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Do zero metal intermediate mass stars experience thermal pulses?

We have studied the evolution of intermediate mass (M.ge.5Mo) zero metal (Z=0) stars with particular attention to the AGB phase. At variance with previous claims we find that these stars experience thermal instability (the so called thermal pulses). The critical quantity which controls the onset of a thermally pulsing phase is the amount of CNO in the envelope during the AGB. For these stars the central He burning starts in the blue side of the HR diagram and the 1^{st} dredge up does not take place. Then the envelope maintains its initial composition up to the beginning of the AGB phase. However, during the early AGB the 2^{nd} dredge-up occurs and fresh He and CNO elements are engulfed in the convective envelope. We find that in stars with M.ge.6Mo the resulting amount of ^{12}C is large enough to sustain a normal CNO burning within the H shell and consequently the star enters the usual thermal pulse phase. In the 5Mo model, owing to the lower ^{12}C enhancement in the envelope after the 2^{nd} dredge-up, the He burning shell suffers weak thermal instabilities. 9 of these thermal oscillation are needed before the He burning luminosity reaches 3x10^5 Lo and a first convective shell develops in between the two burning shells. Later on a second convective shell forms at the base of the H rich envelope. This convective zone cross the H/He discontinuity and partially overlaps the previous one, dredging up fresh ^{12}C. After a huge H flash, a quiescent CNO burning settle on. From this moment a thermal pulse phase starts, which is very similar to the one experienced by the more massive models.