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 !)
Ground-state properties • Audi-Wapstra mass compilation • Mass formulas including deformation and matter densities
Discrete level schemes : J, � , � -transitions, branching ratios • � 2500 nuclei • > 110000 levels • > 13000 spins assigned • > 160000 � -transitions
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) = 600- 700 ¡keV ¡on ¡2149 ¡(Z ¡≥ ¡8) ¡experimental ¡masses
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
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 Acceptable Good description The good Correct description of masses, radii Automatic properties New New Initial New New Final of masses, radii & nuclear matter fit on rms obtained corrections corrections force force force constraints and nuclear props, using 650 known with using D1S � E � � E quad respect matter consistent masses are nearly � E � values to masses properties unchanged 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
BASIC STRUCTURE PROPERTIES (5/5) HFB-Gogny Mass model D1S Comparison with 2149 Exp. Masses • 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
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 ������� = 0.126 MeV r.m.s = 0.798 MeV
THE OPTICAL MODEL Direct (shape) elastic Elastic � Reaction � NC OPTICAL COMPOUND Fission PRE-EQUILIBRIUM MODEL NUCLEUS T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
THE OPTICAL MODEL Direct (shape) elastic Elastic OPTICAL � Reaction � NC MODEL COMPOUND Fission PRE-EQUILIBRIUM NUCLEUS T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
THE OPTICAL MODEL Direct interaction of a projectile with a target nucleus considered as a whole Quantum model � Schrödinger equation � � 2 � � � � � � � � � 2 0 U U E � � � 2 � � Complex potential: U = V + iW Refraction Absorption
THE OPTICAL MODEL Direct interaction of a projectile with a target nucleus considered as a whole Quantum model � Schrödinger equation � � 2 � � � � � � � � � 2 0 U U E � � � 2 � � Complex potential: U = V + iW Refraction Absorption
THE OPTICAL MODEL The optical model yields : Integrated cross sections Transmission coefficients Angular distributions
TWO TYPES OF APPROACHES Phenomenological (Semi-)microscopic Adjusted parameters No adjustable parameters Weak predictive power Usable without exp. data Very precise ( � 1%) Less precise ( � 5-10 %) Important work Quasi-automated Total cross sections
PHENOMENOLOGICAL OPTICAL MODEL - � 20 adjusted parameters - Very precise (1%) - Weak predictive power Total cross section (barn) Neutron energy (MeV)
PHENOMENOLOGICAL OPTICAL MODEL OMP & its parameters Solution of the Schrödinger equation Calculated observables Experimental data � el-inl ��� , A y ( � ), � tot , � reac , S 0 ,S 1 � el-inl ��� , A y ( � ), � tot , � reac , S 0 ,S 1 � Reaction , T lj , � direct
SEMI-MICROSCOPIC OPTICAL MODEL - No adjustable parameters - Based on nuclear structure properties � usable for any nucleus - Less precise than the phenomenological approach
SEMI-MICROSCOPIC OPTICAL MODEL Effective � = Optical potential Radial densities Interaction � = U( � (r’),E) � (r) � U(r,E) = � (r’) Independent of the nucleus Depends on the nucleus Depends on the nucleus
SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering
SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n)
SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n) , (p,p)
SEMI-MICROSCOPIC OPTICAL MODEL Unique description of elastic scattering (n,n) , (p,p) and (p,n)
SEMI-MICROSCOPIC OPTICAL MODEL Enables to give predictions for very exotic nuclei for which there exist no experimental data Experiment performed after calculation
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
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)
THE PRE-EQUILIBRIUM MODEL Shape elastic Elastic � Reaction � NC OPTICAL COMPOUND Fission PRE-EQUILIBRIUM MODEL NUCLEUS T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
THE PRE-EQUILIBRIUM MODEL Shape elastic Elastic � Reaction � NC PRE-EQUILIBRIUM OPTICAL COMPOUND Fission MODEL NUCLEUS T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
TIME SCALES AND ASSOCIATED MODELS (1/4) Typical spectrum shape • Always evaporation peak • Discrete peaks at forward angles • Flat intermediate region
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 1976 - link with Compound Nucleus established in 1987 - 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
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 1n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 1n 3n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 1n 3n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 3p-2h 1n 3n 5n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 3p-2h 1n 3n 5n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 3p-2h 4p-3h 1n 3n 5n 7n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F 1p 2p-1h 3p-2h 4p-3h 1n 3n 5n 7n
THE PRE-EQUILIBRIUM MODEL (Exciton model principle) E 0 E F Compound Nucleus 1p 2p-1h 3p-2h 4p-3h time 1n 3n 5n 7n
THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probabilité to find for a given time t the composite Probability 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.
THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probabilité to find for a given time t the composite Probability 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 dP(n,E,t) = Apparition dt - Disparition
THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probabilité to find for a given time t the composite Probability 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 dP(n,E,t) P(n-2, E, t) �� n-2, n (E) + P(n+2, E, t) �� n+2, n (E) = dt - Disparition
THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probabilité to find for a given time t the composite Probability 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 dP(n,E,t) P(n-2, E, t) �� n-2, n (E) + P(n+2, E, t) �� n+2, n (E) = dt [ ] �� n, n+2 (E) + �� n, n-2 (E) + �� n, emiss (E) - P(n, E, t)
THE PRE-EQUILIBRIUM MODEL (Master equation exciton model) P(n,E,t) = Probabilité to find for a given time t the composite Probability 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 dP(n,E,t) P(n-2, E, t) �� n-2, n (E) + P(n+2, E, t) �� n+2, n (E) = dt [ ] �� n, n+2 (E) + �� n, n-2 (E) + �� n, emiss (E) - P(n, E, t) Emission cross section in channel c ∞ � d � c (E, � c ) = � R � P(n, E, t) �� n, c (E) dt d � c n, � n=2 0
THE PRE-EQUILIBRIUM MODEL (Initialisation & transition rates)
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 2 � ℏ �� � � � ω (p,h,E) with p+h=n+2 �� n, n+2 (E) = Original formulation � 2 � ℏ 3 µ c � c � c,inv ��� c � ω (p-p b ,h,E - �� c - � B c ) �� n, c (E) = 2s c +1 ω (p,h,E)
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 2 � ℏ �� � � � ω (p,h,E) with p+h=n+2 �� n, n+2 (E) = � 2 � ℏ 3 µ c � c � c,inv ��� c � ω (p-p b ,h,E - �� c - � B c ) �� n, c (E) = 2s c +1 Q c (n) � � c ω (p,h,E) Corrections for proton-neutron distinguishability & complex particle emission
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 2 � ℏ �� � � � ω (p,h,E) with p+h=n+2 �� n, n+2 (E) = � 2 � ℏ 3 µ c � c � c,inv ��� c � ω (p-p b ,h,E - �� c - � B c ) �� n, c (E) = 2s c +1 Q c (n) � � c ω (p,h,E) State densities 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
THE PRE-EQUILIBRIUM MODEL (State densities) State densities in ESM • 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
THE PRE-EQUILIBRIUM MODEL (State densities) State densities in ESM • 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
THE PRE-EQUILIBRIUM MODEL Cross section Outgoing energy Total 9% Direct Direct 16% 45% 79% 39% Pré-équilibre Pre-equilibrium 12% Statistique Statistical <E Tot >= 12.1 <E Dir >= 24.3 <E PE >= 9.32 <E Sta >= 2.5 (MeV)
THE PRE-EQUILIBRIUM MODEL 14 MeV neutron + 93 Nb without pre-equilibrium without pre-equilibrium d � /dE(b/MeV) � (barn) pre-equilibrium Compound nucleus Outgoin neutron energy (MeV) Iincident neutron energy (MeV)
Nuclear level densities (formulae, tables, codes) • spin-, parity- dependent level densities fitted to D 0 • single particle level schemes • p-h level density tables
THE COMPOUND NUCLEUS MODEL Shape elastic Elastic � Reaction � NC OPTICAL COMPOUND Fission PRE-EQUILIBRIUM MODEL NUCLEUS T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
THE COMPOUND NUCLEUS MODEL Shape elastic Elastic COMPOUND � Reaction � NC NUCLEUS OPTICAL Fission PRE-EQUILIBRIUM MODEL T lj Inelastic (n,n’), ¡(n, � ), Direct components (n, � ), ¡etc…
THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission + � NC � reaction = � dir + � pre-equ N 0 N 0 -dN D N 0 -dN D -dN PE = E Z 0 Z 0 -dZ D Z 0 -dZ D -dZ PE = Z E* 0 E* 0 -dE* D E* 0 -dE* D -dE* PE = E* J 0 J 0 -dJ D J 0 -dJ D -dJ PE = J N,Z,E*,J � (N,Z,E*)
THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission + � NC � reaction = � dir + � pre-equ N 0 N 0 -dN D N 0 -dN D -dN PE = E Z 0 Z 0 -dZ D Z 0 -dZ D -dZ PE = Z E* 0 E* 0 -dE* D E* 0 -dE* D -dE* PE = E* J 0 J 0 -dJ D J 0 -dJ D -dJ PE = J N’,Z’,E’*,J’ N,Z,E*,J � (N,Z,E*) � (N’,Z’,E’*)
THE COMPOUND NUCLEUS MODEL (initial population) After direct and pre-equilibrium emission + � NC � reaction = � dir + � pre-equ N 0 N 0 -dN D N 0 -dN D -dN PE = E Z 0 Z 0 -dZ D Z 0 -dZ D -dZ PE = Z E* 0 E* 0 -dE* D E* 0 -dE* D -dE* PE = E* J 0 J 0 -dJ D J 0 -dJ D -dJ PE = J … N’’,Z’’,E’’*,J’’ N,Z,E*,J � (N,Z,E*) � (N’’,Z’’,E’’*)
THE COMPOUND NUCLEUS MODEL (basic formalism) Compound nucleus hypothesys - Continuum of excited levels - Independence between incoming channel a and outgoing channel b � ab = � (CN) � a P b T b �� � (CN) � a P b = = T a � � T c 2 k a c � Hauser- Feshbach formula T a T b �� � ab = � � T c 2 k a c
THE COMPOUND NUCLEUS MODEL (qualitative feature) Compound angular distribution & direct angular distributions 45 ° 90° 135°
THE COMPOUND NUCLEUS MODEL (complete channel definition) 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 l b • Total parity : � A (-1) = � CN = � B (-1) l a
THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered max Given by OMP I A + s a + l a �� �� �� ( 2J+1 ) �� ab = 2 k a (2I A +1) (2s a +1) �� = � J=| I A – s a |
THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered max I A + s a + l a �� �� �� ( 2J+1 ) �� ab = 2 k a (2I A +1) (2s a +1) �� = � J=| I A – s a | J + I A j a + s a J + I B j b + s b �� �� �� �� j a = | J – I A | l a = | j a – s a | j b = | J – I B | l b = | j b – s b | Parity selection rules T J �� T J �� T T a, l a , j a b, l b , j b �� �� (a) �� �� (b) �� T J �� T c, l c , j c c
THE COMPOUND NUCLEUS MODEL (loops over all quantum numbers) In realistic calculations, all possible quantum number combinations have to be considered max I A + s a + l a �� �� �� ( 2J+1 ) �� ab = 2 k a (2I A +1) (2s a +1) �� = � J=| I A – s a | Width fluctuation correction factor J + I A j a + s a J + I B j b + s b �� �� �� �� to account for deviations 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 | T J �� T J �� T J �� T T a, l a , j a b, l b , j b W �� �� (a) �� �� (b) a, l a , j a , b, l b , j b �� T J �� T c, l c , j c c
THE COMPOUND NUCLEUS MODEL (width fluctuation correction factor) Breit-Wigner resonance integrated and averaged over an energy width Corresponding to the incident beam dispersion < > � a � b < > � 2 � � ab = � tot 2 k a D < > � �� 2 �� T �� � Since D � a � b � < > � � ab = W ab � ��� c 2 k a c � � � a � b � a � b � W ab = with � tot � � tot
THE COMPOUND NUCLEUS MODEL (main methods to calculate WFCF) • Tepel method 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 ?
THE COMPOUND NUCLEUS MODEL (the GOE triple integral)
THE COMPOUND NUCLEUS MODEL (flux redistribution illustration)
THE COMPOUND NUCLEUS MODEL (multiple emission) Target Compound Nucleus fission E S �� n’ S p S �� S p �� �� S n n (2) S �� n’ �� S n + Loop over CN spins and parities n p S n Z d J �� S p n S �� S p Z c S n S �� S p S �� �� S n S n Z c -1 N N c -2 N c -1 N c
REACTION MODELS & REACTION CHANNELS n + 238 U Optical model Cross section (barn) + Statistical model + Pre-equilibrium model � � R = � d + � PE + � CN ��� � �� ��� � � = � nn’ + � nf + � n �� ������ Neutron energy (MeV)
THE COMPOUND NUCLEUS MODEL (compact expression) � NC � = � ab où b = �� , ¡n, ¡p, ¡d, ¡t, ¡…, ¡fission ¡ b < > J � T b ��� 2 � � J � � 2J+1 � � � ab = T lj ��� W ��� � 2s+1 �� 2I+1 � < > � J � J, � ���� k a T d ��� �� l �� and � = � -1 � � A J = l � + s � + I A = j � + I A with < > and T b ( � ) = transmission coefficient for outgoing channel � associated with the outgoing particle b
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