A&A 396, 337-344 (2002)
DOI: 10.1051/0004-6361:20021350
B. Saha1 - P. K. Mukherjee1 - G. H. F. Diercksen2
1 - Department of Spectroscopy, Indian Association for the Cultivation of Science,
Jadavpur, Kolkata-700 032, India
2 - Max-Planck-Institut für
Astrophysik, Karl-Schwarzschild Strasse 1, 85741 Garching,
Germany
Received 10 June 2002 / Accepted 16 September 2002
Abstract
Time dependent variation perturbation calculations have
been performed for estimating the transition energies, oscillator
strengths and transition probability values for a few dipole
allowed states of compressed hydrogen atom confined in a weakly
coupled plasma. The compression is obtained by embedding the atom
at the centre of an impenetrable spherical box. The dipole
polarizability of the atom is evaluated at each confinement radius
with respect to different plasma screening parameters. The effect
of pressure due to spatial confinement on the dipole
polarizability and other atomic properties is analyzed. Results
obtained are useful for the diagnostic determination of
astrophysical and laboratory plasmas and for the calculation of
collision rate coefficients needed for computing opacity of
stellar envelopes - a quantity of importance in the context of
stellar structure and pulsations.
Key words: atomic data - atomic processes
In recent years the analysis of the spectral properties of
confined atomic systems such as those of atoms embedded in liquid
helium has gained considerable momentum (Tabbert 1997)
firstly, because of the availability of current implantation
technique like ion beam method (Gordon 1974,
1983, 1993), laser sputtering and ablation
technique (Bauer 1990; Yabuzaki 1992; Arndt
1993; Tabbert 1994; Beijersbergen
1993; Kinoshita 1995) and secondly due
to accurate methods available for estimating the spectral line
shifts (Günther 1995; Tabbert 1995;
Kanorski 1994). Although a host of experimental data
for different atoms are currently available, very few methods
exist for theoretical estimation of the spectral line shifts and
other properties under such confinement. In the so called standard
bubble method (SBM), the total Hamiltonian is expressed as the sum
of pair interaction between helium atoms, between impurity atom
and each of the helium atoms and the Hamiltonian of the free atom
itself. The total impurity helium interaction is derived from
individual impurity-helium interaction term. First order
perturbation theory is being utilized to calculate the energy
shift. The method has been applied in a number of cases (Hiroike
1965; Hikman 1975; Bauer 1990; Beau
1996). Basically the problem is similar to that of studying
the spectra of compressed atoms like that trapped in zeolite,
fullerenes or under high pressure (Jaskolski 1996;
Connerade 2000; Gupta 1982). As ab-initio
calculations for atoms in such an environment is very complicated,
the problem of the impurity atom embedded in liquid helium may
alternatively be looked upon using a suitable model in the
following manner. Experimental observations indicate that (i)
electrons in superfluid helium are highly mobile and (ii) because
of the inert nature of helium the atomic charge density has
extremely low penetration probability beyond the surrounding
liquid helium cage due to strong Pauli repulsion. Hence one can
tentatively assume the foreign atom to be embedded in a charge
neutral environment which mimics a plasma with a finite cage
boundary. With this viewpoint we propose a model of studying atoms
inside such an environment which, on one hand assumes screening of
the potential due to plasma, and secondly uses an altered boundary
condition which suits the physical description. We consider at
present the surrounding plasma to be sufficiently weak so as to
produce screened coulomb interaction between point charges.The
effect of this so called Debye plasma on atomic energy levels and
other properties has been analyzed earlier in a limited manner
(Winkler 1996; Ray 1998a,1998b). The overall
effect of this screened Coulomb potential is to reduce the binding
energy and to push the system towards gradual instability with
increase of screening. The screening constant is a function of
temperature and number density of the plasma and different plasma
conditions can be simulated by changing suitably the screening
parameters. Study of laser produced and Tokamak plasmas requires
an understanding of the electronic spectrum of an ion as a
function of electron density and temperature of the plasma. The
screening is important in interpreting the disappearance of
spectral lines near the series limit in astrophysical
observations, in laboratory and astrophysical plasma diagnostics,
in calculating partition functions in thermodynamics, in the
calculation of collision rate coefficients, all of which require
the shifted energy levels and connected properties. Such
calculations have also applications in astrophysical phenomena
like the mass radius relation in the theory of white dwarfs and in
the determination of the rate of escape of stars from galactic and
globular clusters, understanding the interior of giant planets
(Varshni 1997, 1998) and in estimating stellar opacities (Seaton 1987). The study of the properties of a confined
atom in an impenetrable spherical box has, however, a long
history. Such studies originated with a model due to Michels et al.
(1937) who studied the effect of pressure by enclosing an
atom in a spherical box. Sommerfeld & Welker (1938)
carried out investigations on the energy levels of hydrogen in a
spherical box. Subsequently, a number of calculations (Ley-Koo
1979; Ludena 1977,1978; Marin & Cruz 1991a,1991b,1992; Zicovich-Wilson 1994; Fowler
1984; Aquino 1995; Dineykhan 1999;
Singh 1984; Montgomery 2002; Laughlin
2002) have been performed for studying the energy
levels of compressed atoms. In the current communication we have
performed a systematic analysis
of the effect of a plasma screening and that of a finite
confining radius on the dipole
polarizabilities and 2p, 3p and 4p energy levels for the hydrogen atom,
their
oscillator strengths and transition probabilities with a view to
estimate the order of magnitude
of the spectral line shifts under such confinements. Hydrogen atom
is chosen as a prototype as it is free
from correlation effects and admits of accurate results in a
first hand analysis and calculations have already been performed
for the hydrogen donor states in spherical quantum dots (Zhu
1990). While studies on the energy levels and other
properties of hydrogen using a cutoff radius have been performed
by a number of authors (Sommerfeld 1938; Ley-Koo
1979; Zicovich-Wilson 1994; Fowler
1984; Aquino 1995; Varshni 1997,1998;
Dineykhan 1999), only a single limited calculation by
Singh & Varshni (1984) was performed for the bound states
of static screened Coulomb and cut-off Coulomb potentials. Very
recently the dynamic dipole polarizability of compressed hydrogen
was calculated by Montgomery (2002) for different radii
of compression and 1s2p transition energy was determined. The
recent calculation of Laughlin et al. (2002) on compressed
hydrogen atom using a variety of analytical and algebraic methods
yields accurate estimate of the ground and excited state energies
and wave functions. A brief description of the current method is
described in Sect. 2 followed by a discussion of results in
Sect. 3.
The hydrogen atom subjected to a weakly coupled plasma which
admits of a Debye type of screening (Akhiezer 1975) in
the nuclear potential. In addition the charge cloud is assumed
confined in spherical box of radius R which produces an altered
boundary condition such that wave function vanishes at the
boundary. We assume that the potential energy function (atomic
unit is used)
![]() |
Figure 1:
Plot of polarizability
![]() ![]() ![]() |
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R | ![]() |
Dipole | -E0 | Transition | Transition | Oscillator | Transition |
(a.u.) | (a.u.) | polarizability (a.u.) | (a.u.) | energy (a.u.) | strength | probability
![]() |
|
![]() |
0.00 | 4.4997 | 0.50000 |
![]() |
0.37500 | 0.4160 | 1.87(+9) |
4.5a | 0.5b | 0.37500b | 0.4162c | 1.88(+9)c | |||
![]() |
0.44444 | 0.0790 | 0.50(+9) | ||||
0.44444b | 0.0791c | 0.50(+9)c | |||||
![]() |
0.46875 | 0.0290 | 0.20(+9) | ||||
0.46875b | 0.0290c | 0.20(+9)c | |||||
0.01 | 4.5014 | 0.49008 | ![]() |
0.37482 | 0.4150 | 1.86(+9) | |
![]() |
0.44392 | 0.0780 | 0.49(+9) | ||||
![]() |
0.46776 | 0.0280 | 0.20(+9) | ||||
0.05 | 4.5412 | 0.45182 | ![]() |
0.37107 | 0.4000 | 1.76(+9) | |
![]() |
0.43325 | 0.0660 | 0.39(+9) | ||||
![]() |
0.44922 | 0.0150 | 0.94(+8) | ||||
0.08 | 4.6026 | 0.42457 | ![]() |
0.36545 | 0.3766 | 1.61(+9) | |
![]() |
0.41823 | 0.0470 | 0.26(+9) | ||||
![]() |
0.42581 | 0.0062 | 0.38(+8) | ||||
0.10 | 4.6577 | 0.40705 | ![]() |
0.36051 | 0.3561 | 1.48(+9) | |
![]() |
0.40546 | 0.0298 | 0.16(+9) | ||||
0.15 | 4.8423 | 0.36544 | ![]() |
0.34433 | 0.2854 | 1.08(+9) | |
0.20 | 5.0936 | 0.32674 | ![]() |
0.32263 | 0.1773 | 0.59(+9) | |
0.25 | 5.41 | 0.29076 | ![]() |
0.29479 | 0.0383 | 0.11(+9) | |
1.00 | 8.23 | -0.16598 | ![]() |
0.01907 | 0.0034 | 0.397(+5) | |
20 | 0.00 | 4.4628 | 0.50000 |
![]() |
0.37500 | 0.4091 | 1.84(+9) |
![]() |
0.44838 | 0.1068 | 0.69(+9) | ||||
0.05 | 4.5005 | 0.45181 | ![]() |
0.37107 | 0.3925 | 1.73(+9) | |
![]() |
0.43908 | 0.1059 | 0.65(+9) | ||||
0.1 | 4.6059 | 0.40703 | ![]() |
0.36053 | 0.3487 | 1.45(+9) | |
![]() |
0.41746 | 0.1095 | 0.61(+9) | ||||
0.15 | 4.7724 | 0.36540 | ![]() |
0.34448 | 0.2825 | 1.07(+9) | |
0.20 | 4.9975 | 0.32666 | ![]() |
0.32360 | 0.2025 | 0.68(+9) | |
0.25 | 5.2815 | 0.29063 | ![]() |
0.29890 | 0.1287 | 0.37(+9) | |
1.00 | 49.3300 | -0.0213 | |||||
10 | 0.00 | 4.2343 | 0.499810 | ![]() |
0.38094 | 0.4382 | 2.03(+9) |
3.977f | 0.499999d | 0.38114e | |||||
0.05 | 4.2652 | 0.45160 | ![]() |
0.37769 | 0.4299 | 1.96(+9) | |
0.10 | 4.3516 | 0.40676 | ![]() |
0.36906 | 0.4091 | 1.78(+9) | |
0.15 | 4.4875 | 0.36303 | ![]() |
0.35636 | 0.3890 | 1.55(+9) | |
0.20 | 4.6702 | 0.32627 | ![]() |
0.34062 | 0.3488 | 1.29(+9) | |
1.00 | 18.9700 | -0.029178 | |||||
7 | 0.00 | 3.8079 | 0.49871 | ![]() |
0.41122 | 0.5438 | 2.94(+9) |
0.05 | 3.8304 | 0.45046 | ![]() |
0.40889 | 0.5399 | 2.88(+9) | |
0.10 | 3.8935 | 0.40551 | ![]() |
0.40261 | 0.5300 | 2.75(+9) | |
0.15 | 3.9923 | 0.36361 | ![]() |
0.39329 | 0.5158 | 2.55(+9) | |
1.00 | 11.26 | -0.04309 | |||||
6 | 0.00 | 3.4652 | 0.49707 | ![]() |
0.44151 | 0.6063 | 3.78(+9) |
0.05 | 3.4825 | 0.44878 | ![]() |
0.43957 | 0.6036 | 3.73(+9) | |
0.10 | 3.5312 | 0.40375 | ![]() |
0.43433 | 0.5966 | 3.60(+9) | |
0.15 | 3.6074 | 0.36170 | ![]() |
0.42648 | 0.5865 | 3.41(+9) | |
1.00 | 8.4200 | -0.04907 | |||||
5.50 | 0.00 | 3.2176 | 0.49536 |
![]() |
0.46581 | 0.6438 | 4.46(+9) |
0.05 | 3.2319 | 0.44706 | ![]() |
0.46409 | 0.6416 | 4.42(+9) | |
1.00 | 7.0000 | -0.05586 | |||||
5.00 | 0.00 | 2.9051 | 0.49240 | ![]() |
0.49999 | 0.6855 | 5.48(+9) |
0.50401e | |||||||
0.80 | 4.7600 | -0.00304 | |||||
2.00 | 0.00 | 0.3344 | 0.12210 | ![]() |
1.69812 | 0.9643 | 8.88(+9) |
0.338f | 0.125g | 1.70102e | |||||
0.20 | 3.3600 | -0.06197 | |||||
1.00 | 0.00 | 0.0291 | -2.38187 | ![]() |
5.84138 | 0.9920 | 1.08(+12) |
5.85915e |
We have chosen confined hydrogen atom to make a detailed analysis
on the ground and excited state energy levels, polarizabilities,
oscillator strength and transition probabilities. The confinement
effect due to (i) Debye screening (ii) finite boundary conditions
by using a cut off potential, is analyzed. The energy levels and
other properties are calculated systematically first by choosing
and
which correspond to free atom case. Then
we decreased R step by step to get the effect of reduced cut off
parameter and for each R, we have chosen different sets of
screening parameter
,
always starting with
.
In each
case the ground state energy under confined condition has been
evaluated with a two parameter Slater type representation, the
coefficients are obtained from diagonalisation of the Hamiltonian
matrix in an iterative manner described earlier. For confined
hydrogen atom a two parameter representation is expected to be
sufficient for energy convergence. The polarizabilities and
transition properties in each case are obtained from a time
dependent perturbative calculation. An eight parameter
representation of the perturbed wave function is consistently
chosen to estimate the transition properties. The length of the
basis set and exponents are so chosen as to reproduce the static
limit of dynamic polarizability, ground state energy, transition
energies to different dipole allowed excited states, oscillator
strengths and the transition probabilities for the free hydrogen
atom. Results are listed in Table 1. For
and
corresponding to the free atom case, our table shows
(a.u.) corresponding to the exact value 4.5 a.u.
(Miller 1977),
,
,
,
a.u.
respectively which are exact up to the places shown (Moore
1949). The oscillator strengths and transition
probabilities for 2p, 3p and 4p excitations are also the same as
listed by Wiese et al. (1966). Several features of the
different transition properties can be noted from a close look at
the Table 1. The dipole polarizability value increases
continuously with increase of screening parameter
for a
given value of the truncation radius R, whereas for a given
,
the polarizability value decreases with decrease of Rwhich simulates stronger confinement.
Figure 1a shows plots of
against R for three different values of
.
With
decrease of R, the system becomes more compressed and hence the
polarizability decreases. For a given R, with increase of
,
both the kinetic energy and potential energy diminish resulting in
an increase of
value. When
increases for a given
R or when R diminishes for a given
,
the total energy
monotonically tends towards positive value from bound negative
value indicating instability in the system. The cause is entirely
different. For example for the free atom case (
)when
increases from zero value, the kinetic energy as well as the
absolute value of potential energy diminishes but the potential
energy diminishes faster than the kinetic energy resulting in
eventual zero energy configuration. As
increases further the
total energy becomes positive and the system becomes unbound. The
polarizability looses its precise meaning. On the other hand for a
given
(say
),
with decrease of confinement radius R, the atom gets more compressed
and the absolute value of the kinetic and potential energy increases but
the increase of kinetic energy is more than the increase of
potential energy. Eventually at a given value of R, the total
energy becomes zero resulting in pressure ionization. Further
decrease of R results in unbound states, for which
polarizability again looses its meaning. With decrease of
truncation radius R the charge cloud gets pressurized. We have
calculated the pressure on the atom due to truncation of the wave
function at finite radius R by using the relation (Hirschfelder
1954)
![]() |
Figure 2:
Variation of change in kinetic energy (
![]() |
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![]() |
Figure 3:
Plot of ![]() |
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![]() |
Figure 4:
Transition wave length for 2p state of compressed
hydrogen atom under different confinement radii and
Debye shielding (![]() |
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![]() |
Figure 5:
Variation of transition probability (scaled) with a) ![]() ![]() |
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![]() |
Figure 6:
Plot of charge density for 2p state a) under
different Debye screening (![]() ![]() |
Open with DEXTER |
![]() |
Figure 7:
Plot of charge density at different confinement with
![]() |
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![]() |
Figure 8: Plot of charge density at different screening with confinement radii a) R=30 and b) R=10 a.u. respectively. |
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For a given ,
the transition energy to different excited
states increases with decrease of R, i.e., with more and more
compression. However, for given R, the transition energy
diminishes with increase of
value. The wavelength in Å for the 2p transition is schematically shown in Fig. 4 for
different R and
.
The general trend in the change of wave
length is clearly observed. A typical example for R= 10 a.u. and
a.u. the shift
nm.
Experiments performed for different atoms in liquid helium show
shifts of this order (Tabbert 1995). The oscillator
strength and transition probability for dipole allowed transitions
have been evaluated using standard formula (Bethe 1954).
The oscillator strength and transition probability values diminish
along increasing
for a given truncation radius R, while
they gradually increase with decrease of R values for a given
.
Figure 5a shows the plot of transition probability for
the excitation
against
for three different
confinement radii R, while in Fig. 5b, the same is plotted
against R for
three different values of screening parameters
.
Different
trend is observed for the two cases. As the energy becomes less
negative either when
is increased for a given R or when
R is decreased for a given
the ionization potential gets
lowered and the number of excited states becomes finite. To have
an idea about the behavior of the excited state wave functions
under confinement, we plotted in Fig. 6a the charge density
for the 2p wave function calculated by our method against r for
sets of Debye parameters for
while in Fig. 6b the
plots for different confinement radii for the Debye parameter
are given. An overall flattening of the charge density is
noted for increase of
value for
while the charge
density is squeezed for decrease of truncation radius R for
.
Similar plots are given in Figs. 7, 8a and 8b for
finite R and
values. The figures show interesting feature
of the charge densities. The trend is similar for charge
densities of the excited 3p and 4p states.
Acknowledgements
PKM would like to thank the Max-Planck Institute for Astrophysics, Garching, for financial support of his research visit to the institute where part of the work was performed. He would also like to thank the Council of Scientific and Industrial Research (CSIR), Govt. of India for the research grant No. (03)/(0888)/99/EMR II.