Themes > Science > Physics > Solid State Physics > Electrons in Metals > Fermi-Dirac Distribution

After dealing with bosons, fermions will seem relatively simple. Fermions are particles with half-integer spin. By the Pauli exclusion principle, only one fermion can occupy a given state at a time. That means that the occupation number for state $ s$ can be only $ n_s = 0$ or $ n_s = 1$. Furthermore, conservation of particle number requires that $ \sum_s n_s = N$. Following our discussion of boson statistics, we introduce the fermions' chemical potential $ \mu$ and write down their partition function
$\displaystyle Z$ $\displaystyle = \sum_{n_1 = 0}^1 e^{-\beta n_1 (\epsilon_1 - \mu)} \; \sum_{n_2 = 0}^1 e^{-\beta n_2 (\epsilon_2 - \mu)} \times \cdots$ (45)
  $\displaystyle = \prod_s \left( 1 + e^{-\beta(\epsilon_s - \mu)} \right).$ (46)

The mean number of particles in state $ s$ is then
$\displaystyle \bar n_s$ $\displaystyle = - \frac{1}{\beta} \frac{\partial \ln Z}{\partial \epsilon_s}$ (47)
  $\displaystyle = - \frac{1}{\beta} \frac{\partial}{\partial \epsilon_s} \left[ \ln \left(1 + e^{-\beta(\epsilon_s - \mu)}\right) + \cdots \right]$ (48)
  $\displaystyle = \frac{e^{-\beta(\epsilon_s - \mu)}}{1 + e^{-\beta(\epsilon_s - \mu)}}$ (49)

Finally we have an expression for the mean number of fermions in state $ s$ with energy $ \epsilon_s$:
$\displaystyle \shadowbox{ $ \displaystyle \bar n_s = \frac{1}{e^{\beta(\epsilon_s - \mu)} + 1}. $ }$ (50)

This last result is known as the Fermi-Dirac Distribution and is extremely useful for describing the behavior or electrons in solids. Once again, dropping the $ s$ subscript yields an expression for the distribution function $ n(\epsilon)$.

The results described in this document provide the foundation for the rest of this course. Many-body systems are capable of an extraordinary range of surprising and counterintuitive behaviors. A great many of these can be explained using just a few ideas from statistical mechanics.


Information provided by: http://griergroup.uchicago.edu