Miller Indices
Miller Indices:
Consider a square lattice as shown in Fig 1(a).
Figure 1 (a) A few planes in a lattice. 1 (b) The coordinate system.
In this case, an atom occupies each lattice point. The coordinate system used here is also shown in Fig 1 (b). Several lines can be drawn in this lattice such that at least one (or more) atoms lie on the lines. A few examples are shown. The unit cell for the Lattice in Fig 18.3 (a) is a square. Lines 1 and 2 are similar in that atoms lie on these at the same separations. Line 3 is less "densely packed", i.e., the distance between adjacent atoms is larger than that in lines 1 and 2. We need a system to label or characterize each of these lines. In three dimensions our interest would be the different planes of atoms.
fig..(2)
In Fig 2the plane containing the atoms intersects the x axis at a. This plane does not intersect with the y and z axes at all. We may rephrase this by saying that the intersections with y and z axis at y = 
and z = 
. The x, y and z intersections are at ( a, 
, 
). Taking the reciprocals of these, we get ( 1/a, 0, 0 ) and since a, b and c are characteristics of the crystal, we can simply refer to it as (100)
These are the Miller indices of this plane. Using the same procedure verify that the Miller indices of the other planes in the figure (2) are (110), (111). In a simple cubic lattice, the (100) plane is identical to the (1/2, 0 0 )plane, because it has exactly the same density and the relative positions of atoms / ions. Therefore all planes parallel to the (100) planes may be referred to as the (100) planes. In a "body centered" cubic lattice such as the CsCl lattice, the (100) planes containing the Cl- ions would be different from the (200) planes containing the Cs ions. Now that we can label all these planes accurately, let us see how the distance between the planes can be determined using Bragg's law.