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

▶
2 diffraction at periodic structures
SMART_READER_LITE
LIVE PREVIEW

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

Jan 02, 2023 725 likes •804 views

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 -

slide-1
SLIDE 1

58

  • 2. Diffraction at periodic structures
  • 2. Diffraction

Scattering geometry

59

Patterson function

  • 2. Diffraction

60

Patterson function

  • 2. Diffraction

61

Reciprocal lattice

  • 2. Diffraction

Real space lattice Reciprocal lattice 1d - chain 2d - lattice

slide-2
SLIDE 2

62

Brillouin zones

  • 2. Diffraction

The first Brillouin zone is defined as the Wigner-Seitz cell of the reciprocal lattice. All k vectors at the Brillouin zone boundary fulfil the Laue condition. k0 k All k0, k satisfying the Laue condition lie on the bisecting plane perpendicular to Ghkl k0 G k k – k0 = G

63

Brillouin zones

  • 2. Diffraction

bcc

fcc hexagonal

64

Ewald construction

  • 2. Diffraction

i) Die Richtung von k0 ist durch das Experiment vorgegeben (einfallender Strahl). ii) Zeichne k0 so, dass die Spitze an einem Punkt des reziproken Gitters endet, hier (000). iii) Zeichne einen Kreis (eine Kugeloberfläche) mit Radiusvektor k0 iv) Für alle Punkte auf dem Kreis (der Kugeloberfläche) ist G = k – k0 erfüllt, hier G = (340) – (000) = (340). Bemerkung: Der Anfang von k0 liegt in der Abbildung nur zufällig nahe einem Punkt des reziproken Gitters.

65

X-ray diffraction

  • 2. Diffraction

Laue method

Diffraction of collimated, broadband X-ray beam at single crystal sample:

transmission reflection

  • k’ · k0 · k”
  • analysis of crystal symmetry
  • rientation of single crystals
slide-3
SLIDE 3

66

X-ray diffraction

  • 2. Diffraction

Debeye-Scherrer method

Monochromatic X-ray diffraction from polycrystalline powder sample

film polycrystline sample X-ray

  • Diffraction rings
  • 0 ≤ 2Θ ≤ 180

Ghkl ≤ 2k0

  • not verry precise, but lattice constant and crystal structure

67

X-ray diffraction

  • 2. Diffraction

Rotating crystal method

Monochromatic X-ray diffraction from single crystal

  • simultaneous rotation: sample by ϑ, detector by 2ϑ
  • resolution Δ λ / λ ≈ 10-5 (X-rays or neutrons)

3-axis goniometer (monochromator, sample, detector)

68

Photons, electrons, neutrons, He atoms

  • 2. Diffraction

Wavelength λ ≈ lattice constant a

69

Low-energy electron diffraction

  • 2. Diffraction

Electrons impinging on the crystal are elastically back-reflected and imaged by a phosphor screen. The short inelastic mean free path of electrons in conducting materials (only a few lattice planes) makes LEED a surface sensitive technique.

slide-4
SLIDE 4

70

Brillouin zones

  • 2. Diffraction

The first Brillouin zone is defined as the Wigner-Seitz cell of the reciprocal lattice. All k vectors at the Brillouin zone boundary fulfil the Laue condition. k0 k All k0, k satisfying the Laue condition lie on the bisecting plane perpendicular to Ghkl k0 G k k – k0 = G

71

Miller indices

  • 2. Diffraction

1

a r

2

a r

3

a r

1 1a

n r

2 2a

n r

3 3a

n r

Some lattice planes and Miller indices

  • f the cubic Bravais lattice

72

Laue condition and Bragg reflection

  • 2. Diffraction

73

Structure factor and atomic form factor

  • 2. Diffraction

Atomic form factor fFe for iron Structure factor

slide-5
SLIDE 5

74

Debye Waller factor

  • 2. Diffraction

Intensity of X-ray diffraction (h00) spots from Al temperature (K)

Mean square displacement for harmonic oscillator : Debye-Waller factor: