Photon strength functions in Gd isotopes studied from radiative capture of resonance neutrons

The experimental spectra of γ rays following radiative neutron capture on isolated resonances of stable 152,154−158Gd targets were measured by the DANCE calorimeter installed at the Los Alamos Neutron Scattering Center in New Mexico, USA. These spectra were analyzed within the extreme statistical model to get new information on the photon strength functions. Special emphasis was put on study of the scissors vibrational mode present in these isotopes. Our data show that the scissors-mode resonances are built not only on the ground states but also on the excited levels of all studied Gd isotopes. The scissors mode strength observed in 157,159Gd products is significantly higher than in neighboring even-even nuclei 156,158Gd. Such a difference indicates the existence of an odd-even effect in the scissors mode strength. Moreover, there exists no universal parameter-free model of the electric dipole photon strength function describing the experimental data in all of the Gd isotopes studied. The results for the scissors mode are compared with the (γ, γ) data for the ground-state transitions and with the results from 3He-induced reactions.


Introduction
In medium and heavy mass nuclei detailed information on the properties of nuclear levels and transitions between them exists usually only at low excitation energies above the ground state.In this region, the level spacing is sufficiently high to clearly observe individual transitions.As the nuclear level density (NLD) increases with excitation energy, it is almost impossible to resolve transitions to or from the individual levels.Obtaining reliable spectroscopic information on these levels, which form the so-called quasicontinuum, becomes very difficult.It is believed that properties of the nucleus in the quasicontinuum can be described by the extreme statistical model in terms of the NLD and a a e-mail: kroll@ipnp.troja.mff.cuni.czset of photon strength functions (PSFs) for different types and multipolarities of transitions.These statistical quantities are very important for the correct description of various reaction rates that are especially needed in nuclear astrophysics and in the development of advanced nuclear reactors.One of the ways to examine PSFs and the NLD at excitation energies up to the neutron separation energy S n is via study of the γ rays following the radiative neutron capture at isolated neutron resonances.Results of the analysis of the γ-ray spectra measured for 152,154−158 Gd targets at isolated s-wave neutron resonance are presented below.

DANCE Experiment
The measurement was performed using the highly segmented Detector for Advanced Neutron Capture Experiments (DANCE) detector array [1,2] installed at the pulsed neutron beam at the Los Alamos Neutron Science Center (LANSCE) at Los Alamos National Laboratory [3] which produces a white spectrum of neutrons with energies from subthermal up to several MeV with the repetition rate of 20 Hz.These neutrons are sent to flight path 14 at the Manuel Lujan Jr. Neutron Scattering Center.
The DANCE detector array [1,2] is installed at 20.2 m on this flight path and consists of 160 BaF 2 scintillation crystals surrounding a sample and covering a solid angle of 3.5π.A 6 LiH shell about 6-cm thick is placed between the sample and the BaF 2 crystals in order to reduce the background from capture of scattered neutrons on Ba isotopes in the crystals.The DANCE acquisition system [4] is based on waveform digitization of signals from all 160 detectors using Acqiris DC265 digitizers with a sampling rate of 500 megasamples per second.
The energy calibration of the DANCE crystals was performed with a combination of natural γ-ray sources at low γ-ray energies and the intrinsic α-radioactivity of 226 Ra and its daughters present in the BaF 2 crystals.The latter calibration was conducted on a run-by-run basis to provide the energy alignment of all crystals in the off-line analysis.The identification of α particles is possible from the ratio of the fast and slow components of the BaF 2 signal [2].
Measurements were performed using isotopically enriched Gd targets.The 155−158 Gd targets were prepared at Oak Ridge National Laboratory as self-supporting metal foils.The 152 Gd and 154 Gd targets were prepared at Lawrence Livermore National Laboratory by the electroplating of enriched Gd on a Be foil which was glued to an aluminium ring.The isotopic composition of the targets together with their average thicknesses is listed in Ref. [5,6].

Data Reduction
Only events from well-resolved resonances, that can be identified using time-of-flight (TOF) technique, were considered.Examples of TOF spectra for 154 Gd and 158 Gd targets are given in Fig. 1.Only s-wave neutron resonances were observed at low neutron energies in this mass region.
Usually a γ cascade consists of several emitted γ rays.The signals were assumed to originate from a single γ cascade if they were detected within a 10 ns coincidence window.An emitted single γ ray does not usually deposit its full energy in one crystal.Thus the number of crystals that fire is often higher than the physical multiplicity of the capture event.Therefore all contiguous crystals that have fired during a capture event are combined into clusters and considered as the response of the detector array to a single γ ray.The number of clusters observed in a capture event is called the cluster multiplicity.The sum-energy spectra for different cluster multiplicities from several resonances of the 158 Gd target are shown in Fig. 2(a).
Each sum-energy spectrum consists of (i) a peak near the full energy available from the neutron capture reaction -the full-energy peak -which is close to the neutron separation energy S n for a given EPJ Web of Conferences 00018-p.2nucleus and (ii) a low-energy tail that corresponds to γ cascades for which a part of the emitted energy escaped the detector array.The shape of the spectrum at low sum energies, approximately below 3 MeV, is strongly influenced by the background from natural β activity in the BaF 2 crystals, especially for low multiplicities.In practice, this background is not important for us as only events from the full-energy peak were used in our analysis.The events above S n in the sum-energy spectra come from two sources.The dominant contribution, especially at low multiplicities, is from the capture of scattered neutrons in the barium of the BaF 2 scintillation crystals, with neutron separation energies between S n = 4.723 and 9.468 MeV for 139 Ba and 134 Ba, respectively.For higher multiplicities there is also a small contribution from the radiative neutron capture on 155,157 Gd sample impurities in the sum-energy spectra of even Gd targets.
The main subject of our analysis were so-called multistep cascade (MSC) spectra.These are constructed only from those events which fall into the full-energy peak in the sum-energy spectra.The ranges of the energy sums used in the construction of MSC spectra are illustrated by the grey areas in Fig. 2(a).The MSC spectrum for multiplicity M is understood to be a spectrum of energies belonging to a single cascade that are deposited in M detector clusters.Experimental MSC spectra constructed in this way contain a background.This background is usually very small for strong neutron resonances and was subtracted using MSC spectra from off-resonance regions.Experimental background-free MSC spectra from selected resonances of 158 Gd are shown in Fig. 2

(b).
To facilitate the comparison of the experimental MSC spectra with the model predictions given below, all sum-energy and MSC spectra for a given resonance were normalized using one common factor.Specifically, we assume that the integral of the sum-energy spectra for multiplicities M = 2 − 7 takes a value common for all resonances.As the spectrum for M = 1 is strongly dominated by the background contributions, it is omitted from the analysis.Virtually no events for M ≥ 6 or 7 were measured.For comparison with simulations the MSC spectra with 250 keV wide bins were used.This width is comparable with the detector energy resolution.Use of these relatively wide energy bins allows reduction of the experimental uncertainties and simultaneously suppresses the fluctuations predicted by the simulations.

Modelling of γ decay
MSC spectra are products of a complicated interplay between PSFs and the NLD.As a result, it is not straightforward to get information on these quantities from measured spectra.To learn about PSFs and the NLD we adopt a trial-and-error approach in which we compare the experimental MSC spectra with their simulated counterparts obtained from the generation of γ cascades using the DICEBOX algorithm [7] for various models of the PSFs and the NLD followed by simulation of the detector response to these cascades using the GEANT4 code [8].Such a comparison can tell us which models of PSFs and the NLD are most likely to be valid and which are incorrect.The main feature of the DICEBOX algorithm is that it takes into account all expected fluctuations involved in the γ decay.Due to the presence of these fluctuations, which are dominated by the Porter-Thomas fluctuations of partial radiation widths [9], there exists an infinite number of so-called nuclear realizations (NRs) that differ from each other even for fixed models of PSFs and the NLD.About 20 independent NRs with given combination of PSFs and NLD were generated for each J π of the initial s-wave resonances with 10 5 cascades in one NR.Generated cascades were then subject to a GEANT4-based simulation of the detector system response.The MSC spectra from simulations were constructed under the same condition on the detected energy sum as applied to the experimental spectra.

Electric dipole transitions
It is well known that for γ-ray energies above the neutron separation energy S n the electric dipole (E1) transitions play a dominant role.The E1 PSF at these energies in axially deformed nuclei seems to be fully consistent with the sum of the two Lorentzian terms of the giant dipole electric resonance (GDER).The model is known as the Standard Lorentzian (SLO) model.On the other hand, the shape of the E1 PSF below S n is not well known.There are many available parametrizations of the E1 PSF in this region of γ-ray energies that modify the Lorentzian shape of the low-energy tail of the GDER.
Usually, either the model of Kadmenskij, Markushev, and Furman (KMF) [10] or the Enhanced Generalized Lorentzian (EGLO) [11] model is used at these energies.Many other models of the E1 EPJ Web of Conferences Figure 3. Left: Some of the E1 + M1 composite PSF models used in our simulations for the 159 Gd product.Two curves for the KMF, MGLO and MLO2 models indicate how these models change as a function of nuclear temperature -the solid curve corresponds to T f = 0 MeV, while the dashed one to where a is the level density parameter and Δ is the pairing energy [39,40,42].The parametrization of the GDER is taken from Ref. [43] for the 160 Gd nucleus.The M1 PSF consists of a single-particle component f (SP) M1 = 1 × 10 −9 MeV −3 , a spin-flip part represented by the double-humped Lorentzian with the parametrization E SF,1 = 6 MeV, Γ SF,1 = 0.8 MeV, σ SF,1 = 0.7 mb and E SF,2 = 8 MeV, Gamma SF,2 = 1.8 MeV, σ SF,2 = 1.1 mb, and the scissors mode f (SM)  M1 with the parametrization E SM = 3.0 MeV, Γ SM = 1.0 MeV and σ SM = 0.7 mb.The experimental data for f E1 from the (n,γ) reaction are from the RIPL database [44] ( 155,157 Gd at 5.9 and 6 MeV, respectively), and from Ref. [45] ( 159 Gd at 5.1 MeV).Data from the ( 3 He,α) reaction are from Ref. [16].Right: The NLD models of 159 Gd summed over the spin range J = 1/2 − 21/2 and both parities.The parameters of the CT and BSFG models are from [39] and [40].The microscopic calculations are from RIPL-3 [12].The experimental data are taken from the 162 Dy 3 He, αγ 161 Dy reaction measured by the Oslo group [16].
PSF, which have been tested in our analysis, can be found in the literature.We tested the following models which can be found in the RIPL-3 database [12]: the Hybrid model (GH) [13], the Generalized Fermi Liquid (GFL) model [14,15], a family of three modified Lorentzian (MLO) models, and a PSF model originating from Hartree-Fock-Bogoliubov plus quasi-particle random-phase approximation (HFB-QRPA) microscopic calculations [12].In addition to these models, we tested also the KMF model with constant nuclear temperature T f (KMF-T) which was often used in the analysis of data from 3 He-induced reactions measured in the Oslo Cyclotron Laboratory [16][17][18][19][20] and the Modified Generalized Lorentzian (MGLO) model defined in Ref. [6] which is a modification of the EGLO model.There are two (ad-hoc) parameters, k 0 and E γ0 , in the EGLO and MGLO models.
For a complete description of γ decay one needs information on the PSFs at all excitation energies.In some models, the dependence on any quantity other than E γ is neglected in accord with the Brink hypothesis [21].Experimental data from average resonance capture [22] and from 3 He-induced reaction [17] seem to confirm at least the approximate validity of this hypothesis in the region of excitation energies below S n .From the above list of E1 PSF models, only the SLO, KMF-T, and models based on HFB-QRPA microscopic calculations were assumed to follow the strict form of the Brink hypothesis.The remaining models predict a weak dependence of the PSF on the nuclear temperature of the final state T f (which is equivalent to dependence on the excitation energy).

M1
for the M1 PSF.The parameters of the SM and the value of f (SP) M1 are indicated in the figures together with the energies, spin J and parity π of the s-wave neutron resonances used in the analysis.The experimental spectra are the same in both parts of the figure.The gray band characterizes the result from simulations of 20 different NRs and has the width of two sigma -an average of +/one sigma, see the text.The simulated MSC spectra with E SM = 2.7 MeV (a) nicely reproduce the experimental data while for E SM = 3.1 MeV (b) the description of the bump near E γ = 2.5 MeV is problematic as the bump has a different shape, especially for multiplicities M = 3 and 4.

Magnetic dipole transitions
Magnetic dipole (M1) transitions also play an important role in the decay of highly excited nuclear states below S n .In the mid 70's an isovector M1 collective vibrational mode at E γ ≈ 3 MeV was predicted in deformed nuclei [23][24][25].This mode, known as the scissors mode (SM), was observed for the first time for ground-state transitions in 156 Gd [26] in high-resolution electron inelastic scattering at low momentum transfer.
A systematic study of ground-state transitions for E γ ≤ 4 MeV in even-even rare-earth nuclei was performed with nuclear resonance fluorescence (NRF) experiments using the (γ, γ ) reaction [27].The low density of J = 1 states allowed observation of a majority of the dipole transition strength at these energies.In these experiments the total observed reduced M1 strength B(M1)↑ in the energy range E γ ≈ 2.5 − 4.0 MeV was proportional to the square of the nuclear deformation [28,29] and for well deformed nuclei reached B(M1)↑≈ 3μ 2 N .This strength is usually distributed over only a few states.All of the M1 strength in this energy range was usually attributed to the SM.
In odd nuclei, the spacing between nuclear levels near the excitation energy E x = 3 MeV is much smaller, which causes the fragmentation of the SM strength into many weak transitions.The limited experimental sensitivity prevents observation of all these transitions in the NRF experiments.This fact led to smaller observed SM strength in odd nuclei compared to even-even neighbors [30][31][32][33].Attempts to estimate the missing strength in NRF reaction in odd nuclei were made using a fluctuation analysis [33,34].However, it seemed that this method did not work properly in some cases [33].
Later, the analysis of data in two-step cascades (TSC) following thermal neutron capture in 162 Dy suggested that a resonance-like structure of M1 character with the strength of B(M1)↑≈ 6.2 μ 2 N couples not only to the ground state, but also to all levels in 163 Dy [35].This observation gave strong support for the validity of the Brink hypothesis for the SM.A similar finding was corroborated EPJ Web of Conferences 00018-p.6 by the TSC measurement of the 159 Tb(n, γ) reaction [36], and also by data on 3 He-induced photon production [37].
The analysis of TSC spectra in Ref. [35] also indicated that the reduced SM strength in odd 163 Dy ( B(M1)↑≈ 6.2 μ 2 N ) is higher compared to that deduced from (γ, γ ) measurements on neighboring even-even nuclei.Contrary to this, comparable strength of the SM was reported for 3 He-induced reactions on 160−164 Dy, 167 Er, 171,172 Yb isotopes [16,17,19,20].Interestingly, data obtained from the same reaction yielded about 1.7 times lower strength of the SM in even-even 166 Er than in odd 167 Er [19].
The data from TSC measurements and 3 He-induced reactions indicate that the SM is a resonancelike structure with a halfwidth of ≈ 0.7 − 1.5 MeV.The maximum of the resonance structure is close to 3 MeV in the case of the 163 Dy product [35], while it is slightly shifted down to about 2.7 − 2.8 MeV in 160 Tb [36] and even more in data from the 3 He-induced reactions [16,17,19,20].
The M1 strength is likely more complex than just the contribution of the SM.Almost exclusively one of two models is used for the remaining part of the M1 strength.In the spin-flip (SF) model the f (SF)  M1 (E γ ) is assumed to be a Lorentzian resonance function with the energy position at about 7 MeV and a width of about 4 MeV [12], while in the single-particle (SP) model f (SP)  M1 is a constant independent of γ-ray energy.In some cases a composite model for the M1 strength, M1 , was used for the M1 PSF.The absolute values of the SP and SF models were adjusted to obtain the ratio of R = f E1 / f M1 ≈ 7 at 7 MeV in our analysis.This value seems to be reasonably well determined in rare-earth nuclei from average resonance capture (ARC) experiments [22].In models adopting the sum of SP and SF contributions, we varied f (SP)  M1 and adjusted the size of the SF contribution to reproduce the ratio R. The strict validity of the Brink hypothesis was assumed for the SP and SF M1 models.

Higher multipolarities
In addition to dipole transitions, electric quadrupole (E2) transitions might also play a role in neutron resonance decay.In our analysis we either postulate the shape of the f E2 PSF as the standard 00018-p.7 Lorentzian form [38] or we adopt a single-particle model for the E2 PSF, in which f (SP)  E2 is a constant independent of the γ-ray energy.In our analysis the E2 strength was taken to reproduce the ratio of partial radiation widths at about 7 MeV measured in ARC experiments in even-even deformed nuclei, that is Γ(E1)/Γ(E2) ≥ 100 [22].

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The contributions of higher multipolarities, i.e., M2, E3, . . ., are considered to be marginal in the decay of highly excited states of the compound nucleus and we completely neglect them in our simulations.This is justified by the fact that only a few very weak transitions of these types have been detected so far, mainly in the region of low excitations.

Nuclear level density
Two phenomenological NLD models given by closed-form formulas were tested: (i) the Back-Shifted Fermi Gas (BSFG), and (ii) the Constant-Temperature (CT) model [39].There are two adjustable parameters in each of these models.Two different sets of parameters taken from papers of von Egidy and Bucurescu [39,40] were tested.The spin distribution f (J, σ) of the NLD for both of the models was adopted in the standard form [12,39,40] with different expressions introduced for the spin-cutoff parameter σ in Refs.[39,40].The parity distribution function was usually considered in the simplest possible form f (π) = 1/2 in our calculations.
In addition to these closed-form models, we also tested the microscopic NLD model based on the Hartree-Fock-Bogoljubov (HFB) method with the BSk14 effective Skyrme force [41].The HFB microscopic NLDs are available as tables for different excitation energies, spins, and both parities.These tables can be found in the RIPL-3 database [12].The NLD calculated in the HFB approach usually suffers from difficulties in reproducing the average neutron resonance spacing.In order to bring the calculations into agreement with experimental data, the tabulated HFB level densities are proposed to be normalized to reproduce a resonance spacing at the neutron separation energy [12].Some of the PSF and NLD models tested in our simulations for 159 Gd isotope are shown in Fig. 3.These models are compared with the experimental PSF and NLD values extracted by the Oslo method from the 3 He, α data on 161 Dy [16].  157,15Gd.Contrary to this the SM parametrization in part (a) is unable to reproduce experimental MSC spectra in 157,159 Gd, see Refs.[5,6].The MGLO model (with k 0 = 2) was used for E1 PSF.For further details see Fig. 4. Almost identical results were obtained for MSC spectra from J π = 2 − resonances of 158 Gd as well as for MSC spectra from resonances of both possible spins of 156 Gd.

Results from MSC spectra
Several hundred model combinations of PSFs and NLD were simulated and compared with the experimental MSC spectra for each Gd isotope studied.To quantify precisely the agreement between the simulations and the experimental spectra extremely time-consuming simulations would be needed as the individual bins in MSC spectra are mutually correlated in a complicated way and the corresponding correlation matrix is not a priori known.As a consequence, the degree of agreement was checked only visually.
As mentioned in Sec. 4, DICEBOX simulations of nuclear electromagnetic decay correctly incorporate the Porter-Thomas fluctuations of partial radiation widths, as well as fluctuation properties of NLD.Results of individual NRs are together characterized by their mean and sigma.For the sake of clarity the standard confidence interval with a width of two sigma (the average value ± one sigma) is plotted as a grey band in Figs.4-8.The size of fluctuations among the experimental spectra for different resonances is reasonably well reproduced by the simulations which justifies applicability of the statistical model for description of the γ decay of studied nuclei at energies below S n .
It was found that the influence of the E2 PSF on the shape of simulated MSC spectra is negligible.The constant f (SP)  E2 PSF function with the strength f (SP) E2 = 3 − 5 × 10 −11 MeV −5 was usually used in simulations of the MSC spectra.

Nuclear level density
We were not able to reproduce experimental MSC spectra using the CT model of the NLD in combination with any of the tested PSF models for any Gd isotope.Simulations with the CT model give a clear shift toward lower values in the multiplicity distribution as compared to those obtained with the BSFG model.This feature might be expected from the energy dependence of NLD models shown in Fig. 3(b), where the ratio of levels available at very low excitation energy with respect to energies near 00018-p.9CNR*13 3-5 MeV is higher than for the CT model.Such an effect suppresses transitions with lower γ ray energies and leads to an evident preference for lower multiplicities in the CT model, in disagreement with the experimental results.On the other hand, for all studied nuclei we were able to find several models of PSFs which reproduce data in combination with the BSFG model.In the case of the NLD adopted from the HFB theoretical calculations, we obtained acceptable agreement with the experimental data only for 153,155,158 Gd products.

Electric dipole PSF
It appears that the strong influence of the M1 SM on the γ decay of studied Gd nuclei, discussed in Sec.5.3, significantly suppresses our sensitivity to the E1 PSF.From the comparison of the shapes of experimental and simulated MSC spectra, we are not able to identify one particular E1 PSF model as the most acceptable one.
In all Gd isotopes we obtained an acceptable agreement between simulated and experimental MSC spectra for the GH, KMF, KMF-T (with T f ≈ 0.25−0.35MeV), and MGLO (with variable k 0 ) models.The allowed values of the k 0 parameter in the MGLO model seem to increase with the mass of the isotope from about 1.5 − 2.0 (for 153 Gd) to about 4.0 − 5.0 (for 157,159 Gd).A few models, specifically the SLO, MLO1, MLO2 and MLO3 ones, were acceptable only for specific isotopes.On the other hand, we completely failed using the GFL, EGLO, as well as the microscopic E1 PSF models in all nuclei.The list of acceptable models given above indicates that we were unable to confirm or reject the necessity of a temperature dependence of the E1 PSF.

Magnetic dipole PSF
The MSC spectra for all studied Gd isotopes clearly show a pronounced peak (in odd nuclei) or double-peak (in even-even nuclei) structure present in the middle of the M = 2 spectrum.A clear bump is present also between 2 and 3 MeV in the M = 3 MSC spectrum.These maxima in the MSC spectra indicate the presence of a resonance structure in a PSF.Indeed, all models that do not include a resonance near 3 MeV in a PSF are unable to reproduce the observed shapes of the MSC spectra in all Gd isotopes.This effect is more pronounced in heavier odd 157,159 Gd isotopes, where the observed peak structure is much stronger than in the other nuclei studied.
We found that the resonance cannot be purely of E1 character, see Refs.[6,46,47].The sensitivity of the predicted MSC spectra to the type of transitions comes from the fact that γ cascades start from s-wave resonances which have a specific parity.As spin and parity selection rules for different types and multipolarities of transitions are fully taken into account in the simulations, the multiplicity distribution as well as the shape of the MSC spectra for different multiplicities are sensitive to the ratio of f E1 / f M1 below S n .We need f E1 / f M1 ≈ 1 at E γ between about 2 and 4 MeV.If the resonance structure at 3 MeV was in the E1 PSF, the ratio would be f E1 / f M1 1 and the peak structure observed in the M = 2 MSC spectra would not be reproduced by the simulations, as one of the two γ rays in the two-step cascade has to be of M1 (or E2) character in all Gd isotopes.On the other hand, a weak contribution of the E1 strength to the resonance structure cannot be fully excluded.
In reality, we are unable to unambiguously distinguish between the M1 and E2 character of the resonance as the MSC spectra calculated with both multipolarities often have a very similar shape.This is due to the similar spin and parity selection rules for these two types of transitions.Nevertheless, simulations with a resonance structure postulated in the E2 PSF yielded slightly worse agreement in all tested cases.This fact combined with the expectation of the presence of the M1 SM near 3 MeV in heavy well-deformed nuclei and no prediction of strong transitions of E2 character with E γ ≈ 3 MeV EPJ Web of Conferences lead us to identify the structure responsible for the peaks observed in the experimental MSC spectra with the M1 scissors mode.
The MSC spectra are sensitive to the parameters of the SM postulated in the M1 PSF.In all cases a Lorentzian shape was assumed with three independent parameters, energy E SM , width Γ SM , and the maximum cross section σ SM of the SM.The minimum and maximum values of these parameters, together with the corresponding values of the reduced transition probability B(M1) ↑, that lead to the agreement between the shape of experimental and simulated MSC spectra, can be found in Ref. [6].The comparison of simulated MSC spectra with experimental ones for models that nicely reproduce the measured spectra can be found in Refs.[6,[46][47][48].In this contribution we would rather illustrate the sensitivity of simulations to the change in different SM parameters.
We found that the energy of the SM, E SM , must be very close to 3 MeV in 153,156,157,158,159 Gd but it is shifted to lower values in 155 Gd.This is illustrated in Figs. 4 and 5 for 155 Gd and 159 Gd, respectively.The energy E SM = 2.7 MeV in 155 Gd is very close to the value obtained from the analysis of data from 3 He-induced reactions in the Dy isotopes [16].As illustrated in Fig. 5, our data for the remaining isotopes are in contradiction with the position of the SM measured in Dy.
The damping width of the SM, Γ SM , can reach a rather broad range of values between about 0.7 and 1.3 MeV in all studied isotopes, see Ref. [6].Restricted sensitivity to Γ SM is illustrated in Fig. 6 for 159 Gd.
Both the E SM and Γ SM are almost independent of the E1 PSF model used in the simulations.In addition, the allowed values of σ SM are only slightly correlated with E SM and Γ SM .On the other hand, one can expect a scaling of the size of the bump near E γ = 3 MeV (which can be described by σ SM ) produced in simulations of MSC spectra with the absolute size of the E1 PSF model.Such a dependence is really observed.It means that the allowed values of σ SM in combination with the KMF, GH, or MGLO (with k 0 ≈ 1.5) models are smaller than with the SLO, MLO2, or MGLO(with k 0 ≈ 4 − 5) models.For detailed discussion of this effect, see Ref. [6].
We have found that the values of σ SM obtained for even-even 156,158 Gd, odd 153 Gd, and partly also for 155 Gd, are significantly lower than those for 157,159 Gd -the strength in 157,159 Gd isotopes is about three times higher than in 156,158 Gd and 153 Gd and approximately two times higher than in 155 Gd nucleus.This effect persists with a fixed E1 PSF model.The significant difference in σ SM is apparent for 155 Gd and 158 Gd from Figs. 7 and 8, respectively.In these figures, the higher values of σ SM correspond to those needed for reproduction of MSC spectra in 157,159 Gd.We have no explanation of this odd-even difference in the SM strength at the moment.
In reality, more complex model for the M1 PSF than just the SM seem to be needed.In particular, a "non-resonant" single particle M1 component f (SP)  M1 ≈ (1 − 2.5) × 10 −9 MeV −3 had to be added to the M1 PSF to reach a reasonable reproduction of experimental MSC spectra in the 153,155,156,158 Gd isotopes [6,48].On the other hand, we are virtually insensitive to the exact parametrization of the spin-flip PSF f (SF)  M1 postulated in our simulations.To reach reasonable agreement between simulated and experimental MSC spectra, the SM has to be postulated on all excited levels that are accessible in the γ decay of the studied Gd isotopes.More precisely, as illustrated in Refs.[5,6], we need the SM to be built on all levels up to at least 5 or 6 MeV of excitation energy.For higher excitation energies the sensitivity of our method decreases as the SM resonance structure built on levels above 4-5 MeV in odd and 6-7 MeV in even-even studied Gd nuclei does not influence the decay of neutron resonances in these isotopes.Our sensitivity is also much better for 157,159 Gd products, where the SM is two or three times stronger than in the other Gd isotopes studied.We cannot fully guarantee that the parameters of the SM are completely independent of the excitation energy.However, we tried to change systematically -usually linearlythe SM parameters (E SM , Γ SM , σ SM ) with the excitation energy and we found that the best agreement 00018-p.11

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Figure 8.The same as in Fig. 7 but for MSC spectra in 155 Gd.The MGLO model (with k 0 = 3) was used for the E1 PSF.For further details see Fig. 4. for all nuclei is obtained with the SM following exactly the Brink hypothesis.The SM is thus a resonance structure present in the M1 PSF following the Brink hypothesis [21].

Comparison with other experimental data
Several measurements provide information on the PSFs in the low-energy tail of the GDER in the region of A ≈ 150 − 175.In addition to the outcome of the analysis of MSC spectra presented in this contribution, there are other data from (n, γ) reactions: (i) the absolute values of PSFs were obtained from the intensities of primary transitions following the resonance neutron capture in 154,156,158 Gd target nuclei [44,45], (ii) two-step cascades (TSC) measurements on 162 Dy [35] and 159 Tb [36] targets provided information on the PSFs with the emphasis put on the properties of the scissors mode, (iii) information about the shapes of PSFs near S n was also obtained from the average resonance neutron capture experiments on 156,158 Gd product nuclei [14,22] and (iv) the experimental values of the total radiation width of s-wave neutron resonances are available for all stable isotopes [12,50].

Total radiation width
The DICEBOX code can simulate the γ cascades which are used for production of the MSC spectra, as well as any other quantity related to the radiative neutron capture at low neutron energies.Almost all of these quantities depend only on the energy-dependent relative ratios of the PSFs for different types and multipolarities of transitions.One of the quantities that depends on the absolute values of the PSFs and the NLD and is easily comparable to experimentally measured values is the total radiation width of s-wave neutron resonances Γ γ .The value of Γ γ is a simple sum of contributions of different types and multipolarities of transitions, in our case  2) Experiment [12] 54( 5) 75( 6) 108( 10) 88( 12) 97( 10) 105( 10) Experiment [50] 55(3) 74( 3) 110( 3) 88( 12) 99( 6) 90 (6) E1 PSF models to Γ γ are listed in Tab. 1.It appears that the fraction of Γ γ that originates from the E2 PSF is less than 5%, the part from the SF M1 PSF is up to 10% for 153,155 Gd and below 5% for the other Gd nuclei studied.As the single-particle component of the M1 PSF contributes up to about 17% to Γ γ , the remaining 75-90% of Γ γ comes from the E1 PSF and the M1 SM strength.The value of Γ γ also strongly depends on the model of the NLD.As discussed above only the BSFG model of the NLD leads to the reproduction of experimental spectra in all studied Gd nuclei.Therefore further discussion assumes the use of this NLD model.The requirement on the reproduction of Γ γ can serve as an additional constraint on the acceptability of the PSFs models.For instance, the absolute size of f E1 for the SLO model in the energy region of the SM is much higher than in other E1 PSF models.As discussed above, this means that the reproduction of experimental MSC spectra requires a much higher strength of the SM (and thus also the contribution of the SM to Γ γ ) than for all other E1 PSF models.In odd Gd isotopes, where the SLO model could reproduce the shape of MSC spectra, we would need σ SM = 0.6 − 0.8 mb and σ SM = 1.6 − 2.0 mb for the 153,155 Gd and 157,159 Gd isotopes, respectively.As a consequence, the value of Γ γ for the model which would reproduce the MSC spectra would be at least a factor two higher than the experimental value.We can thus definitely exclude the SLO model for the E1 PSF in these nuclei.We should add here that in even-even Gd products the SLO model reproduces neither the experimental values of Γ γ nor the shape of the measured MSC spectra.
As discussed already in Ref. [6], the experimental value of Γ γ significantly increases with the mass of the odd Gd isotopes, see Tab. 1.On the other hand, the calculations performed with the same E1 PSF model lead to very similar values of Γ γ in all odd Gd isotopes.The predicted difference in Γ γ is thus mainly due to different SM contributions.But the SM alone can be responsible for less than a half of the increase of the experimental Γ γ value in odd Gd isotopes.This fact indicates that the E1 PSF is responsible for a remaining part of this increase, which further implies that there is no universal parameter-free E1 PSF model in the chain of odd Gd isotopes.The only tested E1 PSF model that is able to reproduce the shape of the experimental MSC spectra and the experimental Γ γ 00018-p.13CNR*13 values simultaneously in all Gd isotopes is the MGLO model.The adjustment of the free parameter k 0 introduced in the MGLO E1 PSF model is able to mimic the anticipated change of the E1 PSF.The parameter has to vary from k 0 ≈ 2 for 153 Gd, to k 0 ≈ 4 − 5 for 159 Gd.

Intensities of E1 primary transitions
Additional data on the E1 PSF come from the measurement of intensities of primary E1 transitions following the capture of slow neutrons in the 154,156,158 Gd isotopes [44,45].Experimental data, which are shown in Fig. 3(a) indicate an increase of the E1 PSF at E γ = 5 − 7 MeV with the mass of the isotope.These data correspond perfectly to our finding about the E1 PSF discussed above.We would like to point out here that the E1 PSF should match the Lorentzian shape at energies above S n .This condition requires a rather steep increase of the E1 PSF at E γ ≈ 5 − 9 MeV in lighter Gd isotopes.

Scissors mode
The values of the SM parameters obtained from our analysis of the MSC spectra of Gd isotopes can be compared to values deduced from TSC experiments, 3 He-induced reactions and (γ, γ ) experiments.
As discussed above, there is a discrepancy in the extracted energy of the SM.The position of the mode which comes from 3 He-induced reactions is usually too low to be acceptable for description of the MSC spectra.The energy which comes from our analysis of MSC spectra perfectly agrees with results from (γ, γ ) experiments for the even-even rare-earth nuclei.We have no explanation for the difference in position deduced from different experimental techniques at the moment.
The damping width of the SM is Γ SM ≈ 1.0 MeV in all studied Gd isotopes.This value is comparable to results from previous TSC experiments [35,36] as well as to data from 3 He-induced reactions [16,17,19,20].For completeness, we should notice that it seems to be difficult to say anything about the width of the SM from (γ, γ ) experiments.In even-even nuclei, the observed data from (γ, γ ) indicate that the SM is fragmented into only a few states that seem to be concentrated in a rather narrow region of less than half an MeV but the observed pattern in odd nuclei varies significantly.
Comparison of the strength of the SM from different experiments is rather difficult.Specifically, from the (γ, γ ) measurements one knows only the total M1 reduced strength given by a sum of the strengths of individual M1 transitions observed for the even-even rare-earth nuclei in the restricted energy interval E γ = 2.7 − 3.7 MeV.In addition, the (γ, γ ) data on odd rare-earth nuclei must be nontrivially corrected for the unobserved strength and the result depends on the way the correction is performed [33,34,52].Data from the Oslo method provide information on the SM and SF components of the M1 PSF [16][17][18][19][20], while our data indicate that the total M1 strength consists of contributions from the SM, SF and SP components of the M1 PSF.
Therefore we can compare, for instance, the data from the (γ, γ ) experiments with the sum of SM, SF and SP components in our case, or the sum of SM and SF components from the Oslo method, integrated over the energy range E γ = 2.7 − 3.7 MeV.Or we can compare the pure SM strength, B tot (SM) ↑, obtained from the analysis of MSC spectra and from the Oslo method integrated over the whole energy range assuming a Lorentzian shape for the SM.The comparison of the latter quantity from different reactions for all nuclei as a function of the fractional number P = N p N n /(N p + N n ), where N n and N p is the number of valence neutrons and protons, respectively, see also Refs.[53][54][55], is shown in Fig. 9.The total M1 strength observed in (γ, γ ) experiments at energies between 2.7 and 3.7 MeV was considered to be from the SM for the data plotted in the Fig 9 .A comparison of the SM strength defined in a different way as a function of the neutron number N or the square of the nuclear deformation parameter β 2 is discussed in detail in Refs.[6,48,56].Comparison of the total reduced SM strength B tot (SM) ↑ for rare-earth nuclei as a function of the fractional number P which is defined in the text.Our results [6,46,47] are plotted together with the NRF data [27,33], the data from the 3 He-induced reactions [16][17][18][19][20] and the values obtained in the TSC measurements [35,36].The corrected value for 165 Ho was obtained from the fluctuation analysis [33].

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As discussed above, we see a significant difference between the SM strength in well-deformed odd and even-even Gd isotopes.Our results for 157,159 Gd are in very good accord with the Oslo data for odd Dy, Er and Yb isotopes, and with the values from the TSC experiments on 163 Dy and 160 Tb.However, we observe significantly smaller SM strength in even-even 156,158 Gd product nuclei compared to neighboring odd 157,159 Gd isotopes.Such an odd-even effect seemed to be observed also for the 166,167 Er isotopes studied in Oslo [19], while virtually no odd-even asymmetry was measured in 3 He-induced reactions for Dy and Yb isotopes.Unfortunately, the SM parameters, and thus also the SM strength, determined from the Oslo method strongly depend on the nuclear temperature assumed for the E1 PSF model used in the fit to the Oslo experimental data.The total SM strength is directly proportional to the Γ SM × σ SM product, which can change by a factor of more than two for thee different temperatures used in the fit to the same data [16].Oslo values presented for the Dy isotopes correspond to the KMF-T model of the E1 PSF with the temperature T = 0.3 MeV.

Summary
Measurements of γ-ray spectra from strong isolated s-wave neutron resonances on isotopically enriched 152,154−158 Gd targets were performed using the DANCE detector array installed at the LAN-SCE spallation neutron source.The MSC spectra obtained from these resonances were used to test the validity of various PSF and NLD models.Taking into account additional available information on PSFs and the NLD, the main results of our analysis can be summarized as: (i) The energy dependence of the NLD is well described with the BSFG model.The energy dependence predicted by the CT model is highly unlikely.(ii) The MGLO model with a variable parameter k 0 is the only tested E1 00018-p.15 CNR*13 PSF model that simultaneously reproduces the shapes of the measured MSC spectra and the experimental values of total radiation widths for all studied Gd nuclei.The validity of this model is also supported by the data on intensities of primary transitions from (n, γ) reactions.(iii) The experimental total radiation widths of the s-wave neutron resonances clearly show that the E1 PSF in the studied Gd nuclei cannot be described by the Lorentzian extrapolation of the GDER.(iv) The shapes of the experimental MSC spectra indicate the presence of a resonance structure in the M1 PSF near 3 MeV, that can be described by a Lorentzian curve.This structure was identified with the scissors mode.To reproduce the experimental spectra, the SM has to be built on all excited levels of the studied nuclei.The total reduced SM strength B tot (SM) ↑ observed in odd Gd nuclei increases with the mass number.However, its dependence on the nuclear deformation, see Ref. [56], seems to be at variance with the β 2 2 dependence observed in (γ, γ ) measurements on even-even rare-earth nuclei [27][28][29].(v) The SM observed in the 157,159 Gd isotopes is significantly stronger than that in the even-even 156,158 Gd nuclei.Similar odd-even staggering seems to be present in Oslo data for the 166,167 Er isotopes, while no odd-even asymmetry was observed in the Oslo results for Dy and Yb nuclei.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences,

Figure 1 .
Figure 1.Time-of-flight spectra for 154 Gd (a) and 158 Gd (b) targets after transformation to a neutron-energy scale.There are 400 (exponentially distributed) neutron energy bins per decade.Only events corresponding to cluster multiplicities M = 2 − 7 and a detected energy sum in the region of 5.8 − 6.6 MeV (for 154 Gd) and 5.2 − 6.2 MeV (for 158 Gd) were considered.Resonances used in the analysis of the MSC spectra are labeled.Many weak resonances, especially in the spectrum for 154 Gd target, come from 155,157 Gd impurities.

Figure 2 .
Figure 2. Experimental sum-energy (a) and MSC (b) spectra for cluster multiplicities M = 2 − 5 for J π = 1/2 + s-wave neutron resonances of the 159 Gd product nucleus.Energies of the neutron resonances are indicated.All resonances were normalized to the same number of captures that contribute to the gray areas.Only events from these gray areas were used for the construction of MSC spectra.

Figure 4 .
Figure 4. Illustration of the influence of different postulated energies of the SM on the predicted MSC spectra for 155 Gd.The MGLO model (with k 0 = 3) was used for the E1 PSF and the composite model f (SM)M1 + f (SP) M1 + f(SF)

Figure 5 .
Figure5.The same as in Fig.4but for 159 Gd.In this case, the MGLO model (with k 0 = 5) was used for the E1 PSF.The MSC spectra predicted for E SM = 2.6 MeV (a) -the energy that leads to reproduction of the Oslo data for Dy isotopes[16] -completely fail in describing the measured MSC spectra for multiplicities M = 2 − 4, while the spectra for E SM = 3.0 MeV (b) nicely agree with the experimental data.

Figure 6 .
Figure 6.Illustration of the low sensitivity of predicted MSC spectra to different postulated widths of the SM for 159 Gd.Experimental data are reasonably reproduced with Γ SM = 0.7 MeV (a) as well as Γ SM = 1.3 MeV (b).The MGLO model (with k 0 = 5) was used for the E1 PSF.For further description see the caption of Fig. 4.

Figure 7 .
Figure 7. Illustration of the sensitivity of predicted MSC spectra to different postulated strength of the SM for J π = 1 − resonances of 158 Gd product.The SM parametrization in part (b), which does not reproduce experimental MSC spectra for M = 3 and 4, nicely reproduces the MSC spectra in157,159 Gd.Contrary to this the SM parametrization in part (a) is unable to reproduce experimental MSC spectra in157,159 Gd, see Refs.[5,6].The MGLO model (with k 0 = 2) was used for E1 PSF.For further details see Fig.4.Almost identical results were obtained for MSC spectra from J π = 2 − resonances of 158 Gd as well as for MSC spectra from resonances of both possible spins of 156 Gd.

Figure 9 .
Figure 9.Comparison of the total reduced SM strength B tot (SM) ↑ for rare-earth nuclei as a function of the fractional number P which is defined in the text.Our results[6,46,47] are plotted together with the NRF data[27,33], the data from the 3 He-induced reactions[16][17][18][19][20] and the values obtained in the TSC measurements[35,36].The corrected value for165 Ho was obtained from the fluctuation analysis[33].

Table 1 .
Contributions of different E1 PSF models and the SM to the total radiation widths of s-wave neutron resonances of studied Gd nuclei.The isotopes correspond to the product nuclei in (n,γ) reaction.The energy of the SM was E SM = 3.0 MeV for all isotopes except 155 Gd, where E SM = 2.6 MeV.The SM width was Γ SM = 1.0 MeV in all cases and the maximum cross section was σ SM = 0.2, 0.3, 0.2, 0.7, 0.2, 0.7 mb for153,155,156,157,158,159Gd, respectively.Uncertainties in the simulated values of Γ γ originate from different NRs.