Symmetric relaxation function

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

The existence of molecules often creates permanent intramolecular optical anisotropy. The optical anisotropy of the liquid is then due to fluctuations in the orientations of the molecules or molecular subunits. If we assign a symmetric traceless anisotropy tensor a to each molecule or molecular subunit in the scattering volume, then the relaxation function for collective optical anisotropy fluctuations can be expressed as... 

In case of anisotropic motions (an anisotropic fluid), the quadrupolar interaction is partially time-averaged to a residual tensor. This residual interaction A modulates the time relaxation function via a reversible dephasing MYt)=exp[-t/T2]cosAt. For motions uniaxial around a symmetry axis, the effect on the spectrum is to split the resonance line into a symmetric doublet with a splitting (in frequency units) ... 

The expression (1.24) allows obtaining the distribution function of relaxation times for all empirical laws (1.23). In Fig. 1.9, we show the relaxation time distribution functions, obtained in Ref. [31] with the help of Eq. (1.24). The distribution functions have been obtained for the laws of Cole-Cole k = 0.2), Davidson-Cole (P = 0.6) and Havriliak-Nagami at a = 0.42 when it corresponds to KWW law. It is seen that only C-C law leads to symmetric dishibution function while DC and KWW laws correspond to essentially asymmetric one. The physical mechanisms responsible for different forms of distribution functions in the disordered ferroelechics had been considered in Ref. [32]. It has been shown that random electric field in the disordered systems alters the relaxational barriers so that the distribution of the field results in the barriers distribution, which in turn generates the distribution of relaxation times. Nonlinear contributions of random field are responsible for the functions asymmetry, while the linear contribution gives only symmetric C-C function. 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

It is known for a long time (Wetton et al. 1978) that the relaxation function measured for a miscible blend is considerably broadened compared to the spectra of the pure polymers (Colmenero and Arbe 2007). To be more precise the broadening is more or less symmetric. As an example this is shown for a miscible blend of polystyrene (PS) and poly(vinyl methylether) (PVME) in Fig. 12.15 (Colmenero and Arbe 2007 Katana et al. 1992 Zetsche and Fischer 1994). Compared to PVME the dipole moment of PS is weak, and therefore, the contribution of PS to the dielectric loss of the blend is negligible. In other words the fluctuations of PVME are selectively monitored by dielectric spectroscopy, whereas the fluctuations of the PS segments are dielectrically invisible. For the blend (see Fig. 12.15b), the loss peak is much broader than that for the single component PVME (see Fig. 12.15a). Moreover, the loss peak narrows as temperature increases. For the PVME/PS blend system, it was proven by a combination... 

An approximate method to analyze viscoelastic problems has been outlined by Schap-ery.(30) jn this method, the solution to a viscoelastic problem is approximated by a correspond ing elasticity solution wherein the elastic constants have been replaced by time-dependent creep or relaxation functions. The method may be applied to linear as well as nonlinear problems. Weitsman D used Schapery s quasi-elastic approximation to investigate the effects of nonlinear viscoelasticity on load transfer in a symmetric double lap joint. By introducing a stress-dependent shift factor, he observed that the enhanced creep causes shear stress relief near the edges of the adhesive joint. 

Here e , is the high frequ y limit of s, So is the static dielectric constant (low frequency limit of s ). So - Soo = A is the dielectric increment, fR is the relaxation frequency, a is the Cole-Cole distribution parameter, and P is the asymmetry parameter. The relaxation frequency is related to the relaxation time by fa = (27It) A simple exponential decay of P (oc,P = 0) is characterised by a single relaxation time (Debye-process [1]), P = 0 and 1 < a < 0 describe a Cole-Cole-relaxation [2] with a symmetrical distribution function of t whereas the Havriliak-Negami equation (EQN (4)) is used for an asymmetric distribution of x [3]. The symmetry can be readily seen by plotting s versus s" as the so-called Cole-Cole plot [4-6]. 

The symmetry-breaking of the HF function occurs when the resonance between the two localized VB form A+...A and A...A+ is weaker than the electronic relaxation which one obtains by optimizing the core function in a strong static field instead of keeping it in a weak symmetrical field. If one considers for instance binding MOs between A and A they do not feel any field in the SA case and a strong one in the SB solution. The orbitals around A concentrate, those around A become more diffuse than the compromise orbitals of A+ 2 and these optimisations lower the energy of the A. A form. As a... 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

Here a = X and y(a, z) = Jo dt e is the incomplete gamma function. It can be noted that for the axially symmetric potential with a longitudinal field, the only dependence on X is the trivial one in Xp, while in the nonaxially symmetric potential obtained with a transversal field the relaxation rate will strongly depend on X through F(oc), which is plotted in Figure 3.6. 

Fig. 2.49 Dynamic structure factor at two temperatures for a nearly symmetric PEP-PDMS diblock (V = 1110) determined using dynamic light scattering in the VV geometry at a fixed wavevector q = 2.5 X 10s cm-1 (Anastasiadis et al. 1993a). The inverse Laplace transform of the correlation function for the 90 °C data is shown in the inset. Three dynamic modes (cluster, heterogeneity and internal) are evident with increasing relaxation times.

Fig. 3.14. Plot of the spectral density functions for dipolar relaxation in the presence of an axially symmetric g tensor. Conditions gn = 2.3, gj = 2.0, xc = 2 x 10-9 s, 6 — 0° (upper curve) and 0 = 90° (middle curve) compared with the Solomon behavior (lower curve).

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