1. HT2005: Rector PhysicsT09: Thermalisation1 SLOWING DOWN OF NEUTRONS Elastic scattering of neutrons. Lethargy. Average Energy Loss per Collision. Resonance Escape Probability Neutron Spectrum in a Core.

2. HT2005: Rector PhysicsT09: Thermalisation2 Chain Reaction ν β γ γ β ν n Moderator

3. HT2005: Rector PhysicsT09: Thermalisation3 Why to Slow Down (Moderate)?

4. HT2005: Rector PhysicsT09: Thermalisation4 Principles of a Nuclear Reactor n/fission N1N1 N2N2 Leakage Fast fission Resonance abs. Non-fuel abs. Leakage Non-fissile abs. Fission Slowing down Energy E 2 MeV 1 eV 200 MeV/fission ν ≈ 2.5

5. HT2005: Rector PhysicsT09: Thermalisation5 Breeding

6. HT2005: Rector PhysicsT09: Thermalisation6

7. HT2005: Rector PhysicsT09: Thermalisation7 Energy Dependence

8. HT2005: Rector PhysicsT09: Thermalisation8 Breeding

9. HT2005: Rector PhysicsT09: Thermalisation9 Space and Energy Aspects r x y z Double differential cross section

10. HT2005: Rector PhysicsT09: Thermalisation10 Differential Solid Angle x y z ezez eyey exex r Ω d3rd3r φ θ

11. HT2005: Rector PhysicsT09: Thermalisation11 Hard Sphere Model r Total scattering cross section σ = 2πr 2 n r θ

12. HT2005: Rector PhysicsT09: Thermalisation12 r b(θ) θ impact parameter cross section σ(θ ) dθ n(r)n(r) σ(θ) n is the number of neutrons deflected by an angle greater than θ Hard Sphere Scattering

13. HT2005: Rector PhysicsT09: Thermalisation13 Unit sphere r = 1 n

14. HT2005: Rector PhysicsT09: Thermalisation14 Detector n Differential Cross Section

15. HT2005: Rector PhysicsT09: Thermalisation15 Elastic Scattering u0u0 U0U0 U u vcvc v  

16. HT2005: Rector PhysicsT09: Thermalisation16 Energy Loss θ = 0θ = 180

17. HT2005: Rector PhysicsT09: Thermalisation17 E v E+dE v+dv Change of Variables Energy Velocity

18. HT2005: Rector PhysicsT09: Thermalisation18 E E-dE E E0E0 E0E0 p(E;E 0 ) ??

19. HT2005: Rector PhysicsT09: Thermalisation19 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

20. HT2005: Rector PhysicsT09: Thermalisation20 E E0E0 E0E0 Post Collision Energy Distribution

21. HT2005: Rector PhysicsT09: Thermalisation21 Average Logarithmic Energy Loss

22. HT2005: Rector PhysicsT09: Thermalisation22 Average Logarithmic Energy Loss

23. HT2005: Rector PhysicsT09: Thermalisation23

24. HT2005: Rector PhysicsT09: Thermalisation24 Number of collision required for thermalisation: For non-homogeneous medium: Average cosine value of the scattering angle in CM-system

25. HT2005: Rector PhysicsT09: Thermalisation25 Average Cosine in Lab-System

26. HT2005: Rector PhysicsT09: Thermalisation26 MaterialAα 00 1H1H100.667 2D2D20.1110.333 4 He40.3600.167 6 Li60.5100.095 9 Be90.6400.074 10 B100.6690.061 12 C120.7160.056 238 U2380.9380.003 H2OH2O**0.037 D2OD2O**0.033

27. HT2005: Rector PhysicsT09: Thermalisation27 ModeratorξNξΣ s ξΣ s /Σ a H2OH2O0.92719.71.3662 D2OD2O0.510360.1805860 Be0.209870.153138 C0.1581150.060166 U.00842170.00400.011 N - number of collision to thermal energy  s - slowing down power  s /  a - moderation ratio (quality factor) Slowing-Down Features of Some Moderators

28. HT2005: Rector PhysicsT09: Thermalisation28 k B = 1.381×10 -23 J/K = 8.617×10 -5 eV/K Velocity space: v+dv v 4πv 2 dv Probability that energy level E=mv 2 /2 is occupied: Neutron Velocity Distribution

29. HT2005: Rector PhysicsT09: Thermalisation29 The most probable velocity: and corresponding energy: Maxwell Distribution for Neutron Density

30. HT2005: Rector PhysicsT09: Thermalisation30 Don’t forget : Maxwell Distribution for Neutron Flux

31. HT2005: Rector PhysicsT09: Thermalisation31

32. HT2005: Rector PhysicsT09: Thermalisation32 Neutron flux distribution: For thermal neutrons Average Energy of Neutrons

33. HT2005: Rector PhysicsT09: Thermalisation33 Average cosine of scattering angle: CM : LAB-system: The consequence of µ 0  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 µ 0  0 i.e.  90 o in average. This corresponds to isotropic scattering. tr is defined as effective mean free path for non-isotropic scattering.

34. HT2005: Rector PhysicsT09: Thermalisation34    s cos  s cos   s t r Transport Mean Free Path Information regarding the original direction is lost

35. HT2005: Rector PhysicsT09: Thermalisation35 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/(cm 2 ×s × eV)] Φ(E)dE = number of neutrons with energies in dE about E

36. HT2005: Rector PhysicsT09: Thermalisation36 E t dE dt assumed slowing-down real slowing-down Continues Slowing-Down

37. HT2005: Rector PhysicsT09: Thermalisation37 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 Energy E q(E)q(E) Slowing-Down Density E0E0 Q 0

38. HT2005: Rector PhysicsT09: Thermalisation38 Lethargy Variable

39. HT2005: Rector PhysicsT09: Thermalisation39 Lethargy Scale 1 collision

40. HT2005: Rector PhysicsT09: Thermalisation40 Energy Lethargy E ref 0 Energy Dependence Eu q(u) u+duE+dE E/ α Infinite medium, no losses, constant Σ s

41. HT2005: Rector PhysicsT09: Thermalisation41 Neutron spectrum E  (E) u  (u) E 0.025 eV 20 15 10 5 0 10 MeV

42. HT2005: Rector PhysicsT09: Thermalisation42 Probability for absorption per collision: Number of collisions per a neutron in du or dE: Probability for absorption in du or dE: Absorption in du causes a relative change in q: Resonance Absorption Energy u E Lethargy u+du E/α u–lnα -1 E+dE

43. HT2005: Rector PhysicsT09: Thermalisation43 Resonance Escape

44. HT2005: Rector PhysicsT09: Thermalisation44  (u) E u  t  s  c  c   0 (u) q0q0  (u) q

45. HT2005: Rector PhysicsT09: Thermalisation45 How long time does the neutron exist under slowing-down phase respectively as thermal? Slowing-down in time - t s : Number of collisions in du: Number of collisions in dt: v (1 eV) = 1.39 · 10 6 cm/s v (0.1 MeV ) = 4.4 · 10 8 cm/s Thermal life-length - t t : Life Time

46. HT2005: Rector PhysicsT09: Thermalisation46 Materialt fast (  s) t thermal (  s) H2OH2O1200 D2OD2O8 1.5  10 5 Be104300 C25 1.2  10 4 Neutrons Slowing-Down Time and Thermal Life-Time

47. HT2005: Rector PhysicsT09: Thermalisation47 (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.) Under the Neutron Life-Time 10 MeV0.1 MeV1 eV0 (1)(2)(3) E

48. HT2005: Rector PhysicsT09: Thermalisation48 The END

49. HT2005: Rector PhysicsT09: Thermalisation49 θ = 0θ = 180 E v E+dE v+dv