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SLOWING DOWN OF NEUTRONS

SLOWING DOWN OF NEUTRONS. Elastic scattering of neutrons. Lethargy. Average Energy Loss per Collision. Resonance Escape Probability Neutron Spectrum in a Core. Chain Reaction. n. ν. β. Moderator. γ. ν. γ. Moderator. β. Why to Slow Down (Moderate)?. Principles of a Nuclear Reactor.

gretchen-herrmann
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SLOWING DOWN OF NEUTRONS

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  1. SLOWING DOWN OF NEUTRONS • Elastic scattering of neutrons. • Lethargy. Average Energy Loss per Collision. • Resonance Escape Probability • Neutron Spectrum in a Core. T09: Thermalisation

  2. Chain Reaction n ν β Moderator γ ν γ Moderator β T09: Thermalisation

  3. Why to Slow Down (Moderate)? T09: Thermalisation

  4. Principles of a Nuclear Reactor Leakage E N2 2 MeV N1 Fast fission n n/fission Energy Slowing down Resonance abs. ν ≈ 2.5 Non-fissile abs. Non-fuel abs. 1 eV Fission 200 MeV/fission Leakage T09: Thermalisation

  5. Breeding T09: Thermalisation

  6. T09: Thermalisation

  7. Energy Dependence T09: Thermalisation

  8. Breeding T09: Thermalisation

  9. Space and Energy Aspects z r y x Double differential cross section T09: Thermalisation

  10. Differential Solid Angle ez Ω θ d3r z ey r φ y ex x T09: Thermalisation

  11. Hard Sphere Model θ r Total scatteringcross section σ = 2πr2 n r T09: Thermalisation

  12. dθ Hard Sphere Scattering impact parameter cross sectionσ(θ) θ b(θ) r n(r) σ(θ) nis the number of neutrons deflected by an angle greater than θ T09: Thermalisation

  13. Unit sphere r = 1 n T09: Thermalisation

  14. Differential Cross Section Detector n T09: Thermalisation

  15. Elastic Scattering vc u v u0  q U0 U T09: Thermalisation

  16. Energy Loss θ = 180 θ = 0 T09: Thermalisation

  17. Change of Variables Energy Velocity E+dE v+dv v E T09: Thermalisation

  18. ?? p(E;E0) E0 aE0 E E-dE E T09: Thermalisation

  19. Quantum mechanics + detailed nuclear physics analysis conclude • Elastic scattering is isotropic in CM system for: • neutrons with energies E< 10 MeV • light nuclei with A < 13 T09: Thermalisation

  20. Post Collision Energy Distribution aE0 E0 E T09: Thermalisation

  21. Average Logarithmic Energy Loss T09: Thermalisation

  22. Average Logarithmic Energy Loss T09: Thermalisation

  23. T09: Thermalisation

  24. Number of collision required for thermalisation: For non-homogeneous medium: Average cosinevalue of the scattering angle in CM-system T09: Thermalisation

  25. Average Cosine in Lab-System T09: Thermalisation

  26. T09: Thermalisation

  27. Slowing-Down Features of Some Moderators N - number of collision to thermal energy xSs - slowing down power xSs/Sa - moderation ratio (quality factor) T09: Thermalisation

  28. Neutron Velocity Distribution kB = 1.381×10-23 J/K = 8.617×10-5 eV/K v+dv v Velocity space: 4πv2dv Probability that energy level E=mv2/2 is occupied: T09: Thermalisation

  29. Maxwell Distribution for Neutron Density The most probable velocity: and corresponding energy: T09: Thermalisation

  30. Maxwell Distributionfor Neutron Flux Don’t forget : T09: Thermalisation

  31. T09: Thermalisation

  32. Average Energy of Neutrons Neutron flux distribution: For thermal neutrons T09: Thermalisation

  33. Average cosine of scattering angle: LAB-system: CM : The consequence of µ0 ¹ 0 in the laboratory-system is that the neutron scatters preferably forward, specially for A = 1 i.e. hydrogen and practically isotropic scattering for A = 238 i.e. Uranium, because µ0 » 0 i.e. Y = 90o in average. This corresponds to isotropic scattering. ltr is definedas effective mean free path for non-isotropic scattering. T09: Thermalisation

  34. Transport Mean Free Path Y Information regarding the original direction is lost Y Y ls lscosY lscosY2 ltr T09: Thermalisation

  35. Slowing-Down of Fast Neutrons • Infinite medium • Homogeneous mixture of absorbing and scattering matter • Continues slowing down • Uniformly distributed neutron source Q(E) Φ(E) = [n/(cm2×s×eV)] Φ(E)dE = number of neutrons with energies in dE about E T09: Thermalisation

  36. Continues Slowing-Down assumed slowing-down E real slowing-down dE dt t T09: Thermalisation

  37. Slowing-Down Density Energy • q(E) - number of neutrons, which per cubic-centimeter and second pass energy E. If no absorption exists in medium, so:q(E)= Q; Q- source yield (n×cm-3 s-1) • Assuming no or weak absorption (without resonances) • Neutrons of zero energy are removed from the system Q E0 E q(E) 0 T09: Thermalisation

  38. Lethargy Variable T09: Thermalisation

  39. Lethargy Scale 1 collision T09: Thermalisation

  40. Energy Dependence Energy Lethargy 0 Eref E/α q(u) E u E+dE u+du Infinite medium, no losses, constant Σs T09: Thermalisation

  41. Neutron spectrum F(E) E F(u) u 0 5 20 10 15 E 10 MeV 0.025 eV T09: Thermalisation

  42. Resonance Absorption Energy Lethargy Probability for absorption per collision: E/α u–lnα-1 Number of collisions per a neutron in du or dE: E u E+dE u+du Probability for absorption in du or dE: Absorption in ducauses a relative change in q: T09: Thermalisation

  43. Resonance Escape T09: Thermalisation

  44. st=ss+sc~sc s F(u) F0(u) F(u) q0 q E u T09: Thermalisation

  45. Life Time How long time does the neutron exist under slowing-down phase respectively as thermal? Slowing-down in time - ts: Number of collisions in du: Number of collisions in dt: v(1 eV) = 1.39 · 106 cm/s v(0.1 MeV ) = 4.4 · 108 cm/s Thermal life-length - tt : T09: Thermalisation

  46. Neutrons Slowing-Down Time and Thermal Life-Time T09: Thermalisation

  47. Under the Neutron Life-Time (3) (2) (1) E 0 1 eV 0.1 MeV 10 MeV (1) Fission neutrons - fast neutrons (10 MeV-0.1 MeV) (2) Slowing-down neutrons – resonance neutrons (0.1MeV - 1 eV) (3) Thermal neutrons (1eV - 0.) T09: Thermalisation

  48. The END T09: Thermalisation

  49. E+dE v+dv v E θ = 180 θ = 0 T09: Thermalisation

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