A semi-rigorous method Modified single center additivity rule msc-ar for calculating various total cross sections

1. A semi-rigorous method Modified single center additivity rule msc-ar for calculating various total cross sections Minaxi Vinodkumar Department of Physics and Astronomy, The Open University, Milton Keynes, MK7 6AA, UK. V. P. & R. P. T. P. Science College, Vallabh Vidyanagar – 388 120, INDIA

2. Outline of the talk • Why this work? • Theoretical Methods Employed • SCOP & CSP-ic and DM formalism • Theory • Results • Summary & Conclusion • Thanks

3. Why this work ? • Applications of e-atom / molecule CS to, • atmospheric sciences (ozone, climate change etc.) • plasma etching • understanding & modeling plasmas in fusion devices • In radiation physics (medical science) etc. • Electrons: an effective source • Difficulty in performing experiments • Expensive • Limitation to targets • Time consuming • Limitations to accurate theoretical methods: • Slow and tedious calculations • Limitation to energy • Limitation to targets Need for simple, reliable and fast calculations

4. SCOP Method • Formulation of the Complex Optical Potential, Vopt = VR + iVI Complex Optical Potential Real Imaginary Short Range Long Range Absorption Static Exchange Polarization Modified Model RHF WF Hara Buckingham Energy Dependent Final Form of the Complex Optical Potential Vopt = Vst + Vex + Vpol + i Vabs

5. Various Model Potentials Real Potentials Static Potential:The potential experienced by the incident electron upon approachingafield of an undisturbed target charge cloud. The static charge density can be calculated using HF wave functions given in terms of STO. Cox and Bonham gave analytical expression for static potential involving the sum of Yukawa terms

6. Exchange Potential: This potential arises due to exchange of the incident electron with one of the target electrons. It is short ranged potential. Hara adopted free electron gas exchange model. He considered the electron gas as a Fermi gas of non interacting electrons when the total wave function is antisymmetrised in accordance with Pauli’s exclusion principle. Polarisation Potential: This potential arises due to the transient distortion produced in the target charge cloud due to the incoming incident electron. We used the correlation polarization potential at low energy and dynamic polarization potential given by Khare et al at high energies.

7. Where rc is the energy dependent cut off parameter. Absorption Potential : This potential accounts for the removal or absorption of incident particles into inelastic channel. The imaginary part of the absorption potential accounts for the total loss of the scattered flux into all the allowed channels of electronic excitation and ionization. We use the quasi free, Pauli blocking, dynamic absorption potential given by Staszewska which is function of charge density local kinetic energy and the raadial distance r. We have modified the absorption potential to account for screening of inner electrons by the outer ones.

8. SCOP method continued… SCOP method1,2 for QT Formulate Schrödinger eqn using the SCOP Solve this eqn numerically to generate the complex phase shifts using the “Method of Partial Waves” Obtain the Qel and Qinel (Vibrationally & rotationally elastic) Then the QT is found through, The grand TCS, QTOT is, Present energy range  From ionization threshold to 2keV 1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211 2 K N Joshipura et al, Phys. Rev. A, 69 (2004)022705

9. Various Additivity Rules Simple Additivity Rule (AR): The total cross section for a molecule AB is given by Q(AB) = Q(A) + Q(B) This is crude approximation and works for few molecules with larger separation between the atoms. Modified Additivity Rule(MAR): The individual cross sections are modified to incorporate the molecular properties such as structure and ionization energy and the polarizability of the target.

10. Various Additivity Rules Single Center Approach (SC): Additivity methods do not take into account the bonding between the molecules. Single center approach takes into account the bonding of the atoms. The molecular charge density which is major input for obtaining the total cross section. For the diatomic molecule AB, the simplest additivity rule for the charge density of the molecule is This again does not include the bonding of atoms in molecule. For the hydride AH, the charge density is made single center by expanding the charge density of lighter H atom at the centre of heavier A atom for e.g. C, N or O.

11. Various Additivity Rules When diatomic molecule is formed by covalent bonding there is partial migeration of charge across the either atomic partners. For the polyatomic complex molecules, we use group additivity rule MSC-AR. The number of centres and their position will depend on the structure of the molecule. In case of C2H6 molecule, we identify two scattering centers at the center of each carbon atom. The charge density of all three hydrogen atoms is expanded at the centre of Carbon atom and the total charge density is then renormalised to get total number of electrons in the molecule.

12. SPU-VVN CSP-ic Method The Complex Scattering Potential-ionization contribution, CSP-ic method1,2 for Qion In CSP-ic method the main task is to extract out the total ionization cross section from the total inelastic cross section. The first term on RHS is total cross section due to all allowed ionization processes while the second term mainly from the low lying dipole allowd transistions which decreases rapidly at high energies. 1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211 2 K N Joshipura et al, Phys. Rev. A, 69 (2004)022705

13. SPU-VVN CSP-ic Method The Complex Scattering Potential-ionization contribution, CSP-ic method1,2 for Qion The CSP-ic originates from the inequality, Now we will define a ratio, Using R(Ei ) we can determine the Qion from Qinel This method is called CSP-ic 1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211 2 K N Joshipura et al, Phys. Rev. A, 69 (2004)022705

14. SPU-VVN CSP-ic method continued… Above ratio has three conditions to satisfy: where subscript ‘p’ denotes the value at the peak of Qinel This ratio proposed to be of the form, 1 – f (U), ; & where This is the method of CSP-ic. Using this energy dependant ratio R ,Qionfrom Qinel can be extracted.

15. SPU-VVN Results Figure 1: e – CH41 1 K N Joshipura et al, Phys. Rev. A, 69 (2004)022705

16. Results Continued… Figure 2: e – O3 at 100 eV1 1K N Joshipura et al, J Phys. B: At. Mol. Opt. Phys. 35 (2002) 4211

17. Results Continued… Figure 3: Plasma molecules

18. Summary & Conclusion on SCOP & CSP-ic • Results on most of the molecules studied shows satisfactory agreement with the previous investigations where ever available. • First estimates of the Qion for many aeronomic, plasma & organic molecules are also done. We believe it to be reliable from our previous results. • Advantages • Quantum mechanical approximation • Calculating different CS from the same formalism • Simple and fast method • First initiation to extract Qion from Qinel • Disadvantages • Spherical approximation • Lower energy limit of ~10eV • Semi empirical method to find Qion

19. DM Formalism The original concept of Deustch and Meark formalism was developed for the calculation of the atomic ionization cross sections. It was then modified for the molecular targets. In DM formalism only direct ionization processes are considered. That is prompt removal of a single electron from the electron shell by the incoming electron therefore it is not possible to distinguish between single and multiple ionization where inner shell ejection occurs. The semiclassical formula used for the calculation of the ionization cross sections for the atoms was given as

20. Where is the mean square radius of the (n,l) subshell and are the appropriate weighing factors given by Deustch et al. F(u) is the energy dependence of ionization cross section while zeta depends on the orbital angular momentum quantum number of the atomic electrons. DM formalism can easily be extended to the case of molecular ionization cross section provided one carries out a Mulliken or other molecular orbital population analysis which expresses the molecular orbitals in terms of atomic orbitals. Where summation is now carried out over molecular orbitals j.

22. Thanks Professor K N Joshipura Bobby, Chetan, Bhushit and Chirag Department of Physics, Sardar Patel University, Vallabh Vidyanagar 388 120, India Professor N J Mason Director, CeMOS, Open University, Milton Keynes, United Kingdom Professor Jonathan Tennyson Head, Dept of Physics & Astronomy, University College London, United Kingdom