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When dealing with crystalline materials, it often becomes necessary to specify a particular point within a unit cell, a crystallographic direction, or some crystallographic plane of atoms. Sometimes it is necessary to specify a lattice position within a unit cell. Lattice position is defined in ter...

When dealing with crystalline materials, it often becomes necessary to specify a particular point within a unit cell, a crystallographic direction, or some crystallographic plane of atoms. Sometimes it is necessary to specify a lattice position within a unit cell. Lattice position is defined in terms of three lattice position coordinates, which are associated with the x, y, and z axes—we have chosen to label these coordinates as Px, Py, and Pz. Coordinate specifications are possible using three point indices: q, r, and s. These indices are fractional multiples of a, b, and c unit cell lengths—that is, q is some fractional length of a along the x axis, r is some fractional length of b along the y axis, and similarly for s. In other words, lattice position coordinates (i.e., the Ps) are equal to the products of their respective point indices and the unit cell. To illustrate, consider the unit cell in Figure 3.6, the x-y-z coordinate system with its origin located at a unit cell corner, and the lattice site located at point P. Note how the location of P is related to the products of its q, r, and s point indices and the unit cell edge lengths. Figure 3.6 The manner in which the q, r, and s coordinates at point P within the unit cell are determined. The q index ‫ادراج الشكل‬ (which is a fraction) corresponds to the distance qa along the x axis, where a is the unit cell edge length. The respective r and s indices for the y and z axes are determined similarly. 1 A crystallographic direction is defined as a line directed between two points, or a vector. The following steps are used to determine the three directional indices: 1. A right-handed x-y-z coordinate system is first constructed. As a matter of convenience, its origin may be located at a unit cell corner. 2. The coordinates of two points that lie on the direction vector (referenced to the coordinate system) are determined—for example, for the vector tail, point 1: x1, y1, and z1; whereas for the vector head, point 2: x2, y2, and z2.5 3. Tail point coordinates are subtracted from head point components—that is, x2 − x1, y2 − y1, and z2 − z1. 4. These coordinate differences are then normalized in terms of (i.e., divided by) their respective a, b, and c lattice parameters—that is, which yields a set of three numbers. 5. If necessary, these three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. 6. The three resulting indices, not separated by commas, are enclosed in square brackets, thus: [uvw]. The u, v, and w integers correspond to the normalized coordinate differences referenced to the x, y, and z axes, respectively. In summary, the u, v, and w indices may be determined using the following equations: V= In these expressions, n is the factor that may be required to reduce u, v, and w to integers. For each of the three axes, there are both positive and negative coordinates. Thus, negative indices are also possible, which are represented by a bar over the appropriate index. For example, the direction has a component in the −y direction. Also, 2 changing the signs of all indices produces an antiparallel direction; that is, is directly opposite to. If more than one direction (or plane) is to be specified for a particular crystal structure, it is imperative for maintaining consistency that a positive– negative convention, once established, not be changed. The , , and directions are common ones; they are drawn in the unit cell shown in Figure 3.7. Figure 3.7 The , , and directions within a unit cell The orientations of planes for a crystal structure are represented in a similar manner. Again, the unit cell is the basis, with the three-axis coordinate system as represented in Figure 3.5. In all but the hexagonal crystal system, crystallographic planes are specified by three Miller indices as (hkl). Any two planes parallel to each other are equivalent and have identical indices. The procedure used to determine the h, k, and l index numbers is as follows: 1. If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell. 2. At this point, the crystallographic plane either intersects or parallels each of the three axes. The coordinate for the intersection of the crystallographic plane with each of the axes is determined (referenced to the origin of the coordinate system). These intercepts for the x, y, and z axes will be designed by A, B, and C, respectively. 3 3. The reciprocals of these numbers are taken. A plane that parallels an axis is considered to have an infinite intercept and therefore a zero index. 4. The reciprocals of the intercepts are then normalized in terms of (i.e., multiplied by) their respective a, b, and c lattice parameters. That is, 5. If necessary, these three numbers are changed to the set of smallest integers by multiplication or by division by a common factor 6. Finally, the integer indices, not separated by commas, are enclosed within parentheses, thus: (hkl). The h, k, and l integers correspond to the normalized intercept reciprocals referenced to the x, y, and z axes, respectively. In summary, the h, k, and l indices may be determined using the following equations: h= k= l= In these expressions, n is the factor that may be required to reduce h, k, and l to integers. An intercept on the negative side of the origin is indicated by a bar or minus sign positioned over the appropriate index. Furthermore, reversing the directions of all indices specifies another plane parallel to, on the opposite side of, and equidistant from the origin. Several low-index planes are represented in Figure 3.10. 4 Figure 3.10 )Representations of a series each of the (a) (001), (b) (110), and (c) (111) crystallographic planes(. 5 3.2.1 Single Crystals :- For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they can also be produced artificially. They are ordinarily difficult to grow because the environment must be carefully controlled. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal assumes a regular geometric shape having flat faces, as with some of the gemstones; the shape is indicative of the crystal structure. An iron pyrite single crystal is shown in Figure 3.18. Within the past few years, single crystals have become extremely important in many modern technologies, in particular electronic microcircuits, which employ single crystals of silicon and other semiconductors. 3.2.2 Polycrystalline Materials Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline. Various stages in the solidification of a polycrystalline specimen are represented schematically in Figure 3.19. Initially, small 6 Figure 3.18 )An iron pyrite single crystal that was found in Navajún, La Rioja, Spain.( Figure 3.19 Schematic diagrams of the various stages in the solidification of a polycrystalline material; the square grids depict unit cells. (a) Small crystallite nuclei. (b) Growth of the crystallites; the obstruction of some grains that are adjacent to one another is also shown. (c) Upon completion of solidification, grains having irregular shapes have formed. (d) The grain structure as it would appear under the microscope; dark lines are the grain boundaries. (Adapted from W. Rosenhain, An Introduction to the Study of Physical Metallurgy, 2nd edition, Constable & Company Ltd., London, 1915.) 7 3.2.4 Anisotropy graphic direction in which measurements are taken. For example, the elastic modulus, the electrical conductivity, and the index of refraction may have different values in the and directions. This directionality of properties is termed anisotropy, and it is associated with the variance of atomic or ionic spacing with crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic. The extent and magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the crystal structure; the degree of anisotropy increases with decreasing structural symmetry—triclinic structures normally are highly anisotropic. The modulus of elasticity values at , , and orientations for several metals are presented in Table 3.4. For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random. Under these circumstances, even though each grain may be anisotropic, a specimen composed of the grain aggregate behaves isotropically. Also, the magnitude of a measured property represents some average of the directional values. Sometimes the grains in polycrystalline materials have a preferential crystallographic orientation, in which case the material is said to have a ―texture.‖ The magnetic properties of some iron alloys used in transformer cores are anisotropic—that is, grains (or single crystals) magnetize in a 〈100〉-type direction easier than any other crystallographic direction. Energy losses in transformer cores are minimized by utilizing polycrystalline sheets of these alloys into which have been introduced a magnetic texture: most of the grains in each sheet have a 〈100〉-type crystallographic direction that is aligned (or almost aligned) in the same direction, which is oriented parallel to the direction of the applied magnetic field. 8 Historically, much of our understanding regarding the atomic and molecular arrangements in solids has resulted from x-ray diffraction investigations; furthermore, x-rays are still very important in developing new materials. We now give a brief overview of the diffraction phenomenon and how, using x-rays, atomic interplanar distances and crystal structures are deduced 3.3.1 The Diffraction Phenomenon Diffraction occurs when a wave encounters a series of regularly spaced obstacles that (1) are capable of scattering the wave, and (2) have spacings that are comparable in magnitude to the wavelength. Furthermore, diffraction is a consequence of specific phase relationships established between two or more waves that have been scattered by the obstacles. Consider waves 1 and 2 in Figure 3.20a, which have the same avelength (𝜆) and are in phase at point O–Oʹ. Now let us suppose that both waves are scattered ‫ادراج اشكال‬ Figure 3.20 (a) Demonstration of how two waves (labeled 1 and 2) that have the same wavelength 𝜆 and remain in phase after a scattering event (waves 1′ and 2′) constructively interfere with one another. The amplitudes of the scattered waves add together in the resultant wave. (b) Demonstration of how two waves (labeled 3 and 4) that have the same wavelength and become out of phase after a scattering event (waves 3′ and 4′) destructively interfere with one another. The amplitudes of the two scattered waves cancel one another. 9 3.3.2 X-Ray Diffraction: Determination of Crystal Structures in such a way that they traverse different paths. The phase relationship between the scattered waves, which depends upon the difference in path length, is important. One possibility results when this path length difference is an integral number of wavelengths. As noted in Figure 3.20a, these scattered waves (now labeled 1 and 2′) are still in phase. They are said to mutually reinforce (or constructively interfere with) one another; when amplitudes are added, the wave shown on the right side of the figure results. This is a manifestation of diffraction, and we refer to a diffracted beam as one composed of a large number of scattered waves that mutually reinforce one another. Other phase relationships are possible between scattered waves that will not lead to this mutual reinforcement. The other extreme is that demonstrated in Figure 3.20b, in which the path length difference after scattering is some integral number of half- wavelengths. The scattered waves are out of phase—that is, corresponding amplitudes cancel or annul one another, or destructively interfere (i.e., the resultant wave has zero amplitude), as indicated on the right side of the figure. Of course, phase relationships intermediate between these two extremes exist, resulting in only partial reinforcement. 3.3.3 Ray Diffraction and Bragg’s Law X-rays are a form of electromagnetic radiation that have high energies and short wavelengths—wavelengths on the order of the atomic spacings for solids. When a beam of x-rays impinges on a solid material, a portion of this beam is scattered in all directions by the electrons associated with each atom or ion that lies within the beam’s path. Let us now examine the necessary conditions for diffraction of x-rays by a periodic arrangement of atoms. Consider the two parallel planes of atoms A–Aʹ and B–Bʹ in Figure 3.21, which have the same h, k, and l Miller indices and are separated by the interplanar spacing dhkl. Now assume that a parallel, monochromatic, and coherent (in- phase) beam of x-rays of wavelength 𝜆 is incident on these two planes at an angle 𝜃. Two rays in this beam, labeled 1 and 2, are scattered by atoms P and Q. Constructive interference of the scattered rays 1ʹ and 2ʹ occurs also at an angle 𝜃 to the planes if the path length difference between 1–P–1ʹ and 2–Q–2ʹ (i.e., SQ + QT) is equal to a whole number, n, of wavelengths—that is, the condition for diffraction is n𝜆 = SQ + QT 10  Figure 3.21) Diffraction of x-rays by planes of atoms (A–Aʹ and B–Bʹ).( n𝜆 = dhkl sin 𝜃 + dhkl sin 𝜃 = 2dhkl sin 𝜃 is known as Bragg’s law; n is the order of reflection, which may be any integer (1, 2, 3,...) consistent with sin 𝜃 not exceeding unity. Thus, we have a simple expression relating the x- ray wavelength and interatomic spacing to the angle of the diffracted beam. If Bragg’s law is not satisfied, then the interference will be nonconstructive so as to yield a very low-intensity diffracted beam. The magnitude of the distance between two adjacent and parallel planes of atoms (i.e., the interplanar spacing dhkl) is a function of the Miller indices (h, k, and l) as well as the lattice parameter(s). For example, for crystal structures that have cubic symmetry, 11  dhkl = √ in which a is the lattice parameter (unit cell edge length). , should be present. Specific sets of crystallographic planes that do not give rise to diffracted beams depend on crystal structure. For the BCC crystal structure, h + k + l must be even if diffraction is to occur, whereas for FCC, h, k, and l must all be either odd or even; diffracted beams for all sets of crystallographic planes are present for the simple cubic crystal structure (Figure 3.3). These restrictions, called reflection rules, are summarized in Table Figure 3.3 )For the simple cubic crystal structure, (a) a hard-sphere unit cell, and (b) a reduced-sphere unit cell.( Crystal Structure Reflections Present Reflection Indices for First Six Present BCC (h + k + l) even 110, 200, 211, 220, 310, 222 FCC h, k, and l either, 111, 200, 220, all odd or all even 311, 222, 400 Simple cubic All 100, 110, 111, 200, 210, 211 Table 3.5 ) X-Ray Diffraction Reflection Rules and Reflection Indices for Body-Centered Cubic, Face-Centered Cubic, and Simple Cubic Crystal( 12 13 14 15