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Temperature dependence of the interference function of liquid cadmium
PHYSICS
Volume 30A, n u m b e r 8
tial and piezo-electrical potential interaction. Interaction with the deformation potential a l o n e g i v e s n = + 1 [5], t h e p r e s e n c e of t h e p i e z o e l e c t r i c p o t e n t i a l l e a d s to n < 1 [5,6]. In o u r m e a s u r e m e n t s n = 0.9 w a s o b t a i n e d , w h a t m e a n s that combined interaction with the deformation potential and the piezoelectric potential is present.
TEMPERATURE
15 December 1969
LETTERS
References 1. H Neumann, Z. Naturf. 23a (1968) 204. 2. H. Ncumann. Ann. Phys. 22 (1968) 40. 3. K Yamamoto, S.Yano and K. Abe, Japan. J Appl. Phys. 6 (1967) 1222. 4. M. Onuki and K. Shiga, Proc. Intern. Conf. Phys. Semieond., Kyoto 1966, 427. 5. R.S. Crandall. Phys. Rev. 169 (1968) 577. 6. M. Sodha and S.Sharma, Phys. Star. Sol. 30 (1968} 239. 7. E.M. Conwell, Phys. Rcv. 90 (1953) 769.
DEPENDENCE OF THE INTERFERENCE OF LIQUID CADMIUM *
D. M. N O R T H $ a n d C. N. J . W A G N E R Harnrnond L a b o r a t o r y , Yale University, New Haven, Connecticut,
FUNCTION
USA
Received 11 November 1969
The t e m p e r a t u r e dependence of the i n t e r f e r e n c e function (or s t r u c t u r e factor) I(K) of liquid cadmium has been m e a s u r e d between 350 ° and 650°C. The e l e c t r i c a l r e s i s t i v i t y and the t h e r m o - e l e c t r i c power were calculated and a c o m p a r i s o n made with e x p e r i m e n t a l data.
T h e t h e o r y of e l e c t r o n t r a n s p o r t p r o p e r t i e s [1,2] m a k e s u s e of t h e e x p e r i m e n t a l l y a c c e s s i b l e i n t e r f e r e n c e f u n c t i o n I(K) ( a l s o c a l l e d s t r u c t u r e factor) defined as: I(K) = Ia(K)/f2
= 1 +
f 4~r2po[g(r)-
1] -s ~i n K r
dr ,
where Ia(K) is the elastically scattered X-ray intensity per atom expressed in electron units, f is t h e X - r a y s c a t t e r i n g f a c t o r , Po i s t h e a t o m i c density, and g(r) is the pair probability function. S i g n i f i c a n t c h a n g e s in t h e m a g n i t u d e of t h e f i r s t p e a k i n I (K) w i t h t e m p e r a t u r e h a v e b e e n o b s e r v e d f o r t r i a n d t e t r a v a l e n t m e t a l s [3,4], w h i c h l e d to good a g r e e m e n t b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l t e m p e r a t u r e c o e f f i c i e n t s a p of t h e electrical resistivity. However, some-divalent metals and some alloys possess negative or weakly t e m p e r a t u r e d e p e n d e n t c o e f f i c i e n t s w h i c h h a v e b e e n e x p l a i n e d [2,5] a s a c o n s e q u e n c e of t h e p r o x i m i t y of t h e v a l u e s of t h e F e r m i d i a m e t e r 2k F t o t h e p o s i t i o n K 1 of t h e f i r s t p e a k in I ( K ) .
o
350
o
I
2
K o 4,,{s~.e)A.
* R e s e a r c h supported by the United States Atomic Energy Commission. P r e s e n t a d d r e s s : Brookhaven National Laboratory, Upton, N.Y. 440
3
i'h-' ]
Fig. 1. F i r s t peaks of the i n t e r f e r e n c e functions l(/O of liquid Cd m e a s u r e d at 350, 450, 550 and 650°C. 2k F is the F e r m i d i a m e t e r a n d K o is the node of U(//) of Cd.
PHYSICS
Volume 30A, n u m b e r 8
LETTERS
Table 1 The v a r i a t i o n of the position K 1 and height I(K1) of the f i r s t peak in the i n t e r f e r e n c e function, the F e r m i d i a m e t e r 2k F, and the t h e o r e t i c a l and e x p e r i m e n t a l values of r e s i s t i v i t y p and t h e r m o - e l e c t r i e power O with t e m perature. T
K1
2 kF
(Oc) (/~-1) I(K1) (~-1) 350 450 550 650
2.57 2.56 2.55 2.54
2.55 2.45 2.38 2.24
2.726 2.714 2.702 2.690
P th
P ex
(/_t~cm) 23.5±0.6 24.2±0.6 24.8~:0.6 26.8±0.6
Qth
Oex
(]2V/°K)
34.5 2.5 ±0.1 34.7 2.0 ~-0.1 35.6 0.98±0.1 37.5 -0.36i0.1
0.50 0.57 0.65 0.73
In t h i s l e t t e r we r e p o r t t h e i n t e r f e r e n c e f u n c t i o n s of l i q u i d Cd w h i c h w e r e m e a s u r e d w i t h M o Kol r a d i a t i o n in t r a n s m i s s i o n [6] in t h e r a n g e f r o m K = 4 ~ s i n 0/)~ = 0 . 5 / ~ - 1 t o 12.4 ~ - 1 a n d a t t e m p e r atures between 350°C and 650°C. F i g u r e 1 S h o w s t h e f i r s t p e a k of I(K) f o r l i q u i d Cd f o r a l l t e m p e r a t u r e s of m e a s u r e m e n t . The d a t a a r e a n a l o g o u s i n m a n y r e s p e c t s to r e c e n t r e s u l t s on l i q u i d Z n [4,7]. T h e p o s i t i o n K 1 a n d h e i g h t I(K) a r e r e l a t i v e l y t e m p e r a t u r e i n s e n s i t i v e . E x p r e s s e d a s t h e r a t i o X 1 = K1/2k F, o n e f i n d s X 1 = 0.945 to b e i n d e p e n d e n t of t e m p e r a t u r e . I(/O for all temperatures are asymmetric, in t h a t t h e low a n g l e s i d e of t h e f i r s t p e a k i s l e s s s t e e p t h a n the high angle side. A possible explanation for t h i s r e s u l t m a y b e f o u n d in r e c e n t a r g u m e n t s d u e to H e i n e a n d W e a i r e [8]. The electrical resistivity p and thermo-electric power Q were evaluated using the AnimaluH e i n e m o d e l p o t e n t i a l [9], o n a s s u m i n g t h a t it i s energy independent. Corrections for volume c h a n g e s w e r e a p p l i e d , a n d t h e r e s u l t s (Pth) a r e c o m p a r e d w i t h e x p e r i m e n t a l d a t a (Pex)[10] in t a b l e 1. T h o u g h t h e r e s u l t s a r e i n q u a l i t a t i v e a g r e e ment with experiment, inasmuch as both are p r a c t i c a l l y t e m p e r a t u r e i n d e p e n d e n t up to 4 5 0 ° C , a n d t h e s l o p e s A p / A T a g r e e up to 6 5 0 o c , t h e temperature coefficient (defined as = {p (550) - p ( 3 5 0 ) } / 2 0 0 p ( 3 5 0 ) ) a ~ x ~ P . 3 × 1 0 - 4 / ° C
15 D e c e m b e r 1969
i s s m a l l e r t h a n ~pth = (2.8 + 2.5) x 1 0 - 4 / o c , b u t l i e s w i t h i n t h e e r r o r of t h e t h e o r e t i c a l v a l u e . T h e t e m p e r a t u r e b e h a v i o r of Q t h , h o w e v e r , i s in d r a s t i c d i s a c c o r d w i t h Qex [11]. A discrepancy between ~th and sex for liquid 1 2 N a h a s b e e n i n t e r p r e t e d a~ e v i d e n c e f o r the b r e a k d o w n of t h e Z i m a n t h e o r y . H o w e v e r , due to t h e l a r g e u n c e r t a i n t y in a t h f o r Cd no s u c h c o n clusion can be drawn here. Furthermore, the theory has had overall success for some other p o l y v a l e n t m e t a l s a s m e n t i o n e d e a r l i e r [3,4]. The large departure from even qualitative a g r e e m e n t f o r t h e t h e r m o p o w e r f o r Cd ( a s f o r o t h e r d i v a l e n t m e t a l s l i k e Zn [7] a n d Hg [2]) w o u l d s u g g e s t t h a t t h e e n e r g y d e p e n d e n c e of t h e p s e u d o p o t e n t i a l c a n n o t b e i g n o r e d . A s h i f t in t h e n o d e K o of t h e p s e u d o p o t e n t i a l e l e m e n t s s e e m s to b e r e a l e f f e c t in d i s c u s s i n g e n e r g y d e p e n d e n t p o t e n tials, and this factor alone would most certainly i n f l u e n c e t h e r e s u l t s f o r Qth, a n d q u i t e p o s s i b l y t h e t e m p e r a t u r e d e p e n d e n c e of Pth"
References 1. J. M. Ziman, Phil. Mag. 6 (1961) 1013. 2. C. C. Bradley, T . E . F a b e r , E.G. Wilson and J. M. Ziman, Phil. Mag. 7 (1962) 865. 3. N. C. Halder and C. N. Wagner, J. Chem. Phys. 45 (1966) 482. 4. D.M. North, J . E . Enderby and P. A. Egelstaff, J. Phys. C. 1 (1968) 1075. 5. G. Busch and H. -J. GUntherodt. Phys. Kondens. Materie 6 (1967) 325. 6. D. M. North and C. N. J. Wagner, J. Appl. Cryst.. to be published. 7. B. F. Wingfield and J. E. Enderby, Phys. L e t t e r s 27A (1968) 704. 8. V. Heine and D. Weaire, Phys. Rev. 152 (1966) 603. 9. A . O . E . Animalu and V. Heine. Phil. Mag. 12 (1965) 1249. (Values for Cd were c o r r e c t e d in Technical Report No. 4, Solid State Theory Group, Cavandish Lab, Cambridge, England.) 10. A. Roll and H Motz, Z. Metallkunde 48 (1957) 272. 11. A. S. Marwaha and N. E. Cusack, Phys. L e t t e r s 22 (1966) 556. 12. A. J. Greenfield, Phys. Rev. L e t t e r s 16 (1966) 6.
*****
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Volume 30A, n u m b e r 8
tial and piezo-electrical potential interaction. Interaction with the deformation potential a l o n e g i v e s n = + 1 [5], t h e p r e s e n c e of t h e p i e z o e l e c t r i c p o t e n t i a l l e a d s to n < 1 [5,6]. In o u r m e a s u r e m e n t s n = 0.9 w a s o b t a i n e d , w h a t m e a n s that combined interaction with the deformation potential and the piezoelectric potential is present.
TEMPERATURE
15 December 1969
LETTERS
References 1. H Neumann, Z. Naturf. 23a (1968) 204. 2. H. Ncumann. Ann. Phys. 22 (1968) 40. 3. K Yamamoto, S.Yano and K. Abe, Japan. J Appl. Phys. 6 (1967) 1222. 4. M. Onuki and K. Shiga, Proc. Intern. Conf. Phys. Semieond., Kyoto 1966, 427. 5. R.S. Crandall. Phys. Rev. 169 (1968) 577. 6. M. Sodha and S.Sharma, Phys. Star. Sol. 30 (1968} 239. 7. E.M. Conwell, Phys. Rcv. 90 (1953) 769.
DEPENDENCE OF THE INTERFERENCE OF LIQUID CADMIUM *
D. M. N O R T H $ a n d C. N. J . W A G N E R Harnrnond L a b o r a t o r y , Yale University, New Haven, Connecticut,
FUNCTION
USA
Received 11 November 1969
The t e m p e r a t u r e dependence of the i n t e r f e r e n c e function (or s t r u c t u r e factor) I(K) of liquid cadmium has been m e a s u r e d between 350 ° and 650°C. The e l e c t r i c a l r e s i s t i v i t y and the t h e r m o - e l e c t r i c power were calculated and a c o m p a r i s o n made with e x p e r i m e n t a l data.
T h e t h e o r y of e l e c t r o n t r a n s p o r t p r o p e r t i e s [1,2] m a k e s u s e of t h e e x p e r i m e n t a l l y a c c e s s i b l e i n t e r f e r e n c e f u n c t i o n I(K) ( a l s o c a l l e d s t r u c t u r e factor) defined as: I(K) = Ia(K)/f2
= 1 +
f 4~r2po[g(r)-
1] -s ~i n K r
dr ,
where Ia(K) is the elastically scattered X-ray intensity per atom expressed in electron units, f is t h e X - r a y s c a t t e r i n g f a c t o r , Po i s t h e a t o m i c density, and g(r) is the pair probability function. S i g n i f i c a n t c h a n g e s in t h e m a g n i t u d e of t h e f i r s t p e a k i n I (K) w i t h t e m p e r a t u r e h a v e b e e n o b s e r v e d f o r t r i a n d t e t r a v a l e n t m e t a l s [3,4], w h i c h l e d to good a g r e e m e n t b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l t e m p e r a t u r e c o e f f i c i e n t s a p of t h e electrical resistivity. However, some-divalent metals and some alloys possess negative or weakly t e m p e r a t u r e d e p e n d e n t c o e f f i c i e n t s w h i c h h a v e b e e n e x p l a i n e d [2,5] a s a c o n s e q u e n c e of t h e p r o x i m i t y of t h e v a l u e s of t h e F e r m i d i a m e t e r 2k F t o t h e p o s i t i o n K 1 of t h e f i r s t p e a k in I ( K ) .
o
350
o
I
2
K o 4,,{s~.e)A.
* R e s e a r c h supported by the United States Atomic Energy Commission. P r e s e n t a d d r e s s : Brookhaven National Laboratory, Upton, N.Y. 440
3
i'h-' ]
Fig. 1. F i r s t peaks of the i n t e r f e r e n c e functions l(/O of liquid Cd m e a s u r e d at 350, 450, 550 and 650°C. 2k F is the F e r m i d i a m e t e r a n d K o is the node of U(//) of Cd.
PHYSICS
Volume 30A, n u m b e r 8
LETTERS
Table 1 The v a r i a t i o n of the position K 1 and height I(K1) of the f i r s t peak in the i n t e r f e r e n c e function, the F e r m i d i a m e t e r 2k F, and the t h e o r e t i c a l and e x p e r i m e n t a l values of r e s i s t i v i t y p and t h e r m o - e l e c t r i e power O with t e m perature. T
K1
2 kF
(Oc) (/~-1) I(K1) (~-1) 350 450 550 650
2.57 2.56 2.55 2.54
2.55 2.45 2.38 2.24
2.726 2.714 2.702 2.690
P th
P ex
(/_t~cm) 23.5±0.6 24.2±0.6 24.8~:0.6 26.8±0.6
Qth
Oex
(]2V/°K)
34.5 2.5 ±0.1 34.7 2.0 ~-0.1 35.6 0.98±0.1 37.5 -0.36i0.1
0.50 0.57 0.65 0.73
In t h i s l e t t e r we r e p o r t t h e i n t e r f e r e n c e f u n c t i o n s of l i q u i d Cd w h i c h w e r e m e a s u r e d w i t h M o Kol r a d i a t i o n in t r a n s m i s s i o n [6] in t h e r a n g e f r o m K = 4 ~ s i n 0/)~ = 0 . 5 / ~ - 1 t o 12.4 ~ - 1 a n d a t t e m p e r atures between 350°C and 650°C. F i g u r e 1 S h o w s t h e f i r s t p e a k of I(K) f o r l i q u i d Cd f o r a l l t e m p e r a t u r e s of m e a s u r e m e n t . The d a t a a r e a n a l o g o u s i n m a n y r e s p e c t s to r e c e n t r e s u l t s on l i q u i d Z n [4,7]. T h e p o s i t i o n K 1 a n d h e i g h t I(K) a r e r e l a t i v e l y t e m p e r a t u r e i n s e n s i t i v e . E x p r e s s e d a s t h e r a t i o X 1 = K1/2k F, o n e f i n d s X 1 = 0.945 to b e i n d e p e n d e n t of t e m p e r a t u r e . I(/O for all temperatures are asymmetric, in t h a t t h e low a n g l e s i d e of t h e f i r s t p e a k i s l e s s s t e e p t h a n the high angle side. A possible explanation for t h i s r e s u l t m a y b e f o u n d in r e c e n t a r g u m e n t s d u e to H e i n e a n d W e a i r e [8]. The electrical resistivity p and thermo-electric power Q were evaluated using the AnimaluH e i n e m o d e l p o t e n t i a l [9], o n a s s u m i n g t h a t it i s energy independent. Corrections for volume c h a n g e s w e r e a p p l i e d , a n d t h e r e s u l t s (Pth) a r e c o m p a r e d w i t h e x p e r i m e n t a l d a t a (Pex)[10] in t a b l e 1. T h o u g h t h e r e s u l t s a r e i n q u a l i t a t i v e a g r e e ment with experiment, inasmuch as both are p r a c t i c a l l y t e m p e r a t u r e i n d e p e n d e n t up to 4 5 0 ° C , a n d t h e s l o p e s A p / A T a g r e e up to 6 5 0 o c , t h e temperature coefficient (defined as = {p (550) - p ( 3 5 0 ) } / 2 0 0 p ( 3 5 0 ) ) a ~ x ~ P . 3 × 1 0 - 4 / ° C
15 D e c e m b e r 1969
i s s m a l l e r t h a n ~pth = (2.8 + 2.5) x 1 0 - 4 / o c , b u t l i e s w i t h i n t h e e r r o r of t h e t h e o r e t i c a l v a l u e . T h e t e m p e r a t u r e b e h a v i o r of Q t h , h o w e v e r , i s in d r a s t i c d i s a c c o r d w i t h Qex [11]. A discrepancy between ~th and sex for liquid 1 2 N a h a s b e e n i n t e r p r e t e d a~ e v i d e n c e f o r the b r e a k d o w n of t h e Z i m a n t h e o r y . H o w e v e r , due to t h e l a r g e u n c e r t a i n t y in a t h f o r Cd no s u c h c o n clusion can be drawn here. Furthermore, the theory has had overall success for some other p o l y v a l e n t m e t a l s a s m e n t i o n e d e a r l i e r [3,4]. The large departure from even qualitative a g r e e m e n t f o r t h e t h e r m o p o w e r f o r Cd ( a s f o r o t h e r d i v a l e n t m e t a l s l i k e Zn [7] a n d Hg [2]) w o u l d s u g g e s t t h a t t h e e n e r g y d e p e n d e n c e of t h e p s e u d o p o t e n t i a l c a n n o t b e i g n o r e d . A s h i f t in t h e n o d e K o of t h e p s e u d o p o t e n t i a l e l e m e n t s s e e m s to b e r e a l e f f e c t in d i s c u s s i n g e n e r g y d e p e n d e n t p o t e n tials, and this factor alone would most certainly i n f l u e n c e t h e r e s u l t s f o r Qth, a n d q u i t e p o s s i b l y t h e t e m p e r a t u r e d e p e n d e n c e of Pth"
References 1. J. M. Ziman, Phil. Mag. 6 (1961) 1013. 2. C. C. Bradley, T . E . F a b e r , E.G. Wilson and J. M. Ziman, Phil. Mag. 7 (1962) 865. 3. N. C. Halder and C. N. Wagner, J. Chem. Phys. 45 (1966) 482. 4. D.M. North, J . E . Enderby and P. A. Egelstaff, J. Phys. C. 1 (1968) 1075. 5. G. Busch and H. -J. GUntherodt. Phys. Kondens. Materie 6 (1967) 325. 6. D. M. North and C. N. J. Wagner, J. Appl. Cryst.. to be published. 7. B. F. Wingfield and J. E. Enderby, Phys. L e t t e r s 27A (1968) 704. 8. V. Heine and D. Weaire, Phys. Rev. 152 (1966) 603. 9. A . O . E . Animalu and V. Heine. Phil. Mag. 12 (1965) 1249. (Values for Cd were c o r r e c t e d in Technical Report No. 4, Solid State Theory Group, Cavandish Lab, Cambridge, England.) 10. A. Roll and H Motz, Z. Metallkunde 48 (1957) 272. 11. A. S. Marwaha and N. E. Cusack, Phys. L e t t e r s 22 (1966) 556. 12. A. J. Greenfield, Phys. Rev. L e t t e r s 16 (1966) 6.
*****
441