I'm reading a paper which says that the interband transitions ($\pi_1^* \rightarrow \pi_2^*$) involving phonons at $q= 0 $ and $ q = K$ in Bilayer Graphene are prohibited by energy and momentum conservation laws.

The explanation is the following: The contour of integration from Fermi's Golden rule never crosses (or comes close to) the peaks at the points $\Gamma$ and $K$ observed in the matrix element of Bilayer-Graphene. The estimated minimal distance between this curve (which can be approximated by a circle) and the point $q = K$ is $0.034(2π/a)$ for emission and $0.014(2π/a)$ for absorption. For comparison, the analogous distance in case of inter-valley scattering due to $TO$ phonons in MLG is much smaller, $\approx 10^{−7}(2π/a)$, which ensures its dominant role in electron scattering.

How do they calculate the distances and how exactly does this line of reasoning work? Thanks in advance!


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