Quadrupolar operator

Fig. 59. (a) The Dyson equation [eq. (AlO)] for the phonon propagator, (b) The RPA approximation for the quadrupolar susceptibility with respect to the quadrupolar interaction Kp, see eq. (A7). Circles represent the quadrupolar operators Op. 

The deformed charge distribution is generally axially symmetrical and this fact has an important consequence. It permits to characterize the charge distribution asymmetry by means of only one quantity, Q (called the quadrupolar moment), even if the quadrupolar operator, is a 3x3 matrix (in classical physics, a quadrupole is a second rank tensor). The definition of Q and the explicit form of are given in references 2 and 3. 

The rank 0 term, referred to as the quadrupolar-induced shift (QIS) operator, is independent of the molecular and rotor orientation (Bp0 = Bq0). Using (11), (13), and (18), this term can be written... 

Since the second-order quadrupolar broadening is inversely proportional to the Larmor frequency [(35) and (36)], one obvious means for improving the resolution is to employ the highest available magnetic field strength [94], The use of several spectrometers operated at different magnetic field strengths can be helpful in... 

The terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

Two factors contribute to r K. One is the ratio of the magnetogryric ratios of the two different spins, and the other depends on relaxation mechanisms. Provided that the relaxation mechanism is purely dipole-dipole, other relaxation mechanisms affect spin I, then 4> may approach zero. Assuming that the dipolar mechanism is operational (no quadrupolar nuclei with I > 1/2 are present), r has the value ys/ 2y and is regarded as rimax. In the homonuclear case we have r max = 1/ 2. Usually one chooses nuclei where ys > y/ to ensure that the NOE is significant. For observation of 13C for instance, if the protons in the molecule are double irradiated, the ratio is 1.99 and 1 + r max equals approximately 3. To repeat a statement made above, proton broad-band irradiation enhances the intensity of the 13C nucleus, which otherwise has very low receptivity. 

The NMR experiments were performed using the quadrupolar echo pulse sequence 7i/2x—Ti—7i/2y—T2—acquisition with phase-cycling and quadrature detection. A Bruker MSL 400 spectrometer was used for the high pressure studies operating at a resonance frequency of 61.4 MHz. In the liquid-crystalline phase, perdeuterated lipids display NMR spectra, which are superpositions of axially symmetric quadrupolar powder patterns of all C-D bonds.From the sharp edges, the quadrupolar splittings... 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

The forced electric dipole mechanism was treated in detail for the first time by Judd (1962) through the powerful technique of irreducible tensor operators. Two years later it was proposed by Jorgensen and Judd (1964) that an additional mechanism of 4/-4/ transitions, originally referred to as the pseudo-quadrupolar mechanism due to inhomogeneities of the dielectric constant, could be as operative as, or, for some transitions, even more relevant than, the forced electric dipole one. 

The dielectric constant is a macroscopic property of the material and arises from collective effects where each part of the ensemble contributes. In terms of a set of molecules it is necessary to consider the microscopic properties such as the polarizability and the dipole moment. A single molecule can be modeled as a distribution of charges in space or as the spatial distribution of a polarization field. This polarization field can be expanded in its moments, which results in the multipole expansion with dipolar, quadrupolar, octopolar and so on terms. In most cases the expansion can be truncated to the first term, which is known as the dipole approximation. Since the dipole moment is an observable, it can be described mathematically as an operator. The dipole moment operator can describe transitions between states (as the transition dipole moment operator and, as such, is important in spectroscopy) or within a state where it represents the associated dipole moment. This operator describes the interaction between a molecule and its environment and, as a result, our understanding of energy transfer. 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

The matrix elements for the operator //ges = HIot + Hq, which describes the problem of a rotating molecule containing a quadrupolar nucleus, are obtained by simple addition of the matrix elements of the operators Hrot and Hq. This arises from the fact that Hrot operates only on the coordinates of the one system described by the rotational coordinates, i.e. the Euler s angles. 

The calculations presented here are based on the density operator formalism using the Liouville-von-Neumann equation and the theoretical approach is confined to quadrupolar nuclei subjected to EFG as well as CSA-interactions. Following the approach of Barbara et al.,20 the Hamiltonian for an N-site jump may be written as... 

The theoretical approach will take use of these operators. The effective Hamiltonian will be described for a single site, site v, but for simplicity the formalism omits the index v unless it is absolutely necessary as the expressions are equivalent for all sites in the N-site jump process. For a system including both CSA-and quadrupolar interactions, the effective Hamiltonian for a single site during a pulse is... 

The faithful representation of the shape of lines broadened greatly by dipolar and, especially, quadrupolar interactions often requires special experimental techniques. Because the FID lasts for only a very short time, a significant portion may be distorted as the spectrometer recovers from the short, powerful rf pulse. We saw in Section 2.9 that in liquids a 90°, t, 180° pulse sequence essentially recreates the FID in a spin echo, which is removed by 2r from the pulse. As we saw, such a pulse sequence refocuses the dephasing that results from magnetic field inhomogeneity but it does not refocus dephasing from natural relaxation processes such as dipolar interactions. However, a somewhat different pulse sequence can be used to create an echo in a solid—a dipolar echo or a quadrupolar echo—and this method is widely employed in obtaining solid state line shapes (for example, that in Fig. 7.10).The formation of these echoes cannot readily be explained in terms of the vector picture, but we use the formation of a dipolar echo as an example of the use of the product operator formalism in Section 11.6. 

Most simulation studies of nitrogen adsorption have been made at 77 K. A higher operational temperature is likely to have the advantage that the effects due to > quadrupolar interaction will become less important as the kinetic energy is increased. In the work of Kaneko et al. (1994), GCMC simulation was used to study the adsorption of nitrogen in slit-shaped graphitic micropores at 303 K. 

Supercriticial fluids look set to continue to their invasion of the territory of traditional organic solvents in all areas of chemistry. Perhaps the most interesting area for exploration for the organometallic NMR spectroscopist is the use of SCFs for homogeneous catalysis. The presence of a single phase simplifies operation and may lead to enhanced reaction rates, while low viscosity aids observation of quadrupolar nuclei, which includes many of the transition metals. NMR studies in situ could allow observation of reaction intermediates and new insights into the mechanisms of these important reactions. 

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