K-shell ionization cross sections of atoms by muons and pions

Eur. Phys. J. D (2019) 73: 120 https://doi.org/10.1140/epjd/e2019-90623-1 THE EUROPEAN PHYSICAL JOURNAL D Regular Article K-shell ionization cross sections of atoms by muons and pions? Takeshi Mukoyamaa and Károly Tőkési Institute for Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Bem tér 18/c, 4026 Debrecen, Hungary Received 14 November 2018 / Received in final form 27 March 2019 Published online 11 June 2019 c The Author(s) 2019. This article is published with open access at Springerlink.com Abstract. The K-shell ionization cross sections of copper and silver atoms by muons and pions with negative and positive charge have been calculated with the classical trajectory Monte Carlo method and the planewave Born approximation with corrections for the Coulomb-deflection and binding-energy effects. Both results are in good agreement with each other. The obtained cross sections for muons on copper are also compared with the coupled-channels calculations. The dependence of the K-shell ionization cross sections on the sign of the projectile as well as on the projectile mass is discussed. 1 Introduction The inner-shell ionization processes by charged-particle impact have been extensively studied both theoretically and experimentally as a basic process in atomic physics as well as applications to various fields, such as solid-state physics, radiation physics, astrophysics, plasma physics and chemistry [1]. It is well known that the experimental ionization cross sections in high-energy region can be described with the theoretical models based on the firstorder Born approximation (FBA), such as the plane-wave Born approximation (PWBA) [2] and the semi-classical approximation (SCA) [3]. In the FBA, the ionization cross section is proportional to Z12 , where Z1 is the projectile charge. This fact means that the cross section does not depend on the charge of the projectile and the ionization cross sections for antiparticles are same as those for particles. However, for low- and intermediate-energy projectiles the experimental values deviate from the FBA results and higher-order corrections should be introduced for accurate description of experimental observations. In the lowenergy region the projectile trajectory is deflected due to the Coulomb field of the target nucleus. In addition, the initial bound states of the target electrons are distorted due to the presence of the projectile and their binding energies are changed. On the other hand, at the intermediate and high velocities the polarization effect of the target electron orbital caused by the projectile becomes important. Owing to these effects the inner-shell ionization cross sections deviate from the simple Z12 scaling law and change when the sign of the projectile changes. It is ? Contribution to the Topical Issue “Many Particle Spectroscopy of Atoms, Molecules, Clusters and Surfaces (2018)”, edited by Károly Tőkési, Béla Paripás, Gábor Pszota, and Andrey V. Solov’yov. a e-mail: interesting to study the dependence of inner-shell ionization cross sections on the sign of the projectiles. Earlier experimental investigations for comparison of inner-shell ionization cross sections between particles and antiparticles have been performed by the use of electrons and positrons [4–6]. The large difference in the K-shell ionization cross sections between electrons and positrons was observed at low energies and it was explained by the trajectory of the projectile in the Coulomb field [7]. Andersen et al. [8] measured K-shell ionization cross sections of helium by proton and antiprotons at the LowEnergy Antiproton Ring (LEAR) at CERN. They found large difference in the ratios of double to single K-shell ionization cross sections between protons and antiprotons, but their single K-shell ionization cross sections for positively and negatively charged particles are same within the experimental errors. Since then the experimental and theoretical studies on particle and antiparticle collisions have been reported and the results for proton-antiproton pair as well as electronpositron pair have been discussed in many reviews [9–12]. It is well known that the difference in ionization cross sections depends on the energy and mass of the projectile as well as the target atomic number. However, almost all of the single ionization cross sections cited in the reviews are for the K-shell ionization cross sections of low-Z targets, such as H and He atoms. There have been reported the data for rare gases, for example Ne, Ar, Kr and Xe [13], but they are only for total single ionization cross sections, i.e. the sum of ionization cross sections of all shells, and contributions from each shell are not separately measured. It is interesting to study K-shell ionization of high-Z elements by different massive particle-antiparticle pairs, such as muon (µ± ) and pion (π ± ). They are considered to be more favorable projectiles to study the dependence of the ionization cross sections on the projectile charge than the electron-positron and proton-antiproton pairs. In the case of electrons, the incident particle is identical to Page 2 of 5 2 Theoretical models In the present work, we calculate the K-shell ionization cross sections by positive and negative muons and pions with two different methods: (1) the CTMC method [20] and (2) the PWBA corrected for the binding-energy change of the target electron and the Coulomb-deflection effect of the projectile [21]. 2.1 Classical trajectory Monte Carlo method The CTMC method is a non-perturbative method based on the classical dynamics and all the interactions between the colliding particles are automatically taken into consideration. In order to apply the CTMC to the atoms other than hydrogen, we used the screened hydrogenic model and the screening constant was determined according to ^ y P (rP) vp B C T (rT) ^ x b the target electron and the exchange effect is important in low-energy region. Practically the electron scattering is described as Møller scattering, while the positron scattering is known as the Bhabha scattering. On the other hand, the charge transfer channel is important for lowenergy proton-atom collisions and this process reduces the number of protons available for ionization by positively charged particles. The number of theoretical calculations for inner-shell ionization cross sections by muons and pions is rather scarce. Martir et al. [14,15] calculated the K-shell ionization cross sections for positive and negative muons on copper in the energy range from 1 to 2 MeV/amu by the use of the coupled-channels (CC) method. They showed that the Coulomb-deflection effect is quite large in comparison with the case for protons and antiprotons. Cohen [16] used the classical trajectory Monte Carlo (CTMC) method and calculated the ionization and capture cross sections for negative muons on hydrogen atom in the energy region between 3 eV and 100 keV. The similar CTMC cal (...truncated)