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Space Groups. Roya Majidi 1393. Glides: Reflection + translation Screw Axes: Rotation + translation. Screw Axis. A screw (axis) operator rotates a point/object and then moves it a fraction of the repeat distance in one go.
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Space Groups Roya Majidi 1393
Glides: Reflection + translation • Screw Axes: Rotation + translation
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)
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) (240,4/31/3) (360,6/32)… • This is equivalent to a left handed screw (LHS) of pitch 1/3 Note: these are not fraction calculations! m
The 43 axis is a RHS with a pitch of 3/4 • The set of points generated are: (0,0) (90,3/4) (180,6/42/41/2) (270,9/41/4)… • The effect of 43 axis can be thought of as a LHS with a pitch of 1/4 Note: these are not fraction calculations!
The 42axis generates the following set of points:(0,0) (90,1/2) (180,2/21) (270,3/21/2) (360,4/22) • 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!
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) →
Complete set of symmetry operators 3 numbers each 3 numbers each 3 numbers each
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”
