NUCLEAR MAGNETIC RESONANCE 109
tions and their relaxations may be between the same or different type of
nucleus, but in either case they are chemically shifted from the spin I. At
equilibrium, some relative spin I populations exist and the fractional differ-
ence in populations between the two energy states can be written (remember-
ing thatex is approximated by 1 + x ) as
()NN
NkTII
I−+I
−−
=
γB 0
(3.29)A similar expression for the difference of the two spin S populations can be
written as
()NN
NkTSS
S−+I
−−
=
γB 0
(3.30)Now if S is strongly irradiated, then it is saturated and S is no longer at its
Boltzmann equilibrium. Therefore it cannot maintain the Boltzmann equilib-
rium of spinsI and the intensity of the I signal is changed. Equalizing S popu-
lations produces a proportional change inI populations such that equation
3.31 can be written in which ηIS is called the nuclear Overhauser enhancement
(NOE) factor.
Intensity with irradiated
Normal intensityS
IS
IkT kTI
=+
γφγBB 00γγφγ
γ
ηIB
I
I01
1
kT
SS=+
=+ (3.31)
Two factors contribute to ηIS. One is the ratio of the magnetogryric ratios of
the two different spins, and the other depends on relaxation mechanisms.
Provided that the relaxation mechanism is purely dipole – dipole, φ has the
value 1/2. If other relaxation mechanisms affect spin I , then φ may approach
zero. Assuming that the dipolar mechanism is operational (no quadrupolar
nuclei withI > 1/2 are present), η has the value γS /2 γI and is regarded as ηmax.
In the homonuclear case we haveηmax = 1/2. Usually one chooses nuclei where
γS > γI to ensure that the NOE is signifi cant. For observation of^13 C for instance,
if the protons in the molecule are double irradiated, the ratio is 1.99 and
1 + ηmax equals approximately 3. To repeat a statement made above, proton
broad - band irradiation enhances the intensity of the^13 C nucleus, which other-
wise has very low receptivity.
Practically speaking, NOEs occur only between atoms that are quite close
together in space (within 4 – 5 Å ) because the effect occurs with a radius depen-
dence proportional to 1/ r^6. Experimentally, one speaks of strong (atoms closer