  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 13C pocket in the top ~1/10th of the He-rich intershell. All
of the models show surface increases of [Ge/Fe] less than about 0.5, except the
2.5Msun, 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.
