The Astrophysical Journal, 716:701–711, 2010 June 10
C 2010.
doi:10.1088/0004-637X/716/1/701
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
KINETIC ENERGY DISTRIBUTION OF H(1s) FROM H2 X 1 Σ+g –a 3 Σ+g EXCITATION AND LIFETIMES AND
TRANSITION PROBABILITIES OF a 3 Σ+g (v, J )
Xianming Liu1,2 , Paul V. Johnson1 , Charles P. Malone1,3 , Jason A. Young1 , Isik Kanik1 , and Donald E. Shemansky2
1
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA; Xianming@jpl.nasa.gov, xliu@spacenvironment.net,
Paul.V.Johnson@jpl.nasa.gov, Isik.Kanik@jpl.nasa.gov
2 Planetary and Space Science Division, Space Environment Technologies, Pasadena, CA 91107, USA; Dshemansky@spacenvironment.net
Received 2010 February 23; accepted 2010 April 22; published 2010 May 21
ABSTRACT
Dissociative excitation of molecular hydrogen plays an important role in the heating of outer planet upper
thermospheres. This paper addresses the role of one of the triplet states involved in the process. H2 excited to the a 3 Σ+g
state, or higher triplet-ungerade states, is dissociated via the a 3 Σ+g −b 3 Σ+u continuum. The kinetic energy distribution
of H(1s) produced from direct X 1 Σ+g –a 3 Σ+g (v, J ) excitation by electrons is investigated by an accurate theoretical
evaluation of spontaneous transition probabilities of the a 3 Σ+g (v, J )−b 3 Σ+u continuum transition. It is shown that the
X 1 Σ+g (0)–a 3 Σ+g (v, J ) excitation primarily produces H(1s) atoms with kinetic energies lower than 2 eV. In addition
to the continuum a 3 Σ+g (v, J )−b 3 Σ+u transition probabilities, spontaneous emission lifetimes of the a 3 Σ+g (v, J )
(v = 0–20, J 14) levels have been calculated by considering both the a 3 Σ+g −b 3 Σ+u and a 3 Σ+g –c 3 Πu transitions.
The calculated lifetimes show a moderately strong rotational dependence, and the lifetimes for the J = 0 rotational
level of the low v levels agree well with previous calculations and experimental measurements. Calculations
of the a 3 Σ+g −b 3 Σ+u continuum emission spectra from electron impact X 1 Σ+g –a 3 Σ+g excitation are included.
Key words: molecular data – molecular processes
Online-only material: color figures
only planet showing a remarkably confined low latitude region
in the sunlit atmosphere away from auroral influence from which
a large fraction of outflowing hydrogen originates. The auroral
zones are not a significant source of escaping atomic hydrogen
and are not a measurable source on the scale of the low latitude
outflow. It has also been found that the emission spectra of H2
singlet-ungerade states in the primary atomic hydrogen source
region strongly deviate from local thermodynamic equilibrium
(LTE; Shemansky et al. 2009). The H2 spectra collected, along
with the H Lyman-α, into the image mosaic show a distinctive
H2 X 1 Σ+g vibrational excitation correlated with the location of
the H Lyman-α plume. Examination of the range of dissociation
processes is essential to understand and model the observed
atomic hydrogen plume.
Although H2 triplet states can be excited by collisions of H2
X 1 Σ+g with heavy ions such as Ar+ and Kr+ (Brandt & Ottinger
1979) and fast atoms such as Ar (Lishawa et al. 1985) and H
(Petrovic & Phelps 2009), the most important triplet excitation
channel in planetary atmospheres is by electron impact. In
the absence of collisional deactivation, all triplet excitations
eventually lead to dissociation. The excitation of H2 to the
bound levels of the triplet-gerade states results in the dissociative
spontaneous emission to the repulsive b 3 Σ+u state. Excitation to
the continuum and the predissociative levels of the triplet states
leads directly to dissociation (Helm et al. 1984; Dinu et al.
2004). Excitation to the quasi-bound levels of the triplet states
is either followed by barrier tunneling to dissociative continuum
or spontaneous emission to lower triplet levels and eventual
dissociative molecular emissions (Helm et al. 1984; de Bruijin
& Helm 1986; Koot et al. 1989a; Wouter et al. 1997). For the
rotational levels of the c 3 Πu (0) state that lie below the a 3 Σ+g (0)
state, spontaneous emission to the a 3 Σ+g is not possible. The
1. INTRODUCTION
Dissociative excitation of H2 is an important process in
stellar and planetary atmospheres. Since the work of James &
Coolidge (1939), it has been well known that the a 3 Σ+g −b 3 Σ+u
continuum emission is an important contributor to opacity in
stellar atmospheres. In addition, transitions from the discrete
levels of the singlet-ungerade states, such as B 1 Σ+u and C 1 Πu ,
to the continuum levels of the X 1 Σ+g state are a major destruction
mechanism of H2 in the interstellar medium (Stephens &
Dalgarno 1972; Abgrall et al. 1997). In general, the excitation
of H2 to its dissociative or predissociative states by photons
and electrons produces kinetically hot hydrogen atoms. The
collision of these fast hydrogen atoms with H2 and other minor
atmospheric gases results in the heating of the atmospheres of
the outer planets. Thus, dissociative excitation of H2 converts
electronic energy into heat.
Images of H Lyman-α in the atmosphere of Saturn, recently
obtained by the Cassini Ultraviolet Imaging Spectrograph
(UVIS) instrument, show atomic hydrogen flowing out of the
top of the sunlit thermosphere in a localized, distinct plume
in ballistic and escaping orbits (Shemansky et al. 2009). The
images also reveal a continuous distribution of atomic hydrogen
from the top of Saturn’s atmosphere, measurable to at least 45
Saturn radii (RS ) in the satellite orbital plane, and measurable
to at least 30 RS latitudinally above and below the plane.
The phenomenon of escaping atomic hydrogen is unique to
Saturn in two ways. First, the gravitational potential of Saturn is
small enough to allow the loss of atomic dissociation products
produced in the activated thermosphere. Secondly, Saturn is the
3 Also affiliated with the Department of Physics, California State University,
Fullerton, CA 92834, USA.
701
702
LIU ET AL.
c 3 Π+u (0) state rotational levels, however, can be predissociated
by the b 3 Σ+u state via 3 Σ+u –3 Π+u coupling and have lifetimes
shorter than 1.4 ns (Chiu & Bhattacharyya 1979). The rotational
levels of the c 3 Π−
u (0) state are metastable, with lifetimes
ranging from 1 ms to ∼200 μs (Johnson 1972; Berg & Ottinger
1994). However, even these metastable H2 can be dissociated
by coupling to the b 3 Σ+u state via spin–spin and spin–orbit
interaction or by dissociative transitions to b 3 Σ+u via magnetic
dipole and quadrupole radiation (Chiu & Bhattacharyya 1979;
Berg & Ottinger 1994).
The spectra of H2 triplet states have been experimentally
investigated for well over 80 years. The early experimental
investigations by discharge emission spectroscopy have been
well documented in the book by Richardson (1934) and the
extensive wavelength tables of Dieke by Crosswhite (1972).
Because of the spin forbidden nature of the X 1 Σ+g –triplet
transitions, techniques such as electron induced microwave
optical magnetic resonance (Freund & Miller 1973; Miller
et al. 1974) and electron–photon or photon–photon delayed
coincidence (Imhof & Read 1971; King et al. 1975; Mohamed
& King 1979; Kiyoshima et al. 2003) were used to investigate
the triplet states. High-resolution laser spectroscopy has been
widely used since the pioneering work of Eyler & Pipkin (1981),
who produced metastable triplet H2 by electron impact, and
Helm et al. (1984), who obtained metastable triplet H2 by
reaction of an H+2 beam with Cs vapor. Extensive investigations
of dissociation dynamics of the n 3 triplet Rydberg series
have been carried out with H+2 –Cs fast beam photofragment
spectroscopy (de Bruijin & Helm 1986; Bjerre et al. 1988;
Lembo et al. 1988, 1990; Koot et al. 1989a; Schins et al.
1991; Siebbeles et al. 1992; Wouter et al. 1996, 1997). Other
techniques such as Fourier-transform infrared spectroscopy
(Herzberg & Jungen 1982; Jungen et al. 1989, 1990) and infrared
laser spectroscopy (Davies et al. 1988, 1990a, 1990b; Uy et al.
2000) have also been employed to study, primarily, the high l
Rydberg transitions.
The two principal theoretical methods used to calculate the
structures of excited electronic states of H2 are traditional
ab initio, with at most a few coupled electronic states, and
multichannel quantum defect theory (MQDT), which treats the
whole family of Rydberg states. Bishop & Cheung (1981), Kolos
& Rychlewski (1977, 1990a, 1990b, 1994, 1995), Orlikowski
et al. (1999), and Staszewska & Wolniewicz (1999, 2001) have
carried out extensive ab initio calculations. Ross & Jungen
(1994), Ross et al. (2001), Matzkin et al. (2000), and Kiyoshima
et al. (2003) have performed a number of MQDT investigations
of the triplet structure of H2 . Of particular relevance, James
& Coolidge (1939) carried out one of the earliest theoretical
calculations of the a 3 Σ+g −b 3 Σ+u transitions for the first five
vibrational levels. Kwok et al. (1986) and Fantz et al. (2000)
extended the calculation to the J = 0 rotational levels of the
higher a 3 Σ+g vibrational levels. Guberman & Dalgarno (1992)
obtained transition probabilities and transition moments for a
number of triplet electronic transitions. Fantz & Wünderlich
(2006) provided extensive tabulation of transition probabilities,
Franck–Condon factors, and radiative lifetimes for various
electronic band systems of H2 and its isotopomers. More
importantly, the extensive and accurate calculations of triplet
adiabatic potential energy curves and electronic transitions
moments have been performed by Staszewska & Wolniewicz
(1999, 2001). Transition moments among triplet states and
fine structure spin–spin constants for a number of triplet states
have also been computed by Spielfiedel et al. (2004a, 2004b).
Vol. 716
Finally, accurate energy term values of rovibrational levels of
the a 3 Σ+g state have been recently obtained by Wolniewicz
(2007).
Electron impact excitation cross sections and excitation
functions of triplet states have also been investigated by many
experimental and theoretical studies. The absolute cross sections
of low-lying states, such as the a 3 Σ+g , b 3 Σ+u , and c 3 Πu states,
have been almost exclusively measured by electron energy
loss (EEL) spectroscopy (Hall & Andrić 1984; Nisimura &
Danjo 1986; Khakoo & Trajmar 1986; Khakoo et al. 1987;
Khakoo & Segura 1994; Wrkich et al. 2002). Other experimental
investigations (Böse 1978; Mason & Newell 1986; Ottinger
& Rox 1991; Ajello & Shemansky 1993; Furlong & Newell
1995; Harries et al. 2004) measure only the relative values of
cross sections or excitation functions and require normalizations
to certain standards to obtain absolute values. Theoretical
calculations include those by Chung et al. (1975), Stibbe &
Tennyson (1998), Trevisan & Tennyson (2002), Laricchiuta
et al. (2004), da Costa et al. (2005), and Taveira et al. (2006).
The present work examines transition probabilities of the
a 3 Σ+g −b 3 Σ+u continuum and calculates the kinetic energy distribution of H(1s) produced from X 1 Σ+g –a 3 Σ+g electron impact
excitation. It also calculates the a 3 Σ+g −b 3 Σ+u continuum emission profile resulted from X 1 Σ+g –a 3 Σ+g electron impact excitation. All these calculations and examinations are carried out
at the rotational level. The a 3 Σ+g −b 3 Σ+u transition is important
because it determines the kinetic energy distribution of H(1s)
produced from H2 directly excited to the a 3 Σ+g state, as well as
H2 excited to higher-lying triplet-ungerade states that cascade to
the a 3 Σ+g state, and then undergo the a 3 Σ+g −b 3 Σ+u dissociative
transition. Moreover, the a 3 Σ+g −b 3 Σ+u continuum transition has
also been proposed to be a diagnostic for the vibrational population distribution of the X 1 Σ+g state and for the electron impact
dissociation rate of H2 in hydrogen plasmas (Lavrov et al. 1999).
The present work shows that H(1s) atoms produced from electron impact excitation to the a 3 Σ+g state, and triplet-ungerade
states higher than the b 3 Σ+u state cannot be responsible for the
hot hydrogen atoms observed to escape from the atmosphere
of Saturn (Shemansky et al. 2009). The present work has also
obtained accurate a 3 Σ+g −b 3 Σ+u continuum transition probability
profiles and radiative lifetimes for v = 0–20 and J 14 of the
a 3 Σ+g state. Results obtained in this work allow a more accurate
modeling of plasmas in non-LTE conditions.
2. THEORY
In the present work, the subscript indices i and j will be used
to denote the appropriate quantum numbers of the lower and
upper states of H2 . Electron spin interaction in H2 is very small,
so that both X 1 Σ+g and triplet states are well described by Hund’s
case (b), and electron spin is neglected here. J is therefore used
instead of the conventional N to represent the rotational angular
momentum of the triplet states.
2.1. Kinetic Energy Distribution of H(1s) from X 1 Σ+g –a 3 Σ+g
Excitation
Electron impact excitations for dipole allowed or dipole
forbidden transitions have been treated in previous publications
(Abgrall et al. 1997; Liu et al. 2002, 2003). After defining kinetic
energy distribution of dissociative emission, the present work
extends the previous formalism to the X 1 Σ+g –a 3 Σ+g excitation, a
spin and dipole forbidden excitation.
No. 1, 2010
ENERGY DISTRIBUTION OF H(1s) FROM H2 X 1 Σ+g –a 3 Σ+g EXCITATION
The volumetric production rate of H(1s) atoms with kinetic energy, Ek , arising from the a 3 Σ+g (vj , Jj )−b 3 Σ+u discretecontinuum transition is
R(vj , Jj , Ek )dEk = g(vj , Jj )
AEk (vj , Jj , Ek )dEk
,
A(vj , Jj )
(1)
where AEk (vj , Jj , Ek ) is the differential transition probability of the a 3 Σ+g (vj , Jj )−b 3 Σ+u transitions and A(vj , Jj ) is total transition probability of the (vj , Jj ) level, which formally
includes transitions to the lower c 3 Πu (vi , Ji ) levels. The calculations of AEk (vj , Jj , Ek ) and A(vj , Jj ) are discussed in detail in
Section 2.2. The volumetric excitation rate to the (vj , Jj ) level
is represented by g(vj , Jj ), which normally includes direct excitation from the X 1 Σ+g state and cascade from higher tripletungerade states. Only direct X 1 Σ+g –a 3 Σ+g excitation is considered in the present work.
The direct excitation rate is proportional to the population
of the initial state, Ni , the excitation cross section, σij , and the
electron flux, Fe :
g(vj , Jj ) = Fe
N(vi , Ji )σ (vi , vj , Ji , Jj ).
(2)
vi ,Ji
The H(1s) kinetic energy distribution for the X 1 Σ+g –a 3 Σ+g
band excitation, PXa (Ek ), can be defined as the ratio of the rate
of production of H atoms with kinetic energy Ek to the rate of
dissociation. The kinetic energy distribution can be written as
vj ,Jj R(vj , Jj , Ek )
PXa (Ek ) = vj ,Jj g(vj , Jj )
ij N (vi , Ji )σ (vi , vj , Ji , Jj )AEk (vj , Jj , Ek )
=
,
ij N(vi , Ji )σ (vi , vj , Ji , Jj )A(vj , Jj )
(3)
where mono-energetic electron excitation has been assumed.
Following the approach of Liu et al. (2002) and Liu et al.
(2003), the cross section, σij , can be expressed as a product of
electronic (F), vibrational (Q), and rotational (Sr ) terms:
σ (vi , vj ; Ji , Jj ) = Fi,j (E)Qvi ,vj ,Ji ,Jj Sr (Ji , Jj ).
(4)
The electronic term, Fi,j , accounts for the magnitude and energy
dependence of electronic band cross section. The vibrational
term, Qvi ,vj ,Ji ,Jj , is the rotational dependent Franck–Condon
factor |vi , Ji |vj , Jj |2 . The rotational term, Sr (Ji , Jj ), represents the relative cross section of the excitation from Ji to various Jj levels. X 1 Σ+g –a 3 Σ+g excitation takes place via electron exchange. Apart from the symmetry restriction, where the change
in rotational quantum numbers is even, there are no other rigorous constraints on ΔJ . However, if the electron–H2 collision
time is sufficiently short (<1 fs), the rotational motion can be
considered essentially frozen during the collision, and the adiabatic rotation approximation can be used. The first two leading
terms of the rotational interaction can be written as
3(Jj + 1)(Jj + 2)
Sr (Ji , Jj ) = βδJi ,Jj + (1 − β)
δJ ,J +2
2(2Jj + 3)(2Jj + 5) i j
Jj (Jj + 1)
3Jj (Jj − 1)
+
δJi ,Jj +
δJi ,Jj −2 ,
(2Jj − 1)(2Jj + 3)
2(2Jj − 1)(2Jj − 3)
(5)
703
where δJi ,Jj and δJi ,Jj ±2 are the Kronecker δ-function. The parameter β (0 β 1) measures the relative contribution of
the isotropic and anisotropic rotational terms. In the case of
H2 X 1 Σ+g –EF 1 Σ+g excitation, combined experimental measurements and modeling have yielded values of 0.55 ± 0.1 and
0.7 ± 0.1 for β at 20 eV and 100 eV, respectively (Liu et al.
2002; Abgrall et al. 1999). Note that the summation of Sr over
either Ji or Jj is unity.
The electronic form factor, Fi,j (E), is represented by a set
of collision parameters. The functional form and parameters
originate with Ajello & Shemansky (1993):
Fi,j (X)
Ry
1
1
=
− 3
C0
E
X2
X
π a02
4
+
Cm (X − 1) exp(−mC5 X) ,
(6)
m=1
where Ry and E, both in units of eV, are the Rydberg constant
and the electron excitation energy, respectively. X is the excitation energy in units of transition energy (i.e., X = E/(Ej −Ei )),
and a0 is the bohr radius. Ck (k = 0–5) are collision strength
parameters, whose absolute values have been given by Ajello &
Shemansky (1993). Note that Fij decreases with an E −3 dependence in the high-energy asymptote, with its magnitude primarily determined by C0 .
2.2. Spontaneous Continuum Transition Probabilities
The transition probability for a spontaneous emission of
energy Eph = hcν from a discrete level, (vj , Jj ), to the
continuum levels with energy Ek , in units of s−1 /cm−1 , is given
by
Aν (vj , Jj , ν) =
64π 4 ν 3 Hj i (Jj , Ji )
3h
2Jj + 1
Ji
2
× χEk ,Ji (R)|D(R)|χvj ,Jj (R) ρ(Ek ), (7)
where Hj i (Jj , Ji ) and D(R) are, respectively, the Hönl–London
factors and the electric dipole transition moment. χvj ,Jj (R) and
χEk ,Ji (R) are the radial wave functions of the initial discrete
level j and the continuum radial level i, respectively. ρ(Ek ) is
the density of states normalization factor at energy Ek = hcνk
above the dissociation limit of state j:
δ(Ek − Ek )
χEk ,Ji (R)|χEk ,Ji (R) =
ρ(Ek )
(8)
Ek = E(vj , Jj ) − Vi (R → ∞) − hcν,
(9)
where Vi (R → ∞) is the asymptotic potential energy of state i.
Internuclear distance, R, is converted to the dimensionless
quantity z = R/R0 where R0 is a selected scale length. The
amplitude of the continuum wave function is asymptotically
normalized to unity:
lim χEk ,Ji (z) = sin[kz + ηJi (Ek )],
z→∞
(10)
√
where ηJi (Ek ) is the phase shift and k = 2π R0 2μcνk / h, with
μ being the nuclear reduced mass. The conversion and
√ normalization give a state density factor of ρ(Ek ) = 2R0 2μc/ hνk ,
704
LIU ET AL.
−1
in units of states per cm . Equation (7) can be re-written as
2cμ Hj i (Jj , Ji )
128π 4 ν 3
Aν (vj , Jj , ν) =
R0
3/2
3h
νk J
2Jj + 1
i
× |χEk ,Ji (z)|D(z)|χvj ,Jj (z)|2 .
(11)
When both νk and ν are in units of cm−1 , R0 in Å, D(z) in
debye, μ in unified atomic mass units (u), and Aν in units of
s−1 /cm−1 , Equation (11) becomes
Hj i (Jj , Ji )
μ
Aν (vj , Jj , ν) = 2.43133 × 10−8 ν 3
νk J
2Jj + 1
i
2
(12)
× χEk ,Ji (z)|D(z)|χvj ,Jj (z)
.
The total transition probability, A(vj , Jj ), is given by
A(vj , Jj ) =
Em <Ej
Aν (vj , Jj , ν)dν +
A(vj , vm , Jj , Jm ),
m
(13)
where the integration is from ν = 0 to ν = [E(vj , Jj ) −
Vi (∞)]/ hc, and the index m refers to the c 3 Πu state. Summation
is carried out only for the (vm , Jm ) levels that are lower than
(vj , Jj ) in energy.
The discrete and continuum nuclear wave functions, χvj ,Jj (z)
and χEk ,Ji (z), are obtained by Numerov numerical solution of
the Schrödinger equation. For both a 3 Σ+g and b 3 Σ+u states, the
Born–Oppenheimer (BO) potentials with adiabatic corrections
calculated by Staszewska & Wolniewicz (1999) are used.
Relativistic and radiative correction of the a 3 Σ+g state is set
to the sum of the radiative and relativistic correction of H+2
X 2 Σ+g and the relativistic correction for H(2s). The corrections
for the b 3 Σ+u state are on the same basis except that H(1s) is
involved in this case. The calculation by Bukowski et al. (1992)
for the H+2 X 2 Σ+g state is used for the radiative correction. The
relativistic correction of H+2 obtained by Howells & Kennedy
(1990), instead of that by Wolniewicz & Poll (1985) (used
by Wolniewicz 2007), is utilized in the present work. The
relativistic correction for H(2s) is taken to be −0.456 cm−1 .
The relativistic correction for H(1s) is taken to be −1.18 cm−1 ,
a value selected such that the sum of the relativistic and radiative
corrections of the b 3 Σ+u state is equal to the corresponding
sum of the X 1 Σ+g state obtained by Wolniewicz (1993) at large
internuclear distance. The asymptotic limits of the a 3 Σ+g and
b 3 Σ+u states are H(1s)+H(2p) and H(1s)+H(1s), respectively.
The experimental dissociation energy of H2 X 1 Σ+g (vi = 0, Ji =
0) → H(1s)+H(1s) has been accurately determined to be
36118.06962 ± 0.00037 cm−1 by Liu et al. (2009). Based
on the NIST CODATA 2006 (Mohr et al. 2008) hydrogen
energy levels of Jentschura et al. (2010), the averaged value
of energy for H(1s)+H(2p) (over 2 P1/2 and 2 P3/2 ) relative to
that of H(1s)+H(1s) is 82259.163039 cm−1 . Using the derived
asymptotic energies, V(R → ∞), and the last three points of the
calculated potential energies (R = 43.6 − 44a0 ), the a 3 Σ+g and
b 3 Σ+u potentials at R > 44a0 (23.178 Å) can be extrapolated
according to the functional form
V (R) = V (∞) +
C1 C 2
C3
+ 8 + 10 ,
6
R
R
R
(14)
Vol. 716
where, in units of cm
apply:
−1
for V and Å for R, the following constants
state
V (∞)
C1
a 3 Σ+g 118377.2327 −1.306207 × 109
b3 Σ+u 36118.0696
7.446289 × 107
C3
−2.874701 × 1013
1.048959 × 1013 .
C2
5.056772 × 1011
−5.438430 × 1010
Both potentials are relative to the energy of the (v = 0,
J = 0) level of the X 1 Σ+g state. The a 3 Σ+g −b 3 Σ+u dipole
transition moment calculated for R = 0.6–25a0 by Staszewska
& Wolniewicz (1999) is used for the calculation of the transition
probabilities after extrapolation to the separated atom limit
(2.677725 debye or 1.053498 au).
To obtain reliable spontaneous emission lifetimes for a 3 Σ+g ,
transition probabilities of the a 3 Σ+g –c 3 Πu band system have
also been calculated within the adiabatic approximation. The
BO potential energy curve with adiabatic correction for the
c 3 Πu state and the a 3 Σ+g –c 3 Πu electronic transition moment
calculated by Staszewska & Wolniewicz (1999) are used. The
relativistic and radiative corrections need not be applied to the
a 3 Σ+g –c 3 Πu transition because the corrections to each state
cancel in the transition frequency.
The X 1 Σ+g BO potential of Wolniewicz et al. (1998), along
with the adiabatic, relativistic, and radiative corrections of Wolniewicz (1993), is used for the calculation of the a 3 Σ+g –X 1 Σ+g
Franck–Condon factors, |vi , Ji |vj , Jj |2 . a 3 Σ+g –c 3 Πu transition probabilities and a 3 Σ+g –X 1 Σ+g Franck–Condon factors are
calculated with a modified LEVEL 8 program originally developed by Le Roy (2007).
3. RESULTS
3.1. Transition Probabilities of the a 3 Σ+g (vj , Jj ) State
Table 1 lists a portion of the calculated total dipole transition
probabilities of the a 3 Σ+g –c 3 Πu band system for Jj 10. The
calculated total transition probabilities, obtained from summing
over the (vm , Jm ) quantum numbers of the lower c 3 Πu state,
range from 1.3 × 102 to 3.4 × 105 s−1 . For comparison, the
second column also presents the total transition probabilities of
the Jj = 0 level calculated by Guberman & Dalgarno (1992),
indicating good agreement. The total transition probabilities
show significant variation with rotational quantum number,
primarily caused by the significant variation of P-, Q-, and
R-branch transition frequencies with J. The potential energy
curves of the a 3 Σ+g and c 3 Πu states are very similar. As a result,
the a 3 Σ+g –c 3 Πu Franck–Condon factor is close to unity for
Δv = 0, with the values for the Δv = 0 being at least an order
of magnitude smaller. Transitions with Δv = 0 have higher
transition frequencies but smaller values of dipole transition
matrix elements. In contrast, the Δv = 0 transitions have
larger values of dipole matrix elements but very low transition
frequencies. Thus, the small dipole transition probabilities of the
a 3 Σ+g –c 3 Πu transition can be largely attributed to the similarity
and propinquity of the potential curves.
The a 3 Σ+g state has a total of 21 discrete vibrational levels.
Table 2 provides total transition probabilities of the a 3 Σ+g −b 3 Σ+u
band system for vj = 0–20 and Jj 14. The second column
lists transition probabilities for the Jj = 0 levels obtained by
ENERGY DISTRIBUTION OF H(1s) FROM H2 X 1 Σ+g –a 3 Σ+g EXCITATION
No. 1, 2010
705
Table 1
Total Spontaneous Transition Probabilities of H2 a 3 Σ+g (vj , Jj )–c 3 Πu Transitions (Unit: s−1 )
vj
J j = 0a
Jj = 0
Jj = 1
Jj = 2
Jj = 3
Jj = 4
Jj = 5
Jj = 6
Jj = 7
Jj = 8
Jj = 9
Jj = 10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.9E2b
3.0E4
6.2E4
9.4E4
1.3E5
1.6E5
1.9E5
2.2E5
2.2E5
2.4E5
2.2E5
2.0E5
1.6E5
1.1E5
6.2E4
2.4E4
1.0E4
5.6E3
3.2E3
1.5E3
6.4E2
1.29E2
2.86E4
6.00E4
9.31E4
1.26E5
1.58E5
1.86E5
2.09E5
2.24E5
2.28E5
2.20E5
1.97E5
1.60E5
1.12E5
6.22E4
2.48E4
8.79E3
6.32E3
3.62E3
1.74E3
7.10E2
3.75E2
4.37E4
9.14E4
1.41E5
1.92E5
2.40E5
2.82E5
3.16E5
3.37E5
3.43E5
3.30E5
2.96E5
2.39E5
1.67E5
9.20E4
3.61E4
1.28E4
9.30E3
5.26E3
2.48E3
9.85E2
4.16E2
4.09E4
8.55E4
1.32E5
1.79E5
2.24E5
2.63E5
2.94E5
3.14E5
3.19E5
3.06E5
2.73E5
2.20E5
1.52E5
8.22E4
3.12E4
1.12E4
8.23E3
4.54E3
2.06E3
7.63E2
5.10E2
4.00E4
8.34E4
1.29E5
1.74E5
2.17E5
2.55E5
2.85E5
3.04E5
3.08E5
2.94E5
2.61E5
2.08E5
1.42E5
7.46E4
2.67E4
1.00E4
7.30E3
3.86E3
1.64E3
4.90E2
6.59E2
3.99E4
8.27E4
1.28E5
1.72E5
2.14E5
2.51E5
2.80E5
2.97E5
3.01E5
2.86E5
2.52E5
1.99E5
1.32E5
6.67E4
2.19E4
9.04E3
6.26E3
3.10E3
1.17E3
8.78E2
4.02E4
8.30E4
1.28E5
1.72E5
2.13E5
2.49E5
2.77E5
2.93E5
2.95E5
2.79E5
2.43E5
1.89E5
1.22E5
5.80E4
1.68E4
8.27E3
5.09E3
2.24E3
6.51E2
1.19E3
4.08E4
8.38E4
1.28E5
1.72E5
2.13E5
2.49E5
2.75E5
2.90E5
2.90E5
2.72E5
2.34E5
1.78E5
1.11E5
4.82E4
1.18E4
7.49E3
3.86E3
1.34E3
1.64E3
4.18E4
8.52E4
1.30E5
1.74E5
2.14E5
2.49E5
2.74E5
2.87E5
2.85E5
2.65E5
2.25E5
1.66E5
9.86E4
3.73E4
7.87E3
6.07E3
2.52E3
2.25E3
4.32E4
8.70E4
1.32E5
1.76E5
2.16E5
2.49E5
2.73E5
2.85E5
2.80E5
2.57E5
2.14E5
1.53E5
8.46E4
2.52E4
6.02E3
3.92E3
3.07E3
4.49E4
8.94E4
1.35E5
1.78E5
2.18E5
2.50E5
2.73E5
2.82E5
2.75E5
2.49E5
2.02E5
1.39E5
6.90E4
9.49E3
4.17E3
4.70E4
9.22E4
1.38E5
1.81E5
2.20E5
2.51E5
2.72E5
2.79E5
2.69E5
2.39E5
1.89E5
1.23E5
5.12E4
Notes.
a Obtained by Guberman & Dalgarno (1992) for J = 0. Entries in all other columns are obtained in the present work.
j
b 1.9E2 reads as 1.9 × 102 .
Table 2
Total Spontaneous Transition Probabilities of H2 a 3 Σ+g (vj , Jj )–b 3 Σ+u Transitions (Unit: s−1 )
vj
J j = 0a
Jj = 0
Jj = 1
Jj = 2
Jj = 3
Jj = 4
Jj = 5
Jj = 6
Jj = 7
Jj = 8
Jj = 9
Jj = 10
Jj = 11
Jj = 12
Jj = 13
Jj = 14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
8.60E7b
8.49E7
9.73E7
1.08E8
1.18E8
1.28E8
1.37E8
1.46E8
1.56E8
1.68E8
1.82E8
2.01E8
2.28E8
2.70E8
3.38E8
4.58E8
6.83E8
1.03E9
1.12E9
1.18E9
1.22E9
1.24E9
8.53E7
9.76E7
1.09E8
1.18E8
1.28E8
1.37E8
1.46E8
1.56E8
1.68E8
1.83E8
2.02E8
2.29E8
2.71E8
3.40E8
4.62E8
6.92E8
1.04E9
1.13E9
1.19E9
1.22E9
1.24E9
8.59E7
9.82E7
1.09E8
1.19E8
1.28E8
1.37E8
1.47E8
1.57E8
1.69E8
1.83E8
2.03E8
2.31E8
2.73E8
3.44E8
4.71E8
7.13E8
1.06E9
1.13E9
1.19E9
1.22E9
1.24E9
8.68E7
9.90E7
1.10E8
1.20E8
1.29E8
1.38E8
1.47E8
1.58E8
1.70E8
1.84E8
2.04E8
2.33E8
2.77E8
3.51E8
4.84E8
7.47E8
1.08E9
1.14E9
1.20E9
1.23E9
1.24E9
8.80E7
1.00E8
1.11E8
1.21E8
1.30E8
1.39E8
1.48E8
1.59E8
1.71E8
1.86E8
2.06E8
2.36E8
2.82E8
3.60E8
5.03E8
7.97E8
1.10E9
1.16E9
1.21E9
1.23E9
8.96E7
1.01E8
1.12E8
1.22E8
1.31E8
1.40E8
1.49E8
1.60E8
1.72E8
1.88E8
2.09E8
2.40E8
2.89E8
3.73E8
5.30E8
8.71E8
1.12E9
1.17E9
1.22E9
1.24E9
9.14E7
1.03E8
1.13E8
1.23E8
1.32E8
1.41E8
1.51E8
1.61E8
1.74E8
1.90E8
2.12E8
2.45E8
2.97E8
3.89E8
5.69E8
9.67E8
1.12E9
1.19E9
1.23E9
9.35E7
1.05E8
1.15E8
1.25E8
1.34E8
1.43E8
1.52E8
1.63E8
1.76E8
1.93E8
2.16E8
2.51E8
3.08E8
4.10E8
6.26E8
1.07E9
1.14E9
1.21E9
9.58E7
1.07E8
1.17E8
1.26E8
1.35E8
1.44E8
1.54E8
1.65E8
1.79E8
1.96E8
2.21E8
2.58E8
3.21E8
4.39E8
7.22E8
1.14E9
1.18E9
9.84E7
1.09E8
1.19E8
1.28E8
1.37E8
1.46E8
1.56E8
1.67E8
1.85E8
2.00E8
2.26E8
2.68E8
3.39E8
4.79E8
9.10E8
1.14E9
1.01E8
1.12E8
1.21E8
1.30E8
1.39E8
1.48E8
1.58E8
1.70E8
1.85E8
2.04E8
2.33E8
2.79E8
3.61E8
5.40E8
1.13E9
1.04E8
1.14E8
1.24E8
1.33E8
1.41E8
1.51E8
1.61E8
1.73E8
1.88E8
2.09E8
2.41E8
2.93E8
3.91E8
6.78E8
1.07E8
1.17E8
1.26E8
1.35E8
1.44E8
1.53E8
1.64E8
1.76E8
1.93E8
2.16E8
2.51E8
3.11E8
4.34E8
1.11E8
1.20E8
1.29E8
1.38E8
1.46E8
1.56E8
1.67E8
1.80E8
1.97E8
2.23E8
2.63E8
3.35E8
5.09E8
1.14E8
1.23E8
1.32E8
1.41E8
1.49E8
1.59E8
1.70E8
1.84E8
2.03E8
2.31E8
2.78E8
3.67E8
9.82E7
1.09E8
1.19E8
1.28E8
1.37E8
1.45E8
1.56E8
1.68E8
1.82E8
2.01E8
2.28E8
2.69E8
3.35E8
4.49E8
5.49E8
9.59E8
1.09E9
1.09E9
1.00E9
7.87E8
Notes.
a Obtained by Kwok et al. (1986) for J = 0. Entries in all other columns are obtained in the present work.
j
b 8.60E7 reads as 8.60 × 107 .
Kwok et al. (1986). The two sets of transition probabilities
agree with each other within 3%, with the exception of the
vj = 15, 16, 18, 19, and 20 levels. Differences at the vj = 16
and 18 levels, while 7.2% and 7.9%, respectively, are within the
expected uncertainties of both calculations. Differences for the
vj = 15, 19, and 20 levels, at 20%, 18%, and 36%, however,
are outside of the combined expected error limits. The probable
cause for the large difference will be discussed in Section 4.
For a given vj , the total a 3 Σ+g −b 3 Σ+u transition probabilities
increase with Jj . For a given Jj , the total transition probabilities
likewise increase monotonically with vj . From the Jj = 0 to
Jj = 10 levels, the increment ranges from 8% for the vj = 6
level to 147% for the vj = 14 levels. Except for the vj = 12–16
levels, the total transition probabilities generally change by
no more than 6% from the Jj = 0 to Jj = 6 levels. Since
the Franck–Condon overlap integrals between the X 1 Σ+g (0)
706
LIU ET AL.
3.0E+011
3.0E+008
2.5E+008
v=0, J=12
v=1, J=8
v=2, J=6
v=3, J=3
v=4, J=1
2.0E+008
dA(v,J; Ek )/dEk (s -1 eV-1)
dA(v,J; Ek )/dEk (s -1 eV-1)
Vol. 716
1.5E+008
1.0E+008
v=15, J=0
v=15, J=1
v=15, J=2
v=15, J=3
v=15, J=4
v=15, J =8
2.3E+011
1.5E+011
7.5E+010
5.0E+007
0.0E+000
0.0E+000
0
1
2
3
4
Ek (eV/atom)
Figure 1. H2 a 3 Σ+g (vj , Jj )−b 3 Σ+u continuum transition probabilities as a
function of the kinetic energy of outgoing H(1s). The light solid, dot, dotdashed, dash, and heavy solid lines denote the continuum profile of the
(vj = 0, Jj = 12), (vj = 1, Jj = 8), (vj = 2, Jj = 6), and (vj = 3, Jj = 3)
and (vj = 4, Jj = 1) levels, respectively. Note that very few H(1s) atoms
with Ek > 3.5 eV are produced. dA/dEk has been multiplied by a factor of 2
because the range of Ek per atom is half of the two atoms.
(A color version of this figure is available in the online journal.)
0.000
0.005
0.010
0.015
0.020
0.025
Ek (eV/atom)
Figure 2. H2 a 3 Σ+g (vj = 15, Ji )−b 3 Σ+u continuum transition probabilities as
a function of the kinetic energy of outgoing H(1s). The light solid, dot, dotdashed, dash, heavy solid, and dot-dash-triple-dot lines denote the continuum
profile of the Ji = 0, 1, 2, 3, 4, and 8 levels, respectively. Note that very few
H(1s) atoms with Ek > 0.025 eV are produced, and a factor of 1000 increase
in the vertical scale from Figure 1. dA/dEk has been multiplied by a factor of
2 because the range of Ek per atom is half of the total.
(A color version of this figure is available in the online journal.)
level and these high-vj levels are negligible, the rotational
dependence of the levels that are significantly populated by
electron impact excitation at room temperature is negligible.
The total a 3 Σ+g (vj , Jj )–c 3 Πu transition probabilities are, at
most, 0.2% of their counterparts of the a 3 Σ+g (vj , Jj )−b 3 Σ+u transitions. The total a 3 Σ+g (vj , Jj )−b 3 Σ+u transition probabilities
listed in Table 2 are also equivalent to the total spontaneous
dipole transition probabilities of the a 3 Σ+g (vj , Jj ) states.
3.2. Kinetic Energy Distribution of the X 1 Σ+g –a 3 Σ+g Excitation
In principle, the maximum kinetic energy that each outgoing
H(1s) atom can carry ranges from 3.65 eV for (vj = 0, Jj = 0)
to 5.09 eV for vj = 20. The averaged kinetic energy of
H(1s) is much lower. Figure 1 shows the continuum transition
probabilities as a function of kinetic energy of H(1s) for several
rovibrational levels. The kinetic energy distribution is primarily
determined by the vibrational quantum number (vj ) of the a 3 Σ+g
state, although it is also modified by the rotational quantum
number. For the transition from the vj = 0 level, more than
80% of H(1s) atoms are produced with Ek between 1 and 2
eV. While some hydrogen atoms with Ek > 3 eV can be
produced from vj 4 levels, most atoms carry much lower
kinetic energy. The average value of Ek (Ēk ) decreases with the
increase of either vj or Jj quantum numbers. For vj 14 levels,
Ēk < 0.1 eV atom−1 . Figure 2 displays the transition probability
profiles for several low Jj levels of the vj = 15 state. The kinetic
energy distribution sharply peaks in a narrow range near Ek ∼ 0
eV and the rotational quantum number has a moderate effect
on the location of the peak. The radiative continuum emission
is the most probable path for the a 3 Σ+g −b 3 Σ+u transition. The
preference is caused by the overlap of rovibrational wave
functions, and dominantly by the ν 3 term in the spontaneous
transition probability (Equation (12)).
Figure 3 shows the kinetic energy distribution, PXa (Ek ), of
H(1s) from a direct X 1 Σ+g –a 3 Σ+g electron impact excitation at
room temperature for various excitation energies. A value of
0.7 for β is assumed. Each shade of gray (or color) region in
the figure represents 25% of the maximum value of PXa (Ek ).
Figure 3. Kinetic energy of H(1s) from direct excitation of the a 3 Σ+g state from
the X 1 Σ+g state at 300 K and various electron energies. A β-value of 0.7 in
Equation (5) is assumed. The angle between the line of sight and PXa (Ek ) axis
is 20◦ . Each shade of gray or color region represents 25% of the maximum
PXa (Ek ) value.
(A color version of this figure is available in the online journal.)
Several characteristics of the kinetic energy distribution of the
a 3 Σ+g state are worth mentioning. First, the value of PXa (Ek )
for Ek > 3 eV is negligible at all excitation energies, to the
consequence of spontaneous emission as the preferred route.
In addition, as the electron impact energy increases, both Ēk
and the peak of PXa (Ek ) steadily shift to the lower kinetic
energy values. Temperature variation has a small effect on the
PXa (Ek ) distribution. At 25 eV impact energy, the difference
between the kinetic energy distribution at 300 K and 1300 K
is very small, although the difference between 300 K and
2000 K becomes significant. The relatively small temperature
dependence of PXa (Ek ) remains as long as the excitation energy
is above the threshold region. In the threshold region, an
increase of temperature makes the contribution of excitation
from higher (vi , Ji ) levels of the X 1 Σ+g state more important.
ENERGY DISTRIBUTION OF H(1s) FROM H2 X 1 Σ+g –a 3 Σ+g EXCITATION
No. 1, 2010
emission, which primarily occurs on the blue side of 1650 Å
and is not included in the figure, is much stronger than the
continuum emission (Liu et al. 1995). The discrete Lyman band
emission cross section at 25 eV is about 3 times that of the
Lyman continuum emission cross section (Abgrall et al. 1997).
The electron impact induced a 3 Σ+g −b 3 Σ+u continuum emission
spectrum in Figure 4 is similar to that induced by the collision
of H2 with Ar+ obtained by Brandt & Ottinger (1979).
e+H2 Continuum Emission (300K)
1.0
Relative Intensity (arb. unit)
0.9
B - X continuum (25 eV) x 0.1
a - b continuum (22 eV)
a - b continuum (25 eV)
a - b continuum (35 eV)
x 0.1
0.8
707
0.7
0.6
0.5
0.4
0.3
4. DISCUSSION
0.2
The energy term values of the a 3 Σ+g (vj , Jj ) levels are up to 5.6
cm−1 higher than those obtained by Wolniewicz (2007) because
non-adiabatic coupling is not considered here. Wolniewicz
(2007) has also shown that the non-adiabatic coupling is
largely homogenous in nature and is primarily caused by the
nearby h 3 Σ+g and g 3 Σ+g states. Consequently, the shifts caused
by non-adiabatic coupling have a very significant vibrational
dependence but small rotational dependence. The non-adiabatic
term values of Wolniewicz (2007) differ by no more than
0.28 cm−1 from all the available experimental values. The
maximum uncertainty in the transition frequencies calculated
in the present work is 5.9 cm−1 .
Uncertainties in relativistic corrections also contribute to error. As indicated by Wolniewicz (2007), no reliable relativistic corrections for the triplet states have been published. Wolniewicz (2007) used relativistic corrections of H+2 X 2 Σ+g obtained by Wolniewicz & Poll (1985) and the correction of
H(2s), Δrel = −0.42 cm−1 for the a 3 Σ+g state. The present work
utilizes the more accurate and extensive H+2 X 2 Σ+g relativistic correction of Howells & Kennedy (1990) and the H(2s)
correction of Δrel = −0.46 cm−1 . The energy term values
of a few a 3 Σ+g (vj , Jj ) levels obtained by Wolniewicz (2007)
are lower than experimental measurements (Crosswhite 1972;
Jungen et al. 1990), so the relativistic effect was overestimated,
at least, for some levels. Since the present work uses an even
higher value for the H(2s) state, the present relativistic overcorrection will be larger.
Even with the error from neglecting the non-adiabatic coupling and overestimation of the relativistic effect, the maximum error in the a 3 Σ+g −b 3 Σ+u transition frequency is less than
5.9 cm−1 . The maximum error for the a 3 Σ+g –c 3 Πu transition frequency is presumably similar or smaller. Thus, the error in transition frequencies has a negligible effect on the uncertainty of the
calculated transition probabilities. The error in the a 3 Σ+g −b 3 Σ+u
electronic transition moment, from R = 0.6 to 25a0 , has been
estimated by Staszewska & Wolniewicz (1999) to be less than
10−4 au. While it has been extrapolated to the corresponding
value of the separated atomic limit, the transition moment at
R = 25a0 is already nearly identical to the limit. Based on the
magnitude of the transition moment, the maximum error in the
transition probabilities due to the inaccuracy of the transition
moment is estimated to be smaller than 1%. Finally, the fact that
the calculated integrated transition probabilities of the Jj = 0
rotational level of the vj = 0–14 vibrational levels agree with
those of Kwok et al. (1986) to within 3%, also suggests the
uncertainty in the electronic transition moment is insignificant.
Excluding possible uncertainties arising from the neglect of
the non-adiabatic coupling, the error in the calculated transition
probabilities is estimated to be less than 4%–5%. The accuracy
of the wave functions is the primary source of the error of the calculated transition probabilities. In the case of the total transition
probabilities, the error in the numerical integration also needs
0.1
0.0
1300
1600
1900
2200
2500
2800
3100
3400
3700
4000
Wavelength (Å)
Figure 4. Comparison of relative continuum emission spectral intensities by
electron impact excitation at T = 300 K. The dot, dot-dashed, and dash-dash lines
represent calculated a 3 Σ+g −b 3 Σ+u continuum emission at excitation energies of
22, 25, and 35 eV, respectively. For comparison, the emission to the X 1 Σ+g
continuum from electron impact excitation of the B 1 Σ+u , C 1 Πu , B 1 Σ+u , and
D 1 Πu states at 25 eV is shown as solid line. Note that the intensity of the B 1 Σ+u ,
C 1 Πu , B 1 Σ+u , and D 1 Πu −X 1 Σ+g continuum emissions has been reduced by
a factor of 10.
(A color version of this figure is available in the online journal.)
As a result, a significant variation of PXa (Ek ) is expected.
Finally, as the excitation energy increases above 15.5 eV, a
very narrow peak in PXa (Ek ) around EK ∼ 0 eV appears in the
figure. This secondary peak reflects the contribution of high vj
levels (such as vj = 15 shown in Figure 2) whose continuum
emission profiles sharply peak around EK ∼ 0. Even though the
excitation cross sections from X 1 Σ+g (0) to these high vj levels
are very small, the extremely large value of AEk at EK ∼ 0
causes the distribution to be very significant.
3.3. Electron Impact Induced a 3 Σ+g −b 3 Σ+u Continuum
Emission Profile
Figure 4 shows the relative intensities of the a 3 Σ+g −b 3 Σ+u
continuum emission spectrum in the 1250–4000 Å region. The
dot, dot-dash, and dash-dash lines represent the a 3 Σ+g −b 3 Σ+u
room temperature (300 K) continuum emission spectra at 22,
25, and 35 eV excitation energies. Due to the rapid decrease in
the a 3 Σ+g –X 1 Σ+g excitation cross section with energy, the relative
intensities of the a 3 Σ+g −b 3 Σ+u transitions also decrease quickly
with the energy. For comparison, the singlet-ungerade–X 1 Σ+g
continuum emission, due to the excitation of the B 1 Σ+u , C 1 Πu ,
B 1 Σ+u , and D 1 Πu states at 25 eV, based on the work of Abgrall
et al. (1997) and Liu et al. (1998), is also shown as a solid
trace. Because the continuum emission from the B 1 Σ+u state
far-dominates these from the other three states (∼95% versus
∼5%), the singlet-ungerade–X 1 Σ+g continuum is also called the
B 1 Σ+u –X 1 Σ+g or Lyman continuum (Stephens & Dalgarno 1972;
Abgrall et al. 1997).
The intensity of the solid trace in Figure 4 has been scaled
down by a factor of 10. Figure 4 thus shows that the Lyman
continuum emission completely dominates the a 3 Σ+g −b 3 Σ+u
emission on the blue side of ∼1650 Å. However, the Lyman
continuum rapidly diminishes above 1700 Å and its contribution
to the total continuum emission becomes negligible above
1750 Å. It should be noted that the discrete Lyman band
708
LIU ET AL.
to be considered. Non-adiabatic coupling leads to the mixing
of the wave function of the a 3 Σ+g state and other triplet-gerade
states. In addition, uncertainty in the amplitude of the b 3 Σ+u state
continuum wave function, which is used to normalize χEk ,Jj (z)
of Equation (10), also introduces uncertainty into the calculated
transition probabilities. The amplitude, A, is considered to be
converged if the ratio of the difference in two adjacent amplitudes to the amplitude itself, (An+1 − An )/An , is smaller than
5 × 10−6 for three contiguous outward propagations. Depending on the Jj values, the amplitude convergence of the
b 3 Σ+u levels associated with νk from 0 to 0.3–2.5 cm−1 could
not be fully reached. For many a 3 Σ+g (vj , Jj )−b 3 Σ+u transitions, the dipole matrix elements have negligible values in
the νk = 0 − 0.3(2.5) cm−1 region. The problem of convergence does not lead to any significant difference in differential
or integrated transition probabilities. However, transitions from
Jj = 0–2 of the vj = 15, 16, 17, and 18, and Jj = 0 − 3 of
the vj = 19 and 20 states have very large values in this region.
Uncertainty in the amplitudes can have a significant impact on
the accuracy of the calculated transition probabilities for these
levels. The effect has been examined and minimized in several
ways. The first is to expand the R range for the propagation of
the continuum wave functions. The second is to compute the
differential transition probabilities at very small Ek increments
in the peak regions (down to 0.1 cm−1 ) so that contribution in
the intense and narrowly peaked regions such as those in Figure 2 can be accurately integrated. Both approaches result in
more accurate integrated transition probabilities. In many cases,
the error in the transition probabilities can be estimated from
the calculated amplitudes for several points prior to the end of
propagation. An additional estimate can also be obtained by examining the rotational dependence of the integrated transition
probabilities. As Table 2 indicates, the total transition probabilities of Jj = 0–4 differ only a few percent. The variation of
rotational quantum numbers leads to the change of centrifugal
potentials, which, in turn, shifts the location for the peak of the
differential transition probability (see Figure 2). As a result, the
νk = 0 − 0.3(2.5) cm−1 region contributes very significantly
to some Jj transitions but not so significantly to the others. A
comparison of the integrated transition probabilities of different
Ji levels provides a reliable estimate of the uncertainty.
The present values of the vj > 14 levels are consistently
greater than the corresponding values of Kwok et al. (1986).
In particular, the present values for the vj = 15, 19, and 20
levels are 20%, 18%, and 36% greater than the previous calculation. The differential transition probabilities of these three
levels are very strong and peak sharply in the low Ek region.
Calculations with insufficiently small Ek increments by Kwok
et al. (1986) is the probable reason for the underestimation
of the integrated transition probabilities. As noted, the present
Aνk (vj , vj , ν) is calculated over 150,000 intervals with the Ek
increment as small as 0.1 cm−1 in the important regions. Therefore, the A(vj , Jj ) of the present work is more accurate. Fantz
& Wünderlich (2006) also obtained the a 3 Σ+g −b 3 Σ+u total transition probabilities for the Jj = 0 and vj = 0–17 levels. While
their calculated values for the vj = 0–12 levels are in good
agreement with both Kwok et al. (1986) and the present calculations, their values for vj = 13–17 are significantly lower
than those of either calculation. In the case of the vj = 17 level,
the Fantz & Wünderlich (2006) transition probability is smaller
than the present value by more than a factor of 5. Since Fantz &
Wünderlich (2006) and the present calculations both utilize the
transition moment and potential energy curves of Staszewska
Vol. 716
& Wolniewicz (1999), the difference must be primarily
caused by an insufficiently small Ek increment in the former
calculation.
Table 3 compares the presently calculated lifetimes with
several experimental measurements. The calculated values in
the second column refer to appropriate Jj = 1 levels. In the
case of a range of vibrational quantum numbers, the calculated
value refers to the average lifetime obtained from the corresponding Jj = 1 level weighted by normalized Q(1)-branch
Franck–Condon factors of the a 3 Σ+g –X 1 Σ+g transitions. The lifetimes of the Jj = 1 level of both vj = 0 and 1 states of
Wedding & Phelps (1988), obtained from the extrapolation of
accurate collisional quenching rates, are the most reliable measured values. The remaining experimental values unavoidably
refer to those averaged over several rotational or vibrational levels. In any case, the presently calculated lifetimes agree with the
Wedding & Phelps (1988) values within 6%. They also agree
with electron-phaseshift measurements of Smith & Chevalier
(1972) within experimental uncertainty. The lifetimes obtained
from the electron–photon delayed coincidence measurements of
Imhof & Read (1971) differ by less than 7% from the present
values. The coincidence of the radiative decay from the vj = 0
and 1 levels of the a 3 Σ+g state, excited by electron impact, was
explicitly demonstrated using EEL measurements. While these
two a 3 Σ+g (vj ) levels overlap with the c 3 Πu and B 1 Σ+u states in
the EEL spectrum, the spontaneous emission characteristics of
the c 3 Πu and B 1 Σ+u states are vastly different from that of the
a 3 Σ+g state emission. Consequently, there is no ambiguity in the
vibrational quantum number of the emitting levels. However,
in photon–photon delayed coincidence measurements by King
et al. (1975), the emitting levels were poorly defined. Good
agreement with the present calculation is achieved if the King
et al. (1975) value, 10.45 ± 0.25 ns, refers to the average of
the first three or four vibrational levels (10.3 or 9.96 ns). Subsequent photon–photon delayed coincidence measurements by
Mohamed & King (1979) attempted to mitigate the uncertainty.
Experimental values for the vj = 0 and 1 levels reported by
Mohamed & King (1979) are about 15% and 11% shorter than
the presently calculated values.
Table 4 compares various calculated spontaneous emission
lifetimes of the Jj = 0 levels for the a 3 Σ+g (vj ) states. For
the vj = 0–14 levels, there is very good agreement among
the Kwok et al. (1986), Fantz et al. (2000), and the present
values. The lifetimes obtained by Lavrov et al. (1999) were
based on integration and interpolation of calculated transition
probabilities at a few dozen wavelength regions and are thus not
very accurate. Not shown in the table are radiative lifetimes of
the vj = 0–17 levels recently calculated by Fantz & Wünderlich
(2006). As discussed above, their lifetimes for the vj = 13–17
levels are significantly overestimated.
The X 1 Σ+g –a 3 Σ+g excitation function of Ajello & Shemansky
(1993) is a combined result of the a 3 Σ+g −b 3 Σ+u emission and
EEL measurements. Ajello & Shemansky (1993) measured
relative emission cross sections of the transitions between 1800
and 1900 Å, normalized to the Khakoo & Trajmar (1986) EEL
measurement above 20 eV. Ajello & Shemansky (1993) assumed
that spectral intensities in the region were exclusively from
the first five vibrational levels of the a 3 Σ+g state. In addition,
they assumed that the cascade excitation of the a 3 Σ+g state is
negligibly small and therefore does not significantly change
the shape of the excitation function. The Ajello & Shemansky
(1993) cross section near 17 eV is 150% larger than the EEL
result obtained by Wrkich et al. (2002).
ENERGY DISTRIBUTION OF H(1s) FROM H2 X 1 Σ+g –a 3 Σ+g EXCITATION
No. 1, 2010
709
Table 3
Comparison with the Measured Lifetimes of the a 3 Σ+g (v) State (Unit: ns)
v
0
1
2–3
0–3
0–4
Presenta Imhof & Read (1971) Smith & Chevalier (1972) King et al. (1975) Mohamed & King (1979) Bretagne et al. (1981) Wedding & Phelps (1988)
11.0 ± 0.4
10.6 ± 0.6
11.7
10.2
8.89b
9.96b
9.73b
11.9 ± 1.2
10 ± 2
9.94 ± 0.39
9.1 ± 1.0
10.8 ± 1.1
10.45 ± 0.25c
11.4 ± 0.8
11.1 ± 0.3
10.4 ± 0.3
9.62 ± 0.20
Notes.
a Calculated lifetimes of the J = 1 level unless otherwise noted.
b Averaged lifetimes obtained from those of the J = 1 levels weighted by the appropriate Q(1)-branch Franck–Condon factors of the a 3 Σ+ v−X 1 Σ+ (v = 0) transitions.
g
g
c Average lifetime of a number of unspecified vibrational levels.
Table 4
Comparison Between Calculated Radiative Lifetimes of the a 3 Σ+g (v, J = 0) Levels (Unit: ns)
v
Present
Fantz et al. (2000)
Lavrov et al. (1999)
Kwok et al. (1986)
James & Coolidge (1939)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
11.8
10.3
9.23
8.46
7.84
7.32
6.85
6.40
5.96
5.48
4.97
4.38
3.71
2.96
2.18
1.46
0.968
0.889
0.845
0.820
0.807
11.6
10.2
9.17
8.41
7.82
7.32
6.87
6.42
5.99
5.51
4.99
4.39
3.70
3.06
2.22
12.4
11.5
10.3
9.29
8.64
8.07
7.63
11.6
10.2
9.17
8.40
7.81
7.30
6.90
6.41
5.95
5.49
4.98
4.39
3.72
2.99
2.23
1.82
1.04
0.917
0.917
1.00
1.27
11.9
11.0
10.1
9.7
The apparent a 3 Σ+g −b 3 Σ+u excitation function of Ajello &
Shemansky (1993) is used in the present work as if it were
the X 1 Σ+g –a 3 Σ+g excitation function. Although the X 1 Σ+g –a 3 Σ+g
cross section within ∼13 to ∼19.5 eV is overestimated, the
impact on the kinetic energy distribution is small because the
PXa (Ek ) in Equation (3) involves the ratio of the cross sections.
Since the threshold energies of the vj = 0–4 levels are very
close, the overestimation of the band cross section should have
insignificant effect on the kinetic energy distribution shown in
Figure 3.
The present calculation shows that none of the a 3 Σ+g (vj , Jj )
levels are capable of producing more than 1.2% of H atoms
with Ek > 3.5 eV. The excitation of H2 to the triplet-ungerade
states that are higher than the b 3 Σ+u state is not efficient in
the production of H atoms with Ek > 3.5 eV, regardless of
the initial vibrational level of the X 1 Σ+g state. This is in sharp
contrast with the excitation of triplet-gerade states higher than
the a 3 Σ+g state, such as g 3 Σ+g , i 3 Πg , and j 3 Δg , where a large
number of H atoms with Ek > 3.5 eV have been observed in
spontaneous decay to the b 3 Σ+u and c 3 Πu states (Koot et al.
1989b). The predissociation of the c 3 Π+u state by the b 3 Σ+u state
is another way to produce hydrogen atoms with Ek ∼ 4.2 eV
(Koot et al. 1989b; Wouter et al. 1997).
Spontaneous emission is the preferred pathway for the
a 3 Σ+g −b 3 Σ+u band system to release excess energy. As men-
tioned, the preference is caused by the overlap of rovibrational
wave functions, and, more importantly, by the ν 3 term in the
spontaneous transition probability. Indeed, the present calculation shows that none of the a 3 Σ+g (vj , Jj )−b 3 Σ+u ro-vibronic
transitions are capable of generating hydrogen atoms with average kinetic energies higher than 1.5 eV. Since the excitation of
H2 to the high-lying triplet-ungerade states also produce slow
H(1s) atoms, excitation of a 3 Σ+g and the triplet-ungerade levels cannot be responsible for the production of the H atoms
that were observed to escape from the magnetosphere of Saturn
by the Cassini UVIS (Shemansky et al. 2009), although they
contribute to atmospheric heating.
The a 3 Σ+g −b 3 Σ+u transition is similar to the Lyman continuum transition in that both are from higher discrete levels to
the lower continuum levels resulting in slow H(1s) atoms. Thus,
energetic hydrogen atoms observed to escape from the magnetosphere of Saturn are produced in the dissociative excitation
processes. Excitation from vibrationally excited X 1 Σ+g (vi ) levels to the b 3 Σ+u state, and from X 1 Σ+g to the doubly excited and
dissociative ionic states by photoelectrons are primarily responsible for the production of energetic hydrogen atoms on Saturn
(Shemansky et al. 2009; Liu et al. 2010).
The a 3 Σ+g −b 3 Σ+u continuum emission profile has been proposed as a spectroscopic diagnostic of hydrogen-containing
nonequilibrium plasmas (Lavrov et al. 1999; Fantz et al. 2000;
710
LIU ET AL.
Petrovic & Phelps 2009). In these low-pressure plasmas, the
electronic states of H2 are populated predominantly by direct electron impact excitation and deactivated primarily by
spontaneous emission. Under these conditions, the relative
a 3 Σ+g −b 3 Σ+u intensity distribution provides information on the
non-LTE population of the H2 X 1 Σ+g state while the absolute
intensity provides an estimate of the dissociation rate of H2 .
So far, however, only the rotationless a 3 Σ+g −b 3 Σ+u continuum
transition probabilities have been utilized for plasma modeling.
These plasma models also neglect the energy dependence of the
cross section. Since the spectral intensity distribution in a given
wavelength region also strongly depends on the rotational quantum number, the present a 3 Σ+g (vj , Jj )−b 3 Σ+u continuum profile,
along with the Ajello & Shemansky (1993) excitation function,
enables a more precise estimate of the non-LTE population and
dissociation rate. The a 3 Σ+g (vj , Jj ) spontaneous emission lifetimes obtained in the present work and listed in Tables 1 and
2 are a significant addition to the determination of H2 radiative
lifetimes (Astashkevich & Lavrov 2002).
The work reported in this paper was carried out at the
Jet Propulsion Laboratory (JPL), California Institute of Technology, under contract with NASA, and at Space Environment Technologies. We gratefully acknowledge financial support through NASA’s Outer Planets Research, Planetary Atmospheres Research, and Cassini Data Analysis programs. X.L.
acknowledges the support of the NASA/JPL Senior Fellowship, which is administered by Oak Ridge Associated Universities through a contract with NASA. D.E.S. acknowledges the
supported by the Cassini UVIS contract with the University of
Colorado. The authors thank Professor Lutoslaw Wolniewicz
for making results of his ab initio calculations accessible, and
Professor Robert Le Roy for providing the LEVEL 8 computer
program.
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