1. INTRODUCTION
In inertial confinement fusion (ICF) driven by heavy ion beams, the fuel pellet consists of a fuel, a pusher, a tamper, and a radiator. A foamed metal such as the pusher and the radiator is considered as structural materials in the fuel pellet (Atzeni & Meyer-ter-vehn, Reference Atzeni and Meyer-Ter-vehn2004). The fuel pellet is rapidly imploded by irradiation from the energy driver. The fuel becomes dense plasma at the center of the fuel pellet, and causes thermonuclear fusion reactions. In order to obtain effective nuclear fusion reactions, it is a key issue to understand implosion dynamics of the fuel pellet.
The structural materials in the fuel pellet become dense plasma through the warm dense matter (WDM) (Dewald et al., Reference Dewald, Constantin, Udrea, Jacoby, Hoffmann, Niemann, Weiser, Tahir, Kozyreva, Shutov and Tauschwitz2002; Constantin et al., Reference Constantin, Dewald, Niemann, Hoffmann, Udrea, Varentsov, Jacoby, Funk, Neuner and Tauschwitz2004; Hoffmann et al., Reference Hoffmann, Blazevic, Ni, Rosmej, Roth, Tahir, Tauschwitz, Udrea, Varentsov, Weyrich and Maron2005; Ng et al., Reference Ng, Ao, Perrot, Dharma-Wardana and Foord2005; Sasaki et al., Reference Sasaki, Yano, Nakajima, Kawamura and Horioka2006; Drake, Reference Drake2009; Redmer & Röpke, Reference Redmer and Röpke2010) region with implosion time-scale (several-10 ns). The WDM region corresponds to densities from about 1021 to 1024 cm−3 and to temperatures from 103 to 105 K. To estimate on the accurate state of the fuel pellet, the effects of ion–ion correlations and degenerate electrons should be taken into account in the WDM region. However, the WDM is a complex region, because of the unclear theoretical model, and lacked experimental evaluations. Therefore, the properties in WDM should be clear to design the fuel pellet. To understand the properties of matter, we should design evaluation methods for the properties of the WDM.
In previous studies, a short pulse laser (Yoneda et al., Reference Yoneda, Morikami, Ueda and More2003; Glenzer et al., Reference Glenzer, Landen, Neumayer, Lee, Widmann, Pollaine, Wallace, Gregori, Höll, Bornath, Thiele, Schwarz, Kraeft and Redmer2007) and a pulsed-power discharge (DeSilva & Kunze, Reference Desilva and Kunze1994; Saleem et al., Reference Saleem, Haun and Kunze2001; Sasaki et al., Reference Sasaki, Suzuki, Amano, Miki, Kikuchi, Harada and Horioka2011; Clérouin et al., Reference Clérouin, Noiret, Blottiau, Recoules, Siberchicot, Renaudin, Blancard, Faussurier, Holst and Starrett2012) were used for the measurement of WDM. However, these experiments were difficult to compare due to the different time-scale and achievable parameters for the generation of WDM. To understand the relationship of these different experimental evidences, we considered the generation of WDM with an intense pulsed-power generator.
The intense pulsed-power generator had some opportunities with a large volume and/or dense state of the WDM. In order to investigate the properties of the WDM in the fuel pellet with the implosion process, the evaluation method for the WDM with isochoric heating (Amano et al., Reference Amano, Miki, Takahashi, Sasaki, Kikuchi and Harada2012; Miki et al., Reference Miki, Saito, Takahashi, Sasaki, Kikuchi and Harada2014; Sasaki et al., Reference Sasaki, Miki, Tachinami, Saito, Takahashi, Anzai, Kikuchi, Aso and Harada2014) using the intense pulsed-power generator ETIGO-II (~1 TW, ~50 ns) (Jiang et al., Reference Jiang, Sakagami, Masugata and Yatsui1993) has been considered. The features of the method are possible to generate WDM state with the implosion time-scale, isochoric condition, and direct spectroscopic measurement using a rigid-wall capillary having transparency, and avoiding the skin effect with a foamed metal as a sample. Despite the fact that the achievable parameters of the WDM depend on the output parameters of the intense pulsed-power generator, it was limited to change the output parameters such as voltage and current. For this reason, the impedance control using a load is required to control the energy input into the sample, because the output power of ETIGO-II is enough to generate the WDM state. A simple scheme for impedance control should be considered.
A diode has been studied for some applications in pulsed-power system (Yatsui et al., Reference Yatsui, Tokuchi, Tanaka, Ishizuka, Kawai, Sai, Masugata, Ito and Matsui1985; Niu, Reference Niu1997; Devyatkov et al., Reference Devyatkov, Koval, Schanin, Grigoryev and Koval2003; Tarasenko et al., Reference Tarasenko, Shunailov, Shpak and Kostyrya2005; Li et al., Reference Li, Chang, Zhang, Liu, Chen and Wen2012; Abdullin et al., Reference Abdullin, Ivanov, Losev and Morozov2013). In this study, the use of an electron beam diode to control the impedance was investigated. A pulsed-power system with capacitive energy storage achieves its power multiplication by current amplification corresponding to the evolution of the load impedance. The electron beam diode was placed at the output terminal of ETIGO-II, and was used as adjustable impedance for the pulsed-power system. The space-charge limited current was expected to serve as the simple scheme for controlling the load impedance with changing the gap distance of the diode. The energy input into the sample was expected to be affected by the output current. Consequently, to investigate the controllability of the input energy into the sample on the variable gap distance, the output current was measured.
2. EXPERIMENTAL SETUP
The intense pulsed-power generator ETIGO-II, as shown in Figure 1, has been used to generate intense light-ion beams or to produce the dense ablation plasma (Yatsui et al., Reference Yatsui, Grigoriu, Masugata, Jiang and Sonegawa1997; Jiang et al., Reference Jiang, Hashimoto, Shinkai, Ohtomo and Yatsui1998; Kashine et al., Reference Kashine, Yazawa, Harada, Jiang and Yatsui2002). To obtain the large output current, the impedance-conversion line was replaced, and its nominal output parameters of ETIGO-II were 1 MV, 590 kA, and 50 ns [full width at half maximum (FWHM)]. In the Marx generator (2 MV, 70 kJ), 20 stages capacitors were charged in parallel to a voltage of ±20 kV.
Fig. 1. Outline of intense pulsed-power generator ETIGO-II.
Figure 2 shows the experimental arrangement of the electron beam diode, which was placed at the output terminal of ETIGO-II. The electrodes of the electron beam diode consisted of a disk-shaped cathode (ø105 mm, 304 stainless steel) and a disk-shaped anode (ø210 mm, 304 stainless steel), in which gap distance could be adjusted. To prevent the electrical breakdown, the pressure in the chamber was controlled to be less than 0.02 Pa. The time evolution of the output current I K(t) and output voltage V(t) were measured using a Rogowski coil at the cathode and a capacitive divider in the pulse-transmission line, respectively. The output current was measured for gap distances of 10, 15, and 20 mm. These measurements were performed five times for each gap distance.
Fig. 2. Experimental setup of electron beam diode.
Figure 3 shows the typical output waveforms for a gap distance of 10 mm. The pulse width was ~100 ns (FWHM), with peak voltage and current values of −1100 kV and −70 kA, respectively. The energy input into the sample was affected by the output current, which was depended on the impedance of the diode. In this study, the input energy was estimated without considering the sample, because the impedance of the sample was expected to be sufficiently smaller than the impedance of the diode. Therefore, the disk-shaped anode was grounded to an outer feeder of ETIGO-II and the impedance of the sample was assumed to be constant value for the estimation of the energy input into the sample.
Fig. 3. Typical output waveform for 10 mm gap distance.
3. EXPERIMENTAL RESULTS
Figure 4 shows the typical output current waveforms at each gap distance. Figure 5 shows the absolute value of the peak current as a function of the gap distance. As shown in Figures 4 and 5, the peak current decreases with an increase in the gap distance. The peak current is compared with the space-charge limited current (solid line in Fig. 5). The space-charge limited current (Child–Langmuir current) is given by Niu (Reference Niu1989)
$$I = \displaystyle{{4\varepsilon _0} \over 9}\sqrt {\displaystyle{{2e} \over {m_{\rm e}}}} \displaystyle{{V^{{3 / 2}}} \over {d^2}} S $$where ε 0 is the permittivity of a vacuum, e is the electron charge, m e is the electron mass, V is the peak voltage, d is the gap distance, and S is the cross-sectional area of the discharge.
Fig. 4. Typical output current waveforms at cathode for (a) 10 mm, (b) 15 mm, and (c) 20 mm gap distances.
Fig. 5. Absolute value of peak current as function of gap distance for V = 1.1 MV and S=π(40 × 10−3)2 = 5 × 10−3 m2. The error bars indicate the maximum and minimum values.
As a result, the space-charge limited current also decreased with an increase in the gap distance. Therefore, the space-charge limited current was considered to be the cause of the decrease in the output current with the change in the gap distance.
The input energy E(t) can be estimated as follows:
$$E(t) = \int_{t_{\rm o}} ^{t_{\rm f}} {RI_K (t)^2 dt} $$where t o is the rise time, t f is the time after 50 ns from t o, R is the impedance of the sample, and I K(t) is the output current at the cathode. Figure 6 shows the input energy as a function of the gap distance. The input energy was normalized by the impedance of the sample, because the sample impedance is much less than the impedance of the diode. To estimate the input energy with the implosion time-scale, the input energy was estimated after 50 ns from the rise time. The beginning time to calculate the integral of the output current was defined as the response of the output current to a rise to 10% of its peak value. As a result, the normalized input energy decreased with an increase in the gap distance, because of the decrease in the output current.
Fig. 6. Normalized input energy as function of gap distance. The error bars indicate the maximum and minimum values.
Consequently, the ability to control the input energy by changing the gap distance was indicated for pulse duration of several tens of nanoseconds.
4. CONCLUSION
The use of an electron beam diode to control the impedance was proposed for applications such as the generation of WDM using a pulsed-power discharge with isochoric heating. To investigate the dependence of the input energy on the gap distance, the output current was measured for gap distances of 10, 15, and 20 mm, and was compared with the space-charge limited current. As a result, the peak current and the space-charge limited current were found to decrease with an increase in the gap distance. In addition, the input energy, which was estimated from the output current, decreased with an increase in the gap distance. It is expected that the input energy of the pulsed-power generator could be controlled using the proposed method with the implosion time-scale.
ACKNOWLEDGMENTS
This work was supported by MEXT Grant-in-Aid for Scientific Research, and by Program for High Reliable Materials Design and Manufacturing in Nagaoka University of Technology, and by JSPS Grant-in-Aid for Challenging Exploratory Research No. 25630418.
