This document discusses the concept of a neutron chain reaction in nuclear reactors. It explains that a chain reaction can occur when fission produces additional neutrons that cause further fissions. For a reactor to be critical, each fission on average must produce exactly one additional fission. It also discusses factors that influence the multiplication factor like leakage probability and the presence of neutron absorbers. The document covers other key concepts like critical mass, nuclear fuels, moderation of neutrons, and the use of reflectors.

1. Neutron Chain ReactionNeutron Chain Reaction
SystemsSystems
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 Nf
(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
fission 2.5 n etc… etc…
 CHAIN REACTIONCHAIN REACTION
235 1
92 0 2.5U n X Y n+ → + +

5. Concept of chain reaction
Chain reaction
235
U

6. Concept of chain reaction

7. Concept of chain reaction
• If “few” neutrons leak out, or parasitically absorbed:
→ exponentially run-away chain reaction
 super critical reactor k > 1
• If “too many” neutrons leak out, or parasitically
absorbed:
→ exponentially dying-out chain reaction
 sub critical reactor k < 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
kk = multiplication factor= 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
f
a
v n
σ
σ
 
≡ ÷
 

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 p thermalized 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 f thermal neutrons absorbed in the
fuel

12. Multiplication factor
• Hence, after the next generation:
multiplication factor in medium
n p f
k f p
n
k
η ε
η ε∞
∞
= =
= ∞

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
PAF = prob that neutron will be absorbed in the fuel
F F
a a
F remainder
a a a
f
Σ Σ
= =
Σ Σ Σ
≡
+
“thermal utilization
factor”

16. Multiplication factor
i. Mono-energetic infinite reactor

17. Multiplication factor
Pf = prob that an absorbed neutron in the fuel
leads to fission
F F
f f
FF F
a a
P
v
σ η
σ
Σ
= = ≡
Σ
2 1 1f AFN vP P N fNη= =
2
1
F
f
a
vN
k f
N
η∞
Σ
≡ = =
Σ
Number of neutrons in next generation:

18. Multiplication factor
ii. Moderation in infinite thermal reactor
Now η identified with absorption of thermal
neutrons
Also f defined for thermal neutrons
→ reasons for name “thermal utilization factor”

total number of fission numbers
=
number of fission neutrons
Define
Define p = r
caused by
esonance es
thermal
cape pr
ne
ob
utro
ab
ns
ility
ε
"four factor formula"k f pη ε∞ =

19. Multiplication factor
iii. Moderation in finite thermal reactor

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

21. Multiplication factor
2. Finite reactor
 A critical reactor always has kA critical reactor always has keffeff = 1= 1
Influencing factors of keff :
- leakage probability : geometry
- amount of fuel: composition
- presence/absence
strong absorbers: composition
eff NLk k P∞= 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
3
2
4
volume 3e.g., for sphere:
surface 4 3
R R
R
R
π
π
= = µ

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 H2O
-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

25. Critical Mass

26. 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 neutrons Pu-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
Conversion factor C
# of fissile isotopes formed
# of fissile isotopes "consumed" by fission or absorption
≡

32. Nuclear Fuels
* In a U-235 / U-238 reactor, Pu-239 production
consumption of N U-235 atoms
→ NC Pu-239 atoms produced
* In a Pu-239 / U-238 reactor, Pu-239 production
consumption of N Pu-239 atoms
→NC Pu atoms produced
→(NC)C Pu atoms produced
→ (NC²)C Pu atoms produced
→etc.2 3
1
NC
NC NC NC
C
+ + + =
−
K

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

34. Nuclear Fuels
Ref: Duderstadt & Hamilton
1
f
a
v
v
σ
η
σ α
≡ =
+
η(E) for
U-233, U-235, Pu-239 &
Pu-241

35. Slowing down (“moderation”) of
neutrons
• Fission neutrons are born with <E> ~ 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 moderator as low as possible
→ moderator preferably low neutron-absorption
cross section
( ) ( )1 1 1
1 1 0H is the perfect moderator : m H m n;

37. Slowing down (“moderation”) of
neutrons
Hence:
* H2O -good moderator (contains much )
-but absorbs considerable amount of neutrons
→U to be enriched
-can also serve as coolant
* D2O -still small mass: good moderator
-absorbs fewer n than H2O
→can operate with natural U: CANDU
-can also serve as coolant
1
1H

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
12
6 C

39. Slowing down (“moderation”) of
neutrons
• Time to “thermalize” from ~ 2 MeV → 0.025 eV
in H2O: tmod ~ 1 μs
tdiff ~ 200 μs = 2 x 10-4
s
time that a neutron, after having slowed down,
will continue to “random walk” before being absorbed.
tgeneration ~ 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)