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Fundamentals of
Nuclear Engineering
CRITICALITY
CRITICALITY AND NEUTRON POPULATION
Criticality means to produce a self sustained chain reaction
Neutrons play a fundamental role in initiating nuclear reactions
A chain reaction may be initiated in principle by a single fission
which yields more than one neutron
that may be assumed to be captured entirely by fuel nuclide
releasing 2nd generation, which in a similar way, give birth to third generation and so on.
Hence self-sustained chain reaction is possible only when neutron production and
neutron losses are balanced such that sufficient number of neutrons would remain
still available to continue chain reaction
The number or neutron released per thermal fission cannot be increased
therefore the only alternate is to reduce various causes responsible for the neutron losses in the given assembly of fissionable material
IT IS BENEFICIAL TO CONSIDER INITIALLY THE FACTORS THATCONTRIBUTE POSITIVELY OR NEGATIVELY TOWARDS GROWTH
OF NEUTRON IN A MULTIPLYING MEDIUM
The neutrons escaping the resonance absorption region are available for furtherinteraction with the fissile fuel
as probability of fission with decreasing En & approaches 580 barn at 0.0253 eV (figure).
There is a good probability that these are absorbed in fissile material and cause fission
Dependence of fission cross-section on energy for U-235 & Pu-239
POSITIVE TERMS (TEND TO INCREASE THE NEUTRON POPULATION)
( THE THERMAL REPRODUCTION )
(# fast neutrons produced due to thermal fission)Reproduction factor = ------------------------------------------------------------------------
(# thermal neutrons absorbed in the fuel)
POSITIVE TERMS (TEND TO INCREASE THE NEUTRON POPULATION)
( THE FAST FISSION )
When neutron are released in fission they have an average energy of 2MeV (fast neutrons)
Since energies > threshold energy of U-238 (1.1 MeV)
these fast neutron may cause fission in U-238
causing a release of neutron hence contributing positively in neutron population growth
Usually fission probability is not that much as comp. to inelastic scattering or absorption
therefore contribution is not substantial as maximum probability obvious from the
figure is in the range of about 0.4 to 2 barns.
Dependence of fission cross-section on energy for U-238 & others
= (#n emitted by fast fissions + #n emitted by thermal fissions) / (#n emitted by thermal fissions ) = (Total Fissions) / (Thermal Fissions)
Neutron of any energy region may be captured by U-235 or U-238 without causing fission
The types of such an interactions are those such as (n, ), (n,p), (n, ) or (n, D)
NEGATIVE TERMS (TEND TO DECREASE THE NEUTRON POPULATION)
( THE CAPTURE TO FISSION RATIO )
The probability of these reaction to occur is however very small.
Reproduction Factor is the average number of fast fission neutrons released as a
result of capture of one thermal neutron in fissionable material
is slightly smaller then the average number of neutrons emitted per fission as all
the thermal neutrons absorbed in fissionable material do not cause fission
Thus neutrons are absorbed according to their absorption probability a and out of
these a major part created fission according to their fission probability f
Since f is lesser than a therefore the fraction f / a which is less than one, would
reduce the value of
the relation between and may be given as = f/ a
(# fast neutrons produced due to thermal fission)Reproduction factor = ------------------------------------------------------------------------
(# thermal neutrons absorbed in the fuel)
When energy reduces below about 1 MeV
neutrons are absorbed in very large fraction in resonance range without causing fission
As can be seen from the plot of total absorption as a function of energy in figure
A sketch of total microscopic cross section for 238U92
NEGATIVE TERMS (TEND TO DECREASE THE NEUTRON POPULATION)
( THE RESONANCE ABSORPTION )
This results in a substantial negative contribution in neutron population growth
Neutron may simply escape from the reactor core usually termed as leakage from the core
Leakage may occur at any energy but when two main energy groups are considered
such as thermal and fast neutrons then the probability is defined for these energies
The leakage probability primarily depends upon size and shape of the core assembly
NEGATIVE TERMS (TEND TO DECREASE THE NEUTRON POPULATION)
( THE FAST AND THERMAL NEUTRON LEAKAGES )
Neutrons may be lost due to absorption in moderator, coolant, structural & control material.
which contribute negatively in the neutron population growth
A good fraction of neutrons may be absorbed in these materials
NEGATIVE TERMS (TEND TO DECREASE THE NEUTRON POPULATION)
( THE ABSORPTION IN CONTROL AND STRUCTURAL MATERIALS )
f = (Thermal neutrons absorbed in the fuel) /( thermal neutron absorbed in the whole system)
MAJOR DOMINATING FACTORS
Normally the major dominating factor out of these are neutron absorption
due to resonance peaks in epithermal region
In order to counter its effect an option is to reduce the quantity of U-238 so that
lesser atoms may result in lesser number of neutrons absorbed in them
Thus increase in enrichment may be an alternate to look into in order to cater the
excessive absorption in resonance peaks.
The other alternate is to use a moderator and thus decrease neutron energy
A moderator slows down the fast fission neutrons very quickly to thermal energies
Absorption in U-238 resonance peaks is thus appreciably reduced due to large
energy loss of neutron per collision with moderating nuclei.
A much larger fraction of fission neutrons are hence thermalized and cause fission in U-235
The presence of high scattering cross section moderating material such as D2O may even
compensates for the low content of U-235 available in natural uranium and makes a divergent
chain reaction possible.
LIFE HISTORY OF A FAST NEUTRON IN A NATURAL URANIUM ASSEMBLY
(NEUTRON MULTIPLICATION IN CRITICAL SYSTEMS)
In order to assess these positive and negative terms various possible events may be
considered that may happen to fast neutron during its life time.
The life history of a fast neutron in a natural uranium assembly from time it was
created to time it is finally absorbed or escaped from the core should be considered.
Suppose there are no thermal neutrons initially available for capture in fissionable material
If is the average number of fast fission neutrons released as a result of capture of one
thermal neutron in fissionable material, no fast neutrons will be produced as a result of
absorption of no thermal neutrons.
A total of no fast neutrons released in fission will be available now for further interaction.
Fuel U-235
= 1.33
100
thermal
neutrons
no
133
fast
neutrons
no
Fuel U-235
= 1.33
100
thermal
neutrons
no
U-238 in
the Fuel
= 1.05
133
fast
neutrons
no
140
fast
neutrons
no
Fuel U-235
= 1.33
100
thermal
neutrons
no
U-238 in
the Fuel
= 1.05
133
fast
neutrons
no
140
fast
neutrons
no
Fast Non
Leakage
Probability
PFNL=0.95
7
fast leakage
Probability
PFL= 0.05
133
fast
neutrons
no PFNL
Fuel
U-235
= 1.33
100
thermal
neutrons
no
U-238
in the
Fuel
= 1.05
133
fast
neutrons
no
140
fast
neutrons
no
Fast Non
Leakage
Probabilit
y
PFNL=0.95
7
fast leakage
Probability
PFL= 0.05
133
fast
neutrons
no PFNL
Resonance
Escape
Probability
p=0.90
120
Thermal
neutrons
no p PFNL
Thermal
Non
Leakage Probability
PTNL=0.95
6Thermal
leakage
PTL= 0.05
114
thermal
neutrons
no pPFNPTNL
Fuel
U-235
= 1.33
100
thermal
neutrons
no
U-238
in the
Fuel
= 1.05
133
fast
neutrons
no
140
fast
neutrons
no
Fast Non
Leakage
Probabilit
y
PFNL=0.95
7
fast leakage
Probability
PFL= 0.05
133
fast
neutrons
no PFNL
Resonance
Escape
Probability
p=0.90
114
Thermal
neutrons
no p PFNL
Thermal
Non
Leakage Probability
PTNL=0.95
6Thermal
leakage
PTL= 0.05
114
thermal
neutrons
no pPFNPTNL
Thermal Utrilization
Factor
f=0.90
103
Thermal
neutrons
no pPFNPTNLf