Nucleon energy levels

Fig. 2. The sequence of nucleonic energy levels with spin-orbit coupling, redrawn, with small changes, from 3, p. 58.

The differences between Z/N = 1 and 1.04 are of a minor nature, like those between 0.62 and 0.58. In fact, the arrangement at 1.0 can be interpreted as inverse to 0.62. The points at Z/N = 0 have been interpreted in terms of nucleon energy levels. 

The sequence of nucleon energy levels (overlapping ranges), with assignment to successive layers (inner core, outer core, mantle) on the basis of the principal quantum number. 

Nuclear stability is associated with filled nucleon energy levels. Certain heavy nuclei undergo a decay series to reach stability. (Section 23.1)... 

One model of nuclear structure that attempts to explain the stability of even values of N and Z postulates that protons and neutrons lie in nucleon energy levels, and that greater stability results from the pairing of spins of like nucleons. (Note the analogy to electron energy levels and the stability from pairing of electron spins.)... 

Another difference between nucleons and electrons is that nucleons pair whenever possible. Thus, even if a particular energy level can hold more than two particles, two particles will pair when they are present. Thus, for two particles in degenerate levels, we show two particles as II rather than II. As a result of this preference for pairing, nuclei with even numbers of protons and neutrons have all paired particles. This results in nuclei that are more stable than those which have unpaired particles. The least stable nuclei are those in which both the number of neutrons and the number of protons is odd. This difference in stability manifests itself in the number of stable nuclei of each type. Table 1.3 shows the numbers of stable nuclei that occur. The data show that there does not seem to be any appreciable difference in stability when the number of protons or neutrons is even while the other is odd (the even-odd and odd-even cases). The number of nuclides that have odd Z and odd N (so-called odd-odd nuclides) is very small, which indicates that there is an inherent instability in such an arrangement. The most common stable nucleus which is of the odd-odd type is 147N. 

Representation of nuclear energy levels showing the energy levels of nucleons and the magic numbers corresponding to filled shells. Shaded areas represent the gaps between the shells. 

Nature s helping hand in this matter came from the fact that the energy levels of the three crucial nuclei could be fine-mned to boost the reaction probability, with the fortunate consequence that carbon and the heavier nuclei could then be produced in significant quantities. Since energy levels are purely quantum features of nucleon systems, one could only rejoice at nature s delicate touch. 

With a general understanding of the form of nuclear potentials, we can begin to solve the problem of the calculation of the properties of the quantum mechanical states that will fill the energy well. One might imagine that the nucleons will have certain finite energy levels and exist in stationary states or orbitals in the nuclear well similar to the electrons in the atomic potential well. This interpretation is... 

Figure 6.3 Energy level pattern and spectroscopic labeling of states from the schematic shell model. The angular momentum coupling is indicated at the left side and the numbers of nucleons needed to fill each orbital and each shell are shown on the right side. From M. G. Mayer and J. H. D. Jenson, Elementery Theory of Nuclear Shell Structure, Wiley, New York, 1955.

Figure 6.19 A schematic version of the potential energy well derived from the Fermi gas model. The highest filled energy levels reach up to the Fermi level of approximately 32 MeV. The nucleons are hound hy approximately 8 MeV, so the potential energy minimum is relatively shallow.

Nuclei with certain even numbers of protons and of neutrons are the most stable. One model of the nucleus is the collective model, which pictures nucleons as occupying quantized energy levels and interacting with one another strongly. 

Nucleons have a quantum ordering, analogous to the quantum energy levels that electrons occupy. However, the quantum rules for protons and neutrons are much more complicated and heyond our purposes here. 

T, F The nucleons (protons and neutrons) are located in the nucleus of an atom. An atom s electrons are located in the principal energy levels. 

Detailed study of the chart of nuclides makes evident that for certain values of P and N a relatively large number of stable nuchdes exist. These numbers are 2, 8, 20, 28, 50, 82 (126, only for N). The preference of these magic numbers is explained by the shell structure of the atomic nuclei (shell model). It is assumed that in the nuclei the energy levels of protons and of neutrons are arranged into shells, similar to the energy levels of electrons in the atoms. Magic proton numbers correspond to filled proton shells and magic neutron numbers to filled neutron shells. Because in the shell model each nucleon is considered to be an independent particle, this model is often called the independent particle model. 

The collective model, in which nucleons are considered to occupy quantized energy levels and to interact with each other by the strong force and the electrostatic (coulomb) force... 

What evidence exists to support the theory that nucleons are arranged in shells or energy levels within the nucleus ... 

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