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The interaction of charged particles with matter has been an issue of extensive investigations throughout the whole last century. Its theoretical treatment starts with the classical description of the energy loss of fast projectiles considered by Bohr [24]. Later a quantum mechanical treatment of the energy transfer to bound electrons was established by Bethe [18] and refined by Bloch [23]. Further considerable improvements of the theoretical description have been achieved by Fermi and Teller [41] and finally by Lindhard [79]. The present status of the theory has, e.g. been reviewed in the monographs by Sigmund [118, 119]. Till nowadays an enormous number of publications are dedicated to specific questions on the energy loss for a variety of possible projectile and target conditions. Recent applications are the energy transfer to pellets for inertial confinement fusion, electron cooling of heavy ion beams as well as the deceleration of particle beams in traps.

There are basically two complementary approaches to describe the energy loss of charged particles in matter. In the dielectric treatment (DT) the response of the target’s charge and current densities to the perturbation caused by the passing projectile is calculated. Such a continuum theory requires cut-offs to exclude hard collisions of close particles, but the collectivity of the excitation can be taken into account. In the binary collision approximation (BC), on the other hand, the energy loss of the projectile is the aggregate of subsequent pairwise interactions with the target particles. This requires cut-offs at large distance to account for screening. We will first treat the case of an unmagnetized target plasma. This will illustrate the basic ideas of both approaches and serves to highlight the difficulties introduced by the presence of an external magnetic field.

In the presence of an external magnetic field B even the nonrelativistic problem of two charged particles cannot be solved in a closed form as the relative motion and the motion of the center of mass are coupled to each other. There exists no closed solution of this problem that is uniformly valid for any strength of the magnetic field and the Coulomb force between the particles. The classical limit of a hydrogen or Rydberg atom in a strong magnetic field also falls in this category (see, e.g., [56] and references therein) but in contrast to the free-free transitions (scattering) the total energy is negative there.

In this section we analyze expression (2.60) in the case when a projectile ion moves in an anisotropic two-temperature electron plasma (the susceptibility of the ions is neglected and the electronic index e is suppressed) without magnetic field. In the limit of vanishing magnetic field equation (2.49) takes the form X ( t ) = Y ^2 t ^2, where Y = [ μ ^2 + τ (1− μ ^2)]^1/2 with μ = cos β . In this limit the plasma dielectric function from (2.48) after changing the integration variable, t → t / Y , now reads (4.2) $$ \begin{array}{*{20}c} {\varepsilon (k,\omega ) = 1 + \frac{1} {{k^2 \lambda _{D\parallel }^2 }}\left\{ {1 + \frac{{i\zeta \sqrt 2 }} {Y}\int_0^\infty {dt} e^{i(\zeta /Y)t\sqrt 2 - t^2 } } \right.} \\ {\left. { + 2(1 - \tau )\frac{{\sin ^2 \beta }} {{Y^2 }}\int_0^\infty {dt t e^{i(\zeta /Y)t\sqrt 2 - t^2 } } } \right\},} \\ \end{array} $$ where ζ = ω / kv _th‖. The t -integrals in (4.1) can be evaluated in close form and finally the dielectric function is represented as (4.2) $$ \varepsilon (k,\omega ) = 1 + \frac{1} {{k^2 \lambda _{D\parallel }^2 }}\frac{1} {{Y^2 }}W\left( {\frac{\zeta } {Y}} \right). $$ Here W ( ζ ) = g ( ζ ) + i f ( ζ ) is the plasma dispersion function [43], (4.3) $$ g (\zeta ) = 1 - \zeta \sqrt 2 Di\left( {\frac{\zeta } {{\sqrt 2 }}} \right), f(\zeta ) = \sqrt {\frac{\pi } {2}} \zeta e^{ - \zeta ^2 /2} , $$ where (4.4) $$ Di(\zeta ) = e^{ - \zeta ^2 } \int_0^\zeta {dt e^{t^2 } } $$ is the Dawson integral [43] which has for large arguments ζ the asymptotic behavior Di( ζ ) ≃ 1/(2 ζ ) + 1/(4 ζ ^3). Notice that the real and imaginary parts of plasma dispersion function, g ( ζ ) and f ( ζ ), are related to the corresponding functions G and ℱ for the magnetized plasma (see equation (2.48)) according to G | B →0 = Y ^-2 g (ζ/ Y ) and ℱ|_ B →0= Y ^-2 f(ζ/Y) .

In Chap. 4 we have considered the energy loss of a test ion in a magnetized plasma where the motion of the projectile ion as well as the plasma have been treated classically. Here we use a quantum mechanical description of the beam-plasma interaction rather than a classical dielectric function. Again we assume a weak coupling, Z ≪1, between projectile ion and plasma, where the coupling parameter Z , within quantum description is given by Z =| Z | e ^2/ħ v _ r . (see, e.g., [134]). Hence the dielectric formalism in linear response becomes accurate in the limit of high test particle velocities. We describe the plasma in the random-phase approximation (RPA) and are therefore restricted to the weak-coupling limit of the interparticle interactions. Unlike its classical counterpart the RPA dielectric function due to the wave nature of particles guarantees the convergence of the k -integral for short-range interactions and avoids the cut-off procedure. Furthermore, we assume that the electrons give the main contribution to the stopping power.

Examples of applications where the energy loss of ions in a magnetized plasma plays a prominent role are the stopping of the alpha particles created by fusion events in the hot plasma of magnetic confinement fusion devices and the cooling of ion beams or bunches of ions by magnetized electrons in electron cooler sections of storage rings or in traps. In a strong magnetic field the cyclotron motion of the target electrons sets the smallest relevant length and time scale. In that sense the fields usually applied in magnetic confinement fusion are still moderate, while stronger fields are employed to guide the electrons in the cooling sections of storage rings and even more so for the cooling in traps. We will therefore focus on these latter cases.

In the wide field of the interaction of charged particle beams with matter this monograph is primarily concerned with a special topic, namely the influence of external magnetic fields on the passage of heavy ions or antiprotons through an electron plasma. Observables are the stopping power which determines the range of the particles and the time until they come to rest as well as their straggling. The independent variables are the strength of the magnetic field B , the velocity v _i of the projectile (both its magnitude and the angle α with B ), the charge Ze of the projectile and properties of the target like its density n _e and the velocity distribution f ( v _e) of the target electrons. Theoretical estimates of the stopping power and the straggling in terms of these variables are desired for applications in nuclear fusion, electron cooling of charged particle beams in storage rings and the deceleration of particles in traps.