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X-RAY DIFFRACTION The structure of a crystalline material, including most minerals, consists of a periodic, ordered, three-dimensional array of atoms. Crystal structures can therefore be described in terms of absolutely identical unit cells, containing one or a few formula units of the chemical compound, stacked in three dimensions. The crystal structure is therefore determined by the arrangements of the atoms within a unit cell, together with the size and shape of the unit cells (the lattice parameters), and the manner of the stacking of the unit cells. The arrangement of the unit cells in three dimensions is often described in terms of an array of points, termed lattice points, which identify points within the crystal structure which have identical environments in identical orientations. There is always at least one lattice point per unit cell (a 'primitive unit cell'), although there may be more. The unit cells of a crystalline compound therefore form a periodic three-dimensional array whose typical dimensions are of the order, for minerals, of a few Angstroms to a few tens of Angstroms. This is comparable in magnitude to typical wavelengths of X-rays, which range from 0.56 A for AgK. to 2.29 A for CrKa· Therefore, when a beam of X-rays from an X-ray source (a sealed X-ray tube or an electron storage ring such as a synchrotron) impinges upon a crystal, the X-rays are diffracted, giving rise to a diffraction pattern. The direction in which the various diffracted X-rays emerge from the crystal is governed by the Bragg equation: .l. = 2d sin(8)
(Xl)
where }, is the wavelength of the X-rays, 8 is the Bragg angle, and d is the separation of consecutive planes in a parallel set that together pass through all of the lattice points within the crystal. Each crystal has an infinite number of increasingly closely spaced (i.e. with decreasing d) sets of lattice planes, while the largest spacings are equal to the spacings of the primitive unit cell faces. The Bragg equation predicts that diffraction will only occur when the incident X-ray beam is at a certain glancing angle of incidence 8 to a set of planes, controlled also by the wavelength of the X-rays, A. The
diffracted beam emerges from the crystal in the plane defined by the incident beam and the normal to the set of lattice planes, and at an angle 8 to the surface of the planes. Note that in this sense the Bragg equation treats diffraction as a process of reflection of the X-rays by the lattice planes. Since the spacing of the lattice planes is dependent upon the unit cell parameters of the crystal, the set of d spacings and therefore the set of 8 values for a particular X-ray wavelength (often reported as 28 values) are characteristic of a given crystal structure. The intensities of the diffracted beams are determined by the arrangement of the atoms (and thus the electron density) within a unit cell of the crystal, and can be calculated through the structure factor equation. Together, the set of d spacings obtained from the positions of diffracted beams and their
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