Calculating Nuclei Shell Energy Levels: No Bra-Ket Math Needed

[bluestar]

In the nuclei Shell Model I understand the nomenclature for the shell sequence but I don’t know how to calculate the respective energy levels for each shell.

For example how do you calculate the energy level for
1g(7/2) or 3d(5/2)

Pointing me to an online reference will also be helpful. However, I have not learned bra-kets or Hamiltonian math yet.

 

[malawi_glenn]

what is Hamiltonian math?

The energy levels are different for different nuclei, you first take an approriate form of the central potential (V_c), e.g Wood Saxon.

Then you that the potential includes an $$\vec{l}\cdot \vec{s}$$ term (spin-orbit coupling), your potential is thus:

$$V(r) = V_c(r) + V_{ls} \vec{l}\cdot \vec{s}$$

where, of course:

$$V_{ls} = const. \dfrac{1}{r}\dfrac{\partial}{\partial r}$$

Then you take your nuclei, find the excited levels and their energies, fit to the parameters of the V_c and starts to solve the Schrödinger equation (nummerically), and thus you obtain all the energies.

This is a VERY sketchy idea how to do it, Nuclear many body physics is quite complicated..

 

[sir-pinski]

Calculating energy levels for nuclei shells can be done using the nuclear shell model, which is based on the concept of energy levels being determined by the number of nucleons in a specific shell. This model does not require the use of bra-ket or Hamiltonian math.

To calculate the energy levels for a specific shell, you can use the formula:

E = E0 + a(n + 1/2)

Where E is the energy level, E0 is the energy of the last filled shell, a is a constant specific to the type of nucleus, and n is the number of nucleons in the shell.

For example, to calculate the energy level for 1g(7/2), we would use n = 6 because the 1g shell has 6 nucleons. The value of a can be found in tables or online references specific to the type of nucleus. Plugging in these values, we can calculate the energy level for 1g(7/2).

Similarly, for 3d(5/2), we would use n = 14 (3d shell has 14 nucleons) and find the value of a for the specific nucleus to calculate the energy level.

I recommend checking out online resources such as the Nuclear Data Center or the National Nuclear Data Center for specific values of a and other relevant information for different types of nuclei. Additionally, textbooks on nuclear physics or the nuclear shell model can also provide helpful explanations and examples.

Keep in mind that these calculations are simplified and do not take into account other factors that can affect energy levels in nuclei. But they can provide a basic understanding of the concept and help you to get started in learning more about the nuclear shell model.