New results on proton-induced reactions on Vanadium for $^{47}$Sc production and the impact of level densities on theoretical cross sections

New data for the $^\mbox{nat}$V(p,x) reactions have been measured in the range 26-70 MeV, with production of the nuclides $^{47}$Sc, $^{43}$Sc, $^{44m}$Sc, $^{44g}$Sc, $^{46}$Sc, $^{48}$Sc, $^{42}$K, $^{43}$K, $^{48}$V, $^{48}$Cr, $^{49}$Cr, and $^{51}$Cr. The focus is on the production of $^{47}$Sc, a $\beta^-$-emitter suitable for innovative radiotheranostic applications in nuclear medicine. The measured cross sections for this radionuclide and its contaminants are compared with the theoretical excitation functions calculated with the TALYS code. In view of novel radiopharmaceutical applications, it is essential to accurately describe these cross-sections for the evaluation of yields, purities, and dose releases. Hence, we optimize the level-density parameters of the microscopic models in the TALYS code to obtain the best possible descriptions of the new data. We consider different irradiation conditions to estimate the production yields from the cross sections determined in this work.

obtained by using state-of-art nuclear models that will be discussed futher on. The results will allow to accurately estimate the production yields of 47 Sc and contaminants and to design an irradiation experiment for the production of 47 Sc with the highest isotopic and radionuclidic purities. A preliminary assessment of the optimal conditions was presented in [12] in which the energy interval [19][20][21][22][23][24][25][26][27][28][29][30] MeV was identified as the most promising by performing a parametric extrapolation at low energy of the available data: in this analysis we follow a different approach by considering a microscopic theoretical model and, by introducing the tuning of the level densities, we get an excellent reproduction of the data in the region (mass and energy) of interest. Thus, without any parametric-fit extrapolation of data at low energies, we obtain an optimized theoretical prediction of the yields, as well as of the contamination by stable isotopes, such as 45 Sc, which typically are not measurable by γ-spectroscopy. The advantage is that a complete theoretical model has built-in the accurate constraints (centrifugal and Coulomb repulsion, and mass-threshold conditions, etc.) that are needed for a correct threshold behaviour, while a simple fit extrapolation may be not so accurate.
Nuclear reaction codes incorporate a statistical evaporation module, generally based upon the Hauser-Feshbach theory [20]. Amongst the various aspects that enter in that formalism, a crucial role is played by the Nuclear Levels Density (NLD), characterizing the excitation structure of the compound nuclei. Up to the last century, the description of NLD was based mainly upon phenomenological models that could lead to analytical forms quite flexible but depending largely on the adjustment of parameters to the available data. Many phenomenological NLD models have been developed along these lines, starting from the basic Fermi-Gas model, to the composite Gilbert-Cameron model [21], or to Generalized-superfluid model [22].
It was only starting from the years 2000 that the microscopic descriptions became sufficiently accurate in the global NLD description, for an extremely large set of nuclides (about 8000/8500), to compete with the phenomenological models for practical applications. One of the first works was developed in Ref. [23], where effective Skyrme-type nucleon-nucleon interactions have been employed in the context of mean-field theories. The method used the Hartree-Fock plus Bardeen-Cooper-Schrieffer model (HF-BCS) to describe the many-body structure and to incorporate the pairing force. NLD was then calculated using a microscopic statistical method, as described also in Ref. [24].
Further advancements in microscopic NLD calculations were achieved in Refs. [25,26], by resorting to a more consistent (axially symmetric) Hartree-Fock-Bogoliubov (HFB) scheme with Skyrme interaction, with the NLD constructed using a shell-model combinatorial approach [27]. This lead to quite a few improvements, such as the determination of the parity dependence of the NLD, together with the inclusion of the collective aspects of both rotational and vibrational character in the theoretical description.
More recently, a new study has been completed [28] with improvements in the microscopic description of the collective effects, both vibrational and rotational, based on the use of a new Gogny-type nucleon-nucleon effective interaction in the HFB approach. Also, the introduction of a temperature-dependent generalization allowed to take into account the evolution of the nuclear deformation with increasing excitation energy, an important aspect that has been disregarded in previous treatments.
Other important aspects of the theoretical description of nuclear reactions are the optical potential model used to describe the interaction and the preequilibrium model. Talys provides state-of-art implementations for both and we have made calculations with all the available options. We have obtained excellent results with the microscopic model JLM (Jeukenne-Lejeune-Mahaux) for the optical potential, as explained in the following, and with the exciton model with numerical transition rates (default option) for preequilibrium.
In all the calculations, the optical potential and preequilibrium models are used "as provided", while for the NLD we have obtained a significant improvement with the scaling procedure described in Section III B.
This nuclear reaction study is devoted to a specific practical application of nuclear medicine: the evaluation of the production route of the theranostic 47 Sc radionuclide, along with its main contaminants, using the cyclotron-based nat V(p,x) channel. This applicationoriented problem demands a model description of cross sections as close as possible to the experimental data, since many quantities (yields, radionuclidic purities, etc.) are needed with high accuracy for the multidisciplinary evaluation of the production route. Indeed, a precise determination of the relevant quantities is important for the prediction of the amount of radio-pharmaceutical compound that can be produced in view of pre-clinical and clinical trials, for the evaluation of the dosimetric impact to the patient's organs, and for the appropriate radiochemistry separation techniques.
These practical aspects imply that a good reproduction of the relevant cross sections is much more important than the overall theoretical consistency of models. For this reason we have adjusted the NLD parameters for an optimal reproduction of the cross sections, even if it increases the discrepancy between observed and calculated nuclear-level cumulatives.
This work presents the final and complete data set obtained by the PASTA experiment.
In Sect.II, we illustrate the experimental techniques and procedures employed for the acquisition of the new data. However, for details of the apparatus and methods, we refer also to previous publications [7,29] where partial and preliminary data of the experiment have been published. In Sect.III, we illustrate the main considerations upon the models implied by the nuclear reaction code TALYS, and how we have used the NLD parameters to optimize the reproduction of the measured cross-sections. The experimental/theoretical results are presented in Sect.IV, and in Sect.V we discuss their implications, including an evaluation of 47 Sc yield obtainable with a standard irradiation-condition set-up. Conclusive remarks are given in Sect.VI.

II. EXPERIMENTAL SETUP AND MEASUREMENTS
Since the beam-line devoted to cross section measurements for medical radionuclides production is not ready yet at INFN-LNL, the experiments of the PASTA project were performed at the ARRONAX facility [30]. Six irradiation runs were carried out using the low current (100 nA) proton beam with tunable energy , the target-holder and collimator described in a previous work [29]. Stacked-foils targets, composed by a set of thin metallic nat V foils (purity > 99.8%, thickness=20 µm) interchanged with monitor foils and nat Al energy degraders, were used in the experiments [7]. The nat Ni(p,x) 57 Ni and the 27 Al(p,x) 24 Na cross sections, recommended by the IAEA [31], were considered as monitor reactions respectively for energies lower and higher than 40 MeV. nat Ni (purity > 99.95%, thickness=10 µm) or nat Al (purity > 99.0%, thickness=10 µm) foils were thus added in the stacked-structure after each nat V target foil (see Fig. 1). The target was placed under normal atmosphere downstream the end of the beam line, kept under vacuum and closed with a 75 µm thick Kapton foil. The distance from the target holder to the Kapton foil, ranging from 10 cm to 15 cm, was accurately measured for each run. The proton beam energy entering each layer of the stacked-foils target was calculated using the SRIM code [32]. In order to calculate the energy losses in the Kapton foil, in the air and across each target foil, the input parameters were the proton beam energy extracted from the cyclotron and each layer's thickness. At the end, at each foil corresponds one energy value, calculated as the mean energy of the input and output energies in the specific layer. The uncertainty of the proton beam energy was obtained considering the uncertainty of the beam energy extracted from the cyclotron (about 500 keV) and by calculating with the SRIM code the energy straggling through each layer of the stacked-target. All samples were measured with the same high-purity germanium (HPGe) detector (model Canberra GC1020, energy resolution FWHM 1.0 keV at 122 keV, relative efficiency 10%), previously calibrated with 152 Eu point-like solid source. The sample-detector distance was fixed at 19 cm to keep the dead time during measurements below 10%. At least five acquisitions were carried out for each sample, in order to better follow the decay of the radionuclides of interest. The first measurement of each nat V foil, typically 15 minutes long, started about 2 hours after the End of Bombardment (EOB); the following acquisitions, usually 3 hours long, were performed once per day starting from the day after the EOB (up to 5 days). An additional acquisition of each nat V foil was also carried out about 2 months after the EOB, to further measure the activity of the long-lived 46 Sc. Table I  Data analysis, including uncertainty calculations, was performed following the procedure described in [33]. For the case of 44g Sc, the activity was obtained by using the formulation presented in [33], that takes into account the 44m Sc decay correction and the 1157 keV γline interference as shown in Tab. I. The final cross section value related to each target foil was calculated as the weighted average of all single values associated to each counting; the uncertainty of the monitor cross section was added at the end of this calculation.

III. THEORETICAL CALCULATIONS
The theoretical analysis of cross sections for the production of medical radionuclides is important, since a good predictive capability is essential to efficiently guide the experimental activities. Conversely, new measurements are the key ingredient to validate the theoretical models and to optimize the models free parameters. The topic has been thoroughly explored in the past and an exhaustive literature exists on the theory of nuclear reactions, ranging from phenomenological models that provide analytical expressions of important observables to modern microscopic approaches that aim to a more accurate description of the reaction mechanisms.
Modern nuclear reaction codes are today the best tools to perform theoretical calculations in this field: in this work we have used the TALYS simulation package [34] which is actively developed and maintained. The aim of the authors is to provide a complete and high quality set of models [35] for the different reaction mechanisms that are relevant in the energy range of medical interest, namely nuclear scattering, compound nucleus formation and preequilibrium emission.
Recently, a great effort was made to implement different microscopic models, both for the optical potential and for the nuclear level densities. For the optical potential model, we have selected the so-called JLM (Jeukenne-Lejeune-Mahaux) semi-microscopic nucleonnucleus spherical optical model potential as described in [36]. This model relies on the folding of a nucleon-nucleon effective interaction in nuclear matter. It is based upon the Brückner-Hartree-Fock (BHF) theory to describe the mass operator in infinite nuclear matter, and then applied to finite nuclei via the improved local density approximation. To take into account the preequilibrium emission we have used the default option, which is based on the exciton model with transitions rates evaluated numerically. Regarding nuclear levels density, we focus in particular on the microscopic approaches denoted as Hartree-Fock-Models (HFM) which includes both HF-BCS and HFB descriptions, able to provide detailed information about nuclear energy levels, as well as masses, spin, parity, and levels density. The last parameter has been studied starting from the seminal work of Bethe [37] in the context of the compound nucleus theory and in many other phenomenological approaches, but from the physical point of view the microscopic approach provides a modern and flexible description of the level densities. Indicating with N (E) the cumulative number of levels with energy smaller than E, the nuclear level density is defined as [38]: This quantity can be inferred experimentally from different sources, such as spectroscopic studies, neutron resonances, evaporation spectra and Ericson fluctuations studies [39]. Levels density has been calculated in the HFM frameworks and the comparison is usually done by inspecting the cumulative distributions N (E) [40].

A. Microscopic models in the TALYS code
Amongst the various options for the choice of the Optical Model in TALYS, the JLM microscopic optical model, derived according to the work reported in [41,42], was then adjusted phenomenologically [36] to improve the agreement of calculated finite-nuclei crosssections with a large set of experimental data.
The three models are based on the HFM approach and represent somehow the time evolution of the TALYS code with respect to the microscopic level density models, as suggested by the referenced papers. None of them is perfect, but they complement each other being able to describe different aspects of reaction mechanisms, possibly on the entire nuclide chart.
Each option provides different tabulated level densities, but the code allows to overrule them to improve the agreement with data within a restricted Z and A range. In this work we have found good results with the option ldmodel 4 and with an optimized set of parameters that were applied to the calculations by a scaling procedure. The preequilibrium model is fixed in all calculations with the option preeqmode 2, except for the 51 Cr cross section where we have sought for an improvement with a more recent addition as discussed in the following.

B. Nuclear levels density scaling procedure
The approach used in TALYS offers a certain level of flexibility since, in the HFM description, the level densities are tabulated at different excitation energies for both parities and different values of spins: the values in these tables can be re-scaled by simple multiplication to fit experimental data. This technique has been successfully adopted in [43] to find a global best fit on the entire nuclide chart.
In general the original HFM microscopic level densities have not been adjusted to experimental data and the tabulated values can be rescaled by using two parameters (c and p) according to the transformation: The parameter p represents a "pairing shift" and allows to obtain the level density from the table at a different energy. Parameter c acts to the level density in similar way to that of the parameter a of phenomenological models and provides an overall normalisation of the level density. Adjusting p and c together provides flexibility in fitting at both low and high energies, and this implies that both low-energy discrete levels and average experimental resonance separations can be described within the same procedure. This procedure allows to modify the original HFM level density and to obtain different "customized" tables, as shown in the TALYS manual and on the RIPL library [40].

Grid search and optimization procedure
The first step is a grid search on the c and p parameters in the allowed range [-10,10] [44], but we have restricted our search in a narrower interval [-2,4] as suggested by Fig.39 in Ref. [40]. The grid search is multidimensional, since it concerns the c and p parameters of all the nuclides involved. A preliminary solution is found by a first qualitative agreement between theory and measurements, obtained by visual inspection of the cross sections for which new data are available. With this trial and error approach, we define the initial values of the c and p parameters for the next step, characterized by the optimization procedure.
The second step is to refine the solution by evaluating a global chi-square of all the involved cross-sections: a Minuit-based chi-square minimization procedure [45] has been implemented by using the c and p values as free parameters, with initial values identified by the previously described grid search. The minimization algorithm used is the Minuit recommended one, which is based on the combination of gradient and simplex optimization.
The level density distributions and cumulatives associated to the theoretical cross sections that correspond to the minimum chi-square are analyzed and compared with the default HFB and the TALYS proposed ones.
The goal of this work is to achieve the most accurate parametrization of the selected

IV. RESULTS
This work provides new nuclear reaction data for cross-sections induced by protons on nat V targets, within the energy range 26-70 MeV.   [13,15], probably due to the use of old nuclear and monitor values in those analysis. The largest discrepancy with previous data was found for the 43 Sc case (Fig.3) that includes also 43 K production ( Fig. 9) [29]. Since the nat V(p, x) 47  to 60 MeV. The value at 70 MeV obtained by Hontzeas and Yaffe is lower than our result, and the same discrepancy can be found for the 44m Sc case (Fig. 5). For the nat V(p, x) 44m Sc cross section, the results obtained by Michel et al. in 1985 [17] confirm the trend that we have measured.
In the data analysis of 48 Sc production, only the 1037.52 keV γ-line was considered, since  the 1312.12 keV and 983.526 keV lines are emitted also by 48 V (see Table I). Fig.7 reports the new data for the nat V(p, x) 48 Sc cross section, in excellent agreement with the literature [17,19] except for data published in 1963 [15]. The nat V(p, x) 48 V cross section, obtained considering the 944.13 keV γ-line, is shown in Fig. 10: also in this case a very good agreement can be noted among the new and previous data in the entire energy range [16,17,19], also including the oldest values [13,15].   Fig. 11 reports the cross section for 48 Cr production: in the entire energy range an excellent agreement can be seen among our results and the ones published in 2016 [19], that at low energy (E < 45 MeV) are also matching with data by Michel et al. [16]. Instead, oldest values are quite scattered [13,15]. In the case of 49 Cr (Fig. 12) the maximum peak value that we obtained is lower than the one published by R. Michel et al. (1979) [16], even if the peak energy seems to correspond. Concerning the Levkovski data, as already discussed  Table I) is too small to explain our peak value of about 40% lower.
The nat V(p, x) 51 Cr cross section measured in this work is in excellent agreement with the extensive literature (Fig.14). Also for the nat V(p, x) 42  previous data published by Ditroi et al. and Michel et al. 1985 [17, 19]; the previous values at 60 MeV are much lower [13,15].

B. On theoretical models
The new calculations for 47 Sc and 46 Sc with the TALYS-modified code (shown in Figs. 2 and 6) greatly improve the agreement with the new measurements, particularly around the peak, where the level density models have an important role. The goodness of the data reproduction is essential in both cases.
In Figs. 3, 8, and 9, the new calculation provides a larger cross-section than the default one, and for the two potassium productions, 42 K and 43 K, it reproduces the measured data very well. For 43 Sc, both calculations support the current measurements, however the trend of these new data exhibits an oscillatory behavior that cannot be reproduced by the models and that may be ascribed to statistical fluctuations originating from the low counts of the measurements [29].
Additional scandium radionuclides with higher energy thresholds are shown in Figs It is worth noting that the level density parameters have been fitted to the new data obtained with this work, but, in Fig. 12, the resulting cross section better reproduces the peak value measured in Ref. [16].
Finally, in Fig. 13, we provide also the results for 51 Cr production. The new calculation improves the reproduction of data, particularly in the range 15-20 MeV. This is confirmed in the same region also if we compare the two calculations with the IAEA recommended data [46]. In the peak range and at lower energies, both calculations show slight discrepancies when compared with the IAEA recommended evaluation. For this reaction, it is interesting to consider the range at higher energy, where the new data have been collected and where the pre-equilibrium mechanisms play a more significant role. This region is analyzed in detail in Fig. 14, where it is evident that none of the two calculations (TALYS default and modified) describe correctly the data. We have considered additional new calculations with different pre-equilibrium models in order to seek a better description in this specific channel. The recent TALYS-1.95 GDH version [47], in which the Geometry Dependent Hybrid model [48,49] was added, has been exploited: this model was previously implemented in other codes, such as ALICE [50], and has been included in the TALYS package by the authors [47]. The calculated excitation function with the GDH model better reproduces the experimental data and this supports the good performance of this newly added pre-equilibrium feature in the TALYS framework. Another reaction code, known to give good description at intermediate energy, is FLUKA [51,52]. In this code the production of residual nuclei is based on the PEANUT (PreEquilibrium Approach to NUclear Thermalisation) model, which couples a classical intranuclear cascade model, supplemented with quantum corrections, with a preequilibrium framework inspired by the GDH approach. In this framework the produced radionuclides emerge directly from the inelastic hadronic interaction models that are based on a microscopic description of the nucleon-nucleon interaction. This approach is not recommended for the calculation of cross section at very-low energies, close to the reaction thresholds, but, generally, provides reliable results at higher energies. As shown in Fig. 14, FLUKA is able to reproduce the 51 Cr cross section high-energy tail with an excellent agreement with the new experimental data.
The importance of the theoretical modeling strikingly emerges when one considers the production of stable nuclides, since their cross sections cannot be measured with standard γ-spectroscopy techniques. Such is the case for the 45 Sc isotope, shown in Fig. 15 This parametrization leads to a solution that optimizes the production of 47 Sc for theranostic applications in agreement with the measurements presented in this work, and allows to accurately calculate the yields and isotopic/radionuclidic purities for realistic irradiation conditions. Our modified nuclear level density is however different from the previous ones and it is important to assess the effect of the transformation given by Eq. 2 by comparing the theoretical cumulative level distribution with the experimental one deduced from RIPL.
This comparison is presented in Fig. 16, where the cumulatives from the HF-BCS model are compared with Goriely's global analysis and with the fit obtained in this work. We stress that the parametrizations given in Tab. II does not improve substantially the agreement between theoretical and experimental cumulatives: that was not our scope. Our aim is to find the best description for the relevant cross sections in view of their importance in radiopharmaceutical applications, and for this reason we have derived the new parameters reported Tab. II.

C. Thick Target Yield (TTY)
We have calculated the 47 Sc production yield in the same irradiation conditions identified in [12]. The yield calculations are developed along the lines discussed in Ref. [20] This study supports the validity of the approach for nuclear medicine applications and provides an accurate estimation for the yields of 47 Sc and its main contaminants, leading to more reliable assessments of the production route and associated dosimetric analyses. In spite of some minor differences shown in Tabs.IV, this work confirms the proton-vanadium route in the 19-30 MeV range as a viable method for producing 47 Sc, with a low activity of 46 Sc and very minimal contamination of stable 45 Sc. It would be interesting, at this point, to confirm these findings with a dedicated TTY experiment with a thick Vanadium target to be irradiated with protons in the suggested energy range. The produced 47 Sc quantities are sufficient for pre-clinical and clinical phase trial. To get higher activities, work must be done either on other production routes (either using neutrons or photo-production) or on targetry to accommodate mA proton beams.
It is important to observe that the tuning of the theoretical level density parameters to obtain accurate cross section is of general applicability for analyzing possible production routes also for other innovative radionuclides for medical applications. [1] National Nuclear Data Center (NNDC), NuDat 2.8 database., https://www.nndc.bnl.gov/.,