J. Plasma Fusion Res. SERIES, Vol.7 (2006) 45-49

Study of Carbon Ion Behavior by Using Collisional Radiative Modelin the GAMMA 10 Tandem Mirror
KOBAYASHI Takayuki, YOSHIKAWA Masayuki, KATO Takako1 , KUBOTA Yuusuke,
SAITO Masashi, MATAMA Ken, ITAKURA Akiyosi and CHO Teruji
Plasma Research Center, University of Tsukuba, Ibaraki 305-8577, Japan
1 National Institute for Fusion Science, Toki 509-5292, Japan（Received: 4 October 2004 / Accepted: 28 April 2005）

Abstract
In a plasma experiment, collisional radiative model (CRM) is very useful model to evaluate impurity behaviors and plasma parameters with line emission from a plasma. CRMs for carbon and oxygen have beendeveloped. However verification and application of the model for analysis of experimental results are notenough. Then we applied CRM calculation results to observed impurity spectra in the GAMMA 10 tandem
mirror to evaluate the impurity density profile and the particle balance of each charge state of carbon ion. We
calculated the eﬀective ionization rate for each charge state of carbon ion and obtained the density profile ofeach ion. Moreover, we calculated absolute emission intensities from all carbon ions.
Keywords:
collisional radiative model, impurity ion, spectroscopy, GAMMA 10
1．Introduction
Impurity emission spectra from a fusion plasma havea lot of important information of the plasma, such astemperatures and densities of impurities, radiation loss
and rotation velocity of ions. These informations arevery useful to evaluate the impurity transfer and energy loss from a plasma. Moreover the radial profile
of electron density, temperature and electric field in
the plasma are obtained by using a collisional radiativemodel (CRM) and the relation between the profile ofspectra and plasma parameters [1]. By using the CRM,
ratio of an excited state population density to groundstate in an impurity ion is described as a function of
the electron temperature and the electron density under an assumption of the ionizing plasma. CRMs forcarbon and oxygen have been developed [2,3]. However verification and application of the model to experimental results are not enough. Then we applied CRM
calculation results to the analysis of observed spectra in
the GAMMA 10 tandem mirror to evaluate the impurity
density profile and the particle balance of each charge
state of carbon.
In the GAMMA 10 tandem mirror, space and time
resolved spectra are observed and analyzed in orderto evaluate the plasma parameters and impurity transfer. Emission lines from the C II (C+ ) , C III (C2+ ),O II (O+ ), O III (O2+ ), O IV (O3+ ) and O V (O4+ ) ionsdominate the observed spectra in GAMMA 10. In any

experimental setup in GAMMA 10, spectra from highly
ionized carbon ions, such as C IV (C3+ ), C V (C4+ ) andC VI (C5+ ) are not observed by any spectroscopic measurement systems [4]. Therefore it has been believedthat highly ionized carbon ions run away in the confinement time. Then we calculated C II and C III density profiles using both absolute calibrated spectra andCRM calculations under an assumption of the ionizing
plasma. We estimate the all carbon ions density profiles by particle balance of each charge state of carbon
ion with considering the impurity ion confinement andcalculated absolute emission intensity from all carbon
ions. In this paper, we present the results of CRM calculation for C II and C III and experimental results ofcarbon ion emission in the GAMMA 10 tandem mirror.
Moreover we discuss the particle balance of each charge
state of carbon ion and the confinement of carbon in theplasma.
2．Collisional radiative modelIn fusion plasmas, the population densities of the excited states must be determined by CRM. In CRM, bothcollision and radiation eﬀects are considered. Each effect is introduced with the electron impact (de) excitation, ionization, recombination rate and radiative transition rate, and expressed as C z , S z , αz and Az , respecC 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research
Corresponding author’s e-mail: tkobaya@prc.tsukuba.ac.jp

45Study of Carbon Ion Behavior by Using Collisional Radiative Model in the GAMMA 10 Tandem Mirror

tively, in the following rate equation (1).
⎧
⎫
⎪
⎪
⎪

⎪
⎪
dnz (i) ⎪
⎨
⎬
=⎪
C
(
j,
i)n
n
(
j)
+
A
(
j,
i)n
(
j)
⎪
z
e z
z
z
⎪
⎪
⎪
⎪
dt
⎩ ji
⎭
j>i
⎫
⎧
⎪
⎪
⎪
⎪

⎪
⎪
⎬
⎨
C
(i,
j)n
n
(i)
+
A
(i,
j)n
(i)
−⎪
⎪
z
e
z
z
z
⎪
⎪
⎪
⎪
⎭
⎩ i j
j<i
⎫ ⎧
⎫
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬ ⎪
⎬
⎨
⎨
S z−1 (l, i)ne nz−1 (l)⎪S z (i, k)ne nz (i)⎪
+⎪
−⎪
⎪
⎪
⎪
⎪
⎭ ⎩
⎭
⎩
l
k
⎫ ⎧
⎫
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬ ⎪
⎨
⎬
⎨
α
(k,
i)n
n
(k)
−
α
(i,
l)n
n
(i)
,
+⎪
⎪
⎪
⎪
z+1
e
z+1
z
e
z
⎪
⎪
⎪
⎭ ⎪
⎩
⎭
⎩
k

We calculated eﬀective population rate coeﬃcients for
C II – C VI. Figure 1 shows eﬀective ionization rate coeﬃcients vs. T e with ne = 1 × 1012 cm−3 for C II – C VI.
ef f
ef f
is smaller than S CIV, significantly.
We can see that S CV
Because C V is helium like ion, the ionization energyof C V is very high (392 eV), compared to that of C IV
(64 eV).
3．Experimental apparatus
3.1 GAMMA 10

GAMMA 10 is a 20 m long tandem mirror consisting of a 5.6 m long axisymmetric central cell, anchor
cells for suppressing MHD instabilities and axisymmetric end mirrors for forming the plug/thermal barrier potentials [5]. A plasma in the central cell is produced andheated by ECR heating and/or by ICRF heating. The
magnetic field at the center is 0.4 T. The central cell
mirror ratio is about 5. In the central cell, the electron
density is measured by the 4 mm microwave interferometer with movable horns which directly measure theline averaged density and an X-ray absorption method,an X-ray pulse height analyzer and a semiconductor detector are used to measure the electron temperature. The
typical ne is 2 × 1012 cm−3 and electron temperature, T e ,is 60 eV at the center of plasma in the central cell.

l(1)
where nz (i) is population density of i level of z charged
ion and j, k and l is a level of z charged ion, z−1 charged
ion and z + 1 charged ion, respectively. We assume
quasi-steady-state and use ionizing plasma approximation. Then, we can obtain the population densities ofthe excitation states with the following expression.
nz (i) = Rz−1 (i)ne nz+1 (0) + Rz (i)ne nz (0)+ Rz+1 (i)ne nz+1 (0) Rz (i)ne nz (0)= Rz (i)ne nz ,(2)
3.2 Spectroscopic system

where Rz (i) is an eﬀective population rate coeﬃcient for
the i state (i = 0 : ground state) and nz is z charged ion
density. An eﬀective ionization rate coeﬃcient is shownas follows:
S z (i)nz (i)iS ze f f =nz (i)i,nz (i) = nz .

In GAMMA 10, two absolutely calibrated ultraviolet(UV)/visible spectroscopic systems are installed
on both the central and the anchor cells [6,7]. The
UV/visible spectroscopic measurement system views
the plasma column horizontally in the central cell andvertically in the anchor cell. The plasma emission froma cylindrical plasma is collected by a quartz lens andthen transferred to a spectrometer through an 8 m long
bundled optical fiber array. Each optical fiber consistsof a quartz core with a diameter of 400 µm. The lightcollection system covers the range x = −20 cm tox = 20 cm with a channel separation of 1 cm. The
optical fibers are connected to the entrance slit of a 1 mlong Czerny-Turner spectrometer with a grating of 2400
grooves/mm. The observable wavelength is in the range
from 200 to 700 nm. The upper limit is determined by
the grating of the spectrometer, and the lower limit isdetermined by the transmission of the quartz fiber. At
the exit plane the light from each fiber gives the spectrum from each spatial position. The image of the exit
plane is focused onto an image intensifier tube coupledwith a CCD TV camera. The frame rate of the CCD TV
camera is 30 frames/s and an exposure time of 10 ms.
The output of the CCD camera is a standard TV composite video signal, which is digitized by an 8 bit video(3)i

We can calculate the source term to z + 1 charged ionfrom z charged ion as follows:
S ourcez+1 = S ze f f ne nz .

(4)

Fig. 1 Effective ionization rate coefficient of each charge
state of carbon.
46

Study of Carbon Ion Behavior by Using Collisional Radiative Model in the GAMMA 10 Tandem Mirror

capture board installed in a personal computer. The
wavelength resolution (full width of half maximum of
Hg I spectrum) and dispersion are 0.066 nm and 0.016
nm/pixel, respectively, at λ = 436 nm. The absolutecalibrated soft X-ray and vacuum ultra-violet spectrometers are observing the wavelength range 3 − 30 nm and30 − 110 nm, respectively [8,9].
4．Experimental result and analysis
4.1 Carbon ions spectra

We measured spectra of C II (426.7 nm :2s2 4f (2 F) − 2s2 3d (2 D)) and C III (464.7 nm :2s3p (3 P) − 2s3s (3 S)) at ICRF start-up plasma inthe central cell of GAMMA 10. Figure 2 shows line
integrated emission intensities of C II and C III. Figure

Fig. 4 Solid lines are density profiles of C II and C III obtained by spectroscopic measurements and CRM
calculations and broken line is C III density calculated with particle balance equation.

3 shows that emissivities of them which are obtained by
the Abel inversion technique. Upper levels population
density of each ion is obtained by the emissivity andtransition probability as follows:nz (i) =4πEz (i, j)λz (i, j),
Az (i, j)hc

(5)where Ez (i, j), λz (i, j), Az (i, j), h and c are emissivity,
wavelength and transition probability connected withtransition to j level from i level, Planck constant andthe velocity of light, respectively. Then, we can obtain
density profiles of C II and C III by eq. (2). In Fig. 4
solid lines show obtained radial density profile of C II
and C III.

Fig. 2 Line integrated emission intensity of C II and C III.

4.2 Particle balance

We used particle balance equatin as follow:def fnz =S z−1
ne nz−1 − S ze f f ne nz
dtff+ αez+1
ne nz+1 − αez f f ne nz+ Γzin − Γzout ,(6)

where S e f f , αe f f and nz are eﬀective ionization rate coeﬃcients and eﬀective recombination rate coeﬃcientsand density of z charged ion, respectively, and Γ in andΓ out are in-flux and out-flux corresponding to impurity
transfer. We assumed steady-state condition and madeleft hand side of eq. (6) equals to zero.
At first, we considered particle balance of C III. In

Fig. 3 Emissivities of C II and C III obtained by the Abelinversion technique.

47

Study of Carbon Ion Behavior by Using Collisional Radiative Model in the GAMMA 10 Tandem Mirror

Fig. 5 (a) shows ionization time and (b) shows mirror confinement time of each charge state of carbon in the central cellof GAMMA 10.

GAMMA 10, recombination process can be neglected.
Moreover, the ionizing time of C III is the order of
10−3 − 10−4 s and magnetic mirror confinement time,that is dominant in the magnetic mirror device, is theorder of 10−2 − 10−3 s. Then we can consider the netinout− ΓCIII, is small enough.
flux of CIII by transfer, ΓCIII
Therefore, eq. (6) for C III is changed as follows:
ef f
ef f
ne nCII − S CIII
ne nCIII
0 = S CII

(7)

We calculated C III density profile by C II density obtained in Sec. 4.1 and eq. (7) and it is shown as broken
line in Fig. 4. Both density profiles of C III, obtainedin Sec. 4.1 and above, are consistent with each otherin the error range. Moreover, diﬀerence of each C III
density profile in the range of the error means that net
flux corresponding to transfer of C III has small positive
value.
Next, we considered particle balance of all charge
state of carbon ion. In this consideration, we used C II
density profile obtained on Sec. 4.1 and estimate density profiles of other charge states. Contribution of recombination is neglected again. While contribution of
impurity transfer can not be ignored, because impurity
ion confinement time in the magnetic mirror of central
cell is shorter than the ionization time of highly ionizedcarbon ions. Figure 5 (a) shows ionization times (τzS )and Fig. 5 (b) shows mirror confinement times (τzM ) of
each carbon ion vs. radial position in the central cell.
Where mirror confinement time is defined as below:
τzM ≡ τih ln R.

Fig. 6 Density profiles of electron and each charge stateof carbon ion estimated by the particle balance.

Then our particle balance equation is changed as follows:
ef f
nz−1
S z−1
.
(9)
nz = e f f
S z + 1/(τzM ne )Figure 6 shows density profile of each charge state byeq. (8), C II density profile and electron density profile.
We can see that C V is dominant at the center of plasma
and C IV is comparable to C III at the edge region. Thisestimation is greatly diﬀerent with that is believed as
before. In our estimation, in-flux corresponding to impurity transfer is neglected. If in-flux can not be neglected, the value of highly ionized ion density become
more large.
At last, we calculated line emissions from all charge
state of carbon ions. By using CRM, the eﬀective emission rate coeﬃcient is obtained as follows:(8)

where R is mirror ratio and τih is impurity ion – hydrogen ion collision time. We calculated τih with typicalelectron density and ion temperature in the GAMMA
10 central cell [10].

ze f f (i, j) =48

Az (i, j)nz (i)
.
nz ne

(10)

Study of Carbon Ion Behavior by Using Collisional Radiative Model in the GAMMA 10 Tandem Mirror

and estimate carbon ions density profiles with ionizing
plasma approximation with considering the mirror confinement time. The result of our estimation showed thatC V is dominant in the center of plasma and C IV iscomparable to C III in edge region. Then C IV spectrum (154.8 nm) is expected to emit the strongest in all
carbon spectra.
Acknowledgements
The authors would like to thank members of
GAMMA 10 group of the Plasma Research Center,
University of Tsukuba and Dr. Goto (National Institutefor Fusion Science) for significant discussions.

Fig. 7 Absolute line emission intensity from all charge
states of carbon ions.

Then the line emission, Iz (i, j), is obtained as follows:
Iz (i, j) = ze f f (i, j)ne nz .

(11)

References
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(1998).
[4] M. Yoshikawa et al., Trans. Fusion Technol. 35,
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[5] K. Yatsu et al., Nucl. Fusion 39, 1707 (1999).
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[8] Y. Okamoto et al., Jpn. J. Appl. Phys. 39, 3609
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(1996).
[1]

Figure 7 shows absolute line emission from the center of plasma corresponding to carbon ions. C V is
helium like ion. It’s excitation energy is several hundred eV. The electron temperature in the central cellof GAMMA 10 is under 100 eV. Then C V is not excited enough and not emit enough, even if C V ions
are most abundant. Radiation from C IV (154.8 nm :1s2 2p(2 P) − 1s2 2s(2 S)) is the strongest. However, ourspectrometers can not observe this wavelength region
so that CIV spectra are not observed. It is necessary tomeasure the wavelength range 100 − 250 nm to study
impurity transfer, radiation loss and Ze f f , and we are
now planning.
5．Summary
We calculated the CRM of carbon ions to evaluate
impurity density profiles. We obtained C II and C
III density profiles by measured spectra and eﬀective
population rate coeﬃcients. Moreover, we consideredparticle balance of each charge state of carbon ion,

49