{"id":255,"date":"2018-11-12T04:03:52","date_gmt":"2018-11-12T04:03:52","guid":{"rendered":"http:\/\/phyp04.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=255"},"modified":"2019-04-29T09:41:52","modified_gmt":"2019-04-29T09:41:52","slug":"nuclear-models-6","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp04\/chapter\/nuclear-models-6\/","title":{"rendered":"Nuclear Models-6"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n\u00a0 \u00a0 1. Woods-Saxon Potential<\/strong>\r\n\r\n \r\n\r\n \r\n

In order to explain the existence of magic numbers different forms of potentials have been used for the calculations namely the square well potential and the harmonic oscillator potential etc. These forms of potential could reproduce only first two magic numbers. This suggests that none of these correspond to the actual potential; it has probably a shape intermediate between the two. Hence the Woods-Saxon form of potential has also been assumed for the calculation. In the following section, we shall assume a Woods-Saxon potential.<\/p>\r\n \r\n

Woods-Saxon potential has the form of<\/p>\r\n\"\"\r\n\r\n \r\n

where V<\/em>0<\/em><\/sub> is a potential depth of the order of 30 to 60 MeV and R is the radius of the nucleus R \u2248 1.2 A1\/3<\/sup> fm<\/em>, and a<\/em> is the skin thickness and are chosen compatible with experimental results \u2248 0.5 fm<\/em>.<\/p>\r\n \r\n

A realistic form of the nuclear shell model potential for the Woods-Saxon is given in fig. 1, and corresponding energy levels are represented in fig. 2.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
N<\/strong><\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n3<\/td>\r\n3<\/td>\r\n4<\/td>\r\n4<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
nl<\/em><\/strong><\/td>\r\n1s<\/td>\r\n1p<\/td>\r\n1d<\/td>\r\n2s<\/td>\r\n1f<\/td>\r\n2p<\/td>\r\n1g<\/td>\r\n2d<\/td>\r\n3s<\/td>\r\n<\/tr>\r\n
Degeneracy<\/strong><\/td>\r\n2<\/td>\r\n6<\/td>\r\n10<\/td>\r\n2<\/td>\r\n14<\/td>\r\n6<\/td>\r\n18<\/td>\r\n10<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
States with E <\/strong>\u2264 Enl<\/sub><\/em><\/strong><\/td>\r\n2<\/td>\r\n8<\/td>\r\n18<\/td>\r\n20<\/td>\r\n34<\/td>\r\n40<\/td>\r\n58<\/td>\r\n68<\/td>\r\n70<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

It can be seen from the above table that the first three magic numbers (2, 8 and 20) can then be understood as nucleon numbers for full shells. This simple model however does not work for the higher magic numbers as other magic numbers could not be obtained with this potential. For them it is necessary to include the spin-orbit coupling effects which further split the nl<\/em> shells.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Fig. 1: A realistic form for the shell-model potential for the Woods-Saxon potential.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

Fig. 2: Different nuclear energy levels of Woods-Saxon potential<\/p>\r\n \r\n

A comparison of energy levels of different types of potential for harmonic oscillator, infinite square well and Woods-Saxon are given in fig. 3.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Fig 3: Energy levels for different types of potential<\/p>\r\n \r\n\r\n2. Spin-orbit interaction<\/strong>\r\n\r\n \r\n

One can see from these results that a central force potential is able to account for the first three magic numbers only, 2, 8, 20, but not the remaining four, 28, 50, 82, 126. This situation does not change when other shapes of potential forms are used. For higher levels there are discrepancies in the magic numbers thus we need a more precise model to obtain a more accurate prediction. The implication is that something very fundamental about the single-particle interaction picture is missing in the description.<\/p>\r\n \r\n

So we have to improve the previous potential forms in order to reproduce the magic numbers correctly. We should not want to make radical changes because this would destroy the physical content of this potential. Hence in order to predict the higher magic numbers, we need to take into account other interactions between the nucleons. The first interaction we analyze is the spin-orbit coupling.<\/p>\r\n \r\n

In order to explain the disagreement at the higher magic numbers, M. G\u00a8oppert Mayer, and independently D. Haxel J. Jensen and H. Suess in 1949 suggested to add a spin-orbit interaction term for each nucleon to the central potential V(r)<\/em><\/strong> which solved the problem of finding the magic numbers and gave the suitable separation between the shells.<\/p>\r\n \r\n

The spin-orbit potential, which is non-central, can be written as<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Here l\u045b<\/em><\/strong> and s\u045b<\/em><\/strong> are the azimuthal and spin angular momenta of the nucleon under consideration. f (r) <\/em>is a spherically symmetric function giving the profile of the potential (generally Woods-Saxon shape). It is weaker than V(r)<\/em>. ???<\/sub> is a constant.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

The S-O (spin-orbit) term in nuclei reduces the energy of states with spin oriented parallel to the orbital angular momentum l<\/em> while increasing the energy of states with spin oriented opposite to the orbital angular momentum. We can also see from the above equation that this is opposite of S-O interaction in atoms where states with spin oriented opposite to l<\/em> are lower in energy.<\/p>\r\n \r\n

In nuclei the spin-orbit effect comes from an attractive interaction between the orbital angular momentum and the intrinsic spin angular momentum of the nucleon. In case of atoms it arises from the EM interaction of the magnetic moment of an electron with the magnetic field resulting from orbiting a charged nucleus.<\/p>\r\n \r\n

We assume strong coupling between the spin and orbital angular momenta of each individual nucleons giving rise to a total angular momentum j<\/em><\/strong> for each so that we can write<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Since s<\/em> =1\/2 for each nucleon, the two possible values of j<\/em><\/strong> are j<\/em><\/strong> =<\/strong> l<\/em><\/strong> + \u00bd<\/strong> and j<\/em><\/strong> =<\/strong> l<\/em><\/strong> \u2013<\/strong> \u00bd<\/strong>. These two different levels now have different energies because of the strong spin-orbit coupling. The splitting of the two levels can be calculated by computing the expectation values of the spin orbit potential in the two states of the different j<\/em>.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\r\n
\r\n\r\nThis results in energy splitting of individual levels for given\u00a0j <\/em><\/strong>is shown in fig. 4.\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\n\"\"\r\n
\r\n

Fig. 4: Energy splitting of individual levels for given j<\/em><\/strong>.<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nThe corresponding energy splitting becomes<\/span>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\nj\u00a0 <\/em><\/strong>degeneracy<\/strong><\/span>\r\n\r\n \r\n

The effect of adding this spin-orbit term is to split the subshells according to the j<\/em><\/strong> degeneracy. Each subshell now can contain up to 2j<\/em><\/strong> + 1<\/strong> protons or neutrons.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n2.1 Nuclear Potential with Spin-Orbit Term<\/strong>\r\n\r\n \r\n\r\nThe nuclear potential with spin-orbit term is\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n

The spin-orbit term makes the nuclear potential well wider for nucleons with spin parallel to the orbital angular momentum and less wide for nucleons with spin opposite to the orbital angular momentum.<\/p>\r\n \r\n

The wider well results in states of lower energy, and without S-O term the energy of state does not depend on total angular momentum.<\/p>\r\n \r\n\r\nThe effect of spin-orbit interaction on the nuclear shell model is shown in fig. 5.\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Fig. 5: The effect of spin-orbit interaction on the shell model potential.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Fig. 6: Level scheme for a harmonic oscillator with spin-orbit coupling.<\/p>\r\n \r\n

The magic numbers now come out correct. In many cases, the angular momentum of the single-particle states also explains the nuclear angular momentum near magic nuclei. There are large deviations for nuclei between magic shells, so nuclear deformation has to be added.<\/p>\r\n \r\n

2.2 The Harmonic Oscillator with S-O interaction term<\/strong><\/p>\r\n \r\n

When spin-orbit interaction term is considered then the energy levels are labeled as <\/span>nlj.<\/em> Here the radial quantum number <\/span>n<\/em>, orbital angular momentum <\/span>l<\/em>, and total angular momentum <\/span>j<\/em>.<\/span><\/p>\r\n \r\n

The <\/span>nlj<\/em> level is (2<\/span>j<\/em> + 1) times degenerate.<\/span><\/p>\r\n \r\n

The nuclear potential with S-O interaction term for harmonic oscillator can be written as<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nThe energy of harmonic oscillator with S-O interaction term is\r\n\r\n<\/div>\r\n\"\"\r\n
\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n

Fig. 7: Energy levels of nucleons in a smoothly varying potential well with a strong spin-orbit coupling.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0 \u00a0 3.\u00a0<\/strong>Applications of Shell Model 3.1 Spin and Parity<\/strong>\r\n\r\n \r\n

One of the most successful applications of the single particle shell model is the explanation of the ground spins (I) and parity of odd-A nuclei (?). The model correctly predicts excitation energies, spin\/parities, magnetic and quadrupole moments for the ground state and low-energy excited states.<\/p>\r\n \r\n

The intrinsic spin s<\/em> of each nucleon couples with its orbital angular momentum l<\/em> to form the total angular momentum j = l + s<\/em>. The total angular momentum I<\/em> of the system of nucleons is then obtained by the coupling of the j<\/em> vectors of the individual nucleons in the nucleus: I<\/em> = \u2211 ji<\/sub>. This coupling is known as j-j<\/em> coupling. In nuclear physics it is customary to denote the total angular momentum by the symbol I<\/em> in place of J<\/em>.<\/p>\r\n \r\n

In case of nucleus to find out the ground state spins of the nuclei we make the following assumption: Only the last unpaired nucleon dictates the properties of the nucleus, and an even number of nucleons of any kind in the same state j<\/em> always combines to give the resultant spin 0 and even<\/em> parity.<\/p>\r\n \r\n\r\nOn the basis of the above assumptions we get the following results:\r\n