The Fermi-Dirac Distribution In Physics

A number of people sitting in a room or a number of students sit in a class follow a particular seating arrangement. Similar to this, assume that we put some particles in a closed container. What do we now expect? We assume that those particles will arrange themselves in a manner to reside easily inside the box. Let us call this arrangement of particles as "Distribution". In Today's article, I shall explain a famous distribution in quantum mechanics known as Fermi-Dirac Distribution.

Distribution Function

The distribution function describes the probability with which one can expect particles to occupy the available energy levels in a given system. A distribution function 'f (E)' refers to the probability of a particle to exist in an energy state E.

In nature, there are three distinct forms of distribution functions. Now, the form of distribution function or the arrangement of particles in a given system depends upon various factors:

  • Properties of particles: The choice of distribution depends upon the kind of particles. This refers to whether the particles are Bosons or Fermions.
  • Indistinguishability: This means to identify whether the particles in a system are distinguishable of not. In other words, we choose a particular form of distribution for that system depending upon whether in a system of particles each particle can be separately identified.
Standard Model of Particle Physics
The Standard Model of Particle Physics

Fermi Dirac Distribution

In 1926, physicists Enrico Fermi and Paul Dirac developed the Fermi-Dirac distribution. The Fermi-Dirac (F-D) distribution is a quantum distribution that explains the behavior of a collection of fermions in thermodynamic equilibrium. Fermions are particles having a half-integer spin such as electrons, protons, neutrons, and neutrinos. These particles follow Pauli's Exclusion principle. This principle forbids any two particles to occupy the same energy level. We can say that the Fermi-Dirac distribution is applicable to indistinguishable particles having half-integral spin such that no two particles may stay in the same energy level.

The expression for Fermi-Dirac distribution function that depicts the average number of particles in an energy state 'E' is given as:

Fermi-Dirac distribution
Fermi-Dirac distribution

For low temperatures, the probability of energy states below the Fermi energy (E_f) is 1 and of those levels above the Fermi energy, the probability is zero. This means that at absolute zero, fermions fill up all available energy states below the Fermi energy with only one particle occupying each level. At high temperatures, some Fermions may occupy energy states above the Fermi level.

Applications of Fermi-Dirac distribution

  • Electron gas model in metals: The properties of metals are adequately described in terms of the electron gas model. This model views metals as a collection of electrons contained in a three-dimensional box in the presence of a neutralizing positive background. In such metals, the Fermi-Dirac distribution reveals the actual physics behind the conduction phenomenon.
  • In astrophysics: Electrons and neutrons are Fermions and they obey the Fermi-Dirac distribution. This very fact is the reason why white dwarf stars and the neutron stars do not collapse into a black hole. Consider the case of white dwarfs. When a sun like mid sized star runs out of nuclear fusion reactions, the gravity tries to crush it. The electrons will start occupying the lowest energy levels. Due to Fermi Dirac Distribution and the Pauli's principle, not more than two electrons can occupy the same quantum state. They will apply an opposing force against gravity. This is known as the electron degeneracy pressure which keeps the star from collapsing. Thus a white dwarf star, which has no means of nuclear fusion, stays stable because of the electron degeneracy pressure. Analogous to this is the neutron degeneracy pressure which keeps a neutron star from collapsing under its own gravity.

Author's message

Statistical mechanics has a lot of applications in studying the nature of systems at the quantum levels. It is an elegant branch of physics that acts as a bridge between classical and quantum physics. If you wanna be an Astrophysicist and study the stars, do master statistical mechanics.

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