A FRESH LOOK AT THE SCISSION CONFIGURATION Fedir A. Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Shape parameterisations The variational principle for liquid drop shapes Two point boundary problem, the relaxation method The scission configuration Mass-asymmetric shapes Applications: the barriers of heavy nuclei Summary and outlook

The shape parameterisations Expansion around sphere in terms of spherical harmonics (Distorted) Cassinian ovaloids Koonin-Trentalange parameterisation (modified) Funny-Hills parameterisation Two smoothly connected spheroids The two center shell model

Cassini ovaloids

Parameteization of Moeller et al

The two center shell model J. Maruhn and W. Greiner, Z. Phys, 1972

V.M.Strutinsky et al, Nucl. Phys. 46 (1963) 659

Numerical results, V.M.Strutinsky et al, Nucl. Phys. 46 (1963) 659

The two point boundary value problem

Optimal shapes

Deformation energy, (R 12 ) crit = 2.3 R 0

R.W.Hasse, W.D.Myers, Geometrical Relationships of Macroscopic Nuclear Physics:

The scission point: the stiffness with respect to neck is sero U.Brosa, S.Grossmann and A.Muller, Phys. Rep. 197 (1990) 167—262.

Cassini ovaloids

FH: M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky and C. Y. Wong, Rev. Mod. Phys. 44, 320 (1972). MFH: K. Pomorski and J. Bartel, Int. J. Mod. Phys. E 15, 417 (2006).

How unique are the „optimal“ shapes ?

Q 2 - constraint

Mass-asymmetric shapes

Mass asymmetric shapes, x = 0.75

Deformation energy

The scission shapes, R neck =0.2 R 0

Optimal/Cassini shapes

(z-z*)/octupole constraint

K.T.R.Davies and A.J.Sierk, Phys.Rev.C 31 (1985) 915

Businaro-Gallone point

The barriers of heavy nuclei, surface curvature energy Leptodermous expansion: ETF = E vol + E surf + E curv + E Gcurv

The LSD barrier heights F.A.Ivanyuk and K.Pomorski, Phys: Rev. C 79, (2009) K.Pomorski and J. Dudek, Phys. Rev. C 67, (2003) The rms dev.for 35<Z< 105, 0<I< 0.3 is 150 keV

The barrier heights, topological theorem W. D.Myers and W. J. Swiatecki, Nucl. Phys. A601, 141 (1996): the “barrier will be determined by a path that avoids positive shell effects and has no use for negative shell effects. Hence the saddle point energy will be close to what it would have been in the absence of shell effects, i.e., close to the value given by the macroscopic theory!” For E micr see P. Moeller, J. R. Nix, W. D. Myers and W. J. Swiatecki, At. Data and Nucl. Data Tables, 59, 249 (1995).

Summary and outlook 1. The relaxation method allows to solve the variational problem for the shapes of contiional eqilibrium with a rather general constraints 2. The extension of this method to separated shapes and account of the surface diffuseness, attractive interaction (eventually) shell corrections would result in a very accurate method for the calculation of the potential energy surface