A second dredge-up occurs in our 
models, also leading to
surface abundance changes. Contrarily to the first dredge-up event, the HBS
extinguishes. For our 
models, the convective envelope
penetrates so deep that it pushes down the H-He discontinuity, implying
that when it will ignite again, the HBS will be located deeper than before
the second dredge-up.
Table 3:
Surface abundances and isotopic ratios consecutive to the first and
second dredge-up episodes (i.e. respectively at the top RG and time
defined in the text)
is produced at the beginning of the T Tauri phase from the
initial 
burning. On the main sequence, a peak of 
appears, due to the competition of the reactions 
and
. During the first dredge-up, the surface
abundance of 
thus increases. However, the higher the stellar
mass, the lower the 
peak, and the lower the surface enhancement
of 
after the first dredge-up. In the stellar mass range
considered here, the surface mass fraction of 
increases by
factors between 2 and 6.
The first dredge-up is also responsible for the Li and Be surface
depletion. Indeed, these elements are burned while the star is on the
main sequence in the radiative regions where the temperature is higher
than 
K and 
K, respectively.
On the main sequence, 
is burned for the benefit of
. A peak of 
builds up in the region where the CN
cycle operates at a slower rate compared to its equilibrium one. Deeper in
the star, in the region where the ON cycle occurs, 
is burned to
(the abundance profile of 
in the stellar interior
presents a two-step profile), and 
is slightly burned to
. A peak of 
is also present. The chemical profiles
depend both on mass and metallicity. The more massive a star, the larger the
temperature at a given mass, and hence the more external the 
and 
peaks. The shift of the chemical profiles towards more
external region with increasing stellar mass appears for all the chemical
species. Moreover, for a given stellar mass, a decrease of the metallicity
reduces the opacity and increases the temperature, pressure and density at a
given depth. Thus the 
and 
peaks and the other
chemical profiles are also shifted outwards when the metallicity is smaller.
Last but not least, in the stellar mass range we consider, the convective
envelope at its maximum expansion on the RGB reaches less deep regions when
the stellar mass increases. The first dredge-up then leads to a decrease of
the 
/
, 
/
,
/
and 
/
isotopic ratios, the
extent of which depends on both stellar mass and metallicity.
/
ratio
decreases to 
. The higher the stellar mass, the lower the
final carbon isotopic ratio.
This ratio slightly further decreases after the second dredge-up.
profile, for 
stars.
This implies that 
/
, as well as
/
have almost the same value in these models after
the first dredge-up. On the contrary, for the Z = 0.005 objects,
the lower the stellar mass, the less efficient the dredge-up, and
consequently, the less 
is up-heaved to the surface. This
explains the trend for the final 
C/
N ratio versus stellar
mass (see Charbonnel 1994 for more details).
After the second dredge-up, whatever Z, the convective envelope of
stars penetrates deeper for increasing stellar mass as
attested by both 
/
and 
/
ratios.
/
increases with the stellar mass. Moreover, since for Z = 0.005 the
deepening convective envelope of our 4 and 
stars does not reach
the regions where the ON cycle operated, the post dredge-up values of
/
are higher than for Z = 0.02.
After the second dredge-up however, 
enriched matter is
up-heaved in the convective envelope leading to very low
/
.
/
slightly
decreases with increasing stellar mass.
Our 
surface abundance increases are in rather good agreement
with the predictions by Bressan et al. (1993).
Concerning the CNO elements, our results are in general good agreement with
the predictions by Schaller et al. (1992), Bressan et al. (1993) and
El Eid
(1994). The only important discrepancy between theoretical predictions of
the different groups for intermediate-mass stars concerns the post dredge-up
/
ratio, which value highly depends on the adopted
and 
reaction
rates. We use the rates given by Landré et al. (1990), as do
Schaller et
al. (1992), and we obtain 
/
ratios very similar to
those of the Geneva group. On the other hand, Bressan et al. (1993) use the
lower 
proton capture rates given by Caughlan & Fowler (1988),
and obtain much lower 
/
ratios [see also
El Eid
(1994) for a discussion of the influence of the adopted 
destruction rate on the resulting 
/
ratio].
In our intermediate-mass stars, the theoretical post dredge-up values of the carbon isotopic ratio are slightly lower than the observations by Gilroy (1989) in galactic cluster giants.
Red giants in this evolutionary phase present 
/
ratios between 300 to 1000 and 
/
ratios in the range
400 to 600 (Harris et al. 1988; Smith & Lambert 1990a). This is in very
good agreement with our predictions. In particular, our prediction is in
perfect agreement with the observed value in 
UMa (Harris et al.
1988), which estimated mass is roughly 
. This point is in favor of
high 
destruction rates.
This agreement between standard models and observations confirms that no extra-mixing (diffusion, rotation-induced mixing, ...) is expected to occur in intermediate-mass stars, as already discussed in Charbonnel (1994, 1995).
Very complete tables containing information about the center and surface
evolution of the structure and chemical composition for our 3, 4, 5, 6 and
models with Z = 0.005 and 0.02, from the beginning of the PMS
phase up to the end of the E-AGB phase, can be found through the internet
network at the following address and path:
http://www-laog.obs.ujf-grenoble.fr/liens/starevol/evol.html
More specifically, we present, for each star, the evolution of
, 
, 
and 
;
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
, 
,
, 
, 
, 
,
, 
, 
, 
,
.
