Emission of Scission Neutrons: Testing the Sudden Approximation N. Carjan Centre d'Etudes Nucléaires de Bordeaux-Gradignan,CNRS/IN2P3 – Université Bordeaux I, B.P. 120, Gradignan Cedex, France M. Rizea National Institute of Physics and Nuclear Engineering, P.O.Box MG-6, Bucharest, Romania

I. Halpern's sudden approximation and our mathematical formulation of it: a short reminder. More details plus numerical results for symmetric fission in: N. Carjan,P. Talou,O. Serot; Nucl. Phys. A792 (2007) 102 II. Dependence on the mass asymmetry of the fission fragments: N. Carjan,M. Rizea; Nucl. Phys. A805 (2008) 437 III.Beyond the sudden approximation: in reality the transition takes place in a short but finite time interval. We need to follow the time evolution of each neutron state by solving numerically the two-dimensional time- dependent Schrodinger equation: M. Rizea,N Carjan; in Exotic Nuclei and Nuclear Astrophysics, AIP Conference Proceedings 972 (2008) 526.

Halpern’s cartoon ‘sudden approximation’ Equipotentials V 0 /2 for the two scission configurations studied in this work Z [fm]  [fm]

The formalism of the sudden approximation (1/3)‏ The single-particle wave functions for an axially-symmetric fissioning nucleus have the general form Where u( ,z) and d( ,z) contain the spatial dependence of the two components, spin up and down respectively.  is the projection of the total angular momentum along the symmetry axis and is a good quantum number. Since we are dealing with symmetric fission, the parity  is also a constant of motion. If the scission is characterized by a sudden change of nuclear deformation, an eigenstate of the “just before scission” hamiltonian will be distributed over the eigenstates of the “immediately after scission” hamiltonian: One can notice that only |  f > states the same ( ,  ) values as |  i > will have non-zero contributions. These states are mainly bound states but contain also a few discrete states in the continuum that will spontaneously decay.

Therefore the emission probability of a neutron that had occupied the state |  i > is In this way we do not need to use the states in the continuum that are less precise in the numerical diagonalization. The P em i ’s will be referred as partial probabilities since they only consider one occupied state. The limit between bound and unbound states is the barrier for neutron emission, which is zero if the centrifugal potential and the energy needed to break a pair are not taken into account. Summing over all occupied states one obtains the total number of scission neutrons per fission event where v i 2 is the ground-state occupation probability of |  i >. The formalism of the sudden approximation (2/3)‏

We have also calculated the part of the initial wave function that was emitted Since it gives access to the spatial distribution of the emission points Of course, we have Finally, the occupation probabilities after the sudden transition are of interest since they show the degree of excitation in which the fragments are left: The formalism of the sudden approximation (3/3)‏

Typical neutron state favored for emission

Typical neutron state hindered for emission

Low density high-  states: the emitted part contains only one state (the next to the initial state)‏

Scission neutron multiplicity calculated with different assumptions concerning the scission shapes and the emission barrier heights  Emission strongly depends on the quantum numbers (  ) of the state which the neutron occupies just before scission  We predict, on the average, one neutron emitted for every 3 fission events

Distribution of the emission points  The distribution of the emission points are strongly peaked in the region between the nascent fragments  Very few scission neutrons are emitted in the direction of the fission fragments

Fission fragments residual excitation energies After the sudden transition, the primary fragments are left in an excited state: neutron states below the Fermi level (-5 MeV) are depopulated at the expense of neutron states above.

Scission Neutron Multiplicity as a function of Heavy Fragment Mass: Case of 235 U(n th,f)‏ Calculations performed with only  =1/2 states

Scission Neutron Multiplicity as a function of Isospin: Case of Pu Isotopes Calculations performed with only symmetric configurations

 Contributions to the Total Scission Neutron Multiplicity: Cases of 236 Pu and 256 Pu

Conclusions We have presented a dynamical model for the emission of scission neutrons that may lead to quantities characterising the nuclear configuration at scission (minimum neck radius, excitation energy of primary fission fragments, number of emitted neutrons and their angular distribution, etc) that are not available from other sources. In the sudden approximation the model is relatively simple and produces results that are easy to interpret microscopically in terms of the quantum numbers of the states involved. Calculations including the mass asymmetry and for a long series of isotopes open new perspectives calling for new improved measurements of neutrons emitted during nuclear fission. More realistic calculations using the 2 dim TDSE have shown that the sudden approximation is a good approximation (20% error) and also gave time scales for adiabatic and extreme diabatic processes.

Using scission configurations predicted by theory: Hartree Fock or liquid drop estimates Calculating the scission neutron spectrum and the final angular distribution Study the influence of scission neutrons on the emission of prompt neutrons (chronology). Possible extensions