(i) Note down the coordinates of the lattice site nearest to the origin corresponding to a given direction.
(ii) Divide the coordinates by appropriate unit translations.
(iii) If fractions result, multiply each of them by the smallest common divisor.
(iv) Put the resulting integers in a set of square brackets, that is, [hkl] to get the
required indices of that and all other parallel directions.
Miller indices of some principal directions are shown in Fig. 4.3.
Fig. 4.1 a Orthogonal axes, b site indices and principal directions, c site indices in a cube
Table 4.1 Unit translation
for low index directions of
cubic system
Family of directions
Unit translation
p
I
F
<100>
a
a
a
<110>
ffiffi ffi
2
p a
ffiffi ffi
2
p a
a ffiffi
2
p
<111>
ffiffi ffi
3
p a
ffiffi
3
p a
2
ffiffi ffi
3
p a
Table 4.2 Unit translation
for low index directions of a
tetragonal system
Family of directions
Unit translation
P
I
<100>
a
a
<001>
c
c
<110>
ffiffi ffi
2
p a
ffiffi ffi
2
p a
<101>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ða 2 þ c 2 Þ
p
<111>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2a 2 þ c 2 Þ
p
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2a 2 þ c 2 Þ
p
138
4 Unit Cell Representations of Miller Indices