Magic Nuclei and Nuclear Shell Model
| In the previous lectures we have seen that liquid drop model gives a quite clear picture on the nuclear stability and later on (in module III) we will see that nuclear fission can be explained on the basis of the same. However, there are some other properties of nucleus, which can’t be explained on the basis of this model. It has been observed by nuclear physicists that nuclei with number of protons or neutrons equal to 2,4,8, 20 50, 82 and 126, the B.E./A values are higher as compared to their neighbours , giving rise to discontinuities to the binding energy curves. This indicates higher stability of these nuclei as compared to their neighboring nuclei of the B.E. curve. For example 58Ce140, which has 82 neutrons has higher (almost 4 MeV) B.E. compared to its neighbours. The numbers 2,4,8,20,50,82 and 126 are known as magic numbers as they were not understood for quite a long time with existing models such as LDM . | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Explanation of Magic numbers— Shell model of a nucleus | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Whatever it may be, let me tell you that there is a resemblance between occurance of magic numbers and higher ionization potential of the noble gases (element with specific atomic number, Z), which is basically a consequence of atomic shell structure. For these atoms the electron shells are completely filled. Therefore, natuarally question arises whether nucleus too have a shell structure and if it is so, can it explain the magic numbers? It was not an easy job to frame a shell model of a nucleus similar to atomic shell model as there is no obvious central attractive force in a nucleus, like Coulomb force in an atom. Secondly, the density of nucluear matter is extremely high (~1017 kgm-3) and hence free movement of nucleons in a nucleus like electrons in atom is difficult to imagine. However, Barlett, Guggenheimer et al. proposed an independent particle model (shell model) for the nucleus in 1933 and could able to explain lower magic numbers (2,4, 8). Later on, Maria G. Mayer in 1948 and independently Haxel et al. proposed shell structure of the nucleus and closed shell in nuclei having A corresponding to magic numbers. Their models were able to explain all magic numbers with a consideration of strong spin-orbit forces of nucleons and their coupling. Before discussing the shell structure of the nucleus given by Mayer, let us first discuss the concept of atomic shells and spin-orbit coupling. Atomic shells are basically some energy levels where electrons are allowed to stay following some quantum mechanical conditions.* As electrons move in quantized orbits, it has orbital angular momentum associated with it. Apart from orbital angular momentum it also has spin angular momentum, arises due to its spin motion, which with much simplicity can be understood as a motion about its own axis.# Not only electrons, but every elementary particles with such rotational condition have two types of angular momentum, i.e. spin angular momentum and orbital angular momentum. The first one is denoted by symbol s and the second one by symbol l. The particles like electron, proton, neutron etc. have the value of s as +1/2 or -1/2. There are several values of l, corresponding to various orbitals where the particle is moving. In case of electron, the atomic orbitals are denoted by, s, p, d, f, g, h, i, etc. and corresponding l values are 0, 1, 2, 3, 4, 5, 6 respectively. Now the l and s values of an electron (similarly, neucleons) can vectorially couple giving rise to a resultant angular momentum, j, where j = l /plusminus.png){kind=small} s, i.e. l 1/2. For example, if we consider spectroscopic term s, then, l = 0, s = 1/2 and j = 1/2. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Now the next step is to consider the potential configuration within which the nucleons move inside the nucleus. The first assumption made in nuclear shell model is single particle concept with an idea that the spin, parity and moments are determined by the unpaired nucleon alone and hence essentially reducing many nucleon problems to a single particle problem. It is also assumed that each nucleon experiences a central attractive force due to the average effect of the remaining nucleons (core) in the nucleus. This is something similar to the atom where nucleus acts as a central attractive force of the electrons in the orbits. In this central field each nucleon is considered to move in a shell having specific energy and angular momentum. Another assumption of shell model is a weak inter nucleon interaction. The model of Mayer considers that the l and s values of a nucleon couples giving a distinct j value of a nucleon. Nucleons having different j values couple via j-j coupling mode, which eventually explains the origin of magic numbers | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| In order to describe shell model a detail quantum mechanical calculation is required by considering some standard potential. Various potential configuration have been chosen by various authors. M. Mayer and Haxel et al. independently showed that harmonic oscillator potential or square well potential can be used to obtain reasonable results. The form of these potentials are: | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (A) V = -V0 +1/2Mω2r2 ———- (m1.42) (linear harmonic oscillator). Where V0 and ω are constants. (B) V = -V0 for r< R ———- (m1.43) = 0 for r>R. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Of late, more realistic potential configuration have been chosen by other Nuclear Physicists. A detailed mathematical formulation is required to find the energy of the electrons in various shells. However, in this lecture we will just show the nucleons distribution in a closed shell and the associated magic numbers. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Shells are defined by principal quantum number n whose value runs from 0 to integer numbers. The sequence of nuclear spins with increasing n values as per standard spectroscopic notation is shown in table m1. 1. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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TABLE m1.1 Filling up of nuclear shells with nucleons corresponding to magic numbers. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Now it is clear from the above table that in order to understand nuclear property like magic numbers concept of shell model is must. This model has great implication to understand nuclear structure and hence nuclear spectroscopic studies. Such description is out of scope of this lecture module and readers are suggested to read advance books on nuclear structure. |