Analysis of the tensor force in shell evolution and current establishments
Atomic structure is characterized by the specific shell structure. The magic numbers corresponding to the shell closures depend on the N/Z ratio. This happens when we have nuclei in the beta stability line and which shift towards the particle driplines. Since the shell structure plays a major role in determining the shell closures, it is important to understand the underlying processes in more details.
Recent developments in work of lighter nuclei on terms of the nucleon-nucleon(NN) interaction, the shell structure evolution has been widely defined. Otsuka et al, have published that central spin-isospin exchange term of the NN interactions play a major role in shell evolution.
However, the analysis of heavier nuclei have lead to another idea on importance of the tensor force. The is based on experimental analysis of one particle or one hole states in nuclei in addition to semi magic configurations with the Effective Single Particle Energies(ESPE’s). The ESPE are governed by the one nucleon separation energies for the unoccupied orbital plus monopole part of the two body residual interaction[ref]. The spacing between the ESPE’s define the energies for the excitation of the individual nucleons and effective shell gaps are produced. ESPE as a function of proton or neutron number is one of the contributions of the monopole interactions in regarding the change in shell gaps.[ref]
The monopole Hamiltonian is a spherical mean field from the interacting shell model. It’s ESPE’s provide the base for the formation of shells, their configuration and deformation in nuclei. Certain magic numbers can be explained on the basis of large shell gaps from a monopole Hamiltonian. A decrease in the shell gaps may lead to a deformed ground state.
From the experimental datas, an extra attraction is seen between the generalised spin-orbit partners of proton and neutrons defined by their angular momentum. This validates the necessity of the pure tensor force in the interaction. However, it has been readily studied and mainly in an empirical way[ref]. Since there have been no systematic calculations for many body systems within the shell model even if it indicates that the inclusion of the three nucleon forces can help define the microscopic effective interaction. Thus study of the two nucleon case becomes more relevant.
The role played by the tensor force may still be not well determined but there are positive signs. Pion exchanges mainly generates the tensor force and can be discussed as one pion exchange forces[ref]. Since there is some contribution due to the rho meson as well, it is defined in terms of pion+rho exchange forces.
The tensor force due to the one pion exchange effects are due to the coupled dipole operators of spins, coupling of two operators of angular momentum, spherical harmonics of Euler angles of relative coordinate and as a function of radial distance. Since, the dipole operators of the spin have values coupled to rank 0,1,2 the total spin of the interacting nucleons is given by the value of the ranks. This plays an important role in the shell evolution. Theoretically, the tensor forces change the spin-orbit splitting significantly which thus leads to tensor force guided shell evolution theoretically. Also, the effective role of the tensor forces can be studied in co-relation with the central forces. Infact, the terms of tensor force, central forces and vector forces result in the cumulative total of the nuclear forces.
Effect of tensor interaction on the structure of the doubly shell closed nucleus and the utility of the tensor force in the evolution of the shell structure in proton or neutron rich nuclei through the shell model have been studied my many physicists. Zheng and Zamick[ref] constructed a general interaction where they could determine the effects of the two body spin-orbit and tensor interactions in nuclei. In works by Otsuka et al[ref], relative movement of single particle levels was explained interms of absolute separation of energy. A Gaussian tensor force was added to a standard form of Gogny force after adjusting long range part of one pion and rho exchange potential. An addition of zero range tensor force was added in the Skymen interaction in the Hartree-Fock formalism for reproduction of single particle spectrum near the Ferne surface[Brown et al, ref]. This attempt was though unsuccessful in describing the angular momentum dependence of the spin-orbit splitting. The calculation of the energy difference in shell model introducing the tensor force in Skymen’s model was done by Colo et al, Brink and Stance[ref].This was done for Z=50 isotopes and N=82 isotones.
Further, the presence of the tensor force leading to strong rearrangement of spin-orbit force has larger impact on the particle spectra[Lewinsky et al,ref]. Moreover the main effect of the tensor force lies in the evolution of spin-orbit splitting with the N and Z. Moreno-Torres et al[ref] have also addressed the effect of the tensor force in shell evolution. Where the magic numbers 8 and 20 are suitable to be described by fitting tensor parameters in Mean Field Approach. Zou et al, have shown that when one moves form Ca48 to Ar46 , the reduction of the neutron spin-orbit splitting can be explained by the inclusion of the tensor terms.
All these works are focused on showing how the tensor force affects the evolution of the shell model of the nucleon based on different interactions. Although substantive research analysis still have to be done in proving the real contribution of the tensor forces in the evolution if the shell model these works will lay a foundation for works in the future. Detailed analysis can be found on the evolution of the tensor force and its contribution to the shell model and collective interaction with the central and vector forces be will be parts of the further studies and research and thus an approach to generalizing the nature of nuclear forces might be possible.
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