Difference function, atomic pair correlation

FIGURE 13.2 Difference function AG(rz) between the atomic pair correlation functions of the deuterated GD(rz) and hydrogenous GH(rz) samples. 

FIGURE 13.5 Difference pair correlation functions AG(rz) between the atomic pair correlation functions of samples with deuterated and hydrogenous PEO. In the case of the 0.1 M salt concentration (solid line), the butylammonium chains were deuterated in both samples, whereas for the 0.03 M salt concentration (dashed line), both samples contained hydrogenous counterions. The volume fraction of PEO was 4% in all cases. 

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. 

The best way to see the differences between the EXAF.S spectroscopy and the X-ray scattering is to compare the expression for the EXAFS function and the distinct part. But first we have to introduce the atom pair correlation function into equation (10.4). Equation (10.4) is a good approximation for liquids with a high degree of local order. If the degree of disorder is large. Xik) must be represented by the more general equation ... 

At the present time, of all EXAFS-like methods of analysis of local atomic structure, the SEES method is the least used. The reason is that the theory of the SEES process is not sufficiently developed. However the standard EXAES procedure of the Fourier transformation has been applied also to SEES spectra. The Fourier transforms of MW SEES spectra of a number of pure 3d metals have been compared with the corresponding Fourier transforms of EELFS and EX-AFS spectra. Besides the EXAFS-like nature of SEES oscillations shown by this comparison, parameters of the local atomic structure of studied surfaces (the interatomic distances and the mean squared atomic deviations from the equilibrium positions [12, 13, 15-17, 21, 23, 24]) have been obtained from an analysis of Fourier transforms of SEES spectra. The results obtained have, at best, a semi-quantitative character, since the Fourier transforms of SEES spectra differ qualitatively from both the bulk crystallographic atomic pair correlation functions and the relevant Fourier transforms of EXAFS and EELFS spectra. 

The pair correlation function g(r) in a monoatomic fluid is different from unity only within a range of about five atomic diameters from an atom at the origin. Roughly the same statement applies to the atom-atom pair correlation fimctions in polyatomic systems. This observation is put to use in simulations by algorithms that perform discrete sums over atom pairs out to a cutoff distance ro from a given atom and replace the remainder of the interactions out to infinity... 

A reasonable approximation for the pair correlation function of the j8-process may be obtained in the following way. We assume that the inelastic scattering is related to imcorrelated jumps of the different atoms. Then all interferences for the inelastic process are destructive and the inelastic form factor should be identical to that of the self-correlation function, given by Eq. 4.24. On... 

The theoretical understanding of xenon chemical shifts was placed on a firm footing by the work of Jameson and co-workers, who were able to reproduce the chemical shifts for xenon clusters of different size in NaA and AgA zeolites remarkably well [13-17]. However, the procedure is not trivial, as it requires a knowledge of the structure, the shielding functions for all possible Xe-framework atom pairs, the Xe-framework atom potential functions, and a calculation over the accessible dynamic states of the xenon atoms in the cage. To work back from chemical shifts to pore size clearly is not trivial, so the establishment of some general shift - size correlations still is extremely useful. 

In a solution containing n atomic species, p, q, etc., the number of different pair interactions is n(n + l)/2. The pair correlation function, gm(r), measures the probability of finding an atom q at a distance r... 

Amorphous ice has been studied in some detail by both X-ray and neutron diffraction [738-740]. The O- -O pair-correlation functions are similar to those of liquid water, except that on condensing on very cold surfaces, i.e., 10 K, there is an extra sharp peak at 3.3 A. This indicates some interpenetration of the tetrahedral disordered ice-like short-range structures. It appears that none of the many proposed atom-atom potential energy functions can simulate a structure for liquid water that predicts pair-correlation functions which are a satisfactory fit to the experimental data [741, 742]. Opinions seem to differ as to whether the discrepancy is in the theory or the experiments. 

An alternative approach to this question is to investigate the effect in the disordered solid phase where the spatial correlations are more pronounced and higher argon concentrations can be achieved. Fig 5a shows the two curves for a sample of codeposited amorphous ice with 8 atomic per cent of argon. A difference analysis yields a composite pair correlation function in the form... 

All the scattering expressions discussed above are relevant for single component materials. But most glasses are multicomponent materials and consist of several atoms with different scattering factors. In a material consisting of n different atoms, there are n n- )l2 pair correlation functions and each of them contribute to the observed scattering intensities. 

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