Nuclear Fission
J. Frýbort, L. Heraltová
Department of Nuclear Reactors
October 31, 2014
Chapter 4
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Content
1
2
3
4
5
6
7
8
Process of Fission
Fission Reaction and Critical Energy
Fissile and Fissionable Nuclei
Fission Cross-Section
Fertile Material
Fission Products
Fission Neutrons
Prompt and Delayed Neutrons
Fission Spectrum
Energy from Fission
Fission Chain Reaction
Multiplication Factor
Neutron Moderation
Neutron Balance
Infinite Reactor
Finite Reactor
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Process of Fission
Binging Energy
A more stable configuration is obtained when a heavy nuclide is
split into two parts
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Process of Fission
Process of Fission
There is an equilibrium in a nucleus between Columbian repulsion
and nuclear forces among nucleons
When a heavy nucleus absorbs a neutron, a compound nucleus is
formed with an extra energy corresponding to the binding energy
of the neutron
Excess of energy received by the compound nucleus formation
breaks the balance between forces and causes vibrations of the
nucleus
There are two possible processes:
1
2
Neutron is captured in the nucleus and γ-ray is emmited to reach
the ground state
Nucleus is split into two lighter parts and large amount of energy is
released
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Process of Fission
Liquid Drop Model of a Nucleus
A nucleus has a similar structure like a drop of liquid
When a drop receives some energy, the equilibrium state is
disturbed
The drop can either after shaking and vibrating find a new
equilibrium or split into two smaller drops
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Process of Fission
Fission Reaction and Critical Energy
Fission Reaction
235
92 U
˚
97
136
` n Ñ p236
92 Uq Ñ 36 Kr ` 56 Ba ` 3n ` E
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Process of Fission
Fission Reaction and Critical Energy
Fission Reaction (cont’d)
Minimal required energy necessary to deform the nucleus to the
point where it splits is called critical energy – Ecrit
When the kinetic energy of the incident neutron together with
binging energy of the compound nucleus is equal or higher then
the corresponding critical energy a fission can occur
If the binding energy alone is higher than Ecrit , fission can occur
with a neutron having essential no kinetic energy
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Process of Fission
Fission Reaction and Critical Energy
Critical Energy for Fission
Nuclide
Ecrit [MeV]
Binding Energy [MeV]
232 Th
5.9
6.5
5.5
4.6
5.75
5.3
5.85
5.5
5.5
4.0
*
5.1
*
6.6
*
6.4
*
4.9
*
6.4
233 Th
233 U
234 U
235 U
236 U
238 U
239 U
239 Pu
240 Pu
* Neutron binding energy is not relevant, because these nuclides cannot be
formed by absorption of a neutron by a nucleus
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Process of Fission
Fissile and Fissionable Nuclei
Fissile and Fissionable Nuclides
Nuclei that can be fissioned by essentially zero-energy neutrons
are called fissile
For example: 233 U, 235 U, 239 Pu, 241 Pu
Note that nuclei that in fact undergo fission are compound nuclei
234
U, 236 U, 240 Pu, and 242 Pu
Even nuclides such as 238 U can be fissioned, but the incident
neutron has to have kinetic energy at least 0.6 MeV
Nuclei that can undergo fission after being struck by an energetic
neutron are not fissile, they are called fissionable
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Fission Cross-Section
Fission Cross-Section
A nuclear fission can occur after a compound nucleus formation.
Two different shapes of σf depending on the nucleus
characteristic:
Fissile Isotopes Similar shape as the radiative capture – 1/v region,
resonances, smooth function in higher energies, for
example 235 U
Fissionable Isotopes Zero up to the threshold energy which occurs
above the resonance region, for example 238 U
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Fission Cross-Section
Fission Cross-Section of Fissile Nuclides
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Fission Cross-Section
Fission Cross Section of Fissionable Nuclides
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Fission Cross-Section
Capture-to-Fission Ratio
During interaction of a low energy neutron with a fissile nuclide,
there are only three possible processes
1
2
3
Neutron is captured and a compound nucleus is formed, it can
return to the ground state by γ-ray emission
Compound nucleus undergoes fission
Potential elastic scattering of the incident neutron, but σe is much
smaller than σf and σγ
Important factor in designing of nuclear reactors is the ratio
between radiative capture and fission cross-section –
capture-to-fission ratio
σγ
α“
σf
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Fission Cross-Section
Fertile Material
Fertile Capture
Radiative capture can lead in special cases to production of a new
fissile material
This type of reaction is especially important for 232 Th and 238 U
nuclides
Decay of compound nuclei proceed according to the following
reactions:
232
β´
β´
21,83 min
26,98 d
Th ` n ÝÑ 233 Th ÝÝÝÝÝÝÑ 233 Pa ÝÝÝÝÑ 233 U
238
β´
β´
23,45 min
2,356 d
U ` n ÝÑ 239 U ÝÝÝÝÝÝÑ 239 Np ÝÝÝÝÑ 239 Pu
(4-2)
(4-3)
Results of these reactions are 233 U and 239 Pu, respectively, which
are fissile materials
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Fission Products
Fission Products
235
92 U
˚
93
` n Ñ p236
X ` 140 Y ` 3n ` E
92 Uq Ñ
Fission is almost always asymmetric
Two fission fragments have different masses
Despite of expectation based on energy balance calculations the
symmetric fission is rare
Ternary fission – the nucleus is split into three fission fragments. It
was observed but is a very rare event. This ternary fission is a
source of tritium.
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Fission Products
Fission Products (cont’d)
Fission products have more neutrons that would correspond to the
line of stability (they are to the right from the line of stability)
They undergo β ´ decay accompanied by γ radiation
Also daughter products of decay can be unstable
115
β´
β´
β´
Pd ÝÝÑ 115 Ag ÝÝÑ 115 Cd ÝÝÑ 115 In (stable)
Fission products yields are usually plotted as a function of mass
number
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Fission Products
Fission Products Yields for 235 U
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Fission Products
Fission Products Yields for Thermal Fission
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Fission Neutrons
Prompt and Delayed Neutrons
Number of Neutrons per Fission
Fission can occur in many different ways
Number of neutrons released during the fission depends on the
way of fission
Each form of fission has a probability of occurrence and a number
of emitted neutrons
Fission neutron yield – ν
Average number of neutrons produced by fission
The usual value depending on fissile isotope and incident neutron
energy is 2-3 fissile neutrons
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Fission Neutrons
Prompt and Delayed Neutrons
Number of Neutrons per Fission (cont’d)
It is a characteristic value for an isotope
Fission neutron yield increases with energy of the incident neutron
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Fission Neutrons
Prompt and Delayed Neutrons
Prompt Neutrons
235
92 U
˚
93
140
` n Ñ p236
92 Uq Ñ 37 Rb ` 55 Cs ` 3n ` E
More than 99 % of all neutrons emitted as a result of a fission
event are emitted almost instantly as the fission occurs (within
10´13 sec after the fission)
These neutrons are called prompt neutrons
Prompt neutrons originate with kinetic between 0.1 MeV and
10 MeV
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Fission Neutrons
Prompt and Delayed Neutrons
Delayed Neutrons
Some of the fission products are neutron rich and reach the
stability by a neutron emission – delayed neutrons precursors
Neutrons emitted during decay of precursors are called delayed
neutrons
Precursors are grouped by their half-life into 6 groups
Delayed neutrons are emitted with lower energies than prompt
neutrons
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Fission Neutrons
Prompt and Delayed Neutrons
Delayed Neutrons Fractions
Fraction of ith group of delayed neutrons – βi
It is a fraction of all fission neutrons that appear as delayed neutrons in
the i th group of delayed neutrons
Total fraction of delayed neutrons – β
The total fraction of delayed neutrons β is sum of βi
β“
νd
ν
“ d
νp ` νd
ν
The fraction of delayed neutrons is different for different fissile
isotopes
Delayed neutrons are important for reactor kinetics and control
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Fission Neutrons
Prompt and Delayed Neutrons
Delayed Neutrons Data for 235 U
Group Precursors half-life [s]
1
2
3
4
5
6
Energy [keV]
Fraction βi
55.72
250
22.72
560
6.22
405
2.30
450
0.610
410
0.230
–
Total delayed neutron fraction
0.000215
0.001424
0.001274
0.002568
0.000748
0.000273
0.0065
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Fission Neutrons
Fission Spectrum
Prompt Neutron Spectrum
Energy distribution depends on the involved isotopes
Prompt neutron spectrum can be described by an empirical
formula:
?
χpEq “ 0.453e´1.036E sinh 2.29E
The average energy of prompt
neutrons:
ż8
s“
E
EχpEq dE “ 1.98 MeV
0
The most probable energy
Ep = 0.73 MeV
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Fission Neutrons
Fission Spectrum
Delayed Neutron Spectrum
The average energy of delayed neutrons is approximately 0.5 MeV
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Energy from Fission
Energy from Fission
Huge amount of energy is released during the fission
However only part of the energy can be used – recoverable energy
Form
Fission fragments
Fission product decay
β-rays
γ-rays
neutrinos
Prompt γ-rays
Fission neutrons (kinetic energy)
Capture γ-rays
Total
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Emitted Energy
[MeV]
Recoverable Energy
[MeV]
168
168
8
7
12
7
5
–
207
8
7
–
7
5
3–12
198–207
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Fission Chain Reaction
Multiplication Factor
Multiplication Factor
Two important phenomena occur during the fission:
release of energy
production of new neutrons
Production of new neutrons allows to make a self sustainable
chain reaction providing the continuous energy release
Chain reaction could be described quantitatively in terms of multiplication factor
k“
number of neutrons in one generation
n
“ i
number of neutrons in the preceding generation
ni´1
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Fission Chain Reaction
Multiplication Factor
Fission Chain Reaction
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Fission Chain Reaction
Multiplication Factor
Reactor States
k ą 1 Reactor is supercritical and number of fissions is
exponentially increasing
k “ 1 Reactor is critical and number of fissions is constant in
time
k ă 1 Reactor is subcritical and number of fissions is decreasing
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Fission Chain Reaction
Multiplication Factor
Nuclear Reactor
Nuclear reactors are devices where the fission chain reaction can proceed in a controlled manner. Control of the number of fissions (reactor
power) is realized by changing the value of the multiplication factor
Increase of power is realized by increasing k above 1 Ñ reactor is
supercritical
When reactor reaches desired power the multiplication factor is
set back to 1
Decrease of power is a similar process. At the beginning k has to
be bellow 1 Ñ subcritical reactor
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Fission Chain Reaction
Multiplication Factor
Nuclear Reactor (cont’d)
The change of the multiplication factor is realized by controlling
the rate of neutron absorption in the reactor core
Absorbing material is usually in the form of control rods
Materials with high absorption cross-section in the desired energy
range must be used
For thermal reactors – B, Cd, Gd
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Neutron Moderation
Thermal Fission
The fission cross-section is higher in the thermal energy range
Most of the reactors are operated in the thermal neutron energy
range, thus they are called thermal reactors
Based on the previous discussion: Average energy of neutrons
released by fission is 2 MeV Ñ fast neutrons
For higher probability of fission reaction it is necessary to
decrease the energy of fission neutrons to the thermal energy
0.0253 eV
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Neutron Moderation
Neutron Slowing Down and Diffusion
Neutrons lose their energy by scattering reactions with variety of
target nuclei – slowing down, moderation
Effectiveness of energy decrease depends on the character of
target nucleus
Slowing down neutrons in a few collisions reduces leakage from
the core and resonance absorption (it will be discussed later)
When a neutron reaches energy lower than 1 eV, elastic
scattering reactions are the most important – diffusion
Properties of an ideal moderating material
Large scattering cross-section
Small absorption cross-section
Large energy loss per collision
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Neutron Moderation
Loss of Neutron Energy by Elastic Collision
Lethargy is convenient to express the loss of kinetic energy of
neutrons by elastic collisions due to the nature of such a reaction
Ď
Average change of neutron lethargy for one elastic collision – ∆u
– similar to average fractional loss of kinetic energy does not
depend of the initial neutron kinetic energy prior the collision
Ď due to its frequent use – ξ and it is
There is new symbol for ∆u
called average logarithmic loss of energy
ξ is quantity suitable for assessment of moderating properties of
materials
There is an approximate simple formula (the uncertainty for nuclei
with A ă 10 is approximatelly 3 %):
ξ“
J. Frýbort, L. Heraltová (CTU in Prague)
2
A`
Nuclear Fission
(4-5)
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3
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Neutron Moderation
Number of Collisions to Thermalization
ξ is the average logarithmic energy loss
Number of collision required to slowing down from higher energy
to lower energy is:
´
¯
Ehigh
ln
lnpEhigh q ´ lnpElow q
Elow
“
N“
ξ
ξ
For example: Slowing down a neutron with an average energy from fission energy
2 MeV to the thermal energy 0.0253 eV
by H2 O – 19 collisions
by D2 O – 35 collisions
by Be – 86 collisions
by C – 114 collisions
by Fe – 510 collisions
by U – 2168 collisions
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Neutron Moderation
Macroscopic Slowing Down Power
This value characterises the moderating capability of all the nuclei
in 1 cm3
Indicates how rapidly a neutron will slow down in the material
With increasing value of MSDP increases also the capability of
slowing down of neutrons
MSDP “ ξΣs
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Neutron Moderation
Moderating Ratio
Characterises the overall effectiveness of a moderating material
taking into account not only scattering reactions, but also possible
neutron absorption
Even moderators have possibility to absorb neutrons during
slowing down
Effective moderator has low neutron absorption cross-section
Higher MR means higher effectiveness of the moderating material
MR “
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ξΣs
Σa
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Neutron Moderation
Moderating Properties of Selected Light Materials
Material
ξ
NCT˚
MSDP [cm´1 ]
MR
H2 O
D2 O
Be
B
C
0.927
0.510
0.207
0.171
0.158
19
35
86
105
114
1.425
0.177
0.154
0.092
0.083
62
4830
126
0.00086
216
*Number of collisions required to reach thermal energy
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Neutron Balance
Neutron Leakage
If a finite reactor is considered, neutrons are lost not only by
absorption, but there is also probability of their escape from the
reactor
It will be shown later that reactor is critical if there is balance
between neutron production by fission on the one hand and
neutron disappearance realized by absorption and leakage on the
other hand
Fission, absorption, and leakage rate depend on the reactor
composition and dimensions
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Neutron Balance
Neutron Balance in the System
Suppose a homogeneous uniform mixture of uranium fuel,
moderator, coolant, absorbing material and structural materials
inside the reactor core
Prompt and delayed neutrons are not distinguished in this
simplified case
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Neutron Balance
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Neutron Balance
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Neutron Balance
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Neutron Balance
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Neutron Balance
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Neutron Balance
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Neutron Balance
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Neutron Balance
Probability of Events
PNL – Non-leakage Probability Probability that a neutron will not
escape from the reactor
Pa – Probability of Absorption in the Fuel Probability that a neutron is
absorbed in the fuel material
Pf – Probability of Fission Probability that a neutron induces fission
What does it mean fuel?
It has to be clarified at the beginning of the calculation. Fuel can
mean the whole mass of uranium and plutonium in the system, but
it can also mean only fissile isotopes of uranium or plutonium.
In our case, fuel means all compounds with isotopes of uranium or
plutonium
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Neutron Balance
Neutron Generation
Initial number of neutrons in one generation is N1
Number of neutrons in the following generation will be N2 :
k“
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N2
“ ν PNL PAF Pf
N1
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Neutron Balance
Infinite Reactor
Neutron Balance in Finite Thermal Reactor
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Neutron Balance
Infinite Reactor
Fast Fission Factor – ε
Fuel is a mixture of fissile and fissionable isotopes
Some fissions can occur also by fast neutrons with fissionable
nuclides
number of neutrons from fissions by neutrons of all energies
ε“
number of neutrons from fissions by neutrons with thermal energy
Fast fission increases the number of neutrons in the system by neutrons
released by fission of fissionable materials realised by fast neutrons
Fast fission is influenced by configuration of the core, enrichment
and type of moderator
In thermal reactors, fast fission factor is between 1.03 and 1.15
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Neutron Balance
Infinite Reactor
Resonance Escape Probability – p
Absorption reactions occur in the whole energy range, but the
highest probability of absorption for some isotopes is in the
resonance region
High value of absorption cross-section for certain resonance
energies
p“
number of neutrons that reach thermal energy
number of fast neutrons that start to slow-down
Resonance escape probability increases as the fraction of 238 U
decreases
The value p increases with higher enrichment
Value p is also influenced by the ratio between moderator and fuel
volume
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Neutron Balance
Infinite Reactor
Radiative Capture
The microscopic cross-section close to an resonance can be
described by Breit-Wigner formula
The formula is valid for narrow isolated resonances
Breit-Wigner Formula
σγ “
Γn Γγ
λ2 g
4π pE ´ Er q2 ` Γ2 {4
(4-9)
λ is wavelength of neutron with energy of the resonance Er
g is statistical constant
Γ “ Γn ` Γγ is width of the resonance, Γn is neutron width and Γγ
is radiation width
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Neutron Balance
Infinite Reactor
Thermal Utilization Factor – f
Neutrons reaching the thermal energy region can be absorbed in
the fuel, moderator, or by other materials
It is probability of absorption in the fuel
f “
number of thermal neutrons absorbed in fuel
total number of absorbed thermal neutrons
Thermal utilization depends on the fuel nature, type of moderator,
and ratio between their volumes
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Neutron Balance
Infinite Reactor
Probability of Fission – Pf
Thermal neutron absorbed in the fuel can induce fission or can be
captured
number of neutrons inducing fission in fuel
number of neutrons absorbed in fuel
ΣFf
Pf “ F
Σa
Pf “
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Neutron Balance
Infinite Reactor
Reproduction factor – η
Characterizes average number of neutrons released per one
thermal neutron absorbed in the fuel
ΣF
η “ ν fF
Σa
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Neutron Balance
Infinite Reactor
Infinite Multiplication Factor
In an infinite system all neutrons released during fission must be
absorbed
Number of neutrons in a successive generation is a product of
neutron population in the preceding generation, probability of
neutron surviving to the thermal energy range, probability of
causing fission, and number of neutrons released during fission
Using the formulas defined before, it can be expressed as:
k8 “ ε p f
ΣFf
ν
ΣFa
Four-Factor Formula – Infinite Multiplication Factor
k8 “ ε f p η
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(4-10)
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Neutron Balance
Finite Reactor
Finite System
There is a probability that a neutron can escape during slowing
down or during the following diffusion
Leakage from the reactor depends on the mean free path of
neutrons:
1
λ„
σ
High Energy – small value of cross-sections Ñ long mean free path
and high probability of escape
Thermal Energy – high value of cross-sections Ñ short mean free
path and neutrons are usually absorbed before reaching
the system boundary
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Neutron Balance
Finite Reactor
Non-Leakage Probability
PNL “ PFNL PTNL
The overall probability of non-leakage is a product of PFNL fast
neutron non-leakage probability and PTNL thermal neutron
non-leakage probability
Neutron escape is another mean of neutron disappearance from
the system
It must be reflected in the multiplication factor:
Six-Factor Formula – Effective Multiplication Factor
keff “ ε f p η PFNL PTNL “ k8 PFNL PTNL
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Neutron Balance
Finite Reactor
Neutron Balance in Finite Thermal Reactor
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Neutron Balance
Finite Reactor
Reactivity
Reactivity can be understood as a reactor capability to multiply
neutron population
It is a relative difference from the critical state
Reactivity is defined as:
k ´1
(4-12)
ρ “ eff
keff
Subcritical reactor keff ă 1 Ñ ρ ă 0
Critical reactor keff “ 1 Ñ ρ “ 0
Supercritical reactor keff ą 1 Ñ ρ ą 0
Value of reactivity is usually very small, therefore auxiliary units
are defined, these include:
% – 10´2
pcm – 10´5
example:
keff “ 1.0005 Ñ ρ “ 0.0004998 “ 0.04998 % “ 49.98 pcm
J. Frýbort, L. Heraltová (CTU in Prague)
Nuclear Fission
October 31, 2014
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