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How to define a crystallographic direction with Miller indices

  1. Find two points lying on the given crystallographic direction.
  2. Determine the coordinates of the two points using a right hand coordinate system which is defined by crystallographic axes a, b and c.
  3. Subtract the coordinates of the "tail" point form the coordinates of the "head" point to obtain the number of lattice parameters traveled in the direction of each axis of the coordinate system.
  4. Clear fraction and/or reduce the results obtained from the subtraction to lowest integers
  5. Enclose the number in square bracket [uvw]. If a negative sign is produced, represent the negative sign with a bar over the number

Example: Find the Miller indices of direction d shown in the figure

  1. Fine two arbitrary points on direction d: P1 and P2
  2. The coordinates of the two points are: P1 (1/2, 1, 0) and P2 (0, 0, 1)
  3. Subtraction: P2 (0, 0, 1) - P1 (1/2, 1, 0) = (-1/2, -1, 1)
  4. Fractions to clear: 2(-1/2, -1, 1) = (-1, -2, 2)
  5. Result: [-1-22]
example 1
How to define a crystallographic plane with Miller indices?

  1. Find the intercepts between the crystallographic plane and the three crystallographic axes in terms of number of lattice constants. If the plane passes through the origin, the origin of coordinate system should be moved.
  2. Take reciprocals of these intercepts.
  3. Clear the fractions
  4. Enclose the resulting numbers in parentheses (hkl). The negative number should be written with a bar over the number.


Example: Determine the Miller indices for the indicated plane

  1. Lattice constants (axial lengths): 4A, 8A, 3A
  2. Lengths of the intercepts: 1A, 4A, 3A
  3. Intercepts in terms of number of lattice constants: ¼, ½, 1
  4. Reciprocals: 4, 2, 1
  5. Clear the fractions: 4, 2, 1
  6. Result: (421)
example 2
Four indices representation

In hexagonal system, a crystallographic plane is usually expressed in form of four basic vectors a1, a2, a3 and c and is written as (hkil). The index i is the reciprocal of the fractional intercept on the a3 axis. The value of i depends on the values of h and k. The relation is

h + k = -i

The advantage of (hkil) representation is that it can give the similar indices to similar planes. For example, the six side planes of the hexagonal prism in the following figure are similar and symmetrically located. Their indices can be expressed as, in (hkil) representation, (10-10), (01-10), (-1100), (-1010), (0-110) and (1-100) which clearly demonstrate the symmetrical relationship of the six planes. On the other hand, the (hkl) representation: (100), (010), (-110), (-100), (0-10) and (1-10) do not immediately suggest the symmetrical relationship.

hexagonal system

Directions in a hexagonal system are best expressed as [UVW] in the form of three basic vectors a1, a2, and c. The four indices representation [uvtw] does not provide any advantage over [UVW]. The relation between these two representations is as follows:
U = u - t
u = (2U - V) / 3
V = v - t
v = (2V - U) / 3
W = w
t = -(u + v) = -(U + V) / 3
--
w = W
For example, [210] and [10-10] represent the same directions as illustrated in the above figure.
Basic of Texture  
Definition of the Crystal Reference Frame  
Miller Indices  
Euler Angle  
Stereographic Projection  
   
   
   
   
   
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