Volume and temperature dependence of the Grüneisen parameter of hcp 4He

Volume 24.A. n u m b e r 3

PHYSICS

VOLUME

LETTERS

AND TEMPERATURE GRtINEISEN PARAMETER

30 J a n u a r y 1967

DEPENDENCE OF hcp

THE

OF

4He

G. A H L E R S

Bell Telephone Laboratories. Incorporated Mt¢rray Hill, Neu' Jersey R e c e i v e d 31 D e c e r n b c r 1966

Values of the Grtineisen p a r a m e t e r 7' have been derived for hop 4He as a function oi volume and t e m p e r a t u r e . 7' d e c r e a s e d l i n e a r l y with volume at a rate of 0.083 (cm3/mole) -1 at 0°K and is t e m p e r a t u r e dependent. A c c u r a t e d e t e r m i n a t i o n s of a G r t i n e i s e n p a r a m e t e r 7' f r o m p r e c i s i o n m e a s u r e m e n t s of t h e h e a t c a p a c i t y Cv a s a f u n c t i o n of v o l u m e a n d t e m p e r a t u r e f o r h c p 4He a r e r e p o r t e d . No p r e vious experimental data over a sufficient volume range to establish accurately the volume depend e n c e of 7' a t low t e m p e r a t u r e s h a v e b e e n p u b lished for any solid. For this work, the definition 7'

24

( ~ I n C v / ~ in V) T = (~ in C v / 0 In T) V

(~Ci ~

. ~Ci .

(, Yr-)v

(2)

where Ci is the heat capacity contribution from the/th mode. The more conventional parameter ? = aV/(KCz,) a n d 7 ' a r e r e l a t e d by

7 = C~) 1 .f 7 ' ( O C v / O T ) v d T .

O

~

~

2

28

.4 T (°l'k)

I

•

~

14

52

~ j.J~"T3 - ~ ] ~73 li 6

!

DUGDALE u JARV~S AND MEYER o THIS WORK

"

~

u

•

o

/I

~7/~

~

3

2 + 0,083V

I

i

(3) w-

I

15

,

17 V (CCIMOLE)

k

19

(4)

Cv of h c p 4He w a s m e a s u r e d

a t 12 v o l u m e s b e t w e e n 13.7 a n d 20.5 c m 3 / m o l e a n d b e t w e e n 1.38°K and the melting line. Random errors in Cv d i d not e x c e e d ± 0 . 1 % . E r r o r s in 7' a r e b e l i e v e d not to e x c e e d 4-0.05 o r a b o u t ± 2%. 7 ' i s p r e s e n t e d i n fig. l a . B e l o w 19 c m 3 / m o l e 7' c a n b e e x t r a p o l a t e d to 0 ° K 7 o ( 7 ' at 0°K) i s v o l u m e d e p e n d e n t . It i s s h o w n in fig. l b a n d i s g i v e n by Yo = 1.02 + 0 . 0 8 3 V.

I

f

i

13

7 = 7' = - ( V / O o ) ( d O o / d V ) .

' - JARVIS AND MEYER

" ' " * ° ° ~ ~ ' - " ' ~ ' ~ " "

24

At 0°K,

152

.

(1)

V dzzi

....... T

z,,, 27

w a s a d o p t e d . In t e r m s of t h e v o l u m e d e p e n d e n c e of t h e n o r m a l m o d e s , t h i s i s e q u i v a l e n t t o

:

,20.72 / -~b.Yz \\ /~" 2007

3c

(5)

Fig. 1. a) 3" as a function of volume and t e m p e r a t u r e for hep 4tte. The n u m b e r s a r e the m o l a r volumes in em3. The dashed lines a r e from ref. 1. b) 7o(7' at 0°K) for hep 4He. The solid c i r c l e s a r e ealcutated values by Dugdale, [71. The s q u a r e s a r c from ref. 1. At t h e s m a l l e s t v o l u m e s 7' i s a l m o s t i n d e p e n d e n t of T , a n d t h e r e f o r e n e a r l y e q u a l to 7, e x c e p t n e a r t h e m e l t i n g l i n e . At l a r g e r v o l u m e s 7 ' b e comes temperature dependent over the entire experimental temperature range. M e a s u r e m e n t s of ( 3 P / 3 T ) V f o r h e p 4He by J a r v i s a n d M e y e r [1] a b o v e 18.8 e m 3 / m o l e y i e l d

J

Volume 24A, number 3

PHYSICS LETTERS

v a l u e s of V g e n e r a l l y in a g r e e m e n t with ~' f r o m this work. (fig. la). T h e i r m i n i m u m v a l u e s of a r e shown in fig. lb. T h e s e constitute a smooth e x t r a p o l a t i o n of the p r e s e n t data to l a r g e v o l u m e s . Vashchenko and Z u b a r e v [2] c a l c u l a t e d the v o l u m e dependence of V f o r a r g o n on the b a s i s of four d i f f e r e n t a p p r o x i m a t i o n s . In each c a s e ~ d e c r e a s e d with d e c r e a s i n g v o l u m e . R e c e n t l y a v o l u m e dependence of ~ v e r y s i m i l a r to that r e p o r t e d h e r e a l s o has been o b s e r v e d f o r hcp 3He [3]. It a p p e a r s that the v o l u m e dependence of for hcp 4He is s i m i l a r to what is known and e x p e c t e d about the v o l u m e dependence of ~ for other s i m p l e solids. B a r r o n [4] and G r a h a m [5] pointed out that f o r the h e a v i e r i n e r t gas s o l i d s 7 should be mildly t e m p e r a t u r e dependent at t e m p e r a t u r e s of about ~ 0 o . Even at the s m a l l e s t v o l u m e s ~' for hcp 4He is t e m p e r a t u r e independent at m o s t up to T = = 0.08 0o. S o m e of the t e m p e r a t u r e dependence of

30 January 1967

~' n e a r the m e l t i n g line m ay be c a u s e d by t h e r m al e x c i t a t i o n s other than l a t t i c e v i b r a t i o n s . H o w e v e r , r e c e n t e x p e r i m e n t a l r e s u l t s on a r g o n have shown that ~ f o r this solid is a l s o t e m p e r a t u r e dependent below {0 [6]. The author is indebted to J . F . M a c r e f o r data processing.

References 1. J . J a r v i s and H.Meyer, Proc. 10th Int. Conf. Low Temp. Phys., Moscow, to be published. 2. V. Ya. Vashchenko and V. N. Zubarev, Soviet Phys. Solid State 5 (1963) 653. 3. H.H.Sample and C. A. Swenson (Dept. of Physics, Iowa State University, Ames, Iowa, USA), private communication. 4. T.H.K. Barron, Phil. Mag. 46 (1955) 720. 5. G.M.Graham, Can. J. Phys. 42 (1964) 563. 6. O.G.Peterson et al., Phys. Rev. 150 (1966) 703. 7. J.S.Dugdale, Nuovo Cimento 9, Supplement {1956) 27

*****

VACANCY-TIN

BINDING

IN DILUTE

ALUMINUM-TIN

ALLOYS*

J. BASS Michigan State University, East Lansing, Michigan, USA Received 4 January 1967

The most nearly direct experiments here-to-fore used to justify large vacancy-tin binding in aluminum are shown to be equally consistent with small binding. A cautionary comment is also made concerning less direct experiments.

In v i e w of r e c e n t quotation by v a r i o u s a u t h o r s [1,2] of a binding e n e r g y g r e a t e r than 0.4 eV b e tween v a c a n c i e s and tin a t o m s in a l u m i n u m , it a p p e a r s worthwhile to point out that the two m o s t n e a r l y d i r e c t m e a s u r e m e n t s p u r p o r t i n g to y ie ld such a binding e n e r g y [3,4] not only w e r e o r i g i nally i n t e r p r e a t e d in a mutually i n c o n s i s t e n t m a n n e r , but a l s o y i e l d data which a p p e a r to be i n t e r p r e t a b l e in t e r m s of a much s m a l l e r , p o s s i b l y even z e r o , binding e n e r g y . K i m u r a and Hasiguti [3] c o m p a r e d the slope of an A r r h e n i u s plot of r e s i s t a n c e quenched into p u r e a l u m i n u m with the slope of a s i m i l a r plot of r e s i s t a n c e r e m a i n i n g in an A1-0.006%Sn alloy, * This work was supported by the US Atomic Energy commission.

annealed f o r 5 m i n u t e s at 0°C subsequent to the quench. They hoped by this additional annealing p r o c e d u r e to i n s u r e binding of the q u e n c h e d - i n v a c a n c i e s to i m p u r i t i e s . As shown in the a c companying f i g u r e , they obtained alloy data which a r e l e s s than those for pure a l u m i n u m , with l a r g e r d i f f e r e n c e s o c c u r i n g at h i g h e r t e m p e r a t u r e s . It is well known that v a c a n c i e s anneal in p u r e a l u m i n u m at 0°C, and our own st u d i es of such annealing s u g g e s t that p u r e aluminum a n n e a l e d f o r 5 m i n u t es at 0°C subsequent to quenching will give r e s i s t a n c e l o s s e s of the s a m e s i z e as obs e r v e d by K i m u r a and Hasiguti in the alloy. We t h e r e f o r e s u g g e s t that t h e i r data a r e c o n s i s t e n t with z e r o v a c a n c y - t i n binding en er g y , and with a v a c a n c y - t i n r e s i s t a n c e when p a i r e d which is the s a m e as when they a r e s e p a r a t e d . 153