BEB-based models for ionisation cross sections of electron and positron impact with diatomic molecules

Eur. Phys. J. D (2024) 78 :56 https://doi.org/10.1140/epjd/s10053-024-00852-4 THE EUROPEAN PHYSICAL JOURNAL D Regular Article BEB-based models for ionisation cross sections of electron and positron impact with diatomic molecules V. Gravesa School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK Received 10 January 2024 / Accepted 19 April 2024 / Published online 10 May 2024 © The Author(s) 2024 Abstract. The ionising interactions of high-energy particles with molecules have applications in many areas. Despite this, some areas, including positron scattering, lack experimental data due to difficulties in performing experiments. Here, quick and simple methods for computing direct electron and positron impact ionisation cross sections are presented. These calculations, performed using the open-source software RAPID-CS, can provide a complete data set as a first approximation for when experimental, or more detailed computational work is not available. The cross-sectional data set includes the total, partial/fragment-specific, single-differential, average secondary electron energy and stopping power cross sections. The molecules N2 , O2 and CO were chosen to study due to the availability of positron scattering data. An overall good agreement with experimental and other computational results is presented. 1 Introduction High-energy electron and positron interactions with molecules have applications in many areas including in medical settings [1], plasma physics [2], atmospheric science [3,4] and interstellar chemistry [5]. An important high-energy scattering interaction is direct impact ionisation. In lepton scattering, this is when a molecule is ionised due to a direct impact of a lepton. In electron scattering, this is the only straightforward mechanism that can lead to ionisation. However, for positron scattering positronium formation and annihilation can also lead to ionisation. In this work, the focus is on direct impact ionisation and dissociative ionisation, if the collision energy is sufficiently high. A popular and frequently used method for computing electron impact ionisation cross section is the Binary-Encounter-Bethe method (BEB) proposed by Kim et al. [6] This is a semiempirical method that has been found to give reliable results for a range of molecules [7]. As the BEB method computes the total ionisation cross section (TICS), various modifications have been proposed which can be used to compute partial, fragmentation-specific, ionisation cross section (PICS) as well. These frequently utilise branching ratios from a mass spectrum at a given energy and generalise them over an energy range [8–11]. However, all of these methods rely on some experimental data as an input. A method proposed by Huber et al. [12] uses a simple relationship between the formation energy of the fragment and the PICS to get good approximations without the a e-mail: (corresponding author) need for experimental input. This method was applied to polyatomic molecules where, after careful selection of the fragment pathways, the method was shown to still produce good first approximations to the branching ratios [10]. This method forms the basis of the work presented here. Alternative methods to BEB are available including Deutch-Mark [13,14], Spherical Complex Optical Potential (SCOP) approaches [15] and the Jain-Khare method [16]. All of these methods focus on computing the TICS. However, a modification to the Jain-Khare method (mJK) [17] can be used to compute the PICS directly and then sum them to get the TICS. Difficulty in the mJK method arises from the need to know the dipole oscillator strength. If this is not known experimentally then it can be calculated from photoionisation cross section. However, this still relies on experimental, or computationally reliable, photoionisation cross sections to be known. Beyond electron scattering, the BEB method has been modified to compute direct positron impact ionisation. This was first done by Fedus and Karwasz [18] who devised the BEB0 and BEBW approaches but various modifications (BEBA and BEBB) were followed by Franz et al. [19]. Here, the PICS method by Huber et al. is combined with the positron scattering BEB0 and BEBA methods to produce positron PICS. The focus on BEB0 and BEBA was chosen as BEB0 seems to work best for polar molecules and BEBA for non-polar [19]. However, investigations with more molecules are needed to confirm this. More details about the ionisation event can be gained from the single-differential cross section (SDCS) which 123 56 Page 2 of 17 Eur. Phys. J. D (2024) 78 :56 is a function of the ejected electron energy. Garkoti et al. [20] proposed a method for computing the BEBSDCS. The SDCS can be used to compute the average secondary electron energy (ASEE) and the stopping cross section (SCS) [1,20]. The SCS can, for example, be used to probe the extent of potential damage to DNA [1]. Here, partial -SDCS, -ASEE and -SCS are presented for electron and positron scattering. The benefit to this dataset is that the cross sections are quick and easy to compute but also that all the cross sections are computed at the same level. As a result, their accuracy should be comparable and can be used to fill in gaps in databases. The historical lack of research into positron scattering means that the only measurements for direct positron impact ionisation are for H2 [21,22], N2 [23, 24], O2 [23], CO [23,25] and tetrahydrofuran [26]. This is unlikely to change in the coming years due to the difficulty in distinguishing between Ps formation and direct ionisation. [19] The work presented here could be used to infer the Ps formation channel from the measured total cross section. A comprehensive review of all data available for positron scattering has been done by Brunger et al. [27]. In this work, the molecules studied are N2 , O2 and CO as, aside from the positron cross sections, there is also a reasonable amount of experimental data available for electron scattering. The remaining paper is outlined as follows: In the next section, there is an explanation of the theory for all of the BEB-based methods used here. This is followed by computational details including a short description of the software implementation that enabled this work. Results for the three molecules studied are then discussed and finally, conclusions are presented. 2 Theory 2.1 Fragment formation energies The dissociation pathway and associated formation energy due to electron or positron impact ionisation are shown below. This is for an arbitrary diatomic molecule AB and leads to a single ion, AB + e−/+ → A+ + B + e− + e−/+ DA+ = E(A+ ) + E(B) − E(AB + ) + IPAB (1) the charged fragment A+ is produced with the related formation energy DA+ . Here, the formation energy is the dissociation threshold. The ground state energy of species X (where here X = A+ , B or AB + ) is shown as E(X) and IP is the ionisation potential of the parent species (AB). (...truncated)