Neutron induced resonance reactions

1. Frank Gunsing CEA/Saclay DSM/IRFU/SPhN F-91911 Gif-sur-Yvette, France gunsing@cea.fr Neutron induced resonance reactions

2. Reaction: • X + a  Y + b• X(a,b)Y • X(a,b) Examples of equivalent notations: Reaction cross section , expressed in barns, 1 b = 10–28 m2 Neutron induced nuclear reactions: • elastic scattering(n,n) • inelastic scattering(n,n’) • capture (n,) • fission (n,f) • particle emission (n,), (n,p), (n,xn) Total cross section tot: sum of all reactions Neutron-nucleus reactions 10B + 1n 7Li + 4He 10B + n 7Li +  10B(n,) 238U + n 239U* 238U + n 239U +  238U(n,)

3. Reaction: • X + a Y + b • X(a,b)Y Cross section:function of the kinetic energy of the particle a Differential cross section: function of the kinetic energy of the particle aand function of the kinetic energy or the angleof the particle b Double differential cross section:function of the kinetic energy of the particle aand function of the kinetic energy and the angleof the particle b  a b X Y Neutron-nucleus reactions

4. Cross sections sgsn et sf cross section (barn) neutron energy (eV)

5. Interference of spotential and sn transmissionT = exp(–n.T)0<T<1

6. nuclear state The nucleus as a quantum system shell model representation: configuration of nucleons in their potential level scheme representation:excited states of a nucleus(shell model and other states) 0 excitation energy potential depth 3rd excited state 2nd excited state 1st excited state ground state AX –V0 neutrons protons

7. nuclear state The nucleus as a quantum system shell model representation: configuration of nucleons in their potential level scheme representation:excited states of a nucleus(shell model and other states) 0 excitation energy potential depth 3rd excited state 2nd excited state 1st excited state ground state AX –V0 neutrons protons

8. nuclear state The nucleus as a quantum system shell model representation: configuration of nucleons in their potential level scheme representation:excited states of a nucleus(shell model and other states) 0 excitation energy potential depth 3rd excited state 2nd excited state 1st excited state ground state AX –V0 neutrons protons

9. nuclear state The nucleus as a quantum system shell model representation: configuration of nucleons in their potential level scheme representation:excited states of a nucleus(shell model and other states) 0 excitation energy potential depth 3rd excited state 2nd excited state 1st excited state ground state AX –V0 neutrons protons

10. state with a life time t: definition (Heisenberg): Fourier transform gives energy profile: Decay of a nuclear state eigen state,transitition E0Eflife time t E0 Ef

11. Neutron-nucleus reactions Conservation of probability density: Solve Schrödinger equation of system to get cross sections. Shape of wave functions of in- and outgoing particles are known, potential is unknown. Two approaches: • calculate potential (optical model calculations, smooth cross section) • use eigenstates (R-matrix, resonances)

12. cross section: + + entrance channel compound nucleus exit channel partial incoming wave functions: partial outgoing wave functions: related by collision matrix: External region (r>ac, well separated particles):• no interaction, Schrödinger equation solvable. Internal region (r<ac compound nucleus):• wave function is expansion of eigenstates . R-matrix formalism

13. external region internal region match value and derivate of separation distance Internal region: very difficult, Schrödinger equation cannot be solved directlysolution: expand the wave function as a linear combination of its eigenstates.using the R-matrix: Find the wave functions External region: easy, solve Schrödinger equationcentral force, separate radial and angular parts.solution: solve Schrödinger equation of relative motion: • Coulomb functions • special case of neutron particles (neutrons): fonctions de Bessel

14. The R-matrix formalism

15. En + En D=10 eV s AX D=100 keV Sn =10 MeV A+1X Compound neutron-nucleus reactions n + compound nucleus reaction

16. The R-matrix formalism is adapted to describe compound nucleus reactions. Typically used for neutron-induced reactions at low energy (En<10 MeV, resonance region). The resonance parameters are properties of the excited nuclear levels:- in the resolved resonance region (RRR), to each level (resonance) corresponds a set of parameters:- in the unresolved resonance region (URR)average parameters are used: The R-matrix formalism

17. A same set of resonance parameters is used to produce all resonant reactions, at low energies mainly elastic scattering and capture (and fission). In a measurement, one does not measure a cross section, but a reaction yield or transmission factor. The measured reaction yield is not equally sensitive to all parameters, additional constraints can be necessary to extract RP from measurement. Resonance parameters

18. measurements calculations capture cross section neutron flux stellarspectrum thermalspectrum fissionspectrum whitesource neutron energy (eV) Neutron-nucleus reactions R-matrix resonance description • fine structure (resonances) • highly fluctuating cross sections • resonance parametrization • low energy, few channels Optical model calculations • gross structure • average cross sections • optical model potential • high energy, many channels

19. Compound nucleus reactions resonances, R-matrix thermal RRR URR OMP 197Au total cross section (b) neutron energy (eV)

20. 6Li 27Al 55Mn total cross section (b) 197Au 208Pb 241Am neutron energy (eV)

21. The nucleus in the vicinity of Sn is described by the Gaussian Orthogonal Ensemble (GOE) The matrix elements are random variables with a Gaussian distribution. • Consequences: • The partial width have a Porter-Thomas distribution. • The spacing of levels with the same Jπ have approximately a Wignerdistribution. Statistical model Porter-Thomas distribution

22. Distribution of the spacing of two consecutive levels

23. Distribution of neutron widths

24. Sn Level density • Separated states are available • - at low excitation energy • - in the region just above Sn

25. Known nuclei

26. stable Known nuclei

27. stable – EC, +  Known nuclei

28. Level spacing D0

29. Level spacing D0

30. Level spacing D0

31. 4 - 10 MeVpour noyaux stables Neutron separation energy Z N

32. Neutron separation energy

33. fissionactivation fissionactivation n + 235U 6.2 MeV n + 6.6 MeV 238U Sn = 6.5 MeV Sn = 4.8 MeV 236U 239U Fission of 235U+n and 238U+n

34. Measurement of (n.xn) by gamma-ray spectroscopy excitationbyneutron gammasignature 0+ 180W 6.7 MeV 9/2+ 181W 8.1 MeV 0+ 182W 6.2 MeV 1/2– 183W example: 182W(n,3n)180W

35. Simulated gamma-ray spectrum

36. moderator H2O Moderation of neutrons Maxwell-Boltzmann velocity distribution: thermal neutrons:

37. Libraries• JEFF - Europe• JENDL - Japon• ENDF/B - US• BROND - Russia• CENDL - China Common format:ENDF-6 Contents:Data for particle-induced reactions (neutrons, protons, gamma, other)but also radioactive decay data Data are indentified by “materials” (isotopes, isomeric states, (compounds) )ex. 16O: mat = 825natV: mat = 2300242mAm: mat = 9547 Evaluated nuclear data libraries

38. The library JEFF-3.1

39. Files for a materialfrom report ENDF-102 1 General information 2 Resonance parameter data 3 Reaction cross sections 4 Angular distributions for emitted particles 5 Energy distributions for emitted particles 6 Energy-angle distributions for emitted particles 7 Thermal neutron scattering law data 8 Radioactivity and fission-product yield data 9 Multiplicities for radioactive nuclide production 10 Cross sections for photon production 12 Multiplicities for photon production 13 Cross sections for photon production 14 Angular distributions for photon production 15 Energy distributions for photon production 23 Photo-atomic interaction cross sections 27 Atomic form factors or scattering functions for photo-atomic interactions 30 Data Covariances obtained from parameter covariances and sensitivities 31 Data covariances for nubar 32 Data covariances for resonance parameters 33 Data covariances for reaction cross sections 34 Data covariances for angular distributions 35 Data covariances for energy distributions 39 Data covariances for radionuclide production yields 40 Data covariances for radionuclide production cross sections

40. Example: part of an evaluated data file

41. Ex A+1X A+1Xradioactive Neutron Capture Gamma-Ray Detection • Activation - cross sections integrated over known neutron spectrum - applicable to some nuclei only - no time of flight • Level population spectroscopy - applicable to some nuclei only - feasible with HPGe detectors, • Total energy detection - cEx, requires weighting function - neutron insensitive detector example: C6D6 liquid scintillator • Total absorption detection - requires  = 4π, efficiency 100% - capture/fission discrimination in possible, example BaF2 total absorption calorimeter

42. Measuring a reaction yield using the time-of-flight technique reaction productdetector production target,neutron source sample Pulse of charged particles flight length L time zero

43. Measuring a reaction yield using the time-of-flight technique reaction productdetector production target,neutron source sample Pulse of charged particles flight length L

44. Measuring a reaction yield using the time-of-flight technique reaction productdetector production target,neutron source sample Pulse of charged particles flight length L

45. Measuring a reaction yield using the time-of-flight technique reaction productdetector production target,neutron source sample Pulse of charged particles flight length L time of flight t

46. Measuring a reaction yield using the time-of-flight technique reaction productdetector production target,neutron source sample Pulse of charged particles flight length L time of flight t Kinetic energy of the neutron by time-of-flight

47. time-energy relation Resolution Neutron time-of-flight experiment distribution L neutronstart neutronstop t • neutron time-of-flight:• flight length:• neutron kinetic energy: The resolution can be expressed equivalenty in time, distance and energy:

48. Measured reaction yield isolated Breit-Wigner resonance, decaying quantum state with half-life =h/

49. Measured reaction yield isolated Breit-Wigner resonance, decaying quantum state with half-life =h/ Doppler broadened

50. Measured reaction yield isolated Breit-Wigner resonance, decaying quantum state with half-life =h/ Doppler broadened resolution broadened andshifted