Enhancement of the Triple Alpha Rate in a Hot Dense Medium

In a sufficiently hot and dense astrophysical environment the rate of the triple-alpha ($3\alpha$) reaction can increase greatly over the value appropriate for helium burning stars owing to hadronically induced deexcitation of the Hoyle state. In this Letter we use a statistical model to evaluate the enhancement as a function of temperature and density. For a density of ${10}^6\mathrm{g}{\mathrm{cm}}^{-3}$ enhancements can exceed a factor of 100. In high temperature or density situations, the enhanced $3\alpha$ rate is a better estimate of this rate and should be used in these circumstances. We then examine the effect of these enhancements on production of ${}^{12}\mathrm{C}$ in the neutrino wind following a supernova explosion and in an x-ray burster.

Figure 1

The radiative width for typical helium burning stars results from the gamma ray cascade from the Hoyle state through the 4.44 MeV $2^{+}$ state and the (much weaker) pair decay of the Hoyle state. The additional contributions we estimate, shown as solid downward arrows, are mediated by inelastic scattering of protons, neutrons, and alpha particles leading from the Hoyle state to the $2^{+}$ state and to the ground state.

Figure 2

Inelastic scattering cross sections for neutrons (top panel) and protons (bottom panel). Results for the default OMPs described in the text are shown as solid lines, and the maximum and minimum cross sections for other potentials noted in the text are included within the shaded areas. For each projectile results are shown for scattering leading from the ground state to the 4.44 MeV $2^{+}$ state, the ground state to the 7.65 MeV Hoyle state, and the 4.44 MeV $2^{+}$ state to the Hoyle state. Available experimental results are shown as discrete points, for neutrons from Refs. [21, 22], for protons from Refs. [8, 23, 24, 25].

Figure 3

Ratios of the rate induced by the indicated transitions to the measured ($\mathrm{gamma}+\text{pair decay}$) rate. The ratios were calculated for a particle density of $10^6\mathrm{g}{\mathrm{cm}}^{-3}$. The alpha ratios are plotted on the expanded scale on the right-hand ordinate.

Figure 4

On the left-hand ordinate are shown, as a function of time, the neutron, proton, and alpha particle densities calculated following the adiabatic model. Initially, the temperature was set at $T_9=9$ to insure nuclear statistical equilibrium, the entropy at $S=90k_B$, $Y_e=0.45$, and $\mathrm{velocity}=7500\mathrm{km}{\sec }^{-1}$. The calculated overall enhancement, owing to induced deexcitation of the Hoyle state by protons, neutrons, and alpha particles is shown on the right-hand ordinate.