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http://uw.physics.wisc.edu/~himpsel/551/Lectures/E_versus_k.pdf

Look at first picture

If you see this picture of a bloch wavefunction, you see that it has two types of periodicities involved, one with the lattice constant(the bloch part), but what is the periodicity of the enveloppe.

Has it something to do with Born-Von Karman cyclic boundary conditions?

If you see this video

They speak also of 2 periodicities, but I dont get it

big K is supposed to be 2*pi / a where a is the lattice constant

and small k is 2*pi / L

This "L" is the L from the Born-Von Karman cyclic boundary conditions and is the length of the total crystal?

Because a bloch function is a convolution between a free electron that is confined only bye the size L (and it has this periodicity because it repeats itself in a loop over the crystal again? ) of the metal and a periodic bloch part, periodic with the lattice (ions)

Im not sure why the Born-Von Karman conditions are cyclic over a crystal, why does it form a loop?

Is my overall interpretation correct?