Relaxation mechanisms operator

There is greatly renewed interest in electron solvation, due to improved laser technology. However it is apparent that a simple theoretical description such as implied by Eq. (9.15) would be inadequate. That equation assumes a continuum dielectric with a unique relaxation mechanism, such as molecular dipole rotation. There is evidence that structural effects are important, and there could be different mechanisms of relaxation operating simultaneously (Bagchi, 1989). Despite a great deal of theoretical work, there is as yet no good understanding of the evolution of free-ion yield in polar media. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

The 113Cd Ti values estimated for the various peaks varied from 10 to 50 ms and obeyed the qualitative dependence upon 1/R6 (R = Mn-Cd distance) of the dipolar relaxation mechanism expected to be operative. The broad line widths were also shown to have significant contributions from the T2 relaxation induced by Mn++, with both dipolar and contact terms contributing. The 113Cd shifts of the peaks assigned to different shells were measured as a function of temperature, and observed to follow a linear 1/T dependence characteristic of the Curie-Weiss law, with slopes proportional to the transferred hyperfine interaction constant A. 

Two factors contribute to r K. One is the ratio of the magnetogryric ratios of the two different spins, and the other depends on relaxation mechanisms. Provided that the relaxation mechanism is purely dipole-dipole, other relaxation mechanisms affect spin I, then 4> may approach zero. Assuming that the dipolar mechanism is operational (no quadrupolar nuclei with I > 1/2 are present), r has the value ys/ 2y and is regarded as rimax. In the homonuclear case we have r max = 1/ 2. Usually one chooses nuclei where ys > y/ to ensure that the NOE is significant. For observation of 13C for instance, if the protons in the molecule are double irradiated, the ratio is 1.99 and 1 + r max equals approximately 3. To repeat a statement made above, proton broad-band irradiation enhances the intensity of the 13C nucleus, which otherwise has very low receptivity. 

The double commutator [[g, Tr /) (/], Tlp q may form new operators different from Q, and some of these new operators may not even be physical observables. When the double commutator conserves the operator Q, one speaks of the auto-correlation mechanism. Otherwise, one speaks of the cross-relaxation process. In other words, cross-relaxation is independent of the nature of the relaxation mechanism, but involves the interconversion between different operators. To facilitate such a possibility, it is desirable to write the density operator in terms of a complete set of orthogonal basis... 

When r s, one has interconversion between operators Br and Bs, and Rrs is a cross-relaxation rate. Note that the cross-relaxation may or may not contain interference effects depending on the indices l and /, which keep track of interactions Cyj and C,. Cross-correlation rates and cross-relaxation rates have not been fully utilized in LC. However, there is a recent report41 on this subject using both the 13C chemical shielding anisotropy and C-H dipolar coupling relaxation mechanisms to study a nematic, and this may be a fruitful arena in gaining dynamic information for LC. We summarize below some well known (auto-)relaxation rates for various spin interactions commonly encountered in LC studies. 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

It should be self-evident that the time dependencies of the fluorescent emissions may well be very complicated, because they must reflect the simultaneous operation of a large number of relaxation mechanisms. A complete understanding of the transient luminescent emissions must therefore involve an intimate knowledge of a large number of factors. 

The 13C nuclei in formic and acetic acids relax faster and show larger nuclear Over-hauser enhancements than the 13C nuclei of the methyl esters (Table 3.18) [183], Hence a methyl ester is more mobile than its parent carboxylic acid, which is dimerized via hydrogen bonding. It may also be seen that a pure DD mechanism operates in formic acid, owing to the directly bonded proton (tjc = 2.0), whereas spin rotation also contributes to 13C relaxation in methyl formate (tjc = 1.55). 

This result suggests, if it is assumed that a C-H heteronuclear dipolar relaxation mechanism is operative, that methyl protons dominate the relaxation behavior of these carbons over much of the temperature range studied despite the 1/r dependence of the mechanism. The shorter T] for the CH as compared to the CH2 then arises from the shorter C-H distances. Apparently, the contributions to spectral density in the MHz region of the frequency spectrum due to backbone motions is minor relative to the sidegroup motion. The T p data for the CH and CH2 carbons also give an indication of methyl group rotational frequencies. 

Under continuous uv irradiation, the observed steady-state polarization (whether by cw or by FT spectrometers) may be substantially modified by various nuclear relaxation processes. For example, Closs and Czeropski (35,36) have demonstrated that CIDNP can be transferred from a group of polarized nuclei to another group not originally polarized. Both the dipolar and the scalar relaxation mechanisms (of the nuclear Overhauser effects) can be operative. The extremely interesting case of intramolecular dipolar nuclear cross relaxation reported by Closs and Czeropski (35) involves the thermal reaction of... 

In homo-dinuclear systems, such as two copper(ll) ions, no large effects are expected on the electron relaxation rates as the two metal ions relax at the same rate. However, some other relaxation mechanisms are operative, giving rise to faster electron relaxation rates (dementi and Luchinat, 1998). Consequently, nuclear relaxation is slower than in single copper(ll) systems. Several examples from model complexes are available (Brink et al., 1996 Murthy et al., 1997), as well as from a copper(ll)-substituted zinc enzyme, the aminopeptidase from Aeromonas proteolytica (Holz et al., 1998). In contrast, few NMR studies on native copper proteins containing two coupled copper (II) ions have been reported so far (Bubacco cf a/., 1999). 

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