@@ -223,7 +223,7 @@ In the lecture, it has been explained that the density of states (DOS) of the fr
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?
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.
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).
**Hint:** you will need the area of an $n$-dimensional sphere and this can be found on [Wikipedia](https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area)(blue box on the right).
