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Deformation of the Crystal Lattice of a Metal in the Vicinity of the Surface
Glauberman, A. E.
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NATIONAL RESEARCH COUNCIL OF CANADA T e c h n i c a l T r a n s l a t i o n TT-293 By: D e f o r m t i o n o f t h e c r y s t a l B a t t f e e of a m e t a l f n t h e v f e i n f t y sf t h e s u r f a c e , ( D e f o r m a t s f y a k r f s t a l l f e h e s k o f r e s h e k k i m e t a l l a v b l f z f p o v e r k h n o s t i , ) A . GBauberman, L e n i n g r a d P o l y t e e h n f e a l I n s t i t u t e , R e f e r e n e e : Z h u r n a l E k s p e r f m e n t a l ? n o L f T e o ~ e t i e h e s k o i F i z i k f ,
--
19 : 300==303, 1 9 4 9 , T r a n s l a t e d by: G B e l k o v , Read by: J, A , M o r r i s o n ,DEFOFUATION OF THE CRYSTAL LATTICE OF A METAL I N THE V I C I N I T Y O F THE SURFACE
Formulae a r e o b t a i n e d f o r t h e l o e l a t i v e d i s - p l a c e m e n t o f i o n i c p l a n e s of a f a c e - c e n t e r e d m e t a l l a t t i c e i n t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e boundary s u r f a c e u s i n g a s f m p l f f i e d r e - p r e s e n t a t f o n o f t h e mean d e n s i t y of t h e e l e e - t r o n - g a s i n e a c h e l e m e n t a r y c e l l sf a m e t a l c r y s t a l and a n e x p r e s s i o n f o r t h e a d d i t i o n a l e l e c t r i c p o t e n t i a l a r i s i n g b e c a u s e o f t h e boundary suaaface, The c o r r e c t i o n f o r t h e s u r - f a c e t e n s i o n of t h e m e t a l , a s s o c i a t e d w i t h t h e d e f o r m a t i o n o f t h e c r y s t a l l a t t i c e i n t h e v i - c i n i t y of t h e s u r f a c e , i s c a l c u l a t e d , When d i v i d i n g a n i n f i n f t e m e t a l i n t o two e l e c t r i - c a l l y n e u t r a l p a r t s t h e r e i s a change in t h e i n t e r - p l a n a r d i s t a n c e , i , e , , t h e m e t a l l a t t i c e becomes deformed i n t h e d i r e c t 1 on p e r p e n d i c u l a l o t o t h e boundary p l a n e , The Coulomb f o r c e s o f t h e i n t ; e r a c t . i o n c a u s e t h e d e f o r m a t i o n , and e q u i l i b r i u m o c c u r s b e c a u s e of t h e c o m p e n s a t i n g a c t i o n of Cculomb f o r c e s a c t i n g as r e p e l l i n g f o r c e s which become t h e p r e s s u r e of t h e e l e c t r o n g a s of a m e t a l ,
By examining a m e t a l a c c o r d i n g t o F r e n k e l v s model, one c a n e a s i l y c a l c u l a t e t h e d e c r e a s e i n i n t e r - p l a n a r d i s t a n c e s i n t h e v i c i n i t y of t h e s u r f a c e of a m e t ' a l and one c a n e v a l u a t e t h e c o r r e c t i o n f o r t h e s u r f a c e e n e r g y of a m e t a l ,
The f a c e - c e n t e r e d c u b i c i n f i n i t e s e m i - l a t t i c e of a m e t a l w i l l be examined and a p r i s m a t , i c s e c t i o n , t h e b a s e o f which l i e s i n t h e boundary p l a n e
z
= 0, w i l l be s e p a r a t e d from t h i sl a t t i c e , The e n t i r e i n f i n i t e s e m i - l a t t i c e w i l l be c o n s i d e r e d a s c o n s i s t i n g of s i m i l a r p r i s m a t i c s e c t i o n s ( F i g . 1)" If nk i s t h e mean d e n s i t y of t h e e l e c t r o n gas i n a k - t h cube, s f t h e s e c t i o n under s t u d y , a f t e r d e f o r m a t i o n , t h e f o l l o w i n g c a n be w r i t t e n : \ "k =
~ d ( l
9 - - h d k ) , 61) where no i s t h e d e n s i t y s f t h e e l e c t r o n g a s i n a n i n f i n i t e undeformed c r y s t a l , and A d k i s t h e v a l u e of t h e r e l a t i v e change i n t h e edges of t h e cube p a r a l l e l t o t h e a x i s z,By exami n i n g t h e e l e c t r o n gas i n t h e Thomas-Femi a p p r o x i
-
mation t h e f o l l o w i n g w i l l be o b t a i n e d f o r t h e p r e s s u r e of t h e e l e c t r o n gas:where
EF
s i g n i f i e s t h e k f n e t i c energy of e l e c t r o n s , andEA
t h e t e r m c o r r e s p o n d i n g t o t h e exchange e n e r g y of e l e c t r o n s and(1) t h e c o r r e l a t i o n c o r r e c t i o n o f Wigner
Knowing t h e e x p r e s s i o n f o r t h e p r e s s u r e of a n e l e c t r o n g a s one can w r i t e t h e volume d e n s i t y of p e p e l l i n g f o r c e s , f o e , , t h e volume d e n s i t y of t h e p r e s s u r e of a n e l e c t r o n gas on t h e " w a l l s " o f t h e k - t h cube i n t h e form Fk =
-
p P k ; on t h e o t h e r hand t h e e l e c t r o - s t a t i c f o r c e of d e f o ~ m a t i o n a c t i n g on t h e i o n s ~ f a m e t a l having a c h a r g e e i s e x p r e s s e d by t h e fonamula f =-
e ~ Q k , wherep
i s t h e e l e c t r o s t a t i c p o t e n t i a l which a r i s e s i n t h e p r e s e n c e of t h e s u r l a c e ,T h i s a d d i t i o n a l p o t e n t i a l c a n be found e a s i l y by examining - 2 L a P l a c e q s e q u a t i o n
v
= 0 which i s c o r r e c t e x t e r n a l l y i n r e l a - t i o n t o t h e s e m i - l a t t i c e of t h e r e g i o n and P o i s s o n r s e q u a t i o nv2(f
=-
4 TP
f o r t h e i n t e r n a l r e g i o n , where ? =P +
+ P --
when p = c o n s t a n t i s t h e d e n s i t y of t h e e l e c t r o n i c c h a r g e+
andF
= e6
( r-
r k ) i s t h e d e n s i t y ~ f t h e p o s i t i v e charge of t h e i o n s of t h e l a t t i c e , The c o n d i t i o n of c o n t i n u o u s poten- t i a l and i t s f i r s t d e r i v a t i v e a c c o r d i n g t o z on t h e p l a n ez
= 0, when d e n s i t y?
+ i s p r e s e n t e d i n t h e form o f F o u r i e r q s t r i p l e s e r i e s , a l l o w s one t o f i n d t h e s o l u t i o n f o r . a p o t e n t i a l i n t h e e x t e r n a l and t h e i n t e r n a l r e g i o n s i n t ha form o f s e r i e s w i t h good convergence, The a c t u a l p o t e n t i a l of t h e i n t e ~ n a l r e g i o n c o n s i s t s of a p u r e l y p e r i o d i c p a r t and t h e p a r t t h a t v a n i s h e s e x p o n e n t i a l l y w i t h d i s t a n c e from t h e boundary p l a n e z = 0 , The l a t t e r p a r t of t h e p o t e n t i a l is an a d d i t i o n a l ( b y comparison w i t h t h e p o t e n t i a l i n a n i n f i n i t e l a t t i c e ] p o t e n t i a l a s s o c i a t e d w i t h t h e g e s e n e e of t h e boundary and has t h e f o l l o w i n g form:W R
Xcos
-
X cosd
q
Y-
&
-
-
r
2
-
-
+
mf
where d i s t h e i n t e r - p l a n a r d i s t a n c e i n an i n f i ~ a - i t e l a t t i c e o and m a r e p o s i t i v e whole numbers which do not become z e r o
-
d--
m2 z- . - c o s
"L
x c,, irm y ( 3 )( y / 2 ) b j 2 + m2 + =, ( ~ / 2 ) \ f 2 + m2
d 7f
- 4 -
s i m u l t a n e o u s l y . The f a c t o r x i s e q u a l t o 1/2 when
l=
0 andm
#
0 a n d ) 0 , m = 0 and i s e q u a l t o 1 when!#
0 and rn#
0 . T h e r e f o r e t h e e q u i l i b r i u m c o n d i t i o n i n t h e l a t t i c e can bew r i t t e n a s f o l l o w s :
where t h e c o n s t a n t
P
= e/2d3 andC
k i s t h e d i s p l a c e m e n t of a n i o n having t h e number k from a p o i n t h a v i n g c o o r d i n a t ezk
which c o r r e s p o n d s t o t h e e q u i l i b r i u m s t a t e i n a n i n f i n i t e m e t a l . From e q u a t i o n ( 4 ) t h e e q u i l i b r i u m c o n d i t i o n i s o b t a i n e d i n t h e f o l l o w i n g form: s i n c e t h e e q u i l i b r i u m c o n d i t i o n ( 4 ) r e l a t e s t o t h e e n t i r e i n f i n i t e s e m i - l a t t i c e of t h e m e t a l . C o n s i d e r i n g A d k a s a s m a l l v a l u e and e x p r e s s i n g p r e s s u r e pk + and pk t b o u g h t h e d e n s i t y of t h e e l e c t r o n g a s nk, a f t e r d i v i d i n g i n t o a s e r i e s i n terms of s m a l l A d t h e f o l l o w i n g i s o b t a i n e d .fop t h e l e f t hand p o r t i on of e q u a t i o n ( 5 ) :.+
pk 1 " pk = 10/9 xFn 5 / 3 0 ( i ' \ d k-Adk
+ I )-
419 xAnO 4 / ~ k 4 / 3 ( . l . G - a b + ~ ) k n i / 3 + 1 ) - 2, ( 6 ) ( - l d-Adk
++
1/9 xknO 2 7t.2 113 where XF = e aH 6 / 5 (3/87r) 2/3 9 X ~ e 2 = 3/2 ( 3 1 8 ~ ) xk = OU63xA, 4 = 5 , 1 ( 3 / 4 7 r ) 113 98~ i s t h e r a d i u s of t h e f i r s t B o h o r b i t and e i s t h e e l e c t r o n i c
charge. By introducing t h e f o l l o w i n g :
and k e e p i n g i n mind t h a t t h e v a l u e of t h e r e l a t i v e change i n t h e i n t e r - p l a n a r d i s t a n c e i n t h e d i r e c t i o n p e r p e n d i c u l a ~ t o t h e boundary p l a n e a d k , 1'1 dk = ( r k
- c k
-
1 )/d, t h e f o l - lowing e q u a t i o n i s o b t a f n e d : By s u b s t i t u t i n g e x p r e s s i o n ( 3 ) f o r p o t e n t i a l p t h e f o l l o w i n g i s o b t a i n e d : i-a1l1k
+
m2 k 3 k - 1 =- 2 c k+
5 k + 1='b
2TzE + (-
ilk
where f o r a b b r e v i a t i o n 4 e @ / ~ i s denoted a s
Y
and t h e f u n c t i o nF
( z k +t k )
i s g i v e n by a s e r i e s i n t e r m s of t h e s m a l l d i s -placement
c
.
hen
k > I i n t h e r i g h t hand p o r t i o n , t h e t e r m ~ p o ~ o r t i o n a l t ock
i s dropped and i n p l a c e of z k its v a l u e(2k
-
l ) d / 2 i s s u b s t i t u t e d , E q u a t i o n ( 8 ) i s t h e g e n e r a l form of an i n f i n i t e system correspondLng t o v a r i a t i o n o f k fromI n s o l v i n g t h i s i n f i n i t e system o f e q u a t i o n s t h e f o l l o w - i n g s u b s t i t u t i o n i s made:
.then two systems of e q u a t i o n s w i l l be o b t a i n e d r e s p e c t i v e l y
'lrn
w i t h even and odd v a l u e s of and m. Thefala v a l u e s :>
s o l u t i o n of t h e i n f i n i t e system of e q u a t i o n s ( 8 ) i s a s f e l l o w s :
where oonstqant
COO
c a n be determined by s u b s t i t u t i n g t h e s o l u t i o n of" e q u a t f o ~ ~ ( 9 ) i n t o t h e eqslation c o r r e s p o n d f n g t~ k = 1, which is t h e boundary c o n d i t i o n o f t h e examined s y s t e n ,The n a m e r i c a l d e t e r m i n a t i on of
[,
i s of no i n t e r e s t ; sin=~ C Gc h a r a c t e r i z e s t h e d i s p l a c e m e n t of t h e c r y s t a l a s a whole.
For t h e v a l u e of t h e change i n d i s t a n c e between t h e f i r s t and second i o n i c l a t t i c e s
C l
= 0.1% d i s o b t a i n e d ; d i s p l a c e - ments<
v a n i s h e x p o n e n t i a l l y i n p r o p o r t i o n t o t h e d i s t a n c e from t h e boundary,Knowing t h e v a l u e of t h e change i n t h e i n t e r - p l a n a r d i s t a n c e s it; i s e a s y t o e v a l u a t e t h e c o r r e c t i o n f o r s u p f a c e
t e n s i o n of t h e m e t a l . A c t u a l l y , if one p r o c e e d s from t h e approximate r e p r e s e c t a t i o n c o n s i d e r i n g t h e mean d e n s i t y of t h e e l e c t r o n gas i n t h e k - t h cabe and if" one e x p r e s s e s t h i s
d e n s i t y nk by t h e f o l l o w i n g formula:
-
nk
-
no (1 = A d k ) r ( 1 0 )by e x p r e s s i n g a l l t h e p a r t s of t h e energy of t h e non- e l e c t r o s t a t i c f o r c e s (EF and
EAj
t h r o u g h t h e e l e c t r o n d e n s i t y and keeping i n mind t h e s m a l l n e s s of t h e i n c r e a s eIn d e n s i t y n n k =
-
n o A d k , t h e f o l l o w i n g e x p r e s s i o n i s o b t a i n e d :C x j
where i s t h e f u l l change i n k i n e t i c energy a s s o c i a t e d w i t h deformation of t h e l a t t i c e , By u s i n g t h e V i r i a l theorem which e s e x p r e s s e d by t h e follcrwfng F a l a t i o n :
where
AE,
is t h e change, i n p o t e n t i a l energy of Coulomb intar- a c t i o n w i t h o u t c o r r e c t i o n f o r exchange and s e l f - a c t i o n of ales- t r o n s and a l s o w i t h o u t t a k i n g i n t o a c c o u n t t h e c o r r e l a t i c n c o r - r e c t i o n of Wfgner ( t h i s c o r r e c t i o n i s i n c l u d e d i n t h e t e r mAE*)
t h e n u m e r i c a l c a l c u l a t i o n can be made. The c a l c u l a t i o n o f t h e d e c r e a s e I n s u ~ f a e e t e n s i o n 6 of a m e t a l ( c o p p e r ) r e s u l t -c o r r e c t i o n