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.
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.
Historyedit
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 densitiesedit
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.
Featuresedit
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.
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.
Notesedit
^From Nuclear To Sub-Hadronic Physics : A Global View Of Indian Efforts by Asoke N Mitra (preprint - Nov. 18, 2006)
Referencesedit
Lawson, R. D. (1980). Theory of the nuclear shell model. Oxford: Clarendon Press. p. 195. ISBN0-19-851516-2. OCLC 6938483. OSTI 6688143. (formula 3.68)
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.
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.
April 11, 2024
pandya, theorem, this, article, multiple, issues, please, help, improve, discuss, these, issues, talk, page, learn, when, remove, these, template, messages, article, lead, section, need, rewritten, please, help, improve, lead, read, lead, layout, guide, decemb. This article has multiple issues Please help improve it or discuss these issues on the talk page Learn how and when to remove these template messages The article s lead section may need to be rewritten Please help improve the lead and read the lead layout guide December 2021 Learn how and when to remove this template message This article provides insufficient context for those unfamiliar with the subject Please help improve the article by providing more context for the reader December 2021 Learn how and when to remove this template message Learn how and when to remove this template message 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 Contents 1 Description 2 History 3 Shell model Monte Carlo approaches to nuclear level densities 4 Features 5 Bibliography 6 Notes 7 ReferencesDescription editPandya 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 editIt was first published in 1956 as follows Nucleon Hole Interaction in jj CouplingS P Pandya Phys Rev 103 956 1956 Received 9 May 1956A 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 editSince 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 editSome 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 ofFigure 3 shows the results where the discrepancy between the calculated and observed spectra is less than 25 keV 1 Bibliography editPandya 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 editLawson 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 Kaminski W A Zareba 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 b 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 Stoffl Wolfgang Sheline Raymond K Lanier Robert G 1984 02 01 Low lying states in 96Nb from the t a 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 Retrieved from https en wikipedia org w index php title Pandya theorem amp oldid 1191591505, wikipedia, wiki, book, books, library,
