The shell model of atomic nuclei has been in intensive use since the middle of the previous century. This simple model of very complex nuclei, offers a quantitative description of its many features. Other features follow from small deviations from the extreme picture. Our friend and colleague Akito Arima made seminal contributions to this field starting with his famous paper with Horie on the magnetic moments of nuclei [Prog. Theor. Phys. 11, 509 (1954)]. In the following, a detailed description of a simple example is considered. It is the 1f7/2 shell of the neutrons in the nuclei between 40Ca and 48Ca and of the protons in the nuclei between 48Ca and 56Ni. The results demonstrate the power and elegance of the shell model. They show how simplicity arises out of complexity. It is also shown how small deviations from the simple shell model lead to effects, in which valence neutrons act as if they carry electric charge.
W. M. Elsasser, Sur le principe de Pauli dans les noyaux, J. de phys. et radium 4, 549; II 5, 389; III 635 (1934); Energies de liaison des noyaux lourds, 6, 473 (1935)
G. Racah, Theory of complex spectra (I), Phys. Rev. 61, 186; Theory of complex spectra (II), Phys. Rev. 62, 438 (1942); Theory of complex spectra (III), Phys. Rev. 63, 367 (1943) https://doi.org/10.1103/PhysRev.62.438
M. G. Mayer, On closed shells in nuclei (II), Phys. Rev. 75, 1969
11
O. Haxel, J. H. D. Jensen and H. E. Suess, On the “magic numbers” in nuclear structure, Phys. Rev. 75, 1766 (1949), where earlier references are listed. https://doi.org/10.1103/PhysRev.75.1766.2
12
M. G. Mayer, Nuclear configurations in the spin-orbit coupling model (I): Empirical evidence, Phys. Rev. 78, 16 (1950) https://doi.org/10.1103/PhysRev.78.16
13
M. G. Mayer, Nuclear configurations in the spin-orbit coupling model (II): Theoretical considerations, Phys. Rev. 78, 22 (1950) https://doi.org/10.1103/PhysRev.78.22
I. Talmi, nuclear spectroscopy with harmonic oscillator wave functions, Helv. Phys. Acta 25, 185 (1952)
18
K. Ford and C. A. Levinson, Independent-particle model of the nucleus (I): Inter particle forces and configuration mixing, Phys. Rev. 99, 742 (1955); Independent-particle model of The nucleus (III): The calcium isotopes, Phys. Rev. 100, 13 (1955)
D. Amit and A. Katz, Effective interaction calculations of energy levels and wave functions in the nuclear 1pshell, Nucl. Phys. 58, 388 (1964) https://doi.org/10.1016/0029-5582(64)90549-8
B. H. Wildenthal, Empirical strengths of spin operators in nuclei, Prog. Part. Nucl. Phys. 11, 5 (1984), where earlier references are listed. https://doi.org/10.1016/0146-6410(84)90011-5
E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, The shell model as a unified view of nuclear structure,Rev. Mod. Phys. 77, 427 (2005) https://doi.org/10.1103/RevModPhys.77.427
29
T. Otsuka, et al., Monte Carlo shell model for atomic nuclei, Prog. Part. Nucl. Phys. 47, 319 (2001); N. Shimizu, et al., New-generation Monte Carlo shell model for the K computer era, Prog. Theor. Exp. Phys. 2012, 01A205 (2012) https://doi.org/10.1016/S0146-6410(01)00157-0
30
B. H. Flowers, Studies in jj-coupling (I): Classification of nuclear and atomic states, Proc. Roy. Soc. A 212, 248 (1952) https://doi.org/10.1098/rspa.1952.0079
31
G. Racah, Nuclear levels and Casimir operator, in: Farkas Memorial Volume, Res. Council of Israel, Jerusalem, 1952
32
A. de Shalit and I. Talmi, Nuclear Shell Theory, Academic Press, 1963, reprinted by Dover Publications, 2004
33
I. Talmi, Simple Models of Complex Nuclei, Harwood, 1993
34
E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959
E. P. Wigner, Unpublished widely circulated MS. Reprinted in: Quantum Theory of Angular Momentum, Eds. L. C. Biedenharn and H. van Dam, Academic Press, 1965
37
I. Talmi, Effective interactions and nuclear models, in: Proc. Int. School Phys. Enrico Fermi, Varenna (1984)
I. Talmi, On the odd-even effect in the charge radii of isotopes, Nucl. Phys. A 423, 189 (1982)
43
A. Andl, K. Bekk, S. Göring, et al., Isotope shifts and hyperfine structure of the 4s21S0–4s4p1P1 transition in calcium isotopes, Phys. Rev. C 26, 2194 (1982) https://doi.org/10.1103/PhysRevC.26.2194
44
A. Arima and F. Iachello, Collective nuclear states as representations of a SU(6) group, Phys. Rev. Lett. 35, 1064 (1975)
45
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1951