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Figure 3(a) exhibits the systematics of the
$0 _{1}^{+} $ ,$2 _{1}^{+} $ ,$4 _{1}^{+} $ ,$6 _{1}^{+} $ and$8 _{1}^{+} $ in$ ^{90} {\rm{Zr}}$ ,$ ^{92} {\rm{Mo}}$ ,$ ^{94} {\rm{Ru}}$ and$ ^{96} {\rm{Pd}}$ even-even nuclei[6, 21, 26, 35, 37] as well as the$9/2 _{1}^{+} $ ,$13/2 _{1}^{+} $ ,$17/2 _{1}^{+} $ ,$21/2 _{1}^{+} $ and$25/2 _{1}^{+} $ states in$ ^{91} {\rm{Nb}}$ ,$ ^{93} {\rm{Tc}}$ ,$ ^{95} {\rm{Rh}}$ and$ ^{97} {\rm{Ag}}$ odd-$ A $ nuclei[17, 24, 56- 57]. The level structures between the$ N = 50 $ even-even nuclei and the neighboring odd-$ A $ nuclei, shown for comparison in Fig. 3(a), are similar up to the$4 _{1}^{+} $ state. For example, the level energies of$13/2 _{1}^{+} $ and$17/2 _{1}^{+} $ levels in$ ^{93} {\rm{Tc}}$ are close to the energies of$2 _{1}^{+} $ and$4 _{1}^{+} $ levels in the$ ^{92} {\rm{Mo}}$ ($ ^{94} {\rm{Ru}}$ ) core. The systematic of the energy levels in the$ N = 48 $ odd-$ A $ and the even-even isotones is presented in Fig. 3(b). The$13/2 _{1}^{+} $ and$17/2 _{1}^{+} $ states in$ ^{89} {\rm{Nb}}$ [4],$ ^{91} {\rm{Tc}}$ [6],$ ^{93} {\rm{Rh}}$ [58] and$ ^{95} {\rm{Ag}}$ [59] are close in energies with the$2 _{1}^{+} $ and$4 _{1}^{+} $ states in$ ^{88} {\rm{Zr}}$ [5],$ ^{90} {\rm{Mo}}$ [60],$ ^{92} {\rm{Ru}}$ and$ ^{94} {\rm{Pd}}$ [61], respectively. The above features could be explicated by the weak coupling model. Based on the weak coupling simplification of expressions, the low spins of an odd-$ A $ nucleus can be interpreted as a nucleon in a single-$ j $ orbit coupled to an even-even core. The$ \psi_I^{\dagger} | 0 \rangle $ denotes the wave function of the low level$ I_1^{+} $ ($ E_I^{} $ ) of the even-even core. The$ \psi_I^{\dagger} | 0 \rangle $ coupled to the single nucleon$ a_j^{\dagger} $ can generate the multiplet states with spin$ J $ for the odd-mass nucleus:Figure 3. (a) The low-energy levels in the
$ N = 50 $ isotones; (b) the same as (a) but for$ N = 48 $ isotones.$$ \left(\psi_I^{\dagger} \times a_j^{\dagger}\right)_J^{} | 0 \rangle, $$ (1) where
$J = |I-j|,|I-j|+1,\cdots,I+jI+j $ , represents the angular momentum, We employ$ E_J^{}(I,j) $ to represent the excitation energies of the multiplet states. The relation between$ E_I^{} $ and$ E_J^{}(I,j) $ is expressed by$$ E_I^{} = \frac{ \sum_{J}^{}(2J+1)E_J^{}(I,j) }{\sum_{J}^{}2J+1}. $$ (2) The configuration
$ \pi $ $ g_{9/2}^{} $ $ \otimes $ ($2 _{1}^{+} $ ,$ ^{92} {\rm{Ru}}$ ) could generate multiplet states with$5/2 _{1}^{+} $ (621-keV),$7/2 _{1}^{+} $ (240-keV),$9/2 _{2}^{+} $ (1630-keV),$11/2 _{1}^{+} $ (894-keV) and$13/2 _{1}^{+} $ (852-keV) in$ ^{93} {\rm{Rh}}$ . Using Eq. (2), the calculated energy of$2 _{1}^{+} $ level in$ ^{92} {\rm{Ru}}$ is 892-keV, which is consistent with the experimental one (865-keV). It is noteworthy that high spins are not in compliance with the results of the weak coupling framework, for example the energy of the$8 _{1}^{+} $ state in$ ^{88} {\rm{Zr}}$ is much higher than the$ 25_{1}^{+} $ state in$ ^{89} {\rm{Nb}}$ . This might be due to the fact that the core excitations in high-spins of the even-even core may not be negligible, and configuration admixtures become increasingly significant.
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摘要: 在原子核高自旋态的能级结构研究中, 处于
$A\approx 90$ 质量区的原子核一直是核结构研究的热点,其质子数和中子数分别接近$Z = 38$ 半幻数与$N = 50$ 幻数。 本工作基于近年来$A = 90$ 核区实验结果, 研究 该质量区原子核的单粒子激发, 核心激发, 高自旋侵入态和同质异能态的特征。通过调研发现,奇-$A$ 核素的较低自旋能级可以解释为其相邻偶偶核与一个价核子耦合而成。 该核区$2_{1}^{+}$ 态能级与${\rm{E}}_{4_{1}^{+}}$ /${\rm{E}}_{2_{1}^{+}}$ 比值的演化特征表明,$N = 56$ 半闭壳在$Z = 40$ (41)核素对能级结构影响显著, 随着质子数的增加$Z \geqslant 42$ 时影响减弱甚至消失。此外, 低自旋及中等自旋能级结构中强${\rm{E}}2$ 跃迁主要涉及($f_{5/2}^{}$ ,$p_{3/2}^{}$ ,$p_{1/2}^{}$ ,$g_{9/2}^{}$ )轨道耦合, 而对于高自旋态能级结构强$M1$ 跃迁主要由($f_{5/2}^{}$ ,$p_{3/2}^{}$ ,$p_{1/2}^{}$ )轨道质子激发与$g_{9/2}^{}$ 轨道中子跨越$N = 50$ 闭壳跃迁到$d_{5/2}^{}$ 轨道耦合产生。奇-$A$ 核素的$N = 50 $ (51)同中子素中的一些同质异能态来源于单中子或单质子与$g_{9/2}^{}$ 轨道质子对顺排耦合。-
关键词:
- core excitation /
- shell model /
- coupling model
Abstract: In the investigations of the level structure of$A \approx 90$ nuclei, whose numbers of protons and neutrons are close to the$Z = 40$ semimagic number and$N = 50$ magic number, have become a hot spot in nuclear physics. The aim of this work is to further probe the characteristics of single-particle excitation, core breaking, high-$j$ intruder states and isomeric states in the$A\approx 90$ mass region based on the existing experimental results. Investigations show that the low energy levels of the odd-$A$ nuclei originate from their neighboring even-even nuclei coupled to a valence nucleon. The systematics of the$2_{1}^{+}$ excitation energies and the values of${\rm{E}}_{4_{1}^{+}}$ /${\rm{E}}_{2_{1}^{+}}$ indicate that the$N = 56$ subshell closure may appear at$Z = 40$ (41) and disappear for$Z > 42$ nuclei. Furthermore, in this mass region, the strong${\rm{E}}2$ transitions at low or medium spins are interpreted as the recoupling of the pure protons in ($f_{5/2}^{}$ ,$p_{3/2}^{}$ ,$p_{1/2}^{}$ ,$g_{9/2}^{}$ ) orbits, and the strong$M1$ transitions are explicated by moving proton from the ($f_{5/2}^{}$ ,$p_{3/2}^{}$ ,$p_{1/2}^{}$ ) orbits to the$g_{9/2}^{}$ orbit, coupling to a neutron excitation from the$g_{9/2}^{}$ orbit across$N = 50$ closed shell into the$d_{5/2}^{}$ orbit. The isomeric states in odd-$A$ nuclei with$N = 50 $ (51) can be interpreted as a spin-aligned configuration in which a single neutron or proton couples with a fully aligned proton pair in the$\pi g_{9/2}^{}$ orbit.-
Key words:
- core excitation /
- shell model /
- coupling model
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Figure 1. Partial energy levels in
$ N = 50 $ nuclei. (a) odd-$ A $ isotones$ ^{85} {\rm{Br}}$ ,$ ^{87} {\rm{Rb}}$ ,$ ^{89} {\rm{Y}}$ ,$ ^{91} {\rm{Nb}}$ ,$ ^{93} {\rm{Tc}}$ ,$ ^{95} {\rm{Rh}}$ , and (b) even-even isotones$ ^{86} {\rm{Kr}}$ ,$ ^{88} {\rm{Sr}}$ ,$ ^{90} {\rm{Zr}}$ ,$ ^{92} {\rm{Mo}}$ ,$ ^{94} {\rm{Ru}}$ ,$ ^{96} {\rm{Pd}}$ . The neutron core-excited states from the$ N = 50 $ core are denoted by quadrilaterals, and the proton excited across the$ Z = 40 $ subshell are denoted by circles.Figure 6. The evolution of positive parity states in even-even N = 56 isotones
$ ^{96} {\rm{Zr}}$ ,$ ^{98} {\rm{Mo}}$ ,$ ^{100} {\rm{Ru}}$ ,$ ^{102} {\rm{Pd}}$ , and$ ^{104} {\rm{Cd}}$ ; The evolution of positive parity states in odd-A N = 50 isotones$ ^{97} {\rm{Nb}}$ ,$ ^{99} {\rm{Tc}}$ ,$ ^{101} {\rm{Rh}}$ ,$ ^{103} {\rm{Ag}}$ , and$ ^{105} {\rm{In}}$ . -
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