Symmetry energy and its correlations with the nuclear structure properties of Z=40 isotopic series at finite temperature

Symmetry energy is an important quantity in studying the nuclear structure, dynamics of heavy-ion reactions, and physics of neutron stars. The equation of state of asymmetric nuclear matter is limited due to the undetermined symmetry energy and lack of initial constraints on the nuclear matter (NM), leading to a study of the symmetry energy of NM. In this work, we have studied the temperature-dependent nuclear symmetry energy (S) of Zr isotopic series with N=34-86 along with its volume ($S_V$) and surface components ($S_S$) using the Coherent Density Fluctuation Model (CDFM). The nuclear densities used as input to CDFM are calculated along with the bulk properties at finite temperature within the temperature-dependent Relativistic Mean Field Model using NL3 and IOPB-I interactions. The ground state bulk properties are in good concurrence with the available experimental data. The binding energy per nucleon and neutron pairing energy decrease, and the radial density distribution increases with a rise in temperature (T). The deformed nuclei with an increase in T become spherical at and beyond the critical temperature. The bulk properties exhibit the magicity of N=40 along with the standard neutron magic numbers in this range at T=0 MeV. The nuclear symmetry energy, its surface and volume components, and their ratios $\kappa$ show a similar behavior change as that occurring in the nature of deformation with the neutron number N. As IOPB-I interaction predicts a softer equation of state than NL3 (stiffer), the symmetry energy of the nuclei corresponding to IOPB-I is comparatively less. The symmetry energy of Zr isotopic series is found to be correlated with nuclear structure properties at finite temperature. The symmetry energy is maximum at N=50, even at finite temperature, as this isotope possesses the highest stability in the Zr series. This study will help in the production of exotic nuclei and the understanding of heavy-ion reactions.