Fermi-Liquid - amount of electrons available for the interaction

  • #1
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Main Question or Discussion Point

I have attached the pages in Kittel's book (pages 417-420) regarding my question. My question is simply based off of the second to last photo, where e_f = 5*10^4 K and e_1 = 1K.

e_2<e_f and |e_2|<e_1. So how can (e_1/e_f)^2 be less than 1? The energy of the free flowing electron is assumed to be greater than e_f.
 

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Answers and Replies

  • #2
Dr_Nate
Science Advisor
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If you read the bottom of the first page, you will see that they are measuring ##\epsilon_1## and ##\epsilon_2## from ##\epsilon_F##, which they set to zero. Then on the third page they make a shell argument and use the actual value of ##\epsilon_F##.
 
  • #3
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Either way e_1 > e_f, though.
 
  • #4
Dr_Nate
Science Advisor
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Either way e_1 > e_f, though.
I can understand your difficulty, but ##\epsilon_1## is still a small quantity.

It would be more clear if he first introduced variables that are measured from the lowest valence electron level, such as ##\epsilon'_1##, ##\epsilon'_2##, and ##\epsilon_F##, which are all large quantities. Then he could introduce the electron energies measured from the Fermi energy: ##\epsilon_1=\epsilon'_1 - \epsilon_F##, and ##\epsilon_2 =\epsilon'_2 -\epsilon_F,##. That way the shell argument would still use the ratio ##\frac{\epsilon_1}{\epsilon_F}##, which you can see also equals ##\frac{\epsilon'_1-\epsilon_F}{\epsilon_F}##.
 
  • #5
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Ah, got it. So e_1 is just the small difference, and not the complete distance from the origin, right?

Another question, what does this fraction actually tell us? I would assume it would give us the amount of electrons that have enough energy to interact with e_1 to get then e_3 and e_4 outside the fermi sphere, but it is a number much smaller than one. Or is it simply the percentage of electrons that could allow for e_3 and e_4 to be outside of the sphere?
 
  • #6
Dr_Nate
Science Advisor
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Ah, got it. So e_1 is just the small difference, and not the complete distance from the origin, right?
Yes, that is correct, but with a small caveat. This is in reciprocal space and thus the points are vectors, and we know energy is a scalar. For this model, the mapping is $$E=\frac{\hbar^2 k^2}{2m},$$ where ##k## is the radial distance from the origin.

Another question, what does this fraction actually tell us? I would assume it would give us the amount of electrons that have enough energy to interact with e_1 to get then e_3 and e_4 outside the fermi sphere, but it is a number much smaller than one. Or is it simply the percentage of electrons that could allow for e_3 and e_4 to be outside of the sphere?
I don't immediately see the difference between your two options. Upon a first reading, they both seem equivalent (except for a conversion from a fraction to a percentage) and basically correct . The final term ##(\frac{\epsilon_1}{\epsilon_F})^2## gives us an idea of the fraction of the total valence electrons that an electron of energy ##\epsilon_1## can interact with. Thus showing us why there isn't much electron-electron scattering in crystalline conductors.
 

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