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
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with the net effect being a + X →Y +b
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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 • •
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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.
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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.
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This result would tend to indicate a net mass loss to the system, an uncomfortable concept.
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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 .
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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 .
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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.
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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.
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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
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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.
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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.
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A more probable reaction involving U-238, particularly with low energy neutrons, is 238 239 92 U(n , γ ) 92 U
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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
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U-239 is radioactive and decays by emission of a beta particle. 239 239 0 92 U→ 93 Np+ −1e
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Np-239 is also radioactive and decays by beta emission. 239 239 0 93 Np→ 94 Pu+ −1e
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Pu-239 is radioactive, but has a half-life of about 24,000 years and can thus be considered stable for our purposes
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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.
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U-238 is called a fertile material as it can be used to create the fissile Pu239.
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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 • •
• •
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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.
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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
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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.
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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.
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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 •
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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
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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 •
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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
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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.
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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.
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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")
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The effective multiplication factor is k = Lη.
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A value of k =1 is possible for infinitely many values of η and L
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For pure U-235, the neutrons will have very little opportunity to slow down, and η for fast neutrons is approximately 2.2.
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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.
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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.
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Liquid Metal Reactors (LMRs) Utilize liquid metal (Sodium) as the coolant. Contains no moderator. Is the most common type of reactor for breeders.
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Gas Cooled Reactors (GCRs) Thermal reactors utilizing solid graphite as the moderator and Helium Gas as the coolant.
