2. Methods

To study the effect of a transverse magnetic field on the gas-discharge breakdown and plasma properties in helium, experiments and numerical simulations are conducted. The experimental set-up comprises a discharge chamber, a gas pumping system, permanent magnets, two high-voltage rectifiers and a synchronization and start-up unit. Figure 1 displays the structure of the gas-discharge chamber. The gas-discharge chamber constitutes stainless steel hemispherical electrodes with a diameter of 8 mm that are located at gaps of 4 mm inside a quartz tube. The permanent magnets located over the gap yield a magnetic field with a value of $B = 0.4\;\textrm{T}$ between the electrodes. The dimensions of the magnets are as follows: width of 80 mm, length of 100 mm and thickness of 20 mm. The dimensions of the magnets significantly exceed the discharge gap size, and the movement of magnets in different directions within 10 mm does not affect the discharge characteristics, which indirectly confirms the uniformity of the magnetic field in the discharge region. Ashurbekov et al. comprehensively studied the effect of impurities on the characteristics of a pulsed discharge in helium using mass spectrometry for monitoring the ion composition. They experimentally showed that even the presence of an insignificant amount of impurities significantly decreases the densities of metastable atoms and molecular helium ions (Ashurbekov et al. Reference Ashurbekov, Kurbanismailov, Omarov and Omarova2000). In this study, the experiments are conducted under well-controlled laboratory conditions. The discharge chamber is pumped out by a pre-vacuum pump with 10⁻³ Torr pressure and then by a turbomolecular high-pressure pump with 10⁻⁴ Torr pressure. Moreover, to prevent the effect of the residual gas, the discharge chamber is repeatedly pumped with spectrally pure helium before studying the discharge characteristics.

Figure. 1. Structure of the gas-discharge chamber.

A numerical two-dimensional (2-D) fluid model in COMSOL Multiphysics (2023) is established to analyse the effect of a transverse magnetic field on the gas-discharge characteristics and to study the dynamics of the initial stage of discharge development. The numerical model of the gas-discharge chamber corresponds both in size and design to the real geometry shown in figures 1 and 2.

Figure. 2. Geometry of the numerical model of the gas-discharge chamber.

The continuity equation for electron density ${n_e}$ (2023; Hagelaar and Pitchford Reference Hagelaar and Pitchford2005)

(2.1)\begin{equation}\frac{{\partial {n_e}}}{{\partial t}} - \boldsymbol{\nabla }\boldsymbol{\cdot }(({\mu _e}\boldsymbol{\cdot }\boldsymbol{E}){n_e} + \boldsymbol{\nabla }({\boldsymbol{D}_e}{n_e})) = {R_e},\end{equation}

where ${\mu _e}$ is the electron mobility, $\boldsymbol{E}$ is the electric field, ${\boldsymbol{D}_e}$ is the diffusion coefficient for electrons and ${R_e}$ is the electron source term. For electron swarm/transport parameters, look-up tables from the solution of the Boltzmann equation for EEDF in the two-term approximation (Hagelaar and Pitchford Reference Hagelaar and Pitchford2005; Starikovskiy et al. Reference Starikovskiy, Aleksandrov and Shneider2021) with the IST-Lisbon database are employed (Alves Reference Alves2014). Herein, the two-term approximation is used to solve the Boltzmann kinetic equation as the anisotropy degree of EEDF does not increase even under a strong magnetic field. More details about the applicability of the two-term approximation for describing EEDF in a magnetic field are discussed in Starikovskiy et al. (Reference Starikovskiy, Aleksandrov and Shneider2021). In the simulation, the coefficients of direct and stepwise ionization are calculated by integrating the corresponding cross-sections of the processes with EEDF.

To describe the transport of heavy particles in COMSOL Multiphysics (2023), the simplified form of the Maxwell–Stefan equations is used, which correctly considers diffusive transport due to mole fraction and pressure and temperature gradients. In numerous situations, the Maxwell–Stefan equations are the only formulation conserving the total mass in the system. Conservation of mass dictates that the sum of the mass fractions of all the species must equal one. The transport equation for the jth component of the heavy non-electron particles is as follows:

(2.2)\begin{equation}\rho \frac{{\partial {\omega _j}}}{{\partial t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }\left( {{\mu_{j,m}}\rho {\omega_j}\boldsymbol{E} - {D_{j,m}}\rho \frac{{\boldsymbol{\nabla }({M_n}{\omega_j})}}{{{M_n}}}} \right) = {R_j},\end{equation}

where ${M_n}$ is the mean molar mass of the mixture, $\rho$ is the density of the mixture, ${\omega _j}$ is the mass fraction of the $j\textrm{th}$ component, ${\mu _{j,m}}$ is the mobility of the $j\textrm{th}$ component, ${D_{j,m}}$ is the mixture-averaged diffusion coefficient and ${R_j}$ is the source term of the $j\textrm{th}$ component.

The mean electron energy $\langle \varepsilon \rangle = {n_\varepsilon }/{n_e}$ is computed by solving the drift–diffusion equation for the electron energy density ${n_\varepsilon }$ (2023; Hagelaar and Pitchford Reference Hagelaar and Pitchford2005)

(2.3)\begin{equation}\frac{{\partial {n_\varepsilon }}}{{\partial t}} - \boldsymbol{\nabla }\boldsymbol{\cdot }({n_\varepsilon }({\mu _\varepsilon }\boldsymbol{\cdot }\boldsymbol{E}) + {\boldsymbol{D}_\varepsilon }\boldsymbol{\cdot }\boldsymbol{\nabla }{n_\varepsilon }) - \boldsymbol{E}\boldsymbol{\cdot }(({\mu _e}\boldsymbol{\cdot }\boldsymbol{E}){n_e} + \boldsymbol{\nabla }({\boldsymbol{D}_e}{n_e})) = {S_{en}},\end{equation}

where ${n_\varepsilon }$ is the electron energy density, ${\mu _\varepsilon }$ is the energy mobility, ${\boldsymbol{D}_\varepsilon }$ is the energy diffusivity and ${S_{\textrm{en}}}$ is the energy loss or gain due to inelastic collisions.

The electric potential V is calculated by solving the Poisson equation

(2.4)\begin{equation}{\varepsilon _0}{\nabla ^2}V ={-} q\left( {\sum\limits_{i = 0}^n {{N^ + } - {n_e}} } \right),\end{equation}

where ${\varepsilon _0} = 8,85 \cdot {10^{ - 12}}\;\textrm{F}\;{\textrm{m}^{ - 1}}$ is the electric constant and ${N^ + }$ is the density of the positive particle. In our work, positive particles are the atomic ions $\textrm{H}{\textrm{e}^ + }$and molecular ions $\textrm{He}_2^ +$.

For a special case describing the behaviour of a weakly ionized plasma under a not strong magnetic field (until the ions are magnetized), if the frequency of elastic collisions does not depend on the electron energy at a fixed gas pressure, then the ionization frequency in a magnetic field only depends on the effective electric field ${\boldsymbol{E}_{\textrm{ef}}}$ (Tonks Reference Tonks1937; Golant, Zhilinsky & Sakharov Reference Golant, Zhilinsky and Sakharov1980; Scholfield, Gahl & Shimomura Reference Scholfield, Gahl and Shimomura1999; White et al. Reference White, Robson and Ness1999; Starikovskiy et al. Reference Starikovskiy, Aleksandrov and Shneider2021). Under these conditions, the magnetic and electric fields in the kinetic equation are only in combinations ${E^2}/(1 + {\omega ^2}/\nu _{\textrm{ea}}^2)$ and the ionization frequency ${v_i}$ depends only on the effective electric field ${\boldsymbol{E}_{\textrm{ef}}}$, since it is determined by the symmetric part of the EEDF, i.e. ${v_i}(\boldsymbol{E},\boldsymbol{B}) = {v_{i0}}({\boldsymbol{E}_{\textrm{ef}}})$. Therefore, to account for the effect of the magnetic field in (2.1)–(2.4), the electric field $\boldsymbol{E}$ is replaced by the effective electric field ${\boldsymbol{E}_{\textrm{ef}}}$

(2.5)\begin{equation}{\boldsymbol{E}_{\textrm{ef}}} = \frac{\boldsymbol{E}}{{\sqrt {1 + {\omega ^2}/\nu _{\textrm{ea}}^2} }},\end{equation}

where $\omega = qB/m$ is the electron cyclotron frequency in the magnetic field and ${v_{\textrm{ea}}}$ is the electron transport collision frequency. Equation (2.5) is known as Tonks’ theorem (Tonks Reference Tonks1937), and according to it, the application of a magnetic field to a discharge qualitatively corresponds to an increase of the neutral density (pressure), which leads to a decrease of the mean energy. At high values of $E/N$ and low values of $B/N$, the agreement between the results of Tonks’ theorem and exact Monte Carlo calculations is better (Dujko et al. Reference Dujko, Raspopovic and Petrovic2005). This procedure enables the consideration of the ionization relaxation processes in the presence of a transverse magnetic field. At low average electron energies (in the afterglow) for recombination relaxation, the introduction of an effective electric field may be incorrect due to the violation of the constancy of the elastic collision frequency under fixed gas pressure.

Therefore, based on the above assumptions, the ionization relaxation processes with and without a magnetic field can be described by the same equations; in the presence of a magnetic field, the electric field $\boldsymbol{E}$ is replaced by ${\boldsymbol{E}_{\textrm{ef}}}$ and the electron mobility across the magnetic field is replaced by ${\mu _e} = {\mu _{e0}}/(1 + {\omega ^2}{\tau ^2})$, where ${\mu _{e0}}$ is the electron mobility without the magnetic field. Thus, under the considered approximation, the influence of the magnetic field on the discharge characteristics can be described by changing the heating and kinetic coefficients of the electrons. Then, these kinetic and transport coefficients are employed in the fluid model.

To solve (2.1)–(2.4), they need to be supplemented with the initial and boundary conditions. The following initial and boundary conditions are employed herein:

• • Initial electron density: ${n_e}(t = 0,x,y) = {10^{14}}\;{\textrm{m}^{ - 3}}$. As the experiments are conducted in the frequency-periodic mode with a frequency of 50–100 Hz, the electrons with a density of approximately ${\sim} {10^{14}}- {10^{15}}\;{\textrm{m}^{ - 3}}$ from the previous discharge pulses remain in the form of a charge deposited on the walls of the discharge chamber and in a discharge volume where the chemoionization processes are taking place, which plays the role of preliminary ionization (Ashurbekov, Iminov & Ramazanov Reference Ashurbekov, Iminov and Ramazanov2017). In such conditions, the choice of a large initial electron density in the simulation is justified.

• • Initial mean electron energy: $\varepsilon = 4\;\textrm{eV}$.

• • Boundary condition for the cathode: $V = 0$. The secondary electron emission at the cathode is also considered; it is ${\gamma _i} = 0.15$.

• • Boundary condition for the anode: $V = U(t)$. The potential at the anode walls is the same as that in the experiment (figure 4).

In the model, the following species are considered: helium atom in the ground state: $\textrm{He}({1^1}S)$; two metastable atoms: $\textrm{H}{\textrm{e}^\ast }(1)({2^3}S),\;\textrm{H}{\textrm{e}^\ast }(2)({2^1}S)$; two radiative atoms: $\textrm{H}{\textrm{e}^\ast }(3)({2^3}P),\;\textrm{H}{\textrm{e}^\ast }(4)({2^1}P)$; atomic ion: $\textrm{H}{\textrm{e}^ + }$; two excimer molecules: metastable $\textrm{He}_2^\ast (1){(^3}\sum _u^ + )$ and radiative $\textrm{He}_2^\ast (2){(^1}\sum _u^ + )$; and molecular ion: $\textrm{He}_2^ +$. Table 1 presents the 56 plasma-chemical reactions, together with their cross-section or rate constants. Note that the model considers collisions with not only electrons but also heavy particles, including molecules; these are the processes of ion conversion, molecular conversion and Penning ionization. At high pressures, processes involving molecules can significantly affect the helium plasma discharge (Kutasi, Hartmann & Donko Reference Kutasi, Hartmann and Donko2001). In the table, the superelastic collisions of electrons with excited atoms leading to the de-excitation of the level are indicated by reverse arrows (reactions 2–16). In the afterglow of pulsed discharges in helium, the proportion of molecular ions $\textrm{He}_2^ +$ can reach tens of per cent of the total number of charged particles, but in the discharge itself, their proportion is small due to their dissociation via electron impact (Ashurbekov et al. Reference Ashurbekov, Kurbanismailov, Omarov and Omarova2000). Therefore, associative ionization is not considered in this study.

Table 1. Plasma-chemical reactions used in the helium plasma model (Kutasi et al. Reference Kutasi, Hartmann and Donko2001; Belmonte et al. Reference Belmonte, Cardoso, Henrion and Kosior2007; Bogdanov et al. Reference Bogdanov, Kapustin, Kudryavtsev and Chirtsov2010; Alves Reference Alves2014; Santos et al. Reference Santos, Noel, Belmonte and Alves2014). Here, T _e is the electron temperature in eV and T _g is the gas temperature in K.

To solve the system of partial differential equations (2.1)–(2.4), a discrete grid that covers the entire computational domain is manually plotted. Figure 3 displays the discrete grid in the simulation region and the electron density at t = 200 ns, calculated without an external constant magnetic field. A large density gradient of charged particles and a strong reduction of the potential are observed near the cathode surface. Therefore, as shown in figure 3, to avoid the occurrence of singularities and solve the problem, a very fine grid is constructed around the electrodes. Figure 3 shows that the mesh quality around the electrodes is high (right legend bar). The mesh comprises 85 076 elements, where the minimum quality of the elements is 0.44 and the minimum mesh size is 0.04 mm (minimum element size).

Figure. 3. Discrete grid in the simulation region and the electron density at t = 200 ns.