Документ взят из кэша поисковой машины. Адрес оригинального документа : http://shg.phys.msu.ru/educat/vinogradov/Lesson4.pdf Дата изменения: Wed Mar 12 16:11:34 2008 Дата индексирования: Mon Oct 1 20:38:42 2012 Кодировка:

- , eff ij = 0 (1) xi x j eff ij , , . . , , . -.


k- E0 = E , hi = i / k , E ( r ) = - E0

(r ) =
i ( r ) = 1 ( r ) = 0 i m ( h1 , ... , hk


i =1

k

i i ( r

)

i-
-1 k

)

= eff / k ,

h = ( r ) / k =


i =1

hi i ( r

)

, k- , :

( ) = 0

(2)

= 0 = L, d = 0 dn


: ( ) = 0 (3) ef h = 1 D0 = ij f E0 D ( r ) = - D0
D0 1 Dn dS = S h n dS = D S
S

= D = D0 h = D0 = eff E0

"" 2 1 1 D2 (r) 1 E (r) eff = dV eff ij = dV ( r ) 2 ij V ( r ) D02 V E
0

2 1 m ( h1 , ... , hk m ( h1 , ... , hk -1 ) = h ( ) dV V

-1

)

11 2 = = h ( ) dV mV


, ­ , , , ( = k ), . 2 2 11 k2 k =m = h ( ) dV = h ( ) dV = mk 2 mV V , , h = ( h ) - ( h ) = ( h )
1 1 m = ( h ) dV = V V = ( L ) / ( L )

(

h ) dS =

( L

)(

h V

)

x= L

dS

=

( L L

)




, . h > 0 . (2) (3) , , . , m m . , (2) (3) . , , .. , (2). .


m ( h1 ) , - m ( h1 ) . ti = - Eni , , . hi +1ti +1 = hi ti ti = tk / hi . , i dti = 0 . i , , ht1 = t2 t1D1 =- t2 D2 , Di 1 i- . h1 = - D1 / D2 p 1 = - mat 1- p p = D1 / ( D1 + D2 ) .
1 2 1 2 1 2 1


, , Dh1 + D2 m ( h1 ) = 1 = 1 - p + ph1 D h /(1 - p ) 1 , m ( h1 ) = h1 + p /(1 - p ) . . , , . h 0 dS , n , . , 0 . m ( h1 ) , m ( h1 ) .


0 = * ( h )dV = ( *h ) - h ( * ) dV = = h dS
* x=L x =0

-

(

1h +
2

2

)

(

* ) dV =

=* ( L) Dn - = m* L Dn -

m = 0 , ( 1h +2 ) ( * ) dV = 0 h .

( 1 ( 1h

h+ +
2

)

(

* ) dV =

)

(

* ) dV


, m . m h1 / h1 . h = 1/ h = 1h1 + 2

()

h ( h h ) dV = h h = 1 m= V ( h m 1 2 h = dV ( h ) + h h1 V h1 h1
2

V = =1 1 2 = dV 1 ( h ) + 2h h ( h ) = V h1 =0 1 2 = dV 1 ( h ) + 2 ( h ) - 2 ( h ) = V h1 h1 =0
2

(
h

h ) dV

2

)

1 2( L) hdS 1 2 2 = dV 1 ( h ) + = dV 1 ( h ) V V V h h = 1


, m h1

()

, m ( h1

m h1 / h1 > 0 , m h1 . .., m ( h1
h1 =

()

)

()

. ,

h1 = 0 , .

)



.


, . m h1 , m ( h1 ) . .

()




eff

= p11 + (1 - p1 )

2



m(h) = p1h + (1 - p1 )


p1 - (1 - p1 )

2 (1 - 2 ) eff = 2 + 3 p (1 + 2 2 ) - p (1 - 2 ) p1 ( h - 1) m( h) = 1 + 1- p1 + (1 - p1 ) ( h - 1) / 3

1 1 1 = p1 + (1 - p1 ) eff 1 2 h m( h) = p1 + (1 - p1 )h

1



2

1



2

1



2

1

m( h) =

3 (1 - p1
2

)
1/ 2

1 = ( dp1 - 1) h + ( dp2 - 1) 2 ( d - 1)



( dp1 - 1) h + 2(d - 1 + d p1 p2 )h + ± 2 + ( dp2 - 1)
2 2




s = 1/(1 - h ) = 2 /( 2 - 1 ) F ( s ) = 1 - m( h ) = ( 2 - eff ) / 2 F(s) m(h) , s [0,1) F(s) F (s) =




B s - s

Bn > 0,

1 > s0 > s1 > ... > sn




F (s) =



B(x

)

s-x

dx =

p s - (1 - p ) / 3

B ( s ) = p ( s - (1 - p ) / 3

)

1 F (s) = 3 s + sp - 1 4s 1 -9 x 2 + B ( x ) = 4x

{

- 9 s - 6 (1 + p ) s + (1 - 3 p )
2

2 1/ 2

}
]

(

6 (1 + p ) x - (1 - 3 p 0

)

2 1/ 2

)

x1 x x2 x [ x1 , x2

x1,2 =

1 1 + p ± 2 2 p (1 - p 3

(

)

)


p = 0.1

p = 0.2

p = 0.3

­, - ­ , ­ (Y. Kantor, D. J. Bergman "The optical properties of cermets from the theory of electrostatic resonance" J. Phys. C. 15, (1982) 2033-2042)


h=0

m(0) 0

1 h = 1- s

F (1) =




B =1 - m(0) 1 1 - s

h =1 ( s = ) m=1, F ( ) = 1 - m = 0 2 1 ( ) dV h=1 = =- x m ( h1 ) = h1 V

(

)

(

1dV V

)

=p

m(h) =- F ( s ) =- s 2 F ( s ) = s h h s

2




(

B s - s

)

2 s






B




B = p


B , s F ( s ) =
F ( s ) =







B B s + s - s ( s - s )

2



B s - s

,
B = p


B




,

B0 =-



( s - s0 ) B F ( s ) = B + s ( s - s ) ( s - s ) ( s - s0 ) >0 s > 1, s < 0 <0 >0 Bn > 0, 1 > s0 > s1 > ... > sn F ( s ) B . , B = 0 (> 0)




B = p B0 = p


.




B 1 , 1 - s p F (s) < s - (1 - p

p 1 1 - s0

s0 1 - p

)

, s = s0 s0 = 0




B = p B0 = p p F (s) > s

, -


2m ( h ) h 2 2 p(1 - p ) = d




h =1

p(1 - p ) s B = d

p (1 - p ) s B = d ( s B + B s ) = 0 B0 , s0 Bn > 0, 1 > s0 > s1 > ... > 2 B ( s0 - s ) ( s0 + s - 2 s ) ( s - s0 ) F ( s ) = B + s 2 2 2 >0 ( s - s ) ( s - s0 ) ( s - s ) ( s - s0 ) s > 1, >0 <0

sn

, , F ( s ) > 0 . B = 0 (> 0)





B = p B0 = p p (1 - p ) s B = = s0 B0 s0 = (1 - p ) / d d
p F (s) > ( s - (1 - p ) / d




)

, . t = 1 - s, G (t ) = (1 - sF ( s )) /(1 - s ) G (t ) >

(

(1

-p

)

t - p/d

)



1- p 1 + < 1 / ( 2 - 1 ) + p /( d 1 ) 1 <
2 eff

p < 2 + , 1 / ( 1 - 2 ) + (1 - p ) /(d 2 )


1. - ., , , 1984 2. . ., . . , 1984 3. Bergman D. J. // Phys. Rep. 1978 V. 43, 9, P. 377 4. Bergman D. J. // Ann. Phys. 1982. V. 138, P. 78 5. Bergman D. J. in Homogenization and Effective Media eds. J.L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions SpringerVerlag, NY 1986 p.27 6. Bergman D. J. // Phys. Rev. B. 1989 V. 39, P. 4598 7. Bergman D. J., Stroud D. // Solid State Phys 1992. V. 46, P. 147 8. Bergman D. J. Stroud D. // Phys. Rev. B 1980. V. 22, P. 3527 9. Milton G. W. // Appl. Phys. A 1981. V. 26, P. 207 10. Milton G. W. // J. Appl Phys. 1981. V. 52, P. 5294