Q) Determine the Miller indices of a plane that makes ...
SOLVED EXAMPLES: MILLER INDICES
Q) Determine the Miller indices of a plane that makes intercepts of 4Å, 3Å and 8Å on the coordinate axes of an orthorhombic lattice with the ratio of the axial lengths as:
a : b : c = 2 : 3 : 1
A) This problem is about the very basic definition of Miller indices and how to derive them.
| |X |Y |Z |
|Intercepts |4Ǻ |3Ǻ |8Ǻ |
|Lattice parameters |2x |3x |1x |
|Ratio |[pic] |[pic] |[pic] |
|Reciprocal |[pic] |x |[pic] |
|Factoring common factors |[pic] |1 |[pic] |
|Integer form |( 4 |8 |1) |
Q) What are the miller indices of the line of intersection of a [pic] and [pic] plane in a cubic crystal?
A) We shall arrive at the solution by a geometrical method and by an analytical method.
Geometrical Method
Note: [pic], [pic]
[pic]
Analytical Method-1
Weiss zone law states that if a direction [u v w] is contained in plane (h k l) then:
[pic]
Let [u v w] be the intersection of [pic] and [pic]
[pic] ( the indices of the direction are of the form [pic].
As miller indices are written after factoring out common factors the miller indices are: [pic] or the opposite direction [pic].
Analytical Method-2
Using equation of a line in intercept form (a, b, c are intercepts along x, y, z axis respectively):
[pic]. Substituting the intercepts for the two planes and solving simultaneously we get the equation for the line of intersection:
[pic]
[pic] ( [pic] (slope = (1 and intercept = (1)
[pic]
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