Pairing, quasi-spin and seniority
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000047-2
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Review
Pairing, quasi-spin and seniority B.K. Agrawal1,2,a , and Bhoomika Maheshwari3 1 2 3
Saha Institute of Nuclear Physics, Kolkata, India Homi Bhabha National Institute, Mumbai India University of Malaya, Kuala Lumpur, Malaysia Received 23 March 2020 / Accepted 28 August 2020 Published online 23 October 2020 Abstract. We present our concise notes for the lectures and tutorials on pairing, quasi-spin and seniority delivered at SERB school on Role of Symmetries in Nuclear Physics, AMITY University, 2019. Starting with some generic features of residual nucleon–nucleon interactions, we provide detailed derivation of the matrix elements for the δ-interaction which is the basis for the standard pairing Hamiltonian. The eigen values for standard pairing Hamiltonian are then obtained within the quasi-spin formalism. The algebra involving quasi-spin operators is performed explicitly using the annihilation and creation operators for single nucleon together with the application of Wick’s theorem. These techniques are expected to be helpful in deriving the meanfield equations for the Hartree–Fock, Bardeen–Cooper–Schrieffer and Hartree–Fock Bogoliubov theories.
1 Introduction Role of symmetries is important when no exact solutions to a physical problem are known, such as nuclear force. More often, our problem is analogous to the atomic and molecular structure. However, things are simpler there, as Coulomb forces are well known. In case of the nuclear force, the knowledge of its mathematical form is still an open question. Fortunately, due to symmetry properties of the basic interactions in most physical systems, qualitative features of the composite system are not too sensitive to the details of the interaction itself. Symmetries of physical laws may lead to the laws of conservation of spin, isospin and energy. For example, the orbital, spin and isospin quantum operators usually commute with the nuclear interaction Hamiltonian. This means the interaction Hamiltonian commutes with all rotations in orbital, spin and isospin spaces. Therefore, the angular momentum coupling and spherical harmonics play a vital role in understanding various properties of nuclei. One of the most important inputs to the system of interacting nucleons is the matrix elements for the appropriate nucleon–nucleon interaction. These matrix elements can be evaluated by decomposing a given nucleon-nucleon interaction in to its radial and angular parts. The evaluation of two-body matrix elements, thus, reduces to the calculation of the radial and the angular matrix elements. The procedure for the calculation of the two-body matrix elements for different interactions mainly differ a
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The European Physical Journal Special Topics
in their radial matrix elements. The angular part usually depends on the product of the spherical harmonics corresponding to the two nucleons. The angular part of the matrix
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