$\gamma$-ray Strength Function for Barium Isotopes

TL;DR

This study measures photoneutron cross sections for $^{137}$Ba and $^{138}$Ba to constrain the gamma-ray strength function, improving understanding of neutron capture processes relevant to barium nucleosynthesis.

Contribution

It provides experimentally constrained gamma-ray strength functions for barium isotopes using laser Compton-scattering data and theoretical modeling, advancing nuclear astrophysics models.

Findings

Photoneutron cross sections measured below two-neutron threshold.

Constrained gamma-ray strength functions using experimental data and theoretical models.

Discussed implications for s-process nucleosynthesis of barium.

Abstract

Photoneutron cross sections were measured for $^{137}$Ba and $^{138}$Ba at energies below two-neutron threshold using quasi-monochromatic $\gamma$-ray beams produced in laser Compton-scattering at the NewSUBARU synchrotron radiation facility. The photoneutron data are used to constrain the $\gamma$-ray strength function on the basis of the Hartree-Fock-Bogolyubov plus quasi-particle random phase approximation using the Gogny D1M interaction. Supplementing the experimentally constrained $\gamma$-ray strength function with the zero-limit E1 and M1 contributions which are unique to the deexcitation mode, we discuss radiative neutron capture cross sections relevant to the s-process nucleosynthesis of barium isotopes in the vicinity of the neutron magic number 82.

Tables

Table 1. Table 1: The unfolded cross sections σ 𝜎 \sigma evaluated at the maximum γ 𝛾 \gamma -ray energy and the total (statistical plus systematic) uncertainty Δ ​ σ Δ 𝜎 \Delta\sigma .

137Ba	$E_{\gamma }$ [MeV]	$\sigma$ [mb]	$\Delta \sigma$ [mb]
	7.01	3.14	0.33
	7.26	6.97	0.44
	7.51	6.90	0.47
	7.76	8.51	0.40
	8.00	12.81	0.62
	8.26	13.07	0.72
	8.50	17.18	0.80
	8.75	18.99	0.87
	9.00	21,71	1.08
	9.25	27.22	1.30
	9.51	32.71	1.60
	9.74	25.38	1.16
	10.01	29.40	1.40
	10.25	39.46	1.89
	10.49	42.70	2.06
	10.76	42.14	1.95
	11.01	49.51	2.37
	11.26	59.97	2.79
	11.51	61.64	2.88
	12.00	88.85	4.11
	12.49	111.26	5.10
	13.00	148.72	6.84
138Ba	$E_{\gamma }$ [MeV]	$\sigma$ [mb]	$\Delta \sigma$ [mb]
	8.75	11.93	0.27
	9.00	18.62	0.62
	9.25	25.97	1.00
	9.51	29.72	1.21
	9.74	31.28	1.39
	10.01	35.16	1.63
	10.25	39.08	1.82
	10.49	40.57	1.90
	10.76	45.16	2.15
	11.01	51.48	2.47
	11.26	61.69	2.93
	11.51	66.46	3.20
	12.00	90.04	4.35
	12.49	114.18	5.52
	13.00	151.95	7.36

Equations

$$\overleftarrow{f_{X1}}({\epsilon }_{\gamma })={\epsilon }_{\gamma }^{-3}\frac{⟨{\Gamma }_{X1}({\epsilon }_{\gamma })⟩}{D_{\ell }}.$$

$$\overrightarrow{f_{X1}}({\epsilon }_{\gamma })=\frac{{\epsilon }_{\gamma }^{-1}}{3(\pi \hslash c{)}^2}⟨{\sigma }_{X1}({\epsilon }_{\gamma })⟩.$$

$${\sigma }_{\exp }^{E_{\max }}={\int }_{S_n}^{E_{\max }}{D}^{E_{\max }}(E_{\gamma })\sigma (E_{\gamma })d{E}_{\gamma }=\frac{N_n}{N_t{N}_{\gamma }\xi {ϵ}_{n}g}.$$

$${\sigma }_{\mathrm{f}}=\mathbf{D}\sigma ,$$

$$\begin{gathered} {\left(\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ ⋮\\ {\sigma }_N\end{array}\right)}_{\mathrm{f}} \\ \text{mbox}=\left(\begin{array}{ccccc}{D}_{11}& D_{12}& \cdots & \cdots & D_{1M}\\ D_{21}& D_{22}& \cdots & \cdots & D_{2M}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ D_{N1}& D_{N2}& \cdots & \cdots & D_{NM}\end{array}\right)\left(\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ ⋮\\ ⋮\\ {\sigma }_M\end{array}\right). \end{gathered}$$

$${\sigma }^1={\sigma }^0+({\sigma }_{\exp }-{\sigma }_{\mathrm{f}}^0).$$

$${\sigma }_{\mathrm{f}}^i$$

$${\sigma }^{i+1}$$

Full text

††preprint: APS/123-QED

I Introduction

The $\gamma$-ray strength function ($\gamma$SF) Bartholomew73 ; Lone85 is a statistical quantity employed in the Hauser-Feshbach model of the compound nuclear reaction. The $\gamma$SF in the de-excitation mode is a key quantity to determine radiative neutron capture cross sections that are of direct relevance to the s-process nucleosynthesis of elements heavier than iron. The downward $\gamma$SF for dipole radiation with a given energy $\varepsilon_{\gamma}$ is defined RIPL3 by

[TABLE]

Here $X$ is either electric ($E$) or magnetic ($M$), $\langle\Gamma_{X1}(\varepsilon_{\gamma})\rangle$ is the average radiation width, and $D_{\ell}$ is the average level spacing for s-wave ($\ell$=0) or p-wave ($\ell$=1) neutron resonances.

In contrast, the $\gamma$SF in the excitation mode for dipole radiation is defined RIPL3 by the average cross section for $E1/M1$ photoabsorption $\langle\sigma_{X1}(\varepsilon_{\gamma})\rangle$ to the final states with all possible spins and parities Lone85 :

[TABLE]

Above neutron separation energy except at energies near neutron threshold, the total upward $\gamma$SF can be determined by substituting $\langle\sigma_{X1}(\varepsilon_{\gamma})\rangle$ with experimental photoneutron cross sections that dominate photoabsorption cross sections.

A recent systematic study across the chart of nuclei has formulated the zero-limit behavior of both E1 and M1 strengths in the analytical form Goriely18a , the latter of which (referred to as M1 upbend) was experimentally observed Voin04 ; Gutt05 ; Algi08 and theoretically supported by the shell-model calculation Schw13 ; Brow14 ; Siej17a ; Siej17b ; Kara17 ; Schw17 . The presence of the zero-limit strengths which correspond to $\gamma$-ray transitions between high-lying states is unique to the downward $\gamma$SF, showing that the Brink hypothesis of the approximate equality of $\overleftarrow{f_{X1}}$ and $\overrightarrow{f_{X1}}$ Brink ; Axel is violated.

The radiative neutron capture in the s-process path which proceeds along the valley of $\beta$ stability successively produce heavier isotopes by adding one neutron at a time until it is intervened by $\beta^{-}$ decay which increases the atomic number of the element by one. When the s-process flow reaches a magic neutron number, the nuclear binding energy increases by an amount of the order of 1 - 2 MeV, leading to a drop of the neutron separation energy of a compound nucleus which is formed by the neutron capture. The drop results in large level spacing $D$ in Eq. (1), and significantly decreases the downward $\gamma$SF and radiative neutron capture cross section of neutron-magic nuclei BW79 . As a result, the s-process nucleosynthesis forms the 1st, 2nd, and 3rd peaks of elements around the mass number $A$ = 86–90, 138–142 and 208, corresponding to the magic number 50, 82, and 126, respectively.

In this paper, we discuss radiative neutron capture cross sections for barium isotopes including 138Ba with the neutron magic number 82 in terms of the deexcitation $\gamma$SF based on the $\gamma$-ray strength function method Utsu18a ; Utsu19 .

The $\gamma$SF from two relatively different nuclear models are studied, namely the Simple Modified Lorentzian (SMLO) model Goriely18b and the Hartree-Fock-Bogolyubov plus quasi-particle random phase approximation (QRPA) based on the Gogny D1M interaction (hereafter denoted as D1M+QRPA) Martini16 ; Goriely16b ; Goriely18a for both E1 and M1 components is constrained to the present photoneutron cross section measured for 138Ba and, for the first time, 137Ba. The experimentally constrained $\gamma$SF is further supplemented with the zero-limit E1 and M1 contributions which are unique to the deexcitation mode.

Based on the same nuclear ingredients, radiative neutron capture cross sections are calculated with the TALYS code TALYS for other stable barium isotopes including 138Ba and compared with experimental data. The Ba isotopes of the present research interest along the s-process path are shown in Fig. 1.

In Sect. II, our experimental procedure is described and in Sect. III data are analyzed. Our resulting photoneutron cross sections are discussed in Sect. IV and compared with D1M+QRPA calculations. Radiative neutron capture cross sections calculated for stable barium isotopes are compared with experimental data. Finally conclusions are drawn in Sect. V.

II Experimental procedure

The photo-neutron measurements on 137,138Ba took place at the NewSUBARU synchrotronic radiation facility. Quasi-monochromatic pencil-like $\gamma$-ray beams were produced through laser Compton scattering (LCS) of 1064 nm photons from the Nd:YVO4 laser in head-on collisions with relativistic electrons. The electrons were injected from a linear accelerator into the NewSUBARU storage ring with an initial energy of 974 MeV, then subsequently decelerated to nominal energies in the region from 620 and 694 MeV to 849 MeV, providing LCS $\gamma$-ray beams corresponding to the neutron separation energy $S_{n}$ up to 13 MeV for 137Ba and 138Ba respectively. In total, 22 and 15 individual $\gamma$ beams were produced for both 137Ba and 138Ba, respectively.

The 137,138Ba targets were made form isotopically enriched barium carbonate (BaCO3). The material was in pressed together and enclosed in an Al-cylinder with a thin cap. The targets had areal density of $2.236$ g/cm2 and $5.642$ g/cm2, for 137Ba and 138Ba, respectively. The presence of the Al-cap made it necessary not to increase the $\gamma$-ray beam energy above 13 MeV due to $S_{n}$=13.056 MeV for 27Al. The corresponding enrichment of the two isotopes were $85.0\%$ and $99.9\%$. The main contaminant of the 137Ba-target was 138Ba ($\sim$ 15$\%$). This was taken into account by using only the number of 137Ba nuclei below $S_{n}$ of 138Ba, 8611.72 keV, and by considering the neutron contribution from 138Ba above.

The LCS $\gamma$-ray beam produced at NewSUBARU is well calibrated in absolute energy Utsu14 , energy profile Ioana_thesis ; Fili14 ; Utsu15 , and flux Kondo2011 ; Utsu18b . One can find more details of generation and calibration of the LCS $\gamma$-ray beam, and detection of neutrons with a high-efficiency moderation-based detector neutrondet in Ref. Utsu19 .

The measured photo-neutron cross section for an incoming beam with maximum $\gamma$-energy $E_{\rm max}$ is given by the convoluted cross section,

[TABLE]

Here, $D^{E_{\rm max}}$ is the normalized,$\int_{S_{n}}^{E_{\rm max}}D^{E_{\rm max}}dE_{\gamma}=1$, energy distribution of the $\gamma$-ray beam obtained from a GEANT4 geant4ref simulation analysis of experimental response functions of a LaBr3(Ce) beam-profile monitor. The simulated profiles of the $\gamma$ beams, $D^{E_{\rm max}}$, used to investigate 137Ba are shown in Fig. 2. Furthermore, $\sigma(E_{\gamma})$ is the true photo-neutron cross section as a function of energy. The quantity $N_{n}$ represents the number of neutrons detected, $N_{t}$ gives the number of target nuclei per unit area, $N_{\gamma}$ is the number of $\gamma$ rays incident on target, $\epsilon_{n}$ represents the neutron detection efficiency, and finally $\xi=(1-e^{-\mu t})/(\mu t)$ gives a correction factor for self-attenuation in the target. The factor $g$ represents the fraction of the $\gamma$ flux above $S_{n}$.

We have determined the convoluted cross sections $\sigma^{E_{\rm max}}_{\rm exp}$ given by Eq. (3) for $\gamma$ beams with maximum energies in the range $S_{n}\leq E_{\rm max}\leq$ 13 MeV. The convoluted cross sections $\sigma^{E_{\rm max}}_{\rm exp}$ are not connected to a specific $E_{\gamma}$, and we choose to plot them as a function of $E_{\gamma\rm max}$. The convoluted cross sections of the two Ba-isotopes, which are often called monochromatic cross sections, are shown in Fig. 3. The error bars in Fig. 3 represent the total uncertainty in the quantities comprising Eq. (3) and consists of $\sim 3.2\%$ from the efficiency of the neutron detector, $\sim 1\%$ from the pile-up method that gives the number of $\gamma$-rays, and the statistical uncertainty in the number of detected neutrons. The statistical error ranges between $\sim$ 13 $\%$ close to neutron threshold and 4.6 $\%$ for the highest maximum $\gamma$-ray beam energies. Except for the first few data points close to separation energy, the total error is dominated by the uncertainty stemming from the pile-up method and from the simulated efficiency of the neutron detector. For the total uncertainty, we have added these uncorrelated errors quadratically.

III Data analysis

Now we extract the deconvoluted, $E_{\gamma}$ dependent, photo-neutron cross section, $\sigma(E_{\gamma})$, from the integral of Eq. (3) Rens18 . Each of the measurements characterized by the beam energy, $E_{\rm max}$, corresponds to folding of $\sigma(E_{\gamma})$ with the measured beam profile, $D^{E_{\rm max}}$.

By approximating the integral in Eq. (3) with a sum for each $\gamma$-beam profile, we are able to express the problem as a set of linear equations

[TABLE]

where $\sigma_{\rm f}$ is the cross section folded with the beam profile D. The indexes $i$ and $j$ of the matrix element $D_{i,j}$ corresponds to $E_{\rm max}$ and $E_{\gamma}$, respectively. The set of equations is given by

[TABLE]

Each row of $\bf{D}$ corresponds to a GEANT4 simulated $\gamma$ beam profile belonging to a specific measurement characterized by $E_{\rm max}$. See Fig. 2 for a visual representation of the response matrix $\bf{D}$. It is clear that $\bf{D}$ is highly asymmetrical.

As mentioned, we have used $N=15$ beam energies when investigating 137Ba and 18 for 138Ba, but the beam profiles above $S_{n}$ is simulated in steps of 10 keV in $\gamma$-ray energy. As the system of linear equations in Eq. (5) is under-determined, the true $\sigma$ vector cannot be extracted by matrix inversion. In order to find $\sigma$, we utilize a folding iteration method. The main features of this method are as follows:

As a starting point, we choose for the 0th iteration, a constant trial function $\sigma^{0}$. This initial vector is multiplied with $\bf{D}$, and we get the 0th folded vector $\sigma^{0}_{\rm f}={\bf D}\sigma^{0}$.

• 2)

The next trial input function, $\sigma^{1}$, can be established by adding the difference of the experimentally measured spectrum, $\sigma_{\rm{exp}}$, and the folded spectrum, $\sigma^{0}_{\rm f}$, to $\sigma^{0}$. In order to be able to add the folded and the input vector together, we first perform a Piecewise Cubic Hermite Interpolating Polynomial (pchip) interpolation on the folded vector so that the two vectors have equal dimensions. Our new input vector is:

[TABLE]

• 3)

The steps 1) and 2) are iterated $i$ times giving

[TABLE]

until convergence is achieved. This means that $\sigma^{i+1}_{\rm f}\approx\sigma_{\rm exp}$ within the statistical errors. In order to quantitatively check convergence, we calculate the reduced $\chi^{2}$ of $\sigma^{i+1}_{\rm f}$ and $\sigma_{\rm{exp}}$ after each iteration. Approximately four iterations are usually enough for convergence, which is defined when the reduced $\chi^{2}$ value approaches $\approx 1$.

We stopped iterating when the $\chi^{2}$ started to be lower than unity. In principle, the iteration could continue until the reduced $\chi^{2}$ approaches zero, but that results in large unrealistic fluctuations in $\sigma^{i}$ due to over-fitting to the measured points $\sigma_{\rm exp}$.

We estimate the total uncertainty in the unfolded cross sections by calculating an upper limit of the monochromatic cross sections from Fig.3 by adding and subtracting the errors to the measured cross section values. This upper and lower limit is then unfolded separately, resulting in the unfolded cross sections shown in Fig. 4.

In Fig. 4, the unfolded cross sections for 137Ba and 138Ba are evaluated at the maximum energies of the incoming $\gamma$ beams. The error bars represent the difference between the upper and lower limit of the unfolded cross sections.

IV Discussion

The present experimental results are now analyzed in light of the recent systematics of the $\gamma$SF obtained within the phenomenological Lorenztian approach with the SMLO model Goriely18b and within the mean field plus QRPA calculations with the D1M+QRPA model Martini16 ; Goriely16b ; Goriely18a . While the SMLO E1 strength is essentially fitted on available photoabsorption data, the D1M+QRPA has only globally be renormalized to reproduce the bulk of experimental data. Such a renormalization includes a broadening of the QRPA strength to take the neglected damping of collective motions into account as well as a shift of the strength to lower energies due to the contribution beyond the 1 particle - 1 hole excitations and the interaction between the single-particle and low-lying collective phonon degrees of freedom. Such phenomenological corrections have been applied to the present Ba isotopes, as described in Ref. Goriely18a . In addition, in order to reproduce the present photoneutron cross section in the low-energy tail of the giant dipole resonance, we find that a global energy shift of 0.5 MeV of the overall $E1$ strength is required in the specific case of 138Ba. Such renormalizations are within the uncertainties affecting the $\gamma$SF predictions Goriely18a . The resulting D1M+QRPA and SMLO $\gamma$SF for ${}^{137}\rm{Ba}$ and ${}^{138}\rm{Ba}$ are shown in Fig. 5 and compare with the present and previous data. In the case of ${}^{138}\rm{Ba}$, nuclear resonance fluorescence measurements Tonchev10 are also available separately for the $E1$ and $M1$ modes. Details on the extraction of the $\gamma$SF from measured data can be found in Ref. Goriely19 .

Both the D1M+QRPA and SMLO models have proven their capacity to reproduce relatively well all types of experimental data bearing relevant information on $\gamma$SF, such as those extracted from photoabsorption, nuclear resonance fluorescence, Oslo method, radiative neutron or proton captures or inelastic proton experiments as shown in Ref. Goriely19 . We show in Figs.  5-6 that this is also the case for the present Ba photoneutron data in the whole energy range covered by our experiment. In the ${}^{137}\rm{Ba}$($\gamma$,n)${}^{136}\rm{Ba}$ case, the D1M+QRPA M1 spin-flip strength at 8.5 MeV is however clearly overestimated, while the SMLO model underestimates the $E1$ strength in the 9-11 MeV region.

Another way of testing our photoneutron data and the $\gamma$SF deduced out of those is to consider the reverse radiative neutron capture cross sections. Those are available for ${}^{136}\rm{Ba}$ and ${}^{137}\rm{Ba}$ and therefore are sensitive to the low-energy $\gamma$SF of ${}^{137}\rm{Ba}$ and ${}^{138}\rm{Ba}$, respectively. We have considered both the SMLO and D1M+QRPA $E1$ and $M1$ strengths as described above and applied them to the TALYS calculation of the (n,$\gamma$) cross section. It should however be stressed that the photoabsorption strength needs to be complemented by the zero-limit correction when considering the de-excitation of the compound nucleus formed by the neutron capture. Inspired from shell model studies, this low-energy limit has been approximated in Ref. Goriely18a for D1M+QRPA (the model is then referred to as D1M+QRPA+0lim) and in Ref. Goriely18b for SMLO. The radiative cross sections are also sensitive to the adopted nuclear level density model. For these reasons, different prescriptions Koning08 ; Goriely08 have been considered in the TALYS calculations.

We compare in Fig. 7 the ${}^{136}\rm{Ba}$(n,$\gamma$)${}^{137}\rm{Ba}$ and ${}^{137}\rm{Ba}$(n,$\gamma$)${}^{138}\rm{Ba}$ measured cross sections with the TALYS Hauser-Feshbach calculation based on the D1M+QRPA+0lim and SMLO $\gamma$SF and different nuclear level density prescriptions. All nuclear level densities Koning08 ; Goriely08 are normalized to the existing $s$-wave spacing data at the neutron binding energy Capote09 . As shown in Fig. 7, the calculated cross sections are in rather good agreement with available experimental data in the keV region, although D1M+QRPA+0lim tend to underestimate the ${}^{136}\rm{Ba}$(n,$\gamma$)${}^{137}\rm{Ba}$ cross section and SMLO to overestimate the ${}^{137}\rm{Ba}$(n,$\gamma$)${}^{138}\rm{Ba}$ cross section.

To further compare the predictive power of the D1M+QRPA+0lim and SMLO models, we compare in Fig. 8 the 30 keV experimental Maxwellian-averaged cross sections Bao00 with the TALYS predictions based again on both $\gamma$SF models and different prescriptions for the nuclear level densities Koning08 ; Goriely08 . For the 7 stable isotopes of Ba for which experimental values exist, the agreement is relatively good, D1M+QRPA+0lim model giving a root-mean-square deviation on the cross section ratio of 1.26 and SMLO of 1.36. The experimentally unknown Maxwellian-averaged cross sections for 131Ba and 133Ba can therefore be estimated within a typical uncertainty of 30%. We find for 131Ba a 30 keV Maxwellian-averaged cross section of $1100\pm 300$ mb and for 133Ba of $500\pm 200$ mb

V Conclusion

We presented a new experimental determination of the $(\gamma,n)$ cross section for 137Ba and 138Ba performed at the NewSUBARU synchrotron radiation facility. Our new measurements cover the low-energy tail of the giant dipole resonance above the neutron threshold and are found for the 138Ba case to be in excellent agreement with previous measurements. In the 137Ba case, photoneutron cross sections were measured for the first time in this experiment. The new cross sections were used to constrain the E1 and M1 strength functions obtained within the semi-microscopic D1M+QRPA and phenomenological SMLO approaches. We have further confirmed the relevance of the experimentally constrained D1M+QRPA dipole $\gamma$-ray strength function by analyzing the radiative neutron capture cross sections for Ba isotopes considering in addition the zero-limit systematics for both the de-excitation E1 and M1 strengths. Finally, the present analysis was used to estimate the Maxwellian-averaged ${}^{131}\rm{Ba}$(n,$\gamma$)${}^{132}\rm{Ba}$ and ${}^{133}\rm{Ba}$(n,$\gamma$)${}^{134}\rm{Ba}$ cross sections.

VI Acknowledgments

The authors are grateful to H. Ohgaki of the Institute of Advanced Energy, Kyoto University for making a large volume LaBr3(Ce) detector available for the experiment. H.U. acknowledges the support from the Premier Project of the Konan University. S.G. acknowledges the support from the F.R.S.-FNRS. G.M.T. acknowledges funding from the Research Council of Norway, Project Grant Nos. 262952. This work was supported by the IAEA and performed within the IAEA CRP on “Updating the Photonuclear data Library and generating a Reference Database for Photon Strength Functions” (F41032).