1. The Muppet’s Guide to: The Structure and Dynamics of Solids XRD

2. Qualitative understanding Atomic shapeSample Extension C. M. Schleütz, PhD Thesis, University of Zürich, 2009 X-ray atomic form factor Finite size of atom leads to sin  / fall off in intensity with angle

3. ∂ Scattering in Reciprocal Space Peak positions and intensity tell us about the structure: POSITION OF PEAK PERIODICITY WITHIN SAMPLE WIDTH OF PEAK EXTENT OF PERIODICITY INTENSITY OF PEAK POSITION OF ATOMS IN BASIS

4. ∂ Practical Realisation A 4-circle diffo such as in this example gives access to either vertical or horizontal scattering geometries but not both. Limited access due to the  circle. Alternative designs possible (kappa) Typically use a 4-circle machine with sample manipulator to align the sample and move in reciprocal space. Ultimate precision depends on calibration of axes against known standards. Check periodically!

5. ∂ Scattering – Q space  /2  22  22 Scanning the different axes allows reciprocal (q) space to be probed in different directions. A coupled scan of  and 2  (1:2) moves the scattering vector normal. Individual q or 2  scans move in arcs. On a symmetric reflection, a rocking curve (  ) measures the in-plane component.

6. ∂ Laboratory vs. Synchrotron Synchrotron: High flux with polarisation and energy control Complex sample environments Flexible scattering geometries Optimised control software Competitive access and time delays Laboratory Easy access Limited by flux, energy, available geometries, software, resolution and proprietary constraints

7. ∂ Sphere of Confusion Diffractometers / goniometers are mechanical systems engineered to rotate about a fixed point in space. All axes must be concentric otherwise the sample will precess about the focus. This can cause Different parts of the sample to be measured The sample to move in and out of the beam Limits sample environments More general systematic errors Modern laboratory and synchrotron systems have a sphere of confusion of <30  m, but this can cause problems if focused beams and/or small samples are used.

8. ∂ Height Errors Height errors are the main cause of systematic errors in XRD. The surface is displaced from rotation axis and this subtends an incorrect angle and an offset in 2  is introduced. Will result in incorrect values of the lattice parameter X-ray Beam Goniometer Critical that the diffractometer/goniometer rotation axis is well aligned to the incident x-ray beam.

9. ∂ Limitations and Traceability Any diffractometer must be calibrated against a standard to ensure traceability and identify systematic errors (type B). Measurements are limited by: Energy dispersion – set by the monochromator (Si 111 most common which has  E/E~10 -4 ). Angular resolution – set by slits, collimators and angular dispersion. Mechanical and thermal stability Electronics (noise) Number of peaks in a refinement Calibration (consider relative measurements) Routine measurements can give a precision of between 10 -3 and 10 -4 Å in bulk materials. Accuracy much harder to quantify.

10. ∂ Powder Diffraction It is impossible to grow some materials in a single crystal form or we wish to study materials in a dynamic process. Powder Techniques Allows a wider range of materials to be studied under different sample conditions 1.Inductance Furnace  290 – 1500K 2.Closed Cycle Cryostat  10 – 290K 3.High Pressure  Up-to 5 million Atmospheres Phase changes as a function of Temp and Pressure Phase identification

11. ∂ Powder Apparatus Bragg-Brentano uses a focusing circle to maximise flux.  /  system with the specimen fixed Tube fixed with specimen and detector scanned in 1:2 ratio (  /2  ) Parallel Beam method collimates the beam and uses a fixed incident angle. Detector scanned to measure pattern. Counts lower than B-B but penetration and hence probe depth constant.

12. ∂ Peak Widths Instrumental resolution Angular acceptance of detector Slit widths (hor. & vert.) Energy dispersion Collimation These are often summarised as the UVW parameters: Additional terms such as the Lorentz factor relate to how the reciprocal lattice point is cut by the scan type (2  or  /2  ). Peak width/shape also depends on detector slits.  /2  22

13. ∂ Peak intensities can be affected by a large range of parameters: Preferential orientation (texture), Beam footprint, surface roughness, sample volume, temperature etc. For accurate determination of strain one ideally need a large number of well defined peaks and a refinement, checking for offsets Peak positions determined from the translation symmetry of the lattice Peak intensities determined from the symmetry of the basis (i.e. atomic positions) Image courtesy J. Evans, University of Durham

14. ∂ Search and Match Powder Diffraction often used to identify phases Cheap, rapid, non-destructive and only small quantity of sample JCPDS Powder Diffraction File lists materials (>50,000) in order of their d- spacings and 6 strongest reflections OK for mixtures of up-to 4 components and 1% accuracy  Monochromatic x- rays  Diffractometer  High Dynamic range detector

15. ∂ Peak Broadening Diffraction peaks can also be broadened in q z by: 1.Grain Size 2. Micro-Strains OR Both The crystal is made up of particulates which all act as perfect but small crystals Number of planes sampled is finite Recall form factor: Scherrer Equation

16. ∂ Particle Size The crystal is made up of particulates which all act as perfect but small crystals but with a finite number of planes sampled. Ni x Mn 3-x O 4+  (400 Peak) AFM images (1200 x 1200 nm) R. Schmidt et al. Surface Science (2005) 595[1:3] 239-248 

17. ∂ Peak Shape Peaks are clearly NOT Gaussian! What can we learn from the peak shape? Nano-catalyst material in a matrix

18. ∂ ‘Grain Size’ As the scattering profile is the Fourier transform of the scattering profile that makes up the ‘Grain’ one can calculate the inverse Fourier Transform based on the fit to get the real space correlation function and the correct value of . Fit to a Pearson VII function, transform into reciprocal space and inverse FT

19. ∂ Peak Broadening Diffraction peaks can also be broadened in q z by: 1.Grain Size 2. Micro-Strains OR Both The crystal has a distribution of inter-planar spacings d hkl ±  d hkl. Diffraction over a range,  of angles Differentiate Bragg’s Law: Width in radians Strain Bragg angle

20. ∂ Peak Broadening Diffraction peaks can also be broadened in q z by: 1.Grain Size 2. Micro-Strains OR Both Total Broadening in 2  is sum of Strain and Size: Rearrange Williamson-Hall plot

21. ∂ Other contributions to width The total broadening will be the sum of size and strain dispersion. As the two contributions have a different angular dependence they can be separated by plotting: Williamson-Hall analysis Notes on W-H analysis  Likely to be noisy  Slope MUST be positive  Need to be careful if looking at non-cubic systems as the strain dispersion will depend on hkl. Warning! If extracting widths from lab sources – remember there are 2 peaks at each condition (K  1 and K  2 incident energies)

22. ∂ Grain size = 30±2nm Strain Dispersion = 0.005±0.001 Powder Diffraction Lattice Parameter Grain Size Strain Dispersion Calibration

23. ∂ Strain Peak positions defined by the lattice parameters: Strain is an extension or compression of the lattice, Results in a systematic shift of all the peaks