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2

The problem is that you are looking at the gap at the $\Gamma$ point ($k=0$) and not at the real gap at the Brillouin zone boundary. Here is the plot for the phonon dispersion relation of a diatomic linear chain. You can see that the real gap at the $k=\pi/a$ point is vanishing for $m_1=m_2$. This plot looks different as typical plots for a monoatomic ...

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In my answer attempt, I take a detour of analysing TIPT via TDPT to get the concept of transition. With great care, I believe one can derive time independent perturbation theory from time dependent perturbation theory by considering a infinitely slowly varying perturbation. Starting at t=$-\infty$ with zero strength and reaching nominal strength at t=0. ...

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There's an important subtlety: in the Schwarzschild metric, the impact parameter $b$ is not equal to the radius of closest approach. Let's start with the geodesic equation $$\frac{1}{2}\dot{r}^2 + \frac{L^2}{2r^2}\left(1-\frac{2GM}{c^2r}\right) = \frac{1}{2}E^2.$$ At the radius of closest approach $R_0$ we have  \frac{1}{2}E^2 = ...

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