[email protected]

[email protected]@tandem.nipne.ro

Dan FILIPESCUDan FILIPESCU11, Mihaela SIN, Mihaela SIN22 1) Horia Hulubei - National Institute for Physics and Nuclear Engineering,

2) Bucharest University, Romania

Dan FILIPESCUDan FILIPESCU11, Mihaela SIN, Mihaela SIN22 1) Horia Hulubei - National Institute for Physics and Nuclear Engineering,

2) Bucharest University, Romania

Study of sub-barrier fissionStudy of sub-barrier fissionresonances with gamma beamsresonances with gamma beams

Study of sub-barrier fissionStudy of sub-barrier fissionresonances with gamma beamsresonances with gamma beams

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 fission is still a hot topic?

Why photofission?

• Photofission experiments done on available targets Photofission experiments done on available targets may produce additional information to the existing may produce additional information to the existing available data from experiments done with neutronsavailable data from experiments done with neutrons

- - differences in compound nucleus states population differences in compound nucleus states population must be taken into accountmust be taken into account

• Characteristics of ELI-NP Gamma Sorce offer for Characteristics of ELI-NP Gamma Sorce offer for the first time a viable alternative to the study of sub-the first time a viable alternative to the study of sub-barrier fission resonances with neutrons, even below barrier fission resonances with neutrons, even below SSnn

- very high gamma intensity and excellent energy - very high gamma intensity and excellent energy resolution are requiredresolution are required

Fission processFission 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

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 From the initial state to scission

potential energy surface – potential energy as function of deformation V({q})

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)

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.

q2

q1

LOW

LOW HIGH

HIGH

V

fission path

deformation alongthe fission path

Ef

- 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

- 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 barrierFission barrier

)()()( shLD VVV Strutinsky’s procedure

The value of the shell correction is +, - depending on whether the density of single-particle states at Fermi surface is great or small.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

Negative corrections for actinides - g.s. - permanent deformation - vicinity of macroscopic saddle point – second well

double-humped fission barrierdouble-humped fission barrier


V (LD

)

V()

vsh ( )


Vf

Vf

Vf

light actinides(Th)

medium actinides(Pu)

heavy actinides(Cf)

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 fission barrier from nucleus to nucleus:

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 fission 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).

- 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).

Fission barrierFission barrierEarly 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

elongation

axialasymmetry

reflectionasymmetryThese 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.

Symmetry of inner barrierSymmetry of inner barrierShell 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)


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).

The multi-modal fission

The different barrier characteristics give rise to 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.

V

elongation

SL

ST II

ST I


The triple-humped fission barrier (Th anomaly)The triple-humped fission barrier (Th anomaly)The triple-humped fission barrier (Th anomaly)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 experimental inner barrier height and also the structure in the fission cross section of non-fissile Th, Pa and light U isotopes.

Vf

Fission barrier - descriptionFission barrier - description

• parabolic barriers

• numerical barriers Static fission barriers extracted from full 3-dimensional HFB energy surfaces as function of quadrupole deformation

Fission barrier - descriptionFission barrier - description• parabolic barriers

humps only described by decoupled parabolas

Vf

222 )(2

1iififi EV

Vf

Vf

Vf

Vf

Vf

Vf

-13/5 MeV054.0 A

222 )(2

1iififi EV

entire fission path described by smoothly joined parabolas

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 barriersFission barriers Transition states, class I, II, …states Transition states, class I, II, …states

elongation

class Istates

class IIstates

transition states(fission channels)

Vf

Fission barriersFission barriers Transition states, class I, II, …statesTransition states, class I, II, …states Transition states, class I, II, …statesTransition 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 barriersFission barriers Transition states, class I, II, …states Transition states, class I, II, …states

Vf Vf

V1

V1

VII VII

V2

V2

VIII

V3

WII WII WIII

ei (KJ )p

r pi (E*J )

r pi (E*J )

ei (KJ )p

E (J )ci pE (J )ci p

)]1()1([2

)()(2

KKJJKEKJEi

ifiipep- discrete

Fission barriersFission barriers Transition states Transition states

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 )*( pr JEi

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 - descriptionFission barrier - description• numerical barriers

0

2

4

6

8

10

0 0.5 1 1.5 2

Cm252Cm254Cm256Cm258Cm260Cm262Cm264Cm266Cm268

E-E

GS [

MeV

]

β2

0 0.5 1 1.5 2 2.5

Cm270Cm272Cm274Cm276Cm278Cm280

β2

- Static fission barriers extracted from full 3-dimensional HFB energysurfaces as function of quadrupole deformation

- Global microscopic nuclear level densities within the HFB pluscombinatorial method.

Initial stateInitial state- neutron (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 (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)

E

Sn

CN

E*

n+T

n'(n,f)

s f

s n,f

E

Initial stateInitial state- neutron induced fission: fertile and fissile nuclei gamma-ray induced fission: no such classification

- neutron induced fission: fertile and fissile nuclei gamma-ray induced fission: no such classification

En

En

E*6

E*9

Sn9

Sn6

238U

235U

239U

239U*

236U

236U*

(m +m52

n)c

(m +m82

n)c

m c62

m c92

),(),()( ppssp

JEPJEEJ

ff The fission cross section:

- cross section of the initial state formation),( ps JE

- fission probability

- entrance channel

’ - outgoing competing channels

E - energy of the incident particle inducing fission

Jπ - spin, parity of the CN state

'' ),(),(

),(),(

ppp

pJETJET

JETJEP

f

ff

Fission in CN statistical modelFission in CN statistical model

Hill-Wheeler formula for the transmissioncoefficient through a parabolic barrier:Hill-Wheeler formula for the transmissioncoefficient through a parabolic barrier:

)E(E

ωπ *

fii

iT2

exp1

1

Transmission coefficientsTransmission coefficients Vf

Efi ħωi

• parabolic barrier

dVEM i

b

a

i

i

i

2/12 ]/)(*2[

WheelerHill)2exp(1

1 parabola

i

i MT

The coefficients Ti are expressed in thefirst-order WKB approximation in termsof the momentum integrals for the humps:

The coefficients Ti are expressed in thefirst-order WKB approximation in termsof the momentum integrals for the humps:

Transmission coefficientsTransmission coefficients

• non-parabolic barrier• non-parabolic barrier

Vf

E* a b

Vf

21

21

212,1 11

1

TTTT

TTT

Vf

321

3,21

3,213,1 111

1

TTTTT

TTT

h

h

N

i iN TT

11

,1

Nh humps

Transmission coefficients - decoupled humpsTransmission coefficients - decoupled humps

h

VEN N

T f

h

1*,1

Vf

Transmission coefficients – entire barrierTransmission coefficients – entire barrier

)1()1()2cos()1()1(21 212/1

22/1

1

2112

TTTT

TTTdir

Vf

)1()1()2cos()1()1(212323

23

12/12/1

1

1

13dirdir

dirdir

TTTT

TTT

WKB approximation

23

12)(

dM 2/12* /)]([2)( VEM

N. Fröman, P. O. Fröman; JWKB Approximation, North-Holland, Amsterdam (1965)

Fission mechanisms Fission mechanisms

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.

imnac mnnm|||

,

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:

Fission mechanisms Fission mechanisms class I states class II states

complete damping

no (low) damping

medium damping

complete damping

lo w d a m p in g

m e d iu m d a m p in g

c o m p le te d a m p in g

Vf

Pf

E

Fission mechanisms 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 Fission mechanisms – optical model for fission

222 )(2

1)1( ii

ififi EV

Complex potential

- real part: 3 (5) parabolas smoothly joined

- imaginary part:

]*)[(jfjj VEEW

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

WII

Vf

WII WIII

Fission mechanisms within optical model for fission Fission mechanisms within optical model for fission

II

II

IIabsdirf TTT

RT

TTT

TTTT

2121

2

TdirTabs

T1 T2

Vf

TII

W

TTT

iso

f

iso

iso

fR2/12/1

2/1

Transmission through the complex double-humped barrier Transmission through the complex double-humped barrier - direct - via the vibrational states - indirect - reemission after absorption in the isomeric well

1

0

TT

T

abs

dir

E*

0IIT

21

21 TT

TTT f

Tdir, Tabs are derived in WKB approximation for complex potential

Tdir, Tabs are derived in WKB approximation for complex potential

])1)(1()2cos()1()1[(2 221

2/12

2/11

221

eTTTTe

TTTdir

2

22

22 )1(

T

TeTeTT dirabs

2,

,1)(

II

IIdM

2/12* /)]([2)( VEM

2,

,1

2,

,1

2/1*2/1

2/1*

2/1

)]([2)]([

)(

2

II

II

II

II

dVEdVE

W

Fission coefficients within optical model for fission Fission coefficients within optical model for fission

Double-humped barrier Double-humped barrier TdirTabs

T1 T2

Vf

W

1,II II,2

ciEfi

i

iconti

EE

dJJET

)(2

exp1

)()(

*ep

eperp

Transmission coefficientsTransmission coefficients

.)2exp(1

),,(),,(

iEc i

iconti M

dJJET

eperp

)()()( ppp JETJETJET conti

disii

JK

idis

i JKETJET )()( pp

Vf

T1

e1 ( )KiJpr1( )E*Jp

JK

dirdir JKETJET ),(),( pp

JK

Ef

absabs

c

EE

dJJKETJET

1

)(2

exp1

)()()(

*1

1

1**

ep

eperp

Full K-mixing:

Fission coefficients within optical model for fission Fission coefficients within optical model for fission

II

II

IIabsdirf TTT

RT

TTT

TTTT

2121

2

Double-humped barrier Double-humped barrier

Based on these relations, a recursive method was developed to describe transmission through triple- and n-humped barriers.Based on these relations, a recursive method was developed to describe transmission through triple- and n-humped barriers.

Transmission through multi-humped barriers Transmission through multi-humped barriers

w=1w=1

h=1 h=1

w=2w=2

h=2 h=2

w=3w=3 w=4w=4

h=3 h=3h=4 h=4

Ta(1,3)

Ta(2,3)

Ta(4,2)

Td(1,4)

Td(2,4)

Td(3,4)

Td(3,2)

Td(3,1)

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 thederived coefficients are represented by full arrows.

Transmission through multi-humped barriers Transmission through multi-humped barriers

w=1 w=1

h=1 h=1

w=2 w=2

h=2 h=2

w=3 w=3

h=3 h=3Ta(1,2)

Ta(2,3)

Ta(2,3)Ta(3,2)

Ta(3,2)

Ta(1,3)

Td(3,3)

Td(3,3) Td(2,3)

Td(2,3) Td(2,1)

Td(2,1)

Td(1,1)

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) M. Sin, R. Capote, Phys.Rev.C 77, 054601 (2008)

Fission coefficientFission coefficientVf

Vf

Vf

Vf

Vf

)](/2exp[1

1

VEhT f

p

BA

BAf TT

TTT

BA

Babsdir

inddirf

TT

TTT

TTT

BCA

BCabsdir

inddirf

TT

TTT

TTT

21 indinddirf TTTT

E

sf

low damping

medium damping

complete damping

231Pa(n,f)231Pa(n,f)σ(b)

Very good descriptive power Very good descriptive power

233U(n,f)233U(n,f)

E (MeV)

σ(b)

238U(n,f)238U(n,f)σ(b)

E (MeV)

EMPIRE calculations performed byEMPIRE calculations performed by

Mihaela SIN – Bucharest UniversityMihaela SIN – Bucharest University

Region II Region III

Fission coefficientFission coefficient

Tf

E xcitation energy (M eV)

hII

Region V

hI II

232232Th(n,f)Th(n,f)

Triple humped barr.Triple humped barr.

Fission coefficientFission coefficient

Excita tion energy (M eV )

Region II Region III Region V

Tf

hII

hI II

232Th1.4 1010 y

234U2.4 105 y

234Pa1.2 m; 6.8 h

233Th22 min

233Pa27 d

231Th26 h

232Pa1.3 d

231Pa3 104 y

230Th7.5 104 y

230Pa17 d

233U1.6 105 y

232U69 y

n, 3n

n, f

n, f

n, f

n, f n, f

n, f

n, f

n, 3n

n,

n,

n,

n,

n,

n,

n,

n, 2n

n, 2n

n, 2n

n, 2n

n, 2n

n, xn

e

Th-U fuel cycle- Protactinium effect -

Possible use in ADS hybrid systems

n + 233Pa

234Pa

233Pa

232Pa

231Pa

230Pa

229Pa

228Pa

233Th

232Th

231Th

230Th

229Th

228Th

230Ac

229Ac

228Ac

227Ac

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

d

d

d

d

d

Primary and secondary Primary and secondary

chains forchains for n + n + 233233PaPa

reaction up to 50 MeVreaction up to 50 MeV

Fission cross section Fission cross section 223333Pa(n,f) Pa(n,f) G.Vlăducă, F.J.Hambsch, A.Tudora, S.Oberstedt, F.Tovesson, D.FilipescuNucl. Phys. A 740(2004)3

Gamma-ray strength functions

• Gamma transmission coefficient: Gamma transmission coefficient:

Closed forms for the E1 strength functionsClosed forms for the E1 strength functions

• Standard Lorentzian model (SLO)

• Kadmenskij-Markushev-Furman model (KMF)

• Enhanced Generalized Lorentzian model (EGLO)

• Hybrid model (GH)

• Generalized Fermi Liquid model (GFL)

• Modified Lorentzian model (MLO)

12)(2)( eepe L

XLXL fT

XL

LJ

LJJ

EL

KKKL

K

JEfdJET

*

0

*12* ))1(,(),()(),( ereee

322222

20

1 )()(

MeVE

Kf XLE

ee

ese

Normalization:Normalization:

J

LJ

LJJ

BL

KKKnL

XLXLs

s

K

n

JBfdpD 0

12 ))1(,(),()(2 ereeep

Study of gamma capture resonances

Porter-Thomas (=1) widths distribution

Wigner spacing distribution- Gamma-ray strength functions normalization

- Level density parameterization check and tuning

- Statistical model hypothesis check

U-238(g,non)

EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011

U-238(g,abs)

EXFOREmpire

U-238(g,f)

EXFOREmpire

U-238(g,f)


U-238 photo-reaction cross sections: experimental data, evaluations, preliminary calculations

U-238(g,n)

EXFOREmpire

U-238(g,2n)EXFOREmpireENDF-VII.1TENDL-2011

U-238(g,*)

(g,abs)(g,f)(g,g)(g,n)(g,2n)

U-238 photo-reaction cross sections: experimental data, evaluations, preliminary calculations

U-238(g,non)


U-238(g,f)

EXFOREmpire

U-238(g,f)


U-238(g,n)

EXFOREmpire

U-238(g,f)

EXFOREmpireENDF-VII.1TENDL-2011

U-238(g,*)


Thank you !Thank you !

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