Transcript Miller indices and crystal directions

Miller indices and crystal directions
How to describe a particular crystallographic plane and direction ?
Primitive translation vectors a1, a2, a3 :
not necessarily of equal length
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
•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
Miller indices (h,k,l)
specify set of
equivalent planes
Simplified example for cubic system
n1  1, n2  , n3  
1/ n1  1, 1/ n2  0, 1/ n3  0
(1,0,0)
n1  1, n2  1, n3  1
1 / n1  1, 1 / n2  1,
1 / n3  1
n1  1, n2  1, n3  
1/ n1  1, 1 / n2  1, 1 / n3  0
(1,1,0)
(1,1,1)
•negative intercept is denoted by a bar :
•the symbol
h,k,l
( h,k,l)
denotes all planes equivalent to (h,k,l)
•example for a plane that cuts the a-axis at
1
a
2
Set of four Miller indices for hexagonal crystals
first three Miller indices add up to zero
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