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

2. ∂ Characterisation Over the course so far we have seen how thermodynamics plays an important role in defining the basic minimum energy structure of a solid. Small changes in the structure (such as the perovskites) can produce changes in the physical properties of materials Kinetics and diffusion also play a role and give rise to different meta- stable structures of the same materials – allotropes / polymorphs Alloys and mixtures undergo multiple phase changes as a function of temperature and composition BUT how do we characterise samples?

3. ∂ Probes Resolution better than the inter-atomic spacings Electromagnetic Radiation Neutrons Electrons a b

4. ∂ Probes Treat all probes as if they were waves: Wave-number, k: Momentum, p: Photons ‘Massive’ objects

5. ∂ Xavier the X-ray E x (keV)=1.2398/ (nm) Speed of Light Planck’s constant Wavelength Elastic scattering as E x >>k B T

6. ∂ X-ray Sources http://pd.chem.ucl.ac.uk/pdnn/inst1/xrays.htm

7. ∂ Synchrotrons Electrons at GeV in a storage ring. Magnetic field used to accelerate in horizontal direction giving x-rays (dipole radiation)

8. ∂ Norbert the Neutron E n (meV)=0.8178/ 2 (nm) De Broglie equation: mass velocity Non-relativistic Kinetic Energy: Strong inelastic scattering as E n ~k B T

9. ∂ Fission Thermal Neutron 235 U 2.5 neutrons + heavy elements + 200 MeV heat ILL: Flux Density of 1.5x10 13 neutrons/s/mm 2 at thermal power of 62MW

10. ∂ Spallation 800 MeV Protons excite a heavy nucleus Protons, muons, pions.... & 25 neutrons Pulsed Source - 50Hz ISIS - Rutherford Appleton Lab. (Oxford). ESS being built in Lund.

11. ∂ Eric the Electron Eric’s rest mass: 9.11 × 10 −31 kg Eric’s electric charge: −1.602 × 10 −19 C No substructure – point particle De Broglie equation: mass velocity E e depends on accelerating voltage :– Range of Energies from 0 to MeV

12. ∂ Probes Electrons - Eric quite surface sensitive Electromagnetic Radiation - Xavier Optical – spectroscopy X-rays sensitive to electrons: VUV and soft (spectroscopic and surfaces) Hard (bulk like) Neutrons – Norbert Highly penetrating and sensitive to induction Inelastic

13. ∂ Interactions 1. Absorption 2. Refraction/Reflection 3. Scattering Diffraction Eric Xavier Norbert

14. ∂ a* b* 100 300 200 400 010 120 110 0-10 020 030 0-20 130 230 210 330 310 220320 1-10 4-10 3-10 2-10 Diffraction – a simple context 3D periodic arrangement of scatterers with translational symmetry gives rise to a real space lattice. The translational symmetry gives rise to a reciprocal lattice of points whose positions depend on the real space periodicities http://pd.chem.ucl.ac.uk/pdnn/diff1/recip.htm Real SpaceReciprocal Space a b

15. ∂ Interference View Constructive Interference between waves scattering from periodic scattering centres within the material gives rise to strong scattering at specific angles. a* b* 100 300 200 400 010 120 110 0-10 020 030 0-20 130 230 210 330 310 220320 1-10 4-10 3-10 2-10

16. ∂ Reciprocal Lattice of Si  rotation about [001] (010) plane(110) plane

17. ∂ Basic Scattering Theory The number of scattered particles per second is defined using the standard expression Unit solid angle Differential cross-section Defined using Fermi’s Golden Rule

18. ∂ Spherical Scattered Wavefield Scattering Potential Incident Wavefield Different for X-rays, Neutrons and Electrons

19. ∂ BORN approximation: Assumes initial wave is also spherical Scattering potential gives weak interactions Scattered intensity is proportional to the Fourier Transform of the scattering potential

20. 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:

21. ∂ 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

22. 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

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

24. ∂ 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

25. ∂ 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

26. ∂ 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:

27. ∂ 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:

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

29. ∂ 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

30. Qualitative understanding Atomic shapeSample Extension C. M. Schleütz, PhD Thesis, Univerity of Zürich, 2009 X-ray atomic form factor