Nuclear Fission
• Q for 235U + n  236U is 6.54478 MeV.
• Table 13.1 in Krane: Activation energy EA for 236U  6.2 MeV
(Liquid drop + shell)  235U can be fissioned with zero-energy
neutrons.
• Q for 238U + n  239U is 4.??? MeV.
• EA for 239U  6.6 MeV  MeV neutrons are needed.
• Pairing term:  = ??? (Fig. 13.11 in Krane).
• What about 232Pa and 231Pa? (odd Z).
• Odd-N nuclei have in general much larger thermal neutron
cross sections than even-N nuclei (Table 13.1 in Krane).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
1
Nuclear Fission
• 235U + n  93Rb + 141Cs + 2n
• Q = ????
• What if other fragments?
• Different number of neutrons.
• Take 200 MeV as an average.
66 MeV
Heavy
fragments
98 MeV
Light
fragments
miscalibrated
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
2
Nuclear Fission
• Mean neutron energy  2
MeV.
•  2.4 neutrons per fission
(average)   5 MeV
average kinetic energy
carried by prompt neutrons
per fission.
• Show that the average momentum carried by a neutron is only 
1.5 % that carried by a fragment.
• Thus neglecting neutron momenta, show that the ratio between
kinetic energies of the two fragments is the inverse of the ratio of
their masses.
E1 m2 66 95
1st
Nuclear and Radiation Physics, BAU,
Semester, 2006-2007
(Saed Dababneh).
E2


m1 98 140
3
Nuclear Fission
Enge
Distribution of fission energy
Krane
sums
them up
as 
decays.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
Lost … !
4
Nuclear Fission
Segrè
Lost … !
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
5
Controlled Fission
Fast second generation neutrons
• 235U + n  X + Y + (~2.4)n
• Moderation of second generation neutrons  Chain reaction.
• Net change in number of neutrons from one generation to
the next  k (neutron reproduction factor).
Infinite medium (ignoring leakage at the surface).
• k  1  Chain reaction.
• Water, D2O or graphite moderator.
• k < 1  subcritical system.
• k = 1  critical system.
• k > 1  supercritical system.
• For steady release of energy (steadystate operation) we need k =1.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
Chain reacting pile
6
Controlled Fission
235U
• Assume natural uranium:
99.2745% 238U, 0.7200% 235U.
Thermal f = 0 b
Thermal a = 2.75 b
584 b
97 b
mainly (n,)
• f = (0.992745)(0) +
(0.0072)(584)
= 4.20 b.
• a = (0.992745)(2.75) +
(0.0072)(97)
= 3.43 b.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
238U
7
Controlled Fission
• Probability for a thermal neutron to cause fission
4.20
 0.55
• For natural uranium 
4.20  3.43

f
 f a
• If each fission produces an average of  = 2.4 neutrons, then the
mean number of fission neutrons produced per thermal neutron =
f
 = 2.4 x 0.55  1.3
 
 f a
• This is close to 1. If neutrons are still to be lost, there is a danger
of losing criticality.
• For enriched uranium (235U = 3%)  = ????? (> 1.3).
• In this case  is further from 1 and allowing for more neutrons to
be lost while maintaining criticality.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
8
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 1000 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 and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
9
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 and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
Fractions lost at surface
10
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 and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
x 1.03
Fast fission
factor “”
11
Controlled Fission
Time scale for neutron multiplication
• Time constant  includes moderation time (~10-6 s) and diffusion
time of thermal neutrons (~10-3 s).
Time
Average number of thermal neutrons
t
N
t+
kN
t + 2
k2 N
• For a short time dt
• Show that
dN kN  N

dt

N (t )  N 0e
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
( k 1) t 
12
Controlled Fission
N (t )  N 0e
( k 1) t 
• k = 1  N is constant (Desired).
• k < 1  N decays exponentially.
• k > 1  N grows exponentially with time constant  / (k-1).
• k = 1.01 (slightly supercritical)  e(0.01/0.001)t = e10 = 22026 in 1s.
• Cd is highly absorptive of thermal neutrons.
• Design the reactor to be slightly
subcritical for prompt neutrons.
• The “few” “delayed” neutrons
will be used to achieve criticality,
allowing enough time to
manipulate the control
Cd control rods
rods.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
13
Fission Reactors
Essential elements:
• Fuel (fissile material).
Core
• Moderator (not in reactors using fast neutrons).
• Reflector (to reduce leakage and critical size).
• Containment vessel (to prevent leakage of waste).
• Shielding (for neutrons and ’s).
• Coolant.
• Control system.
• Emergency systems (to prevent runaway during failure).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
14
Fission Reactors
Types of reactors:
Used for what?
• Power reactors: extract kinetic energy of fragments as
heat  boil water  steam drives turbine  electricity.
• Research reactors: low power (1-10 MW) to generate
neutrons (~1013 n.cm-2.s-1 or higher) for research.
• Converters: Convert non-thermally-fissionable material
to a thermally-fissionable material.
_
U  n U  Np    
238
Fertile
239
23min

239


2.3d
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
239
_
Pu    

f,th = 742 b
15
Fission Reactors
_
Th  n233Th 22
min
233Pa     
232
Fertile
d
27


_
U    
233
f,th = 530 b
• 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 and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
16
Fission Reactors
What neutron energy?
• Thermal, intermediate (eV – keV), fast reactors.
• Large, smaller, smaller but more fuel.
What fuel?
• Natural uranium, enriched uranium, 233U, 239Pu.
How???
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
From converter or
breeder reactor.
17
Fission Reactors
What moderator?
1. Cheap and abundant.
2. Chemically stable.
3. Very low mass (~1).
4. High density.
5. Minimal neutron capture cross section.
• Graphite (1,2,4,5) increase amount to compensate 3.
• Water (1,2,3,4) but n + p  d +   enriched uranium.
• D2O (heavy water) has low capture cross section 
natural uranium, but if capture occurs, produces
tritium.
• Be and BeO, but poisonous.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
18
Fission Reactors
What assembly?
• Heterogeneous: moderator and fuel are lumped.
• Homogeneous: moderator and fuel are mixed together.
• In homogeneous systems, it is easier to calculate p and
f for example, but a homogeneous natural uraniumgraphite mixture can not go critical.
What coolant?
• Coolant prevents meltdown of the core.
• It transfers heat in power reactors.
• Why pressurized-water reactors.
• Why liquid sodium?
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
19
Boiling water reactor
Pressurized
water
reactor
• Light water reactors.
• Both use “light” water as
coolant and as moderator,
thus enriched (2-3%)
uranium is used.
• Common in the US.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
20
CANDU
reactor
• Canada has D2O
and natural uranium.
• D2O as moderator,
D2O or H2O as
coolant.
• Most
power
reactors in
GB are
graphite
moderated
gascooled.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
Gas
cooled
reactor
21
• Liquid sodium cooled, fast breeder reactor.
• Blanket contains the fertile 238U.
• Water should not mix with sodium.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
Breeder
reactor
22