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Electron and photon dissociation cross sections of the H2 singlet ungerade continua
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2012 J. Phys. B: At. Mol. Opt. Phys. 45 015201
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IOP PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
doi:10.1088/0953-4075/45/1/015201
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201 (15pp)
Electron and photon dissociation cross
sections of the H2 singlet ungerade
continua
Xianming Liu1 , Donald E Shemansky1 , Paul V Johnson2 ,
Charles P Malone2,3 , Murtadha A Khakoo3 and Isik Kanik2
1
Planetary and Space Science Division, Space Environment Technologies, 1676 Palisades Drive,
Pacific Palisades, CA 90272, USA
2
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena,
CA 91109, USA
3
Department of Physics, California State University, Fullerton, CA 92834, USA
E-mail: xliu@spacenvironment.net
Received 17 October 2011, in final form 10 November 2011
Published 9 December 2011
Online at stacks.iop.org/JPhysB/45/015201
Abstract
Photodissociation cross sections and oscillator strengths for H2 from the X 1 g+ (vi , Ji ) levels
to the continuum levels of the B 1 u+ , C 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states have been
calculated. The (vi , Ji ) state-specific electron impact dissociation cross sections to the
continuum levels of these states have been obtained for the first time over a wide energy range
using calculated continuum oscillator strengths along with previously published excitation
functions of the Lyman and Werner bands. Estimated cross sections to the higher (n 5)
npσ 1 u+ and npπ 1 u continua are also provided. Both photon and electron impact excitation
cross sections show strong dependences on the initial (vi , Ji ) quantum numbers. Thermally
averaged electron impact cross sections of all singlet ungerade states increase monotonically
with temperature. While excitation to the B 1 u+ continuum is the dominant dissociation
channel at room temperature, the C 1 u and B 1 u+ continua become more important at high
temperature (>5000 K). This work, along with the previous calculation of the B 1 u+ and
D 1 u states by Liu et al (2009a J. Phys. B: At. Mol. Opt. Phys. 42 185203), provides the
complete electron impact dissociation cross section of H2 through the singlet ungerade
continua. Electron dissociation cross sections of the singlet ungerade continua are provided
for the purpose of modelling atmospheric heating, analysis of occultation measurements and
the hot atomic hydrogen plume observed at Saturn.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
B 1 u+ and C 1 u continua are expected to increase rapidly
with increasing vibrational quantum number of the X 1 g+
state. These calculated cross sections, along with the estimated
values for higher Rydberg states, give a reliable estimate of
the total dissociation cross sections of the singlet ungerade
continua.
In H2 -dominated atmospheres, photon and electron
impact dissociations of H2 are the primary mechanisms for
the production of kinetically hot atomic hydrogen fragments.
These processes efficiently convert electron and photon energy
into heat. The dissociation cross section of H2 is vital for
This paper presents (vi , Ji ) state-specific and thermally
averaged electron and photon dissociation cross sections of
the B 1 u+ , C 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ continua.
Photon and electron impact dissociation cross sections of H2
X 1 g+ (vi , Ji ) to the continuum levels of B 1 u+ and D 1 u
states and the predissociative levels of the D 1 +u state were
presented in a previous paper (Liu et al 2009a). While
dissociation via the B 1 u+ continuum dominates other singlet
ungerade continua at room temperature, the importance of the
0953-4075/12/015201+15$33.00
1
© 2012 IOP Publishing Ltd
Printed in the UK & the USA
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
a quantitative understanding of heating of the interstellar
medium (ISM) and the atmospheres of comets and outer
planets. The fact that H2 is homonuclear results in significant
deviation from local thermodynamic equilibrium (LTE) (Liu
et al 2007, Shemansky et al 2009, Habart et al 2005).
Modelling planetary and astrophysical observations generally
requires state-specific excitation, emission and dissociation
cross sections.
Photoexcitations from the X 1 g+ state to the continuum
and predissociative levels of singlet ungerade states and to the
X 2 g+ ground ionic state are the two dominant mechanisms for
photo-destruction of H2 . While dissociation and ionization via
excited ionic states and doubly excited states also take place,
the cross sections of these transitions are very small, as they
all require a change of two electron configurations and take
place via electron correlation. Experimental measurements by
Chung et al (1993) and He et al (1995) have shown that the
cumulative photoionization cross sections of the excited ionic
states and doubly excited states are very small.
Electron impact destruction of H2 differs from photon
destruction in several ways.
In addition to excitation
to the dissociative and predissociative levels of singlet
ungerade states and to the X 2 g+ ionic state, electron impact
dissociation also takes place by excitation to the continuum
and predissociative levels of the triplet states and singlet gerade
states. Electron impact dissociation occurs via a larger number
of doubly excited states and excited ionic states than photon
excitation and thus is generally more important than its photon
dissociation counterpart. The contribution of the additional
electron exchange channels is either small or important only
in the low energy region. In fact, when electron impact energy
is higher than 25 eV, dissociation via the X 2 g+ ionic state
and the dissociative and predissociative levels of the singlet
ungerade states is more probable than that via all other states
combined (i.e. triplet, singlet gerade, dissociative ionic and
doubly excited states). The dominance of the former becomes
progressively more prominent with the increase of electron
impact energy. The fundamental reason for such dominance
is that excitation to both the X 2 g+ state and singlet ungerade
states is dipole and spin allowed, whereas other channels are
either dipole or spin forbidden.
Corrigan (1965) carried out the only existing published
electron impact total dissociation measurement by observing
the rate of H2 pressure decrease in a closed system where
the dissociation fragments were trapped on a molybdenum
trioxide (MoO3 ) surface. MoO3 absorbs hydrogen atoms with
100% efficiency. The H+ and H+2 produced by ionization
recombine with electrons rapidly on the surface to form
hydrogen atoms, which are then absorbed by the surface.
Corrigan also derived the so-called total triplet cross section
by subtracting the ionization cross section of Tate and Smith
(1932) from the measured total dissociation cross section.
His approach, however, encountered a number of difficulties.
First, the Tate and Smith (1932) ionization cross section
becomes larger than his dissociation cross section when E
is above ∼90 eV. More importantly, dissociation via singlet
ungerade and gerade states was completely neglected in the
Corrigan treatment. The dissociation cross section of singlet
Figure 1. Singlet ungerade adiabatic potential energy curves and
corresponding dissociation limits. Energies are relative to the (vi =
0, Ji = 0) level of the X 1 g+ state, which is not shown. Note that the
dissociation limit of the B B̄ 1 u+ state is H(1s)+H(3). See table 1
for more accurate values of the dissociation energies. All potential
energy curves are based on the calculations of Staszewska and
Wolniewicz (2002), and Wolniewicz and Staszewska (2003a,
2003b).
ungerade continua obtained in this work is an important step
towards the examination of the accuracy of the Corrigan (1965)
measurement and, ultimately, extraction of the total triplet
excitation cross section from the experimental measurement.
The vacuum ultraviolet spectrum of H2 arises from
transitions between the X 1 g+ state and the npσ 1 u+ and
npπ 1 u Rydberg series. The emission spectrum is dominated
by the first two members of the series: the B 1 u+ and C 1 u
states. Below 1000 Å, the contributions from the higher states,
such as B 1 u+ and D 1 u , become significant. 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. At room temperature,
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), while the dissociation via the B 1 u+ continuum is very
significant (Glass-Maujean et al 1985a, Glass-Maujean 1986).
Glass-Maujean (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 have shown that predissociation of
the npσ 1 u+ and npπ 1 +u states is primarily caused by direct
or indirect coupling to the B 1 u+ continuum (Glass-Maujean
1979, Glass-Maujean et al 1979, 1984, 1985b, 1987).
2
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
The electron excitation and emission cross sections of
the discrete levels of the B 1 u+ and C 1 u states have been
determined over a wide energy range from the excitation
function measurement by Liu et al (1998) using the Abgrall
et al (1993a, 1993b, 1993c) transition probabilities. 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 at
100 eV (Jonin et al 2000, Glass-Maujean et al 2009). The
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. A
recent work by Liu et al (2009a) has obtained excitation,
emission and dissociation cross sections of the B 1 u+ and
D 1 u states by using the B 1 u+ and C 1 u shape functions
of Liu et al (1998) and theoretically calculated oscillator
strengths. In addition, the predissociation cross section
of D 1 +u by the B 1 u+ continuum was obtained. Other
investigators have also calculated photodissociation of various
singlet ungerade states. 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. Glass-Maujean (1986) has also examined
vibrational dependence of the Ji = 0 dissociation cross section.
Burciaga and Ford (1991) have investigated the C 1 u state
vibrational shape resonance arising from the small hump in
its potential energy curve and examined the effect of the
B 1 u+ −C 1 +u nonadiabatic coupling on the photodissociation
cross section of the C 1 u state. Beswick and Glass-Maujean
(1987) have considered nonadiabatic coupling between the
B 1 u+ and B 1 u+ continua and demonstrated quantum
interference between photodissociation cross sections of
the B 1 u+ and B 1 u+ states. Cheng et al (1998) have
experimentally investigated the photodissociaton resonance
near the H(1s)+H(2) threshold and reveal a large 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.
The singlet ungerade states of H2 have been extensively
investigated experimentally including electron excitation
(Ajello et al 1984, Khakoo and Trajmar 1986, Liu et al 1995,
2000, 2002, 2003, Abgrall et al 1997, 1999, Jonin et al
2000, Dziczek et al 2000, Wrkich et al 2002, Kato et al
2008, 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, 1995, 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, 2011, Gabriel et al 2009, Ivanov et al
2011). The simultaneous absorption, dissociation, emission
and ionization investigations with synchrotron radiation by
Glass-Maujean et al (2007a, 2007b, 2007c, 2008a, 2008b), and
corresponding theoretical calculations (Glass-Maujean and
Jungen 2009, Glass-Maujean et al 2010) have produced many
physical parameters such as predissociation, autoionization
and emission yields as well as transition probabilities of
singlet ungerade levels. A recent synchrotron photoabsorption
investigation using a Fourier transform technique has yielded
an accurate predissociation cross section of the D 1 +u state
(Dickenson et al 2010).
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 calculations, which deal
with a few coupled electronic states at most, 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 continued to develop
for several decades. Accurate electronic potential energy
calculations, including the adiabatic, diagonal nonadiabatic,
relativistic and radiative corrections, have been carried out
(Dressler and Wolniewicz 1986, Wolniewicz and Dressler
1992, 1994, Wolniewicz 1993, 1995a, 1995b, Wolniewicz
et al 1998, Staszewska and Wolniewicz 2002, Wolniewicz and
Staszewska 2003a, Wolniewicz 2007). Calculations of the H2
transition moment functions (Wolniewicz 1995c, Dressler and
Wolniewicz 1995, Wolniewicz and Staszewska 2003a, 2003b)
and nonadiabatic coupling of the first several members of the
singlet ungerade Rydberg series have been reported recently
(Wolniewicz et al 2006). Accurate adiabatic and nonadiabatic
corrections of the X 1 g+ state have been obtained recently by
Pachucki and Komasa (2009), and very accurate dissociation
energies of the X 1 g+ (vi , Ji ) levels of H2 and D2 have been
reported by Komasa et al (2011). Since its 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 ro-vibronic 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, Glass-Maujean et al
2010). Of particular relevance, Abgrall et al (1993a, 1993b,
1993c, 1994, 1997, 2000) have carried out extensive semi-ab
initio calculations of the nonadiabatic 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 highresolution synchrotron photoabsorption measurements. GlassMaujean et al (2009) also performed nonadiabatic calculations
of the B 1 u+ , B B̄ 1 u+ , 5pσ 1 u+ , D 1 +u and D 1 +u states
for Ji 3 levels. The transition probabilities, along
with dissociation, ionization and emission yields, of the
1 +
npπ 1 −
u −X g band systems have been obtained with
MQDT calculations by Glass-Maujean and Jungen (2009) and
Glass-Maujean et al (2010).
3
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
2. Theory
where Eph and Ek are in hartree, D(z) in au, R0 in bohr (a0 )
and mH is the mass of a hydrogen atom (Allison and Dalgarno
1969).
Differences in the definition of the Hönl–London factors
(see Liu et al 2009a) lead to a degeneracy factor, Gij ,
usually for – transitions, to appear or not appear in
equations (1), (5) and (6). The variation in the definition
of Hönl–London factors in the literature have been described
by Hansson and Watson (2005). In this work, the definition
of Hj i (Jj , Ji ) proposed by Hansson and Watson (2005) is
followed. Consequently, Gij is not needed.
The discrete and continuum nuclear wavefunctions,
χvi ,Ji (R) and χEk ,Jj (R), are obtained by numerical solution
of the Schrödinger equation.
The potential energy
curves consist of Born–Oppenheimer potentials (VBO (R)),
plus adiabatic (Vadi (R)), relativistic (Vrel (R)) and radiative
(Vrad (R)) corrections:
Following the earlier work by Liu et al (2009a), calculations
of the photodissociation cross sections are outlined here.
Throughout this paper, dissociation strictly refers to the nonradiative process of H2 immediately following the excitation
to the continuum levels. Indices i and j denote appropriate
levels of the X 1 g+ and singlet ungerade states, respectively.
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
8π 3 ν Hj i (Jj , Ji )
ρ(Ek )
σ (vi , Ji ; Eph ) =
3hc J
2Ji + 1
j
× |χEk ,Jj (R)|D(R)|χvi ,Ji (R)|2 ,
V (R) = VBO (R) + Vadi (R) + Vrel (R) + Vrad (R).
(1)
where Hj i (Jj , Ji ) and D(R) are the Hönl–London factors and
the electric dipole transition moment (Le Roy et al 1976).
χvi ,Ji (R) and χEk ,Jj (R) are the radial wavefunctions of initial
level i and the continuum level j , respectively. ρ(Ek ) is the
density-of-states normalization factor at energy Ek = hcνk
above the dissociation limit of state j , with
δ(Ek − Ek )
ρ(Ek )
(2)
Ek = Eph + E(vi , Ji ) − Vj (R → ∞),
(3)
χEk ,Jj (R)|χEk ,Jj (R) =
The VBO (R), Vadi (R), Vrel (R) and Vrad (R) calculated by
Wolniewicz (1993) and Wolniewicz et al (1998) are used for
the X 1 g+ potential. The VBO (R) and Vadi (R) of the B 1 u+ ,
C 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states are all taken
from ab initio calculations of Staszewska and Wolniewicz
(2002), and Wolniewicz and Staszewska (2003a, 2003b). The
Vrel (R) of the B 1 u+ and B B̄ 1 u+ states are based on results of
Wolniewicz (1995b) and De Lange et al (2001), respectively.
Since relativistic corrections for other singlet ungerade states
are not available, they are assumed to be R-independent
and are derived from corresponding corrections of correlated
hydrogen atoms at R → ∞. Specifically, the Vrel (R) of the
C 1 u , D 1 u and 5pσ 1 u+ states are −1.673, −1.469 and
−1.622 cm−1 , respectively. The Vrad (R) of all the singlet
ungerade states are assumed to be equal to the radiative
correction of H+2 X 2 g+ calculated by Bukowski et al (1992).
The potential energy curves of the B 1 u+ , C 1 u , B 1 u+ ,
D 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states are displayed
in figure 1.
The photodissociation cross section depends on the
amplitude of the continuum radial wavefunction.
The
converged amplitude is determined by normalization of the
wavefunction via equation (4) in the asymptotic region. The
cross section is then appropriately scaled by the amplitude.
In some cases, propagation of the continuum wavefunction
beyond the available ranges of internuclear distance for V (R)
and D(R) is required to achieve the convergence. Accurate
evaluation of the cross section thus requires appropriate
extrapolation of V (R) and D(R) in the asymptotic region.
The following extrapolation procedure is used in this work.
First, a value of 255 475.6131 cm−1 is added to V (R) in
equation (7) so that all potentials are relative to the v = 0 and
J = 0 level of the X 1 g+ state. This added value is based
on the updated H atom ionization potential and measured
dissociation energy of X 1 g+ (vi = 0, Ji = 0) recently
obtained by Liu et al (2009b), and is slightly different from the
255 475.612 cm−1 value obtained by Wolniewicz (1995a).
V (R) is then extrapolated according to
α
β
γ
V (R) = V (∞) + 6 + 8 + 10 .
(8)
R
R
R
and
where Vj (R → ∞) is the asymptotic potential energy of
state j .
In the present calculation, the internuclear distance, R,
is converted to a dimensionless quantity z = R/R0 , where
R0 is an arbitrarily selected scale length. The amplitude of
the continuum wavefunction 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 2μcνk / h,
with μ being the reduced mass of H2 . The conversion
and √
normalization give a density-of-states factor of ρ(Ek ) =
2R0 2μc/ hνk (states per cm−1 ). The photodissociation cross
section in equation (1), in units of Mb, can be re-written as
2 H (J , J )
ji j
i
ph
−2 μν
σ (vi , Ji ; Eph ) = 3.2270 × 10
νk J
2Ji + 1
z→∞
j
× |χEk ,Jj (z)|D(z)|χvi ,Ji (z)| ,
2
(5)
−1
where νk and ν are in units of cm , D(z) in debye, μ in
unified atomic mass units and R0 in Å (Liu et al 2009a).
In atomic units, the photodissociation cross section can
be alternatively written as
2 4μEph
Hj i (Jj , Ji )
ph
σ (vi , Ji ; Eph ) = 25.936
mH Ek J
2Ji + 1
j
× |χEk ,Jj (z)|D(z)|χvi ,Ji (z)|2 ,
(7)
(6)
4
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
Table 1. Parameters for asymptotic potentials and transition moments.
State
αa
βa
γa
V (∞)a,b
D(∞)c
X 1 g+
B 1 u+
C 1 u
B B̄ 1 u+
D 1 u
5pσ 1 u+
2.861 991E6
8.916 562E11
8.381 670E9
1.650 635E10
9.906 735E9
3.666 206E10
−6.172 744E8
−4.252 095E15
−1.355 295E13
−4.648 399E13
−3.085 039E13
−1.630 490E14
3.400 870E10
5.125 551E18
0.0
3.341 543E16
2.607 558E16
1.858 102E17
36 118.0696
118 377.2327
118 377.2327
133 610.3531
138 941.9834
133 610.2913
–
1.053 498
1.053 498
0.421 875
0
0
The units for α, β and γ are such that V (R) and V (∞) are in cm−1 and R in Å (see
equation (8)).
b
The values of V (R) and V (∞) are relative to the vi = 0 and Ji = 0 level of the X 1 g+ state
(see text).
c
In au, for transition between X 1 g+ and the indicated singlet ungerade state.
a
The second to fifth columns of table 1 list the values of
α, β, γ and V (∞) for the X 1 g+ , B 1 u+ , C 1 u , B B̄ 1 u+ ,
D 1 u and 5pσ 1 u+ states when the potential is in units of
cm−1 and the internuclear distance in Å.
Ab initio transition moments, D(R), calculated by
Wolniewicz and Staszewska (2003a, 2003b) are utilized for
the calculation of photodissociation cross sections. Beyond
the available calculated R values, the transition moment is
extrapolated with the form
2.2. Electron impact dissociation
The electron impact excitation cross sections of this
investigation are obtained from the modified Born
approximation developed by Shemansky et al (1985a, 1985b).
In this formulation, the energy dependence of the cross section
is represented by an excitation shape function, which is
often obtained from the measurement of the relative emission
intensity of a discrete–discrete transition as a function of
the impact energy from threshold to a few keV. If the
electron excitation is dipole-allowed, application of the Born
approximation in the high-energy asymptotic region enables
the determination of the absolute value of the excitation cross
section by normalizing to the optical oscillator strength. When
the shape function is expressed in terms of electron energy
in threshold units, an entire electronic band system can be
characterized with an emission measurement over a few rovibronic levels. The cross section for excitation (vi , Ji ) to
(vj , Jj ) is given by
δ
,
(9)
Rn
where δ and n are parameters. The last column of table 1 gives
the asymptotic values, D(∞), in au for relevant transitions
between X 1 g+ and singlet ungerade states.
In the absence of nonadiabatic coupling, the calculated
photodissociation cross section can be considered as nearly
exact. The calculated eigenvalues of the J 27 levels of the
X 1 g+ state, when combined with the nonadiabatic correction
derived from the polynomial of Wolniewicz (1995a), agree
within fractions of 1 cm−1 to the derived experimental values
of Dabrowski (1984). When compared with the more accurate
theoretical calculation of Komasa et al (2011), the calculated
values differ less than 0.1 cm−1 for J 25 levels. However,
the difference increases with J, with the largest, ∼0.4 cm−1 ,
at J = 31 of v = 0. The use of polynomial representation
of nonadiabatic correction, which is less accurate at high J, is
primarily responsible for the difference in the calculated term
values. Moreover, the calculated eigenvalues of the C 1 −
u rovibrational levels agree with those measured and calculated by
Abgrall et al (1993b, 1993c, 1994) within fractions of 1 cm−1 .
The calculated energy values of the D 1 −
u levels differ by no
more than 1 cm−1 from those measured by Namioka (1964b),
Takezawa (1970), Herzberg and Jungen (1972), Dehmer and
Chupka (1976) and Roncin and Launay (1994).
The photodissociation oscillator strength is related to the
photodissociation cross section by
∞
σ ph (vi , Ji ; Ek ) dνk , (10)
f (vi , Ji ; j ) = 1.1296 × 10−6
D(R) = D(∞) +
σ (vi , vj ; Ji , Jj ) =
with
C0
Sij (X) =
C7
πf (vi , vj ; Ji , Jj )
Sij (X)
Eij E
1
1
− 3
X2
X
× exp(−mC8 X) +
(11)
4
Cm
(X − 1)
C7
m=1
C5 C6 1
+ ln(X),
+
C7 C7 X
(12)
and
C7 (vi , vj ; Ji , Jj ) =
2π(2Ji + 1)
f (vi , vj ; Ji , Jj ),
Eij
(13)
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 normally 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. Note that the
shape function, Sij (X), characterizes the energy dependence
0
where σ is in Mb and νk is in cm−1 . Under the irradiation
of a uniform photon field, the oscillator strength, f (vi , Ji ; j ),
apart from a field strength constant, gives a direct measurement
of the photodissociation rate.
ph
5
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
of the cross section while C7 , which is proportional to the
photoabsorption oscillator strength, determines the absolute
magnitude of the cross section.
For H2 singlet ungerade states, Liu et al (1998) found that
the Sij (X) 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
higher singlet ungerade states are not available. In this study,
the Sij (X) obtained by Liu et al (1998) for the B 1 u+ and
C 1 u states are used for these high singlet ungerade states.
Equation (11) is for a discrete to discrete excitation.
Electron excitation cross sections to the continuum levels,
σ (vi , Ji ; E), can be obtained by dividing the continuum
into very small intervals (1–8 cm−1 ).
The integrated
photodissociation oscillator strength and averaged Eph over an
interval are taken as fij and Eij of equation (11) for the interval
to calculate its contribution to the electron excitation cross
section. The excitation cross section arising from continuum
transitions is then obtained by summing the contribution of
each interval.
The thermally averaged dissociation cross section for a
band system in this study is defined as the population-weighted
average of the ro-vibrational cross-section components:
1 σij (E) =
σ (vi , Ji ; E)gI (2Ji + 1)
QT i,j
× exp[−E(vi , Ji )/kT ],
Figure 2. Comparison of photodissociation cross sections from the
Ji = 3 level of H2 X 1 g+ (vi = 0–7) to the continuum level of the
C 1 u state. The cross sections are in units of Mb. Ek refers to the
total kinetic energy of the two dissociating hydrogen atoms. Note
that the photodissociation cross sections from vibrationally excited
levels are much greater than from the vi = 0 level.
levels. The present cross sections generally agree with those
of Glass-Maujean (1986) within ∼5%. In the high kinetic
energy region, where the cross section drops by a factor
between 20 and 100 from that in the peak region, the difference
can approach ∼15%. Both calculations produce virtually
identical photodissociation oscillator strengths for the R(0)
branch excitation.
The photodissociation cross sections of the R(1) branch
transition from X 1 g+ (0) to the B B̄ 1 u+ , D 1 u and 5pσ 1 u+
continua have been presented in graphic form by GlassMaujean et al (2007a, 2008a). The present cross sections agree
with those shown by Glass-Maujean et al if their values are
multiplied by appropriate Hönl–London factors for the R(1)
branch transition. It appears that Glass-Maujean et al (2007a,
2008a) intentionally left out or implicitly assumed a unity
value for the Hönl–London factors. The implicit assumption
is clear from their definition of the photodissociation cross
section (Glass-Maujean et al 2007a), where the Hönl–London
factor is absent from their equation for the (v , J )−
(E, J ) photoabsorption cross section, σE,J ,J . Moreover, both
measured and calculated P- and R-branch discrete transition
probabilities in the tables of Glass-Maujean et al (2007b,
2008a, 2008b) are presented in the form of equivalent band
transition probabilities (i.e. branch transition probabilities
divided by appropriate Hönl–London factors).
Figure 2 shows the C 1 u photodissociation cross section
by comparing excitation from the Ji = 3 of vi = 0–7 levels
of X 1 g+ . It is clear that the cross section has very strong
vibrational dependence. Moreover, the C 1 u dissociation
cross sections from the vi > 0 levels are significantly larger
than that from the vi = 0 level. Thus, any significant
vibrational excitation of the X 1 g+ state will lead to a very large
enhancement of the C 1 u state dissociation cross section.
(14)
where QT is the partition function at temperature T, gI is the
nuclear spin statistics and k is the Boltzmann constant.
If two states have identical shape functions and similar
threshold energies, equation (11) shows their relative cross
section is in the ratio of the oscillator strengths. As
the principal quantum number increases, the npσ 1 u+ and
npπ 1 u Rydberg series converge to the X 2 g+ state of H+2 .
The energy gap between two adjacent Rydberg states decreases
according to (n∗ )−3 , where n∗ is the effective principal
quantum number. The threshold energies of ro-vibrational
levels of high n states are thus very similar. Moreover, their
potential energy curves also become very similar as they are
related to the energy of the X 2 g+ state by the standard Rydberg
formula (Glass-Maujean and Jungen 2009, Glass-Maujean
et al 2009). As section 2.1 shows, the photodissociation
oscillator strength is proportional to the photodissociation
cross section, which in turn, is proportional to the square of
the dipole matrix elements. The square of the dipole matrix
element of a Rydberg state is also proportional to (n∗ )−3
(Glass-Maujean et al 2009). Thus, relative electron impact
excitation cross sections of high n states, beyond the threshold
region, are inversely proportional to the cubic power of the
ratio of the effective principal quantum number. This relation
is utilized to estimate the dissociation cross section of states
higher than D 1 u and 5pσ 1 u+ .
3. Results and discussion
3.1. Photodissociation cross section
Glass-Maujean (1986) has also calculated B 1 u+ and C 1 u
photodissociation cross sections from Ji = 0 of various vi
6
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
Figure 4. Comparison of photodissociation oscillator strengths
from Ji = 1 of the vi = 0–14 levels to various singlet ungerade
continua. Note that the oscillator strength of the 5pσ 1 u+ state
shown in the figure has been multiplied by a factor of 10. The
dominance of the photodissociation via the C 1 u and B 1 u+
continua from excited vi levels is clear. Thin lines connecting the
discrete points are to guide viewing.
Figure 3. Comparison of photodissociation oscillator strengths of
the Ji = 0–31 levels of H2 X 1 g+ (0) to various singlet ungerade
continua. The oscillator strength of the 5pσ 1 u+ state has been
scaled up by a factor of 10 and values of the B 1 u+ and D 1 u states
are from the previous calculation of Liu et al (2009a). Note that the
values for both C 1 u and D 1 u increase rapidly with Ji . The
dominance of the photodissociation via the C 1 u continuum at high
Ji levels is obvious. Thin lines connecting the discrete points are to
guide viewing.
by the B 1 u+ −C 1 +u coupling and small differences in the
two potential energy curves. Burciaga and Ford (1991) also
suggested that some resonances in the C 1 u cross section
might be observable experimentally. However, the presence
of the much larger B 1 u+ cross section in the same energy
region makes the detection of the resonances unlikely, at least
for excitation from the X 1 g+ (0) level.
The potential energy curve of the 5pσ 1 u+ state also has
a hump of ∼1316 cm−1 that peaks near R = 5.6 a0 (see
figure 1). The photodissociation cross section also shows
large and sharp shape resonance features. As noted by GlassMaujean et al (2007a), extensive resonance peaks also exist in
the photodissociation of the B B̄ 1 u+ state.
In addition to the shape resonances, quasi-resonance
transitions that correspond to excitation to the quasi-bound
levels also occur in the threshold region. While the energies of
these levels are above their singlet ungerade dissociation limits,
V (∞), they can be temporarily stabilized by the centrifugal
potentials. Since the quasi-resonances of the B 1 u+ and D 1 u
cross sections have been discussed in detail elsewhere (Liu
et al 2009a), the quasi-resonance of other singlet ungerade
states will not be repeated here.
While other states, such as B 1 u+ , D 1 u and 5pσ 1 u+ ,
also show strong vibrational dependence, the increase in
the absolute cross sections of these states is not as large as
that of the C 1 u state. Singlet ungerade dissociation cross
sections also show strong Ji dependence. For a given vi level,
photodissociation cross sections very often, but not always,
monotonically increase with Ji .
Since the oscillator strength directly reflects the
cumulative effect of the photodissociation cross section in a
uniform photon radiation field,4 the (vi , Ji ) dependence of
the photodissociation cross section can be more conveniently
shown in terms of ro-vibrational dependence of oscillator
strength. Figure 3 shows the rotational dependence of the
photodissociation oscillator strength from the X 1 g+ (0) level
while figure 4 shows the vibrational dependence from the
Ji = 1 level.
The potential energy curve of the C 1 u state has a
101.6 cm−1 hump (above V (∞)) which peaks near R =
9.1 a0 . Consequently, the photodissociation cross section
of the C 1 u state shows extensive resonance features.
Burciaga and Ford (1991) have investigated these C 1 u
shape resonances caused by the hump. They also examined
the effect of B 1 u+ −C 1 +u nonadiabatic coupling on the
photodissociation cross section of the C 1 u state. The
present adiabatic calculation has confirmed the presence of
these resonances. However, the location and the magnitude
of the resonance cross sections differ slightly from those of
Burciaga and Ford (1991). The difference is likely caused
3.2. Electron impact dissociation cross sections
The shapes of excitation functions for the singlet ungerade
continua are implicitly understood to be identical to their
discrete counterparts, which, in turn, are assumed to be
the same as those of the B 1 u+ −X 1 g+ and C 1 u −X 1 g+
bands obtained by Liu et al (1998). Since the continuum
level can be considered as an extension of the discrete level
within a common electronic state, the first assumption is
expected to be valid. Additionally, it is very difficult, if
4 Note that electron–matter interaction at high impact energy is equivalent to
photon–matter interaction with a constant photon field strength with respect
to frequency.
7
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
not impossible, on a state-to-state specific basis, to measure
the excitation or emission intensity of a discrete-continuum
transition as a function of excitation energy. The first
assumption is thus necessary. In the case of the second
assumption, several experimental observations suggest that
the cross section shapes of singlet ungerade states are similar.
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+ , D 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. Emissions from the B B̄ 1 u+ and 5pσ 1 u+ states are
either very weak or from levels that are strongly perturbed.
While no definitive conclusion can be provided on the
similarity of the shape of the B B̄ 1 u+ and 5pσ 1 u+ excitation
functions, the observed relative emission intensities do not
suggest that their shapes are significantly different from the
low-lying states. Although suffering from spectral overlap,
excitation function measurements by Ajello et al (1984) at
a resolution of δλ = 5 Å from threshold to 350 eV also
suggest that the shapes of the B 1 u+ , B 1 u+ , C 1 u and
D 1 u excitation functions are similar. The effect of the
second assumption on the accuracy of the singlet ungerade
dissociation cross sections will be discussed later.
Tables 2–4 list electron impact excitation cross sections
from the X 1 g+ state to the various singlet ungerade continua
from threshold to 1000 eV at temperatures of 300, 1400 and
5000 K, respectively. For comparison, the B B̄ 1 u+ and D 1 u
cross sections based on the earlier work of Liu et al (2009a)
are also presented. The difference between the excitation and
emission cross sections of the B 1 u+ state in table 1 of Liu
et al (2009) corresponds to the B 1 u+ dissociation cross
section listed in the fourth column of table 2. The D 1 u
dissociation cross section in the sixth column of the tables in
this work corresponds to the values under the heading σ3p in
table 2 of Liu et al (2009)5 . Note that B 1 u+ and D 1 u cross
sections at 300, 1000 and 1500 K were tabulated in the earlier
work while the dissociation cross sections at 300, 1400 and
5000 K are given in this work.
The ninth column under the heading of Others represents
the sum of the excitation cross sections into the continuum
levels of npσ 1 u+ and npπ 1 u Rydberg states that are higher
than those listed in columns 2–8. The cross sections for
excitation into these continuum levels are estimated on the
basis of the ratio of effective principal quantum number as
described in section 2.2. At 300 and 1400 K, the contribution
of these other states accounts for less than 6% of the total
dissociation cross section. At 5000 K, the contribution is
about 7% of the total. The aggregate contribution from other
singlet ungerade states that do not belong to the npσ and
npπ Rydberg series (such as 4B 1 u+ , 6B1 u+ and V 1 u ) is
completely negligible even at 5000 K. Therefore, the total cross
section listed in the last column of the tables 2–4 also represents
the dissociation cross section via all singlet ungerade continua.
While tables 2–4 list the cross sections up to 1000 eV excitation
energy, they can be easily extended to the non-relativistic Born
limit.
Beswick and Glass-Maujean (1987) investigated the effect
of nonadiabatic coupling between B 1 u+ and B 1 u+ around
R ∼ 15 a0 on the photodissociation cross sections of these
two states. The calculations show 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. This effect is caused by couplinginduced transitions between the two states. Nonadiabatic
coupling is expected to have a smaller effect on individual state
electron dissociation cross sections than the corresponding
photodissociation cross sections. This is because the oscillator
strength is an integration of photodissociation cross section
over Ek and the effect of the oscillation in each partial cross
section tends to cancel out over a wide Ek range. In particular,
the total electron impact dissociation cross section should not
be affected by nonadiabatic coupling.
There are three primary sources of uncertainty for the
individual state cross sections. First is experimental error
of the B 1 u+ −X 1 g+ and C 1 u −X 1 g+ excitation functions.
The estimated error of the B 1 u+ and C 1 u discrete excitation
cross sections above threshold (>17 eV) is 16%, while the
relative error between the B 1 u+ and C 1 u cross sections
is 10% (Liu et al 1998). The second source of error is
the possibility that the particular excitation shape function
is somewhat different from those of the B 1 u+ and C 1 u
states. Based on the experimental observations mentioned
previously, an upper error limit of 10% can be attributed to
the possible difference in the shape of the excitation functions.
The final source of the error, ∼6%, represents the imprecision
of the oscillator strength, which arises from inaccuracy of the
transition moments and the neglect of nonadiabatic coupling
among the singlet ungerade states. On the basis of the square
root of the sum of squares for the individual components, the
error in the B 1 u+ and C 1 u dissociation cross section is
about 17% when the excitation energy is above 17 eV. The
corresponding error of the B 1 u+ , D 1 u , B B̄ 1 u+ , D 1 u
and 5pσ 1 u+ states is estimated to be 20% at electron impact
energy >20 eV. The relative error among the dissociation cross
sections of the B 1 u+ , C 1 u , B 1 u+ , D 1 u , B B̄ 1 u+ , D 1 u
and 5pσ 1 u+ states, however, is less than 15%.
Estimation of the higher npσ 1 u+ and npπ 1 u continua
with the ratio of the effective principal quantum numbers
introduces an additional source of error for these cross sections.
If a generous value of 10% is assumed for this additional
error, the overall error of the cross sections listed in the
ninth column of tables 2–4 is 22%. The overall error for
the total dissociation cross section of the singlet ungerade
continua, however, is lower for two reasons. First, the
B 1 u+ and C 1 u cross sections, which have lower error
limits, are significantly greater than the sum of the higher
npσ 1 u+ and npπ 1 u states, which has a higher error
limit. In addition, the electronic transition moments of
the B 1 u+ −X 1 g+ , C 1 u −X 1 g+ and B 1 u+ −X 1 g+ band
systems are presumably more accurate than those of the higher
band systems, and nonadiabatic coupling, while resulting in
the repartitioning of the photodissociation cross sections (and
5
Although the D 1 u state is also designated as 3pπ 1 u , which is true
for small R, it actually correlates with the H(1s)+H(3d) configuration in the
separate atom limit.
8
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
Table 2. Electron impact dissociation cross sections via singlet ungerade continua (T = 300K).a,b
E (eV)
B 1 u+
C 1 u
B 1 u+ c
B B̄ 1 u+
D 1 u c
D 1 u
5pσ 1 u+
Othersd
Total
14.8
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20
22.5
25
27.5
30
35
40
45
50
60
70
80
90
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
0.00
0.02
0.08
0.15
0.22
0.28
0.35
0.41
0.47
0.52
0.57
0.62
0.82
0.98
1.10
1.19
1.32
1.39
1.43
1.45
1.45
1.43
1.39
1.35
1.31
1.14
1.02
0.92
0.84
0.77
0.71
0.66
0.62
0.58
0.55
0.52
0.49
0.47
0.45
0.43
0.41
0.39
0.38
0.01
0.04
0.20
0.38
0.56
0.74
0.90
1.06
1.21
1.35
1.48
1.61
2.14
2.55
2.86
3.10
3.43
3.63
3.73
3.78
3.78
3.71
3.62
3.51
3.41
2.97
2.65
2.40
2.18
2.00
1.85
1.72
1.61
1.51
1.42
1.34
1.28
1.21
1.16
1.11
1.07
1.02
0.99
0.03
0.14
0.69
1.40
2.16
2.89
3.60
4.27
4.90
5.50
6.07
6.60
8.89
10.63
11.98
13.01
14.43
15.27
15.74
15.96
15.98
15.71
15.32
14.88
14.45
12.61
11.25
10.17
9.27
8.51
7.86
7.31
6.83
6.41
6.04
5.71
5.42
5.16
4.93
4.72
4.53
4.35
4.19
0.00
0.00
0.00
0.00
0.00
0.01
0.03
0.06
0.09
0.11
0.14
0.16
0.26
0.33
0.40
0.44
0.51
0.55
0.58
0.60
0.61
0.60
0.59
0.58
0.57
0.50
0.45
0.41
0.37
0.34
0.32
0.30
0.28
0.26
0.25
0.23
0.22
0.21
0.20
0.19
0.19
0.18
0.17
0.00
0.00
0.00
0.00
0.00
0.01
0.04
0.06
0.09
0.11
0.14
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
0.42
0.38
0.35
0.33
0.30
0.28
0.27
0.25
0.24
0.23
0.22
0.21
0.20
0.19
0.18
0.18
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.03
0.04
0.04
0.08
0.11
0.13
0.15
0.18
0.19
0.20
0.21
0.21
0.21
0.21
0.21
0.20
0.18
0.16
0.15
0.13
0.12
0.11
0.11
0.10
0.09
0.09
0.08
0.08
0.08
0.07
0.07
0.07
0.06
0.06
0.00
0.00
0.00
0.00
0.00
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.22
0.29
0.34
0.38
0.44
0.48
0.50
0.51
0.52
0.52
0.51
0.50
0.49
0.43
0.39
0.35
0.32
0.30
0.28
0.26
0.24
0.23
0.21
0.20
0.19
0.18
0.17
0.17
0.16
0.15
0.15
0.00
0.00
0.00
0.00
0.00
0.01
0.04
0.09
0.15
0.21
0.28
0.33
0.55
0.72
0.86
0.97
1.12
1.22
1.28
1.32
1.35
1.34
1.32
1.29
1.26
1.11
1.00
0.91
0.83
0.77
0.71
0.66
0.62
0.58
0.55
0.52
0.50
0.47
0.45
0.43
0.42
0.40
0.39
0.04
0.21
0.96
1.93
2.93
3.95
4.98
6.00
6.99
7.92
8.82
9.65
13.22
15.96
18.07
19.71
21.96
23.30
24.07
24.45
24.54
24.15
23.57
22.92
22.26
19.44
17.36
15.71
14.33
13.17
12.17
11.32
10.57
9.93
9.36
8.85
8.41
8.01
7.65
7.32
7.02
6.75
6.50
Unit is 10−19 cm2 .
The absolute error of the B 1 u+ and C 1 u states is 17% for E 17 eV. The corresponding error
for the B 1 u+ , D 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states is less than 20% when E is above 20 eV.
The error for the total dissociation cross section is ∼19% also for E > 20 eV. See text for discussion
of various errors. Cross section entries are intentionally shown in two digits after the decimal point to
allow meaningful comparison of the magnitude of various states and of different temperature.
c
From Liu et al (2009a).
d
Sum of npσ 1 u+ with n 6 and npπ 1 u with n 5.
a
b
thus oscillator strengths) among the interacting states, does not
alter the total oscillator strength (Beswick and Glass-Maujean
1987). In other words, the ∼6% error, attributed to possible
inaccuracy of transition moments and the negligence of
nonadiabatic coupling, does not apply to the total dissociation
cross section. For this reason, the error limit for the total
dissociation cross section is ∼19%, slightly smaller than the
limit of individual states such as B 1 u+ , D 1 u , B B̄ 1 u+ ,
D 1 u and 5pσ 1 u+ .
3.3. Discussion
A common feature of photoexcitation from the X 1 g+ state
to the singlet ungerade continua is that the dissociation cross
sections and oscillator strengths very often, but not always,
increase with the initial rotational quantum number. The
dependence of the dipole transition moment on internuclear
distance may be partially responsible for the increase. In
addition, the small reduced mass of H2 leads to a very large
rotational constant. Consequently, the number of bound or
discrete vibrational levels decreases fairly rapidly with Jj .
9
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
Table 3. Electron impact dissociation cross sections via singlet ungerade continua (T = 1400K).a,b
E (eV)
B 1 u+
C 1 u
B 1 u+ c
B B̄ 1 u+
D 1 u c
D 1 u
5pσ 1 u+
Othersd
Total
14.5
14.7
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20
22.5
25
27.5
30
35
40
45
50
60
70
80
90
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
0.01
0.01
0.04
0.12
0.21
0.29
0.37
0.45
0.52
0.59
0.65
0.71
0.77
1.01
1.20
1.34
1.45
1.60
1.69
1.73
1.75
1.75
1.72
1.67
1.62
1.58
1.37
1.22
1.11
1.01
0.92
0.85
0.79
0.74
0.69
0.65
0.62
0.59
0.56
0.53
0.51
0.49
0.47
0.45
0.01
0.03
0.11
0.31
0.54
0.76
0.97
1.17
1.36
1.54
1.71
1.87
2.02
2.66
3.15
3.52
3.81
4.20
4.43
4.55
4.61
4.60
4.51
4.40
4.27
4.14
3.61
3.21
2.90
2.64
2.43
2.24
2.08
1.94
1.82
1.72
1.62
1.54
1.47
1.40
1.34
1.29
1.24
1.19
0.01
0.05
0.26
0.90
1.68
2.49
3.28
4.04
4.75
5.43
6.07
6.68
7.25
9.69
11.55
12.98
14.08
15.59
16.47
16.95
17.19
17.19
16.89
16.46
15.99
15.52
13.53
12.07
10.91
9.94
9.13
8.43
7.83
7.31
6.86
6.47
6.12
5.81
5.53
5.28
5.06
4.85
4.66
4.49
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.05
0.08
0.11
0.13
0.16
0.19
0.30
0.39
0.46
0.51
0.59
0.63
0.66
0.68
0.69
0.69
0.68
0.66
0.64
0.57
0.51
0.46
0.42
0.39
0.36
0.34
0.32
0.30
0.28
0.26
0.25
0.24
0.23
0.22
0.21
0.20
0.20
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.05
0.08
0.12
0.15
0.18
0.20
0.32
0.42
0.49
0.55
0.63
0.68
0.71
0.73
0.74
0.74
0.72
0.71
0.69
0.60
0.54
0.49
0.45
0.42
0.39
0.36
0.34
0.32
0.30
0.28
0.27
0.26
0.24
0.23
0.23
0.22
0.21
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.04
0.05
0.06
0.10
0.13
0.16
0.18
0.21
0.23
0.24
0.25
0.26
0.26
0.25
0.25
0.24
0.21
0.19
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.11
0.10
0.10
0.09
0.09
0.08
0.08
0.08
0.07
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.03
0.06
0.08
0.11
0.13
0.15
0.24
0.31
0.37
0.41
0.48
0.52
0.54
0.55
0.56
0.56
0.55
0.54
0.52
0.46
0.41
0.38
0.34
0.32
0.30
0.28
0.26
0.24
0.23
0.22
0.21
0.20
0.19
0.18
0.17
0.17
0.16
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.04
0.10
0.17
0.24
0.32
0.38
0.62
0.81
0.96
1.08
1.25
1.36
1.42
1.46
1.49
1.49
1.46
1.43
1.39
1.23
1.10
1.00
0.92
0.85
0.79
0.73
0.69
0.64
0.61
0.58
0.55
0.52
0.50
0.48
0.46
0.44
0.43
0.03
0.10
0.42
1.33
2.43
3.55
4.69
5.83
6.97
8.06
9.09
10.09
11.01
14.94
17.96
20.28
22.07
24.54
26.00
26.82
27.23
27.29
26.85
26.19
25.46
24.72
21.58
19.26
17.42
15.89
14.59
13.49
12.53
11.71
10.99
10.36
9.80
9.31
8.86
8.47
8.10
7.77
7.47
7.20
Unit is 10−19 cm2 .
The absolute error of the B 1 u+ and C 1 u states is 17% for E 17 eV. The corresponding error
for the B 1 u+ , D 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states is less than 20% for E 19 eV. The error
for the total dissociation cross section is ∼19% for E > 20 eV. See text for discussion of various
errors. Cross section entries are intentionally shown in two digits after the decimal point to allow a
meaningful comparison of the magnitude of various states and different temperature.
c
Based on the photodissociation cross section of Liu et al (2009a).
d
Sum of npσ 1 u+ with n 6 and npπ 1 u with n 5.
a
b
sections. Nevertheless, the thermally averaged cross sections
shown in tables 2–4 are a convenient way to illustrate the
general trend of the cross section variation with ro-vibrational
quantum number. All individual state cross sections listed in
tables 2–4 increase with temperature. The increase is largely
a consequence of the rotational and vibrational dependence
of the dipole matrix elements as illustrated in the variation
of the photodissociation oscillator strengths with Ji and vi
in figures 3 and 4. In general, the increase of the Jj quantum
These missing discrete vibrational states for the higher Jj
levels are shifted into the continuum and thus contribute to
the dissociation cross section and oscillator strength.
The population distribution of H2 has been found to be
highly non-LTE in cometary atmospheres (Liu et al 2007)
at the top of the thermosphere of Saturn (Shemansky et al
2009), in certain regions of the ISM (Habart et al 2005) and in
low-density hydrogen plasma environments. The modelling
of these observations generally requires state-to-state cross
10
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
Table 4. Electron impact dissociation cross sections via singlet ungerade continua (T = 5000K).a,b
E (eV)
B 1 u+
C 1 u
B 1 u+ c
B B̄ 1 u+
D 1 u c
D 1 u
5pσ 1 u+
Othersd
Total
14
14.2
14.4
14.7
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20
22.5
25
27.5
30
35
40
45
50
60
70
80
90
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
0.55
0.70
0.86
1.14
1.44
1.94
2.42
2.89
3.33
3.74
4.13
4.49
4.84
5.16
5.47
6.75
7.71
8.43
8.98
9.69
10.08
10.26
10.31
10.19
9.93
9.63
9.32
9.03
7.82
6.94
6.24
5.66
5.18
4.77
4.42
4.12
3.86
3.63
3.44
3.26
3.10
2.96
2.83
2.72
2.61
2.51
0.97
1.25
1.58
2.14
2.76
3.82
4.86
5.84
6.78
7.66
8.49
9.27
10.00
10.69
11.35
14.09
16.15
17.70
18.88
20.42
21.26
21.66
21.78
21.56
21.03
20.40
19.75
19.13
16.58
14.72
13.24
12.02
10.99
10.13
9.39
8.75
8.21
7.73
7.30
6.93
6.60
6.29
6.02
5.78
5.55
5.34
0.27
0.37
0.50
0.79
1.20
2.09
3.07
4.06
5.01
5.93
6.79
7.60
8.37
9.10
9.79
12.70
14.91
16.60
17.89
19.64
20.64
21.17
21.39
21.32
20.89
20.32
19.72
19.12
16.64
14.82
13.37
12.17
11.15
10.29
9.56
8.92
8.37
7.88
7.46
7.08
6.74
6.43
6.16
5.91
5.68
5.47
0.00
0.00
0.00
0.01
0.01
0.03
0.05
0.11
0.18
0.26
0.34
0.42
0.49
0.56
0.63
0.92
1.14
1.31
1.45
1.64
1.75
1.82
1.86
1.88
1.86
1.82
1.77
1.72
1.51
1.35
1.23
1.12
1.03
0.95
0.89
0.83
0.78
0.74
0.70
0.66
0.63
0.60
0.58
0.55
0.53
0.51
0.01
0.01
0.01
0.02
0.04
0.08
0.17
0.29
0.44
0.60
0.75
0.90
1.04
1.17
1.29
1.83
2.24
2.56
2.81
3.16
3.36
3.49
3.55
3.58
3.53
3.45
3.36
3.26
2.86
2.55
2.31
2.11
1.94
1.80
1.67
1.56
1.47
1.38
1.31
1.24
1.19
1.13
1.08
1.04
1.00
0.96
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.05
0.08
0.13
0.19
0.24
0.29
0.34
0.38
0.58
0.73
0.85
0.95
1.08
1.16
1.21
1.24
1.26
1.25
1.22
1.19
1.16
1.02
0.91
0.83
0.76
0.70
0.65
0.60
0.56
0.53
0.50
0.47
0.45
0.43
0.41
0.39
0.38
0.36
0.35
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.05
0.09
0.14
0.19
0.24
0.28
0.33
0.37
0.55
0.69
0.80
0.89
1.01
1.08
1.13
1.15
1.17
1.16
1.13
1.11
1.08
0.94
0.85
0.77
0.70
0.65
0.60
0.56
0.52
0.49
0.46
0.44
0.42
0.40
0.38
0.36
0.35
0.33
0.32
0.00
0.00
0.00
0.00
0.00
0.01
0.03
0.09
0.24
0.41
0.62
0.82
1.00
1.16
1.32
1.98
2.50
2.90
3.21
3.66
3.93
4.10
4.19
4.25
4.21
4.12
4.02
3.91
3.43
3.07
2.79
2.55
2.35
2.18
2.03
1.90
1.78
1.68
1.59
1.51
1.44
1.38
1.32
1.27
1.22
1.17
1.79
2.33
2.97
4.11
5.46
7.98
10.64
13.37
16.15
18.86
21.49
23.97
26.31
28.51
30.59
39.40
46.08
51.17
55.06
60.29
63.27
64.83
65.48
65.19
63.84
62.09
60.23
58.41
50.81
45.22
40.77
37.09
33.99
31.36
29.11
27.17
25.49
24.01
22.71
21.55
20.52
19.59
18.75
17.98
17.29
16.65
Unit is 10−19 cm2 .
The absolute error of the B 1 u+ and C 1 u states is 17% for E 17 eV. The corresponding error
for the B 1 u+ , D 1 u , B B̄ 1 u+ , D 1 u and 5pσ 1 u+ states is less than 20% for E 19 eV. The error
for the total dissociation cross section is ∼19% for E > 20 eV. See the text for discussion of various
errors.
c
Based on the photodissociation cross section of Liu et al (2009a).
d
Sum of npσ 1 u+ with n 6 and npπ 1 u with n 5.
a
b
number pushes more vibrational levels from the discrete region
to the continuum region, which leads to the increase in the
dissociation cross section. The increased temperature also
leads to more population at higher Ji and vi levels. These
higher Ji and vi levels have smaller threshold energies (Eij )
to the continuum levels than the lower levels, which results
in a larger dissociation cross section in the threshold energy
region. It can be noted that 1400 K is on the higher end of
the range for the Jovian thermosphere, while 5000 K may be
close to the temperature of some hydrogen plasmas. When
the population distribution does not deviate significantly from
LTE, the averaged cross sections in tables 2–4 provide a quick
way to estimate the loss of H2 by singlet ungerade continua.
Unless a very large population is present in very high
Ji (>23) levels, dissociation through excitation from the vi = 0
level takes place primarily via the B 1 u+ continuum. At room
11
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
∼63%, respectively, greater than the tabulated ones. Finally,
the good agreement between Borges et al (1998) and present
cross sections in the high-energy region, to a very large extent,
reflects the good agreement in the dissociation oscillator
strengths and electronic transition moments.
Dissociation by excitation to the continuum levels is
not the only path for the fragmentation of H2 through the
singlet ungerade states. Autoionization, dissociative emission,
predissociation of the discrete levels of singlet ungerade states
and dissociative ion-pair formation are additional mechanisms
for the break up of H2 . Dissociative emission arises from
excitation of H2 to discrete singlet ungerade levels, followed
by spontaneous emission to the continuum levels of the
X 1 g+ state. In general, the dissociative emission from the
B 1 u+ state far dominates over the emission from other states.
The C 1 u , B 1 u+ and D 1 u states contribute marginally
to the continuum emission (Abgrall et al 1997). For this
reason, dissociative emission is usually referred to as the
Lyman continuum (Stephens and Dalgarno 1972). Other
npσ 1 u+ and npπ 1 u states contribute very little because
they have much smaller excitation cross sections and because
they have either poor overlap with the X 1 g+ continuum
or their emission branching ratio to the X 1 g+ state is
significantly reduced by predissociation and autoionization.
As stated above, the predissociation of the singlet ungerade
levels is largely caused by the coupling with the B 1 u+
continuum. The D 1 +u levels above the H(1s)+H(2) limit
are directly predissociated by the B 1 u+ continuum (GlassMaujean et al 1979). The predissociation of npσu 1 u+
(n > 4) takes place by homogeneous coupling with the B 1 u+
continuum levels (Glass-Maujean 1979). The predissociation
of npπu 1 +u (n > 3) takes place by either npπu 1 +u −D 1 +u
homogeneous coupling followed by D 1 +u −B 1 u+ Coriolis
coupling or npπu 1 +u − npσu 1 u+ Coriolis coupling followed
by npσu 1 u+ −B 1 u+ homogeneous coupling. The D 1 −
u and
1 +
npπu 1 −
states
are
not
coupled
to
the
B
state
or
other
u
u
npσu 1 u+ states. They can only couple to a dissociating 1 −
u
state. Among the npπu 1 −
u states below the H(1s)+H(n =
1 −
3) limit, C 1 −
u is the only dissociative u state. Since
1 −
npπu u states are only weakly coupled to the C 1 −
u state,
their predissociation rates are negligibly small. Autoionization
arises from the coupling of npσ 1 u+ and npπ 1 u to the X 2 g+
ionization continuum. Since the classical work of Dehmer and
Chupka (1976), significant progress has been made towards
obtaining the autoionization yield of many singlet ungerade
levels. In particular, recent experimental and theoretical
investigations by Glass-Maujean et al (2007b, 2008a, 2008b,
2009, 2010) and by Glass-Maujean and Jungen (2009) have
made it possible to reliably calculate predissociation and
autoionization cross sections of major singlet ungerade rovibrational levels that are accessible at room temperature.
When combined with the dissociation cross section obtained
in this work, a reliable estimate of the total cross section of H2
fragmentation through singlet ungerade excitation, by photon
or electron, will be obtained. Excitation to ion-pair states
and to Rydberg series that converge to the ion-pair states is
further fragmentation channel of H2 . An important feature
of these Rydberg series is a much smaller Rydberg constant
Table 5. Comparison of electron impact dissociation cross sections
of the B 1 u+ and C 1 u statesa
B 1 u+
E (eV)
Present
Borges et al
100
200
300
400
500
800
1000
1.31
1.02
0.839
0.711
0.617
0.446
0.379
1.60
1.08
0.843
0.686
0.591
0.485
0.347
a
b
C 1 u
b
Present
Borges et al b
3.41
2.65
2.18
1.85
1.61
1.16
0.986
4.66
3.17
2.45
2.01
1.71
1.21
1.03
Unit is 10−19 cm2 .
Borges et al (1998).
temperature, dissociation via the B 1 u+ continuum dominates
dissociation through other singlet ungerade continua. Since
the predissociation of the singlet ungerade states primarily
occurs by direct or indirect coupling with the B 1 u+
continuum, predissociation gives additional importance of
the B 1 u+ continuum. However, for excitation from a
vibrationally excited level, dissociation via the B 1 u+ and
C 1 u continua is more important than via the B 1 u+
continuum (see figure 4). At 5000 K, approximately 1/3 of
H2 population is at the vi > 0 levels. Table 4 shows that the
dissociation cross section of the C 1 u is comparable to that
of the B 1 u+ state.
If the H2 population, along with the photon and electron
energy distributions, is known, the cross sections obtained in
this work make it possible to accurately evaluate the conversion
of photon and electron energies into the kinetic energy of
hydrogen atoms by way of singlet ungerade continua. While
the energy transfer from electrons and photons to hydrogen
atoms can take place by other dissociation mechanisms, the
cross sections obtained in this work represent a set of basic
physical parameters for a quantitative understanding of the
heating of H2 -dominated atmospheres by solar radiation and
photoelectrons.
Unlike the B 1 u+ state, not many theoretical calculations
have been carried out for the electron dissociation cross
sections of the other singlet ungerade states. Table 5 compares
the B 1 u+ and C 1 u dissociation cross section obtained in
this work for T = 300 K to those calculated by Borges et al
(1998) on the basis of first Born approximation. Except for
B 1 u+ and C 1 u cross sections at 100 eV and C 1 u cross
section at 200 eV, all other calculated values agree with the
present values within 12%. At 100 eV, the calculated B 1 u+
and C 1 u cross sections are about 22% and 37% larger than
the present cross sections. Similarly, the C 1 u cross section
is nearly 20% greater than the present value at 200 eV. The
large difference at 100 and 200 eV is consistent with fact that
the Born approximation almost always overestimates the lowenergy cross sections. In the case of the dissociation cross
sections of the B 1 u+ state, which have been calculated by
many theoretical methods, Liu et al (2009a) found that the
Borges et al (1998) cross sections have the best agreement
with the values listed in the fourth column of table 2, though
the calculated values at 100 and 200 eV are also ∼38% and
12
J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 015201
X Liu et al
and Bohr radius (by a factor of ∼918.6) than that of H or H2 .
Because they are stable only at large internuclear distances
(R > 12a0 ), only states with n greater than ∼100 have been
observed. Although many investigations have been performed
(Chupka et al 1975, Kung et al 1986, Pratt et al 1992, Vieitez
et al 2008, 2009, Ekey and McCormack 2011), the magnitude
of ion-pair production cross section is not well known.
However, since the formation of Rydberg series that converge
to ion-pair states takes place in the long-range regime (R >
12a0 ) (Vieitez et al 2008, Vieitez 2010), the dissociation cross
section of this channel from the low vi levels is probably very
small.
Photodissociation of H2 in the ISM primarily takes place
by dissociative emission to the X 1 g+ continuum, especially
by the B 1 u+ −X 1 g+ continuum. However, in the case of
dissociation by the singlet ungerade continua, the variation of
the B 1 u+ , C 1 u and B 1 u+ dissociation cross sections with
vi has a profound implication for the photodissociation of H2
in the ISM. In contrast to the normal laboratory condition,
photodissociation of H2 in the ISM, even in a very cold
environment, is more likely dominated by the C 1 u and B 1 u+
continua than by the B 1 u+ continuum transition. The primary
reason is the shielding of the interstellar photon radiation field
by the ionization continuum of the H atom. The strength of
the field becomes very weak once the frequency is higher than
109 679 cm−1 . Since dissociation of H2 from vi = 0 and
Ji = 0 requires a minimum photon energy of 118 377 cm−1 ,
photodissociation requires an internal energy of more than
8698 cm−1 . This means that any significant photodissociation
must take place from following ro-vibrational levels: (a) vi =
0 and Ji 13, (b) vi = 1 and Ji 9, (c) vi = 2 and Ji 3 and
(d) vi 3. For the cases (c)–(d), figure 4 suggests that B 1 u+ ,
and, especially, C 1 u are more important than the B 1 u+
continuum. Only when most of dissociated H2 is originally
between Ji = 13 and Ji = 22 of the vi = 0 level will the B 1 u+
dissociation be dominant. Even in this case, the contribution
of the B 1 u+ and C 1 u continua is very significant.
As noted, the uncertainty of the electron impact
dissociation cross sections beyond threshold is 16–20%. A
substantial portion of this uncertainty is due to the lack of
measured excitation functions for the B 1 u+ , D 1 u , B B̄ 1 u+ ,
D 1 u and 5pσ 1 u+ states and the uncertainty in the measured
Lyman and Werner band excitation functions of Liu et al
(1998). A reduction of error to 14–15% is achievable by
measuring the excitation functions of B 1 u+ , D 1 u , B 1 u+ ,
D 1 u and 5pσ 1 u+ , along with a few small improvements in
the electron gun and data acquisition as described by Young
et al (2010).
the B 1 u+ continuum, and, especially, the C 1 u continuum
becomes more important.
The calculated continuum oscillator strengths, along with
previously published Lyman and Werner band excitation
functions, are utilized to determine electron impact
dissociation cross sections through the continuum levels of the
singlet ungerade states. In addition to the state-to-state cross
sections, thermally averaged cross sections from threshold to
1000 eV at various temperatures are presented for the first time.
Significant ro-vibrational dependence of the photodissociation
cross sections and oscillator strengths results in temperature
dependence in the electron dissociation cross sections.
Acknowledgments
The authors wish to thank Professor Lutoslaw Wolniewicz
for making results of his ab initio calculations accessible.
The analysis described in this paper was carried out at
Space Environment Technologies (SET) and Jet Propulsion
Laboratory (JPL), California Institute of Technology. Work
performed at SET is supported by the Cassini UVIS contract
with the University of Colorado, by NSF AGS-0938223, and
NASA Cassini Data Analysis program. XL, PVJ and CPM
acknowledge financial support through NASA’s Planetary
Atmospheres Research programs.
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13
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