Weiss Indices
Weiss Indices
According to the law of rational indices, we know that the intercepts of any plane of a crystal along crystallographic axes are simple multiple of unit intercepts. The unit intercepts (or length) are shown as (a, b, c). The OA = a, OB = b and OC = c are the unit intercepts. A crystal plane LMN has the intercepts OL, OM and ON along x, y and z axis where OL = 2a, OB = 4b and OC = 3c.The coefficients of unit intercepts are called Weiss indices of a plane. Weiss indices are not always integers. They may have fractional as well as infinity values. Therefore, Weiss indices are not easy to be used. Hence, these are being replaced by Miller indices.
The Miller indices of a plane are obtained by taking the reciprocals of Weiss indices and multiplying throughout by the lowest possible number in order to make all reciprocals as integers. For KLM crystal plane, we have-
Intercepts: 2a 4b 3c
Weiss Indices: 2 4 3
Reciprocal of Weiss Indices: 1/2, 1/4, 1/3
Multiplying by 12: 6, 3, 4 i.e. Miller Indices.
A plane intercepts the crystal axes at ∞a, 2b and c, then-
Intercepts: ∞a 2b c
Weiss Indices: ∞ 2 1
Reciprocal of Weiss Indices: 1/∞, 1/2, 1 or, 0, 1/2, 1
Multiplying by 2: 0, 1, 2 i.e. Miller Indices.
Hence, the mentioned crystal plane is designed as (012) plane. Such crystal planes are called (h,k,l) planes. In (012) plane, h = 0, k = 1 and l = 2.
If a plane is parralel to x and y axes and cuts z axes at unit intecepts, then (hkl) of the plane is 001. If a plane cuts x, y and z axes at unit length, the Miller indices are 111 and hence, the plane is designed as 111. The distance between the parallel planes in a crystal are shown as dhkl.
For different cubic lattices-
where, a = length of the cubic side.
Miller Indices
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