Spatio-temporal evolution of a transversally excited electrical discharge in Nitrogen

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Spatio-temporal evolution of a transversally excited electrical discharge in Nitrogen

N. Spyrou, Ch. Manassis

To cite this version:

N. Spyrou, Ch. Manassis. Spatio-temporal evolution of a transversally excited electrical discharge in Nitrogen. Journal de Physique II, EDP Sciences, 1991, 1 (9), pp.1021-1031. �10.1051/jp2:1991203�.

�jpa-00247572�

Classification

Physics

Abstracts

52 80 42 55H 51 50

Spatio-teInporal evolution of

_a

transversally excited electrical

discharge in Nitrogen

N.

Spyrou

and Ch. Manassis

Electrotechmc Matenals Laboratory, University of Patras, Greece

(Received19

June 1990, rev~ed 5 June 1991,

accepted13

June 1991)

Rksumk.-Un moddle

thkonque spatio-temporel

_a ktd _{nils au} point afin de dktermlner l'influence des terries de transport dans _une

dbcharge

transverse dans l'Azote moldcu1alre. Los rdsultats obtenus montrent que

l'hypothdse

de l'exlstence d'un

champ klectrique homogdne

low de l'kvolutlon de la

dkcharge

n'est pas valable et que la

rkgton

de chute

cathodlque

est

rapidement

formde. Ce moddle

s'apphque

bien £ la

dkcharge

du laser I Azote.

Abstract. -A one-dimensional

time-dependent

model _is descnbed _m order to determine the influence of the transport terms _m a transversally excited discharge _m molecular Nitrogen. ^The results obtained indicate that the assumption of _a

homogeneous

electnc field is _not valid and that

a cathode fall regton is

rapidly

formed An

application

of this model _m nitrogen laser is realised.

1. InUoduction.

The transverse excited

discharges

_are of

high

interest for laser

applications.

The active

medium,

which is the

discharge plasma,

is created _very

rapidly

in _a few nanoseconds and occupies a

relatively large

volume. A representative

example

is the

N2-laser

where

population

inversion _is obtained

by

direct

impact

ionization collisions with molecules from the

ground

state

according

to the Frank-Condon

principle

In this _case

populauon

_{inversion occurs}

between thw

C~H~

and B ~H~ electronic _states with _an induced _emission at 337.I _nm

parttcularly.

Because the full width at half maxhnum

(FHWM)

of the Laser

pulse corresponding

to the

following

reaction

C~ H~(0)

_-

^B~ H~(0)

is very short

(about

8

ns) [I-11]

the

population

inversion

ought

to settle

by

_{a very} fast

discharge

_in which the electron

density

_is about

10'~cm~~

_{and the electnc current of the}

discharge

_is a

large

number of KA.

In the 70's _numerous theoretical models have been established _in order to investigate the

behaviour of the

discharge

and relate it with the laser _power.

Unfortunately

this _is not

yet

understood

clearly.

This _is due _to the fact that _an

expenmental

observation is difficult

dunng

the very short

maintaining

time. On the other

hand,

the theoretical

approaches

also _remain within _a

homogeneous discharge

model in which the

plasma

parameters and the reduced electnc field _m the laser gap are assumed to be _constant.

1022 JOURNAL DE PHYSIQUE II N 9

Nevertheless it has been

pointed

out

[12-17] that,

in such _a

discharge,

the _non-

homogeneities

_appear

rapidly

and

modify

the

discharge

behaviour.

In order to elucidate the establishment of the

discharge

and

particularly

the forrnauon of the cathode

region

we elaborate _a

spatio-temporal

model which describes the

discharge

evolution _{m a}

transversally

excited

Nitrogen

Laser. The

key

words of this model _are the

«

non-homogeneities

_» and the _use of the flux corrected

transport

»

algorithm

^of Boris and Book

[18].

The

contmulty

equation

including

_transport terms _is treated

numerically

_m

conjunction

with the Poisson equation for the electric field.

2. The

discharge.

The

discharge

_is

produced

between two very

long

electrodes

(70 cm) especially designed

in order to avoid discontinuities _m the1nltial field distnbution

The gap distance _is fixed at I _cm and the mtrogen ^{flow is insured}

by

_a

pumping

system which _is

capable

to evacuate the vessel

population

between two

discharge pulses

This _is done to avoid effects

produced by

rotational and vibrational excited molecules.

A

high voltage

is

applied

to the gap using a

simple

electncal apparatus which _is described with detail in

[10, 11] (Fig. I).

This

discharge

circuit _is charactenzed

by

_{a very} low

inductance,

a very fast

voltage

and _current nsetimes and

high peak

currents. In

figure I,

the

spark-gap

S _is

represented by

_a resistance

1~

and _an inductance

L~. C~

is the pnmary energy storage

capacitor,

C~

the

secondary

_{energy storage}

capacitor _Vc~

and

Cc~

^{denote the time}

dependent voltages

_across

Cs

and

C~ respectively,

while _t denotes the

discharge

current and I

is the current in the electnc circuit

r~~~~~~~~~~~

~'~ _{' Re}

Le '

v

L--- _~---j

fi~

I

t ~

cs _v~~ vc~ c~

ⁱ

i

~

~l _§____

2

Fig

I

Equivalent

electnc circuit apparatus

(R~

= 0.I n, _L~

= 5 nH,

Cs

=

C~

= 60 nF and V is the

extemal

applied voltage).

3.

Discharge theory.

Consider _a

slightly

ionized collision-dominated

plasma

which _is

adequately

descnbed

by

the one-dimension

continuity equations

These

equations

are.

(

+

) _{(ne ~e)}

"

S(ne)

L

(ne) (I)

(

+

) _(n,

_V,)

=

S(ne)

L

(~e) (2)

where n~, ni, v~, v,, are the electron and _ion number densities and

velocities, respecuvely.

S(n~)

and L

(n~)

denote

production

and loss terms. Poisson's

equation

is used _to calculate the electnc field distnbution. This

equation

_{in our} one-dimensional _{case is given}

by

_:

fl

"

(eie0)(~, ne) (3)

where _e, so are the electron

charge

and the

permittivity

of free space.

In such _a

discharge,

for the range of electnc fields considered

here,

the _{main source} of the

charged parudes dunng

the breakdown

penod

_is the

impact

ionization and its rate _is given

by

S(n~)

= n~ av~

(4)

where _a is the first Townsend _ionization coefficient which is _a function of the reduced electnc field

E/p

~p is the gas

pressure)

Losses of

charged particles

_are due to electron-ion radiative

recombination

[19]

with _a coefficient

p

L(n~)

=

fIn~

_n,

(5)

where

p

_may be

expressed

in terms of

E/p. Thus,

_{swarm parameters} such _as

velocity,

ionlzauon and recombination coefficients _are

expressed

_as functions of the reduced electnc field

E/p,

while the

longitudinal

electron diffusion term _is

neglected. Analytical

expressions

are obtained

by assuming

that the electrons _{are m}

equilibrium

with the electnc field nJs assumption

permits

us to express swarrn parameters as functions of the reduced electnc field

E/p.

Primary

reduced ionization and recombination coefficients _as well _as electron dnft

velocity

are given m the literature with _a very

good approximation Thus, Bayle

_gives

[20].

0 ~

< 2.76

V/cmtorr

P

8.633 _exp ^{~~~ ~} 2.76 _« ~

< 44

V/crntorr f

=

E P

~ l 17 _x

10~~

^~ _32.2 ^~ ₄₄

«

~

< 176

V/cmtorr

P P

3.66 ₊ 0.4583 ~ ~~~ ~

m 176

V/cmtorr

P P

for the first ionization coefficient _{m a}

nitrogen discharge

and Felsenthal _et al.

[21]

give for the velocities of electrons and _ions

respectively

v~

(cm/sec)

₌ 2.9 _x

10~

^{~ ~} _in

V/cmtorr

p p

v,(cm/sec)

=

67 _x

10~

^~

p

The recombination coefficient and the electronic

temperature

are

expressed

_as functions of

E/p [19, 9]

_as follows

p (cm~

sec~ ^'

)

=

6.3636 _x 10~ ^{'~ ~}

°~

E

KT~(eV)

= o ii E 08 ~

p ^'

j

^~ll

^V/cmtorr

iu~4 JOURNAL DE

PHYSIQUE

II M 9

In order to calculate the

discharge

current i the formula of Sato

[22] ~as

_been

_{used, adapted}

_to

the electrode geometry of _our

problem.

The

expression

obtained is _:

where S and

Eo

are the emlssive electrode surface and static electnc

field, respectively.

Formula

(6)

indicates that the evolution of the

discharge

_is

strongly dependent

on the electric circuit behaviour. The latter _is described

by

the

following

set of

equations

:

~(

)

~ _~

/~ ^~~~ /~ ^~~~

^~~~

dvc~

_i

I

_C~ ^{~ ~~~}

dvc~

_i

j "$(I-I). ⁽⁹⁾

Although secondary

cathode processes, such _as electron emission

by photon

incidence and

positive

ion

bombardment,

_are

important,

since the _aim of this work is to

danfy

the transport effects _in the

pre-breakdown penod

of the

discharge,

these processes are

neglected

in a first

approximauon

4. Electric field calculation.

The electric field in the gap is calculated _at the

beginning

of each ume step

by solving

Poisson's

equation.

The gap geometry

(electrode length

» electrode

separation)

_imposes the choice of the

appropriate

model used to calculate the space

charge

field. In this

work,

the electron and _ion

population

is distnbuted

uniformly

_in

parallel planes (Fig 2)

with _a surface

charge density

^win

C/m2

The emissive surface of the cathode _is

approximated by

_a

plane.

The

spatial

cell size _m the field calculation is

larger

^{than the}

Debye length A~ (crn) given by

Delcroix

[23].

~ = 743

(

_T~

/n~

^~'2

(10)

where _{T~ is} the electronic

temperature

in eV and _{n~ in}

cm~3

In each

cell, dE/dx

is assumed to be _constant and

change

_in the electnc field in the

j-th

cell

depends

_on the

charge

in the

rema1nlng cells and is given

by

the

following

_{equation :}

~ i _~ b ²

~ l/2

~~

E0_all

,~

^{~~ ~ ~~~}

^~'~

~ ^~ ~

~'~

C a A

~

II

Fig

Equation (I I)

shows that electnc field _vanations in the

parallel planes

_are small

compared

to the field variation _on the

discharge

_axJs which _is

perpendicular

to the

planes.

It should be noted that this

space-charge

electnc field calculation method is ngorous

only

_in the case of _a

uniform

discharge

wide

spread

_{in a} direction

parallel

to the electrodes.

5. Ini6al and

boundary

conditions.

The electrodes of the

discharge

have been considered _as

nearly perfectly absorbing

surfaces.

To avoid _a

singular

behavlour which would

complicate

the numencal treatment of the

problem

_in the

neighborhood

of the

electrodes, they

_are assumed to have _a

charged particle density boundary

condition no due to the ambient radiation. Since the

charge density produced

_m this _{way is very} small

compared

with those

produced by

the electnc

field,

the

electrodes _conserve their

perfectly absorbing _properues.

The

voltage applied

to the gap is determined _{at any time,}

by

the electric circuit behavlour.

The electnc field _across the gap is

iniually homogeneous

and determined

by

the

gap-voltage

and the separation of the electrodes

6. Nunlerical soludon.

A numencal code _was wntten in order _to evaluate the

spatio-temporal

evolution of _a

discharge strongly homogeneous iniually

The continuity equations ^{for electrons and} ions were solved

nurnencally

_{on a} non-uniform mesh which had _a very fine

spatial

resolution _in both the cathode and anode fall

regions

On _this

mesh,

the convective terns of

equations (I)

and

(2)

_were

computed using

the Phoenical flux

corrected-transport algonthm

of Boris and Book

[18]

^{modified for} a use on a non-uniform mesh

by

Morrow and Cram

[24].

The time

step

taken for the computations was limited

by

the

following

relation

[18]

:

At _« 0.5

All(

v~

As denoted

by

Morrow

[24, 25]

the

following

_{points are}

important

when the electnc field

approaches

zero due to

space-charge

effects and also at the

boundary.

(I)

The FCT

algorithm

_uses drift velocities _vi

~ jj2 evaluated at points _x~

~ jj2

lying midway

between mesh

points.

These velocities must be obtained

using

the value of the electric field

E,

~ j/~

computed

at the mid

points.

(2)

For

stability

and accuracy the

integration

of _source term must be second order in -time.

Thus,

intermediate solutions _are

computed

half _a time step

ahead,

source terms and velocities

computed

at t + At

/2

are then used _to advance the solution _a full time

step,

^from t to

t + At.

At any

step

^of the numencal

integrauon

the value of the

discharge

current is calculated and

injected

into the system ^of equations which describes the electnc circuit behaviour. This system is

integrated numencally

_using a 4-th order

Runge-Kutta

method.

Results _are

presented

_in the form of

spatial

distribution of electrons and _{ions as} well _as

spatial

electnc field distribution at different _{time instants, in} order to

emphasize

the role of transport terms

7. Results.

At the

beginning

it is necessary ^{to note} that the numencal method used

(Flux

Corrected

Transport)

_is not the

only

method

capable

_{to represent transport} terms. However _vanous tests

including sham gradients,

have been realized and

they

have proven

[24, 25]

a very

good

behaviour of the FCT

algonthm.

1026 JOURNAL DE

PHYSIQUE

II N 9

Thus it is able _to describe electnc

discharge phenomena especially

the cathode and anode region formauon.

7 COMPUTATIONAL coNDiTtoNs The

computational

conditions _are chosen

according

to

the laser

discharge

^of

Spyrou [10]

which _is descnbed in

paragraph

2. The electrode separation is I _cm and the electrode

length

is 70 _cm. The electnc circuit

parameters

are

li~=0.lQ, Le=5nH, C~=C~=60nF.

The _emlssive cathode surface is taken

equal

to 35

cm2

and _it has been used for the field

calculation.

For the results

presented here,

the 1nltial electron and ion densiues _are 10~ _cm~

uniformly

distnbuted _in the _gap.

7.2 CATHODE _{REGION FORMATION.}

Figure

3 shows the evolution of the electronic

density

distribution _across the gap for _an

applied voltage

of12 kV and for different _pressures

(30

and 50

torrs) respectively.

The electrons _{move to} the anode under the influence of the

homogeneous

field before volume ionization becomes

Significant

This removal of electrons _away from the cathode reveals positive ions which

produce

_{a positive} net

charge density (Fig. 4)

in the

neighborhood

of the cathode. This _is

equivalent

to _a

space-charge

effect which _is

responsible

for the cathode fall _region. As _can be observed _m

figure 4,

the

positive

ions move towards the cathode inside tbJs

high-field

_region. On the other

hand,

_a negative space

charge density

_occurs at the

anode,

due to _a s1mllar process.

Figure

5 shows the electric field distribution _across the gap at different _urne instants for _an

applied voltage

of 12 kV and different _pressures. For the 30 torr case, the formation of _a disturbance to the

homogeneous

electnc field distnbution _appears 24.3 ns after the

voltage application

across the gap and becomes

significant

I _ns later A similar

mechanism, although slightly slower,

is observed at a pressure of 50 torr.

7 3 THE QUASI-NEUTRAL REGION. It _can be _seen from

figures 3,

4 and 5 that the electric field remains

homogeneous

wlthln most of the _space

occupied by

the

discharge,

with _a

relatively

low field

strength (3-5 kV/cm).

This low

field, quasi-neutral

_region is s1mllar to the

positive ^{column of} a

glow discharge.

8. Discussion.

In 1963 Head

ill presented

his work concerning the

N~-laser Meanwhile,

_many studies have been

perfornled,

_aiming at

contnbuung

to the

understanding

of its

physical

mechanism and at

building expenmental

structures in order to optimize ^the laser

output efficiency.

These studies _are of

great

importance ^{because the results} obtained _can be used to describe the

exclrner laser

discharge. However,

the

knowledge

of the

N~-laser operation

is not yet

satisfactory, especially regarding

the

discharge

evolution. This _is due to the fact that the short

maintaining

time and the

high voltage

and current values of the

discharge

do not

permit

us to obtain accurate current measurements On the other

hand,

theoretical studies have been realized based _on the

assumption

that the electric field distribution remains

homogeneous dunng

the

discharge

Thus transport terms have been

neglected

and the

discharge

parameters have been determined

by singly

time

dependent

models.

However,

_{as can} be _seen in

figures

4 and

5,

the

discharge, initially govemed by

_a

homogeneous

electric

field, rapidly

evolutes

towards _a classical

glow discharge

which _is characterized

by

_{a very}

sharp

field

gradient

_near

the cathode. As the electrons leave the cathode _zone,

they

_are accelerated in the

high

field

$

©

0^~

l B ~

x (cm)

lB~~

^{~~~~~~~ ~~~~~}

~1

E u

j

i o^~

a ^°

o-B .1 .2 .3 .4 .5 .6 .7 .8 .9 1-B

x (cm)

Fig. 3 Electron

density

distnbution

along

the gap ^at different times for V

= 12 kV. a) 30 torr t.

5 5, 12.9, 165, 192, 21 4, 23 4, 25 2, 25.7 _ns b) 50 tom

,

t 64, 15 4, 20, 23.3, 26 2, 28.7, 31.1, 32 9 ns

io28 JOURNAL DE

PHYSIQUE

II N 9

net

cha~ge density

5

a 4

3

2

I m

E

f

-.

~ 2

-. 4

B.a .1 .2 .3 .4 .5 .6 .7 .8 .3 1.B

x <cm)

net

cha~ge density

5

4

3

2

I m

e o

«

-.

~ 2

3

-. 4

-.5

B-a .1 .2 .3 .4 5 6 .7 .8 .3 1-B

x (cm)

Fig

4 Net

charge density

distnbutlon

along

the _{gap at} different times for V

= 12 kV

a)

30 torr

,

t 25 4, 25 8, 26 2 _ns.

b)

50 tom t 32.3, 32 5, 32.9 _ns

48

kvcm"~ ~

38

o

z

~~~.0

cm

.1 .2 .3 .4 .5 .G ? .8 .9 1.

~~ ^{-i b}

28

1o

z

-iB

o-o .1 .2 .3 4 .5 .G ? .8 .9 1.

Fig.

5. Electric field dlstnbution

along

the gap at different times for V

= 12 kV. a) 30 tom, t 24.3, 24.7, 25, 25.2, 25.4, 25.8, 26 2 _ns b) 50 tom, t 31 1, 31.6, 32, 32.3, 32 5, 32.9 _ns.

iu3u JOURNAL DE

PHYSIQUE

II N 9

region

of net

positive

_space

charge (Fig. 4).

These

high

_energy electrons at the exit of the cathode _region

they

collide with the

nitrogen

molecules

causing

iomzation and excitation,

losing rapidly

their energy. The _major

part

^of the

discharge

has _a

quasi-neutral

behavlour charactenzed

by

low

homogeneous

^field values. Inside this _zone the ionization _is maintained

by

an electronic

temperature

^{of about 4 eV.}

Regarding

the

positive

ions inside the electnc

field

drop

_zone,

they gain

energy and _move towards the cathode

Because laser operauon is

strongly dependent

_on the circuit

behaviour,

_we

present (Fig. 6)

a set of results

concerning

^the

applied voltage

in the gap and the

discharge

current

pulse

for different pressures.

Obviously,

the current

pulses

obtained

satisfy

the conditions for _a

N~-laser [10-12].

The current grows very

rapidly

with _{a nse} ume of _many kA.

Figure

6 shows that the _maximum value of the

discharge

current _is obtained _m the 30 torr pressure

region.

This _means

that,

inside this region, the electnc

efficiency

is better than in the other pressure

regions,

because the electnc energy transfer between the extemal circuit and the

discharge

_is

optimum [10, 11].

According

to the above results the _maximum laser output ^is

expected

to occur inside the above pressure region This

assumption

is in agreement ^with

experimental

results

[10-12]

and it will be treated _m another future

study

kV kA

ns

o 25 50

Fig. 6

Voltage

(

Vc~)

^applied to the gap and discharge current

(i)

for 12 kV at lo, 30, 50 and 70 tom

9.

Summary.

A

computer

^{simulation of} a

transversally

excited

discharge

_in

Nitrogen

has been realized

including _transport

terms. These terms

rapidly

_{cause a} disturbance _to the

initially

homo- geneous electnc field distnbution and the appearance of a cathode and _an anode fall _region

Thus

high

electnc field values

(~30kV/cm)

_are obtained _near the cathode causing a

displacement

of ions towards the cathode. The _main

part

^{of the}

discharge

is

quasi-neutral plasma

state with _an electronic temperature distnbution of about 4eV. An interaction

between the extemal circuit and the

discharge

has been introduced _in the calculation and the

discharge

current is obtained.

References

[Ii

HEARD H G., Nature 200

(1963)

667.

[2] GODARD B, IEEE J

Quant

Elec QE lo

(1974)147.

[3] NAGATA I., KIMURA Y, J Appl. Elec Sci Inst 6 (1973) l193

[4] GIRARDEAU-MONTAUTJ P, GtRARDEAU-MONTAUT C., Nouv Rev

Opt

5

(1974)179

[5] ALt A W, KOLB A C, ANDERSON A. D,

Appl Opt

6

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2115

[6] BASTINGS J, SCHAFFER F. P, STEVER B.,

Opt

Elec 4

(1972)

43 [7] SHIPMANN J D,

Appl Phys

Lett 10

(1967)

3

[8] SWAB A J, HOLLINGER F W, IEEE J

Quant

Elec. QE12

(1976)

183

[9] FITzSIMMONS W. A, ANDERSON L W, RIEDHAUSER C. E, WATIEK J M

,

IEEE J

Quant

Elec.

QE12 (1976) 624

[10]

SPYROU N., Thesis, Unlversitd de Paris-Sud, Centre

d'orsay (1979)

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[1Ii

SPYROU N., LEPRINCE P and MILLEON H., Rev

Phys Appl

15 (1980) 1459

[12]

LESPtNASSE G, PIGNOLET P, HELD B, Rev. Phys

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