Strongly screening on electron capture for Nuclides $^{52, 53, 59, 60}$Fe by the Shell-Model Monte Carlo method in pre-supernova

The death of the massive stars due to supernova explosion is a key ingredient in stellar evolution, stellar population synthesis. The electron capture (EC) plays a vital role in supernovae explosions. According to the Shell-Model Monte Carlo (SMMC) method, basing onthe Random Phase Approximation (RPA) and Linear Response Theory Model (LRTM), we study the strongly screening EC rates of nuclides $^{52, 53, 59, 60}$Fe in presupernova. The results show that the screening rates can decrease about 18.66\%. We compare our results with those of Fuller et al. (FFN), Aufderheide et al. (AUFD), and Nabi et al. (NKK)in the case with and without strong electron screening(SES). For the case without SES, our calculations are in very good agreement with those of AUFD in relatively high density surroundings and the maximum error is within 0.35\% at $\rho_7=100$ (e. g., even-even nuclei $^{60}$Fe). However, for odd-A nuclei$^{59}$Fe, our rates are close to one, one, two order magnitude smaller than those of FFN, AUFD, and NKK. For the case with SES, our screening results are about three, two orders magnitude, and 7.27\% lower than those of FFN, AUFD, NKK for $^{60}$Fe, respectively, and it is lower about two, two orders magnitude, and 12.42\% than those of FFN, AUFD, NKK for odd-A nuclide $^{59}$Fe.