X-ray Diffraction: Principles and Practice

1. X-ray Diffraction: Principles and Practice Ashish Garg and Nilesh Gurao Department of Materials Science and Engineering Indian Institute of Technology Kanpur

2. Layout of the Lecture • Materials Characterization • Importance of X-ray Diffraction • Basics • Diffraction • X-ray Diffraction • Crystal Structure and X-ray Diffraction • Different Methods • Phase Analysis • Texture Analysis • Stress Analysis • Particles Size Analysis • ……….. • Summary

3. Materials Characterization • Essentially to evaluate the structure and properties • Structural Characterization • Diffraction • X-ray and Electron Diffraction • Microscopy • Spectroscopy • Property Evaluation • Mechanical • Electrical • Anything else

4. Time Line • 1665: Diffraction effects observed by Italian mathematician Francesco Maria Grimaldi • 1868: X-rays Discovered by German Scientist Röntgen • 1912: Discovery of X-ray Diffraction by Crystals: von Laue • 1912: Bragg’s Discovery

5. Electromagnetic Spectrum

6. Generation of X-rays

7. Commercial X-ray Tube

8. X-ray Spectrum from an Iron target • Short Wavelength Limit • Continuous spectrum • Characteristic X-ray Moseley’s Law λSWL

9. Use of Filter • Ni filter for Cu Target

10. Crystal Systems and Bravais Lattices

11. Structure of Common Materials • Metals • Copper: FCC • -Iron: BCC • Zinc: HCP • Silver: FCC • Aluminium: FCC • Ceramics • SiC: Diamond Cubic • Al2O3: Hexagonal • MgO: NaCl type

12. Scattering Interaction with a single particle Diffraction Interaction with a crystal Diffraction • A diffracted beam may be defined as a beam composed of a large number of scattered rays mutually reinforcing each other

13. Scattering Modes • Random arrangement of atoms in space gives rise to scattering in all directions: weak effect and intensities add • By atoms arranged periodically in space • In a few specific directions satisfying Bragg’s law: strong intensities of the scattered beam :Diffraction • No scattering along directions not satisfying Bragg’s law

14. Diffraction of light through an aperture d Intensity

15. Minima Maxima Intensity n = 1, 2,.. n = 0, 1,..

16. Young’s Double slit experiment Constructive Interference d sinθ = mλ, m = 1,2,3….. d sinθ = (m+½)λ, m = 1,2,3….. Destructive Interference

17. Phase Difference = 0˚ Phase Difference = 90˚ Phase Difference = 180˚ Interference

18. Interference and Diffraction

19. in out 2 Bragg’s Law n=2d.sin n: Order of reflection d: Plane spacing = : Bragg Angle Path difference must be integral multiples of the wavelength in=out

20. Braggs Law

21. Geometry of Bragg’s law • The incident beam, the normal to the reflection plane, and the diffracted beam are always co-planar. • The angle between the diffracted beam and the transmitted beam is always 2 (usually measured). • Sin  cannot be more than unity; this requires nλ < 2d, for n=1, λ < 2d λ should be less than twice the d spacing we want to study

22. Order of reflection • Rewrite Bragg’s law λ=2 sin d/n • A reflection of any order as a first order reflection from planes, real or fictitious, spaced at a distance 1/n of the previous spacing • Set d’ = d/n • An nth order reflection from (hkl) planes of spacing d may be considered as a first order reflection from the (nh nk nl) plane of spacing d’ = d/n λ=2d’ sin *The term reflection is only notional due to symmetry between incoming and outgoing beam w.r.t. plane normal, otherwise we are only talking of diffraction.

23. Reciprocal lattice vectors • Used to describe Fourier analysis of electron concentration of the diffracted pattern. • Every crystal has associated with it a crystal lattice and a reciprocal lattice. • A diffraction pattern of a crystal is the map of reciprocal lattice of the crystal.

24. Real space Reciprocal space Crystal Lattice Reciprocal Lattice Crystal structure Diffraction pattern Unit cell content Structure factor x’ y’ y x’ y’ x

25. Reciprocal space Reciprocal lattice of FCC is BCC and vice versa 001 a 010 b c 100

26. Ewald sphere k' k Ewald sphere Limiting sphere

27. Ewald sphere J. Krawit, Introduction to Diffraction in Materials Science and Engineering, Wiley New York 2001

28. Two Circle Diffractometer • For polycrystalline Materials

29. Four Circle Diffractometer For single crystals

30. 2 Circle diffratometer 2 and  • 3 and 4 circle diffractometer 2θ, ω, φ, χ • 6 circle diffractometer θ, φ, χ and δ, γ, µ www.serc.carleton.edu/ Hong et al., Nuclear Instruments and Methods in Physics Research A 572 (2007) 942

31. NaCl crystals in a tube facing X-ray beam

32. Powder Diffractometer

33. (110) (100) (221) (210) (211) (111) (410) (330) (321) (200) (310) (311) (320) (220) (222) (400) Calculated Patterns for a Cubic Crystal

34. Structure Factor Intensity of the diffracted beam  |F|2 • h,k,l : indices of the diffraction plane under consideration • u,v,w : co-ordinates of the atoms in the lattice • N : number of atoms • fn : scattering factor of a particular type of atom

35. Systematic Absences Permitted Reflections

36. Diffraction Methods

37. Zone axis Transmission Zone axis Reflection crystal crystal Film Film Incident beam Incident beam Laue Method • Uses Single crystal • Uses White Radiation • Used for determining crystal orientation and quality

38. Determination of unknown crystal structures Rotating Crystal Method

39. Sample Incident Beam Film Powder Method • Useful for determining lattice parameters with high precision and for identification of phases

40. Indexing a powder pattern Bragg’s Law n = 2d sin For cubic crystals

41. Indexing But what is the lattice parameter?

42. Diffraction from a variety of materials (From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley)

43. Crystallite size can be calculated using Scherrer Formula Reality (From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley) Instrumental broadening must be subtracted

44. polarization factor structure factor (F2) multiplicity factor Lorentz factor absorption factor temperature factor Intensity of diffracted beam • For most materials the peaks and their intensity are documented • JCPDS • ICDD

45. Name and formula Reference code: 00-001-1260 PDF index name: Nickel Empirical formula: Ni Chemical formula: Ni Crystallographic parameters Crystal system: Cubic Space group: Fm-3m Space group number: 225 a (Å): 3.5175 b (Å): 3.5175 c (Å): 3.5175 Alpha (°): 90.0000 Beta (°): 90.0000 Gamma (°): 90.0000 Measured density (g/cm^3): 8.90 Volume of cell (10^6 pm^3): 43.52 Z: 4.00 RIR: - Status, subfiles and quality Status: Marked as deleted by ICDD Subfiles: Inorganic Quality: Blank (B) References Primary reference: Hanawalt et al., Anal. Chem., 10, 475, (1938) Optical data: Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107 Unit cell: The Structure of Crystals, 1st Ed.

46. Stick pattern from JCPDS http://ww1.iucr.org/cww-top/crystal.index.html

47. Actual Pattern Lattice parameter, phase diagrams Texture, Strain (micro and residual) Size, microstructure (twins and dislocations) Bulk electrodeposited nanocrystalline nickel

48. Powder X-ray diffraction is essentially a misnomer and should be replaced by Polycrystalline X-ray diffraction

49. Information in a Diffraction Pattern • Phase Identification • Crystal Size • Crystal Quality • Texture (to some extent) • Crystal Structure

50. Intensity (a.u.) Analysis of Single Phase I1: Intensity of the strongest peak