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As stated in this review article:

Mechanical modes are long compared to the interatomic spacing. It is natural to make the distinction between nanomechanical modes and phonons: The former are lowfrequency, long-wavelength modes strongly affected by the boundary conditions of the nanodevice, whereas the latter are vibrational modes with wavelengths much smaller than typical device dimensions. Phonons are relatively unaffected by the geometry of the resonator and [...] are essentially identical in nature to phonons in an infinite medium

My feeling is, however, that "phonon" is used in all sorts of different contexts, is usually interchanged with mechanical mode, and is even used where there is no vibrating solid (e.g. in harmonic oscillators). Could you help me clarify a bit the difference and the nomenclature?

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I think they are probably trying to characterize small mechanical devices. One part of this analysis would be analyzing their motion.

One motion it can do is to vibrate. If the wavelength of the vibration is much smaller than the size of the object, then the vibration can't tell how small the object is, and the vibration will behave the same way it would if the object were infinitely big. This type of vibration is what he calls a phonon.

On the other hand, if the wavelength is about as big as the object, it will behave differently than it would for an infinite object, and the way it behaves will depend on shape. This is what he calls a mechanical mode. For example, if you have a long thing bar, then there will be a bending mode associated with the bar bending so that its long axis is curved. This mode is qualitatively different from anything that would occur in a bulk material. It is important to understand these mechanical modes.

And I haven't really heard phonon used in all sorts of context like a harmonic oscillator.

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  • $\begingroup$ So is there any fundamental difference between a phonon and a mechanical mode? $\endgroup$
    – nabla
    Commented Nov 25, 2015 at 15:32
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    $\begingroup$ I would say no. You just take your microscopic hamiltonian and find all the normal modes. Then you make a somewhat arbitrary distinction that the ones that look like plane waves are phonons, and the ones that look different because they see the boundary conditions are what he calls mechanical modes. $\endgroup$
    – Brian Moths
    Commented Nov 25, 2015 at 17:06
  • $\begingroup$ I agree that in many cases, the distinction is kind of arbitrary. However, in some cases, the distinction is pretty clear cut. E.g. a freestanding beam has torsional modes (where the beam twists), dilatational modes (where the cross sectional area increases and decreases), Rayleigh surface waves, etc. Infinite media simply has no analogous modes. In contrast, both beams and infinite media have modes that look pretty much identical, like transverse and longitudinal modes. That said, in some contexts (like very small nanostructures) all vibrations are usually called phonons. $\endgroup$
    – lnmaurer
    Commented Nov 25, 2015 at 19:26

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