I What Happens to Energy When Phonons are Damped?

[dRic2]

TL;DR Summary

Where does the energy of a phonon go when it gets damped?

If you go beyond the harmonic approximation, phonons can not be thought as independent quasiparticles anymore and phonon-phonon interactions are taken into account. This eventually translates into the fact that phonons frequencies get renormalized ( $\omega \rightarrow \omega^′ +i\nu$) acquiring a width which gives a damped amplitude $e^{-\nu t}$ with a characteristic time $\tau = 1/{\nu}$, like all quasi-particles.

Now my, possibly trivial, question: if phonons represent lattice oscillations, and one phonon gets dumped... where does the energy carried by the oscillation go? Is it redistributed to the electronic structure? Or is it lost in heat generation?

Thanks,
Ric

 

[hutchphd]

It goes to all the other degrees of freedom as dictated by the gods of thermodynamics. There are plenty of other degrees of freedom, and the finite lifetime indicates the strength of the coupling. Mostly other phonons for most solids. Phonons are the "heat".

 

[dRic2]

Thank you for the answer.

One more question: suppose we start with the simple Born-Oppenheimer approximation to separate electron and ion dynamics. In doing so we obtain a phonon hamiltonian like $H = K + E$ where $K$ is the ion kinetic energy and $E$ is the potential in which the move. In this simple picture, we are neglecting all the electron-phonon scatterings. So since phonons do not interact with electrons, the only possible mechanism to lose energy is through heat generation, right?

 

[hutchphd]

If it isn't in the Hamiltonian, it doesn't exist.
We are neglecting a host of things when writing the simple harmonic Hamiltonian. The ways we include them are various and depend upon what we need to calculate. Rigorously one includes them and shows which ones can be ignored or for a thermal reservoir which can be treated in an average (perhaps complex-valued) potential. The details depend upon how we convert the otherwise intractable problem into a useful solution.
What you are asking in essence is "If we ignore all the other interactions then the energy loss will be due to "X"". The answer to that question is tautologically yes every time.

 

[dRic2]

"If we ignore all the other interactions then the energy loss will be due to "X"". The answer to that question is tautologically yes every time.

Yes, I apologize if you feel like I am wasting your time... but I want to be as clear as possible and I appreciate the time you took to answer.

Btw, I noticed that it is not really correct to say this:

So since phonons do not interact with electrons, the only possible mechanism to lose energy is through heat generation

If the system is in thermal equilibrium after a single phonon is "dumped", another one must be created inside the systems or inside the thermal bath (if I am using a grand canonical approach), so there is no net flow of heat.
Heat is lost only if the system is coupled with the environment somehow, so I really chose bad wording.

 

[hutchphd]

I am not worried about the statements by themselves. In my experience, folks who worry too much about exact semantics about physics often do not focus their energies on how to do physics. It is a trap to be avoided.