Debye analysis

Zero angle extrapolated Intensities and size parameter a from Debye analysis. Gelation of PBLG-138,000/toluene (c 0.49 wt% Initial temp 70 C Quench Temp 29.2 C) 1(0) Increases much more than a. 

Figures 45 and 46 shows the logarithm of intensity count as a function of scattering angle (2 theta in minutes) for as-cast and as-postcured epoxies respectively. Specimens of the same thickness were used (0.825 mm). Guinier and Debye analysis were then carried in the regions indicated.

Hartmann et al. (1994) Debye analysis Pt/Si02 Structural dynamics during rate oscillations, redox chemistry + + + CO oxidation... 

The data converted from the Debye analysis to the equivalent radii agree very well with the results from the Porod analysis over most of the composition range, although at some compositions a discrepancy is observed, connected, probably, with the imperfectness of the models used for calculations. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The above approximation, however, is valid only for dilute solutions and with assemblies of molecules of similar structure. In the event that concentration is high where intemiolecular interactions are very strong, or the system contains a less defined morphology, a different data analysis approach must be taken. One such approach was derived by Debye et al [21]. They have shown tliat for a random two-phase system with sharp boundaries, the correlation fiinction may carry an exponential fomi. 

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

It is noteworthy that it is the lower cross-over temperature T 2 that is usually measured. The above simple analysis shows that this temperature is determined by the intermolecular vibration frequencies rather than by the properties of the gas-phase reaction complex or by the static barrier. It is not surprising then, that in most solid state reactions the observed value of T 2 is of order of the Debye temperature of the crystal. Although the result (2.77a) has been obtained in the approximation < ojo, the leading exponential term turns out to be exact for arbitrary cu [Benderskii et al. 1990, 1991a]. It is instructive to compare (2.77a) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. 

The foregoing conclusions are further supported by a refined X-ray analysis of pyrid-2-one, which indicated that the mobile hydrogen atom is attached to the nitrogen atom in the solid state and that individual molecules are bound into helices by N—H- -0 hydrogen bonds. An oxo structure is also indicated by the molar refractivity of pyrid-2-one. The dipole moment of 4-methoxypyridine is ca. 3.0 debyes in dioxane, whereas the values for pyrid-4-one and its 1-methyl derivative are much higher, ca. 6.0 debyes indicating the... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Gorin has extended this analysis to include (1) the effects of the finite size of the counterions in the double layer of spherical particles [137], and (2) the effects of geometry, i.e. for cylindrical particles [2]. The former is known as the Debye-Huckel-Henry-Gorin (DHHG) model. Stigter and coworkers [348,369-374] considered the electrophoretic mobility of polyelectrolytes with applications to the determination of the mobility of nucleic acids. 

Induction forces, the so-called Debye forces ind> occur in the interaction between a permanent dipole of a solute or a polar solvent and an induced dipole in another compound. They are weak and appear during the analysis of the nonpolar polarized compounds, such as those with multiple... 

Abbas Z, Gunnarsson M, Ahlberg E, Nordholm S. 2002. Corrected Debye-Hiickel (CDH) analysis of surface complexation. J Phys Chem B 106 1403-1420. 

Gunnarsson M, Abbas Z, Ahlberg E, Nordhohn S. 2004. Corrected Debye-Hiickel analysis of surface complexation HI. Spherical particle charging including ion condensation. J Coll Interf Sci 274 563-578. 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

It may seem that the prospeets are bleak for the GvdW approach to electrolytes but, in fact, the reverse is the ease. We need only follow Debye and Hiickel [18] into their analysis of the sereening meehanism, almost as successful as the van der Waals analysis of short-range fluids, to see that the mean-field approximation can be applied to the correlation mechanism with great advantage. In fact, we can then add finite ion size effects to the analysis and thereby unify these two most successful traditional theories. 

In their analysis of electrolyte screening Debye and Huckel assumed that the screening mechanism was weak, i.e.,... 

Melroy and co-workers88 recently reported on the EXAFS spectrum of Pb underpotentially deposited on silver (111). In this case, no Pb/Ag scattering was observed and this was ascribed to the large Debye-Waller factor for the lead as well as to the presence of an incommensurate layer. However, data analysis as well as comparison of the edge region of spectra for the underpotentially deposited lead, lead foil, lead acetate, and lead oxide indicated the presence of oxygen from either water or acetate (from electrolyte) as a backscatterer. 

If the g- and hyperfine anisotropies are known from analysis of a solid-state spectrum, the line-width parameters (1, and yt can be used to compute the rotational correlation time, tr, a useful measure of freedom of motion. Line widths in ESR spectra of nitroxide spin labels, for example, have been used to probe the motional freedom of biological macromolecules.14 Since tr is related to the molecular hydrodynamic volume, Ft, and the solution viscosity, r, by a relationship introduced by Debye 15... 

In contrast, a real challenge is the analysis of scattering images from anisotropic materials, and in this subarea many scientists surrender and resort to the interpretation of peak positions and peak widths in raw data (cf. citation of P. Debye on p. 1). So after having advanced by learning how to analyze curves, in the field of anisotropic materials we are now in a similar situation as science has been in 1931 in respect to isotropic data. 

A single-crystal study of oxidized plastocyanin demonstrated that the absence of Cu-S(Met) EXAFS was due to Debye-Waller damping (30). The analysis benefits from the three-fold enhancement of Cu-S(Met) scattering amplitude expected for... 

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