P r i n c e t o n , New J e r s e y
T h e o r y of a P r o b e in a Strong Magnetic F i e l d by
Juan R. S a n m a r t i n
MATT -599 J u l y , 1968
J u a n R. S a n m a r t i n t
U n i v e r s i t y of C o l o r a d o , Boulder, C o l o r a d o
ABSTRACT
A kinetic a p p r o a c h i s u s e d to develop a t h e o r y of
e l e c t r o s t a t i c p r o b e s in a fully ionized p l a s m a in the p r e s e n c e of a m a g n e t i c field. A c o n s i s t e n t a s y m p t o t i c expansion is . obtained a s s u m i n g that the e l e c t r o n L a r m o r r a d i u s is s m a l l c o m p a r e d to the r a d i u s of the p r o b e . The o r d e r of magnitude of neglected t e r m s i s given. It i s found that the e l e c t r i c p o t e n t i a l within the tube of f o r c e defined by the c r o s s s e c t i o n
of the p r o b e d e c a y s n o n - m o n o tonic ally f r o m the p r o b e ; this bump d i s a p p e a r s at a c e r t a i n p r o b e voltage and the t h e o r y i s valid up to this v o l t a g e . The t r a n s i t i o n r e g i o n , which extends beyond p l a s m a p o t e n t i a l , i s not e x p o n e n t i a l . The p o s s i b l e
s a t u r a t i o n of the e l e c t r o n c u r r e n t i s d i s c u s s e d . R e s t r i c t e d n u m e r i c a l r e s u l t s a r e given; they s e e m to be useful for w e a k e r m a g n e t i c fields down to the z e r o f i e l d l i m i t . E x t e n
-sions of the t h e o r y a r e c o n s i d e r e d .
P a r t l y b a s e d on a P h . D . t h e s i s p r e s e n t e d to the U n i v e r s i t y of C o l o r a d o , 1967, r e f e r r e d to a s I ( A F F D L - T R - 6 7 - 1 9 0 , J a n u a r y 1968).
I. INTRODUCTION
Since L a n g m u i r ' s w o r k , p r o b e t h e o r y h a s been e x t e n s i v e l y developed for 2 - 4 the c a s e of z e r o m a g n e t i c field (B = 0). T h e t h e o r y i s p r a c t i c a l l y c o m p l e t e i n the dilute l i m i t when X « R « X {A , R, and X being t h e Debye length, p r o b e r a d i u s , and m e a n f r e e path, r e s p e c t i v e l y ) . E x t e n s i v e c o m p u t e d r e s u l t s a r e a v a i l a b l e in Ref. 5, including v a l u e s of An/ R > 0(1). T h e continuum c a s e
6 7 X « A « R was s u c c e s s f u l l y t r e a t e d by Su and L a m and Cohen. T h e m o r e
Q difficult p r o b l e m of an i n t e r m e d i a t e X h a s been a p p r o a c h e d r a t h e r c r u d e l y ,
9 10 although l a t e l y s o m e i m p r o v e m e n t s have been obtained. *
When a m a g n e t i c field i s p r e s e n t no r e l i a b l e r e s u l t s e x i s t . The p r o b e c h a r a c t e r i s t i c or c u r r e n t - v o l t a g e (I-V ) d i a g r a m h a s l e s s definite f e a t u r e s : roughly, a c e r t a i n d e c r e a s e in the c u r r e n t and a b l u r r i n g of the p l a s m a p o t e n t i a l kink. P a p e r s on the subject a r e c r u d e and s p a r s e . T h r e e b a s i c difficulties a r e the c a u s e of t h i s . F i r s t , the p r e s e n c e of B m a k e s the space a n i s o t r o p i c . Second, for l a r g e B, c o l l i s i o n s will c o m e into play; u n l e s s X i s the s m a l l e s t r e l e v a n t length, the c h a r a c t e r of the equations will change s u b s t a n t i a l l y in s p a c e . Finally, for fully ionized p l a s m a s , t r a n s p o r t coefficients a r e s p a t i a l l y dependent b e c a u s e the d e n s i t y e x p e r i e n c e s l a r g e c h a n g e s ; in fact, for e i t h e r B = 0 o r B ^ 0, only one p a p e r i s known to the author w h e r e Coulomb c o l l i s i o n s a r e c o n s i d e r e d , although a s a weak effect.
12
Spivak and R e i c h r u d e l studied the c a s e of a weak field. As in l a t e r 13,14 ,
the. u n p e r t u r b e d p l a s m a . M o r e o v e r , the effect of the flux along B i s not c o n s i d e r e d at a l l .
15
B e r t o t t i t r e a t e d the c a s e of a p r o b e p e r p e n d i c u l a r to B and a v e r a g e d m a g n i t u d e s o v e r the c r o s s s e c t i o n . An unspecified diffusion p r o c e s s w a s a s s u m e d and in t h i s way a p h e n o m e n o l o g i c a l i n t e g r o - d i f f e r e n t i a l equation w a s found and i n t e g r a t e d n u m e r i c a l l y . The r e s u l t s a r e in c l e a r c o n t r a d i c t i o n
to a l l e x p e r i m e n t a l e v i d e n c e ,
Bohm e s t a b l i s h e d a balancing between fluxes along and a c r o s s B .
AM
B e s i d e s the known depletion of the p l a s m a h e found the r e s u l t s to be i n -s e n -s i t i v e to t h e -shape of the p r o b e along B. However, -s e v e r a l defect-s should be pointed out. F i r s t , the diffusion equation i s a s s u m e d to be v a l i d up to a vaguely-defined s u r f a c e : no p r o p e r m a t c h i n g i s m a d e , the r e s u l t s a r e quantitatively u n r e l i a b l e , and t h e r e ' i s no way of knowing the e r r o r of the a p p r o x i m a t i o n s o r of i m p r o v i n g the t r e a t m e n t . Second, the value of the c u r r e n t , I, does not depend on V and so it i s u s e l e s s (the c u r v e c e r t a i n l y i s not flat a r o u n d s p a c e potential, V = 0). The i o n - e l e c t r o n t e m p e r a t u r e
ir
17
r a t i o w a s c o n s i d e r e d n e g l i g i b l e . A c o r r e c t i o n by Sugawara i s w r o n g ; the involved I-V dependence p r o d u c e d by an o v e r s h o o t i n g of the potential i s m i s s e d . M o r e o v e r , in both p a p e r s L a p l a c e ' s equation w a s solved i n c o r r e c t l y : the d e n s i t y w a s a s s u m e d constant on the a b o v e - i n d i c a t e d
s u r f a c e . They w e r e c o n c e r n e d with weakly ionized g a s e s .
in S e c . I I . In S e c . Ill a b a s i c p h y s i c a l p i c t u r e i s obtained, and the equations a r e p r o p e r l y solved in S e c . IV. In Sec. V we p r e s e n t the a c t u a l study of t h e I-V d i a g r a m . E x t e n s i o n s of the t h e o r y and d i s c u s s i o n s of r e s u l t s a r e given in t h e l a s t s e c t i o n .
II. STATEMENT O F THE P R O B L E M
Two s h a p e s of p r o b e s will b e c o n s i d e r e d s i m u l t a n e o u s l y : a thin s t r i p of width 2R and infinitely long, and a d i s c of r a d i u s R, both p e r p e n d i c u l a r to
B. T h e y give r i s e to t w o - d i m e n s i o n a l and a x i s y m m e t r i c p r o b l e m s , r e s p e c t i v e l The s t r i p will be found to d i s t u r b e n t i r e l y the p l a s m a , a s i s frequent w h e n e v e r diffusion i s i m p o r t a n t . T h e p r o b e i s n o r m a l l y cold and a c t s a s a Bink of
p a r t i c l e s ; it will be a s s u m e d p e r f e c t l y a b s o r b i n g . This condition i s e a s i l y r e l a x e d (see Sec. VI).
T h e equations to be studied a r e
***** ox m . ox o w m . *** *» d w k
(j - e, i for e l e c t r o n s and i o n s , r e s p e c t i v e l y )
V2V = 4 i r e ( Ne - Z. N1) (2)
N* = j F* d w
T h e boundary conditions a r e V = V on the p r o b e
P
and the F"^ 's given at infinity; in particular, t h e r e ,
N = Z.N. = N e 1 1 oo
T = T „ = T
e OO
T. = T. = B Z T 1 1 0 O J OO
$ being T. / T Z. and Z.e the charge of the ions. Space (or plasma)
" i°o' e°° 1 1
potential is taken as origin of voltages.
We now define non-dimensional variables:
e 3 i 3 F U . F U.
fe _ e i _ i_
^ '
N oyz.
eV w
x =
^Z ' ~
=u:
( j = e , i )OO j
where U m. = kT. . Then Eqs. (1), (2) become explicitly (choosing the } J J°°
z axis along B)
L vz 9z v f 9£ 9z 3 v + 8£ dvy { £ J? M £ 9 v n 9 v , 'J
* ' z ^ £ * e ^ TJ q
= X_ 1[ c ( fe, fS) + c ( fe, f1)] { 3 )
r a x 3 1 3X 3 1 3X 3 / l j l A * w 9 a \1 **
v — - 4- v — - - •— —**• -——- - — —•"•• ' „ f ——J- + —— if v — - — - v ) I f
L z 3 z § 9 £ B Bz 9 vz j3 8£ 3 v l £ i . " | 8 v ^ v ; J = X^f^f1, f1) + c(f\ fe)]
(4)
2
v 2 r 9 , -s 9 , s 9 -i e i ,c,
X
D
*77
+* ^ « af
] x = n"
n t5)dz
j q. B J
k'Tc o 1/2 ^ T J2
XD = ( " 1 ) X = - 4 (6)
4 l Nwe 2*1*^6 l n A ^
3kT kT , /•)
00 00 If L
A c o = — T " [ — 2 T ]
-e 4ffNw.e (1+0 )
1
s - 0 for the s t r i p and 1 for the d i s c . In t h i s c a s e £ i s t h e r a d i a l d i s t a n c e in c y l i n d r i c a l c o o r d i n a t e s ; for s = 0 the z '- £ p l a n e i s p e r p e n d i c u l a r to t h e
s t r i p .
T h e definition of A i s s o m e w h a t a r b i t r a r y but, of c o u r s e , the solution w i l l not depend on i t . T h e point i s t h a t t h e c ' s a r e now n o n - d i m e n s i o n a l q u a n t i t i e s of o r d e r unity for j v) = O(l), a s a r e the t e r m s on the left-hand
s i d e s of (3)-{4) if e x c l u s i o n i s m a d e of s p a t i a l d e r i v a t i v e s or l e n g t h f a c t o r s . The i d e n t i t i e s
u
2A"1 L c ( fj, fk) = - J - £ C(^, Fk)
k Njoo k
define the c ' s . It i s i m p o r t a n t to o b s e r v e t h a t we c a n w r i t e
c(fe, f1) = nP(ie) + f i P j f6, f1) (7)
r
w h e r e P(f ) = 0 if f i s i s o t r o p i c and }XP - 0(ju), [i ~ (m / m . ) ' .
Lit X
the L a r m o r r a d i u s j . In o t h e r c a s e s the p r o p e r n o n - l o c a l o p e r a t o r s will be 2
a s s u m e d ; u n d e r the weak r e s t r i c t i o n that e / k T should be the s m a l l e s t length p r e s e n t , t h e c ' s a r e s t i l l O(l) with X given by (6).
T h e b o u n d a r y conditions a r e
X = X_ = e V / k T on the p r o b e F p oo
X = 0 a t infinity
f i f given at infinity w h e r e n = n = 1.
F i v e c h a r a c t e r i s t i c l e n g t h s a p p e a r explicitly in o u r e q u a t i o n s : Xn, X, R,
$. t $.. * We c a n define four n o n - d i m e n s i o n a l p a r a m e t e r s e i
\l = £ /£. < 0{l) ( i . e . , fx« 1) e x
(7 - SL / R < O ( l )
e v
e =
XD/ R< o(D
% - R / X < O ( l ) .
The f i r s t i s a n a t u r a l s m a l l p a r a m e t e r of our p l a s m a [0 i s a r b i t a r y but 0(1)3. a < 0(1) will be c o n s i d e r e d a s our definition of a " s t r o n g " m a g n e t i c
field; in Sec. VI the u s e of the p r e s e n t r e s u l t s for w e a k e r fields down to the l i m i t B -* 0 (CT —•- «>) w i l l be d i s c u s s e d . We r e q u i r e a l s o the e x c l u s i o n of the c a s e a/{X < 0(1).
T h e condition £ < 0(1) i s frequently satisfied in a c t u a l p l a s m a s ; the compli-c a t i o n s a r i s i n g f r o m the r e l a x a t i o n of t h i s compli-condition will b e compli-c o n s i d e r e d .
In the l a s t r e s t r i c t i o n we exclude the c a s e X « R but not X < R. F u r t h e r m o r e ,
It i s w o r t h pointing out h e r e t h a t w r i t i n g E q s , (3) and (4) e x c l u d e s the p o s s i b l i t y of a n o m a l o u s diffusion due to fluctuating f i e l d s . T h e c o n s e q u e n c e s of t h i s s t r o n g r e s t r i c t i o n will c o m e out c l e a r l y f r o m o u r a n a l y s i s . It c a n be g u e s s e d a t p r e s e n t t h a t a c e r t a i n d e g r e e of s m o o t h n e s s in the p l a s m a i s to be r e q u i r e d . T h e a l l o w a n c e for T = 0(1) m a k e s our t r e a t m e n t useful for a
r e l a t i v e l y l a r g e r a n g e of c a s e s . In itself, t h e i n t e r a c t i o n of body, p l a s m a , and m a g n e t i c field i s of i n t e r e s t .
III. THE PERTURBATION IN SPACE
C o n s i d e r a c y l i n d r i c a l p r o b e in the a b s e n c e of a m a g n e t i c field. T h e d i r e c t e d flux t o w a r d t h e p r o b e d e c a y s a s r w h e r e r i s the d i s t a n c e to the a x i s of t h e c y l i n d e r . T h i s r e s u l t s in a n i m m e d i a t e , obvious f i r s t i n t e g r a l of t h e c o n t i n u i t y equation. If the p r o b e i s s m a l l enough the s p r e a d i n g m a k e s n e g l i g i b l e a l l effects a t d i s t a n c e s s t i l l " m i c r o s c o p i c "
H o w e v e r , when a m a g n e t i c field i s p r e s e n t , no such i n t e g r a l i s a v a i l a b l e . i h e c o n t i n u i t y equation i s not e l i m i n a t e d and c l e a r l y will be of f o r e m o s t
i m p o r t a n c e . F o r l a r g e enough B, e l e c t r o n s a r e inhibited f r o m flowing a c r o s s f i e l d l i n e s so t h a t the flux along B will v a r y slowly and t r a n s p o r t c o e f f i c i e n t s w i l l c o m e into play. M o r e o v e r , in t h e p l a n e o£ the p r o b e (z = 0) a n d o u t s i d e i t s a r e a , the p a r a l l e l flux i s z e r o by s y m m e t r y but i t i s not so i n s i d e . T h e r e f o r e s t r o n g g r a d i e n t s will a p p e a r on the b o u n d a r y of the
the two half-spaces produced by the plane of the probe the problem is
equivalent to that of a plasma bounded by an infinite wall where a d i s
-continuity on the reflection coefficient occurs over a small c i r c l e .
To obtain an initial picture of the situation we apply briefly a multiple
scales method in this section. F i r s t , we define
D = v
9
+2 X _£_
z z 3z 9z 9 v z 9 9v 9
v„ = v, cos 6 , v = v . sin rt
4 1 r T/ 1 r
so that
d d _d_
V v " Vrj 9V. = ' 9f '
Then, multiplying Eqs. (3) and (4) by #. and rearranging t e r m s we get
— + 4 D f + £ c o s f D . f -H sin * — ^ + — -f- —
of e z e 4 e £ v. 9^ of
= err [c(fe, fe) + ^ P t f6) + ^ Pn( fe, f1)]
* O 1 *
l e D ^ + f cos f D* f1 - (X~- - H s i n f (—- - ^ - f j ) | ~
z e £ " o f e *; vi "I of
= <JT[c(f\ f1) + c(f\ f6)] .
(8)
(9)
(10)
i i -1
(For D , D the factor (-/3) multiplies the second t e r m s in the operators z §
We define next a set of non-dimensional variables
z
Lk
where the L, ' s are unknown combinations of the five characteristic lengths k
of our problem. We assume that from an equation such as
3A
°
Zk
= E (Ua)
we can obtain
lim E = 0. . (lib)
zn — oo
k
This results from requiring boundedness for A and non-oscillatory behavior
for E.
We define also £ = £/R. Although one should allow for strong gradients
around £ = R and therefore define £, = (ij -RJ/K/ where Ly / R — 0 as
(fi, 0", £ ) —- 0, in this limit
for any fixed ^ H . Thus the region over which £ is finite or d/d^ & 0
[see Eqs. (11a, b)]shrinks to zero. Its importance will be estimated later.
Now we make the expansions
f1 = £ f j , x = £ v
m m A m Am
and retain terms in (9), (10), and (5) of dominant order. It is possible to show
that the smallest length L cannot be shorter than \ . In effect, then, ( W LZ)2 » land from (5),
— — = 0 o r V = a z + b
2 Ao o
az
o
b u t f r o m l i m ^ 9 / 9 z = 0 w e g e t a - 0. C o n s e q u e n t l y f r o m E q . (10), o
9f 1/dz = 0 a n d f r o m (9), Z
e e o o >
f = f U ~ ~r — •
B u t t h i s f u n c t i o n c a n n o t b e p e r i o d i c i n 0 if 3f / 9 z —• 0 a s z ~— °°. T h u s
^ r o ' o o
8f / 9 0 = 8f w/ 8 z = 0. T h i s c a n b e s e e n i n a n o t h e r w a y : t h e e l e c t r o n s
o ' r o ' o '
e n t e r t h e z l a y e r w i t h a d i s t r i b u t i o n f u n c t i o n s y m m e t r i c a r o u n d t h e z a x i s
(to z e r o o r d e r ) , b e c a u s e f o r l a r g e z , 9f /Bz = 0. T h i s c a n n o t b e a l t e r e d
° o o ' o
b y t h e s p i r a l i n g a r o u n d f i e l d l i n e s .
T h e r e f o r e , L = X _ . W e g e t o u
^o e i
———- = n - n 2 o o 9 z
o e
8 £o 8 * e 8* o 6 , .
-ef
=
°'
(v
,sr
+
r IT 5T
,f
o = ° <
I 2 a
'
b
»
r o D o z
P 9v
, _ £ _ _ £ I —° -2-}
f* - o
(Vz 8z ' X Q 9 Z 9 V ; o o D ^ o z
9 e a n d a l s o • j = 0
9 z Jo
o
, , e z . , e i * w h e r e j = / f v d v = n u
If w e l e t z -» °°, (12a) r e m a i n s v a l i d b u t (12b) v a n i s h e s i d e n t i c a l l y . W e
9f,e £ ^ a
- — + — D f + ~- C O S 0 D , fe
= CTT[c<f e, f e) + n* P(f 6) ] .
o o o o T h i s equation i s of t h e type
8fe
"87
= A (^
+ Cand f r o m the r e q u i r e m e n t of p e r i o d i c i t y of f in <(>:
C = 0, — = A W.
T h e r e f o r e ,
— = (JT or JL. = A.
Ll
D f e = c(f e, f e) + n1 P ( f e) (1
z o o o o * o
6 . f- = - a sin 6 D . f + 6 , f,
1 1 t o lc lc
' r
w h e r e the index c m e a n s c o n s t a n t in 0. U n l e s s o t h e r w i s e s t a t e d , x. s u b s t
for x i n E <1 ' (8) •
6
f g i v e s an r\ flux but no £ flux. Again we obtain 9 . ez „ 9 . ez
r
J
0 =°»
JT
1 o
a«,
jo
=°» aT"
ji
=° '
If we now l e t z — °°, Eq. {12) b e c o m e s c(f e, f e) + n1 P(f e) = 0.
Not only is an i s o t r o p i c Maxwellian d i s t r i b u t i o n a solution to E q . (14) but i t i s the unique solution a s p r o v e d in App. B. Thus T - T . M o r e o v e r , T = T b e c a u s e the e l e c t r i c field i s s m a l l ( c o l l i s i o n s a r e dominant) and
e ©»
h e a t i n g c a n be n e g l e c t e d to z e r o o r d e r . T h u s - v2/ 2
T r e e £ e e
lirn f = n f, , = n
z~«> o o M ~ "0 . . . 3 / 2 '
i (2?r) '
It should be e m p h a s i z e d that the d e c o m p o s i t i o n of Eq. (7) i s valid if the a v e r a g e ion velocity i s m u c h s m a l l e r than that of e l e c t r o n s . T h e r e f o r e , if
1/2
the ions had a c q u i r e d an a v e r a g e z velocity of o r d e r (kT / m ) the above c o n c l u s i o n would be w r o n g ; in p a r t i c u l a r , u = 0 would not follow.. How-e v How-e r , t h i s would r How-e q u i r How-e an How-e n How-e r g y gain for thHow-e ions of o r d How-e r
kT l/ 2 2 _2
m . (—2) ] * kT M . i m e ^ e
16 T h e r e i s no s o u r c e of such a l a r g e e n e r g y . In the c a s e of B o h m ' s c r i t e r i o n for ion collection, an e n e r g y gain i s n e c e s s a r y of o r d e r
kT , 2 m . [ ( ~ — £ )1 / Z] = k T
i m, e
I
and m o r e o v e r it a p p l i e s when T, << T and Y i s l a r g e enough so that a l l e l e c t r o n s a r e r e p e l l e d . Then a f r a c t i o n of V ,
P - i - V = kT ,
*p P
a c c e l e r a t e s the i o n s o u t s i d e the s h e a t h . F r o m Eq. (14) t h e r e r e s u l t s
lirn, j 6 Z < 0(1)
and j (z = 0) < O(l) since j is conserved in both z and z to zero order.
But on the probe all particles travel toward it; thus n (z = 0) < O(l) and also
n {z —*<»)/n (z = 0) < O(l). Now the z-momentum equation in % and z. is
9 e „ez e o „ ,
0 = ' TT nn En + n n J 7 > k = o, l .
a z , o o O C T z ,
(Because j < O(l), the collision term does not contribute to this equation
ez e ~1 e e o 2 •*•**
for k = 1. ) Therefore if E = (n ) f f (v - u ) d v is order unity
every-o o J o z z ' '
where, \ > O(l); in particular it is so in the limit z -*<».
Hence n < O(l) in these layers and the results L. = A , L. = X
e
based on n = O(l) are not true (the local values for A and A have to be
used but they are unknown).
The importance of \ a t these large distances from the probe follows
also from the analysis of the following scale; so does the order of magnitude
of n and x i n the limit z -*•<*>. This and the detailed solution of the problem
will be made in the next section. Here we consider briefly the limit z —«> ,
Then f = f n and from the terms of next order in Eq. (9) there results o M o
an equation of the type
Tf -
A2<*>
+ cz
and again C = 0 , Q f^/3 0 = A . dt Ct L*
It is found that j*l^ ^ 0 and is 0(CT T ). Also, j ^Z ^ 0 and is 0{<X).
If L = AR/£ , the continuity equation allows now a spreading such that at
u e
e e i
infinity (in this scale) u =0 and n (= n ) = i, \ = 0 . Thus in this scale
IV. SOLUTION TO THE EQUATIONS A. The O u t e r z L a y e r
In this s e c t i o n we s h a l l be c o n c e r n e d w i t h a p r o p e r solution of E q s . (5), (9), and (10), a s c o m p l e t e as n e c e s s a r y to d e t e r m i n e the e l e c t r o n c u r r e n t to the p r o b e to f i r s t o r d e r . (To z e r o o r d e r in a , I v a n i s h e s . )
It w a s found in the l a s t s e c t i o n that o v e r d i s t a n c e s of o r d e r X R / i and R, along and a c r o s s B , r e s p e c t i v e l y , the conditions at infinity could b e
s a t i s f i e d . We define t h e s e " o u t e r " v a r i a b l e s :
z e
t _ i
2 " R X ' * r " R '
E q u a t i o n s (5), (9), a n d (10) b e c o m e
e
Z r > - s 9 j . s 9 2 2 9 , e i nc\[
tv n~K
H~
+
°
T
7^
] x = n
"
n (
'
sr br 9 z
r-— + CTcos 0 D . fe + <J T D f
s v e
V 1
sr
r1 n/*e\ . -^ ,,& A*
+ n P(fc) + juP (fe,f )] U6)
r 9 d- i 9 Y e l ri af1 „ **£ af1
+ a2T Di f1 = g ^ c ^ f1) + c ( f \ f6) ] . (17)
Next, we i n t r o d u c e the expansion fe = fe + a. f,e + e,f* +
o 1 1 2 2 f1 = f1 + Of i\ +
o 1 1
x = x
0+ s f x ^ .
Then we obtain f r o m E q . (16), 9fe
° = 0
so that
d(j>
5l l A "+ 0 c o a* Df fo
r
c(fe , fe) + n1 P(fe) = 0
O O O O
S.ff = - a sind> Dt fe + 6, ff
1 1 r £ o lc lc
r
,e e . e, - 3 / 2 v
o o M o r 2
(see A p p . B).
To following o r d e r ,
9 f2 X d \ di&
r r i
r i ^r r
= a T[CT (f , - s i n 0 Dt fe) + n1 P f - s i n ^ D . fe)l
a2x D £e = a T [ 6 . {c_ {fe,ff U n1 P(ff )}
z„ o l cl L o 1c o lc J
2
TD
,i ! ^ „e> . „ ,„e „i
+ 61n1P ( fo)+ M P /0. f0) l (19b)
e e e e
cT i s t h e r e s u l t of l i n e a r i z i n g c(f + 6, f, , f + 6, f, ) . It w i l l be s e e n b e l o w
Li O 1 1 O I 1 i ,
i 2 t h a t f i s M a x w e l l i a n . T h e n u P =* 0 ( u ) a n d s h o u l d b e r e m o v e d f r o m t h e
o ^ fj, v r
l a s t e q u a t i o n [for a r b i t r a r y 0 ; i t i s k n o w n i n p a r t i c u l a r t h a t t h e m e a n f r e e
p a t h f o r e n e r g y e x c h a n g e i s 0 ( X / i ) ] . Of c o u r s e , 6 n P(f ) v a n i s h e s .
T h e r e f o l l o w s
6. = <7,v f6 r ^ ( h n e - X ) ~ c . ( fe, f f ) + n1P ( f f ) (20)
lc z o oz o rto L o l c o l c
a n d i, c a n b e o b t a i n e d . Jl
It i s e a s y to i n t e g r a t e E q . (19a) g i v i n g
9fe
J
?~ ^ d * •
e f
H o w e v e r , w e s h a l l n e e d o n l y j ; t h e o n l y c o n t r i b u t i o n c o m e s f r o m t h e
r i g h t - h a n d s i d e of E q . (19a).
fij*£ = aZT \ d v v [c_ (fe, sin<f»De f6} -f n1 P ( s i n < A Dt fe) ] .
*1 2 J rj L o T £ o o r t o
' ' r ^ r (21)
e i
T h e n f r o m t h e e l e c t r o n c o n t i n u i t y e q u a t i o n a r e l a t i o n b e t w e e n n , n , a n d o o
X. i s o b t a i n e d ,
3 . 9 . e z „ - s 9 „s .e£4
2 r
From Eq. {I5),ne ^ n and from Eq. (17),
^ . o o
3Y , - Of1 crsv 3f'
s sr r ^ r
L o o o o
Because of the form of f6 , c (fi , £ ) = 0{p) and it should be dropped out of
o o o
Eq. (19b). It is shown in App. B that the unique solution for fQ is then
2 i-z ^
v (v - u )
iXQ = A exp(-)3 Xo)(27T)" / exp [- — - ~z " J * " '
where A and ui 0 a r e a r b i t r a r y functions of z Taking the limit ^ - « > z
• ^ f- J „- ,r t D &= i , ii Z = n Thus while t& is a local Maxwellian,
with fixed z gives A= J-, UQ - u. inu&, wimc J.Q
2 . _ |
f1 is a global Maxwellian distribution and nQ = n^ = exp(- 0 XQ). K XQ ~
in the * - z plane, n* * 1 and no diffusion would exist. Only in the limit
^ r Z °
G
/3 —0 is it possible to have X ~ ° w h i l e n 0 * l'
Equation (22) involves only XQ after these substitutions. If boundary
conditions on z = 0 are known Xn = X <*_. *,> c a n b e obtained. In general,
7 O u J. t*
a relation of the type G (X Q, 9X0/9^ ) = ° i s n e e d e d o n Z2 = ° ' " W i U ^
seen l a t e r that this relation involves <J; and XQ [= XQ(Z2 = ° ' *r < X' J 1 S
slightly l a r g e r than 0(1) for Xp = O(l). However, it is neither worth while
nor easy in the partial differential equation to separate terms of almost
equal o r d e r . Moreover, it is not clear that 9X0/9^2 i s n o t °^'
On the probe, z = 0, the expansion for f6 breaks down since all particles
t r a v e l toward the probe there, and f* cannot be isotropic. A boundary layer(s)
will exist of the type classical in fluid mechanics where gradients a r e strong
in fluid mechanics it is possible, however, that the "outer" problem is a
complete one (except for the condition right at the body surface). To see
this, consider as — 0 (£ < 1). Assume that x s 0(1« A ) the re; then
La X. O
n = O(A), with A unknown yet. This results in 6, f, li - O(0"/A). Thus, o ' lc lc' o '
cr/A < O(l) is a condition for our expansion ; since A will be found to decrease
with Y there is a maximum value of probe potential for which our procedure
e i i -»1
is valid. The results n = n and n = exp(-6 x ) are still correct.
o o o ir\ f /\Q>
Assuming <j/A < O(l) we consider the limit £ ~*1. While over "most"
i
(asymptotically) of the probe the characteristic length of £ gradients should
be R , it is possible that at £ =1 much larger gradients are needed. F a r
e e from the probe,where n = 0(l),this is not the case, but where n = O(A)
(at small z2 and £ < 1), j6^ = O(a2 r A2) while for £ > 1, f^ = 0((JZT). The
ez
first term of Eq. (22) has to produce this change. Since j = 0{cr) there
results
2 2 2 , ,
<JVT = 0TA/d <7/6
t £ £
or A = 6, where 6 = L. /R ; cr/6 = S. / L , and L « R is the characteristic
length over which this fast change across B exists. 6, f, /f = O(0"/A) and ~» 1 1 o
CT/A < O(l) is again the condition for the validity of our expansion. However,
only a A fraction of the probe area is subject to this large gradient and
sensibly an error 0(<7) is produced in this case.
The result for f is still unmodified. As for Poisson's equation, the
2 -2
left-hand side is now 0 ( £ A )i and since the right-hand side is O(A) the
2 3
Z 3 e i 8 /A = O(l), we should use instead of n = n ,
rt2 ..-s 3 .s 3 e , rt-l . i-> A \
C
*r W
'*r W~
XO = % " 6 X P (^
V '
( 2 4 ) r ' rg
This equation, together with Eq. (20), determines both X a n^ n • Some
problem in finding boundary conditions for both could exist then, but this
2/3
only over a region on the boundary of order C that for elliptic equations
2/3
seems only important to 0(£ ). In the numerical results presented in
2 3
Sec. V, C /A < O(l) will be assumed.
As soon as we find the relation Gfv > (9x A21-.)] = 0 on the line z_ = 0,
o o L c
e 1
we have results for v , n , and n on the whole £ - z- space valid if
0 0 o r 2
2 3 2/3 3 2
<j/A and t /A for t ' in case A = 0(£ )] < O(l). However, we have
yet to determine which is the order of the t e r m s neglected when writing
6 7 6 2 a C Q c
3 ^ j i > 3 ^ Jy ' ^ t n e t e r m s of next order a r e taken in Eq. (16) it is
possible to find that the order of both 5, j - / 6 . j . > 6, j _ /67 j7 is the
lowest of C / A , 6 ^ / A I and 0^. 6J/A can be seen to be 0(£ /A ) from
Poisson's equation if quasineutrality is taken; in case Eq. (24) is adopted,
it is of higher o r d e r . Thus, we come back to the same critical inequalities X
to be satisfied in our theory. As for 6^ , analysis of Eq. (15) seems to
2 3 rule out the possibility of its being l a r g e r than both 0"/A and Z /A
It can be seen that collisions a r e local in the whole £ - z . plane; r Z hence, a collision operator other than F - P or B-L. needs to be
considered only if Si < X ; that operator, which includes B , is already
e XJ ***•
fe
}/Z < i J = RA if fi2 « A3. known. (Where n = O(A) and at f = 1 , X „ / A < L = RA if t « A •
o r D ' 2 3 £
but even if C « A collisions a r e local because St < Le and B r e s t r i c t s
e *•*
Then both j and j ? can be found from Eqs. (20) and (21):
A l l
!^?_
^ M e
(e
0 9z_ >9v Ll IM' h e ' + * lc'
•e« A l l /2 Xo /Xo f p.9 fM .
r V
The term c (f , 9f /9v ) does not contribute to j _ since the electrons do
X-i &A- JVL Tj L*
not gain average momentum from themselves (collisions are local). There
result,, in the present non-dimensional form,
y(Z.) 0 , . ,7/2 In A 3x
•e z i j3 + I 2 / _ °Q Ao
Jl = Z. 0 1/2 In A 8z_ ( 2 5 a )
h = z
i ,T 7T72 inT~~
e xP < — > 9T~ •
( 2 5 b )•y is the factor given by Spitzer to correct the Lorentz conductivity; In A
e
is spatially dependent since n i s . When & < X no results are known to
the author for i, (dc conductivity along B when B enters the collision
e£
process), although results exist for j for £ « X . Even if X < i ,
L e u D e
the above simple expression for In A may not be correct since the local
Debye length and not Xn should be compared,
In what follows we shall assume an average value ( In A) • This is
basically because of the imperfect knowledge of transport coefficients in
the presence of a strong magnetic field (some review is made in I). Also,
the dominant effect is the exponential term in Eq„ (2 5b) while the variation
due to In A can be negligible for large enough lnA^ . Finally, the results
Using Eqs» (25a) and (25b) in E q . (22) and defining
*
sT* '
y*
z2 ^7772 ^ T T T '
x =^r
K (12 y) ' *>
w e obtain finally
2
*~ + x r — x i ^ r ^ ^ O (26)
_ 2 9x 8x 8y
* l " - <W + «< f ^ | * • <»J
T h i s e l l i p t i c equation for ^// should be solved on the h a l f - s p a c e y S 0 w i t h the b o u n d a r y conditions
\p - 0 at infinity
G ( l / / , f ^ ) = 0 at y = 0 , (28)
t h i s l a s t r e l a t i o n being yet unknown.
Before going to the a n a l y s i s of the i n n e r l a y e r s , s o m e c o m m e n t s should be m a d e . An i m p o r t a n t r e s u l t i s that the e l e c t r i c field is non-negligible v e r y far f r o m the p r o b e ; i n fact, x i s n o t ^0C_) a n <* ^e field i s not
r e s t r i c t e d to d i s t a n c e s OfA^) f r o m the p r o b e . Only in the c a s e |3 ~*0 i s X s m a l l in this " o u t e r " s p a c e ,
N e a r the p r o b e , \ h a s to decay to negative v a l u e s or positive v a l u e s of o r d e r unity; the l a r g e p o t e n t i a l hill r e s u l t i n g will simplify the obtainment of a c o r r e c t solution. H o w e v e r , s t r o n g r e s t r i c t i o n s a r e i m p o s e d by the l a r g e p e n e t r a t i o n of the p o t e n t i a l . F i r s t , the p l a s m a cannot be s h o r t e r
on the p l a s m a . F l u c t u a t i o n s w i l l s e t in and a s t e a d y s t a t e s e e m s not p o s s i b l e ; a n o m a l o u s t r a n s p o r t p r o c e s s e s will a p p e a r . Second, even if the p l a s m a i s long enough, m a c r o s c o p i c v a r i a t i o n s should be v e r y weak; o t h e r w i s e the m e a s u r e m e n t would be n o n - l o c a l and the p r o b e would i n t e r f e r e w i t h the whole p l a s m a .
If any of t h e s e conditions a r e v i o l a t e d it s e e m s i m p r o b a b l e that p r o b e s a r e of any u s e . R e s u l t s would not be u n i v e r s a l but would depend on the p a r t i c u l a r p l a s m a , and s p a t i a l r e s o l u t i o n could not be obtained. T h u s , u s e of p r o b e s with s t r o n g m a g n e t i c field i s s t r o n g l y r e s t r i c t e d . On the o t h e r hand, if t h e s e conditions a r e m e t , the p r e s e n t t h e o r y is a c o r r e c t one in an a s y m p t o t i c s e n s e and r e s u l t s c a n e a s i l y be a c c u r a t e to 10%. If p o o r e r a c c u r a c y is not r e j e c t e d the t h e o r y should be useful even if the above c o n d i -tions a r e not quite s a t i s f i e d . M o r e o v e r , in S e c . VI it w i l l a p p e a r that r e s u l t s c a n be u s e d for w e a k e r fields down to the l i m i t B -*-0 ; then the r e s t r i c t i o n s a r e w e a k t o o . F i n a l l y , a c o m p a r i s o n could be m a d e with the c a s e B = 0 a s f a r a s the p e r t u r b a t i o n of the p l a s m a i s c o n s i d e r e d . In the c a s e of the disc (the s t r i p cannot be u s e d ) the c u r r e n t d e n s i t y , which i s a m e a s u r e of the p e r t u r b a t i o n s , is 0(CT) at d i s t a n c e s 0 ( A R / £ ); in the c a g e
B = 0 for a c y l i n d e r which i s frequently u s e d , the c u r r e n t d e n s i t y i s O(aT) at such d i s t a n c e s . T h u s , if T = O(l) the c o m p a r i s o n i s not b a d ; m o r e o v e r , at s m a l l e r and l a r g e r d i s t a n c e s the c y l i n d e r d i s t u r b s m o r e of the p l a s m a , and t r a n s v e r s e l y to B the p e r t u r b a t i o n of our c a s e d e c a y s a s
B , The Inner z L a y e r s
To satisfy the condition on the p r o b e an i n n e r v a r i a b l e i s defined
z, = Z/XJ, such that the t e r m D f i s now c o m p a r a b l e to the c o l l i s i o n t e r m . 1 ' 1 ^
Next we i n t r o d u c e the e x p a n s i o n s
e r e * e ,
f = A[g + \ gL + . . . ]
X = I B A P + XQ + 6* Xx + ••
so that
d(ji
.* Bl . . -1 > ~ . e
r
61 - ^ - + A acostf* D^ A go = 0 (29)
£
- 2 - D g^ = a r f c c f e ^ + n ^ P f e " ) ] . (30)
Ll L
z -1
Hence L = X A ; P o i s s o n ' s equation s t i l l r e d u c e s to the q u a s i n e u t r a l i t y condition since XA — O(R) . As for f o u r f o r m e r r e s u l t r e m a i n s
o
obviously valid if XA -> O(R); we shall c o m m e n t l a t e r on the o t h e r c a s e .
H e r e A = l i m n (z_,) a s obtained f r o m the a n a l y s i s of the z_ l a y e r and
z ~*"0 o 2 2 thus depends on | . (We a r e p r e s e n t l y c o n c e r n e d only with the r a n g e
e Xo
F r o m E q . (30) we obtain s e n s i b l y n ~ e ; t h i s behavio.r is i n c o m -patible with the knowledge we have that a s z, -— °o , n ~ e a s the
i o
X = 0 s o t h a t i n t h e i n n e r r e g i o n X = l n A + 0^ \ +• . , , , T h i s r e s u l t
s h o w s t h a t t h e f l a t t e n e d p a r t of t h e p o t e n t i a l a r o u n d t h e h i l l c o v e r s t h e z1
l a y e r , w h i c h b a s i c a l l y w i l l p r o d u c e o n l y a c h a n g e in f a s a f u n c t i o n of v ,
T h e r e h a s t o b e a n e w l a y e r w h e r e a f a s t d e c a y of x to i t s v a l u e on t h e p r o b e
w i l l o c c u r . G r a d i e n t s a l o n g B w i l l b e c o m e l a r g e a n d e l e c t r o n s w i l l e x p e r i
-e n c -e f r -e -e - s t r -e a m i n g m o t i o n t o w a r d t h -e p r o b -e . B -e f o r -e s t u d y i n g t h i s r -e g i o n
w e s h a l l t a k e t h e l i m i t z - * 0 i n o u r s o l u t i o n i n t h e o u t e r s p a c e , t o o b t a i n
t h e m a t c h i n g c o n d i t i o n a t z. -» <» ; b y r e w r i t i n g i t f i r s t i n i n n e r v a r i a b l e s ,
X = Xo ( A2!* * ^ X ^ / A ^ ) +
9X
*x
z j + 6ZAX1(0) +
v°
~P + T Q z, + 6 *
= In A ^ + - q Zj + 6J- xt( 0 ) + . . . (31)
w h e r e q ( £ ) = B\ /dz„\ . A l s o r o 2 _
V°
z br 77
(32) w h e r e c i s s u c h t h a t
di 9f 9f
cL(f
M'
ca^T
) + p ( c*T>
=eT
-z -z -z
In t h e z r e g i o n , o °
and t h e r e r e s u l t s
D g6 = 0 (33)
z o o
\ g* = - A"1 a s i n 0 D g* A + J g l e c
*r
*
e. - - . - « » :
1 9e
eo
R D ?' + 6* ~ ^ ^ } = A a r [ c { ? , ? ) +
u l c z 1c l o z o v •* o o
+ >(/ip(Jj)]
n should be m u c h l a r g e r than A so that 6, = 0 ( L n /A). o lc o o
^e The solution for g i s
i ^ = g ( v ^ - J V ^ f x , , - ^ - . . ) (34)
w h e r e g i s an a r b i t r a r y function and h(x) = 1 x > 0 h(x) = 0 x < 0 .
B e c a u s e d h / d x = 6(x) ( D i r a c ' s delta function) and x6(x) = 0 , (34) s a t i s f i e s i d e n t i c a l l y E q . (33) and the condition of p e r f e c t a b s o r p t i o n on the p r o b e , t o o . We r e w r i t e
g ( vz - 2X Q) = e fM g( vz - 2X o, v± )
to put in evidence s o m e e x p e c t e d exponential b e h a v i o r in g ; g i s again a r b i t r a r y .
z flux does not change in z and z, ; the c u r r e n t at z = 0 should then be o 1 o m a d e equal to the c u r r e n t at z = 0. T h i s will be the condition (28).
It i s n e c e s s a r y , h o w e v e r , to d e t e r m i n e A a nd g f i r s t . It should be
noted that b e c a u s e of the l a r g e o v e r s h o o t i n g of x > the discontinuity in the d i s t r i b u t i o n function h a s m o v e d to the positive h i g h - v e l o c i t y tail in v in the
z
™e Y-fc *
l i m i t z -*oo ; g (z -*•*>) is even in v to z e r o o r d e r . Also 6^ \. < O(l)
O O JL Z Li,
and we would have in z , splitting E q . (30): 9 e
6
lc ' , ST-
+"f ST 8 ^ '
Slc [
ei > X >
+«*<£»
+l
p< > •
i 1 z(36) g being even in v .
o z
T h e equation for g i s p a r a b o l i c with z acting a s a t i m e - l i k e v a r i a b l e , e e
H e n c e , if g (z = 0 , v ) i s given, g i s a u t o m a t i c a l l y obtained in z. . The
o 1 *•* o 1 m a t c h i n g condition at z, ~*0 gives not only g (z. = 0) but the b e h a v i o r a s
z1 ~*0 , in z . . H o w e v e r , it i s e a s i l y found that the l i m i t i n g f o r m of f
a s z •* » s a t i s f i e s E q s , (29) and (30), as should be expected if the p r o p e r s e l e c t i o n of l a y e r s h a s been m a d e . T h e r e f o r e only the function g (z, = 0, v )
o 1 ~~ i s given r e a l l y a s independent d a t a ; i . e . , A and the function g .
v t h e s e c o n d t e r m in t h e l e f t - h a n d s i d e of E q . (30) i s of t h e o r d e r of t h e z
f i r s t o n e a n d n o t s m a l l . S t r o n g n o n l i n e a r e f f e c t s a r e p r o d u c e d i n t h i s r e g i o n
b y t h e c o l l i s i o n t e r m . T h e d i f f u s i o n i n v e l o c i t y s p a c e t h a t c o l l i s i o n s p r c d u c e
d o e s n o t a l l o w t h e c u t to p e r s i s t , a n d a p r o g r e s s i v e s m o o t h i n g w i l l o c c u r .
I n t h e l i m i t z -»eo , t h e r e f o r e , t h a t s p l i t t i n g w i l l be v a l i d .
T h e f o r m f o r f a s z, -~°o g i v e n b y (32) h a s to c o m e o u t a u t o m a t i c a l l y
a n d i s n o t a b o u n d a r y c o n d i t i o n . It c a n b e o b s e r v e d i m m e d i a t e l y t h a t if
e
g = f ,(35) i s s a t i s f i e d . F r o m E q . (31), a s z -*•» ,
5X * J l „ or
1 9 Z1 A
a n d
* e , q M |3 + 1 - I M x cr
6l cgl c = (- T ~Zl + V q ° 9v~}X
z
i s s e e n t o s a t i s f y ( 3 6 ) . T h e r e f o r e (32) i s a s o l u t i o n of ( 2 9 ) - ( 3 0 ) i n t h e l i m i t
z -""oo. T h i s o n l y m e a n s , of c o u r s e , t h a t no l a y e r e x i s t s b e t w e e n z a n d z . L L &
* e
T h e r e a r e o t h e r p o s s i b l e f o r m s f o r 6 g a s z • * « j i n p a r t i c u l a r ,
(37) 6\ g = q f z - = 5 * \ r d z f
lc l c ^ M 1 A l j o z 1 M
- * e a -1 9 fM , n 4
6l c gl c = A q C 1 7 " * <38>
z
(The f i r s t o n e v a l i d f o r a n a r b i t r a r y \ . ) H o w e v e r , it c a n b e s h o w n t h a t
t h e s e l i m i t i n g f o r m s a r e r u l e d out b y E q . (30) i t s e l f . Define
• I
<
ln
H = \ (Si In g") vz dv
dH
dz, J (In geQ + l)[c(g*,g*) + P(g*)] dv < 0 . (39) 1
The inequality follows f r o m the u s u a l H - t h e o r e m for a l l known C ' s d H / d z1 cannot v a n i s h s i n c e f r o m (30), t o o ,
8 f e
J go
e- i * - v_ dv = 0 ,
and b e c a u s e f g v dv = c o n s t a n t ;* 0 at z, = 0 the s q u a r e b r a c k e t in
3 °o z -~ 1
e
(39) can n e v e r v a n i s h . T h u s , a s z —*•<» , g has to depend on z and to have a n o n - z e r o z flux, which r u l e s out both (37) and (38).
Of c o u r s e , one s t i l l h a s g a s an a r b i t r a r y d a t u m chosen such that a s e
z —*oo the d e s i r e d b e h a v i o r for f i s obtained. Our p r o b l e m i s to find g ; in the end, to show that g = 1. It s e e m s that the difficulty i s not to find
e a p r o p e r , unique g but that for any g , c o l l i s i o n s will have m a d e f Maxwellian to z e r o o r d e r in the l i m i t z -*°o and (32) can follow. To c l a r i f y and e s c a p e t h i s difficulty, c o n s i d e r f given in (32) at z —»•*> . We a r e going to u s e the o v e r s h o o t i n g in x. now, i . e . , we s h a l l take
advantage of \ « In A o r , b e t t e r , e « A . F r o m the value for A obtained below it can be s e e n that the condition e « A is the s a m e a s <J » A and our f o r m e r r e s t r i c t i o n C / A < O(l) c o m e s b a c k a g a i n .
In the l i m i t of an infinite o v e r s h o o t i n g | Y - lnA - * - « ) , the condition
at z. - 0 i s one of p e r f e c t r e f l e c t i o n . F r o m the a r g u m e n t in App. B, t h e r e
e Y*
c o n d i t i o n a t z = 0 w i l l a f f e c t to z e r o o r d e r o n l y t h e p o s i t i v e v t a i l of t h e
d i s t r i b u t i o n f u n c t i o n . H e n c e , i t i s p o s s i b l e t o o b t a i n i m m e d i a t e l y t h a t
-3
g = 1 , A A H = A .
T h e c u r r e n t a r r i v i n g i n t o t h e p r o b e i s
je z * A£+ 1 eP \ f h( - v ) v d v J M z z *•»
- -
^
le**
W
X'
Z.
(40)
S i n c e A = li*n_ e w e o b t a i n f o r £ < 1 , f r o m t h e e q u a l i t y of (27)
z _ ~ * - 0 ' r 1 / '
a n d (40),
a e "b^ = . | # " a t H, = 0 , £ < I (41a)
8 y 2 * r
w h e r e b = 0 + l ) / 2 , a = [ a (0 + 1) e ^ J "1 ( 3 / H y )1'2 . E q u a t i o n (4ia) c a n
b e w r i t t e n
8* c
U^H-^)=^)L^a-*
1w h i c h s h o w s t h a t x i n z ? i s n o t O ( x ) b u t Ofln or"™'" ) . A l s o , t h e
c o n d i t i o n < J / A < O(l) b e c o m e s
„MM
e
y
+ i A
.
1/P+l < O0)
1/2
w h e r e A = -(9i///ay)(04-1) (16y/3) ' . B e c a u s e a s A de c r e a s e s A c a n
b e c o m e a p p r e c i a b l y l a r g e r t h a n u n i t y , r e s u l t s s h o u l d b e a c c u r a t e r i g h t u p
In the r a n g e £ > 1 , the condition a t z = 0 i s one of p e r f e c t reflection s r 1 * and the condition (28) r e s u l t s in
^ = 0 at z = 0 , £ > 1 . (41b) dy 2 r
In I the d e n s i t y at z ~*0 w a s m a t c h e d to its value at z. ~* » ; b e c a u s e of this i m p r o p e r m a t c h i n g (the d e n s i t y i s not c o n s e r v e d in z to z e r o and f i r s t o r d e r s ) an additional f a c t o r a p p e a r e d in (41a) which l i m i t e d m o r e the valid r a n g e of Y and p r o d u c e d s o m e a n o m a l i e s in the p r o b e c h a r a c t e r i s t i c
(see S e c . V - A ) .
Some c o m m e n t s should be m a d e now. F i r s t , it i s obvious why the shape of the p r o b e along B h a s no s e n s i b l e influence on the c u r r e n t a s pointed out by Bohm. If i t s d i m e n s i o n along B i s , say, 0 ( R ) , t h e p r o b e l i e s e n t i r e l y in the z = 0 p l a n e ; only i t s c r o s s section a p p e a r s in the f o r m u l a t i o n . This would not be the c a s e if the p r o b e w e r e not p e r f e c t l y a b s o r b i n g .
Second, in the i n n e r l a y e r s the p r o b l e m i s o n e - d i m e n s i o n a l and P o i s s o n ' s equation i s u n i m p o r t a n t , a s s u g g e s t e d by Spivak for the whole p r o b l e m .
T h i r d , both the d e c r e a s e and the b l u r r i n g in the I-V d i a g r a m b e c o m e P
c l e a r . B e c a u s e of the inhibition of t r a n s v e r s e e l e c t r o n flux, any e l e c t r o n c u r r e n t is m a i n t a i n e d o v e r long d i s t a n c e s along the field. Since the ions a r e m o t i o n l e s s to O(jLt) with r e s p e c t to the e l e c t r o n s , any s e n s i b l e flux would e x p e r i e n c e a friction o v e r such a long d i s t a n c e that only a s m a l l e l e c t r o n c u r r e n t i s p o s s i b l e .
a l a r g e e l e c t r i c p o t e n t i a l can p r o d u c e l a r g e d i f f e r e n c e s in ion d e n s i t y , i . e . , in e l e c t r o n d e n s i t y . H e n c e , an o v e r s h o o t i n g i s built up inside the " s h a d o w . " T h e decay f a r f r o m the p r o b e of t h i s p o t e n t i a l h i l l d r a w s the e l e c t r o n c u r r e n t .
As Y - - » , I —0 and the h i l l d i s a p p e a r s ; thus A ~*"1 and if T = O(l) the r e s u l t u s e d for f i s not valid in zn , But then the whole z l a y e r
o 1 i
c o l l a p s e s . At Y = 0 no kink in the I-V d i a g r a m should be expected b e c a u s e for the i n n e r l a y e r s the effective p l a s m a potential i s given by the o v e r s h o o t i n g . As v i n c r e a s e s t h i s w i l l d i s a p p e a r , a s shown in S e c . V, and the p r e s e n t t h e o r y b r e a k s down.
F i n a l l y , although difficult of proof without a m u c h m o r e d e t a i l e d a n a l y s i s , it s e e m s that the e r r o r involved i n stating the equality of t o t a l c u r r e n t to
the p r o b e to that r e a c h i n g the plane z = 0 i s 0(tf). A l s o , when A < O(l), the ion c u r r e n t i s expected to d e c r e a s e f r o m i t s value for B = 0 at the
V
V. THE PROBE CHARACTERISTIC A . A n a l y s i s of the I-V D i a g r a m
P
We p r o c e e d now to d i s c u s s the d e t e r m i n a t i o n of e s s e n t i a l p a r t s of the I-V d i a g r a m . F r o m now on, w h e n e v e r the ion c u r r e n t , I , i s taken into a c c o u n t , a c y l i n d r i c a l or s p h e r i c a l p r o b e will be a s s u m e d ; a l s o £. > R .
The floating p o t e n t i a l , X = Xf • w h e r e I v a n i s h e s , i s the m o s t
a c c e s s i b l e r e s u l t f r o m e x p e r i m e n t s ; m o r e o v e r , the p l a s m a i s d i s t u r b e d the l e a s t . T h e r e f o r e an a c c u r a t e t h e o r e t i c a l d e t e r m i n a t i o n of Y. 1 S
r e s u l t s r e l a t e s the p l a s m a p o t e n t i a l to the o t h e r unknowns of o u r p r o b l e m . To find X, we w r i t e
F o r l a r g e , negative X • t h e effect of B on both I and I1 b e c o m e s s m a l l .
We s h a l l find then that t h e r e is a s m a l l change in I for l a r g e j x , I f r o m the value for B = 0; and a m u c h s m a l l e r change in I , for which we can then u s e r e s u l t s by o t h e r a u t h o r s for B = 0. The change in Xr i s s m a l l but n o n - n e g l i g i b l e .
e
L e t us c o n s i d e r f i r s t I . If in E q . (41a) we let x "** ~°° > t h e r e r e s u l t s lf/(x < 1, 0)~*0 . We can expand Xp = xp + i//. + . . . and obtain l i n e a r equations for \J/. , j = 0,1 . . . . F o r xp we get
dZ\j/ • a BxU
^ o , - s 9 s ^ o
+ X -— X - r — = 0
9 x 8 x
with the
9y
boundary^ conditions
xp = 0
&xp
- -a
8 y
at infinity
at y = 0 , x > 1
at y = 0, x < I .
A s soon a s we know tj/ (x < 1, y = 0) we obtain f r o m the left-hand side of E q . (41a),
k T 1/2 e x p x rl Q
It i s obvious that the t w o - d i m e n s i o n a l p r o b l e m i s not w e l l posed s i n c e L a p l a c e ' s equation h a s a l o g a r i t h m i c d i v e r g e n c e a t infinity. F r o m now on we shall take s - 1.
To solve for* \f/ we u s e an i n t e g r a l t r a n s f o r m a p p r o a c h j we obtain
^ (x,y> = \ l#q) e '•qy 4 y J ( x q ) dq
"o T h e conditions a t y = 0 give
.Boo
\ ^/(q) q JQ(xq) dq = a x < 1
° = 0 x > 1 , By i n v e r t i n g the Hankel t r a n s f o r m ,
"* -1 \{/(q) = a q Jx(q)
and
-1 . , . Z ^ 2 _ _ . + ( l . xZ) k K ( k ) , Y Ao0 . t f )
a
^ o
( x' y
) = :l n r -
E ( k ) +, i/2
+2 "
y x < 12 7TX '
2 xl / 2E { k ) ( l - x2) k K { k ) y Ao ^ ' g )
=
—Fk.
+, 1/2 " ~ T ~
X > 12 7T x
2 x 2 2 - l / 2 k = s i n « = — r-r-TT- , sin 0 = y[y + {1 - x) ] '
[ySfx + l)
2]
1/
2and K, E , and A a r e the c o m p l e t e and H e u m a n ' s e l l i p t i c i n t e g r a l s . F o r y = 0 , x < l , \f/ = 2ir~ a E(x) and thus
kT l / 2 ^ \ >
ie
=
e N- < ^ >
2*
R ZrZi7S- ^"T^iTS—
]• <
42>
Let us now c o n s i d e r I . The point to be m a d e i s that if C < O(l) and X i s negative enough the ion c u r r e n t i s somehow i n s e n s i t i v e to c e r t a i n m o d i f i c a t i o n s . A s s u m e f i r s t r < O(l) . We r e f e r the r e a d e r to F i g s . 2 0 , 21,27b, and 39 in Ref. 5. It can be i n f e r r e d that (if roughly t < 0.1 and
i v < - 3 . 5 ) for c h a n g e s of o r d e r unity in Y (and even in the f o r m of f at infinity), A I / l = O(10 ); for changes in the g e o m e t r y of the p r o b e , again I i s modified only slightly and it does not change at a l l by a s s u m i n g
e v p e r f e c t l y r e p e l l e d e l e c t r o n s (n = eA) .
At X w ¥ we found a s m a l l c o r r e c t i o n for I with r e s p e c t to the value for B = 0. The c o r r e c t i o n i s connected with a s m a l l o v e r s h o o t i n g of X. (amounting to a d e c r e a s e in x ) a nd a d i s t o r t i o n on the s p h e r i c a l s y m
-m e t r y (a-mounting to a change in shape of the p r o b e ) . A l s o , n will not have 6 X
the s a m e s p a t i a l dependence a s for B = 0; but n K e . T h e r e f o r e , a l l t h e s e
i
s m a l l effects will p r o d u c e h i g h e r - o r d e r m o d i f i c a t i o n s in I a s deduced
f r o m the c o n s i d e r a t i o n s of the p r e c e d i n g p a r a g r a p h . We can u s e the r e s u l t s for I f r o m Ref. 5.
In the c a s e T = O(l), the i n f o r m a t i o n f r o m Ref. 9 i s v e r y i n c o m p l e t e . H o w e v e r , in F i g . 11 of Ref. 9 it can be seen that u n d e r the s a m e conditions for C and V> d l / d \ i s v e r y s m a l l . Although i t i s not so c l e a r for the effect of g e o m e t r i c a l d i s t o r t i o n , i t is quite p r o b a b l e that again a s s u m i n g
e x i
n = eA w i l l not modify I . T h u s the r e s u l t s for t h e c a s e B = 0 can be
i
It should be pointed out that a s £J d e c r e a s e s the above s t a t e m e n t s on I b e c o m e l e s s t r u e . T h u s , the conditions on /3,T, and £ , a l r e a d y t a k e n advantage of in S e c . IV, have a l s o this m o t i v a t i o n of allowing the u s e of r e s u l t s for I obtained with B = 0.
A second f e a t u r e of the c h a r a c t e r i s t i c which is i m m e d i a t e l y obtained e x p e r i m e n t a l l y i s the slope at z e r o c u r r e n t . Again we can u s e for dl /dV the v a l u e s for B = 0. In fact, d In I / d Y « d In I /dX and the a p p r o x i m a -tion i s still b e t t e r . As for d l / d V , we obtain it f r o m E q . (42).
P
F o r X < - X, a^ *n e a r g u m e n t s about I given above r e m a i n v a l i d .
We have both components of the c u r r e n t for x_ UP to -°°- *n p a r t i c u l a r ,
the b e h a v i o r of I a s Y """* -°° can be taken f r o m the l i m i t B - * 0 . "" i
When Y > - 3 , I d e c r e a s e s rapidly and the d e c r e a s e depends s t r o n g l y on the p a r t i c u l a r c o n d i t i o n s . H o w e v e r , I i n c r e a s e s then, making the
r a t i o 1 / I l a r g e . To solve for I , n u m e r i c a l c o m p u t a t i o n s a r e n e c e s s a r y , but a c e r t a i n amount of information can be obtained a n a l y t i c a l l y .
Let us a s s u m e that the boundary condition for x < I , y = 0 w e r e \f/ ~ a w h e r e 0! is a given c o n s t a n t . 3^/(x < 1, y = 0 ) / 9 y would depend on a and on x . We can talk, h o w e v e r , in s o m e a v e r a g e s e n s e and n e g l e c t the x d e p e n d e n c e . Of c o u r s e what follows h a s only an a p p r o x i m a t e , but
meaningful, v a l u e . Then for the c u r r e n t e l e c t r o n d e n s i t y on the p r o b e we have
,e e^p e * , , y v
J = — ^ 7 7 5 = a b* < ^ (43)
* / l/2
where a = (32y/37T) ' <j and ${a) = - S\f//dy at x < 1, y = 0, $(a) depends
only on the differential equation for \p ; if this were linear f&(a) = Eoc with
E a constant. For small a this is certainly t r u e . For all ot, d&/da > 0.
We obtain next an implicit equation for j :
e * . *p " l n j 6- l n ( 2 7 r ) l / 2
j = a b * ( - * - )
For \ ""* -00» $ is linear ; thus
j6 = a* E f ) ^ - I n je- ln(27T)l/2]
- 3/^^/2
The dependence on b disappears and for |Y I large enough, j - e P/(27T)
If <& were linear for all a it would come out for Y -*•+<» ,
e * e *
j + OF- E Inj = cr E v + ct (44)
o r
je - cr*E<x^ - lnXp) ,
with ff E = 0(0") as a limiting slope. However, we can show that this result
is not valid. In effect,
\ > b o i x , .
e r « e $i(a)
so that da/dx_ > ° • T h e n d$/d^ > 0 o r
d_
i . e . ,
*
j * - < |3/fl . < 1 d xp + 1
where X = X(x "** *• v = °) a n a r e p r e s e n t s the overshooting. Hence, the
overshooting tends to disappear or, what is the s a m e , A ~^0(a) and the
formulation of the last section breaks down. Because of the exponential
character of the left-hand side of Eq. (43),
*
dXp J3+1
and for |3 -*oo (cold electrons or formal limit Z. ~*0) the overshooting
p e r s i s t s . For small p* it does not change; we saw it was negligible.
Therefore the limit X -*•+<* for fixed <T cannot be studied with the
present formulation.
Moreover, it seems that d<fr/da should decay as a grows because of
the exponential factor in the second t e r m of Eq. (26). The slope dj /dv
would fall fast as \ grows and it is possible that a zero slope is practically
reached before the overshooting has disappeared; j will depend now on b
e I e *
and dj /db > 0. We give below the dependence of j on CT , \ , and b :
(45a)
S je
db ' da
Lil
8
V
.e .1. e «y
.e
J
.e , d $ *
J da
,e ... i . .e d<& -.*
J da
*
, rr* tih
CT < * -, e
ll > Q
cr
d $ *
-r- a > 0
da
d*,v> : a J . d a
(45b)
B . N u m e r i c a l R e s u l t s
Equation (26), t o g e t h e r with t h e b o u n d a r y conditions (41a, b ) , w a s solved u s i n g the U n i v e r s i t y of C o l o r a d o CDC 3600 c o m p u t e r . The b e h a v i o r of fy
at infinity w a s obtained a n a l y t i c a l l y Using an a s y m p t o t i c c o o r d i n a t e e x p a n s i o n 2 2 l / 2
for l a r g e p = (x + y ) ' . Thus the domain of i n t e g r a t i o n w a s f i n i t e , F o r l a r g e p ,\p is s m a l l , a n d , expanding ^ = £ ^/ w h e r e ^ / ^ 1 ~* °
k a s p -* « , we get '
F o r the h o m o g e n e o u s p a r t , t h i s r e s u l t s in
1—1 a
U/„ = ) P (cos0) [~~^-r-+ b pnJ rH LJ n ' Lpn + i n ^ J
n
w h e r e P a r e L e g e n d r e p o l y n o m i a l s and y = p cos# . Obviously b = 0 for a l l n and a l s o a = 0 for odd n , b e c a u s e y - 0 i s a plane of s y m m e t r y ,
Thus
a a
ty-n - "^"P + - r P , + . . .
rH P o 3 Z
P W e obtain next for \f/. ,
Id ao 9 ao
* 1 I ro x 9x p 9x p
Using s p h e r i c a l c o o r d i n a t e s in both s i d e s of t h i s equation and expanding 0. in L e g e n d r e p o l y n o m i a l s r e s u l t s in
2 ZQ
a cos Q
xf/,
=
Hence for large p ,
2 2„
a a cos 8 ~
P P
A rectangular area was used as the domain of integration, x = 0 and y = 0
being two sides. On the other two, the last equation was taken as a boundary
condition, eliminating a between t// and V ^ .
Although no use was made of it, a relation between the behavior of lj/
on separate regions of the domain was available as a criterion of convergence;
an over-relaxation factor could be obtained from it. To derive that relation
we rewrite Eq. ,{26) as
* • * * • ; £ * £ < • • * • « •
e•
This is a Poisson-like equation; after integrating over the whole space,
the right-hand side vanishes. Hence
\ V-VtydV = \ V ^ - dS = 0
from which
s
x d x™-ay .
+ a = 0 y=0 °
Since the spatial "charge, " ©, gives a zero total result, a , which o
represents the total charge in a solution to the Poisson equation, must be
equal to the charge on the disc.
Equation (26) was changed into a finite difference equation and solved
value of a , for which the solution w a s found a n a l y t i c a l l y a b o v e . An a p p r o -p r i a t e scaling w a s u s e d a s i n i t i a l i z a t i o n for the next l a r g e r value of a .
In F i g . 1 the n o n d i m e n s i o n a l c u r r e n t I {m / k T ) (e N R ) i s given a s a function of v for /3 = 1 and s e v e r a l <7 . Only v a l u e s of V < 0 a r e given and the v a r i a t i o n of I with (3 i s not i n c l u d e d . "When the c o m p u t a t i o n s w e r e m a d e it w a s unfortunate that an i m p r o p e r m a t c h i n g w a s u s e d and a factor 2[1 + e r f (^\j//Z - v ) ' ]~ a p p e a r e d in the left-hand side of E q . (41a). It p r o d u c e d two s p u r i o u s effects: f i r s t , it gave 9 l / 9 b < 0 a s p r e s e n t e d in I , c o n t r a r y to what i s expected f r o m (45c). T h i s is b e c a u s e if the above f a c t o r is included the d e n o m i n a t o r of (45c) i s s t i l l p o s i t i v e but a new and s t r o n g negative t e r m adds to ($ - (d<J>/dO!) OL ] . Second, I ( \ ) t u r n e d u p w a r d s when Y w a s a p p r o a c h i n g \ = fict/Z . The r e a s o n i s that the e r r o r function c a u s e s 91/9 Y to grow to infinity for \ ~ 0Ot/Z; [d/dt erf t
-t l / 2
e /Zt )] , M o r e o v e r , the s p u r i o u s factor r e s t r i c t e d the r a n g e of
validity of the t h e o r y s i n c e it can be seen that the o v e r s h o o t i n g d i s a p p e a r s for l a r g e x when that is not p r e s e n t . Some p r o b l e m of s t o r a g e w a s found a l s o in the 3600 c o m p u t e r for x > 2 . It i s intended to p r e s e n t in the future m o r e c o m p l e t e n u m e r i c a l r e s u l t s , extended to positive v a l u e s of Y that would allow d e t e r m i n a t i o n of the c h a r a c t e r of d<&/dot for l a r g e a, and give the v a r i a t i o n with /3 , Some e x t e n s i o n s d i s c u s s e d in the next
VI. DISCUSSION A . E x t e n s i o n s of the T h e o r y
A n u m b e r of effects can be included in the p r e s e n t t h e o r y in a s t r a i g h t -f o r w a r d w a y .
20 , ., N o n - u n i f o r m w o r k f u n c t i o n s w e r e c o n s i d e r e d by M e d i c u s and a r e e a s i l y
allowed for h e r e if c y l i n d r i c a l s y m m e t r y i s p r e s e r v e d . (Uniform w o r k f u n c -tions a r e taken into account by shifting the o r i g i n of p o t e n t i a l s . )
Second, n o n - p e r f e c t l y a b s o r b i n g p r o b e s can be included in the t h e o r y . B e c a u s e of the w e l l - d e f i n e d motion of the e l e c t r o n s t o w a r d the p r o b e , t h i s a m o u n t s again to a modification of E q . (41a) w h e r e x will a p p e a r e x p l i c i t l y . H e r e the shape of the p r o b e along B w i l l be of i m p o r t a n c e . The r e f l e c t i o n coefficient should not be n e a r 1 .
T h i r d , r e c o m b i n a t i o n is e a s i l y allowed for; in the z - £ s p a c e , n ~ n ~ e ^ ' and the continuity equation (2 6) would r e q u i r e only a t e r m coe ^ w h e r e CO would be a n o n - d i m e n s i o n a l r e c o m b i n a t i o n coefficient. In the i n n e r % l a y e r s , g r a d i e n t s a r e s t r o n g e r , d e n s i t i e s w e a k e r , so that r e c o m -b i n a t i o n can -be n e g l e c t e d .
In this c a s e we can t r e a t the c y l i n d r i c a l p r o b l e m (s = 0). E v e n if CO = 0 it s e e m s simple to obtain the r e s u l t s for an elongated e l l i p s o i d a l p r o b e .
such a t h e o r y would be useful h e r e b e c a u s e the m e a s u r e m e n t would p r o b a b l y be n o n - l o c a l . As for w e a k l y ionized g a s e s , no difficulties s e e m to a p p e a r ,
They will be c o n s i d e r e d in future w o r k .
As for the s t r e n g t h of 33 the f o r m u l a t i o n of S e c . IV r e q u i r e d only \i/a < O(l) (which, of c o u r s e , allows i. < R but not i. « R ). The l i m i t <7 - * 0 h a s a
p r o p e r b e h a v i o r in o u r f o r m u l a t i o n but ( i - » 0 a t the s a m e t i m e . In S e c . V, when I w a s c o n s i d e r e d , the m o r e r e s t r i c t i v e JL > R w a s n e e d e d . B e c a u s e
1
of £ < O(l) , I. » R i s not r e q u i r e d .
To allow for Z ~ 1 h a s the a d d i t i o n a l , but p e r h a p s s o l v a b l e , c o m p l i c a t i o n of lacking s o m e b o u n d a r y condition in the z - £ s p a c e for n o r % .
The p r e s e n t t h e o r y r e q u i r e d (7 < 0(1). H o w e v e r , in the l i m i t <7 -**> o u r solution goes back to the known r e s u l t
when only negative Y a r e c o n s i d e r e d . Thus it s e e m s that the p r e s e n t s o l u -tion i s v a l i d for w e a k e r magnetic fields down to the l i m i t B = 0 for v a l u e s of X up to that for which X ~ X (for B = 0, X ~ 0 ) . The o v e r s h o o t i n g thus h a s the p r o p e r t y of extending to positive X the r a n g e for which the s i m p l e
