close

Вход

Забыли?

вход по аккаунту

?

The interaction of guided microwaves with a plasma in a transverse magnetic field

код для вставкиСкачать
THE IN TERACTIO N O F GUIDED MICROWAVES W ITH A PLASM A
IN A TRANSVERSE M AGNETIC F IE L D
A D is se rta tio n
By
G E R A L D S T E A R N S HARM O N
S u b m itte d to th e G r a d u a t e School of th e
A g r i c u l t u r a l a u d M e c h a n i c a l C o lle g e of T e x a s in
p a r t i a l f u l f i l l m e n t of th e r e q u i r e m e n t s f o r th e d e g r e e of
D O C T O R O F P H IL O S O P H Y
A u g u st,
M a j o r S u b je c t:
1962
P h y sic s
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: DP16609
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
о
UMI
UMI Microform DP16609
Copyright 2009 by ProQuest LLC.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 E. Eisenhower Parkway
PO Box 1346
Ann Arbor, Ml 48106-1346
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
THE INTERA C TIO N O F GUIDED MICROWAVES WITH A PLASM A
IN A TRANSVERSE M AGNETIC F IE L D
A D isse rta tio n
By
G E R A L D S T E A R N S HARM ON
A p p r o v e d a s to s t y l e a n d c o n te n t by;
( C h a i r m a n of C o m m i t t e e )
( H m d of D e p a r t m e n t )
A ugust,
85
1962
18
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
A CKNOW L E D G M E N T S
T h e a u t h o r w i s h e s to e x p r e s s h is s i n c e r e th a n k s to D r ,
M e lv in E i s n e r f o r h i s i n v a lu a b le g u id a n c e t h r o u g h o u t th e c o u r s e of
th is w o rk .
He a l s o w i s h e s to e x p r e s s h is a p p r e c i a t i o n to D r .
Jam es
P o t t e r f o r h is m o s t h e lp f u l s u g g e s t i o n s d u r in g th e w r i t i n g of th e
m a n u sc rip t.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
TABLE
OF
CONTENTS
Page
L i s t of F i g u r e s
C h ap ter
I.
II.
.......... л . . . .............
I n tr o d u c ti o n
v
.............
1
T h e o r y . . . . , . , . ..............
A,
B,
C,
8
M e th o d s o f D e s c r i p t i o n of a P l a s m a
M ic r o w a v e I n t e r a c t i o n w ith th e
P l a s m a ..............
E le c tro n T e m p e r a tu r e
8
12
28
III.
E q u ip m en t.
33
IV,
E x p e rim e n ta l P r o c e d u re and D a t a . , . . , . .
41
A.
B.
C.
D.
V.
R e s u lts and D is c u s sio n ,. . . . . . . .
A,
B,
C,
D,
'VI.
R eflec tio n C o e f f i c i e
.......................
M a g n e tic F i e l d
C o llisio n F re q u e n c y
E le c tro n T e m p e ra tu re
n
..................
t
s
41
41
43
44
51
The D i s c h a r g e
E le c tro n T e m p e ra tu re . . . . . . . . . . . . .
C o llisio n F re q u e n c y . . . . . . . . . . . . . . .
E le c tr o n D e n sity
51
58
59
60
Sum m ary л, ллллл. . л л л л л л . . , . , . , л л , , л , л
63
B ib lio g ra p h y . . . . . a . . . . . . . . . . л . ╗ л ? . . ╗ л л .
iv
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
65
LIST O F F IG U R E S
F ig u re
1
2
3
4
5
6
7
8
9
10
11
Fuge
R e f l e c t i o n of g u id e d m i c r o w a v e s b y a p l a s m a s l a b ,
w ith i n c r e a s i n g e l e c t r o n d e n s i t y , f o r s e v e r a l
v a lu e s of e l e c t r o n c o l l i s i o n f r e q u e n c y . . . , . , . , . . .
25
C i r c u i t f o r d o u b le p r o b e t e m p e r a t u r e m e a s u r e н
m e n ts
...................
30
W a v e g u id e s e c t i o n u s e d f o r p r o d u c t i o n of th e
p l a s m a ......................
37
B lo c k d i a g r a m of e q u i p m e n t f o r m e a s u r e m e n t s
of r e f l e c t i o n a n d a
b
s
o
r
p
t
i
o
n
39
V a r i a t i o n in m i c r o w a v e p o w e r r e f l e c t e d f r o m a
p l a s m a - f i l l e d w a v e g u id e w ith i n c r e a s i n g
......................
t r a n s v e r s e m a g n etic fie ld
42
C y c l o t r o n r e s o n a n c e a b s o r p t i o n of g u id e d
m i c r o w a v e s in a p l a s m a . ............
45
V a r i a t i o n of e l e c t r o n p l a s m a f r e q u e n c y a n d
e l e c t r o n c o l l i s i o n f r e q u e n c y w ith o s c i l l a t o r
p o w e r i n p u t .................
46
C h a ra c te ris tic c u rre n t-v o lta g e curve fo r m e a s u r e н
m e n t of e l e c t r o n t e m p e r a t u r e
47
V a r i a t i o n of e l e c t r o n t e m p e r a t u r e w ith o s c i l l a t o r
in p u t c u r r e n t , in th e p r e s e n c e of a t r a n s v e r s e
m a g n etic f i e l d .
...................
49
V a r i a t i o n of e l e c t r o n t e m p e r a t u r e w ith i n c r e a s i n g
m a g n e t i c f lu x d e n s i t y ...........................................................
50
R - F e l e c t r i c f i e l d w ith in a p l a s m a - f i l l e d w a v e н
g u id e s e c t i o n ; p e r p e n d i c u l a r o r i e n t a t i o n . . . . . . . . .
51
v
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
12
13
O sq illa to r c u rre n t, and re fle c tio n and tra n s m is s io n
c o e f f i c i e n t s a s a f u n c tio n of o s c i l l a t o r v o lta g e ;
u n s t a b l e d i s c h a r g e .............................
55
O sc illato r c u rre n t, and re fle c tio n and tra n s m is s io n
c o e f f i c i e n t s a s a f u n c tio n of o s c i l l a t o r v o lta g e ;
stab le d is c h a rg e
...........................
57
vi
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER
I
IN T R O D U C T IO N
A p l a s m a m a y b e d e f in e d r o u g h l y a s t h a t p a r t of a n i o n i z e d
g a s w h ic h i s e l e c t r i c a l l y v e r y n e a r l y n e u t r a l ,
Thus,
c a l d i s c h a r g e s in g a s e s w i l l h a v e a p l a s m a r e g i o n .
m ost e le c triн
(T h e i n t e r i o r
r e g i o n of a m e t a l m a y a l s o b e d e s c r i b e d a s a p l a s m a if th e d e f i n i н
tio n i s b r o a d e n e d s l i g h t l y , )
A s a c o n s e q u e n c e of the c o n d itio n of
e l e c t r i c a l n e u t r a l i t y , t h e r e a r i s e c e r t a i n c h a r a c t e r i s t i c s of th e
i o n i z e d g a s w h ic h a r e n o t d i r e c t l y a t t r i b u t a b l e to th e in d iv id u a l io n s
and e le c tro n s,
of p a r t i c l e s .
b u t r a t h e r to th e c o l l e c t i v e b e h a v i o r of th e s y s t e m
F o r e x am p le,
ch arg e n e u tra lity a r is e s ,
suppose a sm a ll sca le d e p a rtu re fro m
i.e .,
th e e l e c t r o n s in a s m a l l r e g i o n
b e c o m e m o m e n t a r i l y d i s p l a c e d w ith r e s p e c t to th e i o n s ,
(We s h a l l
a t a l l t i m e s ta k e th e io n s to be s t a t i o n a r y ; s i n c e th e y a r e m u c h
h e a v i e r th a n th e e l e c t r o n s ,
p a riso n . )
t h e i r t h e r m a l v e lo c i ty i s s m a l l b y c o m н
T h e n th e e l e c t r i c f o r c e a r i s i n g b e tw e e n th e c h a r g e s w ill
a c t so a s to r e s t o r e e q u i l i b r i u m ,
th e e l e c t r o n s .
a n d t h e r e b y s e t u p o s c i l l a t i o n s of
T h i s ty p e of o s c i l l a t i o n i s c l e a r l y a c o l l e c t i v e p h e н
nom enon.
S ince a n e l e c t r o n p l a s m a o s c i l l a t i o n d o e s in f a c t o c c u r ,
th e re
m u s t b e s o m e l a r g e s t v o lu m e in th e p l a s m a w ith in w h ic h th e c h a r g e
s e p a r a t i o n w ill o c c u r s p o n t a n e o u s l y .
W ith in t h i s v o l u m e ,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
th e n , th e
2
c o n d itio n of e l e c t r i c a l n e u t r a l i t y d o e s n o t a p p ly .
The a c tu a l m a g n iн
tu d e of th e ( a v e r a g e ) c h a r g e s e p a r a t i o n d i s t a n c e w ill c l e a r l y b e a
f u n c tio n of th e e l e c t r o n t h e r m a l e n e r g y .
If th is e n e r g y ( p e r u n it
v o lu m e ) i$ s e t e q u a l to th e m a x i m u m p o t e n t i a l e n e r g y of th e s e p a н
ra te d ch arg e,
sep a ra tio n ,
o n c e c a n f in d a m a x i m u m p r o b a b l e v a lu e h f o r t h i s
?<i?m i
T he r e s u l t of s u c h a c a l c u l a t i o n g iv e s (in r a t i o n a l i z e d
M KS u n its )
w h e r e ?Q, k, a n d T a r e the p e r m i t t i v i t y of a v a c u u m ,
th e B o l t z н
m a n n c o n s t a n t , a n d th e a b s o l u t e t e m p e r a t u r e , r e s p e c t i v e l y ,
th e e l e c t r o n d e n sity * a n d e the e l e c t r o n i c c h a r g e .
i s o r d i n a r i l y c a l l e d th e D e b y e le n g th .
sh o w n th a t ,
n is
T he d i s t a n c e h
B o h m a n d P i n e s ( (1) h a v e
if i t i s d e s i r e d to o b s e r v e th e i n t e r a c t i o n of a w a v e
w ith th e c o l l e c t i v e ( p l a s m a ) o s c i l l a t i o n s , th e w a v e le n g th m u s t b e
g r e a t e r th a n th e D e b y e le n g th .
. 05 c m o r l e s s .
c e n tim e te rs,
(In o u r c a s e ,
w e s h a l l f in d h to be
A s th e m i c r o w a v e w a v e le n g th i s of th e o r d e r of
no d if f ic u l ty w ill a r i s e f r o m t h i s l i m i t a t i o n . )
A
r e a s o n a b l y p r e c i s e d e f in itio n of a p l a s m a m a y b e g iv e n in t e r m s of
th e D e b y e le n g th .
If th e d i s t a n c e h i s s m a l l c o m p a r e d w ith th e d i н
m e n s i o n s of th e s y s t e m ,
Thus,
th e n th e i o n i z e d g a s is c a l l e d a p l a s m a .
d i s c h a r g e s of v e r y low d e n s i t i e s a n d / o r v e r y h ig h t e m p e r a н
t u r e s a r e n o t p l a s m a s in th e s t r i c t a p p l i c a t i o n of th e d e f in itio n .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
3
R e t u r n i n g to th e s u b j e c t of e l e c t r o n o s c i l l a t i o n s , o n e m a y a l s o
?how t h a t th e ( a n g u la r ) f r e q u e n c y of o s c i l l a t i o n ,
0> 2 = ^
P
OOp, is g iv e n by
,2
?0m
?
w h e r e m is th e m a s s of th e e l e c t r o n .
Now f o r o r d i n a r y g a s e o u s
d i s c h a r g e s in th e l a b o r a t o r y , n ^ l O ^ m , " ^ .
A c a l c u l a t i o n th e n
s h o w s t h a t the c o r r e s p o n d i n g v a lu e of th e p l a s m a f r e q u e n c y
a p p r o x i m a t e l y 5 x 10^ c p s .
2 IT
is
One w o u ld th e n e x p e c t th e i n t e r a c t i o n
w ith th e p l a s m a of e l e c t r o m a g n e t i c w a v e s in t h i s f r e q u e n c y r a n g e
to e x h i b i t m a r k e d v a r i a t i o n s a s th e e l e c t r o n d e n s i t y i s c h a n g e d .
Suqh i s in d e e d th e c a s e ,
and it is fo r th is r e a s o n th a t m ic ro w a v e s
a r e c o m m o n l y e m p l o y e d a s a p r o b e to s tu d y th e p r o p e r t i e s of
p la sm a s.
It s h o u ld be p o in te d o u t t h a t t h e r e w ill b e a c o r r e s p o n d i n g
p l a s m a f r e q u e n c y f o r th e i o n s ,
b u t i t w ill be s m a l l e r t h a n th e
e l e c t r o n f r e q u e n c y b y the f a c t o r ( m e / m j ) ^ ^ ,
u s u a l l y w e l l b e lo w th e m i c r o w a v e r a n g e ,
T h e io n f r e q u e n c y is
a n d in o u r c a s e i t is
a p p r o x i m a t e l y 10? c p s .
A n a d d i t i o n a l c o n s i d e r a t i o n in m a n y p l a s m a s t u d i e s is the
p r o b l e m of c o n ta in in g th e p l a s m a in a w e l l r d e f i n e d r e g i o n of s p a c e .
T h e d if f u s io n of b o th e l e c t r o n s a n d io n s w ill n a t u r a l l y t e n d to d ilu te
th e p l a s m a .
W h e n e v e r h ig h e l e c t r o n a n d ion c o n c e n t r a t i o n s a r e
d e s i r e d (a s in f u s i o n p l a s m a s ) ,
s o m e m e c h a n i s m m u s t be fo u n d to
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
4
m i n i m i z e t h i s d if f u s io n .
T h is i s a c c o m p l i s h e d b y a p p ly in g a s t r o n g
m a g n e t i c f i e l d ( s e v e r a l th o u s a n d g a u s s ) a lig n e d p a r a l l e l to th e d e н
s ir e d boundary.
Then a p a rtic le tr a je c to r y in itia lly p e rp e n d ic u la r
to th e b o u n d a r y w o u ld b e c o m e c i r c u l a r ,
a n d the d if f u s io n t h r o u g h
th e b o u n d a r y w o u ld b e r e d u c e d a c c o r d i n g l y .
Now a s s o c i a t e d w ith
th e c i r c u l a r o r b i t i s th e c y c l o t r o n ( a n g u la r ) f r e q u e n c y C0c of th e
e le c tro n s,
w h ic h i s e a s i l y sh o w n to b e g iv e n by
w h e r e B is the m a gn etic; f i e l d s t r e n g t h .
F o r a f i e l d of 1000 g a u s s
th e f r e q u e n c y w o u ld be f = Wc /2 ir ~ 5 x 10*о c p s , a n d w o u ld a l s o be
in th e m i c r o w a v e r a n g e .
A s w o u ld be e x p e c t e d , th e p r e s e n c e of
s u c h a m a g n e t i c f i e l d i n f l u e n c e s th e p r o p a g a t i o n of m i c r o w a v e s
t h r o u g h th e p l a s m a .
O nce a g a i n , m i c r o w a v e s c a n s e r v e a s a c o n н
v e n i e n t p r o b e to s tu d y th is i n t e r a c t i o n .
P r e v i o u s t h e o r e t i c a l s t u d i e s h a v in g a b e a r i n g u p o n t h i s w o r k
c a n b e c l a s s i f i e d in to two g e n e r a l c a t e g o r i e s ,
In th e f i r s t ,
th e
p l a s m a i s a s s u m e d to be a h o m o g e n e o u s m e d i u m of i n f i n ite e x t e n t
(a t l e a s t in tw o d i r e c t i o n s ) ,
p la n e w a v e .
(2),
(3),
(4)
a n d th e m i c r o w a v e i s t a k e n to b e a
AH e s s e n t i a l d e t a i l s c o n c e r n i n g th e p r o p a н
g a tio n of th e w a v e in the p l a s m a a r e r e a d i l y o b ta in e d f r o m th e
th e o ry .
T h e s e s t u d i e s c a n be u s e d o n ly a s a g e n e r a l g u id e in
t r e a t i n g th e p r o b l e m f o r a m o r e r e a l i s t i c p l a s m a ,
how ever,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
5
In th e s e c o n d c a t e g o r y th e p l a s m a is b o u n d e d .
It i s e i t h e r
h o m o g e n e o u s , o r in h o m o g e n e o u s in s u c h a w a y t h a t th e e l e c t r o n d e n н
s i t y i s a know n f u n c tio n of p o s i t i o n .
T h e m i c r o w a v e i s ta k e n to be
a p r o p a g a t i n g w a v e s a t i s f y i n g th e a p p r o p r i a t e b o u n d a r y c o n d itio n s .
Jt is th e n g e n e r a l l y p o s s i b l e to c a l c u l a t e s m a l l c h a n g e s in th e
p r o p a g a t i o n c o n s t a n t due to th e p l a s m a ,
if i t i s s u f f i c i e n t l y d ilu te .
One m a y t e s t e x p e r i m e n t a l l y th e g r o s s f e a t u r e s of th e t h e o r y
in e i t h e r of t h e s e c a s e s ; e n t i r e l y r e a s o n a b l e v a l u e s of e l e c t r o n
d e n s i t y a n d c o l l i s i o n f r e q u e n c y a r e r e a d i l y fo u n d .
T h e a p p r o a c h to b e e m p lo y e d h e r e i s s o m e w h a t d i f f e r e n t ,
We
s h a l l t r e a t in d e t a i l th e i n t e r a c t i o n of m i c r o w a v e s w ith a h o m o g e н
n e o u s p l a s m a w h ic h c o m p l e t e l y f i l l s a s e c t i o n of w a v e g u id e , a n d
w h ic h is i m m e r s e d in a s t e a d y , u n i f o r m ,
t r a n s v e r s e m a g n e tic fie ld .
So m e of th e r e s u l t s h a v e a l r e a d y b e e n fo u n d b y a n u m b e r of a u t h o r s ,
th o u g h f r o m a d i f f e r e n t s e t of i n i t i a l c o n d itio n s .
(5),
(7),
We s h a l l f i r s t o b ta in e x p r e s s i o n s f o r th e m i c r o w a v e c o n d u c н
t i v i t y f o r a p l a s m a in a u n i f o r m m a g n e t i c f i e l d , fo llo w in g c l o s e l y
th e m e t h o d of A l l i s (2), w ith n e c e s s a r y m o d i f i c a t i o n s a s i n d i c a t e d
b y M o lm u d (8).
T h e p r o p a g a t i o n c o n s t a n t s w ill th e n b e o b ta in e d f o r
a w a v e p r o p a g a t i n g w ith in a w a v e g u id e th r o u g h a p l a s m a in a t r a n s i
v e r s e m a g n e tic fie ld ,
T h e r e a s o n f o r c h o o s in g th e d i r e c t i o n of
p r o p a g a t i o n to b e t r a n s v e r s e to th e f i e l d i s t h a t , w h e n u s e d a s a
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
6
d i a g n o s t i c to o l f o r p l a s m a s ,
m a d e to p r o p a g a t e t h i s w a y .
th e m i c r o w a v e s a r e m o s t c o n v e n ie n tly
The e x p re ssio n s fo r tr a n s m is s io n and
r e l f e c t i o n c o e f f i c i e n t s w ill b e fo u n d in t e r m s of t h e e l e c t r o n d e n s i t y
and c o llisio n fre q u e n c y ,
w h e n th e m i c r o w a v e is p o l a r i z e d in th e
d i r e c t i o n of the m a g n e t i c f i e l d .
A lso , a p p ro x im a te e x p re ss io n s
w i l l be fo u n d f o r th e s i t u a t i o n w h e n the m i c r o w a v e i s p o l a r i z e d
p e r p e n d i c u l a r to th e m a g n e t i c f i e l d .
So f a r a s i s k n o w n ,
th e e x н
p r e s s i o n o b ta in e d f o r th e r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s
( s e e a q u a ti o n s
[ 4 0 ] a n d [41 ] , w ith [ 3 7 ] , p a g e 22) h a v e n o t b e e n
fo u n d in a n y p r e v i o u s i n v e s t i g a t i o n .
T h e e x p e r i m e n t a l p o r t i o n of t h i s s tu d y is p r i n c i p a l l y c o n c e r n e d
w ith th e m e a s u r e m e n t s of r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s .
T h e p l a s m a is f o r m e d by a r a d i o f r e q u e n c y d i s c h a r g e in a s e c t i o n
of w a v e g u id e , a n d th e t r a n s m i t t e d a n d r e f l e c t e d p o w e r m e a s u r e d
u n d e r v a rio u s d is c h a rg e co n d itio n s.
T h e r e s u l t s of t h e s e m e a s u r e н
m e n t s a r e th e n e m p lo y e d in c o n ju n c t io n w ith th e t h e o r e t i c a l r e s u l t s
to f in d th e e l e c t r o n d e n s i t y w ith in th e p l a s m a ,
S u b s i d i a r y e x p e r i m e n t s in v o lv e the m e a s u r e m e n t of e l e c t r o n
co llisio n freq u e n cy and e le c tro n te m p e ra tu re .
We s h a l l th e n c o n s i d e r a m e t h o d f o r o b ta in in g th e e l e c t r o n
te m p e ra tu re .
It w i l l b e s e e n in th e n e x t s e c t i o n t h a t th e e x p r e s s i o n
f o r th e c o n d u c ti v ity r e q u i r e s t h a t th e e l e c t r o n t e m p e r a t u r e b e
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
7
r e l a t i v e l y low .
T h u s , to e s t a b l i s h th e v a l i d i t y of th e r e s u l t s ,
e s s e n t i a l to m e a s u r e th e t e m p e r a t u r e .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
it is
8
CHAPTER
II
THEORY
A.
M e th o d s of D e s c r i p t i o n of a P l a s m a
T h e s i m p l e s t c o n c e p tu a l a p p r o a c h to th e p r o b l e m of a d e н
s c r i p t i o n of a p l a s m a is to a n a l y z e th e b e h a v i o r of a s i n g l e a v e r a g e
p a rtic le .
D if f ic u lty ,
of c o u r s e ,
w h a t is m e a n t by " a v e r a g e . "
a r i s e s in a t t e m p t i n g to e s t a b l i s h
F u rth e rm o re ,
a n a p p r o a c h of th is
s o r t m u s t ig n o r e a n y c o l l e c t i v e b e h a v i o r of th e s y s t e m of p a r t i c l e s .
N e v e r t h e l e s s , b y p la c i n g s u i t a b l e r e s t r i c t i o n s on th e n a t u r e of th e
p l a s m a , w e w ill s e e th a t th e m e t h o d i s of s o m e v a lu e ,
la r,
if th e p l a s m a is d ilu te (1% i o n i z a t i o n o r l e s s ) ,
ly d e n s e ( 1 0 ^
p a r t i c l e s p e r c u b ic c e n t i m e t e r ) ,
a p p r o a c h is u s e f u l.
In p a r t i c u н
th o u g h r e l a t i v e н
th e s i n g l e p a r t i c l e
It h a s th e a d v a n ta g e th a t th e r e s p o n s e of th e
p l a s m a to e x t e r n a l f o r c e s is e a s i l y v i s u a l i z e d .
We d e m o n s t r a t e
th e m a n n e r in w h ic h i t is e m p lo y e d in s e c t i o n B .
S in c e i t i s i m p o s s i b l e to fo llo w th e m o tio n of a p a r t i c l e e x p e r н
im e n ta lly ,
a m o r e r e a l i s t i c a p p r o a c h is to e m p lo y th e d i s t r i b u t i o n
f u n c tio n f f o r th e s y s t e m .
If t h i s f u n c tio n is k n o w n , th e a v e r a g e
v a lu e of a n y q u a n tity Q(v) m a y be fo u n d by th e r e l a t i o n s h i p
1
w here
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
[1 ]
9
n ( r , t) = J f d 3v
i s th e p a r t i c l e d e n s i t y ,
t is tim e ,
[2 ]
r and v a r e p o sitio n and v e lo city c o o rd in a te s ,
a n d th e i n t e g r a t i o n i s o v e r a l l v e l o c i t y s p a c e .
m a y b e d e s c r i b e d by n o tin g t h a t fd
Thus f
r d v r e p r e s e n t s th e n u m b e r of
p a r t i c l e s in the e l e m e n t of v o lu m e d J r d J v of p h a s e s p a c e .
A p p lic a -
tio n of th e L douville t h e o r e m to th e c h a n g e of f w ith t i m e y i e l d s th e
B o l t z m a n n e q u a ti o n (9):
df
-TT + v ?
ot
H ere a =
7-?f + a * 7-* f =
v
A^f'N
( л. . }
^ - 'c o l l i s i o n s
r? i
L3 J
a c c o u n t s f o r th e c h a n g e in th e n u m b e r of p a r t i c l e s in
d 3 r d 3 v d u e to th e a p p l i c a t i o n of a n e x t e r n a l f o r c e m a .
T h e l e f t s id e
of [ 3 ] r e p r e s e n t s th e d i f f e r e n c e b e tw e e n the p a r t i c l e d e n s i t y a t the
t i m e (t + dt) a t ( r + v d t ) ,
In th e a b s e n c e of c o l l i s i o n s ,
(v + a d t ) ,
a n d th a t a t th e t i m e t a t r , v.
t h i s d i f f e r e n c e m u s t be z e r o .
s o lu tio n to [ 3 ] g iv e s a n e x p r e s s i o n f o r f in t e r m s of r ,
e x p lic itly .
v, a n d t,
If th e c o l l i s i o n s m a y b e r e g a r d e d a s e n c o u n t e r s b e tw e e n
tw o p a r t i c l e s ,
th e n th e M a x w e ll d i s t r i b u t i o n i s th e s o l u t i o n of [ 3 ]
f o r a n y s y s t e m of p a r t i c l e s in t h e r m a l e q u i l i b r i u m ,
th e d e t a i l s of th e c o l l i s i o n a l p r o c e s s .
lib riu m ,
T he
r e g a r d l e s s of
F o r s y s t e m s n o t in e q u i н
h o w e v e r , th e t e r m on th e r i g h t in [ 3 ] , w h ic h a c c o u n t s f o r
th e e f f e c t s of c o l l i s i o n s ,
m u s t be e x p r e s s e d in t e r m s of th e f o r c e
of i n t e r a c t i o n b e t w e e n th e p a r t i c l e s .
Since th e a p p l i c a t i o n of a n
e x t e r n a l e l e c t r i c f i e l d to th e p l a s m a w ill p r e v e n t th e s y s t e m f r o m
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
10
re a ch in g th e rm a l e q u ilib riu m ,
An a lte r n a te p ro c e d u re ,
th is f o r c e m u s t b e know n.
in th e a b s e n c e of s u c h i n f o r m a t i o n ,
a n d o n e w h ic h i s a p p l i c a b l e f o r s m a l l d i s t u r b a n c e s , i s to a p p r o x i н
m a t e th e c o l l i s i o n t e r m b y th e e x p r e s s i o n - (f - f 0 )^c .
th e M a x w e ll d i s t r i b u t i o n ,
H e r e f Q is
a n d Vc i s a c o n s t a n t h a v in g th e c h a r a c t e r н
i s t i c s of a c o l l i s i o n f r e q u e n c y .
T h e n f o r a g iv e n e x t e r n a l f o r c e , f
m a y b e fo un d .
In th e e v e n t t h a t th e d i s t u r b a n c e s a r e n o t s m a l l ,
a n d if the
d e t a i l s of th e c o l l i s i o n s a r e n o t k no w n , i t i s s t i l l p o s s i b l e to o b ta in
s o m e i n f o r m a t i o n a b o u t th e s y s t e m t h r o u g h th e u s e of e q u a ti o n s
and [2 ].
[l]
F o r i f <*[3] i s m u l t i p l i e d b y s o m e q u a n t i t y Q(v) p e r t a i n i n g
to a p a r t i c l e ,
a n d i n t e g r a t e d o v e r v e lo c i ty s p a c e ,
+ 7 r ? (H^Q) + j Q ( v ) 7v* ( a f ) d 3v = J*Q(v)
T h is i s th e B o l t z m a n n t r a n s p o r t e q u a ti o n .
one o b ta in s
d3v .
c o ll
[4]
B y f i r s t s e t t i n g Q (v )= l,
a n d th e n Q(v) = v , a n d d e fin in g VQ by
[5 ]
o ne m a y c o m b in e th e r e s u l t i n g e x p r e s s i o n s a n d o b ta in a n e q u a ti o n
f o r v w h ic h is e s s e n t i a l l y th e s a m e a s t h a t o b ta in e d b y th e s i n g le
p a r t i c l e m e th o d ( s e e e q u a ti o n [ 6 ] of th e n e x t s e c t i o n ) .
F o r p l a s m a s h a v in g r e l a t i v e l y h ig h p e r c e n t a g e i o n i z a t i o n s ,
e l e c t r o n - e l e c t r o n a n d e l e c t r o n - i o n c o l l i s i o n s m u s t be c o n s i d e r e d .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
11
T h is m e a n s th a t c o l l i s i o n s a r e no' l o n g e r t w o - p a r t i c l e e v e n t s ;
p a rtic le s a r e in te ra c tin g sim u lta n e o u sly .
m any
A llis (2) s h o w s t h a t th e
n e c e s s a r y m o d i f i c a t i o n of th e B o l t z m a n n e q u a tio n l e a d s to th e F o k k e r - P l a n c k e q u a ti o n , w h ic h m a y be c o n s i d e r e d a s e r i e s a p p r o x i m a н
tio n f o r th e B o ltz m a n n e q u a tio n in p o w e r s of th e v e l o c i t y d e f l e c t i o n s
o c c u r rin g d u rin g a c o llisio n .
In th e e x p e r i m e n t s w h ic h w e h a v e c o n d u c te d ,
w e r e 100 m i c r o n s o r g r e a t e r ,
or le ss.
th e g a s p r e s s u r e s
a n d th e e l e c t r o n d e n s i t y 1 0 ^ c m " ^
T h u s th e p e r c e n t a g e i o n i z a t i o n w a s v e r y lo w , a n d w e a r e
j u s t i f i e d in t r e a t i n g c o l l i s i o n s a s t w o - p a r t i c l e e v e n t s .
H ow ever, a s
th e s i n g le p a r t i c l e a p p r o a c h i s m o r e r e a d i l y v i s u a l i z e d t h a n th e
m e t h o d of th e B o l t z m a n n e q u a ti o n , w e w ill e m p lo y i t a n d i n d i c a t e
th e m o d i f i c a t i o n s i n t r o d u c e d b y th e m o r e r i g o r o u s m e th o d .
T h e f r e q u e n c y of c o l l i s i o n s i s t a k e n to b e s u f f i c i e n t l y s m a l l so
t h a t th e e l e c t r o n v e l o c i t i e s ,
a n d t h e r e f o r e th e c u r r e n t ,
r e c t l y r e l a t e d to th e a p p lie d e x t e r n a l f i e l d s ;
i.e.,
m a y be d i н
th e v e l o c i t i e s
a r e u n a l t e r e d by c o l l i s i o n s in a t i m e c o m p a r a b l e to th e m i c r o w a v e
o r c y c lo tro n p e rio d .
W ith t h i s r e s t r i c t i o n i t i s p e r m i s s i b l e to r e -
l a t e th e c u r r e n t d e n s i t y j to th e a p p l i e d e l e c t r i c f i e l d E by
?
W
-╗*?
j - O' ? E ,
M
w h e r e (X i s th e c o m p l e x c o n d u c ti v ity t e n s o r .
T h e in f l u e n c e of the m i c r o w a v e on the p l a s m a i s a s s u m e d to
be n e g lig ib le .
T h e a c t u a l m i c r o w a v e p o w e r w a s of th e o r d e r of a
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
12
fe w m i l l i w a t t s , w h e r e a s th e p o w e r in th e d i s c h a r g e w a s s e v e r a l
t e n s of w a t t s ;
henqe,
the m i c r o w a v e s d id n o t a l t e r th e e l e c t r o n
v e l o c i t y d i s t r i b u t i o n in a n y r a d i c a l f a s h i o n .
B.
M ic r o w a v e I n t e r a c t i o n w ith th e P l a s m a
A n a n a l y s i s of th e i n t e r a c t i o n of m i c r o w a v e s w ith a p l a s m a is
c o n c e r n e d p r i n c i p a l l y w ith th e m a n n e r in w h ic h th e p r o p a g a t i o n of a
w a v e i s in f l u e n c e d b y t h r e e p l a s m a p a r a m e t e r s ,
d e n sity ,
c o llisio n fre q u e n c y ,
v iz . , e l e c t r o n
and c y clo tro n fre q u e n c y .
If th e e l e c н
t r o n t e m p e r a t u r e i s v e r y h ig h , i t m u s t a l s o b e t a k e n in to a c c o u n t.
In t h i s w o r k ,
th e t e m p e r a t u r e e f f e c t s w ill be s h o w n to b e s m a l l
e n o u g h to b e u n i m p o r t a n t .
The e x p e r im e n ta l m e a s u r e m e n ts re la tin g
to th e w a v e p r o p a g a t i o n w ill be c o n c e r n e d w ith th e a b s o r p t i o n ,
fle c tio n ,
a n d p h a s e s h i f t of th e m i c r o w a v e s .
rer
T h e m a i n p u r p o s e of
t h i s s e c t i o n is to e s t a b l i s h a c o n n e c tio n b e tw e e n th e f i r s t tw o of
t h e s e q u a n t i t i e s a n d th e t h r e e p l a s m a p a r a m e t e r s .
T h e c o n n e c tio n
i s m a d e by w a y of th e p r o p a g a t i o n c o n s t a n t in th e p l a s m a .
T h is
c o n s t a n t is in t u r n e x p r e s s e d in t e r m s of th e m i c r o w a v e c o n d u c t i v н
i ty , w h ic h f i n a l l y is r e l a t e d to th e p l a s m a p a r a m e t e r s .
t r a c e t h i s p a t h in th e r e v e r s e d i r e c t i o n ,
c o n d u c t i v i t y to th e e l e c t r o n d e n s i t y ,
We s h a l l
s t a r t i n g b y r e l a t i n g th e
c o llisio n fr e q u e n c y ,
and c y c lo -
tro n frequency.
1.
M ic r o w a v e C o n d u c tiv ity .
A d e tailed p ro c e d u re fo r
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
13
d e t e r m i n i n g th e e l e m e n t s of th e c o n d u c ti v ity t e n s o r ,
m a n n e q u a ti o n , i s g iv e n b y A llis (2).
th o u g h s o m e w h a t l e s s r i g o r o u s ,
u s i n g th e B o ltZ '
An a lte rn a tiv e m e th o d ,
i s to e m p lo y th e L a n g e v in e q u a ti o n ,
w h ic h i s e s s e n t i a l l y th e e q u a ti o n of m o tio n of a n a v e r a g e e l e c t r o n ,
in c lu d in g a t e r m r e p r e s e n t i n g th e e f f e c t s of c o l l i s i o n s .
We fo llo w
th i s p r o c e d u r e h e r e .
*?
е
A n e l e c t r o n in a c o m b in e d e l e c t r i c f i e l d E a n d a c o n s t a n t ,
u n i f o r m m a g n e t i c f i e l d B i s s u b j e c t to th e L o r e n t z f o r c e
F = - e ( E + v X B ),
[1]
w h e r e v i s the e l e c t r o n v e l o c i t y .
L e ttin g
~m
C2]
eJii
'm 9
[3 ]
and
we have
v = г + 0)c x v
[4 ]
In o r d e r to i n c o r p o r a t e th e e f f e c t s of c o l l i s i o n s , w e i n t r o d u c e
?
е
a t e r m A (t), w h ic h f l u c t u a t e s b e c a u s e of th e r a n d o m n a t u r e of the
co llisio n s.
Thus we have
v = г + 10C X v +
If
A (t)
[5 ]
n o w th e t e r m s of t h i s e q u a ti o n a r e a v e r a g e d o v e r a
lo n g c o m p a r e d w ith th e t i m e i n t e r v a l b e tw e e n c o l l i s i o n s ,
Zd = 2 + Z
+ x
we
.
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
tim e
o b ta in
[6 ]
14
H e r e vd i s th e d r i f t v e l o c i t y ,
d e n o tin g th a t p a r t of th e v e l o c i t y
w h ic h d o e s n o t a v e r a g e to z e r o ,
a n d is a t t r i b u t a b l e d i r e c t l y to th e
ap p lie d f o r c e s ,
T h e l a s t t e r m w ill b e w r i t t e n a s
i
= -Vc v d
[7 ]
w h e r e VQ m a y be d e s c r i b e d a s a s o r t of c o l l i s i o n f r e q u e n c y of
e l e c t r o n s w ith n e u t r a l p a r t i c l e s .
It i s e s s e n t i a l to n o te t h a t Vc
m u s t in g e n e r a l b e c o m p le x .
F o r e l e c t r i c f i e l d s v a r y i n g w ith ti m e a s S = E 0 e " * ^ ,
v a r i e s a s e " 1^ ,
vd a lso
and [6 ] becom es
i00vd
= 6 + Vc v d ,
[8]
w h e r e w e h a v e o m i t t e d th e t e r m in v o lv in g th e m a g n e t i c f i e l d f o r
p u r p o s e s of s i m p l i c i t y .
vd =
Then
( .~
c . ?? + ..... 2^ - - ^ г .
\ V C2 + 0)Z
vcz +
J
Now a r i g o r o u s t r e a t m e n t u s in g th e B o l t z m a n n
[9]
e q u a ti o n r e l a t e s
vd
a n d ^ by th e g e n e r a l e x p r e s s i o n
vd =
(B - iD) г ,
[1 0 ]
w h e r e B a n d D a r e f u n c tio n s of the c o l l i s i o n f r e q u e n c y f o r m o m e n н
t u m t r a n s f e r l>m .
T h e q u a n tity
i s r e a l a n d is d e f in e d b y
55 j N S ( v ) v f d 3 v,
[11]
w h e r e N i s the n e u t r a l p a r t i c l e d e n s i t y a n d S(v) i s th e c o l l i s i o n
c r o s s s e c tio n .
T h e f u n c tio n a l f o r m of S(v) w ill d e p e n d u p o n th e
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
15
c h o ic e of p l a s m a g a s .
p r o p o r t i o n a l to v,
F o r a p l a s m a f o r w h ic h S(v) i s i n v e r s e l y
M o lm u d (8) s h o w s th a t Vc = Vm , a n d th e v a lu e s
of B a n d D rn a y b e fo u n d b y e q u a tin g [ 9 ] a n d [ l O ] .
tr u e fo r o th e r p la s m a s ,
p ressu res,
H ow ever,
a ir,
p r o v i d e d Vr n / U i л l
T h i s is a l s o
in th e l i m i t of v e r y low
o r f o r Vm / U i ╗ 1 in th e l i m i t of v e r y h ig h p r e s s u r e s .
e x p e r i m e n t a l s t u d i e s h a v e s h o w n th a t , f o r a d i s c h a r g e in
S(v) i s p r o p o r t i o n a l to V (10).
A s w e a r e i n t e r e s t e d in th e
in te rm e d ia te p r e s s u r e ra n g e , we m u s t u se d iffe re n t v a lu es fo r B and
D.
T h e s e a r e fo u n d by M o lm u d .
T h e n u po n e q u a tin g [ 9 ] w ith [ l O ] ,
one f in d s t h a t Vc m u s t be c o m p le x .
Vc
**
W r itin g
Vr + iVi >
h e fin d s (fo r V/tO < 1)
VT
and
= 5 /3 ^m (l + 4 . 2 2 I^m 2 /C02 )
,.2
Vi = 1 0 / 9 i ^
..2
(1 + 5 ^ _ )
T h e e q u a ti o n [ 8 ] i s s t i l l v a l i d ,
[1 2 a]
.
^
th o u g h w e m u s t b r o a d e n th e s i m p l e
p h y s i c a l i n t e r p r e t a t i o n Vc a s a c o l l i s i o n f r e q u e n c y .
L e ttin g J d e n o te th e c u r r e n t d e n s i t y , w e h a v e
J = -n ev j ,
A ls o ,
[1 3 ]
by d e f in itio n of (T,
"?*
**
?*
J = a ? e
T h e n c o m b in in g [ 3 ] ,
r
-1
[ 14]
[6 ],
[7 ],
[1 2 ],
[13],
a n d [ 1 4 ] , w e o b ta in
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
16
th e c o n d u c ti v ity t e n s o r :
(a i k )
=
|
<r0
'<y'
o
-o'
aQ
0
0
0
a2
|
,
[1 5 ]
w h e r e w e h a v e c h o s e n th e d i r e c t i o n of the z - a x i s to lie in th e d i r e c '
tio n of B , a n d
o
(05 - V- ) + i v r
CT0 = i ? 0 0)p Z ----------------1---------- E--------- -[(05 - Vi) + ilfr ] 2 - 0)c 2
C l6 a ]
O' = ?o 05 2
[16b]
F
CTZ =
i e 0 W p2
F
^ ----- ? ----------[(05 - l^i) + i ^ r ] 2 - 0JC2
n -------- ~? r
[(05 r l/j) + :?
iVr r] ?
?
t l6 c ^
H e r e w e h a v e u s e d th e d e f i n i t i o n of th e p l a s m a f r e q u e n c y ,
v
p
2,
?0m ?
P r o p a g a t i o n C o n s ta n t,
T h e r e a r e a n u m b e r of g e n e r a l
t r e a t m e n t s f o r th e p r o b l e m of fin d in g th e p r o p a g a t i o n c o n s t a n t s .
5,
7, 11)
We c o n s i d e r h e r e o n ly th o s e a s p e c t s w h ic h a p p l y
sp ec ific p ro b le m a t hand,
and we
s h a l l fo llo w th e m e t h o d
(4,
to th e
of M o w e r ,
w ith a fe w c h a n g e s in n o ta tio n .
W ith th e r e s t r i c t i o n s on th e p l a s m a w h ic h w e r e m e n t i o n e d
e a rlie r,
M a x w e l l 's e q u a t i o n s g o v e r n in g E ( r , t) a n d H ( r , t) a r e a p p l i c н
ab le:
7 X E = - L2.
ot
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[1 7 ]
17
7
x H -
?
at
+ J
[18]
7 *
d
=p
[ 19]
7?
B =0
[20]
T h e r e is e s s e n t i a l l y n e i t h e r m a c r o s c o p i c m a g n e t i c m o m e n t
n o r e le c tric a l p o la riz atio n ,
D =
and we m a y th e re fo re w rite
?0E
[2 1 ]
B = H0H
[2 2 ]
It i s n e c e s s a r y to f in d a p r o p a g a t i n g w a v e s o lu tio n i n c o r p o r a t н
ing th e fo llo w in g ;
( 1) th e m e t a l l i c w a l l s of th e w a v e g u id e a r e
a s s u m e d p e r f e c t l y c o n d u c tin g ;
h e n c e th e t a n g e n t i a l c o m p o n e n t of th e
e l e c t r i c f i e l d a t th e w a lls m u s t v a n is h ;
( 2 ) th e t i m e d e p e n d e n c e of
2 is e TlWt; (3) th e s t e a d y m a g n e t i c f i e l d i s to be t r a n s v e r s e to the
d i r e c t i o n of p r o p a g a t i o n .
b e th e x - d i r e c t i o n .
We c h o o s e th e d i r e c t i o n of p r o p a g a t i o n to
F o llo w in g th e u s u a l p r o c e d u r e , w e ta k e the
c u r l of [ 1 7 ] a n d e m p lo y [ 1 8 ] a n d [ 2 2 ] to o b ta in
7 X (7 X E ) = - i l 0 7 X ( l T + ^
U sin g [ 1 4 ] a n d [21 ] ,
7 X (7
?
*-2 3 -*
th is m a y b e w r i t t e n a s
X E ) = pio ?o C02E
It i s h e lp fu l h e r e to e x p r e s s
+ itio 00CT ? E ,
[2 4 ]
[ 2 4 ] in c o m p o n e n t f o r m ;
m
i ? \ _ '7 2 - i t
(7 .? E
) - 7 ^ E x - k 0 2 E x -i|J.o W(0,oE x + CT'Ey ) = 0 ,
[2 5 a]
? ?(7 ? g ) - 7 2E y - k o 2E y - iH Qt0(<ToE y - <r'Ex ) = 0 ,
[25b]
_ l -
3x
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
18
J L ( 7 ? E ) - 72E z - k 0 2E z - m o w a sSE z =
w h e r e k 0 2 = U0 C0 (o2.
0 .
Now in th e a b s e n c e of
in th e g u id e i s t r a n s v e r s e ,
a n d 7 ? E = 0;
[2 5 c]
th e p l a s m a , th e w a v e
a lso ,
crQ * crz a n d or* = 0.
T h e n th e g e n e r a l s o lu tio n of [ 2 5 ] i s e x p r e s s i b l e in t e r m s of n o r m a l
m odes,
a l l b u t th e lo w e s t of w h ic h c a n be sh o w n tb b e a t t e n u a t e d in
th e u s u a l o p e r a t i o n of th e w a v e g u id e .
T h is l o w e s t m o d e m a y be
r e p r e s e n t e d e i t h e r by
E v = A s in 1 5 e ^ kxx ? w t), E x = E
y
dz
= 0
[2 6 a]
or
E z = B s in -I X e i(k x x ~w t ), E x = E y ? 0
,
[26b]
dy
w h e r e d z a n d dy a r e th e c r o s s s e c t i o n a l d i m e n s i o n s of th e w a v e н
g u id e in th e z a n d y d i r e c t i o n s ,
re sp ec tiv e ly .
[ 2 6 a ] if d z > dy , a n d [ 2 6 b ] if d z < dy.
T h e c h o ic e w ill b e
A ls o ,
kx is
fo u n d f r o m
kx 2 = ko 2 - (ff/ d )2 ╗
[27 3
a n d i s th e p r o p a g a t i o n c o n s t a n t w ith in th e g u id e .
firs t case
If n o w w e t r y th e
[ 2 6 a ] to r e p r e s e n t th e s i t u a t i o n w ith a p l a s m a p r e s e n t ,
w e s e e i m m e d i a t e l y th a t e q u a ti o n s
[2 5 a ] and [2 5 b ] cannot be s a t i s н
f i e d s i m u l t a n e o u s l y in th e a b s e n c e of E x ,
T h is a r i s e s p h y s i c a l l y
b e c a u s e the c r o s s e d e l e c t r i c a n d m a g n e t i c f i e l d s s e t u p a c u r r e n t
and,
hand,
th e re fo re ,
a n e l e c t r i c f i e l d in th e x - d i r e c t i o n .
On th e o t h e r
th e c o n d u c t i v i t y d o e s n o t c o u p le E z w ith th e o t h e r c o m p o n e n t s ,
a n d [ 2 6 b ] d o e s s a t i s f y [ 2 5 ] a s w e l l a s th e b o u n d a r y c o n d itio n s .
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
19
F o r t h i s r e a s o n we f i r s t c h o o s e to o r i e n t th e w a v e g u id e in the
m a g n e t i c f i e l d so t h a t [ 2 6 b ] i s th e l o w e s t m o d e ,
p a ra lle l.
and E and HQ a r e
(T he e f f e c t s on th e p l a s m a of th e t i m e v a r y i n g m a g n e t i c
f i e l d of th e m i c r o w a v e a r e n e g l i g i b l e in c o m p a r i s o n w ith th o s e of
th e s t e a d y f i e l d H 0 , a n d a r e a c c o r d i n g l y i g n o r e d . )
Then in se rtin g
[ 2 6 b ] in to [ 2 5 ] , w e f in d th e fo llo w in g e x p r e s s i o n f o r the p r o p a g a н
tio n c o n s t a n t kx :
kx 2
= k J_2 ? ( ff/ d )2 *
[2 8 ]
k j2
- k Q2 + i | i o C0 Crz .
[29]
w here
E s s e n t i a l l y th e s a m e r e s u l t i s
g iv e n b y E p s t e i n (7) f o r a g u id e d
w a v e in a g y r o m a g n e t i c m e d i u m ;
?qE a n d B = |i ? H.
ti v i t y ,
i.e .,
a m e d i u m f o r w h ic h D =
H e r e th e p e r m e a b i l i t y ,
r a t h e r th a n th e c o n d u c н
i s t a k e n to b e a n i s o t r o p i c ,
T h e d i f f ic u l ty w h ic h a r i s e s if w e c h o o s e th e o t h e r o r i e n t a t i o n
f o r the w a v e g u id e i s e s s e n t i a l l y t h a t ,
b o u n d a r y c o n d itio n s ,
in a t t e m p t i n g to s a t i s f y th e
the two c o m p o n e n t s E x a n d E y a r e i n e v i t a b l y
c o u p le d
b y a c o n s t a n t w h ic h d o e s n o t v a n is h a s th e p l a s m a
g o e s to
zero.
d e n sity
H e n c e th e z e r o t h a p p r o x i m a t i o n of th e c u s t o m a r y
p e r t u r b a t i o n t r e a t m e n t c a n n o t be e x p r e s s e d by one of th e n o r m a l
m odes.
N ow f o r a n u n b o u n d e d p l a s m a ,
the p r o p a g a t i o n c o n s t a n t kj^
f o r a w a v e p o l a r i z e d p e r p e n d i c u l a r to th e m a g n e t i c f i e l d c a n be
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
20
sh o w n to b e g iv e n by (4)
2 = (kQ2 + m o 0XTn ) 2 - dq 0 coq)2
[30]
(k 0 2 + i |io W0ro )
We s h a l l m a k e th e a s s u m p t i o n , f o r low p l a s m a d e n s i t i e s ,
th a t
th e p r o p a g a t i o n c o n s t a n t w ith in th e w a v e g u id e is v e r y n e a r l y th a t
g iv e n b y [ 2 8 ] ,
w ith kj^ d e f in e d b y [ 3 0 ] .
That is ,
we a r e settin g
th e l o n g itu d in a l c o m p o n e n t of th e w a v e e q u a l to z e r o .
S in c e i t m u s t
b e r a t h e r s m a l l , th e a s s u m p t i o n i s n o t e n t i r e l y u n r e a s o n a b l e .
We
s h a l l c a l l th e o r i e n t a t i o n of th e g u id e , f o r w h ic h [ 2 9 ] a p p l i e s ,
th e
p a r a lle l o rie n ta tio n (e le c tr ic an d m a g n e tic fie ld s p a r a lle l ) ,
f o r w h ic h [ 3 0 ] a p p l i e s ,
3.
a n d th a t
th e p e r p e n d i c u l a r o r i e n t a t i o n .
T r a n s m is s io n and R e fle c tio n C o effic ien ts,
It i s n ow p o s н
s i b l e to f in d th e t r a n s m i s s i o n a n d r e f l e c t i o n c o e f f i c i e n t s f o r m i c r o н
w a v e s in c i d e n t up on a s e c t i o n of w a v e g u id e of le n g th t ,
f i l l e d w ith a p l a s m a .
g e n e o u s th r o u g h o u t .
c o m p le te ly
T h e p l a s m a i s a s s u m e d to b e p e r f e c t l y h o m o н
T h e m e t h o d i s e s s e n t i a l l y o n ly a s l i g h t m o d i f i н
c a t i o n of t h a t g iv e n by S tr a t to n (12).
o r i e n t a t i o n of th e w a v e g u id e .
We c o n s i d e r f i r s t th e p a r a l l e l
T he w a v e in c i d e n t u p o n th e f i r s t
p l a s m a b o u n d a r y is
E = E j sin H i
d
T h e r e f l e c t e d w a v e is
" W t) .
E = A s i n f f X e - i(k x ░ x +
d
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[3 .1 a ]
[3 1b]
21
W ith in th e p l a s m a ,
th e f o r w a r d a n d r e v e r s e w a v e s a r e
E ? B sin I I e - ^ x P *
d
? w t)
[3 1 c]
and
E = C sin J X
d
e ? 1(kxP * + л * > .
B e y o n d th e s e c o n d b o u n d a r y ,
th e t r a n s m i t t e d w a v e i s
E = D sin J D L e ^ ^ P x
d
In t h e s e e x p r e s s i o n s ,
[31d]
- w t) .
[3 1 e]
kx ░ a n d kx P a r e th e p r o p a g a t i o n c o n s t a n t s in
the e m p ty w a v e g u id e a n d p l a s m a - f i l l e d w a v e g u id e ,
re sp e c tiv e ly .
T h e n th e b o u n d a r y c o n d itio n s on E?- a n d H y ie ld th e fo llo w in g e x н
p re ssio n s:
A
=
[( k x P )2 -
E
( kx Q)2 ] (1 - e ~ 2 ik;x p ^
(kxP + kx ░ ) 2e 2 i k x P 't - (kx P - k x 0 )2
D
4kxpk x ░e
?ik x ░ ' t
[33]
(k^P + kx ░ ) 2 e - ik x ░ ' t - (kx P - k x 0 )2e +ikx p ^
We now d e fin e
.2 -
w here c =
W r itin g
1
? ......
C l - f 2c 2
0
h C
SO ' )
V ? d
?
[34]
i s th e v e lo c i ty of lig h t in a v a c u u m .
1 ^ o ?o
kx P
- = a + ib
,
[35]
kx
a n d n o tin g th a t
(k^ ) 2 = ? i s 2 ,
cc
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[36]
22
w e fin d the fo llo w in g e x p r e s s i o n s f o r a a n d b:
. ? K . j iv , -
2
*? L v
r
+ ( 1
V
V
y
U>
' 2 ( 1 + V- ' )
^ 2
^
.
'
[3 7 a]
CO ' Z V
,2
{ r f i - .1 . I
+ r - Z 5 _______ - ___ +
i L v i;r '2 + (1 + t╗.')2 y ^ | / r '2 + (l + i^╗)2
2
-,1 /2
v ir2 + ( 1+1^.,)2 y J
г0'S2 ( 1 + V i ') N ' 11 /2
rr
^
f
Vr 'Z +(1 + Ui ')Z '/ J
r r- ^
\/T "
b = li_
?v 2 t v 2
i/'r 2 + ( l + t y ) 2^
. 2
1 /2
y J
,
CO ' 2 (1
')
>. J / 2
- ( l - ? 4 ---------- ? r ) }
v
tv
+ (i + 1^) y J
[37b]
In t h e s e e x p r e s s i o n s w e h a v e u s e d e q u a ti o n s [ l 6 c ] a n d [2 9 3, t o н
g e t h e r w ith th e fo llo w in g d e f in itio n s 5
IV = V ^ U i ,
[38^]
VT ' -
[38b]
Vr / U ,
ws ' = Wp/sCO .
[3 8 c]
T h e r e f l e c t i o n c o e f f i c i e n t R a n d th e t r a n s m i s s i o n c o e f f i c i e n t T
a r e d e f in e d by
-
AA*
╗
T =
PD*
?
P
[3 9 a ,b ]
L e n g th y b u t s t r a i g h t f o r w a r d m a n i p u l a t i o n th e n y i e l d s th e fo llo w in g
e x p re ssio n s:
p = 2 [ ( a - l ) Z+ b Z ] [ ( a + l ) 2+ b Z ] ( c o s h Z $ l - c o s Z d l )
K
_ _ 1 6 ( a 2 + b 2)
T
g
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
.
.
[41 ]
23
w here
K = [ ( a + l ) 2 + b 2 ] 2 e 2 p 't + [ ( a - l ) 2 + b 2 ] 2 e
-
T 2 c o s ZOLt [ ( a 2 + b 2 - l ) 2 - 4 b 2 ] +
[42]
+ 8b (a 2 + b 2 - 1) s in 2 a t ,
and
a =
>
Now th e p h y s i c a l s i g n i f i c a n c e of th e v a r i o u s t e r m s in th e a b o v e
e x p r e s s i o n s is r a t h e r o b s c u r e .
In o r d e r to f a c i l i t a t e th e i n t e r p r e н
t a t i o n , w e f i r s t s e t UT = l>i = 0, a n d w r i t e th e s i m p l i f i e d e x p r e s н
s i o n s f o r th e c a s e of v a n is h i n g c o l l i s i o n f r e q u e n c y .
tio n th e e x p r e s s i o n s f o r a a n d b ,
ous.
e q u a ti o n s [ 3 7 ] ,
F o r th is s i t u a н
becom e a m b ig u н
T h e c h o ic e of s ig n f o r th e s q u a r e r o o t in th e s q u a r e b r a c k e t s
m u s t be d e cid e d fr o m p h y s ic a l c o n s id e ra tio n s .
F o r a c o llisio n le ss
p l a s m a , w e w ill h a v e
and
a 2 - b 2 = {1 - C0S ' 2 )
[4 3 ]
a b = 0.
[4 4 ]
T h e n if tOs ,гj2 < 1 , w e m u s t c h o o s e b = 0 f o r a to b e r e a l ,
COs 12 > 1 , w e m u s t c h o o s e a = 0 f o r b to be r e a l .
and fo r
Then we have
th e fo llo w in g two c a s e s :
(1) f o r C0S,2 < 1 ,
s i n 2 [k x ░ ( l t C0s ' 2 ) 't ]
[45a]
s
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
24
(2 ) f o r tos l2 > l ,
* ------------*inh2cV (< - : 1)l ]---------r.}) + s i n h 2 [k x ░(tos .2 - ! )
4
S ince V - 0 ,
t h e r e i s no a b s o r p t i o n ,
I
[45b]
]
and T = 1 - R.
We n o te th a t
f o r m i c r o w a v e f r e q u e n c i e s a b o v e th e r e d u c e d p l a s m a f r e q u e n c y tos ,
th e r e f l e c t i o n (and t r a n s m i s s i o n ) c o e f f i c i e n t w ill go th r o u g h a s e r i e s
of m a x i m a a n d m i n i m a a s th e w a v e le n g th is i n c r e a s e d ,
in m u c h th e
s a m e w a y a s f o r w a v e s in th e o p t i c a l r e g i o n t r a v e r s i n g a th in f i l m .
E x p ressio n s
[ 4 0 ] a n d [41 ] a r e i l l u s t r a t e d in F i g u r e 1 f o r s e v e r a l
d i f f e r e n t v a lu e s of th e c o l l i s i o n f r e q u e n c y a n d f o r a s e c t i o n of w a v e н
g u id e 14 c m long.
We m a y n o te f r o m th e p r e v i o u s d i s c u s s i o n t h a t th e p r e s e n c e of
th e b o u n d a r y e f f e c t i v e l y r a i s e s th e p l a s m a f r e q u e n c y f r o m
The e x p re ss io n s
top to tos .
[ 4 0 ] a n d [41 ] a r e i d e n t i c a l to th o s e f o r a n u n н
b o u n d e d p l a s m a if we s e t s = 1 a n d ,
In p r i n c i p l e ,
th e re fo re ,
t o g
*=
t o p .
one m a y now u s e th e c u r v e s of F i g u r e 1 w ith
m e a s u r e d v a lu e s of th e c o l l i s i o n f r e q u e n c y a n d r e f l e c t i o n o r t r a n s н
m i s s i o n c o e f f i c i e n t s to d e t e r m i n e th e p l a s m a d e n s i t y .
F o r th e p e r p e n d i c u l a r o r i e n t a t i o n of th e g u id e ,
th e p r i n c i p a l
f e a t u r e s of th e v a r i a t i o n of the r e f l e c t e d p o w e r w ith m a g n e t i c f i e l d
m a y b e u n d e r s t o o d by s e t t i n g V = 0.
T h e n th e p r o p a g a t i o n c o n s t a n t
in th e g u id e b e c o m e s
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
-
1. 0
of the copyright ow ner.
v'=. 1
v ' = . 15
Further reproduction
ei
0)
o
*rt
?H
(U
o
U
prohibited
o
U
<D
without p e r m issio n .
1.0
F i g u r e 1.
.2
1. 5
2.0
R e f l e c t i o n of g u i d e d m i c r o w a v e s b y a p l a s m a s l a b w i t h i n c r e a s i n g e l e c t r o n d e n s i t y
f o r s e v e r a l v a l u e s of th e e l e c t r o n c o l l i s i o n f r e q u e n c y V1 = P / t o .
26
1/2
(l-o>c | 2 ^ s ' 2 ) - w s ' 4 wc ' 2
= a kx ░ .
k x P = k x ░ L (1 -C0S ,2 -Wc ,2 ) (1 -w c ,2 )
[4 6 ]
T h e r e f l e c t i o n c o e f f i c i e n t h a s th e f o r m
R
.
[4 7 ]
? ? ~7 + s in ^ a I
( 1 - a 2 )2
2
?
T h e r e a r e two p r i n c i p a l p o in ts of i n t e r e s t : (1) w h e n C0c ' ? 1-Wg ' ,
a
, a n d R = 0;
(2) w h e n Wc ' = 1~C0S ' 2 , a = 0, a n d w e h a v e
(kx ░-t-)2
R =
4+(kx ░ I ) 2
In o u r w o r k ( k ^ - t ) ╗ 4 ,
a n d t h e r e f o r e th e s e c o n d c a s e a b o v e
c o n stitu te s a m a x im u m a t R ^ 1 ,
(O ne c a n sh o w t h a t th e e x t r e m u m
v a l u e s of R o c c u r w h e n
ta n a l _ (1 + a 2 )
at
~ (l -a.2)
( f o r V - 0), w h ic h i s s a t i s f i e d f o r a = 0. )
F o r s m a l l v a lu e s of Ws ' ( C0S1< . 1), th e s e p a r a t i o n b e tw e e n th e
m a x i m u m a t <dc ' ? l - t O g '2 a n d th e m i n i m u m a t Wc ?2 = 1-COg'2 is
AWC'
c
** ?4 ----
2
.
[48]
T h u s f o r v e r y lo w c o l l i s i o n f r e q u e n c i e s ,
one m a y c a l c u l a t e th e
e le c tr o n d e n sity fr o m a m e a s u r e m e n t of th is s e p a ra tio n ,
4.
C o llisio n F re q u e n c y ,
As h a s been in d ic a ted ,
it is g e n e r н
a l l y e s s e n t i a l to know th e e l e c t r o n c o l l i s i o n f r e q u e n c y b e f o r e the
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
27
r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s c a n be u s e d to f in d th e
e le c tro n d en sity ,
A v e r y s i m p l e a n d c o n v e n ie n t m e t h o d f o r do ing
t h i s m a k e s u s e of th e t r a n s m i s s i o n m e a s u r e m e n t s f o r m i c r o w a v e s
w h e n th e m i c r o w a v e p o l a r i z a t i o n i s p e r p e n d i c u l a r to th e s t e a d y m a g н
n e tic fie ld H0 ,
A s th e m a g n e t i c f i e l d i s i n c r e a s e d ,
th e p o w e r a b н
s o r b e d by th e p l a s m a r e a c h e s a m a x i m u m v e r y n e a r l y a t toc ' -=1.
K e lly ,
M a r g e n a u , a n d B r o w n h a v e sh o w n th a t th e h a l f - w i d t h of th e
c y c l o t r o n r e s o n a n c e a b s o r p t i o n c u r v e i s e q u a l to th e c o l l i s i o n f r e н
q u e n c y f o r a v a r i e t y of g a s e s (13),
In p a r t i c u l a r ,
th e y h a v e sh o w n
t h a t th e m e th o d i s v a lid w h e n th e c o l l i s i o n f r e q u e n c y i s p r o p o r t i o n a l
to v^ (v th e e l e c t r o n v e l o c i t y ) , w h ic h i s the c a s e f o r a d i s c h a r g e in
a ir.
H o w e v e r , w e s h a l l b e m e a s u r i n g n o t j u s t th e a b s o r p t i o n b u t
r a t h e r th e to ta l e f f e c t of a b s o r p t i o n a n d th e a t t e n u a t i o n w h ic h r e н
s u l t s s o l e l y f r o m th e i m a g i n a r y p a r t of th e c o l l i s i o n f r e q u e n c y ,
d isc u sse d e a r lie r .
as
D odo (14) h a s sh o w n t h a t th e m e t h o d f o r m e a s н
u r i n g th e c o l l i s i o n f r e q u e n c y w ill s t i l l a p p ly f o r P ' < , 3 a n d tos ' < . 3 .
He h a s f u r t h e r sh o w n th a t th e m a x i m u m in th e a b s o r p t i o n c u r v e
w i l l s h i f t f r o m toc ' = 1 to a v a lu e g iv e n by
( л C') m a x * ( l
A s (Wc ') m a x an<*
- V
2 - I '?2 ) 1 /2
.
a r e n ow e a s i l y m e a s u r e d ,
[4 9 ]
top'2 a n d th e e l e c t r o n
d e n s i t y a r e r e a d i l y c a l c u l a t e d f r o m th is e q u a tio n .
In o u r w o r k w e
s h a l l a s s u m e th a t top1 w ill b e r e p l a c e d b y Ws '.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
28
C.
E le c tro n T e m p e ra tu re
T h e m e a n i n g of e l e c t r o n t e m p e r a t u r e is p r e c i s e o n ly if the
e l e c t r o n d i s t r i b u t i o n f u n c tio n is M a x w e llia n .
F o r such a p la sm a ,
th e e l e c t r o n t e m p e r a t u r e i s d e f in e d by
T = i = ^ ,
e
2
k
[5 0 ]
w h e r e w is the m o s t p r o b a b l e e l e c t r o n v e l o c i t y a n d k th e B o l t z m a n n
c o n sta n t.
P r o v i d e d t h a t th e m e a n f r e e p a t h f o r c o l l i s i o n s is s h o r t
e n o u g h f o r e q u i l i b r i u m to be m a i n t a i n e d o v e r p e r i o d s of i n t e r e s t
(c y c lo tro n ,
w ellian .
m ic ro w a v e p e rio d ),
th e d i s t r i b u t i o n f u n c tio n w ill b e M a x н
T h is c o n d itio n i s n o t s a t i s f i e d in the p r e s e n c e of a m icron-
w ave e le c tr i c fie ld .
That is,
t h e r e w ill b e s e l e c t i v e a b s o r p t i o n of
e n e r g y b y e l e c t r o n s in a p a r t i c u l a r e n e r g y r a n g e .
In o u r w o r k ,
th e r a d i o f r e q u e n c y f i e l d w h ic h p r o d u c e s th e p l a s m a h a s a f r e q u e n c y
w e l l b e lo w the c o l l i s i o n f r e q u e n c y a n d th u s w ill n o t a f f e c t th e v e l o c н
ity d i s t r i b u t i o n .
On th e o t h e r h a n d ,
th e s a m e o r d e r of m a g n itu d e .
th e m i c r o w a v e f r e q u e n c y i s of
T h e re fo re th e re is a n o th e r re a s o n
f o r k e e p in g th e m i c r o w a v e p o w e r low ;
d e p a r t u r e s f r o m th e M a x н
w e l l i a n d i s t r i b u t i o n m u s t b e s m a l l in o r d e r t h a t th e c o n c e p t of
e le c tr o n te m p e r a t u r e be v alid .
T h e re , a r e s e v e r a l d i f f e r e n t m e t h o d s f o r d e t e r m i n i n g th e
e l e c t r o n t e m p e r a t u r e in a p l a s m a (15).
T h e s i m p l e s t of t h e s e is
th e p r o b e m e th o d , f i r s t d e v e lo p e d b y L a n g m u i r (16), a n d s u b s e н
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
29
q u e n tly i m p r o v e d a n d m o d if ie d by a n u m b e r of w o r k e r s .
We s h a l l
b r i e f l y d e s c r i b e th e d o u b le f lo a tin g p r o b e m e t h o d w h ic h w a s u s e d
in o u r w o r k a n d w h ic h w a s d e v e lo p e d b y J o h n s o n a n d M a l t e r (17).
C o n sid e r a s m a ll,
c y l i n d r i c a l p r o b e i m m e r s e d in a p l a s m a in
w h ic h e q u i l i b r i u m h a s b e e n e s t a b l i s h e d .
h ig h e r v e lo c itie s ,
B e c a u s e of t h e i r m u c h
e l e c t r o n s w i l l i n i t i a l l y s t r i k e th e p r o b e m u c h
m o r e f r e q u e n t l y th a n th e i o n s ,
a n d th e p r o b e w ill a c c o r d i n g l y s o o n
a c q u i r e a n e g a t i v e p o t e n t i a l w ith r e s p e c t to th e p l a s m a .
T h is n e g a н
tiv e p o t e n t i a l w ill in t u r n r e d u c e the e l e c t r o n flo w to th e p r o b e .
T h e r e f o r e t h e r e w ill a r i s e a s m a l l r e g i o n c a l l e d a s h e a t h ,
su rro u n d н
ing th e p r o b e , w h ic h c o n ta i n s p r i m a r i l y p o s i t i v e io n s (a n d n e u t r a l s ) .
A s h e a t h w ill c l e a r l y be p r e s e n t w h e n e v e r a p l a s m a e n c o u n t e r s a
m a t e r i a l b o u ndary.
It m a y b e s e e n f r o m e n e r g y c o n s i d e r a t i o n s t h a t
th e t h i c k n e s s of th is s h e a t h m u s t b e r o u g h ly th e D e b y e le n g th .
P r o v i d e d th a t th e s h e a t h t h i c k n e s s is s m a l l in c o m p a r i s o n
w ith th e e l e c t r o n m e a n f r e e p a th ,
re g io n .
F u rth e r,
t h e s h e a t h w ill b e a c o l l i s i o n - f r e e
th e p r o b e a n d s h e a t h t o g e t h e r m u s t be s m a l l s o
a s n o t to in f l u e n c e th e e l e c t r o n d i s t r i b u t i o n a t d i s t a n c e s f r o m the
p r o b e of th e o r d e r of m a g n itu d e of t h e m e a n f r e e p a th .
T h e n the
e l e c t r o n s r e a c h i n g th e p r o b e w i l l h a v e th e s a m e v e l o c i t y d i s t r i b u t i o n
a s in th e p l a s m a r e g i o n .
T h e e l e c t r o n c u r r e n t a t the p r o b e i s th e n
g iv e n by
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
30
i e л A jQe " V e / k T e ,
w h e r e A i s th e p r o b e a r e a ,
[5 1 ]
j 0 th e r a n d o m e l e c t r o n c u r r e n t d e n s i t y ,
a n d V th e p o t e n t i a l d r o p a c r o s s th e s h e a th .
Now s u p p o s e th a t t h e r e a r e tw o s u c h p r o b e s ,
s h o w n in F i g u r e 2.
c o n n ected a s
P la sm a
D e n o tin g th e m a g n i t u d e s
of th e e l e c t r o n c u r r e n t s to
V / / / //
th e p r o b e s by i 0 j a n d i ^ ,
a n d th e p o s i t i v e io n c u r r e n t s
by ip j and ip 2 ╗ we have
(i p l " i e l ) +(i p 2 ',ie 2 ) 1B0F u r t h e r m o r e , th e c u r н
r e n t to th e e x t e r n a l c i r c u i t is
i (j= ie 2 - i p 2 л T h e p o s i t i v e ion
c u r r e n t to a p r o b e i s e s s e n t i a l l y
co n sta n t,
0 - 5 0 0 V o lts
F i g u r e 2. C i r c u i t f o r d o u b le
probe te m p e ra tu re m e a s u r e н
m e n ts.
s i n c e th e p o t e n t i a l a c r o s s th e s h e a t h i s a lw a y s a t t r a c t i v e ;
a n d o u ts id e th e s h e a t h the io n s a r e u n in f l u e n c e d by i t s p r e s e n c e .
Then
S i P = *pl + i p2 = (A j o ) i e ? V l e / k T e + ( A j 0 )2e " V 2 e / k T e ,
F u rth e r,
[5 3 ]
d e n o tin g th e a p p l i e d p o t e n t i a l d r o p a c r o s s th e p r o b e s by
Vjj, w e h a v e (if V j > V 2 > s a y ) ,
Vd + V2 = Vx ,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[54]
31
a s s u m i n g th e p l a s m a p o t e n t i a l is e v e r y w h e r e the s a m e .
E q u atio n s
[ 5 4 ] a n d [ 5 3 ] l e a d to th e e x p r e s s i o n
-t-nr = -
kTe
+ ftna
[55]
w here
r * S i p / i e2 - 1,
[56]
and
or *
[57]
.
(A j 0 )2
T h u s a p l o t of -tnT v s .
V<j w ill be a s t r a i g h t li n e ,
t e m p e r a t u r e m a y b e c a l c u l a t e d f r o m th e s l o p e .
is in c re a s e d ,
a n d th e e l e c t r o n
A s th e p o t e n t i a l V<i
i e2 w ill c o n tin u e to i n c r e a s e u n til i e2 = ZJip;
l a r g e r v a lu es a r e p o s sib le .
com e h o riz o n ta l.
Then a t
By sy m m e try ,
th e
th is p o in t th e c u r v e w ill
m a x im u m c u r r e n t
s i t e d i r e c t i o n w ill o c c u r f o r i e j = S i p .
no
beн
in th e o p p o н
T h e n f r o m th e c o r r e s p o n d н
ing v a l u e s of i^ one m a y r e a d i l y c o m p u te S i p .
A d i s c u s s i o n of th e
a p p l i c a t i o n of t h i s m e th o d i s g iv e n in C h a p t e r IV,
If th e p l a s m a i s to b e in a s t r o n g m a g n e t i c f i e l d ,
p r e c e d i n g d i s c u s s i o n n e e d s to b e r e e x a m i n e d .
th e e l e c t r o n m e a n f r e e p a th ,
p a r t of th e
F o r in t h i s f i e l d ,
in a d i r e c t i o n p e r p e n d i c u l a r to th e
f i e l d , i s r e d u c e d to a v a lu e r o u g h l y e q u a l to th e L a r m o r r a d i u s .
B i c k e r t o n a n d V on E n g e l (18) h a v e sh o w n th a t if m a n y c o l l i s i o n s
ta k e p l a c e w ith in th e s h e a t h s u r r o u n d i n g th e p r o b e ,
th e e l e c t r o n
c u r r e n t to a p la n e p r o b e is s t i l l e x p r e s s i b l e in th e f o r m
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
32
i ~ e TeV/ k T e ^
T h u s th e p r e c e d i n g t r e a t m e n t i s u n a l t e r e d e x c e p t f o r a p r o p o r t i o n s
a lity co n sta n t,
a n d th e r e s u l t s of th e t e m p e r a t u r e m e a s u r e m e n t s
a r e s t i l l v a lid .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
33
CHAPTER
III
E Q U IP M E N T
F o r a q u a l i t a t i v e i n v e s t i g a t i o n of th e i n t e r a c t i o n of a g u id e d
w a v e w ith a p l a s m a , a s e c t i o n of w a v e g u id e of know n le n g th , c o m н
p l e t e l y f i l l e d w ith a p l a s m a of u n i f o r m d e n s i t y is d e s i r e d .
AH
p r e v i o u s e x p e r i m e n t s on p l a s m a s w ith in w a v e g u id e s h a v e u s e d s o m e
f o r m of d i s c h a r g e tu b e i n s e r t e d in a w a v e g u id e s e c t i o n , w ith th e
tu b e o r i e n t e d e i t h e r a lo n g th e a x i s of th e g u id e o r p e r p e n d i c u l a r to
it.
It is c l e a r t h a t th e p l a s m a d id n o t c o m p l e t e l y f i l l th e g u id e .
F u rth e rm o re ,
e v e n w ith th e d i s c h a r g e off, th e tu b e i t s e l f i n t r o d u c e d
d e p a r t u r e s f r o m th e t h e o r e t i c a l v a lu e s of the m i c r o w a v e f i e l d , a n d
i t is t h i s d i s t o r t e d f i e l d w h ic h i s p r e s e n t e d to th e p l a s m a .
The
e l e c t r o n d e n s i t y w ith in the p l a s m a i t s e l f w a s a l s o f a r f r o m u n i f o r m .
In h o p e s of o b ta in in g a p l a s m a w h ic h w o u ld b e m o r e n e a r l y
h o m o g e n e o u s a n d w h ic h w o u ld c o m p l e t e l y f i l l th e w a v e g u i d e , two
d iffe re n t m eth o d s w e re tr ie d .
In th e f i r s t m e t h o d ,
th e p l a s m a w a s
to b e f o r m e d b y a h i g h - f r e q u e n c y e l e c t r o d e l e s s d i s c h a r g e ,
in th e t a n k c i r c u i t of a h ig h p o w e r o s c i l l a t o r ,
A c o il
o p e r a t i n g a t 56 m e ,
w a s w r a p p e d a r o u n d a n e v a c u a t e d w a v e g u id e s e c t i o n .
Now in o r d e r
t h a t th e p o w e r f r o m th e o s c i l l a t o r p e n e t r a t e in to t h i s s e c t i o n ,
w a v e g u id e w a l l s s h o u ld n o t b e c o n d u c tin g .
th e
On th e o t h e r h a n d , if
th e m i c r o w a v e s a r e to p r o p a g a t e a lo p g th e g u id e u n a t t e n u a t e d ,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
the
34
w a l l s m u s t be c o n d u c tin g .
It w a s d e c id e d ,
th e re fo re ,
to d e p o s i t a
th in l a y e r of s i l v e r on th e i n t e r i o r of a d i e l e c t r i c w a v e g u id e .
T he
f i e l d f r o m th e o s c i l l a t o r s h o u ld th e n p e n e t r a t e t h r o u g h th e s i l v e r
w ith o u t c o n s i d e r a b l e l o s s 1;o a d i s t a n c e r o u g h ly e q u a l to the s k in
d e p th (in th is c a s e a b o u t 30 X 1 0 ?^ c m ) .
On th e o t h e r h a n d ,
s k in d e p th f o r the m i c r o w a v e s w o u ld be a b o u t 2 X 1 0 "^ c m .
f o r a l a y e r of s i l v e r 10
c m th ic k ,
th e
T hus,
a l a r g e f r a c t i o n of th e e n e r g y
f r o m th e c o il s h o u ld p e n e t r a t e th e l a y e r ,
w h e r e a s th e m i c r o w a v e s
w o u ld r e m a i n n e a r l y c o m p l e t e l y c o n ta in e d .
The d ie le c tr ic sh e ll w as c o n s tru c te d fr o m epoxy r e s in b a r s :
th e to p a n d b o tto m h a l v e s w e r e m a c h i n e d to th e p r o p e r d i m e n s i o n s
sep a ra te ly .
S ilv e r w a s d e p o s i t e d on th e i n n e r s u r f a c e s a c c o r d i n g
to th e m e t h o d of C h e s t e r e t a l .
s e a l e d t o g e t h e r w ith e p o x y r e s i n .
(18).
T h e tw o h a l v e s w e r e th e n
U n f o r t u n a te ly ,
th e r e s i n s h e l l
a b s o r b e d too l a r g e a f r a c t i o n of the e n e r g y in te n d e d to s u s t a i n th e
d isc h a rg e ,
a n d th e m e t h o d w a s a b a n d o n e d in f a v o r of a s e c o n d
ap p ro ach .
It w a s n o te d t h a t th e m o d e o f p r o p a g a t i o n m o s t c o m m o n l y e m н
p lo y e d in a r e c t a n g u l a r w a v e g u id e i s one f o r w h ic h the w a l l c u r r e n t
i s z e r o a t th e c e n t e r of the lo n g d i m e n s i o n of th e g u id e c r o s s s e c н
tio n .
T h u s a n a r r o w s l o t m a y be c u t a lo n g t h e g u id e a t th e c e n t e r
w ith o u t a p p r e c i a b l e d i s t u r b a n c e in th e p r o p a g a t i o n of th e w a v e .
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
A
35
w a v e g u id e s p l i t dow n th e m i d d l e of b o th the w id e f a c e s w a s t h e r e f o r e
e m p lo y e d to p r o d u c e th e p l a s m a .
T he two s i d e s of th e w a v e g u id e
w e r e c o n n e c te d to th e o u tp u t of a h ig h p o w e r o s c i l l a t o r ,
c h a r g e m a i n t a i n e d b e tw e e n t h e m .
c e n tim e te rs,
and a d i s н
The le n g th of th e s e c t i o n w a s 14
o r r o u g h ly s i x t i m e s th e m i c r o w a v e le n g th .
S in ce the
m a g n i t u d e of th e p l a s m a d e n s i t y w a s n o t kijown p r i o r to th e m e a s н
u r e m e n t s , i t w a s h o p e d t h a t t h i s le n g th w o u ld b e s u f f i c i e n t to sh o w
s i g n i f i c a n t a b s o r p t i o n if th e d e n s i t y w e r e lo w , o r s o m e t r a n s m i s s i o n
if i t w e r e h ig h .
S u b s e q u e n t m e a s u r e m e n t s i n d i c a t e d t h a t th e c h o ic e
of le n g th w a s q u ite s a t i s f a c t o r y .
In th e c o n s t r u c t i o n of the w a v e g u id e s e c t i o n , p r o v i s i o n h a d to
b e m a d e f o r i t s e v a c u a t i o n , a n d f o r e l e c t r i c a l i s o l a t i o n f r o m th e
r e s t of th e m i c r o w a v e s y s t e m .
Epoxy re s in w as f i r s t p o u red around
th e e x t e r i o r of a s e c t i o n of w a v e g u id e a n d a llo w e d to h a r d e n .
The
c e n t r a l s l o t in th e m e t a l l i c g u id e w a s th e n c u t o u t a n d f i l l e d w ith
re sin .
T h e r e s i n w a s th e n c u t a w a y f r o m th e e x t e r i o r of th e g u id e
a t a l l p l a c e s e x c e p t a lo n g th e s l o t a n d a t the e n d s .
T h is w a s done
to f a c i l i t a t e the r e m o v a l of h e a t g e n e r a t e d d u r in g th e d i s c h a r g e , a n d
to p r o v i d e e l e c t r i c a l a c c e s s to th e tw o s i d e s .
t h i c k w e r e s e a l e d to th e e n d s of th e g u id e .
M ic a w in d o w s . 0 0 5 "
N o z z le s w e r e s e a l e d in
p l a c e a t th e s l o t to p e r m i t e v a c u a t i o n a n d i n t r o d u c t i o n of a i r in to
th e w a v e g u id e s e c t i o n .
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
36
T h e c o m p l e t e d s e c t i o n i s i l l u s t r a t e d in F i g u r e 3.
T h e p r o b e s u s e d to m e a s u r e th e t e m p e r a t u r e w e r e m a d e f r o m
.0 1 5 " n ic h ro m e w ire ,
i n s e r t e d th r o u g h h o le s d r i l l e d a t b o th e n d s of
o n e s l o t a n d s e a l e d in p l a c e .
T h e s e p r o b e s e x te n d e d a b o u t one
m i l l i m e t e r in to th e p l a s m a .
T h e f i n i s h e d w a v e g u id e s e c t i o n s e r v e d v e r y w e ll in a l l b u t one
re sp e c t.
T h e f le x in g of th e m i c a w in d o w s w h ic h o c c u r r e d w h e n th e
s y s t e m w a s e v a c u a t e d a n d t h e i r o c c a s i o n a l e x c e s s i v e h e a t i n g b y th e
p l a s m a i n t r o d u c e d s m a l l l e a k s in th e s y s t e m .
w in d o w s g r a d u a l l y b e c a m e d i s c o l o r e d in u s e ,
v i s u a l o b s e r v a t i o n of the d i s c h a r g e ,
F u rth e rm o re ,
the
th u s h i n d e r i n g c l e a r
P ro v isio n w as m ad e fo r som e
c o o lin g b y a b l o w e r , in h p p e s o f r e d u c i n g the e f f e c t of h e a t i n g on
th e w in d o w s.
T h e m e a s u r e m e n t of t r a n s m i s s i o n a n d r e f l e c t i o n c o e f f i c i e n t s
fo llo w e d c o n v e n ti o n a l m i c r o w a v e m e t h o d s .
s y s t e m i s sh o w n in F i g u r e 4.
m icro w av e b rid g e,
A b lo c k d i a g r a m of th e
T he m i c r o w a v e s o r i g i n a t e d f r o m a
a s th is w a s a c o n v e n ie n tly a v a i l a b l e s o u r c e .
T h e r - f o s c i l l a t o r u s e d f o r p r o d u c i n g th e d i s c h a r g e w a s a
H a r t l e y o s c i l l a t o r w ith th e o u tp u t c i r c u i t m a i n t a i n e d a t d c g r o u n d
p o te n tia l.
T h e f u n d a m e n t a l f r e q u e n c y w a s 1 .3 m e ;
som e pow er
w a s p r e s e n t in th e h i g h e r h a r m o n i c s .
T h e m a g n e t i c f i e l d w a s p r o v i d e d b y a 1 2 - i n c h m a g n e t c a p a b le
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
38
of p r o v id i n g flu x d e n s i t i e s in e x c e s s of 1 2 ,0 0 0 g a u s s .
The h o m o н
g e n e i t y of th e f i e l d f a r e x c e e d e d th e d e m a n d s of t h e s e e x p e r i m e n t s .
A s w a s i n d i c a t e d b y e q u a ti o n [ 2 9 ] in th e p r e c e d i n g s e c t i o n ,
t h e r e s h o u ld be n o c o u p lin g b e tw e e n th e m i c r o w a v e e l e c t r i c f ie ld
a n d th e m a g n e t i c f i e l d f o r th e p a r a l l e l o r i e n t a t i o n of th e w a v e g u id e ,
a n d n o c y c l o t r o n r e s o n a n c e a b s o r p t i o n w o u ld b e o b s e r v e d .
Thus all
c o l l i s i o n f r e q u e n c y m e a s u r e m e n t s h a d to b e m a d e w ith th e w a v e н
g u id e in th e p e r p e n d i c u l a r o r i e n t a t i o n .
M an y of t h e s e m e a s u r e m e n t s
w e r e m a d e p o in t b y p o in t, a s th e m a g n e t i c f i e l d w a s v a r i e d ,
sin ce
i t w a s n e c e s s a r y to m e a s u r e th e m a g n e t i c f i e l d a t e a c h p o in t v e r y
a c c u ra te ly .
S o m e m e a s u r e m e n t s w e r e m a d e b y m o d u la tin g th e m a g н
n e t p o w e r s u p p ly a t . 0 6 c p s ,
a n d fe e d in g th e o u tp u t s i g n a l f r o m the
p o w e r m e t e r to a r e c o r d e r .
D u r in g th e p o i n t - b y - p o i n t m e a s u r e н
m e n ts,
t h e r e a p p e a r e d to be s o m e s l i g h t a b s o r p t i o n o u t s i d e th e
c y c l o t r o n a b s o r p t i o n r a n g e , b u t th e m i c r o w a v e o u tp u t p o w e r w a s n o t
s u f f i c i e n t l y s t a b l e f o r one to b e c e r t a i n .
a b s o r p t i o n d id in f a c t ta k e p l a c e ,
In o r d e r to s e e if th is
a h ig h f r e q u e n c y m o d u la tio n of th e
m a g n e t c u r r e n t w a s s u p e r p o s e d on th e low f r e q u e n c y s w e e p ,
a n d th e
o u tp u t s i g n a l f e d to a h o m o d y n e u n it, a s s h o w n b y th e d a s h e d l i n e s
in F i g u r e 4.
T h e o u tp u t s i g n a l (w h ic h w a s f e d to a r e c o r d e r ) is
p r o p o r t i o n a l e i t h e r to a r a t e of c h a n g e of a b s o r p t i o n of m i c r o w a v e
p o w e r f w ith m a g n e t i c f i e l d ,
o r to a s h i f t in p h a s e of th e m i c r o -
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
of the copyright ow ner.
1000 cps
S q u a re
W ave
VSWR
M ete r
R -F
O sc illato r
Pow er
M ete r
F req u en cy
M e te r
D e tec to r
20D B
M ic r o w a v e
Source
Further reproduction
S lo tte d
S e c tio n
P la sm a
.S e c tio n s
M agnet
M atch ed
Load
M agnet
Pow er
Supply
prohibited
1000 c p s
O sc illa to r
without p e r m issio n .
Low
F requency
T ria n g u la r
W ave
F ig u r e 4.
H om odyne
U n it
R eco rd er
B lo c k d i a g r a m of e q u i p m e n t f o r m e a s u r e m e n t s of r e f l e c t i o n a n d a b s o r p t i o n .
40
w a v e s a s th e m a g n e t i c f i e l d i s c h a n g e d ,
s i n c e th e h o m o d y n e u n it
r e s p o n d s to b o th .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
41
CHAPTER
IV
E X P E R I M E N T A L P R O C E D U R E AND D A TA
A.
R e f l e c t i o n C o e f f ic ie n ts
T h e r e f l e c t i o n c o e f f i c i e n t s f o r the w a v e i n c i d e n t on th e p l a s m a
s e c t i o n w e r e o b ta in e d f r o m m e a s u r e m e n t s of th e v o lta g e s ta n d in g
w a v e r a t i o (VSWR) in the w a v e g u id e .
by r ,
T h is r a t i o , w h ic h w e d e n o te .
is r e l a t e d to th e r e f l e c t i o n c o e f f i c i e n t R by
r
=
1 + Rl/2
[5 9 ]
1 - R 1 /2
In the a b s e n c e of th e p l a s m a ,
p ro b es,
th e m i c a w in d o w s , t e m p e r a t u r e
a n d o t h e r i r r e g u l a r i t i e s in th e w a v e g u id e b o u n d a r y w ill s e t
up re fle c tio n s.
B efo re any m e a s u r e m e n ts w e re tak en ,
th e se r e f l e c н
t io n s w e r e c a n c e l l e d b y a d j u s t i n g the E f H t u n e r u n til th e r e s i d u a l
VSWR w a s n o g r e a t e r th a n 1 ,0 2 .
T yp ical re fle c tio n c o effic ien t
d a t a a r e sh o w n in F i g u r e 5, f o r th e p e r p e n d i c u l a r o r i e n t a t i o n of
th e w a v e g u id e ,
T he d a ta f o r th e p a r a l l e l o r i e n t a t i o n w e r e n o t s i g н
n i f i c a n t a s w ill be s e e n in th e n e x t s e c t i o n ,
B.
M a g n e tic F i e l d
T h e m a g n e t i c f i e l d w a s m e a s u r e d by m e a n s of a n NMR p r o b e ,
u s in g p r o t o n r e s o n a n c e a b s o r p t i o n in w a t e r .
T h e f r e q u e n c y of th e
p r o b e w a s d e t e r m i n e d b y r a d i o - f r e q u e n c y m o d u la tio n of th e p r o b e
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
lission
-.1 5 0
of the copyright ow ner.
-.1 2 5
- . 100
Further reproduction
u
9)
O
O
-.0 7 5
гo
o
(0
prohibited
o>
Pi
-.0 5 0
without p e r m issio n .
-.0 2 5
.8 0
F i g u r e 5.
.9 0
1 . 00
1.2
1. 3
1 .4
1 .5
V a r i a t i o n i n m i c r o w a v e p o w e r r e f l e c t e d f r o m a p l a s m a - f i l l e d w a v e g u id e w i t h i n c r e a s н
in g t r a n s v e r s e m a g n e t i c f i e l d , ( Ct)c '=
v
W m /
43
fre q u e n c y , and d e te c tio n by a ra d io r e c e i v e r .
The ra d io r e c e iv e r
f r e q u e n c y w a s in t u r n c a l i b r a t e d f r o m th e s i g n a l f r e q u e n c i e s of
WWV.
In m a k in g o b s e r v a t i o n s o v e r a b r o a d r a n g e o f m a g n e t i c f i e l d ,
i t p r o v e d to be m o r e c o n v e n ie n t to u s e th e m a g n e t c u r r e n t a s a
m e a s u r e of th e f i e l d s t r e n g t h .
th e NMR p r o b e m e a s u r e m e n t s .
T h e a m m e t e r w a s c a l i b r a t e d u s in g
T h o u g h th e m a g n e t c u r r e n t g iv e s a
c o n s i d e r a b l y l e s s a c c u r a t e m e a s u r e m e n t of th e f i e l d b e c a u s e of th e
m agnet h y ste re sis,
th e m e a s u r e d v a lu e s of f i e l d s t r e n g t h w e r e r e н
p r o d u c i b l e w e ll w ith in th e r a n g e of a c c u r a c y r e q u i r e d f o r th e b r o a d
range m e a s u re m e n ts.
C.
C o llisio n F re q u e n c y
A s n o te d in C h a p t e r III, th e c o l l i s i o n f r e q u e n c i e s w e r e fo u n d
f r o m th e c y c l o t r o n r e s o n a n c e a b s o r p t i o n ,
The a b so rp tio n m e a s u r e н
m e n t s w e r e t a k e n (1) b y a p o i n t - b y - p o i n t m e th o d ,
in g th e m a g n e t i c f ie ld .
In th e f i r s t c a s e ,
a n d (2) by s w e e p н
m e a su re m e n ts w ere m ade
o v e r a r a n g e of m a g n e t i c f i e l d f r o m 0>c ' = 0 to C0C' = 2 .
second c a se ,
th e r a n g e o f th e s w e e p w a s a b o u t 1200 g a u s s ,
r o u g h l y a b o u t C0C' = 1,
In the
c en te re d
T h e c o l l i s i o n f r e q u e n c i e s V * w e r e th e n fo u n d
d i r e c t l y f r o m th e g r a p h s of T v s ,
a r e not a c tu a lly s y m m e tric ,
C0c '.
The e x p e r im e n ta l c u rv e s
b u t th e h a lf w id th i s s t i l l a r e a s o n a b l y
a c c u r a t e m e a s u r e of the c o l l i s i o n f r e q u e n c y (14),
The tra n s m itte d
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
44
p o w e r i s sh o w n in F i g u r e 6 f o r tw o d i f f e r e n t v a lu e s of o s c i l l a t o r
pow er,
F i g u r e 7 i n d i c a t e s th e v a lu e s of c o l l i s i o n f r e q u e n c y t a k e n
fro m th e se a n d o th e r d a ta,
D.
a s a fu n c tio n of o s c i l l a t o r p o w e r .
E le c tro n T e m p e ra tu re
A s a n e x a m p l e of th e u s e of th e d o u b le p r o b e m e t h o d f o r d e н
t e r m i n i n g th e e l e c t r o n t e m p e r a t u r e ,
c u r r e n t c u r v e , F i g u r e 8,
we c o n sid e r a ty p ic a l v o lta g e -
T h e o r d i n a t e i s th e v o lta g e d r o p a c r o s s
a s e r i e s r e s i s t o r ( F i g u r e 2), r a t h e r th a n th e c u r r e n t i t s e l f .
q u a n tity I \
d e f in e d in e q u a tio n [ 5 6 ] ,
The
d e p e n d s u p o n th e r a t i o of the
s a t u r a t e d - i o n c u r r e n t to th e e l e c t r o n c u r r e n t a t o ne p r o b e ,
h e n c e i s i n d e p e n d e n t of th e c h o ic e of d i m e n s i o n s .
and
The s a tu r a te d
io n c u r r e n t is L i p = i p j + ip2> w h e r e i p i a n d ip2 a r e t a k e n a t th e
p o in ts i n d i c a t e d .
w hat a rb itra ry ,
T h e c h o ic e of lo c a t i o n of t h e s e p o in ts i s s o m e н
b u t th e c a l c u l a t e d e l e c t r o n t e m p e r a t u r e is r e l a t i v e l y
i n s e n s i t i v e to th is c h o ic e .
The ra tio L i p / i g2 is r e a d fo r a n u m b e r
of p o in ts a lo n g th e c e n t r a l s t r a i g h t lin e p o r t i o n of th e g r a p h .
of -tnT v s .
ing to [ 5 5 ] .
A p lo t
( F i g u r e 8) th e n g iv e s th e e l e c t r o n t e m p e r a t u r e a c c o r d н
T h e c u r v e a b o v e th e p o in ts a t w h ic h th e s a t u r a t e d io n
c u r r e n ts a r e tak en is not h o riz o n ta l a s p re d ic te d .
T h is is
p ro b a b ly
due to a n e x p a n s io n of th e s h e a t h a s th e p r o b e p o t e n t i a l b e c o m e s
v e r y n e g a t i v e w ith r e s p e c t to th e p l a s m a .
M o r e io n s a r e a t t r a c t e d
to th e p r o b e , a n d th e c u r r e n t i n c r e a s e s .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
1.0
of the copyright ow ner.
8
Further reproduction
.6
.4
prohibited
tts
1 1 7 .5
without p e r m issio n .
.2
50 w a tts
.7 0
F i g u r e 6.
.8 0
.9 0
1 . 00
I
1.10
1.20
1 .3 0
C y c l o tr o n r e s o n a n c e a b s o r p t i o n of g u id e d m i c r o w a v e s i n a p l a s m a , f o r tw o v a l u e s
of o s c i l l a t o r i n p u t p o w e r .
( CO ' =
V c
m<*>y
ion of the copyright ow ner.
.300
.0 3
o
<u
г
0)
frequency
Further reproduction
3
Collision
2 00
u
V'
prohibited
100
.01
without p e r m issio n .
20
40
60
80
O s c i l l a t o r in p u t p o w e r
100
120
o
F i g u r e 7.
V a r i a t i o n of e l e c t r o n p l a s m a f r e q u e n c y a n d e l e c t r o n c o l l i s i o n f r e q u e n c y with.
o sc illa to r in p u t p o w er.
R eproduced
with perm ission
V o lts
1.2
of the copyright ow n er.
el
Further reproduction
-80
-60
-4 0
-2 0
20
40
60
100
80
v o lts
prohibited
-.4
e2
AVd
without p e r m issio n .
= 2 2 - 9 ev
-
F i g u r e 8.
1.2
-1 5 -1 0
-5
15
C h a r a c t e r i s t i c c u r r e n t - v o l t a g e c u r v e f o r th e m e a s u r e m e n t o f e l e c t r o n t e m p e r a t u r e .
T h e o r d i n a t e i s t h e v o lta g e a c r o s s a s e r i e s r e s i s t o r ( s e e F i g , 2).
4^
-J
48
T h e v a r i a t i o n of e l e c t r o n t e m p e r a t u r e w ith i n c r e a s i n g p o w e r
in p u t to the d i s c h a r g e i s i l l u s t r a t e d in F i g u r e 9 f o r tw o d i f f e r e n t
v a l u e s of m a g n e t i c f i e l d ,
F o r e a c h f i e l d s t r e n g t h th e d a ta t a k e n a t
low d i s c h a r g e p o w e r s i n d i c a t e d t h a t th e e l e c t r o n v e l o c i t y w a s n o t
M a x w e llia n ,
T h e s e d a t a a r e n o t in c lu d e d on th e g r a p h .
F i g u r e 10
i l l u s t r a t e s the v a r i a t i o n of e l e c t r o n t e m p e r a t u r e w ith m a g n e t i c f i e l d
a t a c o n sta n t d is c h a rg e p o w ^r.
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
lission
of the copyright ow ner.
24
ev
21
Further reproduction
18
prohibited
1580 g a u s s
without p e r m issio n .
15
4 94 0 g a u s s
120
F i g u r e 9.
160
200
Os c i l l a t o r C u r r e n t
240
ma
280
Y a r a t i o n of e l e c t r o n t e m p e r a t u r e w ith o s c i l l a t o r i n p u t c u r r e n t , i n t h e p r e s e n c e of a
tr a n s v e r s e m a g n e tic fie ld .
s░
R eproduced
with perm ission
25 e v
of the copyright ow n er.
23
Further reproduction
21
prohibited
without p e r m issio n .
2
6
4
M ag n e tic F lu x D e n sity
--
8
10
k ilo g au ss
Ul
o
F i g u r e 10.
V a r i a t i o n of e l e c t r o n t e m p e r a t u r e w i t h i n c r e a s i n g m a g n e t i c f lu x d e n s i t y .
51
CHAPTER
V
R E S U L T S AND C O N C LU SIO N S
A,
The D isch arg e
It is e s s e n t i a l a t th e o u t s e t to e x a m i n e th e a s s u m p t i o n of a
hom ogeneous p la sm a.
S ince th e e l e c t r o d e c o n f i g u r a t i o n d o e s n o t
g iv e r i s e to a u n i f o r m e l e c t r i c f i e l d ( s e e F i g u r e 11), one w o u ld
c e r t a i n l y e x p e c t th a t th e e l e c t r o n d e n s i t y m i g h t n o t be u n i f o r m ,
v i s u a l e x a m i n a t i o n of th e i n -
A
?
> B
t e n s i t y of th e lig h t f r o m the
d i f f e r e n t p o s i t i o n s w ith in th e
d i s c h a r g e g iv e s a n i n d i c a н
tio n of th e h o m o g e n e i ty .
T h is
m a y b e s e e n f r o m th e f o l l o w in g a r g u m e n t .
F ig u re I I .
rf e le c tric
f i e l d w ith in p l a s m a - f i l l e d
w a v e g u id e s e c t i o n ; p e r p e n н
d ic u la r o rie n ta tio n .
The p rin c ip a l
m e c h a n i s m f o r r a d i a t i o n in th e v i s i b l e r e g i o n i s b y e l e c t r o n - i o n
rec o m b in atio n .
(O th e r p o s s i b l e s o u r c e s w o u ld b e th e b r e m m s t r a h l -
u ng r a d i a t i o n a n d c y c l o t r o n e m i s s i o n .
F o r th e low e l e c t r o n v e l o c i н
t i e s ( n o n r e l a t i v i s t i c ) a n d f o r lo w p e r c e n t a g e i o n i z a t i o n s of t h i s d i s н
charge,
how ever,
th e se s o u rc e s m ay be n eg lec te d , )
The ra d ia tio n
i n t e n s i t y i s p r o p o r t i o n a l to th e d e n s i t y of e l e c t r o n s a n d i o n s .
th e e l e c t r o n a n d io n d e n s i t i e s a r e e q u a l,
a l to n g ^.
S ince
th e i n t e n s i t y is p r o p o r t i o n н
T h e n v a r i a t i o n s of e l e c t r o n d e n s i t y w ill b e m a n i f e s t by
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
52
v a r i a t i o n s in i n t e n s i t y .
T h e v i s u a l o b s e r v a t i o n s i n d i c a t e d t h a t th e d i s c h a r g e w a s f a r
f r o m u n i f o r m f o r th e lo w p o w e r d i s c h a r g e s in s t r o n g m a g n e t i c
fie ld s,
f o r th e p a r a l l e l o r i e n t a t i o n of the w a v e g u id e .
On the o t h e r
h a n d , f o r th e p e r p e n d i c u l a r o r i e n t a t i o n a n d f o r low m a g n e t i c f i e l d
(> 1000 g a u s s ) ,
t h e r e a p p e a r e d to be v e r y l i t t l e v a r i a t i o n in the
l i g h t i n t e n s i t y o v e r th e c r o s s s e c t i o n of th e g u id e .
l a s t tw o c a s e s ,
Thus fo r th e se
th e e l e c t r o n d e n s i t y w a s v e r y n e a r l y u n i f o r m t h r o u g h н
o u t th e p l a s m a .
A ll of t h e s e o b s e r v a t i o n s w e r e u n c h a n g e d a s th e
gas p r e s s u r e w as v a rie d .
T h e la c k of u n i f o r m i t y of th e d i s c h a r g e in th e p a r a l l e l o r i e n н
ta t i o n b e c a m e m o r e a n d m o r e n o t i c e a b l e a s th e m a g n e t i c f i e l d i n н
creased .
In f a c t ,
th e d i s c h a r g e e v e n t u a l l y w a s q u e n c h e d .
R aisin g
th e p o w e r k e p t th e d i s c h a r g e g o in g , b u t the h e a t p r o d u c e d b e c a m e
s o i n t e n s e t h a t th e r e s i n w a v e g u id e s t a r t e d to s m o k e .
The q u e n ch н
ing of th e d i s c h a r g e d id n o t o c c u r f o r th e o t h e r o r i e n t a t i o n .
It i s
p o s s i b l e to e x p la i n t h i s a n n o y in g b e h a v i o r b y c o n s i d e r i n g th e
m e c h a n i s m in v o lv e d in s u s t a i n i n g th e d i s c h a r g e .
a r e a c c e l e r a t e d by the r - f e l e c t r i c f i e l d ,
n e u tra l p a rtic le s ,
C harged p a rtic le s
a n d u p o n c o l l i s i o n w ith
pro du ce new c h a rg e d p a rtic le s .
P r o v i d e d th e r a t e
of p r o d u c t i o n b a l a n c e s th e l o s s due to r e c o m b i n a t i o n a n d d if f u s io n
to th e w a l l s ,
th e d i s c h a r g e w ill b e s u s t a i n e d .
Now w h e n th e w a v e н
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
53
g u id e i s in th e p e r p e n d i c u l a r o r i e n t a t i o n ,
th e m a g n e t i c f i e l d a r e r o u g h l y p a r a l l e l ,
11.
th e r - f e l e c t r i c f i e l d a n d
a s q a n b e s e e n in F i g u r e
T h u s th e m a g n e t i c f i e l d w ill h a v e r e l a t i v e l y l i t t l e e f f e c t on the
e lec tro n a c c e le ra tio n and,
charged p a rtic le s,
not tru e .
th e re fo re ,
on th e p r o d u c t i o n of n e w
F o r th e p a r a l l e l o r i e n t a t i o n ,
ho w ev er,
th is i s
T he v e l o c i t i e s a c q u i r e d by th e e l e c t r o n s f r o m th e r - f
e le c tr ic fie ld m u s t be d im in ish e d ,
s i n c e th e m a g n e t i c f i e l d w ill
c a u s e t h e e l e c t r o n v e l o c i t y to b e d e f l e c t e d a w a y f r o m th e e l e c t r i c
fie ld d ire c tio n .
In f a c t ,
a s th e m a g n e tic fie ld is i n c r e a s e d ,
th e
p a rtic le tr a je c to r ie s a c q u ire an in c re a s in g ly g r e a te r c u rv a tu re .
a r e l a t i v e l y low v a lu e of th e f i e l d ,
At
th e t r a j e c t o r y w ill ta k e th e f o r m
of a p r o l a t e c y c lo i d , w ith th e d i r e c t i o n of d r i f t a lo n g t h e a x i s of
th e w a v e g u id e (20).
T h e n d u r i n g o n e - h a l f of a c y c l e ,
th e e l e c t r o n
a c q u i r e s e n e r g y f r o m th e f i e l d , w h ile d u r i n g th e s e c o n d h a lf it
lo se s en erg y .
T h e r e f o r e i t w i l l n o t be a c c e l e r a t e d to s u f f i c i e n t l y
h ig h e n e r g i e s to s u s t a i n th e d i s c h a r g e .
In a d d itio n to th e o b s e r v a t i o n s on th e h o m o g e n e i t y ,
th e re w ere
a n u m b e r of o b s e r v a t i o n s on th e d i s c h a r g e w h ic h h a v e a b e a r i n g on
th e i n t e r p r e t a t i o n of o t h e r r e s u l t s .
O nce th e d i s c h a r g e w a s e s t a b н
lish e d ,
c h a r a c t e r i s t i c of a i r .
i t h a d th e u s u a l p in k c o l o r ,
p o w e r to th e d i s c h a r g e w a s i n c r e a s e d ,
c h a n g e in th e c o l o r f r o m p in k to w h ite ,
A s th e
th e re o c c u rre d a sudden
p r e s u m a b l y d u e to e*. s e c o n d
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
54
io n i z a t i o n .
O nce th is t r a n s i t i o n to o k p l a c e ,
th e p o w e r to th e d i s н
c h a r g e c o u ld be r e d u c e d c o n s i d e r a b l y b e f o r e th e r e v e r s e t r a n s i t i o n
w o u ld o c c u r .
T h e e f f e c t on th e p r o p a g a t i o n of m i c r o w a v e s t h r o u g h th e
p la sm a,
th r o u g h o u t th e r e g i o n m e n t i o n e d a b o v e ,
ex p ec te d .
w as som ew hat u n н
To i l l u s t r a t e o u r o b s e r v a t i o n s , w e h a v e p l o t t e d in F i g u r e
12 th e m i c r o w a v e t r a n s m i s s i o n a n d r e f l e c t i o n c o e f f i c i e n t s ,
a n d th e
c u r r e n t in p u t to th e o s c i l l a t o r a s f u n c tio n s of th e o s c i l l a t o r v o lta g e .
T h e r e a r e two u n u s u a l f e a t u r e s of t h e s e c u r v e s .
T h e f i r s t i s th e
r e g i o n of i n s t a b i l i t y of th e d i s c h a r g e in th e r a n g e of o s c i l l a t o r
v o l t a g e s b e t w e e n 4 3 0 a n d 530 v o l t s .
n e e d l e s f l u c t u a t e d w ild ly ,
r e a d i n g c o u ld b e m a d e .
O v e r t h i s r e g i o n th e m e t e r
a n d o n ly a r e a s o n a b l e g u e s s of a n a v e r a g e
T h o u g h th e t r a n s i t i o n f r o m th e p in k to th e
w h ite d i s c h a r g e o c c u r r e d in t h i s v o lta g e r a n g e , i t w a s v e r y a b r u p t ,
a n d th u s n o t d i r e c t l y r e l a t e d to th e i n s t a b i l i t y .
The sec o n d fe a tu re
of i n t e r e s t i s t h a t th e t r a n s m i s s i o n c o e f f i c i e n t o v e r a s t i l l b r o a d e r
r a n g e i s g r e a t e r th a n u n ity ,
P r e li m in a r y m e a s u r e m e n ts show ed
t h a t p o w e r r a d i a t e d by th e p l a s m a in th e m i c r o w a v e f r e q u e n c y r a n g e
w a s to o low to b e m e a s u r e d by th e p o w e r m e t e r .
p l a s m a s e r v e d to a m p l i f y th e m i c r o w a v e s .
T h erefo re,
th e
T h e q u e s t i o n of w h e t h e r
o r n o t th e i n c i d e n t a n d th e a m p l i f i e d w a v e a r e c o h e r e n t r e m a i n s to
be s e ttle d .
It s h o u ld a l s o b e r e m a r k e d t h a t th e p l a s m a d id n o t
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
lission
ma
of the copyright ow ner.
1. 50
. 20
200
Further reproduction
1 . 00
150
.7 5
10
? o 100
prohibited
50
without p e r m issio n .
50
. 05
u n stab le
360
440
25
520
O s c i l l a t o r v o lta g e
F i g u r e 12.
600
--
680
760
v o lts
O s c illa to r c u r r e n t an d re f le c tio n a n d tr a n s m is s io n c o e ffic ie n ts a s a fu n c tio n of
o s c illa to r v o lta g e ; u n sta b le d is c h a rg e .
m
56
a lw a y s e x h ib it th e b e h a v i o r i n d i c a t e d by th e c u r v e s of F i g u r e 12.
On o c c a s i o n th e i n i t i a l (b re a k d o w n ) c u r r e n t w a s c o n s i d e r a b l y h i g h e r
a n d w o u ld i n c r e a s e a l m o s t l i n e a r l y w ith v o lta g e .
r e fle c tio n c o effic ien t w as m u ch l e s s ,
of th e m i c r o w a v e s ,
The c o rre sp o n d in g
and t h e r e w a s no a m p lific a tio n
a s s e e n in F i g u r e 13.
T h e r e s e e m e d to be no
w a y o f p r e d i c t i n g w h ic h ty p e of d i s c h a r g e w o u ld ta k e p l a c e .
In th e p r e s e n c e of a s m a l l m a g n e t i c f i e l d (^SOO g a u s s ) ,
a p p e a r e d to be no r e g i o n of i n s t a b i l i t y ,
a b ru p t co lo r change.
th ere
th o u g h t h e r e w a s s t i l l th e
The c u rre n t-v o lta g e cu rv e w as e ss e n tia lly
t h a t of F i g u r e 13.
A s e c o n d o b s e r v a t i o n , p o s s i b l y r e l a t e d to th e f i r s t , w a s of a
s m a l l a n o m a l o u s a b s o r p t i o n o r p h a s e s h if t of th e m i c r o w a v e s in two
m a g n e tic fie ld re g io n s .
In h o p e s of m e a s u r i n g s m a l l v a r i a t i o n s in
a b s o r b e d p o w e r o u ts id e th e c y c l o t r o n r e s o n a n c e a b s o r p t i o n r a n g e ,
h o m o d y n e u n it w a s c o n n e c te d to th e o u tp u t of th e p o w e r m e t e r ,
m e n t i o n e d in C h a p t e r III.
a
as
Q u ite b y a c c i d e n t i t w a s d i s c o v e r e d th a t
t h e r e w a s e i t h e r a p h a s e s h i f t o r a n a b s o r p t i o n of th e m i c r o w a v e s
o c c u r r i n g in a s m a l l m a g n e t i c f i e l d r a n g e a r o u n d 370 g a u s s a n d in
a n o t h e r r a n g e a r o u n d 4 80 g a u s s .
T h i s to o k p l a c e o n ly f o r th e
p a r a l l e l o r i e n t a t i o n of th e g u id e , a s f a r a s c o u ld b e d e t e r m i n e d .
S in c e t h e r e s h o u ld b e no c o u p lin g b e tw e e n th e m a g n e t i c f i e l d a n d th e
m i c r o w a v e s f o r t h i s o r i e n t a t i o n , a c c o r d i n g to th e r e s u l t s of C h a p t e r
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
300
012
of the copyright ow n er.
1. 50
ma
010
1 .2 5
200
Further reproduction
008
1 . 00
006
75
prohibited
?W
g 100
004
without p e r m issio n .
002
u n s t a b l e r e g i o n of F i g .
340
F i g u r e 13.
380
420
460
500
O s c i l l a t o r V o lta g e
50
-
25
12
540
580
O s c i l l a t o r c u r r e n t and. r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s a s a f u n c t i o n of
o s c illa to r v o lta g e ;
sta b le d is c h a rg e .
Ui
-a
58
II, w e f in d i t d i f f i c u l t to g iv e a s a t i s f a c t o r y e x p la n a tio n .
E v e n if
th e w a v e g u id e w e r e n o t e x a c t l y p r o p e r l y a l i g n e d , t h e r e s e e m s to be
n o g oo d r e a s o n f o r e x p e c tin g s u c h a n e f f e c t ,
T h e re is so m e in d ic a н
t io n t h a t th e e f f e c t m a y b e in s o m e w a y a t t r i b u t a b l e to th e i o n s ,
th e r e s o n a n t m a g n e t i c f i e l d s
as
a r e i n v e r s e l y p r o p o r t i o n a l to the io n
m asses.
B.
E le c tro n T e m p e ra tu re
T h r e e r e s u l t s of th e e l e c t r o n t e m p e r a t u r e m e a s u r e m e n t s a r e
of i n t e r e s t .
w a s fo u n d ,
T h e f i r s t of t h e s e is th e r e l a t i v e l y h ig h v a lu e w h ic h
of th e o r d e r of 20 e v , o r a b o u t 2 3 0 ╗ 0 0 0 ░ K ,
T his fig u re
i s a f a c t o r of 10 a b o v e t h a t w h ic h m ig h t b e e x p e c t e d a t th e g a s
p r e s s u r e of th e d i s c h a r g e .
It s h o u ld b e k e p t in m in d t h a t the
p r o b e s w e r e l o c a t e d in th e r e g i o n of a v e r y h ig h e l e c t r i c f i e l d , h o w p
ever.
A l s o , th e t h e o r e t i c a l e x p r e s s i o n s f o r e l e c t r o n t e m p e r a t u r e
u n d e r v a r i o u s c o n d itio n s of p r e s s u r e a n d e l e c t r i c f i e l d a r e s e n s i t i v e
to th e g e o m e t r y of th e s y s t e m ; o t h e r e x p e r i m e n t a l v a l u e s w ith w h ic h
w e c o m p a r e o u r s h a v e b e e n fo u n d f o r q u ite d i f f e r e n t g e o m e t r i e s .
In a n y e v e n t , th e e l e c t r o n t e m p e r a t u r e w a s s t i l l to o lo w to a f f e c t
th e v a l i d i t y of o u r e x p r e s s i o n f o r th e c o n d u c ti v ity .
S e c o n d , in th e a b s e n c e of a m a g n e t i c f i e l d ,
th e p r o b e c u r r e n t -
v o lta g e c u r v e s i n d i c a t e d th a t th e e l e c t r o n v e lo c i ty d i s t r i b u t i o n i s n o t
M a x w e llia n .
T h is w o u ld i m p l y th a t th e p l a s m a e l e c t r o n d e n s i t y is
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
59
s o low t h a t th e m e a n f r e e p a t h f o r c o l l i s i o n i s l o n g e r th a n the
d i m e n s i o n s of th e w a v e g u id e .
T h e n th e e l e c t r o n s do n o t c o llid e
w ith n e u t r a l p a r t i c l e s s u f f i c i e n t l y o f te n to a t t a i n t h e r m a l e q u i l i b r i н
um .
A s th e m a g n e t i c f i e l d is a p p l i e d , th e m e a n f r e e p a th d r o p s to
r o u g h l y th e L a r m o r r a d i u s ,
t h e r e b y i n c r e a s i n g th e li k e l i h o o d of
c o l l i s i o n s b e f o r e th e e l e c t r o n s d if f u s e to th e w a l l s .
F in a lly ,
th e i n c r e a s e in e l e c t r o n t e m p e r a t u r e w ith m a g n e t i c
f i e l d i s th e o p p o s ite of th e e x p e c t e d r e s u l t a s p r e d i c t e d b y th e
T o w n s e n d t h e o r y (21).
H o w e v e r , B e c k a n d D a v is h a v e o b ta in e d
r e s u l t s s i m i l a r to o u r f o r r a d i o - f r e q u e n c y d i s c h a r g e s (2 2, 23).
An
e x p l a n a t i o n f o r th e d e p a r t u r e f r o m th e t h e o r e t i c a l r e s u l t is s t i l l to
b e fo u n d .
C.
C o llisio n F re q u e n c y
A s c a n be s e e n f r o m F i g u r e 7, th e c o l l i s i o n f r e q u e n c y V '
c o u ld be v a r i e d r e a d i l y f r o m a l o w e s t v a lu e of a b o u t 0 .1 to a h i g h н
e s t v a lu e of 0 , 4 .
T h e lo w e r l i m i t o c c u r s j u s t a b o v e b r e a k d o w n ;
th e u p p e r l i m i t i s a l i m i t o n ly u p o n th e m e th o d of m e a s u r e m e n t .
T h a t i s , f o r c o l l i s i o n f r e q u e n c i e s s o m e w h a t l a r g e r th a n 0 . 4 th e
c y c l o t r o n r e s o n a n c e a b s o r p t i o n c u r v e s b e c a m e to o b r o a d f o r a n
a c c u r a t e d e t e r m i n a t i o n of th e h a l f - w i d t h .
P r e s u m a b l y o n e m i g h t be
j u s t i f i e d in s o m e e x t r a p o l a t i o n of th e c o l l i s i o n f r e q u e n c y - p o w e r
in p u t c u r v e tp h i g h e r p o w e r s , a s th e c u r v e i s v e r y n e a r l y l i n e a r
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
60
o v e r th e p o w e r in p u t r a n g e e m p lo y e d .
A lso ,
th e r a n g e c a n b e e x н
te n d e d s l i g h t l y b y e m p lo y in g h i g h e r m i c r o w a v e f r e q u e n c i e s .
V ary н
in g th e g a s p r e s s u r e f r o m 200 to 850 m i c r o n s c a u s e d o n ly s l i g h t
c h a n g e s in th e c o l l i s i o n f r e q u e n c y .
It s h o u ld a l s o b e m e n t i o n e d
t h a t a t th e low p o w e r s th e m e a s u r e m e n t s w e r e n o t s o r e l i a b l e a s
a t h ig h e r p o w e rs.
T h is m a y p o s s i b l y b e r e l a t e d to th e i n s t a b i l i t y
of th e d i s c h a r g e a t low p o w e r s ,
D.
E le c tr o n D e n sity
T h e r e a r e th e t h r e e fo llo w in g m e t h o d s f o r f in d in g t h e e l e c t r o n
d e n sity :
(1) f r o m th e r e f l e c t i o n c o e f f i c i e n t c u r v e s of F i g u r e 1, in
c o n ju n c t io n w ith th e m e a s u r e d v a l u e s of c o l l i s i o n f r e q u e n c y , a n d f o r
th e p a r a l l e l o r i e n t a t i o n of the w a v e g u id e ;
(2) f r o m th e r e f l e c t i o n
co effic ien t fo r th e p e rp e n d ic u la r o rie n ta tio n ,
t h r o u g h th e s e p a r a t i o n
b e t w e e n th e m a x i m u m a n d m i n i m u m a s g iv e n b y [ 4 8 ] ; a n d (3) f r o m
th e s h if t in th e c y c l o t r o n a b s o r p t i o n p e a k a n d th e m e a s u r e d v a lu e s
of th e c o l l i s i o n f r e q u e n c y ,
t h r o u g h e q u a ti o n [ 4 9 ] .
We c o n s i d e r
t h e s e in o r d e r .
(1)
R eflec tio n c o effic ien t, p a r a lle l o rie n ta tio n .
The
a p p l i c a b i l i t y of t h i s m e t h o d i s c l e a r l y l i m i t e d
f o r w h ic h
U i ' ^ > 0 .7 ,
7
V ' > 0 .2 .
t io n s in R f o r C0S , 6 < 1 . 0
s a m e v a lu e of R .
F o r l o w e r v a l u e s of
to c a s e s
V' th e f l u c t u a -
2
sh o w t h a t s e v e r a l v a l u e s of C0S*
F o r C0S
г1.0,
how ever,
g iv e th e
th e m e t h o d w o u ld b e
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
61
ap p lic ab le fo r 0<l ^< . 5.
A l s o , f o r C0S' ^ < . 7 th e f a c t s th a t the
c u r v e s a r e so c l o s e t o g e t h e r a n d th e r e f l e c t i o n s o low w o u ld m a k e
i t d if f ic u l t to s e l e c t th e p r o p e r v a lu e of C0S'^ f r o m a m e a s u r e m e n t
of R.
T h e a c t u a l o b s e r v e d v a lu e s of R i n d i c a t e t h a t
e r a b l y l e s s th a n . 7 ,
CO
9
is c o n s i d -
a n d t h e r e f o r e t h i s m e t h o d w a s n o t e m p lo y e d
in fin d in g th e e l e c t r o n d e n s i t y .
(2)
R eflec tio n c o effic ien t, p e rp e n d ic u la r o rie n ta tio n .
T h e p r o c e d u r e h e r e w a s th e o n ly w a y of m e a s u r i n g e l e c t r o n
d e n s i t y w h ic h y i e l d e d s i g n i f i c a n t r e s u l t s , a n d t h e s e a r e p l o t t e d in
F i g u r e 7.
The v a l u e s of C0S '^ a r e a l l in th e v i c i n i t y of 0 , 0 3 .
T his
w o u ld c o r r e s p o n d to a n e l e c t r o n d e n s i t y of a p p r o x i m a t e l y 2 X 10*^
c m " ?-*,
It i s i n t e r e s t i n g to n o te t h a t th e e l e c t r o n d e n s i t y v a r i e s o n ly
s l i g h t l y a s the in p u t p o w e r to th e o s c i l l a t o r i s r a i s e d ,
c o llisio n fre q u e n c y in c r e a s e s lin e a rly ,
th o u g h th e
A p p a r e n t l y th e i n c r e a s e in
e n e r g y to th e p l a s m a s e r v e s p r i n c i p a l l y to i n c r e a s e th e e l e c t r o n
v e lo city ,
e n t.
w ith o u t s e n s i b l y i n c r e a s i n g th e n u m b e r of e l e c t r o n s p r e s н
T h e e l e c t r o n t e m p e r a t u r e m e a s u r e m e n t s , in th e p r e s e n c e of
r e a s o n a b l y h ig h m a g n e t i c f i e l d s , a l s o i n d i c a t e t h a t th e e l e c t r o n i c
v e l o c i t i e s do i n c r e a s e w ith o s c i l l a t o r p o w e r ( s e e F i g u r e 10, f o r
ex am p le).
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
62
(3)
A b so rp tio n ,
p e rp e n d ic u la r o rie n ta tio n .
T h is m e t h o d s h o u ld b e v e r y u s e f u l f o r e l e c t r o n d e n s i t i e s in
th e r a n g e of o u r e x p e r i m e n t s .
U n f o r t u n a te ly , th e e x p r e s s i o n w h ic h
h a s b e e n d e v e lo p e d f o r C0S '^ (e q u a tio n [ 4 8 ] ) i s v a lid o n ly f o r V = 0,
It i s a p p a r e n t f r o m th e e x p e r i m e n t a l v a lu e s of th e s e p a r a t i o n b e н
tw e e n th e m a x i m u m a n d m i n i m u m of a b s o r b e d p o w e r ( F i g u r e 5)
t h a t th e e x p r e s s i o n f o r C0S'^ m u s t b e s t r o n g l y d e p e n d e n t u p o n th e
c o llisio n freq u e n cy ,
A s e q u a ti o n [ 4 6 ] is b a s e d On [3 0 ] a n d a s
[3 0 ] i s a c t u a l l y a p p r o p r i a t e o n ly f o r a n u n b o u n d e d p l a s m a , i t w a s
n o t f e l t w o r th w h ile to p u r s u e th e a b o v e a r g u m e n t f o r V 1 ^ 0.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
63
CHAPTER
VI
SUMMARY
T h e p u r p o s e of t h i s w o r k h a s b e e n to s t u d y in d e t a i l th e
i n t e r a c t i o n of g u id e d m i c r o w a v e s w ith a p l a s m a in a m a g n e t i c f i e l d .
T h e r e a r e f o u r p r i n c i p a l r e s u l t s of o u r i n v e s t i g a t i o n .
F irst,
we
h a v e d e v e lo p e d e x p r e s s i o n s f o r th e r e f l e c t i o n a n d t r a n s m i s s i o n
c o e f f i c i e n t s f o r m i c r o w a v e s t r a v e r s i n g a p l a s m a - f i l l e d w a v e g u id e ,
in th e c a s e of m i c r o w a v e p o l a r i z a t i o n p a r a l l e l to a n e x t e r n a l ,
tr a n s v e r s e m a g n etic fie ld .
In a d d i t i o n , w e h a v e i n d i c a t e d h o w to
p r o c e e d f o r a p o l a r i z a t i o n p e r p e n d i c u l a r to th e m a g n e t i c f ie ld .
S e c o n d , we h a v e u s e d th e c y c l o t r o n r e s o n a n c e a b s o r p t i o n of m i c r o н
w a v e s to m e a s u r e th e e l e c t r o n c o l l i s i o n f r e q u e n c y a n d th e e l e c t r o n
d e n sity .
T h i r d , we h a v e d i s c o v e r e d a s m a l l m i c r o w a v e a b s o r p t i o n
o r p h a s e s h if t r e s u l t i n g f r o m a n u n e x p la i n e d c o u p lin g of th e m a g н
n e t i c f i e l d w ith th e m i c r o w a v e s , w h e n th e m a g n e t i c f i e l d a n d m i c r o н
w ave p o la riz a tio n a r e p a ra lle l.
F o u r t h , in th e p r e s e n c e of a n u n н
s t a b l e d i s c h a r g e w e h a v e fo u n d a r a t h e r u n e x p e c t e d m i c r o w a v e
a m p lific a tio n .
I n c i d e n t a l to th e m a i n p u r p o s e of th e i n v e s t i g a t i o n w e r e the
o b s e r v a t i o n s of u n u s u a l l y h ig h e l e c t r o n t e m p e r a t u r e s a n d of i n c r e a s н
in g e l e c t r o n t e m p e r a t u r e w ith m a g n e t i c f i e l d ,
b o th of w h ic h a r e a s
y e t u n e x p la i n e d b y t h e o r y .
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
64
T h e l a s t f o u r O b s e r v a tio n s a r e of s u f f i c i e n t i m p o r t a n c e to
w a r r a n t e x ten siv e fu tu re in v e stig a tio n s.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
65
B IB L IO G R A P H Y
1.
B ohm and P in e s,
P h y s , R q v , , 85,
338 (1952).
2.
A llis,
" M o tio n s of Io n s a n d E l e c t r o n s , "
2 1 , 383 ( S p r i n g e r , B e r l i n , 1956).
3.
T urner,
4.
M o w e r , " P r o p a g a t i o n of a P l a n e W ave in E l e c t r i c a l l y
A n iso н
t r o p i c M e d iu m , " S y lv a n ia R e p o r t M P L - 1 ,
1956 ( u n p u b н
li s h e d ) .
5.
G o la n t,
6.
G o ld stein , L a m p e rt, a n d G e ig e r ,
243 (1952).
7.
E p ste in ,
R e v , M od,
8.
M o lm u d ,
Phys. Rev.,
9.
C h a p m a n a n d C o w lin g ,
U n if o r m G a s e s .
C a n . J.
Sov.
P h y s , , 32,
16
(1954).
P h y s .- T e c h . P h y s . , 5,
1197 (1961).
E lec tr.
C o m m u n . , 29,
P h y s . , J28, 3 (1956).
114, 29 (1959).
T h e M a t h e m a t i c a l T h e o r y of N o n C a m b r i d g e U n i v e r s i t y P r e s s , 1939,
10.
C r o m p t o n , H u x le y , a n d S u tto n ,
A 2 1 8 , 507 (1953).
11.
Suhl a n d W a l k e r ,
(1954).
12.
H andbuch d e r P h y s ik ,
P r o c . R o y . Soc.
B e ll S y s te m T e c h .
J. , 3 3 ,
(L o n d o n ),
579, 9 8 7 ,
1133
S t r a t t o n , E l e c t r o m a g n e t i c T h e o r y , M c G r a w - H i l l B o o k C o m p an y ,
Inc. , N ew Y o rk , 1941.
p. 511.
13.
K e lly ,
M arg en a u , and B row n,
14.
Dodo, J
15.
F ra n c is,
" T h e G low D i s c h a r g e a t L o w P r e s s u r e s , "
b u c h d e r P h y s i k , 22 ( S p r i n g e r , B e r l i n , 1956);
pp. 6 1 - 7 0 .
16.
L a n g m u ir an d M o tt-S m ith ,
Phys.
P h y s , R e v , , 108,
1367 (1957).
S o c . J a p a n , 1 6 , 293 (1961).
H andн
see
G e n , E l e c t r . R e v . , _26, 731 (1923).
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
66
17,
Johnson and M a tte r,
P h y s , R e v . , 80,
18,
B i c k e r t o n a n d von E n g e l ,
468 (1956).
58 (1950).
P r o c . P h y s . Soc.
London, B,
19,
C h e s t e r , W a g n e r , C a s t l e , a n d C onn, R e v .
1127 (1959).
20,
S p i t z e r , P h y s i c s of F u l l y I o n iz e d G a s e s ( I n t e r s c i e n c e P u b l i s h н
e r s , N ew Y o r k , 1956),
p . 3,
21 ,
Tow nsend,
22,
B eck,
23,
D a v ies,
Z,
P roc,
Roy.
P h y s . , 97,
P ro c, Phys.
Sci.
69,
I n s t r . , 3 0,
Soe. , A , J}6, 571 (1912),
335 (1935).
Soc.
L o n d o n ,B , 6 6 ,
33 (1953).
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
+ V i ') N ' 11 /2
rr
^
f
Vr 'Z +(1 + Ui ')Z '/ J
r r- ^
\/T "
b = li_
?v 2 t v 2
i/'r 2 + ( l + t y ) 2^
. 2
1 /2
y J
,
CO ' 2 (1
')
>. J / 2
- ( l - ? 4 ---------- ? r ) }
v
tv
+ (i + 1^) y J
[37b]
In t h e s e e x p r e s s i o n s w e h a v e u s e d e q u a ti o n s [ l 6 c ] a n d [2 9 3, t o н
g e t h e r w ith th e fo llo w in g d e f in itio n s 5
IV = V ^ U i ,
[38^]
VT ' -
[38b]
Vr / U ,
ws ' = Wp/sCO .
[3 8 c]
T h e r e f l e c t i o n c o e f f i c i e n t R a n d th e t r a n s m i s s i o n c o e f f i c i e n t T
a r e d e f in e d by
-
AA*
╗
T =
PD*
?
P
[3 9 a ,b ]
L e n g th y b u t s t r a i g h t f o r w a r d m a n i p u l a t i o n th e n y i e l d s th e fo llo w in g
e x p re ssio n s:
p = 2 [ ( a - l ) Z+ b Z ] [ ( a + l ) 2+ b Z ] ( c o s h Z $ l - c o s Z d l )
K
_ _ 1 6 ( a 2 + b 2)
T
g
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
.
.
[41 ]
23
w here
K = [ ( a + l ) 2 + b 2 ] 2 e 2 p 't + [ ( a - l ) 2 + b 2 ] 2 e
-
T 2 c o s ZOLt [ ( a 2 + b 2 - l ) 2 - 4 b 2 ] +
[42]
+ 8b (a 2 + b 2 - 1) s in 2 a t ,
and
a =
>
Now th e p h y s i c a l s i g n i f i c a n c e of th e v a r i o u s t e r m s in th e a b o v e
e x p r e s s i o n s is r a t h e r o b s c u r e .
In o r d e r to f a c i l i t a t e th e i n t e r p r e н
t a t i o n , w e f i r s t s e t UT = l>i = 0, a n d w r i t e th e s i m p l i f i e d e x p r e s н
s i o n s f o r th e c a s e of v a n is h i n g c o l l i s i o n f r e q u e n c y .
tio n th e e x p r e s s i o n s f o r a a n d b ,
ous.
e q u a ti o n s [ 3 7 ] ,
F o r th is s i t u a н
becom e a m b ig u н
T h e c h o ic e of s ig n f o r th e s q u a r e r o o t in th e s q u a r e b r a c k e t s
m u s t be d e cid e d fr o m p h y s ic a l c o n s id e ra tio n s .
F o r a c o llisio n le ss
p l a s m a , w e w ill h a v e
and
a 2 - b 2 = {1 - C0S ' 2 )
[4 3 ]
a b = 0.
[4 4 ]
T h e n if tOs ,гj2 < 1 , w e m u s t c h o o s e b = 0 f o r a to b e r e a l ,
COs 12 > 1 , w e m u s t c h o o s e a = 0 f o r b to be r e a l .
and fo r
Then we have
th e fo llo w in g two c a s e s :
(1) f o r C0S,2 < 1 ,
s i n 2 [k x ░ ( l t C0s ' 2 ) 't ]
[45a]
s
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
24
(2 ) f o r tos l2 > l ,
* ------------*inh2cV (< - : 1)l ]---------r.}) + s i n h 2 [k x ░(tos .2 - ! )
4
S ince V - 0 ,
t h e r e i s no a b s o r p t i o n ,
I
[45b]
]
and T = 1 - R.
We n o te th a t
f o r m i c r o w a v e f r e q u e n c i e s a b o v e th e r e d u c e d p l a s m a f r e q u e n c y tos ,
th e r e f l e c t i o n (and t r a n s m i s s i o n ) c o e f f i c i e n t w ill go th r o u g h a s e r i e s
of m a x i m a a n d m i n i m a a s th e w a v e le n g th is i n c r e a s e d ,
in m u c h th e
s a m e w a y a s f o r w a v e s in th e o p t i c a l r e g i o n t r a v e r s i n g a th in f i l m .
E x p ressio n s
[ 4 0 ] a n d [41 ] a r e i l l u s t r a t e d in F i g u r e 1 f o r s e v e r a l
d i f f e r e n t v a lu e s of th e c o l l i s i o n f r e q u e n c y a n d f o r a s e c t i o n of w a v e н
g u id e 14 c m long.
We m a y n o te f r o m th e p r e v i o u s d i s c u s s i o n t h a t th e p r e s e n c e of
th e b o u n d a r y e f f e c t i v e l y r a i s e s th e p l a s m a f r e q u e n c y f r o m
The e x p re ss io n s
top to tos .
[ 4 0 ] a n d [41 ] a r e i d e n t i c a l to th o s e f o r a n u n н
b o u n d e d p l a s m a if we s e t s = 1 a n d ,
In p r i n c i p l e ,
th e re fo re ,
t o g
*=
t o p .
one m a y now u s e th e c u r v e s of F i g u r e 1 w ith
m e a s u r e d v a lu e s of th e c o l l i s i o n f r e q u e n c y a n d r e f l e c t i o n o r t r a n s н
m i s s i o n c o e f f i c i e n t s to d e t e r m i n e th e p l a s m a d e n s i t y .
F o r th e p e r p e n d i c u l a r o r i e n t a t i o n of th e g u id e ,
th e p r i n c i p a l
f e a t u r e s of th e v a r i a t i o n of the r e f l e c t e d p o w e r w ith m a g n e t i c f i e l d
m a y b e u n d e r s t o o d by s e t t i n g V = 0.
T h e n th e p r o p a g a t i o n c o n s t a n t
in th e g u id e b e c o m e s
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
R eproduced
with perm ission
-
1. 0
of the copyright ow ner.
v'=. 1
v ' = . 15
Further reproduction
ei
0)
o
*rt
?H
(U
o
U
prohibited
o
U
<D
without p e r m issio n .
1.0
F i g u r e 1.
.2
1. 5
2.0
R e f l e c t i o n of g u i d e d m i c r o w a v e s b y a p l a s m a s l a b w i t h i n c r e a s i n g e l e c t r o n d e n s i t y
f o r s e v e r a l v a l u e s of th e e l e c t r o n c o l l i s i o n f r e q u e n c y V1 = P / t o .
26
1/2
(l-o>c | 2 ^ s ' 2 ) - w s ' 4 wc ' 2
= a kx ░ .
k x P = k x ░ L (1 -C0S ,2 -Wc ,2 ) (1 -w c ,2 )
[4 6 ]
T h e r e f l e c t i o n c o e f f i c i e n t h a s th e f o r m
R
.
[4 7 ]
? ? ~7 + s in ^ a I
( 1 - a 2 )2
2
?
T h e r e a r e two p r i n c i p a l p o in ts of i n t e r e s t : (1) w h e n C0c ' ? 1-Wg ' ,
a
, a n d R = 0;
(2) w h e n Wc ' = 1~C0S ' 2 , a = 0, a n d w e h a v e
(kx ░-t-)2
R =
4+(kx ░ I ) 2
In o u r w o r k ( k ^ - t ) ╗ 4 ,
a n d t h e r e f o r e th e s e c o n d c a s e a b o v e
c o n stitu te s a m a x im u m a t R ^ 1 ,
(O ne c a n sh o w t h a t th e e x t r e m u m
v a l u e s of R o c c u r w h e n
ta n a l _ (1 + a 2 )
at
~ (l -a.2)
( f o r V - 0), w h ic h i s s a t i s f i e d f o r a = 0. )
F o r s m a l l v a lu e s of Ws ' ( C0S1< . 1), th e s e p a r a t i o n b e tw e e n th e
m a x i m u m a t <dc ' ? l - t O g '2 a n d th e m i n i m u m a t Wc ?2 = 1-COg'2 is
AWC'
c
** ?4 ----
2
.
[48]
T h u s f o r v e r y lo w c o l l i s i o n f r e q u e n c i e s ,
one m a y c a l c u l a t e th e
e le c tr o n d e n sity fr o m a m e a s u r e m e n t of th is s e p a ra tio n ,
4.
C o llisio n F re q u e n c y ,
As h a s been in d ic a ted ,
it is g e n e r н
a l l y e s s e n t i a l to know th e e l e c t r o n c o l l i s i o n f r e q u e n c y b e f o r e the
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
27
r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s c a n be u s e d to f in d th e
e le c tro n d en sity ,
A v e r y s i m p l e a n d c o n v e n ie n t m e t h o d f o r do ing
t h i s m a k e s u s e of th e t r a n s m i s s i o n m e a s u r e m e n t s f o r m i c r o w a v e s
w h e n th e m i c r o w a v e p o l a r i z a t i o n i s p e r p e n d i c u l a r to th e s t e a d y m a g н
n e tic fie ld H0 ,
A s th e m a g n e t i c f i e l d i s i n c r e a s e d ,
th e p o w e r a b н
s o r b e d by th e p l a s m a r e a c h e s a m a x i m u m v e r y n e a r l y a t toc ' -=1.
K e lly ,
M a r g e n a u , a n d B r o w n h a v e sh o w n th a t th e h a l f - w i d t h of th e
c y c l o t r o n r e s o n a n c e a b s o r p t i o n c u r v e i s e q u a l to th e c o l l i s i o n f r e н
q u e n c y f o r a v a r i e t y of g a s e s (13),
In p a r t i c u l a r ,
th e y h a v e sh o w n
t h a t th e m e th o d i s v a lid w h e n th e c o l l i s i o n f r e q u e n c y i s p r o p o r t i o n a l
to v^ (v th e e l e c t r o n v e l o c i t y ) , w h ic h i s the c a s e f o r a d i s c h a r g e in
a ir.
H o w e v e r , w e s h a l l b e m e a s u r i n g n o t j u s t th e a b s o r p t i o n b u t
r a t h e r th e to ta l e f f e c t of a b s o r p t i o n a n d th e a t t e n u a t i o n w h ic h r e н
s u l t s s o l e l y f r o m th e i m a g i n a r y p a r t of th e c o l l i s i o n f r e q u e n c y ,
d isc u sse d e a r lie r .
as
D odo (14) h a s sh o w n t h a t th e m e t h o d f o r m e a s н
u r i n g th e c o l l i s i o n f r e q u e n c y w ill s t i l l a p p ly f o r P ' < , 3 a n d tos ' < . 3 .
He h a s f u r t h e r sh o w n th a t th e m a x i m u m in th e a b s o r p t i o n c u r v e
w i l l s h i f t f r o m toc ' = 1 to a v a lu e g iv e n by
( л C') m a x * ( l
A s (Wc ') m a x an<*
- V
2 - I '?2 ) 1 /2
.
a r e n ow e a s i l y m e a s u r e d ,
[4 9 ]
top'2 a n d th e e l e c t r o n
d e n s i t y a r e r e a d i l y c a l c u l a t e d f r o m th is e q u a tio n .
In o u r w o r k w e
s h a l l a s s u m e th a t top1 w ill b e r e p l a c e d b y Ws '.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
28
C.
E le c tro n T e m p e ra tu re
T h e m e a n i n g of e l e c t r o n t e m p e r a t u r e is p r e c i s e o n ly if the
e l e c t r o n d i s t r i b u t i o n f u n c tio n is M a x w e llia n .
F o r such a p la sm a ,
th e e l e c t r o n t e m p e r a t u r e i s d e f in e d by
T = i = ^ ,
e
2
k
[5 0 ]
w h e r e w is the m o s t p r o b a b l e e l e c t r o n v e l o c i t y a n d k th e B o l t z m a n n
c o n sta n t.
P r o v i d e d t h a t th e m e a n f r e e p a t h f o r c o l l i s i o n s is s h o r t
e n o u g h f o r e q u i l i b r i u m to be m a i n t a i n e d o v e r p e r i o d s of i n t e r e s t
(c y c lo tro n ,
w ellian .
m ic ro w a v e p e rio d ),
th e d i s t r i b u t i o n f u n c tio n w ill b e M a x н
T h is c o n d itio n i s n o t s a t i s f i e d in the p r e s e n c e of a m icron-
w ave e le c tr i c fie ld .
That is,
t h e r e w ill b e s e l e c t i v e a b s o r p t i o n of
e n e r g y b y e l e c t r o n s in a p a r t i c u l a r e n e r g y r a n g e .
In o u r w o r k ,
th e r a d i o f r e q u e n c y f i e l d w h ic h p r o d u c e s th e p l a s m a h a s a f r e q u e n c y
w e l l b e lo w the c o l l i s i o n f r e q u e n c y a n d th u s w ill n o t a f f e c t th e v e l o c н
ity d i s t r i b u t i o n .
On th e o t h e r h a n d ,
th e s a m e o r d e r of m a g n itu d e .
th e m i c r o w a v e f r e q u e n c y i s of
T h e re fo re th e re is a n o th e r re a s o n
f o r k e e p in g th e m i c r o w a v e p o w e r low ;
d e p a r t u r e s f r o m th e M a x н
w e l l i a n d i s t r i b u t i o n m u s t b e s m a l l in o r d e r t h a t th e c o n c e p t of
e le c tr o n te m p e r a t u r e be v alid .
T h e re , a r e s e v e r a l d i f f e r e n t m e t h o d s f o r d e t e r m i n i n g th e
e l e c t r o n t e m p e r a t u r e in a p l a s m a (15).
T h e s i m p l e s t of t h e s e is
th e p r o b e m e th o d , f i r s t d e v e lo p e d b y L a n g m u i r (16), a n d s u b s e н
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
29
q u e n tly i m p r o v e d a n d m o d if ie d by a n u m b e r of w o r k e r s .
We s h a l l
b r i e f l y d e s c r i b e th e d o u b le f lo a tin g p r o b e m e t h o d w h ic h w a s u s e d
in o u r w o r k a n d w h ic h w a s d e v e lo p e d b y J o h n s o n a n d M a l t e r (17).
C o n sid e r a s m a ll,
c y l i n d r i c a l p r o b e i m m e r s e d in a p l a s m a in
w h ic h e q u i l i b r i u m h a s b e e n e s t a b l i s h e d .
h ig h e r v e lo c itie s ,
B e c a u s e of t h e i r m u c h
e l e c t r o n s w i l l i n i t i a l l y s t r i k e th e p r o b e m u c h
m o r e f r e q u e n t l y th a n th e i o n s ,
a n d th e p r o b e w ill a c c o r d i n g l y s o o n
a c q u i r e a n e g a t i v e p o t e n t i a l w ith r e s p e c t to th e p l a s m a .
T h is n e g a н
tiv e p o t e n t i a l w ill in t u r n r e d u c e the e l e c t r o n flo w to th e p r o b e .
T h e r e f o r e t h e r e w ill a r i s e a s m a l l r e g i o n c a l l e d a s h e a t h ,
su rro u n d н
ing th e p r o b e , w h ic h c o n ta i n s p r i m a r i l y p o s i t i v e io n s (a n d n e u t r a l s ) .
A s h e a t h w ill c l e a r l y be p r e s e n t w h e n e v e r a p l a s m a e n c o u n t e r s a
m a t e r i a l b o u ndary.
It m a y b e s e e n f r o m e n e r g y c o n s i d e r a t i o n s t h a t
th e t h i c k n e s s of th is s h e a t h m u s t b e r o u g h ly th e D e b y e le n g th .
P r o v i d e d th a t th e s h e a t h t h i c k n e s s is s m a l l in c o m p a r i s o n
w ith th e e l e c t r o n m e a n f r e e p a th ,
re g io n .
F u rth e r,
t h e s h e a t h w ill b e a c o l l i s i o n - f r e e
th e p r o b e a n d s h e a t h t o g e t h e r m u s t be s m a l l s o
a s n o t to in f l u e n c e th e e l e c t r o n d i s t r i b u t i o n a t d i s t a n c e s f r o m the
p r o b e of th e o r d e r of m a g n itu d e of t h e m e a n f r e e p a th .
T h e n the
e l e c t r o n s r e a c h i n g th e p r o b e w i l l h a v e th e s a m e v e l o c i t y d i s t r i b u t i o n
a s in th e p l a s m a r e g i o n .
T h e e l e c t r o n c u r r e n t a t the p r o b e i s th e n
g iv e n by
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
30
i e л A jQe " V e / k T e ,
w h e r e A i s th e p r o b e a r e a ,
[5 1 ]
j 0 th e r a n d o m e l e c t r o n c u r r e n t d e n s i t y ,
a n d V th e p o t e n t i a l d r o p a c r o s s th e s h e a th .
Now s u p p o s e th a t t h e r e a r e tw o s u c h p r o b e s ,
s h o w n in F i g u r e 2.
c o n n ected a s
P la sm a
D e n o tin g th e m a g n i t u d e s
of th e e l e c t r o n c u r r e n t s to
V / / / //
th e p r o b e s by i 0 j a n d i ^ ,
a n d th e p o s i t i v e io n c u r r e n t s
by ip j and ip 2 ╗ we have
(i p l " i e l ) +(i p 2 ',ie 2 ) 1B0F u r t h e r m o r e , th e c u r н
r e n t to th e e x t e r n a l c i r c u i t is
i (j= ie 2 - i p 2 л T h e p o s i t i v e ion
c u r r e n t to a p r o b e i s e s s e n t i a l l y
co n sta n t,
0 - 5 0 0 V o lts
F i g u r e 2. C i r c u i t f o r d o u b le
probe te m p e ra tu re m e a s u r e н
m e n ts.
s i n c e th e p o t e n t i a l a c r o s s th e s h e a t h i s a lw a y s a t t r a c t i v e ;
a n d o u ts id e th e s h e a t h the io n s a r e u n in f l u e n c e d by i t s p r e s e n c e .
Then
S i P = *pl + i p2 = (A j o ) i e ? V l e / k T e + ( A j 0 )2e " V 2 e / k T e ,
F u rth e r,
[5 3 ]
d e n o tin g th e a p p l i e d p o t e n t i a l d r o p a c r o s s th e p r o b e s by
Vjj, w e h a v e (if V j > V 2 > s a y ) ,
Vd + V2 = Vx ,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
[54]
31
a s s u m i n g th e p l a s m a p o t e n t i a l is e v e r y w h e r e the s a m e .
E q u atio n s
[ 5 4 ] a n d [ 5 3 ] l e a d to th e e x p r e s s i o n
-t-nr = -
kTe
+ ftna
[55]
w here
r * S i p / i e2 - 1,
[56]
and
or *
[57]
.
(A j 0 )2
T h u s a p l o t of -tnT v s .
V<j w ill be a s t r a i g h t li n e ,
t e m p e r a t u r e m a y b e c a l c u l a t e d f r o m th e s l o p e .
is in c re a s e d ,
a n d th e e l e c t r o n
A s th e p o t e n t i a l V<i
i e2 w ill c o n tin u e to i n c r e a s e u n til i e2 = ZJip;
l a r g e r v a lu es a r e p o s sib le .
com e h o riz o n ta l.
Then a t
By sy m m e try ,
th e
th is p o in t th e c u r v e w ill
m a x im u m c u r r e n t
s i t e d i r e c t i o n w ill o c c u r f o r i e j = S i p .
no
beн
in th e o p p o н
T h e n f r o m th e c o r r e s p o n d н
ing v a l u e s of i^ one m a y r e a d i l y c o m p u te S i p .
A d i s c u s s i o n of th e
a p p l i c a t i o n of t h i s m e th o d i s g iv e n in C h a p t e r IV,
If th e p l a s m a i s to b e in a s t r o n g m a g n e t i c f i e l d ,
p r e c e d i n g d i s c u s s i o n n e e d s to b e r e e x a m i n e d .
th e e l e c t r o n m e a n f r e e p a th ,
p a r t of th e
F o r in t h i s f i e l d ,
in a d i r e c t i o n p e r p e n d i c u l a r to th e
f i e l d , i s r e d u c e d to a v a lu e r o u g h l y e q u a l to th e L a r m o r r a d i u s .
B i c k e r t o n a n d V on E n g e l (18) h a v e sh o w n th a t if m a n y c o l l i s i o n s
ta k e p l a c e w ith in th e s h e a t h s u r r o u n d i n g th e p r o b e ,
th e e l e c t r o n
c u r r e n t to a p la n e p r o b e is s t i l l e x p r e s s i b l e in th e f o r m
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
32
i ~ e TeV/ k T e ^
T h u s th e p r e c e d i n g t r e a t m e n t i s u n a l t e r e d e x c e p t f o r a p r o p o r t i o n s
a lity co n sta n t,
a n d th e r e s u l t s of th e t e m p e r a t u r e m e a s u r e m e n t s
a r e s t i l l v a lid .
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
33
CHAPTER
III
E Q U IP M E N T
F o r a q u a l i t a t i v e i n v e s t i g a t i o n of th e i n t e r a c t i o n of a g u id e d
w a v e w ith a p l a s m a , a s e c t i o n of w a v e g u id e of know n le n g th , c o m н
p l e t e l y f i l l e d w ith a p l a s m a of u n i f o r m d e n s i t y is d e s i r e d .
AH
p r e v i o u s e x p e r i m e n t s on p l a s m a s w ith in w a v e g u id e s h a v e u s e d s o m e
f o r m of d i s c h a r g e tu b e i n s e r t e d in a w a v e g u id e s e c t i o n , w ith th e
tu b e o r i e n t e d e i t h e r a lo n g th e a x i s of th e g u id e o r p e r p e n d i c u l a r to
it.
It is c l e a r t h a t th e p l a s m a d id n o t c o m p l e t e l y f i l l th e g u id e .
F u rth e rm o re ,
e v e n w ith th e d i s c h a r g e off, th e tu b e i t s e l f i n t r o d u c e d
d e p a r t u r e s f r o m th e t h e o r e t i c a l v a lu e s of the m i c r o w a v e f i e l d , a n d
i t is t h i s d i s t o r t e d f i e l d w h ic h i s p r e s e n t e d to th e p l a s m a .
The
e l e c t r o n d e n s i t y w ith in the p l a s m a i t s e l f w a s a l s o f a r f r o m u n i f o r m .
In h o p e s of o b ta in in g a p l a s m a w h ic h w o u ld b e m o r e n e a r l y
h o m o g e n e o u s a n d w h ic h w o u ld c o m p l e t e l y f i l l th e w a v e g u i d e , two
d iffe re n t m eth o d s w e re tr ie d .
In th e f i r s t m e t h o d ,
th e p l a s m a w a s
to b e f o r m e d b y a h i g h - f r e q u e n c y e l e c t r o d e l e s s d i s c h a r g e ,
in th e t a n k c i r c u i t of a h ig h p o w e r o s c i l l a t o r ,
A c o il
o p e r a t i n g a t 56 m e ,
w a s w r a p p e d a r o u n d a n e v a c u a t e d w a v e g u id e s e c t i o n .
Now in o r d e r
t h a t th e p o w e r f r o m th e o s c i l l a t o r p e n e t r a t e in to t h i s s e c t i o n ,
w a v e g u id e w a l l s s h o u ld n o t b e c o n d u c tin g .
th e
On th e o t h e r h a n d , if
th e m i c r o w a v e s a r e to p r o p a g a t e a lo p g th e g u id e u n a t t e n u a t e d ,
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
the
34
w a l l s m u s t be c o n d u c tin g .
It w a s d e c id e d ,
th e re fo re ,
to d e p o s i t a
th in l a y e r of s i l v e r on th e i n t e r i o r of a d i e l e c t r i c w a v e g u id e .
T he
f i e l d f r o m th e o s c i l l a t o r s h o u ld th e n p e n e t r a t e t h r o u g h th e s i l v e r
w ith o u t c o n s i d e r a b l e l o s s 1;o a d i s t a n c e r o u g h ly e q u a l to the s k in
d e p th (in th is c a s e a b o u t 30 X 1 0 ?^ c m ) .
On th e o t h e r h a n d ,
s k in d e p th f o r the m i c r o w a v e s w o u ld be a b o
Документ
Категория
Без категории
Просмотров
0
Размер файла
2 444 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа