kalpakkam m.c. valsakumar & dr. sharat chandra, igcar, india...
TRANSCRIPT
R. RajaramanMaterials Science Division
Indira Gandhi Centre for Atomic ResearchKalpakkam – 603 102, TN, India
Thanks toM.C. Valsakumar & Dr. Sharat Chandra, IGCAR, IndiaDr. G. Amarendra, IGCAR, IndiaDr. A. Bharathi, IGCAR, IndiaDr. C.S. Sundar, IGCAR, IndiaDr. K. Sivaji, Madras University, IndiaProf. M. J. Puska, Helsinki University, FinlandDr. I. Makkonen, Helsinki University, Finland
Angular CorrelationExperimental methodsExamples
Computational Positron PhysicsGeneral methodologyAtomic SuperpositionSelf-consistent computationsExamplesUsing MIKA-dopplerABINIT
22Na (e+ Source)
~100m
(1) Angular Correlation
1/ne-
(3) Lifetime
-ray (1.28MeV)
~ 10-12 s
e+
e-
Sample
-ray (511keV E)
(2) Doppler Broadening
Cm
p yx
yx
0
,
,
2
zCpE
N
Np
ln(N
)
t (ns)
1
2
N
E
Np
S = Np/Ntotal
Nw1 Nw2
W = (Nw1 + Nw2 )/Ntotal
N+
D
E
Positron LifetimeElectron Momentum distribution
Compute these observables?
z
zyx dppconstppN )(),( 2
2D ACAR, we measure projected momentum distribution in px, py plane
e+
e- z
x
y
x z
zxy dpdppconstpN )()( 2
In 1D ACAR measurement
e+
e- z
x
y
Amplifier Amplifier
SCA SCA
Coincidence
NaI (Tl)NaI (Tl)
Data recorder
+y
-y
Motioncontrol
Angular Resolution:slit width / sample detector distance
3mm/3m = 1mrad
Panda, PRB (1997)
Reflecting Micro Prisms
Vertical Plane HorizontallPlane
Laser theodaliteSample mount Position
Source-sample lead chamber
Detector 1 Detector 2
Courtesy Dr. Sivaji, University of Madras
Courtesy Dr. Sivaji, University of Madras
Positron Source : 22 Na positron source 50 mCi Annihilation Gamma : 511 keV Shielding : bi-layer lead Shielding
Inner : moulded lead wellOuter : interlocked lead bricks
Sample mounting : On a X-Y goniometer head
Spectrometer Hardware : Scintillation detector Twin Anger type Gamma Camera (uncollimated) - SIEMENS, USA
Crystal : NaI(Tl) activated, 0.98 cm thick - 375 mm diameterPMT’s : Optically coupled 37 PMTs
Sample distance : 8300 mm
Spatial resolution : 3.7 mm for 511 keV gamma
Detection Efficiency : 27% for 511 keV gamma
Angular Coverage : 45 mrad (-22.5 to + 22.5 mrad)
Angular resolution : 0.35 mrad2
Overall angular : 1.42 mrad2 (FWHM) Resolution measured with quartz
Courtesy Dr. Sivaji, University of Madras
Surface plot of momentum distribution for single crystal -quartz in thec-axis projection after subtracting the core contribution .The narrow positronium peak at low momentum regionsix umklapp peaks each at the boundaries of first and second BZs
px in mrad
py
in m
rad
px in mrad
py
in m
rad
Courtesy Dr. Sivaji, University of Madras
Measurements with single crystals of different orientations
3D fermi surface mapping
Wigner Seitz Cell in Reciprocal Space
Wigner Seitz Cell in Real space
px in mrad
S.W.H. Eijt, A. van Veen, H. Schut, P.E. Mijnarends, A. B. Denison, B. Barbiellini, A. Bansil, Nature Materials 5, 23-26 (2006).A. Puzder, A.J. Williamson, F. Gygi, G. Galli, Phys. Rev. Lett. 92, 217401 (2004).
Courtesy http://www.tudelft.nl/
Is it resolvable with Coincidence Doppler Broadening?
ExperimentComputation
Computational Positron Physics
400 600 800 1000 1200
110
120
130
140
150
160
170
Ti/C=6
Ti free Model alloy
Solution Annealed State
(p
s)
Annealing Temperature (K)
TiC nanoparticles
Where from positrons annihilate?
Rajaraman et al JNM 1994
CW ~ 4-8nm(TEM work in literature)
Ti modified steel
0 10 20 30 400.2
0.4
0.6
0.8
1.0
1.2
1.4
N/N
Ti fr
ee
SS
pL(10
-3m
0C)
Solution annealed D9
Graphite
Titanium
Cold worked D9
D9 - 1073K
Check list for kick start
a PC or laptop
Linux OS (Ubuntu is ideal for windows users) Couple of codes
Imagination to model the problem
<1 hour
Calculate positron density distribution in solids Derive
positron annihilation rate (lifetime) momentum distribution of annihilation pair
Complement & validate experimental observations
Start from first principles without any empirical parametersDensity functional formalism
R. M. Martin, Electronic Structure: Basic Theory and Practical methods (Cambridge University Press, Cambridge, 2004).
M.J. Puska and R.M. Nieminen, Rev. Modern Phys. 66, 841 (1994)
T. Torsti, M. Heiskanen, M. J. Puska, R. M. Nieminen, arXiv:cond-
mat/0205056v1; http://www.fyslab.hut.fi/epm/positron/mikadoppler.html
Positron annihilation rate
);;0()()(1 2
0 nngrnrndrCr
n+, n- - positron and electron densities
g(0;n+;n-) - Exchange correlation functional evaluated at positron
Get n+, n- & have practical description of g
],[)()(
)]()()[(]F[]F[ ],['
''
nnE
rr
rnrndrdrrnrnrdrVnnnnE pe
cext
Total energy functional of positron & electron system
][)()(
2
1]T[ ]F[
'
'' nE
rr
rnrndrdrnn xc
T –kinetic energy functional for non-interacting electron and positronVext – external potential
)()(][)(2
1 2 rrrVr iiieffi
)(
],[
)(
][)(][
rn
nnE
rn
nErrV
pe
cxceff
'
'
0
''' )()()(
)(rr
rnrnrndrr
)()(][)(2
1 2 rrrVr iiieffi
)(
],[
)(
][)(][
rn
nnE
rn
nErrV
pe
cxceff
'
'
0
''' )()()(
)(rr
rnrnrndrr
positron electron
n0 – charge density which gives external potential Vext
N+ = 1
2
)( (r)n
N
i
i r
2
)( (r)n
i
i r
g(0;n+;n-) - Exchange correlation functional evaluated at positron
= (at zero positron density limit)
for independent particle model (IPM) approximation. = 1
within LDA, “Lantto, Boronski and Nieminen” functional form
“Arponen, Pajanne and Barbiellini” functional form
Thomas –Fermi screening length
Variation in electron density within GGA
- adjustable parameter (0.22 is a reasonable value for reproducing experimental lifetimes)
At nth iteration, positron energy eigen value
For the (n+1)
i or j runs over the discretized mesh pointsh- distance between mesh points
r
Use them to obtain positron density and compute lifetime
non-self-consistent
Puska, RMP, 1994
positron
Solid lines : self-consistent LMTO-ASA methodDashed lines : non-self-consistent atomic superposition method
Within Independent Particle Approximation,3-dimensional momentum density
j goes over all occupied single electron states
Improvement with state independent Enhancement factor
State dependent Enhancement factor
and
http://www.fyslab.hut.fi/epm/positron/mikadoppler.html
Atomic superposition method Periodic boundary conditions
Lattice parameter Atomic positions in unit cell Electronic configuration of elements in unit cell 4 -parameters for positron wave function from LMTO-ASA
for each element in unit cell
0.0, 0.0, 0.0
0.5, 0.5, 0.0
0.5, 0.0, 0.5
0.0, 0.5, 0.5
Al
V
(V)ienna (A)b-initio (S)imulation(P)ackage
• is a package for performing ab-initio quantum-mechanical molecular dynamics (MD) and electronic structure calculations using pseudo-potentials.
Relax (both volume and positions of atoms) the given structure by minimizing forces on atoms there by reducing total energy of the system.
MIKA/doppler is a package for the simulation of positron annihilation in solids (real
space Atomic Superposition method)
Port VASP relaxed structure (and electron density) to MIKA/Doppler to get realistic positron parameters
Requires parallel computing facilities
3x3x3 supercell (53 atoms)
Inward relaxation of 1st nearest neighbors
Outward Relaxation of FNNVASP+USP = 0.098ÅXRD result = 0.1Å
3x3x3 supercell (215 atoms)
Positron expands the volume around a vacancy comparable to
the volume contraction associated with structural relaxation.
VASP (Vienna Ab-initio Simulation Package)
Ionic position relaxation (include positron induced forces)self-consistent valance electron density
MIKA-dopplerUse VASP derived atomic structureUse VASP derived valance electron densityUse core electron distribution within IPA
http://www.abinit.org/
Calculates within Density Functional Theory (DFT), using pseudopotentials and a planewave basis total energy charge density electronic structure
optimize the geometry perform molecular dynamics simulations
Array of tools included for calculating various physical propertiesphonon frequenciesforce constantsphonon density of states elastic constants, Piezoelectric coefficientsRaman scattering cross-sections…………..Positron lifetime & angular correlation curves
We are well equipped to “quantitatively” complement positron lifetime, ACAR and CDB experiments
Linux OS for dummies http://www.ubuntu.com
Crystal Structure http://www.cryst.ehu.es http://icsd.ill.fr/icsd ftp://ftp.bam.de/Powder_Cell
Visualisation http://www.xcrysden.org http://www.ks.uiuc.edu/Research/vmd http://www.geocities.jp/kmo_mma/crystal/en/vesta.html
Some self consistent DFT codes http://cms.mpi.univie.ac.at/vasp (license needed) http://www.abinit.org http://www.uam.es/departamentos/ciencias/fismateriac/siesta http://www.wien2k.at http://www.quantum-espresso.org
2p Orbitals 2p Probability Density
1s Orbitals 1s Probability Density
http://winter.group.shef.ac.uk/orbitron/
http://winter.group.shef.ac.uk/orbitron/
Bands are Formed When A Finite Number of Atoms Come Together
(1s2 2s2 2p6) 3s2 3p2
8 Si atoms well separated
E (
eV)
levels
Look at Silicon
6 Si atom cluster
Any property of system of many interacting particles can be viewed as “functional” of ground state electron density n0(r)
The success story of DFT originated from
Self Consistent DFT flow chart
13 361 6933.6 0.05 3 25061 -1 2.0 .TRUE.2 -1 2.0 .TRUE.2 1 2.0 .TRUE.2 -2 4.0 .TRUE.3 -1 2.0 .TRUE.3 1 1.0 .TRUE..FALSE.
atomic number
number of radial points
muffin tin radius
step size
highest main quantum number
number of points to be saved for charge
number of relative electron states
main quantum numbers
quantum number of the relativistic theory (l and -l-1)
occupation of the state
scalar relativistic calc.(always true)
Z name a0 a1 a2 a3 (4 parameters used, calculated with IPM)
1 H 0.0256877 0.039439 1.90665 0.939391 # H in bulk, hcp structure
2 He 0.0226723 0.0445166 1.20352 0.924235 # He in bulk, hcp structure
3 Li 0.0199463 0.114833 2.28663 0.942798 # Li in bulk, bcc structure
4 Be 0.0128035 0.116052 2.05642 1.08467 # Be in Be2O2, wurtzite structure
99 Be 0.00634743 0.0787568 2.61596 1.12897 # Be in BePo, zincblende structure
……………………..
………………..
parameters for positron wave function in core region
Comments out the line
px in mrad
py
in m
rad