The Fermi Gas Model

The nucleus binds $A$ nucleons at a nearly constant density, and each nucleon moves through the volume with only rare collisions because the exclusion principle blocks scattering into occupied states. The simplest quantitative model that respects both facts treats the protons and the neutrons as two independent gases of free spin-$\frac{1}{2}$ fermions confined to a spherical well of volume set by the measured nuclear radius. The gas is fully degenerate: at the low temperatures relevant to a ground-state nucleus every level up to a sharp Fermi surface is filled and every level above it is empty. From the density alone this fixes the Fermi momentum, the average kinetic energy, the depth of the confining well, and, through the difference of the proton and neutron Fermi levels, the asymmetry term of the semi-empirical mass formula.¹

Counting states in the well

A nucleon confined to a cubical box of side $L$ and volume $V=L^3$ has plane-wave states labelled by wavevectors $\vec{k}=(\pi /L)(n_x,n_y,n_z)$ with positive integers $n_i$. Each triple occupies a cell of volume $(\pi /L{)}^3$ in $\vec{k}$-space, so the number of spatial states with wavenumber below $k$ is the positive octant of a sphere divided by the cell,

Each spatial state holds two nucleons of one kind, spin up and spin down. In terms of momentum $p=\hslash k$, the number of neutrons filling every level up to the Fermi momentum $p_F$ is

and the same relation with $Z$ and $p_{F,p}$ counts the protons. Inverting for a single species gives the Fermi momentum directly from its number density ${\rho }_i=N_i/V$,

The Fermi momentum is set by the density, not by the size or the charge of the nucleus.

Fermi energy and momentum for nuclear matter

Charge symmetry of the nuclear force makes symmetric matter, $N=Z=A/2$, the reference case. Each species then has density ${\rho }_i=\frac{1}{2}\rho$, where the total density follows from packing $A$ nucleons into a sphere of radius $R=R_0{A}^{1/3}$,

The density is independent of $A$, the empirical fact of nuclear saturation. With $R_0=1.2\mathrm{fm}$ this is $\rho \approx 0.138\mathrm{f}{\mathrm{m}}^{-3}$, so ${\rho }_i\approx 0.069\mathrm{f}{\mathrm{m}}^{-3}$ for each species. The Fermi momentum is

using $\hslash c=197.3\mathrm{MeV}\cdot \mathrm{fm}$. Nucleons are non-relativistic here, $p_{F}c\approx 0.27M{c}^2$, so the Fermi energy is the kinetic energy at the surface,

Both numbers come entirely from the saturation density: every nucleus, light or heavy, has the same Fermi surface.

The density of states

Turning the count $n(k)$ into a distribution over energy $E=p^2/2M$ gives the number of levels per unit energy. Writing the state count for one species with spin as $N(E)=V(2ME{)}^{3/2}/(3{\pi }^2{\hslash }^3)$ and differentiating,

The density of states rises as $\sqrt{E}$ and is filled up to $E_F$; every state below the Fermi energy is occupied, none above. The total kinetic energy of one species is the first moment of this distribution,

so the average kinetic energy per nucleon is

Even in its lowest state a nucleus carries roughly $20\mathrm{MeV}$ of kinetic energy per nucleon, a purely quantum zero-point motion forced by the exclusion principle in a small volume.

The depth of the well

The Fermi energy is measured from the bottom of the well, not from the free-particle zero. The most weakly bound nucleon sits at the Fermi surface, and removing it costs the separation energy $S$, the binding energy of the last nucleon, about $7$–$8\mathrm{MeV}$ across stable nuclei. The well depth is therefore the sum

This $40\mathrm{MeV}$ is the depth of the average single-particle potential that each nucleon moves in, in agreement with the well depths extracted from optical-model fits to nucleon scattering. The Fermi-gas picture thus turns the measured density and separation energy into the mean field itself.

The asymmetry energy

The two Fermi seas fill to different levels once $N\ne Z$. Adding neutrons beyond $Z$ raises the neutron Fermi energy while the proton sea stays lower, so the total kinetic energy climbs above its symmetric value. This excess is the microscopic origin of the asymmetry term.

At fixed volume $V$, each species contributes total kinetic energy $\frac{3}{5}{N}_i{E}_{F,i}$ with $E_{F,i}=C{N}_i^{2/3}$ and $C=({\hslash }^2/2M)(3{\pi }^2/V{)}^{2/3}$, so

Write $N=\frac{1}{2}(A+D)$ and $Z=\frac{1}{2}(A-D)$ with $D=N-Z$, and expand for $D\ll A$. Using $(1+x{)}^{5/3}+(1-x{)}^{5/3}\approx 2+\frac{10}{9}{x}^2$ with $x=D/A$,

The constant term is the symmetric kinetic energy; the $D^2$ term is the asymmetry energy. Substituting $C(A/2{)}^{2/3}=E_F$ collapses the prefactor,

The kinetic energy alone gives an asymmetry coefficient $a_{\text{sym}}^{\text{kin}}=E_F/3\approx 11\mathrm{MeV}$. The empirical coefficient in the mass formula is roughly $23\mathrm{MeV}$, so the kinetic Fermi-gas contribution accounts for about half. The remainder is a potential effect: the $n$-$p$ force is more attractive than $n$-$n$ or $p$-$p$, so trading a proton for a neutron loses attractive $n$-$p$ bonds as well as raising the Fermi energy. Both push the same way, toward $N=Z$.²

Reach and limits

The Fermi-gas model is the crudest mean-field picture and its successes are therefore instructive. From nothing but the saturation density it delivers a Fermi momentum of $250\mathrm{MeV}/c$, a Fermi energy of $33\mathrm{MeV}$, an average kinetic energy of $20\mathrm{MeV}$ per nucleon, a well depth of $40\mathrm{MeV}$, and half of the asymmetry coefficient. The high internal momenta it predicts are confirmed by quasi-elastic electron scattering, where the knocked-out nucleon carries a momentum distribution that extends to the Fermi surface.

The model ignores everything that distinguishes one nucleus from the next. It has no shell structure, no magic numbers, and no correlation between nucleons beyond the exclusion principle; it treats the mean field as a structureless box. The next refinement keeps the independent-particle idea but replaces the box with a realistic central potential plus a strong spin-orbit term, which quantizes the levels into shells and reproduces the magic numbers the Fermi gas cannot see.