J. Plasma Fusion Res. SERIES, Vol.7 (2006) 54-58

On the Anomalous Characteristics in the P and R Branches
in a Hydrogen Fulcher Band
KADO Shinichiro, YAMASAKI Daisuke1 , XIAO Bingjia2 , IIDA Yohei1 ,OKAMOTO Atsushi∗ , KAJITA Shin1∗∗ , SHIKAMA Taiichi1 ,OISHI Tetsutaro1 and TANAKA Satoru1
High Temperature Plasma Center, The University of Tokyo, Chiba 227-8568, Japan
1 Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
2 Chinese Academy of Sciences, Hefei, Anhui 230031, China（Received: 4 October 2004 / Accepted: 24 May 2005）

Abstract
Anomalous characteristics in the P and R branches in hydrogen Fulcher-α emissions were investigatedwith respect to rotational temperature and population in the excited electronic state (upper-Fulcher state).
The ro-vibrational population distribution of the ground electronic state was deduced by applying the coronalequilibrium to the Q branch, and then the population for the P and R branches was predicted. The anomalies
in P and R branches can be found in the rotational temperature and the branching ratio between the branches.
Our results suggest that the sum of the emission from P and R branches seems to agree with that predictedbased on the Q branch emission.
Keywords:
Fulcher band, intensity anomaly, hollow cathode discharge, rotational structure, vibrational temperature,
rotational temperature
1．Introduction
Fulcher-α band (d 3 Π−u −aΣ+g ) spectra are widely used
to measure ro-vibrational distribution of the hydrogenmolecules in divertor plasmas in fusion devices [1], divertor plasma simulators [2], and laboratory plasmas[3], including negative ion sources [4]. Because it has
been known, from the early stage of molecular spectroscopy, that the P and R branches, which originate
in the d 3 Π+u sublevel, exhibit anomalous intensity compared to the Q branch, which originates in d 3 Π−u [5],usually only the Q branch has been used to determine
the ro-vibrational distribution in the ground electronic
state. The reason for this phenomenon has often beenreferred to as the perturbation between d 3 Π+u and e3 Σ+u[6,7]. However, if the characteristic can be clarified
either theoretically or empirically, data points of theFulcher band could be increased, and then a better fit
to obtain the ro-vibrational structure of the ground electronic state could be provided.
In this study, we investigated the P, Q, and R
branches in the diagonal Fulcher-α transition using thehollow cathode glow discharge of 230 V − 70 mA.

2．Principle of the diagnostics
We have developed an analysis scheme in which theground state (X 1 Σ+g ) populations for both vibrational (v)and rotational (J) distributions are assumed to be deX, rescribed by the Boltzmann temperatures T vXib and T rot
spectively. The electron impact excitation rate from the X(v, J) state to the d(v , J  ) state, RdXvv JJ , and the spontaneous emission coeﬃcient from the d(v , J  ) state to the a(v , J  ) state, AdavvJJ  , are used in a coronal model to
X. Then, the measured intensity
determine T vXib and T rot
ratio of the Fulcher band spectra is fitted to the function
[2],
hc
 
IadvvJJ  = dv J 
λav J 
× ne

 

AdavvJJ  d v J 
Aav J v ,J 





v,J

 

J
RdXvv JJ Cv (2J + 1)gas

F X (J, v) G X (v) − G X (0)
exp −
−
X
kT rotkT vXib

(1)

,

 

where λdavvJJ  is the wavelength of the measured line;
F X (J, v) and G X (v) are the rotational and vibrational energies in the X 1 Σ+g state, respectively; h and c are theCorresponding author’s e-mail: kado@q.t.u-tokyo.ac.jp
Present aﬃliation: ∗ Tohoku University, ∗∗ Nagoya University

C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research54

On the Anomalous Characteristics in the P and R Branches in a Hydrogen Fulcher Bandln (Ndv'J' /(2J'+1)gas ) [arb. unit]

Planck constant and the light velocity, respectively; andCv is the normalization constant with respect to J forJ
standseach v level. The degeneracy of nuclear spin gas
for asymmetric or symmetric, depending on thesymmetry of the level J. In H2 , levels of odd J in d 3 Π−u(Q branches) and X 1 Σ+g and of even J in d 3 Π+u (P andR branches) are asymmetric, so that the wave functionof the nuclear spin is symmetric to satisfy the featureJ= gas = 3, while the opposite caseof fermions, i.e., gasJsrequires gas = ga = 1.
 
The ro-vibronic structure of AaavvJJ  for the diagonalband can be well described by Franck-Condon factor
qv v and the Hönl-London factor for the P, Q, or R
branch, namely,

S QJ J  = (2J  + 1)/2,

(3)
 

The ro-vibrational structure in the excitation rate RdXvv JJ
is evaluated from the adiabatic approximation combined with Gryzinski’s semi-classical electron exchange cross-sections [8]:
 





g J

as
ve > a1J
RdXvv JJ = qdXvv < QGryzinski
0J δg J ,
v ←v
as

⎛ 
⎞2
   ⎜⎜⎜ K r
K ⎟⎟⎟
ΛK

⎜
⎟⎠
aΛK =
Q̄r (2K + 1) ⎝ 

Λ
−
Λ
−Λ
Λ


r
K
K
= 1,

vʼ=0

vʼ=1

vʼ=2

vʼ=3

vʼ=4
vʼ=5

16
15
14
13
12

115000120000
Energy [cm-1 ]125000T e , were 109 cm−3 and a few eV, respectively.
A 1-m Czerny-Turner scanning monochromatorequipped with a 2400 grooves/mm holographic grating
and a photo-multiplier tube detector (PMT: Hamamatsu
R928) was used for the optical measurements. The typical wavelength resolution in the present work was lessthan 0.03 nm at a 50-µm slit width.
We measured ne and T e at the center of the hollow
cathode using a Langmuir probe made of Molybdenum
and covered with a Pyrex glass rod. In the present study,
the working gas pressure was set to 110 Pa. The ne andT e were 3.9 × 109 cm−3 and 3.9 eV, respectively.
states popuMolecular Boltzmann plots of d 3 Π+/−ulation are shown in Fig. 1. Hönl-London factors for
P, Q, and R branches were used. Q branch spectra ofv ≤ 3 were normally used for the evaluation of T vXib
X
and T rot
. Obtained results were T vXib = 2301 ± 292 K
X
and T rot = 380.4 ± 21.7 K. Reconstructed distributions
states using these fitting results are alsoof the d3 Π+/−ushown (lines). The intensity of v ≥ 4 is weaker, since
the dissociation limit to H(1s) and H(2s) lies between
the v = 3 and 4 levels (open circles). The filled circles
in v ≥ 4 represent the corrected data by applying the
lifetime of the state involved in vacuum. One can seethat corrected values are slightly higher in this condition, which is discussed later.
The so-called anomaly in the P and R branches cannot be compensated for by the lifetime [10]. As seen in
the figure, the anomaly in the P and R branches consistsin the diﬀerences in (1) the rotational temperature of thed3 Π+u state, and (2) the population.
For the purpose of investigating the characteristics
Xfrom theof the anomaly, we evaluated T vXib and T rot
3 +Q branch, predicted the populations of d Πu , and thencompared them with the P and R branch emission in(2)

Note that S QJ J  = S PJ J  + S RJ J  = (2J  + 1)/2.

17

Fig. 1 Molecular Boltzmann plot of the Fulcher band inhollow cathode discharge at 110 Pa. Solid lines
represent the reconstruction from the ﬁtting resultof Trot and Tvib in the ground electronic state using the data for v’ ≤ 3. Filled circles represent thecorrected data for the Q branch, taking the relative
lifetime into account.

S PJ J  = J  /2,
S RJ J  = (J  + 1)/2.

18

(4)

where Q̄r is the multi-polar components of the partial cross-section determined in Ref. [9], which, forr = 1 ∼ 4, are 0.76, 0.122, 0.1, and 0.014. Note that thedipolar component of a1J
0J , namely Q̄1 with r = 1, corresponds to the Hönl-London factor for the X to d transition. Kronecker’s delta represents the selection rule
for the symmetry of nuclear spin. The spin multiplicity
is so small that K can be replaced by J in this case.
For the purpose of comparing the accuracy of the fitting, we executed a Boltzmann plot of the d 3 Π−u or d3 Π+ustate. Although the rotational distribution of the excited
vibronic state is not necessarily in thermal equilibrium,d v
, whichlower rotational levels are well described by T rot

is dependent on v .
3．Experimental results
Plasma was generated in a hollow cathode dischargechamber [10]. The cathode region was 19 mm in diameter. The discharge current and voltage were 70 mA
and 230 − 270 V, respectively. The operating range ofthe gas pressure was 10 − 300 Pa in our present apparatus. The typical electron density, ne , and temperature,55

On the Anomalous Characteristics in the P and R Branches in a Hydrogen Fulcher Banddv
terms of T rot
and the population.

700(Te=5eV, v'=0)

+

-

600

4．Discussion

500

Trotd [K]4.1 Rotational temperature
dv
In Fig. 2, T rot, deduced from the Boltzmann plot
3 +/−method for d Πu states, is shown as a function ofv (markers) together with the calculated values fromX
and T vXib to the Q branch spectra (lines).
the fitted T rot
The data in Fig. 1 were used. There are considerable
discrepancies between branches. In particular, the adiabatic approximation requires that P and R branchesfrom the same Λ-sublevel provide equal values.
X,
There have been experiments to determine T rot
which can be interpreted as the gas temperature in ahigh-temperature regime from the obtained rotationaltemperature in the d 3 Π−u state for a specific v level,d v
. The most commonly used formula for the Q
T rot
branch was proposed by Lavrov [12], as follows:
d
X
(v )
(v = 0) T rot
T rot
∼
,
BvX=0
Bdv

100
0
0

Trotd [K]

500

P br.
R br.

300

100

+
Π

-

Π

200

0

1

2

Vibronic level (v')

3

0

1

2

Vibronic level (v')

Simple Formula
200

400

600

800

TrotX [K]1000

in Fig. 3.
Open diamonds represent the result using r = 1 only.
This calculation agrees with the simple formula. Bytaking the higher multi-polar components into account,however, this conservation breaks. For example, J  = 1
in d3 Π−u (ortho-H2 ) is populated by the odd J levels in
X 1 Σ+g , while J  = 1 in d 3 Π+u (para-H2 ) is populated byX
the even J levels. Consequently, the relation in low T rot
becomes non-linear. The large error-bars in the lowerX
are due to the fact that the excited state distribution
T rot
cannot be described by the Boltzmann temperature. In
the same manner, the multi-polar components make thed3 Π−u state rotational distribution slightly depart from
Boltzmann. This might be the dominant reason for thediﬀerence between the Boltzmann plot and the calculated values seen in Fig. 2. Thus, the above simple
evaluation cannot apply. In other words, it is important
to consider the multi-polar components in determiningX
and to use curves such as those in Fig. 3 obtained
T rotfrom the model, or more desirably, to determine theXto a whole Fulcher band structure based
best-fitted T rot
on the model provided here in eq. (1).
d vdoes not depend on T e and has only
Deduced T rot
a slight dependence on T vXib (ignorable). It should be
noted, however, that by considering the multi-polard v
in d3 Π−u and d3 Π+u become slightly
components, T rot
d v
diﬀerent to each other1 . However, the predicted T rot
in d3 Π+u cannot reproduce the experimental data. This
result indicates that the adiabatic approximation cannot
be applied in the analysis of the rotational structure of
P and R branches.

(5)

400

Tvib = 8000 K (+ )Tvib = 2000 K (+ )Tvib = 8000 K (- )Tvib = 2000K (- )r=1 only ( =HL )dX
X(B v'=0 / B v=0 )T rot

Fig. 3 Relation between (X, v = 0) and (d, v’ = 0) state rotational temperatures. Results based on the simple linear formula [12] and those based on the
dipolar component of the excitation rate are added
as a comparison.

(b) PR branch (110Pa)(a) Q branch (110Pa)

300
200

where BvX=0 and Bdv denote the rotational constants,
which are 59.3 and 29.6 (v = 0) cm−1 , respectively.
dX(v = 0) ∼ 2T rot(v = 0).
Hence T rot
However, our model based on eq. (4) gives diﬀerX
regime, as shownent values, especially in a low T rot
in Fig. 3. The diﬀerence comes from the multi-polar
components of Q̄r . The dipole component, r = 1, provides the Hönl-London factor. Due to the selection rulewith respect to the symmetry of the nuclear spin, rotational quanta do not change in either the excitation tod3 Π−u or in the radiative decay to a3 Σ+g for the Q branch.
Consequently, the Boltzmann distribution in the rotational population conserves between the vibronic states,
which yields eq. (5), shown in the line without markers
600

400

3

Fig. 2 Rotational temperature in the (a) d 3 Πu – and (b)d 3 Πu + state as a function of the vibronic level v .
Lines without markers are the reconstructed values from the ﬁtting result of Trot and Tvib in the
ground electronic state.

dv of d 3 Π+ , which was
5 (b) in Ref. [10] had an error in T rotucorrected here.

1 Fig.

56

On the Anomalous Characteristics in the P and R Branches in a Hydrogen Fulcher Bandd

N fragment

1.0
0.8
0.6
0.4
0.2

(a)
Π

Tvib= 0 K
Tvib= 2000 K

+

-

Π

0.0
0
2.5

200

400

600

TrotX [K](b) Q branch (110Pa)

800

1000

(c) P,R branch (110Pa)
P br.

2.0

N(d,v') / N(d,0,Q br.)relative to that for v ≤ 3, over-estimation occurred forthe corrected vibronic population in v = 4 and 5 in the
Q branch. This correction reproduces the population
without taking the predissociation process into consideration. Collisions to the neutral could shorten the lifetime for v ≤ 3. In fact, this correction scheme shows
better results in the case of 15 Pa [10]. This should be
confirmed in a future study.
There is an obvious discrepancy between the casesof the P and R branches. However, the average ofthese branches coincides with the reproduced population. This suggests that the anomaly consists in the
branching ratio of the P and R branches. In this respect,intensity perturbation caused by the adjacent electronic
state is not obvious, though the rotational temperature
exhibits an anomaly. This tendency was also found in
the diﬀerent discharge condition of 15 Pa and the reanalyzed historical data of Ginsburg [5].

from X fit
1.5

average
R br.

Lifetime
1.0

from X fit

A coef.

0.5

0.0
0

1

2

3

4

Vibronic level (v')

5

0

1

2

3

4

5

Vibronic level (v')

5．Summary
In Fulcher-α band spectra, populations of the vibronic state v ≤ 4 in the Q branch can be corrected
to some extent by taking into account the lifetime of thestates. The 110 Pa case in comparison with our previous
result at 15 Pa indicates that the collisional de-excitationof the d3 Π−u might be taken into account. The ratio of P
and R branch intensities is far from that expected fromthe Hönl-London formulae. The anomaly is also foundin the fact that the rotational temperatures in d 3 Π+u deduced from the P and R branches are diﬀerent from eachother. However, our initial results show that the averageof the upper-Fulcher populations in the vibronic state
deduced from the P and R branches coincides with the
population predicted from the Q branch by taking into
consideration the diﬀerence in the selection rule of excitation from the ground electronic state based on the
adiabatic approximation.

Fig. 4 (a) Calculated portion of the population density ofd 3 Πu – and d 3 Πu + states as a function of rotationaltemperature in the X state for the vibrational temperature of 0 and 2000 K. (b) Population summed
over the rotational structure of the vibronic statesnormalized to that of v’ = 0 of the Q branch. (c)Same plot as (b) for the P and R branches.
4.2 Population density of the Λ-type doublet states

According to the selection rule, the lowest rotational
level in X 1 Σg (para-H2 ) can be excited mainly to J  = 1
in d3 Π+u . Consequently, d 3 Π+u has more population thand3 Π−u in low-density coronal equilibrium plasmas wherecollisional redistribution of the Λ-doublet state is negligible. A collisional-radiative equilibrium including theother electronic states may tend to distribute the population proportional to the statistical weight again. Therefore it is especially important to consider the unbalanceof the excitation fluxes when investigating the intensity
ratio from the Λ-doublet in low-density plasmas.
The calculated ratio of populations in the d 3 Π−u to3 +Xis shown in Fig. 4(a).
d Πu states as a function of T rot
The ratio does not depend on T e , while weak dependence on T vXib can be seen. The diﬀerence between theXandΛ-doublet states is more apparent in the low T rot
X
T vib regime.
statesIn Fig. 4 (b) and 4 (c), the population in d 3 Π+/−udeduced from the experimental data for 110 Pa areshown together with the reproduced population fromX
and T vXib . The population for each v level
fitted T rot
d v
, and
was evaluated by the Boltzmann plot to obtain T rot
then the resultant rotational population was summed up
to J  = 20. Because we applied lifetimes in vacuum

Acknowledgements
This work was supported in part by NIFS cllaborative research program (NIFS04KDAB009).

[1]
[2]
[3]
[4]
[5]
57

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58