neutron chain

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)