Reflection asymmetric nuclear shapes obtained by solving a differential equation

The equilibrium nuclear shapes in fission theory are usually obtained by minimizing the deformation energy for a given surface equation. In the following we present a method allowing to obtain a very general equilibrium (saddle-point) shape as a solution of a differential equation without an a priori introduction of a shape parametrization. In the approach based on a pure liquid drop model (LDM), saddle-point shapes are always reflection symmetric: the deformation energy increases with the mass-asymmetry parameter η = (A1 - A2)/(A1 + A2), where η is replaced by an almost linear dependent quantity (dL-dR)/R0. In this way the well established experimentally fission fragment mass asymmetry can not be explained. By adding the shell corrections δE to the LDM deformation energy, Edef ELDM + δ E, we succeeded to obtain minima as shown. The nuclear surface equation of an axially symmetric body u(x) is a solution of the following differential equation: u'' = 2 + 1/u[u'2 + (x - d + Vs)(4u + u'2)3/2], where d is an input parameter which determines the deformation. In our present approach we included in the deformation energy E(R)=ELD(R) + δE(R) - δE0 a phenomenological shell correction δE, and the above written differential equation is solved iteratively by using Runge-Kutta method. The procedure is repeated until the solution of the variational problem leads to the minimum of the deformation energy which is the sum of the surface and Coulomb energies plus shell corrections. At a given deformation we find the fragment volumes and the corresponding number of protons and neutrons Zi(R), Ni(R) (i=1,2).For every fragment we add contributions from protons and neutrons δE(R) = Σi δEi(R) = Σi [δEpi(R) + δEni(R)] given by δEpi Cs(Zi) ; δEni = Cs(Ni), where s(Z) = F(Z)/[(Z-2/3] - cZ1/3 and a similar equation for s(N), where F(n) = 3/5 [((Ni5/3 - Ni-15/3)/(Ni - Ni-1))(n - Ni-1) - n5/3+ Ni-15/3] in which n in (Ni-1, Ni) is the actual number of protons or neutrons Z or N, and Ni-1, Ni are the neighbouring magic numbers. The parameters c = 0.2, C = 6.2 MeV were determined by fit with experimental masses and deformations. By introducing shell corrections we obtained minima of deformation energy for parent nuclei 238 U, 232,228 Th at a finite mass asymmetry giving for the three nuclei the same mass number of the heavy fragment A1 = 125. (authors)