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Total cross-section calculations for electrons collidingwith molecular nitrogen over an extensive energyrange from meV to keVMinaxi Vinodkumar a , Chetan Limbachiya b & Mayuri Barot aa V.P. and R.P.T.P. Science College, Vallabh Vidyanagar, 388 120 Gujarat, Indiab P.S. Science College, Kadi, 382 715 Gujarat, IndiaAccepted author version posted online: 10 May 2012.Published online: 07 Jun 2012.

To cite this article: Minaxi Vinodkumar , Chetan Limbachiya & Mayuri Barot (2012) Total cross-section calculationsfor electrons colliding with molecular nitrogen over an extensive energy range from meV to keV, MolecularPhysics: An International Journal at the Interface Between Chemistry and Physics, 110:24, 3015-3022, DOI:10.1080/00268976.2012.692824

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Molecular PhysicsVol. 110, No. 24, December 2012, 3015–3022

RESEARCH ARTICLE

Total cross-section calculations for electrons colliding with molecular nitrogen over an

extensive energy range from meV to keV

Minaxi Vinodkumara*, Chetan Limbachiyab and Mayuri Barota

aV.P. and R.P.T.P. Science College, Vallabh Vidyanagar, 388 120 Gujarat, India; bP.S. Science College,Kadi, 382 715 Gujarat, India

(Received 6 March 2012; final version received 17 April 2012)

We report a comprehensive study of the electron impact total cross-sections for molecular nitrogen for impactenergies from 0.01 eV to 2000 eV. Ab initio calculations are performed using the R-matrix formalism at lowimpact energies (up to �15 eV), while the Spherical Complex Optical Potential formalism is utilised beyond thisrange. The two methods are consistent at the transition energy, which enables us to provide data for such anextensive range. The results obtained show overall good agreement with the available data.

Keywords: R-matrix method; Spherical Complex Optical Potential (SCOP); eigenphase; excitation cross section;total cross section

1. Introduction

Molecular nitrogen is the main constituent of theEarth’s atmosphere (78%) up to an altitude of about500 km. The scattering of electrons by nitrogen gas is afavored fundamental process in the atmosphere due tothe abundance of electrons produced as a result ofphotoionisation of the many upper atmospheric con-stituents by solar UV radiation. This important pro-cess of the scattering of N2 molecules by electrons playsa vital role in the ionospheric and auroral phenomenain the upper atmosphere of the Earth. A systematicstudy of this phenomenon forms the basis for under-standing gas-discharge devices, plasma processing,laser kinetic modeling and the physics of planetaryatmospheres [1]. Moreover, N2 is the most favoredsystem for study for both theoreticians and experi-mentalist due to its stability, simple structure and fewelectrons. This is clearly evident from the enormousamount of work carried out on e–N2 scattering bytheoreticians and experimentalist for different rangesof impact energies from low to high.

The total electron scattering cross section is animportant quantity at all ranges of impact energy.Low-energy collision studies are most fundamental asthey help us to understand the physics underlying theoccurrence of many physico-chemical processes. Suchprocesses are prevalent in plasmas and industrialdischarges. The important phenomenon of resonanceoccurs at low impact energies, generally below 10 eV,and is reflected as a clear structure in the total

cross section. Theoretical predictions of low-energyresonance formation are strongly linked to a detailedknowledge of the forces acting on the electrons duringthe scattering process and are a consequence of thestructural properties of the target. At intermediate andhigh energies, electron scattering data provide usefulinput in various fields such as astrophysics, atmo-spheric physics, radiation physics, plasmas, etc. Anadded advantage of the total cross section is that itprovides an upper bound to any type of cross sectionas it is the sum of all cross sections arising either fromelastic or inelastic processes. Hence, in the presentwork, our main impetus is to provide total crosssections for e–N2 scattering from meV to keV, which isperhaps performed for the first time.

As discussed above, the collision of electrons withnitrogen molecules is studied widely by many groups,both theoretically and experimentally. Despite theenormous amount of work that has been carried out,there still exist systematic discrepancies in the magni-tude of the total cross sections at low energies and alsoin the energy range where the peak occurs. A detailedreview of earlier work in terms of the range of impactenergies and the method employed in experiment/theory is presented in Table 1. Looking into the vastliterature and work carried out for e–N2 scattering, itcan be divided into three ranges of impact energies,very low energies (up to 5 eV), low and intermediateenergies (up to 250 eV) and high energies (up to a fewkeV range). Very-low-energy work is carried out

*Corresponding author. Email: minaxivinod@yahoo.co.in

ISSN 0026–8976 print/ISSN 1362–3028 online

� 2012 Taylor & Francis

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specifically to determine the peaks in the N2 scatteringarising from the 2�g shape resonance. Much theoret-ical and experimental work has been carried out in thisregion [1–10]. In the low to intermediate region thereare a handful of experiments [1–13] and not muchtheoretical work. In the region above threshold to thehigh-energy region, again many measurements havebeen carried out [14–17]. However, overall it can beconcluded that, compared with experimental investi-gations, theoretical work is sparse. The significance ofthe present work is that it extends from very lowimpact energy (0.01 eV) to high energy (2000 eV) forthe first time.

The present work was carried out using twoformalisms, since a single formalism cannot beemployed over such a wide range as presented here.At low impact energies below the ionisation thresholdof the target we carried out ab initio calculations usingthe R-matrix formalism through the Quantemol-Npackage [18–20], and beyond threshold the SphericalComplex Optical Potential (SCOP) [21,22] formalismwas employed. The results are promising as there isconsistency in the data, particularly at the transitionenergy (�15 eV), where the two formalisms overlap.Our intention in the present work is twofold, one topresent the results over a wide range of impact energiesand make fruitful comparisons, and, second, tobenchmark our results against vast data sets availablein the literature.

2. Theoretical methodology

Before going into the actual description of the theo-retical methods involved in the present study we

present the target model employed for calculation inthe low-energy regime below ionisation.

2.1. Target model

N2 is a linear molecule with a bond length of 1.08 A.We employed the double zeta plus polarisation (DZP)Gaussian basis set for target wave function represen-tation. The double zeta basis set is important because itallows us to treat each orbital separately when weperform the Hartee–Fock (HF) calculation. This willensure the accurate representation of each orbital inthe target wave function. N2 has D2h symmetry oforder 8. The ground-state electronic configuration isrepresented as 1A2

1g, 1B21u, 2A

2g, 2B

21u, 3A

2g, 1B

23u, 1B

22u.

Of the 14 electrons, we froze four electrons in the 1A21g

and 1B21u molecular orbitals, while the remaining 10

electrons were kept free in the active space of the 2Ag,

3Ag, 1B2u, 1B3u, 2B1u, 1B3g and 1B2g molecular orbitals.To improve the representation of the excited states, weused the molecular orbitals (MOs) obtained from aHartee–Fock self-consistent field (HF–SCF) calcula-tion to build state-averaged pseudo-natural orbitals(NOs). Configuration integration (CI) calculationswere performed for the states included in the close-coupling expansion. A weighted average of the densitymatrices obtained from these states was then producedand the NOs were obtained from diagonalisation. Sixhundred and nine Configuration State Functions(CSFs) are used for accurate representation of thetotal of 16 target states. The number of channelsincluded in the R-matrix calculations is 100.Employing the GAUSPROP and DENPROP modules

Table 1. Review of the literature on e–N2 scattering.

Energy (eV) Author Method (Th: theory; Ex: experiment) Ref.

1–10 Sun et al. Close-coupling method (Th). Complimentary time of flight (Ex) [1]0–5 Chandra and Temkin Close coupling method (Th) [2]0–5 Bruke and Chandra A pseudo-potential method (Th) [3]0.3–1.6 Baldwin Single electron time of flight (Ex) [4]0.3–5 Golden A modified Ramsauer technique (Ex) [5]1–4 Mathur and Hasted Electron transmission (Ex) [6]1–5 Weatherford Non-iterative partial differential equation technique (Th) [7]0.5–5 Huo et al. Schwinger multichannel formulation (Th) [8]0.5–5 Gillan et al. R-Matrix method (Th) [9]0–5 Morgan R-Matrix method (Th) [10]0.5–50 Kennerly Transmission time of flight method (Ex) [11]0.5–60 Hofmann et al. Beam transmission technique (Ex) [12]4–30 Nickel et al. Attenuation technique (Ex) [13]0.5–250 Szmytkowski et al. Linear transmission (Ex) [14]15–750 Blaauw et al. Linear Ramsauer technique (Ex) [15]100–1600 Dalba et al. Attenuation technique (Ex) [16]600–5000 Garcia et al. Transmission beam technique (Ex) [17]

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[18] we obtain the ground-state energy of�109.02 Hartree, which is in good agreement withthe theoretical value of �109.12 Hartree [23], as shownin Table 2. The present calculated first electronicexcitation energy is 8.304 eV, which is in good agree-ment with the theoretical values of 7.63 eV [23] and7.32 eV [24] and the experimental value of 7.355 eV[25]. The rotational constant obtained in the presentcalculation is 1.9985 cm�1, which is also in excellentagreement with the experimental value of 1.9982 cm�1

from CCCBDB [26]. The vertical electronic excitationthresholds for N2 are listed in Table 3.

2.2. Low-energy formalism (0.01 to �15 eV)

Ab initio calculations are the most fundamental calcu-lations where rigorous mathematical computations areinvolved. Of the three popular methods for low-energycalculations, viz. the Kohn vibrational method, theSchwinger vibrational method and the R-matrixmethod [27–29], the R-matrix method is used widely[28,29]. We briefly describe the basic principles under-lying the R-matrix method.

The main idea of the R-matrix method is splittingof the configuration space into two spatial regions, aninner region and an outer region. The inner region or‘near target region’ is the region with the greatestcomplexity and is chosen such that, generally, thecomplete electronic charge of the target is included inthis region and negligible outside it. In the inner regionthe scattering electron is indistinguishable from theelectrons of the target and hence electron–electroncorrelation and exchange effects along with polarisa-tion are dominant. In order to obtain accurate resultsof total cross sections it is necessary to include thevibrational cross section. Morgan [10] carried outextensive studies on the vibrational excitation of N2 onelectron impact. However, the present calculations areperformed with the fixed nuclei approximation at theground-state equilibrium geometry of the target,neglecting nuclear motion (vibrational and rotational).

In the inner region, quantum chemistry methodsare employed to solve the Nþ 1 eigenvalue problem,

where N refers to the target electrons. When the

scattering electron is at a large distance from the center

of mass of the target, the probability of such interac-

tions is negligible, thereby simplifying the problem in

the outer region considerably, and the scattered

electron is assumed to propagate in the multipole

potential of the target.In the inner region, the wavefunction is constructed

using the close-coupling approximation [30], which is

common to other ab initio calculations and accordingly

is written as

Nþ1k ¼ A

Xi

�Ni ðx1, x2, . . . , xNÞ

Xj

�j ðxNþ1Þaijk

þXm

�mðx1, x2, . . . , xNþ1Þbmk, ð1Þ

where A is an anti-symmetrisation operator, xN is the

spatial and spin coordinate of the Nth electron, �Ni is

the ith state of the N-electron target, which is repre-

sented using a CI expansion, and �j is a continuum

orbital spin coupled with the target states. The

coefficients aijk and bmk are variational parameters

that can be determined by solving the Nþ 1 eigenvalue

problem in the inner region by employing standard

bound state quantum chemistry methods. Here the first

integral runs over the target states included in the

present calculation. The second sum runs over the

configurations �m, which are formed by placing all

electrons in the target molecular orbitals. The standard

way of performing a CI target calculation is to use a

complete active space CI (CASCI), as this model

Table 2. Properties of the target.

Ground-state energy (Hartree) First excitation energy, E1 (eV) Rotational constant, B (cm�1)

Target Present Theory Present Exp. Theory Present Exp.

N2 �109.02 �109.12 [23] 8.3942 7.355 [25] 7.63 [23] 1.9985 1.9982 [26]7.32 [24]

Table 3. Vertical excitation energy up to the ionisationthreshold for N2.

State Energy (eV) State Energy (eV) State Energy (eV)

1Ag 0.00 3Au 9.479 1B0u 10.7011B0u 8.304 3Au 10.075 1Au 10.7013B00g 8.728 1B000g 10.108 3B00u 12.0753B000g 8.728 1B00g 10.108 3B000u 12.0753B0u 9.479 1Au 10.500 1B000u 14.255

1B00u 14.255

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maintains a balance between the target and scatteringcalculations. In this model, valence electrons are freelydistributed amongst the subsets of the valence orbitals.

The occupied and virtual target molecular orbitalsare constructed using the Hartree–Fock Self-Consistent Field method with Gaussian-type orbitals(GTOs) and the continuum orbitals of Faure et al. [31]and include up to g (l¼ 4) orbitals. The advantage ofusing Gaussian-type orbitals is that infinite-rangeintegrals are evaluated exactly. We performed a Borncorrection to include higher partial waves. In practice,all the integrals are evaluated in the entire configura-tion space and the tail contribution outside the R-matrix sphere is then subtracted. After generating thewavefunctions, using Equation (1), their eigenvaluesare determined. The R-matrix is constructed at theboundary between the inner and outer regions for achosen set of incident energies. For this purpose theinner region is propagated to the outer region potentialuntil its solutions match with the asymptotic functionsgiven by the Gailitis expansion [32]. Thus, bygenerating the wavefunctions, using Equation (1),their eigenvalues are determined. These coupledsingle-centre equations describing the scattering inthe outer region are integrated to identify the K-matrixelements. Consequently, the resonance positions,widths and various cross sections can be evaluatedusing the T-matrix obtained from the S-matrix, which,in turn, is obtained from the K-matrix elements.

2.3. Higher-energy formalism (15 eV to 2 keV)

For impact energies above the ionisation threshold ofthe target, where the R-matrix formalism cannot beemployed, we employed the spherical complex opticalpotential (SCOP). This is a well-established methodol-ogy used by many groups [33–36] and has also beendiscussed extensively in the literature, hence we willpresent only the important aspects of the theory here.

The electron–molecule interaction potential is com-plex in nature and is in general given by

VoptðEi, rÞ ¼ VRðEi, rÞ þ iVIðEi, rÞ, ð2Þ

where the first term corresponds to all interactions thatare real and can be elaborately represented as

VRðEi, rÞ ¼ VstðrÞ þ VexðEi, rÞ þ VPðEi, rÞ: ð3Þ

Here the first electrostatic term arises due toCoulomb interaction between the incoming projectile(electron) and the target, the second term is theexchange term arising from the antisymmetrisationrequirement on the total wavefunction, and the thirdterm is a result of the correlation polarisation potential

due to momentarily generated induced dipoles. Allthree terms are charge-density-dependent and the lasttwo terms depend on the scattering energy. Weemployed the single-centre expansion at half thebond length of the N2 molecule as it is a homo-nucleardiatomic molecule. The molecular charge density isobtained from the spherically averaged molecularcharge density �ðrÞ, which is determined from theconstituent atomic charge densities derived from theHartree–Fock wavefunctions of Bunge and Barrientos[37]. The charge density so obtained is renormalised toinclude the effect of bonding. We have invoked thefixed nuclei approximation, meaning that the workreported here has been carried out with the nuclei fixedat an inter-nuclear separation of 2.07a0. That is,nuclear motion such as vibrational and rotationalmotion are not considered, as they are not important inthis range of impact energies. For the exchangepotential, we employed Hara’s ‘free electron gasexchange model’, which is energy-dependent and doesnot contain any adjustable parameters [38], and for thepolarisation potential, VP, we used the parameter-freemodel of the correlation–polarisation potential givenby Zhang et al. [39]. Here, various multipolenon-adiabatic corrections are incorporated in theintermediate region which will approach the correctasymptotic form at large ‘r’ smoothly.

The imaginary part VI in Equation (2), also calledthe absorption potential Vabs, takes into account thetotal loss of the scattered flux into all the allowedchannels of electronic excitation and ionisation. ForVabs, we used the model potential of Staszeweska et al.[40], which is a non-empirical, quasi-free, Pauli-blocking, dynamic absorption potential. The form ofthe potential is given as

Vabsðr,EiÞ ¼ ��ðrÞ

ffiffiffiffiffiffiffiffiTloc

2

r�

8�

10k3FEi

� �

� �ð p2 � k2F � 2DÞ � ðA1 þ A2 þ A3Þ: ð4Þ

The local kinetic energy of the incident electron isgiven by

Tloc ¼ Ei � ðVst þ VexÞ: ð5Þ

The absorption potential is not sensitive to long-range potentials like Vpol. In Equation (4), p2¼ 2Ei,

kF¼ [3�2�ðrÞ]1/3 is the Fermi wave vector and D is anenergy parameter. Further, �ðxÞ is the Heaviside unitstep-function, such that �ðxÞ ¼ 1 for x� 0, and zerootherwise. The dynamic functions A1, A2 and A3 inEquation (4) depend differently on �ðrÞ, I, D and Ei.The energy parameter D determines the thresholdbelow which Vabs¼ 0, and ionisation or excitation isprevented energetically. We modified the original

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model by considering D as a slowly varying function ofEi around I [41].

After generating the full complex potential given inEquation (2) for a given electron–molecule system, wesolve the Schrodinger equation numerically usingpartial wave analysis. At low energies, only a fewpartial waves are significant, e.g. at the ionisationthreshold of the target around five to six partial wavesare sufficient, but as the incident energy increases,more partial waves are needed for convergence. Usingthese partial waves, the complex phase shifts areobtained, which are employed to find the total elasticand total inelastic cross sections. Finally, the algebraicsum of these cross sections yields the total cross sectionQT. Below the ionisation threshold of the target it is thesum of the total elastic and electronic excitation crosssections, whereas above the threshold, it consists of thetotal elastic cross sections and total inelastic crosssections.

3. Result and discussion

We report a comprehensive study of e–N2 scattering interms of the eigenphase diagram, electronic excitationcross sections and total cross sections, which arepresented over a wide energy range starting fromabout 0.01 eV through to 2000 eV. For the lower-energy calculations (up to the ionisation threshold ofthe target) we used the ab initio R-matrix code. Beyondthe threshold, the SCOP formalism was used tocalculate the total cross sections. The data producedby these two formalisms are consistent at the transitionenergy (�15 eV). This matching at the overlap of thetwo formalisms provides a detailed database over awide energy range, the first time that this has beendone for molecular nitrogen. The present results areplotted as a function of incident energy in Figures 1, 2and 3 together with available theoretical and experi-mental data for comparison. All the numerical resultsfor the total cross section (in A2) for N2 are presentedin Table 4 and are also displayed in Figure 3.

Figure 1 shows the eigenphase diagram for eightdoublet scattering states (2Ag,

2B2u,2B3u,

2B1g,2B1u,

2B3g,2B2g,

2Au) of the N2 system. It is important tostudy eigenphase diagrams as they detect the positionsof resonances, which are important features for studyof the low-energy regime. Resonances are a commoncharacteristic of electron molecule scattering at lowimpact energy and lead to a spectacular structure forthe pure vibrational excitation cross sections [42]. The2B2g state shows a prominent structure at 47.09 A2,which is reflected as a strong peak in the TCS curve at2.6 eV, arising due to the 2�g shape resonance.

The doublet Au and the doublet B1u have a prominentpeak around 16 eV and this is seen as a structure in theTCS curve also at 16 eV. No other prominent struc-tures are seen in the eigenphase diagram below 12 eVfor any symmetry and the same holds true for the totalcross section curve. The eigenphase diagrams show theimportant channels included in the calculations.

Figure 1. (color online). e–N2 eigenphase sums for eight-state CC calculations. Solid line, 2Ag; dashed line, 2B2u;dotted line, 2B3u; dash dotted line, 2B1g; short dash dottedline, 2B1u; short dashed line, 2B3g; short dotted line, 2B2g; andshort dash dotted line, 2Au.

Figure 2. (color online). e–N2 excitation cross sections for14-state CC calculations from the initial state 1Ag. Solid line,3B1u; dashed line, 3B2g; dotted line, 3B3g; dash dotted line,3B1u; dash dot dot line, 3Au; short dashed line, 3Au; shortdotted line, 1B3g; and short dash dot line, 1B2g.

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Figure 2 shows electronic excitation curves forexcitation of ground-state 1Ag to eight target states(3B1u,

3B2g,3B3g,

3B1u,3Au,

3Au,1B3g,

1B2g) for e–N2

scattering as a function of incident energy. From thecurve it is evident that the first electronic excitationenergy for N2 is 8.304 eV. The largest contributioncomes from the transition 1Ag to 3B1u with a peak of0.35 A2 at 11.39 eV. Another transition that is impor-tant is from 1Ag to 3B1u, which has peak of 0.2 A2 at12 eV. The remaining transitions do not contributesignificantly to the excitation cross sections.

Figure 3 compares the present results for e–N2

scattering with available data from 0.01 eV to 2000 eV.

In the low-energy range the resonance structure isclearly visible. We find a prominent peak of 47.09 A2 at2.6 eV, which is a reflection of the �g shape resonancestate [2]. This is in agreement with earlier work ofBruke and Chandra [3] with a peak value of 52.5 A2 at2.4 eV, Sun et al. [1] with a theoretical peak value of26.27 A2 at 2.2 eV, Sun et al. [1] with an experimentalpeak value of 33.32 A2 at 2.21 eV, Itikawa [43] with apeak value of 34 A2 at 2.4 eV, Hoffmann et al. [12] witha peak value of 30.9 A2 at 2.2 eV, and Kennerly [11]with a peak value of 29.1 A2 at 2.5 eV. It is very clearfrom earlier work that the peak is located around 2 eV,which is well reproduced in the present work. Golden[5] and Kennerly [11] observed a vibrational structureof temporary N2

� ion formation that is reflected as aprominent peak at 2.5 eV in the TCS curve.

Morgan [10] performed an extensive study of thevibrational excitation of N2 molecules on electronimpact considering 19 vibrational states. He alsoshowed that the inclusion of vibrational excitation inthe calculation leads to broadening of the total crosssection due to the formation of a series of resonantpeaks that are in good agreement with experiment[1,4–6,11,12,14]. However, the present study is not ableto reproduce the resonance structures in the vicinity ofthe strong peak due to the neglect of the vibrationalmotion of the target.

Theoretical predictions in the low-energy regimewere carried out by Bruke and Chandra [3] and Sunet al. [1] and the recommended data are presented byItikawa [43]. The present values are in very goodagreement with the data of Sun et al. [1] and Kennerly[11] above 1 eV, below which our results diverge, whichmay be attributed to the avoidance of the vibrationalchannel in our calculations. The present results are alsoin very good agreement with the results of Hofmannet al. [12], except at the peak value. The theoreticalvalues of Bruke and Chandra [3] are also in good

Table 4. Total cross sections for e–N2 scattering.

Energy TCS Energy TCS Energy TCS Energy TCS(eV) (A2) (eV) (A2) (eV) (A2) (eV) (A2)

0.01 10.83 4.0 17.05 14 11.13 500 03.210.1 12.00 5.0 14.71 15 11.01 600 02.830.2 12.39 6.0 13.76 20 11.26 700 02.530.4 12.73 7.0 13.12 30 11.49 800 02.290.6 12.86 8.0 12.60 40 10.67 900 02.090.8 12.90 9.0 12.04 80 08.78 1000 01.921.0 12.91 10 11.79 100 07.95 1500 01.361.5 13.10 11 11.50 200 05.28 2000 01.062.0 15.30 12 11.45 300 04.343.0 33.00 13 11.26 400 03.69

Figure 3. (color online). Total cross section of e–N2 scatter-ing. Solid line, present Q-mol results; dashed line, presentSCOP results; dotted line, Bruke and Chandra [3]; shortdashed line, Sun et al. [1]; dash dotted line, Itikawa [43]; opendiamonds, Garcia et al. [17]; open squares, Dalba et al. [16];open circles, Nickel et al. [13]; open hexagons, Blaauw et al.[15]; spheres, Hofmann et al. [12]; open up triangles,Sun et al. [1]; stars, Kennerly et al. [11] andSzmytkowski et al. [14].

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agreement with the present results throughout therange reported, except below 1.5 eV, which is due tothe different target wavefunction representation.

The experimental results of Nickel et al. [13] are inexcellent agreement with the present results throughoutthe range reported, except that between 4 and 8 eV thepresent results are slightly higher. The present resultsare in excellent agreement with all [11–17] results, boththeoretical and experimental, reported beyond 15 eV,which clearly reflects the success and consistency of theSCOP formalism. Around 20 eV there is a broad humpin the TCS, which is clearly reflected in all the previousmeasurements of Hofmann et al. [12], Szmytkowskiand Maciag [14] and Blaauw et al. [15]. As discussedearlier by Kennerly [11], this may be due to the manycore excited resonances, which could lead to vibra-tional excitation. Moreover, there are many electronicexcitation channels leading to ionisation opening up inthis energy regime, which may lead to an increase inthe total cross section. However, Dehmer et al. [44]pointed to the existence of a broad shape resonancecentered at about 25 eV due to ð�uÞ symmetry, which isvery clearly seen in the present work as a broad peakaround 25 eV with a peak value of 12.9 A2. For brevity,we have not included some of the results of Table 1in Figure 3.

4. Conclusion

The present work reports the total cross sections forelectron scattering from N2 over a wide range ofimpact energies between 0.01 eV and 2 keV. A com-posite theoretical methodology – the R-matrix at lowimpact energies to yield ab initio results and SCOP athigh energies – allows us to present data for a wideenergy range. The numerical method is based on thefixed-nuclei, single-centre, close-coupling formalism atlow energy and simple local and parameter-free modelpotentials in the intermediate- to high-energy region,which yields reliable scattering cross sections withrespect to experimental or more accurate theoreticalresults.

The present eigenphase diagram clearly shows thepositions of the predicted resonance structures. The2Og shape resonance, which has been studied veryextensively by many researchers [1–10], is seen in oureigenphase diagram as well as the TCS curve as astrong peak at 2.69 eV. Another important structureobserved experimentally in Ref. [44] is also reflected asa hump at 25 eV in our TCS curve. Thus all structuresat low energy are well reproduced in our eigenphasediagram and also in the TCS curve. This demonstratesthe consistency of the present work.

Previously we have illustrated the viability of thiscomposite methodology for three polyatomic molecu-lar targets NH3, H2S and PH3 [18], and CH4, SiH4 andH2O [45], for which there exists a good databaseagainst which we can benchmark our results. Themethod has also been extended to simple biomoleculessuch as HCHO and HCOOH [46]. The results arepromising since the two methods are consistent at thetransition energy (�15 eV) and show good agreementwith available data throughout the energy range.Therefore, we have confidence that the methodologyproposed may be used to calculate such cross sectionsin other molecular systems where experiments aredifficult, particularly for radicals and exotic systems.Such data sets are needed in a variety of applicationsfrom aeronomy to plasma modeling. Accordingly, sucha methodology may be built into the design of on-linedatabases to provide a ‘data user’ with the opportunityto request their own set of cross sections for use in theirown research. Such a prospect will be explored by theemerging Virtual Atomic and Molecular Data Centre(VAMDC) [47] (http://batz.lpma.jussieu.fr/www_VAMDC/).

Acknowledgements

MVK thanks the Department of Science and Technology,New Delhi, for financial support through a Major ResearchProject grant (No. SR/S2/LOP-26/2008) and CGL thanksUGC, New Delhi, for a Major Research Project grant(No. 40-429/2011 (SR)) under which part of this work wascarried out.

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