Debye form factor

FIGURE 5 SANS from blend of deuterated and conventional polyisoprene (M = 10 g/mol), plotted in the Gunier form [Eq. (9)] yielding a straight at small angles. The solid line is the fit to the Debye form factor for Gaussian coils [Eq. (lO)j. (From Akcasu, Summerfield, Jahsan et al. [126].)... 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

It is instmctive to consider the Debye form factor in the regions of small and large Q compared with the inverse size of the polymer these approximate form factors have a much simpler form and can easily be used for the analysis of the scattering data. So in the region of small Q, for example, Q < 1/Rg one finds Fob - 1 This Zimm approxima-... 

The inverse structure factor is determined from the sum of the inverse Debye form factors of both chains weighted with their molar volumes and volume fractions. For identical molar volume (Va = Vb] and form factor PobiQ], we get the structure factor of eqn [32]. Equation [48] corresponds to an ideal solution of two components whose mixing energy is zero and therefore there are no thermal composition fluctuations as... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

Below the ODT such a label highlights the polymer-polymer interface. A main peak around Q" =0.02 A" corresponding to a lamellar periodocity 2 n/diain with di j =3l5 A is observed. Its visibility results from the asymmetric nature of the diblock. We note the existence of a second order peak, which is well visible at Todt=433 K. At large Q>Q the scattering is dominated by the form factor of the PEP-label in the environment of the deuterated monomers at the interface. This form factor may be described by a Debye function A)ebye( ) (Eq. 3.23). The absolute cross-section for these labels is given by ... 

T(S) is the Debye-WaUer factor introduced in (2). The atomic form factors are typically calculated from the spherically averaged electrcai density of an atom in isolation [24], and therefore they do not contain any information on the polarization induced by the chemical bonding or by the interaction with electric field generated by other atoms or molecules in the crystal. This approximation is usually employed for routine crystal stmcture solutions and refinements, where the only variables of a least square refinement are the positions of the atoms and the parameters describing the atomic displacements. For more accurate studies, intended to determine with precisicai the electron density distribution, this procedure is not sufficient and the atomic form factors must be modeled more accurately, including angular and radial flexibihty (Sect. 4.2). 

Section 5.5 moves on to an extension of the Rayleigh theory essential for colloid science, namely, the Debye theory for particles of the order of the wavelength of the radiation source. The important concept of interference effects, the form factor, the Zimm plot, and... 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

In view of the calculations considered in Section V and in other publications (VIG), these interactions, giving rise to FIR absorption and to low-frequency Debye loss, resemble interactions pertinent to strongly polar nonassociated liquids. However, if we compare water with a nonassociated liquid (e.g., CH3F), then we shall find that in the latter (i) the R-band is absent (ii) the number mvjb of the reorientation cycles is much less, so that the reduced collision frequency y is substantially greater thus, molecular rotation is more damped and chaotic and (iii) the fitted form factor/is greater. 

In terms of the four parameters in Eq. (316a) the latter alone gives a satisfactory description of the Debye-relaxation and librational bands of various liquids. One can control the width of the librational band of liquid H2O by changing the form factor/. 

Another contribution to variations of intrinsic activity is the different number of defects and amount of disorder in the metallic Cu phase. This disorder can manifest itself in the form of lattice strain detectable, for example, by line profile analysis of X-ray diffraction (XRD) peaks [73], 63Cu nuclear magnetic resonance lines [74], or as an increased disorder parameter (Debye-Waller factor) derived from extended X-ray absorption fine structure spectroscopy [75], Strained copper has been shown theoretically [76] and experimentally [77] to have different adsorptive properties compared to unstrained surfaces. Strain (i.e. local variation in the lattice parameter) is known to shift the center of the d-band and alter the interactions of metal surface and absorbate [78]. The origin of strain and defects in Cu/ZnO is probably related to the crystallization of kinetically trapped nonideal Cu in close interfacial contact to the oxide during catalyst activation at mild conditions. A correlation of the concentration of planar defects in the Cu particles with the catalytic activity in methanol synthesis was observed in a series of industrial Cu/Zn0/Al203 catalysts by Kasatkin et al. [57]. Planar defects like stacking faults and twin boundaries can also be observed by HRTEM and are marked with arrows in Figure 5.3.8C [58],... 

The sum is over the magnetic ions in the magnetic unit cell. The /th ion has effective spin form factor fj(x) and Debye-Waller factor exp[—... 

We have shown, in later sections, how precise INS measurements of the DOS provide the most stringent means of testing the model potential functions that lie at the heart of any LD or MD simulation. In the last a few years, we have systematically studied the vibrational dynamics of a large verity of phases of ice using above instruments at ISIS. These spectra were obtained at very low temperatures (< 15 K) on the recoverable high-pressure phases of ice and a few forms of amorphous forms of ice, in order to reduce the Debye-Waller factor and avoid multiphonon excitations. Hence the one-phonon spectra, g(co), can be extracted from the experimental data for the theoretical simulations. 

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