Revision 2
March 2015
Neutron Life Cycle
Student Guide
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Table of Contents
INTRODUCTION ..................................................................................................................... 1
TLO 1 NEUTRON MODERATION ............................................................................................ 2
Overview .......................................................................................................................... 2
ELO 1.1 Neutron Moderation .......................................................................................... 3
ELO 1.2 Moderator Characteristics ................................................................................. 9
TLO 1 Summary ............................................................................................................ 11
TLO 2 PROMPT AND DELAYED NEUTRONS ......................................................................... 12
Overview ........................................................................................................................ 12
ELO 2.1 Production of Prompt and Delayed Neutrons ................................................. 12
ELO 2.2 Delayed Neutron Fraction ............................................................................... 15
ELO 2.3 Prompt and Delayed Neutrons ........................................................................ 16
ELO 2.4 Delayed Neutrons and Reactor Control .......................................................... 18
TLO 2 Summary ............................................................................................................ 19
TLO 3 NEUTRON FLUX ....................................................................................................... 20
Overview ........................................................................................................................ 20
ELO 3.1 Prompt Neutron Energy .................................................................................. 21
ELO 3.2 Neutron Energy Spectrum ............................................................................... 22
ELO 3.3 Neutron Flux Spectrum Shape ........................................................................ 23
TLO 3 Summary ............................................................................................................ 25
TLO 4 NEUTRON LIFE CYCLE AND REACTOR CONTROL..................................................... 26
Overview ........................................................................................................................ 26
ELO 4.1 Neutron Life Cycle Terms .............................................................................. 26
ELO 4.2 Describe Each Term in the Six-Factor Formula ............................................. 30
ELO 4.3 Core Physical Changes and the Six-Factor Formula....................................... 34
ELO 4.4 Core Operating Parameter Changes and the Six-Factor Formula ................... 39
TLO 4 Summary ............................................................................................................ 46
NEUTRON LIFE CYCLE SUMMARY ...................................................................................... 49
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Neutron Life Cycle
Revision History
Revision
Date
Version
Number
Purpose for Revision
Performed
By
10/31/2014
0
New Module
OGF Team
12/10/2014
1
Added signature of OGF
Working Group Chair
OGF Team
3/4/2015
2

Aligned answer of
knowledge check on PPT OGF Team
slide 28 to IG page 11.
Aligned answer d in
IG/SG to align with PPT.

Corrected fast nonleakage probability
factor (Lf) on PPT slide
95.

Adjusted Lf to Lf on
IG/SG page 33.

Corrected slide title for
PPT slide 113 to read
“Moderator-to-Fuel
Ratio.”
Introduction
In thermal reactors, the neutrons that cause fission are born during the
fission process at a much higher energy level than required. To make
Rev 1
1
fission more probable, these neutrons must be slowed down to what is
known as thermal energy. To reduce high-energy neutrons to thermal
energy levels a process known as moderation must take place. Pressurized
water reactors (PWRs) utilize water as a moderator for thermalizing
neutrons.
During the process of moderating fast neutrons, neutrons are subject to
other events resulting in gains and losses before they cause fission and start
the process all over again. The neutron life cycle describes this process.
The neutron life cycle is important as it provides a tool for explaining the
factors involved in controlling the nuclear fission rate. Proper management
of the neutron life cycle makes control of a nuclear reactor possible.
Objectives
At the completion of this training session, the trainee will demonstrate
mastery of this topic by passing a written exam with a grade of 80 percent
or higher on the following Terminal Learning Objectives (TLOs):
1. Describe the process of neutron moderation in a nuclear reactor and
the characteristics of desirable moderators.
2. Describe the production of prompt and delayed neutrons from fission
and how these neutrons affect nuclear reactor control.
3. Describe the neutron flux spectrum in thermal reactors.
4. Describe the neutron life cycle throughout the lifetime of a thermal
reactor and how core design and variation in operating parameters
affect it.
TLO 1 Neutron Moderation
Overview
Fission neutrons are born at an average energy of 2 mega electron volt
(MeV) or fast neutrons and at high temperature. Fission neutrons
immediately begin to reduce their energy levels as they undergo numerous
scattering reactions with nuclei in the nuclear reactor core. A number of
collisions with nuclei reduce the neutron’s energy to approximately the
same average kinetic energy as its surrounding atoms or molecules. This
energy reduction occurs in a medium known as the moderator.
A neutron in energy equilibrium with its surrounding atoms is a thermal
neutron. Since the kinetic energy depends on temperature (molecular
movement), the energy of a thermal neutron also depends on temperature.
At 68° Fahrenheit (F), the energy of a thermal neutron is 0.025 electron volt
(eV). Energies less than 1eV yield neutrons designated or known as slow
neutrons.
2
Rev 1
Objectives
At the completion of this training session, you will be able to do the
following:
1. Describe the following:
a. Thermalization
b. Moderator
c. Moderating ratio
d. Average logarithmic energy decrement
e. Macroscopic slowing down power
2. Describe the four desirable characteristics of a good moderator and
explain how moderator density affects neutron moderation.
ELO 1.1 Neutron Moderation
Introduction
Lowering the energy level of a neutron is essential because thermal fission
requires a neutron to be at thermal equilibrium. In a nuclear reactor, fast
energy neutrons born from a fission event must be slowed to the thermal
energy region to maintain a chain reaction. Thermalization or moderation is
the process of reducing neutron energy to the thermal range by elastic
scattering.
Moderation Terms
Thermalization
Thermalization or moderation is the process of reducing neutron energy to
the thermal range by elastic scattering.
Moderator
A moderator is the material used to thermalize neutrons. A desirable
moderator is one that reduces the velocity of the fission neutrons, using a
minimum number of scattering collisions with a low probability of neutron
absorption.
Slowing the neutrons in as few collisions as possible reduces their travel
distance, thereby reducing the number of neutrons that leak out of the core.
This shorter travel distance also reduces the number of resonance
absorptions in non-fuel materials. Neutron leakage and resonance
absorption are discussed in detail later in this module.
Average Logarithmic Energy Decrement
The average logarithmic energy decrement is the measure of neutron energy
loss per collision. This term is the average decrease per collision of the
logarithm of the change in neutron energy, denoted by the symbol ξ (Xi).
Rev 1
3
𝜉 = 𝑙𝑛
𝐸𝑖
𝐸𝑓
Where:
ξ = logarithmic energy decrement
Ei = initial energy level of neutron
Ef = final energy level of neutron
Macroscopic Slowing Down Power
The logarithmic energy decrement is a convenient measure of the ability of
a material to slow neutrons, but does not consider the probability of
collisions taking place. Another measure of a moderator is the macroscopic
slowing down power (MSDP), defined as the product of the logarithmic
energy decrement and the macroscopic cross-section for scattering in the
material. The equation below calculates the macroscopic slowing down
power.
𝑀𝑆𝐷𝑃 = 𝜉𝛴𝑠
Where:
ξ = logarithmic energy decrement
𝛴𝑠 = Macroscopic scattering cross-section
Moderating Ratio
Macroscopic slowing down power calculates how rapidly a neutron will
thermalize in the chosen moderator. However, it does not yet fully explain
the effectiveness of the material as a moderator. An element such as boron
has a high logarithmic energy decrement and good MSDP, but it is a poor
moderator because of its high probability of absorbing neutrons.
The moderating ratio considers absorbing probability as well as slowing
down power; and therefore, is a more complete measure of moderator
effectiveness. Moderating ratio is the ratio of the microscopic slowing
down power to the microscopic cross-section for absorption. The higher the
moderating ratio, the more effectively the material performs as a moderator.
This equation shows the calculation for the moderating ratio (MR):
𝑀𝑅 =
𝜉𝜎𝑠
𝜎𝑎
The moderating ratio characterizes the effectiveness of a material as a
moderator. It considers the ratio between the absorption and scattering
cross-sections factoring in neutron energy levels. It does not consider the
4
Rev 1
density of the moderator. The table below compares moderating properties
of different materials.
Moderating properties of different materials are compared in the table
below.
Material
ξ
Number of
Collisions to
Thermalize
Microscopic
CrossSections σa–σs
Moderating
Ratio
Water (H2O)
also known as
light water
0.948
19.0
0.66 – 103.0
148.0
Deuterium
Oxide (D2O)
also known as
heavy water
0.570
35.0
0.001 – 13.6
7,752.0
Beryllium
(Be)
0.209
86.0
0.0092 – 7.0
159.0
Carbon (C)
0.158
114.0
0.003 – 4.8
253.0
A good moderator is one with a high moderating ratio. Any of the materials
shown in the table above make good moderators; however, commercial
nuclear power applications often use light water (H2O) because it is
plentiful, easily obtained, and inexpensive. Another benefit of using water
is that water can serve as both a moderator and a coolant. Heavy water
(D2O) is by far the best performing moderator; however, its higher cost
precludes its use in U.S. commercial PWRs.
Calculate Energy Loss Per Collision
The symbol ξ is called the average logarithmic energy decrement because of
the fact that a neutron loses, on average, a fixed fraction of its energy per
scattering collision. Since the fraction of energy loss per collision for a
given material is constant, ξ is also a constant. Because it is a constant and
does not depend on the initial neutron energy, ξ is a convenient quantity for
assessing the moderating ability of a material.
Varieties of sources have tabulated ξ values for the lighter nuclei. With
atomic mass numbers (A) greater than 10, the formula below is a relatively
accurate approximation for ξ:
𝜉=
2
2
𝐴+3
Rev 1
5
Since ξ equals the average logarithmic energy loss per collision, the total
number of collisions necessary for a neutron to lose a given amount of
energy may be determined by dividing ξ into the difference of the natural
logarithms of the energy range in question. The equations below show this
calculation.
𝑁=
𝑁=
𝑙𝑛 𝐸ℎ𝑖𝑔ℎ − 𝑙𝑛 𝐸𝑙𝑜𝑤
𝜉
𝐸ℎ𝑖𝑔ℎ
𝑙𝑛 ( 𝐸 )
𝑙𝑜𝑤
𝜉
If it is desirable to work with an average fractional energy, loss per collision
as opposed to an average logarithmic fraction the following relationship is
useful:
𝐸𝑁 = 𝐸𝑜 (1 − 𝑥)𝑁
Where:
𝐸𝑜 = initial neutron energy
𝐸𝑁 = neutron energy after N collisions
x = average fractional energy loss per collision
N = number of collisions
Average Logarithmic Energy Decrement
The table below presents the steps to determine the number of collisions
necessary for a neutron to reach certain energy. This information is
important when determining the moderator properties.
Step
Action
1.
Determine the average
logarithmic energy decrement for
A > 10
2.
Determine the average
logarithmic energy decrement for
A < 10
3.
6
Determine the number of
collisions (N) to reach a lower
energy
Calculation
𝜉 = 𝑙𝑛
𝜉=
𝑁=
𝐸𝑖
𝐸𝑓
2
2
𝐴+3
𝐸ℎ𝑖𝑔ℎ
𝑙𝑛 ( 𝐸 )
𝑙𝑜𝑤
𝜉
Rev 1
Step
Action
4.
Determine the energy level after a
number of collisions
Calculation
𝐸𝑁 = 𝐸𝑜 (1 − 𝑥)𝑁
Example 1
How many collisions are required to slow a neutron from an energy of 2
MeV to a thermal energy of 0.025 eV, using water as the moderator? For
water, ξ = 0.948.
𝑁=
𝐸ℎ𝑖𝑔ℎ
𝑙𝑛 ( 𝐸 )
𝑙𝑜𝑤
𝜉
𝑙𝑛 (
𝑁=
2 × 106 𝑒𝑉
)
0.025 𝑒𝑉
0.948
𝑁 = 19.2 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠
Example 2
If the average fractional energy loss per collision in hydrogen (H) is 0.63,
what will be the energy of a 2 MeV neutron after...
a. 5 collisions?
b. 10 collisions?
a)
𝐸𝑁 = 𝐸𝑜 (1 − 𝑥)𝑁
E5 = (2 × 106 eV)(1 − 0.63)5
= 13.9 kilo electron volt (keV)
b)
EN = Eo (1 − x)N
E10 = (2 × 106 eV)(1 − 0.63)10
= 96.2 eV
Rev 1
7
Knowledge Check
A _______________ is a material within a reactor which
is responsible for thermalizing neutrons.
A.
fuel rod
B.
moderator
C.
poison
D.
reflector
Knowledge Check
The process of reducing the energy level of a neutron
from the energy level at which it is produced to an
energy level in the thermal range is known as
_______________.
8
A.
moderating ratio
B.
resonance absorption
C.
thermalization
D.
inelastic scattering
Rev 1
Knowledge Check
The average logarithmic energy decrement is important
because...
A.
it can be used to determine if a material is a good
moderator.
B.
it can be used to determine the amount of energy released
from fission.
C.
it accounts for the change in binding energy change
during fission.
D.
it accounts for the change in mass during fission.
ELO 1.2 Moderator Characteristics
Introduction
The ideal moderator requires the following nuclear properties:




Large scattering cross-section
Small absorption cross-section
Large energy loss per collision
High atomic density
Desirable Moderator Properties
A material with a mass equal to a neutron, but with a large scattering crosssection is desirable as a moderator. A substance having a high absorption
cross-section, acts as a poison, removing available neutrons for fission. A
moderator with a low atomic density has few atoms available to thermalize
neutrons.
Carbon, hydrogen, beryllium, and water (both heavy and light) are
moderators used in nuclear reactors. In the U.S., most reactors use light
water as the moderator. Water is a good moderator because the hydrogen
atoms in water are close in mass (good for elastic scattering) to the
thermalized neutrons.
A neutron loses more energy in a collision with an atom of nearly its own
mass than in a collision with an atom whose mass is much greater than the
incident neutron (inelastic scattering — billiard ball effect). Water
moderators, rather than other materials, thermalize neutrons in fewer
collisions, reducing both the time and distance required for thermalization.
Rev 1
9
Moderator Density Effects
Temperature changes affect moderator density in an operating reactor.
These density changes affect the moderator’s ability to thermalize neutrons.
This in turn affects the number of neutrons available for fission.
Consider a PWR (light water moderator and coolant) as an example. If the
moderator temperature increases, the density of the water decreases.
Decreasing water density means there are fewer water atoms per unit
volume to thermalize neutrons. The neutrons will have to travel further to
thermalize, and this larger travel distance will take longer. Because of this
and other factors discussed later, there is now an increased chance that these
neutrons will not be available for fission. Fewer neutrons available for
fission, means fewer fissions, resulting in less reactor power.
Knowledge Check
Which of the items below is not a desirable property for
a neutron moderator?
A.
Large absorption cross-section
B.
Large scattering cross-section
C.
Large energy loss per collision
D.
High atomic density
Knowledge Check
The density of a moderator is important because...
10
A.
it can affect the number of target nuclei available for
collisions.
B.
it can affect the number of target nuclei available for
absorption.
C.
it can affect the number of collisions.
D.
it can affect the energy loss per collision.
Rev 1
TLO 1 Summary
In this lesson, you learned about neutron moderation: how thermalization
works with a moderator to reduce the velocity of the fission neutrons,
average logarithmic energy decrement, and macroscopic slowing down
power and moderating ratio. The listing below provides a summary of
sections in this TLO.
1. Slowing, or lowering, a neutron’s energy level is impacted by the
following factors:
Thermalization — the process of reducing the energy level of a
neutron from its birth energy to the energy level of the
surrounding atoms.
 The moderator is the reactor material present for thermalizing
neutrons.
 Moderating ratio — the ratio of the microscopic slowing down
power to the microscopic cross-section for absorption. This
ratio characterizes the effectiveness of a material as a moderator.
 The average logarithmic energy decrement, ξ — the average
change in the logarithm of neutron energy per collision.
 Macroscopic slowing down power — the product of the average
logarithmic energy decrement, and the macroscopic crosssection for scattering.
2. There are four desirable characteristics of a moderator:
 Large scattering cross-section
 Small absorption cross-section
 Large energy loss per collision
 High atomic density
— The density of the moderator affects its ability to moderate
neutrons. A less dense moderator results in neutrons taking
longer and traveling further to thermalize.
— The equation below calculates the energy loss after a
specified number of collisions.

𝐸𝑁 = 𝐸𝑜 (1 − 𝑥)𝑁
Now that you have completed this lesson, you should be able to do the
following:
1. Describe the following:
a. Thermalization
b. Moderator
c. Moderating ratio
d. Average logarithmic energy decrement
e. Macroscopic slowing down power
2. Describe the four desirable characteristics of a good moderator, and
explain how moderator density affects neutron moderation.
Rev 1
11
TLO 2 Prompt and Delayed Neutrons
Overview
Not all neutrons are born immediately following fission. Fission releases
most neutrons virtually instantaneously; these are referred to as prompt
neutrons. The remaining neutrons (a very small fraction), are born after the
decay of certain fission products and are referred to as delayed neutrons.
Although delayed neutrons are a very small fraction of the total number of
neutrons, they play an extremely important role in controlling the reactor.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Describe the origin and production of prompt and delayed neutrons.
2. State the approximate fraction of neutrons that are born as delayed
neutrons from the fission of the following fuels:
a. Uranium-235
b. Plutonium-239
3. Define prompt and delayed neutron lifetimes and their generation
times.
4. Explain the effects of delayed neutrons on reactor control.
ELO 2.1 Production of Prompt and Delayed Neutrons
Introduction
There are two ways to classify neutrons. One is to categorize according to
energy, such as fast or thermal. Another is to classify according to birth
time relative to a fission event. Those neutrons born immediately after a
fission event are prompt neutrons, those born later from the decay of certain
fission products are delayed neutrons.
12
Rev 1
Glossary
Prompt Neutrons
Within about 10-14 seconds of a fission event, a majority
(≈ 99.36 percent) of the neutrons are released (born).
These are prompt neutrons. The number of prompt
neutrons emitted during a fission event depends on the
type of fuel used (U-235 averages 2.4 per fission event).
The most probable energy for a prompt neutron is
approximately 1 MeV, and the average energy is
approximately 2 MeV.
Delayed Neutrons
A small portion of the neutrons born of fission, are born
delayed from delayed neutron precursors. Delayed
neutrons are neutrons that are born significantly after the
fission process has taken place. On average, delayed
neutrons are born approximately 12.7 seconds after the
fission event. Delayed neutrons are born fast but at a
lower energy than prompt neutrons (≈ 0.5 MeV).
Delayed Neutron Precursors
A delayed neutron precursor refers to delayed neutrons emitted immediately
following the first beta decay of a fission fragment. An example of a
delayed neutron precursor is bromine-87 (Br), shown below. Br-87 is the
fission product; Kr-87 (Krypton) is the delayed neutron precursor, with Kr86 (Krypton) the result of the delayed neutron birth.
86
𝐵𝑟
35
𝑛
𝛽−
87
86
→
𝐾𝑟
𝐾𝑟
→
36 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 36 𝑠𝑡𝑎𝑏𝑙𝑒
55.9 𝑠𝑒𝑐
Example
It is convenient to combine the known delayed neutron precursors into
groups with appropriately averaged half-life properties. These groups will
vary somewhat depending on the fuel or mixture of fuel in the reactor. The
table below lists the characteristics for the six delayed neutron precursor
groups resulting from the thermal fission of uranium-235.
Group
Half Life
(Seconds)
Delayed Neutron
Fraction
Average Energy (MeV)
1
55.7
0.00021
0.25
2
22.7
0.00142
0.46
3
6.2
0.00127
0.41
Rev 1
13
Group
Half Life
(Seconds)
Delayed Neutron
Fraction
Average Energy (MeV)
4
2.3
0.00257
0.45
5
0.61
0.00075
0.41
6
0.23
0.00027
N/A
Total
N/A
0.0065
N/A
Knowledge Check
Which one of the following types of neutrons has an
average neutron generation lifetime of 12.5 seconds?
A.
Prompt
B.
Delayed
C.
Fast
D.
Thermal
Knowledge Check
Delayed neutrons are the neutrons that...
A.
have reached thermal equilibrium with the surrounding
medium.
B.
are expelled within 10-14 seconds of the fission event.
C.
are produced from the radioactive decay of certain
fission fragments.
D.
are responsible for the majority of U-235 fissions.
Knowledge Check
In a comparison between a delayed neutron and a prompt
neutron produced from the same fission event, the
prompt neutron is more likely to...
14
Rev 1
A.
require a greater number of collisions to become a
thermal neutron.
B.
be captured by U-238 at a resonance energy peak
between 1 eV and 1,000 eV.
C.
be expelled with a lower kinetic energy.
D.
cause thermal fission of a U-235 nucleus.
ELO 2.2 Delayed Neutron Fraction
Introduction
The fraction of all neutrons produced by each delayed neutron precursor is
the delayed neutron fraction for that precursor. The total fraction of all
neutrons born as delayed neutrons is the delayed neutron fraction (β). Each
nuclear fuel has different delayed neutron fraction (β).
Delayed Neutron Fraction (β)
The fraction of delayed neutrons (β) varies depending on the predominant
fissile nuclide in use. The delayed neutron fractions (β) for the nuclides of
most interest are as follows:




Uranium-233 β = 0.0026
Uranium-235 β = 0.0065
Uranium-238 β = 0.0148
Plutonium-239 β = 0.0021
It is significant to note that uranium-235 and plutonium-239 are the two
major fuels in use in PWRs. Although the β values are small, β for
uranium-235 is considerably larger than plutonium-239.
Over core life, uranium-235 concentration will decrease, while plutonium239 increases. This will result in lower delayed neutron fraction over core
life.
Knowledge Check
What is the delayed neutron fraction of uranium-235?
Rev 1
A.
0.0065
B.
0.0148
C.
0.0021
D.
0.0026
15
ELO 2.3 Prompt and Delayed Neutrons
Introduction
Neutron lifetime is the time that a free neutron exists, from its birth until its
loss either from leakage or by absorption. Neutron generation time is the
time for a neutron from one generation to cause fission that produces the
next generation of neutrons.
Prompt Neutron Lifetime
Prompt neutron lifetime is the average time span from prompt neutron birth
until its loss from either leakage or absorption in another nucleus. This time
is the sum of thermalization time and diffusion time. Diffusion time relates
to absorption time of the thermal neutron and is a function of the absorption
mean free path, λa, divided by the average velocity of the thermal neutron.
Thermalization time is small (microseconds) compared to diffusion time
(milliseconds) and is usually ignored in calculations. Simply stated, prompt
neutron lifetime is the time from birth to loss by either leakage or
absorption.
Delayed Neutron Lifetime
Delayed neutron lifetime begins at its birth from a delayed neutron
precursor and ends at loss from leakage or absorption in another nucleus.
We calculate delayed neutron lifetime similarly to prompt neutron lifetime.
The key difference is the time of birth, which is not at the time of fission but
at the time of birth from decay of one of the delayed neutron precursors.
Prompt Neutron Generation Time
The generation time for prompt neutrons (ℓ* — pronounced ell-star) is the
total time from birth of a fast neutron in one generation to birth in the next
generation. Prompt neutron generation time is equal to the prompt neutron
lifetime added to the time required for a fissionable nucleus to emit a fast
neutron after absorption.
In water-moderated reactors, thermal neutrons exist for about 10-4 seconds
before absorption. Taking into account losses due to leakage, the prompt
neutron lifetime is equal to about 10-4 to 10-5 seconds (leakage occurs
quicker). Following absorption, fission, and the birth of fast neutron(s)
occurs in about 10-13 seconds — quickly. Therefore, in water-moderated
thermal reactors, ℓ* is about 10-4 seconds to 10-5 seconds.
Delayed Neutron Generation Time
Similar to prompt neutron generation time, delayed neutron generation time
equals the time of birth of a delayed neutron from a delayed neutron
precursor to the time of birth of a neutron(s) in the next generation.
16
Rev 1
The significant difference between prompt and delayed neutron generation
time is the delay in birth from their delayed neutron precursors. As
previously mentioned, the average time for decay of the delayed neutron
precursors from fission of uranium-235 is 12.7 seconds. With this relatively
large time interval, the delayed neutron lifetime is insignificant. Therefore,
the average delayed neutron generation time is equal to approximately 12.7
seconds. This significance of this time is shown next.
Average (or Effective) Generation Time
Slowing the neutron generation time from 10-4 seconds to a more reasonable
time period is necessary for improved reactor control. Adding delayed
neutrons makes this possible. Delayed neutrons with their long generation
time have a large effect on the overall average neutron generation time.
To determine the average generation time, we calculate a weighted average
taking into consideration the prompt neutron generation time, and the
delayed neutron generation. The following equation shows this
mathematically:
𝑇𝑖𝑚𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑇𝑖𝑚𝑒 𝑝𝑟𝑜𝑚𝑝𝑡(1 − 𝛽) + 𝑇𝑖𝑚𝑒 𝑑𝑒𝑙𝑎𝑦𝑒𝑑(𝛽)
Demonstration
Assume a prompt neutron generation time for a particular reactor of 1 x 10-4
seconds and a delayed neutron generation time of 12.7 seconds. If β is
0.0065, calculate the average generation time.
𝑇𝑖𝑚𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑇𝑖𝑚𝑒 𝑝𝑟𝑜𝑚𝑝𝑡(1 − 𝛽) + 𝑇𝑖𝑚𝑒 𝑑𝑒𝑙𝑎𝑦𝑒𝑑(𝛽)
= (1 × 10−4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠)(0.9935) + (12.7 𝑠𝑒𝑐𝑜𝑛𝑑𝑠)(0.0065)
= 0.0827 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Knowledge Check
__________ begins when it is released from a precursor
and ends when it is absorbed in another nucleus.
Rev 1
A.
Delayed neutron lifetime
B.
Prompt neutron lifetime
C.
Fast neutron fraction
D.
Thermal neutron fraction
17
Knowledge Check
Neutron generation time describes...
A.
time from one generation to the next generation of
neutrons.
B.
time that it takes for a neutron to become thermalized.
C.
time it takes for neutron precursors to emit neutrons.
D.
time that neutrons are born after a fission event.
Knowledge Check
What effect on the average neutron generation time
would a smaller β value produce?
A.
The result would be a longer average generation time.
B.
The result would be a shorter average generation time.
C.
More information is needed to determine the effect on
average generation time.
D.
β does not have any effect on average generation time.
ELO 2.4 Delayed Neutrons and Reactor Control
Introduction
Rapid power excursions result from the prompt neutron generation time of
10-4 seconds or faster (prompt neutrons only), which makes safe control of
the reactor difficult. As seen in the previous section, delayed neutrons
increase overall neutron generation time and slow down the power
excursions.
Delayed Neutrons and Reactor Control
If a reactor was operating using only prompt neutrons (β = 0), the
generation time would be about 1 x 10-4 seconds. This means that a
fractional change in power would occur every 1 x 10-4 seconds.
However, by operating the reactor with delayed neutrons, the average or
effective neutron generation time extends to approximately 0.0827 seconds.
While this seems fast, it is more than eight (8) times slower than the rate on
18
Rev 1
prompt neutrons alone, and leads to a more controllable rate of power
change, which increases operator control.
Although delayed neutrons make up only a small fraction of the total
neutron population, they are important to the safe control of a fission chain
reaction and changes in reactor power level.
Knowledge Check
If a reactor fueled with U-235 was operating and not
dependent on delayed neutrons, the average generation
time would be...
A.
12.5 seconds.
B.
0.0001 seconds.
C.
80 seconds.
D.
0.0065 seconds.
TLO 2 Summary
During this lesson, you learned about prompt and delayed neutrons: their
origin and production, the approximate fraction of neutrons born delayed
from the fission of Uranium-235 and Plutonium-239, the lifetime and
generation times of both prompt and delayed neutrons, and the effects of
delayed neutrons on reactor control. The listing below provides a summary
of sections in this TLO.
1. Neutrons are classified by energy, such as fast or thermal, or by birth
time relative to a fission event, and are considered either prompt or
delayed.
 Prompt neutrons are born directly from fission within 10-14
seconds of the fission event.
 Delayed neutrons are born from the decay of fission products are
called delayed neutron precursors. Delayed neutrons are born on
the average, about 12.7 seconds after the fission event.
 When a delayed neutron precursor undergoes a β- decay, it results
in an excited daughter nucleus which ejects a neutron. Delayed
neutrons are born based on the half-life of the delayed neutron
precursor.
 Delayed neutron precursors are grouped according to half-life.
Half-lives vary from fractions of a second to almost a minute.
2. The fraction of neutrons born as delayed neutrons is different for each
fuel isotope; fractions for two common fuel materials are:
 Uranium-235 — 0.0065
 Plutonium-239 — 0.0021
Rev 1
19
3. Define the prompt and delayed neutron lifetimes and their generational
times.
 The prompt neutron lifetime begins when a prompt neutron is
born, and ends when it is lost through leakage or is absorbed in
another nucleus.
 The delayed neutron lifetime begins when a delayed neutron is
released from a delayed neutron precursor, and ends when it is lost
through leakage or is absorbed in another nucleus.
 Prompt neutron generation time is the sum of the time between a
fissionable nuclide absorbing a neutron and fission neutrons being
released (10-13 seconds), and the prompt neutron lifetime.
 Prompt neutron generation time is about 1 x 10-4 seconds.
 Delayed neutron generation time is the sum of the time between a
fissionable nuclide absorbing a neutron and fission neutrons being
born (10-13 seconds), and the delayed neutron lifetime.
 The half-life of the delayed neutron precursors dominates the
delayed neutron generation time.
 Average (or effective) delayed neutron generation time is about
12.7 seconds.
— Weighted average
— 𝑇𝑖𝑚𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑇𝑖𝑚𝑒 𝑝𝑟𝑜𝑚𝑝𝑡(1 − 𝛽) + 𝑇𝑖𝑚𝑒 𝑑𝑒𝑙𝑎𝑦𝑒𝑑(𝛽)
4. Explain the effects of delayed neutrons on reactor control.
 Delayed neutrons slow the rate of power changes.
 If only prompt neutrons existed, reactor control would not be
possible due to the rapid power changes.
Now that you have completed this lesson, you should be able to do the
following:
1. Describe the origin and production of prompt and delayed neutrons.
2. State the approximate fraction of neutrons that are born as delayed
neutrons from the fission of the following fuels:
a. Uranium-235
b. Plutonium-239
3. Define prompt and delayed neutron lifetimes and generation times.
4. Explain the effects of delayed neutrons on reactor control.
TLO 3 Neutron Flux
Overview
The neutron population in a reactor consists of neutrons at many different
energy levels. This spectrum of energy levels is the neutron flux spectrum.
For graphical purposes, it is plotted either by the fraction of neutrons or by
the neutron flux at a given energy, versus neutron energy levels. Different
types of reactors will have different neutron energy spectrums to match
their design. This lesson will discuss neutron energy spectrums associated
with PWR thermal reactors.
20
Rev 1
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Describe the average energy at which prompt neutrons are produced.
2. Describe the shape of the neutron energy spectrum in a thermal
reactor.
3. Explain the reason for the shape of the neutron energy spectrum for a
thermal reactor, including variable(s) that have the most effect on
thermal neutron velocity.
ELO 3.1 Prompt Neutron Energy
Introduction
All neutrons born of fission are high-energy neutrons (fast neutrons), and
most of them range in energy between 0.1 MeV and 10 MeV. Their birth
energy affects the shape of the neutron energy spectrum.
Prompt Neutron Birth Energy
Plotting the fraction of fission neutrons per MeV as a function of neutron
energy provides a graphic illustration of the neutron energy distribution for
prompt neutrons, or spectrum. The figure below shows the prompt neutron
energy spectrum for uranium-235 thermal fissions. Values vary according
to fuel nuclides.
The figure shows that the most probable neutron energy (highest fraction) is
about 0.7 MeV. By analyzing the curve, we determine that the average
neutron energy is about 2 MeV.
Figure: Prompt (birth) Neutron Energy Spectrum for Thermal Fission of
Uranium-235
Rev 1
21
Knowledge Check
All prompt neutrons are born at the same energy level.
A.
True
B.
False
ELO 3.2 Neutron Energy Spectrum
Introduction
The spectrum of prompt energies at birth varies significantly from the
energy spectrum of all neutrons existing in the reactor at any given time.
The next figure shows neutron flux spectrums for a thermal reactor and a
fast breeder reactor. In either case, the prompt neutron spectrum at birth is
approximately the same. However, overall the energy spectrum of all
neutrons is considerably different between these two reactors due to design
and moderator effects. A fast breeder reactor requires a larger fast neutron
flux, where a thermal reactor needs slower thermal neutrons for fission.
Thermal reactors have a neutron energy spectrum that has two pronounced
peaks, one in the thermal energy region where the neutrons are in thermal
equilibrium with the core materials and another in the fast region at energies
where neutron production occurs.
Figure: Comparison of Neutron Flux Spectra for Thermal and Fast Breeder
Reactor
22
Rev 1
Knowledge Check
A graph of the neutron energy spectrum for a thermal
reactor would… (choose the most correct answer)
A.
have a distinct peak at approximately 2 MeV.
B.
have the same approximate shape as the prompt neutron
energy spectrum.
C.
have the same approximate shape as the delayed neutron
energy spectrum.
D.
have two peaks, one in the thermal energy region and
another in the fast region.
ELO 3.3 Neutron Flux Spectrum Shape
Introduction
The shape of the neutron energy spectrum in a thermal reactor depends on
neutron energy losses during the slowing down process and the temperature
of the moderator.
Thermal reactors have a neutron energy spectrum with two pronounced
peaks. The first peak is in the thermal energy region where the neutrons are
in thermal equilibrium with the core materials. The second peak is in the
fast region at energies where neutron production occurs.
The large number of fast neutrons born from fission combined with delayed
neutrons from precursors that start to slow down explains the initial higher
energy peak. During the neutron thermalization process, elastic collisions
remove a constant fraction or average of neutron energy per collision,
meaning that the neutrons lose larger amounts of energy per collision at
higher energies than at lower energies.
Note the following about the neutron energy spectrum:
1. There is a flat flux level between one (1) eV and 100 kiloelectron volt
(keV). This area represents mostly intermediate range neutrons that
have few losses in this energy range as they slow.
2. The neutron energy losses per collision are smaller at lower energy
levels; this results in neutron flux peaking at lower energies before
absorption occurs. This is also near the energy level where diffusion
occurs before absorption – remember diffusion time is longer than
slowing down time. This results in the lower energy flux peak at
approximately 0.1 eV.
Rev 1
23
Figure: Comparison of Neutron Flux Spectra for Thermal and Fast Breeder
Reactor
Most Probable Neutron Velocities
In the thermal region (0.025eV), neutrons achieve thermal equilibrium with
the atoms of the moderator material. In any given collision, they may gain
or lose energy (velocity), and over successive collisions gain as much
energy as they lose. These thermal neutrons, even at a constant
temperature, do not all have the same energy or velocity.
There is a distribution of energies, known as the Maxwell Distribution,
which determines the most probable neutron velocity (energy) for a given
temperature. Most thermal neutrons remain close to this most probable
energy, but with a spread above and below this value.
The most probable velocity (vp) of a thermal neutron is determined by the
temperature of the medium and is decided by:
𝑣𝑝 = √
2𝑘𝑇
𝑚
Where:
vp = most probable velocity of neutron (centimeter [cm]/second [sec])
k = Boltzmann’s Constant (138 x 10-16 ergon (erg)/ Kelvin [°K])
T = absolute temperature in degrees Kelvin (°K)
m = mass of thermal neutron (1.66 x 10-24 grams)
24
Rev 1
Knowledge Check
Thermal neutrons are at what energy level?
A.
< 1 eV
B.
10 eV
C.
> 1 MeV
D.
0.1 MeV
TLO 3 Summary
During this lesson, you learned about neutron flux: the average energy that
produces prompt neutrons, the shape of the neutron energy spectrum in a
thermal reactor, and the reason for the neutron energy spectrum’s shape for
a thermal reactor, including variable(s) that have the most effect on thermal
neutron velocity. The listing below provides a summary of sections of this
TLO.
Prompt neutrons are born at energies between 0.1 MeV and 10 MeV.
 The average prompt neutron energy is about 2 MeV.
2. The neutron energy spectrum for thermal reactors has two pronounced
peaks, one in the thermal energy region where the neutrons are in
thermal equilibrium with the core materials and another in the fast
region at energies where neutrons are born.
3. The neutron flux spectrum for the fast energy region of a thermal reactor
has a shape similar to that of the spectrum of neutrons born by the
fission process.
 Because of the smaller neutron energy losses per collision at lower
energy levels the neutrons pile up at lower energies before
absorption occurs (diffusion time). This results in the lower
energy flux peak at approximately 0.1 eV.
1.
Now that you have completed this lesson, you should be able to do the
following:
1. Describe the average energy at which prompt neutrons are produced.
2. Describe the shape of the neutron energy spectrum in a thermal
reactor.
3. Explain the reason for the shape of the neutron energy spectrum for a
thermal reactor including variable(s) that have the most effect on
thermal neutron velocity.
Rev 1
25
TLO 4 Neutron Life Cycle and Reactor Control
Overview
Some of the fast neutrons born by fission in one generation cause fission in
the next generation. Fission neutrons travel through a series of events as
they slow to thermal energies, leak, or are absorbed in the reactor. The
neutron life cycle describes these events, and is the topic of this chapter.
For simplicity, the following assumptions provide an outline of the neutron
life cycle.









All neutrons are born as fast neutrons.
Some fast neutrons are absorbed by fuel and cause fast fission.
Some fast neutrons leak out of the reactor core.
Some fast neutrons undergo resonance capture while slowing down.
All remaining fast neutrons become thermalized.
Some thermal neutrons leak out of the core.
Some thermal neutrons are absorbed by non-fuel material.
Some thermal neutrons are absorbed by fuel and do not cause fission.
All remaining thermal neutrons are absorbed by fuel and cause
thermal fission.
Objectives
Upon completion of this lesson, you will be able to do the following:
1. Define the following terms associated with the neutron life cycle:
a. Infinite multiplication factor (k∞)
b. Effective multiplication factor (keff)
c. Subcritical
d. Critical
e. Supercritical
2. Describe each term in the six-factor formula using the ratio of the
number of neutrons present at different points in the neutron life
cycle.
3. Explain how the physical design of the reactor core affects each of the
terms in the six-factor formula.
4. Explain how a change to plant operating parameters affects each of
the factors of the six-factor formula.
ELO 4.1 Neutron Life Cycle Terms
Introduction
Several key terms require understanding in order to relate to the neutron life
cycle, including the following:



26
Infinite multiplication factor (k∞)
Effective multiplication factor (keff)
Subcritical
Rev 1


Critical
Supercritical
Infinite Multiplication Factor
Glossary
Not all of the neutrons produced by fission are available
to cause new fissions. Some are absorbed by nonfissionable material, some are absorbed parasitically in
fissionable material and do not cause fission, and others
leak out of the reactor. Fortunately, to maintain a selfsustaining chain reaction, it is not necessary that every
neutron produced in fission initiate another fission
reaction. The minimum condition required for a selfsustaining chain reaction is that each nucleus undergoing
fission produces at least one neutron that ultimately
causes fission of another nucleus. All of the possible
things that can happen to the neutron from birth to
fission are expressed in terms of a multiplication factor.
If the multiplication factor equals one (1), it means that a
self-sustaining chain reaction is occurring.
The infinite multiplication factor (k∞) is used to consider
a reactor of infinitely large size where no neutron
leakage can occur. k∞ is defined as the ratio of neutrons
produced by fission in one generation to the number of
neutrons lost through absorption in the preceding
generation. It is also known as the four-factor formula.
It is expressed mathematically as:
𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚
𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑖𝑛 𝑜𝑛𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑘∞ =
𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑖𝑛
𝑡ℎ𝑒 𝑝𝑟𝑒𝑐𝑒𝑑𝑖𝑛𝑔 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
Rev 1
27
Effective Multiplication Factor (keff)
Glossary
The infinite multiplication factor only represents a
reactor that is infinitely large assuming no neutrons
leaking out of the reactor. To describe the neutron life
cycle in a real, finite reactor, it is necessary to account
for neutrons that leak.
This factor is the effective multiplication factor (keff). It
is expressed mathematically:
𝑘𝑒𝑓𝑓
𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚
𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑖𝑛 𝑜𝑛𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
=
𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑙𝑒𝑎𝑘𝑎𝑔𝑒
𝑖𝑛 𝑡ℎ𝑒 𝑝𝑟𝑒𝑐𝑒𝑑𝑖𝑛𝑔 + 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑟𝑒𝑐𝑒𝑑𝑖𝑛𝑔
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
Effective Multiplication Factor (keff) Versus Infinite Multiplication
Factor (k∞)
The balance between production of neutrons and their absorption in the core
and leakage out of the core determines the value of the multiplication factor.
If the leakage is small enough to be neglected, the multiplication factor
depends only on the balance between production and absorption. This is
called the infinite multiplication factor (k∞) since by definition an infinitely
large core has no leakage.
The infinite multiplication factor, also called the four-factor formula,
considers the four factors shown below:
𝑘∞ = (𝑓𝑎𝑠𝑡 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)(𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒 𝑒𝑠𝑐𝑎𝑝𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦)
(𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑢𝑡𝑖𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)(𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)
To include leakage, the effective multiplication factor (keff) is used. The
effective multiplication factor (keff) for a finite reactor is expressed
mathematically in terms of the infinite multiplication factor and two
additional factors, which account for neutron leakage as shown below.
𝑘𝑒𝑓𝑓
= 𝑘∞ (𝑓𝑎𝑠𝑡 𝑛𝑜𝑛𝑙𝑒𝑎𝑘𝑎𝑔𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦)(𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑜𝑛𝑙𝑒𝑎𝑘𝑎𝑔𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦)
These multiplication factors are explained in the next section.
Effective Multiplication Factor (keff) Versus Criticality
When the value of keff is 1, a self-sustaining chain reaction of fissions
occurs where the neutron population neither increases nor decreases. This
is referred to as critical or critical reactor and is expressed as keff = 1.
28
Rev 1

When neutron production is greater than the losses due to absorption
and leakage, the reactor is called supercritical. A supercritical reactor
has a keff greater than one (keff > 1), and the neutron flux is increasing
each generation. This is normal on power increases.
 When the neutron production is less than the losses due to absorption
and leakage, the reactor is called subcritical. A subcritical reactor has
a keff less than one (keff < 1), and the neutron flux is decreasing each
generation. This is normal on a power decrease.
When keff is not equal to exactly one, neutron flux and therefore reactor
power will change. For this reason, it is important to understand how
changes in reactor operating conditions affect keff.
Knowledge Check
A thermal neutron is about to interact with a uranium238 (U-238) nucleus in an operating nuclear reactor core.
Which one of the following describes the most likely
interaction and the effect on core keff?
A.
The neutron will be scattered, thereby leaving keff
unchanged.
B.
The neutron will be absorbed and U-238 will not undergo
fission, thereby decreasing keff.
C.
The neutron will be absorbed and U-238 will undergo
fission, thereby increasing keff.
D.
The neutron will be absorbed and U-238 will undergo
radioactive decay to Pu-239, thereby increasing keff.
Knowledge Check
A nuclear reactor is initially subcritical with the effective
multiplication factor (keff) equal to 0.998. After a brief
withdrawal of control rods, keff equals 1.000. The reactor
is currently _______________.
Rev 1
A.
prompt critical
B.
exactly critical
C.
supercritical
D.
subcritical
29
ELO 4.2 Describe Each Term in the Six-Factor Formula
Introduction
As mentioned in the introduction to this chapter, a number of assumptions
can be made regarding the possible paths a fission neutron may take during
its lifetime. This section will take these assumptions and place them into
ratios, the product of which equals keff or k∞.
Infinite Multiplication Factor or Four Factor Formula
A group of fast neutrons produced by fission can enter into several
reactions. Some of these reactions reduce the neutron population and some
increase the neutron population. There are four factors independent of the
size and shape of the reactor and do not consider any neutron leakage from
the reactor. This infinite multiplication factor considers all factors, but
excludes fast and thermal neutron leakage. The equation below states the
infinite multiplication factor.
𝑘∞ = 𝜀𝜌𝑓𝜂
Where:
ε = fast fission factor
ρ = resonance escape probability
f = thermal utilization factor
η = reproduction factor
Each of these four factors represents a process that adds to or subtracts from
the initial neutrons born in a generation by fission.
Six-Factor Formula
Because reactors are finite in size, two additional factors need
consideration, including:


Fast non-leakage probability (Lf)
Thermal non-leakage probability (Lth)
With the inclusion of these last two factors, we can determine the fraction
of neutrons that remain after each of the events that occur as the neutrons
complete their generation time. The effective multiplication factor (keff) can
then be determined by the product of six terms:
𝑘𝑒𝑓𝑓 = 𝜀𝐿𝑓 𝜌𝐿𝑡ℎ 𝑓𝜂
30
Rev 1
Note
Note
A pneumonic device to help you remember the sixfactor formula is “Every Little Person Loved the Funny
Navy.”
Fast Fission Factor (ε)
The first event that the neutrons incur after birth is fast fission. Fast fission
is fission caused by neutrons that are in the fast energy range, and results in
a net increase in the fast neutron population of the reactor core. The crosssection for fast fission in uranium-235 or uranium-238 is small. However,
there are still a significant number of fast neutrons that cause fission in
uranium-235, uranium-238, and plutomium-239.
Even though uranium-235 enrichment is small compared to uranium-238, a
large fraction of fast fissions occur with uranium-235 because of its wider
fission energy spectrum. The fast fission factor (ε) is defined as the ratio of
the net number of fast neutrons produced by all fissions to the number of
fast neutrons produced by thermal fissions. The equation below shows the
mathematical expression of this ratio.
𝜀=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑙𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠
Value of Fast Fission Factor
For a fast fission to occur, fast neutrons must pass close enough to a fuel
nucleus while they are still fast neutrons. The value of ε is affected by fuel
concentration and its physical arrangement proximity to the moderator. The
fast fission factor is essentially 1.00 for a homogenous reactor where the
fuel atoms are surrounded by moderator atoms (rapid moderation).
However, in a heterogeneous reactor, a PWR for example, fuel atoms are
packed closely together in fuel pellets within fuel rods and assemblies.
Because of this, neutrons emitted from the fission of one fuel atom have a
good chance of passing near another fuel atom before substantially slowing
down. This arrangement results in some fast fission. For PWRs, 1.02 is a
good value for ε; most heterogeneous reactors have a ε value in the range of
1.02 to 1.05.
Fast Non-Leakage Probability (Lf)
In a real reactor of finite size, some of the fast neutrons leak out of the
boundaries of the reactor core before they begin the slowing down process.
Of concern are the neutrons that do NOT leak out, as these remain to
contribute to the fission process. The fast non-leakage probability (Lf) is
the ratio of the number of fast neutrons that do not leak from the reactor
Rev 1
31
core to the number of fast neutrons produced by all fissions. The equation
below states this ratio:
𝐿𝑓 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑑𝑜 𝑛𝑜𝑡 𝑙𝑒𝑎𝑘 𝑓𝑟𝑜𝑚 𝑟𝑒𝑎𝑐𝑡𝑜𝑟
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑙𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠
The fast non-leakage probability represents a net loss in neutron population
and has value range of 0.85 to 0.97.
Resonance Escape Probability (ρ)
After fast fissions occur, neutrons continue to diffuse throughout the
reactor. As they travel, they collide with nuclei of fuel, non-fuel material,
and the moderator, losing part of their energy in each collision and slowing
down.
All nuclei within the reactor core have some probability of absorbing
neutrons, as indicated by the microscopic cross-section for absorption (σa)
for each material. The microscopic cross-section for absorption is not a
constant value but is dependent on the energy level of the incident neutron.
Normally, absorption cross-sections increase as neutron energy level
decreases. However, certain nuclei, such as uranium-238 and plutonium240 in particular, show extremely high absorption cross-section peaks for
neutrons at specific energy levels.
At certain neutron energy levels, these absorption cross-sections are as
much as 1,000 times higher than the cross-section for a slightly higher or
lower energy neutron. These peaks in absorption cross-sections are referred
to a resonance peaks or resonance peaking.
For example, while neutrons are slowing down through the resonance peak
region of uranium-238, about 6 eV to 200 eV, there is a chance that some of
the neutrons will be captured (resonance capture). These captured neutrons
are lost to the fission process for that particular generation of neutrons.
For mathematical purposes, rather than considering those captured neutrons,
the six-factor formula considers the number of neutrons that are not
captured. The probability that a neutron will not be absorbed by a
resonance peak is called the resonance escape probability.
The resonance escape probability (ρ) is defined as the ratio of the number of
neutrons that reach thermal energies to the number of fast neutrons that start
to slow down. This ratio is shown below.
𝜌=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑎𝑐ℎ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑠𝑡𝑎𝑟𝑡 𝑡𝑜 𝑠𝑙𝑜𝑤 𝑑𝑜𝑤𝑛
Value of Resonance Escape Probability
32
Rev 1
The value of the resonance escape probability is determined largely by the
fuel-moderator arrangement and the amount of enrichment of uranium-235.
To undergo resonance absorption, a neutron must pass close enough to a
uranium-238 nucleus to be absorbed while slowing down.
In a homogeneous reactor, neutrons slow down in the vicinity of fuel nuclei,
easily meeting this condition. This means that a neutron has a high
probability of being absorbed by uranium-238 while slowing down, making
its resonance escape probability low.
In a heterogeneous reactor, the neutron slows down in the moderator where
there are no atoms of uranium-238 present. This means resonance
absorption is less likely to occur and therefore, resonance escape probability
is high. The value of the resonance escape probability is always less than
one and ranges from 0.75 to 0.90 with a value of 0.87 a good approximation
for a PWR.
Thermal Non-Leakage Probability (Lth)
Neutrons leak out of a finite reactor core after they reach thermal energies.
Like fast leakage, our interest is in the neutrons that do NOT leak out, rather
than in the ones that do.
The thermal non-leakage probability (Lth) is defined as the ratio of the
number of thermal neutrons that do not leak from the reactor core to the
number of neutrons that reach thermal energies. The equation below shows
this ratio:
𝐿𝑡ℎ =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑑𝑜 𝑛𝑜𝑡 𝑙𝑒𝑎𝑘 𝑓𝑟𝑜𝑚 𝑟𝑒𝑎𝑐𝑡𝑜𝑟
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑎𝑐ℎ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠
The thermal non-leakage probability represents a net loss in neutron
population and has a value range of 0.85 to 0.99.
Thermal Utilization Factor (f)
The thermalized neutrons are still dispersed throughout the core where they
are subject to absorption by either fuel or non-fuel material. The thermal
utilization factor describes how effectively thermal neutrons are being
absorbed by the fuel or underutilized by non-fuel materials within the
reactor.
The thermal utilization factor (f) is defined as the ratio of the number of
thermal neutrons absorbed in the fuel to the number of thermal neutrons
absorbed in all reactor materials. The equation below presents this ratio.
𝑓=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠
Rev 1
33
The thermal utilization factor is always less than one because some of the
thermal neutrons absorbed within the reactor are not absorbed by atoms of
the fuel, but are lost to the fission process. The thermal utilization factor
ranges from 0.70 to 0.80.
Reproduction Factor (η)
Most of the neutrons absorbed in the fuel cause fission, but some do not.
The reproduction factor (η) is defined as the ratio of the number of fast
neutrons produced by thermal fission to the number of thermal neutrons
absorbed in the fuel. The reproduction factor is shown below.
𝜂=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
The reproduction factor represents net gain in neutron population and has a
value range of 1.65 to 2.0. The reproduction factor can also be stated as a
ratio of rates as shown below.
𝜂=
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑏𝑦 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
Total Non-Leakage Probability (LT)
The fast non-leakage probability (Lf) and the thermal non-leakage
probability (Lth) may be combined into one term that gives the fraction of
all neutrons that do not leak out of the reactor core. This term is called the
total non-leakage probability and is given the symbol (LT). The equation
below shows the formula for LT.
𝐿𝑇 = 𝐿𝑓 + 𝐿𝑡ℎ
The total non-leakage probability can be substituted for the fast and thermal
non-leakage terms in the six-factor formula.
Knowledge Check
Neutrons that are not absorbed in fuel are insignificant
and do not have any effect on reactor operation.
A.
True
B.
False
ELO 4.3 Core Physical Changes and the Six-Factor Formula
Introduction
34
Rev 1
Core physical design characteristics such as fuel enrichment, fuel
temperature, and moderator-to-fuel ratio affect various factors of the sixfactor formula. This section concentrates on the design attributes of the
reactor, while the next section focuses on operating parameter changes
during power operation, such as power, temperature, poisons, core age, etc.
Since keff is the product of these factors, knowledge of these effects is
important to the operator for safe operation of the reactor.
Design Factors Affecting the Value of the Fast Fission Factor (ε)
Reactor design establishes most parameters that affect the value of the fast
fission factor during plant operation. Variables such as temperature,
pressure, enrichment, or neutron poison concentration have little effect on ε.
Reactor design affects ε in the following ways:
Fuel atomic density — as fuel atomic density decreases, ε decreases.
Fuel pellet diameter — as fuel pellet diameter decreases, ε decreases.
— Fuel pellets are encased in a fuel rod; multiple fuel rods are
assembled to make a fuel element.
 Moderator — ε decreases with the ability to slow fast neutrons more
rapidly.
 Enrichment — a higher concentration of uranium-235 atoms results in
a very slightly higher fast fission factor.
— Impact is relatively small and may change ε from 1.04 for a new
core to 1.03, for a depleted core.


Design Factors Affecting Resonance Escape Probability (ρ)
Moderator-to-fuel ratio, fuel temperature, and fuel enrichment are
parameters that affect the value of resonance escape probability.
Importantly, these parameter are set by design, but are also affected during
plant operation. For example, fuel temperature is affected by power level
and moderator-to-fuel ratio is affected by moderator temperature, which is
discussed further in the next section.
Moderator-to-Fuel Ratio
Moderator-to-fuel ratio has a large impact on the value of the resonance
escape probability and thermal utilization factor in an operating nuclear
reactor. Changing fuel element design or moderator density can modify this
ratio. The fuel element design and loading is set by reactor design and is
not controlled by the reactor operator. However, moderator density in a
pressurized water reactor (PWR) is affected by moderator temperature
changes, which the operator can directly control.
In the case of the resonance escape probability, as moderator temperature
decreases, density increases, neutrons spend less time in the resonance
capture energy range, moderator-to-fuel ratio increases, and ρ increases.
The figure below shows this relationship.
Rev 1
35
Figure: Resonance Escape Probability Versus Moderator-to-Fuel Ratio
Fuel Temperature
The resonance escape probability varies with changes in fuel temperature.
In water moderated, low uranium-235 enrichment reactors, raising the
temperature of the fuel will increase the resonance absorption in uranium238 due to the Doppler Effect (broadening of the normally narrow
resonance absorption peaks due to thermal motion of nuclei). The increase
in resonance absorption decreases the resonance escape probability.
Figure: Resonance Escape Probability Change With Fuel Temperature
36
Rev 1
Fuel Enrichment
Increasing fuel enrichment (concentration of uranium-235 atoms) in the
core will result in a minor increase in resonance escape probability. This is
due to the corresponding decrease in uranium-238 concentration, reducing
the probability that neutrons will undergo resonance absorption by uranium238 nuclei.
Design Factors Affecting Thermal Utilization Factor (f)
Several design factors affect the value of the thermal utilization factors.
The largest is the moderator-to-fuel ratio and the second is uranium-235
enrichment. Other operating parameter affects are discussed in the next
section.
Moderator-to-Fuel Ratio
Moderator-to-fuel ratio has the largest impact on the value of the resonance
escape probability and thermal utilization factor in an operating nuclear
reactor. Changing fuel element design or moderator density can modify this
ratio. The fuel element design and loading is set by reactor design and is
not controlled by the reactor operator. However, moderator density in a
PWR is affected by moderator temperature changes, which the operator can
directly control.
In the case of the thermal utilization factor, as the moderator-to-fuel ratio
increases (temperature decrease) thermal utilization decreases, which results
from the increased total number of thermal neutrons absorbed specifically
by the moderator.
Figure: Thermal Utilization Factor Versus Moderator-To-Fuel Ratio
Rev 1
37
Uranium-235 Enrichment
The amount of enrichment of uranium-235 will affect the thermal utilization
factor by increasing the number of thermal neutrons absorbed in the fuel.
𝑓=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠
In an operating nuclear reactor, higher fuel enrichment is used to allow for
extended power operations (typically 18 months or longer). This increased
enrichment means more thermal neutrons absorbed in the fuel compared to
fuel plus all other absorptions, increasing f. Recall that the thermal neutron
macroscopic cross-section for absorption for uranium-235 is greater than
that for uranium-238.
Design Factors Affecting Reproduction Factor (η)
Uranium-235 enrichment is the design factor affecting the value of the
Reproduction Factor (𝜂). Other operating parameter affects are discussed in
the next section.
Enrichment
The value of η increases with uranium-235 enrichment because there is less
uranium-238 in the reactor making it more likely that a neutron absorbed in
the fuel will be absorbed by uranium-235 and cause fission. Refer to the
equation below.
𝑁 𝑈−235 𝜎𝑓 𝑈−235 𝑣 𝑈−235
𝜂 = 𝑈−235 𝑈−235
𝑁
𝜎𝑎
+ 𝑁𝑈−238 𝜎𝑎 𝑈−238
Design Factors Affecting Fast Non-Leakage Probability (Lf)
The ability for a fast neutron to leak out of a reactor depends on how far the
neutron travels as well as its distance from the core boundary. Because of
this, beyond the core design, Lf is primarily a function of the moderator
density.
Because the physical core size is so large for a commercial nuclear reactor
(nearly infinite for neutrons), moderator density actually has a very minor
effect on the value of Lf. Because of this, Lf is often neglected.
Design Factors Affecting Thermal Non-Leakage Probability (Lth)
The thermal non-leakage probability is affected by the same parameters,
effective core size and moderator density, as the fast non-leakage
probability. The effect of core size and moderator density changes is
reduced because the distance that a neutron travels in the thermal energy
range is much less than that of a fast neutron.
38
Rev 1
As with fast non-leakage probability, this leakage term is often neglected
due to the relative infinite size of a commercial nuclear reactor core.
Knowledge Check
An increase in moderator temperature results in a
decrease in all of the factors in a six-factor formula
except...
A.
resonance escape probability (p).
B.
fast fission factor (ε).
C.
thermal utilization factor (f).
D.
fast non-leakage factor (Lf).
ELO 4.4 Core Operating Parameter Changes and the Six-Factor
Formula
Introduction
This section focuses on operating parameter changes during power
operation, that is power, temperature, poisons, core age, etc., and how they
affect the factors in the six-factor formula. In order to control reactor
power, a reactor operator must be able to control the thermal neutron
population in the core. The operator controls thermal neutron population in
the core by controlling the various factors of the six-factor formula.
This section explains the following summary table:
Figure: Core Parameter Changes Affecting keff - S = Slight Effect
Rev 1
39
Core Life Increasing
Fast fission — the most significant effect to fast fission. Decreases slightly
from depleting uranium-235, reducing a number of fast fissions occurring.
(U-235 has a higher cross-section for fast fissions than U-238.)
Resonance escape probability — as the core ages, some uranium-238 is
converted to plutonium-240, by the following reaction.
𝛽 − 239
238
1
239 𝛽− 239
239
1
240
𝑈+ 𝑛 →
𝑈→
𝑁𝑝 →
𝑃𝑢,
𝑃𝑢 + 𝑛 →
𝑃𝑢
92
0
92
93
94
94
0
94
Uranium-238, which has high resonance absorption cross-section peaks,
depletes and plutonium-240 increases. However, plutonium-240 has even
higher resonance absorption cross-section peaks than uranium-238
(approximately 30 times higher). The result is an increase in resonance
capture over core life or a decrease in resonance escape probability over
core life.
Thermal utilization — as the core ages, fuel enrichment decreases with the
burnup of uranium-235 resulting in a decreasing number of uranium-235
atoms causing a decrease in the value of the thermal utilization factor.
Additionally, as the fuel concentration decreases, the moderator-to-fuel ratio
increases, and the probability of neutron absorption by the moderator
increases. With fewer neutrons available for absorption in the fuel, the
thermal utilization factor decreases.
Furthermore, soluble boron, control rods, and burnable poisons are used in
the nuclear reactor to control the excess amount of reactivity present in the
core from fuel loading. Throughout the life of the core, control rods are
normally maintained fully withdrawn (except during startup and shutdown)
so the thermal utilization factor would not be affected at full power by
control rods.
However, boron concentration and burnable poison concentrations are
reduced to compensate for fuel burnup and fission product poisons over
core life. Additionally, the reactor operator can change boron concentration
to control reactor power.
Lowering the concentration of boron decreases the probability that thermal
neutrons will be absorbed by the boron (soluble in the moderator), resulting
in an increase to thermal utilization factor. Control rods would have the
same affect when used.
There is one more affect with increasing core life. As previously described,
plutonium-239 builds up over core life from the uranium-238 neutron
capture. This results in an increase in fuel concentration, in turn causing a
thermal utilization factor increase.
40
Rev 1
The overall effect over core life of these factors may be a slight increase in
the thermal utilization factor due to the changing boron concentration in the
coolant.
Reproduction factor — as the core ages, plutomiun-239 is produced from
neutron capture by uranium-238. Although neutron yield per fission for
plutomium-239 is slightly higher than for uranium-235, production of
plutonium-239 lags the depletion of uranium-235. The result is a slight
decrease in the reproduction factor over core life.
Non-Leakage factors — no significant changes over core life.
keff — decreases without operator action.
Moderator Temperature Increasing
Fast fission factor — as moderator temperature increases, moderator
density decreases, and neutrons take longer to thermalize, causing a greater
chance of fast fission. Therefore, ε increases slightly with increasing
moderator temperatures.
Non-Leakage, resonance escape probability, and thermal utilization
factors – these factors combine to provide for the most basic inherent safety
feature of PWR thermal reactors. This is the negative moderator
temperature coefficient. Much more detail on this is provided later in the
course, especially in regards to the effect soluble boron concentration has
on the coefficient. For the following explanation, only pure light water as
the moderator is considered, with no soluble boron.
Non-leakage factors — as moderator temperature increases, density
decreases, neutron collisions are further apart, neutrons travel further,
and there is a greater chance of leakage. Therefore, the non-leakage
probabilities decrease. The opposite is true for a temperature
decrease.
 Moderator-to-fuel ratio — moderator temperature affects density,
which affects the moderator-to-fuel ratio. This affects both the
resonance escape and thermal utilization factors. Refer to the
following figure for the following explanation.

Rev 1
41
Figure: keff Versus Moderator-To-Fuel Ratio
Water density decreases as temperature increases. At higher temperatures
(550°F), the decrease in water density is greater per degree change in
temperature than at lower temperatures (120°F), as shown in the figure
below. Restated: for the same change in temperature, the change in water
density is greater at higher temperatures.
Figure: Density of Water Versus Moderator Temperature
The change in moderator-to-fuel ratio (Nmod/Nfuel) versus temperature
change is proportional to the water density change. Therefore, as moderator
temperature increases, the moderator-to-fuel ratio (Nmod/Nfuel) decreases.
As previously stated the magnitude of this change is greater at higher
temperatures.
42
Rev 1
At power, a nuclear reactor is designed to operate at a tight temperature
range (approximately 30°F) and small pressure changes. Significant
moderator temperature changes do occur in a reactor plant during heatup to
operating temperature; however, the reactor will not be made critical until at
normal operating temperatures and pressures.
Under moderation
PWRs are designed to be under moderated — less than ideal moderation of
all neutrons in the core. This condition leads to a negative temperature
coefficient, the inherent safety feature as previously mentioned. In an
undermoderated reactor, as moderator temperature increases and density
decreases (causing the non-leakage factors to decrease), a drop in
moderator-to-fuel ratio also occurs causing resonance escape probability to
decrease (neutrons travel further, greater chance of resonance capture) and
thermal utilization to increase because fewer moderator atoms are available
for absorbing thermal neutrons compared to fuel. The effect on resonance
escape probability is greater than for thermal utilization.
By multiplying each term of six-factor formula, the result is that keff
decreases for an increase in moderator temperature. This can be seen on the
moderator-to-fuel ratio figure to the left of the keff curve peak. An increase
in temperature lowers keff, decreasing the neutron population in the core,
resulting in reactor power level decreasing and the temperature increase
stopping. This, along with the fuel temperature coefficient, provides for the
inherent stability for controlling reactor power.
Over moderation
In a reactor where the moderator-to-fuel ratio is high, an over moderated
condition exists. In this case, on a temperature increase, keff increases.
Unlike the under moderated condition, the decrease in resonance escape
probability is smaller and the increase to the thermal utilization factor is
larger. Refer to the moderator-to-fuel ratio curve.
This along with the non-leakage factors causes an increase in keff for a
moderator temperature increase. This is referred to as a Positive Moderator
Temperature Coefficient. While this condition could exist in a PWR with
high soluble boron concentrations, or directly following a reactor refueling,
this is not a desirable situation and requires close monitoring and
understanding by the operator.
Boron Concentration Decreasing or Control Rods Withdrawn
Fast fission factor — no effect.
Non-leakage factors — no significant effect, rod withdrawal may affect
flux shape that could increase the non-leakage factors very slightly.
Resonance escape probability — no effect.
Rev 1
43
Thermal utilization — concentration of non-fuel neutron absorbing
materials decrease, thermal utilization increases.
Reproduction factor — no effect.
keff — increases from thermal utilization increase.
Note
Note
Effects of control rods and boron described above show
that thermal utilization factor is one of factors that an
operator can manipulate to control keff in an operating
nuclear reactor.
Fuel Temperature Increasing
Fast fission factor — no effect.
Non-leakage factors — no effect.
Resonance escape probability — the resonance escape probability varies
with changes in fuel temperature. In water-moderated, low uranium-235
enrichment reactors, raising the temperature of the fuel will increase the
resonance absorption in uranium-238 due to the Doppler Effect, broadening
the normally narrow resonance absorption peaks due to thermal motion of
nuclei. The increase in resonance absorption decreases the resonance
escape probability.
Figure: Resonance Escape Probability Change With Fuel Temperature
44
Rev 1
Thermal utilization — no effect.
Reproduction factor — no effect.
keff — decreases from the decrease in resonance escape probability.
Pressure Increasing
PWRs operate in a tightly controlled pressure band. Water is an
incompressible fluid as well. Therefore, any moderator density changes are
negligible as would be changes to the terms in the six-factor formula. A
one (1) degree temperature change is equivalent to 100 pounds per square
inch (psi) pressure change.
Poison (Fission Products) Increase
Poisons have the same effect on keff as boron and control rods because they
are all reactivity poisons.
Fast fission factor — no effect.
Non-leakage factors — no effect.
Resonance escape probability — no effect.
Thermal utilization — concentration of non-fuel neutron absorbing
materials increases, thermal utilization decreases.
Reproduction factor — no effect.
keff — decreases from thermal utilization decreases.
Fuel Enrichment Increase
Fuel enrichment is a design issue, something over which the operator has no
direct control. Fuel enrichment is included here for comparison with
operating parameter changes.
Fast fission factor — a higher concentration of uranium-235 atoms results
in a very slightly higher fast fission factor.
Non-leakage factors — no effect
Resonance escape probability — increasing the concentration of uranium235 atoms in the core results in a minor increase in resonance escape
probability due to the corresponding decrease in uranium-238
concentration. This results in fewer neutrons being resonance captured by
uranium-238 nuclei.
Thermal utilization — concentration of fuel neutron absorbing materials
increases, thermal utilization increases.
Rev 1
45
Reproduction factor – increases with uranium-235 enrichment increases
because of less uranium-238 in the reactor making it more likely that a
neutron absorbed in the fuel will be absorbed by uranium-235 and cause
fission. Refer to the equation below.
𝜂=
𝑁 𝑈−235 𝜎𝑓 𝑈−235 𝑣 𝑈−235
𝑁𝑈−235 𝜎𝑎 𝑈−235 + 𝑁𝑈−238 𝜎𝑎 𝑈−238
keff — increases from ρ f 𝜂.
Knowledge Check
In an under moderated reactor, if the temperature of the
moderator is increased, f will ________ and p will
_________.
A.
increase; decrease
B.
increase; remain the same
C.
decrease; increase
D.
decrease; remain the same
TLO 4 Summary
During this lesson, you learned about neutron life cycle and reactor control:
terms associated with the neutron life cycle, describing the terms in the sixfactor formula, how the physical design of the reactor core affects each term
in the six-factor formula, and how a change to plant parameters affects each
part of the six-factor formula. The listing below provides a summary of
sections in this TLO.
1. Define the following terms associated with the neutron life cycle:




46
Infinite multiplication factor (k∞) ratio of number of neutrons
produced by fission in one generation to the number of neutrons
lost through absorption in the preceding generation. Infinitely
sized reactor, no neutron leakage considered.
Effective multiplication factor, keff, ratio of number of neutrons
produced by fission in one generation to the number of neutrons
lost through absorption and leakage in the preceding generation.
Finite sized reactor.
Critical is the condition where the neutron chain reaction is selfsustaining and the neutron population is neither increasing nor
decreasing. keff = 1.
Subcritical is the condition in which the neutron population is
decreasing each generation. keff < 1.
Rev 1

Supercritical is the condition in which the neutron population is
increasing each generation. keff > 1.
2. Describe each term in the six-factor formula:
 𝑘𝑒𝑓𝑓 = 𝜀𝐿𝑓 𝜌𝐿𝑡ℎ 𝑓𝜂. Each of the six factors is defined below.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑙𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠

𝜀 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠

𝐿𝑓 =

𝜌 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑠𝑡𝑎𝑟𝑡 𝑡𝑜 𝑠𝑙𝑜𝑤 𝑑𝑜𝑤𝑛

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑑𝑜 𝑛𝑜𝑡 𝑙𝑒𝑎𝑘 𝑓𝑟𝑜𝑚 𝑟𝑒𝑎𝑐𝑡𝑜𝑟
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑎𝑙𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑎𝑐ℎ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
𝐿𝑡ℎ =

𝑓=

𝜂=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠
𝑡ℎ𝑎𝑡 𝑑𝑜 𝑛𝑜𝑡 𝑙𝑒𝑎𝑘 𝑓𝑟𝑜𝑚 𝑟𝑒𝑎𝑐𝑡𝑜𝑟
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠
𝑡ℎ𝑎𝑡 𝑟𝑒𝑎𝑐ℎ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠
𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠
𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑎𝑙𝑙 𝑟𝑒𝑎𝑐𝑡𝑜𝑟 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑠𝑡 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑓𝑖𝑠𝑠𝑖𝑜𝑛
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙
3. Explain how the physical design of the reactor core affects each of the
terms in the six-factor formula.
 Design factors affecting the value of the fast fission factor include
the following:
— Fuel atomic density
— Fuel pellet diameter
— Moderator
— Enrichment
 Design factors affecting resonance escape probability include the
following:
— Moderator-to-fuel ratio
— Fuel temperature
— Fuel enrichment
 Design factors affecting thermal utilization factor (f)
— Moderator-to-fuel ratio
— Uranium-235 Enrichment
 Design factors affecting reproduction factor (η)
— Enrichment
 Design factors affecting fast non-leaking probability (Lf)
 Design factors affecting thermal non-leaking probability (Lth)
4. Physical core changes most directly from the operator produce the
following changes to the Six-Factor Formula:
Rev 1
47
Figure: Core Operating Changes and the Six-Factor Formula




Fast fission factor and reproduction factor are primarily
determined by reactor design and remain essentially constant in
the temperature range of an operating nuclear reactor.
Moderator-to-fuel ratio has the largest impact on the values of the
resonance escape probability and thermal utilization factor in an
operating nuclear reactor.
— Nuclear reactor operator can control reactor moderator
temperature, affecting density and the moderator-fuel ratio.
This affects the non-leakage terms, the thermal utilization
factor, and the resonance escape probability.
In or with an under-moderated reactor (what we want), as
moderator temperature increases and density decreases,
moderator-to-fuel ratio decreases with an accompanying insertion
of negative reactivity. This provides some inherent stability for
controlling reactor power.
In or with an over-moderated reactor, positive reactivity is inserted
as the temperature increases, which is not a desirable condition for
power control in a PWR.
Now that you have completed this lesson, you should be able to do the
following:
1. Define the following terms associated with the neutron life cycle:
a. Infinite multiplication factor (k∞)
b. Effective multiplication factor (keff)
c. Subcritical
d. Critical
e. Supercritical
2. Describe each term in the six-factor formula using the ratio of the
number of neutrons present at different points in the neutron life
cycle.
3. Explain how the physical design of the reactor core affects each of the
terms in the six-factor formula.
48
Rev 1
4. Explain how a change to plant operating parameters affects each of
the factors of the six-factor formula.
Neutron Life Cycle Summary
This module presented the neutron life cycle because nuclear fission
generates or is controlled by the way atoms splits from neutrons combined
with the neutrons respective thermalization by things such as moderators in
the reactor to reduce the energy level of a neutron from its birth energy to
the energy level of the surrounding atoms. The addition of chemical
elements such as Uranium-235 or Plutonium-239 to neutrons and the
respective results of each, including prompt and delayed neutrons and their
effects. When nuclear fission occurs slowly, the energy is used to produce
electricity.
TLO 1 presented how thermalization works with a moderator to reduce the
velocity of the fission neutrons, average logarithmic energy decrement, and
macroscopic slowing down power and moderating ratio.
TLO 2 presented prompt and delayed neutrons, their origin and production,
the approximate fraction of neutrons born delayed from the fission of
Uranium-235 and Plutonium-239, the lifetime and generation times of both
prompt and delayed neutrons, and the effects of delayed neutrons on reactor
control.
TLO 3 presented neutron flux, the average energy that produces prompt
neutrons, the shape of the neutron energy spectrum in a thermal reactor, and
the reason for the neutron energy spectrum’s shape for a thermal reactor,
including variable(s) that have the most effect on thermal neutron velocity
TLO 4 presented neutron life cycle and reactor control: terms associated
with the neutron life cycle, describing the terms in the six- factor formula,
how the physical design of the reactor core affects each term in the sixfactor formula, and how a change to plant parameters affects each part of
the six-factor formula.
Summary
Now that you have completed this module, you should be able to
demonstrate mastery of this topic by passing a written exam with a grade of
80 percent or higher on the following TLOs:
1. Describe the process of neutron moderation in a nuclear reactor and
the characteristics of desirable moderators.
2. Describe the production of prompt and delayed neutrons from fission,
and how these neutrons affect nuclear reactor control.
3. Describe the neutron flux spectrum in thermal reactors.
4. Describe the neutron life cycle throughout the lifetime of a thermal
reactor and how it is affected by core design and variations in
operating parameters.
Rev 1
49