Transcript Slide 1

Study of sub-barrier fission
resonances with gamma beams
Dan FILIPESCU1, Mihaela SIN2
1) Horia Hulubei - National Institute for Physics and Nuclear Engineering,
2) Bucharest University, Romania
[email protected]
Why fission is still a hot topic?
No theory or model is able yet to predict all fission observables
(fission cross section, post-scission neutron multiplicities and
spectra, fission fragments’ properties like mass, charge, total kinetic
energy and angular distributions) in a consistent way for all
possible fissioning systems in a wide energy range.
New nuclear technologies design requires increased accuracy of the
fission data refinements of the fission models and better predictive
power.
To reach the target accuracies more structural and dynamic features
of the fission process must be included in the theoretical models.
Why photofission?
• Photofission experiments done on available targets
may produce additional information to the existing
available data from experiments done with neutrons
- differences in compound nucleus states population
must be taken into account
• Characteristics of ELI-NP Gamma Sorce offer for
the first time a viable alternative to the study of subbarrier fission resonances with neutrons, even below
Sn
- very high gamma intensity and excellent energy
resolution are required
Fission process
Vf
~ 6 MeV
~ 40 MeV
~ 200 MeV
scission point
elongation
Liquid Drop Model
- only the Coulomb (EC) and surface term (ES) depend on deformation
- near g.s. ES increases more rapidly than EC decreases – V rises
- at larger deformation the situation is reversed – V drops
From the initial state to scission
The fissioning system shape modifies continuously during the motion from the
formation of the initial state (characterized by a small deformation) to the elongated
asymmetric pre-scission shape and even scission (where the nuclear system is
composed of two touching fragments). Several types of parameterization and sets of
deformation (shape) parameters {q} may be required to describe completely the
fissionning nucleus in its various stages.
potential energy surface – potential energy as
function of deformation V({q})
q2
a th
G
HI
fission path – corresponds to the lowest
potential energy when increasing deformation
fission barrier – one-dimensional representation
of V; V as function of one deformation coordinate
(ex. elongation)
LOW
np
LOW fissio
H
HIGH
q1
V
Ef
deformation along
the fission path
Fission barrier
- macroscopic models: LDM
– give only a qualitative account of the phenomenon
- microscopic-macroscopic model: LDM + shell correction
– explained a significant number of experimental data
- microscopic models: HFB
– can not provide accurate results for Vf yet, but provide
the trend and are improving
Fission barrier
Strutinsky’s procedure
V
LD
V (  )  VLD (  )  Vsh (  )
( )
V( )
vsh ()



The value of the shell correction is +, - depending on whether the density of
single-particle states at Fermi surface is great or small.
Negative corrections for actinides
- g.s. - permanent deformation
- vicinity of macroscopic saddle point
– second well
double-humped fission barrier
Fission barrier
Vf
Variation of the shell correction
amplitude with changing Z,N together
with the variation of the LD potential
barrier with changing fissility parameter
(EC/2ES) lead to a variation of the fissionVf
barrier from nucleus to nucleus:
- inner barriers almost constant 5-6 MeV
for the main range of actinides; fall
rapidly in Th region;
- secondary well’s depth around 2 MeV;
- outer barriers fall quite strongly from
the lighter actinides (6-7 MeV for Th) to
the heavier actinides (2-3 MeV for Fm).
light actinides
(Th)

medium actinides
(Pu)

Vf
heavy actinides
(Cf)

Fission barrier
Early calculations assumed a maximum degree of symmetry of the nuclear
shape along the fission path and the theoretical predictions were not in
agreement with the experimental barrier heights and the asymmetric mass
distribution of the fission fragments.
Extensive studies concluded that neutron rich nuclei have axial asymmetry
at the inner saddle point and reflection (mass) asymmetry at the outer
saddle point.
Vf
These results have large
implications for the barrier heights
and for the level densities at the
saddle points and explain the mass
asymmetry of the fission fragments.
reflection
axial
asymmetry asymmetry
elongation
Symmetry of inner barrier
Shell corrections performed by Pashkevich and subsequent by Howard and
Moller prooved that along an isotopic chain exists a mass Atr:
- A > Atr inner barrier asymmetry – AAMS (higher inner barr. than the outer one)
- A < Atr inner barrier symmetry – ASMS (lower inner barr. than the outer one)
0.26
0.24
Ac Th Pa
U Np Pu
236 238
Am Cm AAMS region
240 242
244 247
249 252
0.22
(axial asymmetric
mass symmetric)
tr
=(N-Z)/A
tr
tr
0.20
0.18
0.16
0.14
0.72
ASMS region
225
(axial and mass
symmetry)
218
228 231 233
212
206 209
0.74
0.76
2
0.78
0.80
0.82
2
XF = (Z /A) / 50.883(1-1.7826* )
0.84
Fission barrier
The multi-modal fission
In Brosa model the mass distribution of the fission fragments can be described
considering minimum 3 pre-scission shapes, 3 fission paths which branch from the
standard fission path at certain bifurcation points on the potential energy surface. To
each of them corresponds a fission mode: symmetric super-long (SL) and
asymmetric standard 1 (ST I) and II (ST II).
SL
The different barrier characteristics give rise to V
a separate fission probability along the various
fission paths. The corresponding fission
probabilities should add up to the total fission
probability.
The final distributions of the fission fragment
properties (mass, charge and TKE) are a
superposition of the different distributions
stemming from the various fission modes.
ST I
ST II
elongation
Fission barrier
The triple-humped fission barrier (Th anomaly)
In the thorium region, the second hump appears just under the maximum predicted
by the liquid drop model, therefore its exact shape is very sensible to the shell
effects. It was demonstrated that a shell effect of second-order would split the outer
barrier giving rise to a third very shallow well.
A triple-humped barrier for the actinides in thorium region, allowing the existence of
hyper-deformed undamped class III vibrational states
could explain the
disagreement between the calculated and
Vf
experimental inner barrier height and
also the structure in the fission cross
section of non-fissile Th, Pa and light
U isotopes.

Fission barrier - description
• parabolic barriers
• numerical barriers
Static fission barriers extracted from full 3-dimensional HFB energy
surfaces as function of quadrupole deformation
Fission barrier - description
Vf
• parabolic barriers
humps only described by decoupled parabolas

1 2 2
2
V fi  E fi   i (   i )
2
Vf
Vf


entire fission path described by smoothly joined parabolas
1 2 2
V fi  E fi   i (    i ) 2
2
Vf
Vf

Vf
Vf

  0.054 A5 / 3MeV -1

It is customary to assume that the mass tensor it is diagonal in the deformation
space coord. system and all diagonal elements are equal to a constant which is
very important for the fission barrier parameterization and for the transmission
coefficient calculation.

Fission barriers
Transition states, class I, II, …states
transition states
(fission channels)
Vf
class I
states
class II
states
elongation
Fission barriers
Transition states, class I, II, …states
- transition states – excited states at saddle points
- class I (II,III) states – states of the nucleus with deformation corresponding
to first (second, third) well
Fission barriers
Transition states, class I, II, …states
ri (E*Jp)
Vf
ei (KJp )
Vf
Eci (Jp )
ri (E*Jp)
Eci (Jp )
ei (KJp )
V3
V2
V1
V2
V1
VIII
VII
VII
WII

WII
WIII

Fission barriers
Transition states
2

- discrete E ( J Kp )  E  e ( Kp ) 
[ J ( J  1)  K ( K  1)]
i
fi
i
2i
Quantum structure of the fission channels is important for accurate descriptions of
fission cross sections at the sub-barrier and near-barrier energies; it depends on the
nuclear shape asymmetry and the odd-even nucleus type.
Mirror-asymmetric even-even nuclei have ground-state rotational band levels with
Kπ = 0+, J = 0, 2, 4... - that unify with the octupole band levels Kπ = 0−, J = 1, 3, 5...
in the common rotational band. Additional unification arises for axial asymmetric shapes
and levels of the γ-vibrational band with Kπ = 2+, J = 2, 4...
The quantum number of the corresponding rotational bands for odd and odd-odd nuclei
are estimated in accordance with the angular momentum addition rules for unpaired
particles and the corresponding rotational bands.
- continuum ri ( E * Jp )
Consistent treatment of all reaction channels would require for the transition state
densities the same formulation used for the normal states, adapted to consider the
deformation and collective enhancement specific for each saddle point.
Fission barrier - description
• numerical barriers
- Static fission barriers extracted from full 3-dimensional HFB energy
surfaces as function of quadrupole deformation
10
Cm252
Cm254
Cm256
Cm258
Cm260
Cm262
Cm264
Cm266
Cm268
E-EGS [MeV]
8
6
Cm270
Cm272
Cm274
Cm276
Cm278
Cm280
4
2
0
0
0.5
1
1.5
β
2
2
0
0.5
1
1.5
β
2
- Global microscopic nuclear level densities within the HFB plus
combinatorial method.
2
2.5
Initial state
(γ, p, α…) induced fission:
• CN states populated directly or after gamma-decay - (n,f)
• states in residual nuclei - (n,n’f), (n,2nf), (n,3nf)
- neutron
(n,f)
E
n'
E*
sf
sn,f
E
n+T
Sn
CN
Initial state
- neutron
induced fission: fertile and fissile nuclei
gamma-ray induced fission: no such classification
E*6
236
(m5 +mn)c 2
En
U*
235
U
Sn
6
236
U
m6 c2

239
E*9
(m8 +mn)c 2
U*
En
238
U
Sn
9
239
m9
c2
U

Fission in CN statistical model
The fission cross section:
s f ( E )  s ( E, J p ) Pf ( E, Jp )
Jp
s  ( E, Jp )
Pf ( E , Jp ) 
- cross section of the initial state formation
T f ( E , Jp )
T f ( E , Jp )  T ' ( E , Jp )
- fission probability
'
 - entrance channel
’ - outgoing competing channels
E - energy of the incident particle inducing fission
Jπ - spin, parity of the CN state
Transmission coefficients
Vf
Efi ħωi
• parabolic barrier
Hill-Wheeler formula for the transmission
coefficient through a parabolic barrier:
Ti 
1
 2π
* 
1  exp 
(E fi  E ) 
 ω i


Transmission coefficients
• non-parabolic barrier
Vf
E*
The coefficients Ti are expressed in the
first-order WKB approximation in terms
of the momentum integrals for the humps:
a
b

bi
M i   [2 E * Vi (  )  /  2 ]1 / 2 d

ai
1
parabola
Ti 
 Hill  Wheeler
1  exp( 2M i )
Transmission coefficients - decoupled humps
T2
1
T1,2  T1 

T1  T2 1  1
T1 T2
T2,3
1
T1,3  T1 

T1  T2,3 1  1  1
T1 T2 T3
Vf

Vf

Nh humps
1
T1, N h
Nh
1

i Ti
T1, N h
1
 

Nh
E *V f
Transmission coefficients – entire barrier
Vf
Vf
WKB approximation

Tdir12 
Tdir13 

T1T2
1  2(1  T1 )1 / 2 (1  T2 )1 / 2 cos(2 )  (1  T1 ) (1  T2 )
T1Tdir23
1  2(1  T1 )1/ 2 (1  Tdir23 )1/ 2 cos( 2 )  (1  T1 ) (1  Tdir23 )
 23
   M (  )d
12

M (  )  2 [ E  V (  )] / 
*

2 1/ 2
N. Fröman, P. O. Fröman; JWKB Approximation, North-Holland, Amsterdam (1965)
Fission mechanisms
Hamiltonian of a fissionable nucleus:
H=Hβ+Hi+Hiβ
Hβ describes the fission degree of freedom
Hi describes the other collective and intrinsic degree of freedom
Hiβ
accounts for the coupling between the fission mode and other degrees of freedom.
Wave functions of the total Hamiltonian:
| c  m,n amn |  n  | im 
Below the inner barrier the vibrational states |β n> may be classified as class I and class II
vibrations depending on whether the amplitude is greater in the ground state (I) or secondary
minimum (II). The compound states |c> of the system may be classified as class I and class II,
according to the type of vibrational states dominating in the expansion.
The interaction term Hiβ leads both to a coupling between the vibrational and intrinsic
degrees of freedom (the vibrational damping) and between class I and class II states.
Fission mechanisms
class I states
class II states
complete damping
complete
damping
medium damping
no (low) damping
E
Vf
complete damping
medium damping
low damping
Pf

Fission mechanisms
Models for fission coefficient calculation
- conventional approach – complete damping of vibrational class I and II (III)
- doorway-state model – the elements of the coupling matrix are calculated
- optical model for fission – the absorption out of fission mode is
described by an imaginary potential
Fission mechanisms – optical model for fission
The coupling between the vibrational and non-vibrational class II (III)
states makes possible for the nucleus to use the excitation energy to
excite other degrees of freedom and this is interpreted as a loss, an
absorption out from the flux initially destinated to fission. This
absorption is described by an imaginary potential in the second well.
Vf
Complex potential
- real part: 3 (5) parabolas smoothly joined
i 1
V fi  E fi  (1)  2i2 (    i ) 2
2
WII

Vf
- imaginary part:
W j   j ( E )[ E * V f j ]
WII
WIII

Fission mechanisms within optical model for fission
Transmission through the complex double-humped barrier
- direct - via the vibrational states
- indirect - reemission after absorption in the isomeric well

RTII
T2

T f  Tdir  Tabs

T T T
 1 2 II T1  T2  TII




R
T
iso
f 1/ 2
T  1/ 2  T f 1/ 2
iso
iso
Vf
E*
Tdir  0
Tabs  T1
Tdir
Tabs
T2
T f  T1
T1  T2
T1
TII
T2
TII  0
W

Fission coefficients within optical model for fission
Vf
Double-humped barrier
Tdir
Tabs
Tdir, Tabs are derived in WKB approximation
for complex potential
T1
T2

1,II II,2
Tdir 
W
T1T2
e 2  2[(1  T1)1 / 2 (1  T2 )1 / 2 cos( 2 )  (1  T1)(1  T2 )e  2 ]
e 2  (1  T2 )e 2  T2
Tabs  Tdir
T2

 II , 2
1, II
M (  )d
1 / 2  II , 2

   
2

M (  )  2 [ E  V (  )] / 
*
1 / 2  II , 2
W ( )

d



 
 [ E*  V (  )]1/ 2
2
1, II

2 1/ 2
*
1/ 2
[
E

V
(

)]
d

1, II
Transmission coefficients
Vf
Ti ( E Jp )  Ti dis ( E Jp )  Ti cont( E Jp )
Ti ( E Jp ) 
dis
Ti ( E K Jp )
K J


cont
Ti ( E Jp )  

 1  exp
Eci
Ti
cont
( E, J , p ) 
ri (e J p )de
 2p
* 
  ( E fi  e  E )
 i

ri (e , J , p )de

Ec 1  exp( 2M i ) .
i
r1(E*Jp)
e1 (K Jp)
i
T1

Fission coefficients within optical model for fission
Double-humped barrier

RTII
T2

T f  Tdir  Tabs

T T T
 1 2 II T1  T2  TII
Tdir ( E , Jp ) 




 Tdir ( E , K Jp )
K J
Full K-mixing:


r1 (e Jp )de
*
*

Tabs ( E J  )   Tabs ( E K Jp ) 

 2p
* 
K J
( E f 1  e  E )
 1  exp 
 1

Ec1
Based on these relations, a recursive method was developed to describe
transmission through triple- and n-humped barriers.
Transmission through multi-humped barriers
h=1
h=2
h=3
h=4
h=1
Td(3,4)
h=2
h=3
h=4
Ta(1,3)
Td(2,4)
Ta(2,3)
Td(1,4)
Td(3,2)
Td(3,1)
w=1
w=2
Ta(4,2)
w=3
w=4
w=1
w=2
w=3
w=4
Graphic representation of the recursive method used to calculate some of the
direct and absorption coefficients for a four-humps barrier. The transmission
coefficients entering the calculations are represented by dashed arrows and the
derived coefficients are represented by full arrows.
Transmission through multi-humped barriers
h=1
h=2
h=3
h=1
Td (2,3)
Td (2,1)
h=2
Ta(1,3)
Ta(1,2)
Td (1,1)
Td (2,1)
Ta(2,3)
Td (3,3)
Ta(3,2)
Td (3,3)
w=2
Td (2,3)
Ta(2,3)
Ta(3,2)
w=1
h=3
w=3
w=1
w=2
w=3
The evolution of the flux absorbed in the second and third wells.
Each time new contributions are accumulated to the flux undergoing fission
and to the one absorbed in the first well. The shape changing between the
second and third wells continues till the fractions initially absorbed in these
wells are exhausted.
M. Sin, R. Capote, Phys.Rev.C 77, 054601 (2008)
Fission coefficient
TTf  TTdir  TTind 
f
dir 1Tind
T f  T T T ATTB  T
ind
1T
ind 2
1 fexp[f dir
2pT/ h
(
E
V
)]
B
 TB TBC
TTdir  TTAabs
dir
abs T  T
TAA  TBBC
Vf
Vf

Vf

Vf

Vf


sf
low damping
medium damping
complete damping
E
Very good descriptive power
σ(b)
231Pa(n,f)
σ(b)
233U(n,f)
σ(b)
238U(n,f)
E (MeV)
EMPIRE calculations performed by
Mihaela SIN – Bucharest University
E (MeV)
Fission coefficient
h
III
Tf
hII
Region II
Region III
Region V
Excitation energy (MeV)
232Th(n,f)
Triple humped barr.
Fission coefficient
h
Tf
III
hII
Region II
Region III
Region V
Excitation energy (MeV)
Th-U fuel cycle
- Protactinium effect n, 
234Pa
1.2 m; 6.8 h


n, 
233Th

22 min
27 d
n, 
n, 3n

232Pa
n, f
26 h


n, 
230Th

7.5 104 y

n, f 


n, 3n
y
n, f
n, 
1.6 105 y
n, f
n, 
232U
69 y
n, f
n, xn
231Pa
3 104 y
n, 2n
e
n, f
105
233U
n, 2n
n, 
n, 2n
231Th
1.3 d
2.4
n, 2n
n, 2n
232Th
1.4 1010 y
233Pa
234U
230Pa
17 d
n, f
Possible use in ADS
hybrid systems
Primary and secondary
n + 233Pa
chains for n + 233Pa
f
reaction up to 50 MeV
f
p
233Th
p


232Th
p
f
p


f
231Th
p
p


230Th
p
229Th
p
p

228Th
p
n

p

230Pa

p

f
p

229Pa



n

f
228Pa
p
n
229Ac
p





n

228Ac
p


n

f


n

f
n
d
230Ac
f
n
f
n
f
231Pa
d

f
n
f
n
f
232Pa
d


n
f
n
f
233Pa
d
n

n
d
n
f
234Pa
227Ac
p
n


G.Vlăducă, F.J.Hambsch, A.Tudora,
S.Oberstedt, F.Tovesson, D.Filipescu
Nucl. Phys. A 740(2004)3
Fission cross section 233Pa(n,f)
2.0
Pa(n,f)
M. Petit (2002)
Present
1
F. Tovesson (2002)
10
evaluation
ENDF/B-VI
233
ENDF/B-VI
JENDL 3.3
JENDL
3.3
0
10
Pa
n+
1.5
1.0
0.5
0.0
0
5
10
15
20
Fission cross section (b)
Fission cross section (b)
n+
233
233
Fission chances:
(n,f)
(n,nf)(n,nf)
(n,2nf)
(n,3nf)
(n,4nf)
(n,5nf)
(n,6nf)
-1
10
-2
10
Present
evaluation
Pa
(n,f)
(n,2nf)
233-Th
232-Th
231-Th
230-Th
229-Th
M. Petit (2002)
F. Tovesson (2002)
(n,4nf)
(n,5nf) (n,6nf)
(n,3nf)
231-Th
-3
10
233-Th
230-Th
232-Th
-4
10
2510-5 30
35
40
45
50
5
10
15
20
229-Th
En(MeV)
-6
10
0
25
30
En(MeV)
35
40
45
50
Gamma-ray strength functions
•
Gamma transmission coefficient:
T ( E * , J )  
J L

TXL (e  )  2pf XL (e  )e 2 L1
E*
2 L 1
*
L
d
e
f
(
e
)
e
r
(
E

e
,
J

)

(

,
(

1
)
)

K
K
K
   
XL J K  J  L 0
Closed forms for the E1 strength functions
f E1 (e  )  K XL
• Standard Lorentzian model (SLO)
• Kadmenskij-Markushev-Furman model (KMF)
s 0e  2
(e 2  E2 ) 2  e 22
MeV 3
• Enhanced Generalized Lorentzian model (EGLO)
• Hybrid model (GH)
• Generalized Fermi Liquid model (GFL)
• Modified Lorentzian model (MLO)
Normalization: 2p
s
Ds
p XL  
J L
Bn
  de  f
J  J K  J L 0
XL
(e  )e 2 L 1 r ( Bn  e  , J K  K ) ( K , (1) L )
Study of gamma capture resonances
24
22
238
20
U
13
238
Emax = 4593 eV
12
211 widths
<n0> = 2.17 +/- 0.2711
meV
18
16
Number of spacings
Number of widths
14
14
12
10
8
6
4
U
Emax = 4593 eV
211 levels
<D> = 21.84 eV
10
9
8
7
6
5
4
2
3
0
0.0
0.5
1.0
1.5
2.0
(n /<n >)
0
1/2
0
2.52
3.0
1
0
0.0
0.5
Porter-Thomas (=1) widths distribution
- Gamma-ray strength functions normalization
- Level density parameterization check and tuning
- Statistical model hypothesis check
1.0
1.5
2.0
2.5
3.0
D/<D>
Wigner spacing distribution
U-238 photo-reaction cross sections: experimental data, evaluations,
preliminary calculations
U-238(g,f)
U-238(g,abs)
EXFOR
Empire
EXFOR
Empire
U-238(g,f)
U-238(g,non)
EXFOR
Empire
ENDF-VII.1
JENDL/PD
TENDL-2011
EXFOR
Empire
ENDF-VII.1
JENDL/PD
TENDL-2011
U-238 photo-reaction cross sections: experimental data, evaluations,
preliminary calculations
U-238(g,2n)
U-238(g,n)
EXFOR
Empire
U-238(g,*)
(g,abs)
(g,f)
(g,g)
(g,n)
(g,2n)
EXFOR
Empire
ENDF-VII.1
TENDL-2011
U-238(g,non)
EXFOR
Empire
ENDF-VII.1
JENDL/PD
TENDL-2011
U-238(g,f)
EXFOR
Empire
EXFOR
Empire
ENDF-VII.1
JENDL/PD
TENDL-2011
U-238(g,f)
U-238(g,n)
EXFOR
Empire
EXFOR
Empire
ENDF-VII.1
TENDL-2011
U-238(g,f)
EXFOR
Empire
ENDF-VII.1
JENDL/PD
TENDL-2011
U-238(g,*)
Thank you !