Current sheath formation in the plasma focus

Plasma Science and Applications (ICPSA 2013) International Journal of Modern Physics: Conference Series Vol. 32 (2014) 1460321 (8 pages) The Author DOI: 10.1142/S2010194514603214 Current sheath formation in the plasma focus Y. S. Seng*, P. Lee and R. S. Rawat Natural Sciences and Science Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616 *twinkerblast@yahoo.com.sg Published 13 August 2014 The shaping and formation of the current sheath takes place in the breakdown phase of a plasma focus device. Achieving a clear understanding of the current sheath formation process is important because the plasma focus device performance depends on the quality of this sheath. In this paper, we created and successfully run an electromagnetic particle in cell code to simulate the breakdown phase. Magnetic effects are self-consistently incorporated in this formalism, allowing us to carry the simulation all the way to the point prior to breakdown. Keywords: Electromagnetic PIC Simulation; Plasma Focus; breakdown phase. 1. Introduction The radiation yield performance of a plasma focus (PF) device is critically dependent on the quality of the current sheath (CS). For example, a homogeneous and axisymmetric CS is an essential condition for maximal plasma compression at the axis. A reduction of filaments and an improvement in plasma homogeneity resulted in an increase in the neutron yield where the energy is in the order of kilojoules. 1 After more than 40 years of PF research, the physics and CS formation in the breakdown phase are still not adequately understood. 2 The problem is too complex for theoretical analysis, and experimental diagnostics to investigate the CS formation are challenging. In the area of computational modeling, both particle in cell (PIC) and fluid models have been applied to the pre-breakdown phase recently. 3-6 The PIC method is more preferred for PF breakdown simulations for the following reasons. First, the electron energy density function is not Maxwellian during the ionization growth stage. 4 The second reason is that implementation of collisional dynamics, namely, ionization, elastic and inelastic scatterings, is more direct in the PIC framework. It is also more accurate because unlike fluid modeling which typically assumes a Maxwellian velocity distribution in the computation of the transport coefficients, no distribution function is required in PIC modeling. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. 1460321-1

Y. S. Seng, P. Lee and R. S. Rawat Electrostatic (ES) simulations, coupled with Monte Carlo particle collisions (MCC), have been performed over the years to model the initial stages of the breakdown process. For instance, Yordanov et al. 5 performed ESPIC simulations on the their PF device over a duration of 22 ns with different filling gas pressures over a range of filling gas pressures. A follow up study, which incorporated the photo-effect as an additional ionization mechanism to electron impact ionization, was performed at 400 Pa. 6 This additional ionization mechanism was found to sustain and support the discharge. By its very nature, ESPIC does not take into account of magnetic fields produced, for example, by space charge currents. Space charge effects are negligible in the early stage of ionization growth, but grow in importance during the course in the breakdown phase. For simulation over longer times, it is therefore important and desirable to incorporate magnetic fields self-consistently into the model. In the present paper, we report the successful implementation of electromagnetic (EM) PIC model coupled to the standard MCC framework on a PF configuration at a pressure of 400 Pa. In the EMPIC framework, the electromagnetic fields mutually interact with each other and the charged particles in a self-consistent manner. As such, one may carry out the breakdown phase modeling without worrying about the breakdown of the ES approximation anytime. Using our EMPIC model, the CS formation was traced in detail both in time and space. The CS development process progressed essentially through the following stages: volume ionization, localization, gliding discharge, followed by lift off. Spatial plots in the electron and ion densities characterize the CS, and a cathode sheath region, depleted of electrons, is readily identified near the vertical cathode. The point, at which the CS begins to lift off the insulator surface, is preceded by a sharp increase in plasma density. 2. Model and Methodology Figure 1 shows the PF simulation model adapted from reference 5 with the following dimensions: inner and outer radii of anode are respectively 45 and 50 mm, outer radius of cathode is 75 mm, length of insulator is 10 cm and length of anode section is 15 cm. The space in the device is filled with deuterium gas. Fig. 1. Geometry of the PF device. 1460321-2

Current sheath formation in the plasma focus The gas pressure was chosen to be 400 Pa. Ionization by electron impact is the chief mechanism for the production of electrons and ions in PF devices, but this by itself is insufficient to sustain the discharge at 400 Pa. Electrons were subsequently found to hit the walls and lost from the system at a faster rate than its production, eventually reducing its number to zero. This is consistent with the results initially obtained by Yordanov et.al 5. The inclusion of photoeffect 6,7 creates the important additional mechanism to sustain the electron avalanche. Photons arising from the ( Σ ) and ( Π ) transitions to the ground state ( Σ ) are able to produce photo-electrons with a certain probability (γ), known as the quantum efficiency, from the vertical cathode wall upon impact. In their photo-effect model, a photoelectron was instantly emitted, with probability γ, from the state transition taking place at any location in the simulation volume. This neglects the small, but finite, time taken for the photon to traverse its way to the cathode wall. At a distance, say, 5 cm from the cathode, a photon will take 0.17 ns to reach its destination. Our simulation time step is of the average order of 0.5 ps, which translates to an elapse of 340 time steps between photon and photoelectron emission! This finite traveling time may therefore be significant, and as such was being factored into our simulation. Following Ref. 7, γ = 0.01 was used. The current model was implemented in a 2D (radial and axial) geometry with three particle velocity components. Due to the coaxial symmetry, the only relevant EM fields are the radial and axial electric fields, and the azimuthal magnetic field. Following the standard EMPIC framework, the EM fields were staggered in space and time. At any new time step, their values were updated by the two Maxwell curl equations. These values were linearly interpolated to the particles, which were distributed over the discharge volume. The Lorentz force on each particle was updated, leading to new velocities and positions. At the new position, a charge particle collided with the background neutral gas molecules according to the Monte Carlo Collisional (MCC) framework, resulting in either elastic, excitation, or in the case of an electron, ionizing collision. The list of collisions was taken from Ref. 5, and the corresponding reaction cross sections gathered from various relevant databases. For the boundary conditions, both the anode and cathode surfaces were assumed to be perfectly conducting. A perfectly matched layer was extended at the end of the simulation length to minimize artificial EM reflection. The PF was excited by a voltage source placed across AB (see Fig. 1). This voltage source took on a linear increasing voltage profile, with a rate of 2 10 11 V/s. This profile is a reasonably good approximation till the point of dielectric breakdown, beyond which the voltage takes a dip in value. 7 Particles hitting the free space and material boundaries were absorbed. Secondary electron emission (SEE) by energetic ions hitting the vertical cathode was also taken into account. Finally, in order to simulate the discharge whose plasma density grew by more than ten orders in magnitude over time, a particle merging scheme which conserves charge, momentum and energy before and after merger, was implemented. This ensures a modeling that is manageable both in time and data storage by enforcing a cap on the total number of simulation particles. 1460321-3

Y. S. Seng, P. Lee and R. S. Rawat 3. Results and Discussion The simulation was run over a period of about 110 ns, to the point of CS lift off, beyond which the linear increasing voltage profile is no longer valid. Figure 2 shows the time profile of the charge particle densities in the discharge volume. This density corresponds to the global average, computed by dividing the total amount of electrons/ions over the entire discharge volume. Fig. 2. Time development of the electron (in red) and ion (in green) densities. The density time plot of Fig. 2 provides useful insights into the ionization development of a PF device during the breakdown phase. Beginning with some seed electrons, impact ionization became prominent when the voltage went beyond 1 kv, leading to an avalanche growth of the charge particles. Initially, the rate of ionization exceeded the rate of charges lost to the walls. The growth in the plasma density continued at an exponential rate until about 30 ns, after which the growth rate reached a "plateau" with a density of about 10 11 cm -3. From 30 ns to 103 ns, the plasma density experienced a very slow, non-exponential, growth. At 103 ns, a second exponential growth in density took place. This lasted only for a short while, about 4 ns, after which the growth halted. Within this short 4 ns, however, the density increased by one order of magnitude. In order to get insights into the CS development, we turn to a series of electron plots progressing in time as shown in Fig. 3. At the very beginning, electrons were scattered throughout the volume. Each electron became a seed for a second electron created by impact ionization with a neutral deuterium particle. Consequently, the whole discharge volume became filled with electrons (and ions) within a short time, as shown in Fig. 3a. This volume ionization stage did not last long, as electrons, being negative, were attracted towards the positive anode. Also, because of the PF geometry, the electric field was highest in the region near the vertical cathode just above the insulator surface. As a result, this region experienced a higher degree of ionization, which was further supplemented by the photo-effect. The net result was a high concentration of electrons near the cathode, while the rest gradually got annihilated as they accelerated towards and hit the anode and 1460321-4

Current sheath formation in the plasma focus insulator. This localization stage of CS development lasted until about 29.1 ns (see Fig. 3b), and coincided with the end of the exponential avalanche growth observed in the density time plot. Starting from 30 ns, the ionization wave began to glide across the insulator, creating new electrons and ions along the way. This resulted in a smaller, non-exponential increase in the plasma density indicated by the "plateau" of Fig. 1. This gliding discharge phase continued for 72 ns and stopped when the tip of the ionization wave reached the end of the insulator and touched the anode. At end of 103 ns, the CS had filled up the entire insulator length (Fig. 3e). (a) (c) (e) (b) 12.06 ns 29.10 ns 48.71 ns (d) 83.62 ns 103.42 ns (f) 107.57 ns Fig. 3. Snap shots of electrons in the discharge volume at various times. 1460321-5

Y. S. Seng, P. Lee and R. S. Rawat At this point in time, the CS would be stationary for the next 4 ns, but its density began to grow in value. This is clearly seen from Fig. 1. An explanation for this exponential growth will be given later after we analyze the spatial structure of the CS. Continuing the time domain analysis, the increased ionization lasted until around 107 ns, at which point the section of CS near the cathode started to lift itself off the insulator surface (Fig. 3f). The lifting off of the CS signals the dielectric breakdown of the PF, which causes the voltage to drop. It is therefore physically meaningless to carry our simulation beyond this point, since the linearly increasing voltage condition is no longer valid. By 103 ns, the entire length of the insulator was essentially filled with the CS layer. This CS layer is thin, of the order of 1 mm. It is also relatively homogeneous throughout most of its entire length. Figure 4 shows the axial distribution of the local electron and ion densities in red and green respectively. Figure 4a corresponds to the time when the gliding discharge has reached the end of the insulator while Fig. 4b refers to the CS state prior to lift off, during the short period of enhanced ionization. (a) (b) 103.42 ns 107.57 ns Fig. 4. Spatially resolved number density of electrons in (red) and ions (in green). 1460321-6

Current sheath formation in the plasma focus At the end of the gliding discharge, the ionization wave created a layer of plasma which is relatively uniform with a plasma density of around 10 13 cm -3 between 4 to 10 cm of the insulator length. There is a visible drop in its density below the 4 cm mark, whose value declines gradually towards the cathode end. Below 8 mm, virtually no electrons are present. This is the cathode sheath region, whose charges are essentially made up of ions (see Fig. 5). Fig. 5. The distribution of ions in the discharge volume at 102.22 ns, when the gliding discharge reached the insulator end. The (cathode sheath) region below the 8 mm mark is depleted of electrons (see Fig. 2e in contrast). Figure 4b corresponds to the period of enhanced ionization in the post-gliding discharge phase, prior to lift off. It is readily seen that the increase in plasma density occurred near the anode-insulator interface and extended back to the 6 cm mark. The plasma density is uniform within this region. This increase in ionization may be reasonably attributed to the increased in EM fields near the anode-insulator end. When the ionization wave reached the anode, the CS (consisting of both electrons and ions) would have shorted the electrical connection between the anode and the cathode, creating as a result a local surge in the current and consequently the EM field. Last but not least, Fig. 4 shows that the CS plasma is quasi-neutral, with equal amount of ions and electrons on average. 4. Conclusion An EMPIC coupled with MCC was successfully implemented to simulate the breakdown phase of a PF device. The various stages of CS formation culminating to the initial lifting off from the insulator were identified, namely volume ionization, localization of the electron plasma, sliding plasma discharge and finally lift off when the anode cathode gap is bridged by the current sheath. The spatial structure of the CS was resolved, and a distinct cathode sheath region that was depleted of electrons was found near the vertical wall. 1460321-7

Y. S. Seng, P. Lee and R. S. Rawat Acknowledgment This study was supported by AcRF Tier1 research grant No. RP1/11RSR provided by Nanyang Technological University, Singapore. References 1. A.H. Krompholz, W. Neff, F.Ryhl, K. Schonbach & G. H Herziger, Phys. Lett, 77A, 246 (1980). 2. M. J. Sadowski & M. Scholz, Plasma Sources Sci. Technol. 17, 024001 (2008). 3. M. Scholz & I. M. Ivanova-Stanik, Vacuum, 58, 287 (2000). 4. Y. Yordanov, D. Genov, I. Ivanova-Stanik & A. Blagoev, Vacuum 76, 365 (2004). 5. Y. Yordanov, I. Ivanova-Stanik & A. Blagoev, J. Phys. Conf. Ser. 44, 215 (2006). 6. V. Yordanov, I. Ivanova-Stanik & A. Blagoev, J. Phys. D 40, 2522 (2007). 7. V. Yordanov, A. Blagoev, I. Ivanova-Stanik, E. M. van Veldhuizen, S. Nijdam, J. van Dijk & J. A. M van der Mullen, J. Phys. D 41, 215208 (2008). 1460321-8