Density of states?

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Density of states??

According to C. Kittel the density of states is the "number of orbitals per unit energy range". Alright, that's fine, but what exactly does this mean? I can understand the calculations, finding the totalt number of states by considering the fermi sphere and the volume of a single state in k-space, and then differentiating this expression with respect to the energy. But if the DOS is the number of orbitals (states) per unit energy, what would for example happen if i multiply this DOS by some energy? What would i get? Apparently some number of orbitals, but what would this number tell me?

Please enligthen me, I am sort of confused on this topic. It would certainly be nice if you could give me some examples of applications of DOS as well.

It's really all in the definition. The question you would often like to answer is "how many states are there in a given energy range?" The answer to this question is given by the density of states which is literally just the derivative of the number of states with respect to energy. Therefore, if $$N(E)$$ is the density of states, the answer to the question "how many states are there between $$E_1$$ and $$E_2$$ is simply $$\int^{E_2}_{E_1} N(E) dE$$. An important approximation to this formula obtains when the energy range $$\Delta E = E_2 - E_1$$ is small compared to the scale of variations in $$N(E)$$. In such a situation the number of states is given simply by $$N(E^*) \Delta E$$ where $$E^*$$ something between $$E_1$$ and $$E_2$$. This is actually just the mean value theorem, but you don't usually know what $$E^*$$ is, so you often just choose $$E^* = E_1$$ say, and the error in your approximation is second order in $$\Delta E$$.
Example: At low temperatures, meaning $$kT << E_F$$, the number of "active" electrons in a free Fermi gas is simply the number of electrons in a thin shell of thickness $$kT$$ at the Fermi surface. This number is $$N(E_F) k T$$, per the approximation above. Each of these active electrons carries a thermal energy of order $$kT$$, thus the thermal energy of a free fermi gas at low temperatures is given by $$N(E_F) (kT)^2$$ up to numerical factors of order one. This predicts a specific heat $$C_V = \frac{\partial E}{\partial T} \sim T$$, a result which is observed in experiments.