1 Cluster Models and Nuclear Fission Alberto Ventura (ENEA and INFN, Bologna, Italy ) In...

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 < E n < ~ 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 of radiative capture and fission cross sections.

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

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 V 0k and curvatures k can be either evaluated by a microscopic-macroscopic method (liquid drop model with Strutinskys 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 description of the fission process.

11 Cluster Models - 11 The description of nuclear fission in terms of cluster models dates back to the seventies of past century and is mainly due to the Tbingen School (K. Wildermuth, H. Schultheis, R. Schultheis, F. Gnnewein). See, in particular, the book by Wildermuth and Tang, A unified theory of the nucleus, 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 :

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: 1)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; 2)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 W ith the separation given above, the total energy of the intermediate nucleus becomes

16 Cluster Models - 16 II 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 contribute to the shell correction E

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

18 Cluster Models - 18 T he main results are : 1.the shell correction gives rise to two minima between the spherical shape and the shape corresponding to touching fragments ; 2.the ground-state minimum is associated with the presence of the doubly magic A = 208 cluster; 3.the second minimum is associated with the doubly magic A = 132 cluster; 4.at the barriers in the (R 1 /R 2 ) 2 = 1 case (with R i the radii of the two spherical clusters) the doubly magic clusters are broken up; 5.on the fission path the deformation is symmetric up to the second minimum; 6.the second barrier is lowered by the inclusion of mass asymmetry; 7.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 I n these pioneering works, the shell correction to the fission barrier, albeit in qualitative agreement with Strutinskys prescription, was somewhat oversimplified. I n 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. F rom now on, the nuclear system corresponding to clusters in touching configuration will be defined as Dinuclear Model System (DNS). T he 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 T o begin with, one defines the mass asymmetry coordinate = (A 1 -A 2 )/ (A 1 +A 2 ) (mononucleus: = 1; symmetric fission: = 0) or, more commonly = 1- = 2A 2 / (A 1 +A 2 ) (mononucleus: = 0,2 ; symmetric fission: = 1) and, correspondingly, the charge asymmetry coordinate Z = (Z 1 Z 2 )/ (Z 1 +Z 2 ) Z = 1- Z = 2Z 2 / (Z 1 +Z 2 ) Cluster effects are all included in the II 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) ( = 0,,3), but, for simplicitys sake, we assume an explicit dependence of the nuclear wave function, LM, 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:

23 Cluster Models - 23 Considering the mass asymmetry as a continuous variable, the sum over 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 O n the assumption of a sharply peaked overlap integral the Hill-Wheeler equation can be rewritten in the form of a Schrdinger equation obeyed by the weight function b(,), depending on the collective coordinate

25 Cluster Models - 25 After putting R = R m + R and expanding the Hamiltonian to second order in R 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(,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 Strutinskys 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 ).

30 Cluster Models - 30 T he two-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 (N 1,Z 1 ) and (N 2,Z 2 ) the shell corrections to the energies of two fictitious mononuclei with nucleon numbers (2N 1,2Z 1 ) and (2N 2,2Z 2 ) 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 W hen the symmetry axes of the two spheroidal clusters with major (minor) semiaxes c i (a i ) 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 T he nuclear interaction is calculated in the form of a double-folding potential with Skyrme-type density dependent nucleon-nucleon 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 A s 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 R m c 1 + c 2 + , with 1 fm, owing to the repulsive effect of the Coulomb interaction, superimposed to the attractive nuclear interaction. T he general dependence of the interaction potential on cluster orientations will be discussed later.

34 Cluster Models - 34 T he method outlined above is applicable to only one generator coordinate (mass asymmetry ), but, since more collective coordinates are necessary to describe fission, it would become too cumbersome for practical use. I t 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 A n useful approximation before quantizing the kinetic terms of the cluster Hamiltonian : if the potential energy of the system vs. mass asymmetry has a local minimum at = 0, the motion in is considered a vibration around 0 and the mass parameters associated with collective coordinates are replaced by their values at = 0. The quantized kinetic energy then becomes

36 Cluster Models - 36 T he dinuclear system is then described by 15 degrees of freedom : mass asymmetry , elongation R, 3 Euler angles ( 0 ) for rotation of the system as a whole, 6 Euler angles ( 1, 2 ) for independent rotations of the two clusters, 4 Bohr coordinates ( 1, 1 and 2, 2 ) for intrinsic quadrupole excitations of the two clusters. W e have already assumed for charge asymmetry the values that minimizes potential energy at given mass asymmetry. Further simplifications are possible: I f 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 ( 1 = 0, 1 = 0). In this way, T rot and T intr are greatly simplified.

37 Cluster Models - 37 Potential energy and cluster orientation T he 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 1 and 2.

Cluster Models - 39 39

Cluster Models - 40 Bending approximation A n approximate analytical solution of the DNS Hamiltonian is obtained in the frame of the so-called bending approximation: T he 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 i = ( i, i, i ) ( i = 1,2) defining the orientations of the two clusters reduce to i = ( i, i, 0) if the clusters are stable with respect to deformations. T he mass asymmetry is fixed at the value = 0 of 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 1 of the heavy fragment around its equilibrium position in the DNS.

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Cluster Models - 43 T his 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

Cluster Models - 44 Application to 233 U(n,f) The cross section of the neutron-induced fission of 233 U has been measured by the n_TOF Collaboration in the energy range 0.5 < E n < 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.

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

46 Cluster Models - 46 Calculated collective bands of 234 U at the second saddle point ( bending approximation )

Cluster Models - 47 Most probable dinuclear configurations for 234 U at large deformation Deformation Dinuclear configuration Q 2 (e fm 2 ) Q 3 (e fm 3 ) J ( 2 /MeV) 2 nd hump 40 S + 194 Os48.4532.59241.93 3 rd well 102 Zr + 132 Te69.2015.57298.86 3 rd hump 106 Mo + 128 Sn78.0513.70324.87

Cluster Models - 48 In order to compute also the contributions of second- chance fission, 233 U(n,nf), and third-chance fission, 233 U(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 233 U and 232 U, 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.

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

Cluster Models - 50 V A (MeV) A (MeV) V B (MeV) B (MeV) V C (MeV) C (MeV) V II (MeV) II (MeV) V III (MeV) III (MeV) Empire 3 humps 5.350.905.700.805.590.801.400.502.85 (exp. 3.1 0.4) 0.60 Empire 2 humps 5.350.905.800.80--1.400.50-- RIPL-3 ( Maslov 1997) 4.800.905.500.60------

Cluster Models - 51 The experimental energy of the ground state in the hyperdeformed well of 234 U is taken from A. Krasznahorkay et al., Phys. Lett. B 461 (1999) 15. The heights of humps used in the fit of the fission cross section are (not surprisingly) close to the experimental RIPL-3 systematics (Maslov, 1997), but somewhat different from the values predicted by Strutinsky-type calculations performed in the frame of the present work, as well as from other theoretical predictions in the literature : Go-08 : S. Goriely et al., RIPL-3 (2008) (Hartree-Fock-Bogoliubov approximation) M-09: P. Mller et al., Phys. Rev. C 79 (2009) 064304 (Strutinsky method) Mi-11: M. Mirea and L. Tassan-Got, Cent. Eur. J. Phys. 9 (2011) 116 (Strutinsky method) V A (MeV) A (MeV) V B (MeV) B (MeV) V II (MeV) II (MeV) 5.380.666.150.45--Go- 08 3.80-4.89-3.22-M- 09 7.41-6.12-2.18-Mi- 11

Cluster Models - 52 Conclusions on the 233 U(n,f) reaction O n the basis of the (n,f) cross section of a fissile nucleus like 233 U it is, of course, not possible to decide on the structure of the barrier: evidence for a three-humped structure comes from the (d,pf) measurements (J. Blons et al., Nucl. Phys. A 477 (1988) 231, reanalyzed by A. Krasznahorkay et al., Phys. Rev. Lett. 80 (1998) 2073, and A. Krasznahorkay et al., Phys. Lett. B 461 (1999) 15). T he fit of the (n,f) cross section gives indications in favour of a reflection-asymmetric shape of the transition states built on the main peak of the fission barrier.

Cluster Models - 53 Preliminary study of 240 Pu E ven if the 239 Pu(n,f) cross section is not in the plans of the n_TOF collaboration, we have applied the dinuclear model to the study of 240 Pu as a compound fissioning nucleus, owing to the detailed experimental information on the spectrum in the second well of the barrier (as reviewed by P. G. Thirolf and D. Habs, Prog. Part. Nucl. Phys. 49 (2002) 325 ). T he following figures compare experimental and calculated spectra in the ground-state well and in the isomeric well: in both cases calculated states are mainly superpositions of the 240 Pu mononucleus and the 236 U- 4 He dicluster configurations.

Cluster Models - 54 Ground-state well

Cluster Models - 55 Ground-state well - continued

Cluster Models - 56 Isomeric (superdeformed) well

Cluster Models - 57 But the model predicts also a third (hyperdeformed) well with a spectrum characteristic of a reflection-asymmetric system (treated in bending approximation)

Cluster Models - 58 H ere are the parameters of the predicted three-humped barrier: V A = 5.27 MeV, V II = 3.09 MeV, V B = 6.30 MeV, V III = 2.65 MeV ( I = 307 2 / MeV, most probable DN configuration : 82 Ge + 158 Sm ), V C = 3.30 MeV (most probable DN configuration : 90 Kr + 150 Ce ). In order to evaluate the neutron-induced fission cross section up to E n = 20 MeV, analogous calculations are needed for 239 Pu (second-chance fission) and 238 Pu (third-chance) ; they are in progress.

Cluster Models - 59 Angular distributions of fission fragments in the scission- point model A ngular distributions of fission fragments can be evaluated in the scission-point limit, where the fissioning nucleus can be considered as a system of two separated interacting prefragments in thermal equilibrium, which can be described by the dinuclear model at finite temperature. T he model permits to describe: 1.Change of mass and charge asymmetries by nucleon transfer between the two clusters. 2.Change of deformation of the clusters. 3.Angular oscillations around the equilibrium pole-to-pole configuration. 4.Motion in relative distance of the two clusters. T he basic formulation of the scission-point model was given by B. D. Wilkins, E. P. Steinberg and R. R. Chasman, Phys. Rev. C 14 (1976) 1832.

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Cluster Models - 62 I n the adjacent figure, the mass yield predicted by the scission- point model for the 239 Pu(n th,f) reaction is compared with the experimental (pre-neutron emission) mass yield given by C. Wagemans et al., Phys. Rev. C 30 (1984) 218.

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Cluster Models - 64 Comparison of calculated and experimental average kinetic energy of fragments for neutron-induced fission of some U-Pu isotopes as functions of the mass number of the light fragment, from A. V. Andreev et al., Eur. Phys. J. A 22 (2004) 51.

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Cluster Models - 67 A preliminary calculation of angular anisotropy of fission fragments in the 233 U(n,f) reaction vs. incident neutron energy. Expt. data from J.E. Simmons and R. L. Henkel, Phys. Rev. 120 (1960) 198 and R. B. Leachman and L. Blumberg, Phys. Rev. 137 (1965) B814.

Cluster Models - 68 T he agreement with experiment at low incident energies might be improved by taking into account the contributions of rotational bands built on non- collective few-quasiparticle states. A n alternative to the scission-point model in investigating angular anisotropies of fission fragments is the transition state model, based on the assumption that the quantum numbers of the states responsible for angular distributions are those at the outer saddle point, considered as frozen in the descent from saddle to scission. B oth are static models, unable to describe angular distributions of fission fragments over the broad range of incident neutron energies covered by the n_TOF facility (almost 1 GeV): dynamical models should come into play.

Cluster Models - 69 Planned improvements of the DNS model Introduction of stable triaxial shapes for a more realistic evaluation of fission barrier parameters. Consistent evaluation of the collective enhancement factor of nuclear level densities at large deformations. Planned calculations of fission cross sections Non-fissile actinides measured by the n_TOF collaboration, with particular reference to 232 Th and 234 U.

Cluster Models - 70 That is all. Thank you for your attention !

1 Cluster Models and Nuclear Fission Alberto Ventura (ENEA and INFN, Bologna, Italy ) In...

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