# 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
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## References

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© 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