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

Intermediate Structure in Fission; Consequences on Average Partial Cross Sections Olivier H. Bouland Collaborative work with LANL-T2 (E. Lynn & P. Talou) WONDER 2012, SEPTEMBER THE 28 TH , AIX-EN- PROVENCE. | PAGE 1

AVERAGE CROSS SECTION SIMULATION: CHALLENGE AND PROBLEMATIC Heavy fissile nuclei: 10000 Fission cross sections [b] 243Pu+n (x 1000) 1000 Heavy fertile nuclei: Tovesson 2008 100 ENDF/B-VII.1 241Pu+n (x 100) obvious Intermediate Structure (I.S.) effects JENDL-4.0 JEFF-3.1.2 Present work Present work 244Pu+n (x 1e+7) ENDF/B-VII.1 1e+07 JENDL-4.0 239Pu+n (x 10) 10 Fursov 1997 Tovesson 2008 Laptev 2007 1e+06 Fission cross sections [b] JEFF-3.1.2 237Pu+n 1 1e+05 10000 1e+05 1e+06 242Pu+n (x 1e+6) Incident neutron energy [eV] How large is the difference in average xs 10000 model complexity between fertile and 1000 240Pu+n (x 1e+4) fissile nuclei ? 100 238Pu+n (x 100) And so, will the standard HF formalism 10 sufficient for fissile nuclei ? 236Pu+n 1 10000 1e+05 1e+06 | O. Bouland, Wonder 2012 PAGE 2 Incident neutron energy [eV]

AVERAGE PARTIAL CROSS SECTIONS STANDARD HAUSER-FESHBACH FORMALISM π ( l ′ s ′ ) π ( l ′ s ′ ) π ( ls ) π ( ls ) � � Γ J Γ J Γ J ¯ Γ J ¯ c c c ′ c ′ × W cc ′ = π ( l ′′ s ′′ ) π ( l ′′ s ′′ ) c ′′ ¯ c ′′ Γ J Γ J � � c ′′ c ′′ ¯ Γ c ′ T J π ( ls ) ≈ 2 π With c ′ D J Standard Hauser-Feshbach average cross section with width fluctuation correction factor W cc’ | I ′ + i ′ | | J + s ′ | π ( l ′ s ′ ) T J ( E c ′ ) σ J π � � � c ′ σ cc ′ ( E n ) = ¯ c ( E n ) × W cc ′ π ( l ” s ”) c ′′ T J � ( E c ′′ ) c ′′ J s ′ = | I ′ − i ′ | l ′ = | J − s ′ | | O. Bouland, Wonder 2012 PAGE 3

AVERAGE FISSION CROSS SECTION Strutinsky, Lynn, Weigman have explained the presence of I.S observed by Michaudon, Paya, Migneco (etc.) Migneco et al., NP A112 (1968) Collective nuclear vibrations and single- particle excited states manifest as I.S effects. | O. Bouland, Wonder 2012 PAGE 4

EIGENSTATE CLASSIFICATION: FORMAL R-MATRIX (E. LYNN - 1973) H = H η + H int ( ζ , η 0 ) + H c ( η , ζ ; η 0 ) HX λ = E λ X λ Φ ν , E ν χ µ , E µ � X λ i = C λ ( ν ′ µ ′ ) Φ ν ′ χ µ ′ ν ′ µ ′ Class-I or class-II classification is depending on the location of the largest vibrational amplitude. � ( E ν + E µ − E λ ) C λ ( ν µ ) + C λ ( ν ′ µ ′ ) � Φ ν χ µ | H c | Φ ν ′ χ µ ′ � = 0 ν ′ µ ′ | O. Bouland, Wonder 2012 PAGE 5

CLASS-I AND CLASS-II PROPERTIES λ I : by nature, class-I have very small fission widths and high level density Γ λ I ,tot ≈ Γ λ I ,n + Γ λ I ,n ′ + Γ λ I , γ I λ II : : whereas class-II have much larger fission widths and lower level density at the same excitation energy Γ λ II ,tot ≈ Γ λ II ↓ + Γ λ II ↑ + Γ λ II , γ II We define T B = 2 π � Γ λ II ↑ � II T A = 2 π � Γ λ II ↓ � II and D II D II � λ II | H c | λ I � 2 � � I /D I Γ λ II ↓ = 2 π with | O. Bouland, Wonder 2012 PAGE 6

CLASS-I CLASS-II COUPLING COMMON SITUATIONS σ J π σ J π c ( E n ) × ¯ � ¯ cf ( E n ) = P f × W cf J π ( l ′ s ′ ) | I ′ + i ′ | | J + s ′ | T J ( E c ′ ) σ J π f � � � = c ( E n ) × W cf π ( l ” s ”) c ′′ T J � ( E c ′′ ) c ′′ J s ′ = | I ′ − i ′ | l ′ = | J − s ′ | Statistical approximation (strong damping) – across an outer Bohr channel µ T f ( µ ) = T A T B ( µ ) � T f = T f ( µ ) and T A + T B µ Moderately weak coupling: average sub-barrier penetrability (Lynn and Back formula – 1974) �� − 1 / 2 � � 2 � T I,tot � 2 T I,tot � � T A + T B ¯ P f = coth 1 + + T f T f 2 text | O. Bouland, Wonder 2012 PAGE 7

COUPLING AND FISSION WIDTH CORRELATIONS: W II FACTOR Thus we have to consider the overall fission transmission across the double barrier (depending on the damping strength in the second well) � T A T B ( µ ) � � Γ λ II ↓ Γ λ II ↑ ( µ ) � = 2 π � T f � ( µ ) ≈ T A + T B D II Γ λ II λ II λ II Possible width correlations will perturb the P&T distribution of the final R-matrix eigenstate fission widths. Γ λ II ↓ ¯ ¯ � Γ λ II ↓ Γ λ II ↑ � Γ λ II ↑ ( µ ) � = W II ( µ ) ¯ Γ λ II Γ λ II λ II µ | O. Bouland, Wonder 2012 PAGE 8

FORMAL R-MATRIX AVERAGE FISSION XS π ( l ′ s ′ ) | I ′ + i ′ | | J + s ′ | T J ( E c ′ ) σ J π σ J π f � � � ¯ nf ( E n ) = n ( E n ) × W nf × W II π ( l ” s ”) c ′′ T J � ( E c ′′ ) c ′′ J s ′ = | I ′ − i ′ | l ′ = | J − s ′ | Double barrier effect >> W nf *W nf Double barrier effect < W nf *W nf Energy [keV] 1. 100. Energy [keV] 1. 100. 0.60 0.76 W nf 0.80 0.84 W nf 0.78 0.78 W II 0.86 0.88 W II 0.47 0.59 0.70 0.74 W nf × W II W nf × W II | O. Bouland, Wonder 2012 PAGE 9

ν EFFECTIVE COLLATERAL DAMAGE ON THE STANDARD FLUCTUATION FACTOR W nf Explicit I.S. effect with Lorentzian approximation (Weak coupling) � α | H c | λ II � Γ 1 / 2 Γ 1 / 2 λ II ↑ � � λ i ( λ Ii ) ,f = � λ i | λ I i �� λ I i | α � ( E λ i − E λ II ) + i Γ λ II / 2 α λ II In case of single hump across a single effective outer barrier channel -- > we expect a Porter-Thomas distribution of level fission widths ( ν eff = 1) , In the case of an explicit I.S calculation and defining � 2 �� α � Γ λ f ( α ) � ν eff = � α � Γ λ f ( α ) � 2 240 ∗ Pu ( J π = 0 + ) 241 ∗ Pu ( J π = 1 / 2 + ) ν eff 1 keV 0.72 0.66 100 keV 0.74 0.66 Reduced ν eff values are requested for correct W nf calculations | O. Bouland, Wonder 2012 PAGE 10

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 W II 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 ( Γ f ) 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. 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 of 20 % on cross section magnitude below 500 keV. | O. Bouland, Wonder 2012 PAGE 12

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