Quadrupolar interaction treatment

QEUC, see Quasi-molecular enlarged unit cell Quadricyclene, 20 323 valence isomerization of, 20 304 Quadricyclene, isomerization of, 24 146, 148 Quadrupolar interactions, 33 205-209 Quadrupole splitting, 26 126, 134, 140-142 Qualitative studies of simple reactions, 32 116 Quantitative treatment, structure effects, 29 155-162... 

Fig. 26 MAS spectra of M2AI/CI samples after thermal treatment at a 25 °C, b 150 °C, c 200 °C, d 250 °C, e 275 °C, and f 300 °C. (Black line for M = Ca and grey line for M = Zn). The chemical shifts are not corrected from the secondary effect of the quadrupolar interaction. (Reprinted from [70] with permission from Elsevier)...

Broad Lines. - The width of an n.m.r. line, At>1/2, is defined as the width in Hz at half signal height. Narrow lines, i.e., Ap1/2 < 10 Hz, are desirable in order to make use of chemical shift information and to follow chemical change. N.m.r. line widths in the liquid and the physisorbed state tend to be very narrow, with Ar>1/2 of the order of 10-1Hz. This fortuitous state arises because the molecular motion is sufficiently rapid and random in a liquid to average out the line broadening features present in solids, namely dipolar interactions, chemical shift anisotropy, quadrupolar interactions, and paramagnetic interactions which render the spectrum unusable under conventional or liquid-state experimental conditions. The mechanisms of each of these features will be described. The treatment will perforce be cursory, but an indication will be given to where a full theoretical treatment can be found. 

A general treatment of spin coupled with quadrupolar spins was given using density matrix theory.27 This formulation enables one to calculate, on the same theoretical basis, the lineshapes of the systems with different ratios of the quadrupolar interaction to the Zeeman interaction. Additionally, it includes the spinning sidebands very naturally. 

Realumination can only occur if there are sufficient suitable Si sites in the framework. Treatment of amorphous faujasites containing various kinds of aluminium leads to the formation of NFT aluminium with characteristic chemical shift and quadrupolar interactions. 

Cooper et al. reported that solid-state 23Na NMR is useful for observing the sodium cation in sulfonated polystyrene ionomers (NaSPS).44 Three NMR peaks were detected, corresponding to isolated ion pairs, aggregated ions and hydrated ions (Fig. 14). The distributions of these three types of sodium cations are systematically influenced by hydration treatment, sulfonation level and neutralization level. Fully dried NaSPS at low ion content shows that the isolated and aggregated sodium ions in NaSPS are available for hydration. As the sulfonation level increases, the fraction of sodium ions held in isolated ion pairs decreases while the fraction of ionic species in the ionic aggregates increases. This coincides with a shift in the peak position of the aggregated sodium ions to low frequency, indicative of increased quadrupolar interactions. 

It should be emphasized that the quantitative interpretation of all these methods relies on the molecular field approximation, ie., it neglects any fluctuation effects. The same remark holds for the treatment of the ordered phase, the phase diagram obtained using exchange and quadrupolar interactions (Koetzler et al. 1979, Morin and Schmitt 1983). 

To conclude this section on 2D NMR of liquid crystals, some studies of more exotic liquid crystalline systems are pointed out. Polymer dispersed nematic liquid crystals have attracted much attention because of their applications as optical display panels. Deuteron 2D quadrupole echo experiments have been reported [9.28] in the isotropic and nematic phases of / -deuterated 5CB dispersed in polymers. A similar technique was used [9.29] to study two model bilayer membranes. Both studies allow determination of the lineshape F(u ) due to quadrupolar interactions and the homogeneous linewidth L(u ) of the individual lines [9.28]. The 2D quadrupole echo experiment has also been used [9.30] to separate chemical shift and quadrupolar splitting information of a perdeuterated solute dissolved in a lyotropic liquid crystal. The method was compared with the multiple-quantum spectroscopy that is based on the observation of double-quantum coherence whose evolution depends on the chemical shift but not on the quadrupolar splitting. The multiple-quantum method was found to give a substantial chemical shift resolution. The pulse sequences for these methods and their treatment using density matrix formalism were summarized [9.30] for a spin 1=1 system with non-zero chemical shift. Finally, 2D deuteron exchange NMR was used [9.31] to study ring inversion of solutes in liquid crystalline solvents. 

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

To overcome possible errors in the derived parameters when fitting NMR spectra of powdered samples with large quadrupolar interactions, Perras et al. developed and implemented an exact treatment of the quadrupolar and Zeeman interactions for stationary powdered samples [9]. The software, QUEST (QUadrupolar Exact SofTware), is freely available and produces spectral simulations in a fraction of a second on a modem computer [10]. The importance of this exact treatment was demonstrated in a SSNMR study of a series of organic compounds featuring covalent... 

The first term in (24) was already calculated in (9). The second term can be neglected, as the CSA is usually too weak to require the second-order treatment. The last term is the second-order cross-term between the two interactions. A complete review of the second-order cross-terms and their effect on high-resolution solid-state NMR powder spectra of quadrupolar nuclei was recently published by Ashbrook et al. [31]. 

This is particularly easy for rigid solutes dissolved in uniaxial mesophases. The principal axes of Sij for a rigid solute may be located with certainty based on molecular symmetry. NMR can provide precise measurements of Szz tnd xx yy for solute molecules. However interpretation of the solute order requires modeling solute-solvent interactions [3.22]. As discussed in Section 2.3, both dipolar and quadrupolar couplings are suitable for probing orientational order of solutes and solvent molecules. Early treatment of NMR spectra [3.17, 3.41] have used a single Szz approximation by ignoring the small contribution due to Sxx - Syy. 

Treatments to interpret the frequeney dependenee of Tie and Tje have become inereasingly sophisticated. The multifrequency studies done in solution define a funetion of zero-field splitting (ZFS) or the crystal-field interaction cfi) parameters, denoted as A. The relationship between the parameters D and E, whieh account for the quadrupolar part of the cfr, and is defined as A = 2I3)D + 2E. The cfi parameters for Gd(III) complexes are dealt with in mueh more detail below in Section 3, where we discuss more reeent results on frozen glassy solutions. Here we describe efforts to understand eleetronic relaxation in solution and how this impacts nuclear magnetic relaxation and MRI contrast agents. 

The inter-particle distance dependence of the near-field coupling would therefore reflect the distance decay of the near-field itself. In other words, each particle senses the near-field due to the other particle. By varying the distance of the other particle and monitoring the LSPR response, the spatial profile of the near-field can be deduced. The plot of the LSPR red-shift as a function of inter-particle gap (surface-to-surface separation) shows a much more rapid decay of the near-field than predicted by the dipolar model. This is because the dipolar model does not take into account the multipolar interactions between the particles, which become increasingly important at smaller and smaller inter-particle gaps. Plasmon coupling is therefore a multipolar interaction and its true distance-dependence can be quantitatively reproduced only by a complete treatment that includes all modes of interaction (dipolar, quadrupolar, octupolar). Computational electrodynamics methods such as discrete dipole approximation (DDA) (see Chapter 2) and finite-difference-time-domain (FDTD), which include a full multipolar treatment in addition to finite-size retardation effects, fit experimental trends well. 

In a liquid crystalline system there is a rapid molecular motion which will to some extent average out the quadrupole interaction. The residual interaction gives rise to NMR spectra analoguous to the one shown in Figure 5. A theoretical treatment of the quadrupolar effects in micellar solutions and amphiphilic liquid crystals have been published by Wennerstrom et al. (57), showing that eqs. 16 and 17 will be very much the same for this case as for the rigid solids (eqs. 18 and 19). 

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