Kinetic model for stochastic heating in the INCA discharge

Abstract

A novel electron heating mechanism based on periodically structured vortex fields induced in a plane was first proposed in 2014 [U. Czarnetzki and Kh. Tarnev, Physics of Plasmas 21, 123508 (2014)]. This theoretical concept has now been realized in an experiment which confirms efficient collisionless heating in such array structures [Ph. Ahr, T.V. Tsankov, J. Kuhfeld, U. Czarnetzki, submitted to Plasma Sources Science and Technology, arXiv:1806.02043v1 (2018)]. The new concept is called "Inductively Coupled Array": INCA. Here, the physical mechanism behind the collisionless (stochastic) heating is investigated by two analytical models. Firstly, the electron heating rate in an array field structure with an exponential spatial decay of the field in the direction perpendicular to the plane is investigated by stochastically averaging single electron trajectories. The approach is similar to the Lieberman model for the classical stochastic heating in standard inductively coupled plasmas. This analysis shows that classical stochastic heating by thermal motion along the vertical direction makes a negligible contribution. However, there is a strong collisonless non-local heating effect in the plane. In conclusion, heating is non-local in the plane but local in the vertical direction. This insight allows a straightforward solution of the collisionless Boltzmann equation which not only confirms the results of the Lieberman model but provides also explicit expressions for the complex conductivity. Based on the conductivity an effective stochastic collision frequency, the complex damping coefficient and the related field penetration of the field into the plasma is calculated. Finally, elastic collisions with neutral background atoms are included in the model and a condition for dominance of stochastic heating over Ohmic heating is derived.

Electric field measurements on the INCA discharge

“…The structure of E 2 is far from the one created by ideal circular coils, which was assumed in the theoretical investigations of the INCA discharge 8,11 . Instead the field is strongly asymmetric and shows kidney-like structures.…”

“…Because the differences are still small, the consistency of the diagnostic is confirmed by this comparison. The quantity that is more important for the operation of the INCA discharge is the square of the electric field 11 . Having confirmed, that the B-dot diagnostic is consistent, it provides ex situ (without plasma) access to the structure of the electric field.…”

“…It shows consistently a reliable and efficient operation at pressures as low as 0.1 Pa 9,10 . This work was accompanied by a more in-depth theoretical analysis of this electron heating mechanism 11 . It revealed, that the efficient operation is caused by the fact, that all electrons participate in the heating process, making it more efficient than the stochastic heating in standard ICP discharges.…”

“…It revealed, that the efficient operation is caused by the fact, that all electrons participate in the heating process, making it more efficient than the stochastic heating in standard ICP discharges. By including also the electron collisions with the background gas the theory reveals that for low pressures the collisionless non-local heating prevails and at higher pressures the standard local Ohmic heating dominates 11 . At the pressure for which the mean free path of the electrons λ is comparable to the characteristic length of the array Λ: λ ∼ 0.1Λ, the transition between the two regimes occurs 11 .…”

“…By including also the electron collisions with the background gas the theory reveals that for low pressures the collisionless non-local heating prevails and at higher pressures the standard local Ohmic heating dominates 11 . At the pressure for which the mean free path of the electrons λ is comparable to the characteristic length of the array Λ: λ ∼ 0.1Λ, the transition between the two regimes occurs 11 . Addition-ally to the electric field configuration already described, called the ortho-array, another field configuration was proposed and discussed in the theoretical works on the INCA discharge -the so called para-array, where the phase of the current changes by 180°between the neighboring coils 8 .…”