A new spectroscopic probe to search for magic numbers at high-excitation energies
aa r X i v : . [ nu c l - t h ] M a r A new spectroscopic probe to search for magic numbers at high-excitationenergies
Cebo Ngwetsheni a , José Nicolás Orce a, ∗ a Department of Physics & Astronomy, University of the Western Cape, Bellville-7535, South Africa
Abstract
Empirical drops in ground-state nuclear polarizabilities indicate deviations from the effect of giant dipole reso-nances and may reveal the presence of shell effects in semi-magic nuclei with neutron magic numbers N =
50, 82 and126. Similar drops of polarizability in the quasi-continuum of nuclei with, or close to, magic numbers N =
28, 50and 82, could reflect the continuing influence of shell closures up to the nucleon separation energy. These findingsopen a new avenue to investigating magic numbers at high-excitation energies and strongly support recent large-scaleshell-model calculations in the quasi-continuum region, which describe the origin of the low-energy enhancementof the photon strength function as induced paramagnetism. The nuclear-structure dependence of the photon-strengthfunction asserts the generalized Brink-Axel hypothesis as more universal than originally expected.
Keywords: nuclear dipole polarizability, quasi-continuum, low-energy enhancement, photon-strength function,photo-absorption cross sections, magic numbersThe ability for a nucleus to be polarized is a fortioridriven by the dynamics of the isovector giant dipole res-onance (
GDR ). That is, the inter-penetrating motion ofproton and neutron fluids out of phase [1], which resultsfrom the symmetry energy in the Bethe-Weizsäckersemi-empirical mass formula [2, 3], a sym ( ρ N − ρ Z ) / ρ A ,acting as a restoring force [1, 4]. Respectively, ρ N and ρ Z are the mass densities of the neutron and proton flu-ids, and ρ A the sum of the separate densities. The GDR represents most of the absorption and emission of γ -rayphotons by a nucleus and was the first quantum col-lective excitation ever discovered in mesoscopic sys-tems [5]. The idea of giant resonances was soon bor-rowed by atomic, molecular and solid-state physics (seee.g. [6] and references therein); the GDR motion is akinto the plasmons in graphene, which enables strong con-finement of electromagnetic energy at subwavelengthscales [7].Using the collective variable ρ Z as the potential en-ergy of the liquid drop, Migdal calculated the electricdipole polarizability, α E , for the ground state of nuclei ∗ Corresponding author
Email address: [email protected] (José Nicolás Orce)
URL: https://nuclear.uwc.ac.za (José Nicolás Orce) to be directly proportional to the size of the nucleus [1], α E = e R A a sym = . × − A / fm , (1)where a sym =
23 MeV is the symmetry energy parame-ter and R = . A / fm the radius of the nucleus with A = N + Z . Alternatively, α E is well described bymicroscopic mean-field approaches using the random-phase approximation ( RPA ) with various effective inter-actions [8–10], and can be determined empirically withthe use of second-order perturbation theory, α E = e ∑ n h i k ˆ E k n ih n k ˆ E k i i E γ = ¯ hc π σ − , (2)with | i i being the vector of the ground state connectinghigh-lying | n i states in the GDR region via E σ − the ( − ) moment of the total photo-absorption cross section [11, 12] defined as, σ − : = Z E γ max σ total ( E γ ) E γ dE γ , (3)where the total photonuclear-absorption cross section, σ total ( E γ ) , generally includes all σ ( γ , n ) and σ ( γ , p ) contributions [13]. Preprint submitted to Elsevier March 5, 2019 aturally, total σ − values should include both elec-tric and magnetic polarizability contributions, σ − = π ¯ hc ( α E + χ M ) , (4)where χ M is the static magnetic dipole polarizabilityand considers the sum of the paramagnetic χ para M anddiamagnetic χ dia M susceptibilities of nuclei [14], χ M = χ para M + χ dia M = ∑ n h i k ˆ M k n ih n k ˆ M k i i E γ − Ze mc h r i . (5)According to the independent-particle shell model( IPM ), diamagnetism is dominant for nuclei with A >
60 [15], but has a negligible effect in σ − values. Para-magnetism dominates in light nuclei with the rise ofpermanent magnetic dipole moments and can, in con-trast, contribute substantially to σ − values for nucleiwith A <
20 [15].Because of the 1/E γ energy weighting in Eq. 3, σ − values are extremely sensitive measures – unlike σ total – of low-energy long-range correlations in the nu-clear wave functions, which are common feature for allnucleon-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 fromabove the GDR region (e.g., nucleon resonances at E γ &
140 MeV) have a negligible effect on σ − values [18–21].Below the neutron separation threshold, the pygmydipole resonance ( PDR ) in neutron-rich nuclei [22] – the
PDR is an electric dipole resonance arising from the os-cillation of a symmetric proton-neutron core against theneutron skin – may add a ≃
5% contribution to σ − val-ues [23]. To a lesser extent, soft resonances such as the M σ − values may arise fromthe low-energy enhancement ( LEE ) of the radiative orphoton strength function f ( E γ ) – indicating the abilityof nuclei to emit and absorb photons with energy E γ –observed at E γ / LEE and
GDR cross-section contributions affect σ − values and may provide evidence for the continu-ing influence of shell effects at high-excitation energies.Relevant consequences arise from these findings; for in-stance, the possibility to identify new magic numbers.The physical origin of the LEE remains ambiguousand its observation seems to be generally associatedwith weakly deformed nuclei. It has been observedin nearly-spherical nuclei in the A ≈
50 and 90 mass regions starting at E γ ≈ − Cd [27], , La [28] and , Sm [29], where the
LEE starts at a lower E γ ≈ f ( E γ ) is independent ofthe particular nuclear structure and only depends onE γ [30, 31] – which has been confirmed experimen-tally [32, 33]. The reason for not being observed inother heavy nuclei – studied with the same experimen-tal method – could relate to the unprecedented sensitiv-ity achieved by Simon and co-workers in , Sm us-ing high-purity germanium (
HPGe ) detectors in connec-tion with bismuth germanate (
BGO ) shields [29]. An-other relevant finding is that the
LEE presents a domi-nant dipole radiation [26, 32], but whether its nature iseither electric or magnetic remains unresolved [34]. Therecent polarization asymmetry measurements of γ raysin Fe using
GRETINA tracking detectors yields incon-clusive results, although rather suggests an admixtureof electric and magnetic dipole radiation, with a smallbias towards a magnetic character at E γ = . − . LEE anomaly. On one hand, Litvinovaand Belov propose that the
LEE in f ( E γ ) occurs becauseof E SM calcula-tions predict that the LEE has a predominant magnetic-dipole M LEE arises from activehigh- j proton and neutron orbits near the Fermi surfacewith magnetic moments adding up coherently [36]. Thisis 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 com-plementary picture, Brown and Larsen suggest that the LEE arises because of the large M ℓ orbitals [40]. Additionally, Sieja com-puted both E M Sc on equal footingfrom large-scale SM calculations and also supported the M LEE in the A ∼
50 region against E SM cal-culations of neutron-rich Ni [43] and many other nu-clei [44], using various effective interactions, also sup-port the M LEE .Recently, In principle, the validation of these SM pre-dictions in the quasi-continuum region may be arguableas, for instance, they are structure dependent; hence,posing a fundamental question about the validity of theBrink-Axel hypothesis [30, 31].A priori, the comparison of the LEE and the
GDR built on ground states is somewhat misleading as the2ormer corresponds to γ -ray transitions between ex-cited states in the quasi-continuum, whereas the latterinvolves transitions to the ground state. Nonetheless,the study of (p, γ ) and (n, γ ) reactions for light nucleiand fusion-evaporation reactions for heavy nuclei haveshown that GDR s can also be built on excited states(
GDR exc ) [45–47]. In fact,
GDRs exc present – at leastfor moderate average temperature T and spin J – sim-ilar centroid energies, E exc GDR , and resonance strengths, S exc GDR , relative to the Thomas-Reiche-Kuhn (
TRK ) E GDR g . s . ) [45, 46]. These similar features sug-gest a common physical origin for all GDR s in concor-dance with the Brink-Axel hypothesis, which also in-dicates that a
GDR can be built on every state in a nu-cleus [30, 31]. Moreover, the sum rules in Eqs. 2 and5 can also be applied to final excited states | f i [49–51].Henceforth, we assume similar resonance strengths for GDR s built on the ground and excited states. This mayexplain the nice fit between the high γ -ray energy partof the measured f ( E γ ) and the left tail of the GDR g . s . (see e.g. [52]).In order to combine cross-section contributions fromthe LEE and
GDR regions, we use the well-known rela-tion [53], f ( E γ ) = g J π ( ¯ hc ) σ total ( E γ ) E γ MeV − , (6)where g J = J f + J i + is the statistical factor, with J i and J f being the spins of the initial and final states, respec-tively. The magnitude of g J affects the estimation of σ − values in the LEE region. However, assuming a predom-inant dipole character for the
LEE radiation [26, 32, 34],a value g J = J → J dipole transitions anda good approximation for any ∆ J = J = − h ) in the experi-mental studies of f ( E γ ) [54]. This approximation is notvalid for GDR E g J = GDR region have been obtainedfrom available experimental nuclear reaction data bases,
EXFOR [55] and
ENDF [56]. Data corresponding to the
LEE – in units of MeV − – have been collected from theOslo compilation of level densities and f ( E γ ) [57]. Theresulting σ total ( E γ ) was modeled using a cubic-splineinterpolation – which assumes validity of the Brink-Axel hypothesis – in order to compute the total crosssection and σ − values. Fourth-order polynomial fitsyield similar results to the cubic spline interpolation,with almost negligible differences for the integrated σ − values of <0.5%. Lower and higher-order interpolation E γ [MeV] f ( E γ ) [ M e V - ] ENDF
Sm LEE
Sm GDRExtrapolated LEE Sc LEE Sc GDRExcl. 4 pointsExcl. 2 points S n153 Sm ScS n Figure 1: f ( E γ ) vs E γ on a log scale showing the interpolation to thedata (solid lines) for Sc [58–60] and
Sm [29, 71]. Vertical dashlines indicate the neutron separation energy. See text for additionalinformation. polynomials predict unanticipated structures of the ( γ ,n)cross-section (e.g. pronounced bumps between the LEE and
GDR regions).When available,
ENDF data have been utilized to fillthe typical gap between the
LEE and
GDR data sets, asshown in Fig. 1 for the case of
Sm. Nuclei at differentmass regions are evaluated for a systematic study of the
LEE and
GDR effects on σ − values. The results arelisted in Table 1 and Fig. 1 shows the particular fits tothe Sc and
Sm data. Uncertainties on σ − valuesarise 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 Sc, the large E γ gap be-tween the LEE and
GDR data resulted in unrealistic fitswith a drastic drop of σ − values, as shown in Fig. 1.Additional fits were performed by rejecting either thelast two or the last four GDR data points at lower E γ .Figure 1 shows that the former (solid line) is clearlymore realistic and the resulting σ − 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 forthe rest of nuclei studied in this work do not presentsuch large energy gaps and re-fitting of the data was notfound necessary.Although the work by Jones and co-workers supportsan increasing trend of
LEE for E γ < f ( E γ ) behaves approaching E γ =
0. Hence, the low-energy cut off has arbitrarily been setto 800 keV for the nuclides considered in this work up to
La, which incidentally is the typical energy for strong M γ ( max ) (GDR) E γ ( max ) (LEE) σ − (total) σ − κ [Refs.](MeV) (MeV) ( µ b/MeV) (LEE) (LEE) Sc ∗ V 27.8 3.1 1458(100) 2.9% 0.89(7) [52, 61]
V 27.8 3.1 1472(100) 3.3% 0.87(7) [52, 61] Fe ∗ Ge 26.5 2.3 3189(225) 2.7% 0.97(7) [63, 64]
Zr 27.8 2.2 3131(220) 1.1% 0.70(5) [65–67]
Mo 27.8 2.5 4743(330) 1.7% 1.00(7) [68, 69]
La 24.3 1.9 7983(560) 0.4% 0.90(7) [28, 70]
La 24.3 2.5 8015(560) 0.7% 0.90(6) [28, 70]
Sm 20.0 1.6 9999(700) 2.7% 0.95(7) [29, 71]
Table 1: Contributions of
GDR and
LEE cross-sections to σ − and κ values. Data have been extracted from EXFOR [55],
ENDF [56] and theOslo compilation [57]. An asterisk indicates that the σ − value includes σ ( γ , p ) contributions. For
Sm, a low-energy cut off of 645 keV has beenset from f ( E γ ) data [29]. Because of their instability,there is no available GDR information in
Sm,
Laand V, and, instead,
GDR data from
Sm,
La and V, respectively, have been used in the analysis, underthe assumption that nearby isotopes present equal f ( E γ ) (see e.g. [24] and [72]). This assumption may not beadequate given the rapid shape transition from weaklydeformed in Sm to a well-deformed rotor in
Sm,and the realization of shell closures in
La ( N = V ( N = Mass Number (A) σ − ( µ b / M e V ) Dietrich & Berman evaluation (1988) σ −2 without LEE σ −2 with LEE He H He Li C O Si Ne Mg S Ca Ni N=50 Al Y Pr Zr Mo Zr N=28
N=50 σ − = . A / O Pb N=82 N=126N=28 V Sc Fe (a) (d) Ge Mo Pr Sm Nd La (b) Pb (c) V V Figure 2: σ − vs A on a log-log scale from the photo-neutron cross-section evaluation (solid circles) [13] and σ − data listed in Table 1excluding (squares) and including (diamonds) the LEE contributions.For comparison, Eq. 7 (dashed line) is plotted.
For comparison, Fig. 2 shows overall σ − valuesof ground states as a function of A extracted from photo-neutron cross sections using monoenergetic pho-ton beams and determined above neutron threshold toan upper limit of E γ max ≈ −
50 MeV [13]. The datainclude the
GDR region and are representative for nu-clei above A '
50 (except for Ni [73]), where neu-tron emission is generally the predominant decay mode.This may not be true for nuclei with semi-magic num-ber of neutrons – discussed below – where proton sepa-ration energies may lie lower than neutron thresholds.From Eqs. 1 and 2, Migdal extracted the relation σ − = . A / µ b/MeV, which was qualitatively con-firmed by Levinger [21] and further refined [74] as(dashed line in Fig. 2), σ − = . ( ) κ A / µ b/MeV , (7)where κ is the polarizability parameter and representsdeviations from the actual GDR effects. This result is inexcellent agreement with
IPM predictions using, insteadof ¯ h ω = A − / MeV, E g . s . GDR = A − / MeV as the res-onance frequency [75]. A value of κ = A '
50, and probablyfor even lighter nuclei with A '
20 once σ ( γ , p ) contri-butions are taken into account [21, 73, 74]. In contrast,values of κ > A <
20 [21, 49–51, 74, 76, 77], where paramagnetism isimportant.Sudden drops of σ − (and κ ) values are apparent forthe N =
50, 82 and 126 isotones in the insets (a), (b) and(c) of Fig. 2, respectively. Above both proton and neu-tron separation energies, the photo-absorption cross sec-tion in the lower energy part of the
GDR is controlled bythe statistical competition between σ ( γ , p ) and σ ( γ , n ) contributions, which presents a strong correlation withthe level density ratio N p / N n between the open neutron( N n ) and proton ( N p ) channels [73], σ ( γ , p ) / σ ( γ , n ) ≈ p / N n . This ratio depends on the neutron and protonpenetrabilities, ε n and ε p , respectively, as more energyis needed for protons to overcome the Coulomb barrier.Total photo-absorption cross-sections ( σ ( γ , n ) + σ ( γ , p ) ) are reasonably available in the N =
50 isotones,with the latter being indirectly determined from ( e , e ′ p )measurements [60]. The σ ( γ , p ) contribution is partic-ularly important for Mo, with N p / N n ≈ .
95, and de-creases for the lighter N =
50 isotones, with N p / N n ≈ . , < .
28 and 0.09 for Zr, Y and Sr, respec-tively [68], as the isospin quantum number T z = N − Z increases. The σ ( γ , p ) contribution extracted from the N p / N n ratio only applies to the lower energy half ofthe GDR , and σ ( γ , n ) contributions still remain greater.Once σ ( γ , p ) contributions are taken into account, thetotal photo-absorption cross section satisfies the TRK sum rule [68], σ total ( γ , n )+ σ total ( γ , p ) . NZA − . For the Mo case,there remains ≈ σ ( γ , p ) contribution to the totalphoto-absorption cross section [60], which explains thesharper drop in the σ − value shown in Fig. 2(a). Moreconspicuous are the drops of σ − values in Y, Prand
Pb – where σ ( γ , n ) contributions strongly domi-nate – which could provide evidence for shell effects.Clearly, direct measurements of σ ( γ , p ) contributionsare crucially needed for singly- and doubly-magic nu-clei.Furthermore, Table 1 shows that the LEE has a sub-stantial contribution to σ − values in medium-mass nu-clei ( Sc and Fe) away from the N =
28 shell clo-sure, being largest for Sc with ≈
10% increase. Asillustrated in Fig. 1, this enhancement partly arises be-cause of the inverse mass dependence of E GDR and thefact that the
LEE starts at lower E γ as A increases. Infact, Table 1 shows that the LEE has a negligible con-tribution of .
3% to the total σ − values of heavy nu-clei with A ≧
76. A stronger contribution to σ − valueswould arise if the LEE trend keeps increasing at energiesapproaching E γ =
0, as predicted by SM calculations.This possibility will be explored in detail in a separatemanuscript [78].More intriguing are the small overall contributions to σ − values found in nuclei close to or having a magicnumber. When compared with Eq. 7, these nuclidespresent evident deviations from GDR effects (i.e. κ , κ ≈ .
90 in , V ( N ≈
28) and , La ( N ≈ Zr ( N ≈
50 and Z =
40) with κ = . ( ) . In contrast, heavy nucleiaway from shell closures present polarizability param-eters consistent with κ =
1; except perhaps for
Sm,where we used the
Sm data for the
GDR region anda cut-off of E γ =
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, in-dicates the continuing influence of shell effects in thequasi-continuum region up to the neutron threshold. Asshown in Table 1 and inset (d) in Fig. 2, this is consistentwith the smaller
LEE contribution to the total σ − valuesof , V ( N ≈
28) with respect to the neighboring Scand Fe nuclides. Although there is no σ ( γ , p ) dataavailable for , V, ( γ , p ) contributions will relativelybe much weaker for V because of the much lowerlevel density of the open proton channel (even-even Tiwith N =
28) as compared with the open neutron chan-nel (odd-odd V).Interesting SM calculations of the M LEE for various isotopic and isotonic chains by Midtboand collaborators [44] predict a relatively sharper in-crease of the M E γ = − σ − values ( κ <
1) for sev-eral nuclei with, or close to, neutron magic numbers N =
28, 50, 82 and 126, suggest that the shell model re-mains valid at high excitation energies, from the quasi-continuum to the
GDR region; in agreement with Bal-ashov’s SM interpretation of the GDR as a system of in-dependent nucleons plus the residual interaction [79].These deviations from
GDR effects, because of the na-ture of Eq. 7, are plausibly not related to E M LEE by large-scale SM calculations [36, 40–43]. More-over, the empirical evidence for shell effects suggeststhat the generalized Brink-Axel hypothesis allows forstructural changes and is, therefore, more universal thanoriginally expected. This conclusion is supported by thework of Larsen et al. [32], where f ( E γ ) trends are foundto be preserved for different bin energies.Finally, we confirm the induction of permanent mag-netic dipole moments or paramagnetism in the quasi-continuum region, in agreement with previous SM cal-culations and IPM predictions of an enhanced paramag-netism for the ground states of nuclei with large occu-pation number of the shells determining the magneticproperties [14]. 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