九州大学大学院総合理工学府報告第 45 巻 第 2 号 1-10 頁 令和 6 年 2 月

Engineering Sciences Reports, Kyushu University
Vol. 45, No. 2, pp. 1-10, FEB, 2024

Emission Spectra of He2* Excimer Produced fromthe He2+/2e- Collisional Radiative Recombinationin the He Flowing Afterglow
Masaharu TSUJI*1,2† and Erika Oda−Sako*3
E-mail of corresponding author: tsuji@cm.kyushu-u.ac.jp†(Received November 17, 2023, accepted December 1, 2023)
The He2+/2e- collisional radiative recombination (CRR) was studied by observing He2* excimer
emissions in the 200−990 nm region. He2* emissions from twenty-two triplet states and twelve singlet
states with excitation energies of 19.3−21.8 eV were identified. The relative formation rate constant ofan upper 𝑢 level of He2*, 𝑘 𝑢 , rapidly decreased with increasing the excitation energy of He2*. Major
triplet and singlet He2* states were the lowest observed c3g+ and C1g+ states, which occupied 39.9%
and 7.1% of ∑ 𝑘 𝑢 , respectively. The total formation ratio of triplet/singlet states was 5.1. No
vibrational excitation was observed for all observed He2* states. The rotational distributions off 3u+(𝑣 =0), e3g(𝑣 =0), and c3g+(𝑣 =0) were expressed by double Boltzmann rotational temperaturesof 310–500 K and 580–990 K for low and high rotational levels, respectively, whereas that of d3u+(𝑣 =0)
was expressed by a single Boltzmann rotational temperature of 650 K. Rovibrational distributions of
the f, e, d, and C states indicated that most of the excess energies (95.2−98.5%) were deposited into
relative translational energies of products. The observed vibrational and rotational distributions werelower than those of statistical prior ones calculated assuming a long lived [He2*–e-] intermediate.
Key words: He2+ ion, Collisional radiative recombination, Flowing afterglow, He2* excimer emission,

Relative formation rate constant, Rotational temperature, Statistical prior distribution

1.

Introduction

There has been continuing interest inelectron-ion recombination processes indischarge plasma because they are important
loss processes of charged species from natural
plasmas in space to man-made plasmas such asdischarge plasma.1,2) Since helium is an
important rare gas for the formation of stable
discharge plasma, electron-ion recombination
processes in helium discharge plasma havebeen extensively studied.3-9) The main purposeof previous experimental and theoreticalstudies has been the determination of the
responsible recombination processes and theirrecombination coefficient or cross section at
various electron densities, He pressures, andtemperatures.
When the He gas pressure is low (p < about 1
Torr = 133.3 Pa), it is recognized that the three*1

Institute for Materials Chemistry and Engineering,and Research and Education Center of GreenTechnology*2 Department of Molecular Science and Technology*3 Department of Molecular Science and Technology,
Graduate Student

body process He+ + 2e- → He* + e- is the main
recombination mechanism if the electron
density is not too low or the electron
temperature too high. When the He gaspressure is higher (p > about 5 Torr) and He2+
is the dominant ion, the following electron-He2+
recombination processes can occur:
He2+ + e- → neutral products,
Dissociation recombination (DR),He2+ + e- + e- → He2* + e-,(1)

Collisional radiative recombination (CRR),(2)

He2+ + e- + He → He2* + e-,
Ternary recombination.

(3)

Berlande et al.6) determined recombination
coefficients of processes (1)−(3) at various
electron densities and He pressures at 300 K.
In general DR process (1) is slow and CRR
process (2) is a major process at low He
pressure and ternary recombination (3)becomes important with increasing the He gaspressure.

2

Formation of He2* from the He2+/2e- collisional radiative recombination in the He flowing afterglow

Compared with the large number of kinetic
studies of the He discharge plasma, there are
only a few experimental studies on determining
the final electronic states of neutral He2
molecules in the electron-He2+ recombination
process. Two pioneering flowing-afterglow (FA)optical spectroscopic studies have been carriedout on the electron-He2+ recombination process
leading to He2* excimer by Schmeltekopf andBroida10) and Collins and Robertson11) in
1963−1964. Schmeltekopf and Broida10)
observed several He2* emission systems
resulting from CRR (2). Principal He2*
emission systems were 4650 Å e3g(3pπ)−a3u+and 5733 Å f 3Δu(3dδ)−b3g systems. They foundthat the rotational populations of high and low
rotational levels of the e3g state wereexpressed by double Boltzmann temperaturesof 903 K and 492 K at pdn (downstreampressure)10) = 1.4 Torr and 878 and 278 K at
pdn = 1.7 Torr, respectively. Collins andRobertson11) identified fourteen He2* bands
with the wavelength near the band origin at
3680, 3984, 4400, 4430, 4450, 4540, 4650, 5130,
5730, 5880, 5950, 6100, 6300, and 6400 Å.
Among fourteen emissions, five systems at
3680, 3984, 4450, 4650, and 5130 Å wereassigned to the 4p 3Πg−2s 3Σu+, 5d 3Πu−2p 3Πg,
4d 3Σu+−2p 3Πg, 3p 3Πg−2s 3Σu+, and 3p 1Πg−2s 1Σu+transitions, respectively. Based on the axial
variation and pressure dependence of theintensity of these He2* emissions and He2+
concentration, they identified the dominant
populating mechanism of He2* to be CRR (2).11)
After these pioneering studies, no detailed
optical studies have been reported for CRR
process (2) during the past about 60 years.
We have previously made a systematic
optical spectroscopic studies on the He+/2e-,Ne+/2e-, and Ar+/2e- CRR reactions byobserving He*, Ne*, and Ar* emissions in theFA.12-15) In the He+/2e- CRR reaction, theformation of fifty-one singlet and triplet ns, np,
and nd Rydberg states of He* in the23.01−24.53 eV range was observed. The
electronic-state distribution decreased withincreasing the excitation energy of He*. They
were expressed by double Boltzmann electronic
distributionswitheffective
electronic
temperatures of 0.46 eV in the 22.7−24.4 eV
range and 0.089 eV in the 24.4−24.5 eV range.
In this study, the He2+/2e- CRR reaction (2) isinvestigated by observing ultraviolet-visiblenear infrared emissions of He2* excimer in the
He FA. The relative formation rate constants
are determined. The vibrational and rotational
distributions are determined and comparedwith those predicted from statistical priordistributions. Preliminary results of this paper
were communicated in a review article.16)
Although the contribution of radiative cascadefrom upper He2* states to the lower levels was
not considered in the determination of relative
formation rate constant in the previous
study,16) we take it into account in this study.

2.

Experimental

The FA apparatus used in this study has
been previously described in detail.16) In brief,
the flow reactor, which consists of a quartz
discharge tube (11 mm i. d.) and a stainlesssteel reaction cell (60 mm i. d.), wascontinuously evacuated by means of a 10,000
liter/min mechanical booster pump combinedwith a 1,600 liter/min oil rotary pump. He(23S),
He+, He2+, electrons were generated by amicrowave discharge of high purity helium gas(purity >99.995%) in a discharge flow operated
at 1.6–20 Torr. The mechanical booster pump
was equipped with a continuously variable gate
valve.16) Opening or closing the gate valve
allowed us to vary the flow tube pressure from
1.8 Torr to 20 Torr. Low pressure experiments
at 1.6−1.8 Torr were carried out by completely
opening the gate valve, whereas high pressure
experiments at 2−20 Torr were conducted bypartially closing the gate valve.
The contribution of He+ and He2+ ions to theobserved emissions was examined using an ioncollector grid placed between the discharge
section and the reaction zone. The efficiency ofion collection was found to be >98% byobserving the N2+(C2u+−X2g+) emission
resulting from the He+ + N2 charge-transfer
reaction.17)
The positive He2+ ions were formed by thethree-body reaction of He+ with 2He in a flow
tube:

k4

He+ + 2He → He2+ + He,（4)(k4 = 8.3 × 10-32 cm6 molecule-2 s-1 [Ref. 18]).
A reaction flame, observed 10 cm
downstream from the center of the discharge,
was dispersed in the 200−990 nm region with aSpex 1250M monochromator equipped withacooledphotomultiplier(Hamamatsu
Photonics R376 or R316-02). The wavelength
response of monochromator and the opticaldetection system were corrected using standard
D2 and halogen lamps.

令和 5 年度

3.

第 45 巻九州大学大学院総合理工学報告

Results and Discussion

3.1
Emission spectra of He2* excimer
resulting from the He2+/2e- CRR reaction
Figure 1 shows typical emission spectra
observed in the 330−980 nm region at a He gaspressure of 1.8 Torr. Thirty triplet and twelve
singlet transitions of He2* given in Table 1 are
identified by referring to reported molecular
spectral data of He2*.19) The assignment ofseveral weak transitions of He2* is not shownin Fig. 1 for the sake of clarity. In addition to
many He2* emissions, He*, H*, and N2+(B−X)
emissions are identified.20,21) He* lines are
dominantly formed by the He+/2e- CRR
reaction,12,13) whereas H* lines and N2+
emission arise from residual H2O and N2,
respectively.21,22) The observed He2* emissions
disappeared when charged species were
removed from the He afterglow on applying anelectrostatic potential to the grid. These results
imply that the He2* states are produced
through electron-ion recombination process.
The relative intensities of the observed He2
transitions are listed in Table 1. There are

第2号

three 3Δu(ndδ), 3Πu(ndπ), and 3Σu+(ndσ)
components in the q, m, j, and f triplet stateswith n = 6, 5, 4, and 3, respectively. Although it
was difficult to determine relative intensities ofTable 1. Relative intensities of He2* excimer
bands produced through the He2+/2e- CRR in
the He FA.
Transition* *q3Πu(6dπ)-b3Πg

M-B

i-a

k'-a
l-a

q-b

**

*

*

◇

q3Πu(6dπ)-c3Σg+

◇

m-c
E-A

q3Σu+(6dσ)-c3Σg+p3Πg(6pπ)-a3Σu+

1.4

o3Σu+(6sσ)-c3Σg+

1.1

3

3

+

o Σu (6sσ)-b Πg

*
400
◇

*

*

450

* *

500

550

3.0
M1Δu(5dδ)-B1Πg
5.6

3

+

k3Σu+(5sσ)-c3Σg+

1.3

c - a, C - A

3

1.0

1

+

M Σu (5dσ)-B Πg
3.6+

M1Πu(5dπ)-B1Πg1l3Πg(5pπ)-a3Σu+
k Σu (5sσ)-b Πg

1.5

k'3Σg+(5pσ)-a3Σu+

1.2

3

○

○

L1Πg(5pπ)-A1Σu+

2.4

3

j Δu(4dδ)-b Πgj3Πu(4dπ)-b3Πg
3

+

17.0

3

j Σu (4dσ)-b Πg

f-bi3Πg(4pπ)-a3Σu+

d-b

3

F-B

2.2

H1Σu+(4sσ)-B1Πg

1.3

f3Δu(3dδ)-b3Πg

14.1

F1Δu(3dδ)-B1Πg

3.4

f3Πu(3dπ)-b3Πg

15.0

F1Πu(3dπ)-B1Πg

3.1

630

700

770

840

910

980

Wavelemgth / nm

Fig. 1. Emission spectra of He2* resulting
from the He2+/2e- CRR in the He FA at a He
gas pressure of 1.8 Torr. Lines marked by *and ◇ are He* and H* lines, respectively.

+

+

3

6.3

h Σu (4sσ)-b Πg

3

560

1.1

m3Δu(5dδ)-c3Σg+

3

350

1.9

m Σu (5dσ)-b Πg

m-b
h-b

1.3

q3Δu(6dδ)-c3Σg+

3

*
* **

P1Πg(6pπ)-A1Σu+q3Σu+(6dσ)-b3Πgm3Πu(5dπ)-b3Πg

*

*

RI

4.5

m3Δu(5dδ)-b3Πg

e-a

o-b

Transitionm3Σu+(5dσ)-c3Σg+

*
j-b

RIa)

q3Δu(6dδ)-b3Πgm3Πu(5dπ)-c3Σg+
◇

3

3

1

+

1

f Σu (3dσ)-b Πg

11.9

F Σu (3dσ)-B Πg

1.9e3Πg(3pπ)-a3Σu+

10.5

E1Πg(3pπ)-A1Σu+

1.9

3

+

3

1

+

1

d Σu (3sσ)-b Πg

13.2

D Σu (3sσ)-B Πg

4.6

c3Σg+(3pσ)-a3Σu+

100

C1Σg+(3pσ)-A1Σu+17.7a) Relative intensity

Formation of He2* from the He2+/2e- collisional radiative recombination in the He flowing afterglow

4

the three components for the upper q, m, and jstates because of their heavy overlapping in the
narrow wavelength region. Therefore, total
intensities of the three components in eachsystem are given in Table 1. On the other hand,
three components were observed separately forthe f state, so that their relative intensities are
given in Table 1.
Figures 2 and 3 show energy-level diagramsof triplet and singlet states of He2*, whereobserved emitting levels are shown in red. The
excitation energies of He2* and therecombination energy of He2+ (22.223 eV) wereobtained from Ref. 19. Although all of the He2*
states below the recombination energy of He2+
can be formed in the He2+/2e- CRR reaction, 22
triplet levels from c to q states in the 19.2–21.9
eV region and 12 singlet levels from C to P
states in the 19.5−21.8 eV region were observedin this study. These results indicate that
product He2* are distributed in a wide energy
range in both triplet and singlet states. A
similar wide product-state distribution has
been observed in the formation of He* from theHe+/2e- CRR reaction.13)

He2+
np 3Σg+ ns 3Σu+ np 3Πg nd 3Σu+ nd 3Πu

nd 3Δu10

q He2*
6
5 m

22.22
22.00

6
5
21.00

Energy / eV

4

n
k’

6
5
4

o

10

k

6
5

h

4

d

3

q
6
5 m

p
l
i

4

q
6
5 m

j

4

j

4

g
3

e

3

f

3

3

f

j

f

20.00

3

c

19.00

b

2
18.00

2 a

Fig. 2. Energy-level diagram of triplet statesof He2*. Observed emitting levels in theHe2+/2e- CRR reaction are shown in red.
He2+
np 1Σg+ ns 1Σu+

np 1Π

22.22
78
6
L
5

He2*

22.00

4

H

4

I

g

nd 1Σu+ nd 1Πu nd 1Δu

S
R
P
5
4

M
J

5
4

M
J

5
4

M
J

Energy / eV

21.00
3

D

3

E

3

F

3

F

3

3.2 Relative formation rate constants of He2*
excimer in the He2+/2e- CRR reaction
Relative emission intensities of each
emission system, given in Table 1, areproportional to their emission rate constants.
When there is no nonradiative decay andpumping from the observation region, theemission rate constants correspond to theirformation ones. To the best of our knowledge,
no nonradiative decay processes have been
known for the He2* states. Radiative lifetimesof the d, e, and f states have been measured as
25±5 ns, 57±10 ns, and 19±5 ns, respectively.23)
Although radiative lifetimes of other excitedstates observed in this study are unknown,
they may be the same order as the above states.
Radiative lifetimes of excited He2* molecules(≤57 ns) are expected to be short enough to emit
radiation within the observation region. It istherefore reasonable to assume that therelative emission rate constants reflect theirformation ones.
The overall rate constant of the production ofan upper state of He2*, 𝑘 𝑢 , is given bysummation of direct formation rate constant,𝑘 𝑢 , and the contribution of radiative cascadefrom upper states, 𝑘 𝑢: cascade ,

F

𝑘 𝑢

𝑘 𝑢

𝑘 𝑢: cascade .

(5)

20.00
3

C

19.00
2
18.00

2

B

A

Fig. 3. Energy-level diagram of singlet statesof He2*. Observed emitting levels in theHe2+/2e- CRR reaction are shown in red.

Actually radiative cascade partially contributes
to the formation of the lowest energy c state(Table 1). Its 𝑘 𝑢 value was estimated bysubtracting the contribution of radiative
cascade. Results obtained are given in Table 2,
where 𝑇 or 𝑇 values of He2* states areshown. When vibrational constants of He2*
states are known, 𝑇 values are calculated. On
the other hand, when vibrational constants ofHe2* are unknown, 𝑇 values are given in
Table 2. The dependence of 𝑘 𝑢 on theenergy of He2* state is shown in Fig. 4.

令和 5 年度

九州大学大学院総合理工学報告

values decrease rapidly withThe 𝑘 𝑢increasing excitation energy of He2* for boththe triplet and singlet states. They are largest
for the lowest c3g+ and C1g+ states. The total
formation ratio of triplet/singlet states was 5.1,
which was larger than a statistical ratio of 3.0
by 70%.
3.3
Vibrational distribution of He2* androtational distributions of He2(f, e, d, c: 𝒗
𝟎)in the He2+/2e- CRR reaction
Only (0,0) bands were observed for all theHe2* systems, so that no vibrational excitation
was observed. Equilibrium internuclear

第 45 巻第2号

5

distances of He2* are given in Table 2.19) It isclear that equilibrium internuclear distances ofthe observed He2* states (1.068−1.097 Å) aresimilar to that of He2+ (X2u+:1.081 Å).19) Thus,
one reason for the lack of vibrational excitationis large Franck-Condon factors (FCFs) for theHe2+(X : 𝑣" 0 ) → He2* recombination due tosimilar equilibrium internuclear distance.
In order to confirm the validity of the above
explanation, we calculated FCFs from theHe2+(X:𝑣" 0) to major He2(d, c, and C: 𝑣
0–2)
vibrational
statesusing
Morsepotentials.24) Results obtained are shown in
Table 3. The FCFs for the 0 – 0 transitions are

Table 2. Observed triplet and singlet excited states, energies (𝐸), relative direct formation rate
constants (𝑘 𝑢 ), and equilibrium internuclear distance (𝑟 ) of He2* excimer.
State

𝐸

𝑘 𝑢
(eV)

q3Δu(6dδ)3

21.8 (Te)q3Σu+(6dσ)

21.8 (Te)

(Å)

6.9

1.090

p Πg(6pπ)

21.8

1.5

1.080

o3Σu+(6sσ)

21.8

2.4

1.089

m3Δu(5dδ)21.7

m3Πu(5dπ)

21.7m3Σu+(5dσ)

21.7

3

9.3

1.091

l Πg(5pπ)

21.7

3.9

1.080

k3Σu+(5sσ)

21.6

3.0

1.079

+

k' Σg (5pσ)

21.5

1.3

1.068

j3Δu(4dδ)21.4

3

j Πu(4dπ)

21.4j3Σu+(4dσ)

21.4

3

i Πg(4pπ)

3

State

𝐸

𝑘 𝑢

(eV)

𝑟
(Å)

21.8 (Te)q Πu(6dπ)
3𝑟 a)P1Πg(6pπ)21.8

M1Δu(5dδ)

21.7

M1Πu(5dπ)

21.7M1Σu+(5dσ)

21.7

1

1.4

1.080

1.0

1.091

L Πg(5pπ)

21.7

2.6

1.079

1.081
6.1

1.083

21.3

6.8

1.079

h3Σu+(4sσ)

21.3

2.4

1.077

H1Σu+(4sσ)

21.3

1.4

1.077

f3Δu(3dδ)

20.7

15.2

1.079

F1Δu(3dδ)

20.7

3.7

1.079

3

1

f Πu(3dπ)

20.7

16.2

1.087

F Πu(3dπ)

20.7

3.3

1.085

f3Σu+(3dσ)

20.6

12.8

1.091

F1Σu+(3dσ)

20.7

2.1

1.089

3

e Πg(3pπ)

20.6

11.4

1.075

E Πg(3pπ)

20.7

2.0

1.076

d3Σu+(3sσ)

20.5

14.2

1.071

D1Σu+(3sσ)

20.6

5.0

1.069

19.6

19.1

1.093

3

+

c Σg (3pσ)a) Ref. 19

19.3

100

1.097

1

1

+

C Σg (3pσ)

Formation of He2* from the He2+/2e- collisional radiative recombination in the He flowing afterglow

6

Relative formation rate constant (k0(u)) / %

100

80

Triplet state
Singlet state

60

40

20

0
19.0

19.5

20.0

20.5

21.0

21.5

22.0

Energy of He2* / eV

Fig. 4. Relative formation rate constants of
triplet and singlet states of He2* produced
from the He2+/2e- CRR in the He FA.
Table 3. Franck-Condon factors between
He2+(X2u+: 𝑣" = 0) and He2(d,c,C: 𝑣
0−2)
states.
𝑣" 𝑣
He2+→d3u+He2+→c3g+
He2+→C1g+

0–0
0.995
0.992
0.992

0–1
0.481(-2)a)0.771(-2)
0.771(-2)

0–20.279(-4)
0.165(-3)
0.165(-3)

larger than 0.99 in all the three systems. These
data support why vibrational excitation is
insignificant for the formation of He2* in theHe2+/2e- CRR reaction.
Rotational populations of He2* were
estimated for four typical band systems, forwhich rotational structures are well resolved.
Figures 5a−5d show expanded emission
spectra of the f3u+−b3g, e3g−a3u+,d3u+−b3Πg, and c 3g+−a3u+ systems of He2,
where (0,0) bands of the four systems areobserved. In 4He, the nuclear spin is zero anddue to the symmetry rules for homonuclear
molecules all even or odd numbered rotational
levels are missing.25) Therefore, only odd oreven levels of P, Q, or R branches are observedin the above four transitions.26-28)
Rotational distributions of the f, e, d, and cstates were determined from relative
intensities of Q or P branch. Rotational levelsof He2* which belong to Hund’s case (b) arerepresented by the quantum number 𝑁′. Theband intensity (photons s-1) of a transition froma 𝑣 𝑁 level to a 𝑣′′ 𝑁′′ level is expressedasFig. 5. Rotational structures of the (0,0) bandsof He2(f-b, e-a, d-b, c-a) systems resulting fromthe He2+/2e- CRR in the He FA at a He gaspressure of 1.8 Torr.

令和 5 年度

九州大学大学院総合理工学報告𝑟̄ ′ ′′ 𝑞 ′ ′′ 𝜈 ′ ′ ′′ ′′ 𝑆 ′ ′′ 𝑔 ′ , (6)where 𝑁 ′ ′ is the rotational population in agiven vibrational level, 𝑅𝑒 𝑟̄ ′ ′′ the electronictransition moment, 𝑞 ′ ′′ the FCF, 𝜈 ′ ′ ′′ ′′ thetransition frequency, 𝑆 ′ ′′ the rotational line
strength, and 𝑔 ′ 2𝑁
1. 25) When a fixed𝑣 , 𝑣" band, 𝑅𝑒 𝑟̄ ′ ′′ and 𝑞 ′ ′′ values areconstant. Assuming a Boltzmann distribution,
Boltzmann rotational temperature is obtained
from the following relation,

第2号

7

3.0

(a) f 3Σu+(v’=0)

2.0ln(IN'N"/SN'N"N'N"3)

𝐼 ′ ′ ′′ ′′ ∝ 𝑁 ′ ′ 𝑅𝑒

第 45 巻

1.0
0.0
-1.0
-2.0
-3.0

0

50

100

150

200

N'(N'+1)ln∝ 𝑐𝑜𝑛𝑠𝑡ℎ𝑐𝐵 𝑁′ 𝑁′

1 /𝑘𝑇

,

3.0

(7)

(b) e3Πg(v’=0)

Table 4. Rotational temperatures ofHe2(f,e,d,c) produced from the He2+/2e- CRR
reaction.
Emitting species
He2(f 3u+: 𝑣′=0)He2(e 3Πg: 𝑣′=0)
He2(d 3u+: 𝑣′=0)
He2(c 3Σg+: 𝑣′=0)

𝑇 /K
580±20 (𝑁′=5–13)
310±70 (𝑁′=1–5)
990±190 (𝑁′=10–16)500±20 (𝑁′=2–10)830±20 (𝑁′=7–17)440±50 (𝑁′=1–7)
720±140 (𝑁′=0–12)

1.0
0.0
-1.0
-2.0
-3.0

0

50

100

150

200

250

300

N'(N'+1)
2.0(c) d3Σu+(v’=0)ln(IN'N"/SN'N"N'N"3)

1.0
0.0
-1.0
-2.0
-3.0
-4.0
0

50

100

150

200

250

300

350

N'(N'+1)

3.0(d) c3Σg+(v’=0)

2.5
2.0ln(IN'N"/SN'N"N'N"3)

where 𝑘 and 𝑇
are the Boltzmann constantand the Boltzmann rotational temperature,
respectively.25)InFigs.
6a−6d,⁄𝜈ln 𝐼
𝑆values are plottedagainst 𝑁′ 𝑁
1 for the 𝑣
0 levels of thef, e, d, and c and states. An approximately
linear relationship, i.e., approximately a
Boltzmann distribution, is obtained for theabove four levels. From the slopes, ℎ𝑐𝐵 /𝑘𝑇 ,the effective rotational temperatures, 𝑇 , areestimated to be 580±20 K for 𝑁′= 5−13 of He2(f),990±190 K for 𝑁′= 10−16 of He2(e), and 830±20
K for 𝑁′= 7−17 of He2(d), respectively. However,the populations at lower 𝑁′ levels appear to
deviate from the linear relationship. Their
effective rotational temperatures are 310±70 K
for 𝑁′=1−5 of He2(f), 500±20 K for 𝑁′=2−10 ofHe2(e), and 440±50 K for 𝑁′=1−7 of He2(d). The
rotational population of He2(c) is expressed bya single Boltzmann distribution with aneffective rotational temperature of 720±140 K.
Above results are summarized in Table 4. The
rotational temperatures of He2(e) for low andhigh 𝑁′ levels are in good agreement withln(I N'N" /S N'N" N'N" 3 )

2.0

1.5
1.0
0.5
0.0
-0.5
-1.0

0

40

80

120

160

N'(N'+1)

Fig. 6. Estimation of effective rotationaltemperaturesfromtheslopes⁄𝜈of ln 𝐼𝑆
against 𝑁′ 𝑁
1for the He2(f:𝑣 =0), He2(e:𝑣 =0), He2(d:𝑣 =0),and He2(c:𝑣 =0) states in the He2+/2e- CRR
reaction.

8

Formation of He2* from the He2+/2e- collisional radiative recombination in the He flowing afterglow

previous reported values of 492 and 903 K at
pdn = 1.4 Torr.10)
In order to examine collisional relaxation,
dependence of rotational populations on thebuffer He gas pressure was examined in the
1.8–20 Torr range. The rotational distributionsof He2 f: 𝑣 = 0) and He2 d: 𝑣 = 0) were
essentially independent of the He pressure in
the 1.8–20 Torr range, indicating thatcollisional relaxation within short radiative
lifetimes of f(19±5 ns) and d(25±5 ns) states23)
was insignificant under our experimental
conditions. The rotational distributions of
He2 f: 𝑣0), He2 e: 𝑣0), and He2 d: 𝑣
0)reproduced by double Boltzmann rotationaltemperatures. Similar double Boltzmannrotational distributions have been found for thee state.10) Therefore, double Boltzmannrotational distributions may be generalfeatures for the formation of He2* in theHe2+/2e- CRR reaction.
From
theobserved
rovibrational
distributions of He2(f,e,d,c), we estimated theaverage vibrational and rotational energies ofeach state, denoted as <Ev> and <Er>, and theaverage yields of total available energy into
these degrees of freedom, denoted as <fv> and<fr>, respectively. Here, the total available
energy, 𝐸 , was estimated from the relation at
300 K: 𝐸= -H0o ＋ 5/2RT. From the <Ev>,<Er>, <fv> and <fr> values, the relativetranslational energy <Et> and the average yieldof total available energy into the relativetranslational energy <ft> were determined.
Results obtained are summarized in Table 5.
The <fr> value decreases with increasing the𝐸value. This implies that the total excessenergy is not efficiently converted into therotational energy of He2*. On the basis ofenergetic data given in Table 5, most of total
available energies (95.2−98.5 %) are released torelative translational energies of products. Thisimplies that a third-body electron efficiently
receives an excess energy as its translationalenergy during neutralization reaction.
3.4
Comparison between observed andstatistical prior rotational and vibrational
distributions of He2* excimer
In order to obtain information on reaction
dynamics, statistical prior rotational andvibrational distributions,29) were calculated
assuming a long lived [He2*–e-] complex:

Table 5. Average vibrational, rotational, andtranslational energies (eV) deposited intoHe2(f,e,d,c), total available energy (E tot: eV),and average fractions of vibrational, rotational,
and translational energies (%) deposited intoHe2(f,e,d,c) + e- in the He2+/2e- CRR reaction.
He2(f)

He2(e)

He2(d)

He2(c)

<𝐸 >

0.00

0.00

0.00

0.00

<𝐸 >

0.079

0.056

0.059

0.043

𝐸

1.65

1.65

1.79

2.97

<𝑓 >

0.00

0.00

0.00

0.00

<𝑓 >

4.8

3.4

3.3

1.5

<𝑓 > + <𝑓 >

4.8

3.4

3.3

1.5

<𝑓 >

95.2

96.6

96.7

98.5

He2+ + 2e- → [He2*–e-] → He2* + e-.

(8)

Prior rotational and vibrational distributions
were calculated from the Eqs. (9) and (10),respectively:
𝑃 𝑁′ ∝ 2𝑁′

1 𝐸

𝑃 𝑣′ ∝ 𝐸

𝐸

𝐸
/

.

/

,

(9)
(10)

The prior rotational distributions of He2 f ,He2 e , He2 d , and He2 c are compared withobserved ones in Figs. 7a–7d, respectively. The
prior vibrational distributions of 𝑣 =0−5 levelsof He2(f, e, d, c) states were calculated as follow:
He2(f: 𝑣 =0−5) = 1.00:0.83:0.68:0.55:0.43:0.34,
He2(e: 𝑣 =0−5) = 1.00:0.82:0.66:0.52:0.39:0.28,
He2(d: 𝑣 =0−5) = 1.00:0.84:0.69:0.56:0.44:0.34,
He2(c: 𝑣 =0−5) = 1.00:0.91:0.83:0.75:0.69:0.63.
The
prior
rotational
andvibrational
distributions predict much higher rotationaland vibrational excitation than the observed
ones. On the basis of these facts, He2*
molecules are not formed through long-lived[He2–e-]*, where the excess energies aredeposited into all degrees of freedom
statistically.
The deviation from prior rotational
distribution can often be expressed in the formof surprisal,29)
IR = −ln[P(𝑣′,𝑁′)/Po(𝑣′,𝑁′)] = 𝜃0  + 𝜃 𝑔𝑁′, (11)where 𝑔 = 𝐸 ⁄𝐸.

令和 5 年度

九州大学大学院総合理工学報告

Rotational distribution of He2(f:v'=0)

1.2

Obs.

(a) f3Σu+ :v’=0)

1.0

Prior

0.8
0.6
0.4
0.2
0.0

0

2

4

6

8

10

12

14

第 45 巻第2号

9

Figures 8a–8d show rotational surprisal plotsof He2 f: 𝑣0), He2 e: 𝑣0), He2 d: 𝑣
0),and He2 c: 𝑣0), respectively. When the twobody dissociation model (8) is used for 𝑃 𝑁′ ,good linear relationships are found in the four
surprisal plots. These results imply that an
exponential gap behavior is present between
the experimental and statistical distributions.
From slopes of each surprisal plot, large
positive surprisal parameters of 23.4, 25.6, 27.2,
and 45.8 are obtained for He2 f: 𝑣
0 ,He2 e: 𝑣
0 , He2 d: 𝑣
0 , and He2 c: 𝑣
0 ,respectively.

N'

Rotational distribution of He 2(e:v'=0)

1.2

4.

(b) e3Πg :v’=0)

Obs.

1.0

Prior

0.8
0.6
0.4
0.2
0.0

0

2

4

6

8

10

12

14

16

N'

Rotational distribution of He2(d:v'=0)

1.2

Obs.

(c) d3Σu+ :v’=0)

1.0

Prior

0.8
0.6
0.4
0.2
0.0

0

4

8

12

16

N'

Rotational distribution of He 2(c:v'=0)

1.2

Obs.

(d) c3Σg+ :v’=0)

1.0

Prior

0.8
0.6
0.4
0.2
0.0

0

2

4

6

8

10

12

14

N'

Fig. 7. Observed and statistical priorrotational distributions of the He2(f: 𝑣 =0),He2(e: 𝑣 =0), He2(d: 𝑣 =0), and He2(c: 𝑣 =0)
states in the He2+/2e- CRR reaction

Summary and Conclusion

He2* excimer emissions from twenty-two
triplet states and twelve singlet states withexcitation energies of 19.3−21.8 eV and
19.6−21.8 eV region, respectively, wereobserved by the He2+/2e- CRR reaction in the He
afterglow. It was found that more He2* statesin wide energy range are formed in the He2+/2eCRR reaction than those observed by
Schmeltekopf and Broida10) and Collins andRobertson.11) The relative formation rate
constant of He2*, 𝑘 𝑢 , rapidly decreased withincreasing the excitation energy. A similartendency was observed for He* in the He+/2eCRR reaction,13) indicating that CRR reaction
dynamics is similar between atomic He+ andmolecular He2+ ions. Major triplet and singlet
product He2* states were the lowest observed
c3g+ and C1g+ states, which occupied 39.9%
and 7.1% of ∑ 𝑘 𝑢 , respectively. The total
formation ratio of triplet/singlet states was 5.1.
No vibrational excitation was observed for all
thirty-four states. The rotational distributionsof f 3u+(𝑣 =0), e3g(𝑣 =0), and d3u+(𝑣 =0) wereexpressed by double Boltzmann rotationaltemperatures, whereas that of c3g+(𝑣 =0) was
representedbyasingleBoltzmanntemperature. On the basis of observed
rovibrational distributions, most of the total
excess energy (95.2–98.5%) is deposited into
the relative translational energy of products.
The observed vibrational and rotational
distributions were lower than those ofstatistical prior ones calculated assuming along lived [He2*–e-] intermediate. It wastherefore concluded that He2* molecules are
not formed via long lived [He2*–e-]intermediate, where excess energies are
statistically distributed to all degrees of
freedom.

Formation of He2* from the He2+/2e- collisional radiative recombination in the He flowing afterglow

10

Acknowledgments

1.5-ln(P(N')/Po(N'))1.0(a) f 3Σu+(v’=0)

The authors thank Prof. Kenji Furuya for his
helpful comments and the Mitsubishi
foundation (1996) for financial support.

0.5
0.0

References

-0.5

1)

-1.0

2)

-1.5
0.00

0.02

0.04

0.06

0.08

0.10

g(N')

4)

2.5
2.0

3)

5)
6)

(b) e3Πg(v’=0)-ln(P(N')/Po(N'))

1.5

7)

1.0
0.5

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0.0

9)

-0.5

10)

-1.0
-1.5
0.00

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0.15

g(N')

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3.0

(c) d3Σu+(v’=0)

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2.0
-ln(P(N')/Po(N'))

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1.0

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0.0

-1.0

-2.0
0.00

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0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

18)

g(N')

19)

1.5

(d) c 3Σg+(v’=0)

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0.03

0.04

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g(N')

Fig. 8. Rotational surprisal plots for theHe2(f:𝑣 =0), He2(e:𝑣 =0), He2(d:𝑣 =0), andHe2(c: 𝑣 =0) states in the He2+/2e- CRR
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