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Nankai University http://struchem.nankai.edu.cn Nankai University http://struchem.nankai.edu.cn
Nankai University http://struchem.nankai.edu.cn
7.1 7.1.1
1.
-Long-range-order
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Crystalline Solid Glass(Amorphous Solid)
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c b+c a+b+c 1:2:4
NaCl
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2.
F+V=E+2
(F: V: E: )
NaCl 6 128
3.
4. X
5.
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7.1.2 (lattice)
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A
B
C
X
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1. (one-dimensional lattice)
a
a m = 0, 1, 2, … amT m
a
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bnamT nm,
2.
ab
a
b
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( )( )
( )Nankai University http://struchem.nankai.edu.cn
ab
a
ba
b
a
b
ab
a=b a^b=90
a b a^b=90
a=b a^b=120
a b a^b 90
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ab
1.2.
ab
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4.
6 1414 (Bravias)
(m, n, p = 0, 1, 2, …)
cpbnamT pnm ,,
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a
b
c
a, b, ca,
b, c
= b ^ c = c ^ a =a ^ b
a, b, c,
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cubic a = b = c, = ß = = 90o
P- (Primitive) I- (Body centred) F- (All-face centred)
Phexagonal (P)
a = b c, = ß = 90o,
= 120o
a aa
a aa
cP cI cF
hP
Rhexagonal (R)
a = b c, = ß = 90o,
= 120ohR
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a
a
a
Rhombohedral (R)a = b = c, = ß = 90o
Rhexagonal (R)
a = b c, = ß = 90o,
= 120o
hR
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orthorhombic a b c, = ß = = 90o
C- (C-face centred)
tetragonal (P I) a = b c,= ß = = 90o
a
c
b
a a
c
tP tI
oP oI oC oF
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anorthic (P)(triclinic)
a b c, 90o
monoclinic (P C) a b c, = = 90o, 90o
a b
c
a b
c
mP mC
aPNankai University http://struchem.nankai.edu.cn
c
120
a ba a
a
a a
c
a
c
b a b
c
a b
c
a
a
a
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1.
()
——
7.1.3
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1 ——
a a = 2r
2.
a
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2
CH2n
?
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Nankai University http://struchem.nankai.edu.cn
6 2 C 4 H
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 34 4 4 4 4 4 4
5 5 5 5 5 5 56 6 6 6 6 6 6
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Nankai University http://struchem.nankai.edu.cn
C
a
2C 4H
a
—
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2 C
2
3
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4 NaCl
1Cl-, 1Na+
F cF
4Cl- 4Na+
:
:
:
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5 CsCl
1Cl-, 1Cs+
P cP
1Cl- 1Cs+
:
:
:
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1S=, 1Zn++
4S=, 4Zn++
6. ZnS
F cF
:
:
:
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7. ZnS
2S=, 2Zn++
2S=, 2Zn++
P hP
:
:
:
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8. CaF2
1Ca++, 2F-
4Ca++ 8F-
F cF
:
:
:
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9.
2C
8C
F cF
:
:
:
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3.
6 6 *14 14
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7.1.4
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a, b, c, , ,
( )
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OP= xa+yb+zc
•
••
O
P
ab
c
xa yb
zc
Y
Z
P XY Z zc
P YZ X xaP XZ Y yb
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b
Cl-: 0,0,0; 1/2,1/2,0; 0,1/2,1/2; 1/2,0,1/2
Na+: 1/2,0,0; 0,1/2,0; 0,0,1/2; 1/2,1/2,1/2
S= : 0,0,0; 2/3,1/3,1/2;
Zn++: 0,0,5/8; 2/3,1/3,1/8
Cl 4 Na+ 4 S 2 Zn: 2
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S2 :
Zn2+:
S 4 Zn: 4S: +Zn: 1/8
0,0,0; 1/2,1/2,0; 0,1/2,1/2; 1/2,0,1/2
1/4,1/4,1/4;3/4,3/4,1/4; 3/4,1/4,3/4;1/4,3/4,3/4;
ZnS
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F8
F : 1/4,1/4,1/4;3/4,3/4,1/4;3/4,1/4,3/4;1/4,3/4,3/4;3/4,3/4,3/4;3/4,1/4,1/4;1/4,3/4,1/4;1/4,1/4,3/4;
CaF2
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•
•
• 1/8 ( )
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F1/4,1/4,1/4 0,0,03/4,3/4,1/4 1/2,1/2,03/4,1/4,3/4 1/2,0,1/21/4,3/4,3/4 0,1/2,1/23/4,3/4,3/4 1/2,1/2,1/23/4,1/4,1/4 1/2,0,01/4,3/4,1/4 0,1/2,01/4,1/4,3/4 0,0,1/2
Ca0,0,0 3/4,3/4,3/4 (-1/4,-1/4,-1/4)1/2,1/2,0 1/4,1/4,3/41/2,0,1/2 1/4,3/4,1/4 0,1/2,1/2 3/4,1/4,1/4
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x, y, z 1/2
CaTiO3
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7.1.5
( )
=1/3:1/2:1 =2:3:6=h*:k*:l*
a b
c
a
b
c
O
A
B
C
( ) (h*k*l*)
(236)OA/a = 3 OB/b = 2 OC/c = 1
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(100) (110)
(111) (h*k*l*)( )
—
a b
ca b
c
a b
c
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ab
(110) (210)
(100)
(120)-
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2*2*2*)( ***
lkhad
lkh
2*2*2*)( )()()(
1***
clbkahd
lkh
2*22*2*)(
)()(
1***
clakhd
lkh
2
2*
2
2***2*)(
3)(4
1***
cl
akkhh
dlkh
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(hkl))( lkh
Nankai University http://struchem.nankai.edu.cn
7.2 327.2.1
1.
Cn ˆnC n n L( =2 /n)
ˆ m M i i I
Sn ˆnS n L( )Ii
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2.
( =2 /n)L( ) I or I L( )
n
4'
3'
2'
1'
L( /2) I
1"2"
3"
4"1
2
3
4
4- 4- 4-
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21 Si
633 Si
44 S
336 Sm
12 Sm
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3. nn=1, 2, 3, 4, 6
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4. , i , :
i —i —
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5. m
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7.2.2
c 3 a=b=c = = = 90
h
6 a=b c = = 90 = 120
3
a=b c = =90 = 120
a=b=c = = 90
t 4 a = b c = = = 90o m
2a b c = = = 90
m 2 m a b c = = 90a i a b c 90
6
3
4
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34
43O
T
OhO
i
ThTdT
i4-
CsCi
im
DnhDndDn
mhmdn2 n
n
4-S4
D2h( ) D3h(* ) D4h( ) D6h( )D2d(* ) D3d( )D2( ) D3( ) D4( ) D6( )
n
mhmv
C3+i
Cnh
Cnv
C3iCn
C2h( ) C3h(* ) C4h( ) C6h( )C2v( ) C3v( ) C4v( ) C6v( )C3i( )C1( ), C2( ), C3( ), C4( ), C6( )
C3h, D3h 6D2d 4
__
7.2.3 (32 )
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1. (Schöflies Symbol)2. (Hermann-Mauguin Symbol)
32
•( );
•
7.2.4 32
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c 4 3 // 4 a, b, c a, a + b + c, a + b
c // 6( 6) a, b // 2 m a, b c c, a, 2a + b
c // 3( 3) a, b // 2 m a, b c c, a h
a, b, c 3 3( 3) a + b + c, a - b
t c // 4( 4) a, b // 2 m a, b c c, a, a + b
o a, b, c // 2 m a, b, c m b // 2 ( m), a, c c b a a, b, c 3 a
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Nankai University http://struchem.nankai.edu.cn
1.2.3.
a 4
a+b+c 3a+b mTd m43
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m
D2h
b ca2
m2
m2
Oh
a a+b+c a+b
m4 3 m
2
4 23m m
ac
a
b
c
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baac ,,
C4 C4v C4h D4 D4h D2d
42m 4 4mm 4/m 422 4 2 2 _
_ _ _mmmc
a a+b
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32
Maximal subgroups and minimal supergroups of the three-dimensional crystallographic point groups. Solidlines indicate maximal normal subgroups; double or triple solid lines mean that there are two or three maximalnormal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. Thegroup orders are given on the left
1
2
3
4
6
8
12
16
24
484 2m m3
mmm4 2 2
432 43m 32 m
23
mmm6 2 2
622 62m 6mm m6 3 2
m
6 6 32 3m 3
3
4 222 mm2 m 2
2 m 1
1
422 42m 4mm 4m
2 2 2mmm
4
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N
S
AB
CD
A'B' C'D'
A'
B'
C'D'
7.2.5
S4
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7.2.6
Oh mm234
CaF2( )
3m m
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Td m34
ZnS( )
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O 432
NH4Cl -Mn(A13)
Th
FeS2
32m
T 23
FeSi
3m
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C6 6
KAlSiO4( )
C3h
AgH2PO4
C6h
Ca5(PO4)3OH
6
m6
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- (SiO2)D6 622
C6v 6mm ZnO ( )
D3h 6 2m BaTiSi3O9( )
D6h mmm226 Be3Al2(SiO3)6( )
MoS2( )
Zn (H2O)6/mmm
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mmm224
TiO2( ) D4h
CuFeS2( )
D2d
NiSO4·6H2O
D4 422
m24
4/mmm
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m4C4h
BaTiO3C4v 4mm
4S4
C4 4PbMoO4( )
BPO4
CaWO4( )
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D3d m23
C3 3
C3i 3
D3 32
C3v 3mKAl3(SO4)2(OH)6( )
CaCO3( )Al2O3( )
HgS( ) -SiO2( )
CaMg(CO3)2( )FeTiO3( )
Pb9As4S15( )
3m
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D2hmmm222
(S)
D2 222
C2v 2mm(mm2)
MgSO4·7H2O( )
Na2Al2Si3O10·2H2O( )
(mmm)
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m2 Na2B4O7·10H2O( )
CaSO4·2H2O( )
C2 2
Cs m
C2h
Ca2Zn2Si2O7(OH)2·H2O( )
C4H6O8( )
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1
C1 1
CiNaAlSi3O8( )CuSO4·5H2O( )
Al2Si2O5(OH)4( )CaS2SO3·6H2O
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1. ()
n m i n
2.
7. 3 230
7.3.1
Nankai University http://struchem.nankai.edu.cn
t Tnm
)()2( anmT
nL
NaCl
42
1
23
4
5
1'
a
a/4
41 42 43 4
Nankai University http://struchem.nankai.edu.cn
(T) (TM) ()
o
o
o
o
aa/2
o
o
o
o
o
o
o
o
M T
A
A A
a. a(b c) a(b c) a/2(b/2 c/2)
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42 21
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NaCl 42 NaCl c
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• no
o o
o o
o
oo o
o o
ooa
b
222,
22,
22,
22cbacbcaba
• d( )
o
o o
o o
o
oo o
o
oa
b44
,44
,44
cacbba
Nankai University http://struchem.nankai.edu.cn
41
Nankai University http://struchem.nankai.edu.cn
Nankai University http://struchem.nankai.edu.cn
1
2 3 4 6
3
4
6
21
31 32
41 42 43
61 62 63 64 65
+
2/m 21/m
4/m 42/m
6/m 63/m
i ( 1 )
m( 2 )
a b c ------ .........
n - - - - -
d · · · · · ·
Nankai University http://struchem.nankai.edu.cn
7.3.2 230
cCC
cPC
cPC
mCC
mPC
mPC
mC
h
h
h
h
h
h
h
2
2
2
2
2
2
2
62
152
42
32
122
12
2
C2h 6
P14
21 b, b21 , b
c
Nankai University http://struchem.nankai.edu.cn
7.3.3
• ( )
( )
5 12
2hC P
c
y=1/4 c
21
41
a
c
0 b 0
Nankai University http://struchem.nankai.edu.cn
zyx ,,1 1, ,2 2
x y z
, ,x y z 1 1, ,2 2
x y z
+
_
+
__12
_12
+
_ _
+
+
_
+_12
__12
41
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