1. Neutron Chain Reaction Systems William D’haeseleer

2. Neutron Chain Reaction Systems References: Lamarsh, NRT, chapter 4 Lamarsh & Baratta, chapter 4 Also Duderstadt & Hamilton § 3.I

3. Concept of chain reaction Initially, reactor contains a certain amount of fuel, with initially N f (0) fissile nuclei (e.g. U-235) To get fission process started necessary to have an “external” neutron source → this source initiates fission process

4. Concept of chain reaction The by fission produced neutrons can be absorbed in U-235 → can lead to fission 2.5 n fission2.5 n etc… etc…  CHAIN REACTION

5. Concept of chain reaction Chain reaction 235 U

6. Concept of chain reaction

7. If “few” neutrons leak out, or parasitically absorbed: → exponentially run-away chain reaction  super critical reactork > 1 If “too many” neutrons leak out, or parasitically absorbed: → exponentially dying-out chain reaction  sub critical reactork < 1

8. Concept of chain reaction If after one generation precisely 1 neutron remains, which “activates” again precisely 1 neutron, → stationary regime  critical reactor k = 1 k= multiplication factor k= multiplication factor number of neutrons in one generation number of neutrons in previous generation =

9. Concept of chain reaction must be k=1

10. Multiplication factor 1.Infinite reactor (homogeneous mixture of enriched U and moderator) Assume at a particular moment n thermal neutrons absorbed in fuel These produce n η fission neutrons But sometimes also fissions due to fast neutrons → correction factorε ≥ 1(e.g., 1.03)  in fact n η ε fission neutrons

11. Multiplication factor These n η ε neutrons must be slowed down to thermal energies p≡ resonance escape probability = probability for not being absorbed in any of the resonances during slowing down  n η ε pthermalized neutrons After thermalization, a fraction f will be absorbed in the fuel U-235; the remainder absorbs in structural material, moderator material, U-238, etc  n η ε p fthermal neutrons absorbed in the fuel

12. Multiplication factor Hence, after the next generation:

13. Multiplication factor Note: three-step approach for multiplication factor → mono-energetic infinite reactor → moderation in infinite thermal reactor → moderation in finite thermal reactor

14. Multiplication factor i.Mono-energetic infinite reactor

15. Multiplication factor i.Mono-energetic infinite reactor P AF = prob that neutron will be absorbed in the fuel “thermal utilization factor”

16. Multiplication factor i.Mono-energetic infinite reactor

17. Multiplication factor P f = prob that an absorbed neutron in the fuel leads to fission Number of neutrons in next generation:

18. Multiplication factor ii.Moderation in infinite thermal reactor Now ηidentified with absorption of thermal neutrons Also fdefined for thermal neutrons → reasons for name “thermal utilization factor” 

19. Multiplication factor iii. Moderation in finite thermal reactor

20. Multiplication factor iii.Moderation in finite thermal reactor P NL = non-leakage probability k ≡ k eff = k ∞ P NL k = multiplication factor for finite reactor

21. Multiplication factor 2.Finite reactor  A critical reactor always has k eff = 1 Influencing factors of k eff : - leakage probability :geometry - amount of fuel:composition - presence/absence strong absorbers:composition non leakage probability

22. Critical Mass The larger the surface of a certain volume, the higher the probability to leak away The larger R: –more fissile isotopes in volume –larger leak-through surface → relatively more production of neutrons than leakage But Vol ∕ Surf

23. Critical Mass Critical mass = minimal mass for a stationary fission regime Examples: critical mass of U-235 ≤ 1 kg-when homogeneously dissolved as uranium salt in H 2 O -when concentration of U-235 > 90% in the uranium salt ≥ 200 kg-when U-235 is present in 30 tonnes of natural uranium embedded in matrix of C ! Natural uranium alone with 0.7% U-235 can never become critical, whatever the mass (because of absorption in U-238)

24. Critical Mass

27. Nuclear Fuels * fissile isotopes U-233 U-235 only this isotope is Pu239 available in nature * fertile isotopes Th-232 U-233 U-238 Pu-239 U-235 cannot be made artificially → to increase fraction of U-235 in a “U-mixture” → need to ENRICH “enrichment”

28. Nuclear Fuels * consider reactor with 97% U-238 and 3% U-235 most of the U-235 fissions, “produces” energy, produces n U-238 absorbs neutronsPu-239 an amount Pu-239 fissions…..energy…..n….. an amount Pu-239 absorbs n →Pu-240 …Pu-241 …Pu-242 an amount Pu-239 remains behind

29. Production of Pu isotopes Evolution of 235 U content and Pu isotopes in typical LWR

30. Production of Pu isotopes

31. Nuclear Fuels

32. * In a U-235 / U-238 reactor, Pu-239 production consumption of N U-235 atoms → NCPu-239 atoms produced * In a Pu-239 / U-238 reactor, Pu-239 production consumption of N Pu-239 atoms →NCPu atoms produced →(NC)CPu atoms produced → (NC²)CPu atoms produced →etc.

33. Nuclear Fuels *C < 1convertor C > 1breeder reactor *η > 1for criticality write η = 1+ ζ (in addition to leakage, parasitary absorption) To be used for “conversion”

34. Nuclear Fuels Ref: Duderstadt & Hamilton η(E) for U-233, U-235, Pu-239 & Pu-241

35. Slowing down (“moderation”) of neutrons Fission neutrons are born with ~ 2 MeV Fission cross section largest at low E (0.025 eV) → need to slow down neutrons as quickly as possible = “ moderation” Mostly through elastic collisions (cf. billiard balls)

36. Slowing down (“moderation”) of neutrons Best moderator materials: → mass moderatoras low as possible → moderator preferably low neutron-absorption cross section

37. Slowing down (“moderation”) of neutrons Hence: * H 2 O -good moderator (contains much ) -but absorbs considerable amount of neutrons →U to be enriched -can also serve as coolant * D 2 O -still small mass: good moderator -absorbs fewer n than H 2 O →can operate with natural U: CANDU -can also serve as coolant

38. Slowing down (“moderation”) of neutrons * graphite: -now need for separate cooling medium → other properties of moderator materials -good heat-transfer properties -stable w.r.t. heat and radiation -chemically neutral w.r.t. other reactor materials

39. Slowing down (“moderation”) of neutrons Time to “thermalize” from ~ 2 MeV → 0.025 eV in H 2 O: t mod ~1 μs t diff ~200 μs = 2 x 10 -4 s time that a neutron, after having slowed down, will continue to “random walk” before being absorbed. t generation ~ 2 x 10 -4 s

40. Reflector To reduce the leakage of neutrons out of reactor core → surround reactor core with “n-reflecting” material Usually, reflector material identical to moderator material Note: There exist also so-called “fast” reactors But most commercial reactors are “thermal” reactors (=reactors with thermal neutrons)