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 five
§ 2.5 Indices of crystal planes and
directions
?Ⅰ.What ’s crystal planes and
directions?
The atomic planes and
directions passing
through the crystal are
called (crystal) planes
and directions
respectively.
1,Steps to determinate the plane indices:
? Establish a set of coordinate axes
? Find the intercepts of the planes to be
indexed on a,b and c axes (x,y,z).
a
c
bx y
z
?Ⅱ,Plane indices
? Take the reciprocals of the intercepts 1/x,1/y,
1/z.
? Clear fractions but do not reduce to lowest
integers.
? Enclose them in parentheses,(h k l)
Example,1/2,1,2/3 2,1,3/2 (423)
Plane indices referred to three axes a,b and
c are also called Miller Indices.
Several important aspects of the Miller indices for
planes should be noted:
? Planes and their negatives are identical,
Therefore,
? Planes and their multiples are not identical.
? In cubic systems,a direction that has the
same indices as a plane is perpendicular to
that plane.
)020()020( ?
2,The important planes in cubic crystals
(110)
(112)
(111)
(001)
3,A family of planes consists of equivalent
planes so far as the atom arrangement is
concerned.
)110()0 1 1()110(
)1 0 1()101()1 1 0(}1 1 0{
??
????
Total,6
)111(
)111()111()1 1 1(}1 1 1{ ????
Total,4
)121()112()112()2 11(
)112()121()211()1 21(
)211()211()121()1 12(}1 12{
???
????
?????
)132()123()213()321(
)231()213()123()312()321(
)312()132()213()123()132(
)312()231()213()231()321(
)132()312()321()231()123(}123{
???
?????
?????
?????
??????
Total,12
Total,4× 3！ =24
? Ⅲ,Direction Indices
1,Derivation for the crystallographic direction
① As the first above,set the origin on the direction
to be indexed.
② Find the coordinates of another point on the
direction in questions.
③ Reduce to three smallest integers,u,v,w.
④ Enclose in square brackets [u v w].
2,The important direction in cubic crystals:
<100>, crystal axes
<110>, face diagonal
<111>, body diagonal
<112>, apices to opposite face-centers
3,Family of directions consists of
crystallographically equivalent directions,
denoted <u v w>
e.g.
]100[]010[]001[
]0 0 1[]0 1 0[]1 0 0[1 0 0
??
??????
§ 2.6 Hexagonal axes for hexagonal
crystals
? Ⅰ,Why choose four -axis system?
Four indices has been devised for hexagonal
unit cells because of the unique symmetry of the
system.
a
c
)100(
]110[]100[
)011(
b
?Ⅱ,Plane indices ( hkil)
It can be proved,i ≡－ (h＋ k)
)0110()1 0 0( ?
)0011()011( ?
Important planes,
a1
a2
a3
c
)0110(
)0211(
)2110(
)0001(
)1110(
? Ⅲ, Direction indices [ u v t w ]
To make the indices unique,an additional
condition is imposed,---- Let t＝－ (u＋ v)
Important directions
]0001[
]0101[
]0111[
]0112[
Transformation of indices
Transformation of 3 to 4 indices,or vice versa,Suppose we
have a vector,whose 3 indices [u v w],and 4 indices [u v t w].
We have cwatavauL ????? ????
321
cWaVaU ??? ??? 21
Since )()(
213 vutaaa ?????? ???
or:
cWaVaU
cwaavuavau
???
?????
???
??????
21
2121 ))((
cWaVaU
cwavuavu
???
???
???
????
21
21 )2()2(
wWuvVvuU ?????? 22
)2(31 VUu ??
)2(31 UVv ??
Ww ?
)( vut ???
For example:
]0112[?:]100[
0?w
3
1??t
3
1??v
3
2?u
1,Quick way for indexing the directions in cubic
crystals:
The value of a direction depends on its
feature while the sign on direction.
Examples and Discussions
2,The coordinate origin can be set arbitrarily
(for example on apices,body-center,face-
centers etc.),but never on plane in questions,
otherwise the intercepts would be 0,0,0,
3,The coordinate system can be transferred
arbitrarily,but rotation is forbidden.
c′
)111(a
c
b
a′ b′
)111(
4,The atomic arrangement and planar density of the
important direction in cubic crystal.
plane
indices
BCC FCC
atomic
arrangement planar density
atomic
arrangement planar density
{100}
{110}
{111}
22
14
14
aa ?
?
22
4.1
2
1414
aa ?
??
2
2
58.0
2
3
6
13
aa
?
?
22
214
14
aa ?
??
22
4.1
2
2
12
4
14
aa ?
???
2
2
3.2
2
3
2
13
6
13
aa
?
???
a
a
a
a2
a2a2
a2
a
a
a
a2
a2 a2
a2
5,The atomic arrangement and linear density of the
important direction in cubic crystal.
linear
indices
BCC FCC
atomic
arrangement linear density
atomic
arrangement linear density
<100>
<110>
<111>
aa
12
12
?
?
aa
7.0
2
2
12
?
?
aa
16.1
3
1212
?
??
aa
12
12
?
?
aa
4.1
2
1212
?
??
aa
58.0
3
2
12
?
?
a a
a2 a2
a3 a3
Exercise
1,Calculate the planar density and planar packing
fraction for the (010) and (020) planes cubic polonium,
which has a lattice parameter of 0.334nm.
214
2
a t om s / c m1096.8
334.0
1
f a c eofa r e a
f a c epe ra t om
)010(de n s i t ypl a n a r
??
??
Solution
79.0
)2(
)(1
f a c eofa r e a
f a c epe ra t om sofa r e a
)010(f r a c t i onpa c ki n g
2
2
?
?
?
?
r
r?
However,no atoms are centered on the (020) planes,
There fore,the planar density and the planar packing
fraction are both zero.
Thank you !
5
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 five
§ 2.5 Indices of crystal planes and
directions
?Ⅰ.What ’s crystal planes and
directions?
The atomic planes and
directions passing
through the crystal are
called (crystal) planes
and directions
respectively.
1,Steps to determinate the plane indices:
? Establish a set of coordinate axes
? Find the intercepts of the planes to be
indexed on a,b and c axes (x,y,z).
a
c
bx y
z
?Ⅱ,Plane indices
? Take the reciprocals of the intercepts 1/x,1/y,
1/z.
? Clear fractions but do not reduce to lowest
integers.
? Enclose them in parentheses,(h k l)
Example,1/2,1,2/3 2,1,3/2 (423)
Plane indices referred to three axes a,b and
c are also called Miller Indices.
Several important aspects of the Miller indices for
planes should be noted:
? Planes and their negatives are identical,
Therefore,
? Planes and their multiples are not identical.
? In cubic systems,a direction that has the
same indices as a plane is perpendicular to
that plane.
)020()020( ?
2,The important planes in cubic crystals
(110)
(112)
(111)
(001)
3,A family of planes consists of equivalent
planes so far as the atom arrangement is
concerned.
)110()0 1 1()110(
)1 0 1()101()1 1 0(}1 1 0{
??
????
Total,6
)111(
)111()111()1 1 1(}1 1 1{ ????
Total,4
)121()112()112()2 11(
)112()121()211()1 21(
)211()211()121()1 12(}1 12{
???
????
?????
)132()123()213()321(
)231()213()123()312()321(
)312()132()213()123()132(
)312()231()213()231()321(
)132()312()321()231()123(}123{
???
?????
?????
?????
??????
Total,12
Total,4× 3！ =24
? Ⅲ,Direction Indices
1,Derivation for the crystallographic direction
① As the first above,set the origin on the direction
to be indexed.
② Find the coordinates of another point on the
direction in questions.
③ Reduce to three smallest integers,u,v,w.
④ Enclose in square brackets [u v w].
2,The important direction in cubic crystals:
<100>, crystal axes
<110>, face diagonal
<111>, body diagonal
<112>, apices to opposite face-centers
3,Family of directions consists of
crystallographically equivalent directions,
denoted <u v w>
e.g.
]100[]010[]001[
]0 0 1[]0 1 0[]1 0 0[1 0 0
??
??????
§ 2.6 Hexagonal axes for hexagonal
crystals
? Ⅰ,Why choose four -axis system?
Four indices has been devised for hexagonal
unit cells because of the unique symmetry of the
system.
a
c
)100(
]110[]100[
)011(
b
?Ⅱ,Plane indices ( hkil)
It can be proved,i ≡－ (h＋ k)
)0110()1 0 0( ?
)0011()011( ?
Important planes,
a1
a2
a3
c
)0110(
)0211(
)2110(
)0001(
)1110(
? Ⅲ, Direction indices [ u v t w ]
To make the indices unique,an additional
condition is imposed,---- Let t＝－ (u＋ v)
Important directions
]0001[
]0101[
]0111[
]0112[
Transformation of indices
Transformation of 3 to 4 indices,or vice versa,Suppose we
have a vector,whose 3 indices [u v w],and 4 indices [u v t w].
We have cwatavauL ????? ????
321
cWaVaU ??? ??? 21
Since )()(
213 vutaaa ?????? ???
or:
cWaVaU
cwaavuavau
???
?????
???
??????
21
2121 ))((
cWaVaU
cwavuavu
???
???
???
????
21
21 )2()2(
wWuvVvuU ?????? 22
)2(31 VUu ??
)2(31 UVv ??
Ww ?
)( vut ???
For example:
]0112[?:]100[
0?w
3
1??t
3
1??v
3
2?u
1,Quick way for indexing the directions in cubic
crystals:
The value of a direction depends on its
feature while the sign on direction.
Examples and Discussions
2,The coordinate origin can be set arbitrarily
(for example on apices,body-center,face-
centers etc.),but never on plane in questions,
otherwise the intercepts would be 0,0,0,
3,The coordinate system can be transferred
arbitrarily,but rotation is forbidden.
c′
)111(a
c
b
a′ b′
)111(
4,The atomic arrangement and planar density of the
important direction in cubic crystal.
plane
indices
BCC FCC
atomic
arrangement planar density
atomic
arrangement planar density
{100}
{110}
{111}
22
14
14
aa ?
?
22
4.1
2
1414
aa ?
??
2
2
58.0
2
3
6
13
aa
?
?
22
214
14
aa ?
??
22
4.1
2
2
12
4
14
aa ?
???
2
2
3.2
2
3
2
13
6
13
aa
?
???
a
a
a
a2
a2a2
a2
a
a
a
a2
a2 a2
a2
5,The atomic arrangement and linear density of the
important direction in cubic crystal.
linear
indices
BCC FCC
atomic
arrangement linear density
atomic
arrangement linear density
<100>
<110>
<111>
aa
12
12
?
?
aa
7.0
2
2
12
?
?
aa
16.1
3
1212
?
??
aa
12
12
?
?
aa
4.1
2
1212
?
??
aa
58.0
3
2
12
?
?
a a
a2 a2
a3 a3
Exercise
1,Calculate the planar density and planar packing
fraction for the (010) and (020) planes cubic polonium,
which has a lattice parameter of 0.334nm.
214
2
a t om s / c m1096.8
334.0
1
f a c eofa r e a
f a c epe ra t om
)010(de n s i t ypl a n a r
??
??
Solution
79.0
)2(
)(1
f a c eofa r e a
f a c epe ra t om sofa r e a
)010(f r a c t i onpa c ki n g
2
2
?
?
?
?
r
r?
However,no atoms are centered on the (020) planes,
There fore,the planar density and the planar packing
fraction are both zero.
Thank you !
5
