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I'm not sure if a Condensed Matter book is going to give you what you want: as pointed out by commenters, you cannot derive a Lagrangian, you can only justify it because it represents the correct physics. But here is a simple interpretation of the 3rd order term. For small deformations, Hooke's law holds and the restoring force $F_{a}=−k_{ab}q_b$. (For ...

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Yes. The Seebeck effect, for example, is the direct conversion of thermal gradient to electric voltage. It is used in the small scale in thermocouples to measure temperatures electronically, or in the larger scales in thermoelectric generators for power generation. In fact, many of the long-range space probes launched by NASA get power this way. ...

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Answer to title question because the spatial volume is infinite--- the p values in a finite volume are discrete, but as you take the infinite volume limit, you get a continuum of p's. The p's are Fourier variables, they are inside-out with respect to the space variable. To see this consider a periodic function in one dimension with period L:  ...

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My opinion: Phonons are collective excitations of the crystal lattice vibration. They are massless Goldstone bosons resulting from the violation of the continuous translational symmetry of free space, by the crystal lattice. Phonons must carry momentum because they interact with electrons and change the momentum of the latter. An example is the electronic ...

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This is a great question. The point is we've made a huge structural change in going from Lehmann representation to the, um, other one. In Lehmann representation, we've chosen to write $G$ as a sum over an infinite number of real poles, which is fine of course, but when you start adding infinite numbers of delta functions together, things can get tricky. ...

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Unstable particles are concepts of effective field theories (or few-particle systems) in reduced descriptions where the decay products are ignored. In these reduced descriptions, they appear as particles with complex masses, and their Green's functions have complex poles. In an unreduced description, unstable particles appear as poles of the analytically ...

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Your answer is indeed what's going on in the lecture, but it doesn't explain what was wrong with your initial argument: you'd expect a model with $l_0>0$ to be a closer representation of reality than one with $l_0=0$, wouldn't you? Actually, your initial reasoning was correct: transverse displacements of springs under zero tension do indeed result in ...

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Suppose we take a classical approach and model the crystal lattice as a series of coupled simple harmonic oscillators. if the number of individual lattice sites $N$ is large (but not infinite), we may assume periodic boundary conditions, which result in the quantization of allowed wavevectors. Using Ashcroft and Mermin pg 430 as a reference, the above ...

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Question: Are you completely sure that the phonon wavevector spectrum is continuous? For any finite body, phonon modes are (nominally) quantized in much the same fashion as the fermion seas of electrons in metals. It's quasi-continuous, sure, but a true continuum spectrum for phonons is necessarily a bit of an abstraction.

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Found something today. "Actually the concept of polaritons has been described in words as early as 1946 [F. Bloch, Phys. Rev., 70, 460 (1946)] in nuclear paramagnetic resonance, however without using the then still unkown term 'polariton'." Klingshirn, Claus F., Jul o6 2012, Semiconductor Optics Springer Berlin Heidelberg, Dordrecht, ISBN: 9783642283628

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This figure shows the phonon dispersions of ZnO. It is clear that while some phonons have very roughly linear dispersions many do not (especially close to zone center). The second derivative would be non zero in these regions. I hope this has added to the conversation.

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How about the Piezo-electric effect? if I'm not mistaken, pressure on the crystal is in essence equal to long-wavelength phonons. If that's not what you're after, perhaps read this, or else this paper (both the result of a 5 minute Google scholar session). They're don't seem to contain exactly what you're after (haven't read them thoroughly though), ...

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