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We present barium, carbon, and silicon isotopic compositions of 38acid-cleaned presolar SiC grains from Murchison. Comparison with previous datashows that acid washing is highly effective in removing barium contamination.Strong depletions in $\delta$($^{138}$Ba/$^{136}$Ba) values are found, down to$-$400 permil, which can only be modeled with a flatter $^{13}$C profile withinthe $^{13}$C pocket than is normally used. The dependence of$\delta$($^{138}$Ba/$^{136}$Ba) predictions on the distribution of $^{13}$Cwithin the pocket in AGB models allows us to probe the $^{13}$C profile withinthe $^{13}$C pocket and the pocket mass in asymptotic giant branch (AGB) stars.In addition, we provide constraints on the $^{22}$Ne$(\alpha,n)^{25}$Mg rate inthe stellar temperature regime relevant to AGB stars, based on$\delta$($^{134}$Ba/$^{136}$Ba) values of mainstream grains. We found twonominally mainstream grains with strongly negative$\delta$($^{134}$Ba/$^{136}$Ba) values that cannot be explained by any of thecurrent AGB model calculations. Instead, such negative values are consistentwith the intermediate neutron capture process ($i$-process), which is activatedby the Very Late Thermal Pulse (VLTP) during the post-AGB phase andcharacterized by a neutron density much higher than the $s$-process. These twograins may have condensed around post-AGB stars. Finally, we report abundancesof two $p$-process isotopes, $^{130}$Ba and $^{132}$Ba, in single SiC grains.These isotopes are destroyed in the $s$-process in AGB stars. By comparingtheir abundances with respect to that of $^{135}$Ba, we conclude that there isno measurable decay of $^{135}$Cs ($t_{1/2}$= 2.3 Ma) to $^{135}$Ba inindividual SiC grains, indicating condensation of barium, but not cesium intoSiC grains before $^{135}$Cs decayed.

Barium Isotopic Composition of Mainstream Silicon Carbides from Murchison: Constraints for s-Process Nucleosynthesis in AGB Stars