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What is the meaning of a basis?

  1. Oct 15, 2012 #1
    Hi!
    There is a concept I don't understand and would love to have is cleared...
    What is the meaning of a lattice with a basis?
    What do I need it for?
    Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? )
    why does it become a simple lattice spanned by ax,ay,az
    with a two point basis

    0, a/2(x+y+z)

    or a four -point basis

    0, a/2(x+y), a/2(x+z), a/2(y+z) ?

    I don't understand the whole thing. For a start, how can in be (mathematically) that 0 is a part of a basis? What is this 0?
    Second thing, how can something be spread by either 2 or 4 dimension basis?
    It does not fit what I know about a basis...
    I can't find answers to these questions... and I guess I miss the whole concept...
    Thank you in advance for your help :)
     

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  3. Oct 16, 2012 #2

    cgk

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    Science Advisor

    A basis in the crystallographic sense not a basis in the sense of a vector space. The two are completely disjunct concepts, which just happen to have the same name by accident.

    The unit-cell basis you are describing is effectively just a bunch of points in the unit-cell. The entire lattice is then spanned by translating these points (i.e., all the points separately) by integer-amounts of the lattice vectors.
     
  4. Oct 16, 2012 #3
    Thank you!
     
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