材料科学基础
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Lesson six
§ 2.7 Some Important
Crystallographic Formulas
1,Zone and zone equation
All Planes parellel to common axis [u v w] constitute a
(crystal) zone,the [u v w] zone.
If (h k l) belongs to zone [u v w],then hu＋ kv＋ lw＝ 0
2,If hu＋ kv＋ lw＝ 0,then [u v w] lies in (h k l)
3,For cubic crystals,[h k l] ⊥ (h k l)
4,The zone [u v w] contains two planes (h1 k1 l1)
and (h2 k2 l2),then
)(:)(:)( 122112211221 khkhhlhllklk ???
5,The plane (h k l) belongs to two zones [u1 v1 w1]
and [u2 v2 w2] if
)(:)(:)( 122112211221 vuvuuwuwwvwv ???
?lkh,:
?wvu,:
6,The distance between adjacent plane
( interplanar distance )
d(hkl)＝ f(a,b,c α,β,γ h,k,l)
For orthorhombic:
2
2
2
2
2
2
2
1
c
l
b
k
a
h
d
???
For cubic:
222 lkh
ad
??
?
For hexagonal crystals:
2
2
2
22
2 3
41
c
l
a
khkh
d
????
7,The length of [u v w]
?c o s2)()()( 222][ v w b cwvuL cbau v w ????
?? c o s2c o s2 u v a bu w a c ??
For cubic,222
][ wvuaL u v w ????
8,The angle Ф between (h1 k1 l1) and (h2 k2 l2)
For orthorhombic:
])()()[(])()()[(
c os
222222212121
2
21
2
21
2
21
c
l
b
k
a
h
c
l
b
k
a
h
c
ll
b
kk
a
hh
?????
??
??
For cubic:
))((
c o s
2
2
2
2
2
2
2
1
2
1
2
1
212121
lkhlkh
llkkhh
????
??
??
For hexagonal,??co s
?
?
?
?
?
?
???
?
?
?
?
?
?
???
????
22
2
2
22
2
2
211
2
1
22
1
2
1
122121
2
22121
)(
4
3
)(
4
3
)(
2
1
)(
4
3
khl
c
a
khkhl
c
a
kh
khkhll
c
a
kkhh
9,The angle between [u1 v1 w1] and
[u2 v2 w2]
?
For orthorhombic:
??cos
2
2
2
2
2
2
2
1
2
1
2
1
2
21
2
21
2
21
)()()()()()( cwbvaucwbvau
cwwbvvauu
????
??
?
For cubic:
??cos
2
2
2
2
2
2
2
1
2
1
2
1
212121
wvuwvu
wwvvuu
????
??
?
For hexagonal:
22
22
2
2
2
2
211
22
1
2
1
2
1
1221
2
212121
)()(
)(
2
1
)(
vu
a
c
wvuvu
a
c
wvu
vuvu
a
c
wwvvuu
??????
????
?
??co s
10,The volume of unit cells V
?V ?????? c o sc o sc o s2c o sc o sc o s1 222 ????abc
§ 2.8 Stacking Mode of Crystals
? Ⅰ.A crystal can be considered as
the result of stacking the atomic
layers,say (h k l ),one over
another in a specific sequence.
For simple cubic
(001) aaaa……
(110) abab……
]101[a l o n g,]101[21s h if t ?
This sequence is called the stacking order
? Ⅱ.Comparison of stacking mode of
HCP and FCC
HCP stacking order:
ABABABAB……
1,HCP
2,FCC
Stacking order
of (111)：
ABCABCABC……
A
AA
B
BB
C C
C
? Ⅲ.Stacking fault
? For HCP:
normal order,ABABAB……
fault order,ABCABAB……
? For FCC：
normal order,ABCABCABC……
fault order,ABCACABCA……
ABCACBCABC ……
ABCABABAB ……
ABCACBA ……
? Ⅳ.Transformation of hexagonal to
rhombohedral indices and vice versa
cH
aH
bH
cR
aR bR
HHHR cbaa
????
3
1
3
1
3
1 ???
HHHR cbab
????
3
1
3
2
3
1 ???
HHHR cbac
????
3
1
3
1
3
2 ????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
H
H
H
R
R
R
c
b
a
c
b
a
?
?
?
?
?
?
112
121
111
3
1
or
Examples and Discussions
Exercise
Thank you !
6