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These are two different problems. What Jon was saying is correct. However, it does not explain LO-TO splitting. Like Jon said, because you can tell when you are on a Ga or As atom, the degeneracy of the optical modes are lifted at the Gamma point. This is in regards of 3 different optical modes separating. However, the phenomena Cardona is refers to involves ...

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Just to be clear, the two or more atoms do not have to be of different type. Optical phonons are related to the relative vibrations of atoms within the unit cell, while acoustic phonons describe the relative vibration of different unit cells. Optical phonons arise whenever the unit cell has at least one such degree of freedom, meaning at least two atoms in ...

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Crystal lattices are classified in a way that is not necessarily the most natural one. A first classification is by their lattice class, which is determined by its associated unit cell, which is not necessarily the same as a fundamental domain for the lattice. There can be different Bravais lattices in each lattice class, determined by possible additional ...

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In quantum treatment, phonons are quantum harmonic oscillators, which is studied in any quantum mechanics textbooks. The energy spectrum is readily studied there. https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator The $1/2\hbar\nu_E$ comes from uncertainty principle. (A quantum harmonic oscillator is confined in space so its momentum cannot be zero) ...

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The lack of the $e^{-i \omega t}$ term is just because we're using complex wave notation. If you've ever taken an electrical engineering course, it's the same sort of thing that is used there: We're using $A e^{i \omega t}$ to stand for $A \cos (\omega t - \delta)$, with the implicit assumption that we're only interested in the real part of the quantities ...

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In superfluid helium-4, the phonon excitation spectrum includes a mode which has the same energy and momentum as a neutron with a speed of about 440 m/s (wavelength $\lambda \approx 9\,Å$). You can create a neutron beam which contains only 9 Å neutrons by starting with cold neutrons and being clever with diffraction from crystals. If you send these ...

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I don't know nearly enough QFT to address the background or implications of your question. However, I'd basically answer yes to your first two questions, but it depends a little on your definition. A single phonon mode is not localized in space. However a wave packet can in principle be built up of a small range of frequencies, giving a fairly well defined ...

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