Nuclear alpha-particle condensates: Definitions, occurrence conditions, and consequences

There has been a recent flurry of interest in the possibility of condensates of $\alpha$-particles in nuclei. In this letter we discuss occurrence conditions for such states. Using the quantality condition of Mottelson we show that condensates are only marginally expected in $\alpha$-particle states. We proceed to demonstrate that few-body nuclear condensates are ill-defined, and emphasize the conflict between $\alpha$-localization and $\alpha$-condensate formation. We also explore the connection between Ikeda diagrams, linear chains, and Tonks-Girardeau gases. Our findings show that no new information is contained in the approximations of nuclear states as $\alpha$-cluster condensates. Furthermore, condensates of more than three $\alpha$-particles are very unlikely to exist due to couplings to other degrees of freedom.