16p6 p6mm p31m Hexagonal (symmetry 3m, 6, 6mm) + can only located at positions:Hexagonal primitivep31mHexagonal & 6 related can only fit 3P!
1711 lattice types alreadycubic (isometric)Special case of orthorhombic (222) with a = b = cPrimitive (P)Body centered (I)Face centered (F)Base center (C)Tetragonal (I)?Cubica = b c//Tetragonal (P)
18Another way to look as cubic: Consider an orthorhombic and requesting the diagonaldirection to be 3 fold rotation symmetryP222 P23PrimitiveI222 I23Body centeredF222 F23Face centeredC222 I23
23Next, we can put the point groups to the compatible lattices, just like the cases in 2D space group. 3D Lattices (14) + 3D point groups 3D Space groupThere are also new type of symmetry shows up in 3D space group, like glide appears in 2D space (plane) group!
24The naming (Herman-Mauguin space group symbol) is the same as previously mentioned in 2D plane group!The first letter identifies the type of lattice:P: Primitive; I: Body centered; F: Face centeredC: C-centered; B: B-centered, A: A-centeredThe next three symbols denote symmetry elements in certain directions depending on the crystal system. (See next page)
25Monoclinic a b = 90o; c b = 90o. b axis is chosen to correspond to a 2-fold axis of rotational symmetry axis or to be perpendicular to a mirror symmetry plane.Convention for assigning the other axes is c < a. a c is obtuse (between 90º and 180º).Orthorhombic The standard convention is that c < a < b. Once you define the cell following the convention A, B, C centered
40All translations of R has component on c of 1/3 or 2/3! Case R!AED2/3All translations of R hascomponent on c of 1/3 or 2/3!ED1/3Screw atDesignation ofSpace groupAD’ E’33132c/32c/32/3c/32c/3c2/32c/3c4c/33132332331R3R31R32R3==Hexagonal lattice (P, R) + 3, 31, 32 P3, P31, P32, R3.
42Square lattice P with 4, 41, 42, 43.The translation of P havecomponent on c of 0 or unity!ACBBBCCA4414243c/4c/23c/4B/2 0/2 c/4/2 c/2/2 3c/4B 0 c/2 c 3c/2B4414243B221P4P41P42P43
44Homework:Discuss the cases of I4, I41, I42, I43.
45How to obtain Herman-Mauguin space group symbol by reading the diagram of symmetry elements? First, know the Graphical symbols used for symmetry elements in one, two and three dimensions!International Tables for Crystallography (2006). Vol. A, Chapter 1.4, pp. 7–11.
46Symmetry planes normal to the plane of projection Graphical symbolTranslationSymbolReflection planeNone mGlide plane1/2 along line a, b, or c1/2 normal to planeDouble glide plane1/2 along line & 1/2 normal to plane (2 glide vectors) eDiagonal glide plane1/2 along line, 1/2 normal to plane (1 glide vector) nDiamond glide plane1/4 along line & 1/4 normal to plane d
47Symmetry planes parallel to plane of projection Graphical symbolTranslationSymbolReflection planeNone mGlide plane1/2 along arrow a, b, or cDouble glide plane1/2 along either arrow eDiagonal glide plane1/2 along the arrow nDiamond glide plane1/8 or 3/8 along the arrows d3/81/8The presence of a d-glide plane automatically implies a centered lattice!
48Symmetry ElementGraphical SymbolTranslationSymbolIdentityNone 12-fold ⊥ page 22-fold in page2 sub 1 ⊥ page1/2 212 sub 1 in page3-fold 33 sub 11/3 313 sub 22/3 324-fold 44 sub 11/4 414 sub 2 424 sub 33/4 436-fold 66 sub 11/6 616 sub 2 626 sub 3 63
49Symmetry ElementGraphical SymbolTranslationSymbol6 sub 42/3 646 sub 55/6 65InversionNone 13 bar 34 bar 46 bar 6 = 3/m2-fold and inversion 2/m2 sub 1 and inversion 21/m4-fold and inversion 4/m4 sub 2 and inversion 42/m6-fold and inversion 6/m6 sub 3 and inversion 63/m
50c-glide bn-glide|| c21 c|| an2221b-glidemm c|| b a
51From the point group mmm orthorhombic For orthorhombic: primary direction is (100), secondary direction is (010), and tertiary is (001).latticefor orthorhombic:CShort symbolNo. 17 orthorhombicthat can be derived
52Principles for judging crystal system by space group Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m)Tetragonal – The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc)Hexagonal – The primary symmetry symbol will always be a 6, (-6), 61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm)Trigonal – The primary symmetry symbol will always be a 3, (-3) 31 or 32 (i.e P31m, R3, R3c, P312)
53Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc21, Pnc2)Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P21/n)Triclinic – The lattice descriptor will be followed by either a 1 or a (-1).
54What can we do with the space group information contained in the International Tables?1. Generating a Crystal Structure from its Crystallographic Description2. Determining a Crystal Structure from Symmetry & Composition
55Example: Generating a Crystal Structure Description of crystal structure of Sr2AlTaO6Space Group = Fm 3 m; a= 7.80 Å Atomic PositionsAtomxyzSr0.25Al0.0Ta0.5O
59Bond distances:Al ion is octahedrally coordinated by six OAl-O distanced = 7.80 Å − − − = 1.95 ÅTa ion is octahedrally coordinated by six OTa-O distanced = 7.80 Å − − − = 1.95 ÅSr ion is surrounded by 12 OSr-O distance: d = 2.76 Å
60Determining a Crystal Structure from Symmetry & CompositionExample:Consider the following information:Stoichiometry = SrTiO3 Space Group = Pm 3 m a = 3.90 Å Density = 5.1 g/cm3
61First step:calculate the number of formula units per unit cell :Formula Weight SrTiO3 = (16.00) = g/mol (M)Unit Cell Volume = (3.9010-8 cm)3 = 5.93 cm3 (V)(5.1 g/cm3)(5.93 cm3) : weight in aunit cell( g/mole) / (6.022 1023/mol) : weightof one molecule of SrTiO3
62 number of molecules per unit cell : 1 SrTiO3. (5.1 g/cm3)(5.93 cm3)/( g/mole/6.022 1023/mol) = 0.99 number of molecules per unit cell : 1 SrTiO3.From the space group tables (only part of it)6e4mmx00, -x00, 0x0,0-x0,00x, 00-x3d4/mmm½ 0 0, 0 ½ 0, 0 0 ½c0 ½ ½ , ½ 0 ½ , ½ ½ 01bm 3 m½ ½ ½a000
63Calculate the Ti-O bond distances: Sr: 1a or 1b; Ti: 1a or 1b Sr 1a Ti 1b or vice verseO: 3c or 3dEvaluation of 3c or 3d:Calculate the Ti-O bond distances:d 3c) = 2.76 Å (0 ½ ½) d 3d) = 1.95 Å (½ 0 0, Better)AtomxyzSr0.5TiO