Torey Semi
2010 May 26
Lusk Group Meeting
1
Fuel Pin Modeling Strategy
Multiple Length and Time Scales
Continuum Physics
Lagrangian-MPM
-mechanical stress
-lengths: 10-3 - 10-1 m
-times: 100 - 109 s
MPALE
Mesoscale Physics
Calibrated Monte-Carlo (CMC)
-texture evolution (Potts-Glauber)
-pore/bubble transport (Potts-Kawasaki)
-lengths: 10-7 - 10-4 m
-times: 10-3 - 10-7 s
Atomistic Physics
Diffusion Theory
-energy (conduction)
-atomic structure
-atomic species
-gas clusters
-vacancies
-lengths: 10-11 - 10-8 m
-times: 10-16 - 10-13 s
2
Elucidate the effects of strain on:
•  Vacancies
–  single vacancy diffusion
–  formation/motion of voids
•  Fission Product Gas Atoms (Kr)
–  substitutional diffusion (high temperature)
–  coupled configuration
•  Kr atom diffusion in the presence of vacancies (midtemperatures)
•  Schottky defects with Kr atoms
–  formation/motion of clusters
3
Diffusion Coefficients via DFT
D(ε,T) = D0 (T)exp[−E a (ε) /(k B T)]
DFT provides:
€
•  activation energy, Ea
•  vibrational frequencies used to estimate D0
Goals:
•  study strain effect at atomistic level
•  inform meso-scale models
4
Analysis of Arrhenius Equation
5
Research Strategies
For calculations, use 2x2x2 supercell of UO2:
6
UO2 Magnified
7
UO2 Characteristics
•  crystallizes at room temperature and ambient
pressure in fcc fluorite structure
•  retains paramagnetic phase up to melting point
–  Tmelt = 3140K
•  is an ionically bonded insulator
–  U4+ (5f2) ions present localized magnetic moments
–  low temperature magnetic moment: 1.74µΒ/U ion
•  at TNéel = 30.8K
–  discontinuous transition from paramagnetic to
antiferromagnetic material
•  magnetic exchange interactions
•  quadrupole-quadrupole interactions
–  dynamical 1k J-T distortion to static 3k J-T distortion
8
Considerations for UO2 Electronic
Structure Calculations
•  Relativistic Effects
•  Functionals
•  Magnetism
•  Symmetry
9
DFT: The Actinide Challenge
•  Relativistic Effects
–  actinides are heavy elements
–  incorporation of relativistic effects physically important
•  Koelling-Harmon approximation
•  SOC
•  K-S Exchange-Correlation Functionals
–  no systematic way to improve
–  most problematic in materials with:
•  strongly correlated electrons
•  localized electrons
–  UO2 and other actinide oxides meet these criteria
10
The Actinide Challenge
•  Current Approaches
–  Pseudopotentials
•  include relativistic effects using scalar relativistic
approximation with spin-orbit coupling as
perturbation
•  PAW method
–  electron density derived from projector form of all-electron
wavefunction
Projector functions due to partial
waves:
11
The Actinide Challenge
•  Current Approaches
–  FP-LMTO
•  Linear Method (Linearized APW)
–  K-S equations solved by matching value and energy
derivative at MT boundary
–  non-linear, thus need for linearization
•  as RSPt, is undergoing implementation of spin-orbit
coupling as inherent relativistic effect, with spinpolarization as perturbation
–  major focus of inter-LDRD effort
–  implementation of Dirac Equation (DKS):
12
DKS Equation
where:
13
The Actinide Challenge
•  Current Approaches
–  LDA + U Functional
•  adds Hubbard term to functional to account for localized
electron interactions
•  parametrized by:
–  U: reflects strength of on-site Coulomb interaction
–  J: adjust strength of exchange interaction
•  simplified form of functional:
14
Research Strategy: Vacancies
•  First:
–  Determine efficacy of using GGA vs LDA+U
15
Research Strategy: Vacancies
•  Then:
–  Calculate Eact and D for one U vacancy, no strain
–  Goal:
•  determine diffusivity of vacancy
•  decide which functional method is optimal
–  Method: Transition State Theory
•  Modify vacancy, perform similar calculations
–  Oxygen vacancies
–  Schottky defect
16
Research Strategy: Vacancies
Add strain:
–  account for Soret effect, temperature gradients, other
forces
–  model by lattice dilation
•  equilibrium LC = 5.550 A
•  strain by .1%:
–  stretched LC = 5.555 A
–  compressed LC = 5.544 A
–  repeat goals using TST method
17
Research Strategy: Kr Atoms
•  Introduce into (vacancy) defect
–  determine energetically preferred defect site
Schottky, O vacancies, U vacancy
–  calculate activation energy to hop to adjacent defect site in
unstrained lattice
–  determine diffusivity
•  Apply strain
–  repeat calculations and compare effects to unstrained results
18
Research Strategy: Clusters/Voids
•  Quantify tendency of motion of N=2, 3, 5, 8 and
10 vacancies, Kr atoms
•
•
•
•
Agglomeration
Energy barriers
Temperature dependence
Critical size
•  Calculate diffusivity
•  Apply strain
19
Calculational Methods
•  VASP
–  plane-wave pseudo-potential code
•  treats core electrons as inert
•  calculations done on valence electrons only
–  uses PAW potentials
considered best available
•  RSPt
–  all-electron code (LAPW)
–  FP-LMTO
–  advantage: can massage potentials to work with f-electrons that
may penetrate core
–  disadvantage: settings can be difficult to choose for convergence
•  Codes are complementary
•  Exchange-correlation functionals: LDA, GGA
20
VASP method
•  Establish lattice constant using 7-point method
•  Perform geometry optimization on 2x2x2 UO2 supercell
•  Create endpoint and intermediate images for TST
calculations on U vacancy
•  Perform NEB calculations to determine activation energy
for U vacancy
21
Lattice Constant Determination
Estimated values:
LDA+U: 5.550 Å
GGA: 5.424 Å
22
Lattice Constant Comparison
Relative to LDA+U
value:
Method:
VASP
Lattice Constant, Å
PAW-SP-GGA+U
5.520
.0054
5.490
.0108
5.470
.0144
(Gupta)4
PAW-SP-GGA+U
(Nerikar)4
Experiment
(Villars, Baer)4
5.550 Å
23
Comparison of UO2 Bulk Properties
Author
Method
LC (Å)
Bulk Mod
(GPa)
Coh En
(eV/UO2)
Mag Mom Band Gap
(µΒ)
(eV)
Var. Ref.
Expt
5.47
207
22.31
1.74
2.1
Gupta
GGA
5.38
209
23.58
none
none
this work
GGA
5.42
205
23.09
none
none
Nerikar
GGA+U (D)
5.49
-
-
1.93
1.9
Gupta
GGA+U (D)
5.52
209
21.71
1.94
1.8
Yun
GGA+U (D)
5.44
209
20.26
1.89
1.8
Dorado
GGA+U (D)
5.52
214
21.30
2.00
2.1
this work
GGA+U (D)
5.55
191
21.54
2.00
2.2
this work
GGA+U (L)
5.55
191
21.82
2.00
2.4
24
Magnetism
•  Jahn-Teller Theorem
–  Any non-linear molecular system in an orbitally
degenerate electronic state will be unstable with
respect to a distortion, and will form a system of lower
energy, lifting the degeneracy.
–  Consequences for UO2:
•
•
•
•
z-axis compression
point group from Oh to Dh4
lowers ground state by ~ 50 meV
distortion of O cage by 0.086 Å
25
Magnetism
•  Occupation Matrices
–  describe:
•  magnetic states of 2 U f-electrons
•  orbital occupancies of 2 U f-electrons
–  point group symmetry:
•  completely determines form
•  UO2:
–  from Oh to Dh4 at Neel temperature
–  consistent with z-axis compression
26
Magnetism
•  Occupation Matrices
–  7x7 (l=3)
–  basis: real spherical harmonics
–  matrix entries:
•  diagonal = eigenvectors
•  non-diagonal = linear combinations of eigenvectors
•  sign determines direction of spin
–  issue:
•  LDA + U
–  Dudarev Method
–  Lichtenstein Method
27
UO2 G.S. Occupation Matrix
⎛ a
⎜
⎜ 0
⎜ b
⎜
⎜ 0
⎜ 0
⎜
⎜⎜ 0
⎝ 0
0
b
0
0
0
0
0 1− a 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 1− a 0
0
0
0
0
0
0
−b
0
0 ⎞
⎟
0 ⎟
0 ⎟
⎟
0 ⎟
−b⎟
⎟
0 ⎟
⎟
a ⎠
⎛ .3
⎜
⎜ 0
⎜.4
⎜
⎜ 0
⎜ 0
⎜
⎜⎜ 0
⎝ 0
0 .4 0
.1 0 0
0 .7 0
0
0
0
0
0
0
0
0
0
0
0
0
0
.7
0
0
0
0
0
0
0 0 0
0 −.4 0
0 ≤ a ≤1
-1 ≤ b ≤ 1 b≠0
€
⎞
⎟
⎟
⎟
⎟
0 ⎟
−.4 ⎟
⎟
0 ⎟
⎟
.3 ⎠
0
0
0
28
Magnetism
•  UO2 transitions from dynamical 1k to static 3k
–  at Neel temperature (30.8K)
–  dynamical nk
•  vibronic states tunnel between energetically equivalent
positions in distorted geometry
–  static nk
•  nuclear states confined to single well
–  above Néel temperature
•  dynamical 1k averages to crystal flourite structure
–  below Néel temperature
•  static 3k exhibits geometric (Jahn-Teller) distortion
29
Magnetism
•  Magnetically ordered nk crystal system:
–  involves expansion of the moment distribution
in a Fourier series of ‘star of k’
–  UO2 ground state:
•  magnetic order 3k
•  we use 1k to approximate
–  for U < 4.76 eV, 3k unstable (Lichtenstein)
–  U = 4.5 eV (Kotani)
–  3k and 1k energies very close
»  issue of computational efficiency
30
1x1x1 UO2 Calculations:
From Fm3m Starting Structure
Sym
Relax
O Cage
Atom
Displ
Start
Shape
End
Shape
Occ Mat
G.S. En
(eV)
off
all
no
not sgnf
cubic
tetra
g.s.
-118.42
on
all
no
no
cubic
tetra
g.s.
-118.42
off
ions
no
no
cubic
cubic
g.s.
-118.40
on
ions
no
no
cubic
cubic
g.s.
-118.40
31
Effects of Translational O Atom Displacement
O Atom
Displ
(Å)
Sym
Relax
O
Cage
U Dspl
Start
End
Occ
Mat
G.S.
En (eV)
.01
off
all
sym
sym
cubic
sl. tetra
g.s. off
-115.71
.01
off
ions
sym
sym
tetra
tetra
g.s.
-118.41
.01
off
ions
sym
sym
cubic
cubic
g.s.
-118.40
.02
off
all
sym
sym
cubic
sl. tetra
.02
off
ions
sym
sym
tetra
tetra
.02
off
ions
sym
sym
cubic
cubic
metast. -118.02
.1
off
ions
yes
yes
tetra
tetra
NOT g.s
-116.19
.1
off
ions
yes
yes
cubic
cubic
g.s. off
-118.10
metast. -118.02
g.s.
-118.41
Illustrates that:
–  Fm3m 1k gives ground state
–  symmetric and asymmetric starts do not give significantly different results
32
–  metastable states sometimes reached
Effects of ‘Random’ O Atom
Displacement
O
Atom
Displ
(Å)
Sym
Relax
O
Cage
U
Displ
Start
End
Occ
Mat
G.S.
En (eV)
.01
on
all
sym
sym
cubic
slightly
tetra
g.s. #s
off
-118.48
.01
off
all
sym
sym
tetra
tetra
g.s.
-118.48
33
Initial Structure
•  subsequent calculations based on:
–  ground state structure calculation
•  ions initially positioned symmetrically
•  symmetry ‘turned on’
•  full relaxation allowed
•  1k magnetic ordering
–  emergent structure
•  tetrahedral (J-T distortion)
•  symmetric ion positioning
–  computational efficiency
34
3-Image NEB UO2 Vacancy Calculation
Eact = 4.43 eV
DFT Trends: Uranium Vacancies
Author
Year
Experiment or
DFT
Prediction
Activation
Energy (eV)
Calculated
Diffusion
Coefficient
m2/sec
Matzke 19926
Expt
2.40
~1.4x10-20
3.09
~10-24
4.43
~10-30
Yun 20087
Yun 2008
model:
w/sp po
model:
non-magn
Durinck 20078
model
4.40
~10-30
Grimes 19919
model
2.10
~10-19
Semi Calc
model:
LDA+U
4.43
~10-30
5-Image NEB Cu Vacancy Calculation
Eact = 0.74 eV
37
Actual Result: Cu Vacancy
Author
Year
Experiment or
DFT
Prediction
Activation
Energy (eV)
Calculated
Diffusion
Coefficient
m2/sec
Soisson
200710
Expt
0.7 ± .02
~5.7x10-13
Soisson
200710
model:
SIESTA
0.64
~1.1x10-12
Semi
calculation
model:
VASP
0.74
~3.7x10-13
Current Calculations
TST (NEB) calculations
–  2x2x2 UO2 supercells
•  1 using GGA only
•  1 using LDA+U
–  3 and 5 intermediate images
–  NSW set as projected maximum
–  run until change in activation energy is less than .05
eV with respect to the number of iterations
–  determine whether to use GGA or GGA+U
–  continue on ‘Research Strategy’ path
39
References
40
References
41
Dirac Equation: Connection to RDFT
Scalar Relativistic Approximation
–  an approximation to the Dirac equation for a particle
in a (central) potential (such as an electron near a
nucleus)
–  applicable to calculation of relativistic effects in atoms
and molecules
–  approach is to treat spin-orbit term as perturbation
•  spin-orbit term is of negligible size
•  address perturbation in core (spherically)
42
Pre-exponential Factor
We use:
confirmed in VASP:
10.76 THz < Uvib freq < 17.64 THz
43
Pre-exponential Factor:
Independent of T
Case 1: T > 1675 K
44
DFT: The Actinide Challenge
•  Current Approaches
–  LDA + U Functional
•  adds Hubbard term to functional to account for localized
electron interactions
–  Pseudopotentials
•  include relativistic effects using scalar relativistic
approximation with spin-orbit coupling as perturbation
–  FP-LMTO
•  as RSPt, is undergoing implementation of spin-orbit coupling
as inherent relativistic effect, with spin-polarization as
perturbation
–  major focus of inter-LDRD effort
45
Pre-exponential Factor:
Temperature Dependence
Case 2: 1075 K < T < 1675 K:
where
# of vacancies in the
neighborhood of the Kr atom
is an Arrhenius function of T
46
UO2 2x2x2 NEB Endpoints
47