X-ray analysis. 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods
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1 X-ray analysis 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods
2 Introduction Noble prizes associated with X-ray diffraction 1901 W. C. Roentgen (Physics) for the discovery of X-rays M. von Laue (Physics) for X-ray diffraction from crystals W. H. and W. L. Bragg (Physics) for structure derived from X-ray diffraction C. G. Barkla (Physics) for characteristic radiation of elements K. M. G. Siegbahn (Physics) for X-ray spectroscopy A. H. Compton (Physics) for scattering of X-rays by electrons P. Debye (Chemistry) for diffraction of X-rays and electrons in gases M. Perutz and J. Kendrew (Chemistry) for the structure of hemoglobin J. Watson, M. Wilkins, and F. Crick (Medicine) for the structure of DNA A. Cormack and G. Newbold Hounsfield (Medicine) for computed axial tomography K. M. Siegbahn (Physics) for high resolution electron spectroscopy H. Hauptman and J. Karle (Chemistry) for direct methods to determine structures J. Deisenhofer, R. Huber, and H. Michel (Chemistry) for the structures of proteins that are crucial to photosynthesis.
3 Introduction What s the result of X-ray analyses? Crystal data Formula sum Mg 2 SiO 4 (Olivine) Crystal system orthorhombic Space group P b n m (no. 62) Unit cell dimensions a = 4.75(2) Å, b = 10.25(4) Å, c = 6.00(2) Å Z 4 Atomic coordinates Atom Ox. Wyck. x y z Mg1 +2 4a Mg2 +2 4c (600) (600) Si1 +4 4c (500) (50) O1-2 4c (1000) (100) O2-2 4c (100) (1000) O3-2 8d (1000) (1000) (1000) Structure Chemical information (bonding, composition) Real structure (defects)
4 1. Basic crystallography Lattice, motif and structure Example: structure and lattice in 2D Lattice pattern of points no chemical information, mathematical description no atoms, but points and lattice vectors (a, b, c, α, β, γ), unit cell Motif (characteristic structural feature, atom, group of atoms ) Structure = Lattice + Motif contains chemical information (e. g. environment, bond length ) describes the arrangement of atoms (symmetry of the crystal)
5 1. Basic crystallography Unit cell: interconnection of lattice and structure Definition: Unit cell = parallel sided region of the lattice from which the entire crystal can be constructed by purely translational displacements contents of unit cell represents chemical composition (multiples of chemical formula) primitive cell: simplest cell, contains one lattice point centered cell: more than one point inside unit cell Conventions: 1. Cell edges should coincide with symmetry axes or reflection planes 2. The smallest possible cell which fulfills 1. should be chosen
6 1. Basic crystallography Unit cell: exercise Determine the primitive unit cell and one example for a centered setting
7 1. Basic crystallography Aperiodic structures: e. g. quasicrystals Penrose tiling Forbidden symmetry produced by superposition
8 1. Basic crystallography Unit cells and crystal system millions of periodic structures but 7 types of primitive cells (crystal systems) crystal system = particular restriction concerning the unit cell crystal system = unit cell with characteristic symmetry elements (later) Crystal system Restrictions axes Restrictions angles Triclinic - - Monoclinic - α = γ = 90 Orthorhombic - α = β = γ = 90 Tetragonal a = b α = β = γ = 90 Trigonal a = b α = β = 90, γ = 120 Hexagonal a = b α = β = 90, γ = 120 Cubic a = b = c α = β = γ = 90
9 1. Basic crystallography Indexation of directions in direct space c [uvw] = [110] Procedure in three steps b a 1. Select Mark position of second point 3. Draw vector Convention: right-handed coordinate system middle finger: a thumb: b forefinger: c
10 1. Basic crystallography Fractional coordinates (position of the atoms) c 1 / 21 / 21 / 2 b a possible values for x, y, z: [0; 1], atoms are multiplied by translations atoms are generated by symmetry elements (later) Example: Sphalerite (ZnS) Equivalent points are represented by one triplet only equivalent by translation equivalent by other symmetry elements (later)
11 1. Basic crystallography Centered unit cells Bravais-type The Bravais-types P, F, I, C, A, B, R denote centerings of the unit cells Centering of unit cell = Translation 7 crystal systems, 14 characteristic unit cells (i.e. Bravais-type) 7 types of centerings: P: no centering (0,0,0) F: translation of each point by (1/2,1/2,0);(0,1/2,1/2);(1/2,0,1/2) I: translation of each point by (1/2,1/2,1/2) C: translation of each point by (1/2,1/2,0) A: translation of each point by (0,1/2,1/2) B: translation of each point by (1/2,0,1/2) R: translation of each point by (2/3,1/3,1/3);(1/3,2/3,2/3) All fractional coordinates are multiplied by centerings
12 1. Basic crystallography Bravais-type: example Crystal data Formula sum NaCl Crystal system cubic Space group F m -3 m (no. 225) Unit cell dimensions a = (5) Å Z 4 Atomic coordinates Halite Atom Ox. Wyck. x y z Na +1 4a Cl -1 4b 1/2 1/2 1/2
13 1. Basic crystallography Wyckoff-notation and occupancy factor Crystal data Formula sum Cu 0.8 In 2.4 Se 4 Crystal system tetragonal Space group I -4 2 m (no. 121) Unit cell dimensions a = (3) Å c = (1) Å Z 2 Atomic coordinates Molecules Atom Ox. Wyck. Occ. x y z Cu1 +1 2a In1 +3 4d /2 1/4 In2 +3 2b /2 Se1-2 8i 1.0 1/4 1/4 1/8 Occ. factors Wyckoff number = no. of atoms/unit cell
14 1. Basic crystallography Crystallographic symmetry elements SE in crystallography: Inversion Mirror Rotation axes: 1,2,3,4,6 Translations Coupling and combination Question: Why is e. g. 5 forbidden in crystal structures?
15 1. Basic crystallography Coupling- rotation and inversions Coupling: both SE are applied in one step one of the two coupled SE is an intermedium 3 intermedium intermedium... What about 2?
16 1. Basic crystallography Coupling- rotation and translation Screw axes X n Strategy: X-fold rotation after translation of n/x along the screw axis t
17 1. Basic crystallography Coupling- mirror and translation Gilde planes intermedium real point a, b, c, n, d, e a, b, c: ½ a, ½ b, ½ c n: ½(a + b), ½(a + c), ½(b + c) d: ½ n
18 1. Basic crystallography Combination of mirror and rotation Combination: both SE are applied (two steps) no intermedium 2/m 3/m
19 1. Basic crystallography Directions of characteristic symmetry elements Crystal system Characteristic SE Char. direction / sequence Triclinic 1, 1 - Monoclinic 2 and/or m b Orthorhombic 2 (3 ) and/or m(3 ) a, b, c Trigonal 3 (1 ) c, a, [210] Tetragonal 4 (1 ) c, a, [110] Hexagonal 6 c, a, [210] Cubic 3 (4 ) [111], a, [110] What can we learn from this table? The knowledge of the characteristic SE allows to determine the crystal system Pseudometrics possible, e. g. monoclinic with β = 90 Systematization of SE in space group symbols
20 Repetition Basic terms of 3D crystallography Lattice, motif and structure Unit cell (primitive and centered) Wyckoff notation, occupancy factor, composition of crystals Crystal system Bravais type Crystallographic SE Coupling of SE: rotoinversion, screw axes, glide planes Combination of SE Characteristic SE of the crystal casses
21 1. Basic crystallography Crystal class 32 Crystal classes: Combination of SE without translations Crystal system Restriction Crystal Class Triclinic - 1,1 Monoclinic - α = γ = 90 2, m, 2/m Orthorhombic Trigonal Tetragonal Hexagonal Cubic a = b α = β = γ = 90 a = b α = β = 90, γ = 120 a = b α = β = γ = 90 a = b α = β = 90, γ = 120 a = b = c α = β = γ = , mm2, 2/m 2/m 2/m 3, 3, 32, 3m, 32/m 4, 4, 4/m, 422, 4mm, 42m, 4/m 2/m 2/m 6, 6, 6/m, 622, 6mm, 6m2, 6/m 2/m 2/m 23, 2/m 3, 432, 43m, 4/m 3 2/m
22 1. Basic crystallography Characteristic symmetry elements: example The atoms (A, B and C) of a tetragonal structure are located on: 0 1 / 2 0 (A), 1 / 21 / 2 0 (B) and 000 (C). Assume low symmetry Sketch the unit cell along [001] Determine the composition of a crystal with this structure Describe the environment of A, B and C Repeat the exercise for a cubic structure!
23 1. Basic crystallography Crystal Classes- examples 4mm mm2 4/mmm Crystal System Charact. SE Direction / Sequence Triklin 1, 1 - Monoklin 2 and/or m b Orthorhombisch 2 (3 ) und/oder m(3 ) a, b, c Trigonal 3 (1 ) c, a, [210] Tetragonal 4 (1 ) c, a, [110] Hexagonal 6 c, a, [210] Kubisch 3 (4 ) a, [111], [110]
24 1. Basic crystallography Exercise: restrictions of the crystal systems 1 In tetragonal crystals, the restriction α = β = γ = 90 can not be violated Right Wrong 2 The characteristic symmetry element of cubic crystals is the fourfold axis Right Wrong 3 For monoclinic crystals β must be 90 Right Wrong 4 α = β = γ = 90 is not possible for triclinic crystals Right Wrong 5 A crystal system with α = β = γ = 90 and a = b = c must be denominated cubic Right Wrong 6 A crystal system with α = β = γ = 90 and a b c can be denominated orthorhombic Right Wrong
25 1. Basic crystallography Crystal Classes- exercises Determine the SE of following arrangements Specify the crystal system and the crystal class
26 1. Basic crystallography Space groups: Introduction A crystallographic space group is a set of symmetry operations which describes all periodic patterns (230, 3D) in 3D space Space group notations (H.-M.-notation) first letter: Bravais-type (lattice centering) second and subsequent numbers and letters: symmetry elements along characteristic directions caution: some notations are reduced P
27 1. Basic crystallography Space group tables reciprocal space direct space
28 1. Basic crystallography Space group: Examples (H.M. notation) Example 1: Pm (full notation) P means primitive (no centering) characteristic SE: m monoclinic m perpendicular to [010] Example 2: C2/c (full notation) C means ab-plane (001) is centered characteristic SE: 2 or c monoclinic. 2 along b, c perpendicular to b Example 3: P6 3 /mmc = P6 3 /m 2/m 2/c (full notation) P means primitive (no centering) characteristic SE: 6 3 hexagonal 6 3 along c, m perpendicular c, 2 along a, m perpendicular a, 2 along [210], c perpendicular [210] Information derived from H.M. notation Crystal system, Crystal class Centrosymmetric / non centrosymmetric Reflection conditions ( Extinctions )
29 2. Basic diffraction physics Model for X-ray diffraction (XRD) Scattering can be separated into: scattering by all electrons of the distinct atoms of the structure (atomic form factor) scattering by all atoms of the structure (structure factor, convolution lattice and unit cell) scattering by the whole crystal (finite size effects: broadening of FT) atoms 8 atoms YAxis Title ,0 0,2 0,4 0,6 0,8 1,0 h ,0 0,2 0,4 0,6 0,8 1,0 Mathematical description: Fourier transformation h Object (crystal) real space electron density ρ(r) diffraction pattern reciprocal space structure factor F(r*) FT
30 2. Basic diffraction physics Example: FFTs in electron microscopy 1 nm
31 2. Basic diffraction physics Basic FTs in XRD I: Atomic scattering factor ρ(r) = F(r*) exp (-2iπ(rr*)) dr* V* F(r*) = ρ(r) exp (2iπ(rr*)) dr V Scattering by atoms (FT of U a (r)) f a (r*) = U a (r) (sin(2πrr*))/2πrr* dr ~ Σ f el.(r*), with: U a (r) = 4πr 2 ρ a (r) V Experimental consequences high diffracted intensity at low θ, e.g. left side of powder DP light atoms problem, consequence: alternative methods
32 2. Basic diffraction physics Basic FTs in XRD II: periodic and infinite crystals ρ(r) = F(r*) exp (-2iπ(rr*)) dr* V* F(r*) = ρ(r) exp (2iπ(rr*)) dr V ρ(r) exhibits the periodicity of the crystal: Fourier series ρ(r) = Σ C g exp (2iπ(r*r)), C g : Fourier coefficient = 1/V Σ F hkl exp (2iπ(hx + ky + lz)), hkl: rec. space, xyz: dir. space F hkl = Σf i exp(2πi(hx + ky + lz)), summing up the contributions of all atoms Example: graph of a rectangular f(x) = sinx + sin3x/3 + sin5x/5 + sin7x/
33 2. Basic diffraction physics Scattered intensity: Structure factor F hkl F hkl = Σf i exp(2πi(hx + ky + lz)) imaginary = Σf i [cos (2π(hx + ky + lz)) + i sin (2π(hx + ky + lz))] = ΣA i + i B i ib A φ real Remarks F hkl : summing up the contributions from all atoms All structural information is in one reflection φ: phase of F hkl contains structure information, φ = arctan B/A IF hkl I= (A 2 + B 2 ) 1/2 ~ I = : phase problem, i.e. phase is lost Friedel s law: I hkl = I -h-k-l Symmetry of DP: centrosymmetric (first approximation)
34 2. Basic diffraction physics Calculations of structure factors: examples F hkl = Σf i exp(2πi(hx + ky + lz)) = Σf i [cos (2π(hx + ky + lz)) + i sin (2π(hx + ky + lz))] Primitive (one atom type) Calculation for CsCl BCC (one atom type) Calculation for NaCl
35 2. Basic diffraction physics Example NaCl Each peak corresponds to one set of hkl planes (or equivalent plane) P o w d e rc e ll HALITE
36 2. Basic diffraction physics NaCl vs. KCl P o w d e rc e ll 2. 2 P o w d e r C e ll HALITE KCl HALITE NaCl
37 2. Basic diffraction physics Geometrical approach, Bragg s law (BL) Description by wave vector: k I : incident beam, k D : diffracted beam; Ik I I= Ik D I= 1/λ S2 S1 k I Constructive Interference π/4 A d B θ C hkl plane D Destructive Interference k D AC + CD = nλ = 2dsinθ B 2sinθ B /λ = n/d = nid*i
38 2. Basic diffraction physics hkl: indices of planes in direct space c (hkl) = (110) Procedure in three steps b a 1. Select Mark intercept of the plane, i. e. reciprocal values 1/h on a, 1/k on b, 1/l on c Three points, l = 0 means: plane c 3. Draw plane Convention: right-handed coordinate system
39 2. Basic diffraction physics hkl planes: examples c (112) b a c (110) a b
40 2. Basic diffraction physics Properties / Applications of d/d* d ~ to the normal vector of an hkl plane IdI ~ distance of two individual hkl planes of the same type Square form of BL (e.g. orthorhombic) 1/d 2 = h 2 /a 2 + k 2 /b 2 + l 2 /c 2 sin 2 θ = λ 2 /4 (h 2 /a 2 + k 2 /b 2 + l 2 /c 2 ) Application: indexing of DP Example for manual indexing (e.g. cubic): (1/d) 2 = (1/a) 2 (h 2 + k 2 + l 2 ), d = λ/(2sinθ) Determine d-spacing of each peak from its 2θ value (using Bragg s Law) Create a table of 1/d 2 values for each peak Look for a common factor (1/a 2 ) that can be divided into each of the (1/d) 2 values
41 2. Basic diffraction physics Manual indexation: example 1 2-theta d 1000/d 2 hkl /62.5= /62.5= /62.5= /62.5= /62.5= /62.5= /62.5= /62.5= /62.5=10 310
42 2. Basic diffraction physics Manual indexation: example 2 (extinctions) 2-theta d 1000/d2 hkl /33= /33= /33= /33= /33= /33= /33= /33= Systematic of extinctions (general reflection conditions): translation Integral reflection conditions: unit cell translations (centers) Zonal reflection conditions: glide planes Serial reflection conditions: screw axes
43 2. Basic diffraction physics Extinctions: examples P2 P2 1 : 0k0 : k = 2n Pc: h0l : l = 2n, 00l : l = 2n C2: hkl : h + k = 2n
44 3. Experimental methods Generation of X-rays Atomic scale scenario: inner shell electrons are striked out outer shell electrons fill hole production of X-rays due to energy difference between inner and outer shell electron X-ray tube
45 3. Determination of 3D structures Practical work- Essentials 1. Selection of good samples: Single crystals: diameter < 0.2 mm Homogeneous powders: small crystals of one compound 2. Determination of symmetry and unit cell 3. Measurement of diffracted intensities (automatic procedure) 4. Calculation of possible atomic parameters (structure solution) 5. Refinement of the structure model (PC) Comparison of experimental and calculated data Atomic parameters are optimized until refinement converges 6. Interpretation of the resulting refinement results R-values, convergence, thermal parameters 7. Interpretation of the resulting structure (does it make sense?) Interatomic distances, occupancy factors (reasonable values?) Ionic compounds: compensation of all charges
46 3. Determination of 3D structures Results of diffraction studies- Overview a) Position of the reflections (Bragg s law): Lattice parameters (1/d) 2 = (1/a) 2 [h 2 + k 2 + l 2 ] b) Intensity of reflections Symmetry of the structure: Space group Structure (fractional coordinates) c) Profile of the reflections Crystal size and perfection cf. Scherrer formula: (2θ) = λ/lcosθ Indications for structural disorder
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