1. Large-amplitude oscillations in a Townsend discharge in low- current limit Vladimir Khudik, Alex Shvydky (Plasma Dynamics Corp., MI) Abstract We have developed a regular analytical approach to study oscillations in a Townsend discharge when the distortion of the electric field in the discharge gap due to the spatial charge is small. In presented theory the secondary electron emission coefficient can take any value between zero and one. We have found that the large-amplitude oscillations of the particle current in the discharge gap are accompanied by small-amplitude oscillations of the gap voltage. Surprisingly, for certain impedances of the external electrical circuit, this highly dissipative system is governed by the Hamiltonian equations (so that the amplitude of the oscillations slowly changes in time). Direct Monte Carlo/particle-in-cell simulations confirm the theoretical results.

2. Standard Circuit This circuit is commonly use for experimental and theoretical studies of oscillations and stability of Townsend discharge Usually, the value of the resistor R is such that the major portion of the external voltage drops across the discharge gap ( ). Our consideration also includes the case A.V.Phelps et al., 1993 V.N. Melekhin and N.Yu. Naumov, 1985 External applied voltage V 0 =const ~

3. Barrier Discharge Circuit V t Townsend discharge in barrier discharge geometry is used for addressing micro-discharge cells in plasma televisions L.F. Weber, 1998 V.P. Nagorny et al., 2000 To realize a dc Townsend discharge in this circuit, external applied voltage must change linearly with time

4. Small parameter Basic Equations Perturbation Procedure for oscillations of small amplitude and frequency 0 <  <  e -1 Complementing circuit equation Boundary conditions Electron transit time across the gap First, we derive the equation for perturbation of ion density and solve it by method of successive approximations Then, we immediately obtain the expression for the discharge impedance

5. where is proportional to perturbation of the number of ions in the gap at frequency  coefficients a, b, and c depend on the parameters of steady-state discharge and the shape of ion density perturbation Discharge Impedance The dispersion equation for natural oscillations in the circuit with impedance Z circ : For the standard circuitFor the barrier discharge circuit

6. Spectrum of oscillations In the zeroth approximation in parameter  ( ) there exist two different types of oscillations Oscillations in the system “external circuit + capacitance of empty discharge gap” Oscillations of the discharge current described by the dispersion equation - High-frequency harmonics ( ) quickly damp in time ( ) without perturbation of the total charge in the gap - Low-frequency harmonic oscillates slowly with the frequency Perturbation of the ion density has the same shape in the gap as the stationary ion density does ~

7. (case ) Low frequency oscillations in standard circuit oscillation region Stability triangle in the plane of circuit parameters and

8. Nonlinear Low-Frequency Oscillations Remarkable feature of the system is that large pulses of the particle current cause small oscillations of the electric field Spatial distortion of the electric field due to the volume charge is quite small ~ Ion current in large amplitude oscillations is distributed in the gap almost the same way as the ion current in steady-state discharge ~ We consider the case when R diff is negative and does not depend on discharge current

9. Blue line (P =  separate the regions of stability and instability: oscillations with relatively small amplitude are stable and oscillations of sufficiently large amplitude are unstable. Equation of separatrix in dimensionful parameters: Hamiltonian function on the hypotenuse of the stability triangle

10. Phase curves in general case Phase curve corresponding to the circuit parameters inside the stability triangle Phase curve corresponding to the circuit parameters outside the stability triangle More accurate analysis shows that in the case of circuit parameters outside the stability region close to hypotenuse, phase curves asymptotically approach limiting cycle (and do not “infinitely” depart from the steady state) Size of the limiting cycles is a sharp function of the distance to the hypotenuse.

11. Monte-Carlo/PIC simulations Voltage across the discharge gap is always close to the breakdown voltage so that for not too small secondary electron emission coefficients, the spatial distribution of ions created by the electron avalanche is different from exponential one Secondary electron emission coefficient depends on the properties of the cathode surface, gas pressure, and the magnitude of the electric field. To avoid these complications, we assumed that all electrons emitted from the cathode have zero energy, so that always Simulation parameters: Ne gas with pressure = 500 Torr, gap length = 400  m, V br = 200 V, number of ions used in simulations ~ 10 5

12. Numerical Experiment Standard circuit with R=  C sh =0 (constant current source, j=8  A/cm 2 ). The parameter  = 0.02. Voltage across the discharge gap vs. time; red lines correspond to damping rate predicted by the theory Particle current vs. time; red line is analytical solution (when damping is neglected) Period of nonlinear oscillationsMinimum value of particle current

13. In general, two types of oscillations can be distinguished in Townsend discharge: Summary Short-living high-frequency oscillations If initial total charge in the gap Q initial = Q stationary, oscillations damp in time ~  i -1 Long-living low-frequency oscillations Arbitrary distribution of ion density in the gap in short time ~  i -1 takes the universal shape - the shape of ion density distribution in the steady- state discharge. This fact allows one to eliminate the dependence of ion current and electric field on spatial variable and obtain the equations for their amplitudes. For certain parameters of the circuit the small-amplitude oscillations are self- sustained. For these parameters, the system is almost Hamiltonian, so that the amplitude of nonlinear oscillations changes in time very slowly only due to small non-Hamiltonian terms. All analytical results are obtained for arbitrary secondary emission coefficient Townsend discharge oscillations can be unstable (at large R) even when