2. 3-D Lattices a x z c    y b shortest translation vectors each corner of the unit cell : point with equal surroundings = lattice point cell defined by shortest translation vectors = unit cell, origin can be choosen arbitrarily Coordinate axes placed parallel to unit cell axes (not necessarily Cartesian!!) Motif

3. http://www.tf.uni-kiel.de/matwis/amat/mw1_ge/ Unit cell Similar to the planar lattice, an infinite number of primitive 3-D unit cells can be defined. The choice is dictated by 1) the smallest volume and 2) the orientation of the symmetry elements relative to the unit cell axes. For the monoclinic lattices f.ex. the unit cell axes b is chosen to be parallel to the twofold axis and/or perpendicular to the mirror plane. The other two axes can be chosen freely e.g. there are different settings possible (red, brown, blue and green cells above) mirror plane b-axis

4. Bravais lattices I There are 14 symmetrically distinct 3-D lattices which are called Bravais lattices. Because the axes of the primitive unit cell for some of the lattices have odd orientation relationships to the symmetry elements, centered cells are preferred. Cubic lattices Hexagonal-trigonal latticesTetragonal lattices P: primitive I: body centered ("innenzentriert") F: face cenered ("flächenzentriert) R: rhombohedral There are two set-ups for the trigonal R (rhombohedral) lattice.

5. Bravais lattices II Orthorhombic lattices Monoclinic lattices Triclinic lattices P: primitive I: body centered ("innenzentriert") F: face cenered ("flächenzentriert) C: C-centered P C F

6. Glide mirror planes Translations: symmetry element => combination with mirror planes = new symmetry element:glide mirror planes Translation + Mirror operation = Glide mirror plane g t mirror plane with vertical glide horizontal glide diagonal glide mirror plane Translation Mirror operation g

7. Glide mirror planes in Cu(OH) 2

8. Glide mirror planes II C d n b a b a A glide mirror plane is characterized by: - the orientation of the mirror - the orientation and the size of the translation C dn c a b a C b c n d

9. m c a c/2 a/2 c/2 b/2 b/2+c/2 b/4+c/4 mirror planes symbols in projection n or d b a m a-axis b-axis c-axis Glide mirror planes III Symbols for glide mirror planes: mirror planes symbols in projection

10. Screw axes I Combination of rotation axes with translational symmetry = new symmetry element: screw axes Two-fold screw axis 2121 Foldidness R of rotation axis translation fraction f: translation component = f/R Translation t/2 Rotation 180° t 2121 t 2

11. Screw axes II Three-fold screw axes 3131 33232 t 1/3t 2/3t Rotation 120 3232 Alternative interpretation of a 3 2 -axis: left handed 3 1 axis

12. Screw axes in quartz I 3131 3232 Tetrahedra in quartz form helical spirals. In right-handed quartz around a 3 1 axis, in left handed quartz around a 3 2. Lateral combination of spirals lead to the 3-D tetrahedral framework of quartz. http://www.uwsp.edu/geo/projects/geoweb/participants/dutch/

13. Screw axes in quartz II http://syninfo.com/Crystal The two enantiomorphs of quartz can also be distinguished macroscopically, provided that the trigonal trapezohedron faces x and the trigonal pyramid faces s are developped. In left- handed quartz they are in the lower left of the principal rhombohedron face r, whereas in the right handed quartz they are in the opposite lower corner.

14. Screw axes in garnet Four-fold screw axes http://www.uwsp.edu/geo/projects/geoweb/participants/dutch/ 4 1 screw axis Silicon tetrahedra in the garnet structure. They are related by 4 1 screw axes, the red tetrahedron is at level 0, the dark orange at 1/4, the light orange at 2/4 and the the yellow one at 3/4.

15. 1 1-fold rotationnone 1 1-fold rotoinversionnone 2 2-fold rotationnone 2 1 2-fold screw1/2 c, b or a 33-fold rotationnone 3 1 3-fold screw (left handed)1/3 c 3 2 3-fold screw (right handed)1/3 c 3 3-fold rotoinversionnone 4 4-fold rotationnone 4 1 4-fold screw (left handed)1/4 c 4 2 4-fold screw (neutral)2/4 c 4 3 4-fold screw(right handed)3/4 c 4 4-fold rotoinversion none 6 6-fold rotationnone 6 1 6-fold screw(left handed)1/6 c 6 2 6-fold screw(left handed)2/6 c 6 3 6-fold screw (neutral)3/6 c 6 4 6-fold screw (right handed)4/6 c 6 5 6-fold screw (right handed)5/6 c 6 6-rotoinversionnone SymbolSymmetry Axis Graphic Symbol Type of translation Rotation symmetry

16. Symmetry SymbolGraphic symbolOrientation of Glide vector element normal parallelmirror plane to projection to projection plane plane Mirror planemnone Axial glidea  (010) or  (001)a/2 along [100] planes b  (100) or  (001)b/2 along [010] c  (100) or  (010) c/2 Diagonal n  (100) or  (010) or  (001)(a+c)/2,(b+c)/2, (a+b)/2 glide planes  (110) or  (011) or  (101)(a+b+c)/2, (-a+b+c)/2, (a-b+c)/2, (a+b-c)/2 Diamondd  (100) or  (010) or  (001)(a±c)/4, (b±c)/4, (a±b)/4 glide planes  (110) or  (011) or  (101)(a+b±c)/4, (±a+b+c)/4, (a±b+c)/4, (-a+b±c)/4, (±a-b+c)/4,(a±b-c)/4, Mirror symmetry

17. Space groups Combinations of 14 Bravais Lattices + symmetry inherent to the 32 point groups + symmetry operations contanining translations (screw axes and glide mirror planes) = 230 possible symmetry combinations called space groups. All informations on space groups can be found in the „International Tables for Crystallogarphy“.

18. International tables I Header: 1) short Hermann-Maugin symbol 2) Space group nummer 3) Schoenflies symbol 4) full Hermann-Maugin symbol 5) Point group symbol to which the space group belongs 6) Name of the crystal system. 7) Patterson symmetry (useful for crystal structure Space group diagrams: triclinic, monoclinic and orthorhombic: 4 diagrams: a) 3 orthogonal projections (perpendicular to the 3 axes) with all symmetry elements b) one projection with arrangement of symmetrically equivalent points in general position. All other space groups a) one projection down the c-axis with all symmetry elements b) like above. Origin statments: 1) Site symmetry of the origin 2) All symmetry elements crossing the origin.

19. 13 24 56 7 projection down c-axis equivalent points

20. Assymmetric unit: The limits of that part of the unit cell is given, which contains all symmetrically unrelated atoms. All atom positions are created by applying all symmetry elements on the content of the assymmetric unit. Symmetry operations, Generators selected: Short hand description of the symmetry operations in matrix form (important for computational applications). Positions:A) General positions: These positions are only left invariant by the identity operation. All other symmetry operations applied to a general positon creates a new, symmetrically related position. B) Special positions: These positions are part of one or more symmetry elements. Applying this/these symmetry element(s) will not create a new position, e.g. an atom lying on a mirror plane will be reflected onto itself. 1) Multiplicity: Number of symmetrically equivalent positions within a unit cell, when all symmetry element are applied on the first position given. 2) Wyckoff letter: Coding scheme for positions starting with letter a for the general positions. The higher the symmetry of the equivalent positions the higher the letter. International tables II

22. 3) Site symmetry: Symmetry of the arrangement of atoms belonging to one site 4) Positions: All positions created by applying the symmetry elements to the 1st position. 5) Reflection conditions: Indicate the systematic absences of reflections in diffraction experiments due to the presence of a centered cell or glide planes or screw axes. Symmetry of special projections: self explaining. Subgroups, supergroups: Informations indicating what space group is created by taking or adding symmetry elements to the present space group. These informations are very important in relationship with phase transformations International tables III

24. Crystal system symmetry elements indicated Examples in the symbol full symbol short symbol Triclinic -P1 1 1P1 Monoclinic (se)  orC1 2/m 1C2/m  or  b in 2. position Orthorhombicse  or  to a,b and c,P 2 1 /n 2 1 /m 2 1 /aPnma in that order Tetragonal4-fold axis in 1. positionP 4/m 2/m 2/mP4/mmm se  or  a 1, a 2 in 2. position se  or  opposite edge diagonals (  4) in 3. position. Hexagonal3, 6-fold axis in 1. positionP 6/m 2/c 2/cP6/ccc Trigonalse  or  a 1, a 2 in 2. positionP 3 1 2P32 se  or  opposite edge diagonals in 3. position. Cubicse  or  edge diagonals in 1. position. P231P23 3-fold axis in second position se  or  opposite edge diagonals in 3. position. Hermann-Maugin notation

25. Hermann-Maugin notation II Examples

26. The structure of halite I Atom x y z Na 0 0 0 Cl 0 0.5 0 Stoichiometry.NaClUnit cell dimensions:a: 5.64 Å Crystal class:cubic Point group:4/m32/m Space group: F4/m32/m Lattice: F-centered Atom coordinates Density: 2.165g/cm 3 Z (Formula units in the unit cell): r: density (g /cm 3 ) N: Avogadro‘s number 6.02338 x 10 23 V: Volume of the unit cell (cm 3 ) M: Molecular weigth (g) Z NaCl : 4 Z =  NV M

27. 1/4 Na: on special position 0,0,0 Wyckoff letter a Cl:: on special position 1/2,1/2,1/2) Wyckoff letter: b The structure of halite II assymmetric unit: 0 < x < 1/2, 0 < y < 1/4, 0 < z < 1/4, full unit cell = 32 asymmetric units applying all symmetry elements on the atoms in the asymmetric unit creates all other sites in the unit cell. On assymetric units contains 1/8 Na and 1/8 Cl. A full unit cell has therefore 4Na atoms and 4 Cl atoms e.g. Z = 4. origin 0 0 1/2 0 Only a small fraction of the symmetry elements are shown

28. 0,1 The structure of halite III 0,1 1/2 0 0,1 - The clorine atoms along the edges belong to 1/4th to the unit cell e.g. 12 x 1/4 = 3; whereas the clorine in the center is entirely in the unit cell. Total Cl = 3 +1 - The sodium atoms at the corner count for 1/8th e.g. 8 x 1/8 = 1, the atoms in the center of the faces count for 1/2 e.g. 6 x 1/2 =3. Total Na: 1 + 3 = 4

29. Stoichiometry.Al 2 SiO 5 Unit cell dimensions:a: 7.79800 Å Crystal class:orthorhombicb: 7.90310 Å Point group:2/m2/m2/mc: 5.55660 Å Space group: P 2 1 /n 2 1 /n 2/m Lattice: primitive The structure of andalusite I Atom x y z Al 0.00.0 0.2419 Al 0.3705 0.1391 0.5 Si 0.246 0.252 0.0 O 0.42330.36290.5 O 0.42460.36290.0 O 0.1030.40030.0 O 0.23050.13390.2394 Atom coordinates

30. The structure of andalusite II 1/4 Tetrahedral cations: on general position x,y,z Wyckoff letter: f, site symmetry: 2 Oxygen: on general positions x,y,z Wyckoff letter: f, site symmetry: 2 1/4 0.5 assymmetric unit: 0 < x < 1/2, 0 < y < 1/2, 0 < z < 1/2 applying all symmetry elements on to the atoms in the assymmetric unit creates all other sites in the unit cell. M1-cations: on special position 0,0,z (lies on a 2- fold axis) Wyckoff letter: e, site symmetry: 2 1/4 1/2 1/4 a b 0.5 0.0 0.239 0.24

31. The structure of andalusite III 1/4 0.5 1/2 a b 0.5 0.24 0.0 0.5 0.0 0.74 0.24 0.74 0.24 0.74 0.24 0.74 0.24 0.00.50.0 Cation content of the diopside unit cell Structure representation created by CrystalMaker

32. Symmetry The structure of diopside I Atom x y z Si0.2862 0.0933 0.2293 Mg0.5000 0.4082 0.2500 Ca0.0000 0.3015 0.2500 O10.1156 0.0873 0.1422 O20.3611 0.2500 0.3180 O30.1497 0.0176 0.9953 Stoichiometry.CaMgSi 2 O 6 Unit cell dimensions:a: 9.746 Å Crystal class:monoclinicb: 8.899 Å Point group:2/mc: 5.251 Å  : 105.63° Space group: C2/c Lattice: C-centered Atom coordinates

33. Symmetry The structure of diopside II 1/4 Tetrahedral cations: on general position x,y,z Wyckoff letter: f, site symmetry: 2 Oxygen: on general positions x,y,z Wyckoff letter: f, site symmetry: 2 1/4 0.23 0.14 0.31 0.99 assymmetric unit: 0 < x < 1/2, 0 < y < 1/2, 0 < z < 1/2 applying all symmetry elements on to the atoms in the assymmetric unit creates all other sites in the unit cell. origin: for C2/c is placed in the inversion center located on c glide plane 1/4 3/4 M1-cations: on special position 0,y,1/4 on a 2- fold axis) Wyckoff letter: e, site symmetry: 2 M2-cations: on special position 0,y,1/4 on a 2- fold axis) Wyckoff letter: e, site symmetry: 2

34. 0.23 Symmetry The structure of diopside III 0.77 0.23 0.27 M1-cations Tetrahedral cations M2-cations Cations in the assymetric unit 1/4 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 0.23 0.27 0.77

35. Symmetry The structure of diopside IV Perspective view along the c-axis of the diopside structure Cation content of the diopside unit cell 1/4