1. Computer Modelling of Materials for Optical and Energy Applications Robert A Jackson Lennard-Jones Laboratories, School of Physical & Geographical Sciences, Keele University, Keele, Staffs ST5 5BG http://www.robjackson.co.uk r.a.jackson@chem.keele.ac.uk

2. Talk contents 1.Introduction & motivation 2.Methodology 3.Modelling dopants in mixed metal fluorides & oxides 4.Modelling nuclear fuels 5.Modelling zircon & related materials including radioactive decay products 6.Modelling concentration dependence of dopants 7.Future work 8.Acknowledgements Huddersfield Seminar 30/11/11 2

3. Introduction and motivation We are interested in using computer modelling to assist in the understanding, design and optimisation of new materials for specific applications. Applications of current interest are in optical devices, and materials relevant to nuclear energy generation. We have also applied our methods to some geologically important materials. Huddersfield Seminar 30/11/11 3

4. Methodology The calculations described today are all based on the use of empirically derived potentials to describe interactions between ions, and methods based on energy minimisation to determine structures and lattice properties. We have a long term aim to use quantum mechanics for some specific problems, which will be mentioned at the end of the talk. Huddersfield Seminar 30/11/11 4

5. 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 Huddersfield Seminar 30/11/11 5

6. Empirical fitting 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 and properties they predict corresponds to the experimental values. Huddersfield Seminar 30/11/11 6

7. Empirical fitting case study An example of detailed potential fitting is available: – M S D Read, R A Jackson, Journal of Nuclear Materials, 406 (2010) 293–303 In this paper, the potential is fitted to the structure and lattice properties of UO 2. PDF copies of the paper are available from: http://www.robjackson.co.uk/somepublications.aspx Huddersfield Seminar 30/11/11 7

8. UO 2 Experimental Data 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 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. 8 Huddersfield Seminar 30/11/11

9. How good is the final fit? (More details in paper) 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 9 Huddersfield Seminar 30/11/11 Note that it is unusual to have this amount of data to fit to!

10. Defects in materials Most interesting properties are due to the presence of defects! Huddersfield Seminar 30/11/11 10 The picture shows a sample of amethyst, which is quartz, SiO 2 doped with Fe 3+ ions from Fe 2 O 3. The value of the quartz is drastically increased by the presence of a relatively small number of Fe 3+ ions!

11. Defect calculations We are mainly interested in:  Calculation of energies of formation of defects  Modelling ion migration  Modelling doping in crystals  Calculating substitution and solution energies  Determining location of dopants  Determining dopant concentrations (new) Point defect calculations generally use the Mott-Littleton approximation. Huddersfield Seminar 30/11/11 11

12. Mott-Littleton approximation Region I Ions are strongly perturbed by the defect and are relaxed explicitly with respect to their Cartesian coordinates. Region II Ions are weakly perturbed and therefore their displacements, with the associated energy of relaxation, can be approximated. Region IIa Defect Region I © Mark Read (AWE) 12 Huddersfield Seminar 30/11/11

13. Substitution and solution energies Substitution energies are the energies involved in substituting an ion into the material, but they do not take into account all the energetic terms involved in the solution process. Solution energies include all these terms, so they can be used to determine where the ion will substitute, and what form of charge compensation will occur (if it is needed). Huddersfield Seminar 30/11/11 13

14. Case study: modelling Th 4+ in LiCaAlF 6 * Aim: to illustrate all the steps involved in a study of doping a material. 1.Derive and test a potential for LiCaAlF 6. 2.Derive and test a potential for ThF 4. 3.From defect calculations, determine preferred location of Th 4+, and the charge compensation involved. * Journal of Physics: Condensed Matter 21 (2009) 325403 Huddersfield Seminar 30/11/11 14

15. 1. Modelling LiCaAlF 6 * A potential was previously fitted to the LiCaAlF 6 structure; parameters are given in the reference below. Good structural agreement was obtained: *Journal of Physics: Condensed Matter 15 (2003) 2523–2533 Huddersfield Seminar 30/11/11 15 ParameterExperimentalCalculated% difference a = b (Å)5.015.030.42 c (Å)c (Å)9.649.62-0.24 γ (deg)120.0 0.0

16. 16 Huddersfield Seminar 30/11/11 2. Modelling ThF 4 A potential was fitted to the ThF 4 structure, giving agreement as shown below: [1] G Benner and B G Mueller, Zeitschrift für Anorganische und Allgemeine Chemie 588 (1990) 33-42 Potential parameters are given in the 2009 JPCM reference (slide 14).

17. 3. Where does Th substitute in LiCaAlF 6 ? Whichever cation site Th 4+ substitutes at in this material, charge compensation will be needed. 10 possible reaction schemes were considered, and solution energies were calculated for each one. The lowest energy scheme involves substitution at the Ca 2+ site, with charge compensation by 2 fluorine interstitial ions: ThF 4 + Ca Ca → Th Ca  + 2F i ’ + CaF 2 Huddersfield Seminar 30/11/11 17

18. 18Huddersfield Seminar 30/11/11 Solution energies (eV) for different solution schemes 1.951.962.842.151.201.471.95 Charge compensation by vacancies ½ ½ 0½ 0 0¼ ¼ 0¾ ½ 0 ½ ½ 0 ½ 0 0 ½ ½ 0 ¼ ¼ 0 ½ ½ 0 ¾ ½ 0 ½ 0 0 ¼ ¼ 0 ½ 0 0 ¾ ½ 0 ¼ ¼ 0 ¾ ½ 0 2.362.432.462.240.971.000.831.080.980.96 Charge compensation by interstitials

19. Conclusions on case study From solution energy calculations, Th 4+ is predicted to substitute at the Ca 2+ site, with charge compensation by F - interstitials. This is an important result because of the possible effects of charge compensating defects on the optical properties of the doped material. Crystal growth is in progress, using 232 Th initially. Huddersfield Seminar 30/11/11 19

20. Background: nuclear clocks 229 Th is being investigated for use in ‘nuclear clocks’; its first nuclear excited state is (unusually) only ~ 8 eV above the ground state, and can be probed by VUV radiation. Nuclear clocks promise up to 6 orders of magnitude improvement in precision over next generation atomic clocks, as well as enhanced stability. Th has to be doped into a suitable crystal. 20 Huddersfield Seminar 30/11/11

21. Candidate crystals for Th doping LiCaAlF 6 and LiSrAlF 6 are being investigated, as is CaF 2. This is a collaboration with two groups, in UCLA and Vienna, where crystal growth is being carried out. 229 Th costs $50k/mg, so the cheaper 232 isotope is being used initially! Huddersfield Seminar 30/11/11 21

22. 232 Th doped CaF 2 http://www.thorium.at/?p=481 Huddersfield Seminar 30/11/11 22

23. Modelling nuclear fuels The derivation of a potential for UO 2 has already been discussed. Having previously worked on nuclear materials in the 1980s, interest in nuclear power has returned (at least in some countries!), and there is new motivation for research. We are studying UO 2 and PuO 2, and the mixed oxide MOX. Huddersfield Seminar 30/11/11 23

24. MOX structural prediction: mean field approach Huddersfield Seminar 30/11/11 24

25. MOX structural prediction: supercell approach Huddersfield Seminar 30/11/11 25

26. Huddersfield Seminar 30/11/1126 Zircon, ZrSiO 4, readily accommodates U at the Zr site, and the fully substituted compound, USiO 4, is the mineral coffinite. Starting with zircon and progressively substituting U at the Zr site allows the structure of coffinite to be predicted, and the result can be compared with the experimental structure: Modelling zircon and related materials

27. Structure prediction of coffinite The structure is predicted to better than -2% Structures for the full range of solid solutions can be calculated. Predicted coffinite structure Exp (Å)Calc (Å)% a=b6.9956.874-1.8 c6.2626.371-1.7 Black, interstitial coffinite cementing a sub-angular quartzose sandstone. Schumacher Coll. (Temple Mountain, San Rafael District (San Rafael Swell), Emery Co., Utah, USA) 27

28. Coffinite and radioactive decay U decays radioactively, eventually to Pb. Due to the long t 1/2 of U, the oldest samples of coffinite found so far have around 3% Pb. The structure of the end member, PbSiO 4, can be predicted, as can the full Pb-U solid solution. PbSiO 4 Exp (Å)Calc (Å) % a=b? 6.489 c? 6.102 Attempted synthesis of PbSiO 4 (Keelite) is in progress! Older samples of coffinite are being searched for. 28 Huddersfield Seminar 30/11/11

29. Modelling concentration dependence of doping Motivation – for optical materials, dopants are responsible for their important properties We can predict where they substitute in the lattice, and what form of charge compensation will be preferred. We would like to be able to predict how much dopant can be added! We have developed a method to do this … Huddersfield Seminar 30/11/11 29

30. New method – application to M 3+ doped BaAl 2 O 4 To explain the new method, we move from fluorides to oxides, and consider the formation of M 3+ doped BaAl 2 O 4, which has applications as a phosphor material. Instead of just calculating the solution energy for one M 3+ ion in the structure, we mimic the preparation process for BaAl 2-x M x O 4 : 0.5x M 2 O 3 + BaO + (1-0.5x) Al 2 O 3  BaAl 2-x M x O 4 Huddersfield Seminar 30/11/11 30

31. Concentration dependent solution energies The energy of the reaction is: E sol = E [BaAl 2-x M x O 4 ] - [0.5x E latt (M 2 O 3 ) + E latt (BaO) + (1-0.5x) E latt (Al 2 O 3 )] Where E [BaAl 2-x M x O 4 ] = (1-0.5x) E latt (BaAl 2 O 4 ) + x E (M Al ) Note that the perfect and defective terms have been separated. This gives an expression for E sol as a function of x. Huddersfield Seminar 30/11/11 31

32. Some results We can calculate solution energies for M 3+ ions in BaAl 2 O 4 as a function of x: – Energies in eV, T = 293 K Huddersfield Seminar 30/11/11 32 1% M 2 O 3 2% M 2 O 3 3% M 2 O 3 Max. x M 2 O 3 Ce-0.84661.09693.04041.4356 Pr-0.84261.10483.05221.4327 Nd-0.84161.10683.05521.4319 Sm-0.83941.11123.06181.4303

33. Interpretation of results Negative solution energies imply solution of the dopant in the crystal structure. The procedure is to increase the concentration, x, until the solution energy is zero, and this represents the solution limit. The method is still being developed and compared with experimental predictions. Huddersfield Seminar 30/11/11 33

34. Future work We are interested in calculating the electronic structure of dopants in optical materials, with a view to predicting energy transitions. This has already been done with crystal field methods, but the ultimate aim is to use embedded cluster quantum mechanical approaches. Some crystal field results are given on the next slide: Huddersfield Seminar 30/11/11 34

35. Huddersfield Seminar 30/11/11 35 Crystal field calculations on LaF 3 : Ce 3+ [11] R A Buchanan, H E Rast, H H Caspers, J. Chem. Phys. 44 4063 (1966) Poor agreement for low energy transitions Much better agreement (within 10% or better) for higher energy transitions R A Jackson, M E G Valerio, J B Amaral, M A Couto dos Santos and E M Maddock Phys. Stat. Sol. (c) 4(3) 1185-88 (2007) Energy levels in cm -1

36. Embedded QM calculations The eventual aim is to be able to use embedded quantum mechanics to model the dopant and surrounding ions in detail, using a potentials-based approach for more distant ions. Such methods exist (e.g. ChemShell) but they are at an early stage of development … To be pursued, funding and personnel permitting! Huddersfield Seminar 30/11/11 36

37. Acknowledgements 37 Huddersfield Seminar 30/11/11 Tom Littleford, Scott Walker, Michael Montenari, Richard Darton (Keele) Mark Read, Dave Plant (AWE) Mário Valerio, Jomar Amaral, Marcos Rezende (UFS, Brazil) Eric Hudson (UCLA), Thorsten Schumm (TU Wien)

38. Thank you! 38