(Mentor note: moved here from noon homework thread hence no template) I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC). I wanted to justify the use of PBC, so I came up with this: atoms deep inside the crystal sees an infinite crystal with periodic potential atoms near the surface of the system do not see an infinite crystal nor periodic potential. however there aren't many atoms near the surface so they contribute little to the property of the whole system. therefore PBC can be used since the majority of the atoms sees periodic potential. Is my understanding valid? And are there any mathematical way of justifying the PBC?