# Fermi-Dirac distribution at T->0 and \mu->\epsilon_0

• mcas
In summary, the Fermi-Dirac distribution describes the probability of particle occupancy in a system at thermal equilibrium. At T=0 and &mu;=&epsilon;<sub>0</sub>, it becomes a step function with all particles occupying energy states below &epsilon;<sub>0</sub>. This is important because it determines the ground state energy and provides information about electronic structure and properties of materials. As temperature increases, the distribution becomes less steep and more continuous. As the chemical potential approaches the energy of the highest occupied state, the Fermi energy becomes equal to that energy. As the chemical potential approaches the energy of the lowest unoccupied state, the distribution becomes steeper and more closely resembles a step function.
mcas
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?

Depending on ##\epsilon_k## with comparison to Fermi energy as T ##\rightarrow## 0,
For ##\epsilon_k > \epsilon_f ## ##f \rightarrow ?##
For ##\epsilon_k < \epsilon_f ## ##f \rightarrow ?##
and
For ##\epsilon_k = \epsilon_f ## ##f = 1/2## for any temperature ##T \neq 0##.

Last edited:
mcas and DrClaude

## What is the Fermi-Dirac distribution?

The Fermi-Dirac distribution is a mathematical function that describes the probability distribution of fermions (particles with half-integer spin) in a system at thermal equilibrium. It takes into account the effects of quantum mechanics, such as the Pauli exclusion principle, on the distribution of particles.

## What happens to the Fermi-Dirac distribution at T=0?

At T=0 (absolute zero temperature), the Fermi-Dirac distribution becomes a step function, with all energy states below the Fermi energy being occupied and all energy states above the Fermi energy being unoccupied. This is known as the Fermi-Dirac distribution at zero temperature.

## What is the significance of \mu->\epsilon_0 in the Fermi-Dirac distribution?

The parameter \mu (mu) in the Fermi-Dirac distribution represents the chemical potential, which is a measure of the energy required to add or remove a particle from the system. When \mu is equal to the energy level \epsilon_0, the Fermi-Dirac distribution becomes a step function at T=0, as all energy states below \epsilon_0 are occupied and all energy states above \epsilon_0 are unoccupied.

## How does the Fermi-Dirac distribution differ from the Maxwell-Boltzmann distribution?

The Fermi-Dirac distribution takes into account the effects of quantum mechanics, such as the Pauli exclusion principle, on the distribution of particles. This makes it more accurate for describing the behavior of fermions at low temperatures. The Maxwell-Boltzmann distribution, on the other hand, is based on classical statistical mechanics and is more accurate for describing the behavior of particles with integer spin at high temperatures.

## What are some real-world applications of the Fermi-Dirac distribution?

The Fermi-Dirac distribution is used in many fields of physics, including condensed matter physics, nuclear physics, and astrophysics. It is used to describe the behavior of electrons in metals, the distribution of particles in a neutron star, and the energy levels of atoms in a gas. It is also used in the design of electronic devices, such as transistors and diodes.

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