Pair correlation function, Fourier transform

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)]. 

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... 

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. 

Figure 4 Neutron pair correlation function d r) of water contained in activated carbon, at room temperature, shown by solid lines for 200% (a), 42% (b), and 25% hydration (c). For comparison, the d r) of bulk water at the same temperature is also drawn (d) [20]. The dotted lines show the result of a smoothing of the experimental data before doing the Fourier transform. This demonstrates that the additive oscillations that appear between 3 and 6 A have a physical meaning at the opposite of that appearing at higher r values.

In the general case in which the phase might (or might not) have positional order, one can define an anisotropic pair correlation function, g(x), where x is a position vector relative to a given molecule. The Fourier transform of the pair correlation function, namely. 

The main source of experimental structural results are diffraction experiments, either of X-rays or neutrons, as their wavelength is of the right order of magnitude to probe typical interparticle separations of condensed phase system at moderate to high density. It is worth recalling, however, that pair correlation functions are not a direct experimental data. Rather, raw intensity values as a function of diffraction angle require careful data analysis to extract a structure function that is a combination of Fourier transforms of pair correlation functions. 

Top right Fit of EXAFS region with cnbic spline and snbtraction yields EXAFS oscillations shown is the EXAFS weighted by k. Bottom left XANES spectral region expanded. Bottom right PCF or pair correlation function (also called Radial Structure Function to distinguish it from the X-ray analog RDF) shown is the Fourier transform of the spectram at the top left. 

This power law character of the form factor is related to the power law decay of the pair correlation function of an ideal chain [Eq. (2.121)]. Quite generally, the form factor is related to the Fourier transform of the intramolecular pair correlation function g(r) ... 

This identifies 5 (k) with the Fourier transform of the pair correlation function, so the latter may be obtained from the structure factor by inverting the transfonn. Finally, denoting r = r 121 and using, for an isotropic system, g(r 12) = g(r) we get... 

If the individual partial structure factors Sy k ) can be determined, one can calculate the individual partial pair correlation function gij(r) on the basis of a Fourier transformation ... 

The ultimate goal of the diffraction experiment is to determine each contribution ij(kr>) to the structure function i x(kD)- Then one can determine the individual pair correlation functions g r) by Fourier transformation. 

Once the corrected total interference function was obtained, a Fourier transform had to be done to the experimental total interference function to obtain the total pair-correlation function. 

The pair correlation function is important for various reasons. For one thing, a knowledge of the pair correlation function is sufficient to calculate various thermodynamic quantities (particularly the potential energy and the pressure), assuming that the total interaction potential is a sum of pairwise interactions [1, 189, 190]. For another, the structure factor, which is the Fourier transform of the pair correlation function. 

Here, ko and k are the neutron velocities before and after the scattering process. Looking at Eq. (2.75) immediately reveals the relation to G(r, t). The double differential scattering cross-section is directly related to the Fourier transform of the van Hove pair correlation function, which is given by... 

The quanlily of inieresi is not (lie scallerod inlen-sily. hu[ the atom pair correlation functions. which provide information about (he simeuire of liquids. These luneiions can he obtained from the disiinc part by Fourier transformation. Hut lirsi it is... 

Bui there are further differences between the methods, which enn he seen by comparison of the integrands in equations (10.24) and (10.25). The atom pair correlation function in equation (10.24) is multiplied by expt 2i/a. ). which takes into account the effect of the liniie lifetime of the photo-electron and the hole generated by the absorption of the X-rays. Owing to the mean free path term expt--2r//.,). the pair correlation funciions are asymmetrical and damped with increasing distance. This effect can clearly be seen in ihe Fourier transform of the EXAFS function. 

Slow ly-varying tails of the pair correlation function contribute to FXAI- S data only at low k values. Sharp peaks in the pair correlation function. however, give rise to dominant features in the EXAFS signal which persist to high k values. As the data are Fourier transformed only in a liniie range and the low k data of the FXAI- S signal must he omitted in the Fourier transform. Ihe broad tail in the atom pair correlation function is ol ien lost in the analysis ol ihe li.XAF S dala. A... 

Next we define the time-dependent pair correlation function or van Hove correlation function5 G(r,t) as the inverse Fourier transform of F(q,t) in space, that is,... 

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