localized electrons in the crystals Why electrons in low lying levels of individual atoms stay localized in their own atoms in a crystal? Doesn't this contradict Bloch's theorem?
 A: They do not stay localized actually. It's only that their eigenstates (which are non-local) are very close in energy, and we can choose the basis of localized states and they would be close to eigenstates with high accuracy.
You can consider the tight-binding model, and take it to the limit of zero overlap of orbitals of neighboring atoms, and zero overlap integrals. That would be close to what low-level electrons in a crystal behave like.
A: In an ideal crystal at zero Kelvin, all the electrons in the low-lying states are equally to be found throughout the crystal, i.e. as per Bloch's theorem, probability of finding electrons is periodic in the crystal for a given eigenstate (and they all are almost degenerate, i.e. with a sharp energy band). If you consider tight binding method as a means to solve for eigenstates, the wavefunctions are still linear combination of atomic orbitals such that probability density is periodic in the crystal. However, at a finite temperatures, the electrons are going to be localized at atomic sites. This is because the "coupling" in the tight-binding sense is small, and any atomic vibrations (and electron-electron interactions) will destroy this coupling and localize the elecrton. This will eventually lead to each electron having its own local identity as defined by the particular lattice point. Such localization does not happen for valence electrons because they are losely bound, or they couple with neighboring atoms strongly, or the orbitals have a large spatial overlap. This means any imperfections in the periodic potential are not going to cause localization as it would for low-energy state electrons.
