Can We Extract Production Cross-Sections from Thick Target Yield Measurements? A Case Study Using Scandium Radioisotopes

: In this work, we present an attempt to estimate the reaction excitation function based on the measurements of thick target yield. We ﬁt a function to experimental data points and then use three ﬁtting parameters to calculate the cross-section. We applied our approach to 43 Ca(p,n) 43 Sc, 44 Ca(p,n) 44g Sc, 44 Ca(p,n) 44m Sc, 48 Ca(p,2n) 47 Sc and 48 Ca(p,n) 48 Sc reactions. A general agreement was observed between the reconstructions and the available cross-section data. The algorithm described here can be used to roughly estimate cross-section values, but it requires improvements.

In our recent papers [19,20], we have reported on the production routes of medical scandium radioisotopes as well as extending this data with scandium formed in natural and enriched thick CaCO 3 targets (from around 50 up to 1000 mg/cm 2 ) irradiated with α particles up to 30 MeV, deuterons up to 8 MeV and protons up to 30 MeV. The thick targets were used because we found that it was not feasible to prepare thinner (in the order of 1 mg/cm 2 ) self-supporting CaCO 3 as a homogeneously thick target for our experimental set-up. The significant stopping-power of our targets allowed us to obtain experimental thick target yield (TTY) values for scandium production.  47 Sc). 48 Sc, as a radioactive impurity, is also listed here with reference to the analysis in this paper. In this work, we want to complement our research by evaluating the 43 Ca(p,n) 43 Sc,44 Ca(p,n) 44g Sc,44 Ca(p,n) 44m Sc, 48 Ca(p,2n) 47 Sc and 48 Ca(p,n) 48 Sc cross-sections based on reported TTY measurements (the latter is not medically relevant, but 48 Sc production is important as it is a radioactive impurity). A similar attempt has already been proposed in [21] for the study of 34m Cl production. In this work, we verify this approach for above-mentioned reactions while employing a different, straight-forward numerical algorithm (our Python code is submitted in the Supplementary Materials to this paper).

Materials and Methods
In our recent work [20], we reported TTY for 43 Ca(p,n) 43 [22,23]: where H is target enrichment, N A is Avogadro's number, τ is the mean lifetime of a radioisotope, Z is the ionization number of the projectile, e is the elementary charge, m is the atomic mass of the target, E max and E min are the maximal and minimal energy of the projectile penetrating the target (in case of TTY, E min <= reaction threshold), respectively, σ is the cross-section for the nuclear reaction, and dE/dx is the stopping-power of the projectile according to the aerial density of the target. Here, we describe the attempt to obtain the energy dependence of the cross-section (the excitation function) based on the experimental TTY exp (E) [MBq/µAh] values for different projectile energies E. These data are supplemented by an assumption TTY exp (E thr ) = 0, where E thr denotes the energy threshold for this reaction. The crucial factor is the choice of the function used to describe the TTY energy dependence. The number of parameters of the function used to fit the data should be restricted, as the number of the experimental data points is usually limited. Therefore, we propose a simple shape, which fulfils several important criteria. This function is monotonically increasing, as TTY(E) should be. Most importantly, its derivative is a modified q-Weibull distribution [24], which reflects the global shape of the (p,n) and (p,2n) excitation functions, commonly used in the field of the production of medical radioisotopes. The request TTY exp (E thr ) = 0 provides the condition and limits the number of TTY fit parameters to 3: a, b and c. Once those parameters are obtained, the cross-section values can be estimated as In our case, TTY measurements were obtained on CaCO 3 targets instead of metallic Ca. Therefore, we used dE/dx(E) values corresponding to the energy loss in calcium carbonate (provided by SRIM software [25]), m = 100 u to address the mass of CaCO 3 , and H as the level of enrichment of employed material. We have also calculated the 95% confidence band for TTY fit (E) fit and reconstructed the cross-section. Details of our calculations are shown and explained in the Python code attached to this paper.
Alternatively, in [21], the cross-section was reconstructed after fitting the TTY curve by calculating target yields (TY) for thicknesses corresponding to 0.1 MeV projectile energy loss each 1 MeV and multiplied by projectile range. This method assumes the constant stopping-power in each layer. In our approach, this simplification was not necessary.

Results and Discussion
In Figures 1-5, we show the TTY data and the reconstructed cross-sections for 43 Ca(p,n) 43 Sc,44 Ca(p,n) 44g Sc, 44 Ca(p,n) 44m Sc, 48 Ca(p,2n) 47 Sc and 48 Ca(p,n) 48 Sc reactions (the fit parameters are shown in Table 2 while the reconstructed cross-section values are listed in Table 3). We compare them with the experimental cross-section in [26][27][28][29][30][31][32][33][34][35], with the recommended values from [36], with the predictions of the EMPIRE [37] evaporation code (version 3.2.2 Malta) and with the TENDL-2017 cross-section library [38]. All reconstructions exhibit a similar shape to the model predictions and measured cross-section values, indicating the validity of modified q-Weibull distribution in estimating the global shape of the (p,n) and (p,2n) excitation functions.
We have also checked our reconstruction method by implementing the approach in [20]. We obtained similar values (marked on the plots) with a visible correction near the threshold in the 44m Sc case (Figure 3) but also with the discontinuity fragments due to the numerical approach. Since the mentioned paper does not provide the recommended TTY fit function, we adopted ours. In the case of 43 Sc data (Figure 1), the recent experimental results [34] are significantly lower than other measurements (by a factor of 2 around 10 MeV proton energy). The experimental results for TTY are quite linear in the measured proton energy range and do not reach the expected saturation, so the resulting excitation function is relatively flat and does not reproduce any of the previous measurements.
This reaction might require further validation, as with the extension of TTY measurements up to 30 MeV proton energy.
A general agreement is observed for 44g Sc (Figure 2), both with the theoretical models and experimental results, although again the data by [34] are lower than the measurements. More discrepancies are observed in the case of 44m Sc (Figure 3). The excitation function obtained from TTY measurements does not show the peak seen in the experiments and in model calculations and overestimates the values near the reaction threshold. We suspect that the problem with this reconstruction might be related to the offset of TTY data, as only in case of 44m Sc are the TTY values below model predictions at low energies and above them at higher energies, which causes the reconstructed excitation function to be flatter.
For 47 Sc (Figure 4), the shape of the reconstruction reflects the shape predicted by both model calculations. While our results provide about 10% lower values compared to the models, recent measurements [35] indicate similar values at low energies but about 20% higher values at maximum.
Finally, we decided to adopt the arbitrary value of E thr = 3.0 MeV as a parameter for 48 Sc fit ( Figure 5) to satisfy the visible and significant TTY build-up at this energy rather than the actual threshold (0.51 MeV). This might be explained by the fact that the shape of the function used for the fit does not adequately describe the behavior of the cross-section at energies much below the Coulomb barrier. Since the cross-section values far below the Coulomb barrier are very small, they do not contribute significantly to the TTY values. The extracted cross-section values are in line with the data in [30] at lower energies and in [35] at higher energies.

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A general agreement is observed for 44g Sc (Figure 2), both with the theoretical models and experimental results, although again the data by [34] are lower than the measurements. More discrepancies are observed in the case of 44m Sc (Figure 3). The excitation function obtained from TTY measurements does not show the peak seen in the experiments and in model calculations and overestimates the values near the reaction threshold. We suspect that the problem with this reconstruction might be related to the offset of TTY data, as only in case of 44m Sc are the TTY values below model predictions at low energies and above them at higher energies, which causes the reconstructed excitation function to be flatter.
For 47 Sc (Figure 4), the shape of the reconstruction reflects the shape predicted by both model calculations. While our results provide about 10% lower values compared to the models, recent measurements [35] indicate similar values at low energies but about 20% higher values at maximum.
Finally, we decided to adopt the arbitrary value of Ethr = 3.0 MeV as a parameter for 48 Sc fit ( Figure  5) to satisfy the visible and significant TTY build-up at this energy rather than the actual threshold (0.51 MeV). This might be explained by the fact that the shape of the function used for the fit does not adequately describe the behavior of the cross-section at energies much below the Coulomb barrier. Since the cross-section values far below the Coulomb barrier are very small, they do not contribute significantly to the TTY values. The extracted cross-section values are in line with the data in [30] at lower energies and in [35] at higher energies.        Here, we adopted an arbitrary threshold of 3 MeV. The crosssection data points are taken from [26][27][28]30,35]. The results from [27,28,30] are averaged.   Here, we adopted an arbitrary threshold of 3 MeV. The crosssection data points are taken from [26][27][28]30,35]. The results from [27,28,30] are averaged. Here, we adopted an arbitrary threshold of 3 MeV. The cross-section data points are taken from [26][27][28]30,35]. The results from [27,28,30] are averaged.

Conclusions and Summary
We have presented an attempted numerical method for cross-section evaluation based on the thick target yield (TTY) measurements obtained from the irradiation of thick targets (in which the energy of a projectile is reduced to the reaction threshold). This method is based on fitting a function with three free parameters to TTY data points and using its analytical derivative to obtain the cross-section. The fitting requires the knowledge of the reaction threshold and a sufficient number of experimental points to represent the shape of the TTY curve, including the saturation region.
Using this approach, we were able to obtain a useful estimation of cross-sections for the production of medically important 43 Sc, 44g Sc, 44m Sc, 47 Sc, and 48 Sc radioisotopes via (p,n) and (p,2n) reactions on Ca. The results were compared to the already measured cross-sections and to the model predictions. General agreement is observed; however, not all experimental results confirm our reconstructions, particularly those near the reaction threshold. In conclusion, our algorithm can provide good insights for the (p,xn) excitation function, but improvements are necessary.