Calculation of Astrophysical S-factor and Thermonuclear Reaction Rates for (p,n) Medium Elements Reactions

The cross-sections of (p,n) medium elements reactions as a function of proton energies such as 45Sc(p,n)45Ti, 48Ti(p,n)48V, 51V(p,n)51Cr, 52Cr(p,n)52Mn, 55Mn(p,n)55Fe, 56Fe(p,n)56Co, 59Co(p,n)59Ni, 62Ni(p,n)62Cu, 63Cu(p,n)63Zn, and 66Zn(p,n)66Ga have been interpolated near threshold up to 10 MeV in step of 0.05 MeV using MATLAB program. Weighted averages of cross-sections have been used to calculate the astrophysical S-factor and thermonuclear reaction rates as a function of the center of mass energy, Ec.m. and T9 respectively. Polynomial expressions have been used to fit the calculated astrophysical S-factor and thermonuclear reaction rates to determine the astrophysical S-factor at various Ec.m. and thermonuclear reaction rates at various T9 from best fitting equations with minimum Chi-Square. Empirical formulae of reactions set 45Sc(p,n)45Ti, 48Ti(p,n)48V, 55Mn(p,n)55Fe, 59Co(p,n)59Ni, 66Zn(p,n)66Ga and reactions set 48Ti(p,n)48V, 51V(p,n)51Cr, 59Co(p,n)59Ni, 63Cu(p,n)63Zn, 66Zn(p,n)66Ga have been used to calculate the astrophysical S-factor as a function of Ec.m. and Z and thermonuclear reaction rates as a function of T9 and Z of target nucleus. The results have been compared with the adopted data that have been calculated from the fitting equations which have a good agreement.


Theory
Atomic masses of each medium elements and isotopes related this work has been taken from the nuclear wallet cards published by the National Nuclear Data Center (NNDC) [32]. The Q -value of the reaction ‫,(ܺ‬ ݊)ܻ, is defined as the difference between the initial and the final rest mass energies [33]: Where ( ‫ܯ‬ , ‫ܯ‬ , ‫ܯ‬ , ܽ݊݀ ‫ܯ‬ ) are the atomic masses of the incident, target particles, product nucleus and neutron (outgoing particle), respectively and (ܿ ଶ = 931.494013 MeV/u; where u=atomic mass unit (amu) =1.66×10 -27 kg). From conservation law of energy [33]: ‫ܯ‬ ܿ ଶ + ܶ + ‫ܯ‬ ௫ ܿ ଶ = ‫ܯ‬ ܿ ଶ + ܶ + ‫ܯ‬ ܿ ଶ + ܶ (2) Where ܶ , ܶ , and ܶ are the proton, neutron, and heavy product kinetic energies. In the laboratory system conservations of energy and momentum lead to the following equation [33]: This is called the Q-value equation. If Q is positive, the reaction is said to be exoergic; if Q is negative, it is endoergic. The amount of energy needed for an endoergic reaction is called the threshold energy and can be calculated easily [34].
Where ܼ ଵ and ܼ ଵ are the charges of the projectile and target nuclei, and ‫ݎ‬ and ‫ݎ(‬ = ‫ݎ‬ ଵ ‫ݎ+‬ ଶ ) is their separation, ݁ is the charge of electron (݁ ଶ = 1.44 ‫ܸ݁ܯ‬ ݂݉), and the radius of the nucleus is given by ‫ݎ‬ = 1.3 × 10 ିଵଷ ‫ܣ‬ ଵ/ଷ cm, where ‫ܣ‬ is the mass number (atomic weight) [35]. Then Eq. (5) leads to Where ‫ܧ‬ is the coulomb barrier or coulomb energy in ‫,ܸ݁ܯ‬ ‫ܣ‬ ଵ ଵ/ଷ ܽ݊݀ ‫ܣ‬ ଶ ଵ/ଷ are the mass numbers of the charges of projectile and target nuclei respectively. The data of cross-sections of nonresonant reactions, exhibit a dramatic decrease at low energies due to quantum tunneling, as reflected in the energy dependence of the transmission coefficient through the Coulomb barrier [2]: The astrophysical S-factor, S(E) in unit ‫)ܾ-ܸ݁ܯ(‬ is related to the cross-section by [36]: Where ‫ܧ‬ ‫ݏ݅‬ the energy in the center of mass system ‫ܧ(‬ .. ) in ‫,ܸ݁ܯ‬ ‫)ܧ(ߪ‬ is the cross section of the reaction in (mb), 2ߨߟ is the Gamow factor, and ߟ is Sommerfeld parameter [37]: The reduced mass μ in u (amu) is determined by the equation [38]: Where ݉ ଵ and ݉ ଶ represents the masses of the projectile and target nucleus in units of (amu), respectively. The energy of a pair of particles in their center of mass ‫ܧ‬ .. is related to the laboratory energy, ‫ܧ‬ . of the incident particle by the relationship [33]: The Gamow energy ‫ܧ‬ ீ , in ‫ܸ݁ܯ‬ [39]: Where ߙ = ଵ ଵଷ = మ ℏ is the fine-structure constant.
The thermonuclear reaction rates, ܰ ‫〉ݒߪ〈‬ in unit (ܿ݉ ଷ ‫݈݉‬ ିଵ ‫ݏ‬ ିଵ ) [37]: Where ܰ is the Avogadro's number ( 6.022 × 10 ଶଷ ‫݈݉‬ ିଵ ), ݇ is the Boltzmann's constants (1.38 × 10 ିଵ ‫,)ܭ/݃ݎ݁‬ and ܶ ‫ݏ݅‬ ‫ݐ‬ℎ݁ ‫݁ݎݑݐܽݎ݁݉݁ݐ‬ respectively. Eq. (13) leads to [37]: Where ܶ ଽ is the temperature in units of 10 ଽ ‫ܭ‬ ( ܶ ଽ = 10 ିଽ ܶ ) The weighted averages of the cross sections of light elements whose cross sections ߪ (ܾ݉) and the uncertainty (errors) ∆ߪ (ܾ݉) are expressed by the following Eqs. [40]: Where ߪ is the cross section of ݅ ௧ reference, and ߜ (∆ߪ ) is the errors corresponding to each values of ߪ , The type of formalism has been considered in the present work is the polynomial fit expression of the form:  (18) have been considered in this work. Subsequently, by combining the above equations (17) & (18), the following expression has been obtained: Where Y=ln[S(E)] or ln[NA<σv>], (i=0,1,2,…M), (j=0,1,2,…N), (C00,C01,C02,….) are free parameters (coefficients of polynomials), K is the center of mass energy or T9 according to the S(E) or NA<σv>, and X is atomic number Z. The Excel computer program has been used to obtain the best fit formulae corresponding to different energies ranges from the threshold up to 10 ‫ܸ݁ܯ‬ in the center of mass system or T9 ranges from 1 to 10 10 9 K. The data of these ranges were excluded in each step, till an acceptable value of the coefficient of determination ܴ ଶ ≈ 1 was reached. The best fit adopted data was obtained with increasing order to provide the minimum value of Chi-Square (߯ ଶ ) by using the Eq. [41]: Where ܰ is the data points number, ‫ܯ‬ is the fitting coefficients number, ܻ ௫

Data Reduction and Analysis
The Atomic masses have been taken into consideration to determine the Q-Value, threshold energy, Coulomb barrier, reduced mass, and the ratio between (Ec.m./Elab.) of ‫,(‬ ݊) medium elements reactions using the Eqs. (1, 4, 6, 10, and 11) respectively; the results are shown in the table (1). Eqs. ( 8,9,12, and 7 ) taken into consideration to determine the Sommerfeld parameter(η), Gamow factor G(E), Gamow energy (EG), and the astrophysical S-factor, S(E) of the (p,n) medium elements reactions. The results are shown in table (2).

Astrophysical S-factor Empirical Formulae
The adopted astrophysical S-factor has been used to obtain the fitting parameters by using the polynomial expressions (18), (20) and (21) by the following step: 1. The polynomial expressions which are used in eq. (21) to fit the calculated natural logarithm of astrophysical S-factor, S(E) of the studied medium elements to determine the adopted natural logarithm of astrophysical S-factor from the best fitting with minimum Chi-Square using Eq. (20). The obtained best fitting Equations of the mentioned reactions were presented in equations (   2-The adopted astrophysical S-factor, S(E) has been used as a function of target atomic number Z at the fixed center of mass energies using the Excel computer program to get the fitting expressions and then used to calculate the fitting parameters. The obtained results are presented in Table 3. 3-The obtained free parameters Ci (C0, C1, and C2), as shown in Table (3) are plotted against with the prefixed values of center of mass energies from 6 to 10 MeV in step of 0.25 MeV as shown in Fig(2), and then the obtained free parameters Ci have been fitted to the following polynomial expression: Where (C00, C01, C02, C10, C11…. C22) are free parameters and their values are shown in the below:

Thermonuclear Reaction Rates Empirical Formulae
The adopted thermonuclear reaction rates NA<σv> have been used to obtain the fitting parameter by using the polynomial expressions (18), (20) and (21) by the following steps: 1-The polynomial expressions were used in eq. (21) to fit the calculated natural logarithm of thermonuclear reaction rates NA<σv> of the studied medium elements to determine the adopted natural logarithm of thermonuclear reaction rates NA<σv> from the best fitting with minimum Chi-Square using Eq. (20). The obtained best fitting equations of the mentioned reactions are present in equations (41, Table 4. Comparison between polynomial fitting expressions (Best Fitting) of the adopted astrophysical S-factor with those calculated from Eqs. (36) to (40).  The empirical formulae relating to the thermonuclear reaction rates NA<σv> (cm 3 s -1 mol -1 ) with both T9 and Z were performed as follows: 1-At fixed values of the T9 from 6 to 10 10 9 K in steps of 0.25 10 9 K for the 48 Ti(p,n) 48 V, 51 V(p,n) 51 Cr, 59 Co(p,n) 59 Ni, 63 Cu(p,n) 63 Zn, and 66 Zn(p,n) 66 Ga reactions, the natural logarithm of the thermonuclear reaction rates will vary with the atomic number Z this shown in Fig. (3). The data have been fitted to the polynomial expression as the same as Eq. (32)  2-The adopted thermonuclear reaction rates have been used as a function of Z at fixed T9 using the Excel computer program to obtain the fitting expressions and then used to calculate the fitting parameters. The obtained results are presented in Table (5). 3-The obtained free parameters Ci (C0, C1, and C2), as shown in Table (5) are plotted against with the prefixed values of T9 from 6 to 10 10 9 K in steps of 0.25 10 9 K as shown in Fig.(4),and then the obtained free parameters Ci have been fitted to the polynomial expression:    5. Conclusions 1-The astrophysical S-factor, S(E), was starting with increase and then decrease irregularly (fluctuate) by increasing the center of mass energy, this because of Coulomb barrier penetration exp(2πη). 2-The astrophysical S-factor increased with increasing atomic number Z of target nuclei at fixed center of mass energy. 3-The thermonuclear reaction rates, NA<σv>, were increased with increasing T9 because by increasing the T9 the Charged interacting particles need to overcome the existing Coulomb barrier. 4-The thermonuclear reaction rates decreased with increasing atomic number Z of target nuclei at fixed T9 because as Z increase Coulomb barrier increased. 5-The astrophysical S-factor and Thermonuclear reaction rates calculated in the present work are in good agreement with those measured previously by other works.