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Bloch's theorem (Read 2872 times)
Gerrit-Jan Linker
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Bloch's theorem
17.07.09 at 12:42:43
 
Bloch's theorem
 
In the nearly-free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors.
 
Bloch's theorem:
The eigenfunctions for electrons that move in a periodic potential can be written in the following form:
 
ψnk(r) = eik.runk(r)
 
The corresponding energy eigenvalues are: Єn(k)= Єn(k + K)
 
K lattice vector
k wave vector
n index
 
Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n.  
 
Because the eigenvalues for given n are periodic in k, all distinct values of Єn(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.
 
A Bloch wave or Bloch state is the wavefunction of an electron placed in a periodic potential.
 
The opposite extreme to the nearly-free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms: the tight binding model  
 
See also:
Wave vector
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1247825987
Tight binding
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1275249597
 
Reference:
Bloch wave
http://en.wikipedia.org/wiki/Bloch_wave
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« Last Edit: 16.01.12 at 13:23:14 by Gerrit-Jan Linker »  

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