When observing a proton spectrum, even a very simple one like that of ethanol, we can see that the signals of each proton or group of protons (CH₃, CH₂ and OH) do not resonate to give a simple signal to only one line. The signals observable on an NMR spectrum of the proton show a so-called multiplet structure which is very informative for determining a structure but can be very complex.

Origin of multiplicity:

An H_A proton (spin of ½) has two orientations in the B₀ field (parallel to B₀ level ↑α and antiparallel to B₀ level ↓β). This nucleus locally creates a small magnetic field ΔB that adds to B₀ when the orientation of the spin is parallel to B₀ (α => B₀ = B₀+ΔB) and subtracts from B₀ for the antiparallel orientation (β => B₀ = B₀-ΔB). An H_X proton adjacent to H_A is therefore subject to two field values and resonates at two very close but different frequencies. The observed signal consists of two lines. The multiplicity of the signal is a doublet. In this case it is said that H_A is coupled to H_X. The difference in frequency between the lines is the spin-spin coupling constant (J_AX) and is expressed in Hz. The coupling constant is reciprocal, therefore H_A is coupled to H_X with the same coupling constant J and will resonate as a doublet if its only neighbor is H_X.

The interaction between these two protons is a magnetic interaction that is transmitted via chemical bonds. The values of the coupling constants J (Hz) are independent of the B₀ field and depend on the number of chemical bonds, the dihedral angle (Ψ) and the nature of the bonds separating the coupled protons. They are therefore characteristic of the structure of the compounds.

Multiplicity of signals

Case of an H_A proton (or group of equivalent protons) coupled to several neighbors n with the same coupling constant.

The number of H_A signal lines is given by the formula: 2nI+1 (with n the number of neighbors that couple with H_A and I their spin number). In the case of the proton and the ½ spin nuclei, the formula becomes: n+1.
Example: a proton H_A couples with 3 equivalent protons H_X (e.g. CH₃). The number of lines in the H_A signal is 4, the multiplicity is a quartet and the distance between the 4 lines is identical and represents the coupling constant J_AX.

The relative intensity of the lines of a multiplet resulting from a coupling to n protons of spin ½ and with the same coupling constant is given by by Pascal’s triangle.

Case of a proton (or group of equivalent protons) coupled differently to several neighbors - case of two different coupling constants.

If the above case is taken up (H_A coupled to 3 H_X with the same J_AX coupling constant) and it is considered that H_A is additionally coupled to another H_M proton with a coupling constant J_AM (≠ J_AX), then the multiplicity of the signal of H_A is a quadruple doublet with the doublet constant J_AM and a quartet constant J_AX.

Depending on the value of the two coupling constants the signal can appear very different:

Note: it is possible that, incidentally, J_AX = J_AM. In this case, the H_A signal will be a quintet since H_A is coupled with 4 neighbors with the same coupling constant.

Case of protons or groups of non-equivalent protons