1. Miller indices and crystal directions
How to describe a particular crystallographic plane and direction ?
not necessarily of equal length
Primitive translation vectors a1, a2, a3 :
not necessarily right angles
Determine the intercepts of the face along the crystallographic axes,
in terms of unit cell dimensions
n1,n2,n3
here 1,3,1
Miller indices (h,k,l)
specify set of
equivalent planes
Take the reciprocals
1/n1,1/n2,1/n3
here 1,1/3,1
Clear fractions with the smallest possible integer
here 3,1,3

2. Simplified example for cubic system

3. negative intercept is denoted by a bar :
the symbol
denotes all planes equivalent to (h,k,l)
example for a plane that cuts the a-axis at

5. Set of four Miller indices for hexagonal crystals
first three Miller indices add up to zero

6. Crystal directions
Lattice vector:
T=n1a1+n2a2+n3a3
direction defined by
[n1 n2 n3] If
[n1 n2 n3]
have a common factor, the latter is removed
E.g., [111] instead of [222]
Note: [hkl] is in general not normal to the (hkl) plane