A nucleon-pair and boson coexistent description of nuclei

TL;DR

This paper investigates the coexistence and crossover between nucleon pairs and s-bosons in nuclei, demonstrating the robustness of BCS-BEC phenomena through realistic pf-shell calculations and B(E2) transition predictions.

Contribution

It introduces a combined nucleon-pair and boson model with Pauli blocking effects, showing smooth BCS-BEC crossover behavior in nuclei with realistic interactions.

Findings

Smooth BCS-BEC crossover near half-shell in pf-shell nuclei.

B(E2) transition values are well reproduced for Sn isotopes.

Robustness of coexistence phenomena confirmed with realistic single-particle energies.

Abstract

We study a mixture of s-bosons and like-nucleon pairs with the standard pairing interaction outside a inert core. Competition between the nucleon-pairs and s-bosons is investigated in this scenario. The robustness of the BCS-BEC coexistence and crossover phenomena is examined through an analysis of pf-shell nuclei with realistic single-particle energies in which two configurations with Pauli blocking of nucleon-pair orbits due to the formation of the s-bosons is taken into account. When the nucleon-pair orbits are considered to be independent of the s-bosons, the BCS-BEC crossover becomes smooth with the number of the s-bosons noticeably more than that of the nucleonpairs near the half-shell point, a feature that is demonstrated in the pf-shell for several values of the standard pairing interaction strength. As a further test of the robustness of the BCS-BEC coexistence and crossover phenomena in nuclei, results are given for B(E2; 0^{+}_{g}->2^{+}_1) values of even-even 102-130Sn with 100Sn taken as a core and valence neutron pairs confined within the 1d5/2, 0g7/2, 1d3/2, 2s1/2, 1h11/2 orbits in the nucleon-pair orbit and the s-boson independent approximation. The results indicate that the B(E2) values are well reproduced.

Figures

Tables

Table 1. Table 1: The parameter r 𝑟 r used in fitting the 2 1 + subscript superscript 2 1 2^{+}_{1} level energies (in MeV) of even-even 102-130 Sn, for which the experimental values shown in exp2 are used.

Nucleus	$n$	$r$		$E\left(2_1^{+}\right)$	Nucleus	$n$	$r$		$E\left(2_1^{+}\right)$
102Sn	1	3.885		1.472	104Sn	2	3.544		1.260
106Sn	3	3.270		1.208	108Sn	4	3.025		1.206
110Sn	5	2.792		1.212	112Sn	6	2.579		1.257
114Sn	7	2.366		1.300	116Sn	8	2.127		1.294
118Sn	9	1.862		1.230	120Sn	10	1.585		1.171
122Sn	11	1.318		1.141	124Sn	12	1.063		1.132
126Sn	13	0.822		1.141	128Sn	14	0.591		1.169
130Sn	15	0.367		1.221

Equations

$$\hat{H}=\sum _j{ϵ}_j{\hat{N}}_j-G\sum _{j,j^{\prime }}{S}_j^{+}{S}_{j^{\prime }}^{-}+$$

$$\alpha ({\hat{n}}_s)-rG\sqrt{\Omega }\sum _j(S_j^{+}s+s^{†}{S}_j^{-}),$$

$$|n;\zeta ⟩=\sum _{k=0}^n\sum _{{\xi }_k}{C}_{k,{\xi }_k}^{(\zeta )}|n-k;{\xi }_k,k⟩|n-k{⟩}_s,$$

$$\frac{1}{G}\left(E_k^{({\xi }_k)}+\alpha (n-k)-E_n^{(\zeta )}\right)C_{k,{\xi }_k}^{(\zeta )}=$$

$$\sqrt{\Omega (n-k+1)}\sum _{{\xi }_{k-1}}{C}_{k-1,{\xi }_{k-1}}^{(\zeta )}\times$$

$$⟨n-k;{\xi }_k,k|S^{+}|n-k+1;{\xi }_{k-1},k-1⟩$$

$$+\sqrt{\Omega (n-k)}\sum _{{\xi }_{k+1}}{C}_{k+1,{\xi }_{k+1}}^{(\zeta )}\times$$

$$⟨n-k;{\xi }_k,k|S^{-}|n-k-1;{\xi }_{k+1},k+1⟩$$

$${\rho }_s=\frac{1}{n}⟨n;\zeta =1|{\hat{n}}_s|n;\zeta =1⟩$$

$${\Delta }_s=\sqrt{⟨n;\zeta =1|{\left({\hat{n}}_s-n{\rho }_s\right)}^2|n;\zeta =1⟩}.$$

$$T_{\mu }(E2)=q_2(\frac{1-\chi }{2}(d_{\mu }^{†}s+s^{†}{\tilde{d}}_{\mu })$$

$$+\frac{1+\chi }{2\sqrt{\Omega }}(d_{\mu }^{†}{S}^{-}+S^{+}{\tilde{d}}_{\mu })),$$

Full text

Introduction: The interacting boson model (IBM) 1 , like the collective model 2 of Bohr and Mottelson and the microscopic shell model 31 ; 3 , has been successful in providing a logical framework for studying the structure of atomic nuclei, at least in the low-energy regime. Many studies show that the $J=2$ $d$-bosons of the IBM are similar to the $d$-phonons that emerge naturally from a five-dimensional harmonic oscillator description of the quadrupole vibrations of the collective model. The addition of a $J=0$ $s$-boson to the $d$-boson picture allows the IBM to accommodate couplings to low-energy monopole modes, which expands the U(5) algebra structure that underpins the five-dimensional oscillator to a six-dimensional U(6) algebra 4 . Within this expanded U(6) framework of the IBM, the total number of bosons is regarded as the number of valence particle or hole pairs 5 ; 6 . It is now commonly accepted that there is a close relationship between the $s$-bosons and $J=0$ pairs of like-nucleons, since pairs of fermions in these systems often exhibit boson-like behavior 7 ; 8 . It has been shown that bosons may emerge from fermionic pairing due to spontaneous symmetry breaking of the Bardeen-Cooper-Schrieffer (BCS) type 81 ; 82 . However, a schematic study of the relative roles played by $J=0$ nucleon pairs and $s$-bosons as employed in the IBM in nuclei is still lacking.

Generally, the pairing correlation in Fermi many-body systems can be understood in terms of attractive interactions among fermion pairs manifested by the BCS mechanism. When the attractive interaction between two fermions is strong enough, on the other hand, the two fermions may form a bosonic bound state with Bose-Einstein condensation (BEC) in the ground state of the system. As shown in  9 ; 10 ; 11 ; 12 ; 13 ; duk , a smooth crossover from a Cooper-paired state to a Bose condensate state of tightly bound pairs takes place in the continuum model of a Fermi gas at zero temperature. In 14 , the coexistence of the BCS and the BEC-like pair structures in 11Li was investigated based on phase space analysis. Inspired by the above developments, we consider a mean-field plus standard pairing model including $s$-bosons, where the $s$-bosons are regarded as tightly bound nucleon pairs, which should also emerge naturally in nuclei according to the aforementioned observations.

The model: Since some nucleon-pairs in a nucleus may behave like bosons, we consider a mixture of $s$-bosons and like-nucleon pairs with the standard pairing interaction outside an inert core. A schematic Hamiltonian considered is given by

[TABLE]

where $j$ and $j^{\prime}$ run over $p$ distinct orbits considered, $s^{\dagger}$ ($s$) is the $s$-boson creation (annihilation) operator, $\{\epsilon_{j}\}$ is a set of single-particle energies generated from any mean-field theory, $\hat{N}_{j}=\sum_{m}a^{\dagger}_{jm}a_{jm}$ and $S_{j}^{+}=\sum_{m>0}(-1)^{j-m}a^{\dagger}_{jm}a^{\dagger}_{j-m}$ ($S_{j}^{-}=(S_{j}^{+})^{\dagger}$), in which $a^{\dagger}_{jm}$ ($a_{jm}$) is the creation (annihilation) operator for a nucleon with angular momentum quantum number $j$ and that of its projection $m$, $G>0$ is the overall pairing interaction strength, $r$ is a scale factor used to describe the interaction between $s$-boson and nucleon pairs, $\Omega=\sum_{j}(j+1/2)$ is the maximum pair-occupancy of the $p$-orbit system, and the pure $s$-boson part $\alpha(\hat{n}_{s})$, which is a function of the $s$-boson number operator $\hat{n}_{s}=s^{\dagger}s$, is adjusted to reproduce the ground state energy of the Hamiltonian with the first two terms of the nucleon-pair sector for given $G$. With this assumption, the ground state energy of the system in either the pure BCS phase or the pure BEC phase is exactly the same. In (A nucleon-pair and boson coexistent description of nuclei), the interaction between the $s$-boson and the nucleon pairs is the same as that among nucleon pairs when $r=1$. The factor $\sqrt{\Omega}$ appears in the last term of (A nucleon-pair and boson coexistent description of nuclei) because the nucleon pair operator $S^{+}\sim\sqrt{\Omega}s^{\dagger}$, where $S^{+}=\sum_{j}S^{+}_{j}$ and $S^{-}=(S^{+})^{\dagger}$, when the number of nucleon-pairs $k$ is far smaller than $\Omega$ with $k\ll\Omega$. Here and in the following, we assume that all the nucleons in the system are paired and that there is an even number of nucleons. It is obvious that the total number of the $s$-bosons and that of the nucleon pairs is a conserved quantity in the model.

Since the $s$-bosons originate from nucleon-pairs, many nucleon-pair orbits should be blocked due to the Pauli Exclusion Principle. In a single-$j$ case for example, when there are $n_{s}$ $s$-bosons, the maximal nucleon-pair occupancy should become $\tilde{\Omega}_{j}=j+1/2-n_{s}$. Thus, for given $n=k+n_{s}\leq\Omega$, the eigenstates of (A nucleon-pair and boson coexistent description of nuclei) may be written as

[TABLE]

where $\zeta$ labels the $\zeta$-th excitation state of the system, $|n-k;\xi_{k},k\rangle$ is the eigenstate of the mean-field plus standard pairing model of the nucleon-pair sector with $\tilde{\Omega}=\Omega-(n-k)$.

Generally, if there are $p$ orbits, the maximum number of distinct blocking patterns is equal to the number of partitions of $n-k$ into $p$ integers $[\mu_{1},\cdots,\mu_{p}]$ with $\sum_{i=1}^{p}\mu_{i}=n-k$ and $0\leq\mu_{i}\leq\Omega_{j_{i}}$ for $i=1,\cdots,p$, which results in the effective maximal nucleon-pair occupancy of each orbit for a given partition $[\mu_{1},\cdots,\mu_{p}]$ to be $\tilde{\Omega}_{j_{i}}=\Omega_{j_{i}}-\mu_{i}$ for $i=1,\cdots,p$. We take the $p=2$ case with $j_{1}=3/2$ and $j_{2}=5/2$ as an example. If there is no $s$-boson with $n_{s}=0$, the maximal pair occupancy of each orbit is given by $\Omega_{1}=2$ and $\Omega_{2}=3$, respectively. When $n_{s}=1$, there are two blocking configurations. The first has $\tilde{\Omega}_{1}=1$ and $\tilde{\Omega}_{2}=\Omega_{2}=3$, for which the $s$-boson is formed from two nucleons in the first orbit. The second has $\tilde{\Omega}_{1}={\Omega}_{1}=2$ and $\tilde{\Omega}_{2}=2$, for which the $s$-boson is formed from two nucleons in the second orbit. Let the nucleon-pair product states be denoted as $|n_{1};n_{2}\rangle$, where $n_{1}$ and $n_{2}$ are the number of pairs with $0\leq n_{1}\leq 2$ and $0\leq n_{2}\leq 3$, which should all be needed in diagonalizing the first two terms of (A nucleon-pair and boson coexistent description of nuclei) when $n_{s}=0$. However, $|n_{1}=2;n_{2}=0\rangle$ should be ruled out in diagonalizing the first two terms of (A nucleon-pair and boson coexistent description of nuclei) for $n_{s}=1$ in the first blocking configuration, while $|n_{1}=0;n_{2}=3\rangle$ should be ruled out in diagonalizing the first two terms of (A nucleon-pair and boson coexistent description of nuclei) for $n_{s}=3$ in the second blocking configuration. Namely, the dimension of the subspace spanned by the nucleon-pair product states will be reduced due to Pauli blocking.

The eigenstates of the first two terms of (A nucleon-pair and boson coexistent description of nuclei), $|n-k;\xi_{k},k\rangle$, appear in (2) with a given number of nucleon-pairs $k$ for distinct blocking patterns. However, they are not linearly independent since they are all $k$-pair states. In order to treat this situation more precisely, we need to introduce a mixed state description if the probability of a given partition of the blocking is known. In this work, we still treat this feature within a pure state description for simplicity. Hence, only one of the blocked configurations for a given $k$ is considered.

As a relatively simple example, we set $r=1$ and consider a $pf$-shell system with the $4$ orbitals $0f_{7/2}$, $1p_{3/2}$, $1p_{1/2}$, and $0f_{5/2}$ assigned the single-particle energies deduced in HOB ; that is, we set $\epsilon_{7/2}=-8.624$ MeV, $\epsilon_{3/2}=-5.6793$ MeV, $\epsilon_{1/2}=-4.137$ MeV, and $\epsilon_{5/2}=-1.3829$ MeV in the Hamiltonian (A nucleon-pair and boson coexistent description of nuclei). Then the eigen-equation used to determine the eigen-energies $E^{(\zeta)}_{n}$ of (A nucleon-pair and boson coexistent description of nuclei) and the corresponding expansion coefficients $C^{(\zeta)}_{k,\xi_{k}}$ of (2) are given by

[TABLE]

for $k=0,~{}1,~{}\cdots,~{}n$, where $E^{(\xi_{k})}_{k}$ is the $\xi_{k}$-th $k$-pair excitation energy of the original mean-field plus standard pairing Hamiltonian of the nucleon-pair sector with $\tilde{\Omega}=\Omega-n+k$ for a fixed partition $[\mu_{1},\cdots,\mu_{p}]$ of the blocking, and the matrix elements $\langle n-k;\xi_{k},k|S^{+}|n-k+1;\xi_{k-1},k-1\rangle=\langle n-k+1;\xi_{k-1},k-1|S^{-}|n-k;\xi_{k},k\rangle^{*}$ can be calculated when all excited states of the original mean-field plus standard pairing model for the nucleon-pair sector are obtained. For a given number of nucleon-pairs $k$, we calculate all excitation energies and the corresponding eigenstates of the original mean-field plus standard pairing Hamiltonian of the nucleon-pair sector with $\tilde{\Omega}=\Omega-n+k$ for a fixed partition of the blocking, which can be done by using the the Heine-Stieltjes polynomial approach guan .

After all excitation energies $\{E^{(\xi_{k})}_{k}\}$ and the corresponding eigenstates of the nucleon-pair part of the model are obtained for $1\leq k\leq\Omega$ with $\Omega=10$ for this case, we simply set $\alpha(n_{s})=E^{(\xi_{k}=1)}_{k}$ with $k=n_{s}$ for a given value of $G$. The orbits are then arranged in order according to the values of the single-particle energies with $\epsilon_{j_{1}}<\cdots<\epsilon_{j_{p}}$. Specifically, we consider two blocking configurations. In the first, the blocking to the nucleon-pair orbits results in the configuration with the lowest orbits blocked, namely, we take $\mu_{1}$ to be the largest possible integer, then $\mu_{2}$ to be the remaining largest possible integer, and so on. In the second configuration, the blocking to the nucleon-pair orbits results in the configuration with the highest orbits blocked; namely, we take $\mu_{p}$ to be the largest possible integer, then $\mu_{p-1}$ to be the remaining largest possible integer, and so on. More precisely, in the first blocking configuration, when $n_{s}\leq 4$, we assume the $j_{1}=7/2$ orbit is blocked because it is the lowest in energy. When $5\leq n_{s}\leq 6$, we assume the $j_{1}=7/2$ and $j_{2}=3/2$ orbits are blocked, and so on. Similarly, in the second blocking configuration, when $n_{s}\leq 3$, we assume the $j_{4}=5/2$ orbit is blocked because it is the highest in energy. When $n_{s}=4$, we assume the $j_{4}=5/2$ and $j_{3}=1/2$ orbits are blocked, and so on. In the analysis, the pairing interaction strength $G=1.4$ MeV is fixed. Once (A nucleon-pair and boson coexistent description of nuclei) is solved numerically, we calculate the ground-state occupation probability $\rho_{s}$ of the $s$-bosons

[TABLE]

and the ground-state $s$-boson number fluctuation defined by

[TABLE]

As shown in Fig. 1(a), the $s$-boson occupation probability $\rho_{s}$ is always greater than $40\%$, increasing from $40\%$ to $93\%$ for $n=6$ to $n=10$ in the first blocking configuration. In the second blocking configuration it is always greater than $50\%$, increasing from $50\%$ to $60\%$ over the range of $n$ values shown. The staggering in the occupation probability happens near the half-filling point in the first blocking configuration. In both cases, the $s$-boson occupation probability reaches its maximal value near the shell closure point.

The results for the nucleon-pair orbit and the $s$-boson independent approximation are also shown in Fig. 1 for comparison, with the eigenstates of the mean-field plus standard pairing model of the nucleon-pair sector $\{|n-k;\xi_{k},k\rangle\}$ used in (2) replaced by $\{|0;\xi_{k},k\rangle\}$ for any $n$ and $k$ with the maximal nucleon-pair occupancy of each orbit unchanged, namely, the blocking to the nucleon-pair orbits due to the formation of the $s$-bosons is not considered. In this case, there is a critical region of the BCS-BEC crossover, which tracks with a range of $n$ values near the half-filling of the shell where $\rho_{s}$ reaches its maximal value. Though the BCS-BEC crossover behavior in these three cases are different, the BCS-BEC coexistence seems robust. In any case, once the $s$-bosons emerge in the system with the strong pairing interaction shown in 11 ; 12 ; 13 ; duk , which prefers to occur in a dilute fermion-pair environment 11 ; 12 ; 13 ; duk ; 14 , the $s$-boson content, in general, changes noticeably with increasing $n$ due to the interaction between nucleon-pairs and the $s$-bosons. This reinforces the BCS-BEC crossover even when the number of nucleon-pairs becomes large. In addition, as further shown in Fig. 1(b), the $s$-boson number fluctuation also changes rapidly with $n$, indicating that the nucleon-pair constituent is also significant, which becomes more noticeable near the half-filling point. Therefore, once the $s$-boson emerges in the system, the pure BEC phase never occurs except in the first blocking configuration in the large $\Omega$-limit. Rather, the system always seems to be in a BCS-BEC coexistence phase, which is robust for any finite $\Omega$ and $n$, a feature that is consistent with the conclusion made in 14 , with the $s$-boson content greater than that of the nucleon-pairs in general. Thus, the $s$-boson formation induced by the pairing interaction with fewer nucleon-pairs and the further BCS-BEC crossover enhancement with increasing nucleon-pairs seems to track with the emergence of $s$-bosons in a nucleus.

Since we currently lack probability distribution information for different blocking configurations, the nucleon-pair orbit and the $s$-boson independent approximation is adopted in the following. The $s$-boson occupation probability $\rho_{s}$ and the $s$-boson number fluctuation in the $pf$-shell for $G=0.2$ MeV, $0.6$ MeV, $1.0$ MeV, and $2.0$ MeV with the approximation are shown in Fig. 2, in which the scale factor $r=1$ is still taken. It should be noted that $G$ ranges from $0.4\sim 0.7$ MeV in the $pf$-shell in order to reproduce the GXPF1 $J=0$ and $T=1$ pairing excitation spectrum pan17 . Hence, the results shown in Fig. 2 not only provide information about realistic situations, but also show results for a slightly weaker pairing interaction with $G=0.2$ MeV and a stronger pairing interaction with $G=2.0$ MeV. As shown in Fig. 2(a), similar to the case shown in Fig. 1, the ground-state $s$-boson occupation probability $\rho_{s}$ is always greater than $50\%$. There is a critical region of the BCS-BEC crossover, which tracks with a range of $n$ values near the half-filling of the shell where $\rho_{s}$ reaches its maximal value. The critical region moves towards a larger $n$ interval with $n>\Omega/2$ when the pairing interaction becomes stronger. It is also clear that the maximal value of $\rho_{s}$ increases with increasing $G$, and is about $62\%$ when $G=2.0$ MeV. As can be seen in Fig. 2(b), the largest fluctuation $\Delta_{s}$ also occurs in the critical region, within which $\rho_{s}$ reaches its maximal value. Actually, both the $s$-bosons and the nucleon-pairs are most non-localized in the crossover region in this case.

An example of application: As an example of an application of the theory with the nucleon-pair orbit and the $s$-boson independent approximation, we use the model to fit $2^{+}_{1}$ level energies and then to calculate B(E2; $0^{+}_{1}\rightarrow 2^{+}_{1}$) values of even-even 102-130Sn, which have attracted a lot of attention both experimentally and theoretically be20 ; be21 ; be22 ; be23 ; th1 ; th2 . In our calculation, 100Sn is considered as the core of these nuclei with valence neutron pairs confined to the $1d_{5/2}$, $0g_{7/2}$, $1d_{3/2}$, $2s_{1/2}$, and $1h_{11/2}$ orbits with single-particle energies $\epsilon_{5/2}=0.00$ MeV, $\epsilon_{7/2}=0.08$ MeV, $\epsilon_{3/2}=1.66$ MeV, $\epsilon_{1/2}=1.55$ MeV, and $\epsilon_{11/2}=3.55$ MeV, respectively be20 . Since the low-lying states in the Sn isotopes are almost spherical, the bosonic part of the Hamiltonian of (A nucleon-pair and boson coexistent description of nuclei) in this case is chosen as the IBM U(5) type with $\alpha(\hat{n}_{s},\hat{n}_{d})=\alpha(\hat{n}_{s})+\beta(\hat{n}_{d})$, where $\beta(\hat{n}_{d})$ is a function of the number of $d$-bosons $\hat{n}_{d}$, and $\alpha(n_{s})=E^{(\xi_{n_{s}}=1)}_{n_{s}}$ is still assumed. The total number of bosons and nucleon pairs, $n=n_{s}+n_{d}+k$, is a conserved quantity in this case. For a given $n$, the ground state is given by $|n,0^{+}_{1}\rangle=|n,\zeta=1\rangle$ determined by (2), while the $2^{+}_{1}$ state, for simplicity, is assumed to be one $d$-boson state, of which the $d$-boson is assumed to be formed from two neutrons in the lowest $1d_{5/2}$ orbit. Thus, the $2^{+}_{1}$ state $|n,2^{+}_{1},M\rangle=d^{+}_{M}|n-1,\zeta=1\rangle$, where the Pauli blocking in the $1d_{5/2}$ orbit is considered in $|n-1,\zeta=1\rangle$ for $|n,2^{+}_{1},M\rangle$. Furthermore, the $2^{+}_{1}$ level energy is simply given by $E(n,\,2^{+}_{1})=\beta(1)-\beta(0)+{E}^{(1)}_{n-1}-E^{(1)}_{n}$ for a given $n$, where ${E}^{(1)}_{n-1}$ is determined by (A nucleon-pair and boson coexistent description of nuclei) with the Pauli blocking in the $1d_{5/2}$ orbit considered. In the fitting, the parameters $G$ and $\beta(1)-\beta(0)$ in the model are fixed for all nuclei considered with $G=0.18$ MeV and $\beta(1)-\beta(0)=-10.617$ MeV, while the scale factor $r$ is adjusted to reproduce the experimental $2^{+}_{1}$ level energy exactly, of which the values for these nuclei are given in Table 1. It is clearly shown that $r$ deceases almost linearly as the total number of bosons and nucleon pairs, $n$, increases.

The effective E2 operator in this case is defined as

[TABLE]

where $d_{\mu}^{\dagger}$ is the $d$-boson creation operator, $\tilde{d}_{\mu}=(-1)^{\mu}d_{-\mu}$, in which $d_{\mu}$ is the $d$-boson annihilation operator, $q_{2}$ is the effective quadrupole parameter, and $\chi\in[-1,1]$ is used. In the fitting, $\chi=0.44$ and $q_{2}=0.0827$ eb are chosen from the best fit for all nuclei considered. The B(E2; $0^{+}_{1}\rightarrow 2^{+}_{1}$) obtained from this theory and the corresponding experimental values, together with the results obtained from the large-scale shell model (LSSM) with the same shell model space consideration be20 , are shown in Fig. 3(a), which indicates that the experimental data are well reproduced by this theory with the parameter $r$ determined uniquely by the $2^{+}_{1}$ level energy, except that the B(E2) values of 116Sn and 130Sn are a little larger than the corresponding experimental values. Overall, the B(E2; $0^{+}_{1}\rightarrow 2^{+}_{1}$) values for even-even 102-114Sn obtained from this theory are much better than those obtained from the LSSM, while those for even-even 116-128Sn obtained from the LSSM are a little better. The corresponding ground-state $s$-boson occupation probability $\rho_{s}$ for these nuclei is shown in FIG. 3(b), which indicates that 118,120Sn, according to this theory, are within the critical region of the BCS-BEC crossover with the total number of $s$-bosons and neutron pairs a little larger than the half-filling point value.

Conclusion: In conclusion, the BCS-BEC coexistence and crossover phenomena in the nuclear mean-field plus standard pairing interaction model involving the $s$-bosons is observed from the analysis of the ground state $s$-boson occupation probability. It is shown that, though the BCS-BEC crossover behavior may be different from one Pauli blocking configuration to another, the BCS-BEC coexistence seems robust, as demonstrated in the analysis for the $pf$-shell with realistic single-particle energies in two different blocking configurations or in the nucleon-pair orbit and the $s$-boson independent approximation. The crossover is further enhanced with increasing pairing interaction strength. In the example of the model application to even-even 102-130Sn, the B(E2) values are well reproduced by the model in the nucleon-pair orbit and the $s$-boson independent approximation with parameters determined by the experimental $2^{+}_{1}$ level energy. A mixed state consideration related to the Pauli blocking effect in the model merits further study. Moreover, since BCS-BEC coexistence seems common in nuclei, similar to the interacting Boson-Fermion model for odd-A nuclei ibfm , in which, besides the $s$- and $d$-bosons, only a single nucleon degree of freedom is considered, a configuration which considers both the $s$- and $d$-bosons provided from the IBM and a few interacting valence nucleons in an effective mean-field, such as the shell model, may be a better description of the low-energy structure of nuclei. Further work along these directions is in progress.