Nuclear reactor physics

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https://en.wikipedia.org/wiki/Nuclear_reactor_physics 

 

Pressurized water reactor: Projective representation of the thermal neutron flux of a fuel assembly of the 18×18 array with 300 fuel rods and 24 inserted control rods

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. 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.

The physics of nuclear fission has several quirks that affect the design and behavior of nuclear reactors. This article presents a general overview of the physics of nuclear reactors and their behavior.

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, 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).

Starter sources

Main article: Startup neutron source

See also: Zero power critical

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 remains 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 ${\displaystyle k}$ 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,

${\displaystyle S_0+k\times S_0+k\times k\times S_0+\dots }$

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.

Neutron moderators

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.

Moderators and reactor design

The amount and nature of neutron moderation affects reactor controllability and hence safety. Because moderators both slow and absorb neutrons, there is an optimum amount of moderator to include in a given geometry of reactor core. Less moderation reduces the effectiveness by reducing the ${\displaystyle P_{fission}}$ term in the evolution equation, and more moderation reduces the effectiveness by increasing the ${\displaystyle P_{escape}}$ term.

Most moderators become less effective with increasing temperature, so under-moderated reactors are stable against changes in temperature in the reactor core: if the core overheats, then the quality of the moderator is reduced and the reaction tends to slow down (there is a "negative temperature coefficient" in the reactivity of the core). Water is an extreme case: in extreme heat, it can boil, producing effective voids in the reactor core without destroying the physical structure of the core; this tends to shut down the reaction and reduce the possibility of a fuel meltdown. Over-moderated reactors are unstable against changes in temperature (there is a "positive temperature coefficient" in the reactivity of the core), and so are less inherently safe than under-moderated cores.

Some reactors use a combination of moderator materials. For example, TRIGA type research reactors use ZrH₂ moderator mixed with the ²³⁵U fuel, an H₂O-filled core, and C (graphite) moderator and reflector blocks around the periphery of the core.

Delayed neutrons and controllability

Main article: Delayed neutron

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.

Kinetics

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

Reactor poisons

Main article: Nuclear poison

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.

Long-lived poisons and fuel reprocessing

In practice, buildup of reactor poisons in nuclear fuel is what determines the lifetime of nuclear fuel in a reactor: long before all possible fissions have taken place, buildup of long-lived neutron absorbing fission products damps out the chain reaction. This is the reason that nuclear reprocessing is a useful activity: spent nuclear fuel contains about 96% of the original fissionable material present in newly manufactured nuclear fuel. Chemical separation of the fission products restores the nuclear fuel so that it can be used again.

Nuclear reprocessing is useful economically because chemical separation is much simpler to accomplish than the difficult isotope separation required to prepare nuclear fuel from natural uranium ore, so that in principle chemical separation yields more generated energy for less effort than mining, purifying, and isotopically separating new uranium ore. In practice, both the difficulty of handling the highly radioactive fission products and other political concerns make fuel reprocessing a contentious subject. One such concern is the fact that spent uranium nuclear fuel contains significant quantities of ²³⁹Pu, a prime ingredient in nuclear weapons (see breeder reactor).

Short-lived poisons and controllability

Main article: Iodine pit

Short-lived reactor poisons in fission products strongly affect how nuclear reactors can operate. Unstable fission product nuclei transmute into many different elements (secondary 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 product with a half-life of about 9 hours, is an extremely strong neutron absorber. In an operating reactor, each nucleus of ¹³⁵Xe becomes ¹³⁶Xe (which may later sustain beta decay) 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 9 hours before beginning to decay. The result is that, about 6–8 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 over the next several hours. This temporary state, which may last several days and prevent restart, is called the iodine pit or xenon-poisoning. It is one reason why nuclear power reactors are usually operated at an even power level around the clock.

¹³⁵Xe buildup in a reactor core makes it extremely dangerous to operate the reactor a few hours after it has been shut down. Because the ¹³⁵Xe absorbs neutrons strongly, starting a reactor in a high-Xe condition requires pulling the control rods out of the core much farther than normal. However, if the reactor does achieve criticality, then the neutron flux in the core becomes high and ¹³⁵Xe is destroyed rapidly—this has the same effect as very rapidly removing a great length of control rod from the core, and can cause the reaction to grow too rapidly or even become prompt critical.

¹³⁵Xe played a large part in the Chernobyl accident: about eight hours after a scheduled maintenance shutdown, workers tried to bring the reactor to a zero power critical condition to test a control circuit. Since the core was loaded with ¹³⁵Xe from the previous day's power generation, it was necessary to withdraw more control rods to achieve this. As a result, the overdriven reaction grew rapidly and uncontrollably, leading to steam explosion in the core, and violent destruction of the facility.

Uranium enrichment

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).

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 the 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.

Oklo: a natural nuclear reactor

Main article: Oklo Mine

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.

Neutron moderator

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neutron_moderator 

 

Science with neutrons

In nuclear engineering, a neutron moderator is a medium that reduces the speed of fast neutrons, ideally without capturing any, leaving them as thermal neutrons with only minimal (thermal) kinetic energy. These thermal neutrons are immensely more susceptible than fast neutrons to propagate a nuclear chain reaction of uranium-235 or other fissile isotope by colliding with their atomic nucleus.

Water (sometimes called "light water" in this context) is the most commonly used moderator (roughly 75% of the world's reactors). Solid graphite (20% of reactors) and heavy water (5% of reactors) are the main alternatives. Beryllium has also been used in some experimental types, and hydrocarbons have been suggested as another possibility.

Moderation

Neutrons are normally bound into an atomic nucleus, and do not exist free for long in nature. The unbound neutron has a half-life of 10 minutes and 11 seconds. The release of neutrons from the nucleus requires exceeding the binding energy of the neutron, which is typically 7-9 MeV for most isotopes. Neutron sources generate free neutrons by a variety of nuclear reactions, including nuclear fission and nuclear fusion. Whatever the source of neutrons, they are released with energies of several MeV.

According to the equipartition theorem, the average kinetic energy, ${\displaystyle \overline{E}}$, can be related to temperature, ${\displaystyle T}$, via:

${\displaystyle \overline{E}=\frac{1}{2}{m}_n⟨v^2⟩=\frac{3}{2}{k}_{B}T}$,

where ${\displaystyle m_n}$ is the neutron mass, ${\displaystyle ⟨v^2⟩}$ is the average squared neutron speed, and ${\displaystyle k_B}$ is the Boltzmann constant. The characteristic neutron temperature of several-MeV neutrons is several tens of billions kelvin.

Moderation is the process of the reduction of the initial high speed (high kinetic energy) of the free neutron. Since energy is conserved, this reduction of the neutron speed takes place by transfer of energy to a material called a moderator.

The probability of scattering of a neutron from a nucleus is given by the scattering cross section. The first couple of collisions with the moderator may be of sufficiently high energy to excite the nucleus of the moderator. Such a collision is inelastic, since some of the kinetic energy is transformed to potential energy by exciting some of the internal degrees of freedom of the nucleus to form an excited state. As the energy of the neutron is lowered, the collisions become predominantly elastic, i.e., the total kinetic energy and momentum of the system (that of the neutron and the nucleus) is conserved.

Given the mathematics of elastic collisions, as neutrons are very light compared to most nuclei, the most efficient way of removing kinetic energy from the neutron is by choosing a moderating nucleus that has near identical mass.

Elastic collision of equal masses

A collision of a neutron, which has mass of 1, with a ¹H nucleus (a proton) could result in the neutron losing virtually all of its energy in a single head-on collision. More generally, it is necessary to take into account both glancing and head-on collisions. The mean logarithmic reduction of neutron energy per collision, ${\displaystyle \xi }$, depends only on the atomic mass, ${\displaystyle A}$, of the nucleus and is given by:

${\displaystyle \xi =\ln \frac{E_0}{E}=1+\frac{(A-1{)}^2}{2A}\ln \left(\frac{A-1}{A+1}\right)}$.

This can be reasonably approximated to the very simple form ${\displaystyle \xi \simeq \frac{2}{A+2/3}}$. From this one can deduce ${\displaystyle n}$, the expected number of collisions of the neutron with nuclei of a given type that is required to reduce the kinetic energy of a neutron from ${\displaystyle E_0}$ to ${\displaystyle E_1}$

${\displaystyle n=\frac{1}{\xi }(\ln E_0-\ln E_1)}$.

In a system at thermal equilibrium, neutrons (red) are elastically scattered by a hypothetical moderator of free hydrogen nuclei (blue), undergoing thermally activated motion. Kinetic energy is transferred between particles. As the neutrons have essentially the same mass as protons and there is no absorption, the velocity distributions of both particles types would be well-described by a single Maxwell–Boltzmann distribution.

Choice of moderator materials

Some nuclei have larger absorption cross sections than others, which removes free neutrons from the flux. Therefore, a further criterion for an efficient moderator is one for which this parameter is small. The moderating efficiency gives the ratio of the macroscopic cross sections of scattering, ${\displaystyle {\mathrm{\Sigma }}_s}$, weighted by ${\displaystyle \xi }$ divided by that of absorption, ${\displaystyle {\mathrm{\Sigma }}_a}$: i.e., ${\displaystyle \frac{\xi {\mathrm{\Sigma }}_s}{{\mathrm{\Sigma }}_a}}$. For a compound moderator composed of more than one element, such as light or heavy water, it is necessary to take into account the moderating and absorbing effect of both the hydrogen isotope and oxygen atom to calculate ${\displaystyle \xi }$. To bring a neutron from the fission energy of ${\displaystyle E_0}$ 2 MeV to an ${\displaystyle E}$ of 1 eV takes an expected ${\displaystyle n}$ of 16 and 29 collisions for H₂O and D₂O, respectively. Therefore, neutrons are more rapidly moderated by light water, as H has a far higher ${\displaystyle {\mathrm{\Sigma }}_s}$. However, it also has a far higher ${\displaystyle {\mathrm{\Sigma }}_a}$, so that the moderating efficiency is nearly 80 times higher for heavy water than for light water.

The ideal moderator is of low mass, high scattering cross section, and low absorption cross section.

	Hydrogen	Deuterium	Beryllium	Carbon	Oxygen	Uranium
Mass of kernels u	1	2	9	12	16	238
Energy decrement ${\displaystyle \xi }$	1	0.7261	0.2078	0.1589	0.1209	0.0084
Number of Collisions	18	25	86	114	150	2172

Distribution of neutron velocities once moderated

After sufficient impacts, the speed of the neutron will be comparable to the speed of the nuclei given by thermal motion; this neutron is then called a thermal neutron, and the process may also be termed thermalization. Once at equilibrium at a given temperature the distribution of speeds (energies) expected of rigid spheres scattering elastically is given by the Maxwell–Boltzmann distribution. This is only slightly modified in a real moderator due to the speed (energy) dependence of the absorption cross-section of most materials, so that low-speed neutrons are preferentially absorbed, so that the true neutron velocity distribution in the core would be slightly hotter than predicted.

Reactor moderators

In a thermal-neutron reactor, the nucleus of a heavy fuel element such as uranium absorbs a slow-moving free neutron, becomes unstable, and then splits ("fissions") into two smaller atoms ("fission products"). The fission process for ²³⁵U nuclei yields two fission products, two to three fast-moving free neutrons, plus an amount of energy primarily manifested in the kinetic energy of the recoiling fission products. The free neutrons are emitted with a kinetic energy of ~2 MeV each. Because more free neutrons are released from a uranium fission event than thermal neutrons are required to initiate the event, the reaction can become self-sustaining – a chain reaction – under controlled conditions, thus liberating a tremendous amount of energy (see article nuclear fission).

Fission cross section, measured in barns (a unit equal to 10⁻²⁸ m²), is a function of the energy (so-called excitation function) of the neutron colliding with a ²³⁵U nucleus. Fission probability decreases as neutron energy (and speed) increases. This explains why most reactors fueled with ²³⁵U need a moderator to sustain a chain reaction and why removing a moderator can shut down a reactor.

The probability of further fission events is determined by the fission cross section, which is dependent upon the speed (energy) of the incident neutrons. For thermal reactors, high-energy neutrons in the MeV-range are much less likely (though not unable) to cause further fission. The newly released fast neutrons, moving at roughly 10% of the speed of light, must be slowed down or "moderated", typically to speeds of a few kilometres per second, if they are to be likely to cause further fission in neighbouring ²³⁵U nuclei and hence continue the chain reaction. This speed happens to be equivalent to temperatures in the few hundred Celsius range.

In all moderated reactors, some neutrons of all energy levels will produce fission, including fast neutrons. Some reactors are more fully thermalised than others; for example, in a CANDU reactor nearly all fission reactions are produced by thermal neutrons, while in a pressurized water reactor (PWR) a considerable portion of the fissions are produced by higher-energy neutrons. In the proposed water-cooled supercritical water reactor (SCWR), the proportion of fast fissions may exceed 50%, making it technically a fast-neutron reactor.

A fast reactor uses no moderator, but relies on fission produced by unmoderated fast neutrons to sustain the chain reaction. In some fast reactor designs, up to 20% of fissions can come from direct fast neutron fission of uranium-238, an isotope which is not fissile at all with thermal neutrons.

Moderators are also used in non-reactor neutron sources, such as plutonium-beryllium (using the ⁹
Be(α,n)¹²
C reaction) and spallation sources (using (p,xn) reactions with neutron rich heavy elements as targets).

Form and location

The form and location of the moderator can greatly influence the cost and safety of a reactor. Classically, moderators were precision-machined blocks of high purity graphite with embedded ducting to carry away heat. They were in the hottest part of the reactor, and therefore subject to corrosion and ablation. In some materials, including graphite, the impact of the neutrons with the moderator can cause the moderator to accumulate dangerous amounts of Wigner energy. This problem led to the infamous Windscale fire at the Windscale Piles, a nuclear reactor complex in the United Kingdom, in 1957. In a carbon dioxide cooled graphite moderated reactor where coolant and moderator are in contact with one another, the Boudouard reaction needs to be taken into account. This is also the case if fuel elements have an outer layer of carbon - as in some TRISO fuels - or if an inner carbon layer becomes exposed by failure of one or several outer layers.

The moderators of some pebble-bed reactors are not only simple, but also inexpensive: the nuclear fuel is embedded in spheres of reactor-grade pyrolytic carbon, roughly of the size of tennis balls. The spaces between the balls serve as ducting. The reactor is operated above the Wigner annealing temperature so that the graphite does not accumulate dangerous amounts of Wigner energy.

In CANDU and PWR reactors, the moderator is liquid water (heavy water for CANDU, light water for PWR). In the event of a loss-of-coolant accident in a PWR, the moderator is also lost and the reaction will stop. This negative void coefficient is an important safety feature of these reactors. In CANDU the moderator is located in a separate heavy-water circuit, surrounding the pressurized heavy-water coolant channels. The heavy water will slow down a significant portion of neutrons to the resonance integral of ²³⁸
U increasing the neutron capture in this isotope that makes up over 99% of the uranium in CANDU fuel thus decreasing the amount of neutrons available for fission. As a consequence, removing some of the heavy water will increase reactivity until so much is removed that too little moderation is provided to keep the reaction going. This design gives CANDU reactors a positive void coefficient, although the slower neutron kinetics of heavy-water moderated systems compensates for this, leading to comparable safety with PWRs.^[9] In the light water cooled graphite moderated RBMK, a reactor type originally envisioned to allow both production of weapons grade plutonium and large amounts of usable heat while using natural uranium and foregoing the use of heavy water, the light water coolant acts primarily as a neutron absorber and thus its removal in a Loss of Coolant Accident or by conversion of water into steam will increase the amount of thermal neutrons available for fission. Following the Chernobyl nuclear accident the issue was remedied so that all still operating RBMK type reactors have a slightly negative void coefficient but they now require a higher degree of uranium enrichment in their fuel.

Moderator impurities

Good moderators are free of neutron-absorbing impurities such as boron. In commercial nuclear power plants the moderator typically contains dissolved boron. The boron concentration of the reactor coolant can be changed by the operators by adding boric acid or by diluting with water to manipulate reactor power. The Nazi Nuclear Program suffered a substantial setback when its inexpensive graphite moderators failed to function. At that time, most graphites were deposited onto boron electrodes, and the German commercial graphite contained too much boron. Since the war-time German program never discovered this problem, they were forced to use far more expensive heavy water moderators. This problem was discovered by famous physicist Leó Szilárd.

Non-graphite moderators

Some moderators are quite expensive, for example beryllium, and reactor-grade heavy water. Reactor-grade heavy water must be 99.75% pure to enable reactions with unenriched uranium. This is difficult to prepare because heavy water and regular water form the same chemical bonds in almost the same ways, at only slightly different speeds.

The much cheaper light water moderator (essentially very pure regular water) absorbs too many neutrons to be used with unenriched natural uranium, and therefore uranium enrichment or nuclear reprocessing becomes necessary to operate such reactors, increasing overall costs. Both enrichment and reprocessing are expensive and technologically challenging processes, and additionally both enrichment and several types of reprocessing can be used to create weapons-usable material, causing proliferation concerns. Reprocessing schemes that are more resistant to proliferation are currently under development.

The CANDU reactor's moderator doubles as a safety feature. A large tank of low-temperature, low-pressure heavy water moderates the neutrons and also acts as a heat sink in extreme loss-of-coolant accident conditions. It is separated from the fuel rods that actually generate the heat. Heavy water is very effective at slowing down (moderating) neutrons, giving CANDU reactors their important and defining characteristic of high "neutron economy". Unlike a light water reactor where adding water to the core in an accident might provide enough moderation to make a subcritical assembly go critical again, heavy water reactors will decrease their reactivity if light water is added to the core, which provides another important safety feature in the case of certain accident scenarios. However, any heavy water that becomes mixed with the emergency coolant light water will become too diluted to be useful without isotope separation.

Nuclear weapon design

Main article: Uranium hydride bomb

Early speculation about nuclear weapons assumed that an "atom bomb" would be a large amount of fissile material, moderated by a neutron moderator, similar in structure to a nuclear reactor or "pile". Only the Manhattan project embraced the idea of a chain reaction of fast neutrons in pure metallic uranium or plutonium. Other moderated designs were also considered by the Americans; proposals included using uranium deuteride as the fissile material. In 1943 Robert Oppenheimer and Niels Bohr considered the possibility of using a "pile" as a weapon. The motivation was that with a graphite moderator it would be possible to achieve the chain reaction without the use of any isotope separation. However, plutonium can be produced ("bred") sufficiently isotopically pure as to be usable in a bomb and then has to be "only" separated chemically, a much easier processes than isotope separation, albeit still a challenging one. In August 1945, when information of the atomic bombing of Hiroshima was relayed to the scientists of the German nuclear program, interred at Farm Hall in England, chief scientist Werner Heisenberg hypothesized that the device must have been "something like a nuclear reactor, with the neutrons slowed by many collisions with a moderator". The German program, which had been much less advanced, had never even considered the plutonium-option and didn't discover a feasible method of large scale isotope separation in uranium.

After the success of the Manhattan project, all major nuclear weapons programs have relied on fast neutrons in their weapons designs. The notable exception is the Ruth and Ray test explosions of Operation Upshot–Knothole. The aim of the University of California Radiation Laboratory designs was the exploration of deuterated polyethylene charge containing uranium as a candidate thermonuclear fuel, hoping that deuterium would fuse (becoming an active medium) if compressed appropriately. If successful, the devices could also lead to a compact primary containing minimal amount of fissile material, and powerful enough to ignite RAMROD a thermonuclear weapon designed by UCRL at the time. For a "hydride" primary, the degree of compression would not make deuterium to fuse, but the design could be subjected to boosting, raising the yield considerably. The cores consisted of a mix of uranium deuteride (UD₃), and deuterated polyethylene. The core tested in Ray used uranium low enriched in U²³⁵, and in both shots deuterium acted as the neutron moderator. The predicted yield was 1.5 to 3 kt for Ruth (with a maximum potential yield of 20 kt) and 0.5-1 kt for Ray. The tests produced yields of 200 tons of TNT each; both tests were considered to be fizzles.

The main benefit of using a moderator in a nuclear explosive is that the amount of fissile material needed to reach criticality may be greatly reduced. Slowing of fast neutrons will increase the cross section for neutron absorption, reducing the critical mass. A side effect is however that as the chain reaction progresses, the moderator will be heated, thus losing its ability to cool the neutrons.

Another effect of moderation is that the time between subsequent neutron generations is increased, slowing down the reaction. This makes the containment of the explosion a problem; the inertia that is used to confine implosion type bombs will not be able to confine the reaction. The end result may be a fizzle instead of a bang.

The explosive power of a fully moderated explosion is thus limited, at worst it may be equal to a chemical explosive of similar mass. Again quoting Heisenberg: "One can never make an explosive with slow neutrons, not even with the heavy water machine, as then the neutrons only go with thermal speed, with the result that the reaction is so slow that the thing explodes sooner, before the reaction is complete."

While a nuclear bomb working on thermal neutrons may be impractical, modern weapons designs may still benefit from some level of moderation. A beryllium tamper used as a neutron reflector will also act as a moderator.

Materials used

• Hydrogen, as in ordinary "light water". Because protium also has a significant cross section for neutron capture only limited moderation is possible without losing too many neutrons. The less-moderated neutrons are relatively more likely to be captured by uranium-238 and less likely to fission uranium-235, so light water reactors require enriched uranium to operate.
  • There are also proposals to use the compound formed by the chemical reaction of metallic uranium and hydrogen (uranium hydride—UH₃) as a combination fuel and moderator in a new type of reactor.
  • Hydrogen is also used in the form of cryogenic liquid methane and sometimes liquid hydrogen as a cold neutron source in some research reactors: yielding a Maxwell–Boltzmann distribution for the neutrons whose maximum is shifted to much lower energies.
  • Hydrogen combined with carbon as in paraffin wax was used in some early German experiments.
• Deuterium, in the form of heavy water, in heavy water reactors, e.g. CANDU. Reactors moderated with heavy water can use unenriched natural uranium.
• Carbon, in the form of reactor-grade graphite or pyrolytic carbon, used in e.g. RBMK and pebble-bed reactors, or in compounds, e.g. carbon dioxide. As carbon dioxide contains twice as many oxygen atoms as it does carbon atoms and both have moderating and neutron absorbing effects in a similar range (see above), a significant share of the moderation in a (yet to be built) carbon dioxide moderated reactor would actually come from the oxygen. Lower-temperature reactors are susceptible to buildup of Wigner energy in the material. Like deuterium-moderated reactors, some of these reactors can use unenriched natural uranium.
  • Graphite is also deliberately allowed to be heated to around 2000 K or higher in some research reactors to produce a hot neutron source: giving a Maxwell–Boltzmann distribution whose maximum is spread out to generate higher energy neutrons.
• Beryllium, in the form of metal. Beryllium is expensive and toxic, so its use is limited.
• Lithium-7, in the form of a lithium fluoride salt, typically in conjunction with beryllium fluoride salt (FLiBe). This is the most common type of moderator in a molten salt reactor.

Other light-nuclei materials are unsuitable for various reasons. Helium is a gas and it requires special design to achieve sufficient density; lithium-6 and boron-10 absorb neutrons.

Currently operating nuclear power reactors by moderator

Moderator	Reactors	Design	Country
none (fast)	2	BN-600, BN-800	Russia (2)
graphite	25	AGR, Magnox, RBMK	United Kingdom (14), Russia (9)
heavy water	29	CANDU, PHWR	Canada (17), South Korea (4), Romania (2), China (2), India (18), Argentina, Pakistan
light water	359	PWR, BWR	27 countries