Partial interference functions

One can also try to localize the scattering centers within the centers of the atoms from the very beginning. This is achieved by superposing the partial interference functions... 

A complete solution for a binary compound or alloy can be given in the approximation of the AERDF if one disposes of three different diffraction patterns of the same material taken in conditions under which the two types of atoms have as different scattering factors fi and as possible for the incident radiations. This can be done by using alternatively X-rays of different wave lengths, electrons, and neutrons and/or by changing the isotopic composition of the sample. If one admits that the partial interference functions (see Eq. 2.8) do not depend on composition (cj and Ca), which seems to be true to a first approximation in some alloys, the three measurements can be done with the same radiation, but on three different compositions. 

In the above-mentioned cases all three partial interference functions and, in consequence, the three partial RDFs, namely pn r),4nP p 12(f) and 4 nr P22(f) which characterize the structure completely can be obtained (Steeb and Hezel, 1966). The fourth partial RDF, namely 4nr pi2(r) = (C1/C2) 4nr pi2(r) is not independent. 

Thus three partial interference functions are involved and in order to determine them by experiment three separate diffraction experiments on the same liquid must be carried out. 

Since there are virtually no data available for the partial interference functions for systems other than for certain copper and silver based alloys, the remarks made in this section will necessarily be speculative in character. We shall attempt to squeeze as much information as possible out of the relatively meagre experimental data we have at our disposal. 

The partial interference functions for liquid Cu2le are displayed in Figure 7.17, the vertical hues indicating the residual uncertainty due to the inevitable experimental errors in F(q). Hawker et al. have drawn attention to three features which they consider to be worthy of special note ... 

The extent to which the measured partial interference functions are consistent with a more or less random mixture of Cu and Te ions in which substantial electron transfer has taken place is currently under theoretical investigation. 

The nearly free electron theory developed by Faber and Ziman (1964) is an obvious starting point for discussing liquid alloys of type I. For those cases in which information is available about the three partial interference functions which characterize the structure of binary alloys, close quantitative agreement between theory and experiment has been obtained. We emphasize that a positive da/dT is entirely consistent with metaUic behaviour in Hquid alloys on account of the temperature dependence of the partial interference functions. For this reason many liquid alloys which have in the past been thought of in terms of a semiconducting framework should more properly be regarded as metallic. (It may, in certain cases, be necessary to introduce the Mott g factor but there is little evidence either way on this important point at the present time). Alloys of the second type will form the subject for section 7.7. 

We now turn to. alloys which, though still of type I, are characterised by a positive temperature coefficient of conductivity. A general discussion is hampered by a lack of knowledge of the partial interference functions and for this reason we focus particular attention on the alloy system Cu-Sn. liquid Cu-Sn is of particular interest because ... 

It is clear from the Eqs. (7.35) and (7.39) that, provided we ignore effects due to non-locality of w the composite quantity (3— )/a may be evaluated from a knowledge of F(2kp) this in turn depends on the values of the pseudopotentials and the partial interference function evaluated at 2kp. 

No detailed calculations for da/dT can be presented because the temperature dependence of the partial interference functions has not been investigated. It seems probable, however, that if the peak position for Cu-Sn to smaller values of q as the temperature is increased (Section... 

Partial Interference and Atomic Distribution Function of Liquid Ag—Sn... 

With m atomic species, there are m(m + l)/2 partial pair distribution functions gap(r) that are distinct from each other. When only a single intensity function I(q) is available from experiment, no method of ingenious analysis can lead to determination of all these separate partial pair distribution functions from it. Different and independent intensity functions I(q) may be obtained experimentally when measurements are made, for example, with samples prepared with some of their atoms replaced by isotopes. When a sufficient number of such independent intensity functions is available, it is then possible to have all the partial pair distribution functions gap(r) individually determined, as will be elaborated on shortly. When only a single intensity function is available from x-ray or neutron scattering, however, what can be obtained from a Fourier inversion of the interference function is some type of weighted average of all gap (r) functions. The exact relationship between such an averaged function and gap(r)s is as follows. 

For each of the three interference functions /x( )> h( )> and b(q) determined as mentioned above, Equation (4.19) is applicable, where the iap (q)s [i.e., cc( ) ch( )> and i hh( )] are common among the three cases and are unknown and yet to be determined. The weighting factors wa, on the other hand, depend, as seen from (4.16), on the scattering lengths ba and assume different values in the three measurements. For a given q value, Equation (4.19) therefore constitutes a set of simultaneous linear equations with three unknowns iap(q), whose values can be determined by solving the simultaneous equations. The partial pair distribution functions gap(r) are then obtained from them by Fourier inversion as implied by (4.18). 

The environmental radial distribution function (RDF) for Ni is plotted with a solid curve in fig. 83. The ordinary RDF computed from the interference function, Qj Q), which is shown in fig. 84, is shown as a dotted curve in fig. 83. Although six partial RDFs are overlapped in the ordinary RDF, the environmental RDF for Ni is only the sum of the three partial RDFs of Ni-Ni, Ni-Mg and Ni-La pairs. In terms of three constituent elements of different sizes, the first peak of the ordinary RDF consists of some broad diffuse peaks. It is almost impossible to determine an atomic distance and a coordination number for each atomic pair in the nearest-neighbor region from the ordinary RDF. On the other hand, the first peak in the environmental RDF for Ni becomes an... 

From this model the interference functions and pair-correlation functions for both X-ray and neutron are calculated and compared to the eiqperlmental ones (it is not possible to compare individual partial functions as no such information is available in the literature). Nevertheless, as copper and titanium have neutron scattering lengths of opposite signs, the comparison with the neutron experiments is a rather severe test of the quality... 

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)]. 

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