Isotropic relaxation

In samples with negligible quadmpole interaction, isotropic relaxation with relaxation times around s results in spectra consisting of a single,... 

To have Fel for inhomogeneous cases, we distinguish the Cartesian coordinates of the deformed gel, X = (X, X2, X3), representing the real spatial position, and those in the isotropic, relaxed state, jc0 = (x , x , x ), representing the original position before deformation. We define the deformation tensor,... 

Finally, the last term in Eq. (5.92) describes isotropic relaxation processes in the ground state. 

As can be seen, for any isotropic relaxation process different rank polarization moments change independently. The question arises can the relaxation constants 7 and Tk be arbitrary The following simple example [135] shows that this is not the case. Let us assume that the angular momentum J = 1 and the initial state are such that only level M = 1... 

For an approximate estimate of the mobilities, either a constant, isotropic relaxation time r or a constant, isotropic mean free path X can be assumed. In these two approximations, the components pty of the mobility tensor are found from the Einstein relation (Eq. (8.22)) to be... 

Fig. 2. Theoretical Mossbauer line shape for the isotropic relaxation of the electric field gradient. Here p =p =P. Relaxation time is shown in the figure. x y z...

Carper and co-workers have performed a detailed analysis of the relaxation times of both [C4mim] [25] and [Cgmim]" [26] with [PF0]", which was later extended with more detail to deconvolute the relative contributions of the various relaxation mechanisms [27]. They found that the contribution of CSA to the experimentally observed relaxation time was about half of the contribution from dipolar relaxation. This work raises doubts about the applicability of isotropic relaxation models to ionic liquids. It is important to note that the and measurements of ionic liquids in the literature show different behaviour when attached to the same ion. The random Brownian motion that occurs in most liquids leads to rapid spin diffusion between nuclei bonded to a common ion or molecule, causing them to all exhibit the same T. The lack of such behaviour is a clear indication that the dynamics of ionic liquids are... 

Also in this case, Tp corresponds to a relaxation time which determines the coupling of the modulated variable to the external bath. The pressure scaling can be applied isotropically, whidi means that the factor is the same in all three spatial directions. More realistic is an anisotropic pressure scaling, because the box dimensions also change independently during the course of the simulation. 

In the derivation of equations 24—26 (60) it is assumed that the cylinder is made of a material which is isotropic and initially stress-free, the temperature does not vary along the length of the cylinder, and that the effect of temperature on the coefficient of thermal expansion and Young s modulus maybe neglected. Furthermore, it is assumed that the temperatures everywhere in the cylinder are low enough for there to be no relaxation of the stresses as a result of creep. 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Among these three polybibenzoates, PTEB has a smectic mesophase stable during several days at any temperature below its isotropization point, although the transformation into a three-dimensional crystal can be attained by annealing at the appropriate temperatures, thus making it possible to analyze the effect of the thermal history on the dynamic mechanical relaxations of PTEB [27]. 

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-... 

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. 

In the case of the isotropic spectrum it is useful to consider the functional dependence of Acoi/2 and 5gj on rE, since it is rotational energy relaxation that causes frequency modulation in this spectrum. Such a... 

As can be seen, the difference in behaviour of orientational relaxation times Te,2 in models of weak and strong collisions is manifested more strongly than in the case of isotropic scattering. Relation (6.26) is... 

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... 

Bonamy L., Bonamy J., Robert D., Temkin S. I. Consequences of angular momenta coupling on generalized spectroscopic relaxation cross-sections collisional narrowing in isotropic and anisotropic Raman CO2 branches, Proc. 13th ICORS. (Wiley Sons, New York) (1992). 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

Optical and electro-optical behavior of side-chain liquid crystalline polymers are described 350-351>. The effect of flexible siloxane spacers on the phase properties and electric field effects were determined. Rheological properties of siloxane containing liquid crystalline side-chain polymers were studied as a function of shear rate and temperature 352). The effect of cooling rate on the alignment of a siloxane based side-chain liquid crystalline copolymer was investigated 353). It was shown that the dielectric relaxation behavior of the polymers varied in a systematic manner with the rate at which the material was cooled from its isotropic phase. 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

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