ARTICLE IN PRESS
Journal of Quantitative Spectroscopy &
Radiative Transfer 109 (2008) 445–452
www.elsevier.com/locate/jqsrt
Oscillator strengths and polarizabilities of the hot-dense
plasma-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 nonrelativistic 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–2
1
P, 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 different
distributions in the atomic states compared with the unperturbed atomic states. Plasma effects can be
simulated using different models. In hot-dense and low-density warm plasmas, the interaction between two
localized charged particles can be modeled by replacing the Coulomb potential with an effective screened
Coulomb (Yukawa-type) potential. Such a screened Coulomb potential obtained from the Debye model is
characterized by the Debye length lD proportional to the square root of n/T, with n being the density of the
plasma and T its temperature. For the determination of important fundamental parameters such as electron
temperature, ion density, etc., and for certain atomic processes in hot plasmas, it is necessary to have accurate
atomic data, i.e., energy levels, transition wavelengths, oscillator strengths and polarizabilities available in the
literature. The effect of weakly coupled hot plasmas on the bound states ([1–7], references therein), doubly
excited meta-stable bound states and resonance states ([8,9], references therein) of helium atom has been
Corresponding author. Tel.: +886 2 2366 8274; fax: +886 2 2362 0200.
E-mail address: [email protected] (S. Kar).
0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jqsrt.2007.07.003
ARTICLE IN PRESS
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
446
investigated in recent years. The importance of Debye approach of plasma modeling has been discussed in the
literature [1–15].
In the present study, we carry out an investigation of plasma effect on the oscillator strengths and the
polarizabilities of the He atom based on the Deybe model. In the unscreened case, several theoretical studies
have 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 also
available in the literature ([26–28], references in [25]). Advanced experimental techniques for measuring
oscillator strengths of atomic and ionic transitions in vacuum ultraviolet lines are also described in several
reports [29]. The oscillator strengths of He for the S–P transitions are reported in the literature [1,5] in hot and
weakly coupled plasma environments. The static limit of the frequency-dependent dipole polarizability and the
oscillator strength for S–P transitions of the He atom under Debye screening was also reported for two
electron atoms by ignoring the screening on the electron–electron repulsions terms in a time-dependent
perturbation calculation [10]. However, oscillator strengths for the P–D transition and the quadrupole
polarizability 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 with
recent theoretical developments in hot plasma-embedded He ([1–13], references therein), it is important to
have accurate atomic data available in the literature for the quantities of fundamental interests such as
oscillator strengths and polarizabilities for He atom in model plasma environments. Here we have made a
systematic improvement on the oscillator strengths and dipole polarizability of the He atom immersed in
Debye plasmas.
In the present work, we have investigated the dipole oscillator strengths for the 11S–n 1P (n ¼ 2, 3), 2 1S–2
1
P, 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 quadrupole
polarizabilities of the He atom in its ground state for different screening parameters. Oscillator strengths for
the 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 for
the P–D transitions and quadrupole polarizabilities of He immersed in weakly coupled hot plasmas are first
calculations to our knowledge. A variational highly correlated wave function is used in the framework
Rayleigh-Ritz principle. The convergence of our calculations has been examined with increasing number of
terms in the basis expansions.
2. The method
The non-relativistic spin-independent Hamiltonian H of He atom immersed in Debye plasmas characterized
by the Debye length lD is given by
1 2 1 2
expðr1 =lD Þ expðr2 =lD Þ
expðr12 =lD Þ
þ
,
(1)
H ¼ r1 r2 2
þ
2
2
r1
r2
r12
where r1 and r2 are the radial coordinates of the two electrons and r12 is their relative distance. The parameter
m ( ¼ 1/lD) is called the Debye shielding parameter. Sets of plasma conditions can be simulated with different
choices of lD.
In the present work, we employ the following explicitly correlated wave functions [6,7,12,16–20]:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
X
2L þ 1
C¼
ð1 þ S pn O^ 12 Þ
C i rL1 PL ðcos y1 Þ expðai r1 bi r2 gi r12 Þ,
4p
i¼1
(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, O^ 12 is the
permutation operator on the subscripts 1 and 2 representing two electrons. Here, we use a quasi-random
process ([6,7,12,16–20], references therein) to optimize the non-linear variational parameters ai, bi and gi. The
ARTICLE IN PRESS
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
parameters ai, bi and gi are chosen from the three positive intervals [A1, A2], [B1, B2] and [C1, C2],
pffiffiffi
1
iði þ 1Þ 2
ai ¼
ðA2 A1 Þ þ A1 ,
2
pffiffiffi
1
iði þ 1Þ 3
bi ¼
ðB2 B1 Þ þ B1 ,
2
pffiffiffi
1
iði þ 1Þ 5
gi ¼
ðC 2 C 1 Þ þ C 1 ,
2
447
ð3Þ
where the symbol //ySS designates the fractional part of a real number. To calculate bound-excited
energies, one needs to obtain the solutions of the Schrödinger equation HC ¼ EC, where Eo0 using the
Rayleigh-Ritz variational principle. By employing the quasi-random process (3) on the wave functions (2), the
bound S, P and D states’ energies of plasma-embedded He were obtained in our earlier works [6,7]. Once the
optimum bound S, P and D states’ energies and wave functions, as well as the optimized parameters for such
states, are obtained, one can proceed to calculate the oscillator strengths and other fundamental properties of
the plasma-embedded He atom. In this work, we have used 600-term and 700-term basis functions to obtain
the converged results of the oscillator strengths for S–P and P–D transitions, respectively, whereas we have
employed 600-term basis functions to calculate dipole polarizability, and 700-term of S-states and 900-terms of
D-states to calculate quadrupole polarizability of the He atom. However, our calculated results are convergent
up 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
I nm ¼ jhmjV 1 jnij2 ,
(4)
where V1 is the dipole operator. The optical oscillator strengths for the dipole allowed S–P and P–D
transitions using the lengths form are defined as [16]
f nm ¼ CðE n E m ÞI nm ,
(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), the
cases when fnm40 are for absorption and fnmo0 for emission. The multipole operators are given by
V i ¼ ri1 Pi ðcos W1 Þ þ ri2 Pi ðcos W2 Þ,
(6)
with i ¼ 1 for dipole, i ¼ 2 for quadrupole, etc. Using formula (4) we have calculated oscillator strengths for
the 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 1
and 2 and Figs. 1 and 2. In the unscreened case, our results compare well with the reported results [16] and the
comparisons are made in Tables 1 and 2. As mentioned in our earlier work [7], we have not included the ppterms [16] for the D-state wave functions in Eq. (2). However, it seems that the results for energies are accurate
up to 6–7 significant digits for different bound-excited states [7]. Hence, our calculated oscillator strengths for
the P–D transitions are accurate up to 4–5 significant digits compared with the reported results [16]. Also in
Table 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 2
3
S–2 3P transitions are compared with the reported results of Lopez et al. [1]. It is clear from our results that
the oscillator strengths for the transitions decrease with increasing plasma strengths except for the 2 1S–2 1P
and 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 energy
differences [6].
ARTICLE IN PRESS
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
448
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
0.27617a
0.37644
0.37648a
0.539086
0.5391a
0.073435
0.07343a
0.064461
0.06447a
100
0.275554
0.274512b
0.37730
0.379971b
0.539757
0.541245b
0.072659
0.063568
50
0.273802
0.272802b
0.37975
0.383299b
0.541692
0.543159b
0.070558
0.061123
30
0.269866
0.38530
0.546070
0.066100
0.055925
20
0.262623
0.262059b
0.39565
0.397266b
0.554208
0.555492b
0.058340
0.047050
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
0.230451b
0.44846
0.444807b
0.594592
0.594609b
0.023167
0.013646
9
8
7
6
5
0.217202
0.203112
0.182976
0.152342
0.09950
0.46480
0.48780
0.52189
0.57566
0.65603
0.606584
0.622987
0.646127
0.679158
0.71530
0.013342
0.004828
0.007082
0.001783
a
Ref. [16].
Ref. [1] (the reported results are multiplied by 3).
b
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
0.71017a
0.610067
0.61024a
0.120273
0.12027a
0.122797
0.12285a
100
50
30
20
18
15
12
11
0.706431
0.696242
0.673049
0.626724
0.605941
0.553411
0.428787
0.323398
0.606450
0.596539
0.574055
0.529876
0.511507
0.461802
0.350746
0.261157
0.118599
0.114041
0.102730
0.073235
0.054976
0.120711
0.115090
0.101773
0.070064
0.051729
a
Ref. [16].
4. Polarizability
The generalized static polarizability of multipole order i is defined as [21]
Si ¼ 2
X h0jV i jnihnjV i j0i
n
En E0
ða30 Þ.
(7)
For dipole (S1) and quadrupole (S2) polarizabilities, n implies all the P and D states, respectively, including
the continuum states that are represented by pseudo-states, whereas 0 denotes the ground 1s2 1S state of He.
ARTICLE IN PRESS
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
0.08
0.07
0.6
0.4
0.3
2
3S
-2
Oscillator Strength
Oscillator Strength
0.7
0.5
449
3P
1
2 1S - 2 P
1
1 S - 2 1P
0.2
1
0.06
2
0.05
0.04
1
S
3
S
-3
0.03
-3
1
P
3
P
0.02
0.01
0.1
0.00
0.05
0.10
0.15
0.00
0.000 0.025 0.050 0.075 0.100 0.125
0.20
µ
µ
0.13
0.75
0.70
1
0.65 2 P - 3 1D
0.60
2 3P - 3 3
0.55
D
0.50
0.45
0.40
0.35
0.30
0.25
0.00
0.03
0.12
Oscillator Strength
Oscillator Strength
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].
0.11
3
2 P-43
D
2 1P - 4 1
D
0.10
0.09
0.08
0.07
0.06
0.06
µ
0.09
0.05
0.000
0.014
0.028
0.042
0.056
µ
Fig. 2. Oscillator strengths for P–D transitions of hot-dense plasma-embedded He.
Using Eq. (7) we have calculated the dipole and quadrupole polarizabilities of the ground state He atom for
different Debye lengths. The results are presented in Table 3 and Fig. 3. In the unscreened case, our results are
well comparable to the available theoretical results in the literature [20–25]. Our dipole polarizability result
differs with the best result in the literature by no more than 1 109 a30. We have also compared our results
with the available experimental results [25–28] in Table 3, and with the other calculation by Saha et al. [10] in
Fig. 3. The dipole polarizability reported by Saha et al. [10] was obtained by ignoring the electron–electron
screening. For quadrupole polarizability our result is less accurate compared with our dipole case results. As
was discussed earlier in Section 3, that by not including the pp-terms in the D-state wave functions, the
quadrupole polarizability is correct up to some part in 104 with only employing the sd-terms. As our main
interest is focused on the investigation of the plasma effects on the polarizabilities, it is sufficient for now to
consider D-states by using only the sd-term wave functions. The increasing trend of dipole and quadrupole
polarizabilities with increasing plasma strength indicates that the system would become more polarizable when
the plasma strength is increased. We should also mention that for one-electron systems interacting with Debye
potentials, 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 the
polarizibilities for atoms embedded in Debye plasmas. Consider weakly coupled and partially ionized plasma
as an example. Assuming that the plasma has reached thermal equilibrium, the electric field effects due to the
plasma charges on a plasma-embedded atom has led to a screened Coulomb potential of Debye type. Now if
ARTICLE IN PRESS
450
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
Table 3
Polarizabilities of hot-dense plasma-embedded helium atom in its ground state
D
Dipole polarizability (a30)
N
1.383192173
1.383192174a,b
1.3861c
1.38377(7)d
1.383794e
1.383746(7)f
100
50
30
20
15
10
8
6
5
4
3
2.5
2
1.5
1.0
1.383448194
1.384206190
1.385973937
1.389360958
1.394013270
1.406973146
1.419794461
1.446937353
1.473826802
1.522940615
1.629642204
1.739950329
1.955407309
2.49739410
5.0245812
Quadrupole polarizability (a30)
2.444
2.44508a
2.445
2.447
2.453
2.464
2.479
2.521
2.562
2.650
2.738
2.902
3.270
3.669
4.500
6.878
22.52
a
Ref. [23].
Ref. [24].
c
Ref. [26].
d
Ref. [27].
e
Ref. [28].
f
K. Grohmann and H. Luther (1992) (see Ref. [25]).
1.75
1.70
1.65
1.60
1.55
1.50
1.45
1.40
1.35
1.30
Quadrupole polarizability (a03)
Dipole polarizability (a03)
b
Present work
Ref. [10]
0.0
0.1
0.2
µ
0.3
0.4
3.6
3.4
3.2
3.0
2.8
2.6
2.4
0.0
0.1
0.2
µ
0.3
0.4
Fig. 3. Polarizabilities of ground state helium atom immersed in hot weakly coupled hot plasmas.
we also assume that the atom is further subjected to an external DC electric field, its ground state energy level
will be disturbed by such an external field, and the first and second orders of corrections to the energy are
related to the dipole and quadrupole polarizabilities, respectively. For plasma-embedded atoms, we assume
that such a perturbation treatment is still valid, but the polarizabilities (dipole, quadrupole, etc.) for free atoms
in the pure Coulomb environment are now replaced by those determined under the screened Coulomb
environment, as determined in the present calculations.
ARTICLE IN PRESS
S. 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 our
investigations. 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 quadrupole
polarizability of He atom under the influence of Debye screening are reported for the first time. Our
predictions on these fundamental quantities are much improved than the other reported results. Studies on the
multiple charged two-electron atoms in hot-dense plasmas are important for plasma spectroscopy, and such
investigations are of our future interest. We hope our present work on the polarizabilities and oscillator
strengths for S–P and P–D transitions of He under Debye screening will provide a new insight into future
investigations on these fundamental quantities.
5. Summaries and conclusion
In the present work, we have made an investigation on the static dipole and quadrupole polarizabilities of
helium atom immersed in hot, weakly coupled plasma environments in the framework of Debye screening
using highly correlated wave functions. In such an environment, we have also investigated the oscillator
strengths for the ultraviolet and visible series for the plasma-embedded helium atom. The oscillator strengths
for the P–D transitions are calculated for the first time when the screening effects are included. With the recent
advancement in laser plasmas [29,30], and with the recent activities on the studies on multipole polarizabilities
of the helium atom, we hope our results will provide useful information to the research communities in several
branches of physics and chemistry.
Acknowledgment
This work is supported by the National Science Council of Taiwan, ROC.
References
[1] Lopez X, Sarasola C, Ugalde JM. Transition energies and emission oscillator strengths of helium in model plasma environments. J
Phys Chem A 1997;101:1804–7.
[2] Dai S-T, Solovyova A, Winkler P. Calculations of properties of screened He-like systems using correlated basis functions. Phys Rev E
2001;64:016408.
[3] Saha B, Mukherjee TK, Mukherjee PK, Diercksen GHF. Variational calculations for the energy levels of confined two-electron
atomic systems. Theor Chem Acc 2002;108:305–10.
[4] Mukherjee PK, Karwowski J, Diercksen GHF. On the influence of the Debye screening on the spectra of two-electron atom. Chem
Phys Lett 2002;363:323–7.
[5] Okutsu H, Sako T, Yamanouchi K, Diercksen GHF. Electronic structure of atoms in laser plasmas: a Debye shielding approach. J
Phys B 2005;38:917–27.
[6] Kar S, Ho YK. Bound-states of helium atom in dense plasmas. Int J Quantum Chem 2006;106:814–22.
[7] Kar S, Ho YK. Bound D-states of helium atom under Debye screening. Int J Quantum Chem 2007;107:353–8.
[8] Kar S, Ho YK. Doubly-excited1,3Pe meta-stable bound states and resonance states of helium in weakly coupled plasmas. J Phys B
2007;40:1403–15.
[9] Kar S, Ho YK. Transition wavelengths for helium atom in weakly coupled hot plasmas. JQSRT 2007;107:315–22.
[10] Saha B, Mukherjee PK, Bielińska-Waz D, Karwowski J. Time-dependent perturbation calculations for transition properties of twoelectron atoms under Debye plasmas. JQSRT 2003;78:131–7.
[11] Saha B, Mukherjee PK, Diercksen GHF. Energy levels and structural properties of compressed hydrogen atom under Debye
screening. Astron Astrophys 2002;108:337–44.
[12] Kar S, Ho YK. Electron affinity of the hydrogen atom and a resonance state of hydrogen negative ion embedded in Debye plasmas.
New J Phys 2005;7:141.
[13] Zhang L, Winkler P. Debye-Huckel screening and fluctuations. Chem Phys 2006;329:338–42.
[14] Ichimaru S. Plasma physics. Menlo Park, CA: The Benjamin/Cummings Publishing Company, Inc.; 1986.
[15] Griem HR. Principles of plasma spectroscopy. Cambridge monograph in plasma physics, vol. 2. Cambridge: Cambridge University
Press; 2005.
[16] Cann NM, Thakkar AJ. Oscillator strengths for S–P and P–D transitions in heliumlike ions. Phys Rev A 1992;46:5397–405.
[17] Thakkar AJ. The generator coordinate method applied to variational perturbation theory. Multipole polarizabilities, spectral sums,
and dispersion coefficients for helium. J Chem Phys 1981;75:4496–501.
[18] Bishop DM, Lam B. Ab initio study of third-order nonlinear optical properties of helium. Phys Rev A 1988;37:464–9.
ARTICLE IN PRESS
452
S. Kar, Y.K. Ho / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 445–452
[19] Bishop DM, Pipin J. Static electric properties of H and He. Chem Phys Lett 1995;236:15–8.
[20] Bishop DM, Pipin J. Dipole, quadrupole, octupole, and dipole–octupole polarizabilities at real and imaginary frequencies for H, HE,
and H2 and the dispersion-energy coefficients for interactions between them. Int J Quantum Chem 1993;45:349–61.
[21] Bhatia AK, Drachman RJ. Polarizability of helium and the negative hydrogen ion. J Phys B 1994;27:1299–305.
[22] Bonin KD, Kadar-Kallen MA. Linear electric–dipole polarizabilities. Int J Mod Phys B 1994;8:3313–70.
[23] Yan Z-C, Babb JF, Dalgarno A, Drake GWF. Variational calculations of dispersion coefficients for interactions among H, He, and
Li atoms. Phys Rev A 1996;54:2833.
[24] Pachucki K, Sapirstein J. Relativistic and QED corrections to the polarizability of helium. Phys Rev A 2000;63:012504–13.
[25] Masili M, Starace AF. Static and dynamic dipole polarizability of the helium atom using wavefunctions involving logarithmic terms.
Phys Rev A 2003;68:012508.
[26] Mansfield CR, Peck ER. Dispersion of helium. J Opt Soc Am 1969;59:199–204.
[27] Gugan D, Michel GW. Measurement of polarizability and of the second and third virial coefficients of 4He in the range 402–27.1 K.
Mol Phys 1980;39:783–5.
[28] Gugan D. Improved virial coefficients of 4He from dielectric constant measurements. Metrologia 1991;28:405–11.
[29] Leckrone D, Sugar J, editors. In: Proceedings of the fourth international colloquium on atomic spectra and oscillator strengths for
astrophysical and laboratory plasmas. Phys Scr 1993;T47:149–56.
[30] Nakai S, Mima K. Laser driven inertial fusion energy: present and prospective. Rep Prog Phys 2004;67:321–49.
[31] Friedman M, Rabinovitch A, Thieberger R. Field-dependent polarizability calculation in a Debye potential. J Phys B
1986;19:L727–30.
[32] Zimmermann R. The Green’s function of the Debye potential: evaluation of the ground-state polarizability. J Phys B
1985;18:2817–25.