Effect of a Selective 90 Pulse

The transformation of the density matrix by a 90° pulse and its subsequent evolution as the magnetization precesses freely depend on whether the pulse is applied only to one nucleus or to both I and S. We treat first the situation that would normally occur in a heteronuclear system, where a 90x pulse is applied only to the I spins—a 90pulse. This treatment is, of course, also applicable to a homonu-clear system subjected to a selective rf pulse. 

Equations 11.28 and 11.29 provide the framework, with the matrix manipulations following those in Eq. 11.39, but Ix in Eq. 11.36 must be expressed in the basis functions that is, a 4 X 4 matrix.The elements of lx may readily 

48 we saw that the basis functions for our density matrix are divided into three groups with fz = 1,0, and —1, respectively. As we saw in Chapter 6, transitions between energy levels El - E2, E3 E4, Ex E3, and E2 - E4 each result in Afz = 1 and are called single quantum transitions, while transitions Ex E4 and E2 E3 are termed double quantum and zero quantum transitions, respectively. The usual selection rules from time-dependent perturbation theory show that only single quantum transitions are permitted in such simple experiments as excitation by a 90° pulse. Moreover, for weakly coupled nuclei, the single quantum transitions each involve only a single type of nucleus, I or 5, as indicated in Fig. 6.2. 

In Table 11.1 we sketch the form of the density matrix for the two-spin system to show the significance of the elements. Px —P4 refer to the populations of the four states, I and S represent single quantum I and S transitions, and Z refers to zero quantum transitions and D to double quantum transitions. We saw in Eq. 11.9 that an off-diagonal element pm is nonzero only if there is a phase coherence between states m and n, and in Eq. 10.19 we saw that pmn, evolves with a frequency determined by the difference in energies Em — En. Thus, these off-diagonal elements represent not only transitions, but single quantum, double quantum, and zero quantum coherences, which evolve in free precession at approximate frequencies of v, vs, vt + vs, and vt — vs. In Eq. 11.53 we see that p(r) has 

Chapter 11 Density Matrix and Product Operator Formalisms 

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