Lectures co-financed by the European Union in scope of the European Social Fund

1. Crystallography and Diffraction. Theory and Modern Methods of Analysis Lecture 15Amorphous diffraction Dr. I. AbrahamsQueen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

2. Diffraction from Amorphous solids Unlike crystalline solids, amorphous solids show no regular repeating structure that can be defined by a lattice. In these solids atoms show a distribution of environments, that typically manifest themselves as a broadening of peaks in spectroscopic techniques such as NMR and IR. In diffraction experiments amorphous materials show no Bragg peaks, but they do exhibit scattering that can be analysed. Lectures co-financed by the European Union in scope of the European Social Fund

3. Progressive disorder approach One approach to understanding the diffraction patterns of glasses is to consider what happens to a crystalline solid and progressively introduce disorder. Consider a simple crystalline solid with a primitive unit cell and only one atom per cell. The a-axis of the unit cell is equal to the diameter of the atom. We can stack the unit cells in directions X,Y,Z. If we stack N1 atoms in direction X, N2 atoms in direction Y and N3 atoms in direction Z to give a cubic crystal containing N1N2N3 atoms, then we can calculate the intensity I at angle  as: Lectures co-financed by the European Union in scope of the European Social Fund

4. M is the multiplicity = 8, 4 or 2 (if all Nj > 0, two Nj > 0, one Nj > 0 respectively)  Is the standard deviation in Å in the distribution of inter-atomic distances P is the polarization factor f2 is the square of the atomic scattering factor. The sum is carried out over all values of N1N2 and N3 for which 0 < (N1+N2+N3) < N Lectures co-financed by the European Union in scope of the European Social Fund

5. As we increase  i.e. increase the level of disorder the pattern broadens. This method does not lend itself easily to more complex systems and so other methods of analysis are used. Lectures co-financed by the European Union in scope of the European Social Fund

6. The scattering vector Typical diffraction experiment k0 is the incident wave vector of magnitude 2/ kf = final wave vector of magnitude  magnitude of k0 Q = scattering vector = k0 – kf = 4/ Lectures co-financed by the European Union in scope of the European Social Fund

7. Differential scattering cross section In a diffraction experiment the Intensity I(Q) measured at the detector of angle d is given by: Where  is the scattering cross section,  is the fluxand is the differential scattering cross section which is defined as: The differential scattering cross section has components from distinct and self scattering. For a system containing N atoms: Lectures co-financed by the European Union in scope of the European Social Fund

8. Structure factors and correlation functions Distinct diffraction is the diffraction from different atomic sites and self diffraction is the diffraction from individual atomic sites. For a system with N atoms of n chemical species: F(Q) is the total interference function, c is the fraction of chemical species  and b is the scattering length of species . As we are mostly interested in the distribution of one species () around another () we can define F(Q) as: Where S is known as the partial structure factor. Lectures co-financed by the European Union in scope of the European Social Fund

9. The partial structure factor is given by: Where rij is the radial distance between scatterers i and j and the symbol   denotes thermal average. The partial pair distribution functions gare obtained by Fourier transformation of S Where 0 is the total number density of atoms = N/V (V = volume) Lectures co-financed by the European Union in scope of the European Social Fund

10. The number of  atoms around  atoms in a spherical shell i.e. the partial coordination number is given by integration of the partial radial distribution function. The total pair correlation function G(r) is derived by Fourier transform of the total interference function F(Q). The total correlation function T(r) is given by: where Lectures co-financed by the European Union in scope of the European Social Fund

11. Another correlation function that is often used is the differential correlation fund D(r) For X-ray scattering we need to use the X-ray scattering factor rather than the scattering length. The total interference function for X-rays is given by: GX(r) is obtained by Fourier transformation of FX(Q) as before. GX(r) can also be written as: Where Ki is the effective number of electrons for species i. Lectures co-financed by the European Union in scope of the European Social Fund

12. Neutron diffraction correlation functions for lithium borate glasses Swenson et al. Phys Rev. B. 52 (1995) 9310 Lectures co-financed by the European Union in scope of the European Social Fund

13. Q-ranges In order to get good radial distribution function data the range of Q should be large. Typically neutron data allows Q ranges up to ca. 50 Å-1 while in X-ray data the maximum useable Q value is close to 20 Å-1 For laboratory X-ray data, Cu tubes have maximum Q value around 8 Å-1. Ag tubes increase the Q-max to ca. 20 Å-1 Synchrotron radiation is commonly used for X-ray experiments. As we have seen before it is not just the Q-range that is important but the different sensitivities of X-rays and neutrons to different elements and their isotopes that make the choice of radiation important. Lectures co-financed by the European Union in scope of the European Social Fund

14. Comparison of X-ray and neutron contrast in cobalt, lead and magnesium phosphate glasses. Hoppe et al. J. Non-Crystalline Solids 293-295 (2001) 158 Lectures co-financed by the European Union in scope of the European Social Fund

15. Total correlation functions for phosphate glasses of composition NaM(P3O9) (M = Ca, Sr, Ba) T. Di Cristina PhD Thesis Queen Mary Univ. of London 2004 Fit to correlation functions for phosphate glasses of composition NaSr(P3O9) Lectures co-financed by the European Union in scope of the European Social Fund

16. Lectures co-financed by the European Union in scope of the European Social Fund

17. RMC Modelling of diffraction data Reverse Monte Carlo modelling of diffraction data is a very powerful way of structure elucidation allowing for individual pair correlations to modelled. e.g. Calcium metaphosphate glass. Wetherall et al. J. Phys. C. Condens Mater. 21 (2009) 035109 ND XRD Lectures co-financed by the European Union in scope of the European Social Fund

18. Bibliography For more detailed discussion of the theory of diffraction in amorphous solids see 1. Neutron and x-ray diffraction studies of liquids and Glasses, Henry E Fischer, Adrian C Barnes and Philip S Salmon, Rep. Prog. Phys. 69 (2006) 233–299 2. X-Ray Diffraction Procedures For Polycrystalline and Amorphous Materials 2nd Edition Harold P Klug and Leroy E Alexander, Wiley 1974. Lectures co-financed by the European Union in scope of the European Social Fund