{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Allen_Thomas_Chpt3_part_B

Allen_Thomas_Chpt3_part_B - 3.1 The Crystallography of Two...

Info iconThis preview shows pages 1–20. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 16
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 18
Background image of page 19

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 20
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3.1 The Crystallography of Two Dimensions 107 special position. Only objects whose point symmetry equals or exceeds the symmetry at the special position can occupy the special position. International and Schoenflies Symbols TWO different systems for designating crystal symmetries are‘ in common use: the Hermann-Mauguin symbols and the Schoenflies symbols. While both are used in the International Tables for Crystallography, Hermann-Mauguin symbols, com— monly known as international symbols, predominate. The international symbols are more generally used by materials scientists, and the Schoenflies symbols by chemists and physicists. In both notations, Arabic numerals are used to designate axes of rotational symmetry, with the numeral corresponding to the value of n of the axis (e. g., a repetition angle of 277/3 is designated by the numeral 3 to indicate a three- fold axis of rotational symmetry). In the international notation, plane point group symmetries lacking intersecting mirror lines are designated by the appropriate numeral: 1, 2, 3, 4, or 6. The com- bination of the axis 1 with a single mirror line is designated as point symmetry m. The other combinations with mirror lines are designated as 2mm, 3m, 4mm, and 6mm in international notation. Regarding the notation for the plane point groups containing mirror lines, it should be noted that a double m (mm) appears in the point groups 2mm, 4mm, and 6mm, whereas in the point groups In and 3m, it does not. The double m is required when the mirror lines are interleaved, that is, when the angular interval between the mirrors is half that of the rotational symmetry axis of the point group. Thus, in plane point group 4mm, the angle between mirrors is 77/4. In plane point group 3m, all of the mirrors can be generated simply through the operation of the axis A2,” on the original mirror line; the new mirrors identified by Eq. 3.7 superimpose on the ones generated by Aha alone. The Schoenflies notation uses a capital letter from the set {C, D, T, 0, S} and one or more subscripts. All of the plane point groups are designated by the capital letter C, for “cyclic” (because of the cyclic rotation about a single axis perpendicular to the plane of the pattern).5 All of the “C” plane point groups involve a single rotation axis and are thus said to be “monaxial” groups. The subscripts for the plane point groups can include integers 1, 2, 3, 4, and 6 to indicate the value of n of the axis. When no mirror lines intersect the rotation axis, the plane point group sym— metrics are designated respectively as C ,, C2, C3, C4, and C6. For the combinations of two-, three-, four—, and sixfold axes and mirror lines, the symbols Czu, C3,, C4,” and C6,, are used, respectively. The subscript v indicates a “vertical” mirror plane; that is, in three dimensions, a vertical mirror plane intersects the rotation axis along its length (this reduces to a mirror line perpendicular to the rotation axis in a two- dimensional pattern). The symbol C5 is used for the case of intersection of a onefold axis with a mirror line (the s abbreviates Spiegel, German for “mirror”). Figure 3.18 5Note that there are additional point groups identified with a C that are relevant to three—dimensional symmetries. These are considered in Section 3.2.4. 108 Chapter 3 / Crystalline State includes international and Schoenflies symbols for each of the 10 plane point group symmetries. 3.1.5 The Five Distinct Plane Lattices Now it is possible to make a systematic study of rotational symmetry axes and/or mirror lines combining with translations in a self-consistent manner. In the process, we will describe the five plane lattice types and find how the rotational symmetry axes and mirror lines can become distributed within the unit cell in certain cases. A finite set of five plane lattices arises from the possible combinations of the crystallographic plane point group symmetries with translational symmetry in two dimensions. Different lattice types arise because the presence of special symmetry (i.e., rotation axes or mirror lines) usually places restrictions on the basis vectors t1 and t2 and the interaxial angle 3/. For the most general case, when there are no restrictions, t] and t2 form a parallelogram cell which results in a parallelogram lattice. Consideration of the parallelogram lattice illustrated in Figure 3.3 will reveal that this lattice contains twofold rotation axes. First, remember combining an axis of rotational symmetry with a translation, covered in Section 3.1.3. Figure 3.19 illustrates a rotation axis A“ and the translation t. The line 1 is drawn at an angle 05/2 to the line perpendicular to t, such that the rotation axis An moves the point P1 and the line 1 to the point P2 and the line 2, respectively. The translational symmetry operation t moves the point P2 and the line 2 to the point P3 and the line 3, respectively. Consideration of Figure 3.19 shows that the combination of symmetry operations Au and t is thus consistent with a rotation axis Ba located at the intersection of lines 1 and 3. The rotation axis at B is through the same angle a and in the same direction (counterclockwise in Figure 3.19) as for the axis Ad. The location of the axis B is related geometrically to a and ltl. The general conclusion of this result can be stated as follows (Buerger, 1978, p. 72): A rotation about an axis A through an angle at, followed by a translation t perpendicular to the axis, is equivalent to a rotation through the same angle Figure 3.19 Combination A“ - t is equivalent to rotation axis Ba (after Buerger, 1978, p. 71). 3.1 The Crystallography of Two Dimensions 109 a, in the same sense, but about an axis B situated on the perpendicular bisector ofAA ’ and at a distance d = (ltl/Z) cot(a/2)fr0mAA ’. Symbolically, this combination of symmetry operations is represented by Al - t = Ba (3.8) An n—fold rotation axis A“ is an axis about which rotations of oz (= 277m), 2a, . . . , na are all symmetry operations. Thus, in order fully to apply Eq. 3.8 to an axis Au, an allowance must be made for each of these 21 rotations. We shall demonstrate, for example, that when considering the combination Am - t, Eq. 3.8 will dictate that additional fourfold and twofold axes must be present in the primitive cell. Plane Lattice Nets Arising from Crystallographic Rotation Axes and Translations The set of rotational symmetry axes consistent with lattice translation in two dimen- sions results in restrictions on t1, t2, and 'y and yields three of the five distinct two- dimensional lattice nets. In order to identify all of the rotational symmetry axes contained within the primitive cell, Eq. 3.8 must be applied to each of the translations t1, t2, and t1 + t2 because the end points of these three vectors plus the origin define the vertices of the cell. All of these cases are presented in detail by Buerger (1978, pp. 72—78). Here, to illustrate the basic process of derivation, we consider the com- binations involving two— and fourfold axes. The following treatment ignores the application of Eq. 3.8 to the onefold axis A2,” as the finding of other onefold axes at various points is a trivial result. Consider the combination A,r - t. Figure 3.20 shows the axis A, the vector t, and the axis A’ produced from A by t. Equation 3.8 implies the existence of an axis BW located along the perpendicular bisector (shown solid in Figure 3.20) of the lineAA’. The distance of B7, from the line AA' is proportional to cot(a/2) = cot(7T/2) = 0. Because arbitrary basis vectors t1 and t2 define a parallelogram lattice containing a twofold axis at the origin, a twofold axis must also exist halfway along t1. Similar reasoning results in twofold axes existing halfway along t2 and halfway along t] + t2. Figure 3.21 shows the resulting primitive cell and the locations of all rotation axes within the cell. Note that the twofold axis at the origin puts no restrictions on allowed vectors for t2, thus resulting in a primitive cell that is in the general case a Figure 3.20 Combination A1, - t gives rise to twofold axis 8,, at the point halfway along line AA’. 110 Chapter 3 / Crystalline State Figure 3.21 Primitive cell obtained by combination A17 and perpendicular translations. In addition to twofold axes at cell comers, other twofold axes appear midway along cell edges and at center of cell. parallelogram and which repeats on a parallelogram lattice. (The lenticular symbols appear in four different orientations in Figure 3.21 to indicate that the axes are crystallographically nonequivalent; that is, the environment around each of these four regions in the cell is generally different. Similar convention is used in several of the figures that follow.) Now consider the combination of a fourfold axis and t. The fourfold axis contains the set of rotations A7,,2, Am Aha, and A2". Because of the fourfold axis, the vectors defining the primitive cell, t1 and t2, are perpendicular and of equal magnitude, resulting in a square lattice. Considering first the combination AW - t1, Eq. 3.8 gives the result that a fourfold axis must exist along the perpendicular bisector of AA’, at a distance (AA’IZ) cot(1r/4) = AA’IZ. A moment’s reflection will show that this point is in the center of the primitive cell, shown as Bl in Figure 3.22. The combinations Am; - t2 and Ava - (t1 + t2) do not yield additional results. Next, because the rotation A7, is a subgroup of the fourfold axis, we must consider the combination A1T - t2, which was treated above and illustrated in Figure 3.20. This combination produces a new twofold axis halfway along AA’, shown as B2 in Figure 3.22. Similar reasoning can be used to derive the remaining twofold axes shown in Figure 3.22. Plane lattices consistent with threefold axes are similarly derived with the aid of Eq. 3.8. The resulting cell forms a 120° rhombus lattice. Thus, threefold symmetry forces a unique value to 7 and also requires ltll = ltzl. When sixfold axes are combined with lattice translations, the resulting cell is also a 120° rhombus, but, in addition to the sixfold axes, there arise three- and twofold axes in the cell, due to the fact that the rotations A273 and A7, are a subgroup of the rotation A215. The resulting cells, and the locations of rotational symmetry axes within the cells, are illustrated in Figure 3.23. Figure 3.22 Primitive cell obtained by combination Am; and per- pendicular translations. In addition to fourfold axes at cell corners, another fourfold axis exists at center of the cell and twofold axes appear midway along cell edges. 3.1 The Crystallography of Two Dimensions 111 (a) (b) Figure 3.23 Primitive cells obtained by combination (a) A273 and (b) A",3 with perpendicular translations. Both combinations give rise to a 120° rhombus lattice. Different orientations of symbols indicate symmetry elements are at nonequivalent positions in unit cell. To summarize results so far for the combinations of each of the rotational sym- metries l, 2, 3, 4, and 6 with translations, in the absence of mirror symmetry (the cases discussed immediately above), there are only three two-dimensional arrange— ments of lattice points (called lattice nets): the parallelogram, the 120° rhombus, and the square. Example Problem 6 Superpose the symmetry elements shown in Figure 3.22 on the periodic pattern shown in Figure EP6a so as to outline a primitive cell in the pattern. Convince yourself that all of the four— and twofold axes shown in Figure 3.22 are present in the pattern. % % %%% 8% %% &% &%%% fifi‘fi Elfifigé as; %% %%%E%% as (a) (b) Figure EP6 (a) Pattern containing two- and fourfold rotational symmetry axes. (b) Superposition of symmetry elements within single conventional unit cell of pattern in (a). 112 Chapter 3 / Crystalline State Solution Figure 3.22 is superposed on the pattern in Figure EP6b. Note the two nonequivalent locations of fourfold axes: One set is centered on the small repeated motif, and the other is centered midway between four neighboring motifs. Either location can serve as a suitable origin for the primitive cell of this pattern. Lattice Nets Arising from Mirror Lines and Translations An additional derivation is required when mirrors are present and the highest rota- tional symmetry is a one- or twofold rotation axis. This leads to two additional lattice nets. These two lattices can be developed by first considering the one—dimensional combination of a translation vector tl with a line of reflection symmetry m, as shown in Figure 3.24:1.6 We show in the following section (see Eq. 3.9) that this combi- nation results in a periodic array of lines of mirror symmetry with spacing equal to half the magnitude of the lattice translation, as shown in Figure 3.24b. The dots in Figures 3.24a and b represent lattice points; note that the spacing of lattice points is twice the spacing of the lines of mirror symmetry. Figure 3.24c shows a periodic pattern containing the symmetry elements of Figure 3.24b. Note that while the pat- tern in Figure 3.240 has one-dimensional periodicity (a single translation vector t1), the repeated object (an isosceles triangle) is two dimensional. Adding a second lattice translation t2 to the pattern of Figure 3.24%) will extend the repetition of the pattern from one dimension to two. The goal is to incorporate reflectional symmetry into the two-dimensional pattern; thus, the points of mirror symmetry in the one-dimensional repetition must extend to lines of reflectional sym- '" m P 1.1 E1: 5 :E t —*—'" P (a) (b) (c) Figure 3.24 (a) Combination of lattice translation t1 and perpendicular line of reflectional symmetry. (b) Combi- nation introduces lines of mirror symmetry spaced at half the magnitude of the translational symmetry vector. (c) Portion of periodic pattern that conforms to arrange- ment of symmetry elements illustrated in (b). °ln order that the resulting pattern have one-dimensional periodicity, it is necessary that the lattice trans- lation be perpendicular to the line of reflectional symmetry. A mirror m oriented with respect tot by some angle other than 90° leads to the two—dimensional centered—rectangular lattice. 3.1 The Crystallography of Two Dimensions 113 metry in two dimensions. In order for the additional lines of reflectional symmetry to be present, t2 must be constrained in one of two ways: Either it must be perpen— dicular to t2, or its component parallel to t1 must be equal to t1/2. Both cases are illustrated in Figure 3.25. The first is illustrated in Figure 3.25a, with the arrangement of symmetry elements and a set of lattice points shown on the left and a periodic pattern containing this arrangement on the right. The resulting lattice net is the rectangular lattice and the conventional unit cell is a primitive rectangular cell as overlaid on the right in Figure 3.25a. The second case is presented analogously in Figure 3.25b. Because of the constraint on t2, it becomes possible to define a rec— tangular cell that is a double cell, resulting in the centered rectangular lattice, out- lined in Figure 3.25b in the upper right. This is the conventional unit cell for this arrangement of symmetry elements (i.e., it conforms to the terminology in the In- ternational Tables for Crystallography). An alternative choice of unit cell for the centered rectangular lattice is also illustrated in Figure 3.25b in the lower right. In t T | | 4m.» 3 5 5 55 55 V v V— V V v v V v (a) v v v v I l l | 5533§S§§ l l l Figure 3.25 (a) Development of rectangular lattice by adding second translation to pattern of Figure 3.241). In this case, mirror symmetry is preserved by taking t2 perpendicular to t1. Arrangement of symmetry elements shown on left, and a pattern with this symmetry shown on right with conventional unit cell outlined. (b) Development of centered rectangular lattice. Here, t; is constrained to fall on a mirror line mid- way between lattice points, as shown on left. Conventional cell for this symmetry arrangement is a double cell, as shown in upper right, and lattice net is known as centered rectangular lattice. Primitive cell shown at lower right is an alternative unit cell known as a diamond cell or a rhombus cell. 114 Chapter 3 / Crystalline State this case, the basis vectors are of equal length and the cell is a primitive cell, known as the diamond lattice or alternatively as the rhombus lattice. (Note that there is no constraint on the interaxial angle 3/ of the rhombus) A thorough analysis of the addition of lines of mirror symmetry to the other lattices we have derived previously (square and 120° rhombus) shows that no ad- ditional lattice geometries arise. We have therefore developed the complete set of five types of plane lattices: parallelogram, 120° rhombus, square, rectangle, and cen- tered rectangle. The information obtained concerning the plane lattice types is sum- marized in Table 3.1, which specifies, given the highest point symmetry present in the pattern, the least specialized lattice type consistent with that point symmetry. (Higher symmetry lattices may be present but are not required by symmetry.) The five plane lattices are illustrated in Figure 3.26, along with the conventional choices for basis vectors and their positions relative to special symmetry elements that are present. Note that the 120° rhombus lattice associated with rotational symmetries 3 and 6 is a specialized centered rectangular lattice. The 120° interaxial angle is consistent with rotational symmetries 3 and 6, both of which are higher than the symmetries 2 and m associated with the centered rectangular lattice. When either symmetry 3 or 6 is present, the conventional unit cell is chosen to be the 120° rhombus, rather than the centered rectangle, to emphasize the presence of higher (rotational) symmetry. 3.1.6 Plane Groups So far, we have derived the ten plane point groups and the five associated plane lattices. In addition to deriving the shapes of the plane lattices, we have demonstrated that axes of rotational symmetry are often present within the unit cell and their locations are known. The final task in defining the set of crystallographically distinct and consistent ways to arrange symmetry elements in a plane is to allow systemat— ically for reflectional symmetry and glide (translation—reflection) symmetry in the arrangements of symmetry elements that have been derived so far. The end result of this exercise will be the determination of the 17 plane groups: the complete set Table 3.1 The Five Plane Lattice Types Consistent with the Ten Crystallographic Plane Point Groups __________________._.________—————————- Highest Point Group Symmetry Present Lattice Type"‘ 1, 2 Parallelogram m, 2mmb Rectangle m, 2mmb Centered rectangle 4, 4mm Square 3, 3m, 6, 6mm 120° rhombus _______________________.__.———— “A plane lattice with higher symmetry is possible, but not required. bCrystallographically distinct arrangements result when the centering translation 1/2, 1/2 is added to a pattern with m or 2mm point group symmetry. 3.1 The Crystallography of Two Dimensions 115 (t1+t2)/2 Figure 3.26 Five plane lattices—parallelogram, square, 120° rhombus, rectangle, and centered rectangle—conventional unit cells for each and locations of rotational symmetry axes within unit cells. Lines of mirror symmetry not shown. 116 Chapter 3 / Crystalline State of distinctly different ways of arranging symmetry elements in a two—dimensional crystal. Any pattern that has two‘dimensional translational symmetry (e.g., a two- dimensional crystal, floor tiles, patterned textiles) will have its symmetry elements arranged in one of these 17 plane groups. For two-dimensional crystal structures, the complete description of such a pattern can then be given simply by designating the appropriate symbol for the plane group and specifying the contents (e.g., atom types and coordinates) of the unit cell. Addition of Reflectional Symmetry to Plane Lattices The ad...
View Full Document

{[ snackBarMessage ]}