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
4
2x
3
3x
8
1x
Reciprocal
x
2
x
x
8
Factoring common factors
1
2
1
1
8
Integer form
(4
8
1)
Q) What are the miller indices of the line of intersection of a (1 11) and (1 1 1) plane in a
cubic crystal?
A) We shall arrive at the solution by a geometrical method and by an analytical method.
Geometrical Method
Note: (1 1 1)  (111) , (1 11)  (11 1)
(111)
[1 10]  [110]
(111)
Analytical Method-1
Weiss zone law states that if a direction [u v w] is contained in plane (h k l) then:
hu  kv  lw  0
Let [u v w] be the intersection of (1 11) and (1 1 1)
(1 11)  u  v  w  0 
 w0 
&

  the indices of the direction are of the form [u u 0] .
u  v 

(1 1 1)  u  v  w  0 
As miller indices are written after factoring out common factors the miller indices are: [1 10]
or the opposite direction [ 1 10] .
Analytical Method-2
Using equation of a line in intercept form (a, b, c are intercepts along x, y, z axis
respectively):
x y z
   1 . Substituting the intercepts for the two planes and solving simultaneously we
a b c
get the equation for the line of intersection:
 x  y  z  1
 2x  2 y  2


x

y

z

1

x  y  1  y   x  1 (slope = 1 and intercept = 1)
[1 10] & [110]
O
y = x1
Directions
with same
Miller indices