NMR Tutorial Nuclear Overhauser Effect

The Nuclear Overhauser Effect or NOE is the enhancement of one spin's signal due to magnetisation transfer via cross relaxation in a dipolar coupled spin system. In the steady state NOE one transition between two spins in an AX system is selectively saturated by an rf-pulse. This disturbs the equilibrium population, and the system will start to relax to a new equilibrium state. If dipolar couplings are present the populations will be replenished or depleted through double-quantum transitions W2 (small molecules) or zero-quantum transitions W0 (large molecules).

The transfer of magnetisation between spins is described by the Solomon equations, which can be derived from the relaxation matrix by transformation into a basis of Zeeman eigenstates. For two-spin system they are given by:

Rauto is called the auto relaxation constant which defines the leakage of magnetisation to the lattice.

The cross relaxation constant Rcross expresses the exchange of magnetisation between two dipolar coupled spins.

Depending on the size of the molecules (more precisely, the correlation time τc) the magnetisation of the second spin will be enhanced or reduced. For medium sized molecules the zero- and double-quantum transition probabilities cancel and the NOE is not observable.
In the fast motion limit the efficiency εNOE = γI/2γS is given by the quotient of gyromagnetic ratios of the coupled spins.

The disappearing NOE for medium-sized molecules can be circumvented by measuring relaxation during the application of a spin lock field. Under these conditions the spins are held constant along the axis of an applied rf-field in the x,y-plane of the rotating frame and kept from precessing. Under the influence of this decoupling field the spins relax with a time constant T1ρ towards equal population of α and β states and zero magnetisation.

This is called the rotating frame Overhauser effect (ROE) with the auto and cross relaxation constants

and

Since both W0 and W2 have positive coefficients the ROE never disappears for any correlation time.

Since the dipolar coupling and hence the cross relaxation is proportional to the distance r-6 the NOE and ROE can be used for measuring distances between spins. This is done in the NOESY and ROESY experiments. Furthermore, the NOE is employed for signal enhancement during heteronuclear decoupling.