DIFFERENTIAL SCATTERING CROSS SECTIONS FOR ELASTIC ELECTRON-MAGNESIUM SCATTERING

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1 DIFFERENTIAL SCATTERING CROSS SECTIONS FOR ELASTIC ELECTRON-MAGNESIUM SCATTERING C.V. PANDYA, P.M. PATEL 2, K.L. BALUJA 3 Department of Physics, M.G. Science Institute, Navrangpura, Ahmedabad-38009, India. 2 Department of Physics, V.P. and R.P.T.P. Science College, Vallabhvidyanagar-38820, India. 3 Department of Physics & Astrophysics, University of Delhi, Delhi 007, India cvpandya@rediffmail.com Received July 27, 2009 In the present work, theoretical studies of differential cross sections (DCS) of electron scattering by the Mg atom have been carried out at projectile energies of, 20 and 40 ev. The only experimental parameters required are the first ionization potential and the dipole polarizability of the atom under consideration for the construction of the full optical potential. After generating the full optical potential of the scattering system, we treat it exactly in a partial wave analysis in terms of a set of first-order coupled differential equations for the real and imaginary parts of the complex phase shift functions under the variable phase approach. The calculated DCS are compared with the experimental data and theoretical values. Key words: Electron-magnesium scattering, elastic, inelastic scattering cross sections.. INTRODUCTION Magnesium is a closed shell atom with two 3s electrons, and a core consisting of K and L shells. In order to understand the radiation damage of bio molecules caused by the secondary low-energy electrons, a knowledge of binary electronatom collision processes is required. Magnesium is the central atom in chlorophyll molecules which provides energy for almost all metabolic processes, and is required at a number of steps during the synthesis of DNA, RNA and proteins []. Below any inelastic scattering thresholds, the scattering of electrons from atoms can be well represented as a potential scattering problem by including the static, polarization and exchange potentials. Above these thresholds, the existence of additional exit channels for the incident particle flux means that simple potential scattering models produce an overestimate of the elastic cross sections. A simple way to take into account the open inelastic channels within the framework of a potential scattering problem is to use a simple computational approach of complex optical potential in which the imaginary part represents the absorption of flux. Rom. Journ. Phys., Vol. 56, Nos. 2, P , Bucharest, 20

2 2 Elastic electron-magnesium scattering 73 On theoretical side, most calculations are performed either by first-order perturbative calculations [2, 3] or close coupling models [4 5]. Recently, Zatsarinny et al. [6] have calculated the cross section for electron scattering from magnesium atom using a B-spline R-matrix method in which a B-spline basis is employed to represent the continuum functions in the close-coupling expansion of the scattering wave function. These more elaborate theories take into account the additional channels but at the cost of very substantial increase in the complexity of the problem and the computer resources needed. In the present work, we have employed a very simple calculation technique of complex optical potential approach. The real part of the optical potential consists of static potential [7], the exchange potential [8] which is incorporated by treating the electron cloud as a free gas, and a polarization potential [9]. The imaginary part of the optical potential represents the absorption potential that takes into account the loss of flux due to all energetically possible inelastic channels []. After generating the full optical potential of the scattering system, we treat it exactly in a partial wave analysis in terms of a set of first-order coupled differential equations for the real and imaginary parts of the complex phase shift functions under the Variable Phase Approach [] and the differential cross sections are calculated. Several sets of experimental measurements of differential cross sections (DCS) for electron scattering from neutral Mg are available [2 7]. These provide an excellent opportunity to test various theoretical methods employed for the calculation of electron atom collision processes. 2. THEORY All the major interactions of electron-atom scattering can be represented by a complex, energy dependent, optical potential V opt (r, k) as V opt (r,k) = V st (r) + V ex (r,k) + V pol (r,k) + iv abs (r,k) () where V st (r) is the static potential obtained from the DHFS function [7], V ex (r,k) is the exchange potential obtained from FEG model [8], V pol (r,k) is the polarization potential model [9] and V abs (r,k) is the absorption potential []. The only parameters required for the construction of the full optical potential are the first ionization potential and the dipole polarizability of the atom under consideration. The mean excitation energy is set equal to the first ionization potential []. The parameters used in our calculations are shown in Table. These parameters are taken from NIST Database. Table Parameters used in electron Magnesium scattering calculations Parameters for Mg Average dipole polarizability (α d ) a 0 Ionization potential energy (I.P) ev

3 74 C.V. Pandya, P.M. Patel, K.L.Baluja 3 A partial wave analysis is done by solving the following set of first-order coupled different equations for the real χ and imaginary Imχ l parts of the complex phase shift function under the variable phase approach []. χ / l (kr) = 2/k [2V R (r,k)(a 2 B 2 ) + 2V abs (r,k) AB ], (2) / Imχ l (kr) = 2/k[2V R (r,k) AB 2V abs (r,k) (A 2 B 2 )], (3) where, A = Cosh Imχ l (kr)[cosχ l (kr) j l (kr) Sin χ l (kr) n l (kr)] (4) B = SinhImχ l (kr)[sinχ l (kr) j l (kr) Cos χ l (kr) n l (kr)] (5) here j l (kr) and n l (kr) are the usual Riccati Bessel functions []. The potentials V R (r) and V abs (r) are the real and imaginary parts of the complex optical potential. Equations (2) and (3) are integrated up to a sufficiently large r, different for different l and k values. Thus the final S matrix is written as S l (k)=exp(-2im χ l ) exp (i 2 χ l ) (6) and corresponding DCS s are computed as lmax 2 l= 0 dσ = (2l+ )[ Sl(k) ] Pl(cos θ) dω 4k (7) where P l (cosθ) is a Legendre polynomial of order l RESULT AND DISCUSSION We have calculated differential cross sections for electron impact on Mg at various energies and have compared our calculated values with experimental data of Williams and Trajmar [7] and Predojevic [8], and theoretical results of Mitroy and McCarthy [4] and R-matrix results of Zatsarinny et al. [6]. Mitroy and McCarthy [4] have computed these in the coupled-channels approximation. In these theoretical works the configuration interaction wave functions are used to represent the target states which were included in the trial wave function in the close-coupling expansion. In Fig. we have presented our DCS results along with the experimental results [7, 8] and the close-coupled calculation [4, 6] at ev. Our DCS results show good agreement with the recent experimental measurements of Predojevic et al. [8]. The experimental measurements of Predojevic et al. [8] show two minima in contrast to the observation of Williams and Trajmar [7]. Beyond 20 degree the observed results of Williams and Trajmar [7] lie lower than the other curves shown. It is very encouraging to note that our simple model is capable of reproducing DCS that are in good accord with the observation. Our results are in better agreement with observation than the other theoretical work. In Fig. 2, we have depicted DCS at 20 ev. Our results are again in very good agreement with the observed results of Predojevic et al. [8]. All the curves show two minima which is quite encouraging for theoretical as well as observed results.

4 4 Elastic electron-magnesium scattering 75 Differential scattering cross sections ( -6 Cm 2 /sr) Scattering angle (deg). Fig.. Differential scattering cross sections for electron-mg scattering at ev. Differential scattering cross sections ( -6 Cm 2 /sr) Scattering angle(deg). Fig. 2. Differential scattering cross Sections for electron-mg scattering at 20 ev. Differential scattering cross sections ( -6 Cm 2 /sr) Scattering angle(deg). Fig.3. Differential scattering cross sections for electron-mg scattering at 40 ev.

5 76 C.V. Pandya, P.M. Patel, K.L.Baluja 5 The results shown in Fig. 3 are at 40 ev where all the calculations are in reasonable agreement with each other but our calculations are slightly below the experiment measurements [8]. The differences among all the results here are small due to the log nature of the depicted figures. This is a quite satisfactory picture that implies that our simple model is capable of reproducing a very good set of DCS even at moderate energy of 40 ev. 4. CONCLUSIONS We have carried out a study of electron-impact on Mg at three impact energies of, 20 and 40 ev by employing a computationally simple methodology of complex optical potential. The important ingredients of interaction of the scattering system are taken into account by including local versions of static, exchange, polarization and absorption potentials. The presented results compare favorably with the experiment. This methodology can be further exploited to investigate other electron-atom systems. Acknowledgement. University Grants Commission-India is greatly acknowledged. This research is funded to CVP by UGC under research project F. No /06. We are also thankful to O. Zatsarinny for providing us his data. REFERENCES. L. Sanche, Mass Spectrom. Rev. 2, , (2002). 2. R. Srivastava, R.P. McEachran, A.D. Stauffer, J. Phys. B 34, , (200). 3. L. Sharma, R. Srivastava, A.D. Stauffer, Phys. Rev A 78, 0470 (2008). 4. J. Mitroy, I.E. McCarthy, J. Phys. B 22, (989). 5. I.E. McCarthy, K. Ratnavelu, Y. Zhou, J. Phys. B 22, (989). 6. O. Zatsarinny, K. Bartschat, Phys. Rev A 79, (2009). 7. F. Salvat, J.D. Martinez, R. Mayol, J. Parellada, Phy.Rev. A 36, (987). 8. S. Hara, J. Phys. Soc. Jpn, 22, 7 78 (967). 9. N.T. Padial, D.W. Norcross, Phys. Rev. A 29, (984).. G. Staszewska, D.W. Schwenka, D. Thirumalai, D.G. Truhalar, Phys.Rev. A 28, (983).. A. Jain, K. L.Baluja, Phys. Rev. A 45, (992). 2. D.O. Brown, D. Cvejanovic, A. Crowe, J. Phys. B 36, (2003). 3. D.O. Brown, A. Crowe, D.V. Fursa, I. Bray, K. Bartschat, J. Phys. B 38, (2005). 4. D.M. Filipovic, B. Predojevic, V. Pejcev, D. Sevic, B.P. Marinkovic, R. Srivastava, A.D. Stauffer, J. Phys. B 39, (2006). 5. D.M. Filipovic, B. Predojevic, D. Sevic, V. Pejcev, B.P. Marinkovic, R.Srivastava, A.D. Stauffer, Int. J. of Mass Spectrom. 25, (2006). 6. B. Predojevic, V. Pejcev, D.M. Filipovic, D. Sevic, B.P. Marinkovic, J. Phys. B 4, (2008). 7. W. Williams, S. Trajmar, J. Phys. B, (978). 8. B. Predojevic, V. Pejcev, D.M. Filipovic, D. Sevic, B.P. Marinkovic, J. Phys. B 40, (2007).