On the Determination of Partial RDFs for Amorphous Materials
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ABSTRACT Radial distribution functions (RDFs) for vitreous silica (v-Si0 2 ) have been obtained from energy-filtered electron diffraction data obtained in the HB5 scanning transmission electron microscope. Results have been compared with those obtained from high-resolution neutron diffraction experiments, and are in good agreement within experimental errors. It was found to be impractical to obtain partial RDFs for this material from combined neutron, X-ray and electron diffraction data, because the similarities in characteristics of X-ray and electron scattering cause indeterminacies. A criterion equation has been given to determine feasibility.
INTRODUCTION For disordered materials, the atomic arrangements cannot be determined uniquely as in the case of crystals, because the number of independent positional parameters is practically infinite. In crystals, the number reduces to a finite number due to the constraints imposed by symmetry as given by the characteristic space group of the material. For amorphous materials, radial distribution functions (RDFs) have instead been devised to describe the short range order of amorphous materials on a statistical basis, which nonetheless give the atomic environment surrounding an atom in ensemble average. Though the RDF of a monoatomic material can be deduced from a single diffraction experiment, as was demonstrated long ago [1], there are some difficulties in determining the partial RDFs, which also specify the chemical order, for multiple-component systems since there is more than one unknown to be determined. Early attempts [2] using X-ray diffraction provided powerful insights, and the necessary approximation is still often employed in practice. In general there are u(n + 1)/2 partial structure factors to be determined for an n-component system, which in turn determine the partial RDFs. Since each diffraction experiment can provide 39 Mat. Res. Soc. Symp. Proc. Vol. 321. ©1994 Materials Research Society
such a linear equation containing the unknown partial structure factors, it is therefore quite reasonable to expect that combined electron, neutron, and X-ray diffraction experiments would be able to determine the partial RDFs unambiguously [3] for a binary system. Other methods involve the use of anomalous absorption [4] or the recently developed EXAFS technique [5] using very bright synchrotron sources. In the present paper we report the results of our attempt to determine the partial RDFs for vitreous silica (v-Si0 2 ). The technique is to use three independent sets of diffraction intensity data obtained from neutron, Xray and electron diffraction experiments, respectively.
EXPERIMENTAL Energy-filtered electron diffraction data were collected in the VG HB5 STEM operating at 100 kV using an energy window of about 2 eV in width [6]. Neutron diffraction data were collected from the most readable recent published results [8].
THEORY For a system of N atoms with n types of atoms, each type having Na atoms, the partial atomic density functions are defined by P
1 N. =
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