The theoretical calculations in this study were performed using TALYS-2.0, a comprehensive nuclear reaction code developed for the analysis and prediction of nuclear reactions through an optimized integration of established nuclear models [17]. TALYS has been widely applied in the interpretation of experimental nuclear reaction data and the generation of evaluated nuclear data for scientific, medical, and technological applications. The code is capable of simulating neutron-, photon-, proton-, deuteron-, triton-, ³He-, and α-particle-induced reactions over an incident energy range of 0.001 - 200 MeV. In the present work, TALYS-2.0 was employed to perform a continuous evaluation of the excitation functions of the ¹⁶⁹Tm(p, x) reaction channels up to 200 MeV. The nuclear decay characteristics and Q-values of the possible contributing reaction channels were obtained from the National Nuclear Data Center (NNDC) database [3][19] and are summarized in Table 1.

Table 1. Decay characteristics of the investigated reaction products [3][19].

The sensitivity analysis presented in this work is focused primarily on the role of nuclear level densities. While the nuclear level density model (LDM) was systematically varied across six distinct frameworks, the optical model potentials (OMP) and pre-equilibrium parameters were maintained at their global default settings. Consequently, the observed variations reflect the isolated model dependence on level densities alone, providing a focused baseline for the proton-induced reaction channel.

2.1.1. Optical Model Parameters

In TALYS-2.0, reaction cross-sections are calculated within the Hauser-Feshbach statistical theory, where the optical model potential (OMP) and nuclear level density play fundamental roles in describing the reaction mechanism. The OMP is essential for predicting key nuclear reaction observables, including elastic, inelastic, and total reaction cross-sections, angular distributions, and transmission coefficients that govern compound nucleus decay and pre-equilibrium particle emission through either deformed nuclear potentials or distorted-wave Born approximation (DWBA)-based approaches [18]. In TALYS, optical model calculations are performed using the ECIS-06 code developed by Raynal [20]. The default proton optical model parameters implemented in TALYS-2.0 are based on the local and global parameterizations proposed by Koning and Delaroche [21]. For proton-induced reactions, the energy dependence of the potential depths is described as a function of ( E − E f p ), where E is the incident proton energy and E f p , denotes the proton Fermi energy, defined as the energy midway between the last occupied and the first unoccupied nuclear shell states. For incident protons,

where S p is the proton separation energy with Z being the proton number. The nuclear structure mass tables implemented in TALYS-2.0 are used to determine the particle separation energies required for reaction calculations.

The optical model parameterization for incident proton is given by Equations (2) to (6).

Here, V V and V S O denote the real components of the volume-central ( V ) and spin-orbit ( S O ) potentials, whereas W V , W D , W S O represent the corresponding imaginary components associated with the volume-central ( V ), surface-central ( D ) and spin-orbit ( S O ) terms, respectively. The variable E denotes the incident proton energy in the laboratory reference frame, with the model being applicable over the energy range of 0.001 - 200 MeV. In TALYS-2.0, the local optical model parameters (e.g., v 1 , v 2 , and related coefficients) are automatically retrieved from the built-in nuclear structure and model parameter database [21].

2.1.2. Mass Model

Nuclear mass models play a fundamental role in cross-section calculations, as particle separation energies and Q-values are directly determined by the masses of the participating nuclides and dictate the energetic feasibility of individual reaction channels. TALYS-2.0 provides four nuclear mass model options for reaction calculations [17].

In the present study, the GHFB [22] mass model was adopted for the reaction calculations. The GHFB approach is a microscopic nuclear structure model based on self-consistent mean-field theory using the Gogny effective interaction. One of its important advantages is the explicit treatment of nuclear deformation and quadrupole correlation energies within the framework of the 5D collective Hamiltonian approach. Owing to its strong theoretical foundation and reliable predictive performance, the GHFB mass model was selected in the present work to achieve more accurate calculations of the reaction cross-sections and production characteristics.

2.1.3. Nuclear Level Density Model

Nuclear level density models (LDMs) are essential in statistical reaction calculations, as they significantly influence the prediction of reaction cross-sections, radionuclide yields, and production optimization. TALYS-2.0 provides six LDMs for reaction analysis [17]. The phenomenological models include CTM, BFM, and GSM, all derived from the Fermi Gas Model (FGM), which assumes uniformly spaced single-particle energy levels without explicit consideration of collective excitations. The microscopic approaches consist of the SHFB, SHFB(C), and GHFB model. In the present study, all six LDMs available in TALYS-2.0 were systematically investigated to evaluate their influence on the cross-section profiling of the ¹⁶⁹Tm(p, x) reaction.

The level density, ρ ( E x , J , Π ) represents the number of nuclear levels per MeV around an excitation energy E x , for a certain spin J and parity Π. The total level density ρ t o t ( E x ) yields the total number of levels per MeV around E x which is obtained by summing over J and Π, as shown in Equation (7).

1)Phenomenological Level Density Models

a)Constant Temperature Model (CTM)

The CTM is a phenomenological level density model in which an effective temperature, T , is introduced to describe the exponential dependence of level density on excitation energy within the shell-model framework [23]. Owing to its simplicity and reliable performance in the low-excitation energy region, the CTM has been widely employed in nuclear reaction calculations. Although the model provides an effective description of nuclear level densities at lower excitation energies, its applicability gradually decreases at higher excitation energies. The model, originally proposed by Gilbert and Cameron [24], divides E x , into two regions separated by a matching energy, E M . For E x below E M , the total level density is described by the constant temperature formalism, whereas above E M , the level density is represented by the Fermi-gas approximation [17]. Accordingly, for 0 < E x ≤ E M , the total level density is expressed as shown in Equation (8a):

Above E M , it is given by Equation (8b):

b)Back-Shifted Fermi Gas Model (BFM)

Although the conventional FGM provides a useful framework for describing nuclear level densities, it has several inherent limitations, particularly in accounting for pairing correlations and low-energy nuclear structure effects. Nevertheless, owing to its simplicity and its ability to reproduce systematic trends in experimental level density data through fits to cumulative low-lying discrete states and the average spacing of s-wave neutron resonances near the neutron binding energy, the FGM remains widely used in nuclear reaction studies [25].

To overcome these limitations, the BFM, originally developed by Dilg et al. [26], introduces an adjustable back-shift parameter to incorporate pairing effects into the excitation energy. This modification allows the standard Fermi gas formalism to be extended smoothly down to zero effective excitation energy, U , thereby providing a simple treatment of pairing correlations. However, the original formulation exhibits singular behavior as the effective excitation energy approaches zero. To address this issue, an improved treatment was first proposed by Grossjean and Feldmeier [27], and later reformulated into a practical implementation by Demetriou and Goriely [28], which has subsequently been adopted in the TALYS nuclear reaction code. Accordingly, the total level density is expressed by Equation (9).

where ρ 0 ( t ) is the scaling prefactor of the Fermi‑gas formula, determined by the level density parameter a and the energy shift Δ, which anchors the model to experimental data.

c)Generalized Superfluid Model (GSM)

The GSM, introduced by Ignatyuk et al. [29][30], extends the conventional FGM by explicitly incorporating pairing correlations through the Bardeen-Cooper-Schrieffer theory, treating nuclei as superfluid systems at low excitation energies. In this framework, nucleon pairs reduce the level density compared with that predicted by the normal Fermi gas approximation. As excitation energy increases, these pairing correlations gradually diminish, and the model transitions smoothly to the standard FGM description. The GSM describes the nuclear level density using two distinct formulations: one corresponding to the paired (superfluid) phase below critical excitation energy U c , and the other corresponding to the unpaired (normal) phase above U c . This treatment provides a more physically realistic representation of nuclear structure effects than the CTM or BFM, as it explicitly models the phase transition from superfluid to normal nuclear matter. For excitation energy U ≤ U c , the total level density is expressed by Equation (10).

where S is the entropy and D is the determinant related to the saddle-point approximation.

2)Microscopic Level Density Models

In the Reference Input Parameter Library (RIPL) [31], Goriely has provided an extensive set of microscopic nuclear level densities based on the SHFB model, covering nuclei across the entire nuclear chart, from the proton drip line to the neutron drip line. These level densities were obtained through Hartree-Fock-based calculations for excitation energies up to 150 MeV [32]. Moreover, Hilaire and Goriely [33] have introduced energy-, spin-, and parity-dependent nuclear level densities based on a microscopic combinatorial approach called SHFB(C). In this model, the intrinsic state density is calculated explicitly from single-particle configurations, while collective effects, including rotational and vibrational enhancements, are incorporated in detail and self-consistent manner. The most recent microscopic level density option implemented in TALYS is the GHFB model, which is based on temperature-dependent Hartree-Fock-Bogoliubov calculations using the Gogny effective nucleon–nucleon interaction [34]. Unlike phenomenological approaches, these microscopic level densities, ρ H F M , are derived directly from nuclear structure calculations and are not adjusted to experimental data. To improve flexibility in fitting experimental observables, TALYS-2.0 introduces an additional scaling function for these microscopic models. Accordingly, the level density for the microscopic models is expressed by Equation (11).

In Equation (11), the scaling constant, c , and the pairing shift parameter, δ , are set to their default values of c = 0 and δ = 0 in TALYS-2.0. These parameters serve roles analogous to those used in phenomenological level density models. The parameter δ corresponds to the back-shift parameter, Δ, in the BFM, introducing an effectively shift in excitation energy to account for pairing correlations and odd-even nuclear effects. Similarly, the parameter c plays a role comparable to the level density parameter, a , in the BFM, controlling the overall magnitude and energy dependence of the level density at higher excitation energy region.

For the cross-section calculations, a ¹⁶⁹Tm foil was selected as the target material in TALYS-2.0 to evaluate the excitation functions of proton-induced reactions on ¹⁶⁹Tm for all possible reaction channels up to an incident proton energy of 200 MeV. The target geometry was defined as 0.03353 mm^t × 10 mm × 10 mm [16][35]. The GHFB mass model [22] was adopted in the TALYS-2.0 calculations. The primary nuclear model parameters in the TALYS calculation, including the pre-equilibrium model, gamma-emission functions, and width-fluctuation corrections were maintained at their default TALYS-2.0 values. A flag was enforced to incorporate spherical Optical Model in the TALYS 2.0 input parameters. Cross-section profiling was performed using all six available LDMs to systematically investigate model-dependent variations in the predicted reaction channels. Furthermore, for radionuclide production assessment using the calculated cross section data, the proton beam current was fixed at 5 mA with an irradiation period of 24 h, followed by a cooling period of 24 h. In addition, the activity, production amount, and integral yield of ¹⁶⁹Yb were evaluated for the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction to assess the feasibility of accelerator-based production of this medically relevant radionuclide.

To validate the nuclear models, statistical factor analyses were performed to quantitatively assess the variation among the LDMs and their agreement with the experimental cross-section data for the proton-induced reaction on ¹⁶⁹Tm, as available in the EXFOR database [4].

Three statistical deviation factors were chosen to analyze the predictive reliability of the LDMs [36]-[38]. These are:

1) Mean deviation: F-factor (F = 1: perfect agreement)

2) Mean relative deviation: D-factor (D = 0: perfect agreement)

3) Mean Ratio: R-factor (R = 1: perfect agreement on average)

Here, σ i e x p and σ i c a l represents the experimental cross-section data obtained from the EXFOR database [5][6][15][16] and the cross-sections calculated using TALYS-2.0 for the corresponding experimental energy values, respectively, while N denotes the total number of experimental data points considered in the analysis. Within this statistical framework, the LDM yielding the value 1 for F and R, and 0 for D is considered to provide the best agreement between theoretical predictions and experimental observations [39]. The theoretical excitation functions across all investigated models were generated utilizing a discrete incident proton energy mesh with a uniform step size of 0.1 MeV starting from the respective reaction thresholds. To evaluate the statistical deviation factors, a linear interpolation procedure was implemented to extract precise cross-section values ( σ i c a l ) and corresponding to the exact incident energy points of the experimental data ( σ i e x p ).

The F-factor measures deviations on a logarithmic scale, capturing the overall trend of the excitation function. The D-factor gives the average relative difference between calculated and experimental values, while the R-factor compares values point by point, making it sensitive to local variations. Since each factor measures the agreement from a different perspective, they do not always identify the same model as the best fit. Among these metrics, the F-factor is considered the most appropriate for comparative analyses of different calculations, particularly when experimental data are limited [37].

The Stopping and Range of Ions in Matter (SRIM) code [40] is a Monte Carlo-based simulation package designed to study the stopping and range of ions in matter. It provides a set of computational tools for modeling various aspects of ion transport, with common applications in ion stopping, implantation, and transmission studies. In this work, the SRIM-2013 code was used to calculate the stopping power for proton-induced reaction on ¹⁶⁹Tm target over an energy range up to 200 MeV. The stopping power calculations are based on the Bethe-Bloch formula, which describes the mean energy loss per unit path length of charged particles as they traverse a material [40]. The original Bethe-Bloch relativistic stopping formula for a particle with speed v , charge z , and energy E , traveling a distance x into a target of electron number density n and mean excitation energy I is, in SI units:

where c is the speed of light and ϵ 0 the vacuum permittivity, β = v c , e and m e the electron charge and rest mass respectively. The electron density of the material n can be calculated by n = N A ⋅ Z ⋅ ρ A ⋅ M u , where ρ is the density of the material, Z its atomic number, A its relative atomic mass, N A the Avogadro number and M u the Molar mass constant.

Theoretical integral yields were determined for the production of ¹⁶⁹Yb radionuclide over an incident proton energy range up to 200 MeV. The yield calculation was based on the integration of reaction cross sections with the corresponding stopping power of the target material. The yield for ¹⁶⁹Yb was assessed through reaction ¹⁶⁹Tm(p, n). In this case, the integral yield was evaluated from the maximum incident energy down to the specific threshold energy of the reaction. The production cross sections were calculated using the TALYS-2.0 code with the Hauser–Feshbach nuclear model, while the stopping powers were derived from the SRIM-2013 code using Equation (8). The production yield of ¹⁶⁹Yb radionuclide was calculated using the excitation function [41][42] as:

where Y is the activity (in Bq) of the ¹⁶⁹Yb, H is the isotope abundance of the target ¹⁶⁹Tm, σ ( E ) is the energy-dependent reaction cross-section at energy E calculated from TALYS-2.0, I p is the projectile current, S ( E ) is the stopping power derived from SRIM-2013 code, λ is the decay constant of the product, and t is the time of irradiation.

Figure 1 presents the TALYS-2.0 calculated excitation functions for seven proton-induced reaction channels on the monoisotopic target ¹⁶⁹Tm using CTM, BFM, GSM, SHFB, SHFB(C), and GHFB. Since no experimental cross-section data are currently available in the EXFOR database for these reaction channels, the calculated results were compared only with the TENDL-2023 evaluated nuclear data library. Overall, the six LDMs exhibit similar excitation function trends, indicating that the general reaction mechanisms are consistently reproduced by TALYS-2.0, although noticeable differences in magnitude emerge in specific energy regions.

For the ¹⁶⁹Tm(p, n + α)¹⁶⁵Er reaction (Figure 1(a)), the cross section rises rapidly above the reaction threshold and reaches values of approximately 20 - 35 mb in the intermediate-energy region. Several local maxima and minima are observed between 20 and 180 MeV, reflecting the competition between compound nucleus formation and pre-equilibrium particle emission. The GSM model predicts slightly larger cross sections over most of the energy range, whereas the microscopic models tend to produce lower values at higher energies.

A similar trend is observed for the ¹⁶⁹Tm(p, α)¹⁶⁶Er reaction (Figure 1(b)), where the excitation function exhibits a pronounced low-energy peak followed by a gradual increase toward intermediate energies. Beyond approximately 60 MeV, all models predict relatively stable cross sections with moderate model-dependent variations. The GHFB and SHFB(C) models generally yield larger cross sections in the high-energy region, suggesting a stronger sensitivity of α-particle emission to microscopic level-density descriptions.

The excitation function of the ¹⁶⁹Tm(p, ³He)¹⁶⁷Er reaction (Figure 1(c)) shows a steep increase above threshold, followed by a broad plateau extending from approximately 60 to 160 MeV. The calculated cross sections remain within a relatively narrow range among the six LDMs, indicating that this reaction channel is less sensitive to the choice of LDM.

For the ¹⁶⁹Tm(p, t)¹⁶⁷Tm reaction (Figure 1(d)), the cross section increases sharply and attains values exceeding 200 mb in the 40 - 80 MeV region. A broad plateau is subsequently observed, followed by a gradual decline at higher energies. Among the investigated models, GHFB consistently predicts the largest cross sections, while CTM and SHFB yield comparatively lower values. The differences between the models become more pronounced above 100 MeV, where pre-equilibrium contributions dominate the reaction mechanism.

The ¹⁶⁹Tm(p, n + p)¹⁶⁸Tm reaction (Figure 1(e)) exhibits behavior similar to that of the ³He-emission reaction. The excitation function rises rapidly after threshold and reaches a broad maximum between approximately 30 and 80 MeV. Thereafter, the cross section decreases gradually with increasing proton energy. The six LDMs produce very similar excitation functions, indicating that the calculated reaction probability is relatively insensitive to the level-density prescription. Nevertheless, the GHFB model again predicts slightly larger cross sections in the intermediate-energy region.

A distinctly different behavior is observed for the ¹⁶⁹Tm(p, 2n)¹⁶⁸Yb reaction (Figure 1(f)). The excitation function exhibits a well-defined peak near 15 - 20 MeV with cross sections approaching 103 mb, followed by a continuous decline toward higher proton energies. This trend is characteristic of neutron-emission reactions, where the reaction probability is highest near the optimum excitation energy and decreases as competing reaction channels become increasingly important.

Finally, the ¹⁶⁹Tm(p, γ)¹⁷⁰Yb reaction (Figure 1(g)) displays the smallest cross sections among all investigated channels. The excitation function reaches a

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 1. TALYS-2.0 calculated excitation function of the (a) ¹⁶⁹Tm(p, n + α)¹⁶⁵Er, (b) ¹⁶⁹Tm(p, α)¹⁶⁶Er, (c) ¹⁶⁹Tm(p, ³He)¹⁶⁷Er, (d) ¹⁶⁹Tm(p, t)¹⁶⁷Tm, (e) ¹⁶⁹Tm(p, n + p)¹⁶⁸Tm, (f) ¹⁶⁹Tm(p, 2n)¹⁶⁸Yb, and (g) ¹⁶⁹Tm(p, γ)¹⁷⁰Yb reaction obtained using six nuclear LDMs over the proton energy range of 1 - 200 MeV.

maximum below 1 mb in the low-energy region and decreases steadily with increasing proton energy. The small magnitude of the cross section reflects the relatively low probability of radiative capture compared with charged-particle and neutron-emission processes. The calculated results from all six LDMs are nearly identical, indicating that this reaction channel is only weakly affected by variations in the level-density model.

Overall, the TALYS-2.0 calculations demonstrate that the selected LDMs provide consistent predictions for the investigated reaction channels. While CTM, BFM, and GSM generally reproduce the low-energy structures more closely to the TENDL-2023 evaluation, SHFB, SHFB(C), and GHFB tend to predict somewhat larger cross sections at higher proton energies. These theoretical results provide valuable nuclear data for reaction channels where experimental measurements are currently unavailable and may serve as useful guidance for future cross-section measurements and nuclear data evaluations.

The experimental cross-section data for the ¹⁶⁹Tm(p, 4n)¹⁶⁶Yb reaction [5][6], available in the EXFOR database, show generally similar trends over the investigated energy range, although some differences in magnitude and peak position are observed among the datasets. Most measurements indicate a reaction threshold near 25 MeV and a broad maximum around 35 - 40 MeV reaching a maximum cross-section of about 900 mb (see Figure 2). These variations may originate from differences in experimental conditions such as target thickness, proton beam-energy calibration, irradiation parameters, detector efficiency, and data analysis procedures. In many studies, the activation method followed by off-line γ-ray spectrometry was used for cross-section determination, where uncertainties associated with counting statistics, decay data, and overlapping γ-ray lines could contribute to the observed discrepancies. Nevertheless, the available experimental data exhibit an overall consistent excitation-function behavior for this reaction channel.

Figure 2. Excitation function of the ¹⁶⁹Tm(p, 4n)¹⁶⁶Yb reaction compared with experimental data [5][6] from the EXFOR database and the TNDL-2023 evaluated nuclear data library [18].

To validate the nuclear LDMs employed in the TALYS-2.0 calculations, the statistical deviation factors F, D, R were evaluated using Equations (12)-(14). The calculated factors for the six LDMs and the available experimental datasets with the evaluated TENDL-2023 data [5][6][18] were summarized in Table 2.

Table 2. Statistical factors for different LDMs of ¹⁶⁹Tm(p, 4n)¹⁶⁶Yb reaction.

The GSM and SHFB models exhibit lowerF and D values with respect to the TENDL-2023 evaluation, indicating better agreement with the evaluated excitation functions, while their corresponding R-values remain close to unity, further confirming strong model consistency. In contrast, the GHFB model yields relatively larger deviation factors, indicating weaker agreement with the evaluated benchmark data. However, comparison with the experimental datasets reported by Birattari et al. [5] and Tárkányi et al. [6] reveals that the GHFB model produces the lowest F, D and R-factors, indicating superior predictive performance relative to the phenomenological models.

These results demonstrate that the agreement between theoretical TALYS-2.0 calculations and experimental observations strongly depends on the selected LDM and dataset, implying that no single LDM consistently outperforms the others over the entire energy range. The average values of the statistical deviation factors are presented in Figure 3.

(a)

(b) (c)

Figure 3. Average values of the statistical deviation factors: (a) F-factor, (b) D-factor, and (c) R-factor calculated for the different LDMs, experimental data, and TENDL-2023 evolution for the ¹⁶⁹Tm(p, 4n)¹⁶⁶Yb reaction.

Figure 4 presents the excitation function of the ¹⁶⁹Tm(p, 3n)¹⁶⁷Yb reaction calculated using TALYS-2.0, together with the available experimental data and the TENDL-2023 evaluated library for comparison. Experimental cross-section datasets are available in the EXFOR database from Tárkányi et al. [6] for the proton energy range 24.5 - 35.8 MeV and from Birattari et al. [5] for 19.7 - 42.5 MeV, while the TENDL-2023 evaluated data cover the 18 - 30 MeV energy region [18]. In the present work, the excitation function was extended up to 200 MeV incident proton energy. The reaction channel opens at approximately ~18 MeV, marking the threshold energy. Following the threshold, the cross section increases rapidly and reaches a maximum of 1039 mb at 26 MeV. This peak corresponds to the optimal excitation energy of the compound nucleus. Beyond the peak region, the cross section gradually decreases with increasing proton energy, where pre-equilibrium emission effects may contribute significantly to the reaction mechanism at higher energies.

Figure 4. Excitation function of the ¹⁶⁹Tm(p, 3n)¹⁶⁷Yb reaction compared with experimental data [5][6] from the EXFOR database and the TNDL-2023 evaluated nuclear data library [18].

The statistical deviation factors F, D and R, summarized in Table 3, were evaluated to validate the performance of the six nuclear LDMs. The BFM and SHFB models produce the lowest F and D values relative to the TENDL-2023 evaluation. Their corresponding R-values also remain close to unity, confirming reliable predictive capability. In contrast, comparison with the experimental datasets of Birattari et al. [5] and Tárkányi et al. [6] indicates that the SHFB(C) and GHFB models provide comparatively lower statistical deviation factors, suggesting improved agreement with measured cross sections. Although the CTM and BFM models yield favorableR-values for some datasets, the microscopic SHFB(C) and GHFB approaches generally demonstrate better consistency with the experimental observations. Overall, the phenomenological models remain effective in reproducing the evaluated excitation functions, whereas the microscopic models provide improved agreement with the available experimental data. The average values of the statistical deviation factors are presented in Figure 5.

The excitation function of the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction was calculated using TALYS-2.0 with different nuclear LDMs from the reaction threshold up to 200 MeV incident proton energy. The present work provides an extended cross-section profile over a broad energy range, whereas the available experimental

Table 3. Statistical factors for different LDMs of ¹⁶⁹Tm(p, 3n)¹⁶⁷Yb reaction.

(a)

(b) (c)

Figure 5. Average values of the statistical deviation factors: (a) F-factor, (b) D-factor, and (c) R-factor calculated for the different LDMs, experimental data, and TENDL-2023 evolution for the ¹⁶⁹Tm(p, 3n)¹⁶⁷Yb reaction.

measurements are limited to lower proton energies. Among the investigated models, the GHFB model predicts the maximum cross section of approximately 170 mb at around 10 MeV. All LDMs exhibit similar excitation function trends throughout the investigated energy range, with only minor deviations in the peak magnitudes and high-energy behavior. The CTM model predicts comparatively higher cross-section values at elevated proton energies than the other LDMs.

Figure 6 compares the TALYS-2.0 calculated excitation functions, particularly the GHFB results, with the available experimental datasets. Experimental measurements by Spahn et al. [15] cover the proton energy range from 4.8 to 44.9 MeV, with the maximum cross section observed near 11 MeV. Tárkányi et al. [6] extended the investigation of proton-induced reactions on ¹⁶⁹Tm within the 24.5 - 35.8 MeV energy range. Saito et al. [16] measured the excitation function up to 17.5 MeV, providing detailed low-energy data between 5.3 and 17.5 MeV that clearly describe

Figure 6. Excitation function of the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction compared with experimental data [6][15][16][43] from the EXFOR database and the TENDL-2023 evaluated nuclear data library [18].

the threshold and peak behavior. In addition, Sonnabend et al. [43] investigated the reaction in the context of astrophysical p-process nucleosynthesis, performing measurements at very low proton energies between 3.3 and 7.0 MeV corresponding to the Gamow window relevant to stellar environments.

The statistical factor analysis of the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction cross sections was performed to evaluate the predictive performance of different nuclear LDMs against available experimental datasets and the TENDL-2023 [18] evaluated library. The calculated F-, D- and R-factors are summarized in Table 4. Among the investigated models, the microscopic approaches, particularly the SHFB and SHFB(C) models, generally produce lower F,D and R values for the TENDL-2023 evaluation and the experimental datasets reported by Spahn et al. [15] and Tárkányi et al. [6], indicating comparatively better agreement with the measured cross sections in the low-energy region. In contrast, the GSM and CTM models yield lower statistical deviation factors for the experimental data of Sonnabend et al. [43], while the BFM model provides the lowest F and D values for the dataset reported by Saito et al. [16].

Table 4. Statistical factors for different LDMs of ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction.

The R-values remain close to unity for most models, suggesting that the overall normalization of the TALYS-2.0 predictions is reasonable despite variations in absolute deviations. However, the dataset of Tárkányi et al. [6] exhibits comparatively larger F and D values across all LDMs, which may reflect experimental uncertainties or limitations in reproducing the corresponding excitation functions. Overall, the analysis indicates that no single LDM provides universal agreement for all experimental datasets, although the microscopic SHFB and SHFB(C) models demonstrate the most consistent overall performance. The average values of the statistical deviation factors are shown in Figure 7.

Figure 8 illustrates the time evolution of the activity and production amount of ¹⁶⁹Yb calculated using six different nuclear LDMs. These production parameters were evaluated using the monoisotopic ¹⁶⁹Tm target geometry described in

(a)

(b) (c)

Figure 7. Average values of the statistical deviation factors: (a) F-factor, (b) D-factor, and (c) R-factor calculated for the different LDMs, experimental data, and TENDL-2023 evolution for the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction.

Section 2.2. During the 24-h irradiation period, both quantities increase continuously due to the ongoing production of ¹⁶⁹Yb. The activity exhibits an approximately linear increase and reaches its maximum value of approximately 880 GBq at EOB, after which it gradually decreases during the cooling period as a consequence of radioactive decay. The accumulated production amount follows a similar trend during irradiation, increasing rapidly and attaining its highest value at EOB. Following the cessation of irradiation, the production amount decreases only slightly over the 24 h cooling period because of the relatively long half-life of ¹⁶⁹Yb. Among the investigated LDMs, the BFM model predicts the highest activity and production amount, whereas the SHFB model yields the lowest values. Although the GHFB model predicts a larger absolute peak cross section at its maximum, the BFM model exhibits a broader excitation function profile with a significantly larger integrated low-energy cross-section area across the near-threshold domain, which ultimately drives its higher cumulative thick-target yield and activity calculations. Nevertheless, the differences among the six models remain moderate, indicating a high degree of consistency in the predicted production characteristics. The maximum production amount at EOB is approximately 3.5 × 10¹⁸ atoms for the BFM model. After EOB, a gradual decrease in production is observed, reducing the yield to approximately 3.42 × 10¹⁸ atoms during the cooling period. Overall, a moderate but consistent variation is observed among all LDMs, indicating the stability and reliability of the theoretical predictions for accelerator-based ¹⁶⁹Yb production.

Figure 8. Time-dependent activity (solid line, left axis) and production amount (dashed lines, right axis) of ¹⁶⁹Yb calculated using six nuclear LDMs as a function of irradiation + cooling time. The vertical black line indicated the EOB at 24 h, separating the irradiation and cooling periods.

Furthermore, the calculated integral yield of ¹⁶⁹Yb is shown in Figure 9 as a function of incident proton energy for a monoisotopic ¹⁶⁹Tm target. The yield increased significantly with increasing proton energy. Although the maximum cross-section of the ¹⁶⁹Tm(p, n)¹⁶⁹Yb reaction occurs near 11 MeV, the overall yield continues to rise significantly with increasing incident proton energy. This behavior reflects the cumulative nature of thick-target yields. Higher-energy beams contribute additional reaction events across the full energy range down to the threshold, leading to a steep increase in yield up to about 60 MeV and a slower rise thereafter. The curve eventually approaches a plateau at high energies, indicating fading returns due to competing reaction channels and reduced efficiency. However, from a practical medical production standpoint, selecting an optimum irradiation window requires balancing this maximum yield against strict radionuclidic purity constraints. While higher proton energies maximize the total integral yield, they simultaneously trigger competing multi-neutron emission channel, specifically ¹⁶⁹Tm(p, 2n)¹⁶⁸Yb, ¹⁶⁹Tm(p, 3n)¹⁶⁷Yb and ¹⁶⁹Tm(p, 4n)¹⁶⁶Yb. Because the co-produced ¹⁶⁸Yb contaminant is a stable isotope (Table 1), it acts as an inseparable isotopic carrier that permanently reduces the specific activity of ¹⁶⁹Yb, making impurity suppression a primary limiting constraint alongside the peak cross-section values.

Figure 9. Integral yield of ¹⁶⁹Yb as a function of incident proton energy.

For the energy interval of 15 → 8 MeV, the calculated physical thick-target yield of ¹⁶⁹Yb was found to be 0.912 MBq/µA-h. This value is approximately 41.80% higher than the experimental yield of 0.643 ± 0.050 MBq/µA-h reported by Nadi et al. [44], who produced ¹⁶⁹Yb by irradiating a Tm₂O₃ target with 15 MeV protons at a beam current of 20 μA for 20 min using the AMIRS (Cyclone-30, IBA, Belgium) cyclotron. The difference between the calculated and measured yields may arise from target composition, energy degradation within the target, irradiation geometry, beam-loss effects, and other experimental factors that are not fully accounted for in the theoretical model. Nevertheless, the reasonable agreement between the calculated and experimental yields supports the reliability of the TALYS-2.0 predictions.