Moderation of Neutrons (without absorption)

1. Moderation of Neutrons (without absorption) William D’haeseleer

2. 1. Energy Loss in Elastic Collision Recall from before: l = lab c = center of mass ‘ = after collision direction of incidence

3. Energy Loss in Elastic Collision Because of elastic nature: Apply rule of cosines:

4. Energy Loss in Elastic Collision Now, Therefore, Define parameter

5. Energy Loss in Elastic Collision Then:

6. Energy Loss in Elastic Collision After collision, always But where precisely? from now on, drop subscript “l”

7. Energy Loss in Elastic Collision = probability that a neutron with originally a lab energy E, has an energy between E’ and E’+dE’ after the collision because of one-to-one relationship between scattering angle and energy → probability to have energy between E’ and E’+dE’ after the collision ≡ probability to be scattered between an angle

8. Energy Loss in Elastic Collision Because of we have which leads to P(E→E’) as follows:

9. Energy Loss in Elastic Collision * Special case: isotropic scattering in c.m. system Consequently independent of E’ !! (see Fig. 6.2)

10. Energy Loss in Elastic Collision Fig 6.2 en 6.5

11. Energy Loss in Elastic Collision * backward scattering * forward scattering But always see Fig 6.2

12. Energy Loss in Elastic Collision • Average energy after the collision for isotropic scattering

13. Energy Loss in Elastic Collision • Average energy loss for isotropic scattering

14. Energy Loss in Elastic Collision • Average relative energy loss isotropic scattering for hydrogen α = 0

15. 2. Collision Density & Slowing Down Density - Eventual goal of computations: - For moderation/slowing-down computations, it is handy: interaction rate ≡ collision density # of collisions per m³ and per sec, per unit energy internal

16. Collision Density & Slowing Down Density - Define also: slowing down density # of neutrons per m³ at location that drops below the energy E per second, or, # of collisions per m³ and second at , whereby the energy drops below the value E cfr counting of cars

17. 3. Moderation of neutron in hydrogen * In hydrogen, α = 0 → n can be stopped in a single collision * Assumptions for simplicity: - ∞, homogeneous medium of H - uniformly distributed neutron source - all neutrons E0 - S neutrons per m³ per s

18. Moderation of neutron in hydrogen * Determination of collision density F(E) Consider scattering of neutrons to and from interval dE 1) Source neutrons: S neutr/m³s with E0 - in steady state: no accumulation at E0 precisely S collisions/m³s at E0 - α = 0; hence P (E0 →E)dE =dE/E0 - thus SdE/E0 source neutrons directly scattered into dE

19. Moderation of neutron in hydrogen 2) neutrons between E and E0 - F(E’)dE’ collisions/m³s in dE’ - still because α = 0 = prob. that one of the scattered neutrons ends up with dE - Hence neutrons per s will be scattered from dE’ into dE - The in-scattering in dE –without source neutrons– is therefore

20. Moderation of neutron in hydrogen 3) neutrons scattering out of dE F(E)dE Global balance in dE (after deleting dE) Result:

21. Moderation of neutron in hydrogen * Determination of slowing down density q(E) → consider number of collisions at energies >E, and take fraction that ends up with energy below E i) source neutrons - S collisions per m³ and s - end up uniformly between 0 and E0 - fraction that ends up below E equals E/E0 - source contribution to q(E)=S E/E0

22. Moderation of neutron in hydrogen ii) neutrons between E and E0 - in interval dE’, F(E’)dE’ collisions per m³ and s - fraction that “falls” below E equals E/E’ - contribution to q(E) from dE’ equals - contribution of n between E and E0 equals: → Slowing down density or, also:

23. 4. Lethargy and ξ * Recall collision density just computed  number of interactions increases with decreasing energy ~ about 106 more collisions near 1 eV than at 1 MeV

24. Lethargy and ξ * More handy to use a variable that varies loss rapidly E0 = arbitrary reference energy; usually, E0 > all E → then u > 0 always u = 0 at E = E0 → monotonically increasing function for E < E0

25. Lethargy and ξ * Clearly * At every collision, E decreases → u increases Increase

26. Lethargy and ξ * One defines For isotropic scattering in c.m. system, P = constant all possible end energies

27. Lethargy and ξ Examples: H α = 0 ξ = 1 C α = 0.716 ξ = 0.158 U α = 0.983 ξ = 0.00838

28. 5. Moderation of neutrons for A>1 * For H, α = 0 → complete energy interval accessible for scattered neutrons * Now A > 1, α ≠ 0 → α E0 < E < E0 * F now to be written as a series expansion:

29. Moderation of neutrons for A>1 Recall: F(E)dE = # of collisions per m³ and s with energy between E and E+dE now F1(E) = collision density of all neutrons that need one collision to reach dE (they have then their following collision in dE; hence F1 is part of F(E) ) and Fn(E) = collision density of all neutrons that need n collisions before reaching dE

30. Moderation of neutrons for A>1 i) Computation of F1 F1 originates from source neutrons

31. Moderation of neutrons for A>1 ii) Computation of F2

32. Moderation of neutrons for A>1 a) αE0 < E < E0 By definition: F1(E’)dE’ = # of neutrons that have their 2-nd collision in dE’ (they needed 1 collision to reach dE’) Probability to reach dE about E:

33. Moderation of neutrons for A>1 Hence,

34. Moderation of neutrons for A>1 b) α² E0 < E < αE0 Neutrons must have had their second collision between E and E/α Hence

35. Moderation of neutrons for A>1 c) E < α² E0 Clearly, here In conclusion for F2: See Fig 6.5

36. Moderation of neutrons for A>1 * Discontinuities in F1, and in the derivatives of F2, F3 → lead to same discontinuities in F * Discontinuities are really limited to region E ≥ α³ E0 For E < α³ E0 → F(E) “quite” smooth  asymptotic energy region For E ≤ α³ E0

37. Moderation of neutrons for A>1 Hence, (**) With solution: Constant C cannot be found from (**) → needs to be found through slowing down density q

38. Moderation of neutrons for A>1 * Slowing-down density in asymptotic energy region P(E→E’)= constant for isotropic scattering Hence, probability to pass E equals the ratio / or thus

39. Moderation of neutrons for A>1 Contribution to q(E) from dE’: Total contribution from E → E/α then: But without absorption: with F(E’)=C/E’, and after integration

40. Moderation of neutrons for A>1 * In summary in asymptotic energy region Here for A > 0, but idem as for isotropic scattering in hydrogen (disregarding constant ξ)

41. Moderation of neutrons for A>1 * In terms of u For large u, the asymptotic F tends towards a constant See Fig 6.8 For all A, starting from E ≥ α³ E0, the transient behavior of F(u) disappears!!

42. Moderation of neutrons for A>1 Complete solution for F(u) ! Even for all A, F(u) ~ constant starting from E ≥ α³ E0

43. 6. Non mono-energetic sources - until now: sources mono-energetic - assume now: neutron spectrum S(E’) - Consider Hydrogen: - energy-dependent flux = independent of source energy - recall Eq. (6.19) - then now

44. Non mono-energetic sources - such that - practically, one puts - then, - For A > 1 quite complex But for E ≤ α³ Es idem as H  (but with ξ)

45. 7. Slowing Down in Mixtures of Nuclides - - to be read in Lamarsh - - Look at Eqs. (6-57) → (6-59)

46. 8. Multiscattered Neutrons Consider a multitude of collisions in an ∞ medium At “birth”: u0 = 0 After n collisions: The average u after n collisions For isotropic scattering in m.c. system: increase in lethargy due to i-th collision

47. Multiscattered Neutrons Consequently: Or, equivalently: = the required number of collisions to bring the average to the value of , of which the corresponding energy equals E

48. Multiscattered Neutrons Example: E0 = 2 MeV → E = 1eV in H H2O D C Fe U n 14.5 15.8 20.0 91.3 407 1730

49. Multiscattered Neutrons - After a sufficient number of collisions = probability that the lethargy of a neutron lies between u and u+du, after precisely n collisions - For large n: central limit theorem

50. Multiscattered Neutrons - Spread around average value n ξ ± σ (70% within ± σ interval) - Relative deviation in average value