Coordinate-Dependent Mass and the Validity of the WKB Approximation in Fission Barrier Penetration Calculations

Abstract

It is shown that the quasi-classical condition for the validity of the WKB approximation is satisfied although the mass associated with the spontaneous fission of 240pu varies by a factor of 12. It has been known for quite some time that the mass (or mass parameter) associated with collective nuclear motion may vary substantially with the collective coordinates. For instance, the microscopically calculated l ) diagonal matrix elements of the five-dimensional mass-matrix of 152Sm vary by as much as a factor of 7 as the qu'adrupole deformation /3 varies from 0.0 to 0.5 and as the axial asymmetry variable r varies from 0 to 60'. In a recent calculation 2) of the fission lifetimes of transuranic and of superheavy nuclei, even larger variations (by factors of about 12) have been found. Such large variations are not very model-dependent. They arise from the general variation of Nilsson-type single-particle levels 3 ),4) some of which go up as deformation increases while the others go down. Consequently, the energy denomi­ nator appearing in the cranking-type expression for the mass parameter varies rapidly with deformation, especially near level crossings. Such variations of the collective mass by factors of 12, as the nucleus fissions, seem rather large from the point of view of using the WKB approximation to the barrier penetration probability. Hence, the quasi-classical condition necessary for the validity of the WKB approximation is examined below. The WKB approximation is expected to be valid when the quasi-classical condi­ tion is satisfied. This condition is given by5) IdJ'/drl~l , (1)