Debye function analysis

Another approach to the analysis of a WAXS pattern is called Debye function analysis (DFA) and has been applied by several groups (Reinhard et al. 1997, 1998 Gnutzmann and Vogel 1990). The main difficulty in any diffraction experiment is that a unique structural model cannot usually be extracted from the data. This is obvious with powder diffraction data where for a complex structure there are far more structural... 

Figure 24. Debye function analysis done on a colloid of 2-nm gold particles. Solid line is the fit using icosahedral and dodecahedral nanoparticles. Dotted vertical lines indicate FCC stracture peak positions. Essentially no FCC or cub-octahedral particles are present. The bottom plot shows the difference between calculation and observation. From Zanchet et al. (2000), used with permission of Wiley-VCH.

Figure 7. Diffraction pattern of a RuxSey sample after pyrolysis at 220 °C of Ru4Se2(CO)ii. The full line is the Debye function analysis simulation. Adapted with permission from Ref. 144, Copyright (2007) American Chemical Society.

Powder X-ray di action of the gold partides dispersed on alumina support produced a very broad peak at 20 = 40.0° after subtradion of the background pattern from the alumina (see Fig. 3). Recent Debye Function Analysis (DFA) shows this pattern to be consistent with decahedra (decahedral multiply twinned particles, MTPs) of about 2.2 nm mass-mean diameter [7]. X-ray diffraction from a... 

In some cases, a Debye-Bueche analysis was attempted. In such an analysis, the spatial correlation function is given by ... 

The heat capacity of Th02 was determined in an adiabatic calorimeter from 10.2 to 305.4 K. The heat capacity and entropy at 298.15 K were calculated to be (61.76 + 0.06) and (65.24 + 0.08) JK mol respectively when corrected for the modem atomic weight of thorium, these become 61.74 and 65.23 J K mol. The sample of thoria used was ground from electrically-fused material, and was analysed to have a Th content of (87.54-87.93) mass% (theoretical 87.88). Chemical analysis showed the total content of lanthanide elements to be < 150 ppm, similar to the total content of other metals, measured spectroscopically. 53 heat capacity measurements were made, extending from 10.2 to 305.4 K. For the entropy calculations, the heat capacity was extrapolated to 0 K using a Debye function. The heat capacity and entropy from this excellent study were adopted by the review. 

Next, it is useful to expand this analysis to two dimensions. The frequency disttibution is now linear, as shown in Eq. (6) of Fig. 2.38. The mathematical expression of the two-dimensional Debye function is given in Fig. 2.37. Note that in Eq. (7) for it is assumed that there are 2N vibrations for the two-dimensional vibrator, i.e., the atomic array is made up of N atoms, and vibrations out of the plane are prohibited. In reality, this may not be so, and one would have to add additional terms to account for the omitted vibrations. The same reasoning applies for the onedimensional case of Eq. (4) of Fig. 2.36. 

Figure 2.46 illustrates the completed analysis. A number of other polymers are described in the ATHAS Data Bank, described in the next section. Most data are available for polyethylene. The heat capacity of the crystalline polyethylene is characterized by a T dependence to 10 K. This is followed by a change to a linear temperature dependence up to about 200 K. This second temperature dependence of the heat capacity fits a one-dimensional Debye function. Then, one notices a slowing of the increase of the crystalline heat capacity with temperature at about 200 to 250 K, to show a renewed increase above 300 K, to reach values equal to and higher than the heat capacity of melted polyethylene (close to the melting temperature). The heat capacity of the glassy polyethylene shows large deviations from the heat capacity of the crystal below 50 K (see Fig. 2.45). At these temperatures the absolute value of the heat capacity is, however, so small that it does not show up in Fig. 2.46. After...

The quite complicated temperature dependence of the macroscopic heat capacity in Fig. 2.46 must now be explained by a microscopic model of thermal motion, as developed in Sect. 2.3.4. Neither a single Einstein function nor any of the Debye functions have any resemblance to the experimental data for the solid state, while the heat capacity of the liquid seems to be a simple straight line, not only for polyethylene, but also for many other polymers (but not for all ). Based on the ATHAS Data Bank of experimental heat capacities [21], abbreviated as Appendix 1, the analysis system for solids and liquids was derived. 

The next step in the ATHAS analysis is to assess the skeletal heat capacity. The skeletal vibrations are coupled in such a way that their distributions stretch toward zero frequency where the acoustical vibrations of 20-20,000 Hz can be found, hi the lowest-frequency region one must, in addition, consider that the vibrations couple intermolecularly because the wavelengths of the vibrations become larger than the molecular anisotropy caused by the chain structure. As a result, the detailed molecular arrangement is of little consequence at these lowest frequencies. A three-dimensional Debye function, derived for an isotropic solid as shown in Figs. 2.37 and 38 should apply in this frequency region. To approximate the skeletal vibrations of linear macromolecules, one should thus start out at low frequency with a three-dimensional Debye function and then switch to a one-dimensional Debye function. 

According to a nonlinear Poisson-Boltzmann analysis, the initial radial decay of f from a cylindrical chain segment is much steeper than predicted by a Debye-Hiickel analysis see Figure 3 (28). The steep decay lessens at large distance, and eventually adopts an asymptotic functional form compatible with a Debye-Hiickel approximation. However, to superimpose across this distant region the predictions of the Poisson-Boltzmann analysis onto those formidated... 

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. 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

A different analysis of the scattering pattern uses the Debye correlation function (14), derived for a random two-phase structure with sharp interfaces ... 

Informations on the vibrational and electron mean free path properties. Such analysis is possible only if the interface phase is very well defined, and if temperature dependent measurements are done and compared. Debye Waller effects can be tangled with ordering transformation of the interface phase as a function of temperature and so on. If a single phase interface with order at least to the second nearest neighbour is recognised, then a temperature dependent Debye Waller, and mean free path analysis can be attempted. 

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