Nucleosynthesis of heavy elements is an area of active research in nuclear astrophysics. After the Big Bang, all nucleosynthesis takes place in the stars and produces elements heavier than hydrogen. Helium was produced in the Big Bang and continues to be the main product of stellar burning. Theory has been moderately successful in explaining the general trends, though much more needs to be done to reproduce the observed elemental abundances. Experiments are more difficult because of the extremely low cross sections. Elements heavier than iron are produced in two principal processes, the s(low)- and the r(apid)-process. Both of them require neutrons, which are produced in situ. There is another rarer process, called the p-process, which produces some thirty-five proton-rich stable nuclei. The s-process involves stable or long-lived isotopes and is generally better understood. Low-energy neutron capture reactions by unstable nuclei in light mass regions play a very important part in the r-process that takes place in type-II supernovae. Direct experimental measurements of these reactions are currently unfeasible, and the rates of these reactions are evaluated based on different theoretical models, which may be constrained using certain nuclear structure properties as inputs. Most of the previous rate evaluations for these capture reactions involve potential model-based theoretical calculations, which were only poorly constrained from nuclear data or theoretical values. This creates considerable uncertainties in reaction rates and abundance calculations in reaction networks. In the proposed project, we propose to do away with the potential model for reactions. We will calculate the rate of astrophysically relevant capture reactions at low energy related to the short-lived unstable nuclei in the low and medium mass range that are involved in r-process nucleosynthesis in supernovae. Direct experimental determination of the cross-sections, or equivalently, the rates of these capture reactions, is currently not possible. Hence, the possible approach to evaluate the rates of these capture reactions is based on different models with experimental inputs constraining the various parameters of the models to reduce the uncertainty in calculations. The main advantages of this method are: i) Precise determination of interference effects between different reaction mechanisms; DC & multiple resonances. II) No dependence on the choice of potential model. iii) More straightforward approach to deal with multiple resonances and channels so as to provide more precise results. We will use the R-matrix method in light nuclei to extract the cross sections at low energy for nuclei with mass less than 25. For heavier nuclei, Hauser-Feshbach theory, based on the nuclear statistical model, is used to describe low-energy capture reactions. The input to this calculation includes gamma ray strength functions, nuclear level density and an optical potential. The parameters of the model are not accurately constrained due to a lack of experimental results. The plan of the proposal in this regard is to explore microscopic descriptions of level density and gamma ray strength functions using recent experimental data for nuclei with astrophysical importance and use the extracted values as input in the statistical model codes for the low-energy capture reactions involved in the r-process, and also the p-process nucleosynthesis paths. Finally, the reaction rate will be estimated with the updated cross sections in a network calculation to evaluate more precisely the abundance profiles of the r-process nuclei.