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Can one electron decay into an electron plus a phonon? 
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#1
May714, 12:11 PM

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In a metal, can one electron decay into one lowerenergy electron plus one phonon? (i.e., can the attached Feynman diagram occur?)
If we replace phonons by photons and consider the process in a vacuum, I guess this is prohibited because you can always boost to a frame where the incoming and outgoing electron velocities are the same. Thus, the electron has no ability to transfer energy to the photon because there are no photons with finite q but zero energy. However, in a metal, Galilean and Lorentz invariance are broken by the crystal lattice, so it seems to me that the process ought to be allowed. Is my thinking here correct? 


#2
May714, 12:33 PM

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That is not, strictly speaking, a decay.
It corresponds to an electron losing energy  presumably it gained energy someplace. I'd consider the diagram incomplete. 


#3
May714, 12:49 PM

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For example, a transition in an indirect bandgap semiconductor would require both energy and momentum change, the latter being an interaction with phonons. The electronelectron bound state in Cooper pairing for conventional superconductors requires electronphonon coupling (your Feynman diagram is onehalf of the Feynman diagram for Cooper pairing). Now, whether one can call this as a "decay", that's a different matter and, to me, seems rather odd. Zz. 


#4
May714, 12:53 PM

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Can one electron decay into an electron plus a phonon?
Nomenclature aside, what I am trying to ask is, in a metal or semiconductor, can one electron in a state at energy [itex]\omega[/itex] spontaneously scatter into one electron at a lower energy state [itex]\omega\Omega[/itex], plus one phonon at energy [itex]\Omega[/itex]. If not, then the deeper question is, what are the lowest order Feynman diagrams that can explain the fact that the electrons in any real metal have a finite lifetime? 


#5
May714, 01:13 PM

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#6
May714, 10:13 PM

P: 142

As ZapperZ said, this is precisely what the electronphonon coupling diagrams look like:
https://www.google.com/search?tbm=is...phonon+diagram Feynman diagrams are indeed an appropriate way to describe this interaction (in a very rigorous way!), see e.g. Mahan's Many Particle Physics book. 


#7
May814, 01:28 AM

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Electron phonon scattering is in deed one of the principal mechanisms behind electronic resistivity, however, only at finite temperatures. The resistivity should increase with temperature like ##T^5##. This is called Bloch's ##T^5## law.
At very low temperatures the probability for an electron near the Fermi surface to emit a photon goes to zero due to phase space arguments. This is usually discussed within the framework of Landau's Fermi liquid theory. 


#8
May814, 02:23 AM

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So, suppose you have a metal that is not superconducting at any temperature, like copper or silver. It also is a perfect crystal with no impurities or defects of any kind.
What would be the allowed mechanisms for the remaining resistance at low temperature? Does copper at 0,1 K actually and observably have resistance 100 000 times less than the same copper at 1,0 K? 


#9
May814, 02:34 AM

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Finally the low temperature resistance of very pure metal samples is limited only by scattering from crystal surfaces.



#10
May814, 02:37 AM

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#11
May814, 03:16 AM

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#12
May814, 11:12 AM

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Nevertheless, the comments I see above regarding transport in the limit of zero temperature are very interesting. Snorkack, in practice I think that impurity scattering is often impossible to eliminate, so I would guess that the resistance of copper in the lowtemperature limit approaches a finite value. Actually, however, if you were able to fashion an absolutely perfect crystal and eliminate all sources of scattering (from other electrons, as well as phonons and impurities), then Ohm’s law would break down, and the electrons would oscillate back and forth in time under the influence of an applied electric field. The resistance of the material would increase toward infinity rather than dropping to zero. Such Bloch oscillations have actually been observed in certain very clean systems. 


#13
May814, 11:36 AM

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Most , if not all, resistivity measurement on metals will show a residual resistivity approaching 0K. I remember reading a couple of physical Review papers on this that showed the T^2 dependence at very low temp due to electronelectron scattering as expected from Fermi Liquid Theory. Wish I can find those papers. They all showed a finite, nonzero resistivity when you extrapolate that to T=0K.
Zz. 


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