Plasmon resonance in warm dense matter

Collective Thomson scattering with extreme ultraviolet light or x rays is shown to allow for a robust measurement of the free electron density in dense plasmas. Collective excitations like plasmons appear as maxima in the scattering signal. Their frequency position can directly be related to the free electron density. The range of applicability of the standard Gross-Bohm dispersion relation and of an improved dispersion relation in comparison to calculations based on the dielectric function in random phase approximation is investigated. More important, this well-established treatment of Thomson scattering on free electrons is generalized in the Born-Mermin approximation by including collisions. We show that, in the transition region from collective to noncollective scattering, the consideration of collisions is important.

Figure 1

(Color online) Electronic dynamic structure factor $S_{ee}(k,\omega )$ from collective $(\alpha =2.0)$ up to noncollective $(\alpha =0.5)$ Thomson scattering calculated in the RPA (black-dotted line) for $T_e=0.5\mathrm{eV}$ and BMA ($T_e=0.5$, 2.0, $8.0\mathrm{eV}$) for a fully ionized hydrogen plasma with $n_e={10}^{21}{\mathrm{cm}}^{-3}$, a laser wavelength ${\lambda }_0=4.13\mathrm{nm}$, and a scattering angle ${\theta }_S=160°$.

Figure 2

(Color online) Dielectric function ${ϵ}^{\mathrm{RPA}}(k,\omega )$ of electrons for electron densities: (a) $n_e=1.0\times {10}^{23}{\mathrm{cm}}^{-3}$ $(\alpha =1.1)$ and (b) $n_e=5.0\times {10}^{23}{\mathrm{cm}}^{-3}$ $(\alpha =2.1)$. The electron temperature is $T_e=12\mathrm{eV}$, laser wavelength ${\lambda }_0=0.42\mathrm{nm}$, and the scattering angle ${\theta }_S=40°$.

Figure 3

(Color online) Comparison of the maximum position $\Delta E$ of $S_{ee}(k,\omega )$ in the BMA (solid lines) with the energy shift ${\omega }_{\mathrm{GB}}\left(k\right)$ (dashed lines) as a function of the electron density $n_e$ for a fully ionized hydrogen plasma at electron temperatures $T_e=1\mathrm{eV}$ (black lines) and $T_e=20\mathrm{eV}$ (red lines). The scattering setup for the different wave numbers $k$ is given in Table .

Figure 4

(Color online) Comparison of the maximum position $\Delta E$ of $S_{ee}(k,\omega )$ in the RPA (dotted line) and BMA (solid line) with the energy shift of the IDR (dashed line) in dependence on wave number $k$ and scattering angle ${\theta }_S$, respectively, for Thomson scattering on beryllium plasma with $Z_{\mathrm{eff}}=2.5$, $n_e=(1.0,3.0,6.0)\times {10}^{23}{\mathrm{cm}}^{-3}$, $T_e=12\mathrm{eV}$, and laser wavelength ${\lambda }_0=0.42\mathrm{nm}$.

Figure 5

(Color online) Comparison of the maximum position $\Delta E$ of $S_{ee}(k,\omega )$ in the RPA (green-dashed line) and BMA (black-solid line) with the energy shift in the GB dispersion relation (blue dash-dotted line) and IDR (red dash-dotted line) and local field corrections (LFC: orange dotted line) [35] in dependence on electron density $n_e$ for a beryllium plasma with $Z_{\mathrm{eff}}=2.5$, $T_e=12\mathrm{eV}$, and laser wavelength ${\lambda }_0=0.42\mathrm{nm}$ and scattering angle ${\theta }_S=40°$. The green points are the zeros of $\mathrm{Re}ϵ(k,\omega )$, and the experimental point is taken from [4].