Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

(Figure 2a). The relationship between the TSF and I_coh(k) is given by:

      (1)

      where and . The RDF, D(r), is then given by the Fourier transform of S(k):

      (2)

      The distribution function, D(r) = 4π[ρ(r) − ρ₀], can be converted to the total pair‐correlation function (Figure 2b). The TSF is the sum of the partial structure factors S_ij(k) (PSF) for the different atoms i and j. The PSFs cause the oscillating contributions to the scattering curve and take the form:

      where r_ij is the distance between atoms i and j. The TSF itself is expressed as:

      (3)

      where W_ij = c_i c_j f_i(k, E) f_j(k, E), C_i and C_j are the atomic fractions of species i and j, E the photon energy, and f_i and f_j the atomic scattering factors of species i and j. For GeO₂ glass, the measured TSF shows a first sharp diffraction peak (FSDP) at 1.5 Å⁻¹ (Figure 2a left). From the total correlation function, i.e. the sum of all atom pair correlations (Figure 2b), one can extract (Figure 2c) partial pair‐correlation functions showing the individual contributions to G(r). For a binary compound such as GeO₂, three PSFs are needed to obtain S(k) (Figure 2c). These are S_ii, S_jj, S_ij, which correspond to contributions from Ge─Ge, O─O, and Ge─O bonds. The FSDP is characteristic of structural features in the IRO and longer length scales characteristic of long‐range order (LRO). See Chapter 2.5 on the extended structure of glasses for a full discussion on the ordering ranges found in glasses.

      Diffraction experiments provide information about the average structure of a glass. The positions of the peaks indicate interatomic distances, peak widths indicate bond distance variations, and the areas under the peaks are related to average CN. The RDFs have tall sharp peaks at small radial distances, but broad, low‐intensity peaks at larger ones. This difference indicates that the structure is less variable at small (short‐range) than at larger (intermediate‐range) radii. Consequently, the few first peaks in the correlation function can be assigned to well‐defined interatomic distances whereas peaks at larger radial distance are less easily interpreted because they are composed of more than one atom pair correlation.

      One can minimize the problem of overlapping contributions by collecting scattering data to higher k space to improve the resolution. One achieves this goal by using neutron scattering and collecting data out to Q (k) ranges of ~40–50 Å⁻¹ or hard X‐ray photons (>40 keV) with wavelengths lower than 0.03 nm (k ≥ 30 Å⁻¹). Furthermore, one can partly overcome the difficulty of interpreting peaks at high r values by using techniques that can separate the individual PSFs, such as anomalous X‐ray scattering or isotope substitution. Currently, most glass diffraction experiments involve a combination of two or more techniques.

      2.2 Isotope‐substituted Neutron Diffraction

      As already stated, this technique takes advantage of the dependence of the neutron diffraction on the number of neutrons contained in the atoms undergoing diffraction. By making isotopic substitutions in the sample, it becomes possible to eliminate some PSF in the TSF. This makes determining the various pair‐correlation functions simpler and can lead to unambiguous assignment of features in the total correlation function to specific atom interactions. Basically, two experiments are performed on a sample in which one of the elements is in two different isotopic states. The two TSF can then be subtracted and their difference enables extraction of the PSF involving the element in the different isotopic states. To extract three PSF from a binary system, one requires diffraction experiments on three samples that are identical in every respect, except for the isotopic compositions of one or both of the chemical species.

      The difference function is essentially similar to X‐ray absorption spectroscopy (XAS) but the accuracy of the diffraction data can be improved to higher distances and providing more accurate CN of the element of interest. One should note, however, that the structural role of the two isotopes should be identical, which may not necessarily be the case as observed for light elements such as deuterium and hydrogen. The technique is similar to anomalous wide angle X‐ray scattering (AWAXS) or diffraction anomalous X‐ray scattering (DAXS).

      2.3 DAXS and AWAXS

In these techniques, one determines the individual contributions to the TSF by each of the scattering atom pairs by using sharp changes in the X‐ray scattering factor of an element near its X‐ray absorption edge. The data are qualitatively similar to those of extended X‐ray absorption fine structure spectroscopy (EXAFS), but also include low‐k contributions that contain information on the intermediate‐range structure. Scattering data are collected at two or more different energies, one of them being close to the X‐ray absorption edge of the element of interest. With AWAXS and DAXS, the idea is to cancel the correlations that do not involve the element of interest. This is also true for isotope‐substituted diffraction.

3 X‐ray Absorption Techniques

      3.1 General Features

      These atom‐specific methods provide information on the structural environment of a specific element within a glass. The problem remains, however, that glasses are disordered and lack LRO so the data obtained are broadened relative to those for crystalline materials. Even though one may be looking at the structural environment of a single element, the data in addition represent the “averaged” structure over all the possible sites so that is very difficult to discriminate between distinct sites.

      These methods provide similar information in spite of the different sources used for exciting the atomic interactions of interest. EXAFS or X‐ray absorption fine structure spectroscopy (XAFS) along with X‐ray absorption near‐edge structure (XANES) spectroscopy constitute techniques that fall under XAS. EXAFS provides atom‐specific quantitative data on bond distances, thermal bond‐displacement parameters (i.e. Debye–Waller factors, σ), and CN. XANES can be used to determine qualitatively, the oxidation sate, CN, and the electronic structure of the atom of interest. Both techniques are routinely carried out with synchrotron sources and dedicated XAS beam lines. Energy loss near‐edge structure (ELNES) spectroscopy or electron energy loss spectroscopy (EELS) provides information similar to XANES, but uses instead the electron beam of a high‐resolution transmission‐electron microscope and relies on the energy loss as electrons are transmitted through the sample. X‐ray Raman spectroscopy (XRS) or non‐resonant inelastic X‐ray scattering (NRIXS) is a relatively new technique that also provides similar information to XANES but uses the momentum transfer of hard X‐rays