description
Transcript of [email protected]
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[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
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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?
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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
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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
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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
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- 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
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)()()( 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
Fission barrierFission barrier
V (LD
)
V()
vsh ( )
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Fission barrierFission barrier
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).
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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.
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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)
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Fission barrierFission barrier
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
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Fission barrierFission barrier
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
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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
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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.
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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
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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
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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
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)]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.
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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.
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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
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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
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),(),()( 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
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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
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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
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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
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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)
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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:
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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
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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
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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
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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
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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
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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
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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.
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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.
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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)
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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
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E
sf
low damping
medium damping
complete damping
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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
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Region II Region III
Fission coefficientFission coefficient
Tf
E xcitation energy (M eV)
hII
Region V
hI II
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232232Th(n,f)Th(n,f)
Triple humped barr.Triple humped barr.
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Fission coefficientFission coefficient
Excita tion energy (M eV )
Region II Region III Region V
Tf
hII
hI II
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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
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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
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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
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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
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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
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U-238(g,non)
EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011
U-238(g,abs)
EXFOREmpire
U-238(g,f)
EXFOREmpire
U-238(g,f)
EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011
U-238 photo-reaction cross sections: experimental data, evaluations, preliminary calculations
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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
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U-238(g,non)
EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011
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U-238(g,f)
EXFOREmpire
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U-238(g,f)
EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011
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U-238(g,n)
EXFOREmpire
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U-238(g,f)
EXFOREmpireENDF-VII.1TENDL-2011
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U-238(g,*)
EXFOREmpireENDF-VII.1JENDL/PDTENDL-2011
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Thank you !Thank you !