Momentum-space Calculation Of Electron-impact Molecule In Distorted-wave Born Approximation

Electron emission is one of the important processes that occurs when fast electron beams impact on molecular targets. The investigation of the electron-molecule collision processes plays an essential role in understanding the molecular structure and the motion of the electrons of the molecule. Studying the differential cross sections of these processes can deliver information, which is necessary in larger domains, such as astrophysics, upper-atmosphere, plasma and radiation physics, and charged-particle detector design.Over the past two decades, the extensive studies of low and intermediate energy (e,2e) reaction on atoms has been carried out to calculate the triple differential cross sections(TDCS), such as the DWBA, CCC and BBK methods. But the investigation of molecular species has advanced in a slower way than the atomic one.On the experimental said, it has not been possible yet to prepare the molecular target in a particular rovibrational state. Another great problem to be solved is the finite resolution of the electron beam . In general, the rotational and the vibrational state of the molecules cannot be resolved during a collision experiment. Pioneering measurements of TDCS for electron-impact ionization were performed for molecular hydrogen at intermediate incident energies by Jung et.al. in the coplanar asymmetric geometry in 1975. Subsequently, Cherid et.al reported a set of TDCS in the coplanar asymmetric geometry at much higher impact energies (~4keV) in 1989. Murry presented the TDCS in the coplanar asymmetric and coplanar symmetric geometries at low energies in 2005.On the theoretical side, the major difficulty is due to the nonspherical nature of the incident electron and molecule interaction potential, the description of the continuum electron must be in the field of multi-center coulomb or distorted centers. Some earlier studies based on the first order of the Born series and expressed the continuum electrons by plane waves have been performed for the (e,2e) process with molecular target successfully at high incident electron energies(keV). Examples of these methods are the plane-wave impulse approximation (PWIA) and the encounter approximation (EA). The plane wave approximation is not valid as the incident energy decreased. At low and intermediate energies, the distorted-wave velocity approximation(DWVA),coulomb-wave velocity approximation (CWVA),distorted-wavelength approximation(DWLA)and coulomb-wavelength approximation(CWLA)were developed based on the distorted-wave Born approximation in 1988. These methods have been used to calculate the TDCS for the electron-impact ionization of molecular hydrogen at an incident energy of 250eV, in which only the ejected electron is described by a distorted wave. The distorted-wave approximation (DWA) has been developed by Monzani , in which the distorted wave functions were calculated using the Schwinger- variational iterative method in the static-exchange potential field. The DWA method has been applied to calculate the TDCS of electron-impact ionization of H2 at 250 eV incident energy. Recently, the two-effective center approximation(TEC), the two-center continuum approximation(TCC) and the molecular BBK-type approximation (MBBK) were proposed and performed to calculate the TDCS at incident energies ranging from 100eV to 4keV for electron-impact ionization of H2. In these models, the eject electron wave function is constructed by a product of a plane wave centered in the center of the molecule mass center and a continuum factor. More recently, the distorted wave impulse approximation with orientation averaged molecular orbitals (DWIAOA) and the molecular three-body distorted wave approximation(M3DW) were applied to calculate TDCS for electron-impact ionization of H2 at low incident energies(<100eV), in which the distorted waves were obtained by solving the Shr o?? dinger equations with a spherically averaged static potential and a local exchange potential. Although those different varieties of approximations have been performed to calculate the TDCS for electron-impact ionization of H2, significant disagreements between the experimental and the theoretical results are still exist.In present work, a momentum-space molecular distorted-wave approximation (MDWBA) is developed. In the MDWBA model, the incident, the scattered, and the ejected electron wave functions are obtained from solving the Lippmann-Schwinger Equations. The T-matrix in the Lippmann-Schwinger Equations is an elastic scattering transition matrix that has been calculated in the static-exchange-optical (SEO) model. Using this approach, we calculated the triple differential cross sections for electron-impact ionization of hydrogen molecule in the coplanar asymmetric geometry at incident energies of 100 eV and 250 eV, and compared with the existing experimental and theoretical data.From the comparison, the present calculations are giving an overall good representation of the experimental data. It is seen that the agreement of the position for the binary peaks predicted by MDWBA with experimental measurements is very good, except only slightly shifted to the larger angles with respect to experiment. The positions of the binary peak obtained from the MBBK calculations are slightly shifted to larger angles with respect to experiments, and the positions of TEC model are shift nearly 20o to greater angular values. The shift might be attributing to the interaction of the projectile and the ejected electron. The exchange amplitude has been taken into account in both MBBK and MDWBA methods. The repulsive character of the electron-electron interaction in the final state has been included in the descriptions of the exchange amplitudes, whereas we neglect the post collision correlation of the two outgoing electrons in present model. The absence of multichannel effects in the present calculation is the possible reason that cause the discrepancies between the experiments and present results.In addition, we can find a remarkable feature that is the recoil peak predicted by the present results but not appeared in other results. To our knowledge, there are no other experimental data available for comparison in this domain. Gao had said: the typical fully differential cross sections has a binary peak and a recoil peak and as the energy of the incident projectile decreases, the recoil peak becomes increasingly important. The physical picture of the recoil is a double scattering in which the projectile has a binary collision with the atomic electron and then the atomic electron elastically scatters from the ion to the backward direction. The scattering processes have been described in MDWBA method, thus the MDWBA calculation results can exhibit the recoil peak in TDCS.Reasonable agreement is found between the result calculated by MDWBA method and the measurements. It can been seen that the description of the MDWBA method is feasible. However, there also have some discrepancies between the present results and the experimental results, that should be improved with the development of a complete theory taking all channels into account. We believe that this method must be used extensively in future.With the continuous improvement of the calculation conditions, we believe that this approach will certainly be able to produce a more accurate theoretical results. However, the current simulation of the collision environment in the laboratory is very difficult. So far, the experimental data of electronic and molecular collisions (e, 2e) reaction for most of the molecular system is very little. We hope that with the continuous improvement of experimental conditions, there will be more accurate experimental data to test existing theoretical results.