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Publié parDio BAGT Modifié depuis plus de 8 années
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Basic crystallography Paolo Fornasini Department of Physics University of Trento, Italy
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Overview X-rays Crystals Crystal lattices Some relevant crystal structures Crystal planes Reciprocal lattice Crystalline and non-crystalline materials
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X-rays
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1895 - Discovery of X-rays Würzburg (Germany) November 8,1895 Wilhelm Konrad Röntgen (1845-1923) Paolo Fornasini Univ. Trento
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Electromagnetic waves Speed (in vacuum): Paolo Fornasini Univ. Trento Photon energy Frequency Wavelength Electric field Magnetic field 1 Å = 10 -10 m 1 nm = 10 -9 m
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Wave-vector Paolo Fornasini Univ. Trento Plane wave
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Paolo Fornasini Univ. Trento Complex notation
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Electromagnetic spectrum Photon energy Frequency X-rays Wavelength Paolo Fornasini Univ. Trento I R Micro waves E eV) 10 8 1 10 -12 10 -8 10 -4 10 4 m) 10 -14 10 -6 10 6 10 2 10 -2 10 -10 Hz) 10 22 10 14 10 2 10 6 10 10 18 X UV ELF Radio waves
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Interaction of x-rays with matter Paolo Fornasini Univ. Trento Incoming beam (monochromatic) Elastic scattering Inelastic scattering Photo-electrons Auger electrons Fluorescence Photo-electric absorption Scattering Beam attenuation
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X-RAYS and X-ray techniques Paolo Fornasini Univ. Trento X-rays 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 1 Å 1 m (m) U.V.I.R. ScatteringSpectroscopy Imaging
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Crystals
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Crystals Quartz crystal (SiO 2 ) Macroscopic regularities (e.g. constancy of angles) Classification of crystals Regular packing of microscopic structural units R.J. Haüy (1743-1822) Paolo Fornasini Univ. Trento
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Atoms and crystals Structural units = atoms Example: NaCl Cubic structure 1 cm 3 m = 2.165 g N = 44.6 x 10 21 atoms HYPOTHESIS: CONCLUSION: Inter-atomic distances Atomic dimensions Atomic masses:Na 38.12 x 10 -24 g Cl 58.85 x 10 -24 g 0.28 nm = 2.8 Å X-ray wavelengths Paolo Fornasini Univ. Trento
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X-ray diffraction from crystals Munich, 1912: Max von Laue W. Friedrich & P.Knipping Paolo Fornasini Univ. Trento
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Crystallography William Henry Bragg (1862-1942) William Lawrence Bragg (1890-1971) Cambridge, 1912/13 Bragg spectrometer Paolo Fornasini Univ. Trento
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Crystal structure Paolo Fornasini Univ. Trento 1-D 3-D 2-D Atom Molecule Protein Bravais lattice + Basis 1-D 2-D 3-D
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Bravais lattice + basis Paolo Fornasini Univ. Trento 1-D 2-D 3-D
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Crystal lattices
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Translation vectors (2D) Paolo Fornasini Univ. Trento 2-D Arbitrary origin For every lattice point integersprimitive vectors
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Primitive vectors (2D) Paolo Fornasini Univ. Trento 2-D Different choices of primitive vectors
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Non-primitive vectors (2D) Paolo Fornasini Univ. Trento Not all pairs are primitive 2-D
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Primitive vectors (3D) Paolo Fornasini Univ. Trento Different choices of primitive vectors 3-D
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Primitive unit cells (2D) Paolo Fornasini Univ. Trento 2-D Different choices of primitive unit cells Primitive cell = 1 lattice point
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Conventional unit cells (2D) Paolo Fornasini Univ. Trento 2-D More than 1 lattice point per unit cell
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Non-Bravais lattices Paolo Fornasini Univ. Trento 2-D Atoms Un-equivalent sites
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Bravais lattices Bravais lattice Basis Paolo Fornasini Univ. Trento
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Classification of cells 2-D 3-D latin greek
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Surface Bravais lattice Non-primitive unit cell Paolo Fornasini Univ. Trento 2-D
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3-D Bravais lattices 4 unit cells P = primitive I = body centered F = face centered C = side centered 7 crystal systems 14 Bravais lattices += Paolo Fornasini Univ. Trento 3-D
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Coordinates Paolo Fornasini Univ. Trento Inside cell: fractional coordinates Lattice points: integer coordinates
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Some relevant crystal structures
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Simple cubic lattice Paolo Fornasini Univ. Trento Primitive unit cell (1 lattice point per cell) Coordination number = 6 lattice parameter Bravais lattice 84-Poa=3.35 Å
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Body centered cubic lattice (bcc) Paolo Fornasini Univ. Trento conventional unit cell (2 lattice points per cell) Bravais lattice lattice parameter Coordination number = 8 24-Cra=2.88 Å 26-Fea=2.87 Å 42-Moa=3.15 Å
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Face centered cubic lattice (fcc) Paolo Fornasini Univ. Trento conventional unit cell (4 lattice points per cell) Bravais lattice lattice parameter Coordination number = 12 29-Cua=3.61 Å 47-Aga=4.09 Å 79-Aua=4.08 Å
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Diamond structure conventional unit cell (8 atoms per cell) Paolo Fornasini Univ. Trento Coordination number = 4 Non-Bravais lattice fcc Bravais lattice2-atom basis + 6-Ca=3.57 Å 14-Sia=5.43 Å 32-Gea=5.66 Å
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Zincblende (sphalerite) structure Paolo Fornasini Univ. Trento conventional unit cell (8 atoms per cell) Cordination number = 4 Non-Bravais lattice fcc Bravais lattice2-atom basis + ZnSa=5.41 Å GaAsa=5.65 Å SiCa=4.35 Å
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Rock-salt (NaCl) structure Cordination number = 6 Paolo Fornasini Univ. Trento conventional unit cell (8 atoms per cell) NaCla=5.64 Å KBra=6.60 Å CaOa=4.81 Å Non-Bravais lattice fcc Bravais lattice2-atom basis +
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Simple hexagonal structure 2 lattice parameters Paolo Fornasini Univ. Trento Primitive cellHexagonal symmetry Top view Primitive unit cell (1 lattice point per cell)
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Hexagonal close packed structure Cordination number = 12 4-Bea=2.29 Å 12-Mga=3.21 Å 48-Cda=2.98 Å Paolo Fornasini Univ. Trento Conventional unit cell (2 atoms per cell) Non-Bravais lattice primitive cell + 2-atom basis
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Coordination number Paolo Fornasini Univ. Trento N=4 diamond N=6 cubic N=8 bcc N=12 fcc hcp Close packing
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Close-packing of spheres 1st layer A AAA A AAAA A AAA A AAAA AAAAA 2nd layer BB B B BB B BBBB B B B Paolo Fornasini Univ. Trento 3rd layer A AAA A AAAA A AAA A AAAA AAAAA C CC C C C C C A B AA B C
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hcp versus fcc A AAA A AAAA A AAA A AAAA AAAAA C CC C C C C C Paolo Fornasini Univ. Trento A B A A B C fcchcp
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Crystal planes
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Crystal planes Paolo Fornasini Univ. Trento 2-D
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Miller indices, 2-D (a) (hk) = (21)(hk) = (11) 2-D Paolo Fornasini Univ. Trento
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Miller indices, 2-D (b) Paolo Fornasini Univ. Trento
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Miller indices, 3-D Paolo Fornasini Univ. Trento 3-D
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Miller indices, cubic lattices Paolo Fornasini Univ. Trento bcc fcc sc
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Interplanar distance Paolo Fornasini Univ. Trento Cubic lattices Copper, fcc, a=3.61 Å
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Planes and directions Family of planes Perpendicular direction
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Equivalent planes and directions Equivalent directions Equivalent planes
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Reciprocal lattice
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X-ray diffraction pattern Paolo Fornasini Univ. Trento Direct space 3-dimensional lattice Reciprocal space 2-dimensional projection
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Basic idea Paolo Fornasini Univ. Trento A) Family of planes wave-vector B) Wave-vectors set of points C) Set of points lattice
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Reciprocal quantities Paolo Fornasini Univ. Trento frequencytime positionwave-vector Periodic behaviour
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Harmonics Paolo Fornasini Univ. Trento Periodic behaviour fundamental 2nd harmonic 3rd harmonic time frequency
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1-D Direct space Reciprocal space Paolo Fornasini Univ. Trento
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2-D, rectangular lattice (a) Paolo Fornasini Univ. Trento Direct space Reciprocal space
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2-D, rectangular lattice (b) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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2-D, rectangular lattice (c) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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2-D, rectangular lattice (d) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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2-D, oblique lattice (a) Paolo Fornasini Univ. Trento Reciprocal space Direct space
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2-D, oblique lattice (b) Paolo Fornasini Univ. Trento Reciprocal space Direct space
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Cubic lattices (a) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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Cubic lattices (b) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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Cubic lattices (c) Reciprocal space Direct space Paolo Fornasini Univ. Trento
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Primitive vectors: general rule Direct space Reciprocal space Paolo Fornasini Univ. Trento
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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 2 /d. For any reciprocal lattice vector R*, there is a family of lattice planes normal to R* and separated by a distance d, where 2 /d is the length of the shortest reciprocal lattice vector parallel to R*. Paolo Fornasini Univ. Trento
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Microscopic structure of materials
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Macro and micro-crystals Monocrystalline silicon, 13 cm Cr, electron microscopy Grain structure Paolo Fornasini Univ. Trento
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Effects of temperature Paolo Fornasini Univ. Trento Temperature Spread of atomic positionsThermal motion
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Crystalline and non-crystalline materials Paolo Fornasini Univ. Trento Crystalline solids Non-crystalline systems Long-range orderNo long-range order
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Cooling rate High T: liquid Low T: solid Slow coolingFast cooling No thermodynamic equilibrium Thermodynamic equilibrium Paolo Fornasini Univ. Trento
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Radial Distribution Function Paolo Fornasini Univ. Trento r 0 1 g(r) PDF = Pair Distribution Function average density Short-range order
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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
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