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Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation.

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Annalen der Physik. 7. Folge, Band 46, Heft 1, 1989, S. 21-40
VEB J. $. Barth, Leipzig
Electron Kinetics with Attachment and Ionization
from Higher Order Solutions of Boltzmann's Equation
By R. WINKLER*,G. L. BRAGLIA**,
J. WILHELM*
*) Zentralinstitut fur Elektronenphysik, Greifswald, DDR
**) Dipartimento di Fisica, Universith di Parma, Itnlia
A b s t r a c t . An appropriate approach is presented for solving the Boltzmann equation for electron
swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization
and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved a t any even truncation-order.
The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model
plasma. and a plasma of pure SF,.
The calculations have been performed for increasing approximation-orders, up to the converged
solution of the problem.
I n particular, the transition from dominant attachment to prevailing ionization when increasing
the field strength has been studied.
Finally tho establishment of the hydrodynamic stage for a selected case in the model plasma has
been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation.
Elektronenkinetik mit Anlagerung und Ionisation aus Losungen
der Boltamann-Gleiehung in hoherer Ordnung
I n h a l t s u b e r s i c h t . Ein geeignetes Verfahren zur Losung der Boltzmann-Gleichung fur Elektrononschwarme und fur nichtstationare, schwachionisierte Plasmen im hydrodynamischen Zustand
wird vorgestellt, welches 1onisa.tions- und Anlagerungsprozesse mit enthalt. Unter Verwendung einer
Legendre-Polynom Entwicklung der Elektronen-Geschwindigkeitsverteilungsfunktionwurde das
resultierende Eigenwertproblem in beliebiger gerader Abbruchordnung gelost. Diese Technik fand
zur Untersuchung der Geschwindigkeitsverteilung, der mittleren StoOfrequenzen, der Energieubertragungsraten? des nichtstationaren Verhaltens und der Energiebilanz im hydrodynamischen Zustand
von Elektronen in einem Modellplasma und in reinem SF, Verwendung.
Die Berechnhgen erfolgten fur zunehmende Approximationsordnungen bis zur konvergenten
Losung des Problems.
Insbesondere wurdc der mit wachsender Feldstarke erfolgende Ubergang von dominierender
Anlagorung zu vorherrschender Ionisation studiert.
SchlieSlich erfolgte die Untersuchung der Einstellung des hydrodynamischen Zustandes fur einen
ausgewahlten Fall im Pllodellplasma, indem die nichtstationare, raumlich homogene Boltzmanngleichung in 2-Termnaherung gelost wurde.
22
Ann. Physik Leipzig 46 (1989) 1
1. Int8rodnction
Recently a new approach has been developed t o determine the solution of the
Boltzmann equation and relevant macroscopic quantities for electron swarms and
stationary weakly ionized plasma,s a t high orders of approximation [l,21. This approach
has been applied to study the electron behaviour in different model, inert and molecular
gases. These investigations were limited t o conservative inelastic collisions [I- 61.
A still open problem concerns the study of the additional impact of non-conservative
processes, such as ionization and attachment. Some efforts in this direction have recently
been made for model gases [7, 81 and molecular nitrogen [9], considering mainly the
isolated effect of attachment a t very low fields or ionization a t relat'ively high values
of EIN.
The present paper deals with the extension of our just mentioned t,echniclue t o
include non-conservative inelastic processes, i. e. ionization and attachment. Using this
extended approach, the electron kinetics in a model gas (with attachment and ionization)
and in pure SF, are studied and the results properly interpreted. The main interest of
our investigation is for spatially integrated properties of electron swarms and electron
kinetic properties of nonstationary spatially homogeneous plasmas in hydrodxnamic
stage. I n particular, results relevant to mean collision frequencies and corresponding
energy transfer rates are reported and discussed. This is made for increasing approximation-orders, beyond the conventional two-term approximation, and for a large range of
(time-independent) reduced-field strength BIN. However, the nonstationary behaviour
and the power balance of the electrons is also considered in detail.
I n addition, the relaxation of the velocity distribution function to the hydrodynamic
stage is studied for the mentioned same model plasma, having recourse to the solution
of the nonstationary, spatially homogeneous Boltzrnann equation in two-term approximation. Starting from different initial distributions, the same steady-state velocity
distribution and corresponding macroscopic quantities are obtained as found when
solving the kinetic equations in hydrodynamic stage in two-term approximation which,
in turn, also confirms the validity of our new, more general and complex solution-technique relative to the hydrodynamic stage.
2. Theoretical Background
Our starting point is the Boltzmann equation
aF
- + v . V,F
at
e
+ Ck C i p ( F )+ Ci(F) + C u ( F ) ,
- A E . V,F = Cez(F)
m
for electron swarms, or its spatially uniform version for nonstationary plasmas. Here,
F ( r ,u,t ) is the electron velocity distribution function, E a constant electric field, fie(., t )
the electron density, e, and m the charge magnitude and the mass of the electrons
Moreover, Cel,C i p , Ciand Ca are the collision integrals for elastic collisions, conservative
inelastic collisions, ionization and attachment, respectively.
Substitution of the expansion [ 101
F(r, V , t ) = F(")(v)n,
+ F("(v) . (-V,.) n, + .. .
(2)
of the velocity distribution function (for the hydrodynamic stage) with respect to the
gradient of the electron density into the Boltzmann equation, after integration over the
R. WINKLERe t al.. Electron Kinetics with Sttnchment and Ionization
velocity space yields the continuity equation [ 101
-
arL,
_ - vn, - [jwFf")dv
at
+ J C'(F(l))dv + jC'"(F('')dv]
+ ... = 0 .
X(-V,)
n,
-
y'", yc = J C'(F( 0 )) dw,
Here
-
y IVL -
Fa ==
- J C'&(F("))
dw ,
where G' ancl P are the mean collisioii frequencies for ionization and attachment.
i\'hen substituting (2) antl (3) into ( I ) , the following equation
-
' O E. V,F(O)= Cel(F(0))+ 2 c y ( ~ ( 0 ) )
m
(5)
!
i
!
+ Ci(j7CO))+ C'U(J'(0')
-
FpfO)
for F('), with the natural normalization jFC0)dv = 1, is obtained. Ff")is the velocity
tlistribritioii function of spatially integrated s ~ a r i n sor nonstationary, spatially homogeneous plasmas.
For an electric field E = 73e, the (Iistribution fiinctioii B""' lias the particular velocity depentlence F(")(v,vJv) (cf. [ 21) and can thus br given in the Legeiitlre-polyiiornjals
espaiision form
21-1
F(O)(v,% / V )
=
-
2: f n ( v ) P,(v,/v),
n-0
(ti)
which IS written here for an even number 21 of terms. This representation of F("),as a
generalization of eqs. (18) antl (19) of [a]due t o the inclusion of ionizatioii antl attachment, leads to the ordinary differential equation system
m
with the resultant normalization condition 4n
v2fo
0
dv
= 1 for the fimction
f,.
This
differential equation system contains additional difference terms. It is of order 21 a i d
involves the eigen-value GIN,
Here, &el, &!", Qt and &" are the total collisioii cross sections relewnt to the different
collision processes, whereas mv;/2 and mv:/2 are the energy losses In conservattve-inelastic antl ionizing collisions. SnO is the Kroneckei- symbol.
The neutrals, of mass M and density N , are assumed t o be a t rest. Furthermore,
hierarchy (7) is obtained by an additional espansion of the collision integrals with respect
t o the mass ratio mlM antl by truncating the expansion after the leading term, for each
collision process and each equation of the hierarchy.
24
Anh. Physik Leipzig 46 (1989) 1
Isotropic scattering in all of the collision processes has been assumed. Moreover,
for each ionization event, it is supposed that the two electroils after the collision have the
same kinetic energy.
A transformation from speed 71 t o energy U = ( m / 2 ) u 2 in (7), finally yields the hierarchy of equatiorls
3
5 n 5 21 - 1,
fel
=
0,
with the normalization condition
I’
Lpi2
f
o( u) d U = 1,
0
where
The reduced quantities in (8) are
Kote that detailed expressions of the mean frequencies P / N and ;“IN .for ionization and
attachment can be obtained from (4) using expansion (6). They are given by
and ill determined by the isotropic part f o ( U ) of bhe velocity distribution function
F(”’by averaging over the energy space. Note also that 6 = 2m/M while 1J;’)= ( m / 2 ) $ ,
r“ = (m/2)vf are the energy losses in coiisrrvative-iaelastic and ionization processes.
To determine the velocity distri1)ntion P(O)(w)at a given (even) order of approrimatioii with 22 terms, hierarchy (8) (inrolving 21 equations) has t o be solved for given cross
sections &ez, Qi7’, &“ and &“ and given values of BIN, taking into account the normalization condition (9). However, the solution requires the simultaneous calculation of the
R. WIKKLERet al., Electron Kinetics with Attachment and Ionization
25
reclucect frequency GIN of nonconservative inelastic processes, which is involved in (8).
Since GIN depends on the same solution fo, strictly spiking, the solution of hierarchy (8)
is a nonlinear problem.
Furthermore, appropriate averaging of kinetic equation ( 5 ) for F(O)over the velocity space yields the particle-number and power-balance equations
for electroD swarms, or nonstationary plasmas, with the normalized total mean power
loss
@IN
=
l?/N
2 @IN
f
k
+ $IN + @ I N .
(13)
Here, Ge is the spatially integrated
electron density for swarms or the electron density
for homogeneous plasmas. U is the mean electron energy, P the mean power input
from the electric field and U"l, @,?l and? l are the mean energy losses produced by
the different collision processes. All the mean quantities in (12) and (13) are determined
by the first two terms f o and f l of the Legendre polynomial expansion (6) according t o
the equations
u
do
=
j-
fo(U)d U ,
u3/2
0
lj2
P
N
-=
-
-L(A)
3 m
2
Uel
N
N
-
Ua
eoE
F,
0
00
M o
Ui1)= q I J -,
N
-
U f l ( U )d U .
2 - j" U2QT U)f&U )
F- = ( G )
3
_
-
w
j"
N
3
U&iP(U ) fo( U ) d U ,
0
;i
Ui-,
N
7=
):(
112
m
U2Qa(U)f,,(U)d U .
(14)
0
As one can see from ( l a ) , larger deviations from the stationary behaviours of particle
number and power balance in hydrodynamic stage can be expected if the mean collision
frequency of ionization becomes remarkly different from that of attachment or if
only one of these processes occurs.
3. Blain Features of the Solution Technique
The detailed solution procedure of hierarchy (S), without the inclusion of nonconservative inelastic processes, has been presented in [2 , I]. Now, this procedure
must be extended to include ionization and attachment. Of course important features
of the former approach will remain unchanged.
The basic ideas of our former procedure for solving the simplified version of hierarchy
(8) at an arbitrary even order 21 of approximation are the following. The linear system
of d l ordinary differential equations constitutes a weakly singular system a t small
electron energies and a strongly singular system a t large electron energies, with U = 0
Ann. Pliysik Leipzig 46 (1989) 1
26
and U = 00 as singular points. This can be shown with appropriate power series representations of @ ( U ) a t small U , and QiP(U) immediately above the thresholds U;lr of
(conservative) inelastic processes and with asymptotic series presentations of these
same cross sections a t large U .
Separate considerations on the structure of the general solution of the hierarchy in
the region of small and large energies lead to conclude that its general solution contains
I nonsingular and 1 singular fundamental solutions in both energy regions. The physically
relevant solution can be (uniquely) determined by the nonsingular part of the general
solution (NSPGS) a t low and high energies, by continuous connection of these NSPGS's
a t a n appropriate connection point U,: and by the additional normalization according
to condition (9). Finally, the distribution function 3"") a t a n even-order of approximation
(2Z) is numerically fourid by an Z-fold backward integration of the appropriate simplified version of hierarchy (8) down t o U, (starting from a sufficiently large U,) and by
a 22-fold forward integration up to U , (starting from U = 0). Special measures must be
undertaken to preserve the linear independence of the contributions t o the NSPGS,
particularly during the 2-fold backward integration, and to start the 21 forward integrations directly a t the singular point U = 0.
Now, in the generalized case of interest here, the solution of hierarchy (8) is a nonlinear problem. However, it can be dealt with as an eigen-value problem. The proper
eigen-value i;/N has to be self-consistently found, which can be done in an interative
and solve hierarchy (8), with normaliway. To this'end, we start, with an estimate
zation (9), using this estimated value. This approximate solution is used to improve
( ; / N ) Pand then a further approximate solution of (8) and (9), belonging to this latter
After
estimate, is determined, which in turn yields a further improvcment of
having repeated a sufficient number of these st,eps the solution of (8) and (Y), and the
self-consistent eigen-value ?;/AT,can be obtained. However, t o be able to make such steps,
an extension of our previous procedure of solution is necessary, in order to include nonconservative processes.
Special aspects of this extension, with respect to the procedure detailed in [2, I],
are the following.
As obvious from hierarchy (8), the term containing Y/N involves the square root
of the energy, which leads to a somewhat different structure of (8) when compared to
the former hierarchy (18) of [2, I]. Despite this fact, if hierarchy (8) is properly transformed, i.e. we pass from the electron energy U to the square root U1ls as independent
variable (i.e. to the speed v), for a given value of (Y/N)p,it is found t o be again a weakly
singular system a t small energies and a strongly singular system a t large energies, with
U = 0 and U = co as singular points, if appropriate power and asymptotic series presentation of &"(U) and &i(U)are assumed. Thus it is further found that the general
solution of (8) in the regions of small and large energies contains I nonsingular and 1
singular fundamental solutions as in the simpler case. The nonsingular solutions must
be isolated to construct the NSPGS in both energy regions and, finally to determine the
hy the same
approximate iionsingular solution of (a), for any given estimate of
procedure detailed in [2] for the simpler case. In order t o determine the 1 nonsingular
fundamental solutions f ( m,j ) , 1 5 j 5 I, a t large energies by the mentioned backward
integration from U,, a t this particular enwgy and, repeatedly, a t appropriate lower
energies U* down to U , an approximate representation f (') of the fundamental solutions
of (8) must be calculated using ansatz (42) of [2]. To t,his end, the nontrivial solutions
of the linear and homogeneous equation system (43) of [2] for the expansion coefficients
of the approximate funtlamenta.1 solution f(')has to be determined after having
raplaced in (43) the quantity q( U*) by q( U*) $( U*) @( U*)
( U*)-1/2 .i;/lv.These
approximate fundamental solutions are used as initial values for the backward integra-
+
+
+
27
R. WINKLERe t nl., Electron Kinctics with Attachment and Ionization
tion and in order to preserve the linear independence in the calculation of the nonsinguU ) a t large energies can be
lar fundamental solutions f(”,j). Then, the NSPUS f(“)(
obtained by a superposition of these fundamental solutions according to (45) of [2].
I n order to start the forward integration a t the singular point U = 0, power series
presentations of the Z nonsingular fundamental solutions f(‘)(
U ) and of the I particular
solutions f ( ” j ) ( U ) near U = 0 are used. The particular solutions result from the difference terms in the lowest equation of hierarchy (8), containing fo(U U i p ) and
fo(2U
Ui).
These difference terms are already known when performing the calculat’ion of the NSPGS a t high energies, i.e. from U , down t o the connection point U , 5
Nin(U;[?’,Ui),
and are determined by the functions fO(“*i)(U
Ui7’) and fc,(”*7)(2U Ui).
Thus, they can be dealt with as known inhomogeneities of the differential equation system. Instead of (52) and ( 5 3 ) of [a] the initial part of the nonsingular fundamental
solutions near U = 0, now can be represented as
+
+
+
+
+f ( r ) W +p u + O(U)],
f ‘ ” ( U ) == Ur[@
(15)
with r = 0, . . . , 1 - 1, where the components of the coefficients
are recursively given by the equations
gd+l = O
forO<nI:l-
jp = 1, f g =
4n
1,
+
2n - 1 4n
1 2 r ~- 2r - 2
4 ~. 3 272 2% 1 2r
~
~
+3
+ +
2
jg_.1 = 2 n + 1 2 r + 2 n + 3
-
2n
~
for n = I,
-
Pcr),f ( r )and f(‘)
for n
(r - n
=
I, ..., I - I,
+ 1)
. . ., 1 - 1,
f$~=OforO<n<Z-
- 1
1,
for n = 1, ..., I - I ,
if)= o for r = 0, ..., z - 2,
for r = 0, ..., I - 2 and n = I., ..., 1 - 1,
2 n 2r + 2 n - t
jgo2= !?I
!
2n--12
4n
2n-Ir-n+2
for r = I
-
1 and n
=
1 - 1, -.., 1.
+1
3
(16)
Ann. Physik Leipzig 46 (1989) 1
28
I n (16) the expansions
Q“(U) =
5@U.,
m -
Q“(TJ) =
v=o
3 QZU.
v=o
have been used for small energies so that the quantities p , and q$ are given by
p , = @/(eoE/N) and & = @ / ( e o E / N ) .
Concerning the power series representation of the particular solutions at small U , the
expansions of the cross sections
C Q -
Qp(
U )=
2 Qiy,,(U - lJif’)”,Q ( U ) = 2 Qf( U - Ui)”,
v=l
”=l
+
+
as well as the expansions of the functions fo(*yi)(U
UiP) and f,,(”J)(2U U i ) are
necessary, in analogy to (56) of [ 2 ] . Then, the initial part of the particular solutions can
be given the forin
f (Pd(
u)=
j%?b7)
+ j(~,i)u1/2
+ J’(P,~)U
+ j%,i)U3/2 + O( ,573/2),
I l j l l ,
with
j ~ ~ v=
i )
- =0
fp,j)
=
(18)
for 0
4n+3
n
2 r ~ - 3
-
2n+14n-l
for
YL
=
< n -< 21 - 1 ,
_
n+2
1 - 1, ..., 1.
~
for n = I,
.. ., I - 1,
(19)
Prolongation of the initial parts o f f ” ) and f(I’J), by forward integration of ( 8 ) , u p to
the connection point U, yields the NSPUS f (O)( U ) a t small energies by superposition
of the nonsingular fundamental solutions and the particular solutions, according to
(60) of [2].
The physically relevant solution in the simpler case detailed in [ 2 ] is found by a
continuous connection of the NSPGS’s f P) and f (O)at U , and a concluding normalization.
Because of a special property of the lowest equation of hierarchy (18) of [ 2 ] the requirement of continuous connection a t U , of the components of the NSPGS’s, with the excepautomatically ellforces the continuous connection
tion of t’hat belonging to f‘,oo) and
also of the latter. Thus, by this procedure, one of the 21 superposition constants
remains free and can be determined when applying the normalization condition (21)
of [ a ] , which is identical with condition (9) of this paper.
I n the generalized case considered here, the connection off (w) and f (O)mustbe made
in the same way, however with functions f(”)and f ( O ) obtained with the estimated
( G / N ) 2 , , which does not lead t o an automatical continuity of f{”)(U,) with f{o)(U,),
fy),
R. WINKLFRet al., Electron Kinetics with Attachment and Ionization
29
and thus t o the desired physical solution. This can be seen when integrating (analytically)
the lowest equation of hierarchy (8) over both NSPG’s, i.e. from 0 to U, and from
Uc t o 00, and when taking the sum of the two resulting relations. This leads to write
that
When performing the continuous connection and the normalization in the mentioned
way, the resultant function systems, which are uniquely determined by special superpositions from both NSPGS’s, satisfy the reduced relation
+ j- Uqif6”’ dU
Ufjb”) dU );(
1
Do
3uc [fP’(Uc)- f!”’tUc)l
-
Ui
7
UqajiO)
dU -
0
(21)
=
0.
P
UC
These continuously connected and normalized function systems, in (21) are denoted by
the same symbols as before for simplicity. Equations (10) and (11) make obvious that,
when the estimated (G/N)pconverges to its self-consistent value i / N , again the automatiU,) and flW)(
U,) is established and the physically relevant solution
cal continuity of
is found.
Returning to the remark given above on the iterative improvement of a n estimate
(Y/N), of G/N, eq. (21) indicates the possibility of an iterative control of (G/N)P.When
the parameter ( G / L V )used
~
in (8) approaches the self-consistent eigen-value G/N2the simple
equation
/I0)(
I
G/N - (Y/N)p= 0
(22)
must be satisfied. Solving hierarchy (8) for the estimated (Y/N), and calculating the
corresponding value of G/N with the resulting solution according to (10) and (ll),the
left-hand part of (22) will be non-zero. However, the calculated value of G/N can be
considered to be a function of the estimate (Y/N), used for the solution. Therefore, an
improved estimate (;/N)JJcan be found locking (for example with the Newton method)
for a n improved value of the zero point of equation (22) which is then used for the next
iterative step, i.e. the next solution of (8),as mentioned above. I n this way, after some
steps, the physical relevant solution together with the self-consistent eigen-value can
be obtained.
4. Results and Discussion
The extended technique has been applied to investigate velocity distribution and
resulting macroscopic quantities of electrons in an appropriate model gas and in pure
SB’, .
A range of field values has been considered for which either ionization or attachment,
if not both of these non-conservative processes, contribute to the eigen-value GIN.
The results are calculated for increasing approximation orders (2Z).
30
Ann. Physik Leipzig 46 (1989) 1
4.1. Model Gas
For this first analysis a former model gas, already treated in [I, 21, is extended to
take into account of ionization and attachment. The model includes elastic collisions
and single excitation, ionization and attachment processes. The relevant cross sections
&ez(U)=6.10-16cm2, U > O e V ,
0, u 5 uy,
Q?'(U) = &(U - U ; p ) / A U ,
i
&',
U
U51)
2 Uip + A U ,
U(;p= 1 eV, A U
=
2U5
Uc,l,
+AU,
-
0.2 eV, Q;??= 4 . 10-l6 em2,
are presented in Big. 1.
Furthermore, a mass of four atomic mass units (S = 2 m/M = 2.744. lo-*) is used for
the gas particles. Such a model gas shows the typical features of a n electronegative gas,
namely large attachment at low electron energies, excitation a t intermediate energies and
ionization a t the higher energies. First results for this model have already been reported
in [ 1t]. Table 1reports the mean electron energy 6and the normalized collision frequencies for excitation, ionization and attachment for different EIN values and increasing
approximation orders 21. As obvious, a t low EIN attachment and a t high E / N ionization
dominate in the particle balance (12) and cause a large temporal decrease or increase
of the total electron number in the swarm or of the electron density in the homogeneous
plasnia The change of the dominance of the two non-conservative processes occurs for
a field strength E / N between 200 and 300 Td.
10 7
model
Ir/Qa
0
5
I ;I
b'ig. 1. Collision cross sections for the model gas
dy
Pig. 2. Distributions fo and f, for the model gas as obtained with %term and 8-term approximation
at 50 Td
model
EIN =300 Td
- 8-term
10”
Fig. 3. Distributionsf,, andfi for the model gas as obtained with 2-term and 8-term approximation
a t 300 Td
Ann. Physik Leipzig 46 (1989) 1
32
-
Table 1. Mean electron energy U (in eV) and mean collision frequencies (in cm3/s) for excitation
(Y;I>/N),ionization (?/N) and attachment
in the modcl gas a t different E/N's (in Td) and different approximation orders 21
~____
20
2
4
6
8
50
200
300
500
~
~
-
1.746-8
1.6 7 9 8
1.663-8
1.657-8
4.680-16
2.731-15
2.756-15
2.728-16
l.214-8
1.209-8
6.481-12
1.126-11
1.122-11
1.119-11
6.067-9
6.230-9
6.218-9
6.206-9
6.140-10
6.@29-10
li.6"-10
6.627-10
2.219-9
2.416-9
2.418-9
2.415-9
6.715-8
5.640-8
3.233-9
2.261-9
2.260-9
1.324-9
1.460-9
1.464-9
1.462-9
6.559-9
6.612-9
6.514-9
c1.514-9
7.520-10
8.214-10
8.225-10
5.213-10
6.997-9
6.718-9
6.676-9
6.667-9
1.570-8
6
8
1.236
1.245
1.243
1 .242
2
4
6
8
2.354
2.396
2.395
2.393
2.918-8
2.921-8
2.917-8
2
4
6
4.761-8
8
4.978
4.936
4.937
4.936
2
6.873
4
6
8
6.770
6.771
6.771
5.640-8
2
4
9.636
9.476
6.852-8
6
9.473
Y
9.473
3
1
100
0.7473
0.7365
0.7377
0.7388
1.672-8
1.566-8
1.5V8-8
3.916-8
4.iO7-8
4.706-8
4.706-8
5.639-8
6.i67-8
6.766-8
6.766-8
2.260-9
~
1.204-8
1.201-8
When passing froin the two-term to the eight-term approximation, the normalized
attachment frequency iP/X changes up t o about 10% in the EIN range we hare considered. The normalized ionization frequency PIN, however, changes froin less than 1 94
a t 500 Td to more than 500% at 50 Ttl when increasing the approsirnation order. All
the quantities show a good convergence a t high EIN when increasing the value of 21
and a satisfactory convergence a t the lowest values of EIN. The modification of these
average quantities result from the alteration of the isotropic distribution function f o
with the (increasing) approximation order a t different EIN. Pigs. 2 and 3 show the behaviour of f o and f l as a kuunction the electron energy U , for the 2- and 8-term approximation
a t 50 and 300 Td. I n correspondence of the lower field strength, the ionization region
belongs t o the high energy tail of the distribution, where the largest increase of f o occurs
when passing to higher orders of approximation. This leads to the large increase of
Y,JLVwe have just mentioned. At higher fields, the ionization mainly occurs in the body
of the distribution. This part of the distribution, however, remains nearly unchanged
when increasing 21. Then only small changes of the mean ionization frequency will
result a t high E / X . Since, generally, the electrons in the body of the distribution mainly
contribute to the mean collision frequencies of attachment and excitation and to the
R. WINKLERe t a]., Electron Kinetics with Attachment and Ionization
33
mean electron energy, the smaller variations observed for these quantities are immediately understood. The alteration of the first contribution ( f l ) to the anisotropy of the
velocity distribution, is very similar to that of fo, a t not too small electron energies, when
passing from 2-term to 8-term approximation.
A comparison of these results with those relevant to the model gas with only conservative collision processes [ 1, 21, indicates that the aclditional presence of ionization and
attachment does not change substantially the convergence behaviour of the distributions
fo and fl and the resulting macroscopic quantities. This is due to two main reasons. J n
the ionization region the exciting collisions are also acting so that an additional ionization
only enlarges, to some extent, the effectiveness of the inelastic collision processes in this
energy region. On the other hand, attachment presents a n intense inelastic process in
the model a t small energies which, however, occurs only in a very limited energy range
and with a relatively small energy loss per collision. Therefore, for any approximation
order, this process has a certain impact on the structure of the distribution a t small
energies (particularly for small fields) but does not lead to remarkable corrections to
the distribution when passing from the %term approximation t o higher order treatments,
which can be immediately seen from Figs. 2 and 3.
Table 2. Mean power input PIN, total power loss in collisions U'IN and individual power losses
+'IN, *IN, $/A' and S I N (in eV cn13/s) for the model gas at different E / Y s (in Td) and different
approximation orders 2Z
ElX
10
50
PIN
$IN
$IN
Vy/N
+/s
7 3 9 4 - 12
7.278-12
7.289-12
7.300-12
6.997-9
6.718-9
-
6.67 6-9
-
(i.667-9
-
200
300
500
-
cap
7.634- 9
7.211-9
7.1.51-9
7.134-9
2
1.604-9
4
1.576-9
(i
1.673-9
8
1.571-9
1.464-8
1.394-8
1.383-8
1.381-8
1
4
6
6.853-9
6.669-9
6.671-9
6.67 1-9
2.185-8
2.171-8
2.163-8
2.157-5
1.570-11
1.604-11
1.599-11
1.697-11
1.579-8
1.573-8
1.566-8
1.562-8
4.f;80-15
2.721-14
2.756-14
2.728-14
1.8194
1.769-8
1.770-8
1.770-8
3.2433-8
3.269-8
3.254:8
3.261-5
4.21'1-11
4.378-11
4.373-11
4.368-11
"918-8
2.921-8
3.917-8
"916-8
6.481-11
1.125-10
1.122-10
1.119-10
3.147-9
3 ".)0-9
3.2149
4.708-8
6.605-8
6.510-8
8
4.646-8
4.644-8
4.645-8
5.509-8
1.285-10
1.281-10
1.281-10
1.281-10
4.761-8
4.707-8
4.706-8
4.706-8
6.140-9
6.619-9
6.G29-9
6.627-9
1.174-9
1.275-9
1.277-9
1.276-9
>
4
6
8
8.647-8
8.640-8
8.538-8
8.539-8
8.029-8
7.998-8
7.999-8
7.997-8
2.069-10
2.040-10
2.041-10
2.041-10
5.716-8
3
_._&
"93-8
6.640-8
7.047-10
7.756-10
6.640-8
5.639-8
2.260-8
2.261-8
2.260-8
7.781-10
2
1.908-7
1.875-7
1.875-7
1.875-7
1.349-7
1.336-7
1.336-7
1.336-7
3.415-10
3.358-10
3.358-10
3.368-10
6.852-8
6.767-8
6.766-8
6.766-8
6.559-8
6.512-8
6.514-8
6.514-8
4.017-10
4.385-10
4.395-10
4.392-10
s
100
-
21
2
4
6
8
>
4
6
i
4
6
8
5.509-8
6.044-9
3.97 7 - 9
5.950-9
5.937-9
1
.id
3.208-9
7.784-10
34
Ann. Physik Leipzig 46 (1989) 1
The energetic aspect is analysed in Table 2 where the (normalized) power input
PIN from the electric field and the (normalized) power loss in elastic (i5?/N), exciting
( P I N ) ,ionizing ( @ I N ) and attaching ( P I N ) collisions are reported together with the
total power loss in collisions @/AT given by (13). As one can see, a t low EIN the power
loss in collisions is remarkably larger than the power input from the electric field whilst,
a t high BIN, the situation is reversed. This clearly indicates that, due t o the action of
attachment and ionization, in the power balance (12) for the hydrodynamic stage a
pronounced nonstationary behaviour, i.e. large deviations from the stationary situation
with PIN = @IN, occurs. The power, a t low BIN, is mainly dissipated in attachment
and excitation processes. When increasing EIN the loss by excitation and ionization
becomes dominant and the powei input becomes larger than that lost in collisions.
In addition, it is worth noting that the normalized power losses @IN and @IN
clue to attachment and ionization depend, as for Ga/N aiitl ?IN, on the approximation
order and the corresponding changes of power input PIN and total power loss @IN
in collisions amounts to some per cent.
4.2. Pure SF6
\Ve will report now the results of some calciilations for electrons in SF,, a gas which
is of great importance from the technological point of view. This gas presents a pronounced energy dependence of some cross sections and, particularly, a very intense attachment process a t low electron energy. Fig. 1 reports several collision cross sections,
Fig. 4. Collision cross scctions for SF,
R. WINKLER
et al., Electron Kinetics with Attachment and Ionization
35
as proposed in [12]. The figure reports the cross section for momentum transfer in elastic
collisions (&“I), the total cross sections for lumped vibrational excitation (Q:P), lumped
electronic excitation (QgP), ionization (@) and for the two main attachment processes
(QT, Qg), leading t o the production of SF, and SF, ions.
The energy losses per vibrational excitation, electronic excitation and ionization
are 0.0954, 9.8 and 15.8 eV, respectively. A further attachment process leading to production of F- ions is neglected since the corresponding cross section (Qf) is very small.
Tables 3 and 4 report some results relevant t o mean energy, collision frequencies, power
input and power-loss rates for SF, a t different field strengths and for increasing approximation orders.
Table 3. Mean electron energy (in eV) and mean collision frequencies (in cm3/s) for electrons in
SF, at different E / N s (in Td) and different approximation orders 21
-
Yy/N
E/N
21
U
70
2
4
6
5.07
5.05
5.05
1.47-8
1.48-8
1.48-8
5.64-10
5.60-10
5.60-10
1.29-13
1.50-13
1.49-13
5.74-10
5.67-10
5.66-10
4.77-11
5.10- 11
5.14- 11
100
2
4
6
5.35
5.32
5.31
1.35-8
1.38-8
1.38-8
9.80-10
9.67-10
9.65-10
1.55-12
1.74-12
1.73-12
5.76-10
5.53-10
5.50-10
3.74-11
4.06-11
4.11- 11
300
2
4
6
8
7.53
7.47
7.46
7.46
8.60-9
8.78-9
8.82-9
8.83-9
5.27-9
5.14-9
5.13-9
5.13-9
3.98-10
4.15-10
4.14-10
4.14-10
6.80-10
5.95-10
5.72-10
5.63-10
2.19-11
2.18-11
2.21-11
2.22-11
500
2
4
6
8
9.45
9.35
9.34
9.34
6.52-9
6.63-9
6.64-9
6.66-9
9.13-9
8.85-9
8.84-9
8.84-9
2.05-9
2.09-9
2.09-9
2.09-9
6.08-10
5.50-10
5.23-10
5.11-10
1.69-11
1.65-11
1.63-11
1.64-11
The specific effects found in the model case in the particle and power balance, occur
also in SF,. So, a t low EIN, the first attachment process dominates in the particle balance
and the corresponding total power loss in collisions is remarkably larger than the power
input from the electric field. However, a t large fields the ionization process dominates
in the particle balance and the power input becomes larger than the total power lost
in collisions. The transition between these two situations occurs now a t somewhat larger
fields. The corrections to the collision and power transfer rates produced by increasing
approximation orders are relatively small and amounts t o less than 20% in the EIN
range considered here.
Our previous studies [ 111 with only conservative collisions have shown that larger
corrections to the two-term approximation are mainly observed when the total cross
section for inelastic collisions becomes comparable or even larger than that for elastic
collisions over larger energy regions. As one can deduce from Fig. 4, in SF, such a situation occurs in the region of SF- roduction by attachment and of vibrational excitation,
6 p.
but only in a narrow energy region below 1eV. This explains the relatively small corrections introduced by higher order calculations in SF,. I n addition, it should be mentiolied that, due t o the large maximum value and the strong energy dependence of some
cross sections, the accurate numerical solution of the eigen-value problem in pure SF,
is not a t all simple.
36
Ann. Physik Leipzig 46 (1989) 1
Table 4. Power input, total power loss in collisions and individual power losses (in eV cm3/s)for the
different collision processes in S F , a t various E / N (in Td) and different approximation orders 21
-
-
-
UgPIN
$IN
UYIN
UZ/N
3.72-12
3.71-12
3.71-12
1.40-9
1.41-9
1.42-9
5.53-9
5.49-9
5.49-9
2.04-12
2.36-12
2.36-12
1.25-10
1.24-10
1.24-10
2.58-11
2.75-11
2.76-11
1.11-8
1.10-8
1.10-8
4.06-12
4.03-12
4.03-12
1.29-9
1.31-9
1.32-9
9.60-9
9.47-9
9.46-9
2.45-11
2.74-11
2.74-11
1.22-10
1.18-10
1.17-10
2.02-11
2.20-11
2.22-11
8
5.66-8
5.64-8
5.65-8
5.65-8
5.89-8
5.79-8
5.78-8
5.78-8
7.19-12
7.11-12
7.10-12
7.10-12
8.21-10
8.37-10
8.41-10
8.43-10
5.16-8
5.04-8
5.03-8
5.03-8
6.29-9
6.56-9
6.54-9
6.54-9
1.34-10
1.18-10
1.14-10
1.12-10
1.16-11
1.15-11
1.17-11
1.18-11
2
4
6
8
1.36-7
1.35-7
1.35-7
1.35-7
1.23-7
1.20-7
1.20-7
1.20-7
1.04-11
1.03-11
1.03-11
1.03-11
6.22-10
6.33-10
6.34-10
6.35-10
8.94-8
8.67-8
8.66-8
8.66-8
3.24-8
3.30-8
3.30-8
3.30-8
1.18-10
1.07-10
1.02-10
9.98-11
8.72-12
8.59-12
8.56-12
8.60-12
21
PIN
?IN
70
2
4
6
3.94-9
3.95-9
3.95-9
7.09-9
7.06-9
7.07-9
100
2
4
6
7.79-9
7.81-9
7.82-9
300
2
4
6
500
-
U!PIN
E/N
5. Relaxation to the Hydrodynamic Stage
To give a n example of tendency t o the hydrodynamic stage and to assess the validity
of the results obtained by solving the kinetic equation relevant to the hydrodynamic
stage, we also have solved the spatially homogeneous version of the nonstationary
Boltzmann equation (1)in two-term approximation.
I n this approximation the Legendre polynomial expansion (which we write here in
terms of electron energy U and normalized time N t )
leads (after some approximation to the collision integrals as detailed in sect. 2) to two
partial differential equations with difference terms describing the temporal evolution
of the two distribution parts fo( U , N t ) and fl( U , N t ) [ 131. Using the quasistationary
approximation for fl and the solution technique discussed in [14- 161, the resultant
equation for fo(U,N t ) is numerically solved up t o the hydrodynamic stage, starting
from appropriate initial distribution, f,,( U , 0).
For the latter distribution we have assumed that
Thus the energy UM coxresponding to the maximum and the energy width A U M are
parameters of the initial distribution.
The calculations have been performed a t EIN = 300 Td for the model gas specified
in sect. 4.1, starting from two different initial distributions fo( U , 0), relevant t o the parameter combinations V M= 1eV, A U M = 1 2 eV and UM = A U M = 9 eV. Fig. 5 illustrates these initial distributions and the resultant distribution of hydrodynamic
stage (which is established a t a normalized time of about N t = 3 lo8
s) and which
completely agrees with the function fo( U ) reported in Fig. 3 and obtained in two-term
approximation. Figs. 6 and 7 present the relaxation to the hydrodynamic stage of the
~
-
lo-4 rmode(
EIN =300 Td
IO-~
2-term
F
_ _ _ UM=AUM=9 eV
I
Fig. 6. Initial distributions (-,
hydrodynamic stage
I
2-
/
I
,
- - - ) and resultant distribution (-
,
. -)
from calculations up t o the
,,Yo/"
__--,,
/ -
--+Nt[108cm~3s]
38
Ann. Physik Leipzig 46 (1989) 1
collision frequencies for excitation ($')IN, ionization (?IN) and attachment (Ga/N),
and also of the mean electron energy 0,
the power input (PIN)and the total power loss
(@/hr).Also from the macroscopic point of view, the establishment of hydrodynamic
values for the macroscopic quantities occurs in the mentioned time and yields in any
case, the same values reported in Tables 1and 2, for EIN = 300 Td and two-term approximation. Therefore, in agreement with expansion (2) relevant to the hydrodynamic
stage in a homogeneous plasma, a time independence of mean collision frequencies for
ionization and attachment, mean electron energy, power input and power loss per electron, is obtained after the relaxation period, however, in the presence of a pronounced
nonstationarity of particle number and power balance, because of the remaining temporal evolution of the electron density. This clearly also agrees with Fig. 8 where the growth
of electron density is presented up t o times which are remarkably exceeding that requi16
14
model
EIN = 300 Td
?-term
I
12
eV, AU,=12 eV
---Uu,=l
I
\
I
- - -UM=AUM=9 eV
1
\
10
a
I
I
I
I
-
6
N$lO~cm%]
0.5
1
1.5
2
2.5
Fig, 7. Temporal relaxation of mean energy, power input and total power loss in collisions
R. WINKLERet al., Electron Kinetics with Attachment and Ionization
39
red to establish the hydrodynamic stage. Note that, after the initial relaxation period,
a pure esponential growth of the density is observed in agreement with the hyclrodynamic balance equation (12).
A similar behaviour of the electron velocity distribution function has also been observed in previous studies on the temporal evolution of distribution function and macroscopic quantities, following the disturbations of the electric field strength produced
by rectangular impulses of appropriate period [ 161.
2c
1c
8
6
4
E/N=300Td
2-term
__ U,=l eV, A UM=12 eV
_ _ _ _ U,=AUM=9
2
-
eV
Nt[lO9cm3s]
1
I0
I
I
1
L
. I
I
I
2
3
Fig. 8. Temporal growth of the electron density during the relaxation to, and after the establishment
of, the hydrodynamic stage
References
[l] WINKLER,R.; BRAGLIA,
G. L.; HESS,A.; WILHELM,J.: Beitr. Plasmaphys. 24 (1984) 657.
WINKLER,R.; WILHELM,
J.; BRACLIA,
G. L.: I1 Nuovo Cimento 7 D (1986) 641.
WINKLER,R.; BRAGLIA,
G. L.; HESS,A.; WILHELM,
J.: Beitr. Plasmaphys. 25 (1985) 351.
BR~GLIA,
G. L.; WILHELM,
J.; WINKLER,R.: Lettere a1 Nuovo Cimento 44 (1985) 257.
B~acLr.4,G. L. ; W~LIIELM,
J.;J~INKLER,
R. : Lettere a1 Nuovo Cimento 44 (1985) 365.
BRAGLIA,
G. L.; WINKLER,R.; WILHELM,
J.: I1 Nuovo Cimento 7 D (1986) 681.
BLEVIN,H. A.; FLETCHER,
J.; HUNTER,
S. R.: Phys. Rev. A 31 (1985) 2215.
[2]
[3]
[4]
[5]
[6]
[7]
Ann. Physik Leipzig 46 (1989) 1
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[8] NESS,K. F.; ROBSON,
R. E.: Phys. Rev. A 34 (19%) 2185.
[9] PHELFS,
A. V.; PITCHFORD,
L. C.: Phys. Rev. A 3 1 (1985) 2932.
[lo] KUMAR,
K.; SKULLERUD,
H. R.; ROBSON,
R. E.: Aust. J. Phys. 33 (1980) 343.
[Ill WINKLER,
R.; BRAQLIA,
G. L.; WILHELM,
J.: XVII th ICPIG, Budapest 1985, Invited Papers,
pp. 22.
[12] YOSHIZAWA,
T.; SAKAI,Y.; TACASHIRA,
H.; SAKAMOTO,
S.: J. Phys. D: Appl. Phys. 12 (1979)
1839.
[13] WINKLER,R. ; DILONARDO,
M.; CAPITELLI, M. ; WILIIFLM,J.: Plasma Chem. Plasma Process. 7
(198i) 245.
[I 41 WILHELM,
J.;WINKLER,
R. : Ann. Phys. (7) 37 (1980) 35.
[16] WINKLER,
R.; WILHELM,
J.; HESS,A.: Ann. Phys. (7) 42 (1985) 53i.
[lG] WINKLER,
R.; WILHELM,J.: Comput. Phys. Commun. 20 (1980) 113.
Bei der Redaktion eingegangen am 6. Juni 1988.
Anschr. d. Verf. : Dr. R. WINKLERund Prof. Dr. J. WILHELM
Zentralinstitut fur Elektronenphpsik-Bereich 6,
Robert-Blum-Str. 8- 10
Greifswald
DDR-2200
Prof. Dr. G. L. BRAGLIA
Dipartimento di Fisica, Universita di Parma,
CIEQP - GNEQP, 43100 Parma, Itnlia
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