J. Plasma Fusion Res. SERIES, Vol.7 (2006) 59-63

Vibrational Distribution of H2 / D2 Moleculesin Low Temperature Plasmas
XIAO Bingjia, KADO Shinichiro1 and HE Manli∗Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui 230031, China
1 High Temperature Plasma Center, the University of Tokyo, Japan（Received: 5 October 2004 / Accepted: 27 December 2005）

Abstract
Based on the quasi-steady state and quasi-stationary approximation, the vibrational distributions of hydrogen and deuterium molecules in low temperatures are studied. Newest cross section data in the literature
are applied while the unavailable data are calculated by applying Gryzinski-method. The modeling showsthat the initial absolute slopes of the vibrational distribution curves are increasing functions of the electron
temperature when the electron temperature is less than 10 eV. However, this dependence can not be verified
by the experiment performed in MAP-II (B. Xiao, PSI-16, [1]). The mechanisms behind this discrepancy are
discussed, however, remain unrevealed.
Keywords:
vibrational distribution, hydrogen molecule, low temperature plasma
1．Introduction
Vibrational excitation drastically enhances the negative ion production rate and thus attracts extensive attention in the field of negative ion sources [2,3]. In
the edge plasmas in a fusion device, molecular behavior
must play an important role. It was in fact, theoretically
predicted [4] and experimentally confirmed in a linear
plasma device [5] that molecular assisted recombination(MAR) rate can be very eﬀective at plasma temperature
as high as 3 eV. This MAR eﬀect causes much less particle flux and thus lower heat flux to the divertor target.
The MAR eﬀect is mainly related with the vibrational
excitation of molecules.
In this paper, we applied the quasi-steady state andquasi-stationary approximation to model the density
distribution of molecules at diﬀerent vibrational levels to study the vibrational distributions of hydrogenand deuterium molecules. After the description of themodel and the included main kinetic processes, we discuss the modeled results and the comparison with thepreviously reported experiment [1]. This work is still inprogress and the further improvement of the model andthe applied data is ongoing.

As listed in the Table 1, some of these kinetic processes cause the vibrational excitation or de-excitationof molecules and almost all the processes are influenced
by the molecular vibrational level. When the transport
time scale is much longer than the characteristic timeof the molecules reaching vibrational equilibrium, we
can approximate the vibrational population balance by
a quasi-steady state and quasi-stationary model,  dNv 
= 0.
(1)
dt i
i
The summation must be performed on all processes
which make the generation and depopulation of themolecules in vibrational state v which varies from 1 to
14 for H2 . Taking the vibrational ground state as the reference state of which the density is determined by slow
process - the transport, the vibrational distribution canbe obtained by solving the balance equation (1). Table
1 summarizes all the important processes considered inthis paper and their cross section data sources. Whenthe energy of the electrons is low in the order of several electron volts, hydrogen molecule can attach one
electron to form the H2 − negative ion, mostly in the resonant electronic states of 2 Σg + and 2 Σu + . These negativeions can be further dissociated or detach one electron to
return to the ground state. The latter process (hereafterreferred to e-v) causes the H2 molecules to gain vibrational quanta, or to say, to be vibrationally excited. The

2．Model and the kinetic processes
When hydrogen molecules exist in low temperatureplasmas, they collide with ions, electrons and neutrals and thus will be excited, dissociated and ionized.
Corresponding author’s e-mail: bjxiao@ipp.ac.cn∗ Current address: Anhui Institute of Architecture & Industry, China

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research59Vibrational Distribution of H2 / D2 Molecules in Low Temperature Plasmas

Table 1 Kinetic process of H2 (D2 ) in low temperature plasmas.

e-v

e + H2 (v) → H2 − → e + H2 (w)E-V

H2 (v) + e → H2 (B Σu , C Πu ) + e → H2 (w) + e

p-v

+

1

+

H2 (vi ) + H → H2 (vf ) + H[6]

1[7,23]

+

[8-9]

−

[6]

−

e-DA

e + H2 (v) → H2 → H + H

e-D+e + H2 (v) → H2 (b Σu ) + e → H + H + e[7,23]e-D

e + H2 (v) → H2 (a3 Σg + , c3 Πu , e3 Σg + , d3 Πu )This workIonc

H2 (v) + H+ → H+2 + H

e-DE

H2 (v) + e → H2 (B Σu , C Πu ) + e → H + H + e[7,23]

e-I[7,23]e-ID

H2 (v) + e → H+2 + 2e
H2 (v) + e → H + H+ + 2ev-v

H2 (v) + H2 (w + 1) → H2 (v + 1) + H2 (w)(v <= w)

[11]

v-t∗

H2 (v) + A → H2 (w) + A(v > w)

[11-13]

∗

3

1

+

[9-10]
1[7,23]

v-t relaxation by H and H2 is only considered in this paper.

cross sections of e-v process have been calculated by
Bardsley and Wadehra [6]. It was shown that this process mainly changes the vibrational level by one. Because the existing data for this process are limited tocertain vibrational transitions, they must be extended inorder to be applied to the modeling of the vibrational
distribution. We adopted a scaling law suggested byGorse et al. [13],sections can be found on the website, http://wwwcfadc.phy.ornl.gov/h2mol /molh2.html.
When the molecules gain the vibrational quanta fromthe above mentioned collisions, the quanta can be redistributed via the vibrational-vibrational exchange among
the molecules. There is no net gain or loss of the vibrational quanta but the consequent eﬀect is that themolecule distribute toward higher vibrational levels. If
the molecules are much denser than the electrons, theexchange would drive the molecules to distribute in the
Treanor’s function [14,15],ω̄v ω̄ x v (v + 1),
(3)+
N(v) = N0 exp −kT v
kT

σ1→w = wσ0→w−1 (1 < w < 6) andσv→w = σv−1→w−1 = · · · = σ1→w−v+1 (w >= 2). (2)This scaling approximately reproduces the Bardsley
and Wadehra’s data [6] of the excitation started frominitial states higher than zero at discrete electron energies of 6 and 8 eV.
When the energy of the incident electron is above〜 14 eV, H2 can be electronically excited to the higher
electronic states. Among these electronic excitations,some lower lying singlet states such as B1 Σ+u and C1 Πustates can be eﬀective. These electronic states can radiatively return to the ground state. When returning to theground state, these excitations alone can also make themolecules distribute toward higher vibrational states. In
comparison with the e-v process, by these excitations,the vibrational change can be more than one because of
the characteristics of the Franck-Condon factors. The
contribution of the excitations to the higher electronic
states can be neglected because of their small cross sections in low temperature plasmas. The cross sections of
E-V processes have been calculated by the impact parameter method and recently were summarized by Celiberto et al. [7].
In low temperature plasmas, when the ion temperature is comparable with the electron temperature,
the H+ − H2 (v) collision can be eﬀective. The crosswhere T v and T are the vibrational and translationaltemperatures, respectively. v is the vibrational level andω and ω x are the molecular vibrational constants.
The dissociation via the excitation to the triplet repulsive state b3 Σ+u is the dominant dissociation channel.
Ref. [7] has summarized this dissociation cross section
data which are based on the Gryzinski semi-classicalmethod. In addition, the excitations to the other triplet
states such as a3 Σ+g , c3 Πu , e3 Σ+g , d3 Πu are also eﬀective in low temperature plasmas. For these excitations,the complete set of the vibrationally resolved cross sections are not available in the literature. To obtain thecomplete sets of vibrationally resolved electron impact
cross sections we apply Gryzinski semi-classical theory[16,17]. Gryzinski [16] derived an energy gain diﬀerential cross section for atomic/molecular excitation, whichis expressed as,Ei
Ne × 6.56 × 10−14σ(u) =
u2(Eo + Ei )3
Eo 4Eoln [2.7 + (Ei − u)/Eo ]
× 1−
+
Ei
3u
60

Vibrational Distribution of H2 / D2 Molecules in Low Temperature Plasmas

× [1 − (u/Ei )]Eo /(Eo +u)[cm2 /eV],(4)where Eo and Ei are the kinetic energies of the target
and incident electrons, respectively. Ne is defined as theeﬀective number of mobile electrons, which was discussed in detail in refs. [16,17] and can be adjusted
by the experimentally measured cross sections. For a
specific transition, the cross section can be obtained by
the integration of σ(u) within an energy gain limitation(E1 , E2 ), as follows: E2
Q=σ(u)du.
(5)
E1

The excitation to the triplet states from the groundstate is spin exchange excitations which can occur ina way by exchanging the orbital electron with the incident electron. In this scenario, then the energy gain
limitations E 1 and E2 must be,
E2 = Ei + Eion − Un⎧⎪⎪
⎨ Ei + Eion − Un+1
E1 = ⎪⎪⎩ Eion(Ei ≥ Un+1 )(Un ≤ Ei ≤ Un+1 )Fig. 1 Vibrational distribution of H2. (6)It must also be noted that E i in Eq. (6) must be substituted by E i + Eion for a spin change excitation. For
the molecular excitation, the cross section Q must be
further multiplied by the Franck-Condon factor, whichconnects the two vibrational levels at the excited andoriginal electronic states as




Bv
Bv
= qAv
Q,
QAv

Fig. 2 Vibrational distribution of D2

the electron temperature of 2 – 6 eV. In this experiment,
the rotational temperature was also derived from theFulcher-α band spectroscopy. It was within the rangeof 300 600 K. The fitting to the observed Fulcher-αband emission showed that the H2 vibrational temperature is an increasing function of the electron temperature. We have seen in our previous discussion that thevibrational temperature at least reflects the initial slopesof the distribution curve and that any premises used by
the experiment analysis should not change this relative
dependence of the vibrational temperature on the electron temperatures.
The E-V process is only eﬀective when the magnitude of the electron temperature is higher than ∼ 10 eV.
Although the p-v process plays important role in low
temperature plasmas the exclusion of this process fromthe modeling does not change the predicted dependencyof the vibrational distribution on the electron temperatures [20]. Vibrational-vibrational exchange is only
eﬀective in high vibrational levels with the density ofmolecules much higher than the electron density according to the proposed scaling law by Ref. [11].
It is found that the dominant contribution to the vi-(7)where A and B stand for the lower and upper electronic
states, respectively, and q is the Franck-Condon factor.
In our calculation for the repulsive state b3 Σ+u , weapplied the potential curve by T.E. Sharp [18]. For
the other triplet bonding states, we applied the potential
curves in the calculation by Staszwska and Wolniewicz[19]. For the processes related with the D2 molecules
except the excitations to the triplet states, if not available in the literature, we used the same cross sections
with H2 .
3．Results and discussion
Figures 1 and 2 show the calculated vibrational distributions of H2 and D2 at diﬀerent electron temperatures. It is seen that there are higher distributions underthe medium vibrational levels when the electron temperature is lower both for H2 and D2 . However, theexperiment with H2 discharge performed in MAP-II device, which were report earlier [1], showed the opposite
trends. In our previous work, the typical experimental
conditions are the neutral pressure of 2 – 12 mTorr and61Vibrational Distribution of H2 / D2 Molecules in Low Temperature Plasmas

modeling would not cause much error to the modeling
result.
The left unknowns are the collisions related with H3 ,+H2 and H3 + which may contribute to the vibrational
excitation of H2 , however, the information on this issue
is very limited. To clarify it will be our future objective.
4．SummaryWe have performed the modeling of the vibrational
distribution of hydrogen and deuterium molecules in
low temperature plasmas. It was shown that the vibrational distribution is higher under the medium vibrational levels when the electron temperature is lower.
This does not agree with our previous experiment [1]which showed the opposite dependence. There still exists the gap in our understanding on the mechanisms
behind. Our future eﬀort will concentrate on the clarification of the gap between the experiment and the modeling.

Fig. 3 Comparison of the calculated vibrational distribution based on different e-v cross section scaling.

brational distributions at electron temperatures less than
5 eV is the e-v process. The discrepancy might becaused by the Gorse et al. scaling [13] used by us forthe e-v cross sections. As a cross checking, we change
the scaling to what was suggested by Pigarov [21], theresult at electron temperature of 2 eV is shown in Fig.
3. It can be seen that only a slight modification to the
distribution is made. At other electron temperatures, themodifications are also small. The modification to the ev scaling in the modeling did not change our observed
dependence. If one checks the Bardsley and Wadehra’s[6] cross section data then one find that the nature of e-v
process causes the higher vibrational distribution in thelower electron temperatures.
In the experiment in Ref. [1], the vibrational temperatures were measured after the plasmas reached steadystate. There might be influence by the non-thermal highenergy electrons, however, it was shown in the experiment in Ref. [22] where the experimental conditions
were quite similar to those in Ref. [1], that the high energy tail is low and has negligible eﬀect to the reaction
rates considered in this paper.
In the modeling, we only considered the v-t relaxation caused by H2 and H because they are the main
neutral species in the H2 discharges in MAP-II. The rate
coeﬃcients of H + H2 system are directly from the appendix in Ref. [13]. The experimental determination of
H density in MAP-II is diﬃcult and has not been completed yet, so in our modeling, we arbitrarily set the
H density to be of the same order as the electron density. As shown in Ref. [13], the v-t rate coeﬃcients of
H + H2 collisions below the gas temperature of 1000 K,
are within the orders of 10−11 cm−3 s−1 which is two orders lower than the e-v rates, i.e., 1.9 × 10−9 cm−3 s−1 forthe e-v process from vi = 0 to v f = 1 at T e = 3 eV. So it
is reasonable to expect that the arbitrary H density in the

Acknowledgements
One of the authors, B. Xiao acknowledges the partial support by NSFC (grants No. 10135020 and
10275067).
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