Rates of Nuclear Reactions in Solid-Like Stars

In stellar matter as cool and dense as the interior of a white dwarf, the Coulomb energies between neighboring nuclei are large compared to the kinetic energies of the nuclei. Each nucleus is constrained to vibrate about an equilibrium position, and the motion of the nuclei in the interior of a white dwarf is similar to the motion of the atoms in a solid or liquid. We propose a solid-state method for calculating the rate at which a nuclear reaction proceeds between two identical nuclei oscillating about adjacent lattice sites. An effective potential $U\left(\mathrm{r}\right)$ derived by analyzing small lattice vibrations is used to represent the influence of the Coulomb fields of the lattice on the motion of the two reacting nuclei. The wave function describing the relative motion of the two reacting particles is obtained by solving the Schrödinger equation containing the effective potential $U\left(\mathrm{r}\right)$. From this wave function, we derive an expression for the reaction rate. The rates of the $p+p$ and ${\mathrm{C}}^{12}$+${\mathrm{C}}^{12}$ reactions calculated using this solid-state method are typically 1 to 10 orders of magnitude smaller than those calculated by the method previously suggested by Cameron.