Requirements: Explain steps, please

Chapter 8
Nuclear Fission
Most power reactors today produce power by the neutron-induced fission
of a heavy metals such as uranium or plutonium. This chapter examines
the process of fission; the next chapter examines the design and operation
of systems that use fission to generate power.
8.1 Fissionable, Fissile, and Fertile Nuclides
Table 8.1 lists five nuclides that are commonly used as nuclear fuels: Th-
232, U-233, U-235, U-238, and Pu-239. These nuclides can be described as
fissionable, fissile, or fertile:
⌅ Fissionable material. A fissionable nuclide can be split by absorbing a
neutron that possesses sufficient kinetic energy. All of the nuclides listed in
the table are fissionable.
⌅ Fissile material. A fissionable nuclide is said the be fissile if it is readily
fissioned by slow (thermal) neutrons. The principal fissile materials are U-
233, U-235 and U-239. are uranium-233, uranium-235 and plutonium-239.
The reactions of fissile materials are discussed in Sections 8.4 through 8.9.
⌅ Fertile material. A fertile nuclide is not itself fissile, but can be converted
into a fissile nuclide by capturing a neutron. This process, called
breeding or conversion, is discussed in Section 8.11. The principal fertile materials
are U-238 and Th-232.
8-1
8-2 CHAPTER 8. NUCLEAR FISSION
Table 8.1: Selected Fissionable Nuclides
Nuclide Half-life Type Source (Isotopic abundance)
232Th
90 1.405 ⇥ 1010 y Fertile Natural Th (100%)
233U
92 1.592 ⇥ 105 y Fissile Conversion of Th 232
90
235U
92 7.038 ⇥ 108 y Fissile Natural U (0.72 wt%)
238U
92 4.468 ⇥ 109 y Fertile Natural U (99.28 wt%)
239Pu
94 2.411 ⇥ 104 y Fissile Conversion of U 238
92
It should be noted that the terms fissile and fissionable are sometimes used
interchangeably, especially in the older literature.
8.2 Thermal Neutrons
In Section 6.11, we saw that the neutron absorption cross-section of uranium-
235 is greater with slow (thermal) neutrons than with fast neutrons. Figure
6.4 (reproduced here as Figure 8.1) shows the neutron-absorption cross-section of U-235 as a function of neutron kinetic energy. Three regions are
apparent in the figure:
⌅ Low-speed region. At low neutron energies (below 1 eV for U-235) the
absorption cross-section varies approximately as 1/v. In other words, the
absorption is more likely at lower neutron energies.
⌅ Resonance region. A spike or local maximum in the cross-section is
called resonance. In the resonance region of U-235 (neutron energies between
1 eV and 1000 eV) absorption cross-sections very abruptly with neutron
energy.
⌅ High-speed region. The absorption cross-section of U-235 continues to
the decline for neutron energies above 1 keV, dropping off abruptly above 1
MeV.
8.2. THERMAL NEUTRONS 8-3
Figure 8.1: Neutron absorption cross-sections in U-235. This graph was generated
from ENDF-B-VII.1 data using the online nuclear cross-section plotter XSPlot
(http://xsplot.com).
8-4 CHAPTER 8. NUCLEAR FISSION
Similar graphs may be drawn for other fissile nuclides.
Thermal neutrons have energies determined by the temperature of their surroundings.
At 290 K (17 “C), thermal neutrons move at an average speed of
2200 m/s, giving them average kinetic energy of 0.0253 eV.
Fission neutrons initially have high kinetic energies, in the range from 0.1
to 10 MeV, with an average of about 2 MeV. A 2-MeV neutron has a speed
of 2.8 ⇥ 107 m/s.
As noted before, the NIST Center for Neutron Research lists neutron scattering
lengths and cross-sections for thermal neutrons (0.0253-eV neutrons)
in various target media.1
8.3 Neutron Moderation
Fission neutrons initially have high kinetic energies, in the range from 0.1
to 10 MeV, with an average of about 2 MeV. In general, any neutron having
a kinetic energy greater than 1 MeV is called a fast neutron.
Some reactors are designed to work using fast fission neutrons to sustain a
chain reaction. Such reactors are called fast reactors.
Reactors that use thermal neutrons to initiate fission are called thermal reactors.
Most power reactors in use in the world today are thermal reactors.
A thermal reactor requires a means of slowing down the fission neutrons
to the thermal range. This process is called moderation or thermalization.
A moderator is a substance that thermalizes neutrons by elastic collisions.
If the fission neutrons have an initial average energy K0, their average energy
will decrease to EN after N elastic scatterings:
KN
K0
= exp (−Nx) (8.1)
In this equation, x is the logarithmic energy decrement, which varies with the
moderator’s mass number A according to
x = 1 −
(A − 1)2
2A
ln
✓
A + 1
A − 1
◆
(8.2)
⇡
2
A + 2/3
(8.3)
1www.ncnr.nist.gov/resources/n-lengths/
8.3. NEUTRON MODERATION 8-5
An effective moderator will have a high value of x, which implies a low
value of A. In practice, substances heavier than carbon (A = 12) are rarely
used as moderators.
An effective moderator should also have a high cross section for scattering
and a low cross section for absorption.
The moderating ratio MR provides a convenient criterion for choosing a
moderator:
MR = x
smod
s
smod
a
(8.4)
where
x = logarithmic energy decrement [dimensionless]
smod
s = scattering cross-section [b]
smod
a = absorption cross-section [b]
In practice, three substances have been used as moderators:
⌅ Light water. Hydrogen, the lightest element, offers the largest value
of x. However, hydrogen is a gas and therefore is not dense enough to
serve as a moderator. Hydrogen bonded to oxygen can be a liquid or solid,
depending on the temperature and pressure. Ordinary water is the most
commonly used moderator because it is inexpensive and it can also serve
as the coolant.
Its main disadvantage is that ordinary hydrogen has a relatively high neutron
capture cross-section. Light water reduces neutron flux, and cannot be
used as a moderator in reactors fueled with natural uranium.
A reactor that employs ordinary water as a moderator is called a light-water
reactor (LWR). Most power reactors in use today are LWRs.
⌅ Heavy water. Deuterium has a smaller neutron absorption cross-section
than ordinary hydrogen. Hence, “heavy” water (D2O) absorbs fewer neutrons
than “light” water (H2O).
Heavy water is preferred as a moderator for reactors fueled with natural
uranium. The CANDU2 reactors use natural uranium as fuel and heavy
water as a moderator.
2CANDU is short for Canadian Deuterium Uranium.
8-6 CHAPTER 8. NUCLEAR FISSION
Heavy water has two disadvantages. First, deuterium is twice as heavy
as hydrogen, so the x is lower for deuterium than for hydrogen. Because
the neutrons lose less energy per collision, a larger reactor core or pressure
vessel is required for a heavy water reactor than for a light water reactor.
Second, heavy water is much more expensive than light water. To reduce
the quantity of heavy water and therefore its cost, the CANDU reactors use
light water as a coolant.
⌅ Graphite. Carbon in the form of graphite was used as the moderator
in the world’s first artificial nuclear reactor at the University of Chicago,
which achieved criticality on 2 December 1942. Graphite is still used as a
moderator in advanced gas-cooled reactors (AGRs).
Graphite has two disadvantages. First, because carbon is twelve times
heavier than hydrogen, neutrons lose less energy in a collision with a carbon
nucleus than with a hydrogen nucleus. Consequently, more moderator
and a larger reactor core are required. Second, graphite cannot be used as
a coolant.
A moderator may also be placed around the reactor core to serve as a reflector.
Neutrons that escape from the core may be scattered by the reflector;
some of these will be scattered back into the core.
8.4 Chain Reactions of Fissile Nuclides
In general, a chain reaction is a series of related events, in which each event
causes the next event in the series.
In a chemical or nuclear chain reaction, the products of one reacting step
acts as a reactant in the next step. Thus, the chain reaction tends to be selfsustaining
A thermal fission reactor is designed to maintain a chain reaction in a fissile
material such as 235U.
Figure 8.2 illustrates the concept of a U-235 chain reaction.3 Absorption of
a neutron causes a U-235 nucleus to split, according to the equation
n + 235U ! L +M+ nn + xg (8.5)
3Figure 8.2 is a schematic, meaning it is a simplified drawing that show the key features
of a U-235 fission chain reaction. It omits some details such as the presence of U-238 and
other nuclides, moderation of the neutrons, and emission of gamma rays.
8.4. CHAIN REACTIONS OF FISSILE NUCLIDES 8-7
²³5U
²³5U ²³5U
²³5U ²³5U ²³5U ²³5U
L M
Thermal neutron
Fission fragments
Fission neutron
Figure 8.2: Three generations of a U-235 chain reaction. Each 235U fissioning produces
two or more fission fragments (L and M) and two or three neutrons. The
neutrons produced by one fission can cause nearby nuclei to fission. (Not shown
are the gamma rays that are produced.)
8-8 CHAPTER 8. NUCLEAR FISSION
where
L,M = fission fragments
n = average number of fission neutrons per reaction
x = average number of gamma rays per reaction
Each of the fission neutrons has the potential of triggering fission in other
U-235 nuclei. A chain reaction is possible only if n > 1.
In Figure 8.2, some fissions produce two fission neutrons, some produce
three. On average, n ⇡ 2.42 for U-235.
Even if n > 1 a sustainable chain reaction still may not be possible. Recall
that the absorption of a slow neutron by a 235U nucleus produces the
compound nucleus 236U⇤, which can decay via two output channels:
n + 235U
92 ! U 236
92 ⇤ !
(
M+ L + nn + xg (fission)
235U
92 + g (radiative capture)
(8.6)
The absorption cross section can be written as the sum
sa = sg + sf (8.7)
where
sa = absorption cross section (b)
sg = radiative capture cross section (b)
sf = thermal fission cross section (b)
The probability that an absorbed neutron will result in fission is given by
Pr (fission|absorption) =
sf
sa
=
sf
sf + sg
(8.8)
Multiplying this probability by the average number of neutrons released
per fission yields the reproduction factor or thermal fission factor, h:
h = n
sf
sa
= n
sf
sf + sg
(8.9)
The reproduction factor can be interpreted as the ratio,
h =
fission neutrons produced
neutrons absorbed by fissionable nuclei
(8.10)
Table 8.2 lists sa, sg, sf, (sg/sf), n, and h for the principal fuel nuclides.
8.5. EFFECTIVE MULTIPLICATION FACTOR 8-9
Table 8.2: Thermal-Neutron Properties of Five Fissionable Nuclides
Nuclide sa
barn
sg
barn
sf
barn
sg
sf
n h
Fissile
233U 575 46 529 0.087 2.49 2.29
235U 687 99.3 587 0.169 2.42 2.07
239Pu 1020 271 749 0.362 2.87 2.11
Fertile
232Th 5.13 5.13 – – – –
238U 2.73 2.73 – – – –
Data from Shultis & Faw (2008).
8.5 Effective Multiplication Factor
As a criterion of the feasibility of a chain reaction, h is better than n. However,
even if h > 1, a chain reaction may not be sustainable because some
neutrons escape the reactor core and others are absorbed by nonfissile nuclei.
The neutron gain or effective multiplication factor keff indicates whether a chain
reaction will continue:
keff =
number of fissions in one generation
number of fissions in preceding generation
(8.11)
As used here, a generation is a step in the chain reaction. Equivalently,
keff =
rate of neutron production
rate of neutron loss (absorption plus capture)
(8.12)
The effective multiplication factor can also be expressed in terms of the
number of neutrons:
keff =
Ni+1
Ni
(8.13)
8-10 CHAPTER 8. NUCLEAR FISSION
where Ni is the number of neutrons produced in generation i and Ni+1 is
the number of neutrons produced in generation i + 1.
Three kinds of chain reactions can occur, depending on the value of keff:
⌅ Supercritical reaction. If keff > 1, the number of fissions increases with
each generation. In extreme cases, this can lead to an explosion; this, of
course, is the intended effect of a nuclear weapon. Such weapons require a
high concentration of fissile nuclei.4
⌅ Critical reaction. If keff = 1, the number of fissions is constant from one
generation to the next. The chain reaction occurs at a steady state. This is
called criticality. Power reactors operate at criticality.
⌅ Subcritical reaction. If keff < 1, the number of fissions decreases with
each generation. Eventually, fission ceases. A subcritical reaction may be
used for experimental purposes.
The reactivity r is related to the effective multiplication factor by
r =
keff − 1
keff
(8.14)
8.6 Six-Factor Formula
We can compute the effective multiplication factor keff by considering the
various processes that affect the number and energy of neutrons in a reactor.
Figure 8.3 is an event tree showing what can happen when Ni fast neutrons
are produced in a thermal reactor to start generation i:
1. Fast-n Fission. Some neutrons initiate fission even before they are slowed
down to thermal energies. This immediately increases the number of fast
neutrons in the reactor. The fast fission factor e accounts for this increase:
e = fast fission factor = 1 +
fast-neutron fissions
fast neutrons
(8.15)
Typically, 1.02 < e < 1.05.
4Figure 8.2 shows a supercritical reaction, such would occur in a nuclear explosion.
8.6. SIX-FACTOR FORMULA 8-11
Ni neutrons
Fast-n
Non-leakage
Fast-n
Leakage
Thermalization
Thermal-n
Non-leakage
Slow-n
Leakage
Fast-n
Fission
Non-Fuel
Absorption
Resonance
Capture
Fuel
Absorption
Slow-n
Fission
Slow-n
Capture
!f
!a
Ni “PNL
fast
Ni “PNL p fast
Ni “PNL p PNL
fast
Ni ”
th
Ni “PNL p PNL f
fast th
Ni “PNL p PNL f
fast th
Ni+1 = Ni ” PNL p PNL f #
fast th !f
!a
Ni “PNL p PNL f
fast th ν
Figure 8.3: Event tree for a generation of neutrons in a thermal reactor. At the top
of the diagram, Ni fast neutrons are introduced; at the bottom, Ni+1 fast neutrons
are produced. This completes one generation. The Ni+1 fast neutrons from the
bottom of the diagram are introduced at the top of the diagram, starting a new
generation.
8-12 CHAPTER 8. NUCLEAR FISSION
2. Fast-n Leakage or Non-leakage. A fast neutron may escape the core of
the reactor; this is called fast-neutron leakage (or fast leakage). Alternatively,
the fast neutron can remain in the core; this is called fast-neutron non-leakage,
fast-neutron retention, or fast non-leakage. The probability that a fast neutron
is retained in the core rather than being leaked is denoted Pfast
NL .
3. Thermalization or Resonance Absorption. A fast neutron is slowed
to thermal energies by elastic collisions with the moderator. As it slows, a
neutron may be absorbed at resonance energies of U-238 in the core. The
probability that a neutron undergoes thermalization instead of resonance
absorption is called the resonance escape probability, denoted by p.
4. Slow-n Leakage or Non-leakage. A thermal neutron can either leak
from the core or remain in the core. The probability that a thermal neutron
is retained is denoted Pth
NL.
5. Slow-n Absorption in Fuel or Non-Fuel Material. A thermal neutron
may be absorbed in fissile fuel or by a non-fissile medium. The probability
that a slow neutron is absorbed by fuel material is called the thermal
utilization factor, denoted f .
6. Thermal Fission or Radiative Capture. Once absorbed by fuel material,
a neutron may undergo either radiative capture or fission. As shown previously,
the probability that absorption will lead to fission is given by the
ratio sf/sa.
7. Production of Fission Neutrons. The number of fission neutrons per
reaction is given by n.
Multiplying all of these factors together yields the number of fast neutrons
available to start the next generation:
Ni+1 = Ni e Pfast
NL p Pth
NL f
✓
sf
sa
n
◆
(8.16)
The quantity in parentheses is the reproduction factor h defined in Section
8.4. Hence,
Ni+1 = Ni e Pfast
NL p Pth
NL f h (8.17)
Rearranging, we obtain at last the six-factor formula for the effective multiplication
factor:
keff =
Ni+1
Ni
= e Pfast
NL p Pth
NL f h (8.18)
8.7. FOUR-FACTOR FORMULA 8-13
8.7 Four-Factor Formula
Two of the factors in the six-factor formula equal the probabilities that neutrons
will leak from the reactor:
Pfast
NL = Pr (fast neutron does not leak)
Pth
NL = Pr (slow neutron does not leak)
Both of these factors depend on the size, shape, and design of the reactor
core. At times, it is useful to ignore these design-related details to focus on
the performance of the fuel itself.
If the reactor core were a sphere having an unlimited radius, no neutrons
would be able to leak out. In that case, Pfast
NL = 1 and Pth
NL = 1, and the sixfactor
formula becomes the four-factor formula:
k• = e p f h (8.19)
where k• is the infinite multiplication factor. It can be expressed as a ratio:
k• =
rate of neutron production
rate of neutron absorption
(8.20)
8.8 Fission Fragments
Consider the fissioning of U-235 once again:
n + 235U ! L +M+ nn + xg (8.21)
Here L andMrepresent the two fission products, also called fission fragments.
In most cases, L andMhave different masses; occasionally, a third fragment
may be produced.
Figure 8.4 shows how the fission products are distributed by mass numbers.
As can be seen from the figure, fission produces a wide range of
medium-mass nuclides (A = 70 to 160) with peaks around A = 95 and 135.
The fission products are significant for several reasons:
⌅ Energy source. Fission products tend to be neutron-rich beta (beta/
gamma) emitters. Their decay energy contributes to the overall energy of
the process. This can be good when it occurs inside the reactor, but a problem
when it occurs in the fuel after its removal from the reactor.
8-14 CHAPTER 8. NUCLEAR FISSION
?? ?? ??? ??? ??? ??? ???
?????????????
?????????????????????
??
?
???
????
?????
??????
???????
Figure 8.4: Product yield from the fission of 235U by thermal neutrons. Fission
fragments cover the range from A = 70 to 160, with peaks around A = 95 and 135.
8.9. ENERGY FROM THERMAL FISSION 8-15
⌅ Health hazard. The highly radioactive fission products pose a significant
danger to human health and safety.
⌅ Neutron poisons. Many fission products are neutron poisons, meaning
they readily absorb neutrons without undergoing fission. Neutron poisons
reduce the fuel’s thermal utilization factor f , which equals the probability
that a thermal neutron is absorbed by a fuel nucleus:
f =
thermal neutrons absorbed by fuel
all thermal neutrons absorbed
(8.22)
The thermal utilization factor is one of the components of both the sixfactor
formula and the four-factor formula. Neutron poisons can reduce
f enough to make the reactor subcritical (keff < 1).
Neutron poisons do have their uses. They may be introduced deliberately
in the form of control rods or chemical shims to maintain criticality or shut
down the reactor.
8.9 Energy from Thermal Fission
Table 8.3 summarizes the average energy released by the fissioning of U-
233, U-235, and Pu-239.
Several points to note:
⌅ Prompt energy release. The energy from fission is carried by the fission
fragments, fission neutrons, and gamma rays. Because these are produced
immediately, they are considered prompt energy sources.
⌅ Delayed energy release. As noted before, the fission products tend to
be radioactive. The decay of these products releases considerable energy in
the form of beta and gamma particles and neutrinos. Because radioactive
decay can occur over periods ranging from fractions of a second (while the
fuel is still in the reactor) to thousands of years (long after the fuel has been
removed from the reactor), these are considered delayed energy sources.
⌅ Nonrecoverable energy. The energy carried by the neutrinos is not
recoverable. In principle, the rest of the energy is recoverable; however,
as a practical matter, much of the long-term decay energy is not actually
recovered.
8-16 CHAPTER 8. NUCLEAR FISSION
Table 8.3: Average Fission Energy/MeV
Energy carrier U-233 U-235 Pu-239
Prompt
Fission fragments (kinetic energy) 168 169 176
Fission neutrons (kinetic energy) 5 5 6
Fission (prompt) g-rays 8 7 8
Delayed
Fission product decay b energy 5 6 5
Fission product decay g energy 5 6 5
Antineutrino kinetic energy* 7 9 7
205 210 214
*Unrecoverable
For preliminary design calculations, we will assume that the recoverable
fission energy is 200 MeV/fission.
8.10 Reactions of Fertile Nuclides
Previously, Th-232 and U-238 were described as fissionable but not fissile.
In other words, these nuclides do not readily undergo fission by the absorption
of slow neutrons. Only sufficiently fast neutrons will cause them
to fission.
At first glance, this might be puzzling. Both the fission of Th-232 and the
fission of U-238 are exoergic (Q > 0) and both involve an uncharged projectile
(a neutron). According to Table 7.2 in Section 7.8, neither reaction
should have a threshold energy. Yet these fuels cannot sustain a chain reaction
with slow neutrons.
8.11. CONVERSION AND BREEDING 8-17
To understand why these nuclides are not fissile, consider what happens
when U-238 absorbs a neutron.5 An excited compound nucleus is formed,
which then decays by one of two output channels:
n + 238U
92 ! U 239
92 ⇤ !
(
M+ L + nn + xg (fission)
239U
92 + g (radiative capture)
(8.23)
When we examine the thermal-neutron cross sections for the two output
channels, we see that the radiative capture reaction is 162 500 times more
likely than fission:6
Thermal fission: sf = 1.68 ⇥ 10−5 b
Thermal radiative capture: sg = 2.73 b
The situation changes when the projectile is a fast neutron. For example,
the absorption of 14-MeV neutrons by U-238 is more likely to cause fission
than radiative capture:
Fast (14-MeV) fission: sf = 0.3611 b
Fast (14-MeV) radiative capture: sg = 0.0011 b
Although Th-232 and U-238 are not fissile, they are fissionable by fast neutrons.
A fast reactor uses high-energy neutrons. Such reactors lack moderators.
8.11 Conversion and Breeding
The U-239 formed by radiative capture in U-238 is unstable, having a halflife
of 23.45 seconds. It decays by beta emission to Np-239:
239U
92 ! Np 239
93
+ + b− + ne (8.24)
Np-239 is itself unstable, having a half-life of 2.36 days. It decays by beta
emission to Pu-239:
239Np
93 ! Pu 239
94
+ + b− + ne (8.25)
5A similar analysis applies to Th-232.
6Data from Japan Atomic Energy Agency’s Nuclear Data Center, wwwndc.jaea.go.jp
8-18 CHAPTER 8. NUCLEAR FISSION
Plutonium-239 is one of the fissile nuclides listed in Tables 8.1 and 8.2.
The conversion of uranium to plutonium occurs to a certain extent in all
uranium-fueled reactors. Breeder reactors, however, are purposely designed
to encourage conversion.
The breeding ratio BR (or conversion ratio CR) equals the quantity of fissile
material produced divided by the quantity of fissile material consumed:
BR =
fissile material produced
fissile material consumed
(8.26)
If BR > 1, more fissile material is produced that is consumed.
The breeding gain BG equals the fractional gain in the quantity of fissile material:
BG =
fissile material produced − fissile material consumed
fissile material consumed
= BR − 1 (8.27)
The doubling time DT is the amount of time necessary for the quantity of
fissile material to double due to breeding.
The consumption of N fuel nuclei creates N · BR new fissile nuclei.
8.12 Review
1. What is a fissile nuclide?
2. What is a fissionable nuclide?
3. What is a fertile nuclide?
4. Explain the meanings of the symbols sa, sg, and sf. What is the unit for
these quantities?
5. What is the definition of n as used in this chapter?
6. What is the definition of h as used in this chapter?
7. What is a fast reactor?
8. What is a thermal reactor?
9. What is a breeder reactor?
8.13. PROBLEMS 8-19
10. What is the conversion ratio of a breeder reactor?
11. Why does a fast reactor not have a moderator?
12. What are the criteria for a good moderator?
13. What is the significance of the effective multiplication factor, keff?
14. What does it mean to say a reactor is critical?
15. What is the difference between fast fission and thermal fission?
16. Typically, what is the energy range for fission neutrons?
8.13 Problems
1. Ordinary “light” water (H2O) is the most commonly used moderator.
Moderation is due predominately to the hydrogens; the oxygen contributes
relatively little to the moderating power.
(a) What is the logarithmic energy decrement for a hydrogen nucleus?
(b) If smod
s = 103 barns and smod
a = 0.66 barns, what is the moderating ratio
for hydrogen?
(c) On average, what fraction of its kinetic energy will a fast neutron retain
each time it scatters off a hydrogen nucleus?
(d) On average, how many collisions with hydrogen nuclei will thermalize
a 2-MeV neutron?
(e) If the oxygen does not moderate neurons appreciably, why isn’t hydrogen
H2 use as a moderator instead of light water H2O?
2. “Heavy” water (D2O) is used as a moderator in the CANDU reactors.
Repeat the previous problem for deuterium, assuming smod
s = 13.6 barns
and smod
a = 0.003 barns.
3. The Experimental Breeder Reactor II (EBR-II) was a fast-neutron breeder
reactor that operated at the Idaho National Laboratory from 1965–1994.
EBR-II was cooled by liquid sodium metal.
(a) On average, what fraction of its kinetic energy will a fast neutron retain
each time it scatters off a sodium nucleus?
8-20 CHAPTER 8. NUCLEAR FISSION
(b) On average, how many collisions with sodium nuclei will thermalize
a 2-MeV neutron?
(c) What is the moderating ratio MR for liquid sodium? Assume smod
s =
3.28 barns and smod
a = 0.53 barns.
(d) Is sodium a good moderator? Why would sodium have been chosen
for the EBR-II?
4. Suppose 1000 fast neutrons are introduced into a thermal reactor having
the following characteristics:
e = 1.041
Pfast
NL = 0.894
p = 0.777
Pth
NL = 0.941
f = 0.799
h = 1.881
Compute the following:
(a) Neutrons produced by fast fission;
(b) Fast neutrons leaked;
(c) Fast neutrons moderated to thermal energies;
(d) Thermal neutrons leaked;
(e) Thermal neutrons absorbed in fuel;
(f) Thermal neutrons absorbed in non-fuel materials;
(g) Thermal neutrons undergoing radiative capture;
(h) Thermal neutrons causing fission;
(i) Fast neutrons going on to next generation;
(j) Effective multiplication factor keff;
(k) Reactivity r.
5. A lump of uranium produces fission neutrons at a rate of 9.1 ⇥ 107 n/s.
Neutrons are absorbed in the lump at a rate of 8.9 ⇥ 107 n/s; neutrons leak
at a rate of 6.8 ⇥ 105 n/s. Compute the effective multiplication factor and
reactivity.
8.13. PROBLEMS 8-21
6. Th-232 is a fertile nuclide that about four times more abundant on earth
than uranium.
(a) The absorption of a neutron converts Th-232 into a compound nucleus,
which can decay via two output channels. Write an equation describing
these processes, analogous to Equation (8.23).
(b) Radiative capture of a neutron by Th-232 creates an unstable nuclide.
What is that nuclide, what are its half-life and decay mode, and what product(
s) does it yield? (Hint: Refer to the Chart of the Nuclides.)
(c) The nuclide produced by the previous reaction is itself unstable. What
is that nuclide, what are its half-life and decay mode, and what product(s)
does it yield when it decays? (Hint: Refer to the Chart of the Nuclides.)
(d) The nuclide produced by the previous reaction is fissile. What is that
nuclide, what are its half-life and decay mode, and what product(s) does it
yield when it decays?
7. In Section (8.10), we showed why U-238 is not fissile. Repeat the analysis
to show that Th-232 is not fissile.
8. Compute the Q-value for each of the following processes:
(a) Thermal-neutron fissioning of 235U into 139Xe and 95Sr (one of many
possible fission reactions of U-235);
(b) Conversion of 238U to 239Pu;
(c) Thermal-neutron fissioning of 239Pu into 133I and 95Nb (one of many
possible fission reactions of Pu-239).
9. A research reactor is fueled by natural (unenriched) uranium, In one experiment.
it is observed that for every 1000 neutrons absorbed in 235U, 251
fast neutrons are absorbed in resonances of 238U and 644 thermal neutrons
are absorbed by 238U. There is essentially no leakage of neutrons from the
reactor.
(a) What is the conversion ratio for this reactor?
(b) How much 239Pu is produced when 1 kg of 235U is consumed?
8-22 CHAPTER 8. NUCLEAR FISSION