Magnitude of the energy gap

First of all, I'm using the book "Introduction to Solid-State Physics", by Charles Kittel, 8th edition. At chapter 7, page 167, he is assuming he has a lattice with parameter a. After that he writes: "Let us suppose that the potential energy of an electron in the crystal at point x is: $$U(x)=U\cos(\frac{2 \pi x}{a})$$

To me, this seems wrong, I thought we interpreted the ion-cores as positive and the electrons as negative, what he is saying is that the electrons should have a maximum in potential energy at x=a, I would claim the opposite, shouldn't the electrons have their max at $x=\frac{a}{2}$ ?

• The positive charges are at positions ..., -a, 0, a, ... (with +U) while he electrons are at positions ... -a/2, a/2, ... (with -U) – NaOH May 6 '18 at 9:58
• But shouldnt the potential energy be higher the further i am from the positive ion? – John Skeet May 6 '18 at 10:20
• Why can you say that the electrons already are at -a/2, a/2 etc? – John Skeet May 6 '18 at 10:25
• Perhaps U <0 ? (need to add extra characters...) – my2cts May 6 '18 at 11:32
• You are right in saying that the positive ions should have -U instead, but the interpretation does not affect the math. It is a model to illustrate bandgap from a periodic potential using the central equation, i.e. to model this: upload.wikimedia.org/wikipedia/commons/b/b2/… – NaOH May 6 '18 at 11:51