Nuclear Reactor Physics
Nuclear fission is a nuclear decay process where an unstable nucleus splits into two smaller nuclei (known as 'fission fragments'), and a couple of neutrons and gamma rays are also released. The most common fuel used for nuclear reactors is uranium. Natural uranium is composed of U-235 and U-238. U-235 can be induced to fission by absorbing a low energy neutron (known as a thermal neutron and having a kinetic energy of about 0.025 eV). However, U-238 requires much more energetic neutrons to induce a fission, and hence nuclear fuel is really referring to the U-235 within the uranium.
A nuclear fission typically releases about 200 MeV of energy. This is two hundred million more than chemical reactions, such as burning coal, which only release a few eV per event.
What is an eV?
An energy unit commonly used in nuclear and particle physics is the electron volt (symbol eV). It is defined as the energy gained by an electron accelerated across a potential difference of 1V, 1 eV = 1.6×10-19 J. An MeV is shorthand for one million electron volts.
Where does the significant energy released in fission go? The energy released can be categorised as either prompt or delayed. Prompt energy is released immediately, and delayed energy is released by fission products after the fission has occurred, this delay can vary from milliseconds to minutes.
- The fission fragments fly apart at high speed; their kinetic energy is ≈ 170 MeV. This energy will be deposited locally as heat in the fuel.
- The prompt neutrons will also have a kinetic energy of ≈ 2 MeV. Due to their high energy, these neutrons are also called fast neutrons. On average 2.4 prompt neutrons are released in a U-235 fission, and hence the total energy of prompt neutrons is ≈ 5 MeV. The neutrons will lose this energy within the moderator.
- Prompt gamma rays are emitted from the fission fragments, with an energy ≈ 7 MeV. This energy will be absorbed somewhere within the reactor.
- Most fission fragments are neutron-rich and will beta decay after some time has passed, this is the source of delayed energy.
- Beta particles (fast electrons) are emitted, with an energy of ≈ 8 MeV. This energy is deposited in the fuel.
- Beta decay will also produce neutrinos, with an energy of ≈ 10 MeV. These neutrinos and hence their energy will escape the reactor (and our solar system).
- Gamma rays will then be emitted after these beta decays. These delayed gamma rays carry an energy of ≈ 7 MeV. Like the prompt gamma rays, this energy is absorbed somewhere within the reactor.
As previously mentioned, U-235 can be fissioned by neutrons of any energy. This allows the fission of a U-235 atom to induce fission in surrounding U-235 atoms and set off a chain reaction of fissions. This is qualitatively described by the neutron multiplication factor (k). This factor is the average number of neutrons from a fission reaction that causes another fission. There are three cases:
- k < 1, Subcritical - a chain reaction is unsustainable.
- k = 1, Critical - each fission leads to another fission, a steady state solution. This is desirable for nuclear reactors.
- k > 1, Supercritical - a runaway chain reaction, such as in atomic bombs.
Nuclear reactors are complex pieces of engineering, but there are some important features that are common to most reactors:
- Moderator - A moderator is used to decrease the energy of fast neutrons emitted from fissions. Common moderators are water or graphite. The fast neutrons lose energy through scattering off moderator atoms. This is done to bring the neutrons down to a thermal energy. Moderation is crucial because the U-235 fission cross section increases for lower energies and hence a thermal neutron is more likely to fission U-235 nuclei than a fast neutron.
- Control rods - Control rods are used to control the rate of fission. Control rods are made of materials with a high neutron absorption cross section, such as boron. Hence, as more of the control rods are inserted into the reactor, they absorb more of the neutrons produced within the reactor and reduce the chance of more fissions and hence reduces k. This is a very important safety feature to control the reactor.
- Fuel enrichment - Only 0.72% of natural uranium is U-235. Enrichment refers to increasing this proportion of U-235 in the uranium fuel, this increases the thermal fission factor (see below) and makes achieving k equal to one easier. The increase is significant for low enrichment but not much of an advantage for high enrichments. Reactor grade uranium is usually 3-4% enrichment but an 80% enrichment would typically be for a nuclear weapon (maybe as fuel for a research reactor).
- Coolant - A coolant is used to remove heat from the nuclear reactor core (the part of the reactor where the fuel is stored). Most current reactors use water as a coolant.
Four factor formula
By making major assumptions, a simple four factor formula can be written down for k. This formula assumes that no neutrons escape the reactor (an infinite reactor) and also assumes that the fuel and moderator are intimately mixed. The four factors are different ratios and explained below:
- Thermal fission factor (η) - The ratio of neutrons produced by thermal fissions to the thermal neutrons absorbed in the fuel.
- Fast fission factor (ε) - The ratio of the number of fast neutrons from all fissions to the number of fast neutrons from thermal fissions.
- Resonance escape probability (p) - The ratio of neutrons that reach thermal energy to fast neutrons that start to slow down.
- Thermal utilisation factor (f) - The ratio of the number of thermal neutrons absorbed in the fuel to the number of thermal neutrons absorbed in the reactor.
Six factor formula
By adding two factors to the four factor formula, the leakage of neutrons from the reactor can be accounted for. The two factors are:
- pFNL - The fraction of fast neutrons that don't leak out.
- pThNL - The fraction of thermal neutrons that don't leak out.
Negative void coefficients
When boiling occurs in a water moderated reactor (such as a PWR or BWR design). Steam bubbles replace the water (described as "voids"), reducing the amount of moderator. This in turn reduces the reactivity of the reactor and leads to a drop in power. This response is known as a negative void coefficient, the reactivity decreases with the increase of voids and acts as a self stabilising behaviour. A positive void coefficient means that the reactivity will actually increase with the increase of voids. Modern reactors are specifically designed to avoid positive void coefficients. A positive void coefficient was one of the reactor faults at Chernobyl (more on that later).
In a good reactor design, reactivity should also decrease if temperature increases. This can happen naturally as an increase in moderator temperature will increase the neutron energy, and shift the cross section spectrum towards higher energies (a process known as the spectrum hardening). This elevation in temperature will then decrease the fission cross section and reactivity. This can be seen in the plot below.
Some fission fragments (or daughter nuclei of the fission fragments) have very high neutron absorption cross sections. The presence of these nuclides has a negative effect on the reactivity of the reactor and hence they are known as reactor poisons. Xenon-135 is the most important example of a reactor poison. The neutron absorption cross section is over 2 million barns, a staggeringly large number. The decay chain that leads to xenon-135 is shown below. Under normal operating conditions, the concentration of xenon-135 will be in equilibrium. When the reactor is shutdown, the xenon will continue to be produced and build up in the reactor. This leads to a majorly reduced reactivity and could prevent the reactor from starting up again. After waiting about a day, the xenon will have decayed sufficiently to resume the reactor.
There are other reactor poisons, such as samarium-149. Some poisons can only be removed by "burning" them off with neutrons or in extreme cases refuelling the reactor.
Operators at the Chernobyl power plant in Ukraine began a low power turbine test on the 26th April 1986. For the test the automatic control rod system was disabled and control rods were manually extracted; this was done to overcome an extremely low power level from xenon poisoning. Boiling began due to a slower flow-rate. The reactor was an RBMK soviet reactor which had a positive void coefficient. Therefore, reactivity started to increase. Control rods were automatically inserted but the system only had control of 12 out of 211 control rods. A shutdown was initiated but reactivity was still increasing as the xenon was burnt off the reactor. Additional control rods were inserted but they actually increased the reactivity initially because the control rods had graphite tips which displaced the water.
There was a massive power spike and the core began to overheat. Steam built up and lead to a large explosion that blew a hole in the reactor hall roof. The chain reaction stopped but a graphite fire started and a large amount of radioactive material was blown out of the reactor. The accident was later classified as a level 7 event on the INES scale, a major event, and remains the worst nuclear accident in history. There were three main causes of the accident:
- Reactor design - The RBMK design had multiple faults. It had a positive void coefficient, the control rods had graphite tips and the reactor had no containment shell. Instrumentation was also unable to measure the high levels of radiation after the accident.
- Operation - Safety standards weren't followed and the operators took bad decisions during the test.
- Soviet culture - The accident was initially covered up by the Soviet authorities and this lead to increased exposure for citizens.
The accident at Fukushima in March 2011 is the only other nuclear accident to be rated as a level 7 event. On the 11th March, a magnitude 9 earthquake occurred off the north-east coast of Japan. The reactors at the Fukushima plant were automatically shutdown and the control rods were fully inserted. External power supply was destroyed by the earthquake. Backup diesel generators were started to continue the cooling process. Decay heat is still significant after reactor shutdown because of the delayed decay of fission products. Later on, a tsunami struck and managed to flood the diesel generators. This was due to a negligent placement of the generators and an insufficiently high sea wall. Now the decay heat couldn't be removed and reactor cores started to heat up. Hydrogen gas was vented as an emergency measure which lead to explosions and a significant release of radiation.
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