Debye correlation function, calculation

The intensity distribution of SANS from an isotropic, disordered two-component porous material can be calculated generally from a Debye correlation function r(r) by the following formula [5] ... 

We shall describe a method for calculating the Debye correlation functions used in this paper. The most important property of the correlation function for the bulk contrast case with a sharp boundary between two regions having different scattering length densities is that it has a linear and cubic terms in small r expansion of the form... 

Figure 2 shows the sensitivity of Debye correlation function to the change of parameter a which is defined in Eq. (10). As we explained in Section on Theory of scattering, a = 0 corresponds to an isometric two-component system and x 0 a non-isometric system. When x = 0.1 a two-component system becomes a non-isometric system with volume fractions, cpi = 0.46 and q>2 = 0.54. The Debye correlation functions for two cases, a = 0 and X = 0.1, were calculated with a set of representative values of a, b and c. Figure 2 shows that a small deviation from isometry does not affect shape of the Debye correlation function. All our samples were prepared so that they are isometric at a reference salinity, and the change of an effective volume fraction as a function of salinity is expected less than 10%. Therefore, we treat the parameter a as effectively zero in all data analysis. 

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]... 

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. 

For the RPM, the first term cancels, and the second is dominant. In principle, c (r ) defined in this way may also vary locally. Third, one may also calculate C2g by working backward from the Laplace transformation of the pair correlation function, as done by Lee and Fisher [94] for the pair correlation function of their generalized Debye-Hiickel theory (GDH). 

A pioneer effort to the account for electrostatic interaction effects in dipole reorientations and correlation functions was made by Brot and Darmon (39) in their Monte Carlo simulations for the partially ordered solid phase of 1 2 3 trichloro 4 5 6 trimethyl benzene (TCTMB) using the point charge model already mentioned in 2.4. Calculations of transition rates between 6 fold rotational wells of fluctuating depth as a result of changing neighbor orientations resulted in essentially Debye relaxation at 300 Kt but a second simulation at 186 K for which considerable rotational ordering is present produced very nearly a circular arc with od = 0.28 as compared to the experimental Ad = 0.39. 

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. 

The sum involves the ionic components k > 0 only. Note that the last term of Eq. (5) could be alternatively expressed by exchanging h and c symbols. The bulk OZ equation guarantees the equivalence between both, technically and subtly different, versions. The OZ equation is then coupled with the iO HNC closure [Eq. (3)] which depends on the imposed interface-ion potential. The resolution is iterative as before, simplified by the fact that the OZ Eq. (5) becomes linear in the unknown functions (the bulk ion-ion hik calculated previously, are ivm). The final correlation functions hio leads to the desired local ionic densities at the interface Pi( + hio r)). The planar geometry limit for the interface is obtained either by investigating big spheres, of radius R much larger than any characteristic distance (e.g. Debye length), or by expHcitly writing the OZ equation (5) in ID. 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

Expressions for the force constant, i.r. absorption frequency, Debye temperature, cohesive energy, and atomization energy of alkali-metal halide crystals have been obtained. Gaussian and modified Gaussian interatomic functions were used as a basis the potential parameters were evaluated, using molecular force constants and interatomic distances. A linear dependence between spectroscopically determined values of crystal ionicity and crystal parameters (e.g. interatomic distances, atomic vibrations) has been observed. Such a correlation permits quantitative prediction of coefficients of thermal expansion and amplitude of thermal vibrations of the atoms. The temperature dependence (295—773 K) of the atomic vibrations for NaF, NaCl, KCl, and KBr has been determined, and molecular dynamics calculations have been performed on Lil and NaCl. Empirical values for free ion polarizabilities of alkali-metal, alkaline-earth-metal, and halide ions have been obtained from static crystal polarizabilities the results for the cations are in agreement with recent experimental and theoretical work. 

The applications of this model is mostly restricted to hydrogen and helium like systems. Winkler [25] has calculated the detachment energies of H embedded in a variety of Debye plasmas. He used a correlated description of the two-particle wave functions introducing interparticle coordinate in the expansion of the basis set. The linear variational parameters are determined by solving the generalized secular equation... 

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