An Estimation of Critical Electron Density

at Just Starting Breakdown in Gases

Mase. H

Professor Emeritus of Ibaraki University

Electric Field Threshold E *

for Electron Impact Ionization of Neutral Atoms

cf E* = 30 kV/cm for dielectric strength in atmospheric air

If i = 15 V , i becomes 5x10-4 cm ← reasonable

Ionization Potential

m.f.p. for electron impact ionization 1 nii

i

account. ofout are ionizationfor except lossenergy and loss particleany here

breakdown,or spark length gapfor 0 , i*

dt

dnEE e

iiei

i eeEmvE

*221*

0

* pddEVi

iBD

law sPaschen'

Temporal and Spatial Growth of Electron Density

0

e

ieg

e nv

nRdt

dn

01

e

ie

ge nnv

R

dx

dn

tRe

genn 0

ixe enn

0

0 balanceenergy

account. ofout also isexcitaion for lossenergy

account. ofout is loss particleany and here,

2

iegneREdtd

EE

Structure of Electron Avalanche

,1

0ix

eeei

e ennndx

dn

Length x’ of electron avalanche

0

''' ln

e

eiee n

nxnn

densityelectron intial is where 0en

before and after starting breakdown

cf ions are at restinitial electrons;1.spontaneously generated 2.externally generated (pre-ionization)3.photo-ionization due to light emission from front of avalanche (feedback)4.2ry emission by ion bom- bardment (process, feedback)

ne0

ne’

x’

ion space charge

electron cloud

E*

evDxR '2

*EE ne0

ne*

x*

ion space charge plasma

*EE 0E

Scenario of Starting Electric Breakdown in Gases1. E > E* is applied field across the electrodes. ne0 occurs at any time.2. dne/dt > 0 in front of electron avalanche. E is extremely distorted by the space c

harge. 3. ne = n* = ni n* is the critical electron density at just starting breakdown. >> appearance of micro-plasma in front of electron avalanche or near the anode.4. E is increasing in the space between the cathode and front of the plasma column

>> enhanced ionization >> expansion of plasma column. 5. Front of the plasma column (that is virtual anode) is moving toward the cathod

e. go back # 4. < positive feedback process > (* More detailed discussion is required to introduce the energy minimum principle.)

6. When the front-sheath comes at the cathode, the discharge gap makes it to flashover .

7. If the front–sheath satisfies the self-sustaining condition of discharge , the glow discharge or the arc discharge occurs.

In the case, the front-sheath means so-coaled “cathode sheath”. ( Appendix II )anodecathode

micro-plasma column

front-sheath

moving

E > E*

Evolution of Potential Structure before and after Starting Breakdown

I( ｔ )V

(ｔ

)

V(x

,ｔ)

x

Front of plasma column

Cathode sheath

Starting breakdown

Stable discharge

E* would be shielded electrically by the polarization of plasma particles. It is best that Tev should be confirmed theoretically by the plasma balance equation. Usually TeV is in the range of several eV.

column plasma in the 0E

A Hypothesis for the Appearance of Micro-Plasma at Anode Near Region

TeV electron temperature in eV

D Debye shielding length

D

eV

i

i eTE

*

220

0*

21

0

2*1 ,

pT

e

en

T

eneT

i

i

eV

ie

eV

e

DD

eV

i

i

Critical Electron Density ne*

at Just Starting Breakdown in Gases

cm atm105 V, 10 ,10 4 PT

eii

eV

i

atmin cm 102 -3214* ppne

Relationship between critical electron density and space charge density in front-

sheath

sheath-frontin

,

2020

20*

ii

i

i

i

eV

ie

nee

eT

en

ieV

ie n

T

en

*

iiDe nn *

ne*/ni and D/i as a function of TeV/e

０．０１

１０．１

TeV/ei

D/i

ne*/ni

D/i

ne*/ni

Relationship between Micro-Plasma Column and its Front-Sheath

at Starting Breakdown

*2

0

20

ei

eV

i

i

is

ne

T

e

en

space charge density in front-

sheath

anodecathode

micro-plasma column

front-sheath

moving

E > E*

i

i D = (TeV/ei ) i

TeV

nis ne*= (ei/TeV ) nis

potential structure　　　 at starting breakdown

ne ～ 0

Velocity vfs of Moving Front-Sheath (Streamer).

cm/S 109.52

2

7

**

iiD

fs

ie

i

eV

i

D

i

DeseDe

m

e

tv

m

en

e

Tvn

t

n

The growth of plasma volume is caused by the ionization due to fast electrons having their energy of (1/2)mv2 > ei.

cf : expansion due to electron heating is neglected here recent observation of vfs ～ 108 cm/s

+ee

i, i

front-sheath ionization & electron heatinge

plasma expansionD

Velocity vfs of Moving Front-Sheath (Streamer).

cm/S 109.52 7i

iefs m

evv

Expansion of micro-plasma is caused by the ionization due to fast electrons having their energy of (1/2)mv2 > ei. Velocity of moving front-sheath may be determined by the motion of fast electrons which are passing through the front-sheath without any ionization collision. So, vfs could be estimated to

be the same velocity of electrons accelerated up to ei eV (* More detailed discussion is required to introduce the energy minimum principle.)

+ee

i, i

front-sheath ionization & electron heatinge

plasma expansion

cf : recent observation of vfs ～ 108 cm/s　　　

Critical current density at starting breakdown 1/2

Velocity and density of fast electrons which are passing through the front-sheath

i

eVeVe

ie

i

eVsese e

T

m

Ten

m

en

e

Tevenj

22

density current Critical

***

*

2

ei

eVise

ise

ne

Tnn

mev

Critical current density at starting breakdown 2/2

2-2-

73-1419

2

*

2

*

atmcmkA 0.6

101cm/s106cm102C106.1

2

i

eVeVe

e

T

m

T

p

ne

p

j

Cross-section of micro-discharge in atmospheric pressure

1 mm2 → I * ～ 60 mA

0.01 mm2 → I * ～ 0.6 mA

electron thermal velocity

Summary

D

eV

i

i eTE

*

E* is shielded electrically by the polarization of plasma particles

column plasma in the 0E

Following hypothesis for the appearance of micro-plasma at anode near region, seems to be correct. Breakdown is just starting when the micro-plasma appears at anode near region.

We can estimate the critical electron density at just starting breakdown by using the above formula.

*

220

02

0*

iseV

ie

i

i

eV

i

i

i

eV

ie

nT

en

peT

e

eT

en

sheath-front in the 2

020is

i

i nee

Summary continue

We also discus the propagation of plasma column head toward the cathode (streamer). The propagating speed of plasma column head might be determined by the motion of fast electrons which are passing through the front-sheath without any ionization loss and then vfs could be estimated to be the same velocity of electrons accelerated up to ei eV (* More detailed discussion is required to introduce the energy minimum principle.)

cm/S 109.52 7i

iefs m

evv

Appendix I

Fig. 7 Streak photograph

V ｓ＝ 7.1 mm/nS

Recent Observation of Streamer-Kumamoto Univ.-

sheath in theoccur not does ionizationany

,

, condition sastaining-self

,0

0, Eqs. balance plasma

ics

ics

ie

iicseii

eecse

iii

eei

d

V

jj

jVjeLvn

VjLdt

d

vnLvn

Ldt

dn

Appendix IICathode Sheath in Steady State Glows

cathode sheath

anodeplasmacathode

dcsVcs

je jiL

* Here we consider only the most essential processes