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Contents 1 Shell Model of the Nucleus I PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 10 Monday, September 14, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 Shell Model of the Nucleus I 1 1 Shell Model of the Nucleus I The shell model of the nucleus was proposed to explain a collection of experimental data. The data showed that nuclei have nuclear shells in analogy to the atomic shell model. In 1963 Maria Mayer received the Nobel Prize in Physics for proposing the nuclear shell model of the atomic nucleus. Experiments showed that the binding energies deviated from the liquid drop model with increased binding at N or Z at the so-called magic numbers of 2, 8, 20, 28, 50, 126. The number of stable isotopes/isotones is significantly higher for nuclei with the proton/neutron number equal to the magic number. See Fig. 1. Nucleon capture cross sections are high for nuclei with one nucleon shy from the magic number (single vacancy in a closed shell), but significantly lower for nuclei with number of nucleons equal to the magic number (at the closed shell). See Fig. 2. Electric quadrupole moments should be zero for closed shell nuclei since they are spherically symmetric. Fig. 3 shows this to be so. The figure is based on measurements for odd A nuclei. The measured moments have been normalized with respect to the size and charge of each nucleus and these so called reduced quadrupole moments are plotted against the number of protons or neutrons - depending upon which is odd. It is also clear from the plot that in some cases the moments are relatively large. The quadrupole moments vanish near closed shells and reach their largest values far away from them. This points to some nuclei having shapes which are strongly non-spherical. The plot shows that prolate nuclei (Q > 0) are more common than oblately deformed ones (Q < 0). We could come up with a shell model similar to the shell model of the atom. PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 ) MeV ( 2 neutron 1 last of 0 −1 energy −2 28 50 82 126 Binding 40 60 80 100 120 140 Neutron number N Figure 1: Binding energy of the last neutron against neutron number. The measured binding energy is plotted relative to the value predicted by the semi-empirical mass formula. There are discontinuities of about 2 MeV at the shell closures. We may assume that each nucleon occupies a well-defined energy level. A shell is described as the energy level where particles having the same energy exists. In this model, all the nucleons are paired one-to-one, neutron with a neutron, and proton with a proton. The electrons move in the atom in a central Coulomb potential emanating from the atomic nucleus. In the nucleon, on the other hand, the nucleons move inside a (mean field) potential produced by the other nucleons. That is, the motion of each nucleon is governed by the average attractive force of all the other nucleons. In both cases discrete energy levels arise, which are filled up according to the rules that obey the Pauli principle. Experimental data clearly pointed out to nuclear magic numbers at 2, 8, 20, 28, 50, 126. These are the magic numbers that show the most stable nuclei. See Fig. 4. The paired neutrons and protons in nuclear energy levels are filled when the number of neutrons or protons is equal to the magic number. The doubly magic nuclei, those with both magic proton and magic neutron numbers, are excep- tionally stable. These are the following nuclides 4 16 40 48 208 2He2; 8 O8; 20Ca20; 20Ca28; 82 Pb126 The unpaired nucleons are responsible for the properties of a nucleus. This is similar to the case in which valence electrons are responsible for different chemical properties of elements. 2 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 100 ) mb ( σ 10 section Cross 1 28 50 82 126 20 40 60 80 100 120 Neutron number N Figure 2: Nuclei with magic numbers of neutrons have neutron absorption cross-sections up to two orders of magnitude less than other nuclei with similar masses. This indicates that these nuclei are much less likely to absorb an additional neutron. With the help of the shell model, we can accurately predict the properties of nuclei such as angular momentum and parity. But, for nuclei which are in a highly unstable state, the shell model needs to be modified or replaced with other models such as the collective model, liquid-drop model, and compound nucleus model. However, if we consider a nuclear shell model, with a flat bottom potential, we would get the shell gaps that explain the magic numbers at 2, 8, 20, 40, 70, 112. The first three magic numbers are in agreement with the data. But the consecutive higher ones were not. This led to a serious confusion back in the days. What might be happening? Let us look at the shell model with a flat bottom potential. The simplest of such potentials is the harmonic oscillator potential. Writing the wave function as (~r) = Rkl(r)Ylm(θ; φ); (1) where Rkl(r) is the radial wave function and Ylm are the spherical harmonics, we get the energy 1 2 2 levels, for the potential V (r) = 2 µω r 3 E = (2k + l + ) !: (2) nl 2 ~ The energy is described by a single quantum number n ≡ 2k + l: (3) Since k is a nonnegative integer, for every even n, we have l = 0; 2; ··· ; n − 2; n, and for every odd n we have l = 1; 3; ··· ; n − 2; n. 3 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 167Er 0.30 176 0.25 Lu 10B 0.20 moment 0.15 55Mn 0.10 27Ai 9Be 115In quadrupole 2 H 79 0.05 41K Br 139La 201Hg 0.00 63Cu 209 Reduced Bi 93 −0.05 17O36Cl Nb 227Ac 8 20 28 123 82 126 −0.10 16 Sb 0 20 40 60 80 100 120 140 50 Number of odd neutrons or protons Figure 3: Reduced electric quadrupole moments for nuclei with odd proton number Z or neutron number N plotted against this number. The solid curves are based upon the quadrupole moments of very many nuclei, of which only a few are explicitly shown here. The magnetic quantum number m is an integer satisfying −l ≤ m ≤ +l. The degeneracy at level n is X 1 D = (2l + 1) = (n + 1)(n + 2): (4) n 2 l Let us use the spectroscopic notation for the orbital angular momentum l. That is, l = 0; 1; 2; ··· as s; p; d; f; g; ··· . In Fig. 4 we show a schematic representation of the shell structure of the nuclei. In the figure we have listed the number harmonic oscillator quanta in each set of shells. We have also made use of the fact that in the real nuclear potentials, the state with the same number of quanta but different l are no longer degenerate, but there are groups of shells with big energy gaps between them. This cannot predict the magic numbers beyond 20, and we need to find a different mechanism. Thus the early shell model was wrong but not completely wrong. They forgot to add an important piece. 4 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 126 1/2 3p 1/2 3/2 3p 13/2 13/2 5/2 3/2 5/2 2f 2f 7/2 9/2 9/2 7/2 1h 1h 82 3s 1/2 3s 1/2 3/2 3/2 2d 11/2 2d 11/2 5/2 7/2 7/2 5/2 1g 1g 50 9/2 9/2 1/2 1/2 2p 5/2 2p 5/2 3/2 1f 1f 3/2 26 7/2 7/2 20 3/2 3/2 1d 1/2 1d 1/2 2s 5/2 2s 5/2 8 1/2 1/2 1p 3/2 1p 3/2 1s 1/2 1s 1/2 Figure 4: Single particle energy levels calculated using Eq. (5). (See Ref. [2].) Magic numbers appear when the gaps between successive energy shells are particularly large. This diagram refers to the nucleons in the outermost shells. This was the spin-orbit coupling term. This term captures the coupling between the orbital angular momentum ~l and the spin angular momentum ~s. The spin-orbit coupling term in nuclei reduces the energy of states with spin oriented parallel to the orbital angular momentum while increasing the energy of states with spin oriented opposite to the orbital angular momentum. There is one already known for atoms, which is the so-called spin-orbit splitting. 1 This means that the degeneracy in the total angular momentum (j = l± 2 ) is lifted by an energy term that splits the aligned from anti-aligned case. This is shown schematically in Fig. 4, where we label the states by n; l and j. The gaps between the groups of shells are in reality much larger than the spacing within one shell, making the binding-energy of a closed-shell nucleus much lower than that of its neighbors. Note that this is different from the result of spin-orbit interaction in atoms where states with spin oriented opposite to the orbital angular momentum are lower in energy. The spin-orbit interaction in atoms is understood from the electromagnetic interaction of the magnetic moment of an electron with the magnetic field resulting from orbiting a charged nucleus.
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