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Anomalous thermal conductivity in gadolinium below 4°K
Volume 25A, number 3
ANOMALOUS
PHYSICS
THERMAL
LETTERS
CONDUCTIVITY
14 August 1967
IN G A D O L I N I U M
BE L O W
4°K
A. G. KARAGYOZYAN* and K. ~r. RAO Clarendon Laboratory, Oxford, England Received 29 June 1967
Anomalies at 1.3 and 3.5°K have been o b s e r v e d in the t h e r m a l conductivity of gadolinium. Considering the c o r r e l a t i o n between our data and the specific heat, a tentative explanation is suggested.
In our work on the t h e r m a l conductivity of the r a r e - e a r t h m e t a l s [1], i n t e r e s t i n g effects have been o b s e r v e d in gadolinium. L o u n a s m a a and o t h e r s [2-4] have found in the heat capacity of this m e t a l a n o m a l i e s at 1.1, 1.6 and 3.7°K (fig. 1). They have a t t r i b u t e d t h e s e a n o m a l i e s to s o m e m a g n e t i c t r a n s i t i o n s due to the p r e s e n c e of 0.5% Gd203. The influence of m a g n e t i c t r a n s i t i o n s on e l e c t r i c a l and t h e r m a l conduction i s well known and has been o b s e r v e d e x p e r i m e n t a l l y in all r a r e e a r t h m e t a l s at higher t e m p e r a t u r e s , though they a r e not expected to give exactly s i m i l a r effects. At low t e m p e r a t u r e s the t h e r m a l conductivity i s a m o r e s e n s i t i v e i n d i c a t o r than the e l e c t r i c a l r e s i s t i v i t y , s i n c e the l a t t e r tends to a constant r e s i d u a l value. Our m e a s u r e m e n t s w e r e made in a p o l y c r y s t a l l i n e drawn w i r e 1.24ram d i a m e t e r and 5 cm in length, kindly made a v a i l a b l e to us by M e s s r s . Johnson, Matthey and Company, Limited. The t e m p e r a t u r e g r a d i e n t s w e r e d e t e r m i n e d with c h r o m e l v e r s u s gold (+0.03 at% i r o n ) t h e r m o c o u p les u s e d in a d i f f e r e n t i a l a r r a n g e m e n t . The a c c u r a c y of our m e a s u r e m e n t s is e s t i m a t e d a s about 1% or b e t t e r . The c r y o s t a t design and other e x p e r i m e n t a l d e t a i l s will be published e l s e w h e r e . Our data (fig. 1) show two m a r k e d a n o m a l i e s at about 3.5 and 1.3°K. The a b s o l u t e v a l u e s of the t h e r m a l conductivity e x t r a p o l a t e s to those m e a s u r e d by A r a j s and Colvin [5] above 5°K. An indication of a deviation from_the n o r m a l behaviour n e a r 3.5°K a l s o s e e m s to exist in the t h e r m a l conductivity m e a s u r e m e n t s of Aliev and V o l k e n shtein [6] between 2 and 100°K. However, the data given a r e not a c c u r a t e enough and the a u t h o r s do not m e n t i o n the effect. * On leave f r o m Department of P h y s i c s , Yerevan State University, Yerevan, A r m e n i a n SSR, USSR, under the S o v i e t - B r i t i s h Scientific Exchange P r o g r a m m e .
•0---'17 ~5 ill I--
~3
/i/"i"
I,,-
:
z~9
0 .j7
T
,,..
Cp
-~, .....-'
., .........,''"
"'--...,
. ........
250 o
o
E 150
E
3 50
/
I I I 1 2 3 ~'T°K'I fig.1. THERMAL CONDUCTIVITY O]='G'a" & Sp. ht.data (Iounasmaa 2) .--.
Fig. 1. T h e r m a l conductivity of Gd ( ~ ) heat data ( . . . ) [2].
4
and specific
In the a b s e n c e of a theory of t h e r m a l c o n d u c tivity applicable to the r a r e - e a r t h m e t a l s , which show m a g n e t i c o r d e r i n g , a detailed a n a l y s i s of our r e s u l t s cannot be attempted. As is well known in o r d i n a r y m e t a l s at low t e m p e r a t u r e s the t h e r m a l conductivity is d o m i n a t e d by i m p u r i t y s c a t t e r ing which is p r o p o r t i o n a l to the absolute t e m p e r a t u r e s . It i s obvious f r o m our r e s u l t s that g a d o l i n i u m does not conform to this s i m p l e p a t t e r n and that other f a c t o r s b e s i d e s e n e r g y t r a n s p o r t by e l e c t r o n s may play a p a r t . T h i s i s a l s o evident f r o m the fact that the c a l c u l a t e d L o r e n z n u m b e r has t h r e e t i m e s the t h e o r e t i c a l value [5,6]. The s t r i k i n g f e a t u r e of our r e s u l t s is the close c o r r e lation between the heat conductivity a n o m a l i e s and those in the specific heat. Since on d i m e n s i o -
235
Volume 25A, number 3
PHYSICS LETTERS
nal c o n s i d e r a t i o n s K i s p r o p o r t i o n a l to clv, w h e r e c is the s p e c i f i c heat, l the m e a n f r e e path, and the v e l o c i t y of the c a r r i e r s , it is obvious to look to the f i r s t f o r an explanation of the a n o m a l i e s in K, L o u n a s m a a [7] e s t i m a t e d that in gadolinium at 5°K the total s p e c i f i c heat contains a 39% m a g n e ti c contribution, a 31% e l e c t r o n i c one, while 20% a r e due to the phonons. He explains his r e sult by a t t r i b u t i n g the e x c e s s entropy to the m a g n e ti c o r d e r i n g of Cd3 + ions in the Gd20 3 i m p u r i t y . S u s c e p t i b i l i t y m e a s u r e m e n t s and lately neutron d i f f r a c t i o n s t u d i e s by Child et al. [8] on Gd203 s e e m to show that it has a c o m p l e x a n t i f e r r o m a g n e t i c s t r u c t u r e below the N e l l t e m p e r a t u r e TN = = 1.6°K with c o n s i d e r a b l e amount of o r d e r r e m a i n i n g up to a t e m p e r a t u r e at l e a s t t h r e e t i m e s T N. It is t h e r e f o r e t e m p t i n g to look at m a g nons as r e s p o n s i b l e f o r e f f e c ts in the heat conductivity of gadolinium at low t e m p e r a t u r e s . In addition t h e r e m ay e x i s t m a g n e t i c s c a t t e r i n g of e l e c t r o n s l ead i n g to a d e c r e a s e of e l e c t r o n c o n ductivity. Since c o m p l e x i t y of the t h e r m a l c o n d u c t i v i t y of gadolinium m a k e s it v e r y difficult to c a l c u l a t e an " i d e a l " K e , it is i m p o s s i b l e to s e p a r a t e t h e s e t e r m s . Both may play a p a r t . While t h e r e s e e m s to be l i t t l e doubt that the a n o m a l i e s a r e of m a g n e t i c o r i g i n , it is not c l e a r w h e t h e r the explanation i n v o l v in g Gd20 3 is the c o r r e c t
CONTINUUM
THEORY
AND
STRING
14August 1967
one s i n c e the s p e c i f i c heat of a p u r e r s p e c i m e n of gadolinium a l s o shows the anomaly at 3.7OK [7,4]. It is hoped to c a r r y our m e a s u r e m e n t s of the t h e r m a l conductivity of gadolinium in a m a g n e t i c field. A r ed u ct i o n of the m a g n e t i c conductivity in this way m ay p r o v i d e f u r t h e r i n f o r m a t i o n on the r o l e of m ag n o n s in heat conduction. We a r e g r a t e f u l to Dr. M e n d e l s s o h n f o r his e n c o u r a g e m e n t and d i s c u s s i o n s in this work.
References 1. 2. 3. 4. 5. 6. 7. 8.
K.V.Rao, Phys. Letters 24A (1967) 39. O.V.Lounasmaa. Phys. Rev. 129 (1963) 2460. L.T.Crane, J. Chem. Phys. 36 (1962) 10. Donald, Crane and Zimmerman, J. Phys. Chem, to be published. S.Arajs and N.V. Colvin, J. Appl. Phys. 35 (1964) 1043. N.G.Aliev and N. V. Volkenshtein, Soviet Phys. JETP, 22. 1 Jan. (1966). O.V.Lounasmaa, Phys. Rev. 150 (1966)399. H.R. Child, R.M. Moon, T. Roubenheimer and W. C. Koehler, J. Appl. Phys. 38 (1967) 1381,
MODEL
OF
MOVING
DISLOCATIONS
H. BROSS and G, S T E N Z E L
Sektion Physik der Universittit M~nchen, Germany Received 22 June 1967
Using the basic equation of dislocation dynamics the string model is verified.
It i s well known, that d i f f e r e n t p h e n o m e n a of i n t e r n a l f r i c t i o n in s o l i d s m a y be a t t r i b u t e d to m o v i n g d i s l o c a t i o n s [1]. To explain t h e s e e f f e c t s qua nti t at i v el y , it i s a s s u m e d that the d i s l o c a t i o n line b e h a v e s like a s t r e t c h e d s t r i n g , c h a r a c t e r i z e d by a line t e n s i o n and a m a s s p e r unit length. The a i m of this l e t t e r is to show that this s o c a l l e d s t r i n g m o d e l m a y be explained by the t h e o r y of d i s l o c a t i o n d y n a m i c s f o r m u l a t e d by one of the a u t h o r s s e v e r a l y e a r s ago [2]. A c e n t r a l r o l e in this f o r m u l a t i o n i s played by the p l a s t i c 236
d i s t o r s i o n tiP (r, t) due to the d i s l o c a t i o n . It may be d e t e r m i n e d in the following way: 1) The v e c t o r of total d i s p l a c e m e n t s G may be e x p r e s s e d by the p l a s t i c d e f o r m a t i o n
s ? ( r , t) = 1 = -4-~
f c,ij(r-r', t - t ' ) C j k l m
ak
~Plm
(r', t') dV' dr' ,
2) this e x p r e s s i o n is i n s e r t e d in the c h a r a c t e r i s t i c function
ANOMALOUS
PHYSICS
THERMAL
LETTERS
CONDUCTIVITY
14 August 1967
IN G A D O L I N I U M
BE L O W
4°K
A. G. KARAGYOZYAN* and K. ~r. RAO Clarendon Laboratory, Oxford, England Received 29 June 1967
Anomalies at 1.3 and 3.5°K have been o b s e r v e d in the t h e r m a l conductivity of gadolinium. Considering the c o r r e l a t i o n between our data and the specific heat, a tentative explanation is suggested.
In our work on the t h e r m a l conductivity of the r a r e - e a r t h m e t a l s [1], i n t e r e s t i n g effects have been o b s e r v e d in gadolinium. L o u n a s m a a and o t h e r s [2-4] have found in the heat capacity of this m e t a l a n o m a l i e s at 1.1, 1.6 and 3.7°K (fig. 1). They have a t t r i b u t e d t h e s e a n o m a l i e s to s o m e m a g n e t i c t r a n s i t i o n s due to the p r e s e n c e of 0.5% Gd203. The influence of m a g n e t i c t r a n s i t i o n s on e l e c t r i c a l and t h e r m a l conduction i s well known and has been o b s e r v e d e x p e r i m e n t a l l y in all r a r e e a r t h m e t a l s at higher t e m p e r a t u r e s , though they a r e not expected to give exactly s i m i l a r effects. At low t e m p e r a t u r e s the t h e r m a l conductivity i s a m o r e s e n s i t i v e i n d i c a t o r than the e l e c t r i c a l r e s i s t i v i t y , s i n c e the l a t t e r tends to a constant r e s i d u a l value. Our m e a s u r e m e n t s w e r e made in a p o l y c r y s t a l l i n e drawn w i r e 1.24ram d i a m e t e r and 5 cm in length, kindly made a v a i l a b l e to us by M e s s r s . Johnson, Matthey and Company, Limited. The t e m p e r a t u r e g r a d i e n t s w e r e d e t e r m i n e d with c h r o m e l v e r s u s gold (+0.03 at% i r o n ) t h e r m o c o u p les u s e d in a d i f f e r e n t i a l a r r a n g e m e n t . The a c c u r a c y of our m e a s u r e m e n t s is e s t i m a t e d a s about 1% or b e t t e r . The c r y o s t a t design and other e x p e r i m e n t a l d e t a i l s will be published e l s e w h e r e . Our data (fig. 1) show two m a r k e d a n o m a l i e s at about 3.5 and 1.3°K. The a b s o l u t e v a l u e s of the t h e r m a l conductivity e x t r a p o l a t e s to those m e a s u r e d by A r a j s and Colvin [5] above 5°K. An indication of a deviation from_the n o r m a l behaviour n e a r 3.5°K a l s o s e e m s to exist in the t h e r m a l conductivity m e a s u r e m e n t s of Aliev and V o l k e n shtein [6] between 2 and 100°K. However, the data given a r e not a c c u r a t e enough and the a u t h o r s do not m e n t i o n the effect. * On leave f r o m Department of P h y s i c s , Yerevan State University, Yerevan, A r m e n i a n SSR, USSR, under the S o v i e t - B r i t i s h Scientific Exchange P r o g r a m m e .
•0---'17 ~5 ill I--
~3
/i/"i"
I,,-
:
z~9
0 .j7
T
,,..
Cp
-~, .....-'
., .........,''"
"'--...,
. ........
250 o
o
E 150
E
3 50
/
I I I 1 2 3 ~'T°K'I fig.1. THERMAL CONDUCTIVITY O]='G'a" & Sp. ht.data (Iounasmaa 2) .--.
Fig. 1. T h e r m a l conductivity of Gd ( ~ ) heat data ( . . . ) [2].
4
and specific
In the a b s e n c e of a theory of t h e r m a l c o n d u c tivity applicable to the r a r e - e a r t h m e t a l s , which show m a g n e t i c o r d e r i n g , a detailed a n a l y s i s of our r e s u l t s cannot be attempted. As is well known in o r d i n a r y m e t a l s at low t e m p e r a t u r e s the t h e r m a l conductivity is d o m i n a t e d by i m p u r i t y s c a t t e r ing which is p r o p o r t i o n a l to the absolute t e m p e r a t u r e s . It i s obvious f r o m our r e s u l t s that g a d o l i n i u m does not conform to this s i m p l e p a t t e r n and that other f a c t o r s b e s i d e s e n e r g y t r a n s p o r t by e l e c t r o n s may play a p a r t . T h i s i s a l s o evident f r o m the fact that the c a l c u l a t e d L o r e n z n u m b e r has t h r e e t i m e s the t h e o r e t i c a l value [5,6]. The s t r i k i n g f e a t u r e of our r e s u l t s is the close c o r r e lation between the heat conductivity a n o m a l i e s and those in the specific heat. Since on d i m e n s i o -
235
Volume 25A, number 3
PHYSICS LETTERS
nal c o n s i d e r a t i o n s K i s p r o p o r t i o n a l to clv, w h e r e c is the s p e c i f i c heat, l the m e a n f r e e path, and the v e l o c i t y of the c a r r i e r s , it is obvious to look to the f i r s t f o r an explanation of the a n o m a l i e s in K, L o u n a s m a a [7] e s t i m a t e d that in gadolinium at 5°K the total s p e c i f i c heat contains a 39% m a g n e ti c contribution, a 31% e l e c t r o n i c one, while 20% a r e due to the phonons. He explains his r e sult by a t t r i b u t i n g the e x c e s s entropy to the m a g n e ti c o r d e r i n g of Cd3 + ions in the Gd20 3 i m p u r i t y . S u s c e p t i b i l i t y m e a s u r e m e n t s and lately neutron d i f f r a c t i o n s t u d i e s by Child et al. [8] on Gd203 s e e m to show that it has a c o m p l e x a n t i f e r r o m a g n e t i c s t r u c t u r e below the N e l l t e m p e r a t u r e TN = = 1.6°K with c o n s i d e r a b l e amount of o r d e r r e m a i n i n g up to a t e m p e r a t u r e at l e a s t t h r e e t i m e s T N. It is t h e r e f o r e t e m p t i n g to look at m a g nons as r e s p o n s i b l e f o r e f f e c ts in the heat conductivity of gadolinium at low t e m p e r a t u r e s . In addition t h e r e m ay e x i s t m a g n e t i c s c a t t e r i n g of e l e c t r o n s l ead i n g to a d e c r e a s e of e l e c t r o n c o n ductivity. Since c o m p l e x i t y of the t h e r m a l c o n d u c t i v i t y of gadolinium m a k e s it v e r y difficult to c a l c u l a t e an " i d e a l " K e , it is i m p o s s i b l e to s e p a r a t e t h e s e t e r m s . Both may play a p a r t . While t h e r e s e e m s to be l i t t l e doubt that the a n o m a l i e s a r e of m a g n e t i c o r i g i n , it is not c l e a r w h e t h e r the explanation i n v o l v in g Gd20 3 is the c o r r e c t
CONTINUUM
THEORY
AND
STRING
14August 1967
one s i n c e the s p e c i f i c heat of a p u r e r s p e c i m e n of gadolinium a l s o shows the anomaly at 3.7OK [7,4]. It is hoped to c a r r y our m e a s u r e m e n t s of the t h e r m a l conductivity of gadolinium in a m a g n e t i c field. A r ed u ct i o n of the m a g n e t i c conductivity in this way m ay p r o v i d e f u r t h e r i n f o r m a t i o n on the r o l e of m ag n o n s in heat conduction. We a r e g r a t e f u l to Dr. M e n d e l s s o h n f o r his e n c o u r a g e m e n t and d i s c u s s i o n s in this work.
References 1. 2. 3. 4. 5. 6. 7. 8.
K.V.Rao, Phys. Letters 24A (1967) 39. O.V.Lounasmaa. Phys. Rev. 129 (1963) 2460. L.T.Crane, J. Chem. Phys. 36 (1962) 10. Donald, Crane and Zimmerman, J. Phys. Chem, to be published. S.Arajs and N.V. Colvin, J. Appl. Phys. 35 (1964) 1043. N.G.Aliev and N. V. Volkenshtein, Soviet Phys. JETP, 22. 1 Jan. (1966). O.V.Lounasmaa, Phys. Rev. 150 (1966)399. H.R. Child, R.M. Moon, T. Roubenheimer and W. C. Koehler, J. Appl. Phys. 38 (1967) 1381,
MODEL
OF
MOVING
DISLOCATIONS
H. BROSS and G, S T E N Z E L
Sektion Physik der Universittit M~nchen, Germany Received 22 June 1967
Using the basic equation of dislocation dynamics the string model is verified.
It i s well known, that d i f f e r e n t p h e n o m e n a of i n t e r n a l f r i c t i o n in s o l i d s m a y be a t t r i b u t e d to m o v i n g d i s l o c a t i o n s [1]. To explain t h e s e e f f e c t s qua nti t at i v el y , it i s a s s u m e d that the d i s l o c a t i o n line b e h a v e s like a s t r e t c h e d s t r i n g , c h a r a c t e r i z e d by a line t e n s i o n and a m a s s p e r unit length. The a i m of this l e t t e r is to show that this s o c a l l e d s t r i n g m o d e l m a y be explained by the t h e o r y of d i s l o c a t i o n d y n a m i c s f o r m u l a t e d by one of the a u t h o r s s e v e r a l y e a r s ago [2]. A c e n t r a l r o l e in this f o r m u l a t i o n i s played by the p l a s t i c 236
d i s t o r s i o n tiP (r, t) due to the d i s l o c a t i o n . It may be d e t e r m i n e d in the following way: 1) The v e c t o r of total d i s p l a c e m e n t s G may be e x p r e s s e d by the p l a s t i c d e f o r m a t i o n
s ? ( r , t) = 1 = -4-~
f c,ij(r-r', t - t ' ) C j k l m
ak
~Plm
(r', t') dV' dr' ,
2) this e x p r e s s i o n is i n s e r t e d in the c h a r a c t e r i s t i c function