Computational Solid State Chemistry Workshop: Modeling Ionic Crystals
Explore the fundamentals of computational solid-state chemistry through workshops focusing on modeling structures and properties of ionic crystals. Learn about interatomic potentials, lattice energy minimization, and calculation of crystal properties using examples like the fluorite structure of UO2. Gain insights into the essential elements needed to model a structure, including atomic coordinates, cell parameters, and forces between atoms. Discover the role of ion charges, space groups, and interatomic potentials in describing interactions between atoms in ionic materials.
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Computational Solid State Chemistry 1 SSI-18 Workshop 2011 Rob Jackson www.robjackson.co.uk r.a.jackson@chem.keele.ac.uk
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
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
Example structure Diagram shows the fluorite structure as adopted by UO2. A simplified will be shown later! structure SSI-18 Workshop 3 July 2011 4
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
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
Introduction to interatomic potentials Interatomic mathematical functions that describe the interactions between atoms. For ionic materials we are describing interionic interactions, and the Buckingham potential is usually used, supplemented electrostatic term: V(r) =q1q2/r + A exp (-r/ ) Cr-6 potentials are simple by an SSI-18 Workshop 3 July 2011 7
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
The shell model explained Diagram taken from: http://tfy.tkk.fi/~asf/physics/thesis1/node23.html SSI-18 Workshop 3 July 2011 9
Potential parameters In the Buckingham potential, the parameters A, and C must be provided, and they are normally obtained by empirical fitting. The q1 and q2are charges of the interacting ions. Empirical fitting involves parameters until the minimum energy structure corresponds to the experimental structure. We therefore need to discuss the idea of energy minimisation as well. varying the SSI-18 Workshop 3 July 2011 10
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 minimisation is that the structure is varied until a minimum value of the LE is obtained. lattice energy SSI-18 Workshop 3 July 2011 11
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
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
The GULP code The GULP code (General Utility Lattice Program) is written by Julian Gale, and can be downloaded from: 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
Case study: potential fitting Look at Read and Jackson UO2 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
Experimental Data for Empirical Fitting Elastic Constants / GPa Reference C11 C12 C44 Dolling et al. [1] 401 9 108 20 67 6 Wachtman et al. [2] 396 1.8 121 1.9 64.1 0.17 Fritz [3] 389.3 1.7 118.7 1.7 59.7 0.3 Dielectric Constants / GPa Static High Reference Frequency 0 Dolling et al. [1] 24 5.3 [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 UO2 at 25 C, J. Nucl. Mater. 16 (1) (1965) 39 41. [3] I. J. Fritz, Elastic Properties of UO2 at High-Pressure, J. Appl. Phys. 47 (10) (1976) 4353 4358. S. A. Barrett, A. J. Jacobson, B. C. Tofield, B. E. F. Fender, The Preparation and Structure of Barium Uranium Oxide BaUO3+x, Acta Cryst. 38 (Nov) (1982) 2775 2781. 16
Potential fitting for UO2: procedure adopted The procedure followed in the paper will be described and discussed in the workshop. SSI-18 Workshop 3 July 2011 17
How good is the final fit? Comparison of Model with Experiment % % Parameter Calc. Obs. Parameter Calc. Obs. Lattice Constant [ ] 5.4682 5.4682 0.0 C11 [GPa] 391.4 389.3 0.5 U4+ U4+ Separation [ ] 3.8666 3.8666 0.0 C12 [GPa] 116.7 118.7 -1.7 U4+ O2- Separation [ ] 2.3678 2.3678 0.0 C44 [GPa] 58.1 59.7 -2.7 O2- O2- Separation [ ] 2.7341 2.7341 0.0 Bulk Modulus [GPa] 208.3 204.0 2.1 Static Dielectric Constant High Frequency Dielectric Constant 24.8 24.0 3.3 5.0 5.3 -5.7 See: M S D Read, R A Jackson, Journal of Nuclear Materials, 406 (2010) 293 303 Keele Research Seminar, 24 November 2010 18
Final GULP input dataset for UO2 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
Further use of the potential The main motivation for modelling UO2 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
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. BaAl2O4* * MV dos S Rezende, MEG Valerio, R A Jackson, submitted to Optical Materials SSI-18 Workshop 3 July 2011 21
Potential fitting to BaAl2O4 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 BaAl2O4, and to Ba3Al2O6, Ba4Al2O7, Ba18Al12O36, and Ba2,33 Al21,33O34,33. SSI-18 Workshop 3 July 2011 22
Fitted potential parameters ( ) interaction A(eV) C(eV 6) Ba- O 1316.7 0.3658 0.0 Al O 1398.4 0.3006 0.0 O- O 22764. 0.1490 27.88 In this potential, a shell model has been used for O SSI-18 Workshop 3 July 2011 23
Application of fitted potential References available from paper or on request Reference Lattice Compound Exp. Calc. % diff. parameters a( ) b( ) c( ) a( ) b( ) c( ) a( ) b( ) c( ) a( ) b( ) c( ) a( ) b( ) c( ) a( ) b( ) c( ) 10.449 10.449 8.793 10.449 10.449 8.793 16.494 16.494 16.494 9.8835 9.8835 22.9701 16.5068 16.5068 16.5068 11.3126 11.7045 27.1850 10.583 10.583 8.759 10.580 10.580 8.801 16.518 16.518 16.518 9.812 9.812 22.7449 16.4375 16.4367 16.4367 11.1228 11.6469 26.6992 1.28 1.28 -0.38 1.26 1.26 0.10 0.15 0.15 0.15 -0.72 -0.72 -0.98 -0.42 -0.42 -0.42 -1.68 -0.49 -1.79 BaAl2O4 (P63) (293K) [9] BaAl2O4 (P6322) (396K) [9] Ba18 Al12O36 (293K) [12] Ba2,33 Al21,33O34,33 (293K) [13] Ba3Al2O6 (293K) [10] Ba4Al2O7 (293K) [11] SSI-18 Workshop 3 July 2011 24
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
