Neutron Chain Reaction
Systems
William D’haeseleer
Neutron Chain Reaction Systems
References:
• Lamarsh, NRT, chapter 4
• Lamarsh & Baratta, chapter 4
• Also Duderstadt & Hamilton § 3.I
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
Concept of chain reaction
U 
235
92
1
0
n  X  Y  2.5 n
• 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 REACTION
Concept of chain reaction
Chain reaction
235
U
Concept of chain reaction
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
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
number of neutrons in one generation
=
number of neutrons in previous generation
Concept of chain reaction
must be k=1
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   v f n  fission neutrons

a 
But sometimes also fissions due to fast neutrons
→ correction factor ε ≥ 1 (e.g., 1.03)
 in fact n η ε fission neutrons
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
nηεp
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
nηεpf
thermal neutrons absorbed in the
fuel
Multiplication factor
• Hence, after the next generation:
n  p f
 k   f p 
n
k  multiplication factor in  medium
Multiplication factor
Note:
three-step approach for multiplication factor
→ mono-energetic infinite reactor
→ moderation in infinite thermal reactor
→ moderation in finite thermal reactor
Multiplication factor
i.
Mono-energetic infinite reactor
Multiplication factor
i.
Mono-energetic infinite reactor
PAF = prob that neutron will be absorbed in the fuel

 F
a  
F
a
remainder
a


 f
a
F
a
“thermal utilization
factor”
Multiplication factor
i.
Mono-energetic infinite reactor
Multiplication factor
Pf = prob that an absorbed neutron in the fuel
leads to fission


PF 




v

F
f
F
a
F
f
F
a
Number of neutrons in next generation:
N 2  vPf PAF N1   fN1
v
N2
k 
 f 
N1
a
F
f
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
Define  =
number of fission neutrons caused by thermal neutrons
Define p = resonance escape probability
 k   f p 
"four factor formula"
Multiplication factor
iii. Moderation in finite thermal reactor
Multiplication factor
iii. Moderation in finite thermal reactor
PNL= non-leakage probability
k ≡ keff = k∞ PNL
k = multiplication factor for finite reactor
Multiplication factor
2. Finite reactor
keff  k PNL
non leakage probability
 A critical reactor always has keff = 1
Influencing factors of keff :
- leakage probability : geometry
- amount of fuel:
composition
- presence/absence
strong absorbers: composition
Critical Mass
• The larger the surface of a certain volume, the
higher the probability to leak away
4  R3
volume
R
e.g., for sphere:
 3 2 
R
surface
4 R
3
• The larger R:
– more fissile isotopes in volume
– larger leak-through surface
But Vol ∕ Surf
→ relatively more production of neutrons than leakage
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)
Critical Mass
Critical Mass
Critical Mass
Nuclear Fuels
* fissile isotopes
* fertile isotopes
U-233
U-235
Pu239
Th-232
U-238
only this isotope is
available in nature
U-233
Pu-239
U-235 cannot be made artificially
→ to increase fraction of U-235 in a “U-mixture”
→ need to ENRICH
“enrichment”
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
Production of Pu isotopes
Evolution
of 235U content
and Pu isotopes
in typical LWR
Production of Pu isotopes
Nuclear Fuels
Conversion factor C
# of fissile isotopes formed

# of fissile isotopes "consumed" by fission or absorption
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.
NC
2
3
NC  NC  NC  
1 C
Nuclear Fuels
* C<1
C>1
convertor
breeder reactor
* η>1
for criticality
write
η = 1+ ζ
To be used for “conversion”
(in addition to leakage,
parasitary absorption)
Nuclear Fuels
f
v
v

a 1 
η(E) for
U-233, U-235, Pu-239 &
Pu-241
Ref: Duderstadt & Hamilton
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)
Slowing down (“moderation”) of
neutrons
• Best moderator materials:
→ mass moderator as low as possible
1
1
H is the perfect moderator : m

1
1
H
m
 n
1
0
→ moderator preferably low neutron-absorption
cross section
Slowing down (“moderation”) of
neutrons
Hence:
* H2O
-good moderator (contains much 11 H)
-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
Slowing down (“moderation”) of
neutrons
*
12
6
C 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
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
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)