It has been suggested that both the relativistic Schroedinger and Dirac equations allow the existence of so-called “deep Dirac levels” (DDL) in all atoms of the periodic table. An estimate of the size of the DDL atoms is given, and a physics explanation is proposed for exciting the DDL transitions. Possible secondary nuclear reactions of the atoms on the DDLs are suggested, and preliminary experimental results are presented. A search has begun for some direct experimental evidence supporting the proposed DDL model. So far, in electrolytic experiments, only calorimetric evidence was found.