IOP PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
doi:10.1088/0953-4075/42/18/185203
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203 (13pp)
Electron-impact excitation and emission
cross sections of the H2 B 1Σ+u and D 1Πu
states and rotational dependence of
photodissociation cross sections of the
B 1Σ+u and D 1Πu continua
Xianming Liu1 , Paul V Johnson1 , Charles P Malone1,2 , Jason A Young1 ,
Donald E Shemansky3 and Isik Kanik1
1
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena,
CA 91109, USA
2
Department of Physics, California State University, Fullerton, CA 92834, USA
3
Planetary and Space Science Division, Space Environment Technologies, 230 North Halstead Street,
Pasadena, CA 91107, USA
E-mail: xianming@jpl.nasa.gov, paul.v.johnson@jpl.nasa.gov and isik.kanik@jpl.nasa.gov
Received 8 May 2009, in final form 5 August 2009
Published 9 September 2009
Online at stacks.iop.org/JPhysB/42/185203
Abstract
Rotational and vibrational dependence of photodissociation cross sections and oscillator
strengths to the continuum levels of the H2 B 1 u+ and D 1 u states have been examined. The
electron-impact excitation, dissociation and emission cross sections of the B 1 u+ − X 1 g+ and
D 1 u − X 1 g+ band systems have been obtained for the first time over a wide energy range
by using calculated continuum oscillator strengths along with previously published discrete
transition probabilities and Lyman and Werner bands excitation functions. The present B 1 u+
photodissociation cross section from the Ji = 0 level is in excellent agreement with that
obtained by Glass-Maujean (1986 Phys. Rev. A 33 346–50). Photoexcitation from the
X 1 g+ (vi = 0) state to the D 1 u continuum is found to be weak. The present calculation
shows significant contribution of quasi-resonance, which arises from transitions to the
quasi-bound levels above the dissociation limit but stabilized by the centrifugal potential. The
quasi-resonance is largely responsible for the significant rotational dependence of the
continuum oscillator strength of the B 1 u+ − X 1 g+ (0) transition, which, in turn, leads to the
noticeable temperature dependence of electron-impact excitation and emission cross sections.
B 1 u+ − X 1 g+ and D 1 u − X 1 g+ electron-impact excitation, emission, and dissociation
cross sections, important for modelling dayglow and auroral activity in the atmospheres of the
outer planets, are presented.
(Some figures in this article are in colour only in the electronic version)
solar radiation (in particular, the H Lyman-α line) produces
highly excited H2 X 1 g+ (Liu et al 2007, van Harrevelt and
van Hemert 2000, 2008). The long lifetime of the excited
H2 (Wolniewicz et al 1998) in cometary environments makes
it possible to identify H2 emission arising from solar photon
or photoelectron excitation of the hot H2 . Such phenomena
1. Introduction
Excitation of molecular hydrogen by photon and electron
impact is an important process in molecular clouds, comets
and atmospheres of the outer planets. The photoexcitation
of the X̃ 1 A1 − B̃ 1 A1 transition of water in comet comae by
0953-4075/09/185203+13$30.00
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© 2009 IOP Publishing Ltd
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
have been recently identified in Far Ultraviolet Spectroscopic
Explorer (FUSE) observations of comets C/2000 WM1
(LINEAR) 4 and C/2001 A2 (LINEAR) (Liu et al 2007)
and has also been observed in planetary nebulae NGC
6853 and NGC 3132 where excitation of vibrationally hot
H2 X 1 g+ (vi = 2) by Lyman-α has been identified (Lupu et al
2006). Additionally, the atomic hydrogen plume observed by
the Cassini Ultraviolet Imaging Spectrograph (Cassini UVIS)
in the atmosphere of Saturn is attributed to low-energy electron
excitation (14 eV) of vibrationally excited H2 X 1 g+ to the
dissociative b 3 u+ state (Shemansky et al 2009). The highly
non-local thermodynamic equilibrium (LTE) nature of H2
production in comets and the high-temperature environment
of the planetary nebulae require photon and electron cross
sections from excited ro-vibrational levels. The aurorae
and dayglow emissions in the atmospheres of giant outer
planets both arise from electron-impact excitation of H2 .
Spacecraft observations, including Cassini UVIS, FUSE,
Galileo, Hopkins Ultraviolet Telescope (HUT), Hubble Space
Telescope (HST) and Voyager, have shown the importance
of electron-impact excitation of H2 singlet ungerade states
in aurorae and dayglow processes (Broadfoot et al 1981,
Shemansky and Ajello 1983, Shemansky 1985, Feldman
et al 1993, Clarke et al 1994, Trafton et al 1994, and Kim
et al 1995, Morrissey et al 1997, Wolven and Feldman 1998,
Ajello et al 1998, 2001, 2005, Pryor et al 2001, 2005, Gustin
et al 2004). Reliable electron-impact excitation and emission
cross sections are required to interpret these observations.
Finally, the ongoing Cassini UVIS missions have accumulated
a large observational database of Saturn. The determination
of the Saturn atmospheric structure, such as the H and H2
mixing ratio, also requires accurate excitation, emission and
dissociation cross sections of H2 by photons and electrons.
The vacuum ultraviolet (VUV) emission spectrum of
molecular hydrogen is dominated by the transitions between
the X 1 g+ state and the first two members of the singlet
ungerade Rydberg series, the B 1 u+ and C 1 u states.
Below 1000 Å, the contributions from the B 1 u+ , D 1 u
and higher states are important.
Figure 1 shows the
adiabatic potential energy curves and dissociation limits
for several of the lowest singlet ungerade states. The
B 1 u+ , C 1 u and B 1 u+ states are not predissociated by
other singlet ungerade states. However, excitation to the
H(1s)+H(2) continuum is significant. Dissociative excitation
to the continua of the B 1 u+ and C 1 u states from the
ground X 1 g+ (0) level is weak but not negligible (Allison
and Dalgarno 1969, Glass-Maujean et al 1985a), and the
dissociation via the B 1 u+ continuum is very significant
(Glass-Maujean et al 1985a, Glass-Maujean 1986). GlassMaujean (1986) has also shown that dissociation via the C 1 u
continuum from the vibrationally excited X 1 g+ state is very
significant. Predissociation is possible for the other npσ 1 u+
and npπ 1 +u states. A number of experimental and theoretical
investigations by Glass-Maujean and cowokers have shown
that the predissociation in the npσ 1 u+ and npπ 1 +u states is
20
18
16
Potential Eenergy (eV)
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
R(a0)
Figure 1. Relevant adiabatic potential energy curves and
corresponding dissociation limits. Energies are relative to the (vi =
0, Ji = 0) levels of the X 1 g+ state. All potential energy curves are
based on the calculations of Wolniewicz (1993), Wolniewicz et al
(1998), Staszewska and Wolniewicz (2002), Wolniewicz and
Staszewska (2003a, 2003b).
primarily caused by direct or indirect coupling to the B 1 u+
continuum (Glass-Maujean 1978, Glass-Maujean et al 1979,
1984, 1985b, 1987). The D 1 +u levels above the H(1s)+H(2)
limit are directly predissociated by the B 1 u+ continuum
(Glass-Maujean et al 1979). The predissociation of npσu1 u+
(n > 4) takes place by homogeneous coupling with the B 1 u+
continuum levels (Glass-Maujean 1978). The predissociation
of npπu1 +u (n> 3) takes place by either npπu1 +u − D 1 +u
homogeneous coupling followed by the D 1 +u − B 1 u+
Coriolis coupling or the npπu1 +u − npσu1 u+ Coriolis coupling
followed by the npσu1 u+ −B 1 u+ homogeneous coupling. The
1 −
1 +
D 1 −
u and npπu u states are not coupled with the B u or
1 +
other npσu u states. They can only couple with a dissociating
1 −
u state. Among the npπu1 −
u states below the H(1s)+
1 −
H(n = 3) limit, C 1 −
u is the only dissociative u state.
1 −
Since npπu u states are only weakly coupled with the C 1 −
u
state, their predissociation rates are negligibly small.
The electron excitation and emission cross sections of
the discrete levels of the B 1 u+ and C 1 u states over a
wide energy range have been determined from the excitation
function measurement of Liu et al (1998) using the oscillator
strength calculations of Abgrall et al (1993a, 1993b, 1993c).
4
Stands for Lincoln Near-Earth Asteroid Research program, in which the
comet was discovered.
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
The B B̄ 1 u+ and D 1 u cross sections for the transitions to
the levels below their dissociation limits have been obtained
by Glass-Maujean et al (2009) from threshold to 1000 eV.
However, the cross sections of other singlet ungerade states,
in general, are only available at a few energies, usually 100 eV
(Jonin et al 2000, Glass-Maujean et al 2009). A recent
review article on electron-impact cross sections of H2 by
Yoon et al (2008) stressed the need for cross sections of
higher singlet ungerade states over a wide energy range. The
present work fills the gap in the B 1 u+ and D 1 u cross
section database by using the B 1 u+ and C 1 u shape function
of Liu et al (1998) and theoretically calculated oscillator
strengths. Unlike the B 1 u+ and C 1 u states, dissociation and
predissociation are very significant for the B 1 u+ and D 1 u
states. To accurately evaluate the excitation cross section of
the dissociative and predissociative levels, the calculation is
carried out for photoexcitation from various ro-vibrational
levels of the X 1 g+ state to the B 1 u+ and D 1 u continua
and to the D 1 +u discrete levels above the B 1 u+ continuum.
Note that Dalgarno and Allison (1969) and Glass-Maujean
(1986) have calculated the photodissociation cross section of
B 1 u+ from rotationless levels of the X 1 g+ (0) state. GlassMaujean (1986) has also examined vibrational dependence of
the dissociation cross section of the Ji = 0 level. Burciaga
and Ford (1991) have investigated C 1 u vibrational shape
resonance that is caused by the small hump in its potential
energy curve and examined the effect of the B 1 u+ − C 1 +u
non-adiabatic coupling on the photodissociation cross section
of the C 1 u state. Beswick and Glass-Maujean (1987) have
considered non-adiabatic coupling between the B 1 u+ and
B 1 u+ continua and demonstrated quantum inference between
photodissociation cross sections of the B 1 u+ and B 1 u+
states. Cheng et al (1998) have experimentally investigated
photodissociaton resonance near the H(1s)+H(2) threshold
and have shown a great variation in the dissociation cross
section profiles. Glass-Maujean et al (2007a) have recently
calculated photodissociation cross sections of the B B̄ 1 u+
and D 1 u states and have achieved good agreement with
their high resolution measurement. Until the present work,
however, the rotational dependence of the dissociation cross
section of the B 1 u+ and D 1 u states had not been thoroughly
investigated.
The singlet ungerade states of the hydrogen molecule
have been extensively investigated by many experimental
techniques including electron excitation (Ajello et al 1982,
1984, 1988, Khakoo and Trajmar 1986, Liu et al 1995,
2000, 2002, 2003, Abgrall et al 1997, 1999, Jonin et al
2000, Dziczek et al 2000, Glass-Maujean et al 2009),
photoabsorption (Herzberg and Howe 1959, Namioka
1964a, 1964b, Takezawa 1970, Herzberg and Jungen 1972,
Dabrowski 1984, Glass-Maujean et al 1984, 1985a, 1985b,
1987, 2007a, 2007b, 2007c, 2008a), photoemission (Roncin
et al 1984, Larzillière et al 1985, Abgrall et al 1993a, 1993b,
1993c, 1994, Roudjane et al 2006, 2007), photoionization
(Dehmer and Chupka 1976, 1995) and nonlinear laser
spectroscopy (Hinnen et al 1994a, 1994b, 1995a, Hinnen and
Ubachs 1995, 1996, Hogervorst et al 1998, Reinhold et al
1996, 1997, De Lange et al 2001, Koelemeij et al 2003,
Greetham et al 2003, Ubachs and Reinhold 2004, Hollenstein
et al 2006, Ekey et al 2006, Salumbides et al 2008, Gabriel
et al 2009). The spectral atlas of Roncin and Launay (1994),
in particular, has provided an extensive tabulation of accurate
H2 transition frequencies and spectral assignments obtained
by a series of photoemission studies.
As the simplest neutral molecule, molecular hydrogen
has been theoretically investigated by many researchers. The
two principal theoretical methods used to calculate the excited
states of H2 are traditional ab initio calculation, which deals
at most a few coupled electronic states, and multichannel
quantum defect theory (MQDT), which treats the whole
family of Rydberg states. Since the pioneering work of
Kolos and Wolniewicz (1968), ab initio calculations of the
potential energies and transition moments have been developed
for several decades. Accurate and sophisticated electronic
potential energy calculations, including the adiabatic, diagonal
non-adiabatic, relativistic and radiative corrections, have been
carried out (Dressler and Wolniewicz 1986, Wolniewicz
and Dressler 1992, 1994, Wolniewicz 1993, 1995a, 1995b,
2007, Wolniewicz et al 1998, Staszewska and Wolniewicz
2002, Wolniewicz and Staszewska 2003a). Calculations
of the H2 transition moment functions (Wolniewicz 1995c,
Dressler and Wolniewicz 1995, Wolniewicz and Staszewska
2003a, 2003b) and non-adiabatic coupling of the first several
members of singlet ungerade Rydberg series have been
reported recently (Wolniewicz et al 2006). Since the first
application to interpret a high-resolution H2 photoabsorption
spectrum by Herzberg and Jungen (1972), MQDT has been
developed to treat autoionization, dissociation and the rovibronic structures of singlet and triplet manifolds (Jungen
and Atabek 1977, Ross and Jungen 1987, 1994a, 1994b,
1994c, Jungen and Ross 1997, Matzkin et al 2000, Ross
et al 2001, Kirrander et al 2007, Glass-Maujean and Jungen
2009). Of particular relevance, Abgrall et al (1993a, 1993b,
1993c, 1994, 1997, 2000) have carried out extensive semi-ab
initio calculations of the non-adiabatic transition probabilities
of the B 1 u+ , C 1 u , B 1 u+ and D 1 u − X 1 g+ band
systems. Glass-Maujean et al (2007b, 2007c, 2008a, 2008b)
have calculated adiabatic transition probabilities of higher
npσ 1 u+ and npπ 1 u states and tested the accuracy with
high-resolution synchrotron photoabsorption measurements.
Glass-Maujean et al (2009) also performed non-adiabatic
calculations of the B 1 u+ , B B̄ 1 u+ , 5pσ 1 u+ , D 1 +u and
D 1 +u states for limited rotational levels. The Q-branch
1 +
transition probabilities of the npπ 1 −
u − X g band systems
for n up to 30 have been obtained by a recent MQDT
calculation by Glass-Maujean and Jungen (2009).
2. Theory
In this section, we present calculations of the photodissociation
cross sections and relate the cross sections to the electronimpact dissociative excitation cross section. Throughout the
paper, we use indices i and j to denote appropriate levels of
the X 1 g+ and singlet ungerade states, respectively.
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
Differences in the definition of the Hönl–London factors
lead to a degeneracy factor, Gij , to appear or not appear in
equations (5)–(7) or its counterpart. In Allison and Dalgarno
(1969), Le Roy et al (1976), Glass-Maujean (1986) and Weck
et al (2003), Gij takes a value of 1 for a – transition
and 2 for a – electronic transition. The variation in the
definition of Hönl–London factors in the literature has been
fully described by Hansson and Watson (2005). In the present
work, the definition of Hj i (Jj , Ji ) by Hansson and Watson
(2005) is followed. Consequently, Gij does not appear in
equations (5)–(7).
The discrete and continuum nuclear wavefunctions,
χvi ,Ji (R) and χEk ,Jj (R), are obtained by a numerical solution
of the Schrödinger equation. The Born-Oppenheimer (BO)
potential of Wolniewicz et al (1998), along with adiabatic,
relativistic and radiative corrections of Wolniewicz (1993), is
used for the X 1 g+ potential. For the B 1 u+ state, BO and
adiabatic potentials calculated by Staszewska and Wolniewicz
(2002), and Wolniewicz and Staszewska (2003b) are used. In
addition, R-independent relativistic correction, −1.92 cm−1 ,
and radiative correction, 0.308 cm−1 , both suggested by
Wolniewicz et al (2006) are also used. The BO and adiabatic
potentials of Wolniewicz and Staszewska (2003a), along with
R-independent relativistic and radiative corrections, are used
for the D 1 u state. Finally, ab initio transition moments,
D(R), calculated by Wolniewicz and Staszewska (2003a,
2003b) are utilized for the photodissociation cross sections
of the B 1 u+ − X 1 g+ and D 1 u − X 1 g+ transitions.
In the absence of significant non-adiabatic coupling, the
ab initio potentials yield accurate eigenvalues of discrete
levels. The calculated dissociation energy of X 1 g+ (vi =
0, Ji = 0), using the LEVEL 8 program (Le Roy 2007),
is 36 117.558 cm−1 , which can be compared with a recent
experimental value of 36 118.073 ± 0.004 cm−1 (Zhang
et al 2005a). If the non-adiabatic correction, 0.499 cm−1 ,
calculated by Wolniewicz (1995a) is added, the calculated
dissociation energy would be 36118.057 cm−1 , which is just
0.016 cm−1 lower than the experimental value. For the
1 +
1
+
D 1 −
u state, which is not affected by u − u coupling,
the calculated Q(1) transition frequencies agree with those
measured or calculated by Abgrall et al (1994, 2000) within
1.5 cm−1 . However, depending on the magnitude of the
1 + 1 +
u − u coupling, the calculated P- and R-branch transition
frequencies of the B 1 u+ and D 1 u states can differ from
those of Abgrall et al (1994, 2000) by as much as a few
tens of cm−1 . Finally, the calculated Q-branches of the
D 1 u −X 1 g+ (vi = 0) transition probabilities generally agree
with those of Abgrall et al (1994) within ∼3%, even though
Abgrall et al (1994) utilized the transition moment calculated
by Rothenberg and Davidson (1967).
Finally, the continuum photoabsorption oscillator strength
is related to the photodissociation cross section by (Liu and
Shemansky 2006)
∞
mc
σ ph (vi , Ji , E) dEk
f (vi , Ji ; j ) =
π he2 0
∞
= 1.1296 × 1012
σ ph (vi , Ji ; E) dEk ,
(8)
2.1. Photodissociation cross section
Under irradiation by photons with energy of Eph = hcν, the
dissociation cross section for excitation from (vi , Ji ) to the
continuum level (Ek , Jj ) is (Le Roy et al 1976)
8π 3 ν Hj i (Jj , Ji )
σ ph (vi , Ji ; ν) =
3hc J
2Ji + 1
j
2
(1)
× χEk ,Jj (R)D(R)χvi ,Ji (R) ρ(Ek ),
where Hj i (Jj , Ji ) and D(R) are the Hönl–London factors and
the electric dipole transition moment, respectively. χvi ,Ji (R)
and χEk ,Jj (R) are the radial wave functions of the initial level
i and the continuum radial level j , respectively. ρ(Ek ) is the
density of the state normalization factor at energy Ek above
the dissociation limit of state j :
δ(Ek − Ek )
χEk ,Jj (R)χEk ,Jj (R) =
ρ(Ek )
Ek = Eph + E(vi , Ji ) − Vj (R → ∞),
(2)
(3)
where Vj (R → ∞) is the asymptotic potential energy of
state j.
For programming purpose, it is convenient to convert the
internuclear distance, R, to a dimensionless distance z = R/R0
with R0 being an arbitrarily selected scaling length. In the
present work, the amplitude of the continuum wave function
is asymptotically normalized to unity:
lim χEk ,Jj (z) = sin[kz + ηJj (Ek )]
(4)
√
where ηJj (Ek ) is the phase shift and k = 2π R0 2μcEk / h,
with μ being the reduced mass of H2 . The conversion and
normalization
give a density of the state factor of ρ(Ek ) =
√
2R0 2μc/ hEk , which leads to the photodissociation cross
section in equation (1) being re-written as (Le Roy et al 1976):
2μ Hj i (Jj , Ji )
16π 3 ν
ph
R0
σ (vi , Ji ; ν) =
3
3
h cEk J
2Ji + 1
j
2
(5)
× χEk ,Jj (z)D(z)χvi ,Ji (z) .
z→∞
When Ek and ν are in units of cm−1 , D(z) in debye, μ
in unified atomic mass units and the cross section in units of
cm2 , equation (5) becomes
2 H (J , J )
ji j
i
ph
−20 μν
σ (vi , Ji ; Eph ) = 3.2270 × 10
Ek J
2Ji + 1
j
2
(6)
× χEk ,Jj (z)D(z)χvi ,Ji (z) ,
where R0 = 1 Å has been assumed. When both ν (Eph )
and Ek are in hartree, D(z) in au, and R0 in bohr (a0 ),
equation (6) can be recast as (Dunn 1968, Allison and Dalgarno
1969),
σ ph (vi , Ji ; Eph ) = 2.5936 × 10−17
2 4μEph
Hj i (Jj , Ji )
×
mH Ek J
2Ji + 1
j
2
× χEk ,Jj (z)D(z)χvi ,Ji (z) (7)
where mH is the mass of a hydrogen atom.
0
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
where σ ph is in cm2 and kinetic energy, Ek , is in cm−1 .
The discrete absorption oscillator strength, f (vi , vj ; Ji , Jj ),
is related to the line transition probability, A(vj , vi ; Jj , Ji ), by
(Abgrall and Roueff 2006)
2Jj + 1 A(vj , vi ; Jj , Ji )
,
f (vi , vj ; Ji , Jj ) = 1.4992 ×
2
2Ji + 1 Eph
(vi , vj ; Ji , Jj )
B 1 u+ and D 1 u −X 1 g+ band systems are not available. The
Cm /C7 (m = 0–6) and C8 coefficients obtained by Liu et al
(1998) for the Lyman and Werner band systems are, therefore,
used for the B 1 u+ − X 1 g+ and D 1 u − X 1 g+ transitions.
The excitation cross section for a band system in the
present study will be defined as the statistical average of the
ro-vibrational cross section components:
1 σ (vi , vj ; Ji , Jj )Ni
(12)
σex =
NT i,j
(9)
where Eph (vi , vj ; Ji , Jj ) is the i → j transition frequency
in cm−1 .
where Ni is the population at the (vi , Ji ) level and NT is the
total population of the H2 X 1 g+ state. The corresponding
emission cross section is then given by
1 σ (vi , vj ; Ji , Jj )(1 − ηj )Ni ,
(13)
σem =
NT i,j
2.2. Electron-impact excitation and emission
For electronic states with significant emission yields,
experimental measurement of the relative emission intensity
as a function of electron-impact energy from threshold to a
few keV gives rise to the so-called excitation function. If the
electron excitation is dipole-allowed, application of the Bornapproximation in the high-energy asymptotic region enables
the determination of the absolute value of the excitation cross
section by utilization of the optical oscillator strength. For
modelling and application purposes, it is advantageous to
use analytic functions that accurately reproduce the relative
emission intensities over the measurement region while also
yielding the correct asymptotic limit. The modified Bornapproximation developed by Shemansky et al (1985a, 1985b)
represents the excitation function in terms of electron energy
in threshold units, which allows characterization of an entire
electronic band system with an emission measurement over
a few ro-vibronic levels. In the formulation of Shemansky
et al (1985a, 1985b), the cross section of excitation (vi , Ji ) to
(vj , Jj ) is given by
σ (vi , vj ; Ji , Jj ) = πf (vi , vj ; Ji , Jj )
1 1
Eij E
4
Cm
+
(X − 1)
C7
m=1
C5 C 6 1
+
+ ln(X)
× exp(−mC8 X) +
C7 C7 X
C0
×
C7
1
1
− 3
X2
X
C7 (vi , vj , Ji , Jj ) =
where ηj is the nonradiative yield of the level (vj , Jj ), which
refers to either dissociation or predissociation for the B 1 u+
and D 1 u states.
The nonradiative yield, ηj , of B 1 u+ and D 1 u is either 0
or 1 (Glass-Maujean et al 2009). For all the continuum levels,
it has a value of 1. For the discrete levels of the B 1 u+ and
1 +
D 1 −
u states, it equals 0. For a D u level that lies below
the H(1s)+H(2) continuum, it also equals 0. However, for
the D 1 +u levels above the H(1s)+H(2) continuum, it equals
1 because of strong predissociation by the B 1 u+ continuum.
Finally, transition probabilities of the B 1 u+ and D 1 u
states to the low-lying excited singlet-gerade (such as the
EF 1 g+ and GK 1 g+ ) levels are negligible when compared
with those to the X 1 g+ state. Consequently, the emission
cross sections of the B 1 u+ − X 1 g+ and D 1 u − X 1 g+
band systems can be considered identical to the emission cross
sections of the B 1 u+ and D 1 u states, given by equation (13).
2π(2Ji + 1)
f (vi , vj ; Ji , Jj ),
Eij
3. Results and discussion
3.1. Photodissociation cross section
The calculated B 1 u+ photodissociation cross sections for
Ji = 0 of various vi levels generally agree with those obtained
by Glass-Maujean (1986) within 2–5%. In the high kinetic
energy region, where the cross section drops by a factor
20–100 times from that in the peak region, the difference
can approach 10–15%. Both sets of data produce virtually
identical photodissociation oscillator strengths for the R(0)
excitation.
Figure 2 shows the rotational dependence of the B 1 u+
cross section by comparing excitation from Ji = 0 to 9 of the
X 1 g+ (0) levels. Except for the sharp features in the threshold
region, the dissociation cross section increases monotonically
with Ji at nearly every Ek region. Even when the contributions
of the sharp features are included, the photodissociation
oscillator strength increases monotonically from 0.0257 for
Ji = 0 to 0.0372 for Ji = 9.
The sharp features near the threshold energy region
are due to quasi-resonance transitions that correspond to
excitation to the quasi-bound levels of the B 1 u+ state.
While the energies of these levels are above the H(1s)+H(2)
(10)
(11)
where the cross section σij and collision strength parameter
C7 are in units of a20 . Both threshold energy, Eij , and
electron-impact energy, E, are in hartree. X is the excitation
energy in threshold energy units (i.e. X = E/Eij ), and
f (vi , vj ; Ji , Jj ) is the optical absorption oscillator strength.
The coefficients Cm /C7 (m = 0–6) and C8 are determined by
nonlinear least-squares fitting of the experimentally measured
relative excitation function. It is implicitly assumed that
the parameters Cm /C7 (m = 0–6) and C8 depend only
on the electronic quantum number and are independent of
rotational and vibrational quantum numbers. In the case
of H2 , Liu et al (1998) found that the relative excitation
functions of the H2 B 1 u+ − X 1 g+ and C 1 u − X 1 g+ band
systems are identical within their experimental uncertainties.
Experimentally measured collision strength parameters of the
5
X Liu et al
σ(v =0,J ) (Mb)
σ(v =0,J ) (Mb)
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
E (cm )
E (cm )
Figure 4. Rotational dependence of the photodissociation cross
section of H2 X 1 g+ (0) to the continuum level of the D 1 u state.
See the caption of figure 2 for notation and explanation.
Figure 2. Comparison of photodissociation cross section from the
Ji = 0−9 levels of H2 X 1 g+ (0) to the continuum level of the B 1 u+
state. Note that cross sections are in units of Mb (1 Mb = 1×
10−18 cm2 ). Ek is the total kinetic energy of the dissociating
hydrogen atoms. The sharp features near threshold, whose
magnitude is not fully shown, correspond to the quasi-resonance
transitions (see figure 3). Note that the photodissociation cross
section monotonically increases with Ji .
σ(v =0,J ) (Mb)
the peak, Ek = 125–126 cm−1 . Similarly, quasi-resonance
features at Ek ∼ 7 and ∼16 cm−1 arise from excitation to
the Jj = 5 and 6 levels of the vj = 8 state, whose adiabatic
term values are computed to be 118 363.6 and 118 393.5 cm−1 ,
respectively.
Finally, the calculation also predicts that the peak for
Ji = 6 near Ek ∼ 20 cm−1 corresponds to the transition to
(vj = 7, Jj = 7). However, because of strong non-adiabatic
coupling with the D 1 +u state, the prediction is incorrect as
the semi-ab initio calculation by Abgrall et al (1994) has
shown that the (vj = 7, Jj = 7) level actually lies below the
H(1s)+H(2) continuum. In fact, the non-adiabatic coupling
at the Jj = 6 level is so strong that its energy is actually lower
than that of the Jj = 5 (Abgrall et al 1994, Wolniewicz et al
2006). It is interesting to note that the Jj = 6 level consists of
∼76.3% B 1 u+ and ∼22.6% D 1 +u characters while the Jj =
7 levels contains 87.8% of B 1 u+ and 10.6% D 1 +u characters
(Abgrall et al 2000).
Figure 4 shows dissociative excitation cross sections
to the D 1 u continuum from the Ji = 0–11 levels of
the X 1 g+ (0) state. Once again, the dissociation cross
section increases monotonically with Ji and quasi-resonance
enhancement takes place in the threshold energy region. The
photodissociation oscillator strength increases from 1.23 ×
10−3 for Ji = 0 to 2.58 × 10−3 for Ji = 11. In comparison with
the B 1 u+ continuum, the cross section from the X 1 g+ (0)
level to the D 1 u continuum is about a factor of 20 smaller
Figure 5 shows the vibrational dependence of the
dissociative excitation cross section to the D 1 u continuum
for the R(0) transition. While the X 1 g+ (0)−D 1 u continuum
transition is weak, the cross section increases rapidly as vi
increases. For the vi = 2 level, the dissociative excitation
cross section to the D 1 u continuum is comparable to that
of the X 1 g+ (0) − B 1 u+ continuum. For vibrationally
excited H2 , the dissociation via the H(1s)+H(3p) continuum
is generally not negligible. The vibrational dependence of
the X 1 g+ (vi ) − D 1 u continuum is thus similar to that of
the X 1 g+ (vi ) − C 1 u continuum shown by Glass-Maujean
(1986).
E (cm )
Figure 3. Photodissociation cross section of the H2 X 1 g+ − B 1 u+
continuum transition in the threshold region, showing enhancement
by quasi-resonance transitions. The strong peaks centred at Ek ∼
125 cm−1 arise from the P(11) and R(9) excitation to the vj = 4 and
Jj = 10 level whose adiabatic term value is calculated to be
118 502.9 cm−1 . Likewise, the peaks near Ek ∼ 7 cm−1 and
∼16 cm−1 are due to excitation to the vj = 8 and Jj = 5 and 6
levels, whose adiabatic term values are 118 383.6 and
118 393.5 cm−1 , respectively. The peak for Ji = 6 near Ek ∼
20 cm−1 correspond to (vj = 7, Jj = 7) whose position is incorrectly
calculated because of strong B 1 u+ − D 1 +u coupling (see the text).
continuum, they are temporarily stabilized by the centrifugal
potentials. Figure 3 shows the contribution of the quasiresonance transition to the cross section in the expanded scale.
The strong peaks at Ek = 125–126 cm−1 correspond to the
R(9) and P(11) excitation to the Jj = 10 of the vj = 4 levels.
The LEVEL 8 program predicts an adiabatic term value of 118
502.9 cm−1 for the (vj = 4, Jj = 10) level. The difference
between the adiabatic term value and the (vi = 0, Ji = 0) →
H(1s)+H(2) dissociation energy, 118 377.0 cm−1 (Zhang et al
2005b ), is 125.9 cm−1 , which corresponds to the location of
6
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
σ(v ,J =0) (Mb)
probabilities agree with those of Abgrall et al (1994) within
a few percent. A nearly equivalent set of P- and R-branch
data can be derived from the Q-branch values of Abgrall et al
(1994) with appropriate Hönl–London factors.
Electron excitation cross sections to the continuum levels
is obtained by dividing the continuum into very small intervals
(1 to 25 cm−1 ). The integrated photodissociation oscillator
strength and averaged Eph over an interval are taken as fij and
Eij of equation (10) for the interval to calculate its contribution
to the electron excitation cross section. The excitation cross
section arising from continuous transitions is then obtained by
summing the contribution of each interval.
Tables 1 and 2 show electron excitation and emission cross
sections of the B 1 u+ − X 1 g+ and D 1 u − X 1 g+ transitions
from threshold to 1000 eV at three different temperatures.
While the estimated excitation and emission cross sections at
100 eV and room temperature have been presented previously
(Jonin et al 2000, Glass-Maujean et al 2009), tables 1 and 2
represent a systematic determination of the B 1 u+ and D 1 u
cross sections from threshold to 1 keV for the first time. These
cross sections can be easily extended to the non-relativistic
Born limit.
In the case of the B 1 u+ − X 1 g+ band system, the
emission yield is unity for the B 1 u+ levels below the
H(1s)+H(2) dissociation limit and zero for the levels above
the limit. The difference between the excitation and emission
cross sections thus represents that of the excitation into the
B 1 u+ continuum. The absolute error of the cross sections
above 19 eV is estimated to be 20%, most of which is due to
the uncertainty of the excitation shape function. The relative
error between the excitation and emission cross sections is
less than 10%. Note that the cross sections are intentionally
shown in two digits after the decimal point to allow reliable
calculation of the dissociation cross section.
The excitation cross section of the D 1 u state consists
of three components. The first component arises from the
transition to the discrete levels that consists of all D 1 −
u
levels below the H(1s)+H(3p) continuum and all D 1 +u levels
below the H(1s)+H(2) continuum. The second component
comes from excitation to the D 1 +u levels that are between the
H(1s)+H(3p) and H(1s)+H(2) continua. These D 1 +u levels
are coupled with the B 1 u+ continuum by the + − + Coriolis
interaction and have vanishingly small emission yields (GlassMaujean et al 1978, 1979, 1987, Jonin et al 2000). The
final component is due to excitation into the H(1s)+H(3p)
continuum. While many D 1 u levels are above the ionization
potential of H2 , various experimental measurements have
shown that the autoionization of the D 1 −
u levels that are
above the ionization potential is negligibly small (Dehmer and
Chupka 1976, Glass-Maujean et al 2007b). Furthermore, the
1 +
D 1 −
u level does not couple with the B u state and thus
1 +
is not predissociated by the B u continuum. Consequently,
the emission cross section of the D 1 u state arises entirely
from the first component. Table 2 lists excitation (σex ) and
emission (σem ) cross sections of the D 1 u state, along with
the cross sections to the D 1 u continuum (σ3p ), at 300, 1000
and 1500 K. The contribution to the total excitation cross
section by the D 1 +u levels between the B 1 u+ and D 1 u
E (cm )
Figure 5. Vibrational dependence of the photodissociation cross
sections of the H2 X 1 g+ (vi ) Ji = 0 to the D 1 u continuum (i.e.
R(0)-branch transition). See the caption of figure 2 for notation and
explanation.
3.2. Electron excitation and emission cross sections
As stated, the excitation shape functions of the B 1 u+ −X 1 g+
and D 1 u − X 1 g+ band system are assumed to be identical
to those of Lyman and Werner bands obtained by Liu et al
(1998). The limited experimental evidence suggested that
the shapes are not significantly different. The high-resolution
(δλ ∼ 0.1 Å) emission spectra obtained by Jonin et al (2000)
and Glass-Maujean et al (2009) have shown that the shapes of
the B 1 u+ and D 1 u states are very similar to those of the
B 1 u+ and C 1 u states near 100 eV. Although suffering from
server spectral overlap, excitation function measurements by
Ajello et al (1984) at resolution of δλ = 5 Å from threshold to
350 eV also suggest that the shapes of B 1 u+ , B 1 u+ , C 1 u
and D 1 u excited functions are very similar. However, to the
extent or in the regions that the shapes are different, the B 1 u+
and D 1 u cross sections will deviate from the values obtained
in the present study. For this reason, the absolute error of the
present B 1 u+ and D 1 u cross sections beyond the threshold
region (i.e. >19 eV) can be as high as 20%
1 +
To obtain cross sections of D 1 −
u − X g transition, the
oscillator strengths and transition probabilities calculated by
Abgrall et al (1994, 2000) are used. For the B 1 u+ − X 1 g+
and D 1 +u − X 1 g+ , the non-adiabatic transition probabilities
and oscillator strengths of Glass-Maujean et al (2009) (which
is limited to Jj 5), wherever available, are used. For higher
Jj levels, the values of Abgrall et al (1994) are used. The
oscillator strengths of the P- and R-branch transitions of Jj 5 of X 1 g+ − D 1 +u (vj 2) are not available from either
calculations of Abgrall et al (1994, 2000) or Glass-Maujean
et al (2009). Adiabatically calculated P- and R-branch
oscillator strengths and transition probabilities obtained in the
present work are used (see section 2.1). In doing so, the effect
of the non-adiabatic coupling between the D 1 +u discrete
and B 1 u+ continuum levels on the transition probabilities
and oscillator strengths is neglected. As will be shown
in section 3.3, however, discrete-continuum coupling has
very little effect on the transition probabilities and oscillator
strengths. As noted, the present adiabatic Q-branch transition
7
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
Table 1. Electron-impact excitation and emission cross sections of the B 1 u+ statea,b .
T = 300 K
T = 1000 K
T = 1500 K
E (eV)
σexc
c
σem
σex
σem
σex
σem
13.5
13.7
14
14.2
14.5
15
15.5
16
16.5
17
17.5
18
19
20
22.5
25
27.5
30
35
40
45
50
60
70
80
90
100
150
200
250
300
350
400
500
600
700
750
800
900
1000
0.00
0.00
0.13
0.35
0.90
2.36
4.17
6.08
7.96
9.76
11.47
13.08
16.04
18.68
24.11
28.22
31.35
33.75
36.98
38.82
39.78
40.18
40.01
39.18
38.10
36.96
35.84
31.17
27.75
25.03
22.77
20.87
19.26
16.68
14.74
13.23
12.59
12.02
11.04
10.22
0.00
0.00
0.13
0.35
0.90
2.22
3.48
4.68
5.81
6.87
7.87
8.81
10.54
12.08
15.22
17.59
19.38
20.74
22.55
23.55
24.05
24.22
24.03
23.47
22.79
22.08
21.39
18.57
16.50
14.86
13.50
12.36
11.39
9.86
8.70
7.81
7.43
7.09
6.51
6.02
0.00
0.02
0.16
0.40
0.97
2.45
4.28
6.18
8.06
9.85
11.54
13.15
16.09
18.72
24.11
28.20
31.32
33.70
36.90
38.72
39.67
40.07
39.88
39.05
37.97
36.83
35.71
31.05
27.64
24.93
22.68
20.78
19.17
16.61
14.67
13.17
12.54
11.97
10.99
10.17
0.00
0.02
0.16
0.40
0.97
2.24
3.46
4.61
5.69
6.72
7.68
8.58
10.24
11.72
14.74
17.01
18.73
20.03
21.76
22.72
23.19
23.35
23.15
22.61
21.95
21.27
20.60
17.88
15.89
14.31
13.00
11.90
10.96
9.48
8.37
7.51
7.15
6.82
6.26
5.80
0.02
0.05
0.22
0.48
1.09
2.62
4.47
6.40
8.29
10.10
11.81
13.43
16.40
19.05
24.49
28.60
31.74
34.13
37.35
39.18
40.13
40.52
40.32
39.47
38.37
37.22
36.08
31.38
27.93
25.18
22.90
20.99
19.36
16.77
14.82
13.30
12.66
12.08
11.09
10.27
0.02
0.05
0.22
0.48
1.08
2.34
3.55
4.69
5.77
6.78
7.73
8.63
10.27
11.74
14.72
16.97
18.67
19.96
21.67
22.61
23.07
23.23
23.02
22.47
21.81
21.13
20.47
17.76
15.78
14.21
12.90
11.81
10.88
9.41
8.31
7.45
7.09
6.77
6.21
5.75
Unit is 10−19 cm2 .
The absolute error is ∼20%
for E 16.5 eV. However, the relative error is less than 10%.
Cross section entries are intentionally shown in 2 digits after
the decimal point to allow a reliable calculation of
dissociation cross sections.
c
σex and σem refer to excitation and emission cross sections,
respectively. The difference between σex and σem represents
the excitation cross section to the B 1 u+ continuum.
a
b
distance may be partially responsible for the increase, the
increase in the number of the quasi-bound levels with Jj also
plays an important role. Because hydrogen has the smallest
atomic mass, the number of bound or discrete vibrational levels
decreases fairly rapidly with J , especially for the B 1 u+ state
whose electronic potential energy is not very deep. In the
case of the B 1 u+ state, Jj = 0–1, 3–4, 5–7, 8, 9, 10–12,
13–14, 15–17 and 18–19 appear in 10, 9, 8, 6, 5, 4, 3, 2
and 1 bound vibrational levels, respectively. These ‘missing’
discrete vibrational states for the higher Jj levels are shifted
into the continuum and some of these ro-vibrational levels
continua can be obtained by the difference between σex and
(σem +σ3p ). The estimated absolute and relative uncertainties
are similar to those of the B 1 u+ state
3.3. Discussion
A common feature of photoexcitations from the X 1 g+ state
to the B 1 u+ and D 1 u continua is that their dissociation
cross sections and oscillator strengths increase monotonically
with the initial rotational quantum numbers. While the
dependence of the dipole transition moment on internuclear
8
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
Table 2. Electron-impact excitation and emission cross sections of the D 1 u statea,b .
T = 300 K
E (eV)
σexc
c
σem
13.8
14
14.2
14.4
14.6
14.8
15
15.5
16
16.5
17
17.5
18
19
20
22.5
25
27.5
30
35
40
45
50
60
70
80
90
100
150
200
250
300
350
400
500
600
700
800
900
1000
0.00
0.01
0.14
0.44
0.89
1.49
2.23
4.50
6.97
9.43
11.81
14.07
16.21
20.14
23.64
30.87
36.37
40.56
43.78
48.14
50.65
52.00
52.58
52.44
51.41
50.03
48.56
47.10
41.01
36.53
32.97
30.02
27.53
25.41
22.02
19.46
17.47
15.88
14.58
13.50
0.00
0.01
0.14
0.44
0.89
1.48
2.12
3.87
5.67
7.42
9.08
10.65
12.14
14.86
17.29
22.28
26.05
28.93
31.13
34.10
35.78
36.66
37.02
36.85
36.08
35.08
34.03
32.99
28.70
25.54
23.04
20.96
19.21
17.72
15.35
13.56
12.17
11.06
10.15
9.40
T = 1000 K
c
σ3p
0.00
0.00
0.00
0.00
0.00
0.01
0.04
0.06
0.11
0.16
0.26
0.34
0.40
0.45
0.52
0.57
0.59
0.61
0.62
0.62
0.61
0.59
0.58
0.51
0.46
0.41
0.38
0.35
0.33
0.28
0.25
0.23
0.21
0.19
0.18
σex
σem
0.00
0.04
0.19
0.52
1.03
1.68
2.47
4.83
7.38
9.92
12.35
14.67
16.86
20.89
24.48
31.89
37.51
41.80
45.09
49.54
52.10
53.46
54.05
53.89
52.82
51.39
49.87
48.37
42.10
37.50
33.84
30.81
28.25
26.07
22.59
19.97
17.93
16.30
14.96
13.85
0.00
0.04
0.19
0.52
1.02
1.64
2.32
4.16
6.06
7.90
9.65
11.31
12.87
15.73
18.29
23.53
27.50
30.53
32.84
35.95
37.72
38.64
39.02
38.83
38.01
36.96
35.84
34.75
30.22
26.90
24.26
22.07
20.22
18.65
16.16
14.28
12.81
11.64
10.69
9.89
T = 1500 K
σ3p
0.00
0.00
0.00
0.00
0.00
0.02
0.04
0.07
0.13
0.18
0.29
0.37
0.44
0.49
0.57
0.61
0.64
0.66
0.67
0.67
0.65
0.64
0.62
0.55
0.49
0.45
0.41
0.38
0.35
0.31
0.27
0.24
0.22
0.20
0.19
σex
σem
0.02
0.09
0.27
0.63
1.16
1.85
2.66
5.06
7.65
10.22
12.68
15.02
17.23
21.30
24.92
32.39
38.06
42.39
45.70
50.19
52.75
54.12
54.71
54.52
53.43
51.98
50.44
48.92
42.57
37.92
34.21
31.14
28.55
26.35
22.83
20.18
18.11
16.46
15.12
13.99
0.02
0.09
0.26
0.62
1.14
1.77
2.46
4.32
6.23
8.08
9.84
11.50
13.06
15.94
18.50
23.76
27.74
30.77
33.09
36.20
37.96
38.88
39.25
39.05
38.22
37.16
36.03
34.93
30.38
27.03
24.37
22.17
20.31
18.74
16.23
14.34
12.87
11.69
10.74
9.94
σ3p
0.00
0.00
0.00
0.00
0.00
0.02
0.05
0.09
0.15
0.21
0.33
0.43
0.51
0.57
0.66
0.71
0.74
0.76
0.78
0.77
0.76
0.74
0.72
0.64
0.57
0.52
0.48
0.44
0.41
0.36
0.32
0.28
0.26
0.24
0.22
Unit is 10−19 cm2 .
The absolute error is ∼20%
for E 18.5 eV. The relative error is less than 10%. Cross section entries are
intentionally shown in 2 digits after the decimal point to allow a reliable calculation of
dissociation cross sections.
c
σex and σem refer to excitation and emission cross sections, respectively. σ3p is the
dissociative excitation cross section to the H(1s)+H(3p) continuum. The excitation
cross section of the D 1 +u levels that are predissociated by the B 1 u+ continuum can
be obtained from the difference between σex and (σem +σ3p ).
a
b
It is important to note that quasi-resonance transitions can
also occur in the continuum emission of H2 . The calculation
of Abgrall et al (1997) has shown the enhancement to the
continuum emissions by transitions from ro-vibrational levels
of the B 1 u+ , C 1 u , B 1 u+ and D 1 u states to the quasibound levels of the X 1 g+ state. Le Roy (1971) investigated
the quasi-bound levels of the X 1 g+ state.
Electron-impact cross sections listed in table 1 and 2
generally increase with temperature. The increase is primarily
a consequence of rotational and vibrational dependence of the
dipole matrix elements. Overall, the oscillator strengths of
become quasi-bound levels. The fact that the centrifugal
potential is proportional to J (J + 1) also suggests that a quasibound level is more likely to occur in the high J levels. Thus,
even in the absence of the internuclear distance dependence
of the transition moment, the dissociation cross section and
photodissociation oscillator strength generally increase with
Jj . The quasi-bound B 1 u+ (vj = 4, Jj = 10) level has fairly
large transition probabilities, with 1.04 × 107 and 1.22 ×
107 s−1 for the R- and P-branches to X 1 g+ (0), respectively,
which explains the large cross section near Ek ∼ 126 cm−1 in
figure 3.
9
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
Table 3. Electron-impact dissociation cross sections of the B 1 u+ statea .
E (eV)
Present
Borges et al b
Redmon et al c
Liu and Hagstromd
Chung et al e
100
200
300
400
500
800
1000
1.44
1.12
0.927
0.786
0.682
0.493
0.419
2.34
1.54
1.16
0.946
0.802
0.562
0.473
2.88
1.99
1.54
3.06
2.05
1.57
1.29
1.10
0.777
0.655
3.09
2.10
1.13
0.674
Unit is 10−18 cm2 .
Borges et al (1998).
c
Redmon et al (1984).
d
Liu and Hagstrom (1994).
e
Chung et al (1975).
a
b
vi = 1 and 2 of the D 1 u − X 1 g+ and B 1 u+ transitions
increase with the rotational quantum number. Furthermore,
the excitation cross section of the vi = 1 level tends to be larger
than that of the vi = 0 level. Higher temperature increases the
weight of excitation from higher vi and Ji levels via equations
(12) and (13). The higher vi and Ji levels also tend to have
smaller threshold energy (Eij ), which results in a larger cross
section in the threshold energy region.
Jonin et al (2000) and Glass-Maujean et al (2009) have
obtained 100 eV B 1 u+ and D 1 u excitation and emission
cross sections at 300 K. The excitation and emission cross
sections of Jonin et al (2000) for the B 1 u+ state, 38× 10−19
and 21× 10−19 cm2 , are in good agreement with the present
value of 36 × 10−19 and 21 × 10−19 cm2 . However, their
values for the D 1 u state, 41 × 10−19 and 28 × 10−19 cm2 ,
are somewhat smaller than 47 × 10−19 and 33 × 10−19 cm2
obtained in the present work. The present D 1 u state
excitation and emission cross sections at 100 eV and 300 K
are virtually identical to those of Glass-Maujean et al (2009).
The small difference is due to the neglect of weak excitation
to the H(1s)+H(3p) continuum and truncation error in the
tabulation in Glass-Maujean et al (2009) who separately listed
1 +
D 1 −
u and D u components with two significant figures.
1 +
The B u state excitation cross section in Glass-Maujean
et al (2009) was incorrectly listed as 40 × 10−19 cm2 when
the correct value should have been 36 × 10−19 . While the
excitation to the D 1 u continuum was neglected in GlassMaujean et al (2009), the weak continuum excitation from the
X 1 g+ (0) level results in a very small difference. As a result,
the present D 1 u emission and excitation cross sections at
300 K are in excellent agreement with those of Glass-Maujean
et al (2009).
Table 3 compares the B 1 u+ dissociation cross section
obtained in the present work with those obtained by various
theoretical calculations. It is seen that all calculated cross
sections are larger than the present cross sections. The
cross sections calculated with the first Born approximation by
Borges et al (1998) consistently have the smallest difference
with the present values. Between 300 and 1000 eV, the Borges
et al cross sections agree with the present B 1 u+ cross sections
within the present error limit of 20% . At 200 eV or below,
the difference is more than 20%, which reflects the fact that
the Born approximation almost always overestimates the lowenergy cross sections. Additionally, good agreement in the
high-energy region, to a very large extent, reflects the good
agreement in the dissociation oscillator strengths. All other
calculated cross sections, even at 1000 eV, are significantly
greater than the present value.
Beswick and Glass-Maujean (1987) theoretically
investigate the effect of non-adiabatic coupling between B 1 u+
and B 1 u+ at large (∼15a0 ) internuclear distance on the
photodissociation cross sections of these two states. Because
non-adiabatic coupling leads to the transition between the
two states, their calculation showed an oscillating behaviour
of partial cross sections, with the maxima of the B 1 u+
corresponding to the minima of the B 1 u+ and vice versa,
although the sum of the two partial cross sections is identical
to the sum of the partial cross sections obtained without nonadiabatic coupling. Since the B 1 u+ cross section is several
times that of the B 1 u+ cross section, it is expected that the
coupling will increase the non-adiabatic dissociation oscillator
strength of the B 1 u+ state and decrease the corresponding
quantity of the B 1 u+ state. To the extent the non-adiabatic
coupling is significant, our cross sections of the B 1 u+ state
will be too high.
Coupling between the B 1 u+ continuum and the D 1 +u
discrete levels causes the eigenfunctions of B 1 u+ and D 1 +u
to be a mixture of zero-order discrete and continuum wave
functions. The continuum and discrete coupling manifest itself
as a Fano profile in the photoabsorption lineshape (Fano 1961,
Glass-Maujean et al 1987). The present electron-impact cross
section is calculated under the assumption that the effect of
discrete and continuum mixing on photoabsorption oscillator
strengths is negligible even though the mixing is the cause of
the predissociation of the D 1 +u levels. In this sense, the sum
of the B 1 u+ state dissociation cross section and the D 1 +u
state predissociation cross section has a higher accuracy than
either individual component. Nevertheless, the validity of the
present approach can be illustrated by comparing the D 1 +u
predissociation to the B 1 u+ dissociation cross section ratios
by photons and electrons. The ratio of the photodissociation
cross section of the B 1 u+ state to the predissociation cross
section of the D 1 +u state at room temperature has been
determined to be 1.00:0.91 in the experimental work of GlassMaujean et al (1986). The corresponding ratio for electron
10
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
impact at E 1000 eV is 1.00:0.93. Since high energy
electron-impact excitation is equivalent to photonexcitation,
these two ratios are directly comparable. The fact that the two
ratios obtained from photon and electron excitation are nearly
identical shows that the effect of continuum-discrete mixing
on the individual cross section components is very small.
The fact that D 1 +u (vj 3) levels are strongly
predissociated by the B 1 u+ continuum seems to be
inconsistent with the small effect of continuum-discrete
mixing on the individual cross section. However, both
phenomena are consistent with the fact that the continuumdiscrete mixing has a small effect on the photoabsorption
oscillator strengths but a very large effect on the emission
branching ratio of the predissociated D 1 +u levels, a
consequence of the extremely large difference in the zero-order
decay rates of discrete D 1 u and continuum B 1 u+ levels.
First, the continuum-discrete coupling, while increases the
line width and alters the shape of the D 1 +u (vj 3) − X 1 g+
photoabsorption, has every little effect on the integrated value
of cross section (i.e. photoabsorption oscillator strength).
Indeed, Glass-Maujean et al (1984) have obtained very good
agreement between the measured (integrated) D 1 +u − X 1 g+
photoabsorption transition probabilities and those calculated
with the adiabatic approximation in which D 1 +u − B 1 u+
coupling was neglected. Furthermore, Beswick and GlassMaujean (1987) have shown that the coupling between the
Ji = 2 levels of the D 1 +u (3) and B 1 u+ continuum shifts
the Ji = 2 by only 1 cm−1 . In contrast, the coupling with
the discrete levels of B 1 u+ , several thousands cm−1 below,
shifts the position by 6.5 cm−1 . Moreover, the spontaneous
decay lifetimes of the D 1 +u levels, in the absence of the
discrete-continuum coupling, can be obtained from those of
the D 1 −
u levels to be 2–4 ns (Abgrall et al 1994, 1997, 2000).
The excitation into the continuum levels of the B 1 u+ results
in a direct dissociation. While the authors are unaware of
any experimental lifetimes of the B 1 u+ continuum levels,
the lifetimes are expected to be of the order of fs (10−15 s)
or shorter. The X̃ 1 A1 − Ã1 B1 excitation of H2 O is also
a direct dissociation process. Trshin et al (2009) recently
obtained a liftetime of <1.8 fs for the Ã1 B1 state. If the
lifetime of the B 1 u+ continuum level is assumed to be 1 fs,
a mixing, say 0.1%, between the D 1 +u discrete and B 1 u+
continuum levels, while having very little impact on the
oscillator strengths, is sufficient to shorten the lifetimes of the
D 1 +u levels to ∼10−12 s and makes its emission branching
ratio to be less than 10−3 . Indeed, the lifetimes of the Jj =
1 of the vj = 3–11 levels of the D 1 u state have been
determined to be (1.1–1.7)× 10−12 s (Glass-Maujean et al
1979, 1987). In the case of the Jj = 1–4 of the vj = 12 and
13 levels, Croman and McCormack have obtained
predissociative lifetime of (1.8–13) × 10−13 s.
A recent observation of Saturn by Cassini UVIS has
shown a large number of kinetically hot hydrogen atoms
flowing out of the top of the sunlit thermosphere in confined
ballistic and escaping orbits (Shemansky et al 2009). In
addition to the co-rotation energy with Saturn, it takes 5.5–
7.5 eV of additional energy depending on the latitude for a
hydrogen atom to escape from Saturn. The present calculation,
along with that of Glass-Maujean (1986), shows that the
dissociation of H2 , from the ground or excited levels of the
X 1 g+ state to the continuum levels of singlet ungerade states,
by photons or electrons, is incapable of producing hydrogen
atoms energetic enough to overcome the gravitational potential
of Saturn. While excitation of H2 to doubly excited states
produces fast hydrogen atoms, the small excitation cross
section and the energy requirement for this excitation are
inconsistent with the observed spectral intensities of H2 singlet
ungerade states. Hence, the excitation to triplet states from
the vibrationally excited X 1 g+ level by low-energy electrons
is likely responsible for the production of the observed hot
hydrogen atoms.
4. Conclusions
Photodissociation cross sections and oscillator strengths
for excitations from various ro-vibrational levels of the
X 1 g+ state to the continuum levels of the H2 B 1 u+ and
D 1 u states have been obtained. Photoexcitation from the
X 1 g+ (vi = 0) state to the D 1 u continuum was found to be
weak but increases rapidly with vi . The calculated B 1 u+
photodissociation cross section from the Ji = 0 level is in
excellent agreement with earlier results of Glass-Maujean
(1986). The present work also shows the contribution of quasiresonance excitation, arising from transitions to discrete levels
above the dissociation limit but stabilized by a centrifugal
potential. The quasi-resonance is largely responsible for the
significant rotational dependence of the continuum oscillator
strength of the B 1 u+ − X 1 g+ (0) band.
The calculated continuum oscillator strengths, along
with previously published discrete transition probabilities and
Lyman and Werner band excitation functions of Liu et al
(1998), are utilized to determine electron-impact excitation
and emission cross sections of the B 1 u+ − X 1 g+ and
D 1 u − X 1 g+ band systems over a wide energy range.
Excitation, emission and dissociation cross sections from
threshold to 1000 eV at various temperatures are presented
for the first time. Significant ro-vibrational dependence of
the oscillator strengths results in a noticeable temperature
dependence of electron excitation and emission cross sections.
The present 100 eV cross sections at room temperature are in
good agreement with earlier results of Jonin et al (2000) and
Glass-Maujean et al (2009). These calculated cross sections
are useful for modelling dayglow and auroral activity in the
atmospheres of the outer planets and dayglow in comets.
Acknowledgments
The analysis described in this paper was carried out at
Jet Propulsion Laboratory (JPL), California Institute of
Technology and Space Environment Technologies (SET). We
gratefully acknowledge financial support through NASA’s
Outer Planets and Planetary Atmospheres Research programs.
XL acknowledges the support of the NASA/JPL Senior
Postdoctoral Fellowship, which is administered by Oak Ridge
Associated Universities through a contract with NASA. DES
11
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185203
X Liu et al
acknowledges the supported by the Cassini UVIS contract with
the University of Colorado and NASA-NNG06GH76G issued
to SET through the NASA Planetary Atmospheres Program.
The authors wish to thank Professor Robert Le Roy for the
LEVEL 8 and BCONT 2.2 computer programs.
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