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.

A Transition

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 Transition Pathway

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

A Transition Pathway for an echo
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.

A Different Transition Pathway for an echo

The transition pathway in the density matrix...

The transition pathway in the density matrix

The transition pathway in the rotated density matrix...

The transition pathway in the rotated density matrix

The transition frequency in NMR...

The transition frequency in NMR

Relabel transition symmetry functions...

Relabel transition symmetry functions

Relabel spatial 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...

Rewrite contributions to NMR transition frequencies

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

two fundamental pathways in every NMR experiment

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

Hahn echo in a spin I=1 system

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

Solid echo in a spin I=1 system

This approach can be extended to coupled spins...

extended to coupled spins

Here are the symmetry pathways for HSQC on coupled heteronuclear spins

HSQC on coupled heteronuclear spins

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

quadrupolar nuclei experiencing higher order effects

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

Rotation about a single axis

MQ-MAS is an example of a c4 echo.

MQ-MAS is an example of a c4 echo