Differential scattering in proton collisions with molecular hydrogen

Eur. Phys. J. D (2022)76:129 https://doi.org/10.1140/epjd/s10053-022-00442-2 THE EUROPEAN PHYSICAL JOURNAL D Regular Article – Atomic and Molecular Collisions Differential scattering in proton collisions with molecular hydrogen Corey T. Plowman1,a , Ilkhom B. Abdurakhmanov2 , Igor Bray1 , and Alisher S. Kadyrov1,b 1 Department of Physics and Astronomy and Curtin Institute for Computation, Curtin University, GPO Box U1987, Perth, WA 6845, Australia 2 Pawsey Supercomputing Centre, 1 Bryce Ave, Kensington, WA 6151, Australia Received 7 April 2022 / Accepted 17 June 2022 © The Author(s) 2022 Abstract. The recently developed two-centre wave-packet convergent close-coupling approach to proton collisions with molecular hydrogen is applied to calculate various singly differential cross sections. The approach is based on an effective one-electron description of the H2 target. The angular differential cross sections for elastic scattering, total excitation and electron capture are presented. Furthermore, we calculate the singly differential ionisation cross sections as functions of the ejected-electron energy and angle, as well as projectile scattering angle. Good agreement with available experimental data is observed, providing improvement over previous theoretical investigations into the singly differential cross section for ionisation. Specific mechanisms responsible for electron emission in particular kinematic regimes are identified. It is concluded that the effective one-electron WP-CCC method is capable of providing reasonably accurate results on singly differential cross sections for all included interconnected processes taking place in p + H2 collisions. 1 Introduction Accurately modelling the various processes that take place in ion scattering from molecules is a challenging problem. The simplest example is proton scattering on molecular hydrogen, which remains an active area of research both experimentally and theoretically. One reason for this is the recent emergence of hadron therapy for cancer treatment [1] where the need for accurate stopping cross sections for ion scattering in biologically relevant molecules is of the utmost urgency [2]. In this modern cancer treatment modality, protons (or heavier ions [3]) are used to bombard the tumour site and destroy cancerous cells. This allows for the destruction of harmful tissues while sparing significantly more healthy tissue than traditional X-ray bombardment, resulting in improved effectiveness and reducing patient mortality rates [4]. This is because the majority of energy is deposited in the region of the Bragg peak near the end of the beam path. Consequently, careful planning is required to ensure the beam energy is deposited precisely at the tumour site. Treatment plans for hadron therapy are developed using Monte Carlo simulations, which rely on accurate stop- a e-mail: (corresponding author) b e-mail: 0123456789().: V,-vol ping power cross sections for collisions of the beam ions with biological molecules. The water molecule is used as a reference target in these simulations [1]. Hence, there is an urgent need for accurate stopping power cross sections on proton collisions with H2 O. The path to developing theories that can accurately calculate cross sections for ion collisions with water starts with the simpler H2 molecule target. While many methods have been developed to address scattering from atomic targets (see Refs. [5,6] for two most recent reviews of the field of ion-atom collisions), molecular targets are fundamentally more difficult to describe theoretically. The multicentre nature of molecules significantly complicates their description. The obvious starting place for ion-molecule investigations is proton scattering on the H2 target. This is the simplest homonuclear molecular target. Hasan et al. [7] performed a kinematically complete experiment on ionisation of H2 by 75 keV proton impact. They found large discrepancies between experiment and theory for the fully differential cross section in various kinematic regimes. Furthermore, they found large discrepancies between distorted-wave and a continuum distorted-wave eikonal initial-state calculations. These authors suggested that the reason for these discrepancies could be the absence of strong coupling between the ionisation and capture channels in the aforementioned theoretical approaches. Our ultimate goal is to investigate this problem using the coupledchannel formalism. In this work, we start from the singly differential cross sections. 123 129 Page 2 of 15 Experimental investigations of the singly differential cross sections for proton scattering on molecular hydrogen mainly focus on the energy and angular distribution of secondary electrons produced through ionisation. However, Sharma et al. [8] also presented experimental measurements of the angular differential cross section for electron capture at intermediate incident energies of 25 and 75 keV. In this energy region, the velocity of the projectile is comparable to the orbital speed of the target electrons. Additionally, strong coupling between reaction channels has a significant effect on the scattering outcome. As a result, this is the most difficult energy region to describe theoretically. Angular differential cross sections for single electron capture from H2 by proton projectiles were calculated by Igarashi et al. [9] using the continuum distorted wave eikonal initial state (CDW-EIS), and various other eikonal methods, within an effective one-electron model at 25 and 75 keV. Recently, these authors have extended their method to include the effects of vibrational motion within their distorted-wave model, producing differential electron capture cross sections at 25, 75, and 300 keV [10]. Agreement with the experimental data of Sharma et al. [8] is mixed. Their calculated angular differential cross sections for electron capture into the ground state agree well with the experimental data for scattering angles less than 0.5 mrad. However, at larger scattering angles discrepancies are seen between various approaches based on changing the target description. In particular, using a linear combination of atomic orbitals (LCAO) approach, they were able to deduce information about the final vibrational state of the residual ion; however, the angular differential cross sections for electron capture found using this model are very similar to results using a fixed nuclei (FN) approximation. In fact, they find that using the two-effective-centre (TEC) method gives improved agreement with the experimental data despite a less detailed description of the molecular nature of the target. Adivi [11] also used an effective one-electron target description to calculate the differential cross section for electron capture at 300 keV within the first-order Born approximation with correct boundary conditions (B1B). Ghanbari-Adivi and Sattarpour [12] used the four-body eikonal approximation (EA) at 100 and 300 keV. However, perturba (...truncated)