Chapter 4: Diffraction Georg Roth Institute of Crystallography RWTH Aachen University http://www.ifk.rwth-aachen.de http://www.frm2.tum.de The „crystal palace“ in Aachen-Burtscheid Outstation of the IfK at FRM II in Garching Outline Diffraction part 1: Geometric crystallography in a nutshell • Crystal lattices the lattice concept, unit cells, lattice points, lattice directions, lattice planes • Crystallographic coordinates systems The 7 crystal systems, the 14 Bravais lattices • Crystallographic symmetry-operations and -elements Rotation, roto-inversion, glide planes, screw axes • Crystallographic point groups and space groups The 32 crystallographic point groups (crystal classes), symmetry directions, Hermann-Mauguin symbols, the 230 space groups Matter exists in (at least) 3 states: gas – liquid - solid Crystals are representatives of the solid state Definition of a crystal: Thermodynamically stable solid state with regular 3dim. periodic arrangement of atoms, ions or molecules in space in contrast to other states of condensed matter: liquids, glasses and amorphous solids Halite - NaCl + rock salt Na - and Cl–- c ions b 0 a How to choose a unit cell? (2D-example) hexagonal symmetry 60° 60° 60° 60° 60° 60° b4 As small as possible unit cell b2 As much symmetry as possible a4 a b a a b 2 b3 a3 choice of origin: point of highest symmetry (6-fold rotation point) crystal = lattice motive motive basis vectors a b 3a 2b How to adress a lattice point? Linear combination of a and b = 3a + 2b (+ 0c) General translation vector : Motive here: A 2D-snowflake = ua + vb + wc; u, v, w Z In a crystal: Set of atoms in In 3 dimensions the unit cell How to address atoms in the unit cell? a b r y b 1 1 x1a r1 = x1a + y1b (0 x, y < 1) Positions of atoms in the unit cell: Position vector rj = xja + yjb + zjc (0 x, y, z < 1) (3dim) There is more than just lattice & motive: Symmetry Rotational symmetry 60° a b Translational symmetry 120° 180° 3 dim. periodicity of crystals 3D crystal lattice 3 non-colinear basis vectors a, b and c (in the script a1, a2, and a3) define a parallelepiped, called unit cell of the crystal lattice and the crystallographic coordination system with its origin. angles between basis vectors: angle (a1,a2) = angle (a2,a3) = angle (a3,a1) = faces of unit cell: face (a1,a2) = C face (a2,a3) = A face (a3,a1) = B any lattice point (point lattice) is given by a vector = ua1 + va2 + wa3 (u, v, w ) is also known as translation vector lattice points and lattice directions according to the translation vector = ua1 + va2 + wa3 (u, v, w ) • lattice points are indicated by the corresponding integers uvw • lattice directions or lattice rows by [uvw] [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 (a1), Y (a2) and Z (a3): ma1, na2, oa3 • 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) Lattice spacing d(hkl): Distance between lattice planes (hk0): Solely depends on the lattice parameters Y (100) X d(100) Diffraction of X-rays, neutrons, electrons, ... allows us to measure lattice spacings directly The observable is the scattering angle which may be converted into lattice spacings d(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) The lattice spacing d(hkℓ) can be calculated from the lattice parameters for each crystal system, e.g.: • cubic crystal system (a = b = c, = = = 90°), ex. NaCl a d(hkl) h2 k 2 l 2 • hexagonal crystal system (a = b, c, = = 90°, = 120°), ex. Graphite 1 d(hkl) 4 h2 k 2 hk l 2 2 2 3 a c • orthorhombic crystal system (a, b, c, = = = 90°) 1 d(hkl) h2 k 2 l 2 a2 b2 c2 • tetragonal crystal system (a = b, c, = = = 90°) 1 d(hkl) h2 k 2 l 2 a2 c2 7 different coordinate systems in 3D 7 crystal systems in 3D name of system minimum symmetry conventional unit cell triclinic 1 or -1 a b c; a b c; = = 90°, monoclinic one diad – 2 or m (‖Y) > 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 a = b c; = = 90°, one triad – 3 or -3 (‖Z) (hexagonal cell) = 120° a = b c; = = 90°, hexagonal one hexad – 6 or -6 (‖Z) = 120° four triads – 3 or -3 cubic a = b = c; = = = 90° (‖space diagonals of cube) 7 primitive + 7 centred unit cells 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) 14 Bravais lattices (cont.) tetragonal I 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 centred (multiple) lattices cubic I cubic F A, B, C, I, R and F with 2, 3 and 4 lattice points per unit cell Definition of symmetry: Symmetry operations are isometric transformations (or movements) where one object is transformed into another in a congruent or enantiomorphous manner, i.e. all distances, angles and volumes remain invariant That means: an object and its transformed object superimpose in a perfect manner they are indistinguishable crystallographic rotations: 1-fold (identity), 2-, 3-, 4-, and 6-fold rotations Crystallographic symmetry operations are isometric movements in a 3D-periodic space: 1. Translations a = ua1 + va2 + wa3 (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·a 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 a1/2, a2/2, a3/2, (a1 + a2)/2, ..., (a1 a2 a3)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because of at least one fixed point. Point symmetry operations rotations rotoinversions 1 = identity 1 = inversion 2-fold = 180°-rotation 2-fold rotation combined with inversion = reflection 5 1 3 6 4 2 Crystallographic symmetry operations are isometric movements in crystals: 1. Translations a = ua1 + va2 + wa3 (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·a 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 a1/2, a2/2, a3/2, (a1 + a2)/2, ..., (a1 a2 a3)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because of at least one fixed point. Screw rotations nm = n + m/n·a 120° 1/3a a a 60° 2/6a 31 = 3 + 1/3 a 62 = 6 + 2/6 a 60° a 4/6a + 42, 43 and 65 64 = 6 + 4/6 a Crystallographic symmetry operations are isometric movements in crystals: 1. Translations a = ua1 + va2 + wa3 (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·a 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 a1/2, a2/2, a3/2, (a1 + a2)/2, ..., (a1 a2 a3)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because of at least one fixed point. a m reflection: mirror plane m image plane a a a/2 glide reflection: ‖ to the glide plane a is the glide vector a/2 All possible combinations of point symmetry operations in 3D: 32 crystallographic point groups (crystal classes) The point group symmetries determine the anisotropic (macroscopic) physical properties of crystals: mechanical, electrical, optical, thermal, ... In order to understand point group symbols (Hermann-Mauguin symbols), it is essential to understand the concept of symmetry directions: A maximum of 3 independent main symmetry directions (“Blickrichtungen”) are sufficient to describe the complete symmetry of a crystal.