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Lattice spacing typically Max von Laue (1879-1960) 1914 Nobel prize Laue 1912 Crystal diffraction.

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Presentation on theme: "Lattice spacing typically Max von Laue (1879-1960) 1914 Nobel prize Laue 1912 Crystal diffraction."— Presentation transcript:

1 Lattice spacing typically Max von Laue (1879-1960) 1914 Nobel prize Laue 1912 Crystal diffraction

2 Today X-ray diffraction supplemented by electron and neutron diffration Energies X-ray, electrons and neutrons wave-particle X-ray: Electrons: Neutrons:

3 Typical Laue X-ray diffraction pattern symmetry of the pattern Laue X-ray diffraction YAlO 3 c-axis normal to picture symmetry of the crystal Complementarity of the three types of radiation X-ray diffraction Electron diffractionNeutron diffraction Photon energies 10keV-100keV large penetration depth 3D crystal structure scattering by electron density best results for atoms with high Z Charged particle “strong” interaction with matter low penetration depth Study of: surfaces thin films Interaction with nuclei Improved efficiency for light atoms Inelastic scattering: phonons Magnetic moment interacts with moment of electrons Magnetic scattering: Structure, magnons

4 Law describing the necessary condition for diffraction Applicable for photons, electrons and neutrons n: integer Bragg’s law Condition for efficient specular reflection ( click for java applet ) Bragg Diffraction Law

5 Spacing d hkl between successive (hkl) planes In cubic systems: x y a d 110 Top view d hkl for non cubic lattice later in the framework of the reciprocal lattice

6 Bragg’s law necessary condition structure factor atomic form factor Intensity of particular (hkl) reflection General theory of Diffraction X-ray source R P r B R’ R’-r

7 X-ray source R P r B R’ R’-r Plane wave incoming at P Scattered wave contribution from P incoming at B Electron density at P Total scattering from the entire volume:

8 Diffraction experiment measures the intensity I of the scattered waves whereis the scattering vector Diffracted intensity is the square of the Fourier transform of the electron density In crystals is periodic 1D example

9 Fourier series expansion 2π periodic function decomposed into cos kx and sin kx or where

10 3dimensional case 1dimensional case with translational invariance of with respect to lattice vector Reciprocal lattice vectors

11 Diffracted intensity is the square of the Fourier transform of the electron density Remember: periodic electron density with ( click for information about  -functions )

12 Scattering condition is nothing but Bragg´s law ! decomposition into so far unknown basis vectors with h, k, l integers The reciprocal lattice

13 The basis vectors of the reciprocal lattice are determined by: These fulfill the condition holds, where

14 Examples for reciprocal lattices 3 dimensions

15 2 dimensions

16 Important properties of the reciprocal lattice vectors lies perpendicular to the lattice plane with Miller indices (hkl) simple example for the (111) plane in the cubic structure and span the (111) lattice plane vector (111) plane 0

17 Distance d hkl between lattice planes (hkl) related to according to d 111 G 111

18 Equivalence between the scattering condition and Bragg´s law lattice plane (hkl) Ө k0k0 k -k0-k0 Ө Elastic scattering: k=k 0 1

19 Geometrical interpretation of the scattering condition k0k0 k G 2Ө2Ө (000) reciprocal lattice Ewald construction

20 Crystal in random orientation not necessarily reflection rotation of the crystal polychromatic radiation ( click for animation )

21 incoming monochromatic beam Rotating crystal arrangement Powder method / Debye Scherrer determine unknown structure Precise measurement of lattice constants

22 Laue method transmission reflection Polychromatic X-rays Orientation of crystal with known structure

23 The structure factor Scattering condition ( Bragg’s law ) necessary condition Controls the actual intensity of the (hkl)-reflex Remember: because crystal periodic Fourier-coefficients

24 Majority of the electrons are centered in a small region around the atoms core electrons Scattering from valence electrons can be neglected Atom in n-th unit cell is located at position atomic scattering factor f α Structure factor

25 atomic scattering factor Spherically symmetric whereG, r’

26 atomic scattering factor Maximum at Ө=0 (forward scattering) number of electrons/atom ( Click for calculations of atomic scattering factors )

27 Structure factor of a lattice with basis Structure factor of the bcc lattice: Conventional cell contains two atoms at r 1 =(0,0,0) r 2 =(1/2,1/2,1/2) Both atoms have the same atomic scattering factor f 1 = f 2 = f Reciprocal unit cell:cube with cell side of 2π/a

28 We observe e.g. diffraction peaks from (110), (200), (211) planes but no peaks from (100), (111), (2,1,0) planes

29 Shape and dimension of the unit cell can be deduced from Bragg peaks Content of the unit cell (basis) determined from intensities of reflections If f 1 = f 2 peaks like (100), (111), (2,1,0) appearlike CsI Similar situation in the case of fcc KCl/KBr:f(K + )=f(Cl - ) = f(Br - ) KCl: Non-zero if all indices even KBr: all fcc-peaks present

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