Effects of target heating on experiments using Kα and Kβ diagnostics

We describe the impact of heating and ionization on emission from the target of $K\alpha$ and $K\beta$ radiation induced by the propagation of hot electrons generated by laser-matter interaction. We consider copper as a test case and, starting from basic principles, we calculate the changes in emission wavelength, ionization cross section, and fluorescence yield as Cu is progressively ionized. We have finally considered the more realistic case when hot electrons have a distribution of energies with average energies of 50 and 500 keV (representative respectively of “shock ignition” and of “fast ignition” experiments) and in which the ions are distributed according to ionization equilibrium. In addition, by confronting our theoretical calculations with existing data, we demonstrate that this study offers a generic theoretical background for temperature diagnostics in laser-plasma interactions.

Figure 1

Typical experimental setup for measuring the penetration range of fast electrons through $K\alpha$ spectroscopy.

Figure 2

Average fluorescence yield (in %) of (a) $K\alpha$ and (b) $K\beta$ transition arrays as functions of the charge state $Z^{*}$.

Figure 3

Comparison of $K\alpha$ emission energies as calculated from Eq. (5) with those of flychk [33].

Figure 4

Total K-shell electron impact cross sections of the ground states of ions of charge states $Z^{*}$ plotted as functions of the incident electron kinetic energy.

Figure 5

Ratios with respect to (a) neutral species for the KEII cross section at the 30 keV peak energy, and (b) the $K\alpha$ and (c) $K\beta$ fluorescence yields as functions of charge states $Z^{*}$.

Figure 6

Monte Carlo simulations of $K\alpha$ and $K\beta$ intensities generated within a $10-\mu \mathrm{m}$-thick Cu target assuming an input exponential electron distribution with $T_{\mathrm{hot}}=50\mathrm{keV}$ and 500 keV. $K\alpha$ and $K\beta$ intensities are normalized per J of incident electrons. Full symbols correspond to calculations made for individual $Z^{*}$ while empty symbols correspond to $Z^{*}$-averaged calculations over the realistic ion distribution of Fig. 9.

Figure 7

Evolution of $K\alpha$ and $K\beta$ intensities as functions of $Z^{*}$ with respect to cold matter (i.e., with $Z^{*}=0$). Full symbols correspond to calculations made for individual $Z^{*}$ while empty symbols correspond to $Z^{*}$-averaged calculations over the realistic ion distribution of Fig. 9.

Figure 8

Evolution of $K\alpha /K\beta$ ratio as a function of $Z^{*}$. Full circles correspond to calculations made for individual $Z^{*}$ while empty circles correspond to $Z^{*}$ averaged calculations over the realistic ion distribution of Fig. 9.

Figure 9

Ionic charge $Z^{*}$ distribution as calculated by flychk [33] for different target temperatures $T$ at $N_e={10}^{19}/\mathrm{c}{\mathrm{m}}^3$.

Figure 10

(a) Temperature $T$ vs mean ionization state $\langle Z^{*}\rangle$, as calculated by flychk; (b) $K\alpha /K\beta$ ratio as a function of $Z^{*}$, deduced from Figs. 8 and 10; (c) experimental data for the $K\alpha /K\beta$ ratio as a function of energy density $(\mathrm{J}/\mathrm{m}{\mathrm{m}}^3)$, as taken from [42]; (d) bulk temperature inferred using our calculations (full triangles) compared to results given in [42] (dashed line).