Investigations on the positive column of a medium pressure neon discharge

Investigations on the positive column of a medium pressure

neon discharge

Citation for published version (APA):

Smits, R. M. M. (1977). Investigations on the positive column of a medium pressure neon discharge. Technische

Hogeschool Eindhoven. https://doi.org/10.6100/IR79424

DOI:

10.6100/IR79424

Document status and date:

Published: 01/01/1977

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be

important differences between the submitted version and the official published version of record. People

interested in the research are advised to contact the author for the final version of the publication, or visit the

DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page

numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners

and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

• You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please

follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

providing details and we will investigate your claim.

INVESTIGATIONS ON THE POSITIVE COLUMN

OF A MEDIUM PRESSU~ NEON DISCHARGE

PROEFSCHRIFT

TER VERKR1~GING VAN DE GRAAD VAN DOCTOR IN DE TECHNISC!!E WETEw$CHlIPPEN AAN DE TEC'INISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF_DR_P_ VAN DER LEEDEN, VOOR BEN COMMISSIE AANGE~ZEN DOOR HilT CO~~EGE VAN Do;I\A.NJ;;N ;m liE'r OPo;NB.I\I\R Til VERDIlD.Go;N Of' VRIJDAG 21 OKTOBER 1977 TE 16.00 UUR

DOOR

ROBE~US MATlHEUS MARIA SM~TS

DIT PROU'SC!:lI1IFT IS GOllDGEKEURD DOOR DE; Pl<OMOTOREN

PRUf.DR.IR. H.L. HAGEDOORN BN

Contents

Scope of this study IrttroduotHm 1.1 Int~oduction

1.2 The electron energy distribution function(~.F.) 1.3 A numerical model for the positive column 1.4 The conBtriction and the appearance of striations 1.5 The electron-atom bremsstrahlung continuum 1.6 Fluorescence measurements

The electron energy distribution fUnction in the medium pressure d:i.5charge

Scope Introduction

2.1 The BOltzmann-equation 2.2 cross-sections for rteOn

2.3 Solution of the scalar equation

2.3.1 The elastic region 2.3.2 The inelastic region

2.3.3 Continuity conditions 2.4 The axidl electric fieldstrength 2.5 The excitation frequency

2.6 The distribution function for argon Appendix 2.1 Truncation of the expansion

Appendix 2.~! The radial derivative term and the ambipolar field tern!

Appendix 2. In The approximation J lI: 2 fOr J2

Model calculations on the positive column of a medium~ressure inert gas discharge

Scope Introduction

3.1 The element~~y p~ocesses

3.2 The particle balanCe eqUations 3.2.1 The excibed atoms

3.~.~ The @lect~on balance and the cu~rent equation

4 5 6 6 7 9 9 9 9 12 14 14 16 ' 18 19 21 23 26 28 29 31 31 31 32 ]4 34 35

3.3 Ttl" t"mp"r~tcl!'e equations j . J. 1 '1'('1( .... r~lectron t~mperi:l.ture 3.3.2 Th~ qas temperatu~e

).3.3 The prussllre in thc di.~eharg"

36

.36 )7 37 3.4 ~!'h<? (lUme.:ricul treGl.tm,=,nt 38 3.5 NClmertcal results and comp<lrison with experi,mental dat.a 42 Conclusion

Appenclj.x 3. I 'the p06itive colllmn of the argon discharge ApPclldix 3.11 Data for the excited atom b<llilnces l\ppendix J. 1 I.l Calculation, of D _a and N _eo

'LlIl-1 The am!:';,polar diffus ion coefficient :Lu;r-2 The "''Iuation fer the discharge Gu~:rcnt

4 Philosophy on ttle constri,ction of the posit~ve co 1 um" Sc.::ope

Intt"oduction

'l.1 Phy:=:ical mflchanisms for th~ CQi"l.~t:t"iction

4.1.1 Coulomb rel.;...xation

4. t.) Volume :t"i.;-:'combination ana ambipolar diffu6ion 4. I.} Heat 0i.ssipation

4.1.4 The combinilU"n of the three effects 4.2 Discussion of same model~ for. the positive column

4.2. I Thermal mOdels 4.2.;: loniziltio" mod~l",

4.2_, n more sophisticated model 1.2.4 Discussion

4.0 Va,.j,iltion of some parameters of the model 4 .. 1.1 P"rumeter~ wit.h negligible influenoe 4.3.2 pat·.:uiJ.eters wit.h Q. significant influ0nCr~ 4.3.3 A striation model

4.4 The striation development 4.4.1 Introd~ction

4.4.2 ThE" disperSion relat;ion 1,4.3 EXperiment;s

.:1. S COrlel usion

Th" electron-atom bremsstrahlung continuum S(;ope 43 14 45 47 47 49 51 51 51 52 52 53 54 54 55 56 59 59 59 60 61 64 64 65 69 70 II 71

5.1 'l'he electrOI"l-atOrTI bremsstrahlclng theory

5.2 Determinat~on of th~ electron denSity 5.3 Determination of the electron temperature

6 Fluorescence mea~u~ements Scope

Intro(1)ction

6.1 The measuring equipment

6.2 Measurement of the radial del"lsity prof~les of the p) J 3p _{1 ls-"toms}

"nd

6.3 Excitation transfer betwoen the 2p-levels of neOn by colli-sians with neQ~ atoms

6.3.1 The determination of the coupling coeffici~nt~

6.3.2 The temJ?erature dependence of the coupling coeffi-cients

6.).3 ~ca~ured valu8b of the coupling coefficients

Concluding remarks

Appendix I List of symbols

~ppendix I I compilation of data used for the model R~feTenceS Summary Samenvatting Nawoord Levensloop 71 71 'J2 79 7') 81 82 84 84 86 87 93 9S 10:\ lOS 111 113 t1S 116

S~op~ of this studX

In this th8~i~ some properties of the m~dL~m pressure neon dis-charge ~re investigatec. In some cases th€ rned~~ pressure ~rgon di~­ charge is ~~SD digcussed.

At first a ge.neral introduction is given On tne properties of the positive column in a re9ion where 0.50 < PoR < 10 torr m <tnd' 0.01 <: i/R <

ID

Aim (po

=

filling pre~sure, i

=

di~charg~ current, R

=

tu~ radius). The electron energy distribution function i~ oalculated from the

Boltzmann-equation. This distributiDn function is uSBd for the computa-tion of co~fficients appearing in a model set up for the description of the positive column_ The properties of this model are discussed in relation with those of other mod~ls p~e5ented in the literatu"~. The re$ults of the model are compoll:-ed with mea5urement5. The abrupt transi-tion from the diff~se into the con~tr~cted ~olumD is inve~ti9ated_ The connection between,this cOnstriction ~nd the appe~rancB of ionization waves {striations) is di$CU~sed. Opt~~al measurements On the positive column are presented. ThGse are the measurements on the Gle~tron-atom brGms~trahlun9 continuum radiation and fluorescence mea~urements for the detection of ~toms in an excited state. From the$e measurements

~~di~i prot~les of the electron den~ity, the el~ctron temperature and

3 3

the densitie~ of the metastable P2 (155) and re~onant PI (154) states

a,e

to~nd. ror the fluorescence measurements a dye laser was us~d_ The ooupling coefficients of the 2p-levels in neOn are found in dependence on the gas temperature.

CHAFT£R 1 INTROOUCTIQ~

1.1 IntrOduction

In the F"st·the Fositive celtl.lllo of " neon discharg.:. has beet"! the subject of m€l.ny studies. HQweverl these stuoies 'We~e concerned mostly with the low prcs~ure column (po < 5 tO~~) or the high pressure arc

(po > I atm). Only in recent times the inte~ediate region h~s been examined in mor" detail. This medium pressUre region is mOre complicat-ed, but a"~O mere interesting, bGcause many physical prOcesses are of importance. Recently the understanding of the various processes has been g~owing. The electron energy distribution function (E.F.) which was forme~ly assumed to be maxwellian has now been calculated by sever-al authorS (Kag64, Woj65a, wi170, Lya72, ~in72) from the Boltzmann-equation, including the effects of the applied axial electric field, the elastic and inelastic COllisions of electrons wi th neon ground state atoms and interelectronic collisions. AlSO much work h~~ been don" on the investigation of the striations in the pOsitiVe column

(PGk68). These striations are ioniz~tion waves and appear at sev~r"l pressure-cu~rGnt regions (see fig. 1.1). J::xcept for the fact that the column can be axially homogeneous and stationary or striated

(charaote.-diff~s. striated 10 10' ilRtAlm) oQnstri~ted homogeneous 10 diffuse _ ~omo9' 10

Fi.g. 1.1 5x·ietli'nc$ l'sgions of Sel.liill'ai- datum»

states

in Uls

neon

dis-charge

(fl'om Pfa69).

(Shaded

al'ea:

aul' l'egion

Of

intel'est.!

ized by the appearance of ionization waves), there is also a division .i.ntO th~ diffuse and the cOnstricted stat ... );rl th" diffuse state ;::he column fiUs tile Gntire tUbe ;>.no. the axial elect.ric fieldstrength is high. In the con~tricted state the column is very n~rrow ~nd th~ axi~l

electric fieldstrength is low. From fig. 1.1 i t can be seen tllat the

diEch~rge is constricted only for high values of the filling pressure and the discharg~ current_

In this thesis we investigate tne following pressure-current reg.i.on: 0.50 <: pok <: 10 torr m "nd 0.01 ~ j./R ~ 10 Aim (the shaded "rea in fig.I.l). ('1'h0 fj.lling p~essu~e Po' thc disoharge cur~ent i "nd the t\lbe r"di\lS Rare prescnt only in the combinations POR and i/lt conform the simi larj,ty rules (discussed in Pta69) .) In this region the diffuse column (low currents) is a}{ially homogeneous and stationary, w1'li100 the constricted column (high currents) is Rtriated. The con-strjction occurs abruptly at tl cert"in current value which is slightly depenoent on the tilling pressu!:'e (see fig .1.11. The investigation of the phy",ical ll1echanisms responsible for this sudden constriction is one of the Sub)eCLS of this theSiS.

we present model calculations on the positive column, using the electl:on oI.orgy distribution function (E.F.) calculated from the lloltzmann-oquation. The moo"l is compared with other mooel,g p>:"esenteo in the literature. The connection between the constriction and the onset of thO! striations is discussed. Finally optical measurcments on the electron-atom bremsstrahlung continuum and laser fluorescence meas\u:em"nts are preSented. The caloulation8 are comp"'l:ed with t.he oata obtained from theSe measurements.

The optical me~surement5 ~nd measurements of the axial electric fielostrength have beon performed on tubas which had a cataphoresis P\lrj fication chamb'H Oil the anode and on the cathooe side (see for snch a purif.iC;>.t10n Cham»e)." the thesis of 8"ghnig (8"g74». 'I'he C\lrrent was regulated by a Gt"bilization unit (pri7Sl. The axial el~ctrlc

f.i.(>lo~tl::Gngth was measured with the double-probe methoo (llag74, »r175).

1.2 The e'ect.Qn energy distribution function (E.F.)

The E.F. is calculated from the Boltzmann-equation using the methods given by Shkarofsky et al. (Shk66) and LyaquschencKo (Lya72).

In thi~ calculat.ion the E.F. is deve~oped in sph,<orical harmonics_ After substitution of t.his exp<lnsion intc the Boltz,""nn-equation one Obtains a hie~a~chy of tensor eqUations (Shk66). The expansion is t'un~ated

after the zero-order term, yielding a sc~lar equation. III this equ~t.ion

the radial derivative term is n€arly cancelled by the alIlbipolar U"ld term. Thes€ ·terms therefore are negle.:::ted. Furthennm:e superelast.i" collisions are neglected becaus" th,;, plasma is far from tllermodypamic equilibrium (Lya72)_

With respect to earlier calculations fc~ neon (see for instan"e G0168u) we add the following alterations'

-an energy dependence of the elastic cross-section of neon is taken into aceount, Thi~ dependence is f~ tted to the expe:dm .. "tal data given by Salop and Nakano (Sal70);

-tile gas temperature t~rm is taken into account for th~ calc~ldtion of the s)(citation frequen<;.y from the E.F.;

-the a)(ial electriC' fiel-dstrength i5 calculated in a consistent

way;

-for several correction facto~s, the dep~ndences on the electron

tempe~ature and the electron density ~re taken into account; -the ineta~tic region ~5 split into two regions in stead Of d~~

scribing it as one single region;

-the particular solution in the elastic regiOn is treatod ,n a con",istent way.

The calCu~ations are presented in .:::hapter 2_ The calculated E.F. i~

used for the computation of the ~eactiOn coefficients appearing in our numerieal model for the positive column. presented in chapter 3.

1.3 A num€~ical model for thE positive column

The numerieal model which we hav~ set up for t.he positive column of a medium pressure neon discharge is an extension of the model glven in a former publication (Smi74). In this thes~s we also' adapt the model to the cas,;, of argon (the E.F. for argOn is also calculated io chapter

2). ~he differenees with respect to the former version (Smi74) of the mode1 ar,;, the following'

-fo~ the calc~lation of the rea.:::tion coefficient.s, the E.F. calculated from the Boltzmann-equation is used (chapte~ 21 in steaQ of the Druyvesteyn E.F. used in the former publication,

-a mO['e complex treatment of the balanoe equations fo:r: the "xciteo atorns;

-an improved o.n""ytic,,l apptox.i.mo.tion to the m(>"$urements on t:he el~stic cross-s(>ction of Salop and Nakano (8a170):

-inclusion of t.he radiat.;,on term in the cal"ulation of the w~ll temperat1Jre;

-a consistont. calculation of the axial electric fi . .,ldstrengti1; ~l.nclu5ion of the effect of the presence at dead volumes on t.h~

pre$SUlO"C in the activo dischargc;

-a more ndin"d nUlllerical treatment.

10 Cl\apt",);" 4, th" mOdel is compared "lith other models p\"c~CI't."c'l in th" literatur.~. Also the influence of the variation of some parameter~ (the E.,.FT ~ the stepwise ioniz.~tion frequency I the volume recombination coefficient~ the frequency of interelectronio collisions) is investi-gated.

1.4 Tbe constriction and the appearance of st.riatl.on"

In the pressure-current region of our interest the con~trictj.cn is charact.er.ized by the appeara.nce of striations. In cha.pter 4 we

dj.~C\),S::=' this connection in more detail. Tbere are indication8 that. the OIlSGt dmd the nonlinear growt.h of t.he ~ triat.iono' Ca\lSe an i""tabi H ty which in its turn ~tarts the CQfl.st:r~ ction. This hypothr.=-sis is compar('d with mechanisms propos",,, by 'sever"l other ."lltliO;(S.

1.5 Tho e100t't:"on-dtC)i1~ bretnsstrahlun£ continuum

In the medium pressure inert gas discharge the optj.cal 1ir)'" r"diatl.on 15 "uper-imposed upon a rather strong cont.inuum. While in the

pa~t most authot"r:; belieVed this continuum to be of molCCl..Ll:;.r orlgin, nowadays mttny authors are convinced th.&t the continuum is caused by elecl(on-atom bremsstrahlung .(Rut6B, Gol.73, ve)173, l>fa76). In chapter

.s

we present ffil::!6t5u.rements of the radial profiles of the electron density aod the electron ternper"ture based on th" assurnpt.ion t.h"t th" latt.er. pOint of view is correct. The electron density has he~n obt.ained from the continuum .l.utenbity lit do fixed w~velength. The electron

b.::::[r)per-~ture has b01::!I~ determined from the ratio of the continuum intensities at two fixed wavelengthST Th~ measurements are compared w~,th model cal-culat:Lons.

1.6 Fluorescence me~sur~ments

With the fluo~es~ence teChnique the denSities of the metastable 3P2 (IsS) and the"resonant 3P1 (19

4) "toms have been me<>",ured. For this technique a dye-~a",er (Spect~a-~hY$icS 370, S~oo ~ < A < ~400 ~) has been used (Smi7S, Ste75, Coo76). In thiS way the relative radial

. 3 3

density profiles of the P2 and PI atoms have ~een measured for vari-Ous values o£ the discharge current. The measured profiles are "fompared with calcu~ated ones.

With the fluorescence technique the coupling ooetfi~ients of the 2p-levels have been determined. These ooupling coe~ficientg have been measured on the axis of the discharge as a function of the discharg~

current. With the aid of OUr model the measu~ed dependenoe Of the coupling coefficients on the discharge OUrrent is transformed into a

oependen~e on the qas temperature.

C~PTE~ 2 THE E~ECT~ON ENERGY DISTRIBUTION FUNCTiON IN THE MEDIUM PRE$SURE DISCHARGE

In this eh~p,~r the eleetron ener9Y di~tribution function is

cal-culat~d from the ~ltzma~~-equation fOr the conditions in th~ medi~m

pressure discharge. ~xpressions for the ~~ial electric fieldstr~ngth

and the exeitatio~ f.equ~ncy are d~riVed. The calculations are per-formed for neon (sections 2.2-2.5) and argon (section 2.6).

Introductio~

For the calculatio~ of the coefficient~ appearing i~ the equa-tions of the model described in ohapter 3. the electron en~rgy di~tribu­

tion function (E.F.) must be known. In the m~dium pr~s~ure disCharge the E.F. is determined by the electric field, the elastic a~d inelastiC colli5iO~S with the gas ~toms and by the Coulomb interaotion between the ~lectrons. Wh~n the coulomb interaction prev~iA~, the distribution function is maxwell!.an. When the ~lastic collisiO~s and the Held in-flu@nce are domin~t, one obtains the Druyv~~teyn distribution (Shk66). In our pressure a~d current reg10n (0.50 ~ POR < 10 torr m. 0.01 < i!R

<

10

Aim,

see ch~pter 1) all mentioned prooesses ca~ be of equal

impor-tance. Ther@(ore the E.F. has been c~lculated from the Bolt7.mann-equ~­

ticn.

We app~y the methOd given by Shkarcfsky et al. (Shk66) and Lya9uSch~nckc (Lya72) to our caS8. In sections 2.4 and 2.5 expres$~ons for the axial electric fieldstrength and the ."xci tation freq\lenCy are derived whioh will be used in the model de~cribed in c~apter 3.

2.1 The Boltzm~nn-equation

The eleotron energy distribution function iE.F.) ~8 obtained from the aoltzmann-equ~tion for th~ stationary discharge:

v C(f) , (2.1 )

where f = f(~,v) is the

e.F ..

~ position. ~ v~locity. e ,,"ementary charge, m electro~ mass, ~ ~lectric fieldstrength. C collison operator.

The left-hand side of eq. (2.1) represents the flow .orooo!!sses in phase

spClce, the right-hand side .rcpX'c:::;cnts the collision proC'e!:ise::s.

~~n tho 810ctrlc field i . low. such th~t the dYlft velocity of the elect.rons is much srn<JUer than their tt10rm.;>1 velo,ity. the distri-bution function is nearly isotropiC ir\ veloci t.y ~paco:!: _ 'then the E. F r ~a~ be developed into spheric<ll harIllOnics (Shk66). 1:hi.5 develop",,,,,t co."

be wdt.t.en ~s (Shk66, Joh66):

C (f) (2.2)

Here f£ and

tt

are tensors of the

~th ~ank

and

1

represents the

r~p~QL­

ed inproduct (see Joh66).

Substitution of this expansion iflto the BoltzmClnn-equation re-sults in a hierarchy of tensor equations of increa~ing runk~ BecauSe

th~ ratio of an element of the tensOr f~+l to an element of the tensor til, ~s <)f the oroer of magnitude 1!8m/M (1'1 atomic mass) (see appendix 2.1), we have ~l.l:ff;i.~ient accuracy for our purpose when we replace

f(~,~) by fo(!.'v). 'l'h~ 801tzmann~equation then reduces to .;> gc~lar equa-tion for £0 (See app"ndil> 2,r). In r.lli . .5 "''1''''tion the radial d'aiv8ti.v,", term. netl.rly cancels the term COJ)tair)ing the radial ambipola,[' fit::ld '$t.~E;'! app""di,. 2. U). Th<l~eforc thes" t.wo effects are neglect"d. Fc\ychermore, u.s the conditions in the column ~re f~r: from th~rmodynarnic e:quilibriu(t'l., the super elea6tic Golli~ion8 aXe negligibl" (see Lya72).

lifter a tr~n5fonllat.ion hom veloci.t.y coordinates to the <1i"'e,.,sio"-18~s energy coordintlte E

=

U/U

e (U electron energy, ue scyling par.~m­ eter, see below), the soalar equation for fo(e.) (using the Game symbol

i'O i,t ~hB caSB Of energy (Joordinates.') becomes a(:cordi"g to

Lyaguschencko (Lya72) (where we added th0 t8rm containing the gas

tem-perature according to Shkarofsky \~hk(6)),

·.ri :':" . 1 d [

IE

de F,(c) df o df: :~ = rJ/{J~, F':') = u₁/u

e, rJ electror1 en.ergYI u1 excitb.tiOn en~ygYI

. ~"': ~r""~aling parameter to be chosen as the electrOn t.empe!:rature lsee

~~~~WI ((J, tl

H(t) G(IO) V _ee v_E N e T 2

Ve,/'J (E) + VE(E:)e +

~\J

(E)E: , + v (E) t2, e 9 Ve,l'2(8) g 81Te 4N

1~1T(E:

kT )3/2 e in ('1m: m) 2w3 0 4 e 2E2 1

"3

- 2 -_m - - - 3 NgQoW ( o e iii 11283 ) e Vg or e 2m

M

NgQDw

,

electron density, N

g gas density, ~g gas temperature, momentum transter cross-section,

total inelastic crOss-seotion,

After the transformation from v to t, the scaling parameter Us still is a free paratJleter ,. In section 2. I> this parameter will be t1xed, defining

U

e ~s the electron temperature, according to 3

- eu

2 "

1 2

2

III < v > .

~his def~nition then yield~ a relation between U" and E, 6ecause E ~ N

g (see section 2.4), and the logarithm ~ppearin9 in veo is a weaK function of Ne' the E,F. depends on Ne and N_g ma~nly, via the ratio Ne/N

g as can be seen from eq. (2.3).

In eq. (2.3) the term wLth H(t) desoribes the influence of the electron energy gain proceSses on the E.F, ~hese processes are the Coulomo inte~action, t~e inflUence o~ t~e axial electric field ~d the thermal motion of gas atoms. ~he term wit~ G(E) contain~ the influence of the following electron energy loss prOcesses on the ~.~,: Coulomb in-terection and elastic col11sions. The right-hand S~Qe of eq. (2.3) de-sorioes the 1nelast~c losses and the cr~ation of slow electrons by in-elast~c collisions.

Wh~n vee » V_E' v9' V_in, hence when Coulomb interaction prevails, the solutton of (2,3) is the Maxwell dist~ibution, f (~) = Ne-8. When

a V~,Vg » Vee' '.lin' One obtains the Druyvest~yn distribution:

v [C')

f (e) ~ N exp -

JE:

~ C' df:' (N not'm"lization (:o"stant)

o 0 V

E {I=-:' , )

'1' In the fo11""'in9 o.n<>lyt.ical ualcu1ations fo>: neon. the t.erm ...5L V "

Te 9 is neglected, oeCaUSe Tg « Te and inclusion

ot

thiS term give~

cumbersombe expressions (in section 2.5 a corl":'Gctl.on for this omt~:::.1.0t"l,

:i!=l giVen). Far aY9on, however, the term .-:an be maintained w.ithout. dif-ficUlty (sec section 2.6).

The integrals Al and A2 havi'! oeen ualcul"ted by GollAbOVSkii et ,,1. (GonOa) for variO\lS distribution functions. It. appeau,u thttt for all

r

O,385 E for 8 ~ 2.6

ca5e~. within 15~: Al (e) c< 1'.2(8) ~ Il.(£) " 1 for ,. ,. 2.6 •

?.:1 Cros5-s~ctions for neon. 1'0 calcul1:1.te the frequenct~s 'V

g, \)E and \lin whi(~h appear in eq. '2 _ ,1) r we havp. t.o know the cra.::::e-SeOtions for the elastic ttnd inelastic collision processe~. In sections 2.2-2.5 the case for neon will ~e con-sidered. 'I'he caSe of argon will be dealt wi til in section 2.". f'CO" neon

th~ tot.ElI elastic cro",s-section has been measured by S.:tlop and Nakano (Sa1701. we approximat.e thio cross-section by (See fig. ".1)

O<U::,U l LJ ~ U l n

=

1/4 and u₁

=

15.5 V. (2.41

HC,t''.:.'I.I:'Om the mOmeI'ltum tran~fer cross-sectl.on is t:alculated_ wt:~ take (Mco64) , Q o =(aD!ae1 Qel) 0.85 . (2.5)

=r<

(-1/4 _{for 8 <} E

,

1 V -)/4 F. for > v E c 1 t E:I

=

j

"

<:1/4 _{[or <} v ~ _2'1 Vg g", V9 ( 1/4 I Po ;- 81 4 2 2 with v

'"

_}" ~ G 2 " N un 3 u\

_'"

D 9 e and _Vg

'"

2m a N Un w with 1 (2,6) M D g e n

"4

12

The total inelastic crQS9-section for neon hdS been me3~ured by !>chal"er and S(;he~oner (Sch69). we approxi)l101t.e chese measurement" by (fig. 2.2),

o

Al (U-U₁) A2(U~U2) + A1 (U₂-U₁) -22 2 -1 17.0 V. 1\1 = 6.10 m V I -22 2 -1 0_5 10 '" v 1,.0 r---~·'I - - - T ---,--~----, 20 I 1. . ... measurement (Sal 10) : _-1.8.10-2DU'I~, I I I _~_ ... I .. _ ... '0 1S U, U (V)

Fig. 2.1 UC$t-ir) r)2"'088-aecticm

for

neon

meaeUl'ements (SaUO) ~--

sq.

(2.1).

1

..., J-~ --'" - -

meaSvrem~OIS (Sc~69)

·----<"Q1.·7 _-~---"

--.-...:..---"'" I

'/

I .j .... _._...!...-. _ _ _ ...L.. ____ ... _ ... _ .1 )7 )8 19 UIV)

Fig. 2.2 Inelaetio 02"'Oss-seation for neon.

20

(2.'1)

2.3 solution of th~ ~cala~ equation 2.3.1 The elastic region

For tJ-..e cl.;'.l.stic region F:. .-::: ~"':1 f th~ iIlelastic cross-section Qt 0 (See eq. ~.71. BeCauSe f~~ AI~I = 0, and the~~£Ore H(O)

=

G(O)

=

0,

multiplic~tion of eg. (2.3) by

Ie

~nd integration over c yieldS'

df

H(e) o + Gltlf

o - t (0:) , (2.8)

wh"re t(t) - N w J'-(C'+C1IQ (8'+"I)f (8'+f))d(' (2.~)

g o t . 0 '

"~ the nllmbe~ Df electrons scattered in t.he low energy region by i n-elastic collisions ILy~72). The forma] ~clution Of (2,8) is,

WitJ1 und f (t:)

"

J 1 (f:1 NJ 1 (E:) + J 2(8) , I 1 ( ) Jf: -t(t;') dc', J₂ t: - J E: 0 HIS'IJ 1 lEO') (N is a constant), (2.101 (l,11) (2.1))

For "-n analytical ~ol\ltion we h~ve appxoxim~ted th~ int8gl:"~1$

Al and 1\.2 appe.:ring in G(E) and HIE) by a cOn5tttnt C

=

L Compttnoon wit.h a )1umerio~l solution yields errors in the final .8.P. smaller than 15L WrJ.t.ing ,J

1 (1':) = exp(-M(IC)). we find for M(E),

M (£) .. b ( 2 a 13 / 3

sy

3 4( a

-

11 b "rctg .:.))) /3 where y a l

/3 ~1/4

and

'"

*

\J E \J a

=

b ..!L \J \! ee ee

(The expression fo):" M(E)

10

_{- "7}

4 7 4 y + Y +

1.

/l-

y_+/) I (y + 1,-,

_-73

(~Y(;L.g

l+y

(see eqs. (2.3) and(2 ,6)) ,

in the cases n

=

Q and n

₌

_1/3

~

/3+

₃

in eq,

can be found in the references Go168a and I)lly7S ycspG'ctively.)

(2,1)1

(2.14)

(2,4) ,

For low electron densities (N <: 1015 m-3). a <tnd b become large, and M (e)

becom",~

M (E)

~

i ;

ES

/2

(~J:"uYVest"YIl

limit), Ii'or bigh elect.ron d8n-8ities (Me> IOI9m-3), a ~nd b tend to zero, yielding

(MaxWell limit). -4

...

:;l N_g • ) lo2'm-3 ~ ::;:

_,

_{U. ·3V} -6 I, N ... 'O '6 m'"3 2,

Ne-,O

'7ni' " No,,,,O IS", 3 -B ., N~ .'0'9",0 "

Ne~I020m3

-10 I I 0 8 10 12 16 U(V) hoe· 2.3 -M(U!U e) "" a fUnction of U.

In fig. 2.3 M(€) is plotted for seVeral values

ot

N , with the scaling

3 I 2 e

factor U

e = 3 (taken s~~h th~t

2

eUe =

2

m

<

v ~, s~c section 2.4) _ ~o

calcul~te the con~tant a, use has been made of the relation between the electric fieldstrength and the electron temperature deriVed in ~ection 2.4.

For high electron densities, the distribution funetion be'comes maxwellian indeed. For low eleotrQn densities t\1e Druyvesteyn form is achieV0d.

The constant N in eq. (2.10) i~ obtained from the normalizatiQn con-('ji tion'

Because fo(E) rapidly decreases for increasing ~,

replace this condition by:

1 .

(2.15)

(see section 2.3.3), we

(2.16)

Numerical Calculations show that J~(e) ~< NJ, (~), except for E ~ ~I (see fig.2.4below). There:(ore neglecting J

2, we take:

(2.17)

TO fUlfill the continuity conditions (see section 2.3.3), we need to know

'"

J2(~) in the neighbourhood of ~1' A good approximation J

2 (~) fo< J2(£) at: t .. E, is deriveo in appendix 2. III, whel"El it is shown that:

(2.18)

"'

In fig. 2.4, NJl(~)' J

2(€) and J2(E) al"e plotted for Ue

= ),

17 -3

Ne '0 m ,J~ (tl is indeeo negligibl", compi:lred to NJ 1 (£) for t « <::1' For c ., t

1,J2 ~(£l is a good apl?roximation fOl" J2

«(:).

2S 26 21 ~

'"

,

28 29 30 0 'r - - - - , - - r ' .. '---,'---,---r---E I I I

,

I '

F7:g, 2,4 A comparison of NJ] UJith J., o;1'/d J / N J,]024 m-J _N

₌

]ol?'~m-3,

I)

g e e

2.3.2 The inelastio region

J

v,

In the inelastic region where c> E₁, t:he second term on the dght-hand side of eq. (2.3) is negligible beoause of the rapid decay af totE) in this region, Then eq, (2.3) becomes:

d2f df 0 P(~) ---..£ + S(E)f O

o ,

- - + OE2 dE: (2.19) wh<!:re pre) (~+ de G(e) )/~(r;:) , and S(e:) ( - -oG NgWQt(E:)E)/H(e) .

dt (2.20)

Here vErt) ~nd Vg(e:) have to be taken oo~stant (see eqs, 2.5, 2.6), After the

transfor~ation

f (t) =

~(() exp(-~ J~

P(€')dE') eq_ (2.19) becomes;

o 2 El

where +

s .

Using eqs,(2.19) ~nd (2.20) we get: q(g)

(4vb -a 2+ 1)+4b t+(4a b +2b )~2+2vb 2~3+b 2e: 4

0 0 0 0 0 0 0 0

2boe-IVin(S)/Vee)t + -=---~~-~~­

(l+a E+vb E2)

o 0

wi th v

=

Tg/T

~,a

=

at -1/4, b bS:

1/

4

~ Q 1 0 l '

(2.21)

(2.22)

Numerical ey"luation of q gives a

curve

consisting of two ~early strai9ht lines. Therefore we rewrite (2.22) retaini~g only linear terms (Duy 7 5) ,

2 q(£) ~ - a₁ (£-£1+\)1) 2 where a 1 - a 2 (£-E +b ) 2 2 2 = N_{gA2wU 2} II (£2) b ~ 1 E: > t.'} , - 2 2 ( (l+b oEI ) - 4boe:1 (l+ao£l) 4H (t 1)2 2 2 (1+b£2) -4b£2(I+a(2) 4H(£2)2 V e€ 2

The eXponont in the transformation factor can be written in good app~oximation as (Ouy75);

because P(E) is nearly con5ta~t around

e

₁,

(:<.23)

(2.24)

Then the solution of eq. (2.19) hecomes (with the r"qu\rement

till!

f 0 (EO) ~ 0) :

where _'PI Ai (~, 2/"1 . (E:-C:

I +b1)) U(c) 'P2 Bi

(aI2!3{€-~I+bl»

UIE) ¢3 IIi (a 2/3 2 (8-"2+1.>2)) UtE) U["') exp(-7I P (E t) (e-fl))

(Ai tind Bi are Airy functions, see IIbr68) .

2.3.3 Continuity conaitions

(2.25)

The solutions (2.10) for the elastic region and(2.25) ,(2.26) fo~

the inel<lsti<; region haVe to be matched at e=~'l ((2.10) and 12.25) ctnd E=E₂ ((2.25) and (2.26)). This determines the oonstants A, Band C in (2.25) ana 12.2G) as N ha~ already been given in (2.17). Continuir.y for fo(O;) aMI fo' (E) at 10

1 and E2 giv"s four equ"tions:

NJ! (~1) + J 2 (t 1) - Ml (E I) + B{l2 (~1) , (2.27) NJ 1' (101)+ oJ/IEI)= A<jll'(~l)+

B<t>/(8 1) ,

(2.281 AlP 1 (02) + B¢2 (c 2)= C'1J3 ('~;:>J

,

(1.79) 1\4> l' (E: 2) + 2$2' (E2) =C4>3 , (c2) (2. :10)

When eq. 12.27) is fulfilled, "q. (2.28) is not an independent equation, because i t has been a~S1.1med al1:"eady in the ci!llculation of .12 (8) (og. 2.12) from eq. (2.8). 'thi.", can be Seen by repl"cing tIS) ("q.2.9) vl~ the t.,.~ns·

formcttion f~" ~:' +~ I by, 8,+8 N

g'"

I.,

E" Qt(~h) fo((h)dC'. (2 .. ,1)

N",glecting the last t."rm in (2.3) (see section 2.3.2), this int.e9raJ. can be written (by integration of eq. (2.3)) "5:

18 t(E)=IH«()f O' (eJ+G(f::)fo

1

!:l+ E

- I

H(I::)f a' (,:)+G«:lf0

1

';1 (2.32)

~ec~use of the ~~p~~ decay of fo(~) for E ~ E₁, the first te~m in (2"32) can be negle~ted when ~ ~ E

1. ~hen eq. (2.8) becomes for ~ ~ E1, when (2"25) is substituted in (2.32).

(2.33)

Substitution

of

(2_27) into (2.33) yields (2.28). calculation of JZ(E) from (2.8) with t(E) from (2.9), thus yields (2.28), sO that the Latter equation has already been used in the calculations and is superf~uous.

We then retain eqs.(2.27) ,(2.29) and (2.30) for the determination of ~, ~ "nd c"

We repLace JZ by J

2'" = -t(E1)/G(£) (see appendix 2.III). </1_{12 </>32' -} </1_{12 '4>32} Introducing I = ~~~----~~~~ 4> 32</12/ - 4>32''''22 (Where 4>ij ~ </>~ (tj) G(E 1) p. .. - -H(E! ) B AI, C and

4> i j ' =

4>

i ' (e j) ), we get NJ 1 (~l)

4>11'+4>21'1

(2.34)

With eqs.(2.!0-2.14),(2.!7),(2.23-2.26) and (2.34) the ~istribution

function is determLned for the entire energy region. In fig.2.5 fo(t) is plotte<;l at U_e = 3 for v~rious values of thO electron density. ·I"he

in-el~stic collisions causo ~ de~letion in the tail of E.F.

2.4 The axial electrio f1elo~tren9th

In this ~ection a r~lation between E ~n<;l 0e i6 derive<;l,which will be used in the mo<;lel described in ch~~ter 3.

After the transformation of the Boltzmann-equation, Which containS the parameterS E, Ne ~nd Ng' to tho dimensionl~SS energy coord~nate e=U!U

e, an extra (scaling)parameter U_e was introduced" ey most authors (Go168a ,

Go174, Wcj67) U

e is coupled to E by solving the electron energy equation:

Pel + Pin (2.35)

where the left-hand side is the energy gain per electron (be=electron mobility), ~nd the right-h~n~ side r8p~esents the elastic an~ ineldstic

~c

J

-,

'"

~

4' ~o

Eo,

><

Ng -3.lo<"rri3 U_e ·3v I: Ne _I017

_m

3 2: Ne -iO

IB".;3

3: N" _IOI9m') 4:

Ne

_102°",3 9 0 ... L···L.-·J--+---"----,I"'Z---'---,\,-.ll.l,&-.lL---.J· U (VI

energy lasses p"Y "l"CU·on. This relation is o);>taineo );>y ll1ultiplication

f I 1 1 2 . . 2 72)

o t .... e: Bo t.zmano-eq1J.bt,ion by

"2

rnv and lntegratlon over 4"/(v dv (Lya .

Th~ Coulomb terms then r:ancGl each other. The authors mentioned above Ta 2

solve t.he BOltzmunn-equation w~tllOl.\t the gasterm

T.;

V_g': (in H(~)).

rl1hi$ t.~t"rn i.ntroduces t11e of:!xtra ener9Y 9~in tetm :P ga.s

(2.3&)

Bec"uso (2. 36) i 6 obtained from the 60l. tZll\ann-equ<'tion, su);>st.i tution of

f into

OJ (2. 3,,) y~e"os an id,mti ty. Because the maxwellian distribution function dce. not contain E, i t is cften substituted into (2.35', be-cause E th"n "<')1 b~ S01""d from (2.35). As (2.35) is an identi.ty this procedure is not vali.d. Tll8 E. F. can Only be m<1xwellian when the

elec.)-tri~ field and the inelastic collisions are negligibla. In that case the electrons ):"<2'la"ate to the gas t.emperature (Shk66). The):"efore an ~xp~es­

sian wnich oonnects lle wit.h ~. c<ln pdncipal1y not be obtained from

P.

35). When the E.F. i6 close to the m<lxwellian E.F., still the terms in the Bolczmann-equutian whIch (<luse daviation from the maxwellidn E.F., a~e the ones thllt constit.ute eq. (2.35) (or (2.315)). Ther",for", the

tion of E from (2.~~), thro~gh sUbstitution of the maxwellian £.F. is incorrect. and deviates in general conSiderably from the realistic C~}­

culation.

we obt1lin the relation between E and I.J,;, from the requl.l:ement that. the scaling parlillleter \)", is the electrOn temper"-tu~e, according to the defini tion,

(2.37)

Because the contributions to integrals over the distxibutiOn functioIl of both J

2l,-) o.nd of ;Eo(~) in the inel<lstic r",gion are very omall (see Hq.~.

2.4.2.5), "'q. (2.37) can be wr~tten as,

['-1 E3/2 e -N(t) dE

o

JEI cI/2e-M(E)df.

o

)

'2

(2.38)

When El is J:"Gplaceq by infinity in (2.38), thiS eqU"t.j.on is automatlcctlly satisfied for a Maxwell dist.ributiOrl, corr~spondi.I'lg t.o the fact tJ1Ltt foy that case nO elect.ric field can be I?resent (the electron temp".<>tlll-" then m\lst be eq\l"-l to the gi:lS temperat\lre). In our region the E. F. is st.i 11 tctr from m"xwellian. llq_ (2.38) has been evaluat.ed numerically, computi0g M(C) numerically according to eq.(2_11). so thOle the gas t~rm was included in the calculations. Sub~titution of eq. (2_13) into (2.38) yields

with E = e(N _e

IN )

\~

a N U 5/1 g M p g e I e(N

IN )

J

1 e g

jI.4

fo)"" N /N + 0 " g for N~/Ng + (2 _ 3'3)

The numerical oalculations showed that within our region, e can be approximated within 5t by:

N IN <: 0.33 10-7, N e g N

e

2 ( JO_lg e + 7.48) 10- 7 _~N8~O.D

+15

_N 0.33 g _{g -4} 1.4 t<

IN

> 0.33 10 '" g From (2.39) we have b

a __ 1_ which has been used in

~ig.

2_3_ 3C2

2.5 The excitation freqUency

10 -1

For thB model calculations given in chapter 3 we need the excita-tion freguenoy. This 18 given by;

7, = ( ' N vQ (vi f (v) 4'«V"dv

vi g t 0

2elJ! (with vi, = \ 1'1 )

-p_10a)

Ft'OlJ1 multiplica.tion of ~q_ {2_ 3) by

It

and i.nt.~gration over l" we obt.ain

(lleglecting t_he last. term in eq. (2.3), se" 2.3.2);

t(c) = - 21TW3IH(ll)(::O) c=c +G(E_1)fQ('-11/- (/.4()b) 1

Substitution "f fo(E) <lccording to eq. (l.2S) into (2_10b) yield",

Z = NjJ

l (Cj)G(C1) Flf2 (2.41 )

, 3 <1Tw N

(I, <Pij 3nd <P'ij Elccordirtg to

"q.

(2.34») ,

F2 ~ correotioll fact')r for the gas tt'l'1pG"atu,-e (see bel"w eg_

t

2.43) .

Numerical calc:ulat_ioI"1. of 't.he factor NIP 1 ,-:;hOW~ that N₁P 1 i~ a slowly

varying monot:ora\r: functjC")l~ c.>f 'lle ~)"Id Ne/Ng, t.h~ 1ower. bound being 0.:2

und t:he \lpper hot)nd hei,ng 0,6_ 'l'h;5 f,mction has b"en approxillluted fur our region (1 U ( 4 , 0.33 10-8 ( N

IN

0.33 10-51 within ]0\ by;

e ~ 9

-a_lOr, u 0_1" 0_044(10lg :" + 8.48)'.

e "g

U.42.1

F'urth~rmo,-e il correction factor <'2 for tho replacement of 1'.1 (8) a~d Ai (8) by 1, arld re,y

tl,,,

urni~sion of t.he gaG ternpGral:l.lre term in Ilk) h<\s bee~ calculat-,o,d

< 4

v,

0_:13 10-8 ,

nume)"ically. In the xGgion 0 < Tg < 2000 K, < U e N IN < OT 3.~ 10-~, numerical culculutions show thv.t

c g 6 7

b[~COml~ very large l10 to 10 ) £0):" slYI?Ill v?llU(]~j of N

IN

and U and lilrg" vulues of T The excitation frcquem;y is,

how-e g e g

eve:c,used in t_he numerical model for the positive column, disc:;uss<:d in chapt~} 3. In '-11 ( ~ f.J;::..'r~ i t' ~ V(~ ('.:Q;umn at th~ axis, small values of N

IN

" g arc uoupled to sm~ll v.:a1 IJGe.: of T

9 a~d larg~ values of U e' while large values of N

IN

~1 re (: .... ~upll?d to 1 "n-g(' vnlu~s Q,f T an(l sma,l] U Th~rt=:! ...

E':! g 9 e

fore the value of F2 «t the axis of the column rem"inS Umited to valne, in the neighbourhood of 1. l'. good approximation to th~ numerical value8

in th15 region appe«r~d to be,

exp( N ....!!-1'1 -) +8,:;57) ) . (2.43) g

In tho pos~tive column, F2 can become large for increasing radial di8~

tance (near the wall). However, for these values of the radial distance, the inilusnce of F

Z

on the model result8 is negligible because the e~Ci­

tatlon frequency is almost zero near the tube wall. Therefore we use (2.43) in the entire rag ion in our calculations of chapter 3.

In fig.2.6 tha Gxcitatj,on frequency (which wUl be used fm: the mod@l d~sc~ibed in chapter 3) is giVen ~5 « function of Ne for sever«l vallJe5 of Ur,?- For low electron den~itieSr the coulomb t.erms c~nc:e:l in the Bo~tzmann-equation, and th~ E.F. (henc0 Z) is not dependent on N

e. 1'0'" high ele<;:tron densities, the E.F. becomGS ma)<wellian, "nd Z become« agail) electron density independent. In the inte1:"mcd~"te region Z is a

strO~9 function of Ne"

2.6 The distr~bution function for a~gon

To compare e~p8~imental data on the ~rgon di~charge (Woj67) with model c~lculatione W0 need the electron eI"lCX'gy distribution in the argon discharge,

For argon the momentum transfer cross~section is approxirn~ted by (\'loj65b) ,

u ~ VI

=

11.5 V ,

with

The inelastic cross-section is apprO~imated by (Woj65b),

with A ₁

=

0 _" S 10- 2 ) 2_m

v-

1

N ~2-S

o

.\ _." ····T,---r-N_g• 3.102~rri 3 Ig ·WOK 1. We ·1.5v 2. Ue • •. 0'1 3. U

_e

-21.

v

4, ~-2SV

s.

U~ ·32 V

'

... 1019 Fig. 2.b' (, as a j'uHdtion of N€;.

In this case G(E) and S(c) b.com~:

..

T '" 3 H (~:) V

.v£

+ J v E ee T g e G(>':) = V

e"

'"

t 3 , +vG .

..

..

w1th 'i E and Vc according to

"q.

(2.6),

wi.th n

=

1 lind al) and M fo~ argon.

lrl this case the gas temperature term can easily be maintained in H (t)

for the analytic"l calculations. Then M (!::) l;>ecomes,

where C 1 a

=

24 ub - - + ₃ C₂ -1 (I+a) , C 2

'"

VE

Iv""

b 1 b I+u

3

(1 - 3 ) (in ( ) + C 2

VI -

u+u 2 (vb( 1+01) -1) 1(3 v

"'Iv

g

10"

u C2E «no v

=

T./Te ·

When Tg 0 this expxession reduces to the one given by Goluoovskii et ~l. (80170b)

In the inelastic ~eglon Q

O is constant_ Because Qt is one lJ.near f\l""tion in the entire inelastic kcqian of !nterest, we get far argon one Sl~gle

solution ¢1 in the il1el"stic region. Simila~ to the treatm«nt give,) in 2,3.2 for neon, ~e obtain for argon:

and AJNgwUI H(1':I) . ' 2

I

(l+b813) -<lbS! 2 (l-l-a) 4H(8 1)2 -I

p(el) = (H'(O:ll+G(Sl))H(€:I) (see eq.(2.20». Tl\en z = 21TW3NG(E

I)J1

(~1)FIF2

(se<;:tion 2.5),

with Fl It-

~

~

r hence

H("I) <I\r(~l)

2aI2/3(I~a)Ai'

(al2/3bl)+(Hb€13)l\i(aI2/3bj) Fl

=

2aI2/3(I+a)Ail(aI2/3bl)-(1+b~1 »l\i(~12!3b,)

(see also Go170b).

The factor N)F) is numerically approximated by,

N,FI

~

0.40 + 0.067 (1019

:e

+ 8.3)2 g

The correction F 2 now Go~tains only the A (~) correction as the ga~ temper-ature te:rm in M(n has already b"",n taken into account (5ee Ii<bQve) _ 'rhe correction for A(E) appeared to be 20% in our region On interest. Th\.\$ F2

=

1.2 has been taken"

A similar calculation of E as in section 4 reSl)J.t!3 for ~rq{)n in:

where C is app:roximat~d \S~C section 7..4) loy: G 1.(J ₊ (0.049-0. (JOb I) ) (lO]g

~+')

3)2 e N g . N (for u < 5 ;0.5 10 -~j <: e <"0.5 10-4 ) • e _Ng

Appendix 2 _ 1 T:r-Ullcatian of the exp-ilnsiOll

Wh~rt r<:tdial r1~riv.;3.t.lv<::,::; ~I'e neglected (st..":-8 appendix L.rI), thf: "'quat.j(ll) (0, 9.,

=

(J l1ecom"~ (Shk66):

e elem*ni_ary (:ha)"ger ffi electron mass,

(). 1-1)

E axi ~'11 7.

E=!:l~(.:t~"i., C fi~ldstL\ength I

=-z

urJ.i t vector in axial dirE=!:ct_ion (

C

OCl elust~c collisjan operator, COin inelastic col]j~jOI' OP~l-~!Or/

C':'\F~e co1] .t~tl)n opr:a-rator for the Coulomh i nt.el"~cti("Jr) betw~'C'n the electrons_ (Co Opc:)'"F.l.tes CJn f

o.)

The equation for Q, -= 1 b8G'O)l')es I when the term f.:ont_a.i.ni nq f 2 i~: neg 1(1(: t ('c.i (~e" below) • ~1 (!.I) + £1 (£..1) + £1 (£1 ) el in ee U.

T-n

(9.

1 operates on

i

l) .

For- neon arId argon the elastic cro5s-secti on )_$ r:thQ1.Jt_ a h\~nl~Y(ld ~ )m(-"~·; larg!Zl":" lhtm the inel~stic cro~~-.secij.on in the 8n~r.gy y.cg1. OJ"l Of lpl~_'"r-l.'~~;l'l hence

1':'1 1« 1£.1

I·

The ratio

lSI

I/I~I

I

cun be estimated hy

sllh-il~ el ee el

~t.iL.u.ting .1.1 maxwellian r,:li.$t_r:i.O\)tioa function in the (iXp,t'i:!::i::sion for £1 giVRll by Shk~rofsky et al.

by

.!..

_w'

d cc (Shk6&). Replacement of der iva!:; W,3

dv Velocity,givRs that tho ratio 1£1 I/I~I is of t.hc· order or m~9ni tude W"N

IN

(N dectron density, e" <,)

o 9 0 -5

N

g g-:J.,S Cli.br"lSity) ~ The ioni~~tion hence

1:::.1 1« 1£1 I.

ee el The t"rm £l e1 i~ (Shk6b\: C - V f -1 (:1 in-l wh~r.e \) n\ 26

degree lS ].e~s tl~an 10 in our rE=!:glon,

Then eq. (2.1-2) becomes:

t

,,_.Jl -) v of o dv (2.1-4) rn

For an ord~r of magnitude estimation, we 5Ubst~tute for fo the mdXwelli~ d~stribution function f

o

~

ce-v'/w" (C cOnstant, w = lieU _e

1m,

U

J

elec-_v tron temperature in volts, (see section 2.4». This yi<:lld,> f l .. £.AI) 2

t

rn w 0 With the aJ;lproximate expression for the drift velocity

e£. e1:; v

mNgQDw = mVrn ;; (McD64) this gives for the or<.lo. of m<l.gni tude of

Appraxim~ting the expression far the a~~al eleot~ic fieldstrength of section 2.4 by'

we obt21in;

(M atomic mass).

From the general relation for f R given by Johnston (J"h66), a s<>.rne khld of app~oximation gives for the ratio of an element of the tensor

£2

to "n clement of the tensor

f2_l'lf~l/lf~_11

H

O(~)

£0. all i > 1.

Becilu;;e

vielli-

_M 0<10-2, the repltlcement of f by f ₀ gives

~ufficient

accur"cy in the calculation of th~ coefficient~ needed ~r the discharge model given in chapter 3,. Howe"G):", as will be Shown now, the value of

i

l is neceS'1ary to calculate fo' whGreas we do not neeo f

2'£3'" to calcL\late

!o1'

'l'he value of fo is obtained frcm eq. (2.1-1). In this equation the elastic collis1on term is equal to (Shk66):

1 df C a el

"0

2kT } (v\J (f + ----2. 9-V m o m

~)l

(2.1-5) wi th 'l' 9 gas temperature_

<lere the elastic collision term contains the factor filM, which does not appear in the ela5tic coll~5ion tenn in the ~ : 1 eqll.at.l.an

(2.1-3). 'l'h~s is sO because the di~ection of the velocity c~n change 27

m\lcb dln·ing em elastic collie-ian, whilG th0 magnitude changes Httle, so that the ttnisotropy in velocity spac0 i,; sl)l"ll (8hI<66). ~s th" t~nl) con-(.ainj"9 ~'!1 in eq. (2.1-1) i~ of the' orao>:" of magni.tllde i?:o (see th'"

OX-l"r~.~'3ion for E given "bove), thi~ term is of. the same order as _{Co "l and can} L.i)(,r.;,for", nut be negl.ect.ed l,n the oquation for fa. For all R 1 the term

c'."mt.a~.lii1l9 tQ.+l cun be J.1egl.~(.:t.e(~ in the f£ equation, as C£el do~s not c,oJllairl the fuctor ~ (s",,,, eq. (7,.1-3) (or £ ~ L "n(1 Sr,I<66, Joh66 for

~. 1) •

eqlJation. Th" term

C:Oj, hC'l':"e is u fClctor h1.,lnd:rcd laI'r,JEJ:' than Co 1 in t.he inelasti(; Yegion.

fl e N

The Coulomb term f)ec:omcs; irnp0):'t~nt her~ when e:~ ~ 10 ... 3 =:: lO .... 7 r a

Ng M

n~".ll",·lT't.=:al ionizati(.H1 uegree for our region. 'l'h~r-eforG this term mu~t be

~ak0n iI~to aCCOllnt in our region.

Trlen we obtain, aft",r substi t.llt tot) "f; (2.1-4) it)t.o (2. I-I) the scalar equation for fo:

a

(:i..

rlv \I m M --2.) rlv (: + C + C °el Din nee

(2.1-6)

Appendix 2. II The r<tdial d"rivat:ive term

"no

the .;>mbJ.pol.;>r field

u,rm

In t_hl~ .,:3.ppenc:lix the C'Jrder of m~gnitude of th.e radial derivative term ..;:t.nd thE:!" ctmbil?o!ar field term is compared with th~ order of magnitude c)f olh~r terms.

Wllen the rac1ia.1 (J~r~v.;:d.:.ivc term v·V f is taJ.;:,en into account, the - r

s",ll ar equ.;>tion for f" (eq. (2. I-6) becomes,

v~ e2 E2 2 df 'iI

_"

1 d

(2-

~'":e.)

3v

r: f 0 +

3"

~ dv

v

dv m m 1 ITJ d 2kT df (v'v ( f + ...£. .-2.))

_c

_c

\IT

+ + M dv m 0 m dv 0 0 in ee

~

The f i r s t term is of ltle order of m"gnitud" 3vooq2with r

1 tll", col\lIlll\ radilJc'- The first term <It the right-hand side is of the order of magni tude ~ Vmf",. Thus t.he "atio A of these two terms is of the order,

M 3m (N Q r 1) 1 9 D For neon N z 3.1022p 9 0 '0~2 Po '" (p or1)i For Po

=

lOa torr and r

1 = 0.01 m: Po

~

10-?'.

The ~ctual effect of the d~rivative term is, however, even sm~ller

beGause thiS term cElncels ,,(,cording to the Schottky theory (Sch24) nc'ar-ly the radial ambipol!>.); field term contributing CO the term ~. 'V

,,£

ill (.he Bol tzmann-eq1.lation. Tt.e total contribution of these two term~ is th0"e-fore sm(l,ller than thG value of A estJmated above. 'l'hereth0"e-fore these two effects are negleoted.

The term J₂ (E) in the E.F. for t < 8

1 is onty important. for ~. " C l ' where we need J

2(€) fo;.; the c"lculaHon of A, Band C (see fig. 2.4 ~ncl section 2,.3 .. 3). We thsrcfore derive an e~pre65iOI'l. that is tl.I\ ~pproxima­ tiOn fo~ ,J~ in the neighbourhood of t .. 0.

1. ~'rom eq. (2.12) of section 2.3.1 we have,

(2.IIT-I)

llecEluse forE) is a rapidly d8Cr".;lsing function fo< f~ > "1' t po) assumes its t"inal con6t~nt value very soon. (Numerical calculations

E;

show that t(£) " C((l) Eo for E " Eoand t(s) ~ t(c

1) for ~ > ;-0' wit.1l Co

o.

06€ I .) Thercfol:"e we t"ke t (c) out. of the integr<lt,

(2. IU-2) I('ttl"aducing

and applying partiaJ. i .... tegratiOn. yields

J₂(€) '" -J₁ (E)t(t₁)[G(E);1 (e)

-+0[£

~ ~~,

' \

(~')

dE',

(2.rIl-3)

(Here A(t) = t must be tak~n in G(E), to avoid the difficulty at £ = 0, whid' ~$ caused by taking tic) out of the integral (t(O) = 0)). i)e-cause J

1 (e) « Jt (O)G(O)/a(c) for ~ 7 £1' the SeCOnd term in (2.I{I-3) 1s negligible. The

thi~d

term is negligible i f

~ ~~ «~

(Lya72). Aite); sub-stitutior, of G(E) and H(L), this inequality becomes (wIth V",e"\'1 t"ken COnstant) ,

2v c

9

«

V +V (;2 ee g

In the Maxwell limit (~ee » vE,V

g), this condition is satiBfied~ In th~ Druyvesteyn limit (vee

«

VE,V

g) i t yields:

~

V2 U '"

(6 9

("0C "'qs.(2.14) and (2.39)).

Hence if U" « 16.6/(6 ~ 6.9, the third term can be neglectcd in respect to the fiJ:"5t in eq. (2.III-3). We then obtain i:l6 anapproximl'ltion for J

2(£) in the ne~ghbourhood of E

t:

(2. III-4)

In fig. 2.1 of sectl,on 2.3.1 we see that this apPl:oximation is quite suf-fiCient.

CHAPT6R 3 MODEL CALCULATIONS ON 1~ POS~TIVE COLUMN OF A MED~UM

PRESSURE lNERT GAS DISCHARGE

In sections 3.1-3.4 of this ch~pte~ a model fo. the positive oolumn of the medium pressure neon disOharge is preSented. In appendi~

3.1 the model is adapted to the case of argon. The model is bas~d on the pa~ticle balance equationS, the temperature equations and the equa-tion for the discharge current. ~or the caloulation of the reaction

co-effic~ent5 the elect~on energy distribution funotion calculated in

chap-te~ 2 has been used. ~he equations have been solved numerically. The results of the calcu,ations are compared with experimental data.

);ntroduotion

For the po~itiV~ column of inert gas disoharge. several theoretical models have becn pres~nted in the literatu~e. FOr low pre5SU~e columns

(pol,

~

10-2 ton m) the Schottky theory (Sch24) in a

~lightll'

ge)')er"llzed torm (Fr.,76) is often suitable. For high p):"es!;llreS (PoR -" 10 torr m), the arc model holc1s. At intermed~ate pressures thero are several complications "hich reqUire a more sophistic<>ted mode)" Effects Uke the nOn-uniform gas heating and the dependence of the electron energy dist~ibution function on the electron density have to be taken into <>~count.

In this chapter a model for the medium p.ess~re neOn and a~gon column lS presented. The model is cempared witl"t other models from the literl1tU);e 111 chapter 4.

The poSitive column is ~uppOSed to be infinitely long. axially homogeneous and stationary. Effects due to the electrodes or the detailed tube geOmetry are neglected. It has been expe~irnentally verified that the tIlMn features of th" column dO not ohange as a function of the tube length, p);ovided this l~l1gth is much greater than the tube r3diu5. Secaus€ of the atomic densities considered here (N > 3 1023m-3). several assumptions hold.

g These: are:

-th€ molecular ion density is much greate. than the atomi~ ion density so that the latter can'be neglected. fience the molecul~r ion density eq~als the electron density everyWhere in accordance with q~asi-neut~ality

(I;la974) f

-dir~ct ioni2ation is negligible with respect to 5tepwise ionization (Pfa68) ;

-diffusion of e""i ted aton,s is negligible:

-th8 radi<ltion-d! ffusioJ) is described oy the Holst<:.:in theory in the h1gh pressure limit (80147,51),

-the process of dlssocitltive recombination ma1.nly determines the vaJ,ume recombinatiotl p:r::oces:s (MilS74);

-heat conduction by electrons is negligible (Go169): -51)p~:t-eli1stic celUsions a:t'~ negligible (Lya72) .

. J. I The elementary proCeSses

F·O, ttl" an"lysis of the stepwise ionization process the excited stat.es are divided into several groups (Kttg71). Group I (:onti'ins the four Is-levels (Paschen notatioIl, see fig. 3.1).

Ionization level IV //////f/IUI((I?

215 V

Group III Group ;l;I

1

1

j

e

(conta~ning 2p)

18,6 V

I 4 _PI (s2) T ] (s3) Group p

s

TT'66V

.3 0 (lo:;) 2 PI ($4) R 3 (sS) M i?2 "rOUIld sti:lt.e IS

o

V

0

fi£g. ;,.1 SimpUfled leveZ ,,,,hem,;, of nr,{)n: d{''',:"dcm 7:nto 8e,)eral

1:,m-pOi'~an" ql'OUP$ (s"" t(JI1;t) (<1:<-8.) l:'a8ohen '10taNon).

Gr'oup II cont.~j.ns th~ levels whi<;h can be excluded from ehe ionJ.:;:ation prOCeSS (S\.lCh a" the t."n 2p~levels) because their U te-tl.me is very short (the 2p-1eve1" radi<ltG back to the la-levels in "bout 10-7 s). Fallowing Kugan we introduce ~ group III which COnSists of the h~9her

excited levels in which the quenching by atomic collisions is sO strong, that there is a high rate of associative ionization via these levels

(Kag7l) .

Group IV COnbi~ts of the cOntinuum above the ionization leVel. In fi9. 3.2 a_scheme of the sEveral processes occurring in the meo~um pres5u~e discharge is given.

ambipolar diffu~ion/ wall re~ombin~tion molecul~r ion L...'--_

I

Group e -

~

recOlliliination .---..----~ [Gro1.lp I I N" III

I

e instable molecule

Fig. J.2 Scheme oj' step"lise ionization and necombir-.ation l?2'ocesses

(e-=etect2'on, Ne=g~ound atat$ atom, N~~=~xcited atom).

The production of ions is described by a stepwis~ ionization proce~g in the follo"""g way. The levels in group I are produced by inelastic el"ctron collis~ons with ground state atoms. An atom in group I can be excited to higher levels belonging to groups II, I,r or IV. When such a highG~ level beLongs to group II, i; raoi~tC$ baCk to g,oup I Very rapidly. caloulations show that ata:ns i" gr"oup rIca" be ""'glected i" the ionization p,ocess. 'these atoms are only important for a recll.stJ:"i bution over the s-levels in qro1.lp I. Atoms excited into levels belonging to gro~p III are rapioiy ionl~eo oecause of the high ra~e of a~soc~ative ioniz~tion. Therefore in ou~ mocte~ ex~itation to a level in g~oup I~~ is taken as an ionizabion proeess (Kag7)).

need. b) kn.Ow' the (Jen~itie5 of th-2 Cl.tornr::: in group I. 'I'hi:'!::;0 (1(:·r~5iti~.~'''; dl'"(' deterlr'!ined by the oxci t:at.ion -:)f ground ~~t:dt".C' .=;ltoms, the transfer of energy betweOIl lhese 1 eve l.~ by D.tomil.: and 81e~tronic c(llli~ i .)11.-:::, t,hr~;

, '.'rl'.' 3 p ) , ... y

uestL"ucLi(,r), of t.h~ rn*ta~to.bl>::! .tI.t0m.'~ (-'P;,:' l ~ , rJlrt!e-r)ol1y

c(.LlJ~;ll_Hl~-i with IH:-:~t.r.-:'ll cJ.tO'fr't.'I, arId th(:.' ~k;_;;:un.~(: of. I.hf: r(·'~un,.\r:t :It.'[:\

tt.'~nSf"rm"d VEry rapi'.!ly intO !nOlecular iOnS by tIlr",e-bocly ~ollisioT\"

with gas atoms. This process ~s 50 fast that thO atomic ion d01,~i~y carl be neglected with respect to the moleG~lar iOn d~nsi,ty (Bn.g74).

T'", loss of molEcular iOllG occurs by arobj polar

oi

ffusion and di:!i::.H.>:;irltive .r.ecomtincttion. Th~ latter proce:s8 yields excited f.l.tom::: in <)rcI\\p n (Mas74).

AU pl:'Oc"S~"S are described by parti(,lG b, lance eql)i'!t.i 0115. "',r ~:;lcr'r:IY t:.l~tribution fl.H1CLr..0n c..=:t.l(:ulat.f2d itt (:hf.!pt(~Y" 2 _ TriO:: k.-*ac:t.i0:·L (:uc

r r

i'.:ir::nt:::;: uepend un the r~lectI'On t0t)·Lpey&ture aI"ld t.h~ ga::: t.~r.1p~:.I:.-i:L t".Ul""-r::~_ Thr::!r!":?[or~ the equatioTls for th~ electron tcmperature and the ga~

~.('rllpl.: (" r.L .it-a b,wl.: to be tak~n into account. Fuxthcrmore thl:";!; f;!lectron

,l~~rl~.,.t:"y' :?1":"IJfi18 1T1I.tst be in acct"l!·dtHlce with the I2quation for the

dis--dl~r'"I".l·j:.' '·':'U:rr:-r"lt. 4 cOmplcete :=ill::"t of ~~q:uCltiOI"I~ :r.eS\Jl.ts, Th.i~ ~et: has bf"'P-n

(;OlvQd J)\~!":?rlCall~r.

~~....!2...article...E..~lance eguCltions

3.2.1 Til" "xci ted at.orn"

Th,· balance eq\lat.~on5 for the excito0 atoms of group J arc, (J. I )

3 3 3 I

wt, ... ·(<: "lclex i=1.1,),4 refers to the F

2(sS), 1'1 (~4), . Po(s)) «"d Plisn :-'.'·.(1tj~' "r·e::"5~ectively. In this equd.tion we haVe t'epreS0I1t6'd Lhe lev(, l~: ill grOI.lp II LJ¥ One 8ir1g10 laVel H wittl density N

H• Furt.h0rmore zi = t~le

fr.~quf!nC'y uf excitation to 1evl"::l1 1. by elect;r.,)l"l collJsion~ with ground .::.;t,";l.t~ o.toms, Aj = the frequency with which 1l2"vGl H dr=:c.;':I:Ys to leVE'l 1., "'J.) the coefflcient for coupling due to atomic and electronlc co·! l·I..:,iOL'lS betw0cIJ. lev01~ ~ ("lnd 1 within group I. c:i..2F C"iJ ;j.nd c.:'i4 ~r~:'

Vi

=

th~ oestruction f~equency for level i by otoer processes;

for the metastable atoms (i=1,))

v

= YiN 2 (three body destructLon of i 'if

°

205 \ 1/2

metastabl~ atomB)~ for the ~e~onant atoms (1=2.4) vi - ~ (Ft) (Holstein radiation-diffusion frequency for Po > 10 torr, where T.

L natural life-time of leyel i ~nd ~i

=

wavelength of the ~mitted radia-tion, l\

=

t;'be radius (phe59)).

The density of atoms in level H is obtained from the balance equation,

..

E c. 2N N. + Cill 2 i=l ~ e ~ oQ ~

r

".N ,

i=1 1. II

Wh~rG ~ is tOe volume-recombination coefficient given by (Mas74)' 2.5 10-14

IU7

(1 - exp (_900) ) T g (3.2) (3.3) (T

g

=

gas temperature, Ue = electron temperature in Volts). Se~v1ng Nfi from (3.2) and substitution into (3.1) yields a 4 x 4 matrix ~quat1on for the excited atem densities N

i.

From chapter 2 we have i~l Zi

=

z,

where Z is toe total exci-1 tat ion frequency (eq.2.41). By lack of data we take all

zi

=

4

z.

The

valu~s of k_ij, Vi' c

L2-ci4 and Ai are given in Appendix 3.II. The ~qua­ Cions (3.1) and (,_2) ar'e solved numerically yJ.elding the. values Of N

i• Then the ioniz~tion frequency, necessary for the solution of the electron balance, can be calculated from;

Z _ion

=

(3.4)

From eq. (,. 1) i t can be seen that Z ion is a linear function of the total excJ.ta~ien frequency Z, which is 'calculated from sq. (2.41).

(See ~lsQ ~ppendix 3.1 for argon.)

3.2.2 The electron balance and the current e~uation

The electron~ are produced by stepwise ioni~ation and lost by dissociatiVe r~combination (coa{ficient a) and ambipolar diffusion to the tube wall (coeff~cient D~). The'particle balance equation for the electron density

1.

...i..

(r D r dr a Ne to en becomeS, dN are) + "ionNe where r ~ the radial distance.

- nNe 2. ill 0 I O.5}