Axis of Rotation Crystal Structure

Axis of Rotation Crystal Structure

2 2 1

3 3 3 1 2

120 rotation 1/3 unit cell translation

Higher order screw axes The combination of a rotation axis and a translation parallel to the axis produces a screw axis. The direction of such an axis is usually along a unit cell edge, and the translation must be a subintegral fraction of the unit translation in that direction. Screw axes are designated by an integer n and a subscript m where n = 1,2,3,4 or 6 is the fold of the axis and m is an integer less than n. Thus 31 designates a 3-fold screw axis with a translation between successive points of 1/3 (m/n) of a unit translation. Point 2 is generated from point 1 by rotating +360/3 and advancing +1/3 of the unit translation. Point 3 is generated from point 2 by another rotation of +360/3 and an advance of 1/3. In a similar way, 32 indicates a 3-fold screw axis with a translation of 2/3 of the unit translation. Figure shows the relationships between successive points. Point 2 is generated from point 1 by rotating +360  /3 and advancing 2/3. Point 3’ arises from an additional rotation of +360/3 and an additional translation of +2/3. It is characteristic of an n-fold screw axis that the position of the nth point laid down differs from the initial point by an integral number of unit translations; that is, the positions of these points within their respective cells are identical. Screw 31 Screw 32 41-3, 61-5

Three-fold rotation 360o/3 6 step 1 6 step 3 6 step 2

d 9 n-fold Rotation a Z t 1-fold 2-fold 3-fold 4-fold 6-fold identity

6 6 Inversion In 2D, inversion = 2-fold rotation Role play: difference between In 2D, inversion = 2-fold rotation In 3D, inversion ≠ 2-fold rotation

Rotation + Inversion 3

Rotation + Inversion 3 1

Rotation + Inversion 3

Rotation + Inversion 3 1 2

Rotation + Inversion 3

Rotation + Inversion 3

Rotation + Inversion 3 1 2 3

Rotation + Inversion 3 1 2 3 4

Rotation + Inversion 3 1 2 5

Rotation + Inversion 3 3 5 1 4 2 6

Crystal systems: length/angle relations Klein Fig. 5.27, pg. 196

Crystal System - Symmetry Characteristics

Crystal system - Symmetry characteristics Klein Fig. 5.25, pg. 193

Unit cell and asymmetric units Unique atoms Symmetry-related atoms We must first find out the symmetry

Apply correct symmetry Too low symmetry wrong symmetry correct symmetry

The intensities carry the information about the atomic structure Two different structures can have the same unit cell dimensions. The reciprocal unit cells are the same but the intensities of the diffraction spots differ.

Symmetry is best seen in reciprocal space A square unit cell is necessary but not sufficient for the crystal having 4-fold symmetry. If the atoms in the unit cell are not arranged with 4-fold symmetry (a), the diffraction pattern will not have 4-fold symmetry (b). A crystal with 4-fold symmetry (c) gives rise to a diffraction pattern with 4-fold symmetry (d).

An example of symmetry correction PDB code: 1yup spacegroup (PDB): P1 8 molecules per a.u. spacegroup (true): P21 4 molecules per a.u. Pseudo-symmetry spacegroup: C2 2 molecules per a.u. (because of pseudo-translation)

Monoclinic structures related to 1yup Positions of molecules Crystallographic axes NCS axes Spacegroup and its relation to the structure 1yup C2 Pseudo-symmetry spacegroup P2 False spacegroup P21 True spacegroup

Structure solution and symmetry validation Data processing ( 2/m ) Molecular replacement ( P2 ) Refinement R-free ≈ 0.37 Data processing ( -1 ) Molecular replacement ( P1 ) Refinement R / R-free = 0.24 / 0.31 PDB: 1yup ( P1 ) PDB: 1yup Zanuda ( P21 ) R-free = 0.33