A new spectroscopic probe to search for magic numbers at high-excitation energies

TL;DR

This paper introduces a spectroscopic method to detect magic numbers at high excitation energies by analyzing nuclear polarizability drops, supporting shell-model calculations and challenging the universality of the Brink-Axel hypothesis.

Contribution

It proposes a novel spectroscopic probe based on polarizability drops to identify shell effects at high energies, extending the understanding of magic numbers beyond ground states.

Findings

Polarizability drops indicate shell effects at high excitation energies.

Supports shell-model calculations in the quasi-continuum region.

Challenges the universality of the Brink-Axel hypothesis.

Abstract

Empirical drops in ground-state nuclear polarizabilities indicate deviations from the effect of giant dipole resonances and may reveal the presence of shell effects in semi-magic nuclei with neutron magic numbers $N=50$, 82 and 126. Similar drops of polarizability in the quasi-continuum of nuclei with, or close to, magic numbers $N=28$, 50 and 82, could reflect the continuing influence of shell closures up to the nucleon separation energy. These findings open a new avenue to investigating magic numbers at high-excitation energies and strongly support recent large-scale shell-model calculations in the quasi-continuum region, which describe the origin of the low-energy enhancement of the photon strength function as induced paramagnetism. The nuclear-structure dependence of the photon-strength function asserts the generalized Brink-Axel hypothesis as more universal than originally expected.

Tables

Table 1. Table 1: Contributions of GDR and LEE cross-sections to σ − 2 subscript 𝜎 2 \sigma_{{}_{-2}} and κ 𝜅 \kappa values. Data have been extracted from EXFOR   [ 55 ] , ENDF   [ 56 ] and the Oslo compilation  [ 57 ] . An asterisk indicates that the σ − 2 subscript 𝜎 2 \sigma_{{}_{-2}} value includes σ ​ ( γ , p ) 𝜎 𝛾 𝑝 \sigma(\gamma,p) contributions.

Nucleus	Eγ(max)(GDR)	Eγ(max)(LEE)	${\sigma }_{{}_{-2}}$(total)	${\sigma }_{{}_{-2}}$	$\kappa$	[Refs.]
	(MeV)	(MeV)	($\mu$b/MeV)	(LEE)	(LEE)
${}_{21}{}^{}{}_{}{}^{45}$Sc∗	28.1	3.2	1840(130)	9.7$\%$	1.35(9)	[58, 59, 60]
${}_{23}{}^{}{}_{}{}^{50}$V	27.8	3.1	1458(100)	2.9$\%$	0.89(7)	[61, 52]
${}_{23}{}^{}{}_{}{}^{51}$V	27.8	3.1	1472(100)	3.3$\%$	0.87(7)	[61, 52]
${}_{26}{}^{}{}_{}{}^{56}$Fe∗	40.0	3.8	2231(155)	6.3$\%$	1.13(8)	[62, 26]
${}_{32}{}^{}{}_{}{}^{76}$Ge	26.5	2.3	3189(225)	2.7$\%$	0.97(7)	[63, 64]
${}_{40}{}^{}{}_{}{}^{92}$Zr	27.8	2.2	3131(220)	1.1$\%$	0.70(5)	[65, 66, 67]
${}_{42}{}^{}{}_{}{}^{95}$Mo	27.8	2.5	4743(330)	1.7$\%$	1.00(7)	[68, 69]
${}_{57}{}^{}{}_{}{}^{138}$La	24.3	1.9	7983(560)	0.4$\%$	0.90(7)	[70, 28]
${}_{57}{}^{}{}_{}{}^{139}$La	24.3	2.5	8015(560)	0.7$\%$	0.90(6)	[70, 28]
${}_{62}{}^{}{}_{}{}^{153}$Sm	20.0	1.6	9999(700)	2.7$\%$	0.95(7)	[71, 29]

Equations

$$\alpha_{{}_{E1}}=\frac{e^{2}R^{2}A}{40a_{sym}}=2.25\times 10^{-3}A^{5/3}~{}\mbox{fm${}^{3}$},$$

$${\alpha }_{{}_{E1}}$$

$${\sigma }_{{}_{-2}}:={\int }_0^{E_{{\gamma }_{max}}}\frac{{\sigma }_{total}(E_{\gamma })}{E_{\gamma }^{{}^2}}d{E}_{\gamma },$$

$${\sigma }_{{}_{-2}}=\frac{2{\pi }^2}{\hslash c}({\alpha }_{{}_{E1}}+{\chi }_{{}_{M1}}),$$

$${\chi }_{{}_{M1}}$$

$$f(E_{\gamma})=\frac{1}{g_{{}_{J}}\pi^{2}(\hbar c)^{2}}~{}\frac{\sigma_{{}_{total}}(E_{\gamma})}{E_{\gamma}}~{}\mbox{MeV${}^{-3}$},$$

$$\sigma_{{}_{-2}}=2.4(1)\kappa~{}A^{5/3}\mbox{$\mu$b/MeV},$$

Full text

url]https://nuclear.uwc.ac.za

The ability for a nucleus to be polarized is a fortiori driven by the dynamics of the isovector giant dipole resonance (GDR). That is, the inter-penetrating motion of proton and neutron fluids out of phase [1], which results from the symmetry energy in the Bethe-Weizsäcker semi-empirical mass formula [2, 3], $a_{sym}(\rho_{{}_{N}}-\rho_{{}_{Z}})^{2}/\rho_{{}_{A}}$, acting as a restoring force [1, 4]. Respectively, $\rho_{{}_{N}}$ and $\rho_{{}_{Z}}$ are the mass densities of the neutron and proton fluids, and $\rho_{{}_{A}}$ the sum of the separate densities. The GDR represents most of the absorption and emission of $\gamma$-ray photons by a nucleus and was the first quantum collective excitation ever discovered in mesoscopic systems [5]. The idea of giant resonances was soon borrowed by atomic, molecular and solid-state physics (see e.g. [6] and references therein); the GDR motion is akin to the plasmons in graphene, which enables strong confinement of electromagnetic energy at subwavelength scales [7].

Using the collective variable $\rho_{{}_{Z}}$ as the potential energy of the liquid drop, Migdal calculated the electric dipole polarizability, $\alpha_{{}_{E1}}$, for the ground state of nuclei to be directly proportional to the size of the nucleus [1],

[TABLE]

where $a_{sym}=23$ MeV is the symmetry energy parameter and $R=1.2A^{1/3}$ fm the radius of the nucleus with $A=N+Z$. Alternatively, $\alpha_{{}_{E1}}$ is well described by microscopic mean-field approaches using the random-phase approximation (RPA) with various effective interactions [8, 9, 10], and can be determined empirically with the use of second-order perturbation theory,

[TABLE]

with $|i\rangle$ being the vector of the ground state connecting high-lying $|n\rangle$ states in the GDR region via $E1$ virtual excitations, and $\sigma_{{}_{-2}}$ the $(-2)$ moment of the total photo-absorption cross section [11, 12] defined as,

[TABLE]

where the total photonuclear-absorption cross section, $\sigma_{total}(E_{\gamma})$, generally includes all $\sigma(\gamma,n)$ and $\sigma(\gamma,p)$ contributions [13].

Naturally, total $\sigma_{{}_{-2}}$ values should include both electric and magnetic polarizability contributions,

[TABLE]

where $\chi_{{}_{M1}}$ is the static magnetic dipole polarizability and considers the sum of the paramagnetic $\chi^{para}_{{}_{M1}}$ and diamagnetic $\chi^{dia}_{{}_{M1}}$ susceptibilities of nuclei [14],

[TABLE]

According to the independent-particle shell model (IPM), diamagnetism is dominant for nuclei with $A>60$ [15], but has a negligible effect in $\sigma_{{}_{-2}}$ values. Paramagnetism dominates in light nuclei with the rise of permanent magnetic dipole moments and can, in contrast, contribute substantially to $\sigma_{{}_{-2}}$ values for nuclei with $A<20$ [15].

Because of the 1/E${}^{2}_{\gamma}$ energy weighting in Eq. 3, $\sigma_{{}_{-2}}$ values are extremely sensitive measures – unlike $\sigma_{total}$ – of low-energy long-range correlations in the nuclear wave functions, which are common feature for all nucleon-nucleon potentials, and fundamental for shell-model (SM) calculations of heavy nuclei [16] using low-momentum interactions [17]. Intermediate and short-range correlations to the nuclear wave functions from above the GDR region (e.g., nucleon resonances at $E_{{}_{\gamma}}\gtrsim 140$ MeV) have a negligible effect on $\sigma_{{}_{-2}}$ values [21, 18, 19, 20].

Below the neutron separation threshold, the pygmy dipole resonance (PDR) in neutron-rich nuclei [22] – the PDR is an electric dipole resonance arising from the oscillation of a symmetric proton-neutron core against the neutron skin – may add a $\simeq 5\%$ contribution to $\sigma_{{}_{-2}}$ values [23]. To a lesser extent, soft resonances such as the $M1$ scissors mode and spin-flip may also contribute. A potentially larger effect to $\sigma_{{}_{-2}}$ values may arise from the low-energy enhancement (LEE) of the radiative or photon strength function $f(E_{\gamma})$ – indicating the ability of nuclei to emit and absorb photons with energy $E_{\gamma}$ – observed at $E_{\gamma}\lessapprox 4$ MeV [24, 25, 26]. This Letter shows how the LEE and GDR cross-section contributions affect $\sigma_{{}_{-2}}$ values and may provide evidence for the continuing influence of shell effects at high-excitation energies. Relevant consequences arise from these findings; for instance, the possibility to identify new magic numbers.

The physical origin of the LEE remains ambiguous and its observation seems to be generally associated with weakly deformed nuclei. It has been observed in nearly-spherical nuclei in the $A\approx 50$ and 90 mass regions starting at $E_{\gamma}\approx 3-4$ MeV. For heavy nuclei, it is only found in 105Cd [27], 138,139La [28] and 151,153Sm [29], where the LEE starts at a lower $E_{\gamma}\approx 2$ MeV. These findings assume the validity of the Brink-Axel hypothesis – stating that $f(E_{\gamma})$ is independent of the particular nuclear structure and only depends on Eγ [30, 31] – which has been confirmed experimentally [33, 32]. The reason for not being observed in other heavy nuclei – studied with the same experimental method – could relate to the unprecedented sensitivity achieved by Simon and co-workers in 151,153Sm using high-purity germanium (HPGe) detectors in connection with bismuth germanate (BGO) shields [29]. Another relevant finding is that the LEE presents a dominant dipole radiation [26, 32], but whether its nature is either electric or magnetic remains unresolved [34]. The recent polarization asymmetry measurements of $\gamma$ rays in 56Fe using GRETINA tracking detectors yields inconclusive results, although rather suggests an admixture of electric and magnetic dipole radiation, with a small bias towards a magnetic character at $E_{\gamma}=1.5-2.0$ MeV [34].

Two competing scenarios are proposed theoretically to explain the LEE anomaly. On one hand, Litvinova and Belov propose that the LEE in $f(E_{\gamma})$ occurs because of $E1$ excitations from the hot-quasicontinuum to the continuum region [35]. On the other hand, SM calculations predict that the LEE has a predominant magnetic-dipole $M1$ character. In particular, Schwengner, Frauendorf and Larsen suggest that the LEE arises from active high-$j$ proton and neutron orbits near the Fermi surface with magnetic moments adding up coherently [36]. This is a similar mechanism to the magnetic rotation [37] or two-phonon mixed-symmetry states found in nearly-spherical nuclei at about 3 MeV [38, 39]. In a complementary picture, Brown and Larsen suggest that the LEE arises because of the large $M1$ diagonal matrix elements of high-$\ell$ orbitals [40]. Additionally, Sieja computed both $E1$ and $M1$ strengths in 44Sc on equal footing from large-scale SM calculations and also supported the $M1$ character of the LEE in the $A\sim 50$ region against $E1$ contributions [41, 42]. Recently, large-scale SM calculations of neutron-rich 70Ni [43] and many other nuclei [44], using various effective interactions, also support the $M1$ character for the LEE.

Recently, In principle, the validation of these SM predictions in the quasi-continuum region may be arguable as, for instance, they are structure dependent; hence, posing a fundamental question about the validity of the Brink-Axel hypothesis [30, 31].

A priori, the comparison of the LEE and the GDR built on ground states is somewhat misleading as the former corresponds to $\gamma$-ray transitions between excited states in the quasi-continuum, whereas the latter involves transitions to the ground state. Nonetheless, the study of (p,$\gamma$) and (n,$\gamma$) reactions for light nuclei and fusion-evaporation reactions for heavy nuclei have shown that GDRs can also be built on excited states (GDRexc) [45, 46, 47]. In fact, GDRsexc present – at least for moderate average temperature $T$ and spin $J$ – similar centroid energies, $E^{exc}_{{}_{GDR}}$, and resonance strengths, $S^{exc}_{{}_{GDR}}$, relative to the Thomas-Reiche-Kuhn (TRK) $E1$ sum rule [48], as those found for the ground-state counterparts (GDRg.s.) [45, 46]. These similar features suggest a common physical origin for all GDRs in concordance with the Brink-Axel hypothesis, which also indicates that a GDR can be built on every state in a nucleus [30, 31]. Moreover, the sum rules in Eqs. 2 and 5 can also be applied to final excited states $|f\rangle$ [49, 50, 51]. Henceforth, we assume similar resonance strengths for GDRs built on the ground and excited states. This may explain the nice fit between the high $\gamma$-ray energy part of the measured $f(E_{\gamma})$ and the left tail of the GDRg.s. (see e.g. [52]).

In order to combine cross-section contributions from the LEE and GDR regions, we use the well-known relation [53],

[TABLE]

where $g_{{}_{J}}=\frac{2J_{f}+1}{2J_{i}+1}$ is the statistical factor, with $J_{i}$ and $J_{f}$ being the spins of the initial and final states, respectively. The magnitude of $g_{{}_{J}}$ affects the estimation of $\sigma_{{}_{-2}}$ values in the LEE region. However, assuming a predominant dipole character for the LEE radiation [34, 26, 32], a value $g_{{}_{J}}=1$ is valid for $J\rightarrow J$ dipole transitions and a good approximation for any $\Delta J=1$ spin distribution typically populated (up to $J=8-10\hbar$) in the experimental studies of $f(E_{\gamma})$ [54]. This approximation is not valid for GDR $E1$ transitions in even-even nuclei where $g_{{}_{J}}=3$ applies.

The data spanning the GDR region have been obtained from available experimental nuclear reaction data bases, EXFOR [55] and ENDF [56]. Data corresponding to the LEE – in units of MeV*-3* – have been collected from the Oslo compilation of level densities and $f(E_{\gamma})$ [57]. The resulting $\sigma_{total}(E_{\gamma})$ was modeled using a cubic-spline interpolation – which assumes validity of the Brink-Axel hypothesis – in order to compute the total cross section and $\sigma_{{}_{-2}}$ values. Fourth-order polynomial fits yield similar results to the cubic spline interpolation, with almost negligible differences for the integrated $\sigma_{{}_{-2}}$ values of <0.5%. Lower and higher-order interpolation polynomials predict unanticipated structures of the ($\gamma$,n) cross-section (e.g. pronounced bumps between the LEE and GDR regions).

When available, ENDF data have been utilized to fill the typical gap between the LEE and GDR data sets, as shown in Fig. 1 for the case of 153Sm. Nuclei at different mass regions are evaluated for a systematic study of the LEE and GDR effects on $\sigma_{{}_{-2}}$ values. The results are listed in Table 1 and Fig. 1 shows the particular fits to the 45Sc and 153Sm data. Uncertainties on $\sigma_{{}_{-2}}$ values arise from the $RMS$ deviation,which accounts for a 7% error from the lower and upper loci limits provided by GDR and LEE data [57].

For the particular case of 45Sc, the large Eγ gap between the LEE and GDR data resulted in unrealistic fits with a drastic drop of $\sigma_{{}_{-2}}$ values, as shown in Fig. 1. Additional fits were performed by rejecting either the last two or the last four GDR data points at lower Eγ. Figure 1 shows that the former (solid line) is clearly more realistic and the resulting $\sigma_{{}_{-2}}$ value – with a 4% increase with respect to the fit considering the four GDR data points – is quoted in Table 1. Fits to the data for the rest of nuclei studied in this work do not present such large energy gaps and re-fitting of the data was not found necessary.

Although the work by Jones and co-workers supports an increasing trend of LEE for $E_{\gamma}<1$ MeV [34], there is little evidence on how $f(E_{\gamma})$ behaves approaching $E_{\gamma}=0$. Hence, the low-energy cut off has arbitrarily been set to 800 keV for the nuclides considered in this work up to 139La, which incidentally is the typical energy for strong $M1$ isovector transitions in nearly-spherical nuclei [16]. For 153Sm, a low-energy cut off of 645 keV has been set from $f(E_{\gamma})$ data [29]. Because of their instability, there is no available GDR information in 153Sm, 138La and 50V, and, instead, GDR data from 152Sm, 139La and 51V, respectively, have been used in the analysis, under the assumption that nearby isotopes present equal $f(E_{\gamma})$ (see e.g. [24] and  [72]). This assumption may not be adequate given the rapid shape transition from weakly deformed in 150Sm to a well-deformed rotor in 154Sm, and the realization of shell closures in 139La ($N=82$) and 51V ($N=28$).

For comparison, Fig. 2 shows overall $\sigma_{{}_{-2}}$ values of ground states as a function of $A$ extracted from photo-neutron cross sections using monoenergetic photon beams and determined above neutron threshold to an upper limit of $E_{\gamma_{max}}\approx 20-50$ MeV [13]. The data include the GDR region and are representative for nuclei above $A\gtrapprox 50$ (except for 58Ni [73]), where neutron emission is generally the predominant decay mode. This may not be true for nuclei with semi-magic number of neutrons – discussed below – where proton separation energies may lie lower than neutron thresholds.

From Eqs. 1 and 2, Migdal extracted the relation \sigma_{{}_{-2}}=2.25A^{5/3}\mbox{\mub/MeV}, which was qualitatively confirmed by Levinger [21] and further refined [74] as (dashed line in Fig. 2),

[TABLE]

where $\kappa$ is the polarizability parameter and represents deviations from the actual GDR effects. This result is in excellent agreement with IPM predictions using, instead of $\hbar\omega=41A^{-1/3}$ MeV, $E^{g.s.}_{{}_{GDR}}=79A^{-1/3}$ MeV as the resonance frequency [75]. A value of $\kappa=1$ generally holds for the ground state of nuclei with $A\gtrapprox 50$, and probably for even lighter nuclei with $A\gtrapprox 20$ once $\sigma(\gamma,p)$ contributions are taken into account [21, 73, 74]. In contrast, values of $\kappa>1$ are generally found for light nuclei with $A<20$ [21, 74, 49, 76, 77, 50, 51], where paramagnetism is important.

Sudden drops of $\sigma_{{}_{-2}}$ (and $\kappa$) values are apparent for the $N=50$, 82 and 126 isotones in the insets (a), (b) and (c) of Fig. 2, respectively. Above both proton and neutron separation energies, the photo-absorption cross section in the lower energy part of the GDR is controlled by the statistical competition between $\sigma(\gamma,p)$ and $\sigma(\gamma,n)$ contributions, which presents a strong correlation with the level density ratio $N_{p}/N_{n}$ between the open neutron ($N_{n}$) and proton ($N_{p}$) channels [73], $\sigma(\gamma,p)/\sigma(\gamma,n)\approx N_{p}/N_{n}$. This ratio depends on the neutron and proton penetrabilities, $\epsilon_{n}$ and $\epsilon_{p}$, respectively, as more energy is needed for protons to overcome the Coulomb barrier.

Total photo-absorption cross-sections ($\sigma(\gamma,n)$ + $\sigma(\gamma,p)$) are reasonably available in the $N=50$ isotones, with the latter being indirectly determined from ($e,e^{\prime}p$) measurements [60]. The $\sigma(\gamma,p)$ contribution is particularly important for 92Mo, with $N_{p}/N_{n}\approx 1.95$, and decreases for the lighter $N=50$ isotones, with $N_{p}/N_{n}\approx 0.66,<0.28$ and 0.09 for 90Zr, 89Y and 88Sr, respectively [68], as the isospin quantum number $T_{z}=\frac{N-Z}{2}$ increases. The $\sigma(\gamma,p)$ contribution extracted from the $N_{p}/N_{n}$ ratio only applies to the lower energy half of the GDR, and $\sigma(\gamma,n)$ contributions still remain greater. Once $\sigma(\gamma,p)$ contributions are taken into account, the total photo-absorption cross section satisfies the TRK sum rule [68], $\frac{\sigma_{total}(\gamma,n)+\sigma_{total}(\gamma,p)}{0.06NZA^{-1}}$. For the 92Mo case, there remains $\approx 35\%$ $\sigma(\gamma,p)$ contribution to the total photo-absorption cross section [60], which explains the sharper drop in the $\sigma_{{}_{-2}}$ value shown in Fig. 2(a). More conspicuous are the drops of $\sigma_{{}_{-2}}$ values in 89Y, 141Pr and 208Pb – where $\sigma(\gamma,n)$ contributions strongly dominate – which could provide evidence for shell effects. Clearly, direct measurements of $\sigma(\gamma,p)$ contributions are crucially needed for singly- and doubly-magic nuclei.

Furthermore, Table 1 shows that the LEE has a substantial contribution to $\sigma_{{}_{-2}}$ values in medium-mass nuclei (45Sc and 56Fe) away from the $N=28$ shell closure, being largest for 45Sc with $\approx 10\%$ increase. As illustrated in Fig. 1, this enhancement partly arises because of the inverse mass dependence of $E_{{}_{GDR}}$ and the fact that the LEE starts at lower $E_{\gamma}$ as $A$ increases. In fact, Table 1 shows that the LEE has a negligible contribution of $\lesssim 3\%$ to the total $\sigma_{{}_{-2}}$ values of heavy nuclei with $A\geqq 76$. A stronger contribution to $\sigma_{{}_{-2}}$ values would arise if the LEE trend keeps increasing at energies approaching $E_{\gamma}=0$, as predicted by SM calculations. This possibility will be explored in detail in a separate manuscript [78].

More intriguing are the small overall contributions to $\sigma_{{}_{-2}}$ values found in nuclei close to or having a magic number. When compared with Eq. 7, these nuclides present evident deviations from GDR effects (i.e. $\kappa\neq 1$) with smaller values of $\kappa\approx 0.90$ in 50,51V ($N\approx 28$) and 138,139La ($N\approx 82$), and specially for 92Zr ($N\approx 50$ and $Z=40$) with $\kappa=0.70(5)$. In contrast, heavy nuclei away from shell closures present polarizability parameters consistent with $\kappa=1$; except perhaps for 153Sm, where we used the 152Sm data for the GDR region and a cut-off of $E_{\gamma}=645$ keV. This recurrent behavior to the one previously observed in the photo-neutron cross-section data for the $N=50$, 82 and 126 isotones, indicates the continuing influence of shell effects in the quasi-continuum region up to the neutron threshold. As shown in Table 1 and inset (d) in Fig. 2, this is consistent with the smaller LEE contribution to the total $\sigma_{{}_{-2}}$ values of 50,51V ($N\approx 28$) with respect to the neighboring 45Sc and 56Fe nuclides. Although there is no $\sigma(\gamma,p)$ data available for 50,51V, $(\gamma,p)$ contributions will relatively be much weaker for 51V because of the much lower level density of the open proton channel (even-even 50Ti with $N=28$) as compared with the open neutron channel (odd-odd 50V).

Interesting SM calculations of the $M1$ strength in the LEE for various isotopic and isotonic chains by Midtbo and collaborators [44] predict a relatively sharper increase of the $M1$ strength at $E_{\gamma}=0-2$ MeV for neutron-rich nuclei when approaching shell closure. These results may contradict our findings for stable nuclei and suggest the enhancement of nuclear polarizability with increasing instability.

Conclusively, drops of $\sigma_{{}_{-2}}$ values ($\kappa<1$) for several nuclei with, or close to, neutron magic numbers $N=28$, 50, 82 and 126, suggest that the shell model remains valid at high excitation energies, from the quasi-continuum to the GDR region; in agreement with Balashov’s SM interpretation of the GDR as a system of independent nucleons plus the residual interaction [79]. These deviations from GDR effects, because of the nature of Eq. 7, are plausibly not related to $E1$ transitions, which, together with the continuing influence of shell effects, strongly support the $M1$ interpretation of the LEE by large-scale SM calculations [36, 40, 41, 42, 43]. Moreover, the empirical evidence for shell effects suggests that the generalized Brink-Axel hypothesis allows for structural changes and is, therefore, more universal than originally expected. This conclusion is supported by the work of Larsen et al. [32], where $f(E_{\gamma})$ trends are found to be preserved for different bin energies.

Finally, we confirm the induction of permanent magnetic dipole moments or paramagnetism in the quasi-continuum region, in agreement with previous SM calculations and IPM predictions of an enhanced paramagnetism for the ground states of nuclei with large occupation number of the shells determining the magnetic properties [14]. The origin of this paramagnetism can be inferred from SM calculations, which can distinguish between single-particle spin-flips and collective isovector excitations by decomposing the relevant $M1$ strength into their spin and orbital components [16]. Similar to two-neutron separation energies extracted from atomic mass measurements of ground and isomeric states, this work opens a new research avenue to investigate the evolution of shell closures and the existence of “old” and “new” magic numbers at high-excitation energies from $\sigma_{{}_{-2}}$ measurements.

The authors acknowledge Physics discussions with B. Dey, P. E. Garrett, B. V. Kheswa, S. Triambak, K. Sieja, P. von Neumann-Cosel and M. Wiedeking. This work was supported by the National Research Foundation of South Africa under Grant 93500 and the MaNuS/MatSci Honours/Masters program.