JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A07307, doi:10.1029/2005JA011062, 2005
Evaluation of electron impact excitation of N2
X 12+g (0) into the N+2 X 22+g (v), A 2u(v), and
B 22+u(v) states
Donald E. Shemansky and Xianming Liu
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California, USA
Received 10 February 2005; revised 24 March 2005; accepted 22 April 2005; published 26 July 2005.
[1] Published experimental integrated electric dipole photoionization oscillator strengths
(fij) have been applied to the determination of absolute electron impact excitation cross
sections of the N2 X 1S+g (0) state into the N+2 X 2S+g (v), A 2u(v), and B 2S+u (v) states. The
required relative electron impact cross section shape functions from threshold to energy
values, limited by the imposition of relativistic effects at the high end, are established in
this work by extracting accurate analytic functions from a critical review of extensive
published experimental results. The cross sections examined here are important
benchmarks used in establishing absolute values for critical rate processes in aeronomy.
The determination of the absolute cross sections using the derived fij is accomplished
through the physical relationship to a modified Born approximation analytic function that
defines the electron impact shape functions. The zero order term of the analytic function is
established in value relative to the remaining terms through shape analysis of the
experimental electron impact results. The value of fij fixes the absolute value of the zero
order term, determining the cross sections over the entire electron impact energy range. It
is argued that this methodology is the most accurate approach at this time because of the
existence of the intrinsically more accurate photometrically dermined fij. The critical
quantities in atmospheric process applications are the branching ratios of excitation into
the N+2 X 2S+g , A 2u, and B 2S+u states. The N+2 (X 2S+g , A 2u, B 2S+u ) partitioning within
the total ionization cross section has been a source of disagreement in the literature, in
spite of the fact that substantial effort has been expended in establishing acceptably
accurate values. The present work establishes accurate cross sections for these states over
the entire practical energy range required for research in atmospheric N2 physical
chemistry. The accuracy of the results obtained here are verified by consistency with the
most reliably established past experimental electron impact cross sections, in agreement
with the independently established total N+2 cross section. We provide analytic collision
strengths for the vibrational levels of the N+2 X 2S+g , A 2u, and B 2S+u states, allowing
establishment of rate coefficients for the system, at an estimated error ±8%.
Citation: Shemansky, D. E., and X. Liu (2005), Evaluation of electron impact excitation of N2 X 1S+g (0) into the N+2 X 2S+g (v), A
2
u(v), and B 2S+u (v) states, J. Geophys. Res., 110, A07307, doi:10.1029/2005JA011062.
1. Introduction
[2] The work reported here establishes accurate cross
sections for electron impact ionization-excitation of N2 X
1 +
Sg (0) molecules into the N+2 X 2S+g (v), A 2u(v), and B
2 +
Su (v) states through analysis of existing extensive published experimental data. The methodology is based on
(1) the calculation of accurate analytic electron impact
cross section shape functions based on a modified Born
approximation, and (2) placing the shape functions on an
absolute scale using electric dipole oscillator strengths
derived from photoionization cross section measurements.
It is argued here that this approach produces the most
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2005JA011062$09.00
accurate absolute cross sections from the available pool of
experimental data.
[3] The molecular nitrogen ion (N+2 ) is an important
species in the upper atmosphere of the Earth and the other
nitrogen bearing objects in the solar system such as Titan,
comets and cometary meteoroids [Abe et al., 2005]. The B
2 +
Su X 2S+g band is the lasing transition for molecular
nitrogen ion lasers where the N+2 B 2S+u state is generated by
charge transfer of N2 with He+2 produced in an intense
950 keV electron excitation of a He/N2 mixture [Littlewood
and Webb, 1981]. Both B 2S+u X 2S+g , the first negative
system, and A 2u X 2S+g , the Meinel system, are
prominent spectral features in auroral and dayglow observations. At altitudes below 160 km, the lifetime of N+2 is
very short because of its reaction with atomic oxygen to
form NO+. When sunlit aurora extends to high altitude (up
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to 7001100 km), resonant scattering of solar radiation by
N+2 via the B 2S+u X 2S+g transition produces an abnormally
bright blue color in sunlit aurorae [Bates, 1949; Hunten,
2003]. Charge exchange of N2 with O+(2D) and O+(2P) also
plays an important role in the production of N+2 (B 2S+u )
[Broadfoot and Stone, 1999]. The brightness of the
first negative (0,0) band is frequently observed in the
kiloRayleigh range [Romick et al., 1999]. In auroral precipitation, N2 is readily ionized by both primary and secondary
electrons. Photometric measurement of B 2S+u X 2S+g
emission can be utilized to estimate the energy flux of
precipitating electrons. The intensity ratios of some forbidden neutral N2 transitions to N+2 emission can, in principle,
be used to characterize the energy of auroral electrons. Rees
and Lummerzheim [1989] suggested that the average energy
of precipitating electrons can be inferred from the intensity
ratio of N2 C 3u(0) B 3g(0) to the N+2 B 2S+u (0) X 2S+g (1) emission. Gattinger et al. [1991], however,
questioned the sensitivity of the suggested intensity ratio
by showing that variation with electron energy distribution is
significantly smaller than Rees and Lummerzheim [1989]
originally suggested. An accurate excitation function of the
ionization process will clearly help resolve the issue.
[4] The electron configuration of the N2 X 1S+g state is
1s2g1s2u2s2g2s2u1p4u3s2g. Ionization by removal of valence
electrons in the X 1S+g state is typically divided into two
regions in the excitation energy scale. The first region, from
15.6 to 20 eV above N2 X 1S+g (0), consists of the three
lowest ionic states X 2S+g , A 2u and B 2S+u . All three states
have single hole configurations that are obtained by removing one electron from each of three highest occupied
orbitals of the X 1S+g state. Thus, the X 2S+g state of N+2
has an electron configuration of 1s2g1s2u2s2g2s2u1p4u3s1g, often
2
2 +
abbreviated as 3s1
g . Similarly, the A u and B Su states
1
1
have primary configurations of 1pu and 2su , respectively.
The second energy region, from 20 to 45 eV above N2 X
1 +
Sg (0), contains a number of inner-valence states, in particular, those associated with the removal of a 2sg electron.
Excitation to the inner-valence states of N+2 often, though
not always, leads to dissociation of N+2 . Nondissociative
ionization of molecular nitrogen from the X 1S+g state,
therefore, predominately takes place via the X 2S+g , A 2u
and B 2S+u states. Furthermore, excitation from N2 X 1S+g (0)
to the inner-valence ionic states requires simultaneous
change of multiple electron orbital configurations, and is
forbidden assuming no dipole properties are introduced in
configuration interaction and electron correlation. Transitions are allowed by configuration interaction with single
hole states such as the X 2S+g , A 2u and B 2S+u , or by
electron correlation with N2 X 1S+g [Hiyama and Iwata,
1993]. While N+2 has been observed in some inner-valence
2
2
states such as the D 2g, C 2S+u , 2 2g, 2S
u , Du and 2 u
[Van de Runstraat et al., 1974; Baltzer et al., 1992; Fons et
al., 1994; Yoshii et al., 1997], the cross sections are very
small. As a result, the sum of the cross sections for the
production of the X 2S+g , A 2u and B 2S+u states is often
assumed to be the total cross section for nondissociative
N 2 ! N +2 ionization. Van Zyl and Pendleton [1995]
estimated a 2% contribution by the inner-valence states
to the total N+2 production cross section at 100 eV electron
impact energy. Because of the forbidden nature of the
excitation to the higher states, the contribution of inner
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valance states is expected to decrease at higher energies.
The present paper assumes a negligible contribution of the
inner-valence states to N+2 formation.
[5] The electron impact ionization cross sections of N2,
dissociative and nondissociative, have been satisfactorily
established as a result of many experimental investigations
[Rapp and Englander-Golden, 1965; Schram et al., 1965;
Crowe and McConkey, 1973; Märk, 1975; Krishnakumar
and Srivastava, 1990; Freund et al., 1990; Straub et al.,
1996; Tian and Vidal, 1998; Lindsay and Mangan, 2003].
However, the partial ionization-excitation cross sections for
the X 2S+g , A 2u, B 2S+u states, assumed to be established
earlier, were seriously questioned by Goembel et al. [1994]
and Doering and Yang [1996, 1997], and disputed in a
thorough review by Van Zyl and Pendleton [1995]. Much of
the dispute centers on the magnitude of the B 2S+u state cross
section, particularly at 100 eV. Van Zyl and Pendleton
[1995], based on their analysis and the emission cross
section measurement of Borst and Zipf [1970], suggested
that the B 2S+u state contributes 14.5% to the total N+2 cross
section. Goembel et al. [1994] and Doering and Yang
[1996, 1997] in electron scattering measurements, obtained
much lower values of 9% – 10%. Doering and Yang [1996,
1997] argued that the B 2S+u Borst and Zipf [1970] emission
cross section, extensively considered to be a benchmark
quantity, is about 18% too large. Clearly, an 18% reduction
will have a profound effect on the electron impact cross
sections of many other species whose values are established
relative to the Borst and Zipf [1970] measurement.
[6] The present work utilizes the measured total electron
impact cross section for the N2 ! N+2 process and photoionization oscillator strengths of the X 2S+g X 1S+g ,
A 2u X 1S+g and B 2S+u X 1S+g band systems to
obtain the cross sections for the X 2S+g , A 2u and B 2S+u
states of N+2 . The ionization oscillator strengths are derived
from photoionization experimental measurements of
Samson et al. [1977, 1987], Lee [1977], Plummer et al.
[1977], Stolte et al. [1998], and photoionization data compiled by Conway [1988]. The photoionization measurements are intrinsically more accurate on an absolute scale
than the less easily controlled electron beam experiments.
The electron impact cross sections are then established by
fitting the shape of selected measured electron impact cross
sections with an analytic modified Born approximation
formulation incorporating the ionization oscillator strengths.
In other words, we rely on the measured shapes of the
selected electron impact experiments, and the absolute
values of the oscillator strengths to establish the ionization-excitation cross sections of the X 2S+g , A 2u and B
2 +
Su states. The present result indicates that the emission
cross section of Borst and Zipf [1970] is too high. The
derived cross sections at 100 eV are closer to those of
Goembel et al. [1994] and Doering and Yang [1996, 1997]
than to those of Van Zyl and Pendleton [1995].
[7] In addition to the total electron impact ionization
measurements by Rapp and Englander-Golden [1965],
Krishnakumar and Srivastava [1990], Freund et al.
[1990], Straub et al. [1996], and Tian and Vidal [1998],
many investigations of the partial nondissociative ionization
cross sections have been carried out. Most experimental
studies rely on the measurement of emission cross sections
of the A 2u X 2S+g and B 2S+u X 2S+g transitions,
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SHEMANSKY AND LIU: N2 ! N+2
although other techniques such as electron energy loss
[Wight et al., 1976; Hamnett et al., 1976], electron-electron
coincidence [Goembel et al., 1994; Doering and Yang,
1996, 1997; Flammini et al., 2000; Hussey and Murray,
2003] and laser-induced fluorescence (LIF) [Abramzon et
al., 1999a, 1999b] have been utilized. The B 2S+u state,
having a lifetime of 60 80 ns, has been extensively
measured. McConkey and Latimer [1965] measured excitation functions of the (0,0), (0,1) and (0,2) bands of B 2S+u X 2S+g from threshold to 300 eV and obtained absolute
emission cross sections by reference to the helium emission
cross section. Subsequently, McConkey et al. [1967] extended the measurement to 2000 eV using both standard
quartz iodine and tungsten lamps. A graphic error and
contribution from overlapping second positive band transitions was later corrected by McConkey et al. [1971]. Borst
and Zipf [1970] also measured the absolute (0,0) band
emission cross section using a standard tungsten
lamp. Other emission cross section measurements of the
B 2S+u X 2S+g band include Aarts et al. [1968], Srivastava
and Mirza [1968a, 1968b, 1969], Stanton and St. John
[1969], and Shaw and Campos [1983]. The A 2u state,
having a lifetime of several ms, is more difficult to measure.
Experimental investigations of the A 2u X 2S+g emission
cross section have been reported by Srivastava and Mirza
[1968a], Stanton and St. John [1969], Shemansky and
Broadfoot [1971a, 1971b], Holland and Maier [1972,
1973], Pendleton and Weaver [1973], Skubenich and Zapesochnyy [1981], and Piper et al. [1986]. Other electron
impact studies of nondissociative ionization of N2 include
Van de Runstraat et al. [1974] for the C 2S+u state, and Fons
et al. [1994] for the D 2g state.
[8] The photoionization of molecular nitrogen has also
been extensively investigated and accurate cross sections
have been established. Reilhac and Damany [1977] and
Cole and Dexter [1978] measured photoabsorption and
photoionization cross sections of N2 between 50 and
500 Å. Samson et al. [1977], Plummer et al. [1977],
Woodruff and Marr [1977], Lee [1977], and Peatman et
al. [1978] reported both total and partial photoionization
cross sections. Samson et al. [1987] and Samson and Yin
[1989] subsequently published accurate ionization cross
sections from threshold to 107 eV. Stolte et al. [1998] have
recently extended the measurement to 800 eV. Results of
many experimental investigations of N2 prior to 1998 have
been summarized by Gallagher et al. [1988], and in the
book by Berkowitz [2002]. Recent photoionization investigations include Ehresmann et al. [2000] and Nicholas et
al. [2003] on dissociative ionization, Marquette et al. [1999]
and Kugeler et al. [2004] on the coupling between valenceionized and core-excited states. The break-down of the
Frank-Condon approximation in rotationally and vibrationally resolved partial ionization cross sections, caused by
non-adiabatic coupling and autoionization, has also been
examined by Poliakoff et al. [1995], Rao et al. [1996],
Öhrwall et al. [1998, 1999], Riu et al. [2001], Rathbone et
al. [2004], and Bolognesi et al. [2004].
[9] In parallel with extensive experimental investigations,
many theoretical calculations on the photoionization of
molecular nitrogen have been carried out. Gallagher et al.
[1988] have summarized the results of some early calculations. Recent studies on the photoionization cross section
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of N2 include a multi-channel quantum defect calculation by
Lefebvre-Brion and Raseev [2003], the multi-configuration
linear response method by Carravetta et al. [1993], randomphase-approximation by Cacelli et al. [1998], Semenov et
al. [2000], and Montuoro and Moccia [2003], and the timedependent density function method by Stener and Decleva
[2000]. Very few, if any, ab initio calculations have been
carried out on the ionization of N2 by electron impact. Even
under the first Born approximation, the computation of
generalized oscillator strengths for molecules, with, perhaps, the exception of H2, is an arduous task. Many
theoretical investigations have been concerned with semiempirical and semi-classical formulae. Among these are the
binary-encounter-dipole (BED) and binary-encounter-Bethe
(BEB) models by Kim and Rudd [1994] and Hwang et al.
[1996], improved BED models proposed by Saksena et al.
[1997], Khare et al. [1999], and Huo [2001], and the
Deutsch-Märk model by Deutsch et al. [2000] and Probst
et al. [2001]. While these models are typically capable of
reproducing experimental peak cross sections within 5% –
15%, they are difficult to apply to partial ionization because
binary-encounter models are based on collisions between a
free and a bound electron. Vibrational and electronic excitation of N+2 by electrons have been quantum mechanically
calculated by Orel et al. [1990], Naggy et al. [1999], and
Naggy [2003].
[10] The investigation of N+2 spectroscopic structure was
primarily carried out in the 1930s and 1950s [Lofthus and
Krupenie, 1977]. Klynning and Pages [1982] performed
extensive rotational analysis of transitions among X 2S+g ,
A 2u, B 2S+u , and C 2S+u states and obtained a large set of
molecular constants for over 70 vibrational bands. Their
analysis showed that many vibrational levels are coupled,
typically at high rotational levels. More recent studies with
laser techniques can be found in the articles by Miller et al.
[1984], Boudjarane et al. [1996], and Jungen et al. [2003].
2. Theory
[11] For dipole-allowed ro-vibrational excitation from
electronic state i to state j, the cross section, s, based on
the modified Born approximation is given by [Shemansky et
al., 1985a, 1988b; Liu et al., 2003]
Ry Ry C0 1
s vi ; vj ; Ji ; Jj
1
¼ 4f vi ; vj ; Ji ; Jj
2
Eij E C7 X 2 X 3
pa0
4
X Cm
C5 C6 1
þ
ð X 1Þ expðmC8 X Þþ þ
þ lnð X Þ
C7
C7 C7 X
m¼1
ð1Þ
C7 ¼
4pa20 ð2Ji þ 1ÞRy f vi ; vj ; Ji ; Jj
Eij
ð2Þ
where a0 and Ry are Bohr radius and Rydberg constant,
f (vi, vj; Ji, Jj) is the (integrated) absorption oscillator
strength, Eij is the transition energy from (vi, Ji) to (vj, Jj), E
is the impact energy, and X = E/Eij is the dimensionless
electron energy. The collision strength coefficients Cm/C7
(m = 0 – 6) and C8 can be determined by fitting the
experimentally measured relative excitation function. If the
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SHEMANSKY AND LIU: N2 ! N+2
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absolute excitation function of is available, the oscillator
strength can also be determined. Cm/C7 (m = 0 – 6) and C8
reflect the atomic or molecular electronic properties and are,
usually, assumed to be dependent on electronic state but
independent of rotation and vibration.
[12] In addition to the direct electron excitation described
by equation (1), indirect processes also contribute to the
ionization of N2 [Fox, 1951]. The indirect ionization can
take place via Feshbach resonance and autoionization near
the threshold, and shape resonance and channel coupling
beyond threshold [Riu et al., 2001; Bolognesi et al., 2004].
The largest contribution of indirect ionization occurs near
threshold. For this reason, indirect ionization is often
loosely referred to as resonance ionization. The presence
of the resonance ionization is apparent only when the
energy resolution of the electron beam is high. Resonance
excitation can be conveniently represented by
h
i
sres
Ry
¼ C1
exp C22 ð E E0 Þ2
2
E
pa0
ð3Þ
where C1 and C2 are parameters and E0 is the center energy.
sres vanishes if E is lower than the threshold energy of the
X 2S+g state.
[13] In electron impact ionization cross section measurements, the ro-vibrational states of N+2 are not distinguished.
Equation (1) needs to be summed over the vj, Ji, and Jj,
which can be performed using the relation
X f vi ; vj ¼
f vi ; vj ; Ji ; Jj
ð4Þ
J
q vi ; vj
f vi ; vj ¼ fij P vj q vi ; vj
ð5Þ
where fij is oscillator strength of the electronic band system
i ! j, and q(vi, vj) is the Franck-Condon factor, given by
Gilmore et al. [1992] and updated by Laher and Gilmore
[1999]. Since the present study deals with nondissociative
ionization exclusively, vj of equation (5) refers to discrete
vibrational levels only. Equation (5) also implicitly assumes
that the dependence of transition moment on internuclear
distance is negligible. For excitation from vi = 0 of the N2 X
1 +
Sg state to the X 2S+g , A 2u and B 2S+u states of N+2 , the
contribution of continuum levels is negligible and the
summation of q(vi = 0, vj) over vj is very close to 1. So,
equation (5) can be re-written as
f vi ; vj ¼ fij q vi ; vj
ð6Þ
[14] The electronic oscillator strength, fij, is related to the
partial photoionization cross section, sph
ij [Berkowitz, 2002]
mc
phe2
Z
1
sph ði; jÞd
Z 1
¼ 9:1107 1015
sph ði; jÞd
fij ¼
0
ð7Þ
0
where the cross section, sph, is in cm2 and kinetic energy of
photoelectron, , is in eV. If the ionization branching ratio
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ph
(sph
ij /s ) for i ! j is known, the partial ionization cross
section, sph
ij can be obtained from the measured total
ionization cross section, sph. The oscillator strength
corresponding to sph for the production of N+2 is defined
here as fX.
3. Analysis and Results
3.1. Photoionization Oscillator Strengths
[15] This section describes the source of the derived
values of the oscillator strengths fX, fXX, fXB, and fXA.
3.1.1. fX
[16] The combined meaurements obtained by Samson et
al. [1987] and Stolte et al. [1998] allow the calculation of fX.
Samson et al. [1987] measured both non-dissociative and
dissociative photoionization cross sections of N2 from
threshold to 115 Å. The measurements were extended down
to 15.5 Å by Stolte et al. [1998]. For photon wavelengths
shortward of 15.5 Å, Stolte et al. [1998] provided power
law expressions for the cross section. Numerical integration
of photoionization cross sections of Samson et al. [1987]
and Stolte et al. [1998] via equation (7) produces oscillator
strengths of (fX =) 7.066 ± 0.212, 4.563 ± 0.228, and
0.8586 ± 0.043 for the production of N+2 , N+ + N2+
2 and
N2+, respectively. The uncertainty of the oscillator strength
is assumed to entirely arise from experimental error, which
is 3% for nondissociative ionization and 5% for dissociative
ionization. As discussed in section 4.1, these errors are
likely to be upper limits. The derived total oscillator
strength, from the first ionization potential (15.575 eV) to
1, 12.488 ± 0.483, is consistent with the value of 12.446
obtained by Berkowitz [2002], who used a number of earlier
experimental measurements with a more sophisticated polynomial integration method and carefully treated the structures at and near the K-edge. Conway [1988] listed the
ionization cross section of N2 based on experimental data of
Samson et al. [1987] from threshold to 130 Å and a power
law expression for l < 130 Å. Numerical integration of his
data yields an oscillator strength of fX = 7.187 for the
production of N+2 , which agrees with the present derived
fX = 7.066 ± 0.212 within experimental error. The small
difference (2%) is primarily attributed to a slightly different power law equation utilized by Conway [1988].
3.1.2. fXX, fXB, fXA
[17] Woodruff and Marr [1977], Plummer et al. [1977],
Samson et al. [1977], and Lee [1977] have all reported
partial ionization cross sections of N+2 and ionization
branching-ratios. The data by Plummer et al. [1977] cover
partial ionization cross sections and branching-ratios for the
X 2S+g , A 2u and B 2S+u states from 17 to 39 eV, while the
measurement of Woodruff and Marr [1977] is from 16 to
37 eV. Samson et al. [1977] also tabulated the partial cross
sections and branching-ratios in the 16.74 – 42.6 eV region.
While the work of Lee [1977] dealt with the B 2S+u state
exclusively, the cross section and ionization yield was
obtained over a wider energy range, from 20.7 eV (600 Å)
to 70.9 eV (175 Å), than the other three measurements. For
this reason, the B 2S+u X 2S+g ionization oscillator strength
is most reliably obtained from the Lee [1977] measurement.
The Lee [1977] results must be extended downward to
threshold from 20.7 eV to 18.8 eV, and upward from
70.9 eV to 1. Using sph from Samson et al. [1987] and
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SHEMANSKY AND LIU: N2 ! N+2
ph
Stolte et al. [1998] (see 3.1.1) we find that sph
is
XB/s
essentially constant above 47 eV. (It is important to note
that Lee [1977] defined his production yield, h, as the ratio
of the B 2S+u cross section to the total absorption cross
section. The ionization branching ratio, in contrast, is the
ratio of partial cross section to the total N+2 production cross
ph
section.) Using the value of sph
between 20.7 eV and
XB/s
18.8 eV from Samson et al. [1977], and a constant value of
12.96% above 70.9 eV the value fXB = 0.7636 is obtained.
[18] The oscillator strengths fXX, fXA, fXB, can also be
derived from the partial photoionization cross sections
provided by Conway [1988], who used the ionization
branching ratios of Samson et al. [1977] from threshold to
460 Å, Plummer et al. [1977] from 460 to 318 Å and
Hamnett et al. [1976] from 310 to 248 Å. Shortward of
248 Å, Conway [1988] assumed that the ratios are invariant
with excitation energy. Numerical integration of his partial
cross sections leads to partial oscillator strengths of fXX =
2.761, fXA = 3.642, and fXB = 0.7844. When the same set of
branching ratios and procedure are applied to N2 ! N+2
photoionization cross sections of Samson et al. [1987] and
Stolte et al. [1998], the values fXX = 2.714, fXA = 3.581, and
fXB = 0.7712 are obtained. Note that the derived fXB =
0.7712, differs less than 1% from that (0.7636) determined
from the Lee [1977] data.
3.2. B 22+u X 12+g Electron Impact Cross Section
[19] An important question is whether the B 2S+u X 2S+g
ionization oscillator strength derived from photoionization
data is consistent with that determined from electron impact
data. The (0,0) band emission cross section of the B 2S+u X 2S+g band system reported by Borst and Zipf [1970] has
been used as a benchmark cross section. Borst and Zipf
[1970] measured the absolute emission cross section for the
(0,0) band of the B 2S+u X 2S+g system from 19 to 3000
eV. They estimated a ±15% maximum error in their tabulated cross section and suggested a probable error of less
than ±10%. Borst and Zipf [1970] found that their measured
data from 200 to 3000 eV could be well represented by the
Bethe-Born relation. This is an indication that higher order
terms in equation (1) could be neglected in the higher
energy region relative to the zero order term containing
the coefficients C5 and C7 (equation 2). C7 is directly
determined by fij.
[20] The present approach is to derive a precise analytic
representation of the excitation function for the B 2S+u X 2S+g band system and achieve an accurate determination
of oscillator strength. While Borst and Zipf [1970] analyzed
their measured data between 200 and 3000 eV, the present
work fits the observed emission cross section from 19 to
3000 eV using the data as given in their Table 1. The second
column of Table 1 displays the value of collision strength
parameters obtained by a nonlinear least-squares fit of the
experimental data to equation (1). Cross sections in the high
energy region are given slightly higher statistical weights in
this calculation because of the higher data density at low
energy. As Table 1 shows, the derived apparent oscillator
strength for the (0,0) band is 0.5673 with a standard error of
0.0038. Using the inverse of the emission branching ratio
for the (0,0) band, 1.41, and the Franck-Condon factor for
the (0,0) band of the B 2S+u X 1S+g band system, 0.883,
both calculated by Gilmore et al. [1992] and Laher and
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Table 1. Collision Strength Parameters for N2 X 1Sg+ ! N2+ X
2 +
Sg , A 2u, and B 2S+u Band Systems
Parameters
B 2S+u X 1S+g
X 2S+g X 1S+g
A 2u X 1S+g
C0/C7
C1/C7
C2/C7
C3/C7
C4/C7
C5/C7
C6/C7
C8
fa
fc
0.17387783
0.17164000
0.13548065
1.3224952
2.0493295
0.84334411
0.84334411
0.34614314
0.56731242b
0.7636d
0.10280889
0.70941047
0.11572412
0.23848436
1.9525626
1.4209587
1.4209587
0.76349530
0.61639849
0.27160620
0.12738765
0.26736950
1.0845986
2.0121893
2.0121893
0.15338992
2.6968e
3.5603e
a
Apparent oscillator strength for the (0,0) band emission obtained from
data of Borst and Zipf [1970]. The standard error is 0.38386767 102.
b
The corresponding electronic band system oscillator strength is 0.9040.
c
Electronic band oscillator strength.
d
Fixed to the photoionization value.
e
The corresponding photoionization oscillator strengths for the X 2Sg+
and A 2u states are 2.714 and 3.581, respectively. The total nondissociative ionization oscillator strength derived from the electron impact
data is 7.0208 ± 0.0759 (one standard error).
Gilmore [1999], the ionization oscillator strength of the
B 2S+u X 2S+g band system is 0.9040 with a standard error
of 0.0061.
[21] The value, 0.9040, is about 18% larger than the value
(0.7636) obtained from photoionization measurements. The
discussion of the significance and implication of this difference will be deferred to section 4. The emission cross
section of the (0,0) band at 100 eV corresponding to an
oscillator strength fXB = 0.7636 is sXB(100 eV) = 14.7 1018 cm2. This sXB value is consistent with 14.8 1018 cm2 recommended by Doering and Yang [1996]
and 15.0 1018 cm2 reported by McConkey et al. [1971].
3.3. N2 ! N+2 Electron Impact Cross Section
[22] At this stage, it is important to compare the oscillator
strength fX obtained from the analysis of the electron impact
cross section with that obtained from the photoionization
data. As discussed in section 1, the N2 ! N+2 (sX) cross
section by electron impact has been measured many times
since the pioneer work of Rapp and Englander-Golden
[1965]. The Krishnakumar and Srivastava [1990] and
Straub et al. [1996] measurements extend from threshold
to 1000 eV, while those of Freund et al. [1990] and Tian
and Vidal [1998] are limited in range to 200 and 600 eV,
respectively. The original cross sections reported by Straub
et al. [1996] have been revised down 1% –10% by Lindsay
and Mangan [2003]. In general, the Krishnakumar and
Srivastava [1990] cross section in 30 –400 eV region is
14% higher than the revised value of Lindsay and
Mangan [2003]. Between 600 and 1000 eV, the difference
becomes more significant, with the Krishnakumar and
Srivastava [1990] values 22%– 27% larger than those of
Lindsay and Mangan [2003]. Krishnakumar and Srivastava
[1990] utilized the absolute ionization cross section of
helium and the relative flow method to establish the
absolute value for sX and reported an error of less than
±8%. Straub et al. [1996] directly measured the absolute
partial cross sections by extracting, mass analyzing and
counting ions formed along a known path length. Stebbings
and Lindsay [2001] found that the sum of their partial cross
5 of 14
A07307
SHEMANSKY AND LIU: N2 ! N+2
sections to be fully consistent with the total charge production cross section of Rapp and Englander-Golden [1965].
The error in their data, according to Lindsay and Mangan
[2003], is less than ± 5%. Thus, it can be argued that the
method of Straub et al. [1996] is intrinsically more accurate
than that of Krishnakumar and Srivastava [1990]. The N+2
cross section of Tian and Vidal [1998] lies between those of
Krishnakumar and Srivastava [1990] and Lindsay and
Mangan [2003]. The cross sections of Freund et al.
[1990], measured from threshold to 200 eV, are slightly
lower than those of Straub et al. [1996] but agree well with
those recommended by Lindsay and Mangan [2003].
Itikawa et al. [1986] also obtained sX by subtracting the
dissociative ionization cross section from the total cross
section of Rapp and Englander-Golden [1965]. In any case,
the cross section of Krishnakumar and Srivastava [1990] is
higher than most other values while that of Lindsay and
Mangan is lower than most other reported values.
[23] To reliably derive the oscillator strength, the cross
section at high energy is required. Only the direct measurements of Krishnakumar and Srivastava [1990] and Lindsay
and Mangan [2003] reach 1000 eV. The latter cross section
is accepted here as the most accurate. Liu and Shemansky
[2004] have recently shown that the electron impact ionization cross section of H2 by Lindsay and Mangan [2003]
fully reproduces the H2 photoionization oscillator strength
while the Krishnakumar and Srivastava [1994] H2 cross
section, which is 5% – 20% greater than that of Lindsay and
Mangan [2003], produces an 11% higher oscillator
strength value. The present approach is to combine the
cross sections of Rapp and Englander-Golden [1965] from
threshold to 24 eV with those of Lindsay and Mangan
[2003] between 30 and 1000 eV. While the experiment of
Rapp and Englander-Golden [1965] measured the total
ionization cross section of N2, the first dissociation limit
of N+2 is 24.29 eV [Nicholas et al., 2003]. Two assumptions
have been made to make the analysis more tractable. First,
the shape function (i.e. Ck/C7, k = 1 – 6, and C8) for B 2S+u is
assumed to be identical to those listed in the second column
of Table 1, although the oscillator strength is a free
parameter. Second, the shape function of the A 2u state
is obtainable from the (2,0) band emission cross sections of
Holland and Maier [1972, 1973], which cover from threshold to 200 eV. A calibration error in the original data of
Holland and Maier [1972] has been corrected subsequently
[Holland and Maier, 1973]. While the Holland and Maier
[1972, 1973] emission cross section is higher than many
other measurements [Van Zyl and Pendleton, 1995], it is the
relative values that are required for the establishment of the
A 2u X 1S+g shape function. Furthermore, the (2,0) band
shape of Holland and Maier [1972, 1973] tracks well with
that of Shemansky and Broadfoot [1971b] from threshold to
70 eV. The former data set is used because it covers wider
energy range (from threshold to 200 eV) than the latter
(from threshold to 95 eV).
[24] The third and fourth columns of Table 1 list collision
strength parameters for both X 2S+g X 1S+g and A 2u X 1S+g band systems. These parameters are obtained by
fixing the oscillator strength fXB to 0.7636. The oscillator
strengths fXX and fXA show a large correlation and can not
be independently determined even though their sum can be
obtained reliably. The final values of the oscillator strength
A07307
and collision strength parameters are obtained by setting
fXX/(fXX + fXA) = 0.431, the corresponding photoionization
oscillator strength ratio. The Franck-Condon factors and
vibrational energy levels tabulated by Laher and Gilmore
[1999] are used to derive the parameters. The total
ionization oscillator strength, fX = 7.021, derived from
the electron impact data is in agreement with the photoionization oscillator strength, 7.066, within one standard
error (0.076). The partial electron impact oscillator
strengths, fXX = 2.697 and fXA = 3.560, also agree with
photoionization values, 2.714 and 3.581, obtained from
data compiled by Conway [1988] and scaled to the present
total photoionization oscillator strength. The present partial
oscillator strengths are also close to the calculated values
of 2.251 for X 2S+g , 3.464 for A 2u and 0.739 for B 2S+u
state, obtained by Huo [2001].
[25] Table 2 compares observed and calculated cross
sections. It also lists the calculated partial cross sections
of the X 2S+g , A 2u and B 2S+u states up to 15 keV.
[26] We note that the total oscillator strength, fX, determined from electron impact data is not very sensitive to the
assumed fXX/(fXX + fXA) ratio. If the assumed ratio rises from
0.431 to 0.45, fX decreases from 7.021 to 6.954. Likewise, if
it decreases to 0.41, fX rises to 7.096. In both cases, the
change in total oscillator strength is within one standard
error. Obviously, the cross sections of the X 2S+g and A 2u
states have significant dependence on the assumed oscillator
strength ratio.
4. Discussion
[27] Collision strength parameters and partial oscillator
strengths listed in Table 1 and equations (1), (4) and (6)
establish the electron impact ionization-excitation cross
section of N2 X 1S+g (0) into the N+2 X 2S+g (v), A 2u(v),
and B 2S+u (v) states, and allow accurate calculation of rates
into specific vibrational levels of the N+2 states from threshold to any energy within non-relativistic limit. Rate coefficients for thermalized electrons can therefore be derived
directly from the collision strength coefficients [Shemansky
et al., 1985b]. The absolute magnitude of the cross sections
is determined from calculations of the oscillator strengths
for the transitions from published photoionization cross
sections, as discussed in section 3. In this section, the
accuracy of the derived oscillator strengths and partial cross
sections are assessed and the derived partial cross sections
are compared to various experimental values at 100 eV.
4.1. Accuracies of fX, fXX, fXA, fXB, and Partial Cross
Sections
[28] The oscillator strength fX derived from electron
impact ionization cross section is 7.021 with a standard
error of 0.076. The uncertainty in the value of fX from
photoionization is expected to be less than 3%. The stated
absolute experimental error in the nondissociative ionization
cross section of Samson et al. [1987] is 3%. The experimental error of Stolte et al. [1998] is presumably similar to
or smaller than that of Samson et al. [1987]. Since the
oscillator strength is an integration of the cross section over
a wide energy range, a substantial portion of the random
error is expected to cancel out. The uncertainty, ±0.212, for
the nondissociative photoionization fX (7.066) is, therefore,
6 of 14
SHEMANSKY AND LIU: N2 ! N+2
A07307
Table 2. Nondissociative Electron Impact Ionization Cross
Section of N2a
E(eV)
Obs.b
Model
X 2S+c
g
A 2uc
B 2S+c
u
16.5
17
17.5
18
18.5
19
19.5
20
20.5
21
21.5
22
22.5
23
23.5
24
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
110
120
140
160
180
200
225
250
275
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1500
2000
2500
3000
4000
5000
6000
7000
8000
9000
10000
15000
4.66
7.13
9.85
12.9
16.4
19.9
23
27
30.8
34.4
38
41.8
45.5
49.2
52.8
56.5
92.9
116
137
152
160
166
172
174
178
180
181
182
183
185
185
183
181
178
172
167
161
155
148
141
137
128
120
111
105
99.8
94.3
88
84.4
79.6
76.5
73.8
71.9
69.8
67.6
2.55
4.92
9.06
13.33
17.22
20.93
24.60
28.13
31.57
34.95
38.30
41.64
44.98
48.31
51.64
54.97
92.66
117.93
136.81
150.38
160.00
166.82
171.69
175.21
177.79
179.68
181.08
182.09
182.79
183.22
183.44
183.28
182.47
179.29
174.62
169.07
163.10
155.55
148.26
141.43
135.14
124.22
115.23
107.79
101.54
96.21
91.58
87.50
83.87
80.60
77.62
74.91
72.40
70.09
67.95
52.55
43.30
37.07
32.54
26.37
22.32
19.43
17.26
15.56
14.19
13.06
9.45
2.55
4.17
5.89
7.71
9.60
11.54
13.51
15.51
17.52
19.53
21.53
23.52
25.48
27.41
29.31
31.17
49.78
60.12
66.83
71.06
73.68
75.30
76.30
76.93
77.31
77.55
77.68
77.74
77.74
77.69
77.59
77.27
76.79
75.47
73.78
71.88
69.89
67.38
64.93
62.60
60.39
56.38
52.86
49.78
47.06
44.65
42.50
40.57
38.82
37.24
35.79
34.47
33.25
32.13
31.08
23.71
19.38
16.50
14.43
11.63
9.80
8.51
7.54
6.78
6.18
5.68
4.09
0.00
0.76
3.16
5.62
7.63
9.23
10.57
11.71
12.75
13.71
14.65
15.58
16.52
17.48
18.47
19.48
33.26
44.52
53.88
61.15
66.65
70.77
73.88
76.23
78.02
79.40
80.47
81.29
81.90
82.34
82.64
82.86
82.65
81.22
78.83
75.84
72.57
68.40
64.40
60.70
57.37
51.79
47.46
44.09
41.41
39.22
37.40
35.84
34.47
33.25
32.14
31.13
30.20
29.34
28.53
22.58
18.85
16.28
14.38
11.75
10.01
8.75
7.80
7.05
6.44
5.94
4.33
0.00
0.00
0.00
0.00
0.00
0.16
0.53
0.91
1.30
1.70
2.12
2.55
2.98
3.42
3.87
4.32
9.61
13.28
16.10
18.18
19.67
20.75
21.51
22.06
22.45
22.73
22.93
23.06
23.15
23.20
23.21
23.16
23.03
22.60
22.01
21.34
20.65
19.77
18.93
18.13
17.38
16.05
14.90
13.92
13.07
12.33
11.67
11.09
10.58
10.11
9.69
9.30
8.95
8.63
8.33
6.26
5.07
4.29
3.74
2.99
2.51
2.17
1.92
1.72
1.57
1.44
1.03
A07307
an upper limit. Clearly, the fX derived from electron impact
cross sections, 7.021, agrees with the photoionization fX
well within in the experimental uncertainty. It also agrees
with that derived from photoionization cross sections of
Conway [1988] (7.187) within 2.2 standard errors.
[29] The error in partial oscillator strength depends on the
errors of the nondissociative photoionization cross section
of N2 and the ionization branching ratio. There are two
sources of error in the ionization branching ratio. The first is
uncertainty in experimental measurement. The second is
the lack of branching ratio measurements at high energy
(>71 eV for the B 2S+u state and >47 eV for X 2S+g and A
2
u states) which compels the assumption that the branching ratios remain constant in the high energy region. Based
on Samson et al. [1977], the relative error in ionization
branching ratio of the B 2S+u state is estimated to be <7.4%,
and those of the the X 2S+g and A 2u states are <3.7%. The
error arising from the assumption that the branching ratios
remain constant with energy is probably small. Both measurements of Lee [1977] and Samson et al. [1987] show that
the B 2S+u state ionization branching ratios are nearly a
constant, with values of 12.70, 12.81, 12.98, and 12.96%, at
240, 220, 200 and 180 Å, respectively. Moreover, the
contribution of photoionization in the high energy region
to the oscillator strength decreases rapidly with the energy.
Photoionization with energy >73 eV contributes <20% of
fXB. It is, therefore, reasonable to assume that the error in the
high energy region is the same as that in the measured
regions. Taken together the <3% error in nondissociative
ionization cross section andp<7.4%
inffi branching ratio, the
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
error in fXB can be given as 32 þ 7:42 , or 8%. Both X 2S+g
and A 2u branching ratios show some variations with
photon excitation energy, primarily due to Feshbach resonances near the threshold and shape resonance and channel
coupling beyond threshold. Additionally, their branching
ratios were frozen at much lower energy (47 eV). If 4% of
additional error is allowed for both factors, the overall error
for fXX and fXA, on quadrature basis, is 6.2%. fXB obtained
from the measurement of Lee [1977] and from the data of
Conway [1988] differ less than 3%. Finally, 6.2% for fXX
and fXA and 8% for fXB are probably the upper error limits
because of the cancellation of the random error in the
integration.
[30] The relative error for the present X 2S+g , A 2u and B
2 +
Su electron impact cross sections above 21 eV is estimated
to be 8%. The error in the B 2S+u state is assumed to arise
solely from the uncertainty in the partial oscillator strength.
In other words, the error in the shape function of Borst and
Zipf [1970] is assumed to be negligible. The assumption is
based on the fact that the measurement was carried out to
3000 eV and the shape shows expected dipole-allowed
behavior. As noted, the relative error in the X 2S+g and
A 2u partial oscillator strengths is 6.2%. If 5% error is
given to uncertainty in the derived shape functions of the
Notes to Table 2:
Units are in 1018 cm2.
Observed cross sections from 16.5 to 24 eV are from Rapp and
Englander-Golden [1965], while those from 30 to 1000 eV are from
Lindsay and Mangan [2003].
c
Model partial cross sections. The estimated uncertainty is ±8%.
Resonance excitation for the X 2S+g state, important only below 19 eV,
has not been considered (see text).
7 of 14
a
b
A07307
SHEMANSKY AND LIU: N2 ! N+2
X 2S+g and A 2u state (see section 4.3), the overall error for
the X 2S+g and A 2u cross section, again in quadrature, is
8%.
[31] It should be stated that the expected error in the total
model N+2 cross section beyond 21 eV is significantly
smaller than 8%. Table 2 shows that differences between
the model cross sections and those of Lindsay and Mangan
[2003] are less than 4%. The absolute experimental error,
given by Lindsay and Mangan [2003], is <5%. Since the
error in the N+2 cross section in the high energy region is
primarily determined by the error in fX, the error in the
model N+2 cross section above 1000 eV is 3% 5%.
4.2. B 22+u X 12+g Cross Section
[32] The value of fXB, 0.9040, derived from the nonlinear
least-squares analysis of the Borst and Zipf [1970] measurement, and the Franck-Condon factor and branching
ratio of Gilmore et al. [1992] and Laher and Gilmore
[1999], is 18.4% higher than the counterpart determined
from the photoionization cross section. Since the fXB is
obtained by fitting all the data points between 19 and
3000 eV, the uncertainty in the fXB should be described by
the probable error (<±10%), as given by Borst and Zipf
[1970]. The photoionization oscillator strength, 0.7636, is
outside of the error range for the electron-impact oscillator
strength 0.9040 ± 0.0904. However, if the upper limit of the
8% error in photoionization oscillator strength is also
considered, its value, 0.7636 ± 0.0611, agrees with the
value of Borst and Zipf [1970] within the aggregated error
margins of the two data sets. However, the fact that the
former value is 18.4% lower than the latter has significant
implications because the Borst and Zipf [1970] cross section
has been considered a benchmark.
[33] Table 3 summarizes (0,0) band emission cross sections for the B 2S+u X 2S+g band system at 100 eV. The
cross section of Aarts et al. [1968] has been found to be
16% too high in a subsequent re-measurement by Aarts
[1970]. The revised Aarts [1970] value at 100 eV is 17.8 1018 cm2. If the Aarts et al. [1968] value is excluded, the
average in Table 3 is 15.3 1018 cm2 with a standard
deviation of 1.8 1018. The Borst and Zipf [1970] cross
section has the second highest value after that of Aarts
[1970]. The present cross section, obtained by setting fXB to
the photoionization oscillator strength while retaining the
Borst and Zipf [1970] shape, is very close to that of
McConkey et al. [1971] and the average value. Cross
sections reported by Srivastava and Mirza [1968a] and
Shaw and Campos [1983] are the peak values of the (0,0)
band emission, and are treated as if they were 100 eV values
since the maximum occurs near 100 eV. The difference
between Doering and Yang [1996] and Doering and Yang
[1997] is presumably due to a small difference in branching
ratios and the use of different total nondissociative ionization cross sections.
[34] It is difficult to explain the wide range of cross
section values determined by different experiments.
The work by Doering and Yang [1996, 1997] utilized the
electron-electron coincidence technique and measured the
branching ratios of the X 2S+g , A 2u and B 2S+u state. The
absolute cross section for each state was obtained from the
product of branching ratio and the total N+2 cross section. All
other measurements employed the optical detection method
A07307
Table 3. Emission Cross Sections for the (0,0) Band of N+2 B
2 +
Su X 2S+g Transition at 100 eV (1018 cm2)
References
Cross Section
McConkey and Latimer [1965]
Srivastava and Mirza [1968a]
Aarts et al. [1968]
Aarts [1970]
Stanton and St. John [1969]
Borst and Zipf [1970]
McConkey et al. [1971]
Skubenich and Zapesochnyy [1981]
Shaw and Campos [1983]
Doering and Yang [1996]
Doering and Yang [1997]
This work
Average
14.7
16.8 ± 3.7a
21.2 ± 2.1
17.8 ± 1.8b
15.6 ± 2.8c
17.4 ± 1.7
15.0 ± 0.9d
16.7
15.4 ± 1.5a
12.5 ± 2.5
12.1 ± 2.1
14.7 ± 1.2e
15.3 ± 1.8f
a
Maximum cross section.
Error limit is from Aarts et al. [1968].
c
Value at 120 eV. If the shape of Borst and Zipf [1970] were used,
estimated value at 100 eV would be 15.7.
d
Error limit is from McConkey et al. [1967].
e
Obtained by using photoionization oscillator strength and the shape
function of Borst and Zipf [1970] (see Table 1).
f
The average is obtained by excluding the value of Aarts et al. [1968].
The error limit, ±1.8, refers to one standard deviation.
b
with different methods of calibration. Aarts et al. [1968],
Aarts [1970], and Shaw and Campos [1983] utilized the
helium emission cross section as a secondary standard while
Srivastava and Mirza [1968a, 1968b], and Borst and Zipf
[1970] used a standard tungsten lamp for absolute calibration. McConkey and Latimer [1965] and McConkey et al.
[1967, 1971] used helium, standard iodine lamps, as well as
standard tungsten lamps. It should be pointed out that
anomalous threshold behavior in the cross section figure
of McConkey et al. [1967] was attributed to a graphic
drawing error. McConkey et al. [1971] subsequently provided the tabulated values that were corrected for overlap of
the second positive band.
[35] We are unable to explain the difference between the
photoionization oscillator strength and that inferred from the
Borst and Zipf [1970] measurement. While it is possible that
the (1,1) band may contribute to the observed (0,0) band
intensity, a quick calculation shows that the contribution is
less than 4.6% and the application of an interference filter
should reduce it even further. Similarly, the possible error
in the Franck-Condon factor for the (0,0) band of the N2 X
1 +
Sg N+2 B 2S+u system and the (0,0) band emission
branching ratio of the B 2S+u X 2S+g system, which has
been assigned as 5% by Van Zyl and Pendleton [1995],
is insufficient to explain the difference. Furthermore, the
calculated lifetime of Laher and Gilmore [1999] for the vj =
0 level of the B 2S+u state, 62.3 ns, agrees well with the
measured lifetime, 61.35 ± 0.30 ns by Schmoranzer et al.
[1989], 61.8 ± 0.5 ns by Scholl et al. [1995], or 65.1 ± 0.4 ns
by Fukuchi et al. [1995]. In fact, if the measured emission
branching ratio, 0.63 ± 0.03, of Fukuchi et al. [1995], had
been used, the fXB from the Borst and Zipf [1970] data would
have been 1.02. It is important to note that the emission cross
section measurements into the high energy region by Aarts et
al. [1968] (up to 6 keV) and Srivastava and Mirza [1968b]
(up to 4 keV) both indicate a value of 0.80 for fXB.
[36] Based on the total N+2 cross section of Schram et al.
[1965], Aarts et al. [1968] estimated that the s(B 2S+u )/
s(N+2 ) ratio to be 12.5% in the region from 800 eV to
8 of 14
SHEMANSKY AND LIU: N2 ! N+2
A07307
Table 4. Ratios of Measured Relative Intensity of (v0, v00) Band of
A 2u X 2S+g to the (0,0) Band of B 2S+u X 2S+g and Resulting
s[A 2u]/s[B 2S+u ] Ratios at 100 eV
References
Pendleton and Weaver [1973]
Average
O’Neil and Davidson [1968]
Average
Piper et al. [1986]
Average
Total average
Standard deviation
Present
(v0,v00)
A-X
Measured
Ratio
A/B
s Ratioa
A/B
s Ratiob
(2,0)
(3,1)
(4,1)
0.429
0.287
0.138
(0,0)
(1,0)
(2,0)
1.20
1.40
0.600
(2,0)
(3,1)
(4,1)
0.588
0.400
0.125
2.65
2.83
4.23
3.24
3.85
3.56
3.70
3.63
3.63
3.94
3.83
3.80
3.58
0.52
3.56c
3.07
2.98
4.60
3.55
3.71
3.65
4.30
3.98
4.21
4.15
4.17
4.18
3.87
0.56
a
From Van Zyl and Pendleton [1995], who used the Franck-Condon
factors and emission branching ratios of Gilmore et al. [1992].
b
Same as footnote a, except the A 2u X 2S+g emission branching
ratios of Wu and Shemansky [1976] are used.
c
Note that the emission branching ratio is not required for the present
s[A 2u]/s[B 2S+u ] ratio.
6 keV. Table 2 shows the s[B 2S+u ]/s[N+2 ] ratio changes
from 12.5% to 11.2% in the same energy region. Part of the
difference between the present ratio and that of Aarts et al.
[1968] can be attributed to the small discrepancy in fXB
(0.76 vs 0.80). Between 40 eV and 10 keV, the present ratio
is 11% – 12.9%.
4.3. A 2u X 12+g Cross Section
[37] Optical measurements of the A 2u state emission
cross section were generally carried out with relative
intensity of a vibrational band for the A 2u X 2S+g to
the (0,0) band of the B 2S+u X 2S+g . The cross section of
the A 2u state is then obtained from the absolute emission
cross section of the B 2S+u X 2S+g (0,0) band with the
appropriate Franck-Condon factors and emission branching
ratios. Thus, the reported cross section value of the A 2u
state is directly proportional to the absolute cross section of
the B 2S+u X 2S+g (0,0) band. Furthermore, significant
discrepancies in the emission branching ratios of the
A 2u X 2S+g bands exist. The most widely used branching ratio is based on transition probabilities for the A 2u X 2S+g calculated by Gilmore et al. [1992] and Laher and
Gilmore [1999], who used the theoretical transition moment
of Langoff et al. [1987]. Unfortunately, the calculated
transition moment is significantly different from its counterpart derived from experimental lifetimes and relative band
intensities by Wu and Shemansky [1976], who have shown
that the transition moment cannot be uniquely determined by
using lifetimes alone. The drastic difference between the
theoretical and experimental transition moments can be
illustrated by comparing Figure 9 of Gilmore et al. [1992]
with Figure 4 of Wu and Shemansky [1976].
[38] Van Zyl and Pendleton [1995] have examined three
different experimental measurements and obtained the
100 eV A 2u/B 2 S+u cross section ratio, s[A 2 u ]/
s[B 2S+u ] = 3.58 ± 0.52, using the Franck-Condon factor
and emission branching ratios calculated by Gilmore et al.
[1992]. To obtain the 100 eV ratio, Van Zyl and Pendleton
A07307
[1995] also assumed that s[A 2u]/s[B 2S+u ] ratio is
independent of excitation energy. If the A 2u X 2S+g
emission branching ratios of Wu and Shemansky [1976] are
used, a different set of s[A 2u]/s[B 2S+u ] ratios are
obtained using the assumption and procedure of Van Zyl
and Pendleton [1995]. Table 4 shows that the Wu and
Shemansky [1976] emission branching ratio yields higher
cross section ratios.
[39] The relative intensity measurement of Piper et al.
[1986], made from 2.5 to 6.0 keV, is generally considered to
be the most accurate value. Piper et al. [1986] measured the
relative intensity of three vibrational bands and noted that
their intensity ratios are invariant with excitation energy
within experimental uncertainty. They also derived a band
cross section ratio of 4.1 ± 0.7 for s[A 2u]/s[B 2S+u ],
which is consistent with the average ratio 4.18 listed in
Table 4. Table 2 shows that the model s[A 2u]/s[B 2S+u ]
ratio varies with excitation energy, though the variation
is small. Between 30 eV and 10 keV, the model s[A 2u]/
s[B 2S+u ] ratio is given as 3.62 ± 0.46. It is consistent with
3.58 ± 0.52 derived by Van Zyl and Pendleton [1995] or
3.87 ± 0.56 derived from the Wu and Shemansky [1976]
emission branching ratio. Pendleton and Weaver [1973]
noted that their cross section ratio between 50 and 500 eV
is independent of electron energy. The present ratio is 3.38 ±
0.20 in this range. The ±6% variation is smaller than most
experimental errors. From 2.5 keV to 6.0 keV, the ratio
changes less than 7%, from 3.79 to 4.03. Thus, the calculated s[A 2u]/s[B 2S+u ] between 2.5 and 6 keV is fully
consistent of the ratio of 4.1 ± 0.7 reported by Piper et al.
[1986]. The variation of s[A 2u]/s[B 2S+u ] between 40 and
500 eV is primarily due to the large difference in the C5/C7
collision strength parameters and oscillator strengths of the
A 2u and B 2S+u states. The increase with energy above
500 eV reflects the large oscillator strength of the A 2u
state. Finally, the variation of the ratio below 30 eV is
primarily caused by the difference in the threshold energies.
[40] Table 5 summarizes ionization branching ratios and
partial cross sections at 100 eV. The A 2u state cross
section, (82.6 ± 6.6) 1018 cm2 agrees well within error
limits with (87.9 ± 7.0) 1018 obtained by Doering and
Yang [1997], and (101.1 ± 19) 1018 cm2 obtained by
Van Zyl and Pendleton [1995]. The present A 2u cross
section is significantly smaller than (131 ± 33) 1018 cm2
inferred from the measurement of Holland and Maier
[1972, 1973] using the Wu and Shemansky [1976] emission
branching ratio.
Table 5. 100 eV Branching-Ratios and Partial Cross Sections for
e + N2 ! N2+ + 2e
References
9 of 14
X 2S+g
A 2u
B 2S+u
VZPa
DYb
SLc
Branching Ratio
0.320 ± 0.147
0.535 ± 0.112
0.448 ± 0.033
0.452 ± 0.033
0.423 ± 0.034
0.451 ± 0.036
0.147 ± 0.017
0.099 ± 0.017
0.127 ± 0.010
VZPa
DYb
SLc
Cross Section (1018 cm2)
60.5 ± 27
101 ± 19
86.9 ± 7.0
87.9 ± 7.0
77.6 ± 6.2
82.6 ± 6.6
27.4 ± 2.7
19.2 ± 3.3
23.2 ± 1.9
a
Van Zyl and Pendleton [1995].
Doering and Yang [1997].
Shemansky and Liu, this work.
b
c
A07307
SHEMANSKY AND LIU: N2 ! N+2
A07307
data (solid diamond), after converting to the A 2u X 1S+g
cross section, have been scaled down by 67% to account for
the 35% difference in absolute values.
Figure 1. Comparison of experimental (solid square) and
model (solid line) partial ionization cross sections for the
A 2u state. The experimental cross section is based on the
(1,0) Meinel band in Figure 2 of Skubenich and
Zapesochnyy [1981], scaled by the Franck-Condon factor
of Gilmore et al. [1992] and the emission branching ratio of
Wu and Shemansky [1976], and finally adjusted upward by
41%. The dotted line trace is obtained by scaling the solid
line trace by 50%, and the solid diamond trace is based on
Holland and Maier [1972, 1973] (2,0) band data adjusted
for the Franck-Condon factor and emission branching ratio
and then scaled down by 67%.
[41] The shape of the derived A 2u X 1S+g excitation
function also agrees with that of Skubenich and Zapesochnyy
[1981], who measured a number of A 2u X 2S+g optical
excitation functions from threshold to 400 eV. The solid
square trace in Figure 1 shows the excitation function of the
(1,0) Meinel band digitized from Figure 2 of Skubenich and
Zapesochnyy [1981]. The scale of the solid-square trace
was established using the Franck-Condon factor (0.318) of
the (1,0) band of A 2u X 1S+g from Laher and Gilmore
[1999] and inverse emission branching ratio (1.61/1.22)
for the (1,0) Meinel band of Wu and Shemansky [1976].
The solid-square trace was further raised by 41% to aid
the comparison with the model, shown in solid trace.
Except in the threshold region, the difference between
the two traces is less than 5%. Moreover, Skubenich and
Zapesochnyy [1981] noted that their Meinel band cross
sections peak between 105 and 110 eV. Both Figure 1 and
Table 2 show that the present cross section also peaks near
110 eV. The good agreement between the shapes provides
confidence in the accuracy of the present A 2u X 1S+g
shape.
[42] As noted by Van Zyl and Pendleton [1995], the
maximum cross section of the A 2u state inferred from
Skubenich and Zapesochnyy [1981] measurements is too
low. The A 2u state peak cross section, derived from
Table 3 of Skubenich and Zapesochnyy [1981], is (65 ± 5)
1018 cm2 and 30% lower than the present value.
Figure 1 also compares the present A 2u state shape with
that of the (2,0) Meinel band of Holland and Maier [1972,
1973]. The dot line is obtained from scaling the model trace
(solid line) down by 50%. The Holland and Maier [1972]
4.4. X 22+g X 12+g Cross Section
[43] The 100 eV X 2S+g state cross section obtained in the
present analysis is (77.6 ± 6.2) 1018 cm2. Van Zyl and
Pendleton [1995] obtained the X 2S+g state cross section by
subtracting the A 2u and B 2S+u states from the total N+2
production cross section. As a result, their 100 eV X 2S+g
state cross section, (60.5 ± 27)) 1018 cm2, has a very
large uncertainty (45%). Doering and Yang [1997]
obtained a value of (86.9 ± 7.0) 1018 using their
measured branching ratio and total N+2 cross section of
Straub et al. [1996]. The present value is within the
uncertainty of Van Zyl and Pendleton [1995]. It also agrees
with Doering and Yang [1997] in the sense that both values
are within the aggregated error margins. The total N+2 cross
section of Straub et al. [1996] at 100 eV, 194 1018 cm2,
has been revised down to 185 1018 cm2 by Lindsay and
Mangan [2003]. If Doering and Yang [1997] had used
the value of Lindsay and Mangan [2003], their number
would have been (82.9 ± 6.1) 1018, agreeing with
the present value within either error margin.
[44 ] Abramzon et al. [1999a, 1999b] have directly
measured the electron impact cross section of the X 2S+g
state using the LIF technique. In their work, the N+2 X
2 +
Sg (0) state was excited to B 2S+u (0) with a laser operating
at 391 nm. The induced emission is detected via the (0,1)
band transition of B 2S+u X 2S+g at 428 nm. Abramzon et
al. [1999b] obtained absolute cross section values using
the cross section of helium 1 1S ! 2 3S, probed via LIF of
2 3S ! 3 3P. Their 100 eV value, (74.9 ± 7.5) 1018 cm2,
agrees very well with the present value, (77.6 ± 6.2) 1018 cm2. Abramzon et al. obtained a peak value of 80 1018 at 60 eV. The present X 2S+g cross section reaches the
maximum at 85 eV with a value of (77.7 ± 6.2) 1018. As
Figure 2 shows, the excitation function obtained by
Figure 2. Comparison of experimental (dotted line with
triangle) and model (solid line) partial ionization cross
sections for the X 2Sg+ state. The experimental cross section
is from the laser-induced fluorescence measurement of
Abramzon et al. [1999a, 1999b].
10 of 14
SHEMANSKY AND LIU: N2 ! N+2
A07307
Abramzon et al. [1999a, 1999b] has an abnormal shape in
the low energy (E < 50 eV) region. From 60 to 200 eV, both
shape functions are relatively flat and are consistent within
experimental error. The Abramzon et al. [1999a, 1999b]
excitation function was measured in transitions either with
Jj = 11 – 15 levels or the Jj = 6 level. A number of
experimental studies have shown that the nascent population of the X 2S+g state cannot be described by a single
Boltzmann distribution and the high J-levels tend to be
overpopulated, resulting in so called rotation warming
[Hernandez et al., 1982; Nagata et al., 1987; Zetner et
al., 1988]. Moreover, the rotation warming becomes more
significant at low electron energy. For example, Nagata et
al. [1987] observed that an average rotation energy for the
vj = 0 level of X 2S+g increases from of 2.26 ± 0.16 meV to
4.24 ± 0.27 meV as electron impact energy is reduced from
300 eV to 25 eV. Zetner et al. [1988] noted the rotational
warming is nearly constant when incident electron energy is
above 80 eV. The discrepancy in shape functions below
60 eV can be attributed to the electron energy dependence
of rotational warming.
[45] The contribution of indirect ionization via resonance
and autoionization should also be mentioned. The effect of
the indirect process near threshold can be inferred from
experimental measurements by Fox [1951]. The indirect
ionization cross section is represented by three functions
given by equation (3). Table 6 lists the parameters C1, C2
and E0 for the three gaussian functions. These parameters
are derived from the Fox [1951] results, obtained with an
electron energy width of 60 meV. Figure 3 shows the
ionization cross section by resonance and autoionization
processes, obtained through normalization to the present
cross section.
[46] It should be stated that the agreement between the
model and measured cross sections from threshold to 20 eV
is poor (see Table 2). A number of factors can lead to the
discrepancy. First, the cross section changes rapidly with
energy in this region. A small error in experimental energy
can lead to significant deviation in cross section. The
uncertainty in electron beam energy and energy spread of
the electron beam near threshold obviously complicate the
problem. Indeed, the agreement between different measurements near threshold is very poor. At 17 and 20 eV, for
example, Straub et al. [1996] reported values of 2.4 1018 and 21.8 1018 cm2 while Rapp and EnglanderGolden [1965] obtained values of 7.13 1018 and 27 1018 cm2. Furthermore, indirect ionization via resonance
and autoionization [Fox, 1951; Riu et al., 2001; Bolognesi
et al., 2004] shown in Figure 3 also contributes to the
ionization process. As Figure 3 shows, the indirect excitation cross section is sharp and occurs near threshold. The
contribution to the observed (apparent) cross section, there-
Table 6. Collison Strength Parameters for Near Threshold
Resonance Ionization of N2
Functiona
C1
C2b
E0b
1
2
3
0.021549
0.021549
0.021178
1.6687
1.6687
2.5673
16.8
17.5
18.2
a
See equation (3) for functional definition.
The units for C2 and E0 are eV1 and eV, respectively.
b
A07307
Figure 3. Indirect ionization cross section of N2 obtained
from the measurement by Fox [1951].
fore, strongly depends on the energy shape and width of the
electron beam. Without detailed information on the electron
beam energy profile, it is difficult to separate the direct and
indirect components from the Rapp and Englander-Golden
[1965] data. The model cross section in Table 2 is obtained
without consideration of the resonance and autoionization.
Understandably, the negligence distorts the model cross
section in the threshold region and the largest error occurs
between 16.5 and 17.5 eV, where the resonance ionization is
very significant. Finally, the present model treats the ionization only at the vibrational level. Because of the threshold
difference, consideration of rotational motion is probably
required to satisfactorily reproduce experimental observation in the threshold region.
[ 47 ] In summary, partial photoionization oscillator
strengths for the X 2S+g X 1S+g , A 2u X 1S+g and
B 2S+u X 1S+g transitions have been obtained from
photoionization experimental results. The derived partial
oscillator strengths, along with total ionization cross section
of Straub et al. [1996] and Lindsay and Mangan [2003] and
shape functions of Borst and Zipf [1970] and Holland and
Maier [1972, 1973], enable a partition of partial electron
impact ionization cross sections for the X 2S+g (v), A 2u(v)
and B 2S+u (v) states. The derived A 2u X 1S+g shape
function agrees well with experimental measurement of
Skubenich and Zapesochnyy [1981]. The present X 2S+g X 1S+g excitation function is also consistent with the LIF
measurement of Abramzon et al. [1999a, 1999b] in the
region where energy dependence of rotational warming is
small. In addition, the derived s[A 2u]/s[B 2S+u ] and
s[B 2S+u ]/s[N+2 ] ratios agree with many experimental observations over a wide energy range. Moreover, the present
cross sections at 100 eV agree with those obtained by
Doering and Yang [1996, 1997] and Van Zyl and Pendleton
[1995] within the given error limits. Finally, the present
analysis agrees with the conclusion by Goembel et al.
[1994] and Doering and Yang [1996, 1997] that the Borst
and Zipf [1970] B 2S+u state cross section is too high.
[48] Acknowledgments. This work is supported by the National
Science Foundation ATM-0131210.
11 of 14
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SHEMANSKY AND LIU: N2 ! N+2
[49] Arthur Richmond thanks John P. Doering and E. Krishnakumar for
their assistance in evaluating this paper.
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X. Liu and D. E. Shemansky, Department of Aerospace and Mechanical
Engineering, University of Southern California, Los Angeles, CA 90089,
USA. (xianming@usc.edu; dons@hippolyta.usc.edu)
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