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Concept# Nuclear shell model

Summary

In nuclear physics, atomic 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 levels. The first shell model was proposed by Dmitri Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notably Eugene Paul Wigner, Maria Goeppert Mayer, and J. Hans D. Jensen, who shared the 1963 Nobel Prize in Physics for their contributions.
The nuclear 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 better 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 specific magic quantum numbers of nucleons (2, 8, 20, 28, 50, 82, 126) which are more tightly bound than the following higher number, is the origin of the shell model.
The shells for protons and neutrons are independent of each other. Therefore, there can exist both "magic nuclei", in which one nucleon type or the other is at a magic number, and "doubly magic quantum nuclei", where both are. Due to some variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons, but only 114 for protons, playing a role in the search for the so-called island of stability. Some semi-magic numbers have been found, notably Z = 40, which gives the nuclear shell filling for the various elements; 16 may also be a magic number.
In order to get these numbers, the nuclear shell model starts from an average potential with a shape somewhere between the square well and the harmonic oscillator. To this potential, a spin orbit term is added. Even so, the total perturbation does not coincide with experiment, and an empirical spin orbit coupling must be added with at least two or three different values of its coupling constant, depending on the nuclei being studied.

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Binding energy

In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly used in condensed matter physics, atomic physics, and chemistry, whereas in nuclear physics the term separation energy is used. A bound system is typically at a lower energy level than its unbound constituents.

Interacting boson model

The interacting boson model (IBM) is a model in nuclear physics in which nucleons (protons or neutrons) pair up, essentially acting as a single particle with boson properties, with integral spin of either 2 (d-boson) or 0 (s-boson). They correspond to a quintuplet and singlet, i.e. 6 states. It is sometimes known as the Interacting boson approximation (IBA). The IBM1/IBM-I model treats both types of nucleons the same and considers only pairs of nucleons coupled to total angular momentum 0 and 2, called respectively, s and d bosons.

Semi-empirical mass formula

In nuclear physics, the semi-empirical mass formula (SEMF) (sometimes also called the Weizsäcker formula, Bethe–Weizsäcker formula, or Bethe–Weizsäcker mass formula to distinguish it from the Bethe–Weizsäcker process) is used to approximate the mass of an atomic nucleus from its number of protons and neutrons. As the name suggests, it is based partly on theory and partly on empirical measurements. The formula represents the liquid-drop model proposed by George Gamow, which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients.

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