Callister - An introduction - 8th edition

68 • Chapter 3 / The Structure of Crystalline Solids
EXAMPLE PROBLEM 3.12
Determination of Miller–Bravais Indices for a Plane
within a Hexagonal Unit Cell
Determine the Miller–Bravais indices for the plane shown in the hexagonal
unit cell.
z
H
G
E
F
a 2
c
D
B
a 3
a
C
A
a
a 1
Solution
To determine these Miller–Bravais indices, consider the plane in the figure
referenced to the parallelepiped labeled with the letters A through H at its
corners. This plane intersects the a 1 axis at a distance a from the origin of the
a 1 -a 2 -a 3 -z coordinate axis system (point C). Furthermore, its intersections with
the a 2 and z axes are a and c, respectively. Therefore, in terms of the lattice
parameters, these intersections are 1, 1, and 1. Furthermore, the reciprocals
of these numbers are also 1, 1, and 1. Hence
and, from Equation 3.7,
h 1
k 1
l 1
i 1h k2
11 12 0
Therefore the (hkil) indices are (1101).
Notice that the third index is zero (i.e., its reciprocal q), which means
that this plane parallels the a 3 axis. Inspection of the preceding figure shows
that this is indeed the case.
3.11 LINEAR AND PLANAR DENSITIES
The two previous sections discussed the equivalency of nonparallel crystallographic
directions and planes. Directional equivalency is related to linear density in the sense
that, for a particular material, equivalent directions have identical linear densities.
The corresponding parameter for crystallographic planes is planar density, and
planes having the same planar density values are also equivalent.