Indian Journal of Pure & Applied Physics Vol. 48, September 2010, pp. 621-625
Total ionization cross-sections of atmospheric molecules due to electron impact Yogesh Kumar1, *Neelam Tiwari2, Manoj Kumar3 & Surekha Tomar2 1 2
DAV College, Muzzaffarnagar 251 001, UP, India
Department of Physics, RBS College, Agra 282 002, UP, India 3
Meerut College, Meerut 250 001, UP, India
Email:
[email protected],
[email protected] Received 3 June 2009; revised 6 May 2010; accepted 23 July 2010 The total ionization cross-sections of atmospheric molecule due to electron impact for threshold ionization energy to 10 MeV have been studied. Many researchers like Khare and Wadhera [Phys Lett A, 198 (1995) 212] have successfully employed the Plane Wave Born Approximation (PWBA), corrected for the exchange, the Coulomb and the relativistic effects to calculate the outer shell ionization of molecules including transverse interaction for the inner shell ionization of atoms and molecules The present results have been compared with experimental data and other theoretical data wherever available. Keywords: Electron impact, Atmospheric molecule, Ionization
1 Introduction Total ionization cross-sections of molecules by electron impact are required to study the plasma diagnostics, astrophysical and fusion applications, radiation physics, mass spectrometry, ionization in gas discharge, modeling of fusion plasmas, modeling of radiation effects for both materials and medical research, and astronomy1 etc. The gases like H2S, N2O, CO2 are of interest to atmospheric science. H2S is not only present in the earth atmosphere but have been found on comets2 too. It is used in analytical chemistry for qualitative inorganic analysis of metal ions. N2O is the source of NO and NO2 in the stratosphere. N2O also contributes to the upper atmosphere chemistry by acting as an atmospheric thermal insulator. CO2 and N2O are also known as green house gases and CO2 contributes majority to the global warming. In the recent years, the total ionization crosssections for atmospheric molecules have been the subject of many theoretical and experimental studies3-9. Weinberger and Rudd6 theoretically calculated the total cross-section for H2S from threshold to 1 keV by using Binary Encounter Bethe theory with the vertical ionization potential. Experimentally, total cross- sections for H2S are measured by Lindsay et al.5 and Rao and Srivastava8 for energy range from threshold to 1 keV, whereas Belic and Kurepa9 measured it for energy range started from threshold to 100 eV. For high energy
from 0.1 to 2.7 MeV, the total cross-section measured by Reike and Prepejchal10. For N2O, Kim et al.6 calculated the total cross-section for energy range of threshold to 1 keV by applying the Binary Encounter Bethe theory along with vertical ionization potential. According to them, this improves their theory. They also calculated the ground state ionization crosssection for these molecules. Joshipura et al.7 also calculated total cross-section for N2O from threshold to 2 keV. They have calculated total cross-section for elastic and inelastic collisions and then deduced complex scattering by potential-ionization contribution method. Experimentally, total cross-sections for N2O have been given by Rapp and Englander-Golden3 for energy range of threshold to 1 keV. Lindsay et al.5 measured total cross-section and absolute partial cross-sections of the same. Iga et al.4 also measured total cross section for energy threshold to 1 keV. For N2O there is no experimental and theoretical data available for high energy range 0.1 to 2.7 MeV according to best of our knowledge. Total crosssections for CO2 is also calculated by Hwang et al11. and measured by Lindsay and Mangan12 and Hudson et al.13 for energy range from threshold to 250 eV. Experimental value of total cross-section for high energy range 0.1 to 10 MeV has been given by Reike and Prepejchal10. One of the purposes of this work is to calculate the electron impact ionization cross-sections of the molecules by employing the useful features of Kim
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model with Saksena model to remove the deficiency of the later model at low energy. This modification has been included recently by Khare et al.16 for CH4 molecules where (1−ω/E) was replaced by (E′/E′+U+I), where ω is the energy loss suffered by incident electron in the ionizing collision, E the kinetic energy of incident electron, I the ionization energy, U is the average kinetic energy of bound electron. Here U+I represent the increase in kinetic energy of the incident electron due to its acceleration by the field of the target nucleus. This is the only theory which is applicable for such a wide energy range varies from threshold to several MeV. By the present model, the expression for the Kim et al.6 can be obtained. 2 Theory Saksena et al.14 have proposed a model for the molecular ionization cross-sections. They started with the plane wave born approximation (PWBA) but later on included exchange and relativistic corrections. The transverse interaction through emission and the reabsorption of the virtual photons along with the longitudinal interaction through the static unretarded Coulomb field are also included. However, PWBA requires continuum generalized oscillator strengths (CGOS), which are very difficult to evaluate. Hence, they employ a semi-phenomenological relation of Mayol and Salvet15 which expresses CGOS in terms of the continuum optical oscillator strengths (COOS). The use of the above relation breaks the expression of the ionization cross-section σj for the jth molecular orbital into two terms one representing the Bethe term (soft collision) and other one the Mott term (hard collision). But it is found that their model has been found to underestimate the cross-section at low impact energies. Khare and Wadhera18 have successfully employed PWBA, corrected for the exchange, the Coulomb and the relativistic effects. To remove the deficiency of the former model at low E, another model was developed by Khare et al.16 by combining the useful features of Saksena et al.14 model and the Binary Encounter Bethe models of Hwang et al11. In the present theory, we have replaced (1−ω/E) as done by the Saksena et al.14 model by (E′/E′+U+I) and the exchange part of its Bethe term is neglected. We have calculated the total cross-section for atmospheric molecules, although the present molecules are not calculated by Saksena et al14. Hwang et al.11 have carried out numbers of calculation in Binary Encounter-Bethe-Model. They included the effect of acceleration of the incident
electron by the molecular field through the classicalbinary encounter theory and used a simple representation of the COOS.
df (ω, o) N j I j = 2 ω dω
…(1)
where ω is the energy loss, Nj and Ij are the numbers of electron and ionization threshold. Their total ionization cross-section for the jth molecular orbital for incident energy E is given by: σjT=σjKBB+σjKMB
…(2)
where
σ jKBB =
1 1− 2 I (t + u j + 1) t 2
…(3)
σ jKMB =
t − 1 ln t − 2 I (t + u j + 1) t t
…(4)
AN j
2 j
AN j
2 j
σjKBB and σjKMB are the Bethe’s and Mott’s crosssection, respectively with the following values of t and uj.
t=
Uj E , uj = Ij Ij
where Uj is the average kinetic energy of the bound electron of the jth orbital. A=4πa20R2 , with R and a0 are the Rydberg energy and first Bohr radius, respectively. Following Hwang et al.11 dropping the contribution of the exchange to the Bethe term and the adhoc cutoff factor from it. Furthermore, the effect of the acceleration of the incident electron by the molecular field is included through the classical binary encounter theory and using Eq. (1) for COOS, the present total ionization cross-section is: σjt = σjpBB+σjpMB+σjjt …(5)
σ jpBB =
σ jpMB
E
AN j I j (E + U j
1
+I )∫ω
2
j
Ij
ω ln Q−
dω
…(6)
2 t −1 + 2 1 − t + 1 2t 5 − t2 AN j 1 = − + 2 ( E ′ + U j + I j ) I j 2(t + 1) t (t + 1) − (t + 1) ln t + 1 t 2 2 …(7)
KUMAR et al.: TOTAL IONIZATION CROSS-SECTIONS OF ATMOSPHERIC MOLECULES
σ jjt =
A M j2 {ln(1 − β2 ) + β2 } RE
In Eq. (6), Q− is the recoil energy, M j is equal to the total dipole matrix squared.
t=
Table 1 — Molecular orbital constants19
…(8) 2
E Ij
I: Binding Energy; U: Average Kinetic Energy; N: Electron Occupation Number Molecules
Mol. Orbital
I (eV)
U (eV)
N
N2O
1σ 2σ 3σ 4σ 5σ 6σ 1π 7σ 2π
561.71 431.00 426.85 44.49 39.55 22.41 21.20 18.95 12.89
794.52 601.81 602.05 72.53 73.81 77.25 48.88 60.14 59.95
2 2 2 2 2 2 4 2 4
H2S
2a1 1b1 3a1 1b2 4a1 2b1 5a1 2b2
244.06 181.00 180.96 180.89 26.85 16.34 13.54 10.48
509.15 477.97 478.37 479.07 55.39 35.77 46.09 45.68
2 2 2 2 2 2 2 2
CO2
3σ1g 2σ2u 4σ1g 3σ2u 1πu 1πg
42.04 40.60 21.62 20.27 19.70 13.77
75.72 78.38 74.66 71.56 49.97 64.43
2 2 2 2 4 4
1 E = mv 2 (m = rest mass of electron) 2 β=
v (v = incident velocity, c = velocity of light) c
Reike and Prepejchal10 have expressed their molecular cross-section measured in the energy range 0.1-2.7 MeV in terms of two collision parameters M2j and C is given by: σ jjt = −
A M j2 {ln(1 − β2 ) + β2 } + C RE
…(9)
3 Results In the present paper, the total ionization crosssections have been calculated for the three molecules of interest in atmospheric science i.e hydrogen sulphide (H2S), nitrous oxide (N2O), carbon dioxide (CO2). These molecules are important constituents that found at different altitudes of the atmosphere17. From Eq. (5), the ionization cross-sections σj is calculated for each orbit of the molecules for incident energy E varying from threshold to ionization energy to 10 MeV. The cross-section for each orbital added to obtain the total cross section for the whole molecule. Table 1 presents the values of binding energy (I), average kinetic energy (U), electron occupation number (N) of each orbit of molecules19 under consideration. Table 2 presents the calculated collision parameters C and M2j for considered molecules10 which are obtained by employing the COOS given by Khare et al.16 at large E. Figure 1 shows the comparison of present crosssections for N2O along with the experimental data given by Rapp and Englander-Golden3, Iga et al.4, Lindsay et al.5 and theoretical data set of Kim et al.6, Joshipura et al7. Figure 1 shows that the present cross-sections agrees with experimental data within 20% however for E>70 eV they do not differ by more than 10%. The present theoretical cross-sections overestimate
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Table 2 — C, M2j: Collision parameters Molecules C N2O H2S CO2
Calculated M 2j
58.57 60.94 57.63
5.62 3.92 5.32
Experimental C M 2j ---42.19 57.19
---5.03 5.75
Fig. 1 — Comparison of the present theoretical total cross-section and experimental total cross-sections for N2O. Solid curve, the present work; squares, experimental data by Rapp and EnglanderGolden3; triangles, experimental data by Iga, Rao and Srivastava4; rectangles, experimental data by Lindsay, Rejoub and Stebbings5; medium-dashed curve, theoretical data by Kim, Hwang, Weinberger and Rudd6; long-dashed curve, theoretical data by Joshipura, Gangupadhyay and Vaiswav7
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the cross-section obtained by Joshipura et al.7 for E<40 eV and E>150 eV while underestimate between two energies 40 eV
orbital (3p orbital of S). It is noted that the present cross-section are very close to the cross-section obtained by Kim et al.6, while the expressions for both cross-sections are different. The present total cross-sections are in well agreement with those measured by Rao and Shrivastava8 within 5% for energy E < 200 eV but it is lower for energy E> 200 eV. The experimental cross-section measured by Lindsay et al.5, Belic and Kurepa9 at energy range E> 50 eV disagrees with present theoretical data. The data by Belic and Kurepa9 agree with theory when their peak value is renormalized. Figure 4 shows the present theoretical calculation for H2S at E>10 keV. The total cross-section agrees well with experimental data measured by Reike and Prepejchal10. The calculated values of M2j and C obtained from at 1 MeV are 3.92 and 60.94, respectively. All these values of the collision parameter do not change with the increase of E. Figure 5 shows the total cross-section for CO2, again it exactly matches with that of Hwang et al11. The experimental data is available from the study
Fig. 2 — Present theoretical total cross- section for N2O Fig. 4 — Comparison of the present theoretical total cross-section to experimental data for H2S. Solid curve, present work; squares, experimental data by Reike and Prepejchal10
Fig. 3 — Comparison of the present theoretical total cross- section to experimental total cross-section for H2S. Solid curve, present cross-section with a reduced value by U/3 for the valence molecular orbit; long-dashed curve, present cross-section; medium dashed curve, theoretical data by Kim, Hwang, Weinberger and Rudd6; rectangles, experimental data by Lindsay, Rejoub and Stebbings5; triangles, experimental data by Rao and Shrivastava8; squares, experimental data by Belic and Kurepa9
Fig. 5 — Comparison of the present theoretical total cross-section to experimental total cross-section for CO2. Solid curve, present work; dashed curve, Hwang, Kim and Rudd11; squares, experimental data by Hudson, Vallance and Harland13; triangles, experimental data by Lindsay and Mangan12
KUMAR et al.: TOTAL IONIZATION CROSS-SECTIONS OF ATMOSPHERIC MOLECULES
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energy range. To the best of our knowledge, this is the first calculation for H2S, N2O and CO2 over a wide energy range from threshold to 10 MeV. The application of the present model to the ionization of other molecules and atoms, including inner-shell and dissociative ionizations is of interest.
Acknowledgement Authors are grateful to the University Grants Commission, New Delhi for financial support. Fig. 6 — Comparison of the present theoretical total cross-section to experimental data for CO2. Solid curve, present work; squares, experimental data by Reike and Prepejchal10
undertaken by Lindsay and Mangan12 and Hudson et al13. Present calculations agree very well with the data of Lindsay and Mangan12 within 8%. Figure 6 shows the theoretical calculation for energy range 0.1-10 MeV. The theoretical data with the experimental data of Reike and Prepejchal10 available in the range 0.1-2.7 MeV have been compared in Fig. 6. The present values of M2j and C obtained at 1 MeV are 5.32 and 57.63, respectively. These values are about 6% and 1% lower than the corresponding experimental values of Reike and Prepejchal10. The agreement between the experimental and theory is good, although theory has a tendency to underestimate the cross-sections.
4 Conclusions From the present study, it is concluded that the theoretical predicted and measured value of the total cross-sections for H2S, N2O, CO2 molecules are in good concurrence. Furthermore, we concluded that a slight modification in Saksena et al.14 model have considerably improved the agreement between the experimental and theoretical data at low energy. It is found that σjT is very close to the σjt while σjKBB and σjKMB are different from the present σjpBB and σjpMB. The reason for this is given by Khare et al20. At higher values of energy, there is hardly any difference between the present and the data measured by Reike and Prepejchal10. Thus, the experimental data is in good agreement with the present data over a wide
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