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 Uð Þ k and electron distributions fð Þ v<sup>e</sup> as given by Eq. (7) and later for the more specific situation of the regularized interaction (10) and the velocity distribution (8) as given by Eq. (11).

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

At vanishing magnetic field ð Þ B ! 0 , sin ð Þ ωct =ð Þ! ωct 1 and the argument of the Bessel function in Eq. (7) should be replaced by k⊥ve⊥t. Then, denoting the second-order SP at vanishing magnetic field as S<sup>0</sup> and assuming spherically symmetric potential with U ¼ U kð Þ, one obtains

$$S\_0(\mathbf{v}\_i) = \frac{4(2\pi)^2 Z^2 \mathfrak{f}^4 \mathfrak{n}\_\varepsilon}{m v\_i^2} \quad \mathcal{U} \int\_0^{v\_i} f(v\_\varepsilon) v\_\varepsilon^2 dv\_\varepsilon \tag{15}$$

where U is the generalized Coulomb logarithm:

$$\mathcal{U} = \frac{(2\pi)^4}{4} \int\_0^\infty \mathcal{U}^2(k) k^3 dk. \tag{16}$$

Employing the regularized and screened potential U kð Þ given by Eq. (10), the generalized Coulomb logarithm is U ¼ U<sup>R</sup> ¼ Λ ϰð Þ (see also Refs. [8, 24, 44, 45]), where

$$
\Lambda(\varkappa) = \frac{\varkappa^2 + 1}{\varkappa^2 - 1} \ln \varkappa - 1. \tag{17}
$$

Taking the bare Coulomb interaction with U kð Þ¼ UCð Þ� k 1=k 2 , Eq. (16) diverges logarithmically at k ! 0 and k ! ∞, and two cutoffs kmin ¼ 1=rmax and kmax ¼ 1=rmin must be introduced as discussed in Section 2.4. In this case the generalized Coulomb logarithm takes the standard form U ¼ U<sup>C</sup> ¼ ln ð Þ¼ kmax=kmin ln ð Þ rmax=rmin .

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

$$S\_0(\mathbf{v}\_i) \simeq \frac{4\pi Z^2 \mathfrak{f}^4 n\_e}{m v\_i^2} \text{ } \mathcal{U}. \tag{18}$$

At an infinitely strong magnetic field ð Þ B ! ∞ , the term in Eq. (7) proportional to k 2 <sup>⊥</sup> and 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

$$S\_{\approx}(\mathbf{v}\_{i}) = \frac{2\pi Z^{2}\mathfrak{\phi}^{4}n\_{\varepsilon}}{m} \ \mathcal{U}v\_{i}\sin^{2}\theta \left[\frac{1}{v^{5}}\left(v\_{i\parallel}v\_{\varepsilon\parallel} - 2v\_{\varepsilon\parallel}^{2} + v\_{i}^{2}\right)f\_{\varepsilon}(\mathbf{v}\_{\varepsilon})d\mathbf{v}\_{\varepsilon}.\tag{19}$$

The corresponding high-velocity asymptotic expression is given by

$$S\_{\circ}(\mathbf{v}\_{i}) = \frac{2\pi Z^{2} \mathfrak{f}^{4} n\_{e}}{m v\_{i}^{2}} \ \ \mathcal{U} \sin^{2} \mathfrak{d} \,. \tag{20}$$

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 interaction (see Ref. [47]).

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

Here, we have introduced the dimensionless quantities <sup>v</sup> <sup>¼</sup> vi<sup>=</sup> ffiffiffi

Φð Þ¼ z e

74 Plasma Science and Technology - Basic Fundamentals and Modern Applications

3. Comparison with previous approaches

<sup>2</sup>=<sup>2</sup> � � <sup>1</sup> � <sup>ζ</sup><sup>2</sup> � �=ζ<sup>2</sup>

�<sup>z</sup> <sup>þ</sup> <sup>e</sup>

�ϰ2<sup>z</sup> � <sup>2</sup>

ϰ<sup>2</sup> � 1

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

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 Uð Þ k and electron distributions fð Þ v<sup>e</sup> as given by Eq. (7) and later for the more specific situation of the

At vanishing magnetic field ð Þ B ! 0 , sin ð Þ ωct =ð Þ! ωct 1 and the argument of the Bessel function in Eq. (7) should be replaced by k⊥ve⊥t. Then, denoting the second-order SP at vanishing magnetic field as S<sup>0</sup> and assuming spherically symmetric potential with U ¼ U kð Þ,

> U ðvi 0

f vð Þ<sup>e</sup> <sup>v</sup><sup>2</sup>

<sup>e</sup> dve, (15)

dk: (16)

, Eq. (16) diverges logarithmi-

<sup>ϰ</sup><sup>2</sup> � <sup>1</sup> ln <sup>ϰ</sup> � <sup>1</sup>: (17)

2

regularized interaction (10) and the velocity distribution (8) as given by Eq. (11).

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

4 2ð Þ π 2 Z2 =e<sup>4</sup>ne

mv<sup>2</sup> i

> 4 4

ð∞ 0 U2 ð Þk k 3

Employing the regularized and screened potential U kð Þ given by Eq. (10), the generalized

cally at k ! 0 and k ! ∞, and two cutoffs kmin ¼ 1=rmax and kmax ¼ 1=rmin must be introduced

<sup>U</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>π</sup>

Λ ϰð Þ¼ <sup>ϰ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup>

Coulomb logarithm is U ¼ U<sup>R</sup> ¼ Λ ϰð Þ (see also Refs. [8, 24, 44, 45]), where

S0ð Þ¼ v<sup>i</sup>

Taking the bare Coulomb interaction with U kð Þ¼ UCð Þ� k 1=k

where U is the generalized Coulomb logarithm:

1 z e �<sup>z</sup> � <sup>e</sup> �ϰ2<sup>z</sup> � �

between b and vi, Ψð Þ¼ t; ζ t

where ϰ ¼ λ=d ¼ 1 þ λ=ƛ.

in the next sections.

one obtains

2

, 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>p</sup> <sup>v</sup>th, <sup>α</sup> <sup>¼</sup> <sup>ω</sup>cλ=vth. <sup>ϑ</sup> is the angle

<sup>α</sup><sup>t</sup> sin <sup>α</sup><sup>t</sup> 2 � �<sup>2</sup>

, (14)

, and

, <sup>Θ</sup>ðÞ¼ <sup>t</sup> <sup>2</sup>

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

$$\int\_0^\varkappa \frac{dt}{t} \Phi[\Psi'(t,\zeta)] \equiv \Lambda(\varkappa). \tag{21}$$

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,

sin ð Þ αt =αt ! 1 and G tð Þ! ; ζ 1 and the related t integration (Eq. (21)) are also valid for vanishing magnetic field α ! 0. Integration by parts turns Eq. (11) into

$$S\_0 = \frac{4\pi Z^2 \phi^4 n\_e}{m\upsilon\_i^2} \Lambda(\varkappa) \left[ \text{erf}(\upsilon) - \frac{2}{\sqrt{\pi}} \upsilon e^{-\upsilon^2} \right],\tag{22}$$

In particular, at ϑ ¼ 0 and ϑ ¼ π=2 (i.e., when ion moves parallel or transverse to the magnetic

=e<sup>4</sup>ne

Λ ϰð Þve�v<sup>2</sup>

v2 2 � �

� <sup>v</sup><sup>2</sup> K1 v2 2 � � � � : (28)

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

2 ffiffiffi <sup>π</sup> <sup>p</sup> v v<sup>2</sup>

þ 4 ffiffiffi <sup>π</sup> <sup>p</sup> <sup>v</sup><sup>3</sup> e �v<sup>2</sup> cos <sup>2</sup> ϑ

<sup>i</sup> with the ion velocity. But here, the parallel SP (Eq. (27)) vanishes

� �: (30)

, (27)

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

77

<sup>E</sup><sup>1</sup> <sup>v</sup><sup>2</sup> � � � <sup>e</sup>

� � h i , (29)

�v<sup>2</sup>

mv<sup>2</sup> th

Λ ϰð Þve�v2=<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> � �K<sup>0</sup>

respectively, where Knð Þz (with n ¼ 0, 1) is the modified Bessel function. It is also constructive

Λ ϰð Þ erfð Þþ v

2 ffiffiffi <sup>π</sup> <sup>p</sup> ve�v<sup>2</sup> � �

=e<sup>4</sup>ne 3mv<sup>2</sup> i

ϑ erfð Þ� v

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

The SP given by Eq. (30) (or Eq. (20) with U ¼ Λ ϰð Þ) decays as the corresponding SP

exponentially at ϑ ¼ 0 which is a consequence of the presence of a strong magnetic field, where the electrons move parallel to the magnetic field. If the ion moves also parallel to the field (i.e., ϑ ¼ 0), the averaged stopping force must vanish within the BC treatment for

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

2 v sin ϑ � �

� �

� γ <sup>2</sup> � <sup>1</sup>

<sup>þ</sup> cos <sup>2</sup> ϑ

� �, (31)

ϑ ln

where γ≃ 0:5772 is Euler's constant. Now, it is seen that the SP, S<sup>∞</sup>, leads at low ion velocities v ≪ 1 and for a nonzero ϑ to a term which behaves as � v ln 1ð Þ =v . Thus, the corresponding friction coefficient diverges logarithmically at small v. This is a quite unexpected behavior compared to the well-known linear velocity dependence without magnetic field (see asymptotic expressions above). Finally, at ϑ ¼ 0 the logarithmic term vanishes and the SP behaves as

<sup>S</sup><sup>∞</sup> <sup>¼</sup> <sup>4</sup> ffiffiffi <sup>π</sup> <sup>p</sup> <sup>Z</sup><sup>2</sup>

to obtain the angular averaged stopping power. From Eq. (25) one finds

In the high-velocity limit with ωcλ ≫ vi ≫ vth, the SP (Eq. (25)) becomes

while erfð Þ!v 1 yields the asymptotic expression (Eq. (20)) (for U ¼ Λ ϰð Þ).

Λ ϰð Þ<sup>v</sup> sin <sup>2</sup>

Λ ϰð Þ sin <sup>2</sup>

=e<sup>4</sup>ne

mv<sup>2</sup> th

<sup>S</sup>∞ð Þ <sup>v</sup>; <sup>ϑ</sup> sin <sup>ϑ</sup>d<sup>ϑ</sup> <sup>¼</sup> <sup>4</sup>πZ<sup>2</sup>

where E1ð Þz is the exponential integral function.

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

small ion velocity ð Þ v ≪ 1 in Eq. (25), we arrive at

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

<sup>S</sup><sup>∞</sup> <sup>¼</sup> <sup>4</sup>πZ<sup>2</sup>

<sup>S</sup><sup>∞</sup> <sup>≃</sup> <sup>2</sup>πZ<sup>2</sup>

field, respectively), Eq. (25) (or Eq. (26)) yields

<sup>S</sup><sup>∞</sup> <sup>¼</sup> <sup>2</sup> ffiffiffi <sup>π</sup> <sup>p</sup> <sup>Z</sup><sup>2</sup>

S∞ð Þ¼ v

(Eq. (23)) like � <sup>v</sup>�<sup>2</sup>

symmetry reasons.

<sup>S</sup><sup>∞</sup> � <sup>v</sup>.

where erfð Þ<sup>z</sup> is the error function and <sup>v</sup> <sup>¼</sup> vi<sup>=</sup> ffiffiffi 2 <sup>p</sup> <sup>v</sup>th is again the scaled ion velocity. The SP (Eq. (22)) is isotropic with respect to the ion velocity v<sup>i</sup> and represents the two limiting cases of high velocities at arbitrary magnetic field and arbitrary velocities at vanishing field. Of course, expression (22) can be also obtained by performing the remaining integration in the nonmagnetized SP (Eq. (15)) using the isotropic velocity distribution (Eq. (8)) and U ¼ Λ ϰð Þ.

A further increase of the ion velocity finally yields

$$\mathcal{S}\_0 \simeq \frac{4\pi Z^2 \phi^4 n\_e}{m v\_i^2} \Lambda(\varkappa),\tag{23}$$

which completely agrees with the asymptotic expression (18) in case of U ¼ Λ ϰð Þ. Inspecting Eq. (23) shows that the SP does not depend explicitly on the electron temperature T at sufficiently high velocities, while T may still be involved in the generalized Coulomb logarithm Λ ϰð Þ.

At B ! 0 and small velocities ð Þ vi < vth , the SP (Eq. (22)) becomes

$$S\_0 \simeq \frac{4\pi\sqrt{2\pi}Z^2\xi^4 n\_e}{3mv\_{\text{th}}^3}v\_i\Lambda(\varkappa). \tag{24}$$

Now, we consider the situation when the magnetic field is very strong and the electron cyclotron radius is the smallest length scale, ωcλ ≫ vi ð Þ ; vth , and the SP is only weakly sensitive to the transverse electron velocities and, hence, is affected only by their longitudinal velocity spread. In this limit sin ð Þ <sup>α</sup><sup>t</sup> <sup>=</sup>α<sup>t</sup> ! 0 and G tð Þ! ; <sup>ζ</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> are obtained from Eq. (11) after straightforward calculations:

$$S\_{\nu} = \frac{4\pi\sqrt{\pi}Z^2\ell^4n\_e}{mv\_{\text{th}}^2}v\Lambda(\varkappa)\int\_0^1 e^{-v^2\zeta^2P(\zeta)}\zeta^2d\zeta \left[2\cos^2\theta + P(\zeta)\left(1 - 2v^2\zeta^2\cos^2\theta\right)\right],\tag{25}$$

where <sup>P</sup>ð Þ¼ <sup>ζ</sup> cos <sup>2</sup><sup>ϑ</sup> <sup>þ</sup> sin <sup>2</sup>ϑ<sup>=</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> � �.

After changing the variable <sup>ζ</sup> in Eq. (25) to <sup>x</sup> <sup>¼</sup> <sup>ζ</sup>½ � <sup>P</sup>ð Þ<sup>ζ</sup> <sup>1</sup>=<sup>2</sup> and some subsequent rearrangement, Eq. (25) can be expressed alternatively as

$$S\_{\simeq} = \frac{2\sqrt{\pi}Z^2\mathcal{J}^4 n\_e}{mv\_{\text{th}}^2} v\Lambda(\varkappa)\sin^2\theta \int\_{-\infty}^{\infty} \frac{e^{-v^2\mathbf{x}^2}\mathbf{x}^2d\mathbf{x}}{1+\mathbf{x}^2-2\mathbf{x}\cos\theta} \text{\(\prime\)}\tag{26}$$

Up to the definition of the Coulomb logarithm (i.e., U ¼ Λ ϰð Þ versus U ¼ UC), the expressions are identical to those obtained by Pestrikov [59].

In particular, at ϑ ¼ 0 and ϑ ¼ π=2 (i.e., when ion moves parallel or transverse to the magnetic field, respectively), Eq. (25) (or Eq. (26)) yields

$$S\_{\circ} = \frac{4\sqrt{\pi}Z^2\boldsymbol{\upmu}\_{\epsilon}}{m\upsilon\_{\text{th}}^2}\Lambda(\varkappa)ve^{-v^2},\tag{27}$$

$$S\_{\infty} = \frac{2\sqrt{\pi}Z^2\xi^4n\_e}{m\upsilon\_{\text{th}}^2}\Lambda(\varkappa)\upsilon e^{-\upsilon^2/2}\left[(1+\upsilon^2)K\_0\left(\frac{\upsilon^2}{2}\right) - \upsilon^2K\_1\left(\frac{\upsilon^2}{2}\right)\right].\tag{28}$$

respectively, where Knð Þz (with n ¼ 0, 1) is the modified Bessel function. It is also constructive to obtain the angular averaged stopping power. From Eq. (25) one finds

$$\overline{S}\_{\approx}(\upsilon) = \frac{1}{2} \int\_{0}^{\pi} S\_{\approx}(\upsilon, 8) \sin \theta d\theta = \frac{4\pi Z^{2} \mathfrak{z}^{4} n\_{\text{c}}}{3m v\_{i}^{2}} \Lambda(\varkappa) \left\{ \text{erf}(\upsilon) + \frac{2}{\sqrt{\pi}} \upsilon \left[ \upsilon^{2} E\_{1}(\upsilon^{2}) - e^{-\upsilon^{2}} \right] \right\},\tag{29}$$

where E1ð Þz is the exponential integral function.

sin ð Þ αt =αt ! 1 and G tð Þ! ; ζ 1 and the related t integration (Eq. (21)) are also valid for

Λ ϰð Þ erfð Þ� v

2

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

which completely agrees with the asymptotic expression (18) in case of U ¼ Λ ϰð Þ. Inspecting Eq. (23) shows that the SP does not depend explicitly on the electron temperature T at sufficiently high velocities, while T may still be involved in the generalized Coulomb loga-

(Eq. (22)) is isotropic with respect to the ion velocity v<sup>i</sup> and represents the two limiting cases of high velocities at arbitrary magnetic field and arbitrary velocities at vanishing field. Of course, expression (22) can be also obtained by performing the remaining integration in the nonmagnetized SP (Eq. (15)) using the isotropic velocity distribution (Eq. (8)) and U ¼ Λ ϰð Þ.

<sup>S</sup><sup>0</sup> <sup>≃</sup> <sup>4</sup>πZ<sup>2</sup>

<sup>S</sup><sup>0</sup> <sup>≃</sup> <sup>4</sup><sup>π</sup> ffiffiffiffiffiffi

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

3mv<sup>3</sup> th

Now, we consider the situation when the magnetic field is very strong and the electron cyclotron radius is the smallest length scale, ωcλ ≫ vi ð Þ ; vth , and the SP is only weakly sensitive to the transverse electron velocities and, hence, is affected only by their longitudinal velocity spread. In this limit sin ð Þ <sup>α</sup><sup>t</sup> <sup>=</sup>α<sup>t</sup> ! 0 and G tð Þ! ; <sup>ζ</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> are obtained from Eq. (11) after

After changing the variable <sup>ζ</sup> in Eq. (25) to <sup>x</sup> <sup>¼</sup> <sup>ζ</sup>½ � <sup>P</sup>ð Þ<sup>ζ</sup> <sup>1</sup>=<sup>2</sup> and some subsequent rearrangement,

ϑ ð∞ �∞

Up to the definition of the Coulomb logarithm (i.e., U ¼ Λ ϰð Þ versus U ¼ UC), the expressions

<sup>v</sup>Λ ϰð Þ sin <sup>2</sup>

=e<sup>4</sup>ne

dζ 2 cos <sup>2</sup>

<sup>ϑ</sup> <sup>þ</sup> <sup>P</sup>ð Þ<sup>ζ</sup> <sup>1</sup> � <sup>2</sup>v<sup>2</sup>

x<sup>2</sup>dx 1 þ x<sup>2</sup> � 2x cos ϑ

Þ 3=2

e�v2x<sup>2</sup>

2 ffiffiffi <sup>π</sup> <sup>p</sup> ve�v<sup>2</sup> � �

, (22)

<sup>p</sup> <sup>v</sup>th is again the scaled ion velocity. The SP

Λ ϰð Þ, (23)

viΛ ϰð Þ: (24)

ζ<sup>2</sup> cos <sup>2</sup> ϑ � � � � , (25)

: (26)

vanishing magnetic field α ! 0. Integration by parts turns Eq. (11) into

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

<sup>S</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup>πZ<sup>2</sup>

76 Plasma Science and Technology - Basic Fundamentals and Modern Applications

At B ! 0 and small velocities ð Þ vi < vth , the SP (Eq. (22)) becomes

where erfð Þ<sup>z</sup> is the error function and <sup>v</sup> <sup>¼</sup> vi<sup>=</sup> ffiffiffi

A further increase of the ion velocity finally yields

rithm Λ ϰð Þ.

straightforward calculations:

<sup>S</sup><sup>∞</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> ffiffiffi

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

mv<sup>2</sup> th

where <sup>P</sup>ð Þ¼ <sup>ζ</sup> cos <sup>2</sup><sup>ϑ</sup> <sup>þ</sup> sin <sup>2</sup>ϑ<sup>=</sup> <sup>1</sup> � <sup>ζ</sup><sup>2</sup> � �.

Eq. (25) can be expressed alternatively as

=e<sup>4</sup>ne

<sup>S</sup><sup>∞</sup> <sup>¼</sup> <sup>2</sup> ffiffiffi <sup>π</sup> <sup>p</sup> <sup>Z</sup><sup>2</sup>

are identical to those obtained by Pestrikov [59].

vΛ ϰð Þ

ð1 0 e �v2ζ<sup>2</sup> <sup>P</sup>ð Þ<sup>ζ</sup> ζ<sup>2</sup>

=e<sup>4</sup>ne

mv<sup>2</sup> th In the high-velocity limit with ωcλ ≫ vi ≫ vth, the SP (Eq. (25)) becomes

$$S\_{\simeq} \simeq \frac{2\pi Z^2 \mathfrak{f}^4 n\_{\epsilon}}{m\upsilon\_i^2} \Lambda(\varkappa) \left\{ \sin^2 \mathfrak{S} \left[ \text{erf}(\upsilon) - \frac{2}{\sqrt{\pi}} \upsilon e^{-\upsilon^2} \right] + \frac{4}{\sqrt{\pi}} \upsilon^3 e^{-\upsilon^2} \cos^2 \mathfrak{S} \right\}. \tag{30}$$

With further increase of the ion velocity, we can then neglect the exponential term in Eq. (30), while erfð Þ!v 1 yields the asymptotic expression (Eq. (20)) (for U ¼ Λ ϰð Þ).

The SP given by Eq. (30) (or Eq. (20) with U ¼ Λ ϰð Þ) decays as the corresponding SP (Eq. (23)) like � <sup>v</sup>�<sup>2</sup> <sup>i</sup> with the ion velocity. But here, the parallel SP (Eq. (27)) vanishes exponentially at ϑ ¼ 0 which is a consequence of the presence of a strong magnetic field, where the electrons move parallel to the magnetic field. If the ion moves also parallel to the field (i.e., ϑ ¼ 0), the averaged stopping force must vanish within the BC treatment for symmetry reasons.

Finally, we also investigate the case of small velocities at strong magnetic fields. Considering a small ion velocity ð Þ v ≪ 1 in Eq. (25), we arrive at

$$S\_{\simeq} = \frac{4\pi Z^2 \mathfrak{z}^4 n\_{\epsilon}}{m v\_{\text{th}}^2} \Lambda(\varkappa) v \left\{ \sin^2 \mathfrak{s} \left[ \ln \left( \frac{2}{v \sin \mathfrak{s}} \right) - \frac{\mathcal{V}}{2} - 1 \right] + \cos^2 \mathfrak{s} \right\}, \tag{31}$$

where γ≃ 0:5772 is Euler's constant. Now, it is seen that the SP, S<sup>∞</sup>, leads at low ion velocities v ≪ 1 and for a nonzero ϑ to a term which behaves as � v ln 1ð Þ =v . Thus, the corresponding friction coefficient diverges logarithmically at small v. This is a quite unexpected behavior compared to the well-known linear velocity dependence without magnetic field (see asymptotic expressions above). Finally, at ϑ ¼ 0 the logarithmic term vanishes and the SP behaves as <sup>S</sup><sup>∞</sup> � <sup>v</sup>.
