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Submitted on 1 Jan 1981
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LOW FREQUENCY PHONONS IN SINTERED COPPER
B. Frisken, F. Guillon, J. Harrison, J. Page
To cite this version:
B. Frisken, F. Guillon, J. Harrison, J. Page. LOW FREQUENCY PHONONS IN SINTERED COP- PER. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-858-C6-860. �10.1051/jphyscol:19816254�.
�jpa-00221342�
JOURNAL DE PHYSIQUE
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Loque C6, supplkment au n o 12, Tome 4 2 , dbcernbre 1981 page C6-858LOW FREQUENCY PHONONS I N S I N T E R E D COPPER
B. F r i s k e n , F. G u i l l o n , J.P. H a r r i s o n and J . H . Page
Physics Department, Queen's University, Kingston, Ontario, Canada, K7L 3N6
A b s t r a c t .
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Measurements of Young's Modulus and sound v e l o c i t y of a r a n g e of s i n t e r e d c o p p e r powder samples have d e m o n s t r a t e d t h e e x i s t e n c e o f low f r e - quency s o f t phonon modes t h a t c o u l d e x p l a i n t h e low t e m p e r a t u r e anomaly i n t h e K a p i t z a r e s i s t a n c e between l i q u i d 3 ~ e and s i n t e r e d m e t a l h e a t e x c h a n g e r s below 10 mK.I n t r o d u c t i o n . A l l measurements o f t h e K a p i t z a r e s i s t a n c e , o r t h e r m a l boundary r e s i s t a n c e , between l i q u i d 3 ~ e and s o l i d s i n t h e r e g i o n o f 1 mK have b e e n anomalously s m a l l when compared w i t h t h e p r e d i c t i o n s o f a c o u s t i c t h e o r y . I n a d d i t i o n , t h e measured r e s i s t a n c e s have g e n e r a l l y been p r o p o r t i o n a l t o T - ~ r a t h e r t h a n T - ~ a s g i v e n by a c o u s t i c t h e o r y " ) . T h i s i s a low t e m p e r a t u r e e f f e c t s i n c e t h e a c o u s t i c t h e o r y seems t o d e s c r i b e e x p e r i m e n t a l r e s u l t s q u i t e w e l l i n t h e 30-100 mK tempera- t u r e r a n g e . E i t h e r t h e r e i s a n e x t r a c h a n n e l f o r h e a t f l o w a c r o s s t h e boundary o r t h e r e i s a problem w i t h t h e a c o u s t i c t h e o r y when T 6 10 mK. I t h a s been a r g u e d by o t h e r s t h a t m a g n e t i c c o u p l i n g i s t h e e x t r a c h a n n e l . However, t h i s i s u n l i k e l y s i n c e t h e anomalous r e s i s t a n c e i s i n d e p e n d e n t of m a g n e t i c f i e l d , 3 ~ e p r e s s u r e , p h a s e of t h e 3 ~ e (A, B o r normal) o r m a g n e t i c i m p u r i t y c o n t e n t of t h e s o l i d . We a r e i n s t e a d p u r s u i n g t h e p o s s i b i l i t y t h a t t h e u s u a l a c o u s t i c mismatch t h e o r y i s i n a p p r o p r i a t e a t t h e l o w e s t t e m p e r a t u r e s f o r t h e s o l i d s t h a t have been i n v e s t i g a t e d ; a l l e x p e r i m e n t s h a v e been made w i t h s o l i d s t h a t were e i t h e r s i n t e r e d m e t a l powders o r powdered c e r i u m magnesium n i t r a t e . I n a l l c a s e s t h e powder d i a m e t e r ( d ) was s m a l l e r t h a n o r comparable w i t h t h e dominant phonon wavelength a t 1 mK (% 30 ilm i n b u l k s o l i d s ) . T h e r e f o r e t h e u s u a l Debye s p e c t r u m h a s a l o n g wavelength c u t - o f f a t X % d , w i t h a
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c o n s e q u e n t low e n e r g y c u t - o f f a t E/kg 2. hvD/kBd (2. 100 mK when d = 1 um) where kB i s Boltzmann's c o n s t a n t and vD i s t h e Debye sound v e l o c i t y . A consequence i s t h a t e s s e n t i a l l y no b u l k phonons a r e e x c i t e d a t 1 mK. I n s t e a d t h e r e s h o u l d b e new normal modes a s s o c i a t e d w i t h t h e powder p a r t i c l e s a s lumped masses and t h e n e c k s between p a r t i c l e s a s weak s p r i n g s . We p r e s e n t h e r e t h e r e s u l t s of a n i n v e s t i g a t i o n of t h e s e new modes.
Sample P r e p a r a t i o n . A v a r i e t y of copper and s i l v e r powder d i a m e t e r s , p a c k i n g f a c t o r s and s i n t e r i n g p r o c e s s e s have been used by v a r i o u s g r o u p s i n t h e p a s t " ) . T h i s work was done w i t h 700
61,
copper powde;(2) and w i t h a range of p a c k i n g f a c t o r s and s i n t e r - i n g t e m p e r a t u r e s . The powder was f i r s t c l e a n e d i n t r i c h l o r o e t h y l e n e and t h e n i nArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816254
a c e t o n e (10 m i n u t e s i n a n u l t r a s o n i c b a t h ) . The powder was d r i e d , p l a c e d l o o s e l y i n aluminum f o i l b o a t s i n a f u r n a c e t u b e and t h e t u b e evacuated. When t h e p r e s s u r e had dropped t o t o r r t h e f u r n a c e was s w i t c h e d on and brought up t o i t s working temp- e r a t u r e ( u s u a l l y 2 0 0 ~ ~ ) . Then hydrogen was allowed t o s t r e a m through t h e tube f o r 1 5 minutes a t which t i m e t h e f u r n a c e t u b e was q u i c k l y cooled. A s soon a s p o s s i b l e t h e p r e - s i n t e r e d powder was packed i n t o a c l e a n mold. Apart from v a r y i n g t h e s i n t e r t e m p e r a t u r e t h e s i n t e r p r o c e s s was t h e same a s t h e p r e - s i n t e r p r o c e s s d e s c r i b e d above. The f i n a l samples were i n t h e form of 2 mm t h i c k d i s c s o r beams 70 mm x 5 mm x 2 mm.
Method and R e s u l t s . I n o r d e r t o i n v e s t i g a t e t h e low frequency modes, measurements were made of t h e Young's modulus ( z e r o frequency) and t h e v e l o c i t y of l o n g i t u d i n a l sound ( 1 5 MHz)
.
( i ) Young's Modulus. The s i n t e r s a r e q u i t e s t i f f and t o o b t a i n a measurable d i s - t o r t i o n w i t h r e a s o n a b l e l o a d s t h e s i n t e r was formed i n t o l o n g t h i n beams t h a t were s u p p o r t e d a t t h e ends and loaded i n t h e c e n t r e . The beams were l o a d e d i n 2 gm i n c r e m e n t s t o a b o u t 40 gm and then unloaded i n i n c r e m e n t s w h i l e t h e d e f o r m a t i o n s were measured w i t h a t r a v e l l i n g microscope. Young's Modulus (Y) was determined from a deformation v e r s u s l o a d p l o t . The r e s u l t s a r e shown i n f i g u r e 1, p l o t t e d a s Y/Ycu where Y i s t h e Young's modulus of copper (12.5 x 1 0 ~ ~ ~ m - ~ ) .
cu
( i i ) Sound V e l o c i t y . The u l t r a s o n i c v e l o c i t y was measured i n t h e frequency range 12-20 MHz on t h i n disc-shaped samples. The s i n t e r s were d r y p o l i s h e d f l a t and p a r a l l e l u s i n g f i n e s i l i c o n c a r b i d e paper. The samples were t h e n p l a c e d between two q u a r t z d e l a y r o d s and t h e v e l o c i t y was measured from t h e time d e l a y between a r e f l e c t e d p u l s e , which h a s t r a v e l l e d up and down t h e q u a r t z r o d o n l y , and a t r a n s - m i t t e d p u l s e , which has p a s s e d through t h e specimen and each q u a r t z r o d . The t r a n s - m i t t e d s i g n a l s were g e n e r a l l y weak and o f t e n broad, i n p a r t due t o i n t e r f e r e n c e e f f e c t s caused by i n h o m o g e n e i t i e s i n t h e s i n t e r s , and c o n s e q u e n t l y t h e l e a d i n g edge of t h e p u l s e s was used i n a l l measurements. To p r e v e n t bonding f l u i d from p e n e t r a t - i n g t h e p o r e s of t h e s i n t e r t h i n l a t e x membranes were i n s e r t e d between t h e s i n t e r and t h e q u a r t z r o d , w i t h a l a y e r of Nonaq s t o p c o c k g r e a s e o r g l y c e r o l between t h e q u a r t z and t h e membrane b u t n o t between t h e membrane and t h e s i n t e r . A measured c o r r e c t i o n of 0.07 u s was a p p l i e d f o r t h e d e l a y i n t h e membranes. The t e c h n i q u e worked w e l l f o r b o t h Ag and Cu s i n t e r s o v e r a range of pack.ing f a c t o r s f ; f o r a Ag s i n t e r w i t h f = 55%, we o b t a i n t h e v e l o c i t y vE = (1.73
+
.05) x 103ms-' and f o r Cu s i n t e r s w i t h f = 48% and 42% we measure vE = (1.6 ?r . l ) and (1.2+
.2) x 103ms-I r e s p e c t i v e l y . Measurements were a l s o made on a Cu s i n t e r of 33% f i l l i n g f a c t o r , b u t even w i t h t h e sample t h i c k n e s s reduced t o 0.53 mm, t h e u l t r a s o n i c a t t e n u a t i o n was s o g r e a t t h a t t h e membranes had t o b e removed f o r a t r a n s m i t t e d s i g n a l t o be observed;i n t h i s c a s e , t h e bond was made w i t h Nonaq which i s s u f f i c i e n t l y v i s c o u s t o c a u s e minimal c o n t a m i n a t i o n i n t h e s i n t e r . For t h i s sample vE = 0.75 ( 2 . 2 ) x 1 0 ~ r n s - ~ . The r e s u l t s f o r t h e s e samples have been c o n v e r t e d t o a reduced e l a s t i c modulus pvQ2/(PvL2)bulk where b u l k r e f e r s t o b u l k copper o r s i l v e r .
C6-860 JOURNAL DE PHYSIQUE
D i s c u s s i o n o f R e s u l t s . F i r s t of a l l t h e s i n t e r s a r e e l a s t i c and c o n s i d e r a b l y s o f t e r t h a n b u l k c o p p e r . T h e r e i s a marked, b u t n o t u n e ~ p e c t e d ' ~ ) , dependence upon p a c k i n g f a c t o r . T h i s h a s b e e n c o n f i r m e d i n more r e c e n t work on b o t h c o p p e r and s i l v e r (4)
.
The sound v e l o c i t y and e l a s t i c modulus measurements a r e i n s a t i s f a c t o r y a g r e e m e n t ; t h e a g r e e m e n t is b e t t e r i n t h e more r e c e n t work i n which s a m p l e p r e p a r a t i o n was more c o n s i s t e n t .
For t h e s e s o f t modes t o b e s i g n i f i c a n t i n h e a t t r a n s f e r a t
<
1 0 mK t h e y would have t o e x i s t t o v > l o W 2 ( k g / h ) o r 200 MHz. T h e i r u p p e r l i m i t can i n f a c t be estim- a t e d by a p p l y i n g Debye t h e o r y . With N / V 2. 1018m-3(4) we f i n d v % 500 MHz. There-D
f o r e t h e s e modes s h o u l d c o m p l e t e l y dominate t h e phonon s p e c t r u m below 1 0 mK. They a r e s o f t e r modes and h a v e a d e n s i t y o f s t a t e s up by % 50 when compared t o b u l k c o p p e r and i t is n o t s u r p r i s i n g t h e r e f o r e t h a t a c o u s t i c mismatch t h e o r y w i t h b u l k c o p p e r p a r a m e t e r s f a i l s t o a g r e e w i t h e x p e r i m e n t a l r e s u l t s .
I n c o n c l u s i o n i t i s emphasised t h a t t h e s e a r e continuum modes o f t h e s i n t e r p l u s w h a t e v e r o c c u p i e s t h e p o r e s ( u s u a l l y l i q u i d h e l i u m ) . F o r a f u l l u n d e r s t a n d i n g of h e a t t r a n s f e r below 10 mK a s t u d y is now r e q u i r e d o f t h e c o u p l i n g of t h e s e modes t o t h e e l e c t r o n g a s i n t h e m e t a l and t o t h e q u a s i p a r t i c l e e x c i t a t i o n s i n t h e l i q u i d 3 ~ e o r 3 ~ e / 4 ~ e m i x t u r e s .
Acknowledgements. T h i s work h a s b e e n s u p p o r t e d by N.S.E.R.C. and by C.R.A.D. o f t h e D.N.D. We would l i k e t o t h a n k J o h n R o b e r t s o n f o r p r e p a r i n g t h e s a m p l e s u s e d f o r t h e sound v e l o c i t y measurements and C h r i s t i a n FrGnois f o r a s s i s t a n c e w i t h some o f t h e v e l o c i t y measurements.
1. D e t a i l e d r e f e r e n c e s a r e g i v e n i n t h e r e v i e w a r t i c l e : J . P . H a r r i s o n , J . Low Temp. Phys. 37, 467 (1979).
2. U1-Vac Vacuum M e t a l l u r g i c a l Co. L t d . , Tokyo, J a p a n .
3. David Wood ( p r i v a t e communication), E. Domany, J . A . G u b e r n a t i s and J . A . Krumhansl, J o u r n a l o f G e o p h y s i c a l R e s e a r c h
80,
485 (1975).4. J. R o b e r t s o n , F. G u i l l o n and J . P . H a r r i s o n ( t o be p u b l i s h e d ) .
F i g u r e 1. C i r c l e s a r e t h e beam b e n d i n g r e s u l t s w i t h s i n t e r i n g t e m p e r a t u r e s a s f o l l o w s :
Open c i r c l e s , 175OC Half-open c i r c l e s , 2 0 0 ' ~ C l o s e d c i r c l e s , 225OC.
T r i a n g l e s a r e t h e s o u n d v e l o c i t y r e s u l t s w f t h a s i n t e r i n g tempera- t u r e o f 200°C. The 55% r e s u l t i s f o r s i l v e r .
30 40 50
P a c k i n g F a c t o r ( % )
