Parametrization of the octupole degrees of freedom

The collective description of the octupole degrees of freedom has been a long standing problem in nuclear physics @1#. The phenomenological description of low-energy vibrational states @2,3# discussed them in terms of simple ‘‘surface modes’’ thus allowing for the description of the octupole degrees of freedom in terms of seven collective variables a3m @2,4#. In the 1980s, theoretical calculations @5# predicted the existence of octupole stable deformations and this problem aroused considerable interest, especially in the Ce-Ba and the Rn-Th regions. The level scheme of a few moderately or weakly deformed nuclei, such as Ge @6#, Sm @7#, Ra @8#, or Ra @9# presents features that may be related to octupole instabilities and softness of the nucleus with respect to possible exotic octupole deformations. Lately @10# there has been evidence for the existence of stable octupole deformations in the Rn-Th region making the problem of the collective description of this degree of freedom more actual. It has been shown that this type of collective excitations can significantly change the fusion cross section for heavy ions @11#. Initially, only octupole deformations of the Y 30 type were considered, but the possibility of other type of deformations ~such as Y 32 @12# or Y 31 @13#! has also been discussed. The study of more general structures was intended without much success @14#. There were also several proposals related to the appropriate parametrization of the collective variables. Rohozinski @15# proposed a parametrization based on the symmetries coming from the Oh group of transformations of the frame of reference. Hamamoto @16# wrote octupole shapes in terms of the irreducible representations of the octahedron group A1 ,F1(k) and F2(k) and suggested that a parametrization that covers all possible octupole deformed shapes ~without double counting the same shape! is provided by removing F2(k) ~eliminating in this way three parameters!.