e-e scattering rate in normal fermi liquid and in graphene In Ashcroft/Mermin's solid state physics, in equation (17.64) they argued that:

We expect from lowest-order perturbation theory (Born approximation)
  that $\tau$ will depend on the electron-electron interaction through
  the square of the Fourier transform of the interaction potential.

$\tau$ is the e-e scattering time, which means the average time for a pair of particles to scatter. The inverse of it, $1/\tau$ is the scattering rate.
Then they use the Tomas-Fermi screening potential so that:
$$
\frac{1}{\tau} \sim (k_BT)^2(\frac{4\pi e^2}{k_0^2})^2
$$
After the dimensional analysis, they argued:
$$
\frac{1}{\tau}\sim A(k_BT)^2\frac{\hbar}{E_F}
$$
Where $A$ is dimensionless quantities and $E_F$ is Fermi energy. Then they argued that A is of order of a power or two of ten which I can't understand, why should $A$ be this order?
Another question very related to this is that in graphene at half filling. $E_F$ is zero which cause the the scattering rate to be zero as it seems. However in graphene the independent electron approximation is rather good. So can anyone give a physics picture of this?
 A: I'll comment on the second issue that $E_F$ should always be greater than zero, which leads to a finite scattering rate. However, that's beside the point because in an ideal Fermi liquid at $T\rightarrow 0, |E_F-E|\rightarrow0$, the scattering rate becomes zero, which leads to an infinite lifetime. If in graphene, as you say, the independent particle picture is good then the scattering should be low (or zero in the limit). Whenever the scattering rate is non-zero it means that the single-particle states are not the exact eigenstates of the system and thus the independent particle picture is not entirely valid. The particles are scattering in and out of the single-particle levels, which are only approximately stationary.
I agree that the value of the $A$ factor seems to be taken from thin air in the book. If I follow their insertion of $m/\hbar^7$ into Eq. 17.65, I get $A=\pi^4/2\approx49$. Perhaps the range $A=1\ldots100$ is given as a crude measure of uncertainty in their simplistic approach. They only need it in the next paragraph to make a rough estimation on the importance of e-e scattering.
