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An advanced level version of Nuclear Overhauser effect.

Nuclear Overhauser effect: Noe(Definition) : Change in intensity of a signal when irradiation is carried out at the resonance frequency of a spatially proximal nucleus.

The following discussion is relevant for studies in solution/liquid where the molecules are undergoing rapid isotropic rotational motion.

The Noe enhancement is quantitatively defined as

${\displaystyle \eta ={\frac {-}{}}\qquad Eq.1}$

For a pair of nonidentical spins I and S with dipolar interactions, subject to random perturbation from the environment (lattice), the expressions for the time dependence of the the expectation values of the magnetization calculated by using time dependent perturbation theory are the Solomon equations[1]:

${\displaystyle {\frac {d}{dt}}=-\rho _{I}(-)-\sigma (-)\qquad Eq.2}$
${\displaystyle {\frac {d}{dt}}=-\rho _{S}(-)-\sigma (-)\qquad Eq.3}$
${\displaystyle \sigma }$ is called the cross relaxation rate and is responsible for the Nuclear overhauser effect.
${\displaystyle \rho _{I}={\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}{\big (}J(w_{I}-w_{S})+3J(w_{I})+6J(w_{I}+w_{S}){\big )}\qquad Eq.4}$
${\displaystyle \sigma ={\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}{\big (}-J(w_{I}-w_{S})+6J(w_{I}+w_{S})){\big )}\qquad Eq.5}$

In the steady state ${\displaystyle {\frac {d}{dt}}=0}$; when the resonance frequency of spin I is irradiated , ${\displaystyle =0}$, therefore:

${\displaystyle (-)={\frac {\sigma }{\rho _{S}}}()\qquad (from\quad Eq.3)}$

Assuming that the expectation values of magnetization are proportional to the magnetogyric ratios we obtain:

${\displaystyle \eta ={\frac {-}{}}={\frac {\sigma }{\rho _{S}}}{\frac {\gamma _{I}}{\gamma _{S}}}\qquad Eq.6}$

This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that ${\displaystyle {\frac {\gamma _{I}}{\gamma _{S}}}>1}$, because ${\displaystyle {\frac {\sigma }{\rho _{S}}}\rightarrow 1/2}$ when ${\displaystyle w\tau _{c}<<1}$. However, when ${\displaystyle w\tau _{c}>>1}$, ${\displaystyle {\frac {\sigma }{\rho _{S}}}\rightarrow -1}$ and negative Noe enhancements are obtained.
The sign of ${\displaystyle \eta }$ changes from positive to negative when ${\displaystyle w\tau _{c}}$ is close to one and under such conditions the Noe effect may not be observable. This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.

## References

1. Quantum description of high resolution NMR in liquids. M.Goldman. Oxford.