KJM-MENA 4010 – Module 13 Powder X-ray Diffraction Theoretical Basis

1. Slides borrowed from: KJM-MENA 4010 – Module 13Powder X-ray DiffractionTheoretical Basis David Wragg Fredrik Lundvall, Georgios Kalantzapoulos, Marion Duparc

2. Literature Powder Diffraction Theory and Practice R. E. Dinnebier and S. J. L. Billinge (Eds.) Cambridge, UK, RSC Publishing 2008, 604 pp. (HB) ISBN 978-0-85404-231-9

3. Literature Fundamentals of Powder Diffraction and Structural Characterization of Materials by Vitalij Pecharsky, Peter Zavalij Paperback: 744 pages Publisher: Springer; 2nd ed. edition (November 26, 2008) ISBN-10: 0387095780

4. What is X-ray Diffraction? • X-rays are diffracted by crystalline mateials • Diffraction pattern is unique to crystal lattice • Crystal Structure-Contains all the geometric information about the material! • Crystallography • 30+ Nobel prizes • This is VITAL information!

5. Example 1 • Phase ID and quantification from powder

6. Example 2 • Structural information from powder

7. Example 3: in situ powder XRD Structure and behaviour of methanol in SAPO-34 at room temperature Wragg et al. Micro. Meso. Mater. 2010 3% change in c-axis caused by filling of cages during reaction Wragg et al, J. Catal. 2009 In situ comparison of SAPO-18 and SAPO-34 with PXRD/MS Wragg et al. Micro. Meso. Mater. 2011 Direct Observation of intermediates in SAPO-34 by PXRD. Growth from simultaneous PXRD and Raman

8. Objectives • Know the basics of how X-rays interact with matter • Understand the basic description diffraction works (Bragg’s Law) • Know how the safety hazards in the X-ray lab • Collect the appropriate data for what you want to do • Know what kind of data you can extract from a powder pattern • Know a bit about crystal symmetry

9. Part 1: X-rays and Matter

10. X-rays • Electromagnetic waves with wavelengths similar to the length of chemical bonds • Produced in the lab by bombarding a metal target with electrons in a vacuum tube • X-rays are produced by decay of electrons excited to higher energy levels in the target • Several wavelengths from different transitions

11. X-rays continued • The multiple wavelengths generated in the tube are selected using optics • Simplest is a nickel foil to remove Kβ • In our routine instrument a single wavelength (Kα1) is selected using a monochromator • Single peak for each diffraction peak

12. X-ray diffraction • When X-rays are incident with the crystal lattice of a solid they are diffracted by the planes of atoms in the lattice • The diffracted beams form a diffraction pattern due to interference • This phenomenon is described by the famous Bragg equation: nλ = 2dsinθ • Where d is thelatticespacing, λ is thewavelengthofthediffracted beam and θ is thediffraction angle

13. Powder diffraction • A diffractogram is a plot of the intensities of the diffracted beams vs their diffraction angles • There are several ways of measuring the intensities • Random distribution of particle orientations is essential! • Sample spinning helps with this

14. Diffractometer Set Ups • Simplest setup is Debye- Scherrergemometry • X-rays are fired at the sample and are diffracted as they pass through • Diffracted beams are detected on the other side • Commonly used in old film based diffractometers

15. Bragg Brentano Geometry • Most common geometry for modern powder diffraction with position sensitive detectors • “Parafocussing” geometry, orientation of the focus circle changes with detector position • This type of geometry is used on DIFF5 Videos?

16. Safety • X-ray exposure can cause serious burns and radiation related diseases • All the instruments in the lab have interlocked shielding to prevent you from being exposed to X-rays • The routine instrument has a robotic sample changer with very powerful motors • Do not get in the way of the sample changer or goniometer! • Always alert the operators of the instrument to any serious chemical hazards from your samples • Remember your risk assessments!

17. Part 2: Describing Crystal structures • How do we go from the basic unit to the crystal?

18. Symmetry!

19. The Unit Cell • Smallest repeat unit of the crystal lattice • Not the smallest repeated structural motif (asymmetric unit)! • Unit cells are classified according to their cell dimensions (a, b and c) and angles (α, β, γ) and latticecentring.

20. 7 crystal systems: All crystallinecompoundswithlong range order have a structurewhoseunit cellfitsoneofthese7 crystal systems b ≠ 90o

21. Lattices • Identicalmathematicalpoints • A 3D unit celldescribestheirrepetition in space • The most simple latticecorrespond to the 7 unit cellsshown for the 7 crystal systems, withonelatticepoint in origin • Are there more lattices (according to theirdefinitionofidenticalsurroundings for all latticepoints)? Yes… • 14 altogether

22. Primitive and non-primitive unit cells A primitive lattice contains one lattice point A non-primitive contains more than one lattice point: 2 or 4

23. Crystal systems and Bravais lattices b ≠ 90o

24. Effects of Lattice centring • In a centred lattice some reflection classes systematically have zero intensity • Systematic absences • This is used in indexing to determine the lattice type

25. Origin of Systematic Absences • Caused by interference in the same way as diffraction • Two sets of lattice planes oriented so that, although the Bragg condition is satisfied, they produce reflections which are 180° out of phase! 1 2 Path difference 1-3 = 2d sin (θ) 3 d 4 d/2 Path difference 1-2 or 3-4 = 2(d/2) sin (θ) e.g. BCC (1 0 0) reflection

26. Counting of atoms in 3D • Acorner-atom is shared between 8 cells Þ1/8atom per cell • Anedge-atom is shared between 4 cells Þ1/4atom per cell • Aface-atom is shared between 2 cells Þ1/2atom per cell • An atominside1 cell Þ1atom per cell

27. How to describe a crystal structure? Periodic lattice + Atoms (including symmetry operations) that follow the lattice

30. Symmetry operations • Mirror plane m • Rotation axis n(2,3,4,6) • Inversion axis n(1,2…) • Centrosymmetry 1 • Glide plane n, d, a, b, c • Screw axis 21, 31 Point group symmetry Special symmetry operations involving translations Remember: Mirroring is a left – right hand operation

31. Mirror plane m

32. Rotation axis 4-fold rotation axis  = 360/n

33. Rotation inversion axis n Rotation + inversion  = 360o/n

34. Screw axis, Xy Translation: y/x Rotation: 360º/x regular 6-fold rotation axis screw axis 65

35. Glide plane

36. Space groups • All 3-dimensional symmetry can be described by 230 Space groups • Symmetry elements in each group and relationships between the groups are described in volume A of the International Tables for Crystallography

38. Part 2: Powder Diffraction

39. Powder diffraction vs Single Crystal • A diffractogram is a plot of the intensities of the diffracted beams vs their diffraction angles • There are several ways of measuring the intensities • Random distribution of particle orientations is essential! • Sample spinning helps with this

40. Extracting data from the pattern • Peak positions: unit cell parameters, Miller indices, d-spacings of layers, fingerprint for phases, indexing, spacegroup determination • Intensities: Atom types present in phases, site occupancies, thermal parameters, instrumental factors(!) • Peak broadening: Instrumental factors, Crystallite size and strain, shape, structural defects

41. “Fingerprint” phase identification • Bragg’s law shows us that the diffraction pattern is very characteristic of the crystal lattice for a given phase • We can use the diffraction pattern for phase identification • Visual (if you know the pattern of your phase) • Databases (COD, PDF2, ICDD FindIt) • When we know the phase we can study it further…

42. Quantitative Analysis • If the sample contains more than one phase then the pattern corresponds to the weight percentage of each phase • Can fit using intensities of single characteristic peaks • Now easy to determine from full profile using software (TOPAS is excellent for this) • Common application of XRD in industry • Be aware that preferred orientation (see later) can cause problems

43. Extracting data from the pattern • Peak positions: unit cell parameters, Miller indices, d-spacings of layers, fingerprint for phases, indexing, spacegroup determination • Intensities: Atom types present in phases, site occupancies, thermal parameters, instrumental factors(!) • Peak broadening: Instrumental factors, Crystallite size and strain, shape, structural defects

44. Peak positions:Miller Indices • These are labels for the planes of atoms in the lattice • Each set of Miller planes gives rise to a peak in the diffraction pattern whose diffraction angle is related to the d-spacing by the Bragg equation • Take the reciprocal of the fractional intercepts on each axis and write in round brackets • i.e. - 1/2a, 1/2b, 1/2c - (222) plane • (100), (010), (001) define a, b and c axes

45. Peak Positions:Miller Indices (200) / d = 2.82Å (311) / d = 1.70Å (220) / d = 1.99Å (400) / d = 1.41Å (111) / d = 3.26Å

46. Peak positions:Indexing and cell refinement • Assignment of Miller indices to a pattern according to a specific unit cell • Can be done by hand for high symmetry examples • Use of software is more common (TREOR, TOPAS, DICVOL etc) • If we know the cell (from indexing or phase identification) we can refine the lattice parameters against the data

47. Peak positions:Lattice Parameter refinement • Changes in the lattice parameters can be used to study phase modification, e.g.: • Substitution of atoms in a phase- the substituent usually has a different atomic radius and different bond lengths to the original atom • Lattice parameters change with substitution • Can also be used to look at vacancies • Changes in peak intensities may also be observed (see later…) • Least squares refinement methods are used this can be done with simple scripts and spreadsheets or complex software packages like GSAS and TOPAS

48. Peak Positions:Space Group Determination • Crystal symmetry is described by space groups • These group lattices by the symmetry elements present • Lattice centering • Familiar mirror planes, inversion centres and rotation axes from molecular symmetry • Translational symmetry (crystal lattices only): Screw axes and glide planes • Centring and translational symmetry elements lead to further interference phenomena in the diffraction pattern: Systematic absences • Systematic absences are used to determine the spacegroup from the diffraction pattern • Difficult, limited number of observations in powder data (single crystal is easier) • Usually done with software

49. Peak positions:Space Groups • Space group (Hermann-Maugin) names are symbols describing the symmetry elements • E.g. P21/m (number 11): Primitive lattice with a 21 screw axis and a mirror plane • The 230 3D space groups are listed in volume A of the International Tables for Crystallography which are available to us at UiO through an online site license

50. Extracting data from the pattern • Peak positions: unit cell parameters, Miller indices, d-spacings of layers, fingerprint for phases, indexing, spacegroup determination • Intensities: Atom types present in phases, site occupancies, thermal parameters, instrumental factors(!) • Peak broadening: Instrumental factors, Crystallite size and strain, shape, structural defects