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Crystalography
- 1. Basic crystallography Paolo Fornasini Department of Physics University of Trento, Italy
- 2. Overview • X-rays • Crystals • Crystal lattices • • • • Some relevant crystal structures Crystal planes Reciprocal lattice Crystalline and non-crystalline materials
- 3. X-rays
- 4. 1895 - Discovery of X-rays Paolo Fornasini Univ. Trento Wilhelm Konrad Röntgen (1845-1923) Würzburg (Germany) November 8,1895
- 5. Paolo Fornasini Univ. Trento Electromagnetic waves Speed (in vacuum): c ≈ 3×10 8 m/s λ Electric field Magnetic field € c € € 1 Å = 10-10 m 1 nm = 10-9 m Wavelength € λ Frequency ν = c/λ Photon energy € E = hν = hω € € ω = 2 πν
- 7. Plane wave Paolo Fornasini Univ. Trento r E r r € r k 2π k= λ r r € € r r r r r E(r ) = E 0 cos k ⋅ r ( ) rr ik ⋅r € { } Re e € r r r = E 0 € for k ⋅ r = 2nπ Complex notation
- 8. Electromagnetic spectrum E (eV) Photon energy Wavelength E = hν = hc / λ Frequency 10-12 10-8 10-4 X-rays 1 Paolo Fornasini Univ. Trento ν (Hz) 102 106 1010 1014 λ ≈ 0.01÷10 Å ν ≈ 1017 ÷10 20 Hz λ (m) ELF 106 Radio waves 102 Micro waves 10-2 IR 10-6 UV 104 1018 E ≈ 0.4 ÷ 400 keV 108 1022 X γ 10-10 10-14
- 9. Interaction of x-rays with matter Paolo Fornasini Univ. Trento Auger electrons Photo-electrons EA E hν € € Incoming beam (monochromatic) Fluorescence hν F Beam attenuation € € hν ' Inelastic scattering € Photo-electric absorption hν Elastic scattering Scattering
- 10. X-RAYS and X-ray techniques 10-10 1Å 10-8 I.R. 10-6 10-4 E[keV] = λ (m) 10-2 1 µm Imaging Scattering 12.4 λ[Å] Spectroscopy λ = 2dsin θ Intensity 10-12 U.V. Absorption X-rays Paolo Fornasini Univ. Trento Bragg angle Energy
- 11. Crystals
- 12. Crystals Paolo Fornasini Univ. Trento Macroscopic regularities (e.g. constancy of angles) Classification of crystals Quartz crystal (SiO2) Regular packing of microscopic structural units R.J. Haüy (1743-1822)
- 13. Atoms and crystals Paolo Fornasini Univ. Trento HYPOTHESIS: Structural units = atoms Example: NaCl Atomic masses: Na 38.12 x 10-24 g Cl 58.85 x 10-24 g Cubic structure 1 cm3 m = 2.165 g N = 44.6 x 1021 atoms 0.28 nm = 2.8 Å CONCLUSION: Inter-atomic distances Atomic dimensions ≈ X-ray wavelengths
- 14. X-ray diffraction from crystals Munich, 1912: • Max von Laue • W. Friedrich & P.Knipping Paolo Fornasini Univ. Trento
- 15. Crystallography Paolo Fornasini Univ. Trento Cambridge, 1912/13 William Lawrence Bragg (1890-1971) William Henry Bragg (1862-1942) Bragg spectrometer
- 16. Crystal structure Bravais lattice 1-D 1-D Paolo Fornasini Univ. Trento + Basis Atom Molecule 2-D 2-D 3-D 3-D Protein
- 17. Bravais lattice + basis Paolo Fornasini Univ. Trento 1-D 2-D 3-D
- 19. Translation vectors (2D) Paolo Fornasini Univ. Trento 2-D r R For every lattice point r b r r r R = n1a + n2b r € a Arbitrary origin € € integers primitive vectors
- 20. Primitive vectors (2D) Paolo Fornasini Univ. Trento r r r R = n1a + n2b r R r b € r R r R € r € a 2-D € € Different choices of primitive vectors r r a, b
- 21. Non-primitive vectors (2D) Not all € r b € r € a r r a, b r R Paolo Fornasini Univ. Trento 2-D pairs are primitive r R r R € r r r R ≠ n1a + n2b €
- 22. Primitive vectors (3D) r r r r R = n1a + n2b + n3c € r c r b r a € r r r Different choices of primitive vectors a , b , c € € Paolo Fornasini Univ. Trento 3-D
- 23. Primitive unit cells (2D) Paolo Fornasini Univ. Trento 2-D r b r a Different choices of primitive unit cells € Primitive cell = 1 lattice point
- 24. Conventional unit cells (2D) Paolo Fornasini Univ. Trento 2-D r b € r a More than 1 lattice point per unit cell
- 25. Non-Bravais lattices Paolo Fornasini Univ. Trento Atoms 2-D r r r R ≠ n1a + n2b Un-equivalent sites €
- 27. Classification of cells 2-D 3-D r b r c γ r a r α b γ € € β a € b c latin α β γ greek € € € r a
- 28. Surface Bravais lattice Paolo Fornasini Univ. Trento 2-D Non-primitive unit cell
- 29. 3-D Bravais lattices Paolo Fornasini Univ. Trento 3-D 4 unit cells 7 crystal systems + P = primitive I = body centered F = face centered C = side centered = 14 Bravais lattices
- 30. Coordinates (3, 2) 2 Lattice points: integer coordinates € € r r r R = n1a + n2b 1 1 1 € € Paolo Fornasini Univ. Trento € 2 € 4 0 1 4 3 4 1 2 3 € Inside cell: fractional coordinates € € 1 1 , 4 4 € 3 1 , 4 2 € €
- 32. Simple cubic lattice Paolo Fornasini Univ. Trento 84-Po a=3.35 Å Bravais lattice a lattice parameter € Primitive unit cell (1 lattice point per cell) Coordination number = 6
- 33. Body centered cubic lattice (bcc) Paolo Fornasini Univ. Trento 24-Cr 26-Fe 42-Mo a=2.88 Å a=2.87 Å a=3.15 Å Bravais lattice a lattice parameter € conventional unit cell (2 lattice points per cell) Coordination number = 8
- 34. Face centered cubic lattice (fcc) Paolo Fornasini Univ. Trento 29-Cu 47-Ag 79-Au a=3.61 Å a=4.09 Å a=4.08 Å Bravais lattice a lattice parameter € conventional unit cell (4 lattice points per cell) Coordination number = 12
- 35. Diamond structure Paolo Fornasini Univ. Trento 6-C 14-Si 32-Ge a=3.57 Å a=5.43 Å a=5.66 Å Non-Bravais lattice fcc Bravais lattice + 2-atom basis 1 1 1 , , 4 4 4 a (0, 0, 0) € € € conventional unit cell (8 atoms per cell) Coordination number = 4
- 36. Zincblende (sphalerite) structure Paolo Fornasini Univ. Trento ZnS GaAs SiC a=5.41 Å a=5.65 Å a=4.35 Å Non-Bravais lattice fcc Bravais lattice + 2-atom basis 1 1 1 , , 4 4 4 (0, 0, 0) a € € € conventional unit cell (8 atoms per cell) Cordination number = 4
- 37. Rock-salt (NaCl) structure Paolo Fornasini Univ. Trento NaCl KBr CaO a=5.64 Å a=6.60 Å a=4.81 Å Non-Bravais lattice fcc Bravais lattice + 2-atom basis 1 1 1 , , 2 2 2 (0, 0, 0) a € € € conventional unit cell (8 atoms per cell) Cordination number = 6
- 38. Simple hexagonal structure Paolo Fornasini Univ. Trento Top view c a Hexagonal symmetry Primitive cell 2 lattice parameters Primitive unit cell € lattice point per cell) (1 r a2 a1 = a2 € r a1
- 39. Hexagonal close packed structure Paolo Fornasini Univ. Trento 4-Be 12-Mg 48-Cd a=2.29 Å a=3.21 Å a=2.98 Å Non-Bravais lattice + primitive cell c 2-atom basis 1 1 1 , , 2 2 2 (0, 0, 0) 8 c= a 3 € € a € € Conventional unit cell (2 atoms per cell) Cordination number = 12
- 40. Coordination number Paolo Fornasini Univ. Trento diamond bcc cubic N=4 N=6 fcc N=8 hcp Close packing N=12
- 41. Close-packing of spheres A A A 1st layer A A A A A A A A 2nd layer A A A A A A A A A A A A A B A ABA A B B B B B B C C C A C A A B B B B A A A A A B B A A 3rd layer B A A A A A A A A A Paolo Fornasini Univ. Trento C C C A ABC C
- 42. hcp versus fcc Paolo Fornasini Univ. Trento A B C A B A hcp fcc A A A A A A A A A A A C A C A C C A A A A A A A A A A C C C C
- 43. Crystal planes
- 45. Miller indices, 2-D (a) Paolo Fornasini Univ. Trento 2-D r b r a € € (hk) = (21) (hk) = (11)
- 46. Miller indices, 2-D (b) Paolo Fornasini Univ. Trento (hk) = (01) € (hk) = (1 1 ) € (hk) = (10) (hk) = (2 1 )
- 47. Miller indices, 3-D r c Paolo Fornasini Univ. Trento (hkl) = (100) € (hkl) = (111) r b € r a r a 3-D r € c (hkl) = (114) € € rr€ cc (hkl) = (210) € r b € € € r a r a € r b € r b
- 48. Miller indices, cubic lattices Paolo Fornasini Univ. Trento sc (100) € (111) (110) € € bcc (200) fcc € € € (200) (222) (110) (220) (111)
- 49. Interplanar distance Paolo Fornasini Univ. Trento dhkl Cubic lattices 2 dhkl a2 = 2 2 2 h + k +l Copper, fcc, a=3.61 Å € d200 = a = 1.805 Å 2 d220 = a 2 2 = 1.276 Å d111 = a = 2.084 Å 3
- 50. Planes and directions Perpendicular direction [hkl] Family of planes (hkl) €
- 51. Equivalent planes and directions Equivalent directions < 100 > [001] [010] € [100] € € € (010) (100) € Equivalent planes € {100} (001) €
- 53. X-ray diffraction pattern Paolo Fornasini Univ. Trento Direct space 3-dimensional lattice Reciprocal space 2-dimensional projection
- 54. Basic idea Paolo Fornasini Univ. Trento A) Family of planes → wave-vector r K hkl K hkl dhkl € € € B) Wave-vectors → set of points C) Set of points → lattice r K h'k'l' dh'k'l' 2π = dhkl €
- 55. Reciprocal quantities Paolo Fornasini Univ. Trento Periodic behaviour T ω = 2 π /T € time λ frequency k = 2π / λ €position wave-vector
- 56. Harmonics Paolo Fornasini Univ. Trento Periodic behaviour ω0 T fundamental 2ω0 € 3ω0 2nd harmonic 3rd harmonic time € € frequency Periodic behaviour
- 57. 1-D Paolo Fornasini Univ. Trento Direct space Reciprocal space 2π a = a * r a € r a r* a € € r* a
- 58. 2-D, rectangular lattice (a) Paolo Fornasini Univ. Trento Direct space Reciprocal space 2 π 2 πb a = = a ab 2 π 2 πa b* = = b ab r* r a ⊥b r* r b ⊥a * r R r b € € € r a € r r r R = n1a + n2b € € r* R r* b r € a* r* r* r* € R = m1a + m2b
- 59. 2-D, rectangular lattice (b) Direct space Paolo Fornasini Univ. Trento Reciprocal space
- 60. 2-D, rectangular lattice (c) Direct space Paolo Fornasini Univ. Trento Reciprocal space
- 61. 2-D, rectangular lattice (d) Direct space Paolo Fornasini Univ. Trento Reciprocal space
- 62. 2-D, oblique lattice (a) Paolo Fornasini Univ. Trento Direct space Reciprocal space 2π 2 πb = a sin θ ab sin θ 2π 2 πa b* = = b sin θ ab sin θ r* r a ⊥b r* r b ⊥a a* = r b r R r a € € € € € € € r* b r* a
- 63. 2-D, oblique lattice (b) Direct space Paolo Fornasini Univ. Trento Reciprocal space
- 64. Cubic lattices (a) Paolo Fornasini Univ. Trento Direct space r c Reciprocal space r* c r b r* b r a € r* a € € (100) (110) (100) € € € (111) € € (110) (111)
- 65. Cubic lattices (b) Paolo Fornasini Univ. Trento Direct space r c Reciprocal space r* c r* b r b r a r* a € € € € (200) (200) € € (110) (222)
- 66. Cubic lattices (c) Paolo Fornasini Univ. Trento Direct space Reciprocal space r c r* c r b r a € r* a € € € € (200) r* b (200) € € (220) (111)
- 67. Primitive vectors: general rule Paolo Fornasini Univ. Trento Direct space Reciprocal space r c € r a r € a € ( r c ) ( r b ) (€ ) r b € € r r r b ×c a* = 2π r r r a⋅ b ×c r r r* c×a b = 2π r r r a⋅ b ×c € r r r a×b c * = 2π r r r a⋅ b ×c r r r a⋅ b ×c ( € r* c r* a r* c €* r a ) € € r* b r* b
- 68. Reciprocal lattice and lattice planes For any family of lattice planes separated by a distance d there are reciprocal lattice vectors perpendicular to the planes, the shortest of which have a length 2π/d. For any reciprocal lattice vector R*, there is a family of lattice planes normal to R* and separated by a distance d, where 2π/d is the length of the shortest reciprocal lattice vector parallel to R*. Paolo Fornasini Univ. Trento
- 70. Macro and micro-crystals Monocrystalline silicon, ∅ 13 cm Paolo Fornasini Univ. Trento Cr, electron microscopy Grain structure
- 71. Effects of temperature Paolo Fornasini Univ. Trento Temperature Thermal motion Spread of atomic positions
- 72. Crystalline and non-crystalline materials Crystalline solids Long-range order Paolo Fornasini Univ. Trento Non-crystalline systems No long-range order
- 73. Cooling rate Paolo Fornasini Univ. Trento Slow cooling Fast cooling High T: liquid Low T: solid Thermodynamic equilibrium No thermodynamic equilibrium
- 74. Radial Distribution Function Paolo Fornasini Univ. Trento average density RDF = 4 πr 2 ρ (r ) = 4 πr 2 ρ 0 g(r ) PDF = Pair Distribution Function € g(r) 1 Short-range order 0 r
- 75. Summary • Plane waves and wavevector • Crystal structure = Bravais lattice + basis • Bravais lattices: primitive vectors, unit cells (primitive and conventional), classifications • Crystal structures (sc, bcc, fcc, hcp ...) • Crystal planes and Miller indices • Reciprocal lattice • Crystalline and non-crystalline materials
- 76. Dolomite mountains in winter