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We know that the atomic energy levels are quantised since the nucleus has a potential of its own and solving the Schrödinger equation we can see that. But in gamma emission we see that the origin of gamma rays is that an excited nucleus sheds its energy of excitation by the emission of gamma rays, by which I infer that the energy states of the nucleus itself are quantised too. I want to know the reason behind the quantisation of the energy states of the nucleus and not the atomic energy levels.

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The nuclei are bound by an interplay of the attraction of the strong force with the repulsion of the electromagnetic. There are effective potentials that describe the energy levels using the Pauli exclusion principle in filling them, which lead to the periodic table of elements, with stable and unstable nuclei. This is the shell model, described here.

In nuclear physics and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy level

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The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. When adding nucleons (protons or neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model.

also this :

The electronic shell structure defines the architecture of the periodic table. An analogous effect leads to the so-called ìmagic nuclei,î closed nucleon shells that lead to an enhanced binding of the atomic nucleus, that opposes Coulomb repulsion of protons and governs the landscape of the nuclear chart.

So, yes, there are energy levels.

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