Singular observation of shape and spin hindrance in metastable nuclear decay of $^{197}$Au

Nuclear excited states are ordinarily short-lived, with lifetimes significantly shorter than a nanosecond. However, a small fraction have a metastable character and are referred to as nuclear isomers. Their transition probabilities are much lower, and are associated with a hindrance arising from substantial changes in either shape, spin or spin projection $K$. Such states are referred to as shape, spin or $K$ isomers, respectively [1]. The lack of overlap of the isomeric wavefunction with those of levels at lower energy is central to the realization of such isomers, thereby elucidating the underlying nature of the quantum many-body, strongly interacting nuclear system. Additionally, direct insight can be gained into the mechanisms that hinder electromagnetic decay. Shape isomers result from the rearrangement of nucleons in the transition from initial to final states, which are characterized by associated deformations quite different from each other in either magnitude or sign [2, 3, 4]. Since angular momentum is a conserved quantum number, an excited level undergoing electromagnetic decay through a transition with a high multipole order corresponds to a spin isomer [5, 6, 7]. While a large number of shape and spin isomers have been established across light to heavy nuclei, each of these are specific to either a deformed or a spherical nucleus, and as yet, no metastable states with decay hindered on account of both shape and spin differences had been established.

Shape coexistence has been extensively documented in many nuclei, with clear experimental signatures in some cases and circumstantial evidence in most others [8, 9]. Theoretical predictions support a much larger body of nuclei which are expected to demonstrate shape coexistence [4, 8, 9]. Shape isomerism, though, is realized in a small fraction of these nuclei, when the associated minima in the nuclear potential have comparable energies [3, 10, 11, 12].

The presence of high-$j$, unique-parity subshells, which are interspersed with low-$j$ orbitals below every major closed shell with nucleon numbers 50, 82 and 126, creates conditions favorable for spin isomerism with $M3$, $E4$, $M4$ or even $E5$ decays having been identified [5, 6, 7]. Generally, spin isomers are associated with longer level lifetimes than shape metastable states. In transitional nuclei having several valence nucleons with respect to closed shells, it is possible to realize conditions suitable for both shape and spin hindrance, due to the presence of high-$j$ orbitals and moderately-deformed shapes. Such mixed hindrance mechanisms would provide a sensitive probe of the interplay between collective and single-particle degrees of freedom. The stable nucleus, ${}^{197}\mathrm{Au}$ ($Z = 79$, $N = 118$), belongs to this category, and, therefore, was the subject of this study.

Two experiments were performed to populate long-lived states in ${}^{197}\mathrm{Au}$. The target was, in both instances, a thick, 50 mg/cm$^{2}$ Au foil, while ${}^{209}\mathrm{Bi}$ and ${}^{207}\mathrm{Pb}$ beams with 1450- and 1430-MeV respective energies were used; the beam energies were about 10% above the Coulomb barrier, which optimized the population of high-spin levels [13, 14]. Pulsed beams were delivered by the ATLAS accelerator at Argonne National Laboratory (ANL), and $\gamma$-ray coincidence data were recorded using the Gammasphere array which consisted of 100 Compton-suppressed high-purity germanium (HPGe) detectors at the time of the experiments [15]. During the experiments, Pb absorbers were placed in front of the HPGe detectors to suppress the expected copious amount of target x-rays. To search for possible long-lived states, the pulsed beam was incident on the target, and then deflected away for three specific intervals: (a) 200 $\mu$s on and 800 $\mu$s off, (b) 1 ms on and 3 ms off, and (c) 2 s on and 8 s off, with the data being collected in the beam-off periods. Additional data were collected in a $\approx1$ $\mu$s duration, wherein successive beam pulses were separated by 825 ns. In the $\approx1$ $\mu$s interval, three- and higher-fold coincidence data were recorded, while in the beam-off periods of 800 $\mu$s and higher, the minimum fold was two. Some of the histograms used for the data analysis were: (a) two- and three-dimensional symmetric, $\gamma$-energy histograms for establishing the excited level structures, and (b) time-gated, triple-$\gamma$ energy coincidence histograms to determine long half-lives. Additional information can be found in earlier publications of this group and elsewhere [16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

A new metastable state in ${}^{197}\mathrm{Au}$, with $T_{1/2} > 10$ s, has been established at $E_x = 2102$ keV, as displayed in Figure 1. This state deexcites through a 266.6-keV transition to the 1836-keV level with $I^\pi= (21/2^+)$. Another unobserved decay branch of 5.5 keV to the $(25/2^{+})$ state at $E_x = 2097$ keV is evident through the observation of the 261.1-keV $\gamma$ ray in the coincidence spectra of Figure 2. The half-life of the $E_x = 1836$-keV state has been newly measured from this work to be $T_{1/2} = 8.8(6)$ ns by using the centroid-shift method [26], as illustrated in Figure 3. The level at $E_x = 2532$ keV with $T_{1/2} = 150(5)$ ns reported earlier [27] is confirmed from the present data. It was not possible to unambiguously ascertain the multipolarity of the 435.6-keV $\gamma$ ray deexciting this state. It should be noted that transitions with energies 261.1, 357.7, 429.2, 639.7 and 1069.1 keV had been established in earlier work [27]. The 266.6-keV $\gamma$ ray was found to be quite weak in the 1-$\mu$s data, however, in the longer beam-off periods, it was clearly visible in the data from both experiments. The spectra in Figure 2 correspond to three-fold coincidence events between $\gamma$ rays in the 8-s pulsing period. The summed integral three-fold coincidence counts of all $\gamma$ rays in the deexcitation path of the $E_x = 2102$-keV level are displayed in Figure 3(a); these have been obtained from spectra with successively increasing time bins with 0.5-s time intervals. The regular increase in counts up to time $t = 8$ s indicates a half-life longer than the 8-s beam-off period, with $T_{1/2} > 10$ s implied from the trend in Figure 3(a). An inspection of the data collected when the beam was switched off suggests a half-life longer than several minutes; however, a precise value could not be determined, therefore, a $T_{1/2} > 10$ s lower limit has been adopted.

The 261.1-keV transition was firmly established to have $E2$ multipolarity, from the coincidence spectra in the 1-$\mu$s data, based on intensity balance considerations between the transitions feeding and deexciting the 1836- and 2097-keV levels. Therefore, with $I^\pi= (21/2^{+})$ for the $E_x = 1836$-keV level, the spin-parity of the 2097-keV state is assigned as $I^\pi= (25/2^{+})$. It may be noted that the 2097-keV level, which is deexcited by the 261.1-keV $\gamma$ ray, is fed by a 435.6-keV transition from the decay of a 150(5) ns isomeric state at $E_x = 2532$ keV, as shown in Figure 1 [27]. Using similar considerations in the 8-s pulsing data, the $\gamma$-ray intensity of the 266.6-keV transition is consistent with an $E4$ character; other multipolarities are excluded, as illustrated in Table I. The total intensity ($I_t$) has been calculated from the measured $\gamma$-ray intensity ($I_\gamma$) using total internal conversion coefficients ($\alpha_t$) from the BrIcc tabulation [28], and the relation $I_t = I_\gamma(1 + \alpha_t)$. For the 266.6-keV transition, $I_\gamma= 16(1)$ gives $I_t = 102(6)$ for an $E4$ assignment. This agrees closely with $I_t$ of the subsequent 357.7- and 639.7-keV transitions, which are 104(4) and 100(4), respectively. The corresponding $I_t$ for hypothetical $M2$, $E3$, and $M3$ assignments are 46(3), 30(2), and 124(8), respectively, inconsistent with the observed intensity balance. Multipole orders higher than four, for which the transitions would have much larger $\alpha_t$ values, are categorically ruled out. Thus, the $E4$ assignment for the 266.6-keV line is highly favored by intensity balance considerations. The $M3$ assignment is much less likely; further, it would imply $M1$ character for the unobserved 5.5-keV transition, a conclusion difficult to reconcile with known physical considerations, and the non-observation of such a highly-hindered $M1$ transition across the nuclear chart. Since the 261.1-keV $\gamma$ ray is visible in the 8-s pulsing data, the presence of an unobserved 5.5-keV $E2$ transition is inferred. Thus, $I^{\pi} = (29/2^{+})$ is proposed for the metastable state at $E_x = 2102$ keV, which is deexcited by both a 5.5-keV $E2$ transition and a 266.6-keV $E4$ one. It is pertinent to note that there is only one other instance of an $E4$ isomeric transition in this mass region, viz., the 129.1-keV $\gamma$ ray deexciting the $I^{\pi} = 9^{-}$ state with $T_{1/2} = 3.54(2)$ h in $^{202}\mathrm{Pb}$ [29].

The lower limit on the half-life of the $I^{\pi} = (29/2^{+})$ state allows for the determination of upper bounds for the transition probabilities associated with the 5.5-keV $E2$ and 266.6-keV $E4$ $\gamma$ rays. As displayed in Table II, the reduced $E2$ transition probability, $B(E2) < 6.2 \times10^{-9}$ Weisskopf units (W.u.) is indicative of a highly-suppressed $E2$ transition. The extent of this suppression is comparable with that observed for classic shape isomers in actinide nuclei, e.g., $10^{-7}$ W.u. for the $E2$ decay of the 2558-keV, $0^{+}$ state in $^{238}\mathrm{U}$ [30]. In contrast, $B(E4) < 10^{7}$ W.u. is inferred. A future measurement of $T_{1/2}(29/2^{+})$ would affirm whether the $B(E4)$ probability is enhanced, suppressed or close to the single-particle estimate.

The observation of a highly-suppressed $E2$ transition and an $E4$ $\gamma$ ray as decay branches from the same metastable state is without parallel in the nuclear chart. The only two other instances of $E2$ and $E4$ decays from the same isomer have been found for the $I^{\pi} = 10^{+}$ state in $^{54}\mathrm{Fe}$, with $T_{1/2} = 364(7)$ ns [31], and the $I^{\pi} = 19^{+}$ state in $^{96}\mathrm{Ag}$, with $T_{1/2} = 136(20)$ ns [32]. However, unlike in the $^{197}\mathrm{Au}$ case, $B(E2) = 1.69(4)$ W.u. and $B(E4) = 0.79(8)$ W.u. probabilities were inferred in $^{54}\mathrm{Fe}$, and $B(E2) = 4.7(10)$ W.u. and $B(E4) = 0.9(6)$ W.u. were deduced in $^{96}\mathrm{Ag}$ [31, 32], indicating the single-particle character of these four transitions. No $B(E2)$ suppression is evident in either of the cases. In $^{54}\mathrm{Fe}$ and $^{96}\mathrm{Ag}$, the $E2$ decay branches have energies of 146 and 98 keV, whereas the $E4$ decays proceed through 3578- and 4265-keV transitions, respectively. Thus, the high energy of the $E4$ transitions enables the competition with the $E2$ $\gamma$ rays, unlike the situation in $^{197}\mathrm{Au}$.

The origin of the unusual decay pattern in $^{197}\mathrm{Au}$ may be understood in terms of the simultaneous presence of shape- and spin-related hindrance mechanisms. Strongly hindered $E2$ transitions are commonly associated with changes in intrinsic deformation between the initial and final states, whereas $E4$ transitions with near-single-particle strengths are observed in near-spherical nuclei below major shell closures. These two phenomena are generally encountered in structurally distinct regions of the nuclear chart. In ${}^{197}\mathrm{Au}$, however, the observation of comparable $E2$ and $E4$ decay branches from the same $I^{\pi}=29/2^{+}$ metastable state, together with the pronounced suppression of the $E2$ strength, indicates the coexistence of shape and spin hindrance within a single decay sequence. To investigate these complementary structural aspects, calculations were performed using both the Ultimate Cranker code based on the modified oscillator (Nilsson) potential, which describes deformed quasiparticle configurations [33], and the KSHELL framework with the KHHE interaction, which provides a microscopic description of near-spherical states [34, 35, 36].

The nucleus ${}^{197}\mathrm{Au}$ had been reported to be characterized by a moderate oblate deformation in its ground state ($\epsilon_{2}=0.12$) in a recent global tabulation [37]. This is supported in the present work by the UC calculations, which also predict deformation parameters $\epsilon_{2}=0.13$ and a triaxiality parameter, $\gamma=-73^{\circ}$ for the first $I^{\pi}=25/2^{+}$ level; i.e., suggestive of moderate, near-oblate deformation, as depicted in Figure 4. The $E_{x}=2097$-keV level is, therefore, associated with this shape. The UC calculations indicate similar oblate deformations for the bands built on the $11/2^{-}$ and $21/2^{+}$ isomeric levels displayed in Figure 1. The shell-model description is in qualitative agreement, with respective $B(E2:15/2^{-}\to11/2^{-})=25.4$ W.u., and $B(E2:25/2^{+}\to21/2^{+})=21.1$ W.u., values indicative of the collective nature of these transitions. In contrast, a $B(E2:29/2^{+}\to25/2^{+})=0.6$ W.u. value is calculated, demonstrating an absence of collective enhancement and, thus, the single-particle character of the $I^{\pi}=29/2^{+}$ state. The shell-model predictions of the excitation energies of the $21/2^{+}$, $25/2^{+}$ and $29/2^{+}$ states are within 100-200 keV of the experimental values (Table II). The $25/2^{+}$ energy from experiment and UC calculations exhibit similar agreement. Furthermore, empirical calculations have been performed following a prescription given in earlier work of this group [18, 20], where experimental 1-quasiparticle energies, pair-gap energies from a 5-point mass formula [38], and residual interactions derived from isomeric levels with relatively pure configurations in this mass region have been used. These calculations indicate that the $I^{\pi}=29/2^{+}$ level associated with a $\pi h_{11/2}^{-1}\otimes\nu(i_{13/2}^{-1},f_{5/2})$ configuration should lie at a rather low energy, $E_{x}=1911$ keV, within 200 keV of the experimental value. Although the shell-model calculations reproduce the dominant configurations of the states involved, they do not explicitly incorporate deformation changes that may contribute to the observed large suppression of the $E2$ transition strength in the decay of the $I^{\pi} = 29/2^{+}$ state. Any discrepancy between the experimental and calculated $B(E2)$ values may therefore provide insight into shape-related effects beyond the present model space. While a direct comparison of the experimental and calculated transition probabilities in the decay of the $21/2^{+}$ level is possible, and reasonable agreement is seen, the limiting, rather than a precise value of the $29/2^{+}$ half-life precludes a similar juxtaposition.

Though long-lived states have been established in odd-$A$ isotopes of Ir ($Z = 77$), Au ($Z = 79$) and Tl ($Z = 81$), their decays are distinctly different from those observed in $^{197}\mathrm{Au}$. In $^{191,193}\mathrm{Ir}$, $I^{\pi} = 31/2^{+}$ levels associated with triaxial deformation and half-lives in the $\mu$s-s range, and deexciting transitions of $M2$, $E3$ or $M3$ character, have been established [39]. In lighter Au isotopes, viz., $^{189,191,193,195}\mathrm{Au}$, $31/2^{+}$ and $31/2^{-}$ levels with half-lives in the ns-$\mu$s range have been found. These decay by $E2$ transitions in almost all instances [40, 41]. In $^{203,205}\mathrm{Tl}$, $\mu$s isomers with $E2$ and $E3$ decays and predominantly single-particle character have been identified [20, 42]. Finally, the neutron-deficient $^{188,190}\mathrm{Pb}$ isotopes exhibit isomers with spherical and oblate configurations and ns-$\mu$s half-lives. These levels decay via $E1$, $E2$ or enhanced $E3$ ($\pi i_{13/2} \to\pi f_{7/2}$) transitions and considerable state-mixing is evident in these cases [43, 44].

The coexistence of pronounced shape- and spin-related hindrance in $^{197}\mathrm{Au}$ is highly unusual and has broader implications for the understanding of electromagnetic decay in quantum systems, where strongly suppressed transition probabilities are typically associated with a single dominant mechanism. In contrast, the decay of the $^{197}\mathrm{Au}$ isomer appears to involve the simultaneous action of two distinct hindrance mechanisms. The simultaneous presence of shape- and spin-related hindrance presents a significant challenge for current theoretical approaches, which typically describe either collective deformation effects or detailed single-particle configurations, but not both within a unified framework. Further, the observation is particularly significant because $^{197}\mathrm{Au}$ lies in the vicinity of the doubly magic nucleus $^{208}\mathrm{Pb}$, a benchmark region for nuclear-structure studies in which shell closures and single-particle configurations are generally well established. The present results demonstrate that structural complexity persists even near this closed-shell region and that metastable decay cannot always be understood within a purely single-particle framework. The long-lived isomer in $^{197}\mathrm{Au}$ therefore provides a sensitive probe of the interplay between shell structure, configuration mixing, and collective degrees of freedom. These findings provide new insight into the persistence of shape-related effects near closed shells and establish an important benchmark for modern nuclear-structure models and beyond-mean-field approaches.

In summary, the $I^{\pi}=29/2^{+}$ state in $^{197}\mathrm{Au}$ with $T_{1/2}>10\ \mathrm{s}$ exhibits an unusual coexistence of shape- and spin-related hindrance in the decay of a metastable nuclear state. The observation of a strongly suppressed $E2$ transition competing with a comparable $E4$ branch indicates the simultaneous action of collective and single-particle mechanisms within the same decay sequence. These results suggest that metastable decay in the vicinity of the doubly magic nucleus $^{208}\mathrm{Pb}$ may involve a more intricate interplay between deformation, shell structure, and configuration mixing than previously recognized, with a unified description of all observed properties being challenging. Additional measurements, including long-timescale decay studies and mass spectrometry, are planned at ANL to further constrain the structure of the $29/2^{+}$ isomer in $^{197}\mathrm{Au}$ and establish an exacting yardstick for nuclear theory.

Acknowledgements

The authors would like to thank J. Meng, P. Guo and F. Y. He for useful discussions. S.K.T. acknowledges support from the Shiv Nadar Foundation (SNF). This work is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under award numbers DE-FG02-94ER40848 (UML), DE-FG02-97ER41041 (UNC) and DE-FG02-97ER41033 (TUNL), and contract number DE-AC02-06CH11357 (ANL). The research described here utilized resources of the ATLAS facility at ANL, which is a DOE Office of Science user facility.

References