Energy Devices – Fission (Lecture 7) Eric R. Switzer (http://kicp.uchicago.edu/∼switzer/) Nov. 14, 2009

Lecture outline: • Orientation – energies, mass, the nucleus. • Fission, cross sections, moderation. • CP-1; the physics of the reactor; the neutron cycle. • Modern reactors and statistics. Resources: • Plutonium – Jeremy Bernstein. • Megawatts and Megatons: The Future of Nuclear Power and Nuclear Weapons – R. Garwin and G. Charpak. • Introductory Nuclear Physics – K. Krane • Nuclear Physics: Principles and Applications – J. Lilley. • “Experimental production of a divergent chain reaction” – E. Fermi et al., 1952, Am. J. Phys. • Neutron cross sections from NNDA/ENDF. 1

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http://www.nndc.bnl.gov/sigma/

Image sources: Fig. 6 (right) Fig. 7, 8

Diagram following Introductory Nuclear Physics – K. Krane. “Experimental production of a divergent chain reaction” – E. Fermi et al., 1952, Am. J. Phys.

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Figure 1: Fission of

235

U. From label 1 to 2: a thermal neutron is absorbed and the nucleus becomes U. From label 2 to 3: the nucleus is excited and deformed by the kinetic energy of the neutron and the energy associated with the formation of the compound 236 U nucleus. From label 3 to 4: the excitations of the nucleus drive it to fission. The two product nuclei vary widely from fission to fission, but two possibilities are 147 La and 87 Br plus the emission of two neutrons. At some later time (label 5), the product nuclei will decay, emitting further radiation. The mass change and energy units: Consider the sum of the mass of the reactants (the neutron and 235 U), 1.0087 u + 235.0439 u = 236.0525 u, in atomic mass units u. Now, consider the sum of the mass of the products, 146.9282 u + 86.9207 u + 2 × 1.0087 u = 235.8663 u. The mass of the products is 0.19 u less than the reactants. This is because some of the mass has been turned into energy as suggested by E = mc2 . The atomic mass unit is 1.66 × 10−27 kg, but in nuclear physics is it more convenient to define a new unit of mass in terms of the energy, MeV/c2 using m = E/c2 , where the energy is most conveniently measured in MeV, or million-electron volts. This is the energy that an electron has after it has moved across a potential difference of a million volts (described in lecture 2). In these fun units, c2 = 931.494 MeV/u. Therefore the energy gained in this fission is 177 MeV. The distribution of the fission energy: In general, fission can occur though many possible channels so it is better to evaluate mean quantities rather than those for particular fission paths. The emitted neutrons have a broad energy spectrum with mean energy of ∼ 2 MeV. The mean number of neutrons released per fission is 2.5, so that the emitted neutrons from a fission event carry ∼ 5 MeV of energy. Driven by Coulomb repulsion, the product nuclei will carry most of the kinetic energy, typically around 168 MeV. Some gamma rays are also emitted in the fission, typically carrying 7 MeV. The product nuclei have a surplus of neutrons and so tend to convert them to protons through β − decay to reach more stable nuclei (this the radiation shown by the dashed lines labeled 5 above). The electrons and gamma rays emitted by the radioactive product nuclei carry an additional 15 MeV (12 MeV is carried off in neutrinos, which do not deposit heat). (values from Lilley) The number to take away from this discussion is that ∼ 200 MeV of thermal energy (on average) is available from each fission event. Converting units, it takes 31 billion fissions to produce one Joule of thermal energy; nuclear power stations producing ∼GW (billions of Joules per second) are typical. 236

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Figure 2: Left: A baseball breaking a window as a demonstration of the cross section and thresholds. At low velocity, the ball will bounce off regardless of where it hits. At intermediate velocity, it can break the window if it impacts in the middle, but damage is less probable if it hits the edges. At sufficiently high velocity, the ball will break the window wherever it impacts. If we plot the area over which the window will break when impacted, we see a threshold behavior associated with the minimum velocity needed to break the window. The notion of a cross section and threshold in nuclear physics is somewhat similar. The cross section has units of area and quantifies how probable an interaction is by the effective area of the impacted object. Nuclei are much smaller than windows, so a more natural unit of area is the barn, where 1 barn = 1 × 10−24 cm2 . Right: The energetics of splitting the nucleus. Nuclear forces hold the nucleus together, despite all of the positive charge of the protons. If you start to elongate the nucleus (the deformation region of the plot), it will eventually reach a point where it is energetically favorable for the Coulomb repulsion to split the nucleus and drive two large fragments apart. Thus, some input (activation) energy will drive the nucleus to a state where it splits and liberates significantly more energy. A common way to understand these potential energy diagrams is to imagine them as hills, with the current state as a rolling ball. At zero displacement, there is a local minimum where the ball will rest; by rolling the ball to overcome the activation energy, it will be free to roll down the larger hill, liberating its potential energy.

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Neutron cross sections of U235 and U238 10000 U-235 total fission U-238 total fission U-238 absorption

cross section (barns)

1000 100 10 1 0.1 0.01 0.01

1

100

10000

1e+06

neutron energy (eV)

Figure 3: Relevant cross sections of 235 U and 238 U as a function of energy of (data: NNDA/ENDF). First consider the green line for the uranium-238 fission cross section: Here, when a fast neutron runs into the nucleus, it induces fission and exhibits a threshold effect like the example of the ball breaking a window (Fig. 2). If the incoming neutron has insufficient energy, it will not induce fission through the activation barrier shown in Fig. 2. (The units of energy here are described in Fig. 1.) In the uranium isotopes that we are considering, the activation energy is ∼ 6 MeV. The incoming neutron does not need to have this much energy to induce a fission event, though. This is because it is not the parent nucleus 238 U that splits, but a compound nucleus that has formed with the absorbed neutron. The formation of the compound nucleus provides additional energy – in the case of 238 U, 4.8 MeV. Thus, one needs a neutron with additional kinetic energy greater than ∼ 1 MeV to exceed the total 6 MeV activation energy, consistent with the threshold in the plot. Now consider the red line for the uranium-235 fission cross section: Here, the cross section increases as the inverse of the neutron velocity at low energy – there is no threshold! As the neutron slows down, it is more and more readily absorbed. The reason there is no threshold for fission is that the formation of the compound nucleus liberates more energy than the activation energy (6.5 MeV released versus 6.2 MeV activation)– that is, simply the formation of 236 U by absorbing a neutron is sufficient to drive fission, regardless of the neutron’s kinetic energy! Rather than thinking of the incident neutron as shattering the nucleus, think of it simply as introducing an instability that causes the nucleus to split. A neutron in 235 U is unsafe at all speeds (especially slow ones!). The jagged region around 100 eV is due to quantum-mechanical effects and is called the “resonance” region. The blue line shows the 238 U absorption cross section, which again has a smooth low-energy region and a resonance region.

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Figure 4: The principle of neutron moderation. For the reaction to be self-sustaining, each thermal neutron must produce another thermal neutron through fission. As we have seen in Fig. 3, neutrons have the highest probability of inducing fission in 235 U when they are moving slowly (because they are captured more easily). Yet, as we have seen from Fig. 1, the neutrons emitted from the fission event have very high energies ∼ 2 MeV, or about a factor of at least 1 × 107 too high to be practical for inducing 235 U fission. Some small fraction of these fast neutrons can induce fission in 238 U. However, to achieve a self-sustaining reaction, the neutrons must be slowed down using a moderator. A common point of confusion is that moderators slow neutrons, but that this does not impede the fission; the moderator facilitates fission. The basic principle is that collisions of the fast neutrons and nuclei in a moderator exchange energy and momentum, in a way similar to how a fast pool ball imparts its energy to a stationary pool ball. This is most effective when the mass of the moderator nuclei is low (for high mass moderator nuclei, the situation is more like the ping-pong ball bouncing off of the pool ball). Another desired characteristic of the moderator is that it should not absorb the neutrons. Common moderators: Three commonly used moderators are ordinary water, heavy water, and pure graphite. Ordinary (light) water is practical, and provides lightweight protons for scattering. A downside of light water is that it also absorbs neutrons, making it impossible to achieve fission using uranium at natural 235 U/238 U abundance (∼ 0.72%). Light-water reactors must use enriched uranium where typically ∼ few% 235 U is present in the fuel. A natural uranium reactor (such as the modern Canadian CANDU reactor design) can use heavy water instead. Here, the neutrons come “pre-attached” to the protons so the heavy water much less readily absorbs the fission neutrons. Pure graphite can also be used as a moderator in natural and enriched uranium reactors. The first self-sustaining reaction with natural uranium (CP-1) used graphite – as did subsequent weapons reactors and later (predominantly British and Russian) civilian reactors. Carbons are fairly massive compared to protons, but the neutron absorption cross section is very low. The majority of nuclear power today is produced in light-water, enriched uranium reactors. Thermal neutrons: The moderator itself has a finite temperature, associated with a jiggling of its atoms with characteristic energy scale of 1/40 eV. If neutrons had less average energy, they would be accelerated when they scattered with the nuclei in the moderator. Neutrons that have reached this equilibrium with the temperature of the moderator are called thermal neutrons.

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Figure 5: The total fission and absorption cross sections in natural uranium (data: NNDA/ENDF). Here, we weight the 235 U and 238 U cross sections for their abundance in natural uranium by multiplying by 0.0072 and 0.9928, respectively. The fast neutrons that are emitted from fission need to be slowed down so that they are more easily absorbed by 235 U nuclei. Fig. 4 explains how neutrons are slowed down in a moderator. 238 U is the dominant isotope in natural uranium, and it readily absorbs neutrons from ∼ 10 eV-∼ 1 MeV. If a graphite moderator and natural uranium fuel are mixed homogeneously, the neutrons slow down in the graphite, but as they slow they are readily absorbed in the 238 U resonance region, making them unavailable for 235 U fission. For natural 235 U/238 U abundance, a homogeneous graphite-uranium reactor is not possible. The way to make the reaction proceed is to build a heterogeneous reactor where the fuel and moderator are well-separated. By embedding uranium fuel spheres in a lattice of graphite, the neutrons emitted by the spheres can propagate through pure graphite and slow down to thermal velocities. These thermal neutrons can then diffuse into the other fuel spheres and induce 235 U fission. The regions of pure moderator of a heterogeneous reactor allow the fast neutrons to “cool off” without passing through the perilous energy region of resonant 238 U absorption.

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Figure 6: Left: Neutrons diffusing in the CP-1 pile. Here, the black square represents one layer of a pile, where uranium or uranium oxide pressed spheres (the gray circles) are arranged in a regular lattice. These spheres are separated by 8 41 inches, or roughly the thermalization length of fast neutrons in graphite. Consider several possible outcomes for a neutron emitted in fission. In the path marked 1, the fast neutron is emitted, thermalized in the graphite of the pile, and absorbed by 235 U in a separate fuel sphere, inducing a fission which produces two neutrons. In this case, one of the neutrons produces another fission event and the other is absorbed in the graphite moderator. Denote the average number of fast neutrons produced per thermal neutron as η. (In natural uranium, this is ∼ 1.33.) In the path marked 2, the fast neutron is only partly thermalized, but wanders back to the fuel sphere and is absorbed resonantly by 238 U; it is no longer viable to induce 235 U fission. The probability that it escapes such a fate is the resonance escape probability; call this p. In the path marked 3, the neutron is absorbed in the moderator; again, it can not go on to induce 235 U fission. The probability that it escapes this fate and the thermal neutron is available to induce fission is the thermal utilization factor; call this f . In the path marked 4, the neutron thermalizes but escapes the finite pile. Let L be the probability that neither the fast nor slow neutron will leak from the pile. In the path marked 5, a fast neutron induces fission in 238 U (recall that this requires a significant threshold energy, so needs a fast neutron). In this case, we get a “free” neutron that can induce further fissions. Denote the average gain in fast fission neutrons as ǫ. Right: A schematic of the neutron cycle. Here, the paths described above can be imagined as sources and sinks to some flow of neutrons. A reactor will be self-sustaining if all production terms (neutrons emitted from fission and decays of fission products) counter loss terms (leakage, neutron absorption, etc.). The average number of thermal neutrons produced per thermal neutron is the neutron multiplication factor k. For an infinite pile, one can ignore leakage, and k∞ = ηǫpf , and for a finite pile, k = ηǫpf L. This is called the four-factor formula. For k < 1 the pile is sub-critical; k = 1, critical; k > 1, supercritical. A natural uranium reactor in a lattice arrangement can achieve a neutron multiplication factor k∞ ≈ 1.11; thus, a finite reactor (with leakage) can operate at k = 1.

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Figure 7: “The Italian navigator has arrived” Top: The CP-1 in profile. Left: A layer of the pile. Right: Neutron flux as a function the the layer added. (Images: E. Fermi et al., 1952)

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Figure 8: Top: The CP-1 control rod structure. Here, neutron absorbing boron or cadmiumrich materials are directed into the graphite pile to control the neutron multiplication factor and achieve a stable, self-sustained reaction. A neutron emitted in a 235 U fission event can very rapidly thermalize and induce another fission event. If a reactor were only based on these “prompt” fission neutrons, even a slightly supercritical (e.g. k = 1.001) reactor could rapidly run away through exponentially growing neutron flux. (Unfortunately, this is the point of nuclear weapons.) In practice, the reactor is tamed using delayed neutrons. One of the fission products, 87 Br decays with a halflife of 56 seconds to 86 Kr plus a neutron through β − decay (though 98% of the time it simply decays to 87 Kr without emission of the neutron). This neutron contributes to the overall neutron cycle and is delayed by roughly a minute. If these delayed neutrons complete the criticality, then response times are slow enough (minutes) that mechanical devices like the control rod rail can react and ensure k = 1. The reaction rates can therefore be controlled by human operators over the period of minutes. The delayed neutrons comprise < 1% of the neutron cycle. Bottom: The cadmium-rich safety rods. Here if the neutron rate becomes high, then a solenoid can release a rod through the pile, drawn by a weight. An additional set of control rods could be released if a human operator cut through a rope with and axe. Their title was the “Safety Control Rod Axe Man” (SCRAM), though the term is sometimes more soberly attributed as “Super Critical Reaction Abatement Mechanism”. (Images: E. Fermi et al., 1952)

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