Fermi energy

Fermi energy is a fundamental concept in quantum mechanics. It describes a specific energy difference in a system of fermions. Fermions are a group of particles that include electrons, protons, and neutrons. This concept is vital for understanding how matter behaves at very low temperatures. It helps scientists study metals, superconductors, and even the stars in deep space.

To understand this, we must look at the Pauli exclusion principle. This principle states that two fermions cannot occupy the same quantum state at once. In a system of non-interacting fermions, we can analyze these as stationary states. These states are usually separated by different levels of energy. To find the ground state, we imagine an empty system. We add particles one by one, filling the lowest energy states first. Once all particles are placed, the energy of the highest occupied state is the Fermi energy.

It is important to distinguish Fermi energy from the Fermi level. In semiconductor physics, many people use these terms interchangeably. However, they have distinct scientific meanings. Fermi energy is only defined at absolute zero temperature. It represents an energy difference, usually corresponding to kinetic energy. In contrast, the Fermi level, or electrochemical potential, is defined at any temperature. The Fermi level includes both kinetic energy and potential energy. It remains well-defined even in complex, interacting systems at equilibrium.

Even at absolute zero, fermions are not still. Because they must occupy different states, they continue to move. This movement happens even if we extract all possible heat from the system. The fastest particles move at a speed called the Fermi velocity. This velocity corresponds to the kinetic energy of the Fermi energy. There is also a related concept called the Fermi temperature. This is the temperature where thermal effects become comparable to quantum effects. For most metals, this temperature is much higher than room temperature.

We can see different values for Fermi energy in various environments. In metals, electrons act like a Fermi gas. The number density of conduction electrons is very high. It ranges between 10^28 and 10^29 electrons per cubic meter. This density results in a Fermi energy of 2 to 10 electronvolts. This is a relatively small amount compared to other systems. However, it is essential for the physics of solid matter.

Other systems show much higher energy levels. In a white dwarf star, the density is extreme. These stars have a mass similar to our Sun but a much smaller radius. Their electrons form a degenerate electron gas. The Fermi energy in these stars is about 0.3 MeV. Inside an atom, the nucleons also show high energy. The Fermi energy for nucleons in a nucleus is typically 38 MeV. This high energy helps explain why white dwarf stars do not collapse under gravity.

Scientists use mathematical formulas to calculate these values for specific systems. For a three-dimensional, non-relativistic, non-interacting group of identical spin-fermions, the formula uses several variables. It involves the number of particles, the mass of each fermion, and the volume of the system. It also uses the reduced Planck constant. Other related quantities include Fermi momentum and the Fermi wavevector. These describe the momentum and velocity of a fermion at the Fermi surface. These measurements are crucial for the study of quantum liquids, such as superfluid helium-3.