Space Groups

1. Space Groups Roya Majidi 1393

3. Glides: Reflection + translation • Screw Axes: Rotation + translation

4. Screw Axis • A screw (axis) operator rotates a point/object and then moves it a fraction of the repeat distance in one go. • The faction which the screw axes move is called the Pitch of the screw. • We will only consider (1, 2, 3, 4, 6) - fold rotations (crystallographic) as a part of the screw axes. • The screw axes to be considered are: 21  31, 32  41, 42, 43 61, 62, 63, 64, 65 • The normal and screw axis both give the same effect on the external symmetry of the crystal. • All identity points have the same enantiomorphic form (i.e. all objects created by the screw operator are all either left-handed or all are right-handed)

6. The 32 axis produces a rotation of 120 along with a translation of 2/3. • The set of points generated are: (0,0) (120,2/3) (240,4/31/3) (360,6/32)… • This is equivalent to a left handed screw (LHS) of pitch 1/3 Note: these are not fraction calculations! m

7. The 43 axis is a RHS with a pitch of 3/4 • The set of points generated are: (0,0) (90,3/4) (180,6/42/41/2) (270,9/41/4)… • The effect of 43 axis can be thought of as a LHS with a pitch of 1/4 Note: these are not fraction calculations!

8. The 42axis generates the following set of points:(0,0) (90,1/2) (180,2/21) (270,3/21/2) (360,4/22) • The grey arrowhead maps the (270,3/2) point to (270,1/2)→ to keep points within unit cell Note: these are not fraction calculations!

10. Glide Reflection • A glide (reflection) operator move a point/object by a fraction of the repeat distance and reflects the object in one go. • Kinds of ‘Glides’ are considered in crystallography: Axial Glide (a, b, c) →  Diagonal Glide (n) →  Diamond Glide (d) →

12. Different type of glides

13. Complete set of symmetry operators 3 numbers each 3 numbers each 3 numbers each

18. How do we go from a space group to a crystal? Why space groups at all? Why not work with Lattice + Motif picture? Click here • The Space Groupgives us a distribution of symmetry elements in space.(Given this distribution some points in space have a higher symmetry than others.) • If the Asymmetric Unitis used as a tile, then this tile in conjunction with the space group can fill entire space. Like unit cell (as a tile) in conjunction with basis vectors can fill entire space. • Wyckoff Positions for atomic species distribute (put) the atomic entities with respect to the symmetry operators. Wyckoff positions specify Site Symmetry and Occupancy by entities (usually atomic species) Further values for variables in Wyckoff table (x,y,z) have to be specified. • Obviously there is noScale in ‘symmetry related stuff’→ scale has to be added in via the Lattice Parameters (Unit Cell Parameters → Lengths and Angles consistent with the space group). Making a Crystal Space Group+ Asymmetric Unit + Wyckoff Positions + Lattice Parameters Consistent with the crystal system Site symmetry, Values for variables & Occupancy Asymmetric Unit is that part of the crystal which cannot be generated using symmetry operators→ “Crystal  Symmetry = Asymmetric Unit”