Plasma and Fusion Research: Regular Articles

Volume 11, 2401124 (2016)

Numerical Study of Spectral Line Shapes in High-Density HePlasmas∗)
Hayato KAWAZOME and Shinichi NAMBA1)
Department of Information Engineering, National Institute of Technology, Kagawa College, 551 Kohda, Takuma-cho,
Mitoyo 769-1192, Japan1)
Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashihirosima 739-8527, Japan(Received 30 November 2015 / Accepted 18 October 2016)
In order to determine the densities and temperatures in high-density helium plasmas by emission spectroscopy, a numerical study of He I spectral line shapes have been performed taking into account radiation
trapping. A computational simulation code has been developed, consisting of two parts. The first part solves
coupled rate equations to obtain population densities in each energy level of the He atom. The other describes theabsorption rate, emission rate, and spectral line shapes by solving the equation of radiation transfer. In a homogeneous plasma, the calculated line profile has a central dip caused by photoabsorption by residual cold atoms.
In addition, the results show that radiation trapping significantly alters the population dynamics. The population
densities estimated from the intensity of the line profile considering the photoabsorption are lower by one orderof magnitude than those without the photoabsorption. For application to a He arc plasma, the temporal evolutionof this central dip is also examined as a preliminary calculation.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: spectral line shape, radiation transfer, coupled rate equations, He arc plasma
DOI: 10.1585/pfr.11.2401124

1. Introduction
To determine the electron temperature and density of aplasma, the spectral line intensity ratio of appropriate transitions can be employed with the experimental ratio being compared with a value calculated using a collisionalradiative (CR) model [1,2]. However, in high-density plasmas, such as atmospheric thermal plasmas or divertor plasmas of magnetically confined devices, the optical depth
becomes significantly thick. For example, although atmospheric arc plasmas have been studied extensively sa
a component of fundamental research into arcjet thrusters
and atomic/molecular physics [3], it has been found thatthe radiation trapping of resonance transitions alters thepopulation density distribution [4]. Consequently, we cannot apply the intensity ratio method to determine the temperatures and densities of such plasmas. In this study,
therefore, the coupled rate equations for the He atom, incorporating radiation transport, are solved to derive thepopulation densities and spectral line shapes of the resonance spectra. The optically thick transitions considered
are He I 1s 1 S - 2p 1 P, 1s 1 S - 3p 1 P, and 1s 1 S - 4p 1 P and therespective wavelengths are 58.4, 53.7, and 52.2 nm. The
simulation code developed, which directly solves the coupled rate equations, yields the time evolution of the population densities and the spectral line shapes.
In this paper, the coupled rate equations and the equa-tion of radiative transfer are presented in Sec. 2. In Sec. 3,
the plasma model and simulation code are described. The
simulations for inhomogeneous and homogeneous plasmas
are presented in Sec. 4. The final section provides summary.

2. Radiation Transfer and the Coupled Rate Equations
At a frequency ν, the equation of radiation transfer
corresponding to the transition q ← p is given bydIν = {−kν Iν + ην } dx,(1)where Iν is the radiance and kν and ην are the absorption
coeﬃcient and the emission coeﬃcient, respectively. x is
the coordinate along the optical path [5]. In a spatiallyhomogeneous medium, a formal solution of Eq. (1) is givenby
Iν (x) = Iν (0)exp(−kν x) +ην 
1 − exp(−kν x) .
kν(2)Here, the absorption and emission coeﬃcients are given bykν =hν nq Bqp P (ν) − n p B pq P(ν) ,4π(3)ην =hνn p A pq P(ν),4π(4)

andauthor’s e-mail: kawazome@di.kagawa-nct.ac.jp∗) This article is based on the presentation at the 25th International Toki
Conference (ITC25).

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c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

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respectively, where A pq is the Einstein A coeﬃcient fromlevel p to q; Bqp and B pq are the Einstein B coeﬃcientsfor photoabsorption and induced emission, respectively; his the Planck constant; and n p is the population density ofthe level p. P(ν) and P (ν) are the line profiles, normalized
to 1, for emission and absorption, respectively. Assumingthat P(ν) and P (ν) have the same Gaussian shape, they areexpressed as ν − ν0 21
P(ν) = P (ν) =,(5)√ exp −ΔνΔν πwhereΔν =ν0c2kB T.
mHeFig. 1 A schematic drawing of the cylindrical plasma used in
the radiation transport simulation. The plasma is divided
into (n + 1) segments of length dl.

(6)Here, kB is the Boltzmann constant, T is the temperatureof the emitting or absorbing neutral He atoms, mHe is themass of the He atom, c is the speed of light and ν0 is thecentral frequency of the spectrum. The central frequenciesof the spectra for the resonance transitions 1s 1 S - 2p 1 P, 1s1
S - 3p 1 P, and 1s 1 S - 4p 1 P are 5.13×1015 Hz, 5.58×1015
Hz, and 5.74×1015 Hz, respectively.
For the population density of level p, the coupled rateequations including radiation trapping are [6]dn p=−C pq ne n p + S p ne n p +A pq n pdtqpq<p+B pq Iqp (ν)n p P (ν)dν +q>p+line

B pq I pq (ν)n p P(ν)dνq<p

Cqp ne nq +qpline

Aqp nqq>p+Bqp I pq (ν)nq P (ν)dν +q<pline+ α p ne + β p + βdp ni ne ,Fig. 2 Two photoabsorption processes for the 1s 1 S - 2p 1 P transition. The temperatures of the recombined hot atoms
and the residual cold atoms are T h_i and T c , respectively.
hcand kν_i
are the absorption coeﬃcients for the recomkν_ibined hot atoms and the residual cold atoms, respectively.
In the same manner as described above, for each temperature component, ηhν_i and ηcν_i are the emission coeﬃcientsand Ih and Ic are the radiance values.

Bqp Iqp (ν)nq P(ν)dνq>pline(7)where C pq is the electron impact transition rate coeﬃcient.
S p is the ionization rate coeﬃcient. α p , β p , and βdp are therate coeﬃcients for three-body, radiative, and dielectronic
recombination, respectively. The fourth, fifth, eight, andthe ninth terms on the right side in Eq. (7) represent the absorption rate for the outflow to upper levels, the induced
emission rate for the outflow to lower levels, the absorption rate for the inflow from lower levels, and the induced
emission rate for the inflow from upper levels, respectively.

3. The Plasma Model and Radiation
Transport Simulation
In this study, numerical calculations of radiation transport in high-density He plasmas were performed. Figure 1
shows a schematic drawing of the plasma geometry. Theplasma has a cylindrical shape and is divided into (n+1) regions. The radius of the plasma cylinder is 0.25 cm and its
cross section dS is 0.196 cm−2 . The length of the plasma

cylinder L is L = (n + 1) dl. Here dl is the length of a single volume element. The mesh i (i = 0, 1, · · · , n − 1, n) has
electron temperature T e_i , electron density ne_i , absorption
coeﬃcient kν_i , and radiation coeﬃcient ην_i . The ion density ni_i is equivalent to the electron density ne_i , and thetemperature of the recombined hot atoms T h_i , which aregenerated as a result of the recombination of helium ions,is set equal to the electron temperature. This assumption isjustified in high-density thermal equilibrium plasmas. The
density nc and temperature T c of the residual cold atoms
are constant throughout the volume. Each value of the parameters is as determined in the center of the mesh. Figure 2 shows a schematic drawing of the photoabsorption
processes for the 1s 1 S - 2p 1 P transition in this simulah
is the absorption coeﬃcient of thetion. In the figure, kν_i
recombined hot atoms and is calculated with T h_i . On thecis that for the residual cold atoms and isother hand, kν_i
calculated with T c . Ih and Ic are the radiance values fromthe recombined hot atoms and residual cold atoms, respectively. In the same way, ηhν_i and ηcν_i are emission coeﬃcients for the recombined hot atoms and the residual coldatoms, respectively. For the other resonance transitions,2401124-2

Plasma and Fusion Research: Regular ArticlesVolume 11, 2401124 (2016)

similar procedures are also employed.
The algorithm used in the simulation code is divided
into two sections: the coupled rate equations and the equation of radiation transfer. The computational scheme of thecode is as follows:(a) In each computational mesh, the time evolution of
two sets of the coupled rate equations are solved. Oneof the coupled rate equations is solved with ni_i , ne_i ,T e_i , and nh for the recombined hot atoms, and theother is solved with ne_i , T e_i , and nc for the residual cold atoms. We then obtain the spatial profileof the population densities. The time step size dt is
1.0 × 10−12 s. At the first time step, the induced emission and absorption rates are zero because I pq values
are zero. Whenever an increment in the time reaches
the time step size for the equation of radiation transfer (which will be defined later), (b), (c), and (d) are
carried out.
(b) Using the population densities obtained in (a), the absorption coeﬃcient kν_i and the emission coeﬃcientην_i are calculated (i = 1, 2, . . . n). kν_i is the sum ofhcand kν_i. ην_i is also the sum of ηhν_i and ηnν_i .
kν_i(c) The line profile of Iν_1 is calculated using Eq. (2), i.e.,Iν_1 (dl) = Iν_0 exp(−kν_1 dl)ην_1 
+1 − exp(−kν_1 dl) .
kν_1Fig. 3 Spectral line profiles of the transition 1s 1 S - 2p 1 P for
the inhomogeneous plasmas. The electron temperature is0.5 eV and the electron density and the ion density are1014 cm−3 . The densities of the residual cold atoms are1013 , 1014 , and 1015 cm−3 .

(8)Then, Iν_i is calculated using Iν_i−1 (i = 1, 2, . . . n).
Iν_n corresponds to the experimentally measured spectrum.
(d) Using the line profiles obtained in (c), the induced
emission and absorption rates are calculated in eachsegment, and these values are used in the next time
step.
(e) The processes described above are repeated iteratively until steady-state population densities are obtained (typical time : 10−5 s).
From the plasma geometry (see Fig. 1), the time period for the photons to travel from segment i = 0 to i = n is
L/c. However, in the high-density recombining plasma, atime step size of 1.0 × 10−12 s is required to solve the coupled rate equations appropriately, because collisional processes such as three-body recombination and collisionalde-excitation are very rapid phenomena. In this study,
therefore, a time step size of 1.0 × 10−12 s is used to calculate the coupled rate equations. On the other hand, forthe equation of radiation transfer, the time step size is set
to be greater than L/c s. For example, in the case of L = 5cm, L/c is 1.67 × 10−10 s and we use a time step size of
2.0 × 10−10 s.

4. Numerical Results and Discussion
To investigate the eﬀect of radiation trapping on thepopulation density distribution and spectral line shape, weFig. 4 1s 1 S - 2p 1 P line profile calculated with/without the eﬀectof radiation trapping. The electron temperature is 0.5 eV
and the density of the residual cold atoms is 1014 cm−3 .

performed simulations for a plasma with a mesh count n =5 and a plasma cylinder length of 5.0 cm. The temperatureof the residual cold atoms was T c = 0.026 eV (300 K). Theion and electron densities were ni_i = ne_i = 1014 cm−3 .
The electron temperature was T e_i = 0.5 eV. The spatial
distributions of the electron temperature and the density
were constant throughout the plasma. The time step sizefor the equation of radiation transfer was 2.0×10−10 s. Figure 3 shows the spectral line profiles Iν_5 (see Fig. 1) of the1s 1 S - 2p 1 P transition at time 5.0×10−5 s. The densities of
the residual cold atoms were 1013 , 1014 , and 1015 cm−3 . In
the calculation, a slit function is not considered. In Fig. 3,
the peak heights are normalized. As is clearly seen, theprofiles have a hollow shape, as the central frequency has
the highest absorption probability. For the residual cold
atoms of 1013 , 1014 , and 1015 cm−3 , the absorption coeﬃ-2401124-3

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Fig. 6 Time evolution of the line profile for the 1s 1 S - 2p 1 Ptransition at various times.

Fig. 5 Population density distributions of (a) 2p 1 P, (b) 3p 1 P,and (c) 4p 1 P at an electron temperature of 0.5 eV. The
densities of the residual cold atoms were 1013 , 1014 , and1015 cm−3 .

cients kνc0 are 2.2 × 102 , 2.2 × 103 , and 2.2 × 104 m−1 , respectively, at the central frequency. For the recombined hotatoms, kνh0 is 2.1 × 10 m−1 at the central frequency. The appearance of the central dip in the line profile indicates thatthe resonance line emitted by the recombined hot atoms
is absorbed by the residual cold atoms. Thus, with higher
densities, a deeper dip is observed. Figure 4 shows the 1s1S - 2p 1 P line profiles as calculated with/without accounting for radiation trapping. The density of the residual cold
atoms was 1014 cm−3 . In Fig. 4, the estimated population
densities of 2p 1 P with and without the inclusion of photoabsorption are 6.0 × 106 and 2.7 × 107 cm−3 , respectively.
Figure 5 plots the spatial distributions of the population
densities of 2p 1 P, 3p 1 P and 4p 1 P at time 5.0 × 10−5 s. The
densities of the residual cold atoms are 1013 , 1014 , were1015 cm−3 . In the figure, the horizontal axis corresponds
to the position relative to the origin (see Fig. 1). The population densities of 2p 1 P and 3p 1 P increase in proportion
to the distance x. In the radiation transfer process, as theradiation goes through each mesh, Iν_i increases due to additional radiation as expressed by the radiation coeﬃcient.
The increase in the population density results from an increase in the absorption rate at each mesh. On the otherhand, the population density of 4p 1 P is almost constantthroughout the plasma. The A coeﬃcient for the transi-tions of 1s 1 S to 2p 1 P, 3p 1 P, and 4p 1 P are 1.8 × 109 ,5.6 × 108 , and 2.4 × 108 s−1 , respectively. The B coeﬃcients for the photoabsorption from 1s 1 S to 2p 1 P, 3p 1 P,
and 4p 1 P are 2.7 × 1012 , 6.6 × 1011 , and 2.6 × 1011 J−1 m2
sr Hz, respectively. Because the A coeﬃcient of the 1s 1 S- 4p 1 P transition is smaller than that of the other transitions, the B coeﬃcient of the 1s 1 S - 4p 1 P transition has
the lowest value among these transitions. In addition, inhigh-density plasmas, the electron impact transition plays
a crucial role, and the outflow from 4p 1 P into np 1 P (n isthe principal quantum number) increases. Thus, the eﬀectof photoabsorption on the population density of 4p 1 P issmaller. Further analysis of these excitation and deexcitation flows is necessary, which allows for a more detailed
investigation.
Next, we show the preliminary result for the highdensity He arc plasmas. The mesh number n = 3, andthe length and radius of the plasma cylinder are 1.0 cm
and 0.25 cm, respectively. The spatial profiles of the electron temperature and density are given by T e (x) = T e0 {1 −(x/1.0)2 }2 and ne (x) = ne0 {1 − (x/1.0)2 }2 , respectively.
Here, T e0 is 0.5 eV and ne0 is 1014 cm−3 . The density of therecombined hot atoms is 1012 cm−3 . The temperature anddensity of the residual cold atoms are set to be 0.026 eVand 1014 cm−3 , respectively. The ion density is 1014 cm−3 .
The time step size for the equation of radiation transfer is
5.0 × 10−11 s. Figure 6 shows the time evolution of the line
profile for the 1s 1 S - 2p 1 P transition. The central dip canstill be seen at early time. The He ions begin to recombine
at t = 0, and the recombination flux causes an increasein the population density of 2p 1 P. As a consequence, thecentral dip becomes clearer. At 1.0×10−5 and 5.0×10−5 s ,the line profiles overlap with each other, as the population
densities reached a steady state.

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5. Summary
In order to investigate the eﬀect of radiation trappingon the population density distributions and spectral line
shapes in a low temperature and high-density He plasma, anumerical simulation code has been developed. The calculated line profiles show a central dip created by significant
photoabsorption by residual cold atoms. By comparing
the results with and without photoabsorption, it has beenshown that the population densities estimated from the intensity of the line profile when including photoabsorption
were lower by an one order of magnitude than when photoabsorption was not included. Moreover, radiation trapping significantly alters the population dynamics. For ahigh-density He arc plasma, the preliminary result of thetime evolution of the line profile clearly shows the growthof a central dip. In future work, we plan to compare experimental results with our numerical simulations.
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