Pygmy dipole strength in Zr
Abstract
The dipole response of the nucleus Zr was studied in photon-scattering experiments at the electron linear accelerator ELBE with bremsstrahlung produced at kinetic electron energies of 7.9, 9.0, and 13.2 MeV. We identified 189 levels up to an excitation energy of 12.9 MeV. Statistical methods were applied to estimate intensities of inelastic transitions and to correct the intensities of the ground-state transitions for their branching ratios. In this way we derived the photoabsorption cross section up to the neutron-separation energy. This cross section matches well the photoabsorption cross section obtained from data and thus provides information about the extension of the dipole-strength distribution toward energies below the neutron-separation energy. An enhancement of strength has been found in the range of 6 MeV to 11 MeV. Calculations within the framework of the quasiparticle-phonon model ascribe this strength to a vibration of the excessive neutrons against the neutron-proton core, giving rise to a pygmy dipole resonance.
pacs:
25.20.Dc, 21.10.Tg, 21.60.Jz, 23.20.-g, 27.50.+eI Introduction
Gamma-ray strength functions, in particular dipole-strength functions, are an important ingredient for the analysis of photodisintegration reactions as well as of the inverse reactions like neutron capture. These reactions play an important role for specific processes of the nucleosynthesis. Moreover, an improved experimental and theoretical description of neutron-capture reactions is important for next-generation nuclear technologies.
The measurement of dipole-strength distributions from low excitation energy up to the neutron-separation energy delivers information about magnitude and structure of the low-energy tail of the giant dipole resonance (GDR), in particular about possible additional vibrational modes like the so-called pygmy dipole resonance (PDR). Dipole-strength distributions up to neutron-separation energies have been studied for only few nuclides in experiments with monoenergetic photons (see, e.g., Refs. axe70 (); dat73 (); las87 (); ala87 ()) and in experiments with bremsstrahlung (see, e.g., Ref. kne06 () and Refs. therein). The bremsstrahlung facility sch05 () at the superconducting electron accelerator ELBE of the research center Dresden-Rossendorf opens up the possibility to study the dipole response of stable nuclei with even the highest neutron-separation energies in photon-scattering experiments. In the course of a systematic study of dipole-strength distributions for varying neutron and proton numbers in nuclei around = 90 sch07 (); rus08 () we have investigated the = 50 nuclide Zr.
In earlier experiments, the 2 states at 2186, 3308, 3842, 4120, 4230, 4680, and the 1 states at 4580 and 5504 keV were studied with bremsstrahlung produced by 5 MeV electrons met72 (); met74 (). About 20 further states were found in experiments at higher energies up to 10 MeV can76 (); nds97 (). For the 15 = 1 states above 6 MeV negative parity was deduced from an experiment with polarized photons ber84 ().
In the present study we identified about 190 levels up to 12.9 MeV. We applied statistical methods to estimate the intensities of inelastic transitions to low-lying excited levels and to correct the intensities of the elastic transitions to the ground state with their branching ratios. The dipole-strength distribution deduced from the present experiments is compared with predictions of the quasiparticle-phonon model.
Ii Experimental Methods and results
The nuclide Zr was studied in photon-scattering experiments at the superconducting electron accelerator ELBE of the research center Dresden-Rossendorf. Bremsstrahlung was produced using electron beams of 7.9, 9.0, and 13.2 MeV kinetic energy. The average currents were about 330 A in the measurement at 7.9 MeV and about 500 A in the measurements at higher energy. The electron beams hit radiators consisting of niobium foils with thicknesses of 4 m in the low-energy measurements and of 7 m in the measurement at 13.2 MeV. A 10 cm thick aluminum absorber was placed behind the radiator to reduce the low-energy part of the bremsstrahlung spectrum (beam hardener). The collimated photon beam impinged onto the target with a flux of about s in a spot of 38 mm diameter. The target was a disk with a diameter of about 20 mm to enable an irradiation with a constant flux density over the target area. The target consisted of 4054.2 mg of ZrO enriched to 97.7%, combined with 339.5 mg of B, enriched to 99.52% and also formed to a disk of 20 mm diameter, that was used for the determination of the photon flux. Scattered photons were measured with four high-purity germanium (HPGe) detectors of 100% efficiency relative to a NaI detector of 3 in. diameter and 3 in. length. All HPGe detectors were surrounded by escape-suppression shields made of bismuth germanate scintillation detectors. Two HPGe detectors were placed vertically at 90 relative to the photon-beam direction at a distance of 28 cm from the target. The other two HPGe detectors were positioned in a horizontal plane at 127 to the beam at a distance of 32 cm from the target. Absorbers of 8 mm Pb plus 3 mm Cu and of 3 mm Pb plus 3 mm Cu were placed in front of the detectors at 90 and 127, respectively, in the measurements at 7.9 and 9.0 MeV whereas in the measurement at 13.2 MeV absorbers of 13 mm Pb plus 3 mm Cu and of 8 mm Pb plus 3 mm Cu were used for the detectors at 90 and 127, respectively. A detailed description of the bremsstrahlung facility is given in Ref. sch05 ().
Spectra of scattered photons were measured for 56 h, 65 h, and 97 h in the experiments at 7.9, 9.0, and 13.2 MeV electron energy, respectively. Parts of a spectrum including events measured with the two detectors placed at 127 relative to the beam at an electron energy of 13.2 MeV are shown in Fig. 1.
ii.1 The photon-scattering method
In photon-scattering experiments the energy-integrated scattering cross section of an excited state at the energy can be deduced from the measured intensity of the respective transition to the ground state (elastic scattering). It can be determined relative to the known integrated scattering cross sections of states in B ajz90 ():
(1) |
Here, and denote the measured intensities of a considered ground-state transition at and of a ground-state transition in B at , respectively, observed at an angle to the beam. and describe the angular correlations of these transitions. The quantities and are the numbers of nuclei in the Zr and B targets, respectively. The quantities and stand for the photon fluxes at the energy of the considered level and at the energy of a level in B, respectively.
The integrated scattering cross section is related to the partial width of the ground-state transition according to
(2) |
where is the elastic scattering cross section, , and denote energy, spin and total width of the excited level, respectively, and is the spin of the ground state.
For the determination of the level widths one is faced with two problems. First, a considered level can be fed by transitions from higher-lying states. The measured intensity of the ground-state transition is in this case higher than the one resulting from a direct excitation only. As a consequence, the integrated cross section deduced from this intensity contains a part originating from feeding in addition to the true integrated scattering cross section: . Furthermore, a considered level can deexcite not only to the ground state, but also to low-lying excited states (inelastic scattering). In this case, not all observed transitions are ground-state transitions. To deduce the partial width of a ground-state transition and the integrated absorption cross section one has to know the branching ratio . If this branching ratio cannot be determined, only the quantity can be deduced (cf. Eq. (2)).
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 and differ most at these angles. The expected values are and taking into account opening angles of and of the detectors at and , respectively.
ii.2 Detector response and photon flux
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. For the determination of the dipole-strength distribution described in Sec. III the experimental spectrum has to be corrected for detector response, for the absolute efficiency, and for the absolute photon flux, for background radiation, and for atomic processes induced by the impinging photons in the target material. The detector response has been simulated using the program package GEANT3 cer93 (). The reliability of the simulation was tested by comparing simulated spectra with measured ones as described in Refs. sch07 (); rus08 ().
The absolute efficiencies of the HPGe detectors were determined experimentally up to 1333 keV from measurements with Na, Co, Ba, and Cs calibration sources. The efficiency curve calculated with GEANT3 was scaled to fit the absolute experimental values. To check the simulated efficiency curve at higher energies we used an uncalibrated Co source produced in-house using the FeCo reaction. The relative efficiency values obtained from the Co source adjusted to the calculated value at 1238 keV are consistent with the simulated curve as is illustrated in Ref. sch07 ().
The absolute photon flux was determined from intensities and known integrated scattering cross sections of transitions in B which was combined with the Zr target (cf. Sec. II). For interpolation, the photon flux was calculated using a code hau05 () based on the approximation given in Ref. roc72 () and including a screening correction according to Ref. sal87 (). In addition, the flux was corrected for the attenuation by the beam hardener. This flux was adjusted to the experimental values as is shown in Fig. 2. The transition from the level at 7285 keV could not be used because of superposition by another transition.
ii.3 Experiments at various electron energies
Measurements at various electron energies allow us to estimate the influence of feeding on the integrated cross sections. Ratios of the quantities obtained for levels in Zr from measurements at different electron energies are shown in Fig. 3.
The plotted ratios reveal that (i) only levels below 6 MeV are influenced considerably by feeding and (ii) levels in the range of 4 to 6 MeV are mainly fed by levels above 9 MeV. Transitions found in the measurement at = 7.9 MeV are assumed to be ground-state transitions. Transitions additionally observed up to 7.9 MeV in the measurements at 9.0 MeV and 13.2 MeV are therefore considered as inelastic transitions from high-lying to low-lying excited states. In the same way, transitions in the range from 7 – 9 MeV observed in the measurement at 9.0 MeV are considered as ground-state transitions, whereas transitions in this energy range found additionally at 13.2 MeV are considered as branchings to low-lying states. By comparing the measurements in this way, transitions from high-lying to excited low-lying states may be filtered out. The remaining transitions, assumed as ground-state transitions, have been used to derive the corresponding level energies which are listed in Table 1 together with spin assignments deduced from angular distributions of the ground-state transitions, integrated scattering cross sections, and the quantities .
(keV) ^{1}^{1}1 Excitation energy. The uncertainty in parentheses is given in units of the last digit. This energy was deduced from the -ray energy measured at 127 to the beam by including a recoil and Doppler correction. | ^{2}^{2}2 Ratio of the intensities of the ground-state transitions measured at angles of 90 and 127. The expected values for an elastic pure-dipole transition (spin sequence ) and for an elastic quadrupole transition (spin sequence ) are 0.74 and 2.18, respectively. | ^{3}^{3}3 Spin and parity of the state. | ^{4}^{4}4 Ratio of integrated scattering+feeding cross sections deduced at different electron energies. The deviation from unity is a measure of feeding. | ^{4}^{4}4 Ratio of integrated scattering+feeding cross sections deduced at different electron energies. The deviation from unity is a measure of feeding. | ^{4}^{4}4 Ratio of integrated scattering+feeding cross sections deduced at different electron energies. The deviation from unity is a measure of feeding. | (eV b) ^{5}^{5}5 Integrated scattering cross section. The values up to the level at 6875 keV were deduced from the measurement at = 7.9 MeV, those from the level at 6960 keV up to the level at 8832 keV were deduced from the measurement at = 9.0 MeV, and the values given for levels at higher energies were deduced from the measurement at = 13.2 MeV. | (meV) ^{6}^{6}6 Partial width of the ground-state transition multiplied with the branching ratio . The values deduced from the given for states up to the 5504 keV state exceed the values of previous studies nds97 () outside their uncertainties which may indicate that the present values are influenced by feeding. These values are therefore not given in this table. An estimate of the partial width can be obtained from after correction for a mean branching ratio = 0.9(1)% for resolved ground-state transitions (cf. Sec. III). |
2186.2(1) | 2^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 11.3(33) | 88(22) | 7.7(15) | 19(4) | ||
3308.0(2) | 2^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 10.4(45) | 85(35) | 8.2(17) | 4.7(19) | ||
3842.0(2) | 2^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 2.3(5) | 12.6(20) | 5.5(11) | 26(4) | ||
3932.4(6) | 8.3(32) | ||||||
4507.0(8) | 21(10) | ||||||
4578.3(3) | 1.2(3) | 5.1(12) | 4.2(9) | 16(4) | |||
5183.0(5) | 1.4(6) | 5.4(19) | 3.9(10) | 7.1(24) | |||
5304.5(3) | 0.92(20) | 1.4(2) | 1.5(3) | 57(8) | |||
5503.6(3) | 1.2(3) | 1.9(4) | 1.6(3) | 34(6) | |||
5785.0(4) | 0.78(19) | 0.70(13) | 0.90(20) | 50(8) | 145(22)^{8}^{8}8 Value deduced under the assumption of = 1. | ||
5807.9(3) | 0.90(20) | 1.8(9) | 2.0(11) | 78(10) | 228(30)^{8}^{8}8 Value deduced under the assumption of = 1. | ||
5884.4(4) | 0.70(17) | 1.1(2) | 1.6(4) | 48(8) | 143(23)^{8}^{8}8 Value deduced under the assumption of = 1. | ||
6295.8(2) | 0.75(3) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.70(13) | 0.66(8) | 0.94(18) | 740(63) | 2545(218) |
6389.8(3) | 0.82(8) | 1 | 0.90(23) | 0.84(19) | 0.93(20) | 82(15) | 290(54) |
6424.3(2) | 0.74(4) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.74(14) | 0.70(9) | 0.94(18) | 479(43) | 1716(153) |
6565.7(3) | 0.79(15) | 1 | 0.71(18) | 0.89(17) | 1.3(3) | 66(9) | 245(34) |
6669.2(7) | 0.64(16) | 1 | 0.74(31) | 0.89(30) | 1.2(4) | 29(8) | 111(32) |
6761.4(2) | 0.76(3) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.83(16) | 0.79(10) | 0.95(18) | 644(60) | 2553(237) |
6875.4(2) | 0.73(4) | 1 | 1.09(22) | 0.84(13) | 0.77(15) | 198(22) | 813(91) |
6960.4(7) | 0.77(11) | 1 | 1.4(36) | 44(10) | 183(41) | ||
7042.0(7) | 0.74(19) | 1 | 0.74(25) | 25(6) | 109(28) | ||
7085.6(10) | 0.83(21) | (1) | 1.1(4) | 30(10) | 130(43) | ||
7198.2(6) | 0.49(20) | 1 | 1.1(3) | 45(10) | 201(47) | ||
7249.0(3) | 0.78(7) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 1.1(2) | 99(17) | 450(79) | ||
7280.9(7)^{9}^{9}9 This transition is superimposed by a transition in B. | |||||||
7361.0(6) | 0.57(17) | 1 | 1.0(3) | 33(7) | 153(33) | ||
7387.6(4) | 0.68(10) | 1 | 0.96(21) | 75(14) | 355(65) | ||
7424.5(10) | 1.4(6) | 1.1(5) | 14(5) | 69(24)^{8}^{8}8 Value deduced under the assumption of = 1. | |||
7433.8(8) | 0.52(21) | 1 | 1.3(5) | 19(6) | 93(28) | ||
7468(2) | 1.5(5) | 1.8(6) | 12(4) | 61(18)^{8}^{8}8 Value deduced under the assumption of = 1. | |||
7474.9(3) | 0.96(10) | (1) | 1.2(3) | 127(23) | 617(113) | ||
7685.8(4) | 0.65(10) | 1 | 1.2(3) | 70(13) | 358(68) | ||
7702.9(3) | 0.83(7) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 1.3(3) | 158(28) | 815(142) | ||
7723.1(9) | 0.9(3) | 1.4(9) | 20(6) | 106(31)^{8}^{8}8 Value deduced under the assumption of = 1. | |||
7759.7(6) | 0.82(19) | (1) | 1.6(5) | 38(9) | 198(45) | ||
7779.0(6) | 0.77(17) | 1 | 1.0(3) | 40(9) | 212(47) | ||
7807.9(3) | 0.75(6) | 1 | 1.0(2) | 125(23) | 659(120) | ||
7857.8(7) | 0.84(25) | (1) | 2.0(8) | 34(9) | 183(50) | ||
7935.6(3) | 0.81(6) | 1 | 0.86(17) | 209(36) | 1141(199) | ||
7976.6(4) | 0.67(7) | 1 | 1.1(2) | 125(23) | 691(124) | ||
8006.9(8) | 0.71(25) | 1 | 0.78(27) | 36(9) | 197(49) | ||
8067.4(5) | 0.99(27) | (1) | 0.73(19) | 55(13) | 312(74) | ||
8110.0(8) | 0.71(9) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.96(21) | 124(24) | 704(136) | ||
8131.9(4) | 0.74(9) | (1)^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.93(20) | 154(28) | 884(164) | ||
8144(2) | 20(13) | 115(75)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
8166.7(5) | 0.89(14) | (1) | 0.88(20) | 98(19) | 566(111) | ||
8221.2(8) | 0.80(13) | 1 | 0.92(24) | 57(12) | 332(73) | ||
8235.6(3) | 0.74(4) | 1 | 0.89(18) | 254(44) | 1495(260) | ||
8250.7(5) | 0.68(7) | 1 | 0.98(22) | 85(16) | 504(98) | ||
8295.3(10) | 0.89(29) | (1) | 0.87(25) | 40(11) | 241(67) | ||
8313.0(7) | 0.73(18) | 1 | 1.4(4) | 70(15) | 420(95) | ||
8334.1(5) | 0.72(18) | 1 | 1.2(3) | 90(20) | 539(121) | ||
8357.5(18) | 0.32(18) | 1 | 16(7) | 97(42) | |||
8382.1(10) | 1.1(4) | (1) | 25(6) | 157(32) | |||
8403.7(11) | 43(7) | 263(43)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
8413.5(4) | 0.87(9) | 1 | 1.0(2) | 212(38) | 1300(235) | ||
8440.6(4) | 0.87(10) | 1 | 0.96(20) | 224(40) | 1382(250) | ||
8467.7(15) | 1.0(5) | 31(17) | 193(106) | ||||
8501.2(4) | 0.81(8) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.81(17) | 346(63) | 2166(393) | ||
8518(3) | 1.6(6) | 0.61(33) | 40(16) | 253(98)^{8}^{8}8 Value deduced under the assumption of = 1. | |||
8544(4) | 8(3) | 51(19)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
8553.5(12) | 0.12(6) | 1 | 79(8) | 504(50) | |||
8588.3(7) | 0.58(13) | 1 | 0.57(15) | 93(21) | 597(133) | ||
8598.2(10) | 0.59(22) | 1 | 1.3(4) | 42(12) | 270(78) | ||
8625.6(10) | 0.72(27) | 1 | 1.6(6) | 37(11) | 239(69) | ||
8664.1(5) | 0.68(16) | 1 | 0.93(27) | 59(14) | 385(92) | ||
8716.6(5) | 0.75(6) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 0.94(20) | 176(33) | 1162(220) | ||
8751.0(8) | 0.36(14) | 1 | 62(15) | 410(97) | |||
8760.4(5) | 0.58(10) | 1 | 0.77(17) | 162(31) | 1077(203) | ||
8812.0(13) | 0.76(24) | 1 | 1.5(6) | 37(13) | 246(86) | ||
8833.2(8) | 0.67(15) | 1 | 1.0(3) | 83(20) | 561(134) | ||
8874.9(9) | 0.37(13) | 1 | 41(11) | 280(75) | |||
8903.0(8) | 57(6) | 394(43)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
8927.4(4) | 127(13) | 878(91)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
8978.4(9) | 0.8(4) | (1) | 88(31) | 615(215) | |||
8985(2) | 45(13) | 315(90)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9004.7(5) | 0.45(23) | 1 | 34(11) | 239(78) | |||
9014.0(8) | 24(14) | 170(101)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9034.0(8) | 35(7) | 247(50)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9043.6(4) | 0.44(6) | 1 | 71(10) | 503(68) | |||
9053.5(7) | 38(7) | 270(49)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9085.1(3) | 0.60(9) | 1 | 129(15) | 925(109) | |||
9111.1(6) | 0.64(15) | 1 | 141(20) | 1018(140) | |||
9123.6(7) | 126(17) | 913(126)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9137.5(7) | 185(22) | 1338(163)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9148.5(3) | 0.60(4) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 703(66) | 5103(480) | |||
9164.9(7) | 107(14) | 778(104)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9177.5(5) | 162(18) | 1182(134)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9187(3) | 45(13) | 329(95)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9196.5(3) | 0.72(7) | (1)^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 252(25) | 1849(186) | |||
9260.5(6) | 0.67(9) | 1 | 149(19) | 1109(140) | |||
9292.8(5) | 0.65(7) | 1 | 216(24) | 1619(181) | |||
9309.4(7) | 0.71(10) | 1 | 137(18) | 1033(135) | |||
9333.4(6) | 0.67(10) | 1^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 141(19) | 1064(146) | |||
9373.2(7) | 111(21) | 843(161)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9392.4(8) | 0.45(13) | 1 | 102(19) | 780(145) | |||
9409.4(11) | 71(16) | 543(123)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9424.3(10) | 79(17) | 608(129)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9444.7(4) | 0.65(8) | 1 | 221(28) | 1707(219) | |||
9465.1(5) | 0.66(11) | 1 | 169(25) | 1317(193) | |||
9486.8(4) | 0.75(10) | 1 | 226(32) | 1765(251) | |||
9510.5(13) | 0.88(35) | (1) | 45(16) | 350(124) | |||
9524.1(13) | 0.47(22) | 1 | 44(14) | 348(112) | |||
9539.2(5) | 0.85(13) | 1 | 154(22) | 1213(175) | |||
9551.4(6) | 0.47(9) | 1 | 160(23) | 1267(185) | |||
9563.0(6) | 0.64(17) | 1 | 180(28) | 1424(221) | |||
9609.2(7) | 72(22) | 573(177)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9625.1(8) | 58(16) | 469(129)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9640.4(8) | 0.6(3) | 1 | 56(14) | 456(115) | |||
9666.0(8) | 0.7(3) | (1) | 39(9) | 318(71) | |||
9678.3(7) | 0.67(19) | (1)^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 67(10) | 545(83) | |||
9686.9(6) | 0.66(17) | 1 | 72(11) | 591(89) | |||
9733.2(5) | 0.5(3) | 1 | 55(8) | 450(68) | |||
9741.7(7) | 33(6) | 275(51)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9754.0(6) | 0.46(29) | 1 | 50(9) | 413(71) | |||
9784.6(5) | 1.37(27) | 111(15) | 921(122)^{8}^{8}8 Value deduced under the assumption of = 1. | ||||
9805.4(10) | 1.6(6) | 41(9) | 341(73)^{8}^{8}8 Value deduced under the assumption of = 1. | ||||
9843.4(6) | 0.77(16) | 1 | 84(15) | 702(126) | |||
9855.5(8) | 0.55(17) | 1 | 58(13) | 488(109) | |||
9872.4(4) | 0.49(12) | 1 | 126(24) | 1067(207) | |||
9890.7(13) | 0.7(5) | (1) | 81(34) | 686(289) | |||
9901.9(13) | 70(28) | 592(242)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
9932.1(12) | 0.43(14) | 1 | 123(39) | 1053(330) | |||
9962.8(5) | 0.57(14) | 1 | 172(37) | 1479(322) | |||
9984.1(11) | 69(34) | 593(297)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
10004.2(10) | 0.42(19) | 1 | 70(16) | 606(137) | |||
10019.6(11) | 0.65(16) | 1 | 94(16) | 819(138) | |||
10031(2) | 69(16) | 599(136)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
10042.9(4) | 0.70(8) | (1)^{7}^{7}7 Spin and parity have been known from previous work (cf. Sec. I). | 316(36) | 2762(310) | |||
10083.8(6) | 0.78(13) | 1 | 93(13) | 823(118) | |||
10094.2(7) | 0.74(19) | 1 | 83(14) | 732(120) | |||
10104.9(12) | 0.7(3) | (1) | 49(13) | 433(118) | |||
10123.7(18) | 0.5(3) | 1 | 137(100) | 1219(886) | |||
10146.8(9) | 0.57(27) | 1 | 46(13) | 409(114) | |||
10163.4(8) | 0.50(23) | 1 | 60(16) | 540(147) | |||
10193.0(5) | 0.60(12) | 1 | 168(25) | 1514(220) | |||
10216.8(10) | 0.57(24) | 1 | 76(18) | 693(159) | |||
10233(4) | 47(38) | 427(348)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
10241(2) | 0.9(3) | (1) | 79(34) | 723(305) | |||
10260.9(11) | 23(6) | 212(53)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
10270.0(7) | 34(9) | 308(81)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
10286.2(6) | 0.64(19) | 1 | 42(9) | 388(79) | |||
10298.3(10) | 0.8(3) | (1) | 32(8) | 290(69) | |||
10306.6(9) | 0.71(21) | 1 | 46(9) | 426(80) | |||
10315.1(4) | 0.63(12) | 1 | 103(14) | 952(126) | |||
10334.9(6) | 0.68(21) | 1 | 51(10) | 470(93) | |||
10361(2) | 0.86(29) | (1) | 54(14) | 504(129) | |||
10376.8(4) | 0.46(7) | 1 | 240(28) | 2240(259) | |||
10402.5(9) | 0.72(14) | 1 | 85(16) | 799(150) | |||
10494.5(11) | 0.7(3) | (1) | 43(10) | 411(92) | |||
10507.9(8) | 0.6(3) | 1 | 49(10) | 467(96) | |||
10524.6(4) | 0.72(10) | 1 | 143(18) | 1376(175) | |||
10595.0(7) | 0.80(19) | 1 | 92(14) | 901(136) | |||
10618.7(8) | 0.74(23) | 1 | 67(12) | 651(119) | |||
10638.5(9) | 0.41(20) | 1 | 59(12) | 575(122) | |||
10682.2(6) | 0.54(16) | 1 | 42(10) | 417(95) | |||
10713.2(12) | 0.8(4) | (1) | 37(20) | 369(195) | |||
10728.2(11) | 0.67(20) | 1 | 102(32) | 1022(316) | |||
10827.1(5) | 0.82(15) | 1 | 105(16) | 1068(167) | |||
10914(2) | 0.79(21) | (1) | 113(21) | 1168(214) | |||
10957(2) | 0.35(8) | 1 | 118(19) | 1224(202) | |||
10987.0(10) | 0.66(13) | 1 | 161(23) | 1691(242) | |||
11044(2) | 49(17) | 522(179)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11094.2(15) | 70(10) | 743(106)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11108.0(16) | 39(8) | 422(84)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11120.4(9) | 0.60(16) | 1 | 92(17) | 992(179) | |||
11129.2(17) | 57(18) | 611(196)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11140(2) | 57(9) | 612(95)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11232.4(7) | 0.45(14) | 1 | 88(13) | 963(146) | |||
11243.2(6) | 0.58(15) | 1 | 92(14) | 1010(152) | |||
11337.7(6) | 0.85(13) | 1 | 91(15) | 1010(167) | |||
11417.5(7) | 0.8(4) | (1) | 108(25) | 1217(287) | |||
11452.2(10) | 0.42(12) | 1 | 132(28) | 1501(316) | |||
11479.7(8) | 0.58(15) | 1 | 191(33) | 2181(374) | |||
11501(3) | 66(37) | 756(425)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11510(7) | 33(15) | 382(172)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11531(2) | 0.42(25) | 1 | 74(30) | 848(346) | |||
11627.9(9) | 44(16) | 518(182)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
11651.5(8) | 1.0(3) | (1) | 48(16) | 564(185) | |||
11777.4(10) | 0.46(22) | 1 | 124(40) | 1489(479) | |||
11788(3) | 0.44(29) | 1 | 73(36) | 885(440) | |||
11963.3(18) | 0.8(3) | (1) | 68(14) | 845(179) | |||
11984(2) | 0.57(18) | 1 | 57(13) | 715(167) | |||
12020.6(8) | 0.67(14) | 1 | 155(21) | 1942(260) | |||
12067.8(9) | 0.63(17) | 1 | 124(19) | 1570(237) | |||
12208.3(12) | 0.40(15) | 1 | 72(16) | 929(214) | |||
12243.6(14) | 0.57(19) | 1 | 62(15) | 799(188) | |||
12496.3(18) | 87(18) | 1181(244)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
12880.3(10) | 11(3) | 164(46)^{8}^{8}8 Value deduced under the assumption of = 1. | |||||
In general, the complete level scheme may be constructed by searching for combinations of -ray energies that fit another -ray according to the Ritz principle. However, if one includes all observed transitions one finds a huge number of possible two-fold or even three-fold combinations sch07 (). Because we cannot derive complete information about the many possible branching transitions, we are not able to determine the branching ratios of the ground-state transitions from the present data. Therefore, we applied statistical methods to estimate the intensities of branching transitions and thus to correct the dipole-strength distribution.
Iii Determination of the dipole-strength distribution
In the following we describe the statistical methods used to estimate the intensity distribution of branching transitions and the branching ratios of the ground-state transitions.
By using simulations as described in Sec. II.2 we corrected the spectrum including the two detectors at 127, measured during the irradiation of the Zr target at = 13.2 MeV. In a first step spectra of the ambient background adjusted to the intensities of the 1460.5 keV transition (decay of K) and 2614.9 keV transition (decay of Tl) in the in-beam spectrum were subtracted from the measured spectrum. It turned out that transitions following reactions in the HPGe detectors and in surrounding materials are negligibly small and thus, did not require correction. To correct the spectrum for detector response, spectra of monoenergetic rays were calculated in steps of 10 keV by using GEANT3. Starting from the high-energy end of the experimental spectrum, the simulated spectra were subtracted sequentially. The resulting spectrum including the two detectors at 127 is shown in Fig. 4.
The background produced by atomic processes in the Zr target was obtained from a GEANT3 simulation using the absolute photon flux deduced from the intensities of the transitions in B (cf. Fig. 2). The corresponding background spectrum multiplied with the efficiency curve and with the measuring time is also depicted in Fig. 4.
As can be seen in Fig. 4 the continuum in the spectrum of photons scattered from Zr is clearly higher than the background by atomic scattering. This continuum is formed by a large number of non-resolvable transitions with small intensities which are a consequence of the increasing nuclear level density at high energy and of Porter-Thomas fluctuations of the decay widths por56 () in connection with the finite detector resolution (e.g. 7 keV at 9 MeV).
The relevant intensity of the photons resonantly scattered from Zr 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” continuum. The scattering cross sections derived from this intensity distribution for energy bins of 0.2 MeV from the full intensity distribution are shown in Fig. 5. These values are compared with those given in Table 1 for resolved transitions in Zr. 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 2.7 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. 5 contain ground-state transitions and, in addition, branching transitions to lower-lying excited states (inelastic transitions) as well as transitions from those states to the ground state (cascade transitions). The different types of transitions cannot be clearly distinguished. However, for the determination of the photoabsorption cross section and the partial widths the intensities of the ground-state transitions are needed. Therefore, contributions of inelastic and cascade transitions have to be subtracted from the spectra. We corrected the intensity distributions by simulating -ray cascades rus07 () from the levels with = 0, 1, 2 in the whole energy range analogously to the strategy of the Monte-Carlo code DICEBOX bec98 (). In these simulations, 1000 nuclear realizations starting from the ground state were created with level densities derived from experiments egi05 (). We applied the statistical methods also for the low-energy part of the level scheme instead of using experimentally known low-lying levels in Zr because this would require the 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. bro81 ()). The partial widths of the transitions to low-lying levels were assigned using a priori known strength functions for , , and transitions. Fluctuations of the partial widths were treated the applying Porter-Thomas distribution por56 ().
In the calculations, the recently published parameters for the Back-Shifted Fermi-Gas (BSFG) model obtained from fits to experimental level densities egi05 (), = 8.95(41) MeV and = 1.97(30) MeV, were used. In the individual nuclear realizations, the values of and 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 egi05 (). This assumption has recently been justified by the good agreement of level densities predicted by the BSFG model with experimental level densities of states in the energy range from 5 to 10 MeV obtained from the Zr(He,)Nb reaction kal06 () and with experimental level densities of and states in Zr studied in the Zr and Zr reactions kal07 (). The extended analysis of the Zr(He,)Nb reaction in Ref. kal07 () indicates however fluctuations of the level density of states in Nb around the predictions of the BSFG model.
For the , , and photon strength functions Lorentz parametrizations axe62 () were used. The parameters of the Lorentz curve for the strength were taken from a fit to data ber67 () and are consistent with the Thomas-Reiche-Kuhn sum rule (TRK) MeV mb rin80 (). The parameters for the and strengths were taken from global parametrizations of spin-flip resonances and isoscalar resonances, respectively rip06 ().
Spectra of -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. 6, which reflect the influence of fluctuations of level energies and level widths. Because in the nuclear realizations the levels were created randomly starting from the ground state instead of starting with the known first excited state at 2.2 MeV, the distribution of the branching transitions continues to the energy bin of the ground-state transitions.
These spectra resemble qualitatively the ones measured in an experiment on Zr using tagged photons ala87 (). 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 normalized 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 the low-energy end of the spectrum one obtains the intensity distribution of the ground-state transitions. Simultaneously, the branching ratios of the ground-state transitions are deduced for each energy bin . In an individual nuclear realization, the branching ratio 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 sch07 (); rus08 (). By dividing the summed intensities in a bin of the experimental intensity distribution of the ground-state transitions by the corresponding branching ratio we obtain the absorption cross section for a bin as . 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 1 deviation from the mean has been taken.
To get an impression about the branching ratios, the individual values of 10 nuclear realizations are shown in Fig. 7. The mean branching ratio of the 1000 realizatios decreases from about 80% for low-lying states, where only few possibilities for transitions to lower-lying states exist, to about 65% at the neutron-separation energy. Toward low energy the uncertainty of 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 like the resolved transitions given in Table 1. It turns out from the simulations that the branching ratios of transitions with partial widths like the ones given in Table 1 are in the order of 85% to 99% (cf. also Ref. sch07 ()).
The photoabsorption cross sections obtained for Zr are compared with the data of a previous experiment with monoenergetic photons in the energy range from 8.4 to 12.5 MeV axe70 () in Fig. 8. The data of Ref. axe70 () contain corrections in form of additional constant partial widths for branching and for proton emission, i.e. for the channel. Cross sections obtained in experiments ber67 () and cross sections calculated for the reaction using the Talys code kon05 () with the Lorentz model for the strength function axe62 () are also shown in Fig. 8.
To test the influence of the strength function on the results of the simulation and to check the consistency between input strength function and deduced photoabsorption cross section we performed cascade simulations with modified strength functions. In one case, we applied a Lorentz curve with an energy-dependent width . In another case we assumed a PDR on top of the Lorentz curve with constant width. The PDR had a Breit-Wigner shape with its maximum at 9 MeV and a width of 2 MeV. The cross section of the PDR was chosen 2.2% of the TRK (see discussion in Sec. IV.2). Cross sections corresponding to these input strength functions and the obtained absorption cross sections are shown in Fig. 9. One sees that the resulting absorption cross sections differ only slightly from each other and also from the ones obtained by using the Lorentz curve with constant width (cf. Fig. 8) and overlap within their uncertainties. This shows that the resulting cross sections are not very sensitive to modifications of the input strength functions. However, the comparison of the resulting cross sections with the input strength functions shows that there is consistency in the case of the Lorentz curve with a PDR in addition, whereas there is a large discrepancy in the case of the Lorentz curve with energy-dependent width. Summarizing, the Lorentz curve with a constant width and this curve with a PDR in addition deliver consistency between input and output strength functions and similar results.
The total photoabsorption cross section has been deduced by combining the present data with the data of Ref. ber67 () and calculated cross sections kon05 (). This total cross section is compared with the Lorentz curve described above, which is currently used as a standard strength function for the calculation of reaction data rip06 (), in Fig. 10. It can be seen that the experimental cross section includes extra strength with respect to the approximation of the low-energy tail of the GDR by a Lorentz curve in the energy range from 6 to 11 MeV and is in better agreement with a strength function including a PDR on top of the Lorentz curve (cf. Fig. 9). To investigate the nature of this extra strength we have performed model calculations which are discussed in the following section.
Iv Quasiparticle-phonon-model calculations for the = 50 isotones Sr and Zr
The experimental data on the dipole response of Zr discussed in the present work reveal a strong enhancement of the photoabsorption cross section (see Fig. 10) in the energy range = 6 – 11 MeV and a considerable deviation of its shape from the Lorentz-like strength function used to adjust the GDR. Similar dipole resonance structures have been detected also in our recent study of Sr sch07 (). In the following we investigate the nature of the dipole strength in the isotones Sr and Zr in the framework of the extended quasiparticle-phonon model (QPM) Ts04 (); Nadia08 (). The approach is based on a Hartree-Fock-Bogoljubov (HFB) description of the ground state Ts04 () by using a phenomenological energy-density functional (EDF) Nadia08 (). The excited states are calculated within the QPM Sol76 ().
iv.1 The QPM model
The model hamiltonian is given by:
(3) |
and resembles in structure that of the traditional QPM model Sol76 () but in detail differs in the physical content in important aspects as discussed in Ref. Ts04 (); Nadia08 (). The HFB term contains two parts. describes the motion of protons and neutrons in a static, spherically-symmetric mean field and accounts for the monopole pairing between isospin-identical particles with phenomenologically adjusted coupling constants, fitted to data Audi95 (). The mean-field hamiltonian compromises microscopic HFB effects Hof98 (); Nadia08 () and phenomenological aspects like experimental separation energies and, when available, also charge and mass radii. For numerical convenience the mean-field potential is parametrized in terms of a superpositions of Wood-Saxon (WS) potentials with adjustable parameters Nadia08 ().
The remaining three terms present the residual interaction . As typical for the QPM, we use separable multipole-multipole and spin-multipole interactions both of isoscalar and isovector type in the particle-hole channel and a multipole-multipole pairing in the particle-particle channel. The isoscalar and isovector coupling constants are obtained by a fit to energies and transition strengths of low-lying vibrational states and high-lying GDRs Nadia08 (); Vdo83 ().
The excited nuclear states are described by quasiparticle-random-phase-approximation (QRPA) phonons. They are defined as a linear combination of two-quasiparticle creation and annihilation operators:
(4) |
where is a single-particle proton or neutron state, and are time-forward and time-backward operators, coupling proton and neutron two-quasiparticle creation or annihilation operators to a total angular momentum with projection by means of the Clebsch-Gordan coefficients . The time reversed operator is defined as .
We take into account the intrinsic fermionic structure of the phonons by the commutation relations:
(5) |
The QRPA phonons obey the equation of motion
(6) |
which solves the eigenvalue problem, i.e. it gives the excitation energies and the time-forward and time-backward amplitudes Sol76 () and , respectively.
In terms of the QRPA phonons the model hamiltonian can be expressed as
(7) |
The first part in the Eq. (7) refers to the harmonic part of nuclear vibrations, while the second one is accounting for the interaction between quasiparticles and phonons. The latter allows us to go beyond the ideal harmonic picture by including effects due to the non-bosonic features of the nuclear excitations.
The model hamiltonian is diagonalized assuming a spherical ground state which leads to an orthonormal set of wave functions with good total angular momentum . For even-even nuclei we extend the approach to anharmonic effects by considering wave functions including a mixture of one-, two-, and three-phonon components Gri94 ():
(8) |
where labels the number of the excited states. The energies and the phonon-mixing amplitudes , , and are determined by solving the extended multi-phonon equations of motions.
As discussed in Refs. Rye02 (); Nadia08 () we obtain important additional information from the analysis of the spatial structure of the nuclear response on an external electromagnetic field. That information is accessible by considering the one-body transition densities , which are the non-diagonal elements of the nuclear one-body density matrix. The transition densities are obtained by the matrix elements between the ground state and the excited states . We identify in QPM with phonon vacuum and obtain the excited states by means of the QRPA state operator, Eq. (4), . The analytical procedure for the calculation of the transition densities is presented in details in Ref. Nadia08 ().
iv.2 Calculation of dipole excitations in the = 50 isotones
In Ref. Nadia08 (), a close relationship between skins in the ground-state matter-density distributions and the low-energy dipole response was pointed out. It was shown, that the skin thickness, defined by
(9) |
is indeed directly connected to the non-energy-weighted dipole sum rule, thus being complementary to the well established relation of the GDR strength and the energy-weighted Thomas-Reiche-Kuhn dipole sum rule. From the calculated neutron and proton ground state densities displayed in Fig. 11 it is seen that the two isotones considered here, Sr and Zr, show a neutron skin. The total binding energies and radii are given in Table 2. Of special importance for our investigations are the surface regions, where the formation of a skin takes place. For the investigated Sr and Zr nuclei, the neutron skin decreases with decreasing , i.e. when going from Sr to Zr.
(MeV) | (MeV) | (fm) | (fm) | (fm) | |
---|---|---|---|---|---|
Sr | –8.733 | –8.766 | 4.185 | 4.291 | 0.106 |
Zr | –8.710 | –8.692 | 4.229 | 4.303 | 0.074 |
The QRPA calculations in these nuclei result in a sequence of low-lying one-phonon dipole states of almost pure neutron structure, located below the particle threshold. The proton contribution in the state vectors is less than 5%. These states have been related to the oscillations of weakly bound neutrons from the surface region, known as neutron PDR (see Refs. Rye02 (); Adri05 (); Ts04 (); Nadia08 (); Volz06 () and Refs. therein). The calculations of the dipole response in = 50 up to 22 MeV are presented in Fig. 12. The transition matrix elements were calculated with recoil-corrected effective charges for neutrons and for protons Nadia08 (), respectively. To compare the predicted dipole strengths with the experimental values we calculated the integrated absorption cross sections according to MeV mb with in MeV and in fm Bohr (). When comparing the QRPA results with experimental cross sections one has the general problem of comparing discrete values (cf. Fig. 12) with gradually changing averages of energy bins (cf. Fig. 10). Therefore, the QRPA values are usually folded with Lorentz curves that can be considered as a simulation of the damping of the GDR due to higher-order configurations neglected in QRPA. In the present case, we folded the discrete QRPA cross sections with Lorentz curves of 3.0 MeV width, which is a typical value for the damping width. The results are compared with the experimental cross sections in Fig. 10 for Zr and in Fig. 13 for Sr. The QRPA calculations predict extra strength with respect to the Lorentz curve in qualitative agreement with the experimental findings. However, the experimental cross sections show a clear separation of the PDR and the GDR regions around 12 MeV, whereas the calculations seem to predict an excess of strength up to about 14 MeV.
The nature of the dipole strength can be examined by analyzing dipole transition densities that act as a clear indicator of the PDR and allow us to distinguish between the different types of the dipole excitations Nadia08 (). The QRPA proton and neutron dipole transition densities for Sr and Zr were summed over selected energy regions and are shown in Fig. 14. They can be directly related to the spectrum of 1 states in Fig. 12. In the region of 9 MeV we find an in-phase oscillation of protons and neutrons in the nuclear interior, while at the surface only neutrons contribute. Because these features are characteristic for dipole skin modes, we identify these states as part of the PDR. The states in the region of from 9 MeV to 9.5 MeV carry a different signature and their contribution to the total proton and neutron transition densities in the energy region 9.5 MeV changes completely the behavior of the proton/neutron oscillations related to the PDR. At these higher energies the protons and neutrons start to move out of phase being compatible with the low-energy part of the GDR. At from 9 to 22 MeV a strong isovector oscillation corresponding to the excitation of the GDR is observed.
In the case of isotones the neutron number is fixed and only the proton number changes. This affects the thickness of the neutron skin as well as is seen in Fig. 11 and Table 2. Correspondingly, the total PDR strength decreases with increasing proton number. According to the criteria discussed above we consider the states in the energy region 9 MeV as a part of the PDR. In this energy interval, the QRPA calculations predict a total PDR strength of fm in Zr and fm in Sr, respectively. This accounts for 0.27% and 0.44% of the TRK in Zr and Sr, respectively, and compares with experimental values of 0.55(2)% and 0.74(2)% derived from resolved peaks in Zr and Sr, respectively, whereas the values deduced from the corrected full intensities including also the continuum part are remarkably greater and amount to 2.2(4)% and 2.4(6)% in Zr and Sr, respectively.
Finally, we note that the results are in agreement with the established connection between the calculated total PDR strength and the nuclear skin thickness, Eq. (9), found in our previous investigations in the = 82 isotones Volz06 () and the Sn isotopes Nadia08 () nuclei.
In the multi-phonon QPM calculations the structure of the excited states is described by wave functions as defined in Eq. (8). We now investigate multi-phonon effects using a model space with up to three-phonon components, built from a basis of QRPA states with = 1, 1, 2, 3, 4, 5, 6, 7, and 8. Because the one-phonon configurations up to = 22 MeV are considered the core polarization contributions to the transitions of the low-lying 1 states are taken into account explicitly. Hence, we do not need to introduce dynamical effective charges. Our model space includes one-, two- and three-phonon configurations (in total 200) with energies up to = 12 MeV.
The experimental absorption cross sections in Zr in the range from 5 to 11 MeV are compared with the corresponding QPM results in Fig. 15. The strength in the region = 6 – 7 MeV is related to the PDR as identified from the structure of the involved 1 states. The contribution of the GDR phonons becomes significant at energies 7.5 MeV where a coupling between PDR, GDR, and multiphonon states reflects the properties of the dipole excitations at higher energies. At low energy, the calculated values underestimate the experimental values derived from the full intensity and are close to the values deduced from resolved peaks only, whereas at high energy they are in better agreement with the experimental values including the full intensity. The calculated EWSR from 5 to 9 MeV of 1.3% of the TRK underestimates the experimental value of 2.2(4)%. However, increasing the energy range up to 11 MeV the theoretical value of 3.3% is much closer to the experimental result of 4.0(3)%, indicating a slightly different distribution of the QPM strength with respect to excitation energy.
V Summary
The dipole-strength distribution in Zr up to the neutron-separation energy has been studied in photon-scattering experiments at the ELBE accelerator using various electron energies. Ground-state transitions have been identified by comparing the transitions observed at different electron energies. We identified about 190 levels with a total dipole strength of about 180 meV/MeV. Spin = 1 could be deduced from the angular correlations of the ground-state transitions for about 140 levels including a dipole strength of about 155 meV/MeV.
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 27% of the total dipole strength while the continuum contains about 73%.
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 -ray cascades intensities of branching transitions could be estimated and subtracted from the experimental intensity distribution and the intensities of ground-state transitions could be corrected in average for their branching ratios.
A comparison of the photoabsorption cross section obtained in this way from the present experiments with 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 reaction. In comparison with a straightforward approximation of the GDR by a Lorentz curve one observes extra strength in the energy range from 6 to 11 MeV which is mainly concentrated in strong peaks.
QPM calculations in Sr and Zr predict low-energy dipole strength in the energy region from 6 to 10 MeV. The states at about 6 to 7.5 MeV have a special character. Their structure is dominated by neutron components and their transition strength is directly related to the size of a neutron skin. Their generic character is further confirmed by the shape and structure of the related transition densities, showing that these PDR modes are clearly distinguishable from the GDR. The dipole states, seen as a rather fragmented ensemble at 7.5 MeV mix strongly with the low-energy tail of the GDR starting to appear in the same region. The complicated structure of these states and the high level densities imposes considerable difficulties for a reliable description of the fragmentation pattern.
The present analysis shows that standard strength functions currently used for the calculation of reaction data do not describe the dipole-strength distribution below the threshold correctly and should be improved by taking into account the observed extra strength.
Vi Acknowlegdments
We thank the staff of the ELBE accelerator for their cooperation during the experiments and we thank A. Hartmann and W. Schulze for their technical assistance. Valuable discussions with F. Dönau and S. Frauendorf are gratefully acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft under contracts Do466/1-2 and Le439/4-3.
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