As we have already stated, the NMR spectrum of an AX system (two spin-½ 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 mI 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 mI 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 CH2 protons or the CH3 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 AX2 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 AX3 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 AXn 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 CH2 group interacts with the spins of the protons in the CH3 group. Interaction with three spin-½ nuclei splits the resonance line into four components, with relative intensities 1:3:3:1.
The CH3 group interacts with the protons of the CH2 group, and the interaction with two spin-½ 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.