Crystal Structure and Crystallography of Materials Chapter 11

Crystal Structure and Crystallography of Materials Chapter 11: Space Group 2 (The Dihedral Space Groups)

Combinations involving non-parallel rotations When 2 -fold rotations about axes A and B, which intersect at an angle μ, are combined, the result is a rotation of 2μ about a perpendicular axis C’. 2μ 2μ

Now, consider the general combination of two screws Aa, t 1 and Bb, t 2 Aα, t 1 = t 1 · Aα Bβ, t 2 = Bβ · t 2 ∴ Aα, t 1 · Bβ, t 2 = t 1 · Aα · Bβ · t 2 = t 1 · C’γ · t 2 C” B Now, consider the way the translation t 1 transforms a rotation of γ about C, t 1+t 2 C t 1 C’ μ t 2 t 1 -1 C γ t 1 = C’γ Premultiply both sides by t 1, giving A t 1 -1 Cγ t 1 = t 1 C’γ → Cγ t 1 = t 1 C’γ ∴ Aα, t 1 · Bβ, t 2 = Cγ · t 1 · t 2 Standard form of a rotation, Cγ, followed by a translation t 1+t 2

When α and β of (9) are both π, then axis C is perpendicular to the plane of A and B, so that t 1+t 2 has no component parallel to C. ∴ Aπ, t 1 · Bπ, t 2 = C” 2μ The result of combining two 2 -fold screws whose axes A and B intersect at an angle μ is a rotation of 2μ about an axis C” normal to the plane of A and B. The location of C” with respect to C is found, Effect of Displacing an Axis Now, we generalize the previous results with the screws which do not intersect. → necessary to know how the result of a rotation of α about an axis A compares with the result of a rotation of the same amount about a parallel axis s. A, separating axis.

The effect of displacing a rotation axis Aα by s. Rotation of α about a displaced axis is same as the rotation about an undisplaced axis, followed by a perpendicular translation. s. A s α A α · T ┴ = s. A α α/2 where, A A’

Combinations of Operations of Two non-intersecting 2 -fold Screws Find the results of combining two screw motions about axes which do not intersect, and which makes an angle μ with one another. 1) When the rotation components of both screws are π. → combine Aπ, t 1 with s. Bπ, t 2 = s. Bπ, t 2 · T┴ where,

Space Groups Isogonal with 222: Aπ, t 1 · s. Bπ, t 2 = C”π, 2 s The space groups isogonal with 222 are derived by combining the axial sets of Fig. 4 with the translations of the orthorhombic lattices D, C, I, and F.

Space Groups Isogonal with 222: First, consider the possible combination of 2 m 2 m 2 m with P. → P 222, P 2221, P 21212 and P 212121 Can be made by the combinations in the top and bottom rows of Fig. 4.

Space Groups Isogonal with 222: Next, consider the possible combinations of 2 m 2 m 2 m with a lattice having one face centered → let the centered face be C.

Space Groups Isogonal with 222: When a screw Aα, t 1, is placed at A, its translation equivalent occurs at A” and a screw of Aα, t 1+T 11 arises at A’ → if A is a rotation, A’ is a screw, and vice versa → Thus the A axes of Fig. 9 must be arranged in vertical sheets such that rotation axes and screw axes are alternately encountered from left to right. → A similar discussion leads to a similar relation of B axes. → possibilities: C 222 m, C 2212 m, C 21212 m Note that C 222 = C 2122 = C 2212 = C 21212 C 2221 = C 21221 = C 22121 = C 212121 → m assure 0 or 1.

Space Groups Isogonal with 222:

Space Groups Isogonal with 222: Next, consider the combinations of 2 m 2 m 2 m with the translation of lattice I. Each of the A, B, and C” axes, combined with the translations of I must give rise to Fig. 11 pattern of axes. → It only remains to see how A, B, and C” are related in space.

Space Groups Isogonal with 222: Show that I 222 = I 21212 I 2221 = I 212121

Space Groups Isogonal with 222: Consider the combination of 2 m 2 m 2 m with the translations of F. F 222 = F 2221 = F 212121

Space Groups Isogonal with point group 32: The order of listing the axis is C”AB, where A and B axes are equivalent through the operations of axis C”. → Can be derived by combining 3 m 2 m 2 m with each of the two lattices P and R.

Space Groups Isogonal with point group 32: Through each lattice point, there is a set of three 2 -fold axis along each a axis and 3 more 2 -fold axes along the long diagonals of the diamond meshes.

Space Groups Isogonal with point group 32: Consequently there are two permissible orientations of 32 with respect to lattice P. → the last position of the list of space-group symbols is accepted as the position of the axis along the cell diagonal. 3 2 1 : two fold axes along the cell axis. 3 1 2 : two fold axes along the cell diagonal. P 3 m 2 m 1 and P 3 m 12 m

Space Groups Isogonal with point group 32: The combination of the lattice translations and the 3 m produce a pattern of 3 m’s like P 31. The 2 -fold rotation axes intersect the 3 m screws according to the Fig. 15. Fig. 17 illustrate the result of combining the operation Aπ, with a nonparallel axial translation: A screw Aπ, a/2 occurs halfway along a translation. → screws and pure rotation alternate. (need explanation based on Fig 17) From Fig. 17 and Fig. 15, it becomes clear that any combination P 3 m 21 contains P 3 m 211. → all space groups with a primitive lattice and orientation 321 can be derived by using Fig. 17 and fitting the separation, s, between 2 -fold axis of the upper right of Fig. 15, to the 3 -fold screws of P 3, P 31, P 32. → results P 321, P 3121, and P 3221

Space Groups Isogonal with point group 32: 2/6 1/6 1/6 2/6 2/6

Space Groups Isogonal with point group 32: Consider P 3 m 12: the scheme of combination of any 2 -fold screw A is shown in Fig. 21 → non-parallel translation in (001) cause an alteration of parallel rotations and screws lying in this plane. → P 3 m 12 contains the combination P 3 m 121. → the three space groups P 312, P 3112, and P 3212 result.

2/6 1/6 2/6 5/6 1/6 Where 5/6=2/6. 4/6 5/6 5/6 4/6 1/6 5/6

Space Groups Isogonal with point group 32: The lattice R translation → symmetry 3 m 2 m → has only one set of three 2 -fold axis which occur parallel to the edges of the diamond-shaped triple cell. The possible space groups R 3 m 2 m are obtained by combining the translations of R with each of the possible axial combinations of Fig. 15. 1) Consider upper left axial set, which gives R 32. → (Fig. 25) contains all combinations shown in Fig. 15 ∴ R 32 = R 312 = R 321 = R 3121 = R 3221

Space Groups Isogonal with point group 32: 2/6 1/6 1/6 2/6 1/62/62/6 1/6

Space Groups Isogonal with point group 422, 622, and others: I will leave it up to you.