Symmetry Pathways in Solid-State NMR

We have introduced a new concept into magnetic resonance called Symmetry pathways, which is a generalization of the coherence transfer pathway concept. It provides a simple and consistent framework for understanding (and even designing) many NMR experiments.

The symmetry pathways approach begins by first identifying the energy levels and transitions between various energy levels in the system.

Every NMR experiment involves at least one transition pathway. Below is an illustration of the simplest possible transition pathway involving a single NMR transition.

A slightly more complicated example is the transition pathway for a spin echo involving the same two states.

Because the two transition frequencies are equal in magnitude but opposite in sign this experiment generates a spin echo, that is, an unwinding of the t₁ evolution during t₂.

Of course, the more interesting transition pathways in NMR involve more than two states.

The transition pathway in the density matrix...

The transition pathway in the rotated density matrix...

The transition frequency in NMR...

Relabel transition symmetry functions...

Relabel spatial symmetry functions...

Rewrite contributions to NMR transition frequencies in terms of symmetry functions. Here are the first-order nuclear shielding and quadrupolar coupling contributions...

Thus, there are two fundamental pathways in every NMR experiment...

Here is an example of the Hahn echo in a spin I=1 system...

Here is an example of the Solid echo in a spin I=1 system...

This approach can be extended to coupled spins...

Here are the symmetry pathways for HSQC on coupled heteronuclear spins

This approach can be extended to quadrupolar nuclei experiencing higher order effects...

Rotation about a single axis is an example of a spatial pathway..

MQ-MAS is an example of a c4 echo.