THE FERMI GAS!


Why does the physical world seem "solid" when it is made of pointlike fermions and rather abstract-seeming fields? Thanks to the Uncertainty Relation and the Pauli Principle, whenever a system of identical fermions is compressed, it resists compression with an enormous pressure. No two systems of identical fermions can be merged together; they bounce apart, their probability distributions are not allowed to overlap!



One of the most useful models in all of physics: the Fermi Gas. It can be used for solids, heavy atoms, atomic nuclei, neutron stars, white dwarf stars and even nucleons themselves! The two key parameters are the Fermi energy EF and the Fermi momentum, pF = (2mEF)1/2. [Important result, used in the quasielastic electron scattering from nucleus example:  <K> = (3/5)pF2/(2M), so pF2 = (5/3)<P2>.]







Quasi-elastic scattering of electrons from individual protons in nuclei gives a very direct test of the Fermi gas model, assuming the struck proton is in an eigenstate of momentum... from the analysis, it is simple to extract the Fermi momentum and the depth of the potential in which the proton is bound. Since light nuclei are nearly all "surface" one expects good agreement only for heavier nuclei, where values of 260 MeV/c for the Fermi momentum and 44 MeV for the depth of the potential, extracted from the experimental measurements, are in excellent agreement with a Fermi gas model with parameters matching a realistic nucleus.   As Fermi himself might have said, "That model has no right to be that good!"




Applying the Fermi Gas model to atomic nuclei presents a remarkable picture: despite the strong interaction between nucleons, a useful starting concept is that the nucleons are basically "free" inside a central potential with a depth of about 45 to 50 MeV! In other words in computing scattering of a nucleon from a nucleus, an adequate treatment would involve scattering of the incoming particle from a simple central potential with about the same shape as the nuclear matter density! This leads to the so-called Optical Model for elastic scattering of nucleons and nuclei, introduced circa 1950.  Even more remarkable, this idea leads to the picture of a nucleon in a nucleus interacting with a central potential somehow generated by all the other nucleons in combination, instead of a chaotic picture in which each nucleon interacts with every other nucleon, leading to an impossibly complex many-body problem!  This simplification results in the so-called "independent particle model," the basis of nuclear physics since about 1950.






The Woods-Saxon potential is most often used; typically r0 is 1.2 fm and a is 0.65 fm.




The optical model does not work unless an imaginary term is inserted into the potential so that probability is not conserved... many things can happen to the incident particle other than elastic scattering!


No!!!!

Yes!!

Why does the Fermi gas picture work so remarkably well? Consider interaction between two identical particles. It is improbable that the interaction should be so strong as to transfer one particle to a vacant state above EF, so the only likely possibilities are that the particles wind up in the same states, or exchange states. In the case of p interacting with n, the probability is that both wind up in the same states they started in, since exchange is not a possibility.



S. Chandrasekhar (1910 - 1995) was one of the most famous 20th Century astrophysicists, and a colleague and close friend of Enrico Fermi at the University of Chicago. He used the Fermi Gas model to understand why it is impossible for stars to be stable within certain mass ranges.


Extending the Fermi gas model to relativistic fermions makes it easy to understand why there are various mass limits for ordinary stars in astrophysics, and in particular why white dwarf stars must have low mass, what the mass limits for neutron stars are, and why black holes are an inevitable consequence of the collapse of stars beyond a certain critical mass. We will discuss such topics later in the course. Put simply, using K = pc leads to EF = ℏc[3π2ρ]1/3.



Lectures on the Fermi Gas, with astrophysical applications!

The Bose Gas (a gas consisting of identical bosons) is a much more complicated system to understand!

Fermi Gamma-Ray Orbital Telescope!

THE FERMI BUBBLES!

Chandra X-ray Orbital Telescope!


INDEPENDENT PARTICLE MODEL!

NUCLEAR REACTIONS!