The Astrophysical Journal, 645:1560–1567, 2006 July 10
# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
0 1 þ
A SIMPLE MODEL FOR N2 LINE OSCILLATOR STRENGTHS OF THE b 0 1 þ
u (1), c4 u (0),
þ
þ
þ
þ
b 1 u (4), b 1 u (5), AND c3 1 u (0) X 1 g (0) BANDS
Xianming Liu and Donald E. Shemansky
Planetary and Space Science Division, Space Environment Technologies, 320 North Halstead, Pasadena, CA 91107;
xliu@spacenvironment.net, dshemansky@spacenvironment.net
Received 2006 March 8; accepted 2006 March 22
ABSTRACT
A simple model, based on fitting experimental energy term values, has been developed to calculate the line os0 1 þ
1 þ
1 þ
1 þ
1 þ
cillator strengths ( f ) of N 2 b 0 1 þ
u (1), c4 u (0), b u (4), b u (5), and c3 u (0) X g (0) bands. The strong
rovibronic coupling among these states leads to quantum interference in the transition dipole matrix elements and
results in anomalous P/R branch oscillator strength ratios. The calculated line oscillator strengths are in very good
agreement with experimental values. The calculated values can be used for modeling atmospheric emission. Calculated line oscillator strengths for J 0 up to 30 are provided.
Subject headingg
s: ISM: molecules — molecular data — planets and satellites: general — ultraviolet: general
1. INTRODUCTION
including photoabsorption (Carroll & Collins 1969; Carroll et al.
1970; Yoshino et al. 1975, 1979; Yoshino & Tanaka 1977; Shaw
et al. 1992; Stark et al. 1992, 2000, 2005), emission (Roncin et al.
1998, 1999), electron energy loss (Geiger & Schröder 1969; Chan
et al. 1993), electron-impact-induced emission (Ajello et al. 1989,
1998; James et al. 1990; Shemansky et al. 1995; Liu et al. 2005a),
and nonlinear laser techniques (Kam et al. 1989; Helm & Cosby
1989; Helm et al. 1993; Walter et al. 1993, 1994, 2000; Ubachs
et al. 1989, 2000; Sprengers et al. 2003, 2004, 2005; Sprengers &
Ubachs 2006). These investigations have shown that strong coupling among these states results in shifts of energy position and
deviations in spectral intensities, and significant predissociation
exists in many of these excited levels.
The oscillator strength of N2 for the discrete transitions has
been investigated extensively. Early measurements at the vibronic
level by various photoabsorption experiments obtained significantly different values (Watanabe & Marmo 1956; Huffman et al.
1963; Lawrence et al. 1968; Carter 1972; Gurtler et al. 1977; Shaw
et al. 1992), which were subsequently shown to be dependent on
the sample pressure and spectral resolution (Lawrence et al. 1968;
Chan et al. 1993). For this reason, electron impact techniques are
also used extensively (Geiger & Schröder 1969; Ajello et al. 1989;
James et al. 1990; Chan et al. 1993; Felfli et al. 1997). Discrete
oscillator strengths at electronic and vibronic levels have also
been calculated (Stahel et al. 1983; Kosman & Wallace 1985;
Bielschowsky et al. 1990; Spelsberg & Meyer 2001; Neugebauer
et al. 2004; Lavin et al. 2004). Shemansky et al. (1995) obtained
1 þ
line oscillator strengths of the c40 1 þ
u (0) X g (0) band by fitting
the unresolved rotational contour profiles of the electron-impactinduced emission spectrum. The most important and definitive
investigations so far have been room-temperature high-resolution
photoabsorption measurements by Stark et al. (1992, 2000, 2005),
who obtained line oscillator strengths of many vibrational bands of
1
0 1 þ
1 þ
1
1
the b 0 1 þ
u , b u , c4 u , c3 u , and o3 u X g (0) transitions. The measured line f-values show strong dependence on
the rotational quantum number, and the P/R ratios of many bands
deviate significantly from Hönl-London factors. By solving a
1 1 coupled Schrödinger equation of the b 1 u , c 3 u , o3 u ,
0 3 ,
and
C
states,
Lewis
et
al.
(2005b)
and
Haverd
et al.
C 3 u
u
(2005) have successfully reproduced experimental Q-branch
f-values and predissociation rates for various rotational levels
of the low v 0 levels of the b 1 u , c3 1 u , and o3 1 u states.
Molecular nitrogen is the major component in the atmospheres
of Earth, Titan, and Triton. The airglow emissions of N2 from the
atmospheres of Earth (Morrison et al. 1990; Meier 1991; Strickland
et al. 2004a, 2004b; Bishop et al. 2006) and planetary satellites
(Broadfoot et al. 1989; Liu et al. 2005b) have been extensively
1 þ
observed, and absorption of the N2 c40 1 þ
u (0) X g (0) band has
been identified in the interstellar medium (Knauth et al. 2004). The
interpretation and modeling of atmospheric excitation of nitrogen
by photons and photoelectrons require accurate and comprehensive lists of line positions, oscillator strengths, and predissociation
rates.
Strong transitions of N2 arise from the ground X 1 þ
g state to
dipole-allowed singlet ungerade excited states, between 100800
and 120000 cm1 above the ground state. The excited singlet ungerade states consist of two valence states and three Rydberg
1
series. The valence states are known as b 0 1 þ
u and b u . Two
Rydberg series, np and np, both converging to the X 2 þ
g state
0
1 þ
1
of Nþ
2 , are designated as cnþ1 u and cn u . The third series,
ns, which converges to the A 2 u states of Nþ
2 , is known as the
on 1 u states. The spectrum of the singlet ungerade states displays many deviations in both rovibronic energy levels and the
intensity distribution. For many years, these deviations made the
spectrum difficult to understand. Stahel et al. (1983) showed that
homogeneous interactions, primarily the Rydberg valence type,
1
within the 1 þ
u and u manifolds are the cause of many deviations. Spelsberg & Meyer (2001) carried out ab initio calculations
1
of the three lowest members of the 1 þ
u and u states by introducing internuclear-distance-dependent couplings and achieved
better agreement with experimental results. Helm et al. (1993),
Edwards et al. (1995), Ubachs et al. (2001), and Sprengers et al.
(2003) extended the work of Stahel et al. (1983) by introducing
1
u heterogeneous interactions. More recently, Lewis
the 1 þ
u
et al. (2005a, 2005b) and Haverd et al. (2005) considered additional coupling between the singlet and triplet 1 u states. Carroll
0
1 þ
u
& Yoshino (1972) analyzed the interaction between cnþ1
1
and cn u series with L-uncoupling theory.
1
Transitions to valence states, b 0 1 þ
u and b u , and to the low1
est members (n ¼ 3) of the Rydberg series, c40 1 þ
u , c3 u , and
1
o3 u , carry the largest neutral oscillator strengths. They have
been extensively investigated by many experimental techniques,
1560
1561
OSCILLATOR STRENGTHS OF N2 BANDS
The interpretation and modeling of atmospheric processes of
molecular nitrogen require line oscillator strengths of high J transitions. A recent thermospheric dayglow observation of N2 with
the FUSE (Far Ultraviolet Spectroscopic Explorer) instrument
by Bishop et al. (2006) indicates that the rotational temperature
of N2 in the thermosphere can be as high as 700 K. Emission
0 1 þ
from the coupled b 0 1 þ
u (1) c4 u (0) rotational levels to a num00
ber of excited v levels of the X 1 þ
g state was also observed. Because the line oscillator strengths have so far been measured only
at room temperature, line f-values for the high J levels are generally unavailable. Furthermore, owing to the strong rotational
dependence, line oscillator strengths of lower J levels cannot be
easily extrapolated to the required high J levels. Finally, a good
understanding of the J dependence of the f-values and the deviation of P/R ratios from Hönl-London factors is clearly desirable.
We report a simple analysis of rovibronic coupling of the
1 þ
0 1 þ
1 þ
1 þ
b 0 1 þ
u (1), c4 u (0), b u (4), b u (5), and c3 u (0)
1 þ
X g bands. We derive rovibronic eigenfunctions by a leastsquares fit of experimental energy term values of the b 0 1 þ
u (1),
1
1
1
(0),
b
(4),
b
(5),
and
c
(0)
levels.
In
addition
c40 1 þ
u
u
3
u
u
to obtaining a set of molecular and coupling parameters for these
states, we calculate line oscillator strengths of the b 0 1 þ
u (1),
1 þ
1 þ
1 þ
1 þ
(0),
b
(4),
b
(5),
and
c
(0)
X
(0)
bands
c40 1 þ
3
u
u
u
u
g
up to J 0 ¼ 30. The calculated f-values are in very good agreement with the experimental values of Stark et al. (1992, 2000,
2005), and for the great majority of levels, the differences between the calculated and measured values are smaller than the
experimental error. Rovibronic coupling, which leads to quantum interference, is responsible for the deviation of P/R f-value
ratios from Hönl-London factors and partially responsible for the
J 0 dependence of the line oscillator strength.
2. THEORY
As mentioned in x 1, the vibronic levels of singlet ungerade
states of N2 are strongly coupled. The v 0 ¼ 1 level of the b 0 1 þ
u
state, for instance, is homogeneously coupled to the v 0 ¼ 0 level
of the c40 1 þ
u state (Yoshino et al. 1979; Levelt & Ubachs 1992;
Edwards et al. 1995; Ubachs et al. 2001). The c40 1 þ
u (0) level is,
1 þ
in turn, heterogeneously perturbed by the b 1 þ
u (5) and c3 u (0)
levels (Carroll & Yoshino 1972; Stahel et al. 1983; Levelt &
Ubachs 1992). Finally, the c3 1 þ
u (0) level is also perturbed by
the b 1 þ
u (4) level (Carroll & Collins 1969; Levelt & Ubachs
0
1992) and, possibly, by the b 0 1 þ
u (1) level at J ¼ 28 or 29
(Levelt & Ubachs 1992). Clearly, an appropriate account of any
one of the vibronic levels requires a simultaneous consideration of the other four levels. For convenience, we denote the
1 þ
0 1 þ
1
1
b 0 1 þ
u (1), c4 u (0), b u (4), b u (5), and c3 u (0) vibronic levels as levels 1, 2, 3, 4, and 5, respectively.
The oscillator strength, f, for transition (v 00 ; J 00 ) ! (v 0 ; J 0 ) is
related to the dipole matrix element M by (Abgrall & Roueff
1989)
fv 00 ;v 0 ; J 0 0 ; J 0 ¼
2
½ E ðv 0 ; J 0 Þ Eðv 00 ; J 00 ÞjM j2 ;
3ð2 J 00 þ 1Þ
ð1Þ
where M is the electric dipole matrix element between v0000 J 00 ,
the rovibronic eigenfunction of the lower state, and v0 0 J 0 , the
rovibronic eigenfunction of the upper state. The superscript identifies the P, Q, or R-branch transitions. Both energy E and
dipole matrix element M are in atomic units.
The matrix element, M , is evaluated over rovibronic eigenfunctions of initial (double prime) and final (single prime) states.
0 1 þ
Assuming that the vibronic problem for b 0 1 þ
u (1), c4 u (0),
1 þ
1 þ
1 þ
b u (4), b u (5), and c3 u (0) has been solved, we can
construct the basis function as a product of the vibronic wave
function and the case (a) rotational function,
ð2Þ
00 (v 00 ; J 00 ) ¼ n; 00 ¼ 0; v 00 J 00 00 M 00 ;
0
0 0
0
0 0 0 0
(v ; J ) ¼ n; ¼ 0; v J M ;
ð3Þ
1
0 (v 0 ; J 0 ) ¼ pffiffiffi n; 0 ; v 0 J 0 ; 0 ; M 0
2
þ n; 0 ; v 0 J 0 0 M 0 ;
ð4Þ
where equation (2) applies to the X 1 þ
g state and equation (3)
0 1 þ
and
c
states.
The symmetrized basis
applies to the b 0 1 þ
4
u
u
1 e
function (4) applies to the 1 þ
u ( u ) components.
Since the X 1 þ
g state is well separated from other electronic
states, its eigenfunction is simply given by equation (2). However, because of the strong coupling of the upper ungerade states,
the eigenfunction for excited levels generally cannot be represented by any single term of equation (3) or equation (4). Instead,
it is a linear combination of basis functions of equations (3) and
(4),
k0 ðv 0 ; J 0 Þ ¼
5
X
Ck j j0 vj0 ; J 0 ;
ð5Þ
j¼1
where Ck j are the eigencoefficients that diagonalize the 5 ; 5
symmetric Hamiltonian matrix with elements
Hmm ð J Þ ¼ Tm0 þ Bm J ð J þ 1Þ Dm J 2 ð J þ 1Þ2
þ H m J 3 ð J þ 1Þ 3 ;
Hmm ð J
Þ ¼ Tm0
ð6Þ
þ Bm J ð J þ 1Þ þ qm J ð J þ 1Þ
Dm J 2 ð J þ 1Þ2 þHm J 3 ð J þ 1Þ3 ;
Hmn ¼ hmn ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Hmn ð J Þ ¼ hmn J ð J þ 1Þ;
ð7Þ
ð8Þ
ð9Þ
where equation (6) applies to m ¼ 1 and 2 and equation (7) applies to m ¼ 3, 4, and 5. The off-diagonal element Hmn of equation (8) refers to the homogeneous interaction between the 1 þ
u
levels (m; n ¼ 1, 2 and m 6¼ n) or between the 1 þ
u levels
(m; n ¼ 3, 4, 5 and m 6¼ n). The Hmn element of equation (9)
1 þ
u coupling (m ¼ 1, 2 and
represents the heterogeneous 1 þ
u
n ¼ 3, 4, 5). The -doubling parameter, qm, which arises from a
van Vleck transformation of the L-uncoupling interaction (van
Vleck 1951), is partially taken into account by the heterogeneous
coupling elements in equation (9).
The present investigation obtains parameters in equations
(6)–(9) and eigencoefficients by least-squares fits of eigenvalues
to experimentally determined energy term values. The eigencoefficient is then used to calculate the line oscillator strength,
f (v 00 ; v 0 ; J 00 ; J 0 ). For the P-branch (J 00 ¼ J 0 þ 1), the oscillator
strength is given by (Lefebvre-Brion & Field 1986, p. 273;
Hansson & Watson 2005; Abgrall & Roueff 2006)
2½ E ðv 0 ; J 0 Þ Eðv 00 ; J 00 Þ
f (v 00 ; v 0 ; J 00 ; J 0 ) ¼
3ð 2 J 0 þ 3 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi X
2
; J0 þ 1
Cjk X ; 00 ; v 00 MX k k; k0 ; vk0
k¼1
2
5
pffiffiffiffiffi X
00
0
00
0
þ J0
Cjk X ; ; v MX k k; k ; vk :
ð10Þ
k¼3
1562
LIU & SHEMANSKY
Vol. 645
TABLE 1
1
1
1
Molecular Parameters of the b 0 1 þ
(1),
c40 1 þ
u
u (0), b u (4), b u (5), and c3 u (0) Levels
Parameter
b 0 1 þ
u (1)
c40 1 þ
u (0)
b 1 u (4)
b 1 u (5)
c3 1 u (0)
T0 ............................................
B .............................................
D.............................................
H.............................................
qa ............................................
Jmaxb .......................................
104417.82(6)
1.14834(64)
1.14(16) ; 105
2.4(105) ; 1010
...
32
104322.95(5)
1.89997(1)
1.67(20) ; 105
4.67(28) ; 108
...
29
103548.72(5)
1.42343(45)
4.88(11) ; 105
6.63(68) ; 109
...
33
104699.88(5)
1.42522(84)
4.89(37) ; 105
1.03(43) ; 108
...
21
104138.49(5)
1.50804(47)
4.00(11) ; 105
1.13(7) ; 108
7.35(28) ; 103
33
Notes.—In cm1. Refer to eqs. (6) and (7) for definitions. Numbers in parentheses represent one standard error.
Because of the presence of heterogeneous coupling off-diagonal matrix elements, the value of the q parameter obviously differs from its conventional value.
b
Maximum value of J 0 fitted. The listed values for b 1 u (4), b 1 u (5), and c3 1 u (0) are for the 1 þ
u components. The corresponding Jmax for the
1 u components are 31, 24, and 31, respectively.
a
For the R-branch (J 00 ¼ J 0 1) transition, it is given by
2½ Eðv 0 ; J 0 Þ E ðv 00 ; J 00 Þ
f (v 00 ; v 0 ; J 00 ; J 0 ) ¼
3ð 2 J 0 1Þ
pffiffiffiffiffi X
2
; J0
Cjk X ; 00 ; v 00 MXk k; k0 ; vk0
k¼1
2
5
pffiffiffiffiffiffiffiffiffiffiffiffiffi X
00 00 0 0 0
J þ1
Cjk X ; ; v MXk k; k ; vk ;
k¼3
ð11Þ
where MXk refers to the appropriate electronic transition moment
between the X 1 þ
g and the kth ungerade electronic state.
3. ANALYSIS AND RESULTS
The parameters in equations (6)–(9) are obtained by a leastsquares fit of eigenvalues of the 5 ; 5 Hamiltonian matrix to the
experimentally determined energy term values of the b 0 1 þ
u (1),
1 þ (4), b 1 þ (5), and c 1 þ (0) levels listed by
(0),
b
c40 1 þ
3
u
u
u
u
Roncin et al. (1998), Smith et al. (2003),1 and Levelt & Ubachs
1 (1992). The experimental term values of b 1 u (4), b u (5),
1 and c3 u (0) are also jointly fit to equation (7) with qm ¼ 0.
Standard EISPACK subroutines, TRED2 and TQLI, were used
to diagonalize the symmetric Hamiltonian matrix and obtain
eigenvalues and eigenvectors.
The J 0 ¼ 18 level of the b 1 þ
u (4) state, having a residual of
1.5 cm1, is far beyond the experimental error. Stark et al. (1992,
2005) noted that the J 0 ¼ 18 level of b 1 þ
u (4) is strongly perturbed, although the nature of the localized perturbation is not
clear. A similar local perturbation was also found by Sprengers et
al. (2003) for 15N2 J 0 ¼ 15 levels of the b 1 u (4) state. The term
value of the J 0 ¼ 18 level of the b 1 þ
u (4) state is not used. The
sum of the squares of residuals for the 235 data points is 4.25 cm2.
The largest residual is 0.72 cm1, at J 0 ¼ 29 of the c40 1 þ
u (0)
level. The overall standard deviation is 0.14 cm1, which is smaller
than or comparable to the expected experimental error. Most levels with significant residuals are low J 0 levels of b 1 u (4) and
c3 1 u (0) states, whose term values were derived from blended
transitions. Tables 1 and 2 list the derived molecular parameters
and coupling parameters.
1
See also Harvard-Smithsonian Center for Astrophysics Molecular Database, http://cfa-www.harvard.edu/amdata /ampdata/N2ARCHIVE/n2term.html.
of the vibronic
The00value
transition dipole matrix element,
X ; ; v 00 jMX k jk; k0 ; vk0 , is required to calculate the oscillator
strength. Stahel et al. (1983) obtained vertical transition mo1
1
0 1 þ
0 1 þ
1
ments for b 0 1 þ
u , c4 u , c5 u , b u , c3 u , and o3 u
states from v 00 ¼ 0 of X 1 þ
.
Ab
initio
transition
moments
for
g
these states have been recently calculated by Spelsberg & Meyer
(2001) over a wide range of internuclear distance. While the ab
initio values for vertical transitions agree fairly well with those
1 þ
obtained by Stahel et al. (1983), the signs for the b 0 1 þ
u X g
1
1 þ
and b u X g transition dipole matrix elements are different. More recently, Sprengers et al. (2004) and Haverd et al.
(2005) noted that the Spelsberg & Meyer (2001) transition
moments yield greater oscillator strengths for the b 1 u , c3 1 u ,
and o3 1 u states than the corresponding experimental values of
Stark et al. (1992). The vibronic wave function, jn; ; vi, of
equations (2)–(4) is assumed to be not explicitly dependent on
J 0 . By extension, the vibronic transition dipole matrix element,
hX ; 00 ; v 00 jMXk jk; k0 ; vk0 i, is also assumed to be independent of
0 1 þ
1 þ
1 þ
J. For the b 0 1 þ
u (1) X g (0) and c4 u (0) X g (0) bands,
the assumption is very good one. The vibronic transition dipole
1 þ
matrix element of the c40 1 þ
u (0) X g (0) band is 0.695 au,
obtained from the oscillator strength of the P(1) transition measured by Stark et al. (2000, 2005). The matrix element for the
1 þ
3
au, equated to the
b 0 1 þ
u (1) X g (0) band is 1:65 ; 10
product of the negative value of the Stahel et al. (1983) vertical
1 þ
b 0 1 þ
u X g transition moment (0.8243 au) and the (1; 0)
band vibrational overlap integral (0.002). However, the strong
electrostatic interaction within the valence Rydberg 1 u manifold
TABLE 2
0 1 þ
Coupling Parameters of the b 0 1 þ
u (1), c4 u (0),
1 þ
1 þ
(4),
b
(5),
and
c
(0)
Levels
b 1 þ
3
u
u
u
Parametera
Value
h(b 0 1; c40 0)...........................................
h(b 0 1; c3 0)...........................................
h(c40 0; b4)............................................
h(c40 0; b5)............................................
h(c40 0; c3 0) ..........................................
h(b4; c3 0)............................................
h(b5; c3 0)............................................
5.029(82)
0.0612(48)
1.020(57)
2.08(27)
2.63(20)
8.52(173)
0.241b
Notes.—In cm1. Refer to eqs. (8) and (9) for definitions. Numbers in parentheses represent one standard
error.
a
Parameters of the off-diagonal matrix elements not
listed here are assumed to be zero.
b
Fixed.
No. 2, 2006
1563
OSCILLATOR STRENGTHS OF N2 BANDS
TABLE 3
0 1 þ
1 þ
Line Oscillator Strengths of the b 0 1 þ
u (1) and c4 u (0) X g (0) Bands
b 0 1 þ
u (1)
c40 1 þ
u (0)
J
R(J )a
R(J )b
P(J )a
P(J )b
R(J )a
R(J )b
P(J )a
P(J )b
0.........................................
1.........................................
2.........................................
3.........................................
4.........................................
5.........................................
6.........................................
7.........................................
8.........................................
9.........................................
10.......................................
11.......................................
12.......................................
13.......................................
14.......................................
15.......................................
16.......................................
17.......................................
18.......................................
19.......................................
20.......................................
21.......................................
22.......................................
23.......................................
24.......................................
25.......................................
26.......................................
27.......................................
28.......................................
29.......................................
30.......................................
31.......................................
32.......................................
0.47
0.33
0.33
0.36
0.42
0.53
0.74
1.2
2.7
10.9
15.3
2.5
0.82
0.38
0.21
0.13
0.08
0.06
0.04
0.03
0.03
0.02
0.02
0.02
0.02
0.03
0.05
0.15
14.3
0.10
0.02
0.01
0.00
...
...
...
0.35(7)
0.41(5)
0.53(7)
0.61(9)
1.0(1)
...
10(2)
15(3)
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
0.16
0.19
0.22
0.26
0.31
0.38
0.51
0.75
1.3
2.8
11.9
17.0
2.8
0.93
0.43
0.24
0.14
0.09
0.06
0.04
0.03
0.02
0.02
0.01
0.01
0.00
0.00
0.00
0.01
3.48
0.06
0.02
...
...
0.19(4)
0.25(8)
0.22(4)
0.31(7)
0.40(5)
...
...
1.3(3)
3.6(7)
11(2)
18(3)
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
149
98
87
82
79
76
74
72
70
61
56
68
69
69
69
69
69
69
69
69
70
70
70
70
70
68
65
61
57
20
24
28
32
150(15)
89(13)
75(8)
75(8)
71(11)
69(7)
69(14)
61(12)
64(10)
57(6)
50(10)
56(11)
61(12)
65(13)
65(13)
68(14)
65(13)
68(14)
67(13)
70(14)
62(12)
66(13)
65(13)
...
...
...
...
...
...
...
...
...
...
...
51
62
67
70
72
74
75
76
76
75
66
62
76
78
78
78
78
77
76
75
73
70
67
63
58
52
45
37
30
23
74
79
...
51(10)
56(6)
60(6)
64(13)
63(6)
64(13)
64(6)
69(14)
66(10)
71(11)
68(7)
61(12)
81(8)
80(12)
85(9)
82(8)
94(8)
84(8)
88(9)
79(8)
84(17)
81(12)
76(15)
68(14)
66(13)
...
...
...
...
...
...
...
Note.—All listed f-values have been scaled up by a factor of 1000.
a
Calculated. Calculated values within experimental uncertainties are unmarked. Values outside the error but within twice the error
are shown with an asterisk. Values outside twice the error are denoted with a dagger.
b
Measured value of Stark et al. (2000, 2005). Number in parentheses refers to experimental error.
prevents a good separation of vibronic and rotational wave functions (Haverd et al. 2005). The b 1 u and c3 1 u X 1 þ
g dipole matrix elements, therefore, implicitly depend on the upper
state rotational quantum number. Indeed, the Q-branch oscillator
strengths measured by Stark et al. (2005) for many bands of the
b 1 u , c3 1 u , and o3 1 u X 1 þ
g transitions show very significant J dependence. For the present study, the Q-branch oscillator strength functional forms for the b 1 u (4), b 1 u (5), and
c3 1 u (0) X 1 þ
g (0) bands obtained by Stark et al. (2005) are
used to derive the vibronic transition dipole matrix element for
each J 0 . The function forms are used up to three rotational levels
0
0
) levels. Beyond the (Jmax
þ 3)
above the highest measured (Jmax
levels, the Q-branch f-values are frozen at the extrapolated
0
þ 3) levels. Consistent with ab initio results of Spelsberg &
(Jmax
Meyer (2001), the signs of hX ; 00 ; v 00 jMX k jk; k0 ; vk0 i for all five
vibronic levels are negative. No attempt has been made to optimize the values of the vibronic transition dipole matrix elements.
Tables 3 and 4 display the calculated P- and R-branch oscillator strengths along with the measured values of Stark et al.
(2000, 2005). For convenience, all listed f-values have been
scaled up by a factor of 1000. The calculated values within the
experimental error are unmarked. Values outside the experi-
mental error but within a factor of 2 of the error are shown with
an asterisk. Finally, f-values outside twice the error level are
denoted with both an asterisk and a dagger.
4. DISCUSSION
Molecular parameters for the b 0 1 þ
u (1) level in Table 1 agree
with those obtained by Levelt & Ubachs (1992) within two standard errors. The Levelt & Ubachs (1992) parameters are derived
from measurements for J 0 15. As a result, the higher order of
the centrifugal distortion term, H, was not included. It should be
mentioned that the present value of H, which is about 1/5 of the
standard error, is essentially undetermined. Except for the parameter q, the other parameters of the c3 1 u (0) level are also
consistent with those of Levelt & Ubachs (1992). As mentioned,
the q term arises from perturbative treatment of the L-uncoupling
operator, 1/(2R 2 )J L (van Vleck 1951; Lefebvre-Brion &
Field 1986, pp. 118–223). The L-uncoupling interaction among the
0 1 þ
1 þ
1 þ
1 þ
b 0 1 þ
u (1), c4 u (0), b u (4), b u (5), and c3 u (0) levels
is explicitly accounted for by the heterogeneous coupling terms
1 þ
u coupling outof equation (9). To the extent that the 1 þ
u
side of these five levels is not negligible, the q term in the diagonal element, equation (7), is required. Indeed, it was found
1564
LIU & SHEMANSKY
Vol. 645
TABLE 4
1 þ
1 þ
1 þ
Line Oscillator Strengths of the b 1 þ
u (4), b u (5), and c3 u (0) X g (0) Bands
b 1 þ
u (4)
b 1 þ
u (5)
c3 1 þ
u (0)
J
R(J )a
R(J )b
P(J )a
P(J )b
R(J )a
R(J )b
P(J )a
P(J )b
R(J )a
R(J )b
P(J )a
P(J )b
0.................................
1.................................
2.................................
3.................................
4.................................
5.................................
6.................................
7.................................
8.................................
9.................................
10...............................
11...............................
12...............................
13...............................
14...............................
15...............................
16...............................
17...............................
18...............................
19...............................
20...............................
21...............................
22...............................
23...............................
24...............................
25...............................
26...............................
27...............................
28...............................
29...............................
30...............................
66
33
27
24
22
21
21
20
19
19
19
18
18
18
17
17
17
16c
16
15
15
15
14
14
13
13
12
12
11
11
11
...
...
...
...
...
...
...
...
...
...
...
...
20(5)
20(8)
15(3)
18(3)
17(3)
...
19(3)
17(3)
15(2)
...
14(3)
15(4)
13(3)
...
...
...
...
...
...
...
...
6.5
9.3
11
12
12
12
13
13
13
13
13
13
12
12
12
12
12
11c
11
11
11
10
10
9.6
9.2
8.9
8.5
8.0
7.6
...
...
...
...
...
...
...
...
12(3)
12(3)
12(3)
12(2)
12(2)
12(2)
12(2)
11(1)
11(1)
11(1)
11(1)
...
12(1)
10(1)
11(1)
...
13(3)
...
...
...
...
...
...
2.5
1.1
0.84
0.69
0.60
0.54
0.49
0.46
0.43y
0.40
0.38
0.36
0.33
0.31
0.27
0.24
0.19
0.14
0.08
0.03
0.00
0.04
0.20
0.61
1.4
3.3
6.2
10.2
15
52
47
...
...
...
...
...
0.72(11)
...
0.52(9)
0.59(7)
0.47(13)
0.47(12)
0.43(9)
0.48(10)
0.40(10)
0.34(7)
0.55(16)
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
0.35
0.57
0.76
0.95
1.2
1.4
1.6
1.9
2.3
2.7y
3.2
3.7
4.4
5.2
6.2
7.3
8.7
10.3
12
15
18
22
26
32
40
46
54
61
69
...
...
0.4(1)
0.6(1)
0.8(1)
...
...
1.2(2)
1.4(2)
1.6(2)
2.2(3)
2.0(3)
2.8(4)
3.1(4)
4.6(6)
5.3(7)
5.8(7)
7.4(10)
8.0(9)
10.2(12)
...
14(2)
17(2)
...
...
...
...
...
...
...
...
52
28
24
23
22
22
23
23
24
24
24
25
25
26
26
26
27
27
27
28
28
28
29
29
29
29
30
30
16
30
30
...
...
...
...
...
...
...
19(3)
...
20(3)
23(3)
23(3)
21(3)
23(3)
22(3)
25(4)
25(4)
...
...
...
24(3)
23(4)
25(4)
26(3)
24(3)
24(6)
26(4)
...
...
...
...
...
...
4.2
5.6
6.0
6.1
6.1
5.9
5.7
5.6
5.4
5.3
5.2
5.1y
5.1
5.1
5.1
5.1
5.2
5.3y
5.5
5.6y
5.8y
6.0
6.2
6.5
6.7
7.0
7.3
7.6
4.4
...
...
...
...
6.0(9)
6.2(10)
6.0(9)
6.7(11)
6.9(10)
6.6(11)
6.4(10)
...
5.9(10)
7.2(9)
6.8(8)
6.9(17)
6.3(8)
6.3(7)
6.5(7)
7.4(8)
7.1(8)
7.9(11)
8.8(10)
8.5(25)
...
...
...
...
...
...
...
Note.—All listed f-values have been scaled up by a factor of 1000.
a
Calculated. Calculated values within experimental uncertainties are unmarked. Values outside the error but within twice the error are shown with an asterisk.
Values outside twice the error are denoted with a dagger.
b
Experimental values of Stark et al. (1992, 2005), with experimental error is given in the parentheses.
c
Owing to a perturbation at the J 0 ¼ 18 level for the b 1 u (4) state by an unidentified state, the calculated oscillator strengths given here are not expected to be
accurate. Stark et al. (2005) found that about 40% of the intensities for R(17) and P(19) have been transferred to transitions of the unidentified state.
that retaining the term for the c3 1 þ
u (0) level while neglecting
1 þ
(4)
and
b
(5)
states
is sufficient to achieve a
it for the b 1 þ
u
u
good fit. Obviously, the values of q in Table 1 are necessarily
different from those obtained when the heterogeneous coupling
terms of equation (9) are not explicitly considered.
Both the B and D parameters for the c40 1 þ
u (0) level differ significantly from their counterparts of Yoshino & Tanaka (1977)
and Levelt & Ubachs (1992), obtained from analyzing transitions to J 0 19. Their B constants are significantly larger than
the present value, which is consistent with their positive D-values
and the present negative D-value. In addition, the H centrifugal
term was absent in their representation. They noted that equation (6) alone cannot represent the levels beyond J 0 ¼ 19 well.
The present approach, fitting up to J 0 ¼ 29, yields a better representation of the c40 1 þ
u (0) state. The two largest residuals are
0.72 and 0.15 cm1. The 0.72 cm1 residual occurs at the J 0 ¼
29 level. Roncin et al. (1998) noted that the experimental term
value for the level, derived from only two emission lines, has an
uncertainty of 0.42 cm1. Had the J 0 ¼ 29 level of c40 1 þ
u (0) not
been used, the sum of the squares of residuals would have decreased from 4.25 to 3.7 cm2. As noted, some low J 0 levels of
b 1 u (4) and c3 1 u (0) have residuals greater than 0.14 cm1.
These levels generally had their experimental term values derived from blended transitions. In the case of the low J 0 levels of
c3 1 u (0), the differences in the experimental term values obtained in the measurements of Levelt & Ubachs (1992) and
Smith et al. (2003; see also footnote 1) are often greater than the
residuals.
All the off-diagonal coupling parameters listed in Table 2 are
0 1 þ
positive. The value of the b 0 1 þ
u (1) c4 u (0) homogeneous
1
coupling parameter, 5:029 0:082 cm , agrees very well with
5:027 0:017 cm1 obtained by Levelt & Ubachs (1992) from
their two-state coupling model. The good agreement is apparently
due to the fact that heterogeneous coupling of the b 0 1 þ
u (1) level
1 þ
1 þ
(4),
b
(5),
and
c
(0)
levels
is
negligibly
to the b 1 þ
3
u
u
u
1 þ
small. However, the c40 1 þ
u (0) b u (5) heterogeneous coupling parameter, 2:08 0:27 cm1, differs significantly from the
value of 1.24 cm1 estimated by Levelt & Ubachs (1992). The
1
1
c40 1 þ
u (0) c3 u (0) coupling parameter, 2:63 0:20 cm , is
1
between the expectation value, 3.41 cm , of the L-uncoupling
model of Carroll & Yoshino (1972) and 2.25 cm1 adopted by
Levelt & Ubachs (1992) for their modeling of doubling of the
1
c3 1 u (0) level. As expected, the c40 1 þ
u (0) b u (5) coupling
0 1 þ
1
is about twice as strong as the c4 u (0) b u (4) coupling.
No. 2, 2006
OSCILLATOR STRENGTHS OF N2 BANDS
1 þ
Since the standard error for the b 1 þ
u (5) c3 u (0) coupling parameter is many times its value, it is therefore fixed at
0.241 cm1. We note that this fixed value is much smaller
1 þ
than its b 1 þ
u (4) c3 u (0) counterpart, which has a value of
1
8:5 1:7 cm .
It should be mentioned that energy term values of the 1 u
components were simultaneously fit to equation (7) with qm ¼ 0.
1 This means that possible effects of b 1 u (4) c3 u (0) and
1 1 b u (5) c3 u (0) homogeneous coupling on the eigenvalues
of the 1 u components are not explicitly considered even though
1 þ
1 þ
1 þ
coupling of b 1 þ
u (4) c3 u (0) and b u (5) c3 u (0) is
1 1 þ
considered via equation (8). Since both the u and u components were fit to identical sets of T0, B, D, and H parameters,
1 þ
1 þ
1 þ
the b 1 þ
u (4) c3 u (0) and b u (5) c3 u (0) coupling parameters in Table 2 are best treated as the difference between the
1 þ
u and 1 u components. As noted in x 3, the strong electrostatic interaction within the valence Rydberg 1 u manifold results in very significant J dependence of the vibronic dipole
matrix elements. This approach greatly simplified the derivation
1 þ
of the dipole matrix elements of the b 1 þ
u (4), b u (5), and
1 þ
1 þ
c3 u (0) X g (0) transitions from the functional forms of the
Q-branch oscillator strengths.
Table 3 shows very good agreement between the calculated
and experimental line oscillator strengths for the b 0 1 þ
u (1) and
0 1 þ
1 þ
c40 1 þ
u (0) levels. Of the 15 measured b u (1) X g (0) transitions, 12 are fully reproduced by the model within the experimental error. For the remaining three measured transitions, the
calculated values for R(6) and P(10) are within twice the experimental error, while the R(7) line is at twice the experimental
uncertainty. Among the 31 measured oscillator strengths for the
c40 1 þ
u (0) band, the model is capable of reproducing 24 transitions within the experimental error. All remaining seven transitions are less than twice the experimental error. The model has
clearly reproduced the deviation of the P/R oscillator strength
ratio of the c40 1 þ
u (0) levels from the Hönl-London factors.
Table 4 shows that the calculated oscillator strengths for the
b 1 u (4) X 1 þ
g (0) transitions fully agree with all 26 measured
values within the experimental error. As mentioned in x 3, the
perturbation at J 0 ¼ 18 by the unidentified state was not considered. For this reason, the calculated P(19) and R(17) oscillator
strength for b 1 u (4) is not expected to be accurate. Based on the
experimental measurement of Stark et al. (2005), these values
should be reduced by 40%.
Although the oscillator strength of the b 1 u (5) X 1 þ
g (0)
band is very small, the model achieves very good agreement
with measurements. Of 28 measured transitions, only seven calculated values are outside the experimental error. Notably, R(8),
which is very weak, and P(11) are outside twice the experimental
error. Most importantly, the model has successfully reproduced
the very small oscillator strengths for the R-branch transitions.
For the c3 1 u (0) X 1 þ
g (0) band, the model generally underestimates the oscillator strengths of many P-branch transitions and overestimates those of some R-branches. However, the
agreement between the calculated and observed data is still fairly
good. Of the 34 observed values, the model successfully obtained 16 of them within the experimental errors. For the remaining 18 measured values, only four transitions, all of which
are of P-branch type, differ by more than twice the experimental
error.
The vibronic transition dipole matrix element of the b 0 1 þ
u (1)
X 1 þ
g (0) band is very small, largely due to a very unfavorable
Franck-Condon factor. Thus, the value of its oscillator strength
primarily arises from the coupling with the other four levels.
Table 2 shows that b 0 1 þ
u (1) has a fairly strong coupling with
1565
1 þ
the c40 1 þ
u (0) state, a weak one with the c3 u (0) level, and
1
1
negligible interaction with b u (4) and b u (5). With the exception of the transitions to J 0 ¼ 29 levels, all the other b 0 1 þ
u (1)
X 1 þ
(0)
lines
with
significant
values
gain
their
oscillator
strength
g
1 þ
by ‘‘intensity borrowing’’ from the c40 1 þ
u (0) X g (0) transitions, whose oscillator strength is the largest in the EUV region
(Ajello et al. 1989; Liu et al. 2005a). The large calculated oscillator
1 þ
strength for the R(28) and P(30) lines of the b 0 1 þ
u (1) X g (0)
0
band arises from near degeneracy in J ¼ 29 levels of the
1
1
b 0 1 þ
u (1) and c3 u (0) states, with a separation of only 2.2 cm .
The mixing between the two levels is very strong even with the
h(b 0 1; c3 0) coupling matrix element set to zero. With the coupling
parameter values given in Table 2, each eigenlevel has 45%–46%
of the other.
0 1 þ
The percentage of mixing in the b 0 1 þ
u (1)/c4 u (0) eigen0
functions near the J ¼ 10 level obtained in the present work and
those of Edwards et al. (1995) and Ubachs et al. (2001) are significantly different. Both Edwards et al. (1995) and Ubachs et al.
(2000) predicted that the largest mixing should occur at J 0 ¼
10. In contrast, the present work shows that the strongest
0 1 þ
0
b 0 1 þ
u (1)/c4 u (0) mixing actually takes place at J ¼ 11. For
(0)
state,
the perthe J 0 ¼ 9, 10, and 11 eigenlevels of the c40 1 þ
u
0 1 þ
(1)/c
(0)
character
reported
by
Edwards
centages of b 0 1 þ
u
u
4
et al. (1995) are 8.1%/90.7%, 43.1%/56.0%, and 9.0%/89.4%,
respectively. The corresponding percentages of the present work
are 3.6%/94.9%, 15.0%/83.4%, and 21.8%/76.5%, respectively.
The percentages given by Edwards et al. (1995) are not accurate,
1 þ
because they lead to incorrect b 0 1 þ
u (1) X g (0) oscillator
strengths. Indeed, numbers given by Edwards et al. (1995) would
lead to estimated oscillator strengths of 0.031 and 0.006 for the
R(9) and R(10) transitions and 0.034 and 0.007 for the P(11) and
P(12) transitions. These values are drastically different from
0.010, 0.015, 0.011, and 0.018 given by Stark et al. (2000, 2005;
see Table 3).
Rovibronic coupling has a profound effect on the individual
line oscillator strengths. As noted, all the off-diagonal coupling
parameters listed in Table 2 are positive. The eigenvalues of
1 þ
c40 1 þ
u (0) are higher than their counterparts of the b u (4) and
1 þ
1 þ
c3 u (0) states, but are lower than those of the b u (5) states.
It follows that C23 and C25 [the coefficients for the b 1 þ
u (4) and
c3 1 þ
u (0) basis functions] have a sign the same as C22 but opposite to C24 [for b 1 þ
u (5)] in the eigenfunction expansion of
(0).
Since
five
vibronic
dipole matrix elements are all
c40 1 þ
u
negative, equations (10) and (11) show that the coupling with
1 þ
b 1 þ
u (4) and c3 u (0) enhances the P-branch oscillator
strength while suppressing the R-branch f-values of the c40 1 þ
u (0)
1 þ
X 1 þ
(0)
band.
The
role
of
the
b
(5)
one
is
exactly
opposite.
g
u
However, since the vibronic dipole matrix element of b 1 þ
u (5) is
þ
much smaller than those of b 1 u (4) and c3 1 þ
u (0), the net effect
is that the P-branch oscillator strength of c40 1 þ
u (0) is greater than
that implied by Hönl-London factor, while the R-branch is smaller.
Furthermore, the deviation of the oscillator strength from the value
implied by Hönl-London factors is J dependent, because both the
coefficient C2k and the vibronic transition dipole matrix elements of
1 þ
1 þ
the b 1 þ
u (4), b u (5), and c3 u (0) states are J dependent.
Finally, it should be noted that the coupling parameter, h(c40 0; c3 0),
is greater than h(c40 0; b4), and the c3 1 þ
u (0) level is closer to the
1 þ
(0)
level
than
the
b
(4)
level
in energy. The absolute
c40 1 þ
u
u
value of C25 is thus much greater than that of C23. Hence, the in1 þ
terference effect on the c40 1 þ
u (0) transition by c3 u (0) is greater
(4)
level
even
though
the
former
has
a smaller (abthan the b 1 þ
u
solute value of the) transition dipole matrix element.
A similar analysis leads to the conclusion that the c40 1 þ
u (0)
1 þ
1 þ
b u (4) and c40 1 þ
u (0) c3 u (0) coupling is constructive to
1566
LIU & SHEMANSKY
1
the R-branch transitions of the b 1 þ
u (4) and c3 u (0) levels but
destructive to their P-branch transitions. In spite of a large homogeneous coupling parameter, h(b4; c3 0), the mixing between
1
b 1 þ
u (4) and c3 u (0) is small because of the large energy
1 þ
0 1 þ
separation. For the b 1 þ
u (5) X g (0) band, the c4 u (0)
þ
1
b u (5) heterogeneous coupling is primarily responsible for
constructive interference for the P-branch transitions and destructive interference for the R-branch transitions. The P/R oscillator strength ratio exceeds 1 at J 0 4. The contribution from
0
0
c40 1 þ
u (0) is small at low J but increases rapidly with J . The
1 þ
(5)
X
vibronic transition dipole matrix element of b 1 þ
u
g (0)
also increases with J 0 . Thus, many R(J ) transitions have very
small f-values. By the time J 0 reaches 21, destructive interference
1 þ
from c40 1 þ
u (0) approximately offsets that of b u (5) itself in
the R-branch transition, leading to a vanishingly small f-value.
Beyond J 0 ¼ 21, the destructive contribution of the c40 1 þ
u (0)
level exceeds the value of b 1 þ
u (5), causing the R-branch oscillator strength to rise with J 0 .
Comparison of the calculated and measured oscillator
strengths also shows that the sign of the transition dipole element
1 þ
be identical to those of the
for the b 1 þ
u X g band must þ
1 þ
X
and
c3 1 u X 1 g bands. Had the sign for
c40 1 þ
g
u
1 þ
b 1 þ
u X g been reversed, the difference between the calculated and measured f-values for many transitions of the c40 1 þ
u (0),
1
b 1 u (4), b 1 þ
u (5), and c3 u (0) bands would have been unacceptably large. More importantly, the calculated f-values of the
R-branch transition with J 0 < 29 of the b 1 þ
u (5) band would
have been much greater than their counterparts of the P-branch,
which contradicts the experimental measurement of Stark et al.
(2005). Thus, the result of the present analysis is consistent
with the sign of Spelsberg & Meyer (2001) and Haverd et al.
(2005). The vibronic transition dipole matrix element for the
1 þ
b 0 1 þ
u (1) X g (0) band is so small that the present study
1 þ
cannot establish the sign of the b 0 1 þ
u X g transition moment.
However, switching its sign from negative to positive marginally
improves the agreement in the f-values listed in Table 3 in the
sense that the number of entries with asterisks drops from 10
to 8.
Vol. 645
It should be restated that f-values of the present work are
obtained by using the vibronic dipole matrix elements derived
1 from the Q-branch f-values of the b 1 u (4), b u (5), and
1 þ
1 þ
1 c3 u (0) X g (0) transitions for the b u (4), b 1 þ
u (5),
(0)
levels.
The
fact
that
the
calculated
f-values
for
the
and c3 1 þ
u
P- and R-branches agree well with the experimental values suggests that the rotational dependences of the transition dipole matrix
1 þ
1 þ
1 þ
elements for the b 1 þ
u (4), b u (5), and c3 u (0) X g (0)
1 bands are identical or very similar to those of the b u (4),
1 1 þ
b 1 u (5), and c3 u (0) X g (0) transitions. The calculated
Q-branch f-values of Haverd et al. (2005) are in excellent agreement with the experimental values of Stark et al. (2005). The
strong J dependence of the Q-branch vibronic transition moment
is attributed to the strong electrostatic interaction within the
Rydberg valence 1 u manifold (Lewis et al. 2005b; Haverd et al.
2005). The fact that very little J dependence of the b 0 1 þ
u (1) and
1 þ
c40 1 þ
u (0) X g (0) transition dipole matrix element is observed suggests that electrostatic coupling among the 1 þ
u states
is probably weaker.
In summary, the present investigation has examined rovibronic
0 1 þ
1 þ
1 þ
coupling among b 0 1 þ
u (1), c4 u (0), b u (4), b u (5), and
(0)
levels
and
obtained
molecular
parameters
and
coupling
c3 1 þ
u
parameters for these five levels. The P- and R-branch oscillator
strengths calculated from eigenfunctions are in very good agreement with experimental measurements of Stark et al. (1992, 2000,
2005). The present study has also obtained a quantitative account
of the quantum interference arising from interaction among the
five vibronic levels and the effect on the f-value of individual
transitions. It further provides a simple way to calculate the line
oscillator strength for rotational levels up to J 0 ¼ 30.
This work has been partially supported by NSF ATM-0131210
and the Cassini UVIS contract with the University of Colorado. It
is also supported by the National Aeronautics and Space Administration through the Planetary Atmosphere Program under
the grant to the Planetary and Space Science Division, Space Environment Technologies.
REFERENCES
Abgrall, H., & Roueff, E. 1989, A&AS, 79, 313
Kam, A. W., Lawall, J. R., Lindsay, M. D., Pipkin, F. M., Short, R. C., & Zhao,
———. 2006, A&A, 445, 361
P. 1989, Phys. Rev. A, 40, 1279
Ajello, J. M., James, G. K., & Ciocca, M. 1998, J. Phys. B, 31, 2437
Knauth, D. C., Anderson, B.-G., McCandliss, S. R., & Moos, H. W. 2004,
Ajello, J. M., James, G. K., Franklin, B. O., & Shemansky, D. E. 1989, Phys.
Nature, 429, 636
Rev. A, 40, 3524
Kosman, W., & Wallace, S. 1985, J. Chem. Phys., 82, 1385
Bielschowsky, C. E., Nascimento, M. A. C., & Hollauer, E. 1990, J. Phys. B,
Lavin, C., Velasco, A. M., Martin, I., & Bustos, E. 2004, Chem. Phys. Lett.,
23, L787
394, 114
Bishop, J., Stevens, M. H., & Feldman, P. D. 2006, Geophys. Res. Lett., in
Lawrence, G. M., Mickey, D. L., & Dressler, K. 1968, J. Chem. Phys., 48, 1989
press
Lefebvre-Brion, H., & Field, R. W. 1986, Perturbations in the Spectra of DiBroadfoot, A. L., et al. 1989, Science, 246, 1459
atomic Molecules (Orlando: Academic)
Carroll, P. K., & Collins, C. P. 1969, Canadian J. Phys., 47, 563
Levelt, P., & Ubachs, W. 1992, Chem. Phys., 163, 263
Carroll, P. K., Collins, C. P., & Yoshino, K. 1970, J. Phys. B, 3, L127
Lewis, B. R., Gibson, S. T., Sprengers, J. P., Ubachs, W., Johnsson, A., &
Carroll, P. K., & Yoshino, K. 1972, J. Phys. B, 5, 1614
Wahlstrom, C.-G. 2005a, J. Chem. Phys., 123, 236101
Carter, V. L. 1972, J. Chem. Phys., 56, 4195
Lewis, B. R., Gibson, S. T., Zhang, W., Lefebvre-Brion, H., & Robbe, J. M.
Chan, W. F., Cooper, G., Sodhi, R. N. S., & Brion, C. E. 1993, Chem. Phys.,
2005b, J. Chem. Phys., 122, 144302
170, 81
Liu, X., Shemansky, D. E., Ciocca, M., Kanik, I., & Ajello, J. M. 2005a, ApJ,
Edwards, S. A., Tchang-Brillet, W.-U. L., Roncin, J.-Y., Launay, F., & Rostas,
623, 579
F. 1995, Planet. Space Sci., 43, 67
Liu, X., Shemansky, D. E., & Hallett, J. T. 2005b, AGU Abstr. Fall, P33C-0262
Felfli, Z., Chen, Z., Bessis, D., & Msezane, A. Z. 1997, Chem. Phys., 222, 117
Meier, R. R. 1991, Space Sci. Rev., 58, 1
Geiger, J., & Schröder, B. 1969, J. Chem. Phys., 50, 7
Morrison, M. D., Bowers, C. W., Feldman, P. D., & Meier, R. R. 1990, J.
Gurtler, P., Saile, V., & Koch, E. E. 1977, Chem. Phys. Lett., 48, 245
Geophys. Res., 95, 4113
Hansson, A., & Watson, J. K. G. 2005, J. Mol. Spectrosc., 233, 169
Neugebauer, J., Baerends, E. J., & Nooijen, M. 2004, J. Chem. Phys., 121,
Haverd, V. E., Lewis, B. R., Gibson, S. T., & Stark, G. 2005, J. Chem. Phys.,
6155
123, 214304
Roncin, J.-Y., Launay, F., Bredohl, H., & Dubois, I. 1999, J. Mol. Spectrosc.,
Helm, H., & Cosby, P. C. 1989, J. Chem. Phys., 90, 4208
194, 243
Helm, H., Hazell, I., & Bjerre, N. 1993, Phys. Rev. A, 48, 2762
Roncin, J.-Y., Subtil, J.-L., & Launay, F. 1998, J. Mol. Spectrosc., 188, 128
Huffman, R. E., Tanaka, Y., & Larabee, J. C. 1963, J. Chem. Phys., 39, 910
Shaw, D. A., Holland, D. M. P., MacDonald, M. A., Hopkirk, A., Hayes, M. A.,
James, G. K., Ajello, J. M., Franklin, B., & Shemansky, D. E. 1990, J. Phys. B,
& McSweeney, S. M. 1992, Chem. Phys., 166, 379
23, 2055
Shemansky, D. E., Kanik, I., & Ajello, J. M. 1995, ApJ, 452, 480
No. 2, 2006
OSCILLATOR STRENGTHS OF N2 BANDS
Smith, P. L., Stark, G., & Yoshino, K. 2003, AAS Planet. Sci. Meet. Abs., 35,
16.07
Spelsberg, D., & Meyer, M. 2001, J. Chem. Phys., 115, 6438
Sprengers, J. P., & Ubachs, W. 2006, J. Mol. Spectrosc., 235, 176
Sprengers, J. P., Ubachs, W., & Baldwin, K. G. H. 2005, J. Chem. Phys., 122,
144301
Sprengers, J. P., Ubachs, W., Baldwin, K. G. H., Lewis, B. R., & TchangBrillet, W.-U. L. 2003, J. Chem. Phys., 119, 3160
Sprengers, J. P., Ubachs, W., Johansson, A., L’Huillier, A., Wahlstrom, C.-G.,
Lang, R., Lewis, B. R., & Gibson, S. T. 2004, J. Chem. Phys., 120, 8973
Stahel, D., Leoni, M., & Dressler, K. 1983, J. Chem. Phys., 79, 2541
Stark, G., Huber, K. P., Yoshino, K., Chan, M.-C., Matusi, T., Smith, P.-L., &
Ito, K. 2000, ApJ, 531, 321
Stark, G., Huber, K. P., Yoshino, K., Smith, P. L., & Ito, K. 2005, J. Chem.
Phys., 123, 214303
Stark, G., Smith, P. L., Hubber, K. P., Yoshino, K., Stevens, M. H., & Ito, K.
1992, J. Chem. Phys., 97, 4809
1567
Strickland, D. J., Meier, R. R., Walterscheid, R. L., Craven, J. D., Christensen,
A. B., Paxton, L. J., Morrison, D., & Crowely, G. 2004a, J. Geophys. Res.,
109, A01302
Strickland, D. J., et al. 2004b, Geophys. Res. Lett., 31, L03801
Ubachs, W., Lang, R., Velchev, I., Tchang-Brillet, W.-U. L., Johnsson, A., Li,
Z. S., Lokhngyin, V., & Wahlstrom, C. G. 2001, Chem. Phys., 270, 215
Ubachs, W., Tashiro, L., & Zare, R. N. 1989, Chem. Phys., 130, 1
Ubachs, W., Velchev, I., & de Lange, A. 2000, J. Chem. Phys., 112, 5711
van Vleck, J. H. 1951, Rev. Mod. Phys., 23, 213
Walter, C. W., Cosby, P. C., & Helm, H. 1993, J. Chem. Phys., 99, 3553
———. 1994, Phys. Rev. A, 50, 2930
———. 2000, J. Chem. Phys., 112, 4621
Watanabe, K., & Marmo, F. F. 1956, J. Chem. Phys., 25, 965
Yoshino, K., Freeman, D. E., & Tanaka, Y. 1979, J. Mol. Spectrosc., 76, 153
Yoshino, K., & Tanaka, Y. 1977, J. Mol. Spectrosc., 66, 219
Yoshino, K., Tanaka, Y., Carroll, P. K., & Mitchell, P. 1975, J. Mol. Spectrosc.,
54, 87