Introduction to Nuclear Reactions
Dr. J. Michael Doster
Nuclear Engineering Department
North Carolina State University
Nuclear Reactions
•
Many types of nuclear reactions are possible and are usually represented
symbolically as
a + X → C*
C* → Y + b
•
with the net effect being
a + X →Y +b
•
•
As the compound nucleus disintegrates essentially the moment it is
formed, its effect can be neglected when considering most nuclear
reactions.
The residual nucleus (Y) may be stable, or radioactive and decay further
at a later time. The symbols a & b may stand for the neutron (n), gamma
ray (γ), alpha particle (α), proton (p), etc.
Conservation laws
•
•
•
The sum of the A values (number of neutrons plus protons) on both sides of the
equation are equal.
The sum of the Z values (nuclear charge or number of protons) is the same on both
sides of the equation.
Example:
Consider the reaction
4
14
He
+
2
7N
→ 178O+ 11H
Notice the A values are equal (18) and the Z values are equal (9). Abbreviated this
reaction would be written
14
17
7 N( α , p) 8 O
Note: the alpha particle is simply a helium nucleus and the proton is a hydrogen nucleus. The alpha particle,
being positively charged, is normally repulsed from the nitrogen nucleus by electrostatic forces and must
therefore be accelerated to very high energies for this reaction to occur. This is usually accomplished by the
use of accelerators.
Neutron Reactions
•
In contrast to charged particles (α, p, etc.), the neutron (a neutral
particle) need not overcome electrostatic barriers and therefore need not
have high kinetic energies to penetrate the target nucleus.
•
Neutrons of essentially zero energy can induce nuclear reactions in many
materials.
Example
Consider the reaction to produce the useful radioisotope Co-60
1
59
60
0 n+ 27 Co→ 27 Co +
γ
The conservation of mass-energy is a firm requirement for any valid nuclear reaction.
Compare the masses on each side of the equation.
1
0n
= 1.008665 amu (atomic mass units)
59
27 Co
= 58.9332 amu
60
27 Co
= 59.93344 amu
1
59
0 n+ 27 Co
= 1.008665 + 58.9332 = 59.941865 amu
Δm = 59.941865 - 59.93344 = 0.008425 amu.
•
This result would tend to indicate a net mass loss to the system, an
uncomfortable concept.
•
The "missing" mass has been converted to energy according to Einstein's
formula
E = mc 2 = 931 Mev/amu
and shows up as the kinetic energy of the γ ray. This kinetic energy is
given by
E = 931 Mev/amu x 0.008425 amu = 7.84 Mev.
Example
•
Consider again the reaction
4
14
17
1
2 He+ 7 N→ 8 O+1H
The associated masses of the constituent components are:
14
7N
4
2 He
17
8O
1
1H
= 14.00307 amu
= 4.0026 amu
= 16.99914 amu
= 1.007825 amu.
The mass balance yields
Δm = 14.00307 + 4.0026 - 16.99914 - 1.00782
Δm = -0.001295 amu
with an associated energy of
E = 931 x (-0.001295) = -1.20 Mev.
The negative sign implies energy must be supplied to the reaction for it to
go. This energy would normally be supplied by the kinetic energy of the
incident alpha particle.
Example
•
Consider next the compound nucleus formed by the absorption of a
neutron by Uranium-235.
1
235
236
0 n+ 92 U→ 92 U *
The appropriate masses are
235
92 U
= 235.043925amu
236
92 U
= 236.045562 amu
1
0n
= 1.008665amu
which gives a mass change of
Δm = 235. 043925 + 1. 008665 − 236. 045563 amu
= 0. 007027 amu
= 6.5 Mev .
•
This excess energy is called the
excitation energy. To release this
energy, one possible mechanism
is the emission of a 6.5 Mev
gamma ray, i.e.
236
236
U*
→
92
92 U + γ .
•
A second mechanism for the
release of this energy is for the
atom to fission, i.e., split into two
new atoms of lesser atomic
weight.
•
The two new atoms, in this case Xenon and Strontium, are highly
radioactive and are called fission products or fission fragments. A variety
of fission products are possible. One particular reaction is
1
235
97
137
1
0 n+ 92 U→36 Kr+ 56 Ba +20 n
+γ
with a corresponding mass change of 0.2079 amu or 194 Mev. The
number of neutrons emitted during fission of U-235 ranges from 1 to 7
with an average of about 2.6.
•
No kinetic energy was assumed for the neutron in the analysis of this reaction,
implying that fission can be induced in U-235 by neutrons of essentially zero energy
•
Uranium-235 is the only naturally occurring isotope that will undergo fission this
way. Other heavy isotopes can be made to fission but require much larger
excitation energies to bring the compound nucleus to the required energy level for
fission.
•
Materials that exhibit the characteristic of undergoing fission with low energy
neutrons are called fissile. Materials that will undergo fission with neutrons of
sufficient energy are called fissionable. For example, U-238 is a fissionable
material but requires neutrons of energy above 0.9 Mev.
•
A more probable reaction involving U-238, particularly with low energy neutrons,
is
238
239
92 U(n , γ ) 92 U
•
It is interesting to note the by-products of
this reaction. Consider the following
reaction equations
1
238
239
0 n + 92 U → 92 U + γ
•
U-239 is radioactive and decays by
emission of a beta particle.
239
239
0
92 U→ 93 Np+ −1e
•
Np-239 is also radioactive and decays by
beta emission.
239
239
0
93 Np→ 94 Pu+ −1e
•
Pu-239 is radioactive, but has a half-life of
about 24,000 years and can thus be
considered stable for our purposes
•
Pu-239 is of interest as it is also a fissile material, i.e., will undergo fission
with low energy neutrons. Pu-239 however, does not occur naturally and
must be created through the above neutron absorption reaction in U-238.
•
U-238 is called a fertile material as it can be used to create the fissile Pu239.
•
Another common fertile material is thorium, which can be used to
generate the fissile isotope U-233 through the following sequence of
reactions:
1
232
233
0 n+ 90Th→ 90Th
233
233
0
90Th→ 91 Pa+ −1e
233
233
0
91Pa→ 92 U+ −1e.
Neutron Reactions
•
•
•
•
•
Neutrons interacting with matter are not always absorbed, just as we have seen
that not all absorptions lead to a fission reaction.
A variety of reactions are possible for any given material. For example, if the
neutron is absorbed, the absorption may lead to a fission or the emission of
secondary particles.
Neutrons may also be scattered by the nucleus in a reaction very similar to a
billiard ball type collision.
The probability of each of these events is a function of neutron kinetic energy and
the particular material of interest. These probabilities are given in terms of cross
sections.
Cross sections considered on a unit atom basis are call microscopic cross sections
and are represented by the Greek symbol σ. The cross section for a particular
reaction is designated by an appropriate subscript, such as: σs→ scattering cross
section, σa→ absorption cross section, σf→ fission cross section and σt→ total cross
section.
•
•
Materials such as B-10, Gd and Xe-135
have very large absorption cross sections
relative to their scattering cross sections,
and even large absorption cross sections
when compared to U-235. These
materials are therefore called poisons as
they would tend to poison and inhibit
any fission reaction.
Materials such as carbon and hydrogen
have scattering cross sections much
larger than their absorption cross
sections. Any neutron introduced into
these materials will, on the average,
undergo many scattering collisions
before being absorbed.
Isotope
σs
(barns)
σa
(barns)
12
6C
4.8
0.0034
1
1H
38
0.332
13.8
2.70
15
678
(σf = 577)
10
5B
4.0
3,838
64 Gd
4.0
46,000
135
54 Xe
-
2.6 x 106
238
92 U
235
92 U
•
In general, absorption cross sections tend to increase with decreasing
neutron energies. That is, the absorption and fission cross sections tend to
be the highest near low energies.
•
Materials that exhibit properties of large scattering cross sections with
small absorption cross sections are referred to as moderators. It can be
shown that the "lighter" a nucleus is, the more energy can be transferred
in a single collision.
•
Hydrogen, effects the greatest energy change in one collision and is a very
effective moderator. A neutron interacting with hydrogen can give up all
its energy in a single collision.
Macroscopic Cross Sections
•
The macroscopic cross section is the product of the number
density of the target nuclei times its microscopic cross section
Σn = N × σ n
•
•
The macroscopic cross section represents the probability per unit
length traveled of a reaction of type n
For a mixture of materials, the macroscopic cross section of the
mixture is the sum of the macroscopic cross sections of the
individual components
Σn =
∑Σ
j
nj
Reaction Rates
•
Reaction Rate = Number of neutrons x probability of a reaction per
unit length traveled x length traveled per unit time
Rn = Ν × Σ n × v
= V × nN
× v × Σn
φ
= V × φ × Σn
•
Reaction Rate Density = φ × Σ n
•
Note: The reactor thermal power is proportional to the fission rate
Q =
G
N
190 Mev/fission
× V ×ϕ × Σ f
Photon Reactions
•
Photon reactions of interest to radiation
detection are of three primary types
•
•
•
Compton Scattering
Photoelectric Absorption
Pair Production
Compton Scattering
•
•
•
The incident photon interacts with an orbital
electron
The photon changes both energy and
direction
The Compton electron is ejected with
energy Ee = Eγ − Eγ′
Photoelectric Effect
•
•
The photon is absorbed (vanishes)
An orbital (photo) electron is ejected with
energy Ee = Eγ
Pair Production
•
•
The photon vanishes in the vicinity of the nucleus
A positron/electron pair with energy
Eγ − 1.022 Mev
•
•
is emitted
Once the positron has slowed to essentially zero kinetic
energy, it recombines with an electron emitting two 0.511
Mev annihilation photons in opposite directions
Pair Production is a threshold reaction requiring initial
photon energies greater than 1.022 Mev
Fission Chain Reaction
•
Recall our diagram of the fission
event involving U-235. In this
diagram two neutrons are seen to
be emitted along with the fission
fragments.
•
One to seven neutrons can be
emitted depending on the
particular fission products
generated, with an average
number of about 2.6 per fission.
The average energy of these
fission neutrons is about 2 Mev.
•
•
•
•
•
•
A number of fates are possible for these neutrons, depending on the physical
dimensions of the system and the material composition.
Neutrons may: (1) be absorbed by a non-fissile material, (2) escape (or leak)
through the surface of the system, (3) be absorbed by a fissile or fissionable
material but not induce fission or, (4) be absorbed by a fissile or fissionable
material and induce fission giving rise to (on the average) another 2.6 neutrons.
The required condition for a stable, self-sustained chain reaction in a system
containing fissile and fissionable materials is that, on the average, exactly one
neutron must be produced per fission which eventually succeeds in producing
another fission.
The number of fissions per unit time, or the fission rate, must therefore be
constant. A nuclear system that displays this characteristic is said to be a critical
system.
If more than one neutron, on the average, succeeds in producing another fission,
the fission rate would not be constant, but would grow exponentially. The system is
said to be supercritical.
If less than one neutron, on the average, succeeds in producing another fission the
fission rate would decrease exponentially and eventually die out. The system would
be described as subcritical.
1
Scattering
Remain
Escape
1- L
L
Nonleakage
Probability,
L
Absorption
Fission
Capture
σc L
σa
σf L
σa
Neutron
Production
New fast neutrons
σf ν L
σa
=k
Neutron cycle in a U-235 metal assembly
(from Murray, R.L., "Nuclear Energy, 2nd Edition")
•
The effective multiplication factor is k = Lη.
•
A value of k =1 is possible for infinitely many values of η and L
•
For pure U-235, the neutrons will have very little opportunity to slow
down, and η for fast neutrons is approximately 2.2.
•
To produce a critical assembly, L must be 0.45. This implies that at least
45% of the neutrons must remain within the assembly for the system to
achieve criticality. The non-leakage probability is a function of geometry
and is a major consideration in criticality calculations.
Reactor Concepts
•
Light Water Reactors (LWRs)
Thermal reactors using regular (light) water as both the coolant and moderator.
Reactor designs include Pressurized Water Reactors (PWRs) and Boiling Water
Reactors (BWRs). LWRs are the dominant power producing reactors in the world.
Require enriched uranium in order to achieve criticality.
•
Heavy Water Reactors (HWRs)
Thermal reactors using heavy water as the moderator and in some designs as the
coolant. Reactor design chosen by the Canadians (CANDU). Can achieve
criticality using natural uranium.
•
Liquid Metal Reactors (LMRs)
Utilize liquid metal (Sodium) as the coolant. Contains no moderator. Is the most
common type of reactor for breeders.
•
Gas Cooled Reactors (GCRs)
Thermal reactors utilizing solid graphite as the moderator and Helium Gas as the
coolant.