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Expanding the periodic potentials

  1. Apr 24, 2013 #1
    Could one always write the periodic potentials in the form:
    v(r)=Ʃf(r-G)
    where the sum is over G (reciprocal lattice vectors)?
     
  2. jcsd
  3. Apr 24, 2013 #2
    r-G does not make sense. They are vectors in different spaces. They have different units.
     
  4. Apr 24, 2013 #3
    Excuse me. I wrote wrong.
    Can we write v(r)=Ʃf(r-R) where the sum is over lattice vectors R?
     
  5. Apr 24, 2013 #4
    This looks like superposition. Adding the potentials of each atom, in the coordinate system with origin in the specific atom. Periodicity does not seem to be necessary for this.
    f(r-R) will be the potential "produced" by the atom at location R.
     
  6. Apr 25, 2013 #5
    Ok, thank you. But I like to know that could we earn this formula merely by having periodicity condition (without regarding atoms)?
     
  7. Apr 25, 2013 #6

    cgk

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    yes, if the potential v is periodic, then it can be written in this form. This is the most general form of a function v(r) which fulfills v(r+R) = v(r) for all lattice vectors R (i.e., which is periodic).
     
  8. Apr 25, 2013 #7
    Could you please prove it or give me a reference which has proved it?
     
  9. Apr 25, 2013 #8

    cgk

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    Consider a large, finite lattice with periodic boundary conditions. Note that then v(r)=Ʃf(r-R) hold if you put in f(r) = v(r)/N_R itself, where N_R is the total number of lattice points. This also still holds if you set f(r) = v(r) if r is in one single unit cell, and 0 if not.
     
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