1. Crystal Crystal pattern (crystal structure) Molecule or coordination polyhedron Unit cell Lattice nodes Crystal structure vs. crystal lattice

3. b a O

5. apap bcbc O bpbp acac primitive cell conventional cell

7. b c= p O a c= p conventional cell (primitive)

8. conventional cell bcbc acac

9. b a P cell (conventional) H cell

10. mp b a op b a tp a b bcbc acac bpbp apap oc hp a b Crystal family: monoclinic,orthorhombic, tetragonal, hexagonal Lattice type*: primitve, centred *Lattice whose conventional cell is primitive or centred Summary in two dimensions

11. Crystal families and lattice types in three dimensions ● Symmetry element: inversion centre ● No symmetry directions ● Any centred cell can be transformed into a primitive cell triclinic (anorthic) crystal family aP monoclinic crystal family ● One symmetry direction (usually taken as b axis) ● The conventional cell must have two right angles ● Two types of cells, obey this condition mP mS

12. Lattice types mB et mP are equivalent a A = - c C c A = a C mA mC cBcB bBbB aBaB Lattice types mC, mA, mI and mF are equivalent (mS) mC mF c F = - a C +2c C a B = (a P -c P )/2 ; c B = (a P +c P )/2 mB mP mA mI c I = c A -a A cIcI aIaI bI bAbI bA bP bBbP bB cPcP aPaP cFcF bF bCbF bC aF aCaF aC cCcC bCbC aCaC aAaA cAcA bA bCbA bC

13. Ambiguity of definition the conventional cell for the mS lattice- type International Tables: the conventional cell is C-centred Some authors: the conventional cell the cell with the 90º    120º (either C or I-centred) One degree of freedom in choosing the axes in the (a, c) plane

14. oPoIoSoF orthorhombic crystal family ● Three symmetry directions (a, b, c axes) ● The conventional cell must have three right angles ● Four types of cells obey this condition ● The face-centred cell can be described as A, B or C depending on how the axes are taken (collective symbol: S)

15. tPtI tetragonal crystal family ● Five symmetry directions (c, a&b, the two face diagonals) ● The conventional cell must have three right angles and two equivalent edges ● Two types of cells obey this condition, tP (equivalent to tC) and tI (equivalent to tF) tP tC ½ tF tI ½ ½ ½

16. cPcIcF cubic crystal family ● Thirteen symmetry directions (3 axes; 4 body diagonals; 6 face diagonals) ● The conventional cell must have three right angles and three equivalent edges ● Three types of cells obey this condition, cP, cI and cF

17. hexagonal crystal family A1A1 A2A2 C a1a1 a2a2 a3a3 A1A1 A2A2 a1a1 a2a2 a3a3 Symmetry directionsSymmetry directions A 1, A 2, C hexagonal axes parallel to symmetry directions a 1, a 2, a 3 rhombohedral axes NOT parallel to symmetry directions Two types of lattices in the hexagonal crystal family: hP and hR

18. [110] R [101] R [011] R [101] R [011] R [110] R a1a1 a2a2 a3a3 A3A3 [100] H [010] H [110] H [010] H [100] H A1A1 A2A2 [210] H [120] H [110] H [210] H [120] H z = 0 z = 1/3 z = 2/3 Symmetry difference between hP and hR lattice types 6/mmm 3m3m

19. Lattice systems: classification on the basis of the lattice point symmetry aP mP mS tP tI oP oI oS oF hPhR cP cI cF 1 triclinic 2/m monoclinic 4/m 2/m 2/m (4/mmm) tetragonal 2/m 2/m 2/m (mmm) orthorhombic 6/m 2/m 2/m (6/mmm) hexagonal rhombohedral cubic 3 2/m (3m) 4/m 3 2/m (m3m)

20. Crystal systems: classification on the basis of the morphological (macroscopic) / physical symmetry A1A1 A2A2 3A23A2 A4A4 A3A3 A6A6 4A34A3 triclinic monoclinic tetragonal orthorhombic hexagonal trigonal cubic

21. Crystal families, crystal systems, lattice systems and lattice types in the three-dimensional space 6 crystal families 7 crystal systems (morphological symmetry) 7 lattice systems (lattice symmetry) 14 Bravais lattice-types (**) a = anortic (triclinic, asymmetric, tetartoprismatic...) m = monoclinic (clinorhombic, monosymmetric, binary hemiprismatic, monoclinoedric,...) o = orthorhombic (rhombic, trimetric, terbinary, prismatic, anisometric,...) t = tetragonal (quadratic, dimétric, monodimetric, quaternary...) h = hexagonal (senary, monotrimetric...) c = cubic (isometric, monometric, triquaternary, regular, tesseral, tessural...) trigonal (ternary...) (***) triclinic monoclinic orthorhombic rhombohedral hexagonal cubic aP mP ( mB ) mS ( mC, mA, mI, mF ) oP oS ( oC, oA, oB ) oI oF tP ( tC ) tI ( tF ) hR hP cP cI cF (*) Synonyms in parenthesis. (**) S = one pair of centred faces. In parenthesis the equivalent lattice types (different but equivalent choice of the axial setting). (***) Crystals of the trigonal crystal system may have a rhombohedral or hexagonal lattice triclinic monoclinic orthorhombic tetragona l hexagonal cubic tetragonal no restriction on a ; b ; c, , ,  no restriction on a ; b ; c,   =  = 90º no restriction on a ; b ; c  =  =  = 90º a = b ;  =  =  = 90º no restriction on c a = b = c  =  =  = 90º a = b ;  =  = 90º,  = 120º no restriction on c conventional cell

22. Symmetry directions in three-dimensional space (directions in the same box are equivalent under the lattice symmetry) Lattice system First symmetry direction triclinic monoclinic orthorhombic tetragona l rhombohedral hexagonal cubic [010] [100][010][001] [100] [010]   00  [001] [111] [001] [001] [100] [010]   001  rhombohedra l axes hexagona l axes [001] a ; b ; c ;  ;  ;  a ; b ; c;  =  = 90º a ; b ; c  =  =  = 90º a = b ; c  =  =  = 90º a = b = c  =  =  = 90º a = b ; c  =  = 90º ;  = 120º a = b ; c  =  = 90º ;  = 120º a = b = c  =  =  Parameter s Second symmetry direction Third symmetry direction [110] [011] [101]   0  [110] [011] [101]   0  [100] [010] [110]   00  [100] [010] [110]   00  [111] [111]   1  [110] [110] [011] [011] [101] [101]   0  [110] [120] [210]   0  [110] [110]   0 