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

(rraman@igcar.gov.in)

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