Atomic pair-correlation

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

One important use of X-ray probes is in the study of local order and displacements, but this is not within the scope of the present book. The recent availability in intense synchrotron sources with selectable X-ray energies permits high-precision measurements of chemically specific atomic-pair correlations in solid solution alloys. A recent review of the technique is given by G.E. Ice and C.J. Sparks (Modern Resonant X-ray studies of alloys local order and displacement) in Annual Reviews of Materials Science 1999, 29, 25-52. 

The oxygen atom pair correlation function for crystalline ice Ih is shown in Fig. 7b. Note that the first peak in the function hoo(R) is completely resolved. The NVR analysis of the large s part of the structure function indicates that this... 

Fig. 7 b. Oxygen atom pair correlation functions for polycrystalline ice Ih (top) and for H20(as) (bottom), both prepared and studied at 77 K, derived from X-ray diffraction (from Ref. 27>)... 

Fig. 7 g. Atom pair correlation functions for amorphous solid and liquid D2O. Dotted curves are weighted sums of contributions from OO, OD and DD interactions. Broken curves show contributions from OO interactions alone. Solid curves represent nearly evenly weighted sums of OD and DD atom pair correlation functions. (From Ref. 27>)... 

The g(r) vs r diagrams of ice 1 (top) and liquid water at 277 K (bottom). [Here g(r) represents the oxygen atom-pair correlation function and r represents the distance of an oxygen atom-pair taken from A. H. Narten,... 

Palinkas G, Kalman E, Kovaks P (1977) Liquid water II. Experimental atom-pair correlation functions for liquid deuterium oxide. Molec Phys 34 525-537... 

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. 

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. 

The atom-atom pair correlation functions of TIP4P water have also been calculated by MC simulation over a large temperature range, from ambient to... 

Information can be extracted directly from the PDF in a model-independent way because of its definition as the atom-pair correlation function described in Equation (13). 

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... 

In the second paper the same authors studied phenylmagnesium bromide in diethyl ether with the composition CJRMgBr 4.80 Iit-O. Also here the total atom pair correlation function is calculated explicitly in order to see the individual weight... 

The measurements were performed in the same manner as iho-c with the organomagnesium iodides. The reduced intensity and the total atom pair correlation functions nrc shown in Figures 10.19. The interpretation of the total atom pair correlation function followed the outline given in the previous section. The peak at 2.55 A was assigned to a Mg—Br distance and the shoulder at 3.6 A to a Br-Br distance. From the area of this shoulder—the limits of the integration are not given—the number of... 

Both studies show very clearly the problems when solutions are studied with X-rays. The nuiin problem is that an X-ray experiment provides the sum of all atom pair correlation functions, weighted wilh the mole fraction and the electron number of the corresponding atom pairs. If (he concentration of the solute is not very high, die contribution of Ihc atom pairs, from which the local structure can be deduced, to the iota] atom pair correlation function is too small for an unambiguous assignment of certain peaks to definite atom pairs and for a reliable determination of the coordination number. Nevertheless, the X-ray studies of liquids or solutions have their merits, when they arc combined with other experimental methods and arc compared with theoretical calculations. 

The properties of atom-atom pair correlation functions and the hydrogen bond analysis show that the local water structure is affected by the surface only within the range of the density inhomogeneities (a few Angstrom) in the constant-density region structural isotropy is observed. 

As it can be seen from (his equation, the sum of all atom pair correlation functions, weighted with electron numbers of the corresponding atomic-pair, can be determined by measuring the scattered intensity of X-rays. In case of molecular liquids, this sum, however, includes intra- and intermolec-ular contributions. The scattered intensity of an isolated molecule or a stoichiometric unit can be... 

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 ... 

---