Debye approximation distribution

Figure 4. A bimodal volume weighted distribution (chained line) and the corresponding intensity distributions according to the Mie theory (solid line) and the Rayleigh Debye approximation (dotted line).

For small colloidal particles (a < 1 /mi), with spherically symmetric distribution of scattering material inside its volume, the field amplitude is given by the Rayleigh-Gans-Debye approximation [38]... 

The inadequacy of the Debye approximation in describing the details of the frequency distribution function in a real solid is well known. This results in noticeable disparities between Debye temperatures derived from the results of different experimental techniques used to elucidate this parameter on the same solid, or over different temperature ranges. Substantial discrepancies may be expected in solids containing two (or more) different atoms in the unit cell. This has been demonstrated by the Debye-Waller factors recorded for the two different Mdssbauer nuclei in the case of Snl4,7 or when the Debye-Waller factor has been compared with the thermal shift results for the same Mdssbauer nucleus in the iron cyanides.8 The possible contribution due to an intrinsic thermal change of the isomer shift may be obscured by an improper assignment of an effective Debye temperature. 

The distribution of frequencies for one branch of the vibration spectrum, described in the Debye approximation by treating the crystal as an elastic continuum. The Debye frequency is cod-... 

The overall result i.s typically an exponential distribution of ionic charges in the diffuse layer that results in an exponentially decreasing potential according to the so-called Debye approximation ... 

The integration of equation 1 is evaluated in steps for the various regions of the frequency distributions found for macromolecules. The lowest vibrational frequencies (skeletal) usually follow a quadratic function up to a frequency limit called 0D or 03. This is the well-known Debye approximation (59) of the low temperature heat capacity at constant volume, Cy (B), which in this temperature range... 

To conclude this discussion, Eqs. (8) and (9) of Fig. 5.14 represent the three-dimensional Debye Junction. Now the firequency distribution is quadratic in V, as shown in the bottom figure. The derivation of the three-dimensional Debye model is analogous to the one-dimensional and two-dimensional cases. The three-dimensional case is the one originally carried out by Debye. The maximum fi equency at which 3N, the total possible number of vibrators for N atoms, is reached is V3 or 3. From the frequency distribution one can, again, derive the heat capacity contribution. The heat capacity for the three-dimensional Debye approximation is equal to 3R times >3, the three-dimensional Debye function of 3 divided by T. 

Fig. 3.1 The frequency distribution in real solids, as a function of the temperatme compared with the Einstein (delta function at ve) and the Debye (shaded area) models. In the case of Ag the Debye approximation (dashed line) is very well fulfilled. In the case of complicated solids a quadratic increase at lower temperatures can be seen, too, but the respective vd value is no longer identical with ud as overall fit parameter. The inset illustrates the situation for NaCl [55,64,80]. According to Ref. [48].

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

In the limit of zero ion size, i.e. as o —> 0, the distribution functions and themiodynamic fiinctions in the MS approximation become identical to the Debye-Htickel limiting law. 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

To calculate theoretical intensities, an approximate model of potential is needed. For structure refinement, we need an estimate of cell sizes, atomic position and Debye-Waller factor. In case of bonding charge distribution measurement, crystal structure is first determined very... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

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). 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

A complete solution of eqn. (151) for the time-dependent density distribution does not appear feasible, but Hong and Noolandi [323—325] have found the long-time behaviour, as well as the steady-state solution. The mathematics are very complex since the complications encountered in the analysis of the Debye—Smoluchowski equation (Appendix A) are compounded by the applied electric field. For small electric fields and long times, the survival probability is approximately... 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

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