Nonspherical charge distributions

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

Nuclei with a spin number 7 of one or higher have a nonspherical charge distribution. This asymmetry is described by an electrical quadrupole moment, which, as we shall see later, affects the relaxation time and, consequently, the linewidth of the signal and coupling with neighboring nuclei. In quantum mechanical terms, the spin number 7 determines the number of orientations a nucleus may assume in an external uniform magnetic field in accordance with the formula 27+1. We are concerned with the proton whose spin number 7 is 1/2. 

The quadrupolar nature of deuterium is due to the nonspherical charge distribution at the nucleus, caused by the presence of the neutron next to the proton. The quadrupolar Hamiltonian Hq arises from the electrostatic interaction of the nuclear quadrupole moment with the electric field gradient... 

Almost all theories agree that the deviation from cubic symmetry of the environment is associated with a nonspherical charge distribution of the unshared pair. We take this for granted and enquire in more detail about the relation of the electronic properties of the metal ion and the nature of the anion to the extent of distortion (110). Our arguments follow closely those already outlined for the problem of the distortions associated with d10 ions. 

Radiation-induced decomposition of insulating solids has been the subject of extensive research for many years. Because of their structural simplicity, the alkali and silver halides have perhaps received the widest attention. Studies of radiation-induced decomposition in azides could represent the next logical step in structural complexity. The azides in many respects are similar to the halides. Like the alkali halides, the alkali azides are primarily ionically bonded with band gaps of the order of 8 eV. Like the halides, there are azides with smaller band gaps (less than 4 eV). Important differences between the halides and azides are the presence of the triatomic azide anion and the lattice symmetry differences, which are perhaps a result of the nonspherical charge distribution on the azide ion. The salient questions which arise for the purpose of this chapter when one compares the azides to the hahdes are How does the the presence of the molecular anion influence radiation-induced decomposition are new and/or different kinds of defects produced how does the azide molecular anion influence the defect production process ... 

A nucleus with / > has a nonspherical charge distribution. It thus possesses a quadrupole moment that interacts with nonspherical components of the total charge distribution of its surroundings. This interaction further splits the ENDOR transitions into 21 lines at frequencies (to first-order)... 

The charge distributions are frequently assumed to be spherical, for simplicity.125-128 Directional interactions can be incorporated with nonspherical charge distributions, at some added computational expense.129 131... 

Fig. 4. The nonspherical charge distributions associated with the covalent bonds in diamond and silicon. Pileup of charge is found in the bond and a negative region in the antibond directions (reproduced by permission from Ref. 43 and 44).

The second term (eq, ) describes the ionic field gradient arising from nonspherical charge distribution of open-shell electrons of the resonant atom. It is in most cases dominant over For f transition elements one finds (Bleaney 1967) in the nonrelativistic approximation... 

The electrostatic component is generally the dominant attractive contribution in the case of polar molecules, i.e., systems with nonspherical charge distributions, usually manifested as appreciable permanent dipole and/or quadrupole moments. The classical Coulomb interaction between the respective charge distributions of the interacting molecules is conveniently formulated in terms multipole expansions that consist of dipole/dipole, dipole/quadrupole, quadrupole/quadrupole, etc. type terms, with distance (R) dependences R, R, R, ... (In the case of charged systems the monopole (net charge) terms must of course be also included in the expansion, that give rise to R, ... dependent terms.)... 

The electric field gradient (EFG) is a ground state property of solids that sensitively depends on the asymmetry of the electronic charge density near the probe nucleus. The EFG is defined as the second derivative of the electrostatic potential at the nucleus position written as a traceless tensor. A nucleus with a nuclear spin number / > 1 has a nuclear quadrupole moment (Q) that interacts with the EFG which originates from the nonspherical charge distribution surrounding this nucleus. This interaction... 

O, Na, and Al) and arises from the interaction of the nuclear electric quadrupole moment qQ with the electric field gradient eq produced at the nuclear site by a nonspherical charge distribution around the nucleus. The quadrupole interaction is usually characterized by the quadrupole constant e qQlh and asymmetry parameter rj. The magnitude of the quadrupole interaction is such (up to several MHz) that it completely dominates the spectrum for most quadrupolar nuclei, even in the case of H NMR, which has the smallest quadrupole moment. 

For a nonspherical charge distribution about the phosphorus nucleus, the shielding constant largely results from a local diamagnetic shielding contribution ffj and a local paramagnetic shielding contribution Op (Pople et al., 1959) ... 

With a few exceptions such as (I = 1), which has a small nuclear quadmpole moment, the nonspherical charge distribution of nuclei with a spin number I a 1 have a large electric quadmpole moment that affects the relaxation time, and the width of their NMR signals are broad. These broad lines usually mask the ability to measure useful NMR parameters such as chemical shifts and coupling with neighboring nuclei. As a result, they are not discussed here, as most of the solution NMR studies of polymers were performed on J = 1/2 nuclei. In the solid state, quad-rapolar interactions from measurements of NMR can yield valuable molecular stmcture and dynamics information this is discussed in Chapter 2.07. Some of the NMR characteristics of commonly observed nuclei are summarized in Table 1. 

There is another common cause of line splittings in microwave spectroscopy. Nuclei with spin I > 1 have an electric quadrupole moment. An electric quadrupole moment arises from a nonspherically symmetric charge distribution for example, a uniformly charged ellipsoid of revolution has no electric dipole moment, but does have an electric quadrupole moment. The nuclear electric quadrupole moment is16... 

In the previous section, it was assumed that the distribution of the nuclear charge is spherical [140], However, the charge distribution of a nucleus is not always spherically symmetric. In fact, this is not the case for a nucleus with nuclear angular momentum I > 1/2 in this case, the nucleus shows nonspherical nuclear charge distributions [142,143], The electrostatic potential created by a charge distribution localized inside a radius, Ir I rectangular coordinates as follows [144] ... 

In quadrupole splitting, the existence of a nonspherical nuclear charge distribution produces an electric quadrupole moment, Q, which indicates that the charge distribution in the nucleus is prolate, when Q > 0, or oblate, if Q < 0 [137-140],... 

Often the Townes-Dailey approximation is used rather successfully. This approximation assumes that contributions to the EFG from nonsphericity of core electron and from lattice charge distributions offset each other. Then the EFG results from imbalances in valence electron populations. The calculations also show that the errors introduced by the approximations ofthe Townes Dailey theory do not exceed 5%. Thus Townes Dailey calculations generally do well in showing trends in quadrapole coupling constants or NQR frequencies. 

The 57Fe Mossbauer quadrupole splitting (AEq) arises from the nonspherical nuclear charge distribution in the I = 3/2 excited state in the presence of an electric field gradient (EFG) at the 57Fe nucleus, while the isomer shift (6Fe) arises from differences in the electron density at the nucleus between the absorber (the molecule or system of interest) and a reference compound (usually a-Fe at 300 K). The former effect is related to the components of the EFG tensor at the nucleus as follows [45] ... 

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