Thermodynamics of a magnetically expanding plasma with isothermally behaving confined electrons

From the measured downward and upward 1D eepfs (figures 2(a)–(d)) obtained at 2 cm intervals from 3 cm (the leftmost curve) to 49 cm (the rightmost curve) from the nozzle throat at two nozzle currents (50 and 200 A), spatial evolution of eepfs can be distinguished in each case. At low nozzle current (figures 2(a) and (c)), only the significant decrease in the electron density along axial direction is seen, and noticeable cooling of electrons are not observed in both probe measurements. Unlike the results at low nozzle current, the parallel anisotropy of the measured eepfs represented as the discrepancy between the evolution of downward and upward eepfs near the nozzle throat (region I) is clearly seen at high nozzle current (figures 2(b) and (d)).

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For the front probe (figure 2(b)) at high nozzle current, the measured eepfs have an almost Maxwellian distribution and the cooling of electrons with distance from the nozzle throat (decay in the inverse slope of eepf) occurs over the entire electron energy range. In contrast, the eepfs measured by the back probe (figure 2(d)) are distinctly non-Maxwellian distributions and do not show a significant variation in the slope. This distinctive feature of the eepfs between the downward and upward direction implies that at least two electron groups having different thermodynamic properties coexist near the nozzle throat.

The shape of the upward eepfs at two nozzle currents (figures 2(c) and (d)) resembles a high energy-depleted convex distribution, which was already reported by Takahashi et al [42] for the plasma operating at nearly collisionless plasma. The shape of the eepfs is nearly constant with distance from the nozzle throat, except for the cut-off at low energy electrons corresponding to the plasma potential. Hence, the electrons measured by the back probe can be regarded as total energy-conserving electrons obeying non-local kinetics.

In contrast to the upward eepfs, the downward eepfs near the nozzle throat obey nearly the Maxwellian distribution (strictly, at low magnetic field condition, accumulation of low energy electrons is observed) (figures 2(a) and (b)). However, the shape of these eepfs at high nozzle current (figure 2(b)) is greatly changed compared to that of low nozzle current with increasing distance from the nozzle throat, implying that the electrons collected by the front probe do not obey non-local kinetics. Hence, the spatial change in the inverse slope of the downward eepfs, i.e., electron temperature, at high nozzle current results in an overall cooling of electrons ejected from the MN along the divergent magnetic field. Such a coexistence of two electron groups along parallel or anti-parallel direction to the magnetic field (figures 2(b) and (d)) weaken as the distance from the nozzle throat increases and completely disappears at the far-field region (region II). As shown in figures 2(b) and (d), both the eepfs measured by the front and back probes far from the nozzle throat are almost identical at all axial positions, except for the cut-off in low energy electrons.

Axial variations of properties of the plasma, namely, effective electron temperature 〈T_e〉 and electron density 〈n_e〉 show that the MN can be divided into two regions depending on the spatial variation of the measured plasma parameters (figures 3(a)–(b)). In region I, 〈T_e〉 measured by the front probe (figure 3(b)) decreases rapidly compare to that of low nozzle current (figure 3(a)), although its variation gradually declines with distance (region II), being nearly constant in region II (figure 3(b)). On the other hand, 〈T_e〉 measured by the back probe (1D upward eepfs) is slightly decreases with distance from nozzle throat in region I (figures 3(a)–(b)), showing less pronounced variation compared with parallel measurements. In region II, the electron-cooling behavior as found in both probe data is almost identical at two nozzle currents, indicating near-isothermal characteristics. Interestingly, the result exhibits a rapid spatial changes in n_e only at low current conditions.

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From a fluids approach, expressions relating the variations in plasma properties along the divergent magnetic field obey electron thermodynamics, i.e., a polytropic equation. In general, a linear regression of the log V_p versus log 〈n_e〉 or V_p versus 〈T_e〉 data has been used to extract a value for γ_e [15, 32–34, 43]. From a kinetics perspective, the effective electron pressure and electron density can be directly calculated using the measured 1D eepfs. A polytropic equation of state defined as

$$\begin{eqnarray}&&\langle {p}_{e}\rangle =C({\rm{\Psi }}){\langle {n}_{e}\rangle }^{{\gamma }_{e}}\end{eqnarray} \tag{ 1 }$$

can be easily evaluated from

$$\begin{eqnarray}&&{\rm{log}}\,\displaystyle {\int }_{0}^{\infty }{\varepsilon }^{\mathrm{3/2}}f(\varepsilon ){\rm{d}}\varepsilon \propto {\gamma }_{e}\,{\rm{log}}\displaystyle {\int }_{0}^{\infty }{\varepsilon }^{\mathrm{1/2}}f(\varepsilon ){\rm{d}}\varepsilon ,\end{eqnarray} \tag{ 2 }$$

where C(Ψ) is a constant along the magnetic flux surface and 〈p_e〉 the effective electron pressure. The dynamics of the electrons, which are measured by the front probe near the nozzle throat at 200 A, follows a polytropic law with index γ_e = 1.68 ± 0.02(3σ)in region I obtained using the linear regression method (figure 4(b)). In region II, at high nozzle current (figure 4(b)), 〈p_e〉 continuously decreases with rarely changing downward 〈T_e〉 and eventually manifesting isothermal behavior with γ_e = 1.10 ± 0.01(3σ). In contrast, no significant variation in slope is observed with the upward electrons at high nozzle current; the data show a slightly high γ_e than unity, γ_e = 1.14 ± 0.01(3σ) and γ_e = 1.03 ± 0.01(3σ) in regions I and II, respectively. At low MN current (50 A), the measured γ_e averaged over both the 1D downward and upward eepfs shows nearly isothermal value in the entire region (figure 4(a)). The results shown here indicate that the selectively collected electrons moving upward by back probe, which are regarded as confined electrons, behave nearly isothermally in the MN even though the magnetic field strength varies.

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To clearly observe the spatial change in downward γ_e, we conducted the experiment by varying the applied magnet current. In contrast to previous study [33, 34], the convex-shape eepfs were not observed under all experimental conditions at the nozzle throat with a front probe, and the correlation between the strength of the applied magnetic field and γ_e is clearly observed. As the MN current decreases the isothermal region is expanded to the nozzle throat and γ_e averaging the 1D downward eepfs measured over the entire region of the MN evolves into isothermal (figures 5(a)–(d)).

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The change in γ_e presenting the expansion of isothermal region from the far-field region can be explained by dividing electrons into adiabatic and isothermal groups. The downward electrons measured by the front probe near the nozzle throat at high nozzle current can be regarded as adiabatic electrons according to their motion observed in the spatial evolution of γ_e. The electrons responsible for the adiabatic expansion produce spatial changes in the slope of the downward 1D eepfs along the axial direction, relating γ_e closer to 5/3 near the nozzle throat. In comparison to the adiabatic electrons, the newly observed electrons measured by the back probe, which cannot be explained by adiabatic cooling, can be classified into groups of free and confined electrons according to their energy and magnetic moment values [27]. For electrons, the local maximum magnetic moment μ_e,m(z, E_e) with total energy E_e can be expressed as follows: μ_e,m(z, E_e) = (E_e + eϕ(z))/B_z, where the axial variation of the plasma potential ϕ(z) is calculated using the average knee of measured I–V characteristics with front and back probe. The local maximum magnetic moment can have minimum and maximum values at points, which eventually manifests the bounce motion in a MN. The confined electrons having energy below the total potential drop then bounce back (reflected and trapped electrons) and forth (trapped electrons) in the bounce region. Calculated μ_e,m(z, E_e) based on the magnetic field structure and measured potential structure shows that the bounce region, where the electrons are confined, expands to the nozzle throat as the magnetic field is weaken (figure 6); thus, the population of the isothermally behaving trapped electrons significantly increases at the nozzle throat as MN current decreases, which results in the overall decrease of the measured γ_e averaged over 1D downward eepfs even at the nozzle throat (figure 2).

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As the transition of electron thermodynamic occurs from nearly adiabatic to isothermal, the eepfs measured at the nozzle throat show abrupt increase of low energy electrons, whereas the significant change in the eepf slope is not observed at high energy range (figure 7(a)). The results of this experiment showing the opposite trend with respect to a typical low pressure discharge, at which the thermalization of the electrons (Maxwellianization) occurs with increasing the electron density due to increased electron–electron collisions, can be correlated to the existence of non-locally behaving confined electrons.

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In sections 4.1 and 4.2, we showed that the transition of electron thermodynamic, i.e. changes in the electron thermodynamics, is accompanied by the expansion of the bounce region of non-locally behaving confined electrons with low kinetic energy; the bounce region of the confined electrons is expanded to the nozzle throat at the isothermal condition. Therefore, it can be assumed that the amount of electrons with low kinetic energy is superimposed on the adiabatically expanding Maxwellian eepf during the transition from adiabatic to isothermal.

By adopting the experimentally measured 〈T_e〉 at the nozzle throat and spatial profile of 〈n_e〉 with front probe at nearly adiabatic condition (200 A), the spatial variations of T_e of adiabatic group can be estimated by the polytropic equation with the polytropic index γ_e = 5/3. In addition, the plasma potential structure can be calculated by electron momentum conservation equations e〈n_e〉E = −∇〈p_e〉 as the balance between the effective electron pressure 〈p_e〉 and the electric field E to obtain the axial variation of plasma parameters of the confined electrons from the non-locally behaving truncated Maxwellian eepf. The eepf shape of the non-locally behaving confined electrons is fitted to the truncated Maxwellian distribution as [44]

$$\begin{eqnarray}&&f(\varepsilon )=\left\{\begin{array}{l}2{n}_{e}/{\pi }^{1/2}{T}_{\mathrm{bulk}}^{3/2}\exp (-\varepsilon /{T}_{\mathrm{bulk}}),(\varepsilon \lt {\varepsilon }_{\mathrm{depleted}})\\ 2{n}_{e}/{\pi }^{1/2}{T}_{\mathrm{bulk}}^{3/2}\exp (-\varepsilon /{T}_{\mathrm{tail}})\exp (-{\varepsilon }_{\mathrm{depleted}}(1/{T}_{\mathrm{bulk}}-1/{T}_{\mathrm{tail}})),(\varepsilon \gt {\varepsilon }_{\mathrm{depleted}}).\end{array}\right.\end{eqnarray} \tag{ 3 }$$

The shape of eepfs recorded in the back probe (the confined electrons) at all MN conditions is almost identical; thus, the measured eepf of confined group can be described with a good degree of accuracy with a discontinuity occurring at the measured ion beam energy level ε_depleted ∼ 12.8 eV and a tail temperature T_tail ∼ 3.0 eV (figure 7(b)). The fitted bulk temperature T_bulk ∼ 6.03 eV is determined by the cooled 〈T_e〉 of adiabatic electrons, which was calculated through the polytropic equation (figure 7(b)). This assumption is consistent with the experimental result that 〈T_e〉 of the adiabatic and the confined groups becomes identical at the far-field region.

Similar to the bi-Maxwellianization during the transition of electron thermodynamic to isothermal shown in the experiment, a change in the low energy range was also observed in the calculated eepf as the ratio of confined electrons increased, showing the evolution into a bi-Maxwellian distribution like shape (figure 7(c)). The shape of measured downward eepf at the nozzle current of 50 A is nearly equal to the calculated eepf at a ratio of 1:1 (figure 7(c)). It is also expected that when adiabatically behaving Maxwellian distribution is affected by the nozzle wall leading to eepf depletion at high electron energy, then the total eepf with the truncated Maxwellian eepf can evolve into the Druyvesteyn-like distribution.

The plasma parameters of non-local truncated Maxwellian distribution for the confined electrons at different potential locations are represented by the energy shifted distributions with the amplitude consistency. Finally, the total density and temperature weighted for each electron group is evaluated. As the proportion of the confined group increases, the spatial variation of the total electron density dramatically increases with total electron temperature changing toward isothermal, which can represent the results observed in the experiment (figures 8(a) and (b)). Eventually, the polytropic equation shows significantly reduced γ_e (figure 8(c)) with abrupt changes in 〈p_e〉 with increasing the proportion of confined electrons.

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The spatial evolution of the total eepfs according to the proportion of the confined electrons to the adiabatic electrons can be determined based on the calculated plasma parameters, i.e. calculated 〈n_e〉, 〈T_e〉, and the plasma potential structure (figures 9(a) and (b)). As the proportion of electrons with truncated eepf, which represents the distribution of isothermally behaving confined electrons, is included in the modeling, the axial variation of the calculated eepf is similar to that of low nozzle current (figure 9(a)).

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In the previous sections, we showed that the evaluated γ_e of a MN is affected by the existence of confined electrons, showing change in γ_e which eventually deviates from 5/3 as the MN current decreases due to the expansion of bounce region in the potential well formed by external magnetic field and self-generating ambipolar potential. This finding arises an essential question whether isothermally behaving confined electrons affect the ion acceleration which is the fundamental function of the nozzle. To answer this question, detailed measurements of IEDFs are performed by axially movable RFEA to verify the correlation between the change of γ_e and the presence of ion beam over a wide range of MN current. Measurements are performed at the boundary of two regions (regions I and II) according to the transition of electron thermodynamics. Normalized IEDFs are found to have two peaks exhibiting the well known high energy tail and can be fitted by two Gaussian functions, corresponding to the local plasma potential representing background ion group produced via a charge exchange process and the ion beam (high energy tail) resulting from the plasma potential structure. Although the mean free path of collision with neutral is shorter than the IEDF measurement position, it is expected that the effect on the deceleration will be very small [47]. Assuming that the MN consists of a mono-energetic ion beam, the average beam energy of the accelerated ions at each location (the central ∼29 cm and far regions ∼49 cm) can be estimated as 12.2 ± 0.65 V and 13.3 ± 0.38 V, respectively. This results indicates that the plasma potential structure, which is directly related to the ion acceleration, is not affected by thermodynamic property of a MN influenced by isothermally behaving confined electrons. To investigate the reliability of the measured IEDF showing two peaks, stretched structure of the magnetic field is generated by using additional electro-magnets located at the diffusion region. The measured axial profile of 〈n_e〉 and 〈T_e〉 is nearly constant at the and the ion beam (high energy tail in IEDFs) was not observed under the condition, which points out that the contribution of isothermal electrons on the ambipolar electric field is negligible (figures 10(a) and (b)). Therefore, ion acceleration should not be directly inferred from the value of polytropic exponent γ_e because thermodynamic property of a MN is influenced by isothermally behaving confined electrons as well as adiabatically expanding electrons.

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