Quadrupolar definition

Fundamental constants (Cx), spatial tensors in the principal axis frame ((fi3 m,)F), and spin tensors (Tjm) for chemical shielding (a), J coupling (J), dipole-dipole (IS), and quadrupolar coupling (Q) nuclear spin interactions (for more detailed definition of symbols refer to [50])... 

The linear relationships between the traceless moments 0 and the spherical harmonic moments lmp are obtained by use of the definitions of the functions clmp. For example, for the quadrupolar moment element xx, we obtain the equality (3x2 — l)/2 = a (x2 — y2)/2 + b(3z2 — 1). Solution for a and b for this and corresponding equations for the other moments leads to... 

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. 

While the Overhauser shift is due to delocalised electrons, the ENDOR experiments require more localised electrons. Both groups were able to observe ENDOR on the effective mass donor resonance but not on the deep donor signal. An illustrative ENDOR spectrum is shown in FIGURE 4, in which the 69 71Ga ENDOR lines are observed. The 69Ga line is clearly split by quadrupole interactions just as in the Oveihauser shift measurements, while the quadrupolar interaction is much less resolved in the 71 Ga line, due to its smaller quadrupole moment. In the three samples investigated by the two groups, the linewidths were all too broad to resolve any hyperfine interaction and so no definitive identification of the residual donor was possible. 

Nuclear quadrupolar interaction arises from the coupling between the nuclear quadrupole moment Q and the EFG at the nuclear position. The EFG varies in space and is described by a traceless second-rank tensor. The EFG tensor is diagonal and its three principal components are VXXr Vyy and Rzz with the definition of VZZ > Vyy > VX < Such a principal-axis system for the EFG tensor is defined with the direction of the external magnetic field, as illustrated in Figure 3(A). It is convenient to express such quadrupolar interactions by using the following two parameters ... 

The last member of the quadrupolar Group V nuclei, ° Bi, represents another potentially useful nucleus although, surprisingly, the literature seems to be devoid of any chemical applications of its resonance. ° Bi is the only stable bismuth isotope. It has a spin off, a magnetic moment of 4 039 jUn, and is only six times less sensitive than the proton. Its quadrupolar moment of —0-38 barn definitely confines applications to symmetric moieties. 

The main specificity of the lEF method is that, instead of starting from the boundary conditions as in the DPCM, it defines the Laplace and Poisson equations describing the specific system under scrutiny, here including also anisotropic dielectrics, ionic solutions, liquids with a flat surface boundary, quadrupolar liquids, and it introduces the relevant specifications by proper mathematical operators. The fundamental result is that the lEF formalism manages to treat structurally different systems within a common integral equation-like approach. In other words, the same considerations exploited in the isotropic DPCM model leading to the definition of a surface cheurge density a(s) which completely describes the solvent reaction response, are still valid here, also for the above mentioned extensions to non-isotropic systems. 

FIGURE 7.10 Typical examples of the H-NMR spectra of CBA-w in the nematic LC state and the definition of the molecular axis (see the text) (a) n = 9 (b) n = 10. In the upper diagrams, the quadrupolar splittings (Av ) are assigned to the individual C-D bonds of the spacer. The dipolar ( >hd) quadrupolar splittings (AVo) of the mesogenic core (lower diagrams) are due to the deuteron substituted at the ortho position. (From Abe and Furuya [1988].)... 

In the case of quadrupolar nuclei with non-integral spins (eg. Al, 0) there are very definite advantages in working at the... 

Contrary to the nuclear case, where the transitions are of quadrupolar type, i.e., A/= 2, in the molecular case one usually observes dipolar transitions, i.e., A/= 1. Then the definition of the staggering function, the fourth-order finite derivative of the transition energies, becomes in the molecular case [25] ... 

The theory relevant to the description of the NMR spectra of the quadrupolar halogens has been given previously [2,3]- We present here only some of the useful definitions and key equations, along with a brief discussion on some aspects of the modeling of spectra of powdered samples. 

Relaxation measurements provide another way to study dynamical processes over a large dynamic range in both thermotropic and lyotropic liquid crystals (see Sec. 2.6 of Chap. Ill of Vol. 2A). The two basic relaxation times of a spin system are the spin-lattice or longitudinal relaxation time 7] and the spin-spin or transverse relaxation time T2. A detailed description, however, requires a more precise definition of the relaxation times. For spin 7=1, for instance, two types of spin-lattice relaxation must be distinguished, related to the relaxation of Zeeman and quadrupolar order with rates 7j"2 and Jfg. The relaxation rates depend on spectral density functions which describe the spectrum of fluctuating fields due to molecular motions. A detailed discussion of spin relaxation is beyond the scope of this... 

For nuclei with 1 > 1/2, the charge distribution inside the nucleus is not spherical, and the description of this charge distribution requires the definition of a quadrupolar nuclear moment and also, at least in some cases, of higher moments. In all cases, the dipole moment vanishes (28). The moments are always calculated relative to the center of mass, identical to the center of charge. 

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. 

Although C NMR is a powerful technique for the study of metal carbonyl complexes, its utility is marginal when quadrupo-lar metal nuclei are involved due to metal-carbon spin-spin coupling which broadens resonances and distorts their intensities. Since two bond metal-oxygen couplings do not interfere in a similar fashion, 0 NMR alone can provide definitive structural and dynamic information for many quadrupolar metal carbonyls (17-21). The most complex molecule of this type which has been studied to date is the 03 HFeCo3(C0)j 2 cluster (20), 9. At... 

An analysis of the atomic quadrupolar tensors similar to that described a ve for the APT and the FR atomic charges yields the following definition for the force related atomic dipoles ... 

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