Abstract
Although the bulk properties of nuclei such as the mass and size, and also fission and the compound nucleus reactions suggest that the nucleus behaves like a liquid, there exist characteristic properties which cannot be understood from such a point of view. The existence of magic numbers in various phenomena is the most evident example, suggesting the shell structure. The magic numbers appear in various systems in nature. The existence of the noble gases in the periodic table of the elements is the most popular example. Contrary to those magic numbers for atoms, which are associated with the long range Coulomb interaction, the magic numbers for nuclei which originate from the short range force differ in numbers. Also, the spin–orbit interaction plays a crucial role in the magic numbers for nuclei. In this chapter we discuss how the magic numbers arise, and discuss the spin and parity properties of the nuclei in the vicinity of magic numbers. Although the shell model which is based on the mean-field theory succeeds, it is suggested that it is important to take into account the effects of the residual interaction, i.e., the pairing correlation in order to explain the details of the nuclear phenomena. As an example of the current topics, we also discuss the present status of the research of superheavy elements which are stabilized by shell effects.
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Notes
- 1.
The magic numbers can be thought to be the proton and/or neutron numbers for which the nucleus becomes spherical, or those at which the binding energy per nucleon, i.e., the separation energy of a nucleon, becomes particularly large compared with that in the surrounding nuclei, or the proton and neutron numbers at which the nuclear radius suddenly changes as a function of the proton and neutron numbers. If we consider in this way, the magic number is determined by the combination of the proton and neutron numbers, and the magic numbers for nuclei which are unstable with respect to \(\beta \)-decay such as neutron-rich unstable nuclei or neutron-deficient nuclei , so-called proton-rich nuclei , can be different from the known magic numbers for stable nuclei. The change of magic numbers or the appearance of new magic numbers in the region of unstable nuclei is one of the central research subjects in current nuclear physics.
- 2.
The parameter b which is defined by \(b\equiv 1/\alpha =\sqrt{\hbar /M_N \omega }\) is also often used and is called the oscillator parameter . It gives a measure of the strength of the potential which confines nucleons and has the dimension of length. It gives the extension of the 0s-state and the measure of the size of clusters such as the \(\alpha \)-cluster.
- 3.
For electrons in atoms, the spin–orbit interaction is repulsive. Hence the energy of the \(j=\ell -1/2\) state gets lowered as shown in Fig. 4.1.
- 4.
It is difficult to determine the resonance parameters such as the level position for a resonance state with a large width. In fact, [3] gives values which are very different from those in Fig. 5.2 for the energy splitting between the \(p_{1/2}\) and \(p_{3/2}\) states. For example, the excitation energy of the \(p_{1/2}\) state of \({}^5\)He is 4.6MeV. Reference [1] also gives two different parameter sets depending on the method of analysis. The resonance position and the width given in Fig. 5.2 and mentioned also in the following exercise have been obtained from the pole position of the S-matrix on the complex energy plane (extended R-matrix method; after [4]). On the other hand, the conventional method, i.e., standard R-matrix method, determines the parameters from the poles of the cross section by restricting the energy to be real as experiments are performed. The former has the advantage that the results do not depend on the choices of the boundary condition and the channel radius which appear in the R-matrix theory. See [5, 6] for details of the R-matrix theory.
- 5.
It is necessary to handle both the ground and excited states as resonance states in order to achieve a quantitatively accurate estimate. Here, however, use the harmonic oscillator model and ignore the radial dependence of the spin–orbit force in order to obtain a rough estimate of \(\xi _{\mathrm {LS}}\).
- 6.
The tensor force also strongly affects the magic numbers and the shape of nucleus, and is one of the active research subjects in recent years in connection with, e.g., the structure of unstable nuclei such as the change of the magic number. For example, \({}^{40}\)Ca is spherical, but \({}^{32}\)Mg is deformed among the \(N=20\) isotones (see [7]).
- 7.
We used the convention to omit the core part in writing the configuration, and to make the core part as the reference configuration.
- 8.
We used the lower index n in order to indicate that it is the energy level for neutrons.
- 9.
One can rigorously discuss the level structure in the presence of the pairing correlation in an algebraic way by using the quasi-spin formalism if one simplifies that only the pair whose total angular momentum J is zero is affected by the attractive residual interaction [17].
- 10.
These numbers vary somewhat depending on the theoretical models.
- 11.
The reactions which have a large reaction Q-value and hence synthesize a superheavy nucleus as the evaporation residue after emitting about three neutrons by the evaporation process are called hot fusion, while the reactions which have a small Q-value and emit one neutron in synthesizing a superheavy nucleus is called cold fusion. Since the average binding energy of nucleon is about 8Â MeV, one neutron is emitted for each 8Â MeV of the excitation energy.
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Takigawa, N., Washiyama, K. (2017). Shell Structure. In: Fundamentals of Nuclear Physics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55378-6_5
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