rework drude lecture

src/3_drude_model.md

@@ -158,7 +158,7 @@ We have now derived the necessary equation, the rest is merely applying it.
 
 ### Consequences of the Drude model
 
-For convenience from now on we will omit the average signs, and write $\mathbf{v}$ instead of $⟨\mathbf{v}⟩$
+For convenience from now on we will omit the average signs, and write $\mathbf{v}$ instead of $⟨\mathbf{v}⟩$.
 For a warm-up consider $\mathbf{B} = 0$, and a constant $\mathbf{E}$.
 After we wait long enough, we expect the average electron velocity to become constant, $d\mathbf{v}/dt = 0$, and we immediately get
 $$
@@ -206,7 +206,7 @@ We now consider a conductive wire in a magnetic field $\mathbf{B}$ $⇒$ electro
 $\mathbf{E}_\mathrm{H}$ is the electric field caused by the Lorentz force, leading to a _Hall voltage_ in the direction perpendicular to $\mathbf{B}$ and $\mathbf{j}$.
 
 Once again, we consider the steady state, $d\mathbf{v}/dt = 0$.
-After substituting $\mathbf{v} = \mathbf{v}/ne$, we arrive to
+After substituting $\mathbf{v} = \mathbf{j}/ne$, we arrive to
 $$
 \mathbf{E}=\frac{m}{ne^2τ}\mathbf{j} + \frac{1}{ne}\mathbf{j}\times\mathbf{B}.
 $$
@@ -218,7 +218,7 @@ $$
 E_a = ∑_b ρ_{ab} j_b,
 $$
 with $a, b ∈ \{x, y, z\}$, and $ρ$ the *resistivity matrix*.
-Its diagonal elements are $ρ_{xx}=ρ_{yy}=ρ_{zz}=\frac{m}{ne^2τ}$—the same as without magnetic field.
+Its diagonal elements are $ρ_{xx}=ρ_{yy}=ρ_{zz}=m/ne^2τ$—the same as without magnetic field.
 The only nonzero off-diagonal elements when $\mathbf{B}$ points in the $z$-direction are
 $$
 ρ_{xy}=-ρ_{yx}=\frac{B}{ne}\equiv -R_\mathrm{H}B,
@@ -226,7 +226,7 @@ $$
 where $R_H=-1/ne$ is the *Hall coefficient*.
 So by measuring the Hall voltage and knowing the electron charge, we can determine the density of free electrons in a material.
 
-While most materials have $R_\mathrm{H}<0$, interestingly some materials are found to have $R_\mathrm{H}>0$. This would imply that the charge of the carriers is positive. We will see later how to interpret this.
+While most materials have $R_\mathrm{H}<0$, interestingly some materials are found to have $R_\mathrm{H}>0$. This would imply that the charge of the carriers is positive. We will see later in the course how to interpret this.
 
 ## Conclusions
 
@@ -238,7 +238,8 @@ While most materials have $R_\mathrm{H}<0$, interestingly some materials are fou
 ### Warm-up questions
 
 1. How does the resistance of a purely 2D material depend on its size?
-2. Check that the units of mobility and the Hall coefficient are correct. (As you should always do!)
+2. Check that the units of mobility and the Hall coefficient are correct.  
+   (As you should always do!)
 3. Explain why the scattering rates due to different types of scattering events add up.
 
 ### Exercise 1: Extracting quantities from basic Hall measurements