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Does an Effective E2 Operator have a Two-Body Part?

  • F. Khanna
  • M. Harvey
  • D. W. L. Sprung
  • A. Jopko

Abstract

The question of calculating the effective charge for E2 transitions has been quite annoying to the theoretical physicists. In recent years a general theory for the effective transition operators as needed in the nuclear shell model has been formulated (1). The basic idea is to define a model space of dimension d and an operator \({\rm P}\;{\rm ( = }\;\sum\limits_{{\rm i\varepsilon d}} | \;{\rm \phi }_{\rm i} > < {\rm \phi }_{\rm i} |)\) that projects onto this space. Then we can define an operator Q that projects onto the excluded space \(\matrix{ {{\rm Q}\;{\rm = }\;{\rm 1 - P}\;{\rm = }\;{\rm 1}\;{\rm - }\;\mathop \sum \limits_{{\rm i}\varepsilon {\rm d}} |\;{\rm \phi }_{\rm i} > < {\rm \phi }_{\rm i} |\; = \;_{\rm i} \Sigma _{\rm d} |\;{\rm \phi }_{\rm i} > < {\rm \phi |}} & {{\rm where}} & {{\rm |}\;{\rm \phi }_{\rm i} > } \cr } \) are the eigen-functions of the model Hamiltonian H0. As Bruce Barrett mentioned this morning, an effective Hamiltonian can be defined as
$${\rm H}_{{\rm eff}} = \;{\rm H}_{\rm 0} \; + \;{\rm P}\upsilon {\rm P}$$
where
$$\matrix{ {\upsilon = {\rm V}\;\Omega } & {{\rm and}} & {\Omega = 1 + {{\rm Q} \over {{\rm E - H}_{\rm 0} }}} \cr } \,\upsilon $$
Heff has the same eigenvalue spectrum as the original total Hamiltonian H.

Keywords

Matrix Element Body Part Transition Rate Effective Charge Nuclear Shell Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Press, New York 1972

Authors and Affiliations

  • F. Khanna
    • 1
    • 2
  • M. Harvey
    • 1
    • 2
  • D. W. L. Sprung
    • 1
    • 2
  • A. Jopko
    • 1
    • 2
  1. 1.Chalk River Nuclear LaboratoriesAtomic Energy of Canada Ltd.Canada
  2. 2.McMaster UniversityHamiltonCanada

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