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In Kittel`s book, it says

The band structure of a crystal can often be explained by the nearly free electron model for which the band electrons are treated as perturbed only weakly by the periodic potential of the ion cores."

Does this mean that in electron free model we neglected the potential of ion cores completely? Or is it opposite?

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Does this mean that in electron free model we neglected the potential of ion cores completely?

Yes, that's right. The free electron model simply treats the electrons as being non-interacting free particles, hence the name. If the periodic potential due to the ions is added as a perturbation, we obtain the nearly free electron model.

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The quoted sentence explicitly says that the ion potential is not completely ignored, but considered as "weak". In quantum slang this means that it is accounted perturbatively. What is done in practice is that Bloch states are constructed, as if there were no potential, and then the matrix elements of the potential are calculated near the Brillouin points, where different Bloch states are nearly degenerate. We thus solve for the splitting of the energy bands (i.e., the gap) near these points. Most books on solid state physics furnishe the mathematical details for the one-dimensional case.

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