Beta Decay of Carbon 10 and the Cluster Model

A new toroidal spectrometer has been used to determine an end point of 1865±15 keV for the main positron branch in the decay of ${\mathrm{C}}^{10}$. This end point, together with the previous work on the mass 10 nuclei, leads to a Coulomb energy difference of 4.61 MeV between the mirror nuclei ${\mathrm{C}}^{10}$ and ${\mathrm{Be}}^{10}$. According to the cluster model, developed by Wildermuth and Kanellopoulos, ${\mathrm{C}}^{10}$ is composed of a di-proton cluster and two alpha-particle clusters. A lower bound on the size of the di-proton cluster may be obtained from the ${\mathrm{Li}}^6$ charge radius, while the size of the alpha clusters can be calculated from the charge radius of ${\mathrm{C}}^{12}$. However, these estimates of cluster size imply an upper limit of 3.52 MeV on the ${\mathrm{C}}^{10}$-${\mathrm{Be}}^{10}$ Coulomb energy difference. Hence, it is concluded that this particular cluster model does not describe the spatial properties of these nuclear clusters in a manner consistent with experimental evidence.