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材料学基础5


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.

?

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
z
y

x

b

a

? 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 (020) ? (020) . ? 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.

2. The important planes in cubic crystals

(001) (110)

(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)

Total: 6

{111} ? (111) ? ( 1 11) ? (1 1 1) ? (11 1 )

Total: 4

{112} ? (112) ? ( 1 12) ? (1 1 2) ? (112) ? (121) ? ( 1 21) ? (121) ? (12 1 ) ? (211) ? ( 2 11) ? (2 1 1) ? (21 1 )

Total: 12

{123} ? (123) ? ( 1 23) ? (1 2 3) ? (12 3 ) ? (132) ? ( 1 32) ? (1 3 2) ? (13 2 ) ? (231) ? ( 231) ? (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


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 ?? [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.

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
(10 1 1)

(1120)

(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]

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].

We have

Since

? ? ? ? ? L ? ua1 ? va2 ? ta3 ? wc ? ? ? ? Ua1 ? Va2 ? Wc ? ? ? a3 ? ?(a1 ? a2 ) t ? ?(u ? v)

?

? ? ? ? ? ua1 ? va 2 ?(u ? v)( a1 ? a2 ) ? wc ? ? ? ? Ua1 ? Va2 ? Wc
? ? ? (2u ? v)a1 ? (u ? 2v)a 2 ? wc ? ? ? ? Ua1 ? Va2 ? Wc

?

U ? 2u ? v

V ? 2v ? u

W ?w

u ? 1 3 (2U ? V )

or:

v ? 1 3 (2V ? U )

w ?W t ? ?(u ? v)

For example:
u? 2 3

v??

1 3 1

[100] :
t??

?

[2 1 1 0]

3

w?0

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.

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.
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? 1 ?1 ?
1 2 ? 1.4 2 a
1 2 ? 2.3 2 a

atomic atomic planar density arrangement arrangement
a a

{100}

4? a
4?

1 1 a
2

4 ? 2
?1

a a

4 2 a

2 a
2

1

{110}
2a

a

4 2 2a
1

?

1.4 a
2

a
2a 2a 2a

4?

1 4
1 6

? 2? 2a
2

3?

{111}

2a

2a

2a

6 ? 0.58 2 a 3 2 a 2

3?

? 3? 3 a
2

2a

2

5. The atomic arrangement and linear density of the important direction in cubic crystal.
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 ? a 2a ?1 ?

2 2a 1

1.4 a

<111>

2?
3a

1

2 3a

1.16 a

3a

2?

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.

Solution
planar density (010) ? atom per face area of face
14

?

1 0.334
2 2

? 8.96 ? 10
packing fraction (010) ?

atoms/cm

area of atoms per face area of face 1? (?r )
2

?

( 2r )

2

? 0.79

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


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