1. Crystallography and Diffraction Theory and Modern Methods of Analysis Lectures 1-2 Introduction to Crystal Symmetry Dr. I. Abrahams Queen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

2. States of Matter Everything is made of Matter Matter GasesLiquidsSolids Fairly Random distribution of atoms/molecules Extensive motion High degree of order and symmetry Limited (thermal) motion Useful to observe structures while they are stationary using diffraction techniques Lectures co-financed by the European Union in scope of the European Social Fund

3. The Crystalline State Solids Non-Crystalline (amorphous) Crystalline Local order only No repeating unit Local symmetry only Isotropic properties Long range order Repeating unit High symmetry Anisotropic properties Solids may be categorised in many ways. Two broad classifications are as crystalline and non-crystalline solids. Note: nanocrystalline solids might be considered as non-crystalline when the coherence length is below ca. 10 nm

4. Lectures co-financed by the European Union in scope of the European Social Fund The Crystal Lattice Crystalline materials show a regular repeating structure. This can be represented by an imaginary grid known as the lattice. 2D - lattice Similar in 3-D The points of intersection are known as lattice points Lattice points do not necessarily correspond to atom positions

5. Lectures co-financed by the European Union in scope of the European Social Fund The Unit Cell The unit cell is the basic repeating unit of a crystal lattice. Consider some 2D arrays of atoms. Unit cells are indicated. The examples above show primitive and centred cells. The general case for a 2D lattice is a parallelogram defined by 2 axial lengths and 1 angle. for a 3D lattice is a parallelepiped defined by 3 axial lengths and 3 angles

6. Lectures co-financed by the European Union in scope of the European Social Fund Parallelepiped The unit cell is generally chosen as the smallest repeating unit with the highest symmetry. The unit cell, when repeated in 3D, must cover all the space in the crystal lattice. Different crystal structures have different unit cells. Unit cells are defined by six parameters in 3D. a, b, c are the unit cell edges and ,  and  are the inter-axial angles. (0 is the origin and its position is arbitrary).

7. Lectures co-financed by the European Union in scope of the European Social Fund Fractional Coordinates The location of the origin is arbitrary, but is usually chosen to correspond to a point of symmetry. It need not be an atom position. Atoms positions can be defined with respect to the unit cell using fractional coordinates x, y, z x = X/a where X is the distance parallel to the a-axis y = Y/b where Y is the distance parallel to the b-axis z = Z/c where Z is the distance parallel to the c-axis

8. Lectures co-financed by the European Union in scope of the European Social Fund Crystal Systems There are seven crystal systems. These can be distinguished by the different unit cell shapes and their minimum intrinsic symmetry. Crystal systemUnit cell shape Minimum symmetry Triclinica  b  c;  90  None Monoclinic 1 two fold axis or (standard setting)a  b  c;  =  =90 ,  90  mirror plane Orthorhombic a  b  c;  =  =  =90  3 two fold axes or mirror planes Tetragonal a=b  c;  =  =  =90  1 four fold axis Trigonal 1 three fold axis (rhombohedral setting) a=b=c;  =  =  90  (hexagonal setting) a=b  c;  =  =90  =120  Hexagonal a=b  c;  =  =90  =120  1 six fold axis Cubic a=b=c;  =  =  =90  4 three fold axes Note: The symbol  used here refers to not necessarily equal to. In some cases there is accidental equivalence, but the minimum symmetry is not present.

9. Lectures co-financed by the European Union in scope of the European Social Fund Crystal Symmetry Atoms are related to other atoms within the unit cell by symmetry. Symmetry Point symmetry At least one point is invariant Translational symmetry A point symmetry operation followed by a translation Atoms related by symmetry are termed: equivalent A symmetry operation is defined as a movement of a system which when completed brings every point to a physically indistinguishable position from the original orientation. A symmetry element is a geometrical property that can generate the operation. Thus crystal structures possess symmetry elements that are operated on. The symmetry possessed by a molecule might not be the same as that of its crystal structure.

10. Lectures co-financed by the European Union in scope of the European Social Fund Crystallographic Point Symmetry 1. Rotational axes (operation = rotation). Symbol n (n is an integer) This is a rotation of 2  /n (where n is the order of rotation). e.g. a 2-fold axis along b in a primitive cell. Rotation about the b- axis Equivalent positions x,y,z ; -x, y, -z View down b-axis

11. Lectures co-financed by the European Union in scope of the European Social Fund 2. Mirror Planes (operation = reflection). Symbol m Reflection in an imaginary mirror plane perpendicular to a particular axis. e.g. Mirror plane perpendicular to b-axis 3. Inversion Axes (operation = inversion). Symbol Rotation by 2  /n then invert. A 1-fold inversion axis is equivalent to a centre of symmetry. e.g.

12. Lectures co-financed by the European Union in scope of the European Social Fund Translational Symmetry 1. Screw Axes (operation is a rotation followed by a translation). Symbol n m Rotation by 2  /n followed by translation of m/n e.g. 2-screw axis 2 1 parallel to b-axis A 2 fold rotation around b followed by a translation of half a unit cell 2. Glide Planes (operation is a reflection in a mirror plane followed by a translation). Symbols a, b, c, d, n e.g. c-glide A reflection perpendicular to b followed by a translation parallel to c.

13. Lectures co-financed by the European Union in scope of the European Social Fund The glide plane symbol indicates the glide direction. a - reflection perpendicular to b or c followed by translation parallel to a-axis by a/2 b - reflection perpendicular to a or c followed by translation parallel to b-axis by b/2 c - reflection perpendicular to a or b followed by translation parallel to c-axis by c/2 n - reflection perpendicular to a, b or c followed by translation of ½ cell parallel to a face diagonal e.g. n-glide perpendicular to b = reflection perpendicular to b followed by translation of (a + c)/2 d - reflection perpendicular to a, b or c followed by translation of ¼ cell parallel to a face diagonal e.g. d-glide perpendicular to b = reflection perpendicular to b followed by translation of (a + c)/4

14. Lectures co-financed by the European Union in scope of the European Social Fund Centring In centred cells there are one or more extra lattice points which act as additional points of reference for symmetry operations (extra points of origin). These extra lattice points are either at the centre of the unit cell or at the centre of particular faces.The type of centring is designated by a letter prefix. P Primitive (no centring)0,0,0 A A-face centred (face perpendicular to a) 0,0,0; 0,½,½ B B-face centred (face perpendicular to b) 0,0,0; ½,0,½ C C-face centred (face perpendicular to c)0,0,0; ½, ½,0 F Face centred (all faces)0,0,0; 0,½,½; ½,0,½; ½,½,0 I Body centred0,0,0; ½, ½,½ R- Rhombohedral centred Centring Lattice points

15. Lectures co-financed by the European Union in scope of the European Social Fund Bravias Lattices The seven crystal systems combined with the different types of centring result in 14 unique crystal lattices known as the Bravais Lattices.

16. Lectures co-financed by the European Union in scope of the European Social Fund Note that some lattices might appear to be missing, but these can be generated from other lattices. e.g. Face centred tetragonal can be generated from the body centred tetragonal lattice.

17. Lectures co-financed by the European Union in scope of the European Social Fund Point Groups Initial studies on the external morphology of crystals revealed considerable symmetry (point symmetry). This symmetry arises from combination of the point symmetry operations viz. rotations, reflections and inversions. 32 different combinations of point symmetry elements are possible in 3D. These are known as the 32 Point Groups. There are no translational symmetry elements in point groups. e.g. a crystal with a 4-fold axis, a centre of symmetry and mutually perpendicular mirror planes. Point group = 4/mmm

18. Lectures co-financed by the European Union in scope of the European Social Fund Space Groups Combination of the 32 point groups with the 14 Bravais lattices gives 230 unique combinations known as space groups. These space groups show translational symmetry (screw axes and glide planes) not seen in the external morphology of crystals. Every crystal structure belongs to one and only one space group. A given crystal structure must possess all of the symmetry elements of that space group. Space groups are listed in International Tables for Crystallography Volume A, which represents the main reference book for crystallographers.

19. Lectures co-financed by the European Union in scope of the European Social Fund Equivalent Positions The symmetry elements associated with a space group mean that parts of a crystal structure are related by symmetry to other parts. Consider an octahedral ion sitting on a two-fold axis. The two-fold axis means that atoms on the left hand side have equivalent atoms on the right hand side, i.e. for these atoms there are two equivalent positions. The atom sitting on the 2-fold axis does not have an equivalent position.

20. Lectures co-financed by the European Union in scope of the European Social Fund Any atom that does not sit on a symmetry element is said to be in a General Position. Any atom on the 2-fold axis does not have an equivalent. Atoms that sit on symmetry elements are said to be in Special Positions. The symmetry elements of each space group generate one set of general positions and none of these sit on a symmetry element. In most space groups there are also a number of sets of special positions each sitting on one or more symmetry elements. In real structures atoms are located in general and/or special positions.

21. Lectures co-financed by the European Union in scope of the European Social Fund Space Group Symbols Complete specification of a space group requires a knowledge of the minimum number of symmetry elements of the space group and their relative orientation. Each space group has a short form symbol which summarises the symmetry information. e.g. Pnma is the short form of P2 1 /n,2 1 /m,2 1 /a Primitive cell P n m a n-glide perpendicular to 2 1 axis along a mirror plane perpendicular to 2 1 axis along b a-glide perpendicular to 2 1 axis along c

22. Lectures co-financed by the European Union in scope of the European Social Fund C2/m is short for C1,2/m,1 C 2 / m C-face centred cell 2 fold axis parallel to b mirror plane perpendicular to b centre of symmetry F 2 2 2 Face centred cell Monoclinic Cubic three mutually perpendicular 2- fold axes

23. Lectures co-financed by the European Union in scope of the European Social Fund

24. Reference: International Tables for Crystallography, Vol. A.

25. Lectures co-financed by the European Union in scope of the European Social Fund Reference: International Tables for Crystallography, Vol. A.