Can someone explain peierls distortion/transition to me?

[um0123]

Peierls distortion states that for a 1 dimensional polymer (like polyacyteline) with lattice spacing a they should have a half-filled conduction band, why?

And how does changing the lattice space to 2a cause it to form an energy gap? In my mind it should be the same as the first case...

 

[DrDu]

The point is that in Peierls distortion, the bond length change, getting alternatinly longer and shorter. This opens a band gap and only the lower band will be filled, so you get energetic stabilization. This is especially easy to see for a degenerate band, e.g. think of largely spaced hydrogen atoms. If you pair them up into H2 molecules, you will get a full band of binding MO's and an empy band formed from anti-binding MO's.

 

[um0123]

im still fairly confused... is there a simple derivation somewhere i can follow? i tried looking one up but i can't seem to find anything consistent...

 

[DrDu]

Did you check Peierls book "Quantum theory of solids" chapter 5.3?