5. Stopping profiles and ranges

#### 5.1. General trends

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

velocities as shown in Appendix A (see also Ref. [47] for more details).

78 Plasma Science and Technology - Basic Fundamentals and Modern Applications

by averaging over a large number (typically 105

CTMC results for B ¼ ∞ case are shown by the filled circles.

Figure 2. Same as in Figure 1 but for T ¼ 10eV. The SP is given in units eV/cm.

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 ne ≃1016cm�<sup>3</sup> and the temperatures T ≃1 eV, 10 or 100 eV of the electron plasma, are in the expected range of the envisaged experiments on proton or alpha particles stopping in a magnetized target plasma [46] (see corresponding Figures 1–3). As an example we choose proton projectile for our calculations. In all examples considered below, the regularization parameter <sup>ƛ</sup><sup>0</sup> <sup>¼</sup> <sup>10</sup>�<sup>10</sup> mm thereby meets the condition ƛ<sup>0</sup> ≫ b0ð Þ0 , i.e., ƛ0, and does not affect noticeably the SP (Eq. (11)) at low and medium

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 [37–40]. 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

Figure 1. The SP [in keV/cm] for protons as a function of the ion velocity vi [in units of vth ] and for fixed plasma temperature <sup>T</sup> <sup>¼</sup> 1 eV. The theoretical stopping power (Eq. (11)) is calculated for <sup>ƛ</sup><sup>0</sup> <sup>¼</sup> <sup>10</sup>�10m (see appendix a for details) and for an electron plasma with ne <sup>¼</sup> 1016cm�<sup>3</sup> in a magnetic field of <sup>B</sup> <sup>¼</sup> 0 (black), 45 T (green), 200 T (blue), 103 T (red), 104 T (green), and <sup>B</sup> <sup>¼</sup> <sup>∞</sup> (cyan). The angle <sup>ϑ</sup> between <sup>B</sup> and vi is <sup>ϑ</sup> <sup>¼</sup> 0 (left), <sup>ϑ</sup> <sup>¼</sup> <sup>π</sup>=4 (center), and <sup>ϑ</sup> <sup>¼</sup> <sup>π</sup>=2 (right). The

–106

) of trajectories employing a Monte Carlo

The parameter analysis initiated on Figures 1–<sup>3</sup> at ne <sup>¼</sup> 1016cm�<sup>3</sup> and <sup>T</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup> � 100 eV is implemented for monitoring a possible experimental vindication through a fully ionized hydrogen plasma out of high-power laser beams available on facilities such as ELFIE (Ecole Polytechnique) or TITAN (Lawrence Livermore) [62]. The given adequately magnetized targets (in the 20–45 T range) would then be exposed to TNSA laser-produced proton beams out of the same facilities, in the hundred keV-MeV energy range [62].

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

Fixing ne and varying T (see Figures 1–3) display an ubiquitous and increasing anisotropy shared by the stopping profiles (SP) with increasing B and θ and angle between B! and initial projectile velocity V! .

Moreover, that anisotropy evolves only moderately between <sup>θ</sup> <sup>¼</sup> <sup>π</sup> <sup>4</sup> and <sup>π</sup> 2.

Another significant feature is the extension to any B ¼6 0 of the B <sup>¼</sup> 0 scaling <sup>n</sup><sup>e</sup> <sup>T</sup> . For instance, SP at <sup>n</sup><sup>e</sup> <sup>¼</sup> 1012 cm�<sup>3</sup> and T <sup>¼</sup> 1 eV, at a given <sup>θ</sup>, is equivalent to that for n<sup>e</sup> <sup>¼</sup> 1014 cm�<sup>3</sup> and T <sup>¼</sup> 100 eV.

As expected, B effects impact essentially the low-velocity section ( <sup>V</sup> Vth, Vth = target electron thermal velocity) of the ion stopping profile. One can observe, increasing with B, a shift to the left of SP maxima, as shown in Figure 4 at B ¼ 45 tesla, for the profiles displayed in Figure 3, with <sup>θ</sup>-averaged SP remaining close to <sup>θ</sup> <sup>¼</sup> <sup>π</sup> 2.

5.2. Specific trends

can be observed.

unchanged for <sup>V</sup>

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

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>

increased by four orders of magnitude between 1 and 100 eV while remaining essentially

SP remains quasi-isotropic, hardly θ-dependent, except at extreme magnetization (B ¼ ∞).

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

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>

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>

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>

Discrepancies between B = 0 and 20 T remain visible only for <sup>V</sup>

anymore the highest stopping when <sup>V</sup>

and 20 T data remain everywhere distinguishable.

(n<sup>e</sup> <sup>¼</sup> 1016 cm�3, T <sup>¼</sup> 10 eV) Figure 7.

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

Vth ≪ 1,

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<sup>V</sup>th ≤ 2. B ¼ ∞ does not feature

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

2.

Vth ≤ 1. Here, B ¼ 0

<sup>V</sup>th <sup>≤</sup> 1. Also, B <sup>¼</sup> <sup>10</sup><sup>3</sup> SP exhibits a few top wigglings.

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

<sup>4</sup>, and <sup>π</sup> 2.

Switching now attention to corresponding ranges, down to projectile at rest Ep <sup>¼</sup> <sup>0</sup> , one witnesses on Figure 5 the counterpart of the above-noticed SP behavior.

In a low projectile velocity <sup>V</sup> Vth ≤ 1 , one gets the largest B effects and the smallest proton 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 with θ.

Finally, it can be observed that for θ ¼ 0, the infinite magnetized range looks rather peculiar and reminiscent of the ion projectile gliding on B! ∥ V ! [8, 34].

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

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