User:SunderB/Draft:Modelling Solids/Drude Model

Assumptions
$$\boldsymbol{F}=-e(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B})$$
 * Electrons act as classical charged particles
 * Collisions are random and elastic
 * Probability of scattering during a short period of time dt: $$\frac{\tau}{dt}$$
 * After each scattering event, the mean momentum is zero
 * Between scattering events, electrons experience the Lorentz force:

Ignored details:

 * Electrons may collide or interact with nuclei
 * Obstacles are just lumped into scattering term
 * Charge is ignored during collisions
 * Electrons are quantum particles, so elastic collisions are inaccurate!

Mathematical Description
Consider the change in momentum over dt.

What can happen during dt?


 * Scatter: $$P=\frac{dt}{\tau}$$, avg. momentum = 0
 * Not-scatter: $$P=1-\frac{dt}{\tau}$$, Lorentz force

$$\begin{align} \boldsymbol{p}(t+dt) &= \frac{dt}{\tau}\times0+(1-\frac{dt}{\tau})(\boldsymbol{p}(t)+\boldsymbol{F}dt)\\ \therefore \boldsymbol{p}(t+dt) - \boldsymbol{p}(t) &= -\frac{\boldsymbol{p}(t)}{\tau}dt + \boldsymbol{F}dt\\ \frac{\boldsymbol{p}(t+dt) - \boldsymbol{p}(t)}{dt} &= \boldsymbol{F} - \frac{\boldsymbol{p}(t)}{\tau}\\ \Rarr \frac{d\boldsymbol{p}}{dt}&=-e(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B})-\frac{\boldsymbol{p}(t)}{\tau} \end{align}$$

Predicting Known Relations
For the following scenarios, assume a constant E-field.