Rotational relaxation definition

Triplet—triplet energy transfer from benzophenone to phenanthrene in polymethylmethacrylate at 77 and 298 K was studied by steady-state phosphorescence depolarisation techniques [182], They were unable to see any clear evidence for the orientational dependence of the transfer probability [eqn. (92)]. This may be due to the relative magnitude of the phosphorescence lifetime of benzophenone ( 5 ms) and the much shorter rotational relaxation time of benzophenone implied by the observation by Rice and Kenney-Wallace [250] that coumarin-2 and pyrene have rotational times of < 1 ns, and rhodamine 6G of 5.7 ns in polymethyl methacrylate at room temperature. Indeed, the latter system of rhodamine 6G in polymethyl methacrylate could provide an interesting donor (to rose bengal or some such acceptor) where the rotational time is comparable with the fluorescence time and hence to the dipole—dipole energy transfer time. In this case, the definition of R0 in eqn. (77) is incorrect, since k cannot now be averaged over all orientations. 

In this case, as with all other hydrogen halide lasers, only P branch transitions are observed, indicating that only partial inversion is attained. The vibrational transitions observed are 1 - 0 and 2 -+ 1. There is a definite threshold flash energy, below which no laser action is observed because the chain decomposition is not fast enough. The development in time of the emission spectrum was observed and discussed in terms of rotational relaxation. 

Owing to their definite structures, most biomolecules have an appreciable permanent dipole moment which must lead to dielectric polarization via the rotational mechanism of preferential orientation. Thus pertinent experimental investigation permits a direct determination of the molecular dipole moments and rotational relaxation times (or rotational diffusion coefficients, respectively). These are characteristic factors for many macro-molecules and give valuable information regarding structural properties such as length, shape, and mass. 

The experimental results for the relaxation of internal energy in polyatomic gases have been well summarised in various places, see e.g. [61.C 77.L1]. The basic observations are as follows rotational relaxation rates appear to be commensurate with the collision rate itself and are in general rather poorly characterised, partly because of their great speed, but also because of the intrinsic ambiguity in their definition as I have just shown vibrational relaxation rates, on the other hand, are quite well defined and reasonably well understood [59.H 77.L1], and take place on time scales of from a few tens to a few thousands of collisions. 

In the most common case where the target is the equilibrium dynamics, the analysis has to be carried out on samples at thermal equilibrium. Albeit trivial, the above observation is not always respected in MD studies. Even in some recent papers we may find that equilibration runs limited to a few hundreds of picoseconds are often employed, although it is known that, e.g., molecular rotations relax over much longer timescales [34,119]. A possible definition of the timescale associated with a process modulating a given dynamic property, A t), is its correlation (or relaxation) time ... 

Let us note that this definition of y breaks the limits of the Kielson-Storer model and can cause a few contradictions in interpretation of results. If the measured cross-section oj appears to be greater than oo, then, according to (3.45), the sought y does not exist. To be exact, this assertion is valid relative to the cross-section of the rotational energy relaxation oe = (1 — y2)oot since y2 is always positive. As to oj, taking into account the domain of negative values of y, corresponding to the anticorrelated case (see Chapter 2), formula (3.45) fails to define y when oj > 2co. 

Let us demonstrate that the tendency to narrowing never manifests itself before the whole spectrum collapses, i.e. that the broadening of its central part is monotonic until Eq. (6.13) becomes valid. Let us consider quantity x j, denoting the orientational relaxation time at ( = 2. If rovibrational interaction is taken into account when calculating Kf(t) it is necessary to make the definition of xg/ given in Chapter 2 more precise. Collapse of the Q-branch rotational structure at T = I/ojqXj 1 shifts the centre of the whole spectrum to frequency cog. It must be eliminated by the definition... 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Bloembergen et al. (S) have presented a relationship between the correlation time for molecular rotation in liquids and the relaxation times assuming that relaxation takes place via mechanism (i) of Section II,A,3. Although the theory can be at best semiquantitative when applied to the protons of water molecules adsorbed on silica gel, values of the nuclear correlation time have been calculated 18) from the T data. These correlation times as a function of x/m show a definite change of slope near a monolayer coverage. This result, if corroborated by data on other solids, may provide a rather unique method for the determination of monolayer coverage. 

In such a case, no conclusion about the mechanisms can be reached from the form of 4(t) and the observed rate will be determined primarily by the fastest process. By extension of the argument, one easily sees that marked deviation of any of the parallel processes from exponential decay will be reflected in the overall rate with possible change in the functional form. Thus, if the rotation is described by exp(-2D t) as in Debye-Perrin theory, and the ion displacements by a non-exponential V(t), one finds from eq 5 that 4(t) = exp(-2D t)V(t) and the frequency response function c(iw) = L4(t) = (iai + 2D ) where iKiw) = LV(t). This kind of argument can be developed further, but suffices to show the difficulties in unambiguous interpretation of observed relaxation processes. Unfortunately, our present knowledge of counterion mobilities and our ability to assess cooperative aspects of their motion are both too meagre to permit any very definitive conclusions for DNA and polypeptides. 

Also the term r =x(B.0,-l,0)=W(B,0,-l,0) can be expressed either as a CETO or a WO-CETO, relaxing condition 2) and 3) of CETO Definition 1. In this manner the last CETO representing the operator Tb does not have to be rotated because it only has a radial part. 

The four spin-lattice relaxation mechanisms that are usually considered for nuclei of spin are (i) dipole-dipole interactions (DD) (ii) spin-rotation interactions (SR) (iii) scalar coupling (SC) and (iv) chemical screening anisotropy (CSA). (5) In Si NMR SC is not generally considered, except in the rare cases when Si is spin-coupled to I or other nuclei with resonance frequencies close to that of Si. Table XXX compares the contribution of the three remaining mechanisms with Si relaxation times. Definitions of these relaxation mechanisms can be found in ref. 229. Since a theoretical discussion of Si relaxation and NOE is detailed elsewhere [(5) and references therein] only general observations are made in this section. 

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

The velocity field is statistically homogeneous if all statistics are invariants under a shift in position. If the field is also statistically invariant under rotations and reflections of the coordinate system, then it is statistically isotropic. In chemical reaction engineering these mathematical definitions are usually somewhat relaxed, since turbulence is said to be isotropic if the individual velocity fluctuations are equal in all the three space dimensions. Otherwise it is said to be an-isotropic. Similarly, a flow field where turbulence levels do not change from one point to another is called homogeneous. 

The orientational van Hove self-correlation function corresponding to the relaxation of the backbone (top panel) and side (lower panel) vectors of glucose (see text for definition) at 220 K show that glucose rotation in the glass proceed by big jumps and does not have a continuous diffusion component. The side rotation, which involves the jump of the smaller glucose bead, the "exocyclic" B6, relaxes faster (ts 3 fjs) than the rotation involving the jumps of the big "backbone" beads B4 and B1 (ts 15 /is). 

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