Pair correlation function structure factor

At the bases of the second basic assumption made, e.g., that the fluids behave classically, there is the knowledge that the quantum effects in the thermodynamic properties are usually small, and can be calculated readily to the first approximation. For the structural properties (e.g., pair correlation function, structure factors), no detailed estimates are available for molecular liquids, while for atomic liquids the relevant theoretical expressions for the quantum corrections are available in the literature. 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

Fig. 5.23 Time evolution of the three functions investigated for PIB at 390 K and Q=0.3 A"h pair correlation function (empty circle) single chain dynamic structure factor (empty diamond) and self-motion of the protons (filled triangle). Solid lines show KWW fitting curves. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

From the preceding formulas we see that the structure factor is related to the pair correlation function... 

The liquid-like order present in the pair correlation function manifests itself as a peak in the static structure factor (S(q)). The scaling of the position qm of this maximum with the density has attracted much attention in the literature [40, 51-53]. Scaling arguments suggest [35, 42, 49, 51] that qm obeys the relation qm p1/3 for dilute solutions and qm pv/(3v 1) for semidilute solutions. Here v is the scaling exponent for the end-to-end distance, i.e., RE hT. The overlap threshold concentration is estimated as p N1 3v. As a conse-... 

Figure 16. Structural profiles of LDA and HDA Si obtained from the plane-wave DFT calculations [265], (a) Pair correlation functions g(r) for HDA (at 12GPa, solid lines) and LDA (at 0 GPa, dashed lines), (b) Structure factors S(Q) for HDA (at 12 GPa, solid lines) and LDA (at 0 GPa, dashed lines), (c, d) Atomic configurations for (c) LDA and (d) HDA. Atoms separated by 2.55 A or less are linked by thick lines (covalent-like bonds), whereas those separated by 2.857 A or less are linked by thin lines.

To simplify the interpretation of the structural features observed in reciprocal space, the structure factors were Fourier transformed to atomic pair correlation functions, G(rz), measured in the direction perpendicular to the clay plates (in the z-direction). The structural functions r(G(rz) - 1) derived from G(rz) are shown in Figure 8.3 for the same vermiculites as shown in Figure 8.2. It should be noted that... 

FIGURE 8.3 Atomic pair correlation functions G(r.) obtained from the structure factors shown in Figure 8.2, presented as r(G(rz) - 1), where rz is the distance along the swelling axis of the clay. Each function oscillates around zero and has been displaced in the upper panels to show the functions obtained from gels prepared in a 0.1 M deuterated salt solution and in 0.1 and 0.01 M protonated salt solutions. 

How would pair correlation functions and partial structural factors be written for a binary molten salt, e.g., sodium chloride The corresponding correlation function would be given by... 

Determine and explain the terms radial distribution function, pair correlation function, and partial structural factors. 

In the Reverse Monte Carlo (RMC) method [5], the pair correlation function or the structure factor is calculated after each random move (Ssim(<]) or gsimfr)) and compared to the respective target function obtained from experimental diffraction data (Sexp(q) or gexp(r)). It is possible to calculate Ssm(q) with full periodicity from the atomic positions. This method is best in principle [10], but the computational cost is much greater than for any of the other available methods. It is also possible to obtain Ssm(q) by first calculating gsm(r) from the atomic positions and then Fourier transform this function and calculate Ssim(q). The disadvantage of this approach is that there is an additional computational cost associated with the Fourier transform of gsm(r) after each move. 

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