## Evolution of Woods-Saxon energy levels

The Woods-Saxon potential is $\displaystyle H = \frac{V_0}{1+\exp^{\frac{r-R_0}{a_0}}} - LS \frac{V_{so} \exp^{\frac{r-R_{so}}{a_{so}}}}{(\exp^{\frac{r-R_{so}}{a_{so}}})^2}$

The LS coefficient is $\displaystyle LS = \frac{1}{2} (J(J+1) - L(L+1) - \frac{1}{2}\frac{3}{2})$

So, this is the neutron single particle levels. ## Energies in a nucleus

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. 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.

## Shell model calculation and the USD, USDA, and USDB interaction

Form the mean field calculation, the single particle energies are obtained. However, the residual interaction is still there that the actual state could be affected. Because the residual interaction produces the off-diagonal terms in the total Hamiltonian, and that mixed the single particle state.

The Shell Model calculation can calculate the nuclear structure from another approach. It started from a assumed nuclear Hamiltonian, with a basis of wavefunctions. The Hamiltonian is diagonalized with the basis, then the eigenstates are the solution of the wavefunctions and the nuclear structure, both ground state and excited states. The basis is usually the spherical harmonic with some radial function. Or it could be, in principle, can take from the result of mean field calculation. Thus, the Shell Model calculation attacks the problem directly with only assumption of the nuclear interaction.

However, the dimension of the basis of the shell model calculation could be very huge. In principle, it should be infinitely because of the completeness of vector space. Fro practical purpose, the dimension or the number of the basis has to be reduced, usually take a major shell. for example the p-shell, s-d shell, p-f shell. However, even thought the model space is limited, the number of basis is also huge. “for $^{28}$Si the 12-particle state with M=0 for the sum of the $j_z$ quantum numbers and $T_z=0$ for the sum of the %Latex t_z\$ quantum numbers has dimension 93,710 in the m-scheme” [B. A. Brown and B. H. Wildenthal, Ann. Rev. Nucl. Part. Sci. 38 (1998) 29-66]. Beside the huge dimensions and the difficult for diagonalizing the Hamiltonian, the truncation of the model space also affect the interaction.

We can imagine that the effective interaction is different from the actual nuclear interaction, because some energy levels cannot be reached, for example, the short range hard core could produce very high energy excitation. Therefore, the results of the calculation in the truncated model space must be “re-normalized”.

There are 4 problems in the shell model calculation:

• the model space
• the effective interaction
• the diagonalization
• the renormalization of the result

The shell model can also calculate the excited state with $1\hbar \omega$ (1 major shell). This requires combination of the interactions between 2 major shell.

For usage, say in the code OXBASH, user major concern is the choice of the interaction and model space. The shell model are able to calculate

• The binding energy
• The excitation energies
• The nucleons separation energies
• The configuration of each state
• The magnetic dipole matrix elements
• The Gamow-Teller (GT) transition
• The spectroscopic factor
• …… and more.

The W interaction (or the USD) for the s-d shell was introduced by B.H. Wildenthal around 1990s. It is an parametric effective interaction deduced from fitting experimental energy levels for some s-d shell nuclei. Before it, there are some theoretical interactions that require “no parameter”, for example the G-matrix interaction is the in-medium nucleon-nucleon interaction.

The problem for the USD interaction is the interpretation, because it is a black-box that it can reproduce most of the experimental result better than theoretical interactions, but no one know why and how. One possible way is translate the two-body matrix elements (TBME) back to the central, spin-orbit, tensor force. It found that the central and spin-orbit force are similar with the theoretical interactions, but the tensor force could be different. Also, there could be three-body force that implicitly included in the USD interaction.

In 2006, B.A. Brown and W.A. Richter improved the USD interaction with the new data from the past 20 years [B.A. Brown, PRC 74, 034315(2006)]. The new USD interaction is called USDA and USDB. The difference between USDA and USDB is the fitting (something like that, I am not so sure), but basically, USDA and USDB only different by very little. Since the USDB has better fitting, we will focus on the USDB interaction.

The single particle energy for the USDB is

• $1d_{3/2} = 2.117$
• $2s_{1/2} = -3.2079$
• $1d_{5/2} = -3.9257$

in comparison, the single particle energies of the neutron of 17O of $2s_{1/2} = -3.27$ and $1d_{5/2} = -4.14$.

Can to USD interaction predicts the new magic number N=16?

Yes, in a report by O. Sorlin and M.-G. Porquet (Nuclear magic numbers: new features far from stability) They shows the effective single particle energy of oxygen and carbon using the monopole matrix elements of the USDB interaction. The new magic number N=16 can be observed. 