Simultaneous description of β decay and low-lying structure of neutron-rich even- and odd-mass Rh and Pd nuclei

PHYSICAL REVIEW C 106, 064304 (2022) Simultaneous description of βdecay and low-lying structure of neutron-rich even- and odd-mass Rh and Pd nuclei K. Nomura ,1,*L. Lotina ,1R. Rodríguez-Guzmán ,2and L. M. Robledo 3,4 1Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia 2Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville, E-41011, Spain 3Departamento de Física Teórica and CIAFF, Universidad Autónoma de Madrid, E-28049 Madrid, Spain 4Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Bohadilla del Monte, E-28660-Madrid, Spain (Received 6 September 2022; accepted 29 November 2022; published 7 December 2022) The low-energy structure and β-decay properties of neutron-rich even- and odd-mass Pd and Rh nuclei are studied using a mapping framework based on the nuclear density-functional theory and the particle-boson coupling scheme. Constrained Hartree-Fock-Bogoliubov calculations using the Gogny-D1M energy density functional are performed to obtain microscopic inputs to determine the interacting-boson Hamiltonian employed to describe the even-even core Pd nuclei. The mean-field calculations also provide single-particle energies for the odd systems, which are used to determine essential ingredients of the particle-boson interactions for the odd-nucleon systems, and of the Gamow-Teller and Fermi transition operators. The potential-energy surfaces obtained for even-even Pd isotopes as well as the spectroscopic properties for the even- and odd-mass systems suggest a transition from prolate deformed to γ-unstable and to nearly spherical shapes. The predicted β-decay log ft values are shown to be sensitive to the details of the wave functions for the parent and daughter nuclei and therefore serve as a stringent test of the employed theoretical approach. DOI: 10.1103/PhysRevC.106.064304 I. INTRODUCTION Precise measurements and theoretical descriptions associated with the low-energy nuclear structure are crucial to the accurate modeling and better understanding of fundamental nuclear processes, such as βand double-β(ββ) decays intimately connected to stellar nucleosynthesis. In this context, the low-energy excitations and decay properties of neutronrich nuclei with mass A≈100 and neutron number N≈60 are of particular interest from both the nuclear structure and astrophysical points of view. Those nuclei exhibit a rich variety of phenomena such as shell evolution, onset of collectivity, quantum (shape) phase transitions and shape coexistence. They are also involved in the rapid neutron-capture (r)process responsible for the nucleosynthesis of heavy chemical elements in explosive environments. The βdecay half-lives of heavy neutron-rich nuclei have been extensively measured using radioactive-ion beams at major experimental facilities around the world. For example, the neutron-rich A≈110 nuclei from Kr to Tc [1], and from Rb to Sn [2] have been studied at the RIBF facility at RIKEN. The A≈90 region from Se to Zr isotopic chains has been studied at the NSCL at MSU [3]. Moreover, several A≈ 100–110 nuclei are of special interest, including 96Zr, 96Mo, 100Mo, 100Ru, 110Pd, and 110Cd, since they correspond to the parent or daughter nuclei for the possible neutrinoless ββ decays [4]. *[email protected] From a theoretical point of view, the consistent description of both low-lying nuclear states and β-decay properties represents a major challenge. Theoretical studies of the β-decay process have been carried out within the interacting boson model (IBM) [5–14], the quasiparticle random-phase approximation (QRPA) [15–23], and the large-scale shell model (LSSM) [24–28]. The calculation of β-decay properties serves as a stringent test of a given theoretical approach, since the decay rate of this process is very sensitive to the structure of the wave functions corresponding to the low-energy states of both the parent and daughter nuclei. In this paper, we present a simultaneous description of the low-energy collective excitations and β-decay properties of even- and odd-Aneutron-rich Pd and Rh isotopes in the mass range A≈100–120. They represent a region of interest for future experiments and for astrophysical applications. Calculations are performed within a theoretical framework based on the nuclear density-functional theory and the particlecore coupling scheme. In it even-even nuclei are described using the IBM [29]. The particle-core couplings for the oddmass, and odd-odd nuclei are described using the interacting boson-fermion model (IBFM) [30,31] and the interacting boson-fermion-fermion model (IBFFM) [31,32], respectively. The bosonic-core Hamiltonian is built using microscopic input from self-consistent Hartree-Fock-Bogoliubov (HFB) [33] calculations based on the parametrization D1M [34]ofthe Gogny energy density functional (EDF) [35,36]. Essential building blocks of the particle-boson interactions and of the Gamow-Teller (GT) and Fermi (F) transition operators for the βdecay are also determined with the aid of the same 2469-9985/2022/106(6)/064304(17) 064304-1 ©2022 American Physical Society K. NOMURA et al. PHYSICAL REVIEW C 106, 064304 (2022) Gogny-EDF results. The method has already been applied to study the shape evolution and βdecay properties of the odd-A [11] and even-A[12] nuclei in the mass A≈130 region. It has also been employed to study even- and odd-AAs and Ge nuclei in the A≈70–80 region using microscopic input from relativistic Hartree-Bogoliubov calculations, based on the density-dependent point-coupling interaction [14]. The main goal of this work is to examine the performance of the method mentioned above in the case of neutron-rich nuclei, including those for which experimental information is scarce. The results to be discussed latter on in the paper also illustrate the predictive power of the EDF-based IBM to describe the low-lying structure and βdecay in this region of the nuclear chart where future experiments are expected. To identify the relevance of the low-lying structures of individual nuclei in the βdecay, we perform a detailed analysis of the wave functions obtained for both the parent and daughter nuclei of the decay. In addition, we perform conventional IBM calculations, with the parameters for the even-even boson core Hamiltonians taken from the earlier phenomenological calculation [37]. The corresponding results are compared with those from the EDF-based IBM calculations. Note that the present study is restricted to both types of allowed βdecays, i.e., the transition conserves parity and takes place between states that differ in the total angular momentum Iby I=0 or 1. To support our choice we note that, like other nonrelativistic [38] and relativistic [39,40] EDFs, theoretical approaches based on the parametrizations D1M and D1S [41]ofthe Gogny-EDF both at the mean-field level and beyond have been extensively employed to study the low-energy nuclear structure and dynamics in various regions of the nuclear chart as well as fundamental nuclear processes (see Ref. [36]for a review and references therein). In particular spectroscopic studies involving collective degrees of freedom have been carried out within the symmetry-projected generator coordinate method (GCM) [33] using the Gogny forces and involving different levels of sophistication [36,42–48]. Furthermore, the mapping procedure leading to an IBM Hamiltonian from microscopic Gogny mean-field input has already shown its ability to describe spectroscopic properties associated with shape phase transitions, shape coexistence, and octupole deformations in nuclei [49–56]. The paper is organized as follows: The theoretical framework is briefly outlined in Sec. II. The excitation spectra and electromagnetic transition properties obtained for even-even Pd (Sec. III), odd-APd and Rh (Sec. IV), and odd-odd Rh nuclei (Sec. V) are discussed. The computed log ft values for the βdecays of the odd- and even-ARh into Pd nuclei are discussed in detail in Sec. VI. Finally, Sec. VII is devoted to the concluding remarks. II. THEORETICAL FRAMEWORK In this section, we describe the particle-core Hamiltonian (Sec. II A), and the procedure to build it (Sec. II B). Electromagnetic transition operators are discussed in Sec. II C, and Gamow-Teller and Fermi operators are introduced in Sec. II D. A. Particle-core Hamiltonian In this study, we use the neutron-proton IBM (IBM-2) [57,58]. In this model both neutron and proton monopole (sν and sπ), and quadrupole (dνand dπ) bosons are considered as fundamental degrees of freedom. From a microscopic point of view [57,58], the sν(sπ) and dν(dπ) bosons are associated with the collective Sν(Sπ) and Dν(Dπ) pairs of valence neutrons (protons) with angular momenta and parity 0+and 2+, respectively. In comparison with the simpler IBM-1, in which the neutrons and protons are not distinguished, the IBM-2 appears to be more suitable to treat βdecay, since in this process both proton and neutron degrees of freedom should be explicitly taken into account. For the model space the neutron N=50–82 and proton Z=28–50 major shells are used. Hence for 104–124Pd, the number of neutron bosons, Nν, varies within the range 2 ⩽Nν⩽8, while the number of the proton bosons is fixed, Nπ=2. To deal with even-even, odd-mass, and odd-odd nuclei on an equal footing, both collective and single-particle degrees of freedom are treated within the framework of the neutronproton IBFFM (IBFFM-2). The IBFFM-2 Hamiltonian reads ˆ H=ˆ HB+ˆ Hν F+ˆ Hπ F+ˆ Vν BF +ˆ Vπ BF +ˆ Vνπ ,(1) where ˆ HBis the IBM-2 Hamiltonian representing the bosonic even-even core, ˆ Hν F(ˆ Hπ F) is the one-body, single-neutron (- proton) Hamiltonian, and ˆ Vν BF (ˆ Vπ BF) stands for the interaction between the odd neutron (proton) and the even-even IBM-2 core. The last term ˆ Vνπ represents the residual interaction between the odd neutron and the odd proton. The IBM-2 Hamiltonian takes the form ˆ HB=d(ˆndν+ˆndπ)+κˆ Qν·ˆ Qπ,(2) where in the first term, ˆndρ=d† ρ·˜ dρ(ρ=νor π)isthe d-boson number operator, with dthe single d-boson energy relative to the s-boson one, and ˜ dρμ =(−1)μdρ−μ. The second term stands for the quadrupole-quadrupole interaction between neutron and proton boson systems with strength κ, and ˆ Qρ=d† ρsρ+s† ρ˜ dρ+χρ(d† ρט dρ)(2) represents the bosonic quadrupole operator, with the dimensionless parameter χρ. The single-nucleon Hamiltonian ˆ Hρ Ftakes the form ˆ Hρ F=− jρ jρ2jρ+1a† jρטajρ(0),(3) where jρstands for the single-particle energy of the odd neutron (ρ=ν)orproton(ρ=π) orbital jρ.ajρand a† jρ are annihilation and creation operators of the single particle, respectively. The operator ˜ajρis defined as ˜ajρmρ= (−1)jρ−mρajρ−mρ. In this study, we employ the following boson-fermion interaction ˆ Vρ BF [31] ˆ Vρ BF =ρˆ Vρ dyn +ρˆ Vρ exc +Aρˆ Vρ mon.(4) The first, second, and third terms are dynamical quadrupole, exchange, and monopole interactions, respectively. Within the generalized seniority scheme [31,59], the dynamical and exchange terms are assumed to be dominated by the interaction 064304-2 SIMULTANEOUS DESCRIPTION OF βDECAY AND … PHYSICAL REVIEW C 106, 064304 (2022) TABLE I. Even-even Pd core, and the neighboring odd-NPd, odd-ZRh, and odd-odd Rh nuclei considered in this study. Even-even core Odd-NOdd-ZOdd-odd A 46PdN(58 ⩽N⩽64) A+1 46 PdN+1A−1 45 RhNA 45RhN+1 112 46 Pd66 111 45 Rh66 A 46PdN(68 ⩽N⩽78) A−1 46 PdN−1A−1 45 RhNA−2 45 RhN−1 between unlike particles. On the other hand, the monopole term is assumed to be dominated by the interaction between like particles. The explicit form of the different terms in Eq. (4) then read ˆ Vρ dyn = jρj ρ γjρj ρa† jρטaj ρ(2)·ˆ Qρ,(5) ˆ Vρ exc =−(s† ρט dρ)(2)· jρj ρj ρ10 Nρ(2jρ+1)βjρj ρβj ρjρ :d† ρטaj ρ(jρ)×a† j ρטsρ(j ρ)(2):+(H.c.),(6) ˆ Vρ mon =−ˆndρ jρ2jρ+1(a† jρטajρ)(0),(7) where the coefficients γjρj ρ=(ujρuj ρ−vjρvj ρ)Qjρj ρ, and βjρj ρ=(ujρvj ρ+vjρuj ρ)Qjρj ρare proportional to the matrix elements of the fermion quadrupole operator in the singleparticle basis Qjρj ρ=ρ1 2jρY(2) ρ 1 2j ρ. The operator ˆ Qρ in Eq. (5) is the same boson quadrupole operator as in the boson Hamiltonian (2). In Eq. (6) the notation : (···) : stands for normal ordering. Within this formalism, the single-particle energy jρin Eq. (3) is replaced with the quasiparticle energy ˜jρ. For the residual neutron-proton interaction ˆ Vνπ in Eq. (1), we adopt the form [60] ˆ Vνπ =4πvdδ(r)δ(rν−r0)δ(rπ−r0) +vt3(σν·r)(σπ·r) r2−σν·σπ,(8) where the first and second terms are surface-delta and tensor interactions with strength parameters vd, and vt, respectively. Note that r=rν−rπand r0=1.2A1/3fm. Table Isummarizes the even-even Pd core nuclei, neighboring odd-APd and Rh, and odd-odd Rh nuclei considered in this study. B. Procedure to build the Hamiltonian In the initial step a set of constrained HFB calculations for even-even Pd isotopes based on the parametrization D1M of the Gogny-EDF is carried out to obtain the microscopic input to build the IBFFM-2 Hamiltonian. For each even-even Pd isotope, those calculations provide the corresponding energy surfaces, i.e., the total mean-field energies as functions of the triaxial quadrupole deformations βand γ[61]. For each nucleus, the Gogny-D1M HFB energy surface is mapped onto the expectation value of the IBM-2 Hamiltonian ˆ HB(2)in the boson condensate state [62]. This procedure specifies the parameters of the boson Hamiltonian, i.e., d,κ,χν, and χπ. For more details about the mapping procedure, the reader is referred to Refs. [63,64]. Next, the Hamiltonian ˆ HFof Eq. (3) and the boson-fermion interactions ˆ VBF of Eq. (4) are determined using the procedure of Refs. [65,66]. The single-particle energies jρof the odd nucleon are obtained from HFB calculations constrained to zero quadrupole deformation. Once the single-particle energies are available, the quasiparticle energies ˜jρand occupation probabilities v2 jρare computed within the BCS approximation, separately for neutron and proton single-particle spaces. The empirical pairing gap 12A−1/2is used. We include in the BCS calculations the 2s1/2,1d3/2,1d5/2,0g7/2,0g9/2, and 0h11/2orbitals for the odd neutron, and the 1d5/2,0g7/2,0g9/2, 2p1/2,2p3/2, and 1f5/2orbitals for the odd proton. The corresponding quasiparticle energies ˜jν(˜jπ), and occupation probabilities v2 jν(v2 jπ) for the odd neutron (proton) 2s1/2,1d3/2, 1d5/2, and 0g7/2(1d5/2,0g7/2, and 0g9/2) orbitals are taken as the inputs to ˆ Hν F(ˆ Hπ F) and ˆ Vν BF (ˆ Vπ BF), respectively. The strength parameters ρ,ρ, and Aρfor ˆ Vρ BF are then fixed so that the observed low-energy positive-parity levels for the odd-APd (ρ=ν) or odd-ZRh (ρ=π) nuclei are reproduced reasonably well. Finally, the parameters vdand vtfor the residual neutronproton interaction in Eq. (8) are determined [67] so that the observed low-lying positive-parity states for each odd-odd Rh nucleus are reasonably well reproduced. Note that the same strength parameters as those obtained in the previous step for the neighboring odd-Anuclei are employed in the IBFFM-2 calculations for odd-odd nuclei. On the other hand, the quasiparticle energies and occupation probabilities of the odd particles are independently computed. Figure 1shows the neutron and proton spherical singleparticle energies (jνand jπ), resulting from the Gogny-HFB calculations, and the quasiparticle energies (˜jνand ˜jπ) and occupation probabilities (v2 jνand v2 jπ) used in the IBFM-2 and IBFFM-2 calculations. C. Electromagnetic transition operators Theories with effective degrees of freedom, like the IBFFM, require the definition of transition operators to be used in the evaluation of electromagnetic transition probabilities. For the electric E2 transition the operator ˆ T(E2) to be used in the IBFFM-2 takes the form [31] ˆ T(E2)=ˆ T(E2) B+ˆ T(E2) F,(9) where the first and second terms are the boson and fermion parts, respectively. They are given by ˆ T(E2) B= ρ=ν,π eB ρˆ Qρ,(10) and ˆ T(E2) F=− 1 √5 ρ=ν,π  jρj ρujρuj ρ−vjρvj ρ ×ρ 1 2jρ    eF ρr2Y(2)     ρ 1 2j ρa† jρטaj ρ(2).(11) 064304-3 K. NOMURA et al. PHYSICAL REVIEW C 106, 064304 (2022) FIG. 1. (a), (b) The single-particle energies jρ, obtained from the Gogny-D1M HFB calculations at the spherical configuration, (c), (d) the quasiparticle energies ˜jρ, and (e), (f) the occupation probabilities v2 jρ, obtained from the BCS calculations. Results shown in the left column are for the odd neutron in the odd-APd and even-A Rh nuclei, and those in the right column are for the odd proton in the even- and odd-ARh nuclei. The fixed values eB ν=eB π=0.1eb for the boson effective charges are taken so that the experimental B(E2; 2+ 1→ 0+ 1) transition probabilities are reproduced for even-even Pd isotopes. The standard neutron and proton effective charges eF ν=0.5eb, eF π=1.5eb are employed for all the studied odd-nucleon systems. The M1 transition operator ˆ T(M1) is defined as ˆ T(M1)=3 4π ρ=ν,πgB ρˆ Lρ−1 √3 jρj ρujρuj ρ+vjρvj ρ ×jρgρ ll+gρ ssj ρa† jρטaj ρ(1).(12) The empirical gfactors gB ν=0μN(nuclear magneton) and gB π=1.0μN, are adopted for the neutron and proton bosons. For the neutron (proton) gfactors, the standard Schmidt values gν l=0μNand gν s=−3.82 μN(gπ l=1.0μNand gπ s= 5.58 μN) are used, with gρ squenched by 30% with respect to the free value. D. Gamow-Teller and Fermi transition operators As in the electromagnetic case, the transition operators for allowed βdecay have to be redefined in terms of the relevant degrees of freedom of the model. The Gamow-Teller ˆ TGT and Fermi ˆ TFtransition operators take the form ˆ TGT = jνjπ ηGT jνjπˆ Pjν׈ Pjπ(1),(13) ˆ TF= jνjπ ηF jνjπˆ Pjν׈ Pjπ(0),(14) with the coefficients ηGT jνjπ=− 1 √3ν 1 2jν    σ    π 1 2jπδνπ,(15) ηF jνjπ=− 2jν+1δjνjπ.(16) In Eqs. (13) and (14), ˆ Pjρrepresents one of the one-particle creation operators A† jρmρ=ζjρa† jρmρ+ j ρ ζjρj ρs† ρ˜ dρ×a† j ρ(jρ) mρ ,(17a) B† jρmρ=θjρs† ρ˜ajρmρ+ j ρ θjρj ρd† ρטaj ρ(jρ) mρ ,(17b) and the annihilation operators ˜ Ajρmρ=(−1)jρ−mρAjρ−mρ,(17c) ˜ Bjρmρ=(−1)jρ−mρBjρ−mρ.(17d) The operators in Eqs. (17a) and (17c) conserve the boson number, whereas those in Eqs. (17b) and (17d) do not. The operators ˆ TGT and ˆ TFare expressed as a combination of two of the operators in Eqs. (17a)–(17d), depending on the type of the βdecay studied (i.e., β+or β−) and on the particle or hole nature of the valence nucleons. In the present case, ˆ Pjν=˜ Bjν,mν(N⩽66) ˜ A† jν,mν(N⩾68)(18) for the β−decay of the odd-ARh, while ˆ Pjν=˜ Ajν,mν(N⩽65) ˜ B† jν,mν(N⩾67)(19) for the β−decay of the even-ARh. On the other hand, ˆ Pjπ= ˜ Ajπ,mπfor all the considered β−decays. Note that Eqs. (17a)– (17d) are simplified forms of the most general one-particle transfer operators in the IBFM-2 [31]. By using the generalized seniority scheme, the coefficients ζj,ζjj ,θj, and θjj in Eqs. (17a) and (17b) can be written as [68] ζjρ=ujρ 1 K jρ ,(20a) 064304-4 SIMULTANEOUS DESCRIPTION OF βDECAY AND … PHYSICAL REVIEW C 106, 064304 (2022) FIG. 2. The Gogny-D1M HFB and mapped IBM-2 potential-energy surfaces as functions of the (β,γ ) deformation parameters for the even-even 104–124Pd nuclei. The energy difference between neighboring contours is 200 keV. The global minimum is identified by a solid circle. ζjρj ρ=−vjρβj ρjρ10 Nρ(2jρ+1) 1 KK jρ ,(20b) θjρ= vjρ Nρ 1 K jρ ,(20c) θjρj ρ=ujρβj ρjρ10 2jρ+1 1 KK jρ .(20d) The factors K,K jρ, and K jρare defined as K=⎛ ⎝ jρj ρ β2 jρj ρ⎞ ⎠ 1/2 ,(21a) K jρ=⎡ ⎣1+2vjρ ujρ2ˆnsρ+1ˆndρ0+ 1 Nρ2jρ+1j ρβ2 j ρjρ K2⎤ ⎦ 1/2 , (21b) K jρ=⎡ ⎣ˆnsρ0+ 1 Nρ+2ujρ vjρ2ˆndρ0+ 1 2jρ+1j ρβ2 j ρjρ K2⎤ ⎦ 1/2 ,(21c) where ˆnsρis the number operator for the sρboson and ···0+ 1 stands for the expectation value of a given operator in the 0+ 1ground state of the even-even nucleus. The amplitudes vjρand ujρappearing in Eqs. (20a)–(20d) and (21a)–(21c) are the same as those used in the IBFM-2 (or IBFFM-2) calculations for the odd-mass (or odd-odd) nuclei. No additional parameter is introduced for the GT and Fermi operators. For a more detailed account on β-decay operators within the IBFM-2 or IBFFM-2 framework, the reader is also referred to Refs. [6,31,68]. The β-decay ft values are given by ft =K |M(F)|2+gA gV2 |M(GT)|2 ,(22) where the numeric constant Ktakes the value K=6163 s. The quantities M(F) and M(GT) are the reduced matrix elements of the operators ˆ TFof Eq. (14) and ˆ TGT of Eq. (13), respectively. Here gVand gAare the vector and axial-vector coupling constants, respectively. In this study, we use the free nucleon values, gV=1 and gA=1.27, for the βdecays of both even- and odd-ARh. 064304-5 K. NOMURA et al. PHYSICAL REVIEW C 106, 064304 (2022) III. EVEN-EVEN NUCLEI A. Potential-energy surfaces The Gogny-D1M HFB and mapped IBM-2 potentialenergy surfaces are shown in Fig. 2as functions of the (β,γ ) deformation parameters for the even-even 104–124Pd nuclei. The variation of the HFB potential-energy surfaces as functions of the neutron number suggests a transition from prolate (for N62) to γ-soft (64 N70), and to nearly spherical (N72) shapes. In particular, both 112,114Pd exhibit rather flat potential-energy surfaces along the γdirection. This is what is expected in the γ-unstable O(6) limit of the IBM [29]. In the case of 116Pd, a flat-bottomed potential with a weak γ dependence, characteristic of the E(5) critical-point symmetry [69], is obtained. For each of the considered nuclei, the Gogny-HFB and IBM-2 energy surfaces display a similar topology in the neighborhood of the global minimum (the location of the minimum, and the softness in the βand γdirections are similar). However, the mapped IBM-2 surfaces generally become flat at large βdeformation (β0.4). This difference is a consequence of the fact that in the HFB approach all nucleonic degrees of freedom are taken into account while the IBM-2 is built on the more limited model (valence) space of nucleon pairs. However, since the mean-field configurations most relevant to the low-energy collective excitations are those in the vicinity of the global minimum, the mapping is considered specifically in that region [63,64]. B. Spectroscopic properties The mapped IBM-2 excitation energies of the 2+ 1,4 + 1,0 + 2, and 2+ 2states in the even-even 104–124Pd nuclei are shown in Fig. 3as functions of the neutron number N. Results obtained using the conventional IBM-2 approach (hereinafter referred to as phenomenological IBM-2), with parameters adopted from the earlier phenomenological study [37], are also included in the plot. As can be seen from the figure, the excitation energies decrease toward the middle of the major shell, i.e., N=66. For N⩽64, the mapped IBM−22+ 1 and 4+ 1excitation energies underestimate the experimental ones while the energies of the non-yrast 0+ 2and 2+ 2states are overestimated. In the mapped (phenomenological) IBM-2 approach the ratios R4/2of the 4+ 1to 2+ 1excitation energies are 2.96 (2.43), 2.86 (2.39), and 2.69 (2.34) for 104Pd, 106Pd, and 108Pd, respectively. These values should be compared with the experimental ratios of 2.38, 2.40, and 2.41. Thus, the mapped IBM-2 provides excitation spectra which are more rotational in character than the phenomenological IBM-2 and experimental ones. Around the neutron midshell N=66, both the predicted and experimental 2+ 2levels have the lowest energies, being even below the 4+ 1state. The 2+ 2state is the bandhead of the quasi-γband, and the lowering of this state reflects an emergence of pronounced γsoftness. The IBM-2 parameters obtained for the even-even Pd isotopes from the mapping procedure, and those determined phenomenologically are shown in Fig. 4. The phenomenological IBM-2 parameters are extracted from earlier fitting calculations for Pd and Ru isotopes [37]. In Ref. [37], in N 0.0 0.5 1.0 1.5 2.0 Excitation energy (MeV) 46PdN (a) 2+ 1Calc. (mapped) Calc. (phen.) Expt. N (b) 4+ 1 58 60 62 64 66 68 70 72 74 76 78 N 0.0 0.5 1.0 1.5 2.0 Excitation energy (MeV) (c) 0+ 2 58 60 62 64 66 68 70 72 74 76 78 N (d) 2+ 2 FIG. 3. Excitation energies of the (a) 2+ 1,(b)4 + 1,(c)0 + 2,and (d) 2+ 2states in the even-even 104–124Pd nuclei. Results are obtained within the mapped and phenomenological (Phen.) IBM-2. Experimental data are taken from Ref. [70]. addition to the terms that appear in Eq. (2), the like-boson interactions, and the so-called Majorana terms were included in the model Hamiltonian. These terms were, however, shown to play a minor role [37], and are omitted in the present study. From Fig. 4, one sees that the single-dboson energy dand FIG. 4. Parameters for the even-even boson-core Hamiltonian (2) employed in the mapped and phenomenological (Phen.) IBM-2 calculations for even-even Pd isotopes. 064304-6 SIMULTANEOUS DESCRIPTION OF βDECAY AND … PHYSICAL REVIEW C 106, 064304 (2022) FIG. 5. The reduced transition probabilities B(E2) for the transitions (a) 2+ 1→0+ 1,(b)4 + 1→2+ 1,(c)0 + 2→2+ 1,and(d)2 + 2→2+ 1 in even-even Pd isotopes in comparison with the experimental data [70]. the strength κhave similar nucleon-number dependence for both the mapped and phenomenological IBM-2 models. A notable quantitative difference is that the derived κvalues for the former are ≈1.4 larger in magnitude than for the latter. The behavior of the parameter χνis different in the two approaches for N⩾70. The sign and absolute value of the sum χν+χπreflect the extent of γsoftness and whether the nucleus is prolate or oblate deformed. In both calculations, the sum is negative, χν+χπ<0, for N64, indicating prolate deformation, and takes nearly vanishing values, χν+χπ≈0, around the neutron midshell N=66, reflecting γsoftness. However, for N⩾70, the sum is negative (positive) in the mapped (phenomenological) calculations, implying prolate (oblate) deformation. Note that a fixed value χπ=0.2isem- ployed in the phenomenological IBM-2 calculations, whereas in the mapped approach this parameter exhibits a strong nucleon number dependence. The B(E2) transition probabilities, computed within the mapped and phenomenological IBM-2 models, are plotted in Fig. 5as functions of the neutron number N.The same E2 effective boson charge is used for the quadrupole operators in the two sets of the IBM-2 calculations. The B(E2; 2+ 1→0+ 1) and B(E2; 4+ 1→2+ 1) values obtained in the mapped IBM-2 calculations agree reasonably well with the experiment, exception made of 112Pd. Both the mapped and phenomenological IBM-2 calculations predict B(E2; 0+ 2→2+ 1) and B(E2; 2+ 2→2+ 1) rates with similar trends as functions of N. However, the mapped IBM-2 scheme provides smaller B(E2; 0+ 2→2+ 1) values for Pd isotopes with 58 ⩽N⩽62. The enhancement of the predicted B(E2; 2+ 2→ 2+ 1) transition rates around the midshell N=66 [see Fig. 5(d)] can be considered as another signature of γ-soft deformation. IV. ODD-APdandRhNUCLEI The excitation energies of the low-lying positive-parity states obtained for the odd-APd isotopes 105–123Pd are depicted in Fig. 6. The results obtained within the IBFM-2 model with boson-core Hamiltonian determined by mapping the Gogny-D1M EDF [Fig. 6(a)] and those obtained from phenomenological calculations of Ref. [37] [Fig. 6(b)] are compared with experimental data [70–72]. The two IBFM-2 calculations, using different boson-core Hamiltonian parameters, provide an overall consistent description of the experimental excitation energies. As can be seen from the 59 61 63 65 67 69 71 73 75 77 N 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Excitation energy (MeV) 46PdN (a) Calc. (mapped) 59 61 63 65 67 69 71 73 75 77 N (b) Calc. (phen.) 59 61 63 65 67 69 71 73 75 77 N (c) Expt. 1/2+ 1 3/2+ 1 5/2+ 1 7/2+ 1 9/2+ 1 11/2+ 1 5/2+ 2 7/2+ 2 FIG. 6. Excitation energies of the low-lying positive-parity states obtained for odd-APd isotopes within the IBFM-2 with the boson-core Hamiltonian determined by (a) mapping the Gogny-D1M EDF and (b) using the phenomenological fit. The experimental data included in panel (c) are taken from Ref. [71]for117Pd,fromRef.[72]for119Pd, and from the NNDC database [70] for the other nuclei. 064304-7 K. NOMURA et al. PHYSICAL REVIEW C 106, 064304 (2022) 58 60 62 64 66 68 70 72 74 76 78 N 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Excitation energy (MeV) 45RhN (a) Calc. (mapped) 58 60 62 64 66 68 70 72 74 76 78 N (b) Calc. (phen.) 58 60 62 64 66 68 70 72 74 76 78 N (c) Expt. 1/2+ 1 3/2+ 1 5/2+ 1 7/2+ 1 9/2+ 1 11/2+ 1 5/2+ 2 7/2+ 2 FIG. 7. The same as Fig. 6, but for odd-ZRh nuclei. The experimental data are taken from Ref. [70]. figure, the experimental data display a change in the groundstate spin from N=67 to 69. The corresponding even-even core nuclei, 114Pd and 116Pd, are in the transitional region, for which the potential-energy surfaces are suggested to be considerably γsoft (see Fig. 2). The sudden change in the ground-state spin of the odd-Aneighbor, therefore, reflects the transition that takes place in the even-even core systems from the γunstable shape, which is associated with an O(6)-like potential, to the E(5)-like structure characterized by a flatbottomed potential. The excitation energies of the low-lying positive-parity states obtained for the odd-Aisotopes 103–123Rh are depicted in Fig. 7. Experimentally, the ground states of these isotopes have spin Iπ=7/2+. Exceptions are made of some of the heaviest isotopes, and similar results are predicted within both the mapped and phenomenological calculations. Both theoretically and experimentally, some of the energy levels exhibit an approximate parabolic behavior with a minimum around the middle of the major shell, N≈66. For 103–123Rh, the order of most of the energy levels remains unchanged in the whole isotopic chain within both the mapped and phenomenological IBFM-2 calculations. This situation is in a sharp contrast with the one in the odd-APd (see Fig. 6), in which the structural change along the isotopic chain occurs more rapidly. Note that the low-lying states of the odd-ARh nuclei are accounted for almost purely by the proton π0g9/2single-particle configuration while more than one single-particle orbital is considered for the odd-APd. The occupation number of the odd proton in the π0g9/2orbital is also nearly constant along the whole Rh isotopic chain [see Fig. 1(f)], whereas the occupation probabilities for the odd neutron in the odd-APd vary significantly with N[see Fig. 1(e)]. Furthermore, as shown below, the strength parameters for ˆ VBF are fixed in the case of odd-ARh nuclei while they depend on the boson number for odd-APd isotopes. The strength parameters of the boson-fermion interaction (4) for odd-NPd nuclei are shown in Fig. 8. These parameters are chosen so that the ground-state spin and energies of a few low-lying levels are reproduced reasonably well. The parameters for the two IBFM-2 calculations are rather similar, with an exception made of the monopole strength Aν for 59 ⩽N⩽63. Note that common quasiparticle energies ˜jρand occupation probabilities v2 jρare used for both IBFM-2 calculations. The parameters for the 123Pd77 nucleus, where no experimental data are available, are taken to be the same as those for the adjacent nucleus 121Pd75. As can be seen from the figure, the IBFM-2 parameters turn out to have a strong N dependence that reflects the rapid structural change in the odd- APd isotopes. On the other hand, constant strength parameters π=0.6 (0.0) MeV, π=0.6 (0.75) MeV, and Aπ=0.0 (−0.25) MeV reproduce reasonably well the experimental data for odd-ARh nuclei in the mapped (phenomenological) calculations. N 0.0 0.2 0.4 0.6 0.8 1.0 (MeV) (a) Γν mapped phen. 59 61 63 65 67 69 71 73 75 77 N (b) Λν46PdN 59 61 63 65 67 69 71 73 75 77 N −0.5 −0.4 −0.3 −0.2 −0.1 0.0 (MeV) (c) Aν FIG. 8. Parameters of the mapped and phenomenological IBFM- 2 Hamiltonian (2) for odd-NPd nuclei. 064304-8 SIMULTANEOUS DESCRIPTION OF βDECAY AND … PHYSICAL REVIEW C 106, 064304 (2022) TABLE II. B(E2) rates (in Weisskopf units, W.u.), quadrupole moment Q(I)(ineb), B(M1) rates (in W.u. ×10−3), and magneticdipole moments μ(I)(inμN) obtained for odd-APd nuclei within the mapped and phenomenological IBFM-2 calculations. Experimental data are taken from Refs. [70,73]. Calc. Mapped Phen. Expt. 105Pd B(E2;1/2+ 1→3/2+ 1)25 13 2.0+91 −16 B(E2;1/2+ 1→5/2+ 1) 90 45 2.64(15) B(E2;1/2+ 2→3/2+ 1) 0.6 3.1 0.9+12 −7 B(E2;1/2+ 2→5/2+ 1) 0.04 0.9 8.4(9) B(E2;3/2+ 1→5/2+ 1) 44 40 4.6(7) B(E2;3/2+ 1→3/2+ 3) 0.05 5.1 >0.21 B(E2;3/2+ 3→5/2+ 2) 0.01 2.7 >2.2 B(E2;5/2+ 1→5/2+ 2) 15 29 1.8(4) B(E2;7/2+ 1→5/2+ 1) 24 33 0.30(4) B(E2;9/2+ 1→5/2+ 1) 57 40 14.3(13) B(M1;1/2+ 1→3/2+ 1) 372 280 14.9+20 −21 B(M1;1/2+ 1→1/2+ 2) 0.93 1.5 7.8(8) B(M1;1/2+ 2→3/2+ 1)37 7 45 +6 −5 B(M1;3/2+ 1→5/2+ 1) 31 4.4 20.3(22) B(M1;3/2+ 1→3/2+ 3) 0.012 0.0004 >5.9 B(M1;3/2+ 3→5/2+ 2) 0.0026 2.9 >47 B(M1;5/2+ 1→5/2+ 2) 13 1.6 19(3) B(M1;5/2+ 3→3/2+ 1) 4.7 0.47 >0.40 B(M1;5/2+ 3→7/2+ 2)52 32 >25 B(M1;7/2+ 1→5/2+ 1) 31 3.7 10.6(12) Q(5/2+ 1)−0.54 −0.27 +0.660(11) μ(3/2+ 1)−0.56 −0.64 −0.074(13) μ(5/2+ 1)−1.19 −1.32 −0.642(3) μ(5/2+ 2)−0.67 −0.76 +0.95(20) 107Pd B(E2;1/2+ 1→5/2+ 1) 112 90 0.58(7) μ(5/2+ 1)−1.06 −1.05 0.735(7) 109Pd B(E2;1/2+ 1→5/2+ 1) 97 76 1.36(18) B(E2;3/2+ 1→5/2+ 1)58 48 8(8) B(M1;3/2+ 1→5/2+ 1) 4.4 4.4 2.2(8) B(M1;5/2+ 2→3/2+ 1) 159 142 11.7(19) B(M1;7/2+ 2→5/2+ 1) 3.2 0.13 3.6(4) Experimental data for electromagnetic transitions and moments are available for odd-APd and Rh nuclei with N⩽65. The predicted B(E2) and B(M1) transition strengths as well as the electric-quadrupole Q(Iπ) and magnetic-dipole μ(Iπ) moments for the low-lying positive-parity states in odd-APd are given in Table II. In most of the cases, the mapped and phenomenological calculations provide similar results. Large values are obtained for the B(E2; 1/2+ 1→5/2+ 1)(in105Pd, 107Pd, and 109Pd), B(E2; 3/2+ 1→5/2+ 1)(in105Pd and 109Pd), and B(E2; 9/2+ 1→5/2+ 1)(in105Pd) transitions. The experimental data, however, suggest that these E2 transitions are weaker. The B(E2) and B(M1) rates corresponding to some transitions in odd-ARh nuclei are given in Table III.The large B(E2; 5/2+ 1→7/2+ 1) and B(E2; 5/2+ 1→9/2+ 1) rates obtained for 103Rh overestimate the experimental rates by several orders of magnitude. The deviation of the predicted B(E2) and B(M1) transition rates for odd-Asystems with respect to the experiment TABLE III. The same as in Table II, but for odd-ARh nuclei. Calc. Mapped Phen. Expt. 103Rh B(E2;5/2+ 1→7/2+ 1) 33 31 2.0(6) B(E2;5/2+ 1→9/2+ 1) 28 13 0.107(33) B(M1;5/2+ 1→7/2+ 1) 471 354 40(12) B(M1;9/2+ 1→7/2+ 1) 1.0 1.9 43(12) μ(7/2+ 1) 4.85 4.88 +4.540(11) μ(9/2+ 1) 5.69 5.62 +4.9(8) 107Rh B(E2;3/2+ 1→7/2+ 1) 4.73 1.62 0.16(2) 109Rh B(E2;3/2+ 1→3/2+ 2) 0.14 0.18 1.7×102(5) B(E2;3/2+ 1→7/2+ 1) 4.41 0.01 0.0174(5) B(E2;3/2+ 2→7/2+ 1) 5.3 5.9 26.1(19) B(E2;5/2+ 1→9/2+ 1) 7.9 5.3 >23 B(E2;5/2+ 2→3/2+ 1) 12 5.8 1.7(7) B(E2;5/2+ 2→3/2+ 2)22 9 7.E+1(3) B(E2;7/2+ 2→3/2+ 1) 8 15 131(12) B(M1;5/2+ 1→3/2+ 3) 5.2 8.6 >220 B(M1;5/2+ 1→3/2+ 1) 818 414 >0.40 B(M1;5/2+ 2→3/2+ 1) 37 289 2.4(3) B(M1;5/2+ 2→3/2+ 2) 152 207 2.2(15) B(M1;5/2+ 2→3/2+ 3) 18 210 2.5(4) B(M1;5/2+ 2→7/2+ 1) 231 112 4.1×10−2(6) B(M1;7/2+ 2→9/2+ 1) 318 611 0.25(6) B(M1;7/2+ 1→7/2+ 2) 7.6 8.9 6.6×10−2(8) B(M1;3/2+ 1→3/2+ 2) 27 48 0.58(12) B(M1;3/2+ 1→3/2+ 3) 158 256 1.18(11) B(M1;3/2+ 2→3/2+ 3) 276 32 0.32(10) B(M1;5/2+ 1→7/2+ 1) 233 172 >3.2 B(M1;9/2+ 1→7/2+ 1) 4.3 18 >58 could be interpreted in terms of the structure of the corresponding IBFM-2 wave functions. The components of the IBFM-2 wave functions for the low-lying states of odd-A Pd isotopes are shown in Fig. 9. They are associated with the single(quasi)-particle orbitals ν2s1/2,ν1d3/2,ν1d5/2, and ν0g7/2. Only components obtained within the mapped framework are shown as illustrative examples, while qualitatively similar results are obtained using the phenomenological approach. The states considered for odd-ARh nuclei are almost purely made of the proton 0g9/2configuration (with a weight of ≈99%). Therefore, the corresponding wave-function contents are not shown in the plot. As can be seen from the figure, the neutron 1d5/2configuration accounts for most of the IBFM-2 wave functions for the 1/2+ 1,3/2+ 1,5/2+ 1, and 7/2+ 1 in odd-APd nuclei with N67. However, the description of these wave functions in both the mapped and phenomenological IBFM-2 calculations in the present study may not be adequate, and this leads to some of the considerable disagreements between the calculated and experimental electromagnetic properties, including the B(E2; 1/2+ 1→5/2+ 1) values in 105Pd, 107Pd, and 109Pd (see Table II). The deficiency of the IBFM-2 wave functions could arise from various deficiencies of the present model calculations, such as the choice of the single-particle space, the quasiparticle energies and occupation probabilities of the odd particle, and the effective charges involved in the transition operators, which are kept 064304-9 K. NOMURA et al. PHYSICAL REVIEW C 106, 064304 (2022) responsible for the too small log ft values for the ARh(1+ 1)→ APd(0+ 1) decay with respect to the experimental data. The results of the mapped calculations have been compared with conventional IBM-2 calculations in which the parameters for the boson Hamiltonian have been fit to the experiment. The mapped and phenomenological IBM-2 excitation spectra for even-even, odd-A, and odd-odd systems are similar. However, the two sets of calculations differ in their predictions for electromagnetic and β-decay properties of the odd-nucleon systems. The results obtained in this study could be considered a plausible step towards a consistent simultaneous description of the low-lying states and β-decay properties of atomic nuclei. However, the difference between the predicted and experimental β-decay log ft values might require additional refinements of the employed theoretical framework. In particular, the small log ft values obtained suggest that the role of the effective axial-vector coupling constant gAshould be further studied in future calculations. 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