A Comparison Between the Theoretical Cross Section Based on the Partial Level Density Formulae Calculated by the Exciton Model with the Experimental Data

In this paper, the theoretical cross section in pre-equilibrium nuclear reaction has been studied for the reaction Au (n, p) Hg 78 198 79 197 at energy 22.4 MeV. Ericson’s formula of partial level density PLD and their corrections (William’s correction and spin correction) have been substituted in the theoretical cross section and compared with the experimental data for Au 79 197 nucleus. It has been found that the theoretical cross section with one-component PLD from Ericson’s formula when n = 5 doesn’t agree with the experimental value and when n = 7. There is little agreement only at the high value of energy range with the experimental cross section. The theoretical cross section that depends on the one-component William's formula and on-component corrected to spin PLD formula doesn't agree with the experimental cross section. But in case of theoretical cross section based on two-component Ericson's and William's PLD formulae it has been found that there is acceptable agreement when the exciton number is taken n = 5.


Introduction:
The cross section is an important quantity in studying the nuclear reaction, where it helps to calculate the probability of nuclear reaction, therefore, it became the main concern since the beginning of nuclear reaction studies. Many models have been supposed in order to describe the cross sections theoretically, as an example the compound nucleus model for describing the emission from the nucleus in statistical equilibrium (compound nucleus) (1). Also, when the pre-equilibrium emission was suggested many models were supposed for cross section calculations one of them is the exciton model.
Many studies were made for the cross section of pre-equilibrium emission. For example, in 2007 Sharma et.al. (2) studied the preequilibrium emission mechanism of ∝-induced reactions the excitation functions for ( , ) 96  have been measured in the energy range threshold to ≅ 10 MeV using the activation technique. The measured excitation functions have also been compared with theoretical predictions based on the semi-classical code, which considers compound nucleus as well as pre-equilibrium emission.
Tatar and Tel 2010 ( 3) studied proton emission spectra produced by ( , ) reactions for some nuclear reactors and particle accelerator material 26 56 and 28 60 target nuclei have been investigated by a proton beam up to 50 MeV. In these calculations the pre-equilibrium effects have been investigated, the results are compared with the experimental data from literature. Noori et al. 2016 (4) studied the excitation functions for the reaction between deuteron and light nuclei. The following reactions 3 6 ( , ) ,

Theory
The emission cross section in preequilibrium nuclear reactions is given by ( The quantity P(n, t) represents the occupation probability of the exciton state n with excitation energy E for time t.
( , ) is the emission probability of a particle with emission energy from n excitation state in a nucleus of excitation energy E, it is given by Where the spin of the emitted particle is , is the reduced mass, ( ) is the total cross section of the inverse of the excitation channel for more details see (7), ( , ) is the level density of the excited nucleus with excitation energy E and ( − 1, ), where U is the energy of the residual nucleus.
The partial level density PLD in preequilibrium reactions ( it represents the level density of excitation of some nucleons in nuclei) is used from exciton model that was suggested by J.J. Griffin in 1966 (1 ).
Then the PLD formula for the case of the one-component ( i. e the protons and the neutrons are considered as indistinguishable particles) is (1) (8) Where n is the exciton number = + ℎ, is the particle number, h is the hole number and E is the excitation energy. G is the single particle level density in equidistant spacing model (the model considered the spaces between the levels are equal) and it is given by Where A is the mass number and d represents the spacing between the energy level d=13 MeV -1 .
Eq.3 is called one-component Ericson's formula and it represents the crude formula of level density, but if the protons and the neutrons are taken as distinguishable particles two-component formula must be used , ℎ are particles and holes numbers of protons, , ℎ are particles and holes numbers of neutrons, , are single particles state density of protons and neutrons respectively.

Pauli's correction
This correction was made by adding the factor ℎ which represents the effect of Pauli's principle(1)(8) The Ericson's formula then In case of two components the factor becomes

Spin correction
In this correction the PLD is multiplied by the angular momentum factor (1) The parameter represents the total angular momentum of the target nucleus and is the spin cut off parameter. Then the PLD formula becomes In the results a comparison will be made between the different theoretical cross section formulae based on the different PLD formulae (Ericson's formula and its corrections) with the experimental data to test how each one of them is useful in PLD calculations.

Results and Discussion:
In this section the theoretical cross section given by eq. 1 has been compared with the experimental data for 79 197 nucleus. All partial level density (PLD) formulae are substitute in eq.1 and the effect of each formula of PLD is studied by making a comparison with the experimental data taken from reference (9)   141 component Ericson's formula was used with the experimental data and the exciton number is taken = 5. One can notice that the theoretical cross section magnitude is bigger than the experimental one and the difference between them decreases with increasing the energy.  Figure 2 also demonstrates the theoretical cross section with PLD from Ericson with the experimental data but was taken equal to 7. One can find that the theoretical cross section is closer to the experimental data from the case when = 5 and applies on the experimental data with increasing the energy. The convergence and the agreement of the curve with the experimental data when = 7 can be interpreted as the exciton number = 7 may be the most probable exciton number for emission, therefore it agrees with the experimental curve. Figure 3 gives a comparison of the theoretical cross section with PLD from twocomponent Ericson's formula when = 5 and the experimental data. Good agreement is maintained between them.    In Fig.6. a comparison between theoretical cross section which depends on two-component William's formula of PLD with the experimental data is made, where the exciton number is taken = 5. There is good agreement between the theoretical and the experimental results.    Figure 8 shows a comparison between the theoretical cross section with spin correction PLD formula and the experimental data where it is noticed that the theoretical cross section for the two cases = 5 and = 7is more than the experimental cross section. The increasing in theoretical cross section comes from the multiplication by the factor RJ which represents the spin correction factor.

Conclusion:
In case of the theoretical cross section based on one-component of PLD Ericson's formula with exciton number = 7. It can be stated that it agrees with the experimental data only at the end of the energy range and it is better from the same formula when = 5. In case of the cross section based on the two-component Ericson's and William's PLD formulae when = 5, this gives the best agreement with the experimental data. The theoretical cross section that depends on the onecomponent PLD formulae that are corrected for William correction and spin correction does not agree with the Experimental cross section. Therefore, one can say the theoretical cross section based on two-component formula of PLD is the best to describe the cross section.