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A D V A N C E D M A T E R I A L S & P R O C E S S E S | J A N U A R Y 2 0 2 0 3 3 Fig. 2 — Diffraction of x-rays from a series of lattice planes: (a) illustration of Bragg’s law; and (b) extinction of reflected x-rays due to destructive interference. (a) (b) are mathematical points, not material points (atoms). To specify the crystal lattice in the general case (Fig. 1a), it is necessary to set three vector parameters ( a, b, and c ) and the angles between their directions (α, β, and γ). These six parameters are called lattice or unit cell parameters, and the figure constructed on them is a parallelepiped of repeatability. If the x, y, and z axes are chosen in accordance with the rules accepted in crystallography, then such a box is called the unit cell of the crystal. In descriptions of a crystal lattice, one of its nodes is selected as the origin of coordinates, and all other nodes are numbered in order along the coordinate axes (Fig. 1b). The coordinate of each node is determined by three integers, x, y, and z, called indexes. It is possible in a lattice to set a number of nodal rows and nets of different orientation. The ( hkl ) index is assigned to the series of parallel node rows (index of the closest to the origin node, through which the row intersecting the origin is passing). The slope of the nodal nets is described by the ( hkl ) index, indicating the num- ber of parts into which the edge of the unit cell is divided. X-RAY DIFFRACTION X-rays are a form of electromag- netic radiation. The great utility of x-rays for determining the structure and composition of materials is due to the differential absorption of x-rays by ma- terials of different density, composition, and homogeneity. The higher the atom- ic number of a material, the more x-rays are absorbed. X-rays cause photo- chemical reactions and luminescence, and undergo scattering, reflection, in- terference, and diffraction. The range of wavelengths of x-rays is placed be- tween the ultraviolet region and γ-rays, but the most useful for crystallography purposes ranges between 0.4 and 2.5 Å (0.04–0.25 nm), which is comparable with interatomic distances in crystals and why a periodic crystal structure can be used as a diffraction lattice. In addi- tion, the wavelength depends on the anode material of the x-ray source (for instance, λ Mo = 0.71 Å and λ Cu = 1.54 Å). The diffraction of x-rays by crystal can be described as a reflection of rays from the set of lattice planes. A para- llel, monochromatic beam of x-rays (i.e., characterized by a single wave- length λ ) fall on a set of lattice planes of ( hkl ) indexes separated one from an- other by the spacing of d and making an angle of incidence θ with them. The dif- fraction condition can be derived from the scheme illustrated in Fig. 2(a). An in- terference maximum is observed when both scattered waves remain in phase, so the difference between their path lengths should be equal to an integer multiple n of the wavelength. The re- sulting equation is called Bragg’s law, or the Wulff-Bragg condition: n λ = 2 dsin θ (Eq 1) where d is the spacing between the lat- tice planes, θ is the scattering angle, λ is the wavelength of the incident wave, and n is the order of interference. Diffraction spots are usually called reflections, because these spots are left by the ray reflected from the plane. Each XRD experiment gives a number of spots with two characteristics: the an- gle of diffraction and intensity of the spot. The intensity of the reflection is largely de- pendent on the chemi- cal composition of the crystal, and the angular distribution of the re- flections is the result of the symmetry and size of the unit cell of the crystal. Presence of the translational symmetry elements (such as screw axes and glide planes) and centered Bravais lat- tices results in extinction of a reflection’s intensi- ty: some reflections that should appear according to Bragg’s law are actu- ally absent. The regular absence of the reflec- tions makes it possible to determine the par- ticular space group of the crystalline compound. Figure 2(b) shows a section of crystal with two types of atoms arranged on parallel planes with Δ d spacing. Phases of the x-rays reflected from the series of large- and small-atom planes consid- ered separately are matches, so they should constructively interfere. How- ever, considering the diffraction system together, one can find that the diffract- ed waves from a large-atom plane lag behind those from small atoms. If the plane of large atoms is arranged exactly in the middle of the planes of small at- oms, Δφ = 180°, so the waves would de- structively interfere. If the large atoms are arranged in the same plane as small atoms, then Δφ = 0°, and the waves would constructively interfere. A more detailed analysis of Δ d / d shows that it is equivalent to hx + ky + lz, where h, k, and l are the indexes of the plane, and x, y, and z are the fractional coordinates of the atom. X-rays are electromagnetic waves with a frequency of electric andmagnet- ic vector oscillations at approximately 10 18 Hz. Protons are too massive—they react weakly to such fast oscillations of the electric field of the x-ray, whereas electrons can oscillate with the frequen- cy of incident x-rays, thereby emitting