Having had a somewhat haphazard education, I hadn't heard of the Friedel sum rule until I needed it for my research. As usual, Google yielded nothing remotely pedagogical, just research articles, and neither Ashcroft nor AGD had anything to say about the rule. It turns out, however, that Ziman's book Principles of Solid-State Physics has a nice discussion. It also turns out that, if I'm not being a complete idiot, this rule is a case of a general scattering theory result -- again underrepresented in textbooks -- known as Levinson's theorem.
Levinson's theorem relates the zero-energy scattering phase shift of a short-range potential to the number of bound states in the potential. The Friedel sum rule relates the number of states below the Fermi energy to the scattering phase shift at the Fermi energy. The relation in both cases is that the zero-energy (Fermi-energy) phase shift is pi times the number of states. The idea behind Levinson's theorem goes something like this -- as you tune the energy of your incoming scatterer down from infinity (where there's no phase shift), the scattering phase shift goes through pi iff you hit a scattering resonance associated with a bound state in your potential. Add up all the pi's and you get the number of bound states.
The usual argument for the Friedel sum rule goes like this -- put the system in a box, the scattering phase shift feeds into the quantization condition at the edges of the box, and depending on what the phase shifts are you can change the number of states that are below the Fermi energy. I would assume, though, that you could alternatively prove it by putting the system in a box that extends up to the Fermi energy, then using Levinson's theorem to count the bound states.
I should mention that the Friedel sum rule is true for Fermi liquids as well as for the free Fermi gas, which is not terribly surprising as the states of the two are in 1-to-1 correspondence. The equivalence is proved in the following paper:
Friedel Sum Rule for a System of Interacting Electrons
J.S. Langer and V. Ambegaokar
Phys. Rev. 121, 1090 (1961)
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Just wanted to let you know I appreciate your summary of a difficult topic not covered in any textbook I could find. Its frustrating to bump into those theorems/results that are important yet never distilled from the original literature into a form easily readable for students. Perhaps grad student blogs can fill such a hole.
ReplyDeleteThanks Sarang!
For your enjoyment, you're now at position 3 in a Google search on "Friedel sum rule"... :)
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