Nuclear second moment

As with graphite oxide, there are currently two views as to the structure of carbon monofluoride. Although detailed X-ray diffraction work suggested a chair arrangement of the sp -hybridized, carbon sheets (Ml), second-moment calculations of the adsorption mode of the fluorine nuclear magnetic resonance suggested that a boat arrangement is more plausible iE2). The structures are illustrated in Fig. 3. 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

The great utility of moments is that, although the lineshape cannot be calculated analytically for an arbitrary configuration of nuclear spins, any moment may in principle be calculated to arbitrary precision from first principles (10). In practice, only the lowest moments are calculable because of computer time and precision constraints. In particular, the second moment M2 is the lowest moment containing spacial information... 

If we were to have an isolated polymer chain with a single nuclear spin attached to each segment (the marked chain) crosslinked into an unmarked network, the second moment of the NMR line of that spin species would carry information relating to the separation of chain segments, and to their relative orientation with respect to the field direction. If the network were to be subjected to a bulk deformation, these geometrical parameters would be altered, and hence we would expect a corresponding change in the value of the experimentally measured... 

The principal interaction experienced is the Zeeman interaction (Hz), which describes the interaction between the magnetic moment of the nucleus and the externally applied magnetic field, B0 (tesla). The nuclear magnetic moment, p (ampere meter2) is proportional to the nuclear spin quantum number (/) and the magnetogyric ratio (y, radian telsa-1 second-1) ... 

Does T differ significantly from unity in typical electron transfer reactions It is difficult to get direct evidence for nuclear tunnelling from rate measurements except at very low temperatures in certain systems. Nuclear tunnelling is a consequence of the quantum nature of oscillators involved in the process. For the corresponding optical transfer, it is easy to see this property when one measures the temperature dependence of the intervalence band profile in a dynamically-trapped mixed-valence system. The second moment of the band,... 

Another parameter that one can extract from a Mossbauer spectrum is the quadrupole splitting. The 3/2 state in either iron or tin is degenerate with respect to an asymmetric electrostatic field, and in such a field these levels will be split into dz 3/2 and 1/2 levels. One can observe transitions either to or from these two levels to the ground state, and this is the quadrupole splitting. It is actually e qQ, where eq is the electrostatic field gradient—i,e., the second derivative of the potential with respect to the coordinate—and eQ is the nuclear quadrupole moment. The typical quadrupole split spectrum for iron is shown in Figure 6, in which the cubic (octahedral) symmetry around the iron atom is de-... 

Table 15.7. The centroids of charge implied by the second moment of the charge distribution of the nuclear and a framework. The C and H nuclear positions are those of the 6-3IG SCF equilibrium geometry.

The nuclear shielding is more precisely the second derivative of energy with respect to magnetic field (B) and nuclear magnetic moment (/u) (27). 

As Figure 2 shows, the two models differ only by a vertical displacement, which is a measure of the difference in the mean square interparticle dipolar fields (second moments) operative in the two models. The values of these mean square local fields can be calculated easily from the minimum value of Ti or from the rigid lattice values of T2. Because of the 1000-fold ratio between electronic and nuclear magnetic moments, even... 

The lines in an EPR spectrum can be split by interaction of the electron spin with the nuclear magnetic moment of atoms on which the unpaired electron is located (Parish, 1990). Only atoms with nuclear spin (I) nonzero exhibit this type of interaction, which can be of two types (1) contact interaction that is isotropic and results from the delocalization of the unpaired electron onto the nucleus and (2) dipolar interaction between electron spin and the nucleus. In the second case, the interaction is dependent on orientation and, therefore, anisotropic (Campbell and Dwek, 1984). 

Another important extension of the theory concerns NMR chemical shift. Yamazaki et al. proposed a theory for computing the chemical shift of solvated molecules [17]. The nuclear magnetic shielding tensor o-x of a nucleus X can be represented as a mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mx ... 

A nucleus with a nuclear spin I> may possess a charge distribution that has a nuclear quadrupolar moment, Q. Such a nucleus may interact with a nonhomogeneous electric field possessing an electric field gradient, which can be expressed in terms of the second derivative of the electrostatic potential, V x... 

An electrostatic quadrupole moment is a second-rank tensor characterized by three components in its principal-axis system. Since the trace of the quadrupole moment tensor is equal to zero, and atomic nuclei have an axis of symmetry, there is only one independent principal value, the nuclear quadrupole moment, Q. This quadrupole moment interacts with the electrostatic field-gradient tensor arising from the charge distribution around the nucleus. This tensor is also traceless but it is not necessarily cylindrically symmetrical. It therefore needs in general to be characterized by two independent components. The three principal values of the field-gradient tensor are represented by the symbols qxx, qyy and qzz with the convention ... 

The spacing of the different energy levels studied by NQR is due to the interaction of the nuclear quadrupole moment and the electric field gradient at the site of the nucleus considered. Usually the electric quadrupole moment of the nucleus is written eQ, where e is the elementary charge Q has the dimension of an area and is of the order of 10 24 cm2. More exactly, the electric quadrupole moment of the nucleus is described by a second order tensor. However, because of its symmetry and the validity of the Laplace equation, the scalar quantity eQ is sufficient to describe this tensor. 

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. 

As we shall see, each of these two terms, one for each nucleus, describes a second-rank scalar interaction between the electric field gradient at each nucleus and the nuclear quadrupole moment. De Santis, Lurio, Miller and Freund [44] included two other terms which involve the nuclear spins. One is the direct dipolar coupling of the 14N nuclear magnetic moments, an interaction which we discussed earlier in connection with the magnetic resonance spectrum of D2 its matrix elements were given in equation (8.33). The other is the nuclear spin-rotation interaction, also discussed in connection with H2 and its deuterium isotopes. It is represented by the term... 

The direct contribution (c3)dir arises from the through-space dipolar coupling of the nuclear magnetic moments and, as expected, it decreases as the internuclear distance increases with increasing v, because of its R l dependence. The second contribution (C3)ec is the axial component of the tensorial electron-coupled spin-spin interaction the scalar part of this interaction is given by the value of C4. In the v = 0 level the direct contribution is estimated by English and Zorn [51] to be 1.15 kHz, and the electron-coupled part is -0.23 kHz. 

Only the diamagnetic susceptibility and the second moment of the nuclear magnetic resonance show additive molar properties. 

However, the NMR properties of solid-phase methane are very complex, due to subtle effects associated with the permutation symmetry of the nuclear spin set and molecular rotational tunnelling.55 Nuclear spin states ltotai = 0 (irred. repr. E), 1 (T) and 2 (A) are observed. The situation is made more complicated since, as the solids are cooled and the individual molecules go from rotation to oscillation, several crystal phases become available, and slow transitions between them take place. Much work has been done in the last century on this problem, including use of deuterated versions of methane for example see Refs. 56-59. Much detail has emerged from NMR lineshape analysis and relaxation time measurements, and kinetic studies. For example, the second moment of the 13C resonance is found to be caused by intermolecular proton-carbon spin-spin interaction.60 Thus proton inequivalence within the methane molecules is created. 

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