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When faced with a repeating pattern, the conceptual issue is to identify the underlying Bravais lattice and a possible basis to describe the crystalline structure. Of course, there are many (infinite) possibilities for these elements. Still, the constraint of having the minimal basis and the most symmetric primitive unit cell for the Bravais lattice are good ...

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This question is answered in detail in the paper Phonons as Goldstone Bosons. The question What is the difference between a photon and a phonon? is also closely related. Here, I'll just give some basic intuition. Consider two solids that are identical except that one of them is shifted slightly in space compared to the other. Both of them are equally stable: ...

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"How exactly do you measure the roughness in terms of interatomic crystal spacing?" Perhaps this image will help. It shows the surface of a silicon wafer: (Taken from NIST.) You can see a series of plateaus and steps. Each of these plateaus is a crystal plane, and each step between them has a height equal to the interatomic crystal spacing. So, if ...

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Commensurate means that the ration of two parameters is an integer or, more generally, a ratio of integers. For example, if we consider a one-dimensional lattice with period $a$, and we impose on top of it a perturbation potential $$V(x)=V_0\cos(kx)$$ with spatial period $$\lambda=\frac{2\pi}{k}.$$ The sutuations where $\frac{\lambda}{a}=n$ or \$\frac{\lambda}...

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It has something to do with the Wiener-Hopf method. Basically, if you recall quantum mechanics and how the Green's function for a Schrodinger equation is found there, it is a Fourier transform of an evolution operator. However, the evolution operator has the initial condition, and, typically, you do not know pre-history, which prevents you from making the ...

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