The Heteronuclear Two-Spin System

We shall treat the heteronuclear and homonuclear systems separately. Although there are many features in common in the (wide line) spectra produced, the differences in the form of the Hamiltonian suggest quite different methods for manipulating the spin systems so as to produce narrow lines and high resolution spectra. 

As we noted previously, Jtj is a tensor, not a scalar, but unlike Dv it has a nonvanishing trace, hence it is found in molecules tumbling in liquids. In solids and in liquid crystals it is possible in principle to investigate the off-diagonal elements of the J tensor, but measurements are difficult, and few instances of measurable aniostropy in J have been reported. However, one recent result shows that in benzene the anisotropies for one-, two-, and three-bond C-C couplings are approximately +17, —4, and +9 Hz, respectively, as compared with the isotropic values, %c = 56,2JCC = —2.5, and 3JCC = 10 Hz.82 

The spin part of the Hamiltonian in the solid is identical to that of the AX system, and the solution is the same—a pair of lines for A and a pair for X, as in Fig. 6.2. However, the separation is dependent on orientation of the vector r joining the spins relative to B0, because the effective coupling is D(3 cos2 0—1), rather than D alone. As indicated in Fig. 7.2, the lines cross as 0 is varied, becoming coincident when 0 = 54.7°, the angle for which the term (3 cos2 0 — 1) = 0. This angle that will appear frequently in our later discussion. 

FIGURE 7.2 Illustration of the variation of the dipolar splitting of the two A lines as the internu-clear vector r varies in direction from 0° (parallel to B, ) to 90°. 

For an X nucleus with a spin of /2, the X spectrum is, of course, identical to that of A but centered at vx. As we might expect, for X with spin /2, the treatment of A is qualitatively similar, but the spectrum has more branches, and the existence of a quadrupole moment in X may have significant ramifications, 

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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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