As we have already stated, the NMR spectrum of an AX system (two
spin½ nuclei, A and X, spin coupled together) consist of two
lines, each split into a pair of component lines that are known
as a doublet. If we consider the
resonance of the A nucleus, then it is relatively simple to see
why it should be split into two.
The magnetic moment of the X nucleus is the source of a
small local magnetic field, the direction of which is determined
by the value of the quantum number m_{I} for X. Depending
upon its orientation, this magnetic field will either supplement
or oppose the applied magnetic field (deshield or shield the A
nucleus), and thus two different resonance signals will be obtained
for the A nucleus.
(Note that any individual AX system can only be in one or other
of the two states, A shielded or deshielded, but in a macroscopic
sample equal numbers will be in each state, and so the signals
at each resonance frequency will be of equal intensity.)
By exactly the same argument, we can explain why the X resonance
is split into two components by the A nucleus.
We can extend this argument to an AX
system where the X nucleus is of spin I. The resonance line for
A is then split into 2I + 1 lines (corresponding to the 2I + 1
possible values of m_{I} for X and the 2I + 1 different
orientations that X's magnetic field can thus take).
Note that the spin quantum number of A in this situation (providing
that it is nonzero) makes no difference to the appearance of the
spectrum of A, as all possible energy levels of the A nucleus
are equally spaced and will be shielded or deshielded to the same
extent. Thus transitions between adjacent energy levels of the
nucleus will all be of the same energy and so the lines for all
these transitions lie at the same frequency, overlapping with
each other.
It is also necessary to consider the interactions of a nucleus
A with two or more magnetic nuclei. The other nuclei with which
A interacts can either be equivalent (identical isotopes occupying
equivalent sites in a molecule, such as the CH_{2} protons
or the CH_{3} protons in ethanol) or inequivalent. Inequivalent
nuclei can be of the same isotope, if the positions they occupy
in the molecule are such that they couple differently to any particular
atom.
It should be noted at this point that
interactions between the members of a group of equivalent nuclei
do not lead to splitting of the resonance line of the group (though
the line may of course be split by interactions with other magnetic
nuclei in the molecule).
We have already established that when a nucleus A is coupled to
one spin I nucleus, the resonance line of A is split into 2I +
1 components. This rule can be applied to each nucleus that A
is coupled to in turn to construct the final splitting pattern.
Thus when A is coupled to a single spin ½ nucleus, its resonance
line is split into two, as represented below:
When A is coupled to two spin ½ nuclei, the situation is
a little more complex. If we consider that the effect of one
of the two nuclei is to produce a splitting like that above,
then the second nucleus splits each of the two component lines
into two, giving four component lines in total.
If the two spin ½ nuclei are different (so have different
scalar coupling constants to A) the system is designated AMX,
M and X being the two inequivalent spin ½ nuclei.
The resonance line of A splits as follows (assuming that A
couples more strongly to M than to X):
giving rise to a spectrum of the following appearance:
Note that the splitting is greatly exaggerated in both of the
above diagrams, in fact the four component lines would be much
closer together than indicated. The pattern of the NMR spectrum
is described as a doublet of doublets,
indicating that the original line has been split into two lines,
each of which has then been split into two itself.
If however, the two spin ½ nuclei to which A is coupled are
equivalent (i.e. the coupling constant to A is the same for
both nuclei) then the system is described as an AX_{2}
system, and the pattern obtained is somewhat different:
As a result of the fact that the two X nuclei split the resonance
lines equally, the two central lines now overlap at the position
of the unsplit resonance line, giving rise to a line of twice
the intensity of the other two. The pattern in an NMR spectrum
looks like this:
Again, the splitting has been exaggerated for clarity. This
splitting pattern (three equally spaced components of relative
intensities 1:2:1) is known as a triplet.
A similar procedure can be applied to work out the splitting
pattern for any system of coupled nuclei, and will accurately
predict the relative intensities and positions of all the components
of the split resonance line. Thus the A resonance line for an
AX_{3} system can be predicted to split into a quartet
of equally spaced components, with relative intensities 1:3:3:1.
(It is worth noting that in an AX_{n} system, where
A is coupled to n spin ½ nuclei, the relative intensities of
the component lines follow the order of the numbers in the (n
+ 1)th line of Pascal's Triangle. This can provide a quicker
route to prediction than the process of drawing out splitting
diagrams shown above.)
The use of such splitting diagrams as those above is not limited
to spin ½ nuclei, but can be applied quite generally to spin I
nuclei, remembering that they cause a resonance line to split
into 2I + 1 components. Thus the diagram and associated NMR spectrum
for the A resonance in an AX system where the X nucleus is spin
1 look like this:
It is worthwhile looking again at the high resolution spectrum
of ethanol as an example:
(Note that the OH proton does not contribute to the splitting
observed in the spectrum. The reason behind this is too complicated
to go into now, but is discussed under the subject of linewidths,
here.)
The CH_{2} group interacts with the spins of the protons
in the CH_{3} group. Interaction with three spin½ nuclei
splits the resonance line into four components, with relative
intensities 1:3:3:1.
The CH_{3} group interacts with the protons of the CH_{2}
group, and the interaction with two spin½ nuclei splits the
line into three components, with relative intensities 1:2:1,
just as predicted above. Note that degree of splitting here
is realistic, and much smaller than that depicted above.
