Neutron Life Cycle
Why should we
worry about these?
How?
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
1
Controlled Fission
k = fp(1-lfast)(1-lthermal)
• Thermal utilization factor f can be changed, as an
example, by adding absorber to coolant (PWR)
(chemical shim, boric acid), or
by inserting movable control rods in & out.
• Reactors can also be controlled by altering neutron
leakages using movable neutron reflectors.
• f and p factors change as fuel is burned.
• f, p, η change as fertile material is converted to fissile
material.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
2
Controlled Fission
• Attention should be paid also to the fact that reactor
power changes occur due to changes in resonance
escape probability p. If Fuel T↑, p↓ due to Doppler
broadening of
resonance peaks.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
3
Controlled Fission
• Note that  is greater than 2
at thermal energies and
almost 3 at high energies.
• These “extra” neutrons are
Used to convert fertile into
fissile fuel.
• Efficiency of this process is
determined by neutron
energy spectrum.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
Variations in 
4
Controlled Fission
• Conversion ratio CR is defined as the average rate of
fissile atom production to the average rate of fissile atom
consumption.
• For LWR's CR  0.6.
• CR is called BR for values > 1.
• Fast breeder reactors have BR > 1.
• They are called “fast” because primary fissions
inducing neutrons are fast not thermal, thus η > 2.5 but
σf is only a few barns.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
5
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
dn kn  n

dt

• For a short time dt
• Show that
n(t )  n0 e
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
( k 1) t 
6
Controlled Fission
n(t )  n0 e
( 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 Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
7
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 Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
8
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 and breeders: Convert non-thermallyfissionable material (non-fissile) to a thermallyfissionable material (fissile).
What are neutron generators?
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
9
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 Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
From converter or
breeder reactor.
10
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 Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
11
More on Moderators
What moderator?
1. Cheap and abundant.
2. Chemically stable.
3. Low mass (high  logarithmic energy decrement).
4. High density.
5. High s and very low a.
• Graphite (1,2,4,5) increase amount to compensate 3.
• Water (1,2,3,4) but n + p  d +   enriched uranium.
• D2O (heavy water) (1!) but has low capture cross
section  natural uranium, but if capture occurs,
produces tritium.
• …..
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
12
More on Moderators
 s
a
Moderating ratio 
HW 12
Calculate both
moderating power and
ratio for water, heavy
water, graphite,
polyethylene and boron.
Tabulate your results and
comment.
Nuclear Reactors, BAU, 1st Semester, 2007-2008
(Saed Dababneh).
10
B  n B  Li  
11
*
7
B-10
10
B
1/v region
13
More on Moderators
Recall
n
ln E  ln E  n  n 
\
n
ln( E f / Eth )

\
n
ln( E / E )

After n collisions
After one collision
( A  1) 2 A  1
 E
u    ln \   1 
ln
2A
A 1
 E  av
creation
f
Total mean free path = n s
Is it random walk or there is a
preferred direction???
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
th
14absorp
More on Moderators
HW 13 (or 6\)
 A 1 

 E  E
 A 1
2
\
E
Recall E min
(head-on). Then the
maximum energy loss is (1-)E, or E  E\  E.
For an s-wave collision:
E
1
\
\
\
E P( E  E )dE  1  P( E  E )  (1   ) E
Assumptions:
show that
1.
Elastic scattering. E
E  (1   ) E
\
1
2
2.
3.
Target nucleus at rest.
E
Spherical symmetry in
CM.
  s (E)
|
d


E

E
E

\
s
Obviously  s ( E  E )  |   (1   ) E
dE 
otherwise
 0
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Scattering Kernel?
Dababneh).
15
More on Moderators
HW 13 (or 6\) continued…
(Re)-verify
E \ A2  1  2 A cos  CM 1
CM


(
1


)

(
1


)
cos

E
( A  1) 2
2


cos  

A  sin 
( A  1) 2
2
2


2
For doing so, you need to verify and use
cos  
1  A cos  CM
A2  1  2 A cos  CM
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
16
More on Moderators
HW 13 (or 6\) continued…
• Forward scattering is
preferred for “practical”
moderators (small A).
• If isotropic neutron
scattering (spherically
symmetric) in the
laboratory frame 
average cosine of the
scattering angle is zero.
2
Show that cos( ) 
3A
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
17
More on Moderators
HW 13 (or 6\) continued…
Spherically symmetric in CM
d s
1
CM
  s ( ) 
 s (E)
CM
d
4
 s ( E ) ( A2  2 A1 cos  CM  1)3 2
Show that  s ( ) 
4
1  A1 cos  CM
• Neutron scattering is isotropic in the laboratory
system?!  valid for neutron scattering with heavy
nuclei, which is not true for usual thermal reactor
moderators (corrections are applied).
Distinguish from
• Angular neutron distribution.
Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed
Dababneh).
18