Fermi-Liquid - amount of electrons available for the interaction

[annaphys]

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.

 

[Dr_Nate]

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 $\epsilon_F$.

 

[annaphys]

Either way e_1 > e_f, though.

 

[Dr_Nate]

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 $\frac{\epsilon'_1-\epsilon_F}{\epsilon_F}$.

 

[annaphys]

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?

 

[Dr_Nate]

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.