Introduction

A multitude of medical radioisotopes, including ¹⁸F, ^99mTc, ⁶⁸Ga, and ¹⁷⁷Lu, are currently used for diagnostic and therapeutic purposes. However, in recent years, novel radioisotopes have been proposed that exhibit numerous advantages over existing radioisotopes. In particular, the utilization of radiolanthanides in nuclear medicine has been the subject of numerous in vitro and in vivo studies [1]. In a recent preclinical study, Müller et al. [2] investigated the combined use of ¹⁴⁹Tb, ¹⁵²Tb, ¹⁵⁵Tb, and ¹⁶¹Tb radioisotopes. These radionuclides are distinctive in that they exhibit properties suitable for positron emission tomography (PET), single-photon emission computed tomography (SPECT), and radionuclide therapy, the three principal modalities of nuclear medicine. Müller et al. employed a folate-based targeting agent comprising a dodecane tetraacetic acid (DOTA) chelator to facilitate the binding of Tb to the biomolecule, and reported highly promising results [2]. Imaging of folate receptor (FR) positive human tumors xenografted into mice with both ¹⁵²Tb (PET, 17% β⁺) and ¹⁵⁵Tb (SPECT) has been demonstrated to be of high quality. Furthermore, the same compound, labeled with therapeutic ¹⁴⁹Tb and ¹⁶¹Tb, demonstrated the potential to cure the disease.

The primary production method for ¹⁶¹Tb is the irradiation of ¹⁶⁰Gd with thermal neutrons in a reactor [3]. The production of ¹⁴⁹Tb, ¹⁵²Tb, and ¹⁵⁵Tb may be accomplished using charged-particle accelerators. An overview of ¹⁴⁹Tb production methods can be found in [4-7], whereas an overview of ¹⁵⁵Tb production methods can be found in [8]. The production of ¹⁵²Tb can be achieved through the irradiation of Gd and Eu isotopes with protons [9-11], deuterons [12] and alphas [13, 14], as well as spallation reactions initiated by high-energy protons [15].

Although several experimental studies have been conducted on the production of medical Tb isotopes, an appropriate production mechanism has not yet been fully proposed. In cases where experimental studies are insufficient, the combination of theoretical investigations with experiments can be a helpful approach. To study isotope production theoretically, the reaction cross sections in the region of interest must be well known. Theoretical cross sections can be calculated using the Hauser-Feshbach statistical model. In these calculations, parameters such as the optical model, level densities, and strength function were employed. As these parameters have not yet been determined globally, they vary according to the mass and energy regions of interest. All of these models affect the theoretical cross section calculations, particularly in the low-energy region, where the optical α-nucleus potential is likely to introduce the most significant deviations in the charged particle reactions [16-19]. The parameters proposed thus far are listed in Tables 1, 2, and 3. Consequently, it is essential to determine which models should be used for theoretical cross section calculations. To ascertain which models are most appropriate for use in the region of interest, the ¹⁵¹Eu(α,n)¹⁵⁴Tb reaction cross section was subjected to a detailed analysis and compared with the experimental results of Gyürky et al. [13], which exhibited the lowest energy and cross section uncertainties. A total of 432 different combinations of eight optical potentials, six level densities, and nine strength function models were subjected to reaction cross section calculations, which were then compared with experimental results. The Threshold Logic Unit method [20, 16] was used to determine the best model parameters.

In the present study, the cross sections of the alpha induced reactions of ¹⁵¹Eu(α,3n)¹⁵²Tb and ¹⁵¹Eu(α,2n)¹⁵³Tb and ¹⁵¹Eu(α,n)¹⁵⁴Tb and ¹⁵³Eu(α,2n)¹⁵⁵Tb were calculated with best model parameters for the production of ¹⁵²Tb and ¹⁵⁵Tb. Optimal production parameters for ¹⁵²Tb and ¹⁵⁵Tb have been proposed for commercial cyclotron accelerators. This work can be studied not only in medical physics but also in other fields where knowledge of reaction cross sections is required, such as nuclear astrophysics [16, 21-25] and nuclear technology [26].

Method and calculation

2.1

Cross Section Calculations

The cross sections for the ¹⁵¹Eu(α,n)¹⁵⁴Tb reaction were predicted using the TALYS 1.96 computer code [27], which employs Hauser Feshbach statistical model calculations. Optical model potentials (OMP), level density models (LDM), and strength function models (SFM) play significant roles in theoretical cross section calculations. For further details regarding the models, please refer to the relevant literature. To investigate the sensitivities of these parameters to the reaction cross sections, calculations were performed for combinations of eight OMP, six LDM, and nine SFMs, as presented in Tables 1, 2, and 3 [28-52]. Once the optimal model was identified, the reaction pathways illustrated in Fig. 1 and the synthesis of ¹⁵²Tb and ¹⁵⁵Tb were subjected to further analysis [53].

Fig. 1Full-screen View

(Color online) The relevant part of the isotope table and the production route to be used for the production of the medical isotopes ¹⁵²Tb and ¹⁵⁵Tb

OMPs is a major input parameter for calculating the cross section. The OMPs used in the calculations, labeled with OMP-1 through OMP-8, are normal alpha potential [28], McFadden and Satchler [29], Demetriou et al. [30], Avrigeanu et al. [31] which is the default option of the code, Nolte et al. [32], Avrigeanu et al. [33].

Three microscopic and three phenomenological level density models were included in the calculations. The phenomenological LDMs are constant temperature + Fermi gas model [34], back-shifted Fermi gas model [35, 36] and generalized super-fluid model [37, 38], labeled with LDM-1 through LDM-3. Two microscopic LDMs were chosen using the Skyrme force from Goriely’s (LDM-4) [39] and Hilaire’s (LDM-5) [40] tables. The third microscopic LDM used the Gogny force from Hilaire’s combinatorial tables (LDM-6) [41]. The default option of the code is the Constant Temperature - Fermi Gas Model (LDM-1) [34].

Nine different SFMs were selected for the calculations, labeled SFM-1 through SFM-9, as listed in Table 3. The default option of the SFM is the Brink-Axel Lorentzian model (SFM-2) [44, 45].

Results and Discussion

To study the production of terbium isotopes, it is essential to have a comprehensive understanding of the relevant experimental and theoretical cross sections within the designated energy range. It is regrettable that not all cross sections necessary for the production of terbium isotopes have been experimentally measured at the relevant energies. Consequently, theoretical cross sections may prove to be an efficacious instrument for investigating of radioisotope production. Nevertheless, the accuracy of the calculated theoretical cross sections is not guaranteed, particularly in the case of alpha-initiated reactions. A multitude of parameters are employed to calculate the cross sections. Therefore, it is essential to examine each parameter individually for each nucleus in energy range. In order to ascertain which models are most appropriate for use in the region of interest, the ¹⁵¹Eu(α,n)¹⁵⁴Tb reaction cross section was subjected to a detailed analysis and compared with the experimental results of Gyürky et al. [13], which exhibit the lowest energy and cross section uncertainties. A total of 432 different combinations of eight optical potentials, six level densities, and nine power function models were subjected to reaction cross section calculations, which were then compared with experimental results. Threshold Logic Unit method [16, 20] was used to determine the best model parameters.

The theoretical results of the ¹⁵¹Eu(α,n)¹⁵⁴Tb reaction cross section were calculated with 432 combinations of eight OMP, six LDM, and nine SFM were compared with experimental values [13] measured at thirteen energies. Model Parameters are labeled with MP-ijk, where i, j and k represent OMP-i, LDM-j and SFM-k, respectively, in Tables 1, 2, and 3. In the study employing the TLU method, the optimal combinations (MP-238) were identified, which were in alignment with the experimental results at twelve energies of the thirteen, as illustrated in Fig. 2. The calculated cross section values are in agreement with the experiment at all points except at 11.99 MeV. The calculated and experimental cross section values are listed in Table 5.

Fig. 2Full-screen View

(Color online) The ¹⁵¹Eu(α,n)¹⁵⁴Tb cross sections obtained by TALYS using the best combination (MP-238) selected. The first digit in the legend represents the Optical Model number, the second digit the Level Density model number and the third digit the Strength Function model number (Tables 1, 2, and 3)

Furthermore, the model was tested for medical isotope production reactions, specifically the ¹⁵¹Eu(α,2n)¹⁵³Tb and ¹⁵¹Eu(α,3n)¹⁵²Tb reactions. As illustrated in Figs. 3 and 4, the outcomes produced by the MP-238 model are in agreement with the experimental findings reported by Moiseeva et al. [14]. Unfortunately, there are no experimental measurements in the literature for the ¹⁵³Eu(α,2n)¹⁵⁵Tb reaction, which is used for ¹⁵⁵Tb production. Consequently, the theoretical cross sections could not be compared with the experimental results. For these reactions, MP-238 was assumed to be compatible because of the close mass and properties of the target isotopes of Eu, and this model combination was used in all isotope production calculations.

Fig. 3Full-screen View

(Color online) The ¹⁵¹Eu(α,2n)¹⁵³Tb cross sections obtained by TALYS using the best combination. The explanations same as in Fig. 2

Fig. 4Full-screen View

(Color online) The ¹⁵¹Eu(α,3n)¹⁵²Tb cross sections obtained by TALYS using the best combination. The explanations same as in Fig. 2

3.1

The production of ¹⁵²Tb with ¹⁵¹Eu target

For the production calculation of ¹⁵²Tb, it was assumed that the enriched ¹⁵¹Eu target was irradiated with an alpha beam in 1 MeV steps starting at 30 MeV up to 50 MeV, and the activities and isotopic fractions of the isotopes produced in all reaction channels were calculated. A comprehensive analysis was conducted to ascertain the impact of irradiation time on the calculated values, spanning from one to 24 h. Additionally, the influence of varying target thicknesses on the results was investigated, providing insights into the effects of these variables on the observed outcomes. The results obtained after 8 h of irradiation are shown in Figs. 5 and 6. As Europium is typically present in its oxide form and is known to undergo rapid oxidation, this study assumes that the Eu target is in its oxide form throughout the calculations.

Fig. 5Full-screen View

(Color online) The activity of ¹⁵²Tb (in MBq/μA) is presented as a function of beam energy and target thickness at a beam current of 1 μA and an irradiation time of 8 h with an enriched ¹⁵¹Eu target. The target thickness was expressed in terms of beam exit energy, and the beam profile was assumed to be 1 cm²

Fig. 6Full-screen View

(Color online) The isotopic fraction of ¹⁵²Tb with respect to beam energy and target thickness was determined following irradiation with an enriched ¹⁵¹Eu target at a beam current of 1 μA for 8 h. The target thickness is expressed in terms of the beam energy that leaves the target. The beam profile was assumed to be 1 cm²

As shown in Fig. 6, with a beam energy of 50 MeV and a target thickness corresponding to an output energy of 34 MeV (corresponding to a 0.26 mm thick target), ¹⁵²Tb with an isotopic fraction of 0.83 can be produced in 8 h with an activity of 395 MBq/μA (Fig. 5). As the exit energy decreases below 34 MeV, a notable decline in the isotopic fraction and radionuclide purity is observed (Fig. 6). As the irradiation time was extended, the ¹⁵²Tb fraction was calculated to decrease by approximately 1% every eight hours.

Moiseeva et al. [14] proposed the production of ¹⁵²Tb with a 42 MeV beam current and 34 MeV beam exit energy. However, as illustrated in Figs. 5 and 6, both the isotopic fraction and activity increase in ¹⁵²Tb production with 50 MeV beam energy and 34 MeV exit energy. While the isotopic fraction shows only a modest increase, the activity level rises significantly from 256 to 395 MBq/μA. Furthermore, our calculations based on the combination proposed by Moiseeva et al. [14] are in accordance with the experimental findings. The activity reported by Moiseeva et al. [14], 222 MBq/μA, was calculated to be 256 MBq/μA. Figure 7 illustrates the variation in isotopic fractions as a function of target thickness when irradiated with 50 MeV (solid line) and 42 MeV (dashed line) beams. As evidenced by the region highlighted in yellow in the graph, the isotopic fraction exhibited minimal change for ¹⁵²Tb up to an exit energy of 34 MeV. Figure 8 illustrates the impact of varying target thicknesses on the produced activity when irradiated with 50 and 42 MeV beams. The threshold energy for the ¹⁵¹Eu(α,4n)¹⁵¹Tb reaction is 33.76 MeV. Therefore, the production of ¹⁵¹Tb is not possible at energies below this threshold. Similarly, ¹⁵⁰Tb, which results from the ¹⁵¹Eu(α,5n)¹⁵⁰Tb reaction, cannot be produced below the threshold energy of 42.6 MeV. Therefore, Fig. 8 presents only the activity produced by irradiation with a 50 MeV energy beam. The maximum activity obtained for ¹⁵⁰Tb is 0.8 MBq/μA.

Fig. 7Full-screen View

(Color online) 1 μA beam current and 8 h of irradiation with 42 MeV (dashed line) and 50 MeV (solid line) beam energy and ¹⁵²Tb isotopic fraction with respect to target thickness. Target thickness is given in terms of beam exit energy

Fig. 8Full-screen View

(Color online) 1 μA beam current and 8 h of irradiation with 42 MeV (dashed line) and 50 MeV (solid line) beam energy and ¹⁵²Tb activity with respect to target thickness. Target thickness is given in terms of beam exit energy

3.2

The production of ¹⁵⁵Tb with ¹⁵³Eu target

Upon analyzing the production of ¹⁵⁵Tb by the reaction ¹⁵³Eu(α,2n)¹⁵⁵Tb, it was observed that the isotopic fraction of ¹⁵⁵Tb gradually decreased as the beam energy increased. As illustrated in Fig. 9, the irradiation of the enriched ¹⁵³Eu target for eight hours with a beam energy of 31 MeV and a beam exit energy of 17 MeV results in the production of ¹⁵⁵Tb with a 90% isotopic fraction. Under these conditions, a 37.3 MBq activity will be obtained with a 1 μA beam current. The reaction cross sections for the dominant products formed as a consequence of irradiation with the enriched ¹⁵³Eu target are shown in Fig. 10.

Fig. 9Full-screen View

(Color online) The isotopic fraction of ¹⁵⁵Tb with respect to beam energy and target thickness was determined following irradiation with an enriched ¹⁵³Eu target at a beam current of 1 μA for 8 h. The target thickness is expressed in terms of the beam energy leaving the target. The beam profile was assumed to be 1 cm²

Fig. 10Full-screen View

(Color online) Calculated reaction cross section for the ¹⁵³Eu target. The dashed lines show the cross section of the ground and two metastable states for reaction ¹⁵³Eu(α,3n)^154g,m1,m2Tb

This process allows the production of both ¹⁵²Tb and ¹⁵⁵Tb isotopes simultaneously. By employing a target design analogous to that depicted in Fig. 11, it is possible to generate ¹⁵²Tb at high energies and ¹⁵⁵Tb at low energies, respectively. The utilization of an Al energy degrader is an effective method for minimizing contamination from other isotopes in the intermediate region, thereby reducing both the irradiation time and production costs. The required thickness of the Al energy degrader for decreasing the beam energy from 34 to 31 MeV was determined using the ThiMeT code [56]. Figure 12 illustrates the activities resulting from 8 h of irradiation and 24 h of decay, as represented by the proposed target design.

Fig. 11Full-screen View

(Color online) Target design to produce both ¹⁵²Tb and ¹⁵⁵Tb at the same time

Fig. 12Full-screen View

(Color online) The activity of isotopes formed as a result of 1 μA beam current and 8 h of irradiation using the target design in Fig. 11 and decayed 24 hours after irradiation. (a) For the ¹⁵¹Tb, ¹⁵³Tb, ^154g,m1,m2Tb, ¹⁵⁵Tb, and ¹⁵⁶Tb (b) for the ¹⁵²Tb

Conclusion

A computer code using the statistical Hauser-Feshbach approach was used to determine the most compatible theoretical results with the experimental data. The reaction cross sections were calculated using the TALYS code for a total of 432 different combinations of eight optical potentials, six level densities, and nine strength function models. The calculations closest to the experimental results were those obtained with the combinations of MP-238, which was determined using the TLU method. That is, the optical model potential of McFadden and Satchler [29], Generalised superfluid energy level density model by Ignatyuk et al. [37, 38] and the strenght function model of Gogny D1M HFB+QRPA by Martini et al. [50]. The ¹⁵³Eu(α,2n)¹⁵⁵Tb reaction, for which no experimental cross-section data are available in the literature, was the subject of an in-depth study. The predicted cross-section values allowed us to calculate ¹⁵⁵Tb production for the first time.

The TLU method can identify the most optimal parameter model for predicting reaction cross sections that align with experimental data. Recently, a considerable number of studies have employed the chi-square test in this field [57]. The TLU method represents a distinct and straightforward approach that can be utilized in future studies.

To theoretically determine the medical isotope production parameters, it is necessary to have a good understanding of the cross section, stopping power, and decay parameters. The cross sections derived in this study were adequately corroborated by empirical data, and the parameters governing the observed decay were well documented. Currently, no experimental data are available regarding the stopping power of alpha particles in Eu. Consequently, theoretical calculations must be used as the basis for determining these parameters.

Using ¹⁵¹Eu target with a 50 MeV beam energy and 34 MeV exit energy, ¹⁵²Tb with 83% isotopic fraction and 395 MBq/μA activity can be produced. Similarly, ¹⁵⁵Tb with 90% isotopic fraction and 37.3 MBq/μA activity can be produced using ¹⁵³Eu target with 31 MeV beam energy and 17 MeV exit energy. A target design analogous to that depicted in Fig. 11 was employed to produce ¹⁵²Tb with an activity of 395 MBq/μA and 37.3 MBq/μA of ¹⁵⁵Tb, respectively, utilizing a beam energy of 50 MeV. It should be noted that the aforementioned calculations were conducted with a beam current of 1 μA, and thus, the calculated activities will increase in direct proportion to the increase in beam current. The ratio of ¹⁵²Tb to ¹⁵⁵Tb can be modified by allowing the decay of ¹⁵²Tb.

Further investigations with greater diversity and complexity can be conducted using the proposed model. However, more precise measurements of the experimental cross sections over a broader energy range would enhance theoretical investigations. Therefore, it is imperative to employ the thin target activation method in order to obtain precise cross-section measurements, which are crucial for the analysis of the reactions of interest.