Photoneutron cross sections were measured for $^{203}$Tl and $^{205}$Tl at energies between the one- and two-neutron thresholds using quasi-monochromatic $gamma$-ray beams produced in laser Compton-scattering at the NewSUBARU synchrotron radiation facility. Our new measurement results in cross sections significantly different from the previously reported bremsstrahlung experiment, leading to rather different GDR parameters, in particular to lower GDR peak energies and higher peak cross sections. The photoneutron data are used to constrain the $gamma$-ray strength function on the basis of the Hartree-Fock-Bogolyubov plus quasi-particle random phase approximation using the Gogny D1M interaction. Supplementing the experimentally constrained $gamma$-ray strength function with the zero-limit E1 and M1 contributions for the de-excitation mode, we estimate the Maxwellian-averaged cross section for the s-process branching-point nucleus $^{204}$Tl in the context of the $^{205}$Pb - $^{205}$Tl chronometry.
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