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Bulk expansivity and vacancies in solid argon
Volume 32A, n u m b e r 2
PHYSICS
LETTERS
s t a t e due to the i n t e r n a l r o t a t i o n of m e t h y l a r o u n d t h e C - C b o n d , by w h i c h t h e r o t a t i o n a l c o n s t a n t s for every form have been determined. The barrier height hindering methyl rotation has been d e t e r m i n e d , u s i n g l i n e s in t h e f i r s t e x c i t e d s t a t e of m e t h y l r o t a t i o n , to b e 3080 ± 50 c a l / m o l e a n d 2870 + 50 c a l / m o l e f o r t h e t r a n s a n d t h e g a u c h e forms respectively.
15 June 1970
The technique of relative-intensity measurement has been applied to the temperature dependence of the intensity ratio of the trans and gauche lines, giving the result that the gauche form is more stable than the trans form by 0.29 ± 0.15 kcal~ole.
* * * * *
BULK
EXPANSIVITY
AND
VACANCIES
IN
SOLID
ARGON
R. G. PRITCHARD * and D. GUGAN H. H. Wills Ph~,sics Laboratory, Royal Fort, Bristol 2, UK
Received 5 May 1970 The difference between the bulk and the X - r a y expansivities of Ar limits the vacancy concentration to (~z/N) ~ 4 × 10 -4 at 75°K. Many-body effects are thus much less than r e c e n t m e a s u r e m e n t s on Kr appear to indicate. The difference between bulk and X-ray measu r e m e n t s of l i n e a r e x p a n s i v i t y i s
ro
3kT2
where the vacancy concentration (n/N)=exp(-g/kT), g=hTs, and whereh ands are the enthalpy and e n t r o p y of m o n o v a c a n c y f o r m a t i o n . We h a v e c o m b i n e d n e w m e a s u r e m e n t s of t h e b u l k e x p a n s i v i t y of a r g o n w i t h t h e X - r a y d a t a of eq. (1) to o b t a i n l i m i t s on h a n d s by u s i n g eq. (1). T h e m e t h o d u s e d w a s to g r o w a p o l y c r y s t a l l i n e A r r o d by s l o w s o l i d i f i c a t i o n f r o m l i q u i d , to i n c o r p o r a t e s t e e l b a l l b e a r i n g (4 m m d i a . ) in t h e r o d a b o u t 50 m m a p a r t , a n d to m a k e t h e r o d " f r e e s t a n d i n g " b y b u r n i n g off with an electrical heater the surface between the Ar rod and the glass growth tube. The separation b e t w e e n t h e b a l l s w a s m e a s u r e d r e l a t i v e to a f i x e d s p a c e r , u s i n g a p a i r of d i f f e r e n t i a l , a i r - c o r e m u tual inductors which could be accurately moved o v e r a d i s t a n c e of 2 m m in o r d e r to d e t e c t l e n g t h c h a n g e s w i t h a p r e c i s i o n of 10 - 4 m m . T h e e x p a n s i v i t y of A r c a n b e m e a s u r e d o v e r a 2 d e g r e e i n t e r v a l w i t h a p r e c i s i o n of ½ p e r c e n t , i.e. 0.3 × 1 0 - 5 K -1 (the s a m e a s t h e q u o t e d a c c u r a c y of t h e X-ray expansivities), as was confirmed by trial e x p e r i m e n t s on KC1 c r y s t a l s . The measurements on Ar proved very difficult b e c a u s e t h e l e n g t h of t h e s p e c i m e n o f t e n s h o w e d * Now at the Physics Dept., University of Salford, Salford, U.K. 124
l a r g e d i s c o n t i n u o u s c h a n g e s , p r o b a b l y due to t h e r e l a t i v e l y l a r g e s i z e of t h e e m b e d d e d b a l l b e a r i n g s . T h e b a l l b e a r i n g s a l s o h a v e a n i m p o r t a n t e f f e c t on t h e l e n g t h of A r w h i c h e x p a n d s . T h e " e f f e c t i v e l e n g t h " i s i n t e r m e d i a t e b e t w e e n t h e s e p a r a t i o n of t h e b a l l b e a r i n g c e n t r e s , a n d t h e s e p a r a t i o n of t h e i r c l o s e s t e d g e s , d e p e n d i n g on t h e e l a s t i c p r o p e r t i e s (and on t h e r e l a t i v e s i z e s ) of t h e A r a n d of t h e s t e e l b a l l s . F r o m a n u m b e r of d e t e r m i n a t i o n s of e x p a n s i v i t y o v e r t h e r a n g e 72K to 78K we c a n at o n c e p i c k out a g r o u p h a v i n g i n t e r n a l c o n s i s t e n c y at t h e e x p e c t e d h a l f p e r c e n t l e v e l , t h e y a r e unmistakably different from those affected by the u n p r e d i c t a b l e l e n g t h c h a n g e s . T h e r e s u l t s in t h i s group, (obtained from one specimen, but measured with increasing and decreasing temperatures, u n d e r d i f f e r e n t c o n d i t i o n s , on d i f f e r e n t d a y s ) , l i e b e l o w t h e X - r a y v a l u e s b y a m e a n v a l u e of 0.5 × 10-5K -1, the mean deviation for four points being 0.3 × 1 0 - 5 K -1 i n d e p e n d e n t of t e m p e r a t u r e . T h e s i g n of AC~ i s o b v i o u s l y i n c o n s i s t e n t w i t h t h e v a c a n c y m o d e l , a n d it p r o b a b l y a r i s e s f r o m a n i n a p p r o p r i a t e v a l u e of t h e e f f e c t i v e l e n g t h a s c a l c u l a t e d from a minimum strain energy model. Nevertheless, the data as they stand are better than any p r e v i o u s v a l u e s f o r t h e e x p a n s i v i t y of s p e c i m e n s in b u l k [e.g. 2], a n d we c a n f u r t h e r u s e t h e m to set an absolute upper limit on the bulk expansivity of A r (i.e. u s i n g t h e s h o r t e s t p o s s i b l e e f f e c t i v e l e n g t h of s p e c i m e n ) . We t h e n o b t a i n Ac~ = 1.2 + 0.2 ( s . e . m . ) × 1 0 - 5 K -1 f o r t e m p e r a t u r e s c l o s e to 7 5 K . T h i s i s s u f f i c i e n t to l i m i t h a n d s c o n s i d e r a b l y , a s
Volume 32A, n u m b e r 2
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/ I
o
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,
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LETTERS
15 June 1970
timates derived from various different analyses [3,4] of s p e c i f i c h e a t d a t a [7] a r e s h o w n b y c l o s e d circles; the large variations here underline the u n c e r t a i n t y of s u c h a n a l y s e s . T h e s o l i d s q u a r e shows results from the total relative linear exp a n s i o n of K r [8] h b e i n g s c a l e d to a p p l y to A r , a n d s/k a s s u m e d t h e s a m e in A r a n d Kr. T h i s r e s u l t a p p e a r s to b e i n c o m p a t i b l e w i t h o u r w o r k , a s a l s o w i t h w o r k on b u l k d e n s i t y (2), a n d on s e l f d i f f u s i o n (9). It t h e r e f o r e s e e m s p r e m a t u r e t o c o n c l u d e [8] t h a t t h r e e - b o d y f o r c e s a n d e l e c t r o n r e l a x a t i o n in A r a r e a s l a r g e a s t h e K r r e s u l t s i m p l y , a b o u t 30 p e r c e n t of t h e c o h e s i v e e n e r g y ; a f i g u r e of l e s s t h a n 10 p e r c e n t s e e m s m o r e l i k e l y , w h i c h i s a l s o c o n s i s t e n t w i t h w o r k on t h e e l a s t i c a n i s t r o p y [10,11].
s/k
Fig. 1, Enthalpy, h , f o r monovacaney f o r m a t i o n in A r
v e r s u s the entropy of monovacaney formation, s/k. The solid l i n e , ~ , gives the lower bound on h deduced from this work, a s s u m i n g A ~ u = 1.6 × 10-5K -1. ©, t h e o r e t i c a l values of h, s/k. 0, values of h, s/k b a s e d on the analysis by various methods of specific heat data [7], see [3,4]. , , h and s/k for Ar e s t i m a t e d from m e a s u r e m e n t s on Kr [8]. To a good approximation, the solid line shifts l i n e a r l y by 6.7 × 10 -3 e V / a for a two-fold v a r i a t i o n in A0tu. s h o w n in fig. 1 w h e r e we p l o t h v e r s u s s/k f o r T = 75K, u s i n g eq. (1) a n d a s s u m i n g a n u p p e r l i m i t A ~ u = 1.6 × 1 0 - 5 K -1. O u r r e s u l t s a l l o w c o m b i n a t i o n s of h a n d s/k w h i c h l i e a b o v e t h e s o l i d l i n e . It t u r n s o u t t h a t t h e v a l u e of g a l l o w e d by o u r r e s u l t s i s a l m o s t i n d e p e n d e n t of s/k, g # 0.053 eV/a, i.e. (n/N) ~ 4 × 10 - 4 at 75K, a n d ~: 10 × 10 - 4 at t h e t r i o l e p o i n t . T h e v a r i o u s t h e o r e t i c a l c a l c u l a t i o n s [ 3 - 6 ] of h a n d s/k a r e s h o w n i n fig. 1 b y o p e n c i r c l e s , w h i l e e x p e r i m e n t a l e s -
References [1] O. G. P e t e r s o n , D.N. Batehelder and R. O. Simmons, Phys. Rev. 150 (1966) 703. [2] B. L. Smith and J. A. Chapman, Phil. Mag. 15 (1967} 739. [3] A. J. E. F o r e m a n and A. B. Lidiard, Phil. Mag. 8 (1963) 97. [4] H.R. Glyde, J. Phys. Chem. Solids 27 (1966) 1659. [5] G. F. Nardelli and A. Repanai Chiarotti, Nuovo Cim. 18 (1960) 1053. [6] G. F. Nardelli and N. T e r z i , J. Phys. Chem. Solids 25 (1964) 815. [7] P. Flubacher, A. J. L e a d b e t t e r and J. A. M o r r i s o n , Proc. Phys. Soc. (London) 78 (1961) 1449. [8] D. J. Losee and R. O. Simmons, Phys. Rev. 172 (1968) 934. [9] E. H. C. P a r k e r , H.R. Glyde and B. L. Smith, Phys. Rev. 176 (1968) 1107. [10] G. J. Keeler and D. N. B a t c h e l d e r , J. Phys. C. (Solid State) 3 (1970) 510. [11] I. J. Zucker and G. G. Chell, J. Phys. C. (Solid State) 1 (1968) 1505.
125
PHYSICS
LETTERS
s t a t e due to the i n t e r n a l r o t a t i o n of m e t h y l a r o u n d t h e C - C b o n d , by w h i c h t h e r o t a t i o n a l c o n s t a n t s for every form have been determined. The barrier height hindering methyl rotation has been d e t e r m i n e d , u s i n g l i n e s in t h e f i r s t e x c i t e d s t a t e of m e t h y l r o t a t i o n , to b e 3080 ± 50 c a l / m o l e a n d 2870 + 50 c a l / m o l e f o r t h e t r a n s a n d t h e g a u c h e forms respectively.
15 June 1970
The technique of relative-intensity measurement has been applied to the temperature dependence of the intensity ratio of the trans and gauche lines, giving the result that the gauche form is more stable than the trans form by 0.29 ± 0.15 kcal~ole.
* * * * *
BULK
EXPANSIVITY
AND
VACANCIES
IN
SOLID
ARGON
R. G. PRITCHARD * and D. GUGAN H. H. Wills Ph~,sics Laboratory, Royal Fort, Bristol 2, UK
Received 5 May 1970 The difference between the bulk and the X - r a y expansivities of Ar limits the vacancy concentration to (~z/N) ~ 4 × 10 -4 at 75°K. Many-body effects are thus much less than r e c e n t m e a s u r e m e n t s on Kr appear to indicate. The difference between bulk and X-ray measu r e m e n t s of l i n e a r e x p a n s i v i t y i s
ro
3kT2
where the vacancy concentration (n/N)=exp(-g/kT), g=hTs, and whereh ands are the enthalpy and e n t r o p y of m o n o v a c a n c y f o r m a t i o n . We h a v e c o m b i n e d n e w m e a s u r e m e n t s of t h e b u l k e x p a n s i v i t y of a r g o n w i t h t h e X - r a y d a t a of eq. (1) to o b t a i n l i m i t s on h a n d s by u s i n g eq. (1). T h e m e t h o d u s e d w a s to g r o w a p o l y c r y s t a l l i n e A r r o d by s l o w s o l i d i f i c a t i o n f r o m l i q u i d , to i n c o r p o r a t e s t e e l b a l l b e a r i n g (4 m m d i a . ) in t h e r o d a b o u t 50 m m a p a r t , a n d to m a k e t h e r o d " f r e e s t a n d i n g " b y b u r n i n g off with an electrical heater the surface between the Ar rod and the glass growth tube. The separation b e t w e e n t h e b a l l s w a s m e a s u r e d r e l a t i v e to a f i x e d s p a c e r , u s i n g a p a i r of d i f f e r e n t i a l , a i r - c o r e m u tual inductors which could be accurately moved o v e r a d i s t a n c e of 2 m m in o r d e r to d e t e c t l e n g t h c h a n g e s w i t h a p r e c i s i o n of 10 - 4 m m . T h e e x p a n s i v i t y of A r c a n b e m e a s u r e d o v e r a 2 d e g r e e i n t e r v a l w i t h a p r e c i s i o n of ½ p e r c e n t , i.e. 0.3 × 1 0 - 5 K -1 (the s a m e a s t h e q u o t e d a c c u r a c y of t h e X-ray expansivities), as was confirmed by trial e x p e r i m e n t s on KC1 c r y s t a l s . The measurements on Ar proved very difficult b e c a u s e t h e l e n g t h of t h e s p e c i m e n o f t e n s h o w e d * Now at the Physics Dept., University of Salford, Salford, U.K. 124
l a r g e d i s c o n t i n u o u s c h a n g e s , p r o b a b l y due to t h e r e l a t i v e l y l a r g e s i z e of t h e e m b e d d e d b a l l b e a r i n g s . T h e b a l l b e a r i n g s a l s o h a v e a n i m p o r t a n t e f f e c t on t h e l e n g t h of A r w h i c h e x p a n d s . T h e " e f f e c t i v e l e n g t h " i s i n t e r m e d i a t e b e t w e e n t h e s e p a r a t i o n of t h e b a l l b e a r i n g c e n t r e s , a n d t h e s e p a r a t i o n of t h e i r c l o s e s t e d g e s , d e p e n d i n g on t h e e l a s t i c p r o p e r t i e s (and on t h e r e l a t i v e s i z e s ) of t h e A r a n d of t h e s t e e l b a l l s . F r o m a n u m b e r of d e t e r m i n a t i o n s of e x p a n s i v i t y o v e r t h e r a n g e 72K to 78K we c a n at o n c e p i c k out a g r o u p h a v i n g i n t e r n a l c o n s i s t e n c y at t h e e x p e c t e d h a l f p e r c e n t l e v e l , t h e y a r e unmistakably different from those affected by the u n p r e d i c t a b l e l e n g t h c h a n g e s . T h e r e s u l t s in t h i s group, (obtained from one specimen, but measured with increasing and decreasing temperatures, u n d e r d i f f e r e n t c o n d i t i o n s , on d i f f e r e n t d a y s ) , l i e b e l o w t h e X - r a y v a l u e s b y a m e a n v a l u e of 0.5 × 10-5K -1, the mean deviation for four points being 0.3 × 1 0 - 5 K -1 i n d e p e n d e n t of t e m p e r a t u r e . T h e s i g n of AC~ i s o b v i o u s l y i n c o n s i s t e n t w i t h t h e v a c a n c y m o d e l , a n d it p r o b a b l y a r i s e s f r o m a n i n a p p r o p r i a t e v a l u e of t h e e f f e c t i v e l e n g t h a s c a l c u l a t e d from a minimum strain energy model. Nevertheless, the data as they stand are better than any p r e v i o u s v a l u e s f o r t h e e x p a n s i v i t y of s p e c i m e n s in b u l k [e.g. 2], a n d we c a n f u r t h e r u s e t h e m to set an absolute upper limit on the bulk expansivity of A r (i.e. u s i n g t h e s h o r t e s t p o s s i b l e e f f e c t i v e l e n g t h of s p e c i m e n ) . We t h e n o b t a i n Ac~ = 1.2 + 0.2 ( s . e . m . ) × 1 0 - 5 K -1 f o r t e m p e r a t u r e s c l o s e to 7 5 K . T h i s i s s u f f i c i e n t to l i m i t h a n d s c o n s i d e r a b l y , a s
Volume 32A, n u m b e r 2
PHYSICS
h eY/c
0.10
yF . 0 (4)
--;tom L~Cp
/ 0.05
• ~7)
•
/ I
o
,
,
,
,
I
5
,
,
,
LETTERS
15 June 1970
timates derived from various different analyses [3,4] of s p e c i f i c h e a t d a t a [7] a r e s h o w n b y c l o s e d circles; the large variations here underline the u n c e r t a i n t y of s u c h a n a l y s e s . T h e s o l i d s q u a r e shows results from the total relative linear exp a n s i o n of K r [8] h b e i n g s c a l e d to a p p l y to A r , a n d s/k a s s u m e d t h e s a m e in A r a n d Kr. T h i s r e s u l t a p p e a r s to b e i n c o m p a t i b l e w i t h o u r w o r k , a s a l s o w i t h w o r k on b u l k d e n s i t y (2), a n d on s e l f d i f f u s i o n (9). It t h e r e f o r e s e e m s p r e m a t u r e t o c o n c l u d e [8] t h a t t h r e e - b o d y f o r c e s a n d e l e c t r o n r e l a x a t i o n in A r a r e a s l a r g e a s t h e K r r e s u l t s i m p l y , a b o u t 30 p e r c e n t of t h e c o h e s i v e e n e r g y ; a f i g u r e of l e s s t h a n 10 p e r c e n t s e e m s m o r e l i k e l y , w h i c h i s a l s o c o n s i s t e n t w i t h w o r k on t h e e l a s t i c a n i s t r o p y [10,11].
s/k
Fig. 1, Enthalpy, h , f o r monovacaney f o r m a t i o n in A r
v e r s u s the entropy of monovacaney formation, s/k. The solid l i n e , ~ , gives the lower bound on h deduced from this work, a s s u m i n g A ~ u = 1.6 × 10-5K -1. ©, t h e o r e t i c a l values of h, s/k. 0, values of h, s/k b a s e d on the analysis by various methods of specific heat data [7], see [3,4]. , , h and s/k for Ar e s t i m a t e d from m e a s u r e m e n t s on Kr [8]. To a good approximation, the solid line shifts l i n e a r l y by 6.7 × 10 -3 e V / a for a two-fold v a r i a t i o n in A0tu. s h o w n in fig. 1 w h e r e we p l o t h v e r s u s s/k f o r T = 75K, u s i n g eq. (1) a n d a s s u m i n g a n u p p e r l i m i t A ~ u = 1.6 × 1 0 - 5 K -1. O u r r e s u l t s a l l o w c o m b i n a t i o n s of h a n d s/k w h i c h l i e a b o v e t h e s o l i d l i n e . It t u r n s o u t t h a t t h e v a l u e of g a l l o w e d by o u r r e s u l t s i s a l m o s t i n d e p e n d e n t of s/k, g # 0.053 eV/a, i.e. (n/N) ~ 4 × 10 - 4 at 75K, a n d ~: 10 × 10 - 4 at t h e t r i o l e p o i n t . T h e v a r i o u s t h e o r e t i c a l c a l c u l a t i o n s [ 3 - 6 ] of h a n d s/k a r e s h o w n i n fig. 1 b y o p e n c i r c l e s , w h i l e e x p e r i m e n t a l e s -
References [1] O. G. P e t e r s o n , D.N. Batehelder and R. O. Simmons, Phys. Rev. 150 (1966) 703. [2] B. L. Smith and J. A. Chapman, Phil. Mag. 15 (1967} 739. [3] A. J. E. F o r e m a n and A. B. Lidiard, Phil. Mag. 8 (1963) 97. [4] H.R. Glyde, J. Phys. Chem. Solids 27 (1966) 1659. [5] G. F. Nardelli and A. Repanai Chiarotti, Nuovo Cim. 18 (1960) 1053. [6] G. F. Nardelli and N. T e r z i , J. Phys. Chem. Solids 25 (1964) 815. [7] P. Flubacher, A. J. L e a d b e t t e r and J. A. M o r r i s o n , Proc. Phys. Soc. (London) 78 (1961) 1449. [8] D. J. Losee and R. O. Simmons, Phys. Rev. 172 (1968) 934. [9] E. H. C. P a r k e r , H.R. Glyde and B. L. Smith, Phys. Rev. 176 (1968) 1107. [10] G. J. Keeler and D. N. B a t c h e l d e r , J. Phys. C. (Solid State) 3 (1970) 510. [11] I. J. Zucker and G. G. Chell, J. Phys. C. (Solid State) 1 (1968) 1505.
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