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

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. Diffraction from Amorphous solids

Lectures co-financed by the European Union in scope of the European Social Fund 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 N 1 atoms in direction X, N 2 atoms in direction Y and N 3 atoms in direction Z to give a cubic crystal containing N 1  N 2  N 3 atoms, then we can calculate the intensity I at angle  as: Progressive disorder approach

Lectures co-financed by the European Union in scope of the European Social Fund M is the multiplicity = 8, 4 or 2 (if all N j > 0, two N j > 0, one N j > 0 respectively)  Is the standard deviation in Å in the distribution of inter-atomic distances P is the polarization factor f 2 is the square of the atomic scattering factor. The sum is carried out over all values of N 1 N 2 and N 3 for which 0 < ( N 1 +N 2 +N 3 ) < N

Lectures co-financed by the European Union in scope of the European Social Fund 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 k 0 is the incident wave vector of magnitude 2  / k f = final wave vector of magnitude  magnitude of k 0 Q = scattering vector = k 0 – k f = 4  / Typical diffraction experiment The scattering vector

Lectures co-financed by the European Union in scope of the European Social Fund 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 flux and The differential scattering cross section has components from distinct and self scattering. For a system containing N atoms: is the differential scattering cross section which is defined as: Differential scattering cross section

Lectures co-financed by the European Union in scope of the European Social Fund 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. Structure factors and correlation functions

Lectures co-financed by the European Union in scope of the European Social Fund The partial structure factor is given by: Where r ij 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 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 Another correlation function that is often used is the differential correlation fund D(r) G X (r) is obtained by Fourier transformation of F X (Q) as before. G X (r) can also be written as: Where K i is the effective number of electrons for species i. 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:

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

Lectures co-financed by the European Union in scope of the European Social Fund Total correlation functions for phosphate glasses of composition NaM(P 3 O 9 ) (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(P 3 O 9 )

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

RMC Modelling of diffraction data ND XRD 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)

Lectures co-financed by the European Union in scope of the European Social Fund 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– X-Ray Diffraction Procedures For Polycrystalline and Amorphous Materials 2 nd Edition Harold P Klug and Leroy E Alexander, Wiley 1974.