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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:
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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.
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  The Woods-Saxon potential is most often used; typically r0 is 1.2 fm and a is 0.65 fm. A remarkable finding from electron, proton and neutron scattering from nuclei is that all nuclei have the same density! The characteristic density of nuclei is a fundamental aspect of nuclear matter itself. In useful units, the density of nuclear matter is 0.15 GeV/fm³ or 0.15 nucleons per cubic fermi. Thus the typical distance between nucleons in a nucleus is large compared to the 1.2 fm range of the strong nuclear force! |
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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. The typical distance between nucleons is
actually almost beyond the range of the nuclear force, which
makes it no surprise to discover that the individual
interactions between nucleons are not of primary importance.
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. Use the Fermi gas model
to express the kinetic energy of the system and add the
potential energy holding the system together (gravitational,
etc.). Check for a minimum when the total energy is
differentiated with respect to the size of the system.
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.
