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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 A07307 1 of 14 A07307 SHEMANSKY AND LIU: N2 ! N+2 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 A07307 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, 2 of 14 A07307 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 A07307 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 3 of 14 SHEMANSKY AND LIU: N2 ! N+2 A07307 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 A07307 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 4 of 14 A07307 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 A07307 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. 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