Hybrid model for a cylindrical hollow cathode glow discharge and comparison with experiments

doi:10.1016/S0584-8547(01)00385-8

Spectrochimica Acta Part B: Atomic Spectroscopy

Volume 57, Issue 2, 22 February 2002, Pages 311-326

https://doi.org/10.1016/S0584-8547(01)00385-8 Get rights and content

Abstract

Some of the fundamental characteristics of the hollow cathode glow discharge are presented for different discharge conditions, based on a hybrid Monte Carlo fluid model, and on electrical and spectroscopic measurements. The Monte Carlo model describes the movement of the fast electrons as particles, while in the fluid model, the slow electrons and positive ions are treated as a continuum. The transient continuity equations are solved together with the Poisson equation in order to obtain a self-consistent electric field. The source terms of the continuity equations and the electron multiplication coefficient (used for the determination of the secondary electron emission coefficient) are obtained from the Monte Carlo simulation. These two models are run iteratively until convergence is reached. Typical results are, among others, the charged particles densities, the fluxes, the electric field and potential distribution. It is found that the influence of the bottom of the cylindrical hollow cathode cannot be neglected. A very good agreement between calculations and experimental data was obtained.

Introduction

Hollow cathode discharges (HCD) are used in a wide variety of applications, for example in plasma processing (ion etching, thin film deposition, surface treatment) [1], in ion gas lasers [2], [3], and in spectroscopic analysis, where the hollow cathode is used as an emission source, allowing direct excitation and analysis of samples and as a light source in absorption spectrometry because of its sharp and intense spectral lines [4], [5], [6], [7], [8].

This kind of glow discharge, which was first studied by Paschen [9], [10], comprises two regions: a dark space adjacent to the cathode surface, where the electric field is strong, which is called the cathode dark space (CDS) or sheath, and a rather luminous part beyond it, where the field is weak and which is called the negative glow region (NG) or plasma region. The cathode can be built in different shapes, for example as a spherical segment or as a pair of plane parallel plates or as a hollow cylinder. The anode (or anodes) is usually placed close to the cathode surface at such a distance that no positive column can be formed and that the NG can be confined inside the discharge cavity.

The operating voltage of the HCD is lower than in a glow discharge with single planar cathode for the same current density. Moreover, the voltage–current dependence shows two different regions: one with a steep slope at low current and the other with a smaller slope at higher current. The most specific characteristic of the HCD is the so-called hollow cathode effect [11], i.e. a large increase in the current density and light intensity, which is observed when the distance between the opposite cathode surfaces and the gas pressure are set in a way that the NG regions from facing cathode surfaces overlap. Theses properties are mostly due to the very efficient use of the fast electrons and ions in the HCD [12], [13], [14]. Indeed, most of the ions, which usually get lost at the boundaries of the NG with the anode and with the surrounding walls in the case of a plane cathode, can reach the surface of the cathode and release more electrons. The loss of fast electrons is also considerably reduced: the fast electrons can oscillate between opposite cathode surfaces (‘pendulum effect’) [14], [15], using all their energy for excitation and ionization of the gas atoms. Some of these electrons can penetrate into the dark space opposite to the cathode of their origin and cause ionization collisions there. The new electrons thus created will be accelerated in the high CDS field and they generate new electrons, giving rise to an enhanced ionization rate in the CDS as well as in the NG.

Conventional HCDs usually operate at pressures up to 1.3 kPa, with a cathode voltage ranging from 200 to 500 V and a discharge current from a few mA up to 1 A [4], [5], [16].

In this paper, we will present our results for a cylindrical HCD obtained by a self-consistent numerical simulation and by measurements of the current–voltage dependence and the light emission intensity of some ArI spectral lines. Most studies, i.e. analytical models, experiments or mathematical simulations dealing with cylindrical HCDs are focused only on the radial properties of the discharge, assuming a hollow cathode of infinite length, and hence, a discharge uniform in the axial direction [17], [18], [19], [20], [21], [22], [23]. Here we intend to simulate the cylindrical geometry of a HCD as realistically as possible by taking into account the effect of the bottom of the hollow cathode in the discharge behavior. We investigate how the light emission intensities, the rates, densities, potential profiles and the electron energy distribution change with varying discharge conditions.

Section snippets

Geometry and discharge conditions

The discharge geometry consists of a cylindrical cathode closed at one end and a disc anode at the other end, separated by 0.2 cm (Fig. 1). In a classical HCD, the length of the cylinder is typically 5–10 times larger than the radius in order to reduce the loss of electrons through the open end of the cylinder. We assumed a length of 3 cm and a radius of 0.5 cm.

The discharge conditions assumed in the model were taken from the experiments, i.e. the gas pressure was varied from 40 to 133 Pa, the

Experimental set up

The cylindrical HCD with the same dimensions as in Fig. 1 was placed in a Pyrex envelope. Both anode and cathode were made of high purity copper, and high purity argon was used as the discharge gas. The discharge was operated in d.c. mode and digital multimeters measured the discharge current and the voltage. The background pressure of the vacuum system was approximately 1.33×10⁻⁴ Pa.

The spatial distribution of the light emission intensity in the discharge was scanned in the radial direction.

Numerical model

The hybrid model presented here is a combination of two models: a Monte Carlo and a fluid model [24], [25], [26], [27], [28], [29], [30], [31]. In the CDS, the condition for equilibrium approach is not fulfilled for electrons due to the high electric field and the large field variation over the electron mean free path. Indeed the energy gained by electrons from the field is not locally balanced by the energy lost through collisions; hence a flux of high-energy electrons will enter the NG,

Electrical conditions and emission intensities

The measured and calculated current–voltage dependence (V–I) in the pressure range from 40 to 133 Pa is presented in Fig. 2. For each pressure, approximately 20 V–I points were measured. The symbols denote the conditions for which the calculations were performed (typically at 5–7 V–I values per pressure). In the calculations, the total secondary electron emission coefficient, γ, was calculated in the Monte Carlo model based on the condition for the self-sustained discharge, under the assumption

Summary and conclusions

A cylindrical hollow cathode glow discharge was studied experimentally as well as by means of a hybrid model. The hybrid model was based on the coupling of a Monte Carlo model for fast electrons (which gives the source terms of the continuity equations and the γ coefficient) and a fluid model for the slow electrons and ions (in which these continuity equations are solved together with the Poisson equation in order to obtain a self-consistent electric field). The model, as well as the

Acknowledgements

N. Baguer is financially supported by a New Research Initiative of the University of Antwerp. A. Bogaerts is indebted to the Flemish Fund for Scientific Research (FWO) for financial support. This research is also sponsored by NATO's Scientific Affairs Division in the framework of the Science for Peace Programme, by the EC Thematic Network on Glow Discharge Spectroscopy for Spectrochemical Analysis (SMT4-CT98-7517) and by the Federal Services for Scientific, Cultural and Technical Affairs of the

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View full text

Hybrid model for a cylindrical hollow cathode glow discharge and comparison with experiments

Abstract

Introduction

Section snippets

Geometry and discharge conditions

Experimental set up

Numerical model

Electrical conditions and emission intensities

Summary and conclusions

Acknowledgements

Spectrochim. Acta Part B

Physica

Spect. Acta

Hollow cathode metal ion lasers

IEEE J. Quant. Electron.

Comparison of Cu-II 781 nm lasers using high voltage hollow cathode and hollow anode-cathode discharges

IEEE J. Quant. Electron.

The hollow cathode discharge as a spectrochemical emission source

Appl. Spectr. Rev.

The hollow cathode emission source: a survey of the past and a look into the future

Prog. Analyt. Atom. Spectosc.

Pilot study of the analytical performance of the pulsed HCD emission source

Appl. Spectr.

Spatial distribution of radiant intensity from primary sources for atomic absorption spectrometry. Part 1: HC Lamps

Appl. Spectr.

The use of a hollow cathode glow discharge as an atomic emission spectrometric element specific detector for chlorine and bromine in gas chromatography

J. Anal. At. Spectrom.

Bohrs Heliumlinien

Ann. d. Phys.

Die funkenspektren des Aluminiums Teil 2

Ann. d. Phys.

The hollow cathode effect and the theory of the glow discharge

Proc. R. Soc. Lond. A

Analytical model of the hollow cathode effect

Plasma Sources Sci. Technol.

On the hollow-cathode effect: conventional and modified geometry

Plasma Sources Sci. Technol.

Experimenteller Nachweiss des Pendel-effecktes in einer Zilindrichen niderdruck Hohlkathoden entlandung in Ar

Z. Naturf.

Glimmentladung an Hohlkathoden

Zeitschr. f. Phys.

Hollow cathode discharge for gas lasers

Z. Naturf.

Uber der fallraum in einer Zilindrichen Hohlkathode

Z. Naturf.

Rate of ionization and density of the electrons in a hollow cathode in He

J. Phys. D: Appl. Phys.

Multi-electron temperature models for HCD

Jpn. J. Appl. Phys.

Contributions studies on HCD

Phys. Scripta

Slow electron distribution function in a cylindrical HCD

Sov. Phys. Tech. Phys.

Spatial and energy distributions of fast electrons in discharges with a cylindrical hollow cathode

Sov. Phys. Tech. Phys.