One of the most important branches of physics is nuclear physics, the study of the interactions of atomic nuclei and their structure (Feshbach et al. 1980). It allows us to learn about the creation of the cosmos around us and find some important applications that benefit all of humanity. One of the four fundamental forces in the universe is nuclear force, the primary controller of all elements in nature (Voinov et al. 2010). In general, any nuclear reaction is represented as:

X + a → [ C ∗ ] → Y + b + … … . + Q (1)

or X(a, b)Y, where the projectile a hits the target X, C* is an excited nucleus that can produce a new nucleus Y and a new particle b. This nuclear reaction is associated with the released or absorbed energy; Q-value is the energy absorbed, which is negative for endoergic reactions and positive for exoergic reactions (Grover and Gilat 1967). Proton (p), neutron (n), triton (t), deuteron (d), and α particle represent the emitted particles that are released from nuclear reactions or gamma ray (Garcon and Van Orden 2001). Research in nuclear reactions and structure led to the discovery of nuclear forces and their nature and provided knowledge of important and useful applications in medicine, energy, archeology, geology, and age determination(Goldstein et al. 2006). In general, a single free parameter can represent the angle, energy distribution, and strength of the actual interaction between the target nucleus and the projectile (Sarpün et al. 2024). Feshbach-Kerman-Koonin (FKK) includes compound nucleus (CN), multistep direct (MSD), collective (COLL), and multistep compound (MSC) processes (Hodgson 1971). This theory recognizes two different interactions. As the energy rises, the interactions between the P-chains become more and more important until they take over the reaction. The outcome in forward-directed multiple-step angular distributions and highly energetic tails in the emission spectra (Cerizza 2015). At low energies, P-chain dominance is significant, and the cross-section is roughly (90°) symmetric, which is typical of compound nucleus emission (Herman et al. 1992). P-chain emission (MSD) is known as a multistep direct process; MSC represents the term used for Q-chain (Bonetti et al. 1994). Nucleon cross-sections from 10 MeV to 100 MeV can be calculated using the FKK theory of MS reactions (Herman et al. 1992; Sahan et al. 2018). The FKK hypothesis states that a nuclear reaction occurs in a succession of steps corresponding to the target nucleus’s interaction with the incident particle (Abd alsatar et al. 2021; Xu 2022). The pre-equilibrium reactions at each stage cause nucleons to be stimulated to higher states and possibly emit them (Shafik et al. 2014; Akram and Hajer 2022). This study aimed to infer the spectrometry of angular distribution and the cross-section of (neutron, neutron), (neutron, and proton) reactions for the nuclei of silver when the induced neutrons are ejected with an incident energy under 20 MeV. The angular distributions and energy spectra were calculated during the pre-equilibrium stage of the reaction for light ejectiles (neutrons and protons) emitted from ¹⁰⁷Ag in (n, n), (n, p) reactions at incident energies of 3, 5, 10, and 15 MeV of neutrons using FKK theory with Kalbach systematics (Akram and Hajer 2022). implemented in PRECO-06. MSD and MSC contributions to these cross-sections in the pre-equilibrium region were separated and analyzed. Model validity and data needs were assessed by comparing the behavior for neutrons versus protons and identifying that (n, n) and (n, p) cross-sections on ¹⁰⁷Ag “require further investigation” and “more precise experimental data” to establish clear trends.

The energy spectra of the nuclear reactions caused by nucleons, employing ¹⁰⁷Ag as the target nucleus, and occurring in a pre-equilibrium region, were calculated using various approaches. Primary and secondary nucleon emissions were included in the pre-equilibrium process, which had two components and formed the exciton model. If a s_bis the b particle spin and μ_b is the reduced mass, σ_b is the inverse of the exit channel process total cross-section (i.e., absorption of b-particle on the residual nucleus), and U excitation energy. The emission probability of a b-particle within emission energy ε_b from n exciton state in a nucleus of excitation energy E is given by Eq. (2), (Kurtz et al. 2009):

weff(pπ−Zb,hδ,pv−Nb,hv,U)=∑i=iminhπ∑j=jminhvw(pπ−Zb,i,pv−Nb,j,U) (2)

Equation (3) represents the total residual state density:

weff(pπ−Zb,hδ,pv−Nb,hv,U)=∑i=iminhπ∑j=jminhvw(pπ−Zb,i,pv−Nb,j,U) (3)

Equation 3 is used when the nucleus returns from an excited state to a stable state (Hodgson 1971; Sarpün et al. 2024). Many compound combinations are included in the total residual state density where high exciton numbers (n = p + h), pairing of particle-hole (p-h), such as 2p-2h and 3p-3h, and complex particle-hole excitations appear as configurations where a large number of nucleons are excited from the Fermi sea (holes) into unoccupied energy levels (particles), resulting from multiple nucleon-nucleon scattering interactions within the nucleus. The isospin T value in the residual nucleus and C_b (T, T_B) in the escape channel is the Clebsch-Gordan isospin coupling vector, ε emission energy, Z_b, N_b is the emitted particle’s proton and neutron numbers, and μ_b the reduced mass. The residual excitation energy will be U=E−ε-B_b, as B_b is the binding energy of emitted particles. The isospin efficiency refers to the effectiveness, accuracy, or computational speed of employing the isospin symmetry formalism within nuclear physics calculations. The spin-dependent formulation and total range of energy in the pre-equilibrium model of b-particle at the center-mass system are given by Eq. 4 (Herman et al. 1992; Cerizza 2015; Sarpün et al. 2024):

[dσa,b(ε)dεa]pre=∑T[dσa,b(ε,T)dεa]pre=σa(εa)∑T"Ca(T,TA)"2×∑p∑pπSpre(p,pπ,T)Wb(p,pπ,E,ε,T) (4)

where σ_a.pre models the complex nucleus, which is reduced with direct reaction by a cross-section, and (P, p_π, T) is the average amount of time spent in each class of configuration (Hodgson 1971). The entry channel isospin has Clebsch-Gordan coupling coefficient multiplied by the product of Eq. (2) for each spin (T). Because a second particle can be released at the pre-equilibrium position, the release may alter the energy spectra at high-energy excitation levels (Bonetti et al. 1994). Pre-equilibrium exciton components are a part of the MSD, or forward-peaked component in FKK theory. Cross-sections of nucleons, knockout, and inelastic scattering (IN) involving cluster degrees of freedom are represented by Eqs. (5 and 6):

⌊dσεσ⌋MSD=⌊dσεb⌋pre,1+⌊dσεb⌋pre,2+⌊dσεb⌋NT+⌊dσεb⌋IN (5)

⌊dσεσ⌋MSD=⌊dσεb⌋pre,1+⌊dσεb⌋pre,2+⌊dσεb⌋NT+⌊dσεb⌋KO (6)

where KO only contributes for reactions (C, α), (N, α), and (C, N); N is a nucleon; and C is a complex particle (D, T, ³He, or α particle). Only the primary and secondary evaporation cross-sections are given because they include the associated equilibrium or symmetric component. The MSC spectrum is shown as (Cerizza 2015).

For 47107Ag, the PRECO-06 code update was used within FKK theory to obtain the emission spectra, angular distribution, and data of each particle emitted. First, the binding energies of every emitted particle were included in the code. Linking energies were then applied. Table 1 represents the global parameters used in the code.

The FKK model was applied to (n, x) on the ¹⁰⁷Ag isotope at incident energies of 3, 5, 10, and 15 MeV. In general, the results are clear only when the emitted particle is a neutron or a proton because neutrons and protons are light particles (Mahdi et al. 2014; Scholz et al. 2014). By contrast, the results are not clear when the emitted particle is heavy, such as (²D, ³T, ³He, and α particles), because of its weight. When emitted, heavy particles suffer from many collisions and leave large gaps in the parent nucleus (Jasim 2015). When the energy of the incident particle is small, the target nucleus is excited until it reaches a pre-equilibrium state. Here, the target nucleus begins to release light particles such as neutrons and protons, and part of the energy returns to a stable state (ground state). Each particle (p) that is emitted leaves a hole (h) in its place, forming a pair (p-h). In this case, the results are clear. The incident energy is not sufficient to emit heavy particles; therefore, the theoretical results are not clear. When the incident energy of the particle increases, the energy becomes sufficient to emit heavy particles, leaving behind pairs of (p-h) in the target nucleus. Therefore, the theoretical results in the PRECO-06 program are clear. The results begin to appear clear at 20 MeV as incident energy. This incident energy is sufficient to emit heavy particles, leaving behind pairs of (p-h) in the target nucleus, such as (n, D), (n, T), (n, ³He), and (n, ⁴He).

The incident neutron energy of 3 MeV is presented in Fig. 1. The findings are indicated for a wide variety of neutron emission energies. The peak across sections is 850 mb/MeV at emission energy of 4 MeV, showing that the direct knock component is fully involved, which means it has the biggest effect on the cross-section. The other contributions (direct, knockout, exciton primary, exciton secondary, equilibrium primary, and equilibrium secondary components) are extremely small, approaching zero, and therefore can be neglected. The angular distribution for the same nuclear reaction with different angles (from 0° to 180°) was determined to obtain the total contributions for a wide variety of neutron emission energies. Peaks of angular distribution values appear at certain emission energies where the largest angular distribution or differential cross-section is 1310 mb/sr. MeV for θ=0° participation at the energy emission 4 MeV, but is comparable to 1000 mb/sr.MeV for θ=10° participation at the energy emission (4MeV), and comparable to 850 mb/sr.MeV for total participation. However, there are tiny amounts of angular distribution of 463 mb/sr.MeV at 20° participation and 156 mb/sr.MeV at 30° participation and 42.2 mb/sr. MeV at 60° participation and very small participation is almost negligible at 90, 120, and 180 degrees. All these values can be noted as shown in Fig. 1, below, which shows all cross-section and angular distribution contributions of reaction ¹⁰⁷Ag(n, n) ¹⁰⁷Ag for the incident neutron energy 3 MeV.

At the incident neutron energy of 5 MeV, the findings indicate a wide variety of neutron emission energies. The cross-sectional peak has a value of 638 mb/MeV at the emission energy of 5.5 MeV, which represents the total participation of direct knock components, indicative of the greatest contribution to the cross-section. The contributions of other components are minuscule, nearly negligible, and therefore may be disregarded.

The angular distribution and contributions for the same energy of 5 MeV were calculated for angles of 0°–180°. The differential cross-section has peaks at certain neutron emission energies, which are within a wide range of energies. The biggest angular distribution or differential cross-section is 1299 mb/sr.MeV for θ=0° contribution at 5.5 MeV. This is similar to 974 mb/sr.MeV for θ=10° contribution at 5.5 MeV and 649 mb/sr.MeV for the total contribution. Small values of differential cross-section are obtained at 20° contribution (440 mb/sr.MeV), 30° contribution (135 mb/sr.MeV), and 60° contribution (22.5 mb/sr.MeV). The contributions are almost negligible at θ=90°, 120°, 180°. All these values are shown in Fig. 2, above which are all cross-sections and angular distribution contributions of reaction ¹⁰⁷Ag(n, n) ¹⁰⁷Ag at the incident neutron energy of 5 MeV.

At the incident neutron energy of 10 MeV, the findings indicate a wide variety of neutron emission energies. The cross-sectional peak has a value of 430 mb/MeV at the emission energy of 9.91 MeV. This peak represents the total participation of direct knock components, which are indicative of the greatest contribution to the cross-section. The remaining contributions, such as those from direct natural, exciton primary, exciton secondary, equilibrium primary, and equilibrium secondary, are minuscule, nearly negligible, and therefore may be disregarded. The angular distribution and total contributions for the same reaction at the neutron energy (10 MeV) and angles of 0°–180°. The results show a wide of neutron emission energies, with peaks in the angular distribution appearing at specific energies. The largest angular distribution is 1032 mb/sr.MeV for θ=0° participation at the emission energy of 10.3 MeV. This value is comparable with the angular distribution of 775 mb/sr.MeV for θ=10° participation at the emission energy of 15.2 MeV and the total participation of 449 mb/sr.MeV. Small amounts of angular distribution are recorded at 20° participation (340 mb/sr.MeV), 30° participation (95 mb/sr.MeV), and 60° participation (10.7 mb/sr.MeV). Participation is almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 3, which also presents all cross-sections and angular distributions of reaction ¹⁰⁷Ag(n, n) ¹⁰⁷Ag at an incident neutron energy of 10 MeV.

At the incident neutron energy of 15 MeV, the findings indicate a wide variety of neutron emission energies. The cross-sectional peak has a value of 321 mb/MeV at an emission energy of 14.86 MeV. This peak represents the total participation of direct knock components, which are indicative of the greatest contribution to the cross-section. The remaining contributions, such as those from direct natural, exciton primary, exciton secondary, equilibrium primary, and equilibrium secondary, are minuscule, nearly negligible, and may be disregarded. The angular distribution and total contributions for the same nuclear reaction were also calculated at angles of 0°–180°. The results show that within a wide variety of neutron emission energies, peaks of angular distribution appear at certain emission energies.

The largest angular distribution or differential cross-section at θ=0° is 832 mb/sr.MeV for participation at the emission energy of 15.2 MeV. This value is comparable to the 622 mb/sr.MeV for θ=10° of 15.2 MeV and the obtained total participation of 323 mb/sr.MeV. Small amounts of angular distribution are observed at 20° participation (266 mb/sr.MeV), 30° participation (71.9 mb/sr.MeV), and 60° participation (7 mb/sr.MeV). Participations are almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 4, which also presents all cross-sections and angular distributions of reaction ¹⁰⁷Ag(n, n) ¹⁰⁷Ag at 15 MeV.

When protons are the emission particles in a nuclear reaction, ¹⁰⁷Ag(n, p)¹⁰⁷Pd, it shows a radical difference from nuclear neutron reactions. At 3 MeV, the findings indicate a wide variety of proton emission energies. The cross-sectional peak has a value of 285 mb/MeV at the emission energy of 6.1 MeV, which represents the total participation of the equilibrium primary component, that indicative by the greatest contribution to the cross-section. The contributions of direct natural, exciton primary, are minuscule, nearly negligible, and thus may be disregarded. The angular distribution and total contributions were calculated for 3 MeV and angles of 0°–180°. The results show that within a wide variety of proton emission energies, peaks of angular distribution appear at certain emission energies. The largest angular distribution or differential cross-section is 284 mb/sr.MeV for total participation at the emission energy of 6 MeV. The value is 25.3 mb/sr.MeV for θ=0° participation and 25.5 mb/sr.MeV for θ=10° participation at the emission energy of 6 MeV. Small amounts of angular distribution are observed at 20° participation (25 mb/sr.MeV), 30° participation (24.8 mb/sr.MeV), and 60° participation (22.4 mb/sr.MeV). Participation is almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 5, which also presents all cross-sections and angular distributions of reaction ¹⁰⁷Ag(n, p)¹⁰⁷Pd at an incident energy of 3 MeV.

At an energy of 5 MeV, a wide variety of proton emission energies were produced. The cross-sectional peak has a value of 280 mb/MeV at the emission energy of 6.1 MeV, which represents the greatest contribution to the cross-section of the equilibrium primary component. The other contributions are minuscule and can be negligible and disregarded. The angular distribution and total contribution for angles of 0°–180°, the results show peaks of angular distribution at certain emission energies for a wide variety of proton emission energies. The largest angular distribution or differential cross-section is 279 mb/sr.MeV for total participation at the emission energy of 6.1 MeV. The value is 24.9 mb/sr.MeV for θ=0° participation and 24.8 mb/sr.MeV for θ=10° participation at the emission energy of 6 MeV. Small amounts of angular distribution are observed at 20° participation (24.5 mb/sr.MeV), 30° participation (23.9 mb/sr.MeV), and 60° participation (22 mb/sr.MeV). Participation is almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 6, which also presents all contributions of reaction ¹⁰⁷Ag(n, p) ¹⁰⁷Pd within 5 MeV.

At the incident neutron energy of 10 MeV, the findings indicate a wide variety of proton emission energies; the cross-sectional peak has a value of 171 mb/MeV at the energy of emission 6.94 MeV. This peak represents the total participation of the equilibrium primary component, which is indicative of the greatest contribution to the cross-section. The remaining contributions from direct natural and exciton primaries are minuscule, nearly negligible, and thus may be disregarded. The angular distribution and total contribution for 0°–180°, a wide variety of proton emission energies can be seen. The largest angular distribution or differential cross-section is 171.9 mb/sr.MeV for total participation at the emission of 7.2 MeV. The value is 16.2 mb/sr.MeV for θ=0° participation and 15.9 mb/sr.MeV) for θ=10° participation at the emission energy of 7.2 MeV. Small amounts of angular distribution are observed at 20° participation (15.9 mb/sr.MeV), 30° participation (15.4 mb/sr.MeV), and 60° participation (13.9 mb/sr.MeV). Participation is almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 7, which also presents all cross-sections and angular distributions of reaction ¹⁰⁷Ag(n, p) ¹⁰⁷Pd with energy of 10 MeV.

At the incident neutron energy of 15 MeV, the findings indicate a wide variety of proton emission energies. The cross-sectional peak has a value of 106 mb/MeV at the emission energy of 6.94 MeV as a total participation of the equilibrium primary component, which indicates the greatest cross-section, and can be negligible. The angular distribution and total contribution at angles of 0°–180° were calculated, where the results show that within a wide variety of proton emission energies, peaks of angular distribution appear at certain emission energies. The largest angular distribution is 106 mb/sr.MeV for total participation at the energy of 7 MeV. The value is 10.4 mb/sr.MeV for θ=0° participation and 10.3 mb/sr.MeV for θ=10° participation at the emission energy of 7 MeV. Small angular distribution is observed at 20° participation (10.1 mb/sr.MeV), 30° participation (9.83 mb/sr.MeV), and 60° participation (8.65 mb/sr.MeV). Participation is almost negligible at 90°, 120°, and 180°. All these values are shown in Fig. 8 of reaction ¹⁰⁷Ag(n, p) ¹⁰⁷Pd at 15 MeV.

We note that the cross-sectional values decrease as the emission energy of the particles emerging from the nuclear reaction (nucleons) increases (Kavun and Makwana 2020). Since the neutron carries no charge, it can completely and freely approach the ¹⁰⁷Ag nuclei at close distances and interact with them without suffering (Šalamon et al. 2019). However, the situation changes when the particles emitted from the nuclear reaction are protons. In this case, the cross-section is smaller than that of the response when the emitted particle is a neutron at the same incident energies of the neutrons. The reason is that protons carry positive charges and suffer from Coulomb forces when emitted from the nucleus (Ahmed et al. 2024). The Coulomb barrier has a strong effect on the emitted proton when ¹⁰⁷Ag+n→¹⁰⁷Pd+p occurs. If proton energy is high, then the proton easily overcomes the Coulomb barrier and exits the nucleus; if proton energy is low, then the proton has great difficulty getting out and relies on quantum tunneling. This finding explains why (n, p) interactions require a certain energy threshold for the incident neutron, not because it enters the nucleus, but to provide enough energy for the proton to exit, overcoming the Coulomb barrier. Table 2 shows the difference between the cross-sections and incident energies of neutrons and protons emitted from the nucleus (Abd alsatar et al. 2020). The values of the cross-sections of neutrons and protons decrease with the increase in the incident energies of neutrons, which is a common characteristic. Nevertheless, the probability of the nuclear reaction of neutrons is greater than that of protons at the same incident energies of neutrons.

The energies of neutron emission increase as the energy of the neutrons falling on the nucleus increases. Meanwhile, the energy of proton emission is almost constant for the same energy of the neutrons falling. For this finding, we return to the previous reason, that is, a nucleus of an atom is a self-organizing system made up of protons and neutrons that are held together by the short-range attractive nuclear force (Bertulani and Kajino 2016). As in Table 2, the emission energy of protons increases with the fall energy of neutrons and does not remain constant. We note that the proton emission energy is 6.1 MeV at the neutron incident energy of 3 and 5 MeV. When the incident energy is 10 MeV, the proton emission energy increases to 7.2 MeV.

Protons are affected by Coulomb repulsion, which is long-range, while no Coulomb interaction exists between neutrons (Yakubu 2025). Given that protons are positively charged particles, the Coulomb force is long-range and decreases with distance as 1/r², but never fully disappears. Therefore, protons repel from other protons even across the entire nucleus. In a large nucleus with many protons, this cumulative repulsion becomes significant. Given that Coulomb’s law only applies to charged particles, neutrons do not experience any electromagnetic repulsion or attraction with each other. The nuclear (strong) force connects the nucleons (protons and neutrons) but only at short distances (~10^-15 m). At these close ranges, the strong force overwhelms Coulomb repulsion, which is why nuclei can exist despite proton-proton repulsion. For small nuclei, the strong force easily dominates. For large nuclei (heavy), the long-range Coulomb repulsion accumulates across many protons, eventually destabilizing the nucleus. This phenomenon explains why heavy nuclei tend to be radioactive and undergo fission.

The angular distributions for neutrons and protons decrease with the increase in the energies of emitting neutrons. The angular distribution of the emitted neutrons is large when the angle of emission is zero, which is greater than the sum of the angular distributions of the rest of the angles. This phenomenon is attributed to the lack of charge of the neutrons and their arrival at places close to the nucleus so that their collision is vertical with the nucleus (Capel 2019). This characteristic is not observed in the angular distributions of protons emitted from the nuclear reaction and of the same energies as neutron emission because of the positive charge of protons and their exposure to the strong Coulomb forces surrounding the nucleus.

The d²σ/dΩ.dE of protons and neutrons during neutron-induced reactions was computed for nuclei ¹⁰⁷Ag with an incidence energy less than 20 MeV. Calculations were performed in the pre-equilibrium nuclear reaction zone using Kalbach systematics. Compared with Kalbach for choosing the nucleus, we discovered that the model’s cross-section was within a narrow range. Given that the exciton model was successfully applied to a large number of experimental data, the PRECO-6 analysis conducted during the pre-equilibrium period yields a modest exciton number. However, ambiguity remains in the way PRECO-6 formulates the composite particle emission to explain it. For example, how different shell-model type correlations impact the ground state strength in the ¹⁰⁷Ag scenario and how the transition from the closed to the open shell affects it remains unclear. FKK theory has been used in the framework of theoretical calculations performed by the PRECO-6 computer code.