Computational Solid State Chemistry 1 SSI-18 Workshop 2011 Rob Jackson [email protected].

Computational Solid State Chemistry 1

SSI-18 Workshop 2011Rob Jackson

[email protected]

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Contents and Plan

Modelling structures and properties of ionic crystals

What is needed to model a structure?Derivation of interionic potentialsLattice energy minimisationCalculation of crystal properties

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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

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Example structure

Diagram shows the fluorite structure as adopted by UO2. A simplified structure will be shown later!

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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, 2775-2781cell5.4682 5.4682 5.4682 90.0 90.0 90.0 fractional 4U core 0.00 0.00 0.00 -2.54U 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 space225

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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

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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) =q1q2/r + A exp (-r/) – Cr-6

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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

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The shell model explained

Diagram taken from: http://tfy.tkk.fi/~asf/physics/thesis1/node23.html

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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 q2 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.

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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.

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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.

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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.

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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.

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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.

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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 BaUO3+x, ActaCryst. 38 (Nov) (1982) 2775–2781.

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

ReferenceStatic

e0

High Frequency

e∞

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.

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Potential fitting for UO2: procedure adopted

• The procedure followed in the paper will be described and discussed in the workshop.

Keele Research Seminar, 24 November 2010

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How good is the final fit?

Comparison of Model with Experiment

Parameter Calc. Obs. D% Parameter Calc. Obs. D%

Lattice Constant [Å] 5.4682 5.4682 0.0 C11 [GPa] 391.4 389.3 0.5

U4+ – U4+

Separation [Å] 3.8666 3.8666 0.0 C12 [GPa] 116.7 118.7 -1.7

U4+ – O2-

Separation [Å] 2.3678 2.3678 0.0 C44 [GPa] 58.1 59.7 -2.7

O2- – O2-

Separation [Å] 2.7341 2.7341 0.0 Bulk Modulus [GPa] 208.3 204.0 2.1

Static Dielectric Constant 24.8 24.0 3.3 High Frequency

Dielectric Constant 5.0 5.3 -5.7

See: M S D Read, R A Jackson, Journal of Nuclear Materials, 406 (2010) 293–303

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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, 2775-2781cell5.4682 5.4682 5.4682 90.0 90.0 90.0 fractional 4U core 0.00 0.00 0.00 -2.54U 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 space225

#potential from Read & Jackson, , Journal of Nuclear Materials, 406 (2010) 293–3buckU shel O shel 1025.53 0.4027 0.0 0.0 10.4buck4O shel O shel 11272.6 0.1363 134.0 0.0 1.2 2.1 2.6 10.4 0 0 0 0springU core U shel 93.07O core O shel 296.2

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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.

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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

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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.

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Fitted potential parameters

interaction A(eV) (Å) C(eV Å6)

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

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Application of fitted potential

Compound Reference Lattice parameters

Exp. Calc. % diff.

BaAl2O4 (P63) (293K)

[9] a(Å) 10.449 10.583 1.28 b(Å) 10.449 10.583 1.28 c(Å) 8.793 8.759 -0.38

BaAl2O4 (P6322) (396K)

[9] a(Å) 10.449 10.580 1.26 b(Å) 10.449 10.580 1.26 c(Å) 8.793 8.801 0.10

Ba18 Al12O36

(293K) [12]

a(Å) 16.494 16.518 0.15 b(Å) 16.494 16.518 0.15 c(Å) 16.494 16.518 0.15

Ba2,33 Al21,33O34,33

(293K) [13]

a(Å) 9.8835 9.812 -0.72 b(Å) 9.8835 9.812 -0.72 c(Å) 22.9701 22.7449 -0.98

Ba3Al2O6 (293K)

[10] a(Å) 16.5068 16.4375 -0.42 b(Å) 16.5068 16.4367 -0.42 c(Å) 16.5068 16.4367 -0.42

Ba4Al2O7 (293K)

[11] a(Å) 11.3126 11.1228 -1.68 b(Å) 11.7045 11.6469 -0.49 c(Å) 27.1850 26.6992 -1.79

References available from paper or on request

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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.