We demonstrate that the amount of extra mixing required to fit the observed low C/N and ^ { 12 } { C } / ^ { 13 } { C } ratios in first giant branch ( FGB ) stars is also sufficient to explain the carbon and nitrogen abundances of Galactic asymptotic/second giant branch ( AGB ) stars .
We simulate the effect of extra mixing on the FGB by setting the composition of the envelope to that observed in low-mass ( M \leq 2 { M } _ { \sun } ) FGB stars , and then evolve the models to the tip of the AGB .
The inclusion of FGB extra mixing compositional changes has a strong effect on the C and N abundance in our AGB models , leading to compositions consistent with those measured in Galactic carbon-rich stars .
The composition of the models is also consistent with C and N abundances measured in mainstream silicon carbide ( SiC ) grains .
While our models cover the range of C abundances measured in carbon stars in the LMC cluster 1846 , we can not simultaneously match the composition of the O and C-rich stars at the same time .
A second important result is that our models only match the oxygen isotopic composition of K and some M , MS giants , and are not able to match the oxygen composition of carbon-rich AGB stars .
By increasing the abundance of ^ { 16 } { O } in the intershell ( based on observational evidence ) it is possible to reproduce the observed trend of increasing ^ { 16 } { O } / ^ { 18 } { O } and ^ { 16 } { O } / ^ { 17 } { O } ratios with evolutionary phase .
We also find that some Li production takes place during the AGB and that Li-rich carbon stars ( \log \epsilon ( { Li } ) \gtrsim 1 ) can be produced .
These models show a correlation between increasing Li abundances and C. The models can not explain the composition of the most Li-enriched carbon stars , nor can we produce a Li-rich carbon star if we assume extra mixing occurs during the FGB owing to ^ { 3 } { H } e destruction .
We tentatively conclude that 1 ) if extra mixing occurs during the AGB it likely only occurs efficiently in low metallicity objects , or when the stars are heavily obscured making spectroscopic observations difficult , and 2 ) that the intershell compositions of AGB stars needs further investigation .