I have few simple questions about Brillouin zone.
I will take the example of a 1D lattice with a period of "a".
If I have well understood, the reciprocal lattice shows all the point in "k-space" allowed for any function that will respect the periodicity of the lattice.
So, it shows us all the allowed wave vectors when we do a decomposition in Fourier series.
(I know that we also can proove that the vector of the reciprocal lattice are the ones that will allow diffraction but I would like if possible not consider experimental arguments to explain me what is going on).
Now, what is physically the (first) Brillouin zone ? In my example I know that it will be all the vector in the reciprocal space that will be between $ [-\pi/a ; \pi/a] $.
But why do we distinguish these vectors ? Why are they this important ?
I have read explanation that says it is because if we take a wave with wavelength $ a $ and another with wavelength $ 2a $ an observer will not see the difference between them.
But I dont understand this argument because : First, why should the observer only look at what happens on atoms of the lattice and not between them ? Next, when we explain this we draw a wave of wavelength 2a starting with a node at x=0, then there will also be a node at x=a and x=2a. And then we draw a wave of wavelength a and we say "thoose waves has the same nodes at the same place : the observer will not be able to distinguish them". Ok but if we have started our wave at a peak and not at a node the observer would be able to distinguish them so I dont understand this argument. We just drawed a particular case.
So, are they this important in link with bloch theory (I asked a question about this in another topic, I am really a beginner in solid states physics, so don't be too hard with your explanations please :) ).
Thank you a lot for your answers