Nuclear charge distribution nonspherical

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],... 

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] ... 

The source of the electric field can be an externally applied field, or it can originate in the components of the nuclear potential that are not included in the internal component of the field (that is, the nuclear potential V). Such components arise from the nonspherical nature of the nucleus, the lowest-order term of which is the quadrupole moment. The implementation of a finite-nuclear model is quite straightforward we simply expand the nuclear charge distribution in a series ... 

Ix > 1/2 quadrupolar isotope with an associated electric quadrupole moment Qx which reflects the nonspherical symmetry of the nuclear charge distribution. This differentiation (I > 1/2) will be essential in multinuclear observation, as 87 of the 116 magnetically active nuclei possess a spin value Ix > 1/2 and... 

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... 

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... 

Quadrupolar nuclei, those with 1 > 1/2, have a nonspherical distribution of nuclear charge and therefore interact with the electric field gradient in the solid. Since 74% of all nuclei with spin are quadrupolar, their study is of considerable interest. The quadrupole interaction broadens and shifts the NMR signals and may also affect their relative intensities. Since many chemists working in catalysis are unfamiliar with quadrupolar effects, we give a brief summary of the most important considerations to be borne in mind. In practice, two distinct cases must be considered nuclei with noninteger spin (of which Al is perhaps the most important to catalysis) and nuclei with integer spin, such as H. 

The QS arises from the interaction between the nuclear quadrupole moment and the nonspherical component of the electronic charge distribution described by its effective electric field gradient (EFG) in a simplified model [19,28]. The effective electric field gradient is dramatically reduced due to the spin-pairing transition. The electron shell of the high-spin Fe " " ions in the octahedral coordination is spherically asymmetric with S = 2 and t e t, whereas Fe " " ions in the low-spin state are more spherically symmetric with S = 0 and (the arrows indicate spin up or spin down whereas... 

Some appreciation for the nitrogen nuclear characteristics may be obtained from Table 1. Because the relative sensitivities are comparable, the approximately 300-fold higher natural abundance of would seem to make it the nucleus of choice. It is even more sensitive than at natural abundance. However, like all nuclei with spin quantum number I > 1/2, possesses an electric quadrupole moment that arises from a nonspherical electric charge distribution in the nucleus itself. When placed in an electric field gradient, such as that characteristic of most molecular electron distributions, a quadrupolar nucleus experiences random fluctuating electric fields. The characteristic frequencies of these motions have components at the resonance frequency and hence afford an efficient relaxation mechanism. As a result, spin-lattice relaxation times (Tj ) are very short, 0.1-10 ms. Because Tj = To for in most molecules Lie in solution, linewidths are corres-... 

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. 

A nucleus with I > 1/2 has a nonspherically symmetric charge distribution, which leads to a strong coupling between the nuclear quadrupole moment and the electric field gradients (EFGs), produced by surrounding electrons at the nuclear site. This is the quadrupole coupling interaction and its Hamiltonian can be expressed as... 

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. 

The quadrupolar interaction in term (Id) applies to nuclei having nuclear spin I > 1/2, the so-called quadrupolar nuclei. Such nuclei have a nonspherical distribution of positive electric charge that can be expanded in terms of moments, with... 

Static quadrupole effects in NMR are observed in solids (9) and also in anisotropic liquid crystals (10, 11, 12). For nuclei with spin quantum numbers, I, greater than V2, the distribution of positive charge over the nucleus can be nonspherical and the situation can be described in terms of a nuclear electric quadrupole moment. The interaction between the quadrupole moment, eQ and electric field gradients, eq, shifts the energy levels of the nuclear spin states. 

Vanadium-51 is a spin 7/2 nucleus, and consequently it has a quadrupole moment and is frequently referred to as a quadrupolar nucleus. The nuclear quadrupole moment is moderate in size, having a value of -0.052 x 10 2S m2. Vanadium-51 is about 40% as sensitive as protons toward NMR observation, and therefore spectra are generally easily obtained. The NMR spectroscopy of vanadium is influenced strongly by the quadrupolar properties, which derive from charge separation within the nucleus. The quadrupole moment interacts with its environment by means of electric field gradients within the electron cloud surrounding the nucleus. The electric field gradients arise from a nonspherical distribution of electron density about the... 

The interaction between a nonzero nuclear quadrupole moment and a surrounding nonspherical distribution of electric charges, as measured by the electric field gradient at the nucleus, gives rise to a quadrupole interaction. This hyperfine interaction, which also depends upon a nuclear and an electronic factor, is described by the Hamiltonian... 

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