Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

Shizgal, Bernie D.

doi:10.1103/physreva.106.022805

Cited by 3 publications

(8 citation statements)

References 49 publications

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“…where k B is the Boltzmann constant, T b is the temperature of the background gas and m e is the electron mass. The dependence of the electron distribution function on reduced speed and time, f (x, t), is given by the Fokker-Planck equation [4,12,18,31],…”

Section: Time Dependent Fokker-planck Equationmentioning

confidence: 99%

“…The current paper considers the relaxation of an ensemble of electrons dilutely dispersed in a background of an inert gas, for either Argon, Krypton or Xenon for which the electron momentum transfer cross sections have a deep Ramsauer-Townsend minima. It has been demonstrated that the steady state electron distribution functions in these moderators are bimodal over a small range of external electric field strengths [1][2][3][4]. In this paper, we report on the time evolution of the electron distribution functions as determined from the numerical solution of the appropriate Fokker-Planck equation.…”

Section: Introductionmentioning

confidence: 97%

“…The Kappa distribution, analogous to the bimodal distributions of interest here, arises from the nature of the steady solutions of a particular Fokker-Planck equation given by a Pearson ordinary differential equation [4,33,34]. These nonequilibrium distributions are the result of the dynamical aspects of the Fokker-Planck equation.…”

Section: Introductionmentioning

confidence: 98%

“…The theoretical treatment is based on previous works [2,12,18,31]. For a small range of electric field strengths, the steady nonequilibrium electron energy distribution functions develop bimodal structures [1][2][3][4]. This phenomenon occurs for the heavier inert gases, Ar, Kr and Xe with deep Ramsauer-Townsend minima in the momentum transfer cross sections.…”

Section: Introductionmentioning

confidence: 99%

“…A finite difference method of solution of the time dependent Fokker-Planck equation is employed [32] with concern to the time dependence of the electron distributions in the external electric field. The steady state distributions can be bimodal as dependent on the electric field strength and there exists a direct relationship between the maxima in the distribution functions with the minima in the Ramsauer-Townsend momentum transfer cross sections [4].…”

Section: Introductionmentioning

confidence: 99%

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Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Yin¹,

Shizgal

2023

Phys. Scr.

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The current paper considers the thermalization of an ensemble of electrons under the inﬂuence of an external electric ﬁeld and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric ﬁeld strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature Tb has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form e−λnt where λn are the eigenvalues of the Fokker-Planck operator. Alternatively, a ﬁnite diﬀerence algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ0/xp, where x =with p = 2 is also employed to demonstrate a steady-state Kappa distribution which featuresqmev2/2kBTb is the reduced speed. This model prominently in space physics and other ﬁelds. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary diﬀerential equation.

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Section: Time Dependent Fokker-planck Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Yin¹,

Shizgal

2023

Phys. Scr.

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Historical development of electron swarm physics based on the Boltzmann equation towards in-depth understanding of a low-temperature collisional plasma

Makabe,

Sugawara

2024

Plasma Sources Sci. Technol.

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Theoretical study of the electron kinetics (i.e. the velocity distribution and the transport parameter) in gases is generally conducted using the electron Boltzmann equation. The year 2022 marked 150 years since the formulation of the Boltzmann equation. Even in the last several decades, the historical progress has been made synchronously with the development of innovative technologies in gaseous electronics and in combination with the appearance of computers with sufficient speed and memory. Electron kinetic theory based on the Boltzmann equation has mostly been developed as the swarm physics in the hydrodynamic regime in the dc and radio frequency electric fields. In particular, the temporal characteristics are understood in terms of the collisional relaxation times between electron and gas molecule. There are two main theoretical approaches based on the Boltzmann equation for finding the velocity distribution. One is the traditional description of the electron kinetics, starting from the Boltzmann statistics in velocity space under a uniform density or a small density gradient of electrons. The other most recent approach is based on the phase-space tracking of the velocity distribution where the electron transport parameter is given by the moment of the electron density distribution in position space. In the present paper, we will explore the historical development of the electron Boltzmann equation with respect to three key items: collision term, solution method, and intrinsic electron transport in a hydrodynamic regime involved as the key elements in the low-temperature collisional plasma. The important topics listed in a table are briefly noted and discussed.

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Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

Oylukan,

Shizgal

2023

Phys. Rev. E

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Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

Cited by 3 publications

References 49 publications

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Historical development of electron swarm physics based on the Boltzmann equation towards in-depth understanding of a low-temperature collisional plasma

Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

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