materijali - predavanje
TRANSCRIPT
2.1.
.
„” -
. -
. -
, .
-
. . -
, -
„“ , .
–
.
) ) 2.1 ) , )
. , -
.
, 2.1..
, , -
,
, , 2.1.
( o-
) ( ). -
.
, -
. ,
. , -
, , 2.2.
2.2
, -
.
. -
„", -
. „"
.
.
. -
, c 2.3, - -
: , -
.
18 – .
2.3 ,
2.1.
r ,
p m n
a b E
r r = − +
r = 0,3 nm E = –4 eV.
a b, m = 2 n = 10,
.
:
p m n
a b E
r r = − + (2.1)
1 1
dr r r + +
-19
(0,3 nm) (0,3 nm)
b
20 27,2 10 J nma −= ⋅ ; 25 29,4 10 J nmb −= ⋅
F max,
.
2
dr r r + +
1 1m n
r r
, „ " ,
.
, ,
, : , , , -
, .
.
, –38,9 ° ( ) 3410 ° ( ). -
, -
. , -
.
, ,
. -
, -
, ,
,
.
, -
(2+), „
", (+). -
), . Fe, Ni, W, d- -
d-, , , -
. -
d- .
. -
, -
. ,
„”
. ,
, .
,
. , „ ”, . -
, ,
. , -
(), (- )
. .
.
, -
.
. -
, ,
.
(, ),
, ,
. -
,
2 – 21
, ()
() , .
, , .
. .
, -
, , , . , -
( 10-8 ),
( ) -
. ,
.
.
,
.
, , . . -
.
2.4
. -
,
,
. ,
, , 2.4. -
, .
.
,
, .
.
, ( ) 0r r
E F dr ≈
22 – .
2.2.
-
. -
. ,
( )
.
2.5 2.6.
2.5
ABCABC
ABAB
.
.
... ,
ABCABC... -
.
12 -
( ... ) ( -
... ) 2.7 2.8
(X) () .
2 – 23
2.7. ABAB…
2.8 ABCABC…
X.
X
.
( )
, () 2.9.
: ) : α – Fe, β –
Ti V, Cr, Mo W., ) : γ – Fe, Al, Ni, Cu, Ag Pt; ) X: Be, Mg, Zn, α – Ti Zr
.
.
.
2.9
, AFP
,
atoma
V
⋅ = (2.2)
n − .
() : n = 8 ⋅ (1/8) = 1 /,
() : n = 8 ⋅ (1/8) + 1 ⋅ (1) = 2 /,
() : n = 8 ⋅ (1/8) + 6
⋅ (1/2) = 4 /.
( FP)
( ).
:
, 2.10.
3 4
4 3
a R
a R
⋅ =
=
3.1, n = 2,
3
3
2.10
2.3:
- .
2.11
:
4 2
4 2
R a
a R
=
=
( )
( )
4 / 2
2.4:
(X )
:
X , 2.12., -
2.12.
) ) 2.12. ) , )
, 2.12., -
LM . -
2.13. / .
2.13 ) , )
MH ; JM 2 2
c a R= = = ; / 2
c a c a a c a
a = + = + = = 1,633
2.5:
:
( ) ( ) rogalj centar
2 2 3sin 60 1,633 2 (2 ) 11,31V c a R R R= ⋅ ⋅ ° = ⋅ ⋅ =
( )3 32 4 /3 8,38aV R Rπ = ⋅ =
X
3
3
( )
, AFP , 0 Am M N = -
, , A , -
:
A
⋅ (2.3)
: ρ - , n - , M -
, A N - , V - .
2.6:
0.2866 nm. 55.847 g/mol
:
( )
3 -3 7,882 10 kgm
A
.
. -
2.14. -
-
.
2.15.
, -
,
2.16.
2.14
2.15
.
2.16
G H TS = −
.
p
= −
( )
2
2
p
p
2.7:
– Fe – Fe .
0,2863 nm 0,3591 nm.
:
−⋅=== aV PCK
-Fe -Fe
Fe, . -Fe Fe -Fe
:
3 3 3 3
2 23,467 10 nm V
− −
−
, -
. -
. -
-
, -
.
()
()
, )
)
-
( )
2 – 31
.
, , : ) -
, ) , ) -
) ,
), ) ). -
, 2.14.
, -
. -
.
, ,
.
.
.
)
.
-
. -
-
. , a, b c, -
-
2.14. a, b c -
.
, b c α , β γ , 2.14. 14 -
, 2.15,
7 , 2.1
2.1
a b c= = 90α β γ = = = °
a b c= ≠ 90α β γ = = = °
a b c= = 90α β γ = = ≠ °
a b c= ≠ 90 120α β γ = = ° = °
a b c≠ ≠ 90α β γ = = = °
a b c≠ ≠ 90α γ β = = ° ≠
a b c≠ ≠ 90α β γ ≠ ≠ ≠ °
32 – .
,
, ,
.
.
,
.
.
,
. .
. :
1 y z
a b c + + =
2 – 33
, b, c x, y, z . -
, -
, -
. , b c
,
( ). -
.
1 1 1 h k l
a b c = = =
, b c , x, y z
: 1hx ky lz + + = . (
, h,k l (hkl) -
. , x
(100) , 2.16.
. -
, , .
x ; , -
(100)
34 – .
; (330), (220) (110) „(100) ”,
. , -
. " -
". (100), (010) (001) .
.
: {100} (100), (010), (001),
(100), (010) (001) .
2.8:
A, B C 2.17
2.17
A, B C
:
) = 1, y = 1, z = 1; ) 1/ x = 1, 1/ y = 1,1 / z = 1; ) ; ) (111)
:
) z- x = 1, y = 2 z = ∞; ) 1/ x = 1, 1/ y = 1/2, 1/ z = 0;
) : 1/ x = 2, 1/ y = 1, 1/z = 0 ) (210)
:
)
0,0,0. , y-; ) x =∞ , y = -1 z =∞
.
. , ,
, ,
[ ]111 "111" ,
. -
2 – 35
110 110 , 101 , 011 .
, ,
. .
-
.
:
() -
;
() -
;
() -
;
() [ ].
.
2.9:
A, B i C 2.18.
2.18
A, B C
:
() (1,0,0) (0,0,0); () 1,0,0 – 0,0,0 = 1,0,0; () ; () [100].
C:
() 0,0,1 1/ 2 ,1,0; () 0,0,1– 1/ 2 ,1,0 = – 1/ 2 ,–1,1
- -
12 2 .
:
36 – .
, . - (hkl ) - [uvw] :
1) [uvw] (hkl ) u = h, v = k w = l. , [111] (111).
2) [uvw] (hkl ), . 0hu kv lw+ + = .
3) (h1k 1l 1) (h2k 2l 2)
1 2 1 2 1 2 0h h k k l l + + = .
4) (u1v1w1) (u2v2w2)
1 2 1 2 1 2 0u u v v w w+ + = .
5) θ (h1k 1l 1) (h2k 2l 2)
1/ 2 1/ 2 2 2 2 2 2 2
1 1 1 2 2 2
cos h h k k l l
h k l h k l θ
+ + =
2.10:
[110] [111] . ( ).
:
[110] [111] (2.4)
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
arccos u u v v w w
u v w u v w θ
+ + =
+ + ⋅ + +
1 1 1 1 1 0 2 arccos arccos arccos0,816
3 21 1 1 1 1 0 35,3°θ
⋅ + ⋅ + ⋅ = = = =
⋅+ + ⋅ + +
2 – 37
- h, k, i, l, (hkil). a1 , a2 , a3 c . . . ,
. h k i+ = − .
2.19
2.11:
- A B C D 2.20.
:
:
() 1 2 3 1a a a c= = = ∞ = ; () 1 2 3
1 1 1 1 0 1
a a a c = = = = ; () ; () (0001).
:
1 2
a a c = = = ;
(1121) .
:
() (0,0,1) (1,0,0); () 0,0,1 – 1,0,0 = –1,0,1; () h k i+ = −
() [1011]
D:
() (0,1,0) (1,0,0); () 0,1,0 – 1,0,0 = –1,1,0; () i = –(–1+1) = 0
() [1100]
38 – .
2.20 –
- ,
. h k i+ = −
. - , 2.21.
2.21
2.12:
[111] () () . RW = 0,137 nm RAl = 0,143 nm.
:
() . [1 1 1]. , .
r = 2 RW = 2 ⋅ 0,137 nm = 0,274 nm
1 1
0,274 nm r ⋅ = = 3,65 atoma/nm
() a [1 1 1] .
Al
2 2
2.13:
(111) : ) ) . RW = 0,137 nm a RAl = 0,143 nm.
:
() (111) , 2.22.
2.22 (111)
3 3
2 2 0,316 nm 0,447 nmb a= ⋅ = ⋅ =
(1 1 1)
2 2 21 1 3 3 3 (0, 447 nm) 0, 0867 nm
40 – .
(111) 1-6 ,
1
2
nm
() (111) ,
2.23.
2 2
R a = = ⋅
2 2 0,404 nm=0,572 nmb a= = ⋅
(111)
21 1 3 0,572 nm 0,572 nm=0,142 nm
2 2 2 A b h= ⋅ ⋅ = ⋅ ⋅
(111) 3 1/6
½,
1
2
0,142 nm A− ⋅ + ⋅
= = 2 14,1 atoma/nm
, d. -
, . .
-
0
(hkl ) :
0
2.14:
(111)
0,40786 nm.
. 2.6
111 2 2 2
+ +
- .
, d,
. d
, .
2.24 - : ) , ) )
42 – .
,
, 2.24..
, d , θ ,
2.24.. ( 1, 2.24. )
, , ( 2)
, 2 d 1
MPN. s = MP + NP ,
d , 2d 2s . -
-
. -
n. λ = s, n- , ,
λ- s- ,
. , 2.24
s = 2d sinθ
2 sinn λ θ θ ⋅ = (2.6)
.(2.6) .
, λ ,
, d θ .
2.25 ) , ) X-
-
, d, , λ
2.15:
-
λ = 0.07107 nm ( ) -
, 2θ , :
2θ 2θ
1 20,20 5 46,19
2 28,72 6 50,90
3 35,36 7 55,28
4 41,47 8 59,42
,
.
:
sin 2θ -
0.0308
2 θ ⋅ 2sin θ 2sin 0,0308θ 2 2 2h k l + + ( )hkl
1 20,20 0,0308 1 2 (110)
2 28,72 0,0615 2 4 (200)
3 35,36 0,0922 3 6 (211)
4 41,47 0,1230 4 8 (220)
5 46,19 0,1539 5 10 (310)
6 50,90 0,1847 6 12 (222)
7 55,28 0,2152 7 14 (321)
8 59,42 0,2456 8 16 (400)
2θ . , 8 . 2θ = 59,42
θ = 29,71
nm d
( ) 2 2 2 2
0 400 0,071699 4 0 0a d h k l nm= ⋅ + + = ⋅ + + = 0,2868 nm
.
:
1.1. 0,143 nm 20 °C.
. [0,330 nm]
1.2. 0,125 nm 20 °C.
. [0,289 nm]
1.3. K 0,316 nm. .
[0,137 nm]
1.4. 0,304 nm. .
[0,132 nm]
1.5. 0,352 nm. .
[0,125 nm]
1.6. 0,495 nm. .
[0,175 nm]
1.7. aa a ao a j o 0,128 nm . .
[0,362 nm]
1.8. 0,143 nm . .
[0,405 nm]
1.9. = 0,3209 nm c = 0,5209 nm. .
[0,139 nm3]
1.10. 0,149 nm . 0,74.
[0,112 nm3]
1.11. 0,3615 nm.
1, 0, 0 0, 1/2, 1/2. (0,4427 nm)
2 – 45
. .
[0,183 nm, 0,195 nm; 0,127 nm 0,144 nm]
1.13. (0,0,0), (0,1,0) (1/2, 1, 1/2)
[(1 01) (101 )]
1.14. 1/2, 1/2, 0 1/2, 3/2, 1/2.
[[021]]
1.15. 2.26.
2.26
(A: [1 10]; C: [1 2 1]; E: [1 0 2 ]; G: [61 6])
1.16. 2.27.
2.27
46 – .
1.18. a) z– , x– 0,5 y– 1. ) x– 2, – 2/3 z– 3/2.
[a) (210); ) ( 3 94)]
1.19. [221] (111) . []
1.20. [111] [11 1]. [70,5°]
1.21. : ) [110] [111]; )
[001] [112].
[ 71,02/ =aa ; ) 71,02/2 =aa ]
1.22. a) (110) (111) . ) [110] [123]
.
[a) 35°20` ) 55°30`]
1.23. ) [210] (210) . ) [201] (201) .
[) ; ) ]
1.24. e – 1 = 1, 2 = 1 c = 0,5.
[11 2 2]
1.25. : ) [100] –Cu 0,361 nm ) [100] –Fe 0,286 nm.
[a) 2,77 · 106 Cu/mm; b) 3,50 · 106 Fe/mm]
1.26. –Cu 0,361 nm: a) [100]; ) [110] ) [111].
[) 2,77 · 106 /mm; ) 3,91 · 106 /mm; ) 1,6 · 106 /mm]
1.27. –Fe 0,287 nm: a) [100], ) [110] ) [111]
[) 3,48 · 10 6 /mm; ) 2,47 · 10
6 /mm; ) 4,02 · 10
7 /mm]
1.28. –Al 0,405 nm: a) (100); b) (110) ) (111).
[) 1,22 · 1013 /mm2; ) 8,62 · 1012 /mm2; ) 1,41 · 1013 /mm2]
1.29. –Fe 0,287 nm: a) (100); ) (110) ) (111).
[) 1,21 · 1013 /mm2; ) 1,72 · 1013 /mm2 ) 7 · 1012 /mm2]
1.30. –Pb : ) (100) ) (111). Pb 0,175 nm.
[a) 8,2 · 1012 /mm2; ) 9,4 · 1012 /mm2]
2 – 47
1.31. (102) – . 0,1249 nm.
[5,4 · 1012 /mm2]
1.32. – 0,352 nm 58,71 g mol –1
. [8,94 Mg m –3]
1.33. 0,287 nm 7,87 Mg m –3. .
[55,9 g mol –1]
1.34. a a ooj a 4510 kg m –3. Ti Ti c/a = 1,58. 47,9 g mol –1.
[1,058 · 10 –28m3; a = 0,296 nm; c = 0,468 n]
1.35. ,
0,145 nm. : ) –Ti ) –Ti.
[a) 0,335 nm; b) 4240 kg m –3]
1.36. 5960 kg m –3. 50,9 g mol –1.
[0,132 nm]
1.37. 22400 kg m –3. 192,2 gmol –1.
[0,136 nm]
1.38. 0,636. Ni 0,2492 nm 58,71 g mol –1.
[7,65 Mg m –3]
1.39. . –Ti –Ti 2 %.
. [–2,5 . %]
3.1.
-
.
: -
-, - , --
-. (-
) .
,
.
.
;
;
.
3.2
.
: ( ),
( ), ( )
. -
. , 3.1.
,
,
.
, .
, -
.
, .
-
, . -
3 – 51
: a)
15%,) , ) e-
)
. - .
3.1
, ) -
.
Cu-Au, 3.2.
390º .
3.2
-
, 3.3.
52 – .
3.3 ) ) ()
30 mas% Zn.
Cu-Ni, () - - . -
( Cu Zn, 40% )
( Cu Sn, Cu Sn, 30%).
3.1.
-
? Cu Ni 0,128 nm Ni 0,125 nm, .
:
Cu Ni %
0,128 nm 0,125 nm 100 2,3% ( 15%)
0,128 nm
R
R
Δ − = ⋅ = <
Cu Ni , (), -
, -
- .
3 – 53
. - ,
.
-
. ,
3.4.
3.4 () ()
3.2.
( 4.1)
: ) ; )
) .
3.1.
C H O Cr Co Cu Fe Ca Zn Pt Ni
(), (nm)
0,071 0,046 0,060 0,1249
()
0,1253
()
0,1278
()
0,1241
()
0,197
()
0,1332
()
0,1388
()
0,1246
()
:
) Cu Pt Ni -
± 15 %, -
.
54 – .
) Cr, Co, Fe, Ca, Zn Ni - ± 15 %
.
) C, H Ni
.
0,414 R⋅ 0,225 R⋅ , R – .
, -
-
. ( Fe C,
2 mas% 1140 °C. -
γ – Fe α – Fe .
3.3.
α – Fe. 0,077 nm 0,124 nm, .
:
α – Fe
(1/2, 0, 1/2)
3.5.
int Fe
Fe
4
2 3 R R R R= ⋅ − = = ⋅ 0,0192nm
3.4.
(1/4, 1/4, 1/4) (0, 1/2, 1/4). –Fe 0,3571 nm –Fe 0,2866 nm. 0,071 nm. .
:
–Fe (1/4, 1/4, 1/4) ,
Fe
4 4 PCK
a r R
⋅ ⋅ = = ⋅ = = = 0,0284 nm
PCK r
(0,1/2,1/4) 3.6.
3.6 (0,1/2,1/4) –Fe.
( ) 2 2
4 4
0,071 nm. –Fe –Fe, e - –Fe .
3.6
(0, 1/2, 1/4)
. - . , - . . , . - . -
: G H T S Δ = Δ − Δ , -
. ..
- , . .
: N n
. c n N = . -
, , - , 3.7.
3.7 () ()
v H Δ . -
vn H ⋅ Δ
. - . , - . . , -
. ( n N
). ,W n
lnk S k W Δ = ⋅ , .
( )! ln ln
N n
+ Δ = ⋅ = ⋅ (3.1)
k = 1,38054 ⋅ 10–23 J –1 , W – -
3 – 57
ln ! ln( ) x x x= − x, -
n N n
Δ = − ⋅ + ⋅ + + (3.2)
( )3 lnvS k v v′Δ ≅ v
, v′ .
. n
vn S ⋅ Δ .
( )*
k vG G G H T S n H T S n S Δ = − = Δ − Δ = Δ − Δ + Δ (3.3)
* ln lnv v
N n G G n H kT N n nT S
N n N n
− = Δ + + − Δ + + (3.4)
(3.4) 3.8.
3.8
ln 0v v
n N n
N n k kT
Δ Δ = ⋅ − + (3.5)
(3.5) . -
, N + n N , -
C exp v v
X v – , C – ,
Δ H v – .
,
. (3.6) -
.
, -
.
3.5.
400 °C 2,29 ⋅ 10 –5
.
660 °C ( ).
0,76 V
86,2 ⋅ 10 –6
V k –1
:
C (3.6)
400 °C
v E
= ⋅
-6 -15 0,76eV 86,2 10 eV k 673K 2,29 10 11,2C e− ⋅ ⋅ ⋅= ⋅ ⋅ =
660 °C (3.6)
6 1 0,76eV 86,2 10 eV K 93311,2 K
v X e − −− ⋅ ⋅= ⋅ = -4
() -
. ,
, - .
,
-
. -
. -
,
.
. -
.
-
.
-
: , -
.
- -
(
, -
)
-
.
3.9
: ()
; () " "
, ()
( ) , -
( ). .
-
, t
, b
,
. -
-
. , -
.
, 3.10.
3.10.
-
. ,
. 3.10.
() ;
() ;
() , - . ,
.
, b
, .
-
3.11.
. .
3.6.
0,36151 nm.
,
3.12.
0 110
.
(3.7)
[ ]hkl -
2 2 2
0d a h k l = + +
.
, - .
3.7.
-
α− Fe l . Fe
0,124 nm Al 0,143 nm.
:
3.14.
, , -
α− Fe :
Fe2 2 0,124 nm 0, 248 nmb R= ⋅ = ⋅ =
,
, Al
Al2 2 0,143 nm 0,286 nmb R= ⋅ = ⋅ =
. , τ , -
, δ, ,
3.14.
, , . , -
. , 1 -
2 4 1 2
4 4 -
2. 3.14,
,τ , E p δ,
. -
. 1
4, ,
.
.
3.15
3 – 63
, , , .
,
,
,
-
. -
.
. , -
-
, - .
, -
Pierls-Nabarr- napon,
∼ (3.8)
3.14. )
G – , b – – .
,
, 3.16.
.
: 1)
; 2) 3)
,
.
64 – .
. . -
{ }111 a 110 , 3.17.
3.15
3.16
3.17 { }111
110
3 – 65
( )111 [ ]110 4 ·3 = 12 .
. { }0001 1120 ,
3.18 1 ·3 = 3.
{101 0} {101 1}
c/a.
3.18
. τ , -
: {110}, {112} {123}, 3.19.
3.19
, 3.1. -
.
.
. -
66 – .
. .
3.1
,
..
3 – 67
, 3.20. 20 500 .
3.20
3.8.
Ni () 110 . aa oo oa j a-
o a a a o aa
0,001 mm. -
Ni 0,1243 nm.
:
13.21
[110] : 2 0,2486 nmr =
:
1000 nm
68 – .
- . -
. -
.
3.22
3.22
F, F Aσ = ,
cosr A A φ = cosr F F λ = ⋅ .
cos cos cos cos cos
cos
r
r
φ
⋅ = = = ⋅ ⋅ = ⋅ ⋅ (3.9)
, k σ , -
θ 0 φ 0 . k σ ,, θ 0 φ 0 -
(1) , τ -
. -
. 1. , k σ τ
.
cos coskr kr kr τ σ λ φ = ⋅ ⋅ (3.10)
(2). -
, kr τ ,
3 – 69
() . φ -
(3.10) -
, m,. , -
.
1
φ λ
(3.11)
(0,5) φ = 450
( 0,5τ σ = ⋅ ) -
45 ° .
3.9.
{ }110 111 1 Pa. [ ]001
: ) 101 ( )111 ; ) [ ]110 ( )111
:
) 101 ( )111 [ ]001 -
3.23
cos 0,707
2
a λ = = =
( )111 [ ]111 [ ]001
32.3
. cos 0,577
3
, .
1 MPa 2,45 MPa
cos cos 0,577 0,707
3.10.
1 10
:
110 11 1
( )11 1 :
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
21 2 1 2 1
1/2 1/2 2 2 22 2 2 2 2 2 2 2 2 1 1 1 2 2 2
1 1 1 1 0 1 2 2 cos
2 3 61 1 0 1 1 1
h h k k l l
h k l h k l λ
+ + ⋅ + − ⋅ − + ⋅ − = = = =
⋅+ + ⋅ + + + − + ⋅ + − + −
110 0 11
:
1 0 1 1 0 1 1 1 1 cos
22 22 0 1 1
φ ⋅ + − ⋅ − + ⋅
6 MPa
= = = ⋅ ⋅
-
. 3.24
.
, -
. ,
. ,
, -
.
3.24
,
. , , .
-
-
,
. -
3.25.
. -
-
,
,
, -
. -
, -
3.26.
) ) 3.26. . , )
1–9,
.
: () ; () ; ()
3.27
3.28. -
.
3.28 : ) 9
3 – 73
, - . ,
.
, -
( )111
(111)
( )111
(1 11)
3.29.
3.30 -
, -
( )111 ,
, (1 11)
( )111 .
-
-
.
( )111
[ ]111
74 – .
. -
3.31.
3.31
.
, t
.
: () 1 sinb b θ = ⋅ ()
2 cosb b θ = ⋅ , 3.32.
. ,
, t
,
3 – 75
3.33. 3.33. .
-
, b
90° -
.
. -
.
.
-
,
-
- -
. ,
-
-
.
3.33
76 – .
. -
b
3.34.. -
r , 3.34..
3.34 ()
.
3.34. γ , -
τ .
π
⋅ = ⋅ = (3.13)
G – . (3.13) -
.
.
:
2 dE = × ×
21 2
r r r π
π π π = ⋅ ⋅ ⋅ ⋅ = ⋅ (3.14)
2
1
r r π π = ⋅ = ⋅ ⋅∫ (3.15)
(3.15) .
r 2 ∼ 10 –3
mm r 1 ∼ 10 –9
m (3-4 ),
62
1
r π = ≈
(3.14) -
2 E = Gb l (3.16)
G, b l .
, , . - , .
, , . 2
sT = Gb (3.17)
,
. , θ .
2
)
78 – .
3.35.. , . .
, , ν ≈ 1/3, - 3/2 .
b
2, ,
, - .
3.11.
100 000
.
.
. 0,125 nm, 26 GPa 0,3. - 5 .
:
(3.16)
( ) :
l r π = =
r 1 = 5⋅b r 2 = 100 000⋅b, b = 2 R = 0,25 nm
( )9 2 926 10 Nm 0,25 10 m ln20000
4 sT
9,28 10 N
. .
. w d h⋅ ⋅ , τ
w d ⋅
3.36.
3 – 79
τ , w d τ ⋅ ⋅
| |b
.
sW w d bτ = ⋅ ⋅ ⋅ (3.19)
, - , . - . : F ,
d ,
F d ⋅ ,
w
uW F d w= ⋅ ⋅ (3.20)
3.36
w d b F d wτ ⋅ ⋅ ⋅ = ⋅ &
.
„” -
. -
. -
, .
-
. . -
, -
„“ , .
–
.
) ) 2.1 ) , )
. , -
.
, 2.1..
, , -
,
, , 2.1.
( o-
) ( ). -
.
, -
. ,
. , -
, , 2.2.
2.2
, -
.
. -
„", -
. „"
.
.
. -
, c 2.3, - -
: , -
.
18 – .
2.3 ,
2.1.
r ,
p m n
a b E
r r = − +
r = 0,3 nm E = –4 eV.
a b, m = 2 n = 10,
.
:
p m n
a b E
r r = − + (2.1)
1 1
dr r r + +
-19
(0,3 nm) (0,3 nm)
b
20 27,2 10 J nma −= ⋅ ; 25 29,4 10 J nmb −= ⋅
F max,
.
2
dr r r + +
1 1m n
r r
, „ " ,
.
, ,
, : , , , -
, .
.
, –38,9 ° ( ) 3410 ° ( ). -
, -
. , -
.
, ,
. -
, -
, ,
,
.
, -
(2+), „
", (+). -
), . Fe, Ni, W, d- -
d-, , , -
. -
d- .
. -
, -
. ,
„”
. ,
, .
,
. , „ ”, . -
, ,
. , -
(), (- )
. .
.
, -
.
. -
, ,
.
(, ),
, ,
. -
,
2 – 21
, ()
() , .
, , .
. .
, -
, , , . , -
( 10-8 ),
( ) -
. ,
.
.
,
.
, , . . -
.
2.4
. -
,
,
. ,
, , 2.4. -
, .
.
,
, .
.
, ( ) 0r r
E F dr ≈
22 – .
2.2.
-
. -
. ,
( )
.
2.5 2.6.
2.5
ABCABC
ABAB
.
.
... ,
ABCABC... -
.
12 -
( ... ) ( -
... ) 2.7 2.8
(X) () .
2 – 23
2.7. ABAB…
2.8 ABCABC…
X.
X
.
( )
, () 2.9.
: ) : α – Fe, β –
Ti V, Cr, Mo W., ) : γ – Fe, Al, Ni, Cu, Ag Pt; ) X: Be, Mg, Zn, α – Ti Zr
.
.
.
2.9
, AFP
,
atoma
V
⋅ = (2.2)
n − .
() : n = 8 ⋅ (1/8) = 1 /,
() : n = 8 ⋅ (1/8) + 1 ⋅ (1) = 2 /,
() : n = 8 ⋅ (1/8) + 6
⋅ (1/2) = 4 /.
( FP)
( ).
:
, 2.10.
3 4
4 3
a R
a R
⋅ =
=
3.1, n = 2,
3
3
2.10
2.3:
- .
2.11
:
4 2
4 2
R a
a R
=
=
( )
( )
4 / 2
2.4:
(X )
:
X , 2.12., -
2.12.
) ) 2.12. ) , )
, 2.12., -
LM . -
2.13. / .
2.13 ) , )
MH ; JM 2 2
c a R= = = ; / 2
c a c a a c a
a = + = + = = 1,633
2.5:
:
( ) ( ) rogalj centar
2 2 3sin 60 1,633 2 (2 ) 11,31V c a R R R= ⋅ ⋅ ° = ⋅ ⋅ =
( )3 32 4 /3 8,38aV R Rπ = ⋅ =
X
3
3
( )
, AFP , 0 Am M N = -
, , A , -
:
A
⋅ (2.3)
: ρ - , n - , M -
, A N - , V - .
2.6:
0.2866 nm. 55.847 g/mol
:
( )
3 -3 7,882 10 kgm
A
.
. -
2.14. -
-
.
2.15.
, -
,
2.16.
2.14
2.15
.
2.16
G H TS = −
.
p
= −
( )
2
2
p
p
2.7:
– Fe – Fe .
0,2863 nm 0,3591 nm.
:
−⋅=== aV PCK
-Fe -Fe
Fe, . -Fe Fe -Fe
:
3 3 3 3
2 23,467 10 nm V
− −
−
, -
. -
. -
-
, -
.
()
()
, )
)
-
( )
2 – 31
.
, , : ) -
, ) , ) -
) ,
), ) ). -
, 2.14.
, -
. -
.
, ,
.
.
.
)
.
-
. -
-
. , a, b c, -
-
2.14. a, b c -
.
, b c α , β γ , 2.14. 14 -
, 2.15,
7 , 2.1
2.1
a b c= = 90α β γ = = = °
a b c= ≠ 90α β γ = = = °
a b c= = 90α β γ = = ≠ °
a b c= ≠ 90 120α β γ = = ° = °
a b c≠ ≠ 90α β γ = = = °
a b c≠ ≠ 90α γ β = = ° ≠
a b c≠ ≠ 90α β γ ≠ ≠ ≠ °
32 – .
,
, ,
.
.
,
.
.
,
. .
. :
1 y z
a b c + + =
2 – 33
, b, c x, y, z . -
, -
, -
. , b c
,
( ). -
.
1 1 1 h k l
a b c = = =
, b c , x, y z
: 1hx ky lz + + = . (
, h,k l (hkl) -
. , x
(100) , 2.16.
. -
, , .
x ; , -
(100)
34 – .
; (330), (220) (110) „(100) ”,
. , -
. " -
". (100), (010) (001) .
.
: {100} (100), (010), (001),
(100), (010) (001) .
2.8:
A, B C 2.17
2.17
A, B C
:
) = 1, y = 1, z = 1; ) 1/ x = 1, 1/ y = 1,1 / z = 1; ) ; ) (111)
:
) z- x = 1, y = 2 z = ∞; ) 1/ x = 1, 1/ y = 1/2, 1/ z = 0;
) : 1/ x = 2, 1/ y = 1, 1/z = 0 ) (210)
:
)
0,0,0. , y-; ) x =∞ , y = -1 z =∞
.
. , ,
, ,
[ ]111 "111" ,
. -
2 – 35
110 110 , 101 , 011 .
, ,
. .
-
.
:
() -
;
() -
;
() -
;
() [ ].
.
2.9:
A, B i C 2.18.
2.18
A, B C
:
() (1,0,0) (0,0,0); () 1,0,0 – 0,0,0 = 1,0,0; () ; () [100].
C:
() 0,0,1 1/ 2 ,1,0; () 0,0,1– 1/ 2 ,1,0 = – 1/ 2 ,–1,1
- -
12 2 .
:
36 – .
, . - (hkl ) - [uvw] :
1) [uvw] (hkl ) u = h, v = k w = l. , [111] (111).
2) [uvw] (hkl ), . 0hu kv lw+ + = .
3) (h1k 1l 1) (h2k 2l 2)
1 2 1 2 1 2 0h h k k l l + + = .
4) (u1v1w1) (u2v2w2)
1 2 1 2 1 2 0u u v v w w+ + = .
5) θ (h1k 1l 1) (h2k 2l 2)
1/ 2 1/ 2 2 2 2 2 2 2
1 1 1 2 2 2
cos h h k k l l
h k l h k l θ
+ + =
2.10:
[110] [111] . ( ).
:
[110] [111] (2.4)
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
arccos u u v v w w
u v w u v w θ
+ + =
+ + ⋅ + +
1 1 1 1 1 0 2 arccos arccos arccos0,816
3 21 1 1 1 1 0 35,3°θ
⋅ + ⋅ + ⋅ = = = =
⋅+ + ⋅ + +
2 – 37
- h, k, i, l, (hkil). a1 , a2 , a3 c . . . ,
. h k i+ = − .
2.19
2.11:
- A B C D 2.20.
:
:
() 1 2 3 1a a a c= = = ∞ = ; () 1 2 3
1 1 1 1 0 1
a a a c = = = = ; () ; () (0001).
:
1 2
a a c = = = ;
(1121) .
:
() (0,0,1) (1,0,0); () 0,0,1 – 1,0,0 = –1,0,1; () h k i+ = −
() [1011]
D:
() (0,1,0) (1,0,0); () 0,1,0 – 1,0,0 = –1,1,0; () i = –(–1+1) = 0
() [1100]
38 – .
2.20 –
- ,
. h k i+ = −
. - , 2.21.
2.21
2.12:
[111] () () . RW = 0,137 nm RAl = 0,143 nm.
:
() . [1 1 1]. , .
r = 2 RW = 2 ⋅ 0,137 nm = 0,274 nm
1 1
0,274 nm r ⋅ = = 3,65 atoma/nm
() a [1 1 1] .
Al
2 2
2.13:
(111) : ) ) . RW = 0,137 nm a RAl = 0,143 nm.
:
() (111) , 2.22.
2.22 (111)
3 3
2 2 0,316 nm 0,447 nmb a= ⋅ = ⋅ =
(1 1 1)
2 2 21 1 3 3 3 (0, 447 nm) 0, 0867 nm
40 – .
(111) 1-6 ,
1
2
nm
() (111) ,
2.23.
2 2
R a = = ⋅
2 2 0,404 nm=0,572 nmb a= = ⋅
(111)
21 1 3 0,572 nm 0,572 nm=0,142 nm
2 2 2 A b h= ⋅ ⋅ = ⋅ ⋅
(111) 3 1/6
½,
1
2
0,142 nm A− ⋅ + ⋅
= = 2 14,1 atoma/nm
, d. -
, . .
-
0
(hkl ) :
0
2.14:
(111)
0,40786 nm.
. 2.6
111 2 2 2
+ +
- .
, d,
. d
, .
2.24 - : ) , ) )
42 – .
,
, 2.24..
, d , θ ,
2.24.. ( 1, 2.24. )
, , ( 2)
, 2 d 1
MPN. s = MP + NP ,
d , 2d 2s . -
-
. -
n. λ = s, n- , ,
λ- s- ,
. , 2.24
s = 2d sinθ
2 sinn λ θ θ ⋅ = (2.6)
.(2.6) .
, λ ,
, d θ .
2.25 ) , ) X-
-
, d, , λ
2.15:
-
λ = 0.07107 nm ( ) -
, 2θ , :
2θ 2θ
1 20,20 5 46,19
2 28,72 6 50,90
3 35,36 7 55,28
4 41,47 8 59,42
,
.
:
sin 2θ -
0.0308
2 θ ⋅ 2sin θ 2sin 0,0308θ 2 2 2h k l + + ( )hkl
1 20,20 0,0308 1 2 (110)
2 28,72 0,0615 2 4 (200)
3 35,36 0,0922 3 6 (211)
4 41,47 0,1230 4 8 (220)
5 46,19 0,1539 5 10 (310)
6 50,90 0,1847 6 12 (222)
7 55,28 0,2152 7 14 (321)
8 59,42 0,2456 8 16 (400)
2θ . , 8 . 2θ = 59,42
θ = 29,71
nm d
( ) 2 2 2 2
0 400 0,071699 4 0 0a d h k l nm= ⋅ + + = ⋅ + + = 0,2868 nm
.
:
1.1. 0,143 nm 20 °C.
. [0,330 nm]
1.2. 0,125 nm 20 °C.
. [0,289 nm]
1.3. K 0,316 nm. .
[0,137 nm]
1.4. 0,304 nm. .
[0,132 nm]
1.5. 0,352 nm. .
[0,125 nm]
1.6. 0,495 nm. .
[0,175 nm]
1.7. aa a ao a j o 0,128 nm . .
[0,362 nm]
1.8. 0,143 nm . .
[0,405 nm]
1.9. = 0,3209 nm c = 0,5209 nm. .
[0,139 nm3]
1.10. 0,149 nm . 0,74.
[0,112 nm3]
1.11. 0,3615 nm.
1, 0, 0 0, 1/2, 1/2. (0,4427 nm)
2 – 45
. .
[0,183 nm, 0,195 nm; 0,127 nm 0,144 nm]
1.13. (0,0,0), (0,1,0) (1/2, 1, 1/2)
[(1 01) (101 )]
1.14. 1/2, 1/2, 0 1/2, 3/2, 1/2.
[[021]]
1.15. 2.26.
2.26
(A: [1 10]; C: [1 2 1]; E: [1 0 2 ]; G: [61 6])
1.16. 2.27.
2.27
46 – .
1.18. a) z– , x– 0,5 y– 1. ) x– 2, – 2/3 z– 3/2.
[a) (210); ) ( 3 94)]
1.19. [221] (111) . []
1.20. [111] [11 1]. [70,5°]
1.21. : ) [110] [111]; )
[001] [112].
[ 71,02/ =aa ; ) 71,02/2 =aa ]
1.22. a) (110) (111) . ) [110] [123]
.
[a) 35°20` ) 55°30`]
1.23. ) [210] (210) . ) [201] (201) .
[) ; ) ]
1.24. e – 1 = 1, 2 = 1 c = 0,5.
[11 2 2]
1.25. : ) [100] –Cu 0,361 nm ) [100] –Fe 0,286 nm.
[a) 2,77 · 106 Cu/mm; b) 3,50 · 106 Fe/mm]
1.26. –Cu 0,361 nm: a) [100]; ) [110] ) [111].
[) 2,77 · 106 /mm; ) 3,91 · 106 /mm; ) 1,6 · 106 /mm]
1.27. –Fe 0,287 nm: a) [100], ) [110] ) [111]
[) 3,48 · 10 6 /mm; ) 2,47 · 10
6 /mm; ) 4,02 · 10
7 /mm]
1.28. –Al 0,405 nm: a) (100); b) (110) ) (111).
[) 1,22 · 1013 /mm2; ) 8,62 · 1012 /mm2; ) 1,41 · 1013 /mm2]
1.29. –Fe 0,287 nm: a) (100); ) (110) ) (111).
[) 1,21 · 1013 /mm2; ) 1,72 · 1013 /mm2 ) 7 · 1012 /mm2]
1.30. –Pb : ) (100) ) (111). Pb 0,175 nm.
[a) 8,2 · 1012 /mm2; ) 9,4 · 1012 /mm2]
2 – 47
1.31. (102) – . 0,1249 nm.
[5,4 · 1012 /mm2]
1.32. – 0,352 nm 58,71 g mol –1
. [8,94 Mg m –3]
1.33. 0,287 nm 7,87 Mg m –3. .
[55,9 g mol –1]
1.34. a a ooj a 4510 kg m –3. Ti Ti c/a = 1,58. 47,9 g mol –1.
[1,058 · 10 –28m3; a = 0,296 nm; c = 0,468 n]
1.35. ,
0,145 nm. : ) –Ti ) –Ti.
[a) 0,335 nm; b) 4240 kg m –3]
1.36. 5960 kg m –3. 50,9 g mol –1.
[0,132 nm]
1.37. 22400 kg m –3. 192,2 gmol –1.
[0,136 nm]
1.38. 0,636. Ni 0,2492 nm 58,71 g mol –1.
[7,65 Mg m –3]
1.39. . –Ti –Ti 2 %.
. [–2,5 . %]
3.1.
-
.
: -
-, - , --
-. (-
) .
,
.
.
;
;
.
3.2
.
: ( ),
( ), ( )
. -
. , 3.1.
,
,
.
, .
, -
.
, .
-
, . -
3 – 51
: a)
15%,) , ) e-
)
. - .
3.1
, ) -
.
Cu-Au, 3.2.
390º .
3.2
-
, 3.3.
52 – .
3.3 ) ) ()
30 mas% Zn.
Cu-Ni, () - - . -
( Cu Zn, 40% )
( Cu Sn, Cu Sn, 30%).
3.1.
-
? Cu Ni 0,128 nm Ni 0,125 nm, .
:
Cu Ni %
0,128 nm 0,125 nm 100 2,3% ( 15%)
0,128 nm
R
R
Δ − = ⋅ = <
Cu Ni , (), -
, -
- .
3 – 53
. - ,
.
-
. ,
3.4.
3.4 () ()
3.2.
( 4.1)
: ) ; )
) .
3.1.
C H O Cr Co Cu Fe Ca Zn Pt Ni
(), (nm)
0,071 0,046 0,060 0,1249
()
0,1253
()
0,1278
()
0,1241
()
0,197
()
0,1332
()
0,1388
()
0,1246
()
:
) Cu Pt Ni -
± 15 %, -
.
54 – .
) Cr, Co, Fe, Ca, Zn Ni - ± 15 %
.
) C, H Ni
.
0,414 R⋅ 0,225 R⋅ , R – .
, -
-
. ( Fe C,
2 mas% 1140 °C. -
γ – Fe α – Fe .
3.3.
α – Fe. 0,077 nm 0,124 nm, .
:
α – Fe
(1/2, 0, 1/2)
3.5.
int Fe
Fe
4
2 3 R R R R= ⋅ − = = ⋅ 0,0192nm
3.4.
(1/4, 1/4, 1/4) (0, 1/2, 1/4). –Fe 0,3571 nm –Fe 0,2866 nm. 0,071 nm. .
:
–Fe (1/4, 1/4, 1/4) ,
Fe
4 4 PCK
a r R
⋅ ⋅ = = ⋅ = = = 0,0284 nm
PCK r
(0,1/2,1/4) 3.6.
3.6 (0,1/2,1/4) –Fe.
( ) 2 2
4 4
0,071 nm. –Fe –Fe, e - –Fe .
3.6
(0, 1/2, 1/4)
. - . , - . . , . - . -
: G H T S Δ = Δ − Δ , -
. ..
- , . .
: N n
. c n N = . -
, , - , 3.7.
3.7 () ()
v H Δ . -
vn H ⋅ Δ
. - . , - . . , -
. ( n N
). ,W n
lnk S k W Δ = ⋅ , .
( )! ln ln
N n
+ Δ = ⋅ = ⋅ (3.1)
k = 1,38054 ⋅ 10–23 J –1 , W – -
3 – 57
ln ! ln( ) x x x= − x, -
n N n
Δ = − ⋅ + ⋅ + + (3.2)
( )3 lnvS k v v′Δ ≅ v
, v′ .
. n
vn S ⋅ Δ .
( )*
k vG G G H T S n H T S n S Δ = − = Δ − Δ = Δ − Δ + Δ (3.3)
* ln lnv v
N n G G n H kT N n nT S
N n N n
− = Δ + + − Δ + + (3.4)
(3.4) 3.8.
3.8
ln 0v v
n N n
N n k kT
Δ Δ = ⋅ − + (3.5)
(3.5) . -
, N + n N , -
C exp v v
X v – , C – ,
Δ H v – .
,
. (3.6) -
.
, -
.
3.5.
400 °C 2,29 ⋅ 10 –5
.
660 °C ( ).
0,76 V
86,2 ⋅ 10 –6
V k –1
:
C (3.6)
400 °C
v E
= ⋅
-6 -15 0,76eV 86,2 10 eV k 673K 2,29 10 11,2C e− ⋅ ⋅ ⋅= ⋅ ⋅ =
660 °C (3.6)
6 1 0,76eV 86,2 10 eV K 93311,2 K
v X e − −− ⋅ ⋅= ⋅ = -4
() -
. ,
, - .
,
-
. -
. -
,
.
. -
.
-
.
-
: , -
.
- -
(
, -
)
-
.
3.9
: ()
; () " "
, ()
( ) , -
( ). .
-
, t
, b
,
. -
-
. , -
.
, 3.10.
3.10.
-
. ,
. 3.10.
() ;
() ;
() , - . ,
.
, b
, .
-
3.11.
. .
3.6.
0,36151 nm.
,
3.12.
0 110
.
(3.7)
[ ]hkl -
2 2 2
0d a h k l = + +
.
, - .
3.7.
-
α− Fe l . Fe
0,124 nm Al 0,143 nm.
:
3.14.
, , -
α− Fe :
Fe2 2 0,124 nm 0, 248 nmb R= ⋅ = ⋅ =
,
, Al
Al2 2 0,143 nm 0,286 nmb R= ⋅ = ⋅ =
. , τ , -
, δ, ,
3.14.
, , . , -
. , 1 -
2 4 1 2
4 4 -
2. 3.14,
,τ , E p δ,
. -
. 1
4, ,
.
.
3.15
3 – 63
, , , .
,
,
,
-
. -
.
. , -
-
, - .
, -
Pierls-Nabarr- napon,
∼ (3.8)
3.14. )
G – , b – – .
,
, 3.16.
.
: 1)
; 2) 3)
,
.
64 – .
. . -
{ }111 a 110 , 3.17.
3.15
3.16
3.17 { }111
110
3 – 65
( )111 [ ]110 4 ·3 = 12 .
. { }0001 1120 ,
3.18 1 ·3 = 3.
{101 0} {101 1}
c/a.
3.18
. τ , -
: {110}, {112} {123}, 3.19.
3.19
, 3.1. -
.
.
. -
66 – .
. .
3.1
,
..
3 – 67
, 3.20. 20 500 .
3.20
3.8.
Ni () 110 . aa oo oa j a-
o a a a o aa
0,001 mm. -
Ni 0,1243 nm.
:
13.21
[110] : 2 0,2486 nmr =
:
1000 nm
68 – .
- . -
. -
.
3.22
3.22
F, F Aσ = ,
cosr A A φ = cosr F F λ = ⋅ .
cos cos cos cos cos
cos
r
r
φ
⋅ = = = ⋅ ⋅ = ⋅ ⋅ (3.9)
, k σ , -
θ 0 φ 0 . k σ ,, θ 0 φ 0 -
(1) , τ -
. -
. 1. , k σ τ
.
cos coskr kr kr τ σ λ φ = ⋅ ⋅ (3.10)
(2). -
, kr τ ,
3 – 69
() . φ -
(3.10) -
, m,. , -
.
1
φ λ
(3.11)
(0,5) φ = 450
( 0,5τ σ = ⋅ ) -
45 ° .
3.9.
{ }110 111 1 Pa. [ ]001
: ) 101 ( )111 ; ) [ ]110 ( )111
:
) 101 ( )111 [ ]001 -
3.23
cos 0,707
2
a λ = = =
( )111 [ ]111 [ ]001
32.3
. cos 0,577
3
, .
1 MPa 2,45 MPa
cos cos 0,577 0,707
3.10.
1 10
:
110 11 1
( )11 1 :
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
21 2 1 2 1
1/2 1/2 2 2 22 2 2 2 2 2 2 2 2 1 1 1 2 2 2
1 1 1 1 0 1 2 2 cos
2 3 61 1 0 1 1 1
h h k k l l
h k l h k l λ
+ + ⋅ + − ⋅ − + ⋅ − = = = =
⋅+ + ⋅ + + + − + ⋅ + − + −
110 0 11
:
1 0 1 1 0 1 1 1 1 cos
22 22 0 1 1
φ ⋅ + − ⋅ − + ⋅
6 MPa
= = = ⋅ ⋅
-
. 3.24
.
, -
. ,
. ,
, -
.
3.24
,
. , , .
-
-
,
. -
3.25.
. -
-
,
,
, -
. -
, -
3.26.
) ) 3.26. . , )
1–9,
.
: () ; () ; ()
3.27
3.28. -
.
3.28 : ) 9
3 – 73
, - . ,
.
, -
( )111
(111)
( )111
(1 11)
3.29.
3.30 -
, -
( )111 ,
, (1 11)
( )111 .
-
-
.
( )111
[ ]111
74 – .
. -
3.31.
3.31
.
, t
.
: () 1 sinb b θ = ⋅ ()
2 cosb b θ = ⋅ , 3.32.
. ,
, t
,
3 – 75
3.33. 3.33. .
-
, b
90° -
.
. -
.
.
-
,
-
- -
. ,
-
-
.
3.33
76 – .
. -
b
3.34.. -
r , 3.34..
3.34 ()
.
3.34. γ , -
τ .
π
⋅ = ⋅ = (3.13)
G – . (3.13) -
.
.
:
2 dE = × ×
21 2
r r r π
π π π = ⋅ ⋅ ⋅ ⋅ = ⋅ (3.14)
2
1
r r π π = ⋅ = ⋅ ⋅∫ (3.15)
(3.15) .
r 2 ∼ 10 –3
mm r 1 ∼ 10 –9
m (3-4 ),
62
1
r π = ≈
(3.14) -
2 E = Gb l (3.16)
G, b l .
, , . - , .
, , . 2
sT = Gb (3.17)
,
. , θ .
2
)
78 – .
3.35.. , . .
, , ν ≈ 1/3, - 3/2 .
b
2, ,
, - .
3.11.
100 000
.
.
. 0,125 nm, 26 GPa 0,3. - 5 .
:
(3.16)
( ) :
l r π = =
r 1 = 5⋅b r 2 = 100 000⋅b, b = 2 R = 0,25 nm
( )9 2 926 10 Nm 0,25 10 m ln20000
4 sT
9,28 10 N
. .
. w d h⋅ ⋅ , τ
w d ⋅
3.36.
3 – 79
τ , w d τ ⋅ ⋅
| |b
.
sW w d bτ = ⋅ ⋅ ⋅ (3.19)
, - , . - . : F ,
d ,
F d ⋅ ,
w
uW F d w= ⋅ ⋅ (3.20)
3.36
w d b F d wτ ⋅ ⋅ ⋅ = ⋅ &