investigations on the positive column of a medium pressure ...diech~rge is constricted only for high...
TRANSCRIPT
Investigations on the positive column of a medium pressureneon dischargeCitation for published version (APA):Smits, R. M. M. (1977). Investigations on the positive column of a medium pressure neon discharge. TechnischeHogeschool 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 beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI 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 pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand 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, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Download date: 27. Apr. 2021
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
GEBOREN TE OOSTER)!OU'r
DIT PROU'SC!:lI1IFT IS GOllDGEKEURD
DOOR DE; Pl<OMOTOREN
PRUf.DR.IR. H.L. HAGEDOORN
BN
DR. r.w. SLUnER
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 lI:
The approximation J 2 fOr J 2
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 and N a 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 J
Measurement of the radial del"lsity prof~les of the p)
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 II 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. HQwever l 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.!
po=filling pl'eS8Ul'S. i=discharge CUl'l'ent, R=tube l'adius.
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 it 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).
4
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,
5
-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.
6
1.6 Fluorescence me~sur~ments
With the fluo~es~ence teChnique the denSities of the metastable
3P2 (IsS) and the"resonant 3P1 (194 ) "toms have been me<>",ured. For
this technique a dye-~a",er (Spect~a-~hY$icS 370, S~oo ~ < A < ~400 ~)
has been used (Smi7S, Ste75, Coo76). In thiS way the relative radial . 3 3
density profiles of the P2 and PI atoms have ~een measured for vari-
Ous values o£ the discharge current. The measured profiles are "fompared
with calcu~ated ones.
With the fluorescence technique the coupling ooetfi~ients of the
2p-levels have been determined. These ooupling coe~ficientg have been
measured on the axis of the discharge as a function of the discharg~
current. With the aid of OUr model the measu~ed dependenoe Of the
coupling coefficients on the discharge OUrrent is transformed into a
oependen~e on the qas temperature.
7
C~PTE~ 2 THE E~ECT~ON ENERGY DISTRIBUTION FUNCTiON IN THE MEDIUM
PRE$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$~ons
for the axial electric fieldstrength and the ."xci tation freq\lenCy are
derived whioh will be used in the model de~cribed in c~apter 3.
2.1 The Boltzm~nn-equation
The eleotron energy distribution function iE.F.) ~8 obtained from
the aoltzmann-equ~tion for th~ stationary discharge:
v C(f) , (2.1 )
where f = f(~,v) is the e.F .. ~ position. ~ v~locity. e ,,"ementary
charge, m electro~ mass, ~ ~lectric fieldstrength. C collison
operator.
The left-hand side of eq. (2.1) represents the flow .orooo!!sses in phase
9
spClce, the right-hand side .rcpX'c:::;cnts the collision proC'e!:ise::s.
~~n tho 810ctrlc field i. low. such th~t the dYlft velocity of
the elect.rons is much srn<JUer than their tt10rm.;>1 velo,ity. the distri
bution function is nearly isotropiC ir\ veloci t.y ~paco:!: _ 'then the E. F r
~a~ be developed into spheric<ll harIllOnics (Shk66). 1:hi.5 develop",,,,,t co."
be wdt.t.en ~s (Shk66, Joh66):
C (f) (2.2)
Here f£ and tt are tensors of the ~th ~ank and 1 represents the r~p~QLed inproduct (see Joh66).
Substitution of this expansion iflto the BoltzmClnn-equation re
sults in a hierarchy of tensor equations of increa~ing runk~ BecauSe
th~ ratio of an element of the tensOr f~+l to an element of the tensor
til, ~s <)f the oroer of magnitude 1!8m/M (1'1 atomic mass) (see appendix
2.1), we have ~l.l:ff;i.~ient accuracy for our purpose when we replace
f(~,~) by fo(!.'v). 'l'h~ 801tzmann~equation then reduces to .;> gc~lar equa
tion for £0 (See app"ndil> 2,r). In r.lli . .5 "''1''''tion the radial d'aiv8ti.v,",
term. netl.rly cancels the term COJ)tair)ing the radial ambipola,[' fit::ld '$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/Ue
(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
/ue
, rJ electror1 en.ergYI u1
excitb.tiOn en~ygYI
. ~"': ~r""~aling parameter to be chosen as the electrOn t.empe!:rature lsee
~~~~WI ((J, tl1
, Ue all in volt ~quivillerlts),
H(t)
G(IO)
V ee
vE
N e
T 2 Ve,/'J (E) + VE(E:)e + ~\J (E)E: ,
+ v (E) t 2 , 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- ---3
m NgQoW
( o e
iii 112
83
)
e
Vg
or e
2m M NgQDw
,
electron density, Ng
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
Ue ~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 ~ Ng (see section 2.4), and the logarithm ~ppearin9 in
veo is a weaK function of Ne' the E,F. depends on Ne and Ng ma~nly, via
the ratio Ne/Ng
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 » VE ' v9' Vin , 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:
11
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
rO,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 'Vg
, \)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::,Ul
LJ ~ Ul
n = 1/4 and u 1 = 15.5 V.
(2.41
HC,t''.:.'I.I:'Om the mOmeI'ltum tran~fer cross-sectl.on is t:alculated_ wt:~ take
(Mco64) ,
Qo =(aD!ae1 Qel) 0.85 . (2.5)
=r< (-1/4 for 8 < E ,
1 V -)/4 F.
for > vE c 1 t E:I
= j " <:1/4 [or < v ~ 2'1 Vg g",
V9 ( 1/4 Po ;- 8
1 I
4 2 2
with '" ~ v }" G 2 " N un
3 u\ '" D 9 e
and '" 2m a N Un w with 1 (2,6) Vg M n "4 D g e
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 (U 2-U 1)
-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$t-ir) r)2"'088-aecticm for neon
meaeUl'ements (SaUO) ~-- sq. (2.1).
1 ....., J-~
--'"
-- meaSvrem~OIS (Sc~69) ·----<"Q1.·7 _-~---" -.-.....:..-----"'"
I
'/ I
.j .... _._.....!......-. ___ .....L.. ____ ....... _ ....... _ .1 )7 )8 19
UIV)
Fig. 2.2 Inelaetio 02"'Oss-seation for neon.
20
(2.'1)
13
2.3 solution of th~ ~cala~ equation
2.3.1 The elastic region
For tJ-..e cl.;'.l.stic region F:. .-::: ~"':1 f th~ iIlelastic cross-section Qt 0
(See eq. ~.71. BeCauSe f~~ AI~I = 0, and the~~£Ore H(O) = G(O) = 0,
multiplic~tion of eg. (2.3) by Ie ~nd integration over c yieldS'
df H(e)
o + Gltlf o
- t (0:) , (2.8)
wh"re t(t) - N w J'-(C'+C1IQ (8'+"I)f (8'+f))d(' (2.~) got. 0 '
"~ the nllmbe~ Df electrons scattered in t.he low energy region by i n-
elastic collisions ILy~72). The forma] ~clution Of (2,8) is,
WitJ1
und
f (t:)
" J 1 (f:1
NJ 1 (E:) + J2
(8) ,
I 1 ( ) Jf: -t(t;') dc', J 2 t: - J E: 0 HIS'IJ
1 lEO')
(N is a constant),
(2.101
(l,11)
(2.1))
For "-n analytical ~ol\ltion we h~ve appxoxim~ted th~ int8gl:"~1$
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 ,J1
(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
'" * \JE \J
a = b ..!L \J \!
ee ee
(The expression fo):" M(E)
10 4 7 4 - "7 y + Y +
1. /l- y +/) I (y + 1,-, (~Y(;L.g -73 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 <: 10 15 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
Ng • ) lo2'm-3 ~ ::;: U. ·3V ,
-6 I, N ... 'O '6m'"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!Ue ) "" a fUnction of U.
In fig. 2.3 M(€) is plotted for seVeral values ot N , with the scaling 3 I 2 e
factor Ue = 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)
lS
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 J2
(~) fo< J 2 (£)
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 J 2 (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 ., t1,J
2 ~(£l is a good apl?roximation fOl" J
2 «(:).
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) gee
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:
lEo
d 2 f df 0
P(~) ---..£ + S(E)fO
o , --+ OE
2 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
00 0 000 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 (Duy7 5) ,
2 q(£) ~ - a
1 (£-£1+\)1)
2 where a1
- a 2 (£-E +b ) 2 2 2
= NgA2wU 2 II (£2)
b ~ 1
E: > t.'} , - 2 2
(
(l+bo EI
) - 4bo e:1
(l+ao
£l)
4H(t1
)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 e1
,
(:<.23)
(2.24)
17
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 +b
1)) U(c)
'P2 Bi (aI2!3{€-~I+bl» UIE)
¢3 IIi 2/3 (a2 (8-"2+1.>2)) UtE)
U["') I
exp(-7 P (Et
) (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)
NJ1
' (101
)+ oJ/IE I )= A<jll'(~l)+ B<t>/(81) , (2.281
AlP 1 (02) + B¢2 (c 2 )= C'1J 3 ('~;:>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 it 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 Ng '" 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«()fO
' (eJ+G(f::) fo1
!:l+E
- I H(I::)fa
' (,:)+G«:lf01 ';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 ~ ~ E1 . ~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 J2'" = -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(E1
) p. .. - ----
H(E! )
B AI, C
and 4> i j' = 4> i ' (e j) ), we get
NJ1 (~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 Ue
= 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!Ue
,
an extra (scaling)parameter Ue was introduced" ey most authors (Go168 a ,
Go174, Wcj67) Ue
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
19
~c J -, '" ~ 4'
~o
Eo, ><
Ng -3.lo<"rri3
Ue ·3v
I: Ne _I017m3
2: Ne - iO IB".;3 3: N" _IOI9m')
4: Ne _102°",3
90
... 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.; Vg': (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, it 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 calcula-
20
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 J2
l,-) 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 c I/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 IN ) \~ a N U 5/1 e g M pge 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_ 3C
2
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;
21
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('-1 1/- (/.4()b)
1 Substitution "f fo(E) <lccording to eq. (l.2S) into (2_10b) yield",
Z = NjJ l (Cj)G(C 1 ) 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 10-5
1 within ]0\ by; e ~ 9 -
a_lOr, u 0_1" 0_044(10 lg :" + 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, < Ue
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
22
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 FZ 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 Ne
.
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 = 0 S 10- 2) 2v- 1 1 " m
23
N
~2-S
o
.\
_." ····T,-------r-
Ng• 3.102~rri 3
Ig ·WOK
1. We ·1.5v 2. Ue • •. 0'1
3. Ue -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 C1
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 QO
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(81
)2
-I p(el) = (H'(O:ll+G(Sl))H(€:I) (see eq.(2.20».
Tl\en z = 21TW3NG(E
I)J
1 (~1)FIF2 (se<;:tion 2.5),
with Fl It-~ ~ r hence H("I) <I\r(~l)
2aI2/3(I~a)Ai' (al2 /3bl)+(Hb€1 3 )l\i(aI 2/3bj)
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 (10 19 :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:
and
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
-~j e 10- 4 ) • 10 <: <"0.5 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)"ge r 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 (
COCl
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 fo.)
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.cg 1. 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 £le
1 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,
(:2_1-;\)
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 1m, U elec-e J v
tron temperature in volts, (see section 2.4». This yi<:lld,> fl .. £.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- 0<10-2
, the repltlcement of f by f gives ~ufficient accur"cy M 0
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 il
is
neceS'1ary to calculate fo' whGreas we do not neeo f2
'£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 mom ~)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~ e 2 E2 2 df 'iI " 1 d (2- ~'":e.)
3v f + 3" ~ dv r: 0 v dv m m
1 d 2kT df ITJ
(v'v ( f + ..........£. .-2.)) c c \IT + + M dv m 0 m dv 0 0
in ee
~ The first term is of ltle order of m"gnitud" 3vooq2with r1
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,
28
M
3m (N Q r1
) 1 9 D
For neon N z 3.1022
p 9 0
'0~2 Po '" (p
or
1)i
For Po = lOa torr and r1
= 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'herefore these two
effects are negleoted.
The term J2
(E) in the E.F. for t < 81
is onty important. for ~. " C l'
where we need J2
(€) 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(c1
) 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
J 2 (€) '" -J1 (E)t(t 1)[G(E);1 (e) -
+0[£ ~ ~~, '\ (~') dE', (2.rIl-3)
2')
(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 if ~ ~~ «~ (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,Vg), this condition is satiBfied~ In th~
Druyvesteyn limit (vee « VE,Vg
) it 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 J2
(£)
in the ne~ghbourhood of Et
:
(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) ;
31
-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
32
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 II 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)).
33
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 F2
(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 NH
• 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~:'
ex.citatiou frequencies. from If.:v-(;.L i to group III III ~nd IV respl.:,:.:t1.V'i;)A;, I
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)
(Tg = 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 Ni
.
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 kij , 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) it 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}
35
Here D u ..... CObmi Ue ' where hmi = the mObility of tile moleCD.I.;:t:r ions I and
CD " [anor close to unit.y dependiog 011 the sh~pe of the electron energy
distrn.)\~t.ion function. CD and bmi
3t"e given in .::appendix 3. III.
'l'he so lutiO~ of eq. l3. 5) must satisfy tDe boundary (;O)1di tion,9:
dN N, __ (1<) = 0, (~) =0, "nd N (0) = N
dr r=O ~ eo (3.6)
whf::re Neo is obtained f:tom the equat.ion. for 0. givp.n diSuhay.ge curr.:-nt
(~e", appendlx 3. nI):
N (00
(3.7)
Here i - diRcharge curxent, e = elemontary charge,
M atomic m;;:lSS, U eo
R '1'" lube r.a.dius.
1 2/5 Sl = oj t (;<) n(x) xdx ,
1I?:1ectron temper~tur'.;.' F-It the axis .in Volts,
(J.8)
x = r/R, t(x) - Tg(rJ/Tg(O). n(xl = Ne(r)/N.(D).
eN = correction factor close t.o unity, depending on the shape of
the elect):"Otl f!nergy distritmtion [uner.ion. CN
is given in
Appendix 3.llY.
3.3 Ttle temp~rature e9uatio)1~
3. 3. l TI\fo electron temperature
The equation for th" electr.ic fiel.d"trength de~iv"d in chapter 2
(eq.L"3g) is:
~Mm ~DN U S/4C(N /N ) 9 e (0 g
[3.9)
The a}{l.al electric fieldstr.ength E is r.:;dially constant. Because C is
only weakly dependent on Ne/Ng we rep'L:lce in (3.9) C(Ne/Ng) by
C(N"'Q/NgW)). 'l"h"n we obtain from (3.g):
(T (r) /'J' (0)) 4/5 9 9
(3.10)
where use has been made of t.he ideal gas law.
Th(; \lalu~ of t;.he electron temperature at the axis Us (U) i5 fixed
36
by the dem«nd Ne (t1) = 0 (~ee se<;otJ.on 3.4). The g;'ts temperature p,,"Ofile
Tg(r) i~ obtained from the Heller-~lenba~s equation.
3.3.2 The gas t~rnp~rature
In the HellQ~.:-E1Gnba<ls equation, t.he heat-flux to the wall "quaIs
the heat diSSip<ltion of the electric current:
1.. r
dT d (rA (T ) ---li.)
dr g dr - JE ,
where J = eNe(r)beE is the <;ourrent der, 'ty (be
See Append~x 3.111).
(3.11 )
electron mobility,
Using experimentAL data given by Sah
can be w'~tten as:
(Sax68l tl1e thermal conduct~ vi ty
A (T ) = A T 2/3 gog ,
with Ie = 1O-3wjmK5!3. o
The boundary conditions for eq. (3.11) are:
dT (----.i) dr r=o
o and T (R) 9
0.12)
The w<llL temperaturQ Tw~ll J.5 obtained from the cOndition that
the diSSipated power must be equal to the heatflow by convection and
r«diat.ion from the w«Ll to the environment. Hence'
:i.E
where T enV ct
h
"b L
E
tj
temperature of the environroent l
~ heat transfer coeff~~~ent, equal to:
T -T )l/4
7 -1/4( w<l11 env} ,
I. T +T (J.ic57) wall env
~ tube length,
transmittivity of glasS,
Stefan-Boltzmann const<.
3.3.3 The pre~$ure in the ~~SCharge
(3.13)
The gas density Ng is obt«~ned from the ga5 tem~erature Tg v~a
the ideal g«5 law. When the di$charge is oonnected with « large dea~
volume, with tem~erature equal to the room tem~erature, the operating
E,'(8e,;;;;UXO p 1 is eq u.:s.l t.o t.he fi II ,1 il9' pr~9sur£o P CJ (bnth rl0t t'':l.di~;L 1 d,,-·
,,"lldGr'lL), "1 = Po Ng(rlkTg(rl" (.1_1';)
On ths Oth0'r" hand r when t.h~ d*ad volume is SI'!h>i.ll, we::: g~·t
(j. 1 '.\ I
wt1~re P7, i~ obtained from the equ<1tion fot' t.he CO(l~ervation of thl'
total number of gas atoll/,S:
(J _ 1("
Po
k'l' env
i5 th" g"~ den"ity in the i"lisctlarge-off'" L tuatLc,lL
This yi"ld~
( ,_ 1))
f'or an a~t)i 1','1: :.try v-?.lue of. t.ht" f):'.:s.(:t iOn ot t:.hf~ detlod '/oll·LrnE~ I "'~r1
obLain frorn the i,.1 .. :..:;.l ;lns laW ~ g~:n),e:r..-al rOl::'mul.-:. f'o.r· p Cc;LltL~iTlLrH:.I til!;
volume$ V1
?!:mJ V2
iJf cr~.~ dl,SchM.rge;& r.ubf::! and. t.ht~ dCr~d vol L~m~,'r ,oj,nd ~ ~~l
~verage tempel"'atut"Ds Tl .:.md T2 in the cases of $ma'il r~~p~(:t.lV~)y ~~Il"f"';
dett.d volume as paramer.ers. In our region where T 1 -:.: 3'1'2 t.hi~ p.:xpr(~!'-~:=;ton
can be "pproxirnated within lOt by:
V2 VI p V
1+V
2 PI + \!l+V L "2 -
3_4 The numerical treQtment
The equations used fo[' the nurnf!'t."'ical treF.!t.ment tld,Ve been gi\',,~r, I r!
sect.inns .3.2 arld ).:3. They are the bal.ance e-quations tor tho:::: excitnd IH:
om", (3.1) "nd (:L2) , t_he on& for thG electrons (3.S), ch" c:urr"nt: ~'l"'
ti.on (3.7) and tl'" Heller-E1C'r\baas equation (3" 11) _ The "quat_ions ""..,
,~oup.led, "nd eg. (3_") i f. ~onlineaJ::. Therefoz:-e ttley hav" he"n ~oJve,1 n'.,·
mf,!;t"'ically with an it'.erar.ive Pt"O(:~JdULe. Tbe flow diugra.T!1 Df ~:he p:t:Ogl·,lL'l.
.is given jr) £'i9. J.3. The two second or,ic't" dlffeyer'ltU.=.l equationr::: {\.~lJ
.~r\d (:L 11) h"ve be'~n SOlved e"plicitly wH.), forward dJ.sen!ti"'1tl,,,, wi tt,
k discrete intervals. After the calculatLon it was ch~cked wh~th~r n
Suff1cient IF.!rgc [lumber 0f intervals k was used. If not. k wa~ enlarg~d
and the calculation wct;; repeat",d_
The iter"tion procedure is s.tarted with chosen profiles obt,3inGo
£<om former cal(:1l1aHOIlo. The value V(p+l) of E, N ();'l/N ,'!' (r)/'!' ([II e eO 9 9
IUld N calculated in the p+l ttl iteration loop is compared with the value eo
v(p) of t.he foregoing l teriltion, When
}-____________ ~ A
Ng(l:), Tg(r) , ue(r)
N. (r) 1
(eq. 3. 5)
Ueo ' k. ~e(,)/~eo
Tg(r)/Tg(O)
fi",S. 3.4, J. S. N (r)
e nO
yes (eq. 3.22)
(eq.3.11)
,N (r)/N e eO A
T (r) IT", (0)
yes
-- r-;:r ~ no-~
N (r), U (r), N (r), e e '"
T (r). E. p. Ni(r) Fig.Z.Z Sdh~$ of the numepiaaL
p:t'OO$dUhJ.
]9
t.M j,t.0,-atiOll ,$ stOpped, otherwi5e it i5 repe<lted (see fig.3.,)).
1" t.he loop fo,- t.!J(. oalo,dati<)[\ of eg. (3.5), th0 valu,," of
U(." (=l!" (0)) is obtained 1. t0rat.i voly fJ:'om the. domand Ne (R) ~ 0 in the
t0l1owin(j way (5~f:: fig.:l.4J. When the electron temperature .a.t the €lxis
1100 i.~ loo st"J)6I.ll r the pr>:.-.rile Ne (r) can becoille zero for r ::- R or m~y
(3.19 )
eVCJl have no zero point at all J but increase for increasing r 'see £ i9 ,3 L 4j
When Ueo
is too high th~ zero point of t.he profile is reached iJ.lreLtdy
for r < R. The value of Ueo
is adjusted iterQtively with a regula falsi
procedure such thctt a vct!ue of Ueo
is reauhed WhlCh giVe8 a p)"()ftJe
"(;c;ordlng to pr'oflle 2 ill fig. 3.4. The c"loulation of Ueo
.is e:t.opped
when I N(. (R) /Ne (0) I < 10-2
•
Rowever , when the discha.rge current lind the pressure ct:(e ~o htgh
th~tt the column is ne~r the constriction region l the c:-lcctron dens-tty
p"."H 1e b"""lIles so ntlrrow, that the calculation of this profile becomes
6 ; .. 1.0
'l, z
r .-_________ 1
R ~ .. r
PlF' ~i. 1 Nor l1 laU;;:,;rj deatr'on densUy profH!9$ ()btained in the determi
nal;';'o", ()f Ueo
'
very .'3l:!"nsitive tD e)(tremely s!lw.ll variattons of Ueo
- Then ttl8 disch",rge
is not long"r wall dOrtlimlted, ):Jut- is in the volume regime" (R"k'lO). Then
the m"t.hodc to determine U"o giv"n <l.bove gives great difficllltJ,<ils. We th"r:
apply the following m" thod. The profi I.es I and J in Ug. 3.4 st.:lrt to
differ (~Ortl the ideal profile (profile 2) for values of r fO, whioh
40
oq.3.1i'
I I I I I I I I
0.1 - - - - - --
R -'
r > rl" from num0rical calculations it appeared that when the differenc~
in the values of U b~longing to profiles 3 and I, U (ll_U (1), w~s -4 eo eo eo
smaller than)O ,the profiles 1, 2 and 3 differed not more than l~
from each other fo. r < r l , while for all profiles the normalized
densities had dropp~d under 0.1 at r = rl
. Therefore the normalized
density profile calculated with a value of U for which u (I) < u eO €o eO
U (3) and U (3)_u (1) < 10-4 is nearly equal to profile 2 (the id~al eO eo eo
profile)for r < r1
. According to profile 2, the normalized density if,
small fO. r l ~ r ~ R. In thiS outer region the Ne(rl/NelO) prof~le has
been approximated by'
N (rl e D" 20
where the ""lue of C j is chosen such as to 9ua;r:"ntee contl.Duity wi ttl
the inner region at r=r1
.
When in th0 outer region the ionization frequency
(because N (r) and U (r) are small in that region), and
z. 10n
is small
aN 2 is small, e eel d
eq. (3.5/ reduoes to ;r: Or (Dar dNe/dr) = 0 with the analytical 801u-
tion
N (r) '" -tnr (D "constant). (3 _ 21 e a
We have used eq.(3.20) in stead of (3.21) b~cause more preoise numerical
calculations proved that (3.20) is ~ better approximation to the real
situation in th~t region. Hence in the Quter region the proQuction term
Zion and the volume recombination term -aNe2 still have to be taken in-
41
,------,--------" -
I;DOO 200 1 orr
1500
so 1SO i(mA)
Fig. ,L (I F; (N cl f,,IrwUon of i foY' neon. R ~ 32 mm .
• 200 torr memmY'ed, I) Joa tOri' meacured, x .)0 t01'r measured
-~ oatou"iat1:one, ---- ,-,Or1f;tl'ioud d-ieohar'gii!.
to account in the solution of. eq. (3.5).
In 1111 c"se" tho thus calculated profile fulHUed the criterium
INe
(,,-) I Ne(O) I <: 10-2 (see fig.3.5). In this w"'Y eq. (3.5) could be
"Dlved without numerical aifficulties .
. 3_5 Num~rical results ttnd comparison with experimenta.l dati;!
For a few cases calculated values of the axial el€ctr~c field~
st):'"ngth ~):',} g~v("n together with experiment"l data il\ fig. 3.6 for i:t
naon disch<l):'ge ()l=.,2 mm, po=~O,100,2DO torr, V2
!V1
c O.1S). For the dif
fuSe diseharg<il the <lgreement oet:ween c .. lcul"tions and experiment is -20
very good. TO obtain th1s agreement: we held to take "el .. 1. <; 10 in
Ete~d of u = 1.8 10-20 in eq. (2.1). This corresponds to the lowest el
measured curve for the elastic cross-section (see sa170). );\ccal)Se t:he
axial ~lect:ric fieldstrength E is proportional to ael' the v~lu~ a~l
1.8 10-20 (see eq.2.4l giv~s "alcul~te(l values of E which an" 20%
higher than the measured ones. With the model given here the tY'ansition
into the constricted state can n(ll be (;~lcu.lated. A discussion on the
constriction phenomenon is given in Ghapt(~):" 4. 1)""1 fig. ).7 a f.ew cal
cl.l;l.F.l.ted valUe5 for E are g~ven for argon, togethGt:' with expeX'im~J)t.;:l.l
dat:a and calculations 1-.-y wojaczek (Wojt'l). Here' al"o, our calculations
au" ,in <lg>:-80(I)8nt with ~I<p"ri(tjellt re)t UI') (oj ffus~ disch"rge. (For. a~go~
no alteration of ae1
was neces~~y/.) 'r·t'I~~ mo~~l of Wojaczck is d~scug5ed
42
in ~ection 4.2.2. 1n fig. 3.8 so~e calculated and obse~ved radial elee
tron den~ity profiles are given for neOn (R = 32 mm, Po = 100 torr) fo~
various values of the di$Charge current. The exp~~imental values have
been obtained froQ:! measurements on the electX'Ol1 .... atom brem!?strahlung con-
tinuum (See chapter 5).
COnsidering the e~p~rimental inaccuracy (see chapter 5) here too
we ean conclude to a good agreement between model and expe~1m~t, c~leu
lated electron temperature profiles a~e comp~red with experiments in
chapter 5. In chapter 6 calculated density p~ofile5 for the excited atoms
are compared with observed ones. In that ehapter the model will al~o be
used in the determination of the dependence of the coupling eoefticients
of the 2p-l~vels in neon on the g~s temperature.
5000 \ w
lSOO
o 10
'~.-....... --.. --. .......... .......... , 3 ,
I I , , , \,
" ,,~-
.......
........ - ... ·11: ............... 1..:
20 40
dmAJ
Fig, J.? E in dependence Or! i for argOn, Po'" 76 topI'. '11 ~ 10 mm.
, our model aalcutation (appendix 3.II),
2 measurementa of Wojaazek (Woj8?) (x).
1 oalculations of Wojaczek (Woje?).
Conclusion
The model presented he~e gives a fair description of the diffuse
discharge. The transition to the constricted gt~te, however, is not in
cluded in this model. Possiule mechaniSmS ~e5pon9ible for an onset of
the constriction are discu5sed in the next chapter. There we also ~ive
a comparison with other models preSented in the literature.
43
§ " z
== ~
" " " SOmA " diff "~....,
16
__ , c~I'<JI~lod
- - - - me~~l,.Irro
Fig. 3. 8 Cateu~ated and Obf3'.'F/JBd }'(ldia.Z eZeatron den87:ty pr'ofiles.
A£~",ndix 3.); 'l'hc po"'~t-Jve ,"olumn of the arJl..on discharge
In the "ame way as it. has been done for neOn, a model for the
argon di=charge ca.n be set up. we give her~ some dij,ta which are needed
in that model. Our treatment of the balan~e equation,; of Lhe el<c~t-ed
atoms in the argon discharge diffel:"S hom t-he treatment for the neon
di5~harge. In group I we have (by lack of dat-al only t~ken into account-J
t.he ffil7.t.astablt7!: .P2 (sS) level. As this level has the largest population
wit_hin grau}? r this u5sumption CttTl not ver:y s(~riously affect th~ :r:~S1),lts
For neon this i8 also the case. r!owever ~ for neon we havO ce.).L";"ulated
the ~xci ted atom balances in full det,8lil, to obtain the densities ot the
e~('''tecl l~-atom~, as these denr:;ities: haVe been mea~ured with th~
fll)o,,,"scence met-hod de~cribed in chapter b •
'J'h",n the tot,,1 ionizat~on frequency :for o.rgon is (cf. eq. (:l.ll
for neon):
z. l.on (; .N +yN ~ z ,
Ul.1 e 9
",h0~'" 7. ~ the total eK"i t"tiorl frequency (se8 section 2.6) ,
(3.I-l)
y = three-body (,01 lis ion "oeffi.::ient for the destruction of atoms in
the sS-stat~, Cmt
:.: co~fficient for exc~. tation frol.t'I the a5-level in
group I to levels in t-he groups III and iv.
Group U is ignored here. FOX {leon, group II has been t~en into
acc",mt becaus,;, grO\lp II may be of influence for redi8tribution (after
44
recombination) over the is-leVels. AS we can not measure the IS-levels
in the C~!3e of argon I bec~u::;e of the ~e.st.ricted W.:1ve lEngth region of the
dye~la~er, a det~iled knowledge of the distribution OV0r these lev~lR lS
not of importance here.
The ~oefficient cmi
has been Gstimated using the clas~ical Thomso~
C~OSS-5ecUQn (Mit73). Numerical cvaluaUon then sho>'S that cm1
can be
given by:
o . m1 1
5.3 10- 14
= (LB!) -3.8) e
(Ue~2.5),
7.S 10- 14 (Ue
> 2.5 )
From Massey and Burhop (Mas74) we obtain:
y ~ 3.6 10 -45 m6/s
(3. I-2)
For the oalculation Of Da we refer to appendix 3.IU. The volll.'';' recombi
nation cOGf£icient is (Mas74):
-14 8.6 10
U 2/3 e
From the e~preggion for E given in section 2.6. we have:
ue(r) (T (r)) 1/2 ~~- .. ~ U (0) T (0)
e g
(3. <-3)
(3.1-4)
Because of the different en.;.rgy dependences of the el&stiC croS5-s~ction
of argon We have here the exponent 1/2 in stead of 4/5, as is the ca8c
for neon. Then Neo can be written aE (see appendix 3.111).
N eO
eN (3. I-5)
where 51 = 0[1 t(x)i/4n (x)xdx. and eN is given in appendix 3.111.
~inally we have for A(T), needed in the Heller-Elonbaas equation
(Vr\l76, WojE7):
Appendix 3.11 Data tor the excited atom balances
The results obta!neo with our model are very insensitive to the
treatment of the balance equations for the excited atoms. In fact treat
in9 the CaSe of neon similar to the argon case gives alreaoy very good
r~sults. However, because in our laboratory WQ are ~ble to measure the
densities of the la-atoms with the oye-la5e~. we want a more detdiled
45
knowledge of t.he distribution over th~ ~S:-10Vr.:.'1$r w(::, give-: h~re the co
cfficiellt$ n",cessary to solveeqs. (3.1) atld (.3.2). From data given by
Phelps (Phe59), we obtain'
4 - )/2 V
4 ~ 4 10 R •
l~e coefficients kij
c~n be written us~
with
k .. ].J
~ k 1,i' energy gap betl'e;'rl level i and lc,vel j in
i
2
4
4
4
2
k .. a
'J
1. 40
p.32
O. ]2
0.61
0.104
0.55
kij
e E .. gij 1)
S.O 0.60 600
o.~ 0.20 1~00
0.5 0.33 502
2.0 0.60 2650
2.0 1.0 2040
0.5 3.0 1540
Tabt<~ .l.II-l P=conetel' Vatue8 for' caupl'inr;/ aoeffic'ien~s betwMYI the
1a-~iJV1le (gJ!'ou.p I).
We defit .. , a i = Ai/P\' Prom Cool.,n (C:0076) we obtain,
0.1
= 0.40, aZ'~ 0.27, a3
= 0.08, a 4 = 0.25.
The cross-section for excitation from group I to group IV is
(Dix7 .}) ,
\
1.2 10-20 (ll-5)
= (6.44-0.0444u) 10-20
{ U <: 10 (3. Il-l
10 < U < 100
The coefficients Gi1
are then obt.ained according to {whore we taKc.::.'
1 Qi,4 =4' QI4)'
4&
with ~ a
lmv 2 =~
U a
U U e e
~1 r;:.l" Q14.1') (E:) IE dE a
(for all i 1.2,3,4),
(U", = 4.9),
8tl t\1t~d (1'1 eN. 211'",3) aeeo;>:"dtng to chapter 2. Numerical calculations show
that c 14 can in our region be approximated by;
\
0.7 10-14
= (l.8u -3.8) 10- 14 (3. II-3)
e u :. 2.5
e
Ey lack of data on the cross-sections for excitation f~om group r to the
groups II and III, these crOss-sections have been approximated by the
classical Thomecn formula (Mit73):
. U-u Q 2..Ji.i 10- 18 ( __ 1)
U 1 u2 (3.Il-4)
where U1
is the energy gap in Volt (for the excitation trom group
to IV th~s formula
yields ci2
gives Ql41 ~ 1.5 . Q14' with Q14 from (3.11-1). This
and ~13 ~ ~~i4 with ~i4 from (3.II-3). The inaccurate
knowledge of the above mentioned cross-sections has no serious conse
quenCeS fOr the model results. For instance when c14 is multiplied by
" factor 100, the value of the axial electri~ fieldstrength decreases
with 20~. Other oischarge qu~ntitics sho~ a similar weak dependence on
the values of c i2 ,ci3 ~nJ ~i4'
Appendix J.III Caloulation of Dl and Nco
3.111-1 The ambipolar diffusion eoetficient
The ambipolar diffusion coefficient Da can be w~1tten as (because
the elcctron temperature is much higher than the gas temperature,
(McD64) ,
D
" (3.III-l)
where D e the elec~ron diffusion coefficient, be = the electron mObility . + and bmi = the mobil'ty of the Ne
2 molecular ion, which can be written
as (McD64):
(3.III-2)
47
"'"Lt. b'~io 6.10-4
for neon "nd 2.10-4
for argon. Th~ electron mOhility
is (Shk66):
b C
(3.III-:
with f ()
"Dun (5e~ chapter 2 for the defin~t.ion
of the parameter.l.
E>~or argon n= 1 r and we get.:
For
h e
neon
b e
1£" NI J IT! a N lJ 3/2
o g e we get aft~r pa~tial integr~tiOn of (3.Itl-1):
!!ere J(p) " /"EPe-M(E\lE: , "'nd N __ 1 __ o 1 - J(I/2)
The electron diffusion coefficient is (Shk66),
This C<::LfJ. be written as:
D e
'*'1 3N a lJ D J(1-n).
9 D e
Substituting n=I/4 for neon into (3,111-5) "'nd (3.111-6) we get.
b e
b U e e
(3. III-'
(3. UI-
(3. UJ-I
(for argon we have to use eq. (). nI-4) and eq. (3. III-6) wi th n=l) .
Numerioal calculation6 ~how that in our region CD can be well approxi
mated by:
48
I. 36
1.36+0)
~
where '::.'\
2.2
2.2+°2 --1+<;2
N /1'1 < 3.10-8 e g
N IN e g -,. 3.10
-8
N
0.07(1°19 1'1" + 9.48)4.0
9
N,/Ng
< 3.10-10
N IN e g
~10 > .,.10
9.48) 5.9 •
3.111-2 ~he equation for the discharge current The expression for the dischar~e current is,
fR 'J. 2-rrrdr '" JR Ne(r)
eN Il E ~ 2ITrdr 0 0 eo e e
Hence N i
eo 21TeR 2';;b e(O)SI
lbe(x) N (r)
whGre nIx) " SI =of be(O) n(x)xdx, NTol e
and x = r/fC be(X) 2/5
".om (3"III-4) and (3.1II-5) we obtain tor neon j)(O) ~ t(x) . e
and for argon
(3.111-8)
(J" IU-9)
(3.IIl-1O)
D.III-l1)
substituting E frOm Chapter 2 and be from section 3.111-1 (eqs.(3_111-4)
and (3.111-5)) into (3.111-10) we obtain for neOn,
where
N eo
eN ----~--~~~i , 3'ITe v¥. 1.1 1/2$ R2
M eo 1
J(l/2} CN = C(N IN )J(-1/4)
eo go
and for argon'
N eo 2TTe J12.68
M
J( 1/2)
i ,
The facto~ CN
has been calculated n~erically and could be
approximated by:
N IN < -8
r93 3" 10 , eO go neon CN
= O.93+0.4c1 3.10 -8
1+el N IN > eO go
c1
as in o. III-7) ,
D. II1-12)
49
50
N IN < 3.1U- 10 e" go
N IN '3_10- 10 , eo go:"'-
CHAPTER 4 PHILOSOPHY ON THE CONSTRICTION OF THE POSITIVE COLUMN
In this chapter we discuss the constriction pheno~enon. In section
4.1 some phy~lcal mechanisms which are of importance tor the constric
tion are di9cu~~ed, On the basis of this discussion a review of some
mOdels presented in the l~terature is given in sect10n 4,2. In section
4.3 a number of deviations from our model (given in chapter 3) are dis
cussed. From thGSG calculations arguments follow for a connection be
tween the constriction and the appearance of striations in the consid~red
prGssure/current range. In section 4.4 the development of the striations
i~ ~nvesti9ated.
Introduction
The C'onstric.tion of the positive column of inert gU,s dischdrges
sets in above certain values of the discharge current and the filling
pr0ssure (see fig. 1.1). This means that the con~triction is caused by
prOCeSSes which become more important for h~9her val~~s of the gus den
sity and the electron density. These proe~ss~s are; the maxwellizac~on
of the electron energy distribution function (E.F.) by th~ increasing
Coulomb interaction of the electrons at increasing el~ctron d~nsity,
the increasip-g influence of volume-recombination with rebpect to ambi
pol~r diffusion and the non-uniform gas heating. causing a radial de
pendence of gas and electron temp~rature, These p~ocessGs, which arc
of equal importance in our region l are discus6ed in section 4.1.
In the literature sevG);:al ~o.dels basGd on one or more of the effects
discussed ab0Ve h~ve been set up for the ~onstriction_ ~e discllss these
models in section 4.2.
As none af these models is capable of describing the abrupt
transition of the di'fus€ into the constricted discharge, and this
transition is not described by our model (given in chapter 3) e~ther,
w~ searChed for prOCesSes whiCh may be the cause of the constriction
by a variation of par<lrneters in our model (section 4.3). There !1re !1rgu
ment~ that the onset. of th~ constriction is connected with the abrupt
generation of ionization waVeS (striatiOnS) above certain pre~Sure and
current valueb.
In sectiOn 4.4 we therefore investigate these striation~ in mo~e
detaiL
51
4_1 Pl~sical (l)C(:t1.B.ni!3m!3 for the constriction
The in flu~nc~ of the tkn-eG- processes meI1tiQned in the lntroductiOi)
on the cOllstrictlon phenomellOll is (H~cu5sed in this sectioll.
4.1. I Cov<omo reluxatiOrl
From 0~periments it follow~ that in the constricted. column the
radial Glect-ron d",nsity profile No (r) INe
(0) is mv~h n"-rrowelC than in the
diffu5e ,,()lu~'n at th~ same current. From the equation for tllG oischargo
current (eq. (3.7)) it follows th«t ill the constricteo column at the axis
the electron density NolO) must b~ higher than in the oiffuse column_
,'rom the e l"ctrorl densi ty equatiOrl (eg _ (3.5)) we obtain that the ioniz,,
tion fr'eql(€nc:y on the axis Zion (0) also must be ~nhanced in the canstric
t~~d col\lmn_ Because the ionization frequGncy l!=l neClrly equal to thE' elC
dtation frequency Z in this region (see ~qS_ (J.l), (3_4)), it ±-oll.ows
that Z inc:;.tl'::!ases l:luring the transition of the column from th",-~ dlff\lse
into the con~tricted ~tate&
n,is occurs sjroultaneouSly with an observed decrease of the axial
eJ.'~Gt"i" fieldstrength E (fi9- 3.6). This decrease is cOnne~ted wittl a oe
C!"!.:Q.SC of 1) CQ r because E and U eO are coupled <;l:.ccording to t".:.~q_ (:3 _ 9) :
0; = ~ d N U 5(4C (N IN ) M 0 go eO eo go
AS C increu~e5 with illC~ea$ing Neo (eq. (2--~9)), "nd NgO is higher in the
constricted cOil).ffi(l than in the diffuse one (because: the power "input per
uIII t J.engt.t"l. if: ~s lowet in the constricted column), the do(:rease of E C3f1.
only be expl~ined ~ith a simultaneous decrease in Ueo
'
']'he excitation frequency on the axiS Z(O) can only increase for
(l",cn"a",ing lleo when it depends strong enough OIl Neo When Z(O) depends
~o ~trong on NeD' that
J..,on e o(Z, N) I d (4.1 )
an ins tubili Ly in the electron dell",i ty production i~ amplifiod which
ct\u=-e:s: oCltl aV.;'.l.lc:m.:;:he in the electron density at the axis that can be
balanced by ambipolar difflOsi.or> and volume reCombination only wh'm
Nelr)/i'Je(O) '''' ~ufficientl'l narrow and NeW) is Sufficiently high.
TheS8 t.Wt') condition:::- C:l.J:'"0 coupled via the equation for Ne (0). $ee
",g_ (3.1). 'rl\l.s mea,l'; that the collOmn con5tri~ts.
S2
A model WhiCh dces not take into acco~nt the dependence of the
0xc~tation frequ~noy on the electron densLty is not cap~ble Of de-
5cribing the abrupt transition from the dLffuse into the constricted
column. As for increasing electron dens1ty this dependence i~ caused
by the increasing influence of the Coulomb terms Ln the Baltzmann
equation (see chapter 2), an accur~te calculation of the electron
energy distribution function (E.P.l is necessary for the description
of the constriction (see section 4.1,4 for a further discussion).
4.1.2 Volume recombination and ambipolar diffusion
When ambipolar dlftusion is neglected in the electron density
equation (~q. (3.5)), the electron density profile has the same shape as
the radial profile of the ~onization frequency, ~hich is nearly equal
to that of the excitation frequency:
(4.2)
For large values of the electron density, the expression for Z (see
chapter 2), can be written as (see also Woj66)'
Z
where C. depends solely on Ue , and C1 depends only weakly on Ne and
Ue • Therefore tor large Ne ~q.(4.2) approaohes to
N (r) e
N(()) ~
(4.3)
where we t~ke C2
constant, because its radial dependence is weak.
$ecause of the bound~~y condition Ne(R)-O and the exponent~al dependence
at the right-haruj si<;\e of eq. (4.3) the stable sol~tion i",
r = 0 , (4.4)
O<r.<R.
The condition Ne
' (0)=0 is violated because of the neglect of arnbipolar
diffuaion~ Numerical solution of the mOdel of Ohapt~r
gives indeed electron density pLofiles which rtarrow
With D =0 a
mo;rG and more
when the number of discretization intervals k .is increased. From model
53
calculations for the diffuse column we find Ulat_ the volume r~(;oml).1 nat-ion
r.::.::-p;t'"csent!';;: ~bo1,lt 60% of the electron lossos at. the axis in the rc'gJ.on
whet"c the c."00$t.riction can be expected. It may thus he conjectu:red that
volume-recombination is essential fo)"" tl18 constriction. HoweV~):". oFJ,mbi
polar diffusion neyer may be n€glecteo in t.he constricted column. This
can be seen from the equation for the o,i.s(.:harge current (eq. (3. 7}) ~ from
which we obtain ~s ~n approximation'
NCo r 12
::::1:Ci
WhCt"(' C -= a -cone:tantr r1
- a characteristic radius for the column, ~nd
[) N i = disc/)ar-go current_ ApprDxim<1ting "r' (Da"rNe)r=c by "re~, we have,
" • (D 'iI N ) r a:t' e r=o
1
Hence the dependence is quadrattca11y on Nt;:Q a.s is the voluP1c recomO"ination.
Therefor" as Vr - (DcIVrNe)r=o ano (tNeo ' have the same order" of fll(\gl,jt.\lc']e
in the diffuse discharge near the const~1ctionr this will remain th~
case i.n th~ CO)1sr.ri c:ted discharge as long as the current has not been
alt<'red_ Thi ~ i.,; "onfirm"c'] by the numeric"l c(ll<::ulabOnS.
4.1.3 t!e.;:±,t (iissipi..,tion
At low current_ values ~ the effects of elsctrOr'J.-electron 1 ntP.raction
~nd voluIlle-recombintltion ttre ::;ti 11 negl igiblr::: in t.J"lC (.l.iffuse column. How
eV~t, al~o for ttl€S~ cur~ent values the radial electron density profIle
can differ consider~bly from the Be~gel~l?ro(ile resulting from tl,,, original Schottky-theory (Sch24). This is caused by the radial dcpcnoence
of the electron temperature, in its tur.-n due to the radial dependencc;, of
the gas temperature lsee chaptet' 3). 'rhe Ue
(r.) aepenn-ance causes Ll
strong radi:..l dependence of the ioni:l.ation frequency Zion (r), w~nch is
re5pDn~lt.le for the c'] .. viation from the Bessel-proHle.
4.1. 4 '~he cDmbinat ion of the three effects
The maxwellizCltion of th" E. F. sets in Wh<?O t.he \0" '. zation d",gree
Ne/Ng is sufficiently high. Th" volume-:o::ecombination becom"s important
with respect tD iimbipolar diffl1sion when NeNg
i$ large enough_ The effect
of thegCls henting also sets in at high values of NgNg
_ These thr"'''
e ffe<::t5 are conne~t"d with each ether. When ttle gas density and the
electron density ara high. the non-uniform gasheating causeS ~ strQ~g
radial dependence of the ionization frequency Zion(r). Hence the
&lectrotl density profile in the diffu5~ column narrOws (section 4.1.3),
so that Ne(O) becomes high. For higher values of Ne(O) the volume
recombinatiOn becomes important (section 4.1.2) ,whioh in its turn
causeS an e~tra narrowing of Ne(r), hence an extra increase ot Ne(O)
Then Ne(O) may beoome SO high that the rnaxwellization of the E.F.
sets in (section 4.1.1) ,which gives an extra contribution to the
electron production. fo~ low current values the Zian(Ne ) dependence
is not So strong that this re"ult~ ~n an instability. However, for
high CUr~ents we may have ($ee eq.(4.l»,
d dN
e (ZN +'1 - (D V N ) -aN 2 ) era r e e
so that constriction ~ets in.
::- 0 , r=o
Therefore constriction is expected to start when the three effects
mentioned in sections 4.1.1-4.1.3 are so 9trong that they enhance @ach
other. This is the case when the ionization degree Ne/Ng and the product
NeNg a~e large enough. Because Ne is coupled to the current (eq_ (3_7))
and Ng to the pressure (eq. (3.15»), the const~iction will take place ~t
~uffic~ently h~gh current~ while the p~es8u~e ha~ to be in the right in
terval (see alsa fig.l.l)_
The model described in chapter 3, however. contains th~ three
menUoned effects but does not desc~ibe the constr~ction. It is likely
that the con~trictian i~ an instability effect whi~h can not be described
by a stationary model. In section 4.3 we have laid a connection between
th~instability and the on~et of striations. The striation development
is discussed in section 4.4. First, however, we discuss some models f~orn
the liter~ture usi~g th~ ~on$iderat~on5 given In this section.
4.2 Discussion of some models for the positive column
The models pl'esented in the literature for the medium pressure
column can be divided into three group$.
4.2.1 Thermal models
Many authors (aar69. Rak7Q, Mou71. Uly73), have considered the con
striction as a purely the~mal effect (described in s~ction 4.1.3). They
55
have not ti:\ken. into accou.nt the other eff~ctg mentioned in section 4.1.
All these authors Itssume a maxwellian B.F. r\lrthermore they negle"t
a.mbipoL:1r diffusion and their t:alculatioI1s al:"e based on direct ioniza
tion~ From sections 4~lLl-4.1.4 it is clear that the abrupt transition
from the diffuse to the oonstricted column can not be explained on the
b~sis of these ~8sum~tion:3~ Moreover, these assumptions not- even hold in
the constricted column itself. Because the estimated ion.\Zi;Ltj,on degree
on the axis of the constricted column is about 10-6 to 10- 5 (see section
4.3), the depletl.on of the hi9h ener9Y t.ail of the E.F. ~till is very
large (ge~ fig _ 2 _ S), and. the a~gurnption of direct ionizo!:\tiOr'l calc\llated
with the h~lp of a maxwellian distribution CilTI not be cOL~ect. Further
more, the neglect of ambipolar diffusion is not justifi~d (see section
4.1.2). Therefore the models from this c~tegory can not be expe~ted to
give quantitative ~greement with experiments.
4.2.2 Ionization models
The st~ong dependenc8 of the e"oitation frequency on the electron
density as ~ possible cause for the constriction has been considered by
Kagu.n et ,,1. (Kag65). The ionization has been assumed to take place 5tep~
wise, where the ionization frequency has been taken equQ.l to the exc~ta-
tion frequency. (Near the constriction region this assumption is reason-
abl",. ) Fc.:.::thermOre. the author,. neglect vOlumG-);"ecombLnation and the
radial d.ependence of the gas temperatut"8 and the electron t_e:rnperature.
Thes~ assumptions do not lead to a realistic de5G~iption of the medium
presstue column. A mo):"e sophisticatod version of thIs model is discussed
In se~tion 4.2.3.
woj<>c~ek h"s set up a model fo:;: the constricted argon discharge
WhiCh is of the same kind as the One of Kagan ~t. OIL discussed "bove
(Woj66). In three pul;llications he then considers subsequently the
separ<>te intluence of th~ Tg(~) depcndonce, the Ue(r) dependence and
volume-recombination (Woj67, 68, 69). He concluded that they all three
can be neglected. In reality the effects ~re interrelated and onhance
each othe"(, ,;;c that their combined effect can not be neglected.
4.2.3 A more SO£histicated model
In a .Late~ version Of the mod.;,l of Kagan the effect~ of gas
heating and volume-reCombinatiOn haVe been incOrporated (Gol74). Below
we di$CUSS ~ number of prope.t'ties of their model~
.56
-Kagan et al. solve the Heller-Elenbaas ~qUation ~ssuming ~ p~yarZ
bolic prof~~e ~9(r) Tg(O) (1-~). The OOefficient c h&S been obtained
from the oondition:
ThiS COnditiOn has b~en Obtained from multiplication of the Hell~r
Elenbaas equation by 2TIr and integration from 0 to R. HoweVer, be
cause T (r)' ~~ l'\0t: paraboUc fO>: r " R, (which oan be !>",en from g 5/3
J.E ~ 0 for r '" R sO th<>t Tg(t) ,,- ~nr ~n that region, (see
chapter 3)), in general the thu~ calCu~ated Tg(r) profile differs
considerably from the one obtained in our model from the Heller
Elenbaas equation itself (see fig.4_1)_
1.(1
o 16 rCmmJ
24
Fig, 1.1 CaZoulation of the gas temFe~atu~e profile,
1 Method of GoZuhovskii et aZ.; ~ Ezact soZution.
Po = 100 tor~, R ~ 32 mm, i = 100 mAo
-K&gan et al. take the integral 5) in the equat~on for Neo (eq.
(3.7)) constant (5) = 0.22). ~hi8 is more or l~ss a self-fulfilling
propheoy, because when 51 is kept constant, Neo ~ncrease5 only linear
w1th ehe discharge cu~~ent, so that volume-recombination ~no maxwelli
zation of the E.F. can not ~ncrease rapioly_ Then the Ne(X) pxofil~
will not narrow, so that 51 ODes not ohange muoh ~ith 1ncrensing dis
charge current. In reality, when 51 is not ~ept constant, we get a non-
S7
··r--------
0.20 ' 4.1017 , \ , \ \
OJ~ l1017
Neo
Vi "" '5 010 Sl 2.1017 ]
Po .looI0r,
R ·31mm
005 ~
1.1017
O=----:5:';;O--------,1:'::;;~O----
itmAI
Fig. 1.2 The deru;ll1f;leMe of the inteflt>d S1 (in the (!U),),r.lnt {lquation!
and of N eo On the di(J(Jh=ge dUl'rent, in OUl' model. GO~11bovskii
et al. take .";l~O. 22 ana (Jonaequently, the dashed (!l-"'-O" gives
theil' valueaof Neo ealeulated with S1=O.22.
linea~ dependence of Neo on the discharge current (see ~ig.4_2), which
ca\)~e ~ Sh3rp decrease of 81
het'l.Ce a. still larger incre.i;Lse of Nco with
Ln~rea~ing current.
-The axial .. lectric fieldst~ength E is 'calculated from the electron
~n~rgy equation. Arguments against this procedure have been discussed in
section 2.4.
-In the balance equations for the excited atoms in group I (see 3
s0ction 3.2) on.y the resonant Pi (s4)-state has been taken into account. 3
The metastable P2(s5)-state has, however, a much larger density 50 th~t
the stepwise ionization takes place m~inly over th~s level. Fu~thermore
the effects of redist~lbution of the 1,,-1evelS v1a group II, and the
p~od~ction of e~cited atoms in group ~l via volume-recombination hdV0
been neglected.
-The elastic cross-section of neon has been taken constant in the
calculation of the E.F., which 1s not realistic (s~e fig.2.1).
58
-The gas temperature term has been neglected in the calculation of
the E.F. I which is of import~ce fOr the calcu19t~on of the excitation
frequency (see s:;ction 2. S). At the con~tJ:iction region, where the gas
temperature is h~gh (~ee table 4.1 (section 4.3)), this term becomeS
non-negligible. -ThG wall temper~ture has been taken equal to the room temperature
which is not correct for the currents and pre~5ures near the constriction
region.
-The correction factors CN and CD in Neo and Da haVe not been taken
intO aCCount. (The importanoe of these factors, hOWeVer, is not very
lllrge. )
6speoia11y the first two mentioned assumptions cause the large
deviations between this model and our results. Their calculated T (r) 9
profiles are much too flat, and their calculated values of Neo too low
(see figs.4.1, 4.2).
4.2.4 Discussion
Because of the aS5umptions made in the models de~~ribed in SeC
tions 4.2.1-4.2.3), these moae~5 00 not describe the meaium pressure col
umn accurately. Though our model gives an a~ourate description of the
diffuse diScharge it does not include the transition to the cOnst~icted
state without special precautions. In the next section we investigate
by a variation of some parameters in our model whether we can find
methodS to ac~ount for the constriction.
4.3 Variation of some £arameters of the moael
4.3.1 parameters with negligible influence
In this section we investigate the sensitivity of Our model to
variations in the various reaction coefficients. All results are for
Po = 100 torr, R ~ 32 rom. Num~ri~al caleulat~on5 showed that a variat~on
in the coefficients have the most important influence on the value of the
axial electric fieldstrength. Therefore we discuss in detail the influence
of the variations on the ~-i curve. A variation of the coeffi~~ents of
the balance equation~ for the excited atoms did not have mu~h influence
on the model result~. For instan~e multiplication Of the stepwise
ionizatiOn coefficient ~y a factor 100 resulted in ~ decrease of E of
20t. The stepwise iOnization frequency. has an inaccuracy of abo~t ~
factor of 3. Variation of the elastic ~ro~s-5ection factor dO with a
59
~00C11
w
.. !..".---.•.... 00
.. '.-~---"" ..
Po - 100 10(!
R -32mm
1. _____ .
100 i(r~A)
.. ,J-_ ISO
f'7:g. 4 •• 5 The two-t",mper'atuT'e "lode ~ and influence of the 1JO lwne-l'ecol7lb7:
rL<),don coeffic7:ent.
certain factor. r!8sult,s in ~ sb.J'Ile relative variation of E. The va.ri~nce
of values found iD Lhe literature for ilD is about 15%. Thf':! precise
Valu.e of the inelastic cyoss-s,8ction is less itr'lportant, beca.use of the
s-;hJwly v~rying funct.ion F 1 ill thG" expression for the exci t.tl.tion fre
q"(o()Cy 7, (section 2.5; when Qt is changea ." f<l<:'to~ 10, th~ v"!:'i«tion
af F I is approximately 25%).
1.3.2 Paramctc~s with ~ s~nificant influence
We discuss first th" volum"-r,,combination coefficient (I.. (For a
r"viewof the resul,ts of the v"ri<ltion analysis ,;ee figs.4.3, 4.4 "nd
to.ble 4. I.)
rn(:'rei;:lS~ of. t.he volume-recombino.tion coefficient Ct with a factor
of 3 results ~n an incre~se of Nee (see 5p.ction 4.1.2 and table 4.1)"
con$equently in " "tronger Coulollili-interaCtion in the E.F., which in its
turn r~~"lt.s itt" dec,".;,as" (of E (Section 4.1.1). It app.;,al:"" not to be
poss ible to get. constriction by enlargement of 0..
In previous calculations the number of discretization intervals
in the numeri""l elaboz""tion of the electron density equation (eq. (3.5))
60
w~s rest~i~ted to k = 32. In that case both the sharp constriction and
the hysteresis we~e ~alculated with good values of E. The ~alculated
electron density profiles, however, were much too na~rOW. In the normal
calculations and the variations. it i~ checked whe-theX' a sufficient large
n~mber of intervals k is used. ~he value of k is altered if necessary-
If in stead of using the solution of the 601tzmann-equation, we
U5e a chosen E.F. aooording to a,mi:1xwellian J::;rF~ with. a bulk temperiltur€
and a tail temperatu~e, an abrupt transition of t_he diffuse inta the con
stricted st~te can be obtained. ThiS distribution function has been sug
gested by Vriens ~n another context (vr1.7JJ _ Par our purpose u~ w&:;; cal.
culated not fro~ the method of VriGns but according to
Ue c 1 c 2 + N,,/Ng
Ut c 2 + N,,/Ng
In this model, "1 i9 chosen such th"t the calculilted E Hts the measured
cne at 10 mA. The value of c 2 is cho~en ;;;uch that the constriction takes
plac@ arouno 120 rnA. The values ot c1
and c 2 belongin·g to fig_4.3 wee""
6.5 10-7 . we have introduced this version of the model
to demonstrate that a calculation of the constriction is po"sible with
"- steady state model, Though ·the choice of ueJut
is arbitrary, and this
model does oot calculate the hystere~is loop, it seGms to indicate th~l
"- variation in the pararoeters of the Boltzrrt"nn~eqllation l.s necessary to
explain th~ con$triction.
In fl,g. 4.4 the influence of some variations in the useo E _ F. dre
Shown.
~h~ normal calculation (chapter 3) fits w~ll the experimental
curve fo~ low currents. For high currents the eXperimental values tend
to the Maxwellian case (constricted column). An increase of the Coulomb
terms in the Boltzmann-equation with a factor of six, 9ivc~ a c~rve
Which goes smoothly from the.diffuse to the constricted One. As in the
case of the volume-recombination coefficient, it appears impossible to
obtain a sh&rp const~iction by & change of Vee
4.3.3 A striation model
In section 4.4 a criterium for the on$et of striations is derived
from the dispersion relation for these WavGs. when the striations develop,
the values of Neo E and Ueo "t the top of the striation are related to
the average value5 in the following way:
61
1000
o
r-·····.-.. ·-.·----,----Ti-·
'(rnAI
Ib .1001(1,", R ~32"'m
0<;>""0 r"Ill!'.aSvr~m~l"lts .. _,_.- OruIJve''btel,n --'lee.6 - - Ma'w<olI --_._-_. $Iriation mJdel
normal
!Hg. 4.4 Irlfh«mce of th", P_ F. on the axial jieZdstrength, and (!(l.Z¢U
ration of th", (:onHrir)tion with the striation modd,
N eo top iT"'" eo 0.9 U eo
Thg fOllOws fron) nOn-lin"2I1' calculations >lhieh wo havc carried out
for our case using the "'et.hod of G~abec (Gra74). The value of Neo is
in agreement wit.h elect.r.on density measurements in the constricted
striated column pert'ormeo by us with the method described in chapt.er S,
We therefore introduce tne foUowing st.riation-model. When the
9triat~ons start to develop, the values of Neo' E and Ueo must be re
placed by their value. at the top (see above), With these values, "
region for Neo and Ueo
is reached in which the constriction criterium
(4.1) (given in section 4.1,1) is reached, 50 that constriotion sets
in~ The numericul c~lcultition5 show, however, that th~ onset c~iterium
for tne 8t~~ation development (deriVed in section 4.4), is not ful
fillod fQr the conditions in the diffuse discharge. for our current val
ues", (For the conditions i,)'1 the c;::on~tr.i,ct~d discharge ~ however, the cri-
terium is fulfilled for currents higher than 90 rnA (~o = 100 torr, R =
32 mm), assuming a homogeneous column with changes in the values of N ,E eo
"nd Ueo a9 given above. This corresponds with the obsel"ved fact that tne
constriction disapp~ars below 90 rnA (see fig.4.4ll·Under the conditiOns of
62
i (mA)
Modd
Normal mOdel"
Striation model~
Two temperatures
\J '" 6 ee
a *
EXp. values
Normal model
5tx1ation model
TWO temp",ratures
v eli! ,. 6
C1. 'I- 3
Normal model
Striation model
TWa temperatures
\J .. 6 ee a lI- 3
10
0.96
0.96
0.94
0.96
1. 14
15
16
16
16
16
13
401
401
400
400
409
40 80
diffuse
6.29
6.29
6.13
7.34
12.6
10
10
10
10_5
9
6.5
615
_ 615
619
617
674
19.3
1Sl.]
lSl.0
51. 9
62.4
7
7.5
7.5
7
3.5
3.5
838
838
847
8B
955
110
]5.4
35.4
48.6
233
186
120
41.7
352
445
266
266
r 1 (nnn)
5
6
6
4
2.3
2,3
973
973
1000
929
1144
1.8
5.5
1.8
1.0,
2.3
2.0
1022
1008
1075
955
1207
130 140
constricted
49.3
401
493
274
339
1.8
5.5
1.8
1.0
2.5
1.9
1059
1055
1112
961
1243
56.4
450
589
281
456
1.8
1.8
1-0
2_8
1.6
1099
1076
1138
983
1303
Tab~e 1,; Calculated aoZumn p~ameter8 fov various ~evgions of the
mode~ (po ~ 100 torr, R = 32 mm).
~ ahapte2" 3,
~~ avepage ~aZue.
the diffuse disoha.ge the on~et of the constriction at 120 mA has been
forced on the system .ad hoc substituting the above given changes in Neo
E and UeO ' P~obably local fluctuations in the densities in the column make
it possible to meet the st~iation criterium at 120 mA_
From table 4.1 and [igs.4.3 and 4,4 it appears that the striation
mOdel gives the best agreement with the constricted column experiments_
63
1'.3 a matter of fact for the diffuse discharge the striation model is
identic",l with the normal model l:>ecause there are no striationS in the
diffuSe cOlumn. We thus hav-E;! pr~.!;Ient.ed strong indications that the can
?,triction can tH~ regarded as a striation generated pX'OceSS. Tht=:! stria..
tions wUl be investigated in more detail in the next seotion.
4 . .::\ The fitr iation develo;g!!!.ent
4.4.1 Introduction
The phys~cal meohanism of the striatiOr) development can be under
stood by con~~dering the equations foy the OOnservdtion of electron
de::nsi ty,. moment-um a.:n.d energy ..
aN 8N 3 aN e
(D __ e) __ E) ,
at h +;- ~ (rD + z. N (jN (4.5)
" "X " ar Lon e e
2 [J aN
N v b N (I;: + "3 ~~) - b N I;: (4.6) e e c N ax eo eO 0'
e
ao N U N .., " - N :J'X"" = e
'" VE e e
t1 N Z
e U
1, (1.7 )
wher.-:- x = i:l.xial distanoe, t = time, tl ::..: a characteristic time for
elastic energy losses, v = electron drift v~loc~ty (defined positive
from anode to caChode).
An index zero rf!fers to the- station.:lry and homogeneous situat.:i.o' ..... II'; eq.
(4.7) the time derivatJ.ve hag been neglected, as the striation fre
qu€ncy ("bout 2 t.o 3 k,u) is negligible with respect to the frequencies
of the fundrunental proc~ssc's (about 106 llZ). Furthermote the h"<lt con
d\lction of th~ electrons has been neglected in eg. (4. 7) (GQl69).
For" quali tatl.ve description of th'" scriatior)s We follow tho argu
mentation of Pekarek (Pek68). Let us a~g\lme a local fluctuatiun in the
electron density in the column. Then an amb.ipolar. field will al"Ci8e,
char'lgiT~g the local ~}{ial fieldstrength according to eg .. (.;l. 6). According
to eq.(4.7), the electron temperll.tur02 will also change, but this change
is temporally and spatially delay"d because of til", electron drift. Thi.
will influence the l.on~",at:l.on frequency Zion' generating a new t<i!mporally
~nd spatially delayed ~l~ccrDn dens~ty fluctuation (Gooform ~q. (4.5)
having oppo5ite sign with respect to the initial One. If this ch"in of
pLooe~Seb Lepeatb, the waVe deVelops. The striations may be darnpeOI how
eVer, if the ambipOlar diffuSion is sufficiently importClnt, (~ee eq, (4,5))
64
and/or by too large (in)~lastic energy loeees (see eq. (4.7)).
'l'he eqs. (4.5-4.7) nave been calculated mllnerically by GrabeC (Gra74).
He negl~cts volum~ re~ombination and assumes a maxwellian electron
energy distribution function. 'l'heneq. (4.S) is linear, and allows D" separation, yielding the term - ~ in
1 ~ aN .. stead of ~ 3r (rDa ar-" with
q r
l = R/2.405 according to.a zero order BeSSel function for the ~adi~l
electron density profile. (R is the tube radius.) Substituting (4.6)
into (4.7). and treating the latter as a first order Lnhomogeneous dif
ferential equation. the equations have been solved numerically by
Grabec (G.a74).
In the medium p.essure diScharge volume-recombination can not be neglect
ed (chapter 3 and section 4.1.2). MoreOVer the eleotron energy distribu
tion function is strOngly non-maxwellian. and depends on the electron den
sity (chapter 2). Therefore eq. (4.5) is non-linear ana can not be sepa
rated" (The fact that ths column constricts simultaneou91y .. ith the stri
ation development introduces extra difficulties.) C91culations wh~ch we
performed using the method of Grabec for the medium pressure oischarge 1 a' dNe ~
where the term ~ ar (rOa ~) is replaced by -Da rj2 , with r l a oharac-
teristic co~lmn radius, ana with vol~me-reoombination and the non-maxwell
ian E.F. taken into account give large deviations from the experimental
values for the frequency of the waves. 'l'he calculations are also very
sensitive to the numerical treatment. In fact the possibility th"t
the thus calculated value~ are mere nUmerical instabilities can not be
excluded. 'l'herefore these calculations are not preSented here. tn ste~d
we will derive analytically in the next section the aispersion relation
£0); the waVes by means of a ffi,sarization of eqs. (4.5-4.7).
4.4.2 'l'he disperSion relation
The moment of onset for the striation development is deriveo from a
linearization of eqS. (4.5-4.7). The, linearization yields the dispersion
relation. £.om which the initial growth rate of the waves can be found.
Assuming th9t the radial 1 a
ment. the term ~ a; (r09 eq. (4.6) in eq. (4.7) and
ditio~s:
N
va~iationS a.e not large 3Ne ~) is rep,aced by -Oa
using the station?ry and
Z N eO .... uN o eo
r1
2 eo
in this linear treatNe ---2 here. Substituting rl homogeneous column con-
65
b N E ~ eo eo 0
(4.71,.)
(where Zjon ~ 7. h<ls ))",en t"ken),eqs.(4.5) "nd (4."1) become (D<I is i.\ssumed
to be constant);
and
D
" (Z-Z )N
o e aN (N -N ), e e E~O
(Z-Z o
U e
(4.8)
u)·(4.9) eo
Substituting Ne ~ Neo+IloCXP(i(k,,-wt )) alld Ue
~ ueo+uoel(p(iOo,-ult)), with
flO ~nd llO comp~.cx, into (4.8) and 4.9) J :t"et~in.l.n9 only line~r terms
j.n no and Uo gives twa linear and hom.oge(l€ol~s cql..lat.i Or"l.~ i.n nC") arid uQ
Tliis !i!ystem has Only c1. nontrivial solution when tt.s det.~rrnin.a.nt
v.Q.ni:::hes. Tbis conditlon yields th.e di!:;lpp.Ys)on r~lat_ion:
(4.10)
E 1'. wh-ere "1 - 0
(I -t ~n
(1;u I)) , -U 1',,1 eo
33 Zlz 7,
0 ~.
(l
u at:! n ~ eo eo u N
t;u eo
Z ~n ~Z Z ') 7.. n 0 0
i:lIld P U·I bE:::' an(.l el eo 0
losses respectively.
We t11ke II! = "\ + iWi ctnd k re.,l, ))"cau[;e spatial damping can not
be prese-nt becuusp. the column is Olsgumed to be infinitely long _ TtK~ d18-
persian relution (4.1o} is .~plit into real and imagino!:\.t:y pE'Lrts. Ther"l. (1.\. and 6\ exp't"CS:=l?d in term!=.: of k become:
Tjk W,.
k2+k :2 1
Tiki ar)c"l w. -D k/. I·T -, a 2
k2+kl ~
(4.11)
(4.12)
where T (l Pin
(I;n 2 2
ZuEo(3 + + r.u - 3)) 1 Pel l
66
ON o~d T (2 L + r ~) ~ ~" 2 = 3'u Sn - Z ~o
o
When inelastic ~nergy losses ~re absent (Pin=O), then kl=Eo/Ueo ' which
is the reciprocal of the electron temperature rela~ation length due to
elastic collisions (Gra5S).
~rom eq. (4.12) the onset oriterium for the stri~tions i6 d~riveo,
~equiring Wi > O. This gives:
c •
,n our par~meter domain, b < a and the striation development will st~rt
as soon as b2
becomes larg~r than 4c. The striation9 then sta~t ~ith a
wavenl.llllber k .. 0 . From table 4.2 it can b~ seen that th~ oriterium b2 > 4c is ful-
N (m- 3) eO
b2
_4C(m-4
) -1 w. (s ) ~ 1';\1 sn Pin/Pel k
-1 kl (m )
1.6 10 18 -3.4 )0 8 -5.4 104 8.4 1.0 .02 574
2.5 1018 -2.6 )0 8 -3.2 10
4 7.0 1.0 .03 590
4.4 1018 +9.9 107 +8.S 103 5.7 1.0 .06 2.7 10
3 641
1.0 1019 +7.9 109 +2.8 105 4.6 2.2 .10 4.4 10
3 690
TabtB 4.2 CalcuZat~d values tram th~ dispersion re~ation for p = 100 o ~
torp. The striation deveZopm~nt starts when N > 4.1018m-3 eO
(wi ;> 0). (Tube PadiU8 32 mm.)
From ~q. (4.11) the phaS€ velocity v~ and the g~oup velocity Vg a~e
obtained:
v p
67
v g v
p
In Our c~se k » kl (see table 4.2) thus V ~ 0 (Iv I < Iv I). Accord-g g p-
i"g to ttl", notatiOn of Pekarek (Psk68) , t.he waves are of the I;! type, B
referring to Vp/Vg < 0, t.he minus sign referring to Vg < 0 (opposit~ to
the current direction which i~ from anode to cathod~}~
Eqs_(4.11) and (4.12) are of the same form ~5 those derived by
Pekarek (l'ek68).
~o get some insight in the physical mechanisms determining eqs. (4.11
and (4.12), we discus~ ths case a - 0, Pin = 0, ~n = () (from the d<lta ~n
table 4.2, and oonsidering th<lt aNeo ~ Z , t.his appears to be ~ realistic o Zo 1
approximation). From eq. (3.5) we get app~oximately: D~ ~. BeC<lUSe
EO/Ueo ~ 5po' we get (see also table 4.1), Ueo/EO » r l , hence
(Eo/Ueo l2 « ZelDa in the :<:egion considered here. Then k l2 « T2/D.,' alld
b -2/3 Z U
u eo D
" and
2 Z U u eO c: = ~
a r.
where \:_ = Ueo/Eo ts the electron teroper"ture relax"tion length.
The criterion b2 > 4c then becomes;
z {) u eo A 2
r
This formula shows the ~.ordzation mecha.nism as a creation process and
ambi~olar diffusion and electron temperature relaxation as stri~tion
damping processes. The wavenumber with which the stri<ltion development
start5 is~
k
Thus the wav"length A becomeg,
2IT -I 76 '\ 0., Ng (see chapter 3).
The f~()quency f becomes in thiS case,
w 4£ f't:ll---E.=-,-zu o"\iN
2n 21~ u eO 9
68
~he group velocity Vg b~corne~ finally,
v g _ ~ Zu\leo
147 -\-r
v p
(independent of Ng) .
In table 4.3 the ealC>.11ated values for the gen",ral case (c):, Pin'
~n p 0) a~e compared with ",xperimental data (see next section). The
calculat~d relationship of A, f, and Vg on the gas presSUre i~ in
reasonable agreement with experiment. The large deviations in the
calculated frequen"i~s are attributed to the approx~mations inherent to
the lin",ar treatm",nt. This linear tr",atmsnt 1s given here b~cause we are
only interested in the onset of th~ constriction (see section 4.3"3) and
not in a detailed calculation of the striation development its",lf.
Po (tOl'T) \.102 (m) f(kllz) v(m!s)
expo calc. eXJ? calc. expo calc.v calc_v '9 p
50 2.9 0.86 1.7 33 -.5 102 _2.10 2 +2.B 10
2
100 1.0 0.45 2.8 63 -.3 102 -2.102 +2.8 102
200 0.9 0.22 5.0 132 -.5 102 -2.102 +:<.9 102
1'aNe 4.3 Compa!'iaon of ¢Q'ldUtated values with experimentaZ data.
(Tube zoadiu8 32 mm, dUl'l'ent 1JO mA.)
4.4.3 Exps~iments
For the experimental study of the st~1ations we have us eo the dis
charge tubes "nd the electric ei.~cuit as mentioned in section' 1_ 1. The
striations were observed oJ?tieally. For this purpose, one photodiode,
positioned at a fixed place along the tube axis, was used as trigg~r ~i9-
nal, whereas a second one could be moved "xially. Both signals were di~
played on a double ray oscilloscope. wavelength, frequency and velocity
of the striations were measured (see table 4.3). Using a rnonochrom"tor
with. a photomuLtiplie~, various line transitions were investigated. In
this case a movable photodiode was used as triqqer signal, while the pho
tomultiplier was pOSitioned at a fixed place. The striations did not de
pend Observably on the wavel",ngth of the line radiation. (In the low
pressure col~, Rutscher has observed a slight dependene~ (see 01068).)
The rnOdultition 6f: the line radiation, which is nearly proportional
to t.he ionizal.:ion functio()' :~:;r\Je (see chapter 3), is about 90%. The modu
li:ttion of the continuum intensity, which is nearly p:roportional to the
electron den::::ity (set;!: (:hapter 5), i~ about SOt. (Bence a nonlinear treat
ment is ne~essary to calculate the striation development accurately.)
The w~ves traVel froM cathode to anode l i.e. opposite to the
CI),"t:':t"ent directi.on.
The st<lbility of the W"VI')S dep~nds ruther strongly on the gcometr"
of the tUbe. Therm"l t.\lrbulenOe; disturbed the meCl9Ul:"ements <It higher
pressures {PaR> 5 torr m). In case Of irregular fluctuations of the
frequ~n.;:y ij.nd the amplit.tlde of the Wuve5 arO\lnd their aV""ctge values.
the waVeH; (;Ould be stabi lJ.zed by modulating the refct"("rl(;e V'01 tage of
the current. regulator with the mean frequency (or the f.i yst harm.onic
component) "f the striations.
4.5 <;onclusion
Prom the cttlculations giVen in this chapter. b~s~d on vttrious
versions of:' 0ur model, it is Very likely that Ul~ .:.;on:=.triction is
c.:l.used by thr= Onset of tho st.riations_ 'l'he model which incorporc:s.tes
the changes in the electron density. the el ectxoll t"mperilture and the
"xial eJ.ectric field~trGngttl in the top of the striation. gives good
agreement. With the mea~~~rements on the con::rt.rict.Gd (;Olw"I1n. To give a
full description of the transitiOn of the diffuse into the const:ricted
state , a description in r, Z and t coordinates would be necessary for
the e].ect:ron density, momentwn and energy eql).atiOrlS. As the st.J.tionary
model (only radi<ll dependences) is dlreo.dy complicat.ed enough. such il
complete descxiption would give large numerical complicationS. The
dosoription of the transition with the- ~trie.tion model d.escribed in
SeCtion 4.3.3 is a simple "nd rather accurate way to circtlmvGnt these
difficul ties.
70
CHAPT~R 5 ~BE ELECTRO~-~TOM BREMggTRAHLU~G CONTINUUM
Mea~urements of rad~al profiles of the eleCtron density and the
electron temperature have been pertormed using the electron-atom
bremsstrahlung continuum. The profiles are "co~pared with ~del calcu
lations.
Introduction
The optioal sPectr~ emitted by ~edium pressure ~nert gas dis
charges consists o( a line speotrum superimposed on a rather strong
continuum. In the past this continuum was explained as a molecular
effect (Pri66, Ken67 , Gol66b).More rec~ntly Rutscher and pfau (gut6B)
have ascribed the oontinuum to electron-atom bremsstrdhlung. rn the
last few years many authors have taken the latter pOint of view as the
oorrect one (Gol73, Ven73). Though there is still some reason to
believe that mOleoular radiation gives a non-negligib~e contribution
to the continuum (Smi7Sb) ,in this Chapter we will confine ourselves to
the electron-atom bremsstrahlung continuum. The electron den~ity and
the eleotron temperature are determined from measurements on the
continuum, a$suming that the continuwn is caused by electron-atom
brerossU"ahlung.
5.1 The electron-atom bremsstrahlung theory The 5pectral inten5~ty ot the electron-atom bremsstranlung at
wavelengtn A is (Rut6Bl:
() l/2F(U)dU (5.1)
Here F(U) is the isotropic part of the electron energy distribution
function,and Qs is the electron-atom bremsstrahlung cross-section.
~rom the quantum-mechanical calculation ot Kas'yanov and Starostin
(Kas65, see also Ned32, Fir61l, we have;
(5.2)
WII","e QD
= aDufi
(see s"ction 2.2).
From SeCtiOn 2.1 we have
(s.n
Be~ause th~ electron energies involved her" liB mostly in the elastic
region. WG take (see section 2.3),
f (E) o
with
Substitution of eqs. (5.2-5.4) into eq.(5.1) yields,
N N
C1
j~ ~ J(Ue , Ne/Nn' A) , U . 2A5+n • (5.5)
e with
4e ] j2e "D he
3+n C
1 --N Vm ( ) 311 1 2 e
(. me 0
J /" zn~ (z - t1e-M(XZ)dZ I
(5.6)
and x = ~nd z = dimensionless integration variable.
The integral J depe"d" On Ne/Ng because tne diBtribut:ion funct:ion depends
On Ne/Ng via the exponent M(E) (see cn&pter 2). Tne dependence of J on
Ne/Ng , Ue and A is not very str.ong. TlIerefore in first ord~r We obtain
from eq.(5.S):
i(l-) '" ~ N e g
Hence from the continuum int:ensity the electron density oan be det~rmined.
This is disouSsed in s"ct:ion 5.2. The detennl.nation ot th~ electron tem
perature from the continuum is di~c~gged ~n gect~on 5.3.
5.2 Determination of t:h" electrOn den!3~ty
From eq. (5 .. S) we can de,ive tIle radial profile of the electron den
sity from the measured intensities,
(5.7)
72
with C (r) = o
(5.8)
The r~di~l profiles of the continuum intensity have been measured at
A E 4000 ~. The ~rcfiles i(A,r) were obtained from the measured profiles
t(A,r) by means of the Abel~transformation (Rom67).
The numerical Abel-transformation is rather sensitive to mea~ure
ment inaccuracies. ~herefore the measured profiles were curve-fitted
first with an an~lytical function containing a few parameters.
The gas density profiles Ng(r)!Ng(O) were obtained from the model
calculations (see chapter 3).
Because the f~ctor Co(O)/Co(~) in eq,(5,71 depends rdther weakly
on Ue(r) and Ne(r)/Ng(r), here al~o the profiles following from the nu
merical model (see chapter 3) were substituted. The relative behaviou~
of Co is given in fig. S.l. In fig. 5.2 a measured intensity profile
~(A.r) is given (curve 1). After Abel-transformation curve 2 i~ found.
Finally curve 3 represents the relative electron density p~Qf~le. It can
be seen that the oit~erence between curves 2 and is very small, 50
that the influence of the correction (eq,(5.7») is small. In fig. 5.3
SOme measured prof tIes are compared with madel calculations (see also
chapter 3). The caloulation of the constricted profile is obtained with
the striation model discussed in chapter 4. The agreement between mea
sured and calculated profiles is satisfying. ThiS may, howeVer, not be
regarded as a clear cut confirmation of ehe bremsstrahlung hypothesis,
because the molecular theory can yield ehe same result (Smi75b).
5.3 Determination .ot the electron temperature
The electron temperature can be oetermined from the bremsstrahlung
intensity according to a method given by Golubovskii et al. (Go173).
From eq. (5.5) we get for the ratio Ra of the continuum ~ntensities at
two different wavelengths Ai and A2 '
J(Ue
,Ne
!Ng
,A 1)
J(Ue,Ne!Ng;A~) (5.9)
From the measurement of this ratio the electron temperature Ue
can
be de~ived when the ~l~ctron density Ne is known. we choose Al = 2500 R and A2 = 4000 R. In fig_ 5.4 the relationship between Rand U is given
a e_7 with Ne/Ng as parameter. In the diffuse column, where Ne!Ng < 3,10 ,
73
15
~ 10 " f '"
lie (V)
Fig. 5.1 Cor~ection factor Co (Ue> Ne/Ng> A} fo~ the determination of Ne
f~Gm the bpem66t~ahtung intensity.
o
10 ,(mm)
%-100 10"
R -12""''' , _BOmA
20 30
Fig. 5.2 Mea$ured prOfi~$ of the ~ntinuum intensity (1» p~ofiLe after
Abtl~-tranf!foY'mation (2), and oorreoted p~Gfne, aCQor'ding to
the fac'tor Co an.d Nr/);') eq. (5. 7J (0).
74
16 r'mm}
32
Fig. 5.3 M@asured and ~alouZated ete~tron density profites fop some
current VaZuea at Po = 100 torP, R = 32 mm (s@@ text).
tne dependence of Ra on Ne is weak. The~efore the determination of Ue(r)
with th~~ method is not very sensitive to the precise value of N (r)/ -6 e
Ng(r). In the constricted column Ne/Ng ~ 3.10 on the axis, but Ne/Ng
f~lls off very rapidly ~t increasing radial distance. Then the determi
nation of the electron temperature from the ratio Ra will be less aocu
rate on the axiS, but becomes better with increasing radial distanoe.
We therefore SUb$titute for Ne(r)/Ng(r) in eq. (5.9) the v~lues c~lculated
from the model of chapter 3 for the diffuse ~olumn, and the striation
model discussed in ehapter 4 for the eonst~icted column. (A~ was shown
in sectiOn 5.2 the calculated relative profiles for t'le(r) were in good
agreement with the measured ones.) Measurement of R~ = i(A1l/i(A2l then
yields Ue
.
The measurements for the dete~lnation of the electron temperature
had to be performed with high acouraoy and stability to obtatn r~produc
able results. Temperature stabiliz~tion of the amplifie~ unit was ap
plied. At each radial position the intensity was mea&ured 9t the two
wavelengths 2500 and 4000 ~ with a 25 em "arrell A~h monochromator. This
was more stable than ,scanning the radial position first for 2500 j\ and
then for 4000 ~. A calibration for the relative wavelength sensitiv~ty
of the optical ~ystem was performed with a syn~hrotron radiation cali
brated standard deuterium lamp.
In fig. 5.S the thus obtained electrOn temperature profiles are
75
R,
Ng. ~. 4 CaZcutaud :rBZatio11.flhip bfi't;wfi'IIi11. Ra a:nd U(! (eq. (.5.9)) foY'
dijJe:rBn~ vdw.JS of NeiNg.
given tox a 150 torr neon discharge (tube radius 32 mm), for various
values of the discharge current. They ~re compared with model calcula
tiOns (chapters 3 and 4 l _ The d j,SC);ep3ncies b"tw"en the calculated and
mea;;ured profiles are too b~g for high cun:ents (tb" accu,;acy of tbe
measurements is appro><imately O. \ v)_
The calclliateo values follow fx-om tM rel<ltion U (r) /U (0) ~ 4/5 e e
(Tg(rl/'l'g(O)) (sCG eq. (3.10)). The gas t~mp~rature Tg follows from
the Hell.er-El"I'lbaas equation (eq. (3.11) ). The calculated T 9 (r) ",rofiles
are not very sensitive to the coefficients used in the model of chu~ter
3- further.mor~ the agreement ,between the calculated and measured electron
den~Jty profiles gives us reason to assume that also the calculated
-l'g (r) -p:CO£iles will be in agreement with the actual ones _ Therefore the
deviation can not bo e~plain8d with our mOdel, although this mOdel takes
into account most of the phYbical mechanisffis of import~ncer and iG in
good agreement with other experimental results. A part of the continuum
radiation may be ascribed to molecular radiation {Smi75b). This gives a
possible expL~n~tkon of the above found discrepancies. The discrepancies
76
1.0
~------~-----,
lOrnA
~OmA
mOmA
\. "lOOmA \. 150mA IcorsIOcI<:d!
\.
" 150mA iclf6tnctaJl
~ - - caleulaloo ---- measur'ed
am 6,O:=2-----0=-.'.-b3.------~o.~Q4· ((m)
fig. 5.5 Measured and aa~cutated e~ectron tempe~atu~e profilee, 150
torr, tube radius 32 mm. Curve par(JJflete"l': dischaT'ge current.
in the constricted di~charge may be due to larger experimental errOr~
becaUSe of the appearance of striations and the strOng radial oependen
ces.
77
CHAPTER 6 FLUO~SCENCE MEA8uREM~NTS
3 3 Radi~l density prOfiles of the P
2 and PI Is-states have been
measured in " neOn discharge us.ing the fluorescence technique. The mea
$~red profiles are compar~d with model ca~culation~_
Reaction coefficients for atom induced t;r-an~itions betweeII. the
2p-levels have been determined by measuring the fluore8cenCe speotra
fo~ different laser input wavelengths. Theo~ ooeff1cients have been m0a
sureo at different values of thG discharge cur~ent~ Herefrom the depen
dence of the coupling coefficient::.; on the gas temperature h~::; b€C-il 0')-
tainBd, using th~ dependence of the gas temperature at th~ discharge
axis on the di~charge current, calculated with ou~ discharge model (s€~
chapter 3).
lntrorjuction
~h@ densities of excited atoms in a discharge a~e often measured
by m8Qns of the ~adiation-absorption technique (Lad33, 6eh71) _ H~re, we 3
present rneasu~ements of th~ radial profiles of the densities of the P2
and ·)Pj
Is-atoms Isee fi,..6.!) ~n a neon discharge (po: 100 torr, R:
32 rum), whiCh have been pe~formed with the fluoresoence technique
(smi75a, Ste75, coa76), using a tun~ble CW dye-laser. Tne fluorescence
light was measured ~erpendicular to the lase~ beam with a spatial rGSo-
lution better than rum. ~h€ discharge current W<lS v~ri8d betweon 20 rnA
(diffuoe Column) and 120 rnA (diffuse ~nd constricted column)- Excitation
of the 3P2 1s-atoms with laser light of 5882 g yields an extra contribu
tion to the population of the P2-level (see fig_6.1). The P2-level de
cnys to the four 1s-levels emitting light with wavelengthS 588., 60"0,
6163 and 6598 g. The density of the 3P2 (ss) <evel is determined by mea
s~ring the 6598 R fluorescence signal with a monochromator to eliminate
the scattering of the input light. The intensity of the fluorescence sig~ 3
nal is proportional with the ~2 atom density wnen sufficiently low laser
~hoton fluxes are used (thib haS been treated in Smi75a). A correction
has been appliBd fo~ ·the transition to the other p-levBls disc\lSSed below.
In the same way the dengity of the 3p level h~s been measured by ~5ing !
ldser light of 6030 R. Wh h 3 (3 . en t e P2 Or P
1) state ~s excited to the p,-level by photons
from the dye laser, the (,\loreScence spectrum 1~ not only composed of
79
.~
'-'--'r---"-'-~--'--) ---_.- - .... ~
~ ~
15_9
-~
-7
16.6
fig. G-l En-&J:'gy ~<wd diagrOJ7l of the fi>'et t/.)O ",xcit",d Mnfigur>aho-n.s
of neon (D~~J7).
ttl'" four lines originating from this level (5882, 6030, 6163 i\"c'i (;5~HJ 21, but- "l,~o from lines originatl.ng from othel: 2p-Jevels. This is Ca')5"a by
t-.t:"nsf~r of excitation botween these leVels by (coUisions with gas
.:3.tom~T We have recordr:::d tIle spectra of the fluore.scence light induced
by laser excittttioIl lo each of the 2P2-2P9 states at various values of
the distha:rge (:lJrrent_ Herefr'om the reaction coef{l(.:1.~r"lt:-_S for the C1.tom
induCed trans H_ians b"twee" the 2p-levels hav", boi:eo obtained in dep00-
dence on the dle;charge current. With the aid or ou.'( model for the
positive columflr we know the relQ.t.iOr'). b~tw(;:!en the discharge current Cl.nc.i
the gas tempe:rature ~o that th~ dependf:!nce of the reaction coe.tficic;,'r)"t,S
on the gas teIIlpel'ature C.;3.n be found, The valUES of the t;t:"an~i tion
8D
coetficients for Tg ft 300 K are found by extrapolation from tho
m~asured values to 300 ~. We have compared these valu8$ with those
measu~ed by coolen (Coo76) in a p~oton-gene~ated pl~sma at 300 K,
The derivation of the ga~ temperature dependenoe of the coupling
cQ~fficient5 from the measurements with the aid of our model, can be
considered ag an application of our model to measurements on medium
p~essu~~ discharges.
6.1 The m~asu~1ng ~quipment
The e~pe~~rnental setup which ha6 been uSed in these measurements
is shown in fig.6.2. An argon-ion laser supplies the power input for the
t'ig. 6. j Thlil e1I:plill"imental setup. 1'he al"l"OW indic:ates the direatiO'l of
tl"ans~ation of the disaharge tube fo!" the measurements of the
l'adia ~ pl"OfHes.
CW dye-laser. The dye-laser beam is ohopped periodically for synchronous
detection. The diameter of the laser beam and the optL~al properties
of the detection system are such that the spatial resolution is better
than 1 mm, so that deconvolution is not necessary. The intensity and
the wavelength ot the flUQres~en~e light of the dis~harge are detected
v~a a ~onochromator with a photomUltiplier. In order to check and control
the laser stdbility, h~l£ the laser ~eam intensity is used for ~ refer
ence neon dis~harge. The radial profiles o( the excited atom
81
densities have been meas~red by t(~nslatiOn of the discharge tube per
pendicular to the laser bee:..m.
6.2 3 3
Measurement of the radi<ll density profiles of the P~~l~
atQms 3
In figs. 6.3-6.5 the rel<ltiv€ radial density /?rofiles of the P 2
and 3P1 1~-atoms (P<I"Ch8n Ilotation) are shown for varJ.ous values of t.he
3 p i ' dischaE9~ current. Th~ calc~lated 2 prufiles ~re always ident C~.
t.o tile calculated 3P1 profiles. In the diffuse dis011arge tho measured 3 3
P:;; profi leo ttre equal to the moasun~d PI profiles. For t.he 20 rnA
diffuse di'>Charge the ~greelDent bet.w~en t.he calculat.ed and the measured
profiles is good. FQr tho 120 rnA diffuse discharge the measured profiles
a.;t."c SOmewhat broader than the oalculated ones. For this current v<ilue
'ehe e"Oitation fJ::-~qu~ncy is a strong function of U" (r) and Ne (r) so
that small ~rrOrS in the c"lculat.~on of chesa quantities may be respon
sible for the observed devi~tion. !n the 120 rnA constricted column l the
measured 3p1 profile is less narrow than the measured :J P2 prof~le. The
.. ,,-'" 1.0 20mA diffuse
"lculated - ....... - fTll!aslJrerJ
~ ~ "
"rl -0
~ 0.5 0..'" ,.,
ru \ >
ji, \
~ \ \
\ \
\ , , '"
. - , I B 16 24 3Z
rtrom)
Fig. 6.3 J P2 and JP1 pY'ofUIi!8 fo!:' a 20 mA diffuse rleon d1:aahQY'ge
(po ~ 100 torr, R = 32 mm).
82
120mA diffuse
-- calculated - - - n"o!!asured
\
\ \
" 16
rlmm)
Fig. (1,9 $am~ ¢<mditiol1S as in fig.6.J, but i == 1?0 mA (diffuse).
1.0
~ ... :;3
" M~ ."
" .. OS ",(l·
~ ~ ~
.•. "'( \ \ \ \ \
\ \
I
\ \
\
\ \
\ \
\ \ \
\\ . , '-.,
\ \
\
120 rnA constricted calculated ~Pa and 3p, 3;" IJ\B;'~ u r~ 3~2 'j'7'I ~a~ u. ti!. d.
16 r(mm)
Fig. 6.5 Same (Jondition8 a8 in fig. (i.3, out i
32
120 mA (aon$t~iated).
latter is in good agreement with the c~lculations tor ~mal1 rddial dis
tances. For larger values of r the c~l~~lated ~rofil~ d~creases very
83
rapidly, probably dlle to the same factors as discU8Seod above. The "",a-3 . 3 . . .
Suted PI proh 1e is b:roader than the measllxed P 2 Pl"ofl.le. Th> s dey L-
atiOIl must be ascribed to our treatme);"l.t of the radiat.ion..-diffusiOr'l tOYffi.
In the constricted cU.gcharge the densities of U,e e){cited atoms decreas~
very r1lpidly at. increasing radial di"tancG. ThGn the ~ppro"imations made
in the treatment of tho \lolsteirl-radiation term al;"e not valid, and th"
origina.l ra:..diation-dj ffusiol') equatt.on must. be taken into D.ccount to obJ
taln a reilli5tic calculation of the radial density profile of th~ PI
atomS.
6 . .3 Excitation transfer between the 2p-l"vels of neon by collisions
with neon atoms
T),e eOC1pling between the neon 2p-l"vels by collisiol)8 with neutral
neon atoms has been studied exp~rimGntally by several authors (Pilr65,
Gr,,?S. Coo7l;). !lexe We investigate the temperat\lre d~pendenc" Of eh"
cOupling coeff:i.cients, by rne~suring t.h"se coeff\c1 ents "t V"l'iOllS
val ues of the di::;ch~rge current. 'the: relation betw~eJ) the gas tempera
ture an<.i t.h8 di::;Chtlrge current is obt.ained [rom the model de-scribed in
cho.pter 3.
6.5.1 The deteI'mini!ltion of the coupling coefficients
whGn ~toms in one of the ls-1evels are excited to ~ level in the
2p-state by light from the dye-laser, other 2p-levels are a150 extra
populated beCilU5e of the collisional transfer of excitation. lienee the
[l'lt)l'escenc .. hght. is no t only cOmposed of rani ative transi tlons from
the originally excited 2p-level, hut also from oth~r 2p-l .. velo. To
determine the reaction-coefficient", tor the coupling betWeen the 2p
levels, Coach level has to be excited by thE> l2l.s"r light and all the spec
tral llnes of the levels have to be measured. When there are r:t lev~l:9,
e""h level coup l.'H to \'J-) 'Other levels, so that. th,",-" a>:0 N (N-l) coupling
cOf:-fticients.
When level 2pj
i,; e"cited by the laser, level ?'Pi i.~ ,,1150 populated
vi" collisiQnal coupling, 50 that the balance equatiorl fo~ level 2Pi is
(i ;! j).
134
j j ~ k"N, N + Ni Vi·
9,;li ~"' ~ g
tot~l decay frequency of level i by radiation,
(6. I)
coefficient for COUpling of leVel ~ to level i.
density of ~toms in the 2Pi leVel, arising from laser
excitation to th~ 2pj
level.
Eq.(6.1) is deriVed by subtraction of the de-I1~ity equation for level i for
the laser-off situation trom th~ equation for level i for the la~er-on
situation; N; = the difference of th~ densities of level i in th~ la~~rOn ,,-nd laser-off s~ tuations, mcasuY<l:d with the synchronous d.;-te"tion
technique. In this way the term repreSenting the production of 2p atom. . i
by electrons in the discharge cancels. The optical signa.l 3;0£ all lines
arising from the decay of level 2Pi is,
s j t
\) .N. j 1 1
[b.2)
The values of \). have been obtained from Wiese et "-1. (W1066). Hence • th~ values of N
i) arc obta~ned from the emission spectrum caused by
laser excitat10n to level j. After renumbering the co~pling ~oefficients
kll
, etc., eq. [6.1) can be written as,
wher~ the N(N-I) x N(N-I) maCrix ~ consists of elements Ni ~nd
Che vector _v oonsists of elements-V.N. j. Eq.(6.3) has been inv~rted
1 1
numerically according to;
-1 A v. (6.4)
'l'he measured spectra have been cor-rected for the relative w3ve-
length sens~tivity of the optical system in the w~y desc~tbed by
Coolen (Coo 76) .
Because of the re~tricted wavelength region of the dye-laser,
we were "ot able to excite the 2PI "nd 21'10 levels. Because the COUpH,)g
of the 2P I0 level Co the 2PI-2pg levels is very weak [becau~e of the
large energy gap), and the coupling of the 2P2-2P10 levelS to the 2PI
leVel is also weak [for the same reason), we are able to determine the
coupling ooeffi,cients between levels P2-P10 by laser excitatior> to tho:!
levels P2-P9 (we insert klO
, i = 0 into eq. (6.1). Then we have 64
equations with 64 unknown parameters kti
(as Kii is not appearing in
the equations, beca~$e j r i).
85
In. this way we measu):"oo. the set of value::; k9..1 fox different v"lUEl::'
of t.he cJisctl"fgC current OIl UW axis of a n"on disch"rge (po = 100 ton,
II = :l2 mm). The dcpen(lencc of the coefficients k Q•i
on the (lischilrge
cu;rrq.nt.: must be a::;'C.l::'ibed to i:1 dependence of k.£.~. on the 9as t~mperatuTe,
whir.:h inCI'€d.ses at incre~sing current~ as the el€GtrOn coupli.ng ~ffect
(mentiO(lGd in S~~li'"!:;u.) is very iIi.lprObable ~s prOven by t.he ma.ximum value
of t".llo t.ho<)rotically caloulated orOSs-sectiOn for thIs effect (Mas69)
That the coupling cocf.f.icients may depend on the ga~ temperature is
shoW'!) in the next section. In section 6.3 . .3 the experimentally obtained
dopendences dre given.
6.3.2 The tempe:r;ature ciependence of the oou.eling COofUcients
The cross-sectioll for "xcitat.ion transfer at small eflergy gaps
depends on whether the pot"ntial Gnergy curves 'Of th" initial and
final ~tates of the atoms huve d crossing point or. not". (Mas"!l). In the
ftrst (.:i':I.!3e the cyo.<;:,S-s.ection depends on the v.a.luGS of the derivatives
of t.he pot.e{)t.ial ~t1ergy curves 4tt the Cr.-OS8,'lng point. Without direct
knowledge of the"" potential energy curves for the 2p-levels of neon
no estimation can be giveIl~ fOX this cage. In the case that. t.h~re is no
crossing point sev0ral approximations are possir .... lc (Ma~71) _ As for: the
xenOn molecule both c.ases are possible, we a~S1J.mo that a.lso for neOn
the appr"ximalion that t.here i.", no crossing point will hold in "eve):"a)
relevant situ~tions. In that case Mas5ey gives for the cross-section
(Ma~71) :
Q [00
21f 0 2p()-p) scis ,
wher" p ~ the prOb"biHty Of a transition and
H:er~; p
w.tth
the J.mpact parameter.
2 2 F(S} sin )/ ,
o
F' (6) 2 r (m;l)
sllB ill-I 21"!v )"""T" K
m_
1 -2-
and
86
a m-l nvs
sllE 1w
(6.5)
(6.6)
whe~e 6£ = en~rgy gap, V = rel~tiv~ velocity of the ~tom~, h = Dirac
constant, r gamma f~nction, K ~ modifled Bebsel-function.
~ = a constant oepending on the kind of atom, m = an integer
depending on the kind of transition,
(this approximation holds when ~~ is not so large that P(s) ,< 1),
In our ca~e (neOn 2p-statcs, m = 5), no is so large 'or all relevant
values of s, that 5inz .. o
in the integr"l can be repl,aced by 1 ts me,m 1 Va,ue 2' Then eq. (6_5) Gan be w~itten as,
Q c (C a con .. tant). (6.-1)
The coupling GOefficient for coupling between two levels is:
(6. B)
sub"titution of " maltwellian distribution function for the velocity
distribution of the ga~ atom~, and eq.(6.7) for Q yields,
(6_ 9)
Hence the expected dependence of th~ coupling coefficients on the gas
temperature is of the form (6.9), We point out, however, that this ex
pression is deriv~d f.om approximation (6.5), (6.6) fo~ Q, which is only
valid fo~ the case of no crossing point.
6.3.3 Mea .. ured values of the ~ouplin9 coefficients
~rom the measurements, we obtain atter a numB~ical ~alysis ~G-
cording to eq. (6.4) the values of k~iNg for i = 10, 40, 60, eo and ll0
mh (diffu$e column)_ From the model of chapter 3, we obtain the gas den
sity Ng and the gas temperature Tg in depenoence of the diseharge current
(fig. 6.6). Hence we can derive from the mea~uremBnts, using the model
~alculations, the dependences of the v~riOUs kli on Tg . It appea.ed that
all k£i eould be written as:
(6.10)
:I: The values kli ar~ the (extrapolated) value~ for Tg = 300 ~, which can
b~ compared with thoSe of coolen (Co076) (~e~ table 6.1).
TaMe 6.1 Val!w$ of k .. ~ (300 K). Jt
i n k .. -. 1017
k .. ..
1017
J' J' (;';, 0.3) (this wo~k) (Coo76)
2 2 0.040 + 0.010
4 0.25 + 0.15 0+15 :!.- 0.07
5 0.030 1. 0.030
(, 0.026.:. 0.020
0.034 :!.- 0.020
2 0.10 + 0.03
4 15.0 + 2.0 14.4 .:. 3.2
5 0 0.80 + 0.30 0.62 + 0.20
6 0.50 + 0.20
8 2.5+1.0 0.10 .. 0.07
4 2 2 0.05 + 0.02 0.08 !. 0.04
2.6 :!.- LO ).7 + 0.4
1.'1 + 0.5 2.1 + 0.4
6 1 + S 0.17 .:. 0.07 0.29 + 0.10
7 0_5 0.85 :!:. 0.35 0.22 .. 0.08
8 0 D.3 :!:. 0.2
5 2 2.5 0.015 .: 0.015
1.5 0.15 + 0.05 0.09 ;t 0.04
4 1.S 1.9 !. 1.0 2.3 :!:. 0.4
6 2 0.03 + 0.01
7 0.40 + 0.20 0.54 !. 0.09
6 4 2.5 0.015 + 0.007
7 0.90 + 0 .• 0 0.85 + 0.16
8 0.5 0.30 + 0.20 0.28 + 0.09
9 2.5 0.025 + 0.012
10 0.08 + 0.07
4 0.12 + 0.03
0.12 + 0.03
6 1.4 + 0.4 0.60 :!.- 0.13
8 0.80 + 0.20 0.63 + 0.12
9 0.70 + 0.20
10 0.40 + 0.15 0.27 + 0.09
38
Table 6.1 (continued)
• 1017
k .. '" 1017
n k:J!. J~
(.!. 0.3) (this work) (coo76)
8 4 2.5 0.030 :t o.02S
6 0 0.26 ! 0.05
2.5 0.05 .:!:. 0.03
9 2.6 + 0.6 1.9 .!. 0.4
10 0 0.6 .!. 0.2
9 (, 0.06 .!. 0.01
7 O.~O .!. 0.20
8 1." 1.05 .:!:. 0.15 0.55 .!. 0.10
10 0 4.4 + 0.3 2.6 + 0.4
1000
Tg
Ng
SO 100 i(mA)
Fiqu~B 6.6 The oalauLated gas temperature Tg and the gas density Ng
as
a function of the discharge aurrent in a lOO torr neon dis
oharge at the tube axis. (tube radius 32 mm.)
69
---o:!2~SO----=,-!:-OO::-'"WO ',OO(J Tg (KJ
tn Ugs. 6.7 and 6.8 some dependences are given oeriveo from the measur
pO.i.nts.
Th0 a,;p:eement wJ. th the val\le$ of Cool.en is satisfying (see table
fi.l). In 80me c~ses (k47
, K?6' k9B' k910 ) there exist some oiscrep~ncie
In view of the rather large stundard devi~tions these particular v~l~c$
must be <.lbtained fr<.lnl the weighted average (with weighting fa<::tors choS
according to th~ largeness of tho ~U,noaro devl,ationsl over the vo.lues
coolen ano our values.
The gas temperature dependence of th~ coupling coefticients i.g in
reasonable agreement with the predicted behavi<.lur for many coefficient3
(following eg. (6.9) an exponent n = 1.:; is expected). The deviations m
be Ouel to t.he assumption which has bNln used he!"e, that the potentJ.al
curves have no crossing ~oint.
In conclusion we may say that the d~pendence of the coupling co
efficients on the discharge current lItust be due to this gus temperat_urc,.:..
')0
J.~ .
10'
"1,
~~
"'~
llj'S
! , r ! ~ , _..1 .• _. WJ lOCI) 1000
T9 (K)
Ng. 6.8 Measured dependenM of k.;ji on Tg for j ;; s.
dependence r~ther than the electrOn indUCed transitions suggested by us
in an earlier publicat~on (Smi75~).
91
Concluoing rem&~ks
! The model ~res~nted here give~ a fair deseription of the diffuse
column. Th~ constriction Can be described by incorpordting the stria~
tion development above c~rtain current and pressure thresholds. In
the constricted column the E.F. is st~ll strongly non-maxwellian in
th~ inela~tic region. Furthermore ambipolar diffusion can not be ~e
glected in the constricted column.
I From the caleulateo quantities which haVe been experimentally verified,
the axial electric f~eldstr~ngth 15 most sensitive to v&~iations in the
coefficients of the model. Especially the value of the elastic electron
atom cross-section is impo~tant, because the axial electric field
strength is proportiOn&l to this cross-sectian. To fit the c~lcul~ted
values of the axial electric fieldstrength to the measured ones, it
was n~eessary to diminish the measured curve for the elastic cros~
section of Salop and Nakano (Sal70) with 20~. This co~~e5ponds to th~
lowest meas~~ed values fo~ thi~ cross-section given,in the lite~&tur~
(see sal70). The othe~ data from the literature whiCh have becn used
for the model have l~ss influenc@ on th~ model results, and have not
b~en altered.
1 In the calculation of the ~.~. several effects have been tak~n into
account whieh are commonly. neglected. The influence of the gas temper
ature term in the Boltzmann-eq~ation on the excitation fr~quency ha~
been taken into account. The axial electric fi~ldstrength has been cal
culat~d in a eonsi~tent way via the def1nit10n of the electron temper
ature. Also the particular integral J 2 has been treated more consist~nt
ly than is usually done.
! The radial profiles of the electron density and the eleetron temperature
could be measured from the electron-atom bremsstrahlung continuum. In
the determination of the elect~on temperatu~e there may have been a
distu~bing influence from the molecular radiation.
~ The reaction coeff1cients for atom induced transitions betWeen the 2p
levels on neon were measured as a function of the diSCharge current.
He~efrom the d@penoence on the gas temperature was determined. The re
Lation betw@@n the gas temperature and the discha~ge current was calcu-
93
94
lated with our Ilumcr1 .• ~Ell model. A theoretical estimat.e show-ed that the
coupling coefficients should be proportional to T 3/2 for a special g
CElse. The- measured de'peI~(i(~)'"Ic.:es o.re Tn, with 0 < n .:: 2.5 T
9
Appendix
List 01: '>;(l!lbols
SYMol
A
A(I':)
Ai (E) ')\2 (E;)
A1 ,/l.2
b
c
Designo.tion
1) adaptation COnstant for the 5.F" in th~ inelastic region,
2) ord~r of magnitude of the quotient of the ro.dial deri
vative term and the e~agtic col11sion-te~m in the Boltzmann-
equation.
~pproximat1on for AI (E) and A2 (E).
Coulomb integrals.
constants in the approximation for the inelastic cro~s
:section.
transition probability for line radi~tion from l~vel i.
acceleration due to the applied electric field (a e~. - III
quotient of field f~eq~ency and Coulomb frequency
" (\IE /\Jee)·
"£1-114 .
constant5 in approximation for q(t).
constants in approximation for the momentum transfe~
Gross-~ection and the elastic cross-section.
Ai/~A~ • adaptation constant to~ the E.F. in the inelastic region.
1) quotient-of ela~tic collision frequency and coulomb
frequency (v ~/v )" g ee
2) parameter in dispersion ~elation.
bE: 1/4 1
constants in appro~imation fo~ q(E:).
~lectrDn mob~lity.
mobility ot the molecular ions.
adaptation constant for the E.F. in the inelastic region
collision operator in the Boltzmann~equation.
correction factor in the equation for the axial electric
fieldst~ength .
1) constant in the parabolic approximation for the radial
gas temperature prottle,
2)pa~aweter in dispersion relation,
3) light velocity.
95
c " e1
::-1 '~1 'E:j e 1 in ee
Cl
c £
"'roi
D
"
f: ij
e ---z ali:
" I l' 2
I'(U)
f
collision op"rat.Ol:'i; in t.he ~=O eq\laUOn for ttl'" elastIc-,
j~el"stic ilnd e,""ctroIl-electron colli9jo~~ r""p,-,ct)v('ly.
correction factl1r to Obtain the electron density from th .....
brem$strahlung intcn~1ty.
0011i3ion operl;\tors in the ~.:.:l equation for lhe eLlstlc,
jnela~ci(.': and electran-elect:ron collisions re5pectiv~lYT
I) "a,1stant in the expression fo~ the bremsst.rahlllng
t()tensit.y.
::n co:)sta.nt In th~ numerical approximat,lOr'l. fur N (r} Eor e
r ., l:' 1 (Ollt",. "eg~ot'\).
cyn~t;:ants in dpp,roximat.e expression fot" Z in high electrun
densl ty limit.
1) parameters in expression ~o~ Z for argon.
;:) con.st~Tltc in two-tempe~~tu:t"e model.
corr(~cttOll factor ~n the equi.lit.i(in for the .:JJTlbipo.l.E::I.r
diffusion ccefficient.
reaction C'oeffjcient".$ eor l2x.cit":ttion from -I.~vel t if'.
g.oup I to ~ i.vel in group II, III or IV <eepcctiv.ly by
el~ctl::"on$.
~thorj,,~ t.enso.: i.n the development. of elf).
reacctor). cC1efficient ~o;r t~le-ctron excitat:i..on ,from the
3 \ metasul.ble P 2 Level. 19n)up 1/ te' grollp IV tor I1rgon_
Gorr""tio_1 factor tn tbc equat~on ~or Neo
~ll\bip"ldr. diffu"ion coefficient.
difflPi(',n coefficient of the electrons.
ax1~1 electric fleldstrangth.
e 1 em~:r: (l tary charge.
o·r",>:gy gap betw<een lCVBls i and j.
':..nit voctor in all.1.al ~1i::t"~ttion,
"nergy gap.
f.;..ctO:r in expressiOr'l [0,[ z~
cOrreCtion factor fot' the gas temperi':l,tur~3 in the
~xpression for Z.
factor in the expression for the cross-se<;:t t ('tl for
e~citation transfer between the 2p-1<eve18
isotropic p"rt of the !l.l'. (energy coordinates).
1) electron energy distribution function,
2) striation frequency.
H(n
h
h
I
i(A)
J
J(p)
J(Ue,N/Ng,A)
J 1 (tl
J2 (0
.:r/'(IC)
Ji,J j k
kij
a,kij
e
K (xl n
L
M
M(E)
m
iSotropic pa~t of th~ £.F.
first order term in the development of f.
tth order tensor in the development of f.
expression in the scalar equation.
(2J i +l)/(2Jj+l), where Ji,J
j total quantum numbers of
levels i and j.
expression in the scalar equatlon
Planck constant.
Di~ac Don,;tant.
expression in the inelastic part of th~ E.P.
discharge Current.
measured bremsstrahlung intensity after Abel transforma
tion.
current density
integrals oVer the B.F.
expression in the bremsstrahlung intensity.
homogeneous solution of the scalar equatiOn far E ~ t 1 •
particular solution of the scalar equation for E < E j •
approximation for J~(~) in the neighbourhood of E = ~1. total quantum numbers of levels i and j.
I) Boltzmann constant,
2) number of discretizatiOn intervals,
3l wave number.
pa~ametBr in dispersion ~elatiOn.
coupling coefficient for transfer of excitation b~twB~n
levels i and j (kij* at 300 K) (group I, chapter] and
grOup II, chapter 6) .
atomie ano electronic ~art of kij
(group I, cpapter 3).
modi{i~d Bessel function of nth oroer.
tube length.
~ank of the tensors.
atomic mass.
~"ponent in the expression for J 1 (t)
1) eleotron mass,
2) integer constant in the exp'~~5ion for the cross
section fo~ excitation-transfer betw~en the 2p~level~.
normalization con~tant.
2rrw SN.
97
n ,x)
N (r) e
N eO
Nq
N go
Ngf
NH
N~ ,Nj
N j 1
o(x)
p
p(,,)
Pel,Pin
Pgtt !'1
Q
qU:)
Q8
Q[)
Qel
Qi,4
')8
1) exponent in tho> expre:;:;ion fo~ tl1e "lasti" o-ro",,-
section,
2) O .... pOnei!t in the gas temperature dependence of the
measured coupJ.ing coetficient" for excit;;.ti on lranstc-r
between tl1e 2p-J,evel-'3.
relative electron density profile (N,,(r)/Ne(O».
electron density as a function of the radial distance.
Ne,O) (elect.ron density at the axis).
gas J"nsity.
gas d~n$ity at the axis.
ga~ (lel~$i ty in the discharge-off situation_
donsity of the .:ttorns in level H, representing gl-OUp II~
d0nsitieS of atoms in levels i and j.
d"ns.j.ty of atoms i" the 2Pi-level «rising from laser
eKcitat10n to the 2pj-level.
order of magnitude ~.
1) pressure in the <Icti ve dj. ~ch~rge,
2) argument of v(p),
3) probabUi.ty in tho expression for the "ro"~-,,eGtion
for the cO\lpHng c00fficienC':; of th" 2p-leV<lls.
expression in the scalar-equation tor L > £1'
filling pressure.
pressure j,n the discharge when the dead volwne is ID.rge
respectively smi'll.
elastic and in€:lastic en~rgy 10"5e5 per electron.
gas term tn the electron energy equ~tion.
cr05s-s*ct. ton.
expression In the sCi'lar-cquation for E > E1.
bremsstrahlung cross-::;.ection.
f.l)OlllentUIn transfer crQ5s-s€'ction.
elastic cross~s~ctian_
(::rOSs-section for elect:roIli~ 0xci ta.tion f.rom level i in
group I to group IV.
cros5"'section for electronic ex~i t.atiQrt :f.rom a 1.0vel j n
g~oup J to group IV.
total in~la5tic cross-section.
tubE' >:"adiuS,
radial 1'Ii..t.~nCe.
t
t(x)
t (£)
T o
tl T e T eflV Tg
Tw .... 1l U
U (0
u
v·
v a
v 9
characte~istlc column radius.
ratio of the bremsstrahluflg intensities at the waVe
length$ ,500 ~ and 4000 ~. imp .... ct parameter.
expression in the scalar equation for ~ > £1'
integral in the equation for Neo
integral in the equation for p.
optical signal of all litles a~1$~ng from the decay of
Level 2p~ in the ~ase of laser excitation to level 2pj'
time.
r~lative gas temperature profile (Tg<r)/Tg(O)).
term itl the scalar equation.
gas temperature at the discharge .... xis.
1) average temperatures in the discharge with small
respectively large dead volume,
2) terms in the dispersion relation_
characteristic time constant for ela~tiC collisions.
electron temperature.
temperature of the env1~onment_
gas temperature.
wall temperature.
electron energy.
expression in the calculation of the E.F. for E ~ E1-
parameter in the E.F. for argofl.
etlergy gap between group l ana groups II or III.
energy values in the approximation for the inelastic
cross-s€ct~on of neon.
eflergy gap between gro~p I ana group IV.
electron temperat~re in vo~t equivalents.
electxon temperature at the ais~harge ~xis.
"tail-temperature" in the two temper<Lture models.
1) electron velocity,
2) TiTe' values ot the vo~~e of the discharg~ tube ana the dead
volume respect~ve~y.
electron velo~ity belonging to the value of Ua .
electron drift velooity.
striatiofl group velocity.
99
v. , v
w
"
y
z
z
p
Zion
Z ,Z n u
f: 0., r) (. -·0
E: j ''''2 E; a <1> 1 ,<1>2,<1>3
'1'\ j ,<P ij'
Y ,Y i
A (T ) g
A o
100
electrQrl velecity belonging to th<: val\le of U j •
~crJ.at.t.on phase velocity.
therm~l velocity of the 61~ctrons.
1) r/R,
2) axial distance,
3) ~ in the bremsstr~hlung expression. eU A
"xpr6~sierl in M(eJ.
excitation frequency (total).
dimensionless integration var~able.
excitation frequen~y (for excitation to level i in group I)
ionization frequency.
expressions in the disper~ion relation.
I) volume-recombination coefficient,
2) constant in the expression for the c:r.oss-section for
excit~tion transfer between the 2p-levels.
he~t transfer coefficient.
gClmma-function.
gradient in the geoMetrical respectively velocity space.
1) \J/lle
,
2) emission coefficient.
measured bremsstrahlung intensity without Abeh correction.
permittivi ty of vacuum.
lJ/lIe , lJ/U.:,' lJ/lJ". expressions in the E.F. for E > [I'
<p. (E.), dq,./del e .. 'J 1 J three body collision coefficient for ~rgon re~p~ctively
3 3 the neon P
4 and Po atoms.
heat conductivity.
1) w~velength of light,
2) w~velength of striatio~s.
coefficient in A(T ). 9
2500 resp. 4000 ~. 3,., I
wavelengths of the radiation emitted by the "I and PI
atoms.
relaxation length of the electron ternper~ture.
frequency repre5ent~n9 the field influence in the
Bol tZ\1lann-equ~tj.On.
~oeff1cient in VE
.
Coulomb-~nteracticn frequency.
~lastic collis10n frequency.
coefficient in Vq .
1) dec~y frequency of level i in group I,
2) trequency of the emitted light at the fluorescence
measu~ewents.
inelastic collision frequency.
striation frequency in the dispersion relation.
imaginary part of W.
real part of w.
expression in the cross-sect~on for excit~tion transfer
between.the 2p-levels.
expressions in the dispersion relation.
Stefan-Eoltzmann constant.
decay t!me of level i in group •.
101
A£2endix I" Compilation of data used for the model
Several coefficients and cross-sections have been used for the
calculations on the E.F. and for the numerical model. These values
have been obtain~d fxom measurements reported in th~ literature. Here
we give a com~ilation of these data.
Quantity
Qi,4
Q I ,2' Qi,3
a
A(T ) g
\)i (Ch. 3)
Vi (Ch.E;)
Neon
Value or formUha
0.40, 0.27, o.oe, 0.25
6.10- 4
see appendix 3.11
0.85 Qe1
11.8 10-20Ul/4 U ~ U1 1.8 )0-20U 1/4 U > u
l )
- 16.6v
o a .::.. U .::.. U 1
6.10-22
W-U 1) U1< U:: [1
2" 17.0v
-22 -22 6.)0 (U2-Ul)+0.5 10 (l,l-[)2) U>U2
\
1.2_10- 20 (u-5) 5 <: U <: 10v
(6.44-0.0444U) ,10-20 )0 < U < 100 V
1 <rQ14
0 0 < U < U
9~64 . )0-)6 - 1
(V-VI) V > U1 V2
1 with U " 1
for Qi,2
U " 1 3 for Qi,J
2.5 10- 14 (l-exp( _22Q»)
U 1/2 T e g
10-3T 2/3 9
See appendi,. 3. II
see referen<;e
Reference
Co076
McD64
PheS9
McD64
analytic~l app~oximation
on data given by Sa170
analytical appro~imatiOn
on data given by Sch69
analytical approximation
on data given by Dix73
classical Thomson
formula
(M1t73)
Ma974
sax6B
Phe59
Wi.,66
103
Argon
Quantity value or formula Ref:e):,,~f1(:e
brn'\'o 2.IO~4 McD64
C mj, 15 . 3 10-
14 U <: 2.5v calculated from cl"S!3j~a:
e (1.8tl -3.8) 7.S lIl- 14
tl > 2.5V ThomsOrl Ct"os5 .... sect~orl e e
(Mit73) -2(l
WOjSSb QI) 11.<1 10 "u 8 " U < II - 11, ~v
10- 2O Oj
- 1
1.4 u > UI.
Qt
1~." Q < [) :s. U 1
I'IOjoSb
-21 10 (~J-[Jl) U > U 1
8.0 10- 14 650
Mas74 Ci u 2/3
(l-e"p(~T) ) e g
y 3.6 10 -4" Mas74
Ie (T ) 4. l'1 10- 14'T 2/3 Vru76, wo]67
9 g
lI.J~
Abr68 W.AOramowitz and I.Stegun, Handbook of Mathematioal Funation8,
Dover 1968.
Bag74 L.C.J.Baghuis, th~sis Eindhoven 1974.
Ba~69 V.Yu.BaranOV and K.N.Ul'yanov, SoV.Phys.Te~hn.Phys. 14 (1969) 183.
Beh71 J.Behn~e, Beitr.Plasmaphys. 11 (1971) 23.
Coo76 F.C.N.Coolen, thesis Eindhoven 1976.
Dix57 J.R.Dixon and F.A.Grant. PhyS.Rev. ~ (1957) 118.
Dix73 A.J.Dixon et al., VIII ICPEAC Beograd 197] r, 405.
Duy75 C.J.van Duyn, EUT internal re~ort (in dutch) ~~19S (1975).
Fire5 O.B.Firsov and M,I.ChibisoY, Sov.Phys. J,E,T.P. 12 (1961) 1235.
fra76 R.N. Franklin, prasma Phenomena in GaS Di8o~ges> Oxford 1976.
Go16Sa Yu.B.Goluboyski1 et al., BBitr.plasroaphys. 8 (1968) 423.
Go168b Yu.B.Golubovs~ii et &1., Opt.Spectr. 24 (1968) 286.
Go169 Yu.B.Golubovskii et al., Beitr.PlasmaphyS. ~ (1969) 265.
Go170a
Yu.B.GolubOVskii et al., Sov.Phys. J.E.T.P. ~ (1970) 1204.
Go170b
1u.B.Golubovskii et ai., Beitr.Plasmaphys. 10 (1970) 427.
Gol73 1u.a.Goluoovskii et ai., Opt.Spectr. 35 (1973) 124.
Go174 Yu.~.Golubov5kii et al., Sov.Phys.Techn.Phys. 19 (1974) ))5.
Gra5S w.L.Granoyski, D~~ Etektpieahe Stpom im Gas, Berlin 1955.
105
G:r<l74 I.GrabeC, Beitr.Plasrnaphy •. 11 (1374) 1.
J.p.Gt~ndin et al., J.Physique 36 (1975) 787.
Ho14'/,SI T.Holstein, PhyS.Rev. n (1947) 1212; 83 (1951) 11>9.
Jac57
Jotl66
K<lg64
K.i:>g6S
Kilg71
1<.01865
Kcn67
Lad)]
L"rn73
Lya72
MCD64
Mas69
106
M.Jacob and G.II.Hawkins, ELements of Heat Tl'''l1~f'''I' and
Insulation, New York 1957.
T.W. Johnston , J.Math.Phys. (1966) 1453.
Yu. M. Kagan and R. I. Lyagl1s~h@n<;ko, Sov. Phys .1'echn .Phys . .§. (1964)
627.
YU.M. Kagan ano R. I. Ly.;<gu,>ctleI1Cko, Sov .P\1ys. Techn.Phys. 'J (1965)
1445.
Yu. M. Kagan ano R. I. Lya1uschencko, $ov. Phys. 'l'echn. Phys. 15 (1971)
1568.
V.Kas'yanov, A.St"rostin, Sov.Phys. J.E.T.P. 21 (1965) 193.
C.K~nty, J.Ch~m.Phy5. 41 (19b?) 2545.
R.Ladenhurg, Rev.Mod.l'hys. 5 (19."13) 243.
P.H.M.Lammers, and ~_M_M_Srnits, E.U.T. internal report (in
dutch) NK159 (1973).
R.r.Lyagl1"chen~ko, Sov.l'hyS.Techn.Phy". 17 (1972) 901.
E.w.MCl)aniel, CoZUaic:m Ph~momerw. ir, Iorl"ized Gas,u;, New
York 1964.
H.S.W.Massey and E.H.S.Burhop, t"lectr'On<:c and 10>1-1:c Impact
PherlOmana. Oxford 1969, Vol. 1 .
Mas71
Mas74
Miti3
Mou7t
Ole68
Par65
Pek68
P)1e59
Pfa68
Pfa69
Pri66
Pri75
RaR.70
Rom67
H.S.W.Massey, Etectponic and Ionia Impact Phenomena, Oxford
1971, VoL3.
H. S. W.Massey and H.B.Gilbody, Elect"orda and Tonia Impact
Phenomena, oxford .974, Vol.4.
M.Mitchner and C. H. Kruger, FartiaUy Ionized Gase?, New York
1973.
C.A.M.Mouwen. thesis Eindhoven 1971.
L.Nedelsky, Phys.Rev. 42 (1932) 641.
N.L.Oleson and A.W.Cooper. in Advancea "n E~ectroni(J and
E~eatpon Physics, New York 1968.
J.H.parks and ~.Javan, Phys.Rev. 139 (1965) 1351.
L.Pekarek, Sov.Phys. OSPEKHI 11 (1968) 188.
A.V.Phelps. Phys.ReV. ~ (1959) 1011.
S.Pfau and A.~utBcner, eeitr.Plasmaphys. f (196B) 85.
$.Pfau et al., Beitr.plasmaphys. ~ (1969). 333.
S.Ptau et al .• ~eitr.Plasmaphys. 16 (1976) 317.
J.F.Prince and W.W.Robe~t50n, J.Chem.Phys. 45 (1966) 3577;
46 (1967) 3309.
M.Prins and R.M.M.Smit~, XII ICPIG Eindhoven 1975 I, 67.
A.T.Rakhimov and ~.R.Ulinich, SoV.PhyS. Doklady 14 (1970)
655.
R.Rompke and M.Steenbeck. E~9~bni6se d~~ Plasmaphysik I,
Serlin 1967.
107
Rut.6H
S.,17(1
Smi74
Ulyn
Venn
Vri'lJ
Vn,76
wie66
Wi170
Wi,., 72
woj65"
A.Rutscher and S.Pfau, Bei~r. Plasmaphys. ! (1968) 315.
1\. Sa 1 "f? 'md H. FF. Nak ~no, P!'!ys. Rev. A 2 (1970) I;?
V. K. Sax~n" and S. r.:. SaX"ni:t, J. chern. PhyS. 48 (1 ~Ibfl) r,.,62.
W.SChottky, 1'1Iyo.z. 25 (1924) 635.
M. Scllaper and IL Scheibner., lleitL pl<lsttlaphys. 2. (l969) 45.
I. P. ~hk,,~of"ky ,~t "1., Th~ i'(J,)"tioi.e Kir£et,>:", of Pi.aama3,
[)'Hdrecht 1966.
R.M.M.SmHo et al •• tleitr.l'la"mapllys. 14 (1974) 1.57.
R.M.M.Smit" and M.Prins, Phy~ica aoc (1975) 5)1.
R .. M.M.Smit~ and M.PrinsJ XII ICPI(~ Eil1dhOV~n l: .. ns I, 74.
L.W.G.Ste~nhuye0n and H.a.d.Brugh, I'hyRica 79C (1975) 207.
K.N.Ulyanov, Sov.Phys.Tachn.Phys. 18 (1973) 36U.
lLVYlcm;, J.Appl.PlIys. 44 (197." 398lJ.
P.J.VnI<Jht, thesis r,:lndhoven 1976.
W. L. W.l."",," et 0.1., A tom!:" Trar:l8it-irin PToobab·i U t·ies ~ waShing-ton
196<':,. Vol. 1.
~.Wilhelm et al., Beit".Plasm~phy~. 10 (1970) 171.
R.Winkler, l3"itl:'.Pl~.sm".prlY~. 12 (1372) 193.
K.W<>j"cz"k. Bllitr.1?I"sm"phy.~. 2. (1965) 181.
WOj6Sb K.woJaczek. Beitr.Plasmaphys. 5 (1%5) 307.
Woj66 K.Wojaezek. ~eitr.Pla8maphY5. §.. ( 1966) 211.
Woj67 K.WOjaez~k. Seitr.~la5maphy5. 2- ( 1967) 149.
WojG8 K.Wojaczek, Beitr.Plasmaphys. 8 (1968) 109.
Woj69 K.wojaczek, Beitr.Plasmaphys. 2- ( 1969) 243.
109
S\lIllIIlary
In this thesis we describe a model for the positive column of a
medium preSsure neon disch~rge, together with optical measurements to
test the model. FUrthermore the constr~ction of the positive colwnn
above certain thresholds of the CUrrent and the filling pressure is
studied. The pressure-current reg~on of our cOncern is: 0.50 < PaR <
10 torr m and 0.01 < i!R < 10 A!m (po = filling pressure, i ~ discharge
current, R = tube radius).
We present a numerical ruodel for the positive column of the neon
d~scharge in this parameter doma1~~or this model we need the electrOn
energy distribution function, which is calculated from the Boltzmann
equation. Prom this calculation we obtain an expression for the axial
electric fieldstrength via the definition of the electron temperature
in a consistent way. Furthermore the excitation frequency is calculated,
taking into account the effect of the gas temperature term in the
Boltzmann-equation.
The axial electric fieldstrength and the excltation frequency
are used in the numerical model set up for the positiVe column. The
results based upon the model are in good agre~ment w~th e~peri~ent5 for
the diffuse discharge. The constriction of the column which occurs
abruptly above certain thresholds of the discharge current and the
filling pressure can not be CalcUlated without modifications. From
a variation of the parameters of the model it appea~~ that the con
striction is started by the onset of ionization wave~ (striatiOns).
A comparison with other models presented in the literature is made.
The results of the model are· compared with experimentah values
for the electron density, the electron temperature and the densities 3 3
of the metastable P2 (IsS) and resonant P1
(ls4 ) atoms (Paschen
notation). The radial profiles of the electrOn density and the
e'ectron temperature have been obtained from measurements of the
electron-atom bremSStrahlung continuum. The measured electron density
profiles are in good agreement with the Calculated ones, both tor the
diffuse and for the constricted column (the latter calculated with
the striation model)., Between the calculated and the meaSured
electron temperature profiles there is an increasing discrepancy
with increasing current. Possib.y this discrepancy is due to a con
tribution of molecular radiation to the continuum. The density profiles
111
J 3 of the P 2 and P 1 atom~ have bt:.'cn meQ..su~eo ...... ,1. th the fluore:scen(;e
t.echn:i.que, using i1 dye-laser. The measured profiles al'e in l"easonable
"greement. w~ th tile (:<llcul<lt~d Ones for the di:Hu5C ct~$charge. For the
constricted column, the m~~sured 3P1 p~of1~e i~ le~g narrow than the
c"leulated profile. This must be due to our simplified treatment of
th@ radiation-diffusion term {th:l.s treatment i~, hDwever, :sufficiently
ClCCUl:"ate for the c"lcul<ttion of other d"t" of the coluron) .
1J1he nwl:lericCll model has been used for the evc=..luCltion of Jneasure-
ment5 of t.he reaction cocfHcients for atom induced transitions betcween
t.he 2pu.ulevels 111. neOn. These measurements were performed for severa.l
vall,lt;':!~ of the discharge current. Herefrom the coupling coefficient.s
have been determined as II function of the g"5 temperature, where the
relat;,ion b0t.WGI)(j t.he gas teI'I'lp~rdture and the discharge current was
calculated with the h"lp of our mo4;,1, Theo)::;,tical eo;tit))ates shOw that
th,., 4epe(t(lcIlCE' of the coupling coefficients on t.he gas t.ernper"tu~e Tg 3/2 can in $Qme (::~SGS be written as Tg . The measurements show a dependence
according to Tgn, with 0 -< n < 2.5 (dependent on the levels between
wl"\ich the tJ:"an,;ition tZl.ke9 place).
112
Samenvlttting
In dit p~oefschrift wordt een ~odel voor de p05itieve ko~om van
een neon-ontladin9 van gematigoe dru~ beschreven~ tesamen met optische me
tingen ter toetsing van dit model. Verder wordt de contrnctie van de posi
tieve kolom boven bepaalde waarden van 5troorusterkte en vuldruk bestudeerd.
Bet onderzechte 5treemdrukgeo~ed is, 0.50 < poR < 10 torr m en 0.0) < i/R
~ 10 A/m (po = vuldruk, i = ontladings5troomsterkte, R ~ buisstraal).
Voer dit gebied is een nurneriek model Veor de positieve kelOID van
de neon-ontlading epq@steld. ~en behaeve van dit model is kennis nodig
van de energieverdelingsfunctie veor de electronen. Deze is berekend uit
de Boltzmann-vergelijking. Hieruit wordt op een con5i5tente manier een
uitdrukking voor de elektri5che veldste~kte afgeleid, via de definitie
van de electronentemperdtuur. OOk is de excitatie frequentie berekend,
daarbij wordt rekening geheuden met die t~rm in de Beltzmann-vergelij~
king, die de invloed van de gastemperatuur beschrljft.
De axiale elektrische veldsterkte en de excitatie-frequentie worden
gebruikt in het nuroerieke model dat ~g opgesteld voor de verSchijn5elen
in de positieVe kelom. De ~e5Ultaten van het model zijn in goede oVereen
stemming met experimenten voor de diffu5e ontlading. De abrupte centractie
van de koloID boven bepaalde d~empelwaarden van vuldruk en stroomsterkte
kan niet ~onder modificaties in het model berekend worden. uit een varia-
tie van de modelparameters lijkt de conclus~e gerechtva&rdigd dat de oon
tractie gestart wordt door de entwikkeling van ionizatiegolven (striatie5).
Het model wordt vergeleken met ~ndere modellen uit de liter&t~ur.
De modelresultaten worden vBrgBl~ken met experimentele waa~den van
de eleotronendichtheid, de eleotronentemperatuur en de dichtheden van de
metastabiele 3P2 (lsS) en ~esonante 3P1 (154 ) atomen (PaSChen not&tie).
De r~diale pro~ielen van de electron~ndiOhtheid en de electronentempera
tuur zijn verk~egen utt metingen van het ~lectron-atoom remstralingsoon
tinuum. De gemeten electronendichtheldsprofielen zijn in goede evereen
stemming met de berekende, zowel vOer de diffuse als voor de geoont~a
heerde kolom (de laatste is oerekend met het strlat1e-model). Er is een
toenemende afwijking tu~sen gem0ten en berekende eleetronentemperatuur5-
profieleo o~j toenemende stroomsterkte. Het is rnogelijk dat de,-e afwij
king te verkl~ren valt ult een eventuele bijdrage van moleculaire 5tr~ling
tat net continuum. 3 3
De dichtheidsproti01en van de P2. en P1 atemen zijn gemeten met d",
fluores~entietechniek door middel van een kleurstof laser. DB geffieten pro-
113
fiel-CTI zijn in reOelijke overeenstemming met de: berekende voO,( do::' 11.t('l..1$t-;
on tlading. aij d" 90contraheerde kolorn i8 het gerneten 3p 1 profiel D.~0Je.t: d~n het berekende profiel. Dit is te wi:Jten "an Or\2;8 v~reel1voudi9de be
h~nd~ling van de stralingstr~nsport term (deZe behandeli119 heeft echter
geen OOs"kbl1re invloed op de bBrekening van andere kolorngrootheden).
Het numerieke model i~ gebr~ikt b~j de interpretatie van metingen
van OG- :t"8actie-coefficienten voor atoomgelno\l~e-~rdE;! overgangen l;.ussen de
2p-niveaus in neon. Deze rnetingen z~jn verr~cht bij ver$chillende stroorn
ste;r;kten. Hieruit zijn de koppelingscoefticienten in ~fhankelijkheid v~n
de gasternperatuur bepaald, waarbij de relatie tuso:;en de gasteooperatuur en
de ontladingsstroornsterkte uit het model berekend iO:;. De afhankelijkheid
van de koppeling8coeffi~ienten van dB gasternperatuur T is theoretisch 3/2 9
gesGh11t op Tg . De rnet~ngen tonen een athankelijkheid volgens Tgn, met
o ~ n ~ 2.5 (een en ander afhankelijk v~n de niveaus waartussen de OVer
gang plaatsvindt) .
114
Nawooro
set onoerzaek aan de gematigde druk neon-ontlading is gestart in
het kader van hat prOject "gasontladin9sonderzoek met behulp V<ln tr<lCer
tecnnieken" met het afstlloeeronderzo@k van P.H.M. Lammers en mijz",lJ' in
oe periode juni 1972-jllni 1973. Het in dit proe~~chri~t,be5chreven onder
zoek is uitgevoerd in de periods van oktober 1973 tot augustus 1977, en
is een vervolg op het afstudeerproj.ect.
Gedurende deze periode hesft M. Prins aan het onderzoek meegewerkt.
Verder zijn dee~onoerzoeken verricht ooor de volgende studenten.
C.J. van DUyn (s~ge en afstuderenj, ~.J. van Bommel, R. Buren, ~.B. van
PanthaleOn van ~ck, p. van Hooff, J. Krabbenborg, T. van der Laar en
J. Pennings (allen stage). De gasontladingsbulzen zijn steeds met zorg
gBmaakt en gevulo ooor M. Hoddenbagh, M.J.F. van der Sande, J_C. Holten
en R+ ·neynd~rB.
De numerieke berekeningen hebben naar schatting 100 uur rekentijd
gekost op de 6u~roughs B6700 en B7700 mach~ne~. De dye-Iase~ i~ gedurende
ongevee~ 300 uur gebruikt. De totaLe kosten van apparatuur voor dit on
derzoek bedroegen ongeveer 65 kf.
Ben bijzQnoer woard van dank wil ik allereerst richten tot Maarten
Prins, waarmee de samenwerking deze vier jaa~ uitermat~ plezi~ri9, stimu
lereno en nuttig is geweest. Vervolgens dank ik ~ans van Duyn in hoge
mate voor zijn berekeningen aan de Boltzmann-vergelijking. Tevens Qank ik
H.~. fiagebeuk voor zijn assistentie bij de numerieke berekeningen aan de
striaties. Bet typewerk is ~eer bekwaam verricht door Francien Duifhuis,
terwijl het teken~erk met veel zorg is verricht door Ruth G~uyters. Bei
den hiervoo. oprechte dank.
Tot slot wil ik alle 11. H. -medewerkers bedanken oie direc.t of in
Oirect hebben bijgedragen tot dit proefsohrift en aIle menS en van het
cyclotrongebouw vOOr oe plezierige samenwerking en sfeer binnen het ge
bouw.
115
Levens loop
j\lnl 1967
geboren te Oosterhout (N.~~.).
eind",x",men IL EL S. -b ,,all het MOIlseigneur Frencken
college te OOBterhout.
september 1967 aanvang natuuikunde studie a",n de T_H.E.
juni 1973 doctor",al examen natuurkundig ingenie\lr_
oktober 1973- wetenschappelijk medewerker in de groep cyclotron-
",ugustus 1977 toopassingen, afdeling der Technisehe N",tuurkunde ,
T.Il.E.
v~naf ~ugustus 1977 ler~ar nat~urkUnde ",an het niS5Choppelijk College
te Weert.
116
STELLINGEN
behor~nde bij het ~roefschrift van
R.M.M. SmHs
~~nOhDvenr 21 oktober 1977
.!::i:r zijn principiele .argumeflten a,an te voeren tegen he:t berekenen uit de
10)0"1,, e'18:r:giobalans va.() de axiale elektrische vcldst"rkt_c in CCn '0""'onLl"ding.
Dit proefeoh7'"ft, hoofd9tuk 2.
In 0~tl gocont:r:-aheerde ontlading van gemi:ltigde: druk en stroom (voor neon:
10 ~ POR ~ 100 tor.>- m, 5 < i/j~ <: 20 Aim) zijn de elekt:r:ommverHezen op
de as van de ontlading ten gevo1ge v~n ambipolai:r:e difrus1c en vo,ume-re
coml::linatie van v,,:r:gelijkbare ordc van 'Oroatte_
Dit proefeohY'"ft, hOOfd$-tl<k 4.
De ioni::;atie:golven in een e:delg-8.sontlad,1,.n9 van gematigde druk leiden BOD
ffi<2:chanisID!3 in dat tot cont~~ctie van de po.!;:itieve kolom voert.
J)it p>'O$f$dwift, hoofdfltll-i< 4.
4
Benadering van het gastemperutuurprofiel in €en ontlading van 'Oematigde
druk door cen par"bool of een Lorentzprofiel kan ontoelaatbar" afwijkin
gen ten opz1chte van de numedek bepaalde op10ssing Vl1n de Helle:r:
Elenbaasvergelijking geven. -DeZe atwijkin'Oen lH,bben ten gevolge dat ook
het elektrQnendichtheid5profiel en de axiale elektrische veldsterkte
Verkeerd worden berekend.
5
n" t geb>:uik van de kli'seieke lo'Oica o.oor Wi ttgenstein, Pclppe):" C. S. ter
bestrijding van huns inziens onjuiste theorieen is <I<lnvechtllaar op g.-ond
van de niet in term"n van de klassieke logica forrnali5ee:r:bare k~antum
mech~Tlic", die zijn bruikba"rheid al£ fysische theorl,e bewezen heeft.
L. Wittqens~rdl1, "Tl"'actatus Loqiao-PhitosophiC'lU!".
X. POpp~J::~;, "'What i8 d-iaZ8ctic[;u.
P. Mitt~Z$ta~dt, "PhiZosopnisahe Probleme. de."" mode!'",$", Physi/(".
M. JamJ'Ml"', "The rid Losophlj of quantwn meaham:aa ".
6
"ndie~ kort~evende radio-i6otop~n belangrijk worden veer roedische toe
pa~singen, is een samenwerkingsverband tussen regiOnale oyclotronlabo
ratoria nodiq om te voldoen aan lever1ngsgaranties.
"I
Bij d~ huidige stand van de rekenmdchine-t~chniek i~ het zinvol om het
gebruik van analogonmethoden die duur en omalachtig zijn, af te wegen
tegen een numerieke ~imulatie.
8
De recente ontwikkeling in ~omputer on-line technieken maakt het nood
zakelijk dat studenten in deze technieken steeds meer training krijgen.
Hierlllee dient in deonde,rwijsbegroting rek~ning gehouden te worden.
9
Het i5 wen5elijk om elektronische zakrekenmachines op middelbare school
el<amerts tOe te laten; tevens zou dIm echter het onderwerp "numerieke
schattingsmethoden" in het wiskundepakket verpHcht moeten wordEn behan~
de~d.
10
Aanwezigheid van bewaakte gratis rijwielstall~ngen in de steden heeft als
vcordBsl. dat rijwieLdief"tal wordt bemoeilijkt, zo<;l.at het rijwielgeJy<uLk
wordt bevorderd.