The density of states, denoted by $g(E)$, describes the number of available electron states per unit energy interval per unit volume (or area in 2D). For a non-relativistic electron gas in two dimensions (2D), the number of states $N(E)$ with energy less than or equal to $E$ is proportional to the energy itself.

Deriving Density of States in 2D

The energy of a non-relativistic electron is given by $E = \frac{\hbar^2 k^2}{2m}$, where $k$ is the wavevector magnitude, $\hbar$ is the reduced Planck constant, and $m$ is the electron mass.

In 2D k-space, the number of states within a radius $k$ is proportional to the area of the circle, $\pi k^2$. Specifically, the number of states per unit area within a k-magnitude $k$ is $N_{2D}(k) = \frac{\pi k^2}{(2\pi)^2} = \frac{k^2}{4}$.

Substituting $k^2 = \frac{2mE}{\hbar^2}$ into the expression for $N_{2D}(k)$:

$ N_{2D}(E) = \frac{1}{4} \left( \frac{2mE}{\hbar^2} \right) = \frac{m}{2\hbar^2} E $

The density of states $g(E)$ is the derivative of the number of states $N(E)$ with respect to energy $E$:

$ g(E) = \frac{dN_{2D}(E)}{dE} = \frac{d}{dE} \left( \frac{m}{2\hbar^2} E \right) = \frac{m}{2\hbar^2} $

Determining the Exponent n

The result shows that for a 2D non-relativistic electron gas, the density of states $g(E)$ is a constant value, independent of energy $E$.

The question provides the energy dependence as $g(E) = CE^n$, where $C$ is a constant.

Comparing our derived result $g(E) = \frac{m}{2\hbar^2}$ (a constant) with the given form $g(E) = CE^n$, we can equate them:

$ CE^n = \frac{m}{2\hbar^2} $

For this equation to hold true for all energies $E$, the exponent $n$ must be zero ($E^0 = 1$).

Therefore, $n=0$. The density of states is constant: $g(E) = C$, where $C = \frac{m}{2\hbar^2}$.