Back-n facility at CSNS

The measurements were conducted at the CSNS Back-n. The CSNS comprises an 80 MeV linear accelerator, a 1.6 GeV proton synchrotron (PS), two beam transport lines, a target station, three spectrometers, and the associated instrumentation [12, 13]. At the CSNS, spallation neutrons are generated by a 1.6 GeV/c proton beam with a 41 ns full width at half maximum from the PS incident on a tungsten target. The PS operates in either double- or single-beam cluster mode, delivering pulses at 25 Hz and an average current of 64 μA. A dedicated 15 ° deflection magnet designed by CSNS steers the proton beam before the target, separating the backscattered white neutron beam from the primary proton beam [14].

The Back-n beamline is situated 80 m downstream of the CSNS proton beam. It comprises two experimental stations: 55 m (ES#1) and 76 m (ES#2). ES#2 is outfitted with two different systems for capture experiments: the 4π GTAF-II array of 40 BaF₂ detectors for total absorption calorimetry and a total energy detector consisting of four low-sensitivity C₆D₆ detectors [15]. The neutron flux at the ES#2 sample position is 6.92×10⁵ cm^-2·s^-1 over an energy range of 0.3 eV to 200 MeV distributed over three beam spot sizes: (Ф20 mm, Ф30 mm, and Ф60 mm). The neutron spectrum was characterized using a Li-Si detector and a calibrated fission chamber, exploiting the standard ⁶Li(n,t) and ²³⁵U(n,f) reactions, respectively [16]. Additionally, a silicon flux monitor (SiMon) comprising a thin ⁶LiF converter layer and eight silicon detectors positioned 20 mm upstream of the sample continuously monitored the beam intensity [17-19].

The ¹⁵⁹Tb and Samples

In this study, neutron capture cross-sectional measurements of ¹⁵⁹Tb were performed using a C₆D₆ detector array. A metallic sample of pure terbium with a diameter of SI30.0 mm and area density of 6.27×10^-4 atom/barn was used for the prompt capture γ-ray measurement. A natural lead ^natPb with a diameter and thickness of 0.53 mm was used to evaluate the in-beam γ-ray and neutron-induced background [13]. Additionally, ¹⁹⁷Au sample with thickness of 0.1 mm was used to normalize the neutron capture data. An empty sample holder was placed on the beam to measure the background it generates. The evaluated background spectrum was normalized using the black resonance technique, which made it possible to distinguish between background and true signals. Samples ⁵⁹Co and ¹⁸¹Ta were employed as neutron filters. Sample preparation was performed by the Department of Nuclear Physics of the China Institute of Atomic Energy. Much More details on the experimental setup can be found in Ref. [20-26].

Background Deduction and Data Analysis

Background determination is a major challenge in experimental studies. The experimental data analysis was largely dependent on Monte Carlo simulations conducted using the Geant4 toolkit [27], which accurately reproduced the experimental configuration and formed a critical foundation for subsequent analyses. To maximize the precision of the measured gamma energy spectra, we implemented a rigorous efficiency correction that accounted for gamma energy in detection efficiency. This was achieved using the pulse-height weighting technique (PHWT) [28], which applies weighting functions to ensure that efficiency is independent of the cascade path and proportionality to the known cascade energy.

In addition to simulations, essential experimental measurements were performed using neutron filters composed of ⁵⁹Co and ¹⁸¹Ta [20]. Here, we employed the black resonance technique to normalize the experimental background, enabling the separation of the background contributions from the true spectrum. This systematic approach is crucial for obtaining the required capture yield precision. The experimental neutron capture yield as a function of E_n is calculated as follows $Y_{\text{exp}}\left(E_{\text{n}}\right)=\frac{C^{\text{w}}\left(E_{\text{n}}\right)-C^{\text{b}}\left(E_{\text{n}}\right)}{f_{\text{Au}}\cdot \text{Φ}\left(E_{\text{n}}\right)\times E_{\text{c}}},$ (1) where $C^{\text{w}}\left(E_{\text{n}}\right)$ and $C^{\text{b}}\left(E_{\text{n}}\right)$ are the weighted count and total background spectra, respectively, f_Au is the normalization factor determined using the saturated resonance technique for the 4.9 eV resonance of the Au sample, Φ(E_n) is the neutron flux spectrum, and E_c is the detection efficiency of a capture event.

Resolved Resonance Region

This performs a theoretical reanalysis of ¹⁵⁹Tb neutron capture yield from a previous experiment [13] and interprets its statistical properties. We obtained the fundamental resonance parameters by fitting the measured ¹⁵⁹Tb(n,γ) capture yields in the resonance region using the R-matrix code SAMMY [29], based on the Reich-Moore approximation. The experimental conditions, neutron multiple scattering, self-shielding, and Doppler effects were included in the SAMMY code fitting. The resonance spin J and partial radiative and neutron widths Γ_n and Γ_γ, respectively, cannot be determined accurately using the capture measurement. Only the resonance energy and capture kernel k, which are directly related to the resonance area, can be reliably obtained from these capture measurements [30, 31]. The k defined as $k=g\frac{{\text{Γ}}_{\text{n}}{\text{Γ}}_{\gamma }}{{\text{Γ}}_{\text{n}}+{\text{Γ}}_{\gamma }}\mathrm{,}$ (2) where g indicates the spin (statistical) weighting factor and Γ_n and Γ_γ are the partial neutron and radiative widths, respectively. Usually, g is given by $g=\frac{2J+1}{(2I+1)(2s+1)}\mathrm{,}$ (3) where J is the compound nucleus resonance spin (total angular momentum) and I and s denote the target and neutron spins, respectively. In Fig. 1(a), the measured neutron capture yields at the resonance energy of 11.04 eV are fitted with the different spin J by the SAMMY code. Both J=1 and J=2 are well-fitted to the experiment, as expected. The relationships between the resonance parameters are given in the $\left(\text{Γ}\mathrm{,}2g{\text{Γ}}_{\text{n}}^0\right)$ plane at E_n=11.04 eV resonance for different J, as shown in Fig. 1(b). The solid red and dashed black lines indicate J=1 and J=2, respectively. The two lines of Fig. 1(b) means there are many group of values in Γ_n and Γ_γ with fitted results for one of spin J (eg. J=1 or J=2). Usually, even when both the capture and transmission data are available, accurately establishing the spin factor remains challenging [9].

Fig. 1Full-screen View

(Color online) (a)The SAMMY fit to the measured yield at E_n=11.04 eV with different spins J. (b) Resonance parameters analysis in the $\left(\text{Γ}\mathrm{,}2g{\text{Γ}}_{\text{n}}^0\right)$ plane at 11.04 eV. The red solid line and black dashed line indicate J=1 and J=2, respectively

Experimental studies focused on calculating the resonance spins for ¹⁵⁹Tb are scarce. Among the available methods, transmission measurements using polarized neutrons on polarized ¹⁵⁹Tb nuclei have proven to be more precise in identifying the resonance J values. These measurements were reported in [32]. Given the scarcity of data and the high precision of this method, we adopted resonance spin values from [32] for our R-matrix analysis. Using this approach, we avoided some limitations of parameter extraction while relying on the most accurate spin information available.

We used SAMMY code for the resonance analysis of the ¹⁵⁹Tb neutron capture yield data from 1 eV to 1.2 keV [13]. Each SAMMY fitting iteration provided the initial resonance parameter values, which were iteratively refined until convergence. Below 100 eV, SAMMY was initialized with the parameter set from Table VII in Ref. [13]. Above 100 eV to 1.2 keV, initial parameters were taken from the JEFF-3.3 database [11]. Figure 2 compares the ¹⁵⁹Tb capture yield (black dots) with the SAMMY fit results (red lines). The resonance parameters are listed in Table 3.

Fig. 2Full-screen View

(Color online) SAMMY fits of ¹⁵⁹Tb capture yield data from 1 eV to 1.2 keV

Average Level Spacing and Average Radiative Width

The resonance distribution as a function of the neutron energy in the experimental measurements was directly related to the nuclear-level density ρ of the compound nuclei at the neutron separation energy. Specifically, ρJ for a given spin J can be derived from the number of observed resonances NJ within the neutron energy interval Δ En using ${\rho }^J=\frac{N^J}{\text{Δ}{E}_{\text{n}}}=\frac{1}{\langle D^J\rangle }\mathrm{.}$ (4) where DJ represents the average level spacing of spin J. The latter can be calculated based on the cumulative distribution of the observed resonances NJ (E_n): $N^J\left(E_{\text{n}}\right)=a+\langle D^J\rangle E_{\text{n}}\mathrm{.}$ (5) The spacing between successive resonances, which share the same total angular momentum and parity, was predicted to exhibit random behavior, which was verified experimentally. For a series of N consecutive resonances at energy Ei, the level spacing is defined as Di = Ei - E_i-1. The probability distribution of these level spacings is expected to follow the Wigner-Dyson distribution [33] as Eq.(6). $p(x)\text{d}x=\frac{\pi x}{2}{e}^{\left(-\frac{\pi x^2}{4}\right)}\text{d}x\mathrm{,}x=\frac{D_i}{\langle D\rangle }\mathrm{.}$ (6) The Wigner-Dyson law provides a mathematical prediction of the level-spacing distribution in chaotic systems, accurately reproducing experimental observations. For narrow-spaced resonances, the level repulsion effect leads to a distribution that disappears at small spacings, reflecting the repulsive nature of close levels. The Wigner-Dyson distribution predicts an average spacing of $\langle x\rangle =1$ and variance of ${\sigma }^2=\left(\frac{4}{\pi }-1\right)$. Consequently, the relative uncertainty in the average-level spacing obtained from the N observed resonances can be expressed by Eq.(7): $\frac{\text{Δ}D}{\langle D\rangle }=\sqrt{\frac{1}{N}\left(\frac{4}{\pi }-1\right)}\mathrm{.}$ (7) The cumulative number of experimentally observed levels as a function of neutron energy is shown in Fig. 3. The cumulative number of resonance analyses revealed differences between the data provided in Refs. [34] which is greater than 100 eV. Furthermore, when the neutron energy increased, the percentage of missed resonances increased progressively, a behavior that can be explained by statistical fluctuations. A linear fit was used to determine the average level spacing D₀, which was 3.84(4) electronvolt below 100 electronvolt. The consistency of the observed spin-experimental level spacing distribution was tested by comparing this value with the predicted theoretical Wigner-Dyson distribution.

Fig. 3Full-screen View

(Color online) Cumulative number of levels as a function of neutron energy. The green line correspond to D₀=3.78 eV recommended by Mughabghab [10]. The blue line corresponds to D₀=3.21 eV, which was reported by Mizumoto et.al [6]. The red line corresponds to the adjusted fitting of currently accumulated experimental data below 100 eV, producing D₀=3.84± 0.04 eV

The average radiative width $\langle {\text{Γ}}_{\gamma }\rangle$ was calculated exclusively within the energy range below 100 eV. Figure 4 shows the radiative width obtained by analyzing all resonances obtained from the SAMMY fits. The calculated weighted average of the entire energy spectrum was $\langle {\text{Γ}}_{\gamma }\rangle$=102.6(13) meV. Significant deviations of the individual values from the mean values were primarily attributed to the substantial correlations between the radiative and neutron widths. These divergences do not imply the non-statistical behavior of the radiative width because all resonances at higher energies can be satisfactorily analyzed with a fixed value corresponding to $\langle {\text{Γ}}_{\gamma }\rangle$.

Fig. 4Full-screen View

(Color online) Fitted values for the radiative widths below 100 eV. The red-dashed line corresponds to the weighted average value $\langle {\text{Γ}}_{\gamma }\rangle$=102.6(13) meV

Neutron Width

The neutron width Γ_n of the neutron energy E₀ is correlated with the average lifetime τ_n relative to neutron emission decay. This relationship is given by an expression reported in [35]: ${\text{Γ}}_{\text{n}}=\hslash /{\tau }_{\text{n}},$ (8) where ℏ denotes the reduced Planck’s constant. The width can be approximately considered as the time required for the excitation energy to be concentrated on a specific neutron and the probability that a neutron penetrates the nuclear potential barrier. Thus, the neutron width can be viewed as the probability per unit time that a compound nucleus emits neutrons. Therefore, the neutron width Γ_n is given by ${\text{Γ}}_{\text{n}}=\hslash p/t_0,$ (9) where t₀ is the average lifetime before the energy is concentrated on an individual neutron in a specific excited state and p is the actual potential barrier penetrability. However, the reduced neutron width ${\text{Γ}}_{\text{n}}^0$ at 1 eV for s-wave neutrons can be defined as ${\text{Γ}}_{\text{n}}^0=\hslash p^0/t_0={\text{Γ}}_{\text{n}}/\sqrt{E_0},$ (10) where p⁰ is the number of 1 eV neutrons. Thus, insight into the properties of the nuclear lifetime distribution can be obtained through the reduced neutron width, which provides valuable information independent of energy considerations.

The neutron scattering reaction encompasses only one decay channel (ν = 1). Owing to the extreme complexity and random nature of the initial state [36], the neutron width Γ_n follows a zero-mean normal distribution. Therefore, the reduced neutron width ${\text{Γ}}_{\text{n}}^0$ conforms to the Porter-Thomas (P-T) distribution, which can be expressed as [37] $P_{\text{PT}}\left(x\right)\text{d}x=\frac{1}{\sqrt{2\pi c}}{e}^{-\frac{x}{2}dx}\mathrm{.}$ (11) where $x={\text{Γ}}_{\text{n}}^0/\langle {\text{Γ}}_{\text{n}}^0\rangle$ denotes the fluctuation in the normalized reduced width. However, it is typical for comparisons between the experimental data and the expected P-T distribution to be articulated in terms of the integral of the P-T distribution, interpreted as a function of xi $N\left(x_i\right)=N_0{\displaystyle {\int }_{x_i}^{\infty }}{P}_{\text{PT}}\left(x\right)\text{d}x=N_0\left(1-erf\sqrt{x_i/2}\right)\mathrm{.}$ (12) where N₀ is the total number of resonances.

In the present work, the distribution of neutron widths was only studied between 1 eV and 100 eV because of the lack of resonances. Figure 5 illustrates the experimentally observed and computationally derived resonance numbers with ${\text{Γ}}_{\text{n}}^0>x_i\langle {\text{Γ}}_{\text{n}}^0\rangle$ as functions of xi. Hence, our experimental data exhibit remarkably good agreement with the expected J=1-resonance behavior. This result substantiates the consistency of the resonance parameters within the energy range with the least resonance overlap.

Fig. 5Full-screen View

(Color online) Comparison between the experimental and expected integral distributions of the neutron widths

Neutron Strength Function

In optical models [38], the neutron strength function S₀, radiation width $\langle {\text{Γ}}_{\gamma }\rangle$, and average level spacing $\langle D\rangle$ are crucial for determining high-energy neutron cross-sections. The ratio of the average level spacing to the spacing between the resonances is the neutron strength function or S₀. This function is associated with the average total cross section. The neutron strength function can be expressed as shown in Eq.(13) for the s-wave neutron because of the good agreement between the experimentally reduced neutron widths and the level spacing with the theoretical P-T distribution and Wigner-Dyson distributions. $S_0=\frac{\langle g{\text{Γ}}_{\text{n}}^0\rangle }{\langle D_0\rangle }=\frac{{\displaystyle {\sum }_E^{E+\text{Δ}E}g{\text{Γ}}_{\text{n}}^0}}{\text{Δ}E}\mathrm{.}$ (13) where g is the statistical spin factor. This implies that the neutron strength function can be calculated based on the slope of the cumulative distribution of $g{\text{Γ}}_{\text{n}}^0$, which represents the neutron energy. The uncertainty of the strength function is expressed as $\frac{\text{Δ}{S}_0}{S_0}=\sqrt{\frac{1}{N}\left(\frac{4}{\pi }+1\right)}\mathrm{.}$ (14) S₀ is derived from a linear fit (red line) to the experimental data (black dots) in the region 1 eV - 1.2 keV, as shown in Fig. 6. Comparisons of the present S₀ with the literature data are presented in Table 1. Early neutron capture experiments on ¹⁵⁹Tb had experimental limitations, resulting in the determination of neutron strength functions ranging from 0.9×10^-4 to 1.56×10^-4 within a restricted energy range. The current S₀ value is more reliable for applications due to improved detector performance and a wider neutron energy range in the experiment. The average resonance parameters obtained from this study and in the literature are shown in Table 2.

Fig. 6Full-screen View

(Color online) Cumulative sum of the reduced neutron widths as a function of neutron energy