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