On electron bunching and stratification of glow discharges

On electron bunching and stratification of glow dischargesYuri B. Golubovskii, Vladimir I. Kolobov, and Vladimir O. Nekuchaev Citation: Physics of Plasmas (1994-present) 20, 101602 (2013); doi: 10.1063/1.4822921 View online: http://dx.doi.org/10.1063/1.4822921 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/20/10?ver=pdfcov Published by the AIP Publishing

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On electron bunching and stratification of glow discharges

Yuri B. Golubovskii,1 Vladimir I. Kolobov,2 and Vladimir O. Nekuchaev3

1St. Petersburg State University, St. Petersburg 198504, Russia2CFD Research Corporation, Huntsville, Alabama 35805, USA3Ukhta State Technical University, Ukhta 169300, Russia

(Received 26 February 2013; accepted 13 May 2013; published online 10 October 2013)

Plasma stratification and excitation of ionization waves is one of the fundamental problems in gas

discharge physics. Significant progress in this field is associated with the name of Lev Tsendin. He

advocated the need for the kinetic approach to this problem contrary to the traditional

hydrodynamic approach, introduced the idea of electron bunching in spatially periodic electric

fields, and developed a theory of kinetic resonances for analysis of moving striations in rare gases.

The present paper shows how Tsendin’s ideas have been further developed and applied for

understanding the nature of the well-known S-, P-, and R-striations observed in glow discharges of

inert gases at low pressures and currents. We review numerical solutions of a Fokker-Planck

kinetic equation in spatially periodic electric fields under the effects of elastic and inelastic

collisions of electrons with atoms. We illustrate the formation of kinetic resonances at specific field

periods for different shapes of injected Electron Distribution Functions (EDF). Computer

simulations illustrate how self-organization of the EDFs occurs under nonlocal conditions and how

Gaussian-like peaks moving along resonance trajectories are formed in a certain range of discharge

conditions. The calculated EDFs agree well with the experimentally measured EDFs for the S, P,

and R striations in noble gases. We discuss how kinetic resonances affect dispersion characteristics

of moving striations and mention some non-linear effects associated with glow discharge

stratification. We propose further studies of stratification phenomena combining physical kinetics

and non-linear physics. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4822921]

I. INTRODUCTION

Stratification of DC glow discharges and the formation

of ionization waves (striations) were described in several

reviews and books.1 In noble gases, different types of stria-

tions have been observed depending on the gas pressure and

discharge current. Today, it is well-known that kinetic analy-

sis of electrons is required for most types of striations.2 The

non-hydrodynamic behavior of electrons manifests itself

most strikingly as kinetic resonances of the electron distribu-

tion function (EDF) in spatially periodic electric fields. This

phenomenon was first observed by Ruzicka in the numerical

solution of the Boltzmann equation3 and later confirmed by

theoretical analysis4,5 and by experimental measurements of

the EDF. Kinetic resonances explain the existence of three

varieties of striations (called S, P, and R-striations) in noble

gases at low gas pressures with fixed potential drops over

striation length.6,7

Specifics of noble gases consist of a high value of the

first electronic excitation potential of atoms, e1, and a rela-

tively small gap between the ionization and excitation poten-

tials, ei � e1 � e1. The first excitation potential separates the

elastic and inelastic energy ranges with completely different

mechanisms of electron energy relaxation. The energy relax-

ation of electrons in the elastic energy range (w < e1) is

solely due to small energy loss in quasi-elastic collisions

with atoms, and the corresponding electron energy relaxation

length kT ¼ k=ffiffiffidp

is 2–3 orders of magnitude larger than

the electron mean free path k (due to the small ratio of

electron and atom mass d ¼ m=M). In inelastic energy range

ðw > e1Þ, electrons lose large energy quanta for excitation

and ionization of atoms. The frequency of inelastic collisions

�� is smaller than the elastic collision frequency, �, so that

electron transport is diffusion-dominated, and the EDF is

near isotropic at all energies of interest. Finally, the length

ke ¼ e1=ðeEÞ provides an intrinsic spatial scale. Over this

length, electrons gain kinetic energy equal to the first excita-

tion threshold in the electric field E.

In 1982, Tsendin published a seminal work,8 which pro-

posed an analytical model for kinetic resonances responsible

for plasma stratification in inert gases at low pressures and

currents, at k� � ke � kT (where k�¼kffiffiffiffiffiffiffiffiffiffi��=�

p). Exposition

of this theory can be found in reviews9,10 and the book.11 The

theory was based on the analytical solution of a non-local ki-

netic equation for electrons in spatially periodic electric fields.

Tsendin had demonstrated the existence of resonance condi-

tions for specific spatial periods of the field and the appear-

ance of resonant trajectories to which the electrons are

“attracted.” The resulting EDF was obtained in the form of

narrow Gaussian peaks moving along resonant trajectories.

Figure 1 illustrates qualitatively the mechanism of elec-

tron bunching in the plane of total energy e ¼ wþ euðxÞ and

space coordinate. Electrons diffuse between the curve x1ðeÞon which their kinetic energy w is equal to zero and the curve

x2ðeÞ where their kinetic energy is equal to the first excitation

threshold of atoms. Inelastic collisions taking place to the

right of x2ðeÞ produce jumps along the vertical axis with an

energy quantum comparable with a step e1. The conditions

shown in Figure 1 assume that k��ke � kT , so that electron

energy loss in quasi-elastic collisions with atoms is small

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PHYSICS OF PLASMAS 20, 101602 (2013)


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compared to e1 over the field period, and inelastic collisions

occur near the curve x2ðeÞ because of k� � ke. It is also

assumed that the rate of energy loss in quasi-elastic colli-

sions increases with electron energy, so that electron trajec-

tories in the elastic energy range are close to horizontal lines

and slowly converge towards a resonant trajectory.

There are three completely different mechanisms of dis-

charge stratification in noble gases dominating at different

currents and pressures. The first one is kinetic, related to the

non-local nature of the EDF at low currents and low gas

pressures. The second one is hydrodynamic and is due to an

ambipolar electric field, electron diffusion, and thermodiffu-

sion at low pressures and high currents. The third one is due

to radial plasma constriction and the formation of two-

dimensional waves at high gas pressures. The kinetic mecha-

nism can be further classified into two categories: the first

one is associated with electron bunching and kinetic resonan-

ces at low gas pressures. The second one is due to non-

locality of the EDF “tail” ðTe < w < e1Þ observed at elevated

pressures (at kT < ke) when the electron energy balance is

controlled by quasi-elastic collisions.

This paper is devoted to electron bunching and kinetics

of stratification. We show how Tsendin’s ideas have been fur-

ther developed and applied for understanding the nature of S-,

P-, and R-striations in DC glow discharges of inert gases at

low pressures and currents. Section II introduces basic equa-

tions used for analysis of electron kinetics. Section III reviews

numerical methods for solving a Fokker-Plank equation for

EDF. Numerical simulations are used to illustrate dynamics

of the EDF bunching process and effects of injected EDF on

the formation of resonant trajectories. The calculated EDFs

are compared with experimentally measured EDFs for the S-,

P-, and R-waves. Section IV discusses the effects of kinetic

resonances on stratification and recent efforts towards self-

consistent simulations of striations. Some non-linear effects

are discussed in Sec. V.

II. THEORY OF ELECTRON BUNCHING

The frequency of moving striations is smaller than the

characteristic frequency of the EDF formation. So, in the

original Tsendin’s paper,8 a steady-state kinetic equation for

the isotropic part of the EDF f0ðe; xÞ was used in the varia-

bles total energy–coordinate

@

@x

v3

3�ðvÞ@f0ðe; xÞ@x

!þ @

@em2

Mv3�ðvÞf0ðe; xÞ

� �¼ 0; (1)

with a “black-wall” approximation at the first excitation

threshold

f0ðe; xÞjw¼e1¼ 0: (2)

When the second term in Eq. (1) is small, this equation has

an analytical solution

f0ðe; xÞ ¼ UðeÞðx2ðeÞ

x

�ðx0Þdx0

v3ðx0Þ : (3)

The energy-dependent electron flux, UðeÞ, was found by an

expansion in the small parameter

Kel ¼e1m�2ðe1Þ1

6

M

mðeEÞ2

� LS

kT

� �2

;

where LS is an approximate length of strata and E is an aver-

age electric field. The parameter Kel corresponds to an

energy spread of a mono-energetic electron flux over the

length of the strata. If we completely neglect the energy loss

in quasi-elastic collisions (put Kel ¼ 0), then any injected

EDF will be reproduced with a period ke at an arbitrarily

large distance from the injection point. Due to small energy

loss in quasi-elastic collisions, electron trajectories converge

to resonance trajectories in spatially periodic resonance

electric fields. Accounting for the small energy loss in elas-

tic collisions leads to bunching of electrons and establishing

a “steady-state” periodic solution after a certain number of

field periods. The number of these periods is proportional to

(K�2el ). Tsendin has shown that resonance conditions corre-

spond to wavelengths LS=n, where n is an integer. For reso-

nance conditions, the established differential flux UðeÞ has

the form of narrow peaks concentrated near resonance tra-

jectories. Tsendin called this phenomenon “electron

bunching” and calculated the shape of the flux UðeÞ under

resonance conditions.

Other factors can lead to bunching of electrons in the

elastic energy range. For example, electron heating in colli-

sions with atoms results in an additional term describing dif-

fusion in energy. However, this diffusion effect is of higher

order of smallness (�KelkTe1

, where T is the temperature of

FIG. 1. Bunching of electrons consists of a convergence of non-resonant tra-

jectories II and III to a resonant trajectory I in a spatially periodic resonant

electric field. The corresponding EDFs versus kinetic energy are shown at

two spatial points. Reprinted with permission from Kudryavtsev et al.,Physics of Glow Discharges [in Russian] (2010). Copyright 2010 Authors,

LAN, http://www.lanbook.com.

101602-2 Golubovskii, Kolobov, and Nekuchaev Phys. Plasmas 20, 101602 (2013)


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http://www.lanbook.com

gas atoms), so it is unlikely to play an important role.

Coulomb interactions among electrons result in advection

and diffusion in energy, which affect bunching and gradually

diminish the resonances. Excitations of different levels of

atoms by electron impact provide an important alternative

mechanism of electron bunching.

An analytical model of electron bunching due to excita-

tion of different atomic levels was developed in Ref. 12. The

kinetic equation was used in the form

@

@x

v3


!þ @

@em2


� �

¼X

k

v��kðvÞf0ðe; xÞ�X

k

v0k��kðv0kÞf0ðeþ ek; xÞ; (4)

where ��k is the frequency of excitation of the k level. The

velocities v and v0k are related by the equation: mv2

2¼ mv02k

2þ ek.

The process of “back scattering” (second term in the right

hand side of Eq. (4)) describes re-appearance of electrons in

the elastic region after inelastic collisions.

In inert gases, there is a large energy gap between the

ground state and the first excited level, and all the excited

states are close to the ionization potential. There is then a small

parameter ek ¼ Dek=e1 ¼ ðek � e1Þ=e1, which can be used to

obtain a solution of Eq. (4) by extending the method originally

used by Tsendin. To do this, one separates the excitation of the

first level in the "back scattering" operator and performs a

Taylor expansion in the small parameter Dek=e1 up to quad-

ratic terms. This procedure corresponds to the Fokker-Planck

approximation. The resulting equation (4) takes the form

@

@x

v3


!þ @

@em2


� �þ @

@e

Xk

ðDekÞ��kðv01Þf0ðeþ e1; xÞ þ1

2

@

@e

Xk

ðDekÞ2��kðv0kÞf0ðeþ e1; xÞ" #

¼ vf0ðe; xÞX

k

��kðvÞ � v01f0ðeþ ek; xÞX

k

��kðv01Þ: (5)

Additional terms in the square brackets of Eq. (5) describe

an electron flux along energy due to excitation of different

electronic levels of atoms. This flux adds to the flux caused

by the energy loss in quasi-elastic collisions. The right side

of Eq. (5) describes the inelastic collisions leading to excita-

tion of the first level. Thus, excitations of different electronic

states of atoms and the "back scattering" process produce

effects similar to quasi-elastic collisions, both contributing

to the bunching process. Analytical solution of the kinetic

equation (5) was obtained in Ref. 12.

Shveigert13 conducted detailed computational studies

of electron kinetics in spatially modulated static electric

fields by solving kinetic equation for small EDF perturba-

tions in helium over a wide range of E/N. He took into

account Coulomb collisions and excitation of seven effec-

tive levels: four sublevels with a principal quantum number

qk ¼ 2 and three effective levels with qk ¼ 3; 4; 5 with

energy thresholds ek ¼ 23; 23:7; 24 eV. Shveigert concluded

that in the absence of detailed information about excitation

cross sections for various levels, it is appropriate to use a

two-level model taking into account direct ionization and

lumping all excited states into a single effective level. It

was found that Coulomb interactions among electrons make

substantial contribution to the width and the height of the

resonance for n=N � 10�4. For n=N � 10�4, the resonance

vanishes and a transition to the hydrodynamic regime takes

place.

III. NUMERICAL SIMULATIONS OF ELECTRONKINETICS

Simulations of electron kinetics based on the numerical

solution of the Fokker-Planck kinetic equation for the EDF

have been described in numerous publications (see Refs. 14

and 15 for reviews and further references). The CPU time

required by these models is less than the time of computing

electron kinetics by the Monte-Carlo method, because of

reduced dimensionality of the Fokker-Planck kinetic equa-

tion compared to the full Boltzmann equation. Most of the

developed solvers use “total energy-coordinate” as independ-

ent variables. Notably, Sigeneger and Winkler developed a

one-dimensional solver using total energy as an evolution

coordinate.16 They assumed that the electric field is always

monotonic, and no field reversals are possible in the plasma.

The parabolic equation (4) was solved for a given total

energy by a tridiagonal inversion method using already

known EDF values at higher total energy. The method was

further developed and applied to study EDF relaxation and

electron bunching in spatially periodic electric fields. The

collision processes taken into account included the excitation

of several excited levels of the inert gas atoms and direct ion-

ization. Ionization was treated as an inelastic collision with-

out generation of secondary electrons. It was shown that,

depending on gas pressure, the EDF relaxation process is

dominated by either the small energy loss in quasi-elastic

collisions (at pressures of the order of several Torr) or by

mixing between different excited states (at pressures of a

few tenths of Torr or less). Effects of Coulomb interactions

on the EDF relaxation have also been studied.17 Recently,

Fokker-Planck solvers using kinetic energy as independent

variable have been developed to analyze problems where

field reversals occur.18

Analysis of the kinetic equation in the form (1), made

by Tsendin showed that along with the resonance at the

length LS, integer resonances exist at lengths LS=n, where nis an integer. The exact value of the resonant length obtained



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by Tsendin is LS ¼ e1þDeeE , where De is the energy loss in colli-

sions over the field period. The experimentally observed S-

strata in inert gases have the length which corresponds to the

resonant length LS. Along with the S-strata, P-strata have

been observed with the length LS=2, which correspond to

Tsendin’s model with n¼ 2, and the R-strata with the length

LR ¼ 2LS=3, which were not considered in the original

Tsendin’s model (Figure 2).

For the interpretation of experimentally observed pat-

terns, the EDFs in striation-like fields were calculated using

the method developed in Ref. 19. First of all, one had to

obtain exact values of the resonant length LS for a specific

gas with a particular set of collision cross-sections. For this

purpose, the relaxation of an arbitrary EDF injected into a

homogeneous electric field was calculated. With calculated

EDF, the electron density was obtained in the form of

damped oscillations with a spatial period LS (Figure 3).

Then, the EDF formation in sinusoidally modulated elec-

tric fields EðxÞ ¼ E0ð1þ a sin ð2px=LÞÞ was studied for dif-

ferent spatial periods L and modulation depth a. Withcalculated EDFs, the modulation of electron density, the

mean electron energy, and excitation rates were plotted as

functions of the field period L. Figure 4 shows an example of

such calculations with kinetic resonances at certain lengths

FIG. 2. Three types of striations observed in neon discharge at 1.5 Torr for

different currents: (a) the length of the strata, (b) the voltage drop over stria-

tion length, (c) striation frequency. Tube radius is 1 cm. Reprinted with per-

mission from Golubovskii et al., Phys. Rev. E 72, 026414 (2005). Copyright

2005 The American Physical Society.

FIG. 3. Damped oscillations of the electron density during establishment of

the EDF in a spatially uniform electric field. Neon pressure 0.8 Torr.

FIG. 4. Modulation of the electron density (a), the average electron energy

(b), and excitation rate (c) versus spatial period L for a ¼ 0:9. Open circles

are for real cross-sections of inelastic collisions, filled circles are for an arti-

ficially increased collision cross-sections. Neon pressure 0.5 Torr. Reprinted

with permission from Golubovskii et al., Phys. Rev. E 72, 026414 (2005).

Copyright (2005) The American Physical Society.



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producing sharp peaks. Interestingly, the resonances are

greatly exacerbated if the inelastic collision cross-sections are

artificially increased by an order of magnitude (filled circles

in Figure 4). This procedure illustrates key differences

between the exact solution and the "black wall" approxima-

tion, which corresponds to �� ! 1. Clearly seen in Figure 4

are the S-resonance at the length LS, the P-resonance at the

length LP ¼ LS=2, and the Q-resonance at the length

LQ ¼ LS=3. Along with the integer resonances, one can also

observe a non-integer R-resonance at the length LR ¼ 3LS=2.

Generally, there are non-integer resonances at lengths m/n LS,

where m and n are integer numbers. The non-integer reso-

nance for m¼ 2 and n¼ 3 was first associated with the exper-

imentally observed R-strata in Ref. 19.

A. Dynamics of the EDF bunching process

To analyze the relaxation of an arbitrary injected EDF

to a "steady-state" and the appearance of resonant trajecto-

ries which “attract” electrons, calculations using Eq. (4)

were performed for neon gas at a pressure of 0.8 Torr for the

average field of 3 V/cm for sinusoidal fields corresponding to

the S-, P-, and R- resonances.20 The injected EDF at x¼ 0

had a G-shaped form in the energy range of 0–30 eV. The

obtained potential drop along the S-stratum length was

17.5 V that is larger than the first excitation threshold by

De� 1.1 eV. As noted above, this difference corresponds to

the energy that electrons lose in quasi-elastic collision and

the mixing effect of inelastic collisions during acceleration

in the field from zero energy up to the excitation threshold.

The calculation results are shown in Figure 5 in the form of

3D carpet plots and 2D contour graphs.

For S-strata, the EDF relaxed quickly to a "steady-state."

Figure 5 shows that during the first four periods, the injected

broad distribution loses its rectangular shape and becomes a

Gaussian. After several more periods, a bunched EDF is

formed with a Gaussian peak moving along a resonance

trajectory.

In the case of P-strata, the picture is more complicated

(Figure 6). Since the wavelength of P-strata is two times

smaller than that of S-strata, electrons must pass two field

periods to gain the energy corresponding to the first excita-

tion threshold. Therefore, two independent resonant trajecto-

ries are formed.

Figure 6 (top) illustrates the formation of a clear maxi-

mum from a G-shape EDF, which creates a resonant trajec-

tory. This trajectory can be denoted as the first resonance

trajectory (solid line in Figure 6). In the middle and the bot-

tom of Figure 6, one can observe more clearly the division

into two peaks, the narrowing of the EDF and the formation

FIG. 5. Relaxation of a G-shape injected EDF in an S-strata-like field: near the boundary (top) and far away (bottom).



146.229.56.102 On: Tue, 19 Nov 2013 19:38:12

of a second resonant trajectory (dashed line in Figure 6).

Finally, after a sufficient number of periods, we see the EDF

that takes the form of two peaks, moving along two resonant

trajectories.

In the case of R-strata (Figure 7), electrons have to pass

1.5 field periods to gain kinetic energy equal to the first exci-

tation threshold. The situation is repeated every 2 periods of

S-strata or every 3 periods of P-strata. Figure 7 shows a grad-

ual formation of resonant trajectories and bunching of initial

EDF for R-strata like fields.

B. Effect of injected EDF on the bunching process

During discussions of our simulations at a Tsendin’s

seminar a question came up: will electrons appear at the sec-

ond resonant trajectory, if one injects a d-like EDF on the

first resonant trajectory. The answer to this question is shown

in Figures 8 and 9, which illustrate the relaxation of a d-like

EDF injected at one of the resonance trajectories into P- and

R-striation-like fields.21 It is seen that during a few initial

periods near the boundary, electron density on the resonant

FIG. 6. Relaxation of a G-shape injected EDF in a P-strata-like field: near the boundary (top), intermediate distance (middle) and far from the boundary

(bottom).



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trajectories is negligible, but the secondary peak gradually

builds up and finally establishes a "steady-state," which is

the same as for the relaxation of a G-shaped injected EDF.

It is interesting to compare peculiarities of the bunched

EDFs for integer and non-integer resonances (Figure 10). In

the case of integer resonances, peaks of equal amplitude are

traveling along resonant trajectories. For the non-integer res-

onance, alternating amplitude peaks are observed.

C. Comparison of calculated and measured EDFs

The calculated EDFs have been compared with experi-

mentally measured EDFs. There have been subtle experi-

ments to measure EDFs in striated discharges with high

spatial and temporal resolution. Special difficulties appear

for measurements in R-striations, where EDFs have the most

complex structure and striation frequency is one order larger

FIG. 7. Relaxation of a G-shape injected EDF in an R-strata-like field.



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than that of S- and P-striations. Figures 11–13 compare the

measured and calculated EDF for the strata of different

types.22 The kinetic energy and spatial coordinate variables

are used in these plots.

As can be seen in these figures, the experimental meas-

urements confirm all the peculiar features of the EDFs

obtained in simulations for striations of different types. In

particular, a moving single peak for the S-strata and two

peaks for the P-strata are clearly visible. Should any hardware

distortion measurements improve the resolution of the peaks,

then we can talk about the good quantitative agreement. For

the R-strata, there is a good qualitative agreement of simula-

tions with measurements.

IV. EFFECT OF ELECTRON BUNCHINGON DISCHARGE STRATIFICATION

In the previous section, we have demonstrated that ki-

netic resonances result in the formation of EDF with multi-

ple peaks. Results of numerical solutions of the kinetic

equation for the prescribed electric fields agree well with

experimentally measured EDFs in S, P, and R striations. In

this section, we briefly describe how kinetic resonances

affect the properties of moving striations observed in DC

discharges of noble gases at low pressures and low currents.

Rohlena et al.6 were the first to calculate dispersion

characteristics of the S and P waves with account for

FIG. 8. Relaxation of a d-shape EDF injected in a P-striation-like field. Reprinted with permission from Golubovskii et al., Tech. Phys. 56, 731 (2011).

Copyright (2011) Pleiades Publishing, Ltd.



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kinetic resonances. They have shown that in the vicinity of

resonances, the rapidly changing amplitude and phase rela-

tions modify the hydrodynamic dispersion curves. As a

result, an effective selection of the wavelength with charac-

teristic potential UL takes place. Kinetic resonances explain

key features of the S- and P-waves, in particular, suffi-

ciently high ratios of the group and phase velocities.

However, the ratio of the group and phase velocity obtained

in Ref. 6 was too high due to very sharp resonance peaks.

Later, it was observed that Coulomb interactions among

electrons and excitation of several atomic levels produce

smearing of resonances and merging dispersion curves for

the S and R waves.23,24

Tsendin7 obtained analytical dispersion characteristics of

moving striations assuming that the EDF bunching occurs

solely due to energy loss in quasi-elastic collisions within the

"black wall" approximation. Dispersion characteristics and

amplification of the S and P waves were calculated in the vi-

cinity of resonances. It was shown that the maximum value

of the amplification coefficient for the P and S waves is deter-

mined by the kinetic resonances, whereas the R wave was

assumed to be purely hydrodynamic in nature. Good agree-

ment was observed for absolute values of the wavelengths

and their dependence on the discharge current. For the disper-

sion law, the agreement between calculations7 and experi-

ment was fair. The discrepancies were attributed to additional

FIG. 9. Relaxation of a d-shape EDF injected in an R-striation-like field. Reprinted with permission from Golubovskii et al., Tech. Phys. 56, 731 (2011).

Copyright (2011) Pleiades Publishing, Ltd.



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broadening of the resonances due to processes not included in

the kinetic model.7 Later, the importance of kinetic resonan-

ces for the R-waves was discovered. Dispersion characteris-

tics of the backward waves in a weakly collisional DC

discharge in argon at very low pressures, p¼ 5–30 mTorr, at

discharge current I¼ 100 mA, and tube radius R¼ 1.8 cm

were calculated and measured in Ref. 25. No kinetic resonan-

ces were found under these conditions, which correspond to

k�>ke. As explained above, kinetic resonances manifest

themselves only in a certain range of gas pressures when the

conditions k� � ke � kT are satisfied.

Recently, self-consistent simulations of kinetic stria-

tions by hybrid methods have been reported in the litera-

ture. The methods included coupled solution of a Fokker-

Planck kinetic equation for electrons, fluid models for

heavy plasma species, and Poisson equation for the electric

field. Standing striations have been obtained in one-

dimensional simulations,26 which agreed with the experi-

mental observations of a disturbed positive column in neon

discharges. Notably, standing striations of the P-type were

obtained.

Another hybrid model of striated positive column was

described in Ref. 27. The model is based on coupled simula-

tions of electron kinetics, drift-diffusion model for ion trans-

port, and Poisson solver for the electric field. It was assumed

that electrons enter the positive column from cathode side

with a prescribed EDF. The model predicted self-consistently

non-uniform electric field, spatially resolved EDF, and the

ion density profile. The authors obtained a steady solution for

a self-consistent “resonance” length of strata and the modu-

lated electric field by neglecting ionization and particle

losses. The obtained distribution of the electric field in stri-

ated positive column had the form of spatially damped oscil-

lations with a period inversely proportional to gas pressure

L� 1/p. The field had peaks at the locations of strata, and the

distribution of electron density lagged behind in phase by

FIG. 10. Calculated EDFs for integer (S, P, Q) and non-integer (R) resonan-

ces. Neon p¼ 0.5 Torr. Reprinted with permission from Golubovskii et al.,Phys. Rev. E 68, 026404 (2003). Copyright (2003) The American Physical

Society.

FIG. 11. Comparison of experimentally measured and calculated EDFs

in various phases of S-strata. Neon, pressure 1.5 Torr, current 10 mA, tube

radius 2 cm. Reprinted with permission from Golubovskii et al., Phys.

Rev. E 68, 026404 (2003). Copyright (2003) The American Physical

Society.



146.229.56.102 On: Tue, 19 Nov 2013 19:38:12

half a period. Clearly, ionization processes and the particle

losses neglected in the model must be taken into account to

obtain moving striations.

Recent paper28 presents a two-dimensional hybrid

model of positive column disturbed by a sudden change of

radius of the tube. The properties of calculated standing

striations were found to be in good agreement with the ex-

perimental data under studied conditions.

In summary, in spite of impressive progress in the

theory and computer simulations of electron kinetics in

glow discharges, no self-consistent simulations of moving

kinetic striations have been reported so far. According to the

latest Tsendin’s review10 “a self-consistent theory of S-, P-,

and R-striations is still lacking.” The key differences

between moving and standing striations remain not clear,

even for the simplest case of noble gases. Stratification

kinetics in gas discharges has many similarities with

electron-density stratification in semiconductors.29 Phased

motion of hot electrons in configuration and momentum

spaces causes spatial modulation of the electron concentra-

tion and mean electron velocity with the period ke ¼ e0=eE,

which has been observed in semiconductor structures of fi-

nite size30 (here, e0 ¼ �hx0 is the energy of optical phonons).

The most important manifestation of this phenomenon is the

possibility of terahertz radiation with frequency tuned by

the applied voltage.

V. SYNCHRONIZATION AND VISUALIZATIONOF STRIATIONS

The presence of inherent spatial scale LS defined by gas

type and by discharge conditions poses the following ques-

tion: what are all values of external field period L, which

deliver periodicity of electron kinetics. To answer this ques-

tion, consider a single electron dynamics in a periodically

modulated electric field.19

Assume zn is a spatial position of an electron with the ki-

netic energy w and the total energy en

zn ¼ �en � w

eE0

þ aL

2pcos

2pzn

L

� �: (6)

The next position of the electron with the same kinetic

energy can be written as

znþ1 ¼ �enþ1 � w

eE0

þ aL

2pcos

2pznþ1

L

� �: (7)

FIG. 12. Comparison of experimentally measured and calculated EDFs in

various phases of P-strata. Neon, pressure 1 Torr, current 10 mA, tube radius

2 cm. Reprinted with permission from Golubovskii et al., Phys. Rev. E 68,

026404 (2003). Copyright (2003) The American Physical Society.

FIG. 13. Comparison of measured and calculated EDFs in various phases of

R-strata. Neon, pressure 0.72 Torr, current 20 mA, tube radius 1 cm.

Reprinted with permission from Golubovskii et al., Plasma Sources Sci.

Technol. 18, 045022 (2009). Copyright (2009) IOP Publishing Ltd.



146.229.56.102 On: Tue, 19 Nov 2013 19:38:12

Using the fact that enþ1 ¼ en þ VSL, and introducing

X ¼ LS=L, Eq (7) can be rewritten as a nonlinear map31

Hnþ1 ¼ Hn þ AðHnÞ: (8)

Here, H ¼ zn=L mod1 is the phase position within a single

period of L, and

AðHnÞ ¼ X� a2p

cosð2pHnÞ

� f1� cosð2pAðHnÞÞ þ tanð2pHnÞsinð2pAðHnÞÞg(9)

is a periodic function of Hn, which corresponds to a distance

in horizontal direction between the curves x1ðeÞ and x2ðeÞ in

Figure 1. This map indicates the reason for nonlinearity: the

motion of electrons in physical space is mapped nonlinearly

onto energy space. The map (8) is similar to the circle map

well known in nonlinear physics,32 and can be investigated

for periodic solutions.

Figure 9 shows a contour plot for a set of histograms of

Hn series for different values of X. One can see in this figure

the values of H where more than 25% of points Hn series are

concentrated. The sum over H for a given value of X gives

100% of points Hn. One can clearly see that for some values

of X we obtain cycling values of H: for X¼ 1, 2, 3 it is only

H¼ 0, for X¼ 1.5, 2.5 it is H¼ 0 and 0.5, for X¼ 1.25, 1.75

it is H¼ 0.1, 0.47, 0.85, 0.93, and so on. These values of Xcorrespond to the L=LS ratios shown on the top of Figure 14.

The ratios are given by a so-called Farey tree—a characteris-

tic of a circle map.32 The investigated map gives resonances

at fractions of LS: 1/3 (Q resonance), 1/2 (P resonance), 2/3 (Rresonance), and 1 (S resonance). The latter three fractions cor-

respond to waves which have been observed experimentally.

The S, P, and R striations can exist in self-excited mode.

Other periodic states can be revealed by applying an external

periodic disturbance to the positive column. In particular,

those corresponding to the resonances 1/3 and 2/5 of the

Farey tree have been revealed.32 These waves do not exist

naturally because they are effectively damped. Modulation

of discharge at appropriate frequencies results in visualiza-

tion of moving regular striations, excitation of artificial stria-

tions, and synchronization of chaotic striations.33,34

Figure 15 shows an example of experimental phase dia-

gram33 obtained in Neon discharge at 1.4 Torr in a tube of ra-

dius R¼ 1.1 cm at current i¼ 9.3 mA. These conditions

correspond to P-waves with dominant internal frequency

f1 ¼ 1450 Hz. Modulation of discharge voltage by an exter-

nal signal with amplitude U in the frequency interval 700 <fa < 1500 Hz reveals a series of mode locking and two-

frequency quasi-periodicity. The experimental results of

Figure 15 confirm predictions of the circle-map theory out-

lined above concerning the existence and ordering of

Arnold’s tongues obeying the Faray law. For modulation am-

plitude U< 3 V, the observed Arnold’s tongues are very nar-

row with exception of the 1/1 horn. For U> 3 V, a second

internal signal with f2 ¼ 1150 Hz appeared, corresponding to

second Arnold’s horn shown in Figure 15. The appearance

of this wave is associated with typical hysteresis effects.

Staring from small amplitudes, one can observe the tongues

with winding numbers 6/5 or 5/4 related to f1 until the sys-

tem switches over to the f2-regime at U� 3 V. On the other

hand, coming from higher amplitudes, one observe the 1/1

horn of f1 down to U � 0:3 V.

A number of interesting effects typical to non-linear

physics can be found in the literature devoted to glow dis-

charge stratification. We want to mention here a recent publi-

cation35 devoted to experimental and theoretical studies of

hysteresis of ionization wave modes due to finite length of the

discharge and a feedback from the current oscillations. We do

not agree with the author’s assessment that “full kinetic

description fails to explain the dynamics of saturated wave

modes—on the other hand—a description of macroscopic dy-

namics by bifurcation theory does not self-consistently recover

the interplay of electron kinetics and wave properties.” The

authors imply that it is difficult to bridge the two fields: physi-

cal kinetics and non-linear physics. We see an opportunity for

future work here—stratification of glow discharges opens

unique opportunities for theoretical, computational and experi-

mental studies of physical kinetics and modern non-linear

physics in convenient laboratory conditions.

VI. CONCLUSIONS

Electron bunching in spatially periodic resonant electric

fields is an interesting example of self-organization at the ki-

netic level. Tsendin first illustrated that under certain condi-

tions, the EDF in such fields has the form of Gaussian peaks,

FIG. 14. Histogram of trajectories for map (8) at different X. Spots show

positions where more than 25% of sequential values Hn are concentrated.

The sum over spots lying along vertical lines gives 100% of points.

Reprinted with permission from Golubovskii et al., Phys. Rev. E 72, 026414

(2005). Copyright (2005) The American Physical Society.

FIG. 15. Phase diagram with a number of Arnold tongues. Reprinted with

permission from Albrecht et al., Phys. Scr. 47, 196 (1993). Copyright (1993)

IOP Publishing Ltd.



146.229.56.102 On: Tue, 19 Nov 2013 19:38:12

moving along resonance trajectories. The nature of this effect

is particularly revealing in the “total energy-coordinate” rep-

resentation advocated by Tsendin. We have presented results

of numerical solutions of the Fokker-Planck kinetic equation

in spatially periodic electric fields under the effects of elastic

and inelastic collisions of electrons with atoms. We have illus-

trated the formation of kinetic resonances for specific field

periods in a certain range of discharge conditions. For differ-

ent shapes of injected EDF (a G-shape and a d-shape EDFs),

Gaussian-like peaks moving along resonance trajectories are

formed during the EDF establishment process. The calculated

EDFs agree well with the experimentally measured EDFs for

S, P, and R striations in noble gases at low pressures.

Numerical simulations and experimental studies per-

formed over the last three decades have demonstrated the na-

ture of electron bunching and revealed that any rational

values of the inherent length scale LS can lead to EDF bunch-

ing. The presented computer simulations of the EDF forma-

tion process illustrate how the resonance trajectories are

formed and how self-organization of the EDFs occurs under

nonlocal conditions. Resonances predicted by a non-linear

theory and obtained in numerical simulations have been

experimentally observed by applying external periodic dis-

turbances to DC discharges. Discharge modulation results in

visualization of moving regular striations, excitation of artifi-

cial striations, and synchronization of chaotic striations.

We have briefly discussed how kinetic resonances affect

dispersion characteristics of moving striations in rare gases for

the considered range of discharge conditions. In spite of re-

markable progress, no self-consistent simulations of moving

kinetic striations have been reported so far. Such simulations

could offer a clear test of our understanding of electron

kinetics and elementary processes in gas discharge plasmas.

Striations present a perfect object for studies of physical

kinetics and non-linear physics in convenient laboratory

settings.

ACKNOWLEDGMENTS

This work was partially supported by the Russian

Federal Target Program “Research and scientific-pedagogical

personnel of innovative Russia” and by the project UGTU

2.4501.2011. V.I.K wants to acknowledge the U.S.

Department of Energy Office of Fusion Energy Science

Contract No. DE-SC0001939.

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On electron bunching and stratification of glow discharges