With the mass asymmetry described by the dynamical collective fragmentation coordinate &#958;, and with use of the asymmetric two-center shell model, the fission mass distributions for 226Ra, 236U, and 258Fm (which are typical representatives for triple-, double-, and single-humped distributions) are explained

Greiner, Walter

Maruhn, Joachim

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLCME  32, NUMBER  10  PHYSICAL REVIE\XJ  LETTERS  I  I  MARCH  1974 Theory of Fission-Mass Distributions Demonstrated for '2bRa.  2"UU,  25iiF~n+ Joachim Maruhn and Walter Greiner lizstituf für Theoretische Physik der Lhiversität Fraxkfui?,  Frankfuvf nrli iIlnin,  Certnnnl (Received 29 November  1973) With the mass asjn~metry  described bj the dynamical c*ollectivc  iragmentation coor di- nate ci , ancl  with use of  the asjmmetric two-center  shell model, the fission mas- clistri- butions for ?Ib~a,  2''T,  and '"~m  (ahich are typical representatives for triplc-,  double-, and single-humped distributions) are explained. The understanding of  mass distributions of  fis- sioning nuclei belongs to the most interesting, yet unexplained phenomena in fission physics. Using the concept of  mass asymmetry treated as a dynamical collective coordinate,  based on the asymmetric two-center  shell model (ATCSM), we have calculated,  free of  parameters,  the mass distributions of  fissioning '"Ra,  23fi~,  and 258~m  nuclei. The nuclear shape is defined by five coordi- nates:  h,  the elongation;  bl  and  P„  the fragment deformations;  E,  the necking-in parameter;  and <,  the mass asymmetry defined as Al and A, are the fragment masses obtained from the geometrical size of  the fragments.  The pre- cise geometrical significance of  the parameters is demonstrated in Fig. 1. For each set of  parameters there is an associ- ated single-particle Hamiltonian of  the ATCSM,'?' consisting essentially of  two deformed harmonic- oscillator potentials joined  smoothly at the neck, plus 1. S and i2  corrections.  Single-particle states calculated in this n~odel  are used for ob- taining shell and pairing corrections.  The liquid- drop formula used for renormalization was that of  Myers and S~iatecki,~  with the modification of the surface asymmetry constant K,  as  introduced by Johansson, Nilsson,  and Szyn~anski.~ To obtain the behavior of  the potential energy as  a function of  the two parameters interesting in mass asymmetry calculations,  we performed a full three  -dimensional  minimization in C,  ß „ and ß,  at each pair of  values X  and  5. Since this is very time consuming,  it was feasible only for a small number of  points.  In particular, for each value of  h  we did the full calculation at only five points in [. The collective mass parameters were calcu- lated according to the crünking formula in the BCS formulation.  The minimization has to be taken into account by taking E,  „ and 0, as  functions of  h and  5  and substituting a  a  ata  aß,  d  ---+--+ a~  a~  aha.  C~ai;,  (31 and analogously for a/ag,  in the cranking formu- la  . As in the previous paper3  it is assumed that  5 vibrations are much faster than the relative nlo- tion described by  X  at the stage of  fission con- sidered, i.e.,  right after the completion of  bar- rier penetration.  The potential remains nearly constant in its dependence on asymmetry at later stages (see also the results of  Mustafa,  Mosel, and Schmitt5), so that the main behavior of  the distribution should be fixed at this early time. Regarding X  as  a parameter, the Schrödinger FIG. 1.  Explanation of  the nucletir  shapes occurring in the two-center  model and the associated geoinetrical quantities. 

Theory of fission-mass distributions demonstrated for 226Ra, 236U, 258Fm

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