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Help for supercell, Bloch function

  1. Jul 21, 2010 #1
    I recently have a puzlle that what is the relation between the periodic part of Bloch function in primitive cell u_nk(r) and that in supercell, e.g., three primitive cells in one of the directions. Are they the same one?



    pls give me some hints or explanations!


    thanks~~
     
  2. jcsd
  3. Jul 21, 2010 #2

    DrDu

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    Science Advisor

    Well, the unit cell in the reciprocal lattice is smaller in the case of supercells than in the case of normal cells in direct lattice. That is the vector k for the normal lattice will become k'+K in case of the supercell lattice where K may take on 3 values in the case of a supercell which is 3 times longer than a normal cell.
    Hence
    [tex]u_{nk}(x)\exp(ikx)=u'_{NKk}\exp(ik'x)[/tex]
    or
    [tex]u'_{nKk}(x)=u_{nk}(x)\exp(iKx)[/tex]
    If the length of the original unit cell is d and the length of the supercell is 3d (at least for a space group with orthogonal axes)
    then [tex]K \in \{0,2\pi/3d,4\pi/3d\}[/tex].
     
  4. Jul 21, 2010 #3
    Thanks very much for your useful reply!
     
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