The Astrophysical Journal Supplement Series, 177:388Y407, 2008 July
# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.
THE ELECTRON-EXCITED MID-ULTRAVIOLET TO NEAR-INFRARED SPECTRUM OF H2:
CROSS SECTIONS AND TRANSITION PROBABILITIES
Alejandro Aguilar,1 Joseph M. Ajello, Rao S. Mangina, and Geoffrey K. James
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109
and
Hervé Abgrall and Evelyne Roueff
LUTH & UMR 8102 du CNRS, Observatoire de Paris, 92195 Meudon Cedex, France
Received 2007 March 29; accepted 2008 February 12
ABSTRACT
At high resolving power (k/k 10; 000), we have measured the electron impactYinduced fluorescence spectrum
of H2 from 3300Y10000 8 at 20 and 100 eV. We have analyzed the visibleYopticalYnear-IR ( VOIR) emission spectrum under optically thin, single-scattering experimental conditions. The high-resolution spectrum contains transi1 þ
tions from the gerade Rydberg series of singlet states to the ungerade series of Rydberg states (EF 1 þ
g , GK g ,
þ
1
01 þ
1
,
C
,
B
,
D
)
and
the
Rydberg
series
of
triplet
states
dominated
by
H H̄ 1 g , I 1 g , J 1 g : : : ! B 1 þ
u
u
u
u
band
systems.
A
model
VOIR
spectrum
of
H
from
7500
to
10000
8,
based
on
newly
d 3 u , k 3 u , j 3 g ! a 3 þ
g
2
calculated transition probabilities and line positions including rovibrational coupling for singlet gerade states, is in
excellent agreement with the observed intensities. The rotational line emission cross sections for direct excitation of
the singlet states at 100 eV and triplet states at 20 eV were measured. The absolute cross section values for excitation
to the singlet gerade states at 100 eV was measured to be (4:58 1:37) ; 1018 cm2; the excitation cross section to
the triplet states at 20 eV was found to be (1:38 0:41) ; 1018 cm2. The singlet gerade emission cross sections are
the cascading cross sections to the UV spectrum (including the Lyman and Werner band systems) and the triplet state
3 þ
emission cross sections are the cascade dissociation cross sections of the H2 (a 3 þ
g Yb u ) continuum for the
production of fast hydrogen atoms. Electrons are also effective in exciting high vibrational levels belonging to ungerade symmetry (B, B 0 , C, D) which subsequently emit toward gerade symmetry states in the VOIR domain. Strong
emission from the ungerade-gerade transitions are observed from 7300 to 7400 8.
Subject headingg
s: infrared: general — line: identification — molecular data — molecular processes —
ultraviolet: general
data on the singlet gerade states of H2 (Tawara et al. 1990). The
cross sections for visibleYopticalYnear-infrared ( VOIR) emissions from H2 by electron impact are determined by measuring
1 þ
1 þ
1
the emission intensities of the EF 1 þ
g , GK g , H H̄ g , I g ,
1
1 þ
1
0 1 þ
1
J g : : : ! B u , C u , B u , D u . . . band systems.
These band systems referred to as the gerade-ungerade systems
emit in the VOIR and furnish a significant cascade contribution
to the Lyman and Werner band systems in the UV (Dziczek et al.
2000). Direct excitation of the ungerade states also occurs
0
00
through the two Rydberg series of H2: 1 þ
u 1s, np (B, B , B ;
1
0
00
and
1s,
np
(C,
D,
D
,
D
;
n
¼
2,
n ¼ 2, 3, 4)
X 1 þ
u
g
.
The
close
relationship
between
the
UV
and
3, 4, 5)
X 1 þ
g
VOIR transitions is shown in Figure 1.
The gerade series has been studied earlier in our laboratory
using time-resolved spectroscopy (TRS; Dziczek et al. 2000;
Liu et al. 2002). In the TRS experiment it was observed that
many VOIR rotational lines contribute to each UV rotation line
from the B and C states. In our earlier work we could only estimate the cascade cross section of each member of the gerade
series based on the models (Liu et al. 2002). It is more accurate to
directly measure the cascade cross section contributing to the
UV spectrum by measuring the rotational line emission cross
section of each member of the gerade series in the VOIR. The
total emission cross section of the VOIR transitions is significant. For example, at 20 eV we have estimated the cascade contributions to the UV emission spectrum to be comparable to the
direct excitation cross section of the Lyman (B ! X ) and Werner
bands (C ! X ) (Dziczek et al. 2000; Liu et al. 2002). The energy
dependence of the gerade states has been discussed by Liu et al.
1. INTRODUCTION
As pointed out by Dalgarno (1993, 1995) H2 has a unique and
extraordinary position in astronomy by virtue of its spectroscopic
signature in the UV of diverse energetic environments. The signature of the interaction of the interstellar medium with stars is
found in the intensity of the molecular features in both absorption (Moos et al. 2000; Shull et al. 2000) and emission (Herczeg
et al. 2002, 2004). The accuracy of the analysis of the observations was succinctly stated by Dalgarno et al. (1999), ‘‘the reliability of the models (in astronomy) are determined largely by
the accuracy of the cross sections and oscillator strengths adopted
for excitation, ionization, absorption, and emission processes.’’
The models of ultraviolet (UV) photons generated within molecular clouds from H2 by secondary electrons created by cosmic
rays, X-rays, and supernova remnants depend critically on the electron impact cross sections (Gredel et al.1987,1989; Dalgarno et al.
1999). Verification of the importance of both the fluorescence
excitation and collisional excitation processes has occurred with
the analysis of Hubble Space Telescope (HST ), Far Ultraviolet
Spectroscopic Explorer (FUSE ), and International Ultraviolet
Explorer (IUE ) observations of Herbig-Haro (HH ) objects, T
Tauri stars, and reflection nebulae (Raymond et al.1997a,1997b;
Herczeg et al. 2004).
The most recent review article evaluating the cross sections on
excitation of molecular hydrogen show a dearth of experimental
1
Permanent address: Lawrence Berkeley National Laboratory, Advanced
Light Source, Berkeley, CA 94720.
388
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
Fig. 1.— Partial energy-level diagram for H2 showing the energy regions for
the VOIR and UV transitions. The double minima states are represented by the
dotted lines.
(2003). Liu et al. have produced an effective excitation function
X 1 þ
for the EF; GK; : : : 1 þ
g
g band system by a measurement of dipole-forbidden excitation and cascade emission of the
Lyman system.
The complete single scattering VOIR spectrum of the geradeungerade band systems has not been studied in the laboratory. In
the past we have studied at low resolution (FWHM 17 8) the
electron excited spectrum of H2 from 1000 to 5000 8 at 14, 19 and
100 eV (James et al. 1998). The measured spectra were found to
be dramatically different at each electron-impact energy. At 14 eV
only the H2(aYb) continuum appears in the spectrum above 2000 8.
At 19 eV both the continuum and the transitions between the
bound singlet and triplet states are very strong and at 100 eVonly
the singlet gerade-ungerade states are strong along with the H i
lines from dissociative excitation. These spectra are important in
exploring and interpretation of Jupiter and Saturn aurora. For example, these spectra lead to the first identification of the H2(aYb)
emission continuum in astronomy from the Jupiter aurora (Pryor
et al. 1998). Future higher resolution opportunities by the Space
Telescope Imaging Spectrograph (STIS) may be possible. If the
upcoming servicing mission in 2008 is successful, HST will have
the capability to obtain spectra from the dark side of Jupiter in the
2000Y11000 8 range. During this servicing mission, an attempt
will be made by NASA to repair the STIS, which can normally
obtain spectra matching the resolution and wavelength range reported here.
The most extensive laboratory investigation of the VOIR system
was the subject of the tome, edited by Crosswhite (1972), detailing
the lifetime work of G. Dieke and it was also the subject of several papers (Dieke 1958; Dieke & Cunningham 1965). Dieke determined at very high resolution the spectroscopic constants of
389
most of the electronic states of H2, and its isotopes and resolved
nearly every rotational line in the VOIR spectrum (over 100,000
different lines; Crosswhite 1972). In detail, the rotation-vibration
levels associated with the inner minimum compose the Dieke’s
2A state, which is more recently designated as 1s2s E-state.
The outer minimum is Dieke’s 2K state, which has been established to be the doubly-excited (2p)2 F-state, perturbing the
E-state. Davidson (1961) has shown that these are not two separated states, but a unique state with a double minimum adiabatic
þ
potential curve. This single state is referred to as the EF 1 g
1 þ
state. The members of the singlet-gerade states, such as EF g ,
1 þ
GK 1 þ
g , and H H̄ g , are characterized by double-minimum
potential curves that arise from avoided crossings of the 1s, ns
or 1s, nd (singly-excited) with the 2p, np (doubly-excited)
configuration. Ab initio calculations have shown the singletgerade series, including the ground state, to be strongly coupled
by nonadiabatic interactions (Yu & Dressler1994). A more complete bibliography of ab initio calculations of the singlet-gerade
series was given in our recent publications (Liu et al. 2002, 2003).
The emission cross sections of the gerade-ungerade band systems
have been a few in number. About three decades ago, the University
of Arkansas group (Watson & Anderson1977; Anderson et al.1977;
Day et al. 1979) measured excitation functions for about a dozen
rotational features of the EF, GK, and H H̄ states (out of the thousand
or more observed in this study).
Optical excitation functions of the triplet states have also been
measured by few experimenters in the VOIR. Dieke (Crosswhite
1972) in his discharge emission experiments has shown the presence of many triplet band systems with the strongest and most
extensive to be the Fulcher- band system (3p d 3 u !
2s a 3 þ
g ). The emission cross sections of the Fulcher- diagonal bands (v ¼ 0) have been measured by Möhlmann & de
Heer (1976). Ajello & Shemansky (1993) have studied the
þ
electron excitation function of the H2 (a 3 g Yb 3 þ
u ) continuum in the UV. These processes leading to triplet emissions arise
first from singlet-triplet excitation. The excitation process occurs
by electron-exchange, which is characterized by a fast rise and
decrease in the emission cross section within a few volts of the
threshold. Since our study covers a wide range of the VOIR
spectrum at both 20 and 100 eV, it is possible to identify and
measure the emission cross sections of the other triplet states
observed by Dieke (Crosswhite1972), which are referenced with
more modern nomenclature by Huber & Herzberg(1979) such as
the Beta bands (4p k 3 u ! 2s a 3 þ
g ). Tawara et al. (1990)
have reviewed the excitation cross sections of the 2s a 3 þ
g,
3
3 þ
,
2p
c
,
and
3p
e
states
by
electron
energy2p b 3 þ
u
u
u
loss experimental techniques. The emission transition of the e 3 þ
u
triplet state, e ! a, has been observed by Dieke (1958; Crosswhite
1972) as well as by Dieke in a couple of earlier publications as
referenced in Huber & Herzberg(1979) while the c-state, whose
v 0 ¼ 0 level lies below the a-state, is metastable with respect to
the transitions to the b- and X-state. Thus, there are two Rydberg
series terminating in the two bound triplet states, a and c.
The triplet states are the major source of the dissociation of
hydrogen molecule by electron impact. The b 3 þ
u state, which
is the lowest lying repulsive state (1s)(2p), can be excited by
state or via cascade
direct excitation from the ground X 1 þ
g
3 þ
Yb
continuum.
All
of
the dipole-allowed
from the a 3 þ
g
u
triplet transitions, cited above, making transitions to the a-state
also contribute to the dissociation. The d 3 u state is also pre3 þ
dissociated by the e 3 þ
u state and the e u state is directly
0
dissociated above v ¼ 6 ( Missakian & Zorn 1971; Sharp 1971).
Calculations of the dissociation cross sections to the b-state and
excitation cross sections to the a-, c-, d-, and e-states have been
390
AGUILAR ET AL.
Vol. 177
Fig. 2.— Schematic diagram of the VOIR experimental apparatus showing the collision chamber, the interaction region, a schematic calibration curve, and the CzerneyTurner optics of the grating turret and two mirrors of the spectrometer.
performed by Chung et al. (1975) and by Chung & Lin (1978).
Chung et al. claim that the cascade and direct excitation of the
b-state are of equal importance at the peak cross section energy.
The review of Tawara et al. (1990) indicates incomplete data in
this regard and does not seem to support this assertion.
1.1. Plan for Paper
We have recently reported the analysis of high-resolution UV
emission spectra following electron impact excitation of H2 ( Liu
et al. 1995 2000; Jonin et al. 2000), HD (Ajello et al. 2005a), and
D2 (Abgrall et al. 1999). The discrete and continuum line transition probabilities were calculated by Abgrall et al. (1993a,1993b,
1993c, 1994, 1997, 2000) for H2 and Abgrall et al. (1999) for D2.
In this paper we report a combined experimental and theoretical
investigation of the VOIR emission spectrum of H2. The theoretical study involves detailed calculations of emission transition
probabilities of individual rotational lines of the nine coupled
1 þ
1 þ
1 þ
1 þ
1
1
EF 1 þ
g , GK g , H H̄ g , O g , P g , I g , R g ,
J 1 g , and S 1 g states. The experimental data presents highresolution (0.7 8 FWHM) electron impactYinduced UVemission
spectra from 3300 to 10000 8, and the present model illustrates
the study of H2 from 7500 to 12000 8. Many of our previous
laboratory studies on molecular hydrogen and other astrophysical species have been reviewed by Ajello et al. (2002). Three examples of the application of the laboratory cross sections of H2
to the Jupiter UV aurora can be found in Pryor et al. (1998) and
Ajello et al. (2001, 2005b).
The organization of the paper is as follows. Section 2 describes briefly the experimental apparatus used to measure the
VOIR emission from H2 produced by electron impact and reports the swarm gas measurement of H2 7526.67 8 rotational
line EF(6; 0)P2 cross section relative to the absolute standard H
6562.8 8 cross section at 100 eV. The coupled Schrödinger equations formalism used in the ab initio calculations of continuum and discrete transition probabilities for the first nine coupled
1 þ
1 þ
1 þ
1 þ
1
states EF 1 þ
g , GK g , H H̄ g , O g , P g , I g ,
1
1
1
R g , J g , and S g states is described in x 3. In the model
section (x 4), we provide a concise description of the theoretical
model used to analyze the observed electron impact fluorescence
intensities. In x 5 rotational cross sections are provided at 100 eV
for the singlet states and at 20 eV for the triplet states. Section 6
provides summary and discussion of the results.
2. EXPERIMENTAL APPARATUS
The experimental apparatus has been described in a previous
publication (Ajello et al. 2008). A schematic drawing of the apparatus is shown in Figure 2. It consists of a visible-optical nearinfrared (VOIR) imaging spectrometer operating in air in tandem
with an electron impact collision chamber under vacuum. The spectrometer and collision chamber are vacuum isolated by a quartz window. A magnetically collimated beam of electrons (10Y1000 eV)
with an energy resolution of 1 eV is crossed with a beam of target
gas formed by a capillary array under optically thin conditions at
a background pressure of 1 ; 104 torr. The electron impactY
induced fluorescence spectrum is observed at 90 emission angle using a Spectra Pro 0.5 m spectrometer capable of having a
resolving power of 10,000 at 5000 8 with 50 m entrance slit
size. The spectrometer is equipped with a Princeton Instruments liquid nitrogenYcooled CCD detector array 1340 pixels
wide and 400 pixels tall (each pixel area is 20 m2). The wavelength response of the detector is calibrated using standard
deuterium and tungsten sources from 2000 to 11000 8.
No. 1, 2008
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
391
Fig. 3.— Instrumental slit function as measured by placing the central wavelength on one of the fine-structure lines of the S i 9213 8 multiplet 4s 5 SY4p 5 P
using grating 3 produced from electron excitation of SO2 (Ajello et al. 2008). The
FWHM is given for various entrance slit settings from 10 to 500 m. The resolving power k/k 2 ; 104 is determined for 10 m slit width.
Fig. 4.— Instrument performance in measuring the atomic O fine-structure
features at 7771.94 8 ( 5 S2o Y5 P1 ), 7774.17 8 ( 5 S2o Y5 P2 ), and 7775.39 8 ( 5 S2o Y5 P3 )
for grating 2 produced from electron excitation of SO2 (Ajello et al. 2008). The
instrument is capable of resolving the three fine-structure lines for 10 Y50 m
slit settings. The resolving power k/k 2 ; 104 is determined for 10 m slit
width.
There are three gratings inside the spectrometer: (1) 1800 groves
mm1: 2500 8 blaze wavelength (16.67 8 mm1); (2) 1800 groves
mm1: 5000 8 blaze wavelength (16.67 8 mm1); and (3) 1200
groves mm1: 7500 8 blaze wavelength (11.11 8 mm1). Ordersorting filters are used for gratings 2 and 3 with low-wavelength
cutoffs of 3200 8 and 5900 8, respectively. The three overlapping wavelength responses for the spectrometer are 2000Y3800 8
using grating 1, 3600Y8000 8 using grating 2, and 6000Y11000 8
using grating 3. The full range of a grating is covered during an
experiment by rotating the grating in predetermined small steps.
At a given grating position the wavelength range captured by the
detector array varies from 200 8 to 250 8 depending on the grating and slit sizes used.
Spurious signals in the CCD array arising from cosmic-ray
hits are common. These hits cause an increase in charge in one
or more pixels which are indistinguishable from the arrival of
signal photons. An active detector exposed for a few minutes
might have nearly two dozen cosmic-ray hits. Since the location
of the cosmic-ray hits within the chip is random in space and
time, a commonly used algorithm to remove them without altering the photosignal consists in collecting several frames under
the same conditions and removing the counts due to the cosmicray hits by comparing corresponding pixels in the different images. The algorithm rejects those with aberrantly large values. In
these experiments 10 frames of 3 minutes each were collected
for each grating position. The wavelength overlap for consecutive
grating positions was typically 40 8.
We have measured the instrument resolution with grating 3
by studying one of the fine-structure lines of the S i multiplet
4s5 SY 4p5 P emission at 9213 8 from electron-impact dissociative excitation of SO2. In Figure 3 we show the instrument performance for several slit widths, including the 50 m slit setting
used for this study (0.7 8 FWHM ). Because of the larger grating
constant (smaller groove density) for grating 3, the FWHM for
this grating is slightly larger than for grating 2 for the same slit
width. The instrument is found to be capable of achieving high
resolving power (k /k > 10;000) in the VOIR for both gratings. To show our instrument performance, using grating 2
we have also measured the fine-structure emission lines at
7771.94 8 (5 S2o Y5 P1 ), 7774.17 8 (5 S2o Y5 P2 ), and 7775.39 8
(5 S2o Y5 P3 ) corresponding to the O i multiplet 5 S o Y5 P, excited by
dissociative excitation from SO2. The fine structure for this
multiplet is shown in Figure 4. We found excellent agreement
between the present relative intensities of O i fine-structure
features at 7771.94 8 (5 S2o Y5 P1 ), 7774.17 8 (5 S2o Y5 P2 ), and
7775.39 8 (5 S2o Y5 P3 ) with their values in the National Institute
of Standards and Technology ( NIST) database. It is clearly evident from the figure that using a slit width of 50 m or below
the instrument could completely resolve the three fine-structure
lines up to their base. At smaller slit settings the resolving
power of the instrument is somewhat similar for both gratings
2 and 3.
The apparatus was calibrated with deuterium and tungsten
blackbody spectral irradiance lamps over the spectral range of
2000 Y12000 8 spanned by the three gratings. We show in Figure 5 the normalized inverse sensitivities (S1) for gratings 2
and 3. The inverse sensitivity is defined as the input standard
blackbody spectrum irradiance divided by the measured output spectrum of the spectrometer.
The wavelength scale for our spectra is based on the vacuum
wavelength scale of Dieke (Crosswhite 1972) for the strong singlet and triplet H2 rotational lines. The laboratory wavelengths,
measured in air in this work, were corrected by a linear fit of the
laboratory wavelengths to the Dieke benchmark wavelengths,
which were calibrated against the iron spectrum (Crosswhite
1972). Dieke’s reported wavelength accuracy in the VOIR was
a few hundredths of a wavenumber (0:05 cm1 10 m8). The
uncertainty to our observed wavelengths is less than 500 m8.
The absolute calibration of the emission spectra for H2 at 20
and 100 eVelectron impact energies has been based on using the
known emission cross sections of H at 20 and 100 eV electron
impact energies in a swarm gas experiment. Karolis & Harting
(1978) measured the electron impact dissociative emission cross
section of H2 from threshold (15.6 eV ) to 100 eV. These results
were recently reviewed by Lavrov & Pipa (2002) and were found
to be in excellent agreement (10%) with the data of Karolis &
392
AGUILAR ET AL.
absolute scale. It would not be simple to put this information
conveniently in a long table. However, this technique of depicting cross sections on an absolute scale works well for a highresolution spectrum, where nearly all the rotational lines are
resolved and the line width is instrumental. In other words, the
peak height in a high-resolution experiment is proportional to
the feature area (i.e., the cross section). The values of the peaks,
therefore, represent the absolute cross sections. Similarly, at
20 eV the cross section for the 7526.67 8 rotational line was
measured to be 0.133 Mbarn (1:33 ; 1019 cm2). The line cross
section values at 20 and 100 eVare consistent with the cross section behavior deduced by Liu et al. (2003) for the BYX (0; 4)P3
Lyman H2 line.
The grating 2 and grating 3 spectra were obtained under similar conditions with approximately 100 A electron beam current and 1 ; 104 torr gas pressures. Karolis & Harting (1978)
compare their emission cross sections to the experimental results
of several other groups. We base the uncertainty of the absolute
calibration on the cited 20% variation of the cross sections for
the various experiments. Additional uncertainties of the absolute
cross section can be traced to the correction for the gas pressure
read by a Varian Bayard-Alpert UHV gauge tube of 10%, background offset to the singlet transitions at 100 eV from low-energy
secondary electrons of 10%, and the variation of electron beam
current and the stability of the gas pressure during the spectral
scan of 15%. The relative uncertainty of the instrument calibration is 10% over the range of each grating. The root-sum-square
uncertainty of the absolute cross sections is about 30%.
3. THEORETICAL CALCULATIONS
OF TRANSITION PROBABILITIES
3.1. General Formalism
Fig. 5.— Normalized inverse sensitivity of grating 2 (top) and 3 (bottom) as
determined from a blackbody spectral irradiance lamp with an emitting surface in
the shape of an integrating sphere.
Harting (1978). The measured cross sections of Karolis & Harting
(1978) for H at 20 and 100 eV were found to be 5:2 ; 1019 cm 2
and 9:3 ; 1019 cm2, respectively. A comparison of the integrated
intensity of the H2 7526.67 8 rotational line EFYB(6; 0)P2 relative
to H 6562.8 8 using grating 3 gives the emission cross section Q
for 7526.67 8 at 100 eV,
Q(100 eV) ¼ 1:109 ; 1019 cm2
ð1Þ
and with grating 2
We have calculated the Einstein emission coefficients between
the upper Rydberg rovibrational levels of g-symmetry (EF 1 þ
g,
1 þ
1 þ
1 þ
1
1
1
1
GK 1 þ
g , H H̄ g , P g , O g , I g , R g , J g , S g )
0 1 þ
and the rovibrational levels of u-symmetry (B 1 þ
u , B u ,
1
1
C u , D u ) of H2. As the corresponding electronic potentials
become relatively close both for the upper and lower levels, it is
not possible to assign a single electronic state to a specific upper
or lower rovibrational level. We have then performed two separate coupled treatments for g- and u-symmetry electronic levels.
The treatment is an extension of our previous calculations on the
Lyman and Werner band systems (Abgrall et al. 2000) and UV
transitions arising from the singlet gerade states (Liu et al. 2002),
where the couplings between levels of g-electronic symmetry
had been neglected. The total wave function SvJ is expanded on
the different Born-Oppenheimer ( BO) wave functions which are
coupled via nonadiabatic couplings,
SvJ ¼
X
TJ fST vJ :
ð3Þ
T
Q(100 eV ) ¼ 1:159 ; 1019 cm2
ð2Þ
(which is 5% agreement with that of grating 3).
We use the average value of the above two cross sections, i.e.,
0.113 Mbarn (1 Mbarn ¼ 1 ; 1018 cm2), for normalization of
all H2 data measured with grating 2 and grating 3 at 100 eV.
Thus, grating 2 and grating 3 give nearly identical benchmark
100 eV cross section for 7526.67 8 emission. By using this cross
section the relative values of the remaining 1000 or more rotational lines in the observed H2 spectrum have been placed on the
Each TJ is the product of the electronic BO wave function (T
describes the electronic quantum numbers) and the radial nuclear
wave function describing the rotational motions (J stands for the
rotational quantum number). The nuclear wave function fSTvJ and
the energy levels EvJ are obtained from the eigenfunctions and
eigenvalues of the Schröedinger coupled equations whose diagonal terms are adiabatic potentials and off-diagonal terms are
rotational and radial electronic coupling matrix elements. The
formalism is described in detail by Senn et al. (1988), Quadrelli
et al. (1990a), and Yu & Dressler (1994).
TABLE 1
Calculated Transition Wavelength and Emission Probabilities
Greater than 2 ; 105 s1 between 7500 and 8000 8
from g Upper States toward u Lower States
Line
k
(8)
oYca
A
(s1)
eJYC 0Y0 Q3.........................
eEFYB 6Y0 P1 ......................
fJYC 0Y0 Q4.........................
eEFYC 29Y2 Q1....................
eEFYC 25Y1 R4 ....................
eEFYB 12Y3 P2 ....................
eJYC 3Y2 R3 .........................
eEFYC 21Y0 R4 ....................
eEFYC 28Y2 P1 ....................
eHHYC 2Y2 P6 .....................
fJYC 2Y2 R3 .........................
eEFYB 6Y0 P2 ......................
fJYC 0Y0 Q3.........................
eEFYC 29Y2 R2 ....................
eJYC 2Y2 R4 .........................
eGKYC 5Y2 P3 .....................
eEFYC 20Y0 R4 ....................
eJYC 0Y0 Q2.........................
eJYC 1Y1 R1 .........................
fJYC 1Y1 R1 .........................
eEFYB 6Y0 P3 ......................
fJYC 0Y0 Q2.........................
eEFYB 12Y3 P3 ....................
fJYB 1Y10 Q5 .......................
eJYC 0Y0 R3 .........................
eEFYC 24Y1 Q4....................
eGKYB 5Y10 P5....................
eGKYC 7Y2 Q4 .....................
eEFYC 30Y2 Q2....................
eEFYB 8Y1 R0 ......................
fJYC 2Y2 R2 .........................
eEFYB 8Y1 R1 ......................
eJYC 2Y2 R2 .........................
eJYC 1Y1 Q5.........................
eJYB 3Y12 R3 .......................
eEFYB 6Y0 P4 ......................
eEFYC 22Y0 P6 ....................
eEFYC 30Y2 Q5....................
eJYC 2Y1 Q4.........................
eEFYC 29Y2 P2 ....................
eEFYB 8Y1 R2 ......................
eGKYB 6Y12 P4....................
eIYC 0Y0 R2..........................
fJYC 1Y1 Q4 .........................
eEFYB 12Y3 P4 ....................
eJYC 1Y1 Q3.........................
fJYC 1Y1 Q5 .........................
eEFYC 29Y2 R3 ....................
eEFYB 8Y1 P1 ......................
eEFYC 26Y1 P6 ....................
eGKYC 5Y1 P5 .....................
fJYC 1Y1 Q3 .........................
fJYC 2Y2 R1 .........................
eJYC 2Y2 R1 .........................
eEFYB 22Y8 P6 ....................
eEFYB 8Y1 R3 ......................
eJYC 1Y1 Q2.........................
eEFYC 31Y2 P6 ....................
fJYC 1Y1 Q2 .........................
eJYC 3Y2 Q5.........................
eEFYB 10Y2 P1 ....................
eEFYC 29Y2 Q3....................
7506.888
7508.477
7510.090
7513.645
7518.080
7518.741
7520.725
7521.902
7523.148
7523.645
7524.183
7526.187
7528.238
7530.126
7531.731
7535.704
7538.698
7539.107
7539.846
7540.995
7543.583
7547.249
7549.107
7549.877
7550.560
7553.989
7555.695
7556.312
7557.659
7560.768
7561.860
7563.083
7563.987
7565.859
7566.220
7573.371
7573.744
7575.322
7576.486
7576.624
7577.595
7577.933
7578.352
7582.087
7585.717
7587.541
7589.816
7590.243
7591.561
7592.968
7595.697
7596.262
7600.436
7600.604
7603.516
7604.193
7605.216
7608.393
7609.065
7609.760
7616.917
7622.509
0.030
0.550
0.210
2.950
1.290
0.950
0.310
0.390
3.200
6.050
0.890
0.590
0.140
1.360
2.840
5.590
0.640
0.200
0.580
0.460
0.460
0.110
0.900
0.500
0.180
0.360
5.100
4.420
2.450
0.390
1.040
0.370
0.580
2.600
0.090
0.370
1.820
1.820
3.760
3.000
0.400
5.800
0.200
0.490
0.920
0.380
0.290
1.190
0.360
1.740
5.360
0.570
1.210
1.240
2.110
0.320
0.560
1.840
0.560
0.120
0.630
1.410
2.65E+07
3.78E+06
1.46E+06
1.45E+06
8.94E+05
3.26E+05
1.55E+07
2.54E+06
2.67E+06
2.06E+06
2.75E+07
2.51E+06
2.21E+06
6.93E+05
9.91E+06
2.12E+06
1.35E+06
2.58E+07
1.99E+07
3.73E+07
2.09E+06
3.32E+06
2.80E+05
3.31E+05
2.62E+06
6.50E+06
1.48E+06
5.71E+06
1.53E+06
6.89E+05
2.97E+07
7.20E+05
1.80E+07
1.55E+07
2.73E+06
1.43E+06
5.00E+05
1.90E+06
1.61E+07
1.46E+06
6.47E+05
5.53E+05
1.12E+06
1.18E+06
2.40E+05
2.45E+07
5.27E+05
1.06E+06
2.18E+06
7.59E+05
1.60E+06
1.81E+06
3.43E+07
1.81E+07
6.72E+05
5.11E+05
2.17E+07
3.91E+05
2.66E+06
1.64E+07
6.08E+05
4.35E+06
TABLE 1—Continued
Line
k
(8)
oYca
A
(s1)
eEFYB 8Y1 P2 ......................
eEFYB 6Y0 P5 ......................
eGKYC 6Y2 P4 .....................
eEFYB 8Y1 R4 ......................
eEFYB 10Y2 P2 ....................
eJYC 0Y0 P6 .........................
eJYC 3Y2 Q4.........................
eJYC 1Y0 P5 .........................
eEFYC 24Y1 Q1....................
eEFYC 30Y2 P3 ....................
fJYC 0Y0 P3 .........................
eJYC 0Y0 P4 .........................
fJYC 2Y2 Q3.........................
eEFYC 23Y1 R2 ....................
eJYB 2Y12 R2 .......................
fJYC 2Y2 Q5.........................
eJYC 2Y2 Q3.........................
eEFYC 23Y1 P1 ....................
fJYC 0Y0 P4 .........................
eJYC 2Y2 Q2.........................
fJYC 2Y2 Q4.........................
fJYC 2Y2 Q2.........................
eGKYC 5Y1 P6 .....................
eJYC 2Y2 Q5.........................
eEFYB 8Y1 P3 ......................
eGKYC 4Y1 R3 .....................
eGKYC 4Y1 R4 .....................
eEFYB 10Y2 P3 ....................
fJYC 0Y0 P5 .........................
eJYB 0Y8 P6 .........................
eEFYC 25Y1 Q5....................
eJYC 0Y0 Q4.........................
eIYC 0Y0 Q2 .........................
eEFYC 21Y0 Q5....................
eEFYC 29Y2 R4 ....................
fJYC 0Y0 P6 .........................
fIYC 0Y0 R2..........................
eEFYC 24Y1 Q2....................
fIYC 0Y0 R1..........................
fIYC 0Y0 R3..........................
fIYC 0Y0 R4..........................
fJYB 2Y12 Q3 .......................
eIYC 0Y0 Q1 .........................
eEFYC 20Y0 Q5....................
eEFYC 19Y0 Q1....................
fJYC 1Y1 P3 .........................
eEFYB 10Y2 P4 ....................
eJYC 1Y1 P3 .........................
eEFYB 6Y0 P6 ......................
eEFYC 24Y1 P2 ....................
eJYC 1Y1 P4 .........................
eGKYC 7Y2 P5 .....................
eJYB 2Y10 P5 .......................
fJYC 1Y1 P4 .........................
eEFYB 8Y1 P4 ......................
fIYC 0Y0 Q1 .........................
fJYC 1Y1 P5 .........................
eJYB 2Y12 P4 .......................
eEFYC 28Y2 Q1....................
eIYC 1Y1 R1..........................
eJYC 2Y2 P3 .........................
eEFYC 20Y0 P5 ....................
fJYC 1Y1 P6 .........................
fJYC 2Y2 P3 .........................
eEFYC 27Y2 P1 ....................
7624.403
7631.636
7633.268
7641.452
7643.191
7645.289
7646.102
7647.113
7647.961
7649.657
7649.781
7650.056
7653.961
7654.454
7654.489
7654.583
7657.208
7659.273
7662.465
7663.664
7663.704
7664.885
7668.236
7669.094
7669.183
7669.547
7674.155
7674.426
7674.686
7675.628
7675.929
7685.591
7686.313
7687.021
7687.075
7687.453
7692.604
7693.645
7694.350
7694.995
7700.482
7701.087
7701.497
7704.564
7705.846
7707.753
7708.905
7709.363
7716.472
7717.985
7724.376
7725.193
7725.247
7725.879
7726.237
7739.584
7742.665
7743.959
7753.320
7753.447
7758.277
7759.149
7759.324
7760.625
7762.969
0.420
0.200
5.980
0.340
0.710
0.210
0.290
0.170
2.010
2.420
0.150
0.130
1.150
1.860
0.400
...
0.590
1.790
0.050
1.220
0.800
1.230
3.070
2.830
0.350
1.390
1.540
0.640
0.210
0.490
1.380
0.280
0.050
0.430
1.110
0.250
0.210
1.870
0.090
0.120
0.030
0.940
0.100
0.680
1.170
0.560
0.800
0.580
0.140
1.950
0.410
4.740
3.890
0.660
0.380
0.160
0.470
0.580
1.270
0.760
1.340
1.220
0.460
1.250
1.320
1.41E+06
7.53E+05
1.93E+06
3.19E+05
3.82E+05
2.89E+05
1.98E+07
8.91E+05
6.87E+05
1.04E+06
2.95E+05
3.39E+05
1.24E+06
2.21E+05
3.22E+05
6.69E+05
1.85E+07
5.72E+05
2.88E+05
1.87E+07
8.34E+05
2.11E+06
2.48E+06
5.43E+06
1.17E+06
2.62E+05
2.35E+05
2.90E+05
2.49E+05
2.04E+05
2.40E+05
1.04E+06
1.72E+06
1.33E+06
2.14E+05
2.19E+05
7.16E+05
2.42E+05
8.63E+05
5.79E+05
4.94E+05
2.09E+05
1.91E+07
8.54E+05
8.05E+05
8.51E+05
2.01E+05
4.37E+05
4.36E+05
6.48E+05
3.67E+05
2.31E+06
3.47E+05
9.68E+05
9.74E+05
1.29E+07
9.05E+05
4.48E+05
5.14E+05
4.78E+05
1.25E+06
2.10E+05
8.22E+05
1.50E+06
5.90E+05
394
AGUILAR ET AL.
Vol. 177
TABLE 1—Continued
TABLE 1—Continued
Line
k
(8)
oYca
A
(s1)
eEFYC 29Y2 P4 ....................
eJYC 2Y1 P5 .........................
fIYC 0Y0 Q2 .........................
eEFYC 30Y2 P6 ....................
eIYC 0Y0 P2..........................
eJYC 1Y1 P6 .........................
eEFYC 19Y0 P2 ....................
fJYC 2Y2 P4 .........................
eEFYB 8Y1 P5 ......................
eIYC 0Y0 P3..........................
eEFYC 24Y1 P3 ....................
eEFYC 29Y2 Q2....................
eJYC 2Y2 P4 .........................
fIYC 0Y0 Q3 .........................
fJYC 2Y2 P5 .........................
fIYC 1Y1 R3..........................
fIYC 1Y1 R2..........................
fIYC 1Y1 R4..........................
fIYC 0Y0 P2..........................
fIYC 1Y1 R1..........................
eJYC 3Y2 P5 .........................
eEFYC 28Y2 P2 ....................
eIYC 1Y1 Q2 .........................
fJYC 2Y2 P6 .........................
eIYC 1Y1 Q1 .........................
eIYC 0Y0 P4..........................
eGKYC 4Y1 Q5 .....................
fIYC 0Y0 Q4 .........................
eEFYC 24Y1 R4 ....................
fIYC 1Y1 Q1 .........................
eJYC 0Y0 P5 .........................
eJYC 2Y2 P6 .........................
eEFYC 34Y3 Q1....................
eEFYB 8Y1 P6 ......................
eEFYC 33Y3 P1 ....................
fIYC 0Y0 P3..........................
eGKYC 1Y0 R1 .....................
fIYC 1Y1 Q2 .........................
eEFYB 33Y14 P1 ..................
eEFYC 29Y2 P5 ....................
eEFYC 25Y1 P6 ....................
eGKYC 1Y0 P1 .....................
eEFYC 23Y1 P4 ....................
eEFYC 21Y0 P6 ....................
eEFYC 29Y2 P3 ....................
eGKYC 1Y0 Q1 .....................
eIYC 1Y1 P2..........................
eEFYB 9Y2 R2 ......................
eEFYB 9Y2 R3 ......................
eEFYB 9Y2 R1 ......................
eEFYC 20Y0 P6 ....................
fIYC 1Y1 Q3 .........................
eEFYB 9Y2 R4 ......................
eGKYC 1Y0 R2 .....................
eGKYC 4Y2 Q2 .....................
eEFYB 9Y2 R0 ......................
eGKYC 4Y2 Q1 .....................
eGKYB 4Y10 P6....................
eEFYB 21Y8 P6 ....................
eIYC 1Y1 P3..........................
eEFYC 33Y3 Q1....................
fIYC 1Y1 P2..........................
eGKYC 5Y2 Q3 .....................
eEFYC 32Y3 P1 ....................
fIYC 2Y2 R4..........................
7765.736
7767.069
7769.073
7775.270
7776.358
7777.864
7780.793
7783.858
7795.490
7798.791
7800.245
7800.623
7801.755
7805.268
7805.969
7806.669
7807.961
7809.517
7812.323
7814.643
7819.067
7820.399
7822.590
7827.979
7831.461
7831.719
7838.698
7845.500
7850.977
7865.017
7872.323
7874.090
7874.605
7875.491
7875.584
7877.762
7887.555
7887.766
7893.183
7894.236
7894.236
7896.929
7897.303
7898.270
7898.669
7900.922
7904.901
7905.444
7907.870
7913.315
7916.791
7917.776
7924.119
7924.225
7925.683
7928.014
7928.303
7928.749
7930.655
7932.818
7935.852
7936.174
7936.678
7937.289
7939.098
1.590
3.710
0.100
1.910
0.090
2.760
1.180
1.160
0.350
0.120
1.920
1.230
0.480
0.150
0.940
0.040
0.050
0.160
0.210
0.130
0.180
1.300
0.810
...
1.010
0.070
1.650
0.230
1.460
0.040
0.060
2.970
...
0.300
1.710
0.140
1.520
0.040
1.890
1.450
1.450
1.570
1.970
0.310
1.170
1.590
0.930
0.980
0.800
0.960
0.580
0.060
0.760
1.480
3.240
1.020
4.560
1.720
0.610
0.830
0.990
0.010
2.680
1.130
0.190
4.19E+05
3.79E+05
2.41E+06
3.86E+05
1.90E+07
1.08E+06
7.67E+05
1.96E+06
7.67E+05
2.29E+07
1.63E+06
3.07E+05
2.11E+06
7.78E+05
2.02E+06
1.27E+06
1.57E+06
1.03E+06
1.28E+07
1.77E+06
1.89E+06
4.76E+05
2.31E+06
1.97E+06
1.52E+07
2.30E+07
6.88E+05
3.23E+05
2.51E+05
1.17E+07
2.18E+07
5.71E+06
7.32E+05
5.18E+05
1.03E+06
1.63E+07
3.71E+06
2.59E+06
4.43E+05
2.06E+06
1.22E+06
1.29E+07
5.70E+06
7.21E+06
6.43E+05
5.93E+05
1.52E+07
4.22E+05
5.40E+05
3.08E+05
4.88E+06
9.13E+05
6.43E+05
4.86E+06
2.54E+06
2.02E+05
7.65E+06
3.27E+05
5.30E+06
1.62E+07
5.24E+05
1.17E+07
3.93E+05
5.23E+05
1.56E+06
Line
k
(8)
o Y ca
A
(s1)
fIYC 2Y2 R3..........................
eGKYC 2Y1 R1 .....................
eGKYC 3Y1 P1 .....................
fIYC 2Y2 R2..........................
eEFYC 34Y3 P2 ....................
eEFYB 20Y8 P6 ....................
fIYC 0Y0 P4..........................
fIYC 1Y1 Q4 .........................
eEFYB 32Y14 P1 ..................
fIYC 2Y2 R1..........................
eGKYB 4Y10 P5....................
eEFYB 28Y12 P4 ..................
eGKYC 1Y0 Q2 .....................
eEFYB 9Y2 P1 ......................
eGKYC 1Y0 R3 .....................
eGKYC 5Y2 P1 .....................
eGKYC 0Y0 R1 .....................
eEFYB 28Y12 P5 ..................
eGKYC 1Y0 P2 .....................
eEFYB 21Y9 P1 ....................
eIYC 1Y1 P4..........................
eGKYC 3Y1 R2 .....................
eEFYC 28Y2 R1 ....................
eEFYC 35Y3 Q2....................
eGKYC 7Y2 Q5 .....................
eEFYB 23Y10 P4 ..................
fIYC 1Y1 P3..........................
eEFYB 9Y2 P2 ......................
eGKYC 0Y0 Q1 .....................
eGKYC 4Y2 P2 .....................
eEFYB 7Y1 R4 ......................
7939.337
7939.451
7939.583
7944.421
7946.895
7949.328
7949.650
7951.597
7955.165
7955.361
7959.528
7961.650
7963.114
7969.460
7969.912
7970.648
7971.666
7977.993
7979.731
7982.038
7982.738
7983.554
7984.472
7989.958
7992.787
7993.452
7993.867
7994.877
7998.419
7998.458
7998.777
0.230
4.100
5.040
0.330
...
0.880
0.060
0.060
1.300
0.370
1.440
1.280
1.500
1.000
1.070
6.350
1.730
3.430
1.610
2.730
0.840
4.610
3.000
0.770
4.470
2.120
0.050
1.070
2.010
4.640
0.720
1.92E+06
2.39E+06
5.31E+06
2.29E+06
6.61E+05
4.01E+06
1.78E+07
3.92E+05
3.30E+05
2.39E+06
5.02E+06
2.49E+05
1.72E+06
4.54E+05
5.64E+06
3.87E+06
8.22E+05
3.10E+05
5.90E+05
2.05E+05
1.21E+07
3.14E+06
1.58E+06
2.39E+05
3.07E+05
3.99E+05
1.47E+07
3.09E+05
3.68E+05
7.80E+06
1.85E+06
a
observed (Yu & Dressler 1994)-calculated.
The spontaneous emission probability between two discrete
levels is given by the expression
A vk ; vj ; Jk ; Jj ¼
2
3 4
;
E
M
E
vk Jk
vj Jj 4 3
;v
;
J
v
ð k j k Þ
3 f c ð2Jk þ 1Þ
ð4Þ
where EvJ is the energy of the level (v; J ), M kj is the electric
dipole matrix element between the wave functions of upper and
lower states, and indicates whether the spectroscopic branch
label is P, Q, or R. The k( j) labels refer, respectively, to levels of
gerade (g) (ungerade [u]) symmetry. Exchange between the labels may occur, depending on the symmetry of the upper level.
Tables 1 and 2 show a sample of our calculated gYu and uYg
transition probabilities, respectively, and compares our line wavelengths with those obtained from the difference between the experimental g-states terms cited in Tables 5 and 6 of Yu & Dressler
(1994) and the experimental u-states terms of Abgrall et al. (1993c,
1994).
The dipole matrix elements M(vk ;vj ; Jk ) appearing in equation (4)
are reported in Liu et al. (2002) when the upper gerade level is assumed to be uncoupled. In the present treatment, we explicitly introduce the couplings between gerade upper states. The different
1 þ
1 þ
1 þ
symmetries involved comprise EF 1 þ
g , GK g , H H̄ g , P g ,
1 þ
1
1
1
1
O g , I g , R g , J g , and S g . Both nonadiabatic radial couplings between states of same symmetry and rotational
1 þ
1 þ 1 þ
couplings between 1 þ
g Y g and g Y g take place. One obtains
No. 1, 2008
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
TABLE 2
Calculated Transition Wavelength and Emission Probabilities
Greater than 2 ; 105 s1 between 7500 and 8000 8
from u Upper States toward g Lower States
Line
k
(8)
oYca
A
(s1)
eDYEF 1Y4 P5 ......................
fCYEF 7Y0 Q3......................
eBYEF 23Y1 R1 ....................
eB 0 YEF 1Y0 R2 .....................
eBYEF 23Y1 R2 ....................
eB 0 YEF 1Y1 R3 .....................
eCYEF 12Y8 R5 ....................
eB 0 YEF 1Y0 R1 .....................
eCYEF 7Y0 P4 ......................
eB 0 YEF 1Y1 R4 .....................
eB 0 YEF 4Y6 P2 .....................
eB 0 YEF 1Y0 R0 .....................
eB 0 YEF 4Y7 R3 .....................
eCYEF 7Y1 R1 ......................
eB 0 YEF 4Y7 R4 .....................
eCYEF 7Y1 R5 ......................
eB 0 YEF 1Y1 R5 .....................
fCYEF 7Y1 Q2......................
eB 0 YEF 4Y7 R2 .....................
eB 0 YEF 4Y7 R5 .....................
eCYEF 7Y1 R0 ......................
eCYEF 7Y0 P3 ......................
fCYEF 7Y1 Q1......................
eB 0 YEF 4Y7 R1 .....................
eCYEF 7Y1 P2 ......................
eB 0 YEF 4Y7 R0 .....................
eB 0 YEF 1Y0 P2 .....................
eB 0 YEF 4Y7 P2 .....................
eB 0 YEF 4Y7 P3 .....................
fDYEF 2Y8 Q5......................
eB 0 YEF 1Y1 P3 .....................
eB 0 YEF 4Y7 P4 .....................
fCYEF 13Y9 Q1....................
fCYEF 13Y9 Q2....................
eB 0 YEF 1Y1 P4 .....................
fDYEF 2Y8 Q4......................
eDYEF 2Y8 P5 ......................
eCYEF 8Y3 R5 ......................
eB 0 YEF 1Y1 P5 .....................
eB 0 YEF 2Y3 R2 .....................
eB 0 YEF 2Y4 R3 .....................
eB 0 YEF 2Y3 R1 .....................
eDYEF 2Y8 P4 ......................
eB 0 YEF 2Y4 R4 .....................
eB 0 YEF 2Y3 R0 .....................
eB 0 YEF 6Y9 R4 .....................
eCYEF 8Y3 R4 ......................
eB 0 YEF 4Y8 R5 .....................
eB 0 YEF 2Y4 R5 .....................
eCYEF 7Y2 R5 ......................
fCYEF 8Y3 Q5......................
eB 0 YEF 5Y9 R3 .....................
eB 0 YEF 5Y9 R2 .....................
eB 0 YEF 5Y9 R1 .....................
fCYEF 11Y8 Q5....................
7523.368
7524.358
7529.372
7530.637
7531.198
7533.671
7536.573
7537.010
7538.829
7545.826
7549.096
7553.167
7558.539
7560.722
7562.134
7566.775
7566.929
7570.275
7579.065
7585.722
7594.963
7595.841
7601.551
7608.549
7616.204
7634.410
7663.012
7672.248
7689.161
7691.178
7720.046
7729.821
7732.559
7752.719
7786.767
7796.748
7832.442
7845.950
7862.736
7898.583
7903.245
7905.419
7914.209
7916.159
7922.198
7926.134
7933.095
7937.736
7938.676
7951.591
7977.911
7979.885
7983.241
7990.418
7993.240
0.950
0.170
0.060
0.080
0.140
0.030
0.540
0.110
0.060
0.020
2.620
0.120
0.370
0.230
0.380
0.070
0.060
0.170
0.060
0.710
0.270
0.220
0.230
0.220
0.200
2.900
0.120
2.840
0.020
0.400
0.110
0.390
...
...
0.100
0.310
0.580
0.120
0.050
0.430
0.520
0.500
0.210
0.450
0.490
1.220
0.040
0.450
0.540
0.300
0.110
1.480
1.480
1.230
0.700
8.69E+06
1.99E+06
2.41E+05
3.23E+06
2.60E+05
3.57E+06
2.62E+05
2.74E+06
6.58E+05
3.81E+06
5.79E+05
2.01E+06
1.55E+06
8.99E+05
2.12E+06
1.30E+06
2.68E+06
1.90E+06
8.04E+05
2.17E+06
1.19E+06
5.96E+05
1.83E+06
4.22E+05
5.94E+05
2.37E+05
2.12E+06
3.21E+05
3.06E+05
7.19E+05
1.46E+06
2.39E+05
2.21E+05
2.22E+05
1.07E+06
3.07E+05
5.22E+05
2.44E+05
8.18E+05
3.98E+06
4.50E+06
3.52E+06
2.56E+05
4.83E+06
2.68E+06
2.26E+05
2.20E+05
4.70E+05
5.08E+06
2.95E+05
4.00E+05
4.28E+05
5.39E+05
5.51E+05
2.31E+05
a
observed (Yu & Dressler 1994)-calculated.
395
nine coupled equations for the ‘‘+’’ symmetry electronic levels (five
1 þ
1
þ
g , two 1 þ
g and two g ) and four coupled equations for
1 1 the ‘‘’’ symmetry electronic levels (I 1 g , J g , R g , and
1 S g ) for a specific J-value. Alternatively, the u-symmetry
lower states manifold results in four coupled equations for the
1 þ
‘‘+’’ symmetry states [1 þ
u (B, B’), u (C, D)] and two coupled
equations for the ‘‘’’ symmetry levels [ 1 u (C, D)]. Lower
dimension coupling matrices arise for low-J values; when J ¼ 0,
only 1 electronic states coupled via radial coupling are involved. When J ¼ 1, only 1 and 1 electronic states have
to be considered with both radial and rotational coupling, and
when J 2, the full couplings take place, including 1 electronic states.
We express M(vk ;vj ; Jk ) as in Liu et al. (2002) including both the
expansion on the gerade upper T states and showing the explicit
expansion for the BO states of the ungerade manifold,
h
X
FT; fT vk Jk jMT B jfBvj Jj
M v ;v ; J ¼
ð k j kÞ
T
i
þ fT vk Jk jMTB 0 jfB 0 vj Jj
X
FT; fT vk Jk jMT C j fCvj Jj þ fT vk Jk jMT D j fDvj Jj ; ð5Þ
þ
T
1 þ
where T represents the gerade BO states EF 1 þ
g , GK g ,
1 þ
1 þ
1 þ
1
1
1
1
H H̄ g , P g , O g , I g , R g , J g , and S g . MTB ,
MTB 0 , MTC , and MTD are the real values of the electronic transition moments, which are functions of the internuclear distance
and calculated in the BO approximation. We explicitly write the
formulae of the electric dipole matrix elements for the different
branches in the Appendix.
3.2. Calculations
The discrete eigenvalues and wave functions are obtained by
solving the nuclear Schrödinger coupled differential equations
via the Numerov algorithm, using a step of 0.01 au, up to an internuclear distance of 30 au (Johnson 1978). We have performed
calculations up to the J ¼ 5 rotational level, and we describe
below the required molecular R-dependent data involved in the
equations.
We use the same diagonal electronic potential terms as in
1 þ
Liu et al. (2002) for the upper g states (EF 1 þ
g , GK g ,
1 þ
1 þ
1 þ
1
1
H H̄ g , O g , P g ), I g , J g ) and the lower states
0 1 þ
1
1
(B 1 þ
u , B u , C u , D u ). Two additional gerade states
(R g and S g ) have been introduced and are taken from
Wolniewicz (1995a, 1995b). Radial couplings between the g
states are taken from Wolniewicz & Dressler (1994), and we do
not include the radial couplings for g and g states, as these
values are expected to be small and are not available in the present literature. Only one single rotational coupling between GK
and I states has been calculated by Dressler & Wolniewicz (1984).
We have then calculated the other rotational coupling terms within
the upper Rydberg states manifold (EFYI, GKYI, H H̄YI, EFYR,
GKYR, H H̄YR, IYJ, RYJ ) by using the MOLPRO code (Werner &
Knowles 1988) and the MRCI electronic wave functions used by
Spielfiedel (2003) for these singlet excited states of H2. We have
checked that the GKYI coupling term calculated in this way is very
close to the value given by Dressler & Wolniewicz (1984). Additional PYI, OYI, IYS, RYS terms have been approximated by the
values obtained for GKYI, H H̄YI, RYJ, IYJ. We do not comment
further on the data used for the u-states, which have been used
extensively in our previous work and found to accurately reproduce the experimental VUV spectrum of H2 (see, for example,
396
AGUILAR ET AL.
Vol. 177
Abgrall et al. 2000). For the 20 transitions between the upper
(EF, GK, H H̄, P, O) g states and lower (B, B 0 , C, D) u states, we
have used the matrix elements of Dressler & Wolniewicz (1995);
for the eight transitions connecting the upper g states (I, R) and
g states (J, S ) to the lower u states we have used the data of
Wolniewicz (1996) for the transitions between g (I, R) and
u (B,B 0 ) we have used the data of Spielfiedel (2003).
4. THE SPECTRAL MODEL
We distinguish two different possibilities for the upper states
symmetry.
4.1. Emission Lines of g-States toward u-States
If the upper levels are of g-symmetry, the electronic excitation
model involves a forbidden mechanism and the corresponding
formulae are taken from Liu et al. (2002). The excitation cross
section is given by the product
(vi ; vk ; Ji ; Jk ) ¼ Fi;k (X )Qvi ;vk Sr (Ji ; Jk );
ð6Þ
with X ¼ E/Eki , where E is the electron kinetic energy, and Eki is
the excitation energy Eki ¼ Ek Ei. The label i is kept for the
labeling of the ground X electronic state of H2. The term Sr is
a dimensionless factor accounting for the rotational branching
ratios as expressed in Liu et al. (2002) for Y, Y, and Y
transitions with Jk YJi ¼ 0, 1, 2. Qvi ;vk is the Franck Condon
factor and is a measure of the radial overlap between the upper
(vk ) and lower (vi ) vibrational functions. Fik is an electronic term
which is assumed to be independent of the internuclear distance
and has been derived experimentally in Liu et al. (2003) for the
excitation to EF.
We derive the mean emission cross section for a specific transition between Ek and Ej , corresponding to an electron beam
energy of 100 eV and a hydrogen sample temperature of 300 K,
(v
¯ k ; v j ; Jk ; Jj ) ¼
A(vk ; vj ; Jk ; Jj )
1X
; ð7Þ
Nvi ;Ji (vi ; vk ; Ji ; Jk )
N vi ;Ji
A(vk ; Jk )
P
where A(vk ; Jk ) ¼ vj ;Jj A(vk ; vj ; Jk ; Jj ) is the total emission probability of the level vk ; Jk and Nvi Ji /N is the relative population
of the vi ; Ji level of H2, computed under thermodynamic conditions at 300 K. We estimate the contribution of the other gerade
Rydberg levels from Liu et al. (2002) to the EF-state which was
derived at 20 eV. The values are, respectively, 1: 0:35 : 0:24 :
0:17 : 0:31 : 0:095 : 0:095 for EF : GK : H H̄ : I : J : O : P. This approximation is questionable as the g-states become very strongly
coupled for vibrational quantum numbers larger than 16 in the
EF-state. The absolute cross section obtained in this way for each
gerade state depends on the Liu et al. (2003) measurement of
a single Lyman band rotational line that is cascade-driven at low
energy (method 1).
In this paper we choose to renormalize the model calculations to the 7526.67 8 rotational line EFYB(6; 0)P2, whose cross
section is obtained by direct comparison with the H feature in
the same spectra as described in x 2 (method 2). We find that the
model cross section for the EFYB(6; 0)P2 rotational line from
method 1 has to be multiplied by a factor of 2.17 to agree with
direct method 2 adopted here. The origin of the discrepancy is
not clear. For the spectral model figures, we enter rotational
features whose cross sections lie above 1 ; 1023 cm2.
4.2. Emission Lines of u-States toward g-States
The 100 eV electrons may also excite high vibrational levels
belonging to u symmetry (B, B 0 , C, D), which subsequently emit
Fig. 6.— Cross sections in cm2 for the electron impactYinduced fluorescence
spectrum from 3300Y12000 8 of H2 at 100 eVin three wavelength regions: (1) grating 2 (3300 Y7500 8) in black, grating 3 (7500 Y10000 8) in red and theory
(10000 Y12000 8) in blue. The H feature is off scale.
toward g-symmetry states in the infrared domain. The excitation
from the ground electronic state is now an electric dipole transition, and the excitation function is modified accordingly,
(vi ; vj ; Ji ; Jj ) ¼ Fi;j (X )Qvi ;vj Sr (Ji ; Jj )
ð8Þ
with the selection rule Jj YJi ¼ 0, 1; Fi;j is given in Liu et al.
(1998) for the excitation toward B and C. We have used the same
expressions for excitation toward B 0 and D, as in Liu et al. (2002).
The corresponding expression of the mean emission cross section is now
(v
¯ j ; v k ; Jj ; Jk ) ¼
A(vj ; vk ; Jj ; Jk )
1X
Nvi ; Ji (vi ; vj ; Ji ; Jj )
N vi ;Ji
A(vj ; Jj )
ð9Þ
Avj ; Jj is the total emission probability of the level vj ; Jj , which
emits directly toward the X ground state via VUV photons as
studied by Abgrall et al. (2000).
5. EXPERIMENTAL RESULTS
High-resolution VOIR spectral measurements of electronimpact excited molecular hydrogen under optically thin conditions have been performed over the wavelength range of 3300
to 10000 8 at 20 and 100 eV electron-impact energies. We extend the full spectrum of H2 to 12000 8 by using the spectral
model from 10000 to 12000 8. We show the complete VOIR
100 eV spectrum of H2 extending from 3300 to 12000 8 as an
overview in Figure 6. We identify three separate regions in Figure 6 covered by grating 2 (3300Y7500 8), by grating 3 (7500 Y
10000 8), and by theory (10000Y12000 8). The overview
spectrum shows that the H2 molecular spectrum is dominated
by the rotational features in the near-IR from 3300 Y12000 8,
which include the main ( lowest lying) gerade-ungerade EFYB
band system rotational lines. The strongest EFYB rotational line
besides the strong H i multiplets (H , H, . . .) excited by dissociative excitation is the rotational line at 8901.240 8 EFYB
(3; 0)P2. Figure 6 can be summarized as follows; the total VOIR
emission cross section of the singlet gerade states at 100 eV is
4.58 Mbarn, the H i lines is 1.11 Mbarn, and the cross section of
the triplet states is less than 0.02 Mbarn. More than 1000 rotational lines are observed in the VOIR region that are resolved at
No. 1, 2008
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
397
Fig. 7.— Electron impactYinduced fluorescence spectrum of H2 at 20 and
100 eV from 4500 to 5500 8. The strongest triplet series rotational lines are
labeled 1Y25a and are identified by the rotational branch, vibrational quantum
numbers, and electronic band system in Table 3.
Fig. 8.— Electron impactYinduced fluorescence spectrum of H2 at 20 and
100 eV from 5500 to 6500 8. The strongest triplet series rotational lines are
labeled 26Y69 and are identified by the rotational branch, vibrational quantum
numbers, and electronic band system in Table 3.
the instrument resolution of 0.7 8 FWHM and having a cross
section greater than 0.0005 Mbarn. We plot the peak intensity in
Figure 6 as an absolute cross section. The H cross section is out
of scale in the figure.
The model spectrum of H2 was calculated from 3500Y12000 8.
However, deficiencies in the model only allowed us to compare
model and data from 7500 to 12000 8. This spectral region is
distinguished by strong EFYB transitions and is important as the
most intense region as Figure 6 shows. We discuss the problems
with the emission model of the other gerade transitions in x 6.
Making use of the electronic energies for the d 3 u and a 3 þ
g
states found in Huber & Herzberg (1979) or from the identifications and intensities given in Crosswhite (1972) for Process
15 we find that the Fulcher- band system is expected to be
strong in the wavelength region from 4500Y6500 8. For example,
the threshold for the d 3 u state is 14.0 eV and the Q1 rotational branch of the (0,0) vibrational band is found at 6019.97 8.
Spindler (1969) has reported Franck-Condon factors for the Fulcher band system, based on Morse potentials, and found the diagonal bands to be very strong. Since the R- and P-branch lines of
the d-state v ¼ 0 and 1 are perturbed by levels of the 3s h 3 þ
g
state, it is important to measure at high resolution the intensity of
all the branches (Baltayan & Nedelec 1976). When we directly
excite the d-state, Process 12 occurs through electron exchange
characterized by a maximum in the electron excitation cross section at low-electron energies of 14Y20 eV. The excitation function has been measured by Möhlmann & de Heer (1976). The
peak cross section occurs at 15.6 eV.
The electron exchange process can excite both gerade and
ungerade triplet states as indicated in equations (10) and (11). We
show our electron impactYinduced fluorescence spectra in Figures 7 and 8 of the VOIR spectral region from 3500Y7500 8 at
two energies 20 and 100 eV. Each observed triplet state would
have a large near-threshold cross section, similar to the results
found by Möhlmann & de Heer (1976) for the Fulcher- band
system, and thus a much stronger relative intensity at 20 eV than
100 eV. Thus, each triplet rotational line stands out strongly at
20 eV and is identified with a feature number in Figures 7 and 8.
It should be noted that the singlet gerade-ungerade transitions
also involve a dipole-forbidden excitation process but do not violate the S ¼ 0 selection rule. Möhlmann & de Heer (1976)
found that the Fulcher- triplet system has a cross section that
decreases rapidly with electron-impact energy, E. For the singlet
gerade-ungerade transitions, the University of Arkansas group
( Watson & Anderson 1977; Anderson et al. 1977; Day et al.
1979) have found a slower energy dependance (E0.6 to E0.85)
of the excitation cross sections from 15 to 300 eV. The E1 dependence corresponding to the Born approximation is obtained
at higher energies (Liu et al. 2003).
The triplet band systems associated with the observed rotational lines are identified and their corresponding cross sections
are reported in Table 3. The 20 eV data are normalized to the
5.1. The Triplet Band Systems
It is important to discuss the triplet band systems first, before
considering the singlet band systems for two reasons, even though
the triplet band systems are a minor contributor to the emission
cross section of the VOIR at either 20 or 100 eV (Dziczek et al.
2000). First, at 20 eV electron impact energy the Fulcher- band
system produces a rich spectrum in the wavelength region from
4500 to 6500 8. Second, at 100 eV electron impact energy lowenergy scattered electrons contribute to the measured spectrum in
this spectral range by exciting the Fulcher- and other weaker
triplet band systems.
We employ electron impactYinduced fluorescence to examine
the bands originating from both the triplet ungerade (eq. [10])
and triplet gerade (eq. [11]) Rydberg system by the processes
3 þ
e þ H2 (X 1 þ
g ) ! H2 ( u ;
! H2 (2s a
3
þ
g ;:
3
u ) þ e
: :) þ h þ e;
ð10Þ
and
3 þ 3
3
e þ H2 (X 1 þ
g ) ! H 2 ( g ; g ; g ) þ e
! H2 (2p c 3 u ; : : :) þ h þ e:
ð11Þ
The strongest triplet band system with over 60 rotational lines
found in the VOIR is the Fulcher- bands
3
e þ H2 (X 1 þ
g ) ! H2 (3p d u ) þ e
! H2 (2s a 3 þ
g ) þ h þ e:
ð12Þ
TABLE 3
Wavelengths and Cross Sections of the Fulcher-
(d 3 u ! a 3 þ
g ) and Other Bands at 20 eV Using Grating 2
Feature
Cross Section
(1019 cm2)
1....................................................................
2....................................................................
3....................................................................
4....................................................................
4A.................................................................
5....................................................................
6....................................................................
7....................................................................
8....................................................................
9....................................................................
9a..................................................................
10..................................................................
11..................................................................
12..................................................................
12a................................................................
13..................................................................
14..................................................................
15..................................................................
0.0658
0.102
0.138
0.133
0.125
0.248
0.0903
0.115
0.142
0.201
0.069
0.144
0.112
0.0680
0.0730
0.0650
0.102
0.0572
16..................................................................
0.0427
16a................................................................
17..................................................................
0.0490
0.0572
18..................................................................
0.0469
18a................................................................
18b................................................................
19..................................................................
20..................................................................
20a................................................................
21..................................................................
22..................................................................
23..................................................................
0.0422
0.0315
0.0947
0.0506
0.108
0.157
0.0620
0.0340
24..................................................................
25..................................................................
25a................................................................
26..................................................................
27..................................................................
28..................................................................
29..................................................................
30..................................................................
31..................................................................
32..................................................................
33..................................................................
34..................................................................
35..................................................................
35a................................................................
36..................................................................
0.0930
0.0707
0.042
0.0826
0.0716
0.181
0.050
0.042
0.042
0.0676
0.0936
0.0679
0.0982
0.042
0.0728
37..................................................................
38..................................................................
39..................................................................
0.0826
0.148
0.0677
40..................................................................
0.0898
41..................................................................
42..................................................................
43..................................................................
43a................................................................
44..................................................................
0.0261
0.118
0.138
0.0432
0.313
Identificationa
kDb
4cY2a(0; 0)R1
4cY2a(0; 0)Q1
4cY2a(1; 1)Q1
4cY2a(2; 2)Q1
4eY2c(3; 3)Q4
4cY2a(1; 1)P5
4cY2a(3; 3)Q1
3eY2c(2; 0)Q6
4eY2c(0; 1)Q1
3cY2a(3; 1)Q5
4cY2a(0; 1)R4
4bY2a(1; 1)P3
4eY2c(1; 2)P5
3eY2c(1; 0)R5
4eY2c(1; 2)P6
5cY2a(1; 3)P4
4cY2a(1; 3)Q3
4cY2a(2; 3)Q1
3f Y2c(2; 1)Q3
3aY2c(1; 0)Q3
4cY2a(3; 4)Q1
3bY2a(6; 1)P2
4cY2a(4; 5)Q1
3bY2a(4; 0)R2
3cY2a(1; 0)R1
3bY2a(4; 0)P1
3cY2a(6; 4)Q1
4cY2a(5; 6)Q1
3cY2a(1; 0)Q1
3cY2a(2; 1)R1
3eY2c(2; 1)P5
3cY2a(2; 1)Q1
3cY2a(1; 0)P5
3cY2a(2; 1)Q3
3eY2c(3; 2)Q3
3cY2a(2; 1)P2
3cY2a(2; 1)P3
3cY2a(3; 2)R2
3cY2a(3; 2)R1
3cY2a(3; 2)R0
3cY2a(3; 2)Q1
3cY2a(3; 2)P3
5cY2a(0; 3)R0
3cY2a(4; 3)Q1
3f Y2c(0; 0)R1
3f Y2c(1; 1)R1
3cY2a(5; 4)Q1
3aY2c(2; 2)R1
3cY2a(0; 0)R3
3f Y2c(2; 2)Q4
3eY2c(0; 0)R1; R3
3f Y2c(2; 2)Q2
3f Y2c(3; 3)R1
3f Y2c(3; 3)Q2
3f Y2c(3; 3)Q2
3eY2c(0; 0)P2
3aY2c(2; 2)R2
3dY2c(0; 0)R1
3eY2c(1; 1)P2
3aY2c(1; 1)P2
3cY2a(0; 0)R2
3cY2a((0; 0)R1
4468.395
4491.711
4555.439
4618.820
4628.070
4633.143
4681.738
4720.359
4630.091
4935.941
5005.832
5018.523
5055.366
5071.390
5081.915
5086.695
5116.765
5155.306
5155.925
5197.562
5198.667
5228.236
5240.478
5240.478
5273.769
5273.769
5274.492
5279.814
5304.590
5389.653
5420.148
5421.406
5428.245
5436.328
5436.280
5458.500
5482.606
5497.487
5507.042
5520.022
5539.006
5599.201
5637.497
5657.324
5690.755
5738.466
5776.203
5787.372
5808.759
5813.916
5814.214
5824.370
5837.739
5873.246
5873.574
5880.848
5880.848
5922.455
5926.470
5951.545
5961.476
5977.101
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
399
TABLE 3—Continued
Feature
45.........................................
46.........................................
47.........................................
48.........................................
49.........................................
50.........................................
51.........................................
52.........................................
53.........................................
54.........................................
55.........................................
56.........................................
57.........................................
57a.......................................
58.........................................
59.........................................
60.........................................
61.........................................
62.........................................
63.........................................
63a.......................................
64.........................................
65.........................................
65a.......................................
66.........................................
67.........................................
68.........................................
69.........................................
70.........................................
71.........................................
72.........................................
All Triplet Bands Total .....
dYa Total .........................
73.........................................
Cross Section
(1019 cm2)
0.200
0.772
0.143
0.161
0.530
0.298
0.334
1.043
0.216
0.191
0.133
0.450
0.376
0.261
0.915
0.212
0.236
0.147
0.303
0.352
0.163
0.335
0.075
0.123
0.0985
0.313
0.140
0.0572
0.0273
0.245
13.8
10.40
1.33
Identificationa
kDb
3cY2a(0; 0)R0
3cY2a(0; 0)Q1
3cY2a(0; 0)Q2
3cY2a(0; 0)Q3
3cY2a(1; 1)R1
3cY2a(0; 0)P3
3cY2a(1; 1)R0
3cY2a(1; 1)Q1
3cY2a(0; 0)P4
3cY2a(1; 1)Q3
3cY2a(1; 1)P2
3cY2a(2; 2)R1
3cY2a(1; 1)P3
3cY2a(2; 2)R0
3cY2a(2; 2)Q1
3cY2a(2; 2)Q2
3cY2a(2; 2)Q3
3cY2a(2; 2)P2
3cY2a(3; 3)R1
3cY2a(2; 2)P3
3cY2a(3; 3)R0
3f Y2c(0; 1)R5
3cY2a(3; 3)Q1
3cY2a(3; 3)Q2
3cY2a(3; 3)Q3
3cY2a(3; 3)P2
3cY2a(3; 3)P3
3cY2a(4; 4)Q1
3cY2a(5; 5)Q1
3bY2a(3; 1)R1
3cY2a(3; 4)Q1
5995.732
6019.966
6025.425
6033.579
6082.466
6097.643
6099.923
6123.481
6129.344
6137.093
6171.346
6184.701
6201.112
6202.901
6226.538
6231.982
6240.117
6273.046
6287.122
6301.165
6305.222
6328.017
6328.809
6334.233
6342.328
6373.974
6401.237
6429.896
6529.158
6698.610
7311.620
EF(6; 1)YB(0; 2)
7526.680 (reference line)
a
The identifications are based on Dieke (Crosswhite 1972) notation: 2a (2s a 3 þ
g ) lowest lying triplet
3
3 þ
bound state; 3a (3s h 3 þ
g ) perturbing triplet state to the 3p u and a weak cascade to the 2s a g state; 3b
3 þ
3
(3p e u ) upper state eYa bands; 3c (3p d u ) upper state Fulcher- (dYa) bands; 3e (3d i 3 g ) upper state
iYd, eYc bands; 3 f (3d j 3 g ) upper state jYc bands; 4c (4p 3 u ) upper state bands (kYa).
b
Crosswhite (1972) vacuum wavelengths.
7526.67 8 rotational line EF(6; 2)P2 emission cross section
from the laboratory value of 0.133 Mbarn. We can easily identify
the relatively strong Fulcher- (dYa) and -bands (kYa), using
Dieke’s wavelength tables (Crosswhite1972). We find some other
band systems that are very weakly excited, e.g., the eYa, iYd, iYe,
and iYc bands. From Table 3 the emission cross section at 20 eV
for the triplet states is found to be 1.38 Mbarn and for the Fulcher band system is found to be 1.04 Mbarn. Möhlmann & de Heer
(1976) show that the triplet cross section falls by a factor of 66
from 20 to 100 eV. The 100 eV cross section is estimated to be
about 0.02 Mbarn. At 100 eV the total emission cross section of
the triplet systems is weak and negligible compared to the emission cross section of the singlet systems. At 100 eV spurious signals from the dYa triplet system are stronger in Figure 6 from
low-energy scattered electrons than predicted by the accurate
excitation function indicated by Möhlmann & de Heer (1976).
We subtract the contribution of these spurious signals, roughly
0.1 Mbarn, from our total 100 eVemission cross section for the
VOIR. We also have subtracted a small contribution to the nearconstant background level to the grating 2 signal from the H2
3 þ
(a 3 þ
g Yb u ) continuum that extends from Lyman at 1216
to 6000 8 as described in Ajello & Shemansky (1993). The com-
plete 100 eV VOIR spectrum (3300Y10000 8) is to be more fully
discussed in x 5.2.
The non-Fulcher- bands, including the -bands, total to a scant
0.34 Mbarn at 20 eV. The strongest rotational lines of the Fulcher band system are the dYa(0,0)Q1 at 6019.97 8, dYa(1,1)Q1 at
6123.48 8, dYa(2,2)Q1 at 6226.54 8 and dYa(3,3)Q1 at 6328.81 8
with 20 eV emission cross sections of 0.0772, 0.104, 0.0915, and
0.0345 Mbarn, respectively.
Möhlmann & de Heer (1976) measured the cross sections of
a few of the Fulcher- diagonal bands (v ¼ 0) sequence. However, their resolution was a broad 27 8 and resulted in a blending
of singlet and triplet rotational lines and a likely overestimate of
the triplet emission cross sections. For example, they found the
absolute electronic emission cross section of the Fulcher- bands
to be 4.2 Mbarn at 15.6 eV and 3.2 Mbarn at 20 eV, compared to
our value of 1.04 Mbarn for the Fulcher- band system at 20 eV.
Their 20 eV value for the dYa(0,0)Q1, dYa(1,1)Q1, dYa(2,2)Q1
and dYa(3,3)Q1 rotational lines are 0.47, 0.62, 0.48, and 0.34 Mbarn,
respectively, which are over 5 times larger than the values found
in this work. However, the ratio of the (0,0), (1,1), (2,2), and (3,3)
bands are in good agreement with Mohlmann & de Heer and a
comparison is shown in Table 4. We compare the relative intensities
400
AGUILAR ET AL.
TABLE 4
Relative Emission Cross Section at 20 eV for the Diagonal Bands
of the Fulcher- Band System Normalized to the (1,1) Band
Relative Experimental Diagonal
Band Cross Section
Relative Model Diagonal
Band Cross Section
v 0 Yv 00
This work
Möhlmann
and de Heer 1976
Möhlmann
and de Heer 1976
0Y0.........
1Y1.........
2Y2.........
3Y3.........
0.58
1.0
0.93
0.37
0.75
1.00
0.77
0.54
0.70
1.00
0.85
0.58
of the whole vibrational bands using all the P-, Q-, and R-branch
rotational lines shown in Figures 7 and 8 with respect to the Q1 line
ratios from the experimental work and models found in Möhlmann
& de Heer (1976). The comparisons are in good agreement and
indicate that the (1,1) vibration band at 6123.48 8 is the most
intense.
5.2. The Singlet Gerade-Ungerade Band Systems
The 100 eV emission cross section of the forbidden excitation
to the excited singlet gerade states of H2 and ensuing cascade to
the singlet-ungerade states dominates all other types of VOIR
transitions (triplets and H i lines). At low energy (20 eV ) the
optically forbidden VOIR cascade cross section contribution to
the UVemission cross section is even more important, nearly 50%
(Dziczek et al. 2000). Dziczek et al. (2000) have estimated the
100 eV cascade cross section to be 5:0 0:6 Mbarn by separating the direct and cascade UV processes using TRS. The total
UVemission cross section of 59.9 Mbarn for all singlet ungerade
states was measured at 100 eV by Jonin et al. (2000). We evaluate
the total VOIR singlet gerade cross section in this direct measurement of VOIR emissions and compare the results to previous
UV studies for cascading.
The calibrated 100 eV spectra are shown in Figures 9Y12 for
grating 2 and in Figures 13Y16 for grating 3 along with the
model. These figures represent small distinct spectral regions of
the full spectrum shown in Figure 6. The region from 3600 to
7500 8 contains experimental data only, and it is shown in Figures 9Y12, with each figure covering a range of about 1000 8. The
grating 2 rotational line emission cross sections for 35 strong
singlet gerade transitions in this 3600Y7500 8 region are given in
Table 5. The region from 7500Y10000 8 shown in Figures 13Y16
compares the grating 3 data with the model results. The grating 3
rotational line emission cross sections for a sample of 50 strong
singlet gerade transitions (model and data) from the Figures 13Y16
are given in Table 6 along with the spectroscopic identification
TABLE 5
Emission Cross Section Values at 100 eV for the Selected Features Numbered in Figs. 9Y12
Feature
kexp
(8)
exp
(1020cm2)
Transition
kD
(8)
krep
rep
(1020cm2)a
1..........................
2..........................
3..........................
4..........................
5..........................
6..........................
7..........................
8..........................
9..........................
10........................
11........................
12........................
13........................
14........................
15........................
16........................
17........................
18........................
19........................
20........................
21........................
22........................
23........................
24........................
25........................
26........................
27........................
28........................
29........................
30........................
3701.0
3873.4
3874.2
3890.8
4068.0
4178.2
4206.2
4405.6
4634.8
4720.0
4935.7
5056.2
5357.6
5387.2
5613.3
5714.7
6322.9
6380.6
6573.9
6624.9
6679.4
6745.7
6828.6
6877.1
6965.0
7065.4
7178.0
7197.6
7256.0
7353.2
0.063
0.288
0.475
0.769
0.647
0.560
0.988
0.262
1.31
0.646
0.953
0.895
0.592
0.656
0.808
0.671
0.388
0.484
0.691
1.13
0.123
0.367
0.509
1.16
1.06
0.706
1.58
4.26
8.91
5.30
H H̄e(2; 1) ! B(0; 2)
...
H H̄e(2; 1) ! B(1; 0)
H H̄e(2; 1) ! B(1; 2)
Ie(2; 1) ! B(1; 0)
If (1; 2) ! B(0; 2)
H H̄e(1; 1) ! B(1; 0)
H H̄e(0; 1) ! B(0; 2)
GKe(1; 3) ! B(0; 2)
GKe(3; 3) ! B(2; 2)
Ie(3; 1) ! B(6; 0)
EFe(22; 0) ! B(3; 1)
Oe(0; 1) ! Ce(0; 2)
EFe(31; 1) ! Cf (0; 1)
...
GKe(0; 1) ! B(3; 2)
EFe(12; 1) ! B(1; 2)
EFe(10; 1) ! B(0; 2)
EFe(9; 1) ! B(0; 0)
EFe(9; 1) ! B(0; 2)
EFe(9; 3) ! B(0; 4)
EFe(16; 1) ! B(4; 2)
EFe(12; 1) ! B(2; 0)
EFe(12; 1) ! B(2; 2)
EFe(10; 1) ! B(1; 2)
EFe(15; 1) ! B(4; 2)
EFe(11; 1) ! B(2; 0)
EFe(9; 1) ! B(1; 0)
EFe(9; 1) ! B(1; 2)
...
3701.112
3873.456
3874.166
3890.089
4068.016
4178.284
4206.279
4405.845
4634.908
4720.359
4935.625
5056.504
5357.400
5386.996
5614.114
5715.037
6322.144
6380.969
6573.857
6624.429
6679.804
6745.426
6827.644
6877.177
6964.264
7065.021
7178.277
7197.647
7256.025
7352.741
3702
...
...
3889
...
...
...
...
4634.4
...
...
...
...
...
...
...
6320
...
...
...
6678
...
6826
...
...
...
7176
7196
...
...
0.27
...
...
1.25
...
...
...
...
1.97
...
...
...
...
...
...
...
0.83
...
...
...
0.26
...
0.78
...
...
...
1.86
4.11
...
...
Notes.—Using the model we have been able to identify most of the transitions. The vacuum wavelength for the strongest transition
reported by Dieke for each corresponding feature is indicated in col. (5). Col. (6) and (7) correspond to the wavelength and cross section
values measured by the Arkansas group (see text).
a
Estimated cross section at 100 eV from the cross section measured at 200 eV by Watson & Anderson (1977) or at 50 eV by Day et al.
(1979) or Anderson et al. (1977). The estimated values of this column were obtained by using their reported optical excitation functions.
TABLE 6
Experimental and Model Wavelengths and Cross Sections at 100 eV for the Selected Features Indicated in Figs. 13Y16
Feature
31......................................
32......................................
33......................................
34......................................
35......................................
36......................................
37......................................
38......................................
39......................................
40a ....................................
41a ....................................
42......................................
43......................................
44......................................
45......................................
46......................................
47......................................
48......................................
49......................................
50......................................
51......................................
52......................................
53......................................
54......................................
55......................................
56......................................
57......................................
58......................................
59......................................
60......................................
61......................................
62......................................
63......................................
64......................................
65......................................
66......................................
67......................................
68......................................
69......................................
70......................................
71......................................
72......................................
73......................................
74......................................
75......................................
76......................................
77......................................
78b ....................................
79......................................
80......................................
81......................................
82......................................
83......................................
84......................................
kexp
(8)
exp
(1020cm2)
kmod
(8)
mod
(1020cm2)
Transition
kD
(8)
7509.0
7526.7
7540.4
...
7543.9
7573.5
7598.9
...
7608.2
7624.5
7643.4
7652.5
7722.1
8015.7
8093.9
8225.1
8243.0
8275.4
8332.9
8352.0
8369.5
8374.6
8400.7
8446.2
8447.1
8562.4
8632.1
8673.8
8718.0
8749.9
8778.4
8792.6
8799.3
8810.7
8849.2
8879.0
8887.4
8899.0
8901.1
8950.3
8981.3
9019.9
9037.9
9046.0
9085.4
9090.8
9133.2
9163.2
9222.6
9255.9
9321.8
9372.2
9393.1
9407.1
9435.8
9577.2
4.09
11.3
1.98
...
2.03
1.15
2.91
...
1.56
1.61
1.68
1.53
0.945
1.12
0.995
1.46
2.06
7.77
5.13
17.5
2.92
1.96
1.89
3.02
1.96
1.11
2.29
6.93
1.66
5.24
1.02
3.33
2.54
20.9
1.52
7.71
7.09
5.77
25.7
1.25
1.62
1.77
1.10
3.02
1.08
1.24
5.98
2.25
2.54
1.29
1.72
1.10
2.12
3.90
1.03
1.55
7508.48
7526.19
7539.85
7541.00
7543.58
7573.37
7600.44
7600.60
7609.06
7624.40
7643.19
...
...
8013.80
8092.26
8224.86
8242.56
8274.97
8332.12
8351.25
8368.74
8373.60
8400.28
8445.15
8446.54
8561.11
8630.23
8672.69
8716.61
8748.91
8776.43
8791.11
8797.58
8809.62
8847.51
8878.11
8886.45
8898.69
8900.71
8949.33
8980.23
9019.42
9037.58
9045.77
9084.13
9090.65
9132.92
9161.99
9221.42
...
9320.32
9370.60
9393.00
9405.54
9436.04
9575.67
3.03
11.3
2.40
13.95
2.14
1.60
14.39
2.54
0.38
1.60
1.49
...
...
0.69
0.52
2.31
2.55
7.94
4.24
15.5
3.17
1.83
2.91
2.92
1.73
1.18
2.25
9.08
1.59
5.70
1.30
2.99
2.26
17.5
1.25
12.1
8.78
8.27
34.4
2.11
2.11
3.25
1.30
2.97
1.26
1.23
5.93
2.74
3.83
...
1.72
0.88
1.36
3.50
0.66
1.83
EFe(6; 0) ! B(0; 1)
EFe(6; 1) ! B(0; 2)
Je(1; 2) ! Ce(1; 1)
Jf (1; 2) ! Cf (1; 1)
EFe(6; 2) ! B(0; 3)
EFe(6; 3) ! B(0; 4)
Jf (2; 2) ! Cf (2; 1)
Je(2; 2) ! Ce(2; 1)
Jf (1; 2) ! Ce(1; 2)
EFe(8; 1) ! B(1; 2)
EFe(10; 1) ! B(2; 2)
J (3; 2) ! C(3; 1)
J (3; 2) ! C(3; 2)
Ie(2; 1) ! Cf (2; 1)
GKe(3; 3) ! Cf (1; 3)
EFe(6; 3) ! B(1; 2)
EFe(6; 2) ! B(1; 1)
EFe(6; 1) ! B(1; 0)
EFe(6; 0) ! B(1; 1)
EFe(6; 1) ! B(1; 2)
EFe(6; 2) ! B(1; 3)
EFe(10; 1) ! B(3; 0)
EFe(6; 3) ! B(1; 4)
EFe(10; 1) ! B(3; 2)
EFe(8; 1) ! B(2; 2)
EFe(13; 1) ! B(5; 0)
EFe(13; 1) ! B(5; 2)
EFe(4; 3) ! B(0; 2)
EFe(11; 1) ! B(4; 0)
EFe(3; 2) ! B(0; 1)
EFe(16; 1) ! B(7; 2)
EFe(11; 1) ! B(4; 2)
EFe(9; 1) ! B(3; 0)
EFe(3; 1) ! B(0; 0)
EFe(9; 0) ! B(3; 1)
EFe(4; 3) ! B(0; 4)
EFe(3; 0) ! B(0; 1)
EFe(3; 2) ! B(0; 3)
EFe(3; 1) ! B(0; 2)
EFe(9; 3) ! B(3; 4)
EFe(14; 1) ! B(6; 2)
EFe(7; 3) ! B(2; 2)
EFe(7; 2) ! B(2; 1)
EFe(7; 1) ! B(2; 0)
EFe(12; 1) ! B(5; 0)
EFe(7; 0) ! B(2; 1)
EFe(7; 1) ! B(2; 2)
EFe(12; 1) ! B(5; 2)
EFe(7; 3) ! B(2; 4)
B 0 (1; 0) ! EFe(3; 1)
EFe(10; 1) ! B(4; 0)
EFe(10; 0) ! B(4; 1)
EFe(8; 0) ! B(3; 1)
EFe(10; 1) ! B(4; 2)
B 0 (2; 2) ! EFe(6; 1)
EFe(13; 1) ! B(6; 2)
7509.052
7526.737
7540.409
7541.325
7544.050
7573.686
7599.148
7599.344
7608.462
7624.770
7643.840
7652.889
7721.239
8015.550
8093.902
8225.193
8243.065
8275.556
8332.722
8351.833
8369.230
8374.361
8400.587
8445.903
8446.966
8562.267
8631.404
8673.200
8717.484
8749.442
...
8791.997
8798.758
8810.161
8848.704
8878.519
8886.977
8899.204
8901.240
8950.436
8981.353
9020.036
9037.979
9046.057
9085.458
9090.893
9133.204
9163.341
9222.047
9255.369
9321.219
9371.492
9392.494
9406.453
9435.021
9577.113
Notes.—The identification for these transitions is given in col. (6). Col. (7) indicates the corresponding Dieke’s vacuum wavelengths.
a
The upper states are above the dissociation limit H(1s) þ H(2s; 2p) and are not included in our calculation.
b
There is no feature in the model close to this wavelength.
402
AGUILAR ET AL.
Fig. 9.— Calibrated electron impactYinduced fluorescence spectrum of H2 using
grating 2 at 100 eVelectron impact energy from 3600 to 4500 8. The strongest singlet
features are labeled 1Y8 and are identified by rotational transitions in Table 5. The
rotational features studied by the University of Arkansas group are labeled with
stars, and the ordinate positions give their published absolute cross sections.
of the vibrational and rotational quantum number, the measured
vacuum wavelength, the measured emission cross section, the
model wavelength and the model cross section. The model wavelengths and the laboratory wavelengths differ by varying amounts
up to 6 8. The present model involves the strongest 500 singlet
transitions in the 7500 to 10000 8 region obtained with grating 3.
We measured the total 100 eV emission cross section of the
singlet states using grating 2 in the region 3300Y7500 8 to be
1.38 Mbarn. The total emission cross section of H i lines, which
are all located in the grating 2 region is 1.15 Mbarn with the following emission cross section break down: (1) cross section of
H (6562.88) is 0.93Mbarn ( Karolis & Harting1978), (2) cross
section of H (4861.3 8) is 0.15 Mbarn (Karolis & Harting1978),
(3) cross section of H (4340.5 8) is 0.053 Mbarn ( Vroom & De
Heer1969), and (4) cross section of H (4101.7 8) is 0.0202 Mbarn
(Vroom & De Heer1969). The spectral region from 3300Y 6000 8
1 þ
1
1
is dominated by the GK 1 þ
g , H H̄ g , I g , J g : : : !
1 þ
1
0 1 þ
1
B u , C u , B u , and D u band systems.
Fig. 10.— Calibrated electron impactYinduced fluorescence spectrum of H2
using grating 2 at 100 eV electron impact energy from 4500 to 5500 8. The
strongest singlet features are labeled 9Y14 and are identified by rotational transitions in Table 5.
Vol. 177
Fig. 11.— Calibrated electron impactYinduced fluorescence spectrum of H2
using grating 2 at 100 eV electron impact energy from 5500 to 6500 8. The
strongest singlet features are labeled 15Y18 and are identified by rotational transitions in Table 5. The star indicates a rotational features studied by the University
of Arkansas group, and the ordinate positions gives their published absolute cross
sections for that transition.
Figures 9Y12 show the detailed spectral regions of 300 8 each
for grating 2 comparing the data with the published emission
cross section measurements from University of Arkansas (Watson
& Anderson 1977; Anderson et al. 1977; Day et al. 1979). A comparison between the experimental values of the University of
Arkansas and our experimental data is given in Table 5. The
feature numbers for comparison are 4, 9, 17, 21, 23, 27, and 28.
The agreement is excellent for features 23, 27, and 28. For the
other features at the shorter wavelengths of grating 2 the experimental results differ by a factor of 2, with our work being
the lesser of the two results. The spectral resolutions of the two
experiments are similar at better than 1 8, so we cannot question whether the actual rotational lines were resolved in the
earlier study. However, the absolute calibration of the two works
is fundamentally different with the University of Arkansas work
Fig. 12.— Calibrated electron impactYinduced fluorescence spectrum of H2
using grating 2 at 100 eVelectron impact energy from 6500 to 7500 8. The strongest singlet features are labeled 19Y30 and are identified by rotational transitions
in Table 5. The rotational features studied by the University of Arkansas group are
labeled with stars, and the ordinate positions give their published absolute cross
sections.
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
403
Cross section (x10-18 cm2)
No. 1, 2008
Fig. 13.— Calibrated electron impactYinduced fluorescence spectrum of H2
(black curve) using grating 3 at 100 eV impact energy is compared to a model
from 7500 to 8000 8 (red curve). The strongest singlet features are labeled 31Y 41
and are identified by rotational transitions in Table 6.
Fig. 15.— Calibrated electron impactYinduced fluorescence spectrum of H2
(black curve) using grating 3 at 100 eV impact energy compared to a model from
8600 to 9200 8 (red curve). The strongest singlet features are labeled 55Y79 and
are identified by rotational transitions in Table 6.
based on the weak H atomic emission from dissociative excitation of H2. The H cross section has a corresponding larger uncertainty of a factor of 2 than that of H . For example, we can
compare their standard cross section of 0.0347 Mbarn used in
their analysis to the value discussed above of 0.053 Mbarn from
Vroom & De Heer (1969). The work by the University of Arkansas
group was limited to about 12 rotational lines and the agreement
between the two groups is acceptable within the uncertainties of
the absolute calibration standards. A more interesting comparison between the two experiments arises from the ability of the
University of Arkansas group to measure excitation functions of
the gerade-ungerade cross sections and our capability to study
the 20 and 100 eV absolute cross sections of rotational features.
For example, Watson & Anderson (1977) measured the excitation function of the EFYB(2; 1)R0 rotational line at 8273 8. The
ratio of the 20 eV to 100 eV cross section is 1.25. In our experiment we have measured the same ratio for the EF ! B(6; 0) P2
rotational line at 7526.67 8 to be 1.18. The EF excitation function has a low-energy peak of about 14Y20 eV as shown by
Watson & Anderson (1977) and Liu et al. (2002).
Grating 3 data allows a model-data comparison. Figures 13Y16
show the model fit to the grating 3 (7500Y10000 8) data. This
region is composed of the strongest rotational lines of the EF 1 þ
g
band system. In total, the cross section of this spectral region contributes 2.42 Mbarn to the VOIR emission cross section. Our program has not measured the spectral region from 10000Y12000 8
1 þ
and relies on theory for the remainder of the EF 1 þ
g ! B u
band system which contributes 0.78 Mbarn in this region. We
find the total VOIR (3300Y12000 8) emission cross section at
100 eV for singlet gerade-ungerade + ungerade-gerade cross sections to be 4.58 Mbarn. Our 100 eVemission cross section for the
VOIR region of 4:58 1:37 Mbarn is in excellent agreement
with the UV result of Dziczek et al. (2000), who found an emission cross section of 5:0 0:6 Mbarn.
The model appears to be in excellent agreement with the data
from 8200Y10000 8. Below 6000 8 the model and data cannot
be compared at the present time. Between 6000 and 8200 8 other
gerade-ungerade band systems besides the EFYB become important and additional ungerade-gerade band systems are important.
Fig. 14.— Calibrated electron impactYinduced fluorescence spectrum of H2
(black curve) using grating 3 at 100 eV impact energy compared to a model from
8000 to 8600 8 (red curve). The strongest singlet features are labeled 42Y54 and
are identified by rotational transitions in Table 6.
Fig. 16.— Calibrated electron impactYinduced fluorescence spectrum of H2
(black curve) using grating 3 at 100 eV impact energy compared to a model from
9200 to 1000 8 (red curve). The strongest singlet features are labeled 80Y84 and
are identified by rotational transitions in Table 6.
404
AGUILAR ET AL.
Vol. 177
Fig. 17.— Calibrated electron impactYinduced fluorescence spectrum of H2
at 100 eV electron impact energy of one detailed spectral region from 7400 to
7700 8, showing discrepancies between model and data for the JYC rotational
lines.
Fig. 18.— Calibrated electron impactYinduced fluorescence spectrum of H2
at 100 eV electron impact energy of one detailed spectral region from 7300 to
7400 8 (black curve), compared to the model of both singlet-gerade-ungerade
(red curve) and singlet-ungerade-gerade rotational transitions ( green curve).
We list a sample of the strongest rotational lines found from grating 3 observations in Table 6. The feature numbers in the table
are also identified in Figures 13Y16 by feature number from 31 to
84. The feature numbering continues from Table 5. The theoretical cross sections, which have been normalized to the EF !
B(6; 0)P2 rotational line are in excellent agreement (40%) with
the data for most EFYB features.
However, in the region from 7500 to 8200 8 discrepancies
arise for the data-model comparison. There are cases shown in
Table 6 in which the model wavelength may lie more than 1 8
away from the experimentally measured wavelength. For example, a strong observed feature in Table 6 with a well-shifted
model peak is feature 36 with an observed cross section of 2:91 ;
1020 cm2. In the region 7598:9 1 8 Dieke identifies a strong
3FY2C transition labeled (3d J 1 Y2p C 1 u )(2; 2)R1 branch in
Huber’s notation at 7599.1 8 as shown in Table 6. The model has
a strong feature nearby at 7600.94 8 identified as (3d Jf 1 Y
2p Cf 1 u )(2; 2)R1 and a weaker feature (3d Je1 Y2p Ce1 u )
(2; 2)R1 at 7601.10 8. These model features lie within the wavelength model uncertainty of the observed peak (see Table 1) and
consequently appear separately from the laboratory rotational line
in Figure 13. Identifying the strongest 3d JfY2p Cf feature according to Dieke as JYC, we find the model tends to overestimate many
non-EFYB transitions in this region. Similarly, we find a shift for
feature 42 at 8015.7 8, which Dieke identifies as two blended
rotational lines J YC(2; 2)R2; Q1 (3FY2C; Dieke notation) at
8015.55 8 and another rotational line at 8016.2 8 GKYB(1; 1)Q2
(3DY2C ). The closest model peak occurs in the model calculation
at 8014.3 8 and is identified as IYC(2; 2)Q1. This example is one
case in which the model emission cross section underestimates the
data but the wavelength uncertainty explains the position separation. A closer examination of the model for this wavelength
shows the upper state is a mixed state that is a blend of EFYC
(42%), IYC (47%), and GKYC (11%).
In some cases strong singlet emission is observed from states
other than EF and not reported as strong in the current model.
Undesignated spectral features in Figure 13 at 7815.3 and 7903.1 8
are examples where there is no corresponding nearby strong model
feature. For the first feature Dieke finds strong features at 7814.6 8
and 7815.6 8, which are 3EY2C 3d (I 1 g Y2p C 1 u ) (1,1)R1, R5
rotational transitions. For the second wavelength Dieke finds a
strong rotational transition nearby at 7902.5 8 (3DY2C) (0,0)Q1
1
corresponding to 3d GK 1 þ
g Y2p C u . These transitions are apparently unimportant in the current model. For the first case there
is an IYC transition at 7815.1 8 with a weak model cross section
of 1 ; 1022 cm2 compared to the measured value of 0.007 Mbarn
(7 ; 1021 cm2), which is a difference of a factor of 70. For the
second feature we also measure a cross section of 0.007 Mbarn.
There is a model feature at 7901.5 8 which involves the GKY
C(1; 0)Q1 line with a cross section of 3:25 ; 1022 cm2. Once
again there is a disparity in the emission cross section for a nonEFYB transition of greater than a factor of 10. For the next nearby
line found in Figure 13 at 7905.5 8 EFYB(9; 2)R1 the agreement
between the experiment and the model is better than 20%.
A more general case in which discrepancies arise is where
the model substantially overpredicts the resultant intensity for a
band system. This discrepancy appears to be especially true for
the JYC rotational line transitions. Figure 17 shows the comparison between the experimental data and the model in the
7400Y7700 8 wavelength region. The JYC rotational line identifications in this region are indicated and clear data-model
mismatches are observed. One possibility we considered for the
JYC discrepancy between model and data was that the model was
developed from a 20 eV spectrum found in Liu et al. (2002). We
are modeling a 100 eV spectrum, and it is possible that the EFand J-band systems have different energy dependencies; the relative excitation cross section toward J-states used in the model
may have been overestimated. However, as it is evident from
Figures 14Y16, once the wavelength regime extends beyond
8200 8 the model and data show excellent agreement. This is
the region where the EFYB rotational lines dominate except for
a few remaining high-lying gerade state transitions.
5.3. The Singlet Ungerade-Gerade Band Systems
It is energetically possible for the VOIR spectrum to be ex1
1 þ
cited by the B 0 1 þ
u , D u ! EF g dipole-allowed excitation followed by dipole-allowed emissions. We have calculated a
model for these transitions and show the model versus the grating 2 data in Figure 18 over the most intense wavelength range
No. 1, 2008
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
from 7300Y7400 8 for these transitions. The model is now composed of separate spectra for the g-uY and u-gYband systems. The
total model spectrum of summed gerade-ungerade and ungeradegerade model spectra has been used in Figures 17 and 18. The
u-gYmodel band system in green clearly shows two strong rotational lines at 7330.59 and 7353.00 8 in agreement with the laboratory spectrum. The rotational lines arise from Df YEF(2; 6)Q1
and Df YEF(1; 3)Q1 transitions, respectively. The JYC transitions
in red where the agreement between model and data is lacking
has been pointed out in x 5.2.
6. DISCUSSION
We have contributed significantly to the molecular physics database for H2 through the process of analyzing a high-resolution
VOIR emission spectra of molecular hydrogen excited by electron impact at 20 and 100 eV under optically thin, single-scattering
experimental conditions. The analysis of the 100 eV spectrum is
based on newly calculated transition probabilities with rovibraþ
1 þ
tional coupling for the singlet-gerade EF 1 g , GK 1 þ
g , H H̄ g ,
1
1
1 þ
1
0 1 þ
1
I g , J g : : : ! B u , C u , B u , D u . . . band
systems. The 20 and 100 eV high-resolution laboratory spectra
(FWHM ¼ 0:7 8) cover the wavelength range from 3000 to
10000 8, and contain the singlet gerade-ungerade and ungerade-gerade band systems, the H i lines and the Rydberg
series of triplet states dominated by d 3 u, k 3 u , j 3 g ! a 3 þ
g
band systems.
It has been nearly 50 years since a study of the experimental
H2 many-line spectrum in the VOIR has been extensively done.
We have measured and modeled successfully the line intensities
for the H2 spectrum for wavelenths greater than 8200 8. Between 3000 and 8200 8 wavelengths are generally well reproduced; however, the relative intensity of the peaks shows many
discrepancies. In this wavelength region, emission comes from
nine strongly coupled gerade BO states, and we think that remaining uncertainties in the rotational coupling matrix elements are
responsible for the drawbacks in modeling of both steps, excitation and emission. As shown in equation (5) and equations
(A1)Y(A3), the emission probabilities are obtained by squaring a
sum of up to nine terms. The signs of the vibrational wave function depend on the signs of the BO electronic coupling matrix
elements. It is necessary that all electronic matrix elements (dipole momentun and nonadiabatic coupling) use the same sign convention for the electronic wave functions. The sign convention
405
of the electronic matrix elements calculated by MOLPRO is not
known and may be different from the Wolniewicz convention
(see Wolniewicz (1985, 1996).
The excitation step model could also induce discrepancies for
similar reasons. The gerade-state cross sections are the square
of the sum of nine terms involving a collision operator and each
cross product term is capable of producing interference. The
signs of the phase factors could eventually be obtained from a
close comparison to the high-resolution spectrum. For this analysis all the signs are chosen to be positive. We plan to improve the
model for the higher lying gerade states that produce the highresolution spectrum observed from grating 2 in Figures 9Y12.
Another possible reason of model discrepancies is the fact that
the relative excitation cross sections shared among the various
gerade states estimated at 20 eV relatively to EF excitation cross
section states in the earlier work of Liu et al. (2002, 2003) needs
to be improved (see x 4.1).
The absolute value for the cross section for each rotational line
is shown graphically in Figures 9Y16 as the peak intensity. The
intensity for each rotational line is proportional to the excitation
rate times branching ratio or emission cross section. In a highresolution experiment the rotational lines are all resolved and the
area of each feature which represents the photoemission intensity
is proportional to peak height or emission cross section. We believe
this is a very efficient way to retrieve cross section information.
On the basis of the theory we are able to accurately separate
gerade-ungerade spectra from the ungerade-gerade spectra. The
latter furnish less than 10% of the VOIR emission intensity.
The laboratory measurements described in this text were carried out at the Jet Propulsion Laboratory, California Institute of
Technology. The experimental work was supported by the NASA
Planetary Atmospheres and NASA Outer Planets Research Analysis Program Offices. Aguilar and Mangina acknowledge the
support of a NASA Research Associateship while at the Jet
Propulsion Laboratory. We thank A. Spielfiedel for kindly helping us using electronic wave function of her 2003 paper. We
thank D. Bailly for fruitful discussions. The computations were
done at the computer center of Observatoire de Paris, and at
computer center IDRIS (Université Paris-Sud) for running the
code MOLPRO. We also thank Ronald Cummings for his technical assistance in the experiments.
APPENDIX
EXPLICIT FORMULAE OF THE ELECTRIC DIPOLE MATRIX ELEMENTS
We explicitly write here the formulae of the electric dipole matrix elements for the different branches:
pffiffiffiffiffiffiffiffiffiffiffiffiffih
MðPv ;v ; J Þ ¼ Jk þ 1 fEFvk Jk jMEFYB j fBvj Jj þ fEFvk Jk jMEFYB 0 j fB 0 vj Jj þ fGKvk Jk jMGKYB j fBvj Jj
k j k
E D
E
D
þ fGKvk Jk jMGKYB 0 j fB 0 vj Jj þ fH H̄vk Jk MH H̄YB fBvj Jj þ fH H̄vk Jk MH H̄YB 0 fB 0 vj Jj
i
þ fOvk Jk jMOYB j fBvj Jj þ fOvk Jk jMOYB 0 j fB 0 vj Jj þ fPvk Jk jMPYB j fBvj Jj þ fPvk Jk jMPYB 0 j fB 0 vj Jj
pffiffiffiffiffih
i
þ Jk fI þ vk Jk jMIYB j fBvj Jj þ fI þ vk Jk jMIYB 0 j fB 0 vj Jj þ fRþ vk Jk jMRYB j fBvj Jj þ fRþ vk Jk jMRYB 0 j fB 0 vj Jj
pffiffiffiffiffiffiffiffiffiffiffiffiffih
Jk þ 2 fEFvk Jk jMEFYC j fCþ vj Jj þ fEFvk Jk jMEFYD j fDþ vj Jj þ fGKvk Jk jMGKYC j fCþ vj Jj þ fGKvk Jk jMGKYD j fDþ vj Jj
D
E D
E þ fH H̄vk Jk MH H̄YC fCþ vj Jj þ fH H̄vk Jk MH H̄YD fDþ vj Jj þ fOvk Jk jMOYC j fCþ vj Jj þ fOvk Jk jMOYD j fDþ vj Jj
i
þ fPvk Jk jMPYC j fCþ vj Jj þ fPvk Jk jMPYD j fDþ vj Jj
406
AGUILAR ET AL.
Vol. 177
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E D
E D
E D
Ei
Jk ðJk þ 2Þ hD
fI vk Jk jMI YC j fC vj Jj þ fI vk Jk jMI YD j fD vj Jj þ fR vk Jk jMRYC j fC vj Jj þ fR vk Jk jMRYD j fD vj Jj
þ
ðJk þ 1Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E D
E D
E D
Ei
Jk ðJk Y1Þ hD
þ
fJ vk Jk jMJ YC j fC vj Jj þ fJ vk Jk jMJ YD j fD vj Jj þ fS vk Jk jMSYC j fC vj Jj þ fS vk Jk jMSYD j fD vj Jj ; ðA1Þ
2ð J k þ 1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2Jk þ 1 fI vk Jk jMIYB jfBvj Jj þ fI vk Jk jMIYB 0 j fB 0 vj Jj þ fR vk Jk jMRYB j fBvj Jj þ fR vk Jk jMRYB 0 j fB 0 vj Jj
MQ
(vk ;vj ;Jk )
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih
2Jk þ 1 fEFvk Jk jMEFYC j fC vj Jj þ fEFvk Jk jMEFYD j fD vj Jj þ fGKvk Jk jMGKYC j fC vj Jj þ fGKvk Jk jMGKYD j fD vj Jj
D
E D
E þ fH H̄vk Jk MH H̄YC fC vj Jj þ fH H̄vk Jk MH H̄YD fD vj Jj þ fOvk Jk jMOYC j fC vj Jj þ fOvk Jk jMOYD j fD vj Jj
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i
(2Jk þ 1) h
þ fPvk Jk jMPYC j fC vj Jj þ fPvk Jk jMPYD j fD vj Jj þ
fI vk Jk jMIYC j fC vj Jj þ fI vk Jk jMIYD j fD vj Jj
Jk (Jk þ 1)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i
(2Jk þ 1)(Jk þ 2)(Jk 1) h
þ fR vk Jk jMRYC j fC vj Jj þ fR vk Jk jMRYD j fD vj Jj fJ vk Jk jMJ YC j fC vj Jj
2Jk (Jk þ 1)
i
þ fJ vk Jk jMJ YD j fD vj Jj þ fS vk Jk jMSYC j fC vj Jj þ fS vk Jk jMSYD j fD vj Jj ;
ðA2Þ
h
pffiffiffiffiffi MðRv ;v ;J Þ ¼ Jk fEFvk Jk jMEFYB jfBvj Jj þ fEFvk Jk jMEFYB 0 j fB 0 vj Jj þ fGKvk Jk jMGKYB j fBvj Jj
k j k
E D
E D
þ fGKvk Jk jMGKYB 0 j fB 0 vj Jj þ fH H̄vk Jk MH H̄YB fBvj Jj þ fH H̄vk Jk MH H̄YB 0 fB 0 vj Jj þ fOvk Jk jMOYB j fBvj Jj
i
þ fOvk Jk jMOYB 0 j fB 0 vj Jj þ fPvk Jk jMPYB j fBvj Jj þ fPvk Jk jMPYB 0 j fB 0 vj Jj
pffiffiffiffiffiffiffiffiffiffiffiffiffi
þ Jk þ 1 fI þ vk Jk jMIYB j fBvj Jj þ fI þ vk Jk jMIYB 0 j fB 0 vj Jj þ fRþ vk Jk jMRYB j fBvj Jj þ fRþ vk Jk jMRYB 0 j fB 0 vj Jj
pffiffiffiffiffiffiffiffiffiffiffiffiffih
Y Jk 1 fEFvk Jk jMEFYC j fCþ vj Jj þ fEFvk Jk jMEFYD j fDþ vj Jj þ fGKvk Jk jMGKYC j fCþ vj Jj þ fGKvk Jk jMGKYD j fDþ vj Jj
D
E D
E þ fH H̄vk Jk MH H̄YC fCþ vj Jj þ fH H̄vk Jk MH H̄YD fDþ vj Jj þ fOvk Jk jMOYC j fCþ vj Jj þ fOvk Jk jMOYD j fDþ vj Jj
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E D
E
i
ðJk 1ÞðJk þ 1Þ hD
þ fPvk Jk jMPYC j fCþ vj Jj þ fPvk Jk jMPYD j fDþ vj Jj fI vk Jk jMI YC j fC vj Jj þ fI vk Jk jMIYD j fD vj Jj
Jk
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D
E D
Ei
E
ðJk þ 1ÞðJk þ 2Þ hD
þ fR vk Jk jMRYC j fC þvj Jj þ fR vk Jk jMRYD j fD þvj Jj þ
fJ vk Jk jMJ YC j fC vj Jj
2Jk
D
E D
E D
Ei
þ fJ vk Jk jMJ YD j fD vj Jj þ fS vk Jk jMSYC j fC þvj Jj þ fS vk Jk jMSYD j fD þvj Jj ;
ðA3Þ
and levels are split by nonadiabatic couplings into +, +, and , , in a doublet splitting. + and + levels give rise to
e-parity rovibronic levels and, conversely, and levels give rise to f-parity levels, following the notation of Brown et al. (1975).
REFERENCES
Abgrall, H., Roueff, E., & Drira, I. 2000, BAAS, 141, 297
Crosswhite, H. M. 1972, The Hydrogen Molecule Wavelength Tables of
Abgrall, H., Roueff, E., Launay, F., & Roncin, J. Y. 1994, Can. J. Phys., 72, 856
Gerhard Heinrich Dieke ( New York: Wiley-Interscience)
Abgrall, H., Roueff, E., Launay, F., Roncin, J. Y., & Subtil, J. L. 1993a, BAAS,
Dalgarno, A. 1993, J. Chem. Soc. Faraday Trans., 89, 2111
101, 273
———. 1995, in ASP Conf. Ser. 80, Physics of the Interstellar Medium and
———. 1993b, BAAS, 101, 323
Intergalactic Medium, ed. A. Ferrara (San Francisco: ASP), 37
———. 1993c, J. Molec. Spectrosc., 157, 512
Dalgarno, A., Yan, M., & Liu, W. 1999, ApJS, 125, 237
Abgrall, H., Roueff, E., Liu, X., & Shemansky, D. E. 1997, ApJ, 481, 557
Davidson, E. 1961, J. Chem. Phys., 35, 1189
Abgrall, H., Roueff, E., Liu, X., Shemansky, D. E., & James, G. K. 1999, J.
Day, R. L., Anderson, R. J., & Shaprton, F. A. 1979, J. Chem. Phys., 71, 3683
Phys. B, 32, 3813
Dieke, G. H. 1958, J. Molec. Spectrosc., 2, 494
Ajello, J. M., Aguilar, A., Mangina, R. S., James, G. K., Geissler, P., & Trafton
Dieke, G. H., & Cunningham, S. P. 1965, J. Molec. Spectrosc., 18, 288
L. 2008, J. Geophys. Res. Planets, 113, E03002
Dressler, K., & Wolniewicz, L. 1984, Can. J. Phys., 62, 1706
Ajello, J. M., & Shemansky, D. E. 1993, ApJ, 407, 820
———. 1995, Berichte der Bunsen-Gesellschaft-Physical Chemistry Chem.
Ajello, J., Vatti Palle, P., & Osinski, G. 2002, in Current Developments in
Phys., 99, 246
Atomic Molecular Physics, ed. M. Mohan ( New York: Klewer), 143
Dziczek, D., Ajello, J. M., James, G. K., & Hansen, D. L. 2000, Phys. Rev. A,
Ajello, J., P. Vatti Palle, H. Abgrall, E. Roueff, A. Bhardwaj & J. Gustin, J.
61, 64702
2005a, ApJS, 159, 314
Gredel, R., Lepp, S., & Dalgarno, A. 1987, ApJ, 323, L137
Ajello, J., et al. 2001, Icarus, 152, 151
———. 1989, ApJ, 347, 289
———. 2005b, Icarus, 178, 327
Herczeg, G., Linsky, J. L., Valenti, J. A., Johns-Krull, C. M., & Wood, B. E.
Anderson, R. J., Watson, J., & Sharpton, F. A. 1977, J. Opt. Soc. Am., 67, 1641
2002, ApJ, 572, 310
Baltayan, P., & Nedelec, O. 1976, J. Quant. Radiat. Transfer, 16, 207
Herczeg, G., Wood, B. E., Linsky, J. L., Valenti, J. A., & Johns-Krull, C. M.
Brown, J. M., et al. 1975, J. Mol. Spectr., 55, 500
2004, ApJ, 607, 369
Chung, S., & Lin, C. C. 1978, Phys. Rev. A, 17, 1874
Huber, K. P., & Herzberg, G. 1979, in Constants of Diatomic Molecules, ed.
Chung, S., Lin, C. C., & Lee, E. T. P. 1975, Phys. Rev. A, 12, 1340
K. P. Huber ( New York: Van Nostrand), 240
No. 1, 2008
ELECTRON-EXCITED MID-UV TO NEAR-IR H2 SPECTRUM
James, G. K., Ajello, J. M., & Pryor, W. J. 1998, Geophys. Res., 103, 20113
Johnson, B. R. 1978, J. Chem. Phys., 69, 4678
Jonin, C., Liu, X. Ajello, J. M., James, G. K., & Abgrall, H. 2000, ApJS, 129,
247
Karolis, C., & Harting, E. 1978, J. Phys. B, 11, 357
Lavrov, B. P., & Pipa, A. V. 2002, Opt. & Spectrosc., 92, 709
Liu, X., Ahmed, S. M., Multari, R. A., James, G. K., & Ajello, J. M. 1995,
ApJS, 101, 375
Liu, X., Shemansky, D. E., Abgrall, H., Roueff, E., Dziczek, D., Hansen, D. L.,
& Ajello, J. M. 2002, ApJS, 138, 229
Liu, X., Shemansky, D. E., Ahmed, S. M., Ajello, J. M., Abgrall, H., & Roueff,
E. 2003, J. Phys. B, 36, 173
Liu, X., Shemansky, D. E., Ahmed, S. M., James, G. K., & Ajello, J. M. 1998,
J. Geophys. Res., 103, 26739
Liu, X., Shemansky, D., Ajello, J. M., James, G., & Abgrall, H. 2000, ApJS,
129, 267
Missakian, M., & Zorn, J. C. 1971, Phys. Rev. Lett., 27, 174
Möhlmann, G. R., & de Heer, F. J. 1976, Chem. Phys. Lett., 43, 240
Moos, H. W., et al. 2000, ApJ, 538, L1
Pryor, W. R., et al. 1998, J. Geophys. Res., 103, 20149
407
Quadrelli, P., Dressler, K., & Wolniewicz, L. 1990a, J. Chem. Phys., 93, 4958
Raymond, J., et al. 1997a, ApJ, 482, 881
———. 1997b, ApJ, 489, 314
Senn, P., Quadrelli, P., & Dressler, K. 1988, J. Chem. Phys., 89, 7401
Sharp, T. E. 1971, Atomic Data, 2, 119
Shull, J. M., et al. 2000, ApJ, 538, L73
Spielfiedel, A. 2003, J. Mol. Spectr., 217, 162
Spindler, R. J. 1969, J. Quant. Spectrosc. Radiat. Transfer, 9, 597
Tawara, H., Itakawa, Y., Nishimura, H., & Yoshino, M. 1990, J. Phys. Chem.
Ref. Data, 19, 617
Vroom, D. A., & De Heer, F. J. 1969, J. Chem. Phys., 50, 580
Watson, J., & Anderson, R. J. 1977, J. Chem. Phys., 66, 4025
Werner, H.-J., & Knowles, P. J. 1988, J. Chem. Phys., 89, 5803
Wolniewicz, L. 1985, J. Chem. Phys., 82, 4720
———. 1995a, J. Mol. Spectrosc., 169, 329
———. 1995b, J. Mol. Spectrosc., 174, 132
———. 1996, J. Mol. Sprectrosc., 180, 398
———. 1994, J. Chem. Phys., 100, 444
Yu, S., & K. Dressler, J. 1994, J. Chem. Phys., 101, 7692