Collisionless electron heating in periodic arrays of inductively coupled plasmas

Czarnetzki, Uwe; Tarnev, Kh.

doi:10.1063/1.4903880

Cited by 6 publications

(21 citation statements)

References 39 publications

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“…The first term is the mean energy of a free oscillating electron. The subsequent dimensionless factor has a similar behavior as the G function but due to the additional factor  it peaks already at about 2…”

Section: Stochastic Heating In the Ortho Arraymentioning

confidence: 85%

“…Naturally, diagonal contributions are missing and the axial terms have only a multiplicity of 2 instead of 4 since here only one axis contributes. The overall factor in the heating power (equation (33)) is then 2 1/ 2 1/ 2 2 1    , i.e. it remains unchanged, but the second exponential term in   V ortho G (equation (34)) has to be dropped.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The electric field of the para array is also listed in Appendix A. One of the main consequences is that now the fields of Kinetic Model for INCA, Uwe Czarnetzki Thursday, 07 June 2018 -11 -adjacent coils interfere constructively and not destructively, as in the ortho array, and the overall field structure is slightly altered with fewer voids [2]. As a quantitative measure for comparison one can take the spatial mean square field which is Inspecting the explicit form of the electric field structure (Appendix A) reveals immediately that the general problem is identical to the ortho array but without the parallel resonances, i.e.…”

Section: Stochastic Heating In the Para Arraymentioning

confidence: 99%

“…The linearized equation then reads: The fundamental electrical fields of the ortho and the para array in the x,y-plane without any z-dependence were first introduced in Ref. [2] and are repeated here for convenience, although with a slightly different notation. The term "fundamental" means that in the real field in the experiment also higher spatial modes are present but with much lower amplitudes.…”

Section: Accounting For Elastic Collisionsmentioning

confidence: 99%

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Kinetic model for stochastic heating in the INCA discharge

Czarnetzki¹

2018

Plasma Sources Sci. Technol.

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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.

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Section: Stochastic Heating In the Ortho Arraymentioning

confidence: 85%

Section: Summary and Discussionmentioning

confidence: 99%

Section: Stochastic Heating In the Para Arraymentioning

confidence: 99%

Section: Accounting For Elastic Collisionsmentioning

confidence: 99%

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Kinetic model for stochastic heating in the INCA discharge

Czarnetzki¹

2018

Plasma Sources Sci. Technol.

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“…Recently another possibility was investigated. It is based on a theoretical concept for collisionless resonant electron heating proposed by Czarnetzki and Tarnev 8 . Under collisionless conditions (low pressure) an infinite array of electric RF vortex fields each a distance L apart and oscillating at the frequency ω 0 can couple energy efficiently to the electrons.…”

Section: Introductionmentioning

confidence: 99%

Electric field measurements on the INCA discharge

Stetzkamp,

Tsankov,

Czarnetzki

2021

Preprint

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Developing large-area inductively coupled plasma sources requires deviation from the standard coil concepts and the development of advanced antenna designs. First steps in this direction employ periodic array structures. A recent example is the Inductively Coupled Array (INCA) discharge, where use is made of the collisionless electron heating in the electric field of a periodic array of vortex field. Naturally, the efficiency of such discharges depends on the how well the experimental array realizes the theoretically prescribed field design. In this work two diagnostic methods are employed to measure the field structure of the INCA discharge. Ex situ B-dot probe measurements are compared to in situ radio-frequency modulation spectroscopy (RFMOS) and good agreement between their results is observed. Both diagnostics show systematic deviations of the experimentally generated field structure from the one employed in the theoretical description of the INCA discharge. The subtleties in applying both diagnostic methods together with an analysis of the possible consequences of the non-ideal electric field configuration and ways to improve it are discussed.

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Describing local and non-local electron heating by the Fokker–Planck equation

Czarnetzki

Alves

2022

Rev. Mod. Plasma Phys.

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The common description of kinetic effects in low-pressure plasmas is based on the Boltzmann equation. This applies especially to the description of Ohmic (collisional) and non-local (stochastic/collisionless) electron heating, where the Boltzmann equation is the starting point for the derivation of the corresponding heating operator. Here, it is shown, that an alternative and fully equivalent approach for describing the interaction between electrons and electric fields can be based on the Fokker–Planck equation in combination with the corresponding Langevin equation. Although, ultimately the final expressions are the same in both cases, the procedures are entirely different. While the Fokker–Planck/Langevin approach provides physical insights in a very natural way, the linearized Boltzmann equation allows straightforward calculation but requires some effort to interpret the mathematical structure in terms of physics. The Fokker–Planck equation for the present problem is derived, with particular emphasis on the consistent treatment of velocity-dependent elastic collision frequencies. The concept is tested for a simple case by comparing it with results from an ergodic Monte-Carlo simulation. Finally, the concept is applied to the problem of combined Ohmic and stochastic heating in inductively coupled plasmas. The heating operator is first analyzed for an exponential model field profile. Self-consistent field profiles are determined subsequently. In this context, a generalization of the plasma dispersion function is introduced, which allows for arbitrary forms of the distribution function and velocity dependence of the elastic collision frequency. Combined with the Fokker–Planck heating operator, a fully self-consistent description of the plasma and the fields is realized. Finally, a concept for integrating the operator in a standard local Boltzmann solver and using the local solver for determination of the global electron velocity distribution function in a low-pressure plasma is provided.

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Collisionless electron heating in periodic arrays of inductively coupled plasmas

Cited by 6 publications

References 39 publications

Kinetic model for stochastic heating in the INCA discharge

Kinetic model for stochastic heating in the INCA discharge

Electric field measurements on the INCA discharge

Describing local and non-local electron heating by the Fokker–Planck equation

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