2.1 FKK MSC formalism

The MSC pre-equilibrium emission cross-section is given by the sum of product of the cross-section for the formation of the compound system multiplied by the sum over all stages of the probability of pre-compound emission from the N^th stage and the probability of reaching the N^th stage without pre-equilibrium emission. The double differential cross-section for pre-equilibrium emission by the MSC process is given by^[2]:

Where ${\text{C}}_{\mathcal{l}\text{sJ}}^{\text{λ}}$ is the angular momentum coupling coefficient with λ = 0, 2, 4, … to ensure the symmetric emission characteristic of quasi-equilibrium processes; and $2\text{π}\langle {\Gamma }_{1J}\rangle /\langle D_{1\text{J}}\rangle$ is the strength function for the initial reaction stage, with $\langle D_{1J}\rangle$ being the average level spacing within the class; the total width $\langle {\Gamma }_{mJ}\rangle$ is the sum of the total escape width, $\langle {\text{Γ}}_{mJ}^{\uparrow }\rangle$, and the damping width, $\langle {\text{Γ}}_{mJ}^{\downarrow }\rangle$ referring to emission and internal transitions, respectively; ${\text{ρ}}_{\text{s}}^{\text{ν}}\left(\text{U}\right)$is the density of states in each final state, v, whose excitation energy is U and P_λ(cosθ) are the Legendre polynomials of order zero. Also, l, s and J are the orbital, spin and total angular momentum quantum numbers, respectively.

The escape and damping widths are written in the factorised form as:

To calculate the X and Y functions, the density of a p-particle, h-hole configuration at an excitation energy is required. According to Ref.[35], the state density is expressed as:

where, ɡ = 6ɑ/π² is the single-particle spacing, with ɑ being the level density parameter; ∆ is the pairing energy and B is the binding energy. The step function H(E - ∆- E_pp – iB) is unity for (E - ∆- E_pp – iB) > 0, and zero otherwise. A_ph and α_ph account for Pauli blocking and are given by:

The angular momentum distribution of states at excitation energy is represented by a Gaussian function with a spin cut-off^[1]:

Thus, the total density of states of spin, J, is given by ρ(p, h, E, J) = ω(p, h, E)R_n(J).

2.2 Method of calculation

In this work, it was assumed that the target consisted of a core to which the neutron was bound in a shell model state. The projectile was captured into a state of the residual nucleus, and the neutron was ejected through a residual nucleon-nucleon interaction. The level density parameter, a and the pairing energy were obtained from the table of Ref.[35]. The cross-section for pre-equilibrium emission from each stage is proportional to the square of modulus of the radial matrix element:

where R_1,2(r) are the radial wave functions for the initial bound states, and R₃(r) and R(r) are those of the bound and unbound states, respectively. The distorted wave functions for the projectile, ejectiles and competing ejectiles were calculated using Beccheti and Greenlees (BG)^[37] and Wilmore and Hodgson (WH)^[38] optical potentials for neutrons which are applicable to A > 40 and is determined on the basis of data up to 24 MeV. For protons, the BG and Patterson et al. (P)^[39] optical potentials were used. The harmonic oscillator wave functions were used for the bound nucleons. A two-body residual interaction of zero range given by^[4] was adopted:

where V₀ is the residual interaction strength and r₀ = 1.20 fm is the Bohr radius

The GAMME code is a nuclear reactions code for calculating the MSC double differential cross-section using the FKK theory^[2]. Being made up of five major subroutines, GAMME is flexible and can be adapted for different nuclear reactions. The input file contains information on the projectile particle, target nucleus, ejectile of interest and the competing ones. On the basis of the equations and philosophy of pre-equilibrium nuclear interactions, GAMME calculated the MSC double differential cross-sections for the first three stages of the pre-equilibrium reaction, the evaporation stage and the sum of the cross-sections.

3.1 Performance of the GAMME Code

The GAMME code was written for single particle emission calculations using the MSC FKK formalism and has been successfully used for the calculation of the double differential cross-sections for the ⁵²Cr(p, n)⁵²Mn, ⁵⁶Fe(p, n)⁵⁶Co and ⁶⁰Ni(p, n)⁶⁰Cu reactions. In carrying out the calculations, the input file was used to generate the MSC double differential cross-sections for the pre-equilibrium chain –first three stages of the excitation process, the evaporation or residual stage and the sum of these cross-sections for each of the reactions. The pre-equilibrium stages after the third do not contribute appreciably to the MSC cross-section and were absorbed into the residual stage^[2] to avoid complications without an appreciable effect on the final calculated results.

The reaction cross-section contributions from the first stage were between 30% for Fe (BG optical potential parameters) and 51% for Ni (BG–WH optical potential parameters). For the second stage, the contributions were between 22% for Fe (BG optical potential parameters) and 26% for Ni (combination of BG–WH and that of WH–P). The third stage contributed between 12.5% for Cr and 15% for Ni with the same BG optical potential parameter. The evaporation stage contributed between 10% for Ni (BG–WH optical potential parameters) and 34% for Fe with the optical potential parameters of BG.

It was observed that though the contributions of the first stage to the calculated double differential cross-section are appreciable, the contributions from the second, third and the evaporation stages are also significant, at least for the FKK MSC calculations on the targets considered and at the incident energy. Therefore, the contributions must be added to that of the first stage for a better fit to the experimental data. This is depicted in Fig. 1 where sums of these contributions were in better agreement with the experimental data. This is unlike in the FKK MSD calculations where the contributions from the first stage alone is about 80% of the calculated double-differential cross-section and contributions from higher stages can be neglected with no significant effect on the results^[5].

Fig. 1Full-screen View

(Color online) FKK MSC calculations for ⁵²Cr(p, n) ⁵²Mn, ⁵⁶Fe(p, n) ⁵⁶Co and ⁶⁰Ni(p, n) ⁶⁰Cu reactions at 22.2 MeV using optical-model parameters by (a) the Beccheti and Greenlees [37], (b) Beccheti and Greenlees [37] and Wilmore and Hodgson [38], and (c) Wilmore and Hodgson [38] and Patterson et al [39] for the first three stages of the pre-equilibrium reaction, the evaporation stage and the sum of the cross-sections compared with the experimental data of Biryukov et al. [31]. The double differential cross-section (Mb·MeV⁻¹.Sr⁻¹) is plotted against the outgoing particle energy (MeV)

3.2 Comparison of the Calculated Results with the Experimental Data

The calculated and experimental double differential cross-sections data are compared in Fig. 1. In doing this, it is important to note that the experimental spectrum is a combination of four different processes: the compound nucleus (CN), MSC, direct and MSD reactions. The GAMME code only calculates the MSC contributions by considering the cross-sections at the backward angles where MSC and CN are believed to be dominant^[4]. Some degree of deviations was observed between the calculated and experimental data, especially at the lower outgoing energy regions of the spectra for the three nuclei considered. This is in agreement with the reported dominance of the compound nucleus reactions at energy regions below 10 MeV and pre-equilibrium reactions between 10 and 15 MeV^[4]. The reaction channel, CN, responsible for the underestimation at lower energies was not included in the FKK theory for which the GAMME code used in this work is based. At higher emission energies, there are significant contributions to the double differential cross sections from reactions to low energy collective excited states and/or direct nuclear reaction (DNR). Since the MSD and direct reaction channels for such contributions were not captured in the theory, it is, therefore, believed to be responsible for the level of disagreement between theory and experiment data at higher emission energies.

Figure 1 shows the consistent pattern in the shapes of the spectra between 1.0 and 9.0 MeV energy region for the three nuclei considered and for the various optical potential parameters used. This consistency could be due to the relative closeness of the atomic masses of the nuclei considered or due to the same interaction strength, V₀ used for the theoretical calculations. Between 9.0 MeV and about 13 MeV is the MSC dominant region, depicting the strong agreement between the calculated and experimental results.

Comparing the calculated and experimental results, vis-à-vis the different optical-model potentials (Fig. 1), there were no appreciable visible differences between the spectra obtained except in Fig.1(c) where the combination of W–H and P parameters were used for the ⁶⁰Ni nucleus.

3.3 Sensitivity of the Calculated Results to Optical-model Potentials

In Fig.2, only the sum of contributions from various stages was fitted to the experimental data while considering the sensitivity of the calculated double differential cross-sections to the different optical-model potential parameters. The optical-model potential parameters (OMP) were those of proton –the projectile and neutron –the emitted particle employed in the calculations.

Fig. 2Full-screen View

(Color online) FKK MSC calculations for ⁵²Cr(p, n)⁵²Mn, ⁵⁶Fe(p, n)⁵⁶Co and ⁶⁰Ni(p, n)⁶⁰Cu reactions at 22.2 MeV using optical-model parameters in Refs.[37–39] compared with the experimental data of Biryukov et al. (1979)

From Fig.2, the calculated cross-sections, reflecting the sum of the first three stages and the evaporation stage are relatively insensitive to the OMP for the three nuclei. We observed a total agreement in the spectra originating from the B–G and those of B–G and W–H OMP. Only the combination of the OMP of W–H and P showed some degree of sensitivity of the calculated double-differential cross-sections to the optical parameters. This implies that the OMP of B–G and that of W–H could be used on either chromium or iron nucleus without any significant difference in the cross-section data. The idea of combination of OMP implies different sets of OMP for the entrance and exit channels. The combination of B–G and W–H implies B–G for protons (entrance channel) and W–H for neutrons (exit channel). For W–H and P, we mean W–H for neutrons and P for protons representing the exit and entrance channels, respectively.