2. Diffraction at periodic structures 2. Diffraction Patterson - PowerPoint PPT Presentation

2. Diffraction at periodic structures 2. Diffraction Patterson function 2. Diffraction 58 60 Scattering geometry Patterson function 2. Diffraction Reciprocal lattice 2. Diffraction 59 61 Real space lattice Reciprocal lattice 2d - lattice 1d - chain

Brillouin zones 2. Diffraction Ewald construction 2. Diffraction 64 62 k The first Brillouin zone is defined as the k 0 Wigner-Seitz cell of the reciprocal lattice. All k vectors at the Brillouin zone boundary fulfil the Laue condition. i) Die Richtung von k 0 ist durch das Experiment vorgegeben (einfallender Strahl). ii) Zeichne k 0 so, dass die Spitze an einem Punkt des reziproken Gitters endet, hier (000). iii) Zeichne einen Kreis (eine Kugeloberfläche) mit Radiusvektor k 0 k Für alle Punkte auf dem Kreis (der Kugeloberfläche) ist G = k – k 0 erfüllt, iv) All k 0 , k satisfying the Laue condition hier G = (340) – (000) = (340). lie on the bisecting plane perpendicular to G hkl k – k 0 = G k 0 Bemerkung: Der Anfang von k 0 liegt in der Abbildung nur zufällig nahe einem Punkt G des reziproken Gitters. Brillouin zones 2. Diffraction X-ray diffraction 2. Diffraction 65 63 Laue method bcc fcc hexagonal transmission reflection Diffraction of collimated, broadband X-ray beam at single crystal sample: • k ’ · k 0 · k ” • analysis of crystal symmetry • orientation of single crystals

X-ray diffraction 2. Diffraction Photons, electrons, neutrons, He atoms 2. Diffraction 66 68 Debeye-Scherrer method Wavelength λ ≈ lattice constant a X-ray film polycrystline sample Monochromatic X-ray diffraction from polycrystalline powder sample • Diffraction rings 0 ≤ 2 Θ ≤ 180 � • G hkl ≤ 2 k 0 • not verry precise, but lattice constant and crystal structure X-ray diffraction 2. Diffraction Low-energy electron diffraction 2. Diffraction 67 69 Rotating crystal method 3-axis goniometer (monochromator, sample, detector) Electrons impinging on the crystal are elastically back-reflected and imaged by a phosphor screen. Monochromatic X-ray diffraction from single crystal The short inelastic mean free path of electrons in conducting materials (only a few lattice planes) makes LEED a surface sensitive technique. • simultaneous rotation: sample by ϑ , detector by 2 ϑ resolution Δ λ / λ ≈ 10 -5 (X-rays or neutrons) •

Brillouin zones 2. Diffraction Laue condition and Bragg reflection 2. Diffraction 70 72 k The first Brillouin zone is defined as the k 0 Wigner-Seitz cell of the reciprocal lattice. All k vectors at the Brillouin zone boundary fulfil the Laue condition. k All k 0 , k satisfying the Laue condition lie on the bisecting plane perpendicular to G hkl k – k 0 = G k 0 G Miller indices 2. Diffraction Structure factor and atomic form factor 2. Diffraction 71 73 n r 3 a 3 Atomic form factor r f Fe for iron a 3 r r n r a 1 a 2 1 a n r 1 2 a 2 Structure factor Some lattice planes and Miller indices of the cubic Bravais lattice

Debye Waller factor 2. Diffraction 74 Intensity of X-ray diffraction (h00) spots from Al Mean square displacement for harmonic oscillator : Debye-Waller factor: temperature (K)