J. Plasma Fusion Res. SERIES, Vol.7 (2006) 123-126

Effect of Radiation Trapping on an He I CR Modelfor a Divertor Simulator MAP-II
IIDA Yohei, KADO Shinichiro1 , OKAMOTO Atsushi∗ , KAJITA Shin∗∗ ,SHIKAMA Taiichi, YAMASAKI Daisuke and TANAKA Satoru
School of Engineering, the University of Tokyo, Tokyo 113-8656, Japan
1 High Temperature Plasma Center, the University of Tokyo, Chiba 227-8568, Japan（Received: 4 October 2004 / Accepted: 18 August 2005）

Abstract
An He I collisional-radiative (CR) model with radiation trapping was applied to spectroscopic data obtained in a divertor simulator MAP-II. We first applied the model to ionizing plasmas and examined thecorrection mode by taking into consideration the eﬀect of radiation trapping. In the recombining plasmas, theelectron density obtained from the best-fit of the model to measured n 3 D populations decreased by several
tens percent when the radiation trapping was included. This result shows that the eﬀect of radiation trapping
for resonant transition must be taken into consideration even for the measurement using triplet series.
Keywords:
radiation trapping, collisional-radiative model, He I, optical escape factor, recombining plasma,divertor simulator, MAP-II
1．Introduction
Measurement of electron temperature (T e ) and electron density (ne ) is important to understand plasma dynamics in the divertor region of nuclear fusion devices.
The Langmuir probe method is widely used for parameter measurement in various plasmas. In the application
to divertor plasmas, however, large heat flux can resultin the erosion of the probe tip in high density ionizing
plasmas, while in recombining plasmas, an anomaly inthe probe current (I)-voltage (V) characteristics disturbs
the measurement of T e and ne [1]. Thus, passive spectroscopies are thought to be alternatives for the measurement of the parameters of divertor plasmas.
For the spectroscopic measurement of T e and ne
based on the intensity ratio of He I series, we need aproper model of the excitation and de-excitation processes. A collisional-radiative (CR) model for He I was
first developed by Fujimoto in 1979 [2], and updated byGoto in 2003 [3]. Because neutral density is high (1011 1015 cm−3 ) in the divertor region, the eﬀect of the radiation trapping of the resonance lines must be taken intoconsideration. The optical escape factor, which is used
for describing the eﬀective radiative decay rate, wasfirst formulated by Holstein in terms of optical depth[4,5], and modified by Otsuka et al. [6] and Fujimoto[2] to yield more accurate values over a wider range ofoptical depth. Fujimoto developed a CR model for He I

with radiation trapping by including the optical escapefactor [2].
In the present study, we applied the CR model for
He I with Otsuka’s optical escape factor to ionizing andrecombining plasmas in a divertor simulator MAP-II [7,8]. The former application was aiming at confirming theeﬀect of radiation trapping for known T e and ne , while
the latter was aiming at determining them only from thespectroscopic results.

2．Effect of radiation trapping on an He I CR
model
2.1 Optical escape factor

Under the condition of considerable gas density, thespatial distribution of the resonant state is modified dueto the optical trapping as if the corresponding spontaneous emission coeﬃcient A decreases to Λ(r0 )A. Here,the optical escape factor Λ(r0 ) for the position r0 is dependent on the parameter profiles of the plasmas. Otsuka [6] and Fujimoto [2] formulated the optical escapefactor in the infinite cylindrical plasma for which theline profile of the emission/absorption spectra with respect to the resonant state can be described by Gaussian. Let nk (r0 ) and ni (r0 ) be the population densities atthe position r0 in the media, where k and i denote theupper and lower state for the transition k → i, respec-Corresponding author’s e-mail: iida@flanker.q.t.u-tokyo.ac.jp∗ Present aﬃliation: School of Engineering, Tohoku University, Sendai 980-8579, Japan∗∗ Present aﬃliation: Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan123

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Eﬀect of Radiation Trapping on an He I CR Model for a Divertor Simulator MAP-II

tively. Otsuka analytically deduced the optical escapefactor on the axis of a cylinder (r0 = 0) based on theassumption that nk (r0 ) = βni (r0 ), where β is a constant,as 12 ∞τ̂22
Λik (0) =dx
dt exp −x − √exp(−x ) .
π 001 − t2(1)The optical depth τ̂ for the transition k → i at the centerof the line profile is described as  lMgi
e2
fik 1 − β λki
ni (ρ)dρ,τ̂ =4me c0gk
2πκB T g 0(2)where fik is the absorption oscillator strength (i → k),
λki is the wavelength for the transition k → i, me is themass of the electron, M is the mass of the He atom andT g is the gas temperature. l is the eﬀective radius of theplasma and needs to be left as a free parameter.
In the present study we assumed the uniformity of
the ground and upper state density profiles. The termβ(gi /gk ) in eq. (2), which came from an induced
emission process, was neglected in the calculation becausethe upper state density is smaller than that of theground state by a few orders of magnitude in the presentconditions.
It should be noted that Fujimoto’s formula in ref. [2]assumed that nk (r0 ) is a parabolic function while ni (r0 )is a constant. If we apply Fujimoto’s model for thesame nk (0) and l, the diﬀerence in the resulting Λik (0)becomes 2.5 times larger for the optically thick (τ̂ > 10)

region, which is practically within the ambiguity in thechoice of l.
The optical escape factors Λ for the resonance transition 11 S − n1 P and for the non-resonance transition
21 S − n1 P as a function of the principal quantum number n of the upper states are shown in Fig. 1. Based on
our experimental conditions, T g , n11 S and n21 S of 300
K, 1.1 × 1014 cm−3 and 5.3 × 109 cm−3 , respectively,
are used in the calculation. Because the position of theplasma boundary cannot be determined clearly, two extreme cases for l were chosen. One is the radius of theplasma column, 0.025 m, while the other is that of thevacuum chamber, 0.25 m. As shown in Fig. 1, Λ for
21 S − n1 P are almost equal to unity, whereas those for
11 S − n1 P are smaller than that by a few orders of magnitude. Therefore, Λ of the radiative transition to the11 S state needs to be taken into account in the analysisof our spectroscopic results.
2.2 CR model with radiation trapping

The outflow rates of the population from 21 P and
71 P states due to the electron impact excitation andde-excitation are compared with that due to the spontaneous emission in Table 1. In the case of 21 P, theoutflow rate due to spontaneous emission is larger thanthat due to the electron impact, whereas for 71 P, the outflow rate due to electron impact is larger than that due
to spontaneous emission by two orders of magnitude.
Thus, for determining the population of the excited stateatoms in our conditions, the radiation trapping from thestates of small principal quantum number is important.
In the following CR model calculation, we replaced
A with ΛA for 11 S − n1 P (n = 2 ∼ 7) transitions. We
refer to this as a modified CR model hereafter to distinguish it from the CR model without radiation trapping.
We assumed a quasi-steady-state (QSS) approximationeven for meta-stable levels. This assumption is valid in
the present condition because the decay time due to theelectron collision (∼ 0.41 ms for 21 S and ∼ 0.87 ms for
23 S) is much shorter than the residence time of neutrals(∼ 16 ms). Therefore, the population of each excitedlevel can be expressed as the sum of the ionizing component and the recombining component. In this paper,
only the ionizing component was used for ionizing
plasma while only the recombining component wasFig. 1 Optical escape factor for 11 S – n1 P and 21 S – n1 P
transitions for two different lengths of the system l.

11
Table 1 Comparison of the decay rate of 2 P and 7 P populations due to the electron impact with that due to the spontaneous emission. ΛA is also evaluated. ne = 1.0 × 1012 cm–3 and Te = 6.1 eV were used.

state21 P71 P

decay rate by electron impact
ne C p→q [s−1 ]1.34 × 106
1.52 × 109<

decay rate by spontaneous emission
A [s−1 ]1.80 × 109
4.85 × 107

124


→→

ΛA [s−1 ]8.02 × 106
2.10 × 107

Eﬀect of Radiation Trapping on an He I CR Model for a Divertor Simulator MAP-IIFig. 2 Relative population per degeneracy (g) of excited levels of He I is shown as a function of principal quantum number.
Filled circles show the measured population distribution, solid lines the calculations based on the unmodiﬁed CR
model, and dashed and dotted lines the calculations based on the modiﬁed CR model with l = 0.025 m and 0.25 m,respectively.

used for recombining plasma, because the other component was negligible.

bination (EIR) phase plasmas at the periphery of theplasma column. In this case, a Boltzmann-plot of themeasured population in the Rydberg state yields T e[7]. The deduced value depends on the lowest principal
quantum number usable for the fitting procedure. In ourplasmas, the obtained T e from n3 D for n = 9 − 15 was
0.070 ± 0.003 eV. Note that this value may be slightly
overestimated, since, in partial local thermal equilibrium plasmas, the slope of the Boltzmann plot in thefinite n is smaller than that in the ionization limit.
For the application of the CR model to recombining
plasmas, the 23 P − n3 D series was used because it is
the brightest among the six He-Balmer series. Comparing the measured population ratio of n3 D with theCR model calculation for l = 0.025 m, we obtained
T e and ne as 0.06 eV and 1.22 × 1012 cm−3 , respectively. On the other hand, these values were 0.06 eV and1.16×1012 cm−3 , respectively, for l = 0.25 m. The result
for l = 0.025 m is shown in Fig. 3. It should be notedthat the measured population for n = 6 happened to beirregular in this shot, and thus this point was removed
from the analysis. The sensitivity of the population ofn3 D series to ne is so high that ne can be determined to aconsiderable accuracy by using this small diﬀerence observed in the linear scale of Fig. 3. In comparison, theresults obtained based on the unmodified model were0.06 eV and 1.62 × 1012 cm−3 . One can see that the obtained T e is not largely dependent on the optical escapefactor in the recombining plasmas, while the obtained
ne decreases by several tens percent by using the modified CR model. This result shows that the eﬀect of radiation trapping must be included for the ne measurement
even when we make use of the n3 D series which are notoptically coupled with the singlet states. As seen in the3．Experiment
3.1 Applicability of the modified CR model

The population distribution for six sublevels of He
I in the ionizing plasmas was obtained from the spectroscopic measurement and was compared with that deduced from the CR model calculation as shown in Fig.
2. The three conditions for the calculation were an unmodified CR model, a modified CR model with an absorption length l of 0.025 m, and a modified CR modelwith an absorption length of 0.25 m. The T e and nevalues used for the model calculation were obtained by
the Langmuir probe measurement, and were 6.1 eV and
1.0 × 1012 cm−3 , respectively. It should be noted thatthe calculated populations were normalized to an average value of n = 6 levels over the whole terms obtained
experimentally.
As shown in Fig. 2, the large deviation from theexperimental data observed in the case of the unmodified CR model in n1 P and n1 D was compensated for in
the case of the modified CR model. On the other hand,these diﬀerences were not clear in the n1 S and the three
triplet terms in this logarithmic scale. This is because
the redistribution of the population from n1 P to thesestates was rather small. These results confirm the validity of the modified CR model used in our experiment.
The eﬀect of the small diﬀerence found in the linear
scale on the ne measurement is important for the studyof recombining plasmas as shown in the next section.
3.2 Application to recombining plasmas

We observed the spectra from Electron-Ion Recom-125

Eﬀect of Radiation Trapping on an He I CR Model for a Divertor Simulator MAP-IIfied CR model, which adopts an the optical escape factor and the eﬀective A values, the calculated population
distribution of He I almost reproduced the experimental result. In the recombining plasmas, T e and ne canbe obtained by the best-fit of the n3 D population. By
applying the modified CR model, the best-fitted ne wasdecreased by several tens percent compared with thatfrom the unmodified CR model. The ambiguity of thebest-fitted ne value caused by the ambiguity of l wasestimated to be about ten percent.
This work was supported in part by a NIFS
Collaboretive Research Program (NIFS04KOAB009)
drirected by the second author.

Fig. 3 Best-ﬁt of the n 3 D population to experimental
data to determine ne and Te . Populations for
ne = 0.61 × 1012 cm–3 and 1.83 × 1012 cm–3 for thesame Te are calculated as a comparison.

[1]
dependence of deduced ne on l, ambiguity of l leads toan error of about ten percent in the measurement of ne .
This is because the increased population in the n 1 P state
due to the radiation trapping is redistributed through thecollisional processes.

[2]
[3]
[4]
[5]
[6]

4．SummaryThe eﬀect of radiation trapping on an He I CR
model was investigated.
In the divertor plasma
simulator MAP-II, the eﬀective spontaneous transition probability for 11 S − n1 P was decreased by
the order of magnitude due to the self-absorption
process. In the ionizing plasmas, using the modi-

[7]
[8]

126

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