A one-dimensional diatomic crystal (with two distinct atoms A and B arranged in a line) can exhibit two types of collective motions. In one type, the consecutive atoms move in-phase and in the other, move out-of-phase. The first type of motion corresponds to a low-frequency band called the acoustic branch and the second type corresponds to a high-frequency band called the optical branch.
However, I see no reason that the same two types of motions are not possible for a one-dimensional monatomic crystal, and in fact, are easily imaginable.
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I want to ask if I am correct i.e. whether the out-of-phase motion is indeed possible for the monatomic case too. 'Yes' or 'No'?
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If 'Yes', why don't we have an optical branch in the monatomic case? Doesn't out-of-phase motion imply an optical branch?
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If 'No', why not? Please explain physically. I have worked out the mathematics.
Best Answer
You are correct that out-of-phase motion is indeed possible for the monoatomic case. So "YES" to your first question. However, this is simply a high-$k$ mode of the acoustic branch. Imagine the following: let's say we have a monoatomic chain but we don't know that all our atoms are the same, so we decide to divide it into cells each containing two distinct atoms. We doubled (unnecessarily) the degrees of freedom that we use to describe the system. This can be thought of as 'folding' the BZ. So basically the result is of course valid, but you interpret it in a weird manner, where modes shifted by $\pi/2$ somehow belong to different branches.
The question to your second question is that the name 'optical' comes from the fact that for true diatomic chains, the out-of-phase movement has a dipole moment. Therefore, this mode couples to a an electromagnetic wave and will be excited by it. For a monoatomic chain, even the out-of-phase movement is neutral, and will not couple to such a probe. So this distinction, which seems purely mathematical (a choice of basis and BZ) has physical implications. In a monoatomic chain the only excitations are acoustic ones.