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 *W*_{2} (small molecules)
or zero-quantum transitions *W*_{0} (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:

*R*_{auto} is called the auto relaxation constant which defines the leakage of
magnetisation to the lattice.

The cross relaxation constant *R*_{cross} expresses the exchange of magnetisation between
two dipolar coupled spins.

Depending on the size of the molecules (more precisely, the correlation time *τ _{c}*) 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

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 *T*_{1ρ} 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 *W*_{0} and *W*_{2} 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.