Calculation the Cross Sections for 6Li(α,p)9Be Reaction by Reverse Reaction

In this study light elements 6 Li , 9 Be , 10 Be for 6 Li(α , p) 9 Be reaction with proton energy from ( 27.5) MeV to ( 67.5 ) MeV with threshold energy (2.3626) MeV are used according to the available data of reaction cross sections. The Q-value is equale (2.125MeV) and parity of ( 9 Be =3/2 + ) ,(B =3 + ) and (Li = 1 + ) for the ground state .The more recent cross sections data of 6 Li(α,p) 9 Be reaction is reproduced in fine steps and by using (Matlab-7.6-2008a) program and get the equation from 3-degree for plotted. We deduced that the high probability to produced 9 Be by bombard 6 Li by alpha particle .


1-THEORY:
The cross section of compound nucleus is given by [1]:
 is the kinetic energy of an incident particle. R  is a single isolated resonance energy.
 is the width of the state. k is the wave number which is given by: Where  is the de-Broglie wavelength divided by 2π of incident particle.
 is the Plank constant divided by 2π.
p is the momentum of an incident particle.
v and m are the mass and velocity of an incident particle.
The statistical g-factors is given by [1]: Where I is the total angular momentum of the resonance (compound nucleus) which is given by: Where a S is the spin of the incident particle. X S is the spin of the target. a  is the orbital angular momentum of incident particle.
The total width of the state is the sum of the partial widths [2,3]: Where  is the lifetime of any decay state; The basic assumption of the compound nucleus model is that the compound nucleus has been formed in such a complicated set of interactions that it does not remember the initial stage of formation. The cross sections for the reaction X(a,b)Y can be split into a formation cross section of the compound nucleus [C.N.]* corresponding to the process: And the fractional probability that [C.N.]* breaks up into particles b+Y. We can therefore write [5].  Where σ(α,p) and σ(n,p) represent cross-sections of (α,p) and (p,α) reactions respectively , g is a statistical factor and  is the de-Broglie wave length divided by 2π and is given by The reactions A(α,p)B and B(p, α) can be represented with the compound nucleus C as in the following schematic diagram. It is clear that there are some important and useful relations between the kinetic energies of the proton and alpha particle . One can calculate the separation energies of α-particle (Sα) and proton (Sp) using the following relations:  In fig.(1)[7] we observed that the cross sections were smoothly decreased and the maximum cross section(0.4492mbarn) when proton energy is equal (Tp=27.5MeV).

Combining (23a) , (23b) , (24) and (25) and as the Q-value of the reaction A(α,p )B is given by
By using the compound theory we derived the mathematical formula from 9 Be(p,α) 6 Li reaction for ground state to get the cross sections of 6 Li(α,p) 9 Be reaction: We calculated the cross sections of alpha energy with energy range between (37.6533MeV) to (96.0714MeV) are (0.8452mbarn)to (0.027mbarn) respectively . These data are plotted in fig.(2) and listed in table (2) . We observed that the high probability to produced 9Be by bombard 6Li with alpha energy is (37.6533MeV) and we get semi empirical formula with three-degree as follow: y = -6.1e-010*x^{3} + 1.5e-007*x^{2} -1.3e-005*x + 0.0004 ISSN 2347-3487 552 | P a g e A p r i l 1 7 , 2 0 1 4