Intermediate Structure in Fission; Consequences on Average Partial - - PowerPoint PPT Presentation

Intermediate Structure in Fission; Consequences on Average Partial Cross Sections

WONDER 2012, SEPTEMBER THE 28TH, AIX-EN- PROVENCE.

Olivier H. Bouland

Collaborative work with LANL-T2 (E. Lynn & P. Talou)

| PAGE 1

10000 1e+05 1e+06

Incident neutron energy [eV]

1 10 100 1000 10000

Fission cross sections [b]

Tovesson 2008 ENDF/B-VII.1 JENDL-4.0 JEFF-3.1.2 Present work

243Pu+n (x 1000) 241Pu+n (x 100) 239Pu+n (x 10) 237Pu+n

10000 1e+05 1e+06

Incident neutron energy [eV]

1 10 100 1000 10000 1e+05 1e+06 1e+07

Fission cross sections [b]

Present work ENDF/B-VII.1 JENDL-4.0 Fursov 1997 Tovesson 2008 Laptev 2007 JEFF-3.1.2

244Pu+n (x 1e+7) 242Pu+n (x 1e+6) 240Pu+n (x 1e+4) 238Pu+n (x 100) 236Pu+n

AVERAGE CROSS SECTION SIMULATION: CHALLENGE AND PROBLEMATIC

Heavy fissile nuclei: Heavy fertile nuclei:

• bvious Intermediate Structure (I.S.) effects

| O. Bouland, Wonder 2012 PAGE 2

How large is the difference in average xs model complexity between fertile and fissile nuclei ? And so, will the standard HF formalism sufficient for fissile nuclei ?

AVERAGE PARTIAL CROSS SECTIONS STANDARD HAUSER-FESHBACH FORMALISM

Standard Hauser-Feshbach average cross section with width fluctuation correction factor Wcc’

| O. Bouland, Wonder 2012 PAGE 3

¯ σcc′(En) =

• J

σJπ

c (En) |I′+i′|

• s′=|I′−i′|

|J+s′|

• l′=|J−s′|

T J

π(l′s′)

c′

(Ec′)

• c′′ T J

π(l”s”)

c′′

(Ec′′) × Wcc′

With

• ΓJ

π(ls)

c

ΓJ

π(l′s′)

c′

• c′′ ΓJ

π(l′′s′′)

c′′

• =

¯ ΓJ

π(ls)

c

¯ ΓJ

π(l′s′)

c′

• c′′ ¯

ΓJ

π(l′′s′′)

c′′

× Wcc′

T Jπ(ls)

c′

≈ 2π ¯ Γc′ DJ

AVERAGE FISSION CROSS SECTION

Strutinsky, Lynn, Weigman have explained the presence of I.S observed by Michaudon, Paya, Migneco (etc.)

| O. Bouland, Wonder 2012 PAGE 4

Migneco et al., NP A112 (1968)

Collective nuclear vibrations and single- particle excited states manifest as I.S effects.

EIGENSTATE CLASSIFICATION: FORMAL R-MATRIX (E. LYNN - 1973)

Class-I or class-II classification is depending on the location of the largest vibrational amplitude.

| O. Bouland, Wonder 2012 PAGE 5

HXλ = EλXλ

Φν, Eν χµ, Eµ

Xλi =

• ν′µ′

Cλ(ν′µ′)Φν′χµ′

(Eν + Eµ − Eλ)Cλ(νµ) +

• ν′µ′

Cλ(ν′µ′)Φνχµ|Hc|Φν′χµ′ = 0

H = Hη + Hint(ζ, η0) + Hc(η, ζ; η0)

CLASS-I AND CLASS-II PROPERTIES

: by nature, class-I have very small fission widths and high level density We define

| O. Bouland, Wonder 2012 PAGE 6

ΓλI,tot ≈ ΓλI,n + ΓλI,n′ + ΓλI,γI

λI λII

: : whereas class-II have much larger fission widths and lower level density at the same excitation energy and

TB = 2π ΓλII↑II DII

TA = 2π ΓλII↓II DII

ΓλII,tot ≈ ΓλII↓ + ΓλII↑ + ΓλII,γII

ΓλII↓ = 2π

• λII|Hc|λI2

I/DI

with

CLASS-I CLASS-II COUPLING COMMON SITUATIONS

Statistical approximation (strong damping) – across an outer Bohr channel µ and Moderately weak coupling: average sub-barrier penetrability (Lynn and Back formula – 1974) text

| O. Bouland, Wonder 2012 PAGE 7

¯ σJπ

cf (En)

=

• J

σJπ

c (En) × ¯

Pf × Wcf =

• J

σJπ

c (En) |I′+i′|

• s′=|I′−i′|

|J+s′|

• l′=|J−s′|

T J

π(l′s′)

f

(Ec′)

• c′′ T J

π(l”s”)

c′′

(Ec′′) × Wcf

Tf(µ) = TATB(µ) TA + TB

Tf =

• µ

Tf(µ)

¯ Pf =

• 1 +

TI,tot Tf 2 + 2TI,tot Tf

• coth

TA + TB 2 −1/2

COUPLING AND FISSION WIDTH CORRELATIONS: WII FACTOR

Thus we have to consider the overall fission transmission across the double barrier (depending on the damping strength in the second well) Possible width correlations will perturb the P&T distribution of the final R-matrix eigenstate fission widths.

| O. Bouland, Wonder 2012 PAGE 8

ΓλII↓ΓλII↑ ΓλII

• λII

=

• µ

WII(µ) ¯ ΓλII↓¯ ΓλII↑(µ) ¯ ΓλII

Tf(µ) ≈ TATB(µ) TA + TB

• λII

= 2π DII ΓλII↓ΓλII↑(µ) ΓλII

• λII

FORMAL R-MATRIX AVERAGE FISSION XS

Double barrier effect < Wnf*Wnf

| O. Bouland, Wonder 2012 PAGE 9

¯ σJπ

nf (En) =

• J

σJπ

n (En) |I′+i′|

• s′=|I′−i′|

|J+s′|

• l′=|J−s′|

T J

π(l′s′)

f

(Ec′)

• c′′ T J

π(l”s”)

c′′

(Ec′′) × Wnf × WII

Double barrier effect >> Wnf*Wnf

Energy [keV] 1. 100. Wnf 0.60 0.76 WII 0.78 0.78 Wnf × WII 0.47 0.59

Energy [keV] 1. 100. Wnf 0.80 0.84 WII 0.86 0.88 Wnf × WII 0.70 0.74

ν EFFECTIVE COLLATERAL DAMAGE ON THE STANDARD FLUCTUATION FACTOR Wnf

Reduced νeff values are requested for correct Wnf calculations

Explicit I.S. effect with Lorentzian approximation (Weak coupling)

| O. Bouland, Wonder 2012 PAGE 10

In case of single hump across a single effective outer barrier channel -- > we expect a Porter-Thomas distribution of level fission widths (νeff = 1) , In the case of an explicit I.S calculation and defining

Γ1/2

λi(λIi),f =

• λII
• α

λi|λIiλIi|α α|Hc|λIIΓ1/2

λII↑

(Eλi − EλII) + iΓλII/2

νeff

240∗Pu (Jπ = 0+) 241∗Pu (Jπ = 1/2+)

1 keV 0.72 0.66 100 keV 0.74 0.66

νeff =

• αΓλf (α)

2

• αΓλf (α)2

NESTED DOUBLE BARRIER MONTE CARLO FISSION CROSS SECTION CALCULATIONS

Nested Monte Carlo-type calculations are a powerful alternative to analytical decoupled expressions of sub-barrier and fluctuation effects, The resulting MC fission cross sections are even lower in magnitude than those calculated from analytical formulae.

| O. Bouland, Wonder 2012 PAGE 11

CONCLUSIONS

• 1. Proof is made of the existence of a WII factor (>15% below 100keV),
• 2. The correlation between coupling (Γ) and fission (Γ) class-II widths reduces

significantly the DoF of the final R-matrix eigenstate fission width (Γf) distribution Porter-Thomas hypothesis is invalid

• 3. Sub-barrier and fluctuation effects are strongly nested and Monte Carlo type

calculations are definitively the alternative to common (or exotic) analytical formulae.

| O. Bouland, Wonder 2012 PAGE 12

To the question raised as preamble, Can we calculate average fission cross section from standard HF formulation with only entrance-outgoing channel fluctuation factor? The answer is “NOT EVEN TRUE FOR FISSILE ISOTOPES” with an expected error

• f 20 % on cross section magnitude below 500 keV.

Direction de l’Energie Nucléaire Département d’Etude des Réacteurs Service de Physique des Réacteurs et du Cycle Laboratoire d’Etudes de PHysique Commissariat à l’énergie atomique et aux énergies alternatives Centre de Cadarache | 13108 Saint Paul Lez Durance

• T. +33 (0)4 42 25 49 13 | F. +33 (0)4 42 25 70 09

Etablissement public à caractère industriel et commercial | RCS Paris B 775 685 019

| PAGE 13