- #1

mcas

- 24

- 5

- Homework Statement
- Starting with F-C distrubution for ##T>0##

$$f(\epsilon_\vec{k})=(e^{\frac{(\epsilon_\vec{k} - \mu)}{kT}}+1)^{-1}$$

derive a distrubution at limit of ##T->0## when ##\mu(T)-> \epsilon_F##

- Relevant Equations
- ##f(\epsilon_\vec{k})=(e^{\frac{(\epsilon_\vec{k} - \mu)}{kT}}+1)^(-1)##

##\mu(T=0)=\epsilon_F##

The limit itself is pretty easy to calculate

##lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}##

But I'm very confused about changing ##\epsilon_\vec{k}## to ##\epsilon_F##. Why do we do this?

##lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}##

But I'm very confused about changing ##\epsilon_\vec{k}## to ##\epsilon_F##. Why do we do this?