Solved Examples
Example 1 Show how the atomic positions are located in three cubic unit cells?
Indicate the position coordinates for the atoms in bcc and fcc unit cells.
Solution: The corner atomic positions in the cubic unit cells are located at unit
distances from the origin along the three crystallographic axes as indicated above in
Table 4.3 Unit translation for low index directions of an orthorhombic system
Family of
direction
Unit translation
P
C
I
F
<100>
a
a
a
a
<010>
b
b
b
b
<001>
c
c
c
c
<110>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða 2 þ b
2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða 2 þ b
2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða 2 þ b
2 Þ
q
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða 2 þ b
2 Þ
q
<101>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
<011>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðb
2 þ c 2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðb
2 þ c 2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðb
2 þ c 2 Þ
q
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðb
2 þ c 2 Þ
q
<111>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ b
2 þ c 2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ b
2 þ c 2 Þ
q
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ b
2 þ c 2 Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ b
2 þ c 2 Þ
q
Fig. 4.2 Indices of six faces of a cube
Fig. 4.3 Indices of principal directions in a cubic crystal
4.1 Miller Indices of Atomic Sites, Planes and Directions
139