Solids local fields

In solids, local fields are often relatively large and thus cause very short decays. In order to observe the FID, the spectrometer must recover from the preceding rf pulse before M decays to zero. In most cases, this condition can be satisfied, and FID signals are normally observed. However, in the case of our Pt-cfjtalyst samples, the distribution of local fields is very large. M decays to zero long before the spectrometer can recover from the rf pulse, and no FID signal can be observed. 

With this imaging system it is possible to study virtually all metals and alloys, many semiconductors and some ceramic materials. The image contrast from alloys and two-phase materials is difficult to predict quantitatively, as the effects of variations in chemistry on local field ion emission characteristics are not fully understood. However, in general, more refractory phases image more brightly in the FIM. Information regarding the structure of solid solutions, ordered alloys, and precipitates in alloys has been obtained by FIM. 

To determine the behavior of g(q) for large q, we performed measurements of iS lq, ) of Li for 1.1 a.u. < q < 2.6 a.u. and performed for each spectrum a fit of the g(g)-modified c° to the experimental data. Figure 10 shows the result of this semi-empirical determination of g(q) together with the shape of the local-field correction factor after Farid et al. [7] calculated for different values ofz solid line (z = 0.1), dashed line (z = 0.5) and dash-dotted line (z = 0.7). One clearly sees that the curve for the surprisingly small value of z = 0.1 fits our experimental findings best. 

Dynamic parameters for heterogeneous systems have been explored in the liquid, liquid like, solid like, and solid states, based on analyses of the longitudinal or transverse relaxation times, chemical exchange based on line-shape analysis and separated local field (SLF), time domain 1H NMR, etc., as summarized in Figure 3. It is therefore possible to utilize these most appropriate dynamic parameters, to explore the dynamic features of our concern, depending upon the systems we study. 

Thus, since intramolecular bonding interactions in the solid are much stronger than relatively weak i n termo1ecu1 ar van der Vaals interactions, each molecular unit is essentially an independent source of nonlinear response, arrayed in an acentric cystal structure, and coupled to its neighbors mainly through weak local fields. In the rigid lattice. gas approximation, the macroscopic susceptibility X is expressed as... 

Here, cp (eV) is the barrier height, F is the local field, and a and b are constants. The typical field required for emission from solids is of order 1000 V/pm. The easiest way to create such a high field is by field enhancement at a sharp tip, so that the local field F is many times larger than the applied field F, F = ( /. (1 is a dimensionless geometrical field enhancement factor given by h/r for a tip, where h is the height and r is its radius. 

The next point to realize is that the best emitter is a metal. Many forms of carbon initially studied are semiconductors or even insulators, including nanodiamond [8-11] and diamond-like carbon (DLC) [12-13,4]. Combine this with local field enhancement means that there is never uniform emission from a flat carbon surface, it emits from local regions of field enhancement, such as grain boundaries [8-11] or conductive tracks burnt across the film in a forming process akin to electrical breakdown [13]. Any conductive track is near-metallic and is able to form an internal tip, which provides the field enhancement within the solid state [4]. Figure 13.2 shows the equipoten-tials around an internal tip due to grain boundaries or tracks inside a less conductive region. 

Another feature that is still puzzling scientists is the role of the soft mode in structural order-disorder transitions. As already mentioned in Sect. 3 the energy U in theoretical treatments is based predominantly on short-range interactions and often the local fields are replaced by a mean field to mimic some long-range properties. In the solid solution D-RADP-x the FE soft mode suffers from the lack of translational invariance, so that only for x < 0.32 sufficient coherence can be achieved to provide a successful contribution to... 

The broad line spectra of nuclei with spin I = 1/2 in the solid state are mainly a consequence of the dominant contribution of the dipolar Hamiltonian HD (Eq. (4)), which gives rise to a local field B)oc. Its magnitude varies as a function of the angle 0.. between the intemuclear vector r.j and the applied magnetic field B0. Depending on the nature of spin system, two general types of interactions can be distinguished ... 

In the separated local field technique, dipolar I-S interactions are separated from chemical shifts of nucelus S. As dipolar interactions are highly sensitive to internuclear distances, the obvious use of the method is for the determination of molecular structure in the solid state. An example is provided by the work of Hester et al. (411) and Rybaczewski et al. (414) on... 

This phenomenon explains why resonance lines in nonviscous liquids are much narrower than those for viscous liquids or solids, where local fields are not averaged out by slower motions. 

An individual atom or ion in a dielectric is not subjected directly to an applied field but to a local field which has a very different value. Insight into this rather complex matter can be gained from the following analysis of an ellipsoidal solid located in an applied external field Ea, as shown in Fig. 2.28. The ellipsoid is chosen since it allows the depolarizing field Edp arising from the polarization charges on the external surfaces of the ellipsoid to be calculated exactly. The internal macroscopic field Em is the resultant of Ea and Edp, i.e. Ea — Edp. 

As already mentioned, the above analysis would be valid for a gas for a solid a properly calculated local field would have to be used in Eq. (2.105). Fortunately, doing this does not change the general forms of Eqs (2.110) and (2.111) but leads only to a shift in co0. Furthermore, because the restoring forces are sensibly independent of temperature, so too are the resonance curves. 

Figure 1. 205T1-NMR spectrum (solid line) at 5 K. The intensity is plotted in a linear scale. The thin solid line depict the histogram at particular local fields of the Readfield pattern. The dotted line represents the simulation spectrum convoluted with Lorentzian broadening function. The filled circles show the frequency dependence of 205 f,1 1 at the T1 site. The inset shows the image of the field distribution in the vortex square lattice center of vortex core (A), saddle point (B) and center of vortex lattice (C).

Solids give rise to the "wide-line" or (broad-line) spectra, because the local fields arising from nuclear magnetic dipole interactions contribute significantly to the total field experienced by a nucleus in the solid state. A measure of this direct spin-spin interaction is the spin-spin relaxation time T2 (see Sect. 12.2.1.3), which is much shorter in solids than in liquids, and thus gives rise to broader lines (of the order of 10-6 to 10 4 T, depending on the kind of nucleus). Now the contour of the absorption line provides information as to the relative position of the neighbouring nuclei. 

In NMR work, spin-lattice relaxation measurements indicated a non-exponential nature of the ionic relaxation.10,11 While this conclusion is in harmony with results from electrical and mechanical relaxation studies, the latter techniques yielded larger activation energies for the ion dynamics than spin-lattice relaxation analysis. Possible origins of these deviations were discussed in detail.10,193 196 The crucial point of spin-lattice relaxation studies is the choice of an appropriate correlation function of the fluctuating local fields, which in turn reflect ion dynamics. Here, we refrain from further reviewing NMR relaxation studies, but focus on recent applications of multidimensional NMR on solid-ion conductors, where well defined correlation functions can be directly measured. 

The situation is much more complicated in solids because the intermolecular effects can no longer be ignored, i.e. the approximation EM = 0 inherent in the simple formula for the local field (2.29) is not generally true. Consequently, although we can predict the molecular dipole moment from known group moments, it is not possible to calculate the molar polarisation and thereby the relative permittivity, without further elaboration of the dielectric model. In the case of a polymer there are further complications which arise from the flexibility of the long chains. 

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