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Electron impact ionization of atoms and ions is one of the most fundamental collision processes in atomic and molecular physics. Knowledge of ionization and excitation cross sections is of fundamental importance for understanding collision dynamics and electron -atom interactions, as well as in several applied fields such as radiation science, astrophysics, Auger electron spectroscopy (AES), electron energy loss spectroscopy (EELS). These areas of study need enormous and continuous quantities of data, within a certain accuracy level, for different targets over a wide range of energy values. Electron impact ionization and excitation have been actively studied by many research groups since the 1920’s. Most of the work produced was based on classical collision theory and several first principle theories were developed. The most important work in the field of electron-atom collision was made by Bethe (1930) who derived the correct form of the ionization cross section for high-energy collision using the plane-wave Born approximation (PWBA). Since then, several empirical, semi-empirical and semi-classical approximations have been proposed to describe electron impact ionization of atoms and molecules and several reviews on them were published.
Electron-atom collision can be divided into two broad types: soft or distant collisions with large impact parameters and hard or close collision with small impact parameters. The Mott theory (1930) which describes the collision of two free electrons, accounts for hard collisions well but not the soft collision. Bethe (1930) has shown that soft collision takes place essentially through the dipole interaction between the incident particle and the target electron.
In the last one hundred years, the atomic physics community devoted a lot of effort into the study of particle collision. Many methods and approximations derived throughout the times are still used in the interpretation of particle interactions. Till date a large number of investigations dealing with electron impact single ionization of atoms have been carried out. A number of reviews covering the developments in this field in recent years are available. Different quantum mechanical methods have been employed for electron impact ionization by Younger (1981, 1982). To obtain total ionization cross sections for complex targets over a wide range, a number of semi-classical and semi-empirical formulae have also been developed by Younger (1985). An elegant discussion on electron impact ionization of alkali metals have been given by McDowell (1969), Roy and Rai (1973) and others. There applied plane wave Born approximation (PWBA) to calculate electron impact ionization cross sections for atoms including exchange effect in quintal calculation. In spite of all these successes, the difficulty still lays in the calculation of electron impact single ionization cross sections for heavy atoms in quantal approximations due to mathematical complexities.
The structures observed in the experiments have been clearly explained by different theoretical approaches. In case of electron impact ionization of the alkaline earths, Peach (1966) has performed a quantal calculation for Mg. Vainshtein et al. (1971) used the Born approximation and the classical binary encounter theory (BET) to explain the experimentally observed structures for magnesium, calcium, strontium and barium. Though, these calculations have shown good agreement between theory and experiment, the calculations suffer from two deficiencies. First, the born approximation is not well suited for low incident energy in which the structures are present. Secondly, in case of binary encounter calculations, Vainshtein et al. (1971) has modified Stabler’s (1964) expression by considering the acceleration of the incident electron by the field of neutral atom but have used a δ-function velocity distribution for the bound electron and has not taken exchange into consideration. On the other hand, a binary encounter theory for the investigation of electron impact ionization cross sections of atoms has been found to be suitable as it gives reliable results consistent with the experiments. The earlier classical model did not take into account the in distinguishability of the incident and the bound electron (unsymmetrical collision model) and is not reliable at low impact energies. At low incident energies, exchange plays an important role. To remove these difficulties, a more reliable classical formalism of electron impact ionization including the effect of exchange and interference, has been given by Vriens (1966) (symmetrical collision model).Some more calculations on electron impact ionization have been reported by McFarland (1967), Tripathi et al. (1969) and Mann (1967) for the alkaline earths. Roy & Rai (1973) have used symmetrical collision model including exchange and interference to investigate the structures observed in the ionization curves of heavier alkali atoms and alkaline earths. They have used the correct Hartree-Fock velocity distribution for the bound electron to obtain single ionization cross-sections. The contribution of inner cell as well as excitation auto ionization has been explicitly included. The results obtained are in fairly good agreement with the experimental observations.
Keeping the above-mentioned fact in view we have used this symmetrical collision model of Vriens including exchange and interference the semi-classical BEA has been found useful in explaining contribution of inner shell ionization for electron impact single ionization cross sections of atoms. In the present calculations, momentum distribution function for bound electrons has been formulated using Hartree-Fock radial wave functions reported by Clementi & Roetti (1974).
There are some theoretical, whose contribution to the development of the BEA and theoretical investigations for ion-atom collision processes are particular important. Various processes can contribute to electron impact double ionization of atoms and ions depending on the incident electron energy and on the structure of parent and intermediate atomic states. For direct ejection of two outer shell electrons, two different types of mechanism are identified: shake off and two state mechanism. In addition, many indirect double ionization processes are associated with the formation of auto ionizing states following inner shells ionization or excitation. In the case of direct double ionization (DDI) via shake off process, the incident electron interacts with a bound electron and ejects it with outer bound electrons being left in the state which is not an Eigen state of the residual ion. In the subsequent relaxation process there is finite probability of a second ionization. Electron impact double ionization of atoms and ions is a four particle (one ion and three electrons) problem. In the final channel, these four charged particles interacts with each other via long range Coulomb potential and makes this many body problem extremely difficult. For this reason, it is still impossible to carry out exact calculations for these processes. The full theoretical calculation and detail experimental investigations remain scare in such cases. The most detailed description of the process is given by means of full differential cross sections allowing the analysis of angular and energy distributions for each one of the ejected or scattered electrons.
Experimental investigation of ionization cross section for metals lead to several difficulties and have been carried out by only very few experimental groups for limited number of elements. Accurate experimental measurement of multiple ionization of iron by electron impact have been carried out by Shah et al. (1993) using pulse cross beam technique incorporating time-of-flight spectroscopy of the collision products to study the electron impact ionization of ground state Fe atom within the energy range from respective thresholds to 1250 eV. Experimental data obtained by Shah et al. (1993) would not be compared with previous theoretical calculation of double ionization cross section due to non-availability of the data in the literature.
Freund et al. (1990) performed crossed beam experiment in the presence of meta-stable atom of Fe and measured cross sections that exhibit evidence of contribution from inner shell electrons. Rigorous theoretical calculation of double ionization cross section becomes very complicated as it is related with the consideration of the four charged particles in the final channel interacting through the long range Coulomb potential. Quantal calculation of the double ionization cross sections of atoms/ions by electron impact have not been reported so far. Belenger et al. (1997) have reported a semi empirical formula for evaluation of double ionization cross section of neutral atoms, and positive and negative ions by electron impact and presented results for Cu-target. The shape of the cross section is described by analytical expression and approximation parameters are estimated by fitting the model cross sections to reliable experimental data. Besides this, similar methods have been reported by Deutsch et al. (1996). Few attempts have been made to calculate electron impact double ionization cross section for light target e.g., H+, He and Li+ using Born approximation. In a promising approach the time dependent close coupling method was used by Pindzola et al. (2007) in the calculation of electron impact double ionization of cross section of He. Afterward Pindzola and his coworkers carried out calculations of electron impact double ionization cross section of Mg, Be and B+ ion using a non-perterbative time dependent close coupling method. However, such calculations are restricted so far to essentially two electrons system. Using classical binary encounter approximation (BEA), Gryzinski & Kune (1999) have derived general analytical expression for electron impact double ionization cross section of atoms with atomic number Z ≥ 20 and s or d outer shell with two electrons. They have compared their calculations only with experimental data for Ca, Sr, Ba and Hg atoms and found satisfactory agreement. However, this model is not applicable in case of Fe.
Keeping in view the above-mentioned fact, we have used the semi-classical BE symmetrical collision model of Vriens (1966) including exchange and interference in the present work along with Hartree-Fock velocity distribution for the target electrons. We have also taken into consideration of the inner shells. In the past, the BE approximation has been found successful in the calculation of electron impact single and double ionization of atoms.
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