Controlled Fission
Fast second generation neutrons
• 235U + n  X + Y + (~2.4)n
• Moderation of second generation neutrons  Chain reaction.
• Water, D2O or graphite moderator.
• Ratio of number of “neutrons” (fissions) in one generation to
the preceding  k (neutron reproduction or multiplication
factor).
Infinite medium (ignoring leakage at the surface).
• k  1  Chain reaction.
• k < 1  subcritical.
• k = 1  critical system.
• k > 1  supercritical.
For steady release of energy (steadystate operation) we need k =1.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
Chain reacting pile
1
Controlled Fission
•
•
•
•
•
•
Average number of all neutrons released per fission
  (for thermal neutrons, 0.0253 eV).
233U : 2.492
235U : 2.418
239Pu : 2.871
241Pu : 2.927
Reactor is critical (k = 1): rate of neutrons produced
by fission = rate of neutrons absorbed + leaked.
Size and composition of the reactor.
Controlled Fission
235U
thermal cross sections
fission  584 b.
scattering  9 b.
radiative capture  97 b.
Probability for a thermal neutron to
cause fission on 235U is
f
1


 f   1 
If each fission produces an average of  neutrons, then the mean
number of fission neutrons produced per thermal neutron = 
f
f

 


a
 f   1 
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
 <
3
Controlled Fission
• Assume natural uranium:
99.2745% 238U, 0.7200% 235U.
Thermal f = 0 b
Thermal  = 2.75 b
584 b
97 b
235U
4R 2
   x   y  N x x  N y y
 ( x x   y y ) N
• f / N = (0.992745)(0) +
(0.0072)(584)
= 4.20 b.
•  / N = (0.992745)(2.75) +
(0.0072)(97)
= 3.43 b.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
238U
Doppler effect?
4R 2
Using the experimental elastic
scattering data the radius of the
nucleus can be estimated.
4
Moderation (to compare x-section)
(n,n)
(n,)
2H
(n,n)
1H
(n,)
• Resonances?
Controlled Fission
• Probability for a thermal neutron to cause fission in natural
uranium
4.20

4.20  3.43
 0.55
• If each fission produces an average of  = 2.4 neutrons, then the
mean number of fission neutrons produced per thermal neutron =
 = 2.4 x 0.55  1.3
• This is close to 1. If neutrons are still to be lost, there is a danger
of losing criticality. (Heavy water?).
• For enriched uranium (235U = 3%)  = ????? (> 1.3). (Light
water?).
• In this case  is further from 1 and allowing for more neutrons to
be lost while maintaining criticality.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
6
Controlled Fission
HW 11
• Verify
1

a
 (i)
f
(i )
i
• Comment on the calculation for thermal neutrons
and a mixture of fissile and non-fissile materials,
giving an example.
• Comment for fast neutrons and a mixture of
fissionable materials, giving an example.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
7
Conversion and Breeding
Converters: Convert non-thermally-fissionable material
to a thermally-fissionable material.
U  n U 
238
23min
239
239
_
Np    



2.3d
239
_
Pu    

f,th = 742 b
_
Th  n233Th 22
min
 233Pa     
232
d
27


1st
Nuclear Reactors, BAU,
Semester, 2007-2008
(Saed Dababneh).
_
U    
233
f,th = 530 b
8
Conversion and Breeding
• If  = 2  Conversion and fission.
• If  > 2  Breeder reactor.
• 239Pu: Thermal neutrons ( = 2.1)  hard for breeding.
Fast neutrons ( = 3)  possible breeding  fast
breeder reactors.
After sufficient time of breeding, fissile material can be easily
(chemically) separated from fertile material.
Compare to separating 235U from 238U.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
9
Controlled Fission
• N thermal neutrons in one generation have produced so far N
fast neutrons.
• Some of these fast neutrons can cause 238U fission  more fast
neutrons  fast fission factor =  (= 1.03 for natural uranium).
• Now we have N fast neutrons.
• We need to moderate these fast neutrons  use graphite  for 2
MeV neutrons we need ??? collisions. How many for 1 MeV
neutrons?
• The neutron will pass through the 10 - 100 eV region during the
moderation process. This energy region has many strong 238U
capture resonances (up to ????? b)  Can not mix uranium and
graphite as powders.
• In graphite, an average distance of 19 cm is needed for
thermalization  the resonance escape probability p ( 0.9).
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
10
Controlled Fission
• Now we have pN thermal neutrons.
• Graphite must not be too large to capture thermal neutrons;
when thermalized, neutrons should have reached the fuel.
• Graphite thermal cross section = 0.0034 b, but there is a lot
of it present.
• Capture can also occur in the material encapsulating the fuel
elements.
• The thermal utilization factor f ( 0.9) gives the fraction of
thermal neutrons that are actually available for the fuel.
• Now we have fpN thermal neutrons, could be > or < N
thus determining the criticality of the reactor.
The four-factor formula.

k = fp
k = fp(1-lfast)(1-lthermal)
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
Fractions lost at surface
11
Neutron
reproduction
factor
k = 1.000
x 0.9
Thermal
utilization
factor “f”
x
x 0.9
Resonance
escape
probability ”p”
What is:
• Migration length?
• Critical size?
How does the
geometry affect the
reproduction factor?
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
x 1.03
Fast fission
factor “”
12