Study the Matrix Elements of Pre – Equilibrium Nuclear Reactions

The matrix element have the main role in the transition rates of nuclear reactions. In the present work, different formulae of matrix elements for pre – equilibrium nuclear reactions have been tested wide ranges of energy, which ranged from 0 to 200 MeV. It's found that the two recent formulae of Koning with and without asymptotic value deviate at high energy with old one that was suggested by Kalbach. The differential cross section was calculated using these formulae and comparison between them and with available experimental were made. This comparison showed a good agreement between the studied matrix formulae and the experimental formulae.


Introduction
Pre-equilibrium calculations depend on a wide set of parameters of different types, one of them is the matrix element.The emission rates in terms of a squared matrix element (|M| 2 ) have been used in many exciton model analyses.Slight variations among different formulae of matrix element give similar behavior of the squared matrix element (a decreasing function of both the excitation energy of a composite system and of its mass number).Therefore, there is no exact formulaused in calculations of a specific reaction.
In general, the matrix element is evaluated empiricallyand incorporate in the codes of the exciton model crosssection calculations.The most common form of |M| 2 is empirically in nature and is given for N-N interaction by the most popular form suggested by Kalbach [1].K is a constant has the values [2];K=100-110 MeV -3 which gives best results in the PEQAG code [2],whilein GNASH code [3],K has higher values of 130-160 MeV -3 ,Ethe total excitation energy, and A is the mass number.If excitation energy replaced by average excitation perexciton energy€, the squared matrix element written by [4]: Where E e n  and have the values between 7 and 15 MeV.
As the residual interaction inside the nucleus is not same for like and unlike nucleus, the twobody transition rates are described by the matrices where Dobes and Betak [5] used this different matrix elements depend on the nature of the interacting particles; protonproton, neutronneutron, and protonneutron (pp, nn, pn), respectively.For the purpose of model calculation, one can write it as the average 2 M by [6] : Kalbach [6] found from experimentally calculations of transition rates,with energy up to 100 MeV, the new formula of average matrix element, which was given by: Where Aa projectile mass number and the constants :: Recently, KoningandDuijvestijn [7] adopted two new formulae.These formulae wee suggestedbased on thepotential of the optical model for neutrons and protons at range of energies depend onexperimental database consists of a complete set of continuum emission spectraof reactions for incident energy between 7 and200 MeV.However, the suggested semiempirical expressions for thesquared matrix element can be given by; For asymptotic behavior.Wheren is exciton states and E tot is a composite nucleus energy.
Thesecond formula suggested without asymptoticbeaviorfor energy range 10-60 MeV, and can be given as:

Results and Discussions
In the exciton model, the square matrix element of the residual interaction that depending on the energy and mass number, is used to be regarded as effective as a single parameter for the residual two-body interactions responsible for energy equilibration in the composite nucleus.
These have been determined phenomenologically by some theoretical models used for  5) and ( 6) a comparison made these equations at 120 MeV as in fig ( 1).This is repaired by the asymptotic value in eq. ( 4).The explanation for the difference is that

Conclusions
The analytical solution of transition rates needs a value for the squared matrix elements and different formula have been used at different energies to study its effective on the transitions rats.The differential cross section was calculated using these formulae :900, respectively.The single-particle state density can be found from[5] as; the present study, different formulae of the squared matrix element have been studied, and the comparison between the old and recent formulae with energy dependence.Further, The effects of these formulae of squared matrix element on the cross section of 56 Fe(p,α) reaction at 120MeV have been illustrated.

Vol: 13
No:3 , July 2017 DOI : http://dx.doi.org/10.24237/djps.1303.281AP-ISSN: 2222-8373 E-ISSN: 2518-9255 extending it to incident ( or excitation ) energies outside the range studied.The values of the mean square matrix elements may recover for approximations in other parts of calculation and coupled to the values of other model parameters.Using eq's (3), (

Fig 1 .
Fig 1.The squared matrix element M 2 comparison for exciton number n=3 using different formulafor nucleus 56 Fe.

Fig ( 2 )
Fig (2) 56 Fe ( p,α ) reaction cross sections using different matrix elements in calculation at 120 MeV.Experimental data taken from