Perturbation quadrupolar

The low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

Quadrupolar perturbation reveals nonspherical charge distributions about Ru atoms ... 

Figure 4 Schematic representation of the populations of the nuclear spin energy levels of a quadrupolar nucleus with spin 5/2 (such as Mg) under a strong magnetic field and a perturbative quadrupole coupling showing (A) populations at thermal equilibrium, (B) populations after complete saturation of the satellite transitions, and (C) populations after complete inversion of the satellite transitions, following the order first, inversion of STl and ST4 and then inversion of ST2 and ST3. The numbers at left of each level (named pj in the text) are proportional to the population of that level, with —hVl/ 2k T= 0. ...

Symmetries of local electrical environments of quadrupolar nuclei (/ 1) profoundly influence relaxation times and resonance line shapes of such nuclei (9, 116). Consider a nucleus for which I = % (Br79, Bn). In the absence of quadrupolar perturbation, the nuclear spin levels are evenly spaced, as shown in I below, and the three possible nuclear resonance transitions have equal energies (Am = 1). If, however, eqQ 0... 

Fig. 3.16. Variation of the experimental values of the spectral moment Sq (solid line), and values calculated with the assumption of quadrupolar induction (dashed line), with the square of the polarizability of the perturbing gas. Reproduced with permission from the National Research Council of Canada from [213].

The Zeeman effect must be mentioned in the case of nitrogen it behaves normally when 77 = 0 but, in the general case of a non-zero asymmetry, the Zee-man part of the hamiltonian no longer commutes with the quadrupolar part and there appears to be no first-order Zeeman effect, The second-order treatment of the perturbation yields the following values for the transition frequen cies 81 ... 

The exact calculation of the rotational levels of a molecule with a quadrupolar nucleus by the method given above needs a large amount of computer storage space. In most cases, therefore, the results used are calculated by means of the perturbation theory of quantum mechanics. This gives the following equations for the perturbation energy in first order. 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

Since the effects of a perturbation on the relaxation rate are mechanism-specific, clarity about the active spin relaxation mechanism is essential. While there is no doubt about the overwhelming dominance of quadrupolar relaxation for Na, Rb, and Rb,... 

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 analysis of the splitting pattern can afford the isotropic J value and also the value. Furthermore one can consider that the interplay of the quadrupolar, dipolar and indirect coupling also affects the spinning sideband manifolds. Each component of the sextet will have its own effective anisotropy, and, with the same assumption of the perturbation approach, the effective anisotropy evaluated by the analysis of the SSB manifolds affords the ZY value and the value of the quadrupolar coupling constant %. Furthermore, if the Mn-P intemuclear distance is known by X-ray diffraction, the A/ value can be readily extracted [58]. By using the perturbation approach, the interaction of with metal nuclei having spin 5/2 C Mo, Ru) [61,62], 7/2 ( Co) [58] and 9/2 ( Nb) [63] has been investigated. 

The molecular DFT approach derives an expression J-c[p] expanding the anisotropic excess parts of the perturbative pair interaction potentials into spherical harmonics. That way the inclusion of the anisotropic dispersion and quadrupolar interactions succeeds [7]. 

Figure 2.9. The quadrupole perturbed powder patterns for an I = nucleus. A. The outer satellite transitions to first-order. B. The second-order quadrupolar broadening of the central transition with...

To obtain the lineshape of the central transition the second-order perturbation of the quadrupolar interaction must he calculated on the transition. These give... 

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