2: Fission and Other Neutron Reactions
B. Rouben
McMaster University
Course EP 4D03/6D03
Nuclear Reactor Analysis
(Reactor Physics)
2015 Sept.-Dec.
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Contents
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Concepts:
 Fission and other neutron reactions
 Beam Intensity
 Cross Sections
 Reaction rates
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Go Forth and Multiply!
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Nuclear reactors exploit the neutron-induced
fission chain reaction to release heat energy.
Neutrons are the agents of the reaction.
We need to know what neutrons can do and
what they’re doing.
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Neutron Interactions with Matter
Inelastic Scattering:
Scattered neutron, E2
elec tron
neutron
Incident neutron, E1
p roton
Gamma Photon, E
E1 = E + E2
Scattered neutron, E2
Elastic Scattering:
elec tron
neutron
p roton
Incident neutron, E1
a EA
E1 = EA + E2
Gamma Photon, E
E  ~ 7 MeV
Neutron Absorption:
elec tron
neutron
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Incident thermal neutron, E
p roton
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Neutron Interactions with Matter
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Scattering
The neutron bounces off, with or without the same
energy (elastic or inelastic scattering)
Absorption
Following absorption these outcomes can happen:
Activation: the neutron is captured, & the resulting
nuclide is radioactive, e.g.
10B(n,)7Li [n in, followed by alpha decay]
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16O(n,p)16N [n in, followed by proton emission]
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Special Case, Radiative Capture - decay by gamma emission:
e.g., 238U(n,)239U or 40Ar(n,)41Ar
Fission (happens only with some heavy nuclides)
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The nuclide splits into 2 lighter nuclides and releases energy
Neutrons are also released which continue the chain reaction
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(neutron-induced)
A neutron splits a
uranium nucleus,
releasing energy (quickly
turned to heat) and more
neutrons, which can
repeat the process.
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The energy appears
mostly in the kinetic
energy of the fission
products and in the beta
and gamma radiation. 6
Spontaneous and Neutron-Induced Fission
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Question: Does spontaneous human combustion exist?
Answer: I doubt it, but I 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: The
half-lives of U-235 and U-238 are 7.038*108 years and
4.68*109 years respectively, and most of their decay is
by alpha emission, so spontaneous fission is not a
practical source of energy.
But fission can be made into a practical source of energy
when induced by a neutron collision with a nucleus of
uranium (or a nucleus of a few other heavy elements).
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Multiplying Medium
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Multiplying medium: A material or environment in
which fissionable nuclides are present, i.e., where
neutrons can induce fission, and thereby be
“multiplied”.
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Outcome of Neutron-Induced Fission Reaction
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Energy is released (a small part of the nuclear mass is
turned into energy).
One neutron enters the reaction, 2 or 3 (on the average)
emerge, and can induce more fissions.
The process has the potential of being a chain reaction;
this can be self-perpetuating (“critical”) under certain
conditions.
By judicious design, research and power reactors can be
designed for criticality; controllability is also important.
The energy release is open to control by controlling the
number of fissions.
This is the operating principle of fission reactors.
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Outcome of Neutron-Induced Fission Reaction
~200 MeV of energy is eventually released
per fission
 There is only a small variation in this
quantity, depending on which nuclide is
fissioning.
 Discussion: A fundamental law of physics is
that energy is always conserved. How can
energy be “released” in fission?
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Fission Process
The fission process occurs when the nucleus
which absorbs the neutron is excited into an
“elongated” (barbell) shape, with roughly half
the nucleons in each part.
 This excitation works against the strong force
between the nucleons, which tends to bring the
nucleus back to a spherical shape  there is a
“fission barrier”, ~ 6 MeV
 If the energy of excitation is larger than the
fission barrier, the two parts of the barbell have
the potential to completely separate: binary
fission!
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Radiative Capture and Fission
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If the neutron is absorbed by a very heavy
nuclide (as opposed to being scattered), the two
most important outcomes are:
1) (radiative) capture, where the neutron remains
in the nucleus and a gamma ray is emitted, and
2) fission
Whether there is a real potential for a selfsustained fission chain reaction to take place
depends on the relative probabilities of these
outcomes (“cross sections”), and on the number
of neutrons emerging.
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Concepts of Cross Sections
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To understand the concepts of
macroscopic and microscopic cross
sections, let us first start by considering
neutron beams interacting with targets.
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Intensity of a Monoenergetic Beam
Consider a monoenergetic beam of neutrons: All have the
same speed  and all are moving in the same direction.
Density of neutrons
in monoenergetic
beam = n cm-3
Unit Area
of Target
Speed of neutrons =  [cm/s]
The Beam Intensity I is defined as the number of
neutrons crossing a unit area of the target per unit time.
All neutrons within a distance (*1 s) will cross the target
within 1 s,  I = n [units = n.cm-2.s-1 - or cm-2.s-1]
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Neutron Beam Impinging on a Slice of Target
The monoenergetic beam of intensity I is impinging on a target. Let  be
the “effective area of interaction” presented by a single nucleus to a neutron
(not necessarily the geometric area of the nucleus!).  is called the
miscroscopic cross section; it has units of area, e.g. cm2. Consider a thin
slice of target of unit area and infinitesimal thickness dx ( of volume V =
1*dx=dx).
 = area
presented to
neutron by 1
nucleus
Neutron Beam
(area 1 cm2)
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Thin (dx) slice of target
Density of atoms (nuclei)
in target = N cm-3
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Neutron Beam Impinging on a Slice of Target
Atomic density in target = N
Microscopic cross section of nucleus  
Area of beam and of target = 1
Differential thickness of target = dx
Volume of target = 1 * dx = dx
Bull’s Eye Area in slice = N dx
Beam intensity = I
Reaction Rate (per unit volume of target per s)
= I*N dx/dx = IN = I 
where  = N is called the macroscopic cross
section; it has units cm-1.
 can be identified as the probability of a
beam of unit intensity interacting with the
target per unit distance of beam travel.
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Area = 1 cm2
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Cross Sections and Reaction Rate
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, and , varies with:
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the nuclide
 the type of interaction i (scattering, absorption,
fission, etc.)
 the neutron speed 
If there are several types of nuclei, the ’s add:
i = N11,i+ N22,i + N33,i +…
For each type of interaction,
Reaction Ratei = Ii (1)
This is an all-important equation!
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Cross-Section Databases
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The ’s are obtained by experiment. They must be
measured (by aiming neutrons of various speeds at
various targets). They cannot be calculated from first
principles.
Therefore experimental databases of cross-section
values have been, and continue to be, developed by
international teams of experimenters.
“Evaluated Data Sets” (such as ENDF/B-VI) are sets of
cross-section values established over the ranges of
neutron energy important in reactor applications. They
are of crucial importance to reactor-physics
calculations.
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Energy Instead of Speed
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It is important to remember that neutron energy E
can be used instead of the neutron speed , since
these two quantities are directly related to one
another by
1
E  m 2
2
(in reactors neutron energies are not relativistic)
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Macroscopic & Microscopic Cross Sections
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Again: i = Ni (1), where N = atomic density = number of atoms
of the material per cm3
N can be obtained from the material’s density and atomic or
molecular weight. Remember or rederive the expression:
N0 
, where N 0  Avogadro' s number  0.6023 *10 24 ,
A
  material density , and A  atomic weight of material
N
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(use judiciously when the material is a composite or a solution!)
Note: When the material is composed of several types of nuclides,
denoted say by a subscript k, the macroscopic cross section must
be calculated taking all nuclides into account, i.e.
 i   N k  ik
(1)'
k
therefore the relative abundances of the nuclides, and all
microscopic cross sections, must be known.
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Macroscopic & Microscopic Cross Sections
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Both  and  depend on the material, the neutron energy
(or speed), and the type of reaction.
The cross sections for scattering, absorption, fission and
radiative capture are often denoted by subscript s, a, f, ,
e.g., s, a, f and  respectively.
The total cross section tot measures the total number of
all types of reaction per unit distance: tot = s + a
For nuclear fuels, only radiative capture and fission are
significant. Neglecting other reactions, then:
a = f +  (2)
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Important Quantities in Neutron Production
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Neutrons are produced in fission, but not all
absorptions result in fission.
The probability of radiative capture relative to
fission is given by the parameter
 = /f (3)
Important quantities which can be quoted when
we consider neutron production are:
 = Average number of neutrons per fission (4)
&  = Average number of neutrons per absorption (5)
  and  can be related using the various cross sections
and the relative probability α.
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Show that  
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1 
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Fission Chain Reaction
The chain reaction of neutrons in one
“generation” giving rise to neutrons in a next
generation can be self-perpetuating (“critical”)
under certain conditions.
 By judicious design, research and power reactors
can be designed for criticality; controllability is
very important.
 The energy release is open to control by
controlling the number of fissions.
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Fissionable and Fissile Nuclides
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Only a few nuclides can fission.
Fissionable nuclides are nuclides which can be induced to
fission, e.g., 238U, 235U, 239Pu, and 240Pu.
In some cases, e.g., 238U, the nuclide is fissionable only
by neutrons of energy higher than a certain threshold
A fissionable nuclide which can be induced to fission by
an incoming neutron of any energy is called fissile. There
is only one naturally occurring fissile nuclide: 235U.
Other fissile nuclides are 233U, isotopes 239Pu and 241Pu of
plutonium, and some isotopes of elements with still
higher atomic number. None of these is present in nature
to any appreciable extent.
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Plutonium
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There is no appreciable amount of plutonium in
nature, but plutonium is produced in the reactor by
absorption of neutrons in U-238.
Some of these absorptions result in U-239 betadecaying to Np-239, which then beta-decays to Pu239
Some Pu-240 is produced by neutron absorption in
Pu-239, and some Pu-241 is produced by neutron
absorption in Pu-240
Pu-239 and Pu-241 are very important because they
are fissile and therefore contribute to the chain
reaction
Pu-240 is not fissile, therefore it is mostly an absorber
of neutrons
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Fission Products
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The two large “fission fragments” in the sketch of fission
represent fission products, i.e., elements with roughly half
the mass of uranium.
Most of these elements are radioactive, and decay with
various half-lives (some very long). Irradiated nuclear fuel
must therefore be handled and disposed of very carefully.
Fission products absorb neutrons and therefore represent a
“negative” load on the chain reaction.
Also, the decay of fission products is a source of heat in the
fuel. Decay heat represents ~7% of the energy released in
the reactor, a not-insignificant fraction.
 Nuclear fuel must continue to be cooled even when the
reactor is shut down and also when fuel is discharged.
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Fission-Product Decay
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Discussion:
There are hundreds of fission products (different
nuclides). Why are most of them radioactive?
Hints:
What is the ratio of neutrons to protons in light nuclei
(e.g., 12C or 16O). What is it in heavy nuclei (U)? Why?
What is this ratio in the fission fragments? Why?
You would say that the fission fragments are neutron***. Can this ratio be sustained?
What are ways in which most fission products can
decay?
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END
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