nuclear physics - Explanation of energy levels in molecules, atoms, nuclei and their relationship

Why are the energy levels of molecules, the atoms that form them and the nuclei inside the atoms considered separately? Or phrased in a different way- what is it that makes their energy levels so different? What is the relationship between the energy levels of molecules, atoms and nuclei?

I'd interested to learn different interpretations on this question.

Answer

In physics we distinguish between the physics of "atoms and molecules" and nuclei. Atoms and molecules are described by the same theory, thus I will ignore those molecules here completely and only consider the difference between nuclei and atoms.

I suppose you recognize that an atom is a bound system, so is a nucleus a bound system. Maybe you have seen how the energy levels are computed for an atomic system and now you wonder what is the difference to the energy levels of a nucleus.

The simplest form of a natural atom is the Hydrogen atom. It consists of a nucleus, a proton, and an electron. The energy levels of the Hydrogen atom are described by the theory of quantum mechanics. The Coulomb force between the nucleus and the electron defines the $-1/r$ potential which we use in the quantum mechanical approach to solve for the energy levels of the atomic system. This is a very long story, which I illustrate in this short paragraph, and there exist a lot corrections (fine structure, hyperfine structure, lamb shift, darwin term, ...) to the energy level of the simple Hydrogen atom. But their net effect on the energy levels is a minor correction in our problem here, thus it is not what distinguishes the atomic energy levels from those of a nuclei.

When Goeppert and Jensen firstly described the nuclear shell model, they considered a different potential to solve for the energy levels of the nucleus. If we compare the atom and the nucleus, we'll see that there is a fundamental difference in the potential, that we need to describe the nucleus.

1. The nucleus consists of two different types of particles, neutron (uncharged) and proton (charged)


2. In contrast to the atomic potential, which has a center, that's why we also named the Coulomb force a central force, the potential of the nucleus is generated by the neighboring particles in the nucleus itself.


3. On short distances the interaction force between nucleons (protons, neutrons) is stronger than the Coulomb force

For an analytical description of the bound states of the nucleus, people use the Harmonic Oscillator potential or the potential well. This approach leads to reasonable results, which help us understand and predict the energy levels. Yet, I do not know of a standard potential which describes all nuclei correctly. It is an active field of research.

I will also give you an example, how we can estimate the energy scale in the nucleus and the atom just by considering their dimensions. The Heisenberg Uncertainty principle states, that $\Delta x \Delta p \ge \frac{\hbar}{2}$ . For the electron of an atom, we assume that its location $x$ has the uncertainty on the same order of the diameter $d_{atom}$ of the atom , so $\Delta x_{atom} = d_{atom}$. With this we construct the minimal impulse of the electron in the atom:

$$\Delta p_{atom} \ge \frac{\hbar}{2\Delta x_{atom}}$$

Although you might say, this is only the uncertainty of the impulse, yet it is a measure for the order of magnitude the impulse will be.

For a nucleon (proton or neutron) in a nucleus we can make the similar approach with Heisenberg Uncertainty principle:

$$\Delta p_{nuc} \ge \frac{\hbar}{2\Delta x_{nuc}}$$

Since the diameter of a nucleus is orders of magnitudes smaller, the uncertainty of impulse of the nucleon is also by that order larger and so will be the energy levels. Here are the numbers: $d_{nuc} \simeq 1.75\text{fm}$ and $d_{atom} \simeq 0.1\text{nm}$ or $ 100\, 000\text{fm}$.

To give you a short summary:

The nature of the potential, which leads to the bound states in an atom and a nucleus is different and leads to fundamental differences in the energy levels for both. Since the nucleus is more compact, the energy levels will also be greater.