Pandya theorem
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The Pandya theorem is a good illustration of the richness of information forthcoming from a judicious use of subtle symmetry principles connecting vastly different sectors of nuclear systems. It is a tool for calculations regarding both particles and holes.
Description
[edit]Pandya theorem provides a theoretical framework for connecting the energy levels in jj coupling of a nucleon-nucleon and nucleon-hole system. It is also referred to as Pandya Transformation or Pandya Relation in literature. It provides a very useful tool for extending shell model calculations across shells, for systems involving both particles and holes.
The Pandya transformation, which involves angular momentum re-coupling coefficients (Racah-Coefficient), can be used to deduce one-particle one-hole (ph) matrix elements. By assuming the wave function to be "pure" (no configuration mixing), Pandya transformation could be used to set an upper bound to the contributions of 3-body forces to the energies of nuclear states.
History
[edit]It was first published in 1956 as follows:
Nucleon-Hole Interaction in jj Coupling
S.P. Pandya, Phys. Rev. 103, 956 (1956). Received 9 May 1956
A theorem connecting the energy levels in jj coupling of a nucleon-nucleon and nucleon-hole system is derived, and applied in particular to Cl38 and K40.
Shell model Monte Carlo approaches to nuclear level densities
[edit]Since it is by no means obvious how to extract "pairing correlations" from the realistic shell-model calculations, Pandya transform is applied in such cases. The "pairing Hamiltonian" is an integral part of the residual shell-model interaction. The shell-model Hamiltonian is usually written in the p-p representation, but it also can be transformed to the p-h representation by means of the Pandya transformation. This means that the high-J interaction between pairs can translate into the low-J interaction in the p-h channel. It is only in the mean-field theory that the division into "particle-hole" and "particle-particle" channels appears naturally.
Features
[edit]Some features of the Pandya transformation are as follows:
- It relates diagonal and non-diagonal elements.
- To calculate any particle-hole element, the particle-particle elements for all spins belonging to the orbitals involved are needed; the same holds for the reverse transformation. Because the experimental information is nearly always incomplete, one can only transform from the theoretical particle-particle elements to particle-hole.
- The Pandya transform does not describe the matrix elements that mix one-particle one-hole and two-particle two-hole states. Therefore, only states of rather pure one-particle one-hole structure can be treated.
Pandya theorem establishes a relation between particle-particle and particle-hole spectra. Here one considers the energy levels of two nucleons with one in orbit j and another in orbit j' and relate them to the energy levels of a nucleon hole in orbit j and a nucleus in j. Assuming pure j-j coupling and two-body interaction, Pandya (1956) derived the following relation:
This was successfully tested in the spectra of
Figure 3 shows the results where the discrepancy between the calculated and observed spectra is less than 25 keV.
Bibliography
[edit]- Pandya, Sudhir P. (1956-08-15). "Nucleon-Hole Interaction in jj Coupling". Physical Review. 103 (4). American Physical Society (APS): 956–957. Bibcode:1956PhRv..103..956P. doi:10.1103/physrev.103.956. ISSN 0031-899X.
- Racah, G.; Talmi, I. (1952). "The pairing property of nuclear interactions". Physica. 18 (12). Elsevier BV: 1097–1100. Bibcode:1952Phy....18.1097R. doi:10.1016/s0031-8914(52)80178-8. ISSN 0031-8914.
- Wigner, E. (1937-01-15). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Physical Review. 51 (2). American Physical Society (APS): 106–119. Bibcode:1937PhRv...51..106W. doi:10.1103/physrev.51.106. ISSN 0031-899X.
Notes
[edit]- ^ From Nuclear To Sub-Hadronic Physics : A Global View Of Indian Efforts by Asoke N Mitra (preprint - Nov. 18, 2006)
References
[edit]- Lawson, R. D. (1980). Theory of the nuclear shell model. Oxford: Clarendon Press. p. 195. ISBN 0-19-851516-2. OCLC 6938483. OSTI 6688143. (formula 3.68)
- Muto, Kazuo (2006-10-10). "Double Beta Decay and Spin-Isospin Ground-State Correlations". Journal of Physics: Conference Series. 49 (1). IOP Publishing: 110–115. Bibcode:2006JPhCS..49..110M. doi:10.1088/1742-6596/49/1/024. ISSN 1742-6588. S2CID 250672618.
- Bobyk, A.; Kamiński, W.A.; Zaręba, P. (1998). "Effects Of The Pion Wave Distortion On The Absorption/Emission Mechanism Of The DCX Reaction On 56Fe". Acta Physica Polonica B. 29 (3): 799. Bibcode:1998AcPPB..29..799B.
- Asahi, K; Uchida, M; Shimada, K; Nagae, D; Kameda, D; et al. (2006-10-10). "Structure of unstable nuclei from nuclear moments and β decays". Journal of Physics: Conference Series. 49 (1). IOP Publishing: 79–84. Bibcode:2006JPhCS..49...79A. doi:10.1088/1742-6596/49/1/018. ISSN 1742-6588. S2CID 250672135.
- Molinari, A.; Johnson, M.B.; Bethe, H.A.; Alberico, W.M. (1975). "Effective two-body interaction in simple nuclear spectra". Nuclear Physics A. 239 (1). Elsevier BV: 45–73. Bibcode:1975NuPhA.239...45M. doi:10.1016/0375-9474(75)91132-x. ISSN 0375-9474.
- Cloessner, Paul F.; Stöffl, Wolfgang; Sheline, Raymond K.; Lanier, Robert G. (1984-02-01). "Low-lying states in 96Nb from the (t,α) reaction". Physical Review C. 29 (2). American Physical Society (APS): 657–659. Bibcode:1984PhRvC..29..657C. doi:10.1103/physrevc.29.657. ISSN 0556-2813.