Quadrupolar interaction second-order perturbation theory

As an example, quadrupole nutation NMR of nuclei with half-integer quadrupolar spin in zeolitic materials can distinguish between nuclei of the same chemical element subjected to different quadrupole interactions, the signals of which overlap in conventional spectra. The situation is favourable for half-integer quadrupolar spins since the m=l/2 <-> m= -1/2 transition for these nuclei is broadened by the quadrupole interaction only in second-order perturbation theory. The technique can be usefully applied for the determination of the local environment of A1 in zeolitic catalysts (28). It allows discrimination between species of similar chemical shift but different quadrupolar coupling constants (see Figure 5). The main difficulty in the interpretation is the complex spectmm that results from a nutation experiment since it can consist of many overlapping powder patterns (29). 

The NMR transition frequencies for quadrupolar nuclei are generally discussed in terms of first- and second-order perturbation theory where the quadrupolar interaction is treated as a perturbation of the Zeeman inter-action. Although the central transition (CT) is not perturbed by the first-order quadrupolar interaction, the remaining ZI— 1 single-quantum transitions (i.e., the satellite transitions (STs)) are perturbed by both the first-and second-order quadrupolar interactions (vide infra). 

To first order, the CT for half-integer quadrupolar nuclei is not perturbed by the quadrupolar interaction (Equation (9)) however, second-order perturbation theory is typically required to properly describe the line shape and position of the CT. For the case where the EFG tensor is axially symmetric (i.e., ijq = 0.0) and in the absence of anisotropic magnetic shielding, the frequencies, vq(1I2, —112), for the CT of a stationary powder sample are given by " ... 

Finally, it should be noted that the phonon part in eq. (14) also leads to an effective quadrupolar interaction between 4f ions if the phonon coordinates are eliminated in second-order perturbation theory. This virtual phonon exchange mechanism is important in insulators, see Orbach and Tachiki (1967). In R intermetallics such as TmZn and TmCd, however, the electronic origin of effective interactions is now established, Levy et al. (1979). 

Second-Order Perturbation Theory versus an Exact Treatment of the Quadrupolar Interaction... 

The first attempt to develop a statistical model of the cholesteric phase was by Goossens who extended the Maier-Saupe theory to take into account the chiral nature of the intermolecular coupling and showed that the second order perturbation energy due to the dipole-quadrupolar interaction must be included to explain the helicity. However, a diflUculty with this and some of the other models that have since been proposed is that in their present form they do not give a satisfactory explanation of the fact that in most cholesterics the pitch decreases with rise of temperature. 

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