Frequency-dependent dipolar

Frequency-dependent dipolar relaxation In ordered i stems... 

Because of very high dielectric constants k > 20, 000), lead-based relaxor ferroelectrics, Pb(B, B2)02, where B is typically a low valence cation and B2 is a high valence cation, have been iavestigated for multilayer capacitor appHcations. Relaxor ferroelectrics are dielectric materials that display frequency dependent dielectric constant versus temperature behavior near the Curie transition. Dielectric properties result from the compositional disorder ia the B and B2 cation distribution and the associated dipolar and ferroelectric polarization mechanisms. Close control of the processiag conditions is requited for property optimization. Capacitor compositions are often based on lead magnesium niobate (PMN), Pb(Mg2 3Nb2 3)02, and lead ziac niobate (PZN), Pb(Zn 3Nb2 3)03. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

NMR spectroscopy is a powerful technique to study molecular structure, order, and dynamics. Because of the anisotropy of the interactions of nuclear spins with each other and with their environment via dipolar, chemical shift, and quadrupolar interactions, the NMR frequencies depend on the orientation of a given molecular unit relative to the external magnetic field. NMR spectroscopy is thus quite valuable to characterize partially oriented systems. Solid-state NMR... 

The combined effects of the dipolar and exchange interactions produce a complex frequency-dependent EPR spectrum, which can however be analysed by performing numerical simulations of spectra recorded at different microwave frequencies. When centre A is a polynuclear centre, the value of its total spin Sa = S, is determined by the strong exchange coupling between the local spins S, of the various metal sites. In this case, the interactions between A and B consist of the summation of the spin-spin interactions between Sb and all the local spins S, (Scheme II). The quantitative analysis of these interactions can therefore yield the relative arrangement of centres A and B as well as information about the coupling within centre A. 

The dipolar component arises from diffusion of bound charge or molecular dipole moments. The frequency dependence of the polar component may be represented by the Cole-Davidson function ... 

Electrode polarization, represented by the second term in equation (5), in general is a significant and difficult to account for factor at frequencies below 10 Hz and/or for high values of a usually associated with a highly fluid resin state. The frequency dependence e due to dipolar mobility is generally observed at frequencies in the KHz and MHz regions. For this reason an analysis of the frequency dependence of e, equations 3 and 5, in the Hz to... 

The frequency dependence of the loss e is used first to determine <7 by determining from a computer analysis or a plot of e" u (Figure 4), the frequency region where e" u is a constant. Over this frequency region the value of ct is determined from the 1/u dependence of e", eq. 5. The ionic contribution l is substrated from e measured to determine the dipolar component e" The time at which a peak occurs in the dipolar portion, for a particular... 

In this field, the resolution of DMR is promising. However, experiments on deuterated molecules have just begun, and the nuclear relaxation was not yet analyzed. We can just present here some preliminary ideas that were obtained from proton relaxation experiments (19). Because of the nature of dipolar interaction, we are dealing with a multispin system this entails some complex problems of nuclear spin dynamics which are beyond the scope of this discussion. The quantitative analysis of proton relaxation data is thus far from straight-forward (20). We shall limit ourselves to a qualitative interpretation of the frequency dependence of the relaxation rate that is summarized schematically in Figure 4. Important relaxation effects appear in both high and low frequency regions. 

Some qualitative guidelines can be given to make an a priori estimate of the relative weight of dipolar, contact, and Curie relaxation contributions. Consider first the fast motion limit where Rim = Rim and none of the frequency-dependent terms is dispersed. The equations take the simple form already noted ... 

Fig. 2.46 Frequency dependence of s" for PTHFM (open symbol) at 393 K. Global fit (—), contributions of dipolar relaxation curve (-), and conductive processes...

Depending on the scheme chosen, the birefringence experiments provide [143,144] direct measurements of either Av or (Av)2. To present the theoretical results in a form suitable for comparison with the experimental data, let us consider the orientational oscillations induced in the dipolar suspension by a harmonic held H = Hq cos (at and analyze the frequency dependencies of the spectra of the order parameters (P2) and (Pi)2- As formula (4.371) shows, the latter quantities are directly proportional to Av and (Av)2, respectively. Since the oscillations are steady, let us expand the time-dependent orientational parameters into the Fourier series... 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

Figure 11 a illustrates the frequency dependence of e for Eq. (3-6). Note that e is midway between eu and er when co = l/id. The corresponding plots for s" are more complex, because one must assess the relative contributions of a and the dipole loss. The simplest case is for cr = 0 (Fig. lib), where the characteristic dipolar loss peak of amplitude (sr — eu)/2 is observed at frequency co = l/td. For non-zero ct, however, the 1/co dependence of e" greatly distorts the e" curve from the ideal Debye peak. Log-log scales are helpful, as illustrated in Fig. 12. The ct = 0 case is replotted from Fig. lib also plotted are the frequency dependences of e" for CTTd/Eo having various values relative to er — eu. Asct increases, it becomes increasingly difficult to discern the dipole loss peak. Roughly speaking, for CTTd/Eo greater than about three times er, the observed e" is entirely dominated by ct. (Ideally, even when cr dominates the dipolar contribution to e", it should still be possible to observe the dipolar contribution to e however, when o is large, electrode polarization effects tend to dominate the e measurement as well. See Sec. 3.2.1).

Discussion of the dipolar relaxation involves two issues first, the average dipolar mobility at a given temperature and degree of conversion, as measured by the frequency of the maximum in the loss factor fmax (or by its reciprocal, the typical dipolar relaxation time xd), and, second, the detailed distribution of relaxation times as measured by the frequency dependence of the permittivity and loss factor. In spite of the clear evidence that the dipolar relaxation is associated with the glass transition... 

Freezing of a dipolar liquid is accompanied by a rapid decrease in its electric permittivity [8-10]. Following solidification, dipole rotation ceases and the electric permittivity is almost equal to n, where n is refractive index, as it arises from deformation polarisation only. Investigation of the dynamics of a confined liquid is possible from the frequency dependences of dielectric properties, which allows both the determination of the phase transition temperature of the adsorbed substance and characteristic relaxation frequencies related to molecular motion in particular phases. 

The application of relaxation time measurements to study segmental motion (in polymers) as well as diffusional chain motion is very well documented but is still a subject of study, particularly using the frequency dependence of relaxation times to test the detailed predictions of models (McBriety and Packer 1993). The anisotropy of reorientation can also be studied conveniently, and recent interest in motion of molecules on surfaces (e.g. water on porous silica) has been investigated with great sueeess (Gladden 1993). Since the dipolar interaction is usually both intermolecular and intramolecular, the relaxation of spin- /2 nuclei (e.g. H) in the same molecule as a quadrupolar nucleus (e.g. H) can permit a complete study of reorientation and translation at a microscopic level (Schmidt-Rohr and Spiess 1994). 

T+ being the time-ordering operator. In the derivation of Eq. (168) we assumed that B(t)) = 0. It needs to be stressed that Eq. (168) generalizes previously known master equations to arbitrary time-dependent hamiltonians, Hs t) for the system and Hi(t) for system-bath coupling, [Cohen-Tannoudji 1992], Henceforth, we explicitly consider a driven TLS undergoing decay, whose resonant frequency and dipolar coupling to the reservoir are dynamically modulated, so that... 

The dielectric constant of a material is a measnre of its polarizability in response to an electric field. This polarizability is the resnlt of reorganization of charge, which can be in the form of interfacial or space charge motion, ionic motion, dipolar motion, and electronic motion (see Figure 3.2.3) [14]. The timescale of the charge redistributions determines the frequency dependence of this contribution to the dielectric constant for a given material. In the case of ionic motion, the frequency range is up to 10 Hz. 

We have already seen that the coupled relaxation transitions between the nuclei and electrons, which give rise to the Overhauser effect, are stimulated by fluctuations in the local magnetic fields at frequencies coj and co. The intensities of these fluctuations (denoted by where ft) is the appropriate frequency) depends critically on the correlation time for, and the nature of, the nuclear electron interactions, which may be either dipolar or scalar (see above). 

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