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Allen_Thomas_Chpt3_part_C

Allen_Thomas_Chpt3_part_C - 3.2 The Crystallography of...

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Unformatted text preview: 3.2 The Crystallography of Three Dimensions 127 Figure 3.33 An inversion center is created between right and left hands when they are positioned as illustrated. Rotoinversion We previously discussed proper rotation axes in Section 3.1.3. A proper axis pro- duces a sequence of objects all of the same handedness. A series of objects alternating in handedness is produced through improper rotation axes by combining rotation with either reflection or inversion symmetry operations. In three-dimensional pat— terns, rotation axes can combine with an inversion center to create an improper rotoinversion axis. Additionally, an axis of rotational symmetry can combine with a reflection plane to produce an improper rotoreflection axis (see below). A rotoinversion axis is a compound symmetry operation that combines a rotation axis with an inversion center located at a point 0 on the axis. The sequence of operations in an n-fold rotoinversion axis which produces a series of objects in space related by rotoinversion is as follows: Rotate the object by 27/11, invert the object through 0, and then draw the inverted image of the object; rotate the inverted object by 27r/n and invert through 0; and draw the image of the object (which is now of the same handedness as the original starting object). Continue until the object is repeated at its original location with its original handedness. The object superposes after n rotation intervals if n is even; otherwise it repeats after 2n intervals. An overbar is used to indicate a rotoinversion axis (e.g., 8). Figure 3.34 illustrates the operation of the fourfold rotoinversion axis Z. It also illustrates a view looking down the axis, in the form of a two—dimensional projection. Note that the symmetry operation Z does not have the proper rotation 4 as a subgroup; it does have the proper rotation 2 as a subgroup, however. The rotoinversion axis T is equivalent to the inversion operation, i. Similar projections illustrating repetitions by the crystallographic axes T, E, 3, Z, and 5 are presented in Figure 3.35. These should be compared with the patterns produced by the operation of proper rotation axes, shown in Figure 3.11. The ro- toinversion axes T, 3, Z, and 6 have special symbols to distinguish them from the corresponding proper rotation axes; these are shown in Figure 3.35 (there is no special symbol for the axis 5, as it is equivalent to a horizontal mirror plane, i.e., a mirror perpendicular to the 5 axis). A good example of rotoinversion symmetry is provided by the ethane molecule, CZHG, in the low—energy, staggered conformation, illustrated in Figure 1.6. This 128 Chapter 3 / Crystalline State (a) (b) Figure 3.34 Operation of fourfold rotoinversion axis: (a) perspective sketch; (b) projection along axis. Open and filled triangles are of opposite handedness. Note the open square with solid lens shape inside, which symbolizes the 1 symmetry element. Figure 3.35 Patterns produced by crystallographic rotoin— version axes. Large circles indicate midplane, e.g., the plane of the page. Filled triangles lie above the midplane, and open triangles lie below it. Where filled and open triangles superpose, they are drawn gray. Open and filled triangles are of opposite handedness. Note set of symbols at centers of each illustration, which indicate types of rotoinversion axes. 3.2 The Crystallography of Three Dimensions 129 conformational state of ethane has a 3 axis coaxial with the C—C bond, with the inversion center at the bond’s midpoint. Figure l.6 should be compared with the illustration of the 3 axis in Figure 3.35. Rotoreflection A rotoreflection axis is a compound symmetry operation, and its operation is de- scribed below. An n-fold rotation axis is perpendicular to a reflection (mirror) plane m and passes through m at an origin 0. The object to be repeated is rotated by 277/n, then reflected by m, then drawn. The process is repeated until the object is reproduced in its original location and with its original handedness. If n is even, the object will be superposed in its original location after 11 steps (total rotation 277). If n is odd, the object will not superpose at its original location until a rotation of 477' (2n steps) has been completed. Rotoreflection axes are denoted by an integer with a tilde above it (e.g., 3). Figure 3.36 illustrates the operation of the threefold rotoreflection axis 3. It also illustrates a view looking down the axis in the form of a two— dimensional projection4. Similar projections that illustrate repetitions by the crystallographic axes 1,2, 3,4,and 6 are presented in Figure 3. 37. These should be compared with Figures 3 11 and 3. 35 Comparison of Figures 3.11, 3.35, and 3. 37 will show that the set of rotoreflection axes are equivalent to the set of rotoinversion axes For instance, 1 = 2. These and other aspects of equivalencies of improper rotational symmetry axes with simpler symmetry combinations, are discussed-by Buerger (1978, pp. 27—30), and will be considered below in Section 3.2.4. Because rotoreflectional symmetry can generally be expressed by an equivalent combination of a proper rotation axis and a perpen— dicular mirror plane, there are no special symbols for rotoreflection axes, unlike those introduced above for rotoinversion symmetry (the axis 4 = 4 is the only special case). 53 (a) (b) Figure 3.36 Operation of threefold rotoreflection axis: (a) perspec- tive sketch; (b) projection along axis. Because triangles are super- posed, they are drawn gray. 130 Chapter 3 / Crystalline State 993 <35? Figure 3.37 Patterns produced by crystallographic ro- toreflection axes. Large circles indicate the mirror mid— plane, e.g., the plane of the page. Where filled and open triangles superpose, they are drawn gray. Filled triangles lie above the midplane, and open triangles lie below it. Screw Axes A screw axis is a compound symmetry operation involving a proper rotational sym- metry axis and a translation. Recall that in Sections 3.1.3 and 3.1.5 we considered this type of combination in two dimensions and hence necessarily constrained the translation to lie perpendicular to the rotational symmetry axis. We now consider combinations of rotation axes and translations that lie parallel to the rotation axis. Figure 3.38 demonstrates the combination of a rotation axis A0, and a parallel translation 1'; an object (in this case, a group of atoms) located at P is repeated at Figure 3.38 Combination of rotation axis Am and parallel translation 1'. The first two repetitions of the object at P are shown at Q and R. 3.2 The Crystallography of Three Dimensions 131 Q, then R, and so on. Depending on the sequence of the rotation and the translation, this can be expressed by either of the paths PVQ or PUQ; that is, the component operations of rotation and translation are commutable. Such a combination can be considered as a screw motion. This can be represented symbolically by Aa'T = A,, (3J5) where the symbol A,” represents a screw axis; a is called the rotation component of the screw axis and '7 is called the translation component of the screw axis. When screw symmetry is applied to an object, a helical arrangement of otherwise identical copies of the object results. Helical symmetry is quite common, and many of the three-dimensional crystallographic symmetry groups possess one or more screw axes. The allowed rotational components of crystallographic screw axes are the same as for pure rotation axes, namely 277, 77, 277/3, 77/2, and 77/3. Clearly, the translational component of the screw axis cannot be arbitrary. After n rotational repetitions, the total rotation amount must be 277. This will produce a total translation n7, which must be an integer multiple m of the lattice translation TH parallel to the screw axis. Thus, 711' = MT”, 01' ¢=En (mm n Therefore the only values of m that need to be considered in Eq. 3.16 are those between 0 and n — I (see Buerger, 1978, p. 203). Permissible values of n and Fr for each of the possible crystallographic screw axes, along with the conventional des- ignation of the various screw axes, are presented in Table 3.3. In international no- tation, a screw axis is represented by an integer n representing the symmetry of the rotation, followed by a subscript denoting the value of m in Eq. 3.16 (e.g., a 63 screw axis has n = 6 and m = 3). Figure 3.39 illustrates the process by which patterns showing the crystallograph- ically allowed types of fourfold screw axes can be generated. Figure 3.39a illustrates a 41 screw axis. Beginning with the representation of an object at the top of the axis, a clockwise rotation of 77/2 is performed, followed by a shift '7 parallel to the axis. The process is repeated three times, until the total shift is TH = 41', and the object is returned to its original orientation, which coincides precisely with the requirement that an identical object and orientation exist at all multiple distances of T”. Note that this 41 screw axis produces a helical arrangement of objects, with a right-hand orientation of the screw motion. Figure 3.3% illustrates the process by which a pattern with 4; symmetry is gen— erated. By definition, for a 42 screw axis, the translation parallel to the axis is TH/2. The motif is repeatedly rotated (clockwise as viewed from above) by 77/2 and trans- lated Til/2 downward. Successive positions are indicated as l, 2, 3, 4. Because 132 Chapter 3 / Crystalline State Table 3.3 Allowed Crystallographic Screw Axes ____—_______—__..__————————-———-———-——-_-——- Proper Rotation The Eleven Permissible n Components Axes Crystallographic Screw Axes l a O (or 277) T O (or TH) Designation l '2 a 72' 7T Designation 2 21 3 a §7T §7T %7T Designation 3 31 32 4 a £77 %7r %7 in 7 0 iTH fiTil iTu Designation 4 41 42 43 6 at %’2T §7r §7T 317r §7r §7T T 0 éTll éTH 3TH 3TH 3TH Designation 6 61 62 63 64 65 _______________—_____________.____———-—-———————- Source: Buerger, 1978, p. 204, m = 2 for this screw axis, the full rotation of 360° occurs over a distance of 2TH- But the translational symmetry operation TH applies to the pattern as a whole, and thus it operates on objects 2, 3, and 4, as shown in the lower Figure 3.391? [new objects can be generated in the pattern by applying the fundamental lattice transla- tions to any object(s) generated by a symmetry operation]. Inspection of Figure 3.3919 reveals that the 42 screw axis is neutral, in that there is no sense of rotation of the object; in fact, a 42 screw axis also contains a twofold axis. Figure 3.390 illustrates the process by which a pattern with 43 symmetry is gen— erated. Here the translation parallel to the axis is iTH. Successive positions are again indicated 1, 2, 3, 4. Since m = 3 for this screw axis, the full rotation of 360° is accomplished over a distance of 3TH. Again, translational symmetry is applied to place equivalent objects within the basic interval (0, T“). This results in a pattern similar to the 4, screw axis, except that the direction of rotation of successive objects now forms a left-handed screw. The sense of handedness of the series of objects produced by a screw axis (not of a single object, which may or may not possess handedness and, in either case, 3.2 The Crystallography of Three Dimensions 133 T” = 4T 2T” = 4T 3T”: 41' 41 42 4 3 (a) (b) (C) Figure 3.39 Repetitions of an object (here a radial spoke normal to screw axis) by operation of 41, 42, and 43 screw axes. Bottom figures are the final patterns after applying TH to all objects generated by screw symmetry. Sense of rotation is clock- wise (CW) when viewed from above. Note compressed vertical scale of figures ([7) and (c) relative to (a) [i.e., TH is progressively shorter on going from (a) to (6)]. will have its handedness preserved by the screw symmetry operation) is related to the relative values of m and 11, according to the relations '1' < %TH right-hand screw axis 7 = éT” neutral screw axis 1' > %T” left—hand screw axis The repetitions of an object produced by each of the crystallographic screw axes are presented in Figure 3.40. In addition, at the top of each axis in Figure 3.40 the conventional symbol for the axis is shown. Note that the screw axes are represented by extending the conventional symbol for a pure rotation axis such that it appears to have “spokes” and these spokes have a sense of rotational direction that relates to the pattern of the repetition. 134 Chapter 3 / Crystalline State Proper rotation axes Screw axes of 2 21 ‘ i 3 31 32 4 41 42 43 6 61 62 63 64 65 Figure 3.40 Repetitions of an object by operation of screw axes. Object is indi- cated by an open circle; larger circles are oriented toward reader, i.e., on reader’s side of the rotation axis (Buerger, 1978, p. 205). Polymer molecules often display helical symmetry. For example, polyethylene in its orthorhombic crystal form has a 21 screw axis parallel to the chain axis. Figure 3.41 shows an array of unit cells with the c axis (parallel to the molecular Chain axes) oriented approximately along the viewing direction. The 2i screw axes centered along the molecular axes should be apparent from the staggered orientations of adjacent —CH2——— units. The three crystalline forms8 of isotactic polypropylene all 3.2 The Crystallography of Three Dimensions 135 Figure 3.41 Crystal structure of polyethylene has 21 screw axes coaxial with molecular chain axes. In this View, portion of polyethylene crystal is shown with its orthorhombic unit outlined. Screw axes are viewed approximately end—on, par- allel to the c axis of the unit cell. The international symbol for space group of this crystal is P 21/n 2,/a 2,/m. exhibit a threefold screw axis, again parallel to the chain axis. Another familiar material containing screw axes is graphite, which has a primitive hexagonal Bravais lattice (see Section 3.2.5), the base of the unit cell forming what is termed the basal plane, with the “c axis” normal to the basal plane. Both 63 and 21 screw axes are present in the structure, oriented parallel to the c axis. The same screw axes are also present in the “hexagonal close—packed” structure (e.g., in Mg). The double—helix structure of DNA is perhaps the most well—known of all molecules with screw symmetry. 3.2.2 Techniques for Three-Dimensional Spatial Relationships Specifying points, lines, and planes in three dimensions is straightforward. However, some tools are necessary for geometric analysis in three dimensions, and for this purpose, we introduce some concepts in spherical trigonometry. Visualization of these structures becomes more challenging, especially as two-dimensional figures often must be used to represent three—dimensional objects; we later describe how stereographic projection fulfills this need. Rational Intercept Plane: Miller Indices In Section 3.1.1, we introduced a notation for specifying points and lines in two- dimensional lattices in relation to a coordinate system determined by the lattice translations t1 and t2. In three-dimensional lattices, not only is there a third coor» dinate to contend with, but also there is often a need to specify two-dimensional planes and, in particular, the orientation of families of parallel planes. The orienta— tions of planes are specified by means of Miller indices. A plane that intersects a set of lattice points is called a rational plane. The equa- tion of a plane in three dimensions can be written in terms of its intercepts as follows. Let the basis vectors of the lattice be t1, t2, and t3 and the corresponding lattice constants be a, b, and c. A coordinate system with axes x, y, and z parallel to t1, t2, and t3 is defined. The unit distances along x, y, and z are taken to be the lattice 8A substance with more than one crystalline form is said to be polymorphic. 136 Chapter 3 / Crystalline State constants a, b, and c, respectively. A plane can then be described in terms of its intercepts P, Q, and R along the axes x, y, and z, respectively, as x y z — + — + — = l 3.17 P Q R ( ) The coefficients of x, y, and z in Eq. 3.17 specify the orientation of a plane. For convenience, it is customary to specify these coefficients as integers by multiplying both sides of Eq. 3.17 by PQR, and defining h E QR, k E RP, and l —=— PQ which leads to hx + ky + lz = PQR (3.18) Equation 3.18 is the equation of the rational intercept plane, that is, the plane with intercepts P, Q, and R (proportional to llh, 1/k, and 1/], respectively) along the axes x, y, and z. Planes are commonly specified by writing the numbers h, k, and l in parentheses [i.e., (hkl)]. In this form, (/1 kl) are called the Miller indices of the plane. In the Miller index notation, a negative number is indicated by an overbar. Thus, for the planes with h = —3, k = 2, and l = 5, the Miller indices would be written (3 2 5). Common multiplicative factors among the coefficients h, k, and [are usually divided out when reporting Miller indices. It is therefore preferable to write (2 l 1) rather than (6 3 3). If a plane is parallel to an axis, its intercept with that axis is at infinity; therefore, the corresponding Miller index for that axis is zero. When a unit cell is chosen to have basis vectors that are among the smallest of the lattice translations for the crystal structure, small values of the Miller indices 11, k, and 1 correspond to the more widely spaced planes in the crystal and to planes in which the number of lattice points per unit area on the plane is high. Such planes tend to make up the external faces of naturally occurring faceted crystals and to play a dominant role in plastic deformation of crystals by the motion of dislocations. The definition of Miller indices does not rely on a Cartesian coordinate system. The axes x, y, and z may make arbitrary angles with each other, and the lattice constants along each axis can be different. For example, Figure 3.42 ShOWS a plane with intercepts at A, B, and C along x, y, and z, where A = 3, B = 9, and C = 6. Thus, Eq. 3.17 becomes (3.19) mm \o|‘< cum II as and Eq. 3.18 is 54x + 18y + 272: = 162 (3.20) As it is conventional to divide Miller indices by any common factors, the Miller indices of the plane ABC in Figure 3.42 are (6 2 3). As all three planes shown in 3.2 The Crystallography of Three Dimensions 137 Z Figure 3.42 A stack of parallel rational planes be- C longing to the {6 2 3} family. >y Figure 3.42 are parallel, they have the same Miller indices (6 2 3). Figure 3.43 pro- vides additional examples by illustrating the Miller indices of several planes. A family of planes refers to the set of planes in a crystal which all have identical atomic arrangements but lie in different orientations (i.e., the planes of a family are crystallographically equivalent). A family of planes is denoted by braces, for ex- ample, the hkl family of planes is denoted {hkl}. In a cubic crystal, the {1 l 1} family of planes is comprised of planes in four different orientations: (1 l 1), (1 1T), (1 T 1), and (1 TT). It is important to note that unless a crystal possesses special symmetries, atomic configurations on planes with permuted Miller indices will be different; for instance, in a crystal with unspecialized lattice constants a, b, c, a, [3, and y, the (h kl) plane will be geometrically different from the (h l k) plane. In such cases, the (h kl) and (h l k) planes are not of the same family. In Section 3.1.1, we presented the method of designating points and lines in two- dimensional lattices. The same approach applies in three—dimensional lattices. In particular, lines can be represented by vectors, for example, I' = ”t1 + Ut2 + Wt3 (3.21) and the direction of the line is specified by [u v w], where the numbers u, v, and w are by convention cleared of fractions and reduced by dividing out any common {10 X Figure 3.43 Three-dimensional lattice with several ra- tional planes indicated by Miller indices. 138 Chapter 3 / Crystalline State factors. Families of crystallographically equivalent directions are gi...
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