Assuming an $\alpha$-cluster structure of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, we construct ground-state wave functions (gsw) for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$. We expand the gsw of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ in a product basis of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ ($3\alpha$) states and relative motion of the fourth $\alpha$ particle with respect to ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$. A corresponding expansion is used for the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ ground state. From those expansions, we get the probability amplitudes for intermediate states of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$. Those are compared with the amplitudes extracted from other calculations. Best choices of coordinate systems for the description of the virtual breakups are discussed using the concept of "democracy."

• Received 10 July 1972

DOI:https://doi.org/10.1103/PhysRevC.7.2340