Analytical Studying of Positron Annihilation Equations: دراسة تحليليّة لمعادلات الفناء البوزيتروني

In this study examined the changes in some materials by studying different types of positron annihilations, such as free annihilation, bound annihilation, para and ortho positronium annihilation, and found that the o- Ps annihilation passes through a free- annihilation stage called the "pick- off" annihilation. Also solved the equations of the positron annihilation in two ways, computationally and programmatically (using the scilab- 6.0.1 and Mathematica12) according to specific conditions, at the moment t= 0, positronium was not yet formed, In the first stage studied the free positron annihilation, in the second stage the annihilation of the p- Ps and in the third phase, the annihilation of the o- Ps was studied, and we have found that the coefficient of decrease in the number of free positrons at the same time reflects the probability of positronium formation. Studied the spectrum of the positron and positronium in both para and ortho and determine the positron case by solving the annihilation equations under different conditions and according to the life time of each case, this requires the study of the annihilation spectra of different materials to take advantage of this in the detection of defects in the material and improve the structure.