"vibrations of electrons" isn't probably a good concept for metal electrons. For a vibration to occur, there must be a restoring force. In the simple Drude picture for metallic electrons, a metal is characterized by the absence of a restoring force.

This said, the following is nevertheless valid:

• yes, there is electron phonon interaction in metals
• this interaction lowers the lifetime of both, vibronic and electronic quasiparticles (i.e. phonon and electron excitations) and also modifies the band dispersions compared to the non-interacting case.
• On the electron side, this can be measured using angle resolved photoelectron spectroscopy with good resolution and as a function of temperature.
• An initial read might be a review article by R. Matzdorf, in Surf. Si. Reviews, I believe. Or find out, where this has been quoted. I am sure there is more recent work.
• As to how electron phonon interaction affects both electron and phonon bands (dispersion and/or lifetime), this should be treated either in Kittel's or Ashcroft's book on solid state physics (or both). This text book treatment actually motivated ome of the research on this basic topic as soon as the ARPES instruments improved to be really high resolution instruments as they are (or can be) today.

Hi Kai Fauth,

Thanks for your comments, it was indeed helpful.

I have a query, you mentioned that, "For a vibration to occur, there must be a restoring force". What if we are applying an external source of vibration to metals ? Can this  control the electrons motion?

Yes, of course. This would then be a forced oscillation of the electrons. Since the restoring force is zero, this means that the resonance frequency is zero. Towards zero frequency you obtain then infinite polarizability (as it should, since in a metal there is a steady flow of current if an electric field is being maintained).

All this is well accounted for (in classical terms) in the Drude-Lorentz model of metals. I recommend two texts for studying this. One would be Ashcroft & Mermin's Solid state physics, the other one F.Wooten: Optical properties of Solids. [The latter one is an older book which I still find very useful.]

Note added: when you mentioned "external source of vibration" I understood this to be an electric field (i.e. light).

Hi Kai Fauth,

How would it be if the external source of vibration is an acoustic wave ( phonon excitation)

I try to explain in a different way. In normal conditions (ie. room temperature) phonon interaction with electrons is not much significant ( or electrons are unaffected by phonon energy) Phonon contribution become significant when the temperature is increased or decreased. When temperature is increased, phonon vibrates even faster (in terms of understanding) and interact with electron. That's why for metals there is decrease in conductivity on increasing the temperature. While decreasing the temperature, phonon vibrates ceases, leads to increases the conductivity. 

While coming to your question, i dont thing acoustic wave will affect the vibration. If the frequency is in ultrasound range, it produce heating effect (high frequency ultrasound transfers energy to electron and lattice. the resultant thermal energy is sum of kphonon+kelectron)

I find it not so easy to imagine at what level giving an adequate answer. What is the physical picture or framework in which to discuss your question [such that the answer will be useful to you] ... ? Classical, semi-classical, quantum physical approach? And I reckon that the context of your questino relates to the paper which you are involved with (looks interesting btw).

OK, I'll give it a try and invite whoever else to do better. Beware: it's going to be lenthy, I assume...

The bonding mechanisms of the electrons provide the glue by which atoms stick together in condensed matter. It is therefore clear that motions of atoms and electrons cannot be independent. Nevertheless, they are usually separate topcs in solid state physics courses and there are good reasons for this. Instead of expanding on this here, I recommend two solid state texts that I like: One is the classic Ashcroft & Mermin solid state textbook. The other one is from S. Elliott: The Physics and Chemistry of Solids, which might be a somewhat more accessible initial read (depending on your formation).

When treating vibronic motions, electronic degrees do not enter explicitely, but implicitely in the form of spring constants mediating the coupling between atoms. As in mechanic models, it is assumed for simplicity that the springs are massless. Since the electrons are so much lighter than the nuclei, this is not a bad approximation at all.

The electron system is generally understood in terms of the electron band structure (especially in crystals). Band structure may be introduced as arising due to interactions between atomic electron states (tight binding picture) or as a periodically disturbed (nearly) free electron gas. In both cases, the starting point is to ghave fixed atomic positions defining the location of atomic orbitals or the periodic potential which perturbs the electron gas. That corresponds to zero temperature and neglects zero point fluctuations. [one could also say that infinite mass is attributed to the atoms in this context]

In both, the vibrational and electronic motions we have to do with mutually interacting objects. The stationary solutions of the equations of motion are generally what interests us first since any general state of motion can be thought of as a superposition of these stationary modes of motion. In the context of quantum solid states physics, These elementary excitations are called quasiparticles. (Phonons in the case of vibrations, band states [Bloch electrons] in case of the electronic degrees of freedom) They are associated with an energy and a wave vector. The wave vector is closely linked to and plays the role of the momentum vector. Interactions do respect the laws of energy and momentum (wave vector) conservation. And these elementary excitations usually provide the framework and language in which interactions (which you are interested in) are discussed and formulated.

Rahul, this would have been kind of an elementary introduction. If that sounds like an interesting path to follow, then please let me know and I shall try to carry on as time permits. I'd hate to spend my time writing down stuff which would turn out not to be useful for you.

Hi Kai Fauth,

Yes, as you pointed out I am very much interested to know more about the interactions. I read through a topic on acousto-electronic interaction (AEI). Is that the principle for an acoustic wave interacting with electrons in metals?

Dear Rahul, I would be unable to agree or disagree out of the blue. This is not my specialiization and I am not acquainted with the technical terms. All I can do is to provide you with some background from the perspective of solid state physics on which you can hopefully build and reach your own conclusions and opinions.

Getting back to your question, I'll open a little parenthesis here. I hope it will become clear lateron why I do so.

Think of two pendulums with different (eigen-) frequencies. When uncoupled, They vibrate on their own. With do energy dissipation present they both would do so forever. Their frequencies are well defined. Mathematically, both movements independently are stationary solutions of the differential equation(s) describing the physics of those pendulums.

Now think of introducing some coupling between those pendulums. Physically you may thing of a spring added to the system, connecting the two oscillating masses. Solving the now coupled set of two differential equations for the stationary solutions, we will find two different new solutions. Especially when the original eigenfrequencies were pretty different and the coupling is weak, the following shall be true:

• one of the solutions will have a frequency near one of the original frequencies, the second one shall be close to the other original one. (try to prove this for yourself)
• In both 'new' stationary modes of motion the amplitude of both pendulums is finite (nonzero). But in both cases the amplitude of the pendulum with the nearer-by eigenfrequency will be larger than the other one.
• The individual motion of each pendulum is no longer a stationary motion. If you tip one pendulum while holding the other one, the one being held at rest will acquire a finite amplitude as you release it. It's amplitude will be time-dependent (and hence its movement is said to be non-stationary).

This means that properly speaking, it does not make 100% sense to talk about the motions individually, but they're always coupled. Still, you may characterize the two stationary oscillations as "mainly pendulum 1 moving" or "mainly pendulum 2 moving", depending on wich of the pendulums has the (much) larger amplitude.

We can transfer this to the electron phonon problem: owing to the huge difference in atomic and electron masses, their stationary oscillations have very different frequencies. [At this point I speak of density oscillations. You may think about the proper (eigen-) vibrations of the atomic lattice just as oscillating periodic modulations of density. Likewise, the metal electrons may have density oscillations of their own, In case of vibrations, we speak about phonons, in case of electrons of plasmons when dealing with them quantum mechanically.]

Now, if there is electron phonon coupling, you will (hopefully) immediately understand that in a strict sense, one cannot speac of pure vibronic and pure electronic oscillations any more. There are still two types of motions, one essentially vibronic, the other essentially plasmonic in character. But in both (kinds of) modes there is a finite amplitude of motion on the other side.

In this sense, an acoustic wave in a metal shall automatically entail some electron motion, when looking at the two constituents individually. Nevertheless, both kinds of oscillations (again: in absence of dissipation mechanisms) are stationary in the classical sense, and could be termed quasiparticles in the realm of quantum mechanics. Probably, their names would still just be "phonons" and "plasmons". Only in case their individual frequencies were relatively similar, then so would be their amplitudes in the coupled modes and we migh feel like providing them with new names, highlighting the heavily composite nature of the coupled oscillations.

While this might be a (partial) answer to your original question, I have so far only spoken of stationary modes. In so far, this does not yet cover the list of items I wrote down in my initial post.

Let me (us) know what you think.

Dear Kai,

I found this discussion by accident, but wanted to feedback to you, that your explainations are very enlighting. Thanks for these analogies!

Yes, they do Dear Prof. Rahul Kishor

A classical monography was written by Prof. J. Ziman

"Electrons and Phonons: The Theory of Transport Phenomena in Solids"

J.M. Ziman, ch. 5.

I further elaborate on this post:

First, by using creation creation and annihilation operators for electrons & phonons seen as quasiparticles (second quantization approach) you may find a general hamiltonian for electron-phonon interaction (also you can use instead, the Fermi Golden rule wich states that during a collision the electron gains or loses an amount ω of energy from the crystal lattice by annihilation or generation of a phonon)

Second, one can approximate different electron-phonon couplings, which will depend on an specific approximation to the electron-phonon interaction hamiltonian:

1. the Frohlich hamiltonian coupling for free electrons.

2. the Holstein hamiltonian (several assumptions made here such as Einstein model for phonons-single frequency, also the coupling to a single branch of optical phonons)

3. The Peierls hamiltonian (more difficult to understand: vertex depends on k) Migdal's theorem and electron-phonon vertex corrections.