Electrical diagnosis
An electrical diagnosis is performed on the HV circuit, providing information on the discharge current and the plasma resistance and heat power. The waveform of the current pulse produced by the circuit is acquired by the oscilloscope, as shown in Fig. 4a, while varying the voltage applied by the HV generator, thus allowing to retrieve the characteristic voltage-current curve of the pulser circuit. The reported discharge current waveform is obtained by applying 5 kV pulses to the 3 cm-long 2 mm-diameter ceramic capillary, ionizing around 20 mbar of \({\hbox {N}}_2\)–\({\hbox {H}}_2\) mixture. As shown in Fig. 4b, V–I curves are determined for a short circuit configuration and with the ceramic capillary, with maximum peak current of 1850 A and 1550 A respectively. The lower current produced with the capillary is due to the electrical resistance of the plasma channel, which increases with the capillary length.
By comparing the V–I traces in the two configurations, it is possible to determine the plasma resistance \(R_{p}\) as the difference between the total resistance \(R_{tot}\) (circuit and capillary) and the circuit resistance in short circuit configuration \(R_{sc}\) (without the capillary):
$$\begin{aligned} R_{p} = R_{tot} – R_{sc} = V/I_{plasma}-V/I_{sc} \end{aligned}$$
(4)
given \(R_{tot}\) and \(R_{sc}\) by the ratio between the applied voltage and the peak current measured with the capillary and in short circuit respectively. In particular, in the experimental range of 5–21 kV, the plasma resistance spans from 1.5 to 2.2 \(\Omega\). Moreover, by multiplying the plasma resistance with the measured current intensity, it is possible to determine the instantaneous heat power produced by the plasma discharge through Ohmic heating, according to21:
$$\begin{aligned} P = R_p I_p^{2} \end{aligned}$$
(5)
The instantaneous heat power, reported in Fig. 4b as a function of the applied voltage, spans from 100 kW to 5 MW in the experimental range of 5–21 kV. Therefore, considering the discharge pulse duration of 1 μs FWHM, the heat produced by a single plasma discharge ranges from 100 mJ to 5 J. As a result, the average heat power deposited onto the capillary walls during continuous operation at low voltage (e.g. 5 kV) in the range of 10–100 Hz turns out to be around 1–10 W.
Preliminary characterization
Before high repetition rate tests, a first characterization of the Shapal capillary is performed. 7.5 kV voltage pulses are delivered to the capillary electrodes, ionizing around 20 mbar of pure \({\hbox {H}}_2\) and producing plasma discharges at 1 Hz, with 500 A peak current. The telescopic system of the beam transport line is shifted longitudinally over 1 cm to scan the plasma channel and acquire the plasma-emitted light from different transverse slices, in particular from the capillary exit to 5 mm and 1 cm inside the capillary channel, as schematized in Fig. 3. Hydrogen Balmer lines are then analyzed to retrieve the transverse plasma density distribution in the different slices. Results from Stark broadening method, reported in Fig. 5, show the evolution of the transverse plasma density profile 5 mm from the capillary exit, at different delays with respect to the onset of the HV discharge.
(Top) \({\hbox {H}}_{\beta }\) spectral images and (Bottom) transverse plasma density profiles, measured from the plasma channel slice located 5 mm inside the capillary at different delays with respect to the onset of the HV discharge.
During the plasma formation, the density distribution is characterized by a hollow profile with a depth of 18% with respect to the near-wall density (around ± 1 mm), as observed at a delay of 600 ns at the density peak of around 1.3\(\times\) \(10^{17}\) \({\hbox {cm}}^{-3}\) . In the recombination phase, after the discharge is over (delay beyond 1000 ns), the transverse distribution turns into a gaussian-shaped profile. Measured error bars represent the standard deviation calculated by acquiring 50 spectral images.
Observed transverse density profiles are in good agreement with theoretical models19, showing that, given a uniform plasma pressure, the radial temperature gradient, established between the channel axis and the capillary walls due to heat transfer with the capillary, results in a hollow density profile with an on-axis minimum and a density maximum towards the channel walls. The same behaviour is observed with the transverse slice located 1 cm inside the plasma channel, as reported in Fig. 6.
Transverse plasma density profiles, measured 1 cm inside the capillary at different delays after the discharge onset.
Regarding the plasma composition, Fig. 7 reports the plasma emission spectrum acquired by the 150 grooves/mm spectrometer grating and integrated over the transverse slice located 1 cm inside the capillary. The spectrum is acquired at a delay of 100 ns and shows the hydrogen lines of the Balmer series, together with emission lines of \(N^+\) (centered at 464, 500, 517 and 568 nm) and \(N^{2+}\) (452 nm), thus showing the trace presence of nitrogen in the plasma channel, due to residual air in the gas injection pipe, which then vanishes at higher delays. Concerning the plasma temperature, Eq. 2 yields a temperature of around 3 eV during the plasma formation and a peak of 4.3 eV in correspondence of the 500 A peak current. Such result is confirmed by the relative intensities of \({\hbox {H}}_{\alpha }\) and \({\hbox {H}}_{\beta }\) spectral lines, measured from Fig. 7 and inserted into Eq. 3.
The acquired emission spectra and the measured plasma electron density and temperature are benchmarked with NIST LIBS Database spectra22, tabulated for a hydrogen plasma doped with few percent of nitrogen and with electron density and temperature of \(10^{17}\) \({\hbox {cm}}^{-3}\) and 3–4 eV respectively. As shown in Fig. 7, the acquired spectrum is well overlapped with LIBS tabulated lines with the plasma parameters obtained through spectroscopic analysis, except for the broadening of Balmer lines, which is not taken into account into NIST spectrum.
Plasma emission spectrum, acquired 100 ns after the discharge onset from the plasma channel slice located 1 cm inside the capillary and integrated along the transverse profile. The measured spectrum (black line) is benchmarked with the NIST LIBS Database spectrum tabulated for a hydrogen plasma with few percent nitrogen and density and temperature of \(10^{17}\) \({\hbox {cm}}^{-3}\) and 3 eV respectively. In particular, the Database spectrum shows the hydrogen lines of the Balmer series (red line) and also emission lines of \(N^+\) (NII, orange line) and \(N^{2+}\) (NIII, blue line).
High repetition rate operation
High repetition rate tests are performed in the range of 10–150 Hz, using the \({\hbox {N}}_2\)–\({\hbox {H}}_2\) mixture in continuous flow regime, injected at 80–100 mbar. 5 kV voltage pulses are delivered to the electrodes, producing 400 A peak current plasma discharges, up to a total amount of 20 million shots. During the experimental campaign, laser spot imaging and plasma density measurements are performed regularly to monitor any modification both in the capillary walls and the plasma density distribution. In particular, after a given number of shots, plasma density measurements are performed using 7.5 kV pulses to ionize pure hydrogen in pulsed flow regime, as for the preliminary characterization, and taking as a reference the transverse plasma density profiles acquired 1 cm inside the channel at the density peak (600 ns after the HV discharge onset). Higher voltage pulses are applied, compared to 5 kV pulses in high repetition rate tests, in order to maximize the plasma stability and the output signal, thus minimizing the error in density measurements. Results obtained with Stark broadening method are reported in Fig. 8a. Transverse density profiles, measured in the same experimental conditions after a different number of discharges, are well overlapped within the error bars, which are computed by acquiring 50 images for each measure. Furthermore, as depicted in Fig. 8b, the plasma density, averaged over the transverse profile and normalized to the value obtained in the preliminary characterization, remains approximately constant during the entire experimental campaign at 10–150 Hz.
(a) Transverse density profiles and (b) average plasma density, measured 1 cm inside the plasma channel at a delay of 600 ns, after different number of shots at 10–150 Hz up to 20 million shots.
Regarding the capillary channel cross section, the analysis of laser spot images is reported in Fig. 9. Horizontal and vertical laser spot lineouts, measured during the experimental campaign, are well overlapped within the error bars, which are computed by acquiring 50 images for each measurement. Therefore, no significant alteration is observed in the overall channel profile.
Laser spot vertical and horizontal lineouts progressively measured up to 20 million shots at 10–150 Hz.
In addition, concerning the microscopic analysis, Fig. 10 reports the channel diameter, measured along the capillary after 20 million discharges and normalized to the channel profile acquired before the experimental campaign. For each longitudinal position, the reported diameter is obtained by measuring the channel diameter both in the horizontal and vertical directions and computing the average, in order to take into account possible deformations that would determine a transverse elliptical shape. Measured error bars result from 200 acquired images. As a result, microscopic analysis confirms that the capillary channel is well preserved at the end of the experimental campaign.
Capillary channel diameter, measured by the stereomicroscope along the capillary axis after 20 million shots at 10–150 Hz, normalized with respect to the first measurement performed before the experimental campaign.
In conclusion, plasma measurements highlight the stability of the density distribution inside the capillary during the experimental campaign, while the laser spot imaging and the microscopic analysis show that the capillary integrity is preserved as well. Hence experimental results demonstrate exhaustively the suitability of the Shapal-Macor capillary for long-term plasma discharges operation at high repetition rate (up to 150 Hz).
Heat transfer numerical simulations
A numerical analysis is carried out to evaluate the heat transfer inside the entire plasma source during high repetition rate operation. First, the instantaneous heat power produced by a single plasma discharge through Ohmic heating is considered as:
$$\begin{aligned} P(t) = R_p(t) I_p(t)^{2} \end{aligned}$$
(6)
The discharge current \(I_p\) is directly measured by the oscilloscope during the experimental campaign, while the resistance of the plasma channel \(R_p\) is estimated through Ohm’s law23:
$$\begin{aligned} R_p(t) = \rho _{tot}(t) \frac{L}{\pi r^2}, \end{aligned}$$
(7)
in which the length L and radius r of the capillary channel are respectively 3 cm and 1 mm, and the plasma resistivity \(\rho _{tot}\) is computed as24,25:
$$\begin{aligned} \rho _{tot}(t) = \frac{m_{e}}{n_{e}(t) e^2} (\nu _{ei}(t) + \nu _{ae}(t)) \end{aligned}$$
(8)
Electron-ion and atom-electron collision frequencies are in turn determined as26:
$$\begin{aligned} \nu _{ei}(t) = \frac{4}{3} \sqrt{\frac{2\pi }{m_{e}}}\frac{e^4 n_{e}(t) \ln {\lambda _{ei}(t)}}{(4\pi \epsilon _{0})^2 (k_{B} T_{e}(t))^{3/2}}, \quad \nu _{ae}(t) = \frac{\pi r^2_{a} P_{0}}{\sqrt{m_{e} k_{B} T_{e}(t)}} \end{aligned}$$
(9)
in which \(r_{a}\) and \(P_0\) are the atomic radius and the gas pressure, while the Coulomb logarithm \(\ln {\lambda _{ei}}\) is given by26:
$$\begin{aligned} \ln {\lambda _{ei}(t)} = \ln {\left[ \frac{3}{2\sqrt{2\pi }}\frac{ (4\pi \epsilon _{0})^{3/2} (k_{B} T_{e}(t))^{3/2}}{e^3 [n_e(t)]^{1/2}} \right] } \end{aligned}$$
(10)
In order to compute the Coulomb logarithm and the collision frequencies and, in turn, retrieve the plasma resistivity and the plasma channel resistance, the temporal profiles of the plasma density and the plasma temperature, depicted in Fig. 11, are inserted into Eqs. (8, 9, 10). In particular, the density temporal profile is obtained by averaging the measured transverse profiles from Fig. 6, while the temperature temporal profile is computed by implementing the measured discharge current into Eq. 2.
Temporal profiles of the plasma density (a) and temperature (b), respectively determined by transverse Stark broadening method and Eq. 2.
Analytical results for the plasma resistance and the instantaneous heat power are reported in Fig. 12, together with the discharge current waveform measured during high repetition rate tests. In particular, the obtained plasma resistance and instantaneous heat power are in agreement with experimental results achieved through the electrical diagnosis of the HV circuit and the capillary discharge, as shown in Fig. 4b.
(a) Discharge current waveform, acquired during high repetition rate tests. (b) Resistance of the plasma channel. (c) Resulting instantaneous heat power produced by a single plasma discharge.
Finally, by integrating the instantaneous heat power over the plasma discharge duration, an energy per pulse of \(\approx\) 100 mJ is obtained. Considering repetition rate operation from tens of Hz up to the kHz range, the average heat power deposited onto the capillary walls spans from few Watts (10–50 Hz) to 100 W. Given these reference values, 3D numerical simulations are performed with COMSOL Multiphysics12 to analyze the capillary overheating and the heat removal inside the plasma source. Relying on Fourier’s law of heat conduction, the temperature gradient inside the whole source is computed, given the thermal conductivity k of the different components and the heat flux q, estimated through the analytical model previously described:
$$\begin{aligned} \vec {q} = -k(T)\nabla T \end{aligned}$$
(11)
Heat transfer simulations are performed considering the Shapal-Macor capillary shown in Fig. 1, subject to a constant heat transfer rate of 1-100 W flowing through the capillary channel walls, thus reproducing the average power deposited by HV pulses in the range 10 Hz – 1 kHz. As shown in Fig. 13a, the capillary geometry also includes HV cables and the gas injection pipe, conducting the thermal load from the capillary to the HV pulser and the vacuum chamber, which in turn are considered as heat sinks. In addition, the external surfaces of all the components are thermally insulated, so as to replicate the experimental conditions inside the vacuum chamber. Due to the heat removal from the source to the heat sink, a thermal steady-state is reached within the capillary. For instance, Fig. 13a displays the equilibrium temperature distribution within the whole source, with a maximum temperature of \(840~^{\circ }\)C inside the capillary, reached after three hours of continuous plasma discharge operation at 300 Hz. Moreover, Fig. 13b reports the equilibrium temperature as a function of the repetition rate.
(a) 3D view of the steady-state surface temperature distribution during operation at 300 Hz. The plasma source is modeled with COMSOL Multiphysics 6.112. (b) Equilibrium temperature inside the capillary as a function of the operating repetition rate. The temperature on the channel wall (solid blue line), within the holder (dashed red line) and the electrode (dashed yellow line) are reported together with the melting temperatures of the adopted materials (horizontal dashed lines).
In particular, in the range of 10–150 Hz, assessed during the experimental testing, the equilibrium temperature is kept below the melting temperature of Macor and Shapal, thus preventing any damage in the capillary. For higher repetition rate operation up to the kHz range, the dissipated energy per pulse can be reduced by tuning the discharge duration and peak current, thus keeping the capillary temperature under control. In conclusion, heat transfer simulations are in good agreement with experimental results, confirming the longevity of the Shapal capillary in high repetition rate operation up to 150 Hz. In addition, simulations provide further insight into the operative limit of the source, which can be extended to the kHz range by properly tuning the experimental settings, such as the discharge current intensity and duration. In this regard, considering the operative range of 100–400 Hz foreseen by the EuPRAXIA@SPARC_LAB scientific case, the proposed design of plasma discharge capillaries made in Shapal and Macor represents a reliable solution in terms of longevity and cost-effectiveness.