Transition sum rules in the shell model
An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the nonenergyweighted sum rule (NEWSR), or total strength, and the energy weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the doublecommutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We then apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous ThomasReicheKuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$shell.
 Authors:

^{[1]};
^{[2]}
 Qufu Normal Univ., Shandong (China). College of Physics and Engineering; Shanghai Jiao Tong Univ., Shanghai (China). School of Physics and Astronomy; San Dieog State Univ., CA (United States). Dept. of Physics
 San Diego State Univ., CA (United States). Dept. of Physics
 Publication Date:
 Grant/Contract Number:
 FG0203ER41272; FG0296ER40985; 11225524; 11705100; SC0009971
 Type:
 Accepted Manuscript
 Journal Name:
 Physical Review C
 Additional Journal Information:
 Journal Volume: 97; Journal Issue: 3; Journal ID: ISSN 24699985
 Publisher:
 American Physical Society (APS)
 Research Org:
 San Diego State Univ., CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Nuclear Physics (NP) (SC26); National Natural Science Foundation of China (NNSFC)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; nuclear shell model; electromagnetic transitions; sum rules; electroweak interactions in nuclear physics
 OSTI Identifier:
 1430213
 Alternate Identifier(s):
 OSTI ID: 1430378
Lu, Yi, and Johnson, Calvin W. Transition sum rules in the shell model. United States: N. p.,
Web. doi:10.1103/PhysRevC.97.034330.
Lu, Yi, & Johnson, Calvin W. Transition sum rules in the shell model. United States. doi:10.1103/PhysRevC.97.034330.
Lu, Yi, and Johnson, Calvin W. 2018.
"Transition sum rules in the shell model". United States.
doi:10.1103/PhysRevC.97.034330.
@article{osti_1430213,
title = {Transition sum rules in the shell model},
author = {Lu, Yi and Johnson, Calvin W.},
abstractNote = {An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the nonenergyweighted sum rule (NEWSR), or total strength, and the energy weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the doublecommutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We then apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous ThomasReicheKuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$shell.},
doi = {10.1103/PhysRevC.97.034330},
journal = {Physical Review C},
number = 3,
volume = 97,
place = {United States},
year = {2018},
month = {3}
}