The ${\mathrm{C}}^{12}$ nucleus is studied from the three-$\alpha$-particle point of view based on a wave-function method in which the Hermiticity of the Fredholm kernel is preserved. The interaction between two $\alpha$ particles, obtained from the resonating-group calculation of Okai and Park, is recast in a separable form with the aid of Hille's formula. For the numerical calculation, we have employed three simpler types of the potential: Gaussian, Tabakin, and Yamaguchi. It is shown that Harrington's theory, based on the $T$-matrix method, reduces to a result of the present study, but not to those of other wave-function methods. Contrary to a conclusion of the previous work, we have succeeded in finding an excited state with the same ($J, \pi$) as the ground state. The present method also supplies the mean height and side of the equilateral triangles formed by three $\alpha$ particles. These values are found to imply a root-mean-square radius of the ${\mathrm{C}}^{12}$ nucleus which agrees well with the experimental value.

• Received 3 June 1965

DOI:https://doi.org/10.1103/PhysRev.187.1239