ARTICLE IN PRESS

Journal of Quantitative Spectroscopy &

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Radiative Transfer 107 (2007) 315–322

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Transition wavelengths for helium atom in weaklycoupled hot plasmas

Sabyasachi Kar�, Y.K. Ho

Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 106, Taiwan, ROC

Received 4 November 2006; received in revised form 22 January 2007; accepted 23 January 2007

Abstract

We have investigated the effect of surrounding plasmas on several singly excited and doubly excited meta-stable bound

states of helium atom using highly correlated basis functions for singly excited S, P, D states and CI-type basis functions

for doubly excited meta-stable D states. Plasma effect is taken care of by using a screened Coulomb (Yukawa) potential

obtained from the Debye model that admits a variety of plasma conditions, and such a model plays an important role in

plasma spectroscopy. The wavelengths for transitions from the 1snp 1P1 (n ¼ 2,3)-1s2 1Se, 1snp 3P1 (n ¼ 2,3)-1s2s 3Se,

2pnp 1Pe (n ¼ 3,4)-1s2p 1P1, 2pnp 3Pe (n ¼ 2,3)-1s2p 3P1, 2pnd 1D1 (n ¼ 3,4)-1s3d 1De, 2pnd 3D1 (n ¼ 3,4)-1s3d 3De,

2p3p 1Pe-2pnd 1D1 (n ¼ 3,4), 2pnd 1D1(n ¼ 3, 4)-2p4p 1Pe, 2pnp 3Pe (n ¼ 2,3)-2p3d 3D1, and 2pnp 3Pe (n ¼ 2,3)-2p4d 3D1 of helium atom in plasmas for various Debye lengths are reported.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Transition wavelengths; Debye screening; Meta-stable bound states; Weakly coupled hot plasmas; Redshift; Blueshift

1. Introduction

Effect of external environment produced by a charge-neutral background such as that of plasma on theatomic processes has gained considerable interest in recent years ([1–14] and references therein). An atomimmersed in plasmas experiences various perturbations from the plasmas leading to different distributions inthe atomic states compared to those in the unperturbed atomic states. Due to such perturbations, the atomicstates wave functions will go through considerable changes. For the implementation of atomic processes inhigh-temperature plasmas, and the determination of its useful macro-parameters, for example, iontemperature, electron density, it is necessary to have accurate atomic data, e.g. energy levels, transitionprobabilities, etc. Plasma effect can be modeled by different representation of screened Coulomb potentials. Inweakly coupled hot plasmas in which coupling constant is less than 1, the Debye shielding approach can beconsidered to represent the plasma effect between charged particles. In the Debye shielding approach, thelocalized two-particle interaction is to replace the Coulomb potential by an effective exponentially decayingpotential with a screening parameter that depends on the plasma density (n) and temperature (T). The Debye

e front matter r 2007 Elsevier Ltd. All rights reserved.

srt.2007.01.055

ing author. Tel.: +886 2 2366 8274; fax: +8862 2362 0200.

ess: skar@pub.iams.sinica.edu.tw (S. Kar).

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322316

screening provides new insights in atom-in-plasma spectroscopy with reduced binding energies, modifiedtransition energies, oscillator strength, etc. [12]. Several other applications of Debye screening in plasmaphysics, astrophysics and atomic physics have been highlighted in the literature ([1–17] and references therein).With recent advancement in laser plasmas [18] and with the continued interest in helium abundance inastrophysical plasmas [19], it is important to have accurate atomic data available in the literature for thehelium atom embedded in plasmas.

In the present work, we have calculated the wavelengths of transitions from 1snp 1P1 (n ¼ 2,3)-1s2 1Se,1snp 3P1 (n ¼ 2,3)-1s2s 3Se, 2pnp 1Pe (n ¼ 3,4)-1s2p 1P1, 2pnp 3Pe (n ¼ 2,3)-1s2p 3P1, 2pnd 1D1

(n ¼ 3,4)-1s3d 1De, 2pnd 3D1 (n ¼ 3,4)-1s3d 3De, 2p3p 1Pe-2pnd 1D1 (n ¼ 3,4), 2pnd 1D1(n ¼ 3, 4)-2p4p1Pe, 2pnp 3Pe (n ¼ 2,3)-2p3d 3D1, and 2pnp 3Pe (n ¼ 2,3)-2p4d 3D1 of the helium atom immersed inplasmas. The Debye shielding approach has been used to represent plasma effect between the chargedparticles. For bound S, P and D states with natural parities, we have employed highly correlated basisfunctions, whereas CI-type basis functions have been used for the meta-stable bound P and D states withunnatural parities. A state having a parity of (�1)L is called natural parity state, and of (�1)L+1 the unnaturalparity state, where L denotes the total angular momentum of the two-electron system. For calculations oftransition wavelengths, the 1s3p 1,3P1 and 1,3D1 states energies are our new predictions, whereas for the ground1Se state, the bound-excited 3Se, 1,3P1, 1,3De states, as well as for the doubly excited 1,3Pe meta-stable states,their energies are taken from our earlier works [6,11–14]. The convergence of our calculations has beenexamined with the increasing number of terms in the basis sets, and with different sets of parameters in thebasis expansions. It is also interesting to mention here that the effect of Debye plasmas on the doubly excitedstates of highly stripped ions has been investigated by Sil and Mukherjee [9] without consideringelectron–electron screening. We have initiated the resonance calculations on H�, He and Ps� embedded inDebye plasmas by considering electron–electron screening ([6,8,10,13,14] and references therein).

2. Calculations

The non-relativistic Hamiltonian H (in a.u.) describing the helium atom embedded in Debye plasmascharacterized by a parameter lD, called the Debye length, is given by

H ¼ �X2i¼1

1

2r2

i þ ZV ðri; lDÞ� �

þ V ðr12; lDÞ (1)

with

V ðri; lDÞ ¼expð�ri=lDÞ

ri

and V ðr12; lDÞ ¼expð�r12=lDÞ

r12, (2)

where r1 and r2 are the radial coordinates of the two electrons and r12 is their relative distance. Z ¼ 2 in thepresent work. A set of plasma condition can be simulated for different choice of m, as the Debye screeningparameter m ( ¼ 1/lD) is a function of n and T. The Debye screening parameter m (a�10 ) can be written as[7,16,17]

m ¼e2ðNe þ

Pq2

kNkÞ

�0kBT

� �1=2, (3)

where Ne denotes the electron density and Nk denotes the ion density of the kth ion species having a nuclearcharge qk. Also, it can be observed from the perturbation theory that the screening is a repulsive perturbationfor which all the isolated energy levels are displaced upwards and ultimately in the continuum due to repulsiveperturbation. In general, by the writing the screened Coulomb potential as

�z

rexp

�r

lD

� �� �

z

rþ

z

lD�

zr

2l2Dþ � � � , (4)

z being the nuclear charge, one can observe that all the matrix elements will be reduced due to screening andthe first-order correction upshifts all bound levels equally without changing the wave functions [17]. The effect

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322 317

of the perturbation potential in the lowest order in r/lD is given by DV ¼ �ðzr=2l2DÞ. The effect of first-ordercorrections on the energy levels has been investigated in our earlier works ([13,14] and references therein). Ingeneral, the statement is correct if we approximate all the pairs (not only the electron–electron but also theelectron–nucleus pairs). For one-electron case, it is straight forward as is shown in Eq. (4). For two-electroncase, the first order (after expanding ALL the screened Coulomb terms and keeping just the m ¼ 1 terms) willbe +2Z/lD–1/lD and the net result also has the upshift effect. In our actual calculations, we treat theelectron–nucleus pairs in full terms (no approximations) but approximate the electron–electron pair.

For the S, P and D states calculation, we have considered the wave function

C ¼ ð1þ SpnO21ÞXN

i¼1

CirL1 PLðcos y1Þ expð�air1 � bir2 � gir12Þ, (5)

where ai, bi, gi are the non-linear variation parameters, Ci (i ¼ 1, y, N) are the linear expansion coefficients,L ¼ 0 for S states, L ¼ 1 for P states and L ¼ 2 for D states. Spn ¼ 1 denotes singlet-spin states and Spn ¼ �1indicates triplet-spin states, N is the number basis terms and PL denotes the Legendre polynomial of order L.The operator O21 is the permutation of the two identical particles. In the present work, the wave functionsused for the bound-excited states are of (l1,l2) ¼ (0,0), (1,0) and (2,0) for the S, P and D states, respectively,where l1 and l2 and the angular momentum for electron 1 and electron 2, respectively. Since one of the twoelectrons will always be of an S state, and the angular momentum for S states is related to Y0,0, itself is just aconstant. Usually it is understood that such a constant is included implicitly in the linear constants (the Ci) inEq. (5). The non-linear variational parameters ai, bi and gi are chosen from a quasi-random process asproposed by Frolov [20], and as used in our earlier works [6,8,10–14].

For parity unfavored P and D states calculations, we have considered the following CI-type wavefunctions [21]:

Cðr1; r2Þ ¼ AXlalb

Xij

Cai ;bjZaiðr1ÞZbj

ðr2Þ � Y Llalbðr1; r2ÞSðs1;s2Þ, (6)

where

ZaiðrÞ ¼ rnai e�aai r (7)

is a Slater-type orbital, C’s are the coefficients to be determined, A is the antisymmetization operator, S is two-particle spin eigenfunction, Y is the two-body spherical harmonics, i, j and a, b refer to the channel indices forthe two electrons. The electron–electron correlation is included by using the products of Slater orbitals. Thehighest angular momentum for the individual electron is li ¼ 10 for 1Pe (525 terms), li ¼ 14 for 3Pe (730), andli ¼ 14 for 1,3D1 leading to 730 terms and 1166 terms, respectively, of basis function (7). In the presentcalculations of the unnatural parity P and D states, we have included the electron–nucleus screening explicitly,but approximated the electron–electron screened Coulomb potential in the form of Eq. (4) by the followingexpansion, i.e.,

V ðr12; lDÞ �Xm

n¼0

ð�1Þnrn�112

lnDn!

. (8)

In Eq. (8), m ¼ 0 indicates the omission of electron–electron screening. By inclusion only the m ¼ 1 termwill decrease the energy level, and for m41, it indicates the inclusion of the second and higher ordercorrections on the electron–electron screening. For the present problem, we have used m up to 7. We believethe truncation of the expansion Eq. (8) to m ¼ 7 would lead to reasonably accurate results for mp0.125.Furthermore, for the electron–electron screening term given in Eq. (8), we have approximated r12Er1+r2,assuming the two electrons are located on the opposite sides of the nucleus. This lower order approximation ofr�112 terms in the Hamiltonian (1) is a valid approximation because of exponential cut-off in the basis functions.The details of the validity and accuracy of the approximation has been discussed in our earlier works [13,14].

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322318

3. Results and discussions

We first calculate the singly excited states and doubly excited meta-stable bound states by obtaining thesolutions of the Schrodinger equation HC ¼ EC, where Eo0 following the Rayleigh–Ritz variationalprinciple using the correlated basis (5) and (6), respectively. As mentioned earlier, the doubly excited statesreported here are of unnatural parities (�1)(L+1), and they are lying below the N ¼ 2 threshold of the He+

ion. These states are not quasi-bound states (resonance states) but meta-stable bound states. From thescattering point of view, the system consists of a scattering electron and the target ground state of the He+(1S)ion. The parities for such continuum states are of (�1)L, the so-called natural parities. The doubly excitedstates with unnatural parities will not interact with the continuum that is of natural parities, the Rayleigh–Ritzvariational principle is applicable to such states, and their variationally obtained energies are upper bound tothe exact energies, respectively. The doubly excited states of natural parities are quasi-bound (resonance)states. For example, the 2s2p 1,3P1 states (natural parity) will interact with the scattering continuum and theymight autoionize to the He+(1S) ground state by ejecting a P-wave electron, and as such, the Rayleigh–Ritzvariational principle does not apply to such doubly excited resonance states. For doubly excited states lyingbelow the He+(N ¼ 3) thresholds but above the N ¼ 2 thresholds (2S and 2P), they are quasi-bound(resonance) states for both natural and unnatural parities. For example, the 3s3p 1,3P1 states (natural parity)can autoionize to a lower lying He+(1S or 2S) state by ejecting a P-wave electron. The 3p2 3Pe (unnatural

Table 1

The singly excited bound 1s3p 1,3P1 state energies of helium atom in plasmas for different Debye lengths along with the He+(1S) threshold

energies

lD 1s3p 1P1 1s3p 3P1 He+(1S) thresholdb

N �2.05514636208 �2.05808108426 �2.00000000000

�2.05514636209a �2.05808108427a

100 �2.02582235758 �2.02872364248 �1.98007475170

50 �1.99775506360 �2.00056549845 �1.96029802699

30 �1.96209243842 �1.96471073511 �1.93415761310

20 �1.92006881466 �1.92235747445 �1.90184477572

15 �1.8806206796 �1.88250349387 �1.86992912020

10 �1.8087160 �1.8096166 �1.80726571410

8 �1.761350 �1.761479 �1.76126804403

aRef. [25].bRef. [11].

Table 2

The doubly excited 1,3D1 meta-stable bound state energies of helium atom in plasmas for different screening parameters along with the

He+(2S) threshold energies

lD 2p3d 1D1 2p4d 1D1 2p3d 3D1 2p4d 3D1 He+(2S)d

N �1.12760074 �1.069153 �1.1186560 �1.0653563 �1.000000

�1.12760081a �1.06915203a �1.11865650a �1.106535615a

�1.12760046b �1.06907c �1.11865608b �1.06523b

�1.12743c �1.11842c

100 �1.068714 �1.011186 �1.0598634 �1.0075055 �0.960593

50 �1.012004 �0.957015 �1.0034207 �0.9536427 �0.922346

30 �0.939690 �0.890245 �0.9317031 �0.8874991 �0.873091

20 �0.854533 �0.814855 �0.8476339 �0.8125957 �0.814207

15 �0.775233 �0.7696699 �0.758175

13 �0.729369 �0.7245223 �0.725056

aRef. [23].bRef. [22].cRef. [24].dRef. [11].

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322 319

parity) state might autoionize to the He+(2P state) by ejecting a P-wave electron. In the present work, wereport results for doubly excited states with unnatural parities. For a treatment of doubly excited quasi-bound(resonance) states with natural parities, readers are referred to our earlier works [6,10,13].

The singly excited bound 1s3p 1,3P1 state energies and the doubly excited 1,3D1 meta-stable bound stateenergies obtained from the present calculations are presented in Tables 1 and 2, respectively. It is clear fromTables 1 and 2 that our calculated results in the unscreened case are fairly comparable with the available

Table 4

Transition wavelengths (in A) of helium atom in plasmas below the He+(2S) threshold under Debye screening

lD AA BB CC DD FF GG HH RR

N 27704.9 9976.65 19177.3 83372.0 3014.01 2562.31 53705.0 12968.5

3014.01a 2561.9b

3013.0b

100 27698.1 10077.3 19531.1 83826.4 3019.92 2573.40 53513.9 13133.2

50 27685.9 10366.7 20571.7 85229.6 3037.18 2604.99 53004.2 13607.0

30 27694.1 11065.8 23218.4 89361.9 3077.54 2677.78 52034.0 14765.3

20 27848.1 12586.4 29778.4 100399.3 3157.40 2815.58 50809.1 17202.4

15 28374.8 3274.08 50207.2

10 30930.5

AA: E(2p3p 1Pe)�E(2p3d 1D1); BB: E(2p3p 1Pe)�E(2p4d 1D1); CC: E(2p3d 1D1)�E(2p4p 1Pe); DD: E(2p4d 1D1)�E(2p4p 1Pe); FF:

E(2p2p 3Pe)�E(2p3d 3D1); GG: E(2p2p 3Pe)�E(2p4d 3D1); HH: E(2p3p 3Pe)�E(2p3d 3D1); RR: E(2p3p 3Pe)�E(2p4d 3D1).aRef. [22].bRef. [26].

Table 3

Transition wavelengths (in A) of helium atom in plasmas below the He+(2S) threshold for different Debye lengths

lD A B C D F G H R

N 305.42 299.55 304.50 299.18 295.18 287.68 320.267 291.07

305.4a 304.5a 295.18a 320.27a 291.07a

100 305.42 299.64 304.51 299.27 295.22 287.80 320.27 291.14

50 305.41 299.88 304.53 299.55 295.36 288.15 320.28 291.32

30 305.42 300.44 304.60 300.17 295.68 288.91 320.32 291.75

20 305.48 301.47 304.78 301.25 296.29 290.31 320.40 292.55

15 305.63 305.06 297.13 320.53 293.66

10 305.66 299.50 320.99

A: E(2p3d 1D1)�E(1s3d 1De); B: E(2p4d 1D1)�E(1s3d 1De); C: E(2p3d 3D1)�E(1s3d 3De); D: E(2p4d 3D1)�E(1s3d 3De); F: E(2p3p1Pe)�E(1s2p 1P1); G: E(2p4p 1Pe)�E(1s2p 1P1); H: E(2p2p 3Pe)�E(1s2p 3P1); R: E(2p3p 3Pe)�E(1s2p 3P1).

aRef. [22].

Table 5

Transition wavelengths (in A) of helium atom below the He+(1S) threshold for different Debye lengths

lD AI AJ AK AL

N 584.234 536.937 10831.59 3889.37

100 584.388 537.293 10834.58 3901.31

50 584.838 538.303 10843.19 3935.13

30 585.874 540.558 10863.02 4011.31

20 587.829 544.703 10901.68 4155.38

15 590.485 550.215 10957.62 4357.44

10 597.826 565.205 11136.23 4985.28

8 605.098 580.693 11351.61 5831.35

AI: E(1s2p 1P1)�E(1s2 1Se); AJ: E(1s3p 1P1)�E(1s2 1Se); AK: E(1s2p 3P1)�E(1s2s 3Se); AL: E(1s3p 3P1)�E(1s2s 3Se).

ARTICLE IN PRESS

0.00 0.02 0.04 0.06 0.08 0.10285

288

291

294

297

300

303

306

E(2p4p1Pe)−E(1s2p1P°)

E(2p3p1Pe)−E(1s2p1P°)

E(2p4d1D°)−E(1s3d1De)

Wa

ve

len

gth

(A

ng

str

om

)

µ

E(2p3d1D°)−E(1s3d1De)

0.00 0.02 0.04 0.06 0.08 0.10290

295

300

305

310

315

320

325

E(2p3p 3Pe)−E(1s2p 3P°)

E(2p2 3Pe)−E(1s2p 3P°)

E(2p3d 3D°)−E(1s3d 3De)

E(2p4d 3D°)−E(1s3d 3De)

Wa

ve

len

gth

(A

ng

str

om

)

µ

0.00 0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

E(2p3p3Pe)−E(2p3d 3D°)

E(2p4d1D°)−E(2p4p 1Pe)

E(2p3d1D°)−E(2p4p 1Pe)

Wa

ve

len

gth

(1

04 A

ngstr

om

)

µ

E(2p3p1Pe)−E(2p4d1D°)

0.000 0.025 0.050 0.075 0.100 0.1252.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

E(2p2p 3Pe)−E(2p4d 3D°)

E(2p2p 3Pe)−E(2p3d 3D°)

E(1s3p 3P°)−E(1s2s 3Se)

Wa

ve

len

gth

(1

03 A

ngstr

om

)

µ

0.000 0.025 0.050 0.075 0.100 0.1251.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

E(2p3p 1Pe)−E(2p3d1D°)

E(2p3p 3Pe)−E(2p4d 3D°)

E(1s2p 3P°)−E(1s2s 3Se)

Wa

ve

len

gth

(1

04 A

ngstr

om

)

µ0.000 0.025 0.050 0.075 0.100 0.125

530

540

550

560

570

580

590

600

610

E(1s3p 1P°)−E(1s2 1Se)

E(1s2p 1P°)− E(1s2 1Se)

Wa

ve

len

gth

(A

ng

str

om

)

µ

a b

c e

d f

Fig. 1. Transition wavelengths of helium atom immersed in plasmas in terms of the screening parameter m.

S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322320

results [22–25] in the literature. The singly excited 1s3p 1,3P1 states and the doubly excited 2pnd (n ¼ 3, 4) 1,3D1

meta-stable bound states results presented in Tables 1 and 2 for different Debye lengths are the new results.The He+(1S) and He+(2S) threshold energies are taken from our earlier work [11].

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 315–322 321

To obtain the wavelengths, we then calculate the energy differences in atomic units between each of thedoubly excited meta-stable bound 2pnp (nX2) 1,3Pe and 2pnd (nX3) 1,3D1 states below the He+(2S) threshold,and the bound 1sns (nX1) 1,3Se, 1snp (nX2) 1,3Po and 1snd (nX3) 1,3De states of helium below the He+(1S)threshold for different Debye lengths. The energy differences are then converted from atomic units toAngstrom (A) by using the standard conversion unit (1 a.u. of energy corresponds to a wavelength of455.633 A). We have presented the results in Tables 3–5 and Fig. 1. In the unscreened case, our results arecomparable to the best theoretical [22–24] and experimental [26] results available in the literature.

In physics and astronomy, ‘redshift’ (i.e., any increase in the wavelength) means the shift of the optical/non-optical spectrum of an object towards the red end of the electromagnetic spectrum. Conversely, ‘blueshift’ (i.e.,any decrease in the wavelength) means the shift of the visible light from an object towards the blue side of thespectrum. Our results show that as the plasma screening effect increases (decreasing lD), the transitionwavelengths for the different states are red-shifted to a region of longer wavelengths (see Fig. 1 and Tables3–5) except the transition between the 2p3p 1,3Pe and 2p3d 1,3D1 states. From Table 4 and Fig. 1c, it is clearthat the transition wavelengths between the 2p3p 3Pe and the 2p3d 3D1 states are blue-shifted to a region ofshorter wavelengths. Also it seems from the energy difference E(2p3p 1Pe)�E(2p3d 1D1) (see Table 4) that thewavelengths for this transition are first blue-shifted very slowly and then red-shifted for the values of Debyelength lDo50. However, at present we are not in a position to explain the fact for the transition 2p3d 1D1-2p3p 1Pe.

4. Summary and conclusions

In this work, we have estimated for the first time the wavelengths of transitions between the 2pnd (n ¼ 3, 4)1,3D1 and 2pnp (n ¼ 2, 3) 1,3Pe states, the 2pnd (nX3) 1,3D1 and 1s3d 1,3De states, the 1s3p 1P1 and 1s2 1Se

states, the 1s3p 3P1 and 1s2s 3Se states of helium in plasmas for different Debye lengths. We have alsopresented the transition wavelengths between the 1s2p 1P1 and 1s2 1Se states, between the 1s2p 3Po and 1s2s 3Se

states, and between the 2pnp (n ¼ 2, 3) 1,3Pe and 1s2p 1,3P1 states for various screening parameters. Some ofthe singly excited states and doubly excited meta-stable bound P states energies are taken from our earlierworks. We have considered highly correlated Hylleraas-type wave functions for some S, P, D states withnatural parities and CI-type basis functions for unnatural parities. The transition wavelengths of helium atomin weakly coupled hot plasmas for different Debye lengths will provide a new insight in plasma spectroscopybesides several other applications in plasma diagnostic, astrophysics and atomic physics. However, as theDebye shielding parameter is temperature dependent, for detailed spectroscopic applications, such atemperature-dependent model needs to be studied in the framework of quantum statistics and such a study isbeyond the scope of our present investigations. The Debye shielding approach is a good approximation in theweakly coupled hot plasmas and with the recent development in the laser plasmas we hope our prediction willprovide useful information to the plasma physics community in near future.

Acknowledgment

The work is supported by the National Science Council of Taiwan, ROC.

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