shift anisotropy and sample motion must still be interpreted within the actual experiment.

In their 1976 paper, Jacob Schaefer and Edward O. Stejskal compared carbon-13 spectra of solids obtained with cross-polarization (nuclear magnetic resonance), with and without magic angle spinning, against solution spectra. The polysulfone example showed how the combined method could reveal much more resolved structural detail in a solid.^[ref]

For an original comparison, consider two carbon sites with equal populations but different efficiencies of polarization transfer. Their measured peak areas can differ even when their populations are equal. Changing contact time or relaxation conditions can change that weighting. NMR signal integration in a CPMAS spectrum therefore requires a justified acquisition model before peak areas are treated as direct composition measurements.

Jacob Schaefer was a physical chemistry researcher who developed methods for studying complex solid materials using solid-state nuclear magnetic resonance. After his 1964 doctorate he worked at Monsanto, then joined Washington University in 1986. His contributions included cross-polarization magic-angle spinning with Edward O. Stejskal and rotational-echo double resonance with Terry Gullion.^[ref]

The materials in the 1976 cross-polarization magic-angle spinning demonstration made the method's reach tangible. They included polysulfone, ebony wood and a cylinder cut from an ivory billiard ball. Schaefer's father and grandfather were billiards champions; a familiar object became a specimen for measuring carbon-13 responses in a complex solid. Combining polarization transfer with magic angle spinning made weak and broadened solid-state responses much more accessible.^[ref]

REDOR addresses another measurement problem. Magic angle spinning averages interactions that broaden spectra, but some of those interactions carry structural information. Appropriately timed radio frequency pulses restore sensitivity to NMR dipolar coupling between different nuclear species. Comparing a reference echo with a deliberately dephased echo then probes that coupling, with structural interpretation supplied by a suitable spin model.^[ref]

Schaefer reproduced a laboratory record from 29 May 1987 in which Gullion measured carbon-13 spectra of nitrogen-15-labelled alanine. Carbon remained at natural isotopic abundance. The record compares full and dephased echoes at a carbon frequency of $50.3\mathrm{MHz}$, using magic angle spinning at $3.2\mathrm{kHz}$ and 10 rotor periods of dipolar evolution. The corresponding duration isThe reproduced record does not give sample mass or number of accumulated scans. Later use of XY8 NMR phase cycling suppressed pulse-train amplitude and phase errors, enabling measurements of weaker couplings over longer pulse trains.^[ref]

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.