Non-adiabatic transition of the fissioning nucleus at scission: stationary and time- dependent approaches N. Carjan Centre d'Etudes Nucléaires de Bordeaux-Gradignan,CNRS/IN2P3 – Université Bordeaux I, B.P. 120, 33175 Gradignan Cedex, France and National Institute of Physics and Nuclear Engineering, P. O. Box MG-6, Bucharest, Romania M. Rizea National Institute of Physics and Nuclear Engineering, P.O.Box MG-6, Bucharest, Romania
Why emission of prompt neutrons and -rays is an important research topic Reflecting the properties of the primary fission fragments, they contribute to the the fundamental understanding of the fission dynamics around scission. In addition they play a major role in applications: development of nuclear reactors and transmutation of used fuel . These applications require a very accurate knowledge of the neutron multiplicity and of the neutron spectrum N( ). To estimate the total energy release the -rays are also needed. For instance, the computer codes (deterministic or stochastic) that simulate the transport of neutrons and gammas in reactors use this information as input to calculate the reactor properties: reactivity, power distribution, heating, etc. Evaluation files (JEFF, ENDF, JENDL,etc) present discrepancies, are incomplete and have no predictive power. To compensate, a big effort has been done to make better fits or better simulations of the fragments ’ de-excitation but relatively little was done on a more fundamental level i.e., to understand the emission mechanisms and identify the various energy sources responsible for this emission. My talk is a step into this direction.
1)A microscopic dynamical model of the transition that a fissioning nucleus undergoes at scission will be presented: a) for an infinitely fast transition (the sudden approximation) b) for a diabatic change of the potential at scission (in a short but finite time interval) using the 2-dim time-dependent Schrodinger equation with time-dependent potential. 2)The model allows to estimate: a) the excitation energy of each primary fission fragment as function of its mass (thus avoiding the arbitrary hypotheses employed so far concerning the partition of the excitation energy among the fragments) b) the number of scission neutrons as function of mass asymmetry c) the extra-deformation energy of each primary fragment at scission. Calculations have been performed for 235 U(n th ,f). Plans for the near future: use the results to predict the mass asymmetry dependence of prompt neutron and gamma-ray mutiplicities .
I.Halpern's sudden approximation and our mathematical formulation of it 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 & Phys. Rev. C82 (2010) 014617 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.
‘sudden approximation’ Halpern’s cartoon Equipotential lines V 0 /2 [fm] for the pre and post scission configurations studied in this work Z [fm]
The formalism of the sudden approximation (1/4) The single-particle wave functions for an axially-symmetric fissioning nucleus have the general form Where u( ,z) and d( ρ ,z) contain the spatial dependence of the two components, spin up and down respectively. Ω 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 with 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 (assuming no centrifugal barrier).
The formalism of the sudden approximation (2/4) of a neutron that had occupied the state | i > is Therefore the emission probability 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 is the ground-state occupation probability of | i >. where v i 2
that can be used to test the numerical accuracy.
. The second source being the extra deformation energy of the nascent fragments E def
Equipotential lines V 0 /2 that define the sudden transition for 3 mass asymmetries
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
Occupation probabilities of the neutron states 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.
Contribution of each Ω to E sc*
Excitation energies
Total excitation energy
Contribution of each Ω to ν sc
Neutron multiplicities
3.5 Appelin+Nishio NuTOT 3.0 2.5 Neutron Multiplicity 2.0 1.5 1.0 0.5 0.0 60 70 80 90 100 110 120 130 140 150 160 Mass Number
P. Moller and J. R. Nix, Nucl. Phys. A536 (1992) 20
Pairing gap at scission
Neutron levels at scission
Summary and Conclusions • We have presented a microscopic dynamical model for the emission of scission neutrons that may lead to quantities characterising the nuclear configuration at scission (minimum neck radius, excitation energy and deformation 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. It predicts that up to 30% (the sudden approximation giving only an upper limit) of the prompt neutrons are scission neutrons. •More realistic calculations using the 2 dim TDSE have shown that the sudden approximation is a good approximation (20% error only). They also give the scales of time for adiabatic and extreme diabatic processes.
Summary and Conclusions (cont.) • A strong dependence of the pairing gap at scission ∆ sc on the fragment mass ration A L /A H was found with a pronounced minimum at 1.5 which is the experimental mass division of the main fission mode. This proves the reliability of the neutron energies and wave functions used. • What we usually call ‘prompt neutrons’ contains 2 distinct components well separated in time and having different origins. They are expected to have different behaviours as function of mass asymmetry. This opens new perspectives calling for new improved measurements of neutrons emitted during nuclear fission.
Ongoing extensions •Estimate the nuclear re-absorption by the fragments of the unbound neutrons immediately after scission. This correction has to be taken into account when comparing with experimental data. With Takahiro Wada • Study the influence of the sudden transition at scission on the evaporation of neutrons from fully accelerated fragments (chronology). Note that TXE=E sc + ∆ E def can be calculated for the * light and heavy fragment separately thus avoiding arbitrary hypotheses. With Vitaly Pashkevich
Extra-deformation energy of the primary fragments 71 Ni/ 165 Gd fragmentation
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