Nuclear reactor physics

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, which includes six separate factors, the product of which is equal to the ratio of the number of neutrons in any generation to that of the previous one; this parameter is called the effective multiplication factor k, also denoted by K_eff, where k = Є L_f ρ L_th f η, where Є = "fast-fission factor", L_f = "fast non-leakage factor", ρ = "resonance escape probability", L_th = "thermal non-leakage factor", f = "thermal fuel utilization factor", and η = "reproduction factor". This equation's factors are roughly in order of potential occurrence for a fission born neutron during critical operation. As already mentioned before, k = (Neutrons produced in one generation)/(Neutrons produced in the previous generation). In other words, when the reactor is critical, k = 1; when the reactor is subcritical, k < 1; and when the reactor is supercritical, k > 1.

Reactivity is an expression of the departure from criticality. δk = (k − 1)/k. When the reactor is critical, δk = 0. When the reactor is subcritical, δk < 0. When the reactor is supercritical, δk > 0. Reactivity is also represented by the lowercase Greek letter rho (ρ). Reactivity is commonly expressed in decimals or percentages or pcm (per cent mille) of Δk/k. When reactivity ρ is expressed in units of delayed neutron fraction β, the unit is called the dollar.

If we write 'N' for the number of free neutrons in a reactor core and ${\displaystyle \tau }$ 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)

${\displaystyle {\frac {dN}{dt}}={\frac {\alpha N}{\tau }}}$

where ${\displaystyle \alpha }$ is a constant of proportionality, and ${\displaystyle dN/dt}$ 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 ${\displaystyle \alpha }$, which is just the expected number of neutrons after one average neutron lifetime has elapsed:

${\displaystyle \alpha =P_{impact}P_{fission}n_{avg}-P_{absorb}-P_{escape}}$

Here, ${\displaystyle P_{impact}}$ is the probability that a particular neutron will strike a fuel nucleus, ${\displaystyle P_{fission}}$ is the probability that the neutron, having struck the fuel, will cause that nucleus to undergo fission, ${\displaystyle P_{absorb}}$ is the probability that it will be absorbed by something other than fuel, and ${\displaystyle P_{escape}}$ is the probability that it will "escape" by leaving the core altogether. ${\displaystyle n_{avg}}$ is the number of neutrons produced, on average, by a fission event—it is between 2 and 3 for both ²³⁵U and ²³⁹Pu.

If ${\displaystyle \alpha }$ is positive, then the core is supercritical and the rate of neutron production will grow exponentially until some other effect stops the growth. If ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha }$ is exactly zero, then the reactor is critical and its output does not vary in time (${\displaystyle dN/dt=0}$, from above).

Nuclear reactors are engineered to reduce ${\displaystyle P_{escape}}$ and ${\displaystyle P_{absorb}}$. 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 ${\displaystyle P_{escape}}$.

The probability of fission, ${\displaystyle P_{fission}}$, depends on the nuclear physics of the fuel, and is often expressed as a cross section. Reactors are usually controlled by adjusting ${\displaystyle P_{absorb}}$. 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 ${\displaystyle \alpha }$. ${\displaystyle P_{absorb}}$ is also controlled by the recent history of the reactor core itself (see below).

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 ${\displaystyle k}$ is the neutron multiplication factor of a subcritical core and ${\displaystyle S_{0}}$ 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, number of neutrons in the core will be ${\displaystyle S_{0}}$. After 1 generation, this neutrons will produce ${\displaystyle k\times S_{0}}$ neutrons in the reactor and reactor will have a totality of ${\displaystyle k\times S_{0}+S_{0}}$ neutrons considering the newly entered neutrons in the reactor. Similarly after 2 generation, number of neutrons produced in the reactor will be ${\displaystyle k\times (k\times S_{0}+S_{0})+S_{0}}$ and so on. This process will continue and after a long enough time, the number of neutrons in the reactor will be,

${\displaystyle S_{0}+k\times S_{0}+k\times k\times S_{0}+\ldots }$

This series will converge because for the subcritical core, ${\displaystyle 0<k<1}$. So the number of neutrons in the reactor will be simply,

${\displaystyle {\frac {S_{0}}{1-k}}}$

The fraction ${\displaystyle {\frac {1}{1-k}}}$ is called subcritical multiplication factor.

Since power in a reactor is proportional to the number of neutrons present in the nuclear fuel material (material in which fission can occur), the power produced by such a subcritical core will also be proportional to the subcritical multiplication factor and the external source strength.

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:

${\displaystyle {\frac {dN}{dt}}={\frac {\alpha N}{\tau }}+R_{ext}}$

where ${\displaystyle R_{ext}}$ is the rate at which the external source injects neutrons into the core. In equilibrium, the core is not changing and dN/dt is zero, so the equilibrium number of neutrons is given by:

${\displaystyle N=-{\frac {\tau R_{ext}}{\alpha }}}$

If the core is subcritical, then ${\displaystyle \alpha }$ is negative so there is an equilibrium with a positive number of neutrons. If the core is close to criticality, then ${\displaystyle \alpha }$ is very small and thus the final number of neutrons can be made arbitrarily large.

To improve ${\displaystyle P_{fission}}$ 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.

Fission reactions and subsequent neutron escape happen very quickly; this is important for nuclear weapons, where the objective is to make a nuclear pit release as much energy as possible before it physically explodes. Most neutrons emitted by fission events are prompt: they are emitted effectively instantaneously. Once emitted, the average neutron lifetime (${\displaystyle \tau }$) in a typical core is on the order of a millisecond, so if the exponential factor ${\displaystyle \alpha }$ is as small as 0.01, then in one second the reactor power will vary by a factor of (1 + 0.01)¹⁰⁰⁰, or more than ten thousand. Nuclear weapons are engineered to maximize the power growth rate, with lifetimes well under a millisecond and exponential factors close to 2; but such rapid variation would render it practically impossible to control the reaction rates in a nuclear reactor.

Fortunately, the effective neutron lifetime is much longer than the average lifetime of a single neutron in the core. About 0.65% of the neutrons produced by ²³⁵U fission, and about 0.20% of the neutrons produced by ²³⁹Pu fission, are not produced immediately, but rather are emitted from an excited nucleus after a further decay step. In this step, further radioactive decay of some of the fission products (almost always negative beta decay), is followed by immediate neutron emission from the excited daughter product, with an average life time of the beta decay (and thus the neutron emission) of about 15 seconds. These so-called delayed neutrons increase the effective average lifetime of neutrons in the core, to nearly 0.1 seconds, so that a core with ${\displaystyle \alpha }$ of 0.01 would increase in one second by only a factor of (1 + 0.01)¹⁰, or about 1.1: a 10% increase. This is a controllable rate of change.

Most nuclear reactors are hence operated in a prompt subcritical, delayed critical condition: the prompt neutrons alone are not sufficient to sustain a chain reaction, but the delayed neutrons make up the small difference required to keep the reaction going. This has effects on how reactors are controlled: when a small amount of control rod is slid into or out of the reactor core, the power level changes at first very rapidly due to prompt subcritical multiplication and then more gradually, following the exponential growth or decay curve of the delayed critical reaction. Furthermore, increases in reactor power can be performed at any desired rate simply by pulling out a sufficient length of control rod. However, without addition of a neutron poison or active neutron-absorber, decreases in fission rate are limited in speed, because even if the reactor is taken deeply subcritical to stop prompt fission neutron production, delayed neutrons are produced after ordinary beta decay of fission products already in place, and this decay-production of neutrons cannot be changed.

The rate of change of reactor power is determined by the reactor period ${\displaystyle T}$, which is related to the reactivity ${\displaystyle \rho }$ through the Inhour equation.

The kinetics of the reactor is described by the balance equations of neutrons and nuclei (fissile, fission products).

Any nuclide that strongly absorbs neutrons is called a reactor poison, because it tends to shut down (poison) an ongoing fission chain reaction. Some reactor poisons are deliberately inserted into fission reactor cores to control the reaction; boron or cadmium control rods are the best example. Many reactor poisons are produced by the fission process itself, and buildup of neutron-absorbing fission products affects both the fuel economics and the controllability of nuclear reactors.

While many fissionable isotopes exist in nature, the only usefully fissile isotope found in any quantity is ²³⁵U. About 0.7% of the uranium in most ores is the 235 isotope, and about 99.3% is the non-fissile 238 isotope. For most uses as a nuclear fuel, uranium must be enriched - purified so that it contains a higher percentage of ²³⁵U. Because ²³⁸U absorbs fast neutrons, the critical mass needed to sustain a chain reaction increases as the ²³⁸U content increases, reaching infinity at 94% ²³⁸U (6% ²³⁵U).[2]

Concentrations lower than 6% ²³⁵U cannot go fast critical, though they are usable in a nuclear reactor with a neutron moderator. A nuclear weapon primary stage using uranium uses HEU enriched to ~90% ²³⁵U, though the secondary stage often uses lower enrichments. Nuclear reactors with water moderator require at least some enrichment of ²³⁵U. Nuclear reactors with heavy water or graphite moderation can operate with natural uranium, eliminating altogether the need for enrichment and preventing the fuel from being useful for nuclear weapons; the CANDU power reactors used in Canadian power plants are an example of this type.

The Uranium enrichment is difficult because the chemical properties of ²³⁵U and ²³⁸U are identical, so physical processes such as gaseous diffusion, gas centrifuge, laser, or mass spectrometry must be used for isotopic separation based on small differences in mass. Because enrichment is the main technical hurdle to production of nuclear fuel and simple nuclear weapons, enrichment technology is politically sensitive.

Modern deposits of uranium contain only up to ~0.7% ²³⁵U (and ~99.3% ²³⁸U), which is not enough to sustain a chain reaction moderated by ordinary water. But ²³⁵U has a much shorter half-life (700 million years) than ²³⁸U (4.5 billion years), so in the distant past the percentage of ²³⁵U was much higher. About two billion years ago, a water-saturated uranium deposit (in what is now the Oklo mine in Gabon, West Africa) underwent a naturally occurring chain reaction that was moderated by groundwater and, presumably, controlled by the negative void coefficient as the water boiled from the heat of the reaction. Uranium from the Oklo mine is about 50% depleted compared to other locations: it is only about 0.3% to 0.7% ²³⁵U; and the ore contains traces of stable daughters of long-decayed fission products.

• Critical mass
• List of nuclear reactors
• Nuclear physics
• Nuclear fission
• Nuclear fusion
• Void coefficient

• DOE Fundamentals Handbook: Nuclear Physics and Reactor Theory (PDF). U.S. Department of Energy. January 1993. Archived from the original (PDF) on 2013-12-03. Retrieved 2012-01-02.

Fermi age theory

Notes on nuclear diffusion by Dr. Abdelhamid Dokhane