I am not really understanding, how to view the increasing of lattice parameter a. Lets see it in 1D. Volume of the unit cell is a.

Reciprocal vector is 2 \pi /a , so volume of FBZ will be 2\pi /a . If we increase the lattice parameter a, the volume of a unit cell in real space increases but in reciprocal space decreases.

Lets take a finite crystal with Length L (fixed) = N*a. So if we increase a , the number of unit cells will decrease. Also the size of the FBZ got smaller. Therefore the number of reachable k points in FBZ becomes less. How is it possible to still few the band structure spectrum as a continious fct? I dont get from that point of view how the bands will flatten.

For a free atom, the lattice parameter goes to infinite so the FBZ gets infinite small, therefore there will be no reachable k points anymore. From this point of view there is no possbile way to draw E(k) and the concept doesnt make sense to me.

Any help ?

Cheers