From c7d5b6d9291f781d8478d26f8abf9293f5118f4b Mon Sep 17 00:00:00 2001
From: "T. van der Sar" <t.vandersar@tudelft.nl>
Date: Fri, 25 Jan 2019 22:15:55 +0000
Subject: [PATCH] Update 4_sommerfeld_model.md - fix

---
 src/4_sommerfeld_model.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/4_sommerfeld_model.md b/src/4_sommerfeld_model.md
index 0139a981..c76f8a56 100644
--- a/src/4_sommerfeld_model.md
+++ b/src/4_sommerfeld_model.md
@@ -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).
-- 
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