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Journal of Quantitative Spectroscopy &

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Radiative Transfer 109 (2008) 445–452

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Oscillator strengths and polarizabilities of the hot-denseplasma-embedded helium atom

Sabyasachi Kar�, Y.K. Ho

Institute of Atomic and Molecular Sciences, Academia Sinica, No. 1, Sec. 4, Roosevelt Road, P.O. Box 23-166, Taipei, Taiwan 106, ROC

Received 13 April 2007; received in revised form 6 June 2007; accepted 4 July 2007

Abstract

The effect of weakly coupled hot plasma environment on the oscillator strengths of the ultraviolet and visible series and

the polarizabilities of helium has been investigated using variational highly correlated wave functions within the non-

relativistic framework. The Debye shielding approach that admits a variety of plasma conditions is used to simulate the

plasma effects. For each shielding parameter, dipole oscillator strengths are calculated for the 1 1S–n 1P (n ¼ 2, 3), 2 1S–21P, 2 3S–n 3P (n ¼ 2, 3) and 2 1,3P–n 1,3D (n ¼ 3, 4) transitions. The dipole and quadrupole polarizabilities for the ground

He (1s2 1S) state are also reported for each screening parameter. Results obtained are useful in plasma diagnostic purposes

besides several other applications.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Hot plasmas; Debye shielding; Oscillator strengths; Dipole polarizability; Quadrupole polarizability; Variational wave

functions

1. Introduction

An atom immersed in plasma experiences various perturbations from the plasma, leading to differentdistributions in the atomic states compared with the unperturbed atomic states. Plasma effects can besimulated using different models. In hot-dense and low-density warm plasmas, the interaction between twolocalized charged particles can be modeled by replacing the Coulomb potential with an effective screenedCoulomb (Yukawa-type) potential. Such a screened Coulomb potential obtained from the Debye model ischaracterized by the Debye length lD proportional to the square root of n/T, with n being the density of theplasma and T its temperature. For the determination of important fundamental parameters such as electrontemperature, ion density, etc., and for certain atomic processes in hot plasmas, it is necessary to have accurateatomic data, i.e., energy levels, transition wavelengths, oscillator strengths and polarizabilities available in theliterature. The effect of weakly coupled hot plasmas on the bound states ([1–7], references therein), doublyexcited meta-stable bound states and resonance states ([8,9], references therein) of helium atom has been

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

srt.2007.07.003

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 109 (2008) 445–452446

investigated in recent years. The importance of Debye approach of plasma modeling has been discussed in theliterature [1–15].

In the present study, we carry out an investigation of plasma effect on the oscillator strengths and thepolarizabilities of the He atom based on the Deybe model. In the unscreened case, several theoretical studieshave been performed on the oscillator strengths ([16], references therein) and polarizabilities of He ([17–25],references therein). In the free atom case, few experimental results on dipole polarizability of He are alsoavailable in the literature ([26–28], references in [25]). Advanced experimental techniques for measuringoscillator strengths of atomic and ionic transitions in vacuum ultraviolet lines are also described in severalreports [29]. The oscillator strengths of He for the S–P transitions are reported in the literature [1,5] in hot andweakly coupled plasma environments. The static limit of the frequency-dependent dipole polarizability and theoscillator strength for S–P transitions of the He atom under Debye screening was also reported for twoelectron atoms by ignoring the screening on the electron–electron repulsions terms in a time-dependentperturbation calculation [10]. However, oscillator strengths for the P–D transition and the quadrupolepolarizability of plasma-embedded He have not been reported in the literature to the best of our knowledge.With the recent developments in laser plasmas produced by laser fusion in the laboratories ([29,30] and withrecent theoretical developments in hot plasma-embedded He ([1–13], references therein), it is important tohave accurate atomic data available in the literature for the quantities of fundamental interests such asoscillator strengths and polarizabilities for He atom in model plasma environments. Here we have made asystematic improvement on the oscillator strengths and dipole polarizability of the He atom immersed inDebye plasmas.

In the present work, we have investigated the dipole oscillator strengths for the 11S–n 1P (n ¼ 2, 3), 2 1S–21P, 2 3S–n 3P (n ¼ 2, 3) and 2 1,3P–n 1,3D (n ¼ 3, 4) transitions, and have calculated the dipole and quadrupolepolarizabilities of the He atom in its ground state for different screening parameters. Oscillator strengths forthe S–P transitions in He were reported in the literature [1,5] for weakly coupled hot plasma environments,and the earlier results will be compared with our present results later in the text. The oscillator strengths forthe P–D transitions and quadrupole polarizabilities of He immersed in weakly coupled hot plasmas are firstcalculations to our knowledge. A variational highly correlated wave function is used in the frameworkRayleigh-Ritz principle. The convergence of our calculations has been examined with increasing number ofterms in the basis expansions.

2. The method

The non-relativistic spin-independent Hamiltonian H of He atom immersed in Debye plasmas characterizedby the Debye length lD is given by

H ¼ �1

2r2

1 �1

2r2

2 � 2expð�r1=lDÞ

r1þ

expð�r2=lDÞr2

� �þ

expð�r12=lDÞr12

, (1)

where r1 and r2 are the radial coordinates of the two electrons and r12 is their relative distance. The parameterm ( ¼ 1/lD) is called the Debye shielding parameter. Sets of plasma conditions can be simulated with differentchoices of lD.

In the present work, we employ the following explicitly correlated wave functions [6,7,12,16–20]:

C ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Lþ 1p

4pð1þ SpnO12Þ

XN

i¼1

CirL1 PLðcos y1Þ expð�air1 � bir2 � gir12Þ, (2)

where ai, bi, gi are the non-linear variation parameters, L ¼ 0, 1, 2 for S, P, D states respectively, Ci(i ¼ 1, y,N) are the linear expansion coefficients, for singlet states Spn ¼ 1 and Spn ¼ �1 indicate triplet states, O12 is thepermutation operator on the subscripts 1 and 2 representing two electrons. Here, we use a quasi-randomprocess ([6,7,12,16–20], references therein) to optimize the non-linear variational parameters ai, bi and gi. The

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452 447

parameters ai, bi and gi are chosen from the three positive intervals [A1, A2], [B1, B2] and [C1, C2],

ai ¼1

2iði þ 1Þ

ffiffiffi2p

� �� �ðA2 � A1Þ þ A1,

bi ¼1

2iði þ 1Þ

ffiffiffi3p

� �� �ðB2 � B1Þ þ B1,

gi ¼1

2iði þ 1Þ

ffiffiffi5p

� �� �ðC2 � C1Þ þ C1, ð3Þ

where the symbol //ySS designates the fractional part of a real number. To calculate bound-excitedenergies, one needs to obtain the solutions of the Schrodinger equation HC ¼ EC, where Eo0 using theRayleigh-Ritz variational principle. By employing the quasi-random process (3) on the wave functions (2), thebound S, P and D states’ energies of plasma-embedded He were obtained in our earlier works [6,7]. Once theoptimum bound S, P and D states’ energies and wave functions, as well as the optimized parameters for suchstates, are obtained, one can proceed to calculate the oscillator strengths and other fundamental properties ofthe plasma-embedded He atom. In this work, we have used 600-term and 700-term basis functions to obtainthe converged results of the oscillator strengths for S–P and P–D transitions, respectively, whereas we haveemployed 600-term basis functions to calculate dipole polarizability, and 700-term of S-states and 900-terms ofD-states to calculate quadrupole polarizability of the He atom. However, our calculated results are convergentup to the quoted digits using 500 basis terms of Eq. (2).

3. Oscillator strengths

The relative intensities of radiative transitions from the initial states m to various final states n is given by

Inm ¼ jhmjV 1jnij2, (4)

where V1 is the dipole operator. The optical oscillator strengths for the dipole allowed S–P and P–Dtransitions using the lengths form are defined as [16]

f nm ¼ CðEn � EmÞInm, (5)

with C ¼ 2 for S–P transitions and C ¼ 5/3 for P–D transitions between the states m and n. From Eq. (4), thecases when fnm40 are for absorption and fnmo0 for emission. The multipole operators are given by

Vi ¼ ri1Piðcos W1Þ þ ri

2Piðcos W2Þ, (6)

with i ¼ 1 for dipole, i ¼ 2 for quadrupole, etc. Using formula (4) we have calculated oscillator strengths forthe ultraviolet principal series, the 1 1S–n 1P (n ¼ 2, 3) transitions, the visible principal series 2 1S–n 1P (n ¼ 2)transition, 2 3S–n 3P (n ¼ 2, 3) and 2 1,3P–n 1,3D (n ¼ 3, 4) transitions, and the results are presented in Tables 1and 2 and Figs. 1 and 2. In the unscreened case, our results compare well with the reported results [16] and thecomparisons are made in Tables 1 and 2. As mentioned in our earlier work [7], we have not included the pp-terms [16] for the D-state wave functions in Eq. (2). However, it seems that the results for energies are accurateup to 6–7 significant digits for different bound-excited states [7]. Hence, our calculated oscillator strengths forthe P–D transitions are accurate up to 4–5 significant digits compared with the reported results [16]. Also inTable 1 and Fig. 1(a) our results for oscillator strengths in the screened case for the 1 1S–2 1P, 2 1S–2 1P and 23S–2 3P transitions are compared with the reported results of Lopez et al. [1]. It is clear from our results thatthe oscillator strengths for the transitions decrease with increasing plasma strengths except for the 2 1S–2 1Pand 2 3S–2 3P transitions that increase with increasing plasma strength. With the increasing plasma strengths,the intensities for the 2 1S–2 1P and 2 3S–2 3P transitions are much larger to suppress the decreasing energydifferences [6].

ARTICLE IN PRESS

Table 1

Oscillator strengths of He in plasmas for different Debye lengths

D 1 1S–2 1P 2 1S–2 1P 2 3S–2 3P 1 1S–3 1P 2 3S–3 3P

N 0.276165 0.37644 0.539086 0.073435 0.064461

0.27617a 0.37648a 0.5391a 0.07343a 0.06447a

100 0.275554 0.37730 0.539757 0.072659 0.063568

0.274512b 0.379971b 0.541245b

50 0.273802 0.37975 0.541692 0.070558 0.061123

0.272802b 0.383299b 0.543159b

30 0.269866 0.38530 0.546070 0.066100 0.055925

20 0.262623 0.39565 0.554208 0.058340 0.047050

0.262059b 0.397266b 0.555492b

15 0.253038 0.40963 0.565101 0.048518 0.036382

12 0.241289 0.42720 0.578600 0.036821 0.024873

10 0.227497 0.44846 0.594592 0.023167 0.013646

0.230451b 0.444807b 0.594609b

9 0.217202 0.46480 0.606584 0.013342 0.007082

8 0.203112 0.48780 0.622987 0.004828 0.001783

7 0.182976 0.52189 0.646127

6 0.152342 0.57566 0.679158

5 0.09950 0.65603 0.71530

aRef. [16].bRef. [1] (the reported results are multiplied by 3).

Table 2

Oscillator strengths of He under Debye screening

D 2 1P–3 1D 2 3P–3 3D 2 1P–4 1D 2 3P–4 3D

N 0.710075 0.610067 0.120273 0.122797

0.71017a 0.61024a 0.12027a 0.12285a

100 0.706431 0.606450 0.118599 0.120711

50 0.696242 0.596539 0.114041 0.115090

30 0.673049 0.574055 0.102730 0.101773

20 0.626724 0.529876 0.073235 0.070064

18 0.605941 0.511507 0.054976 0.051729

15 0.553411 0.461802

12 0.428787 0.350746

11 0.323398 0.261157

aRef. [16].

S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452448

4. Polarizability

The generalized static polarizability of multipole order i is defined as [21]

Si ¼ 2X

n

h0jV ijnihnjVij0i

En � E0ða3

0Þ. (7)

For dipole (S1) and quadrupole (S2) polarizabilities, n implies all the P and D states, respectively, includingthe continuum states that are represented by pseudo-states, whereas 0 denotes the ground 1s2 1S state of He.

ARTICLE IN PRESS

0.00 0.03 0.06 0.09

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

2 1P - 3 1D

2 3P - 3 3D

Oscill

ato

r S

trength

µ0.014 0.042

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.122 1P - 4 1D

2 3P - 4 3D

Oscill

ato

r S

trength

µ

0.13

0.000 0.028 0.056

Fig. 2. Oscillator strengths for P–D transitions of hot-dense plasma-embedded He.

0.00 0.05 0.15 0.20

0.2

0.5

1 1S - 2 1P

2 1S - 2 1P

2 3S - 2 3P

Oscill

ato

r S

trength

µ

0.025 0.075 0.125

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

2 3S - 3 3

P

Oscill

ato

r S

trength

µ

1 1S - 3 1

P

0.7

0.6

0.4

0.3

0.1

0.10 0.000 0.050 0.100

Fig. 1. Oscillator strengths for S–P transitions of He under Debye screening. Solid lines denote present works and the dashed lines in (a)

are the reported results (multiplied by 3) of Lopez et al. [1].

S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452 449

Using Eq. (7) we have calculated the dipole and quadrupole polarizabilities of the ground state He atom fordifferent Debye lengths. The results are presented in Table 3 and Fig. 3. In the unscreened case, our results arewell comparable to the available theoretical results in the literature [20–25]. Our dipole polarizability resultdiffers with the best result in the literature by no more than 1� 10�9 a0

3. We have also compared our resultswith the available experimental results [25–28] in Table 3, and with the other calculation by Saha et al. [10] inFig. 3. The dipole polarizability reported by Saha et al. [10] was obtained by ignoring the electron–electronscreening. For quadrupole polarizability our result is less accurate compared with our dipole case results. Aswas discussed earlier in Section 3, that by not including the pp-terms in the D-state wave functions, thequadrupole polarizability is correct up to some part in 10�4 with only employing the sd-terms. As our maininterest is focused on the investigation of the plasma effects on the polarizabilities, it is sufficient for now toconsider D-states by using only the sd-term wave functions. The increasing trend of dipole and quadrupolepolarizabilities with increasing plasma strength indicates that the system would become more polarizable whenthe plasma strength is increased. We should also mention that for one-electron systems interacting with Debyepotentials, the ground state static and dynamic polarizabilities were investigated earlier by Friedman et al. [31]and by Zimmermann [32], respectively. Finally, we now comment on the physical implications of thepolarizibilities for atoms embedded in Debye plasmas. Consider weakly coupled and partially ionized plasmaas an example. Assuming that the plasma has reached thermal equilibrium, the electric field effects due to theplasma charges on a plasma-embedded atom has led to a screened Coulomb potential of Debye type. Now if

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Table 3

Polarizabilities of hot-dense plasma-embedded helium atom in its ground state

D Dipole polarizability (a03) Quadrupole polarizability (a0

3)

N 1.383192173 2.444

1.383192174a,b 2.44508a

1.3861c

1.38377(7)d

1.383794e

1.383746(7)f

100 1.383448194 2.445

50 1.384206190 2.447

30 1.385973937 2.453

20 1.389360958 2.464

15 1.394013270 2.479

10 1.406973146 2.521

8 1.419794461 2.562

6 1.446937353 2.650

5 1.473826802 2.738

4 1.522940615 2.902

3 1.629642204 3.270

2.5 1.739950329 3.669

2 1.955407309 4.500

1.5 2.49739410 6.878

1.0 5.0245812 22.52

aRef. [23].bRef. [24].cRef. [26].dRef. [27].eRef. [28].fK. Grohmann and H. Luther (1992) (see Ref. [25]).

0.1 0.31.30

1.35

1.40

1.45

1.50

1.55

1.60

1.65

1.70

1.75

Present work

Ref. [10]

Dip

ole

pola

rizabili

ty (

a0

3)

µ0.1 0.3

3.0

3.6

Quadru

pole

pola

rizabili

ty (

a0

3)

µ0.0 0.2 0.4 0.0 0.2 0.4

3.4

3.2

2.8

2.6

2.4

Fig. 3. Polarizabilities of ground state helium atom immersed in hot weakly coupled hot plasmas.

S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452450

we also assume that the atom is further subjected to an external DC electric field, its ground state energy levelwill be disturbed by such an external field, and the first and second orders of corrections to the energy arerelated to the dipole and quadrupole polarizabilities, respectively. For plasma-embedded atoms, we assumethat such a perturbation treatment is still valid, but the polarizabilities (dipole, quadrupole, etc.) for free atomsin the pure Coulomb environment are now replaced by those determined under the screened Coulombenvironment, as determined in the present calculations.

ARTICLE IN PRESSS. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452 451

In general discussions, we would like to mention again about the possible improvement of ourinvestigations. It is important to have accurate ab initio results to confirm the expected behaviors in plasmas.In this work, the results obtained for the oscillator strengths for P–D transitions and the quadrupolepolarizability of He atom under the influence of Debye screening are reported for the first time. Ourpredictions on these fundamental quantities are much improved than the other reported results. Studies on themultiple charged two-electron atoms in hot-dense plasmas are important for plasma spectroscopy, and suchinvestigations are of our future interest. We hope our present work on the polarizabilities and oscillatorstrengths for S–P and P–D transitions of He under Debye screening will provide a new insight into futureinvestigations on these fundamental quantities.

5. Summaries and conclusion

In the present work, we have made an investigation on the static dipole and quadrupole polarizabilities ofhelium atom immersed in hot, weakly coupled plasma environments in the framework of Debye screeningusing highly correlated wave functions. In such an environment, we have also investigated the oscillatorstrengths for the ultraviolet and visible series for the plasma-embedded helium atom. The oscillator strengthsfor the P–D transitions are calculated for the first time when the screening effects are included. With the recentadvancement in laser plasmas [29,30], and with the recent activities on the studies on multipole polarizabilitiesof the helium atom, we hope our results will provide useful information to the research communities in severalbranches of physics and chemistry.

Acknowledgment

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

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