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
1
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.
2
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 of radiative capture and fission cross
sections.
3
Cluster Models - 4
4
Cluster Models - 5
5
Cluster Models - 6
In the first experimental campaign (2002-2004) fission
cross section measurements were performed on
actinides of the U-Th fuel cycle (232Th, 233-234-235236U), 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).
6
Cluster Models - 7
Fission cross sections can be calculated with upto-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.
7
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
k

 1
Vk    V0 k 
 2 2    k 2 ,
2
( k  1, N humps  wells ) , with
  0.054A5 / 3 MeV-1
8
Cluster Models - 9
Heights V0k and 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 HartreeFock-Bogoliubov approximation or relativistic
mean-field approximation). In general,
however, theoretical values do not reproduce
experimental fission data and need to be
adjusted.
9
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.
10
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 Tűbingen School (K.
Wildermuth, H. Schultheis, R. Schultheis, F.
Gönnewein). 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 :
11
Cluster Models - 12







h
h 
 r1 ,...,rA0 , t   aijk t  ijk r1 ,...,rA0 , with

h
ijk
h
 
ijk
 



ˆ
r1 ,...,rA0  A i  Ah  j  Al  k R , where

i  Ah  : wave functionof thei th internalstateof theheavycluster
( Ah  Z h  N h )
 j  Al  : wave functionof the j th internalstateof thelight cluster
( Al  Z 0  Z h  N 0  N h )

 k R : wave functionof thek th stateof relativecollectivemotion
 

( R  Rh  Rl , vectordistanceof thecentresof mass of the two

clusters)
Aˆ : ant isymmetrizationoperator
12
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).
13
Cluster Models - 14
On the basis of the above considerations, the two-cluster
expansion can be written as the sum of two terms












 r1 ,...,rA0 ; s   I r1 ,...,rA0 ; s   II r1 ,...,rA0 ; s , where
s  (1 ,  2 ,...) is a set of collective coordinat es describing
the nuclear shape,


 I r1 ,...,rA0 ; s : linear combination of two - cluster st at es


wit h overlapping clusters interiorof theintermediate nucleus,


 II r1 ,...,rA0 ; s : linear combination of two - cluster st at es


wit h clustersin touching configuration  surface region
of theintermediate nucleus.
14
Cluster Models - 15
With the separation given above, the total energy of the intermediate nucleus
becomes
E s    I s  Hˆ  I s    I s  Hˆ  II s 
  II s  Hˆ  I s    II s  Hˆ  II s  , where
 I s  Hˆ  I s   liquid drop energy
 II s  Hˆ  II s   shell correction
 I s  Hˆ  II s  and  II s  Hˆ  I s   small mixed terms
which can be included in theliquid drop energy(wave functions
with large magnitudesin differentpartsof configuration space
 shell effects averagedout ).
15
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
h
h

 II   a00
k max 00 k max ,
h
ΔE   a00
k max ΔE  N h , Z h  ,
2
h
h
wit h ΔE  N h , Z h  theshell energy of thesphericalcluster  N h , Z h  and
h
a00
k max
2
theprobability tofind thecluster configuration  N h , Z h  in st at e II .
In early works ΔE  N h , Z h  was approximated wit h theshell - energy values S  N h , Z h 
of theMyers- Swiatecki mass formula
2
h
estimatedby a variational procedureafterapproximating
and theprobability a00
k max
for a given set s of deformation paramet ers thenuclear surface by a large number
of calot tesof sphericalclusters.
16
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
17
Cluster Models - 18
The 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 (R1/R2)2 = 1 case (with Ri 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.
18
Cluster Models - 19
In these 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.
19
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,
 
 
 II r1 , r2 ,...   a   r1 , r2 ,... , with


( LM )

 
 
( l1 )
( l2 )
l 0  
r1 , r2 ,...   bijk i  A1   j  A2  g k R
LM

, where
ijk
l1 (l2 ) : angular momentumof cluster of mass A1  A2  ,
l0 : angular momentumof relative motion,
L : totalangular momentumwith 3rd componentM .
20
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:
21
Cluster Models - 22



 LM q2 ; r1 ,...,rA0
  a  q ; r ,...,r 
(0)
0 LM
2
A0
1

 2 
   al0l1l2    A1 , 1 , Rm  ; ri  A2 ,  2 , Rm  ; rj 
1
 l0l1l2
Y Ω  Y Ω  Y Ω  , with theconstraint


 ˆ

q ; r ,...,r  Q  q ; r ,...,r   q .
l1
 LM
2
1
A0
1
l2
20
LM
l0
2
2
1
0
A0
LM
2
Here:
(0)
:
LM
wavefunctionof themononucleus, wit h angular momentumL, M
 k 1, 2  : w. f. of cluster of mass Ak , quadrupole deformation  k ,
angular momentumlk and orientation given by Euler angle set Ωk
l0 , Ω0 : angular momentumof relativemotionand orientation

of thedistance vectorR, of (fixed)length Rm .
22
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



 LM q2 ; r1 ,...,rA0
   b ,   ; r ,...,r   d ,
int r
1
A0
with   1 ,  2 ,  0  and   d an infinit esimal volumeelement;
T he weight functionis


b ,     al0l1l2   Yl1 1  Yl2  2  Yl0  0  LM
l0l1l 2
and theintrinsicwave functionis




int r  ; r1 ,...,rA0  1  A1 , 1 , Rm  ; ri 2 A2 ,  2 , Rm  ; rj .


T he weight functioncan be derivedfrom theHill - Wheelerequation




  ; r ,...,r Hˆ  E   ; r ,...,r b ,    d  0 .

int r

1
A0

int r

1
A0

23
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

2
1
d
   d
2M  , E 

db ,   
   
  V  b ,    Eb ,   ,
d



where M  , E  is an effectivemass,      3  , with   
thereduced mass and thepotentialenergyis




V    int r  , r1 ,...,rA0 Hˆ int r  , r1 ,...,rA0 .




Hˆ is thesum of theHamiltonians of the two clustersand of theirinteraction.
One transforms from thelaboratoryframeto theintrinsiccoordinatesystems
of the two clusters:
 
 
~
r  R  D r , (i  1,..., A )
r  R  D ~
r , (i  A  1, ,..., A )
i
1
1
i
1
j
2
2
j
1
0
where D is a Wignerrotation matrix.
24
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
V    E1  , 1 , Rm    E2  ,  2 , Rm    Vint  , Rm  , 1 ,  2 
2
 2 ˆ2
 2 ˆ2
1
ˆ2 ,
l
l2 
l1 
 0   
0
2Rm2  
2 I 2  
2 I1  
2
where thezero - pointenergyof vibrationin thedistance
around the touchingconfiguration is
C  
2 d 2
1
2


int r   ,
R


0    int r  
2
2
2 d R 
2
E1 and E2 are thebinding energies of the two clusters, I1 and I 2
their momentsof inertiaandVint theirinteraction energy.
25
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 ).
26
Cluster Models - 27
27
Cluster Models - 28
In the original version with harmonic oscillator potentials
the Hamiltonian in cylindrical coordinates z,  ,  is
 2 2
H 
 V  , z   VLS (r , p , s )  VL2 ( r , l ) , where
2m N
1
1

2 2
2 2

m

z

m

N z1
N  1  , z  z1

2
2
 f
1
 0 mN  z21 z2 1  c1 z  d1 z2  mN  21  2 1  g1 z2  2 , z1  z  0

2
V  , z    2
f
1
 0 mN  z22 z2 1  c2 z  d 2 z2  mN  2 2  2 1  g 2 z2  2 , 0  z  z2
2
2
1
1

2 2
2
2

m

z

m


, z  z2
N z2
N 2

2
2
with


 z  z1 , z  0
z  
 z  z2 , z  0






28
Cluster Models - 29
T he model has five parameters( collectivecoordinates)
 2
: mass asymmetry
1
 zi
( i  1,2 ) : ellipsoidal deformations of the prefragment s
 i
z 2  z1 : separation of the prefragment s (elongation)
E0
: neck parameter,permittinga variable barrier height
E
1
  1
2
2
 E  mN  z1 z1  mN  z 2 z 2 
2
2


29
Cluster Models - 30
The two-centre shell model is applied as it stands to the
calculation of the Strutinsky shell correction to the liquiddrop 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.
30
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
Z1 Z 2 e 2
s1   s2   S 1 , 2   1 , with
VC d  
d
2
2
c

a
i2  i 2 i and, for prolateellipsoids
d
3  1 1   1  i  3
  2 , (i  1,2)
s i     3  ln
4  i i   1  i  2i
S 1 , 2  
2 j  2k ! 2 j 2 k .
3
3

1
2
j , k 12 j  12 j  3 ( 2k  1)(2k  3) 2 j !2k !

31
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
 F  Fex
VN d   C0  in
  00




 

  
 2   
  r  2 r  d dr   1 r  2 r  d dr
2
1

 

  
 Fex  1 r  2 r  d dr .


Here, thenucleondensities 1 r  and  2 r  are approximated with Woods - Saxon forms,
 00  0.17 fm-3 is thenucleondensityat thecentreof thenucleus,
Finex   f inex   f 'inex 
N1  Z1 N 2  Z 2
,
A1
A2
C0  300 MeVfm3 , f in  0.09, f ex  2.59, f in  0.42, f ex  0.54,
fromfits to experimental data within the frameworkof the theory
of finiteFermisystems.
32
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.
Solving thecollectiveSchrodinger equationsatisfiedby b ,  

2
1
d
db ,   2  2 ˆ 2
2
ˆ2
  

li 
l

0
2






2
M

,
E
d

d

2
I

2

R





i

1
i
m

 E1  , 1 , Rm    E2  ,  2 , Rm    Vint  , Rm  , 1 ,  2 b ,    Eb ,  
one extractsfromb ,   thecoefficients al0l1l2   corresponding
to thecomponentsof thedifferentdinuclear configurations
in theintrinsicwave function.
33
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
T
1
 g  qq q
2 
thequantum - mechanicalkineticoperatoris
(P auli prescription)
2

Tˆ  
2


1

det g q
1
det g g 
and the volumeelementis d 

q
det g  q .
34
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 ξ has a
local minimum at ξ = ξ0 , the motion in ξ is 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
2
2

1



1  2 
3
/
2


Tˆ  



R
3/ 2
2
2 B 0     
 2    R R
R
2 2
ˆ2
3
l
2

 n k
ˆ2 

l
 Tˆrot

0
2
n 
2   R
2 n 1 k 1 I k  n ,  n 

2
2

 1  4 
1
1

 




sin
3


n
n
4
2

 n  n sin 3 n  n
 n 
n 1 Dn  0   n  n
2
1
 Tˆ 
int r
35
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.
36
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 Ω1 and Ω2 .
Vint  A1 , Z1 , 1 ,  1 , 1 ; A2 , Z 2 ,  2 ,  2 ,  2 ; R  
m ll l



1
   Vm m 1 ,  1;  2 ,  2  l1m1; l2 m2
12
1
2
lm Ylm  0 
lm l1l 2 m1m1 m2 m2
 Dml11m1 1 Dml22 m2  2  ,

  are Wigner D - matrix elements.
where D
Neglecting rot at ionaland vibrat ional excit ations of thelight fragmentl2  m2  m2  0
and expandingthe potentialenergy around the minimumcorresponding to theequilibrium
values of the i deformat ions of the clusters one obtainsa simpler formof Vint
and, consequently, of Hˆ .
37
Cluster Models - 38
Hˆ  Hˆ vib  Hˆ rot  Hˆ dns , where
2
2
ˆ2  1

L


1
1


1
2
2 2
3
ˆ


H vib  




C


C


 0 ,
2
2
2 2 

2 D1  
 0    4  0   2
2
Hˆ rot
 2 Lˆ20
 2 ˆ2 ˆ2
m 2
2






L

L

C

1
Y

D
3
rot 
m
0
m 0 1 ,
2
2 R
2I 0
m
Hˆ dns
 2 1  3/ 2   2 1  2 



R
 U 0  , R .
3/ 2
2
2 B  
 2  R R
R


 0 and I 0 refer totheheavycluster,as well as theangular momentumLˆ
and its thirdcomponentLˆ3 . U 0 is thepotentialenergy for clustersin
pole- to - pole configuration (1   2  0).
38
Cluster Models - 39
T he wave functioncan be writtenin theform
lmK  ,  ,  , R;  0 , 1   n  g n K   Al1l0lK  , R   l0 m0 ; l1m1 lm Yml00  0 Dml11K 1  .
l0l1
m0 m1
K is the third componentof theangular momentuml1 of theheavycluster, while
the third componentof the totalangular momentumis not strictlyconserved.
n   and g n K  , which diagonalize the vibrational part of theHamiltonian,
are expressed in termsof Hermiteand Laguerrepolynomial
s, respectively.
T heeigenvalues of Hˆ have the well - known form
vib


K
1

En n K    n      2n 
 1 .
2
2



T he wave functionAl1l0lK  , R  is obtained by numerical diagonalization of Hˆ rot  Hˆ dns .
39
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 = ξ0 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 ε1
of the heavy fragment around its equilibrium position in the DNS.
Cluster Models - 41
Cluster Models - 42
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
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 threehumped 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
234U at ground-state deformation
Both the ground-state band and the
higher bands contain contributions of
the 234U mononucleus and of the 230Th4He dinuclear system.
Intrinsic excitations of the
mononucleus configuration ( beta- and
gamma- bands) are omitted.
Cluster Models - 46
Calculated collective bands of 234U
at the second saddle point
( bending approximation )
46
Cluster Models - 47
Most probable dinuclear configurations for
234U at large deformation
Deformation
Dinuclear
configuration
+ 194Os
Q2 (e fm2)
Q3 (e fm3)
J (ħ2/MeV)
48.45
32.59
241.93
2nd hump
40S
3rd well
102Zr
+ 132Te
69.20
15.57
298.86
106Mo
+ 128Sn
78.05
13.70
324.87
3rd hump
Cluster Models - 48
In order to compute also the contributions of secondchance 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.
Cluster Models - 49
Expt: F. Belloni et al., Eur. Phys. J. A 47 (2011) 2.
Cluster Models - 50
VA
ħωA
VB
ħωB
VC
ħωC
VII
ħωII
VIII
ħωIII
(MeV) (MeV)
(MeV) (MeV)
(MeV) (MeV) (MeV) (MeV)
(MeV) (MeV)
Empire
3
humps
5.35
0.90
5.70
0.80
5.59
0.80
1.40
0.50
2.85
(exp.
3.1±
0.4)
0.60
Empire
2
humps
5.35
0.90
5.80
0.80
-
-
1.40
0.50
-
-
RIPL-3
(Maslov
4.80
0.90
5.50
0.60
-
-
-
-
-
-
1997)
Cluster Models - 51
The experimental energy of the ground state in the
hyperdeformed well of 234U 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)
VA
ħωA
VB
ħωB
VII
ħωII
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
(MeV)
5.38
0.66
6.15
0.45
-
-
Go08
3.80
-
4.89
-
3.22
-
Mö09
7.41
-
6.12
-
2.18
-
Mi11
(Hartree-Fock-Bogoliubov approximation)
Mö-09: P. Möller 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)
Cluster Models - 52
Conclusions on the 233U(n,f) reaction
On the basis of the (n,f) cross section of a fissile nucleus like
233U
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).
The 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 240Pu
Even if the 239Pu(n,f) cross section is not in the plans of the
n_TOF collaboration, we have applied the dinuclear model to
the study of 240Pu 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 ).
The 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
240Pu mononucleus and the 236U-4He 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
Here are the parameters of the predicted three-humped barrier:
VA = 5.27 MeV, VII = 3.09 MeV, VB = 6.30 MeV,
VIII = 2.65 MeV ( I = 307 ħ2 / MeV, most probable DN
configuration : 82Ge + 158Sm ), VC = 3.30 MeV (most probable
DN configuration : 90Kr + 150Ce ).
In order to evaluate the neutron-induced fission cross section up
to En = 20 MeV, analogous calculations are needed for 239Pu
(second-chance fission) and 238Pu (third-chance) ; they are in
progress.
Cluster Models - 59
Angular distributions of fission fragments in the scissionpoint model
Angular 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.
The model permits to describe:
1.
2.
3.
4.
Change of mass and charge asymmetries by nucleon transfer between the two
clusters.
Change of deformation of the clusters.
Angular oscillations around the equilibrium pole-to-pole configuration.
Motion in relative distance of the two clusters.
The 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.
Cluster Models - 60
T heprobability w of formationof a system of two clusters wit h
ci
m asses and charges( Ai , Z i ) and deformations βi  (i  1,2)
ai
at tempera
tureT is proportion
al to a Bolt zmann factor
dependingon theenergyU of thesystem(internalenergiesand interaction
energy of theclusters)


 U A1 , Z1 , 1 , A2 , Z 2 ,  2 , E * 
w A1 , Z1 , 1 , A2 , Z 2 ,  2 , E  exp
,
*
T E


where T is relatedto theexcitationenergy E * by theFermi- gas formula

*

 
E*
A  A2
T
, with theleveldensit y parametera  1
MeV-1.
a
12
E * is takenin thedeepest minimumof thepotentialenergy surface.
Cluster Models - 61
From w one calculatesthe yield of a given pair of clusters
by integration over cluster deformations

*

U
A
,Z
,

,
A
,Z
,

,
E
1 1 1
2
2
2
Y A1,Z1,A2 ,Z 2 , E *  Y0  exp
T

with Y0 a normalization factor.


d d


1
2
T hecharge yield and themass yield are obtainedby furthersummation
over mass and charge,respectively

 U A1,Z1,1 , A2 ,Z 2 ,  2 , E *
Y ( Z1 )  Y0   exp
T
A1


 U A1,Z1,1 , A2 ,Z 2 ,  2 , E *
Y  A1   Y0   exp
T
Z1

d d


1


1
d d
2
2
Cluster Models - 62
In the adjacent figure, the mass
yield predicted by the scissionpoint model for the 239Pu(nth,f)
reaction is compared with the
experimental (pre-neutron
emission) mass yield given by
C. Wagemans et al., Phys.
Rev. C 30 (1984) 218.
Cluster Models - 63
Anotherquantitycomparablewith experiments is theaverage
kineticenergy of fragments(assumed to be equal to theinteraction
energy of thepre - scission configuration)

TK  A1 , Z1 , A2 , Z 2
U Ai , Z i ,  i 
T
T A , Z ,  e



 
d d
e
K
i
i
i
U Ai , Z i ,

T
d1d 2
i
1
,
2
commonlygiven as a functionof themass of one of the two fragments

TK
U Ai , Z i ,  i 
T
T A , Z ,  e

A   

 
d d
e
K
1
Z1
i
i
i
U Ai , Z i ,

T
i
1
2
d1d 2
.
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.
Cluster Models - 65
Let Hˆ dns be theDNS Hamiltonian containingtherotationalenergy
termsof the two clustersand of theirrelativemotion,as well as
the potentialenergy term dependingon cluster orientations
Ω1 and Ω2 in thelaboratoryframe,while Ω0  ( ,  ,  ) is theorientation
of the vectorconnectingthecentres.
Diagonalizationof Hˆ yields eigenfunctions  n
dns
and eigenvalues E nJMK Ai , Z i ,  i .
JMK
Ai , Z i ,  i , i , Ω0 
n
T heprobability of emittingfragments fromstateJMK

between and  d with respect tothenuclear symmetryaxis is
n
JMK
dP
 sin d  
n
JMK
2
dΩ1dΩ2 dd
Cluster Models - 66
T heprobability of finding thefissioningnucleus
n
at thescission configuration ( A1 , Z1 )   A2 , Z 2  in stateJMK
 is
n


E
n
*
JMK
wK Ai , Z i , E  A d1d 2 exp
,
* 
 T E 
with A a normalization factor.
T heangular distribution of fission fragment sin the J , M ,   channel


 
 
 E

 2J 1 
A , Z , E   A 2  exp T E  d  
at tempera
tureT E * is
WJM
*
i
i

 nK
n
JMK
*
J
MK


J
  is a reduced Wignermatrixelement.
where d MK
2
,
One usually measuresangular anisotropies, e. g. rat iosof emission
at   00 with respect to  900.
Cluster Models - 67
A preliminary calculation of angular anisotropy of fission fragments in the 233U(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
The agreement with experiment at low incident energies might be improved
by taking into account the contributions of rotational bands built on noncollective few-quasiparticle states.
An 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.
Both 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 232Th and 234U.
Cluster Models - 70
That is all.
Thank you for your attention !