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In the present work, the $^{204-210}$Tl isotopes have been investigated byperforming large-scale shell-model calculations, including configurationsallowing both neutron and proton core excitations across the $Z$ = 82 and $N$ =126 shell gaps. Inspired by the recent high-spin experimental data, thestructure of Tl isotopes has been studied for a considerably large model space.The KHHE interaction has been used for $^{204-206}$Tl isotopes, KHH7Binteraction for $^{204-210}$Tl isotopes, and additionally KHM3Y interaction hasbeen used for $^{208}$Tl isotope. The core excitation has been performed usingthe KHH7B and KHM3Y interactions. The level spectra of $^{204-210}$Tl isotopesare comprehensively described and explained by multi-nucleon couplings ofsingle-particle-hole orbitals within the valence space and by core excitationsacross $^{208}$Pb core. The well-known isomeric states are analyzed in terms ofthe shell model configurations.

Shell-model study for $^{204-210}$Tl isotopes and core excitations across the $Z = 82$ and $N = 126$ shell gaps