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

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 by averaging over a large number (typically 105 –106 ) of trajectories employing a Monte Carlo sampling for the related initial conditions. For a more detailed description of the method, we refer to Refs. [8, 44, 45]. Both the analytic perturbative treatment and the non-perturbative numerical CTMC simulations are based on the same BC picture and use the same effective spherical screened interaction U rð Þ. The following comparison of these both approaches thus essentially intends to check the validity and range of applicability of the perturbative approach

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

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79

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

Therefore, we are looking for the most conspicuous effect of the applied magnetized intensity

Fixing ne and varying T (see Figures 1–3) display an ubiquitous and increasing anisotropy

<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.

thermal velocity) of the ion stopping profile. One can observe, increasing with B, a shift to the

and initial

<sup>T</sup> . For instance, SP at

Vth, Vth = target electron

<sup>4</sup> and <sup>π</sup> 2.

shared by the stopping profiles (SP) with increasing B and θ and angle between B!

as it has been outlined in the preceding sections.

Figure 3. Same as in Figure 1 but for T ¼ 100 eV. The SP is given in units eV/cm.

of the same facilities, in the hundred keV-MeV energy range [62].

Moreover, that anisotropy evolves only moderately between <sup>θ</sup> <sup>¼</sup> <sup>π</sup>

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>

As expected, B effects impact essentially the low-velocity section ( <sup>V</sup>

5. Stopping profiles and ranges

5.1. General trends

B on the proton stopping.

.

projectile velocity V!

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 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.

Stopping Power of Ions in a Magnetized Plasma: Binary Collision Formulation http://dx.doi.org/10.5772/intechopen.77213 79

Figure 3. Same as in Figure 1 but for T ¼ 100 eV. The SP is given in units eV/cm.

sampling for the related initial conditions. For a more detailed description of the method, we refer to Refs. [8, 44, 45]. Both the analytic perturbative treatment and the non-perturbative numerical CTMC simulations are based on the same BC picture and use the same effective spherical screened interaction U rð Þ. The following comparison of these both approaches thus essentially intends to check the validity and range of applicability of the perturbative approach as it has been outlined in the preceding sections.
