Nuclear level densities and γ-ray strength functions of 180 , 181 Ta and neutron capture cross sections

The γ -ray strength functions and nuclear level densities in the quasi-continuum of 180,181Ta are extracted from particle-γ coincidence events with the Oslo Method, below the Sn . The data were used as input in the TALYS reaction code for calculations of the astrophysical Maxwellian-averaged (n, γ ) cross-sections to investigate nucleosynthesis of nature’s rarest stable isotope 180Ta.


Introduction
A small number of naturally occurring neutron-deficient nuclides with Z ≥ 34 referred to as p-nuclei cannot be produced by stellar neutron-capture processes, while almost all p-nuclei with A > 110 are thought to be produced by the photodisintegration of s-and r -process seed nuclei.However, for some nuclear systems, these processes are not sufficient to explain their observed solar abundance and their origin is still not well understood.Calculations of the 180 Ta production in the universe are often controversial since several processes, sometimes exclusively, could reproduce the observed 180 Ta abundance in the cosmos, making it a particularly interesting case to study.A peculiar feature of 180 Ta is that it is the rarest isotope in the solar system, which exists in a 9 − isomeric state at E x = 77 keV (t 1/2,iso > 10 15 yr), with an isotopic abundance of about 0.012%.Over the years many processes, such as slow and rapid neutron capture reactions (s-process, r -process) in stars and supernova explosions, photon-and neutrino-induced reactions in supernovae, have been proposed to be the production mechanism of 180 Ta.However, no consensus exists and it has been theoretically shown that 180 Ta could be exclusively explained with the (γ, n) p-process reaction [1].The s-process can explain the production of 180 Ta, as well, mostly via branching in 179 Hf through the reaction 179 Hf(β − ) 179 Ta(n,γ ) 180 Ta and/or 179 Hf(n,γ ) 180 m Hf(β − ) 180 Ta [2].
Furthermore, more exotic reactions such as neutrino (υ) processes, which include 180 Hf(υ e , e) 180 Ta and 181 Ta(υ, υ n) 180 Ta, have been proposed to partly explain a e-mail: kgashanel@gmail.comb e-mail: bngkheswa@gmail.comc e-mail: wiedeking@tlabs.ac.za its synthesis [3][4][5].Since the astrophysical sites for the nucleosynthesis of 180 Ta remain unknown, a combination of the above processes is undeniably possible.However, the significance of individual processes cannot be clearly determined, as a result of the uncertainties on the reaction rates for 180 Ta due to unavailability of experimental data, such as the nuclear level density (NLD) and γ -ray strength function (γ SF) [6].The NLD is described as the average number of nuclear energy levels as a function of excitation energy E x , while the γ SF gives a measure of the average transition probability for a γ -ray decay.Both nuclear properties are critical for the Hauser-Feshbach formalism, which is implemented in the statistical nuclear reaction code TALYS [7], which is used here to calculate astrophysical neutron capture reaction rates of 179,180 m Ta.
In the present case study, the 180,181 Ta γ SF and NLD below the neutron separation energy, S n , were investigated using the Oslo Method [8].These results are used to determine the corresponding astrophysical Maxwellianaveraged (n, γ ) cross-sections (MACS) which in turn will be utilized in astrophysical network calculations to investigate nucleosynthesis of 180 Ta.In Sect.2, we present experimental details and an overview of the data analysis.In Sect.3, we discuss the results and use our data to estimate MACS for the 179 Ta(n,γ ) 180gs Ta and 180 m Ta(n,γ ) 181 Ta reactions, and their implications for the 180 Ta nucleosynthesis.

Experimental analysis and results
The particle-γ coincidence experiment was performed at the Oslo Cyclotron Laboratory (OCL) using 34 MeV 3 He beam, with an average intensity of ≈2 nA, to populate excited states in 180,181 Ta through the ( 3 He, 3 He γ ) and ND2016 ( 3 He,αγ ) reactions.A 0.8 mg/cm 2 thick self-supporting 181 Ta foil was used as a target.The charged ejectiles in coincidence with γ -rays were recorded with eight E − E silicon ring particle telescope array (SiRi) [9] and the γ -rays were recorded using the high-efficiency multidetector NaI(Tl) array (CACTUS) [10].
The SiRi array was mounted inside the target chamber 5 cm away from the target and placed at backward angles, covering an mean scattering angular range of θ ≈ 126 • to 140 • in steps of 2 • , with respect to the beam axis.The 8-fold segmented front ( E) and back (E) detectors have thicknesses of ≈ 130 µm and 1550 µm, respectively, giving a total of 64 E − E particle telescopes.A 10.5 µm thick aluminium foil was placed in front of the E − E telescopes, to shield δ-electrons.The average energy resolution 1 of the SiRi array is ≈ 350 keV, for ( 3 He, 3 He γ ) 181 Ta reaction.The CACTUS array consists of 26 collimated cylindrical NaI(Tl) detectors with crystal dimensions of 5 × 5 each.The crystals are surrounded by a 3 mm thick lead shield to reduce crosstalk between neighboring detectors and are positioned 22 cm away from the target.The CACTUS array has a total efficiency and resolution of 14.1% and 7% FWHM for a 1332 keV γ -ray transition, respectively.A valid trigger for the analog-todigital converters (ADCs) is constructed when a E − E Si event is in coincidence with a NaI(Tl) event within the ADC master gate.The measured 3 He and α energies were transformed to E x of residual nuclei 181 Ta and 180 Ta, using reaction kinematics, different Q-values and energy losses.As a result, the respective E x versus γ -ray energy, E γ , matrices can then be extracted from the particle-γ coincidence events spectra.
The γ -ray spectra, extracted for each E x bin, were unfolded using unfolding iterative procedure and then corrected for the known response functions of the CACTUS array [11], to obtain the full-energy γ -ray spectra.At this point, the first-generation method [12] is used to extract the primary γ -rays, from the γ rays that emerge from later steps in the decay cascades at each E x bin of the continuum γ -ray spectra.The resulting experimental first generation matrix, which is a distribution of primary γ -rays as a function of E γ and E x , P(E x , E γ ), is shown in Fig. 1.The two regions that correspond to E γ = 400 and 1300 keV are dominated by low statistics due to over-subtraction of discrete and strong γ -ray transitions during the generation of primary γ -ray matrix.Both nuclei under study had low statistics.
The NLD and γ SF of 180,181 Ta were extracted simultaneously from P(E x , E γ ) through an iterative procedure [8], using the ansatz: where the decay probability, P(E x , E γ ), of a γ -ray with energy E γ to be emitted from a specific initial excited state, with energy E x , is proportional to the NLD ρ(E f ) of the final state, with energy E f = E x − E γ , and the γ -ray transmission coefficient T (E γ ).The relationship in Eq. ( 1) is only appropriate at high NLDs, assuming that the Brink Hypothesis [13] holds for all types of collective decay modes and that the transition 1 The energy resolution of the particle telescope is determined by measuring the full width half maximum (FWHM) of the 3 He beam elastically scattering off the 181 Ta target.Table 1.Parameters used for normalization of ρ(E x ) and T (E γ ) in 180,181 Ta, where σ is the spin cut-off parameter at (S n ).probability for a decay into any specific combination of final states is independent of how the compound nucleus [14] was formed.Henceforth, ρ(E f ) and T (E γ ) can be extracted using an iterative procedure [8], where the theoretical first-generation γ -ray matrices P th (E x , E γ ) are fitted to the experimental first-generation γ -ray matrices P(E x , E γ ) by performing a global χ 2 minimization.A global χ 2 minimum was achieved in the energy regions of E γ > 1634 keV and 2569 keV ≤ E x ≤ 7376 keV for 181 Ta, and E γ > 1734 keV and 2969 keV ≤ E x ≤ 6348 keV for 180 Ta.
Once the ρ(E f ) and T (E γ ) have been simultaneously extracted, there exist infinitely many solutions, for the χ 2 above, of the form: where α, A and B are the normalization parameters, which correspond to physical solutions.The parameters α and A are determined by normalizing ρ to the level density of known discrete states at low E x and to ρ(S n ) (calculated from experimental average neutron resonance spacing, D 0 ) at high E x , and B is calculated from the average total radiative width γ (S n ) .In the case of 180 Ta, neither D 0 nor γ (S n ) are known in the literature, since the target nuclei for (n, γ ) reactions is unstable.Therefore, using the spline fit, as implemented in TALYS [7], γ (S n ) was estimated.The ρ(S n ) was estimated by normalizing both ρ(E x ) and T (E γ ) of 180 Ta on the basis of having the same slope as ρ(E x ) and T (E γ ) of 181 Ta.It has been shown that ρ(E x ) and T (E γ ) of neighboring isotopes have the same slope [15][16][17].The value of ρ(S n ) was then used to calculate D 0 of 180 Ta using equation ( 20) of Ref. [18].The NLD of 180,181 Ta are shown in Fig. 2. Assuming dipole transitions, the experimental γ SF, f (E γ ), is related to γ -ray transmission coefficient by The extracted 181 Ta γ SF is compared to various known data as shown in Fig. 3.The two components of the giant electric dipole resonance, (GEDR) are fitted with enhanced generalized Lorentzian functions (EGLO) [25], f G E D R1 (E γ ) and f G E D R2 (E γ ), at E γ ≈ 12.6 MeV and 15.9 MeV.A constant nuclear temperature of T f = 0.47 MeV, which was treated as a free parameter, was considered for the temperature dependence width γ .This is consistent with the Brink hypothesis assumed in the Oslo method, since T f is constant with increasing E x .In addition to the GEDR, a weaker resonance was also fitted using the Standard Lorentzian functions (SLO), f Res2 (E γ ) at E γ ≈ 6.7 MeV.This resonance was recently observed [26] and was considered as E1 pygmy resonance.The SLO f Res1 (E γ ) was used to fit the additional strength at E γ ≈ 4.8 MeV, although the electromagnetic character is unknown, and f res3 (E γ ) to fit the M1 spin-flip resonance at E γ ≈ 7.5 MeV.Therefore, the total model prediction of the γ SF is given by The fitted functions clearly reproduce the (γ ,x) data together with the measured low-energy data.

Discussion and future outlook
The 180,181 Ta γ SFs show no pronounced features, except for the observed enhancement in the strength function from 6 MeV termed "Res2" resonance in 181 Ta which may be related to E1 pygmy resonance (see Fig. 3).Besides the E1 pygmy resonance, the 181 Ta γ SF is relatively featureless with only a weak resonance at E γ ≈ 4.8 MeV, and certainly no low-energy enhancement.The NLD for odd-odd 180 Ta is higher than that of the even-odd 181 Ta (see Fig. 2).This is expected, due to one extra unpaired neutron  in 180 Ta which increases the number of degrees of freedom.In the region around 2 MeV of the 181 Ta NLD, a small change in the slope is observed which can be explained as Cooper pair breaking.
Assuming the principle of detailed balance to be valid [27], the (n, γ ) cross sections and the reverse photoneutron emission rates of astrophysical relevance, as well as the MACS, were estimated for both 180,181 Ta isotopes.The calculations were achieved using the statistical nuclear reactions code TALYS (version 1.6).Figure 4 shows the final (n, γ ) cross sections, σ (E n ), as a function of incident neutron energies, E n , taking into account the uncertainties affecting the γ SFs and the NLDs.The (n, γ ) cross sections of 180m Ta from Ref. [28] are shown for comparison.Our 180m Ta(n, γ ) cross sections show good agreement with the previously measured 180 m Ta(n, γ ) cross sections [28], within the error bars.
The astrophysical MACS were calculated for both 179,180 m Ta(n,γ ) reactions, at the s-and p-process thermal energies of kT = 30 keV and kT = 215 keV, respectively, using the newly determined NLDs and γ SFs.At kT = 215 keV, the 179,180 m Ta(n,γ ) reaction rates amount to ND2016 σ ν = 793 +241 −186 mb and 574 +49 −53 mb, respectively.It can be noted that the 181 Ta(γ ,n) reaction rates are about 28% less than the destructive 180gs Ta(γ ,n) reaction rates.At kT = 30 keV, the 179,180 m Ta(n,γ ) reaction rates amount to σ ν = 2445 +482 −349 mb and 2047 +129 −146 mb, respectively.These newly calculated 179,180 m Ta(n,γ ) σ ν values are 45% and 28% larger than the MACS from KADoNiS [29], respectively.The possible s-process production of 180 Ta, occurs mostly via beta-decay branching from an excited state in 179 Hf according to Ref. [2].To further investigate the s-process production of 180 Ta, relevant cross sections of neighboring nuclei need to be experimentally investigated as well.
Future measurements of the NLD and γ SF are essential to obtain experimentally constrained (n,γ ) cross sections to investigate galactic production mechanism of 180 Ta from various processes and astrophysical sites.

Figure 1 .
Figure 1.The experimental first generation matrix for 181 Ta.

Figure 2 .
Figure 2. The extracted NLDs of 180,181 Ta.The 181 Ta data points are normalized to known discrete levels (solid line) at low E x and to the level density at the S n (open square) using an interpolation with the Constant Temperature model [22] (dashed line).