1. Computational Solid State Chemistry 1 SSI-18 Workshop 2011 Rob Jackson www.robjackson.co.uk r.a.jackson@chem.keele.ac.uk

2. Contents and Plan Modelling structures and properties of ionic crystals  What is needed to model a structure?  Derivation of interionic potentials  Lattice energy minimisation  Calculation of crystal properties SSI-18 Workshop 3 July 2011 2

3. What is needed to model a structure? In order to get started, we need: – Atomic coordinates and cell parameters – A description of the forces between the atoms in the structure SSI-18 Workshop 3 July 2011 3

4. Example structure SSI-18 Workshop 3 July 2011 4 Diagram shows the fluorite structure as adopted by UO 2. A simplified structure will be shown later!

5. Example dataset : structural information # UO2 Structure parameters obtained from: # Barrett, S.A.; Jacobson, A.J.; Tofield, B.C. and Fender, B.E.F. # Acta Crystallographica B (1982) 38, 2775-2781 cell 5.4682 5.4682 5.4682 90.0 90.0 90.0 fractional 4 U core 0.00 0.00 0.00 -2.54 U shel 0.00 0.00 0.00 6.54 O core 0.25 0.25 0.25 2.40 O shel 0.25 0.25 0.25 -4.40 space 225 SSI-18 Workshop 3 July 2011 5

6. Guide to the structural information On the previous slide is listed: – The reference for the structure (optional but very useful!) – The cell parameters (a, b, c, , ,  ) – The fractional coordinates and ion charges for the unit cell (the latter explained later) – The space group SSI-18 Workshop 3 July 2011 6

7. Introduction to interatomic potentials Interatomic potentials are simple mathematical functions that describe the interactions between atoms. For ionic materials we are describing interionic interactions, and the Buckingham potential is usually used, supplemented by an electrostatic term: V(r) =q 1 q 2 /r + A exp (-r/  ) – Cr -6 SSI-18 Workshop 3 July 2011 7

8. Ion charges: rigid and polarisable ions Ions can be given their formal charges, or …  If the ions being modelled are polarisable (particularly the case for anions), they can be described by the Shell Model*, where each ion consists of a core and shell coupled by a harmonic spring. The charge is distributed between the core and shell. * B G Dick, A W Overhauser, Phys. Rev. 112 (1958) 90–103 SSI-18 Workshop 3 July 2011 8

9. The shell model explained SSI-18 Workshop 3 July 2011 9 Diagram taken from: http://tfy.tkk.fi/~asf/physics/thesis1/node23.html

10. Potential parameters In the Buckingham potential, the parameters A,  and C must be provided, and they are normally obtained by empirical fitting. The q 1 and q 2 are charges of the interacting ions. Empirical fitting involves varying the parameters until the minimum energy structure corresponds to the experimental structure. We therefore need to discuss the idea of energy minimisation as well. SSI-18 Workshop 3 July 2011 10

11. Lattice Energy Minimisation The lattice energy (LE) of a crystal is defined as the sum of the interactions between its constituent ions. Hence we can write: LE =  (Buckingham potentials) =  (V(r)) The principle behind lattice energy minimisation is that the structure is varied until a minimum value of the LE is obtained. SSI-18 Workshop 3 July 2011 11

12. Empirical fitting and lattice energy minimisation Potential fitting can be seen to be the reverse process to energy minimisation, in that the potential parameters are varied until the desired structure is obtained. So a good potential should reproduce the crystal structure without further adjustment. SSI-18 Workshop 3 July 2011 12

13. Fitting to crystal properties By fitting a potential to a structure, we should obtain a potential which can reproduce at least the crystal structure and maybe the properties as well. If experimental values of properties such as elastic and dielectric constants, or phonon modes, are available, they can be included in the fit. SSI-18 Workshop 3 July 2011 13

14. The GULP code The GULP code (General Utility Lattice Program) is written by Julian Gale, and can be downloaded from:Julian Gale http://projects.ivec.org/gulp/ It can be used to fit potentials and calculate perfect and defect properties of crystalline materials. SSI-18 Workshop 3 July 2011 14

15. Case study: potential fitting Look at Read and Jackson UO 2 paper (PDF copies available at the workshop).  M S D Read, R A Jackson, Journal of Nuclear Materials, 406 (2010) 293–303 The procedure used to fit the potential will be described. We start by looking at the data available. SSI-18 Workshop 3 July 2011 15

16. Experimental Data for Empirical Fitting S. A. Barrett, A. J. Jacobson, B. C. Tofield, B. E. F. Fender, The Preparation and Structure of Barium Uranium Oxide BaUO 3+x, Acta Cryst. 38 (Nov) (1982) 2775–2781. Elastic Constants / GPa ReferenceC 11 C 12 C 44 Dolling et al. [1]401 ± 9108 ± 2067 ± 6 Wachtman et al. [2]396 ± 1.8121 ± 1.964.1 ± 0.17 Fritz [3]389.3 ± 1.7118.7 ± 1.759.7 ± 0.3 Dielectric Constants / GPa Reference Static  0 High Frequency  ∞ Dolling et al. [1]245.3 [1] G. Dolling, R. A. Cowley, A. D. B.Woods, Crystal Dynamics of Uranium Dioxide, Canad. J. Phys. 43 (8) (1965) 1397–1413. [2] J. B. Wachtman, M. L. Wheat, H. J. Anderson, J. L. Bates, Elastic Constants of Single Crystal UO 2 at 25°C, J. Nucl. Mater. 16 (1) (1965) 39–41. [3] I. J. Fritz, Elastic Properties of UO 2 at High-Pressure, J. Appl. Phys. 47 (10) (1976) 4353–4358. 16

17. Potential fitting for UO 2 : procedure adopted The procedure followed in the paper will be described and discussed in the workshop. SSI-18 Workshop 3 July 2011 17

18. How good is the final fit? Comparison of Model with Experiment ParameterCalc.Obs. %% ParameterCalc.Obs. %% Lattice Constant [Å] 5.4682 0.0C 11 [GPa]391.4389.30.5 U 4+ – U 4+ Separation [Å] 3.8666 0.0C 12 [GPa]116.7118.7-1.7 U 4+ – O 2- Separation [Å] 2.3678 0.0C 44 [GPa]58.159.7-2.7 O 2- – O 2- Separation [Å] 2.7341 0.0Bulk Modulus [GPa]208.3204.02.1 Static Dielectric Constant 24.824.03.3 High Frequency Dielectric Constant 5.05.3-5.7 See: M S D Read, R A Jackson, Journal of Nuclear Materials, 406 (2010) 293–303 18 Keele Research Seminar, 24 November 2010

19. Final GULP input dataset for UO 2 conp opti compare prop # UO2 Structure parameters obtained from: # Barrett, S.A.; Jacobson, A.J.; Tofield, B.C. and Fender, B.E.F. # Acta Crystallographica B (1982) 38, 2775-2781 cell 5.4682 5.4682 5.4682 90.0 90.0 90.0 fractional 4 U core 0.00 0.00 0.00 -2.54 U shel 0.00 0.00 0.00 6.54 O core 0.25 0.25 0.25 2.40 O shel 0.25 0.25 0.25 -4.40 space 225 # potential from Read & Jackson,, Journal of Nuclear Materials, 406 (2010) 293–3 buck U shel O shel 1025.53 0.4027 0.0 0.0 10.4 buck4 O shel O shel 11272.6 0.1363 134.0 0.0 1.2 2.1 2.6 10.4 0 0 0 0 spring U core U shel 93.07 O core O shel 296.2 SSI-18 Workshop 3 July 2011 19

20. Further use of the potential The main motivation for modelling UO 2 was, as with most materials considered, calculation of defect properties. This will be looked at in the second session. SSI-18 Workshop 3 July 2011 20

21. Other case studies Much of my recent work has involved materials where not much more than the structure is available. We will look at an example of fitting a potential to an example material, e.g. BaAl 2 O 4 * * MV dos S Rezende, MEG Valerio, R A Jackson, submitted to Optical MaterialsOptical Materials SSI-18 Workshop 3 July 2011 21

22. Potential fitting to BaAl 2 O 4 We were interested in this material because of its applications in phosphors when doped with rare earth ions. Only structural information was available, but it was also available for a number of related compounds. A single set of potential parameters were derived by fitting to the two phases of BaAl 2 O 4, and to Ba 3 Al 2 O 6, Ba 4 Al 2 O 7, Ba 18 Al 12 O 36, and Ba 2,33 Al 21,33 O 34,33. SSI-18 Workshop 3 July 2011 22

23. Fitted potential parameters interactionA(eV)  (Å) C(eV Å 6 ) Ba - O1316.70.36580.0 Al – O1398.40.30060.0 O - O22764.0.149027.88 SSI-18 Workshop 3 July 2011 23 In this potential, a shell model has been used for O

24. Application of fitted potential SSI-18 Workshop 3 July 2011 24 References available from paper or on request

25. Conclusions for part 1 The procedure for modelling a structure has been described. Interatomic potentials have been introduced. Lattice energy minimisation and potential fitting have been introduced. Examples of potential fitting to (i) structures and properties, and (ii) structures have been given. SSI-18 Workshop 3 July 2011 25