Electric and magnetic dipole strength in 112,114,116,118,120,124Sn
S. Bassauer, P. von Neumann-Cosel, P.-G. Reinhard, A. Tamii, S. Adachi, C.A. Bertulani, P.Y. Chan, A. D'Alessio, H. Fujioka, H. Fujita, Y. Fujita, G. Gey, M. Hilcker, T.H. Hoang, A. Inoue, J. Isaak, C. Iwamoto, T. Klaus, N. Kobayashi, Y. Maeda, M. Matsuda, N. Nakatsuka, S. Noji, H.J. Ong, I. Ou, N. Pietralla, V.Yu. Ponomarev, M.S. Reen, A. Richter, M. Singer, G. Steinhilber, T. Sudo, Y. Togano, M. Tsumura, Y. Watanabe, V. Werner
EElectric and magnetic dipole strength in , , , , , Sn S. Bassauer, ∗ P. von Neumann-Cosel, † P.-G. Reinhard, A. Tamii, S. Adachi, C. A. Bertulani, P. Y. Chan, A. D’Alessio, H. Fujioka, H. Fujita, Y. Fujita, G. Gey, M. Hilcker, T. H. Hoang, A. Inoue, J. Isaak,
1, 3
C. Iwamoto, T. Klaus, N. Kobayashi, Y. Maeda, M. Matsuda, N. Nakatsuka, S. Noji, H. J. Ong,
10, 3
I. Ou, N. Pietralla, V. Yu. Ponomarev, M.S. Reen, A. Richter, M. Singer, G. Steinhilber, T. Sudo, Y. Togano, M. Tsumura, Y. Watanabe, and V. Werner Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany Institut f¨ur Theoretische Physik II, Universit¨at Erlangen, D-91058 Erlangen, Germany Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429, USA Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan RIKEN, Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, 351-0198 Wako, Saitama, Japan Department of Applied Physics, Miyazaki University, Miyazaki 889-2192, Japan Department of Communications Engineering, Graduate School of Engineering,Tohoku University, Aramaki Aza Aoba, Aoba-ku, Sendai 980-8579, Japan National Superconducting Cyclotron Laboratory, MichiganState University, East Lansing, Michigan 48824, USA Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China Okayama University, Okayama 700-8530, Japan Department of Physics, Akal University, Talwandi Sabo, Bathinda Punjab-151 302, India Department of Physics, Rikkyo University, Tokyo, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, University of Tokyo, Tokyo 113-8654, Japan (Dated: July 14, 2020)
Background:
Inelastic proton scattering at energies of a few hundred MeV and very forward anglesincluding 0 ◦ has been established as a tool for the study of electric and magnetic dipole strengthdistributions in nuclei. The present work reports a systematic investigation of the chain of stableeven-mass tin isotopes. Methods:
Inelastic proton scattering experiments were performed at the Research Center for Nu-clear Physics, Osaka, with a 295 MeV beam covering laboratory angles 0 ◦ − ◦ and excitation energies6 −
22 MeV. Cross sections due to E M E M E Results:
Total photoabsorption cross sections derived from the E M γ, xn ) experiments in the energy region of theIsoVector Giant Dipole Resonance (IVGDR). The widths of the IVGDR deduced from the present datawith a Lorentz parameterization show an approximately constant value of about 4.5 MeV in contrast tothe large variations between isotopes observed in previous work. The IVGDR centroid energies are ingood correspondence to expectations from systematics of their mass dependence. Furthermore, a studyof the dependence of the IVGDR energies on bulk matter properties is presented. The E γ, γ (cid:48) ) experiments on , , , Snin the energy region between 6 and 7 MeV, where also isoscalar E Sn. Athigher excitation energies large differences are observed pointing to a different nature of the excitedstates with small ground state branching ratios. The isovector spin- M a r X i v : . [ nu c l - e x ] J u l Conclusions:
The present results contribute to the solution of a variety of nuclear structure problemsincluding the systematics of the energy and width of the IVGDR, the structure of low-energy E M I. INTRODUCTION
Inelastic proton scattering at energies of a fewhundred MeV and very forward angles includ-ing 0 ◦ has been established in recent years as anew spectroscopic tool for the investigation ofelectric and magnetic dipole strength distribu-tions in nuclei [1]. Although the ( p, p (cid:48) ) reactionis rather non-selective in general exciting elec-tric and magnetic modes alike, in the particularkinematics of very small momentum transfer aselective excitation of E M i ) theincident beam is relativistic and Coulomb ex-citation dominates the cross sections [2], and( ii ) the effective proton-nucleus interaction [3] isdominated by isovector spinflip transitions withorbital angular momentum transfer ∆ L = 0, i.e.the analog of Gamow-Teller (GT) transitions.At present, such experiments at scatter-ing angles very close to zero degrees can beperformed at the Research Center for Nu-clear Physics (RCNP), Japan [4] and at theiThemba Laboratory for Accelerator Based Sci-ences (iThemba LABS), South Africa [5]. Dis-persion matching between the beams and themagnetic spectrometers used to detect the scat-tered particles allows for high-resolution mea-surements of the order ∆ E/E = (1 − × − .Here, we report the results of a study of thestable tin isotopes , , , , , Sn per-formed at RCNP. A decomposition of the dom-inant E M ∗ [email protected] † [email protected] the much simpler measurement of cross sec-tions using an unpolarized beam and employthe MDA thereof is sufficient.The results allow to address a variety ofnuclear structure problems of current interest.Low-energy electric dipole strength in nucleiwith neutron excess, commonly termed PygmyDipole Resonance (PDR), is currently a subjectof intense experimental and theoretical activi-ties [10, 11]. It occurs at energies well belowthe IVGDR and exhausts a considerable frac-tion (up to about 10%) of the photoabsorptioncross sections in nuclei with a large neutron-to-proton ratio [12–15]. The properties of themode are claimed to provide insight into theformation of a neutron skin [13, 16–19] andthe density dependence of the symmetry en-ergy [13, 20–22], although this is questioned[19, 23, 24]. Furthermore, dipole strength in thevicinity of the neutron threshold S n has an im-pact on neutron-capture rates in the astrophysi-cal r -process [25–27]. A study of Sn revealeda dramatic difference of the low-energy isovector E p, p (cid:48) ) [28] and( γ, γ (cid:48) ) [29] reactions, respectively. The presentwork establishes this as a general phenomenonfor the chain of stable even-mass tin isotopesand discusses implications for the structure ofthe PDR.Most of the information on photoabsoptioncross sections in heavy nuclei stems from twomethods, viz. ( γ, γ (cid:48) ) [30] and ( γ, xn ) [31] re-actions. Both rely on the measurement of theemission probability from the excited state andthus on knowledge of the branching ratio of theparticular decay. In contrast, the ( p, p (cid:48) ) crosssections are directly related to the photoab-sorption cross sections. The experiments coveran excitation energy region from well below S n across the IVGDR, thus avoiding the difficultiesof matching results from the two different ex-perimental techniques, particularly pronouncednear S n . The IVGDR in stable tin isotopes wasinvestigated in a series of ( γ, xn ) experiments bydifferent laboratories [31–37]. The present worksheds new light on the significant differences ob-served in the energy region of the IVGDR.The energy region studied in the present ex-periments also covers the major part relevantto a determination of the nuclear electric dipolepolarizability [38]. There is renewed interestinto the polarizability because Energy DensityFunctional (EDF) theory [39] predicts a cor-relation with the neutron skin thickness [40]and leading parameters of a Taylor expansionof the symmetry energy around saturation den-sity [19, 41, 42]. This provides important con-straints for the Equation of State (EoS) ofneutron-rich matter, a major topic of currentnuclear structure research [43] important for anunderstanding of astrophysical events like core-collapse supernovae [44], the formation of neu-tron stars [45], or neutron star mergers [46].The polarizability in the chain of proton-magictin nuclei is of particular interest because theunderlying structure changes little between neu-tron shell closures N = 50 and 82. Two dif-ferent driving agents for the evolution of thedipole polarizability are conceivable, viz. neu-tron excess and the general trend with massnumber A (i.e., the radius) both dependent onthe symmetry energy. Accordingly, a varietyof model calculations have been performed forthis case attempting to explore this connection[16, 17, 27, 41, 47–57]. Including a model-aidedcorrection for the high-energy part of the exci-tation spectrum, the systematics of the dipolepolarizability in the stable tin isotope chain areextracted from the present data. A partial ac-count of this work has been given in Ref. [58].Finally, the data provide new results on M M γ -strength functions utilized for physics of reac-tor design [62] or the modeling of reaction crosssections in large-scale nucleosynthesis networkcalculations [63]. Since the mode is related to transitions between spin-orbit partners, it pro-vides information on the evolution of single-particle properties leading to new shell closuresin neutron-rich nuclei [64]. Furthermore, theIVSM1 resonance is the isospin-analog of theGT resonance [65] and thus provides insight intothe long-standing problem of quenching of theGT strength [66, 67]. Data in heavy sphericalnuclei are scarce, essentially limited so far to Zr [68] and
Pb [69, 70].This article is organized as follows. SectionII yields information on the experiment and thedata analysis including the techniques used todetermine the unknown isotopic enrichment ofsome of the targets. Section III provides detailsof the MDA and the resulting multipole decom-posed cross section spectra. Section IV presentsthe conversion to photoabsoprtion cross sectionsand their comparison to previous work. It alsoincludes an analysis of the sensitivity of theIVGDR centroid energies to bulk parameters ofnuclear matter. The relevance of the new re-sults on E M II. EXPERIMENT AND DATAANALYSISA. Experimental details
The inelastic proton scattering experimentswere performed in 2015 and 2017 at RCNP.In 2015, , , Sn and with lesser statistics , Sn were measured. In the second ex-perimental campaign in 2017, , , Sn weremeasured again to improve statistics. Addition-ally, data on , , , Sn were taken. Themeasurements used the Grand Raiden spec-trometer [71]. Data were taken at central spec-trometer angles of 0 ◦ , 2.5 ◦ and 4.5 ◦ . Typicalbeam currents were 2 −
20 nA, depending on thespectrometer angle. The unpolarized incidentproton beam had an energy of 295 MeV. Dis-
TABLE I. Targets used during the experiments.Given are the areal density ρ x, the enrichment andthe purpose of the corresponding target.Target ρ x Enrichment Purpose(mg/cm ) (%) Sn 3.38 90.2(1.4) main target
Sn 10.3 95.1( <
1) calibration
Sn 7.51 87.1( <
1) main target
Sn 4.98 95.5( <
1) main target
Sn 4.65 97.8( <
1) main target
Sn 4.50 86(7) main target
Sn 6.50 98.4( <
1) consistency check
Sn 5.00 97.0( <
1) main target
Sn 4.67 97.4( <
1) main target
Au 1.68 100 beam tuning Mg 1.16 unknown energy calibration Ni 100.1 unknown sieve slit C 1.01 98.9 energy calibrationC H − beam tuning persion matching and background optimizationat 0 ◦ were performed following the proceduresdescribed in Ref. [4]. The energy resolutionsachieved varied between 30 and 40 keV (FullWidth at Half Maximum, FWHM). At the endof both experimental campaigns sieve slit mea-surements were made with a thick Ni targetto obtain precise angle calibrations. Addition-ally, elastic scattering data for all investigatedtin isotopes were taken in the first experimentalcampaign.A summary of the used targets is given inTab. I. All tin targets were highly enriched self-supporting metallic foils with areal densities be-tween 3.4 and 7.5 mg/cm . The uncertainties ofthe target enrichment are quoted by the supplierto be better than 1 %. However, in some casesthe enrichment was unknown. In the case of Sn, a second thicker target with a known en-richment was used to determine the enrichmentof the thinner target, but could only be mea-sured in achromatic mode with correspondingreduced resolution due to its limited extensionand the high areal density of 10.3 mg/cm . Af-ter folding to obtain the same energy resolution,the abundance was determined to 90.2(1.4)% bynormalization of the two spectra. The enrich- ment of Sn was estimated from the systemat-ics of the IVGDR after conversion to photoab-sorption cross sections. A Lorentzian fit showsa smooth dependence of the centroid energy andwidth on the mass number as discussed belowin Sec. IV C. By interpolating these integratedvalues, the enrichment for
Sn was determinedto 86(7)%. A presumably enriched
Sn targetwas also measured. However, the IVGDR prop-erties deviated significantly from the systemat-ics, and low-energy spectra taken at larger scat-tering angles showed a broad bump in the en-ergy region 2 − + states in Sn. Bothfindings suggested a natural isotopic composi-tion. Thus, the spectra were discarded fromfurther analysis. Data for
Sn were taken tocheck the consistency with a previous experi-ment of the same type [8, 28].After each measurement of a main target forone hour, a short run with C was performedfor energy calibration and to account for possi-ble energy shifts of the beam. Further data forthe energy calibration were taken using Mgand polyethylene (C H ) targets. The arealdensities of the tin targets quoted by the man-ufacturer were remeasured and the correspond-ing uncertainties were determined to be around5 %. B. Particle Identification
A distinction of protons from other ejectilescan be achieved by investigating the depositedenergy in the plastic scintillator trigger detec-tors. Furthermore, the particles can also bediscriminated by their time of flight (ToF).The ToF information is obtained from the trig-ger signal generated by one of the scintillatorsand from the radio frequency of the AzimutallyVarying Field (AVF) cyclotron. To improve theparticle identification, the ToF information waslinearly corrected to make it independent of thehorizontal position x fp in the focal plane and ofthe horizontal scattering angle θ fp .In Fig. 1, the energy loss ∆ E in the plasticscintillator is plotted against the corrected time
400 600 800 1000 1200
ToF c (channel) ∆ E ( c h a nn e l ) P r o t o n s D e u t e r o n s T r i t o n s P r o t o n s D e u t e r o n s T r i t o n s FIG. 1. Particle identification via the correlationof energy loss and corrected time of flight (ToF c ).Two beam bunches are shown. The time differencebetween the two bunches corresponds to a beampulse period of about 60 ns. of flight ToF c . The proton scattering events,framed by a two-dimensional rectangular gate,can be clearly identified. Predicted regions fordeuteron and triton events (e.g. from ( p, d ) and( p, t ) reactions) are also indicated. However,neither deuterons nor tritons were observed inthis experiment. C. Angle calibration
To obtain a precise angle calibration, sieveslit measurements were performed with a thick Ni target under a spectrometer angle of 16 ◦ .In order to investigate the dependence of thescattering angle on the horizontal position atthe focal plane, five different magnetic field set-tings were measured covering the entire momen-tum acceptance of the spectrometer. Addition-ally, the vertical beam position at the target waschanged by 0 and ± θ f p ( d e g ) − −
250 0 250 500 x k (mm) C o u n t s / mm − −
250 0 250 500 x c (mm) FIG. 2. Focal plane spectra of the C( p, p (cid:48) ) reac-tion before (left) and after (right) the aberrationcorrection described in the text. D. Energy calibration
In order to achieve an optimum energy resolu-tion, the correlation between the horizontal po-sition x k and the horizontal scattering angle atthe focal plain θ fp due to the ion-optical prop-erties of the Grand Raiden Spectrometer [71]needs to be corrected. On the left side of Fig. 2data for C are shown in the θ fp − x k plane aswell as their projection on the abscissa. Onecan clearly see the most prominently excitedstates in C at 7.6, 12.7 and 15.1 MeV (fromleft to right). The curvature visible in the two-dimensional correlation leads to an asymmetricline shape in the projected energy spectrum dis-torting the resolution. A two-dimensional least-squares fit was performed to account for this.The result is depicted on the right side of Fig. 2,where x c denotes the corrected position on thefocal plane. The resolution is improved consid-erably from about 180 keV to 30 keV (FWHM).Excitation energies were determined from asecond-order polynomial fit of the focal planeposition of well-known transitions in the cali-bration spectra determined assuming Gaussianline shapes. Using these calibration functions,the reference energies of the known transitionscould be reconstructed to ± E. Background subtraction
The main contribution of the experimentalbackground at very forward angles stems frommultiple scattering of incident protons in thetarget material. Scattering off the beam pipesor slits also contributes occasionally, especiallyduring the 0 ◦ measurements. Due to the statis-tical nature of multiple scattering, a flat distri-bution of background events is expected on thefocal plane in non-dispersive direction y fp , whiletrue events are concentrated around y fp = 0.However, the operation of the GrandRaiden Spectrometer in the so-called underfo-cus mode [72] necessary to improve the resolu-tion of the vertical angle leads to a dependenceof y fp on the vertical scattering angle φ fp as il-lustrated on the left side of Fig. 3. Hence, beforethe background can be determined, a correctionof y fp needs to be carried out to restore the fo-cusing condition at the focal plane. This can beachieved with a multidimensional least-squaresfit as a function of the position and scatteringangles plus a correction for the vertical position, − − φ f p ( d e g ) -1.0-0.50.00.51.0 − −
25 0 25 50 y f p (mm) C o u n t s / mm − −
25 0 25 50 y c (mm) FIG. 3. Correlation of the position ( y fp ) in the non-dispersive direction and the vertical angle ( φ fp ) be-fore (left) and after (right) the restoration of thefocusing condition at the focal plane. The verticalangle was corrected in such a way that the back-ground events are distributed symmetrically around y c = 0. − − −
20 0 20 40 y c (mm) − − φ c ( d e g ) − − −
20 0 20 40 60 y c (mm) − −
20 0 20 40 60 y c (mm) FIG. 4. Background subtraction procedure usingthe correlation of the position y c in non-dispersivedirection and the vertical angle φ fp . The two-dimensional gate including true and backgroundevents is indicted by the black rectangle. To de-termine the background, the data were shifted by aconstant value to the left and right in y c direction. see Ref. [4].The effect of the correction can be seen inFig. 3, where the correlation between y fp andthe vertical angle φ fp before and after therestoration of the focusing condition at the fo-cal plane is compared. The corrected spectrumexhibits the expected flat background distribu-tion.After the correction, the background was de-termined in the following way. Three data setswere generated as illustrated in Fig. 4. In thefirst data set a gate was set to contain the trueplus the background events. The second andthe third data sets were analyzed in exactly thesame way as the first one, except that the datawere shifted along the y c axis by a constantvalue. After the shift, the gate only containedbackground events.The background events from the shifted datasets were then averaged and finally subtractedfrom the first data set. The energy spectra cor-responding to the three data sets are displayedin the upper part of Fig. 5 for the example of Sn measured at 0 ◦ , where the blue histogramcorresponds to true-plus-background and theorange and green histograms to the backgroundspectra after the shift to left and right in Fig. 4,respectively. As expected, the pure backgroundspectra from the two shifted data sets are iden-tical within statistical uncertainties. The lowerpart of Fig. 5 present a background-free spec-trum after subtraction of the averaged contri- C o u n t s / M e V Sn (True + BG)
Sn (BGL)
Sn (BGR)
Energy (MeV) C o u n t s / M e V Sn(p,p’)
FIG. 5. Top: Excitation energy spectra of
Snmeasured at 0 ◦ corresponding to the three datasets from Fig. 4 true-plus-background (blue), back-ground from shift to the left (orange), and tothe right (green). Bottom: Background-subtractedspectrum. bution from the orange and green spectra. F. Cross sections and uncertainties
Absolute double differential cross sectionswere determined from the experimental param-eters: collected charge, target properties, driftchamber efficiency, spectrometer solid angle anddata acquisition dead time. For the proceduresto extract these quantities from the raw datasee Ref. [73].The total cross section uncertainties were cal-culated taking statistical and systematic uncer-tainties in quadrature. Major contributions tothe systematic uncertainties originate from thesolid angle determination (4 − < III. MULTIPOLE DECOMPOSITION
An MDA of the measured spectra has beenperformed to extract E M A. Experimental spectra
The double differential cross sections ex-tracted as described in the previous Section aresummarized in Fig. 6. Data at 4 . ◦ are miss-ing for Sn due to the lack of beam time. For
Sn, data at 4 . ◦ were only taken in the firstexperimental campaign. Therefore, the excita-tion energy spectrum extends only up to about23.5 MeV, due to different magnetic field set-tings. In all isotopes, the GDR can be clearlyidentified around 15 MeV. In the PDR regionbetween 6 and 10 MeV, a structure can be seenbecoming gradually more pronounced for heav-ier isotopes culminating in Sn, where evendistinct peaks are formed. The typical decreaseof the cross section with increasing angle due todominant Coulomb excitation is apparent bothin the PDR and GDR energy regions.
B. Theoretical input
Theoretical angular distributions of the dif-ferential cross sections for different multipolar-ities were calculated using the code DWBA07[80]. Transition amplitudes and single-particlewave functions obtained from QuasiparticlePhonon Model (QPM) calculations of the typedescribed in Refs. [6, 28] were used as input.The parameterization of Love and Franey [3]was employed to describe the effective nucleon-nucleon interaction. An example of the angu-lar distributions of different multipolarities is Sn Sn d σ d Ω d E ( m b / s r / M e V ) Sn Sn Sn Energy (MeV) Sn FIG. 6. Double differential cross sections ofthe , , , , , Sn( p, p (cid:48) ) reactions at E =295 MeV for spectrometer angles Θ = 0 ◦ (blue),Θ = 2 . ◦ (orange) and Θ = 4 . ◦ (green). θ (deg) − − d σ d Ω ( a r b . u n i t s ) Sn(p,p’)
E1(GDR)E1(PDR)M1 E2M2E3
FIG. 7. Angular distributions of different multi-polarities calculated with the code DWBA07 andQPM transition densities for the
Sn( p, p (cid:48) ) reac-tion in the angular range 0 ◦ − ◦ . The maxima ofthe curves are normalised to unity. shown in Fig. 7 for the case of Sn.The shapes suggest that E M E M E
3, are only relevant for larger angles in theexperimentally studied range. The theoreticalcurves of Fig. 7 were also used for the MDA ofall other tin isotopes, since the underlying struc-ture for the tin isotopes studied in this work isvery similar and the angular distributions of col-lective modes show a weak dependence on massnumber. They were, however, corrected for theslightly different recoil term depending on theisotope masses and convoluted with the experi-mental angular resolution.
C. Subtraction of the ISGMR and ISGQR
Since the number of data points available islimited to 15 (5 per spectrometer angle), thenumber of multipolarities in the MDA must alsobe limited to avoid ambiguities. One particularproblem is the excitation of the IsoScalar Gi-ant Monopole Resonance (ISGMR), which hasan angular distribution similar to the E M Energy (MeV) S t r e n g t h (f m / M e V ) ISGMR
Sn( α , α ’)
10 15 20 25 30
Energy (MeV) S t r e n g t h (f m / M e V ) ISGQR
Sn( α , α ’) FIG. 8. Strength distribution of the ISGMR (top)and ISGQR (bottom) in
Sn from α scatteringexperiments [78]. Lorentzian fits in the resonanceregion are shown as orange curves. the MDA. Experimental information on thesemodes is available from ( α , α (cid:48) ) experiments [78]for all tin isotopes in question. The correspond-ing strength distributions for the example of Sn are presented in Fig. 8. The orange curvesare Lorentzian fits in the resonance region. Wenote that only the Lorentzians were used for thesubtraction procedure described below, becauseat higher excitation energies contributions fromcontinuum scattering are included.The contribution of the ISGMR and ISGQRto the proton scattering cross sections can beestimated with the following approach [81]d σ dΩ ( θ, E x ) = d σ dΩ ( θ ) DW BA IS ( Eλ )( E x ) exp IS ( Eλ ) th , (1)where IS ( Eλ )( E x ) exp are the isoscalar strengthdistributions from α scattering and IS ( Eλ ) th the theoretical strength from QPM calculationswith λ = 0 for ISGMR and λ = 2 for IS- θ (deg) d σ d Ω ( m b / s r ) Sn(p,p’)
ISGMR ISGQR
FIG. 9. Theoretical ( p, p (cid:48) ) cross sections of theISGMR and ISGQR in
Sn calculated with theDWBA07 code.
GQR, respectively. Equation (1) makes use ofthe fact that inelastic proton scattering at en-ergies of a few hundred MeV is a direct processand one can assume proportionality betweenthe strength and the cross sections. The the-oretical strength distributions were calculatedwithin the QPM and the strongest E E Sn are presented for two an-gles at 0 . ◦ and 5 . ◦ . They are rather smallfor the very forward angle. The monopole con-tribution is more important but never exceeds5 %. For larger angles however, a considerablecontribution from the ISGQR is found reaching25 % at the maximum, while the ISGMR con-tribution is negligible. After the subtraction ofthe ISGMR and ISGQR contributions a bumparound 13 MeV can still be seen for the 5 . ◦ data. This suggests that the absolute cross sec-tion of the ISGQR might be underestimated,though possible contributions from higher mul-tipolarities, such as M2 and E3, were not con-sidered yet which could possibly explain the re-maining bump. Since all higher multipoles show0 d σ d Ω d E ( m b / s r / M e V ) θ = ◦ Sn(p,p’)
Sn(p,p’),GMR/GQR ex.ISGMRISGQR
Energy (MeV) d σ d Ω d E ( m b / s r / M e V ) θ = ◦ Sn(p,p’)
Sn(p,p’),GMR/GQR ex.ISGMRISGQR
FIG. 10. Spectra of the
Sn( p, p (cid:48) ) reaction before(blue) and after (orange) subtraction of the ISGMR(green) and ISGQR (red) contributions for two dif-ferent angles. a similar angular distribution at larger angles(above 3 ◦ in the present case) in Fig. 7, possibleremaining ISGQR contributions are accountedfor by allowing one representative multipolarity λ > D. Continuum background
Besides the excitation of electric and mag-netic resonances the spectra also contain a con-tinuum part, which dominates the spectra atenergies above the IVGDR and needs to betaken into account for in the MDA. It is be-lieved to result mainly from quasifree scattering (QFS), although other contributions are not ex-cluded. The QFS process occurs only at ener-gies above the particle thresholds. In Ref. [82], aphenomenological parameterization was deter-mined for the ( p, p (cid:48) ) reaction on
Pb and asimilar approach was used in this work based onthe
Sn spectra. The nucleus
Sn was cho-sen because it is the heaviest measured nucleuswith data available for all three measured an-gles in the high excitation energy region, wherepossible contributions from the high-energy tailof the IVGDR are negligible. The data were an-alyzed in 1 MeV bins to reduce statistical fluc-tuations, and angular distributions in the en-ergy region between 22.5 and 25.5 MeV wereextracted. The angular distributions were thenfitted with polynomial functions of second or-der. Since these were identical within error barsfor all bins in the selected energy region, an av-erage polynomial functiond σ dΩ ( θ ) BG = 5 . − . θ + 0 . θ . (2)was determined for the background component.The upper part of Fig. 11 displays the Sndata used. The energy bins chosen for the an-gular distributions are indicated by the verticaldashed lines. In the lower part, the angular dis-tributions for the three energy bins are showntogether with the fit given in Eq. (2). For bet-ter visibility, they are shifted relative to eachother by a constant (2 mb/sr). Note that only4 angular gates were applied to the data takenat finite spectrometer angles because of limitedstatistics. Equation (2) describes all data welland also scales well with the results of a similaranalysis of the
Pb data [7] if the mass ratiois taken into account.
E. Results
For the MDA all spectra were rebinned to200 keV and the ISGMR and ISGQR contribu-tions were subtracted as described in Sec. III C.Experimental angular distributions of the dif-ferential cross sections for each bin were thendetermined and the data were fitted by means1
Energy (MeV) d σ d Ω d E ( m b / s r / M e V ) Sn(p,p’), 0 ◦ Sn(p,p’), 2.5 ◦ Sn(p,p’), 4.5 ◦ θ (deg) d σ d Ω ( m b / s r ) E x =
23 MeVE x =
24 MeVE x =
25 MeV
FIG. 11. Top: Excitation energy spectra of
Snand excitation energy bins (vertical dashed lines)used to determine a parameterization of the angu-lar dependence of the continuum background. Bot-tom: Corresponding angular distributions for dif-ferent energy bins together with the fit of Eq. (2).For better visibility, they are shifted relative to eachother by 2 mb/sr. of a least-squares method with linear combina-tions of the theoretically predicted angular dis-tributions of the differential cross sections via (cid:88) i (cid:18) d σ dΩ ( θ i , E x ) exp − d σ dΩ ( θ i , E x ) th (cid:19) = min , (3)withd σ dΩ ( θ, E x ) th = (cid:88) πλ a πλ d σ dΩ ( θ, E x , πλ ) DWBA + b d σ dΩ ( θ ) BG , (4)where a πλ and b are fit parameters. The fitswere performed using the following criteria and boundary conditions:– For each data set measured at the spectrom-eter angle θ = (0 ◦ , . ◦ , . ◦ ), five data pointsper angle and energy bin were generated by ap-plying gates to the vertical and horizontal an-gles respectively, so that in total 15 data pointsbetween 0 . ◦ and 5 . ◦ were available for theMDA.– In total six E B ( E
1) values in the QPM calculationswere used, since the corresponding angular dis-tributions show sensitivity to the Coulomb-nuclear interference.– Two M B ( M E a πλ and b had to be positive.The least-squares fitting procedure was car-ried out including all possible combinations ofthe theoretical angular distributions satisfyingthe above criteria. For each combination the χ and the reduced χ red = χ / ( p − n ) values werecalculated with p the number of experimentaldata points and n the number of fit parame-ters. Using ω = 1 /χ red as a weighting param-eter, mean cross sections for each contributionwere finally determined (cid:28) d σ dΩ ( θ, E x ) πλ (cid:29) = (cid:80) i ω i d σ dΩ ( θ, E x ) πλi (cid:80) i ω i . (5)The corresponding uncertainty was obtainedfrom the weighted variance σ = (cid:80) i ω i (cid:0) d σ dΩ ( θ, E x ) πλi − (cid:10) d σ dΩ ( θ, E x ) πλ (cid:11)(cid:1) (cid:80) i ω i . (6)In Fig. 12 a typical result of the MDA is dis-played for the example of Sn and three dif-ferent energy bins at 8, 15 and 23 MeV. The up-per part shows the
Sn spectra and the energybins indicated by vertical dashed lines. In thelower part the corresponding experimental an-gular distributions and the results of Eq. (5) for2
Energy (MeV) d σ d Ω d E ( m b / s r / M e V ) Sn(p,p’), 0 ◦ Sn(p,p’), 2.5 ◦ Sn(p,p’), 4.5 ◦ θ (deg) − − d σ d Ω ( m b / s r ) E x = E1E3 M1Sum θ (deg) E x =
15 MeV
E1E3 QFSSum θ (deg) E x =
23 MeV
FIG. 12. Typical results of the MDA for the example of
Sn and three different energy bins at 8, 15and 23 MeV. Top: Spectra and energy bins indicated by the vertical dashed lines. Bottom: Experimentalangular distributions and results of Eq. (5) for different multipoles and their sum. different multipoles and their sums are given. E M ◦ is non-negligible. At larger an-gles some higher-multipole component is neededto account for the data. The energy bin nearthe maximum of the IVGDR (15 MeV) exhibitsthe expected dominance of E ◦ (cf. Fig. 6). Theorange data show the experimental cross sec-tions after subtraction of ISGMR and ISGQRcontributions, respectively. The error bars in-clude statistical, systematical and MDA uncer- tainties added in quadrature. The E M M − M Sn Sn d σ d Ω d E ( m b / s r / M e V ) Sn Sn Sn Energy (MeV) Sn FIG. 13. Results of the multipole decompositionanalysis for the spectra measured at a spectrometerangle θ = 0 ◦ . Orange: Experimental cross sectionsafter subtraction of the ISGMR and ISGQR con-tributions. Blue: E M λ >
1. Purple: Continuum background. The er-ror bars include statistical, systematical and MDAuncertainties.
IV. PHOTOABSORPTION CROSSSECTIONSA. Virtual photon method
The conversion of Coulomb-excitation to pho-toabsorption cross sections is based on the vir-tual photon method described e.g. in Ref. [2].In contrast to the previous results published for
Sn [8, 28], which were based on the semi-classical approximation, here the virtual photonspectrum is calculated in the eikonal approxi-mation [83]. It allows for a proper treatment ofrelativistic and retardation effects and providesmore realistic angular distributions due to tak-ing into account absorption on a diffuse nuclearsurface. Examples of virtual photon spectra forthe case of
Sn and the differences betweenboth approaches can be found in Sec. 3.3 ofRef. [1]. However, the experimental data aregiven for an average scattering angle. To ac-count for this, one needs to average the differ-ential virtual photon number over the experi-mental solid angle.Another point to be considered is the max-imum scattering angle at which the strong in-teraction between projectile and target nucleusstarts to play a role. This can be calculatedfrom relativistic Rutherford scattering using[74] θ maxlab = Z Z e bµβ γ , (7)where Z is the projectile charge, Z the chargeof the target nucleus, e the elementary charge, µ the reduced mass, β the velocity in unitsof speed of light, γ the Lorentz factor and b the impact parameter. The impact parame-ter is taken as the sum of the projectile andtarget nucleus radii b = r p + r A / , where r p = 0 .
87 fm is the proton root mean squarecharge radius [84], r = 1 .
25 fm and A the massnumber. For the investigated tin isotopes, themaximum scattering angle was determined to θ maxlab = 2 . ◦ − . ◦ depending on A . The av-erage differential virtual photon number is then4
10 15 20 250100200300 σ ( m b ) Sn
10 15 20 25 Sn
10 15 20 250100200300 σ ( m b ) Sn
10 15 20 25 Sn Energy (MeV) σ ( m b ) Sn
10 15 20 25 30
Energy (MeV) Sn FIG. 14. Photoabsorption cross sections obtained in this work (blue circles) in comparison to ( γ, xn )experiments by Fultz et al. [32] at Livermore (green left triangles), Leprtre et al. [33] at Saclay (red righttriangles), and Sorokin et al. [34, 35] (orange downward triangles). ( γ, n ) data from Utsunomiya et al. [36, 37]are shown as black upward triangles. The neutron thresholds are indicated by vertical dashed lines. given by (cid:28) d N E dΩ ( E, θ ) (cid:29) = (cid:82) d N E dΩ ( E, θ )dΩ (cid:82) dΩ , (8)where the integration is performed up to themaximum angle. For heavy nuclei, after inte-gration over the relevant angular range, differ-ences of virtual photon numbers from the semi-classical and the eikonal approach are found to be small for the present kinematics, typicallyless than 10%. B. Results and comparison to previouswork
The resulting photoabsorption cross sec-tions (blue circles) are summarized in Fig. 145 σ ( m b ) Sn Sn σ ( m b ) Sn Sn Energy (MeV) σ ( m b ) Sn Energy (MeV) Sn FIG. 15. Same as Fig. 14, but restricted to the energy region 8 −
13 MeV. in comparison to data from previous exper-iments. Photoabsorption cross sections in , , , Sn have been measured in Liver-more [32] (green left triangles) and Saclay [33](red right triangles) with the ( γ, xn ) reaction.Additional ( γ, xn ) data for all isotopes investi-gated here are available from Refs. [34, 35] (or-ange downward triangles). There are also morerecent ( γ, n ) data for , , , Sn from ex-periments with monoenergetic photons at NEWSUBARU [36, 37] (black upward triangles).In the energy region near the resonance max- imum reasonable agreement is found in mostcases except for the significantly lower datapoints of Ref. [35] in , Sn. Also, the Liver-more results for
Sn are below the other threeexperiments. These two data sets tend to besystematically higher than the present resultson the low-energy flank of the IVGDR. TheSaclay results, on the other hand, show a sys-tematic relative shift with increasing A from un-dershooting the present Sn results to slightlyovershooting for
Sn.Around the neutron threshold, however,6larger deviations can be observed as illustratedin Fig. 15, where the energy region between 8and 13 MeV is magnified. The Saclay crosssections are larger than the present ones in , , Sn except close to threshold and agreefor
Sn. The Livermore data closer to thethreshold show smaller cross sections for
Sn,cross sections similar to the present work for , Sn, and larger cross sections for
Sn.The results of Refs. [34, 35] are significantlyhigher for , , Sn but agree fairly well for , , Sn except the region close to thresholdin
Sn. On the other hand, the ( γ, n ) experi-ments of Utsunomiya et al. [36, 37] are in goodagreement with the present work for all studiedisotopes.At high excitation energies, the present re-sults are shown up to 20 MeV only, since thecross section ratio between E Sn are abouttwo times larger than for , Sn, which inturn are larger than for
Sn. The data ofRefs. [34, 35] are on the average more consistentwith each other but show large fluctuations asa function of energy between neighboring iso-topes. These observations point towards prob-lems in the extraction and separation of ( γ, n )and ( γ, n ) events. C. Systematics of the IVGDR
Lorentzian fits to the different data sets arepresented in Fig. 16 and summarized in Tab. II.The parameters for the present experiment wereobtained using data in the energy range 13 − σ GDR in Tab. II agreevery well within the uncertainties for all datasets except the aforementioned reduction of the E G D R ( M e V ) this workSorokin et al. Fultz et al.
Leprˆetre et al.
112 114 116 118 120 122 124 A Γ G D R ( M e V ) this workSorokin et al. Fultz et al.
Leprˆetre et al.
FIG. 16. Centroid energies E GDR (top) and widthsΓ
GDR (bottom) of Lorentzian fits to the IVGDRin tin isotopes determined from the data shown inFig. 14. The purple line shows the phenomenologi-cal mass dependence of the centroid energy, Eq. (9).
Livermore results for
Sn in the IVGDR peakregion. The situation is different with respectto the centroid energies and the widths as il-lustrated in Fig. 16. The expected decrease ofthe centroid energy E GDR with increasing massnumber A is found in all experiments, thoughneither the absolute values nor the slope agreebetween the different sets of data. The centroidenergies determined in this work are found tobe generally higher than in previous work, yetthey yield the best agreement comparing withthe well-known phenomenological formula [31] E GDR = 31 . A − / + 20 . A − / (9)plotted as purple line in the upper part ofFig. 16.The widths Γ GDR differ considerably betweenthe experiments. The values from the presentexperiment are systematically smaller. They7
TABLE II. Lorentzian fits to the IVGDR photoabsorption cross sections in , , , , , Sn fromdifferent experiments. All data were fitted in the excitation energy range 13 −
18 MeV. The results for thedata of Refs. [34, 35] were taken from Table 5 of Ref. [85]. Neither uncertainties nor the fitting range areavailable for these numbers. Sn Sn Sn Sn Sn Sn σ GDR (mb)this work 272(16) 280(16) 279(16) 290(16) 285(16) 286(15)[34, 35, 85] 268 265 260 272 297 270[32] − − − − E GDR (MeV)this work 15.91(5) 15.96(6) 15.81(5) 15.67(8) 15.61(5) 15.46(5)[34, 35, 85] 15 . . . . . . − − − − GDR (MeV)this work 4.51(20) 4.50(22) 4.42(22) 4.47(33) 4.48(19) 4.33(17)[34, 35, 85] 5 . − − . . − [32] − − − − are constant within the uncertainties with anaverage of about 4.5 MeV. Likewise, the data ofFultz et al. [32] show a rather constant widthwith E GDR ≈ . Sn. Thedata of Leprtre et al. [33] exhibit a fluctuatingbehaviour around an average value of about5.1 MeV. The values quoted in Ref. [85] for thedata of Refs. [34, 35] are generally much largerexceeding 5.5 MeV.
D. IVGDR energies and nuclear matterbulk parameters
The tool of choice for the theoretical mod-eling of nuclear giant resonances is since longthe Random-Phase Approximation (RPA). Itis based on a mean-field description in termsof one-particle-one-hole states recoupled by aresidual two-body interaction [86]. Early real-izations of RPA were mostly based on empiri-cal shell-model potentials and separately addedinteractions [87]. Meanwhile, steady progressin nuclear energy density functional theory [39] and in numerical capabilities has made fullyself-consistent RPA calculations a widely usedstandard tool. However, most EDFs are tunedto ground state properties which leaves itsisovector properties to some extent undeter-mined. The RPA predictions for the IVGDRare thus widely varying. A proper tuning of theIVGDR within nuclear EDF theory is a field ofactive research, see e.g., Refs. [43, 88, 89], andany precise new data are highly welcome.Thus we will now compare the present mea-surements with a variety of RPA predictions.To do that in systematic manner, we chose afamily of EDF parameterizations which varycertain nuclear-matter properties (NMP) in sys-tematic manner, i.e., they all describe the samepool of ground-state properties equally well, butdiffer in one of the NMP varied within accept-able bounds while leaving the quality of groundstate properties intact. There exist several suchsets of families from Skyrme EDF as well asfrom relativistic models [57, 89–91]. We confinethe present study to the set from Ref. [89] as itcovers the broadest set of NMP (see below). For
Pb a one-to-one relation between each major8giant resonance and one of the NMP was found[89] and corroborated by statistical correlationanalysis [92]. On the other hand, the quality ofthe description of IVGDR can change dramati-cally with nuclear size [93]. The tin isotopes areabout half the size of lead and it is interestingto see how the RPA description performs in thiscase.RPA is capable to describe the gross proper-ties of giant resonances well, but it fails if onelooks at the detailed profile of the strength dis-tributions. RPA spectra for the IVGDR are toomuch structured while experimental data showusually one broad GDR peak, see e.g. Fig. 14.It requires two-body correlations beyond RPAto describe the spreading width [94]. Practi-cal realizations in terms of the phonon-couplingmodel are, indeed, able to produce realisticallysmooth excitation spectra [95, 96]. These are,however, very expensive to use and contain toomany ingredients for a simple comparison withdata. Before going into details, one has first tocheck the gross properties and this can be donevery well at the simpler level of RPA when com-paring averaged properties. One such quantityis the dipole polarizability discussed in Sec. V Band Ref. [58] which can be obtained from the( −
2) moment of the photoabsorption cross sec-tion distribution. The other prominent featureis the IVGDR peak position scrutinized here.A quick glance at Fig. 14 shows that it is un-safe to read off the peak position directly fromthe strength distribution. A more robust valueis obtained from the average energy in a giveninterval [ E , E ] E = (cid:82) E E dEσ D ( E ) E (cid:82) E E dEσ D ( E ) . (10)The Lorentzian fits to the data in Sec. IV Bhave been performed for an energy region 13 −
18 MeV and the same interval is chosen for thetheoretical results. Before doing that, the RPAspectra are folded to resemble approximatelythe smoothness of the data. To explore theimpact of smoothing, we have used Lorentzianas well as Gaussian folding with widths from1 to 2 MeV. The results are found to be only
TABLE III. NMP for the family of Skyrme pa-rameterizations from [89], where SV-bas is the basepoint of the systematic variation. Variations of theincompressibility K are given in SV-K*, of the ef-fective mass m ∗ /m in SV-mas*, of the symmetryenergy J in SV-sym*, and of the TRK sum ruleenhancement κ TRK in SV-kap*.force
K m ∗ /m a sym κ TRK (MeV) (MeV)SV-bas 234 0.9 30 0.4SV-K218 218 0.9 30 0.4SV-K226 226 0.9 30 0.4SV-K241 241 0.9 30 0.4SV-mas10 234 1.0 33 0.4SV-mas08 234 0.8 26 0.4SV-mas07 234 0.7 20 0.4SV-sym28 234 0.9 28 0.4SV-sym32 234 0.9 32 0.4SV-sym34 234 0.9 34 0.4SV-kap00 234 0.9 30 0.0SV-kap20 234 0.9 30 0.2SV-kap60 234 0.9 30 0.6 weakly dependent on the actual folding recipe.We take the variations of the resulting peak en-ergies as uncertainties of the analysis, shown aserror bars in the following.As said above, we compare data with RPAresults from a family of Skyrme functionalsdervied from SV-bas [89], which varies system-atically the four crucial NMP: incompressibility K , symmetry energy J , isoscalar effective mass m ∗ /m , and isovector effective mass in termsof the Thomas-Reiche-Kuhn sum rule enhance-ment κ TRK . The NMP for all parameterizationsare given in Tab. III. SV-bas was developed asa Skyrme functional which fits the three ma-jor giant resonances and the dipole polarizabil-ity in
Pb together with an excellent descrip-tion of ground state properties. For
Pb, itwas found that each NMP has a one-to-one cor-relation with one giant resonance peak energy[89, 92], viz. the ISGMR with K , the ISGQRwith m ∗ /m , the IVGDR with κ TRK , and thedipole polarizability α D with J . This means,e.g., for the IVGDR peak that variation of K , m ∗ /m , and α D has negligible effect while κ TRK has direct impact on the result. The question is9 I V G D R a v e r a g ee n e r gy ( M e V ) this workm*/m=0.7 m*/m=0.9m*/m=1.0 this workJ=32 MeV J=30 MeVJ=28 MeV
112 114 116 118 120 122 124 A this workK=241 MeV K=234 MeVK=226 MeV
112 114 116 118 120 122 124 A this work κ TRK = κ TRK = κ TRK = κ TRK = FIG. 17. Average IVGDR peak positions, Eq. (10), along the chain of tin isotopes. Compared areexperimental data (blue) with results from various Skyrme parameterizations as indicated (for details seetext). Each panel collects variation of one NMP. The results of the original SV-bas interaction [89] arealways shown in green. how the IVGDR in the tin isotopes behaves inthat respect.Figure 17 compares the average IVGDR peakenergies from the various Skyrme parameteri-zations with those from the strength distribu-tions of the present experiment. At first glance,the behavior is similar to what we have seen in
Pb: SV-bas is still fairly well fitting, varia-tion of κ TRK has a strong impact, and the othervariations change less. Closer inspection, how-ever, reveals remarked deviations from the sim-ple behavior in
Pb. First, variation of J and m ∗ /m is not totally inert (as K still is), buthas some impact, indicating that the near per-fect one-to-one correlation between a giant res-onance and “its” NMP is weakened in the tinisotopes. Second, SV-bas predicts 100 −
200 keVhigher centroid energies than seen experimen-tally, while the description is almost perfect in
Pb. This indicates that the mass dependenceof isovector properties is not yet fully mod- eled by present-day EDFs, much in line withour findings for the isovector polarizability [58]and earlier studies of the A -dependence of theIVGDR [93].The isotopic trend of the data was found toagree with the known phenomenological form,Eq. (9). Most of the theoretical results com-ply with this trend. Only the variation of κ TRK shows slight changes, but these fine details gobeyond the resolution of the present analysisand data. The results along an isotopic chainthus confirm the known trend of IVGDR withmass number A . On the other hand, the largestep from A = 208 to A ≈
120 revealed devi-ations. It is not yet clear whether this is dueto the larger change in A or due to a changein charge number Z . The present data arethus one important entry for a future systematicstudy.0 V. ELECTRIC AND MAGNETICDIPOLE STRENGTH
In this section the systematics of the E M B ( E
1) strength distri-butions are derived from the photoabsorptioncross sections. In Ref. [70] a method to extractthe spin- M M p, p (cid:48) ) experiments hasbeen introduced and successfully tested. Underthe assumption that isoscalar and orbital contri-butions can be neglected one can convert the re-sults to the equivalent electromagnetic B ( M A. E strength below the neutronthreshold Below the neutron threshold, comparison canbe made with data from nuclear resonance flu-orescence (NRF) experiments. Strength dis-tributions from NRF experiments are availablefor , , , Sn [29, 98]. For
Sn, a com-parison between the B ( E
1) strengths deducedfrom proton scattering and from NRF data waspresented already in Ref. [28]. It was foundthat in proton scattering considerably more E Sn and
Sn.As in
Sn, an approximate agreement is seenin the region up to 6.5 MeV, in particular ifinelastic branchings (estimated with statisticalmodel calculations) are included for the NRFdata [28]. Above 6.5 MeV, substantially morestrength is found for both isotopes measured inproton scattering.There are two potential explanations for thesefindings. Due to the high level density inthe tin isotopes much of the strength cannotbe resolved in NRF experiments close to theneutron thresholds [99, 100], which leads to dB ( E ) / dE ( e f m / M e V ) Sn(p,p’)
Sn( γ , γ ’) Energy (MeV)
Sn(p,p’)
Sn( γ , γ ’) FIG. 18. B ( E
1) strength distributions for
Sn and
Sn below the neutron threshold in 200 keV binsfrom the present work (blue) in comparison withresults from NRF experiments [29, 98] (orange). lower B ( E
1) values. Furthermore, excitationstrengths are usually determined under the as-sumption that decays to excited states are neg-ligible. This assumption however is not alwaysjustified [101, 102] and can lead to a severe un-derestimation of the B ( E
1) strength.Figure 19 shows results from four differ-ent experiments studying the electric dipoleresponse in
Sn. While the present ( p, p (cid:48) )and ( γ, γ ’) experiments induce predominantlyisovector transitions, the ( O, O (cid:48) γ ) [103] and( α, α (cid:48) γ ) [104] studies probe the isoscalar re-sponse. As in the cases of , , Sn, a strongincrease of the B ( E
1) strength is found towardsexcitation energies > Sn in contrast to the NRFdata. The structure around 6.5 MeV observedin lighter tin isotopes is even more prominentin
Sn and clearly seen in both experiments.A completely different picture results from the1 dB ( E ) / dE ( e f m / M e V ) Sn(p,p’)
Sn( γ , γ ’) Energy (MeV) S I S ( e f m / k e V ) Sn( O, O’ γ ) Sn( α , α ’ γ ) d σ d Ω ( m b s r − / k e V ) FIG. 19. Electric dipole strength distributionsfor
Sn in 200 keV bins from different experi-ments. Top: B ( E
1) strength distributions for
Snfrom the present work (blue) in comparison withNRF results [98] (orange). Bottom: Isoscalar E O, O (cid:48) γ )experiment [103] (red) and differential cross sectionsfrom an ( α, α (cid:48) γ ) experiment [104] (green). ( O, O’ γ ) and ( α , α ’ γ ) experiments A compa-rable isoscalar E . − E α scatteringdata, thus no quantitative comparison is possi-ble.Above 7 MeV, hardly any isoscalar E B. Dipole polarizability
The electric dipole polarizability α D of a nu-cleus is related to the photoabsorption cross sec-tions, respectively the B ( E
1) strength distribu-tions by inverse moments of the E α D = (cid:126) c π (cid:90) σ abs E d E x = 8 π (cid:90) B(E1) E x d E x . (11)The present data provide photoabsorption crosssections for the determination of α D in the en-ergy region 6 −
20 MeV as discussed in Sec. IV.Below 6 MeV, B ( E
1) strength distributions of , , , Sn have been measured in ( γ, γ (cid:48) )experiments [29, 98]. These contributions aresmall ( < . TABLE IV. Total dipole polarizability α D of , , , , , Sn determined as described inthe text. Partial values are given for the contribu-tions from 6 MeV to the neutron threshold energies S n given in the first column, from S n to 20 MeV,and >
20 MeV. S n (MeV) α D (fm )6 − S n S n − >
20 Total
Sn 10.79 0 . . . . Sn 10.30 0 . . . . Sn 9.56 0 . . . . Sn 9.32 0 . . . . Sn 9.10 0 . . . . Sn 8.49 0 . . . . chosen as 50 MeV, which roughly corresponds tothe single-particle model space of the theoreti-cal results. For further details, see. Ref. [58].Figure 20 displays the evolution of α D asa function of excitation energy (the runningsum) for the investigated tin isotopes. Theerror bands consider statistical and systemat-ical uncertainties, the latter including contri-butions from experiment and from the MDAas discussed above. Note that the relative er-rors are similar at low and high excitation en-ergies. The figure illustrates that the totalpolarizabilities are dominated by the contribu-tion of the IVGDR (blue), but the low-energy(red) and high-energy (orange) parts are non-negligible. The corresponding total and partialvalues are summarized in Table IV. The vari-ation of the low-energy contribution up to theneutron threshold ( S n ) – i.e. the part missed in( γ ,xn) experiments – is driven by two counter-acting factors, viz. the decrease of the IVGDRcentroid energy and of S n with increasing A .The variation of S n between Sn and
Snis more than 2 MeV (cf. Tab. IV). Since thevariation of the IVGDR centroid energy is onlyabout 0.5 MeV and the IVGDR widths are ap-proximately constant (cf. Tab. II), the largestcontribution of 13 % is found in
Sn droppingto 8 % in
Sn. The high-energy contributionfrom the QPM calculations amounts to 9 −
10 %in all isotopes.Above neutron thresholds, results are also Sn Sn α D (f m ) Sn Sn Sn
10 20 30 40 50
Energy (MeV) Sn FIG. 20. Running sums of the dipole polarizabilitydeduced from the present (p,p (cid:48) ) data. Red: Contri-bution from 6 MeV to S n . Blue: Contribution from S n to 20 MeV. Orange: Contribution above 20 MeVfrom QPM calculations, see text for details.
112 114 116 118 120 122 124 A α D (f m ) E x = S n −
20 MeV (p,p’), this work( γ ,xn), Leprˆetre et al. ( γ ,xn), Fultz et al. FIG. 21. Contribution to the dipole polarizabilityin the energy region from S n to 20 MeV deducedfrom the present data (blue circles), Ref. [32] (greendiamonds), and Ref. [33] (orange squares). available from ( γ, xn ) experiments [32–35],which in principle allow to reduce the error barsby averaging over energy regions covered bymore than one experiment or not covered by thepresent data. However, we refrain from usingthem, since they show large variations betweendifferent isotopes and systematically differentisotopic dependence as discussed in Sec. IV Band illustrated in Figs. 21 and 22. Figure 21compares the polarizabilites deduced by the dif-ferent experiments in the energy region from theneutron threshold to 20 MeV. It is obvious thatif one would include the data of Ref. [32], theisotopic dependence of α D would be changedsignificantly. Concerning the isotopic depen-dence extracted from the data of Ref. [33] oneshould note that for , Sn results are avail-able only from about 1 MeV above S n .Data for , , , Sn in the excitation en-ergy region 20 −
30 MeV not covered in thepresent experiments are available from Ref. [32].However, these results again show large vari-ations between different isotopes and no sys-tematic isotopic dependence as illustrated inFig. 22. The problems are aggravated lookingat the energy dependence of the photoabsorp-tion cross sections. Between 20 and 25 MeV,they are about two times smaller for
Sn than
112 114 116 118 120 122 124 A α D (f m ) E x = − QPM( γ ,xn), Fultz et al. FIG. 22. Dipole polarizability of stable even-mass tin isotopes in the energy region from 20 to29.6 MeV deduced from the data of Ref. [32] (greendiamonds) compared with the theory-based esti-mate used for the present results (red pentagons). those for , Sn, which in turn are signifi-cantly smaller than those for
Sn. On theother hand, between 26 and 28 MeV, the crosssections for
Sn are about two times largerthan those for
SnWe note that a larger α D value was publishedfor Sn based on the same type of experiment[8], which after correction for the quasideuteronpart amounted to α D = 8 . . However,as pointed out in Sec. II the difference to thepresent result is not due to the (p,p (cid:48) ) data (crosssections from the previous and present experi-ments agree within error bars). Rather theyresult from averaging with the ( γ ,xn) data ofRefs. [32, 33], whose contributions to α D in theIVGDR region are larger than those from the( p, p (cid:48) ) data as illustrated in Fig. 21 and from theparticularly large photoabsorption strengths ofRef. [32] in the energy region 20 −
30 MeV(cf. Fig. 22).The implications of the isotopic dependenceand absolute values of the polarizabilities sum-marized in Tab. IV are discussed in Ref. [58].4
C. Magnetic dipole strength
The multipole decomposition analysis yieldsapart of the dominant E M B ( M στ ) and with someadditional assumptions also the correspondingelectromagnetic B ( M
1) strength. The analy-sis is based on the so-called unit cross sectionmethod and utilizes isospin symmetry of theisovector spin M M M σ dΩ (0 ◦ ) IVexp = ˆ σ M F ( q, E x ) B ( M στ ) , (12)where ˆ σ M is the unit cross section, F ( q, E x )a kinematic correction factor depending on mo-mentum transfer q and excitation energy E x ,and B ( M στ ) the dimensionless isovector spin M T = T , where T denotes the g.s. isospin).Because of the properties of the effectiveproton-nucleus interaction [3] at small momen-tum transfers, for the inelastic proton scatter-ing experiment discussed in this work the spin M p, n ) reactions at E p ∼ = 300 MeV was investigated in Ref. [111],where a mass dependent formula for the unitcross section (in mb/sr) was derivedˆ σ GT = 3 . − . A / − / )] . (13) The kinematical correction factor was deter-mined by DWBA calculations and an extrap-olation from experimental data at finite an-gles to the cross section at 0 ◦ with the aid ofthe theoretical M M B ( M
1) = 34 π ( g IVs ) B ( M στ ) µ , (14)where g IVs = ( g πs − g νs ) is the isovector gyro-magnetic factor with proton and neutron g fac-tors g πs = 5 .
586 and g νs = − . B ( M
1) strength distri-butions applying the above described method tothe M Sn)and 12.4 MeV (
Sn), respectively. We note,however, that the MDA results for the M S n energies (in-dicated by vertical lines in Fig. 23) due to thesimilarity of the theoretical M E M E χ values. Thus, the χ -weightedaveraging over the different fits performed inEq. (5) becomes questionable and Eq. (6) under-estimates the uncertainties. Accordingly, the B ( M
1) strengths above the respective S n valuesshould be taken with some care and additionalweak contributions at even higher excitation en-ergies cannot be excluded. Table V summarizesthe results.It is instructive to compare the B ( M Sn with re-5 Sn Sn dB ( M ) / dE ( µ N M e V − ) Sn Sn Sn Sn, [8]
Energy (MeV) Sn FIG. 23. B ( M
1) strength distributions extractedwith the method described in the text. Addition-ally, for
Sn results based on the measurement ofspin-transfer observables from Ref. [8] are shown.The vertical lines indicate the neutron threshold en-ergies. TABLE V. Neutron threshold energies S n , B ( M S n , and total B ( M
1) strengths upto energy E max in , , , , , Sn deducedfrom the present data as described in the text. S n (cid:80) S n B ( M E max (cid:80) E max B ( M µ N ) (MeV) ( µ N ) Sn 10.79 13.1(1.2) 11.2 14.7(1.4)
Sn 10.30 9.2(1.0) 12.8 19.6(1.9)
Sn 9.56 8.1(0.7) 11.8 15.6(1.3)
Sn 9.32 8.2(1.1) 11.2 18.4(2.4)
Sn 9.10 4.8(0.5) 12.4 15.4(1.4)
Sn 8.49 5.6(0.6) 11.4 19.1(1.7) sults from an independent decomposition of E M M VI. CONCLUSIONS AND OUTLOOK
In this work the electric and magnetic dipoleresponse of the even-even stable tin isotopes , , , , , Sn was extracted in the ex-citation energy range 6 −
20 MeV from inelas-tic proton scattering experiments at 295 MeVand very forward angles 0 ◦ − ◦ . The individ-ual contributions of different multipoles to thedouble differential cross sections were extractedby means of an MDA.Utilizing the virtual photon method, pho-toabsorption cross sections were extracted fromthe E γ, xn ) experiments [32–35] and significant differences are observed onthe low-energy flank of the IVGDR, particu-larly pronounced near the neutron threshold,while recent measurements of the ( γ, n ) reac-tion [36, 37] show good agreement. Lorentzian6fits in the IVGDR energy region show a smoothcentroid energy dependence as a function of A consistent with phenomenological models anda constant width. A systematic study of thedependence of IVGDR energies on bulk mat-ter properties with an EDF tuned to describethe giant resonances in Pb reveals that themass dependence is not yet fully reproduced bypresent-day models, similar to what was con-cluded for the polarizability [58].The B ( E
1) strength distributions were de-termined and compared below the neutronthreshold to ( γ, γ (cid:48) ) experiments, where dataon , , , Sn are available. Consider-ably more strength was found for all cases inthe present work, confirming previous findingsfor
Sn [28]. Furthermore, an accumulationof strength has been detected between 6 and7 MeV in all tin isotopes being most promi-nent in
Sn. Comparison with results fromisoscalar probes for
Sn demonstrates thatthese transitions are of dominant neutron char-acter as expected for the PDR. At higher exci-tation energies the E E p, p (cid:48) ) and ( γ, γ (cid:48) ) dataindicate the influence of complex wave functionsof the excited states resulting in small branch-ing ratios to the ground state.The evolution of the dipole polarizability inthe chain of stable tin isotopes was determinedcombining the experimental photoabsorptioncross sections up to 20 MeV from the presentwork with a theory-aided correction for the un-observed high-energy part. The implications ofthese results for the development of EDFs aim-ing at a global description of the dipole polariz-ability across the nuclear chart and the result-ing constraints on symmetry energy parametershave been discussed in Ref. [58].Using the unit cross section technique [70], B ( M
1) strength distributions were determinedfrom the M M − , Sn. Below S n they exhibit broad dis-tributions similar to what was found in heavy deformed nuclei [59]. Above S n , the accuracy islimited because of the similarity of the M B ( E
1) and B ( M
1) strength distri-butions at hand, the Gamma Strength Function(GSF) can be determined for the nuclei stud-ied. Below neutron threshold, the GSFs show aspecific evolution with mass number. In combi-nation with compound nucleus γ -decay exper-iments using the Oslo method [113] this canprovide a unique test of the controversially dis-cussed Brink-Axel hypothesis [9, 114–118] stat-ing an independence of the GSF from initial andfinal states. Such an analysis is presently pre-pared [119].Finally, an aspect of the experimental resultsnot discussed here is their high energy resolu-tion of 30 −
40 keV (FWHM), which allows aquantitative analysis of the fine structure of theIVGDR similar to Refs. [108, 120]. Utilizingwavelet analysis techniques [121], informationon the relevance of different mechanisms con-tributing to the width of the IVGDR can beretrieved [122]. The cross section fine struc-ture also permits an extraction of the J π =1 − level density in the IVGDR energy region[9, 108, 115] based on a fluctuation analysis[123, 124]. However, this requires excellentstatistics which were only reached in the presentdata for Sn and
Sn. The results will bepresented elsewhere [119].
ACKNOWLEDGMENTS
The experiments were performed at RCNPunder program E422. The authors thankthe accelerator group for providing excellentbeams. This work was funded by the DeutscheForschungsgemeinschaft (DFG, German Re-search Foundation) under grant No. SFB 1245(project ID 279384907), by JSPS KAKENHI(grant No. JP14740154), and by MEXT KAK-ENHI (grant No. JP25105509). C.A.B. wassupported in part by U.S. DOE Grant No.DE-FG02-08ER41533 and U.S. NSF Grant No.1415656.7 [1] P. von Neumann-Cosel and A. Tamii, Eur.Phys. J. A , 110 (2019).[2] C. A. Bertulani and G. Baur, Phys. Rep. ,299 (1988).[3] W. G. Love and M. A. Franey, Phys. Rev. C , 1073 (1981).[4] A. Tamii, Y. Fujita, H. Matsubara, T. Adachi,J. Carter, M. Dozono, H. Fujita, K. Fu-jita, H. Hashimoto, K. Hatanaka, T. Ita-hashi, M. Itoh, T. Kawabata, K. Nakanishi,S. Ninomiya, A. Perez-Cerdan, L. Popescu,B. Rubio, T. Saito, H. Sakaguchi, Y. Sakemi,Y. Sasamoto, Y. Shimbara, Y. Shimizu,F. Smit, Y. Tameshige, M. Yosoi, and J. Zen-hiro, Nucl. Instrum. Methods A , 326(2009).[5] R. Neveling, H. Fujita, F. D. Smit, T. Adachi,G. P. A. Berg, E. Z. Buthelezi, J. Carter, J. L.Conradie, M. Couder, R. W. Fearick, S. V.F¨ortsch, D. T. Fourie, Y. Fujita, J. G¨orres,K. Hatanaka, M. Jingo, A. M. Krumbholz,C. O. Kureba, J. P. Mira, S. H. T. Murray,P. von Neumann-Cosel, S. O’Brien, P. Papka,I. Poltoratska, A. Richter, E. Sideras-Haddad,J. A. Swartz, A. Tamii, I. T. Usman, and J. J.van Zyl, Nucl. Instrum. Methods A , 29(2011).[6] I. Poltoratska, P. von Neumann-Cosel,A. Tamii, T. Adachi, C. A. Bertulani,J. Carter, M. Dozono, H. Fujita, K. Fu-jita, Y. Fujita, K. Hatanaka, M. Itoh,T. Kawabata, Y. Kalmykov, A. M. Krumb-holz, E. Litvinova, H. Matsubara, K. Nakan-ishi, R. Neveling, H. Okamura, H. J.Ong, B. ¨Ozel-Tashenov, V. Y. Pono-marev, A. Richter, B. Rubio, H. Sak-aguchi, Y. Sakemi, Y. Sasamoto, Y. Shim-bara, Y. Shimizu, F. D. Smit, T. Suzuki,Y. Tameshige, J. Wambach, M. Yosoi, andJ. Zenihiro, Phys. Rev. C , 041304(R)(2012).[7] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertu-lani, J. Carter, M. Dozono, H. Fujita,K. Fujita, K. Hatanaka, D. Ishikawa,M. Itoh, T. Kawabata, Y. Kalmykov, A. M.Krumbholz, E. Litvinova, H. Matsubara,K. Nakanishi, R. Neveling, H. Okamura,H. J. Ong, B. ¨Ozel-Tashenov, V. Y. Pono-marev, A. Richter, B. Rubio, H. Sak- aguchi, Y. Sakemi, Y. Sasamoto, Y. Shim-bara, Y. Shimizu, F. D. Smit, T. Suzuki,Y. Tameshige, J. Wambach, R. Yamada,M. Yosoi, and J. Zenihiro, Phys. Rev. Lett. , 062502 (2011).[8] T. Hashimoto, A. M. Krumbholz, P.-G. Rein-hard, A. Tamii, P. von Neumann-Cosel,T. Adachi, N. Aoi, C. A. Bertulani, H. Fu-jita, Y. Fujita, E. Ganioˇglu, K. Hatanaka,E. Ideguchi, C. Iwamoto, T. Kawabata, N. T.Khai, A. Krugmann, D. Martin, H. Mat-subara, K. Miki, R. Neveling, H. Okamura,H. J. Ong, I. Poltoratska, V. Y. Pono-marev, A. Richter, H. Sakaguchi, Y. Shim-bara, Y. Shimizu, J. Simonis, F. D. Smit,G. S¨usoy, T. Suzuki, J. H. Thies, M. Yosoi,and J. Zenihiro, Phys. Rev. C , 031305(R)(2015).[9] D. Martin, P. von Neumann-Cosel, A. Tamii,N. Aoi, S. Bassauer, C. A. Bertulani,J. Carter, L. Donaldson, H. Fujita, Y. Fujita,T. Hashimoto, K. Hatanaka, T. Ito, A. Krug-mann, B. Liu, Y. Maeda, K. Miki, R. Nevel-ing, N. Pietralla, I. Poltoratska, V. Y. Pono-marev, A. Richter, T. Shima, T. Yamamoto,and M. Zweidinger, Phys. Rev. Lett. ,182503 (2017).[10] D. Savran, T. Aumann, and A. Zilges, Prog.Part. Nucl. Phys. , 210 (2013).[11] A. Bracco, E. Lanza, and A. Tamii, Prog.Part. Nucl. Phys. , 360 (2019).[12] P. Adrich, A. Klimkiewicz, M. Fallot,K. Boretzky, T. Aumann, D. Cortina-Gil,U. D. Pramanik, T. W. Elze, H. Em-ling, H. Geissel, M. Hellstr¨om, K. L. Jones,J. V. Kratz, R. Kulessa, Y. Leifels, C. No-ciforo, R. Palit, H. Simon, G. Sur´owka,K. S¨ummerer, and W. Walu´s (LAND-FRSCollaboration), Phys. Rev. Lett. , 132501(2005).[13] A. Klimkiewicz, N. Paar, P. Adrich, M. Fal-lot, K. Boretzky, T. Aumann, D. Cortina-Gil, U. D. Pramanik, T. W. Elze, H. Em-ling, H. Geissel, M. Hellstr¨om, K. L.Jones, J. V. Kratz, R. Kulessa, C. No-ciforo, R. Palit, H. Simon, G. Sur´owka,K. S¨ummerer, D. Vretenar, and W. Walu´s(LAND Collaboration), Phys. Rev. C ,051603(R) (2007).[14] O. Wieland, A. Bracco, F. Camera, G. Ben- zoni, N. Blasi, S. Brambilla, F. C. L. Crespi,S. Leoni, B. Million, R. Nicolini, A. Maj,P. Bednarczyk, J. Grebosz, M. Kmiecik,W. Meczynski, J. Styczen, T. Aumann,A. Banu, T. Beck, F. Becker, L. Caceres,P. Doornenbal, H. Emling, J. Gerl, H. Geis-sel, M. Gorska, O. Kavatsyuk, M. Kavat-syuk, I. Kojouharov, N. Kurz, R. Lozeva,N. Saito, T. Saito, H. Schaffner, H. J. Woller-sheim, J. Jolie, P. Reiter, N. Warr, G. deAn-gelis, A. Gadea, D. Napoli, S. Lenzi, S. Lu-nardi, D. Balabanski, G. LoBianco, C. Petra-che, A. Saltarelli, M. Castoldi, A. Zucchiatti,J. Walker, and A. B¨urger, Phys. Rev. Lett. , 092502 (2009).[15] D. M. Rossi, P. Adrich, F. Aksouh,H. Alvarez-Pol, T. Aumann, J. Benlliure,M. B¨ohmer, K. Boretzky, E. Casarejos,M. Chartier, A. Chatillon, D. Cortina-Gil,U. Datta Pramanik, H. Emling, O. Er-shova, B. Fernandez-Dominguez, H. Geis-sel, M. Gorska, M. Heil, H. T. Johans-son, A. Junghans, A. Kelic-Heil, O. Kiselev,A. Klimkiewicz, J. V. Kratz, R. Kr¨ucken,N. Kurz, M. Labiche, T. Le Bleis, R. Lem-mon, Y. A. Litvinov, K. Mahata, P. Maier-beck, A. Movsesyan, T. Nilsson, C. Nociforo,R. Palit, S. Paschalis, R. Plag, R. Reifarth,D. Savran, H. Scheit, H. Simon, K. S¨ummerer,A. Wagner, W. Walu´s, H. Weick, and M. Win-kler, Phys. Rev. Lett. , 242503 (2013).[16] J. Piekarewicz, Phys.Rev. C , 044325(2006).[17] N. Tsoneva and H. Lenske, Phys. Rev. C ,024321 (2008).[18] J. Piekarewicz, Phys. Rev. C , 034319(2011).[19] P.-G. Reinhard and W. Nazarewicz, Phys.Rev. C , 051303(R) (2010).[20] A. Carbone, G. Col`o, A. Bracco, L.-G. Cao,P. F. Bortignon, F. Camera, and O. Wieland,Phys. Rev. C , 041301(R) (2010).[21] F. J. Fattoyev, W. G. Newton, J. Xu, andB. A. Li, Phys.Rev. C , 025804 (2012).[22] M. B. Tsang, J. R. Stone, F. Camera,P. Danielewicz, S. Gandolfi, K. Hebeler, C. J.Horowitz, J. Lee, W. G. Lynch, Z. Kohley,R. Lemmon, P. M¨oller, T. Murakami, S. Ri-ordan, X. Roca-Maza, F. Sammarruca, A. W.Steiner, I. Vida˜na, and S. J. Yennello, Phys.Rev. C , 015803 (2012).[23] P.-G. Reinhard and W. Nazarewicz, Phys.Rev. C , 014324 (2013). [24] P.-G. Reinhard, V. O. Nesterenko, A. Repko,and J. Kvasil, Phys. Rev. C , 024321 (2014).[25] S. Goriely, E. Khan, and M. Samyn, Nucl.Phys. A , 331 (2004).[26] E. Litvinova, H. Loens, K. Langanke,G. Martnez-Pinedo, T. Rauscher, P. Ring, F.-K. Thielemann, and V. Tselyaev, Nucl. Phys.A , 26 (2009).[27] I. Daoutidis and S. Goriely, Phys. Rev. C ,034328 (2012).[28] A. Krumbholz, P. von Neumann-Cosel,T. Hashimoto, A. Tamii, T. Adachi, C. Bertu-lani, H. Fujita, Y. Fujita, E. Ganioglu,K. Hatanaka, C. Iwamoto, T. Kawabata,N. Khai, A. Krugmann, D. Martin, H. Mat-subara, R. Neveling, H. Okamura, H. Ong,I. Poltoratska, V. Ponomarev, A. Richter,H. Sakaguchi, Y. Shimbara, Y. Shimizu, J. Si-monis, F. Smit, G. Susoy, J. Thies, T. Suzuki,M. Yosoi, and J. Zenihiro, Phys. Lett. B ,7 (2015).[29] B. ¨Ozel-Tashenov, J. Enders, H. Lenske,A. M. Krumbholz, E. Litvinova, P. vonNeumann-Cosel, I. Poltoratska, A. Richter,G. Rusev, D. Savran, and N. Tsoneva, Phys.Rev. C , 024304 (2014).[30] U. Kneissl, N. Pietralla, and A. Zilges, J.Phys. G , R217 (2006).[31] B. L. Berman and S. C. Fultz, Rev. Mod.Phys. , 713 (1975).[32] S. C. Fultz, B. L. Berman, J. T. Caldwell,R. L. Bramblett, and M. A. Kelly, Phys. Rev. , 1255 (1969).[33] A. Leprtre, H. Beil, R. Bergre, P. Carlos,A. D. Miniac, A. Veyssire, and K. Kernbach,Nucl. Phys. A , 39 (1974).[34] Y. I. Sorokin and B. A. Yurev, Yad. Fiz. ,233 (1974).[35] Y. I. Sorokin and B. A. Yurev, Izv. AN. SSSR,Ser. Fiz. , 114 (1975).[36] H. Utsunomiya, S. Goriely, M. Kamata,T. Kondo, O. Itoh, H. Akimune, T. Yama-gata, H. Toyokawa, Y. W. Lui, S. Hilaire, andA. J. Koning, Phys. Rev. C , 055806 (2009).[37] H. Utsunomiya, S. Goriely, M. Kamata,H. Akimune, T. Kondo, O. Itoh, C. Iwamoto,T. Yamagata, H. Toyokawa, Y.-W. Lui,H. Harada, F. Kitatani, S. Goko, S. Hilaire,and A. J. Koning, Phys. Rev. C , 055805(2011).[38] O. Bohigas, N. V. Giai, and D. Vautherin,Phys. Lett. B , 105 (1981). [39] M. Bender, P.-H. Heenen, and P.-G. Rein-hard, Rev. Mod. Phys. , 121 (2003).[40] B. A. Brown, Phys. Rev. Lett. , 5296(2000).[41] J. Piekarewicz, B. K. Agrawal, G. Col`o,W. Nazarewicz, N. Paar, P.-G. Reinhard,X. Roca-Maza, and D. Vretenar, Phys. Rev.C , 041302(R) (2012).[42] X. Roca-Maza, M. Brenna, G. Col`o, M. Cen-telles, X. Vi˜nas, B. K. Agrawal, N. Paar,D. Vretenar, and J. Piekarewicz, Phys. Rev.C , 024316 (2013).[43] X. Roca-Maza and N. Paar, Prog. Part. Nucl.Phys. , 96 (2018).[44] H. Yasin, S. Sch¨afer, A. Arcones, andA. Schwenk, Phys. Rev. Lett. , 092701(2020).[45] F. zel and P. Freire, Annu. Rev. Astron. As-trophys. , 401 (2016).[46] B. P. Abbott et al. (LIGO Scientific Collab-oration and Virgo Collaboration), Phys. Rev.Lett. , 161101 (2017).[47] N. Tsoneva, H. Lenske, and C. Stoyanov,Phys. Lett. B , 213 (2004).[48] D. Vretenar, T. Niksic, N. Paar, and P. Ring,Nucl. Phys. A , 281 (2004).[49] J. Terasaki and J. Engel, Phys. Rev. C ,044301 (2006).[50] E. Litvinova, P. Ring, and V. Tselyaev, Phys.Rev. C , 014312 (2008).[51] E. G. Lanza, F. Catara, D. Gambacurta,M. V. Andr´es, and P. Chomaz, Phys. Rev. C , 054615 (2009).[52] E. Litvinova, P. Ring, and V. Tselyaev, Phys.Rev. Lett. , 022502 (2010).[53] A. Avdeenkov, S. Goriely, S. Kamerdzhiev,and S. Krewald, Phys. Rev. C , 064316(2011).[54] P. Papakonstantinou, H. Hergert, V. Y. Pono-marev, and R. Roth, Phys. Rev. C , 034306(2014).[55] J. Piekarewicz, Eur. Phys. J. A , 25 (2014).[56] S. Ebata, T. Nakatsukasa, and T. Inakura,Phys. Rev. C , 024303 (2014).[57] E. Y¨uksel, T. Marketin, and N. Paar, Phys.Rev. C , 034318 (2019).[58] S. Bassauer, P. von Neumann-Cosel, P. G.Reinhard, A. Tamii, S. Adachi, C. A. Bertu-lani, P. Y. Chan, G. Col, A. D’Alessio, H. Fu-jioka, H. Fujita, Y. Fujita, G. Gey, M. Hilcker,T. H. Hoang, A. Inoue, J. Isaak, C. Iwamoto,T. Klaus, N. Kobayashi, Y. Maeda, M. Mat-suda, N. Nakatsuka, S. Noji, H. J. Ong, I. Ou, N. Paar, N. Pietralla, V. Y. Pono-marev, M. S. Reen, A. Richter, X. Roca-Maza, M. Singer, G. Steinhilber, T. Sudo,Y. Togano, M. Tsumura, Y. Watanabe, andV. Werner, arXiv:2005.04105 [nucl-ex].[59] K. Heyde, P. von Neumann-Cosel, andA. Richter, Rev. Mod. Phys. , 2365 (2010).[60] K. Langanke, G. Mart´ınez-Pinedo, P. vonNeumann-Cosel, and A. Richter, Phys. Rev.Lett. , 202501 (2004).[61] K. Langanke, G. Mart´ınez-Pinedo, B. M¨uller,H.-T. Janka, A. Marek, W. R. Hix, A. Juoda-galvis, and J. M. Sampaio, Phys. Rev. Lett. , 011101 (2008).[62] M. Chadwick, M. Herman, P. Obloinsk,M. Dunn, Y. Danon, A. Kahler, D. Smith,B. Pritychenko, G. Arbanas, R. Arcilla,R. Brewer, D. Brown, R. Capote, A. Carl-son, Y. Cho, H. Derrien, K. Guber,G. Hale, S. Hoblit, S. Holloway, T. John-son, T. Kawano, B. Kiedrowski, H. Kim,S. Kunieda, N. Larson, L. Leal, J. Lestone,R. Little, E. McCutchan, R. MacFarlane,M. MacInnes, C. Mattoon, R. McKnight,S. Mughabghab, G. Nobre, G. Palmiotti,A. Palumbo, M. Pigni, V. Pronyaev,R. Sayer, A. Sonzogni, N. Summers, P. Talou,I. Thompson, A. Trkov, R. Vogt, S. van derMarck, A. Wallner, M. White, D. Wiarda,and P. Young, Nucl. Data Sheets , 2887(2011).[63] H. P. Loens, K. Langanke, G. Mart´ınez-Pinedo, and K. Sieja, Eur. Phys. J. A , 34(2012).[64] T. Otsuka, A. Gade, O. Sorlin, T. Suzuki,and Y. Utsuno, Rev. Mod. Phys. , 015002(2020).[65] Y. Fujita, B. Rubio, and W. Gelletly, Prog.Part. Nucl. Phys. , 549 (2011).[66] F. Osterfeld, Rev. Mod. Phys. , 491 (1992).[67] M. Ichimura, H. Sakai, and T. Wakasa, Prog.Part. Nucl. Phys. , 446 (2006).[68] G. Rusev, N. Tsoneva, F. D¨onau, S. Frauen-dorf, R. Schwengner, A. P. Tonchev, A. S.Adekola, S. L. Hammond, J. H. Kelley,E. Kwan, H. Lenske, W. Tornow, and A. Wag-ner, Phys. Rev. Lett. , 022503 (2013).[69] R. M. Laszewski, R. Alarcon, D. S. Dale, andS. D. Hoblit, Phys. Rev. Lett. , 1710 (1988).[70] J. Birkhan, H. Matsubara, P. von Neumann-Cosel, N. Pietralla, V. Y. Ponomarev,A. Richter, A. Tamii, and J. Wambach, Phys.Rev. C , 041302(R) (2016). [71] M. Fujiwara, H. Akimune, I. Daito, H. Fu-jimura, Y. Fujita, K. Hatanaka, H. Ikegami,I. Katayama, K. Nagayama, N. Matsuoka,S. Morinobu, T. Noro, M. Yoshimura, H. Sak-aguchi, Y. Sakemi, A. Tamii, and M. Yosoi,Nucl. Instrum. Methods A , 484 (1999).[72] H. Fujita, G. Berg, Y. Fujita, K. Hatanaka,T. Noro, E. Stephenson, C. Foster, H. Sak-aguchi, M. Itoh, T. Taki, K. Tamura, andH. Ueno, Nucl. Instrum. Methods A , 55(2001).[73] S. Bassauer, Doctoral thesis D17, Tech-nische Universit¨at Darmstadt (2019);http://tuprints.ulb.tu-darmstadt.de/9668.[74] M. N. Harakeh and A. van der Woude, GiantResonances (Oxford University Press, NewYork, 2001).[75] T. Wakasa, H. Sakai, H. Okamura, H. Otsu,N. Sakamoto, T. Uesaka, Y. Satou, S. Fujita,S. Ishida, M. Greenfield, N. Koori, A. Oki-hana, and K. Hatanaka, Nucl. Phys. A ,217 (1996).[76] T. Wakasa, H. Sakai, H. Okamura, H. Otsu,S. Fujita, S. Ishida, N. Sakamoto, T. Uesaka,Y. Satou, M. B. Greenfield, and K. Hatanaka,Phys. Rev. C , 2909 (1997).[77] B. Bonin, N. Alamanos, B. Berthier,G. Bruge, H. Faraggi, D. Legrand, J. Lugol,W. Mittig, L. Papineau, A. Yavin, D. Scott,M. Levine, J. Arvieux, L. Farvacque, andM. Buenerd, Nucl. Phys. A , 349 (1984).[78] T. Li, U. Garg, Y. Liu, R. Marks, B. K.Nayak, P. V. Madhusudhana Rao, M. Fuji-wara, H. Hashimoto, K. Nakanishi, S. Oku-mura, M. Yosoi, M. Ichikawa, M. Itoh,R. Matsuo, T. Terazono, M. Uchida, Y. Iwao,T. Kawabata, T. Murakami, H. Sakaguchi,S. Terashima, Y. Yasuda, J. Zenihiro,H. Akimune, K. Kawase, and M. N. Harakeh,Phys. Rev. C , 034309 (2010).[79] M. Itoh, S. Kishi, H. Sakaguchi,H. Akimune, M. Fujiwara, U. Garg, K. Hara,H. Hashimoto, J. Hoffman, T. Kawabata,K. Kawase, T. Murakami, K. Nakanishi,B. K. Nayak, S. Terashima, M. Uchida,Y. Yasuda, and M. Yosoi, Phys. Rev. C ,064313 (2013).[80] J. Raynal, DWBA07, NEA Computer Pro-gram Services, NEA-1209/08.[81] L. M. Donaldson, C. A. Bertulani, J. Carter,V. O. Nesterenko, P. von Neumann-Cosel,R. Neveling, V. Y. Ponomarev, P.-G. Rein-hard, I. T. Usman, P. Adsley, J. W. Brum- mer, E. Z. Buthelezi, G. R. J. Cooper, R. W.Fearick, S. V. F¨ortsch, H. Fujita, Y. Fu-jita, M. Jingo, W. Kleinig, C. O. Kureba,J. Kvasil, M. Latif, K. C. W. Li, J. P.Mira, F. Nemulodi, P. Papka, L. Pellegri,N. Pietralla, A. Richter, E. Sideras-Haddad,F. D. Smit, G. F. Steyn, J. A. Swartz, andA. Tamii, Phys. Lett. B , 133 (2018).[82] I. Poltoratska, Doctoral thesis D17, Technis-che Universit¨at Darmstadt (2011).[83] C. Bertulani and A. Nathan, Nucl. Phys. A , 158 (1993).[84] https://physics.nist.gov/cuu/constants/index.html,accessed 23.08.2019.[85] V. V. Varlamov, B. S. Ishkhanov, V. N. Orlin,and V. A. Chetvertkova, Bull. Rus. Acad. Sci. , 833 (2010).[86] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, New York, Heidel-berg, Berlin, 1980).[87] K. Goeke and J. Speth, Annu. Rev. Nucl.Part. Sci , 65 (1982).[88] N. Paar, D. Vretenar, E. Khan, and G. Col`o,Rep. Prog. Phys. (2007).[89] P. Kl¨upfel, P.-G. Reinhard, T. J. B¨urvenich,and J. A. Maruhn, Phys. Rev. C , 034310(2009).[90] B. K. Agrawal, S. Shlomo, and V. K. Au,Phys. Rev. C , 014310 (2005).[91] W. Nazarewicz, P.-G. Reinhard, W. Satu(cid:32)la,and D. Vretenar, Eur. Phys. J. A , 20(2014).[92] J. Erler and P.-G. Reinhard, J. Phys. G ,034026 (2015).[93] J. Erler, P. Kl¨upfel, and P.-G. Reinhard, J.Phys. G , 064001 (2010).[94] G. F. Bertsch, P. F. Bortignon, and R. A.Broglia, Rev. Mod. Phys. , 287 (1983).[95] G. Col`o and P. F. Bortignon, Nucl. Phys. A , 282c (2001).[96] V. Tselyaev, N. Lyutorovich, J.Speth, S. Kre-wald, and P.-G. Reinhard, Phys. Rev. C ,034306 (2016).[97] M. Mathy, J. Birkhan, H. Matsubara, P. vonNeumann-Cosel, N. Pietralla, V. Y. Pono-marev, A. Richter, and A. Tamii, Phys. Rev.C , 054316 (2017).[98] K. Govaert, F. Bauwens, J. Bryssinck,D. De Frenne, E. Jacobs, W. Mondelaers,L. Govor, and V. Y. Ponomarev, Phys. Rev.C , 2229 (1998).[99] R. Schwengner, G. Rusev, N. Benouaret,R. Beyer, M. Erhard, E. Grosse, A. R. Jung- hans, J. Klug, K. Kosev, L. Kostov, C. Nair,N. Nankov, K. D. Schilling, and A. Wagner,Phys. Rev. C , 034321 (2007).[100] D. Savran, M. Fritzsche, J. Hasper, K. Lin-denberg, S. M¨uller, V. Y. Ponomarev,K. Sonnabend, and A. Zilges, Phys. Rev. Lett. , 232501 (2008).[101] J. Isaak, D. Savran, M. Krtika, M. Ahmed,J. Beller, E. Fiori, J. Glorius, J. Kelley,B. Lher, N. Pietralla, C. Romig, G. Ru-sev, M. Scheck, L. Schnorrenberger, J. Silva,K. Sonnabend, A. Tonchev, W. Tornow,H. Weller, and M. Zweidinger, Phys. Lett. B , 361 (2013).[102] B. Lher, D. Savran, T. Aumann, J. Beller,M. Bhike, N. Cooper, V. Derya, M. Duchne,J. Endres, A. Hennig, P. Humby, J. Isaak,J. Kelley, M. Knrzer, N. Pietralla, V. Pono-marev, C. Romig, M. Scheck, H. Scheit,J. Silva, A. Tonchev, W. Tornow, F. Wamers,H. Weller, V. Werner, and A. Zilges, Phys.Lett. B , 72 (2016).[103] L. Pellegri, A. Bracco, F. Crespi, S. Leoni,F. Camera, E. Lanza, M. Kmiecik, A. Maj,R. Avigo, G. Benzoni, N. Blasi, C. Boiano,S. Bottoni, S. Brambilla, S. Ceruti, A. Giaz,B. Million, A. Morales, R. Nicolini, V. Van-done, O. Wieland, D. Bazzacco, P. Bednar-czyk, M. Bellato, B. Birkenbach, D. Borto-lato, B. Cederwall, L. Charles, M. Ciemala,G. D. Angelis, P. Dsesquelles, J. Eberth,E. Farnea, A. Gadea, R. Gernhuser, A. Grgen,A. Gottardo, J. Grebosz, H. Hess, R. Isocrate,J. Jolie, D. Judson, A. Jungclaus, N. Kark-our, M. Krzysiek, E. Litvinova, S. Lunardi,K. Mazurek, D. Mengoni, C. Michelagnoli,R. Menegazzo, P. Molini, D. Napoli, A. Pullia,B. Quintana, F. Recchia, P. Reiter, M. Salsac,B. Siebeck, S. Siem, J. Simpson, P.-A. Sder-strm, O. Stezowski, C. Theisen, C. Ur, J. V.Dobon, and M. Zieblinski, Phys. Lett. B ,519 (2014).[104] J. Endres, D. Savran, P. A. Butler, M. N.Harakeh, S. Harissopulos, R.-D. Herzberg,R. Kr¨ucken, A. Lagoyannis, E. Litvi-nova, N. Pietralla, V. Y. Ponomarev,L. Popescu, P. Ring, M. Scheck, F. Schl¨uter,K. Sonnabend, V. I. Stoica, H. J. W¨ortche,and A. Zilges, Phys. Rev. C , 064331(2012).[105] X. Roca-Maza, X. Vi˜nas, M. Centelles, B. K.Agrawal, G. Col`o, N. Paar, J. Piekarewicz,and D. Vretenar, Phys. Rev. C , 064304 (2015).[106] K. Schelhaas, J. Henneberg, M. Sanzone-Arenhvel, N. Wieloch-Laufenberg, U. Zurmhl,B. Ziegler, M. Schumacher, and F. Wolf, Nucl.Phys. A , 189 (1988).[107] N. Ryezayeva, T. Hartmann, Y. Kalmykov,H. Lenske, P. von Neumann-Cosel, V. Y.Ponomarev, A. Richter, A. Shevchenko,S. Volz, and J. Wambach, Phys. Rev. Lett. , 272502 (2002).[108] I. Poltoratska, R. W. Fearick, A. M.Krumbholz, E. Litvinova, H. Matsubara,P. von Neumann-Cosel, V. Y. Ponomarev,A. Richter, and A. Tamii, Phys. Rev. C ,054322 (2014).[109] T. Taddeucci, C. Goulding, T. Carey,R. Byrd, C. Goodman, C. Gaarde, J. Larsen,D. Horen, J. Rapaport, and E. Sugarbaker,Nucl. Phys. A , 125 (1987).[110] R. G. T. Zegers, T. Adachi, H. Akimune,S. M. Austin, A. M. van den Berg, B. A.Brown, Y. Fujita, M. Fujiwara, S. Gal`es,C. J. Guess, M. N. Harakeh, H. Hashimoto,K. Hatanaka, R. Hayami, G. W. Hitt,M. E. Howard, M. Itoh, T. Kawabata,K. Kawase, M. Kinoshita, M. Matsub-ara, K. Nakanishi, S. Nakayama, S. Oku-mura, T. Ohta, Y. Sakemi, Y. Shim-bara, Y. Shimizu, C. Scholl, C. Simenel,Y. Tameshige, A. Tamii, M. Uchida, T. Ya-magata, and M. Yosoi, Phys. Rev. Lett. ,202501 (2007).[111] M. Sasano, H. Sakai, K. Yako, T. Wakasa,S. Asaji, K. Fujita, Y. Fujita, M. B. Green-field, Y. Hagihara, K. Hatanaka, T. Kawa-bata, H. Kuboki, Y. Maeda, H. Oka-mura, T. Saito, Y. Sakemi, K. Sekiguchi,Y. Shimizu, Y. Takahashi, Y. Tameshige, andA. Tamii, Phys. Rev. C , 024602 (2009).[112] F. T. Baker, L. Bimbot, C. Djalali,C. Glashausser, H. Lenske, W. G. Love,M. Morlet, E. Tomasi-Gustafsson, J. Van deWiele, J. Wambach, and A. Willis, Phys. Rep. , 235 (1997).[113] A. C. Larsen, M. Guttormsen, N. Blasi,A. Bracco, F. Camera, L. C. Campo, T. K.Eriksen, A. Grgen, T. W. Hagen, V. W. Inge-berg, B. V. Kheswa, S. Leoni, J. E. Midtbø,B. Million, H. T. Nyhus, T. Renstrøm, S. J.Rose, I. E. Ruud, S. Siem, T. G. Tornyi, G. M.Tveten, A. V. Voinov, M. Wiedeking, andF. Zeiser, J. Phys. G , 064005 (2017).[114] L. Netterdon, A. Endres, S. Goriely, J. Mayer, P. Scholz, M. Spieker, and A. Zilges, Phys.Lett. B , 358 (2015).[115] S. Bassauer, P. von Neumann-Cosel, andA. Tamii, Phys. Rev. C , 054313 (2016).[116] M. Guttormsen, A. C. Larsen, A. G¨orgen,T. Renstrøm, S. Siem, T. G. Tornyi, andG. M. Tveten, Phys. Rev. Lett. , 012502(2016).[117] J. Isaak, D. Savran, B. Lher, T. Beck,M. Bhike, U. Gayer, Krishichayan,N. Pietralla, M. Scheck, W. Tornow,V. Werner, A. Zilges, and M. Zweidinger,Phys. Lett. B , 225 (2019).[118] P. Fanto, Y. Alhassid, and H. A. Wei-denm¨uller, Phys. Rev. C , 014607 (2020).[119] M. Markova et. al. , to be published.[120] R. W. Fearick, B. Erler, H. Matsubara, P. vonNeumann-Cosel, A. Richter, R. Roth, andA. Tamii, Phys. Rev. C , 044325 (2018).[121] A. Shevchenko, J. Carter, G. R. J. Cooper, R. W. Fearick, Y. Kalmykov,P. von Neumann-Cosel, V. Y. Ponomarev,A. Richter, I. Usman, and J. Wambach, Phys.Rev. C , 024302 (2008).[122] P. von Neumann-Cosel, V. Ponomarev,A. Richter, and J. Wambach, Eur. Phys. J.A , 224 (2019).[123] Y. Kalmykov, T. Adachi, G. P. A. Berg,H. Fujita, K. Fujita, Y. Fujita, K. Hatanaka,J. Kamiya, K. Nakanishi, P. von Neumann-Cosel, V. Y. Ponomarev, A. Richter,N. Sakamoto, Y. Sakemi, A. Shevchenko,Y. Shimbara, Y. Shimizu, F. D. Smit,T. Wakasa, J. Wambach, and M. Yosoi, Phys.Rev. Lett. , 012502 (2006).[124] Y. Kalmykov, C. ¨Ozen, K. Langanke,G. Mart´ınez-Pinedo, P. von Neumann-Cosel,and A. Richter, Phys. Rev. Lett.99