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Reactor Physics and Reactor Control Dr. Zee, Sung-Kyun
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Introduction to Nuclear PowerDr. Zee, Sung-Kyun
Session 3 : Reactor Physics
• Nuclear reactor theory is used to predict the behavior of neutrons in
nuclear reactors.
• The concepts in nuclear reactor theory such as nuclear reactions,
fission process, neutron cross sections, and the moderation and
diffusion of neutrons will be introduced.
• The neutron diffusion equation can be used to predict the nuclear
power distribution in a nuclear reactor and the neutron multiplication
factor.
• The nuclear reactor theory treats the static and transient behavior of
nuclear reactors as well as nuclear reactor design and operations.
Reactor Physics
– Nuclear Reaction
– Neutron Cross Section
isotopic generators, and so on – depends fundamentally on the way
in which nuclear radiation (neutrons, gamma rays, and charged
particles) interacts with matter.
• This section will present the mechanism of neutron interaction with
matter and various types of them.
Nuclear Reaction
• Nuclear Reaction
– Consider the incident particle a of mass m0 and velocity v0 strikes the
stationary (v=0) target nucleus X of mass M.
– As the result of the reaction, the target nucleus transforms, a new product
nucleus Y and a particle b is emitted.
a + X Y + b or X(a, b)Y
where a = incident particle (p, n, d, α, γ)
X = target nucleus
Y = product nucleus
*
*
0 5 5 2 3n B B Li
• Nuclear Reaction with Accelerator (Cockroft-Walton, 1932)
14 4 17 1 14 17
7 2 8 1 ( , ) ( , )ReactionN O p N p O p
9 4 12 1 9 12
4 2 6 0 ( , ) ( , )ReactionBe C n Be n C n
7 1 4 4 7 4
3 1 2 2 ( , ) ( , )ReactionLi P He Li p He p
Example: Nuclear Reactions
1. Formation of compound nucleus in excited state.
• Its energy equal to kinetic energy of incident neutron + binding energy of the neutron in the compound nucleus.
• It stays in a quasi-stationary state relatively long (10-14 sec), compared to transit time of neutron thru the nucleus.
1. Excited nucleus decay by emitting particles from the nucleus
• Mostly radiative capture often with beta ray accompanied
• Sometime emits least bound nucleons.
n + X C* Y + y (= n, n’, γ, p, α, 2n, 3n, pn)
Neutron Interactions
• Two-step interaction
– The compound decays
• Inelastic scattering : kinetic energy is not conserved
• Multiple neutrons
• Potential scattering
X
Classification of Nuclear Reaction
– Elastic scattering: (n,n)
– Inelastic scattering: (n,n')
– Spallation reaction or fission: (n,f)
– Charged-particle reaction: (n,p) or (n,α)
– Neutron generation reaction: (n,2n) or (n,3n)
n + X Y + y or X(n, y)Y
where n = incident neutron
y = emitted particle( n, n’,γ, p, α, 2n, 3n)
Nuclear Reaction Classification
– Thermal neutron energies ≈ (1/40) eV
– Epithermal neutron energies ≈ 1 eV
– Slow-neutron energies ≈ 1 keV
– Low-energy charged particles ≈ 0.1∼10 MeV
– High energies ≈ 10∼100 MeV
• Classification of reaction by the target nucleus:
– Light nuclei : A ≤ 25
– Heavy nuclei : A ≥ 150
Types of Nuclear Reactions
X(a, a')X*, excitation of the target nucleus
γ ray emitted
– The target nucleus gains the amount of kinetic energy that
the neutron loses
– Nuclei with low atomic number
• Neutron Emission
– (n,2n), (n,3n)…..
– Fast neutron
– 92U 235 92U
(before reaction – after reaction)
Q > 0 : Exothermic Reaction
M EVmMEE
|| Q M
mM E
Neutron Cross Section
• The extent to which neutrons interact with nuclei is described in
terms of quantities known as cross sections.
• In order to describe the neutron balance, cross sections are crucial
information together with neutron flux.
• The multiplication of cross sections and fluxes are the reaction rate
per unit volume and per second.
• The measured cross section data have been evaluated and
collected in ENDF, JEF, JENDL, etc.
• In this section, we will discuss the followings:
– Microscopic Cross Section
• Physical Meaning of Microscopic Cross section
– The probability of a particular reaction occurring between a neutron and a nuclide
– a measure of the relative occurrence probability of a given reaction
– the effective area of the particular nuclear reaction
The larger effective area, the greater the probability for reaction
• Measurement of Cross Section
Incident Neutron Target Nucleus
Intensity of Incident Neutron
Microscopic Neutron Cross Section
• Microscopic cross section units:
– cm2 (units of area)
• Components of total microscopic cross section :
– Summation of Microscopic Scattering Cross section and
Microscopic Absorption Cross section
slowing down neutron velocity?
(U235, Pu239)
– Fission cross section in thermal energy region is much larger than
Fission cross Section in fast energy region
Fast Neutron
Thermal Neutron
• Light nuclei:
– fairly wide resonances in MeV region
– smooth function of energy above
MeV region
• Heavy nuclei:
– smooth function of energy above resonance region
• Heavy nuclei in slowing down neutrons at high energy:
– more effective than light nuclei because of inelastic scattering
– Threshold energy Eth > the energy of the first excited state ε1
H-1
Total XS ~ Absorption XS
• Light nuclei:
– fairly wide resonances in MeV region
– smooth function of energy above MeV region
• Heavy nuclei:
– constant between resonances
ies σσσ
Microscopic

f
p
α
σ : n,γ cross section
σ : n,f cross section
σ : n,p cross section
σ : n,α cross section
Neutron Energy
Resonance Absorption
• Resonance Absorption
– Very large resonance absorption at 1eV ~ 100 keV energy region
– Depends on nuclei and neutron energy
Resonance Absorption Cross section of U235
Resonace Region
Macroscopic Cross Section
• The chance for a reaction is given by the sum of all microscopic
cross sections in the material
Neutron
• Macroscopic cross section: Σ [cm-1]
≡ the probability of neutron interaction per unit track length with a
nucleus X
≡ the average distance that a neutron travels in the substance
before it experiences the nuclear reaction under consideration
≡ the reciprocal of the macroscopic cross section

i
N : the number of a nucleus X per cubic centimeter
σ : the microscopic cross section of a nucleus X
Σ N σ N σ N σ N σ for the mixture
Σ Σ Σ Σ
has a density of 7.86 g/cm3.
The microscopic cross section for absorption of iron is 2.56 barns
and the atomic weight is 55.847 g.
• Solution Calculate the atomic density of iron, Calculate the macroscopic cross section
A
23
3
22
3
cm mole =
g 55.847
cm 1 barn
Reaction Rates
• The neutron balance equation needs to be solved to predict the
neutron population throughout a reactor.
• The neutron balance equation requires expressions for the rates at
which various nuclear events will occur at any given location and
involving neutrons of any given energy.
• In this section, we will discuss;
– Neutron Flux
– Reaction Rates
– Neutron Moderation
Neutron Flux
• Neutron flux
– The number of neutrons of energy E, in the interval dE,
that penetrates a sphere of a 1-cm2 cross section, located at r,
per second.
Neutron Flux
• Neutron flux
one cubic centimeter (1-cm3) during one second (1-sec)
– Scalar sum of the contributions from all neutrons which will be moving in
all directions
– Or, can be considered to be comprised of many neutron beams
traveling in various directions
• In each beam, all neutrons move in a same direction
• No neutron – neutron interaction is assumed
2
3
= nv
n = neutron density (neutrons/cm )
v = neutron velocity (cm/sec)
I = neutron beam intensity (neutrons/cm -sec) = n vn

– The number of neutron-nucleus interactions taking place in a cubic
centimeter in one second
– Note
• The flux is the total path length of all the neutrons in
a cubic centimeter (1-cm3) in a second (1-sec)
• The macroscopic cross section is the probability of having an interaction per
centimeter path length of a neutron
• Therefore, the multiplication of those two is the total number of interactions
in that cubic centimeter in a second – reaction rate!!
3
2
1
= neutron flux (neutrons/cm -sec)
= macroscopic cross section (cm )
• Example
– A one cubic centimeter of a reactor with fission cross section of 0.1 cm-1.
– Thermal neutron flux is 1013 neutrons/cm2-sec.
– What is the fission reaction rate?
13 -1
cm -sec
• Reactor power calculation
– The total number of fissions in a reactor core per second is the
multiplication of the average fission reaction rate per unit volume by the
total volume of the core.
– 3.12x1010 fissions release 1 watt-second of energy
– From flux to power
-6 -71 fission = 200 MeV, 1 MeV = 1.602 x 10 ergs, 1 erg = 1 x 10 watt-sec
10
1 erg 1 MeV 1 fission fissions 1 watt =3.12x10
1 x 10 watt-sec 1.602x10 erg 200 MeV second

rth : crow-flight distance from thermalization to absorption
Fast neutron
2r = 6f
– The process of reducing the energy of a neutron to
the thermal region by (in)elastic scattering
• Fission neutrons are born at an average energy level of 2 MeV
• Fission cross section is low at high energy (~MeV) and high at low energy
(< 1 eV)
• After a number of collisions with nuclei, the speed of a neutron is reduced to
have approximately the same average kinetic energy as the atoms of the
medium in which the neutron is undergoing elastic scattering => thermal
energy (0.025 eV at 20 °C)
• Thermal neutron: Maxwellian distribution
– Ideal moderating material (moderator)
• Large scattering cross section
• Small absorption cross section
Neutron Moderation
– Transfer of kinetic energy of neutron to a target nucleus
energy loss of neutron
A-1 where α = , A = atomic mass number
A+1
2
2
E
E
– Energy loss per scattering collision in logarithmic scale
• Definition
• Independent of the initial energy
– Average number of scattering collisions to thermalize
(from 2 MeV to 0.025 eV)
i i f
E
E = average final neutron energy after scattering collision

2( 1) 1 2 ξ=1- ln( ) (for A > 10)
22 1
• Macroscopic slowing down power (ξΣs)
– The measure of how rapidly a neutron will slow down in the material
– Still not sufficient to represent the effectiveness of moderator
– e.g. Boron has a good LED (ξ) and a good MSDP(ξΣs), but boron is not
a good moderator due to its high neutron absorptions
• Moderating ratio (ξΣs/Σa)
– The most complete measure of the effectiveness of a moderator
Neutron Moderation
Hydrogen 1 0 0.5 1.000 18
H2O 0.920 20 1.425 62
Deuterium 2 0.111 0.725
Helium 4 0.360 0.32 0.427 42 9x10-6 51
Beryllium 9 0.640 0.209 86 0.154 126
Boron 10 0.669 0.165 0.171 105 0.092 0.00086
Carbon 12 0.716 0.14 0.158 114 0.083 216
Oxygen 16 0.779 0.120
Sodium 23 0.840 0.0825
Iron 56 0.931 0.0357
Neutron Diffusion
• In order to design a nuclear reactor properly, we need to know how
the neutrons are distributed throughout the system.
• It is a reasonably good approximation that the neutrons undergo
diffusion in the reactor medium much like the diffusion of one gas in
another.
• This section will introduce the neutron diffusion equation and its
solution.
– Solute diffuse from higher concentration to lower concentration
• Rate of solute flow is proportional to negative of the gradient
• Can be used to approximate the behavior of neutron in reactor
– Neutron diffuses from higher flux region to lower flux region
– Limitations
• Near the isolated source
• Too strong absorbing medium
x
2
x
2
dx
where J = the net number of neutrons passing per a second per cm
perpendicular to the x-direction (neutrons/cm sec)
D = diffusion coefficient (cm)
4 3 2 1 rate of change in
rate of neutron rate of neutron rate of neutron number of neutrons = - -

f a
Divergence theorem
– Balance equation
• RHS : neutron production
– Multiplication factor
• k has been introduced to adjust ν factor in order to make
the loss and source balanced
• Rearranging the equation for k
• Shows the physical meaning of k as a neutron multiplication factor
2 1 a fD
– Must be physical (flux must be real, non-negative, and finite)
– Mathematically an appropriate set of BCs must be set
– or d/dn or a linear combination of those two be specified
• At the surface
• From transport theory
1 1 where d = extrapolation distance
n d
Neutron Diffusion Equation
– Isotropic point source with strength, S (neutrons/sec) at r=0
– Diffusion equation
– General solution
– Boundary conditions
• B.C. #2: source condition
r L
D L

4 4
r L
x x
d S Se S r J r r D DA A r
dr D Dr
moderator
• Will eventually be absorbed
• The probability that the neutron is absorbed between r and r+dr
• Average of the square of the “crow-flight” distance from the source to the
point the neutron is absorbed
• Diffusion length
p r dr S S

– Multiplication Factor
– Measure of change in fission neutron population from any one
generation to subsequent generation
– relationship of multiplication factor and reactor power
• keff < 1 : sub-criticality, power decrease
• keff = 1 : criticality, constant power
• keff > 1 : super-criticality, power increase
number of fissions in any one generation
number of fissions in the immediately preceding generation k
: six factor formula
eff f t f tk k L L pfP P
k pf
– Thermal utilization factor f
– Neutron production factor η
• The average number of neutrons produced per thermal neutron absorbed in
the fuel
the probability that a neutron will be absorbed in fuel
the probability that the neutron will be absorbed in the core
(fuel)
i fi
capture
0
0
the number of neutrons produced by fissions at all energies
the number of neutrons produced by thermal fission
( ) ( ) ( )
• Fraction of non-leaked neutrons from the system during the slowing-down
from fission energy to thermal energy
– Thermal non-leakage factor Pt
• Fraction of the thermal neutrons that do not leak out of the system during
thermal diffusion

tP L B
fpPtPfNo
– If a moderator temperature increases (Mod. Den. decreases)
• f increase
• p decreases ⇒ keff decreases
• Heterogeneous reactor in terms of four factors
– Thermal non-leakage factor Lt • Fraction of the thermal neutrons that do not leak out of the system during thermal
diffusion
homo
homo
homo
• A critical equation determines the condition under which a given
bare reactor is critical.
• The critical equation can be used to find the critical reactor size
when its composition and the amount of fuel are given and/or to find
critical mass when its size is given.
• This is a simplified approach using analytic expression of critical
equation. In real world design, the search for critical mass / critical
size can be done using reactor core design codes.
Critical Equations
reactor
– Rearranging
• B2 (material) < B2 (geometrical) : sub-criticality
• B2 (material) = B2 (geometrical) : criticality
• B2 (material) > B2 (geometrical) : super-criticality
– When the composition in a reactor is given, the critical size of the
reactor can be determined
• Nuclear reactors are not always in critical condition at constant
power.
• It is necessary for a reactor to be supercritical to start it up or raise
its power level, whereas it must be subcritical to shut it down or
reduce power.
• This section will study the behavior of the neutron population in a
noncritical reactor;
• The number of neutrons in the core after n generations
• Reactivity
– Reactivity vs. core power
= reactor: prompt critical
ρ ρ k k
Δk Δk Δk Δk 1 pcm = 0.00001 , 1 % = 0.01 = 1000 pcm, 1 mk = 0.001 = 100 pcm
k k k k
– Important bearing on the operation and safety of reactors
• Temperature coefficients
– “temperature coefficients of reactivity”
T T
Reactivity Coefficients
• Effects of temperature reactivity coefficients
– When αT is positive (>0)
• Increase in T increase in k increase in power level increase in T
…. power keep increasing
• Inherently unstable reactor
– When αT is negative (<0)
• Increase in T decrease in k decrease in power level decrease in T
…. back to original state
– Negative coefficient inherent reactor safety
• P increase fuel temp increase Doppler broadening (more neutron capture by U-238) negative reactivity feedback P decrease : stable
• Moderator Temperature Coefficient
– Positive(+) in an over-moderated core, negative(-) in a under-moderated core
– Provides delayed feedback due to the time for heat to be transferred to moderator
– Negative coefficient inherent reactor safety
• P increase Tm increase mod. Density decrease less moderation, p decrease (more resonance capture in U-238), f increase negative feedback P decrease : stable
T Tf
– Pressure increase mod. Den. Increase more moderation
positive effect P increase
– The magnitude of the coefficient is small in PWR
– More important in BWR due to larger density change associated with
boiling of coolant or moderator
• Void Coefficient
– BWR: P increase more void formation in moderator replace the
volume of moderator less moderation negative feedback in under-
moderated core P decrease
• Power Coefficient of reactivity
• Stationary nuclei absorb only neutron of energy E0
• If the nucleus is moving away from the neutron, the velocity (and energy) of
the neutron must be greater than E0 to undergo resonance absorption
• Likewise, if the nucleus is moving toward the neutron, the neutron needs
less energy than E0 to be absorbed
• Raising the temperature causes the nuclei to vibrate more rapidly within their
lattice structures, effectively broadening the energy range of neutrons that
may be resonantly absorbed in the fuel
C a
p tu
re C
0oK
20oC
1000oC
E0
– As temperature rises
• Neutron flux in resonances increase with temperature (less energy self-
shielding)
• Negative reactivity effect
– The most prompt effect of the power level change
– In LWRs, U-238 is principal contributor over the core life,
Pu-240 becomes important later in core life.
Contribution is small for U-235 and Pu-239
Prompt and Delayed Neutrons
• Prompt neutrons
– The great majority (> 99%) of the neutrons produced in fission are
released within about 10-13 seconds of the actual fission event
– Prompt neutron generation time
– Fission to prompt neutron production ~ 10-13 seconds
• Fast reactor ~ 10-6 seconds

• Delayed neutrons
– Are emitted immediately following the first beta decay of a neutron-rich
fission fragment (delayed neutron precursor)
– Characteristic half-life determined by that of the precursor of the actual
neutron emitter
87 87 86
5 0.610 1.14 - 0.00182 0.000748
6 0.230 3.01 - 0.00066 0.000273
total 0.0158 0.0065
Nucleus fraction
U-233 0.0026
U-235 0.0065
U-238 0.0148
Pu239 0.0021
– Example
• Given that a prompt neutron generation time is 5x10-5 seconds and a
delayed neutron generation time is 12.5 seconds. Calculate the average
generation time (β = 0.0065)
– With delayed neutrons, the reactor power level control becomes easier
(1 )average prompt delayed
= 0.0813 seconds
• Prompt neutron only
– Number of fissions
• The absorption of a neutron from one generation leads to, p sec later, k∞
neutrons in the next generation
– Number of fission after time t when k∞ ≠ 1
( ) ( )
( ) ( ) ( )
F F
dN t N t l N t l
dt
dt l
k
T
– Trial solution
– Graphical solution
– Reactor period
a
dt
1
– For large negative reactivity,
only longest-lived precursors remain
– FP accumulates
– Absorption cross section1/v behavior : important for thermal reactor
– FP poison removes neutrons from the reactor and therefore it will affect
the thermal utilization factor and keff.
• Xe-135
– Formed directly as a fission product
yr102.3
– Equilibrium concentrations
• Increases as flux level increases but there is a limit (0.052dk/k for U-235)
– Xenon after shutdown
• Removal: no absorption, radioactive decay
• Production: no fission yield, radioactive decay of I Xe poisoning effect will change!!
• Peak after ~11 hours
• Cannot restart the reactor !!
– Step change of power
• Cause Xe concentration change
– Equilibrium concentration
• but production from Pm decay
• The amount of Sm will increase to
~0.04 dk/k for high flux

2.212d
• Control rods
– Rods made of neutron-absorbing materials ( Ag, In, Cd, B, Hf ) which can be moved into or out of the reactor core
– Types
• Shutdown rod (safety rod) ⇒ Reactor shutdown (scram, trip)
– (integral/differential) Rod worth ρ = ρout - ρin
• varies depending on the location in the core
• Highest worth when inserted at the highest flux location
– Can compensate for rapid reactivity change
– Increase peak-to-average power density
– Usually the control rods alone is not enough to compensate for the
excess reactivity at the beginning of cycle
– Must be able to satisfy shutdown margin with N-1 control rods
Reactor Reactivity Control
• Burnable poison rods
– High neutron absorption cross section
– Converted into a material of relatively low absorption cross section as a result of neutron absorption
– Compensate for the excess reactivity of the fuel in the beginning of cycle
– Excess reactivity control with soluble boron alone requires too high boron concentration positive MTC
– No adverse effect to moderator temperature coefficients
– Can be used for shape flux profiles (local and global)
– Residuals of burnable absorbers can degrade neutron economy
– Gd, Er, B
– Relatively constant neutron absorption characteristics over core life
– The absorption of a neutron by one isotope in the material produces another isotope also with high absorption cross section
– Power shaping, power peaking reduction near moderator region
– Hf
– Compensate for the fuel burnup, poison buildup, temperature defects
– Spatially uniform effect
– Possible to increase or decrease amount of poison in the core during
reactor operation
concentration in the moderator (more absorption than moderate)
less negative MTC
adjusted to compensate for the reduced
excess reactivity as the core burns
Reactor Operation
• Basic concepts related to the nuclear reactor operations will be
introduced.
– Source neutrons from irradiated fuels and/or installed source
– Subcritical multiplication is used to increase power level in the source
range
• Estimated critical position
– The position of control rods that can result in criticality of reactor
– Take into accounts all of the changes in conditions: time since shutdown,
temperature, pressure, fuel burnup, samarium and xenon poisoning
Reactor Operation
• Need to flatten the power across the assemblies
• Use reflectors, enrichment zoning, burnable poisons
– To lower the radiation damage to reactor vessel
• A core power distribution problem
• Non-symmetrical variation of core power in one quadrant of the core relative
to the others
• Shutdown margin
– The reactivity required to make a reactor subcritical from its present
condition assuming all control rods fully inserted except for the single
rod with the highest integral worth, which is assumed fully withdrawn
Reactor Operation
• Temperature variations
– Temperature change of the reactor has a significant effect to the
reactivity of the core
• Pressure
– The effects are more noticeable at BWR
• Power level
– Once the power level is increased over the point of adding heat, then it
affects the reactivity through temperature variations
• Flow
– For BWR, increasing the flow rate decreases the fraction of steam voids
in the coolant and results in a positive reactivity
Reactor Operation
• Core burnup
– As a reactor is operated, fissile atoms of fuel are consumed
– For PWR, chemical shim concentration must be reduced to compensate
for the negative reactivity effect
– For BWR, control rods must be withdrawn
– As burnup increases, the delayed neutron fraction decreases
• Shutdown
– A reactor is subcritical and sufficient shutdown reactivity exists, no
gaining of criticality
– Following a large negative reactivity insertion, power level undergoes a
rapid drop (prompt drop), then the final rate of decrease will be
determined by the decay of the delayed neutron precursors
Reactor Operation
• Decay heat
– About 7 % of the 200 MeV produced by an average fission is released
at some time after the instant fission, from decay of fission products
– After a reactor shutdown from full power operation, the initial decay heat
is 5 ~ 6 % of the thermal rating of the reactor
– The decay heat generation rate diminishes to less than 1 %,
1 hour after shutdown
– Continued removal of heat is required for an appreciable time after
shutdown