Nuclear level densities and $\gamma$-ray strength functions of $^{87}\mathrm{Kr}$ - First application of the Oslo Method in inverse kinematics

The $\gamma$-ray strength function ($\gamma$SF) and nuclear level density (NLD) have been extracted for the first time from inverse kinematic reactions with the Oslo Method. This novel technique allows measurements of these properties across a wide range of previously inaccessible nuclei. Proton-$\gamma$ coincidence events from the $\mathrm{d}(^{86}\mathrm{Kr}, \mathrm{p}\gamma)^{87}\mathrm{Kr}$ reaction were measured at iThemba LABS and the $\gamma$SF and NLD in $^{87}$Kr obtained. The nature of the low-energy region of the $\gamma$SF is explored through comparison to Shell Model calculations and is found to be dominated by M1 strength. The $\gamma$SF and NLD are used as input parameters to Hauser-Feshbach calculations to constrain $(\mathrm{n},\gamma)$ cross sections of nuclei using the TALYS reaction code. These results are compared to $^{86}$Kr(n,$\gamma$) data from direct measurements.


Introduction
The nuclear level density (NLD) and the γray strength function (γSF) are fundamental properties of the nucleus. The NLD was introduced by Bethe soon after the composition of nuclei was firmly established [1]. When ex-citation energy in a nucleus increases towards the particle separation energy, the NLD increases rapidly, creating a region referred to as the quasi-continuum. The ability of atomic nuclei to emit and absorb photons in the quasicontinuum is determined by the γSF [2]. It is a measure of the average reduced γ-ray decay probability and reveals essential information about the electromagnetic response and therefore the nuclear structure of the nucleus.
With their significant applicability to astrophysical element formation via capture processes [3,4,5,6], NLDs and γSFs have received increased experimental and theoretical attention. They are also relevant to the design of existing and future nuclear power reactors, where reactor simulations depend on many evaluated nuclear reactions [7,8]. The importance of NLDs and γSFs is increasingly being recognized and efforts are currently underway to generate a reference database for γSFs [9]. Nonetheless, challenges remain and nuclear physics properties, such as the NLD and γSF, remain a main source of uncertainty in cross-section calculations. This is either due to the complete lack of experimental data or the associated large experimental uncertainties.
The situation can be improved through accurate experimental neutron capture cross sections, or indirectly by measuring NLD and γSF data. One experimental approach, the Oslo Method [10], has been extensively used to measure the NLD and γSF from particle-γ coincident data. NLDs and γSFs obtained with the Oslo Method have been shown to provide reliable neutron capture cross sections [11,12] and proton capture cross sections [13]. In recent years, the Oslo Method has been extended to extract the γSF and NLD following β decay [14]. Using γSFs and NLDs to determine capture cross sections has several advantages since these properties can be obtained for any nucleus that can be populated in a reaction from which the excitation energy can be experimentally determined. Although the Oslo and β-Oslo Methods provide access to a vast range of stable and radioactive nuclei some species remain inaccessible. Many more nuclei become accessible by using inverse kinematic reactions, from radioactive species to several stable isotopes for which the manufacture of targets is problematic due to their chemical or physical properties.
In this Letter we report on the first application to measure the NLD and γSF with the Oslo Method following an inverse kinematic reaction. This work lays the foundation of new opportunities to study statistical properties of nuclei, which were previously inaccessible, at stable and radioactive ion beam facilities. The results from the d( 86 Kr, p) 87 Kr reaction exhibit a low-energy enhancement of the γSF in 87 Kr, which is discussed in the context of Shell Model calculations. The 86 Kr(n, γ) cross section is obtained from the TALYS reaction code [15] and compared to previous direct measurements to test the robustness of the experimental method.

Experiment
The experiment was performed with a 300 MeV 86 Kr beam from the Separated Sector Cyclotron facility at iThemba LABS. Polyethylene targets with 99% deuteron enrichment were bombarded with a beam intensity of ≈ 0.1 pnA for 80 hrs. Several deuterated polyethylene targets, ranging in thicknesses from 110 to 550 µg/cm 2 , were used. Accounting for the target thicknesses the center-of-mass (CM) energy was 6.44 (40) MeV. The reactions were identified through the detection of light charged particles in two silicon ∆E-E telescopes covering scattering angles between 24 • and 67 • relative to the beam direction (corresponding to CM-angles 50 • to 130 • ). The E detectors were 1 mm thick while the ∆E detectors were 0.3 and 0.5 mm thick. The dimensions of the W1-type doublesided silicon strip detectors [16] were 4.8 × 4.8 cm and they consisted of 16 parallel and perpendicular strips 3 mm wide with an opening angle of ≈ 1.5 • for each pixel. Suppression of δ electrons was achieved by an aluminum foil of 4.1 mg/cm 2 areal density which was placed in front of the ∆E detectors. The γ-rays were measured with the AFRODITE array [17], which at the time of the experiment consisted of eight collimated and Compton suppressed high-purity germanium CLOVER-type detectors. Two non-collimated LaBr 3 :Ce detectors (3.5 " × 8 " ) were coupled to the AFRODITE array and mounted 24 cm from the target at 45 • . The detectors were calibrated using standard 152 Eu and 56 Co sources. The detector signals were processed by XIA digital electronics in time-stamped list mode with each channel self-triggered.
From the time-stamped data particle − γ events were constructed with an offline coincidence time window of 1850 ns. From double fold events, the p − γ coincidences were extracted by placing a gate on the protons in the particle identification spectrum. The selection of correlated events was made with a coincidence time of ∼ 80 ns by appropriately gating the prompt time peak. Contributions from uncorrelated events were subtracted from the data by placing off-prompt time gates. Approximately 100k proton-γ events remained in both LaBr 3 :Ce and CLOVER matrices. In this letter only the data from the LaBr 3 :Ce detectors are included, although data from the CLOVER detectors yield similar results. Kinematic corrections due to the reaction Qvalue, recoil energy of 87 Kr, and the energy losses of the protons in the target and aluminum foils were applied to determine the ex-citation energy of the populated states. The experimental resolution of the excitation energy is of the order of ≈ 500 keV. The γ-rays in coincidence with protons were Doppler corrected by assuming the residual 87 Kr nucleus not being deflected from the beam axis and has a constant velocity of 8.5% of c. Due to these assumptions the error in deflection angle is less than 1.3 • while the error in velocity is less than 0.4% of c. These error are negligible as the major contributor to errors in the Doppler correction is the 17 • opening angle of the LaBr3:Ce detectors. The resulting p − γ coincidence matrix for 87 Kr is shown in Figure 1(a). This matrix is unfolded [18] with response functions of the detectors extracted from a Geant4 [19] simulation of the LaBr 3 :Ce detectors. An iterative subtraction method, known as the First-Generation Method [20], is applied to the unfolded γ-ray spectra, revealing the distribution of primary γ-rays in each excitation bin (bin width 256 keV) and is shown in Figure 1 The NLD ρ(E x ) at excitation energy E x and γ-ray transmission coefficient, T (E γ ), are related to the primary γ-ray spectrum by and are extracted with a χ 2 -method [10] giving the unique solution of the functional shape of the NLD and T (E γ ). These are normalized to known experimental data to retrieve the correct slope and absolute value. The extraction has been performed within the limits 3.2 < E x < 5.2 MeV and E γ > 1.7 MeV of the primary γray matrix where statistical decay is observed to be dominant, as shown by the area enclosed by dashed lines in Figure 1(b).

Normalization
From the primary γ-ray spectrum the NLD ρ(E x ) and γ-transmission coefficient T (E γ ) are extracted. These are related to the physical solution by the following transformation: where A and B are the absolute values and α gives the slope. For the level density, the slope and absolute value are determined by a fit to the level density found from the known discrete levels [21] at low-excitation energy and the level density at the neutron separation energy (S n = 5.5 MeV) interpolated to the experimental data points with the constant temperature (CT) model [22]. The temperature is determined such that it minimizes where ρ i is the level density found from known discrete levels for energies E i < 2.4 MeV and the constant temperature interpolation for energies E i > 2.4 MeV. The ρ(E i ) is given by Eqn.
(2) and ∆ρ(E i ) is the statistical and systematic uncertainties associated with the unfolding and first-generation method. The level density at S n is determined from the average neutron resonance spacing [23] and is found to be 26.2(21) keV for s-wave resonances and 18.8 (14) keV for J = 1/2 p-wave resonances, giving a level density of 1/2 states of 91 (5) MeV −1 . The total NLD at S n is given by ρ(S n ) = ρ(S n , 1/2)/g(S n , 1/2) with a spin distribution The spin cutoff parameter σ(E) is modeled with the following energy dependence [11] where E d is the excitation energy below which the spin cutoff parameter σ = σ d is a constant. The spin cutoff parameter σ d at E d ≤ 2.4 MeV is estimated to be 1.75, based on the spin assignment of the known levels, while the cutoff parameter at the neutron separation energy σ(S n ) is estimated to be 3.95, based on the estimate of the spin cutoff models of Refs. [24,22,25]. The uncertainty of σ(E) is estimated to be 15% for all relevant energies, giving a total NLD of 1472(427) MeV −1 at S n . Since the reaction is sub-Coulomb barrier, most of the reactions are due to neutron capture following deuteron breakup in the Coulomb field of the 86 Kr projectile. Thus the extracted level density in (1) needs to be weighted by a reduction factor as 1/2 states seems to be strongly favoured in the initial population. This is accounted for in The absolute value of the transmission coefficients are normalized to the average radiative width of s-wave resonances Γ γ0 in a process detailed in [27], and converted to γSF by The value of Γ γ0 is estimated to be 0.25(10) eV based on the measured Γ γ of s-wave resonances of [26]. All normalization parameters are listed in Table 1.

Nuclear level densities and γ-ray strength functions
The normalized NLD is shown in Figure 2 and is in excellent agreement with the constant temperature level density and matches well with the known discrete states at lower excitation energies. The normalized γSF is shown in Figure 3 and is consistent with γSFs from 86 Kr(γ, γ ) [28] and 86 Kr(γ, n) [29], with the enhancement seen in the (γ, γ ) data between 6 and 8 MeV caused by a Pygmy resonance [28]. A drop in the γSFs at ∼ 2.1 MeV is caused by the 2123-keV state in 87 Kr, which is strongly populated in the reaction, but less through feeding from the quasi-continuum. This causes the first generation method to    [28] and 86 Kr(γ, n) (green squares) [29]. The solid black line are results from Shell Model calculations with a 78 Ni core (see section 5 for details), while the red line is the microscopic HFB+QRPA prediction [30] for the E1 strength. The error bars include all known statistical and systematic errors. over-subtract in the higher excitation-energy bins, causing an artificial drop in the γSF. This effect has previously been discussed [31]. At low energies we observe a large enhancement in the γSF, similar to what has been observed in several other nuclei [32,33,34,35,36,37,38]. Although the upbend has been independently confirmed [39], little is known of the origin of this feature, except that it is dominated by dipole radiation [40,41,42] and that it can have large effects on neutron capture cross sections [43].

Shell Model calculations
Calculations within the shell-model framework predicts the upbend due to M1 transitions [44]. In this work, large-scale shell-model calculations of the M1 component of the γSF were performed in the model space outside the 78 Ni core, containing f 5/2 p 3/2 p 1/2 g 9/2 -proton and d 5/2 s 1/2 d 3/2 g 7/2 h 11/2 -neutron orbitals. The effective interaction employed here is described e.g. in Refs. [45,46]. The diagonalization of the Hamiltonian matrix in the full configuration space was achieved using the Strasbourg shell-model code NATHAN [47]. The spin-part of the magnetic operator was quenched by a common factor of 0.75 [47]. We computed this way up to 60 states of each spin between 1/2 and 15/2 for both parities. This leads to a total of around 8 · 10 4 M1 matrix elements, among which 14822 connect states located in the energy range E x = 3.4 − 5.4 MeV, as considered in the experiment. To obtain the average strength per energy interval, B(M1) , the total transition strength was accumulated in 200 keV bins and divided by the number of transitions within these bins. The γSF was obtained from the relation is the partial level density at the energy of the initial state (E i ). The γSF, shown in Figure 3, is an average of the f M1 s evaluated for each spin/parity separately. The shape of the shell-model γSF is consistent with experimental data up to ∼ 3MeV. Since the model space does not contain all spin-orbit partners (i.e., νg 9/2 and π f 7/2 orbits) the strength above 4 MeV, due to the spin-flip transitions, can not be accounted for. However, the theoretical γSF exhibits significant strength at E γ = 0, as in the previous shell-model calculations in this mass region [44]. The largest B(M1) contributions at low γ-ray energies in 87 Kr are related to transitions between close-lying negative-parity states with νd 5/2 ⊗ π f −1 5/2 g 1 9/2 and νd 5/2 ⊗ πp −1 3/2 g 1 9/2 components. The magnitude of the theoretical M1 strength is in good agreement with the data as measured in the experiment, however we cannot exclude an additional contribution from E1 strength. Recent experimental results in 56 Fe [42] could suggest a mixture of M1 and E1 radiation in the enhancement region and the addition of a non-zero E1 component without an upbend towards E γ → 0 MeV is predicted from Shell Model calculations [48]. Including the E1 strength calculations from the Hartree-Fock-Bogolyubov + QRPA (HFB+QRPA) model by [30] we observe an overall good agreement between theoretical predictions and experimental results.

Neutron capture cross sections
In a statistical framework the 86 Kr(n, γ) cross section can be determined from the NLD, γSF and a suitable neutron optical model potential (nOMP) for 87 Kr.
Phenomenological nOMPs e.g. from Ref. [49] are observed to give good agreement with the total cross section for nuclei close to the valley of stability. We performed Hauser-Feshbach (HF) [50] calculations with the TALYS 1 code [15], and the optical model potential of Ref. [49]. A semi-microscopic optical model [51] was also tested, and gave virtually the same results. Pre-equilibrium reactions were also taken into account. The 87 Kr states used for the TALYS calculations are described by the known discrete states up to 2.3 MeV, and by the measured NLD above. Beyond 3.7 MeV the NLD is described by the CT model.
The measured γSF are used as input between 1.6 ≤ E γ ≤ 5.2 MeV (excluding the 2.1 MeV data point), as shown in Figure 3, for E γ < 1.6 MeV the results from the Shell Model calculations are used, while the results from microscopic HFB+QRPA calculations [30], as implemented in TALYS, of the E1 strength are used for E γ > 5.2 MeV. The M1 spin-flip contribution is also included as a standard Lorenzian with the TALYS parameterization. Figure 4 shows the resulting neutron capture cross section calculations. The input parameters have been varied in accordance with the statistical and systematic uncertainties to produce the red hashed error-band. We observe an overall good agreement with the direct measurements of Walter et al. [52] and Bhike et al. [53], while somewhat high compared with the activation results of Beer et al. [54].
The Maxwellian average (MACS) at the typical s-process temperature of 30 keV is found to be 7.6(49) mb, which is higher than the evaluated value of 3.4(3) mb found in KaDoNis [55]. This discrepancy can be explained by the fact that HF calculations will give results that overestimate the MACS for low temperatures when the level density is low [56]. A possible resolution could be the emerging "High Fidelity Resonance" method [57,58].

Conclusion
We have presented a novel method for obtaining γSF and NLD using inverse kinematic reactions, which opens opportunities to study a wide range of stable and radioactive nuclei. The d( 86 Kr, pγ) reaction was used to measure the NLD and γSF in 87 Kr. The low-energy part of the γSF is found to exhibit an enhancement. Shell Model calculations were performed and suggest that the enhancement is predominantly due to low-energy M1 transitions in 87 Kr.
The γSF and NLD measurements in 87 Kr were used to calculate (n, γ) cross sections, which are in good agreement with those from direct measurements, and give confidence in the approach using inverse kinematic reactions. This is consistent with the findings of previous work with the Oslo Method and is particularly interesting since direct measurement of neutron capture cross sections over a wide range of incident neutron energies is very challenging. It is clear that γSFs and NLDs provide a viable alternative to obtain reliable capture cross sections.
With the Inverse-Oslo Method, new regions of the nuclear chart become accessible to experiments, which also brings about new challenges. For exotic nuclei, neutron resonance data are not known and the normalizing procedure needs to be revised. One possibility is that the slope of the γSF, and thereby also the slope of the NLD, could be constrained using the Ratio Method [39], leaving the absolute value of the NLD to be determined by the known discrete levels. Unfortunately, this still does not determine the absolute value of the γSF. However, reasonable estimates of the absolute value may be obtained from systematics of the Γ γ0 .
Measuring statistical properties of nuclei from inverse kinematic reactions provides a novel and complementary foundation for ex-ploring the limitations of the current models of statistical behavior in the nucleus. It will allow to further constrain the uncertainties in models which are used in nuclear astrophysics and reactor physics.

Acknowledgements
The authors would like to thank iThemba LABS operations for stable running condi-