MONTE CARLO SIMULATION OF IONIZATION PROCESSES FOR DISCHARGES IN THE LEFT BRANCH OF PASCHEN’S CURVE
A.A. GRISHKOV, V.A. SHKLYAEV, Y.D. KOROLEV
 Institute of High Current Electronics SB RAS, Tomsk, Russia

The paper deals with the development of simulation methods as applied to low-pressure discharges with extremely high reduced electric fields E/p for which it is problematic to correctly describe the ionization processes in classical electron avalanches. In some conditions, the electron free path for ionization can compare with the interelectrode gap width such that the notion of electron avalanches becomes inapplicable at all. Typical examples are discharges in so-called pseudo-spark switches and in electron and ion plasma sources [1].
One of the computer codes that could be applied to such discharges is PIC/MC XOOPIC [2, 3]. This code allows one to compute the spatial and temporal electric field distribution consistently with the time variation of the excess space charge of electrons and ions. The block that takes into account the ionization processes is based on the Monte Carlo method. One of the advantages is that the code is open and we can modify the initial data on cross sections and the calculation algorithm to solve a particular problem [4].
Here, using the above code, we calculate the impact ionization coefficient a, the electron drift velocity, and the electron diffusion coefficient D in electron avalanches in nitrogen for E/p ≤ 1000 V/(cm∙Torr) The calculation of a is exemplified below.

Fig.1. Coefficient a from simulation (nitrogen, 20 Torr) and experiments [5].
Formally speaking, we have a reasonable agreement with the experimental data even for extremely high reduced electric fields. However, in terms of electron avalanches at E/p ≤ 1000 V/cm, the results give an incredibly high electron diffusion coefficient and an electron temperature kTe ≈ 30 eV, which is higher than the ionization potential [4]. The spatial electron distribution in the gap, obtained by Monte Carlo simulation, shows that at moderate E/p values, a spherical electron cloud is located inside the gap. As for high E/p, we can hardly speak of any electron avalanche in its generally accepted sense due to the lack of space and time for electron cloud formation in the gap. In the paper, we analyze in detail the physical cause for this effect and the features of ionization at high E/p.
References
[1]	Y.D. Korolev and N.N. Koval, "Low-pressure discharges with hollow cathode and hollow anode and their applications," J. Phys. D: Appl. Phys., vol.51, Article Number 323001, 2018.
[2]	V.A. Shklyaev, E.Kh. Baksht, S.Ya. Belomyttsev et al., "On the dynamics of a subnanosecond breakdown in nitrogen below atmospheric pressures," J. Appl. Phys., vol.118, Article Number 213301, 2015.
[3]	J.P. Verboncoeur, A.B. Langdon, N.T. Gladd, "An object-oriented electromagnetic PIC code," Comput. Phys. Commun., vol.87, Article Number 199, 1995.
[4]	Yu.D. Korolev and G.A. Mesyats, Physics of Pulsed Breakdown in Gases, Yekaterinburg, UrB of RAS, 1991.
[5]	L.B. Loeb, Fundamental Processes of Electrical Discharges In Gases, New York, 1950.