The neutron capture cross section of the unstable isotope $^{135}\mathrm{Cs}$ was measured relative to that of gold by means of the activation method. The sample was produced by ion implantation in a high resolution mass separator and irradiated with quasimonoenergetic neutrons at $30\phantom{\rule{0.3em}{0ex}}\text{keV}$ and $500\phantom{\rule{0.3em}{0ex}}\text{keV}$, using the $^{7}\mathrm{Li}(p,n)^{7}\mathrm{Be}$ reaction. An additional irradiation with thermal neutrons has been carried out for defining the sample mass and for measuring the half-life of $^{136}\mathrm{Cs}$. The neutron capture cross sections were determined as $164\ifmmode\pm\else\textpm\fi{}10\phantom{\rule{0.3em}{0ex}}\text{mb}$ and $34.8\ifmmode\pm\else\textpm\fi{}3.0\phantom{\rule{0.3em}{0ex}}\text{mb}$ at $30\phantom{\rule{0.3em}{0ex}}\text{keV}$ and $500\phantom{\rule{0.3em}{0ex}}\text{keV}$, respectively, and were used to normalize the theoretically derived cross section shape. Based on these data, refined statistical model calculations were performed to obtain the $(n,\ensuremath{\gamma})$ cross sections of the short-lived isotopes $^{134}\mathrm{Cs}$ and $^{136}\mathrm{Cs}$ as well. Updated Maxwellian-averaged capture cross sections of all unstable Cs isotopes were calculated for a range of thermal energies characteristic of helium burning scenarios for an improved $s$-process analysis of the Xe-Cs-Ba region.
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