Deviation from local thermodynamic equilibrium in a cesium-seeded argon plasma

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Deviation from local thermodynamic equilibrium in a

cesium-seeded argon plasma

Citation for published version (APA):

Stefanov, B., Zarkova, L., & Veefkind, A. (1985). Deviation from local thermodynamic equilibrium in a cesium-seeded argon plasma. (EUT report. E, Fac. of Electrical Engineering; Vol. 85-E-152). Eindhoven University of Technology.

Document status and date: Published: 01/01/1985 Document Version:

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in a Cesium-Seeded

Argon Plasma

By

B. Slefanov, L. Zarkova and A. Veefkind

EUT Report 85-E-152 ISBN 90-6144-152-8 ISSN 0167-9708

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Electrical Engineering Eindhoven The Netherlands

DEVIATION FROM LOCAL

THERMODYNAt~IC

EQUILIBRIUM

IN A CESIUM-SEEDED ARGDN PLASMA

by

B. Stefanov

L. Zarkova

A. Veefkind

EUT Report 85-E-152

ISBN 90-6144-152-8

ISSN 0167-9708

Coden: TEUEDE

Eindhoven

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Deviation from local thermodynamic equilibrium in a cesium-seeded

argon plasma / by B. Stefanov, L. Zarkova and A. Veefkind.

-Eindhoven: University of Technology. - Fig., tab. - (Eindhoven

University of Technology research reports / Department of Electrical

Engineering, 1SSN 0167-9708; 85-E-152)

Met lit. opg., reg.

5150 535 UDC 537.5 UG1 590

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1. Introduction 1

2. Four level model of Cs

3

3.

Balance equations

5

4. Collisional rate coefficients

6

5.

Radiative rate coefficients and escape factors 7

6.

Diffusion 9

7.

Solution of the balance equations 11

8.

Results and discussion 12

9.

Conclusions 14

Acknowledgement 15

References 16

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Deviation from local thermodynamic equilibrium

in a cesium-seeded argon plasma

B. Stefanov and L. Zarkova

Institute of Electronics, Bd. Lenin 72, Sofia 1184, Bulgaria

A. Veefkind

Group Direct Energy Conversion, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract

The possibility of deviations from local thermodynamic equilibrium of a cesium seeded argon plasma has been analyzed. A four level model of cesium has been employed. Overpopulations of the ground state and the first excited state as well as the corresponding reduction of the

elec-tron density are calculated for cylindrical discharge structures by solving stationary rate equations. Numerical results are presented. These results indicate that in a large regime of plasma conditions the LTE assumption is valid for electron temperatures larger than 3000 K.

1. Introduction

The objective of this work is to develop a model to analyze deviations from local thermodynamic equilibrium (LTE) in cesium seeded argon plas-mas. Specific evaluations are carried out considering conditions which generally apply to noble gas MHD generators.

In the case of LTE the charged particles concentration n = n is

de-e i

termined by the Saha equation:

(1 )

(7)

+ 2 3/2

Cs ). A = (2nmk/h) and Ei/k = 45190 K is the ionization potential of Cs in temperature units.

bEi/k is the lowering of the ionization potential (see for example

[1] ):

nCs is the total neutral Cs density (note that the total sity here below is denoted as nC + n ). When

s e o n2 denotes

+

Cs

+

Cs (2 ) den-the equilibri-um (Boltzmann) population density of the £-th level, nCs can be written

as

n =

Cs

Z is the partition function of neutral cesium defined by

o

where E[

=

0 for the ground state.

(3 )

(4 )

Further we define partial local thermodynamic equilibrium (PLTE) as a situation in which the electrons have a Maxwellian distribution at a temperature Te and the population densities are equal to b~ times their equilibrium values (n£ = be n~). The neutral Cs density is then given by n = Cs max I: b 0

l=

1

t

ne

The analogue of the Saha equation becomes

2

n _eInC _s

o

The PLTE equivalent of Z becomes

Z =

(3a)

(ia)

(8)

o

An adequate estimate of Z (or Z ) for sider levels up to

Ee

= 3.80 eV. For

T up to 5000 K requires to

con-e

practical purposes it is conve-nient to average the close-lying levels reducing them to 10 effective levels according to Table 1.

Table 1. Level scheme of Cs

Level State _{Degeneracy, ge}

_{Ee '}

eV

1 6S 2 0 2 6P 6 1.432 3 5D 10 1.805 4 7S 2 2.298 5 7P 6 2.714 6 6D 10 2.804 7 8S+4F 16 3.032 8 8P+7D 36 3.217 9 9S+5F+ ••• +lOP 54 3.446 10 Empirical

-

825

-

3.68

The last numbers g10 and E

10 are adjusted to fit the elaborated calcu-lations [1]. In the case of LTE the contribution of the last effective level to ZO is 6% at 5000 K and less than 2% at 4000 K.

2. Four level model of Cs

When a situation not far from equilibrium is considered (be close to unity), usually the approximation can be made that all levels ~ 2 are in equilibrium with the continuum (be ~ 1) whereas the ground level

e=

1 deviates from equilibrium (b_l

*

1). Therefore a convenient model should include at least three levels: the ground state, the first ex-cited state and the continuum. To have an estimate of the error intro-duced by neglecting levels between

f

= 3 and

l

=

l

we also introduce

max

in the balance equations the contribution of the

l

= 3 level to f = I and

f

= 2 and consider b2

*

1. All levels with

P

~ 3 will further be

(9)

included in eqs. (3a) and (4a) with values of be = 1 (i.e. all high levels are supposed to be in equilibrium with the continuum). Therefore a four-level model of Cs is considered according to Table 2.

Table 2. Four-level model of Cs

Level Configuration Degeneracy

E,

/k,K Transition

pro-bability, s -1 1 65 2 0 2 6p 6 16620 _A21 = 3.59xl0 1 3 50 10 20950 _A32= 0.141xl0' 4 Continuum 1 45190

-The transition probabilities have been found as averages for 6P

1/2 and 6P

3/2 and correspondingly 503/2 and 505/2 according to (see e.g. Rosado

[2])

In this expression Aji is given by

where the oscillator strengths f

ij are given in Table 3 [3J.

Table 3. Oscillator strengths for Cs transitions

Transition Oscillator strength

65 - 6P

\

0.394 65 - 6P3 / 2 0.814 6P\- 503 / 2 0.251 6P3/ 2 - 503/ 2 0.0211 6P 3/2 - 50 5/ 2 0.204

(10)

3. Balance equations

Following Rosado's notation

[2]

we write the balance equations for the ground state and the first excited level under the assumption of a Maxwellian distribution of electrons:

(Sa)

(Sb)

where

6b = b -1

q q '

R

=

nO/no

=

(g /g ) exp [- (£ - £ )/kT ]

=

K /K ,and

rq r q r q r q e qr rq

K are the electron excitation (de-excitation) rates r + q • K i are

t~~

ionization rates r + i;

Ki!)

are the radiative

recombinatio~

rate coefficients and A (indexed corresponding to the transition considered) represents the escape factor for line or continuum radiation. Note that the indices

I,

r, q stand for atomic levels while i stands for the Cs ion.

I f eq. (Sb) is multiplied by R₂₁, added to eq. terms including index 3, then a balance equation

(Sa) while dropping

+

for Cs will follow. In eq. (Sa) the loss of ground state particles (the first term of the L.R.S.) is compensated by what is coming from the excited level, from the continuum and from the inward flow of ground level atoms. For stea-dy state conditions this flow is equal to the outward flow of ions

(11)

represented by Oif > 0 defined by

Oif

In this expression ne~A

=

-nl~l where ~A is the ambipolar diffusion velocity. Eq. (5b) represents the balance of the 6P level and is built the same way, but also the 50 level (index 3) is taken into account and the diffusion of the 6P level particles is neglected because n 2

«

nj.

4.

Collisional rate coefficients

The electronic excitation rate coefficients K (and analogous the rq

ionization rate coefficients K

ri) are given by

[4]:

Qrq being x = £/kT • e K rq the For Qrq Qrq 8 kT

i

= ( e) Trm cross kT e

<

£ = a(£

-f

£ /kT rq e section of rq Q rq can £ ) rq for -x xe Q (£ )dx, rq

the proces r .. q and be approximated by

£

<

£

<

£* rq

= a(£* - £ )

+

a*(£ - £*) for £*

<

£ rq

Then eq. (6) becomes

K

=

B[a(£ /kT + 2) exp (-£ /kT)-rq rq e rq e (a - aX) (£*/kT + 2) exp (-£*/kT )] e e with B (8 kT /Trm)! kT

=

8.58 x 10-20 _T _3/2_J _m/s. e e e (6 ) x being given by (7 ) (8 ) -3

According to the formula of Orawin (see for example

[4])

a - £ Then

2 rq in a eqs. rough approximation K rq (5) K13

«

_{K12 ; Kli}

«

K12

- exp (-£ /kT )/£ • Therefore, in the rq e rq

and K2i

«

K

(12)

electron-ic excitation rate coeffelectron-icients may be calculated according to Table 4.

Table 4. Collisional excitation cross-section characteristics

Transition a, m2

/J

a*, m2

/J

E /k, K E*/k, K Reference

rq

1 - 2 6.45 0.75 16620 23440 [5J

2 - 3 75 0 4330 7580

The figures in the last line of Table 4 are based on an approximation using the Drawin formula (see for example [4J with ~l

=

1 and ~2

=

1.6. The de-excitation rate (2 - 1) is taken as K21 = K12 exp (-\2/kTe)/3 in accordance with detailed balancing.

5. Radiative rate coefficients and escape factors

The radiative two-body recombination rate coefficients are given by

kT /"m) \ w K(2) ₍₈

_f

-x Qiqdx (9a) xe iq e 0

For small electron energies Qiq is proportional to E -1 , Qiq = mC i /2E:

Ki!) = 2(m/2"kT_e)\C_iq (9b)

-16 4 -14 4

For q = 1,2 C_iq

=

1. 77 x 10 m /s and 2.79 x 10 m /s correspond-ingly (see for example [6J).

(2 )

Now using the expression Rqi given in section 3 We come to

K(2)R(2) = (8 m/"k)\ (C

i T /g ) A exp [-(Ei - E )/kT )J =

iq qi q e q q e

(13)

The evaluations show that in the cases of practical interest (arcs and MHD conversion) the terms R(Z)K(Z)A(Z) (A(Z)

~

1) in the right sides of

qi iq iq iq

equations (5) may be neglected. Note that i f we consider the balance equation for Cs+ these terms will be important.

The escape factor for a bound-bound transition is given by

"

A = a/(1Ik r)2

rq qr (lla)

(see for example

[4],

where a

=

1.115 for a cylindrically symmetric plasma with a radius r. The absorption in a center of a line is

Z

k = e f n /Z1I£ me6\1

qr qr q 0 (llb)

and the half half-width 6\1 consists of two parts

(7]:

Van der Waals broadening due to collisions with Ar

(lle)

(Tg is the temperature of the heavy particles) and resonant broadening

(lld)

(A is the transition wavelength). The Van der Waals constant is

rq 1 Z 1

~6 =

3

_{C6(Cs6Pl/Z - Ar 1/ Z, l/Z) +}

3 [3

C6(:;~P3/6

- _{Ar 3/ Z, liZ) +}

3

_{C6 (Cs6P 3 / Z - Ar 3/ Z, 3/Z)]}

=

6.11 x 10 Jm for the first excited level of Cs, the values of C

6 corresponding to the different transi-tions being according to

[8]

6.13, 7.90 and 5.19 x 10-77 Jm6 in the order of their appearance in the above formula.

Finally for the transition 6P - 65 we come to

(14)

where s is the seed fraction, s = (n + n )/n ,

Cs e Ar

and r in m.

The second term in the last· parentheses in eq. (12) is equal to the ratio ~VR/~VW. Under the conditions of close-cycle MHO conversion it is small (- 0.1) so that the resonant broadening is not important and therefore the last factor in eq. (12) may be omitted.

Now let us consider the escape factor for the transition 3-2. In this case ~VR is much smaller than ~VW (compare eqs. (lld) and (llc)) and

therefore should be neglected. The Van der Waals broadening may be

estimated by a comparison with the transition 2-1: ~V (3-2)

w

~V (2-1)

w

(see for example

[9]).

Then

f12 c 0.674, f23

=

0.2 (values averaged over sublevels) and

6. Diffusion

As was mentioned in section 3 the diffusion term in eq. (5a) is

o

Dif = div (n~)/nl. The diffusion flux density is

where DA

=

2 D is the ambipolar diffusion coefficient. ia

(13)

(15)

The equation (14a) is easily transformed into

The radial dependences of n , T , Ti may be approximated by

e e (14b) (15a) (15b) (15c) where n

eo' Teo and Tio are values at the axis, Ti= is the ion tempera-ture outside the arc and p is the equivalent radius of the arc, p = r

o 0

in the notation of section 5. We found that equations (15) fit well the experimental data of [2] and [9] for a wide range of current densities

(104 to 3 x 106 A/m2).

Now we use eq. (14b) and eqs. (15) to find out the diffusion term:

Dif = -1 d- (pn W ) d ~ p p e A

(16 )

The function f(E-) is monotonic and equal to 1.5 at p = 0 and to 1.1 at

Po

p + =. For further estimations the last factor in eq. (16) will be taken at (p/p ) = 0.25: f(p/p ) ~ 0.9.

o 0

Typical values for a closed-cycle

MHD

generator are:

T ITi

~

3;

(2) 2 e

Ti=/T_{i ~} 0.75; taking into account the relation Rli = n/n~ from

sec-tion 3 we come to

(16)

The coefficient of diffusion of ions through neutrals D is determined

+

ia

+

mostly by interactions Cs -Ar. Data for the mobility ~ of Cs through Ar are available [10]: at normal density (N

=

2.69 x 1025 m-3)

2 0

~ = 2.07 cm

IVs.

To calculate D. we use the Einstein relation

o 1a

D = kT ~/e; taking into account the proportionality ~ - N

In

we come

g 0 Ar

to

and therefore

-10 2 2 2

DA = 2 D_ia

=

1.33 x 10 Tg(K )/p(Bar) m

Is

(18)

In obtaining eq. (18) we assumed that the product ~nAr does not depend on T which agrees well with the experimental data. On the other hand

g 3/2

it is well known [11] that pD - T

10.*

where 0* is the dimensionless collision integral. For a purely attractive or repulsive interaction

-6

with a potential energy - R (R is the distance between the particles)

Q* - T-2/6• As the potential well depth Elk of the Cs+-Ar interaction is of the order of 1000 K the attraction prevails; for a pair ion +

-1/2 2

neutral 6

=

4. Therefore 0* - T and pD - T in accordance with eq. (18) •

7. Solution of the balance equations

With small terms neglected the system of equations (5) becomes

(19a)

(19b)

(17)

solving this system of equations.

Simplified expression for obI and n /no can be obtained in the

follow-e e

ing way. We may assume ob

2

«

obI and ObI < 1. Let us denote R21A21A21/neKI2

=

_{QR and Dif/neK12}

=

QD; then ObI = _QR+ QD. If the seed fraction and the pressure are not very low, then a substantial deviation from LTE takes place only at low temperatures (T < 3000 K),

e

where (n

Cs + ne)/n1

=

1 (moderate degree of ionization). Using eq. (8) for K12 (without the second term in the brackets which represents a small correction), eq. (12) for A2l and eqs. (17) and (18) for the diffusion term we estimate the relative contribution of the diffusion to be

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-3

(T is in K, p is in Bar, r is in m and ne is in m ). For QD/QR > 1 the deviation from LTE is dominated by the diffusion. For a radiation es-cape regime QD/Q < 1 and

. R T 0.15 -3 g ObI = 1. 08 x 10 -(-T -s-r.s),,"o-. 5'-(-n-/-10""2"'"0 ) e e (21 ) (if 6b 2 « 6b1, ObI < 1) and (ncs

+

ne)/n1 < 1).

Note that the overpopulation of the ground state corresponds (for mod-erate degrees of ionization) to a decrease in the ion denSity,

Eq. (22) follows from eqs. (I), (4), (1 a) and (4a) provided n

l = nCs. It is consistent with the result of Mitchner and Kruger [4] (their equation 3.6 of Ch. IX).

8. Results and discussions

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charges with radial distribution lSa, lSb and lSc. The figures show

of n , T and e e Ti according to eqs. o n /n as functions e e 6b₁, 6b₂, ne and

of the electron temperature. Parametric variation has been applied involving the radius, the gas temperature, the pressure and the seed fraction according to Table S.

Table S. Parametric variations.

Curve radius gas temperature pressure seed fraction

a 0.1 mm T /2 0.1 bar 0.2 x 10- 3 e b 0.316 mm T /3 0.316 bar 1.0 x 10- 3 e c 1.0 mm T /4 1.0 bar 5.0 x 10-3 e

The general feature of all the figures is that strong deviations from LTE occur at electron temperatures lower than 3000 K. This is caused by the fact that at such low electron temperatures the electron density becomes so small that the electron collisions can no longer dominate the diffusion and radiation processes. The other parameters change the electron temperature below which deviation from LTE occurs, but for the given parameter variation this change is not dramatical.

Figs. 1 - 4 show the influence of the radius of the discharge. A smal-ler radius results in a larger deviation from LTE because of a larger diffusion and a smaller absorption of radiation. The increase of 6b

1 and 6h

2 at high electron temperatures occurs when the neutral cesium concentration considerably decreases due to ionization. The electron density keeps approaching the equilibrium value with increasing elec-tron temperature (Fig. 4). The decrease of the elecelec-tron density between T = 4000 K and T = 5000 K is caused by the determination of the value

e e

of the gas temperature. Because the gas temperature is determined as a fraction of the electron temperature it increases with T •

e

Since the pressure is a fixed number the total cesium density (neutrals

o

plus ions) decreases with Te causing a decrease of ne when its value is close to complete ionization.

(19)

Figs. 5 - 8 exhibit the effect of variation in gas temperature. Larger gas temperatures correspond to larger deviations from LTE which are caused by larger escape factors (eqs. 12, 13) and larger diffusion coefficients (eq. 18). The large differences in the maximum values of ne (Fig. 7) result from. the different values of nCs + ne for the three cases considered following from the three different temperatures all

taken at the same pressure.·

Figs. 9 - 12 demonstrate how the pressure affects LTE. Lower pressures cause a larger deviation than higher pressures due to a larger

diffu-sion. The differences in the curves for n (Fig. 11) are again the

e

result of different values of nC + n • s e

Figs. 13 - 16 represent changes of the nonequilibrium situation due to variations in seed fraction. From the parametric variations considered the seed fraction variation affects the deviation from LTE the most (compare Figs. 1, 5, 9 and 13). A low seed fraction increases the es-cape of radiation according to eqs. 12 and 13. Furthermore a lower seed fraction results in a lower electron density and therefore in lower collisional (de)excitation rates. Fig. 15 exhibits again the different values of nCs + ne for each curve.

9. Conclusions

The assumption of local thermodynamic equilibrium has been analyzed for stationary discharges in cesium seeded argon plasmas. The following conditions have been considered:

- 2000 K

< T

< 5000 K

e - T /2

< T

< T /4

e g e 0.1 mm

<

radius

< 1.0

mm 0.1 bar

<

p

< 1.0 bar

- 0.2 x 10-4

<

seed fraction

< 5.0 x 10-

4

For these conditions a noticable deviation of n from its LTE value is

e

only found when Te

<

3000 K. Deviations of n

l and n2 from their equili-brium values occur also at higher electron temperatures at large degrees of ionization.

(20)

The deviation from LTE is amplified by the following changes of

para-meters:

-

increase of gas temperature.

-

decrease of radius.

-

decrease of pressure.

-

decrease of seed fraction.

Acknowledgement

This work has been carried out as a part of the research program of the Shock tube MHD Project of the Group Direct Energy Conversion at the Eindhoven University of Technology.

The investigations have been accomplished during a visit of the first and second author to the Eindhoven University of Technology. This visit has been financially supported by the Dutch Organization for Pure Scientific Research (Z.W.O.).

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References

[1]

Drawin, H.W. and P. Felenbok

DATA FOR PLASMAS IN LOCAL THERMODYNAMIC EQUILIBRIUM.

Paris: Gauthier-Villars, 1965.

[2]

Rosado, R.J.

AN INVESTIGATION OF NON-EQUILIBRIUM EFFECTS IN THERMAL

ARGON PLASMAS.

Ph.D. Thesis. Eindhoven University of Technology, 1981.

[3]

Smirnov, B.M.

ATOMIC COLLISIONS AND ELEMENTARY PROCESSES IN A PLASMA

(in Russian).

Moscow: Atomizdat, 1968.

[4]

Mitchner, M. and Ch.H. Kruger, Jr.

PARTIALLY IONIZED GASES.

New York: Wiley, 1973.

Wiley series in plasma physics

[5]

~,S.T.

and A.C. Gallagher

ELECTRON EXCITATION OF THE RESONANCE LINES OF THE

ALKALI-METAL ATmlS.

Phys. Rev. A, Vol. 17(1978), p. 551-560.

[6]

Borghi, C.A.

[ 7]

DISCHARGES IN THE INLET REGION OF A NOBLE GAS MHD GENERATOR.

Ph.D. Thesis. Eindhoven University of Technology, 1982.

Hindmarsh, W.R. and J.M. Farr

COLLISION BROADENING OF SPECTRAL LINES BY NEUTRAL ATOMS.

Prog. Quantum Electron., Vol. 2(1972), p. 141-214.

[8]

Mahan,

J.D.

VAN DER WAALS CONSTANT BETWEEN ALKALI AND NOBLE-GAS ATOMS.

II. Alkali atoms in excited states.

J. Chern. Phys., Vol. 50(1969), p. 2755-2758.

[9}

Wetzer, J.M.

SPATIALLY RESOLVED DETERMINATION OF PLASMA PARAMETERS

OF A NOBLE GAS LINEAR MHD GENERATOR.

Ph.D. Thesis. Eindhoven University of Technology, 1984.

[10}

Hasted, J.B.

PHYSICS OF ATOMIC COLLISIONS.

London: Butterworth, 1964.

Butterworths advanced physics series

[11}

Hirschfelder, J .0. and Ch.F. Curtiss, R.B.

~

MOLECULAR THEORY OF GASES AND LIQUIDS.

New York: Wiley, 1954. 4th printing 1967.

Structure of matter series

(22)

1~ 10'

-,

10 -3 10 \

\

~"--~

~

• ---

---b

--

-==---~ 2000 lOOO 4000 5000 Tel KI

Fig. 1. Relative overpopulation of the ground

state (6b₁> as a function of electron tempera-ture (T e). P""O.316 bar; a. raO.l mm; 15 T -T /3; seed fraction-lxlO- 3 • g e

b. raO.3t6 mm; c. r-l.a trim.·

20 _3 n (10 m I

•

10 5

o

2000 Fig. 3. 5000

Electron density (n ) as a function

•

of electron temperature (T e).

p-O.3L6 barj T -T /3; seed fraction-lxlO- 3•

g e a. r-0.1 mm; c. ral.D mm.

\

~~

,--a 2 10

~

_b

'"

ni'

2000 3000

~

----

~

--

---4000 5000 T.IK I

Fig. 2. Relative overpopulation of the first excited state (6b 2) as a function of electron

temperature (T e).

p-O.316 bar; T -T /3; seed fraction-lxlO- 3 •

g •

a. raO.l mm; b. raO.3i6 mm; c. ral.D mm.

1.0 ne /Reo

cJf/

.8

1'1 /

.6

f

,/

.ZL-______ - L ________ L -______ ~ 2000 3000 4000 50 aD T. (K I

Fig. 4. Ratio of electron density to its

o

equilibrium value (De/oe> 8S a function of

electron temperature (T e).

p-O.3i6 bari T -T 13i seed fractloa-ixiO- 3•

g •

(23)

6b, 2 10 10' \ \\ ~\ / '

"

a IIi' b . / -2 10 2000 JOOO 4000 5000 T. IKI

Fig. 5. Relative overpopulation of the gt'ound state (obI> as a function of electron tempera-ture (T ).

e

p-O.316 bar; seed fractlon-lxlO- 3; r-O.316 mm. a. Tg-1/2; b. Tg-T/3; c. T g-r/4. 20 20 -3 "oliO m I I I

I

1 1~

15 10 5

o

2000 FIg. 7. i

I

.

,

. ! JOOO 4000 5000 Tol K I Electron density (n ) as a function of

e

electroD temperature (T e).

p-O.316 bar; seed fractioo-lxlO- 3; r-O.316 mm.

a. T -r /2; b. T -T /3; c. T -T /4. g e g e g e

,

10 -3 10 l -_ _ _ --L _ _ _ _ 1--_ _ _ ---l 2000 JOOO 4000 5000 TolK I

FIg. 6. Relative overpopulation of the first

excited state (ob_{2) as a function of ele<tron}

temperature (Ie)'

p-O.316 bar; seed fractloa-lxlO- 3; r-O.316 mm.

a. Tg-re/2i b. Tg-Te/3; c. I g

-T/4.

1.0 "./n.o

tV

.8

rt

.6 .4 2 0 2000 JOOO 4000 5000 To I K t

Fig. 8. Ratio of the electron density to ita

equilibrIum value (n Ina) as a function of

e e

electron temperature (Ie)'

p-O.316 bar; seed fract1on-lxlO- 3 ; r-O.316 mm.

a. T aT /2; b. T -T /3. c. T -T /4.

(24)

,

_,

, ,

I

1 10' i \

~

1\ \

~"'--

~

,...---3 10 2000

~

3000 b c ~

V

~ 4000 5000

Fig. 9. Relative overpopulation of the ground state (ObI) as a function of electron tempera-ture (r ).

e

T -T 13- seed fractlon-lxlO- 3,' r=0.316 mm.

g e '

a. p-O.l bar; h. p=-O.316 bar; c. p""l bar.

50 I ZD -3) De 1D m 40 30 20 10

I

I o 2000

.!i-/

/

/L

b

k

V

•

---3000 4000 T.IK) 5000

Fig. 11. Electron density (De) as a function of electron temperature (T

e).

T -r /3; seed fraction-lxlO- 3; r-O.JI6 mm.

g •

a. p-O.l bari b. paD. JIb bar; c. pal bar.

2 10

\

l\~

I

•

. /

~

i / '

i

i~

_~

-lL

--~

...,

----lIf' 2000 3000 4000 5000

Fig" 10. Relative overpopulation of the first

excited state (6b₂) as a function of electron temperature (T

e).

T -r /3; seed fractlon-lxlO- 3; raO.Jl6 am.

g e

a. p-O.l bar; b. p-D.316 bar; c. p-l bar.

1.0 .6

1/

j

.4 .2 .0 2000 3000 4000 5000

Fig. 12. Ratio of the electron density to ita equilibrium value (n Ina) 8S a function of

e •

Tg-Te/3; seed fractlon-lxlO- 3; r-O.316 mm.

(25)

lib, 10' I , 10'

I

103 10 '

\

l i I

'"

~

-,

10

~---

.l!---

---~

----

b ~ c 3000 4000 5000 T.' K)

Fig. 13. Relative overpopulation of the ground state (ObI> as a func.tion of electron

ture (1 ). e p-O.316 bar; r-D.316 mm; T -T /3. • e fraction-O.2xlO- 3 ; a. seed b. seed fraction-l,OxlO- 3j c. seed fraction-S.OxlO- 3 , 80 tempera-, 2. -3) n. 10 m

_I

i

I

60

I

V

I

;II

40 20

/1

V

b

~

v---.

o

2000 3000 4000

Fig. 15. Electron density (n

e) as a function of electron temperature (T

e). p-Q.316 bar: r-O.316 mm; Tg-Te/3.

a. seed fractlon-O.2xlO- 3• b. seed fraction-l.OxlO- 3 ; c. seed fraction-S.OxlO- 3• I _I

I

\

~

\"

~

~~'

I

10' 2000

~

3000

.---

---b

---4000 5000

Fig. 14. Relative overpopulation of the first excited state (bb 2 ) as a function of electron

temperature (T

e).

p-O.3Lb bar; r-O.31b 1IIDl; T -T /3.

• e a. seed fractlon-O.2xlO- 3 ; b. seed fraction-l.OxlO- 3; c. seed fractlon-S.OxlO- l • 1.0 .8

'7/

a/

7

.6 .4

I

J .2

1/

o

2000 3000 4000 5000 Tt ' K)

Fig. 16. Ratio of the electron density to its

o

equilibrium value (De/De> as a fUDct~OD of

p-O.316 barj r-O.316 mm; T -T /3 •

• e

a. seed fractlon-O.2x10- l ;

b. seed fractloa-l.OxlO-lj c. seed fraction-S.OxlO-3•

(26)

Eindhoven University of Technology Research Reports (IS5N 0167-9708) :

(138) Nicola, V.F.

~LE SERVER QUEUE WITa MIXED TYPES OF INTERRUPTIONS:

Application to the modelling of checkpolnting and recovery in a transactional system.

EUT Report 83-E-138. 1983. ISBN 90-6144-138-2 (139) Arts, J.G.A. and W.F.H. ~

(140)

TWO-DIMENSIONAL MHO BOUNDARY LAYERS IN ARGON-CESIUM PLASMAS. EUT Report 83-E-139. 1983. ISBN 90-6144-139-0

Willems, F.M.J.

~TION OF THE WYNER-ZIV RATE-DISTORTION FUNCTION. EOT Report 83-E-140. 1983. ISBN 90-6144-140-4

(141) Heuvel, W.M.C. van den and J.E. Daalder, M.J.M. Boone, L.A.H. Wilmes "fNTE'RRUPTION OF A DRYTYPE: TRANSF"""'O'RMERIN NOLOAD BY A VACUUM -CIRCUIT-BREAKER.

(142 )

(143)

EUT Report 83-E-141. 1983. ISBN 90-6144-141-2 Fronczak, J.

DATA COMMUNICATIONS IN THE MOBILE RADIO CHANNEL. BUT Report 83-E-142. 1983. ISBN 90-6144-142-0 Stevens, M.P.J. en M.P.H. van Loon

~TlFUNCTIONELE I/O-Souws'T'EEN.

BUT Report 84-E-t43. 1984. ISBN 90-6144-143-9 (144) Dijk, J. and A.P. Verlijsdonk, J.C. ~

DIGITl>.L TRANSMISSION EXPERIMENTS WITH THE ORBITAl", TEST SATeLLITE. EUT Report 84-E-144. 1984. ISBN 90-6144-144-7

(145) weert, M.J.M. van

~MALISATIE VAN PROG~LE LOGIC ARRAYS. EUT Report 84-E-145. 1984. ISBN 90-6144-145-5

(146) Jochems, J.e. en P.M.C.M. van den Eijnden

~Zm:; IN SE;ltJENI'IELE CIICUITS.

EUT Re!X>rt 85-E-146. 1985. ISBN 90-6144-146-3

(147) Rozendaal, L.T. en M.P.J. Stevens, P.M.C.M. van den Eljnden DE REALtSAT!E VAN BEN MULT"iFiiNCTIONELE r/O-coNTROLLER'"'METBEHULP

VAN EEN GATE-ARRAY.

EUT Report 8S-E-147. 1985. ISBN 90-6144-141-1

(148) Eijnden, P.M.C.M.

A COORSE W FIEll l'KlGRl\!otWlLE IDGIC.

EUT Report 85-E-148. 1985. ISBN 90-6144-148-X (149) Beecknan, P.A.

MILLIMEl'ER-WAVE l\NTENNA MEASlJREMENrS wrrn WE 1IP8510 NE.ThORK

l\Nl\LYZER •

EUT Re!X>rt 85-E-149. 1985. ISBN 90-6144-149-8

( 150) (151 )

Meer, A.C.P. van

E'"X'"AMENRESULTATEN IN CONTEXT MBA.

EUT Report 85-E-150. 1985. ISBN 90-6144-150-1 S. and W.M.C. van den Heuvel

CURRENT INTERRUPTION IN A LOW-VOLTAGE FUSE WITH

ABLATING WALLS.