The development of alpha emitting radiopharmaceuticals for Targeted Alpha Therapy (TAT) is an active area of academic and commercial research worldwide owing to the properties of alpha suitable for the purpose (Shaunak and Ashley 2018; Francisco et al. 2020). This includes the penetration range and the linear energy transfer which are critical factors of the radiopharmaceutical radiation, as the potent pathway of malignant cell destruction is related to the direct interaction between ionization events and the deoxyribonucleic acid string in the nucleus. The emitted alpha particles possess a suitable penetration range such that the neighbouring healthy cells are not affected (Bryce et al. 2021).

The International Atomic Energy Agency (IAEA) has supported series of investigations to identify and quantify the production routes and decay characteristics of such radioisotopes that have existing and emerging importance in nuclear medicine (Nichols 2013). Several candidate isotopes for TAT are currently under clinical and preclinical evaluation, including ²²⁵Ac and ²¹³Bi. ²¹³Bi (t_1/2 = 45.6 min) decays by one alpha and two beta particles to a stable ²⁰⁹Bi. The alpha energy is 8.4 MeV (Kellerbauer et al. 2020). ²²⁵Ac (t_1/2 = 9.9 days) decays to stable ²⁰⁹Bi via four alpha- and two beta-decays. It is used as a generator of ²¹³Bi. Because of the short half-life of ²¹³Bi, ²²⁵Ac/²¹³Bi generator is considered as the best method its production for TAT applications (McDevitt et al. 1999a, 1999b; Deborah et al. 2005; Rose et al. 2005; Melville et al. 2006; Longmena et al. 2023). This method of radionuclide production has a significant benefit by allowing for rapid generation of short-lived radionuclide on site for patient-specific treatments when needed. The few alpha emission by the ²¹³Bi contributes to lower overall systemic toxicity and off-target effects in the patient. The much shorter half-life of 45.6 minutes means it decays faster, resulting in a shorter overall radiation exposure period for patients and healthcare providers. With these advantages, several clinical trials have demonstrated the potentials of ²¹³Bi and ²²⁵Ac radiopharmaceuticals to treat advanced cancers (Ermolaev et al. 2012; Tapas and Pillai 2013; Shaunak and Ashley 2018; Francisco et al. 2020; Randy et al. 2020; Ahenkorah et al. 2021; Bryce et al. 2021). The development of the drugs faces limited supply due to shortage of the production facilities among other challenges that have slowed the progress of the application. These resulted to high cost that makes ²²⁵Ac inaccessible to many researchers.

The present reliable supply of ²²⁵Ac comes entirely from the existing ²²⁹Th-based generator systems in the United States, Germany, and Russia in a total quantity not much to reach the world demand (Engle 2018; Robertson et al. 2018). These sources are relying on the build-up of ²²⁹Th through the decay of ²³³U stockpiles produced between 1954 and 1970. Most of the alternative production methods explored involved the use of ²²⁶Ra as a target material. The ²²⁶Ra(p,2n)²²⁵Ac is the only favorable reaction for the production of ²²⁵Ac on accelerator but it faces challenges of the danger of handling ²²⁶Ra, the target material, because it is radioactive. It is restricted and extremely dangerous at irradiation, chemical processing and target material recovery (Ermolaev et al. 2012). Due to these challenges of handling large quantity of ²²⁶Ra and the low yield of ²²⁹Th, the spallation reactions on naturally occurring ²³²Th and ²³⁸U were the only other alternatives (Engle 2018; Robertson et al. 2018). ²³²Th has a lot of advantages over ²²⁶Ra. It is the preferred chemical form for post-irradiation processing of the target and it is highly available in stockpile in a number of countries and more can be produced in large quantities from the oxide or the nitrate (Andrew et al. 2018). This availability means that recycling of irradiated ²³²Th target material may not be necessary. It is not prohibitively radioactive and poses fewer radiological hazards. Another advantage of this method is that facilities already exist with demonstrated ability to perform target fabrication, irradiation, and processing. Isolation of ²²⁵Ac from irradiated thorium metal has been demonstrated by both American and Russian research groups (Andrew et al. 2018).

While spallation of naturally occurring ²³⁸U will also produce ²²⁵Ac, ²³²Th irradiation is preferred for a number of reasons. The ²³⁸U(p,x)²²⁵Ac reaction cross-section is about10 times lower (as modelled using FLUKA and GEANT4), and due to the higher density and lower melting point of uranium, thorium targets could more safely handle the higher heat load induced by higher beam currents (Robertson et al. 2018). The coproduction of fissile ²³⁹Pu and ²³⁵U is also avoided by ²³²Th irradiation. The challenge with this method is that only a few existing accelerators can produce proton beams with a current and energy sufficient for large-scale ²²⁵Ac production. So, there is need for intense study of the reaction mechanism of the production of ²²⁵Ac from spallation reaction of ²³²Th with proton projectile. Though the direct medical application of ²²⁵Ac produced by this method is problematic because the restrictions for the long-lived impurity ²²⁷Ac may be very strong (Ermolaev et al. 2012). However, ²²⁵Ac obtained by the method is prospective for preparation of a ²²⁵Ac/²¹³Bi isotope generator (Zhuikov et al. 2011). The modelling of the production routes of ²²⁵Ac through thorium bombardment with proton beam (²³²Th(p,x) ²²⁵Ac) are being investigated using different theoretical models. Weidner et al. (2012) in comparison of the theoretical predictions with their experimental data had reasonable-to-good agreement by different theoretical codes (ALICE2010, Bertini+MPM+Dresner+RAL event-generator, and Liege intra-nuclear cascade model INCL). Zhuikov et al. (2011) in the production of ²²⁵Ac reported that their experimental data lie between the theoretical data calculated by ALICE-IPPE and СЕF. Ermoleav et al. (2012) used ALICE and CEF calculations also.

Even though some of these codes such as Talys are robust, Empire has advantage of specialized heavy ion and high-energy performance, flexibility in nuclear level density models, data evaluation pedigree, and advanced fission channel treatment (Herman et al. 2005). Extensive theoretical study of the nuclear reaction is very important for the understanding and guided design for effective and successful improvement on the production process. Studies of excitation functions of charged particle induced reactions are of considerable significance for testing nuclear models as well as for practical applications, especially in accelerators production of radioisotopes. The production of non-conventional radionuclides, however, demands detailed nuclear data work covering both experimental investigations and nuclear model calculations. In this study, the production cross-section of ²²⁵Ac through proton bombardment of Thorium-232 is calculated using a nuclear model code Empire (Herman et al. 2015). The calculation was done with the emphasis on optical models and the optical model potentials.

The purpose of the study is to ascertain the effect of the dispersion and relativistic optical model potentials on the production cross-section of ²²⁵Ac from ²³²Th by proton energy. The result of the research will provide strong information to researchers on the choice of models in production of ²²⁵Ac, hence the production of ²¹³Bi. The theory of nuclear reaction in relation to dispersity and the relativity of the optical model is discussed in the section of theory of the calculation. The result obtained are compared graphically with the experimental data from nuclear data library. The result and the discussion are presented thereafter as well as the conclusion.

The cross-section calculation was done using Empire 3.2.3 (MALTA) Linux version installed on Fedora version 16 operating system 32-bit in an acer Laptop computer. was used for the calculation of the cross-section. The Empire code is a modular system of nuclear reaction codes written to facilitate the computational execution of various nuclear models. Empire 3.2.3 is an improved version which has benefitted from the experience gained in the previous editions. The goal is to obtain a powerful tool for modeling nuclear reactions that can be used in basic research as well as for the generation of evaluated nuclear data files. There for, empire implements in a single system an extensive set of models that cover all the most important nuclear reaction mechanisms. These models provide a complete set of reaction channels and observables such as cross-sections that are of interest in nuclear applications. Such extensive calculations require a considerable number of input parameters. To provide them, empire incorporates a comprehensive library of model parameters, largely based on the IAEA Reference Input Parameter Library III (RIPL-3) project (Herman et al. 2015). Optical model input parameters such as optical model potential and the different calculation methods of the optical model calculation both for the single-channel and coupled-channels are included in the input file. The single-channel is the default option for nucleus with masses up to 220 otherwise coupled-channels are used. Different values are used to select the calculation methods as follows:

The value 0 is used for spherical optical model. In this option, the direct cross-sections are exact but spherical transmission coefficients are used for subsequent pre-equilibrium and Hauser-Feshbach calculations in the whole energy grid (Herman et al. 2015). These results are re-normalized at the transmission coefficient level by taking into account direct component. The total, elastic, absorption and the inelastic cross-sections are taken from the coupled-channel calculations if a coupled-channel optical model potential is being used otherwise only inelastic cross-sections are accepted.

The value 1 is used to calculate coupled-channel transmission coefficients for the incident channel in addition to Distorted-Wave Born Approximation uncoupled (Herman et al. 2015). In this optical model cross-section calculation mechanism, the population of collective levels in the incident channel is calculated using the coupled-channels model for coupled levels and Distorted-Wave Born Approximation method for uncoupled discrete levels and levels embedded in the continuum.

The value 2 is used for coupled-channel transmission coefficients for the incident and outgoing channel in addition to Distorted-Wave Born Approximation uncoupled (Herman et al. 2015). In this optical model mechanism, the population of collective levels both in the incident and outgoing channels is calculated using the coupled-channels model for coupled levels and Distorted-Wave Born Approximation method for uncoupled discrete levels and levels embedded in the continuum. Importantly, the coupled-channel is used consistently, considering the ground state and the coupled levels, to calculate all necessary transmission coefficients for subsequent pre-equilibrium and Hauser-Feshbach calculations. These coupled-channel transmission coefficients define absorption cross-section that is available for the pre-equilibrium and Hauser-Feshbach decay. The absorption cross-section sums up with the direct cross-sections to collective levels to the reaction cross-section. This option is an exact option that considers flux decrease due to the collective excitations at the level of transmission coefficient. It is a recommended option for accurate calculations whenever an appropriate coupled-channels model potential is available though it takes time in the calculation.

The value 3 is used for Distorted-Wave Born Approximation calculation for all collective levels (Herman et al. 2015). This is an optical model cross-section calculation where the population of discrete collective levels in the inelastic scattering is calculated using the Distorted-Wave Born Approximation method providing approximate direct cross-sections. The spherical transmission coefficients are used for the whole energy grid in subsequent pre-equilibrium and Hausa-Feshbach calculations. These results are re-normalized at the transmission coefficient level by taking into account Distorted-Wave Born Approximation calculated direct component.

To determine the effect of the optical model on the production cross-section, the coupled-channel optical models 1, 2 and 3 were used instead of the single-channel because the target Thorium-232 has a massive nucleus capable of deformation. The optical models’ configurations were designated by D1, D2 and D3 respectively. Empire uses ECIS and OPTMAN codes to implement the coupled-channel and the Distorted-Wave Born Approximation for the optical model calculations. The OPTMAN can be invoked by the Empire for D1 and D2 methods of calculation when optical model potentials that use either soft-rotor for spherical nuclei, or rigid-rotor or deformed vibrational rotational model for deformed nuclei are being used (Herman et al. 2015). The different optical model potentials with proton incident particles were used as the RIPL optical model potentials to test the effect of dispersive and relative potential, non-dispersive and relative potential, and the non-dispersive and non-relativistic potentials. The optical model potentials were all taken from the Reference Input Parameter Library (RIPL) as contained in Empire 3.2.3. Table 1 contains the optical model potentials used for the investigation i.e. the optical model potentials with RIPL number 4303, 5600 and 5408 were used for the investigation. They are global, coupled-channels rigid rotor with energy range of 0–200 MeV that are non-dispersive and non-relativistic, non-dispersive and relativistic, and dispersive and relativistic respectively.

To verify the effects of optical models and the dispersive and relativistic potentials to the cross-section calculations, recoils were not calculated. The pre-equilibrium contribution was made by the combination of the Hybrid Monte-Carlo Simulation (HMS) phenomenological pre-equilibrium and the exciton model with mean free path of 3.0. The HFB accounted for the emitted neutrons and protons in both discrete and continuum levels while the exciton model accounted for the gamma emission at the pre-equilibrium level. This model was used because of its capability to account for multiple emission (Herman et al. 2005). For the Hauser-Feshbach contribution, the default Enhanced Generalized Superfluid level density model was used. The Level density parameter in ²³²Th was multiplied by 3.05. The ²²⁵Ac possible production cross-section route through ²³²Th(p,x)²²⁵Ac reaction were calculated with proton energy from 0 to 200 MeV. The possible routes are ²³²Th(p,6n2p)²²⁵Ac, ²³²Th(p,6n2p)²²⁵Ac, ²³²Th(p,5nHe-3)²²⁵Ac, ²³²Th(p,4npt)²²⁵Ac and ²³²Th(p,4nα)²²⁵Ac. The result obtained were compared with the measured data obtained from the experimental nuclear reaction data EXchange FORmat (EXFOR) library contained in the nuclear data section of the IAEA. The method of retrieval is contained in our previous work (Longmena et al. 2023).

The result of the calculation of the production cross-section using the various optical model configurations D1, D2 and D3 to determine the best configuration to be used for the calculation of the production cross-section. The result of the investigation of the optical model relativistic and dispersive potentials is also presented in this section. The calculated production cross-section data are plotted along the experimental production cross-sections and presented in the respective figures below. The results are discussed accordingly. In the calculation, there were missing spin/parity for ²²⁴Ac at different levels of excited state and 0.00 1 was assigned to it by the code. The ground-state of ²²⁶Pa has no assigned spin/parity. The branching ratios and the conversion coefficients for different levels in ²³³Pa, ²³²Th and ²²⁹Ac were missing.

Fig. 2 is the comparison of the calculated cross-section base on ²³²Th(p,6n2p)²²⁵Ac reaction. It shows that the three configurations (D1, D2 and D3) reach the same maximum intensity of approximately 19 mb at the 30 MeV mark. It does not show significant disparity among the models in the low-energy region. There is very small disparity between the three models across most of the energy spectrum. D2 and D3 models are nearly identical throughout the 20–200 MeV energy range. They follow the same trajectory, including the peak at 30 MeV and the subsequent decline. The model D1 shows the most visible divergence, particularly between 70 and 135 MeV, where its cross-section calculation is slightly higher than that of D2 and D3 which tend towards the experimental data. Above 135 MeV, the three models converge completely, dropping to a consistent low cross-section between 1 and 2 mb. This is attributed to the failure of the Hauser-Feshbach and the pre-equilibrium models.

There is significant scattering among the experimental datasets, which is common in thorium spallation studies: Steyn et al. (2021) and Ermolaev et al. (2012) show good agreement in the 40–70 MeV range, defining a clear hump that the models attempt to follow. Older data like Gauvin (1963) are significantly lower than modern measurements, highlighting the evolution of measurement precision over the decades. Moreover, the one point measured by Lefort et al. (1961) is the only older data measured at high energy region.

Similarly, in the comparison of the models based on the cross-section of ²³²Th(p,4nα)²²⁵Ac reaction presented in Fig. 3, the level of disparity between the models is extremely low across the entire energy range. D2 and D3 models are nearly indistinguishable, following an almost identical trajectory from 20 to 200 MeV. Model D1 shows the most deviation, appearing as a slightly higher line compared to D2 and D3 in the energy region between 70 and 90 MeV. above 900 MeV the deviations are negligible dropping to a consistent low-value below 2 mb. Below 70 MeV, the three models converge completely with low-energy peak cross-section of 19 mb around 30 MeV.

From both Figs 2, 3, the three models predict a sharp reaction peak around 30 MeV much earlier than observed in modern experimental datasets like Steyn et al. (2021) and Ermolaev et al. (2012), which show the reaction threshold of 40 MeV. From 90 MeV to 200 MeV, the models capture the oscillatory nature of the cross-section but generally under-predict the experimental values at the Hauser-Feshbach and the pre-equilibrium models significant contribution region. The three model variations are almost indistinguishable across most of the energy range, suggesting that the choice between uncoupled and fully collective DWBA treatments in these specific EMPIRE configurations D1, D2, and D3 have a negligible impact on this specific isotope’s production cross-section in EMPIRE. Also, adding coupled-channel transmission coefficients for the outgoing channel does not significantly alter the predicted cross-sections. Specifically, the models are likely favouring the emission of other particles or fission over the channels been studied as energy increases. However, D1 model tend to diverge towards the experimental data more than D2 and D3 models suggesting that D1 which is coupled channel transmission coefficients for the incident channel and spherical by distorted wave born approximation uncoupled should be used for further calculations.

Haven determined the optical model calculation method, the different optical model potentials of non-dispersive and non-relativistic (Non-Disp. & Non-Rel.), non-dispersive and relativistic (Non-Disp. & Rel.), and dispersive and relativistic (Disp. & Rel.) of the optical model parameters were investigated and the results presented in Figs 4, 5, 6, 7, 8. Fig. 4 shows the comparison of the relativistic and dispersive potentials base on ²³²Th(p,6n2p)²²⁵Ac reaction cross-section. This is the channel where its emissions are all nucleons of 2 protons and 6 neutrons. The figure shows all the models significantly over-predicting the experimental cross-section between 20 and 40 MeV, reaching a peak near 20 mb, whereas experimental data (e.g., Ermolaev et al. 2012) suggest much lower values.

The model with Non-Dispersive & Non-Relativistic effect consistently yields the highest cross-section values across the entire energy range. It is slightly closer to the experimental data in the 50–90 MeV range but contributes most to the over-prediction at the 30 MeV peak. This shows that introducing relativistic effects slightly lowers the calculated cross-section compared to the non-relativistic version. This adjustment brings the model marginally closer to the lower-bound experimental points (like Weidner et al. 2012) between 50 and 95 MeV. The model with Dispersive & Relativistic effect is the most complex model, accounting for the dispersion relation and relativistic effects. In this plot, it produces the lowest cross-section values of the three. While this helps mitigate the over-prediction at 30 MeV, it results in the greatest divergence from the experimental data at energies above 100 MeV.

The models suggest the reaction becomes significant much earlier (25 MeV) than the experimental consensus, which shows a more gradual rise starting around 40–50 MeV. The three lines are most distinct between 40 and 100 MeV. The Non-Dispersive & Non-Relativistic potentials appears to predict the experimental peaks (like those from Steyn et al. near 55 MeV) better, while the Dispersive & Relativistic model is more conservative. The fact that all models drop off after 100 MeV suggests that the chosen optical model potentials or the level density parameters used in these calculations may not be accounting for high-energy pre-equilibrium effects or specific spallation-like mechanisms that dominate at higher incident proton energies.

In the prediction of emission of cluster of deuterons among the nucleons in ²³²Th(p,5ndp)²²⁵Ac reaction, (Fig. 5), just as observed in the previous analysis of ²³²Th(p,6n2p)²²⁵Ac, the theoretical calculations for the (p, 5ndp) component generally exhibit a sharp theoretical peak near 30 MeV. Between 50 and 90 MeV, the models show a structured cross-section that attempts to follow the fluctuations in the data. Here, the model with Non-Disp. & Non-Rel. yields the highest excitation function. In reactions involving deuteron emission, this model often over-predicts the probability because it lacks the damping effects provided by relativistic kinematics or dispersive corrections to the potential depth. In the model with Non-Disp. & Rel. potential, the relativistic correction slightly reduces the cross-section. This is particularly noticeable in the 40–100 MeV range, where the green line sits consistently below the purple line, marginally improving the fit to Steyn et al. (2021) data points. Similar to the (p,6n2p) channels, the model with Disp. & Rel. potential is the most conservative of the three. By using a dispersive optical model potential (DOMP), the imaginary part of the potential is linked to the real part via dispersion relations. This typically results in a lower peak cross-section, which is the closest theoretical representation to the lower-bound experimental data in the mid-energy region.

The (p, 5ndp) channel involves the emission of a composite particle (d). The models likely treat this via a combination of pre-equilibrium emission and evaporation. The under-prediction at high energies could suggests the pre-equilibrium pick-up/knock-out components for deuterons might be under-parameterized.

Similar results are obtained for the production channels where Helium-3 is emitted among the nucleons in ²³²Th(p,5nHe-3)²²⁵Ac reaction (Fig. 6).

The theoretical curves for the (p, 5n³He) channel follow the same systematic behaviour as the previously discussed (p, 6n2p) and (p, 5ndp) channels. All models predict a sharp excitation peak near 30 MeV (about 20 mb) that is entirely absent in the experimental data Fig. 6). The models attempt to simulate the fluctuations in experimental data, showing a local peak near 85 MeV (about 6 mb). This moderately aligns with data points from Grisworld et al., Weidner et al. and Zhuikov et al. in that energy.

The model with Non-Disp. & Non-Rel. parameters yields the highest cross-section values. Without relativistic or dispersive corrections, the model assumes a more transparent nucleus with higher emission probabilities for cluster particles like ³He. In the model with Non-Disp. & Rel. parameters, the inclusion of relativistic kinematics slightly reduces the predicted yields compared to the non-relativistic case. This provides a marginally better fit to the lower experimental bounds seen around 70–80 MeV. The Disp. & Rel. generally provides the lowest and most constrained cross-section. By linking the real and imaginary parts of the optical potential through dispersion relations, it accounts for the energy dependence of the potential depth more rigorously, though it still suffers from the same high-energy under-prediction.

The three models are nearly identical at very high and very low energies but show visible divergence in the 40–100 MeV region. The Non-Dispersive & Non-Relativistic model remains the most optimistic regarding production yields, while the Dispersive & Relativistic model is the most physically conservative.

In the prediction of ²³²Th(p,4npt)²²⁵Ac reaction (Fig. 7), the model with Non-Disp. & Non-Rel. potential consistently predicts the highest cross-section across the entire energy range. In the model with Non-Disp. & Rel. potential, the inclusion of relativistic kinematics results in a slight reduction of the cross-section magnitude compared to the non-relativistic case. This brings the mid-energy calculations slightly closer to the lower data points. The model with Disp. & Rel. potential provides the most conservative (lowest) values. The use of a dispersive optical model potential (DOMP) constraints the excitation function, which helps reduce the low-energy over-prediction but contributes to the under-prediction at higher energies. In the region between 40 MeV and 70 MeV, the models are most sensitive to the specific potential used. The Non-Dispersive & Non-Relativistic potential appears to track the shape of the Steyn et al. data more closely in terms of trend, even if the absolute magnitude remains shifted.

In the comparison of the effect of the relativistic and dispersive optical potentials on ²³²Th(p,4nα)²²⁵Ac reaction cross-sections (Fig. 8), in the 50–90 MeV range, the models show some structure, attempting to follow the fluctuations in the experimental data, with a small local peak near 85 MeV. The model with Non-Disp. & Non-Rel. potential predicts the highest cross-section values. This model typically provides the most optimum production yields but shows the greatest deviation from the experimental threshold. In the model with Non-Disp. & Rel. potential, the inclusion of relativistic kinematics leads to a slight reduction in the calculated cross-section compared to the non-relativistic version, marginally improving the fit to lower-bound experimental points. As seen above, the model with Disp. & Rel. potential generally provides the most conservative (lowest) values. The use of a dispersive optical model potential (DOMP) constraints the excitation function, which helps mitigate the over-prediction at 30 MeV but increases the gap at higher energies. The theoretical output for the (p, 4nα) reaction is identical in shape to the (p, 6n2p) and (p, 5ndp) channels, indicating that the current optical model parameters treat these different A = 225 isobaric contributors with similar probability distributions.

The comparative analysis of the various reaction channels for ²²⁵Ac production reveals a consistent performance pattern across the theoretical models, alongside distinct physical challenges for each. All five reaction channels — (p, 6n2p), (p, 5ndp), (p, 5n³He), (p, 4npt), and (p, 4nα)—share a fundamental theoretical behaviour in the calculations. Every model predicts a dominant, non-physical peak around 30 MeV (about 20 mb). This is due to a failure in the compound nucleus evaporation parameters which miscalculate the competition between these channels at low energies. All models collapse to a baseline below 5 mb after 100 MeV. In contrast, experimental data (e.g., Weidner et al. 2012; Griswold et al. 2016) show cross-section rising toward 20 mb as spallation and pre-equilibrium mechanisms dominate. The channels involving cluster emission (e.g., ³He, t, α) show slightly more model divergence in the 40–90 MeV range, reflecting the difficulty of modelling complex particle pick-up or knock-out processes.

The combination of dispersive and relativistic effects (Disp. & Rel.) tracks almost perfectly with the Non-Disp. & Rel. model. This suggests that the dispersive component of the optical model potential (OMP) provides minimal correction for this specific interaction compared to the relativistic adjustment.

From the discussion and the observation above, all three configurations predict a significant reaction peak at 30 MeV. This shows threshold that is higher than simple (p, xn) reactions. The presence of this peak in the models indicates a potential issue with how the models handle the competition between different A = 225 isobaric contributors at low energy.

All three theoretical curves drop to below 2 mb above 110 MeV. The collapse of the models after 110 MeV highlights a systemic gap in how these optical models handle the transition from compound nucleus evaporation to direct spallation/multi-particle emission regimes. Thess can be attributed to the failure of the Hauser-Feshbach models (the default Enhance Generalised Superfluid model level density) and the pre-equilibrium models (the combination of the HMS and the exciton pre-equilibrium models) to effectively replicate the experimental data.

Though the different optical models’ configurations is to provide physical realities of the nuclear reactions, it was observed that they do not have any significant variant effects on the calculation of the cross-section of the production of ²²⁵Ac by proton bombardment of Thorium within energy range from threshold to 200 MeV.

The relativistic and the dispersive potentials did not show significant effect in the calculations. The optical model with non-dispersive and non-relativistic potentials produces highest yield across the energy range of 20–200 MeV. It tends to move towards the experimental data at mid-energy and high energy regions but contributed most in the overestimation of the low energy peak. Thus, the Non-Dispersive & Non-Relativistic model remains the most optimistic regarding production yields, while the Dispersive & Relativistic model is the most physically conservative.

This shows that any of the models can be used for the calculation of the cross-sections. Though based on the physical accuracy and the alignment with the lower bounds of experimental data, the model with Dispersive & Relativistic (Disp. & Rel.) potentials has more advantage and the most recommended for further development. It utilizes the Dispersive Optical Model (DOM), which uses dispersion relations to link the real and imaginary parts of the potential. This provides a more consistent treatment of nuclear structure and reaction dynamics compared to local, non-dispersive potentials. By linking the real and imaginary parts of the potential via dispersion relations, it accounts for the energy dependence of the potential depth and surface absorption more accurately. It consistently provides the most conservative (lowest) cross-section values. While it still overestimates the 30 MeV peak, its magnitude is lower than the other models, making it closer to the experimental reality in that region.

Even though this study did not show much deviation between the models, proton irradiation of heavy targets like Thorium at energies reaching 200 MeV and beyond requires relativistic kinematics to correctly simulate the scattering wavefunctions and non-elastic cross-sections.

Including relativistic effects is critical for proton energies above 100 MeV to correctly describe the scattering wavefunctions and the absorption cross-section. It offers the best baseline for addressing the high-energy under-prediction. Since it is already the most theoretically rigorous, adjusting its pre-equilibrium parameters or adding a spallation component would yield a more reliable fit than adjusting the mathematically simpler models.