Generalized-seniority pattern and thermal properties in even Sn isotopes

Even tin isotopes of mass number $A=108–124$ are calculated with realistic interactions in the generalized-seniority approximation of the nuclear shell model. For each nucleus, we compute the lowest 10 000 states (5000 of each parity) up to around 8 MeV in excitation energy, by allowing as many as four broken pairs. The lowest 50 eigen energies of each parity are compared with the exact results of the large-scale shell-model calculation. The wave functions of the midshell nuclei show a clear pattern of the stepwise breakup of condensed coherent pairs with increasing excitation energy. We also compute in the canonical ensemble the thermal properties—level density, entropy, and specific heat—in relation to the thermal pairing phase transition.

Figure 1

Collective pair structures $v_j$ (2) in tin isotopes $(A$ is the mass number). They are normalized such that $v_{0g_{7/2}}=1$.

Figure 2

Errors of the generalized-seniority eigen energies for the lowest 50 positive-parity eigenstates in ${}^{110-122}\mathrm{Sn}$. Every panel has 50 data points, each represents one $J$ state. The horizontal coordinate is the exact excitation energy from the shell-model calculation. The vertical coordinate is the error of the eigenenergy from the generalized-seniority calculation, relative to the exact shell-model eigenenergy. The vertical dotted line is $E=E_{s=1}^{<}$ as explained in the text.

Figure 3

Errors of the generalized-seniority eigenenergies for the lowest 50 negative-parity eigenstates in ${}^{110-122}\mathrm{Sn}$.

Figure 4

The generalized-seniority $\left(S=2s\right)$ mean $\overline{s}$ and fluctuation $\mathrm{\Delta }s$, and the level density $\rho$, versus the excitation energy in ${}^{108}\mathrm{Sn}$. The upper (lower) panel plots the lowest 5000 eigenstates with positive (negative) parity. Each black plus (blue circle) symbol represents one state; its horizontal coordinate is the excitation energy, and the vertical coordinate is $\overline{s}$ $\left(\mathrm{\Delta }s\right)$ corresponding to the left axis. The red solid line corresponding to the right axis plots the level density $\rho$ averaged over an energy bin of 0.4 MeV. The two vertical dotted lines are $E=E_{s=1}^{<}$ and $E=E_{s=2}^{<}$, respectively.

Figure 5

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{110}\mathrm{Sn}$.

Figure 6

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{112}\mathrm{Sn}$.

Figure 7

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{114}\mathrm{Sn}$.

Figure 8

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{116}\mathrm{Sn}$.

Figure 9

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{118}\mathrm{Sn}$.

Figure 10

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{120}\mathrm{Sn}$.

Figure 11

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{122}\mathrm{Sn}$.

Figure 12

The generalized-seniority mean and fluctuation, and the level density, versus the excitation energy in ${}^{124}\mathrm{Sn}$.

Figure 13

Amplitudes $P\left(s\right)$ of each generalized seniority $S=2s$ versus the excitation energy in ${}^{108}\mathrm{Sn}$. The left (right) panels plot the lowest 5000 eigenstates with positive (negative) parity. Therefore each panel has 5000 data points. The two vertical dotted lines $E=E_{s=1}^{<}$ and $E=E_{s=2}^{<}$ are at the same positions as those in Fig. 4.

Figure 14

Amplitudes of each generalized seniority versus the excitation energy in ${}^{110}\mathrm{Sn}$.

Figure 15

Amplitudes of each generalized seniority versus the excitation energy in ${}^{112}\mathrm{Sn}$.

Figure 16

Amplitudes of each generalized seniority versus the excitation energy in ${}^{114}\mathrm{Sn}$.

Figure 17

Amplitudes of each generalized seniority versus the excitation energy in ${}^{116}\mathrm{Sn}$.

Figure 18

Amplitudes of each generalized seniority versus the excitation energy in ${}^{118}\mathrm{Sn}$.

Figure 19

Amplitudes of each generalized seniority versus the excitation energy in ${}^{120}\mathrm{Sn}$.

Figure 20

Amplitudes of each generalized seniority versus the excitation energy in ${}^{122}\mathrm{Sn}$.

Figure 21

Amplitudes of each generalized seniority versus the excitation energy in ${}^{124}\mathrm{Sn}$.

Figure 22

Generalized-seniority compositions $P\left(s\right)$ of the $0_1^{+}$ and $2_1^{+}$ states in Sn isotopes.

Figure 23

Generalized-seniority amplitudes $P\left(s\right)$ of the $J=0$ states in ${}^{116}\mathrm{Sn}$. In other words we take the data points from those in Fig. (17) corresponding to $J=0$. The two vertical dotted lines $E=E_{s=1}^{<}$ and $E=E_{s=2}^{<}$ are at the same positions as those in Figs. 8 and 17.

Figure 24

Generalized-seniority amplitudes of the $J=4$ states in ${}^{116}\mathrm{Sn}$.

Figure 25

Generalized-seniority amplitudes of the $J=7$ states in ${}^{116}\mathrm{Sn}$.

Figure 26

Generalized-seniority amplitudes of the $J=10$ states in ${}^{116}\mathrm{Sn}$.

Figure 27

The canonical-ensemble mean-energy $\langle E\rangle$, entropy $S$, and specific heat $C$ versus the temperature $T$ in ${}^{108}\mathrm{Sn}$. $k_B$ is the Boltzmann constant. The three curves correspond to the three different energy cutoffs $E_c$.

Figure 28

The canonical-ensemble mean-energy, entropy, and specific heat versus the temperature in ${}^{116}\mathrm{Sn}$.

Figure 29

The canonical-ensemble mean-energy, entropy, and specific heat versus the temperature in ${}^{124}\mathrm{Sn}$.