J. Plasma Fusion Res. SERIES, Vol.7 (2006) 310-313

Recommended Cross Section Data for Carbon Ions and Atoms:
Electron-Impact Excitation, Ionization and Charge Exchange
SUNO Hiroya∗ and KATO Takako
National Institute for Fusion Science, Toki 509-5292, Japan（Received: 4 October 2004 / Accepted: 6 December 2005）

Abstract
Cross section data have been compiled for electron-impact excitation and ionization of carbon ions C n+ ,as well as charge-exchange process between carbon ions and hydrogen atoms. A large amount of theoreticaland experimental cross section data have been collected from the litterature, and we have critically assessed
their accuracy. The recommended cross sections, the best values for use, are expressed in the form of simple
analytical functions. These are also presented in graphical form. In this work, we have treated 76 electronimpact excitation processes for Cn+ (n = 5, 4, . . . , 0), electron-impact ionization processes for all of them, and26 total and state-selective charge exchange processes in H+C n+ processes.
Keywords:
electron-impact excitation, electron-impact ionization, charge exchange
1．Introduction
Electron-impact excitation and ionization of atomic
carbon ions are important processes in plasma diagnostics and energy-loss calculations in fusion plasmas.
Charge-exchange process of carbon ions with hydrogenatoms also provides a useful plasma diagnostics. In thisreport, we present a complete set of cross sections forthese processes. As a basis for this cross section set
we have used the available experimental and theoretical data from the literature and critically assessed theiraccuracy.

method and R-matrix method provide more reliable results at lower collision energies, so can the distorted
wave method and Coulomb-Born method at high energies. Our recommended cross sections are chosen following this criterion.
Electron-impact excitation cross sections σ are expressed in terms of collision strengths Ω. For the excitation from a state i to f , the cross section is given
by2．Electron-impact excitation
There exist numerous theoretical cross section datafor electron-impact excitation of carbon ions. Previously, Itikawa et al. [1] compiled cross sections
reported before 1985, assessed them for carbon ions
(C+ − C5+ ) as well as oxygen ions (O+ − O8+ ), and fitted their recommended values to an analytical formula.
Theoretical cross sections were reported by McDowell et al. [2], Aggarwal et al. [3], and Zou et al. [4]for C5+ . Fisher et al. [5] have also carried out crosssection calculations. Callaway [6] have reported theoretical rate coeﬃcient data. There also exist results onelectron-impact excitation cross sections for the otherionic states. Except for optically allowed transitions,
all electron-impact excitation cross sections are scattered, and it has been sometimes diﬃcult to determine
the recommended data. However, the close-couplingwhere Ee is the energy of the incident electron, ωi is thestatistical weight of the initial state, V i f is the excitationenergy, and Ωi f is the collision strength. Here, the crosssection is in units of πa0 2 , where a0 is the Bohr radiusand X is the reduced electron energy defined byσi f [πa0 2 ] =

Ωi f
Ωi f
1=,
ωi Ee [Ry] ωi Vi f [Ry] X

X = Ee /Vi f .

(1)

(2)

When the cross section and energy are given respectively in units of cm2 and eV, we have
Ωi f
ωi Ee [eV]Ωi f.
= 11.969 × 10−16 ×
ωi Vi f [eV]Xσi f [cm2 ] = 11.969 × 10−16 ×(3)

With the Maxwellian distribution of electron velocity
for temperature T e , the rate coeﬃcient is calculated byCorresponding author’s e-mail: suno@jamstec.go.jp∗ Present address: Earth Simulator Center, JAMSTEC, Yokohama 236-0001, Japan310

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Recommended Cross Section Data for Carbon Ions and Atoms: Electron-Impact Excitation, Ionization and Charge ExchangeRi f [cm3 s−1 ] =8.010 × 10−8y√
ωi T e [eV] ∞Ωi f (X) = A +dXΩi f (X)e−yX ,

1(4)
(5)

We have carefully chosen reliable electron-impact datafrom the above mentioned references. Once the recommended cross sections were determined, a fit was made,which facilitates their applications. Unless otherwise
stated, we used the formula:5+

C 1s-2s
0.04
Recommended
Zou et al. (1992) T
McDowell et al. (1977) TAggarwal et al. (1991) TFisher et al. (1997)TCollision Strength Ω0.033．Electron-impact ionization
Electron-impact ionization of carbon ions has been
extensively studied both experimentally and theoretically. Expetimental measurements of Cn+ were carriedout by Donets and Ovsyamnikov [7] for ionic statesn = 1, 2, 3, 4 and 5, in the energy range above 0.1 keV.
Cross sections were obtained semi-empirically by Pattard and Rost [8] for C5+ . Experimental measurements
were also performed by Aichele et al. [9] for the samespecies. Theoretical calculation cross sections were alsoreported by Younger [10], Kao et al. [11], and Fanget al. [12].

0.02

0.01

01

10X=E/V

100Fig. 1 Collision strengths for the electron-impact excitation 1s → 2s of C5+ .
5+5+C 1s-2s

C + e --> C-10

10

2

10-12

10

106+-19-11

Cross Section σ (cm )103

Rate Coefficient (cm /s)10(6)where A, B, C, D and E are adjustable coeﬃcients.
However, we should note that we can not use such asimple formula when resonant eﬀects dominate. In sucha case, we need to indicate the boundary of the energy region where resonances dominate. The expression for the rate coeﬃcient can be obtained analytically by using eq.(4). For example, we have obtained
A = 2.446 × 10−2 , B = −1.567 × 10−2 , C = 1.010 × 10−2
and D = E = 0 for the 1s→2s transition of the C5+ion. The recommended collision strength and the original ones from the references are graphically shown in
Fig. 1. The rate coeﬃcients are shown in Fig. 2. In
this work, we have treated 76 electron-impact excitation
processes for C5+ − C0+ , as shown in Table 1. These results will be published in Atomic Data and Nuclear DataTables.

wherey = Vi f /T e .

B C
D+++ E ln X,
X X2 X3-13

10-20

10-21

Recommended
Pattard et al. (1999)TAichele et al. (1998) E
Donets et al. (1981) E
Younger (1980) T
Kao et al. (1992) T
Fang et al. (1995) T

Recommended
Aggarwal et al. (1991) T
Callaway et al. (1994) T-14

10110210310104Te (eV)Fig. 2 Rate coefﬁcients for the electron-impact excitation
1s → 2s of C5+ .

-22102310Electron Energy (eV)104Fig. 3 Cross sections for the electron-impact ionizationof C5+ .
311

Recommended Cross Section Data for Carbon Ions and Atoms: Electron-Impact Excitation, Ionization and Charge Exchange

Table 1 List of processes treated in the work.

excitation

ionization
chargeexchange

C5+ (H-like): 1s→2s, 1s→2p, 1s→3s, 1s→3p, 1s→3d, 1s→4s, 1s→4p, 1s→5p
C4+ (He-like): 1s2 1 S→1s2s 3 S, 1s2 1 S→1s2s 1 S, 1s2 1 S→1s2p 3 P,
1s2 1 S→1s2p 1 P, 1s2 1 S→1s3d 3 D, 1s2 1 S→1s3p 1 P, 1s2s 3 S→1s2s 1 S,
1s2s 3 S→1s2p 3 P, 1s2s 3 S→1s2p 1 P, 1s2s 3 S→1s3p 3 P, 1s2s 1 S→1s2p 3 P,
1s2s 1 S→1s2p 1 P, 1s2p 3 P→1s2p 1 P, 1s2p 3 P→1s3s 3 S
C3+ (Li-like): 2s 2 S→2p 2 P, 2s 2 S→3s 2 S, 2s 2 S→3p 2 P, 2s 2 S→3d 2 D,
2s 2 S→4p 2 P, 2p 2 P→3s 2 S, 2p 2 P→3d 2 D, 2p 2 P→4s 2 S
C2+ (Be-like): 2s2p 3 P→2p2 3 P, 2s2p 3 P→2p2 1 D, 2s2p 3 P→2p2 1 S,
2s2p 3 P→2s3s 1 S, 2s2p 3 P→2s3p 1 P, 2s2p 3 P→2s3p 3 P, 2s2p 3 P→2s3d 3 D,
2s2p 3 P→2s3s 1 D, 2s2p 1 P→2p2 3 P, 2s2p 1 P→2p2 1 D, 2s2p 1 P→2p2 1 S,
2p2 3 P→2p2 1 D, 2p2 3 P→2p2 1 S, 2p2 1 D→2p2 1 S
C+ (B-like): 2s2 2p 2 P→2s2p2 4 P, 2s2 2p 2 P→2s2p2 2 D, 2s2 2p 2 P→2s2p2 2 S,
2s2 2p 2 P→2s2p2 2 P, 2s2 2p 2 P→2s2 3s 2 S, 2s2 2p 2 P→2s2 3p 2 P,
2s2 2p 2 P→2s2 3d 2 D
C: 2s2 2p2 3 P→2s2 2p2 1 D, 2s2 2p2 3 P→2s2 2p2 1 S, 2s2 2p2 3 P→2s2p3 5 S,
2s2 2p2 3 P→2s2 2p3s 3 P, 2s2 2p2 3 P→2s2 2p3s 1 P, 2s2 2p2 3 P→2s2p3 3 D,
2s2 2p2 3 P→2s2 2p3p 1 P, 2s2 2p2 3 P→2s2 2p3p 3 D, 2s2 2p2 3 P→2s2 2p3p 3 S,
2s2 2p2 3 P→2s2 2p3p 3 P, 2s2 2p2 3 P→2s2 2p3p 1 D, 2s2 2p2 3 P→2s2 2p3p 1 S,
2s2 2p2 3 P→2s2p3 3 P, 2s2 2p2 3 P→2s2 2p3d 1 D, 2s2 2p2 3 P→2s2 2p3d 3 F,
2s2 2p2 3 P→2s2 2p3d 3 D, 2s2 2p2 3 P→2s2 2p3d 1 F, 2s2 2p2 3 P→2s2 2p3d 3 P,
2s2 2p2 3 P→2s2p3 1 D, 2s2 2p2 3 P→2s2p3 3 S, 2s2 2p2 3 P→2s2p3 1 P,
2s2 2p2 1 D→2s2 2p2 1 S, 2s2 2p2 1 D→2s2p3 3 D, 2s2 2p2 1 D→2s2 2p3s 3 P,
2s2 2p2 1 D→2s2 2p3s 1 P
C5+ →C6+ , C4+ →C5+ , C3+ →C4+ , C2+ →C3+ , C+ →C2+ , C+ →C3+ , C→C+
C6+ +H→C5+ (nl)+H+ (nl=1s,2s,2p,3s,3p,3d,4s,4p,4d,4f,5s,5p,5d,5f,5g),
C6+ +H→C5+ (n)+H+ (n=1,2,3,4,5), C6+ +H→C5+ +H+ , C5+ +H→C4+ +H+ ,C4+ +H→C3+ +H+ , C3+ +H→C2+ +H+ , C2+ +H→C+ +H+ , C+ +H→C+H+
4．Charge exchange in collisions between carbon ions and hydrogen atoms
There exist numerous theoretical and experimental
results for charge exchange processes in collisions ofcarbon ions with hydrogen atoms. The C6+ +H collision
is the most studied process: theoretical cross sections
for charge exchange were reported by Panov et al. [13],Fritsch et al. [14], Kimura et al. [15], Janev et al. [16],Fritsch [17], Olson et al. [18] and Belkic [19]. The
recommended cross sections are fitted to the followinganalytic functions:⎧a1 exp[−(a2 /E)a3 ]
⎪⎪
⎪
⎪
⎪
⎪
1 + (E/a4 )a5 + (E/a6 )a7
⎪
⎪
⎪
⎪
⎪
⎪
⎪
a1 exp[−(a2 /E)a3 ]
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1 + (E/a4 )a5 + (E/a6 )a7 + (E/a8 )a9,
σ [cm2 ] = 10−16 ×⎪⎪⎪
a1 exp[−(a2 /E)a3 ]
⎪
⎪
⎪
⎪
⎪
⎪
1 + (E/a4 )a5 + (E/a6 )a7 + (E/a8 )a9
⎪
⎪
⎪
⎪
⎪
⎪
⎪
a10 exp[−(a11 /E)a12 ]⎪⎪⎪⎩ +1 + (E/a13 )a14(9)where the collision energy E is expressed in eV/amu
units and ai , i = 1−14 are fitting parameters. For example, we have found a1 = 2.301 × 102 , a2 = 5.983 × 103 ,The recommended cross section for electron-impact
ionization is parametrized by the expression
⎧⎫N

⎪
⎪
10−13 ⎪I i−1 ⎪⎨⎬2Ai 1 −σ [cm ] =A1 ln(E/I) +,⎪⎪
⎪
⎪
⎭
IE ⎩
Ei=2(7)
where the collision energy E and ionization potential I
are expressed in eV units and Ai are fitting coeﬃcients.
The coeﬃcient A1 can be related to the continuum oscillator strength d f /d by ∞1 dfd(8)A1 = 8.39 × 10−2 I [eV]E+  d0where  is the energy of ejected electrons. For example,we have obtained I = 490.0, A1 = 2.489 × 10−1 , A2 =1.847 × 10−1 , A3 = 4.475 × 10−2 , A4 = −9.432 × 10−2
and A5 = 5.122 × 10−1 for the C5+ →C6+ transition.
The recommended cross section is shown graphicallyin Fig. 3, together with the cross sections from the literature. In this work, we have treated the electron-impact
ionization processes for all ionic species C5+ − C0+ as
well as the electron-impact double ionization processesof C+ .

312

Recommended Cross Section Data for Carbon Ions and Atoms: Electron-Impact Excitation, Ionization and Charge Exchange

6+5+collisions between carbon ions and hydrogen atoms.
The recommended cross sections are expressed in termsof simple analytic fit functions. These can immediately
lead to applications in fusion science. In this paper, wehave shown only examples of our results. All our results
will be published in Atomic Data and Nuclear Data Tables.

+C +H --> C +H
1022cm )100Cross Section (10-161011010-2

1010-1

Recommended
Panov et al. (1983) T
Fritsch et al. (1984) TKimura et al. (1985) TJanev et al. (1988) T
Fritsch (1989) T
Olson et al. (1989) TBelkic (1991) T-3-4

100

10110231010E (eV/amu)

4[1]
[2]

105

10[3]
[4]
[5]
[6]

6Fig. 4 Cross sections for the charge exchange process
C6+ + H → C5+ + H+ .

[7]

a3 = 3.117 × 10−1 , a4 = 1.300 × 104 , a5 = 6.613 × 10−1 ,a6 = 7.299 × 104 , a7 = 4.985, a8 = 2.100 × 104 ,a9 = 1.557, a10 = 3.355 × 10−2 , a11 = 1.800 × 10+5 ,a12 = −9.768 × 10−2 , a13 = 4.500 × 103 and a14 = 3.584
for the C6+ + H → C5+ + H+ transition. The fitted crosssection as well as the original cross sections form theliterature are graphically shown in Fig. 4. In this work,
we have treated 26 total and state-selective charge exchange processes in H+Cn+ processes, as shown in Table 1. These results will be published in Atomic Data
and Nuclear Data Tables.

[8]
[9]
[10]
[11]
[12]

5．SummaryWe have compiled and critically assessed the crosssections for electron-impact excitation and ionizationof atomic carbon ions, as well as charge exchange in

[17]
[18]
[19]

[13]
[14]
[15]
[16]

313

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