@@ -222,7 +222,7 @@ The Sommerfeld model provides a good description of free electrons in alkali met
In the lecture, it has been explained that the density of states (DOS) of the free electron model is proportional to $1/\sqrt{\epsilon}$ in 1D, constant in 2D and proportional to $\sqrt{\epsilon}$ in 3D. In this exercise, we are going to derive the DOS of the free electron model for an arbitrary number of dimensions.
Suppose we have an $n$-dimensional hypercube with length $L$ for each side and contains free electrons.
1. Assuming periodic boundary conditions, what is the distance between nearest-neighbour points in $\mathbf{k}$-space? What is the density of points in n-dimensional $\mathbf{k}$-space?
1. Assuming periodic boundary conditions, what is the distance between nearest-neighbour points in $\mathbf{k}$-space? What is the density of $\mathbf{k}$-points in n-dimensional $\mathbf{k}$-space?
2. The number of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ is given by $g(k)dk$. Using the answer for (1), find $g(k)$ for 1D, 2D and 3D.
3. Now show that $g(k)$ for $n$ dimensions is given by
$$g(k) = \frac{1}{\Gamma(n/2)} \left( \frac{L }{ \sqrt{\pi}} \right)^n \left( \frac{k}{2} \right)^{n-1},$$ where $\Gamma(z)$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
