How can one intuitively understand formulas of the form $χ\sim\sum_{\bf k}{f_{\bf k}-f_{\bf k+q}\over ε_{\bf k+q}-ε_{\bf k}}$? When calculating various susceptibilities, we get below formula again and again.
$$\chi( {\bf q},0) \sim \sum\limits_{\bf{k}} {\frac{{{f_{\bf{k}}} - {f_{{\bf{k}} + {\bf{q}}}}}}{{{\varepsilon _{{\bf{k}} + {\bf{q}}}} - {\varepsilon _{\bf{k}}}}}}  \sim {N_0}$$
How can one intuitively understand this formula? Why does this mean susceptibility? What does $N_0$ mean? (the lecture didn't say).
 A: When you take the limit $q \rightarrow 0$, your formula becomes
$$\chi( 0,0) \sim  - \sum\limits_{\bf{k}} \frac{d f(\varepsilon_k)}{d \varepsilon_k}$$
where $f$ is the occupancy number of the electronic states. This susceptibility represents the response of the system to some external perturbation. Usually, this perturbation tends to displace electrons to low-energy nearby states.
If at some wavevector $k$ the occupancy function $f$ is very steep, you will have a lot of accessible states very close to the state of energy $\varepsilon_k$ ; and $\frac{d f(\varepsilon_k)}{d \varepsilon_k}$ will be very large. This makes the system very "susceptible" to an external perturbation, and hence $\chi$ very large.
A: The susceptibility quantifies the response of a material to an external electric field due to the redistribution of electronic charge within the material. The formula you give for the charge in a particular electronic band (in practice the total susceptibility comes from the contributions from all bands). It is known as the Lindhard model and is derived from first order quantum perturbation theory. If you are familiar with perturbation theory, the form of the Lindhard formula may look familiar. 
Both k and q are wavevectors (so proportional to electron momentum), so the sum is over all electrons (in each k-state) in the band. The N0 is the total number of electrons in the band. The external potential may be described in terms of its Fourier components (labelled by q), so chi(q) is the response to the qth component of the potential. 
Intuitively, consider that as the electronic charge in the material shifts, the electrons change state and therefore momentum. A change in the electronic charge distribution by q corresponds directly to the q component of the potential. One might think of this in terms of scattering. The qth component of the field causes the electrons to scatter by a change in momentum q. 
