1 Cluster Models and Nuclear Fission Alberto Ventura (ENEA and INFN, Bologna, Italy ) In...
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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
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
Slide 2
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.
Slide 3
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.
Slide 4
4 Cluster Models - 4
Slide 5
5 Cluster Models - 5
Slide 6
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).
Slide 7
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.
Slide 8
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
Slide 9
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.
Slide 10
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.
Slide 11
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 :
Slide 12
12 Cluster Models - 12
Slide 13
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).
Slide 14
14 Cluster Models - 14 On the basis of the above
considerations, the two-cluster expansion can be written as the sum
of two terms
Slide 15
15 Cluster Models - 15 W ith the separation given above, the
total energy of the intermediate nucleus becomes
Slide 16
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
Slide 17
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
Slide 18
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.
Slide 19
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.
Slide 20
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,
Slide 21
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:
Slide 22
22 Cluster Models - 22
Slide 23
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
Slide 24
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
Slide 25
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
Slide 26
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 ).
Slide 27
27 Cluster Models - 27
Slide 28
28 Cluster Models - 28
Slide 29
29 Cluster Models - 29
Slide 30
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.
Slide 31
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
Slide 32
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
Slide 33
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.
Slide 34
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
Slide 35
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
Slide 36
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.
Slide 37
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.
Slide 38
38 Cluster Models - 38
Slide 39
Cluster Models - 39 39
Slide 40
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.
Slide 41
Cluster Models - 41
Slide 42
Cluster Models - 42
Slide 43
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
Slide 44
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.
Slide 45
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.
Slide 46
46 Cluster Models - 46 Calculated collective bands of 234 U at
the second saddle point ( bending approximation )
Slide 47
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
Slide 48
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.
Slide 49
Cluster Models - 49 Expt: F. Belloni et al., Eur. Phys. J. A 47
(2011) 2.
Slide 50
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------
Slide 51
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
Slide 52
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.
Slide 53
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.
Slide 54
Cluster Models - 54 Ground-state well
Slide 55
Cluster Models - 55 Ground-state well - continued
Slide 56
Cluster Models - 56 Isomeric (superdeformed) well
Slide 57
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)
Slide 58
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.
Slide 59
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.
Slide 60
Cluster Models - 60
Slide 61
Cluster Models - 61
Slide 62
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.
Slide 63
Cluster Models - 63
Slide 64
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.
Slide 65
Cluster Models - 65
Slide 66
Cluster Models - 66
Slide 67
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.
Slide 68
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.
Slide 69
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.
Slide 70
Cluster Models - 70 That is all. Thank you for your attention
!