Constraints on the dipole photon strength for the odd uranium isotopes

PHYSICAL REVIEW C 105, 024618 (2022) Constraints on the dipole photon strength for the odd uranium isotopes J. Moreno-Soto,1,*S. Valenta,2,†E. Berthoumieux,1A. Chebboubi,3M. Diakaki,3W. Dridi,4E. Dupont,1F. Gunsing,1,‡ M. Krtiˇ cka,2O. Litaize,3O. Serot,3O. Aberle,5V. Alcayne,6S. Amaducci,7J. Andrzejewski,8L. Audouin,9V. Bécares,6 V. Babiano-Suarez,10 M. Bacak,5,11,1M. Barbagallo,5,12 Th. Benedikt,13 S. Bennett,14 J. Billowes,14 D. Bosnar,15 A. Brown,16 M. Busso,17,18 M. Caamaño,19 L. Caballero-Ontanaya,10 F. Calviño,20 M. Calviani,5D. Cano-Ott,6A. Casanovas,20 F. Cerutti,5 E. Chiaveri,5,14 N. Colonna,12 G. Cortés,20 M. A. Cortés-Giraldo,21 L. Cosentino,7S. Cristallo,17,22 L. A. Damone,12,23 P. J. Davies,14 M. Dietz,24 C. Domingo-Pardo,10 R. Dressler,25 Q. Ducasse,26 I. Durán,19 Z. Eleme,27 B. Fernández-Domínguez,19 A. Ferrari,5P. Finocchiaro,7V. Furman,28 K. Göbel,13 A. Gawlik-Rami˛ega,8S. Gilardoni,5 I. F. Gonçalves,29 E. González-Romero,6C. Guerrero,21 S. Heinitz,25 J. Heyse,30 D. G. Jenkins,16 A. Junghans,31 F. Käppeler,32,§Y. Kadi,5A. Kimura,33 I. Knapová,2M. Kokkoris,34 Y. Kopatch,28 D. Kurtulgil,13 I. Ladarescu,10 C. Lampoudis,35 C. Lederer-Woods,24 S. J. Lonsdale,24 D. Macina,5A. Manna,36,37 T. Martínez,6A. Masi,5C. Massimi,36,37 P. Mastinu,38 M. Mastromarco,5E. A. Maugeri,25 A. Mazzone,12,39 E. Mendoza,6A. Mengoni,40 V. Michalopoulou,5,34 P. M. Milazzo,41 F. Mingrone,5A. Musumarra,7,42 A. Negret,43 R. Nolte,26 F. Ogállar,44 A. Oprea,43 N. Patronis,27 A. Pavlik,45 J. Perkowski,8L. Piersanti,17,22 C. Petrone,43 E. Pirovano,26 I. Porras,44 J. Praena,44 J. M. Quesada,21 D. Ramos-Doval,9 T. Rauscher,46,47 R. Reifarth,13 D. Rochman,25 M. Sabaté-Gilarte,21,5A. Saxena,48 P. Schillebeeckx,30 D. Schumann,25 A. Sekhar,14 A. G. Smith,14 N. V. Sosnin,14 P. Sprung,25 A. Stamatopoulos,34 G. Tagliente,12 J. L. Tain,10 A. Tarifeño-Saldivia,20 L. Tassan-Got,5,34,9P. Torres-Sánchez,44 A. Tsinganis,5J. Ulrich,25 S. Urlass,31,5G. Vannini,36,37 V. Variale,12 P. Vaz,29 A. Ventura,36 D. Vescovi,17 V. Vlachoudis,5R. Vlastou,34 A. Wallner,49 P. J. Woods,24 T. Wright,14 and P. Žugec15,¶ (n_TOF Collaboration) 1CEA Irfu, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France 2Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic 3CEA Cadarache, DES/DER/SPRC, F-13108 Saint-Paul-lez-Durance, France 4Laboratory on Energy and Matter for Nuclear Sciences Development (LR16CNSTN02), Technopark Sidi Thabet, 2020 Ariana, Tunisia 5European Organization for Nuclear Research (CERN), Geneva, Switzerland 6Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 7INFN Laboratori Nazionali del Sud, Catania, Italy 8University of Lodz, Lodz, Poland 9Institut de Physique Nucléaire, CNRS-IN2P3, Université Paris-Sud, Université Paris-Saclay, F-91406 Orsay Cedex, France 10Instituto de Física Corpuscular, CSIC - Universidad de Valencia, Valencia, Spain 11Technische Universität Wien, Austria 12Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy 13Goethe University, Frankfurt, Germany 14University of Manchester, Manchester, United Kingdom 15Department of Physics, Faculty of Science, University of Zagreb, Zagreb, Croatia 16University of York, York, United Kingdom 17Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia, Italy 18Dipartimento di Fisica e Geologia, Università di Perugia, Perugia, Italy 19University of Santiago de Compostela, Santiago de Compostela, Spain 20Universitat Politècnica de Catalunya, Barcelona, Spain 21Universidad de Sevilla, Seville, Spain 22Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Teramo, Teramo, Italy 23Dipartimento di Fisica, Università degli Studi di Bari, Bari, Italy 24School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 25Paul Scherrer Institut (PSI), Villigen, Switzerland *[email protected] †[email protected] ‡[email protected] §Deceased. ¶www.cern.ch/ntof. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 2469-9985/2022/105(2)/024618(14) 024618-1 Published by the American Physical Society J. MORENO-SOTO et al. PHYSICAL REVIEW C 105, 024618 (2022) 26Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany 27University of Ioannina, Ioannina, Greece 28Joint Institute for Nuclear Research (JINR), Dubna, Russia 29Instituto Superior Técnico, Lisbon, Portugal 30European Commission, Joint Research Centre (JRC), Geel, Belgium 31Helmholtz-Zentrum Dresden-Rossendorf, Dresdan, Germany 32Karlsruhe Institute of Technology, Campus North, IKP, 76021 Karlsruhe, Germany 33Japan Atomic Energy Agency (JAEA), Tokai-mura, Japan 34National Technical University of Athens, Athens, Greece 35Department of Physics, Aristotle University of Thessaloniki, University Campus, GR-54124, Thessaloniki, Greece 36Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy 37Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy 38Istituto Nazionale di Fisica Nucleare, Sezione di Legnaro, Legnaro, Italy 39Consiglio Nazionale delle Ricerche, Bari, Italy 40Agenzia nazionale per le nuove tecnologie (ENEA), Bologna, Italy 41Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy 42Dipartimento di Fisica e Astronomia, Università di Catania, Catania, Italy 43Horia Hulubei National Institute of Physics and Nuclear Engineering, Magurele, Romania 44University of Granada, Granada, Spain 45Faculty of Physics, University of Vienna, Vienna, Austria 46Department of Physics, University of Basel, Basel, Switzerland 47Centre for Astrophysics Research, University of Hertfordshire, Hatfield, United Kingdom 48Bhabha Atomic Research Centre (BARC), Mumbai, India 49Australian National University, Canberra, Australia (Received 24 August 2021; accepted 31 January 2022; published 24 February 2022) Background: The photon strength functions (PSFs) and nuclear level density (NLD) are key ingredients for calculation of the photon interaction with nuclei, in particular the reaction cross sections. These cross sections are important especially in nuclear astrophysics and in the development of advanced nuclear technologies. Purpose: The role of the scissors mode in the M1 PSF of (well-deformed) actinides was investigated by several experimental techniques. The analyses of different experiments result in significant differences, especially on the strength of the mode. The shape of the low-energy tail of the giant electric dipole resonance is uncertain as well. In particular, some works proposed a presence of the E1 pygmy resonance just above 7 MeV. Because of these inconsistencies additional information on PSFs in this region is of great interest. Methods: The γ-ray spectra from neutron-capture reactions on the 234U,236U,and238Unuclei have been measured with the total absorption calorimeter of the n_TOF facility at CERN. The background-corrected sum-energy and multi-step-cascade spectra were extracted for several isolated s-wave resonances up to about 140 eV. Results: The experimental spectra were compared to statistical model predictions coming from a large selection of models of photon strength functions and nuclear level density. No combination of PSF and NLD models from literature is able to globally describe our spectra. After extensive search we were able to find model combinations with modified generalized Lorentzian (MGLO) E1 PSF, which match the experimental spectra as well as the total radiative widths. Conclusions: The constant temperature energy dependence is favored for a NLD. The tail of giant electric dipole resonance is well described by the MGLO model of the E1 PSF with no hint of pygmy resonance. The M1 PSF must contain a very strong, relatively wide, and likely double-resonance scissors mode. The mode is responsible for about a half of the total radiative width of neutron resonances and significantly affects the radiative cross section. DOI: 10.1103/PhysRevC.105.024618 I. INTRODUCTION Nuclear level densities (NLDs) and photon strength functions (PSFs), also called γ-ray or radiation strength functions, represent average properties of the nucleus in the regime of excitation where individual levels and transition probabilities by γdecay are not readily accessible by experimental or theoretical means. They are key ingredients for statistical calculations of the reaction cross sections involving γrays via the Hauser-Feshbach approach [1], like inelastic scattering or neutron capture reactions. These cross sections are important for nuclear network calculations, for example in stellar nucleosynthesis models [2,3] or in nuclear technology applications, 024618-2 CONSTRAINTS ON THE DIPOLE PHOTON STRENGTH … PHYSICAL REVIEW C 105, 024618 (2022) usually relying on evaluated nuclear data libraries [4,5]or astrophysical databases [6]. Individual levels can be observed and their properties (spin, parity, widths, and eventually decay paths) determined almost exclusively only in two regions: at the lowest excitation energies and in the neutron resonance region just above the neutron separation energy Sn. The concept of NLD thus has to be used already from an excitation energy well below 1 MeV in odd U isotopes as the rapidly increasing number of levels prevents their observation. Neutron resonances then serve as an anchor point for the determination of the NLD parameters. The PSFs for different transition types and multipolarities describe the average transition probability between nuclear levels. Detailed information on PSFs below the neutron separation energy in actinides (needed for cross section calculations) has been a subject of several experimental studies during the last years. They included mainly data from the Oslo technique [7–9] and neutron capture experiments [10,11]. Moreover, there are also data from other experimental techniques including average resonance capture (ARC) [12]or nuclear resonance fluorescence (NRF) [13,14]. The two features that strongly influence the γdecay and neutron capture cross section are the energy dependence of the giant electric dipole resonance (GEDR) tail with a possible presence of the pygmy resonance and the collective M1 excitation known as the scissors mode (SC) [15–19]. A detailed understanding of these features is important also for the calculation of cross sections for short-lived nuclei. There are many available models of both the PSFs and NLD in the literature [20,21] and their validation as well as obtaining further information on the properties of these quantities is important. At present, literature sources differ on the properties of the scissors mode. In particular its integrated strength in 235Udeduced from NRF is about 3μ2 N[13], where μNstands for the nuclear magneton, while the analyses of Oslo data [7–9] provide strengths of 8–11 μ2 Nfor several actinides (consistently for even-even, odd, and odd-odd nuclei). The DANCE data [11] from the Los Alamos Neutron Science Center indicated a lower limit of 11μ2 Nfor odd uranium isotopes. The calculation therein showed that the presence of a SC with a strength of ≈15μ2 N is responsible for about half of the total neutron capture cross section. In the present paper we explore data from the 4πtotal absorption calorimeter (TAC) of the n_TOF facility at CERN for three uranium isotopes. Data for these nuclei have been taken previously and analyzed in terms of neutron capture cross sections for 234U[22,23], 236U[24,25], and 238U[26,27]. Most of the used experimental procedures are detailed in those references and we only summarize the main aspects here. We reanalyzed these data and extracted sum-energy and multistep cascade (MSC) spectra. These spectra were then confronted with predictions of statistical model simulations of γ-ray cascades based on several models for NLDs and PSFs. Such a comparison provides significant constraints on the NLD and PSF models. We found that models available in the literature needed to be improved. Based on extensive simulations we were able to find PSFs common for all three nuclei. A first intermediate report showed preliminary results on the nucleus 234U[28] while work on all three nuclei is reported FIG. 1. One hemisphere of the TAC showing the BaF2crystals and photomultipliers mounted in the support structure. The vacuum time-of-flight tube traversing the detector is split at the center of the TAC where the sample holder in air is positioned, surrounded by the neutron absorber of which only the lower half is shown. in Ref. [29]. Since then additional models have been used as well and the final results are presented here. An important conclusion of our analysis is a verification of results obtained from the analysis of MSC spectra from the DANCE detector [11], the 4πcalorimeter with 160 BaF2crystals operated at the Los Alamos Neutron Science Center. As the MSC spectra from the independent n_TOF experiment are used for the present extensive PSF analysis, such a verification forms an important confirmation of the employed method. The paper is organized as follows: the experimental setup and data reduction are described in Sec. II while simulations within the statistical model in Sec. III. Sections IV and V present results coming from the comparison of experimental spectra with their simulated counterparts and the comparison of integral quantities to available literature data, respectively. The conclusions are then given in Sec. VI. II. EXPERIMENT AND DATA REDUCTION A. Experimental setup The n_TOF neutron-time-of-flight facility at CERN [30,31] uses a white neutron spectrum produced by a pulsed proton beam of 20 GeV/csupplied by the Proton Synchrotron impinging on a lead spallation target. Neutron capture experiments on 234U,236U, and 238Uwere performed with the total absorption calorimeter (TAC) [27,32] consisting of 40 BaF2crystals, lined with carbon shells doped with 10B,ina 4πconfiguration surrounding the capture sample placed in the neutron beam. The high-efficiency calorimeter is placed at a distance of about 185 m from the spallation target in the time-of-flight station EAR1 [31]. In Fig. 1the detector setup shows one hemisphere of the TAC with the vacuum neutron flight tube traversing it. A sample holder is placed in the center of the TAC where the vacuum tube is interrupted by mylar windows. The lower 024618-3 J. MORENO-SOTO et al. PHYSICAL REVIEW C 105, 024618 (2022) TABLE I. Summary of the characteristics of the samples and experimental conditions for each measurement. Additional details are given in the text. 234U236U238U Mass (mg) 32.7 338 6125 Areal density (10−4atoms/b) 1.07 10.9 9.56 Canning Ti Al Al Sn≈Q(MeV) 5.297 5.126 4.806 Resolution (%) (0.9 MeV) 14.5 16.5 16.5 half of the surrounding neutron absorber, intended to reduce the main source of background produced by sample-scattered neutrons, is shown as well. The absorber for the 234Uexperiment consisted of an inert nonflammable 6Li-enriched salt produced from carbonic and dodecanedioic acids encapsulated by 0.5 mm of Al, while the absorber used for the 236Uand 238Uexperiments was produced as a self-supporting borated polyethylene containing 5% of 10B. Energy calibration and resolution determination of individual BaF2crystals were performed using spectra from 137Cs and 88Yradioactive sources. The crystal resolution was on average 14.5% for 0.9 MeV γrays at the time of the 234U(n,γ) measurement and slightly worsened to 16.5% during the 236U(n,γ) and 238U(n,γ) experiments. All three uranium samples were highly enriched (>99%). No further information on the isotopic composition of the 32.7 mg 234Usample was available, the measured spectra did not show any isotopic impurity. For the 338 mg 236Usample the enrichment was 99.85% and contained small amounts of 235U (0.05%) and 238U(0.1%). Both samples were provided by the Institute of Physics and Power Engineering in Obninsk as 10 mm diameter disk-shaped encapsulated pressed pellets of uranium oxide. The 6.125 g 238Usample supplied by JRCGeel was a highly enriched (<1 ppm234U,<11 ppm235U, and <1 ppm236U) metallic foil, encapsulated in 60 μmof aluminum and about 75 μm of Kapton, with a nearly rectangular shape of about 53.9×30.3mm 2, therefore fully covering the neutron beam diameter of approximately 20 mm. The effective area of 1621.22 mm2was determined with a microscope-based measurement system. Given the cross sections and the selected neutron energy ranges we have used in this work, we assumed that the impurities have a negligible impact on the extracted spectra. A summary of the most relevant sample and detector parameters is given in Table I. B. Data processing The signals from each individual BaF2crystal were sampled using digitizers at a rate of 5e8 samples/s. The signal consists of two components with decay times of 0.7 ns (fast) and 630 ns (slow). The fast and slow components allow the precise determination of the arrival time and the deposited energy, respectively. The ratio of their amplitudes then allows the discrimination of γrays from background αparticles caused by the natural radioactivity of the Ra contaminant in the BaF2crystals [22,33]. In the analysis we considered only signals corresponding to a deposited energy above a 250 keV threshold, which is slightly above the hardware one. The γ-ray signals within a coincidence window of 20 ns were considered to belong to the same TAC event. Each TAC event is characterized by (i) a corresponding neutron energy determined by the time-of-flight technique from the time of signal arrival, (ii) a number of firing crystals called crystal multiplicity m, and (iii) energies deposited in individual crystals, Ei. The time-of-flight spectra for the (n,γ) reactions on 234U,236U, and 238Uare shown in Fig. 2. 6 10 6 10×26 10×36 10×46 10×56 10×66 10×77 10 Time of Flight (ns) 2− 10 1− 10 1 )γU(n, 238 2− 10 1− 10 1 )γU(n, 236 2− 10 1− 10 1 ) -1 sμCount rate ( 234567891020304050607080 2 10 2 10×2 Neutron energy (eV) )γU(n, 234 FIG. 2. The time-of-flight spectra for the (n,γ) reactions on 234U,236U,and238Ufor m⩾2 TAC events. The top axis shows the corresponding neutron-energy scale. The blue regions are used for the analysis while the red ones correspond to the windows for the background subtraction; see text for details. 024618-4 CONSTRAINTS ON THE DIPOLE PHOTON STRENGTH … PHYSICAL REVIEW C 105, 024618 (2022) 0246 0 2 4 3 10× Counts/bin )γU(n, 234 0246 )γU(n, 236 0246 Sum energy (MeV) )γU(n, 238 m = 1 m = 2 m = 3 m = 4 5≥m FIG. 3. Sum energy spectra for the first resonance in (n,γ) reaction on 234U,236U,and238U, not corrected for background, for different crystal multiplicity criteria. The counts for m=1, attributed mainly to background events, go up as high as 120 ×103counts/bin. The Q values of the 234,236,238U(n,γ) reactions are 5.3, 5.1, and 4.8 MeV respectively, in practice equivalent to the Snvalues listed in Table I. Only cascades originating from well-resolved s-wave (Jπ=1 2 +) resonances with sufficient statistics were further used. To minimize the event pileup due to high local count rate and a strong dead time we excluded the peak regions of the resonances and used only regions indicated in Fig. 2. The eight resonances at 5.16, 31.1, 48.6, 77.4, 94.3, 111.1, 146.3, and 152.2 eV, seven at 5.45, 29.8, 34.1, 43.9, 71.5, 86.5, and 124.9 eV, and six at 6.67, 20.9, 36.7, 66.0, 102.6, and 116.9 eV from 234,236,238U(n,γ), respectively, were analyzed. In reality, the selected neutron-energy regions are still slightly influenced by the dead-time effects. The correction for dead-time and pileup effects for the TAC are far from trivial [34]. However, as this impact is very similar for all the regions we have decided not to correct the experimental spectra but instead consider this effect in the simulations. For this purpose we modeled the average observed distribution of consecutive detected events, which in the ideal case contains a step function representing the dead time, with a parameterizable sigmoid function and an exponential [22,29]. For a given neutron resonance two kinds of spectra were constructed from individual cascades for a given multiplicity mcriterion: the energy spectrum summed over all crystals Es=iEi, hereafter called sum-energy spectrum, and the multistep cascade (MSC) spectrum, which corresponds to the energies Eideposited in each of the mindividual crystals within a TAC event. The sum-energy spectra before background subtraction are shown in Fig. 3for the first resonance of each nucleus. In the case of complete cascade detection the Eswould be equal to the Qvalue of the reaction, Q= Sn+En≈Sn, with Enbeing negligible relative to Sn.The Compton scattering and the finite efficiency of the detectors makes a significant fraction of the cascades contribute to lower energies. At least for several multiplicities a peak near the Qvalue (with a width of a few hundreds keV given by detector energy resolution) with a sharp fall above is visible. The MSC spectra were constructed only from TAC events with deposited energy sum near Sn, specifically for Es=5.0–5.6MeV,4.9–5.3 MeV, and 4.5–4.8MeVfor the 234,236,238Usamples, respectively. Those windows encompass the resolution-broadened Qvalues and were also adjusted in order to have sufficient statistics in the MSC spectra. A bin width of 130 keV, which is close to the energy resolution of crystals for low Eγ, was used. For a background subtraction of spectra from individual resonances we used linear interpolation from neighboring off-resonance regions shown in Fig. 2; for details on background subtraction see Ref. [29]. All spectra for a given resonance were normalized by one common factor given by the integral of the m⩾2 sum-energy spectrum in the corresponding aforementioned Es range. The background corrected spectra from 234U(n,γ)resonances are shown in Fig. 4. Analogous figures for 236,238U(n,γ) can be found in the Supplemental Material [35]. The spectra may not necessarily be consistent within their counting uncertainties since there is an additional spread due to the Porter-Thomas fluctuations [36] of individual transition intensities. This additional spread is clearly observed and confirmed by a maximum likelihood fit assuming a normal distribution of the MSC intensity. The effect is mainly visible in the m=2 MSC spectra, for example in 234U(n,γ)inthe peaks around 0.7 and 4.5 MeV as shown in Fig. 4. In other energy domains it is to most extent masked by the uncertainties of individual spectra. For a comparison of experiment with predictions we decided not to use spectra from individual resonances. The restricted number of resonances does not allow one to reliably determine the properties of the distribution from a maximum likelihood fit as in Ref. [37]. We thus calculated the unweighted average and standard deviation of the set; see Fig. 4.Them=1 spectra are dominated by the background (see Fig. 3) and are not considered in the analysis. Spectra for m>4 do not show any interesting structures (see Fig. 4), and the intensity is rapidly decreasing with m; there are practically no TAC events with m⩾7. 024618-5 J. MORENO-SOTO et al. PHYSICAL REVIEW C 105, 024618 (2022) 0246 Sum energy (MeV) 0 10 20 30 40 m=5 20 40 60 80 m=4 50 100 m=3 20 40 60 3− 10× Counts/bin m=2 024 -ray energy (MeV)γ 0 50 100 4 e rgy ( MeV ) 0 5 0 m=5 Average spectra = 111.06 eV r E = 146.25 eV r E = 152.16 eV r E 50 100 50 10 0 20 40 60 80 = 5.16 eV r E = 31.13 eV r E = 48.56 eV r E = 77.38 eV r E = 94.29 eV r E m=4 20 40 60 V = 5 16 eV V V V V E E 2 0 4 0 m=3 5 10 15 20 3− 10× Counts/bin m=2 2 0 t s / bi n )γU(n, 234 FIG. 4. The sum-energy and MSC spectra for individual resonances and multiplicities m=2–5 after background subtraction for 234U(n,γ). The label “γ-ray energy” refers to the deposited energy in any crystal. The “Average spectra” band represents the average of resonances plus/minus one standard deviation representing the fluctuation among the resonances. The shaded interval depicts the Esrange used for the construction of MSC spectra. From the available information on TAC events we can construct also other observables, e.g., the commonly used multiplicity distribution of detected TAC events; see experimental results in Fig. 5. However, as this quantity is inherently contained in both the sum-energy and MSC spectra and was found to be less sensitive to the changes in NLD and PSF models, we do not concentrate on its comparison with simulated 2468 0 0.1 0.2 0.3 0.4 Probability )γU(n, 234 2468 )γU(n, 236 2468 Multiplicity )γU(n, 238 FIG. 5. Crystal multiplicity distribution for each uranium isotope from the same Esenergies as used for the construction of MSC spectra. The individual resonances after background subtraction are plotted following the color scheme in the legend of Fig. 4.The average spectra and the standard deviation are represented by the band. multiplicity distributions. Nevertheless, a few figures can be found in the Supplemental Material [35]. III. SIMULATIONS The sum-energy and MSC spectra arise from a complex interplay between the PSFs, NLD, and detector response. This fact prevents a direct extraction of the PSFs and/or NLD from our data. However, we can learn about these quantities by comparing the experimental spectra with their simulated counterparts based on various PSF and NLD models. This trial-and-error approach is the only reasonable way to test the available models. Attempts to fine tune the models then usually require extensive simulations. The resulting model combinations will in that case provide the best match with the experimental data but do not necessarily form a unique or optimum solution. A. Algorithms Individual γ-ray cascades were simulated within the statistical approach utilizing the Monte-Carlo DICEBOX code [38]. The fluctuations between artificial nuclei as generated by DICEBOX are depicted by the color ±1σbands when comparing the simulated spectra to their experimental counterparts. We simulated 20 artificial nuclei with 105cascades within each. Many different combinations of models of NLD and E1, M1, and E2 PSFs were used. As the description of γ decay within the statistical approach is surely inadequate at the lowest excitation energies, the information on discrete levels and their decay was taken from the ENSDF database [39–41] up to excitation energies of 820, 760, and 830 keV for the compound nuclei 235U,237U, and 239Urespectively. The TAC response to these cascades was obtained using the code based on the GEANT4 package [42]. We implemented the full geometry of the supporting structures, beam pipes, detectors, neutron absorber, and uranium samples in detail [43]. We implemented also the energy resolution of BaF2 crystals and the dead time process following the experimentally determined distribution of consecutive pulses [22,29]. Simulated spectra thus should be directly comparable to the measured ones. The normalization of simulated spectra was done in the same way as for the experimental ones. A precise statistical quantification of the agreement between the simulated and experimental spectra cannot be made without time-consuming simulations due to apriori unknown highly nontrivial correlations between individual bins of the MSC spectra. We tested a simple numerical scoring function of the spectral similarity or goodness of fit, but it was found to be inconclusive mainly because of the large associated uncertainties, except for obvious discordant cases. In this paper we mostly kept the PSF and NLD model combinations which were not in undebatable disagreement with the measured spectra. It is to be stressed that the predicted spectra are not sensitive to the absolute values of PSFs if the Eγ-dependent ratios of PSFs for different transition types are kept the same. So, we can rather probe the Eγdependence of the PSFs and their relative contributions than the absolute PSF values. The only 024618-6 CONSTRAINTS ON THE DIPOLE PHOTON STRENGTH … PHYSICAL REVIEW C 105, 024618 (2022) TABLE II. Parameters of PSF models used in simulations. The E1 PSF parameters correspond to the double-peaked GEDR energy E, width , and maximum cross section σ. The same quantities then characterize the double resonance SC and the SF resonances in the M1 PSF. MGLO(k) is used with a T=0.3MeV. E1 PSF M1 PSF Model EσEσEσEσEσ combination (MeV) (MeV) (mb) (MeV) (MeV) (mb) (MeV) (MeV) (mb) (MeV) (MeV) (mb) (MeV) (MeV) (mb) RIPL-3 [21] 11.11 1.12 243.3 13.41 4.98 426 6.61 4.00 2.35 Osloa,b[7] 11.40 4.20 572 14.40 4.20 1040 2.15 0.80 0.45 2.90 0.60 0.40 6.61 4.00 7.00 Osloa,c[7] 11.40 4.20 572 14.40 4.20 1040 2.00 0.80 0.40 2.80 1.20 0.30 6.61 4.00 7.00 DANCE [11] 11.28 2.48 325 13.73 4.25 384 2.15 0.80 0.60 2.90 0.60 0.53 6.61 4.00 1.50 MGLO(1.8) 10.90 2.30 358.0 13.96 4.75 459.0 2.15 0.80 0.98 2.90 0.60 0.82 6.61d4.00d3.05 aE1 PSF contained an additional Lorentzian at 7.3 MeV with width of 2 MeV and maximum cross section of 15 mb. b235,237U. c239U. dThe value was taken from systematics in the RIPL-3 database [21] and was not adjusted. quantity from simulations that depends on the absolute PSF values is the total radiative width γ. B. Tested models There are many models of PSFs and NLD available in the literature; see, e.g., reviews [20,21,44]. We decided to check our spectra against predictions based on a few of them which are either proposed in the recent reviews or describe relevant experimental data, namely, (i) one of the widely used model combinations available in the RIPL-3 database [21] consisting of the analytical generalized Lorentzian (GLO) model for E1 PSF in combination with the spin-flip (SF) Lorentzian for M1 PSF coupled with the constant-temperature (CT) NLD model [45], hereafter called RIPL-3, (ii) the PSF models proposed in the recent review [20] based on microscopic calculations combined with some phenomenological parts used in conjunction with microscopically Hartree-Fock-Bogoliubov (HFB) based plus combinatorial NLD [46], hereafter called IAEA-19, (iii) the original model interpretation of the Oslo PSF [7] and NLD [47], based on the (d,t)237Uand (d,p)239Udata, hereafter called Oslo, consisting of the enhanced generalized Lorentzian (EGLO) E1 model, the Lorentzian SC and SF M1 modes, and the CT NLD model from Ref. [47], (iv) the model combination, hereafter called DANCE, that reasonably described the MSC spectra in the U isotopes from DANCE experiment [11] consisting of the modified generalized Lorentzian (MGLO) E1 PSF model, the Lorentzian SC and SF M1 modes, and the CT NLD [45]. The parameters of the PSF resonance structures can be found in Table II. The parameters of the phenomenological parts of the IAEA-19 model combination are f0=1×10−10 MeV−4,E0=4MeVforE1 PSF and C=1×10−8MeV−3 and η=0.8MeV −1for M1 PSF and correspond to one of the options proposed in [20,48]. The PSF and NLD models for 239Uare shown in Figs. 6 and 7, respectively. The PSFs for two of these model combinations—Oslo and DANCE—were chosen such that they follow the exact form of the Brink hypothesis [49] saying that the PSFs are only a function of Eγ. This is achieved by describing the M1 modes by Lorentzians and by using a constant value of temperature Tin the E1 models. The GLO model of the RIPL-3 combination contains a dependence on nuclear temperature T, which is considered a function of the excitation energy in accord with Ref. [21]. The E1 PSF of IAEA-19 shows a dependence, albeit very weak, on the excitation energy through the phenomenological part. As is evident from the comparison in Sec. IV, the description of our experimental spectra is not perfect with any of these models. We thus decided to make an extensive search to find a model combination which would ideally provide a perfect agreement with the experimental data. Based on the promising results of the MSC spectra analysis from the DANCE experiment [11] and on the adjustability of the MGLO model [52] we decided to dominantly exploit this E1 PSF model in combination with the SF and SC M1 modes (each M1 resonance term was described by a Lorentzian in the PSF). This combination of PSFs was tested in conjunction with different NLD models: the CT NLD model using two different parametrizations [45,50], the Back-shifted Fermi gas (BSFG) NLD model [50], as well as the microscopic NLD [46]; see Fig. 7. The other E1 model possibilities were tested as well, albeit not in such detail; see the Supplemental Material [35]. There are free parameters in both the E1 and M1 PSF models. In the latter case we considered the energies, widths, and maximum cross sections of the SC resonance terms, as well as the maximum cross section of the SF resonance, as the free parameters. We almost exclusively used a doubleresonance SC; simulations with a single-resonance SC did not lead to an agreement comparable to that reached with the double-resonance one for all three isotopes. The MGLO model depends on a few parameters besides the GEDR energies E, widths , and maximum cross sections σ.The parameter k, see the Supplemental Material [35], was considered a free one in our analysis. As mentioned above, the 024618-7 J. MORENO-SOTO et al. PHYSICAL REVIEW C 105, 024618 (2022) FIG. 6. Photon strength functions of 239Uas a function of γ-ray energy for some of the models used in our simulations. The panels (a) and (b) display the E1andM1 PSFs, respectively. For the model parameters see Table II. The IAEA-19 E1andM1 PSFs were calculated by Goriely et al. [20] and the average resonance capture data were compiled by Kopecky [12]. For the temperature dependent GLO model the lower and upper curve correspond to excitation above ground state and decay from the capturing state, respectively. The panel (c) shows the sum of the E1 and the corresponding M1 PSF [from panels (a) and (b)] compared to the Oslo data from the (d,p)239Ureaction [7]. MGLO model further contains a dependence on nuclear temperature T, which can be either considered a free parameter or dependent on the excitation energy in accord with the original formulation [52]. Both options were considered in our analysis with a few values of constant Ttested. The GEDR parameters could be taken from the literature. However, with such a parametrization the MGLO model does not usually reproduce the photoabsorption data. We thus decided to fit the photoabsorption data from Ref. [53] in the range 9–16 MeV by the MGLO model for each set of kand Tparameters. The resulting GEDR parameters are listed in Table II and the FIG. 7. Spinand parity-summed level density for 239Uaccording to the CT and BSFG models and the HFB calculations [46]. The parameters of the CT and BSFG NLDs were taken from Refs. [45,50], denoted as (vEB06) and (vEB09) respectively. The experimental data from the Oslo method [7] are shown. Note that the differences at Snstem from different spin distributions in the models. The point corresponding to the s-wave resonance spacing [51]was converted using the spin distribution with the spin-cutoff parameter from Ref. [50]. The significant deviation of the Oslo data at higher energies hints to the normalization issue, which was discussed in detail by Ullmann et al. [11]. Supplemental Material [35]. These parameters slightly differ from the available ones coming from the fits of photoabsorption data with different E1 PSF models. The E2 transitions are believed to play a marginal role in the statistical decay of the nucleus. In all model combinations we used an E2 PSF representing the giant electric quadrupole resonance by a single Lorentzian with parameters from systematics in Ref. [54]. IV. COMPARISON OF SPECTRA The common features of the experimental spectra strongly indicate, under an assumption of smooth behavior of the NLD, a presence of a resonant structure in the PSF peaked between 2 and 3 MeV. These common features are namely (i) the shape of the continuum of m=2 sum-energy spectra with the maxima between 2 and 3 MeV, (ii) the change of the slope of m=3 sum-energy spectra at ≈3 MeV, (iii) the shape of m=2 MSC spectra showing an accumulation of intensity around their midpoints, and (iv) the bump at ≈2MeVin m=3 MSC spectra; see Fig. 4. Note also that the features (i) and (iii) change with different Snwhile features (ii) and (iv) are relatively stable. Within our trial-and-error approach we tested the model combinations from the literature (Sec. IV A), tried to fine-tune the PSFs to match the experimental spectra (Sec. IV B) and performed a number of tests to investigate the sensitivity of our data, (Sec. IV C). As mentioned above, we decided not to compare m=1 spectra. Although we checked the simulated spectra against their experimental counterparts for m=2–6, 024618-8 CONSTRAINTS ON THE DIPOLE PHOTON STRENGTH … PHYSICAL REVIEW C 105, 024618 (2022) 0246 Sum energy (MeV) 0 50 100 m=4 50 100 m=3 20 40 60 3− 10× Counts/bin m=2 024 -ray energy (MeV)γ 0 50 100 4 e rgy ( MeV ) 0 50 100 m=4 RIPL-3 Oslo IAEA-19 DANCE n_TOF data 20 40 60 RIPL 3 2 0 40 m=3 5 10 15 3− 10× Counts/bin m=2 1 5 n ts / bi n )γU(n, 234 FIG. 8. Comparison of 234U(n,γ) sum-energy (left) and MSC (right) spectra using the model combinations RIPL-3, IAEA-19, Oslo, and DANCE. The label “γ-ray energy” refers to the deposited energy in any crystal. for the sake of clarity we show only comparison for m=2–4. The comparison between the experimental and simulated multiplicity distributions for several model combinations can be found in the Supplemental Material [35]. A. Model combinations from literature The average experimental sum-energy and MSC spectra are for 234U(n,γ) compared to their simulated counterparts for the model combinations RIPL-3, IAEA-19, Oslo, and DANCE in Fig. 8. Analogous figures for 236,238U(n,γ) can be found in the Supplemental Material [35]. The figures indicate that predictions from none of the model combinations from literature match our spectra globally, although the DANCE and IAEA-19 combinations present the best results. In reality, the results depend on the specific nucleus and it might (accidentally) happen that a good agreement is achieved for some spectra. More pronounced differences are usually found in the MSC rather than the sum-energy spectra. A significant disagreement for the RIPL-3 model combination, namely an absence of structures in MSC spectra [aforementioned features (iii) and (iv)] is evidently a consequence of the complete absence of the SC. This is not really surprising as previous experiments, including analogous DANCE experiment [11], already prove that the SC is necessary for a reasonable description of γdecay of highly excited uranium compounds. Although the model combinations IAEA-19, Oslo, and DANCE contain the SC, their overall disagreement is mostly (but not only) caused by a too low contribution of the SC. In practice, the predicted MSC spectra are very sensitive to the SC position and strength distribution. We can conclude that the available model combinations do not provide a consistent satisfactory description of all our experimental spectra. We thus decided to search for a model combination better reproducing our spectra. Results of this search are presented below. B. Search for optimal PSFs Altogether, hundreds of simulations with different MGLO and scissors mode parametrizations in combination with the above-mentioned NLD models have been compared to the average experimental spectra. The search led to common E1 and M1 PSFs for all three nuclei, which globally describe all the spectra much better than any of the literature model combinations. The achieved level of agreement between simulations and experiment is presented in Fig. 9for one particular parameter set of the MGLO model with k=1.8 and fixed T=0.3 MeV in combination with the CT NLD from [50]. The remaining parameters of the PSFs are listed in Table II. As mentioned in Sec. III A simulated spectra are completely independent of absolute PSF scales providing that the Eγ-dependent ratios of the PSFs for the involved transition types are fixed. An agreement comparable to that in Fig. 9 can thus be achieved by an infinite number of PSF sets. One possibility is to multiply dipole PSFs by the same constant factor,1but such a scaling affects the ability to describe the photoabsorption data as well as the total radiative width γ, which is discussed below. Another possibility is to change the parameter kof the MGLO E1 model with a simultaneous adjustment of the SC parameters. The photoabsorption data can be well fitted by the MGLO model with T=0.3MeV for a range of k≈1.0–4.0.2The influence of these changes on γvalues is presented below. Additional comparisons of simulated to experimental spectra with different values of k, together with the PSF parameters, are available in the Supplemental Material [35]. There is apriorino reason for using T=0.3MeV.However, data from the Oslo method usually yield a value of T≈ 0.2–0.4MeV[7,8,47] and the NLD calculations of Hilaire et al. [55] result in similar Trange for 238U. We checked T in this range in the initial stage of our analysis and found that a value close to T=0.3 MeV is reasonable. Hence we fixed it and varied the parameter kof the MGLO model and parameters of the SC peaks. In addition, similar quality of agreement was achieved also with the MGLO model using excitation energy dependent T; see the Supplemental Material [35]. With this choice the MGLO model does not exactly follow the Brink hypothesis and the PSF shapes (for the decay from different excitation energies) are thus not necessarily similar to the constant-T MGLO model. For different values of ka good reproduction 1The corresponding adjustment of E2 PSF is not a simple scaling. 2For k=1.0 the MGLO model is identical to the GLO model. 024618-9