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A microwave acoustic investigation of the spin -lattice interaction for F centers in quartz

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RICE UNIVERSITY
A MICROWAVE ACOUSTIC INVESTIGATION OP THE
SPIN-LATTICE INTERACTION FOR P CENTERS IN
QUARTZ
by
R ic h a r d N . C l a y t o r
A THESIS SUBMITTED
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
H o u s to n , T e x a s
May, 1 9 6 1
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
UMI Number. 3079652
UMI*
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ABSTRACT
An I n v e s t i g a t i o n w as made o f t h e s p i n - l a t t i c e i n t e r ­
a c t i o n o f P c e n t e r s i n q u a r t z t h r o u g h t h e u s e o f 9 .3 Gc
a c o u s tic w aves.
An u p p e r l i m i t to th e a c o u s t i c - a t t e n u a t i o n
•3
c o e f f i c i e n t o f 7 . 7 x 10
n e p e r s p e r c e n t i m e t e r was f o u n d .
T h is c o r r e s p o n d s to a c r o s s s e c t i o n o f l e s s t h a n 3 .9 x 10
sq u a re c e n tim e te r s .
th a t
C o m p ariso n w i t h t h e o r y i n d i c a t e s
f o r F c e n t e r s i n q u a r tz i s g r e a t e r t h a n 8 .9
m illis e c o n d s .
R eproduced with perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission.
_20
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TABLE OF CONTENTS
I.
I n t r o d u c t i o n ..................................................................................
1
II.
T h e o r y ...........................................................................................
7
III.
P i e z o e l e c t r i c G e n e r a ti o n o f H y p e rs o n ic W a v e s.
IV .
E q u i p m e n t ............................................................................................20
V.
R e s u l t s ...........................
V I.
D i s c u s s i o n o f R e s u l t s .............................................................. 32
.
16
29
A.
C a lc u la tio n o f
................................................... 32
B.
C a lc u la tio n o f
...................................................35
C.
S a t u r a t i o n E f f e c t s ...................................................36
B i b l i o g r a p h y .................................................................................................. 39
A c k n o w l e d g m e n t s ........................................................................................ 4 1
i
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
1
I.
I n tro d u c tio n .
The im p o r ta n c e o f l a t t i c e v i b r a t i o n s i n d e te r m in in g
t h e r e l a x a t i o n tim e o f a p a r a m a g n e tic s p i n sy s te m h a s lo n g
been re c o g n iz e d .
I t i s c l e a r t h a t p h o n o n s p l a y th e dom i­
n a n t r o l e i n p a r a m a g n e tic r e l a x a t i o n , a s o p p o s e d to t h e
i m p u r i t y - c o n t r o ll e d n u c le a r-m a g n e tic -re s o n a n c e r e l a x a t i o n .
A g e n e r a l l y a p p lic a b le th e o r y o f th e s p i n - l a t t i c e i n t e r a c ­
t i o n f o r p a r a m a g n e ti c s u b s ta n c e s w o u ld b e o f g r e a t v a l u e .
N ot o n l y w o u ld i t p r o v i d e f u n d a m e n ta l i n f o r m a t i o n a b o u t
p r o c e s s e s i n s o l i d s , b u t i t w o u ld h a v e th e m ore Im m ed iate
a d v a n ta g e o f a i d i n g m a s e r a n d l a s e r r e s e a r c h a n d d e v e lo p m e n t.
I n 1 9 4 0 , v a n V le c k ^ i n t r o d u c e d t h e f i r s t th e o r y o f
p a r a m a g n e ti c r e l a x a t i o n .
He e n v i s i o n e d a p a ra m a g n e tic i o n
s u r r o u n d e d b y a n o n - p a r a m a g n e tic m a t r i x a n d c o u p le d t o t h a t
m a trix , th e l a t t i c e , by e l e c t r i c f i e l d s .
m a g n e tic i o n a n d t h e s u r r o u n d in g l a t t i c e
In d eed , a p a ra ­
c a n b e lo o k e d
u p o n a s a s y s te m o f c o u p le d o s c i l l a t o r s .
n o t a t a l l d i r e c t , h o w e v e r.
th e a c tio n o f i t s
The c o u p lin g i s
A v ib ra tin g l a t t i c e
c a n , th ro u g h
o s c i l l a t i n g e l e c t r i c f i e l d , b rin g ab o u t
v i r t u a l t r a n s i t i o n s b e tw e e n th e o r b i t a l l e v e l s o f th e p a r a ­
m a g n e tic i o n .
T h ro u g h t h e s p i n - o r b l t c o u p l i n g , t h e n ,
tra n s i­
t i o n s c a n o c c u r b e tw e e n th e s p i n l e v e l s .
The s u c c e s s o f van V l e c k 's t h e o r y h a s b e e n s l i g h t .
i s n o t a t a l l g e n e ra l, nor p a r tic u la r ly a c c u ra te .
t h e m a in f a i l i n g o f t h e t h e o r y l i e s i n i t s
It
H ow ever,
c o m p le x ity :
it
i s v e r y d i f f i c u l t t o make p r e d i c t i o n s fro m t h e t h e o r y , a n d
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
2
t h e r e f o r e v e r y d i f f i c u l t to t e s t .
A u s e f u l r e f o r m u l a t i o n o f v an V l e c k 's t h e o r y h a s been
p r e s e n t e d b y M a ttu c k a n d S t r a n d b e r g .
2
W h ile n o t a new t h e o r y ,
i t h a s t h e a d v a n ta g e s o f g r e a t e r g e n e r a l i t y a n d re d u c e d com­
p le x ity .
T h ese a d v a n ta g e s h a v e b e e n o b t a i n e d th r o u g h a
s im p le r e a r r a n g e m e n t o f t h e te r m s i n t h e H a m il t o n ia n .
M a ttu c k a n d S tr a n d b e r g e n d e a v o r e d to g ro u p th e te rm s in to
t h o s e i n v o l v i n g th e l a t t i c e ,
n a te s s e p a r a te ly .
s p i n , and i n t e r a c t i o n c o o rd i­
The r e g r o u p i n g l e a d s t o a c o n s id e r a b le
s i m p l i f i c a t i o n , i n t h a t a p e r t u r b a t i o n c a l c u l a t i o n need
n o t b e c a r r i e d t o so h i g h a n o r d e r to g e t t h e m a t r i x elem en t
r e p r e s e n tin g th e s p i n - l a t t i c e
i n t e r a c t i o n t o goo d a c c u ra c y .
M o re o v e r, t h e r e s u l t s o f c a l c u l a t i o n s b a s e d on M a ttu c k and
S t r a n d b e r g 's r e f o r m u l a t i o n c a n be s im p ly a n d d i r e c t l y In­
te rp re te d p h y s ic a lly .
I t is
t h e n a r e l a t i v e l y s im p le m a tte r
t o p r o p o s e e x p e r im e n ts t o t e s t t h e t h e o r y .
a n d S t r a n d b e r g s u g g e s te d two d i r e c t t e s t s ,
I n f a c t , M attuck
th e s a tu r a tio n
o f a p a r a m a g n e tl c - r e s o n a n c e s i g n a l b y p h o n o n s on th e Larmour
f r e q u e n c y , a n d t h e a t t e n u a t i o n o f so u n d w aves a t th e Larmour
f r e q u e n c y i n a p a ra m a g n e tic s u b s t a n c e .
E x p e r im e n ts o f b o th
t y p e s h a v e b e e n p e rfo rm e d a n d a t l e a s t p a r t i a l l y co n firm
th e th e o ry .
The e x p e r i m e n t a l i n v e s t i g a t i o n o f t h e s p i n - l a t t i c e
i n t e r a c t i o n h a s b e e n c o n d u c te d u s i n g new t e c h n i q u e s f o r
th e p ro d u c tio n o f c o h e re n t l a t t i c e
v i b r a t i o n s , o r "h y p er­
s o n ic w a v e s ," a t m icrow ave f r e q u e n c i e s .
L a t t i c e w aves o f
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
3
d e f i n i t e f r e q u e n c y , p o l a r i z a t i o n , a n d d i r e c t i o n c a n now be
u sed to p ro v id e a n a r t i f i c i a l ,
s i m p l i f i e d phonon d i s t r i b u ­
t i o n a t f r e q u e n c i e s n o r m a lly e m p lo y e d f o r e l e c t r o n p a r a ­
m a g n e tic r e s o n a n c e w o rk .
I t was t h e d e v e lo p m e n t o f h y p e r ­
s o n ic so u n d a n d M a ttu c k an d S t r a n d b e r g 's t h e o r e t i c a l t r e a t ­
m ent o f th e s p i n - l a t t i c e i n t e r a c t i o n w h ich s t i m u l a t e d t h i s
re sea rc h .
L a t t i c e v i b r a t i o n s a t m icro w av e f r e q u e n c i e s w ere f i r s t
o b s e r v e d i n 1958 b y Bflmmel a n d D r a n s f e l d . 3
C o n tin u o u s waves
a t 1 ,0 0 0 Me w ere i n t r o d u c e d i n t o a q u a r t z r o d b y t h e p i e z o ­
e l e c t r i c e f f e c t , w h ic h we s h a l l p r e s e n t l y e x p l a i n .
The
w a v e le n g th o f t h e v i b r a t i o n s i n t h e q u a r t z was m e a su re d
o p tic a lly ;
t h e f r e q u e n c y was fo u n d t o c o r r e s p o n d to t h a t o f
th e r f e l e c t r i c
f i e l d a p p lie d to th e end o f th e ro d .
Btfmmel a n d D r a n s f e l d l a t e r fo u n d 4 t h e e f f e c t t o b e r e c i p r o ­
c a l , so t h a t th e s o u n d w aves c o u ld a l s o be d e t e c t e d p i e z o e le c tric a lly .
T h e i r e x p e r im e n ts w e re e x te n d e d t o 10 Gc b y
J a c o b s e n , S h i r e n , a n d T u c k e r, who w ere a l s o a b le t o do echo
r a n g in g e x p e r im e n ts i n q u a r t z r o d s , u s i n g p u l s e d r f fro m a
m a g n e tro n . 5 *6
The l a t t e r a u t h o r s h a v e a l s o h a d c o n s i d e r a b l e
s u c c e s s i n g e t t i n g t h e h y p e r s o n ic w av es i n t o m a t e r i a l s
o t h e r t h a n q u a r t z v i a t h i n b o n d s o f in d iu m m e t a l .
T h e ir
i n t e r e s t h a s b e e n d i r e c t e d to w a rd m ea su re m e n t o f t h e s p i n 7
la ttic e in te ra c tio n .
T h e ir f i r s t e f f o r t a lo n g t h i s l i n e
was th e s a t u r a t i o n o f t h e 10-Gc EPR s i g n a l s fro m t r i v a l e n t
m anganese an d fro m F c e n t e r s i n q u a r t z th r o u g h t h e u s e o f
l a t t i c e v i b r a t i o n s a t th e L arm our f r e q u e n c y .
H e re , t h e
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
q u a r t z s e r v e d b o th a s t r a n s d u c e r a n d m a t r i x f o r th e p a r a m a g n e tic i o n s •
The same e x p e rim e n t w as l a t e r p e rfo rm e d
Q
b y S h i r e n a n d T u c k e r o n Cr®4, i n A I 2 O3 ( r u b y ) a n d Mn2*
a n d F e ^ * i n MgO a s a f u n c t i o n o f t h e o r i e n t a t i o n o f t h e
DC m a g n e tic f i e l d w i t h r e s p e c t t o t h e d i r e c t i o n o f th e
l a t t ic e v ib ra tio n s .
The e x p e rim e n t c o n f ir m e d t h e p r e d i c t i o n
o f M a ttu c k a n d S tr a n d b e r g t h a t s p i n - r e s o n a n c e t r a n s i t i o n s
in d u c e d b y l a t t i c e v i b r a t i o n s o b ey q u a d r u p o le s e l e c t i o n
ru le s .
T he a t t e n u a t i o n o f 10-G c h y p e r s o n i c w aves i n r u b y
i n t h e r e s o n a n c e m a g n e tic f i e l d was m e a s u re d b y Tucker®
f o r th e th r e e l i n e s i n ru b y .
T h is g a v e a d i r e c t m e a su re ­
m ent o f t h e c r o s s s e c t i o n f o r phonon a b s o r p t i o n by a
re so n a n t s p in .
The e x p e r im e n t c l e a r l y d e m o n s tr a te d t h e f e a s i b i l i t y
o f m ak in g d i r e c t a t t e n u a t i o n m e a su re m e n ts a s s u g g e s te d by
M a ttu c k a n d S t r a n d b e r g .
r e l a x a t i o n tim e b y O rb a ch
A c a l c u l a t i o n o f th e s p i n - l a t t i c e
10
fro m t h e m e a s u re d v a lu e o f th e
a t t e n u a t i o n c o e f f i c i e n t g a v e a v a lu e i n a g re e m e n t w ith th e
a c c e p te d s p i n - l a t t i c e r e l a x a t i o n tim e .
th is th e s is ,
I n S e c tio n I I o f
i t w i l l b e show n t h a t t h e t h e o r i e s o f O rbach
a n d o f M a ttu c k a n d S t r a n d b e r g l e a d t o i d e n t i c a l d ep en d en ce
o f t h e t r a n s i t i o n p r o b a b i l i t y upon t h e a c o u s t i c a t t e n u a t i o n
1
due t o p a r a m a g n e tic i o n s .
The e x p e r i m e n t a l l y d e te rm in e d
a t t e n u a t i o n c o e f f i c i e n t w as s e v e r a l o r d e r s o f m a g n itu d e
s m a l l e r t h a n t h e ro u g h p r e d i c t i o n o f M a ttu c k a n d S t r a n d b e r g .
The p r o c e s s o f t h e g e n e r a t i o n a n d d e t e c t i o n o f h y p e r s o n i c
lI
I1
I
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5
w av es i s d i s c u s s e d i n S e c t i o n I I I o f t h i s t h e s i s , a n d th e
e q u ip m e n t r e q u i r e d f o r t h e a c o u s t i c a l m e a su re m e n t o f t h e
s p i n - l a t t i c e i n t e r a c t i o n i n S e c t i o n IV .
F c e n t e r s i n q u a r t z w ere c h o s e n f o r t h i s r e s e a r c h i n
t h a t t h e y a r e p a r a m a g n e tic io n s w h ich c a n be e a s i l y i n t r o ­
d u c e d b y m eans o f i o n i z i n g r a d i a t i o n .
The u s e o f q u a r t z
a s a m a t r i x f o r th e i o n s e l i m i n a t e s t h e n e c e s s i t y f o r b o n d in g
a t r a n s d u c e r to t h e p a ra m a g n e tic c r y s t a l ; a q u a r t z r o d w i t h
F c e n t e r s a c t s a s b o t h t r a n s d u c e r a n d p a r a m a g n e tic s a m p le .
The p a ra m a g n e tic r e s o n a n c e s p e c tru m o f F c e n t e r s i n
q u a r t z was f i r s t o b s e r v e d b y G r i f f i t h s , Owen, a n d W ard^"
in 1954.
A c o u s tic s a t u r a t i o n o f t h e p a r a m a g n e tic r e s o n a n c e
o
s i g n a l h a s b e e n o b s e r v e d b y M a ttu c k a n d S t r a n d b e r g
a t 13 Me
7
a n d b y J a c o b s e n , S h i r e n , a n d T u c k e r a t 1 0 Gc, s u g g e s t i n g
th e p o s s i b i l i t y o f an a c o u s tic a tt e n u a t io n e x p e rim e n t.
O 'B r ie n
12
p ro p o s e d a s t r u c t u r e f o r t h e F c e n t e r s w h ic h
c o n s i s t s o f an alum inum s u b s t i t u t i o n a l i m p u r i t y a t a s i l i c o n
s ite .
The alum inum i s p re su m e d t o h a v e a c c e p t e d a n a d d i t i o n a l
e l e c t r o n t o p e r m it t e t r a h e d r a l b o n d in g t o t h e s u r r o u n d i n g
oxygen I o n s .
U nder t h i s schem e, i r r a d i a t i o n re m o v e s t h e
l o o s e l y b o u n d e x t r a e l e c t r o n fro m th e alu m in u m .
b o r i n g o x y g e n Io n t h e n h a s one u n p a i r e d e l e c t r o n .
th e n c o n s t it u t e s th e F c e n t e r .
The n e i g h ­
T h is
The o b s e r v e d EPR a n d o p t i c a l
a b s o r p t i o n s p e c t r a c a n b e e x p la in e d on t h e b a s i s o f t h i s
m o d e l.
I n many r e s p e c t s , i t b e a r s c o n s i d e r a b l e s i m i l a r i t y
t o t h e de B o e r o r v a c a n c y m odel d i s c u s s e d i n d e t a i l b y
13
G o u r a r y a n d A d r ia n .
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
The n e g a t i v e r e s u l t s o f a n i n v e s t i g a t i o n on F c e n t e r s
In q u a rtz a re p re s e n te d i n S e c tio n V o f t h is t h e s i s , and
a r e d i s c u s s e d i n S e c t i o n V I.
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I
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“ 1
7
II.
T h e o ry
As s t a t e d i n S e c t i o n I o f t h i s t h e s i s * M a ttu c k a n d
g
S t r a n d b e r g b a s e d t h e i r t h e o r y u p o n t h e r e c o g n i z e d m odel
o f a p a r a m a g n e tic i o n c o u p le d to th e l a t t i c e v i a th e c r y s ­
ta llin e
e le c tric f ie ld .
C e rta in r e s t r i c t i o n s *
s till a l­
lo w in g a g r e a t d e a l o f g e n e r a l i t y , a r e p l a c e d u p o n th e
th e o ry .
a .)
T h ese a r e s
O n ly m a g n e t i c a l l y d i l u t e c r y s t a l s a r e c o n s i d e r e d .
T h e r e f o r e , t h e e f f e c t s o f t h e s p i n - s p i n i n t e r a c t i o n may be
n e g le c t e d * e x c e p t f o r l i n e b r o a d e n in g .
b .)
The p a r a m a g n e ti c i o n s a r e p re s u m e d t o f i t w e ll
in to th e l a t t i c e ,
so t h a t th e e f f e c t s o f l o c a l l a t t i c e
t o r t i o n may be n e g l e c t e d .
fro m t h e t r e a t m e n t .
d is ­
T h is c o n d i t i o n e x c l u d e s F c e n t e r s
F - c e n t e r e l e c t r o n s a r e n o t w e ll l o c a l i z e d .
The wave f u n c t i o n i s n o t p u r e l y s - t y p e , a n d t h e c h a rg e c lo u d
o v e r l a p s n e i g h b o r i n g i o n s t o some e x t e n t .
c .)
The c r y s t a l l i n e
f i e l d p o t e n t i a l I s assum ed t o be
s m a l l e r t h a n t h e f r e e I o n p o t e n t i a l , b u t l a r g e r th a n t h e
s p in - o r b it e n e rg y .
T h is c o n d itio n r e s t r i c t s
th e p a ra m a g n e tic
io n s c o n s id e re d to th o s e o f th e ir o n g ro u p .
d .)
The t h e o r y e x c l u d e s S - s t a t e i o n s i n t h e a p p ro x im a ­
t i o n to w h ic h M a ttu c k a n d S t r a n d b e r g h a v e c a r r i e d i t .
S in c e
a n S - s t a t e i o n h a s z e r o o r b i t a l a n g u l a r momentum, th e c o u p li n g
i
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o f th e o r b i t to th e l a t t i c e
i s e x tr e m e l y s m a l l .
The t h e o r y
w o u ld a p p l y i f t h e p e r t u r b a t i o n c a l c u l a t i o n w ere c a r r i e d t o
a s u f f i c i e n t l y h i g h o r d e r so a s t o ta k e a c c o u n t o f th e s l i g h t
I
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8
d i s t o r t i o n o f th e s p h e r ic a l S - s t a t e o r b i t s by th e c r y s t a l l i n e
e le c tric f ie ld .
U n d er t h e s e a s s u m p tio n s , t h e H a m ilto n ia n t a k e s th e fo rm :
*• V*
1 •)
w h e r e I s
S •H
t h e B ohr m a g n e to n ,
A 1_ • 5
L•H ,
A t h e s p i n - o r b i t c o u p lin g
p a r a m e t e r , S a n d L th e s p i n a n d o r b i t a l a n g u la r m om ents o f
th e p a r a m a g n e tic i o n , H t h e e x t e r n a l m a g n e tic f i e l d , V th e
e n e r g y o f th e i o n i n th e c r y s t a l l i n e e l e c t r i c f i e l d , “>£0 th e
f r e e I o n H a m ilto n ia n , a n d
The l a t t i c e
u th e l a t t i c e e n e rg y .
e n e r g y c a n b e w r i t t e n i n te rm s o f e l a s t i c
wave d i s p l a c e m e n t s , i n te r m s o f a s e t o f n o rm a l c o o r d i n a t e
o p e r a t o r s , o r i n te rm s o f p h o n o n a n n i h i l a t i o n a n d c r e a t i o n
o p e ra to rs .
th e o ry .
The a u t h o r s c h o s e t h e l a s t a l t e r n a t i v e f o r t h e i r
The p h o n o n v ie w p o in t s i m p l i f i e s t h e t h e o r y t o a
c o n s i d e r a b l e e x t e n t , b u t m ay l e a d to some c o n f u s i o n i f n o t
tre a te d c a re fu lly .
A s u b s t i t u t i o n o f t h e l a t t i c e H a m ilto n ia n ,
, in
te rm s o f th e c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s , a n d th e
c r y s t a l l i n e f i e l d p o t e n t i a l , V, i n te rm s o f th e s e t o f
n o rm a l d i s p l a c e m e n t s o f t h e p a r a m a g n e tic i o n 's n e a r e s t
n e i g h b o r s , g i v e s t h e t o t a l H a m ilto n ia n i n t h e fo rm :
2 .)
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9
w here VQ I s t h a t p a r t o f V w h ic h does n o t d e p en d u p o n th e
p o s i t i o n s o f th e n e a r e s t n e ig h b o r s a s e x p r e s s e d i n t h e n o rm a l
f1
f*f I
d is p la c e m e n ts ;
and VAA a r e t h e f i r s t a n d s e c o n d p a r t i a l s
o f V w i t h r e s p e c t to t h e n o rm a l d i s p l a c e m e n ts ; th e q u a n t i t i e s
Pp a n d
a r e r e l a t e d t o t h e phonon a n n i h i l a t i o n a n d c r e a t i o n
o p e r a to r s ; and
an d A*>t(l a r e c o e f f i c i e n t s i n t h e e x p a n s io n
o f t h e o r d i n a r y n e a r e s t - n e i g h b o r d is p la c e m e n ts i n te rm s o f
t h e p h o n o n a n n i h i l a t i o n a n d c r e a t i o n o p e r a t o r s .**’
F o r t h e p u rp o s e o f c o n s t r u c t i n g a u s e f u l t h e o r y ,
it
i s d e s i r a b l e t o b r e a k t h e H a m ilto n ia n i n t o p a r t s w h ic h h a v e
a c o n v e n i e n t p h y s i c a l i n t e r p r e t a t i o n b e f o r e a p p ly in g p e r ­
t u r b a t i o n t h e o r y to th e p r o b le m .
T hus, t h e H a m ilto n ia n i s
s e p a r a t e d I n t o p a r t s w h ic h in v o lv e th e l a t t i c e ,
s p i n , an d
I n t e r a c t i o n c o o rd in a te s s e p a r a t e l y ;
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o f M a ttu c k a n d S tr a n d b e r g i n t h e e x p a n s io n o f t h e d i s p l a c e ­
m en ts I n te r m s o f th e p h o n o n a n n i h i l a t i o n an d c r e a t i o n o p e r a ­
t o r s , The f a c t o r (a p ♦ a p ) i n t h e e x p a n s io n s h o u ld b e r e ­
p l a c e d b y ( a p ♦ a l p ) , ^ ^ The e r r o r s h o u ld h a v e l i t t l e e f ­
f e c t u p o n t h e o r d e r - o f - m a g n i t u d e fo rm u la s l a t e r d e r i v e d .
R eproduced with perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission.
10
^
i n t e r a c t i o n silo u ^ c* be c o n s i d e r e d a s in d u c in g e n e r g y c o n s e r v in g e x c h a n g e s o f q u a n ta b e tw e e n
In t h i s fo rm u la tio n ,
s p in
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and
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la ttic e .
s p i n 118,3 t h e
a d v a n ta g e t h a t i t l e a d s d i r e c t l y t o t h e s p i n H a m ilto n ia n
w h ic h d e s c r i b e s t h e e n e r g y l e v e l s o b s e r v e d i n p a r a m a g n e ti c reso n an c e e x p e rim e n ts .
I n f a c t , t h e s p i n H a m ilto n ia n r e s u l t s
fro m a d l a g o n a l i z a t i o n o f
to se co n d o rd e r, th e f i r s t
s p in
s te p i n c a l c u l a t i n g th e s p i n - l a t t i c e i n t e r a c t i o n .
One th e n f i n d s th e e n e r g y l e v e l s a n d c o r r e s p o n d in g wave
fu n c tio n s f o r th e d ia g o n a liz e d
^
Sp i n » a n d com putes t h e a p ­
p r o p r i a t e m a t r i x e le m e n t s o f ® ^ n t e r a c t i o n l3etw eei1 s i m u l ­
ta n e o u s e i g e n s t a t e s o f
^ ap i n
^ la ttic e *
Tlie Pr o c e s s
t h u s y i e l d s t h e d e s i r e d m a t r i x e le m e n ts d i r e c t l y .
M a ttu c k a n d S t r a n d b e r g w orked o u t t h e m a t r ix e le m e n ts
f o r Raman a n d d i r e c t i n t e r a c t i o n s , b o t h f o r phonons fro m
t h e Debye l a t t i c e
s p e c tr u m and f o r m o n o c h ro m a tic p h o n o n s .
F o r a t r e a tm e n t o f t h e a t t e n u a t i o n o f m o n o ch ro m atic h y p e r ­
s o n i c waves a t t h e L arm o u r f r e q u e n c y , o n l y th e d i r e c t p r o c e s s
i s re q u ire d .
T h is i s t h e e a s i e s t m a t r i x e le m e n t t o c a l c u l a t e ,
s i n c e o n ly a o n e -p h o n o n o p e r a t o r I s r e q u i r e d by t h e d i r e c t
i n t e r a c t i o n , a n d t h e p h o n o n s a r e n e c e s s a r i l y a l l o f t h e same
f r e q u e n c y , p o l a r i z a t i o n , a n d d i r e c t i o n , t o m eet t h e c o n d i t i o n s
f o r a h y p e r s o n ic wave p r o p a g a tin g i n th e c r y s t a l l a t t i c e .
I n a d d i t i o n t o c a l c u l a t i n g th e we x a c t n ( t o s e c o n d o r d e r )
i n t e r a c t i o n m a t r i x e l e m e n t s , th e a u t h o r s a l s o em p lo y ed a
s im p le p i c t u r e o f t h e c r y s t a l l a t t i c e t o o b t a i n o r d e r - o f -
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11
m a g n itu d e f o r m u la s f o r th e i n t e r a c t i o n s .
t o th e i n v e s t i g a t i o n s o f t h i s t h e s i s i s
The one r e l e v a n t
^ d ire c t*
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th e l a t t i c e ;
e is
t h e v e l o c i t y o f so u n d i n
th e e le c t r o n i c c h a rg e ; r 0 i s th e io n ic
r a d iu s ; R i s th e a v erag e in te r a to m ic s e p a r a tio n ;
S ta rk s p l i t t i n g
i n th e c r y s t a l l i n e e l e c t r i c
i s th e s p i n a n tic o m m u ta to r ,
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c a lc u la te d once
d ire c t
o b ta in e d ; fo r th e d i r e c t i n ­
t e r a c t i o n w i t h a m o n o c h ro m a tic p h o n o n i t i s :
5 ->
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H a m ilto n ia n a n d p , p f two e n e r g y e i g e n s t a t e s o f t h e l a t t i c e
i
H a m ilto n ia n ; a n d
q (v p - v r )
i s th e lln e -s h a p e f a c t o r f o r
i
;
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an d
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th e re s o n a n c e fre q u e n c y .
m a g n itu d e fo rm u la f o r
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E m p lo y in g th e o r d e r - o f tiie t r a n s i t i o n p r o b a b i l i t y
p e r u n i t tim e b e c o m e s :
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3 6 e + r +
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maximum.
* tiie lin e w id th a t h a l f
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T he e x p e c te d a c o u s t i c - a t t e n u a t i o n c o e f f i c i e n t c a n a l s o
be c a l c u l a t e d .
I f on e assu m e s no s a t u r a t i o n o f th e s p i n
s y s te m b y th e i n c i d e n t a c o u s t i c p o w er,
t h e n th e a t t e n u a t i o n
c o e f f i c i e n t f o r N s p in s p e r c u b ic c e n t i m e t e r i s g iv e n b y :
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a t t e n u a t i o n c o e f f i c i e n t from a p h e n o m e n o lo g ic a l p o i n t o f
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WDM a Sa i n > O r b a c h 's fo rm u la i s :
w h ere G i s th e " m a g n e to e l a s ti c c o u p l i n g c o n s t a n t 1';
th e e l a s t i c
s t r a i n a m p litu d e ;
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s p i n o p e r a t o r b e tw e e n th e l e v e l s i n v o l v e d .
c o e ffic ie n t is
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e x p r e s s i o n f o r t h e a t t e n u a t i o n c o e f f i c i e n t c a n be e a s i l y
seen.
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T he two fo r m u la s a r e th u s e q u i v a l e n t .
O rb ach d e r i v e d a f o rm u la f o r c a l c u l a t i n g t h e s p i n l a t t i c e r e l a x a t i o n t im e , T.^, fro m a m e a su re d v a lu e o f V»DM<
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III,
P i e z o e l e c t r i c G e n e r a t i o n o f H y p e rs o n ic W aves
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s t i l l o b t a i n s a t m icro w av e f r e q u e n c i e s .
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volum e V i n t o w h ich t h e e n d o f t h e r o d i s i n s e r t e d .
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17
c o n c e n t r a t e d I n th e r o d , a n d t h a t t h e c a v i t y i s u n i t y c o u p le d .
A t t h e e n d o f a n X - c u t q u a r t z r o d , t h e s t r e s s m ust go
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th e a p p ro p ria te p i e z o e l e c t r i c c o n s ta n t.
The
a c o u s t i c e n e r g y d e n s i t y o f t h e wave w h ic h p r o p a g a t e s i n
th e r o d i s :
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C x i
so t h a t t h e pow er flo w I n th e r o d w o u ld b e :
s.)
IP =
R e w r i ti n g , t h e e x p r e s s io n b eco m es:
u s in g th e r e l a t i o n
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* 9
The e n e r g y c o n v e r t e d p e r s e c o n d i s
th e n :
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
w h ere q I s th e c r o s s s e c t i o n a l a r e a o f t h e r o d .
th e p i e z o e l e c t r i c c o u p li n g f a c t o r
8 .)
-
p ( c / q .
Cx
a
*
/ - s j r v )
u s i n g t h e e x p r e s s i o n f o r th e e l e c t r i c
D e f in in g
,
f i e l d we o b t a i n e d
in E qn. 2 . ) .
The d e t e c t i o n p r o c e s s i s t h e i n v e r s e o f th e g e n e r a ­
tio n p ro c e ss.
An e l e c t r i c f i e l d i s p r o d u c e d i n s i d e t h e
c a v i t y b y th e v i b r a t i o n s o f th e r o d .
I t c a n be shown
fro m a th erm o d y n am ic a rg u m e n t t h a t t h e e f f i c i e n c y f o r
g e n e r a t i n g and f o r d e t e c t i n g th e w aves i s t h e sam e.
T h e r e f o r e , th e r f p o w er o u tp u t from t h e a c o u s t i c a l w aves
is :
^ o u tp u t
™
(
^ x
^ ^
) •
The above e q u a t i o n assu m es i d e a l c o n d i t i o n s , w h ic h
c a n s c a r c e l y be r e a l i z e d a t 10 G c.
In th e f i r s t p la c e ,
a l l th e e l e c t r i c f i e l d I s c e r t a i n l y n o t c o n c e n tra te d in
th e e n d o f th e q u a r t z r o d .
Nor i s
th e d i r e c t i o n o f t h e
\
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
19
e l e c t r i c f i e l d u n if o r m ; t h e h o l e i n t h e c a v i t y t h r o u g h
w h ich th e r o d i s i n s e r t e d c a u s e s a m a jo r p e r t u r b a t i o n o f
t h e e l e c t r i c f i e l d i n th e g a p .
A f u r th e r d e crea se in
e f f i c i e n c y r e s u l t s fro m th e m e c h a n ic a l i m p e r f e c t i o n s o f
th e q u a r t z r o d s .
E v en th o u g h t h e e n d s o f a r o d may be
p o l i s h e d f l a t t o w i t h i n a q u a r t e r w a v e le n g th o f sodium
lig h t,
th e y a r e s t i l l
o f so u n d a t 10 G c.
f l a t to o n l y a q u a r t e r w a v e le n g th
The r f o u t p u t i s a m easure o f th e
a c o u s t i c wave a m p litu d e I n t e g r a t e d o v e r th e end f a c e o f
th e c r y s t a l , so t h a t t h e d e t e c t e d o u t p u t i s e x tr e m e l y
s e n s i t i v e to a n y d e v i a t i o n s o f t h i s f a c e from p e r f e c t
fla tn e s s .
I n d e e d , i f p r e c i s e l y o n e h a l f th e a r e a o f
t h e f a c e w ere p r e c i s e l y one q u a r t e r w a v e le n g th h i g h e r
t h a n th e o t h e r h a l f ,
a n d i f th e o t h e r en d f a c e w e re
p r e c i s e ly f l a t and p a r a l l e l to
t h e f i r s t f a c e , no r f
o u tp u t w ould be s e e n fro m t h e c r y s t a l .
The d e v i a t i o n s
i n f l a t n e s s a l s o l e a d t o t h e n o n e x p o n e n t ia l d e c a y s a l ­
w ays o b s e r v e d a t 1 0 G c.
O f f - a x i s p r o p a g a t io n a n d c o n ­
v e r s i o n t o o t h e r m odes w i t h i n t h e c r y s t a l a l s o l e a d t o
a r e d u c t i o n i n t h e e f f i c i e n c y o f th e p r o c e s s .
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
20
IV .
E q u ip m e n t
As d e m o n s tr a te d i n S e c t i o n I I I , a m icrow ave e l e c t r i c
f i e l d i s r e q u ir e d f o r th e p i e z o e l e c t r ic p ro d u c tio n o f th e
I
h y p e r s o n i c w aves n e e d e d f o r t h e p r e s e n t e x p e r im e n t.
e le c tric f ie ld
The
i n t e n s i t y s h o u ld b e a s l a r g e as p o s s i b l e ,
a n d w i l l be l i m i t e d b y e l e c t r i c a l o reax d o w n .
m eth o d f o r o b t a i n i n g h i g h e l e c t r i c
A good
f i e l d s a t m icrow ave
f r e q u e n c i e s i s th ro u g h t h e u s e o f a r e e n t r a n t r e s o n a n t
c a v ity , e . g . P ig . 1 .
The c a v i t y i s e s s e n t i a l l y a s h o r t
p ie c e o f c o a x ia l tr a n s m is s io n l i n e te rm in a te d by a
c a p a c ito r.
The end o f a q u a r t z r o d c a n be p l a c e d w i t h i n
t h e c a p a c i t o r , o r " g a p ."
Eqn. 9 .) o f S e c tio n I I I i n d ic a te s
t h a t th e Q o f th e c a v i t y s h o u l d b e a s h i g h a s p o s s i b l e .
F o r b e s t r e s o l u t i o n o f t h e e c h o e s , h o w e v e r, th e r f p u l s e s
s h o u ld b e a s s h o r t a s p o s s i b l e , r e q u i r i n g a low -Q c a v i t y .
Some co m p ro m ise i s th e n n e c e s s a r y .
S in c e a m a g n e tro n
was a v a i l a b l e w h ic h was c a p a b le o f d e l i v e r i n g 1 , 0 0 0 w a tt
p u ls e s o f
i
m ic ro s e c o n d d u r a t i o n , th e maximum p r a c t i c a l
c a v i t y Q was f i x e d t o g i v e a b a n d w id th o f a b o u t 10 Me a t
10 G c, i . e . ,
i
n e a rly
a Q o f a b o u t 1 ,0 0 0 .
The ^ s h o u ld a l s o be a s
i n d e p e n d e n t o f t e m p e r a t u r e a s p o s s i b l e i f m e a s u re ­
m e n ts a r e t o b e made a s a f u n c t i o n o f t e m p e r a t u r e .
re s tric ts
T h is
th e c h o ic e o f m a t e r i a l s f o r th e c a v i t y t o th o s e
whose r e s i s t i v i t y I s n o t a s t r o n g f u n c t i o n o f t e m p e r a t u r e .
The f i r s t s u c h c a v i t y c o n s t r u c t e d h e r e was made o f
I
j
i
b ra ss.
Its
fo rm i s p i c t u r e d i n F i g . 2 .
The c a v i t y h a d a
j
I
.
.
.
.
.
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
b a n d w id th o f a b o u t 8 Me a t room te m p e r a tu re when u n i t y c o u p le d .
The b a n d w id th was r a t h e r a s tr o n g f u n c t i o n o f
te m p e ra tu re .
R ough c a l c u l a t i o n s i n d i c a t e d t h a t a s t a i n l e s s -
s t e e l c a v i t y w o u ld h a v e a b o u t th e p ro p e r b a n d w id th , b u t
when one was c o n s t r u c t e d , i t was fo u n d to h a v e a b a n d ­
w id th o f a b o u t 50 Me.
F u r t h e r , i t was im p o s s ib le t o make
th e c o u p lin g h o l e l a r g e enough t o u n i t y - c o u p l e t h e c a v i t y ;
e a c h i n c r e a s e i n th e s i z e o f t h e h o le b ro u g h t a b o u t a drop
i n Q w h ic h e r a s e d t h e a d v a n ta g e o f t h e l a r g e r h o l e .
The f i n a l c a v i t y was c o n s t r u c t e d o f G erm an s i l v e r ,
i n t h e fo rm show n i n F i g . 3 .
I t s b a n d w id th was a b o u t
14 Me a t room te m p e r a t u r e a n d 10 Me a t l i q u i d n i t r o g e n
te m p e r a tu r e a n d b e lo w .
I n o r d e r t o c o n s e rv e s p a c e i n s i d e
th e D ew ar, th e c a v i t y was c o n s t r u c t e d to f i t i n s i d e a
p i e c e o f s t a n d a r d s t a i n l e s s - s t e e l w a v e g u id e .
in th r e e p ie c e s ,
I t was made
sc re w e d t o g e t h e r so t h a t i t c o u l d be
e a s i l y d i s a s s e m b le d f o r c le a n i n g a h d p o l i s h i n g .
T h is i s
n e c e s s i t a t e d b y th e i n e v i t a b l e c o n d e n s a tio n o f w a t e r on
low t e m p e r a t u r e e q u ip m e n t.
B e c a u se a t u n e a b l e m a g n e tro n was n o t r e a d i l y a v a i l a b l e ,
i t was n e c e s s a r y to make th e c a v i t y t u n e a b l e .
I
!
!
F u r t h e r , due
t o d i f f e r e n t i a l e x p a n s io n b e tw e e n t h e q u a r tz a n d th e German
s i l v e r , th e t u n i n g r a n g e h a d t o b e r a t h e r l a r g e .
I t was
a l s o d e s i r a b l e t h a t n e i t h e r t h e c o u p lin g n o r t h e Q change
r a d i c a l l y o v e r th e tu n in g ra n g e .
T h is was a c c o m p lis h e d
th r o u g h u s e o f a t e f l o n r o d w i t h a b r a s s i n s e r t ,
so t h a t
th e t u n i n g was i n p a r t e l e c t r i c a n d I n p a r t m a g n e t i c .
i
i
ii
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
The t u n i n g ra n g e o b t a i n e d i n t h i s m a n n e r was a b o u t 500
Me, w i t h o u t o b j e c t i o n a b l e change i n ft o r c o u p li n g ,
The r f pow er was s u p p l i e d b y a Bomac B L-233 raagne-
j
i
tro n .
T h is i s a r e l a t i v e l y l o w - c o s t , l i g h t - w e i g h t u n i t
d e l i v e r i n g ^ - m ic r o s e c o n d , 1 , 0 0 0 - w a t t p u l s e s w ith t h e
m o d e s t pow er s u p p ly r e q u ir e m e n ts o f 2 ,8 0 0 v o l t s a t 1 .5
a m p s.
A t h y r a t r o n m o d u la to r was u s e d t o pow er th e m agne­
tr o n b e ca u se o f th e s im p lic ity o f th e c i r c u i t r y .
A lth o u g h
t h e c a t h o d e - f a u l t - c u r r e n t r a t i n g o f 1 0 amps i s e x c e e d e d
b y 1 5 0 ^ , a 2D21W t h y r a t r o n was c h o s e n f o r i t s
w id e a v a i l a b i l i t y .
low c o s t an d
A m ore d e p e n d a b le c h o ic e w ould b e t n e
T u n g - s o l 1 2 5 8 , b u t t h e 2D21W h a s s u r v i v e d w e l l o v e r 100
ru n n in g h o u rs w ith o u t d i f f i c u l t y .
A 6 0 -cp s r e p e t i t i o n
r a t e w as c h o se n so t h a t hum i n t h e r e c e i v e r c o u ld be
l a r g e l y ig n o re d .
The m a g n e tro n -p o w e r s u p p ly i s i l l u s t r a t e d
in F ig . 4 .
The m icrow ave c i r c u i t f o r t h e e x p e r im e n t i s i l l u s t r a t e d
in F ig . 5 .
I t em p lo y s th e minimum o f c i r c u i t com ponents
fo r p ro p e r o p e ra tio n .
The u s e o f e a c h com ponent s h a l l b e
d e s c r i b e d i n some d e t a i l b e lo w .
E l e c t r i c a l b re a k d o w n i n th e c a v i t y p r e v e n t s o p e r a t i o n
j
o f t h e m a g n e tro n a t f u l l pow er u n d e r some c o n d i t i o n s .
An
I
!
a t t e n u a t o r , p l a c e d im m e d ia te ly a f t e r t h e m a g n e tro n , r e d u c e s
j
th e p o w e r i n c i d e n t u p o n th e c a v i t y w hen n e c e s s a r y .
l
i
j
p u ls e th e n t r a v e l s th ro u g h th e c i r c u l a t o r ,
p a t h shown b y th e a r r o w s .
The r f
f o l lo w in g th e
The c i r c u l a t o r p e r m i ts o n l y
i
p o w er r e f l e c t e d fro m t h e c a v i t y o r o r i g i n a t i n g i n th e
I
!
i
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
1
23
c a v i t y to p a s s t o t h e d e t e c t o r s .
|
A s lia e -s c re w tu n e r
a s s u r e s t h a t th e r f p u l s e r e f l e c t e d from th e c a v i t y c a n
h e r e d u c e d t o some s m a ll v a l u e , I f t h e c a v i t y c o u p li n g
d i f f e r s s l i g h t l y fro m u n i t y .
R f p u l s e s o f po w er g r e a t e r t h a n 1 0 0 m i l l i w a t t s a r e
s t r o n g l y a t t e n u a t e d b y t h e T-R t u b e .
Power i n t h e e c h o e s ,
h o w e v e r, i s much l e s s t h a n 1 0 0 m i l l i w a t t s ; t h e y t r a v e r s e
t h e T-R tu b e v i r t u a l l y u n a t t e n u a t e d .
w as a S y lv a n ia 1B24A, p i c k e d f o r i t s
its
The T-R tu b e c h o s e n
f a s t re c o v e ry tim e ,
100-Mc t u n i n g r a n g e a n d 30-M c b a n d w id th , a n d i t s
p ric e .
A 1 , 0 0 0 - v o l t i g n i t o r s u p p ly f o r th e tu b e i s
m o d est
in ­
c lu d e d on t h e same c h a s s i s a s t h e m a g n e tro n p u l s e r , F i g .
4.
Any power r e f l e c t e d fro m th e T-R tu b e i s a b s o r b e d i n
t h e m atch e d l o a d a f t e r p a s s i n g
th r o u g h th e c i r c u l a t o r .
The e c h o e s t h e n p a s s t h r o u g h a p r e c i s i o n a t t e n u a t o r ,
p e r m i t t i n g r e l a t i v e a t t e n u a t i o n m e a su re m e n ts t o b e made
t o 1 0 . 1 db w i t h th e p o w er i n c i d e n t u p o n th e c a v i t y h e l d
c o n s ta n t.
We now t u r n t o t h e p r o b le m o f d e t e c t i n g t h e e c h o e s .
E q n . 9 .) o f S e c tio n I I I i n d ic a t e s t h a t fo r th e a r r a n g e ­
m en t d e s c r i b e d ( $ s 1 , 0 0 0 , t r a n s m i t t e r pow er = 1 , 0 0 0 w a t t s ,
I
I
j
;
!
O
a r e a o f ro d = 0 .0 3 cm , v o lu m e o f r o d I n s i d e c a v i t y ■
2
0 .0 0 3 cm , p i e z o e l e c t r i c c o u p li n g c o n s t a n t ■ 1 0 - 2 ) , t h e
o u t p u t pow er fro m t h e f i r s t e c h o w o u ld b e a p p r o x i m a t e ly a
-1 o
-7
f a c t o r o f 10
o f t h e i n p u t p o w e r, o r 10
w a tts . T h ere f o r e , a good v id e o d e t e c t o r h a v in g a s e n s i t i v i t y o f 1 0
I
w a t t s s h o u ld be more t h a n a d e q u a t e .
• ft
W ith t h i s i n m in d ,
I
I
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
a
24
M icrow ave A s s o c i a t e s MA 408B c r y s t a l w as o b t a i n e d and a
s u i t a b l e v id e o a m p l i f i e r was b u i l t .
The c i r c u i t o f th e
a m p l i f i e r a n d th e c r y s t a l m a tc h in g t r a n s f o r m e r and b i a s
a r r a n g e m e n t a r e i l l u s t r a t e d i n P i g . 6 . U n f o r t u n a t e l y , th e
-7
e s tim a te o f 10
w a t ts o f r f p o w er i n th e f i r s t echo
p ro v e d o p t i m i s t i c b y f o u r o r d e r s o f m a g n itu d e .
E choes
w ere n o t s e e n u n t i l a s u p e r h e te r o d y n e d e t e c t i o n scheme
c a p a b le o f d e t e c t i n g p o w ers on th e o r d e r o f 1 0
was t r i e d .
-1 2
w a tts
The m icrow ave c i r c u i t f i n a l l y u s e d i s i l l u s ­
tr a te d in F ig . 5.
W a v e lin e d e t e c t o r m o u n ts w ere ch o sen
f o r t h e i r lo w c a p a c i t a n c e a n d b e c a u s e t h e s e d e t e c t o r s may
b e m a tc h e d th r o u g h t h e u s e o f i n c o r p o r a t e d t u n in g s c r e w s .
A V a r ia n A s s o c i a t e s X -13 k l y s t r o n was u s e d a s a l o c a l
o s c i l l a t o r b e c a u s e o f i t s low a m p litu d e a n d fr e q u e n c y
m o d u la tio n n o i s e . The u s e o f a b a la n c e d m ix e r f u r t h e r
c a n c e l l e d n o i s e from th e l o c a l o s c i l l a t o r , i n a d d i t i o n t o
p r o v i d i n g a n o th e r 3 db o f s i g n a l .
R f b i a s on th e 1N23E
c r y s t a l s w as o p tim iz e d t h r o u g h u s e o f t h e a t t e n u a t o r .
S in c e t h e l o c a l o s c i l l a t o r p o w e r c o u ld b e i n c r e a s e d by
1 0 db w i t h o u t a m e a s u ra b le i n c r e a s e i n n o i s e a f t e r
r e a c h i n g maximum c o n v e r s io n g a i n , i t may be assu m ed t h a t
l o c a l o s c i l l a t o r n o i s e d o e s n o t d e g ra d e t h e p e rfo rm a n c e
o f th e r e c e i v e r .
The IP a m p l i f i e r i s a T r i d e a m odel 1QA, w i t h a 10 Me
i
b a n d w id th a b o u t a 60 Me c e n t e r f r e q u e n c y .
i s 1 .5 d b .
I t s n o is e fig u re
The e n t i r e r e c e i v e r th u s h a s a t o t a l n o is e
f i g u r e o f 7 . 5 d b , u s in g th e p u b l i s h e d n o i s e f i g u r e f o r
j
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
25
1N23E c r y s t a l s .
T h is i s t h e b e s t w hich c a n b e a c h ie v e d
w i t h p r e s e n t t e c h n i q u e s , w i t h o u t th e u s e o f lo w - n o is e
m icrow ave p r e a m p l i f i e r s .
T h ere i s a n o th e r a d v a n ta g e i n u s in g a s u p e r h e te r o d y n e
d e te c to r.
B e c a u s e t h e m a g n e tro n r i s e a n d d e c a y tim e s a r e
so s h o r t , u n i t y c o u p lin g c a n n o t be o b t a i n e d f o r a l l f r e ­
q u e n c y c o m p o n e n ts o f th e m ain p u l s e i f t h e c a v i t y h a s o n ly
a 1 0 Me b a n d w id th ( s e e P i g . 7 ) .
The u n c a n c e l l e d com ponents
a r e r e c t i f i e d i n a v id e o d e t e c t o r and t e n d t o o v e r lo a d th e
v id e o a m p l i f i e r .
A s u p e r h e te r o d y n e r e c e i v e r i s s e n s i t i v e
o n ly to b e a t f r e q u e n c i e s f a l l i n g w i t h i n i t s b a n d w id th , so
t h a t u se o f a 1 0 Me IP b a n d w id th e f f e c t i v e l y e l i m i n a t e s
t h e o v e r lo a d p r o b le m .
A f u r t h e r a s s i s t i s g iv e n by s a tu r a ­
t i o n i n t h e c r y s t a l m ix e r s .
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o b t a i n e d t h r o u g h th e u se o f a " b o x c a r i n t e g r a t o r . "
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d e v ic e s h a v e b e e n u s e d in r a d a r w ork f o r many y e a r s , b u t
h a v e s u f f e r e d fro m d r i f t an d i n s t a b i l i t y .
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f o r t h i s e x p e r i m e n t , d e s ig n e d b y D r. P . L . D onoho, ta k e s
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a c h ie v e r e m a r k a b ly s t a b l e o p e r a t i o n a f t e r a warmup p e r i o d
o f o n ly one h o u r .
P i g . 8 show s t h e c i r c u i t o f t h e d e v ic e .
The g e n e r a l i d e a o f a b o x c a r i n t e g r a t o r i s
g a te i s opened f o r a s h o rt tim e , l e t t i n g
p l u s , p r e s u m a b ly , a s i g n a l .
th is :
a
th r o u g h n o i s e
The n o i s e an d s i g n a l a r e th e n
s t o r e d i n a n RC n e tw o rk w ith a tim e c o n s t a n t much l a r g e r
t h a n th e p e r i o d o f th e g a ti n g p u l s e .
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a v e r a g e s t o t h e DC l e v e l o f th e g a t e p e d e s t a l , a n d th e
s i g n a l t o some h i g h e r DC l e v e l .
S in c e th e s i g n a l i s a lw a y s
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a v e r a g e s t o a c o n s t a n t DC l e v e l in d e p e n d e n t
o f th e i n t e g r a t i n g
tim e c o n s t a n t .
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n o i s e , h o w e v e r, a v e r a g e s to a v a l u e w h ic h a p p r o a c h e s t h e
l e v e l o f th e g a t e p e d e s t a l a s t h e tim e c o n s t a n t a p p r o a c h e s
in fin ity .
By v a r y i n g th e tim e a t w h ic h th e g a t i n g p u l s e
o c c u rs, i t
i s p o s s ib le
o f t im e .
to t r a c e o u t a s i g n a l a s a f u n c t i o n
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c o n s t a n t , a c c u r a t e m e a su re m e n ts c a n b e made o f t h e v a r i a t i o n
o f th e s i g n a l a m p l i t u d e a s v a r i o u s p a r a m e te r s a r e v a r i e d .
I n t h e c i r c u i t o f F i g . 8 , a 2D21W t h y r a t r o n p r o v i d e s
th e g a t i n g p u l s e .
I t I s t r i g g e r e d fro m t h e " G a te A" o u t p u t
o f a T e k t r o n ix 545A o s c i l l o s c o p e , w i t h th e h o r i z o n t a l d i s ­
p l a y i n th e t
I n t e n s i f i e d b y A" p o s i t i o n .
The t i m i n g o f
th e g a t i n g p u l s e c a n th e n b e v a r i e d w i t h th e " D e la y T im e"
an d t h e " D e la y Tim e M u l t i p l i e r " c o n t r o l s , o r b y e x t e r n a l l y
r e p l a c i n g th e " D e la y Time M u l t i p l i e r " c o n t r o l w i t h a lo w fre q u e n c y f u n c t i o n g e n e r a to r su c h a s th e H e w le tt-P a c k a rd
m odel 202A.
The p u l s e from t h e 2D211»V i s
th e n u s e d to
b r i n g t h e 6 6 8 8 g a t e tu b e o u t o f c u t o f f f o r a p e r i o d o f
0 .1 m ic r o s e c o n d .
The s i g n a l a n d n o i s e a r e f e d fro m th e IF
a m p l i f i e r t o t h e c o n t r o l g r i d o f t n e 6 6 8 8 th r o u g h a c a th o d e
f o l l o w e r , so t h a t l i n e a r o p e r a t i o n c a n be m a i n t a i n e d e v e n
when t h e 6 6 8 8 i s
d ra w in g g r i d c u r r e n t .
The s i g n a l - p l u s - n o i s e
c u r r e n t p u l s e fro m t h e 6 6 8 8 i s a v e r a g e d , a lo n g w i t h t h e
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27
p e d e s t a l fro m t h e g a t i n g p u l s e , i n th e RC c i r c u i t c o n s i s t i n g
o f a 1 0 0 K p r e c is io n r e s i s t o r and an e x te rn a l o i l - f i l l e d
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o f th e t u b e .
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t r i b u t o r t o th e i n s t a b i l i t y o f a b o x c a r i n t e g r a t o r , a c o n ­
s i d e r a b l e a d v a n ta g e i s g a in e d th r o u g h use o f t h e d i o d e .
The v o l t a g e d e v e lo p e d a c r o s s t h e s t o r a g e c a p a c i t o r
i s f e d d i r e c t l y to a T ex a s I n s t r u m e n t s S e r v o - r i t e r .
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th e i n p u t t o th e r e c o r d e r t o b a l a n c e o u t tn e c o n s t a n t DC
v o l t a g e fro m th e g a ti n g p u l s e .
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s t o r a g e c a p a c i t o r f o r a 10 s e c o n d tim e c o n s t a n t .
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C are w as t a k e n to m in im ize t h e h e a t
l e a k i n t o t h e b a t h sp a c e th r o u g h u s e o f t h i n - w a l l s t a i n l e s s - s t e e l w av eg u id e a n d a t h i n - w a l l s t a i n l e s s - s t e e l tu b e
f o r th e c a v i t y tu n in g a r r a n g e m e n t.
A s ta in le s s -s te e l
vacuum c a n was c o n s t r u c t e d t o s u r r o u n d the a p p a r a t u s i n !
s i d e t h e D ew ar, i n o r d e r t o p r e v e n t e l e c t r i c a l b reak d o w n
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in th e c a v i t y .
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28
!
i
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c o u ld be u se d a t a tm o s p h e r ic p r e s s u r e w i t h th e c a v i t y
c o o le d to l i q u i d h e liu m t e m p e r a t u r e , m aking th e vacuum
can u n n e c e ssa ry .
A c o p p e r s h e a th s u r r o u n d s th e sam ple
to p r o v i d e a d e q u a te c o o li n g ev en when n o t i.um ersed i n
l i q u i d h e liu m . A t a p r e s s u r e o f 1 0 ” ° mm, a rough c a l c u l a 16
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i n c i d e n t upon t h e c a v i t y .
t h e sam p le i s p l a c e d i n th e f i e l d o f a
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w ith a V a r ia n
1 . 4 amp pow er s u p p l y , t h i s p r o v id e s f i e l d s up to 3600
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(th o s e
o v e r 300 g a u s s w id e , w i t h a g - f a c t o r o f t w o ) .
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29
V.
R e s u lts
A.
Q u a r tz
E c h o e s i n a c y l i n d r i c a l q u a r t z r o d 3 cm lo n g a r e se e n
re p ro d u c e d i n F ig . 1 0 .
W hile a l a r g e number o f e c h o e s was
o b t a i n e d , t h e i r r e g u l a r i t y o f th e d e c a y o b v i a t e s m ea n in g ­
f u l a b s o l u t e a t t e n u a t i o n m e a s u re m e n ts .
The r e a s o n f o r
th e n o n e x p o n e n t i a l d e c a y i s t h e i m p o s s i b i l i t y o f o b t a i n i n g
s u f f i c i e n t f l a t n e s s and p a r a l l e l i s m
o f th e e n d s .
W ith a
6,0C 0 £ w a v e le n g th , e v en a p o l i s h t o l / 5 w a v e le n g th o f
sodium l i g h t i s o n l y a p o l i s h t o 2 / 3 w a v e le n g th o f t h e
a c o u s tic a l v ib r a tio n s .
o f c o n s ta n t p h a s e .
The e n d s a r e s im p ly n o t s u r f a c e s
T h a t t h i s i s t h e m a jo r c o n t r i b u t i o n
to th e n o n e x p o n e n t i a l d e c a y , r a t h e r th a n o f f - a x i s p r o p a g a ­
t i o n o r c o n v e r s i o n t o o t h e r m o d es, h a s b e e n d e m o n s tr a te d
17
by S h ire n
w i t h th e a i d o f a " d o u b le quantum " d e t e c t o r .
The g r e a t s e n s i t i v i t y o f t h e m ethod i n d e t e c t i n g
c h an g e s i n d i s l o c a t i o n d e n s i t y i s i l l u s t r a t e d i n F i g . 1 1 .
The two p h o to g r a p h s a r e e ch o t r a i n s
i n th e sam e q u a r t z
r o d , ta k e n s e v e r a l w eeks a p a r t , b e f o r e an d a f t e r h a n d l i n g .
U n f o r t u n a t e l y , p e rf o r m in g a n e x p e r im e n t on d i s l o c a t i o n s
a t h y p e r s o n i c f r e q u e n c i e s w o u ld h a v e t o be d o n e La s i t u
b e o a u se a b s o l u t e a t t e n u a t i o n m e a su re m e n ts c a n n o t b e m ad e.
S u f f i c i e n t dam age i s done b o th t o t h e b o d y a n d t h e s u r f a c e s
o f th e r o d s i n re m o v in g them fro m t h e e q u ip m e n t t o make
r e l a t i v e a t t e n u a t i o n m ea su re m e n ts m e a n in g le s s u n l e s s th e
ro d s a r e k e p t i n p l a c e b e tw e e n t h e m e a s u re m e n ts .
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
30
B.
Q u a r tz w i t h F c e n t e r s
One s o r t o f r e l a t i v e a t t e n u a t i o n m e a su re m e n t w hich
c a n b e made w ith th e p a r a m e t e r s ch an g e d i n s i t u
is th a t o f
t h e e v a l u a t i o n o f th e s p i n - l a t t i c e i n t e r a c t i o n f o r p a r a ­
m a g n e tic i o n s i n a n o n - p a r a m a g n e tic m a t r i x , e . g . F c e n t e r s
13
in q u a r tz .
T h is e x p e r i m e n t was a t t e m p t e d w i t h a n a t u r a l
q u a r t z r o d 3 cm l o n g .
F c e n t e r s w ere i n t r o d u c e d b y s u b 7
j e c t i n g th e sam ple t o 1 0 r a d o f 1 Mev gamma r a y s fro m a
60
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bom b. D a rk e n in g o f t h e r o d was u n if o r m a n d e x tr e m e .
A C a ry o p t i c a l s p e c t r o p h o t o m e t e r was u n a b le t o g i v e m ore
—5
t h a n a lo w e r l i m i t o f 1 0
t o th e t r a n s m i s s i v i t y b e tw e e n
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s p e c tr u m c o u ld n o t b e o b t a i n e d a t room t e m p e r a t u r e , s e t s
36
3
th e c o n c e n t r a t i o n o f F c e n t e r s a t a b o u t 1 0 /c m .
The r o d was p l a c e d i n t h e a p p a r a t u s w h ic h h a s b e e n
p re v io u s ly d e s c rib e d .
F u l l m a g n e tro n p o w e r c o u ld n o t
b e u s e d b e c a u s e o f e l e c t r i c a l b reak d o w n ; 2 5 0 w a t t s was
t h e maximum u s e a b l e i n c i d e n t p o w e r.
c o u ld b e m o n ito r e d t o * 1 d b .
The f i r s t
two e c h o e s
No change i n t h e h e i g h t o f
e i t h e r e c h o w as n o t e d a s t h e m a g n e tic f i e l d w as v a r i e d ,
s e ttin g
t h e u p p e r l i m i t f o r t h e a t t e n u a t i o n a t l / 6 d b /cm .
The e x p e r im e n t w as r e p e a t e d u n d e r m ore a d v a n ta g e o u s
c o n d itio n s .
A 0 .2 2 i n c h l o n g X -c u t q u a r t z r o d o f 0 . 1
i n c h d i a m e t e r was i r r a d i a t e d i n a th o riu m i n t e r n a l i r r a d i a 7
t o r . T he d o se was 10 r a d * B la c k e n in g w as c o n s i d e r a b l y
m ore s e v e r e i n t h i s r o d t h a n i n th e p r e v i o u s r o d .
A G ary
m odel 14 MS o p t i c a l s p e c t r o p h o t o m e t e r w as u s e d to e v a l u a t e
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31
t h e m a g n itu d e a n d w id th o f t h e F - b a n d a b s o r p t i o n l i n e .
E m p lo y in g a n o s c i l l a t o r s t r e n g t h o f 0 .7 5
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±
p e r c u b ic c e n t im e te r b y t h e m e th o d o u t l i n e d b y M o tt a n d
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G u rn e y .
F u l l m a g n e tro n pow er c o u ld be em ployed i n t h i s
e x p e rim e n t.
F u r t h e r , th e b o x c a r i n t e g r a t o r was u s e d i n
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The echo t r a i n i s show n i n F i g . 1 2 .
W ith in
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o b se rv ed .
The a t t e n u a t i o n due t o t h e F c e n t e r s p i n s a t
r e s o n a n c e m ust t h e n be l e s s t h a n 0 .0 6 7 db p e r c e n t i m e t e r ,
o r 0 .0 0 7 7 n e p e r s p e r c e n t i m e t e r .
W ith 5 .1 x 10
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p e r c u b ic c e n t i m e t e r , t h i s i s a c r o s s s e c t i o n f o r a b s o r p t i o n o f l e s s t h a n 3 .9 x 10
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32
V I.
D is c u s s io n o f R e s u lts
From E q n . 2 0 . ) o f S e c t i o n I I a n d f ro m t h e u p p e r l i m i t
o b t a i n e d t o t h e a t t e n u a t i o n c o e f f i c i e n t , a lo w e r l i m i t
t o t h e s p i n - l a t t i c e r e l a x a t i o n tim e
c a n be c a l c u l a t e d
a n d co m p ared w i t h th e e x p e c te d v a lu e fro m o t h e r m e a su re ­
m e n ts .
F u r t h e r , a lo n g w ith E q n . 1 1 . ) o f S e c t i o n I I an d
t h e u p p e r l i m i t to th e a t t e n u a t i o n c o e f f i c i e n t , know ledge
o f th e a c o u s t i c pow er i n p u t to t h e c r y s t a l p e r m i t s a c a l c u ­
l a t i o n o f an u p p e r l im i t f o r W ^ , th e t r a n s i t i o n p r o b a b il­
i t y p e r s e c o n d f o r a s p i n t o a b s o r b a p h o n o n a t t h e L arm our
f r e q u e n c y fro m t h e l a t t i c e .
U n f o r t u n a t e l y , T u c k e r made no
e s t im a t e o f t h e a c o u s t i c p o w er i n p u t t o h i s c r y s t a l , so
t h a t no c a l c u l a t i o n o f
can b e made fro m h i s o b s e r v a ­
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A.
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E q n . 2 0 . ) o f S e c ti o n I I s t a t e s t h a t :
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to q u e s t i o n i s
g(-v) * 5 n,„ K's ’
th e l i n e - s h a p e f a c t o r , g ( v ) .
<ne t a k e
, t h e l i n e w i d t h a t h a l f maximum.
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i n a c a l c u l a t i o n o f T^ f o r F c e n t e r s
i n KC1 fro m s a t u r a t i o n m e a s u re m e n ts , u s e d a c o n s i d e r a b l y
s m a l l e r l i n e w i d t h th a n th e m e a s u re d v a lu e (o n e c a l c u l a t e d
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33
b y K l t t e l and A braham s
19.
) , r e a l i z i n g t h e im p o r ta n c e o f
h y p e r f i n e b r o a d e n in g i n t h e s a t u r a t i o n p r o c e s s .
H e re,
t h e l i n e - s h a p e f a c t o r e n t e r s th ro u g h t h e a t t e n u a t i o n c o ­
e ffic ie n t,
0" ; th e e f f e c t
o f h y p e r f i n e b r o a d e n in g i s to
d r a s t i c a l l y re d u c e t h e num ber o f s p i n s a b l e t o a c c e p t a
m o n o c h ro m a tic p h o n o n a t a n y tim e .
p l o y t h e m e a su re d v a l u e o f
v ±
c a l c u l a t i o n s w h ich i n v o l v e
(T .
H en ce, one s h o u l d em­
(1 0 0 g a u s s ) i n m ak in g a n y
The r e d u c t i o n i n t h e
num ber o f r e s o n a n t s p i n s i s th e r e b y a c c o u n te d f o r .
F o r a t w o - l e v e l s y s te m su c h a s t h a t o f F c e n t e r s i n
q u a r t z , th e p o p u l a t i o n d i f f e r e n c e p e r u n i t v o lu m e , n ,
is
g iv e n b y :
2 .)
n -
N_ - N +
,
w h ere N_ i s th e nu m b er o f s p i n s p e r u n i t volum e i n t h e
lo w e r e n e r g y s t a t e a n d N+ t h e number p e r u n i t v o lu m e i n
th e h ig h e r e n e rg y s t a t e .
tu re " i s
Ts
I f th e " e f f e c t i v e s p i n te m p e r a ­
, th e n
i
i
E q u i l i b r i u m c a n s u r e l y be assum ed b e tw e e n t h e s p i n s a n d
th e l a t t i c e ,
s in c e t h e r e w ere no o b s e r v a b le t r a n s i t i o n s ,
so t h a t Tg may be r e p l a c e d b y
4 .2 °K .
, th e l a t t i c e
te m p e ra tu re ,
Thus f o r a f r e q u e n c y o f 9 .3 6 7 G c, th e p h o n o n
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w ith o u t perm ission.
34
fre q u e n c y ,
4. )
n »
( 0 . i 0 3 ) M_
.
B u t v /ith H ■ S _ t N+ « 5 . 1
5 .)
( 1.8 9 7 ) N _
*
x 10
16
s p in a p e r c u b ic c e n t i m e t e r ,
5 . 1 x l O 16
,
or
t
6.)
Thus
M_ s
n * 2. 7 6
x IO 1*
2 . G S
.
x |Ol 5 .
The v a lu e f o r
o f 100 g au ss c o rre sp o n d s to a l i n e *
cps a t 1 0 G c.
8
v /id th o f 2 .8 x 10
n a l w aves i n X - c u t q u a r t z i s
The v e l o c i t y o f l o n g i t u d i 5
5 .7 x 10 c e n t i m e t e r s p e r
se c o n d , an d t n e a v e r a g e o f t h e l o n g i t u d i n a l a n d t r a n s v e r s e
5
v e l o c i t i e s , v a v g , i s 5 .2 x 10 c e n t i m e t e r s p e r s e c o n d .
The u p p e r l i m i t t o
r
, a s g i v e n i n S e c t i o n V, i s
n e p ers p e r c e n tim e te r .
fo r T
7 .)
0 .0 0 7 7
T h u s th e c a l c u l a t e d lo w e r l i m i t
is :
T,
£
&•
9
m illfa a c o n d s
.
The v a lu e i s p r o b a b l y a t l e a s t two o r d e r s o f m a g n itu d e
!
i
s m a ll e r t h a n t h e t r u e v a l u e .
T h e re i s no d e t e r m i n a t i o n o f
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission.
35
f o r P c e n te rs in q u a rtz in th e l i t e r a t u r e , b u t P c e n te r s
i n o t h e r c r y s t a l s u s u a l l y h a v e T ^ 's o f th e o r d e r o f one
s e c o n d a t 4 ,2 ° K .
T u c k e r 's c a l c u l a t e d v a lu e o f
in ru b y
was 500 m i l l i s e c o n d s , i n good a g re e m e n t w i t h some d e t e r m i ­
n a tio n s o f
B.
b y o t h e r m ean s.
C a lc u la tio n o f
E qn. 1 1 . ) o f S e c t i o n I I g i v e s
a s a fu n c tio n o f o ' :
or b e tte r,
9 .)
w h ere
W „M -
* ? " v
r\ h V
<£
5
■
t h e a c o u s t i c e n e rg y d e n s i t y .
j
The r f in p u t pow er t o th e c a v i t y was 10 w a t t s , a n d
t h e r f o u tp u t pow er fro m th e c a v i t y due t o th e f i r s t ech o
w as a b o u t lO "1^ w a t t s .
The a c o u s t i c pow er i n i t i a l l y i n
t h e h y p e r s o n ic sound p u l s e in d u c e d i n t h e q u a r t z r o d was
-4
t h u s 10
w a t t s . T h is c o r r e s p o n d s , a t 1 0 Gc, to a n e n e r g y
-2
d e n s i t y o f 1 .6 4 x 10
e r g s p e r c u b ic c e n t i m e t e r i n a
q u a r t z r o d o f 0 .1 in c h d i a m e t e r .
We a r e p e r m i t t e d to
u s e t h e g e o m e tr i c a l m ean o f t h e r f i n p u t and o u t p u t
p o w e rs s in c e l o s s e s i n t h e r o d a r e s m a l l .
T h at t h i s i s
t r u e c a n be s e e n fro m t h e l a r g e num ber o f e c h o e s i n F i g .
12
.
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36
The d e n s i t y o f q u a r t z i s 2 .6 5 gram s p e r c u b ic c e n t i ­
m e t e r , t h e v e l o c i t y o f l o n g i t u d i n a l w aves i n X - c u t q u a rtz
i s 5 .7 x 10
5
c e n t i m e t e r s p e r s e c o n d , a n d th e p o p u l a t i o n
X6
d i f f e r e n c e p e r u n i t v o lu m e, n , i s 5 .4 3 x 10
p e r c u b ic
c e n tim e te r.
T hus f o r
^
7 .7 x 10
n e p ers p e r c e n ti­
m e te r,
2
DM
^
* 1°
Pe r se co n d .
T h is i s th e t r a n s i t i o n p r o b a b i l i t y p e r u n i t t i m e , w hich
c a n b e g r e a t e r t h a n u n i t y , s i n c e i t i s o n ly v a l i d f o r an
e x tr e m e ly s h o r t t im e .
C.
S a tu ra tio n E ffe c ts
I f th e P c e n t e r s p i n s h a d b e e n s a t u r a t e d d u r i n g th e
e x p e r im e n t,
i t w o u ld h a v e b e e n I m p o s s ib le to make a m eaning­
f u l m e a su re m e n t o f th e a t t e n u a t i o n c o e f f i c i e n t .
One m ust
t h e r e f o r e d e m o n s tr a te t h a t th e s p i n s w ere n o t s a t u r a t e d
b y t h e a c o u s t i c e n e r g y i n th e h y p e r s o n i c w a v es.
We s h a l l
do t h i s b y c a l c u l a t i n g t h e maximum v a lu e w hich T^ c o u ld
j
i
h a v e s u c h t h a t t h e th e r m a l e q u i l i b r i u m o f th e s p i n s w ith
th e l a t t i c e w o u ld n o t h a v e b e e n s i g n i f i c a n t l y d i s t u r b e d
b y th e a c o u s t i c in p u t p o w e r.
S in c e t h e p u l s e r e p e t i t i o n r a t e
i s f a s t c o m p ared w ith
T1 , t h e s p i n s y s te m r e s p o n d s o n l y t o t h e a v e ra g e pow er
i n th e a c o u s t i c w aves
!
!
The p h o n o n f i e l d h a d a p o w e r i n p u t o f 10
.4
w a tts fo r
0 .5 m ic r o s e c o n d 60 tim e s a s e c o n d d u r i n g th e e x p e r im e n t.
T h is c o r r e s p o n d s t o a n e n e r g y p e r p u l s e o f :
I
I
J
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37
E s 5 x 1 0 "4 e r g s .
I f we assum e a r e a s o n a b l y
e x p o n e n t ia l d e c a y f o r th e e c h o e s ,
t h e a v e r a g e e n e r g y o f t h e a c o u s t i c w aves I s E 7 ^ -
, w here
i s th e tim e b e tw e e n s u c c e s s i v e m a g n e tro n p u l s e s and i s
much g r e a t e r t h a n
T
F o r t h i s e x p e rim e n t,
m ic r o s e c o n d s .
, t h e d e c a y tim e o f t h e e ch o t r a i n .
T* »
17 m i l l i s e c o n d s an d
'TT S ' 20
T hus t h e a v e r a g e e n e rg y o f t h e a c o u s t i c
w aves i s :
E s 5 .9 X 10
-7
e rg s.
The a v e r a g e num ber o f p h o n o n s i s th e n :
n
P
s _ J L ■ 9 ,5 x 10 .
hv
P
We h a v e a n e q u i l i b r i u m s i t u a t i o n w i t h i n t h e r o d ,
w i t h n ^ p h o n o n s a c t i n g u p o n a sy s te m o f N s p i n s w hich
can r e la x to a l a t t i c e
o f te m p e r a tu r e T ^
i n a tim e T ^ (
The p r o c e s s i s d e s c r i b e d b y t h e d i f f e r e n t i a l e q u a t io n :
w here N+ i s t h e e x c e s s o f s p i n s p e r u n i t volum e i n th e
h ig h e r e n erg y s t a t e
d u e t o e x c i t a t i o n b y t h e phonons o f
t h e h y p e r s o n i c w av e, a n d w h e re
i s th e t r a n s i t i o n
p r o b a b i l i t y p e r u n i t tim e t h a t a phonon w i l l c a u s e a
s p i n i n th e lo w e r e n e r g y s t a t e
t o make a t r a n s i t i o n t o
i
I
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
T
38
th e h ig h e r e n e rg y s t a t e .
We c a n g e t t h e s t e a d y - s t a t e
) s o l u t i o n t o t h i s e q u a t i o n by n o t i n g t h a t e q u i ­
l i b r i u m c o r r e s p o n d s t o d N ^ /d t s 0 .
1 1 .)
r
n+ / wdm
Thus
np - N_f/ T 1 , o r :
nP*
To b e s a f e , we c a n u s e t h e u p p e r l i m i t f o r W
d e riv e d
g
DM
fro m t h i s e x p e r i m e n t , 7 .9 x 10 . E q n . 6 . ) o f t h i s s e c t i o n
g i v e s t h e num ber N__ o f s p i n s p e r c u b ic c e n t i m e t e r i n th e
lo w e r s t a t e a t 4 .2 °K .
I f 10/i s a t u r a t i o n i s a ll o w e d , so
t h a t N,
s 0 .1 N - 2 .6 8 x 10^"® a c o u s t i c a l l y s a t u r a t e d
"’’max
~
s p i n s p e r c u b ic c e n t i m e t e r ,
T
i max
z
3 .7 3 x 1 0
3
seco n d s.
T h is i s t h e maximum T^ w h ich t h e F c e n t e r s c a n h a v e ,
u s i n g th e d e r i v e d u p p e r l i m i t t o t h e t r a n s i t i o n p r o b a b i l i t y
p e r u n i t tim e , a n d s t i l l r e m a in o n l y
a c o u s t i c pow er e m p lo y e d .
10%
s a t u r a t e d w i t h th e
I t I s t h e r e f o r e c o n c lu d e d t h a t
th e s p i n s y s te m c o u l d n o t h a v e b e e n s a t u r a t e d d u r i n g th e
e x p e rim e n t•
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
]
39
BIBLIOGRAPHY
1.
v a n V le c k , J . H ., P hys Rev 5 7 , 426 (1 9 4 0 )
2.
Mat t u c k , R . D ., a n d S t r a n d b e r g , M. W. P . , Fhys Rev
1 1 9 . 1204 (1 960)
3.
Bflmmel, H . E « , an d D r a n s f e l d , K ., F hya Rev L e t t e r s 1 ,
234 (1 9 5 8 )
4.
B0mmel, H . E . , an d D r a n s f e l d , K ., P hys Rev L e t t e r s 2 ,
298 (1 9 5 9 )
5.
J a c o b s e n , E . H ., Phys Rev L e t t e r s 2 , 249 (1 9 5 9 )
6.
J a c o b s e n , E . H ., Quantum E l e c t r o n i c s . C olum bia U n iv .
P r e s s , ( I 9 6 0 ) , p . 4&8
7.
J a c o b s e n , E . H ., S h l r e n , N. S . , a n d T u c k e r , E . B .,
Phys Rev L e t t e r s 3 , 81 (1 9 5 9 )
8.
S h l r e n , N. S . , and T u c k e r, E . B ., P h y s Rev L e t t e r s 6 ,
105 (1 9 6 1 )
9.
T u c k e r , E . B . , Phys Rev L e t t e r s 6 , 1 8 3 (1 9 6 1 )
1 0 . O rb a c h , R . L . , T h e s i s , U n i v e r s i t y o f C a l i f o r n i a (u n ­
p u b lis h e d ) '
11* G r i f f i t h s , J . H. E . , Owen, J . , a n d W ard , I . M«, N a tu re
1 7 3 . 439 (1 9 5 4 )
1 2 . O’B r i e n , M. C. M ., P ro c Roy Soc (L o n d o n ) A 231. 404
(1 9 5 5 )
------1 3 . G o u r a r y , B . S . , a n d A d r i a n , P. J . , P hys Rev 1 0 5 .
1180 (1 9 5 7 )------------------------------------------------------------ ----1 4 . Z im an, J . M ., E l e c t r o n s a n d P h o n o n s. O x fo rd U n iv .
P r e s s ( I 9 6 0 ) , C h a p te r I .
1 5 . BBmmel, H . E . , an d D r a n s f e l d , K ., P hys Rev 1 1 7 . 1245
(1 9 6 0 )
----1 6 . W h ite , G. K ., E x p e rim e n t a l T e c h n iq u e s i n L o w -T em p eratu re
P h y s i c s . O x fo rd U n iv . P r e s s ( 1 & 5 9 ) , p . 179
1 7 . S h i r e n , N. S . , Phys Rev L e t t e r s 6 , 1 6 8 (1 9 6 1 )
;i
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
1 8 . P o r t i a , A. M ., P hys Rev 91, 1071 ( 1 9 5 3 ) .
1 9 . M o tt, N. P . , a n d G u rn e y , R. W ., E l e c t r o n i c P r o c e s s e s
i n I o n i c C r y s t a l s . O x fo rd U n iv . P r e s s ( 1 9 5 ^ ) , p . 1 0 3 .
2 0 . K i t t e l , C ., a n d A b rah am s, £ . , Phys Rev 9 0 , 238 (1 9 5 3 )
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
ACKNOWLEDGMENTS
The a u t h o r w is h e s t o e x p r e s s h i s g r a t i t u d e to t h e
e n t i r e f a c u l t y a n d s t a f f o f th e P h y s ic s D e p a rtm e n t o f
R ic e U n i v e r s i t y f o r t h e i r c o - o p e r a t i o n a n d a s s i s t a n c e .
He I s p a r t i c u l a r l y I n d e b te d t o D r s . P . L . D onoho a n d Ben
J o s e p h s o n o f th e f a c u l t y , t o M r. C la r e n c e B e l c h e r f o r th e
p r o d u c t i o n o f l i q u i d h e liu m , a n d t o M r. E d g a r S u r l e s f o r
c o n s tr u c tio n o f th e a p p a r a tu s .
H is c o l l e a g u e s M r. R o b e rt
H e m p h ill a n d M r. M organ W ald ro p a r e a l s o d u e t h a n k s f o r
t h e i r a id .
D r. R o b e r t S h a l e k and M r. E a r l v a n n o o s e n b e e k o f M.
D. A n d e rso n H o s p i t a l h e lp e d i n i r r a d i a t i n g
th e q u a r t z
r o d s , a n d D r . M. G . E t t l i n g e r o f t h e R ic e C h e m is tr y
D e p a rtm e n t h e l p e d i n o b t a i n i n g t h e o p t i c a l a b s o r p t i o n
s p e c tra .
T h is r e s e a r c h w as s u p p o r te d b y th e N a t i o n a l A e ro ­
n a u tic s and S pace A d m in is tr a tio n .
Some o f t h e e q u ip m e n t
w as p r o v id e d b y t h e R o b e rt A . W elch F o u n d a t io n .
I
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
MAGNETIC
FIELD L IN E S
R EC TAN G U LA R
W A VEGUIDE
CO U PLIN G
ELEC TR IC
FIELD
HOLE
LINES
GAP
J
'
0 0
\
0 9®
ELECTRIC
© © ©
LIN ES
®
M AGNETIC
© © ©
C ENTER
F IE L D
* ♦ f M 0 4) ©
\II I I I / s
\
P O ST
2ZZZ2
F IE L D
L IN E S
R EC T A N G U LA R
W AVEGUIDE
FIG. I
reentrant
c y l in d r ic a l
ca v ity
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MILL FACE TO VARY
COUPLING HOLE
HOLES TO FIT X -B A N D
WAVEGUIDE FLANGE
FIG. 2
TEST CAVITY
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FIG. 3
SAMPLE
CAVITY
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I
=*
MAGNETRON PULSER
AND TR TUBE SUPPLY
a-
> *u .
o.i
..uuu.
ww'»w*wwnnr
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OSCILLOSCOPE
SAWTOOTH
GEN.
RECORDER
CIRCULATOR
MAGNETRON
IF
OC AMPLIFIER
ATTENUATOR
AMPLIFIER
INTEGRATOR
MAGIC
TEE
MATCHED LOAD
TR TUBE
ATTENUATOR
PRECISION
ATTENUATOR
I
^
IN 23E M R _^
CRYSTALS
ISOLATOR
KLYSTRON
CAVITY
MAGNET
FIG. 5
APPARATUS BLOCK OIAGRAM
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S Sn
^ uj
CDS
£T
-> o
<x u.
i—cn
< n z
>- <
(ro :
(0 O h­
s' a fo
n- UJz
t i
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0 - h2<
< 2
o o
^2 Z
<
>
AMUAIIM.IU.I*
mmtrnfm
<
KaH'
ymyvtm
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FIS. 7
REFLECTED
PULSE
FROM
CAVITY
I
ii
I
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n
BOXCAR
FIG. 8
I
IN T E G R A T O R
Ss
>
I
I'l—1
i
1
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PRESSURE WINDOW
WAVEGUIDE FLANGE
STAINLESS STEEL
WAVEGUIDE
TUNING SCREW
FILLING TUBE
TOP PLATE
N EED LE VALVE
PUMPING COLLAR
TO
MANOMETER
\_
HELIUM DEWAR
BALL
VALVE
STOPCOCK
<9
1—— BT
0
O
*IIU
NITROGEN
DEWAR
CAVITY
SAM PLE HOLDER
MAGNET
ON ROTATING TABLE
COPPER "HEAT LEAK1
TUNING
PLUNGER
FIG. 9
LOW TEMPERATURE APPARATUS
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I
FIG. 10
ILLUSTRATION OF THE EFFECT OF NON-PARALLEL
AND NON-FLAT ENDS
_
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
i
(B )
FIG. II
COMPARISON SHOWING EFFECT OF DISLOCATION
DENSITY ON ATTENUATION OF SOUND IN QUARTZ
( A ) BEFORE HANDLING, (B ) AFTER HANDLING
I
1
I
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FIG. 12
ECHO TRAINS IN 0 .2 2 " QUARTZ
WITH F CENTERS
ROD
i
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
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