1 Fission & the Neutron Cycle

8/3/2019 1 Fission & the Neutron Cycle

http://slidepdf.com/reader/full/1-fission-the-neutron-cycle 1/59

2011 September 1

1: Fission & The Neutron Cycle

B. Rouben

McMaster University

Course EP 4D03/6D03Nuclear Reactor Analysis

(Reactor Physics)

2011 Sept.-Dec.



2011 September 2

Table of Contents

Multiplying medium Neutron reactions with matter

Fission

Neutron density, neutron flux, neutron current Reaction rates

Reactor multiplication constant, reactivity, critical

mass

Thermalizing neutrons, Maxwellian distribution

Neutron cycle



2011 September 3

Go Forth and Multiply!

Interactive Discussion/Exercise: What is a “multiplying medium” in reactor

physics?

What is being “multiplied”?

What is the basic criterion for a medium to be

multiplying?

Is there a one-to-one relationship between the

concepts of “multiplying medium” and“criticality”?



2011 September 4

Multiplying Medium

Multiplying medium: A material or environment inwhich fissionable nuclides are present, i.e., where

neutrons can induce fission, and thereby be

“multiplied”.

Note: The degree of multiplication is not at issue.

In other words, “multiplying medium” does not

necessarily signify criticality. Unlike rabbit

populations, neutron populations in a multiplyingmedium need not be growing, or even be self-

sustaining.



2011 September 5

Spontaneous and Neutron-Induced Fission

Question: Does spontaneous human combustion exist?

Answer: I doubt it, but don’t really know. Question: Does spontaneous fission exist? Answer: Yes.

Nuclei of uranium sometimes split spontaneously,releasing energy.

However, this happens with very low frequency: Thehalf-lives of U-235 and U-238 are 7.038*108 years and4.68*109 years respectively, and most of their decay isby alpha emission, so spontaneous fission is not apractical source of energy.

But fission can be made into a practical source of energywhen induced by a neutron collision with a nucleus of uranium (or a nucleus of a few other heavy elements).

Discussion/Exercise: From general knowledge, howmany things can you list about what happens in neutron-

induced fission?



2011 September 6

Neutron Reactions with Matter

Scattering: the neutron

bounces off, with or withoutthe same energy (elastic orinelastic scattering – what isthe difference?)

Activation: the neutron is

captured, & the resultingnuclide is radioactive, e.g.

16O(n,p)16N

10B(n,)7Li Radiative Capture: the

neutron is captured and agamma ray is emitted from stainless steel

40Ar(n,)41Ar

Fission (follows absorption)

electron neutron

proton

Incident neutron, E1

Scattered neutron, E2

a EA

E1 = E + E2

Inelastic Scattering:

electron neutron

proton Elastic Scattering:

Incident neutron, E1

Scattered neutron, E2

E1 = EA + E2

electron neutron

proton

Neutron Absorption:

Incident thermal neutron, E

Gamma Photon, E

Gamma Photon, E E ~ 7 MeV



Neutron Absorption in Nuclear Fuel

When a neutron is absorbed in a fuel nuclide, the2 most important (although not the only)

consequences which can follow are neutron

capture and fission. The competition between neutron capture and

fission, along with the neutron reactions with

other materials in the reactor, determines

whether the fission chain reaction can be self-

sustaining.

2011 September 7



2011 September 8

A neutron splits a

uranium nucleus,releasing energy (quickly

turned to heat) and more

neutrons, which can

repeat the process.

The energy appears

mostly in the kinetic

energy of the fission

products and in the beta

and gamma radiation.

(neutron-induced)



2011 September 9

Outcome of Neutron-Induced Fission Reaction

Energy is released (a small part of the nuclear mass isturned into energy).

One neutron enters the reaction, 2 or 3 (on the average)

emerge, and can induce more fissions.

This chain reaction can be self-perpetuating (“critical”) if at least one of the neutrons released in fission is able

to induce more fissions.

By judicious design, research and power reactors can be

designed for criticality; controllability is also important. This is the operating principle of fission reactors.



Energy from Fission

Neutrons + U (or other fissionable nuclide) Fission (sometimes)

Fission Rate * 200 MeV/fission Power Rate

Number of neutrons Fission rate

Control reactor power by controlling number

of neutrons (or fission rate)

2011 September 10



Production of Plutonium

Not all absorptions in nuclear fuel produce afission. Some absorptions end up in a neutron

capture.

Neutron capture in 238U gives 239Pu, which is a

fissile nuclide.

Thus 239Pu is created in the fuel, and it

participates in the fission chain reaction.

Neutron capture in 239Pu gives 240Pu (non-fissile), and neutron capture in 240Pu gives 241Pu

(which is fissile and participates in fission).

2011 September 11



A Nuclear Generating Station

2011 September 12



Components of a Nuclear Plant

What are the basic components of a nucleargenerating station?

They consist of the nuclear reactor and the

Balance of Plant. The reactor must contain:

Nuclear fuel

Coolant (Heat-Transport System) Moderator (in thermal reactors only)

Control and Shutdown Mechanisms cont’d

2011 September 13



Components of a Nuclear Plant

The Balance of Plant must contain: One or more Steam Generators (Boilers) to turn

water into steam (unless the primary coolant is

turned into steam in the reactor itself, and unless a

gas coolant is used)

A Turbine-Generator to turn mechanical energy

into electricity

Connections to the outside electrical grid.

2011 September 14



Reactor Components

Nuclear fuel/Multiplying medium: Only very heavy

nuclei are fissionable; these are isotopes of uranium and

of plutonium and other transuranics.

Some nuclides can be fissioned by neutrons of any

energy; these nuclides are called fissile; e.g.,235

U,239

Pu,241Pu, 233U. Note: 235U is the only naturally occurring

fissile nuclide.

Fissionable but non-fissile nuclides, e.g., 238U, can be

fissioned by neutrons of energy greater than somespecific threshold. cont’d

2011 September 15



Reactor Components (Cont’d)

Fissile nuclides are easier to fission than non-fissile nuclides, and furthermore the fission cross

section of fissile nuclides is much much greater

for slow (thermal) neutrons. Therefore it ismuch easier to build a reactor which relies on

fissions induced by thermal neutrons.

Such a reactor is called a thermal reactor. It

requires a moderator, which is a light material,

with atoms of low mass number, used to slow

neutrons down to thermal energies. cont’d

2011 September 16



Reactor Components (Cont’d)

Fission processes transform some small fractionof the mass of the fuel to energy (E = mc2).

In a nuclear reactor, most of this energy is turned

very quickly into heat (random kinetic energy). Therefore a coolant is required to take away the

heat and turn water into steam to feed the

turbine-generator.

Finally, any reactor needs control mechanisms to

control the fission chain reaction. Some reactors

have independent shutdown systems.

2011 September 17



Intensity of a Neutron Beam

Consider the concept of a neutron beam, i.e., a number of

neutrons all moving in the same direction towards a target

of some material.

The intensity I of the beam represents the number of

neutrons crossing a unit area in a plane perpendicular tothe beam direction per unit time.

Typical units for I are neutrons.cm-2.s-1.

If the “density of neutrons in the beam is n neutrons.cm-3

and we imagine them all to be travelling at the samespeed v, i.e., the beam is monoenergetic, then it is easy to

see (figure next slide) that the neutrons crossing the area

per s will be those within a distance v from the target, i.e.,

I = nv.2011 September 18



Intensity of a Neutron Beam

2011 September 19

Unit Areaof Target

Density of neutrons in beamis n per cm3

Speed of neutrons = v

All neutrons within adistance (v*1 s ) will crossthe area within 1 s,

i.e., I = nv



Reaction Rate

Recall (from nuclear physics) the concept of macroscopic cross section (units cm-1) - for a

given reaction type.

This is the probability of reaction of 1 particle in

the beam (1 neutron here) with nuclides of the

target per distance travelled into the target (note:

this really applies to infinitely small distances).

Since the intensity I counts all the neutrons in thebeam and the distance they travel per s, we can

see that the total rate R of reactions (of the type

considered) will be R = I (reactions.cm-3

.s-1

)2011 September 20



2011 September 21

Review: Macroscopic & Microscopic Cross Sections

Macroscopic cross section for a neutron reaction of type i

with a certain material (e.g., scattering, absorption, orfission), i: The probability per unit distance of travel that

a neutron will undergo a reaction of type i - units are

inverse length (e.g., cm-1).

Microscopic cross section, i: Effective area for reaction of

type i which seems to be presented to the neutron by 1

nucleus of the material – units are area

(e.g., barn (b) = 10-24 cm2, or kb = 103 b) Relationship between microscopic and macroscopic cross

sections: i = Ni

where N = number of atoms of the material per cm3

’



2011 September 22

Neutron Beam Impinging on a Slice of Material

This is the microscopic

cross section

This is the macroscopic

cross section. It takes

account of the density

of nuclides in the

material.



2011 September 23

Macroscopic & Microscopic Cross Sections (cont’d)

The microscopic cross section is a basic physical

quantity which is determined by experiments of neutronbeams of various energies on target materials.

Once is known, then the macroscopic cross section can be obtained from N and .

Both and depend on the material, the neutron energyor speed, and the type of reaction.

The cross sections for scattering, absorption, fission aredenoted by a subscript s, a, f, e.g. s, a, and f respectively.

The total cross section tot measures the total number of all types of reaction per unit distance: tot = s + a . . . (Note that the fission cross section is included in the

absorption cross section, since it occurs following aneutron absorption.)



2011 September 24

Neutron Flux

Now say that you do not have a beam of neutrons, but that

you have a neutron density n (number of neutrons/unit

volume), all moving at speed v in different directions.

Consider each neutron as if it is in a “beam” of its own, of

intensity 1*v. Imagine “adding up” the intensity of all these beams -

even if they are not parallel; then the total “beam

intensity” is still I = nv neutrons.cm-2.s-1.

The reason that it makes sense to add the intensities thisway, even if the areas that the neutrons are crossing are at

different angles, is that the nuclides don’t really care from

which direction the neutrons are coming.



2011 September 25

Neutron Flux

The neutron flux for speed v is denoted f( v) and is

defined as the total intensity of all these disparate beams,

i.e., f (v) = nv

If the neutrons have different speeds (energies), then wecan define a total flux

(Or, if we are interested in only a range of neutron

energies, we can customize the range of integration.)

(

0

dvvf f



Neutron Flux

An equivalent way to define the neutron flux isto visualize an arrow associated with eachneutron. The arrow shows the direction of motion of the neutron, and its length denotes the

neutron’s speed. The sum of all the arrow lengths is the flux f

(see figure in next slide). It is also the sum of the distances (path lengths)

which would be traversed by the neutrons perunit time.

A flux f has units of neutrons.cm-2.s-1, also

abbreviated as of n.cm-2

.s-1

. 2011 September 26



Neutron Flux

2011 September 27

Unit

Volume

Total flux f =sum of allarrow lengthsin unit volume



Reaction Rate

The Reaction Rate R of neutrons with the nuclidesof the material, for a given reaction type, is a very

important quantity.

Since the nuclides don’t care about the direction of

motion of the neutrons, then as shown for a beam

of neutrons of speed v, R is given by:

R(v) = (v)f (v), where (v) is the material’s

macroscopic cross section for neutrons of speed v. If the neutrons are not monoenergetic, then the

total reaction rate is

2011 September 28

( (

0

dvvv R f



Energy Instead of Speed

It is important to remember that in any and all of the treatment in the previous slides, neutron

energy E can be used as the independent variable

instead of the neutron speed v, since these two

quantities are directly related to one another by

2011 September 29

2

2

1mv E



Neutron-Production Rate

If the average number of neutrons produced in a fission

is (don’t confuse this with neutron speed), we can

define a new quantity, the “production” (or “yield”)

cross section f ( E , r ).

ThenProduction rate of neutrons at r = f ( E , r )f ( E , r )

This can also be called the “volumetric source” of

neutrons.

The total neutron production rate in the reactor can beobtained by integrating the above quantity over r .

It is of course important to distinguish between

fission rate and yield rate (volumetric source).2011 September 32



2011 September 33

Note on Calculating Reaction Rates

To calculate reaction rates, we need the macroscopic cross

sections and the neutron flux.

These are calculated with the help of computer programs:

The cross sections are calculated from international databases

of microscopic cross sections

The neutron flux distribution in space (the “flux shape”) iscalculated with specialized computer programs, which solve

equations describing the transport or diffusion of neutrons

[The diffusion equation is an approximation to the more

accurate transport equation.] The product of these two quantities (as per previous slides) gives

the distribution of reaction rates, but the absolute value of the

neutron flux is tied to the total reactor power.



2011 September 34

General Definition of Neutron Flux(es)

The most general neutron flux is the angular flux,

i.e., the product of neutron density n and speed v.

By summing this product over various subsets of energies E and/or

directions , we get various of the following:

Angular flux for 1 energy and 1 direction of motion [Eq.(1)] Total flux over all angles, for 1 energy E: [integrate Eq.(1) over

angle ] (this can be thought of as summing neutron beams over

all directions)

Total angular flux over all energies: [integrate Eq. (1) overenergy E]

Total angular flux or total flux over a range of energies (e.g.,

“thermal” flux, “fast” flux, etc.)

Total flux over all energies

( ( )1(,,,, v E r n E r f



2011 September 35

New Concept: Neutron Current

If you make a vector from the angular flux bymultiplying it by the unit vector of the direction of

motion , and then (vector) “add”, i.e., integrate, that

product over all directions, you get a new, vector,

quantity, the neutron current:

Just as for the flux, the current is defined for a neutron

energy (or speed). Or it can be integrated over a range

of energies, or over all energies (in the latter case, to

get the total current).

( )2(,,),( d E r E r J f



2011 September 36

Components of Current

Because the current is a vector, we can alsofocus on components of the vector in any

direction, for instance the components J x, J y, J z,

along the positive x, y, and z axes.

We can also focus on partial currents, i.e.,

partial components of the current, e.g., J z+ and

J z- would be the components of J z for directions

of motion pointing generally along +z and – z

respectively, so that, for instance

J z = J z+ + J z-



2011 September 37

Difference Between Flux & Current

Flux is a scalar quantity; current a vector quantity. Since the “unit volume” over which flux is defined can be

as small as we like, we can think of flux as the totalnumber of particles emerging from, or, more usefully,converging to a point (or single nuclide), per unit area perunit time. We just add up all the arrows.

But to calculate the number of neutrons crossing a unitarea of a given plane at a given solid angle (i.e., thecurrent crossing that unit area at an angle ), we need to

take into account the angle between the direction andthe plane, i.e., the dot product of with the normal to theplane, just as we do when we consider the number of raysof sunlight warming a given area on the earth’s surface(see next slide).



Aside on Mathematics Review

I would like to take an aside right now to do areview of some basic mathematics in different

co-ordinate systems, as we will need to consider

these in our analysis of reactors of various

geometries.

To “Mathematics Review”

2011 September 38



Reactor Multiplication Constant

Several processes compete for neutrons in a nuclearreactor: “productive” absorptions, which end in fission “non- productive” absorptions (in fuel or in structural

material), which do not end in fission leakage out of the reactor

Self-sustainability of the chain reaction depends onrelative rates of production and loss of neutrons.

The self-sustainability of the fission reaction in the finite

reactor is measured by the reactor multiplicationconstant:

)( leakagesabsorptionlossneutronof Rate

productionneutronof Ratek eff

2011 September 39



Reactor Multiplication Constant

Another common definition of k eff is: ratio of number of

neutrons born in one “generation” to number born in theprevious generation. It can be shown that the twodefinitions of k eff are equivalent.

Three possibilities for k eff : k eff < 1: Fewer neutrons being produced than lost.

Chain reaction not self-sustaining, reactoreventually shuts down. Reactor is subcritical.

k eff = 1: Neutrons produced at same rate as lost.

Chain reaction exactly self-sustaining, reactorin steady state. Reactor is critical.

k eff > 1: More neutrons being produced than lost.Chain reaction more than self-sustaining,

reactor power increases. Reactor is supercritical.2011 September 40



Critical Mass

Because leakage of neutrons out of the reactor increases as

the reactor size decreases, the reactor must have a minimumsize for criticality.

Below minimum size (critical mass), leakage is too high andk eff cannot possibly be equal to 1.

Critical mass depends on: shape of the reactor composition of the fuel other materials in the reactor.

The shape with lowest relative leakage, i.e. for which critical

mass is least, is the shape with smallest surface-to-volumeratio: a sphere.

All these concepts will be made more concrete as we goalong in the course.

2011 September 41



2011 September 43

Units

The multiplication constant k eff is a ratio of like

quantities, therefore it is a pure number and has no units. Similarly, the reactivity r is also a pure number, and has

no units. However, because in reactor physics we seldom deal

with k values extremely different from 1, reactivity isoften written in “units” of a small fraction of unity. Twoof the common units are 1 milli-k (or mk) 0.001

1 pcm 10-5

= 0.01 mk Thus, e.g., a reactivity of +3 mk means r = +0.003, and

a reactivity of -50 mk means r = -0.050 1 mk may seem small, but one must consider the time

scale on which the chain reaction operates.

Fi i N



2011 September 44

Fission Neutrons

Neutrons from fission

have a distribution of

energies, mostly in the

1-to-a-few-MeV range.

This distribution has amaximum at ~1 MeV

(neutron speed ~13,800

km/s!).

Energy Distribution of Fission Neutrons

F d Th l N



2011 September 45

Fast and Thermal Neutrons

• Neutrons may lose energy by collisions with

materials in the reactor, i.e., they may be slowed

down.

• Maximum slowing down is to a distribution of

energies in thermal equilibrium with the ambientenvironment.

• For a temperature of 20o C, these “thermal”

energies are of order of 0.025 eV, i.e. neutronspeed = 2.2 km/s, orders of magnitude slower than

fast neutrons, but still ~ speed of bullet.

Wh Th li N ?



2011 September 46

Why Thermalize Neutrons?

Because they can be fissioned by neutrons of any

energy, 235U and 239Pu are the main fissioning

nuclides in most reactors [although 235U is in

small abundance (0.72% of natural uranium)], and239Pu is created in the reactor].

The probability of a neutron inducing fission in a

fissile nuclide is very much greater (by orders of

magnitude) for very slow neutrons than for fastneutrons (see next Figure).

Schematic View of a Typical Cross Section,



2011 September 47

Schematic View of a Typical Cross Section,

Showing Resonances

Th l R



2011 September 48

Thermal Reactors

Thermal reactors are designed to take advantage of the

much greater probability of inducing fission of fissilenuclides at low neutron speeds.

Therefore, a moderator is used to thermalize neutrons. However, as neutrons slow down from ~MeV energies to

thermal energies, they may be absorbed in fuel. In the “resonance” energy range, ~ 1 eV - 0.1 MeV, the

probability of non-productive capture in fuel is great.Capture resonances “rob” neutrons from the chainreaction.

Therefore, ways are sought to minimize resonancecapture. Lumping the fuel into channels separates themoderator from the fuel and helps to reduce resonancecapture.



Speed Distribution in a Population of Particles

In a population of free particles, there will be a

distribution of particle speeds (or energies).

The temperature of a gas is a measure of the

energy of motion of the gas molecules.

The molecules are flying around at various

speeds, and exchanging energy in collisions.

Maxwell showed that in a gas at steady

temperature T (on the absolute, Kelvin scale), the

molecules will have a distribution of speeds

given by a function which he derived.

cont’d 2011 September 49



The Maxwellian Distribution of Speeds

Let m be the mass of the particles and T (K ) be

the absolute temperature.

If we write the fraction of particles with speed in

an interval dv about speed v as n(v)dv, then the

Maxwellian distribution of speeds is:

2011 September 50

(

0

23

22

2 / 3

)1:(

/ 10*380658.1tan

)(24

)(

2

dvvnthat Show Exercise

K J t cons Boltzmannk where

slidenext graphseedvevm

kT dvvn kT

mv

M lli Di t ib ti i



Maxwellian Distribution in v

2011 September 51

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

n(v)

n(v)

v (m/s)



The Maxwellian Distribution of Energies

The kinetic energy E of a particle is related to itsspeed v by E = mv2 /2. The Maxwellian

distribution can therefore be (and often is)

written in terms of E instead of v.

(See the graph on the next slide)

2011 September 52

( kT E e E kT

mv

vn

dvdE

vn

dE

dvvn E n

exercisederivecanwedvvndE E nwritewe If

/ 2 / 12 / 32)(

/

)()()(

)!(,)()(

M lli Di t ib ti i E



Maxwellian Distribution in E

2011 September 53

0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01

n(E)

n(E)

E (eV)



Maxwellian Distribution for Thermal Neutrons

In a nuclear reactor, neutrons are “thermalized”

by the moderator, i.e., they are slowed down to

the “thermal” energy range (<~ 0.6 eV), where

they are in thermal equilibrium with the ambient

environment. As a consequence, thermal neutrons in a reactor

are almost in a Maxwellian distribution – the

distribution is slightly perturbed by theabsorption of neutrons in the fuel.

2011 September 54



2011 September 55

Exercises in Thermal-Neutron Distribution

Exercises: From the form of the Maxwellian in v and in E ,

show that the most probable kinetic energy in the

Maxwellian distribution is E peak =kT/2 (≈ 0.0125 eV at room temperature),

Show on the other hand that the peak speed

v peak corresponds to an energy of kT (≈0.025 eV atroom temperature), and that this gives a value

v peak = 2,200 m/s (not that slow, faster than a bullet)

N t C l



2011 September 56

Neutron Cycle

The next slide illustrates the cycle of neutrons in fission, slowing down,

absorptions and leakage in a thermal

reactor.

We will come back to this cycle more

quantitatively as we go on in the course.

N t C l i Th l R t



2011 September 57

Neutron Cycle in Thermal Reactor

Neutrons Born in Fission

Fast-Neutron Leakage Fast Fissions

Neutrons Slowing Down

Neutrons Captured in

Fuel Resonances

Thermal-Neutron Leakage Non-Productive Thermal-

Neutron Absorptions in

Fuel & Other Materials

Thermal Fissions

E i



2011 September 58

Exercise

From anything you knew previously, or anythingyou think you knew from general knowledge, or

anything that you have learned from this learning

module:

List as many things which are important in

running a nuclear plant as you can think of, and

for each item on the list indicate whether that

item is closely, not so closely, or not at all,related to reactor physics.



END

1 Fission & the Neutron Cycle

Documents

Transcript of 1 Fission & the Neutron Cycle