In the AX proton spin system, proton A can sense the two spin states of a coupled proton X, resulting in a doublet NMR signal with two peaks of equal (1:1) intensity. When proton A is coupled to two equivalent protons (AX2 spin system), the spin states of each X can be aligned with or against the external field, creating three possible scenarios. This results in a 1:2:1 triplet signal, where the central peak corresponds to the chemical shift of A and is twice as large or intense as the others. Similarly, if A has three neighbors (AX3 spin system), four possible electronic environments result in a quartet of peaks with the relative intensities 1:3:3:1.
Empirically, a proton coupled to n equivalent neighbors yields a multiplet signal split into n + 1 peaks. The relative intensities of the peaks in a multiplet can be predicted using Pascal's triangle, which is an array where each entry is the sum of the entries to its left and right in the row above it.
