Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 189

Neutron and X‐ray diffraction are two of the principal techniques used to probe the bulk structure of glasses through determination of average bond lengths, coordination numbers (CN), and angles over both the short (nearest neighbors) and intermediate (next‐nearest neighbors) length scales. Data analysis and interpretation are comparable for X‐ray and neutron diffraction so that I will differentiate between the two methods only when needed. In passing, note that in earlier studies the interaction between the X‐rays and the sample was termed scattering, and not diffraction as made now.

For X‐rays, diffraction experiments are readily made but they are not well suited to discriminate between neighbor atoms in the periodic table, such as silicon and aluminum, because of the similar electronic clouds with which X‐rays interact in these cases. Thanks to their lack of electrical charge, neutrons are (with neutrinos) the only particles that can penetrate a condensed phase. They do not interact with electronic clouds, but are diffracted by atomic nuclei because they are strongly sensitive to nuclear forces. If both methods are available, the choice of X‐ray or neutron diffraction depends to some extent on the chemical composition of the glass. Neutrons are, for instance, better suited than X‐rays for investigating light elements such as H and they have the advantage of providing better spatial resolution and allowing more detailed information to be obtained through isotope substitution methods, which rely on the fact that neutron diffraction is sensitive to the neutron content of a nuclei. Practical factors have also to be taken into account: intense sources are much more widely available for X‐rays than for neutrons, whereas the required sample sizes are of the order of 1 mg and 50 g, respectively. The samples themselves may be in the form of powders, glass chips, or shaped glass chips.

The “diffraction pattern” of a glass is very different from that of a crystal (cf. [3]). A crystal will produce a series of “spots” or “reflections” on a two‐dimensional diffraction image or sharp diffraction lines on a powder diffraction trace. These cannot be seen in the diffraction image of a glass. This contrast is illustrated by the area electron‐diffraction images of a crystalline material (Ba₂TiGe₂O₈, Ge‐substituted fresnoite) and of an amorphous material (Ba₂TiSi₂O₈, fresnoite glass) shown in Figure 1a, b. Clearly, no “reflections” are observed in the glass image. A similar contrast is seen between an X‐ray powder diffraction 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, I_coh(k) (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.