User:SunderB/Draft:Modelling Solids/Drude Model

Assumptions

• 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:

${\boldsymbol {F}}=-e({\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}})$

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

$${\begin{aligned}{\boldsymbol {p}}(t+dt)&={\frac {dt}{\tau }}\times 0+(1-{\frac {dt}{\tau }})({\boldsymbol {p}}(t)+{\boldsymbol {F}}dt)\\\therefore {\boldsymbol {p}}(t+dt)-{\boldsymbol {p}}(t)&=-{\frac {{\boldsymbol {p}}(t)}{\tau }}dt+{\boldsymbol {F}}dt\\{\frac {{\boldsymbol {p}}(t+dt)-{\boldsymbol {p}}(t)}{dt}}&={\boldsymbol {F}}-{\frac {{\boldsymbol {p}}(t)}{\tau }}\\\Rightarrow {\frac {d{\boldsymbol {p}}}{dt}}&=-e({\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}})-{\frac {{\boldsymbol {p}}(t)}{\tau }}\end{aligned}}$$

Predicting Known Relations

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