Stopping power of dense plasmas: The collisional method and limitations of the dielectric formalism

We present a study of the stopping power of plasmas using two main approaches: the collisional (scattering theory) and the dielectric formalisms. In the former case, we use a semiclassical method based on quantum scattering theory. In the latter case, we use the full description given by the extension of the Lindhard dielectric function for plasmas of all degeneracies. We compare these two theories and show that the dielectric formalism has limitations when it is used for slow heavy ions or atoms in dense plasmas. We present a study of these limitations and show the regimes where the dielectric formalism can be used, with appropriate corrections to include the usual quantum and classical limits. On the other hand, the semiclassical method shows the correct behavior for all plasma conditions and projectile velocity and charge. We consider different models for the ion charge distributions, including bare and dressed ions as well as neutral atoms.

Figure 1

Distribution functions for a $n={10}^{23}{\mathrm{cm}}^{-3}$ and $T=10\mathrm{eV}$ plasma. FD denotes the Fermi-Dirac, MB the Maxwell-Boltzmann, and MB* the modified Maxwell-Boltzmann.

Figure 2

Distribution functions and their corresponding SPWS stopping, for two plasma conditions: (a) $n={10}^{23}{\mathrm{cm}}^{-3}$ ($\theta =1.274$) and (b) $n=5\times {10}^{23}{\mathrm{cm}}^{-3}$ ($\theta =0.436$). In both cases the temperature is $T=10\mathrm{eV}$ and the projectile is a stripped ion with $Z=5$. FD, MB, and MB* refer to Fermi-Dirac, Maxwell-Boltzmann, and the modified Maxwell-Boltzmann, respectively.

Figure 3

Comparison of the Brandt-Kitagawa (BK) and Molière (M) projectile models. In (a) we show the potentials, for neutral atoms ($\eta =0$), as a function of the radial coordinate, and in (b) we show the charge density in momentum space, for different ionization degrees.

Figure 4

Comparison of the stopping power for different calculation methods. The plasma conditions are $n={10}^{18}{\mathrm{cm}}^{-3}$ and $T=100\mathrm{eV}$ (classical plasma). In (a) we can observe the concordance between the SPWS, DF, and PMtV calculations. In (b) we show that the plain DF calculation yields a very large overestimation, but when the Coulomb classical cutoff is applied (DF-$k_{\mathrm{cl}}^c$) it becomes in good agreement with the SPWS and PMtV calculations.

Figure 5

Comparison of the stopping power for different calculation methods. The plasma conditions are $n={10}^{23}{\mathrm{cm}}^{-3}$ and $T=10\mathrm{eV}$ (partially degenerate plasma). In (a) both models (SPWS and DF) are in agreement, but in (b) we can observe that the Coulomb $k_{\mathrm{cl}}^c$ cutoff approach fails at low velocities.

Figure 6

Comparison between $k_{\mathrm{qm}}$, $k_{\mathrm{cl}}^c$, and $k_s$. In (a) we show them as a function of plasma density, for $T=10\mathrm{eV}$ and $v=0$. In (b) we show the same $k$ limit values for a fixed plasma density $n={10}^{23}{\mathrm{cm}}^{-3}$ (and $T=10\mathrm{eV}$) but varying the projectile velocity $v$. In both cases the $k_s>k_{\mathrm{cl}}^c$ condition is not physically correct.

Figure 7

$k_s\sim k_{\mathrm{cl}}^c$ condition in the static limit ($v\ll v_s$) and for different charges $Z$ (bare ions). The curves separate the regions in which the Coulomb $k_{\mathrm{cl}}^c$ cutoff can be used.

Figure 8

Comparison of the SPWS and DF stopping with other formalism: a particle-in-cell code (PIC) and the BPS stopping (see the text). Here $u=v/\sqrt{3T/m}$ is a normalized velocity. The plasma conditions are (a) $n=1.1\times {10}^{20}{\mathrm{cm}}^{-3}$, $T=14\mathrm{eV}$ and (b) $n=1.4\times {10}^{20}{\mathrm{cm}}^{-3}$, $T=11\mathrm{eV}$.

Figure 9

Stopping of carbon for different ionization degrees $\eta$. We compare projectile models (BK and Molière) and stopping calculation (SPWS and DF) methods.

Figure 10

Typical frequency integrand of the dielectric stopping power, given by Eq. (21) and using the quantum dielectric function. The figure shows how the Bethe limit is implicitly incorporated in this description.