Electrostatic fluctuations in collisional plasmas

We present a theory of electrostatic fluctuations in two-component plasmas where electrons and ions are described by Maxwellian distribution functions at unequal temperatures. Based on the exact solution of the Landau kinetic equation, that includes electron-electron, electron-ion, and ion-ion collision integrals, the dynamic form factor, $S(\vec{k},\omega )$, is derived for weakly coupled plasmas. The collective plasma responses at ion-acoustic, Langmuir, and entropy mode resonances are described for arbitrary wave numbers and frequencies in the entire range of plasma collisionality. The collisionless limit of $S(\vec{k},\omega )$ and the strong-collision result based on the fluctuation-dissipation theorem and classical transport at $T_e=T_i$ are recovered and discussed. Results of several Thomson scattering experiments in the broad range of plasma parameters are described and discussed by means of our theory for $S(\vec{k},\omega )$.

Figure 1

Dynamical form factors for argon plasma at $n_e={10}^{17}{\text{cm}}^{-3},T=2\text{eV},Z=1,A=18$. The probe wavelength is ${\lambda }_0=10.6\mu \text{m}$ and the scattering angle $\theta =6^{\circ }$. Dashed line is obtained using Eq. (54) and the continuous black line corresponds to the full theoretical $S(k,\omega )$ of our theory Eq. (51) for $T_e=T_i=T$.

Figure 2

Dynamical form factors for argon plasma at $n_e={10}^{17}{\text{cm}}^{-3},T=5\text{eV},Z=1,A=18$. The probe wavelength is ${\lambda }_0=10.6\mu \text{m}$ and the scattering angle $\theta =6^{\circ }$. Dashed line is obtained using Eq. (54) and the continuous black line corresponds to the full theoretical $S(k,\omega )$ of our theory Eq. (51) for $T_e=T_i=T$.

Figure 3

Dynamic form factors for Thomson scattering probe at ${\lambda }_0=5270\mathring{\mathrm{A}}$, scattering angle $\theta ={117}^{\circ }$, $T_e=100\mathrm{eV}$, $n_e=5.6{10}^{18}{\mathrm{cm}}^{-3}$, and $\alpha =1.58$. Only red shifted ion acoustic peaks are shown for three different ion temperatures in panels (a), (b), and (c). Results of the full collisional theory Eq. (51) of $S(\vec{k},\omega )$ are depicted by continuous lines, the collisionless $S^V(\vec{k},\omega )$ expression Eq. (52) gives results depicted by dashed lines. Panel (d) shows all three cases calculated using Eq. (51).

Figure 4

Dynamic form factors for $n_e={10}^{19}{\text{cm}}^{-3},{\lambda }_0=5320$, $\theta ={90}^{\circ }$ in nitrogen plasma. Only one, red shifted peak is shown out of two symmetric resonances. Results for collisional plasma with $k{\lambda }_{ei}=5.67$, $k{\lambda }_{ii}=0.27$, $Z=4$, $T_e=2T_i=20\mathrm{eV}$ are shown in (a), results for $T_e=2T_i=160\mathrm{eV}$, $Z=7$, $k{\lambda }_{ei}=129$, $k{\lambda }_{ii}=2.23$ are shown in panel (b). Dashed lines correspond to the collisionless theory Eq. (52), continuous lines correspond to the complete theory of the dynamical form factor Eq. (51).

Figure 5

Dynamic form factors for $n_e=5.6{10}^{18}{\text{cm}}^{-3},{\lambda }_0=5270,T_e=50\text{eV}$. (a) $\theta ={10}^{\circ }$, (b) $\theta ={60}^{\circ }$. Dashed lines correspond to the collisionless RPA theory Eq. (52), continuous lines correspond to the complete theory of the dynamic form factor, Eq. (51) or (55) and the BGK model Eq. (59) is shown by dotted lines, marked with BGK.

Figure 6

Dynamic form factors in the high frequency regime corresponding to H plasmas from x-ray probe scattering experiment with ${\lambda }_0=2.25\mathring{\mathrm{A}}$, $n_e={10}^{23}{\mathrm{cm}}^{-3}$, $T_e=20\mathrm{eV}$, $\theta ={10}^{\circ }$, $\alpha =1.97$. Dashed lines correspond to the collisionless theory Eq. (52), continuous lines correspond to the complete theory of the dynamical form factor, Eq. (51) or (55) and the BGK model Eq. (59) is shown by dotted lines. For comparison, we also show results from the Born-Mermin (BM) approximation [46].

Figure 7

Dynamic form factors in the high frequency regime, red shifted peak, corresponding to Be ($Z=2,A=9$) plasmas from x-ray scattering experiments with ${\lambda }_0=4.19\mathring{\mathrm{A}}$, $n_e={10}^{23}{\mathrm{cm}}^{-3}$, $\theta ={30}^{\circ }$. Dashed lines correspond to the collisionless theory Eq. (52), continuous lines correspond to the complete theory of the dynamical form factor, Eq. (51) or (55) and the BGK model Eq. (59) is shown by dotted lines. (a) $T_e=10\mathrm{eV}$, $k{\lambda }_{ei}=0.28$, $\alpha =1.74$; (b) $T_e=20\mathrm{eV}$, $k{\lambda }_{ei}=1.12$, $\alpha =1.23$; (c) $T_e=30\mathrm{eV}$, $k{\lambda }_{ei}=2.51$, $\alpha =1$; (d) $T_e=40\mathrm{eV}$, $k{\lambda }_{ei}=4.46$, $\alpha =0.87$; For comparison, we also show results from the BM approximation [46].