TALYS toolkit

The latest version, TALYS-1.96 [33, 37], was adopted in this study to guide the construction and verification of SPAGINS formulas. The optical reaction model governs the basic concepts in the TALYS toolkit[38]. TALYS toolkit includes nuclear reactions induced by n, p, d, t, ³He, α, γ, and with incident energies ranging from 1 keV to approximately 200 MeV [39]. The valid range of target nuclei was between 12 and 339 (12 < A < 339). In TALYS, the nuclear reaction process is divided into three stages: (1) the independent particle stage, in which the incident particle is partially scattered and partially absorbed, similar to light waves passing through a translucent glass sphere; (2) the compound system stage, which follows the time of the incident particle being absorbed by the target nucleus, energy exchange occurs between them, and a compound system is formed; and (3) the end stage, in which the compound system is decomposed into outgoing particles and residual fragments. The calculation of the cross section of a specific residual fragment is based on the manual of the TALYS toolkit [40]. The default input parameters were adopted in TALYS calculations.

Emprical EPAX and SPACS formulas

The cross sections of the fragment productions with the mass number A and charge number Z for spallation reactions can be divided into three terms: the total reaction cross section, mass yield distribution, and charge distribution, which were first proposed by Rudstam [42, 41], the five-parameter fitting formula, and the development of empirical EPAX and SPACS formulas. The five parameters in the five-parameter fitting formula (named the Rudstam formula in this article) were obtained by fitting with the nonlinear least-squares method; thus, it cannot be used to predict nuclides.

The main characteristics of fragment production in spallation reactions were included in the EPAX parameterization (a universal empirical parameterization of fragmentation crosssections), which was proposed by Sümmerer et al. in 1990. The EPAX formulas inherit the ideas of Rudstam and Silberberg [43], who aimed to describe the fragmentation of medium-to-heavy mass projectiles. The fragment cross section was independent of the incident energy of the reaction system above 140 MeV/u. The updated versions of EPAX2 and EPAX3 successfully describe the production of fragments in projectile fragmentation reactions above 100 MeV/u [44, 45]. The improved formulas for fragments produced in nuclear spallation reactions based on EPAX have been proposed by Schmitt et al. in 2014, which is named SPACS (a semiempirical parameterization for isotopic spallation cross sections) [46, 47]. In the SPACS formulas, more than 40 parameters were adopted to reestablish the mass yield and explicitly state the dependence of the yields of the residual fragments on the bombardment energy [48]. The charge distribution of the fragments was given by EPAX [44, 45, 49]. The dependence of the fragment cross section on the collision energy, shell structure, and even-odd effect was further considered in the SPACS formulas. The readers can refer to Refs. [44, 45] for detailed descriptions of fragments cross sections in EPAX and to Ref. [47] for SPACS parameterizations. The main formulas in both EPAX and SPACS were adopted to construct the SPAGINS formulas in this study, which are introduced in Sect. 2.3.

Phenomenological Isotopic Distributions in PNSR

In this subsection, the fragment isotopic cross sectional distributions are compared to determine the basic ideas for developing the SPAGINS formulas. In the first step, we illustrated the similarity in fragment production between γ and proton-induced nuclear spallation reactions. The TALYS-1.96 was adopted to predict the isotopic distributions for the γ + ⁶³Cu and p + ⁶²Ni reactions at 100 and 200 MeV, where the mass and charge numbers of the reaction systems were the same. In Fig. 1(a) and (b), the isotopic cross section distributions in the γ + ⁶³Cu reaction (open symbols) share the same pattern as those in the p + ⁶²Ni reaction except they have smaller magnitudes. The quantity is defined as the ratio of the fragment cross section in proton-induced spallation ($\sigma {\left(f\right)}_{\text{p}}$) to γ induces spallation ($\sigma {\left(f\right)}_{\gamma }$), $R_{\text{p}/\gamma }\equiv \sigma {\left(f\right)}_{\text{p}}/\sigma {\left(f\right)}_{\gamma }$. The $R_{\text{p}/\gamma }$ values for the isotopic crosssections in p + ⁶²Ni and γ + ⁶³Cu reactions are shown in Fig. 1(a) and (b), respectively. The $R_{\text{p}/\gamma }$ values for isotopes in the reactions at an incident energy of 100 MeV were approximately 150 and 400 at an incident energy of 200 MeV. This indicates that the formula for fragment cross sections in the proton-induced nuclear spallation reaction can be adopted for γ-induced ones by further considering the incident energy dependence.

Fig. 1Full-screen View

(Color online) The calculated isotopic cross section for fragments by TALYS-1.96 in the γ + ⁶³Cu and p + ⁶²Ni reactions at Eγ= 100 MeV [in (a)] and 200 MeV [in (b)], respectively. The ratio of the isotopic cross sections in the p + ⁶²Ni and γ + ⁶³Cu reactions are plotted in the inserted figure.

Rudstam Formula

The Rudstam formula has two types of distributions for determining the fragment crosssection in PNSR. One uses charge distribution and mass yield distribution (CDMD), and the other uses isotopic distribution and elemental distribution (IDED)[41]. In this study, the CDMD in Refs. [4] is adopted, and the Rudstam formula reads, $\sigma \left(Z\mathrm{,}A\right)=\frac{\widehat{\sigma }P{W}^{2/3}}{1.79\left(e^{P{A}_{\text{t}}}-1\right)}\exp \left(PA-W|Z-SA+T{A}^2{|}^{3/2}\right)\mathrm{,}$ (1) where P, W, S, T and $\widehat{\sigma }$ are free parameters. P defines the slope of the mass yield curve, and W defines the charge distribution width. S and T describe the most probable charge and define the peak locations of the charge distribution (or isotopic distribution), respectively. $\widehat{\sigma }$ denotes the total inelastic yield of the reaction. The parameters were determined by performing a nonlinear least-squares fit to the measured fragments in γ+^natCu PNSR within the range of Eγ from 100 to 1000 MeV [4] and adopted in this study to investigate the incident energy dependence of the fragment cross section (see Sect. 3.1).

In the second step, with the help of the Rudstam formula, the fragment cross sections of chromium isotopes were compared for the γ + ⁶⁴Cu reaction at incident energies ranging from Eγ= 100 to 1000 MeV.

Developing SPAGINS Formulas

For a residual fragment (Z, A) produced in fragmentation or spallation reactions, the cross section can be described as $\sigma \left(A\mathrm{,}Z\right)={\sigma }_{\text{R}}Y\left(A\right)Y\left(Z_{\text{prob}}-Z\right){|}_A\mathrm{,}$ (2) where ${\sigma }_{\text{R}}$ is the normalized total reaction cross section [50, 51], and Y(A) is the mass yield. The third term represents the isobaric yield for a given mass A. The construction of SPAGINS formulas started with this formula to describe fragment production. In this study, modifications were made to the mass and charge distribution terms.

The terms that influence σ_R are the masses of the projectile A_proj and target A_tar. The energy-dependent function δE considers the effects of transparency, Pauli blocking, and the Coulomb barrier B. The energy of the collision system is in the center-of-mass framework, and the quantity χm corrects the intensity of the optical model interaction at low energies [46, 47]. Because photonuclear reactions involve only electromagnetic interactions, the Coulomb barrier is zero. The mathematical expressions for these quantities are given in the SPACS formalism [5, 6].

The measured fragment cross sections in the γ +Cu spallation reaction [4] were used as an example to perform the analysis. Natural copper has two stable isotopes: 69.17% ⁶³Cu and 30.83% ⁶⁵Cu. The findings of this study revealed that the isotopic effects in the fragment cross sections were minimal. Thus, ⁶⁴Cu was adopted as the spallation target instead of natural copper, which is mainly composed of ⁶³Cu and ⁶⁵Cu. To maintain the same masses and charge numbers in spallation systems, a proton-induced system of p + ⁶³Ni was selected for comparison.

The calculated fragment cross sections were compared to the measured cross sections according to two main aspects. The first method considers the incident energy dependence of the fragments. With an increase in energy, the cross sections of the residual fragments produced by PNSR increase exponentially. The second factor is the charge number Z. For the generation of different types of residual fragments, the difference between the calculated residual fragment cross section and experimental value constantly changes with the change in charge number. To reduce the difference between the section value calculated using the empirical formula and the experimental value, the corrections for energy and charge are necessary.

Schmitt et al. found that the mass distributions of the spallation and fragmentation reactions cannot be described by the same mathematical expression [46]. Importantly, in spallation reactions, the energy dependence of Y(A) cannot be ignored[44, 45, 48]. Schmitt et al. added an energy dependence related to Y(A) to the SPACS formalism [43, 45, 49]. The energy dependence of Y(A) is discussed in Sect. 3.2.

The third term in Eq. (2), $Y\left(Z_{\text{prob}}-Z\right){|}_A$ describes the isotopic distribution, which is insensitive to the interaction mechanism but is mainly controlled by the level density. In this study, the application ranges for certain quantities were modified. The magnitude of the brute-force factor was changed, and the charge dependence was modified, as shown in Sect. 3.2.

Main Formulas in SPAGINS

Mass yield Y(A) in Eq. (2) is divided into two parts considering the fragment contributions of the central collisions Y(A)_cent and peripheral collisions Y(A)_prph, which are expressed as $Y\left(A\right)=Y{\left(A\right)}_{\text{cent}}+Y{\left(A\right)}_{\text{prph}}\mathrm{.}$ (3) Y(A) depends exponentially on the incident energy. Particularly, the production of Cr isotopes was studied to determine their dependence on the incident energy. To observe how the fragment cross section depends on Eγ, the Rudstam formula in Eq. (1) with the parameters according to Ref. [4] is adopted to estimate the Cr isotopic productions in γ + ⁶⁴Cu reactions with Eγ from 100 MeV to 1 GeV (see Fig. 2). Within the range of 100 MeV≤E_γ≤500 MeV, the isotopic distribution increases with the incident energy, whereas it becomes very similar when Eγ > 500 MeV. Thus, the trend of Y(A) is parameterized as the following equations for different incident energy ranges: $Y^{\prime }\left(A\right)=G\cdot Y\left(A\right)\mathrm{.}$ (4) G depends on the incident energy in the form of $G=\{\begin{array}{cc}{10}^{\left(E_{\gamma }-100\right)/100}\mathrm{,}& 100\le E_{\gamma }<220;\\ {10}^{\left(E_{\gamma }-220\right)/300}\mathrm{,}& 220\le E_{\gamma }<500;\\ {10}^{\left(E_{\gamma }-500\right)/2000}\mathrm{,}& 500\le E_{\gamma }\le 1000.\end{array}$ (5) Eγ is sorted into three different ranges: (1) from 100 to 220 MeV, (2) from 220 to 500 MeV, and (3) from 500 to 1000 MeV, which empirically reflect the energy dependence of the isotopic distributions shown in Fig. 2.

Fig. 2Full-screen View

(Color online) The Rudstam fitting formula according to Eq. (1) for production cross sections of Cr isotopes in γ + ⁶⁴Cu reactions from 100 to 1000 MeV.

In the EPAX formulas, Z_prob describes the deviation of the most probable charge from the position of β^-stability valley (Zβ) by the quantity Δ. For photonuclear spallation reaction, the parameters Δ used in EPAX are as follows: $\Delta ={\Delta }_5{A}^2\mathrm{,}$ (6) ${\Delta }^{\prime }=\Delta \left[1+d_1{\left(A/A_{\text{proj}}-d_2\right)}^2\right]\mathrm{.}$ (7) The SPACS formula modifies the parameter R in EPAX, which is also the formula used: $\begin{array}{l}R=R_0^{\text{n,p}}\cdot R^{\text{phy}}\\ R_0^{\text{n,p}}=r_0\exp \left[r_4\left(Z_{\text{proj}}-Z_{\beta \text{p}}\right)\right]\mathrm{,}\\ R^{\text{phy}}=\exp \left\{-\ln \left[R_1\left(A/t_2\right)\right]\right\}\mathrm{.}\end{array}$ (8) The residual fragments in the spallation reaction can be divided into proton- and neutron-rich fragments, which are parameterized in different forms. The SPACS refined the normalization of the charge-dispersion curve based on EPAX3, and the so-called “brute-force factor” f_n (f_p) is used to adjust the distribution of neutron-rich (proton-rich) fragments. For $\left(Z_{\beta }-Z\right)>\left(Z_{\text{proj}}-Z_{\beta \text{p}}+b_2\right)$, $f_{\text{n}}{=10}^{\left[-b_1|Z_{\text{proj}}-Z_{\beta \text{p}}|{\left(Z_{\beta }-Z+Z_{\text{proj}}-Z_{\beta \text{p}}+b_2\right)}^3\right]}\mathrm{.}$ (9) Otherwise, f_n=1. For Z>Z_exp $f_{\text{p}}=1/{\left[{10}^{\text{d}F/\text{d}Z}\right]}^{\left(Z-Z_{\text{exp}}\right)}\mathrm{.}$ (10) When the charge of the residual fragment was less than that of the target nucleus, the SPACS predictions agreed with the measured data, whereas when the charge of the residual fragment was close to that of the target nucleus, the SPACS predictions differed significantly from the measured values. In addition, f_n and f_p determine the peak position of the Gaussian distribution of the isotopes. The following conditions of “brute force factors” are formulated for those residual fragments on the valley of β stability. When Zβ>Z-0.75, f_n is used, and f_p is used in all other cases. For these reasons, in γ + ⁶⁴Cu spallation reactions, the residual fragments with charge numbers ranging from 21 to 29 are divided into three groups. The first group consisted of residual fragments with Z< 21, second group consisted of those with 21 <Z < 26, and last group consisted of those with Z> 26. The parameterized brute force factor is described in Sect. 6.

Validation for SPAGINS formulas

In Fig. 3, the predicted isotopic cross section distributions by the SPAGINS formulas in γ + ⁶⁴Cu reactions and solid line used to guide the eye is the five-parameter fitting formula from Rudstam. The cross sections for fragments in the corresponding γ + ^natCu results are plotted for comparison in solid symbols, for which the incident energies Eγ range from 100 MeV to 1 GeV at 100, 130, 160, 220, 310, 400, 500, 800, and 1000 MeV. The mass (charge) number of the fragments ranged from 38 (19) to 64 (29). Owing to the similarities in the types of graphs, only Eγ values at 100, 130, and 160 are displayed in this study. The SPAGINS formula reproduced the measured data well, except for those near the target isotopes.

Fig. 3Full-screen View

(Color online) The isotopic cross section in the 100 MeV (a), 130 MeV (b), and 160 MeV (c) γ + ⁶⁴Cu reactions. The SPAGINS predictions are plotted as open symbols, and full lines represent the fitting lines by the Rudstam formula, which are used to guide the eye. The measured fragments in the γ + ^natCu reactions (taken from Ref. [4]) are plotted in full symbols.

The excitation function of the residual fragment, that is, the dependence of the fragment crosssection on the incident energy of the reaction, reflects how the probability changes with Eγ. Figure 4 shows the cross sections of fragments from ⁴²K to ⁶¹Cu, which are produced in the γ+⁶⁴Cu reaction within Eγ from 100 MeV to 1 GeV, in which both the SPAGINS predictions and Rudstam fitting data are compared to the measured results of the γ+^natCu reaction. Both the SPAGINS and Rudstam formulas can reproduce the experimental excitation functions in γ+^natCu reactions, except that the SPARGINS prediction underestimates the measured function for ⁵⁷Ni. In the SPAGINS formula, Eq. (26) in the APPENDIX, and the brute force factors f_n and f_p the fitting data of the Rudstam formula are plotted in depend on Eγ and influence the fragment cross section [46] when Eγ is changed. In the Rudstam formula, all five parameters depend on incident energy [41]. The good agreement between the SPAGINS predictions and measured results for the excitation curves of fragments suggests that the parameterization of the incident energy dependence of fragment crosssections reflects the inner mechanism of fragment production in PNSR. The mean relative errors are shown in Figs. 5. The dashed line indicates that σ_exp/σ_cal is equal to 1.

Fig. 4Full-screen View

(Color online) Excitation functions for fragments ⁴²K, ⁴⁴Sc, ⁴⁸V, ⁵¹Cr, ⁵²Mn, ⁵⁹Fe, ⁵⁶Co, ⁵⁷Ni, and ⁶¹Cu produced in reactions of γ + ⁶⁴Cu with Eγ from 100 MeV to 1 GeV. The measured data in γ + ^natCu reactions (solid circles) are taken from [4]. The predictions by SPAGINS and the fitting data of the Rudstam formula are plotted in open squares and open triangles, respectively.

Fig. 5Full-screen View

(Color online) Mean relative error of ^42,43K, ^44,46∼48Sc, ⁴⁸V, ^49,51Cr, ^52,54,56Mn, ⁵⁹Fe, ^56∼58,60Co, ⁵⁷Ni, and ^60,61,64Cu produced in the reactions of γ + ⁶⁴Cu for which the incident energies Eγ ranged from 100 MeV to 1 GeV. The dashed lines denote σ_exp/σ_cal is 1.

To further verify the SPAGINS formulas, the predicted cross sections for fragments using the TALYS toolkit and SPAGINS formulas in the 100 MeV γ +⁶⁴Cu reaction are compared in Fig. 6. The mean relative error between the predictions of the SPAGINS formula and experimental value in the 200 and 900 MeV γ+⁵⁹Co reactions [52] is shown in Figure 7. In general, the data predicted by both TALYS and SPAGINS are consistent with the measured data, except for the neutron-rich fragments, in which TALYS predicts relatively lower data. If this phenomenon is valid for neutron-rich fragments, then the SPAGINS predictions agree with the measured data in PSNR. Based on comparisons between the SPAGINS, Rudstam formula, and TALYS toolkits, the SPAGINS formulas provided reasonable predictions for fragments produced in PSNR.

Fig. 6Full-screen View

(Color online) Production cross sections for Cr, Mn, Fe, Co, Ni, and Cu isotopes in the 100 MeV γ + ⁶⁴Cu reaction. The measured data for γ + ^natCu reactions (solid circles) are taken from Ref. [4]. The predicted results by the TALYS model and SPAGINS formulas are in open triangles and open squares, respectively.

Fig. 7Full-screen View

(Color online) Mean relative error of ^34,38,39Cl, ^42,43K, ^44,46∼48Sc, ^48∼51Cr, ^52,54,56Mn and ^52,53Fe produced in the 200 MeV and 900 MeV γ+⁵⁹Co reactions. The measured data are taken from [52]. The dashed line means that σ_exp/σ_cal is 1.