5.2. Specific trends

left of SP maxima, as shown in Figure 4 at B ¼ 45 tesla, for the profiles displayed in Figure 3,

Switching now attention to corresponding ranges, down to projectile at rest Ep <sup>¼</sup> <sup>0</sup> , one

ranges attributed to the highest B. The fan of B ranges then merges on a given point, located between 10 keV and 100 keV at ne <sup>¼</sup> <sup>10</sup>16cm�3, and then inverts itself with increasing B featuring now increasing ranges. Moreover, the aperture of the fan of ranges increases steadily

Finally, it can be observed that for θ ¼ 0, the infinite magnetized range looks rather peculiar

Figure 4. Same as in Figure 3 restricted to B ¼ 45 T, featuring θ-dependent and θ-averaged SP in eV/cm.

∥ V !

[8, 34].

, one gets the largest B effects and the smallest proton

2.

witnesses on Figure 5 the counterpart of the above-noticed SP behavior.

Vth ≤ 1 

80 Plasma Science and Technology - Basic Fundamentals and Modern Applications

with <sup>θ</sup>-averaged SP remaining close to <sup>θ</sup> <sup>¼</sup> <sup>π</sup>

and reminiscent of the ion projectile gliding on B!

Figure 5. Ranges, down to zero energy pertaining to SP in Figure 3.

In a low projectile velocity <sup>V</sup>

with θ.

The projected experimental setup [62] could manage constant, static, and homogeneous B values up to 45 T. So, we are let to investigate n<sup>e</sup> range limits within which significant B effects can be observed.

Obviously, n<sup>e</sup> <sup>¼</sup> 1012 cm�<sup>3</sup> is expected to show quantitatively larger B impact than 1018 cm�3.

Giving attention to proton ranges of T dependence in a low-density plasma n<sup>e</sup> <sup>¼</sup> <sup>10</sup>12cm�<sup>3</sup> at <sup>T</sup> <sup>¼</sup> 1 and 100 eV, respectively, (Figure 6), one witnesses the smallest ranges for <sup>V</sup> Vth ≪ 1, increased by four orders of magnitude between 1 and 100 eV while remaining essentially unchanged for <sup>V</sup> Vth <sup>≥</sup> 1. Turning now to n<sup>e</sup> <sup>¼</sup> 1018cm�<sup>3</sup> at T <sup>¼</sup> 1 eV, one can see that the given SP remains quasi-isotropic, hardly θ-dependent, except at extreme magnetization (B ¼ ∞). Discrepancies between B = 0 and 20 T remain visible only for <sup>V</sup> <sup>V</sup>th ≤ 2. B ¼ ∞ does not feature anymore the highest stopping when <sup>V</sup> <sup>V</sup>th <sup>≤</sup> 1. Also, B <sup>¼</sup> <sup>10</sup><sup>3</sup> SP exhibits a few top wigglings. Upshifting T at 10 eV yields back n<sup>e</sup> <sup>¼</sup> <sup>10</sup>18cm�<sup>3</sup> SPs very similar to these displayed on Figure 2 (n<sup>e</sup> <sup>¼</sup> 1016 cm�3, T <sup>¼</sup> 10 eV) Figure 7.

Corresponding proton ranges (n<sup>e</sup> <sup>¼</sup> 1018 cm�3, T <sup>¼</sup> 1 eV) are shown in Figure 8.

Experimentally, accessible and very small ranges are thus documented for <sup>V</sup> Vth ≤ 1. Here, B ¼ 0 and 20 T data remain everywhere distinguishable.

Figure 6. Proton ranges down to the rest of the target with n<sup>e</sup> <sup>¼</sup> 1012 cm�3, T <sup>¼</sup> 1, and 100 eV at <sup>θ</sup> <sup>¼</sup> <sup>π</sup> 2.

Figure 7. Same as in Figure 1 for n<sup>e</sup> <sup>¼</sup> 1018 cm�<sup>3</sup> and T <sup>¼</sup> 1 eV with <sup>θ</sup> <sup>¼</sup> 0, <sup>π</sup> <sup>4</sup>, and <sup>π</sup> 2.

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 B. ~ It should be pointed out that we restricted the target plasma to its electron component. It therefore remains to include the target ion contribution to proton stopping [63], thus featuring

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

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

83

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 exper-

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

Our results (Eq. (11)) were derived by using the screened interaction URð Þr . As already mentioned, the use and the modeling of such an effective two-body interaction are a major but indispensable approximation for a BC treatment where the full ion-target interaction is replaced by an accumulation of isolated ion-electron collisions. The replacement of the complicated real non-spherically symmetric potential, like the wake fields as shown and discussed in Ref. [60], with a spherically symmetric one is, however, well motivated by earlier studies on a BC treatment at vanishing magnetic field (see Refs. [53–55]). It was shown by comparison with 3D self-consistent PIC simulations that the drag force from the real nonsymmetric potential induced by the moving ion can be well approximated by an BC treatment employing a symmetric Debye-like potential with an effective velocity-dependent screening length λð Þ vi . In these studies also a recipe was given how to derive the explicit form of λð Þ vi , which turned out to be not too much different from a dynamic screening length of the simple form

the electron plasma frequency, and vth is a thermal velocity of electrons. Although no systematic studies about the use of such an effective interaction with a screening length λð Þ vi have been made for ion stopping in a magnetized electron plasma, the replacement of the real interaction by a velocity-dependent spherical one should be a reasonable approximation also in this case. The introduced dynamical screening length λð Þ vi also implies the assumption of a weak perturbation of the electrons by the ion and linear screening where the screening length is independent of the ion charge Ze, which coincide with the regimes of perturbative BC (see,

e.g., Ref. [54]). Therefore, we do not consider here possible nonlinear screening effects.

. Here, λ<sup>D</sup> ¼ vth=ω<sup>p</sup> is the Debye screening length at vi ¼ 0, ω<sup>p</sup> is

imental facts in the field of nonrelativistic ion stopping in magnetized target plasmas.

Bundesministerium für Bildung und Forschung (BMBF) under contract 06ER9064.

Appendix A: Adjustment of the effective interaction

a complete low-velocity ion slowing down.

Acknowledgements

λð Þ¼ vi λ<sup>D</sup> 1 þ ð Þ vi=vth

<sup>2</sup> h i<sup>1</sup>=<sup>2</sup>

Figure 8. Proton ranges in electron target n<sup>e</sup> <sup>¼</sup> 1012 cm�<sup>3</sup> and T <sup>¼</sup> 1 eV with <sup>θ</sup> <sup>¼</sup> 0 and <sup>π</sup> 4.

Figure 9. Very-low-velocity proton slowing down on target protons at B <sup>¼</sup> <sup>10</sup>3T (upper straight line) contrasted to target electron stopping (any B, lower straight lines).

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

Up to now we limited our investigation to proton stopping by target electrons. In the verylow-velocity regime V ≤ Vthi, the target protons can also contribute significantly as evidenced on Figure 9. This topic will be more thoroughly addressed in a separate presentation.

### 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 B. ~ It should be pointed out that we restricted the target plasma to its electron component. It therefore remains to include the target ion contribution to proton stopping [63], thus featuring a complete low-velocity ion slowing down.

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.
