Numerical and experimental analysis of registration of small ammonia impurities using the PLES method

1 Introduction

Nonequilibrium gas-discharge plasma in organic and inorganic gases of various compositions is used in several fields of modern science and technology [1]. An interesting area of recent research is the use of ammonia impurities as precursors or catalysts in the deposition of thin films [2–4] and coating in spacecraft engines [5]. In addition, plasma methods for the synthesis of ammonia [6–8] make it possible to obtain it with particularly high purity, which is subsequently extremely important for the preparation of nitric acid, nitrogen fertilizers, dyes, explosive materials, polymers, fuels, pharmaceuticals and cleaning products [9–13].

However, ammonia is a toxic substance [14]. Based on its physiological effect on the body, it belongs to the group of substances with a suffocating and neurotropic effect, which, when inhaled, can cause toxic pulmonary edema and severe damage to the nervous system. Ammonia has both local and resorptive effects [15]. In addition, since the breakdown of protein produces ammonia, its presence in natural waters is a sign of their pollution [16]. It is therefore necessary to develop methods for the rapid detection and quantification of ammonia impurities on site in real time.

For these purposes, the method of plasma electron spectroscopy (PLES) has been developed and used in recent years [17–21]. The PLES method is based on the ionization of impurity atoms or molecules by metastable atoms, a process commonly known as Penning ionization [17],where A * is a metastable atom and X is an impurity atom (molecule) in its ground state in Equation 1. A and X + are the ground state atom and the impurity ion, respectively. Helium is typically chosen as the carrier gas for the detection of impurity gases due to its large ionization energy of 19.8 eV. Its metastable states accordingly provide enough energy to ionize any other gas except neon. For each X, the ionization of an excited atom via the Penning ionization reaction results in the production of a Penning electron, the kinetic energy of which is given by:

A * + X → A + X + + e ε x (1)

ε x = E m − E x (2)

Here, in Equation 2 x designates the reactant, which is specific to each substance and does not depend on experimental conditions or the presence of other substances. E m is the energy of the metastable helium atom, and E x is the ionization energy of the impurity, both of which are atomic or molecular constants. The electron energy spectrum resulting from reaction (1) contains a number of characteristic peaks, each corresponding to the energy of the electrons involved. To record the characteristic peaks corresponding to Penning electrons, two fundamental conditions must be met: (a) The electron energy distribution function (EEDF) must be non-local (in such a plasma, fast electrons in the free diffusion mode reach the boundaries of the discharge volume without losing energy, and different groups of electrons behave independently of each other) [22]. This criterion represents the characteristic scale of the plasma, whereby the mean free range of the electrons is taken into account. In the elastic collision region energy ε < E m , the electron energy relaxation length is λ ε = δ − 1 / 2 λ > 100 λ, where λ is the mean free range of the electron. (b) It is necessary to maintain a low main electron temperature.

Plasma probe diagnostics techniques are used to record the spectra of fast electrons [23–26]. The PLES method has been empirically validated for the detection of a range of impurities at low gas pressure. The following substances have been identified as appropriate for the PLES method: elemental gases such as Ar, Kr, N₂, and O₂ [19–21, 27], organic compound gases such as C₂H₅OH, and inorganic compound gases such as CO, CO₂, CH₄, C₂H₄ and NH₃ [28–30], also solids vapor such as Pt, NH₄NO₃ (NH₄)₂SO₄, AgNO₃, and AgCl [31–34].

Despite the fact that the PLES method showed the possibility of detecting ammonia impurities, issues related to the minimum and maximum detection limits remain unexplored. These parameters can be found by a thorough experimental study, which, however, may be limited by the sensitivity of the measuring equipment and by its inherent errors [23–26]. In particular, this applies to the probe system, which is actually main tool for detecting fast electrons and therefore detecting various impurities. On the other hand, using self-consistent models to conduct numerical simulations is a convenient tool that allows one to describe and also predict all the main parameters and physical processes occurring in the discharge cell.

Following up on the above insights, the purpose of the work reports here was to study, both numerically and experimentally, the processes of registration of fast electrons formed as a result of Penning ionization reactions of ammonia impurities in the negative glow plasma of a short glow discharge. It should be possible to determine the minimum detection limits for ammonia impurities in helium plasma through numerical experiments.

2 Description of the model

The self-consistent hybrid model employed in this work comprises two coupled components: the first describes the electron kinetics via the Boltzmann equation, and the second describes the heavy plasma constituents (ions and excited atoms) via fluid balance equations. This hybrid approach overcomes the limitations of fluid models in capturing non-equilibrium electron kinetics [35] by utilizing a self-contained solution to the spatially inhomogeneous electron Boltzmann equation [36], following the framework of self-consistent hybrid kinetic-fluid models [37]. The model is based on the kinetic approach developed in our previous studies [30, 38, 39] and implemented in the COMSOL Multiphysics software environment. A detailed derivation of the governing equations and boundary conditions can be found in those references; here we outline the essential aspects relevant to the present analysis.

To characterize the spatial and energy evolution of the EEDF, we solve the Boltzmann kinetic equation. Its general form iswhere f r , υ , t is the electron energy distribution function depending on spatial coordinates r, velocity υ, and time t in Equation 3. The left-hand side sequentially comprises the time-derivative, convective, and electric field acceleration terms. The right-hand side collision integral, C f, encompass all electron-particle interactions, such as elastic, inelastic (excitation, ionization), superelastic, and Penning ionization processes. Here e, is the elementary charge, m is the electron mass, and E is the electric field.

∂ f r , υ , t ∂ t + υ · ∇ f + e m E · ∇ υ f = C f (3)

For collision-dominated plasmas, the collision integral is primarily governed by the elastic collision operator. Accordingly, the distribution function can be expanded into a series of its eigenfunction, namely, the spherical harmonics. Truncating this expansion to retain solely the isotropic zero-th-order component f 0 (corresponding to random thermal motion) and the weakly anisotropic first-order component f 1 (corresponding to drift motion) yields the two-term approximation.where υ = υ and f 1 = f 1 < < f 0.

f r , υ , t = f 0 r , υ , t + υ υ · f 1 r , υ , t (4)

Substituting Equation 4 into the Boltzmann kinetic equation and taking the spherical average of both sides (i.e., integrating over the velocity solid angle to eliminate the anisotropic first-order term), the Boltzmann equation decomposes into

w ∂ f 0 ∂ t + χ w 3 ∇ · f 1 + ∂ G ∂ w = I r , w + I f a s t (5)

f 1 = − λ ∇ f 0 − E ∂ f 0 ∂ w (6)

Here, w = m υ 2 / 2 e is the kinetic energy of the electron in electronvolts; χ = 2 e / m; λ = υ / ν m = 1 / N 0 σ m is the electron mean free path in Equation 6, ν m is the momentum relaxation frequency, σ m is the momentum transfer cross sections, and N 0 is the gas density.

The second term in Equation 5 is the divergence of the spatial phase flux

Φ = χ w f 1 / 3

The term ∂ G / ∂ w in Equation 5 represents the flux of electrons in energy space, which includes contributions from the electric field (Equation 7), elastic collisions (Equation 8), and electron–electron Coulomb collisions (Equation 9):

G = G E + G e l + G e e

These contributions are given by where M is the mass of the neutral target particles, N is the their number density, σ c is the elastic cross section, k B is the Boltzmann constant, T is the gas temperature;with γ = 2 e / m, α = m / M being the electron-to-target mass ratio, and F 0 u being the isotropic component of the target particle distribution (for electron–electron collisions, F 0 = f 0 , Z = 1 , α = 1), The coefficient Y is defined aswhere Z is the charge of the target particles in units of e, ε 0 is the dielectric constant in Equation 10 andis the Coulomb logarithm, with n e being the electron density and T e the electron temperature in Equation 11.

G E = − 2 e m w 3 E · f 1 (7)

G e l = − 2 e m 2 m M + m N w 2 σ c f 0 + k B T e ∂ f 0 ∂ w (8)

G e e = 2 Y α γ 3 ∫ 0 w / α F 0 u u 1 / 2 d u f 0 + 4 Y 3 γ 3 ∫ 0 w / α F 0 u u 3 / 2 d u + w 3 / 2 ∫ w / α ∞ F 0 u d u ∂ f 0 ∂ w (9)

Y = Z 2 e 4 ⁡ ln Λ / 4 π ε 0 2 m 2 (10)

ln ⁡ Λ = 12 π ε 0 e T e 3 / 2 / e 3 n e 1 / 2 (11)

In the numerical implementation, the integrals in Equation 9 are not computed directly, but are evaluated by solving auxiliary differential equations for the integral moments, as described in our previous work [37]. This approach ensures numerical accuracy and stability.

The kinetic Equation 5 is solved numerically using the finite-element method in the COMSOL Multiphysics. In the numerical implementation, the electron distribution function is represented in velocity-space form. Specifically, the dependent variable in the COMSOL model iswhere f 0 v is the isotropic component of the velocity distribution function, Exponentiating both sides of Equation 12 yields Equation 13 and the physical distribution function is reconstructed as

ϒ x , y = ln f 0 v (12)

f 0 v = exp ϒ · 1 s 3 / m 6 (13)

The velocity-space representation is normalized according to the standard convention:

n e = 4 π ∫ 0 ∞ f 0 v v 2 d v (14)

To facilitate the solution of the kinetic equation, the simulation is performed on a two-dimensional phase-space grid with coordinates x , y, where x is the spatial coordinate and y = w / w 0 is the dimensionless kinetic energy, with w 0 being the maximum kinetic energy considered in the simulation (w 0 = 400 eV).

The electron density n _*e*​ and electron temperature T _*e*​ are computed from the velocity-space distribution function as where we have introduced the energy-space representation f ₀​(w) related to the velocity-space distribution by

n e = 4 π 2 w 0 m 3 / 2 ∫ 0 ∞ f 0 w w 1 / 2 d w (15)

T e = 8 π 2 3 n e w 0 m 3 / 2 ∫ 0 ∞ f 0 w w 3 / 2 d w (16)

f 0 w d w = f 0 v 4 π v 2 d v , w = m v 2 2 e (17)

Substituting the Equation 17 into the normalization condition (Equation 14), and rearranging the resulting (Equation 15), we obtain the normalization of the energy-space distribution, Equation 18. Thus, the energy-space representation f ₀​(w) is normalized aswhich is consistent with the normalization used in our previous works [30, 39].

∫ 0 ∞ f 0 w w d w = n e 4 π 2 w 0 m 3 / 2 − 1 (18)

In this work, we use the term EEDF to denote the electron distribution function as a function of kinetic energy. In the theoretical formulation, the EEDF is represented as f ₀​(w) with dimensions m⁻³⋅eV^-3/2 and normalization given by Equations 15 and 16. In the COMSOL implementation, the EEDF is stored in the velocity-space representation f ₀​(v) with dimensions s³/m⁶; when plotted as a function of energy w = mv ²/(2e), it is referred to as the EEDF in velocity-space representation. This convention is adopted for consistency with the numerical implementation and for direct comparability with our previous works [30, 39].

The right term of represents inelastic collision integrals, including collisions leading to excitation and ionization, and also the second type of collisions. The corresponding terms for the excitation, de-excitation, and ionization processes are expressed in Equation 19 as follows:where N is the density of the target heavy particles, σ k is the cross section for the kth process, and ε k is the energy threshold. The parameter a and the shifted energy w ∼ are defined as follows:

I r , w = − χ N σ k w w f 0 w − a σ k w ∼ w ∼ f 0 w ∼ (19)

• - For excitation processes: a = 1 , w ∼ = w + ε k
• - For ionization processes: a = 4 , w ∼ = 2 w + ε k
• - For superelastic collisions: a = 0 if w < − ε k and a = 1 otherwise,
• - For recombination processes a = 0.

The source term for fast electrons produced by Penning ionization and other processes is written aswhere, R p , k is the reaction rate of the kth process producing fast electrons, and θ k w is the normalized energy spectrum of the produced electrons in Equation 20. In the COMSOL implementation, the spectrum θ k w is represented by a rectangular function (using the COMSOL Rectangle function) centered at the characteristic energy of the Penning electrons, with a width corresponding to the energy spread of the process.

I f a s t = ∑ k R p , k θ k w , ∫ 0 ∞ θ w d w = 1 (20)

The densities of helium ions n _*i*​ and metastable atoms n _*m*​ are obtained from the drift–diffusion continuity equations in Equation 21:with the ion and metastable fluxes given by Equations 22, 23, respectively: where D i and D m are the diffusion coefficients for He +, and He T, respectively, and μ i is the ion mobility for He +.

∂ n k ∂ t + ∇ · Γ k = S k (21)

Γ i = − D i ∇ n i + μ i E n i (22)

Γ m = − D m ∇ n m (23)

The sources included in Equation 21, are written as Equations 24, 25: where K p e n n i n g = 1.5 × 10 − 15 m 3 / s and n a is the density of ammonia. Reaction rates R k were calculated by convolution of EEDF f 0 with the corresponding cross-section σ k (Equation 26) where N k = N 0 for direct ionization and excitation reactions; and N k = n m for stepwise ionization and superelastic collision reactions. In the COMSOL implementation, these integrals are evaluated numerically using the general projection operator genproj1.

S m = R e x c − R sup ⁡ e r r l a s t i c − R s t e p w i s e − 2 K p e n n i n g · n m 2 − K p e n n i n g · n m · n a (24)

S i = R i o n + R s t e p w i s e + K p e n n i n g · n m 2 (25)

R k = 8 π w / m 2 N k ∫ 0 ∞ f 0 u σ k w u u d u (26)

The electric potential φ is determined from the Poisson equation in Equation 27:

∇ · ∇ φ = − e ε 0 ∑ k = i n k − n e , E = − ∇ φ (27)

A detailed description of the boundary conditions for the kinetic equation, continuity equations and Poisson equation is given in previous works [30, 37]. To describe the negative glow plasma of a short glow discharge, a reduced set of elementary processes was considered, as presented in Table 1.

TABLE 1. The set of plasma-chemical processes in helium with NH₃taken into consideration.

Finite element numerical simulations were conducted using the Foundation (Mathematics) module of the COMSOL Multiphysics package. Here, the implicit (BDF) time-stepping method was employed. The MUMPS direct solver was utilized to determine all unknown quantities at each time step. All cross-section data for reactions 1–5 were obtained from the LXCAT database. To prevent the occurrence of negative values of the EEDF, the kinetic equation as well as the equilibrium equations for the ion and excited state ion densities are written in logarithmic form. It is crucial to highlight that only the triplet state of metastable helium atoms is considered in the reactions that occur, based on previous work [39], which indicates that the density of excited state helium atoms in the singlet state is significantly lower than that of excited state helium atoms in the triplet state. Furthermore, excited state helium atoms in the singlet state exert minimal influence on plasma parameters in the negative glow region.

3 The results of numerical simulation

A series of numerical simulations were conducted to investigate the characteristics of short glow discharges of hollow cathodes with an electrode diameter of 1.5 cm and a spacing of 5 mm. The simulations were carried out at a pressure of 2.5 Torr. The distributions of electron density, ion density (NH 4 + and He +), excited state atom density, and electron temperature are illustrated in Figure 1. Figure 2 illustrates the electric field and potential distributions when only triplet excited state helium atoms are considered in the framework of the hybrid model. The voltage drop across the discharge gap is set to 400 V. It is evident from Figure 1 that the cathode layer has a thickness of approximately 1.78 mm, while the anode layer is considerably thinner, with a thickness of only 0.1 mm. The length range from 1.78 mm to 4.9 mm corresponds to the negative glow region. The electron temperature is very low, about 0.4 eV. Figure 2 illustrates that the electric field at the cathode surface is particularly high (approximately 3.82 × 10⁵ V/m), and that the electric field decreases almost linearly in the cathode layer. A reversal of the electric field can be observed around 1.7 mm, which leads to a positive field in the negative glow region.

Figure 3 provides a three-dimensional plot of the electron energy fraction function at a pressure of 2.5 Torr. It can be observed that the electron beams near the cathode are accelerated in the cathode sheath to an energy W = eU ₀ (corresponding to a decrease in cathode potential) and subsequently move to the negative glow region of the plasma at a low electric field. In the negative glow plasma region situated behind the fast electron beam, a pronounced decline in the EDF can be discerned. The EEDF in the plasma can be approximated as a function of two distinct electron temperatures. The portion of the EEDF in the low-energy region from 0 to 1.5–1.7 eV corresponds to the lower-temperature main body electron (slow electron) population. The portion of the EEDF in the energy range 1.7 eV and above corresponds to the higher-energy fast electrons. The low electron temperature of the main electron population in the negative glow region of the plasma and the non-local nature of the EEDF create conditions for recording the fast electron energy spectra produced in reactions six and 7.

The high-energy portion of the EEDF and differential flux of a mixture of helium and doped ammonia with varying densities was magnified for observation, and the results are presented in Figures 4, 5.

• The peak at 19.8 eV corresponds to electrons produced by superelastic collisions.
• The peak at 15 eV corresponds to electrons produced by Penning ionization between two helium atoms in the triplet excited state.
• The peak at 9.6 eV corresponds to the electrons produced in the Penning reaction of a metastable helium atom with an impurity gas molecule (ammonia).

As illustrated in Figure 5, the peak at 15 eV gradually diminishes in intensity as the density of the impurity gas ammonia increases. In contrast, the peak at 9.6 eV becomes more pronounced and sharper. A comparison of Figures 4, 5 reveals that the observed phenomenon can be attributed to the inverse relationship between the density of ammonia and the density of metastable helium atoms.

As the density of ammonia increases, the density of stable helium atoms decreases, resulting in a reduction in the number of electrons produced by Penning ionization between two triplet excited helium atoms. Concurrently, the density of the ammonia molecular ion increases, which is indicative of the production of electrons in the Penning reaction between the metastable helium atom and the impurity gas molecule (ammonia). Furthermore, it has been demonstrated in published work that the area bounded by the characteristic peaks and transverse axes corresponding to impurity molecules or atoms is proportional to the density of the impurity. The peak energy of the impurity gas is equal to the difference between the excitation energy of the triplet metastable helium atom and the ionization energy of the impurity gas. Numerical simulation results show that the minimum of ammonia impurity that can be detected is 5 ppm.

To quantitatively evaluate the effect of impurity content on the electron generation process, we calculated the integrated area of the characteristic peak around 9.6 eV at different ammonia densities, and the results are shown in Figures 6, 7. At low density levels, when the density increased from 5 ppm to 1000 ppm, its peak flux jumped from about 5.2 × 10 6 s / m 4 to 4.1 × 10 7 s / m 4, an increase of nearly 8 times. At the same time, clear secondary peaks can be observed at 15.5 eV and 20 eV, indicating that high-energy electrons were involved in specific excitation processes. The integrated area shows an almost linear increase with the increase in ammonia density. This linear characteristic is consistent in trend with the proportional model proposed by Sheverev [43]. In this plasma system, the excitation or ionization of ammonia is achieved through Penning ionization. Therefore, when the ammonia density is low, an excess of metastable particles generated by the carrier gas increases, and as the ammonia density rises, the rate of energy transfer increases linearly, enhancing the intensity of the characteristic peak.

At high density levels (2000–10,000 ppm), when the density increases from 2000 ppm to 10,000 ppm (a fivefold increase), the total flux only increases by about 5%, and the curves at each density almost coincide. The flux evolution shows obvious convergence characteristics, with the growth of the integral area tending towards saturation. We speculate that there may be two reasons. First, when the density of ammonia molecules is sufficiently high, it causes all the metastable particles generated by the carrier gas to be consumed in a very short time. At this point, the factor limiting the intensity of the characteristic peak is no longer the impurity density, but the rate at which metastable particles are produced by the carrier gas (which is, however, limited by the input power); therefore, adding more ammonia does not further increase the intensity. Second, because ammonia molecules have abundant vibrational and rotational degrees of freedom, the probability of electrons undergoing inelastic collisions with ammonia molecules increases significantly, transferring energy to the vibrational levels of ammonia. This frequent energy dissipation results in a decrease in the number of high-energy electrons, hindering the growth of electrons capable of reaching an excitation threshold of around 9.6eV, thus causing saturation. Ultimately, this leads the plasma electron energy distribution function to enter a steady state dominated by collisional quenching.

4 Experimental verification

To verify the EEDF predictions derived from the hybrid model, probe measurements were conducted in the negative glow region of the plasma. The experimental setup consisted of a 20-mm-diameter cylindrical quartz glass tube, equipped with three- and four-way stainless steel KF fittings at each end. Molybdenum hollow electrodes (cathode and anode), separated by a 15 mm gap, were utilized. Cylindrical probes, constructed from tantalum wire (0.2 mm diameter, 2.5 mm exposed tip), were employed to determine the electron energy distribution function. The unexposed probe shafts were shielded by capillary glass tubes. A schematic diagram of the experimental setup is provided in Figure 8.

To observe a stable short glow discharge, the gas pressure was set to 2.5 Torr, and the probe was positioned in the negative glow region at the center of the hollow cathode. Furthermore, a current-limiting resistor of an appropriate value was connected in series with the discharge circuit. Prior to the experiment, the discharge chamber was evacuated to a base pressure of Torr using a vacuum pump. The desired operating pressure was then maintained under a continuous flow regime. Specifically, the gas inflow was controlled by a needle valve at the inlet, while the pumping speed was regulated by a ball valve at the exhaust. Probe measurements were conducted using an automated Multi-Functional Plasma Probe Analyser (MFPA).

During the experimental study using the probe system, the high-energy part of the second-order derivative of the probe current was obtained using a mixture of He (500 ppmNH₃) and He (1000 ppmNH₃) gases, as shown in Figure 9. The anode is used as a reference electrode. The spectra of the characteristic electrons corresponding to reactions 5, 6, and seven in Table 1 were observed in the region of 19.8 eV, 15 eV, and 9.6 eV, respectively. It can be noted that the density of the metastable helium atoms decreases with increasing density of ammonia, which is in qualitative agreement with the calculations.

As shown in Figure 10, we also performed EEDF measurements of He + 1000 ppm NH₃ at different currents. The results show that the shape of the characteristic peaks of impurities obtained from the measurements is “taller and fatter” when the current used for the discharge is higher. The reason for this is that when the voltage across the plasma is increased, the energy available for the Penning ionization reaction increases, as does the number of characteristic electrons produced and the frequency of collisions between particles.

Following is a comparative analysis of the results of numerical calculations and experimental values of electron temperatures. The temperature of the main group of electrons was determined from the transient part of the probe current-voltage characteristic (CVC), as well as from the linear part of the logarithm of the first derivative of the probe CVC with respect to the applied potential, according to the formula and the linear part of the probability function of the electron energy with respect to the energy, which is proportional to the second derivative of the probe CVC [32].

Thus, Table 2 presents a comparative analysis of the results of numerical calculations and experimental data on the electron temperature. It can be seen that the results of numerical calculations of electron temperature determined from the linear part of the logarithm of the first-order derivative of the probe current-voltage characteristic agree quite well with the results of experimental studies. In the process, the experimentally determined electron temperature is systematically higher than calculated using the hybrid model.

TABLE 2. Electron temperature in a discharge with a hollow cathode at a pressure of 2.5 Torr with an admixture of NH₃.

A possible factor is the resistance of the probe circuit and the probe system as a whole, resulting in a slight increase in the probe potential. As a result, the slope of the transition region of the single probe CVC decreases and the electron temperature increases slightly. Moreover, due to the inherent increase in the error associated with the differentiation procedure, even a relatively minor distortion of the probe CVC can result in a significant distortion of the EEDF between the floating potential and the plasma potential [30].

With increasing ammonia concentration from 500 ppm to 1000 ppm, the experimental electron temperature decreases from 0.98 eV to 0.69 eV, while the calculated temperature remains nearly constant (0.40 eV and 0.39 eV). This difference may be attributed to the fact that the calculated temperature is obtained from the slope of the EEDF in the low-energy region, whereas the experimental temperature is derived from the transition region of the probe characteristic, which is sensitive to probe surface conditions and measurement errors [30]. Nevertheless, both experimental and calculated temperatures are in the same range (sub-eV) and show qualitative agreement in terms of the overall plasma behavior.

It should be noted that the one-dimensional model inevitably neglects the radial diffusion and wall recombination losses existing in the two-dimensional experimental device. Therefore, although the absolute value of the plasma density may deviate from the experiment, the one-dimensional model accurately predicts the qualitative evolution law of the characteristic peak on the EEDF and its formation mechanism.

We need to discuss the limitations of the model. On the one hand, in high-density discharges with an electron density of 10¹⁸ m⁻³, the electron dissociation of ammonia molecules cannot be ignored, and it will generate free radical fragments such as NH₂ and NH. The ionization energies of these free radicals are lower than that of NH₃. They can undergo Penning ionization with metastable helium, generating electrons with energies different from the 9.6 eV peak. The superposition of these additional signals may cause the main characteristic peak to broaden or distort, thereby reducing resolution and affecting the predicted detection sensitivity. On the other hand, our model assumes that the discharge is radially uniform, and ignores the three-dimensional radial diffusion of fast electrons as well as wall loss. In the actual discharge, the fast electrons generated in the negative glow region can drift towards the tube wall and be absorbed. As a result, the fast-electron density at the center is lower than the value obtained from one-dimensional simulation. This overestimation of the number density of fast electrons is likely to result in a higher peak, thereby causing the theoretical predicted minimum detection density to be low (that is, the detection limit of 5 ppm should be regarded as a relatively optimistic lower limit). Nevertheless, the one-dimensional model can still well reproduce the qualitative experimental laws of EEDF and the variation of peak area with impurity density at low doping densities. Therefore, it still has reference value for trend prediction and mechanism analysis.

Furthermore, the 5 ppm value obtained through numerical simulation is a theoretical estimation based on the idealized assumptions of a one-dimensional model. However, our experimental verification was conducted under conditions of 500 ppm and 1000 ppm. This disparity has practical reasons. At lower densities, the characteristic peak signals are relatively weak and are easily masked by background current fluctuations and electromagnetic interference. After the probe surface is exposed to the discharge environment for a long time, it may be contaminated due to ammonia adsorption or film deposition, thereby altering its current-voltage characteristics and making the extraction of weak signals more difficult. Additionally, the slight drift of the discharge current can introduce negligible measurement noise. These factors work together, making it challenging to achieve direct and stable detection at the 5 ppm level. Therefore, conducting experiments under the conditions of 500 ppm and 1000 ppm to verify the qualitative trend of the model is reasonable.

It should be noted that the electronic temperature deviation between calculation and experiment has a limited impact on the conclusion of the 5 ppm detection limit. On one hand, the theoretical detection limit of 5 ppm is derived from the distinguishability of the 9.6 eV characteristic peak produced by the Penning ionization reaction (Reaction 7). The reaction rate mainly depends on the density of metastable helium atoms and ammonia molecules, rather than the main electron temperature. The main electron temperature mainly affects the energy distribution of low-energy electrons and has no direct decisive influence on the formation mechanism of the 9.6 eV characteristic peak. On the other hand, even if the actual electron temperature is higher than the calculated value, this mainly reflects an increase in the average energy of the main electron population, rather than a change in the fast electron population. In the PLES method, the identification of the characteristic peak depends on the energy spectrum structure of the fast electrons, while an increase in the main electron temperature will increase the probability of inelastic collisions between electrons and ammonia molecules, which to some extent weakens the signal intensity of the 9.6 eV peak, but this effect is not sufficient to completely suppress the characteristic peak corresponding to 5 ppm. Therefore, the theoretical detection limit of 5 ppm remains a valuable lower limit estimate at higher actual electron temperatures, rather than an unrealistic ideal value.

It should be noted that there is a difference between the 5 mm electrode distance used in the simulation and the 15 mm hollow cathode spacing used in the experiment. This is due to two reasons. Firstly, the one-dimensional model aims to capture the physical processes along the axial direction of the discharge gap, especially the generation and transport mechanism of fast electrons in the negative glow region. In short-gap discharges, a 5 mm spacing is sufficient to establish a stable negative glow region, while reducing the computational load and improving convergence efficiency, facilitating the systematic study of the evolution law of characteristic peaks under different ammonia densities. Secondly, the 15 mm spacing hollow cathode structure used in the experiment is to ensure that the probe can be smoothly inserted into the negative glow region for measurement, and to provide sufficient spatial stability for the discharge. Therefore, the difference in spacing between 5 mm and 15 mm does not affect the core conclusions of this paper. Firstly, the physical basis of the PLES method is the non-local electron energy distribution function and the Penning ionization reaction. This mainly depends on gas pressure, electric field distribution, and collision processes, rather than the absolute spacing value. Secondly, the one-dimensional model predicts the axial electron energy distribution characteristics, and the experimental probe measurement is also taken at the axial position of the negative glow region center, and the physical mechanisms reflected by both are consistent. Of course, we admit that radial diffusion and wall loss do exist in the three-dimensional hollow cathode structure. The main value of the one-dimensional model lies in capturing qualitative trends and revealing physical mechanisms, rather than precisely replicating the absolute numerical values of the experiment.

5 Conclusion

This study introduced and used a self-consistent theoretical model of plasma kinetic based on a hybrid model. It was used to refine the current theoretical models for ammonia detection using plasma techniques. The applicability of detecting small ammonia impurities using plasma electron spectroscopy was studied both numerically and experimentally. The results show that the lowest detectable impurity density is about 5 ppm at low pressure. This provides an important guideline for detection of trace impurities in gases. The simulation suggests an idealized theoretical detection limit of about 5 ppm under the assumptions of the 1D model, while the experimental verification was conducted at 500 ppm and 1000 ppm, demonstrating the practical feasibility of the PLES method for higher densities. Achieving the theoretical sensitivity of 5 ppm in actual experiments would require further improvements in signal-to-noise ratio, in situ probe cleaning, and noise suppression. The effects of varying the ammonia density in helium on the characteristic peaks of the electron energy distribution function were studied. In experiment, the density of the metastable helium atoms decreases with increasing density of the applied ammonia molecules. This is in qualitative agreement with the calculations. We found that the hybrid model describes relatively accurately the formation of peaks associated with fast electrons generated in Penning ionization and superelastic collision reactions. When the impurity content was 500 ppm or 1000 ppm, three peaks were observed: from electrons resulting from the Penning ionization reaction of the impurity, from electrons resulting from the Penning ionization reaction of metastable helium atoms and from superelastic collisions. It is concluded that the hybrid model can better describe the plasma parameters in the negative glow region, particularly an electron temperature of about 0.4 eV. Experimental tests were also carried out with different densities of ammonia gas using the PLES, and the numerical simulations are in good agreement with the experimental results. The reliability of the model has been verified. In addition, the reliability of plasma electron spectroscopy for impurity detection is further demonstrated by the experimental analysis of the numerical simulations.

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Footnotes

Edited by: Jayr Amorim, Aeronautics Institute of Technology, (ITA), Brazil

Reviewed by: Krishan Kumar, West Virginia University, United States

Yashuang Zheng, South China University of Technology, China