- Email: [email protected]
Dynamical model for lattice vibrations in metals
Volume 10, number 1
PHYSICS
LETTERS
w h e r e H is the f i e l d in t h e j u n c t i o n and a is t h e t h i c k n e s s s o l u t i o n in eq. (5) we o b t a i n 1 sin(2d(~ln B = [A21 {[Al12}~ ~.~ , ~~[eHa)
15 May 1964
of the i n s u l a t o r .
- ella)]
S u b s t i t u t i n g the A b r i k o s o v
exp {-eHl(Xl-J~n--~H1) 2}
(6)
1
In o r d e r to s e e t h e d i f f r a c t i o n p a t t e r n the f i e l d H should be of the o r d e r of (A~)~H1/a w h i c h is f a i r l y high w h e r e A is the p e n e t r a t i o n l e n g t h and ~ the c o r r e l a t i o n l e n g t h of the c o n d e n s e d p a i r s . Though in r e a l j u n c t i o n s the i r r e g u l a r i t y of t h e s u r f a c e as w e l l as the p r e s e n c e of the b o u n d a r y s c a t t e r i n g m a y c o m p l i c a t e the s i t u a t i o n , we b e l i e v e the e s s e n t i a l f e a t u r e s t i l l r e m a i n s u n c h a n g e d . The a u t h o r w i s h e s to thank Mr. M a s u o Suzuki f o r p l e a s a n t c o n v e r s a t i o n s on r e l a t e d t o p i c s .
References 1) 2) 3) 4) 5)
A.A.Abrikosov, Zh. Eksp. i Teoret. Fiz. 32 (1957) 1442; translation: Soviet Phys. - J E T P 5 (1957) 1174. B.D.Josephson, Physics Letters 1 (1962) 251. V.Ambegaokar and A. Haratoff, Phys. Rev. Letters 10 (1963) 486; 11 (1963) 104 (erratum). K.Maki, to be published. J.M.Rowell, Phys. Rev. Letters 11 (1963) 200.
DYNAMICAL
MODEL
FOR
LATTICE
VIBRATIONS
IN
METALS
K. K R E B S
Department of Reactor Physics, Euratom C. C.R., Ispra Varese , Italy Received 18 April 1964
T h e new m e t h o d to i n v e s t i g a t e l a t t i c e d y n a m i c a l p r o p e r t i e s by i n e l a s t i c n e u t r o n s c a t t e r i n g can g i v e p r e c i s e i n f o r m a t i o n on f r e q u e n c y v e r s u s w a v e v e c t o r d i s p e r s i o n r e l a t i o n s in m e t a l s 1). It is thus of i n t e r e s t to r e c o n s i d e r the t h e o r e t i c a l s i d e of t h i s p r o b l e m . So f a r f i t t i n g of e x p e r i m e n t a l d i s p e r s i o n c u r v e s by e x i s t i n g m o d e l s h a s b e e n only p a r t i a l l y s u c c e s s f u l . One d i f f i c u l t y with t h e known m o d e l s f o r m e t a l s of De L a u n a y ' 2 ) , S h a r m a and J o s h i 3), o r B h a t i a 4), is the f a c t that all t h r e e v i o l a t e symmetry requirements, as recently again pointed out by L a x 5). T h e d i s a d v a n t a g e of B o r n - V o n K a r m a n m o d e l s is that no e x p l i c i t a c c o u n t f o r the e l e c t r o n s is m a d e and that one n e e d s too m a n y p a r a m e t e r s to fit e x p e r i m e n t a l d i s p e r s i o n c u r v e s . In the p r e s e n t l e t t e r I a m r e p o r t i n g on c a l c u l a t i o n s for a s i m p l e m o d e l w h i c h is f r e e of t h e s e d e f i c i e n c i e s of e x i s t i n g t h e o r i e s . As in B h a t i a ' s t h e o r y 4) t h i s m o d e l is b a s e d on a s c r e e n e d C o u l o m b i n t e r a c t i o n b e t w e e n i o n s and in a g r e e m e n t with L a x 5) t h e s y m m e t r y r e q u i r e m e n t s a r e t a k e n into a c c o u n t by i n c l u d i n g U m k l a p p p r o c e s s e s . The e l e c t r o n s i n f l u e n c e the ion m o t i o n v i a the s c r e e n ing p a r a m e t e r of t h e C o u l o m b p o t e . i t i a l . T h e int e r a c t i o n b e t w e e n c l o s e d ion s h e l l s is d e s c r i b e d by a c e n t r a l i n t e r a c t i o n b e t w e e n f i r s t and s e c o n d n e i g h b o u r s. 12
F o r a b c c l a t t i c e we o b t a i n t h e n the f o l l o w i n g e q u a t i o n f o r t h e d i s p e r s i o n r e l a t i o n s (q):
I Iti .
n2 M y -
2--~- u2 6xyl = 0
(1)
where
Mxx = 1 - c o s ~qx c o s ~rqy c o s 7rqz +
C l l - c12 2c44 sin2~qx + Ixx '
Mxy = sin 7rqx sin ~qv sin 7rqz + Ixy ,
...
(M = m a s s of l a t t i c e a t o m s , ~ = H o o k e ' s law f o r c e constant for nearest neighbour central interaction b e t w e e n c l o s e d ion s h e l l s , q = ak/27r, a = l a t t i c e c o n s t a n t , k = phonon w a v e v e c t o r , Cik = e l a s t i c constants). The t e r m s Ixy a r i s e t h r o u g h a s c r e e n e d C o u l o m b i n t e r a c t i o n b e t w e e n the m e t a l ions e m b e d d e d in a s e a of B l o c h e l e c t r o n s :
lxy(q) = ×
c12 - c44 a2X2 8c44
I
(qx +hx) (qy + by)
h I q +hi 2 + (a2X2/4~2)f(tl) g 2 ( u l )
hxhy )l h 2 + (a2X2/4g2)f(t2) g2(u2
Volume 10, n u m b e r 1
PHYSICS
LETTERS
[1oo,]~ ~
0(000
(~00)
15 May 1964
[7111 [1101 /
(1ti ~~
(100)°(111)
(000)
(~2
0 O)
Fig. 1. D i s p e r s i o n c u r v e s for sodium a t 90OK in the 3 s y m m e t r y d i r e c t i o n s . points is very satisfactory and better than for T o y a ' s q u a n t u m m e c h a n i c a l m o d e l 13,14). A full a c c o u n t of t h i s w o r k , t o g e t h e r w i t h c a l c u l a t i o n s for other metals, will be published elsewhere.
(h = aK/2% K= reciprocal lattice vector). The screening parameter X depends on the electron w a v e v e c t o r k F at t h e F e r m i s u r f a c e . A c c o r d i n g t o P i n e s 6) = 0.353 (Ts/ao) 2 k F (7 s = i n t e r - e l e c t r o n i c The function
spacing, a o = Bohr radius).
T h e a u t h o r w o u l d l i k e to t h a n k M r . D e b r u y n from the C.E.T.I.S. computational center for his v e r y a b l e a s s i s t a n c e in p e r f o r m i n g t h e n u m e r i c a l c a l c u l a t i o n s on t h e I B M 7090.
1 1-t 2 t l+t /(t) = g + ~ in Y : 7
I
(tI = 7:Iq +bI/ak F, t2 = ?~h/akF) describes the decrease in screening with increasing t 7) and g(u) : 3
sinu
- u c o s ~t u3
(u 1 = 2 ~ Y s ] q + h i / a , u 2 = 2 ~ s h / a ) i s u s e d to d i m i n i s h t h e i n f l u e n c e of t h e I x y - t e r m s f o r h i g h e r K - v a l u e s 8,9). F o r s o d i u m at 9 0 ° K g o o d n e u t r o n r e s u l t s f o r t h e d i s p e r s i o n c u r v e s in s y m m e t r y d i r e c t i o n s a r e a v a i l a b l e 10). W e h a v e t h e r e f o r e s o l v e d eq. (1) f o r t h i s s p e c i a l c a s e (fig. 1). T h e e l a s t i c c o n stants for sodium as given by different authors ( r e f s . 3, 11, 12) s h o w c o n s i d e r a b l e d i s c r e p a n c i e s . F o r t h e p r e s e n t c a l c u l a t i o n we h a v e c h o s e n t h e v a l u e s of B e n d e r 12) f o r C l l a n d c1~. F o r c44 a v a l u e of 0.554 × 1011 d y n / c m 2 w a s - u s e d whict~ g i v e s good a g r e e m e n t w i t h n e u t r o n d a t a f o r t h e p o i n t (100). T h e a g r e e m e n t w i t h e x p e r i m e n t a l
1) Inelastic s c a t t e r i n g of Neutrons in Solids and Liquids, P r o c e e d i n g s I. A. E.A. (I. A. E. A., Vienna, 1961, 1963). 2) J . D e Launay, J. Chem,, Phys.,21 (1953) 1975. 3) P . K . S h a r m a and S . K . J o s h i , J. Chem. Phys. 39 (1963) 2633. 4) A. B. Bhatia, Phys. Rev. 97 (1955) 363. 5) M. Lax, P r o e . Int. Conf. on Lattice Dynamics, Copenhagen 1963, paper A 24. 6) D . P i n e s , Solid State Phys. 1 (1955) 367. 7) J . S . Langer and S. H. Vosko, J. Phys. Chem. Solids 12 (1959) 196. 8) L . J . Sham and J. M. Ziman, Solid State Phys. 15 (1963) 221. 9) E . J . Woll and W. Kohn, Phys. Rev. 126 (1962) 1693. 10) A .D.B. Woods, B.N. Brockhouse, R.H. March and A . T . S t e w a r t , Phys. Rev. 128 (1962) 1112. 11) S. L. Ouimby and S.Siegel, Phys. Rev. 54 (1938) 293. 12) O. Bender, Ann. Physik 34 (1939) 359. 13) T . T o y a , J. R e s e a r c h Inst. Catalysis Hokkaido Univ. 6 (1958) 183. 14) A . D . B . Woods, B.N. Brockhouse, R. H. March and R. Bowers, P r o c . Phys. Soc. (London) 79 (1962) 440.
* * * * *
13
PHYSICS
LETTERS
w h e r e H is the f i e l d in t h e j u n c t i o n and a is t h e t h i c k n e s s s o l u t i o n in eq. (5) we o b t a i n 1 sin(2d(~ln B = [A21 {[Al12}~ ~.~ , ~~[eHa)
15 May 1964
of the i n s u l a t o r .
- ella)]
S u b s t i t u t i n g the A b r i k o s o v
exp {-eHl(Xl-J~n--~H1) 2}
(6)
1
In o r d e r to s e e t h e d i f f r a c t i o n p a t t e r n the f i e l d H should be of the o r d e r of (A~)~H1/a w h i c h is f a i r l y high w h e r e A is the p e n e t r a t i o n l e n g t h and ~ the c o r r e l a t i o n l e n g t h of the c o n d e n s e d p a i r s . Though in r e a l j u n c t i o n s the i r r e g u l a r i t y of t h e s u r f a c e as w e l l as the p r e s e n c e of the b o u n d a r y s c a t t e r i n g m a y c o m p l i c a t e the s i t u a t i o n , we b e l i e v e the e s s e n t i a l f e a t u r e s t i l l r e m a i n s u n c h a n g e d . The a u t h o r w i s h e s to thank Mr. M a s u o Suzuki f o r p l e a s a n t c o n v e r s a t i o n s on r e l a t e d t o p i c s .
References 1) 2) 3) 4) 5)
A.A.Abrikosov, Zh. Eksp. i Teoret. Fiz. 32 (1957) 1442; translation: Soviet Phys. - J E T P 5 (1957) 1174. B.D.Josephson, Physics Letters 1 (1962) 251. V.Ambegaokar and A. Haratoff, Phys. Rev. Letters 10 (1963) 486; 11 (1963) 104 (erratum). K.Maki, to be published. J.M.Rowell, Phys. Rev. Letters 11 (1963) 200.
DYNAMICAL
MODEL
FOR
LATTICE
VIBRATIONS
IN
METALS
K. K R E B S
Department of Reactor Physics, Euratom C. C.R., Ispra Varese , Italy Received 18 April 1964
T h e new m e t h o d to i n v e s t i g a t e l a t t i c e d y n a m i c a l p r o p e r t i e s by i n e l a s t i c n e u t r o n s c a t t e r i n g can g i v e p r e c i s e i n f o r m a t i o n on f r e q u e n c y v e r s u s w a v e v e c t o r d i s p e r s i o n r e l a t i o n s in m e t a l s 1). It is thus of i n t e r e s t to r e c o n s i d e r the t h e o r e t i c a l s i d e of t h i s p r o b l e m . So f a r f i t t i n g of e x p e r i m e n t a l d i s p e r s i o n c u r v e s by e x i s t i n g m o d e l s h a s b e e n only p a r t i a l l y s u c c e s s f u l . One d i f f i c u l t y with t h e known m o d e l s f o r m e t a l s of De L a u n a y ' 2 ) , S h a r m a and J o s h i 3), o r B h a t i a 4), is the f a c t that all t h r e e v i o l a t e symmetry requirements, as recently again pointed out by L a x 5). T h e d i s a d v a n t a g e of B o r n - V o n K a r m a n m o d e l s is that no e x p l i c i t a c c o u n t f o r the e l e c t r o n s is m a d e and that one n e e d s too m a n y p a r a m e t e r s to fit e x p e r i m e n t a l d i s p e r s i o n c u r v e s . In the p r e s e n t l e t t e r I a m r e p o r t i n g on c a l c u l a t i o n s for a s i m p l e m o d e l w h i c h is f r e e of t h e s e d e f i c i e n c i e s of e x i s t i n g t h e o r i e s . As in B h a t i a ' s t h e o r y 4) t h i s m o d e l is b a s e d on a s c r e e n e d C o u l o m b i n t e r a c t i o n b e t w e e n i o n s and in a g r e e m e n t with L a x 5) t h e s y m m e t r y r e q u i r e m e n t s a r e t a k e n into a c c o u n t by i n c l u d i n g U m k l a p p p r o c e s s e s . The e l e c t r o n s i n f l u e n c e the ion m o t i o n v i a the s c r e e n ing p a r a m e t e r of t h e C o u l o m b p o t e . i t i a l . T h e int e r a c t i o n b e t w e e n c l o s e d ion s h e l l s is d e s c r i b e d by a c e n t r a l i n t e r a c t i o n b e t w e e n f i r s t and s e c o n d n e i g h b o u r s. 12
F o r a b c c l a t t i c e we o b t a i n t h e n the f o l l o w i n g e q u a t i o n f o r t h e d i s p e r s i o n r e l a t i o n s (q):
I Iti .
n2 M y -
2--~- u2 6xyl = 0
(1)
where
Mxx = 1 - c o s ~qx c o s ~rqy c o s 7rqz +
C l l - c12 2c44 sin2~qx + Ixx '
Mxy = sin 7rqx sin ~qv sin 7rqz + Ixy ,
...
(M = m a s s of l a t t i c e a t o m s , ~ = H o o k e ' s law f o r c e constant for nearest neighbour central interaction b e t w e e n c l o s e d ion s h e l l s , q = ak/27r, a = l a t t i c e c o n s t a n t , k = phonon w a v e v e c t o r , Cik = e l a s t i c constants). The t e r m s Ixy a r i s e t h r o u g h a s c r e e n e d C o u l o m b i n t e r a c t i o n b e t w e e n the m e t a l ions e m b e d d e d in a s e a of B l o c h e l e c t r o n s :
lxy(q) = ×
c12 - c44 a2X2 8c44
I
(qx +hx) (qy + by)
h I q +hi 2 + (a2X2/4~2)f(tl) g 2 ( u l )
hxhy )l h 2 + (a2X2/4g2)f(t2) g2(u2
Volume 10, n u m b e r 1
PHYSICS
LETTERS
[1oo,]~ ~
0(000
(~00)
15 May 1964
[7111 [1101 /
(1ti ~~
(100)°(111)
(000)
(~2
0 O)
Fig. 1. D i s p e r s i o n c u r v e s for sodium a t 90OK in the 3 s y m m e t r y d i r e c t i o n s . points is very satisfactory and better than for T o y a ' s q u a n t u m m e c h a n i c a l m o d e l 13,14). A full a c c o u n t of t h i s w o r k , t o g e t h e r w i t h c a l c u l a t i o n s for other metals, will be published elsewhere.
(h = aK/2% K= reciprocal lattice vector). The screening parameter X depends on the electron w a v e v e c t o r k F at t h e F e r m i s u r f a c e . A c c o r d i n g t o P i n e s 6) = 0.353 (Ts/ao) 2 k F (7 s = i n t e r - e l e c t r o n i c The function
spacing, a o = Bohr radius).
T h e a u t h o r w o u l d l i k e to t h a n k M r . D e b r u y n from the C.E.T.I.S. computational center for his v e r y a b l e a s s i s t a n c e in p e r f o r m i n g t h e n u m e r i c a l c a l c u l a t i o n s on t h e I B M 7090.
1 1-t 2 t l+t /(t) = g + ~ in Y : 7
I
(tI = 7:Iq +bI/ak F, t2 = ?~h/akF) describes the decrease in screening with increasing t 7) and g(u) : 3
sinu
- u c o s ~t u3
(u 1 = 2 ~ Y s ] q + h i / a , u 2 = 2 ~ s h / a ) i s u s e d to d i m i n i s h t h e i n f l u e n c e of t h e I x y - t e r m s f o r h i g h e r K - v a l u e s 8,9). F o r s o d i u m at 9 0 ° K g o o d n e u t r o n r e s u l t s f o r t h e d i s p e r s i o n c u r v e s in s y m m e t r y d i r e c t i o n s a r e a v a i l a b l e 10). W e h a v e t h e r e f o r e s o l v e d eq. (1) f o r t h i s s p e c i a l c a s e (fig. 1). T h e e l a s t i c c o n stants for sodium as given by different authors ( r e f s . 3, 11, 12) s h o w c o n s i d e r a b l e d i s c r e p a n c i e s . F o r t h e p r e s e n t c a l c u l a t i o n we h a v e c h o s e n t h e v a l u e s of B e n d e r 12) f o r C l l a n d c1~. F o r c44 a v a l u e of 0.554 × 1011 d y n / c m 2 w a s - u s e d whict~ g i v e s good a g r e e m e n t w i t h n e u t r o n d a t a f o r t h e p o i n t (100). T h e a g r e e m e n t w i t h e x p e r i m e n t a l
1) Inelastic s c a t t e r i n g of Neutrons in Solids and Liquids, P r o c e e d i n g s I. A. E.A. (I. A. E. A., Vienna, 1961, 1963). 2) J . D e Launay, J. Chem,, Phys.,21 (1953) 1975. 3) P . K . S h a r m a and S . K . J o s h i , J. Chem. Phys. 39 (1963) 2633. 4) A. B. Bhatia, Phys. Rev. 97 (1955) 363. 5) M. Lax, P r o e . Int. Conf. on Lattice Dynamics, Copenhagen 1963, paper A 24. 6) D . P i n e s , Solid State Phys. 1 (1955) 367. 7) J . S . Langer and S. H. Vosko, J. Phys. Chem. Solids 12 (1959) 196. 8) L . J . Sham and J. M. Ziman, Solid State Phys. 15 (1963) 221. 9) E . J . Woll and W. Kohn, Phys. Rev. 126 (1962) 1693. 10) A .D.B. Woods, B.N. Brockhouse, R.H. March and A . T . S t e w a r t , Phys. Rev. 128 (1962) 1112. 11) S. L. Ouimby and S.Siegel, Phys. Rev. 54 (1938) 293. 12) O. Bender, Ann. Physik 34 (1939) 359. 13) T . T o y a , J. R e s e a r c h Inst. Catalysis Hokkaido Univ. 6 (1958) 183. 14) A . D . B . Woods, B.N. Brockhouse, R. H. March and R. Bowers, P r o c . Phys. Soc. (London) 79 (1962) 440.
* * * * *
13