The Homonuclear Two-Spin System

For a pair of magnetically equivalent nuclei (e.g., two protons), symmetrized wave functions must be used, as in Eqs. 6.42 and 6.44, and the resulting calculation gives eigenvalues and transitions as described in Section 6.11, except that the adjacent symmetric energy levels are separated by 

FIGURE 7.6 Powder patterns for homonuclear dipole coupling. Dashed line represents a two-spin system. Solid line shows broadening from other nearby nuclei in a multispin system. 

If the nuclei are of the same species but are not magnetically equivalent (e.g., because of chemical shift differences), the four basis functions of Eq. 6.1 must be used. However, because the dipolar coupling is normally much larger than chemical shift differences, the resulting spectra and powder pattern are little changed from those in Fig. 7.6. 

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The raising operator I+, acting as an operator, raises the a state of a single spin to the p state. All of these operators can be represented as matrices. In the case of the homonuclear two-spin system (Ha and Hb), these are 4 x 4 matrices. For example, the raising operator 1+ can be represented by the following matrix, which acts on the vector that describes the aa state to give a new vector that describes the Pa state ... 

For the heteronuclear two-spin system we found that we could create an average of the two X spin states by decoupling. Clearly such heteronuclear decoupling is inapplicable to the homonuclear two-spin system, but examination of the Hamiltonian of Eq. 7.6 suggests another approach—manipulate the spin system to make the two spin terms equal on the average so that pulse sequences. Prior to our discussion of pulse sequences in Chapters 9—11, we are not prepared to treat this rather complex process in detail, but a simple pictorial presentation gives the essential features. 

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