Transcript Ground State and Resonance States of Ps in Weakly Coupled

Ground state and resonance states
of Ps interacting with screened
Coulomb (Yukawa) potentials
Sabyasachi Kar and Y. K. Ho
Institute of Atomic and Molecular Sciences
Academia Sinica, Taipei, Taiwan
[18th International IUPAP Conference
on Few-Body Problems in Physics]
August 21-26, 2006
Outline
I. Introduction
II. Screened Coulomb (Yukawatype) Potentials

III. Bound States of Ps and Ps
IV. Stabilization Method for
Resonance Calculations
1 e 1 o 3 o
V. Doubly-excited S , P , P

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,
 (ra , rb )  Za Zb exp( ra  rb ) / ra  rb ,
The screening parameter μ (   1/ D ,
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.
Introduction
Bound 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
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]
Ps
Calculations
Hamiltonian:
1
e
r31
3 q q exp(r /  )
1 3 1
i j
ij
D
H     i2  
2 i 1 mi
rij
i  j 1
Debye length:
D 
1
r12
e+

r32
e
2
1

2

k
T
  0 k BT 

0 B
D   2    2
2
2
e ( N e   zk N k )
 en 


k
1
2
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 zk , and Z* is the effective charge.
For two-component plasmas: D  4 (Z *  1)e2n / kBT 
1/ 2
.
1
2
13
Ne  5.4210  5.421016 cm3 , T  103  104 K
z
z z
zr
 ........
From perturbation theory:  exp(r / D )    
r
r D 2D 2
Frank-Thomson criteria: N e  (k BT / 4a0 )3 , kBT is in terms of large rydbergs
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 r / D is given by V  
zr
2D
2
.
Calculations
N
Wavefunction:
  (1  S pnOˆ12 ) Ci r31L PL (cos1 ) exp( i r32  ir31   i r21 ) 
i 1
Schrödinger equation: H  E, where E  0
  |  (rij )
The two-body cusp:  ij 

| 
rij
  |  (rij ) |  
 0.5, (ij )  (31)
,
 qi q j

mi  m j  0.5, (ij )  (21)
mi m j
where  (rij ) is the  -function , (ij ) denotes (21) and (31) , q i and q j are the charges, and
mi and m j 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
(k)
(k)
(k)
(k)
 i [A1(k) , A (k)
],


[B
,
B
]
,


[C
,
C
2
i
1
2
i
1
2 ]
k  mod(i,3)  1, 1  i  N
 1
(k )
(k )
(k ) 
i   
(i (i  1) 2 ( A2  A1 )  A1 
 2

1
(k ) 
(k )
(k )
(k ) 
i  2 
(i (i  1) 3 ( B2  B1 )  B1 
 2

1
(k ) 
(k )
(k )
(k ) 
 i  3 
(i (i  1) 5 (C2  C1 )  C1 
 2

(k )
1
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

 21
 31
EPs
EPs
0.2620050702325
0.2500000000000 0.5000005275 0.49994404
0.5 (exact)
(exact)
0.26200507023298*
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]
0.012
Electron affinity (a.u.)
0.00
-0.10

-0.20
S)
-0.15
Ps
(1
E (a.u.)
-0.05
0.00
0.008
0.006
0.004
0.002
Ps
-0.25
0.010
0.14
0.28

0.42
0.56
0.000
0.00
0.14
0.28
0.42
0.56


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
E ( )  E n ( i 1 )
 n ( E )  n i 1
 i 1   i 1
where the index
i
is the i -th value for
Lorentzian form yields resonance energy
y0
,
En ( i )  E

and the index n is for the n-th resonance. Fitting to the
Er
and a total width
 n ( E )  y0 
where
1
A
,
with
 2
 ( E  Er ) 2   22
,
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]
-0.026
600
22
-0.028
700
-0.030
Density
400
-0.032
-0.034
-0.036
0.3
300
200
100
15
E(a.u.)
500
0.4
0.5
0.6
0.7
-1
(a0 )
0.8
0.9
0
-0.0320
-0.0315
-0.0310
-0.0305
E(a.u.)
-0.0300
-0.0295
The 2s2 1Se resonance energies and widths of Ps under Debye screening
S. Kar and Y. K. Ho, Phys. Rev. A 71 (2005) 052503
4.5
-0.01
4.0
 (10 a.u.)
0.00
5
Er(a.u.)
-0.02
-0.03
-0.04
Ps(
2S)
3.5
3.0
2.5
2.0
-0.05
1.5
-0.06
1.0
-0.07
10
20
30
40
50
60
70
80
90
0.5
100
20
D
0.00
4.5
-0.01
4.0
s(2
P
-0.03
D
60
80
100
3.5
3.0
5
Er (a.u.)
 (10 a.u.)
S)
-0.02
40
-0.04
-0.05
2.5
2.0
-0.06
1.5
-0.07
1.0
-0.08
0.00
0.02
0.04
0.06

0.08
0.10
0.12
0.5
0.00
0.02
0.04
0.06

0.08
0.10
The lowest 1,3Po resonance energies of Ps under Debye screening
S. Kar and Y. K. Ho, Phys Rev A 73 (2006) 032502
1
-0.01
0.000
0
P (1)
0
P (1)
Ps(2S)
1
-0.03
-0.015
Er (a.u.)
Er (a.u.)
-0.02
-0.04
o
P (1)
3 o
P (1)
Ps(2S)
3
-0.030
-0.045
-0.05
-0.060
-0.06
-0.07
20
40
60
80
100 120 140 160 180 200
D(a0)
-0.075
0.00
0.02
0.04
0.06

0.08
0.10
The 1,3Po resonance widths of Ps embedded in Debye plasma
[Kar and Ho, PRA 73 (2006) 032503]
1.5
1.6
1.2
3
a.u.)
1.0
1.3
3
0
P (1)
o
P (1)
 (10
1.2
1.1
5
4
0.8
0.6
0.4
o
P (2)
1.5
 (10 a.u.)
1
6
 (10 a.u.)
1.4
1.4
1.3
1.2
1.1
1.0
0.2
80
100
120
140
160
180
200
10
20
30
D(a0)
1
o
70
80
90
60
1.2
100
120
140
D(a0)
160
180
200
1.65
1.0
0.8
1.0
0.6
3
0.4
o
P (1)
0.2
0.9
0.008

0.012
0.016
3
1.35
o
P (2)
5
4
 (10
1.1
0.004
80
1.50
1.3
0.000
1.0
100
1.2
P (1)
a.u.)
a.u.)
60
1.4
6
 (10
50
D(a0)
1.5
1.4
40
 (10 a.u.)
60
0.0
0.00
1.20
1.05
0.02
0.04
0.06

0.08
0.10
0.000
0.005
0.010

0.015
0.020
The 2s2 1Se resonance energies of Ps embedded in Debye Plasmas
[S. Kar and Y. K. Ho, Phys Rev A 71 (2005) 052503 ]
λD

100
70
50
40
30
20
15
10
9
Er

0.076030
0.07603044 a
0.076029 b
0.066260
0.062231
0.057025
0.052633
0.045674
0.033172
0.022642
0.007736
0.004629
4.31[-5]
4.3034[-5]a
4.3[-5]b
4.31[-5]
4.29[-5]
4.26[-5]
4.21[-5]
4.09[-5]
3.64[-5]
2.95[-5]
1.16[-5]
6.32[-5]
The 1,3Po resonace energies and widths of Ps under Debye Screeing
λD
1Po(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]
*Bhatia and Ho, PRA 42 (1990) 1119
†Igarashi et al , NJP 2 (2000) 17
0.0060526
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/λ D), 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 “notextremely-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 onephoton 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.