A mathematical model has been introduced to investigate the effect of nuclear reaction constant ( A ), probability of the BEC ground state occupation Ω i, nD is the number density of deuteron (d) and the overall number of nuclei ND on the total nuclear d-d fusion rate (R). Under steady-state of the condensates of Bose-Einstein, the postulate of quantum theory and Bose-Einstein theory were applied to evaluate the total nuclear (d-d) fusion rate trapping in Nickel-metal The total nuclear fusion rate trapping predicts a strong relationship between astrophysical S-factor and masses of Nickel. The reaction rate trapping model was tested on three reaction d(d,p)T, d(d, n)3He and d(d, 4He)Q = 23.8MeV respectively. The reaction rate has described with astrophysical S -factor 110, 110 x 106 and 110 x 1013 (KeV. barn) for three reactions respectively. The masses of Nickel in the range (1-10) g can be taken to reach a small region for D-D trapp in metal. Results show that the reaction rate can be increased with an increase in metal masses and astrophysical S-factor.
