Electron Temperature and Ionization degree

After having specified the treatment of energy losses in plasma, the influence of tar- get electron temperature and ionization degree can be studied. Since the electron temperature was not measured in the experiments, some work has to be dedicated to find reasonable constraints to this quantity from published data and theoretical considerations.

Estimating the electron temperature is quite straightforward for conducting (i.e. metal) targets. Since the charge transported by the relativistic hot electron beam can be compensated by cold return currents, no large space-charge fields can be sustained by the target. Thus, the target heating can be described by a purely collisional model for electron stopping in cold matter [56], and the internal energy in eV per target atom is simply given by (in the 1-D case):

Ei(x) = j(x) nat · dE dx(x) ! e . (4.10)

Here, j(x) is the electron current density in electrons/cm, nat the atom density in atoms/cm3 _{and (dE/dx(x))}

e the electron energy loss in eV/cm at a given target depthx. The situation gets more complex for non-conducting targets. Here, there are no free electrons available to set up a return current, so a large space-charge field can be built up, acting against a further transport of electrons. This leads

to a varying degree of electric (space-charge) inhibition, depending on the target conductivity, electron beam charge and current density [52,43]. This effect cannot be modeled easily, but fortunately some measurements exist [56], which allow a very crude extrapolation to the conditions prevailing in the present experiment.

Tikhonchuk’s model [55] for field-inhibited electron propagation suggests a smaller electron range in the target, leading to an enhanced stopping for electrons in the target and therefore to a larger energy deposition of the electrons along a given trajectory. However, the estimate presented here agrees pretty well with the nu- merical results given by Gremillet [52, 43].

10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5 101 102 103 104 105 106 107 108 109 1010 C Al CH CD₂ H₂O temperature [eV] in te rn a le n e rg y [J /g ]

Figure 4.5: EOS curves taken from SESAME [57] for carbon, H2O, CD2, and aluminum.

After having determined the local internal energy of the target, its local tem- perature can be inferred from Equation-of-State (EOS) tables like SESAME [57]. Fig. 4.5 shows the temperature plotted against the internal energy for carbon, water (as substitute for D2O), CD2 and aluminum.

From the local temperature, the ionization degree of the target matter was determined using the Thomas-Fermi ionization model [58]. This simple model provides an estimate of the electron temperatures and mean charge states in the target at a time before the ions are penetrating. The values calculated from it are surprisingly close to detailed 3-D calculations done by Gremillet [43].

In Fig.4.6 typical results for the local electron temperature and effective charge state are shown, along with modeled neutron time-of flight spectra for electron- heated and cold targets. The spectra were simulated with MCNEUT (described below). The laser parameters used in the upper row (corresponding to the Jena experiment in chapter 7) of Fig. 4.6were EL=0.6J, I=3×1019 W/cm2,λ=0.79µm, which assuming a conversion efficiency of η=0.2 leads to 5.2_×1011 _{electrons with} a temperature of Te=1.45 MeV. One can see that the temperature in the vicinity

of the focus reaches 40 eV, leading to an ionization degree of 5, corresponding to 50% of all electrons. This amount of heating is sufficient to ionize a large enough fraction of the target material to have an effect on the stopping and therefore the neutron output. However, the main effect is a small increase in stopping power due to the ionization. This in turn leads to a small reduction in neutron output, since a given ion ranges out more quickly and therefore encounters less fusion partners. The spectral shape remains unchanged. The heating is not strong enough to raise the thermal velocity of the bulk electrons to a value sufficient to have a decreasing effect on the stopping power.

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Figure 4.6: Neutron time-of-flight spectra, electron temperature and effective charge state as calculated by the simple model used here. The steps in the graphs are artefacts from the numerical treatment. They are negligible for the overall result. (upper row: Jena case, D2O droplets, lower row: LULI case, CD2 target.)

This effect is only encountered for the typical LULI case (described in chapter

EL=15J, I=5×1019 W/cm2, λ=1.064µm, η=0.3, leading to 1013 electrons with Te=2.81 MeV. This leads to a local electron temperature of∼1 keV, and to almost complete ionization close to the focus. Now the heated target produces a slightly higher neutron output than the cold one, indicating a reduced ion energy loss. Still, the overall effect is small and this consideration shows that the effect of target heating is of minor importance.