Structure of Odd-A Ag Isotopes Studied via Algebraic Approaches

Abstract

The structure of the odd-A silver isotopes ${}^{103–115}\mathrm{Ag}$ is discussed within the frame of the interacting boson–fermion model (IBFM). An overview of their key properties is presented, with a particular attention paid to the “J-1 anomaly”, represented by an abnormal ordering of the lowest ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states. By examining previously published data and newly performed calculations, it is demonstrated that the experimentally known level schemes and electromagnetic properties of ${}^{103–115}\mathrm{Ag}$ can be reproduced well within IBFM-1 by using a consistent set of model parameters. The contribution of different single-particle orbitals to the structure of the lowest-lying excited nuclear states in ${}^{103–115}\mathrm{Ag}$ is discussed. Given that the J-1 anomaly brings down the ${\mathrm{7/2}}^{+}$ level from the $j^{-3}$ multiplet to energies, which can be thermally populated in hot stellar environments, the importance of low-lying excited states in odd-A silver isotopes for astrophysical processes is outlined.

1. Introduction

The atomic nuclei in the A∼100 mass region exhibit a wide variety of phenomena. From a nuclear structure point of view, they manifest a complex interplay between single-particle and collective degrees of freedom. Furthermore, low-lying excited states in these nuclei could significantly contribute to astrophysical processes and nucleosynthesis mechanisms. Such states are thermally populated in stellar environments and the lower their excitation energy is, the higher the probability for their population [1]. This could significantly impact the astrophysical r-process—one of the primary mechanisms for the synthesis of nuclei heavier than iron [1]. Recent studies suggest that neutron star mergers (NSMs) are an active site where this mechanism takes place, and NSMs account for a large portion of the r-process synthesized heavy nuclei [2]. It is essential to note that the alteration of the r-process production rate depends on the half-lives of the ground and excited states [3,4]. Thus, complete and accurate low-energy nuclear data are crucial for astrophysical models and nuclear reactions network calculations, in particular.

A vast amount of nuclei relevant to the astrophysical r-process are placed close to the shell gaps, where single-particle and core-excited states emerge. Despite the fact that these states are relatively easy to predict, their detection is often experimentally difficult. Hence, the ordering of the single-particle levels away from the line of beta-stability is not easy to establish experimentally. This impacts our knowledge about isomerism near the shell closures and complicates the correct determination of the size of the shell gaps. In addition to the phenomena discussed above, effects such as spin traps, shape coexistence, and others, can also give rise to isomers. Being more difficult to predict, should they exist with half-lives significantly different from the known states, these isomers will also perturb the r-process pace.

Ag nuclei along the isotopic chain have a broad astrophysical importance. Heavier Ag isotopes lie along the r-process path, while lighter ones are engaged in the post r-process flow. Their origin is subject to an extensive discussion about the s- and r-process nucleosynthesis [5,6,7,8]. In the context of these mechanisms, fine nuclear structure details are important for the process paths and the expected Ag abundances.

The internal structure of the Ag isotopes also exhibits some unique characteristics. The Ag nuclei are among a few systems where signatures for three-particle-hole cluster–vibration excitation modes are observed, leading to more complex configurations than the ones present in neighboring nuclei. Due to this coupling, the $(9/2^{+},7/2^{+})$ doublet of the split $\pi g_{9/2}^{-3}$ multiplet is rearranged. The net result is that the $7/2^{+}$ state becomes the lowest-lying positive-parity state. This effect represents the so-called “J-1 anomaly”. Because of the proximity of the $7/2^{+}$ and $1/2^{-}$ levels, they are connected via $E3$ transitions instead of $M4$ transitions, and the isomeric states have shorter lifetimes than the ones they would have if the strong cluster–vibration interaction did not exist at the observed intensity. The configuration of low-lying $9/2^{+}$, $7/2^{+}$, and $1/2^{-}$ levels within a small energy interval is common along the Ag isotopic chain. It is also identified in heavy odd-A Ag nuclei, where experimental information is generally sparse [9]. Therefore, the exact arrangement of these states could be highly relevant for precise reaction network calculations.

2. Nuclear Data and Systematics

Exhibiting such unique properties, the Ag nuclei present an excellent playground to apply different theoretical models. The available data spanning the region between the semi-magic ${}_{47}{}^{97}\mathrm{Ag}_{50}$ and ${}_{47 }{}^{129}\mathrm{Ag}_{82}$ allow to test the robustness of the shells close to the magic numbers [10], the rise of quadrupole collectivity in the region, as well as the interplay between single-particle and collective degrees of freedom.

It should be mentioned that, even though the neutron mid-shell silver nuclei have a large number of valence particles, and hence collectivity might be expected to dominate, this cannot explain their structure alone. For example, in the simplest weakly interacting particle–core model, multiplets of states should appear in the odd-A system near the energies of the core phonon excitations, and the $E2$ reduced transitions have to be approximately similar to these of the core. This approach oversimplifies the picture, given that the odd-A Ag levels are more scattered and their B($E2$) values do not exactly follow those of the even-A cores.

2.1. Excited States

The positive-parity states which appear at low excitation energies are of particular interest. Those can be formed only by taking into account the $\pi g_{9/2}$ occupation. Due to the spin–orbit interaction, this positive-parity single-particle orbit appears just below the Z = 50 shell gap.

Some of the particular properties of the odd-A Ag level schemes which can be crucial to understand their structure are as follows:

• The low excitation energy of the $7/2_1^{+}$ states which become the lowest-lying positive parity states around the middle of the isotopic chain.
• In several nuclei (including some in the middle of the mass chain), the sequence based on $7/2^{+}$ is interrupted at $21/2^{+}$ by nearly equidistant levels, connected via $M1$ transitions (often interpreted as magnetic rotational band).

2.2. Transition Probabilities

Lifetimes, along with $\gamma$-ray multipolarities and mixing ratios, are physical quantities that are essential for the calculation of the electromagnetic transition rates. However, lifetime data for the positive-parity states in Ag nuclei are generally missing. Experimental information is available mainly for the $9/2_1^{+}$ and/or $7/2_1^{+}$ states. A summarized set of lifetime information for positive-parity states in the stable ${}^{107,109}\mathrm{Ag}$ isotopes is shown in

Table 1. Experimental nuclear data for properties of ^107,109Ag [9].

Isotope	Ei	${\mathit{J}}_{\mathit{i}}$	${\mathit{T}}_{\mathbf{1}/\mathbf{2}}$	${\mathit{E}}_{\mathit{f}}$	${\mathit{J}}_{\mathit{f}}^{\mathit{\pi }}$	$\mathit{B}\left(\mathit{M}\mathbf{1}\right)$	$\mathit{B}\left(\mathit{E}\mathbf{2}\right)$
	[keV]			[keV]		[W.u.]	[W.u.]
107Ag	93	$7/2^{+}$	44.3 (2) s
	126	${(9/2)}^{+}$	2.85 (10) ns	93	$7/2^{+}$	0.018 (1)	81 (29)
	773	${(11/2)}^{+}$	<15 ns
	991	${(13/2)}^{+}$	<15 ns
	1577	${(15/2)}^{+}$
	2054	${(17/2)}^{+}$
	3148	${(21/2)}^{+}$
109Ag	88	$7/2^{+}$	39.79 (21) s
	133	$9/2+$	2.60 (12) ns	88	$7/2^{+}$	0.0165 (17)	130 (120)
	773	$11/2^{+}$
	931	$13/2^{+}$
	1703	$15/2^{+}$
	1894	$17/2^{+}$	0.57 (5) ps	931	$13/2^{+}$		39 (4)
	2567	$19/2^{+}$	0.39 (4) ps	1894	$17/2^{+}$	0.08 (3)	14 (+18 −14)
	2567	$19/2^{+}$	0.39 (4) ps	1703	$15/2^{+}$		50 (20)
	2841	$21/2^{+}$	0.82 (8) ps	2567	$19/2^{+}$	(0.39 7)	(4 +21 −4)

. It should be noted that there are no firm lifetime values for the $11/2^{+}$ and $13/2^{+}$ states in either of the two nuclei.

In addition to the poor lifetimes systematics, many of the transitions between the lowest-lying excited states (such as the $9/2^{+}\to 7/2^{+}$ transitions) have a mixed character. Thus, measurements of mixing ratios are required to experimentally determine the transition probabilities. Data for mixing ratios are often sparse, or the measured values have a large uncertainty. This is also the case with nuclei along the Ag isotopic chain [9].

Despite the lack of complete information, the experimental data are somewhat sufficient to perform a systematic study of the $9/2_1^{+}\to 7/2_1^{+}$ transition probabilities along part of the Ag isotopic chain. Under the assumption of a three-particle cluster single-j occupancy, the magnetic dipole transitions connecting the the positive-parity states from the $\Delta J=1$ sequence based on $9/2_1^{+}$ or $7/2_1^{+}$ states are forbidden. As such, they have to be hindered with respect to the single-particle estimates. It is instructive to study their evolution with the neutron number.

The $9/2^{+}\to 7/2^{+}$ experimental reduced $M1$ transition strengths are, indeed, hindered by two to three orders of magnitude with respect to the single-particle estimates. At the same time, the $B\left(E2\right)$ values are enhanced by several orders of magnitude. Because in many cases the half-lives are not known and given that the $T_{1/2}$ contribution negates, the R-value ($R=B\left(M1\right)/B\left(E2\right)$) was introduced in Ref. [11] for the purpose of studying the systematic trends. It is observed that the typical ratio for the $9/2^{+}\to 7/2^{+}$ transitions is $R=B\left(M1\right)/B\left(E2\right)\sim {10}^{-4}$ (Figure 1). Knowledge of the mixing ratio is still required for such evaluations. The values of $\delta$ for ${}^{111}\mathrm{Ag}$ and ${}^{115}\mathrm{Ag}$ were recently extracted from intensity balances to the $9/2^{+}$ state [11]. In ${}^{111}\mathrm{Ag}$, $\delta$ was determined from $\beta$-decay [12] data and intensity balance to the ${\mathrm{9/2}}^{+}$ level and by assuming that there is no direct feeding from the $5/2^{+}$ parent state. Despite the large uncertainty $\Delta \delta$, the value fits the overall trend shown in Figure 1. In ${}^{115}\mathrm{Ag}$, the mixing ratio of the $9/2^{+}\to 7/2^{+}$ transition was evaluated from a double-gated $E_{\gamma }-E_{\gamma }-E_{\gamma }$ spectrum and intensity balance to the $9/2^{+}$ level [11]. The uncertainty is large, but the R-value nicely contributes to the systematics. The $\beta$-decay data for ¹⁰³Ag are discrepant, not allowing to unambiguously determine $\delta$. Still, the systematic presented in Figure 1 shows a gradual decrease of R from 0.00045 (18) in ${}^{105}\mathrm{Ag}$ to 0.00009 (8) in ${}^{115}\mathrm{Ag}$, leading to a strong correlation between R and $\Delta E=E_{9/2^{+}}-E_{7/2^{+}}$ [11].

The evolution of the transition strengths with spin is more difficult to analyze due to the scarce data, but the information available for ¹⁰⁹Ag shows that the $M1$ components of transitions of mixed multipolarity remain hindered by two orders of magnitude even at the high-spin part of the $j^{-3}$ multiplet. Similar trends are also observed in other three-hole systems. For example, ⁸³Kr and ⁸⁵Sr have remarkable similarities to the Ag isotopes. These nuclei are three holes away from the neutron magic number N = 50. The spin and parity of their ground states is $9/2^{+}$, and sequences of levels are built on top of them, in a manner similar to the Ag spectra. The B($M1$) values are two to three orders of magnitude hindered with respect to the single-particle estimates.

2.3. Static Moments

The magnetic moments for a $j^n$ system are [13]

$$\mathit{\mu }=\sum _{i=1}^{n}g{\mathit{j}}_{\mathit{i}}=g\sum _{i=1}{\mathit{j}}_{\mathit{i}}=g\mathit{J}.$$

This means that the g-factor of all states in a $j^n$ configuration of identical nucleons should be equal to the g-factor of a state with a single nucleon on j-orbit. Therefore, the g-factor alone cannot distinguish between seniority $\upsilon$ = 1 and $\upsilon$ = 3 states, but it gives a deeper insight into the occupied single-particle orbits. The measured magnetic moments of low-lying states in odd-A Ag nuclei are shown in

Table 2. Magnetic dipole and electric quadrupole moments of states in some odd-A Ag isotopes.

Isotope	${\mathit{J}}^{\mathit{\pi }}$	$\mathit{\mu }\left[{\mathit{\mu }}_{\mathit{N}}\right]$	$\mathit{g}\left[{\mathit{\mu }}_{\mathit{N}}\right]$	Q [b]
97Ag	$(9/2^{+})$	6.13 (12) [14]	1.362 (27)
99Ag	${(9/2)}^{+}$	5.81 (3) [15]	1.291 (4)
101Ag	$9/2^{+}$	5.627 (11) [16]	1.2504 (24)	+0.35 (5) [16]
103Ag	$7/2^{+}$	+4.47 (5) [17]	1.277 (14)	+0.84 (9) [16]
105Ag	$1/2^{-}$	+0.1013 (13) [18]	0.2026 (26)
105Ag	$7/2^{+}$	+4.414 (13) [19]	1.261 (4)	+0.85 (11) [16]
107Ag	$1/2^{-}$	−0.11352 (5) [20]	−0.22704 (1)
107Ag	$7/2^{+}$	(+)4.398 (5) [21]	1.2566 (14)	+0.98 (11) [22,23]
109Ag	$1/2^{-}$	−0.130696 (2) [20]	−0.261392 (4)
109Ag	$7/2^{+}$	+4.400 (6) [24]	1.2571 (17)	(+)1.02 (12) [22,23]
${}^{111}\mathrm{Ag}$	$1/2^{-}$	−0.146 (2) [25]	−0.292 (4)
${}^{113}\mathrm{Ag}$	$1/2^{-}$	+0.159 (2) [26]	0.318 (4)
${}^{113}\mathrm{Ag}$	$7/2^{+}$	+4.447 (2) [27]	1.2706 (6)	+1.03 (9) [27]
${}^{115}\mathrm{Ag}$	$1/2^{-}$	−0.1704 (9) [27]	−0.3408 (18)
${}^{115}\mathrm{Ag}$	$7/2^{+}$	+4.44223 (9) [27]	1.26921 (3)	+1.04 (8) [27]
117Ag	$1/2^{-}$	−0.1752 (8) [27]	−0.3504 (16)
117Ag	$7/2^{+}$	+4.43897 (8) [27]	1.26828 (2)	+1.05 (8) [27]
119Ag	$1/2^{-}$	−0.1707 (9) [27]	−0.3414 (18)
119Ag	$7/2^{+}$	+4.434 (1) [27]	1.2669 (3)	+0.93 (8) [27]
121Ag	$7/2^{+}$	+4.447 (1) [27]	1.2706 (3)	+0.85 (8) [27]
121Ag	$1/2^{-}$	−0.1797 (4) [27]	−0.3594 (8)

, along with the extracted g-factors.

It is instructive to compare the experimental g-factors of the $9/2^{+}$ and $7/2^{+}$ states in Ag nuclei to the g-factor of the $9/2^{+}$ single-particle state in ₄₉In. The lightest indium isotope for which g-factor data are available [19] is ¹⁰⁵In. There, g = 1.2611 (11) ${\mu }_N$, which is similar to the g-factors of the levels of interest in the Ag nuclei, suggesting a leading $\pi g_{9/2}$ or $\pi g_{9/2}^{-3}$ contribution to their wave functions.

Furthermore, the electric quadrupole moments (Q) provide valuable information about the charge distribution inside atomic nuclei. Table 2 also shows the measured values of some quadrupole moments in the odd-A Ag isotopes. Experimental data are available mainly for the $7/2_1^{+}$ state. Similarly to the trend observed for $\mu$ (${\mathrm{7/2}}^{+}$), the Q ($7/2_1^{+}$) values appear to be consistent along the Ag isotopic chain. This might indicate that no sudden onset of deformation of the cores, or drastic changes in the core polarization, is present along the isotopic chain.

3. Theoretical Approaches to the Structure of Odd-$\mathbf{A}$ Ag

The structure of the odd-A Ag isotopes, and particularly the properties of their lowest-lying excited states, have been a long-standing subject of investigations over the years. Given that $\pi g_{9/2}$ is the only positive-parity orbital bellow the Z = 50 shell in this region, it is natural to expect that the structure of low-lying positive parity excitations in such nuclei is dominated by $\pi g_{9/2}$. Indeed, as discussed, a low-lying ${\mathrm{9/2}}^{+}$ state is experimentally observed in all well-studied Ag isotopes. It is accompanied by the low-lying $7/2^{+}$ level which drops in energy even below ${\mathrm{9/2}}^{+}$ in the Ag nuclei with A ≥ 103 [9]. The energy of approximately 4 MeV required for excitations across the shell gap excludes the possibility that this ${\mathrm{7/2}}^{+}$ state emerges from the $\pi g_{7/2}$ orbital.

Exhibiting such intriguing examples, the J-1 anomaly was studied within the framework of multiple theoretical approaches. Breaking the $jj$-coupling approximation and introducing $g_{7/2}^n$ admixtures to the wave functions [28], the $j^{-3}$ coupling scheme [29], large-scale shell model calculations with effective Q·Q and surface delta (SDI) interactions [30,31], three-valence holes–vibrator coupling model calculations [32], particle-plus-rotor model [33], and IBFM works [34,35], are just a few examples of these studies. Such anomalous behavior was also observed in other isotopic and isotonic chains [30,36,37], but it is particularly prominent in the Ag nuclei.

The stable ^107,109Ag isotopes are often regarded as an example of the core–excitation weak-coupling model. Within the framework of this model, a particle–core interaction is necessary to remove the J-degeneracy of levels which would otherwise arise if no interaction is considered. However, it was observed that the quasi-degenerate $7/2^{+}$, $9/2^{+}$ doublet in these isotopes is anomalous for a standard weak-coupling model [38]. A good reproduction of experimental data for the Ag J-1 anomaly is achieved, though, within the Alaga model. Its application to ^107,109Ag by coupling three-hole proton valence shell cluster moving in the shell model configuration space to low-frequency quadrupole vibrational field yields a satisfactory description of a large set of low-lying states [39]. It was observed that higher-spin states are better reproduced within a deformation-based approach [40]. Also, the positive-parity bands in the heavier silver isotopes generally do not resemble the weak particle–core coupling scheme from Ref. [41]. They are rather showing fingerprints of deformed systems [42].

A key problem for the interpretation of odd-A Ag structures is understanding the role of the residual interaction. In general, as noted in Ref. [43], using the quadrupole residual interaction as the driving force to make nuclei slightly deformed, one obtains a ${\mathrm{7/2}}^{+}$ level at low excitation energy. This origin of the low energy as due to the quadrupole residual interactions was also verified by IBFM calculations [43,44,45,46,47,48,49]. The present work aims to summarize some general results from earlier IBFM approaches to the structure of odd-A Ag nuclei, as well as to present a systematic study of ^103–115Ag performed in the framework of IBFM-1.

4. ^103–115Ag IBFM-1 Calculations

The applications of the algebraic interacting boson–fermion model (IBFM) [50,51] to odd-A Ag isotopes are particularly interesting with regard to the J-1 anomaly in these nuclei. An approach based on the coupling of an odd particle in the $\pi$g_9/2 orbital to a core with a structure close to SU(5) was found to successfully describe the low-lying positive parity levels in Rb, Tc, and Ag isotopes [52]. It was observed that the correct level order could be obtained through the exchange term in the particle–core coupling. Furthermore, important anomalies of the 5/2⁺ states point to a possible influence of the $\pi$d_5/2 orbital [52]. While the single-j approach has the advantage of its relative simplicity, more detailed theoretical predictions in the Ag isotopes could be obtained via multi-shell calculations [45]. A similar observation in was reported by Vanhorebeeck et al. [48] who could reproduce the exact position of the 5/2⁺ state in ⁹⁷Rh within IBFM only after taking into account the $\pi$d_5/2 orbital. However, a single-j configuration of a $\pi$g_9/2 proton coupled to a core seems sufficient for the low-level positive parity states in odd-A ^97–103Tc nuclei within IBFM [46,47]. These observations show that the inclusion of the $\pi$d_5/2 orbital in IBFM calculations becomes more important when approaching the Z = 50 shell closure.

In addition to the aforementioned weak-coupling model predictions [38], Wood et al. also compared experimental data for some gyromagnetic ratios in ^107,109Ag to IBFM calculations. While the weak-coupling model cannot reproduce the energy separation of the ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states, a stronger quadrupole force results in a much better global fit, describing well not only the negative-parity spectra, g-factors, and $B\left(M1\right)$ rates but also the energies of the ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ levels. Based on the level schemes, and especially $M1$ observables, it was concluded that the particle–core mixing in these nuclei is not weak [38].

One-nucleon transfer reactions provide sensitive information about nuclear structure, and such experimental data were also used to investigate the reliability of IBFM calculations in the odd-A Ag isotopes [53]. Considering a multiconfiguration space, including the $\pi$f_5/2,$\pi$p_3/2, $\pi$g_9/2, $\pi$p_1/2, and $\pi$d_5/2 orbitals, a good agreement with experimental spectroscopic factors and the the energies of the ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states was obtained.

Intruder states in heavier Ag isotopes were also studied via IBFM. This was achieved by coupling to particles placed at orbitals above Z = 50 [54].

Although the IBFM framework seems to generally have success for Ag nuclei, more detailed approaches were also developed to study certain Ag isotopes. For example, the interacting boson–fermion plus broken pair model was found to be successful in reproducing multiple features of ¹⁰¹Ag [55].

Recently, the structure of ^111,113Ag was experimentally studied in induced fission reactions [34]. The results of the measurements were compared to IBFM-1 calculations including the $\pi$p_3/2, $\pi$f_5/2, $\pi$p_1/2, $\pi$g_9/2, and $\pi$d_5/2 orbitals. Even–even cores of ^112,114Cd were considered in the calculations, represented within the interacting boson model (IBM-1) [56,57,58,59]. A relatively good reproduction of the experimental level schemes of ^111,113Ag and electromagnetic properties of ¹¹¹Ag was achieved. The theoretical approach of Ref. [34] was further extended in subsequent IBFM-1 calculations applied to ¹¹⁵Ag [35]. The present work expands these investigations towards the lighter ^103–109Ag nuclei by using a similar approach.

4.1. IBM-1 Calculations of Even-A Cd Isotopes

Consistent with previous related works [34,35], Cd nuclei were considered even-A cores of ^103–109Ag. The core properties were described using the extended consistent-Q formalism (ECQF) to IBM-1, in a manner similar to the one discussed in Refs. [60,61]. In order to fully present the systematic study in a comprehensive way, the following discussions involve both the published data for ^111–115Ag [34,35] and newly obtained results for ^103–109Ag.

The EQCF Hamiltonian used in the current IBM-1 approach [62] is

$$H_B=\epsilon n_d-\kappa Q^2-{\kappa }^{\prime }{L}^2.$$

(1)

where

$$\begin{array}{c}{n}_d=\sqrt{5}{T}_0,L=\sqrt{10}{T}_1,\\ Q=(d^{†}s+s^{†}\tilde{d})+\chi {\left(d^{†}\tilde{d}\right)}^{\left(2\right)}=(d^{†}s+s^{†}\tilde{d})+\chi T_2,\\ \widetilde{d_{\mu }}={(-1)}^{\mu }{d}_{-\mu }.\end{array}$$

(2)

The Hamiltonian represented in this form has a physical connotation in terms of angular momentum and quadrupole operators, while the term $\epsilon$ is equivalent to assigning an energy to the d-boson. It is important to note that these operators act on boson states, not in the fermion space. A detailed description of the IBM model parameters can be found in Ref. [58].

The present IBM-1 calculations were performed using the PHINT program package [63], with the parameters presented in

Table 3. IBM-1 sets of parameters used to calculate the properties of the even-A ^104–116Cd cores.

A	$\mathit{\epsilon }$	$\mathit{\kappa }$	${\mathit{\kappa }}^{\prime }$	$\mathit{\chi }$	${\mathrm{e}}_{\mathit{B}}$
104	0.68	0.0210	−0.0030	−0.411	0.125
106	0.68	0.0240	−0.0060	−0.268	0.109
108	0.71	0.0220	−0.0075	−0.246	0.100
110	0.72	0.0150	−0.0095	−0.224	0.097
112	0.66	0.0065	−0.0050	−0.089	0.103
114	0.63	0.0075	−0.0050	−0.089	0.103
116	0.69	0.0208	−0.0045	−0.411	0.095

. Their values were set such that the energies of the low-lying states in ^104–116Cd, as well as the $E2$ strengths of the transitions between them, are well reproduced. The parameters are generally similar to the ones used in ECQF IBM-1 descriptions of even–even Ru and Pd nuclei [60,61].

The $E2$ transition operator

$$T\left(E2\right)=e_B[(s^{†}\tilde{d}+d^{†}s)+\chi {\left(d^{†}\tilde{d}\right)}^{\left(2\right)}]=e_{B}Q,$$

(3)

was used to calculate transition probabilities, taking into account the relation

$$B(E2;J_i\to J_f)={\displaystyle \frac{1}{2J_i+1}}{\langle J_f\parallel T\left(E2\right)\parallel J_i\rangle }^2.$$

(4)

where $J_i$ and $J_f$ indicate the spins of the initial and final states.

Calculated and experimental energies of low-lying states in the ground-state bands of even-A Cd isotopes are compared in Figure 2. In addition, theoretical and experimental probabilities for transitions between them are shown in Figure 3. Although the set of experimental data is not fully complete, the properties of the ground-state bands in the even-A isotopes are relatively well described. The established IBM-1 core parameters were further included in the subsequent set of ${}^{103–115}\mathrm{Ag}$ IBFM-1 calculations.

4.2. IBFM-1 Calculations of Odd-A Ag Isotopes

The Hamiltonian used in this odd-A Ag IBFM-1 approach is in the form

$$H=H_B+H_F+V_{BF},$$

(5)

where ${\mathrm{H}}_B$ represents the even-A core IBM-1 Hamiltonian, while the fermionic part is

$$H_F=\sum _j{E}_j{n}_j.$$

(6)

The quasiparticle energies of the single-particle shell model orbitals are labeled with $E_j$.

The boson–fermion interaction ${\mathrm{V}}_{BF}$ is described by several interactions that are sufficient to phenomenologically study different properties [49,50]:

$$V_{BF}=\sum _j{A}_j{n}_d{n}_j+\sum _{j{j}^{\prime }}{\Gamma }_{j{j}^{\prime }}(Q·{\left(a_j^{†}{\tilde{a}}_{j^{\prime }}\right)}^{\left(2\right)})+\sum _{j{j}^{\prime }{j}^{″}}{\Lambda }_{j{j}^{\prime }}^{j^{″}}:{({\left(d^{†}{\tilde{a}}_j\right)}^{\left(j^{″}\right)}\times {\left(\tilde{d}{a}_{j^{\prime }}^{†}\right)}^{\left(j^{″}\right)})}_0^{\left(0\right)}:$$

(7)

Only three terms are retained in this form of the boson–fermion interaction, namely monopole, quadrupole, and exchange terms [51].

The number of parameters can be decreased on the basis of microscopic considerations, therefore leading to [44]:

$$\begin{array}{c}{A}_j=A_0,\\ {\Gamma }_{j{j}^{\prime }}={\Gamma }_0(u_j{u}_{j^{\prime }}-{\upsilon }_j{\upsilon }_{j^{\prime }})〈j∥Y^{\left(2\right)}∥j^{\prime }〉,\\ {\Lambda }_{j{j}^{\prime }}^{j^{″}}=-2\sqrt{5}{\Lambda }_0{\beta }_{j{j}^{″}}{\beta }_{j^{″}{j}^{\prime }}/{(2j^{″}+1)}^{1/2}(E_j+E_{j^{″}}-\hslash \omega ),\end{array}$$

(8)

where

$$\begin{array}{c}{\beta }_{j{j}^{\prime }}=〈j∥Y^{\left(2\right)}∥j^{\prime }〉(u_j{\upsilon }_{j^{\prime }}+{\upsilon }_j{u}_{j^{\prime }}),\\ u_j^2=1-{\upsilon }_j^2.\end{array}$$

(9)

The occupation probabilities of the the single-particle orbitals j are noted with ${\upsilon }_j^2$. $A_0$, ${\Gamma }_0$, and ${\Lambda }_0$ are the strengths of the monopole, quadrupole, and exchange interactions. A comprehensive discussion with more details about the IBFM parameters is available in Ref. [51].

The ODDA program package [64] was used to calculate the properties of ${}^{103–115}\mathrm{Ag}$ within IBFM-1. An initial set of single-particle energies (s.p.e.) was determined using the approach presented in Ref. [65]. These values, as well as the boson–fermion interaction parameters, were further varied to better reproduce the experimental data available for each of the Ag isotopes. The variation of the parameters was performed keeping in mind the initial sets of s.p.e., as well as the experimentally observed similar characteristics of the isotopes at low excitation energies.

The occupation probabilities and quasiparticle energies of the orbitals in ${}^{103–115}\mathrm{Ag}$ were calculated using a BCS procedure. A pairing gap of $\Delta$ = 1.5 MeV was considered. The results of the BCS procedure applied to all Ag nuclei are shown in

Table 4. BCS calculated quasiparticle energies $E_j$ and occupation probabilities ${\upsilon }_j^2$ of the proton single-particle orbitals in the studied odd-A Ag nuclei.

	103Ag	105Ag	107Ag	109Ag	111Ag	113Ag	115Ag
$E_j\left(p_{3/2}\right)$	3.16	3.12	3.14	3.15	3.14	3.14	3.40
$E_j\left(f_{5/2}\right)$	2.90	2.86	2.80	2.89	2.80	2.80	3.05
$E_j\left(p_{1/2}\right)$	1.91	1.77	1.73	1.69	1.73	1.73	1.57
$E_j\left(g_{9/2}\right)$	1.74	1.77	1.78	1.79	1.78	1.78	1.78
$E_j\left(d_{5/2}\right)$	2.68	2.71	2.69	2.69	2.69	2.69	2.46
${\upsilon }_j^2\left(p_{3/2}\right)$	0.94	0.94	0.94	0.94	0.94	0.94	0.95
${\upsilon }_j^2\left(f_{5/2}\right)$	0.93	0.93	0.92	0.93	0.92	0.92	0.94
${\upsilon }_j^2\left(p_{1/2}\right)$	0.81	0.77	0.75	0.73	0.75	0.75	0.64
${\upsilon }_j^2\left(g_{9/2}\right)$	0.75	0.77	0.77	0.77	0.77	0.77	0.77
${\upsilon }_j^2\left(d_{5/2}\right)$	0.09	0.08	0.08	0.09	0.08	0.08	0.10

.

The properties of both positive-parity and negative-parity states in each of the ${}^{103–115}\mathrm{Ag}$ isotopes were calculated using the same boson–fermion interaction parameters (see

Table 5. Boson–fermion interaction parameters used in the ^103–115Ag IBFM-1 calculations.

Isotope	${\mathit{A}}_{\mathbf{0}}$	${\mathbf{\Gamma }}_{\mathbf{0}}$	${\mathbf{\Lambda }}_{\mathbf{0}}$
103Ag	−0.30	0.20	3.8
105Ag	−0.34	0.24	3.9
107Ag	−0.36	0.24	3.8
109Ag	−0.30	0.20	4.0
${}^{111}\mathrm{Ag}$	−0.30	0.20	3.8
${}^{113}\mathrm{Ag}$	−0.30	0.20	3.8
${}^{115}\mathrm{Ag}$	−0.48	0.38	3.5

). Partial experimental [9] and IBFM-1 calculated level schemes of the odd-A Ag nuclei are compared in Figure 4.

Reduced $M1$ and $E2$ transition probabilities for transitions in the odd Ag nuclei were also calculated within IBFM-1. Operators in the following forms were considered:

$$T\left(M1\right)=\sqrt{{\displaystyle \frac{90}{4\pi }}}{\mathrm{g}}_d{\left(d^{†}\tilde{d}\right)}^{\left(1\right)}-{\mathrm{g}}_F\sum _{j{j}^{\prime }}(u_j{u}_{j^{\prime }}+{\upsilon }_j{\upsilon }_{j^{\prime }})·\langle j\parallel {\mathrm{g}}_{l}l+{\mathrm{g}}_{s}s\parallel j^{\prime }\rangle \times [{\left(a_j^{†}{\tilde{a}}_{j^{\prime }}\right)}^{\left(1\right)}+c.c.],$$

(10)

$$T\left(E2\right)=e_B({(s^{†}\tilde{d}+d^{†}s)}^{\left(2\right)}+\chi {\left(d^{†}\tilde{d}\right)}^{\left(2\right)})-e_F\sum _{j{j}^{\prime }}(u_j{u}_{j^{\prime }}-{\upsilon }_j{\upsilon }_{j^{\prime }})\langle j\parallel Y^{\left(2\right)}\parallel j^{\prime }\rangle \times [{\left(a_j^{†}{\tilde{a}}_{j^{\prime }}\right)}^{\left(2\right)}+c.c.].$$

(11)

The effective boson ($e_B$) and fermion ($e_F$) charge in each nucleus were set equal to the effective boson charge of the respective IBM-1 calculated core. The d-boson $\mathrm{g}$-factors used in the calculations were determined from magnetic moments of the $2_1^{+}$ states in the neighboring even-A Cd. Also, values of ${\mathrm{g}}_s$ = $4.0 {\mu }_N$ and ${\mathrm{g}}_l$ = 1.0 were applied.

The calculated B($M1$) and B($E2$) values for sets of several transitions between states in the odd-A Ag nuclei are compared to experimental data (where available) in

Table 6. Experimental and IBFM-1 calculated transition probabilities for several transitions in the odd-A ^103–115Ag isotopes. The experimental data are taken from Ref. [9].

Isotope	${\mathit{J}}_{\mathit{i}}^{\mathit{\pi }}$	${\mathit{J}}_{\mathit{f}}^{\mathit{\pi }}$	$\mathit{B}{\left(\mathit{M}1\right)}_{\mathit{exp}}$ [W.u.]	$\mathit{B}{\left(\mathit{E}2\right)}_{\mathit{exp}}$ [W.u.]	$\mathit{B}{\left(\mathit{M}1\right)}_{\mathit{th}}$ [W.u.]	$\mathit{B}{\left(\mathit{E}2\right)}_{\mathit{th}}$ [W.u.]
103Ag	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$			0.009	24
103Ag	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.008	35
103Ag	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				7.1
103Ag	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.069	36
105Ag	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$	0.0241 (10)	53 (+21 −18)	0.013	28
105Ag	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.013	40
105Ag	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				7.8
105Ag	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.087	39
105Ag	${\mathrm{9/2}}_2^{+}$	${\mathrm{7/2}}_1^{+}$	0.00051 (+32 −16)		0.079
105Ag	${\mathrm{9/2}}_2^{+}$	${\mathrm{9/2}}_1^{+}$	0.00014 (6)		0.013
105Ag	${\mathrm{7/2}}^{-}$	${\mathrm{3/2}}^{-}$		0.71 (+49 −29)		41
105Ag	${\mathrm{7/2}}^{-}$	${\mathrm{5/2}}^{-}$	0.0014 (+9 −7)		0.046
105Ag	${\mathrm{9/2}}^{-}$	${\mathrm{5/2}}^{-}$		2.6 (+17 −9)		46
107Ag	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$	0.018 (1)	81 (29)	0.014	31
107Ag	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.018	45
107Ag	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				8.2
107Ag	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.11	43
107Ag	${\mathrm{3/2}}^{-}$	${\mathrm{1/2}}^{-}$	0.12 (2)	42 (4)	0.039	34
107Ag	${\mathrm{5/2}}^{-}$	${\mathrm{1/2}}^{-}$		43 (3)		34
107Ag	${\mathrm{5/2}}^{-}$	${\mathrm{3/2}}^{-}$	0.033 (4)	11.1 (13)	0.0018	1.6
107Ag	${\mathrm{3/2}}_2^{-}$	${\mathrm{1/2}}^{-}$	0.11 (3)	0.5 (3)	0.028	1.2
107Ag	${\mathrm{3/2}}_2^{-}$	${\mathrm{3/2}}^{-}$	0.23 (8)		0.014
107Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{1/2}}^{-}$		2.3 (4)		0.70
107Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{3/2}}^{-}$	0.024 (4)	4.2 (10)	0.004	5.4
107Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{5/2}}^{-}$	0.053 (7)	9.5 (26)	0.013	22
109Ag	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$	0.0165 (17)	130 (120)	0.0199	28
109Ag	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.017	41
109Ag	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				7.6
109Ag	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.101	44
109Ag	${\mathrm{17/2}}^{+}$	${\mathrm{13/2}}^{+}$		39 (4)		32
109Ag	${\mathrm{19/2}}^{+}$	${\mathrm{15/2}}^{+}$		50 (20)		37
109Ag	${\mathrm{19/2}}^{+}$	${\mathrm{17/2}}^{+}$	0.08 (3)	14 (+18 −14)	0.029	20
109Ag	${\mathrm{21/2}}^{+}$	${\mathrm{17/2}}^{+}$		20.4 (23)		42
109Ag	${\mathrm{21/2}}^{+}$	${\mathrm{19/2}}^{+}$	0.39 (7)	4 (+21 −4)	0.41	9.0
109Ag	${\mathrm{3/2}}^{-}$	${\mathrm{1/2}}^{-}$	0.117 (8)	37 (4)	0.035	33
109Ag	${\mathrm{5/2}}^{-}$	${\mathrm{1/2}}^{-}$		40.5 (17)		33.3
109Ag	${\mathrm{5/2}}^{-}$	${\mathrm{3/2}}^{-}$	0.0316 (21)	4 (4)	0.0013	1.1
109Ag	${\mathrm{3/2}}_2^{-}$	${\mathrm{1/2}}^{-}$	0.18 (5)	0.25 (14)	0.03	0.73
109Ag	${\mathrm{3/2}}_2^{-}$	${\mathrm{3/2}}^{-}$	0.26 (7)	60 (30)	0.012	31
109Ag	${\mathrm{3/2}}_2^{-}$	${\mathrm{5/2}}^{-}$	0.14 (4)		0.007
109Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{1/2}}^{-}$		2.6 (4)		0.55
109Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{3/2}}^{-}$	0.034 (5)	7.3 (18)	0.005	6.3
109Ag	${\mathrm{5/2}}_2^{-}$	${\mathrm{5/2}}^{-}$	0.089 (13)	9 (5)	0.014	26
109Ag	${\mathrm{9/2}}^{-}$	${\mathrm{5/2}}^{-}$		68 (11)		52
${}^{111}\mathrm{Ag}$	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$			0.028	27
${}^{111}\mathrm{Ag}$	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.022	43
${}^{111}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.102	51
${}^{111}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				5.6
${}^{113}\mathrm{Ag}$	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$			0.018	41
${}^{113}\mathrm{Ag}$	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.018	59
${}^{113}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.087	65
${}^{113}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				5.0
${}^{115}\mathrm{Ag}$	${\mathrm{9/2}}^{+}$	${\mathrm{7/2}}^{+}$			0.004	59
${}^{115}\mathrm{Ag}$	${\mathrm{11/2}}^{+}$	${\mathrm{9/2}}^{+}$			0.048	78
${}^{115}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{11/2}}^{+}$			0.187	72
${}^{115}\mathrm{Ag}$	${\mathrm{13/2}}^{+}$	${\mathrm{9/2}}^{+}$				19

. A coparison between experimental and IBFM-1 electromagnetic moments is presented in

Table 7. Experimental and IBFM-1 calculated magnetic dipole and electric quadrupole moments of low-lying states in the odd-A Ag isotopes. The experimental data are taken from Refs. [27,66,67,68].

	103Ag	105Ag	107Ag	109Ag	111Ag	113Ag	115Ag
$\mu$ (${\mathrm{1/2}}^{-}$)exp [${\mu }_N$]		0.1013 (10)	−0.11352 (5)	−0.1306906 (2)	−0.146 (2)	0.159 (2)	−0.1704 (9)
$\mu$ (${\mathrm{7/2}}^{+}$)exp [${\mu }_N$]	4.426 (2)	4.408 (13)	4.392 (5)	4.394 (6)		4.447 (2)	4.4223 (9)
q (${\mathrm{7/2}}^{+}$)exp [b]	0.84 (9)	0.85 (11)	0.98 (11)	1.02 (12)		1.03 (9)	1.04 (8)
$\mu$ (${\mathrm{1/2}}^{-}$)th [${\mu }_N$]	0.004	0.005	0.009	0.005	0.009	0.011	0.013
$\mu$ (${\mathrm{7/2}}^{+}$)th [${\mu }_N$]	4.893	4.824	4.761	4.852	4.810	4.711	4.188
q (${\mathrm{7/2}}^{+}$)th [eb]	0.465	0.562	0.667	0.555	0.631	0.844	1.298

.

Given that the transitions between the ${\mathrm{9/2}}_1^{+}$ and ${\mathrm{7/2}}_1^{+}$ states of the odd-A Ag isotopes are of major interest for the J-1 anomaly studies, they were investigated in additional details. A plot with the systematics of $E2$/$M1$ mixing ratios for these transitions is presented in Figure 5. The experimental data are taken from Refs. [9,11].

5. Discussion

5.1. Structure of ${}^{103–115}\mathit{Ag}$

The IBFM-1 approach summarized in the present work shows a reasonably good reproduction of the experimental properties of odd-A Ag isotopes near the line of beta stability, as well as in the mid-shell region. It was observed that the characteristics of the odd-A Ag are well described considering even-A cores with structure close to the even Cd isotopes [34,35]. In the framework of dynamical symmetries, this relates the cores closer to the U(5) limit rather than $\gamma$-soft O(6) structures (such as Pd nuclei).

Both the model parameters used in the discussed IBFM-1 calculations and the experimental properties of the investigated nuclei indicate similarities between all odd-A ${}^{103–115}\mathrm{Ag}$ isotopes. Smooth evolution of the energies of excited states and electromagnetic properties as a function of the neutron number is observed. This statement is also valid for their even–even cores. It is instructive to note the slight decrease in the energy gaps between the ground-state band levels of the even-A Cd with A > 110. The effect is reproduced by the IBM-1 calculations and is consistent with the gradual increase of the B($E2;2_1^{+}\to 0_1^{+}$) values observed experimentally.

The ordering and characteristics of the lowest-lying positive-parity states in ${}^{103–115}\mathrm{Ag}$ are well described, and the J-1 anomaly is reproduced within the calculations. It is recognized that the correct energy ordering of the ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states can be achieved through the strength of the exchange term in the particle–core coupling. This is consistent with similar observations in previous IBFM-1 calculations [52]. As also noted in Refs. [34,35], the $\pi g_{9/2}$ orbital dominates the structure of positive-parity states at low energies. Within the current IBFM-1 approach, this orbital has a contribution of >95% to the wave functions of the first excited ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states in all studied Ag isotopes (see

Table 8. Contributions of different single-particle components to the wave functions of the lowest-lying states in ^103–115Ag as calculated within IBFM-1.

${\mathit{J}}^{\mathit{\pi }}$	${\mathit{E}}_{\mathit{level}}^{†}$ [keV]	$\mathit{\pi }{\mathit{p}}_{3/2}$ [%]	$\mathit{\pi }{\mathit{f}}_{5/2}$ [%]	$\mathit{\pi }{\mathit{p}}_{1/2}$ [%]	$\mathit{\pi }{\mathit{g}}_{9/2}$ [%]	$\mathit{\pi }{\mathit{d}}_{5/2}$ [%]
			103Ag
${\mathrm{7/2}}^{+}$	0.0	0.0	0.0	0.0	97.80	2.20
(9/2)+	27.54	0.0	0.0	0.0	98.93	1.07
${\mathrm{1/2}}^{-}$	134.45	1.96	3.95	94.09	0.0	0.0
			105Ag
${\mathrm{1/2}}^{-}$	0.0	3.03	5.89	91.08	0.0	0.0
${\mathrm{7/2}}^{+}$	25.468	0.0	0.0	0.0	97.58	2.42
${\mathrm{9/2}}^{+}$	53.138	0.0	0.0	0.0	97.92	2.08
			107Ag
${\mathrm{1/2}}^{-}$	0.0	3.44	7.20	89.36	0.0	0.0
${\mathrm{7/2}}^{+}$	93.125	0.0	0.0	0.0	97.53	2.47
(9/2)+	125.59	0.0	0.0	0.0	97.38	2.62
			109Ag
${\mathrm{1/2}}^{-}$	0.0	2.44	4.62	92.94	0.0	0.0
${\mathrm{7/2}}^{+}$	88.0337	0.0	0.0	0.0	97.39	2.61
${\mathrm{9/2}}^{+}$	132.762	0.0	0.0	0.0	98.02	1.98
			${}^{111}\mathrm{Ag}$
${\mathrm{1/2}}^{-}$	0.0	2.98	6.32	90.70	0.0	0.0
${\mathrm{7/2}}^{+}$	59.82	0.0	0.0	0.0	97.42	2.58
${\mathrm{9/2}}^{+}$	130.28	0.0	0.0	0.0	97.91	2.09
			${}^{113}\mathrm{Ag}$
${\mathrm{1/2}}^{-}$	0.0	3.64	7.75	88.61	0.0	0.0
${\mathrm{7/2}}^{+}$	43.5	0.0	0.0	0.0	97.25	2.75
${\mathrm{9/2}}^{+}$	139.30	0.0	0.0	0.0	97.14	2.86
			${}^{115}\mathrm{Ag}$
${\mathrm{1/2}}^{-}$	0.0	7.25	13.61	79.14	0.0	0.0
${\mathrm{7/2}}^{+}$	41.16	0.0	0.0	0.0	97.29	2.71
${\mathrm{9/2}}^{+}$	166.56	0.0	0.0	0.0	95.18	4.82

^† from [9].

). This could be easily explained given that the $\pi g_{9/2}$ is the only positive-parity orbital below Z = 50 in this mass region. It is particularly important to note that the energy gap between the ${\mathrm{9/2}}_1^{+}$ and ${\mathrm{7/2}}_1^{+}$ states is strongly correlated with the excitation of the $2_1^{+}$ states in the even cores [36].

A more complete and comprehensive understanding of the structure of the low-lying states in the Ag isotopes can be obtained by exploring their electromagnetic properties. As shown in Table 6, experimental data for transition probabilities in the odd-A Ag are sparse and rather incomplete. Still, the measurements in ^105–109Ag show consistent values of the ${\mathrm{9/2}}_1^{+}\to {\mathrm{7/2}}_1^{+}$ transition probabilities in all three nuclei. The hindered $M1$ components of this transition are well described by the IBFM-1 calculations. The theoretical approach also reproduces the enhanced $E2$ components of the ${\mathrm{9/2}}^{+}\to {\mathrm{7/2}}^{+}$ transitions, suggesting the involvement of a significant degree of collectivity.

The systematic study of the $\delta (E2/M1)$ mixing ratio evolution with the mass number provides additional insight to the structure of the ${\mathrm{9/2}}_1^{+}$ and ${\mathrm{7/2}}_1^{+}$ states. The calculation results agree with the experimental observations (see Figure 5), and a generally increasing trend toward heavier nuclei is observed.

Recent experiments [27] provide a rather complete set of information for magnetic dipole and electric quadrupole moments of the first two excited states in the odd-A Ag isotopes. The comparison presented in Table 7 confirms that the IBFM-1 calculations capture the trends of these observables along the isotopic chain. Both the measured data and the calculations show rather consistent values as the neutron number increases. It is worth noting that the experimental measurements of $\mu (1/2^{-})$ indicate different signs for certain isotopes. The IBFM-1 calculations clearly relate the ${\mathrm{1/2}}_1^{-}$ states to the structure dominated by the $\pi p_{1/2}$ orbital but the experimental data indicate that different configurations might be realized as well. In principle, three negative-parity orbitals are included in the present IBFM-1 approach ($\pi p_{3/2}$, $\pi f_{5/2}$, and $\pi p_{1/2}$). Their modeled behavior, along with the selected boson–fermion interaction parameters, is deemed sufficient to describe the low-lying negative-parity excitations in ${}^{103–115}\mathrm{Ag}$.

The present work does not extend the study towards very neutron-rich Ag isotopes but it is worth mentioning that large-scale shell model calculations were performed in the past for heavier Ag isotopes [69]. While the theoretical spectra obtained using the jj45pna interaction reproduce well the level schemes of the studied nuclei, certain electromagnetic features are particularly interesting. For example, the shell model calculations predict lower B($E2$; ${\mathrm{9/2}}^{+}$→ ${\mathrm{7/2}}^{+}$) values compared to a single-j approach [69].

Besides the vast amount of experimental data for nuclear moments in odd-A Ag nuclei, Ref. [27] reports results from calculations performed with density functional theory (DFT) for multiple nuclei along the Ag isotopic chain. The calculations show good agreement with the measured electromagnetic moments but also a larger energy gap between the lowest-lying I = 1/2 and I = 7/2 states compared to the experimental data and various IBFM calculations.

The present systematic IBFM-1 approach suggests that the odd-A ${}^{103–115}\mathrm{Ag}$ isotopes generally exhibit the features of a proton hole coupled to even–even cores with properties similar to the U(5) characteristics. Still, the cores (described as even-A ^104–116Cd here) show some deviations from the exact U(5) dynamical symmetry. This is observed both experimentally and theoretically, particularly through the electromagnetic properties of the nuclei. The structure of the even–even cores cannot be also undoubtedly connected to $\gamma$-soft, O(6)-like nuclei. The vanishing electric quadrupole moments are a particularly important feature of $\gamma$-soft nuclei which are strongly related to the O(6) dynamical symmetry. The present IBM-1 calculations outline a fairly simplified approach of the properties of the Cd even-A cores, mainly focusing on the lowest-lying states in the ground state bands. These states play a crucial role in the coupling to the proton hole in silver nuclei and the configuration of the lowest-lying odd-A Ag excitations. It is evident from the experimental measurements and theoretical calculations that the structure of the odd-A Ag isotopes cannot be fully enclosed within simplified dynamical symmetry interpretations. Nevertheless, the presented IBFM-1 approach seems to describe well certain features of these nuclei, such as the J-1 anomaly characteristic for odd-A Ag isotopes.

5.2. Astrophysical Relevance

As discussed earlier, the silver nuclei in this mass region are part of complex nucleosynthesis processes on astrophysical scales. Along with the Pd isotopes, they are key components of the weak r-process studies [6,7]. Recent works [70] indicate major r-process contribution to the stable ${}^{107,109}\mathrm{Ag}$ component in the solar system isotopic composition.

Furthermore, some of the heavier unstable Ag isotopes (^113,115Ag) are tightly related to the rather complex nucleosynthesis in the Cd-In-Sn region. That involves multiply-branched reaction flows which result from numerous long-lived $\beta$-isomers [71]. In the framework of these processes, the ${\mathrm{7/2}}^{+}$ states in these nuclei complicate the post-r-process flow. In addition to the isomeric $\gamma$-transitions, the ${\mathrm{7/2}}^{+}$ levels in ^113,115Ag have a significant $\beta$-decay branching. Due to the large spin difference compared to the ${\mathrm{1/2}}^{-}$ ground states, they have significantly different $\beta$-decay patterns and population of states in the daughter nuclei, ^113,115Cd. Along with their half-lives being considerably different from the ones of the ${\mathrm{1/2}}^{-}$ ground states, the presence of the ${\mathrm{7/2}}^{+}$ levels in ^113,115Cd changes the dynamics of the post-r-process flow and the respective abundance contributions. It should be also noted that the thermalization in stellar conditions additionally affects the Ag branching, and the $\beta$-decay via the short-lived isomer is enhanced. This effectively reduces the production of some isotopes along the flow chain (e.g., ¹¹³In) [71].

Although the presented IBFM-1 calculations do not give quantitative measures of the ${\mathrm{7/2}}^{+}$ isomer $\beta$-decay properties, they provide valuable structural information for the ${\mathrm{7/2}}^{+}$ states which is relevant (i.e., single-particle orbital contributions to the wave functions). While the astrophysical abundances of nuclei in this odd-A Ag mass region are a result of the complex interplay of multiple processes, it should be underlined that the presence of the low-lying isomeric states definitely plays an important role, and this is an active field of research. For example, the significance of the isomeric state in ¹¹⁹Ag was recently outlined in the context of the astrophysical r-process [72], showing its properties of an “astromer” (isomer retaining its metastable character in pertinent astrophysical environments [73]).

As new studies explore in detail nuclei towards the neutron-rich part of the Ag isotopic chain, it becomes evident that the silver isotopes are heavily involved in multiple astrophysical processes. Therefore, new experimental data are essential to build a coherent understanding of their role at astrophysical scale.

6. Conclusions

The odd-A Ag isotopes between the line of $\beta$-stability and the mid-shell neutron-rich region exhibit a variety of properties which challenge our understanding of nuclear structure in this mass region. Furthermore, the characteristics of some of their low-lying excited states can be crucial for processes at astrophysics sites, given that Ag isotopes are part of several nucleosynthesis mechanisms. The present work continues previous efforts to describe the structure of odd-A Ag nuclei within IBFM-1, extending the study toward lighter isotopes.

The results show that the known level schemes and electromagnetic properties of the Ag nuclei can be reproduced well by using relatively constant properties of the single-particle orbitals and boson–fermion interaction parameters along the isotopic chain. The calculations suggest that ^103–115Ag can be represented by a proton hole coupled to near-U(5) even–even cores. This approach also reproduces the well-known J-1 anomaly in the odd-A Ag nuclei, manifesting as a change of the order of the first ${\mathrm{7/2}}^{+}$ and ${\mathrm{9/2}}^{+}$ states. The low-lying positive-parity excitations are mainly associated with the role of the $\pi g_{9/2}$ orbital. Although a lot of new experimental data have become recently available in this mass region [69,72,74,75], information about many properties of Ag nuclei is still scarce. This is particularly valid for the lifetimes and electromagnetic properties of excited states. Therefore, acquiring such experimental data is essential to understand the structure of silver isotopes.