$^{137,138,139}$La($n$, $\gamma$) cross sections constrained with statistical decay properties of $^{138,139,140}$La nuclei

TL;DR

This study measures nuclear level densities and gamma-ray strength functions of La isotopes to accurately calculate neutron capture cross sections relevant for astrophysical processes.

Contribution

It provides new experimental data on level densities and gamma-ray strengths of La isotopes, improving cross section calculations for astrophysical nucleosynthesis.

Findings

Good agreement with previous measurements of $^{139}$La(n,γ) cross sections

Observation of low-energy enhancement in gamma-ray strength functions

Enhanced understanding of nuclear properties in the A~140 region

Abstract

The nuclear level densities and $\gamma$-ray strength functions of $^{138,139,140}$La were measured using the $^{139}$La($^{3}$He, $\alpha$), $^{139}$La($^{3}$He, $^{3}$He$^\prime$) and $^{139}$La(d, p) reactions. The particle-$\gamma$ coincidences were recorded with the silicon particle telescope (SiRi) and NaI(Tl) (CACTUS) arrays. In the context of these experimental results, the low-energy enhancement in the A$\sim$140 region is discussed. The $^{137,138,139}$La($n, \gamma)$ cross sections were calculated at $s$- and $p$-process temperatures using the experimentally measured nuclear level densities and $\gamma$-ray strength functions. Good agreement is found between $^{139}$La($n, \gamma)$ calculated cross sections and previous measurements.

Tables

Table 1. Table 1: Structure data and normalization parameters for 138,139,140 La.

Nucleus	$I_t^{\pi }$	$D_0$	$S_n$	${\sigma }_{BSFG2}\left(S_n\right)$	${\sigma }_{BSFG1}\left(S_n\right)$	${\rho }_{BSFG2}\left(S_n\right)$	${\rho }_{BSFG1}\left(S_n\right)$	${\rho }_{HFB}\left(S_n\right)$	$⟨{\Gamma }_{\gamma }(S_n,J_T,{\pi }_T)⟩$
		[eV]	[MeV]			[${10}^4$ MeV-1]	[${10}^4$ MeV-1]	[${10}^4$ MeV-1]	[meV]
138La	$7/2^{+}$	20.0 $\pm$4.4a	7.452	5.7$\pm$0.6	6.7$\pm$0.7	52.3$\pm$12	68.1$\pm$18.6	74.2$\pm$ 17.0	71.0$\pm$13.6b
139La	$5^{+}$	31.8$\pm$7.0	8.778	5.8$\pm$0.6	6.9$\pm$0.7	30.1$\pm$7.0	37.8$\pm$9.7	25.5 $\pm$ 7.0	95.0$\pm$18.2
140La	$7/2^{+}$	220$\pm$20	5.161	5.0$\pm$0.5	6.2$\pm$0.6	4.1$\pm$0.4	5.5$\pm$1.0	6.2 $\pm$ 0.7	55.0$\pm$2.0

Table 2. Table 2: Astrophysical Maxwellian-averaged cross-sections.

Reaction

$(n,\gamma )$138La

$(n,\gamma )$139La

$(n,\gamma )$140La

$(n,\gamma )$138La

$(n,\gamma )$139La

$(n,\gamma )$140La

Temperature (keV)

30

30

30

215

215

215

MACS (mb)

277.5$\pm$101

298$\pm$81

30.5 $\pm$6

86 $\pm$34

26.5$\pm$10

8.5 $\pm$2

Equations

$$\text{centering}P(E_{\gamma },E_x)\propto \rho (E_f){\mathcal{T}}_{if},\text{@add@centering}$$

$$\text{centering}P(E_{\gamma },E_x)\propto \rho (E_f)\mathcal{T}(E_{\gamma }).\text{@add@centering}$$

$${\chi }^2=\frac{1}{N}\sum _{E_x}\sum _{E_{\gamma }}{\left(\frac{P_{th}(E_{\gamma },E_x)-P(E_{\gamma },E_x)}{\Delta P(E_{\gamma },E_x)}\right)}^2,$$

$$P_{th}(E_{\gamma },E_x)=\frac{\rho (E_f)\mathcal{T}(E_{\gamma })}{\sum _{E_{\gamma }}\rho (E_f)\mathcal{T}(E_{\gamma })}.$$

$$\tilde{\rho }(E_f)=A\rho (E_f)e^{\alpha E_f}$$

$$\tilde{\mathcal{T}}(E_{\gamma })=B\mathcal{T}(E_{\gamma })e^{\alpha E_{\gamma }},$$

$$\rho (S_n)=\frac{2{\sigma }^2}{D_0(J_T+1)e^{[-(J_T+1{)}^2/2{\sigma }^2]}+e^{\left(-J_T^2/2{\sigma }^2\right)}{J}_T},$$

$${\sigma }^2=0.0146A^{\frac{5}{3}}\frac{\sqrt{1+4a(E_x-E_1)}}{2a},$$

$$\rho (E_x)=\frac{1}{T}{e}^{\frac{E_x-E_0}{T}},$$

$$\rho (E_x)=\frac{1}{12\sigma \sqrt{2}}\frac{e^{2\sqrt{a(E_x-\delta )}}}{a^{\frac{1}{4}}(E_x-\delta {)}^{\frac{5}{4}}},$$

$${\sigma }^2=0.01389A^{\frac{5}{3}}\frac{\sqrt{a(E_x-\delta )}}{\bar{a}},$$

$$f(E_{\gamma })=\frac{B\mathcal{T}(E_{\gamma })e^{\alpha E_{\gamma }}}{2\pi E_{\gamma }^3}.$$

$$\begin{array}{r}⟨{\Gamma }_{\gamma }(S_n,J_T,{\pi }_T)⟩=\frac{B}{4\pi D_0}{\int }_0^{S_n}\mathcal{T}(E_{\gamma })\rho (S_n-E_{\gamma })d{E}_{\gamma }\\ \times \sum _{J=-1}^{1}g(S_n-E_{\gamma },J_T\pm \frac{1}{2}+J),\end{array}$$

Full text

I Introduction

At relatively low excitation energies, $E_{x}$, well resolved quantum states are available to which a nucleus can be excited. The $E_{x}$, spins and parities ($J^{\pi}$) of these states, as well as the electromagnetic properties of $\gamma$-ray transitions can be measured using standard particle and $\gamma$-ray spectroscopic techniques. In contrast, as $E_{x}$ approaches the neutron separation energy ($S_{n}$) the number and widths of levels increases dramatically and create a quasi-continuum. In this region states cannot be resolved individually to measure their decay properties. Instead of using discrete spectroscopic tools, a broad range of techniques has been developed to extract statistical properties, below or in the vicinity of $S_{n}$, such as the nuclear level density (NLD) and $\gamma$-ray strength function ($\gamma$SF) which are measures of the average nuclear response. Some of the commonly used experimental methods include (i) ($\gamma$,$\gamma$’) scattering using mono-energetic beams Tonchev2010 ; Angell2012 or Bremsstrahlung photon sources Schwengner09 ; Ozel2014 , (ii) ($n,\gamma$) measurements with thermal/cold neutron beams Firestone2007 ; Kudejova2000 , average resonance capture arc1 , (iii) two-step cascade methods using thermal neutrons Becvar92 or charged particle reactions Wiedeking12 , and (iv) isoscalar sensitive techniques savran2013 ; krumbholz2015 ; pellegri2014 ; negi2016 .

At the University of Oslo a powerful experimental method, known as the Oslo Method AS00 , was developed. It is based on charged particle-$\gamma$ coincidence data from scattering or transfer reactions and allows for the simultaneous extraction of the NLD and $\gamma$SF up to $S_{n}$. The $\gamma$SF extracted with the Oslo Method can not only be used to identify and enhance our understanding of resonance structures on the low-energy tail of the giant electric dipole resonance, but also to obtain sensitive nuclear structure information such as the $\gamma$ deformation from scissors resonances Guttormsen2014 ; Laplace2016 . The $\gamma$SF has the potential to significantly impact reaction cross sections and therefore astrophysical element formation Arn2003 ; Arn2007 and advanced nuclear fuel cycles AFC . Measurements of the NLD provides insight into the evolution of the density of states for different nuclei GuttormsenNLD and can be used to determine nuclear thermodynamic properties such as entropy, nuclear temperature, and heat capacity as a function of $E_{x}$ Gia2014 ; Moretto2015 .

In the present paper we report on the details of the NLDs and $\gamma$SFs, extracted using the Oslo Method, of 138,139,140La and the corresponding ($n$, $\gamma$) cross sections. The 138,139La experimental results have already been used to investigate the synthesis of 138La in $p$-process environments BVK2015 and were able to reduce the uncertainties of its production significantly. The findings do not favour the 138La production by photodisintegration processes, but rather the theory that 138La is produced through neutrino-induced reactions Woosley1990 ; Kajino2014 , with the $\nu_{e}$-capture on 138Ba as the largest contributor Goriely2001 ; Byelikov2007 .

II Experimental Details

Two experiments were performed at the cyclotron laboratory of the University of Oslo, over two consecutive weeks, with a 2.5 mg/cm2 thick natural 139La target and 3He and deuterium beams. The excited 138,139La nuclei were produced through the 139La(3He, $\alpha$) and 139La(3He, 3He′) reaction channels at a beam energy of 38 MeV, while 140La was obtained from 139La(d, p) reactions at 13.5 MeV beam energy. The $\alpha$-$\gamma$, 3He-$\gamma$ and $p$-$\gamma$ coincident events were detected with the SiRi a and CACTUS b arrays within a 3 $\mu$s time window and recorded. During the offline analysis the time gate was decreased to 50 ns for 138,139La and 40 ns for 140La. The SiRi array consists of 64 $\Delta$E-E Si detector telescopes (130 and 1550 $\mu$m thick $\Delta$E and E, respectively) and was positioned 50 mm from the target at $\theta_{Lab}$ = 47*∘* with respect to the beam axis, covering a total solid angle of $\approx$ 6$\%$. CACTUS comprised 26 collimated 5*′′x5′′* NaI(Tl) detectors mounted on a spherical frame, enclosing the target located at the center, with a total efficiency of 14.1$\%$ for 1.3 MeV $\gamma$-ray transitions.

The measured $\alpha$, 3He and $p$ energies were converted to $E_{x}$ for each of the compound nuclei 138,139,140La. Kinematic corrections due to the geometry of the setup and the Q-values of 11800 and 2936 keV NNDC of the respective reactions (3He, $\alpha$) and ($d$, $p$) were taken into account. A typical $E_{x}$ vs $E_{\gamma}$ matrix is shown in Fig. 1 for 140La, and similar matrices were extracted for 138,139La. Above $S_{n}$ there is a significant decrease in the number of events due to the dominating neutron emission probability.

III Oslo Method

A brief outline of the analytical methodology is given here, but a more detailed description of the Oslo Method can be found in Ref. AS00 . The $\gamma$-ray spectra of 138,139,140La nuclei were unfolded using the detector response functions and iterative unfolding method e . Thus the contributions from pair production and Compton scattering were eliminated and only the true full-energy spectra were obtained. From these, the primary $\gamma$-ray spectra were extracted according to the first generation method f .

The $\gamma$SF and NLD of all three La isotopes were extracted from the corresponding primary $\gamma$-ray matrices, $P(E_{\gamma},E_{x})$, referred to as the first-generation matrices AS00 . According to Fermi′s golden rule Dirac1927 ; Fermi1950 , the probability of decay from an initial state $i$ to a set of final states $j$ is proportional to the level density at the final state, $\rho(E_{f})$ where $E_{f}=E_{i}-E_{\gamma}$, and the transition matrix element, $|\langle f|H^{\prime}|i\rangle|^{2}$. The first-generation matrix is proportional to the $\gamma$-ray decay probability and can be factorized according to Fermi*′*s golden rule equivalent expression

[TABLE]

where $\mathcal{T}_{if}$ is a $\gamma$-ray transmission coefficient for the decay from state $i$ to state $f$. Assuming the validity of the Brink hypothesis w and generalizing it to any collective excitation implies that $\mathcal{T}_{if}$ is only dependent on the $\gamma$-ray energy ($E_{\gamma}$) and not on the properties of the states $i$ and $f$ and equations (1) becomes

[TABLE]

The $\mathcal{T}(E_{\gamma})$ and $\rho(E_{f})$ are simultaneously extracted by fitting the theoretical first generation matrix $P_{th}(E_{\gamma},E_{x})$ to the experimental $P(E_{\gamma},E_{x})$ according to AS00

[TABLE]

where $N$ and $\varDelta P(E_{\gamma},E_{x})$ are the degrees of freedom and the uncertainty in the primary matrix, respectively. The theoretical first-generation matrix can be estimated from

[TABLE]

The $\chi^{2}$ minimization was performed in the energy regions of 1 MeV $\leq$ $E_{\gamma}$ $\leq$ 7.1 MeV and 3.5 MeV $\leq$ $E_{x}$ $\leq$ 7.1 MeV for 138La, 1.7 MeV $\leq$ $E_{\gamma}$ $\leq$ 8.5 MeV and 3.5 MeV $\leq$ $E_{x}$ $\leq$ 8.5 MeV for 139La, and 1 MeV $\leq$ $E_{\gamma}$ $\leq$ 5 MeV and 2.8 MeV $\leq$ $E_{x}$ $\leq$ 5 MeV for 140La. The ranges were determined by inspection of the matrices and exclude non-statistical structures. The goodness of fit between $P(E_{\gamma},E_{x})$ and $P_{th}(E_{\gamma},E_{x})$ is illustrated for 140La, at various bins of $E_{x}$, in Fig. 2. This comparison is equally good for all spectra and demonstrates the excellent agreement between the theoretical and experimental first-generation matrices. Hence it allows for the extraction of the correct $\rho(E_{f})$ and $\mathcal{T}(E_{\gamma})$. Similar fits are also obtained for 138,139La.

IV Results

The procedure outlined in Sec. III yields a functional form for $\rho(E_{f})$ and $\mathcal{T}(E_{\gamma})$ which must be normalized to known experimental data to obtain physical solutions. It can be shown that infinitely many solutions of Eq. (3) can be obtained and expressed in the form AS00 :

[TABLE]

[TABLE]

where the $\alpha$ parameter is the common slope between $\tilde{\rho}(E_{f})$ and $\tilde{\mathcal{T}}(E_{\gamma})$ and $A$, $B$ are normalization parameters. The values of $\alpha$ and $A$ are obtained by normalizing $\tilde{\rho}(E_{f})$ to $\rho(S_{n})$ and to the level density of known discrete states.

IV.1 Nuclear Level Densities

Two theoretical models were used to obtain different values of $\rho(S_{n})$ for each isotope. These are the i) Hartree-Fock-Bogoliubov + Combinatorial (HFB + Comb.) h and ii) Constant Temperature + Fermi Gas (CT + FG) model with both parities assumed to have equal contributions. In the latter case, two spin cut-off parameter prescriptions were considered. Thus we explored three different normalizations for each La isotope.

The HFB + Comb. model is a microscopic combinatorial approach that is used to calculate an energy-, spin-, and parity-dependent NLD. It uses the HFB single-particle level scheme to compute incoherent particle-hole state densities as a function of $E_{x}$, spin projection on the intrinsic symmetry axis of the nucleus, and parity. Once the incoherent state densities have been determined, the collective effects such as rotational and vibrational enhancement are accounted for. As shown in Ref. h , these microscopic NLDs can be further normalized to reproduce the experimental neutron resonance spacing at $S_{n}$, hence determining $\rho(S_{n})$, and to the level density of known discrete states.

The first normalization with the CT + FG model is based on the spin cut-off parameter of Ref. Edigy2005 and we calculate $\rho(S_{n})$ according to AS00 :

[TABLE]

where $D_{0}$, $\sigma$, and $J_{T}$ are the $s$-wave resonance spacing, spin cut-off parameter, and spin of a target nucleus in $(n,\gamma)$ reactions. The spin cut-off parameter is given by Edigy2005 :

[TABLE]

where $a$, $E_{1}$ and $A$ are level density parameter, excitation energy shift and nuclear mass. In addition to $\rho(S_{n})$, the NLD for other $E_{x}$ regions was computed with the constant temperature law TEricson1959 :

[TABLE]

where $T$ and $E_{0}$ are the nuclear temperature and energy-shift parameter, respectively. The FG spin distribution was assumed for all $E_{x}$.

In the second approach, $\rho(E_{x},J)$ was calculated with the spin cut-off parameter equation as implemented in the TALYS code Kon2008 . Here the excitation energy is divided into two regions separated by the matching energy $E_{M}$, the point where values from different models and their derivatives are equal. For 0 $<$ $E_{x}$ $<$ $E_{M}$ the Constant Temperature (CT) model is used, while for $E_{x}$ $>$ $E_{M}$, including $S_{n}$, the FG model is used:

[TABLE]

where $a$ and $\sigma$ are the level density parameter and width of the spin distribution, respectively. The energy $\delta$ accounts for breaking of nucleon pairs that is required before the excitation of individual components. The spin cut-off parameter at $S_{n}$ was calculated from TALYS with Kon2008 :

[TABLE]

where $\bar{a}$ is the asymptotic level density parameter that would be obtained in the absence of any shell effect. For the remainder of this contribution we refer to the CT + FG model that is based on Eq. (8) as the BSFG1 + CT, and that from Eq. (11) as the BSFG2 + CT model.

The normalized $\rho(E_{x})$ from models HFB+Comb, BSFG1 + CT, and BSFG2 + CT are shown in Figs. 3, 4, and 5, respectively. In each figure these $\rho(E_{x})$ are superimposed with their corresponding theoretical NLDs for comparison. In the case of 138La there is no $D_{0}$ measurements from ($n$, $\gamma$) resonance experiments due to the unavailability of 137La target material. Hence, we used the estimated value which was taken from our previous work BVK2015 . Similarly the experimental average radiative width $\langle\varGamma_{\gamma}(S_{n},J_{T},\pi_{T})\rangle$, used for the normalization, was estimated with a spline fit as implemented in the TALYS reaction code. For 139La, $D_{0}$ and $\langle\varGamma_{\gamma}(S_{n},J_{T},\pi_{T})\rangle$ are averages of experimental values taken from p ; q , while for 140La they were obtained from Ref. p only. The experimental NLD does not reach energies above $S_{n}$ - $E_{\gamma}^{min}$, where $E_{\gamma}^{min}$ is the minimum $\gamma$-ray energy considered in the extraction of the $\gamma$SF and $\rho(E_{x})$, as discussed in Sec. III. As a result, the interpolation between experimental data to $\rho(S_{n})$ is accomplished using the models discussed (see Figs. 3, 4 and 5). The normalization parameters for the three La isotopes are provided in Tab. 1.

IV.2 $\gamma$-ray Strength Function

With the assumption that statistical decays of the residual nuclei are dominated by dipole transitions LarsenPRL2013 , the $\gamma$SF can be calculated from the $\gamma$-ray transmission coefficient according to:

[TABLE]

The absolute normalization parameter $B$ is calculated from $\langle\varGamma_{\gamma}(S_{n},J_{T},\pi_{T})\rangle$ according to JKopecky1941 :

[TABLE]

where $J_{T}$ and $\pi_{T}$ are the spin and parity of the target nucleus in the ($n$, $\gamma$) reaction, and $\rho(S_{n}-E_{\gamma})$ is the experimental level density. The spin distributions $g(E_{x}$,$J)$ were assumed to follow Gaussian distributions with energy-dependent $\sigma$ which were obtained separately from the HFB + Comb., BSFG1 + CT, and BSFG2 + CT models. These were normalized such that $\sum_{J}g(E_{x},J)\approx 1$. The $\gamma$SF normalized with all three spin distributions are individually compared for each La isotope in Figs. 6 and 7. For 139La these are further compared to the giant electric dipole resonance data taken from Utsunomiya2006 ; Beil1971 . The normalization parameters for the three La isotopes are provided in Table 1.

V Discussion

For 138,140La our measurements provide the first data of the $\gamma$SF and NLD below $S_{n}$. For 139La data are available from $(\gamma,\gamma^{\prime})$ measurements Mak2010 for $E_{x}>$ 6 MeV where a broad resonance structure has been observed for 6 MeV $<E_{x}<$ 10 MeV and interpreted as an E1 pygmy dipole resonance. This is consistent with our data (Fig. 6) where the $\gamma$SF exhibits a broad feature for 6 MeV $<E_{x}<$ 9 MeV. Overall the three spin distributions from the HFB + Comb., BSFG1 + CT, and BSFG2 + CT models yield very similar $\gamma$SFs for each isotope (see Figs. 6 and 7). The $\gamma$SF of 138La exhibits a low-energy enhancement for $E_{\gamma}$ $<$ 2 MeV, (Fig. 7 (a)) for all tested spin-distributions. For 139La the strength function (Fig. 6) could not be extracted for $E_{\gamma}$ $\leq$ 1.7 MeV due to non-statistical (discrete) features in the first-generation matrix. However, it is obvious that the $\gamma$SFs of 139La exhibits a plateau behavior for $E_{\gamma}$ $<$ 3 MeV, similar to 138La which may be indicative of the development of a low-energy up-bend at energies below the measurement limit. A similar plateau structure is also observed in the $\gamma$SF of 140La for $E_{\gamma}$ $\leq$ 3 MeV (Fig. 7(b)) but no clear enhancement can be identified within the available $E_{x}$ range.

The low-energy enhancement has been a puzzling feature since its first observation in 56,57Fe voinov . Its existence was independently confirmed using a different experimental and analytical technique in 95Mo Wiedeking12 which triggered the study into the consistency of this feature with several $\gamma$SF models Kritcka2016 . Experimentally, the composition of the enhancement remains unknown, although it has been shown to be due to dipole transitions LarsenPRL2013 ; LarsenIoP2016 . Three theoretical interpretations have been brought forward to explain the underlying mechanism. According to Ref. Sch2013 this low-energy structure is due to M1 transitions resulting from a reorientation of spins of high-$j$ nucleon orbits, or due to 0$\hbar\omega$ M1 transitions Alex2014 . It has also been suggested that the up-bend could be of E1 nature due to single particle transitions from quasi-continuum to continuum levels Lit2013 .

The emergence of the low-energy enhancement in the La isotopes is interesting and unexpected due to its prior non-observation for A $\geq$ 106 nuclei Larsen2013 . The appearance of this structure in La suggests that it is not confined to specific mass regions but may be found across the nuclear chart, an assumption that has recently received support through its observation in 151,153Sm Simon2016 .

The Brink hypothesis w states that the $\gamma$SF of collective excitations is independent of the properties of initial and final nuclear states and only exhibits an $E_{\gamma}$ dependence. The validity of the Brink Hypothesis was experimentally verified for $\gamma$-ray transitions between states in the quasi-continuum guttormsen2016 . The independence of the set of quantum states from which the enhancement is extracted was confirmed for 138La where two non-overlapping $E_{x}$ regions have been independently used to measure the $\gamma$SF, as shown in Fig. 8. It is apparent that the overall shape of the $\gamma$SF is indeed very similar for both excitation energy regions.

The presence of the low-energy enhancement in the A$\sim$ 140 region emphazises the need for systematic measurements to explore the extent and persistence of this feature, not only for nuclei near the line of $\beta$ stability but also for neutron-rich nuclei where the enhancement is expected to have significant impact on $r$-process reaction rates Larsen2010 . Establishing its electromagnetic character will also improve our understanding of the underlying physical mechanism of the enhancement and should be a priority for future measurements.

The calculated NLDs using different models for the spin distribution (Figs. 3, 4, and 5) are in good agreement with experimental data for all measured $E_{x}$ and for all La isotopes. The measured $\rho(E_{x})$ for 138,139,140La have very similar slopes, but are reduced for 139La compared to 138,140La. This behavior is due to odd-odd 138,140La nuclei having one extra degree of freedom that generates an increase in $\rho(E_{x})$ compared to odd-even 139La. The horizontal difference between NLDs of odd-odd and odd-even nuclei has been related to the pair gap parameter, while the vertical difference is a measure of entropy excess for the quasiparticle Moretto2015 . The constant temperature behavior of the NLDs (above the pair-breaking energy) is a consistently observed feature GuttormsenNLD , that is also confirmed by the HFB + Comb predictions, and has been interpreted as a first-order phase transition Moretto2015 .

According to the Hauser-Feshbach formalism Hauser1952 implemented in the TALYS code Kon2008 , the A-1X$(n,\gamma)^{A}$X cross-sections are proportional to the $\gamma$-ray transmission coefficient, $\mathcal{T}(E_{\gamma})$, of a compound nucleus AX. This $\mathcal{T}(E_{\gamma})$ can in turn be determined from $\rho(E_{x},J^{\pi})$ and $f(E_{\gamma})$, obtained from our measurement, and from which the 137La$(n,\gamma)$, 138La$(n,\gamma)$ and 139La$(n,\gamma)$ cross sections (see Fig. 9) were computed. The statistical uncertainties of the experimental NLDs and $\gamma$SFs have been modified to include uncertainties in $D_{0}$ and $\langle\varGamma_{\gamma}(S_{n},J_{T},\pi_{T})\rangle$, as discussed previously BVK2015 . These modifications to the uncertainties resulted in up to 69$\%$ and 34$\%$ uncertainties in the $\gamma$SFs and NLDs, respectively. For each La isotope we performed three cross-section calculations, in a consistent way, using the $\gamma$SFs and NLDs corresponding to the three adopted models (HFB + Comb., BSFG1 +CT and BSFG2 + CT), resulting in very similar cross sections. The NLDs calculated with theoretical models were used in the excitation energy regions where they agree with the present experimental data, while our data points were interpolated and used in regions where they do not agree with calculated NLDs and discrete states (typically for $E_{x}<$ 2 MeV). In addition, the GSF was assumed to be of $E1$ character for these ($n$, $\gamma$) calculations. However, the effect of having the up-bend and pygmy resonance as M1 was also explored and this resulted in no change in the cross-sections.

Fig. 9(c) shows the 139La$(n,\gamma)$ cross sections which are compared to the directly measured data taken from MIgashira2007 ; rTer2007 ; vhTan2008 ; JVoignier1992 ; DCStupegia1968 ; VAKonks1963 . These are in excellent agreement and support the use of statistical nuclear properties to extract $(n,\gamma)$ cross sections, as previously discussed Laplace2016 ; Larsen2016 ; Renstrom2016 . The comparison of the present cross-section data with those from direct measurements tests the reliability of using statistical decay properties to obtain ($n,\gamma$) cross sections and lends credibility to using this approach to also obtain reliable neutron-capture cross sections for 137La and 138La or for neutron-rich nuclei Spyrou2014 ; Liddick2016 for which no direct measurements are available

Futhermore, the normalized NLDs and GSFs were used to calculate the stellar Maxwellian-averaged cross-sections (MACS) at 30 and 215 keV which are the $s$- and $p$-process temperatures, respectively. These are shown in Tab. 2 for the 137La$(n,\gamma)$, 138La$(n,\gamma)$ and 139La$(n,\gamma)$ reactions. The present MACS for 137La$(n,\gamma)$ and 138La$(n,\gamma)$ are lower than those that were reported in BVK2015 by up to a factor of 2. This is due the newly determined $\gamma$SFs that are correspondingly lower than the previously at $E_{\gamma}$ $<$ 5 MeV due to the different normalization parameters. Nonetheless, for 138La at 215 keV the destructive 137La$(n,\gamma)$ MACS are three times the MACS of the producing reaction 138La$(n,\gamma)$. From these cross sections, it can be deduced Goriely2001 that the synthesis of 138La through photodisintegration processes cannot be efficient enough to reproduce observed abundances, which is consistent with our previous results BVK2015 .

VI Summary

The NLDs and $\gamma$SF of 138,139,140La have been measured below $S_{n}$ using the Oslo Method. Three spin distributions, calculated with HFB + Comb. and the FG Model with two spin cut-off parameters, were used for each La isotope for the normalization of these statistical nuclear properties. The NLDs were further compared with theoretical level densities obtained with HFB + Comb. and CT + FG approaches and are in reasonable agreement with the data. The excitation-energy independence of the low-energy enhancement of 138La has been verified in two different $E_{x}$ regions of the quasi-continuum which is consistent with the Brink hypothesis. Furthermore, the $\gamma$SFs of 139,140La are suggestive of the development of this low-energy structure as well. None of the considered spin distributions, used for the normalization, can unambiguously eliminated it. The 137,138,139La$(n,\gamma)$ cross sections have been computed with the Hauser-Feshbach Model using consistently the NLDs and $\gamma$SFs data which are based on three distinct spin distributions. The 139La$(n,\gamma)$ cross sections were compared to available data and found to be in excellent agreement, giving confidence in the approach to obtain $(n,\gamma)$ cross sections from NLDs and $\gamma$SFs. The new MACSs calculated at 215 keV, for 138La$(n,\gamma)$ and 137La$(n,\gamma)$ reactions, confirm the underproduction of 138La in the $p$-proces.

Acknowledgements.

The authors would like to thank J. C. Müller, A. Semchenkov, and J. C. Wikne for providing excellent beam quality throughout the experiment and N.Y. Kheswa for manufacturing the target. This material is based upon work supported by the National Research Foundation of South Africa under grant nos. 92789 and 80365, by the Research Council of Norway, project grant nos. 205528, 213442, and 210007, by US-NSF grants PHY-1204486 and PHY-1404343, by the US Department of Energy under contract no. DE-AC52-07NA27344, and the Department of Energy National Nuclear Security Administration under Award Number DE-NA0000979 through the Nuclear Science and Security Consortium. S.G. grants the support of the F.R.S.-FNRS. A.C.L. acknowledges funding from the Research Council of Norway, project grant no. 205528 and from ERC-STG-2014 Grant Agreement no. 637686. G.M.T gratefully acknowledges funding of this research from the Research Council of Norway, Project Grant no. 222287.