There are many kinds of energies, such as single particle energy, potential energy, kinetic energy, separation energy, and Fermi energy. How these energies are related?

I summarized in the following picture that the occupation number as a function of kinetic energy.

Picture1.png

Since the nucleus is a highly interactive system, although the temperature of the ground state of a nucleus should be absolute Zero, but the Fermi surface is not sharp but diffusive.

The Fermi energy of a nucleon , which is the maximum kinetic energy of a nucleon, is approximately ~35 MeV.

The potential energy is approximately ~ 50 MeV per nucleon.

There is an additional energy for proton due to Coulomb force, which is a Coulomb barrier.

The separation energy  is the different between the potential energy and Fermi energy.

The single particle energy is the energy for each single particle orbit.

The binding energy for a nucleon is the energy requires to set that nucleon to be free, i.e. the energy difference between the single particle energy and the potential energy.

There is a minimum kinetic energy, using the 3D spherical well as an approximation. The n-th root of the spherical Bessel function can give the energy of n-orbit with angular momentum l, so that.

j_{l}(kR) = 0, k^2=\frac{2 m E}{\hbar^2}

For l=0, j_0(x) = \frac{sin(x)}{x}, the 1-st root is x = \pi/2, then

k=\frac{\pi}{2R}

\frac{2 m E}{\hbar^2} = \frac{\pi^2}{4 R^2}

use R = 1.25 A^{1/3},

E = \frac{\pi^2 \hbar^2}{8 m 1.25^2 A^{2/3}}

use \hbar c = 200 MeV \cdot fm, mc^2 = 940 MeV

E = \frac{33.6}{A^{2/3}} MeV

we can see, for ^{16}O, the minimum KE, which is the 1s-orbit, is about 5 MeV.

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