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

2. ∂ Kinematical Diffraction Theory Violates Energy Conservation Far-Field, Fraunhofer regime Assumes weak scattering Works surprisingly well!

3. ∂ What Happens when an EM wave meets a free electron? Electron: Dipole Moment, P Spin, S Need to consider the force that the electric and magnetic components exert on the electron..... An EM wave incident on a free electron will induce motion of both the charge and spin

4. ∂ Electric component Electric Component: Force on Electron: Force along  with  phase change

5. ∂ Electric component The electron follows the oscillating electric field creating a electric dipole moment along  No polarisation changes

6. ∂ Magnetic component Magnetic Component: Force on Electron: Zeeman Effect – tries to rotate away from H producing a torque: Force along  with  phase change

7. ∂ Magnetic component The electron oscillates creating a dipole moment along  Rotation of incident polarisation

8. ∂ Re-radiation Acta Cryst. A37, 314 (1981) F. de Bergevin and M. Brunel We have considered to two cases which produce E-dipole radiation, but what are the relative strengths?

9. ∂ Force Equation Electron motion is elliptical from the sum of the two forces: Ratio Amplitude of Forces: Magnetic force (amplitude) much weaker than charge force - x-rays measure charge ‘Soft’ x-rays, 500ev ‘Normal’ x-rays, 10keV

10. Atomic scattering factor Atomic scattering factor: Sum the interactions from each charge and magnetic dipole within the atom ensuring that we take relative phases into account: Atomic scattering factor - neutrons:

11. ∂ X-ray scattering from an Atom To an x-ray, an atom consist of an electron density,  (r). In coherent scattering (or Rayleigh Scattering) The electric field of the photon interacts with an electron, raising it’s energy. Not sufficient to become excited or ionized Electron returns to its original energy level and emits a photon with same energy as the incident photon in a different direction

12. Resonance – Atomic Environment In fact the electrons are bound to the nucleus so we need to think of the interaction as a damped oscillator. Coupling increases at resonance – absorption edges. The Crystalline State Vol 2: The optical principles of the diffraction of X-rays, R.W. James, G. Bell & Sons, (1948) Real part - dispersionImaginary part - absorption Real and imaginary terms linked via the Kramers-Kronig relations

13. ∂ Anomalous Dispersion Ni, Z=28 Can change the contrast by changing energy - synchrotrons

14. ∂ Scattering from a Crystal As a crystal is a periodic repetition of atoms in 3D we can formulate the scattering amplitude from a crystal by expanding the scattering from a single atom in a Fourier series over the entire crystal Atomic Structure Factor Real Lattice Vector: T=ha+kb+lc

15. ∂ The Structure Factor Describes the Intensity of the diffracted beams in reciprocal space hkl are the diffraction planes, uvw are fractional co-ordinates within the unit cell If the basis is the same, and has a scattering factor, (f=1), the structure factors for the hkl reflections can be found

16. ∂ The Form Factor Describes the distribution of the diffracted beams in reciprocal space The summation is over the entire crystal which is a parallelepiped of sides:

17. ∂ The Form Factor Measures the translational symmetry of the lattice The Form Factor has low intensity unless q is a reciprocal lattice vector associated with a reciprocal lattice point N=2,500; FWHM-1.3” N=500 Deviation from reciprocal lattice point located at d* Redefine q:

18. ∂ The Form Factor The square of the Form Factor in one dimension N=10N=500

19. ∂ 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

20. 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

21. ∂ 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.

22. ∂ 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.

23. ∂ 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

24. ∂ 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.

25. ∂ Alignment X-ray Beam Goniometer Critical that the diffractometer/goniometer rotation axis is well aligned to the incident x-ray beam.

26. ∂ 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.

27. ∂ 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

28. ∂ 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.

29. ∂ Powders Powder diffractometers often only have limited sample manipulation and sample preparation is key to obtaining reliable data. 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

30. ∂ 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

31. ∂ 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

32. ∂ 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

33. ∂ 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

34. ∂ 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 

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

36. ∂ ‘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

37. ∂ 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

38. ∂ 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

39. ∂ 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)

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

41. ∂ 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