Debye correlation function

Light scattering on chaotical structure heterogeneities is of great importance, these features are described by a correlation function (Debye et ah, 1949, 1957 Kerker, 1969). 

G(r, t) - space- and time-dependent pair correlation function = Debye-Waller temperature factor Ge(r) = equilibrium spatial pair cor relation function for atoms Go(r) == instantaneous spatial pair correlation function... 

The function y(r)—named correlation function (Debye, 1949)— is the volume average of the product of Ap(r) in two volume elements dv, located at i and 2, connected by a vector r. The function y(f) can directly be determined from an experimental scattering intensity function I( ) by a Fourier transformation ... 

From measurements of the angular distribution of light scattering from a gel it should be possible to calculate by Fourier inversion a correlation function (Debye, 1945) characteristic of the structure, which could be compared with functions calculated for various hypothetical models. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

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. 

Table 4.3. Water dimer properties the interaction energy (Ei t) in kcal/mol, the intermolecular distance (R00) in A, and the dipole moment p. in Debye, calculated using the B88/P86 exchange-correlation functional and different basis sets.

Several structure sizes caused by microphase separation occurring in the induction period as well as by crystallization were determined as a function of annealing time in order to determine how crystallization proceeds [9,18]. The characteristic wavelength A = 27r/Qm was obtained from the peak positions Qm of SAXS while the average size of the dense domains, probably having a liquid crystalline nematic structure as will be explained later, was estimated as follows. The dense domain size >i for the early stage of SD was calculated from the spatial density correlation function y(r) defined by Debye and Buche[50]... 

The correlation function C(t) is purely phenomenological. Interpretation of its time evolution is often based on theory in which the longitudinal relaxation time, tl, is introduced. This time is a fraction of the Debye relaxation time ... 

Although more complex pair-correlation functions are available, the Debye-Huckel expression is adequate for our present purpose. It is valid when the work required to bring the reactants... 

For a two-phased system where domains are randomly dispersed, Debye [37,38] showed that the scattered intensity can be expressed in terms of a correlation function,, in a simple exponential... 

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

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

To study the self-image effect on the interaction between two plates, we will develop a linear theory with respect to the plasma parameter with kt1 = (SncoPBe2 / e)1/2, where k-1 is the Debye length, and cq is the electrolyte concentration in solution. According to this approximation, all the higher distribution functions (gapy, etc.) are represented via the correlation function as (see, for example, [19])... 

In order to explain the non-Debye response (134) it is possible to use the memory function approach [22,23,31,266-268]. Thus, the normalized dipole correlation function k(f) (22) corresponding to a nonexponential dielectric relaxation process obeys the equation... 

Flere (1 - f) and (1 - S) are the losses brought about by the corresponding process and gfastj/3 (t -> oo) = 0 (see Fig. 6). The factor / can be regarded as a generalized non-ergodicity parameter and, hence, it is expected to show a similar anomaly as the Debye-Waller factor /g (see Fig. 5). Such decomposition of the correlation function is useful in spin-lattice relaxation studies, as will be discussed in Section 3.2.4. 

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

Notice that equation 13 Is essentially a Gulnler type equation, and that equation 12 Is simply a special case of the correlation function form of equation 1, given by Debye and coworkers (40,41). As such, when non-llnearlty In a semi-log plot Is observed then the methods of Stoll et al. (38) and Effler (17) should prove useful. 

Constructs are generated from theoretical descriptors representing a certain medium. The correlation function for a Debye random medium [8] is... 

The important feature of this expression is the reproducibility of the electrostatic term in the Debye-Hiickel form by the contact value, pf(0) [58], Since the bulk correlation functions within the AMSA theory at a low ion concentration can be presented in the Debye-Hiickel-like form (64), the modified version... 

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