Computational Solid State Chemistry 1 SSI-18 Workshop 2011 Rob Jackson ...

Computational Solid State Chemistry 1

SSI-18 Workshop 2011Rob Jackson

[email protected]

http://www.robjackson.co.uk/

mailto:[email protected]

SSI-18 Workshop 3 July 2011 2

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


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


Example structure

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


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


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


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


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


The shell model explained

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


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.


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.


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.


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.


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.

http://nanochemistry.curtin.edu.au/people/julian.cfm

http://projects.ivec.org/gulp/


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.

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 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.


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

18

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


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


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.


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

http://www.elsevier.com/wps/find/journaldescription.cws_home/522512/description#description


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.


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


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


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.

Computational Solid State Chemistry 1 SSI-18 Workshop 2011 Rob Jackson ...

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