Computer modelling of materials:
from nuclear fuels to nuclear clocks
Rob Jackson
24 November 2010
A wide range of materials …
From:
To:
2
Plan of talk
•
•
•
•
Why do computer modelling of materials?
Types of problem
What techniques do we use?
Examples:
– Nuclear fuels
– Optical materials
– Geological materials
– Materials for nuclear clock development
Keele Research Seminar, 24 November
2010
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Role of Computational Chemistry, and
where Materials Modelling ‘sits’
Computational Chemistry
Materials Modelling can:
• Calculate material
structures and properties.
•Help explain/rationalise
experimental data.
Fundamental calculations to:
Predict parameters often unavailable
from experiment.
Elucidate ‘mechanistic’ information.
•Predict material structures
and properties.
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Types of problem
• Modelling the structures of nuclear fuels (UO2,
PuO2, MOX)
• Modelling optical materials (YLiF4, BaMgF4)
– Predicting the location of dopant ions
– Calculating and predicting optical transitions
• Modelling geological materials (e.g. zircon,
ZrSiO4 and related materials )
– USiO4 → PbSiO4
• Nuclear clocks: 229Th doping in LiCaAlF6
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Techniques
• The main technique employed is atomistic
modelling.
• The material structure (lattice parameters, ion
positions) is provided, and interactions between
ions are defined by interionic potentials:
– These are simple mathematical functions that
represent the important interactions between atoms:
• Van der Waals forces
• Electron repulsion
• A well-known example is the Lennard-Jones
potential.
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Sir John Lennard-Jones (1894-1954)
http://www.quantum-chemistry-history.com/Le-Jo1Ue.htm#continue
• Sir John Lennard-Jones was a mathematical
physicist who became the first professor of
theoretical chemistry in the UK, in
Cambridge, where he worked from 19321953.
• He was born John Jones; Lennard was his
wife’s surname.
• In 1953 he was appointed 2nd Principal of
the
University
College
of
North
Staffordshire, which later became Keele
University.
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The Lennard-Jones potential
• Lennard-Jones developed his potential in
1931, 22 years before coming to Keele:
V = Ar-12 – Cr-6
• The first term represents electron repulsion,
and the second van der Waals attraction. A
potential is thus defined for the interaction
between each pair of atoms. How the
parameters are obtained could fill another
seminar!
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More on techniques used
• The basis of atomistic simulation is energy
minimisation: structures are calculated
corresponding to an energy minimum and
properties are calculated for that structure.
• We are interested in defects; they destroy the
periodicity of the unit cell, and need special
treatment, and a method called the MottLittleton approximation is used.
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Mott-Littleton approximation
© Mark Read (AWE)
Region I
Ions are strongly perturbed
by the defect and are relaxed
explicitly with respect to
their Cartesian coordinates.
Defect
Region I
Region IIa
Region II
Ions are weakly perturbed
and
therefore
their
displacements, with the
associated
energy
of
relaxation,
can
be
approximated.
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Modelling nuclear fuels
• Motivation: understanding the effect of the
fission process on the structure and properties
of UO2, PuO2 and other actinide oxides.
• This work forms part of a collaboration with
AWE, and is the basis of Scott Walker’s PhD
project.
• In addition, Gemma Turner (3rd year project
student) is modelling MOX (mixed oxide fuel,
UO2/PuO2).
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Why Study Uranium Dioxide?
Understanding Corrosion
Understanding factors limiting or inducing uranium corrosion is of
interest to a variety of industrial activities. [2]
Extreme affinity of pure uranium for oxygen is well documented. At
least 16 oxides are known between UO2 and UO3 and are the principal
products of uranium metal corrosion.
Once formed as a layer on the surface of metallic uranium, the oxides
act as a passive barrier to further corrosion. [2,3]
[1]
Thus it is the generally accepted view that the reactivity of uranium
towards various gases is primarily affected by the properties of its
native oxide layer. For example, in the case of uranium–hydrogen
systems, the surface oxide layer prevents rapid concentration of
hydrogen at the metal surface and, as a result, provides a limiting
influence on the onset of the gas–solid reaction that forms pyrophoric
uranium hydride (UH3). [3]
[1] R. J. Pearce, I. Whittle, D. A. Hilton, The Oxidation of Uranium in Carbon Dioxide and Carbon Monoxide (A Review), J. Nucl. Mater. 33 (1969) 1-16.
[2] J. R. Petherbridge, T. B. Scott, J. Glascott, C. Younes, G. C. Allen, I. Findlay, Characterisation of the surface over-layer of welded uranium by FIB, SIMS and
Auger electron spectroscopy, J. Alloys Compd. 476 (1-2) (2009) 543–549.
[3] R. M. Harker, The influence of oxide thickness on the early stages of the massive uranium-hydrogen reaction, J. Alloys Compd. 426 (1-2) (2006) 106–117.
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Simulation of Uranium Dioxide
Simulation of the bulk lattice →
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Experimental Data for Empirical Fitting
Elastic Constants / GPa
Reference
Dolling et al. [1]
Wachtman et al. [2]
Fritz [3]
C11
C12
C44
401 ± 9
108 ± 20
67 ± 6
396 ± 1.8
121 ± 1.9
64.1 ± 0.17
389.3 ± 1.7
118.7 ± 1.7
59.7 ± 0.3
Dielectric Constants / GPa
Static
Reference
Dolling et al. [1]
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.
e0
High
Frequency
e∞
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.
14
How good is the 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+ – O2Separation [Å]
2.3678
2.3678
0.0
C44 [GPa]
58.1
59.7
-2.7
O2- – O2Separation [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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Some results from UO2 modelling
• Formation energies for defects (vacancies,
dopants) in the structure can be obtained.
• Location of dopant ions in the structure, and
atoms formed from fission processes (e.g. Xe)
can be predicted.
• Surface energies can be calculated and crystal
morphology predicted:
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Surface Simulations
Morphology
If UO2 crystallites attain thermodynamic
equilibrium, the morphology will be dominated by
the (111) surfaces, forming an octahedron
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Modelling optical materials
• Motivation: we are interested in helping to
develop new materials for optical applications,
including solid state lasers and scintillators for
detection of ionising radiation.
• Interesting (and useful) optical properties can
be added to metal fluorides and metal oxides
by doping, usually with lanthanide elements.
• This topic is the theme of Tom Littleford’s PhD
project (also funded by AWE).
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Blue John: CaF2 with F-centres
• The picture shows a
sample of Blue John,
CaF2 coloured by the
presence of F-centres
(electrons trapped at
vacant F- sites in the
crystal).
• There is a Blue John
mine at Castleton in
Derbyshire.
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Amethyst: SiO2 with Fe3+ impurities
• The picture shows a
sample of amethyst, which
is quartz, SiO2 doped with
Fe3+ ions from Fe2O3.
• The value of the quartz is
drastically increased by the
presence of a relative
small number* of Fe3+
ions!
http://www.gemstone.org/gem-by-gem/english/amethyst.html
*’As much iron as would fit on the head of a pin can colour one cubic foot of quartz’
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More on amethyst
• The colour is due to the Fe3+
ions occupying the Si4+ sites,
so a charged [FeO4]4- centre
results.
• The amount of iron present
is very small, about 40 parts
per million!
Brazilian amethyst, value $94.50
(June 2007)
http://www.mineralminers.com/html/ameminfo.htm#items
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Doping for technological applications
• For most applications, doping with rare earth
cations is carried out:
The rare earth
elements
are
chosen because of
their
emission
wavelengths
as
dopants (in the
m range).
http://perso.univ-rennes1.fr/martinus.werts/lanthanides/ln_descr.html
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Host Materials: mixed metal fluorides
E M Maddock, PhD
thesis (2010)
KY3F10
Cubic (Fm3m)
a = b = c = 11.543 Å
KYF4
Hexagonal (P31)
K2YF5
a = b = 14.060 Å
c = 10.103 Å
Orthorhombic (Pna21)
Plus K3YF6
Monoclinic (P21/n)
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a = 10.791 Å
b = 6.607 Å
c = 7.263 Å
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Structural modelling studies of KYF
materials (E M Maddock, PhD thesis 2010)
• A common set of interatomic potentials was fitted to
all 4 materials, giving reasonable agreement with
structures to within a few % (better than 1% for KYF4
shown below).
KYF4
a=b
c
Exp (Å)
14.060
10.103
Calc (Å)
13.953
10.185
Keele Research Seminar, 24 November 2010
% Diff
-0.76
0.81
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Solution energies for RE doping
• Solution energies give the total energy
needed for doping to take place.
• Potentially 2 sites are available, Y and K.
• Solution energies were calculated for doping
at the Y3+ site (and the K+ site with various
forms of charge compensation).
• As expected the lowest energy site is the Y3+
site (no charge compensation needed).
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Crystal morphology and RE doping
•
We are interested in the answers to 2
questions here:
1. What is the crystal morphology of the pure
materials and how is it affected by doping?
2. Do the dopants have a tendency to
segregate to the crystal surface?
• In both cases there are implications for the
use of doped materials in devices.
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Morphology: Wulff plots
• Wulff plots can be
constructed to give
morphologies based
on surface energies,
& also issues like low
indices & interplanar
spacing.
• An example is shown
for KY3F10:
Miller index
Esurface/Jm-2
1 -1 1
0.8764
200
1.9114
2 0 -2
1.1617
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Surface segregation of dopants
• If a material is doped, it is important to know if
the dopant ion remains in the bulk or moves to
the surface.
• The segregation energy (Eseg) of a dopant is
defined as the difference between the energy to
substitute it at the surface and in the bulk:
Eseg = E (dopant, surface) – E (dopant, bulk)
• A negative value of the segregation energy indicates that
there will be a tendency for surface segregation to occur
for a particular dopant.
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Segregation energies to dominant
surfaces in KY3F10 / eV
dopant
1 -1 1
200
2 0 -2
dopant
1 -1 1
200
2 0 -2
La
-2.97
-2.11
-2.86
Tb
0.01
0.29
0.11
Ce
-2.52
-1.58
-2.32
Dy
0.54
0.89
0.63
Pr
-2.05
-1.75
-1.79
Ho
0.73
1.42
1.178
Nd
-1.65
-0.98
-1.80
Er
1.05
1.54
1.30
Sm
-1.13
-0.60
-0.90
Tm
1.11
1.64
1.35
Eu
-0.25
0.03
-0.52
Yb
1.37
2.01
1.23
Gd
-
0.29
0.03
Lu
1.80
2.156
1.87
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Modelling optical properties
• As well as understanding what happens to the
structure and morphology, we are interested
in trying to predict the optical transitions of
dopant ions.
• This can be done in 2 ways:
– Crystal field calculations
– Quantum mechanics (embedded clusters)
• Some results from crystal field calculations will
be shown:
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Calculation of energy levels for LaF3: Ce3+
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)
Exp. [11]
Calculated
0
151
280
0
428
567
Term symbol
Energy levels in cm-1
2
F5/2
Exp. [11]
Calculated
2160
2240
2635
2845
2212
2555
2891
2973
Poor agreement for low
energy transitions
Term
symbol
2
F7/2
Much better agreement (within
10% or better) for higher energy
transitions
[11] R A Buchanan, H E Rast, H H Caspers, J. Chem. Phys. 44 4063 (1966)
Keele Seminar - 13 June 2007 (KPA)
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Modelling geological materials: Zircon
and coffinite
• Zircon readily accommodates U at the Zr site,
and the fully substituted compound, USiO4, 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:
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Coffinite
Predicted coffinite
structure
Exp (Å) Calc (Å) %
a=b 6.995
6.874
-1.8
c
6.371
-1.7
6.262
• The structure is
predicted to better
than -2%
• Structures for the
full range of solid
solutions can be
calculated.
Black, interstitial coffinite cementing a sub-angular quartzose sandstone. Schumacher Coll.
(Temple Mountain, San Rafael District (San Rafael Swell), Emery Co., Utah, USA)
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Coffinite and radioactive decay
• U decays radioactively,
PbSiO4
eventually to Pb.
Exp (Å) Calc (Å) % • Due to the long t of U,
1/2
the oldest samples of
6.489
a=b ?
coffinite found have
6.102
c
?
around 3% Pb.
Older samples of coffinite are
• The structure of the end
being searched for.
member, PbSiO4, can be
predicted,
as
can
the
full
Attempted synthesis of PbSiO4
Pb-U solid solution.
(Keelite) is in progress!
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Development of nuclear clocks
•
229Th
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!
• They also have advantages of improved
stability over existing atomic clocks.
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Practical considerations
• The 229Th nucleus needs to be embedded in a
VUV-transparent crystal for use in devices.
• Metal fluorides, e.g. LiCaAlF6/LiSrAlF6 have been
identified as being suitable.
• A modelling study was therefore carried out, to
find where the Th ions substitute in the lattice.*
* Details in ‘Computer modelling of thorium doping in LiCaAlF6 and
LiSrAlF6: application to the development of solid state optical
frequency devices’ by Jackson et al, Journal of Physics: Condensed
Matter 21 (2009) 325403
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Results of modelling study and
planned experimental study
• The modelling showed that the Th4+ ions
preferentially substitute at the Ca2+ site, with
charge compensation by F- interstitial ions.
• Crystal growth experiments are in progress,
but hindered by the difficulties of growing
fluoride systems, plus the cost (and location)
of 229Th ($50k/mg!).
• This is a collaboration with UCLA and LANL.
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Pyrochlores and Defect Fluorite
Materials (with Richard Darton)
BiIII2ZrIV2O7
Defect Fluorite
BiIII2TiIV2O7
Pyrochlore
BiIII2HfIV2O7
Pyrochlore
Bi2Zr2O7 : where modelling can help
• Can Bi2Zr2O7 exist as a pyrochlore phase ?
• Can we predict intermediate structures ?
BiIII2ZrIV2O7
BiIII2TiIV2O7
BiIII2HfIV2O7
• Can the structure be doped with +2, +3 and +4 cations ?
• e.g. SrTiZr2O7
• (Doping will change structure and therefore properties)
• e.g. dielectrics, nuclear waste storage materials
• Can we predict new materials ?
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Plan for pyrochlores project
• Model Bi2Ti2O7 (known structure, but …).
• Substitute Zr for Ti and calculate the energy
minimised structure.
• Compare with the structure synthesised by
Luke Daniels (predict powder pattern and
compare with experimental pattern).
• We can then model intermediate structures
and doped materials.
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Summary
• The technique of materials modelling has
been introduced and set in the overall context
of computational chemistry.
• Some current examples have been considered,
both of complete and ongoing projects.
• I have (hopefully) given you an idea of the
scope of the technique, and what can be
achieved.
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Acknowledgements
Liz Maddock, Tom Littleford, Scott Walker, Michael Montenari,
Richard Darton (Keele)
Mark Read, Dave Plant (AWE)
Mário Valerio, Jomar Amaral, Marcos Rezende (UFS)
Eric Hudson (UCLA)
Keele University Centre for the Environmental, Physical and
Mathematical Sciences (iEPSAM)
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