Periodicity in 3 dim smallest repeated unit cell

Periodicity in 3 -dim. – smallest repeated unit cell Symmetry in two-dimension 2 D unit cell

Symmetry in two-dimension

Symmetry elements

Periodicity in 3 -dimetions ---Smallest repeated Unit --- Unit cell Symmetry elements, symbols, matrix representation： Basic symmetry elements ☆ proper rotation Cn → n; symbol in ‘point group → space group’ e. g. 4 z matrix representation To suit inside a repeated unit in the space 1, 2, 3, 4, 5, 6 fold

☆ mirror planes v, h, d --- m (a, b, c, d, n) ☆ center of symmetry ☆ improper rot. s→ ☆ translation along edges of the cell by fractions of the edge length

☆ translation along edges of the cell by fractions of the edge length ☆ screw axis rot. + tr. ☆ glide planes m + tr along a, b, c, diagonal, a, b, c, n, d + tr along c

Derived Symmetry within the lattice Unit cell or crystal lattice formed by 3 -non-planar vectors Limitation of symmetry by periodicity n t t

Lattice Centering --- Pure translational rot: + translation

Lattice Centering – pure translational 1 2 3 4 P I, A, B, C R 4 F A F P I B C R

Unit cell classifications Lattice centering Crystal system Min. sym. P Triclinic 1 a b c , 90 P, I Monoclinic 2 a b c , 90 P, I, F, B Orthorhombic 222 a b c , 90 P, I Tetragonal 4 a b c , 90 P Hexagonal 6 a b c , 90 , 120 P Trigonal 3 a b c , 90 , 120 , V R Rhombohedral 3 P, I, F Cubic 23 Max. sym. 3 m Cell parameters a’ b’ c’ , ’ ’ ’ 90 , V’ = 1/3 V a b c , 90

Other of Positions in the Symbols of the Three-dimensional Point Groups as applied to Lattices System and Point Group Poles of directions Position in Point-group Symbol Primary Secondary Tertiary Triclinic Only one symbol which denotes all directions in the crystal. Monoclinic Orthorhombic Tetragonal Trigonal and hexagonal Cubic The symbol gives the nature of the unique diad axis (rotation and/or inversion). 1 st setting: z-axis unique (001) 2 nd setting: y-axis unique (010) Diad (rotation and/or inversion) along xaxis (100) Diad (rotation and/or inversion) along yaxis (010) Diad (rotation and/or inversion) along z-axis (001) Tetrad (rotation and/or inversion) along z-axis (001) Diads (rotation and/or inversion) along x- and y-axes (100) or (010) Diads (rotation and/or inversion) along [110] and [1 0] axes (110) (1 0) Triad or hexad (rotation and/or inversion) along xaxis (001) Diads (rotation and/or inversion) along x-, y- and uaxes (100) ____ Diads (rotation and/or inversion) normal to x-, y -, u-axes in the plane(001); (100) …… Diads or tetrads (rotation and/or inversion) along (100) axes (100) Triads (rotation and/or inversion) along (111) axes (111) Diads (rotation and/or inversion) along (110) axes (110) Stereographic for primary position ditto representation Secondary Tertiary ditto z y x Triclinic z y x x Monoclinicnd 1 st setting 2 setting x Orthorhombic z y x Tetragonal u z y x Trigonal and hexagonal z x Cubic y

Symmetry Operations and Space Groups The 14 Bravais lattices a b a c b a c b b c a c c b a a c b

續上頁 c 1 c a a 1 b b 1 or P 3 m 1 c c b b a 120 c c b b a a c a

Laue symmetry unique part of sphere Triclinic Monoclinic Orthorhombic 1/2 2/m 1/4 (m m m) 1/8 Tetragonal 1/8 1/16 Hexagonal 1/12 1/24 Trigonal 1/12 1/6 Cubic 1/24 1/48

Space Group Definition 1. a i aj = ak where ak must be an element in the group 2. must have an identity element, I, so that ai I = ai 3. The inverse of every element must also be an element in the group 4. associative law (a i aj ) ak = ai (a j ak)

1, 2, 3, 4, 6 inverse rot. 5 (Sn) rot. 5 Cn : 32 Point Gourps m 3 Cnh: rot. + m 1 Cnv rot. + m 3 m 6 Dn 3 rot. 222 , 32(2) , 422 , 622 , 23 , 432

mm 2 Dnh 4 mm 6 mm rot. +m + m 9 3

m

(7) (32) (230)

( 7) Crystal system (32) point group (230) space groups

Space group P 21/c basic sym elements origin shift rot. tr.

after origin shift to (0 0 0) after origin shift of (0, ¼, ¼)

P 21/c 2/m P 121/c 1 UNIQUE AXIS b, CELL CHOICE 1 Monoclinic Patterson symmetry P 12/m 1

P 21/c

Space group Pnc 2 Fig Completed worksheet

basic sym = derived sym n

P 6 mm Hexagonal Patterson symmetry P 6 mm

P 6 mm

P C PT R ( Trigonal Rhombohedral Cell ) [ det ] 2 [ det ] 3 R (0, 0, 0) ; (2/3, 1/3) ; (1/3, 2/3) F I [ det ] 3 [ det ] 2 R (0, 0, 0) ; (1/3, 2/3, 1/3) ; (2/3, 1/3, 2/3)

P 21/c UNIQUE AXIS b, DIFFERENT CELL CHOICE 1 P 121/c 1 UNIQUE AXIS b, CELL CHOICE 1 2/m Monoclinic

P 121/n 1 UNIQUE AXIS b, CELL CHOICE 2

P 121/a 1 UNIQUE AXIS b, CELL CHOICE 3

direct space Inverse transpose reciprocal space Inverse transpose

Cell Transformation Cell 1 (a, b, c) (h. k. l) Cell 2 (x, y, z) ; (a*, b*, c*) ; (u, v, w) u x, v y, w z where u, v, w integer

Transformation between c 1 a 2 a, b, c c 2 reverse h, k, l transpose a*, b*, c* x, y, z u, v, w reverse

I P Trigonal T F P F S rhombohedral cell trigonal cell obverse (positive) reverse (negative) I

Trigonal lattices As for the hexagonal cell, in the conventional trigonal cell the threefold axis is chosen parallel to c, with a b, unrestricted c, 90 , and 120. Centred cells are easily amenable to the conventional P trigonal cell. Because of the presence of a treefold axis some lattices can exist which may be described via a P cell of rhombohedral shape, with unit vectors a. R, b. R, c. R such that a. R b. R c. R, R R R, and the three fold axis along the a. R b. R c. R direction. Such lattices may also be described by three hexagonal cells with basis vectors ahex, bhex, chex defined according to ahex a. R b. R ahex b. R c. R ahex c. R a. R bhex b. R c. R or bhex c. R a. R or bhex a. R b. R chex a. R b. R c. R These hexagonal cells are said to be in obverse setting. Three further triple hexagonal cells, said to be in reverse setting, can be obtained by changing ahex and bhex to ahex and bhex. The hexagonal cells in obverse setting have centring point at (0, 0, 0), (2/3, 1/3), (1/3, 2/3) While for reverse setting centring points are at (0, 0, 0), (1/3, 2/3, 1/3), (2/3, 1/3, 2/3) It is worth nothing that a rhombohedral description of a hexagonal P lattice is always possible. Six triple rhombohedral cells with basis vectors a’R, b’R

Trigonal T Rhombohedral Cell Obverse (positive) Trigonal Cell Reverse (negative)

In direct space: ☆ x, y, z fractional coordinates In reciprocal space: ☆ h, k, l plane, miller indices