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Introductory Nuclear Physics Problem Set 05

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Introductory Nuclear Physics Problem Set 05 UNIVERSITY OF CAPE TOWN EEE4106Z: Introductory Nuclear Physics Problem Set 05 Due 1400 Tuesday 2 June 2015 This problem set should prove valuable in preparation for the final examination. The questions allow you to possibly gain a fuller understanding of important concepts that were not familiar to you at the time of preparing your answers to the preceding problem set. At Koeberg hand in to Z Isaacs. At UCT place in Aschman mailslot in mailboxes opposite room 508, top floor RW James Building UCT. Late answers will incur a penalty. Students may work together on the problems, and discuss the results together, but a handed-in script must be each student's own work. Copied work gets zero. Discuss the problems before tackling them! 1) Explain how the fuel temperature coefficient of reactivity is largely determined by the resonances in the cross-section for the 238U(n; γ) capture of neutrons. Provide a clear explanation of the Doppler broadening of the resonance peaks. SOLUTION: At higher temperatures there is greater thermal motion of the fuel nuclei, in particular U-238 nuclei. This increases the spread in the velocities, and hence the energies, with with the neutrons approach the U-238 nuclei. This causes Doppler broadening of the neutron capture resonances in the 238U(n; γ) cross-section in the epi-thermal energy region, so there is a greater chance of the a neutron being captured in the fuel as the neutron energy gets stepped down, from fast to thermal, by the moderator. This reduces the resonance escape factor p, and hence k. 2) Describe the role of delayed neutrons in enabling the control of a nuclear reactor. Explain clearly what is meant by prompt criticality, and why this will prove catastrophic. SOLUTION: Here is an overly full and length answer . The term \prompt critical" means that the reactor can achieve criticality, k = 1, with prompt neutrons alone. The results in a very short reactor period, of the order of a frac- tion of a second, which is too fast for reactor control by control rod insertion. From Wikipedia article on \prompt critical" Criticality: An assembly is critical if each fission event causes, on average, exactly one additional such event in a continual chain. Such a chain is a self-sustaining fission chain reaction. When a uranium-235 (U-235) atom undergoes nuclear fission, it typically releases between 1 and 7 neutrons (with an average of 2.4). In this situation, an assembly is critical if every released neutron has a 1/2.4 = 0.42 = 42% probability of causing another fission event as opposed to either being absorbed by a non-fission capture event or escaping from the fissile core. The average number of neutrons that cause new fission events is called the effective neu- tron multiplication factor, usually denoted by the symbols k-effective, k-eff or k. When k-effective i is equal to 1, the assembly is called critical, if k-effective is less than 1 the assembly is said to be subcritical, and if k-effective is greater than 1 the assembly is called supercritical. Critical versus prompt-critical: In a supercritical assembly the number of fissions per unit time, N, along with the power production, increases exponentially with time. How fast it grows depends on the average time it takes, T , for the neutrons released in a fission event to cause another fission. The growth rate of the reaction is given by: kt=T N(t) = N0e Most of the neutrons released by a fission event are the ones released in the fissioning pro- cess itself. These are called prompt neutrons, and strike other nuclei and cause additional fissions within a very short time. This time can be tens of nanoseconds if the neutrons stay fast and find another nucleus to fission in the fuel itself. If the neutrons must first be thermalized by the moderator, and then diffuse at thermal velocities to find a fissile nucleus in the fuel, this time can be a hundred microseconds or so. [. ] a supercritical assembly is said to be prompt-critical if it is critical without any con- tribution from delayed neutrons and super-prompt-critical if it is supercritical without any contribution from delayed neutrons[clarification needed]. In this case the time between successive generations of the reaction, T , is only limited by the lifetime of the prompt neutrons, and the increase in the reaction will be extremely rapid, causing a rapid release of energy within a few milliseconds. Prompt-critical assemblies are created by design in nuclear weapons and some specially designed research experiments. Should a PWR core reach prompt criticality, it will be catastrophic as power will rise by factors of twenty thousand or so within a second. There is no time to provide negative reactivity by inserting slow mechanical control rods. The coolant can not remove the heat that quickly. The coolant will flash to steam, the fuel may melt, and the core will probably undergo what is euphemistically termed \disassembly". DELAYED NEUTRONS: See Lilley Ch 8. Or, from Wikipedia: However a small additional source of neutrons is the fission products. Some of the nuclei resulting from the fission are radioactive isotopes with short half-lives, and nuclear reac- tions among them release additional neutrons after a long delay of up to several minutes after the initial fission event. These neutrons, which on average account for less than one percent of the total neutrons released by fission, are called delayed neutrons. The relatively slow timescale on which delayed neutrons appear is an important aspect for the design of nuclear reactors, as it allows the reactor power level to be controlled via the gradual, mechanical movement of control rods. Typically, control rods contain neutron poisons (substances, for example boron or hafnium, that easily capture neutrons without producing any additional ones) as a means of altering k-effective. With the exception of experimental pulsed reactors, nuclear reactors are designed to operate in a delayed-critical mode and are provided with safety systems to prevent them from ever achieving prompt criticality. In a delayed-critical assembly, the delayed neutrons are needed to make k-effective greater than one. Thus the time between successive generations of the reaction, T , is dominated by the time it takes for the delayed neutrons to be released, on the order of seconds or minutes. Therefore the reaction will increase slowly, with a long time constant. This is slow enough to allow the reaction to be controlled with electromechanical control sys- tems such as control rods, and as such all nuclear reactors are designed to operate in the delayed-criticality regime. [ . ] When differentiating between a prompt neutron versus a delayed neutron, the difference between the two has to do with the source from which the neutron has been released into the reactor. The neutrons, once released, have no difference except the energy or speed which have been imparted to them. 3) Explain fully the phenomenon of xenon poisoning, including a) the reason that, after a shutdown, a reactor usually can not be started for many tens of hours b) why running a reactor at low power for a long time is not safe. SOLUTION: A neutron poison (also called a neutron absorber or a nuclear poison) is a substance with a large neutron absorption cross-section, in applications such as nuclear reactors. In such applications, absorbing neutrons is normally an undesirable effect. However neutron- absorbing materials, also called poisons, are intentionally inserted into some types of reactors in order to lower the high reactivity of their initial fresh fuel load. Some of these poisons deplete as they absorb neutrons during reactor operation, while others remain relatively constant. Some of the fission products generated during nuclear reactions have a high neutron ab- 6 sorption cross-section, such as xenon-135 with cross-section σc = 2×10 b, and samarium- 3 149 with σc = 74:5 × 10 b. Because these two fission product poisons remove neutrons from the reactor, they will have an impact on the thermal utilization factor and thus the reactivity. The poisoning of a reactor core by these fission products may become so serious that the chain reaction comes to a standstill. Xenon poisoning can decrease k by 0:03. The xenon-135 builds up after shutdown (no neutron flux to remove it), and thus acts negative reactivity. Until it decays, the reactor may be hard to start-up. This make take several tens of hours. Lilley Ch 10.8 has the full story, as follows here . In short, then . Short-lived reactor poisons in fission products strongly affect how nuclear reactors can operate. Unstable fission product nuclei transmute into many different elements (sec- ondary fission products) as they undergo a decay chain to a stable isotope. The most important such element is xenon, because the isotope 135Xe , a secondary fission prod- uct with a half-life of about 9 hours, is an extremely strong neutron absorber. In an operating reactor, each nucleus of 135Xe is destroyed by neutron capture almost as soon as it is created, so that there is no buildup in the core. However, when a reactor shuts down, the level of 135Xe builds up in the core for about 11 hours before beginning to decay. The result is that, for about 25 hours after a reactor is shut down, it can become physically impossible to restart the chain reaction until the 135Xe has had a chance to decay. This is one reason why nuclear power reactors are best operated at an even power level around the clock. They are thus suited for baseload power and not for peaking power.
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