Study of interaction of Positronium with light atoms: H, He and Li - … · 2011. 9. 26. · H 2.126 a0 [1] 2.8 a0 He 1.566 a0 [1] 2.4 a0 Li 3.8-4.1 a0 [2] 5.8 a0* [1] Zhang et al.,

A. Zubiaga, F. Tuomisto and M. Puska

Department of Applied Physics, Aalto University, P.O. Box 11100, FIN-00076 Aalto Espoo, Finland

Financial support also from the Academy of Finland

Study of interaction of Positronium with light atoms: H, He and Li

10th International Workshop on Positron and Positronium Chemistry (PPC10)

• Point defects in metals and semiconductors (vacancies, vacancy clusters, substitutional ions)• Voids in polymers or irradiated materials• Pore size and density in porous materials• Study of bulk samples or thin films/quantum structures• Lifetime spectroscopy, Doppler broadening spectroscopy, positron induced Auger spectroscopy,Coincidence Doppler Broadening, 2D-ACAR, Positron Microscopy ...

Positron(ium) Annihilation Spectroscopy

Very good capabilities to study open volume in materials: vacancies, voids, pores...


Pore/void size & distribution


Ps in molecular materials e+ e-

Metals, semiconductors• Positrons in molecular materials annihilate in both states: isolated positron and positronium• Annihilation of o-Ps clearly distinguished from e+ + p-Ps experimentally• Polymers and bio-structures have non-homogeneous composition and structure• Distribution of positron(ium) non-homogeneous • Lifetime depends on the annihilation site

Identifying the annihilation site important for the interpretation of the experimental data


Relevant processes to understand o-Ps annihilation

- Pick-off annihilation with electrons in the matter - Main reason for the vacuum lifetime (142 ns) be shortened to 1-10 ns- Low energy o-Ps thermalized or quasi-thermalized

- o-Ps --> p-Ps spin quenching - Processes: electron exchange and spin-orbit coupling- Could be relevant in some materials ???- Net effect in lifetime spectra is to lower the total lifetime (increase the annihilation rate)

- Binding to individual molecules- Ps can bind several atoms (H, Li,...)- Not known if it binds molecules- Could be relevant in materials ???- Probably annihilation rate increases

- ...


Ps modelling

- Adiabatic approximation completely fails & positron density is far from being point like (unlike atoms) - Simplified models for Ps in solids:

- Ps in square potential wells and spherical shape (W. Brandt et al., Phys. Rev. 120 1289, 1960)

- Ps in infinite square potential wells (S. J. Tao, J. Chem. Phys. 56 & M. Eldrup et al., Chem. Phys. 63)

- Chemical dependencies of relevant quantities (annihilation rates, open volume/lifetime relation) are not described

Very light atom difficult to model in atomistic models (beyond qualitative behaviour)

Atomistic models including a realistic Ps-matter interaction desirable, but ...

- e-p correlation not well known for Ps (studied for positron Nieminen et al.)- Two component DFT very expensive for large systems (even worse for higher accuracy methods)


One-particle Ps model (Free volume model)

Polymers H. Schmitz, JCP 112 1040 (2000)• Path Integral Monte Carlo for the CM• VPs(r)= Ae-r/b-C/r6 Buckingham potential• Electron density parametrized ne-(r) ≈ Ne-r/a

Quartz surface R. Saniz, PRL 99 096101 (2007)• One dimensional calculation for the CM• VPs(r)= Ae-r/b-Cf(r)/r3

Previous attempts

- o-Ps thermalized or quasi-thermalized at annihilation - One (point) particle approximation: at low energies Ps (EB=6.8 eV) not much distorted by the interaction with the matter (Brand et al., Phys. Rev. 120 1289, (1961))

- One-particle potential --> atomistic system computationally reachable- Interaction with the matter two parts:

- Short range repulsion dominated by e-e repulsion- Long range dispersion interaction (van der Waals)


One-particle Ps model

- What can be described?- Potential energy landscape inside molecular materials- Distribution of o-Ps (not forming a bound state)- Annihilation rate- Determination of lifetime vs open volume relations ??

- What cannot be described?- Bound states- Spin quenching- Excited states describing non-thermalized states- Polarized Ps (q-Ps) might be difficult to describe


Atom-Ps systems

Accessible to SVM-ECG method

Ps not bound

H-Ps (triplet)

He-Ps

Li-Ps (triplet)

Characteristic of studied systems:- Few particles (HPs 4 particles, LiPs 6 particles)- L=0 (radial symmetry)- Spin for Ps (S=1)

Studied cases

Comments on HPs and LiPs:- HPs and LiPs have a bound (singlet) and an unbound (triplet) configuration- Select the triplet configuration:

S(Ps)= 1, MS(Ps)= +1;

S(Li)= 1/2, MS(Li)= +1/2

S(e-)= 1, MS(e-)= +1


Many-body wavefunction

Ψ =N�

i=1

ciψSMs(x,Ai) ψSMs(x,Ai) = A{e−xAix/2χSMs}

Jacobi coordinates

xi = ri+1 −i�

j=1

mjrj/i�

j=1

mj

xN = XCM

ci

(Ai)jklinear coefficients

non-linear coefficients

- Exact exchange interaction and full correlations included- Polarization interaction included (the wavefunction has to be well converged)- ECG basis adequate to account for the particle-particle correlation- Matrix elements analytical --> speed, large basis sets feasible

- Basis size 600 functions- Energy accuracy >= 10-20 meV- Wavefunction “quality”: Virial theorem 2V

T+ 1 ∼ 10−3 − 10−1


Stochastic Variational method

{< ψi|H|ψj > −E < ψi|ψj >}ci = 0

Exact diagonalization for determination of the linear coefficients and the energies

Stochastic variational optimization method for the basis functions (Ai)jk

Minimization of the lower eigenvalue is the criterion for the selection of the best basis set

H =N�

i=1

p2i

2mi+

N�

i


Non-bounded systems

- Technical difficulty: the minimum energy state is always at infinite --> Position of Ps needs to be bound during the calculation

- Solution: Set a minimum value for the basis coefficients of the nucleus-positron pair wavefunction

- Mitroy et al. Phys. Rev A 65, 012509 (2001)- The maximum nucleus-positron distance is controlled this way (effective box) without introducing an external potential

- Lower quality (Virial theorem) of the wavefunction at short distances, due to the slower convergence

50-60 configurations calculated for each atom-Ps system varying the nucleus-positron mean distance between 2au and 100au

Distance dependence of the repulsive Atom-Ps interaction can be studied


Densities of the unbound electron and the positron overlap when the mean positron distance is larger than 5 au

Densities and e-p correlation

The correlation of electron and positron shows the formation of Ps (the peak at 2 au)

When the positron-atom electron density overlap increases the “Ps” is strongly deformed (distance < 5 au)


Benzene-Benzene interaction energy from E. R . Johnson et al., J. Phys. Org. Chem. 22, 1127 (2009)

Interaction energyEI = EXPs − EX − EPs

Model for the atom-Ps description needed

- Interaction energy increases very fast for separations smaller than 5 au - More repulsive than Benzene-Benzene inter-molecular interaction


Ps model- Ps a point particle with mass M=2me-- Ps CM density approximated as the positron density - Polarization effects not included --> low energy approximation- Wavefunction square root of the density- Effective potential obtained fitting EI - Only systems not showing polarization included in the fitting

EKin =−�22M

�

Vd�r ψ∗∇2ψ

EI = EKin +

�

Vd�r ψ2(�r) V (�r)


Ps model

- Repulsive (exchange mainly) term modeled with a single exponential and two parameters (A intensity and b range parameter)

- Attractive dispersion (van der Waals) term modeled with the C6 term from (Mitroy et al., Phys. Rev. A 68 035201, 2003)

- Cut-off function g(r) accounts for the saturation of the dispersion interaction at short distances

V (r) = Ae−br − g(r)C6r6

g(r) = 1− e−r6/ρ6

Range parameter ρ sets the cutt-off distance for the vdW interaction

ρH 2-5 au

He 1,5-2,4 au

Li 3-5 au

- H, He: Mitroy et al., Phys.Rev. A 65 012509 (2002)- Li: Chakraborty et al., Phys. Rev. A 65 062504 (2002)


Energy analysis

Kinetic energy from the one-particle distribution smaller than the many-body wavefunction

Excess kinetic energy due to correlation effects --> the one-particle potential has to describe them

The triplet states (H+Ps, Li+Ps) have larger kinetic energy than He+Ps

Potential energy of H+Ps can be negative but the system is not bound because of the higher kinetic energy of the triplet configuration

Kinetic and potential energies defined as the difference between the interacting system and the isolated elements


One particle potential energiesExact Diagonalization + Explicitly Correlated Gaussian basis + Stochastic Variational optimization of the basis + Ps one particle modeling

- The potential energy is highly repulsive- The polarizability of single atoms too small and the van der Waals is not strong enough to counterbalance the short range repulsive term at any distance

- The potential of Ps interacting with H and He are similar- The Ps interacting with Li has stronger repulsive potential

Benzene-Benzene interaction energy from E. R . Johnson et al., J. Phys. Org. Chem. 22, 1127 (2009)


Repulsive potentialExact Diagonalization + Explicitly Correlated Gaussian basis + Stochastic Variational optimization of the basis + Ps one particle modeling

The repulsive potential and the valence electron density linearly related?

Can the repulsive potential described as a density functional ???

More work necessary to further check this possibility


Repulsive potentialExact Diagonalization + Explicitly Correlated Gaussian basis + Stochastic Variational optimization of the basis + Ps one particle modeling

The decay constant of the repulsive term also depends on the excitation energy of the atom

(HOMO-LUMO gap of molecules) H, He: 1s->2s

Li: 2s->2p

Non-linear but monotonic dependency of the spatial decay constants of the potential and the electron density


- Scattering Length is calculated from the s wave using the effective range approximation valid at low scattering energies (k < 0.1 au-1)

- Solution approximates to a spherical wave with a phase shift at large distances- Finite difference method to solve the Schrödinger equation

Scattering length

Phase shift related to the scattering length A

ψs(r) ∼sin(kr + δ(k))

kr

δ(k) = −kA

Low energy scattering length calculated and compared to benchmark calculations (SVM, CI, Hylleraas)


Scattering length

Benchmark Present work

H 2.126 a0 [1] 2.8 a0

He 1.566 a0 [1] 2.4 a0

Li 3.8-4.1 a0 [2] 5.8 a0*

[1] Zhang et al., PRA 78 012703 (2008)[2] Chakraborty et al., PRA 65 062504 (2002)

- Scattering lengths reproduce the trend of the benchmark values- Calculated values 1-2 au larger than the benchmark values- Calculated potentials are too repulsive

- Including the polarization effects would reduce the scattering length

- But one-particle approximation breaks ☹ - May be an effective potential for the positron density can be still defined ???


Conclusions & future plan

- Positronium, a very light “atom”, lifetime spectroscopy a powerful technique to study open volume (pores/voids, phase transitions, transport properties...) in molecular materials (polymers,liquids, biostructures...)

- Study of (ortho-)Positronium in matter a quantum many-body problem

- Many-body wavefunctions of light atoms (H, He, Li) + Ps have been calculated for different nucleus-atom distances

- Low energy unpolarized Ps modeled using a one-particle treatment

- The short range repulsive potential is calculated

- The resulting repulsive potential has monotonic dependence with the electron density and the atom excitation energy

- Scattering lengths 1-2 au larger --> description needs to include the polarization

- Further work: extend the study to more atom-Ps systems, include polarization effects, study the annihilation rate

Documents

Study of interaction of Positronium with light atoms: H, He and Li - … · 2011. 9. 26. · H 2.126 a0 [1] 2.8 a0 He 1.566 a0 [1] 2.4 a0 Li 3.8-4.1 a0 [2] 5.8 a0* [1] Zhang et al.,