Basic crystallography Paolo Fornasini Department of Physics University of Trento, Italy

Overview X-rays Crystals Crystal lattices Some relevant crystal structures Crystal planes Reciprocal lattice Crystalline and non-crystalline materials

 X-rays

Discovery of X-rays Würzburg (Germany) November 8,1895 Wilhelm Konrad Röntgen ( ) Paolo Fornasini Univ. Trento

Electromagnetic waves Speed (in vacuum): Paolo Fornasini Univ. Trento Photon energy Frequency Wavelength Electric field Magnetic field 1 Å = m 1 nm = m

Wave-vector Paolo Fornasini Univ. Trento Plane wave

Paolo Fornasini Univ. Trento Complex notation

Electromagnetic spectrum Photon energy Frequency X-rays Wavelength Paolo Fornasini Univ. Trento I R Micro waves E  eV)  m)  Hz) X UV  ELF Radio waves

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

X-RAYS and X-ray techniques Paolo Fornasini Univ. Trento X-rays Å 1  m (m) U.V.I.R. ScatteringSpectroscopy Imaging

 Crystals

Crystals Quartz crystal (SiO 2 ) Macroscopic regularities (e.g. constancy of angles) Classification of crystals Regular packing of microscopic structural units R.J. Haüy ( ) Paolo Fornasini Univ. Trento

Atoms and crystals Structural units = atoms Example: NaCl Cubic structure 1 cm 3 m = g N = 44.6 x atoms HYPOTHESIS: CONCLUSION: Inter-atomic distances Atomic dimensions Atomic masses:Na x g Cl x g 0.28 nm = 2.8 Å  X-ray wavelengths Paolo Fornasini Univ. Trento

X-ray diffraction from crystals Munich, 1912: Max von Laue W. Friedrich & P.Knipping Paolo Fornasini Univ. Trento

Crystallography William Henry Bragg ( ) William Lawrence Bragg ( ) Cambridge, 1912/13 Bragg spectrometer Paolo Fornasini Univ. Trento

Crystal structure Paolo Fornasini Univ. Trento 1-D 3-D 2-D Atom Molecule Protein Bravais lattice + Basis 1-D 2-D 3-D

Bravais lattice + basis Paolo Fornasini Univ. Trento 1-D 2-D 3-D

 Crystal lattices

Translation vectors (2D) Paolo Fornasini Univ. Trento 2-D Arbitrary origin For every lattice point integersprimitive vectors

Primitive vectors (2D) Paolo Fornasini Univ. Trento 2-D Different choices of primitive vectors

Non-primitive vectors (2D) Paolo Fornasini Univ. Trento Not all pairs are primitive 2-D

Primitive vectors (3D) Paolo Fornasini Univ. Trento Different choices of primitive vectors 3-D

Primitive unit cells (2D) Paolo Fornasini Univ. Trento 2-D Different choices of primitive unit cells Primitive cell = 1 lattice point

Conventional unit cells (2D) Paolo Fornasini Univ. Trento 2-D More than 1 lattice point per unit cell

Non-Bravais lattices Paolo Fornasini Univ. Trento 2-D Atoms Un-equivalent sites

Bravais lattices Bravais lattice Basis Paolo Fornasini Univ. Trento

Classification of cells 2-D 3-D latin greek

Surface Bravais lattice Non-primitive unit cell Paolo Fornasini Univ. Trento 2-D

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

Coordinates Paolo Fornasini Univ. Trento Inside cell: fractional coordinates Lattice points: integer coordinates

 Some relevant crystal structures

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

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

Face centered cubic lattice (fcc) Paolo Fornasini Univ. Trento conventional unit cell (4 lattice points per cell) Bravais lattice lattice parameter Coordination number = Cua=3.61 Å 47-Aga=4.09 Å 79-Aua=4.08 Å

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

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

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 +

Simple hexagonal structure 2 lattice parameters Paolo Fornasini Univ. Trento Primitive cellHexagonal symmetry Top view Primitive unit cell (1 lattice point per cell)

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

Coordination number Paolo Fornasini Univ. Trento N=4 diamond N=6 cubic N=8 bcc N=12 fcc hcp Close packing

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

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

 Crystal planes

Crystal planes Paolo Fornasini Univ. Trento 2-D

Miller indices, 2-D (a) (hk) = (21)(hk) = (11) 2-D Paolo Fornasini Univ. Trento

Miller indices, 2-D (b) Paolo Fornasini Univ. Trento

Miller indices, 3-D Paolo Fornasini Univ. Trento 3-D

Miller indices, cubic lattices Paolo Fornasini Univ. Trento bcc fcc sc

Interplanar distance Paolo Fornasini Univ. Trento Cubic lattices Copper, fcc, a=3.61 Å

Planes and directions Family of planes Perpendicular direction

Equivalent planes and directions Equivalent directions Equivalent planes

 Reciprocal lattice

X-ray diffraction pattern Paolo Fornasini Univ. Trento Direct space 3-dimensional lattice Reciprocal space 2-dimensional projection

Basic idea Paolo Fornasini Univ. Trento A) Family of planes  wave-vector B) Wave-vectors  set of points C) Set of points  lattice

Reciprocal quantities Paolo Fornasini Univ. Trento frequencytime positionwave-vector Periodic behaviour

Harmonics Paolo Fornasini Univ. Trento Periodic behaviour fundamental 2nd harmonic 3rd harmonic time frequency

1-D Direct space Reciprocal space Paolo Fornasini Univ. Trento

2-D, rectangular lattice (a) Paolo Fornasini Univ. Trento Direct space Reciprocal space

2-D, rectangular lattice (b) Reciprocal space Direct space Paolo Fornasini Univ. Trento

2-D, rectangular lattice (c) Reciprocal space Direct space Paolo Fornasini Univ. Trento

2-D, rectangular lattice (d) Reciprocal space Direct space Paolo Fornasini Univ. Trento

2-D, oblique lattice (a) Paolo Fornasini Univ. Trento Reciprocal space Direct space

2-D, oblique lattice (b) Paolo Fornasini Univ. Trento Reciprocal space Direct space

Cubic lattices (a) Reciprocal space Direct space Paolo Fornasini Univ. Trento

Cubic lattices (b) Reciprocal space Direct space Paolo Fornasini Univ. Trento

Cubic lattices (c) Reciprocal space Direct space Paolo Fornasini Univ. Trento

Primitive vectors: general rule Direct space Reciprocal space Paolo Fornasini Univ. Trento

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

 Microscopic structure of materials

Macro and micro-crystals Monocrystalline silicon,  13 cm Cr, electron microscopy Grain structure Paolo Fornasini Univ. Trento

Effects of temperature Paolo Fornasini Univ. Trento Temperature Spread of atomic positionsThermal motion

Crystalline and non-crystalline materials Paolo Fornasini Univ. Trento Crystalline solids Non-crystalline systems Long-range orderNo long-range order

Cooling rate High T: liquid Low T: solid Slow coolingFast cooling No thermodynamic equilibrium Thermodynamic equilibrium Paolo Fornasini Univ. Trento

Radial Distribution Function Paolo Fornasini Univ. Trento r 0 1 g(r) PDF = Pair Distribution Function average density Short-range order

 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

Dolomite mountains in winter