October 13: Fermi Energy
Meaning of Equation:
Fermi Energy represents the energy of the highest occupied energy level in a system of non-interacting fermions (such as electrons) at absolute zero of temperature. This quantity, thus, solely depends on the number of particles per unit volume in the system (i.e. the number density).
After equations on quantum mechanics, electrostatics and relativity, I now move on to one of the most important branches of physics, the bridge between the classical and quantum world: statistical mechanics. No other equation is as important as the Fermi energy.
Before moving on to the meaning of this mathematical equation above, I will first define what are fermions. All the particles in nature can be classified into two types: Bosons (named after the Indian physicist, S.N. Bose) and Fermions (named after the Italian physicist, E.Fermi). In quantum mechanics, every particle is associated with what is known as the spin.
This spin can be half-integral (1/2, 3/2 and so on including negative values) or integral (1,2,3,4…). Particles with half-integral spin are called fermions (eg. electrons, protons) and particles with integral spin are called bosons (eg. photons). So far so good.Now without any mathematical details, there is one last piece of information I will add in this. Fermions obey Pauli’s exclusion principle and bosons do not.
So in a given state, no more than one particle can have the same set of quantum numbers thereby excluding the possibility of more than one particle to occupy the same quantum state. When I say the quantum state, here I mean the energy state. In the case of electrons, two of them occupy the same quantum state because one will have spin 1/2 and others will have spin -1/2. In the case of bosons, there’s no such restriction. All of them can come down and occupy the same quantum state.
That’s all. We have developed the concept we need for understanding the meaning of Fermi energy. Imagine you have vacant shelves placed above each other with some gap. Your task is to put red and green colored balls on these shelves. The only rule: No two balls of the same color can be on the same shelf. You start arranging the balls and by following the rule you have a final configuration in which each energy level is occupied. The balls on the top have more energy than the balls in the first shell at the bottom.
Now suppose you want to add more balls in this system. Even if you try to put them in the lowest energy states, you cannot, because those levels are already occupied. You have to follow the rule. The balls that have occupied the lower energy levels will say, “No you cannot even forcefully occupy this state. If you try to, you’ll have to face a counter outward pressure from us.
You have to follow the rule”. With no option left, the remaining balls will occupy the higher energy shelves. This is the concept! The energy difference between the highest and the lowest occupied energy levels is known as the Fermi energy.
Replace the balls by electrons, shelves by energy levels and the rule of the game by Pauli’s exclusion principle and you’ll enter the world of Physics. So basically, Fermi energy is the kinetic energy of the electron in the highest state.
I will describe one such application of this system in astrophysics: the stability of white dwarf stars. In white dwarfs, all the electrons have already occupied the lowest quantum states that were available. The gravitational collapse is halted by this outward pressure by electrons. Gravity tries to crush the star and put all the electrons in the lowest quantum state but due to Pauli’s exclusion principle, this can’t happen and an outward pressure, known as the degeneracy pressure saves it.
You see, statistical mechanics is so important to study astrophysics. It is one such beautiful branch of physics that elegantly connects quantum and classical mechanics. It has applications in most of the solid state physics, plasma physics and astrophysics. So if you want to become an astrophysicist, master statistical mechanics and the road will be smooth ahead.