Miller indices and crystal directions

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Miller indices and crystal directions
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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
here 3,1,3

• Clear fractions with the smallest possible
  integer

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Simplified example for cubic system
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• negative intercept is denoted by a bar

• the symbol

denotes all planes equivalent to (h,k,l)
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Set of four Miller indices for hexagonal crystals

first three Miller indices add up to zero
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Crystal directions
n1 n2 n3
direction defined by
E.g., 111 instead of 222
Note hkl is in general not normal to the (hkl)
plane