Indian J. Phys., Vol. 85, No. 12, pp. 1761-1766, December, 2011
Calculations of total ionization cross sections for halogen compounds on electron impact from threshold to 2 keV 1
M Vinodkumar*1, H Bhutadia1, R Dave2 and B Antony3
Department of Electronics, V P & R P T P Science College, Vallabh Vidyanagar-388 120, Gujrat, India 2 Department of Physics, Anand Agricultural University, Anand-388 110, Gujarat, India 3 Department of Applied Physics, Indian School of Mines, Dhanbad-826 004, Jharkhand, India E-mail :
[email protected]
Abstract : Calculation for electron impact total ionization cross sections on halogen compounds (BF, SiF, BCl, SiCl) are performed employing Spherical Complex Optical Potential and Complex Optical Potential – ionization contribution (CSP-ic) formalisms. In this article we are presenting data for energies ranging from above threshold to 2000 eV. Our results are compared with available experimental and theoretical data wherever available. It is found that the present result gives a better account of the ionization cross sections. Keywords : Halogen compounds; ionization cross sections; excitation cross sections; complex spherical optical potential; complex scattering potential-ionization contribution. PACS Nos. : 34.50.-s, 34.80.Bm, 34.50.Gb
1. Introduction Elastic scattering and inelastic scattering phenomena are the most important tool to probe into the target and get inferences about the important characteristics of the target. In the present work, we are interested in the inelastic process and hence will restrict our discussion to this channel only. Inelastic process is a measure of the loss of incident flux into the inelastic channel, which results into two most important processes namely ionization and electronic excitations which are widely studied both experimentally and theoretically. Ionization is one of the most fundamental processes in the collision physics which deals with the dynamics and kinetics of collision. Owing to its importance in many applied fields, lot of emphasis is put on the study electron impact ionization process. Electron impact ionization cross sections for molecules and radicals are important quantities in a variety of applications as they play key role in low temperature plasma processing, fusion edge plasma, gas discharges, planetary, stellar *Corresponding Author
© 2011 IACS
1
%$
M Vinodkumar, H Bhutadia, R Dave and B Antony
and cometary atmosphere, radiation chemistry, mass spectrometry and chemical analysis [1]. In addition, quantitative modeling of gas discharges requires accurate determination of ionization cross sections. The electron impact ionization cross section of SiCl and BCl is very relevant since, along with SiCl4 and BCl3 these are the two main etch product in chlorine-based etching of silicon [2]. Moreover the electron impact ionization data are very important quantities for the understanding and modeling of the interaction of silicon-chlorine plasmas with materials. SiF finds applications in microelectronics and organic synthesis. BF is applied as dopant in ion implantation and is used in sensitive neutron detectors in ionization chambers and devices to monitor radiation levels in the Earth’s atmosphere. Electron impact ionization cross sections for SiCl is measured by Mahoney et al [2] and the only theoretical data is provided by Deustch et. al. [3] from threshold to 200 eV. For BCl there is no experimental data and theoretical data is provided by Kim et. al. [4]. For the fluorine compounds there is great difficulty in performing experiments due to its high electronegativity and hence there is dearth of experimental data. As a consequence for BF there are no experimental data to best of our knowledge in the literature and theoretical data is provided only by Kim [4]. Measurement for SiF is reported by Hayes et. al. [5] from threshold to 200 eV. In the next section we describe the theoretical methodology depicting the salient features. A detailed discussion may be obtained from our earlier papers [6–9]. 2. Theoretical methodology The electron-molecule system is represented by a complex potential,
V(r, Ei) = VR(r, Ei) + iVI (r, Ei),
(1)
VR (r , E i ) = Vst ( r ) + Vex (r , E i ) + Vpol (r , E i )
(2)
where represents various real potentials arising from the electron target interaction namely, static, exchange and polarization respectively. To evaluate these potentials we use spherically averaged molecular charge-density H(r ), determined from the constituent atomic charge densities of Bunge and Barrientos [10]. Then the total charge density (single center, renormalized incorporating the covalent bonding) is formulated by expanding the charge density at the centre of mass of each molecule [6]. The exchange potential may be derived from the static potential thus obtained employing Hara’s parameter free and energy dependent ‘free electron gas exchange model’ [11]. Finally, the imaginary part VI of the complex potential is derived from the well-known, non-empirical, quasi-free model form given by Staszeweska et. al. [12],
Vabs (r , E i ) = -H( r )
Tloc 2
æ 8F × ççç çè 10k F3 E i
ö ÷÷÷ ´ G( p 2 - k F2 - 2,) ´ ( A1 + A2 + A3 ) ÷ø
(3)
Calculations of total ionization cross sections for halogen compounds etc.
%$!
where Tloc = E i - (Vst + Vex ) , A1, A2 and A3 are dynamic functions [9] depends differently on G(x), I, , and Ei , where I is the ionization threshold of the target. p 2 = 2Ei, kF = [3F2 H(r)]1/3 is the Fermi wave vector and , is an energy parameter. Further, q(x) is the Heaviside unit step-function and , is the energy parameter which determines a threshold below which Vabs = 0, and the ionization or excitation is prevented energetically. We have modified the original model by considering as a slowly varying function of Ei around I as, D E i = 0.8 I + > E i - I . This is meaningful since , fixed at I would not allow excitation at incident energy E i £ I . After generating the full complex potential given in Eq. (3), we solve the Schrödinger equation numerically using partial wave analysis to obtain complex phase shifts that are the main ingredients used to find cross sections. Since Qinel cannot be measured directly, the measurable Qion is of more practical importance, which is contained in the Qinel. Hence, to obtain Qion, Qinel is partitioned as,
Qinel E i = å Qexc E i + Qion E i
(4)
where, the first term is the sum over total excitation cross sections for all accessible electronic transitions. The second term is the total cross sections of all allowed ionization processes induced by the incident electrons. In order to extract Qion from Qinel, a reasonable approximation can be evoked by using a dynamic ratio function,
R (E i ) =
Qion (E i ) Qinel (E i )
such that, 0 < R X 1. We assign three physical conditions that when the incident energy is less than or equal to the target, this ratio is zero as the ionization process has not energy, almost five times the energy at peak, the only ionization and hence the ratio R(Ei) approaches almost 1.
(5) to this ratio. It is apparent ionization threshold of the started. Also, at very high dominant process is the Thus,
R E i = 0 for E i £ I = RP at E i = E P @ 1 for E i >> E P
(6)
Rp is the value of R at Ei = Ep. The general observation is that, at energies close to peak of ionization, the contribution of Qion is about 70–80% of the total inelastic cross sections Qinel. This behavior is attributed to the smaller values of SQexc compared to Qion with the increase in energy beyond Ep value. However the choice of Rp in Eq. (6) is not rigorous and introduces uncertainty in the final results. It has been by now tested for large number of atoms and molecules and it is observed that the proposed uncertainty is found to be ~10% [7]. For calculating the Qion from Qinel we use the following analytical form.
1
%$"
æ C ln U ö÷ ÷÷ R E i = 1 - C1 ççç 2 + U ø÷ èç U + a
M Vinodkumar, H Bhutadia, R Dave and B Antony
(7)
where, U is the dimensionless variable defined by, U = (Ei/I ). The reason for adopting a particular functional form of f (U ) in Eq. (7) can be understood as follows. As Ei increases above I, the ratio R increases and approaches 1, since the ionization contribution rises and the discrete excitation term in Eq. (4) decreases. The discrete excitation cross sections, dominated by dipole transitions fall off as their contribution decreases at higher energies, while the contribution of total ionization cross sections increases as energy increases beyond ionization threshold. Accordingly the decrease of the function f(U ) must also be proportional to ln(U )/U in the high range of energy. However, the two-term representation of f(U ) given in Eq. (7) is more appropriate since the first term in the brackets ensures a better energy dependence at low and intermediate energy, Ei and the second term is the Born Bethe term and governs the situation at high energies. The three conditions stated in Eq. (6) are used to determine these three parameters and hence the ratio R. This method is called the Complex Scattering Potential –ionization contribution, (CSP-ic). Having obtained Qion through CSP-ic, the summed excitations cross sections SQexc can be easily calculated using Eq. (4). We note that in view of the approximations made here, no definitive values are claimed, but by and large our results fall within the experimental error limits in most of the cases. 3. Results and discussions The theoretical approach of SCOP along with our CSP-ic method discussed above offers the determination of the total ionization cross sections, Qion along with a useful estimate on electronic excitations in terms of the summed cross section 5Qexc. Present data of total ionization sections results for the halogen compounds are presented and compared with available theoretical as well as experimental results through Figs. 1–4. In Fig. 1 we have depicted electron impact total ionization cross section for BF molecule and compared with the lone theoretical values of Kim et al [4]. Overall both data maintain same shape and value till peak beyond which present result seems to be shifted slightly towards left and underestimates in comparison with the data of Kim et. al. [4]. In absence of other data, no more conclusions could be drawn for this target. Figure 2 shows the comparison of present Qion for SiF molecule with the only measurement of Hayes et. al. [5] and the theories of Kim et. al. [4] and Deustch et. al. [3]. Present results finds very good comparison with measurement of Hayes et. al. [5] till around 100 eV and then falls below the experiments. However, present data compares very well with the theoretical results after 100 eV. The theoretical data of Deustch et. al. [3] underestimates almost in the entire energy range.
Calculations of total ionization cross sections for halogen compounds etc. 3.0
8
e - BF Present Kim
2.5
%$# e - SiF Presetn Kim Deustoh Hayes
7 6
Q ion[Â2 ]
Q ion[Â2 ]
2.0
1.5
5 4 3
1.0
2 0.5 1 0.0 10
100 Ei (eV)
Figure 1. Total ionization cross sections for e-BF scattering in Â2. Solid line ® Present Qion, Dash line ® Kim et. al. [4]. 6
0
1000
10
10
Present Kim
e - SiCI
9
Presetn Deustoh Mahoney
8 7
4
6 3
Q ion[Â2 ]
Q ion[Â2 ]
1000
Figure 2. Total ionization cross sections for e-SiF scattering in Â2. Solid line ® Present Qion, Dash line ® Kim et. al. [4], Dash dot dot line ® Deustch et. al. [3], Star ® Hayes et. al. [5].
e - BCl
5
100 Ei (eV)
2
5 4 3 2
1
1 0 10
100 Ei (eV)
1000
Figure 3. Total ionization cross sections for e-BCl scattering in Â2. Solid line ® Present Qion, Dash line ® Kim et. al. [4].
0 10
100 Ei (eV)
1000
Figure 4. Total ionization cross sections for e-SiCl scattering in Â2. Solid line ® Present Qion, Dash line ® Deustch et. al. [3], Star ® Mahoney et. al. [2].
We report our Qion for BCl molecule along with the measurement of theoretical data of Kim et. al. [4] in Fig. 3. It is worth noting that the comparison looks very similar to that of the plot for BF molecule except at the energies below the peak of ionization. Here our value underestimates the data provided by Kim et. al. [4]. Figure 4 represents the comparison of the present total ionization cross sections for e-SiCl scattering along with other available data. Unlike earlier targets, here we find quite large deviation between the data available from the literature. The experimental results from Mahoney et. al. [2] gives surprisingly large cross section and over estimate both the theoretical values by about 1.6 times at the peak. A present result
1
%$$
M Vinodkumar, H Bhutadia, R Dave and B Antony
gives a better comparison with the other theoretical data of Deustch et. al. [3]. However, it is interesting to note that peak of ionization cross sections falls at same incident energy for all the data reported here. The jump in Qion by Mahoney et. al. [2] is not clearly warranted for. Since no other experimental data is available, it is quite difficult to make any conclusions. 4. Conclusion We have performed calculations to obtain total ionization cross sections for halogen compounds as discussed above. The well known SCOP and CSP-ic formalisms were employed to perform these computations. The results obtained are presented and are compared with other available measurements and theories. Unavailability of required data set, especially reliable measurements makes this study very imperative, since most of the previous studies are fragmented. The values presented here can be considered reliable since the method employed here has been successfully tested for variety of targets from atoms to radicals and to heavier molecules. In view of the fact that, present targets are halogen compounds, we are quite sure that the data presented here are consistent and can be further utilized to perform modeling in technological systems. Also, we hope that our effort will encourage experimentalists to perform more measurements of these important targets. Acknowledgment MVK is thankful to University Grants Commission, New Delhi for major research project under which this work has been carried out. References [1]
T D Mark and G H Dunn (eds.) Electron impact Ionization (Wein : Springer) (1985)
[2]
J Mahoney, V Tarnovsky and K H Becker Eur. Phys. J. D46 289 (2008)
[3]
H Deutsch, K Becker and T D Mark Int. J. Mass Spec. 167/168 503 (1997)
[4]
http://physics.nist.gov/PhysRefData/ Ionization/molTable.html (2005)
[5]
T R Hayes, R C Wetzel, F A Baiocchi and R S Freund J. Chem. Phys. 88 823 (1988)
[6]
M V Kumar, K N Joshipura, C G Limbachiya and B K Antony Eur. Phys. J. D37 67 (2006)
[7]
M Vinodkumar, C Limbachiya, K Korot and K N Joshipura Eur. Phys. J. D48 333 (2008)
[8]
M Vinodkumar, C Limbachiya, B Antony and K N Joshipura J. Phys. B : At. Mol. Opt. Phys. 40 3259 (2007)
[9]
M Vinodkumar, C Limbachiya, K Korot, K N Joshipura and N Mason Int. J. Mass Spec. 273 145 (2008)
[10]
C F Bunge, J A Barrientos and A V Bunge At. Data Nucl. Data Tables 53 113 (1993)
[11]
S Hara J. Phys. Soc. Japan 22 710 (1967)
[12]
G Staszewska, D W Schewenke, D Thirumalai and D G Truhlar Phys. Rev. A28 2740 (1983)
