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Chapter B1: Crystal Structures and Symmetries Georg Roth Institute of Crystallography RWTH Aachen University

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Chapter B1: Crystal Structures and Symmetries Georg Roth Institute of Crystallography RWTH Aachen University Chapter B1: Crystal Structures and Symmetries Georg Roth Institute of Crystallography RWTH Aachen University http://www.xtal.rwth-aachen.de http://www.frm2.tum.de The „crystal palace“ in Aachen-Burtscheid Outstation of the IfK at FRM II in Garching Symmetry Principles are widely used in Solid State Science Mathematics, Physics, Chemistry, Crystallography, … Why do Crystallographers use symmetry? A: To understand the direction dependence of macroscopic physical properties: Anisotropy B: To write down crystal structures in a concise manner 1 cm3 of matter consists of (roughly) 1023 atoms. Write down 3x1023 atom-coordinates? Or better use the symmetry concept and write down only very few atoms …one to a few hundred at best… 1 Outline: • Crystal lattices The lattice concept: points, directions, planes • Crystallographic coordinate systems Unit cell, origin choice, 7 crystal systems, 14 Bravais-lattices • Symmetry operations and symmetry elements Translation, rotation, roto-inversion, screw-axes, glide-planes • Crystallographic point groups and space groups 32 point groups, symmetry directions, Hermann-Mauguin symbols, 230 space groups in 3D • Quasicrystals Ordered solids without translational symmetry • Application of symmetry: Crystal structure of YBa2Cu3O7 2-dim. periodic pattern of snowflakes Hexagonal symmetry 60° 60° 60° 60° 60° 60° b4 Unit cell b2 a4 a b a a 2 b b3 a3 Choice of origin: point of highest symmetry (6-fold rotation point) 2 2-dim symmetry: 6-fold, 3-fold & 2-fold rotation axes: 60° a b 120° 180° Crystal = Lattice Motive Motive Basis vectors, translational symmetry a b 3a 2b =3a+2b(+0c) general translation vector : =ua+vb+wc; u, v, w Z (3dim.) 3 How to describe the contents of the unit cell? …usually: atoms, here: snowflakes… a b y b r1 1 x1a r1 = x1a + y1b (0 x, y < 1) Positions of atoms in the unit cell, expressed as fractional coordinates: Positional vector rj = xja + yjb + zjc (0 x, y, z < 1) (3D, atom label: j) 3 dim. periodicity of crystals crystal lattice 3 non-linear basis vectors a, b and c define a parallelepiped, called unit cell of the crystal lattice and the crystallographic coordinate system with its origin angles between basis vectors: angle (a,b) = angle (b,c) = angle (c,a) = faces of unit cell: face (a,b) = C face (b,c) = A face (c,a) = B any lattice point (point lattice) is given by a vector = ua + vb + wc (u, v, w ) is also known as translation vector 4 Lattice points and lattice directions according to the translation vector = ua + vb + wc (u, v, w ) • lattice points are indicated by the corresponding integers → uvw • lattice directions or lattice rows by → [uvw] [square brackets] [001] [-2-3-1] or [2 3 1 ] [231] [010] [100] Lattice planes (crystal faces are special lattice planes) 3 non-collinear lattice points define a lattice plane: • interceptions of a lattice plane with the axes X (a), Y (b) and Z (c): ma, nb, oc • reciprocal values: 1/m, 1/n, 1/o (with smallest common denominator no/mno, mo/mno, mn/mno) • Miller indices: h = no, k = mo, l = mn Miller indices (hkl) describe a set of equally spaced lattice planes. I: 1 1 1 1/1 1/1 1/1 h=1 k=1 l=1 (111) II: 1 2 2 1/1 1/2 1/2 h=2 k=1 l=1 (211) 5 Projection of the lattice of graphite (hexagonal) down the Z-axis on the XY-plane to illustrate the decrease of the d(hkl)-spacing between lattice planes (hk0) as their indices h and k increase: Y (100) X d(100) Hexagonal crystal system a=b, c, ==90°, =120° 1 d(hkl) = 4 h2 +k 2 +hk + l 2 3 a2 c2 Lattice spacings are directly accessible by experiments: Diffraction of X-rays, neutrons, electrons, ... = coherent elastic scattering For a crystal: Interference between waves scattered from a set of lattice planes (hkl) Bragg equation for the reflection condition 2d(hkl)·sin(hkl) = d(hkl): interplanar distance of a set of lattice planes (hkl) (hkl): scattering angle, angle between the incident beam and the lattice plane (hkl) : wavelength of the radiation [hkl] (hkl) (hkl) (hkl) d(hkl) 6 Coordinate systems of crystals = 7 crystal systems in 3 dim. Name of system Minimum symmetry Conventional unit cell triclinic 1 or -1 a b c; monoclinic one diad – 2 or m (‖Y) a b c; ==90°, >90° three mutually perpendicular orthorhombic a b c; ===90° diads – 2 or m (‖X, Y and Z) tetragonal one tetrad – 4 (‖Z) a = b c; ===90° trigonal one triad – 3 or -3 (‖Z) a = b c; ==90°, =120° (hexagonal cell) hexagonal one hexad – 6 or -6 (‖Z) a = b c; ==90°, =120° four triads – 3 or -3 cubic a = b = c; ===90° (‖space diagonals of cube) || means: parallel to, diad means: 2-fold rotation, triad: 3-fold etc. Are the 7 crystal systems in 3D, corresponding to the 7 significantly different, symmetry adapted coordinate systems all we need? Not quite: There are good arguments (again based on symmetry) to define 7 additional lattices with more than one lattice point per unit cell The 7 non primitive “centered” lattices Altogether: The 14 Bravais-lattices 7 The 14 Bravais lattices (represented by their unit cells) triclinic P monoclinic P monoclinic A orthorhombic P monoclinic axis‖c (0,0,0 + 0, ½, ½) • orthorhombic I orthorhombic C orthorhombic F tetragonal P (0,0,0 + ½, ½, ½) (0,0,0 + ½, ½,0) (0,0,0 + ½, ½,0 ½,0, ½ + 0, ½, ½) The 14 Bravais lattices (cont.) tetragonal I Trigonal/hexagonal P hexagonal/ cubic P rhombohedral 14 Bravais lattices: 7 primitive lattices P for the 7 crystal systems with only one lattice point per unit cell + 7 centered (multiple) lattices cubic I cubic F A, B, C, I, R and F with 2, 3 and 4 lattice points per unit cell 8 Diffraction geometry: Concept of the reciprocal lattice: Reminder: The crystal lattice 'direct lattice' is composed of the set of all lattice vectors generated by the linear combination of the basis vectors a1, a2, a3 with coefficients u, v, and w (positive or negative integers, incl. 0). a = u a1 + v a2 + w a3. The Fourier-transform which is occurring during a diffraction experiment, transforms this direct lattice into the so called ’reciprocal lattice’, with basis vectors 1, 2, 3 and the integer Miller indices h,k,l as the coefficients. Diffracted intensity I() is only observed at the nodes of this reciprocal lattice addressed by the vectors: = h 1 + k 2 + l 3 of the reciprocal lattice. Example: Monoclinic cell a1, a2, a3, > 90° . Direct and reciprocal basis d001 vectors satisfy the following a3 conditions: 3 1a1 = 2a2 = 3a3 = 1 . d100 This means that |ai| = 1 / |i| and * 1a2 = 1a3 = 2a1 = ... = 0 1 a2||2 a1 This means that each i is perpendicular to aj and ak: * = * = = = 90° = (a a )/V * = 180°– i j k c with V = a (a a ) as the d : lattice spacing for the c 1 2 3 hkl volume of the direct cell set of lattice planes (hkl) 9 Crystallographic symmetry operations All types listed systematically: 1. Translations = ua + vb + wc (u, v, w ) properties: no fixed point, shift of entire point lattice 2. Rotations: 1 (identity), 2 (rotation angle 180°), 3 (120°), 4 (90°), 6 (60°) properties: line of fixed points which is called the rotation axis 3. Rotoinversions (combination of n-fold rotations and inversion): 1(inversion), 2 = m (reflection), 3 , 4 , 6 properties: (exactly one) fixed point 4. Screw rotations nm (combination of n-fold rotations with m/n· translations ‖ to rotation axis) properties: no fixed point 5. Glide reflections a, b, c, n, d (combination of reflection through a plane (glide plane) and translation by glide vectors a/2, b/2, c/2, (a + b)/2, ..., (a b c)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because they leave at least one point fixed. Point symmetry operations rotations rotoinversions 1=identity inversion 2-fold = 180°-rotation 2-fold rotation combined with inversion = reflection 10 1 3 4 2 How many distinct combinations of point symmetry operations (rotation & roto-inversion) are possible in 3 D? The answer is: 32 32 crystallographic point groups (“crystal classes”) The point group symmetries determine the anisotropic (macroscopic) physical properties of crystals: Mechanical, Electrical, Optical, Thermal, ... Tensorial Crystal Physics 11 Hermann-Mauguin symbols of point groups e.g. m m 2 and symmetry directions Basis vectors are conventionally chosen parallel to important symmetry directions of the crystal system. Example: In the cubic lattice, a, b and c are parallel to the 4-fold rotation axes. A maximum of 3 independent main symmetry directions (“Blickrichtungen”) are sufficient to describe the complete point group symmetry of a crystal. These symmetry directions are specific for each of the 7 crystal systems and are essential to understand the Hermann-Mauguin symbol symmetry directions in the orthorhombic lattice a b c; = = = 90° z z z y y y x x x [100] [010] [001] 2 m 12 symmetry directions in the tetragonal lattice a = b c; = = = 90° z z z y y y x x x [001] [100] [110] 4 m symmetry directions in the cubic lattice a = b = c; = = = 90° z z z y y y x x x [100] [111] [110] 42 m 3 13 All possible combinations of point symmetry operations in 3 dim. lead to 32 crystallographic point groups (crystal classes) Plotted: „stereographic projections“ : Point on upper hemisphere : Point on lower hemisphere Nomenclature: Left: Schoenfliess- Symbol Right: Hermann- Mauguin-Symbol 14 Example: Orthorhombic system mb Crystallographic point group: mm2 2 ‖ to [001]: b symmetry operation − x −1 0 0 x ma represented by − y = 0 −1 0 y a rotation matrix z 0 0 1 z -x,-y,z a b x a y x,y,z − x −1 0 0 x m ⊥ b: x 1 0 0 x ma ⊥ a: b y = 0 1 0 y − y = 0 −1 0 y z 0 0 1 z z 0 0 1 z Crystallographic point groups which have a centre of symmetry 11 Laue classes Crystal systems (7) Laue classes (11) triclinic -1 monoclinic 1 2/m 1 = 2/m orthorhombic 2/m 2/m 2/m = 2/m m m 4/m tetragonal 4/m 2/m 2/m = 4/m m m -3 trigonal -3 2/m = -3 m 6/m hexagonal 6/m 2/m 2/m = 6/m m m 2/m -3 = m -3 cubic 4/m -3 2/m = m -3 m By diffraction methods, only the 11 Laue classes can be distinguished and not all the 32 crystal classes.
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