defect structure in zirconia-based materials
DESCRIPTION
Defect structure in zirconia-based materials. Jan Kuriplach Department of Low Temperature Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic. Cooperation. J. Čížek, O. Melikhova, I. Procházka W. Anwand, G. Brauer (Dresden). Outline. - PowerPoint PPT PresentationTRANSCRIPT
Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Defect structure in zirconia-based materials
JanJan KuriplachKuriplach
Department of Low Temperature PhysicsFaculty of Mathematics and Physics
Charles University, Prague, Czech Republic
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
CooperationCooperation
J. Čížek, O. Melikhova, I. Procházka
W. Anwand, G. Brauer (Dresden)
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
OutlineOutline
Introduction to positron annihilation
Positron calculations
Zirconia (ZrO2)
Conclusions
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
A positron is the antiparticle of an electron: i.e. mass = me , charge = +e , spin = ½ ,
rest energy (mec2) = 511 keV.
In solids positrons reside in the interstitial region due to their repulsion from ionic cores.
When a positron and an electron meet, they can annihilate producing two -quanta.
Positrons can be effectively used for defect studies.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
Positrons are self-seeking probes of various defects:
e+
Positrons can get trapped in this well.
e+e-
PA properties are defect specific.
e+
Many defects represent a potential well for positrons.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
Ni3Al bulk = 104 ps
Al vacancy = 174 ps
Al+Ni vacancy = 193 ps
GB (210) = 126 ps
GB+Al vacancy = 191 ps
Perfect vs imperfect Ni3Al lattice
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
Positrons sensitive to: open volume defects
• vacancies and their agglomerates (+loops)
• dislocations (+loops)• grain boundaries• stacking faults (+tetrahedra)
some precipitatesneutral and negatively charged defects
Positrons insensitive to:isolated impuritiesinterstitialssome precipitatespositively charged defects
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
How to mesure positron lifetime?
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
How to measure momentum distribution of e-p pairs?
coincidence
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron annihilationPositron annihilation
Comparison with other techniques:
STM=scanning tunneling microscopy, AFM=atomic force microscopy, NS=neutron scattering, OM=optical microscopy, TEM=transmission electron microscopy
After Howell et al., Appl. Surf. Sci. 116 (1997) 7
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Density functional theory (DFT) = theoretical background for electronic structure calculations.
A many-electron system is considered as a collection of many one-electron subsystems.
Basic DFT statement: the ground-state energy of a system composed of electrons (and nuclei) is the functional of the electron density (and nuclear positions).(Hohenberg and Kohn, Phys. Rev. (1964))
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Mathematical formulation:
Vext is the (electrostatic) potential of nuclei,T is the kinetic energy functional,Exc is the exchange-correlation functional.
][|'|
)'()('
2
1
][][
)()(][][
nEnn
dd
nTnF
nVdnFnE
xc
ext
rr
rrrr
rrr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Kohn-Sham equations (Kohn and Sham, Phys. Rev. (1965)) for one-electron ‘wave functions’ and ‘energies’:
where
are the electrostatic potential generated by electrons and the electron density, respectively.
)()()()()(
][)(
2
1rrrr
rr
ext
xc Vn
nE
F
nn
d
2|)(|2)(and|'-|
)'(')( rr
rr
rrr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Practical calculations:
1. Start from a reasonable electronic density (e.g. the superposition of atomic densities).
2. Generate the Coulomb potential () and exchange-correlation potential (Vxc) using this density.
3. Solve the Kohn-Sham equations.
4. Find the Fermi level and determine the new electron density.
5. Mix the new and old electron densities; go to point 2 until self-consistency is reached.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Positrons are included from now on two component DFT (B. Chakraborty, Phys. Rev. B 24, 7423 (1981)).
Basic TCDFT statement: the ground-state energy of a system composed of electrons and positrons (and nuclei) is the functional of the electron and positron density (and nuclear positions).
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Mathematical formulation:
Ece-p is the electron-positron correlation
functional.
],[|'-|
)'()('
2
1
)]()()[(
][][],[
nnE
nndd
nnVd
nFnFnnE
pec
ext
rr
rrrr
rrrr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
One-particle equations:
where
)()()(
],[)()(
)(
][)(
2
1rr
rrr
rr
n
nnEV
n
nE pec
extxc
)()()(
],[)()(
)(
][)(
2
1rr
rrr
rr
n
nnEV
n
nE pec
extxc
|'-|
)'()'(')(
rr
rrrr
nn
d
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
One has to solve a system of coupled equations for electrons and positrons.
This requires knowledge of the Ece-p[n-,n+]
functional (in addition to the Exc one).
The knowledge of the Ece-p functional is far from
being complete.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
The TCDFT is quite time and computationally demanding.
When n+ 0 (delocalized positron state), TCDFT can be simplified considerably.
Then we obtain the zero positron density limit or the so-called ‘conventional scheme.’
In this case, electronic structure (n-) is not influenced by the presence of positrons.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Electronic structure is calculated first not considering positrons.The Schrödinger equation for positrons is then simplified as follows:
with
.0for)(
],[))((
nn
nnEnV
pec
corr rr
)()())(()()()(2
1rrrrrr nVV corrext
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
The positron potential can be written as
where VCoul is the Coulomb potential for electrons (obtained from electronic structure calculations).
Correlation energy and enhancement: Boroński and Nieminen, PRB 34 (1986) 3820; correction for incomplete positron screening: Puska et al., PRB 39 (1989) 7666.
Vxc is effectively cancelled by the positron part of (due to self-interaction).
))(()()( rrr nVVV corrCoul
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Summary:
The electronic structure of a given system is determined.
The positron potential is calculated using the electron Coulomb potential and density.
The Schrödinger equation for positrons is solved (+, +).
Further positron quantities of interest are calculated.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Positrons with intensity I+ [m-2s-1] are coming into the sample with volume V [m3]. The number N of annihilation events per unit time is
where n– is the positron density and
is the cross section for positron annihilation.
VnIN
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Cross section for 2 annihilation (QED)
where (Lorentz factor)
and (electron radius)
1
31ln
1
14
1 2
22
220
2g
ggg
g
gg
g
r
22 /1/1 cg v
m103 150
r
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Non-relativistic approximation
Furthermore
Annihilation rate (number of annihilation events per unit time when only one positron is in the sample) is as follows:
1/ cv
vc
r 202
IPMIPM
IPM ncrVnnVnN
/1
)1( 202
v
v nI
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
So far the independent particle model (IPM) has been considered: e– and e+ interact at the moment of the annihilation only.
However, in reality positrons attract electrons:
where is the so-called enhancement factor.
Assumption: depends on n– .
)(/1 20 nncr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
When the system is non-homogeneous
supposing
Assumption: is the same as for homogeneous electron gas.
rrrr dnnncr )())(()(/1 20
.1)( rr dn
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Practical expressions: Enhancement factor:
The Boroński-Nieminen parametrization (of Lantto’s results) is widely used at present.
)6/3286.026.1
8295.023.11(32/52
2/320
sss
ssBN
rrr
rrncr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Practical expressions:
The annihilation rate (lifetime) approaches 2 ns-1 (500 ps) as electron density decreases [or rs increases].
PsoPsp 4
3
4
1
13
4 3 nrs
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Positron binding energy to a defect:
Positron affinity:
)defect()bulk( EEEb
.)(
ΦΦ
EEA F
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
Positron calculationsPositron calculations
Momentum distribution of e-p pairs = MD of annihilation -quanta:
where
is the state dependent enhancement factor and
2
,2 )()().exp()(
Fj
di jj
rrrrpp
jIPM
jENHj
,
,
.)())((|)(| 2,
20, rrrr dnncr XjjX
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
ZrO2 (zirconia):high melting point (2700 C)low thermal conductivitygood oxygen-ion conductivity (higher temperatures)high strengthenhanced fracture toughness
For applications stabilization of the tetragonal or cubic phase is necessary.Zirconia is often stabilized by yttria (Y2O3) yttria stabilized zirconia (YSZ)
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
cubic structure (CaF2)above ~1380 C
tetragonal structureabove ~1200 C
monoclinic structure
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Zr has +4 charge state (Zr’’’’) and Y is +3 only (Y’’’), i.e. -1 with respect to the lattice compensation by O vacancies having +2 charge state (VO
)
there are two Y’’’ per one VO
stability ranges: > 8 mol% of Y2O3 cubic phase
> 3 mol% of Y2O3 tetragonal phase
large amount of vacancies in YSZ materials
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Identification of defects in ZrO2 is of large importance. Positrons may help to identify open volume defects.Cubic YSZ single crystal (8 mol% of Y2O3) 175 ps positron lifetime.Before we studied the following vacancy-like defects:
VZr, VO, VO-2Y, 2VO-4Y (VO’s along [111] direction)
positron trapping at VZr only (but too long lifetime)
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
H is everywhere …
First NRA results indicate that we have appreciable amount of H in our sample.
Now we look at positions of H in ZrO2 lattice.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
ZrO2 (CaF2) structure:
Zr, fcc (8 fold)
O, sc (4 fold)
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ZrOZrO22
Defect configurations:
obtained using VASP-PAW, LDA & GGA
96 atom based supercells
total energy of studied defect configurations relaxed with respect to atomic positions
cubic structure unstable (without Y) !!
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Positron induced forces:can be treated within the scheme developed by Makkonen et al., PRB 73 (2006) 035103 conventional scheme for positron calculations consideredHellman-Feynman theorem used to calculate forces
V+ is constructed using atomic (Coulomb) potentials and densities and force calculation is thus very fastsuch forces added to ionic forces calculated by VASP and atomic positions where total forces vanish are found though the method is not fully self-consistent and does not use TCDFT, it is sufficient to get reliable results
|}){,(| ijj V RrF
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Configurations examined:
Interstitial hydrogen
Hydrogen in VZr
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Interstitial hydrogen:
z
y
x
-0.03[1,-1,1]
-0.03[1,1,1]
-0.03[1,-1,-1]
0.01[1,0,0]
0.01[0,0,1]
0.01[-1,0,0]
0.01[0,0,-1]
0.01[0,1,0]
-0.03[-1,-1,-1]
-0.03[-1,1,-1]
-0.03[-1,1,1]
-0.03[1,-1,1]
0.01[0,-1,0]
z
y
x
-0.23[1,-1,1]
0.22[-1,1,1]
-0.05[-1,2,2]
-0.05[2,2,-1]
-0.14[-1,-1,2]
-0.05[2,-1,2]
-0.11[-1,-1,-1]
0.22[1,-1,1]
-0.23[-1,1,1]
0.22[-1,1,-1]
-0.14[-1,2,-1]
-0.53[1,1,1]
0.68[-1,-1,-1]
-0.14[2,-1,-1]
O
Zr
H-0.05[-1,1,1]
-0.05[-1,1,-1]
-0.05[-1,-1,-1]
-0.05[1,-1,-1]
-0.05[1,1,-1]
-0.05[1,-1,1]
-0.05[1,1,1]
-0.05[-1,-1,1]
0.06[0,-1,-1]
0.06[-1,0,-1]
0.06[-1,-1,0]
z
y
x
cubic tetragonal
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ZrOZrO22
Hydrogen in VZr:
cubic, EB = 2.3 eV tetragonal, EB = 5.8 eV
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ResultsResults
Positron lifetimes and binding energies (LDA/GGA):
(ps) Eb (eV)
ZrO2 bulk 133/151 --
VZr 220/287 2.8/2.3
VZr+1H-c 167/195 0.9/1.1
VZr+1H-t 171/175 1.4/1.8
H in VZr can in principle explain lifetime data !!
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
ConclusionsConclusions
We have studied H positions in the zirconia lattice.In the case of the interstitial H the lowest energy configuration is for H bound to an O atom near the center of the interstitial space (tetragonal, O-H bond formed).As for the VZr , H prefers position close to a neighboring O atom (tetragonal, O-H bond formed).
The latter defect traps positrons and could be responsible for the observed positron lifetime.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
OutlookOutlook
More H-related configurations needed.
The question of the cubic YSZ structure needs to be solved.
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Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010
TT hh aa nn k k yy oo uu ! !