Volume 57B, number 5 PHYSICS LETTERS 4 August 1975 QUANTUM MECHANICAL AND SEMICLASSICAL DESCRIPTION OF A TWO- DIMENSIONAL FISSION MODEL "~ H. MASSMANN*, P. RING** and J.O. RASMUSSEN Lawrence Berkeley Laboratory, University of California, Berkeley, Calif. 94710, USA Received 21 May 1975 The penetration through a two dimensional fission barrier is investigated by a fully quantum mechanicalcoupled channel calculation and by a new semiclassical method. One finds a quantitative agreement. In recent years much progress has been made in the determination of collective Hamiltonian describ- ing the fission process [1,2]. One finds that in order to understand better the physics behind the fission process, one has to take into account more than one degree of freedom, and the inertial tensor seems to depend strongly on the coordinates. A fully quan- tum mechanical solution of this multidimensional barrier penetration problem is possible by the meth- od of coupled channels. We discuss in the first part of this letter how to carry out such a calculation and apply it to a simple two dimensional model. However, for computational reasons such an approach may break down for realistic surfaces with several minima and saddles. Therefore one usually look for approximations. The common approach is to make a one dimensional problem by introducing a suitable fission path [3]. The traditional choice of the path is along the bottom of the valley in the potential surface. The potential energy along the valley is taken as the potential energy of the one-dimensional prob- lem; some suitable expression for the mass parameter, from the hydrodynamic or the cranking model, is taken, and the one dimensional WKB formula is ap- plied. Starting with one dimensional wavefunctions t Work performed under auspices of the U.S. Energy Re- search and Development Administration. * On leave from Faeulted de Ciencias, U. de Chile with a fellowship from the Convenio U. de Chile - U. of California. ** On leave from Physik Department der Technischen Universitat Munchen, West Germany, supported by the Deutsche Forschungsgemeinsehaft. of this type one can take into account the other de- grees of freedom in a fully quantum mechanical way by some kind of DWBA approach if the coupling is not too strong [4]. For realistic cases with winding valleys and variable inertial tensor the Strutinsky- Pauli group [1] applied the one dimensional WKB formula for many paths, looking for the minimum action integral. In this two dimensional approach however, one still ignores the kinetic energy tied up in the motion orthogonal to the fission path. Based on the formulation of quantum mechanics by path integrals given by Feynman [5] together with the correspondence principle, a unified semiclas- sical theory has been developed in the field of mole- cular reactions [6,9]. In the second part of this let- ter we will describe briefly this method and apply it to the same model for which the quantum mechani- cal calculation was done. In the following we use a model for a collective Hamiltonian which depends on the fission coordinate x and a coordinate y perpendicular to that, which describes collective excitations along the fission path: H = p2x/2mx + p2/2m + V 0 exp(-x2/a 2) + ~C(1 + a exp(-x2/a2))y 2. (1) The barrier in the x-direction has a simple Gaussian shape, and the potential in the y-direction is harmon- ic but with an x-dependent spring constant. This al. lows a coupling between the two degrees of freedom. The parameters are chosen so as to reproduce rough- ly a typical fission barrier: m x = 500 MeV-1 , my = 4.7 MeV-1, V 0 = 7 MeV, a = 0.185, C = 5.1 MeV 417