Transcript An Estimation of Critical Electron Density at Just

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
fi
E 
li
*
fi

1
2
m v  eE li  efi
2
e
*
Ionization Potential
li   i n 1
m.f.p. for electron impact ionization
dne
 0 for li  gap length  spark or breakdown,
dt
hereany particleloss and energyloss exceptfor ionizationare out of account.
E  E*,
cf E* = 30 kV/cm for dielectric strength in atmospheric air
If fi = 15 V , li becomes 5x10-4 cm ← reasonable
f
VBD  E *d   i   pd  Paschen' s law
 li 0 
Temporal and Spatial Growth of Electron Density
v
dne
 Rg ne    ne  0
dt
 li 
ne  n0e
Rg t
1
dne  Rg 
ne   ne  0
 
dx  v 
 li 
ne  n0e
x li
here, E  E  and any particleloss is out of account.
energyloss for excitaionis also out of account.
energy balance d  E 2  eRg nefi  0
dt
Structure of Electron Avalanche
dne 1
 ne ,
dx li
ne  ne 0e x li
where ne0 is intialelectrondensity
Length x’ of electron avalanche
'


n
'
e
x  li ln 

n
 e0 
ne  ne'
ne0
cf ions are at rest
initial 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 bombardment (g process, feedback)
before and after starting breakdown
ne’ electron
*
*
E

E
EE
cloud
ne0
ion space charge
x’
2 R  Dx '
E*
ion space charge
ve
x*
ne* E  0
plasma
Scenario of Starting Electric Breakdown in Gases
1. 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
charge.
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 cathode.
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 )
cathode
E > E*
anode
front-sheath
moving
micro-plasma column
V(ｔ)
Evolution of Potential Structure
before and after Starting Breakdown
V(x,ｔ)
Front of plasma column
Starting breakdown
Stable discharge
x
Cathode sheath
I(ｔ)
A Hypothesis for the Appearance of Micro-Plasma
at Anode Near Region
fi TeV e
E  
li
lD
*
TeV electron temperature in eV
lD
Debye shielding length
E  0 in theplasmacolumn
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.
Critical Electron Density ne*
at Just Starting Breakdown in Gases
 ne*e 2
 
lD   0TeV
fi TeV e

,
li
lD
1



1
2
 0  efi  fi 2
 2 p
n  
e  TeV  li 0
*
e
efi
 10, fi  10 V, li  5 104 Patm cm
TeV
n  2 10 p cm
*
e
14
2
-3
p in atm
Relationship between critical electron density and
space charge density in front-sheath
 efi
n  
 TeV
*
e
0
  0 fi

,
2
 e li
 0 fi
f
 ni in front- sheath
2
e
e li
2
 ef i
n  
 TeV
*
e

 ni

n l  ni li
*
e D
ne*/ni and lD/li as a function of TeV/efi
ne*/ni
ne*/ni
lD/li
lD/li
０．０１
０．１
TeV/efi
１
Relationship
between Micro-Plasma Column and its Front-Sheath
at Starting Breakdown
cathode
anode
E > E*
front-sheath
micro-plasma column
moving
space charge density
in front-sheath

nis  0  2f
e

 0 fi  TeV  *
ne
 2  
e li  efi 
potential structure
at starting breakdown
TeV
fi
f
nis
ne*= (efi/TeV ) nis
ne～ 0
li lD = (TeV/efi ) li
Velocity vfs of Moving Front-Sheath (Streamer).
The growth of plasma volume is caused by the ionization due to fast electrons
having their energy of (1/2)mv2 > efi.
front-sheath
ionization & electron heating
+
e
li, fi
e
e
plasma expansion
lD
ne*lD nse ve lD  lD  TeV  * 2efi

   ne
t
li
m
 li  efi 
2efi
lD
v fs 

 5.9 107 fi cm/S
t
m
cf : expansion due to electron heating is neglected here
recent observation of vfs ～108 cm/s
Velocity vfs of Moving Front-Sheath (Streamer).
v fs  ve 
2efi
m
 5.9  10 7 fi cm/S
Expansion of micro-plasma is caused by the ionization due to fast electrons
having their energy of (1/2)mv2 > efi. 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 efi eV
(* More detailed discussion is required to introduce the energy minimum
principle.)
ionization & electron heating
front-sheath
+
e
li, fi
e
e
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
vse 
2efi
m
 TeV
nse  ni  
 efi
 *
ne

Criticalcurrent density
 TeV
j  ense vse  e
 efi
*
 * 2efi
* 2TeV
ne
 ene
m
m

TeV
efi
Critical current density at starting breakdown 2/2
electron thermal velocity
ne*
j*
e 2
2
p
p
2TeV
m
 TeV 


 ef 
i 



 1.6 1019 C 2 1014 cm-3  6 107 cm/s  1 
 10 
 0.6 kA cm-2atm-2
Cross-section of micro-discharge in atmospheric pressure
1 mm2 → I *～ 60 mA
0.01 mm2 → I *～ 0.6 mA
Summary
Following hypothesis for the appearance of micro-plasma at anode near
region, seems to be correct. Breakdown is just starting when the microplasma appears at anode near region.
fi
TeV e
E 

li
lD
*
E  0 in theplasmacolumn
E* is shielded electrically by the polarization of plasma particles
We can estimate the critical electron density at just starting breakdown by
using the above formula.
 efi   0 fi  efi   0 fi
2



n  


p
2
 T  e l2
T
e
l
i
i0
 eV 
 eV 
 efi 
*
nis   f   f  n in thefront- sheath
ne  
e
e l
 TeV 
*
e
0
2
0
i
2
i
is
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 frontsheath without any ionization loss and then vfs could be
estimated to be the same velocity of electrons accelerated
up to efi eV (* More detailed discussion is required to introduce
the energy minimum principle.)
v fs  ve 
2efi
m
 5.9  10 7 fi cm/S
Appendix I
Recent Observation of Streamer
-Kumamoto Univ.-
Vｓ＝7.1 mm/nS
Fig. 7 Streak photograph
Appendix II
Cathode Sheath in Steady State Glows
plasma balance Eqs.  
dni
nvL
 L  e e  ni vi  0,
dt
li
nvL
d
 L  jeVcs  e e efi  jeVcs  jifi  0,
dt
li
self - sast ainingcondit ion
je  g  ji ,
fi
,
g
d cs  li any ionizationdoes not occur in thesheath
Vcs 
* Here we consider only the most essential processes
je
anode
plasma
cathode
ji
dcs
cathode sheath
L
Vcs