2. Theoretical model

cooling of heavy ion beams by electrons [3–12], as well as many very dense systems involved in

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

magnetized target fusions [11], or heavy ion inertial confinement fusion (ICF).

68 Plasma Science and Technology - Basic Fundamentals and Modern Applications

### 2.1. Binary collision (BC) formulation

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 r ¼ r<sup>1</sup> � r<sup>2</sup> is the relative coordinate of the colliding particles. For two isolated charged particles, this interaction is given by the Coulomb potential, i.e., UCð Þ¼ r 1=r. In plasma applications UC is modified by many-body effects and the related screening and turns into an effective interaction. In general, this effective interaction, which is related to the wake field induced by a moving ion, is non-spherically symmetric and depends also on the ion velocity. For any BC treatment, however, this complicated ion-plasma interaction must be approximated by an effective two-particle interaction Uð Þr . This effective interaction U may be modeled by a spherically symmetric Debye-like screened interaction uDð Þ¼ <sup>r</sup> <sup>e</sup>�r=<sup>λ</sup>=<sup>r</sup> with a screening length λ, given, e.g., by the Debye screening length λ<sup>D</sup> (see, e.g., [16]), in case of low ion velocities and an effective velocity-dependent screening length λð Þ vi for larger ion velocities vi (see [53–55]). Further details on the choice of the effective interaction U rð Þ are given in Ref. [47].

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., M ≫ m, the equations of motion can be simplified by treating the cm velocity vcm as constant and equal to the ion velocity v<sup>i</sup> , i.e., vcm ¼ v<sup>i</sup> ¼ const. Then, introducing the velocity correction through relations δvð Þ<sup>t</sup> ¼ veð Þ� t ve0ðÞ¼ t vð Þ� t v0ð Þt , where vðÞ¼ t r tðÞ¼ veð Þ� t v<sup>i</sup> is the relative electron-ion velocity ve0ð Þt and v0ðÞ¼ t r\_0ðÞ¼ t ve0ð Þ� t v<sup>i</sup> are the unperturbed electron and relative velocities, respectively, the equation of relative motion turns into

$$r\_0(t) = \mathbf{R}\_0 + \mathbf{v}\_r t + a[\mathbf{u}\sin\left(\omega\_c t\right) - [\mathbf{b} \times \mathbf{u}]\cos\left(\omega\_c t\right)],\tag{1}$$

$$
\delta\dot{\mathbf{v}}(t) + \omega\_{\mathbf{\dot{c}}}[\delta\mathbf{v}(t) \times \mathbf{b}] = -\frac{\mathbf{Z}\dot{\boldsymbol{\varphi}}^2}{m}\mathbf{f}[\mathbf{r}(t)].\tag{2}
$$

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 λ, velocities in units of the initial relative velocity v0, and time in units of λ=v0. Then, it is seen (see Ref. [47] for details) that the perturbative treatment is essentially applicable in

is large compared to the characteristic potential energy ∣Z∣=e <sup>2</sup>=λ in a screened Coulomb potential. Or, expressed in velocities, the initial relative velocity v0 must exceed the characteristic

perturbative regime. If this condition is met not only for a single ion-electron collision but in the average over the electron distribution, e.g., by replacing v0 with the averaged initial ionelectron relative velocity h i v<sup>0</sup> , i.e., h i v<sup>0</sup> ≳vd, we are in a regime of weak ion-target or, here, weak ion-electron coupling, which allows the use of perturbative treatments (besides BC also, e.g., linear response (LR)). For nonmagnetized electrons this is discussed in much detail in Refs. [53, 54]. Even though the particle trajectories are much more intricate in the presence of an external magnetic field, the given definitions and demarcations of coupling regimes are basically the same for magnetized electrons. That is, the applicability of a perturbative treatment is essentially related to the charge state Z of the ion and the typical range λ of the effective interaction, but not directly on the strength B of the magnetic field. The latter may affect the critical velocity

The equation for the first-order velocity correction is obtained from Eq. (2) replacing on the right-hand side of the exact relative coordinate rð Þt by r0ð Þt with the solutions v1ðÞ¼ t r\_1ð Þt and

vd only implicitly via a possible change of the effective screening length λ with B.

Q∥ðÞ¼ t

1 iω<sup>c</sup> ðt �∞

Q⊥ðÞ¼ t

ðt �∞

f r½ � <sup>0</sup>ð Þτ e

We now consider the interaction process of an individual ion with a homogeneous electron plasma described by a velocity distribution function fð Þ v<sup>e</sup> and a density ne. We assume that the ion experiences independent binary collisions (BCs) with the electrons. The total stopping force, F vð Þ<sup>i</sup> , acting on the ion is then obtained by multiplying the binary force <sup>Z</sup>=e<sup>2</sup>f r½ � ð Þ<sup>t</sup> by the

velocity distribution of the electrons. The impact parameter s introduced here in the electron flux is defined by s ¼ R<sup>0</sup><sup>⊥</sup> ¼ R<sup>0</sup> � n<sup>r</sup> n<sup>r</sup> ð Þ � R<sup>0</sup> and is the component of R<sup>0</sup> perpendicular to the

<sup>0</sup>λ < 1, that is, when the (initial) kinetic energy of relative motion mv<sup>2</sup>

, that is, vd here demarcates the perturbative from the non-

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

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

<sup>m</sup> �bQ∥ð Þþ <sup>t</sup> Re½ � b bð Þ� � <sup>Q</sup>⊥ð Þ<sup>t</sup> <sup>Q</sup>⊥ð Þþ <sup>t</sup> <sup>i</sup>½ � <sup>b</sup> � <sup>Q</sup>⊥ð Þ<sup>t</sup> � �: (3)

b � f r½ � <sup>0</sup>ð Þτ ð Þ t � τ dτ,

<sup>ω</sup><sup>c</sup> ð Þ <sup>t</sup>�<sup>τ</sup> � <sup>1</sup> h id<sup>τ</sup>

sdt, integrating with respect to time and folding with

<sup>0</sup>=2,

71

(4)

cases where ∣Z∣=e <sup>2</sup>=mv<sup>2</sup>

velocity vd <sup>¼</sup> j j <sup>Z</sup> <sup>=</sup><sup>e</sup> <sup>2</sup>=m<sup>λ</sup> � �<sup>1</sup>=<sup>2</sup>

r1ðÞ¼ t

2.3. Second-order stopping power

element of the flux relative flux nevrd<sup>2</sup>

Z=e<sup>2</sup>

Here, we have introduced the following abbreviations:

and have assumed that all corrections vanish at t ! �∞.

Here, �Z=<sup>e</sup> <sup>2</sup>f r½ � ð Þ<sup>t</sup> ð Þ <sup>f</sup> ¼ �∂U=∂<sup>r</sup> is the force exerted by the ion on the electron, <sup>ω</sup><sup>c</sup> <sup>¼</sup> eB=<sup>m</sup> is the electron cyclotron frequency, and δvðÞ!t 0 at t ! �∞. In Eq. (1) u ¼ ð Þ cosφ; sinφ is the unit vector perpendicular to the magnetic field; the angle φ is the initial phase of the electron's helical motion; v<sup>r</sup> ¼ ve∥b � v<sup>i</sup> is the relative velocity of the guiding center of the electrons, where ve<sup>∥</sup> and ve<sup>⊥</sup> (with ve<sup>⊥</sup> ≤ 0) are the unperturbed components of the electron velocity parallel and perpendicular to b, respectively; and a ¼ ve⊥=ω<sup>c</sup> is the cyclotron radius. In Eq. (1), the quantities u and R<sup>0</sup> are defined by the initial conditions. In Eq. (2) rðÞ¼ t reð Þ� t vit is the ion-electron relative coordinate.

#### 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 r and v in a perturbative manner r ¼ r<sup>0</sup> þ r<sup>1</sup> þ …, v ¼ v<sup>0</sup> þ v<sup>1</sup> þ ⋯, where r0ð Þt , v0ð Þt are the unperturbed ion-electron relative coordinate and velocity, respectively, and rnð Þt , vnð Þt ð Þ n ¼ 1; 2; ⋯ are the nth-order perturbations of rð Þt and vð Þt , which are proportional to Z<sup>n</sup>.

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 λ, velocities in units of the initial relative velocity v0, and time in units of λ=v0. Then, it is seen (see Ref. [47] for details) that the perturbative treatment is essentially applicable in cases where ∣Z∣=e <sup>2</sup>=mv<sup>2</sup> <sup>0</sup>λ < 1, that is, when the (initial) kinetic energy of relative motion mv<sup>2</sup> <sup>0</sup>=2, is large compared to the characteristic potential energy ∣Z∣=e <sup>2</sup>=λ in a screened Coulomb potential. Or, expressed in velocities, the initial relative velocity v0 must exceed the characteristic velocity vd <sup>¼</sup> j j <sup>Z</sup> <sup>=</sup><sup>e</sup> <sup>2</sup>=m<sup>λ</sup> � �<sup>1</sup>=<sup>2</sup> , that is, vd here demarcates the perturbative from the nonperturbative regime. If this condition is met not only for a single ion-electron collision but in the average over the electron distribution, e.g., by replacing v0 with the averaged initial ionelectron relative velocity h i v<sup>0</sup> , i.e., h i v<sup>0</sup> ≳vd, we are in a regime of weak ion-target or, here, weak ion-electron coupling, which allows the use of perturbative treatments (besides BC also, e.g., linear response (LR)). For nonmagnetized electrons this is discussed in much detail in Refs. [53, 54]. Even though the particle trajectories are much more intricate in the presence of an external magnetic field, the given definitions and demarcations of coupling regimes are basically the same for magnetized electrons. That is, the applicability of a perturbative treatment is essentially related to the charge state Z of the ion and the typical range λ of the effective interaction, but not directly on the strength B of the magnetic field. The latter may affect the critical velocity vd only implicitly via a possible change of the effective screening length λ with B.

The equation for the first-order velocity correction is obtained from Eq. (2) replacing on the right-hand side of the exact relative coordinate rð Þt by r0ð Þt with the solutions v1ðÞ¼ t r\_1ð Þt and

$$r\_1(t) = \frac{\mathbf{Z}\dot{\boldsymbol{\varphi}}^2}{m} \left\{-\mathbf{b}\,\mathcal{Q}\_{\parallel}(t) + \text{Re}[\mathbf{b}(\mathbf{b}\cdot\boldsymbol{\mathcal{Q}}\_{\perp}(t)) - \mathcal{Q}\_{\perp}(t) + i[\mathbf{b}\times\boldsymbol{\mathcal{Q}}\_{\perp}(t)]]\right\}.\tag{3}$$

Here, we have introduced the following abbreviations:

$$\begin{aligned} \mathcal{Q}\_{\parallel}(t) &= \int\_{-\infty}^{t} \mathbf{b} \cdot \mathbf{f}[\mathbf{r}\_{0}(\tau)](t-\tau)d\tau, \\ \mathcal{Q}\_{\perp}(t) &= \frac{1}{i\omega\_{c}} \int\_{-\infty}^{t} \mathbf{f}[\mathbf{r}\_{0}(\tau)] \left[e^{\omega\_{c}(t-\tau)} - 1\right] d\tau \end{aligned} \tag{4}$$

and have assumed that all corrections vanish at t ! �∞.

#### 2.3. Second-order stopping power

r ¼ r<sup>1</sup> � r<sup>2</sup> is the relative coordinate of the colliding particles. For two isolated charged particles, this interaction is given by the Coulomb potential, i.e., UCð Þ¼ r 1=r. In plasma applications UC is modified by many-body effects and the related screening and turns into an effective interaction. In general, this effective interaction, which is related to the wake field induced by a moving ion, is non-spherically symmetric and depends also on the ion velocity. For any BC treatment, however, this complicated ion-plasma interaction must be approximated by an effective two-particle interaction Uð Þr . This effective interaction U may be modeled by a spherically symmetric Debye-like screened interaction uDð Þ¼ <sup>r</sup> <sup>e</sup>�r=<sup>λ</sup>=<sup>r</sup> with a screening length λ, given, e.g., by the Debye screening length λ<sup>D</sup> (see, e.g., [16]), in case of low ion velocities and an effective velocity-dependent screening length λð Þ vi for larger ion velocities vi (see [53–55]). Further details on the choice of the effective interaction U rð Þ are

70 Plasma Science and Technology - Basic Fundamentals and Modern Applications

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., M ≫ m, the equations of motion can be simplified by treating the cm velocity vcm as constant and equal to

relations δvð Þ<sup>t</sup> ¼ veð Þ� t ve0ðÞ¼ t vð Þ� t v0ð Þt , where vðÞ¼ t r tðÞ¼ veð Þ� t v<sup>i</sup> is the relative electron-ion velocity ve0ð Þt and v0ðÞ¼ t r\_0ðÞ¼ t ve0ð Þ� t v<sup>i</sup> are the unperturbed electron and

<sup>δ</sup>v\_ð Þþ <sup>t</sup> <sup>ω</sup>c½ �¼� <sup>δ</sup>vð Þ� <sup>t</sup> <sup>b</sup> <sup>Z</sup>=<sup>e</sup> <sup>2</sup>

Here, �Z=<sup>e</sup> <sup>2</sup>f r½ � ð Þ<sup>t</sup> ð Þ <sup>f</sup> ¼ �∂U=∂<sup>r</sup> is the force exerted by the ion on the electron, <sup>ω</sup><sup>c</sup> <sup>¼</sup> eB=<sup>m</sup> is the electron cyclotron frequency, and δvðÞ!t 0 at t ! �∞. In Eq. (1) u ¼ ð Þ cosφ; sinφ is the unit vector perpendicular to the magnetic field; the angle φ is the initial phase of the electron's helical motion; v<sup>r</sup> ¼ ve∥b � v<sup>i</sup> is the relative velocity of the guiding center of the electrons, where ve<sup>∥</sup> and ve<sup>⊥</sup> (with ve<sup>⊥</sup> ≤ 0) are the unperturbed components of the electron velocity parallel and perpendicular to b, respectively; and a ¼ ve⊥=ω<sup>c</sup> is the cyclotron radius. In Eq. (1), the quantities u and R<sup>0</sup> are defined by the initial conditions. In Eq. (2) rðÞ¼ t reð Þ� t vit

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 r and v in a perturbative manner r ¼ r<sup>0</sup> þ r<sup>1</sup> þ …, v ¼ v<sup>0</sup> þ v<sup>1</sup> þ ⋯, where r0ð Þt , v0ð Þt are the unperturbed ion-electron relative coordinate and velocity, respectively, and rnð Þt , vnð Þt ð Þ n ¼ 1; 2; ⋯ are the nth-order perturbations of rð Þt and vð Þt , which are propor-

relative velocities, respectively, the equation of relative motion turns into

, i.e., vcm ¼ v<sup>i</sup> ¼ const. Then, introducing the velocity correction through

r0ðÞ¼ t R<sup>0</sup> þ vrt þ a½ � u sin ð Þ� ωct ½ � b � u cos ð Þ ωct , (1)

f r½ � ð Þt : (2)

m

given in Ref. [47].

the ion velocity v<sup>i</sup>

is the ion-electron relative coordinate.

2.2. The perturbative treatment

tional to Z<sup>n</sup>.

We now consider the interaction process of an individual ion with a homogeneous electron plasma described by a velocity distribution function fð Þ v<sup>e</sup> and a density ne. We assume that the ion experiences independent binary collisions (BCs) with the electrons. The total stopping force, F vð Þ<sup>i</sup> , acting on the ion is then obtained by multiplying the binary force <sup>Z</sup>=e<sup>2</sup>f r½ � ð Þ<sup>t</sup> by the element of the flux relative flux nevrd<sup>2</sup> sdt, integrating with respect to time and folding with velocity distribution of the electrons. The impact parameter s introduced here in the electron flux is defined by s ¼ R<sup>0</sup><sup>⊥</sup> ¼ R<sup>0</sup> � n<sup>r</sup> n<sup>r</sup> ð Þ � R<sup>0</sup> and is the component of R<sup>0</sup> perpendicular to the relative velocity vector v<sup>r</sup> with n<sup>r</sup> ¼ vr=vr. As can be inferred from Eq. (1), s represents the distance of the closest approach between the ion and the guiding center of the electron's helical motion.

The resulting stopping power, <sup>S</sup>ð Þ¼� <sup>v</sup><sup>i</sup> <sup>v</sup><sup>i</sup> vi � F vð Þ<sup>i</sup> , then reads

$$S(\mathbf{v}\_i) = -\frac{Ze^2n\_\epsilon}{v\_i} \left[ d\mathbf{v}\_\epsilon f(\mathbf{v}\_\epsilon) v\_r \int d^2\mathbf{s} \int\_{-\infty}^\infty \mathbf{v}\_i \cdot \mathbf{f}[\mathbf{r}(t)] dt,\tag{5}$$

f vð Þ¼ <sup>e</sup>

where the thermal velocity vth is related to the electron temperature by v<sup>2</sup>

tions (see Ref. [56]), the stopping power

ð Þ 2π 4 4

ð∞ 0 dk<sup>∥</sup> ð∞ 0 U2

8Z<sup>2</sup> =e<sup>4</sup>ne mωcvi

S vð Þ¼<sup>i</sup>

where <sup>d</sup>�<sup>1</sup> <sup>¼</sup> <sup>λ</sup>�<sup>1</sup> <sup>þ</sup> <sup>ƛ</sup>�<sup>1</sup>

S vð Þ¼<sup>i</sup>

1 ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>=<sup>2</sup> v3 th e �v<sup>2</sup> <sup>e</sup> =2v<sup>2</sup>

temperature is measured in energy units). Inserting Eq. (8) into expression (7) and assuming now a spherically symmetric potential U ¼ U kð Þ yields after performing the velocity integra-

> ð∞ 0 e �t 2 2 k2 ∥a2 e �k 2

� <sup>k</sup>⊥ai<sup>⊥</sup> cos <sup>k</sup>∥ai∥<sup>t</sup> � �J1ð Þþ <sup>k</sup>⊥ai⊥<sup>t</sup> <sup>k</sup>∥ai<sup>∥</sup> sin <sup>k</sup>∥ai∥<sup>t</sup> � �J0ð Þ <sup>k</sup>⊥ai⊥<sup>t</sup> � �:

ð Þk k⊥dk<sup>⊥</sup>

screened interaction U rð Þ¼ URð Þ¼ <sup>r</sup> <sup>1</sup> � <sup>e</sup>�r=<sup>λ</sup> � �e�r=<sup>λ</sup>=<sup>r</sup> with the Fourier transform

ð Þ 2π 2

ered as a given constant or as a function of the classical collision diameter [47].

URð Þ¼ <sup>k</sup> <sup>2</sup>

arrive, after lengthy but straightforward calculations, at

mv<sup>2</sup> th

=e<sup>4</sup>ne

v ð∞ 0 dt t ð1 0

<sup>P</sup>1ð Þ¼ <sup>t</sup>; <sup>ζ</sup> 2 cos <sup>2</sup>

G tð Þ ; ζ

sin <sup>2</sup>ϑ

<sup>P</sup>2ð Þ¼ <sup>t</sup>; <sup>ζ</sup> <sup>2</sup>

4 ffiffiffi <sup>π</sup> <sup>p</sup> <sup>Z</sup><sup>2</sup>

where P tð Þ¼ ; <sup>ζ</sup> cos <sup>2</sup><sup>ϑ</sup> <sup>þ</sup> sin <sup>2</sup>ϑ=G tð Þ ; <sup>ζ</sup> and

Here, we have introduced the thermal cyclotron radius of the electrons a ¼ vth=ωc, and ai<sup>⊥</sup> ¼ vi⊥=ωc, ai<sup>∥</sup> ¼ vi∥=ωc, where vi<sup>⊥</sup> and vi<sup>∥</sup> are the ion velocity components transverse and parallel to b, respectively. For the Coulomb interaction U kð Þ¼ UCð Þk , the full two-dimensional integration over the s-space results in a logarithmic divergence of the k integration in Eqs. (7) and (9). To cure this, cutoff parameters kmin and kmax must be introduced (see, e.g., Refs. [8, 24, 47] for details). These cutoffs are related to the screening of the interaction in a plasma target and the incorrect treatment of hard collisions in a classical perturbative approach. As an alternative implementation of this standard cutoff procedure, we here employ the regularized

1

which is additionally regularized at the origin [51, 52] and thus removes the problems related to the Coulomb singularity in a classical picture and prevents particles (for Z > 0) from falling into the center of the potential. The parameter λ related to this regularization is here consid-

Substituting the interaction potential (10) into Eq. (9) and performing the k<sup>∥</sup> integration, we

<sup>d</sup><sup>ζ</sup> exp �v<sup>2</sup>

� <sup>P</sup>1ð Þþ <sup>t</sup>; <sup>ζ</sup> sin ð Þ <sup>α</sup><sup>t</sup>

ζ2

<sup>ϑ</sup> <sup>þ</sup> P tð Þ ; <sup>ζ</sup> <sup>1</sup> � <sup>2</sup>v<sup>2</sup>ζ<sup>2</sup> cos <sup>2</sup>

G tð Þ ; <sup>ζ</sup> <sup>þ</sup> P tð Þ ; <sup>ζ</sup> <sup>1</sup> � <sup>v</sup><sup>2</sup>ζ<sup>2</sup> sin <sup>2</sup><sup>ϑ</sup>

αt

P tð Þ ; <sup>ζ</sup> � �Φ½ � <sup>Ψ</sup>ð Þ <sup>t</sup>; <sup>ζ</sup>

P2ðt; ζÞ � � <sup>ζ</sup><sup>2</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> � �

G tð Þ ; ζ � � � � : (13)

G tð Þ ; <sup>ζ</sup> ,

ϑ � �, (12)

<sup>2</sup> <sup>þ</sup> <sup>λ</sup>�<sup>2</sup> � <sup>1</sup> k <sup>2</sup> <sup>þ</sup> <sup>d</sup>�<sup>2</sup>

: UR represents a Debye-like screened interaction UD (see Section 2.1)

k

th , (8)

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

<sup>∥</sup> þ k 2 ⊥ sin t t � �tdt

<sup>⊥</sup>a2ð1� cos <sup>t</sup> k<sup>2</sup>

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

� �, (10)

th ¼ T=m (here, the

(9)

73

(11)

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 U rð Þ, and the binary force f rð Þ is written using Fourier transformation in space. Furthermore, the factor e<sup>i</sup>k�rð Þ<sup>t</sup> in the Fourier transformed binary force is expanded in a perturbative manner as <sup>e</sup><sup>i</sup>k�rð Þ<sup>t</sup> <sup>≃</sup>e<sup>i</sup>k�r0ð Þ<sup>t</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>i</sup>ð Þ <sup>k</sup> � <sup>r</sup>1ð Þ<sup>t</sup> , where r0ð Þt and r1ð Þt are the unperturbed and the first-order corrected relative coordinates (Eqs. (1) and (3)), respectively. Next, we consider only the second-order binary force f<sup>2</sup> and the corresponding stopping force F<sup>2</sup> with respect to the binary interaction since the averaged first-order force F<sup>1</sup> (related to f1) vanishes due to symmetry reasons [8, 24, 44, 45, 47]. Within the second-order perturbative treatment, the stopping power can be represented as

$$S(\mathbf{v}\_i) = -\frac{Z\dot{\varphi}^2 n\_\epsilon}{v\_i} \left[ d\mathbf{v}\_\epsilon f(\mathbf{v}\_\epsilon) v\_r \int d^2 \mathbf{s} \int d\mathbf{k} U(\mathbf{k}) (\mathbf{k} \cdot \mathbf{v}\_i) \int\_{-\epsilon v}^{\epsilon v} [\mathbf{k} \cdot \mathbf{r}\_1(t)] e^{i\mathbf{k} \cdot \mathbf{r}\_0(t)} dt. \tag{6}$$

From Eq. (6) it is seen that the second-order stopping power is proportional to Z<sup>2</sup> . Inserting now Eqs. (1) and (3) into Eq. (6), assuming an axially symmetric velocity distribution fð Þ¼ v<sup>e</sup> f ve∥; ve<sup>⊥</sup> � �, and performing the s integration, we then obtain

$$\begin{split} S &= -\frac{(2\pi)^4 Z^2 \rho^4 n\_\epsilon}{m\upsilon\_i} \int\_{-\infty}^\infty d\upsilon\_{\epsilon \parallel} \int\_0^\alpha f(\upsilon\_{\epsilon \parallel}, \upsilon\_{\epsilon \perp}) \upsilon\_{\epsilon \perp} d\upsilon\_{\epsilon \perp} \\ &\times \int d\mathbf{k} |\mathcal{U}(\mathbf{k})|^2 (\mathbf{k} \cdot \mathbf{v}\_i) \int\_0^\alpha \left[ k\_\parallel^2 + k\_\perp^2 \frac{\sin\left(\omega\_c t\right)}{\omega\_c t} \right] \\ &\times J\_0 \left( 2k\_\perp a \sin\frac{\omega\_c t}{2} \right) \sin\left(\mathbf{k} \cdot \mathbf{v}\_r t\right) dt d\omega \end{split} \tag{7}$$

where Jn is the Bessel function of the nth order; k<sup>∥</sup> ¼ ð Þ k � b and k<sup>⊥</sup> are the components of k parallel and transverse to b, respectively; and ve<sup>∥</sup> and ve<sup>⊥</sup> are the electron velocity components parallel and transverse to b, respectively. This general expression (7) for the stopping power of an individual ion has been derived within second-order perturbation theory but without any restriction on the strength of the magnetic field B.

#### 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

Stopping Power of Ions in a Magnetized Plasma: Binary Collision Formulation http://dx.doi.org/10.5772/intechopen.77213 73

$$f(\upsilon\_{\epsilon}) = \frac{1}{\left(2\pi\right)^{3/2} \upsilon\_{\text{th}}^3} e^{-\upsilon\_{\epsilon}^2/2\upsilon\_{\text{th}\_{\text{th}\_{\text{th}}}}^2} \tag{8}$$

where the thermal velocity vth is related to the electron temperature by v<sup>2</sup> th ¼ T=m (here, the temperature is measured in energy units). Inserting Eq. (8) into expression (7) and assuming now a spherically symmetric potential U ¼ U kð Þ yields after performing the velocity integrations (see Ref. [56]), the stopping power

$$\begin{split} S(\boldsymbol{v}\_{i}) &= \frac{8Z^{2}\boldsymbol{\theta}^{4}\boldsymbol{n}\_{\varepsilon}(2\pi)^{4}}{m\omega\_{\varepsilon}\boldsymbol{v}\_{i}} \frac{(2\pi)^{4}}{4} \int\_{0}^{\boldsymbol{\alpha}} d\boldsymbol{k}\_{\parallel} \int\_{0}^{\boldsymbol{\alpha}} \boldsymbol{L}^{2}(\boldsymbol{k}) \boldsymbol{k}\_{\perp} d\boldsymbol{k}\_{\perp} \int\_{0}^{\boldsymbol{\alpha}} e^{-\frac{2}{\lambda}\boldsymbol{k}\_{\parallel}^{2}\boldsymbol{a}^{2}} e^{-\boldsymbol{k}\_{\perp}^{2}\boldsymbol{a}^{2}(1-\cos\boldsymbol{t})} \Big(\boldsymbol{k}\_{\parallel}^{2} + \boldsymbol{k}\_{\perp}^{2} \frac{\sin\boldsymbol{t}}{t}\Big) \text{t} dt \\ & \quad \times \left[\boldsymbol{k}\_{\perp}\boldsymbol{a}\_{\perp}\cos\left(\boldsymbol{k}\_{\parallel}\boldsymbol{a}\_{\parallel}\boldsymbol{t}\right)\boldsymbol{I}\_{1}(\boldsymbol{k}\_{\perp}\boldsymbol{a}\_{\perp}\boldsymbol{t}) + \boldsymbol{k}\_{\parallel}\boldsymbol{a}\_{\parallel}\sin\left(\boldsymbol{k}\_{\parallel}\boldsymbol{a}\_{\parallel}\boldsymbol{t}\right)\boldsymbol{I}\_{0}(\boldsymbol{k}\_{\perp}\boldsymbol{a}\_{\perp}\boldsymbol{t})\right]. \end{split} \tag{9}$$

Here, we have introduced the thermal cyclotron radius of the electrons a ¼ vth=ωc, and ai<sup>⊥</sup> ¼ vi⊥=ωc, ai<sup>∥</sup> ¼ vi∥=ωc, where vi<sup>⊥</sup> and vi<sup>∥</sup> are the ion velocity components transverse and parallel to b, respectively. For the Coulomb interaction U kð Þ¼ UCð Þk , the full two-dimensional integration over the s-space results in a logarithmic divergence of the k integration in Eqs. (7) and (9). To cure this, cutoff parameters kmin and kmax must be introduced (see, e.g., Refs. [8, 24, 47] for details). These cutoffs are related to the screening of the interaction in a plasma target and the incorrect treatment of hard collisions in a classical perturbative approach. As an alternative implementation of this standard cutoff procedure, we here employ the regularized screened interaction U rð Þ¼ URð Þ¼ <sup>r</sup> <sup>1</sup> � <sup>e</sup>�r=<sup>λ</sup> � �e�r=<sup>λ</sup>=<sup>r</sup> with the Fourier transform

$$MI\_R(k) = \frac{2}{\left(2\pi\right)^2} \left(\frac{1}{k^2 + \lambda^{-2}} - \frac{1}{k^2 + d^{-2}}\right),\tag{10}$$

where <sup>d</sup>�<sup>1</sup> <sup>¼</sup> <sup>λ</sup>�<sup>1</sup> <sup>þ</sup> <sup>ƛ</sup>�<sup>1</sup> : UR represents a Debye-like screened interaction UD (see Section 2.1) which is additionally regularized at the origin [51, 52] and thus removes the problems related to the Coulomb singularity in a classical picture and prevents particles (for Z > 0) from falling into the center of the potential. The parameter λ related to this regularization is here considered as a given constant or as a function of the classical collision diameter [47].

Substituting the interaction potential (10) into Eq. (9) and performing the k<sup>∥</sup> integration, we arrive, after lengthy but straightforward calculations, at

$$S(v\_i) = \frac{4\sqrt{\pi}Z^2\rho^4 n\_e}{m v\_{\rm th}^2} v \left[\int\_0^\infty \frac{dt}{t}\right]\_0^1 d\zeta \exp\left[-v^2\zeta^2 P(t,\zeta)\right] \Phi[\Psi'(t,\zeta)]$$

$$\times \left[P\_1(t,\zeta) + \frac{\sin(at)}{at}P\_2(t,\zeta)\right] \frac{\zeta^2\left(1-\zeta^2\right)}{G(t,\zeta)},\tag{11}$$

where P tð Þ¼ ; <sup>ζ</sup> cos <sup>2</sup><sup>ϑ</sup> <sup>þ</sup> sin <sup>2</sup>ϑ=G tð Þ ; <sup>ζ</sup> and

relative velocity vector v<sup>r</sup> with n<sup>r</sup> ¼ vr=vr. As can be inferred from Eq. (1), s represents the distance of the closest approach between the ion and the guiding center of the electron's helical

� F vð Þ<sup>i</sup> , then reads

ð d2 s ð∞ �∞

dkUð Þ k ð Þ k � v<sup>i</sup>

f ve∥; ve<sup>⊥</sup> � �ve⊥dve<sup>⊥</sup>

� �

sin ð Þ ωct ωct

ð∞ �∞

½ � k � r1ð Þt e

<sup>i</sup> <sup>k</sup>�r0ð Þ<sup>t</sup> dt: (6)

. Inserting

(7)

v<sup>i</sup> � f r½ � ð Þt dt, (5)

vi

the second-order perturbative treatment, the stopping power can be represented as

From Eq. (6) it is seen that the second-order stopping power is proportional to Z<sup>2</sup>

ð∞ �∞ dve<sup>∥</sup> ð∞ 0

ð Þ k � v<sup>i</sup>

2

� �, and performing the s integration, we then obtain

4 Z2 =e<sup>4</sup>ne

<sup>d</sup>kj j <sup>U</sup>ð Þ <sup>k</sup> <sup>2</sup>

� <sup>J</sup><sup>0</sup> <sup>2</sup>k⊥<sup>a</sup> sin <sup>ω</sup>ct

mvi

� �

now Eqs. (1) and (3) into Eq. (6), assuming an axially symmetric velocity distribution

ð∞ 0 k2 <sup>∥</sup> þ k 2 ⊥

where Jn is the Bessel function of the nth order; k<sup>∥</sup> ¼ ð Þ k � b and k<sup>⊥</sup> are the components of k parallel and transverse to b, respectively; and ve<sup>∥</sup> and ve<sup>⊥</sup> are the electron velocity components parallel and transverse to b, respectively. This general expression (7) for the stopping power of an individual ion has been derived within second-order perturbation theory but without any

For an electron plasma with an isotropic Maxwell distribution, the velocity distribution rele-

sin ð Þ k � vrt dtd,

ð d2 s ð

dvefð Þ v<sup>e</sup> vr

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 U rð Þ, and the binary force f rð Þ is written using Fourier transformation in space. Furthermore, the factor e<sup>i</sup>k�rð Þ<sup>t</sup> in the Fourier transformed binary force is expanded in a perturbative manner as <sup>e</sup><sup>i</sup>k�rð Þ<sup>t</sup> <sup>≃</sup>e<sup>i</sup>k�r0ð Þ<sup>t</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>i</sup>ð Þ <sup>k</sup> � <sup>r</sup>1ð Þ<sup>t</sup> , where r0ð Þt and r1ð Þt are the unperturbed and the first-order corrected relative coordinates (Eqs. (1) and (3)), respectively. Next, we consider only the second-order binary force f<sup>2</sup> and the corresponding stopping force F<sup>2</sup> with respect to the binary interaction since the averaged first-order force F<sup>1</sup> (related to f1) vanishes due to symmetry reasons [8, 24, 44, 45, 47]. Within

ð

motion.

The resulting stopping power, <sup>S</sup>ð Þ¼� <sup>v</sup><sup>i</sup> <sup>v</sup><sup>i</sup>

Sð Þ¼� v<sup>i</sup>

fð Þ¼ v<sup>e</sup> f ve∥; ve<sup>⊥</sup>

Z=e<sup>2</sup>ne vi

ð

<sup>S</sup> ¼ � ð Þ <sup>2</sup><sup>π</sup>

� Ð

restriction on the strength of the magnetic field B.

vant for the averaging in Eq. (7) is given by

2.4. The SP for a regularized and screened coulomb potential

dvefð Þ v<sup>e</sup> vr

Sð Þ¼� v<sup>i</sup>

72 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Ze<sup>2</sup> ne vi

$$P\_1(t,\zeta) = 2\cos^2\theta + P(t,\zeta)\left(1 - 2\nu^2\zeta^2\cos^2\theta\right),\tag{12}$$

$$P\_2(t; \zeta) = \frac{2}{G(t, \zeta)} \left[ \frac{\sin^2 \theta}{G(t, \zeta)} + P(t, \zeta) \left( 1 - \frac{v^2 \zeta^2 \sin^2 \theta}{G(t, \zeta)} \right) \right]. \tag{13}$$

Here, we have introduced the dimensionless quantities <sup>v</sup> <sup>¼</sup> vi<sup>=</sup> ffiffiffi 2 <sup>p</sup> <sup>v</sup>th, <sup>α</sup> <sup>¼</sup> <sup>ω</sup>cλ=vth. <sup>ϑ</sup> is the angle between b and vi, Ψð Þ¼ t; ζ t <sup>2</sup>=<sup>2</sup> � � <sup>1</sup> � <sup>ζ</sup><sup>2</sup> � �=ζ<sup>2</sup> , G tð Þ¼ ; <sup>ζ</sup> <sup>Θ</sup>ð Þ<sup>t</sup> <sup>ζ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> , <sup>Θ</sup>ðÞ¼ <sup>t</sup> <sup>2</sup> <sup>α</sup><sup>t</sup> sin <sup>α</sup><sup>t</sup> 2 � �<sup>2</sup> , and

$$\Phi(z) = e^{-z} + e^{-\varkappa^2 z} - \frac{2}{\varkappa^2 - 1} \frac{1}{z} \left( e^{-z} - e^{-\varkappa^2 z} \right),\tag{14}$$

as discussed in Section 2.4. In this case the generalized Coulomb logarithm takes the standard

The asymptotic expression of Eq. (15) at high ion velocities can be easily derived using the

argument of the Bessel function vanish since the cyclotron radius a ! 0. In this limit, denoting the SP as S∞ð Þ vi and assuming a spherically symmetric interaction potential, we arrive at

> 2πZ<sup>2</sup> =e<sup>4</sup>ne mv<sup>2</sup> i

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 U ¼ UC. Using instead a regularized interaction potential and thus the Coulomb logarithm, U<sup>R</sup> allows closed analytic expressions and converging integrals and avoids any introduction of lower and upper cutoffs "by hand" in order to restrict the domains of integration. Moreover, employing the bare Coulomb interaction may, as pointed out by Parkhomchuk [58], result in asymptotic expressions which essentially different from Eqs. (19) and (20), which is related to the divergent nature of the bare Coulomb interac-

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 highvelocity limit where vi > ð Þ ωcλ; vth , only small t contributes to the SP (Eq. (11)) due to the short time response of the electrons to the moving fast ion. In this limit we have sin ð Þ αt =αt ! 1. The

Here, the function Φð Þz is determined by Eq. (14), and Λ ϰð Þ is the generalized Coulomb logarithm (Eq. (17)). The remaining expressions do not depend on the magnetic field, i.e., ωc, as a natural consequence of the short time response of the magnetized electrons. In fact,

remaining t integration can be performed explicitly. This integral is given by [47].

ð∞ 0 dt t

=e<sup>4</sup>ne mv<sup>2</sup> i

<sup>v</sup><sup>5</sup> vi∥ve<sup>∥</sup> � <sup>2</sup>v<sup>2</sup>

U sin <sup>2</sup>

<sup>e</sup><sup>∥</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> i

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

Φ½ �� Ψð Þ t; ζ Λ ϰð Þ: (21)

U: (18)

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

� �<sup>f</sup> <sup>e</sup>ð Þ <sup>v</sup><sup>e</sup> <sup>d</sup>ve: (19)

ϑ: (20)

2 <sup>⊥</sup> and the 75

<sup>S</sup>0ð Þ <sup>v</sup><sup>i</sup> <sup>≃</sup> <sup>4</sup>πZ<sup>2</sup>

At an infinitely strong magnetic field ð Þ B ! ∞ , the term in Eq. (7) proportional to k

Uvi sin <sup>2</sup> ϑ ð 1

S∞ð Þ¼ v<sup>i</sup>

The corresponding high-velocity asymptotic expression is given by

form U ¼ U<sup>C</sup> ¼ ln ð Þ¼ kmax=kmin ln ð Þ rmax=rmin .

S∞ð Þ¼ v<sup>i</sup>

tion (see Ref. [47]).

3.2. Some limiting cases of Eq. (11)

normalization of the distribution function which results in

2πZ<sup>2</sup> =e<sup>4</sup>ne m

where ϰ ¼ λ=d ¼ 1 þ λ=ƛ.

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