The resistivity of liquid aluminium

Volume 14, number 1

PHYSICS LETTERS

THE

RESISTIVITY

OF

1 January 1965

LIQUID

ALUMINIUM

N. W. ASHCROFT * and LORNA J. GUILD **

Cavendish Laboratory, Cambridge, England Received 2 December 1965 Z i m a n [1] and B r a d l e y et al. [2] have a p p l i e d the p s e u d o p o t e n t i a l a p p r o a c h to a c a l c u l a t i o n of the t r a n s p o r t p r o p e r t i e s of liquid m e t a l s . T h e i r r e s u l t for the r e s i s t i v i t y i s :

(3rr/e21i) ( 1 / v 2) (aV2) ~ where

a(K)

(1)

2k F

(aV2} = (4k4) -1 f

2,0

1.0

a(K) IV(K)] 2 K 3 dK. (2)

O

is the volume p e r a t o m , and kF, v F the m o m e n t u m and v e l o c i t y at the F e r m i level, a(K) is the c o r r e l a t i o n function. T h e s e f o r m u l a e have b e e n e v a l u a t e d with much g r e a t e r c a r e than p r e v i o u s l y f o r a l u m i n i u m at 700°C. The a(K) is t a k e n f r o m X - r a y m e a s u r e m e n t s [3] and is shown in fig. 1. V(K), the p s e u d o p o t e n t i a l , can be d e t e r m i n e d by i n t e r p o l a t i n g b e t w e e n the K ~ 0 l i m i t w h e r e i t s value is fixed by v e r y g e n e r a l c o n s i d e r a t i o n s [4] and two e x p e r i m e n t a l l y d e t e r m i n e d p o i n t s at K equal to the (111) and (200) r e c i p r o c a l l a t t i c e v e c t o r s . The l a t t e r have been d e d u c e d f r o m the shape of the F e r m i s u r f a c e a s e l u c i d a t e d by de H a a s - v a n Alphen and o t h e r e f f e c t s [5, 6]. A s m a l l c o r r e c t i o n is n e c e s s a r y due to the change in a t o m i c volume between s o l i d at 4°K and liquid at 700oc. T h e s e p a r a m e t e r s give a m e a s u r e of the s c a t t e r i n g a m p l i t u d e of the p o t e n t i a l for the s c a t t e r i n g of e l e c t r o n s confined to the F e r m i s u r f a c e which is the s a m e quantity a s r e q u i r e d in eq. (1) and (2). (Although (1) and (2) a r e w r i t t e n in the B o r n a p p r o x i m a t i o n the f a c t that we a r e u s i n g the e x p e r i m e n t a l s c a t t e r ing a m p l i t u d e s m e a n s that we a r e a l r e a d y including the m o s t i m p o r t a n t h i g h e r o r d e r t e r m s . ) The i n t e r p o l a t i o n is a c h i e v e d v i a the m o d e l p o t e n t i a l of Heine and A b a r e n k o v [7]: a c a l c u l a * Present address: Institute for the Study of Metals, The University of Chicago, Chicago, Illinois 60637 U.S.A. !Work supported in part under Contract No. NONR2121(22). ** Present address: Lorna J. Sundstr~m, Teknillisen Fysiikan Osasto Otaniemi, Helsinki, Finland.

i

0,0

0.2

0.4

0.6

I

0.8

1.0 .(K/2kF~ .~

Fig. 1. The correlation function a(K) for aluminium at 700oc. tion [8, 9] of V(K) ab initio f r o m the m o d e l p o t e n t i a l g i v e s a c u r v e going within 0.01 r y of the e x p e r i m e n t a l p o i n t s at K = (111) and (200). Only a v e r y s l i g h t a d j u s t m e n t of the m o d e l p o t e n t i a l p a r a m e t e r s is r e q u i r e d to m a k e the c u r v e go t h r o u g h the two points e x a c t l y , and this then should give a e x c e l l e n t i n t e r p o l a t i o n (fig. 2). In p a r t i c u l a r , V(K) is w e l l d e t e r m i n e d at l a r g e K w h e r e (since a (K) is l a r g e and K 3 is r i s i n g steeply) m o s t of the i n t e g r a l (3) is g e n e r a t e d . E x p r e s s i o n s (1) and (2) a l s o include a m i n o r e f f e c t i v e m a s s c o r r e c t i o n (1.06) r e s u l t i n g f r o m the d e t a i l e d band s t r u c t u r e of the m e t a l [6]. E l e c t r o n - e l e c t r o n i n t e r a c t i o n s a r e e x p e c t e d to y i e l d a n e g l i g i b l e m a s s e n h a n c e m e n t [10]. At the t e m p e r a t u r e s we a r e d e a l i n g with h e r e the o v e r a l l d e n s i t y of s t a t e s is unaffected by the electron-phonon interaction. The c a l c u l a t e d and e x p e r i m e n t a l v a l u e s for the r e s i s t i v i t y of liquid a l u m i n i u m at 700 ° in IZohm c m a r e as follows: expt.

24.7

theor.

24.5

The s i g n i f i c a n c e of the c l o s e a g r e e m e n t with e x p e r i m e n t is twofold. F i r s t , it d e m o n s t r a t e s 23

Volume 14, number 1

C.2

PHYSICS LETTERS

v(K) Rydbergs

0.i

0.0

I

I

I 8

1.0

._(_K_/ 2 k

F)

-0.I

-0.2

1 January 1965

be r e l a t e d to the f o r m factor V(K). It f u r t h e r shows how i n f o r m a t i o n f r o m one p r o p e r t y can be applied to another. Second, since both V(K) and a(K) a r e taken f r o m e x p e r i m e n t the c a l c u l a t i o n p r o v i d e s a f i r m check on the basic c o r r e c t n e s s of Z i m a n ' s theory and r e m o v e s doubts about a s y s t e m a t i c d i s c r e p a n c y suggested by the rough e s t i m a t e s of Bradley et al. [2]. The r e s u l t s of f u r t h e r applications to a s e r i e s of m e t a l s will be given l a t e r [11]. We wish to thank Dr. V. Heine for valuable d i s c u s s i o n s .

References -0.3

1. J.M.Ziman, Phil.Meg. 6 (1961) 1013. 2. C.C.Bradley, T . E . F a b e r , E.G.Wilson and J.M. Zi~nan, Phil.Mag. 7 (1962) 865. 3. C.Gamertsfelder, J.Chem,Phys. 9 (1941)450. 4. L.J.Sham and J.M. Ziman, Solid State Physics 15 (1963) 221. 5. N.W. Ashcroft, Phil. Meg. 8 (1963) 2055. 6. N.W. Ashcroft, ThesiS, Cambridge University

-0.4

-0.5

(1964).

-0.6

Fig. 2. The pseudopotential form factor for aluminium at 700oc. the basic point of the s c r e e n e d ion pseudopotential method that all p r o p e r t i e s of the m e t a l c a n

7. V. Heine and I. Abarenkov, Phil.Mag. 9 (1964) 451. 8. A.O.E.Animalu, Phil. Mag. (to appear). 9. V.Heine, Prec. 9th Intern. Conf. o- Low temperature physics, Columbus, Ohio, 1964 (to appear). 10. T.M.Rice, Annals of Physics (1964, to be pub~ lished). 11. L.J. Sundstr~m, Phil.Mag. (to appear),

DYNAMIC NUCLEAR POLARISATION IN LIQUIDS AT 13 000 GAUSS R. VAN STEENWINKEL and K. H. HAUSSER

Max-Planck Institut, Heidelberg Received 3 December 1964

We have m e a s u r e d the d y n a m i c n u c l e a r p o l a r i s a t i o n , due to the O v e r h a u s e r effect, of s o l v e n t p r o t o n s in f r e e r a d i c a l solutions at m a g n e t i c fields of about 13 000 gauss. The e n h a n c e m e n t f a c t o r V = (qz> - I0)/I0 of d y n a m i c n u c l e a r p o l a r i s a t i o n d e c r e a s e s with i n c r e a s i n g m a g n e t i c field [1]. The function V(H0) has been d e t e r m i n e d f r o m m e a s u r e m e n t s of the d i s p e r s i o n of the n u c l e a r m a g n e t i c r e l a x a t i o n in the r a n g e 33 G - 38kG [2]. In o r d e r to check t h e s e r e s u l t s by a d i r e c t m e a s u r e m e n t of V we have used s o l u t i o n s of v a r i o u s c o n c e n t r a t i o n s of 1 , 3 - b i s d i p h e n y l e n e - 2 - p - c h l o r p h e n y l a l l y l (10-C1BPA) in b e n z e n e at r o o m t e m p e r a t u r e ; p r e l i m i n a r y m e a s u r e m e n t s have also b e e n p e r 24

f o r m e d with the r a d i c a l 2, 4, 6 - t r i t e r t . - b u t y l phenoxyl (TBP). The dynamic nuclear polarisation was detected using an autodyne oscillator with 6 kc/s field modulation and phase sensitive detection [3]. The magnetic field was provided by a Varian 12"-magnet. The microwave power (from an Elliot type 8 T F K 2 klystron) required for saturating the electronic Z e e m a n levels was concentrated in the sample using a helix of 1.9 m m internal diameter. The m a x i m u m power input into the helix was of the order of 4 watt but we are not sure if all of this power was actually available in the sample for saturation. W e have determined the concentrations of the radicals by