1. Introduction
The molybdenum isotopes have rich nuclear structures and have been studied both experimentally and theoretically over many years, for example, for 92Mo and 94Mo [1–3] and 96Mo and 98Mo [4–9]. The theoretical methods include the variation-after-projection (VAP) calculations in conjunction with Hartree-Bogoliubov (HB) ansatz applied to A = 100–108 molybdenum isotopes [10], the triaxial projected shell model (TPSM) approach to the band structures for neutron-rich 102–108Mo [11], and algebraic model IBM [12–16]. One of the greatest interests that attract physicists lies in the shape-related topics, such as shape coexistence, shape evolution and shape phase transition from the even–even spherical magic-N-nucleus 92Mo to deformed nucleus 108Mo along the chain of Mo isotopes [17–19].
Shape coexistence has been observed in many parts of the nuclear chart and related theoretical works were carried out for a long time [20–26]. In or near the shell or sub-shell closures, the monopole interaction of nuclear force tends to stabilize the nucleus into a spherical shape. However, in around mid-shell regions, the correlation energy gain tends to drive the nucleus into a deformed shape. The competition between the two opposite actions is the resource of the shape coexistence phenomenon [27]. For 94Mo, the 0+ state at 1.742 MeV is the third excited state above the ground state. But for 96Mo, the 0+ state at 1.148 MeV is the second excited state. The 0+ state at 0.735 MeV becomes the first excited state in 98Mo. The appearance of low-lying 0+ state provides a clue for the coexistence of different shapes in some Mo isotopes [6, 27, 28]. The theoretical description of shape coexistence is mainly based on shell model and the mean field theory. In the shell model, the intruder configurations originated from the valence nucleons across the shell or sub-shell gaps are responsible for the mechanism of shape coexistence. In the mean field theory, one can calculate the potential energy surface (PES) and different shapes correspond to a different minimum in the PES. The interacting boson model (IBM) provides another tool to deal with the shape coexistence. The different shapes correspond to the different algebraic structure group of IBM, therefore, one can discuss the shape-related phenomenon by investigating the group structure. It has been found that there exists three dynamical symmetries characterized by the first algebra in the chain originating from group structure, that is, U(5), SU(3), O(6), corresponding to spherical, axially deformed, and γ-unstable shapes, respectively [29–37].
The IBM-2, which makes a distinction between neutrons and protons, predicts the existence of MS states, which are not completely symmetric under the exchange of the neutron-proton bosons [38, 39]. The totally symmetric states under this exchange are called full symmetry (FS) states. The experimental signatures of MS states are characterized by the strong M1 transitions between FS and MS states with transition matrix element of about one μN and weakly collective E2 transitions between FS and MS states, which strengths can only reach a few percent of $B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$. In addition, there exists strongly collective E2 transitions between MS states, with typical strength comparable to the collective E2 transitions between symmetric states [40, 41].
The properties of 98Mo nucleus have been discussed in many previous papers. In [8], the authors employed the IBM-1 to describe the experimental data available at that time and found that the triaxial ground state and the prolate first excited state coexist in 98Mo. In [12], the authors discussed the effect of configuration mixing on Mo isotopes, but they only calculated the E(2) transitions and did not consider the M(1) transitions and MS states. Both of these two works were based on the previous experiments, the data were not enough at that time. Recently, the excited low-spin states in 98Mo were studied in γγ angular correlation experiment [4], their new experimental data show that both shape coexistence and MS states exist in 98Mo. In [28], the authors found the the IBM-2 could reproduce the correct excitation energy levels of the Mo isotopes without resorting to the configuration mixing by taking the energy of neutron boson ϵν different from that of proton boson ϵπ. In this study, the purpose is to apply the method of [28] to describe the properties of shape coexistence and MS states in 98Mo in a unified framework.
The paper is structured as follows. In section 2 we briefly introduce the theoretical framework of IBM-2 model and its parameters fitted in our calculation. In section 3 , we discuss the calculated results of the low-lying levels in comparison with its corresponding experimental data. In section 4 , we investigate the electromagnetic transition probabilities including $E2,M1$ and E0 properties. Finally, a conclusion is given in section 5 .
2. Theoretical model
The microscopic picture of the IBM originates from the idea that the collective fermion pairs of valence nucleons play an important role in the nuclear structure, where only pairs coupled to L = 0 (s boson) and L = 2 (d boson) are considered. The connection between the IBM and shell model was first pointed out in [42–44]. This is one of the reasons for the success of IBM. In the IBM-2, one assumes that the valence neutrons or protons outside a closed shell can be treated as neutron or proton bosons. For a full description of IBM-2, we refer the reader to [45]. The explicit expression of the Hamiltonian in this calculation is as follows1 ) stand for the dipole proton-neutron and neutron-neutron interaction, respectively. The angular momentum operator is defined as $\hat{{L}_{\rho }}=\sqrt{10}{[{d}_{\rho }^{\dagger }\cdot \tilde{{d}_{\rho }}]}^{(1)}$. The sixth term represents the Majorana interaction, having expression as ${\hat{M}}_{\pi \nu }={\lambda }_{2}{\left({s}_{\pi }^{\dagger }{d}_{\nu }^{\dagger }-{s}_{\nu }^{\dagger }{d}_{\pi }^{\dagger }\right)}^{(2)}\cdot {\left({s}_{\pi }{\tilde{d}}_{\nu }-{s}_{\nu }{\tilde{d}}_{\pi }\right)}^{(2)}$ $+\,{\sum }_{k=\mathrm{1,3}}{\lambda }_{k}{\left({d}_{\pi }^{\dagger }{d}_{\nu }^{\dagger }\right)}^{(k)}\cdot {\left({\tilde{d}}_{\pi }{\tilde{d}}_{\nu }\right)}^{(k)}$. The Majorana parameters λk (k = 1, 2, 3) only influence the excitation energies of states having MS components, which will be fixed by the known experimental data. The last term is the interaction between like bosons. The E2 and M1 transition operators are written as
$\begin{eqnarray}\begin{array}{rcl}\hat{H} & = & {\varepsilon }_{\pi }{\hat{n}}_{{d}_{\pi }}+{\varepsilon }_{\nu }{\hat{n}}_{{d}_{\nu }}+{\kappa }_{\pi \nu }{\hat{Q}}_{\pi }\cdot {\hat{Q}}_{\nu }\\ & & +{\omega }_{\pi \nu }{\hat{L}}_{\pi }\cdot {\hat{L}}_{\nu }+{\omega }_{\nu \nu }{\hat{L}}_{\nu }\cdot {\hat{L}}_{\nu }+{\hat{M}}_{\pi \nu }\\ & & +\displaystyle \frac{1}{2}\sum _{\rho =\pi ,\nu }\sum _{L=0,2,4}{\left(2L+1\right)}^{\tfrac{1}{2}}{c}_{\rho }^{(L)}{\left({[{d}_{\rho }^{\dagger }{d}_{\rho }^{\dagger }]}^{(L)}\cdot {[{\tilde{d}}_{\rho }{\tilde{d}}_{\rho }]}^{(L)}\right)}_{0}^{(0)},\,\end{array}\end{eqnarray}$
where ${\hat{n}}_{d\rho }={d}_{\rho }^{\dagger }\cdot \tilde{{d}_{\rho }}\,$ is the d-boson number operator, and ${\hat{Q}}_{\rho }={\left({s}_{\rho }^{\dagger }{\tilde{d}}_{\rho }+{d}_{\rho }^{\dagger }{s}_{\rho }\right)}^{(2)}+{\chi }_{\rho }{\left({d}_{\rho }^{\dagger }{\tilde{d}}_{\rho }\right)}^{(2)}\,$ the quadrupole operator for the neutron (ρ = ν) and proton (ρ = π), respectively. The parameters ϵπ, ϵν denote the energy of d bosons relative to the s bosons. χπ(χν) represents the quadrupole structure parameter of proton (neutron) bosons, having only a minor effect on the excitation energy but contributing a lot to the electromagnetic properties. For U(5)-like nuclei, their spectrum is dominated by the value of ϵρ, which is relatively much larger than other parameters. For O(6)-like nuclei, the value of κπν large in comparison with ϵρ characterizes the main properties of their spectrum. The fourth and fifth terms on the right-hand side of equation ( $\begin{eqnarray}{\hat{T}}^{(E2)}={e}_{\pi }{\hat{Q}}_{\pi }+{e}_{\nu }{\hat{Q}}_{\nu },\ \end{eqnarray}$
$\begin{eqnarray}{\hat{T}}^{(M1)}=\sqrt{3/4\pi }({g}_{\nu }{\hat{L}}_{\nu }+{g}_{\pi }{\hat{L}}_{\pi }),\ \end{eqnarray}$
where the eρ and gρ are the effective quadrupole charges and the gyromagnetic ratios, respectively.The reduced E2 transition probability $B(E2)$ is obtained by1 ). More detailed explanation about the Hamiltonian and the transition operators can be seen in [45]. The Hamiltonian was diagonalized using NPBOS code [47].
$\begin{eqnarray}B(E2,J\to J^{\prime} )=\displaystyle \frac{1}{2J+1}| \langle J^{\prime} \parallel {\hat{T}}^{(E2)}\parallel J\rangle {| }^{2},\end{eqnarray}$
with J and $J^{\prime} $ being the initial and final spin angular momenta, respectively. The reduced M1 transition probability $B(M1)$ reads $\begin{eqnarray}B(M1,J\to J^{\prime} )=\displaystyle \frac{1}{2J+1}| \langle J^{\prime} \parallel {\hat{T}}^{(M1)}\parallel J\rangle {| }^{2}.\end{eqnarray}$
The monopole strength ρ2 in the IBM-2 is defined as follows [46] $\begin{eqnarray}{\rho }^{2}(E0,J\to J^{\prime} )=\displaystyle \frac{{Z}^{2}}{{e}^{2}{R}^{4}}{[{\beta }_{0\nu }\langle J^{\prime} \parallel {\hat{n}}_{d\nu }\parallel J\rangle +{\beta }_{0\pi }\langle J^{\prime} \parallel {\hat{n}}_{d\pi }\parallel J\rangle ]}^{2},\end{eqnarray}$
where $R=1.2{A}^{1/3}$ fm, and ${\beta }_{0\pi (\nu )}$ is the proton (neutron) monopole boson effective charge in unit of efm2, the d-boson number operator ${\hat{n}}_{d\rho }$ is the same as in equation (We choose the doubly closed-shell ${}_{\ 50}^{100}{\mathrm{Sn}}_{50}$ as the inert core in this paper. For nucleus ${}_{42}^{98}$Mo${}_{56},{N}_{\pi }=4$ hole-like proton bosons from the Z = 50 closed shell and Nν = 3 particle-like neutron bosons from the N = 50 closed shell can be easily determined. Generally, the IBM-2 calculations assumed επ = εν for the sake of restricting the number of free parameters. However as pointed out in [28], the valence neutrons and protons do not fill the same orbitals when they are added to the inert core. Therefore, the excitation energy of neutron d boson ${\epsilon }_{\nu }$ should be different from that of proton d boson επ. This is our staring point in this calculation.
The interaction between like proton bosons for the Mo isotopes can be negligible, the parameters ${c}_{\pi }^{(0,2,4)}$ is set to zero. We use ${c}_{\nu }^{(2)}=0$, ${c}_{\nu }^{(4)}=0.0$ to minimize the number of free parameters in the Hamiltonian (1 ). By a least-square fitting to the experimental data for 98Mo, we obtain the final IBM-2 parameters in our analysis: ϵν = 0.501 MeV, ϵπ = 0.815 MeV, ${\kappa }_{\pi \nu }=-0.06\,\mathrm{MeV}$, ωπν = 0.1 MeV, ωνν = 0.14 MeV, χπ = −χν = 0.6, λ1 = 1.318 MeV, λ2 = 0.072 MeV, and λ3 = 0.47 MeV ${c}_{\nu }^{(0)}=-0.49$.
3. The energy level structure
Figure 1 shows the experimental and calculated energy levels. The explicit results are listed in table 1. As discussed above, by noting that the energy of neutron boson ϵν is not equal to that of proton boson ϵπ, namely ${\varepsilon }_{\nu }\ne {\varepsilon }_{\pi }$, we can reproduce the ordering of ${0}_{1}^{+},{0}_{2}^{+},{2}_{1}^{+},{2}_{2}^{+}$ and ${4}_{1}^{+}$ states very well. Especially, the level energies of the first four states are almost equal to the corresponding experimental data. The energy of ${2}_{2}^{+}$ state is slightly lower than ${4}_{1}^{+}$ state, they constitutes a pair, the ${0}_{2}^{+}$ state is far from them, this is a character of γ-soft spectrum.
Table 1. Experimental level energies reported in [4–6]) are compared to the results from IBM-2 calculations near Z = N = 50 doubly shell closure. |
| ${E}_{\mathrm{Expt}}$ | ${E}_{\mathrm{IBM}-2}$ | F | R | |
|---|---|---|---|---|
| ${0}_{1}^{+}$ | 0.000 | 0.000 | 2 | 99.2 |
| ${0}_{2}^{+}$ | 0.735 | 0.735 | 2 | 68.7 |
| ${1}_{1}^{+}$ | 3.258 | 3.243 | 1 | 55.2 |
| ${1}_{2}^{+}$ | 3.405 | 3.668 | 1 | 51.5 |
| ${2}_{1}^{+}$ | 0.787 | 0.787 | 2 | 96.3 |
| ${2}_{2}^{+}$ | 1.432 | 1.453 | 2 | 94.3 |
| ${2}_{3}^{+}$ | 1.758 | 1.628 | 2 | 65.8 |
| ${2}_{4}^{+}$ | 2.207 | 2.031 | 2 | 68.0 |
| ${2}_{5}^{+}$ | 2.333 | 2.068 | 1 | 57.6 |
| ${2}_{6}^{+}$ | 2.419 | 2.479 | 2 | 60.8 |
| ${2}_{7}^{+}$ | 2.526 | 2.649 | 1 | 54.7 |
| ${2}_{8}^{+}$ | 2.562 | 2.886 | 2 | 76.1 |
| ${3}_{1}^{+}$ | 2.105 | 2.364 | 2 | 94.0 |
| ${3}_{2}^{+}$ | 2.485 | 2.845 | 1 | 53.8 |
| ${4}_{1}^{+}$ | 1.510 | 1.727 | 2 | 82.8 |
| ${4}_{2}^{+}$ | 2.224 | 2.523 | 2 | 83.8 |
| ${6}_{1}^{+}$ | 2.343 | 2.722 | 2 | 61.1 |
A new quantum number called F-spin first introduced by Arima et al can give a good description of MS states [42]. The FS states have the maximum eigenvalue of F-spin, namly ${F}_{\max }=({N}_{\pi }+{N}_{\nu })/2$, where Nν (${N}_{\pi }$) is the total number of neutron (proton) bosons. The MS states are characterized by other eigenvalues: $F={F}_{\max }-1$, ${F}_{\max }-2,\,\ldots ,\,{F}_{\min }=\left|{N}_{\pi }-{N}_{\nu }\right|/2$ [48]. An important quantity indicating the F-spin nature of a given state $| s\rangle $ is the ratio R defined by
$\begin{eqnarray}R=\langle s\parallel {F}^{2}\parallel s\rangle /{F}_{\max }({F}_{\max }+1),\ \end{eqnarray}$
where $\begin{eqnarray}| s\rangle =\alpha | {F}_{\max }\rangle +\beta | {F}_{\max }-1\rangle ,\hspace{5mm}{\alpha }^{2}+{\beta }^{2}=1,\ \end{eqnarray}$
then we obtain $\begin{eqnarray}\langle s\parallel {F}^{2}\parallel s\rangle ={\alpha }^{2}{F}_{\max }({F}_{\max }+1)+{\beta }^{2}{F}_{\max }({F}_{\max }-1).\ \end{eqnarray}$
R = 1 for an FS state. The smaller the R is, the lower the F-spin becomes. Therefore, R indicates the main component of a state wavefunction in the F-spin space. If R is less than sixty percent, one can consider this state as an MS state. Table 1 shows that the ${1}_{1}^{+},{1}_{2}^{+},{2}_{5}^{+},{2}_{7}^{+},{3}_{2}^{+}$ are some candidates for MS states. The experimental energy of ${2}_{5}^{+}$ state lies at 2.333 MeV, the calculated energy at 2.068 MeV is a little lower than the experimental value. The lowest MS state is the ${2}_{5}^{+}$ state. Further identification of MS states needs to analyze the information about the electromagnetic transitions, we will discuss these properties later.The energy ratio ${R}_{4/2}=E({4}_{1}^{+})/E({2}_{1}^{+})$ is an ideal observable of measuing the extent of quadrupole deformation, which has different characteristic values for three kinds of typical nuclei: R4/2 = 2 for spherical nuclei, R4/2 = 2.5 for γ-unstable nuclei and R4/2 = 3.33 for rigid rotors. The experimental value of R4/2 is 1.92, the calculation gives R4/2 = 2.19, both of them indicate that it is a near-harmonic U(5) spectrum. But only the R4/2 value can not uniquely determine the shape character of a nucleus, one needs other information such as the electromagnetic transition properties to give more decisive discussions. Figure 1 shows that the experimental levels can be perfectly described by IBM-2 including the MS states.
4. The electromagnetic transitions
In this study, we let the effective quadrupole charge of neutron boson eν = 0 for simplicity, and determine the effective charge eπ by fitting to the experimental result of $B(E2,{2}_{1}^{+}\to {0}_{1}^{+})=21.4$ W.u. for 98Mo nucleus. At last, we obtain the eπ = 0.1211 eb. For the boson g factors, we take the naked values gν = 0 and gπ = 1.0 μN.
In tables 2 and 3, we compare the calculated electromagnetic $B(E2)$ and $B(M1)$ transition strengths for 98Mo with the recent experimental values. Tables 2 and 3 indicate that most of the calculated B(E2) transition strengths can reproduce the experimental results very well. For the $B(M1)$ transitions, we list the calculated results that have the correspondence with the experimental data. In table 3, we use the relative values of B(E2) transition strengths as reported in [5]. For U(5)-like nuclei, E2 transition strengths must obey the selection rule $\bigtriangleup {n}_{d}=0,\pm 1$. But M1 transitions can only occurs between states which have the same d boson number nd [49]. In the O(6) limit the ${0}_{2}^{+}$ state is the head of a band with τ = 3 and $\sigma ={\sigma }_{{\rm{m}}}$, the selection rules only allow the transition between ${2}_{2}^{+}$ state and ${0}_{2}^{+}$ state , but this transition is forbidden in the U(5) limit. As shown in table 2, the calculated transition $B(E2,{2}_{2}^{+}\to {0}_{2}^{+})=0.513$ W.u. agrees with the experimental one ${2.5}_{-6}^{+8}$ W.u. within the experimental uncertainty. This displays that 98Mo has the O(6) feature. The experimental value of $B(E2,{2}_{1}^{+}\to {0}_{2}^{+})$ transition is too large and needs further theoretical investigation.
Table 2. Experimental E2 and M1 transition strengths for 98Mo are compared to the calculated ones, where E2 strengths are given in unit of W.u., and the M1 transition are given in unit of ${\mu }_{N}^{2}$. The experimental data are adopted from [4–6]. |
| ${J}_{i}^{\pi }\to {J}_{f}^{\pi }$ | $B(E2)$ | $B(M1)$ | ||
|---|---|---|---|---|
| Expt. | IBM-2 | Expt. | IBM-2 | |
| ${2}_{1}^{+}\to {0}_{1}^{+}$ | 21.4 | 21.4 | ||
| ${2}_{1}^{+}\to {0}_{2}^{+}$ | 280(40) | 0.250 | ||
| ${2}_{2}^{+}\to {0}_{1}^{+}$ | ${1.0}_{-1}^{+2}$ | 0.251 | ||
| ${2}_{2}^{+}\to {0}_{2}^{+}$ | ${2.5}_{-6}^{+8}$ | 0.513 | ||
| ${2}_{2}^{+}\to {2}_{1}^{+}$ | ${47.8}_{-100}^{+132}$ | 23.93 | 0.0119(2) | 0.019 99 |
| ${2}_{3}^{+}\to {0}_{2}^{+}$ | ${7.8}_{-34}^{+286}$ | 9.27 | ||
| ${2}_{3}^{+}\to {2}_{1}^{+}$ | ${3.2}_{-16}^{+134}$ | 2.73 | 0.0179(4) | 0.111 |
| ${2}_{3}^{+}\to {2}_{2}^{+}$ | ${4.7}_{-23}^{+189}$ | 1.12 | ||
| ${2}_{4}^{+}\to {2}_{1}^{+}$ | 1.7(2) | 0.45 | ${0.059}_{-6}^{+7}$ | 0.0599 |
| ${2}_{5}^{+}\to {2}_{2}^{+}$ | ${1.6}_{-4}^{+8}$ | 0.635 | ${0.153}_{-41}^{+87}$ | 0.1467 |
| ${2}_{7}^{+}\to {2}_{2}^{+}$ | 0.0043(6) | 0.06 | 0.018(12) | 0.005 14 |
| ${4}_{1}^{+}\to {2}_{1}^{+}$ | ${49.1}_{-4.5}^{+5.5}$ | 33.64 | ||
| ${6}_{1}^{+}\to {4}_{1}^{+}$ | 10.1(4) | 34.10 | ||
Table 3. Same as table 2, but normalized with respect to the largest $B(E2)$ transition strength of initial states ${3}_{1}^{+}$ and ${4}_{2}^{+}$, respectively. The experimental data are taken from [5]. |
| ${J}_{i}^{\pi }\to {J}_{f}^{\pi }$ | $B(E2)$ | |
|---|---|---|
| Expt. | IBM-2 | |
| ${3}_{1}^{+}\to {2}_{1}^{+}$ | 0.04(3) | 0.004 |
| ${3}_{1}^{+}\to {2}_{2}^{+}$ | 1 | 1 |
| ${3}_{1}^{+}\to {4}_{1}^{+}$ | $\lt 0.40$ | 0.24 |
| ${4}_{2}^{+}\to {2}_{1}^{+}$ | 0.04(1) | 0.01 |
| ${4}_{2}^{+}\to {2}_{2}^{+}$ | 0.88(11) | 1.45 |
| ${4}_{2}^{+}\to {4}_{1}^{+}$ | 1 | 1 |
Another observable characterizing the shape character is $B(E2,{4}_{1}^{+}\to {2}_{1}^{+})$ transition. The calculated value of $B(E2,{4}_{1}^{+}\to {2}_{1}^{+})$ is 33.64 W.u., a little lower than the experimental value of ${49.1}_{-4.5}^{+5.5}$ W.u.. As shown in table 4, the calculated ratio ${B}_{4/2}={\text{}}B(E2,{4}_{1}^{+}\to {2}_{1}^{+})/{\text{}}B(E2,{2}_{1}^{+}\to {0}_{1}^{+})=1.57$ closely matches the corresponding experimental value of 2.29, although the recent experimental uncertainty is large. For $N\to \infty $, the well-known results are B4/2 = 2 for U(5) limit and 10/7 for O(6) limit. For the finite number of boson N = 7, the IBM gives B4/2 = 1.71 for the U(5) limit and B4/2 = 1.34 for the O(6) limit [49, 50]. Both the calculated and experimental values of B4/2 indicate that 98Mo nucleus appears to be closer to the U(5) limit, but the calculation contains a little more O(6) components.
Table 4. The characteristic $B(E2)$ ratios for low-lying states in the three dynamical limits (taken from the [49, 50]). Both of the experimental and calculated values are listed. |
| $\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{2}_{2}^{+}\to {0}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$ | $\tfrac{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ | $\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$ | |
|---|---|---|---|---|
| U(5) | 1.71 | 0.011 | 1.4 | 1.0 |
| SU(3) | 1.34 | 0.70 | 0.02 | 6.93 |
| O(6) | 1.34 | 0.07 | 0.79 | 1.84 |
| Exp. | 2.29 | 0.02 | 2.23 | 1.03 |
| IBM-2 | 1.57 | 0.01 | 1.12 | 1.41 |
The computed E2 transition strength of $B(E2,{2}_{2}^{+}\,\to {2}_{1}^{+})\,=23.93$ W.u. is in agreement with the experimental value ${47.8}_{-100}^{+132}$ W.u. within the experimental uncertainty. In order to investigate the shape character, we introduce another characteristic ratio ${B}_{2/2}={\text{}}B(E2,{2}_{2}^{+}\to {2}_{1}^{+})/{\text{}}B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$. The experimental value gives B2/2 = 2.23. But the available experimental data have some uncertainties in 98Mo. Our calculation obtains B2/2 = 1.12. Both the calculated and experimental values of B2/2 imply that 98Mo nucleus demonstrates the U(5) character. Table 4 also gives the results of $B(E2,{2}_{2}^{+}\to {0}_{1}^{+})/B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ and $B(E2,{4}_{1}^{+}\,\to {2}_{1}^{+})/B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$, which reveal similar behavior as above.
The E0 transition strength ${\rho }^{2}(E0)$ can provide the valuable information on shape coexistence [51]. Therefore, we calculate the ${\rho }^{2}(E0,{0}_{2}^{+}\to {0}_{1}^{+})$ transition probability in order to further understand the structure character in 98Mo. Because the E0 operator is proportional to ${\hat{n}}_{d}$, there are no E0 transitions in the U(5) limit [37]. We adopt the same parameters ${\rho }_{0\nu }=0.25$ efm2 and ${\rho }_{0\pi }=0.10$ efm2 as in [46]. The calculated transition strength ${\rho }^{2}(E0,{0}_{2}^{+}\to {0}_{1}^{+})=0.0159$. The experimental value of ${\rho }^{2}(E0,{0}_{2}^{+}\to {0}_{1}^{+})$ is 0.027(2) [6]. The theoretical calculation agrees with the experimental value. Combining with the discussion on the level structure, we conclude that the character of shape coexistence can be well described in the framework of IBM-2 in 98Mo.
From table 2, it is shown that the agreement between the theoretical calculation and the experimental data for M1 transitions appears to be satisfactory. Especially, the strongest M1 transition from ${2}_{5}^{+}$ state to ${2}_{2}^{+}$ can be well reproduced. The calculated $B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ and $B(E2,{2}_{4}^{+}\to {2}_{1}^{+})$ are almost equal to the experimental values. Both the experimental $B(E2,{2}_{5}^{+}\to {2}_{2}^{+})$ value and the calculated one demonstrate that this is a weakly collective E2 transition. Combining with the R value in table 1, one can determine the ${2}_{5}^{+}$ state has an MS character. This is the first MS state in 98Mo. In the same way, the ${2}_{7}^{+}$ state is also an MS state.
5. Conclusion
In summary, we have calculated the low-lying levels and E2 and M1 electromagnetic transitions for a number of transitions in 98Mo using IBM-2 by considering the energy difference of neutron boson and proton boson, ${\varepsilon }_{\nu }\ne {\varepsilon }_{\pi }$. The comparison between the recent experimental data and calculated ones shows that they are in good agreement. In particular, the strongest M1 transition from ${2}_{5}^{+}$ state to ${2}_{2}^{+}$ state can be well reproduced. The calculation indicates that ${2}_{5}^{+}$ and ${2}_{7}^{+}$ states belong to the mixed-symmetry states. We also calculated the electric monopole strength ${\rho }^{2}(E0,{0}_{2}^{+}\to {0}_{1}^{+})$, the result agrees with the experimental data. Our calculation indicates that shape coexistence and MS states can be simultaneously described very well in the framework of IBM-2.