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This work mainly investigates the properties of the low-energy quadrupole strength in Ni isotopes, especially the evolution of the pygmy quadrupole states with the increase of neutron number. And the effect of shell evolution on the pygmy resonance is also discussed in detail. Based on the Skyrme Hartree-Fock+Bardeen-Cooper-Schrieffer (HF+BCS) theory and the self-consistent quasiparticle random phase approximation (RPA) method, the evolution in the nickel isotope chain with the increase of neutron number is studied. And in the calculations, three effective Skyrme interactions, namely SGII, SLy5 and SKM*, and a density-dependent zero-range type force are adopted. The properties of the first 2+ state in Ni isotopes are studied. A good description on the experimental excited energies of the first 2+ states are achieved, and the SGII and SLy5 can well describe the reduced electric transition probabilities for $^{58-68}{\rm{Ni}}$. It is found that the energy value of the first 2+ state for $^{68}{\rm{Ni}}$ and $^{78}{\rm{Ni}}$ are obviously high than those of other nuclei, reflecting the obvious shell effect. In addition to the first 2+ states, pygmy quadrupole states between 3 MeV and 5 MeV with relatively large electric transition probabilities are evidently found for $^{70-76}{\rm{Ni}}$ in the isoscalar quadruple strength distribution. The pygmy quadrupole states have the energy values decreasing with the number of neutrons increasing, but their strengths increase gradually. Therefore, they are more sensitive to the change in the shell structure. This is due to the fact that the gradual filling of the neutron level $1{{\mathrm{g}}}_{9/2}$ has a significant effect on the pygmy quadrupole states of $^{70-76}{\rm{Ni}}$, and it leads to switching from proton-dominated excitations to neutron-dominated ones. The pygmy quadrupole states for $^{70-76}{\rm{Ni}}$ are sensitive to the proton and neutron shell gaps, so they can provide the information about the shell evolution in neutron-rich nuclei.
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States $^{64}{\rm{Ni}}$ $^{68}{\rm{Ni}}$ $^{72}{\rm{Ni}}$ $^{76}{\rm{Ni}}$ $E_{{\mathrm{q.p}}.}$ $\upsilon^{2}$ $E_{{\mathrm{q.p}}.}$ $\upsilon^{2}$ $E_{{\mathrm{q.p}}.}$ $\upsilon^{2}$ $E_{{\mathrm{q.p.}}}$ $\upsilon^{2}$ $1{\rm{f}}_{7/2}$ 7.43 0.98 8.89 0.99 10.47 0.99 11.47 1.00 $2{\rm{p}}_{3/2}$ 2.51 0.86 3.60 0.96 5.17 0.98 6.20 0.99 $1{\rm{f}}_{5/2}$ 1.95 0.55 2.66 0.89 4.34 0.95 5.48 0.98 $2{\rm{p}}_{1/2}$ 1.70 0.47 2.04 0.86 3.53 0.95 4.57 0.99 $1{\rm{g}}_{9/2}$ 4.30 0.05 2.59 0.12 1.84 0.44 1.68 0.80 $2{\rm{d}}_{5/2}$ 8.45 0.00 6.65 0.01 4.91 0.01 3.61 0.01 $\lambda_{n}$ –9.34 –7.98 –6.66 –5.84 
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$^{64}{\rm{Ni}}$ $^{70}{\rm{Ni}}$ $^{76}{\rm{Ni}}$ Configurations ${{E}}_{\rm{conf.}}$ Percentage/% ${{b}}_{{{cd}}}$ Configurations ${{E}}_{\rm{conf.}}$ Percentage/% ${{b}}_{{{cd}}}$ Configurations ${{E}}_{\rm{conf.}}$ Percentage/% ${{b}}_{{{cd}}}$ 第一个2+态 1.46 MeV 2.52 MeV 2.08 MeV $\nu 1{\rm{f}}_{5/2}-\nu 2{\rm{p}}_{1/2}$ 3.65 27.85 –7.45 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 4.07 68.71 17.43 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 3.37 71.34 –16.45 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 3.89 24.67 –9.02 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.12 15.53 7.55 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 5.30 11.88 –8.64 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 4.83 18.58 –10.52 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 7.68 4.39 3.00 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.54 8.92 –6.30 $\nu 2{\rm{p}}_{3/2}-\nu 2{\rm{p}}_{1/2}$ 4.22 12.75 –5.10 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 6.15 2.40 1.49 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 5.79 2.21 –1.49 $\nu 2{\rm{p}}_{3/2}-\nu 1{\rm{f}}_{5/2}$ 4.46 3.27 –1.27 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 7.22 1.42 0.84 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{7/2}$ 8.75 0.70 –0.83 $\nu 2{\rm{p}}_{3/2}-\nu 2{\rm{p}}_{3/2}$ 5.03 2.35 –1.50 $\nu 1{\rm{f}}_{5/2}-\nu 2{\rm{p}}_{1/2}$ 6.48 1.13 0.53 矮四极共振态 5.16 MeV 4.98 MeV 4.11 MeV $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 4.83 61.35 –10.38 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.12 57.89 10.52 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 5.30 45.03 –13.35 $\nu 2{\rm{p}}_{3/2}-\nu 2{\rm{p}}_{3/2}$ 5.03 22.68 3.22 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 4.07 28.57 –8.98 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 3.37 26.99 7.85 $\nu 2{\rm{p}}_{3/2}-\nu 2{\rm{p}}_{1/2}$ 4.22 6.20 1.97 $\nu 1{\rm{f}}_{5/2}-\nu 2{\rm{p}}_{1/2}$ 6.48 3.85 1.23 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.54 20.62 –6.82 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 3.89 4.72 2.14 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 7.68 3.77 2.46 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 5.79 3.27 –1.29 $\nu 2{\rm{p}}_{3/2}-\nu 1{\rm{f}}_{5/2}$ 4.46 2.78 0.70 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 7.22 1.75 1.18 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{7/2}$ 8.75 0.64 –0.71 $\nu 1{\rm{f}}_{5/2}-\nu 2{\rm{p}}_{1/2}$ 3.65 1.62 0.93 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 6.15 1.15 0.80 矮四极共振态 6.89 MeV 6.46 MeV 6.31 MeV $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 6.59 90.06 –5.54 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 6.15 75.16 –5.15 $\pi 1{\rm{f}}_{7/2}-\pi 1{\rm{f}}_{5/2}$ 5.79 51.09 –4.34 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 8.60 2.14 –1.48 $\nu 1{\rm{f}}_{5/2}-\nu 2{\rm{p}}_{1/2}$ 6.48 12.00 1.83 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 5.29 30.23 9.29 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 4.83 2.06 1.49 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.12 4.39 2.44 $\pi 1{\rm{f}}_{7/2}-\pi 2{\rm{p}}_{3/2}$ 5.54 16.37 –5.49 $\nu 2{\rm{p}}_{3/2}-\nu 2{\rm{p}}_{1/2}$ 4.22 1.07 0.82 $\nu 1{\rm{g}}_{9/2}-\nu 2{\rm{d}}_{5/2}$ 7.68 3.47 –2.21 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{7/2}$ 8.75 1.04 –0.63 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 3.89 0.9 1.02 $\nu 1{\rm{f}}_{5/2}-\nu 1{\rm{f}}_{5/2}$ 7.22 1.42 –0.88 $\nu 1{\rm{f}}_{7/2}-\nu 1{\rm{f}}_{5/2}$ 9.37 0.9 –0.45 $\nu 1{\rm{g}}_{9/2}-\nu 1{\rm{g}}_{9/2}$ 4.07 1.38 1.69
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]
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