Observations of planetary nebulae ( PNe ) by Sterling , Dinerstein and Bowers have revealed abundances in the neutron-capture element Germanium ( Ge ) from solar to factors of 3 – 10 above solar .
The enhanced Ge is an indication that the slow -neutron capture process ( s process ) operated in the parent star during the thermally-pulsing asymptotic giant branch ( TP-AGB ) phase .
We compute the detailed nucleosynthesis of a series of AGB models to estimate the surface enrichment of Ge near the end of the AGB .
A partial mixing zone of constant mass is included at the deepest extent of each dredge-up episode , resulting in the formation of a ^ { 13 } { C } pocket in the top \sim 1 / 10 ^ { th } of the He-rich intershell .
All of the models show surface increases of [ Ge/Fe ] \lesssim 0.5 , except the 2.5 { M } _ { \sun } , Z = 0.004 case which produced a factor of 6 enhancement of Ge .
Near the tip of the TP-AGB , a couple of extra TPs could occur to account for the composition of the most Ge-enriched PNe .
Uncertainties in the theoretical modeling of AGB stellar evolution might account for larger Ge enhancements than we predict here .
Alternatively , a possible solution could be provided by the occurrence of a late TP during the post-AGB phase .
Difficulties related to spectroscopic abundance estimates also need to be taken into consideration .
Further study is required to better assess how the model uncertainties affect the predictions and , consequently , if a late TP should be invoked .