trace of quartz (crystalline SiO2) and that of the glass plate on which the quartz sample was mounted (Figure 1c, d). Note the very weak intensity, lack of diffraction peaks (reflections), and low signal‐to‐noise ratio of the glass trace. Where there is a “diffraction peak,” it has weak intensity and is extremely broad and diffuse, which is characteristic of a material lacking periodic structure.
Figure 1 Structural differences between crystals and glasses revealed by diffraction images and patterns. (a) Selected area electron diffraction image of crystalline Ba2TiGe2O8 (fresnoite). (b) Similar image for Ba2TiSi2O8 glass. (c, d) Powder diffraction traces for crystalline SiO2 (quartz) and the glass slide on which the sample was mounted. Glass diffraction trace included in (c) to give an indication of the difference in intensity between a glass and a crystalline sample.
Each reflection in a crystal image represents coherent diffraction from a plane of atoms whose intensity depends on the nature of these atoms. The “diffraction pattern” of a homogeneous glass will exhibit no such reflections but simply a series of diffuse rings or peaks reflecting those interatomic distances that prevail throughout the glass structure (Figure 1b, d). One does need to be aware that if the material is polycrystalline, it will also exhibit a series of rings but they will be better defined than those characteristic of glasses. Nevertheless, this type of diffraction image can be analyzed and interpreted to obtain average interatomic bond distances and angles, characteristic of the short‐range order (SRO, nearest‐neighbor distances), and to a lesser extent, the intermediate‐range order (IRO). An example of SRO (typically up to ~3 Å) would be around a Si atom with the four oxygens of its SiO4 tetrahedron, whereas IRO (typically up to ~3–5 Å) would be the linkage of this tetrahedron with others to form groups such as rings.
Figure 2 Information drawn for GeO2 glass from diffraction data. (a) Measured total structure factor. (b) Total correlation function. (c) Pair‐correlation functions showing the contribution from the individual atom pairs. Comparison with the earlier data of [4] is also shown.
(Source: After [5].) Results corrected for a number of instrumental effects before derivation of pair distributions serving to identify the individual atomic pairs contributing to the bulk diffraction pattern of the sample.
The most common information derived for glasses from the diffraction peaks shown in Figure 2 are radial distribution functions (RDF), which represent the probability of finding a given atom at any distance r from some atom considered to be at the center of the system. A RDF thus is a one‐dimensional representation of the three‐dimensional structure of a glass that is averaged over the entire system. Because these functions represent the sum of individual pair distributions for every atom of the material, they are widely used for comparisons with theoretical results to check the accuracy of the simulated glass structure.
If two theta (2θ) is the angle between incident and scattered beams and λ the X‐ray or neutron wavelength, the wave or scattering vector is 4π sin θ/λ. It is denoted as Q in neutron scattering, and as s or k in X‐ray diffraction. In the latter case, it is common to extract the X‐ray weighted Faber‐Ziman or total structure factor (TSF), S(k) [6], from the experimentally measured corrected coherent scattering intensity, Icoh(k) (Figure 2a). The relationship between the TSF and Icoh(k) is given by:
(1)
where
(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 Sij(k) (PSF) for the different atoms i and j. The PSFs cause the oscillating contributions to the scattering curve and take the form:
where rij is the distance between atoms i and j. The TSF itself is expressed as:
(3)
where Wij = ci cj fi(k, E) fj(k, E), Ci and Cj are the atomic fractions of species i and j, E the photon energy, and fi and fj 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 Sii, Sjj, Sij, 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
