Proton Spin Pairs

In the solid state, in the absence of very fast magic angle spinning or homonuclear multiple pulse decoupling schemes, the ¹H NMR spectrum of a typical solid is a broad featureless line greater than 50 kHz in width. This is due to the homonuclear dipolar coupling interactions between the many protons present in the system. The situation is different for an isolated pair of protons. For an isolated pair of protons, there is only one dipolar interaction between the protons and the energy level diagram for the system has only three levels corresponding the combination of spin states among the two protons and the dipolar coupling between them. There are two transitions and therefore two resonances. The separation between the resonances depends on the magnitude of the dipolar coupling constant, R, and the orientation of the internuclear vector with respect to the applied magnetic field. For powdered samples where all orientations with respect to the applied magnetic field are represented, one observes a "Pake" doublet. This situation is very similar to the solid state NMR of ²H where, in that case, the three energy levels arise from the Zeeman states of a single ²H nucleus and their coupling to an electric field gradient.

Isolated proton pairs occur naturally in the waters of hydration of inorganic salts and the solid state ¹H NMR spectrum is a Pake doublet. The separation between the inner peaks of the Pake doublet is 3/2 R and between the two shoulders is 3R, where R is the dipolar coupling constant. The dipolar coupling constant is directly proportional to the inverse cube of the distance between the protons. Therefore from a single spectrum, one can measure the internuclear separation, r. For the case of the waters of hydration, one can measure the H-O bond length with knowledge of the H-O-H bond angle. The figure below illustrates the Pake doublet spectrum obtained for CaSO₄^. 2 H₂O. The asymmetry in the spectrum is the result of chemical shielding anisotropy and the broadening is the result of dipolar coupling to distant protons.