Rotational dephasing

Because of the complex spin-dynamics of the CP transfer involving quadrupolar nuclei, CP-based methods may not be quantitative. An alternative approach to CP spectral editing, introduced by Fernandez et al. [70], combines MQMAS with (quadrupolar spin)-observe (spin l/2)-dephase Rotational Echo Double Resonance (REDOR) [90]. The method relies on reintroducing the het-... 

This is no longer the case when (iii) motion along the reaction patir occurs on a time scale comparable to other relaxation times of the solute or the solvent, i.e. the system is partially non-relaxed. In this situation dynamic effects have to be taken into account explicitly, such as solvent-assisted intramolecular vibrational energy redistribution (IVR) in the solute, solvent-induced electronic surface hopping, dephasing, solute-solvent energy transfer, dynamic caging, rotational relaxation, or solvent dielectric and momentum relaxation. 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

Physically the independence reflects the fact that dephasing is performed by weak long-range interactions, and rotational relaxation results mainly from short-range, repulsive forces. In other words the rotational state is changed solely when the distance between molecules becomes rather short, while the phase is frustrated in all cases and the contribution of frontal collisions is not so significant. 

Condition (3.19) is usually satisfied in processes of vibrational dephasing [41, 130, 131, 132], Because of this condition the dephasing is weak and the effect of rotational structure narrowing is pronounced. A much more important constraint is imposed by inequality (3.18). It shows that perturbation theory must be applied to a rather dense medium and even then only the central part of the spectrum (at Aa> < 1/t , 1/tc) is Lorentzian. 

Hence the dephasing rate is directly proportional to the rotational relaxation rate 1/xj, while w0 in (3.23) is inversely proportional to it. 

In the conclusion of the present chapter we show how comparison of NMR and Raman scattering data allows one to test formulae (3.23) and (3.24) and extract information about the relative effectiveness of dephasing and rotational relaxation. In particular, spectral broadening in nitrogen caused by dephasing is so small that it may be ignored in a relatively rarefied gas when spectrum collapse proceeds. This is just what we are going to do in the next sections devoted to the impact theory of the isotropic Raman spectrum transformation. 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Fig. 3.8. The Q-branch Raman width alteration with condensation of nitrogen. The theoretical results for the strong (A) and weak (B) collision limits are shown together with experimental data for gaseous [89] ( ) and liquid nitrogen [145] ( ) (point a is taken from the CARS experiment of [136]). The broken curves in the inset are A and B limits whereas the intermediate solid curve presents the rotational contribution to line width at y = 0.3. The straight line estimates the contribution of vibrational dephasing [143], and the circles around it are the same liquid data but without rotational contribution.

It implies that relaxation times obey relation (3.47) even after gas condensation, although both 1/te and ydP become nonlinear in density. The contribution of the rotational broadening represented by the first component may be estimated to a rather high accuracy via the value of y found in (3.45). Subtrrcting it from the width observed, we obtain the dephasing contribution which is linear in T (see inset in Fig. 3.8). The... 

Fig. 3.13. Density-dependence of the Qo, branch line width y of methane (the dashed line is for pure vibrational dephasing, supposed to be Unear in density), (o) experimental data (with error bars) [162] Top part rotational contribution yR and its theoretical estimation in motional narrowing limit [162] (solid line) the points were obtained by subtraction of dephasing contribution y

Vibrational broadening in [162] was taken into account under the conventional assumption that contributions of vibrational dephasing and rotational relaxation to contour width are additive as in Eq. (3.49). This approximation provides the largest error at low densities, when the contour is significantly asymmetric and the perturbation theory does not work. In the frame of impact theory these relaxation processes may be separated more correctly under assumption of their statistical independence. Inclusion of dephasing causes appearance of a factor... 

Fig. 3.15, The CARS spectrum rotational width versus methane density for various values of parameter y (1) y = 0, (2) y = 0.3, (3) y = 0.5, (4) y = 0.7, (5) y = 0.75, (6) y = 0.9, (7) y = 0.95, (8) y = 1. Curves (4) and (6) are obtained by subtraction of the dephasing contribution from the line width calculated taking account of vibrational broadening. The other dependences are found assuming purely rotational broadening (vibrational relaxation neglected).

The contribution of dephasing may be easily extracted, if the experiment provides high enough pressures. When the contribution of rotational broadening (3.68) is relatively small, Acoi/2 lAdP is linear over density and the slope is OdP v). On the other hand the true relaxation time of rotational energy xE = W(1 — A) may be expressed via xe and TdP found experimentally as follows ... 

In [162] experiments on methane provided a linear pressure dependence of the contour width. This made it possible to find the dephasing cross-section and to discriminate between contributions of rotational and vibrational relaxation to the contour width. This was done under the above-mentioned simplifying assumption that they are additive. (Let us note that processing of experimental data on linear molecules was always performed under this assumption.) The points found by this method are shown in Fig. 3.15, curves (4) and (6). 

The relaxation operator f carries out rotational broadening as well as vibrational dephasing between 0 and 1 states. 

It is reasonable when both vibrational and rotational dephasings are negligible. Using this approximation SCS estimations of the rate coefficients of line broadening were compared in [191] with the experimental y -dependence of kj in the S-branch of ty-Ar mixture obtained in [214],... 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

The rotational phase shift 5, which cannot exceed a mean angle of a molecular rotation during collisional time (anc), is certainly small in the case of non-adiabatic collisions. This condition is exactly that needed for anisotropic scattering (or IR absorption) spectrum narrowing, just as vibrational dephasing must be weak for an isotropic spectrum to narrow. 

Q-branch non-broadening 117 rotational cross-sections 193 spin-lattice relaxation 47 weak dephasing, collisional narrrowing 93... 

Iwata K, Ozawa R, Hamaguchi H (2002) Analysis of the solvent- and temperature-dependent Raman spectral changes of S1 trans-stilbene and the mechanism of the trans to cis isomerization dynamic polarization model of vibrational dephasing and the C=C double-bond rotation. J Phys Chem A 106 3614—3620... 

Fig. 2 (a) DRAMA pulse sequence (using % = t/2 = rr/4 in the text) and a representative calculated dipolar recoupled frequency domain spectrum (reproduced from [23] with permission), (b) RFDR pulse sequence inserted as mixing block in a 2D 13C-13C chemical shift correlation experiment, along with an experimental spectrum of 13C-labeled alanine (reproduced from [24] with permission), (c) Rotational resonance inversion sequence along with an n = 3 rotational resonance differential dephasing curve for 13C-labeled alanine (reproduced from [21] with permission), (d) Double-quantum HORROR experiment along with a 2D HORROR nutation spectrum of 13C2-2,3-L-alanine (reproduced from [26] with permission)... 

Figure 41 shows the absorption spectrum for the 24-mode model of pyrazine. As was done by Raab et al. [277], we have included a phenomenological dephasing time of T2 = 150 fs to model the experimental broadening due to hnite resolution and rotational motion. It can be seen that the inclusion of all 24 normal modes of the pyrazine molecule leads to a shape of the spectrum which is in good agreement with the experimental result (Fig. 38b). The semiclassical result is seen to be in fairly good agreement with the quantum result. The spurious structure in the semiclassical spectrum is presumably due to the statistical error.

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