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Cluster Models and Nuclear Fission

Cluster Models and Nuclear Fission. Alberto Ventura (ENEA and INFN, Bologna, Italy ). In collaboration with Timur M. Shneydman and Alexander V. Andreev (BLTP, JINR Dubna, Russian Federation) Cristian Massimi and Gianni Vannini (University of Bologna and INFN, Bologna, Italy)

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Cluster Models and Nuclear Fission

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  1. Cluster Models and Nuclear Fission Alberto Ventura (ENEA and INFN, Bologna, Italy) In collaboration with Timur M. Shneydman and Alexander V. Andreev (BLTP, JINR Dubna, Russian Federation) Cristian Massimi and Gianni Vannini (University of Bologna and INFN, Bologna, Italy) Debrecen, March 27, 2012

  2. Cluster Models - 2 Motivation of theoretical research Analysis of neutron-induced fission cross sections and angular distributions of fission fragments measured by the n_TOF (neutron TIME-OF-FLIGHT) Collaboration at CERN, Geneva, since 2002. The n_TOF facility is dedicated to the measurement of neutron capture and fission cross sections, the former of main interest to nuclear astrophysics, the latter to reactor physics.

  3. Cluster Models - 3 The n_TOF Facility Neutrons with a broad energy spectrum ( ~10-2 eV < En < ~ 1 GeV) are produced by 20 GeV/c protons from the CERN Proton Synchrotron impinging on a lead block surrounded by a water layer acting as a coolant and a moderator of the neutron spectrum. Neutron energies are measured by the time-of-flight method in a ~ 187 m flight path; hence the name of the collaboration. The neutron beam is used for measurements ofradiative capture and fission cross sections.

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  6. In the first experimental campaign (2002-2004) fission cross section measurements were performed on actinides of the U-Th fuel cycle (232Th, 233-234-235-236U), natural lead, 209Bi and minor actinides (237Np, 241-243Am and 245Cm). In the current campaign, started in 2008, cross section measurements are planned for 240-242Pu and minor actinides (231Pa) , as well as on angular distributions of fission fragments (232Th(n,f), 234-236U(n,f)) up to high incident neutron energies ( ~ 1 GeV). Cluster Models - 6

  7. Cluster Models - 7 • Fission cross sections can be calculated with up-to-date versions of nuclear reaction codes, such as Empire-3.1 (www.nndc.bnl.gov) and Talys-1.4 (www.talys.eu), whose fission input admits multiple-humped fission barriers and barrier penetrabilities depending on discrete as well as continuum (level densities) spectra at the humps and in the wells of the barriers.

  8. Cluster Models - 8 • In particular, fission barriers can be given either in numerical form or parametrized with a set of smoothtly joined parabolas, as functions of an appropriate coordinate along the fission path

  9. Cluster Models - 9 • Heights V0kand curvatures ħωk can be either evaluated by a microscopic-macroscopic method (liquid drop model with Strutinsky’s shell and pairing corrections) or by a fully microscopic method (non-relativistic Hartree-Fock-Bogoliubov approximation or relativistic mean-field approximation). In general, however, theoretical values do not reproduce experimental fission data and need to be adjusted.

  10. Cluster Models - 10 • In addition to barrier parameters, also discrete states and level densities at the humps and in the wells of the barrier are basic ingredients of the statistical model of nuclear fission and can be evaluated by microscopic-macroscopic or fully microscopic methods (at least in principle, in the latter case). • Purpose of this work is to investigate the possible use of nuclear cluster models in the descriptionof the fission process.

  11. Cluster Models - 11 • Thedescription of nuclear fission in terms of cluster models dates back to the seventies of past century and is mainly due to the Tűbingen School (K. Wildermuth, H. Schultheis, R. Schultheis, F. Gönnewein). See, in particular, the book by Wildermuth and Tang, A unified theory of thenucleus, Vieweg, Braunschweig, 1977. • Starting point of the formalism is the representation of the time-dependent wave function of the fissioning nucleus as a linear superposition of two-cluster wave functions :

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  13. Cluster Models - 13 • The two-cluster expansion given above allows for any overlap of clusters. If the overlap is strong, the antisymmetrization operator washes out the effects of cluster decomposition. • There are two regions where antisymmetrization effects play a minor role: • clusters well separated in momentum space → strong overlap in coordinate space ( R ≈ 0 ) → no connection with nuclear shape → peculiar role of Z = 82 and N = 126 shell closures in actinide ground states; • clusters well separated in coordinate space → higly excited state of relative motion → clusters in low-lying internal states of excitation → directly connected with nuclear shape (reflection asymmetry).

  14. Cluster Models - 14 • On the basis of the above considerations, the two-cluster expansion can be written as the sum of two terms

  15. Cluster Models - 15 • With the separation given above, the total energy of the intermediate nucleus becomes

  16. Cluster Models - 16 • ΦII basically contains contributions from spatially separated clusters in their ground states and, therefore, in the highest excited state of relative motion allowed by the excitation energy of the intermediate nucleus and only upper single-nucleon states contributing to <ΦII |H| ΦII >contribute to the shell correction ΔE

  17. Cluster Models - 17 • An application to the fission barrier of 236U is given by H. Schultheis, R. Schultheis and K. Wildermuth, Phys. Lett. 53B (1974) 325

  18. Cluster Models - 18 • The main results are : • the shell correction gives rise to two minima between the spherical shape and the shape corresponding to touching fragments ; • the ground-state minimum is associated with the presence of the doubly magic A = 208 cluster; • the second minimum is associated with the doubly magic A = 132 cluster; • at the barriers in the (R1/R2)2 = 1 case (with Ri the radii of the two spherical clusters) the doubly magic clusters are broken up; • on the fission path the deformation is symmetric up to the second minimum; • the second barrier is lowered by the inclusion of mass asymmetry; • between the second minimum and the scission point the path of minimum energy corresponds to those asymmetric deformations which leave the doubly magic A = 132 cluster largely unbroken.

  19. Cluster Models - 19 • Inthese pioneering works, the shell correction to the fission barrier, albeit in qualitative agreement with Strutinsky’s prescription, was somewhat oversimplified. • In present day applications of cluster models to fission, one usually adopts a hybrid procedure in which the Strutinsky approach to shell and pairing corrections is applied to the mononucleus configuration dominant in early stages of fission (up to about the second minimum of the barrier) as well as to the separated clusters appearing at larger deformations. • From now on, the nuclear system corresponding to clusters in touching configuration will be defined as Dinuclear Model System (DNS). • The mononucleus configuration can be included in the DNS on the formal assumption that it is coupled with a light cluster of zero mass.

  20. Cluster Models - 20 • To begin with, one defines the mass asymmetry coordinate • η = (A1-A2)/ (A1+A2) (mononucleus: η = ± 1; symmetric fission: η = 0) • or, more commonly • ξ = 1- η = 2A2 / (A1 +A2) (mononucleus: ξ = 0,2 ; symmetric fission: ξ = 1) and, correspondingly, the charge asymmetry coordinate • ηZ = (Z1 –Z2 )/ (Z1 +Z2 ) → ξZ = 1- ηZ = 2Z2 / (Z1 +Z2 ) • Cluster effects are all included in the ΦII function ; neglecting antisymmetrization,

  21. Cluster Models - 21 • In order to compute the fission barrier and the collective excitations of the fissioning nucleus at the humps and in the wells we need the wave function of the DNS at given elongation (separation of the cluster centres). • In general, the DNS will be described by a set of mass and charge multipole moments, Q(c,m)λμ(λ = 0,…,3) , but, for simplicity’s sake, we assume an explicit dependence of the nuclear wave function, ΦLM, on quadrupole moment only. • Moreover, the simplifying assumptions are made: • The quadrupole deformations of the clusters are chosen so as to minimize the energy of the DNS. • Intrinsic excitations of the clusters are not allowed. • The relative distance, R, is not an independent variable and is fixed, for a given mass asymmetry, at the touching configuration of the clusters. Thus:

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  23. Cluster Models - 23 • Considering the mass asymmetry ξ as a continuous variable, the sum over ξ is replaced with an integral and the wave function of the intrinsic state can be written in the Hill-Wheeler form

  24. Cluster Models - 24 • On the assumption of a sharply peaked overlap integral the Hill-Wheeler equation can be rewritten in the form of a Schrödinger equation obeyed by the weight function b(ξ,Ω), depending on the collective coordinate ξ

  25. Cluster Models - 25 • After putting R = Rm + δR and expanding the Hamiltonian to second order in δR one obtains the potential energy in the form

  26. Cluster Models - 26 • The moments of inertia of the clusters can be calculated by means of the Inglis formalism. • A similar formalism can be adopted for the effective mass, M(ξ,E), • and is presented in : • G. G. Adamian et al., Nucl. Phys. A 584 (1995) 205. • The binding energies of the (deformed) clusters are evaluated in the Strutinsky’s microscopic-macroscopic approach, with shell and pairing energy corrections computed with the two-centre shell model, suited to the description of nuclei with large deformations ( J. Maruhn and W.Greiner, Z. Phys. 251 (1972) 431).

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  30. Cluster Models - 30 • Thetwo-centre shell model is applied as it stands to the calculation of the Strutinsky shell correction to the liquid-drop energy of the deformed mononucleus configuration. • For a configuration of two different clusters with neutron and proton numbers (N1,Z1) and (N2,Z2) the shell corrections to the energies of two fictitious mononuclei with nucleon numbers (2N1,2Z1) and (2N2,2Z2) are computed separately and the results divided by two in order to get the values of the shell corrections of the single clusters. In this way it is possible to treat clusters with different N/Z ratios.

  31. Cluster Models - 31 • Coulomb interaction between clusters • When the symmetry axes of the two spheroidal clusters with major (minor) semiaxes ci(ai) coincide with the line connecting the centres, at distance d (pole-to-pole configuration) the Coulomb interaction energy is

  32. Cluster Models - 32 • Nuclear interaction between clusters • The nuclear interaction is calculated in the form of a double-folding potential with Skyrme-type density dependent nucleon-nucleon δ forces (G. G. Adamian et al., Int. J. Mod. Phys. E 5 (1996) 191). • For separated clusters momentum and spin dependence of the nucleon-nucleon interaction are neglected. The final result is

  33. Cluster Models - 33 • As a function of elongation (distance of cluster centres) the interaction potential has a minimum at a value slightly larger than the sum of the two major semiaxes • Rm≈ c1 + c2 + Δ , with Δ ≈ 1 fm, owing to the repulsive effect of the Coulomb interaction, superimposed to the attractive nuclear interaction. • The general dependence of the interaction potential on cluster orientations will be discussed later.

  34. Cluster Models -34 • The method outlined above is applicable to only one generator coordinate (mass • asymmetry ξ), but, since more collective coordinates are necessary to describe fission, it would become too cumbersome for practical use. • It is more convenient to write down the classical Hamilton function appropriate to the model and then quantize it by standard procedures. If the classical kinetic energy is of the form

  35. Cluster Models - 35 • An useful approximation before quantizing the kinetic terms of the cluster Hamiltonian : if the potential energy of the system vs. mass asymmetry ξhas a local minimum at ξ = ξ0 , the motion in ξis considered a vibration around ξ0 and the mass parameters associated with collective coordinates are replaced by their values at ξ = ξ0 . The quantized kinetic energy then becomes

  36. Cluster Models - 36 • The dinuclear system is then described by 15 degrees of freedom : mass asymmetry ξ, elongation R, 3 Euler angles (Ω0) for rotation of the system as a whole, 6 Euler angles (Ω1, Ω2) for independent rotations of the two clusters, 4 Bohr coordinates (β1,γ1 and β2,γ2) for intrinsic quadrupole excitations of the two clusters. • We have already assumed for charge asymmetry the values that minimizes potential energy at given mass asymmetry. Further simplifications are possible: • If we are interested in the lowest-lying excitations of the system, we can either neglect intrinsic excitations of the two clusters, or, limit ourselves to the small oscillations of the heavier cluster around its equilibrium shape • (β1 = β0 , γ1 = 0). In this way, Trot and Tintr are greatly simplified.

  37. Cluster Models - 37 • Potential energy and cluster orientation • The interaction potential, previously given for co-linear clusters in a pole-to-pole configuration, depends in general on the mutual orientation of the two clusters and can be expanded into multipoles of their Euler angles Ω1and Ω2.

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  40. Cluster Models -40 • Bending approximation • An approximate analytical solution of the DNS Hamiltonian is obtained in the frame of the so-called bending approximation: • The Hamiltonian is written in the DNS-fixed coordinate system, with z axis along the vector R of separation of the two centres, and the Euler angles Ωi= (φi , εi , αi) ( i = 1,2) defining the orientations of the two clusters reduce to Ωi = (φi , εi , 0) if the clusters are stable with respect to γ deformations. • The mass asymmetry is fixed at the value ξ = ξ0of the most probable dinuclear configuration corresponding to a minimum or a maximum of the fission barrier. • On the above approximations, the lowest-collective modes correspond to the rotation of the DNS as a whole and the oscillations in the bending angle ε1 of the heavy fragment around its equilibrium position in the DNS.

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  43. Cluster Models - 43 • This seems to be a good approximation for the collective states at the humps of a fission barrier (transition states), not for the states in the wells

  44. Cluster Models - 44 • Application to 233U(n,f) • The cross section of the neutron-induced fission of 233U has been measured by the n_TOF Collaboration in the energy range 0.5 < En < 20 MeV (F. Belloni et al., Eur. Phys. J. A 47 (2011) 2) • and has been studied by means of the Empire-3 code (M. Herman et al., Nucl. Data Sheets 108 (2007) 2655), using in the fission input of the code the parameters of the three-humped fission barrier predicted by the DNS approach as a first guess, together with the collective bands computed in the same model for the secondary wells and the humps of the barrier. • In the latter case (transition states) use is made of the bending approximation, valid for reflection-asymmetric shapes.

  45. Cluster Models - 45 • Calculated collective bands of 234U at ground-state deformation • Both the ground-state band and the higher bands contain contributions of the 234U mononucleus and of the 230Th-4He dinuclear system. • Intrinsic excitations of the mononucleus configuration ( beta- and gamma- bands) are omitted.

  46. Cluster Models - 46 • Calculatedcollective bands of 234U at the second saddle point • ( bending approximation )

  47. Cluster Models - 47 • Most probable dinuclear configurations for 234U at large deformation

  48. Cluster Models - 48 • In order to compute also the contributions of second-chance fission, 233U(n,n’f), and third-chance fission, 233U(n,2nf), the DNS model has been applied to the evaluation of the fission barriers and collective spectra at barrier humps and wells for the fissioning nuclei 233U and 232U, respectively. • The theoretical spectra have been kept fixed, but the calculated humps and wells have been adjusted so as to reproduce the experimental fission cross section.

  49. Cluster Models - 49 • Expt:F. Belloni et al., Eur. Phys. J. A 47 (2011) 2.

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