2. The TS experiment

A schematic diagram of the experimental set-up for TS is shown in figure 1. In the red dashed box is the ICP generation system to be studied. A laser beam with a wavelength of 532 nm, a repetition frequency of 30 Hz, a maximum laser pulse energy of 300 mJ and a pulse width of 10 ns is emitted by a Nd:YAG laser. After passing through mirrors 1 to 3, the laser beam enters the ICP cavity through a Brewster quartz window. When passing through the plasma generated by the ICP system, the laser light scattered by electrons, ions and atoms in the plasma is collected by an optical fibre bundle and transmitted to a triple grating spectrometer and intensified CCD (ICCD) camera for imaging or forming a spectrum to complete TS signal detection. Controller DG645 is a time controller that synchronizes the laser pulse with the shutter of the ICCD camera and optimizes the detected signal by changing the exposure time of the ICCD camera. The laser beam passing through the plasma eventually enters the laser stack and terminates its transmission.

Figure 1. Schematic diagram of the TS experiment. In the upper-right corner is a schematic diagram of the optical fibre bundle end face.

To avoid the influence of stray light on the TS signal, Brewster windows are used in the incident and exit windows. To avoid Rayleigh scattering from the air and reflections from other surfaces, the collection end of the optical fibre is placed in a dark chamber. The collection direction of the optical fibre is at an angle of 90° from the laser transmission direction. The detection optical fibre bundle consists of 19 optical fibres arranged linearly along the axis of the ICP. A schematic diagram of the optical fibre bundle end face is shown in the upper-right corner of figure 1. Through a plano-convex lens, the laser–plasma interaction region is demagnified by half and the image is converged on the end of the detection optical fibre bundle. When collecting data, the optical path composed of the plano-convex lens and the optical fibre bundle does not move. The ICP system is driven by two stepper motors to perform axial translation in the x direction and radial translation in the y direction. The TS signals of the measured point are collected simultaneously by nine adjacent fibre channels in the gap of the inductor coil helical antenna, and the final experimental data are obtained by averaging them through data processing software. Each fibre is 200 μm in diameter, giving an axial spatial resolution of 0.4 mm. Affected by the curvature and thickness of the quartz tube, the radial spatial resolution is 2 mm.

When the laser beam is injected into the plasma, the free electrons in the plasma oscillate in the laser electric field. These electrons accelerated by the laser electric field emit electromagnetic radiation, which can be interpreted as scattering of incident laser radiation. The total scattering spectrum of plasma to incident laser consists of three components: TS component generated by free electron scattering; Rayleigh scattering components produced by bound electrons in atoms and ions; and stray light component reflected by the chamber walls and windows. The stray light can be suppressed by a range of methods such as using Brewster windows and special baffles. Compared with the electron cloud around atoms and ions, the free electron velocity is higher, which leads to the Doppler broadening of TS spectral components much wider than that of Rayleigh scattering.

For low-temperature plasmas such as ICP, the total intensity of TS light is proportional to the electron density n_e. The total intensity of Rayleigh scattering light is proportional to the density of neutral particles n ₀. Therefore, after calibrating the TS results with Rayleigh scattering, n_e can be determined from the measured spectra by accurately calibrating the absolute sensitivity of the detection system. When the plasma generator chamber is filled with gas at a given pressure and there is no plasma, the calibration of the intensity of TS light can be completed by measuring the intensity I_g of Rayleigh scattering light.

The Rayleigh scattering light intensity is expressed as (Muraoka, Uchino & Bowden Reference Muraoka, Uchino and Bowden1998)

(2.1)\begin{equation}{I_g} = {n_0}{\sigma _R}\Delta {\lambda _g}{f_{\textrm{sys}}}.\end{equation}

After the gas is excited in the plasma reactor cavity to generate the plasma, the intensity I_p of the scattering light of the plasma is measured by the same experimental system as

(2.2)\begin{equation}{I_p} = {n_e}{\sigma _{\textrm{Th}}}\Delta {\lambda _p}{f_{\textrm{sys}}},\end{equation}

where ${\sigma _{\textrm{Th}}}$ and ${\sigma _R}$ are the differential cross-sections for TS and Rayleigh scattering (for the calibration gas), respectively. Parameters Δλ_p and Δλ_g are the spectral widths of the plasma and gas scattering spectra, respectively, and f _sys is a function of the laser energy and the efficiency of the detection system.

Since the same laser and detection system are used for both measurements, f _sys is the same in each case, so the electron density is

(2.3)\begin{equation}{n_e} = {n_0}\frac{{{I_p}}}{{{I_g}}}\frac{{{\sigma _R}\Delta {\lambda _g}}}{{{\sigma _{\textrm{Th}}}\Delta {\lambda _p}}}.\end{equation}

For a low-temperature plasma such as ICP, the TS spectrum is directly related to the free electron velocity distribution function. When the electron velocity distribution function is a Maxwell function, the TS spectrum is a Gaussian distribution. The electron temperature can be characterized by its half-width, Δλ _1/e (nm), as

(2.4)\begin{equation}{T_e} = \frac{{{m_e}{c^2}}}{{8k{{\sin }^2}(\theta /2)}}{\left( {\frac{{\Delta {\lambda_{1/e}}}}{{{\lambda_0}}}} \right)^2}.\end{equation}

Here, λ ₀ is the laser wavelength, θ is the scattering angle and k, c and m_e are the Boltzmann constant, the speed of light and the electron mass respectively.

Figure 2(a) shows a schematic diagram of the ICP generator used in the experiment, which is mainly composed of a 15 cm long quartz tube, a six-turn inductor coil, inlet and outlet, etc. The inductor coil is made of hollow copper tube, which is filled with circulating water to take away the heat generated by the plasma in time. The vacuum pump draws out the gas in the chamber from the outlet. During the experiment, the argon to be ionized is slowly released into the chamber from the inlet. A vacuum gauge is placed on the side of the inlet to monitor the pressure in the chamber. When the required pressure is reached, Rayleigh scattering and TS are measured, and the data of electron density and electron temperature are obtained through data processing.

Figure 2. (a) Schematic diagram of ICP generator experimental set-up with experimental results of electron density distribution and (b) FEM simulation results of electron density distribution inside ICP model (gas pressure: 65 Pa; absorbed power: 350 W).

3. The FEM simulation

In this study, COMSOL software is used to model and analyse the ICP. There are seven types of chemical reactions in the chemical process of ICP argon ionization, as shown in table 1.

Table 1. List of plasma chemical reactions.

According to the types of reactions listed in table 1, reaction 1 is an elastic collision between an electron and an argon atom without energy exchange. However, reactions 2, 4 and 5 need to consume different amounts of electromagnetic energy when they occur. Reaction 2 is an excitation reaction, in which an electron collides with an argon atom and consumes 11.5 eV of energy to excite the argon atom into the metastable state. Reaction 3 is a super-elastic collision, in which an electron collides with a metastable-state argon atom and regains 11.5 eV of energy released by a metastable-state argon atom (Demidov, DeJoseph & Kudryavtsev Reference Demidov, DeJoseph and Kudryavtsev2005; Shen et al. Reference Shen, Wang, Wang, Liu, Liu, Vourlidas, Miao, Ye, Liu and Zhou2012; Liu, Guo & Pu Reference Liu, Guo and Pu2015). At the same time, the metastable-state argon atom returns to a ground-state argon atom after losing energy. Reaction 4 means that an electron consumes 15.8 eV of energy to ionize a ground-state argon atom to release a new electron, and the argon atom becomes an argon ion, which is a typical ionization reaction. Reaction 5 means that an electron only needs 4.24 eV of energy to ionize a metastable argon atom from which a new electron is released, while this metastable argon atom becomes an argon ion. The last column of table 1 shows the maximum reaction rates of the various reactions at 350 W absorbed power and 75 Pa gas pressure. It is evident that the maximum reaction rate of reaction 5 is greater than that of reaction 4 in ionization reactions. Therefore, reaction 5 plays an important role in maintaining the low-pressure argon discharge process (Lymberopoulos & Economou Reference Lymberopoulos and Economou1995; COMSOL 2020). Reactions 6 and 7 respectively indicate that the metastable argon atom is suppressed by another metastable argon atom and ground-state argon atom, returning to a ground-state argon atom and argon ion.

In addition to the bulk reactions, the process also involves two surface reactions (table 2). When metastable argon atoms and argon ions come into contact with the wall, there will be a certain probability (adhesion coefficient) of conversion to ground-state argon atoms.

Table 2. List of surface reactions.

In conclusion, the FEM generation model of ICP has been established by COMSOL. The inductive coil couples the electromagnetic energy into the argon gas in the tube to cause seven kinds of chemical reactions and two kinds of surface reactions, thus ionizing the argon gas to obtain the plasma. The plasma in the tube is mainly composed of neutral argon atoms, electrons and ions. The electron density n_e, electron energy density n_ɛ and electron temperature T_e can be obtained by solving two fluid equations of electron density and electron energy density by FEM. These two fluid equations describe the electron density and the electron energy density as a function of configuration space and time t. The R_e is either a source or a sink of electrons which can be described by (COMSOL 2020)

(3.1)\begin{equation}{R_e} = \frac{\partial }{{\partial t}}({n_e}) + \boldsymbol{\nabla }\boldsymbol{\cdot }[ - {n_e}({\mu _e}\boldsymbol{\cdot }{\boldsymbol{E}_{\textrm{in}}}) - {\boldsymbol{D}_e}\boldsymbol{\cdot }\boldsymbol{\nabla }{n_e}].\end{equation}

The energy loss or gain due to inelastic collisions, R_ε, can be described by

(3.2)\begin{equation}{R_\varepsilon } = \frac{\partial }{{\partial t}}({n_\varepsilon }) + \boldsymbol{\nabla }\boldsymbol{\cdot }[ - {n_\varepsilon }({\mu _\varepsilon }\boldsymbol{\cdot }{\boldsymbol{E}_{\textrm{in}}}) - {\boldsymbol{D}_\varepsilon }\boldsymbol{\cdot }\boldsymbol{\nabla }{n_\varepsilon }] + {\boldsymbol{E}_{\textrm{in}}}\boldsymbol{\cdot }{\Gamma _e}.\end{equation}

The electron diffusivity ${\boldsymbol{D}_e}$, energy mobility ${\mu _\varepsilon }$ and energy diffusivity ${\boldsymbol{D}_\varepsilon }$ are computed from the electron mobility μ_e and electron temperature T_e using ${\boldsymbol{D}_e} = {\mu _e}{T_e}$, ${\mu _\varepsilon } = {\textstyle{5 \over 3}}{\mu _e}$ and ${\boldsymbol{D}_\varepsilon } = {\mu _\varepsilon }{T_e}$. Parameter ${\Gamma _e}$ is the electron flux and ${\boldsymbol{E}_{\textrm{in}}}$ is the induced electric field.

According to the specifications of the ICP reactor used during the experiment, the dimensions of the ICP model in COMSOL have been designed. The length of the ICP tube is 15 cm. The inner diameter is 2.6 cm. The wall thickness is 0.2 cm. The inductor coil has six turns, the wire diameter of the coil is 0.4 cm and the turn spacing is 1 cm. During simulation, the temperature parameter is specified as 300 K, the electron transport characteristics are limited to the specified mobility and the electron mobility for isotropic reduction is set to 4 × 10²⁴ (V m s)⁻¹. The thickness of the air layer outside the tube is set to 2 cm. The established FEM simulation model is shown in figure 2(b). An extra-fine mesh is used to fill the triangular mesh with 658 632 elements, and the minimum element is set to 11.2 μm. Boundary layers are created near the tube walls, and each boundary layer is divided into eight sublayers to ensure a smooth transition from the tube wall mesh to the tube inner mesh. Since the plasma ignition time is usually about 1 μs, the maximum ionization state is reached at about 30 μs. To ensure that the plasma reaches a steady state, the transient settings are calculated from 10⁻⁶ to 0.1 s.