Evaluation of Excitation Functions of Reactions Used in Production of Some Medical Radioisotopes

In this work, reaction cross-sections were calculated and Excitation Functions were evaluated for productions of Bi, At, Po isotopes using EXIFON code in the energy range from 0 MeV to 30 MeV. The code is based on an analytical model for statistical multistep direct and multistep compound reactions (SMD/SMC model). This work also investigates the shell structure effect on the reaction cross-section, the results obtained show that the cross-sections of (a, na) reaction for both with shell correction and without shell correction are zeros at energies range considered, this shows that the energy of the incident particle is below the threshold of this reaction due to the present of coulomb repulsive force between the projectile and target nucleus.


Introduction
Studies of excitation functions of particle-induced reactions are of considerable significance for testing nuclear models as well as for practical applications, especially in cyclotron production of radioisotopes [1].Nuclear reaction in the intermediate-energy region is a matter of interest in some fields of technology and science such as reactor technology, radiation therapy in nuclear medicine, medical radionuclide production, diagnostic and therapeutic studies, Accelerator Driven Systems , fusion and fission reactor.The artificially produced radioactive isotopes are important for many different applications [2] [3] [4] [5].Radioactive isotopes play an important role in the field of medical science in terms of beneficial applications in both diagnosis and therapy purposes [6].In radioiso-tope production programmers, nuclear reactions data are mainly needed for optimization of production routes [7] [8] [9].The cross section data for different nuclide was intensively investigated and up to now, the nuclear databases are accessible online [10]- [22].
A nuclear reaction is a process that occurs when a nuclear particle (nucleon or nucleus) gets into close contact with another [23].In the general case, an arbitrary number of particles may emerge.The probability of the reaction processes as a function of the energy of the incident particle, in the energy and the direction of the outgoing particles is usually interested in the whole set of reactions.
The first two reactions (1.1) are distinguished by the fact that the projectile re-emerges after the reaction.The first of these represents elastic scattering.The second reaction represents inelastic scattering.
To calculate the reaction cross section, it is necessary to compute the number of particles that disappear from the elastic channel, what is measured by the flux of the current of probability vector through a spherical surface of large radius centered at the target [24] ( ) ( ) A radial wave function inside the nucleus should connect to the external function with a continuous function and its derivative at r R = .
The function must have identical values if calculated with the internal or the external function and this condition creates a relationship between l f and l η .
Hence, the knowledge of l f leads to the knowledge of the cross sections [24].
( ) We have an equation that is useful when we study the presence of resonances in the excitation function (cross section as a function of the energy).

Statistical multistep model
Statistical multistep models are very successful in describing nuclear reactions at energies up to about 100 MeV [25].These models enable the description of direct, pre-equilibrium, and equilibrium processes in a consistent way for a wide mass number range and various reaction channels, e.g.neutrons, protons, alpha-particles, and gamma-particles.
In the statistical multistep model, the total emission spectrum of the process ( ) , a xb is divided into three main parts [14] [26], The first term on the right hand side of Equation ( 1 The following relations between the optical model (OM) reaction cross section and the energy-integrated partial cross sections should be satisfied (at each incident energy ( ) The SMD cross section is a sum over s-step direct processes given by: [29] ( ) ( ) The SMD cross section has the form where N τ satisfies the time-integrated master equation and The multiple particle emission is expressed as: To keep the model tractable, a simple two-body interaction is assumed: [27] ( ) ( ) ( ) ( ) taken from nuclear structure considerations.
The factor ( )  contains the wave function at the nuclear radius The single-particle state density of particles , , ) is equal to the nuclear volume [26].
The single-particle state density of bound particles (at Fermi energy) is then defined by ( ) Where the factor 4 considers the spin and isospin degeneracy

Procedure
EXIFON code was used which is computer program package for computational nuclear Data physics which is based on an analytical model for statistical multistep direct and multistep compound reactions (SMD/SMC model).It predicts emission spectra, angular distributions, and activation cross sections for neutrons, protons, alpha particles, and photons.Multiple particle emissions are considered for up to three decays of the compound system.EXIFON is a fast, easy-to-handle code which predicts cross sections from one global parameter set.The only adjustable quantity is the pairing shift.The INPEXI code creates input files for EXIFON2.0 from mass and shell-correction tables.The MAKE6 code transforms EXIFON output into an ENDF-6 format file [26].It is tested and recommended code by international atomic energy agency (IAEA).
The model is based on random matrix physics with the use of the Green's function formalism [29] [30].All calculations are performed without any free parameters.Results were presented for bombarding energies below 30 MeV [31] [32].

Cross section Calculations
The program was run and the input and output directory was defined, and then the target nucleus is specified.The neutron was chooses as incident particle followed by selecting the target nucleus and excitation function in the general option section for this calculation.
The number of incident energy was specified followed by the first incident energy, and then the incident energy step is also specified.The cross section correspond to each particular energy was obtained.

Shell Structure Effects
The shell structure effects are considered in SMC processes.Under such a situation, the single-particle state density g, in Equation (1.19) is multiplied by the factors ( ) as the shell correction energy taken from tables [33] where the quantity or which denotes the excitation energy of the composite or residual systems respectively.
The calculations in this study were performed with ( ) and without ( ) shell corrections.The procedures in 2.1 and 2.2 were repeated several times and the results of cross sections were obtained.

Results and Discursions
The cross section obtained was tabulated in hot-pot Figures 1-3 and the excita-Figure 1. Cross sections at different incident energies for (a, a), (a, na), (a, ag), (a, an) reactions.In hot-pot Figure 1, One can observe that the cross sections for 208 Bi(a, na) 207 Bi and 208 Bi(a, an) 207 Bi reactions are zeros.This shows that these two reactions would not occur at the incident energy of (0 -30) MeV.Figures 5-8 are graphs of cross section against incident energy of the alpha particle, 208 Bi(a, g) 212 At is a compound nucleus formation of Astatine(212) followed by gamma emission. 208Bi(a, ng) 211 At is a knock out reaction for the nucleus formation of Astatine (211) followed by gamma emission. 208Bi(a, pg) 211 Po is also a knock out reaction for the nucleus formation of Polonium (211) followed by gamma emission. 208Bi(a, 2ng) 210 At is also a knock out reaction for the nucleus formation of Astatine(210) followed by gamma emission.All the results shows that shell structure correction does not much changes at the range of energies from (0 -30) MeV.
Figures 10-13 are graphs of cross section against incident energy of the alpha particle, 208 Bi(a, n) 211 At is a knock out reaction for the nucleus formation of Astatine (211). 208Bi(a ,2n) 210 At is also a knock out reaction for the nucleus formation of Astatine (210) 208 Bi(a, pn) 210 Po is also a knock out reaction for the nucleus formation of Polonium (210). 208Bi(a, 3n) 211 At is a knock out reaction for the nucleus formation of Astatine (209) All the results shows that shell structure correction does not much changes at the range of energies from (0 -30) MeV except for 208 Bi(a, 3n) 212 At reaction although the cross section value is very small.science in terms of beneficial applications in both diagnosis and therapy purposes.The reaction cross-sections were calculated and Excitation Functions were evaluated for productions of 208 Bi, 212,211,210 At, 211,210 Po isotopes in the energy range from 0 MeV to 30 MeV.We also investigate the shell structure effect on the reaction cross-section, the results obtained show that the cross-section of (a, na) reaction for both with shell correction and without shell correction are zeros at energies range considered, this shows that the energy of the incident particle is below the threshold of this reaction due to the present of coulomb repulsive force between the projectile and target nucleus.The application in precise evalu-ation of the Excitation Functions as well as the production of isotopes is necessary to embrace the current and future needs for medical radionuclide.

. 5 )
represents the statistical multistep direct (SMD) part which contains from single-step up to five-step contributions.The second term represents the statistical multistep compound (SMC) emission which is based on a master equation.Both terms together (SMD + SMC) represents the first-chance emission process [27] [28].The last term of Equation (1.5) represents the multiple particle emission (MPE) reaction which includes the second-chance, third-chance emissions, etc.These terms are summarized below:

≠
first-chance emission.In this context, activation cross sections are given by For example, the (n,p)-activation cross sections have the form

Figure 2 and
Figure 2 and Figure 3 are graphs of cross section against incident energy of