X-ray diffraction

Wilhelm Röntgen discovered X-rays in 1895. Seventeen years later, Max von Laue suggested that they might be diffracted when passed through a crystal, for by then he had realized that their wavelengths are comparable to the separation of lattice planes. This suggestion was confirmed almost immediately by Walter Friedrich and Paul Knipping and has grown since then into a technique of extraordinary power. The bulk of this section will deal with the determination of structures using X-ray diffraction. The mathematical procedures necessary for the determination of structure from X ray diffraction data are enormously complex, but such is the degree of integration of computers into the experimental apparatus that the technique is almost fully auto mated, even for large molecules and complex solids. The analysis is aided by molecular modelling techniques, which can guide the investigation towards a plausible structure. X-rays are electromagnetic radiation with wavelengths of the order of 10−10 m. They are typically generated by bombarding a metal with high-energy electrons (Fig. 20.14). The electrons decelerate as they plunge into the metal and generate radiation with a continuous range of wavelengths called Bremsstrahlung.1 Superimposed on the continuum are a few high-intensity, sharp peaks (Fig. 20.15). These peaks arise from collisions of the incoming electrons with the electrons in the inner shells of the atoms. A collision expels an electron from an inner shell, and an electron of higher energy drops into the vacancy, emitting the excess energy as an X-ray photon (Fig. 20.16). If the electron falls into a K shell (a shell with n = 1), the X-rays are classified as K radiation, and similarly for transitions into the L (n = 2) and M (n = 3) shells. Strong, distinct lines are labelled Kα, Kβ, and so on. Increasingly, X-ray diffraction makes use of the radiation available from synchrotron sources (Further information 13.1), for its high intensity greatly enhances the sensitivity of the technique. von Laue’s original method consisted of passing a broad-band beam of X-rays into a single crystal, and recording the diffraction pattern photographically. The idea behind the approach was that a crystal might not be suitably orientated to act as a diffraction grating for a single wavelength but, whatever its orientation, diffraction would be achieved for at least one of the wavelengths if a range of wavelengths was used. There is currently a resurgence of interest in this approach because synchrotron radiation spans a range of X-ray wavelengths. An alternative technique was developed by Peter Debye and Paul Scherrer and independently by Albert Hull. They used monochromatic radiation and a powdered sample. When the sample is a powder, at least some of the crystallites will be orientated so as to give rise to diffraction. In modern powder diffractometers the intensities of the reflections are monitored electronically as the detector is rotated around the sample in a plane containing the incident ray (Fig. 20.17). Powder diffraction techniques are used to identify a sample of a solid substance by comparison of the positions of the diffraction lines and their intensities with diffraction patterns stored in a large data bank. Powder diffraction data are also used to determine phase diagrams, for different solid phases result in different diffraction patterns, and to determine the relative amounts of each phase present in a mixture. The technique is also used for the initial determination of the dimensions and symmetries of unit cells.

Fig. 20.15 The X-ray emission from a metal consists of a broad, featureless Bremsstrahlung background, with sharp transitions superimposed on it. The label K indicates that the radiation comes from a transition in which an electron falls into a vacancy in the K shell of the atom.

Fig. 20.16 The processes that contribute to the generation of X-rays. An incoming electron collides with an electron (in the K shell), and ejects it. Another electron (from the L shell in this illustration) falls into the vacancy and emits its excess energy as an X-ray photon.

Fig. 20.17 X-ray powder photographs of (a) NaCl, (b) KCl and the indexed reflections. The smaller number of lines in (b) is a consequence of the similarity of the K⁺ and Cl⁻ scattering factors, as discussed later in the chapter.

The method developed by the Braggs (William and his son Lawrence, who later jointly won the Nobel Prize) is the foundation of almost all modern work in X-ray crystallography. They used a single crystal and a monochromatic beam of X-rays, and rotated the crystal until a reflection was detected. There are many different sets of planes in a crystal, so there are many angles at which a reflection occurs. The complete set of data consists of the list of angles at which reflections are observed and their intensities. Single-crystal diffraction patterns are measured by using a four-circle diffractometer (Fig. 20.18). The computer linked to the diffractometer determines the unit cell dimensions and the angular settings of the diffractometer’s four circles that are needed to observe any particular intensity peak in the diffraction pattern. The computer controls the settings, and moves the crystal and the detector for each one in turn. At each setting, the diffraction intensity is measured, and background intensities are assessed by making measurements at slightly different settings. Computing techniques are now available that lead not only to automatic indexing but also to the automated determination of the shape, symmetry, and size of the unit cell. Moreover, several techniques are now available for sampling large amounts of data, including area detectors and image plates, which sample whole regions of diffraction patterns simultaneously.

Fig. 20.18 A four-circle diffractometer. The settings of the orientations (φ, χ, θ, and Ω) of the components is controlled by computer; each (hkl) reflection is monitored in turn, and their intensities are recorded.

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مواضيع ذات صلة

Energetics

Structure

Close packing

Structure refinement

The phase problem

Bragg’s law

Lattices and unit cells

Surface pressure

Membrane formation

Micelle formation

The electrical double layer

Classification and preparation