Consolidating the concept of low-energy magnetic dipole decay radiation
J. E. Midtbø, A. C. Larsen, T. Renstrøm, F. L. Bello Garrote, E. Lima
CConsolidating the concept of low-energy magnetic dipole decay radiation
J. E. Midtbø, ∗ A. C. Larsen, T. Renstrøm, F. L. Bello Garrote, and E. Lima Department of Physics, University of Oslo, N-0316 Oslo, Norway
We have made a thorough study of the low-energy behaviour of the γ -ray strength function within the frame-work of the shell model. We have performed large-scale calculations spanning isotopic and isotonic chains overseveral mass regions, considering 283 nuclei in total, with the purpose of studying the systematic behavior ofthe low-energy enhancement (LEE) for M I. INTRODUCTION
The atomic nucleus is an extremely complicatedmany-body quantum system [1]. Despite intensescrutiny over many decades, many of its facetsare still poorly understood. This is especially truewhen a significant amount of energy is put intothe nuclear system, placing it in a highly excitedstate. Since the number of accessible quantumlevels grows approximately exponentially with en-ergy [2, 3], a region of high excitation energy is onewhere many quantum levels are packed closely to-gether. It is a question of fundamental scientific in-terest how the quantum-mechanical wave functionof such levels is composed, and what degree of cor-relations exist between the levels [4].Two basic experimental quantities revealing in-formation on the structure of the nuclear wave func-tions are excitation-energy levels and their corre-sponding transition strengths. However, when theexcitation energy becomes large, it is experimen-tally difficult to separate individual levels and tran-sitions, and one instead works with average quan-tities, such as the energy level density and γ -ray strength function . Our focus in this article is onthe strength function, more specifically on the M γ -ray strength function ex-hibits an enhancement towards zero γ -ray energy( e.g. Refs. [5, 6]). This low-energy enhancement(LEE) has been shown to be of dipole order [7–10].However, its electromagnetic character is, so far,experimentally undetermined although recent mea-surements indicate a small bias towards M γ -ray strength functionhave an important application in calculations of ( n , γ ) capture cross sections ( e.g Ref. [11]). Ra-diative neutron capture is responsible for the syn-thesis of most elements heavier than iron, mainly ∗ [email protected] through the slow ( s ) and rapid ( r ) neutron-captureprocesses. The latter process involves neutron-richnuclei far from stability, close to the neutron dripline. While we are still far from a complete under-standing of the r process, which has been singledout as one of the eleven science questions for the21st century [12], huge strides were made recentlywith the discovery of a neutron-star merger eventwhich seemingly produced r -process elements [13–15]. In such a neutron-rich, low-entropy environ-ment, an ( n , γ ) − ( γ , n ) equilibrium cannot be main-tained at all times [11, 16, 17]. Thus, ( n , γ ) reac-tion rates become important not only at freeze-outbut also for the nucleosynthesis at earlier stages.It has been shown that the presence of an LEE inthe γ -ray strength function can impact the ( n , γ ) cross sections by orders of magnitude, especiallyfor neutron-rich nuclei [18]. Hence it is importantto obtain an understanding of the prevalence andproperties of the LEE. II. THE HISTORY OF THE LOW-ENERGYENHANCEMENT
In Fig. 1 we have charted the nuclei that havebeen studied using the Oslo or β -Oslo methods, andindicated whether the experiment saw a low-energyenhancement or not. It must be stressed that exper-imental limitations make it difficult to extract thevery low- E γ strength function using the ( β -)Oslomethod. This is mainly due to the uncertainties in-troduced by unfolding of the Compton-scatteringevents, which induce large uncertainties the low- γ energy spectrum at high excitation energies. Typi-cally, the lower limit on E γ is set at about 1.5 MeV.An exception is , Sm [8], where Compton sup-pression allowed extraction all the way down to E γ =
700 keV. In these experiments, they did see asizable LEE. It could thus be that the LEE is presentin some or all of the nuclei marked off with circlesand diamonds in the figure.Over the last several years, different theoretical a r X i v : . [ nu c l - t h ] F e b Figure 1. (Color online) Map detailing where an LEE has been seen using the Oslo method. Yellow stars indicate yes,red circles no. Blue diamonds denote cases where it is difficult to say whether there is an LEE or not. Note that anegative result cannot rule out the presence of an LEE at lower E γ energies than was experimentally accessible (see textfor more details). The nuclear chart is made using Ref. [19], while the experimental data used are from Refs. [5–9, 20–56]. interpretations have been put forward to explain theLEE. In fact, the terminology varies, and the phe-nomenon has been variously referred to as LEE, up-bend [6], LEMAR [57, 58] and zero limit [59]. Ifa phenomenon with more than three names can beconsidered a “hot topic”, then this clearly qualifies.In the following, we make an attempt to summarizethe theoretical work that has been done on explain-ing the LEE.Perhaps the first line of demarcation should bedrawn between those works explaining the LEEas M E et al. used thethermal-continuum quasiparticle random-phase ap-proximation to demonstrate a low-energy enhance-ment in the E M E h ω transitions, due to theparity change in the E h ω excitations requires a large model space;hence the dimensions of the calculation quickly blow up. It can however be done in some cases, forexample by Schwengner et al. [61] and Sieja [62].Still, most shell-model work related to the quasi-continuum strength function to date has been donefor M h ω .The first shell-model study was done bySchwengner et al. [57], who studied Zr and Mo iso-topes and compared calculations to strength func-tion data from the Oslo group. They obtained goodagreement with the low-energy ( E γ ≤ γ -ray strength, and were able to explain almost thecomplete strength for E γ < M B ( M ) values as a function of E γ and the strengthfunction f M ( E γ ) can be well fitted by an exponen-tial function, B exp ( − E γ / T B ) , with T B ∼ . − . T B ∼ . B ( M ) and f M , re-spectively. Further, the mechanism behind the LEEwas explained as being due to a recoupling of thespins of high- j protons and neutrons, analogous tothe shears-band phenomenon.Brown and Larsen [63] investigated the strengthfunction of , Fe, and were also able to explainit as an M j , in thiscase from the f / orbital.In a subsequent work, Schwengner et al. studiedthe LEE in a series of Fe isotopes extending into themiddle of the neutron shell [58]. They found evi-dence for a bimodality in the M E γ ∼ i.e. M et al. [65] presented an interestingstudy using a “toy model” where only the f / or-bital was included, for both protons and neutrons.With this model space they studied , Cr and V.They again found evidence for a low-energy en-hancement, and they showed that its slope is depen-dent upon the strength of the (in isospin formalism) T = B ( M ) distribution to an exponential function,but found a much larger T B of 1.33 MeV, i.e. a sig-nificantly gentler incline.Sieja [62] considered the nuclei , Sc, , Ti,and obtained both E M E M E M III. SYSTEMATIC SHELL MODELCALCULATIONS
The present work follows the tradition of usingthe shell model. We employ KSHELL [66], a veryefficient M -scheme shell model code able to cal-culate levels and transition strengths within verylarge model spaces. All the calculations presentedhere have been made publicly available throughZenodo [67]. As interaction and model space istaken JUN
45 [68], which comprises the orbitals ( f / pg / ) atop a Ni core. The valence spaceallows up to 22 protons and neutrons. To facili-tate computation, the model space is truncated byturning off proton excitations to the g / orbital.We have checked that this does not have an ef-fect on Cu isotopes, but cannot rule out that itcould impact nuclei with higher Z . Calculationsare performed for the entire isotopic chains of Ni,Cu, Zn, Ga, Ge and As that are within the modelspace, as well as some neutron-rich Se isotopes.For each nucleus, we calculate 100 levels of eachparity and each spin between J = J = /
2) and J =
14 ( J = /
2) for even (odd) A , respectively.We then calculate B ( M ) transition strengths for all E γ ( MeV ) − − f ( M e V − ) Ga Ga Ga Ga Ga Ga Ga
35 67 Ga Ga Ga Ga Ga Ga Ga
42 74 Ga Ga Ga Ga Ga Ga Ga Figure 2. (Color online) Calculated M γ -ray strengthfunctions of Ga isotopes using the JUN
45 interaction. allowed transitions and compile the γ -ray strengthfunction using Eq. (A1). A bin size of ∆ E = . JUN
45, we use the recommended effective g s val-ues of g s , eff = . g s , free [68]. The dependence of thestrength function on E x , J and π is removed by av-eraging. The average includes all calculated statesand transitions. We observe that the strength func-tion is remarkably similar for different choices ofthese parameters, except for statistical fluctuations– hence averaging them out is justified, in accor-dance with the Brink hypothesis [69]. As an exam-ple, we show the calculated M N =
28 being very steep, flattening out towardsmid-shell before increasing back again approach-ing the N =
50 closure. The same effect is presentin the other isotopic chains that we have studied.To see this clearly, we have taken the ratio of theintegrated strength in the intervals E γ ∈ [ , ] MeVto E γ ∈ [ , ] MeV, respectively. This is shown inFig. 3 for all the isotopic chains. The overall trendof increasing low-energy strength towards the shellclosures is present for all isotopes.One could worry that some or all of these effectsare due to the particulars of the model space, suchas the choice of Ni as closed core. In Fig. 4, weshow the chain of Ni isotopes calculated both in the Ni model space and in a different model space,namely using a Ca core with the CA MH G in-teraction [6, 70], truncated so that two protons can
30 35 40 45 50 N R e l . a m o u n t o f l o w - e n e r gy s t r e n g t h NiCuZn GaGeAs SeNi (ca48mh1g)Cu (ca48mh1g)
Figure 3. (Color online) The amount of strength between0 and 2 MeV relative to the strength between 2 and 6MeV, plotted as function of neutron number for isotopicchains calculated with the
JUN
45 and CA MH G inter-actions. See text for details. excite from the f / orbital. Details of the Ca cal-culations are given in Ref. [6]. The trend of thestrength functions is clearly the same, with morelow-energy strength and steeper slope at the shelledges. The inclusion of the proton f / orbitaldoes however change the strength function, notablyby inducing what could be a spin-flip resonanceat higher E γ for some of the isotopes. The abso-lute values are also affected, becoming less vari-able and generally larger than with the Ni core. Itis not so surprising that the calculation with onlyneutrons in the model space gives lower B ( M ) values when we consider the structure of the M (cid:99) M ∝ g l (cid:126) l + g s (cid:126) s . Since g pl = g nl = CA MH G -calculated Ni isotopes. In this case, the increase atlow and high neutron number are complemented byan additional, large bump in the middle, peaking at Ni. The Ni isotopes in the middle of the neutronshell are known to exhibit shape coexistence includ-ing spherical components [71]. This shape coex-istence would involve proton excitations from the f / orbital, which means that it should not appearwhen using the Ni closed core. The CA MH G interaction reproduces features attributed to shapecoexistence in Ni [6]. Hence, this mid-shell LEEbump can be interpreted to be consistent with thesystematic trends.Among the
JUN N . Since Cu has onlyone proton on top of the Ni core, it is possible thatthe linear trend is an artifact of the restricted model − − f ( M e V − ) (a) Ni Ni Ni Ni Ni Ni
36 65 Ni Ni Ni Ni Ni Ni
42 71 Ni Ni Ni Ni Ni E γ ( MeV ) − − f ( M e V − ) (b) Figure 4. (Color online) γ -ray strength functions of iso-topic chains of Ni calculated with Ni (a) and Ca (b)closed cores, respectively. See text for details. space. To check this, we again used the CA MH G interaction and calculated , , , , , Cu, allow-ing up to two proton excitations from the f / aswas done for the Ni isotopes. Interestingly, thelinearity remains, as shown by the dashed line inFig. 3. This seems to indicate that the LEE variationwith neutron number is hindered in nuclei with oneproton atop magicity. We also note that the samelinear trend is present in the fluorine isotopes shownbelow.We have made similar calculations as the onesdescribed above in a different mass region, namelythe sd shell on top of a O closed core, using theUSD A interaction [72]. For this model space, weare able to calculate all isotopes without any trun-cation. With this interaction, B ( M ) strengths arecalculated using g s , eff = . g s , free [72]. In Fig. 5,we show the results for the isotopic chain of Al.These strength functions are generally much moreflat, but reveal the same trend of increase towards E γ ( MeV ) − − f ( M e V − ) Al Al Al Al Al
12 26 Al Al Al Al
16 30 Al Al Al Al Figure 5. (Color online) Calculated M γ -ray strengthfunctions of Al isotopes using the USD A interaction. N R e l . a m o u n t o f l o w - e n e r gy s t r e n g t h OFNeNa MgAlSiP SClAr
Figure 6. (Color online) Correlation between relativesum of low-energy strength and neutron number in the sd region. Note the logarithmic scale. magicity. Fig. 6 displays the relative amount oflow-energy strength for all isotopic chains. Thereis less change in the LEE as function of N in themiddle of the neutron shell compared to the JUN Z (a)05101520253035 Z (b)0 5 10 15 20 25 30 35 40 45 50 N Z (c) 0.51.02.04.06.010 − − − − Figure 7. (Color online) Integrated γ -ray strength from(a) 0 to 2 MeV and (b) 2 to 6 MeV, respectively, and(c) the fraction of the integrated γ -ray strength from 0 to2 MeV relative to the 2 to 6 MeV range, i.e. panel (a)divided by panel (b). lations indicate that the low-energy enhancement ismore pronounced near shell closures. Furthermore,the overall steepness of the strength is much higherin the f / pg / region than the sd region. Lastly,in both model spaces, the southeastern corner is en-hanced relative to the southwestern one. This isinteresting, because it is consistent with the shearsband picture advocated in Ref. [58], as discussed inSection II. We note that the same feature is apparentalso in the northern corners of the sd shell, wherethe north western corner has the constructive align-ment of proton holes with neutron particles. Look-ing at Fig. 1, this is consistent with the experimen-tal evidence for nuclei with A ≤ Sn and
Pb. However, it is seem-ingly at odds with the data for − Pd, , Cdand − , , Sn, where no LEE is seen, despitetheir proximity to the Z =
50 shell closure. Therecould be several explanations for this. It could bethat the LEE is very steep, and thus pushed to lower E γ than experimentally accessible. It could also bethat the proton shell closure is not a major drivingfactor for the LEE by itself, or there could be someother mechanism suppressing LEE in this region.Turning away from the question of relative steep-ness, it seems, from the present calculations like the M sd nuclei. This isimportant, because it implies that an M E E Si. Inaddition, we have considered Sc, located in the f p shell. The nickel mass region is unfortunatelynot accessible to E SDPF - MU interaction [73], which comprises the sd and f p shells, allowing the cross-shell excitations essentialfor E h ω trun-cation, meaning that the single-particle basis con-figurations are limited to ones where at most oneparticle is excited across the sd - f p shell gap. TheLawson method [75–77] with β =
100 MeV is usedto push the spurious centre-of-mass states up to en-ergies outside the considered range. For the E e p eff = ( + χ ) e , e p eff = χ e , with χ = − Z / A [74].In both cases, we obtain an E M Si it only serves to change the slope of the GLO,while for Sc it completely dominates the low-energy part of the strength function, demonstratingan LEE.Incidentally, we can compare our results withSieja’s calculations for the E Sc. Wefind a steeper slope on the low-energy tail of thestrength function compared to Fig. 6 in Ref. [62].This has a large influence on the summed dipolestrength function at E γ ≈ E M B ( E / M ) values (see Appendix A). Bothcalculations are consistent with the shape of theexperimental γ -ray strength function of Sc fromRef. [27], but Sieja’s provide the best match for theabsolute value. − − − f ( M e V − ) (a) Si M Si E Si tot.0 5 10 15 20 E γ ( MeV ) − − − − f ( M e V − ) (b) Sc M Sc E Sc tot.
Figure 8. (Color online) Calculated total dipole strengthfunctions for Si (a) and Sc (b).
IV. COMPARISONS WITH DISCRETEEXPERIMENTAL DATA
Many nuclei are so well studied that we have ac-cess to experimental information about levels, life-times and branching ratios up to quite high exci-tation energy. It is interesting to see if this infor-mation can be used to compile a strength function,and how it compares to shell model calculations.To this end, we extract experimental informationfrom the RIPL library [79]. We choose it over otherdatabases due to the ease with which it allows dataparsing, despite its lacking transition multipolarityinformation. We thus extract a strength functionof presumed
M1 transitions by selecting transitionsbetween levels where | J i − J f | ≤ π i π f = +
1. Thisdoes not rule out E M apriori expected to dominate. As such, this givesan impression of how the low-excitation M E x ∈ [ , ] MeV that pass the aforementioned re-quirement and that have a known lifetime and mea-sured γ -ray branching ratios. From this informa-tion we obtain partial decay widths, which we aver-age over ( E x , E γ , J , π ) bins. The strength function isthen obtained by multiplying by the level density at E γ ( MeV ) − − − − − f M ( M e V − ) Fe RIPL-3SM discrete SM quasicontinuumExponential fit
Figure 9. (Color online) Low-energy “ M
1” strength func-tion of Fe compiled from discrete experimental data.The bin width is ∆ E = . the corresponding ( E x , J , π ) , which we obtain con-sidering all known levels, not just the ones withknown lifetimes. This is important to get the correctabsolute value of the strength function (otherwise itwould be too low, see Appendix A). By compar-ing the level density from the discrete levels to thatfrom shell-model calculations, we verify that theexperimental level scheme seems to be complete upto the excitation energies we consider , as shown inFig. 12. Finally, we average over ( E x , J , π ) to ob-tain the average strength function depending onlyon E γ .We demonstrate this for the case of Fe in Fig. 9.The wealth of available experimental informationenables us to construct a strength function based on90 transitions selected according to the criteria de-scribed above. We compare to shell model calcu-lations done using the
GXPF A [80] interaction, aswas used in Ref. [63]. The agreement between ex-periment and calculations is excellent, both in termsof slope and absolute value. The results for a va-riety of nuclei in the sd shell and f / pg / shellregions are shown in Figs. 10 and 11, respectively.For these regions, we compare to the previouslydiscussed shell model calculations. The dotted linein each strength function panel shows the “quasi-continuum” strength function for that nucleus, bywhich we mean the strength function compiled us-ing all calculated levels, in the same was as wasdone for the systematics above. We have also ex-tracted a strength function from the shell modeldata by selecting discrete transitions similar to theRIPL ones. Specifically, for each RIPL level usedin the construction of the strength function, we havetaken the lowest-energy shell model level with the If the total level density from RIPL falls below the shell modellevel density before the “RIPL used” density dies off, thiswould indicate that we are compiling a strength function us-ing too low level density. This does not seem to be the casehere. same spin and parity, and included all transitionsfrom this level in the discrete SM strength function.(We also tried an alternative method selecting theclosest-in-energy shell model level, but this givesmuch poorer results.)In an attempt to quantify the differences betweenthe mass regions considered, we make a fit to an ex-ponential function f ( E γ ) = B exp ( E γ / T ) . To maxi-mize statistics, we fit the average strength functionin each of the regions (the green line shown in thelast panel of each of the figures). We have also fitted Fe separately. The results for the fit are listed inTable I. With all the assumptions that go into this fit,we should refrain from drawing strong conclusions,but it is striking that the sd fit displays almost factor3 gentler slope than f / pg / . This is compatiblewith the trend from the systematic calculations. B ( − MeV − ) T (MeV) sd f / pg / Fe 0.94 2.07Table I. Fit parameters for experimental RIPL strengthfunctions. See text for details.
V. SUMMARY AND OUTLOOK
In this work we have performed large-scale shellmodel calculations of M γ -ray strength functionsfor many isotopic chains in different major shells,focusing on the low-energy behaviour. We observesystematic trends in the calculations. The slope ofthe strength functions is generally steeper in the f / pg / than in the sd shell. This correlates withthe availability of high- j orbitals. Furthermore, theslope is steeper near the shell closures and gentler inthe mid-shell region for both model spaces. This isespecially pronounced in the region northwest andsoutheast of a doubly-magic nucleus, where, in theshears-bands picture, proton and neutron magneticmoments align to generate strong magnetic transi-tions.The present findings consolidate several insightsfrom previous studies – such as the dependence onhigh- j orbitals, the coupling of protons and neu-trons, and the relation to shears bands – and showsthat rather than being separate, incompatible ex-planations of the low-energy enhancement, theymay be complementary pieces of the same puzzle.Based on this and previous studies, we propose thatlarge low-energy magnetic decay strength is a fea-ture inherent to nuclei when they are excited to highenergies. The slope of the LEE seems to correlatewith the availability of high- j orbitals, which alsocorrelates with nuclear mass. While the slope ofthe M − − − − − f M ( M e V − ) Al RIPL-3 SM discrete SM quasicontinuum Al Mg Mg10 − − − − − f M ( M e V − ) Si Si Si P E γ ( MeV ) − − − − − f M ( M e V − ) S E γ ( MeV ) S E γ ( MeV ) S E γ ( MeV ) S E γ ( MeV ) − − − − − f M ( M e V − ) Cl E γ ( MeV ) Cl E γ ( MeV ) Cl E γ ( MeV ) Ar0 2 4 6 E γ ( MeV ) − − − − − f M ( M e V − ) Ar 0 2 4 6 E γ ( MeV ) K 0 2 4 6 E γ ( MeV ) E γ ( MeV ) AverageRIPL-3 averageSM discrete average SM quasicontinuum averageExponential fit
Figure 10. (Color online) M1 strength function of different sd shell nuclei. The bin width is ∆ E = . for the lightest nuclei, but merely turns flat. Hence,in phenomenological terms, an M E M n , γ ) reaction rates relevant tothe r process.Whilst there are experimental difficulties pre-venting definitive exclusions of the LEE with theOslo method, the data that exist support our presentfindings. It would be very interesting to study othernuclei in mid-shell regions, and preferably employ-ing experimental techniques enabling the extractionof the strength function to low gamma-ray energy.It is equally interesting to consider nuclei in the“shears regions”, where we expect the LEE to bemost significant. Neutron-rich Xe isotopes are a promising case in this regard, located as they arejust northwest of the doubly-magic Sn. An ex-periment has recently been carried out on
Xeat iThemba LABS, and analysis using the Oslomethod in inverse kinematics is underway [81]. Weeagerly await these experimental results.
ACKNOWLEDGMENTS
J.E.M., A.C.L., T.R., and F.L.B.G. gratefully ac-knowledge financial support through ERC-STG-2014 under grant agreement no. 637686. A.C.L.acknowledges support from the ChETEC COSTAction (CA16117), supported by COST (EuropeanCooperation in Science and Technology). Calcula-tions were performed on the Stallo and Fram high-performance computing clusters at the Universityof Tromsø, supported by the Norwegian ResearchCouncil. − − − − − f M ( M e V − ) Ni RIPL-3 SM discrete SM quasicontinuum Ni Ni Ni10 − − − − − f M ( M e V − ) Cu Cu Cu Zn0 2 4 6 E γ ( MeV ) − − − − − f M ( M e V − ) Zn 0 2 4 6 E γ ( MeV ) Ga 0 2 4 6 E γ ( MeV ) Ge 0 2 4 6 E γ ( MeV ) AverageRIPL averageSM discrete average SM quasicontinuum averageExponential fit
Figure 11. (Color online) M1 strength function of different f / pg / -shell nuclei. The bin width is ∆ E = . All calculations have been made publicly avail-able on Zenodo [67]
Appendix A: Issues with conversion of B(M1) valuesto strength function
We recently became aware of an issue with howshell model calculations are converted to γ -raystrength functions [82]. The conventional defini-tion of the strength function, as found in Ref. [83],is f M ( E γ , E i , J i , π i )= π h c (cid:104) B ( M ) (cid:105) ( E γ , E i , J i , π i ) ρ ( E i , J i , π i ) , (A1)where ρ ( E i , J i , π i ) is the partial level density and (cid:104) B ( M ) (cid:105) is the average transition strength of statesat excitation energy E i , spin J i and parity π i . Using that µ N = ( e ¯ h ) / ( m p c ) , the constant in front worksout to 16 π h c = . × − µ − N MeV − . (A2)However, in some works, the total level density hasbeen used in place of the partial. Since the totallevel density is ρ tot ( E x ) = ∑ J , π ρ ( E x , J , π ) , this in-troduces ( i ) an artificial overall enhancement of thestrength function and ( ii ) an arbitrary scaling de-pending on how many J , π combinations were in-cluded in the calculations. 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