## Abstract

In this chapter, we investigate the stopping power of an ion in a magnetized electron plasma in a model of binary collisions (BCs) between ions and magnetized electrons, in which the two-body interaction is treated up to the second order as a perturbation to the helical motion of the electrons. This improved BC theory is uniformly valid for any strength of the magnetic field and is derived for two-body forces which are treated in Fourier space without specifying the interaction potential. The stopping power is explicitly calculated for a regularized and screened potential which is both of finite range and less singular than the Coulomb interaction at the origin. Closed expressions for the stopping power are derived for monoenergetic electrons, which are then folded with an isotropic Maxwell velocity distribution of the electrons. The accuracy and validity of the present model have been studied by comparisons with the classical trajectory Monte Carlo numerical simulations.

### Keywords

- ion stopping
- magnetized plasma target
- binary collisions

## 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 cooling of heavy ion beams by electrons [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], as well as many very dense systems involved in magnetized target fusions [11], or heavy ion inertial confinement fusion (ICF).

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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 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, 38, 39, 40, 41, 42, 43, 44, 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

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, 38, 39, 40, 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

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, 49, 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.

## 2. Theoretical model

### 2.1. Binary collision (BC) formulation

Let us consider two point charges with masses

In the presence of an external magnetic field, the Lagrangian and the corresponding equations of particle motion cannot, in general, be separated into parts describing the relative motion and the motion of the center of mass (cm) [8]. However, in the case of heavy ions, i.e.,

Here,

### 2.2. The perturbative treatment

We seek an approximate solution of Eq. (2) in which the interaction force between the ion and electron is considered as a perturbation. Thus, we are looking for a solution of Eq. (2) for the variables *n*th-order perturbations of

The parameter of smallness which justifies such kind of expansion can be read off from a dimensionless form of the equation of motion Eq. (2) by scaling lengths in units of the screening length

The equation for the first-order velocity correction is obtained from Eq. (2) replacing on the right-hand side of the exact relative coordinate

Here, we have introduced the following abbreviations:

and have assumed that all corrections vanish at

### 2.3. Second-order stopping power

We now consider the interaction process of an individual ion with a homogeneous electron plasma described by a velocity distribution function

The resulting stopping power,

which is an exact relation for uncorrelated BCs of the ion with electrons. We evaluate this expression within a systematic perturbative treatment (see Ref. [47] for more details). First, we introduce the two-particle interaction potential

From Eq. (6) it is seen that the second-order stopping power is proportional to

where

### 2.4. The SP for a regularized and screened coulomb potential

For an electron plasma with an isotropic Maxwell distribution, the velocity distribution relevant for the averaging in Eq. (7) is given by

where the thermal velocity

Here, we have introduced the thermal cyclotron radius of the electrons

where

Substituting the interaction potential (10) into Eq. (9) and performing the

where

Here, we have introduced the dimensionless quantities

where

Eq. (11) for the SP is the main result of the outlined BC treatment which will now be evaluated in the next sections.

## 3. Comparison with previous approaches

Previous theoretical expressions for the stopping power which have been extensively discussed by the plasma physics community (see, e.g., Refs. [3, 8] for reviews) basically concern the two limiting cases of vanishing and infinitely strong magnetic fields. We therefore investigate the present approach for these two cases, first for arbitrary interactions

### 3.1. General SP Eq. (7) at vanishing and infinitely strong magnetic fields

At vanishing magnetic field

where

Employing the regularized and screened potential

Taking the bare Coulomb interaction with

The asymptotic expression of Eq. (15) at high ion velocities can be easily derived using the normalization of the distribution function which results in

At an infinitely strong magnetic field

The corresponding high-velocity asymptotic expression is given by

Eqs. (15) and (19) and their asymptotic expressions for high velocities in Eqs. (18) and (20), respectively, agree with the results derived by Derbenev and Skrinsky in Ref. [57] in case of the Coulomb interaction potential, i.e., with

### 3.2. Some limiting cases of Eq. (11)

Next, we discuss some asymptotic regimes of the SP (Eq. (11)) where the regularized interaction (Eq. (10)) and the isotropic velocity distribution (Eq. (8)) have been assumed. In the high-velocity limit where

Here, the function

where

A further increase of the ion velocity finally yields

which completely agrees with the asymptotic expression (18) in case of

At

Now, we consider the situation when the magnetic field is very strong and the electron cyclotron radius is the smallest length scale,

where

After changing the variable

Up to the definition of the Coulomb logarithm (i.e.,

In particular, at

respectively, where

where

In the high-velocity limit with

With further increase of the ion velocity, we can then neglect the exponential term in Eq. (30), while

The SP given by Eq. (30) (or Eq. (20) with

Finally, we also investigate the case of small velocities at strong magnetic fields. Considering a small ion velocity

where

## 4. Features of the SP (Eq. (11)) and comparison with CTMC simulations

In this section we study some general properties of the SP of individual ions resulting from the BC approach by evaluating Eq. (11) numerically. We consider the effect of the magnetic field on the SP at various temperatures of the plasma. The density

For a BC description beyond the perturbative regime, a fully numerical treatment is required. In the present cases of interest, such a numerical evaluation of the SP is rather intricate but can be successfully implemented by classical trajectory Monte Carlo (CTMC) simulations [37, 38, 39, 40]. In the CTMC method, the trajectories for the ion-electron relative motion are calculated by a numerical integration of the equations of motion (Eq. (2)). The stopping force is then deduced by averaging over a large number (typically

## 5. Stopping profiles and ranges

### 5.1. General trends

The parameter analysis initiated on Figures 1–3 at

Therefore, we are looking for the most conspicuous effect of the applied magnetized intensity B on the proton stopping.

Fixing

Moreover, that anisotropy evolves only moderately between

Another significant feature is the extension to any

As expected, B effects impact essentially the low-velocity section (

Switching now attention to corresponding ranges, down to projectile at rest

In a low projectile velocity

Finally, it can be observed that for

### 5.2. Specific trends

The projected experimental setup [62] could manage constant, static, and homogeneous B values up to 45 T. So, we are let to investigate

Obviously,

Giving attention to proton ranges of T dependence in a low-density plasma

Corresponding proton ranges (

Experimentally, accessible and very small ranges are thus documented for

### 5.3. Very-low-velocity proton slowing down

Up to now we limited our investigation to proton stopping by target electrons. In the very-low-velocity regime

## 6. Summary

We developed and extensively used a kinetic approach based on a binary collision formulation and suitably regularized Coulomb interaction, to numerically document for any value of the applied magnetization B, the stopping of a proton projectile in a fully ionized hydrogen plasma target. Both ion projectile and target plasma parameters have been selected in order to fit a planned ion-plasma interaction experiment in the presence of an applied magnetic field

More generally, we expect that the present investigation, experimentally geared as it is, could help to bridge a long-standing and persisting gap between theoretical speculations and experimental facts in the field of nonrelativistic ion stopping in magnetized target plasmas.

## Acknowledgments

The work of H.B.N. has been supported by the State Committee of Science of the Armenian Ministry of Education and Science (Project No. 13-1C200). This work was supported by the Bundesministerium für Bildung und Forschung (BMBF) under contract 06ER9064.

Our results (Eq. (11)) were derived by using the screened interaction

Next, we specify the parameter