Nuclear reactor physics is the field of physics that studies and deals with the applied study and engineering applications of chain reaction to induce a controlled rate of fission in a nuclear reactor for the production of energy.van Dam, H., van der Hagen, T. H. J. J., & Hoogenboom, J. E. (2005). Nuclear reactor physics. Retrieved from https://www.janleenkloosterman.nl/reports/ap3341.pdf
Most nuclear reactors use a chain reaction to induce a controlled rate of nuclear fission in fissile material, releasing both energy and free neutrons. A reactor consists of an assembly of nuclear fuel (a reactor core), usually surrounded by a neutron moderator such as regular water, heavy water, graphite, or zirconium hydride, and fitted with mechanisms such as control rods which control the rate of the reaction.
Criticality
In a nuclear reactor, the neutron population at any instant is a function of the rate of neutron production (due to fission processes) and the rate of neutron losses (due to non-fission absorption mechanisms and leakage from the system). When a reactor's neutron population remains steady from one generation to the next (creating as many new neutrons as are lost), the fission chain reaction is self-sustaining and the reactor's condition is referred to as "critical". When the reactor's neutron production exceeds losses, characterized by increasing power level, it is considered "supercritical", and when losses dominate, it is considered "subcritical" and exhibits decreasing power.
The "Six-factor formula" is the neutron life-cycle balance equation and takes the form . The parameter is known as the effective multiplication factor (also denoted by ), and defined to be . As indicated by its name, the Six-factor formula accounts for six factors in the fission reaction process:
: Thermal fission factor : Thermal fuel utilization factor : Resonance escape probability : Fast fission factor : Fast non-leakage probability : Thermal non-leakage probability
When , the reactor is said to be critical; when , the reactor is subcritical; and when , the reactor is supercritical.
Reactivity, expressed as either or and given by the equation , is an expression of the departure from criticality. When , the reactor is critical. When , the reactor is subcritical. When , the reactor is supercritical. Reactivity is commonly expressed in decimals, percentages, or pcm (per cent mille) of . When reactivity is expressed in units of the delayed neutron fraction , the unit is called the dollar.
If we write for the number of free neutrons in a reactor core and for the average lifetime of each neutron (before it either escapes from the core or is absorbed by a nucleus), then the reactor will follow the differential equation (evolution equation).
where is a constant of proportionality, and is the rate of change of the neutron count in the core. This type of differential equation describes exponential growth or exponential decay, depending on the sign of the constant , which is just the expected number of neutrons after one average neutron lifetime has elapsed:
Here, is the probability that a particular neutron will strike a fuel nucleus, is the probability that the neutron, having struck the fuel, will cause that nucleus to undergo fission, is the probability that it will be absorbed by something other than fuel, and is the probability that it will "escape" by leaving the core altogether. is the number of neutrons produced, on average, by a fission event—it is between 2 and 3 for both 235U and 239Pu (e.g., for thermal neutrons in 235U, = 2.4355 ± 0.0023 ).
If is positive, then the core is supercritical and the rate of neutron production will grow exponentially until some other effect stops the growth. If is negative, then the core is "subcritical" and the number of free neutrons in the core will shrink exponentially until it reaches an equilibrium at zero (or the background level from spontaneous fission). If is exactly zero, then the reactor is critical and its output does not vary in time (, from above).
Nuclear reactors are engineered to reduce and . Small, compact structures reduce the probability of direct escape by minimizing the surface area of the core, and some materials (such as graphite) can reflect some neutrons back into the core, further reducing .
The probability of fission, , depends on the nuclear physics of the fuel, and is often expressed as a cross section. Reactors are usually controlled by adjusting . Control rods made of a strongly neutron-absorbent material such as cadmium or boron can be inserted into the core: any neutron that happens to impact the control rod is lost from the chain reaction, reducing . is also controlled by the recent history of the reactor core itself (see below).
Starter sources
The mere fact that an assembly is supercritical does not guarantee that it contains any free neutrons at all. At least one neutron is required to "strike" a chain reaction, and if the spontaneous fission rate is sufficiently low it may take a long time (in 235U reactors, as long as many minutes) before a chance neutron encounter starts a chain reaction even if the reactor is supercritical. Most nuclear reactors include a "starter" neutron source that ensures there are always a few free neutrons in the reactor core, so that a chain reaction will begin immediately when the core is made critical. A common type of startup neutron source is a mixture of an alpha particle emitter such as 241Am (americium-241) with a lightweight isotope such as 9Be (beryllium-9).
The primary sources described above have to be used with fresh reactor cores. For operational reactors, secondary sources are used; most often a combination of antimony with beryllium. Antimony becomes activated in the reactor and produces high-energy gamma photons, which produce photoneutrons from beryllium.
Uranium-235 undergoes a small rate of natural spontaneous fission, so there are always some neutrons being produced even in a fully shutdown reactor. When the control rods are withdrawn and criticality is approached the number increases because the absorption of neutrons is being progressively reduced, until at criticality the chain reaction becomes self-sustaining. Note that while a neutron source is provided in the reactor, this is not essential to start the chain reaction, its main purpose is to give a shutdown neutron population which is detectable by instruments and so make the approach to critical more observable. The reactor will go critical at the same control rod position whether a source is loaded or not.
Once the chain reaction is begun, the primary starter source may be removed from the core to prevent damage from the high neutron flux in the operating reactor core; the secondary sources usually remain in situ to provide a background reference level for control of criticality.
Subcritical multiplication
Even in a subcritical assembly such as a shut-down reactor core, any stray neutron that happens to be present in the core (for example from spontaneous fission of the fuel, from radioactive decay of fission products, or from a neutron source) will trigger an exponentially decaying chain reaction. Although the chain reaction is not self-sustaining, it acts as a multiplier that increases the equilibrium number of neutrons in the core. This subcritical multiplication effect can be used in two ways: as a probe of how close a core is to criticality, and as a way to generate fission power without the risks associated with a critical mass.
If is the neutron multiplication factor of a subcritical core and is the number of neutrons coming per generation in the reactor from an external source, then at the instant when the neutron source is switched on, the number of neutrons in the core will be . After 1 generation, these neutrons will produce neutrons in the reactor and the reactor will have a totality of neutrons considering the newly entered neutrons in the reactor. Similarly after 2 generations, the number of neutrons produced in the reactor will be and so on. This process will continue and after a long enough time, the number of neutrons in the reactor will be,
This series will converge because for the subcritical core, . So the number of neutrons in the reactor will be simply,
The fraction is called subcritical multiplication factor (α).
As a measurement technique, subcritical multiplication was used during the Manhattan Project in early experiments to determine the minimum critical masses of 235U and of 239Pu. It is still used today to calibrate the controls for nuclear reactors during startup, as many effects (discussed in the following sections) can change the required control settings to achieve criticality in a reactor. As a power-generating technique, subcritical multiplication allows generation of nuclear power for fission where a critical assembly is undesirable for safety or other reasons. A subcritical assembly together with a neutron source can serve as a steady source of heat to generate power from fission.
Including the effect of an external neutron source ("external" to the fission process, not physically external to the core), one can write a modified evolution equation:
where is the rate at which the external source injects neutrons into the core in neutrons/Δt. In equilibrium, the core is not changing and dN/dt is zero, so the equilibrium number of neutrons is given by:
If the core is subcritical, then is negative so there is an equilibrium with a positive number of neutrons. If the core is close to criticality, then is very small and thus the final number of neutrons can be made arbitrarily large.
Neutron moderators
To improve and enable a chain reaction, natural or low enrichment uranium-fueled reactors must include a neutron moderator that interacts with newly produced fast neutrons from fission events to reduce their kinetic energy from several MeV to thermal energies of less than one eV, making them more likely to induce fission. This is because 235U has a larger cross section for slow neutrons, and also because 238U is much less likely to absorb a thermal neutron than a freshly produced neutron from fission.
Neutron moderators are thus materials that slow down neutrons. Neutrons are most effectively slowed by colliding with the nucleus of a light atom, hydrogen being the lightest of all. To be effective, moderator materials must thus contain light elements with atomic nuclei that tend to scatter neutrons on impact rather than absorb them. In addition to hydrogen, beryllium and carbon atoms are also suited to the job of moderating or slowing down neutrons.
Hydrogen moderators include water (H2O), heavy water (D2O), and zirconium hydride (ZrH2), all of which work because a hydrogen nucleus has nearly the same mass as a free neutron: neutron-H2O or neutron-ZrH2 impacts excite rotational modes of the molecules (spinning them around). Deuterium nuclei (in heavy water) absorb kinetic energy less well than do light hydrogen nuclei, but they are much less likely to absorb the impacting neutron. Water or heavy water have the advantage of being transparent liquids, so that, in addition to shielding and moderating a reactor core, they permit direct viewing of the core in operation and can also serve as a working fluid for heat transfer.
Carbon in the form of graphite has been widely used as a moderator. It was used in Chicago Pile-1, the world's first man-made critical assembly, and was commonplace in early reactor designs including the Soviet RBMK nuclear power plants such as the Chernobyl plant.
Moderators and reactor design