Chapter 4
Unit Cell Representations of Miller
Indices
4.1 Miller Indices of Atomic Sites, Planes and Directions
(a) Miller Indices of Atomic Sites
In order to locate the atomic sites (positions) in cubic or orthogonal unit cells,
rectangular axes are used. The positive x-axis is taken as the direction coming out
of the paper, positive y-axis along the right side of the paper and positive z-axis
towards the top as shown in Fig. 4.1a. Negative directions are just opposite to their
counterparts. The atomic positions in the unit cells are located at unit distances
along the three axes as indicated in Figs. 4.1b and c. For example, the position
coordinates of the eight corners are:
[[000]], [[100]], [[010]], [[001]], [[111]], [[110]], [[101]], [[011]].
(b) Miller Indices of Planes
Miller indices of planes are obtained by adopting the following procedure:
(i) Determine the intercepts of the plane along x, y and z-axes in terms of lattice
parameters.
(ii) Divide the intercepts by appropriate unit translations (see Tables 4.1, 4.2 and
4.3).
(iii) Note their reciprocals.
(iv) If fractions result, multiply each of them by the smallest common divisor.
(v) Put the resulting integers in parenthesis, that is, (hkl) to get the required
indices of that and all other parallel planes.
Miller indices of six faces of a cube are shown in Fig. 4.2.
(c) Miller Indices of Directions
Miller indices of directions are obtained by adopting the following procedure:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_4
137
Unit Cell Representations of Miller
Indices
4.1 Miller Indices of Atomic Sites, Planes and Directions
(a) Miller Indices of Atomic Sites
In order to locate the atomic sites (positions) in cubic or orthogonal unit cells,
rectangular axes are used. The positive x-axis is taken as the direction coming out
of the paper, positive y-axis along the right side of the paper and positive z-axis
towards the top as shown in Fig. 4.1a. Negative directions are just opposite to their
counterparts. The atomic positions in the unit cells are located at unit distances
along the three axes as indicated in Figs. 4.1b and c. For example, the position
coordinates of the eight corners are:
[[000]], [[100]], [[010]], [[001]], [[111]], [[110]], [[101]], [[011]].
(b) Miller Indices of Planes
Miller indices of planes are obtained by adopting the following procedure:
(i) Determine the intercepts of the plane along x, y and z-axes in terms of lattice
parameters.
(ii) Divide the intercepts by appropriate unit translations (see Tables 4.1, 4.2 and
4.3).
(iii) Note their reciprocals.
(iv) If fractions result, multiply each of them by the smallest common divisor.
(v) Put the resulting integers in parenthesis, that is, (hkl) to get the required
indices of that and all other parallel planes.
Miller indices of six faces of a cube are shown in Fig. 4.2.
(c) Miller Indices of Directions
Miller indices of directions are obtained by adopting the following procedure:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_4
137