4/2003 Rev 2 I.4.6 – slide 1 of 63

Session I.4.6

Part I Review of Fundamentals

Module 4 Sources of Radiation

Session 6 Basic Reactor Physics Theory

IAEA Post Graduate Educational CourseRadiation Protection and Safety of Radiation Sources

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Overview

In this session we will discuss fission and fusion reactions

We will also discuss criticality

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Fission

gammafree

neutron

fissionfragment

fissionfragment

freeneutron

beta

alpha

energy

nucleus

freeneutron

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Fission

Sample fission reactions:

235U + n 141Ba + 92Kr + 3n + 170 MeV

235U + n 94Zr + 139La + 3n + 197 MeV

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Fission

Follows neutron capture

Thermal neutrons fission 233U, 235U, 239Pu which have odd number of neutrons

For isotopes with even number of neutrons, the incident neutron must have energy above about 1 MeV

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Fission

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Fission

Neutron Name/Title Energy (eV)

Cold Neutrons 0 < 0.025

Thermal Neutrons 0.025

Epithermal Neutrons 0.025 < 0.4

Cadmium Neutrons 0.4 < 0.6

Epicadmium Neutrons 0.6 < 1

Slow Neutrons 1 < 10

Resonance Neutrons 10 < 300

Intermediate Neutrons 300 < 1,000,000

Fast Neutrons 1,000,000 < 20,000,000

Relativistic Neutrons >20,000,000

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Fission

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fission products

andtransuranics

from neutron capture

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Fission

Source of energy released during fission:

Kinetic energy of fission fragments Gamma rays Kinetic energy of neutrons emitted

Prompt Delayed

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Criticality

Neutrons ejected during fission equal

neutrons producing more fissions+ neutrons absorbed + neutrons lost from system

Criticality is constant if balance exists. Fission rate (power) can be changed by varying the number of neutrons absorbed and/or controlling the number lost

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Criticality

Multiplication Factor (4 factor formula)

Keff =Nf+1

Nf

Nf+1 is the number of neutrons produced in the “f+1” generation by the Nf neutrons of the previous “f” generation

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Criticality

Sub-Critical (keff < 1) – more neutrons lost by escape from system and/or non-fission absorption by impurities or “poisons” than produced by fission.

Critical (keff = 1) – one neutron per fission available to produce another fission

Super-Critical (keff > 1) – rate of fission neutron production exceeds rate of loss

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Criticality

Keff depends on the availability of neutrons with the required energy and the availability of fissile atoms

As a result, keff depends on composition, arrangement and size of fissile material

If assembly is infinitely large, no neutrons are lost and Keff = L x k , where L is the non-leakage probability and K depends on 4 factors

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Criticality

Let’s follow “n” fission neutrons through their life cycle

η is mean number of neutrons emitted per absorption in uranium

if “n” fission neutrons are captured,n x η fission neutrons will be produced

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Criticality

ν is the mean number of neutrons emitted per fission which depends on the fuel (2.5 for 235U and 3 for 239Pu)

not every uranium absorption results in fission (238U can absorb thermal neutrons without fission)

mean number of fission neutrons per absorption is less than ν

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Sample Calculation

For 100% enrichment of 235U, = 2.1. What is for natural uranium?

= x f

a

= macroscopic cross section = microscopic cross sectionn = average number of neutrons per fissionN = atoms per cm3

a = absorption (5 = 235U)

f = fission (8 = 238U)

f = (N5 x f 5)a = (N5 x a 5) + (N8 x a 8)

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= x 2.5 = 1.32 for natural U549

650 + (139 x 2.8)

f 5 = 549 barnsa 5 = 650 barnsa 8 = 2.8 barns = 2.5 for 235U

N8

N5

For natural U, = 139

f

a

(N5 x f 5)

(N5 x a 5) + (N8 x a 8)

(f 5)

(a 5) + ( x a 8)N8

N5

= =

Sample Calculation

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Criticality

238U has a small cross section for fission by fast neutrons (not thermal)

= 0.29 barns

Fast fission factor is

ε =total number of fission neutrons

number of thermal fission neutrons

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Criticality

ε depends on 3 factors:

ratio of moderator to fuel ratio of inelastic scattering cross

section to fission cross section geometrical relationship between fuel

and moderator

capture of n thermal neutrons will producen x x ε fission neutrons

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Criticality

For unmoderated pure U metal, ε = 1.29 which is the maximum value

For homogenous fuel (such as a solution)ε is very close to 1

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Criticality

while fast neutrons are being slowed, they may be captured by 238U without fission

resonance for this capture occurs between 5 and 200 eV

p is the probability that a neutron will escape this resonance capture

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Criticality

p = resonance escape probability (fraction of fast, fission produced neutrons that become thermalized)

p depends on ratio of moderator to fuel

for high ratio of moderator to fuel, p 1

for a low ratio of moderator to fuel, p small

for pure unmoderated natural U, p = 0

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Criticality

from the original n thermal neutrons we have n x x ε x p thermal neutrons

some of the thermal neutrons are absorbed by non-fuel atoms

some are absorbed by 235U without fission (only 84% of thermal neutrons absorbed by 235U cause fission)

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Criticality

f = thermal utilization factor which is the fraction of all the thermal neutrons which are absorbed by the fuel (all of the U)

the total number of new neutrons produced by the original n thermal neutrons is n x x ε x p x f

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Criticality

f =aU

aU + aM + ap

aU = macroscopic cross section for UaM = macroscopic cross section for moderatorap = macroscopic cross section for other stuff

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Criticality

K = = = εpf

Nf+1

Nf

nεpfn

depends only on the fuel

ε, p and f depend on the composition and arrangement of the fuel

ε varies from 1.29 for unmoderated U to almost 1 for homogeneous dispersion of fuel and moderator

p is about 0.8 to 1 (for pure 235U, p = 1)

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in a nuclear reactor the factors are combined to produce a controlled, sustained chain reaction

excess reactivity (k) is an increase in the multiplication factor (k) above 1

k = k - 1

Reactivity and Reactor Control

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Reactivity and Reactor Control

For n neutrons in one generation, the number of additional neutrons in the next generation is nk

If the lifetime of a neutron generation is L sec, the time rate of change of neutrons is

dndt

nkL

=

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Reactivity and Reactor Control

dndt

nkL

=

when integrated from no to n we get

The reactor period (T) is the time during which the neutrons (power level) increase by factor of “e”

nno

kL= e

t

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Reactivity and Reactor Control

or the reactor period is:

The mean lifetime of a neutron (birth to absorption) in pure 235U is about 0.001 sec

kL =

1T

kL

T =

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Reactivity and Reactor Control

The mean lifetime of a neutron (birth to absorption) in pure 235U is about 0.001 sec

Assume excess reactivity = 0.1% (k = 0.001)

0.0010.001

T = = 1 sec

And the power level would increase by 2.718 each second

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Reactivity and Reactor Control

0.0050.001

T = = 0.2 sec

If k is increased to 0.5%

The power level increase each second would be

tTn

no

= e = e = 15010.2

This would be hard to control

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Reactivity and Reactor Control

The actual mean generation time in a reactor is much greater than 0.001 because 0.6407% of fission neutrons are delayed from 0.3 seconds to 80 seconds. This delay permits the reactor to be controlled.

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Reactivity and Reactor Control

Group ni (%) Ti (sec) ni x Ti

1 0.0267 0.33 0.009

2 0.0737 0.88 0.065

3 0.2526 3.31 0.836

4 0.1255 8.97 1.125

5 0.1401 32.78 4.592

6 0.0211 80.39 1.688

ni = 0.6407 nixTi = 8.315

Delayed Neutrons

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Reactivity and Reactor Control

The mean generation time for all fission neutrons is

If k 0.006407 the reactor is prompt critical – the reaction can be sustained by prompt neutrons alone

If k < 0.006407 the reactor is delayed critical – the delayed neutrons are needed to sustain the reaction

T = = = 0.084 secniTi

ni

8.315 + (99.359 x 0.001)100

delayed prompt

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Sample Calculation

What is the reactor period and the increase in power level in 1 second for a generation time of 0.084 sec for excess reactivity (k) of 0.1% and 0.5%

For k = 0.001

0.0010.084T = = 84 sec

1

84nno

= e = 1.012

For k = 0.005

0.0050.084T = = 16.8 sec

1

16.8nno

= e = 1.06

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Reactivity and Reactor Control

Excess reactivity is measured in units of “dollars” and “cents” (one dollar = 100 cents) and “inhours” (inverse hours)

One “dollar” worth of reactivity will cause the reactor to go prompt critical

One “inhour” is the amount of excess reactivity which results in a reactor period of 1 hour

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Fission

Control of Fission

Fission typically releases 2-3 neutron (average 2.5)

One is needed to sustain the chain reaction at a steady level of controlled criticality

The other 1.5 leak from the core region or are absorbed in non‑fission reactions

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Fission

Control of Fission

Boron or cadmium control rods absorb neutrons

When slightly withdrawn the number of neutrons available for fission exceeds unity and the power level increases

When the power reaches the desired level, the control rods are returned to the critical position

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Fission

Control of Fission

ability to control the reaction is due to presence of delayed neutrons

without delayed neutrons, change in the critical balance of the chain reaction would lead to a virtually instantaneous and uncontrollable rise or fall in the neutron population

safe design and operation of a reactor sets strict limits on departures from criticality

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Fission

Control of Fission

fission neutrons initially fast (energy above 1 MeV)

fission in 235U most readily caused by slow neutrons (energy about 0.02 eV)

moderator slows fast neutrons by elastic collisions

For natural (unenriched) U only graphite and “heavy” water suitable moderators

For enriched uranium “light” water may be used

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Fission

Control of Fission

commercial power reactors are designed to have negative temperature and void coefficients

if temperature too high or excessive boiling occurs rate of fission, and hence temperature, are reduced

238U absorbs more neutrons as the temperature rises, pushing neutron balance towards subcritical

steam within water moderator reduces its density which pushes neutron balance towards subcritical

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Fission

Control of Fission

fuel gradually accumulates fission products and transuranic elements which increases neutron absorption (control system has to compensate)

after about three years, fuel is replaced due to: build‑up in neutron absorption metallurgical changes as a consequence of the

constant neutron bombardment

fuel burn‑up is effectively limited to about half of the fissile material

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Fission Summary

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In 1920 Arthur Eddington suggested that the energy of the sun and stars was a product of the fusion of hydrogen atoms into helium

In the core of the sun at temperatures of 10‑15 million degrees Celsius, hydrogen is converted to helium

Since the 1950's, great progress has been made in nuclear fusion research: however, the only practical application of fusion technology to date has been the "hydrogen" or thermonuclear bomb

Fusion

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Fusion has an almost unlimited potential

The hydrogen isotopes in one gallon of water have the fusion energy equivalent of 300 gallons of gasoline

A fusion power plant would have no greenhouse gas emissions and would not generate high level radioactive waste

Experts predict the world is still at least 50 years and billions of dollars away from having fusion generated electricity largely due to the enormous size and complexity of a fusion reactor

Fusion

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Hydrogen atoms merged to create helium

Helium mass is slightly less (1%) than the original mass with the difference being given off as energy

Rather than using hydrogen atoms, it is easier to promote fusion by using two isotopes of hydrogen, deuterium and tritium

Deuterium is a naturally occurring isotope of hydrogen which has one extra neutron

One hydrogen atom in 6700 occurs as deuterium and can be separated from the rest

Fusion

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Fusion

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FusionA 56

FissionA 56

Atomic Mass Number0 25020015010050

0

-2

-4

-6

-8

-10

En

erg

y p

er N

ucl

eon

(M

eV)

Fusion

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Tritium is very rare because it is naturally radioactive and decays quickly

Tritium can be made by bombarding the naturally occurring element lithium with neutrons

Tritium could be created by having a "blanket" made of lithium surrounding a fusion containment vessel (this would result in a breeder reactor)

Fusion can only be accomplished at temperatures typical of the centre of stars, (~100 million degrees Celsius)

Fusion

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The fusion components exist in the form of a plasma, where atoms are broken down into electrons and nuclei

No known solid material could withstand the temperatures involved in nuclear fusion, so that a powerful confinement system is required to keep the plasma away from the walls of the vessel in which it is contained

Fusion

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Deuterium can be extracted from water (If all the world's electricity were to be provided by fusion, deuterium would last for millions of years)

Tritium does not occur naturally and will be manufactured from lithium within the machine

Lithium, the lightest metal, is plentiful in the earth's crust (if all the world's electricity were to be provided by fusion, known reserves would last for at least 1000 years)

Even though fusion occurs between Deuterium and Tritium, the consumables are Deuterium and Lithium

Fusion Fuels

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For example, 10 grams of Deuterium which can be extracted from 500 litres of water and 15g of Tritium produced from 30g of Lithium would produce enough fuel for the lifetime electricity needs of an average person in an industrialised country

Fusion Fuels

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Fusion research was big news in 1989 when it was reported that scientists had achieved fusion at room temperatures with simple equipment

Unfortunately, the scientists involved could not prove their claims and the experiments were not repeatable

Currently, two methods of confining the hot plasma are being studied around the world, "magnetic confinement" and "inertial confinement"

Current Fusion Research

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Magnetic confinement has the greatest potential and most research is now based on the "TOKOMAK" system (Tokomak is an acronym for the Russian words "torroidal magnetic chamber“)

The tokomak system was developed in the former U.S.S.R. and has been taken to an advanced stage in the ITER project whose construction began in 2007

A torroidal magnetic chamber is a doughnut-shaped steel structure in which the fusion plasma is confined by means of powerful coils of super‑conducting material which create a strong magnetic field

Current Fusion Research

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The other method is inertial confinement (small amounts of a deuterium‑tritium mixture are rapidly heated to extremely high temperatures with a high powered laser beam or a beam of charged particles)

Very high-power lasers are needed in the inertial confinement method. Biggest test facility is NIF. Less advanced than magnetic confinement.

To be useful the energy produced must be many times that required to sustain the reaction

Even the most optimistic researchers feel this will not be achieved before second half of the century

Current Fusion Research

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A fusion reactor capable of generating 1000 MW of electricity would be very large and complex

While fission reactors can be made small enough to be used in submarines or satellites, the minimum size of a fusion reactor would be similar to that of today's largest commercial nuclear plants

The difficult part is creating a sustainable fusion reaction - capturing the energy to generate electricity is very similar to a fission reactor

A 1000 MW fusion generator would consume only 150 kg of deuterium and 400 kg of lithium annually

Fusion Power Plants

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The fuels required for fusion reactors, deuterium and lithium, are so abundant that the potential for fusion is virtually unlimited

Oil and gas fired power plants as well as nuclear plants relying on uranium will eventually run into fuel shortages as these non‑renewable resources are consumed

Unlike fossil plants, fusion reactors have no emission of carbon dioxide (contributor to global warming) or sulphur dioxide (cause of acid rain)

Advantages of Fusion

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Barriers to the widespread use of nuclear power have been public concern over operational safety, and the disposal of radioactive waste

Accidents such as Chernobyl are virtually impossible with a fusion reactor because only a small amount of fuel is in the reactor at any time

It is also so extremely difficult to sustain a fusion reaction that, should anything go wrong, the reaction would invariably stop

Advantages of Fusion

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Long lived highly radioactive wastes are generated by conventional nuclear plants

Radioactive wastes generated by a fusion reactor are the walls of the vessel exposed to neutrons

Although the quantity of radioactive waste produced by a fusion reactor might be slightly greater than that from a conventional nuclear plant, the wastes would have low levels of short lived radiation, decaying almost completely within 100 years.

Advantages of Fusion

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The major disadvantages of nuclear fusion are the vast amounts of time and money which will have to be expended before any electricity is generated, even assuming that the technical/materials problems can eventually be solved

Development of other electricity supply technologies such as photovoltaic cells, which convert sunlight directly into electricity, could potentially eliminate the need for fusion before it is operational

Disadvantages of Fusion

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Where to Get More Information

Cember, H., Johnson, T. E., Introduction to Health Physics, 4th Edition, McGraw-Hill, New York (2008)

Martin, A., Harbison, S. A., Beach, K., Cole, P., An Introduction to Radiation Protection, 6th Edition, Hodder Arnold, London (2012)

Glasstone, S., Sesonske, A., Nuclear Reactor Engineering, 4th Edition, Dordrecht:Kluwer Academic Publishers (1995)

Friedberg, J., Plasma Physics and Fusion Energy, Cambridge University Press, Cambridge (2007)