材料科学基础-课件 清华skja_05

Total: 12
{123} (123) ( 1 23) (1 2 3) (12 3 ) (132) ( 1 32) (1 3 2) (13 2 ) (231) ( 2 31) (2 3 1) (23 1 ) (213) ( 2 13) ( 2 1 3) (21 3 ) (312) ( 3 12) (3 1 2) (31 2 ) (321) ( 3 21) (3 2 1) (32 1 )
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].
a a
atomic atomic planar density arrangement arrangement
4
a a
1 1 a
2
1
{100}
1
a
4
4 2
4 2 a
4 1 4 2a
2
2 a
2
1 4
{110}
2a
a
1
2

1.4 a
2
a
2a 2a 2a
2
1 2 1.4 2 a
In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane.
2. The important planes in cubic crystals
(001) (110)

Ⅱ. Plane indices
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). c
100 [100] [010] [001] [ 1 00] [0 1 0] [00 1 ]
§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.
(111) (112)
3. A family of planes consists of equivalent planes so far as the atom arrangement is concerned.
{110} (110) ( 1 10) (101) (10 1 ) (011) (0 1 1)
c c´
(11 1 )
b a
( 1 1 1)
a´
b´
4. The atomic arrangement and planar density of the important direction in cubic crystal.
plane indices
BCC
FCC
planar density
4
材料科学基础
Fundamental of Materials
Prof: Tian Min Bo
Tel: 62795426 ，62772851 E-mail: tmb@ Department of Material Science and Engineering Tsinghua University. Beijing 100084
Important directions
[01 1 0]
[2 1 1 0]
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].
1.4 a
2a 2 1
<111>
3a
1.16 a
3a
2 0.58 a 3a
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.
2 3
u
v
1 3
[100] :
t
1 3

[2 1 1 0]
w0
Examples and Discussions
1. Quick way for indexing the directions in cubic crystals:
The value of a direction depends on its feature while the sign on direction.
1 2 2.3 2 a
2a
3
1
{111}
2a
2a
2a
6 0.58 2 a 3 2 a 2
3
1 6
3 3 a
2
2a
2
5. The atomic arrangement and linear density of the important direction in cubic crystal.
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 (020) (020) . Planes and their multiples are not identical.
(10 1 1)
(11 2 0)
(10 1 0)

Ⅲ. Direction indices [ u v t w ]
To make the indices unique, an additional condition is imposed. ---- Let t＝－(u＋v)
[0001]
[ 1 011]
c
(1 1 0)
(100)
b a
[100]
[110]

Ⅱ. Plane indices (hkil)
It can be proved: i ≡－(h＋k)
(100) (10 1 0)
(1 1 0) (1 1 00)
Important planes :
(10 1 2)
c
(0001)
a3 a2 a1
linear indices
BCC
atomic arrangement
a
FCC
atomic arrangement
a
linear density
2 1
linear density
2 1
<100>
2 1 a a
1 0.7
2 1 a a
1
<110>
2
2a
2
2a
1 2
2 a 2a 2 1 2 3a 1
L ua1 va2 ta3 wc Ua1 Va2 Wc a3 (a1 a2 ) t (u v)
We have
Since

ua1 va 2 (u v)( a1 a2 ) wc Ua1 Va2 Wc
2. The coordinate origin can be set arbitrarily (for example on apices, body-center, facecenters 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.
Thank you !
5
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.