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The light Sn isotopes between the closed shells with
$N=50$ and 82 is the longest chain that can be calculated by the shell model[16]. With the increase in size and number of valence nucleons and the broken Cooper pairs, the space size of the exact shell model will also increase. For heavy nucleus, the dimension will increase even more, which will bring certain difficulties to the calculation. The introduction of seniority truncation can effectively reduce the matrix dimension[17-18]. By using the fast algorithm of generalized seniority and modern realistic interaction, the real CD-Bonn nucleon-nucleon potential[19] has been calculated, and the perturbed G matrix method is used to normalize the interaction, considering the core polarization effect[20]. The lowest 10 000 states (5 000 of each parity) of$ ^{108-124} {\rm{Sn}}$ in the subspace of unpaired nucleons are calculated when the generalized seniority truncates the quantum number 8.For the model space, the orbits with proton number 50 and neutron number between 50~82 are selected, and the isospin symmetry condition is considered in the effective Hamiltonian[16]. Set subspace
$ \left| S,\; M \right \rangle $ , where$S~(S = S_{{{\pi }}}+S_{{\nu}}$ ,${\rm{\pi }}$ ,${\rm { \nu}}$ represent protons and neutrons, respectively) is the total generalized alteration,$M (M = M_{{\rm{\pi }}} + M_{\nu}$ ) is the total magnetic projection. The Hamiltonian H contains three parts:$H = H_{{\rm{\pi }}} + H_{\nu} + H_{{\rm{\pi }}\nu}(H_{{\rm{\pi }}\nu}$ is proton neutron interaction). For this article, there are no valence protons, so the proton part is not considered, only valence neutrons are discussed. Partition calculation is used in the calculation, which is more effective and faster than the accurate shell model. The truncation of Generalized seniority is consistent with the idea of condensation of coherent pairs with the same pair of structures and has a smaller size[21]. The Hamiltonian is taken from Ref. [16]. Common practice in the calculation of the full configuration-interacting shell model is to express the effective Hamiltonian with the energy of a single particle and the numerical value of the two-body matrix elements, see the following formula:$$ \begin{split} H =& \sum\limits_{\alpha} \epsilon _{\alpha}\hat{N}_{\alpha}+\\ &\frac{1}{4} \sum\limits_{\alpha \beta \delta \gamma JT} \left \langle j_{\alpha} j_{\beta} \right| V \left| j_{\gamma} j_{\delta} \right \rangle _{JT} A^+_{JT;j_{\alpha}j_{\beta}} A_{JT;j_{\delta}j_{\gamma}}\,,\end{split} $$ (1) where
$ \alpha = \{ nljt \} $ represents the single particle orbitals,$ \epsilon _{\alpha} $ represents the corresponding single particle energy;$ \hat{N} _{\alpha} $ is the particle number operator;$ \left \langle j_{\alpha} j_{\beta} \right| V \left| j_{\gamma} j_{\delta} \right \rangle $ is a two-body matrix element coupled to a good spin J and isospin T;$ ( A_{JT} ) A^{+}_{JT} $ is Fermion's annihilation (creation) operator.For the calculation of energy level density, different models have been established before. Back at least in 1936, Bethe proposed the Fermi gas model (FGM), which is only applicable to high-energy areas[22]. The first model tested, the back-shifted Fermi-gas model (BSFGM) contains three parameters
$ a $ ,$ \sigma $ and E1. U and J represent excitation energy and spin, respectively[see Eq. (2)][23-27]. Then developed the standard Bethe formula independent of a specific model to the more generalized Bethe formula. The generalized formula for double-closed shell$ ^{208}{\rm Pb} $ is better than the standard one[28]. Later, the constant-temperature model (CTM) was proposed[29-30]. There are two parameters$ E_0 $ and$ T $ , which can very well represent the experimental data of Sn below the the neutron separation energy. Considering pairing correlation, shell effect, and collectiveness, generalized superfluid model has appeared, and the energy density is shown. However, this model ignores the rotating collective effect[31]. The Oslo cyclotron group extracted the nuclear level density by measuring$ \gamma $ -ray spectra below low spin[32-35].$$ \rho(U,J) = \frac{\exp\big\{2\sqrt{[a(U-E_1)]}\big\}f(J)}{24\sqrt{2}\sigma a^{\frac{1}{4}}(U-E_1)^{\frac{5}{4}}}\,. $$ (2) The thermodynamic quantity can be extracted by the energy level density, and the microcanonical entropy can be derived. The microcanonical entropy
$ S(E) $ is a measure of the number of ways to arrange quantum systems under a given excitation energy. The microcanonical entropy can be calculated according to Eq. (3)[36], where$ \varOmega (E) $ represents the multiplicity [see Eq. (4)],$ \rho _0 $ is determined by the ordered system in which the ground state entropy of even-even nuclei is zero. For$ ^{116} {\rm{Sn}}$ ,$ \rho_0 = 0.135 $ MeV–1[14] ,$$ \ S(E) = k_{\rm{B}}\ln\varOmega(E)\,, $$ (3) $$ \varOmega(E) = \frac{\rho(E)}{\rho_0}. $$ (4)
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摘要: 能级密度在研究核物理的动力学、热力学性质中具有重要的物理意义。本文利用壳模型和截断算法计算了Sn同位素的最低10 000个状态。
$^{116}{\rm{Sn}}$ 在Sn同位素中空穴数和粒子数都为8,能级最为复杂,根据$^{116}{\rm{Sn}}$ 能级密度研究了能级密度与角动量的关系和体系的微正则熵。研究发现在偶宇称时的最低平均能级出现明显的角动量奇偶效应,用泡利不相容原理得到合理解释。进一步,又研究了Sn同位素的此性质,得到相同的结论。$^{116}{\rm{Sn}}$ 的微规范熵在低能量段受奇宇称的波动,这与中子对的断裂有关。Abstract: Energy level density plays an important role in the study of kinetic and thermodynamic properties of nuclear physics. In this paper, shell model and the truncation algorithm are used to calculate the lowest 10 000 states of Sn isotopes. The number of holes and the number of particles in the$^{116}{\rm{Sn}}$ isotope of Sn is 8, and the energy level is the most complex. According to the$^{116}{\rm{Sn}}$ energy level density, the relationship between energy level density and angular momentum and the microcanonical entropy of the system are studied. It is found that the lowest average energy level under even parity has an obvious odd-even effect of angular momentum, which can be reasonably explained by Pauli exclusion principle. Furthermore, the properties of different Sn isotopes are studied and a similiar conclusion is obtained. The microcanonical entropy of$^{116}{\rm{Sn}}$ is fluctuated by odd parity in the low energy band, which is related to the fracture of neutron pairs.-
Key words:
- 116Sn /
- level densities /
- Pauli exclusion principle
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