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I Miller indices for hexagonal crystal systems

  1. Aug 8, 2017 #1
    Hi everyone, to find the draw the direction for a given miller index say, [1234] we first convert this miller index consisting of 4 indices into one containing 3 indices.
    To do so, we have a set of formulae prescribed in almost every book. Sadly I haven't been able to come across a single book the gives the derivation of those formulae!
    I thought that i could use vector-component method to get the results but that gives totally weird formulae not even close to the ones i see in my textbooks. (have a look at the attached image)

    So, can anyone suggest me a textbook, a link or anything that can help me understand the derivation? I'm not finding the enthusiasm for rote-memorising the formulae if i don't know where they come from...

    Thanks!
     

    Attached Files:

  2. jcsd
  3. Aug 8, 2017 #2
    The 4 Miller indices often used with hexagonal crystals are just a convenience. The 3rd index can be dropped. 4 indices are used to make planes with the same symmetry also "look alike" in the 4-notation.
    The physical reason is that the a- and b- axes form a 120 degree angle. There is a third axis within the basal plane that has the exact same symmetry. You get this axis, let's call it "d", by rotating the b-axis by another 120 degrees. So instead of using a and be as basis vectors, you could just as well use b and d, or d and a. Absolutely nothing would change, as the 120 degree rotation is the defining feature of a hexagonal crystal.
    The redundant 3rd Miller index corresponds to the reciprocal space "d*" axis. The 4th index is along the c-axis.
     
  4. Aug 9, 2017 #3
    yes, i get that but still, how do we arrive at those formulae?
    i tried using vectors and their components method but it doesn't work
     
  5. Aug 10, 2017 #4

    DrDu

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