X-ray Diffraction & Crystal Structure Basic Concepts

1. X-ray Diffraction & Crystal Structure Basic Concepts T. P. Radhakrishnan School of Chemistry, University of Hyderabad Email: tprsc@uohyd.ernet.in Web: http://chemistry.uohyd.ernet.in/~tpr/

2. This powerpoint presentation is available at the following website http://chemistry.uohyd.ernet.in/~ch521/ Click on x-ray_powd.ppt

3. Outline • Crystals • symmetry • classification of lattices • Miller planes • Waves • phase, amplitude • superposition of waves • Bragg law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of Powder diffraction

4. Crystals • Waves • Bragg Law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of powder diffraction

5. Molecular Structure Optical spectroscopy – IR, UV-Vis Magnetic resonance – NMR, ESR Mass spectrometry X-ray diffraction High resolution microscopy

6. A B 5 Å 5 Å C D 20 Å 5 Å Molecular Structure Resolved by Atomic Force Microscopy Pentacene on Cu(111) B. STM image C, D. AFM images (tip modified with CO molecule) A. Molecular model of pentacene Gross, Mohn, Moll, Liljeroth, Meyer, Science 2009, 325, 1110

7. Crystal and its structure 3-dimensions Anthony, Raghavaiah, Radhakrishnan, Cryst. Growth Des. 2003, 3, 631

8. STM image of 1,3-diheptadecylisophthalate on HOPG (with a model of two molecules) Plass, Kim, Matzger, J. Am. Chem. Soc. 2004, 126, 9042

9. 2-dimensional square lattice

10. Point group symmetries : Identity (E) Reflection (s) Rotation (Rn) Rotation-reflection (Sn) Inversion (i) In periodic crystal lattice : (i) Additional symmetry - Translation (ii) Rotations – limited values of n

11. Translation

12. Translation

13. Translation

14. Translation

15. Rotation

16. Rotation

17. Rotation

18. a a a n 3 2 1 0 -1 qo180 120 90 60 0 Rotation 2 3 4 6 1 Restriction on n-fold rotation symmetry in a periodic lattice q q na (n-1)a/2 cos (180-q) = - cos q = (n-1)/2

19. Crystal Systems in 2-dimensions - 4 square oblique hexagonal rectangular

20. Crystal Systems in 3-dimensions - 7 Cubic Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Trigonal

21. Bravais lattices in 2-dimensions - 5 square rectangular centred rectangular oblique hexagonal

22. Bravais Lattices in 3-dimensions (in cubic system) Body centred cube (I) Primitive cube (P) Face centred cube (F)

23. Bravais Lattices in 3-dimensions - 14 Cubic - P, F (fcc), I (bcc) Tetragonal - P, I Orthorhombic - P, C, I, F Monoclinic - P, C Triclinic - P Trigonal - R Hexagonal/Trigonal - P

24. Point group operations 7Crystalsystems Point group operations + translation symmetries 14 Bravais lattices

25. Lattice(o) + basis (x) = crystal structure

26. C4 Spherical basis C4 Non-spherical basis

27. Lattice+ SphericalBasis Lattice+ NonsphericalBasis Point group operations 7Crystalsystems 32 Crystallographic point groups Point group operations + translation symmetries 230 space groups 14 Bravais lattices

28. Miller plane in 2-D a a Distance between lines = a y (01) x (10)

29. Miller plane in 2-D Distance between lines = a/2 = 0.7 a y x (11)

30. Distance between lines = a/(2)2+(3)2 = 0.27 a Miller plane in 2-D (2, 3, 0) y Take inverses (23) x In 3-D: intercepts = 1/2, 1/3, 

31. z y x Miller plane in 3-D (100) Distance between planes = a a

32. z y x Miller plane in 3-D (010) Distance between planes = a

33. z y x Miller plane in 3-D (110) Distance between planes = a/2 = 0.7 a

34. z y x Miller plane in 3-D (111) Distance between planes = a/3 = 0.58 a

35. a h2+k2+l2 dhkl = Spacing between Miller planes for cubic crystal system

36. Crystals • Waves • Bragg Law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of powder diffraction

37. p 2p 0 0 l/2 l Phase Displacement l = wavelength u = frequency A = amplitude A sin{2p(x/l - ut)} sin (0) = sin (np) = 0 sin ([n+1/2]p) = +1 n even -1 n odd

38. Superposition of Waves amplitude = A amplitude = 2A Constructive interference

39. l/4 Superposition of Waves amplitude = A amplitude = 1.4A

40. Superposition of Waves l/2 amplitude = A amplitude = 0 Destructive interference

41. x 1 x+ l/2 2 x+ l 3 Waves 1 and 2 interfere destructively Waves 1 and 3 interfere constructively

42. Crystals • Waves • Bragg Law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of powder diffraction

43. Wavelength =l q q d h k l h k l p l a n e q l 2 d s i n = n h k l

44. Crystals • Waves • Bragg Law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of powder diffraction

45. Collection of several small crystals Single crystal Cones intersecting a film

47. Powder diffraction setup

48. Powder x-ray diffractogram (sodium chloride) Counts 2q (degree)

49. NaCl - powder x-ray data source Cu-Ka (l = 1.540598 Å) Indexing a = d(h2+k2+l2)½

50. Crystals • Waves • Bragg Law • Powder diffraction • Systematic absences, Structure factor • Single crystals - Solution and Refinement • Diffraction line width • Applications of powder diffraction