The Free Electron Gas

Orion Ciftja

The homogeneous electron gas model has been quite successful to predict the bulk properties of systems of electrons at various densities. In many occasions, a simplified free homogeneous electron gas model represents a powerful first approximation to a real system. Despite our considerable knowledge on the bulk properties of a homogeneous electron gas, advances in nanoscience and nanotechnology call for a greater effort to understand the opposite limit of small finite systems of electrons with size-dependent properties. In this work, we provide a detailed description of the properties of a finite fully spin-polarized (spinless) free homogeneous one-dimensional electron gas, the simplest of the free homogeneous electron gases. We derive exact analytical results for various quantities such as the one-particle density function, two-particle density function, one-particle density matrix, pair correlation function and energy of finite systems with an arbitrary number of electrons. The results obtained provide a detailed view on how various quantities corresponding to a finite system approach their bulk (thermodynamic limit) value.

Gerald Dennis Mahan

D. C. Johnston

A free-electron Fermi gas is a gas of electrons with spin https://www.w3.org/1998/Math/MathML" display="inline"> S = 1 / 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_1.tif "/> . A fermion is a particle with half-integer spin S such as the electron. A boson is a particle with integer spin https://www.w3.org/1998/Math/MathML" display="inline"> S ≥ 0 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003606857/19698a40-dbd9-4f9a-9cb7-890a1d39adce/content/math8_2.tif "/> . By a gas is meant that there are no interactions between the particles making up the gas. Thus a Fermi gas can exhibit no phase transitions by itself. On the other hand, a free boson gas can exhibit an interesting quantum phase transition into a superfluid state, qualitatively similar to that of the superconducting state of a metal, called Bose-Einstein condensation . In contrast, an electron Fermi liquid would be a system containing electrons that do interact with each other. We will see in the last two chapters of this book that when such interactions occur, magnetic or superconducting transitions associated with the electrons can happen.

Philippe A. Martin, FRANÇOIS ROTHEN

Philippe A. Martin, FRANÇOIS ROTHEN