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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 of the nucleus of 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.)The calculated excited energy values of the first 2+ states can accurately accord with the experimental values, 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 states, reflecting the obvious shell effect. In addition to the first 2+ states, pygmy quadrupole states between 3 and 5 MeV with relatively large electric transition probabilities are evidently found for $^{70-76}{\rm{Ni}}$ in the isoscalar quadruple strength distribution [see Fig. (b)]. 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$g_{9/2}$ 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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图 2 (a)图: 利用SGII, SLy5和SkM*相互作用计算的镍同位素链原子核的第一个2+态激发能与实验值的对比. (b)图: 对应的电磁跃迁强度与实验值的对比. 实验数据取自文献[46]
Figure 2. Panel (a): The energies of the first 2+ state in Ni isotopes are obtained by using SGII, SLy5, and SkM*interactions, compared with the experimental data. Panel (b): Corresponding electromagnetic transition strengths. The experimental data is taken from Refs.[46].
图 3 镍同位素链原子核的同位旋标量四极强度分布. 计算采用了SGII相互作用. 图 (a): $^{60-68}{\rm{Ni}}$的结果. 图 (b): $^{70-78}{\rm{Ni}}$的结果
Figure 3. The isoscalar quadrupole strength distributions in Ni isotopes. The SGII interaction is employed in the calculations. Figure (a): The results for $^{60-68}{\rm{Ni}}$. Figure (b): The results for $^{70-78}{\rm{Ni}}$.
图 4 镍同位素链原子核低能区跃迁强度. 计算采用了SGII相互作用
Figure 4. The transition strength for the low-energy region in Ni isotopes. The SGII interaction is employed in the calculations.
图 5 组态$\nu 1\text{g}_{9/2}\longrightarrow\nu 1\text{g}_{9/2}$和$\nu 1\text{g}_{9/2}\longrightarrow\nu 2\text{d}_{5/2}$对$^{64-76}{\rm{Ni}}$矮四极共振态的贡献百分比与跃迁概率幅. 图 (a): 贡献百分比. 图 (b): 跃迁概率幅. 计算采用了SGII相互作用
Figure 5. The contribution percentage and reduced transition amplitudes ${b}_{cd}$ of configurations $\nu 1\text{g}_{9/2}\longrightarrow\nu 1\text{g}_{9/2}$ and $\nu 1\text{g}_{9/2}\longrightarrow\nu 2\text{d}_{5/2}$ contributed to the pygmy quadrupole states in $^{64-76}{\rm{Ni}}$. Figure (a): The contribution percentage. Figure (b): The reduced transition amplitudes ${b}_{cd}$. The SGII interaction is employed in the calculations.
表 1 利用SGII相互作用计算的$^{64, 68, 72, 76}{\rm{Ni}}$费米面附近中子态的准粒子能$E_{q.p.}$ (MeV)、占据几率$\upsilon^{2}$以及中子费米面$\lambda_n$ (MeV)
Table 1. The quasi-particle energies ($E_{q.p.}$ in MeV), occupation probabilities ($\upsilon^{2}$) of neutron states around the Fermi level and neutron Fermi energies ($\lambda_n$ in MeV) in $^{64, 68, 72, 76}{\rm{Ni}}$, which are calculated by using SGII interaction.
States $^{64}{\rm{Ni}}$ $^{68}{\rm{Ni}}$ $^{72}{\rm{Ni}}$ $^{76}{\rm{Ni}}$ $E_{q.p.}$ $\upsilon^{2}$ $E_{q.p.}$ $\upsilon^{2}$ $E_{q.p.}$ $\upsilon^{2}$ $E_{q.p.}$ $\upsilon^{2}$ $1{\rm{f}}_{7/2}$ 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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表 2 对$^{64, 70, 76}{\rm{Ni}}$的第一个2+态以及矮四极共振态做出主要贡献的准粒子组态的组态能量${{E}}_{{\rm{conf}}.}$ (MeV), 贡献百分比以及对应的跃迁概率幅${{b}}_{cd}$ (fm2). 计算采用了SGII相互作用. 此处, π和ν分别代表质子态和中子态.
Table 2. The quasiparticle configurations giving the major contribution to the first 2+ and pygmy quadrupole states in Ni isotopes. For each transition, configuration energies (${{E}}_{{\rm{conf}}.}$ in MeV), their contribution to the norm of the state (in percentage) and the corresponding reduced transition amplitudes (${{b}}_{cd}$ in fm2) are given for $^{64}{\rm{Ni}}$, $^{70}{\rm{Ni}}$, and $^{76}{\rm{Ni}}$, respectively. The SGII interaction is employed in the calculations. Herein, the superscripts π and ν refer to the proton and neutron states, respectively.
$^{64}{\rm{Ni}}$ $^{70}{\rm{Ni}}$ $^{76}{\rm{Ni}}$ 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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