Introduction

Photonuclear data describes the interactions between photons and atomic nuclei. They are primarily composed of photoneutron cross-section (PNCS) data and photocharged-particle cross-section data. They play a vital role in a wide range of current and emerging applications [1]. These applications include radiation shielding design, radiation transport analysis [2], absorbed dose calculations in the human body during radiotherapy, fission and fusion reactor physics and technology [3, 4], activation analyses, safeguards and inspection technologies, medical isotope production [5, 6], nuclear waste transmutation, and astrophysical nucleosynthesis [7, 8]. In the field of medical isotope production, the PNCS data can guide the production of radioisotopes for diagnostic and therapeutic purposes [9, 10]. With the advent of new facilities that produce brilliant and intense photon beams, the photoproduction of neutron-deficient isotopes could become a competitive alternative to traditional methods that use neutrons generated by nuclear reactors or charged particle beams [11-13].

PNCS data have been experimentally obtained from various types of measurements, with bremsstrahlung and quasi-monoenergetic photons from positron annihilation in flight, and more recently, from laser Compton scattering (LCS) [14-16]. Most existing experimental data correlate with studies on PNCSs. Although PNCSs are obtained by counting the total number of emitted neutrons, the determination of partial PNCSs requires neutron multiplicity sorting [17]. One advantage of bremsstrahlung measurements [18] is the high photon beam intensity. However, the technique has several disadvantages. First, it is necessary to know the bremsstrahlung spectrum sufficient for all electron energies. Next, measuring the reaction yield curve in a small energy step requires a stable accelerator and large counting statistics. Third, the process of subtracting the yield curves in the unfolding procedure may introduce correlations between the experimental data points, which can lead to fluctuations in the unfolded cross-sections. In contrast to bremsstrahlung measurements employing a mathematical approach, positron annihilation in flight and LCS offered an apparatus method for producing them with variable energies [19, 20]. However, the effective spectrum of the incident photon beam should be known in advance. Consequently, all these measurements require a precise scan of the incident photon energy and accurate knowledge of the incident photon spectrum and beam intensity.

Regarding the evaluation of the PNCS, the International Atomic Energy Agency has organized two coordinated research projects and photonuclear data of 219 isotopes were evaluated successfully because of the growing need for photonuclear data [21, 22]. Despite the available experimental and evaluation efforts on photonuclear data, there is still an urgent need to study PNCS data for the following three reasons: (a) there is a lack of data for a number of cases; (b) it is not possible to produce a complete PNCS data file based on measured PNCSs alone; and (c) the experimental PNCS data often suffer from systematic discrepancies that are not easy to resolve [1].

In this study, we propose an effective method for the extraction of PNCS distributions using gamma activation and the reaction yield ratio method. Note that gamma activation can be accomplished through either laser plasma-based bremsstrahlung radiation or LCS photon irradiation of a solid target. This extraction method was verified by reasonably reproducing the experimental/evaluated and calculated PNCS data and can potentially be used to predict the PNCS distributions of 30 stable isotopes. The remainder of this study is organized as follows: In Sect. 2, the extraction method is introduced to obtain the peak cross-sections and further predict PNCS distributions. The benchmark results obtained using the extraction method are presented in Sect. 3. The merits of the extraction method and the possibility of predicting PNCS distributions for 30 isotopes are discussed in Sect. 4. The conclusions and outlook are presented in Sect. 5.

Extraction of PNCS distribution

Relativistic laser-plasma interactions are bright, incoherent sources of gamma radiation with energies up to several tens of megaelectronvolts. They can be produced by compact tabletop laser systems that deliver laser pulses with energies of less than 100 fs. The spectra of the laser-accelerated electrons and bremsstrahlung photons can be described by the Boltzmann distribution [23] $n_{\text{e}\mathrm{,}\gamma }\left(E\right)=\frac{n_{\text{e}\mathrm{,}\gamma }\left(0\right)}{k_{\text{B}}{T}_{\text{e}\mathrm{,}\gamma }}\exp \left\{-E/k_{\text{B}}{T}_{\text{e}\mathrm{,}\gamma }\right\}\mathrm{,}$ (1) where T_e,γ is the electron or photon temperature, k_B is the Boltzmann’s constant, and n_e,γ(0) is the number of electrons or photons within the energy bin at E_e,γ = 0 MeV. It can be seen that the intensity of laser-accelerated electrons or bremsstrahlung photons decreases exponentially with energy. The threshold energies E_th are generally a few MeV for photoneutron reactions. To efficiently trigger such reactions, the photon temperature Tγ must reach the MeV/k_B.

For nuclides with a single giant dipole resonance (GDR) peak, the PNCS distribution can be described by a Lorentzian-like shape [24]. $\sigma \left(E\right)=\frac{{\sigma }_{\text{m}}}{1+\left[{\left(E^2-E_{\text{m}}^2\right)}^2/E^2{\Gamma }^2\right]}\mathrm{,}$ (2) where the peak energy E_m, peak cross-section σ_m and width $\Gamma$ are the three Lorentzian distribution parameters. These parameters can be acquired from TALYS calculations, TENDL evaluations, and experimental data available in the EXFOR database. Figure 1 shows that the Lorentzian curves accurately reproduce the experimental PNCSs for the four representative nuclides: ⁶³Cu, ⁸⁸Sr, ¹²⁷I, and ¹⁹⁷Au. The TALYS calculation underestimated the measured σ_m values for ⁶³Cu and ¹²⁷I, whereas the TENDL evaluations aligned with the experimental data. In addition, visible discrepancies were found between the experimental data and the TENDL evaluations.

Fig. 1Full-screen View

(Color online) PNCS distributions of four nuclides ⁶³Cu, ⁸⁸Sr, ¹²⁷I and ¹⁹⁷Au. Experimental data in the figure are taken from Refs. [24-27]. The hollow black circle and hollow pink triangle indicate the (γ, n) cross-sections obtained with TALYS (version 1.8) calculations and TENDL (2021) evaluations [28], respectively. The red line shows the Lorentzian curve fitted with Eq. (2).

The reaction yield Y depends on the temperature Tγ and cross-section σ(E). The expression for the reaction yield Y can be written as: $Y=\frac{nd}{k_{\text{B}}{T}_{\gamma }}{\displaystyle \int }\sigma \left(E\right)n_{\gamma }\left(0\right)\exp \left\{-E/k_{\text{B}}{T}_{\gamma }\right\}\text{d}E\mathrm{,}$ (3) where n is the number of target nuclei per unit area, and d is the thickness of the target. For a specific target comprising isotopes A and B irradiated by a laser-generated bremsstrahlung photon beam, the residual nuclei A’ and B’ are produced through photoneutron reactions. When isotopes A and B have very similar mass numbers, their PNCS distributions could be very close in terms of peak energy E_m and width $\Gamma$ but vary in terms of the peak cross-section σ_m. Figure 2 shows the PNCS distributions of Cu, Sr, I, and Au nuclei, whose isotopes have three adjacent mass numbers. The three Lorentzian parameters of their PNCS distributions are listed in Table 1. It can be seen that, although the peak cross sections σ_m for these nuclides with adjacent mass numbers have large relative differences of 10%-30%, the peak energies E_m are very close to each other because their relative differences are lower than 3%, and the relative deviation of the width $\Gamma$ is approximately 10%. Substituting Eq. (2) into Eq. (3), the isotope yield ratio can be written as $\begin{array}{c}\frac{Y_{A^{\prime }}}{Y_{B^{\prime }}}=\frac{n_A}{n_B}\cdot \frac{{\displaystyle \int }{\sigma }_A\left(E\right)\exp \left\{-E/k_{\text{B}}{T}_{\gamma }\right\}\text{d}E}{{\displaystyle \int }{\sigma }_B\left(E\right)\exp \left\{-E/k_{\text{B}}{T}_{\gamma }\right\}\text{d}E}\\ =\frac{n_A}{n_B}\cdot \frac{{\sigma }_{\text{m}A}}{{\sigma }_{\text{m}B}}\cdot \frac{{\displaystyle \int }\frac{\exp \left\{-E/k_{\text{B}}{T}_{\gamma }\right\}}{1+\left[{\left(E^2-E_{mA}^2\right)}^2/E^2{\Gamma }_A^2\right]}\text{d}E}{{\displaystyle \int }\frac{\exp \left\{-E/k_{\text{B}}{T}_{\gamma }\right\}}{1+\left[{\left(E^2-E_{mB}^2\right)}^2/E^2{\Gamma }_{\text{B}}^2\right]}\text{d}E}\\ \approx \frac{n_A}{n_B}\cdot \frac{{\sigma }_{\text{m}A}}{{\sigma }_{\text{m}B}}.\end{array}$ (4) Because of the complexity of the integral ratio in Eq. (4), we cannot deduce the integral ratio directly; thus, an approximate expression is provided. To verify the validity of the approximation, the integral ratio was calculated using the summation method (i.e., the integral was replaced by the summation). Considering the slope temperature of the photon spectrum Tγ=5 MeV/k_B (which can be readily achieved by state-of-the-art laser plasma-based bremsstrahlung sources [29, 30]), the integral ratio is approximated to be 1.0 and the root mean square deviation is 0.067, which is calculated over a wide range of A and Z (note that 26 < Z < 81 and for a certain nuclide, four nuclides with mass numbers adjacent to A are considered). This deviation correlates with the relative differences between E_m and $\Gamma$. Consequently, we consider 6.7% as the systematic uncertainty of the proposed extraction method.

Fig. 2Full-screen View

(Color online) PNCS distributions of ^62-64Cu isotopes (a), ^87-89Sr isotopes (b), ^126-128I isotopes (c), and ^196-198Au isotopes computed with TALYS software (version 1.8). Note that experimental data for these PNCS distributions are not available, and the TALYS calculations are used for showing the difference of the PNCS distributions of adjacent isotopes.

After irradiating a target composed of isotopes A and B with laser-generated bremsstrahlung photons, γ-ray counting was performed for residual nuclei A’ and B’. The γ lines unique to the residual nuclei are clearly identified in the γ-ray spectra, which are generally measured using an energy- and efficiency-calibrated high-purity germanium (HPGe) detector. The reaction yields $Y_{A^{\prime }}$ and $Y_{B^{\prime }}$ were obtained using the γ-ray spectrum. For example, the expressions for $Y_{A^{\prime }}$ or $Y_{B^{\prime }}$ are as follows [31]: $Y_{A^{\prime }\mathrm{,}{B}^{\prime }}=\frac{N_{\text{det}}\left(t_{\text{r}}/t_{\text{m}}\right)}{I_{\gamma }\epsilon k_{\text{e}}\left(1-e^{-\lambda t_{\text{m}}}\right)e^{-\lambda t_{\text{c}}}}$ (5) where N_det is the peak count of characteristic γ rays used to identify the residual nuclei, t_r is the time for real measurement, t_m is the measurement time, t_c is the cooling time, Iγ is the branching intensity, ε is the source-peak detection efficiency, and $k_{\text{e}}=\frac{1-e^{-\mu d}}{\mu d}$ is the correction factor for the self-absorption of the characteristic γ rays in the sample thickness d with absorption coefficient μ. According to Eq. (4), the peak cross sections σ_m,A can be obtained as follows: ${\sigma }_{\text{m}\mathrm{,}A}=\frac{Y_{A^{\prime }}}{Y_{B^{\prime }}}\cdot \frac{n_B}{n_A}\cdot {\sigma }_{\text{m}\mathrm{,}B}\mathrm{.}$ (6) This indicates that when the distribution of the PNCS is known for isotope B (or A), the distribution of the PNCS for isotope A (or B) can be readily extracted if their peak energies E_m and widths $\Gamma$ are very close to each other. The above method used to extract the PNCS distribution is valid only for nuclei with a single GDR peak. In the case of deformed nuclei, the GDR peak may split into two, where each corresponds to the major or minor axis of the ellipsoid [19], and the resulting σ(E) is given by the sum of the two GDR peaks, which complicates the proposed extraction method.

Benchmarking Results

To verify the aforementioned extraction method using gamma activation and the isotope yield ratio, experiments for generating ^{62, 64}Cu and ^{85m, 87m}Sr isotopes via laser-induced photonuclear reactions were performed at the XingGuangIII laser facility of the Laser Fusion Research Center (LFRC) at Mianyang. Large-charge megaelectronvolt electron (e^-) beams were generated using 100 TW picosecond (ps) laser pulses. The e^- beams then impinge on metal stacks composed of a Ta foil and activation plates (Cu or SrCl target) of interest, producing high-energy bremsstrahlung radiation and isotopes ^{62, 64}Cu or ^{85m, 87m}Sr. The bremsstrahlung radiations produced have a Boltzmann-like distribution with T_γ ≈ 4.6 MeV/k_B. The obtained reaction yields of $Y_{{}^{\mathrm{62,64}}\text{Cu}}$ are obtained 4.86 × 10⁶ and 2.65 × 10⁶ per laser shot, respectively [32]. The reaction yields $Y_{{}^{\text{85m,87m}}\text{Sr}}$ were determined to be 1.8 × 10⁴ and 2.1 × 10⁵ per laser shot, respectively. In the following section, we introduce how to obtain the peak cross-sections of the isotopes of interest and then predict the PNCS distributions for two exemplary isotopes, ⁶³Cu and ⁸⁶Sr.

3.1

PNCS distribution of ⁶³Cu

The half-lives of ^{62, 64}Cu isotopes are 9.7 min and 12.7 h, respectively. They have principle characteristic emissions Eγ = 511.0 keV with branching intensity Iγ= 195.7% for ⁶²Cu and of Iγ = 35.2% for ⁶⁴Cu. Although the ^{62, 64}Cu isotopes have the same characteristic emissions at 511 keV, their reaction yields can be reasonably extracted because of the significant differences in their half-lives [32]. According to the experimental reaction yields $Y_{{}^{\mathrm{62,64}}\text{Cu}}$ and the natural abundance of Cu isotopes, the ratio of the peak cross-section σ_m,⁶³Cu/σ_m,⁶⁵Cu was obtained as 0.82. Experimental cross-sectional data [17] and evaluated data (IAEA-2019) [33] are available for ⁶⁵Cu(γ, n) reaction, as shown in Fig. 3(a). The Lorentzian-like shape of Eq. (2) was used to fit the data. The Lorentzian parameters ($E_{\text{m}{\mathrm{,}}^{\mathrm{65,63}}\text{Cu}}$, σ_m,^65,63Cu, and ${\Gamma }_{{}^{\mathrm{65,63}}\text{Cu}}$) were then obtained (see Table 2). Using experimental and evaluated PNCS data, the σ_m,⁶⁵Cu are fitted to be 96.63 mb and 93.03 mb, respectively. According to Eq. (6), the peak cross-sections σ_m,⁶³Cu were obtained as 79.24 ± 5.44 mb and 76.28 ± 5.24 mb, respectively. Finally, the PNCS distribution for ⁶³Cu can be predicted by the three Lorentzian parameters $E_{\text{m}{\mathrm{,}}^{65}\text{Cu}}$, σ_m,⁶³Cu, and ${\Gamma }_{{}^{65}\text{Cu}}$ (or ${\Gamma }_{{}^{63}\text{Cu}}$ when available). Figure 3(b) shows two PNCS distributions for ⁶³Cu (i.e., PNCS #1 and PNCS #2), which were extracted from the Lorentzian parameters fitted from the experimental and evaluated data of ⁶³Cu, respectively. The experimental and evaluated PNCS data for ⁶³Cu are shown in Fig. 3(b). The uncertainty of the predicted PNCS distribution for ⁶³Cu was obtained using the error propagation formula, considering a statistical uncertainty of 1.5% of the measured $Y_{{}^{\mathrm{62,64}}\text{Cu}}$ and a systematic uncertainty of 6.7% induced by the proposed extraction method. PNCS #1 and PNCS #2 have an uncertainty of 6.9%. The benchmark results show that the extracted PNCS distribution is consistent with the evaluated data (IAEA-2019). Consequently, the validity of the proposed extraction method was confirmed.

Table 2Full-screen View

Lorentzian parameters fitted using experimental and evaluated PNCS data.

Reaction	Cross section data	Fitted Lorentzian parameters
		Em (MeV)	$\Gamma$ (MeV)	σm (mb)
65Cu(γ, n)64Cu	1995 V. V. Varlamov	16.53 ± 0.04	4.52 ± 0.08	96.63 ± 1.09
65Cu(γ, n)64Cu	2019 IAEA evaluation	16.39 ± 0.10	4.81 ±0.27	93.03 ±3.53
63Cu(γ, n)62Cu	2019 IAEA evaluation	16.86 ±0.08	5.51 ± 0.22	84.77 ± 2.17

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Fig. 3Full-screen View

(Color online) (a) Experimental and evaluated PNCS data for ⁶⁵Cu [25, 33] and the resultant fitted Lorentzian-like distributions, and (b) Predicted PNCS distributions for ⁶³Cu and the experimental and evaluated PNCS data for comparison. PNCS #1 and PNCS #2 represent the PNCS distributions extracted from the Lorentzian parameters fitted from experimental and evaluated data, respectively. The shadow area indicates the uncertainty of the extracted PNCS distributions for ⁶³Cu.

3.2

PNCS distribution of ⁸⁶Sr (leading to ^85mSr isotope production)

The half-lives of ^{85m, 87m}Sr isotopes are 67.6 min and 2.8 h, respectively. They have principle characteristic emissions Eγ = 231.9 keV with branching intensity Iγ = 83.9% for ^85mSr and Eγ = 388.5 keV with Iγ = 82.2% for ^87mSr. Similarly, the ratio of the peak cross-section σ_m,⁸⁶Sr/σ_m,⁸⁸Sr was 0.72. Because experimental PNCS data were not available for ⁸⁸Sr(γ, n)^87mSr and ⁸⁶Sr(γ, n)^85mSr reactions, TALYS calculations were performed, and the calculated curves are shown in Fig. 4(a). These curves were then fitted to a Lorentzian-like shape, and the obtained Lorentzian parameters ($E_{\text{m}{\mathrm{,}}^{\mathrm{86,88}}\text{Sr}}$, σ_m,^86,88Sr, and ${\Gamma }_{{}^{\mathrm{86,88}}\text{Sr}}$) are listed in Table 3. The fitted σ_m,⁸⁸Sr is 163.13 mb. According to Eq. (6), the peak cross-section σ_m,⁸⁶Sr was obtained to be 117.11 ±7.96 mb, respectively. Finally, the PNCS distribution of ⁸⁶Sr can be predicted using the following three Lorentzian parameters: $E_{\text{m}{\mathrm{,}}^{88}\text{Sr}}$, σ_m,⁸⁶Sr and ${\Gamma }_{{}^{88}\text{Sr}}$. The results are presented in Fig. 4(b). The cross-sectional data for the ⁸⁶Sr(γ, n)^85mSr reaction were calculated using the TALYS software and are also shown in Fig. 4(b). Similarly, the uncertainty in the predicted PNCS distribution for ⁸⁶Sr is 6.8%. The extracted PNCS distribution was consistent with the TALYS calculations, validating the proposed extraction method.

Table 3Full-screen View

Lorentzian parameters fitted using TALYS-calculated PNCS data.

Reaction	Cross section data	Fitted Lorentzian parameters
		Em (MeV)	$\Gamma$ (MeV)	σm (mb)
88Sr(γ, n)87mSr	TALYS calculation	16.72 ±0.04	3.71 ± 0.10	163.13 ± 3.08
86Sr(γ, n)85mSr	TALYS calculation	16.51 ± 0.05	4.74 ±0.14	117.74 ± 2.36

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Fig. 4Full-screen View

(Color online) (a) TALYS-calculated PNCS data for ⁸⁸Sr [34] and the resultant fitted Lorentzian-like distributions, and (b) Predicted PNCS distributions for ⁸⁶Sr and the TALYS-calculated PNCS data for comparison. PNCS represents the PNCS distribution extracted from the Lorentzian parameters fitted from TALYS data. The shadow area indicate the uncertainty of the predicted PNCS distribution for ⁸⁶Sr.

Discussions

The proposed extraction method has the following advantages: First, an accurate knowledge of the bremsstrahlung spectrum induced by intense lasers is not required because the reaction yield ratio method is used. When employing intense laser-accelerated electrons to produce bremsstrahlung radiation, they can still be subtracted using our extraction method although shot-to-shot variations exist in the bremsstrahlung spectrum and intensity. Second, it is not necessary to scan the incident photon energy, which generally happens in both the apparatus method using a quasi-monoenergetic photon beam and the mathematical method using classical bremsstrahlung measurements. Third, PNCS distributions can be successfully extracted within a very short time, within seconds, when using a state-of-the-art laser plasma-based bremsstrahlung source [29, 30]. Finally, it should be noted that any gamma source with a continuous spectrum covering the central GDR region is suitable for gamma activation. In general, gamma rays can be produced using either bremsstrahlung or LCS. When the spectral distribution of the gamma source is $n_{\gamma }(E)=n_{\gamma }(0)f(E)$, where f(E) is an energy-dependent function, Eq. (4) can then be written as $\frac{Y_{A^{\prime }}}{Y_{B^{″}}}=\frac{n_A}{n_B}\cdot \frac{{\displaystyle \int }{\sigma }_A\left(E\right)f\left(E\right)\text{d}E}{{\displaystyle \int }{\sigma }_B\left(E\right)f\left(E\right)\text{d}E}$. The approximations given in Eq. (4) remain valid because of the similar values of E_m and $\Gamma$ for the same nuclides as adjacent A, as discussed above.

However, the PNCS distributions of the studied isotopes must have Lorentzian-like shapes. This was a prerequisite for validating the proposed extraction method. Here, it is interesting to discuss how precisely the PNCS distribution can be described by a Lorentzian shape. Generally, a Lorentzian-like shape accurately describes the PNCS distribution in the central GDR energy range but fails in the high-energy tail [16]. In our study, the relative difference between the predicted PNCS (integrated over the 10–25 MeV energy range) and the benchmarked PNCS was determined to be 3.2% for ⁶³Cu and 3.9% for ⁸⁶Sr. However, it increases to more than 50% in the high-energy range of 20–25 MeV. Meanwhile, successful extraction of the PNCS distribution relies on the fact that the residual nuclide should have a suitable half-life (note that a relatively long half-life is also acceptable), which is helpful for detecting characteristic emissions after target activation. To obtain a high reaction yield and reduce statistical uncertainty, the offline-detected characteristic emissions should have relatively large branching intensities and detection efficiencies. In addition, isotopes undergoing photonuclear reactions should have acceptable natural abundances because natural metal targets are commonly used for irradiation. Based on these considerations, the PNCS distributions of 30 isotopes were predicted using the proposed extraction method. Figure 5 shows the natural abundances of these isotopes and half-lives of the residual nuclei. Approximately half of them had PNCS distributions, leading to isomeric states of the residual nuclei. More detailed information regarding the target and residual nuclides is presented in Table 4. Their experimental PNCS data for 18 isotopes exhibited large uncertainty (>10%). The PNCS data for ⁸⁵Rb, ¹⁰⁷Ag, ¹³⁶Xe, and ¹⁵³Eu have acceptable uncertainties because these isotopes are categorized as either medical or other types of materials [1], which requires an uncertainty of less than 10%.

Fig. 5Full-screen View

(Color online) 30 isotopes with corresponding PNCS distributions that are suitable for extraction using our proposed method. The transversal and longitudinal coordinates indicate the name of these isotopes and their natural abundances, respectively. The color bar presents the half-lives of the residual nuclei.

Conclusion

We proposed an effective method for obtaining the peak cross-section and extracting the PNCS distribution using the gamma activation and reaction yield ratio method. The PNCS distributions of ⁶³Cu and ⁸⁶Sr isotopes (leading to ^85mSr isotope production) were successfully extracted using the proposed extraction method. The uncertainty of this method was maintained within 7%. The extracted PNCS distributions and the available PNCS data (or TALYS calculations) were compared to validate the proposed extraction method. The prerequisites and merits of the extraction method and the possibility of predicting PNCS distributions for 30 isotopes were also discussed. The proposed extraction method could be complementary to the available sophisticated methods, including the mathematical method using bremsstrahlung measurements and the apparatus method with positron annihilation in flight and LCS photon spectrum. In the near future, we plan to perform gamma activation experiments using laser plasma-driven bremsstrahlung sources and extract the PNCS distributions for a list of isotopes with relatively large abundances and significant characteristic emissions after short-term irradiation.