Pygmy dipole strength in86Kr and systematics ofN=50isotones

PHYSICAL REVIEW C 87, 024306 (2013) Pygmy dipole strength in 86 Kr and systematics of N = 50 isotones R. Schwengner,1 R. Massarczyk,1,2 G. Rusev,3,4,* N. Tsoneva,5,6 D. Bemmerer,1 R. Beyer,1 R. Hannaske,1,2 A. R. Junghans,1 J. H. Kelley,4,7 E. Kwan,3,4,† H. Lenske,5 M. Marta,1,‡ R. Raut,3,4,§ K. D. Schilling,1 A. Tonchev,3,4,† W. Tornow,3,4 and A. Wagner1 1 Institut für Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany 2 Technische Universität Dresden, 01062 Dresden, Germany 3 Department of Physics, Duke University, Durham, North Carolina 27708, USA 4 Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA 5 Institut für Theoretische Physik, Universität Gießen, 35392 Gießen, Germany 6 Institute for Nuclear Research and Nuclear Energy, BAS, 1784 Sofia, Bulgaria 7 Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA (Received 26 November 2012; published 8 February 2013) The dipole strength of the N = 50 nucleus 86 Kr was studied in photon-scattering experiments using bremsstrahlung produced with electron beams of energies of 7.9 and 11.2 MeV delivered by the linear accelerator ELBE as well as using quasimonoenergetic and linearly polarized γ rays of 10 energies within the range from 4.7 to 9.3 MeV delivered by the HIγ S facility. A high-pressure gas target was used. We identified 39 levels up to an excitation energy of 10.1 MeV. Simulations of γ -ray cascades were performed to estimate intensities of inelastic transitions and to correct the intensities of the ground-state transitions for their branching ratios. The photoabsorption cross section derived in this way up to the neutron-separation energy is combined with the photoabsorption cross section obtained from a (γ , n) experiment at HIγ S. The enhanced E1 strength found in the range from 6 to 10 MeV is compared with the ones in the N = 50 isotones 88 Sr, 90 Zr, and 92 Mo and with predictions of calculations within the quasiparticle-phonon model. DOI: 10.1103/PhysRevC.87.024306 PACS number(s): 25.20.Dc, 21.60.Jz, 23.20.−g, 27.50.+e I. INTRODUCTION Gamma-ray strength functions, in particular electric dipole (E1) and magnetic dipole (M1) strength functions, are an important ingredient for the calculation of rates of photonuclear reactions as well as of the inverse radiative-capture reactions on the basis of statistical reaction models. Radiative neutron capture, for example, is an important reaction for the synthesis of heavy nuclei in stellar environments. Moreover, an improved experimental and theoretical description of neutron capture is important for next-generation nuclear technologies, such as transmutation of nuclear waste. The dipole strength function f1 is connected with the photoabsorption cross section σγ via the relation f1 = σγ /[g(π h̄c)2 Eγ ] with g = (2Ji + 1)/(2J0 + 1), where J0 and Ji are the spins of the ground state and the excited state, respectively. At high excitation energy above the neutronseparation energy Sn , the photoabsorption cross section is dominated by the isovector giant dipole resonance (GDR) observable in (γ , n) experiments. To approximate the shape of the GDR, σγ has been phenomenologically described by * Present address: Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. † Present address: Physics Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA. ‡ Present address: GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany. § Present address: UGC-DAE Consortium for Scientific Research, Kolkata Centre LB-8 Sector-III Bidhannagar, Kolkata 700098, India. 0556-2813/2013/87(2)/024306(12) a standard Lorentz curve (SLO) [1,2] or by a generalized Lorentz curve (GLO) including terms taking into account nuclear temperature [2–4]. Double humps or a widening of the GDR caused by quadrupole and triaxial deformation of the nuclei can be reproduced with combinations of two or three Lorentz curves [5,6], which are adjusted to (γ , n) data [2]. An alternative description [7] uses the energies of a triaxial oscillator for the centroids of the three Lorentz curves and a global expression for the widths [8,9] instead of an adjustment to individual (γ , n) data. A special feature under discussion for a long time [10] is enhanced strength that is observed in the low-energy tail of the GDR. This strength reflects nuclear structure properties and is often referred to as the pygmy dipole resonance (PDR). In the framework of the quasiparticle random-phase approximation (QRPA) this PDR has been ascribed to an oscillation of excessive neutrons versus the symmetric (N = Z) neutronproton system [11–14]. Recent experiments may indicate that the PDR splits into a low-lying isoscalar component and a higher-lying more isovector component [15]. Photon scattering from nuclei, also called nuclear resonance fluorescence (NRF), is an ideal tool to study dipole strength functions below Sn because predominantly states with spin J = 1 and, to a lesser extent, states with J = 2 are excited from the ground state in an even-even nucleus. NRF experiments allow an unambiguous determination of σγ and f1 on an absolute scale. We have performed systematic studies of dipole strength distributions up to Sn for varying neutron numbers in the chain of stable even-mass Mo isotopes [16,17] and for varying proton numbers in the chain of stable N = 50 isotones [18–20] by 024306-1 ©2013 American Physical Society PHYSICAL REVIEW C 87, 024306 (2013) R. SCHWENGNER et al. means of photon scattering using the bremsstrahlung facility [21] at the superconducting electron accelerator ELBE [22,23] of the Helmholtz-Zentrum Dresden-Rossendorf (HZDR). The present work describes the first photon-scattering study of the lightest stable N = 50 isotone 86 Kr. We performed experiments with bremsstrahlung at ELBE as well as with monoenergetic γ radiation at the High-Intensity γ -ray Source (HIγ S) [24] operated by the Triangle Universities Nuclear Laboratory (TUNL). We identified 39 levels up to 10.1 MeV and assigned spins and parities to most of them. We performed simulations of γ -ray cascades to estimate intensities of inelastic transitions to low-lying excited levels in the experiments with bremsstrahlung. The dipole strength distribution deduced for 86 Kr from the present experiments is compared with the ones in the other stable even-even N = 50 isotones and with predictions of the quasiparticle-phonon model (QPM). II. EXPERIMENTAL METHODS AND RESULTS the ground-state transition is greater than the one resulting from a direct excitation only. As a consequence, the integrated scattering cross section Is+f deduced from this intensity contains a portion If originating from feeding in addition to the true integrated scattering cross section Is . In the case of inelastic scattering, inelastic and subsequent cascade transitions appear in the measured spectrum in addition to ground-state transitions. To deduce the partial width of a ground-state transition Ŵ0 and the absorption cross section one needs to know the branching ratio b0 = Ŵ0 / Ŵ. Spins of excited states can be deduced by comparing experimental ratios of intensities, measured at two angles, with theoretical predictions. The optimum combination are angles of 90◦ and 127◦ because the respective ratios for the spin sequences 0–1–0 and 0–2–0 differ most at these angles. The expected values are W (90◦ )/W (127◦ )0–1–0 = 0.74 and W (90◦ )/W (127◦ )0–2–0 = 2.18 by taking into account opening angles of 16◦ and 14◦ of the detectors placed at 90◦ and 127◦ , respectively, in the setup at ELBE. A. The photon-scattering method In photon-scattering experiments, the energy- and solidangle-integrated scattering cross section Is of an excited state at energy Ex can be deduced from the measured intensity of the respective transition to the ground state. It can be determined relative to known integrated scattering cross sections. In the present experiments, we used the integrated scattering cross sections Is (ExB ) of states in 11 B [25] and their angular correlations including mixing ratios [26] as a reference:   Iγ (Eγ , θ ) Is (Ex )   = W (Eγ , θ )γ (Ex )NN Is ExB   −1  Iγ EγB , θ     × . (1) W EγB , θ γ ExB NNB Here, Iγ (Eγ , θ ) and Iγ (EγB , θ ) denote the measured intensities of a considered ground-state transition at Eγ and of a groundstate transition in 11 B at EγB , respectively, observed at an angle θ to the beam. W (Eγ , θ ) and W (EγB , θ ) describe the angular correlations of these transitions. The quantities NN and NNB are 86 11 the numbers of nuclei in the Kr and B targets, respectively. The quantities γ (Ex ) and γ (ExB ) stand for the photon fluxes at the energy of the considered level and at the energy of a level in 11 B, respectively. The integrated scattering cross section is related to the partial width of the ground-state transition Ŵ0 according to    π h̄c 2 2Jx + 1 Ŵ02 Is = σγ γ dE = , (2) Ex 2J0 + 1 Ŵ where σγ γ is the elastic scattering cross section, Ex , Jx , and Ŵ denote energy, spin, and total width of the excited level, respectively, and J0 is the spin of the ground state. The determination of the level widths is complicated by two problems. First, a considered level can be fed by transitions from higher-lying states; second, a considered level can deexcite to low-lying excited states (inelastic scattering) in addition to the deexcitation to the ground state (elastic scattering). In the case of feeding, the measured intensity of B. The target The target used was a high-pressure gas target as described in Ref. [27]. The spherical container made of stainless steel with an inner diameter of 20 mm and a wall thickness of 0.5 mm contained 1012.6 mg of Kr enriched to 99.41% in 86 Kr. In the experiments at ELBE, the 86 Kr target was combined with 150.5 mg of 11 B, enriched to 99.5% and shaped into a disk of 20 mm in diameter, to determine the photon flux from known scattering cross sections of levels in 11 B. C. Detector response For the determination of the integrated scattering cross sections according to Eq. (1) the relative efficiencies of the detectors and the relative photon flux are needed. The determination of the absorption cross section described in Sec. III requires a correction of the experimental spectrum for detector response, for the absolute efficiency and the absolute photon flux, for atomic processes induced by the impinging photons in the target material, and for ambient background radiation. The detector response was simulated using the program package GEANT4 [28]. The reliability of the simulation was tested by comparing simulated spectra with measured ones as described in Refs. [16,18,29]. The absolute efficiencies of the HPGe detectors in the setup at ELBE were determined experimentally up to 2.4 MeV from measurements with 137 Cs, 154 Eu, and 226 Ra calibration sources. For interpolation, an efficiency curve calculated with GEANT4 and scaled to the absolute experimental values was used. A check of the simulated efficiency curve up to about 9 MeV was performed via various (p, γ ) reactions at the HZDR Tandetron accelerator up to about 9 MeV. The efficiency values deduced from these measurements agree with the simulated values within their uncertainties [30]. Similar results were obtained for the resonances at 4.44 and 11.66 MeV in 12 C populated in the 11 B(p, γ ) reaction at the TUNL van de Graaf accelerator [31]. 024306-2 PYGMY DIPOLE STRENGTH IN 86 Kr AND . . . PHYSICAL REVIEW C 87, 024306 (2013) 4401 1000 86 11 kin B Ee 4038 4000 4500 5000 5500 5924 5788 5571 4867 4932 5517 B 6000 o θ = 127 6532 0 = 11.2 MeV 11 500 0 6000 6500 7000 7570 7675 7028 6818 6679 6329 6432 6463 B 7745 7798 7846 7874 7958 11 500 6160 6213 Number of counts 7500 8000 500 8000 8500 9000 Kr + steel steel 9500 10116 9477 9452 B 9068/9086 11 9014 8621 8651 0 8841 8802 86 8428 The nuclide Kr was studied in two experiments at ELBE. Bremsstrahlung was produced using electron beams with kinetic energy of 7.9 and 11.2 MeV. The average currents were about 550 μA in the measurement at 7.9 MeV and about 720 μA in the measurement at 11.2 MeV. The electron beams hit a niobium foil of 7 μm thickness acting as a radiator. A 10-cm-thick aluminum absorber was placed behind the radiator to reduce the low-energy part of the bremsstrahlung spectrum (beam hardener). The photon beam, collimated by a 2.6-m-long pure-aluminum collimator with a conical borehole of 8 mm in diameter at the entrance and 24 mm in diameter at the exit, impinged onto the target with a flux of about 109 s−1 in a spot of 38 mm in diameter. Scattered photons were measured with four high-purity germanium (HPGe) detectors that have an efficiency of 100% relative to a NaI detector of 7.6 cm in diameter and 7.6 cm in length. All HPGe detectors were surrounded by escape-suppression shields made of bismuth germanate (BGO) scintillation detectors of 3 cm in thickness. Two HPGe detectors were placed vertically at 90◦ relative to the photon-beam direction and a distance of 28 cm from the target. The other two HPGe detectors were positioned in a horizontal plane at 127◦ to the beam and at a distance of 32 cm from the target. Absorbers of 8 mm of Pb plus 3 mm of Cu and of 3 mm of Pb plus 3 mm of Cu were placed in front of the detectors at 90◦ and 127◦ , respectively, in the measurement at 7.9 MeV, whereas absorbers of 13 mm of Pb plus 3 mm of Cu and of 8 mm of Pb plus 3 mm of Cu were used for the detectors at 90◦ and 127◦ , respectively, in the measurement at 11.2 MeV. Spectra of scattered photons were measured for 145 and 51 h in the experiments at 7.9and 11.2-MeV electron energy, respectively. To identify γ rays from 86 Kr, measurements with an empty steel container were carried out for comparison. Gamma-ray spectra of the empty container were measured for 55 and 45 h at electron energies of 7.9 and 11.2 MeV, respectively. Peaks observed in the spectrum measured with 86 Kr, but not observed in the spectrum of the empty container, were assigned to transitions in 86 Kr. Parts of the spectra including events measured with the two detectors placed at 127◦ relative to the beam at an electron energy of 11.2 MeV are shown in Fig. 1 for the steel container filled with 86 Kr and for the empty container. The absolute photon flux at ELBE was determined from intensities and known integrated scattering cross sections of transitions in 11 B. For interpolation, the photon flux was calculated using a code [32] based on the approximation given in Ref. [33] and including a screening correction according to Ref. [34]. In addition, the flux was corrected for the attenuation by the beam hardener. This flux curve was adjusted to the experimental values obtained at the energies of levels in 11 B. Measurements at various electron energies allowed us to estimate the influence of feeding on the integrated cross sections. Ratios of the quantities Is+f obtained for levels in 86 Kr from the measurements at the two electron energies are shown in Fig. 2. The plotted ratios reveal that only levels below Ex ≈ 6 MeV are influenced considerably by feeding. Transitions found in the measurement at Eekin = 7.9 MeV are assumed to be ground-state transitions. Transitions additionally observed up to 7.9 MeV in the measurement at 11.2 MeV are consequently 7234 7304/7314 D. Experiments with bremsstrahlung at ELBE 10000 E γ (keV) FIG. 1. (Color online) Parts of a spectrum of photons scattered from 86 Kr in a steel container combined with 11 B, measured during the irradiation with bremsstrahlung produced by electrons of an energy of Eekin = 11.2 MeV. This spectrum is the sum of the spectra measured with the two detectors placed at 127◦ relative to the beam. Transitions assigned to 86 Kr are marked with their energies in keV. For comparison, the corresponding parts of a spectrum measured under the same conditions with an empty container are shown. considered as inelastic transitions from high-lying to lowlying excited states. By comparing the respective spectra, these inelastic transitions were sorted out. The remaining ground-state transitions were used to derive the corresponding level energies which are listed in Table I together with spin assignments deduced from angular distributions of the groundstate transitions and integrated scattering cross sections. For states with Ex < 5 MeV, integrated scattering cross sections are not given because they may be influenced by feeding even in the experiment with 7.9-MeV electron energy. E. Experiments with monoenergetic and linearly polarized γ radiation at HIγ S Monoenergetic photon beams are produced at HIγ S by Compton backscattering of a high-intensity free-electron laser (FEL) beam from an intense electron beam in the Duke storage ring. Presently, the energy of the backward scattered photons can be tuned in a wide energy range, from about 1 to 100 MeV, by changing the energy of the electron beam and the FEL wavelength [24]. The polarization of the FEL photons, defined by the magnetic field of the undulators, is mostly preserved 024306-3 PHYSICAL REVIEW C 87, 024306 (2013) R. SCHWENGNER et al. TABLE I. Levels assigned to 86 Kr. Is+f(11.2 MeV) / Is+f(7.9 MeV) 9 8 86 Ex (keV)a Kr(γ,γ’) 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 Ex (keV) 86 FIG. 2. Ratios of integrated cross sections Is+f of transitions in Kr obtained at electron energies of 7.9 and 11.2 MeV. during the Compton backscattering due to a negligible recoil effect, leading to the production of intense photon beams with a degree of polarization of nearly 100%. The measurements at HIγ S were carried out at γ -ray energies of 4.7, 5.1, 5.6, 6.1, 6.5, 7.0, 7.5, 8.0, 8.4, 8.7, 9.0, and 9.3 MeV. The energy spread (full width at half maximum) of the beam was about 3% of the energy using a 30.5-cm-long lead collimator with a cylindrical hole of 1.9 cm in diameter positioned 60 m downstream from the collision point of the electrons with the FEL photons. The measuring time was about 7 h for each selected energy. The photon beam impinged onto the target with a flux of about 5 × 106 s−1 for the lowest energies up to 3 × 107 s−1 for the highest energies. Scattered photons were measured with two HPGe detectors taken from the detector setup at ELBE (see Sec. II D) without BGO shields. The detectors were placed at polar angles of 90◦ to the beam, one vertically on top of the beam tube and one horizontally downstream right of the beam tube at distances of 10 cm from the beam axis. Lead absorbers of 6 mm in thickness were placed in front of the detectors. The type of radiation (E1 or M1) was deduced from a comparison of the intensities of the transitions measured at different azimuthal angles [26,36]. In the present setup, E1 radiation is detected preferentially in the vertical detector and M1 radiation in the horizontal detector. As an example, spectra measured at a beam energy of 6.5 MeV are shown in Fig. 3. The spectra allow a clear distinction between E1 and M1 radiation. Azimuthal asymmetries A = (Iγ H − Iγ V )/(Iγ H + Iγ V ) deduced from the intensities Iγ H measured in the horizontal detector and Iγ V measured in the vertical detector are given in Table I together with the resulting parities assigned to the emitting states. All the states listed in Table I have negative parity. Several peaks observed with the horizontal detector were assigned to M1 transitions in 56 Fe, which is a main component of the steel container. Some small peaks could not be uniquely assigned and it cannot be excluded that these 1564.8(1) 2349.4(2) 3782.7(4) 4038.5(3) 4400.7(1) 4867.4(6) 4932.4(2) 5517.5(2) 5571.0(12) 5788.2(3) 5924.1(4) 6160.0(2) 6213.1(18) 6328.6(3) 6431.9(2) 6462.9(3) 6531.7(2) 6678.6(5) 6818.3(4) 7028.1(4) 7234.3(4) 7304.2(5) 7314.3(3) 7569.6(4) 7675.3(4) 7745.4(4) 7797.5(4) 7846.2(5) 7873.8(7) 7958.0(4) 8428.2(4) 8621.2(8) 8650.8(3) 8802.0(6) 8841.1(8) 9013.9(6) 9067.6(10) 9085.6(8) 9452.3(5) 9477.4(18) 10115.6(8) a Iγ (90◦ ) b Iγ (127◦ ) 1.07(7) 1.08(9) 0.92(19) 1.0(2) 0.79(6) 0.75(12) 0.91(9) 0.72(8) 0.72(5) 0.86(12) 0.83(7) 0.81(10) 0.5(2) 0.65(9) 0.85(7) 0.75(7) 0.74(6) 0.53(16) 0.76(8) 0.81(13) 0.87(13) 0.62(8) 0.79(10) 0.73(12) 0.68(15) 0.79(15) 0.63(13) 0.79(18) 0.71(16) 0.72(13) 0.74(16) 0.8(2) 0.65(9) 0.66(15) 1.0(5) 1.05(19) 0.6(2) 0.59(14) 0.83(12) 1.3(5) 0.77(14) Ac Jxπ d Is (eV b)e 2+f 2+f −0.87(15) −0.84(5) −0.98(2) −0.93(7) −0.95(6) −0.98(3) −0.94(4) −0.95(2) −0.99(9) −0.99(2) −0.75(31) −0.96(4) −0.95(6) −0.90(10) −0.94(8) −0.84(12) −1.00(1) −0.99(1) −0.98(3) −0.82(6) −0.90(15) −0.77(24) −0.93(9) 1 1(−) 1− 1 (1) 1− 1− 1 1− 1− 1− 1− 1 1− 1− (1) 1− 1− 1− 1 1 1− 1− 1− 1− 1− 1− 1− 1 1− 1− 1 1− 1 1 154(9) 20(6) 56(5) 129(8) 149(10) 49(9) 117(8) 199(11) 166(10) 454(23) 57(6) 138(9) 103(13) 119(15) 95(11) 260(30) 156(20) 105(16) 220(31) 211(30) 107(16) 89(14) 219(29) 216(32) 148(25) 336(42) 246(48) 214(54) 172(24) 84(17) 118(19) 265(40) 157(30) 233(31) Excitation energy. The uncertainty of this and the other quantities in the table is given in parentheses in units of the last digit. This energy value was deduced from the γ -ray energy measured at 127◦ including a recoil and Dopppler-shift correction. b Ratio of the intensities measured at angles of 90◦ and 127◦ . The expected values for an elastic dipole transition (spin sequence 0–1–0) and for an elastic quadrupole transition (spin sequence 0–2–0) are 0.74 and 2.15, respectively. c Azimuthal asymmetry A = (Iγ H − Iγ V )/(Iγ H + Iγ V ) of the intensities Iγ H and Iγ V measured with the detectors placed in the horizontal and vertical planes, respectively. A negative asymmetry indicates E1 radiation and a positive asymmetry indicates M1/E2 radiation. d Spin and parity deduced from angular correlation and azimuthal asymmetry, respectively, of the ground-state transition. e Energy-integrated scattering cross section. The value is an average over the values obtained at 90◦ and 127◦ . Below an excitation energy of 7.0 MeV the value was deduced from the measurement at 7.9 MeV electron energy; otherwise it is from the measurement at 11.2 MeV. f Spin and parity taken from previous work [35]. 024306-4 PYGMY DIPOLE STRENGTH IN 86 Kr AND . . . PHYSICAL REVIEW C 87, 024306 (2013) 10 E beam = 6.5 MeV Kr + steel horizontal detector − M1 vertical detector − E1 9 7 σ γγ’ (mb) 6432 40 Peaks + Continuum 5 4 Peaks 3 20 2 1 0 6200 Kr 6 6329 Counts 60 86 8 6463 80 86 6532 100 6300 6400 6500 0 6600 4 E γ (keV) 5 6 7 8 9 10 E γ (MeV) FIG. 3. (Color online) Parts of spectra of photons scattered from Kr in a steel container, measured during the irradiation with quasimonoenergetic polarized γ rays of 6.5 MeV. The spectrum plotted in red was measured with the vertical detector and contains E1 radiation, whereas the spectrum plotted in blue was measured with the horizontal detector and contains M1 radiation. Transitions in 86 Kr are labeled with their energies in keV. 86 86 belong to M1 transitions in Kr that are below the detection limit in the experiments at ELBE. III. DETERMINATION OF THE DIPOLE-STRENGTH DISTRIBUTION The determination of the dipole-strength distribution and the related photoabsorption cross section requires knowledge of the ground-state transitions and their branching ratios. As these cannot be derived directly from the measured spectra, we applied statistical methods described in the following. First, the in-beam spectrum measured with the empty steel container at Eekin = 11.2 MeV and containing signals of the two detectors at an equal observation angle was subtracted from the respective spectrum measured with the container filled with 86 Kr. Transitions following (n, γ ) reactions in the HPGe detectors and in surrounding materials are negligibly small and thus did not require correction. To correct the spectra for detector response, spectra of monoenergetic γ rays were calculated in steps of 10 keV using GEANT4. Starting from the high-energy end of the experimental spectrum, the simulated spectra were subtracted sequentially. The background radiation produced by atomic processes in the 86 Kr target was obtained from a GEANT4 simulation using the absolute photon flux deduced from the intensities of the transitions in 11 B. As found in previous studies [16–20, 37] the continuum in the spectrum of γ rays scattered from 86 Kr is considerably higher than the background due to atomic scattering. This continuum is formed by a large number of nonresolvable transitions with small intensities which are a consequence of the increasing nuclear level density at high energy in connection with the finite detector resolution (e.g., E ≈ 7 keV at Eγ ≈ 9 MeV). FIG. 4.(Color online) Scattering cross sections in 86 Kr, derived as σγ γ ′ =  Is / for energy bins of  = 0.2 MeV, not corrected for branching, as derived from the difference of the experimental spectrum and the atomic background (“Peaks + Continuum”; triangles) and from the resolved peaks only (“Peaks”; circles). The relevant intensity of the photons resonantly scattered from 86 Kr is obtained from a subtraction of the atomic background from the response-corrected experimental spectrum. The remaining intensity distribution includes the intensity contained in the resolved peaks as well as the intensity of the nuclear quasicontinuum. The scattering cross sections σγ γ ′ derived for energy bins of 0.2 MeV from the full intensity distribution are shown in Fig. 4. These values are compared with those derived from the integrated scattering cross sections of the resolved transitions given in Table I. One sees that the two curves have similar structures caused by the prominent peaks. However, the curve including also the continuum part of the spectrum contains altogether a strength that is by a factor of about 5 greater than the strength of the resolved peaks only. The full intensity distribution (resolved peaks and continuum) and the corresponding scattering cross sections shown in Fig. 4 contain (elastic) ground-state transitions and, in addition, branching transitions to lower-lying excited states (inelastic transitions) as well as subsequent transitions from those states to even lower states or to the ground state (cascade transitions). The different types of transitions cannot be clearly distinguished. However, for the determination of the photoabsorption cross section the intensities of the ground-state transitions are needed. Therefore, contributions of inelastic and cascade transitions have to be subtracted from the spectra. To correct the intensity distributions we performed simulations of γ -ray cascades from levels with J = 0, 1, 2 excited in the whole energy range up to the neutron-separation energy. The Monte Carlo code used is described in Ref. [38]. It works analogously to the code DICEBOX developed for γ -ray cascades following neutron capture [39] but in addition it includes the excitation from the ground state as a first step. In the present simulations, 1000 nuclear realizations, each starting with an excitation from the ground state, were 024306-5 PHYSICAL REVIEW C 87, 024306 (2013) R. SCHWENGNER et al. 1.0 86 Kr Ground−state transitions Iγ (arb. units) created with level densities derived from experiments [40]. We applied the statistical methods also for the low-energy part of the level scheme instead of using experimentally known low-lying levels in 86 Kr because this would require knowledge of the partial decay widths of all transitions populating these fixed levels. Fluctuations of the nearest-neighbor spacings were taken into account according to the Wigner distribution (see, e.g., Ref. [41]). The partial widths of the transitions to low-lying levels were assigned using a priori known strength functions for E1, M1, and E2 transitions. Fluctuations of the partial widths were treated by applying the Porter-Thomas distribution [42]. In the calculations, parameters of the back-shifted Fermigas (BSFG) model obtained from fits to experimental level densities [40], a = 9.83(14) MeV−1 and E1 = 0.97(6) MeV, were used. In the individual nuclear realizations, the values of a and E1 were varied within their uncertainties. As usual in the BSFG model, we assumed equal level densities for states with positive and negative parities of the same spin [40]. This assumption has been justified by the good agreement of level densities predicted by the BSFG model with experimental level densities of 1+ states in the energy range from 5 to 10 MeV obtained from the 90 Zr(3 He, t)90 Nb reaction [43] and with experimental level densities of 2+ and 2− states in 90 Zr studied in the 90 Zr(e, e′ ) and 90 Zr(p, p′ ) reactions [44]. The extended analysis of the 90 Zr(3 He, t)90 Nb reaction in Ref. [44] indicates however fluctuations of the level density of 1+ states in 90 Nb around the predictions of the BSFG model. For the E1, M1, and E2 photon strength functions Lorentz parametrizations [1] were used. The parameters of the Lorentz curve for the E1 strength were determined according to the prescription given in Ref. [7], resulting in E0 = 17.1 MeV and Ŵ = 4.7 MeV at zero deformation. The integral over this curve is consistent with the Thomas-Reiche-Kuhn (TRK) sum rule π σ Ŵ = 60NZ/A MeV mb [45]. The parameters for the M1 2 0 and E2 strengths were taken from global parametrizations of M1 spin-flip resonances and E2 isoscalar resonances, respectively [2]. Spectra of γ -ray cascades were generated for groups of levels in 100-keV bins in each of the 1000 nuclear realizations. For illustration, the distributions resulting from 10 individual nuclear realizations populating levels in a 100-keV bin around 9 MeV are shown in Fig. 5. The levels were created randomly starting from the ground state, where however the strength function and level density are small. In contrast, starting with the known first excited state at 1.565 MeV would cause a gap in the spectra between Ex = 1.565 MeV and the considered excitation energy Ex . A spectrum simulated for a given energy bin is comparable to a spectrum measured using monoenergetic γ rays of this energy. For the present case, this comparison is however difficult because of the contribution of the steel container to the measured spectrum. Starting from the high-energy end of the experimental spectrum, which contains ground-state transitions only, the simulated intensities of the ground-state transitions were adjusted to the experimental ones in the considered bin and the intensity distribution of the branching transitions was subtracted from the experimental spectrum. Applying this procedure step-by-step for each energy bin moving toward 0.5 Branching transitions 0.0 0 1 2 3 4 5 6 7 8 9 10 Eγ (MeV) FIG. 5. Simulated intensity distribution of transitions depopulating levels in a 100-keV bin around 9 MeV in 86 Kr. The squares depict the intensities obtained from 10 individual nuclear realizations. the low-energy end of the spectrum one obtains the intensity distribution of the ground-state transitions. Simultaneously, the branching ratios b0 of the ground-state transitions are deduced for each energy bin . In an individual nuclear realization, the branching ratio b0 is calculated as the ratio of the sum of the intensities of the ground-state transitions from all levels in  to the total intensity of all transitions depopulating those levels to any low-lying levels including the ground state [16,18]. The absorption cross section for a bin is obtained by dividing the summed intensities in a bin of the experimental intensity distribution of the ground-state transitions by the corresponding branching ratio as σγ = σγγ ′ /b0 . Finally, the absorption cross sections of each bin were obtained by averaging over the values of the 1000 nuclear realizations. For the uncertainty of the absorption cross section a 1σ deviation from the mean has been taken. The individual branching ratios of 10 nuclear realizations are shown in Fig. 6. The mean branching ratio of the 1000 realizations decreases from about 90% for low-lying states, where only few possibilities for the deexcitation to lower-lying states exist, to about 40% at the neutron-separation energy Sn = 9.9 MeV. Toward low energy the uncertainty of b0 increases due to level-spacing fluctuations and the decreasing level density. The large fluctuations below about 6 MeV make these values useless. Note that the mean branching ratio is not representative for transitions with large intensities such as the resolved transitions given in Table I. It turns out from the cascade simulations that the branching ratios of ground-state transitions deexciting states with integrated scattering cross sections like the ones given in Table I are in the order of b0 ≈ 85% to 99% which is in agreement with experimental findings. This behavior was discussed in our study of the neighboring N = 50 isotone 88 Sr [18]. In measurements with monoenergetic γ rays, the branching ratios can be deduced directly from the experimental intensity distributions of the elastic and inelastic transitions and do not need to be simulated. This is however difficult in the present case because of the 024306-6 PYGMY DIPOLE STRENGTH IN 86 Kr AND . . . PHYSICAL REVIEW C 87, 024306 (2013) 25 100 86 Kr 86 σγ (mb) 60 Δ b0 (%) 80 40 20 0 (γ,n) Kr 20 15 (γ,γ’) 10 5 Sn 4 5 6 7 8 9 10 0 Ex (MeV) 4 5 6 7 8 9 10 11 12 Ex (MeV) FIG. 6. Branching ratios of ground-state transitions as obtained from simulations of γ -ray cascades for 86 Kr. The squares represent the values of 10 individual nuclear realizations. contributions of the radiation from the steel container. We deduced branching ratios from spectra measured at HIγ S for the cases of 98 Mo [46] and 136 Ba [29]. The analysis of these spectra was analogous to the one just described in connection with the experiments at ELBE; i.e., the spectra were corrected for detector response, the atomic background was subtracted, and the intensity in the continuum was also considered. This procedure is illustrated in Ref. [29]. The experimental branching ratios deduced in this way from spectra measured at HIγ S for 98 Mo [46] and 136 Ba [29] are in good agreement with the ones obtained from the cascade simulations. This agreement proves the reliability of the simulations of γ -ray cascades as used in the present analysis. In a recently published study of 142 Nd at HIγ S [47] branching ratios are shown that were deduced from resolved peaks and do not include strength in the continuum (cf. Fig. 1 of Ref. [47]), in contrast to the just mentioned procedure applied in Refs. [29,46]. That causes large branching ratios of up to 100% at energies up to 6 MeV, as usual for strong transitions (cf. values just given and Ref. [18]), and produces pronounced kinks in the curve of b0 versus Eγ instead of a smooth behavior. As pointed out in our investigation of 88 Sr [18], these strong transitions do not obey statistical characteristics such as Porter-Thomas distributions. In Ref. [47] the b0 values deduced from transitions in 142 Nd were compared with the ones obtained from cascade simulations with DICEBOX using various assumptions for the input strength functions. A reproduction of the discontinuous behavior of the experimental b0 was achieved with the very specific assumption of a small soft pole of 0.6-mb maximum cross section at Eγ = 1.0 MeV, which was applied to states in a narrow energy window from 4.9 to 6.3 MeV. This energy window is below the PDR region in 142 Nd and does not affect the absorption cross section above 6 MeV. Indeed, the absorption cross section of 142 Nd presented in Ref. [47] resembles the one deduced for 136 Ba from our experiments at ELBE and HIγ S [29] in magnitude and location, although the double-hump structure seen in 142 Nd FIG. 7. (Color online) Photoabsorption cross section deduced from the present 86 Kr(γ , γ ′ ) experiments at ELBE after correction for branching transitions (red circles) in comparison with 86 Kr(γ , n) cross sections obtained by using monoenergetic γ rays at HIγ S (green squares) [48]. The black dashed line is a Lorentz curve with parameters given in the text. is less pronounced in 136 Ba. This similarity confirms our earlier finding that the cross section corrected for branching on the basis of cascade simulations is not very sensitive to variations of the shape of the input strength function, which we studied for the case of 90 Zr [19]. The photoabsorption cross sections derived from the present experiments for 86 Kr are shown in Fig. 7. In addition, the values obtained from a 86 Kr(γ , n) experiment at the HIγ S facility [48] are shown. The total photoabsorption cross section was deduced by combining the present (γ , γ ′ ) data with the (γ , n) data of Ref. [48]. The values averaged over energy bins of 500 keV are shown in Fig. 8. IV. DISCUSSION The experimental absorption cross section shown in Fig. 8 has a peak at about 6.5 MeV and a resonance-like structure in the energy range from about 7 to 11 MeV. Such a structure has also been found in the neighboring N = 50 isotones and is shown for 88 Sr [18], 90 Zr [19], and 92 Mo [17] in Fig. 9. This structure strongly resembles the PDR observed in other neutron-rich nuclei, which is explained as a vibration of the excessive neutrons against the symmetric N = Z system and is expected to correlate with the ratio N/Z [12–14]. To examine the behavior of the strength with varying N/Z we deduced the sums of the measured absorption cross sections weighted with the corresponding energy bins in the energy range from 6 to 10 MeV for the even-even isotones. The  MeV resulting quantities 10 6 MeV σi Ei are shown in Fig. 10. The ratio N/Z varies from 1.39 for 86 Kr to 1.19 for 92 Mo. There is no clear tendency discernible for this series of isotones. However, the closed proton subshells, π (1p3/2 ) at Z = 38 (88 Sr) 024306-7 PHYSICAL REVIEW C 87, 024306 (2013) R. SCHWENGNER et al. 100 60 88 Sr 90 86 Kr (MeV mb) 50 σ γ (mb) 10 Zr QRPA Kr EXP 40 92 Mo 2ph 30 3ph Σ σ ΔE i 2ph−QPM 86 i 3ph−QPM QRPA 1 20 6 MeV < Ei < 10 MeV 10 N = 50 Sn 0 4 5 6 7 8 9 10 11 12 36 38 40 42 Z E x (MeV) FIG. 8. (Color online) Total photoabsorption cross section of 86 Kr including (γ , γ ′ ) and (γ , n) data (red circles) compared with results of QRPA (black line) and two-phonon QPM (blue line) and threephonon QPM (green line) calculations. The QRPA and QPM solutions were folded with Lorentz curves of 0.5 MeV width. The black dashed line is a Lorentz curve with parameters given in Sec. III. and in addition π (1p1/2 ) at Z = 40 (90 Zr), seem to cause larger strength. In fact, there are intense isolated resonances in the absorption cross sections of these nuclides [18,19]. For the chain of stable N = 82 isotones with ratios N/Z varying from 1.52 for 136 Xe to 1.32 for 144 Sm an increase of the strength summed up to an energy of 8 MeV with increasing N/Z was observed [49,50]. This feature was attributed at least in part to the fact that there is strength missing because of the sensitivity limit of the experiments and it tends to vanish when applying a 100 88 90 σγ (mb) 92 10 Sr Zr Mo 1 4 5 6 7 8 9 10 11 12 Ex (MeV) FIG. 9. (Color online) Total photoabsorption cross sections of Sr (blue squares), and 90 Zr (red circles), 92 Mo (green triangles) and a Lorentz curve (black dashed line) adjusted to 88 Sr(γ , n) cross sections. The data were taken from Refs. [17–19]. 88 FIG. 10. (Color online) Energy-weighted sums of photoabsorption cross sections for the excitation energy region from 6 to 10 MeV for the even-even stable N = 50 isotones. The data for 88 Sr, 90 Zr, and 92 Mo were taken from Refs. [17–19]. corresponding correction [49,50]. Experiments on the N = 82 nuclides 139 La at ELBE [37] and 138 Ba at HIγ S [51] as well as on the N = 80 neighbor 136 Ba at ELBE and HIγ S [29] take into account the strength in the quasicontinuum and the branching ratios of the ground-state transitions. However, there is information on further N = 82 isotones needed for a systematic investigation of the PDR strength in that mass region. For the interpretation of the present data for 86 Kr in connection with the findings for the N = 50 isotones just mentioned we use a self-consistent theoretical approach based on the density-functional theory [52] and the QPM [53] as described in detail in Ref. [13]. The results for 86 Kr calculated using the QRPA and the three-phonon QPM [54] are compared with the experimental results in Fig. 8 and are discussed in the following. In the case of N = 50 isotones the neutron number is fixed and the proton number changes, which affects the thickness δr of the neutron skin as well. This can be seen in ground-state proton- and neutron-density distributions and in (relative) differences of proton and neutron root-mean-square (rms) radii,  δr = r 2 n − r 2 p , (3) defining the thickness of the neutron skin [19]. The tendencies of δr and of the total B(E1) strength up to 9 MeV with varying N/Z in the considered N = 50 isotones are illustrated by results of Hartree-Fock-Bogoliubov calculations shown in Figs. 11 and 12, respectively. The QRPA calculations of low-energy dipole states with excitation energies up to Ex = 22 MeV in 86 Kr, 88 Sr, 90 Zr, and 92 Mo nuclei show as a common feature a sequence of almost pure neutron 1− states located in the energy range Ex < 9 MeV, which is in agreement with our previous results for 88 Sr and 90 Zr presented in Ref. [19]. These states correspond to excitations of least bound neutrons from a valence orbit 024306-8 PYGMY DIPOLE STRENGTH IN 86 Kr AND . . . 0.15 86 Kr 88 δr (fm) PHYSICAL REVIEW C 87, 024306 (2013) Sr 0.1 90 Zr 92 QRPA 0.05 Mo E i < 9 MeV N = 50 0 36 38 40 42 Z FIG. 11. Calculated differences of neutron and proton rms radii defining the neutron skin thickness for N = 50 isotones. to a higher orbit, such as 0f5/2 → 1d5/2 , 1p1/2 → 2s1/2 , 1p3/2 → 1d5/2 , and 0g9/2 → 0h11/2 , but with only a minor proton contribution of less than 1% and they are related to neutron skin oscillations and the PDR. The structure of these states is dominated by one two-quasiparticle (2qp) neutron component, showing that these excitations are of noncollective character. Similar results were found in Z = 50 and N = 82 nuclei [13,55]. Insight into the properties of the excitations is gained from an inspection of transition densities calculated according to the procedure described in Ref. [14] and shown in Fig. 13. The transition densities of the 1− states with Ex < 9 MeV display clear signals of a PDR mode, in agreement with our previous studies [13,19]. Namely, we observe in-phase oscillations of protons and neutrons in the nuclear interior, whereas at the surface only neutrons contribute. These features are characteristic for dipole skin vibrations [13]. QRPA  calculations of the sum strength 90 MeV MeV B(E1)↑ related to 86 Kr 88 Sr 2 B(E1) (e fm ) 0.2 2 90 Zr QRPA Σ 0.1 92 Mo Ei < 9 MeV N = 50 0 36 38 40 42 Z FIG. 12. Total B(E1) strength up to 9 MeV obtained from QRPA calculations for N = 50 isotones. the PDR in N = 50 isotones are presented in Fig. 12. As can be seen, the total PDR strength decreases with increasing proton number, i.e., with decreasing N/Z. Finally, we note that the results are in agreement with the established connection between the calculated total PDR strength and the nuclear skin thickness defined in Eq. (3). The analysis of the evolution of dipole transition densities with increasing excitation energy allows us to distinguish between the PDR and other types of dipole excitations. In this respect, the states in the energy region of 9 < Ex < 10 MeV carry a signature different from the PDR. The protons and neutrons start to move out of phase, being compatible with the low-energy part of the GDR. A strong argument in this direction is that the observed amount of dipole strength located in the region 9 < Ex < 10 MeV cannot be directly connected to N/Z ratios and corresponding skins, as can be done for the dipole strength below 9 MeV. A further increase of energy in the range of 10 < Ex < 11 MeV leads to dynamic processes of collective excitations of different neutron and proton subshells of the nuclear interior, which in some cases could be in-phase as shown in Fig. 13. The presence of in-phase collective dipole excitations closely above the neutron threshold is also observed in collective model approaches [56], in which attempts are made to relate that to the PDR spectral component. However, it is clear that such more or less collective excitations including a considerable contribution of inner-shell neutrons and protons should not be interpreted in the same way as the genuine PDR mode explained by neutron skin oscillations. At Ex = 11 to 22 MeV a strong isovector oscillation corresponding to the excitation of the GDR is found. The comparison of QRPA with two-phonon (2ph) and three-phonon (3ph) QPM calculations which include even- and odd-parity states with spins 1  J  5 and excitation energies Ex < 11 MeV indicates that for the PDR region the coupling of PDR- and GDR-QRPA phonons and multiphonon states is very important. This coupling causes a fragmentation of the strength that results in a smoothing of the usual QRPA fluctuations and a shift of E1 strength toward lower energy, as is well visible in Fig. 8. Consequently, the QPM calculations predict about two times greater total E1 strength than the QRPA calculations do in the energy range 0 < Ex < 9 MeV and are in better agreement with the experimental findings. The enhanced strength in 86 Kr observed in the region above Ex = 6 MeV is related to the PDR as identified from the structure of the involved 1− states. The contribution of the GDR dynamics becomes significant at energies Ex > 9 MeV where a coupling among PDR, GDR, and multiphonon states reflects the properties of the dipole excitations at higher energies. With increasing excitation energy toward the GDR region, the sum B(E1) strengths obtained from QRPA and QPM calculations are of comparable amount, which suggests the predominance of one-phonon excitations of collective GDR type. The experimental total energy-integrated cross sections in  MeV the region of Ex = 6 to 10 MeV, 10 6 MeV σi Ei , are compared with sums over energy-integrated cross sections obtained  MeV 10 MeV according to 10 6 MeV Is /(MeV mb) = 6 MeV [4.03Ex /MeV B(E1)↑ /(e2 fm2 )] [5] from the QRPA and QPM calculations for the considered N = 50 isotones in Fig. 10. This comparison shows that the Is deduced from the one-, two-, and 024306-9 PHYSICAL REVIEW C 87, 024306 (2013) R. SCHWENGNER et al. 0.2 86 Kr 0.5 Ex = 9 − 10 MeV 0.1 0 0 Ex < 9 MeV −1 r ρ(r) (fm ) −0.1 Ex = 10 − 11 MeV Ex = 11 − 22 MeV −0.5 0.2 88 0.5 Ex = 9 − 10 MeV 2 Sr 0.1 0 0 −0.1 Ex < 9 MeV Ex = 10 − 11 MeV −0.5 0 5 0 5 0 5 Ex = 11 − 22 MeV 0 5 r (fm) 0.2 90 Zr 0.5 Ex = 9 − 10 MeV 0.1 0 0 Ex < 9 MeV −1 r ρ(r) (fm ) −0.1 Ex = 10 − 11 MeV Ex = 11 − 22 MeV −0.5 0.2 92 0.5 Ex = 9 − 10 MeV 2 Mo 0.1 0 0 −0.1 Ex = 10 − 11 MeV Ex < 9 MeV Ex = 11 − 22 MeV −0.5 0 5 0 5 0 5 0 5 r (fm) FIG. 13. Transition densities of neutrons (solid lines) and protons (dashed lines) in N = 50 isotones. three-phonon calculations of 1− states with energies up to Ex = 10 MeV reproduce the experimental values and the systematic behavior of the total E1 strength with varying N . The comparison demonstrates that the relative differences between the E1 strengths calculated within QRPA and QPM for the energy region above Ex = 9 MeV decrease with the increase of the excitation energy toward the GDR, which is expected because of the weak dependence of the GDR strength on the N/Z ratios in the N = 50 isotones. Hence, the tendency shown in Fig. 12 for the QRPA calculations is washed out. Similar results were found for tin nuclei [13]. In general, both the QRPA and QPM calculations are found to be 024306-10 PYGMY DIPOLE STRENGTH IN 86 Kr AND . . . PHYSICAL REVIEW C 87, 024306 (2013) reliable for the description of the experimental values of the total photoabsorption cross sections in N = 50 isotones and their isotonic dependence at high energy. V. SUMMARY The dipole-strength distribution in 86 Kr up to the neutronseparation energy has been studied in photon-scattering experiments at the ELBE accelerator using various electron energies. Ground-state transitions were identified by comparing the transitions observed at different electron energies. We identified 39 levels. Spin J = 1 was deduced from angular correlations of ground-state transitions for 33 levels. The parities of 22 states were determined from azimuthal asymmetries of intensities measured in an experiment with monoenergetic and polarized γ radiation at the HIγ S facility. The intensity distribution obtained from the measured spectra after a correction for detector response and a subtraction of atomic background in the target contains a continuum part in addition to the resolved peaks. It turns out that the dipole strength in the resolved peaks amounts to about 16% of the total dipole strength whereas the continuum contains about 84%. An assignment of inelastic transitions to particular levels and, thus, the determination of branching ratios was in general not possible. To get information about the intensities of inelastic transitions to low-lying levels we have applied statistical methods. By means of simulations of γ -ray cascades intensities of branching transitions were estimated and subtracted from the experimental intensity distribution and the intensities of ground-state transitions could be corrected on average for their branching ratios. A comparison of the photoabsorption cross section obtained in this way from the present (γ , γ ′ ) experiments with (γ , n) data shows a smooth connection of the data of the two different experiments and gives new information about the extension of the dipole-strength function toward energies around and below the threshold of the (γ , n) reaction. In comparison with a straightforward approximation of the GDR by a Lorentz curve one observes extra E1 strength in the energy range from 6 to 11 MeV which is mainly concentrated in strong peaks. [1] P. Axel, Phys. Rev. 126, 671 (1962). [2] R. Capote et al., Nucl. 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