-
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2.1
,
,
, ,
, .
.
, ,
. .. , ,
. .
, ..
, .
, ,
,
,
. , ,
,
.
(. 2.1).
. 2.1: ()
.
.
-
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46
, .
,
, :
1) :
,
.
2) :
, .
:
, .
, ,
:
,
, .
3) :
( )
(), .
4) :
(,
).
,
, . , , ().
-
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47
(, )
. ,
.
, .
,
. ,
,
,
, ,
.
,
. Hay, 1784,
,
. , ,
, (. 3 ). , ,
, ,
.. ,
. Hay, ,
,
.
-
, :
, . ,
, , ,
,
.
,
. ,
-
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48
, .
(. 2.2,).
( )
. ,
, . , (, m
)
-.
. 2.2: -
(m, ).
. 2.2:
-
( .)
,
.
,
.
,
. , ,
, .
(. 2.3).
.
, ,
.
, . ,
-,
2/ .
, 2.
(. 2.4).
-
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49
. 2.3: (i).
.
. 2.4: , 2, 3, 4
6, .
,
. ,
56. ()
() .
,
1.
-
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50
1: . (: 1, 2,
3, 4, 6 ) ,
.
: 6.
6
2= 60
: 4
.
4
2= 90
: 3.
3
2= 120
: 2.
2
2= 180
-
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51
(. 2.5). , , ,
. , ..
2,
.
. 2.5: 4 , 4 2
, 5 (m).
, ,
., -
-
,
( ).
( ).
, , ,
,
.
, (proper
rotation),
, (improper rotation).
(1, 2, 3, 4, 6)
(m). 1/m, 2/m, 3/m, 4/m
6/m, . , 1
(. 2.6), 2
(. 2.6) , , 6 3/m.
, 1 , 2 6
.
-
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52
() ()
. 2.6: () 1( 1 ),
360
. () 2 (2) 180
.
(m) 2 .
, ,
,
.
, . ,
(low symmetry).
, (high symmetry). ,
: 6
4,
3... ,
, .
. 2.7: (
).
, ,
.
-
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53
, , .
4, 3, 2, (. 2.7)
. , , ,
.
.
.
.
, ,
, .. , ,
, , .
, 6
,
.
,
,
, 32
. 32
(point group). .
,
.
, ,
: .
, .
.
, . ,
, .
(32 )
. .
32
. ,
-
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54
-,
.
, ,
.
, 1.
-,
, .
- - .
.
( ),
x,y,z
a, b, c (
) .
. ..
4
. ,
90
z,
x, a
, y, b = a
(. 2.8). , , xy.
, , z,
. c/a z
x y, a/b = 1, . ,
(a=bc)
.
. 2.8:
(a, b = a,
c)
4(,
z).
.
.
1.
-
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55
32 (, )
, .
.
,
,
(. 2).
1, , ,
.
.
2:
. 2,
, . , ,
- /3
a, b, (a+b).
1
, . , .
-
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56
32
,
.
,
-.
.
, ,
,
( ),
() .
, , ,
.
.
, ,
. :
+ = =
, .
.
.
, ,
.
(. 2.9).
,
,
.
,
, .
,
. , ,
-
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57
,
.
() ()
. 2.9: () () .
ab.
. , 5
6
. 2.10
. (,
) , . ,
. , ,
, , .
-
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58
. 2.10: . , ,
2, 3, 4 6 ,
. , .
-
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59
,
1, T,
a, b, c:
T= n1a + n2b + n3c
ni . ,
abcab.
() 2
.
()
rr:
r= r + T= r+ n1a + n2b + n3c
(r, r)
()
r, n1, n2, n3.
, a, b, c .
(),
, ni.
,
. , a, b, c,
, ,
r, r
n1,n2, n3(. . 2.11). ,
ni,
, .
(.. ) a,
b, c,. ,
90,
, .
,
.
1.2
-
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60
. 2.11: c
(n1,n2, n3) = (2, 3, 5). a, b,
c .
,
. ,
. , ..,
, , ,
, (.
, ).
.
, ()
. ,
, ,
(unit cell).
a, b, c
(. 2.12).
, . 1, (:
primitive unit cell), .
,
a
(b
c). . 2.13. , ,
, .
1. , . ()
, 1/8 , .
-
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61
. 2.12: . a, b, c
a, b, c.
, , .
.
. 2.13:
(Primitive) (Non-Primitive) .
(P) .
.
,
1
, .
. , (,
.)
.
1, , . , , ,
, .
-
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62
BRAVAIS
, 2,
, ,
, .
.
,
,
, , (
32 ).
(
).
, ,
.
(centering)
(.. 4 ,
1.10).
,
. ,
,
1.
.
,
. ,
:
(face centered, F):
.
1
. , . , ,
(, ). - -.
-
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63
(body centered, I):
.
- (side base end- centered, C):
.
.
(.
).
, ( )
, .
.
Bravais. 3 . 2.14 ,
,
,
.
-
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64
3
(,
)
()
()
(P, R)*
(I, F, C)*
()
a=b=c
a===90
4 3 P, I, F
a=b
===90
1 4 P, I
===90 3 2
P, C, I, F
a=b, ==90
=120
1 6
P
()
a=b=c
==90
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66
-
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67
: )
) .
( ),
, ,
.
(1/3a, 2/3b, 2/3c) (2/3a, 1/3b, 1/3c)
a = b = c, = = 90
-
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68
,
. , ,
, . .
,
, ,
(P, I, F C) ,
.
3(3 3 ). , 4
. 3
,
2
. , ,
23.
(), .
, , :
:
4/m 3 2/m 1 432 4 3 m 2/m 3 2 3
6
3 (3 3 ) ,
. 3 ( 3 ) .
, , 2
.
. , :
:
6/m 2/m 2/m 622 6mm 6 2/m 6/m 6 6
() :
3 2/m 32 3m 3 3
1
Bravais, . , 23 (23P, 23I, 23F ). .
-
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69
( Hermann
Mauguin), , ,
, .
,
, 4:
4
() 4, 4/m, 4 , 2, 2/m 3, 3 2, 2/m, m
- 6, 6/m, 6 , 3, 3 2/m, m 2, 2/m, m
4, 4/m, 4 2, 2/m, m 2, 2/m, m
2, 2/m, m 2, 2/m, m 2, 2/m, m
2, 2/m, m
1, 1
, ,
, , (
). 3
,
5444 (). , ,
2
, , 45
(2 ). , 23
2
( ) 3
5444.
,
(). ,
, 2, 120
,
120, .
, , 2
.
. ,
. ,
, 4/m3 2/m,
, .. . ,
(crystal forms). ,
,
-
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70
. , ,
, , , .
( ) , .
,
, , (
4/m3 2/m). , ,
. ,
, (
4 3m).
. ,
.
, , ,
.
(-, , )
,
.
, .
-
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71
-
.
(P, I, F, C),
,
(.) Bravais.
,
.
, .
, , .
.
, (Cu) Cu
. Cu ,
(
) ,
,
. ,
. ,
.
. , ,
,
. , ,
, - -
. (. . 2.15 ).
( ) ,
,
. ,
,
(. 2.16). ,
,
. , , , :
,
,
-
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72
. . ,
4 , , ,
4
(. . 2.10). 3
6
, , ,
(. . 2.10). , 2
,
, 2 (. 2.10),
. , NaCl
(. 2.17).
. 2.15: , , 6mm.
,
33 ()
3m ( ).
(6mm, 6, 3m, 3, 2mm, 2, 1m).
-
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73
. 2.16: (),4
(4mm),
. (2mm)
, (
, ).
. 2.17: NaCl
(4mm), (.. Cl).
, ,
, , .
-
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74
,
, ,
32 ,
. , 32
, : 32
.
, , 32
, 14 Bravais.
(space
groups). , 2/m .
, P C, P2/m
C2/m . , P4 ,
4
. , F4/m3 2/m
,
.
Bravais
72 . , ,
(-
),
,
.
.
,
, ,
(. 2.18, 2.19,
2.20).
-
.
-
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75
. 2.18: ) , ) (
) , )
.
. 2.19: , t,
.
, (/)
(.. 31t/3 /3).
-
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76
. 2.20: O
() . PP
.
PP ty/2
, PP .
,
, . ,
().
.
.. ,
, .
,
. , , 1.18
1.20 , ,
,
. , ,
. ,
, ,
. ,
P4/m P42/m, 4/m.
-
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77
,
(. ). , ,
. , ..
(. ),
,
,
.
,
, (. . 2.19)
m.
:
1)
2)
3)
4)()
5)
6)
()
,
,
.
,
, .
,
,
. ,
,
,
. , -
,
. , 11
.
-
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78
1 14 Bravais
230 ,
. 230
2. :
( ), 230 230
, .
, 17 , 17
.
230 .
,
.
.
:
)
,
)
.
.
.
,
,
, .
,
. ,
Bravais
,
.
1 , 32 .2,
32 . , Bravais . .. (. . 4)
2
. , 3, 3132.
-
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79
-
. , , ,
-
,
. ,
.
xyz
,
: xa, yb, zc. ,
, a, b, c,
, (x, y, z),
(. 2.21 & 2.23).
1, ,
. >1 < 0,
, , . ,
.
, .. (2/3, 1/2, 7/8) (1/3, 1/2, 7/8).
,
(x, y, z). ,
1.
(n1a + n2b + n3c, ni),
(. 2.22).
1
: , , , . , , .
-
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80
. 2.21: .
,
, x
.
. 2.22: .
-
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81
. 2.23: (
z).
.
,
,
.
, , .
,
. ,
, .
,
().
, (x, y, z),
192 , (x,
y, z). , , , () , , 192 16.
-
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82
,
, , , .. [hkl],
(.
224). : ,
, ,
.
x, y, z, h, k, l,
x, y, z,
[(x/2, y/2, z/2), (2x, 2y, 2z), (3x, 3y, 3z) ...].
.
. 2.24: () ,
OA =1a+3b+2c [132].
() ab
.
() . 2.24.
:
- (1/2, 0, 1), (1, 0, 2),
[102]. ,
: 1a+0b+2c = a+2c
xz.
-
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83
- [3
2 3
1 1] [213]
- (-1, 2, 0) ]201[ .
. 2.24, [120] (OC) [240] (OD) . ,
(.. , , ). . 2.24, a-b,
[hk0]. ,
0,
(). ,
, 0, . ,
.. [010] b (. 2.25).
. 2.25: .
,
.
. , , ,
,
.
, ,
.
.
-
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84
:
. , .. . ,
, ,
. -
.
, ,
. ,
.. [100], [010], [001] (,
. . 2.25) 100 . ,
( ]111[ , ]111[ , ]111[ , ]111[ , .) 111 1.
,
,
, , .. (hkl),
Miller.
Miller, () , :
1)
,
2)
,
3) .
, .. (210) (. . 2.26) a, b, c,
, 1 (c). ,
2, 1 0, Miller.
, .. )( 221 . {}
. , .. (100), (010), (001), )001( ,
)010( )100( , ,
{100}.
1, . ,
.. ]111[ ]111[ , ]100[ ]001[ , . (. . 2.25).
-
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85
. 2.26:
() .
(hkl) ,
,
(hkil), Miller-Bravais. ,
, , i
h k: (h+k = i). , ,
.. (0110)
(01.0) (010). , , [hkl]
(hkl), ,
.
Miller
.
d, ,
, . Bragg. , ,
-
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86
(.. )
,
/ ,
d.
, ,
.
(, , , = =
= 90), d ,
Miller (hkl), :
2
2
2
2
2
2
2hk cb
k
a
h
d
1 l
l
++= ,
a, b, c (
).
a = b ( ) a = b = c
(). :
2
222
2hk a
kh
d
1 l
l
++=
a ,
(hkl) , d .
, :
3da111
=
, :
2
222
2
hka
)c
a()khkh(
3
4
d
1 l
l
+++
= .
, , ..:0.11d2a= 2.00d2c=
-
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87
d. , V , :
2)d
V( = h2b2 c2 2+ k2 a2 c22+ l
2a2b22+
+ 2hl ab2 c(- )
+ 2hkabc2 (- )
+ 2kl a2bc(- ).
:
( ). (.. )
. , ,
.
, ,
. ,
. ,
. , , Bravais,
- . 2.27. ,
.
. 2.27:
:
1 2 3 4 5.
, ,
. , ..
1/ 2 , 1 2/3, 2 , 1, 3/2 . ,
-
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88
().
,
, ,
. ,
.
, ,
, . ,
,
, /
. , ,
,
,
(. 2.28).
. 2.28:
{001} a - {111} o .
.
-
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89
, ,
.
. , ,
, (. 2.29).
. 2.29:
().
().
().
, , , .
, ,
:
:
8 (1/8) + 6 (1/2) = 4
:
8 (1/8) + 1 (1) = 2
:
8 (1/8) + 2 (1/2) = 2
,
. , ,
. ,
-
7/25/2019 Crystalography (1)
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90
-Fe Fe, .
Fe, . Fe.
, , .
, ( : CsCl)
, ,
[Cs+Cl-] (. 2.30). (:
NaCl)
, [Na+Cl-]. ,
(CaF2) [Ca2+
2F-].
, . .
.
. 2.30: (CsCl)
, .. ,
. .
,
, .. NaCl
4 [Na+Cl-], 4 Na+4 Cl-(1:1),
CaF24 Ca2+ 8 F-(1:2). ,
,
.
, ,
, , (
)
.
-
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91
:
A
mol1
D
=
=
=
Avogadro.
VZ, : V = (
), :
ANV
ZD
=
V:V = a
(b
c).
, :
) ()
, )
, )
(D)
( ) /
,
.
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