Single shot measurement of NRA

The experimental setup is shown in Fig. 1 and the details are described in METHODS. We obtained neutron energy spectra with the resonance absorption of the the Ag (thickness = 0.2 mm) and Ta (thickness = 0.1 mm) foils, where the Ta foil was heated up to T = 361, 413, 474, 573, and 617 K but the Ag foil was kept to the room temperature for using as a reference. Figure 2a shows the signal of epithermal neutrons, which transmitted through the Ag and Ta foils at the room temperature, recorded by the ⁶Li-TOF detector. In the time region of 50–60 μs, two distinct troughs are seen in the continuous neutron signal. The flight time t_raw measured by the detector is expressed by t_raw = (t*D)(t), where t is the flight time of neutrons that is deconvolved from the systematic pulse broadening D(t) (see Methods). The neutron kinetic energy E is obtained by E = m(d/t)²/2, where d = 1.78 m is the flight distance and m is the mass of the neutron. The energies of the two dips correspond to the resonance peaks in Fig. 2b. The neutron absorption cross sections at 300 K are ~12,500 and 23,000 barns at 4.28 eV (¹⁸¹Ta) and 5.19 eV (¹⁰⁹Ag), respectively.

The target setup shows spatial position relationship of the laser beam, CD foil, Be and neutron moderator. The laser-target chamber was kept at high vaccum during the experiment. The beamline setup including samples and detectors are shown in the figure.

a The epithermal neutron signal. Two troughs can be checked in a continuous curve which should be exponential. b The neutron absorption cross sections of ¹⁰⁹Ag and ¹⁸¹Ta at 300 K, obtained from JENDL4.0 data base³⁵. c Experimental and theoretical neutron absorption. d JEDNL4.0 cross section data and σ_BW(E) calculation of a ¹⁰⁹Ag resonance at 5.19 eV.

In present analysis, we evaluate the background level for each laser shot using the reference foil (Ag) in the following procedure similar to black resonance³⁴. The baseline [the blue dashed line in Fig. 2a] is obtained from a least-square fitting for the raw signals excluding the two resonance dips. The background (the green solid line) is obtained from the baseline by subtracting the depth of the resonance peak of ¹⁰⁹Ag, because the Ag target is enough thick to absorb almost all neutrons around the resonance energy. From the difference between the raw signal and the baseline, the experimental absorption rate R_exp(E) normalized to the difference between the baseline and the background is obtained as the blue dashed line in Fig. 2c. The absorption rate R₀(E) for the Ag and Ta foils is theoretically analyzed by

where n is the volumetric number density, l is the thickness of the targets and σ(E) is the NRA cross section (JENDL4.0 data base³⁵) at 300 K [Fig. 2b]. The black line in Fig. 2c shows the theoretical absorption rate when t_raw = t without considering the effect of D(t). R_exp(E) exhibits two peaks at 4.3 and 5.2 eV which are also found in the analytical model R₀(E). However, the detailed shape of the experimental peaks is not well reproduced by R₀(E), especially for the thicker target (Ag), where the saturated absorption observed in R₀(E) is not exhibited in R_exp(E). This difference indicates the presence of another effect, F(t), that causes further pulse broadening in addition to D(t).

F(t) is considered to originate from random neutron scattering in the beamline and involved in a Gaussian form. We determine F(E) as a function of E by fitting the experimental absorption rate R_exp(E) with the following equation:

Fig. 3a shows R_exp(E) (gray) fitted with the model R₁(E) (red) by using an nonlinear-curve-fitting tool with a weighted region around resonances. The resonance peaks are well reproduced as the errors shown in the lower frame. F(t) has a half-width of ~100–200 ns, which is sufficiently shorter than D(t) (~0.5 μs). This result indicates that the advantage of the miniature size of the moderator is not offset by the pulse broadening caused along the beamline.

a Experimental neutron absorption results and model fitting by R_T(E, T) [Eq. (5)]. The temperature of Ag was kept at 296 K and Ta was heated to T = 297, 361, 413, 474, 573, and 617 K. b Theoretical Doppler width and experimental results. The error bars of Doppler width depend on the fitting error and the noise level of the original signal. The temperature of each data point was measured by the thermocouple in the experiment.

Although the two resonance peaks are close to each other in the present experiment, this overlapping region was already considered in the FWHM analysis as explained later. When we chose the reference material whose resonance energy is enough far from the resonance energy of the material to be measured, the uncertainty caused from the overlapping between the two resonance energies becomes negligibly small.

Temperature dependence on resonance Doppler broadening

We obtained temperature dependence on the NRA of the Ta foil. The Ta foil was heated to T = 361, 413, 474, 573, and 617 K, whereas the Ag foil was kept to the room temperature for using as a reference. Every temperature case was measured in single laser pulse.

To analyse the temperature dependence of the resonance absorption, we introduce a Breit-Wigner (BW) single-level formula⁴, as seen in refs. ^10,36, for the neutron absorption cross section at 0 K:

(3)

where E_r is the resonance energy, E is the kinetic energy of the incident neutron, is the de Broglie wavelength of incident neutron (divided by 2π), and g_j is the statistical factor determined by the angular momentum. Γ_n and Γ_γ represent the resonance width for the neutron and decay width for γ-ray, respectively. For the ¹⁸¹Ta resonance at E_r = 4.28 eV, Γ_n = 1.74 meV and Γ_γ = 55 meV; For the ¹⁰⁹Ag resonance at E(r) = 5.19 eV, Γ_n = 8.34 meV and Γ_γ = 136 meV³⁵. We have confirmed the σ_BW(E) formula by fitting it to the latest analytical result of JENDL4.0 data base³⁵ for 0 K and 300 K [Fig. 2d]. The averaged differences on cross sections were evaluated as <10 barns.

When the thermal motion of target nuclei is sufficiently slower than the incident neutron, the Doppler broadening effect can be well approximated by a Gaussian function^8,10. Therefore, we developed an analytical cross-section σ_T(E, T) for the NRA involving the target temperature as follows

where A is a fitting parameter and \({E}^{{\prime} }\) is the relative neutron energy for integration variable. σ_BW and the following Doppler broadening term are convoluted in the energy axis. The Doppler width Γ_D(T) broadens with the temperature T. Then, the temperature-dependent absorption rate R_T(E) is developed as follows:

According to the model given by Eq. (5), we analyse the absorption rate R_exp(E) measured for different target temperature T, as shown in Fig. 3a.

Although the Ag target was kept at room temperature, it can be seen that the resonance widths of ¹⁰⁹Ag vary from shot to shot. This is due to the fluctuation of F(E) caused by statistical processes including neutron scattering on the beamline. Therefore, the ¹⁰⁹Ag resonance serves as a reference to evaluate F(E) for each measurement. In Fig. 3b, we plot the Doppler width Γ_D(T) obtained by fitting R_exp(E) as a function of T. The error bars are determined by the least mean square of the difference between the fitted curves and experimental data. The Doppler width increases as the square root of T. We show a calculated result using a free gas model by Bethe³:

where m and M are the masses of the neutron and the target nucleus, respectively, and k_B is Boltzmann’s constant [the red dashed line in Fig. 3b]. We also introduce a proportionality coefficient B in the right side of Eq. (6) to better reproduce the measured widths. The function obtained by least-square fitting is presented by the red solid line in Fig. 3b. This model can well reproduce the experimental results.