Nucleons orbital angular momentum

The spin of a proton is, like an electron, a neutron also has spin In addition to spin angular momentum, the nucleons in a nucleus have orbital angular momentum the orbital angular-momentum quantum number of a nucleon can be 0,1,2, The spin angular momentum of each nucleon... 

Quantum Number (Orbital). A quantum number characterizing the orbital angular momentum of an electron in an atom or of a nucleon in the shell-model description of the atomic nucleus. The symbol for the orbital quantum number is l. 

To form an a particle within a nucleus, two protons and two neutrons must come together with their spins coupled to zero and with zero orbital angular momentum relative to the center of mass of the a particle. These four nucleons are likely to come from the highest occupied levels of the nucleus. In odd A nuclei, because of the odd particle and the difficulty of getting a partner for it, one pah of nucleons is drawn from a lower lying level, causing the daughter nucleus to be formed in an excited state. 

We assume that each nucleon has a pseudo-spin i and pseudo-orbital angular momentum k. These couple to form the single particle angular momenta J,J (in [j]) of the two interacting nucleons. The wavefunction of a pair of nucleons coupled to a total angular momentum L (and z component p) is then given by ... 

For composite systems, like an electron in an atom or a nucleon in a nucleus, the (magnetic) spin quantum number m may have two values, +1/2 or —1/2, because the spin vector has two possible orientations (up or down) with regard to the orbital angular momentum. 

The even-Z-even-N nuclei have even parity in their ground states. The parity of the ground state of odd-A nuclei is determined by (—1), where I is the orbital angular momentum of the unpaired nucleon (see in the shell model, Sect. 2.3.1.2). In the case of the odd-odd nuclei the parity of the ground state is described by the product of the odd proton and odd neutron parities. 

The situation is complicated for nucleons as they carry both orbital angular momentum and intrinsic spin, leading to complex expressions for the g-factors. However, they can be calculated readily, and then compared to the observed values to make structural assignments. We will refer to the g-factors due to the individual nucleons as g, and to the g-factor due to the nucleus rotating as a whole as gR. Since only the protons contribute to the magnetic moment of the rotating nucleus we use gi Z/A as a good approximation [30]. 

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.

Since there exist two types of angular momentum, one caused by orbital movement of the individual nucleons and the other due to the intrinsic spin of the nucleons (internal angular momaitum), a more practical formulation of (4.8) is... 

As an exanq>le, consider die level designation lij t/2> This has the following interpretation the principal quantum number is 1 i indicates that the orbital quantum number / is 6 the angular momentum quantum number j is 11/2 (/ = / — 1/2). The number of permitted nucleons in each level is + 1, thus 12 for j = 11/2. 

The fine structure of atomic line spectra and the hyperfine splittings of electronic Zeeman spectra are non-symmetric for those atomic nuclei whose spin equals or exceeds unity, / > 1. The terms of the spin Hamiltonian so far mentioned, that is, the nuclear Zeeman, contact interaction, and the electron-nuclear dipolar interaction, each symmetrically displace the energy, and the observed deviation from symmetry therefore suggests that another form of interaction between the atomic nucleus and electrons is extant. Like the electronic orbitals, nuclei assume states that are defined by the total angular momentum of the nucleons, and the nuclear orbitals may deviate from spherical symmetry. Such non-symmetric nuclei possess a quadrupole moment that is influenced by the motion of the surrounding electronic charge distribution and is manifest in the hyperfine spectrum (Kopfer-mann, 1958). 

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