ADVANCES IN
r-KlNETIC 1HE0RYAND QOMPUTING
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Series on Advances in Mathematics for Applied Sciences - Vol. 22
Klli:;t;c THEORYAND OMPUTING Selected Papers
Editor
B. Perthame Universite Pierre et Marie Curie France
1II1»
World Scientific Singapore· New Jersey· London· Hong Kong
Published by World Scientific Publishing Co. Pie. Lid. P O Box 128. Fairer Road. Singapore 9128 USA office: Suite I B , 1060 Main Street. River Edge, NJ 07661 UK office: 73 Lymon Mead. Totteridge, London N20 SDH
A D V A N C E S IN K I N E T I C T H E O R Y AND C O M P U T I N G : S E L E C T E D PAPERS Copyright© 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved This book, or parts thereof, may not be reproduced in any form orby any means, electronic or mechanical includingphotocopying, recording or any information storage and retrieval system now known or to be Invented, without written permission from the Publisher.
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ISBN: 981 -02-1671-8
Printed in Singapore by Utopia Press.
V
FOREWORD
T h i s b o o k contains a collection of papers dealing w i t h the k i n e t i c theory o f gases. C o m p u t a t i o n a l a s p e c t s a r e d i s c u s s e d as w e l l as t h e a p p l i c a t i o n s a n d m o d e l i s a t i o n or macroscopic properties related to the kinetic structures.
The
u n d e r l y i n g m o d e l s a r e t h o s e o f V l a s o v - P o i s s o n o r B o l t z m a n n e q u a t i o n s as u s e d i n t h e m o d e r n s c i e n c e s ( p l a s m a p h y s i c s , s e m i c o n d u c t o r s , h y p e r s o n i c flows e t c . ) . T h e idea o f c o l l e c t i n g these papers came o u t after a one-day o r g a n i z e d b y J. M o s s i n o for physicists a n d m a t h e m a t i c i a n s
workshop
in Orleans.
Al-
t h o u g h t h e b o o k c o n t a i n s a l a r g e r n u m b e r o f p a p e r s , i t s m a i n b o d y is t h a t o f the conference
a n d I w o u l d l i k e t o t h a n k J . M o s s i n o f o r t h e t i m e she
spent
i n o r g a n i z i n g t h i s successful conference a n d M . Feix for the h o s p i t a l i t y o f t h e C N R S in Orleans.
B.
Pertkame
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vii
CONTENTS
Foreword
I.
v
Vlasov-Poisson
in P l a s m a
Physics
T h e C h i l d - L a n g m u i r Law i n the Kinetic Theory of
Charged-Particles.
P a r t 1, E l e c t r o n F l o w s i n V a c u u m P.
Degond
3
Eulerian Codes for the Vlasov E q u a t i o n M.
R. Feix,
P
Bertrand
and A.
Ghizzo
45
M a t h e m a t i c a l Models o f Ion E x t r a c t i o n a n d Plasma Sheaths S.
II.
Mas-Gallic
and
P
A. Ravtart
Quantum Mechanics and
T r a n s p o r t E q u a t i o n s for Q u a n t u m G. Manfredi
and M.
82
Semiconductors Plasmas
R. Feix
109
M a t h e m a t i c a l T h e o r y o f K i n e t i c E q u a t i o n s for T r a n s p o r t M o d e l l i n g in F.
III.
Semiconductors
Poupaud
Boltzmann
141
Equations and Gas
O n Zero Pressure Gas F.
Dynamics
Dynamics
Bouckut
171
A Remark Concerning the Chapman-Enskog L . Dtsvillettts
and
Asymptotics
F. Golse
I n t r o d u c t i o n to the T h e o r y of R a n d o m Particle
191 Methods
for B o l t z m a n n E q u a t i o n B.
Perthame
204
3
T H E CHTLD-LANGMUIR L A W I N T H E K I N E T I C THEORY OF C H A R G E D - P A R T I C L E S . P A R T 1, E L E C T R O N F L O W S I N V A C U U M
PIERRE DEGOND Math&natiques pour ('Industrie el la Physique C.N.R.S. UFR MIC, University Paul Sabalier US, route de Narbonne 31062 Toulouse Cedex, FYance
This paper is the first part of a series of three papers reviewing the mathematical theory of the Child-Langmuir law in the kinetic theory of charged-particle beams and various of its applications-
1.
Introduction
T h e C h i l d - L a n g m u i r law goes back t o the early studies of C h i l d , L a n g m u i r and C o m p t o n [19] i n the early 30's. I t states t h a t the m a x i m a l current w h i c h can flow t h r o u g h a plane v a c u u m diode cannot exceed a l i m i t i n g value, independent on the way the electrons are e x t r a c t e d from the cathode (and i n p a r t i c u l a r on how m u c h of t h e m are e x t r a c t e d ) , and w h i c h o n l y depends on the length of the diode and on the applied p o t e n t i a l . T h e C h i l d - L a n g m u i r f o r m u l a for the current J is the following :
(1.1) where e
u
is the v a c u u m p e r m i t t i v i t y , e and m are the electron charge and mass,
(A)to zero at the cathode and to the applied potential $ / , at the anode. We denote the electron d i s t r i b u t i o n function by F{X, V),X g [0, LJ, V € IR. T h e flow of electrons between the electrodes is governed by the Vlasov-Poisson system :
Z^.= ±N(X), F(X,V)dV, -oo
A-e[0,L],
X E [O.LJ.
(2.2)
(2.3)
5
F ( 0 , V ) = G(V),
F{L,V)
${0)
= 0,
= 0,
V>0,
(2.4)
V 0.
L
(2.6)
where N{X) is the electron density and G(V), V > 0, is the d i s t r i b u t i o n of i n jected electrons at the cathode. T h e existence of solutions of the non-linear p r o b l e m (2.1)-(2.6) is proved i n [17], A n extension t o the m u l t i d i m e n s i o n a l case can be found i n 121], T h e t y p i c a l energy associated w i t h the b o u n d a r y value G[V) energy \mV£ where
V
2
= ( Jo
C
V G{V)dV
I / Jo
G(V)dV)i
is the t h e r m a l
(2.7)
T h r o u g h o u t t h i s paper, we shall investigate w h a t happens when t h i s t h e r m a l energy is small compared w i t h the applied p o t e n t i a l energy
|mVg
< eft
(2.8)
We thus i n t r o d u c e a small parameter e by •
and we shall be concerned w i t h the l i m i t s —> 0. I n physical terms, V is a t y p i c a l value o f the velocity of the injected electrons inside the d o m a i n . F o r m u l a (2.8) implies t h a t such a velocity is very small compared w i t h the t y p i c a l velocity Vr, t h a t the elctrons can acquire due t o t h e i r acceleration by the electric field, and w h i c h can be measured by G
1 -mVl
;2e4>i = e*i.,
V
L
= yj
I f velocities are scaled by V/,, t h i s means t h a t V
. G
(2.10)
is small of the order o f e. I f
s i m u l t a n e o u s l y t h e density associated w i t h the b o u n d a r y value G : N
=
G
I
G{V)dV
(2.11)
is left of order 1, t h e injected current i n the diode N Vq
= O(e) is very s m a l l ,
G
and very few interesting phenomena are likely t o happen. Therefore, i n our asympt o t i c s , w e ' d better impose the injected current : J
G
=
/ Jo
VG{V)dV
= 0(1)
to be of order 1, and the density t o be large of order
(2.12) l/e.
6
T o make all these statements more precise, i t is necessary t o introduce a scaling of the Vlasov-Poisson problem (2.1 }-(2.6). Wc shall use the length of the diode L , the applied p o t e n t i a l tpL, and the typical anode velocity Vj, as length, p o t e n t i a l and velocity units. The units of density N and d i s t r i b u t i o n function F follow from N = 5$£^ —
(2.13)
TV
F = ~
(2.14)
T h e density N is the one w h i c h is needed to achieve a potential d r o p of order 4>L w i t h i n a distance L . T h e n , we introduce scaled variables : X = Lx, N = iVn,
V = V v, F = Ff,
* = $L0,
=
0,
0,
(2.29) (2.30)
v < 0,
(2.31)
/
in
M (\0,l\x]&) h
u? ->y>
in
of solutions
weak star
C'([0,1]) strong
of (2.28)-
(2.34)
(2.35)
where Mb([0,1] x IR) denotes the space of bounded measures on (0,1) x IR. The limit distribution function f is given by . f(x,v)
=
J.
6{v-
^pjx)),
(x,v)
€ |0,1] x IR.
(2.36)
ip is the solution of % « dx 2
n(x) = - T L = , ^fipjx~
^(0) = . 0 , ^ ( 1 } = 1,
(2.37)
and j = lim f where
is given by
=
titfa,^
(2-38)
(2.25).
C o m m e n t s o n t h e o r e m 2.1 : F o r m u l a (2.36) shows t h a t the l i m i t d i s t r i b u t i o n function is t h a t of a monokinetic beam of particles (i.e. a delta function i n the v variable), e m i t t e d from the cathode w i t h velocity 0 (because <Jtp(x) — 0, for £ = 0). Before giving a rigorous proof, we show how t o understand i t intuitively. I t is well k n o w n t h a t any solution / of (2.28) is constant along the characteristics. These are curves ( X ( t ) , V ( f ) ) , solutions of the differential system :
2
T h e energy invariant W(x,v) (2.39)
= v -
2
jch and £ small, tp is negative close t o 0 (see figure 3). Since
0), but, due to the non-linearity of the problem, its effects are s t i l l present i n the saturation of the current j to the value JCL • I n the process S —• 0, an " i n f i n i t e s i m a l " potential barrier a u t o m a t i c a l l y adjusts so as to allow a transmission of the current exactly at the value j e t , whatever the injection profile 7 is. T h e analysis of the "infinitesimal" potential barrier is made possible by a boundary layer analysis that, w i l l be summarized i n section 3. c
c
E
c
c
F i g u r e 4 : Phase p o r t r a i t o f characteristics o f the s t a t i o n a r y V l a s o v e q u a t i o n associated w i t h the p o t e n t i a l E
(p w h e n x
E c
>0
12
2.3.
Sketch of proof of theorem 2.1
I t is interesting t o give a sketch o f the proof of theorem 2.1, because i t explains w h y the m a t h e m a t i c a l analysis is still incomplete in many cases of application of t h i s theory. T h e first proof of theorem 2.1 was given i n [15] by use of explicit solutions of the p r o b l e m (2.28)-(2.32). T h e n , another proof based on functional a n a l y t i c a l arguments was given i n [10], [11], and used for extending the m a t h e m a t i c a l analysis to the case of cylindrical or spherical diodes. T h i s second proof uses an uppersolution concept which first appeared i n [21], I t can be used t o extend the theory to more complex situations like magnetized flows, collisional problems, etc, as we shall see i n [5] and [6] (see also references cited i n section L ) . However, the first proof is more i n t u i t i v e , and gives a better physical understanding of the p r o b l e m . Since b o t h methods are usefull, we shall outline b o t h of t h e m . F i r s t m e t h o d : p r o o f o f t h e o r e m 2.1 v i a e x p l i c i t s o l u t i o n s o f t h e p r o b lem
(2.28)-(2.32)
Existence of solutions for (2.28)-(2.32) follows from [17] or [21], under very m i l d assumptions on the b o u n d a r y d a t a j(v). Furthermore, i t is an easy m a t t e r to show t h a t 'p is s t r i c t l y convex. Therefore, i t reaches its m i n i m u m value
S k e t c h o f p r o o f o f L e m m a 2.2 c
F r o m the explicit expressions (2.54) and (2.55) o f t ,
i t is fairly easy t o prove
t h a t i t is bounded i n L ° ° 11^ 111* CPU)) < C. Therefore, since k
c
(2.60) c
defined by (2.57) is independent o f x, dip /dx
w i l l be uni-
formly bounded as soon as we show t h a t i t is bounded at one p o i n t . More precisely we show t h a t there exists a constant C, independent of e, such t h a t for any £, there r
exists one p o i n t x - i n [0, l j , such t h a t . < C s
(2.61) 1
L e t us suppose first t h a t i f = 0 (i.e. t h a t f is non decreasing). T h e n , since ip is convex and reaches the value 1 at x — 1, we obviously have : | ^ ( 0 ) |
< 1,
(2.62)
w h i c h shows t h a t i ' = 0 is a convenient choice i n t h i s case. O n the other h a n d , i f i f > 0, t h e n *p reaches its m i n i m u m at i f and we have : !
w h i c h shows t h a t i
£
c
= x
c
is the convenient choice i n the second case.
F r o m (2.57) i t follows immediately t h a t
15
I ^ H t ~ < | o , i ] ) < C,
(2.64)
T h e n . Poisson equation (2.29) yields : -1 J2 I K I t a M i - l X 2 f * l - t f « - f « I «
M B
• E s t i m a t e (2.59) allows t o show t h a t f sures. Indeed, we have :
fn
£
(p
1
converges i n the weak t o p o l o g y o f mea-
-
f
in
M ([0,l\xtR)
-*
n
in
M (\0,1])
—
jet. Oforx £ ( 0 , 1 ] . •
such that ip(x)
>
WS: S 3CL- problem (2.37) has a unique solution suck that Oforx (0,1].
€
• Furthermore, j = jcL dtp/dx(0) = 0. Proof.
problem (2.37) has no solution
is the unique value of j such that the solution
satisfies
s k e t c h of t h e p r o o f of l e m m a 2.4
M u l t i p l y i n g equation (2.37) by dip/dx(x)
leads t o a first integral of the equation: * W - t , dx
("4)
w i t h 5 — dip/dx(0). T h e n , equation (2.74) can easily be integrated, and the value o f S is found by m a t c h i n g the boundary value ip(l) = 1. T h i s leads to the equation :
s: .
, -**
= 1.
(2.75)
I t is readily seen t h a t the m a p p i n g 5 — i j is monotonically decreasing from [0,1] onto [ O . j c i , ] - T h e conclusion follows.
•
!
T h e proof of theorem 2.1 is now almost complete. Since the l i m i t of j as e —* 0 exists, we necessarily have, by v i r t u e of l e m m a (2.4) : 0 < j < jet- We also have, from (2.73) : 0 < j < j (since obviously j ' < j - , ) . Therefore, we have : 7
4
17
0 < j < Mm(j , 7
J C L
)
(2.76)
To show t h a t j is a c t u a l l y equal to Min(j-f,j L), we need a technical l e m m a , the p r o o f of w h i c h can be found i n [15] and w h i c h w i l l be o m i t t e d i n the present paper. C
L e m m a 2.5 End
The convergence
of tp' to ip holds in Ike C ' f l O , 1]) topology.
o f t h e p r o o f o f t h e o r e m 2.1
Let us Erst assume t h a t dtp/dx(0) > 0. T h e n , by L e m m a 2.5, for e s m a l l , we have : d
0. I t follows t h a t ip\ - 0, and by the aid of the explicit f o r m u l a (2.52), t h a t j = j y , Therefore, by l e t t i n g e —• 0, we get j = j - , < JCL- L e t us now assume t h a t d 0 and V > 0 such t h a t : 0
0 < G(u) < M,
2
SuppG
C { u , 0 < u < V }.
(2.79)
0
T h e n , the function
**(*,«)
v
= G (
2
~ £
l
x
)
)
(2.80)
is a function of the energy invariant and, as such, is an exact solution of the Vlasov equation (2.28). Furthermore, i t satisfies :
1
2
v
£
=
-^G(^)
e
=
i c ( ^ ) > / ' a ^ 0 ,
F (0,v) F (l,v)
= f(0,v),
V>0,
(2.81) D O .
(2.82)
T h u s , F'- is an upper-solution of the Vlasov equation (2.28). T h e s o l u t i o n if* of the Vlasov-Poisson problem (2.28)- (2.32) is not unique i n general, b u t i t can be shown (see (21]) t h a t there exists at least one solution / such t h a t : £
c
c
f {xj)
< F (x,v),
V(x,v)
e [0,1] x IR.
(2.83)
We shall consider one of these. F r o m (2.79) and (2.83), we immediately o b t a i n bounds on / 2
Suppf
c {(x,u), 0 < v
s
2
E
and on its support:
2
-
is guaranteed as soon as 5 is i n the range of the left hand side of (3.28). T h i s c o n d i t i o n can be e x p l i c i t e l y found and coincides w i t h the c o n d i t i o n j > jch£
E
y
L e t us now assume t h a t = #g = 0. We can the previous case and integrate the energy i d e n t i t y m u s t be careful t h a t (di/> /d£J(0) is n o t necessarily 0. (dil> /d^)(0) , we o b t a i n t / i ( 0 for £ > 0 i m p l i c i t l y by £
£
E
ftfsv^tf))
= &
use the same m e t h o d as i n (3.16) on [ 0 , ] , Simply, we Therefore, i n t r o d u c i n g 5 = the formulae : c
E
eafti
(3.29)
(3.30) i-OO
Gi(S\ip)
= 5
e 2
+ 4 /
Jo
t?7C) ( ^
+ V< - 9).t" .
(3-33)
5 jcL and i f ip is (uniquely) defined by : y
c
e~,(B)d9
£
= jcL-
(3.36)
s
A s y m p t o t i c b e h a v i o u r o f i/> , tp a n d j'~ £
€
£
We are now ready t o study the asymptotic behaviour of ^ , ip , and j " as s —• 0, in the case j > jcLy
P r o p o s i t i o n 3.7
We suppose that j - , > jcL0(f),
Then, we have :
C - 6 +
O'e),
(3.37)
and = 0(f) + 0(e), uniformly for § belonging to any compact subset of
(3.38) fft . +
Proof. S k e t c h of p r o o f of p r o p o s i t i o n 3.7 I t can be shown t h a t the equations (3.28) for ipl and (3.36) for ip can be w r i t t e n in the form : c
H W £ , e ) = p,
(3.39)
25
£
such t h a t , ip appears as the regular branch of solutions of an i m p l i c i t equation passing t h r o u g h t h e p o i n t (e~,ip) = (0, T/J ). T h e n , (3.37) for ib follows from the i m p l i c i t function t h e o r e m and the estimate of £ is an easy consequence. T h e same a r g u m e n t s can be applied for V> (£) by recasting (3.20), (3.23), (3.33) and (3.34), i n the framework of the i m p l i c i t function t h e o r e m i n a similar way. c
c
£
£
•
2
2
e
R e m a r k 3.8 G o i n g back t o the original variables x% = £^ (.l and ifi' = £ ip , we o b t a i n t h a t the b o u n d a r y layer length i f of order 0(e ' ), and the p o t e n t i a l barrier height, of order 0(e ). A n a p p r o x i m a t i o n of the derivative at the origin is p r o v i d e d by the q u o t i e n t 'p\fx\ — 0 ( e ) , which shows t h a t t h i s derivative tends t o 0 as e ^ 0. T h i s is consistent w i t h the fact t h a t the C h i l d - L a n g m u i r c u r r e n t is associated w i t h a zero cathode electric field : (d-p/dx)(x — 0) ~ 0. e
c
3 2
2
1 / 2
T h e constant i n front of the O(e) t e r m i n the estimate (3.7) can be made explicit. e
I t is more usefully reformulated i n terms of the " unresealed" p o t e n t i a l 2). However, many open problems remain t o be solved as we shall see i n section 5. (ii) : We shall also see from these examples t h a t i t is too much t o expect t h a t j = Min(j ,jcL)Indeed, there are situations where a current j larger t h a n the C h i l d L a n g m u i r current jcL can be found (the C h i l d - L a n g m u i r current jcL w i l l be defined as the current w h i c h produces a vanishing normal derivative of the p o t e n t i a l at the b o u n d a r y ) . T h i s example shows t h a t conjectures for the general d-dimensional p r o b l e m are not easy t o formulate, and even less easy t o prove. y
4.2.
The cylindrical diode
We first consider a cylindrical diode which consists of two coaxial metallic cylinders of circular section and of infinite length. We denote by R, and R- the cathode and anode r a d i i respectively. T h u s , we have Ri < R$ i f the cathode is surrounded by the anode and Ri > Ri i n the converse s i t u a t i o n . T h e scaling is chosen similarly as i n the plane diode case except for the length scale t h a t wc choose equal to the radius of the cathode H j , and for the density and d i s t r i b u t i o n function scales N and F t h a t are defined by formulae (2.13) and (2.14) w i t h L replaced by R . T h e scaled c d i s t r i b u t i o n function S (t,vt,v$,v:) now depends on the radial distance r, and on the r a d i a l , azimuthal and l o n g i t u d i n a l velocities, (v , v$, t i ) . T h e scaled cathode radius is (by definition) r = 1, while the scaled anode radius is r = p, where p is the aspect r a t i o of the diode p = Rq/Ri. Therefore, the p r o b l e m is set u p i n the interval N o t e t h a t p > 1 i f the cathode is surrounded by the anode and p < 1 2
x
T
z
c
if the cathode is outside the anode (see figure 5). T h e dimensionless density n electrostatic p o t e n t i a l tp* now depend on r. T h e diineiisiouless Vlasov-Poisson problem is w r i t t e n :
and
27
1
df
c
v*
1 d 0 or < 0 express the conditions for the velocities t o p o i n t inwards the d o m a i n , whatever the configuration of the cathode and the anode are. As i n the plane case, the cathode p o t e n t i a l is set equal t o zero, and the anode p o t e n t i a l is chosen as reference p o t e n t i a l . T h u s , the scaled anode p o t e n t i a l is equal t o 1. T h i s gives : r
e
* (l)
-
0,
y'ip)
-
I-
(4-5) c
N o w , the conservation of the current implies t h a t the current intensity
i (r)
flowing t h r o u g h any cylinder of radius r between the t w o electrodes does not depend on r : E
i (r)
s
= r
v f (r,v)dv
independent
r
E
A g a i n , we are interested i n the l i m i t of f , T h e o r e m 4.11 (4.1)-(4.5)
([10], [11])
E
of r, £
rS[l,p],
(4.6)
E
tp , n , i , w h e n e —• 0 !
f
There exists a sequence (/ ,^ )
of solutions
of
such that , as e —> 0, vie have : c
f
3
— finM ([\,p\
x I R ) weak star
b
c
(4.7)
l
0, P * I .
(4.22)
2
Indeed, icL.
problem
(4-11)
has no solution such that 0 forr
£ (p, 1].
• / / i < i c / , , problem (4-il) has a unique solution such that Oforr
£
(PM • Furthermore, i = icL dtp/dr{l) = 0.
is the unique value of i such that the solution
satisfies
T h e surprising result is i n the case p > 1 : L e m m a 4 . 1 3 Let p > I (cathode inside the anode), •
•
There exists i {p),
0 < i {p)
max
0
mal
— ifi
satisfies
— ifi
satisfies i > t
< i ax(p). m
m 0
then ;
< + o o , such that: problem
(4-11)
i ( p ) , problem (4-H)
has at least one
has no
solution.
solution.
There exists Po > I such that W * { / > ) = icdp).
Vp e [1,
P o
\,
(4.23)
and W •
> tc/.- in(po.oo).
There exist values of p € [ l . o o ) and i E [0, i solution exist.
m n l
(4.24)
( p ) ) such that more than one
These Lemmas are proved i n [ l l j . T h e i r proof is much more technical t h a n t h a t of the corresponding lemma for the plane case ( L e m m a 2.4], Indeed, due t o the weight r i n the Laplace operator, no first integral is available any longer. T h e existence proof relies o n the use of the fixed p o i n t theorem. T h e complicated p a r t is t o o b t a i n a p r i o r i i n f o r m a t i o n on the behaviour of the solution near the singular p o i n t r = 1, so as t o fix the correct functional s e t t i n g for the fixed p o i n t t h e o r e m .
L e m m a 4.13 has an i m p o r t a n t physical consequence. I t states t h a t , i n the l i m i t e —• 0, the C h i l d - L a n g m u i r c u r r e n t (i.e. the current associated w i t h the s o l u t i o n w i t h a vanishing d e r i v a t i v e at r = 1) is n o t necessarily the m a x i m a l possible curr e n t i n the diode. I n [11], n u m e r i c a l s i m u l a t i o n s are presented, w h i c h s u p p o r t t h i s conclusion, b u t w h i c h also show t h a t the relative difference between ici(p) and i m a i ( c ) is v e r y s m a l l . T h e e x p e r i m e n t a l verification of t h i s difference must be difficult. However, from a t h e o r e t i c a l v i e w p o i n t , i t is i m p o r t a n t t o know t h a t the regime o f m a x i m a ! current is n o t necessarily associated w i t h a vanishing cathode electric field. I n d e e d , i n the physical l i t t e r a t u r e on space-charge l i m i t e d flows, t h i s fact is usually taken for g r a n t e d . T h e m e r i t of the present analysis is t o e x h i b i t a counter-example. T h e t w o L e m m a s 4.12 and 4.13 allow t o precise the value of i i n t h e o r e m 4 . 1 1 . We have : L e m m a 4.14 • Let p < 1 (cathode outside the anode), orp > 1 (cathode inside the anode) with p < po where p is defined in Lemma 4-13 (small aspect ratio). Then we have: 0
i = Afin^JctG*))•
Let p > po (cathode inside
(4-25)
the anode with large aspect ratio),
i
-
Mm(i„i
i
e
{ i - , , i c 7 i ( p ) } , ifiy
c t
then !
( p ) ) , if i, f [ i i , ( r t , * « « ( / > ) ] , C
€ [ict(p),imax(p)]-
(4.26) (4-27)
We notice t h a t L e m m a 4.14 o n l y gives a p a r t i a l conclusion concerning the value of i i n the case p > po (cathode inside the anode w i t h large aspect r a t i o ) a n d h £ [ i c t ( p ) , i m a r ( p ) ] (injected current i n the i n t e r v a l comprised between the C h i l d L a n g m u i r c u r r e n t and the m a x i m a l allowed c u r r e n t ) . I n t h i s case, we do not k n o w , up t o now, how t o d i s c r i m i n a t e between the values i and I ' C L ( P ) - Indeed, since z < i {p), both and ICL{P) are allowed values of the current i. I f i = z , the associated p o t e n t i a l ip satisfies (dip/dr)(r = 1) > 0, and i f i = icL(p), i t satisfies [dtp/dr)(r = 1) = 0. I t m a y be possible t h a t these two solutions are b o t h l i m i t p o i n t s of the sequence ip 7
7
max
7
c
4.3.
T h e spherical
diode
I t is i n t e r e s t i n g t o investigate i f the same results extend t o the spherical case. We o b t a i n the s u r p r i s i n g result t h a t the p a t h o l o g y w h i c h appears for p > po (cathode inside the anode w i t h large aspect r a t i o ) i n the c y l i n d r i c a l case does not show u p in the spherical case, and e v e r y t h i n g behaves p r o p e r l y like i n the plane case. L e t us investigate the spherical case i n more details. W e consider a spherical diode w h i c h consists of t w o concentric m e t a l l i c spheres. We denote by Ri and R? the cathode and anode r a d i i respectively, and by p the
32 aspect ratio p = KijR\. A g a i n , we have p < 1 i f the cathode is surrounded by the anode and p > 1 i n the converse situation {see figure 5). T h e scaling is chosen s i m i l a r l y as for the cylindrical diode. T h e scaled d i s t r i b u t i o n function f (r, v ,a) now depends on the radial distance r , on the radial velocity r y and on the squared n o r m of the angular m o m e n t u m a = \x x v\ 6 [0, co). c
r
2
T h e dimensionless Vlasov-Poisson problem is w r i t t e n
+
* £ ^ d r ar {
r
2
+
£
5£>J£-*
d f ar
_
}
«
n
(
= ^TCJ-^)-
= g (v ,a)
r
= _L / r(nW.,,a)
