1. Introduction

There is an ongoing in the theory of interaction of charged particle beams with plasmas. Although most theoretical works have reported on the energy loss of ions in a plasma without magnetic field, the strongly magnetized case has not yet received as much attention as the field-free case. The energy loss of ion beams and the related processes in magnetized plasmas are important in many areas of physics such as transport, heating, magnetic confinement of thermonuclear plasmas, and astrophysics. The range of the related topics includes ultracold plasmas [1, 2], the

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

cooling of heavy ion beams by electrons [3–12], as well as many very dense systems involved in magnetized target fusions [11], or heavy ion inertial confinement fusion (ICF).

with corresponding Larmor radius indicates that to experience a strong influence of the

Stopping Power of Ions in a Magnetized Plasma: Binary Collision Formulation

endeavors to lead to the very first unambiguous and genuine identification of an experimental

Motivated by these recent developments, our purpose is to investigate the SP of an ion moving in a magnetized plasma in a wide range of the value of a steady magnetic field. The present paper is based on our earlier studies in Refs. [8, 24, 44, 45] where the second-order energy transfers for individual collisions of electron-ion [8, 24, 44] of any two identical particles, like electron-electron [44], and finally of two gyrating arbitrary charged particles [45] have been calculated with the help of an improved BC treatment. This treatment is—unlike earlier approaches of, e.g., Refs. [9, 42]—valid for any strength of the magnetic field. As the first application of the theoretical BC model developed in Refs. [8, 24, 44, 45], we have calculated in Ref. [47] the cooling forces on the heavy ion beam interacting with a strongly magnetized and temperature anisotropic electron beam. It has been shown that there is a quite good overall agreement with both the CTMC numerical simulations and the experiments performed at the ESR storage ring at GSI [48–50].

In Section 2 we introduce briefly a perturbative binary collision formulation in terms of the binary force acting between an ion and a magnetized electron and derive general expressions for the second-order (with respect to the interaction potential) stopping power. In contrast to the previous investigations in Refs. [8, 24, 44, 45], we here consider the (macroscopic) stopping force which is obtained by integrating the binary force of an individual electron-ion interaction with respect to the impact parameter and the velocity distribution function of electrons. That is, the stopping force for monoenergetic electrons is folded with a velocity distribution. The resulting expressions involve all cyclotron harmonics of the electrons' helical motion and are valid for any interaction potential and any strength of the magnetic field. In Section 2.4 we present explicit analytic expressions of this second-order stopping power for the specific case of a regularized and screened interaction potential [51, 52] which is both of finite range and less singular than the Coulomb interaction at the origin and which includes as limiting cases the Debye (i.e., screened) and the Coulomb potentials. For comparison of our expressions with previous approaches, we consider in Section 3 the corresponding asymptotic expressions for large and small ion velocities and strong and vanishing magnetic fields. The analytical expressions presented in Section 2.4 are evaluated numerically in Section 4 using parameters of the envisaged experiments on ion stopping [46]. In particular, we compare our approach with the CTMC simulations. The results are summarized and discussed in Section 5. The regularization parameter and the screening length involved in the interaction potential are briefly specified and discussed in Appendix A.

Let us consider two point charges with masses m, M and charges �e, Ze, respectively, moving in a homogeneous magnetic field B ¼ Bb. We assume that the particles interact with the potential �Z=<sup>e</sup> <sup>2</sup>U rð Þ with <sup>=</sup><sup>e</sup> <sup>2</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>=4πe0, where <sup>e</sup><sup>0</sup> is the permittivity of the vacuum and

. We expect these

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http://dx.doi.org/10.5772/intechopen.77213

magnetic field, the electron density should be comparable with a few 1016 cm3

magnetic signature for nonrelativistic ion stopping in plasmas.

2. Theoretical model

2.1. Binary collision (BC) formulation

For a theoretical description of the energy loss of ions in a plasma, there exist some standard approaches. The dielectric linear response (LR) treatment considers the ion as a perturbation of the target plasma, and the stopping is caused by the polarization of the surrounding medium. It is generally valid if the ion couples weakly to the target. Since the early 1960s, a number of calculations of the stopping power (SP) within LR treatment in a magnetized plasma have been presented (see Refs. [13–37] and references therein). Alternatively, the stopping is calculated as a result of the energy transfers in successive binary collisions (BCs) between the ion and the electrons [37–45]. Here, it is necessary to consider appropriate approximations for the screening of the Coulomb potential by the plasma [8]. However, significant gaps between these approaches involve the ion stopping along magnetic field B and perpendicular to it. In particular, at high B values, the BC predicts a vanishingly parallel energy loss, which remains at variance with the nonzero LR one. Also, challenging BCLR discrepancies persist in the transverse direction, especially for vanishingly small ion projectile velocity vi when the friction coefficient contains an anomalous term diverging logarithmically at vi ¼ 0 [23, 24]. For calculation of the energy loss of an ion, two new alternative approaches have been recently suggested. One of these methods is specifically aimed at a low-velocity energy loss, which is expressed in terms of velocity-velocity correlation and, hence, to a diffusion coefficient [34]. Next, in Ref. [27] using the Bhatnagar-Gross-Krook approach based on the Boltzmann-Poisson equations for a collisional and magnetized classical plasma, the energy loss of an ion is studied through a LR approach, which is constructed such that it conserves particle number locally.

An alternative approach, particularly in the absence of any relevant experimental data, is to test various theoretical methods against comprehensive numerical simulations. This can be achieved by a particle-in-cell (PIC) simulation of the underlying nonlinear Vlasov-Poisson Equation [10, 31]. While the LR requires cutoffs to exclude hard collisions of close particles, the collectivity of the excitation can be taken into account in both LR and PIC approaches. In the complementary BC treatment, the stopping force has been calculated numerically by scattering statistical ensembles of magnetized electrons from the ions in the classical trajectory Monte Carlo (CTMC) method [7, 10, 37–41]. For a review we refer to a recent monograph [8] which summarizes all theoretical and numerical methods and approaches also discussing the ranges of their validity.

The very recent upheaval of successful experiments involving hot and dense plasmas in the presence of kilotesla magnetic fields (e.g., at ILE (Osaka), CELIA (Bordeaux), LULI (Palaiseau), LLNL (Livermore)) remaining nearly steady during 10–15 ns strongly motivates the fusion as well as the warm dense matter (WDM) communities to investigate adequate diagnostics for their dynamic properties. This opens indeed a novel perspective by allowing magnetic fields to play a much larger if not a central role both in ICF and WDM plasmas. In this context proton or any nonrelativistic ion stopping is likely to provide an option of choice for investigating genuine magnetization features such as anisotropy, when the electron plasma frequency turns significantly lower than the cyclotron one [46]. In addition, an experimental test of proton or alpha particle stopping in a magnetized plasma is currently envisioned (see, e.g., Ref. [46] for a preliminary discussion). The parameters at hand are a fully ionized hydrogen plasma with a density up to 10<sup>20</sup> cm<sup>3</sup> and temperature between 1 and 100 eV. The steady magnetic field can be up to 45 T strong. A preliminary examination based on comparing electron Debye length with corresponding Larmor radius indicates that to experience a strong influence of the magnetic field, the electron density should be comparable with a few 1016 cm3 . We expect these endeavors to lead to the very first unambiguous and genuine identification of an experimental magnetic signature for nonrelativistic ion stopping in plasmas.

Motivated by these recent developments, our purpose is to investigate the SP of an ion moving in a magnetized plasma in a wide range of the value of a steady magnetic field. The present paper is based on our earlier studies in Refs. [8, 24, 44, 45] where the second-order energy transfers for individual collisions of electron-ion [8, 24, 44] of any two identical particles, like electron-electron [44], and finally of two gyrating arbitrary charged particles [45] have been calculated with the help of an improved BC treatment. This treatment is—unlike earlier approaches of, e.g., Refs. [9, 42]—valid for any strength of the magnetic field. As the first application of the theoretical BC model developed in Refs. [8, 24, 44, 45], we have calculated in Ref. [47] the cooling forces on the heavy ion beam interacting with a strongly magnetized and temperature anisotropic electron beam. It has been shown that there is a quite good overall agreement with both the CTMC numerical simulations and the experiments performed at the ESR storage ring at GSI [48–50].

In Section 2 we introduce briefly a perturbative binary collision formulation in terms of the binary force acting between an ion and a magnetized electron and derive general expressions for the second-order (with respect to the interaction potential) stopping power. In contrast to the previous investigations in Refs. [8, 24, 44, 45], we here consider the (macroscopic) stopping force which is obtained by integrating the binary force of an individual electron-ion interaction with respect to the impact parameter and the velocity distribution function of electrons. That is, the stopping force for monoenergetic electrons is folded with a velocity distribution. The resulting expressions involve all cyclotron harmonics of the electrons' helical motion and are valid for any interaction potential and any strength of the magnetic field. In Section 2.4 we present explicit analytic expressions of this second-order stopping power for the specific case of a regularized and screened interaction potential [51, 52] which is both of finite range and less singular than the Coulomb interaction at the origin and which includes as limiting cases the Debye (i.e., screened) and the Coulomb potentials. For comparison of our expressions with previous approaches, we consider in Section 3 the corresponding asymptotic expressions for large and small ion velocities and strong and vanishing magnetic fields. The analytical expressions presented in Section 2.4 are evaluated numerically in Section 4 using parameters of the envisaged experiments on ion stopping [46]. In particular, we compare our approach with the CTMC simulations. The results are summarized and discussed in Section 5. The regularization parameter and the screening length involved in the interaction potential are briefly specified and discussed in Appendix A.
