Reactions, Channel Widths

Transcription

1 Statistical Theory of Nuclear Reactions, Channel Widths and Level Densities S. Hilaire - CEA,DAM,DIF TRIESTE 2014 S. Hilaire & The TALYS Team 23/09/2014

2 Content - Introduction - General features about nuclear reactions Time scales and associated models Types of data needed Data format = f (users) - Nuclear Models Basic structure properties Optical model Pre-equilibrium model Compound Nucleus model Miscellaneous : level densities, fission, capture - From in depth analysis to large scale production with TALYS General features about TALYS Fine tuning and accuracy Global systematic approaches - What remains to be done? TODAY

3 Content - Introduction - General features about nuclear reactions Time scales and associated models Types of data needed Data format = f (users) - Nuclear Models Basic structure properties Optical model Pre-equilibrium model Compound Nucleus model Miscellaneous : level densities, fission, capture - From in depth analysis to large scale production with TALYS General features about TALYS Fine tuning and accuracy Global systematic approaches - What remains to be done? TOMORROW

4 CEA 10 AVRIL 2012 INTRODUCTION

5 The bible today

6 Why do we need nuclear data and how much accurate? Nuclear data needed for Understanding basic reaction mechanism between particles and nuclei Astrophysical applications (Age of the Galaxy, element abundances ) Good accuracy if possible good understanding or room for improvements Existing or future nuclear reactor simulations Medical applications, oil well logging, waste transmutation, fusion, Finite number of experimental data (price, safety or counting rates) Complete measurements restricted to low energies ( < 1 MeV) to scarce nuclei Predictive & Robust Nuclear models (codes) are essential

7 Why do we need nuclear data and how much accurate? Nuclear data needed for Understanding basic reaction mechanism between particles and nuclei Astrophysical applications (Age of the Galaxy, element abundances ) Existing or future nuclear reactor simulations Good accuracy if possible good understanding or room for improvements Medical applications, oil well logging, waste transmutation, fusion, Predictive power important sound physics (first principles) Finite number of experimental data (price, safety or counting rates) Complete measurements restricted to low energies ( < 1 MeV) to scarce nuclei Predictive & Robust Nuclear models (codes) are essential

8 Why do we need nuclear data and how much accurate? Nuclear data needed for Understanding basic reaction mechanism between particles and nuclei Astrophysical applications (Age of the Galaxy, element abundances ) Existing or future nuclear reactor simulations Medical applications, oil well logging, waste transmutation, fusion, Good (Excellent) accuracy required reproduction of data, safety Predictive power less important Reproductive power Finite number of experimental data (price, safety or counting rates) Complete measurements restricted to low energies ( < 1 MeV) to scarce nuclei Predictive & Robust Nuclear models (codes) are essential

9 Why do we need nuclear data and how much accurate? Nuclear data needed for Understanding basic reaction mechanism between particles and nuclei Astrophysical applications (Age of the Galaxy, element abundances ) Existing or future nuclear reactor simulations Medical applications, oil well logging, waste transmutation, fusion, Finite number of experimental data (price, safety or counting rates) Good accuracy required reproduction of data Predictive power less important Reproductive power Complete measurements restricted to low energies ( < 1 MeV) to scarce nuclei Predictive & Robust Nuclear models (codes) are essential

10 Why do we need nuclear data and how much accurate? Nuclear data needed for But Understanding basic reaction mechanism between particles and nuclei Astrophysical applications (Age of the Galaxy, element abundances ) Existing or future nuclear reactor simulations Medical applications, oil well logging, waste transmutation, fusion, Finite number of experimental data (price, safety or counting rates) Complete measurements restricted to low energies ( < 1 MeV) to scarce nuclei Predictive & Robust Nuclear models (codes) are essential

11 GENERAL FEATURES ABOUT NUCLEAR REACTIONS CEA 10 AVRIL 2012 PAGE 12 2 DÉCEMBRE 2014

12 Content - Introduction - General features about nuclear reactions Time scales and associated models Types of data needed Data format = f (users) - Nuclear Models Basic structure properties Optical model Pre-equilibrium model Compound Nucleus model Miscellaneous : level densities, fission, capture - From in depth analysis to large scale production with TALYS General features about TALYS Fine tuning and accuracy Global systematic approaches - What remains to be done?

13 TIME SCALES AND ASSOCIATED MODELS (1/4) Typical spectrum shape Always evaporation peak Discrete peaks at forward angles Flat intermediate region

14 TIME SCALES AND ASSOCIATED MODELS (2/4) d 2 / d de Compound Nucleus Low emission energy Reaction time s Isotropic angular distribution Reaction time Emission energy

15 TIME SCALES AND ASSOCIATED MODELS (2/4) d 2 / d de Compound Nucleus Direct components High emission energy Reaction time s Anisotropic angular distribution - forward peaked - oscillatory behavior spin and parity of residual nucleus Reaction time Emission energy

16 TIME SCALES AND ASSOCIATED MODELS (2/4) d 2 / d de Compound Nucleus Pre-equilibrium Direct components MSC MSD Intermediate emission energy Intermediate reaction time Anisotropic angular distribution smoothly increasing to forward peaked shape with outgoing energy Reaction time Emission energy

17 TIME SCALES AND ASSOCIATED MODELS (3/4) Direct (shape) elastic Elastic Reaction NC OPTICAL MODEL PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

18 TIME SCALES AND ASSOCIATED MODELS (4/4)

19 TYPES OF DATA NEEDED Cross sections : total, reaction, elastic (shape & compound), non-elastic, inelastic (discrete levels & total) total particle (residual) production all exclusive reactions (n,nd2a) all exclusive isomer production all exclusive discrete and continuum -ray production Spectra : elastic and inelastic angular distribution or energy spectra all exclusive double-differential spectra total particle production spectra compound and pre-equilibrium spectra per reaction stage. Fission observables : cross sections (total, per chance) fission fragment mass and isotopic yields fission neutrons (multiplicities, spectra) Miscellaneous : recoil cross sections and ddx particle multiplicities astrophysical reaction rates covariances informations

20 DATA FORMAT Trivial for basic nuclear science : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file

21 DATA FORMAT : ENDF file Trivial for basic nuclear science : x,y,(z) file Content nature ( ) Complicated (even crazy) for data production issues : ENDF file

22 DATA FORMAT : ENDF file Trivial for basic nuclear science : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file Content type (n,2n)

23 DATA FORMAT : ENDF file Trivial for basic nuclear science : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file Material number

24 DATA FORMAT : ENDF file Target Trivial identification for basic nuclear ( 151 science Sm) : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file

25 DATA FORMAT : ENDF file Trivial for basic nuclear science : Target x,y,(z) file mass Complicated (even crazy) for data production issues : ENDF file

26 DATA FORMAT : ENDF file Trivial for basic nuclear science : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file Number of values

27 DATA FORMAT : ENDF file Trivial for basic nuclear science : x,y,(z) file Complicated (even crazy) for data production issues : ENDF file Values

28 NUCLEAR MODELS CEA 10 AVRIL 2012 PAGE 36 2 DÉCEMBRE 2014

29 Content - Introduction - General features about nuclear reactions Time scales and associated models Types of data needed Data format = f (users) - Nuclear Models Basic structure properties Optical model Pre-equilibrium model Compound Nucleus model Miscellaneous : level densities, fission, capture - From in depth analysis to large scale production with TALYS General features about TALYS Fine tuning and accuracy Global systematic approaches - What remains to be done?

30 BASIC STRUCTURE PROPERTIES (1/5) What is needed Nuclear Masses : basic information to determine reaction threshold Excited levels : Angular distributions (depend on spin and parities) Decay properties (branching ratios) Excitation energies (reaction thresholds) Target levels deformations : Required to select appropriate optical model Required to select appropriate coupling scheme Many different theoretical approaches if experimental data is missing Recommended databases (RIPL!)

31 Ground-state properties Audi-Wapstra mass compilation Mass formulas including deformation and matter densities

32 Discrete level schemes : J,, -transitions, branching ratios 2500 nuclei > levels > spins assigned > transitions

33 BASIC STRUCTURE PROPERTIES (2/5) Mass models Reliability Accuracy Macroscopic-Microscopic Approaches Liquid drop model (Myers & Swiateki 1966) + + Droplet model (Hilf et al. 1976) + + FRDM model (Moller et al. 1995) KUTY model (Koura et al. 2000) Approximation to Microscopic models Shell model (Duflo & Zuker 1995) ETFSI model (Aboussir et al. 1995) Mean Field Model Hartree-Fock-BCS model Hartree-Fock-Bogolyubov model EDF, RHB, Shell model Typical deviations for the best mass formulas: rms(m) = kev on 2149 (Z 8) experimental masses

34 BASIC STRUCTURE PROPERTIES (3/5) Mass models predictive power Comparison between several mass models adjusted with 2003 exp and tested with 2012 exp masses Current status rms < 1 MeV (masses GeV) micro macro micro more predictive Microscopic models

35 BASIC STRUCTURE PROPERTIES (4/5) HFB Mass models Most advanced theoretical approach = multireference level Methodology : E = E mf + E + E bmf 1 month 4-5 years Initial force Automatic fit on 650 known masses New force Correct rms with respect to masses New constraints Acceptable description of masses, radii and nuclear matter properties New corrections E Good description of masses, radii & nuclear matter props, using consistent E values New corrections E quad The good properties obtained using D1S are nearly unchanged Final force Mean field level Beyond mean field * Additional filters - Collective properties (0+,2+, BE2), RPA modes, backbending properties, pairing properties, fission properties, gamma strength functions, level densities

36 BASIC STRUCTURE PROPERTIES (5/5) HFB-Gogny Mass model Comparison with 2149 Exp. Masses D1S E th = E HFB r.m.s ~ 4.4 MeV E th = E HFB - r.m.s ~ 2.6 MeV E th = E HFB - - quad r.m.s ~ 2.9 MeV

37 BASIC STRUCTURE PROPERTIES (5/5) HFB-Gogny Mass model Comparison with 2149 Exp. Masses r.m.s ~ 2.5 MeV r.m.s ~ 0.95 MeV = MeV r.m.s = MeV

38 THE OPTICAL MODEL Direct (shape) elastic Elastic Reaction NC OPTICAL MODEL PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

39 THE OPTICAL MODEL Direct (shape) elastic Elastic OPTICAL Reaction NC MODEL PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

40 THE OPTICAL MODEL Direct interaction of a projectile with a target nucleus considered as a whole Quantum model Schrödinger equation U E 0 Complex potential: U = V + iw Refraction Absorption

41 THE OPTICAL MODEL Direct interaction of a projectile with a target nucleus considered as a whole Quantum model Schrödinger equation U E 0 Complex potential: U = V + iw Refraction Absorption

42 Integrated cross sections THE OPTICAL MODEL The optical model yields : Transmission coefficients Angular distributions

43 TWO TYPES OF APPROACHES Phenomenological Adjusted parameters Weak predictive power Very precise ( 1%) Important work (Semi-)microscopic No adjustable parameters Usable without exp. data Less precise ( 5-10 %) Quasi-automated Total cross sections

44 PHENOMENOLOGICAL OPTICAL MODEL - Very precise (1%) - 20 adjusted parameters - Weak predictive power Total cross section (barn) Neutron energy (MeV)

45 PHENOMENOLOGICAL OPTICAL MODEL OMP & its parameters Solution of the Schrödinger equation Calculated observables el-inl, A y ( ), tot, reac, S 0,S 1 Experimental data el-inl, A y ( ), tot, reac, S 0,S 1 Reaction, T lj, direct

46 SEMI-MICROSCOPIC OPTICAL MODEL - No adjustable parameters - Based on nuclear structure properties usable for any nucleus - Less precise than the phenomenological approach

47 SEMI-MICROSCOPIC OPTICAL MODEL Optical potential = Effective Interaction Radial densities = U(r,E) = U( (r ),E) (r ) (r) Depends on the nucleus Independent of the nucleus Depends on the nucleus

48 SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering

49 SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n)

50 SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n), (p,p)

51 SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n), (p,p) and (p,n)

52 SEMI-MICROSCOPIC OPTICAL MODEL Enables to give predictions for very exotic nuclei for which there exist no experimental data Experiment performed after calculation

53 Average neutron resonance parameters average s-wave spacing at B n level densities neutron strength functions optical model at low energy average radiative width -ray strength function

54 OMP for more than 500 nuclei from neutron to 4 He standard parameters (phenomenologic) deformation parameters (levels from levels segment) energy-mass dependent global models and codes (matter densities from mass segment)

55 THE PRE-EQUILIBRIUM MODEL Shape elastic Elastic Reaction NC OPTICAL MODEL PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

56 THE PRE-EQUILIBRIUM MODEL Shape elastic Elastic OPTICAL MODEL Reaction NC PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

57 TIME SCALES AND ASSOCIATED MODELS (1/4) Typical spectrum shape Always evaporation peak Discrete peaks at forward angles Flat intermediate region

58 THE PRE-EQUILIBRIUM MODEL (quantum vs semi-classical approaches) Semi-classical approaches - called «exciton model» - «simple» to implement - initially only able to describe angle integrated spectra (1966 & 1970) - extended to ddx spectra in link with Compound Nucleus established in systematical underestimation of ddx spectra at backward angles - complemented by Kalbach systematics (1988) to improve ddx description - link with OMP imaginary performed in 2004 Quantum mechanical approaches - distinction between MSC and MSD processes MSC = bound p-h excitations, symetrical angular distributions MSD = unbound configuration, smooth forward peaked ang. dis. - MSD dominates pre-equ xs above 20 MeV - 3 approaches : FKK (1980) - 3 approaches : TUL (1982) - 3 approaches : NWY (1986) - ddx spectra described as well as with Kalbach systematics

59 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F

60 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n

61 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n

62 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n

63 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n

64 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n 3p-2h 5n

65 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n 3p-2h 5n

66 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n 3p-2h 5n 4p-3h 7n

67 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n 2p-1h 3n 3p-2h 5n 4p-3h 7n

68 THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F Compound Nucleus 1p 1n 2p-1h 3n 3p-2h 5n 4p-3h 7n time

69 THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probability Probabilité to find for a given time t the composite system with an energy E and an exciton number n. a, b (E) = Transition rate from an initial state a towards a state b for a given energy E.

70 dp(n,e,t) dt THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probability Probabilité to find for a given time t the composite system with an energy E and an exciton number n. a, b (E) = Transition rate from an initial state a towards a state b for a given energy E. Evolution equation = - Apparition Disparition

71 dp(n,e,t) dt THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probability Probabilité to find for a given time t the composite system with an energy E and an exciton number n. a, b (E) = Transition rate from an initial state a towards a state b for a given energy E. Evolution equation = - P(n-2, E, t) n-2, n (E) + P(n+2, E, t) n+2, n (E) Disparition

72 dp(n,e,t) dt THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probability Probabilité to find for a given time t the composite system with an energy E and an exciton number n. a, b (E) = Transition rate from an initial state a towards a state b for a given energy E. Evolution equation = P(n-2, E, t) n-2, n (E) + P(n+2, E, t) n+2, n (E) - P(n, E, t) [ ] n, n+2 (E) + n, n-2 (E) + n, emiss (E)

73 dp(n,e,t) dt THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probability Probabilité to find for a given time t the composite system with an energy E and an exciton number n. a, b (E) = Transition rate from an initial state a towards a state b for a given energy E. Evolution equation = P(n-2, E, t) n-2, n (E) + P(n+2, E, t) n+2, n (E) - P(n, E, t) [ ] n, n+2 (E) + n, n-2 (E) + n, emiss (E) Emission cross section in channel c d c (E, c ) = R 0 n, n=2 P(n, E, t) n, c (E) dt d c

74 THE PRE-EQUILIBRIUM MODEL (Initialisation & transition rates)

75 THE PRE-EQUILIBRIUM MODEL (Initialisation & transition rates) Initialisation P(n,E,0) = n,n0 with n 0 =3 for nucleon induced reactions Transition rates n, n-2 (E) = 2 ħ ω(p,h,e) with p+h=n-2 n, n+2 (E) = 2 ħ ω(p,h,e) with p+h=n+2 n, c (E) = 2s c+1 µ 2 ħ 3 c c c,inv c ω(p-p b,h,e- c - B c ) ω(p,h,e) Original formulation

76 THE PRE-EQUILIBRIUM MODEL (Initialisation & transition rates) Initialisation P(n,E,0) = n,n0 with n 0 =3 for nucleon induced reactions Transition rates n, n-2 (E) = 2 ħ ω(p,h,e) with p+h=n-2 n, n+2 (E) = 2 ħ ω(p,h,e) with p+h=n+2 n, c (E) = 2s c+1 µ 2 ħ 3 c c c,inv c ω(p-p b,h,e- c - B c ) ω(p,h,e) Q c (n) c Corrections for proton-neutron distinguishability & complex particle emission

77 Initialisation THE PRE-EQUILIBRIUM MODEL (Initialisation & transition rates) P(n,E,0) = n,n0 with n 0 =3 for nucleon induced reactions Transition rates n, n-2 (E) = 2 ħ ω(p,h,e) with p+h=n-2 n, n+2 (E) = n, c (E) = 2s c+1 µ 2 ħ 3 c c c,inv c ω(p-p b,h,e- c - B c ) ω(p,h,e) State densities 2 ħ ω(p,h,e) with p+h=n+2 ω(p,h,e) = number of ways of distributing p particles and h holes on among accessible single particle levels with the available excitation energy E Q c (n) c

78 State densities in ESM THE PRE-EQUILIBRIUM MODEL (State densities) Ericson 1960 : no Pauli principle Griffin 1966 : no distinction between particles and holes Williams 1971 : distinction between particles and holes as well as between neutrons and protons but infinite number of accessible states for both particle and holes

79 State densities in ESM THE PRE-EQUILIBRIUM MODEL (State densities) Ericson 1960 : no Pauli principle Griffin 1966 : no distinction between particles and holes Williams 1971 : distinction between particles and holes as well as between neutrons and protons but infinite number of accessible states for both particle and holes Běták and Doběs 1976 : account for finite number of holes states Obložinský 1986 : account for finite number of particles states (MSC) Anzaldo-Meneses 1995 : first order corrections for increasing number of p-h Hilaire and Koning 1998 : generalized expression in ESM

80 THE PRE-EQUILIBRIUM MODEL Cross section Outgoing energy 39% 16% 45% Total Direct Direct Pré-équilibre Pre-equilibrium Statistique Statistical 79% 12% 9% <E Tot >= 12.1 <E Dir >= 24.3 <E PE >= 9.32 <E Sta >= 2.5 (MeV)

81 THE PRE-EQUILIBRIUM MODEL 14 MeV neutron + 93 Nb without pre-equilibrium d /de(b/mev) without pre-equilibrium (barn) pre-equilibrium Compound nucleus Outgoin neutron energy (MeV) Iincident neutron energy (MeV)

82 Nuclear level densities (formulae, tables, codes) spin-, parity- dependent level densities fitted to D 0 single particle level schemes p-h level density tables

83 THE COMPOUND NUCLEUS MODEL Shape elastic Elastic Reaction NC OPTICAL MODEL PRE-EQUILIBRIUM COMPOUND NUCLEUS Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

84 THE COMPOUND NUCLEUS MODEL Shape elastic Elastic COMPOUND OPTICAL MODEL Reaction NUCLEUS NC PRE-EQUILIBRIUM Fission T lj Direct components Inelastic (n,n ), (n, ), (n, ), etc

85 THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission reaction = dir + pre-equ + NC N 0 Z 0 E* 0 J 0 N 0 -dn D Z 0 -dz D E* 0 -de* D J 0 -dj D N 0 -dn D -dn PE Z 0 -dz D -dz PE E* 0 -de* D -de* PE J 0 -dj D -dj PE = E = Z = E* = J N,Z,E*,J (N,Z,E*)

86 THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission reaction = dir + pre-equ + NC N 0 Z 0 E* 0 J 0 N 0 -dn D Z 0 -dz D E* 0 -de* D J 0 -dj D N 0 -dn D -dn PE Z 0 -dz D -dz PE E* 0 -de* D -de* PE J 0 -dj D -dj PE = E = Z = E* = J N,Z,E*,J (N,Z,E*) N,Z,E *,J (N,Z,E *)

87 THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission reaction = dir + pre-equ + NC N 0 Z 0 E* 0 J 0 N 0 -dn D Z 0 -dz D E* 0 -de* D J 0 -dj D N 0 -dn D -dn PE Z 0 -dz D -dz PE E* 0 -de* D -de* PE J 0 -dj D -dj PE = E = Z = E* = J N,Z,E*,J (N,Z,E*) N,Z,E *,J (N,Z,E *)

88 Compound nucleus hypothesys THE COMPOUND NUCLEUS MODEL (basic formalism) - Continuum of excited levels - Independence between incoming channel a and outgoing channel b ab = (CN) P b a (CN) = a k a T a 2 P b = T b T c c Hauser- Feshbach formula ab = 2 k a T a T b T c c

89 THE COMPOUND NUCLEUS MODEL (qualitative feature) Compound angular distribution & direct angular distributions

90 Channel Definition THE COMPOUND NUCLEUS MODEL (complete channel definition) a + A (CN )* b+b Incident channel a = (l a, j a =l a +s a, J A, A, E A, E a ) Conservation equations Total energy : E a + E A = E CN = E b + E B Total momentum : p a + p A = p CN = p b + p B Total angular momentum : l a + s a + J A = J CN = l b + s b + J B Total parity : A (-1) = CN = B (-1) l a l b

91 THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered ab = k a 2 I A + s a + l a max J= I A s a = Given by OMP (2J+1) (2I A +1) (2s a +1)

92 THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered ab = k a 2 I A + s a + l a max J= I A s a = (2J+1) (2I A +1) (2s a +1) Parity selection rules J + I A j a = J I A (a) (b) j a + s a l a = j a s a J T a, l a, j a c J T c, l c, j c J + I B J T b, l b, j b j b = J I B j b + s b l b = j b s b

93 THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered ab = k a 2 I A + s a + l a max = (2J+1) (2I A +1) (2s a +1) J= I A s a Width fluctuation correction factor J + I A to account j a + s a for J deviations + I B j b + s b from independance hypothesis j a = J I A l a = j a s a j b = J I B l b = j b s b (a) (b) J T a, l a, j a c J T b, l b, j b J T c, l c, j c T J W a, l a, j a, b, l b, j b

94 THE COMPOUND NUCLEUS MODEL (width fluctuation correction factor) Breit-Wigner resonance integrated and averaged over an energy width Corresponding to the incident beam dispersion < ab > = 2 k a 2 D < a b > tot Since T 2 < > D < ab > = 2 k a with W ab = a b c c W ab a b a b tot tot

95 Tepel method THE COMPOUND NUCLEUS MODEL (main methods to calculate WFCF) Simplified iterative method Moldauer method Simple integral GOE triple integral «exact» result Elastic enhancement with respect to the other channels Inelastic enhancement sometimes in very particular situations?

96 THE COMPOUND NUCLEUS MODEL (the GOE triple integral)

97 THE COMPOUND NUCLEUS MODEL (flux redistribution illustration)

98 E Z c THE COMPOUND NUCLEUS MODEL (multiple emission) Z S n (2) + Loop over CN spins and parities J S p S S n Target d S p S S n n S p S S n n n n Compound Nucleus p S p S S n S p S S n fission Z c -1 S n N c -2 N c -1 N c N

99 REACTION MODELS & REACTION CHANNELS n U Cross section (barn) Optical model + Statistical model + Pre-equilibrium model R = d + PE + CN = nn + nf + n Neutron energy (MeV)

100 THE COMPOUND NUCLEUS MODEL (compact expression) NC = ab où b =, n, p, d, t,, fission b ab = 2 k a J, 2J+1 2s+1 2I+1 J T lj < > T b < > T d J J W with J = l + s + I A = j + I A < > and = -1 A and T b ( ) = transmission coefficient for outgoing channel l associated with the outgoing particle b

101 Possible decays THE COMPOUND NUCLEUS MODEL (various decay channels) Emission to a discrete level with energy E d < > J T b ( ) = T lj ( ) given by the O.M.P. Emission in the level continuum E + E < > T b ( ) = J T lj ( ) (E,J, ) de E (E,J, ) density of residual nucleus levels (J, ) with excitation energy E Emission of photons, fission Specific treatment

102 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) Two types of strength functions : - the «upward» related to photoabsorption - the «downward» related to -decay Spacing of states from which the decay occurs f [D.Brink. PhD Thesis(1955); P. Axel. PR 126(1962)] 2 E Гr ( E Er ) E Гr f Standard Lorentzian (SLO) ~ 0 E 0

103 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) Two types of strength functions : - the «upward» related to photoabsorption - the «downward» related to -decay Spacing of states from which the decay occurs BUT f [D.Brink. PhD Thesis(1955); P. Axel. PR 126(1962)] 2 E Гr ( E Er ) E Гr f Standard Lorentzian (SLO) ~ 0 E 0

104 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) T k (E, ) = 2 k ( ) (E) de E E+ E = 2 f(k, ) 2 +1 k : transition type EM (E ou M) : transition multipolarity : outgoing gamma energy f(k, ) : gamma strength function (several models) Decay selection rules from a level J i i to a level J f f : Pour E : f =(-1) i J Pour M : i - J f J i + f =(-1) i (XL 10-3 XL-1) Renormalisation method for thermal neutrons 0 B n <T >= C T k ( ) (B n -,J f, f ) S(,J i, i J i, f ) d = 2 < > (B n ) J i, i k J f, f

105 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) T k (E, ) = 2 k ( ) (E) de E E+ E = 2 f(k, ) 2 +1 k : transition type EM (E ou M) : transition multipolarity : outgoing gamma energy f(k, ) : gamma strength function (several models) Decay selection rules from a level J i i to a level J f f : Pour E : f =(-1) i J Pour M : i - J f J i + f =(-1) i Renormalisation method for thermal neutrons (XL 10-3 XL-1) experiment 0 B n <T >= C T k ( ) (B n -,J f, f ) S(,J i, i J i, f ) d = 2 < > (B n ) J i, i k J f, f 1 D 0

106 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) Improved analytical expressions : - 2 Lorentzians for deformed nuclei - Account for low energy deviations from standard Lorentzians for E1. Kadmenskij-Markushef-Furman model (1983) Enhanced Generalized Lorentzian model of Kopecky-Uhl (1990) Hybrid model of Goriely (1998) Generalized Fermi liquid model of Plujko-Kavatsyuk (2003) - Reconciliation with electromagnetic nuclear response theory Modified Lorentzian model of Plujko et al. (2002) Simplified Modified Lorentzian model of Plujko et al. (2008)

107 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules)

108 MISCELLANEOUS : THE PHOTON EMISSION (strength function and selection rules) Improved analytical expressions : - 2 Lorentzians for deformed nuclei - Account for low energy deviations from standard Lorentzians for E1. Kadmenskij-Markushef-Furman model (1983) Enhanced Generalized Lorentzian model of Kopecky-Uhl (1990) Hybrid model of Goriely (1998) Generalized Fermi liquid model of Plujko-Kavatsyuk (2003) - Reconciliation with electromagnetic nuclear response theory Modified Lorentzian model of Plujko et al. (2002) Simplified Modified Lorentzian model of Plujko et al. (2008) Microscopic approaches : RPA, QRPA «Those who know what is (Q)RPA don t care about details, those who don t know don t care either», private communication Systematic QRPA with Skm force for 3317 nuclei performed by Goriely-Khan (2002,2004) Systematic QRPA with Gogny force under work (300 Mh!!!)

109 MISCELLANEOUS : THE PHOTON EMISSION (phenomenology vs microscopic) See S. Goriely & E. Khan, NPA 706 (2002) 217. S. Goriely et al., NPA739 (2004) 331.

110 MISCELLANEOUS : THE PHOTON EMISSION (phenomenology vs microscopic) Capture cross E n =10 MeV for Sn isotopes Weak impact close to stability but large for exotic nuclei

111 MISCELLANEOUS : THE FISSION PROCESS (static picture exhibiting fission barriers) Surface 238 U

112 MISCELLANEOUS : THE FISSION PROCESS (fissile or fertile?) Energy Fission barrier with height V Energy Fission barrier with height V B n B n V V B n < V elongation elongation Fertile target ( 238 U)

113 MISCELLANEOUS : THE FISSION PROCESS (fissile or fertile?) Energy Fission barrier with height V Energy Fission barrier with height V B n B n V V B n < V elongation B n > V elongation Fertile target ( 238 U) Fissile target ( 235 U)

114 MISCELLANEOUS : THE FISSION PROCESS (fissile or fertile?) Fission barrier

115 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-1) 2 nd chance B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

116 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-1) 2 nd chance B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

117 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-2) 3 rd chance Nucleus (Z,A-1) 2 nd chance B n B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

118 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-2) 3 rd chance Nucleus (Z,A-1) 2 nd chance B n B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

119 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-2) 3 rd chance Nucleus (Z,A-1) 2 nd chance B n B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

120 MISCELLANEOUS : THE FISSION PROCESS (multiple chances) Energy Nucleus (Z,A-2) 3 rd chance Nucleus (Z,A-1) 2 nd chance B n B n Nucleus (Z,A) V B n V 1 st chance fission (barn) elongation Incident neutron energy (MeV)

121 MISCELLANEOUS : THE FISSION PROCESS (Fission penetrability: Hill-Wheeler) Energy Fission barrier ( V, ħω ) E B n Transmission T hw (E) = 1/[1 + exp(2 (V-E)/ħ )] Hill-Wheeler for one barrier! elongation + transition state on top of the barrier! Bohr hypothesys

122 MISCELLANEOUS : THE FISSION PROCESS (Fission transmission coefficients) Energy T hw (E) = 1/[1 + exp(2 (V-E)/ħ )] Hill-Wheeler E+B n V Discrete transition states with energy d elongation T f (E, J, ) = discrets J, T hw (E - d ) + Es E+B n (,J, ) T hw (E - ) d

123 Energy MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Fission barrier ( V, ħω ) B n elongation

124 MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Energy Fission barrier ( V, ħω ) B n elongation + transition states on top of the barrier!

125 MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Energy Barrier A ( V A, ħω A ) Barrier B ( V B, ħω B ) B n elongation + transition states on top of the barrier!

126 Energy MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Barrier A ( V A, ħω A ) Barrier B ( V B, ħω B ) B n elongation + transition states on top of each barrier!

127 Energy MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Barrier A ( V A, ħω A ) Barrier B ( V B, ħω B ) B n elongation + transition states on top of each barrier! + class II states in the intermediate well!

128 Energy MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Barrier A ( V A, ħω A ) Barrier B ( V B, ħω B ) B n elongation + transition states on top of each barrier! + class II states in the intermediate well!

129 Energy MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Barrier A ( V A, ħω A ) Barrier B ( V B, ħω B ) B n elongation + transition states on top of each barrier! + class II states in the intermediate well!

130 MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers) Two barriers A et B Resonant transmission T f = T A T B T A + T B Energy Three barriers A, B and C T f = T A T B T A + T B T A T B x T C T A + T B + T C 0 1 T f T f = More exact expressions in Sin et al., PRC 74 (2006) T A T B T A + T B 4 T A + T B

131 MISCELLANEOUS : THE FISSION PROCESS (multiple humped barriers with maximum complexity) See in Sin et al., PRC 74 (2006) Bjornholm and Lynn, Rev. Mod. Phys. 52 (1980) 725.

132 MISCELLANEOUS : THE FISSION PROCESS (Impact of class II states) 239 Pu (n,f) With class II states Cross section (barn) 1 st chance 2 nd chance Neutron energy (MeV)

133 MISCELLANEOUS : THE FISSION PROCESS (impact of class II and class III states) Case of a fertile nucleus

134 MISCELLANEOUS : THE FISSION PROCESS (impact of class II and class III states) Case of a fertile nucleus

135 MISCELLANEOUS : THE FISSION PROCESS (Hill-Wheeler?) For exotic nuclei : strong deviations from Hill-Wheeler.

136 MISCELLANEOUS : THE FISSION PROCESS (Microscopic fission cross sections)

137 MISCELLANEOUS : THE LEVEL DENSITIES (Principle)?

138 MISCELLANEOUS : THE LEVEL DENSITIES (Qualitative aspects 1/2) n+ 232 Th N(E) 56 Mn 57 Fe 58 Fe Total cross section (b) E (MeV) Exponential increase of the cumulated number of discrete levels N(E) with energy dn(e) (E)= increases exponentially de odd-even effects Mean spacings of s-wave neutron resonances at B n of the order of few ev (B n ) of the order of levels / MeV Incident neutron energy (ev)

139 MISCELLANEOUS : THE LEVEL DENSITIES (Qualitative aspects 2/2) Iljinov et al., NPA 543 (1992) 517. Mass dependency Odd-even effects Shell effects D 0 1 = (B n,1/2, t ) for an even-even target = (B n, I t +1/2, t ) + (B n, I t -1/2, t ) otherwise

140 (U, J, ) = MISCELLANEOUS : THE LEVEL DENSITIES (Quantitative analysis 1/2) exp ( 2 au ) a 1/4 U 5/4 2J exp - ( ) J+½ U = I rig a

141 (U, J, ) = MISCELLANEOUS : THE LEVEL DENSITIES (Quantitative analysis 1/2) exp ( 2 au ) a 1/4 U 5/4 2J exp - ( ) J+½ 2 2 odd-even effects Masse

142 (U, J, ) = MISCELLANEOUS : THE LEVEL DENSITIES (Quantitative analysis 1/2) exp ( 2 au ) a 1/4 U 5/4 2J exp - ( ) J+½ 2 2 Odd-even effects accounted for U U*=U - = 0 12/ A 24/ A odd-odd odd-even even-even Masse Shell effects

143 MISCELLANEOUS : THE LEVEL DENSITIES (Quantitative analysis 2/2) ~ a (N, Z, U*) = a (A) 1 + W(N,Z) 1 - exp ( - U* ) U*

144 MISCELLANEOUS : THE LEVEL DENSITIES (Summary of most simple analytical description) N(E) E (MeV) Discrete levels (spectroscopy) Temperature law (E)=exp E E 0 ( ) T Fermi gaz (adjusted at B n ) (E) = exp ( 2 au* ) a 1/4 U* 5/4

145 MISCELLANEOUS : THE LEVEL DENSITIES (More sophisticated approaches) Superfluid model & Generalized superfluid model Ignatyuk et al., PRC 47 (1993) 1504 & RIPL3 paper (IAEA) More correct treatment of pairing for low energies Fermi Gas + Ignatyuk beyond critical energy Explicit treatment of collective effects (U) = K vib (U) * K rot (U) * int (U) a eff A/8 Several analytical or numerical options a A/13 Collective enhancement only if int (U) 0 not correct for vibrational states

146 MISCELLANEOUS : THE LEVEL DENSITIES (More sophisticated approaches) Superfluid model & Generalized superfluid model Ignatyuk et al., PRC 47 (1993) 1504 & RIPL2 Tecdoc (IAEA) More correct treatment of pairing for low energies Fermi Gas + Ignatyuk beyond critical energy Explicit treatment of collective effects Shell Model Monte Carlo approach Agrawal et al., PRC 59 (1999) 3109 Realistic Hamiltonians but not global Coherent and incoherent excitations treated on the same footing Time consuming and thus not yet systematically applied Combinatorial approach S. Hilaire & S. Goriely, NPA 779 (2006) 63 & PRC 78 (2008) Direct level counting Total (compound nucleus) and partial (pre-equilibrium) level densities Non statistical effects Global (tables)

147 THE LEVEL DENSITIES (The combinatorial method 1/3) See PRC 78 (2008) for details - HFB + effective nucleon-nucleon interaction single particle level schemes - Combinatorial calculation intrinsic p-h and total state densities (U, K, )

148 THE LEVEL DENSITIES (The combinatorial method 1/3) See PRC 78 (2008) for details Level density estimate is a counting problem: (U)=dN(U)/dU N(U) is the number of ways to distribute the nucleons among the available levels for a - HFB fixed + excitation effective energy nucleon-nucleon U interaction single particle level schemes - Combinatorial calculation intrinsic p-h and total state densities (U, K, )

149 THE LEVEL DENSITIES (The combinatorial method 1/3) See PRC 78 (2008) for details - HFB + effective nucleon-nucleon interaction single particle level schemes - Combinatorial calculation intrinsic p-h and total state densities (U, K, ) - Collective effects from state to level densities (U, J, ) : 1) folding of intrinsic and vibrational state densities 2) construction of rotational bands for deformed nuclei (U, J, ) = K 2) spherical nuclei JK (U-E rot, K, ) (U, J, ) = (U, K=J, ) - (U, K=J+1, ) - Phenomenological transition for deformed/spherical nucleus

150 THE LEVEL DENSITIES (The combinatorial method 2/3) Structures typical of non-statistical feature

151 THE LEVEL DENSITIES (The combinatorial method 3/3) D 0 values ( s-waves & p-waves) Back-Shifted Fermi Gas HF+BCS+Statistical HFB + Combinatorial f rms = 1.79 f rms = 2.14 f rms = 2.30

152 THE LEVEL DENSITIES (The combinatorial method 3/3) D 0 values ( s-waves & p-waves) Back-Shifted Fermi Gas HF+BCS+Statistical HFB + Combinatorial f rms = 1.79 f rms = 2.14 f rms = 2.30 Description similar to that obtained with other global approaches

153 CONCLUSIONS & PROPECTS Nuclear reaction modeling complex and no yet fully satisfactory pre-equilibrium phenomenon must be improved fission related phenomena (fission, FF yields & decay) must be improved Formal and technical link between structure and reactions has to be pushed further pre-equilibrium and OMP efforts already engaged computing time is still an issue Fundamental - interaction knowledge (and treatment) has to be improved Ab-initio not universal (low mass or restricted mass regions) Relativistic aspects not included systematically Human & computing time is still an issue