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I c e i s a m a t e r i a l which n o r m a l l y e x i s t s a t a t e m -
p e r a t u r e c l o s e t o tk-,at a t which i t m e l t s . T h i s f a c t h a s t o be k e p t i n mind when a t t e m p t s a r e made t o c a l c u l a t e how i c e behaves u n d e r l o a d i n problems such as t h e p r e - d i c t i o n o f t h e b e a r i n g c a p a c i t y o f a n i c e s h e e t o r t h e p r e s s u r e s which i c e c a n e x e r t on c l a m s , b r i d g e p i e r s and o t h e r e n g i n e e r i n g s t r u c t u r e s . I n r e c e n t y e a r s , t h e m e c h a n i c a l b e h a v l o u r o f m a t e r i a l s I n t h i s t h e r m a l s t a t e h a s been r e c e l v l n g greater a t t e n t i o n , b u t t h i s knowledge h a s n o t d e v e l o p e d t o t h e s t a t e where t h e m e c h a n i c a l be- h a v i o u r o f i c e c a n b e p r e d i c t e d a c c u r a t e l y w i t h c o n f i d e n c e .
The Snow nnd I c e S e c t i o n o f t h e D i v i s i o n o f B u i l d i n g R e s e a r c h i s c o n c e r n e d w i t h making a v a i l a b l e t o Canadian s c l e n t i s t s and e n g i n e e r s Itnowledge on hoW t h i s u n i q u e m a t e r i a l behaves when i t i s s u b j e c t t o s t r e s s , and how
t h i s b e h a v i o u r depends o n t h e t e m p e r a t u r e , r a t e o f l o a d i n g , geometry o f loacllng, s t r u c t u r e and o t h e r v a r i a b l e s . T h i s p a p e r , t r a n s l a t e d from t h e German, g i v e s I n f o r m a t i o n on how t h e modulus o f e l a s t i c i t y o f s i n g l e i c e c r y s t a l s de- pends on t h e t e m p e r a t u r e , f r e q u e n c y o f t h e s t r e s s and c r y s t n l l o ~ ~ a ~ h i c o r i e n t a t i o n 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 t h e s t r e s s . The D i v i s i o n r e c o r d s i t s a p p r e c i a t i o n f o r t h i s t r a n s l a t i o n t o M r . D . A . S i n c l a i r . O t t a w a May 1960 R.F. L e g g e t D i r e c t o r
NATIONAL RESEARCH COUNCIL
OF
CANADA Technical T r a n s l a t l o n 890 M e c h a n i c a l r e l a x a t i o n i n p u r e m o n o c r y s t a l s o f i c e ( D i e m e c h a n i s c h e R e l a t i o n * i n r e l n e n E i s e i n k r i s t a l l e n ) P. S c h i l l e r Z. P h y s i k , 1.53: 1-15, 1958 A u t h o r : R e f e r e n c e : T r a n s l a t o r : C.A. S l n c l a i r , T r a n s l a t i o n s S e c t i o n , N.R.C. L i b r a r y*
T r a n s l a t o r ' s Note: Assumed t o be a m i s p r i n t f o r " R e l a ~ a t l o n ~ ~ .MECI-IAIIIChL RELAXAT I O N I N I-'URE FIONOCRYSTAL OF ICE ---.
A b s t r a c t
U s i n c t h e n a t u r a l v t b r a t i o n s o f combined r e s o n a t o r s
a rnecl~unic;ll r e l a x a t i o n i s investigated i n monocry st;als
o f i c e i n t h e f r e q u e n c y r a n g e between 8CO and 6000 c p s and t h e t e m p e r a t u r e I n t e r v a l between O 0 and - l O O ° C . The phenomenon (Iepends on t h e k i n d o f o s c l l l a t i o l l a n d t h e c r y s t z i l l o g r a ~ ~ h l c o r i e n t a t i o n . I t c a n be r e g a r d e d as a t r a n s f o r m a t i o n produced by t h e d : . s t r i b u t i o n o f de- f l c i e n c i e s i n t h e P a u l i n e c n n f i g u r a . t i o n s .
I n t r o d u c t i o n
D i e l e c t r i c r e l a x a t i o n phenomena i n i c e have a l r e a d y been known f o r some time''). I n r e c e n t y e a r s t h e e a r l y l n v e s t l g a t i c n s h a v e been supplemented and e x t e n d e d t o c o v e r heavy i c e , e.g. by A ~ l t y and C o l e ( 2 ) , CrAnicher
(''I
and ~ d e r ( " ) . To g e t R c l e a r p i c t u r e of t h ep r o c e s s e s i t a p p e a r e d o b v i o u s t o a p p l y m e c h a n i c a l methods of i n - v e s t i g a t i o n . A p i o n e e r p a p e r ( 5 ) produced e n t i r e l y s a t i s f a c t o r y r e s u l t s , even thou& i t w a s a l r e a d y a p p a r e n t h e r e t h a t 1.n t h e m e c k ~ a n i c a l c a s e t h e e f f e c t i s d e f i n i t e l y s m a l l e r t h a n i n t h e d i - e l e c t r i c e x p e r i ~ e n t s . I n t h e p r e s e n t p a p e r a n a t t e m p t w i l l be made t o c o n t r i b u t e t o t h e c l a r i f i c a t i o n o f t h e phenomena by p r e s e n t i n g as e x t e n s i v e e x p e r i m e n t a l e v i d e n c e a s p o s s i b l e . With t h i s In view m e c l ~ a n l c a l r e l a x a t i o n h a s been s t u d i e d a s a f u n c t i o n of t e m p e r a t u r e , f r e q u e n c y , ~ r y s t a . l l o ~ a p l 7 i C o r i e n t a s t i o n 2nd t y p e o f d e f o r m a t i o n . The v a l u e s f o r c?.ctlva.tion e n e r g y and r e l a x a t i o n time o f t h e p r o c e s s a p - e e w i t h i n t h e l i m i t s o f m e a s u r i n g a c c u r a c y w i t h t h e r e s u l t s o f t h e e l e c t r i c a l i n v e s t i g a t i o n s . The e x t e n t o f t h e m e c h a n i c a l e f f e c t h a s been c n l c u l ~ ~ t e d by
ass'^),
u s i n g a thermodynamic method, and shows good agreement w i t h t h e e x p e r i m e n t s . The t e m p e r a t u r e and f r e q u e n c y dependence o f t h e l o g a r i t h m i c d e c r e m e n t , t h e most s e n s i - t i v e o f t h e measured v a l u e s , obey, i n f i r s t a p p r o x i m a t i o n , t h e l a w s which h o l d f o r a s i m p l e r e l a x a t i o n p r o c e s s . A more c a r e f u la n a l y s i s o f t h e t e s t r e s u l t s , however, c a l l s f o r a c o r r e c t i o n which w i l l be a t t e m p t e d a t t h e end o f t h e p a p e r . A p p a r a t u s The r e s u l t s of p r e v i o u s i n v e s t i g a t ; i o r l s ( 5 ) s u g g e s t e d t h e u s e o f t h e n a t u r a l f r e q u e n c i e s o f combined r e s o n a t o r s f o r t h e measurement o f t h e r e l a x a t i o n phenomena i n i c e ( ~ i g . 1). I n t h i s way i t i s p o s s i b l e t o s c a n t h e e n t i r e f r e q u e n c y i n t e r v a l i n which we were I n t e r e s t e d . The e x c i t a t I o n and measurement o f t h e n a t u r a l f r e - q l i e n c l e s o f t h e s e r e s o n n t o r s were c a r r i e d o u t e l e c t r o s t a t i c a l l y . Tile method d e s c r i b e d by Bordoni and Nuovo (7) p r o v e d e s p e c i a l l y
s11it;able for* t h i s . The l o g a r i t h m i c d e c r e m e n t , d e p e n d i n g on i t s
v a l u e , was d e t e r m i n e d from t h e damping t i n e o f t h e r e s o n a t o r o s c i l - l a t i o n o r from i t s r e s o n a n c e w i d t h .
F o r c a l c u l a t i n g t h k n a t u r a l f r e q u e n c i e s o f t h e o s c i l l a t o r s shown i n F i g . 1 t h e f o l l o w i n g a p p r o x i m a t i o n g i v e s good r e s u l t s f o r t h e d i m e n s i o n s i n v o l v e d . The r e s o n a t o r i s assumed t o c o n s i s t o f two w e i g h t s J o i n ~ . d by a s p r i n g o f z e r o muss. The r o l e of s p r i n g i s p l a y e d by t h e bar of i c e i n t h e c e n t r e of t h e r e s o n a t o r . The s p r i n g c o n s t a n t i s d e t e r m i n e d from t h e g e o m e t r i c and e l a s t i c d a t a o f t h e i c e . I n t h e c a s e o f a f i n i t e i n t e r n a l damping t h e phenomenon c a n b e d e s c r i b e d by a complex modulus o f t h e i c e . I n t h e s e c a l c u l a t i o n s , o f c o u r s e , t h e a n i s o t r o p y o f t h e e l a s t i c c o n s t a n t o f t h e i c e must be talcen i n t o a c c o u n t . U s i n g Voigt i n d i c e s f o r t h e e l a s t i c i t y v a l u e s , t h e f o l l o w i n g e x p r e s s i o n s a r e o b t a i n e d f o r t h e n a t u r a l f r e q u e n c y of t h e r e s o n a t o r s , d e p e n d i n g on t h e o r i e n t a t i o n o f t h e a x i s i n t h e p i e c e o f i c e and d e p e n d i n g on t h e t y p e o f o s c i l l a t i o n : F o r l o n g i t u d i n a l o s c i l l a t i o n s o f r e s o n a t o r s i n which t h e i c e s e c t i o n s h a v e a c y l i n d e r a x i s ( a ) p a r a l l e l t o t h e h e x a g o n a l ax- of t h e i c e :
( b ) p e r p e n d i c u l a r t o t h e h e x a g o n a l a x i s o f t h e i c e :
S l m l l u r l y , f o r t o r s i o n a l o s c i l l a t i o n s :
( c ) c y l l n ~ ~ r l c a l and h e x a g o n a l a x e s p a r a l l e l :
( d l c y l i n d r l c a l and h e x a g o n a l a x e s m u t u a l l y . p e r p e n d i c u l a r :
I n t h e s e e x p r e s s i o n s
m
and @ are t h e mass and moment o f I n e r t i a , r e s p e c t i v e l y , of t h e t e r m i n a l m a s s e s , a i s t h e r a d i u sand h t h e l e n g t h of t h e p i e c e o f i c e , w h i l e
silt
and cik a r e t h e e l a s t i c c o n s t a n t s and e l a s t i c moduli of t h e I c e , r e s p e c t i v e l y .E x p e r i m e n t s
Mat-erlal and a c c u r s c y
-
The l o g a r i t h m i c decrement and t h e f r e q u e n c y of r e s o n a t o r s o f v a r i o u s g e o m e t r i c d i m e n s i o n s were measured as a f u n c t i o n o f t h e t e m p e r a t u r e between 100° and 273OK. The a x e s o f t h e i c e c y l i n d e r were o r i e n t e d p a r a l l e l l y and p e r p e n d i c u l a r l y t o t h e h e x a g o n a l a x i s . T o r s i o n a l and l o n g i t u d i n a l o s c i l l a t i o n s were i n v e s t i g a t e d i n b o t h o r i e n t a t i o n s . A t d i f f e r e n t t i m e s s e v e r a l f r e q u e n c i e s were employed on t h e same p i e c e s o f i c e , t h e frequenc'jr b e i n g v a r i e d by c h a n g i n g t h e g e o m e t r i c a l d i m e n s i o n s o r by i n t e r c h a n g i n g t h e end w e i g h t s . The o v e r a l l f r e q u e n c y t n t e r v a l scanned w a s from 800 c p s t o 6 kcps. The c r y s t a l s employed were from t h e s t o c k o f t h e i n s t i t u t e . I n p o l a r i z e d l i g h t no p a i n b o u n d a r i e s c o u l d be f o u n d ' l n t h e i c e
employed. The c o n d u c t i v i t y o f t h e w a t e r o f m e l t i n g w a s
- 1 - 1
x = 2 1 0 - ~ 5 2 cm
.
The p i e c e s o f i c e c o u l d b e o r i e n t e d t o a na c c u r a c y o f ?lo. The a c c u r a c y o f t h e f r e q u e n c y measurement. was t h a t o f t h e damplng measurement b e t t e r t h a n 5 5 . R e s u l t s F i g . 2 shows a t y p i c a l p l o t o f t h e l o g a r i t h m i c decrement w i t h t h e t e m p e r a t u r e . T h e r e i s a marked maximum of t h e c u r v e , t h e p o s l - t i o n o f which v a r i e s w i t h t h e f r e q u e n c y o f t h e o s c i l l a t i o n w h i l e i t s h e i g h t depends on t h e k i n d of o s c l l l n t i o n and t h e c r y ~ t n l l o g r a p h -
i c o r l e r ~ t a t i o n . The v e r y low r e s i d u a l damping beyond t h e maximum I s due i n p a r t t o t h e m a t e r i a l i t s e l f and i n p a r t t o t h e a i r . The a i r damping i n t h e c a s e of l o n g i t u d l n u l o s c i l l a t i o n i s a p p r o x i m a t e l y of t h e o r d e r o f
T h e v a l u e s from a l l t h e measurements a r e a s s e m b l e d i n T a b l e s I and 11.
R e s o n a t o r no. 1 2 was n o t s y m m e t r i c a l , b u t had two d i f f e r e n t end w e i g h t s . A l l t h e o t h e r r e s o n a t o r s were s y m m e t r i c a l l y d e s i g n e d and t h e v a l u e s f o r t h e mass o r t h e moment of i n e r t i a , as t h e c a s e may
b e , a p p l y t o one o f t h e end we:ghts. The ,a' a n d h v a l u e s a p p l y t o
t h e r a d i u s and t h e l e n g t h of t h e i c e s e c t i o n , r e s p e c t i v e l y .
The p o s i t i o n of t h e damping maximum w a s d e t e r m i n e d g r a p h i c a l l y f r o n the. measured c u r v e s . The t e m p e r a t u r e f i g u r e s o f t h e maximum t h e r e f o l - e do n o t p o s s e s s t h e a c c u r a c y o f t h e o t h e r measured v a l u e s . The e r r o r i s a b o u t 20.5OC. The maximum o f t h e l o n g i t u d i n a l
o s c i l l a t i o n s p a r a l l e l t o t h e c a x i s i s however c o n s i d g r a b l y s m a l l e r t h a n t h e o t h e r maxima, and t h e r e f o r e t h e d a t a o n i t s t e m p e r a t u r e
-
i s e v e n more I n a c c u r a t e . The p o s i t i o n of t h e s e maxima w a s t h e r e f o r e n o t t a k e n i n t o a c c o u n t and o n l y t h e i r h e i g h t i s a r e e n t e r e d i nT a b l e 111. A c r i t i c a l - e x m i n a t i o n o f T a b l e s I and I1 l e a d s t o tihe f o l l o w i n g c o n c l u s i o n s :
1. The h e i g h t o f t h e damping maximum depends on t h e o r l e n t a - t l o n . C o n s i d e r , t h e measurements o f b a r s no. 1 7 and 1 9 ( F i g . 3 ) .
Both h a v e a l o n g i t u d i n a l r e s o n a n c e a t a b o u t 2.9 kcps. Now,, bar
no. 1 7 i s p e r p e n d i c u l a r t o t h e c a x i s I n t h e c r y s t a l and bar no. 19 I S p a r a l l e l t h e r e t o . G m a X . i s 24
l o m 3
f o r no. 17 and 4f o r no. 19. The d i f f e r e n c e f o r t h e t o r s i o n a l oscillations i s some- what l e s s ( 1 0 % ) .
2. The h e i g h t o f t h e damping maxima v a r i e s w i t h t h e t y p e o f o s c i l l a t i o n . A s a n example w e may c i t e t h e measurements o f no. 13
and 1 7 . F o r similar c r y s t a l l o g r a p h i c o r i e n t a t i o n t h e t o r s i o n a l o : ; c I l l a t I o n 1 s somewhat more s t r o n g l y damped t h a n t h e l o n g i t u d i n a l . E s p e c i a l l y m a r k e 4 i s t h e d i f f e r e n c e f o r r e s o n a t o r s p a r a l l e l t o t h e c a x i s . The o b s e r v a t i o n s made a t p o i n t s 1 and 2 a r e a s s e m b l e d i n
T a b l e 111. 3 . W i t h i n t h e l i r n $ t s o f m e a s u r i n g a c c u r a c y t h e h e i g h t o f t h e maximum i s i n d e p e n d e n t o f t h e t e m p e r a t u r e . I f t h e r e i s a n y depend- 1 e n c c a t a l l t h e n i t must be s m a l l e r t h a n - T o r N - T * 4. No m e a s u r a b l e r e l a t i o n s h i p between t h e f r e q u e n c y and t h e h e i g h t o f t h e maximum was found.
5. I n F i g . 4 t h e l o g a r i t h m o f t h e f r e q u e n c y i s p l o t t e d a g a i n s t 1 / ~ i n d e p e n d e n t l y o f the k i n d o f o s c i l l a t i o n and t h e o r i e n t a t i o n . W i t h i n t h e l i m i t s o f a c c u r a c y a l l p o l n t s l i e on a s t r a i g h t l i n e . From a smoothing c a l c u l a t i o n we o b t a i n t h e c o n s t a n t s o f t h e c o r r e s - p o n d i n g A r r h e n i u s e q u a t i o n f = foexp.[
-
&i ] a s : f o = (5.3 t 4)1014 s e c-
1.
6. I n a d d i t i o n t o t h e t e m p e r a t u r e dependence o f t h e l o g a r i t h m i c-
d e c r e m e n t , t h e dependence on t h e n a t u r a l f r e q u e n c y w a s a l s o s t u d i e d f o r some r e s o n a t o r s . These i n v e s t i ~ a t i o n s were r a t h e r d i f f i c u l t , s i n c e t h e bar d i m e n s i o n s , and h e n c e t h e n a t u r a l f r e q u e n c i e s , v a r i e d u n c o n t r o l l a b l y as t h e r e s u l t o f e v a p o r a t i o n . F i g . 5 shows t h et e m p e r a t u r e . I t w i l l b e n o t e d t h a t i n t h e r e g i o n of the damping maximum a d i s ~ e r s i o n s t e p i s superimposed on t h e normal t e m p e r a t u r e dependence o f t h e n a t u r a l f r e q u e n c y o r o f t h e modulus. The, s o l i d c u r v e s i n I : i , g . 5 h o l d f o r a s i m p l e r e l a x a t i o n p r o c e s s and confol-m
t o t h e f u n c t i o n s
and
I n computing them, t h e v a l u e s f o r Q , f T and Smax were i n s e r t e d . The measured p o i n t s a g r e e w e l l w i t h t h e c u r v e s . I t must be p o i n t e d o u t , however, t h a t a s m a l l , s y s t e m a t i c d e v i a t i o n became n o t i c e a b l e i n t h e d i r e c t i o n o f low t e m p e r a t u r e s , t h e measurement v a l u e s i n g e n e r a l on t h e low t e m p e r a t u r e s i d e o f t h e maximum b e i n g s i t u a t e d below t h e v a l u e s vrhlch would be e x p e c t e d a c c o r d i n g t o e q u a t i o n (1) f o r t h e s i m p l e r e l a x a t i o n p r o c e s s ( F i g . 9 ) . 7. Because o f t h e l a r g e nurnber o f r e s o n a t o r s i n v e s t i g a t e d i t i s p o s s i b l e t o p l o t t h e l o g a r i t h m i c decrement as a f u n c t i o n o f t h e f r e q u e n c y . F i g . G shot,s t h e s t ~ r n c l a r d i z e d v a l u e s o f t h e v a r i o u s r e s o n a t o r s f o r t = - 1 7 O C , Here, a g a i n , t h e s o l i d c u r v e c o r r e s p o n d s t o t h e v a r i a t i o n i n a s i m p l e r e l a x a t i o n p r o c e s s ( F i g . 1). To sum up: ( a ) There i s a r e l a x a t i o n p r o c e s s i n i c e which i s e x p r e s s e d as a maximum o f t h e dumping and n d i s p e r s i c n o f t h e modulus,
( b ) The magnitude o f t h e e f f e c t depends on t h e c r y s t a l l o g r a p h i c L : r * l e n t a t i o n and o n t h e - t y p e o f d e f o r m a t i o n .
( c ) W i t h i n the measubing r a n g e i t i s i n d e p e n d e n t of f r e q u e n c y and t e m p e r a t u r e .
( d ) The p r e c e s s c a n b e d e s c r i b e d i n good a p p r o x i m a t i o n by a s i m p l e r e l a x a t i o n p r o c e s s w i t h a s i n g l e r e l a x a t i o n t i m e , w h i c h d e - p e n d s on t h e t e m p e r a t u r e a c c o r d i n g t o a n A r r h e n i u s e q u a t i o n . ( e ) The c o n s t a n t s o f t h e A r r h e n i u s e q u a t i o n a r e : Q = 1 3 . 4 k c a l = 0.58 e v ; To = 3 1 0 - l 6 sec. R e l a x a t i o n Time The s t r u c t u r e o f t h e
ice
The p o s i t i c n o f t h e oxygen a t o m s c a n be d e t e l n ~ l n e d d i r e c t l y i n i c e by means o f X-ray methods. The oxygen a t o m s form a h e x a g o n a l l a t t i c e o f t h e k i u r t z i t e t y p e . I n t h i s arrangement; e a c h o x y g e n atom i s s u r r o u n d e d , i n good a p p r o x i m a t i o n , t e t r a h e d r i c a l l y by i t s f o u r n e a r e s t n e i g h b o u r s . The X-ray s t u d i e s c a n g i v e n o i n f o r m a t i o n con- c e r n i n g t h e p o s l t i ~ n of. t h e p r o t o n s i n t h i s l a t t i c e e l e m e n t ,V a r i o u s o t t - e r s t r u c t ~ l r a l i n v e s t i g ~ t i c n s , u s i n g t h e r e f r a c t i o n o f n e u t r o n and e l e c t r o n beams s u r g e s t a n arr-angement o f t h e p r o t o n s o f t l , e k i n d s u g g e s t e d b y P a u l i n g 'lo) a n d B e r m 1 a n d F o w l e r (11) I t ctrn b e s t b e described by t h e f o l l o w i n g f o u r r u l e s . 1. The p r o t o n s are s i t u a t e d on t h e l l n e s c o n n e c t i n g t h e n e a r e s t n e i g l ~ b o u r l n g o x y g e n atoms. 2. T h e r e are -two s t a . b l e p o s i t i o n s o f t h e p r o t o n s a l o n g t h e s e c o n n e c t i n g 1 l n c s. 3. On e a c h o f t h e s e c o n n e c t i n g l i n e s t h e r e i s o n l y o n e p r o t o n . 4. On t h e f o u r c o n n e c t i n g l l n e s b e l o n g i n g t o a s i n g l e oxygen atom t h e f o u r p r o t o n s
a r e
s o d i s t r i b u t e d t h a t on two o f t h e l i n e s t h e y t a k e t h e p o s i t i o n n e a r e s t t h e atom i n q u e s t i o n and o n t h e O t h e r two t h e p o s i t i o n r e m o t e s t t h e r e f r o m . I n c o n s l d e ~ ~ a t l o n o f t h e s e r u l e s A m u l t i p l i c i t y o f d i f f e r e n t c o n f i g u r a t i o n s c a n be c o n s t r u c t e d . C o m h i n a t o r i a l c o n s i d e r a t i o n s rJ y i e l d ( 3 / 2 ) p o s s i b l e c o n f l g u r n t i o n s f o r N m o l e c u l e s . The r e s u l t - i n g v a l u e o f 0.82 c r i l p e r mole O C f o r t h e z e r o p o l r ~ t e n t r o p y i s a l s o d e r i v e d from t h e m e a s u r e m e n t s ( 1 2 ).
T h l s a g r e e m e n t i s a b a s i cconfirmation o f t h e model. I n a n i d e a l c r y s t a l , however, t h e
s i n c e I n such a c a s e t h e p r o t o n s on a r i n g of a t l e a s t s i x m o l e c u l e s would have t o t r a n s f e r s i m u l t a n e o u s l y from o n e p o s i t i o n t o a n o t h e r . A p a r t from t h e f a c t t h a t t h i s r i n g would h a v e t o s a t i s f y c e , r t a i n c o n d i t i o n s f o r t h e I n i t i a l p o s i t i o n , such a c o l l e c t i v e jump would r e q u i r e a v e r y h i g h a c t i v a t i o n e n e r g y , and i s t h e r e f o r e h i g h l y
improbable. However, when we t a k e i n t o a c c o u n t t h e d e f e c t s p r e s e n t i n e a c h c r y s t a l i t i s q u i t e a s i m p l e m a t t e r t o c o n s t r u c t p r o c e s s e s which c o u l d l e a d t o a t r a n s f o r m a t i o n o f t h e c o n f i g u r a t i o n . A s l a t t i c e d e f e c t s s u i t a b l e f o r o u r problem t h e r e a r e i n f r a c t i o n s o f t h e a b o v e - c i t e d c o n d i t i o n s f o r t h e p r o t o n d i s t r i b u t i o n (13) An i n f r a c t i o n of Rule 3 l e a d s t o a 0-0 c o n n e c t i n g l i n e o c c u p i e d e i t h e r by two p r o t o n s , D - d e f i c i e n c i e s
,
o r t o d e f i c i e n c i e s l a c l r i n g t h e p r e s c r i b e d p r o t o n , L - d e f i c i e n c i e s . Both d e f i c i e n c i e s a r e p r e s e n t i n e q u a l c o n c e n t r a t i o n . They a r i s e by r e a s o n o f a p r o t o n jumping from t h e l i n e a,ssigned t o i t t o a c o n n e c t i n g l i n e of one o f t h e n e i g h b o u r i n g o w g e n atoms. A s a r e s u l t a n L - d e f i c i e n c y r e m a i n sa t t h e abandoned p o s i t i o n and a D - d e f i c i e n c y a r i s e s a b o u t t h e new p o s i t i o n . P i g . 7 a i l l u s t r a t e s how such p r o t o n jumps may produce a
d i f f u s i o n o f t h e s e two deficiencies. It i s a p r c g r e s s i o n of t h e p a r t i c l e s from one p o t e n t i a l w e l l t o t h e n e x t , a s h a s been d e a l t w i t h t h e o r e t i c a l l y i n v a r i o u s p a p e r s (14'15). The jumping o f a
p r o t o n a l o n g i t s 0-0 c o n n e c t i n g l i n e produces two d e f i c i e n c i e s which c o r r e s p o n d t o a n i o n i z a t i o n o f t h e oxygen atoms j o i n e d by t h e l i n e . Moreover, t h e s e v a c a n c i e s c a n be d i f f u s e d by a r e p e t i t i o n o f t h e
same p r o t o n jump ( P i g . 7 b ) . Common t o b o t h t y p e s o f d e f i c i e n c y i s
t h e f a c t t h a t t h e y l e a v e b e h i n d them m o l e c u l e s which t h e y h a v e touched i n t h e c o u r s e o f t h e i r d i f f u s i o n i n a changed p o s i t i o n , ,i .e
.
,
R c o n f i g u r a t i o n t r a . n s f o r m a t i o n r e s u l t s from t h e i r m i g r a t i o n .R e l a x a t i o n time
According t o Bass t h e c h e m i c a l - r e l a x a t i o n c a n be a t t r i b u t e d t o a change o f c o n f i g u i - a t i o n . The & p e e m e n t between t h e r e s u l t s of t h e m e c h a n i c a l measuremerlt wit11 r e s p e c t t o a c t i v a t i o n e n e r g y and t h e f r e q u e n c y o f t h e p r o c e s s w i t h t h e v a l u e s o f t h e d i e l e c t r i c
be a s s o c i a t e d w i t h a p r o c e s s which would a l s o e x p l a i n t h e d i e l e c t r i c phenomena t h a t h a v e b e e n d e s c r i b e d . The d i f f u s i o n o f t h e D- and L-deficiencies a r e foremost among t h e s e , s i n c e t h e y r e s u l t ' i n a
r o t a t i o n of t h e m o l e c u l a r d i p o l e t o w a r d s t h e f i e l d . F u r t h e r , i t
w i l l now be assumed t h a t t h e r a t e o f d l f f u s i o n of t h e L - d e f i c i e n c i e s i s c o n s i d e r a b l y l e s s t h a n t h a t o f t h e D - d e f i c i e n c i e s , s o t h a t now we may c o n c e n t r a t e on t h e l a t t e r . Such a r e s t r i c t i o n appe:Ws r e a s o n - a b l e , s i n c e t h e e f f e c t o f a n L - d e f i c i e n c y on t h e p o t e n t i a l o f t h e p r o t o n s I n v o l v e d i n a c o n t i n u e d d i f f u s i o n n u s t c e r t a i n l y b e l e s s , owing t o t h e i r g r e a t e r s e p a r a t i o n , t h a n t h e p o t e n t i a l d i s t u r b a r l c e which 1;he two p r o t o n s of t h e D - d e f l c i c n c y e x p e r i e n c e by r e a s o n o f t h e i r c l o s e n e s s t o e a c h o t h e r . T h i s means t h s t t h e p r o b a b i l i t y o f a Jump, and hence t h e d i f f u s i o n c o n s t a n t s , d i f f e r p e a t l y from e a c h o t h e r . L e t t h e mean d i f f u s i o n d i s t a n c e o f t h e D - d e f i c i e n c y from g e n e s i s t o a n n i h i l a t i o n be d m I t i s now p o s s i b l e t o d i v i d e u p t h e c o r i f i g u r - a t i o n t r a n s f o r m a - t i o n i n t o s t a t i s t i c a l l y i n d e p e n d e n t p r o c e s s e s . These p r o c e s s e s are i d e n t i c a l w i t h t h e h i s t o r i e s o f t h e d e f i c i e n c i e s , i . e . , t h e y are p r o c e s s e s made up o f t h e g e n e s i s o f t h e d e f i c i e n c y , i t s d i f f u s i o n t h r o u g h t h e c r y s t a l and i t s a n n i h i l a t i o n by a r e c l p r o c a l d e f i c i e n c y . The r e l a x a t i o n t i m e i s a measure o f t h e time r e q u i r e d f o r t h e forma- t i o n o f a d e f e c t a t a n i n ( l i v i d u a 1 m o l e c u l e and i t s a n n i h i l a t i o n by d l f f u s i o n . Thus the r e l a x a t i o n time i s d e t e r m i n e d f o r s m a l l
d i f f u s i o n times ( i . e . , t i m e s which c o r r e s p o n d t o h i g h r a t e s o f d i f - f u s i o n and small d i s t a n c e s ) b a s i c a l l y by t h e f i r s t p r o c e s s , l e e . , t h e f o r m a t i o n o f t h e d e f i c i e n c y . I n o t h e r words, i n t h e l i m i t i n g c a s e w e h a v e a s1mpl.e r e l a x a t i o n p r o c e s s and t h e d i s t a n c e o f t h e ' d l f f u s i o n i s talren i n t o a c c ~ u n t o n l y as a c o r r e c t i o n . I n t h e model t l l i s i s clone by I n t r o d u c i n g n d i f f u s i o n d i s t a n c e o f l e n g t h d between t h e two p o t e n t i a l w e l l s f o r which a s i m p l e r e l a x a t i o n p r o c e s s i s d e s c r i b e d ( F i g . 8a a n d b ) .
The model may t h u s b e d e s c r i b e d n s f o l l o w s : I n t h e c r y s t a l t h e r e a r e N/dt o f t h e above-mentioned d i f f u s i o n d i s t a n c e s , where
d ' Is t h e number o f l a t t i c e p o i n t s s i t u a t e d a l o n g t h e p a t h d . A l l
expanded model. I f t h e y a l l r u n i n o n e d i r e c t i o n t h e n t h e e n t i r e c r y s t a l 1s f o l d e d o v e r . The particle^^^ which c o r r e s p o n d t o t h e expanded model and which l1,e i n t h e two p o t e n t i a l w e l l s
are
t h u s t h e i n d i v i d u a l d i f f u s i o n p r o c e s s e s . The t r a v e r s i n g o f such a d i f - f u s i o n d i s t a n c e b r i n g s a b o u t t h e f o l d i n g o v e r o f t h e d ' m o l e c u l e s l y i n g b e t w e e n them i n t o a n o t h e r o f t h e p o s s i b l e p o s i t i o n s i n t h e i c e l a t t i c e c o n f o r m i n g t o t h e P a u l l n g model. The expanded t r u n s p o s l t i o n m o d a J u s t as i n t h e s i m p l e t r a n s p o s i t i o n n o d e l (see f o r example r e f e r e n e e (18) the eeeupaeien n u m k a ~ s a f Cha ewe )lelsl?r and t h a l rchange w i t h time u n d e r a n e x t e r n a l f o r c e can a l s o b e c a l c u l a t e d f o r t h i s expanded model. I n d o i n g s o , i t must a l w a y s b e borne i n mind
t h a t t h e " p a r t i c l e s " now r e p r e s e n t t h e i n d i v i d u a l d i f f u s i o n d l s - t a n c c s , They b e g i n w i t h t h e f a c t t h a t a t some l a t t i c e p o i n t o r
o t h e r a d e f e c t a r i s e s which b e g i n s t o migrate t h r o u g h t h e c r y s t a l u n t i l f i n a l l y i t i s a n n i h i l a t e d a t a d e f e c t o f o p p o s i t e t y p e .
The p r o b n b l l l t y f o r t h e origination o f a d e f e c t d i s t a n c e i s wo and d e p e n d s on t h e t e m p e r a t u r e a c c o r d i n g t o a n e x p o n e n t i a l f u n c t i o n o f t h e forrn A s i n t h e s i m p l e c a s e , i t i s i n f l u e n c e d by a n e x t e r n a l f i e l d and f o r a p o t e n t i a l v a r i a t i o n q i t h a s , I n f i r s t a p p r o x i m a t i o n , t h e form The d i f f u s i c n problem a s s o c i a t e d w i t h t h e f o r m a t i o n o f t h e d e f i c i e n c y c o r r e s p o ~ l d s t o h e a t c o n d ~ c t i o n i n a unilaterally, i n f i n i t e l y c x t e n d - ed medium, I f w e a t t e m p t , a s u s u a l , t o draw up t h e o c c u p a t i o n b a l a n c e s h e e t o f t h e two h o l e s , w e must f i r s t t a k e i n t o a c c o u n t t h e " p a r t i c l e s " which l e a v e t h e h o l e . The o t h e r h o l e a c t s as i f w e w e r e i n a u n i l : r t e r a l l y , i n f i n i t e l y e x t e n d e d medium t h r o u g h t h e s u r f a c e o f which p a r t i c l e s a r e c o n t i n u a l l y b e i n g i n t r o d u c e d which
t h e n b e g i n t o d i f f u s e . T h i s f l u x o r i g i n a t i n g from t h e s e c o n d h o l e h a s t h r e e components. I t c a n h e c a l c u l a t e d by t h e method d e s c r i b e d i n r e f e r e n c e (17). ( a ) A c o n s t a n t f l u x woNo, which c o n s t i t u t e s t h e e q u i l i b r i u m f l u x . A t a d i s t a n c e d from t h e s u r f a c e , i n t h e u n i l a t e r a l l y e x t e n d - e d medium i t h a s t h e v a l u e : d FAF = Nowo e r f c
-
2 6'
where x i s t h e d i f f u s i o n c o e f f i c i e n t . ( h ) A p e r i o d i c f l u x f o r which t h e p o t e n t i a l v a r i a t i o n s are r e s p o n s i b l e . Here, fit t h e f i i s t a n c e d : ( c ) The p e r i o d i c v a r i a t i o n s o f Ni a l s o r e s u l t i n a f l u x : ' The o c c u p a t i o n i n d i c e s Ni c o n s i s t o f : From t h e s e e q u a t i o n s we o b t a i n t h e f o l l o w i n g h y p o t h e s i s f o r t h e c h a n g e i n t i m e of t h e number o f p a r t i c l e s : d N 1- - -
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3 w, N , + Nowo e r f c-
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q l u t e - ( l + i ) a + d t 2 m W bkT
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kT N 1 e AM, vsoe - ( l + l ) a S u b t r a c t i o n of t h e two e q u a t i o n s g i v e s : d-
d t ( N , - N , ) =-
w o ( l + e - ( l + i ) a )f
N - N + ~l ( I J , + N 2 ) e The I n t e g r a l of t h l s e q u a t i o n i s : -9
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14, - KT (N, + N 2 ) e i o t 1.
l+i W w o ( l + eThe Imaglnnry p a r t o f t h l s e x p r e s s i o n c o n s t i t u t e s t h e v a l u e wi:ich i s measured by t h e l o @ r I t h m i c d e c r e m e n t . F o r
a
<< 1 and employing,
and a f t e r a few c o m p u t n t l o n s , t h e t h e a b b r e v i a t i o n T =-
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f r e q u e n c y and t e m p e r a t u r e dependence of 6 I s found t o he:
I n a c c o r d a n c e w l t h t h e a s s u m g t i o n t h a t tkic? e f f e c t o f t h e d i f f u s i o n i s c o n s i d e r e d s m ~ : l l , t h i s r e s u l t I s p r e s e n t e d I n t h e form of a s i m p l e p r o c e s s m o d i f i e d by a c o r r e c t l c n term. F o r l a r g e
x
ands m ~ l l (1, e q u a t i o n ( 5 )
,
as r e q u i r e d , becomes t h e co1'1-esponding exprWt+s- ; i o n of t h e s i m p l e p r o c e s s , A b r i e f g l a n c s shows t h a t t h e m o d i f i e d cul-ve 1-01- WT < 2.4 i s above t h e c u r v e o f t h @ s i m p l e model and f o rU ) T > 2 . ~ I S below i t . I f w e a g a i n starlt!~.i-dize the two c u r v e s a t
t h e s : m c h e l p b t , t h e y c ~ l r ~ c l d e up t o a b o u t W T = 1 and from WT = 1 the m o d l f i e ~ ? c u r v e i s s-ituatecl above t h e s i m p l e o n e . T h i s asjrmrnetry co1-r-es)1onr1s t o t h e e x p e r i m e n t a l l y obsel-ved d e v i u t l o n s .
A q u a n t i t a t i v e comparison o f e q u a t i o n ( 5 ) w i t h t h e e x p e r i m e n t a l r e s u l t s r e q u i r e s a knowledce o f t h e v a l u e of a and i t s dependence 011 t h e t e m p e r a t u r e . N o w , accorcllng t o v a r i o u s es timates t h e a c t i v a - t i o n e n e r g y o f t h e d i f f u s i o n c o n s t a n t x I s a b o u t 2 k c a l p e r mole. I t s a b s o l u t e v a l u e a t t h e t e m p e r a t u r e s of measurement i s a p p r o x i - m a t e l y 1 0 " l a t t i c e steps p e r second. d 1 i s p r o p o r t i o n a l t o
%,
i . e . , a p p r o x i m a t e l y 10.. - 5 0 l a t t i c e d i s t a n c e s , w i t h a n a c t i v a t i o n 1 e n e r g y /3 t h a t o f 7;. I n g e n e r a l w hail a v a l u e i n t h e v i c i n i t y ofl o 4 .
\ l i t 1 1 t h e s e rough f i g u r e s we g e t a v a l u e o f '17 f o r a. The t e m p e r a t u r e depend-e n c e o f a i s c o n s i d e r a b l y l e s s t h a n t h a t o f ?;, s o t h a t t o b e g i n -
w i t h i t a p p e a r s , l u s t i f i e 4 t o p u t n c o n s t a n t i n t h e t e m p e r a t u r e
i n t e r v a l r e r r e s e n t e d by a c u r v e o f t h e t y p e found i n fpig. 9 a n d 10. I n F i g . 9 and 1 0 t h e comparison o f t h e s i m p l e and t h e m o d i f i e d t h e o r y i s car.rlied o u t w i t h a n e x p e r i m e n t a l c u r v e , where a v a l u e o f
'/5 i s assumed f o r a . I n FIE. 9 a d i s c r e p a n c y w a s c l e a r l y evi(1ent
betr;eer~ t h e measured and t h e t h e o r e t i c a l c u r v e t o t h e r i g h t o f t h e maximum. This c l l f f e r e n c e showed up i n a l l t h e measurements.
Wow~ver, t h i s i s J u s t what i s e x p l a i n e d b y t h e c o r r e c t i o n i n t h e m o d i f i e d t h e o r y and e v e n i t s o r d e r of magnitude I s p r e d i c t e d
( F i g . 1 0 ) .
I wish t o t h a n k P r o f . Cr. 11.0. Kneser f o r s u g g e s t i n g t h i s work and f o r h i s c o n s t a n t i n t e r e s t i n i t s p r o g r e s s .
I a l s o wish t o t h a n k Dip1.-Ing. S. Magun f o r h e l p i n g I n many ways d u r i n g i t s e x e c u t i o n .
P h y s i c s c a n d l d a t e E. Vals w a s k i n d enough t o grow t h e r e q u i r e d c r y s t a l s i n a n a p p a r a t u s f i n a n c e d from t h e f u n d s of t h e R e s e a r c h C o r p o r a t i o n . My t h a n k s t o them a r e a l s o e x p r e s s e d h e r e . I a m p ; i r t l c u l u r l y o t ~ l l g e d t o Dip1
.
-llhys. W. Pechhold for many v a l u a b l e d i s c u s s i o n s . The Ileutsche F o r s c ~ i u n ~ s ~ m e i r i s c h n f t a l s o s u p p c r t e d me I n a v e r y k i n d manner.R e f e r e n c e s
1. E r r e r a , M . J . J. P h y s , ' R a d i u m , 5: 3 0 4 , 1924.
2. Auty, P. and C o l e ,
R.N.
J. Chem. Phys. 20: 1 3 0 9 , 1952. 3, G r a n i c h e r , H . , J a c c a r d , C . , S c h e r r e r , P. and S t e i n e m a n n , A. F a r n d a y D i s c u s s i o n s . Amsterdam, 1957. 4. E d e r , F . X . Ann. P h y s i k , 1: 381, 1947. 5. K n e s e r , H.O., Magun, S. a n d Z i e g l e r , G. ~ a t u r w i s s . 42: 437, 1955. 6. B a s s , R . P 1 s . s r r t a t i c n . S t u t t g a r t , 1 9 5 8 , R e p r i n t e d i n 2 . P h y s i k , v o l . 1 5 3 , 1958. 7. B o r d o n i , P.G. and Nuovo, M. R i c . S c i . 24: 560, 1954.8. K n e s e r , H.0. Z . angew. Phys. 3: 113, 1951. A u s e f u l estimate
o f t h e a i r damping o f l o n g i t u d i n a l o s c i l l a t i o n s i s o b t a i n e d by r e g a r d i n g t h e b a r s u r f a c e a s a p i s t o n o s c i l l a t o r and e s t i m a t i n g t h e e n e r g y r B a d 1 a t e d from i t .
9. Z e n e r , C . E l a s t i c i t y a n d a n e l a s t i c i t y i n m e t a l s . 1 9 4 8 , 10. P a u l i n g , L. The n a t u r e o f t h e c h e m i c a l bond. 1940.
11. H e r n a l , J . C . and F o w l e r , H . H . J. Chem. Phys. 1: 515, 1933. 1 2 . G i a u q u e , '1f.F. and A s h l e y , M. Phys. Rev. 4 3 : 81, 1933.
13. B j e r r u m , N. Dan. Mat. F y s . Medd. 27 ( l ) , 1951. 1.4. e r , A . Phys. Rev, 7 9 : 6 0 1 , 1950. 15. S e e g e r , A. fIandbuch d e r P h y s i k , V I I ( l ) , 1 9 5 5 . 1 6 . F r b h l i c h , H. T h e o r y of d i e l e c t r i c s . 1949. l'?. C a r l s l a w , H.S. and J i i g e r , J . C . C o n d u c t i o n o f h e a t i n : ; o l l d s . 1948
-
T a b l e
I
R e s o n a t o r s with I c e c y l i n d e r s o r i e n t e d p e r p e n d i c u l a r l y t o the hexagonal a x i s d 0 no. 1 2 . 4 5 6 7 S C) 1 0 1 1 12 13 11 15 16 1 7 I ,l C l l l 12,3 1 1 5 1 1 ,(I 6 , s 0 . 5 lo,.: 1 1 , O I l , O (;),Ss 5,4 1 1 , o 0,35 11,9 10,s 0 5 9,75 0,7 Hernarks From no, F r o m no. F r o m no. F r o m no. r'rom no. F r o m no. F r o m no, Darnpi ag m a x i m u m 1 I 1 dr,t:~x ~ i i1
I I L i in1i
26.3 .<(?I1 259,O 2Sc ) c ) 25.3,7 ' 1073 262.5 ! 4 0 1 0 2 0 sj
0027 250 2 1 1 s 250 21 1 0 2 5 0 , s 2287 26.31
3040 2671
5485 25!?, 5i
2600 SGS gcn1"5(.),2!
2090 25G 255,5 204,
2( )( )(-
837 200 297{) Q 4 .rl 2 5 2 5 2 5 2 6 2 0 3-5 24,5 2 5 2-1,s 2 4 2 5 32,s 3 1 2 4 3 1 2 5 3 2 2 4 , s 3 2 24 32,s 2 4 n C l I 1 O , S 1 0,785 0 4 0,73 0 , 0 , 5 5 0 , 0,502 0 , 5 0 2 0,375 0,725 0,47 0,935 0,74 0,73 0 , 7 2 0 6 0 4 r: d m a .-I .d .d :4 0 ~n o I I- 1- 1. 1, I- I 1. 1- 1- 1. 7' T L 7' T 1. '1' .! 11
I I L 0 1 o -18,O g 0 4 , s ,g 9 4 , s g 74.5 g 2 6 , s g 9 4 , 5 g 75,4 g 6 6 , 3 g 23,g g 1 1 , 9 g 0 4 , s g 17,3gcm"40,5 1 5 , s gcm2 1 5,4 3 5 > 4 g 15,4 g c m 2 35,4 g 15.4 gcm2 .35,4 h' 1 5 , 4 g c n l V 5 0 , 5 35,4 g 15,4 g c m b 4 S 3 5 , 4 gT a b l e I1 R e s o n a t o r s w i t h i c e c y l i n d e r s o r i e n t e d p a r a l l e l t o t h e h e x a g o n a l a x i s T a b l e 111 cl 0 (H .r(
Dependence of a b s o l u t e v a l u e of damping maximum on
o r i e n t a t i o n a n d t y p e of d e f o r m a t i o n
I
I
41
260,
4291) 1 3,GI
( I ( 1 1 1 o,72 o,53 0,79 0,61 0,4s5 Hod 110. 1s 1 0 3 ) 21 2 2 -~ 0 4 . (d -??A t n D a m p i n g maximum G d .r( .A Hemarks :4 0 V) G-
39,s g 262 4oGO L 39'8 g 25s 200() 3 O r i e n t a t i o n 11 CLC
I1 ( ' 1 1 1----
13,45 1 3 , ~ ~ I 1 , o 1 0 , O 6 , 3 7'-
'1' 7' 7' Type o f d e f o r m a t i o n I 5,4 gcm2 3 5 , 4 6 1S,4gcm2 35,4gcn12 I 5,4L
3.5 2 5 T-
30 33 • 356,s 20 5 252 252 gcn1"50,5 21 70 30 5130 3,5 1330 1 3 0 400 71
3,O I100 1 30 F r o m no.Pig* 1 Resonators
Pig* 2
-
Bar 8, longitudinal oscillations perpendicular to c axis.
F i g . 3
Dependence of logarithrn&c decrement on t h e d i r e c t i o n . L o n g i t u d i n a l o s c i l l a t i o n p a r a l l e l and p e r p e n d i c u l a r t o c a x i s , f = 2.9 kcps Fig. 4
-
Dependence of t h e r e l a x a t i o n frequency on t h e t e m p e r a t u r e , o-
l o n g i t u ( 1 i n a l o s c i l l a t i o n p n r p e n d i c u l a r t o c a x i s , +-
t o r s i o n a l o s c i l l a t i o n pel-pendicular t o c a x i s , 0-
t o r s i o n a l o s c i l l a t i o n p a r a l l e l t o c a x i s , A c t i v a t i o n energy: 0.5@ ev-
13.4 kcal/mole-
1 f = 5.3 1014 s e cF i g . 5
B a r 17. T o r s i o n a l o s c l l l ~ i t i o n t o c a x i s . f z 8 4 0 c p s
Pependence of t h e -1o.iprithmic decrement on t h e frecluency:
Fig. 7a
D- and L-def lclencies
Fig. 7b
Positive and negative deflciencies
-
Fig. 8a and b(a) Simple transposition model
F i g * 9
Bar 13. Comparison o f s i m p l e t h e o r y w l t h t e s t r e s u l t s
-
F i g * lo
