Ground state and resonance states of Ps interacting with screened Coulomb (Yukawa) potentials

Sabyasachi Kar and Y. K. Ho

Institute of Atomic and Molecular SciencesAcademia Sinica, Taipei, Taiwan

[18th International IUPAP Conference on Few-Body Problems in Physics]

August 21-26, 2006

Outline

I. Introduction II. Screened Coulomb (Yukawa-

type) Potentials III. Bound States of Ps and Ps IV. Stabilization Method for Resonance Calculations V. Doubly-excited 1Se, 1Po, 3Po Resonance States of Ps . VI. Summary and Conclusions

INTRODUCTION

Atomic Structures and Collisions in Astrophysical and Laser Plasmas

With the recent developments in laser plasmas (laser produced hot and dense

plasmas) produced by laser fusion in laboratories, and the continued interest of

helium abundances in astrophysics plasmas, it is important to have accurate atomic

data available in the literature for atoms/ions in various plasma environments. In

the Debye-Hückel model for plasmas, the interaction potential between two

charged particles a and b is represented by a Yukawa-type potential,

babababa rrrrZZrr /)exp(),( ,

The screening parameter μ ( D /1 , D : Debye length) is a function of temperature

(T) and the charge density (n) of the plasma.

For laser plasmas, T ~ 1 keV, n ~ 1022 cm-3 and μ ~ 0.1 – 0.2.

S. NaKai and K. Mina, Rep. Prog. Phys. 67 (2004) 321.

D Leckrone and J. Sugar (eds) 1993 Proceedings. of the 4th Int. Colloquium. on Atomic

spectra and Oscillataory strengths for Astrophysical and Laboratory plasmas, Phys Scr. T47.

IntroductionBound and resonance states of Ps in dense plasmas

• Ps, a simplest three lepton system (e+, e, e) Experiment : A P Mills, PRL 46 (1981)717; 50 (1983) 671

P Balling et al, NIMB 221 (2004) 200,

D. Schwalm et al, 221(2004) 185

D. B. Cassidy et al, PRL 95 (2005) 195006 Theory : Y K Ho, PRA 48 (1993) 4780; G W F Drake et al, PRA 65 (2002) 054501; A M Frolov Phys. Lett. A 342 (2005) 430.

• Ground-state of Ps embedded in Debye plasmas B Saha et al, CPL 373 (2003) 218 Kar and Ho, PRA 71 (2005)052503, CPL 424 (2006) 403

• The 1Se and 1, 3Po resonance states of plasma-embedded Ps Kar and Ho, PRA 71 (2005) 052503; PRA 73 (2006) 032502.

• The Stabilization Method V A Mandelshtam et al, PRL 70(1993) 1932 Ho and co-workers [J Phys B 38 (2005) 3299 and references therein]

Calculations Hamiltonian:

3

1

3

1

2)/exp(1

2

1

ji ij

Dijji

ii

i r

rqq

mH

Debye length:

1D

2

1

2220

2

1

20

)(

k

kke

BBD NzNe

Tk

ne

Tk

e+

ee

r31 r32

r12

where n being the plasma density, T being the plasma- temperature, Ne

indicates electron density, Nk denotes the ion-density of k-th ion species having the nuclear charge kz , and Z* is the effective charge.

For two-component plasmas: 2/12* /)1(4

TkneZ BD .

Frank-Thomson criteria: 30 )4/(

2

1aTkN Be , TkB is in terms of large rydbergs

,1042.51042.5 31613 cmNe KT 43 1010

From perturbation theory: ........2

)/exp( 2 DD

D

zrz

r

zr

r

z

The first-order correction upshifts all bound levels equally but does not change the wave function. The effect of the perturbation potential in the

lowest order in Dr / is given by 22 D

zrV

.

Calculations

Wavefunction: )(exp)(cos)ˆ1( 2131321311

12 rrrPrCOS iiiLL

N

iipn

Schrödinger equation: , EH where 0E

The two-body cusp:

)21()(,5.0

)31()(,5.0

|)(|

|)(|

ij

ij

mm

mmqq

r

rr

ji

jiji

ij

ijij

ij

,

where )( ijr is the -function , )(ij denotes )21( and )31( , iq and jq are the charges, and

im and jm are the masses of the particle i and j.

Some references H. Okutsu, T. Sako, K. Yamanouchi , G. H. F. Diercksen, J. Phys. B 38 (2005) 917. S. Kar,Y. K. Ho, Int. J. Quant. Chem. 106 (2006) 814, 107 (2007) (in press). H R. Griem , in Principle of Plasma Spectroscopy : Chembridge Univ Press 2005 G. J. Iafrate, L. B. Mendelsohn Phys Rev 182 (1969) 244 S. Kar,Y K. Ho, Chem. Phys. Lett. 424 (2006) 403

Quasi-random process A. M. Frolov, Phys Rev E 64 (2001), 036704

Selection of non-linear parameters

] C,[C , ] B,[B ], A,[A (k)2

(k)1i

(k)2

(k)1i

(k)2

(k)1i

,1)3,mod( ik Ni 1

)(1

)(1

)(2

)(3

)(1

)(1

)(2

)(2

)(1

)(1

)(2

)(1

)(5)1((2

1

)(3)1((2

1

)(2)1((2

1

kkkki

kkkki

kkkki

CCCii

BBBii

AAAii

The symbol ... designates the fractional part of a real number.

Ground-state energies of Ps (EPs ) and Ps (EPs) in a.u., electron-positron cusp ( 31 ) & electron-electron cusp ( 21 ). [Kar and Ho, CPL 424 (2006) 403]

λD EPs EPs 31 21

0.2620050702325 0.26200507023298*

0.2500000000000 (exact)

0.5000005275 0.5 (exact)

0.49994404 0.5 (exact)

100 0.2521260115714 0.2401480529919 0.5000005414 0.49994391

50 0.2424865663745 0.2305848184067 0.5000005778 0.49994373

20 0.2149738357888 0.2035290153067 0.5000006891 0.49994306

10 0.1736181600010 0.1634042556846 0.5000010905 0.49994153

8 0.1549479742756 0.1454597937606 0.5000017562 0.49994029

6 0.1266427187516 0.1184163351348 0.5000013658 0.49992319

4 0.0798460757487 0.0740585109450 0.5000016594 0.49988734

3 0.0449254118079 0.04122367056 0.5000039360 0.49975111

2 0.005965669 0.00514 0.49995635 0.4992069

1.8 0.0011185 0.0008 0.4992365 0.50214

*Best results [G W F Drake and M. Grigorescu, J Phys B 38 (2005) 3377 A M Frolov, Phys Lett A 342 (2005) 430]

31

0.00 0.14 0.28 0.42 0.56

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

E (

a.u

.)

Ps(1

S)

Ps

0.00 0.14 0.28 0.42 0.560.000

0.002

0.004

0.006

0.008

0.010

0.012

Ele

ctro

n a

ffin

ity (

a.u

.)

The ground state energy of Ps ion and the electron affinity (EPs EPs ) of Ps atom for the different values of the screening parameter . [S. Kar and Y. K. Ho, Phys Rev A 71 (2005) 052503; Chem. Phys. Lett. 424 (2006) 403]

Determination of resonance energies and widths by the

stabilization method

V A Mandelshtam, T. R. Ravuri and H.S. Taylor, Phys. Rev.Lett. 70 (1993) 1932

Calculations of density of resonance states 1

)(11

11 )()()(

EEii

ininn

in

EEE

,

where the index i is the i -th value for and the index n is for the n-th resonance. Fitting to the

Lorentzian form yields resonance energy rE and a total width , with

220

2)(

2)(

r

nEE

AyE

,

where 0y is the baseline offset, A is the total area under the curve.

Some recent references:

S. Kar and Y. K. Ho, J. Phys. B 37 (2004) 3177; 38 (2005) 3299; 39 (2006) 2425 . S. Kar and Y. K. Ho, Phys. Rev. E 70 (2004) 066411. L. B. Zhao Y. K. Ho, Phys. of Plasmas 11 (2004) 1695. S. Kar and Y. K. Ho, Chem. Phys. Letters 402, (2005) 544 S. Kar and Y.K. Ho, Phys. Rev. A 71 (2005) 052503; 72 (2005) 010703 (R). S. Kar and Y.K. Ho, New J. Phys.7 (2005) 141. A. Yu and Y. K. Ho, Phys. of Plasmas 12 (2005) 043302.

The Stabilization diagram for the lowest 3Po states of Ps and the fitting of density of states for λD=20. From the fit, we have obtained

Er =0.0306920 and = 0.0000970[S. Kar and Y. K. Ho, PRA 73 (2006) 032502]

-0.0320 -0.0315 -0.0310 -0.0305 -0.0300 -0.0295

0

100

200

300

400

500

600

700

De

nsi

ty

E(a.u.)

0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.036

-0.034

-0.032

-0.030

-0.028

-0.026

15

E(a

.u.)

(a0

-1)

22

The 2s2 1Se resonance energies and widths of Ps under Debye screening S. Kar and Y. K. Ho, Phys. Rev. A 71 (2005) 052503

10 20 30 40 50 60 70 80 90 100-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00E

r(a.u

.)

D

Ps(2S)

20 40 60 80 1000.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

D

(1

05

a.u

.)

0.00 0.02 0.04 0.06 0.08 0.10 0.12-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

Er (

a.u

.)

Ps(2S)

0.00 0.02 0.04 0.06 0.08 0.100.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

(1

05

a.u

.)

20 40 60 80 100 120 140 160 180 200-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01 1P0(1)

3P0(1) Ps(2S)

Er (

a.u

.)

D(a

0)

0.00 0.02 0.04 0.06 0.08 0.10-0.075

-0.060

-0.045

-0.030

-0.015

0.000

1Po(1)

3Po(1) Ps(2S)

Er (

a.u

.)

The lowest 1,3Po resonance energies of Ps under Debye screeningS. Kar and Y. K. Ho, Phys Rev A 73 (2006) 032502

60 80 100 120 140 160 180 200

1.0

1.1

1.2

1.3

1.4

1.5

(1

06

a.u

.)

D(a

0)

1Po(1)

0.000 0.004 0.008 0.012 0.016

0.9

1.0

1.1

1.2

1.3

1.4

1.5

(1

06

a.u

.)

1Po(1)

10 20 30 40 50 60 70 80 90 100

0.2

0.4

0.6

0.8

1.0

1.2

(1

04

a.u

.)

D(a

0)

3P0(1)

0.00 0.02 0.04 0.06 0.08 0.100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(1

04

a.u

.)

3Po(1)

60 80 100 120 140 160 180 2001.0

1.1

1.2

1.3

1.4

1.5

1.6

(1

05a.

u.)

D(a

0)

3Po(2)

0.000 0.005 0.010 0.015 0.020

1.05

1.20

1.35

1.50

1.65

(1

05

a.u

.)

3Po(2)

The 1,3Po resonance widths of Ps embedded in Debye plasma[Kar and Ho, PRA 73 (2006) 032503]

The 2s2 1Se resonance energies of Ps embedded in Debye Plasmas [S. Kar and Y. K. Ho, Phys Rev A 71 (2005) 052503 ]

λD Er

0.0760300.07603044 a

0.076029 b

4.31[-5]4.3034[-5]a

4.3[-5]b

100 0.066260 4.31[-5]70 0.062231 4.29[-5]50 0.057025 4.26[-5]40 0.052633 4.21[-5]30 0.045674 4.09[-5]20 0.033172 3.64[-5]15 0.022642 2.95[-5]10 0.007736 1.16[-5]9 0.004629 6.32[-5]

aY K Ho, Chin J Phys 35(1997) 2345, bA. Igarashi et al, J Phys B 37, (2004) 4221.

The 1,3Po resonace energies and widths of Ps under Debye Screeing

λD1Po(1)

Er

3Po(1)

Er

3Po(2)

Er EPs(2s)

0.0631559 8.87[-7] 0.0631553* 1.0[-6]* 0.063155† 9.92[-7] †

0.07332659 1.274[-4]0.07332735* 1.278[-4]*

0.063097 1.64[-5]0.063097* 1.6[-5]*

0.062500

200 0.0582334 9.58[-7] 0.06838366 1.273[-4] 0.058173 1.61[-5] 0.0576466

100 0.053499 1.31[-6] 0.06356607 1.268[-4] 0.053437 1.51[-5] 0.0530742

70 0.0496161 1.42[-6] 0.0595471 1.259[-4] 0.049557 1.35[-5] 0.0493657

60 0.047535 1.5[-6] 0.0573612 1.25[-4] 0.047482 1.23[-5] 0.0473860

50 0.044723 0.0543595 1.24[-4] 0.044686 0.00001 0.0447073

40 0.04078 0.0499896 1.21[-4] 0.04079 0.0408856

30 0.03489 0.0430748 1.15[-4] 0.03491 0.0350118

25 0.0306 0.037886 1.09[-4] 0.0306 0.0307323

20 0.0248 0.0306920 9.70[-5] 0.0249 0.0249641

15 0.0168 0.020313 7.23[-5] 0.017 0.0169025

10 0.0059 0.00602 1.4[-5] 0.0060526

*Bhatia and Ho, PRA 42 (1990) 1119 †Igarashi et al , NJP 2 (2000) 17

The 1Se and 3Po states in Ps are ‘+’ states, and the two electrons are located on opposite sides of

the positron. The movements of the two electrons are moving toward the positron ‘in phase.’

The autoionization of such a state is through the momentum transfer, as one of the electrons is

‘knocked out’ by the other via the positron. Apparently, when the electron-positron screening is

increased (decreasing λD, increasing 1/ λD), the movement of the electrons will be slowed down.

As a result, the lifetime of the autoionization process will be prolonged, leading to the narrowing of

the resonance width, a consequence of the uncertainty principle.

The 1Po state in Ps is a ‘ ’ state, and one of the electrons is located further away from the excited

positronium. The interaction between the outer electron and the ‘parent positronium’ is an attractive

dipole potential of order r -2. When such a quasi-bound state interacts with the scattering continuum

the outer electron would autoionize. When the screening effect increases (increasing 1/λD), the

strength of the attractive dipole potential will diminish and the outer electron hence autoionize more

rapidly, leading to broadening of the resonance width, a result from again the uncertainty principle.

Summary and conclusions

---- We have investigated the electron affinity of plasma-embedded Ps atom. ---- We have made the first investigation on the 1Se, 1,3Po resonance states of Ps in plasmas for different Debye lengths using the stabilization method. ---- The stabilization method is a very simple and powerful method to extract resonance parameters of the few-body systems for pure Coulomb and screened Coulomb (Yukawa-type) potentials. ---- The screened Coulomb (Yukawa-type) potential is widely used in plasma modeling and is a good approximation for weakly coupled hot plasmas. ---- Our results will provide useful information to plasma physics, astrophysics and atomic physics research communities.

Acknowledgements

• The work is supported by the National Science Council of Taiwan, R.O.C.

• I am thankful to Dr Y. K. Ho for suggesting me the problems, and for his immense help and cooperation to solve the problems.

We next discuss the possible decay routes for the doubly excited states of Ps . (I) Autoionization: The doubly excited state, denoted by (Ps )**, may decay to

the ground state of Ps by ejecting an electron. The process is called autoionization: (Ps )** (2snp 1,3Po) Ps(1S) + e For the present case the 1,3Po states lie below the Ps (n=2) thresholds, there is only one open channel, and the widths we have reported here are the “total widths” as far as the autoionization processes is concerned.

(II) Radiative decay: For the singlet-spin state, the system may cascade to the ground state of Ps ,

(Ps )** (2s3p 1Po) (Ps )**(1s2 1Se)+ , by emitting a photon. Alternatively, it may cascade to a lower-lying S-wave doubly excited state, i.e., (Ps )** (2s3p 1Po) (Ps )**(2s2 1Se )+ . For the triplet-spin case, the system may cascade to an S-wave doubly excited state, i.e., (Ps )** (2s2p 3Po) (Ps )** (2s3s 3Se )+ . The processes described in the last two equations are very similar to the “dielectronic recombination” for electron-ion collisions in atomic cases. Without the plasma environments (D ), for a resonance with a “not-extremely-narrow” width, the autoionization rate usually is much larger than the radiative rates for low-Z ions.

(III) Annihilation: A unique decay route for atomic systems involving positrons is that the positron may annihilate with one of the electrons. For Ps , the positron and an electron annihilate with each other leaving one electron behind, (Ps )** n + e .

Usually, the two-photon case (n=2) would be the dominate annihilation process, but one-photon and three photon, etc., are also possible, but with considerably smaller rates. In an earlier study [36] of two-photon annihilation for the doubly excited 1Se states in Ps (again without the plasma environments), it was found that the autoionization rates are much larger than the annihilation rates. Even for a narrow resonance such as the 2s3s 1Se state, the autoionization rate is about 3000 times lager than the annihilation rate. Here we assume that for the P-waves, the autoionization would still be the dominant process. The above discussion is valid for the unscreened (D ) positronium ions. The situation may change when the Ps ion is embedded in a plasma, as the radiatively decay line may be broadened and shifted by the plasma effect. The subject of line shifts and broadening is an active research area in plasma physics. However, while there has been considerable among of work for bound singly-excited states, investigations for the line broadening for doubly excited states are very scared. As we have predicted that the autoionization width would decrease for some states (i.e. the 3Po states) when the screening effect increases (increasing 1/D), the radiative decay may become more competitive as the decay line is broadened for increasing plasma effect. As for positron annihilation in plasma environments, again to the best of our knowledge, no such investigation has been reported in the literature. Overall, the above three decay processes are competing, and interfering, with each other. The autoionization widths we have reported here represent the partial widths as far as the overall decay is concerned. In the present work, we have not considered the radiative decay process and the positron annihilation process. Studies of such processes for doubly excited positronium ions are of interest and it is worthwhile for future investigations. Our present work is a first step to shed light on an intriguing problem involving doubly excited positronium ions embedded in plasmas.