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The accurate calculation of the isotope shift factors is helpful in extracting the mean-square charge radius of the nucleus,which is an important nuclear parameter to investigate the nuclear properties and improve nuclear structure theories.However,for atomic systems with many electrons the uncertainties of the calculated isotope shift factors are difficult to evaluate accurately,since high sensitivity of the isotope shift factor to the electron correlation and limitation of the computational resource.Based on the calculations of the isotope shift factors of the 3s2 1S0→ 3s3p 3,1P1o transitions in Al+by using the multi-configuration Dirac-Hartree-Fock method,the convergences of these physical quantities with the expansion of the configuration space are investigated in detail.In our calculation,the electron correlations are divided into the first-order correlation and the higher-order correlations according to the perturbation theory,and captured by using the active space approach.The effect of the first-order correlation are considered by including configuration state functions (CSFs) that are generated by the single and double substitutions from the occupied orbitals in the single reference configuration set.After the first-order correlation effect are taken into account adequately,the reference configuration sets are augumented by adding the dominant CSFs from the first-order correlation configuration space,in order to consider the higher-order correlation effect.We find that the convergence of the mass shift factors (including the normal shift factor and the specific mass shift factor) is linearly correlated with the convergence of the level energies in our computational model.For the transitions,the linear correlation of the convergence between the mass shift factors and the transition energies is not so good as that for the levels involved in the transitions due to the limited computational resource,but it can be improved with the expansion by including more higher-order correlation related 2s and 2p core electrons.Furthermore,we made use of the linear correlation to estimate the uncertainties of our isotope shift factors, and obtain the reasonable value of error.The authors hope that the linear correlation between the convergence of the mass shift factors and the level or transition energies can be proved and explained in more atomic systems,and the linear correlation can be used to evaluate accurately the uncertainties of the mass shift factors for the atoms and ions with many electrons in the near future.
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Keywords:
- multi-configuration Dirac-Hartree-Fock method /
- electron correlation /
- isotope shift /
- Al+ ion
[1] Campbell P, Moore I D, Pearson M R 2016 Prog. Part. Nucl. Phys. 86 127
[2] Cheal B, Cocolios T E, Fritzsche S 2012 Phys. Rev. A 86 042501
[3] Carette T, Godefroid M R 2011 Phys. Rev. A 83 062505
[4] Carette T, Godefroid M R 2010 Phys. Rev. A 81 042522
[5] Carette T, Godefroid M R 2013 J. Phys. B: At. Mol. Opt. Phys. 46 095003
[6] Carette T, Godefroid M R 2016 arXiv: 1602.06574.
[7] Liu J P, Li J G, Zou H X 2017 Chin. Phys. B 26 23104
[8] Grant I P 2007 Relstivistic Quantum Theory of Atom and Molecules (New York: Spinger)
[9] Li J G, Jönsson P, Godefroid M R, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523
[10] Tupitsyn I I, Shabaev V M, Crespo López-Urrutia J R, Draganić I, Soria Orts R, Ullrich J 2003 Phys. Rev. A 68 022511
[11] Filippin L, Beerwerth R, Ekman J, Fritzsche S, Godefroid M R, Jönsson P 2016 Phys. Rev. A 94 062508
[12] Palmer C W P 1987 J. Phys. B: At. Mol. Opt. Phys. 20 5987
[13] Torbohm G, Fricke B, Rosén A 1985 Phys. Rev. A 31 2038
[14] Filippin L, Godefroid M R, Ekman J, Jönsson P 2016 Phys. Rev. A 93 062512
[15] Filippin L, Bieroń J, Gaigalas G, Godefroid M R, Jönsson P 2017 Phys. Rev. A 96 042502
[16] Gaidamauskas E, Nazé C, Rynkun P, Gaigalas G, Jönsson P, Gaigalas G 2011 J. Phys. B: At. Mol. Opt. Phys. 44 175003
[17] Li J G, Godefroid M R, Wang J G 2016 J. Phys. B: At. Mol. Opt. Phys. 49 115002
[18] Zhang T X, Xie L Y, Li J G, Lu Z H 2017 Phys. Rev. A 96 012514
[19] Jönsson P, He X, Fischer C F, Grant I 2007 Comput. Phys. Commun. 177 597
[20] Jönsson P, Gaigalas G, Bierón J, Fischer C F, Grant I 2013 Comput. Phys. Commun. 184 2197
[21] Nazé C, Gaidamauskas E, Gaigalas G, Godefroid M R, Jönsson P 2013 Comput. Phys. Commun. 184 2187
[22] Kramida A, Ralchenko Y, Reader J 2016 NIST Atomic Spectra Database (Version 5) https://www.nist.gov/pml/atomic-spectra-database
[23] Godefroid M R, Fischer C F, Jönsson P 2001 J. Phys. B: At. Mol. Opt. Phys. 34 1079
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[1] Campbell P, Moore I D, Pearson M R 2016 Prog. Part. Nucl. Phys. 86 127
[2] Cheal B, Cocolios T E, Fritzsche S 2012 Phys. Rev. A 86 042501
[3] Carette T, Godefroid M R 2011 Phys. Rev. A 83 062505
[4] Carette T, Godefroid M R 2010 Phys. Rev. A 81 042522
[5] Carette T, Godefroid M R 2013 J. Phys. B: At. Mol. Opt. Phys. 46 095003
[6] Carette T, Godefroid M R 2016 arXiv: 1602.06574.
[7] Liu J P, Li J G, Zou H X 2017 Chin. Phys. B 26 23104
[8] Grant I P 2007 Relstivistic Quantum Theory of Atom and Molecules (New York: Spinger)
[9] Li J G, Jönsson P, Godefroid M R, Dong C Z, Gaigalas G 2012 Phys. Rev. A 86 052523
[10] Tupitsyn I I, Shabaev V M, Crespo López-Urrutia J R, Draganić I, Soria Orts R, Ullrich J 2003 Phys. Rev. A 68 022511
[11] Filippin L, Beerwerth R, Ekman J, Fritzsche S, Godefroid M R, Jönsson P 2016 Phys. Rev. A 94 062508
[12] Palmer C W P 1987 J. Phys. B: At. Mol. Opt. Phys. 20 5987
[13] Torbohm G, Fricke B, Rosén A 1985 Phys. Rev. A 31 2038
[14] Filippin L, Godefroid M R, Ekman J, Jönsson P 2016 Phys. Rev. A 93 062512
[15] Filippin L, Bieroń J, Gaigalas G, Godefroid M R, Jönsson P 2017 Phys. Rev. A 96 042502
[16] Gaidamauskas E, Nazé C, Rynkun P, Gaigalas G, Jönsson P, Gaigalas G 2011 J. Phys. B: At. Mol. Opt. Phys. 44 175003
[17] Li J G, Godefroid M R, Wang J G 2016 J. Phys. B: At. Mol. Opt. Phys. 49 115002
[18] Zhang T X, Xie L Y, Li J G, Lu Z H 2017 Phys. Rev. A 96 012514
[19] Jönsson P, He X, Fischer C F, Grant I 2007 Comput. Phys. Commun. 177 597
[20] Jönsson P, Gaigalas G, Bierón J, Fischer C F, Grant I 2013 Comput. Phys. Commun. 184 2197
[21] Nazé C, Gaidamauskas E, Gaigalas G, Godefroid M R, Jönsson P 2013 Comput. Phys. Commun. 184 2187
[22] Kramida A, Ralchenko Y, Reader J 2016 NIST Atomic Spectra Database (Version 5) https://www.nist.gov/pml/atomic-spectra-database
[23] Godefroid M R, Fischer C F, Jönsson P 2001 J. Phys. B: At. Mol. Opt. Phys. 34 1079
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