We present the evolutionary models of metal-free stars in the mass range from 0.8 to 1.2 Msun with up-to-date input physics. The evolution is followed to the onset of hydrogen mixing into a convection, driven by the helium flash at red giant or asymptotic giant branch phase. The models of mass M >= 0.9 Msun undergo the central hydrogen flash, triggered by the carbon production due to the 3-alpha reactions. We find that the border of the off-center and central ignition of helium core flash falls between 1.1 and 1.2 Msun; the models of mass M <= 1.1 Msun experience the hydrogen mixing at the tip of red giant branch while the models of M = 1.2 Msun during the helium shell flashes on the asymptotic giant branch. The equation of state for the Coulomb liquid region, where electron conduction and radiation compete, is shown to be important since it affects the thermal state in the helium core and influences the red giant branch evolution. It is also found that the non-resonant term of 3-alpha reactios plays an important role, although it has negligible effect in the evolution of stars of younger populations. We compare our models with the computations by several other sets of authors, to confirm the good agreement except for one study which finds the helium ignition much closer to the center with consequences important for subsequent evolution.
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