3. Results and Discussion

A typical energy-loss measurement spectrum of 100 keV proton penetrating through the hydrogen plasma at different discharging time (the initial gas pressure was about 0.81 mbar, and the voltage was 3 kV) is shown in Figure 1. Figures 1(a) and 1(b) show the measured positions of ions in the detector at 0 μs and 3 μs time, respectively. The systematical energy loss can be obtained by measuring the position shift at different discharging time. Here, that energy loss increases by 4.07 keV comparing to the cold gas can be found in Figure 1.

Figure 1: A typical energy-loss measurement spectrum of 100 keV proton penetrating through hydrogen gas discharging plasma at different discharging time (a) was at 0 μs time and (b) was at about 3 μs time (the initial gas pressure was about 0.81 mbar, and the voltage was 3 kV).

The plasma state has been diagnosed by Kuznetsov, and in our experiment, the plasma state can be determined by the initial gas pressure and discharge voltage based on the results presented in [Reference Kuznetsov, Byalkovskii and Gavrilin38]. The initial energy loss ΔE for 100 and 90 keV protons was measured to be 5.02 and 7.73 keV before discharge. The gas pressure is determined to be 0.81 mbar and 1.25 mbar according to the measured ΔE (see [Zhang et al., 2020]. for details). With the discharge voltage of 3 kV, the linear free-electron density n _f and average ionization degree of the plasma are found to be 3.35∗10¹⁷ cm⁻² and 3.75∗10¹⁷ cm⁻², and 0.76 and 0.44, respectively, at the peak stage of discharge (around 3 μs). The linear free- and bound-electron density can be obtained by ref [Reference Kuznetsov, Byalkovskii and Gavrilin38]. Figure 2 shows the free- and bound-electron density (the initial gas pressure was about 0.81 mbar, and the voltage was 3 kV), where n _b and n _f denote the linear bound- and free-electron density, respectively. n _f gradually rises up until the onset of the discharge peak stage. Then, it gradually decreases with the discharge time. Meanwhile, n _b evolves in the opposite tendency.

Figure 2: Linear free-electron (solid line) and bound-electron densities in plasma as a function of discharge time (the initial gas pressure was about 0.81 mbar, and the voltage was 3 kV).

The energy-loss change in the whole plasma lifetime was recorded as a function of time after the discharge, which is shown in Figure 3 where the theoretical prediction is also shown for comparison. In our experiment, the total uncertainly of the energy loss is about 10% mainly from the broadening of the ion beam spot and the detector itself. Figures 3(a) and 3(b) represent energy losses of 100 keV and 90 keV protons in plasma (the discharging voltage was 3 kV), respectively. The initial gas pressures were estimated to be 0.81 mbar for 100 keV proton incident and 1.25 mbar for 90 keV proton incident. It should be noted that a similar trend of the change of discharge current, free-electron density, and energy loss of protons as a function of time can be observed in [Reference Cheng, Zhou and Wang31]. Both the discharging current and energy loss are mainly dependent on the free-electron density in the plasma, and for the first 1 microsecond, the discharge current and energy loss are not very stable (see ref. [Reference Cheng, Zhou and Wang31] for details), which is probably due to the fast changing of the electromagnetic field in the beginning. It results in decrease of energy loss at the beginning of discharge (0-1 μs). The phenomena, however, have not been clear yet, and a similar case is also found in refs. [Reference Frank, Blažević and Grande39, Reference Frank, Blažević and Bagnoud40]. When the discharging current reaches the maximum at around 3 μs after discharging, the temporal gradients of the electromagnetic field and the plasma parameters are minimum. Thus, the energy loss reaches the maximum at around 3 μs where the plasma reaches its most stable state. We choose the experimental data from the relatively stable plasma state from 2 to 4 μs to carry out the discussion below.

Figure 3: Energy losses of protons penetrating through hydrogen gas discharging plasma. (a) 100 keV proton in plasma (the initial gas pressure was about 0.81 mbar, and the voltage was 3 kV); (b) 90 keV proton in plasma (the initial gas pressure was about 1.25 mbar, and the voltage was 3 kV). The symbols represent experimental data. The solid line represents the theoretical predictions of the Bethe model with Z _eff = 1.

In a partially ionized plasma target, the incident ions lose their energy through cascade collisions with free electrons and/or bound electrons. Considering the homogeneity of the plasma target and the (nearly zero) slope of the stopping power function at this energy regime, a stepwise integration is not necessary and the total energy loss ΔE can be expressed as follows:

(1) $\begin{array}{}\Delta E=\left({\left(\frac{\text{d}E}{\text{d}x}\right)}_{\text{free}}+{\left(\frac{\text{d}E}{\text{d}x}\right)}_{\text{bound}}\right)\times L,\end{array}$

where L = 15.6 cm is the plasma target length and [dE/dx]_free and [dE/dx]_bound represents the stopping power of the free electrons and bound electrons, respectively.

According to the Bethe model, the stopping power from the aspects of free electrons and bound electrons can be represented as follows:

(2) $\begin{array}{c}{\left(\frac{\text{d}E}{\text{d}x}\right)}_{\text{free}}=-\frac{Z_{eff}^2{e}^2{\omega }_p^2}{v_p^2}\ln \left(\frac{2m_e{v}_p^2}{\hslash {\omega }_p}\right),{\left(\frac{\text{d}E}{\text{d}x}\right)}_{\text{bound}}=-\frac{4\pi Z_{eff}^2{e}^4{n}_{be}}{m_e{v}_p^2}\ln \left(\frac{2m_e{v}_p^2}{I}\right),\end{array}$

where ${\omega }_p=\sqrt{4\pi n_{fe}{e}^2/m}$ is the plasma frequency, Z _eff is the projectile effective charge state, v_p denotes the projectile velocity, m _e and e are the electronic mass and charge, and n _be and n _fe are the density of free electrons and bound electrons, respectively, in which the degree of ionization has been considered. I = ħϖ is the average excitation energy of target atoms, which is 15 eV for hydrogen atom [Reference Belyaev, Basko and Cherkasov41].

In the present work, when Z _eff is chosen as 1, the experimental data for 100 keV proton incident can be well reproduced by Bethe theoretical predictions, which is consistent with our previous results [Reference Cheng, Zhou and Wang31]. However, for 90 keV proton incident, the theoretical calculations obviously overestimate the experimental data by a factor of about 2, which may be attributed to the charge-state evolution of protons in the plasma [Reference Olson and Salop42].

In [Reference Olson and Salop42], the classical trajectory calculations were used to predict the charge-transfer and impact-ionization cross sections for collisions of H⁺-H in the velocity range of 2–7 × 10⁸ cm/sec, which is equivalent to the ion velocity in our experiment. Here, the charge-exchange cross section corresponds to a capture into any of the bound states of the ions, that is, a total capture cross section rather than a capture into the ground state only. The projectile charge state is determined by the total electron-loss cross sections (sum of charge exchange and impact ionization). One can clearly see that impact-ionization cross sections for 100 keV and above 100 keV H⁺-H collisions are dominated. This means that, in this case, the charge-transfer cross sections can be ignored, in which the cross sections decrease with the increase of incident energy, while for below 100 keV H⁺-H collisions, the total electron-loss cross sections are determined by both the charge-transfer cross sections and the impact-ionization cross sections. Therefore, the charge-state evolution effect needs to be taken into account for below 100 keV H⁺-H collisions. Based on the discussion above, when Z _eff is equal to 1, the theoretical predictions for 90 keV proton incident overestimate the experimental data, which can be explained by the charge-state evolution effect. The experimental phenomenon concerned with the variation of the effective charge of protons in the plasma has not been reported so far. In our experiment, for the low-energy regime, when the incident energy is 100 keV, the effective charge of protons is equal to 1. While for 90 keV proton incident, the charge-state evolution needs to be considered, Z _eff should be less than 1 [Reference Olson and Salop42].

In the present work, the effective charge of 90 keV protons in the hydrogen plasma can be calculated through some empirical formulae. Kreussler et al. [Reference Kreussler, Varelas and Brandt43] suggested that the equilibrium charge state of projectile ions can be used to estimate the energy loss. The equilibrium charge state relies on the relative velocity of the projectile v to the electrons of the target v_e, in which all the possible orientations of vector v − v_e are considered, which is given by

(3) $\begin{array}{}{\nu }_r=\left(v-v_e\right)=\frac{{\nu }_e^2}{6\nu }\left({\left(\frac{\nu }{{\nu }_e}+1\right)}^3-{\left(\frac{\nu }{{\nu }_e}-1\right)}^3\right).\end{array}$

In the case of plasma, the electron velocity is determined by its corresponding Fermi velocity and the thermal velocity of free electrons:

(4) $\begin{array}{}{\nu }_e={\left(2\cdot \frac{3}{5}{E}_{\text{F}}+3k_{\text{B}}T\right)}^{1/2},\end{array}$

where T is the plasma temperature, E _F is Fermi energy, and k _B is the Boltzman constant. The effective charge state is then calculated by

(5) $\begin{array}{}{Z}_{\text{eff}}=Z-Z{e}^{-{\nu }_r/Z^{2/3}{\nu }_0},\end{array}$

where Z is the projectile atomic number.

Moreover, Gus’kov et al. [Reference Gus’kov, Zmitrenko, Il’in, Levkovskii, Rozanov and Sherman23] proposed a similar model, in which the effective charge is defined by the following relationship:

(6) $\begin{array}{}{Z}_{\text{eff}}=Z\ast \gamma .\end{array}$

Here, the typical parameter is given by

(7) $\begin{array}{}\gamma =1-\exp \left(-0.92\ast Z^{-2/3}\ast \left(\left(v-v_e\right)\right)\right).\end{array}$

In the Kreussler model, Z _eff is equal to 0.861. In the Gus’kov model, the value of Z _eff is 0.832. Figure 4 shows the theoretical calculations from two empirical models, in which they all underestimate the experimental data. The main reason is that the parameters of only incident ions in two empirical formulae are considered, while target properties are ignored. In our experiment, the plasma is partially ionized, and ionization degree should be taken into account [Reference Zhang, Liu, Cheng, Zhao and He32].

Figure 4: Energy losses of 90 keV protons in hydrogen gas discharging plasma. Theoretical model calculations are also shown for comparison.

Compared to neutral matter, plasmas have different components (atoms, ions, and electrons), and the component density depends on plasma temperature and density. In order to describe the interaction between incident ions and plasma, the rate constants of the processes are employed, i.e., N〈υ σ〉(s⁻¹) [Reference Tolstikhina and Shevelko44]. The constant quantities averaged over a Maxwellian distribution of particle velocities v; here, N is the particle density of the plasma. Therefore, in the present work, the ionization degree of plasma needs to be considered for the calculation of the effective charge of the projectile. Moreover, for the empirical formulae, it is necessary that the mean values of the relative velocity and fluctuations of the quantities also need to be taken into account [Reference Gus’kov, Zmitrenko, Il’in, Levkovskii, Rozanov and Sherman23].

In this work, the value of Z _eff is found to be about 0.92, and the theoretical predictions give a good description with the experimental data, as shown in Figure 4. In [Reference Ziegler, Biersack and Littmark45, Reference Brown and Moak46], to achieve better agreement with the experimental data, various typical parameters of expression (5) have been used. Here, expression (5) can be modified as follows:

(8) $\begin{array}{}\gamma =1-\exp \left(-1.31\ast Z^{-2/3}\ast \left(\left(v-v_e\right)\right)\right).\end{array}$

Thus, our experimental results can be reproduced by the theoretical predictions.

Similar calculations for 100 keV protons are also applied. Basing on the Gus’kov and Kreussler models, the values of Z _eff are 0.846 and 0.864, respectively. They all underestimated the experimental data, as shown in Figure 5. The effective charge calculated by the modified empirical model is equal to 0.93. The value is consistent with the result presented in [Reference Zhang, Liu, Cheng, Zhao and He32], in which electrons are all assumed to be captured into the projectile ground state. Figure 5 represents the theoretical predictions from the modified empirical model, which are in agreement with the experimental data in the range of errors. It implied that the effective charge for 100 keV protons in plasmas also needs to be considered, as described in [Reference Zhang, Liu, Cheng, Zhao and He32]. It is not in accord with the case of neutral matter in [Reference Olson and Salop42]. Meanwhile, in our previous work, that the effective charge for 100 keV protons is chosen as 1 is arbitrary. However, the detailed and further experimental measurements are necessary, and how the target properties are added in the empirical formulae still needs further theoretical investigation.

Figure 5: Energy losses of 100 keV protons in hydrogen gas discharging plasma. Theoretical model calculations are also shown for comparison.