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Understanding X-ray Diffraction and Bragg's Law in Crystal Analysis

X-ray diffraction is a powerful technique for probing atomic structures in solids. Typical interatomic distances in solids are on the order of an angstrom, necessitating X-ray wavelengths of similar scale. The energy associated with these wavelengths is around 12,000 eV. This article discusses Bragg's Law, which describes the conditions for constructive interference in diffraction, and details the role of reciprocal lattice vectors in understanding crystal structures. Key concepts such as Brillouin zones and lattice planes are also explored, highlighting their significance in material sciences.

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Understanding X-ray Diffraction and Bragg's Law in Crystal Analysis

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Presentation Transcript

  1. Don’t Ever Give Up!

  2. X-ray Diffraction Typical interatomic distances in solid are of the order of an angstrom. Thus the typical wavelength of an electromagnetic probe of such distances Must be of the order of an angstrom. Upon substituting this value for the wavelength into the energy equation, We find that E is of the order of 12 thousand eV, which is a typical X-ray Energy. Thus X-ray diffraction of crystals is a standard probe.

  3. Wavelength vs particle energy

  4. Bragg Diffraction: Bragg’s Law

  5. Bragg’s Law The integer n is known as the order of the corresponding Reflection. The composition of the basis determines the relative Intensity of the various orders of diffraction.

  6. Many sets of lattice planes produce Bragg diffraction

  7. Bragg Spectrometer

  8. Characteristic X-Rays

  9. Brehmsstrahlung X-Rays

  10. Bragg Peaks

  11. X-Ray Diffraction Recording

  12. von Laue Formulation of X-Ray Diffraction

  13. Condition for Constructive Interference

  14. Bragg Scattering =K

  15. The Laue Condition

  16. Ewald Construction

  17. Crystal and reciprocal lattice in one dimension

  18. First Brillouin Zone: Two Dimensional Oblique Lattice

  19. Primitive Lattice Vectors: BCC Lattice

  20. First Brillouin Zone: BCC

  21. Primitive Lattice Vectors: FCC

  22. Brillouin Zones: FCC

  23. Near Neighbors and Bragg Lines: Square

  24. First Four Brillouin Zones: Square Lattice

  25. All Brillouin Zones: Square Lattice

  26. First Brillouin Zone BCC

  27. First Brillouin Zone FCC

  28. Experimental Atomic Form Factors

  29. Reciprocal Lattice 1

  30. Reciprocal Lattice 2

  31. Reciprocal Lattice 3

  32. Reciprocal Lattice 5

  33. Real and Reciprocal Lattices

  34. von Laue Formulation of X-Ray Diffraction by Crystal

  35. Reciprocal Lattice Vectors • The reciprocal lattice is defined as the set of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice. • Let R denotes the Bravais lattice points;consider a plane wave exp(ik.r). This will have the periodicity of the lattice if the wave vector k=K, such that exp(iK.(r+R)=exp(iK.r) for any r and all R Bravais lattice.

  36. Reciprocal Lattice Vectors • Thus the reciprocal lattice vectors K must satisfy • exp(iK.R)=1

  37. Brillouin construction

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