Theory of the shell structure of the nucleus

The number of nucleons ( protons or neutrons ) in the nucleus, at which the nuclei have a higher binding energy than nuclei with the closest (more or less) number of nucleons, is called the magic number ^[1] . Particularly stable are atomic nuclei containing magic numbers 2, 8, 20, 50, 82, 114, 126 , 164 for protons and 2, 8, 20, 28, 50, 82, 126, 184, 196, 228, 272, 318 for neutrons . ( Bold magic numbers are marked out twice, that is, magic numbers that exist for both protons and neutrons).

Note that the shells exist separately for protons and neutrons, so we can talk about a “magic nucleus” in which the number of nucleons of one type is a magic number, or about a “twice magic nucleus” in which magic numbers are the numbers of nucleons of both types. Due to fundamental differences in the filling of the orbits of protons and neutrons, further filling occurs asymmetrically: the magic numbers for neutrons are 126 and, theoretically, 184, 196, 228, 272, 318 ... and only 114, 126 and 164 for protons. This fact is important in the search for the so-called " islands of stability ." In addition, several semi-magic numbers were found, for example, Z = 40 ( Z is the number of protons).

“Double magic” nuclei are the most stable isotopes , for example, the lead isotope Pb-208 with Z = 82 and N = 126 (N is the number of neutrons).

Magical nuclei are the most resilient. This is explained in the framework of the shell model: the fact is that the proton and neutron shells in such nuclei are filled - like the electronic ones in noble gas atoms.

According to this model, each nucleon is in the nucleus in a specific individual quantum state , characterized by energy , angular momentum (its absolute value j, and also the projection m onto one of the coordinate axes) and orbital angular momentum l.

The energy of the level does not depend on the projection of the moment of rotation on the external axis. Therefore, in accordance with the Pauli principle, at each energy level with moments j, l there can be (2j + 1) identical nucleons forming a “shell” (j, l). The total rotation moment of the filled shell is zero. Therefore, if the nucleus is composed only of filled proton and neutron shells, then its spin will also be zero.

Whenever the number of protons or neutrons reaches a number corresponding to the filling of the next shell (such numbers are called magic), there is the possibility of an abrupt change in some quantities characterizing the nucleus (in particular, the binding energy). This creates a semblance of periodicity in the properties of nuclei depending on A and Z, similar to the periodic law for atoms. In both cases, the physical cause of periodicity is the Pauli principle, which prohibits two identical fermions from being in the same state. However, the shell structure of nuclei manifests itself much weaker than in atoms. This happens mainly because in nuclei the individual quantum states of particles (“orbits”) are perturbed by their interaction (“collisions”) with each other much more strongly than in atoms. Moreover, it is known that a large number of nuclear states does not at all resemble the totality of nucleons moving independently in each other, that is, cannot be explained within the framework of the shell model.

In this regard, the concept of quasiparticles — elementary excitations of the medium that effectively behave in many respects like particles — is introduced into the shell model. In this case, the atomic nucleus is considered as a Fermi liquid of finite size. The core in the ground state is regarded as a degenerate Fermi gas of quasiparticles that do not interact effectively with each other, since any collision act that changes the individual states of quasiparticles is forbidden by the Pauli principle. In the excited state of the nucleus, when 1 or 2 quasiparticles are at higher individual energy levels, these particles, having released the orbits occupied by them earlier inside the Fermi sphere , can interact with each other and with the hole formed in the lower shell. As a result of interaction with an external quasiparticle, a transition of quasiparticles from filled to unoccupied states can occur, as a result of which the old hole disappears and a new one appears; this is equivalent to the transition of a hole from one state to another. Thus, according to the shell model based on the theory of quantum Fermi liquid, the spectrum of the lower excited states of nuclei is determined by the motion of 1-2 quasiparticles outside the Fermi sphere and their interaction with each other and with holes inside the Fermi sphere. Thus, the explanation of the structure of a multinucleon nucleus at low excitation energies actually reduces to the quantum problem of 2–4 interacting bodies (a quasiparticle — a hole or 2 quasiparticles — 2 holes). The difficulty of the theory, however, is that the interaction of quasiparticles and holes is not small, and therefore there is no certainty that the appearance of a low-energy excited state is caused by a large number of quasiparticles outside the Fermi sphere.

In other variants of the shell model, an effective interaction is introduced between quasiparticles in each shell, leading to mixing of the initial configurations of individual states. This interaction is taken into account by the method of perturbation theory (valid for small perturbations). The internal inconsistency of such a scheme is that the effective interaction required by the theory to describe experimental facts is by no means weak. In addition, the number of empirically selected model parameters is increasing. Also, shell models are sometimes modified by introducing various additional interactions (for example, interactions of quasiparticles with vibrations of the surface of the nucleus) to achieve better agreement between the theory and experiment.

The shell model of the nucleus is actually a semi-empirical scheme that allows one to understand some of the laws in the structure of nuclei, but is not able to sequentially quantify the properties of the nucleus. In particular, in view of the above difficulties, it is not easy to theoretically find out the order of filling the shells, and therefore the "magic numbers" that would serve as analogues of the periods of the periodic table for atoms. The order of filling the shells depends, firstly, on the nature of the force field, which determines the individual states of quasiparticles, and, secondly, on the mixing of configurations. The latter is usually taken into account only for unfilled shells. The experimentally observed magic numbers common to neutrons and protons (2, 8, 20, 28, 40, 50, 82, 126) correspond to the quantum states of quasiparticles moving in a rectangular or oscillatory potential well with a spin-orbit interaction (precisely because of this, numbers 28, 40, 82, 126)

• Atomic nucleus
• Nucleon
• Electronic configuration
• The Hartree-Fock-Bogolyubov Method
• Stability island

• Gapon E., Iwanenko D. , Zur Bestimmung der isotopenzahl, Die Naturwissenschaften, Bd. 20, s. 792-793, 1932.
• Geppert-Mayer M. , Jensen I., Elementary Theory of Nuclear Shells, Foreign Literature, M., 1958.
• The atomic nucleus is an article from the Great Soviet Encyclopedia .