Quadrupolar nuclei theory

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. 

We now consider relaxation of quadrupolar ions in greater detail. The theory presented applies to any spin- quadrupolar nucleus, though the experimental examples discussed are of 7Rb+ in agarose gels. 

Probably of more relevance to the practising organic chemist is the influence quadrupolar nuclei have on the spectra of spin- /2 nuclei, by virtue of their mutual scalar coupling. Coupling to a quadrupolar nucleus of spin 1 produces, in theory, 21 -f- 1 lines so, for example, the carbon resonance of CDCI3 appears as a 1 1 1 triplet ( H has 1 = 1) by virtue of the 32 Hz coupling. 

To a good approximation, powder NMR line shapes for the STs of quadrupolar nuclei may be understood by first-order perturbation theory thus, we first consider these transitions. To first order, the transition frequencies, —1), expected for a quadrupolar nucleus, are given by... 

When the quadrupolar nucleus is bromine, effects are important, and NMR spectra of C-Br carbons appear as asymmetrically distorted quartets (Figure 7). Further, the ratio (xl o is in this case large (Tables 3 and 4), and second-order theory cannot be applied. Thus, full-matrix calculations were used to account for the observed spectra, as well as a so-called inverse first-order theory, in which the Zeeman term is considered as a small perturbation... 

There are two conceptually different theories for quadrupolar relaxation of ionic nuclei in solution. Deverell (22) rationalized the electric field gradients at the site of the nucleus as arising from distortions of the closed-shell orbitals in the ion due to collisions with solvent molecules and, at higher concentrations, also counter-ions. In another theory developed by Valiev (23) and by Hertz and his coworkers (24) it is assumed that the electric field gradients are caused by the electric dipoles of the surrounding solvate molecules. It is certainly Hertz to whom we owe the detailed understanding of ionic quadrupole relaxation, and because of the fundamental implications that his work has on ionic solvation the important results are briefly summarized here. 

Another approach to deducing the structures of materials from their solid state NMR spectra is to make an ab initio theoretical calculation of the electric field gradient at the nucleus of an atom in a known crystal structure environment (Figure 1.3, pathway 1), from which the nuclear quadrupolar parameters and hence the expected NMR spectrum can be calculated and compared with the experimental spectrum (Figure 1.3. pathway 2). The difficulties inherent in making such ab initio calculations are such that relatively few real compounds have so far been solved completely, but developments in ab initio theory, and mathematical algorithms and computing techniques should make this approach much more widely accessible in future. 

For an ensemble of crystallographically equivalent nuclei with spin /, 21 NMR transitions take place. For most elements, the most abundant spin-active isotope is quadrupolar with half-integer spin. For these nuclei, spectro-scopists usually focus on the mj = 1/2 — mi = -1 /2 transition since it is not perturbed by the first-order quadrupolar interaction (i.e., to a first approximation, this transition behaves like that for a spin-1/2 nucleus, see below). Hence, an overview of the theory for half-integer spin quadrupolar nuclei is presented here. For a thorough description of the theory for the quadrupolar interaction, readers are referred to numerous texts and reviews on the topic. ... 

In quadrupolar nuclei, the situation differs notably the quadrupolar interaction only affects spins with I>% and is created by electric field gradient resulting from the asymmetry of charge distribution around the nucleus of interest. The quadrupolar interaction is characterized by the nuclear quadrupolar coupling constant Cq (from 0 in symmetrical environments to tens or hundreds of MHz) and an asymmetry parameter T]q. NMR spectra are usually recorded when Cq Vl the Larmor frequency of the quadrupolar spin. In such a case, the NMR spectrum can easily be simulated First, the first-order quadrupolar Hamiltonian, which is the quadrupolar interaction Hamiltonian truncated by the Larmor frequency, has to be taken into account. The first-order quadrupolar interaction (or the zeroth-order term in perturbation theory) is an inhomogeneous interaction and is modulated by MAS and does not affect symmetrical transition —m +m. Therefore, in half-integer spins, the single-quantum central transition (CT, i.e., —1/2 +1/2) is not affected by the first-order quadrupolar inter-... 

Fig. 4.2. The NMR spectrum (94.1 MHz, room temp.) of the IfJ cation in HF solution. The signal is split into a sextet by coupling of the equivalent fluorines to (I = 5/2). The coupling constant Jjp = 2730 15 Hz. The lines in the sextet have equal intensities but unequal line widths and thus heights. This unequal broadening is caused by the quadrupolar relaxation of the iodine nucleus which effects the gp n states differently as predicted by theory [236]. Inverted...

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