I am taking an introductory course in condensed matter physics, and am absolutely stumped by the concept of the Brillouin zone and backfolding of the dispersion curve in the nearly free electron model.
Specifically, when we assume that the lattice is periodic IN REAL SPACE with a period a, then we can represent this as a fourier series and the significant k vectors (or G vectors) i.e. the reciprocal lattice vectors, are spaced $2\pi /a$ apart. Now with my understanding, we have no fourier components which have k vectors intermediate to these G vectors. These are the permitted wavevectors to describe any function (e.e the electron wavefunction) in order to ensure that it has the correct periodicity. It should have no components with k smaller than the first reciprocal lattice vector $2\pi /a$ (or if we centre our reciprocal space intermediate to two lattice points, we can go from $\pm \pi /a$.
Why then do we backfold the free electron dispersion curve into the first Brillouin zone when we introduce periodicity? This si the biggest mystery to me, and I'm certainly not getting something important here. To me, periodicity in real space means that all of the information is contained in a fucntion defined on the small interval which is the space occupied by a unit cell. In reciprocal space, small intervals in real space translate to large distances in reciprocal space. So it makes absolutely no sense to me as to why we then say that all of the information is contained within a small interval of the first Brilluin zone in k space. It should be the other way around! If the only unique structure in real space is that contained within the unit cell, then the description of this in k space should require the fourier components at all of the recirprocal lattice points going to infinity!
I would greatly appreciiate if someone could clarify why we talk of all of the information being contained in the first Brillouin zone, and why supposedly k vectors outside of the first brillouin zone are equivalent to those within the first brillouin zone.