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First, yes the speed of sound is the limit of the group velocity at small wave vectors. In the limit of small wave vectors (long wavelengths), the material acts like the continuum material we see at a macroscopic scale. (You don't see any atoms with your eye. Macroscopic materials appear to be infinitely divisible.) The speed of these long-wavelength waves ...

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You need to find the $k$ vector in the medium. $$k = \frac{2\pi f}{v_m}$$ where $v_m$ is the speed of sound in the medium, which can be found from the dispersion relationship $$v_m = \frac{1}{\hbar}\frac{\mathrm{d}E}{\mathrm{d}k}$$ The intensity has nothing to do with phonon energy. Intensity provides the total energy incident in a unit area. Typically ...

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For an infinite chain with periodic boundary conditions, you have $n$ translation symmetry. This means you can look for a basis of solutions whose $n$ dependence is $e^{ikn}$. The boundary conditions constrain you to $k=m\frac{2\pi}N$ with $m=0$..$N-1$

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When you have a great number of interacting particles (such as electrons in a solid), it becomes impossible to try to describe each electron individually. We introduce the concept of quasiparticle to describe whan can be the low-lying excitations of the ensemble of particles. For instance, If you take a metal at 0K, the ground state of the system is the ...

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Think about the wave-particle mechanism in quantum mechanics. Everytime you have a wave travelling in a medium, you can think of it as a particle with a given propagation law (dispersion law). In the case of phonons, you can show that they propagate with the sound speed on the given medium. So, you can think of it as a particle (a set of variable values ...

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Because of the way they vibrate? Acoustic phonons are basicaly longer wavelenghts and neighbouring sites are in phase while for optical ones you have shorter wavelenght and oposite phase in neighbouring sites of lattice.

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For very basic understanding, because optical phonon has larger energy than accoustic phonon. If you have larger energy, you will have more ways to decay, like into two lower energy accoustic phonon. If you have low energy like an accoustic phonon, there are fewer decay paths and hence is more stable.

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