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The density of states function, $$DOS(E)$$, is defined as the number of electronic states per unit energy for electron energies between $$E$$ and $$E + dE$$. It is represented as

$$DOS(E)=\frac{dN}{dE}$$ where $$N= N(E)$$ is number of states occupied at energy $$E$$. The energy $$E = E_F$$ is the highest occupied energy level, a.k.a. Fermi Energy (citation for Kittel).

The derivation of $$DOS(E)$$ for the particle in 3D box model at 0K can be used as an approximation for electrons in the metal. The energy of a state at $$n=(n_x,n_y,n_z)$$ is given by $$E =\frac{h^2}{8m_eL^2}(n_x^2+n_y^2+n_z^2)$$, where $$h$$ is Planck’s constant, $$m_e$$ is the mass of electron, and $$L$$ is the box length. The number of states with $$E \leq E_F$$is given by twice the volume of a positive octant of the sphere with radius $$n$$:

$$N(E) = 2*\frac{1}{8}*\frac{4}{3} \pi n^3 = \frac{8\pi}{3} (\frac{2m_eE}{h^2})^{3/2} L^3= \frac{8\pi}{3}(\frac{2m_eE}{h^2})^{3/2}V$$,

where $$V=L^3$$ volume of the box. Using the definition of density of states:

$$DOS(E) = \frac{dN}{dE}=4(\frac{2m_e}{h^2})^{3/2}E^{1/2}V=\frac{V}{2\pi^2}(\frac{2m_e}{\hbar} )^{3/2}E^{1/2}V$$

The function can also be defined as $$g(E) = \frac{DOS(E)}{V}$$, or density of states per unit volume. Thus, the $$g(E)$$ at 0K is proportional to $$E^{1/2}$$ (see Fig.). At finite temperatures, $$g(E)$$ is modulated by Fermi-Dirac distribution.