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

2. 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’. Combinations involving non-parallel rotations 2μ2μ 2μ2μ μ 2μ2μ

3. Now, consider the general combination of two screws A  t1 and B  t2 t 1 +t 2 C” C C’ B A t1t1 t2t2 μ A α,t1 = t 1 · A α B β,t2 = B β · t 2 ∴ A α,t1 · B β,t2 = t 1 · A α · B β · t 2 = t 1 · C’ γ · t 2 Now, consider the way the translation t 1 transforms a rotation of γ about C, t 1 -1 C γ t 1 = C’ γ Premultiply both sides by t 1, giving t 1 t 1 -1 C γ t 1 = t 1 C’ γ → C γ t 1 = t 1 C’ γ ∴ A α,t1 · B β,t2 = C γ · t 1 · t 2 Standard form of a rotation, C γ, followed by a translation t 1 +t 2

4. 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. ∴ A π,t1 · B π,t2 = 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.

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

6. 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 A π,t1 with s B π,t2 B π,t2 = s B π,t2 · T ┴ where,

7. Space Groups Isogonal with 222: A π,t1 · s B π,t2 = C” π,2s 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.

8. Space Groups Isogonal with 222: First, consider the possible combination of 2 m 2 m 2 m with P. → P222, P222 1,P2 1 2 1 2 and P2 1 2 1 2 1 Can be made by the combinations in the top and bottom rows of Fig. 4.

9. 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.

10. Space Groups Isogonal with 222: When a screw A α,t1, is placed at A, its translation equivalent occurs at A” and a screw of A α,t1+T11 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: C222 m, C22 1 2 m, C2 1 22 m, C2 1 2 1 2 m Note that C222 = C2 1 22 = C22 1 2 = C2 1 2 1 2 C222 1 = C2 1 22 1 = C22 1 2 1 = C2 1 2 1 2 1 → m assure 0 or 1.

11. Space Groups Isogonal with 222:

12. 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.

13. Space Groups Isogonal with 222: Show that I222 = I2 1 2 1 2 I222 1 = I2 1 2 1 2 1

14. Space Groups Isogonal with 222: Consider the combination of 2 m 2 m 2 m with the translations of F. F222 = F2221 = F2 1 2 1 2 = F2 1 2 1 2 1

15. 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.

16. 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.

17. 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. P3 m 2 m 1 and P3 m 12 m 3 2 1 : two fold axes along the cell axis. 3 1 2 : two fold axes along the cell diagonal.

18. 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 P3 1. The 2-fold rotation axes intersect the 3 m screws according to the Fig. 15. Fig. 17 illustrate the result of combining the operation A π, with a nonparallel axial translation: A screw A π,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 P3 m 21 contains P3 m 2 1 1. → 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 P3, P3 1, P3 2. → results P321, P3 1 21, and P3 2 21

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

20. Space Groups Isogonal with point group 32: Consider P3 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. → P3 m 12 contains the combination P3 m 12 1. → the three space groups P312, P3 1 12, and P3 2 12 result.

21. 5/6 1/6 2/6 4/6 5/6 1/6 5/6 4/6 Where 5/6=2/6.

22. Space Groups Isogonal with point group 32: → 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 R3 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 R32. → (Fig. 25) contains all combinations shown in Fig. 15 ∴ R32 = R3 1 2 = R3 2 2 = R32 1 = R3 1 2 1 = R3 2 2 1 The lattice R translation

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

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