Theory shape

The steady-state methods involve theoretical analysis of magnetic resonance spectra observed under steady-state conditions. This typically involves assumptions regarding the adequacy of magnetic resonance line shape theory, some model for molecular motions and distances of closest approach on collision, and a comparison of calculated spectra for various assumed diffusion constants, and observed spectra. In general, the agreement between diffusion constants calculated using the transient and steady-state methods has been excellent. 

The LB model. Levine and Birnbaum have developed a classical line shape theory, assuming straight paths for the molecular encounters and a dipole model of the form [232]... 

The theory of line shapes of systems involving one or more molecules starts from the same relationships mentioned in Chapter 5. We will not repeat here the basic developments, e.g., the virial expansion, and proceed directly to the discussion of binary molecular systems. It has been amply demonstrated that at not too high gas densities the intensities of most parts of the induced absorption spectra vary as density squared, which suggests a binary origin. However, in certain narrow frequency bands, especially in the Q branches, this intensity variation with density q differs from the q2 behavior (intercollisional effect) the binary line shape theory does not describe the observed spectra where many-body processes are significant. In the absence of a workable theory that covers all frequencies at once, even in the low-density limit one has to treat the intercollisional parts of the spectra separately and remember that the binary theory fails at certain narrow frequency bands [318],... 

Rate process with activation energies between 6 and 25 kcal mol-1 can be studied conveniently by the NMR method. Line shape theories have been well reviewed.103 In most cases a single rate constant k is estimated at the coalescence temperature, and AG is obtained from the Eyring equation. For exchange between two unequally populated sites, AG for the forward and reverse reactions are different. This is an important point to be considered in the case of nitrogen inversion phenomena. 

Vol. 1283 S. Mardesib, J. Segal (Eds.), Geometric Topology and Shape Theory. Proceedings, 1986. V, 261 pages. 19S7. 

U. Balucani, V. Tognetti, and R. Vallauri. Line shape theory and molecular dynamics in collision-induced light scattering. Phys. Rev. A, 79 177-186(1979). 

J. Segal, Shape Theory Notes. Mimeographed notes. University of Washington, 1976. 

S. Mardesic, The Foundations of Shape Theory. Lecture notes. University of Kentucky, 1978. 

R. R. Summerhill, A Categorical Description of Shape Theory. Mimeographed Notes. Kansas State University, Manhattan, 1979. 

According to Ref. 51, the status of the AB approach in the low-temperature studies can be described as follows. Beyond the obvious instance where (i) c = kt/k approaches 1, use of the AB model is fully legitimate in the case where (ii) A —> oo (it is the latter instance that is addressed in the already mentioned line shape theory by Heuer" ). 

Proposed Shape Theory of Liquid Clathrate Formation... 

The shape of a molecule determines many of its physical and chemical properties. Often, shapes of reactant molecules determine whether or not they can get close enough to react. Electron densities created by the overlap of the orbitals of shared electrons determine molecular shape. Theories have been developed to explain the overlap of bonding orbitals and can be used to predict the shape of the molecule. 

At least in principle, can be calculated based on standard line shape theories (e.g., in Appendix A we calculate for our working example... 

Appendix C. The limit T -> oo has, of course, special interest since it is used in standard line shape theories, and does not depend on an assumption of whether the frequency modulations are slow. The exact expression for Q in the limit of T oo is given in Eqs. (A.47) and (A.48). These equations are one of the main results of this chapter. It turns out that Q is not a simple function of the model parameters however, as we show below, in certain limits, simple behaviors are found. 

Remark 1 In the limit of F 0, which corresponds to the limit of Kubo s line shape theory, we get... 

On the other hand, if Td is shorter than T2, the diffusion tends to increase the width of the resonance line from T2 to Td When the conductivity, a increases, the skin depth, S decreases and then the diffusion time, Td also decreases according to Eqs. (7.34) and (7.35). As a result, the condition with Td < T2, is realized. The line shape theory by Dyson is then applicable. The exact line shape can be derived as functions of Td, T2, and Tx where Tj is the time it takes for the electron to pass through the sample. Eeher and Kip [49] calculated the line shapes for different ratios of the diffusion time Td to the relaxation time T2, which are shown in Fig. 7.44 in the case of Tx Td, Tj T2. A typical Dysonian line shape is that depicted in the... 

The first group of papers are relative to the study of band moments in presence of the Coriolis interaction. The main papers are due to Gilbert, Nectoux et al. (25) and to St Pierre and Steele (26). Rymmptric tops, spherical tops and linear molecules are carefully examined. These theories are elaborated much in the same spirit as other band shape theories and contain similar restrictions. 

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