effective multiplication factor

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In
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
, a nuclear chain reaction occurs when one single
nuclear reaction In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic phy ...
causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The specific nuclear reaction may be the fission of heavy
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbe ...
s (e.g.,
uranium-235 Uranium-235 (235U or U-235) is an Isotopes of uranium, isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile ...
, 235U). A nuclear chain reaction releases several million times more energy per reaction than any
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
.

# History

Chemical
chain reaction A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. In a chain reaction, positive feedback leads to a self-amplifying chain of events. Chain reactions are one way that syst ...
s were first proposed by German chemist
Max Bodenstein Max Ernst August Bodenstein (July 15, 1871 – September 3, 1942) was a German physical chemistry, physical chemist known for his work in chemical kinetics. He was first to postulate a chain reaction mechanism and that explosions are branched ch ...
in 1913, and were reasonably well understood before nuclear chain reactions were proposed. It was understood that chemical chain reactions were responsible for exponentially increasing rates in reactions, such as produced in chemical explosions. The concept of a nuclear chain reaction was reportedly first hypothesized by Hungarian scientist
Leó Szilárd Leo Szilard (; hu, Szilárd Leó, pronounced ; born Leó Spitz; February 11, 1898 – May 30, 1964) was a Hungarian-German-American physicist and inventor. He conceived the nuclear chain reaction In nuclear physics, a nuclear chain ...
on September 12, 1933. Szilárd that morning had been reading in a London paper of an experiment in which protons from an accelerator had been used to split lithium-7 into alpha particles, and the fact that much greater amounts of energy were produced by the reaction than the proton supplied.
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
commented in the article that inefficiencies in the process precluded use of it for power generation. However, the neutron had been discovered by
James Chadwick Sir James Chadwick, (20 October 1891 – 24 July 1974) was an English physicist who was awarded the 1935 Nobel Prize in Physics for his discovery of the neutron in 1932. In 1941, he wrote the final draft of the MAUD Committee, MAUD Repo ...
in 1932, shortly before, as the product of a nuclear reaction. Szilárd, who had been trained as an engineer and physicist, put the two nuclear experimental results together in his mind and realized that if a nuclear reaction produced neutrons, which then caused further similar nuclear reactions, the process might be a self-perpetuating nuclear chain-reaction, spontaneously producing new isotopes and power without the need for protons or an accelerator. Szilárd, however, did not propose fission as the mechanism for his chain reaction, since the fission reaction was not yet discovered, or even suspected. Instead, Szilárd proposed using mixtures of lighter known isotopes which produced neutrons in copious amounts. He filed a patent for his idea of a simple nuclear reactor the following year. In 1936, Szilárd attempted to create a chain reaction using
beryllium Beryllium is a chemical element with the Symbol (chemistry), symbol Be and atomic number 4. It is a steel-gray, strong, lightweight and brittle alkaline earth metal. It is a divalent element that occurs naturally only in combination with other ...
and
indium Indium is a chemical element with the Symbol (chemistry), symbol In and atomic number 49. Indium is the softest metal that is not an alkali metal. It is a silvery-white metal that resembles tin in appearance. It is a post-transition metal that ma ...
, but was unsuccessful.
Nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
was discovered by
Otto Hahn Otto Hahn (; 8 March 1879 – 28 July 1968) was a German chemist who was a pioneer in the fields of radioactivity and radiochemistry. He is referred to as the father of nuclear chemistry and father of nuclear fission. Hahn and Lise Meitner dis ...
and
Fritz Strassmann Friedrich Wilhelm Strassmann (; 22 February 1902 – 22 April 1980) was a German chemist who, with Otto Hahn in December 1938, identified the element barium as a product of the bombardment of uranium with neutrons. Their observation was the key ...
in December 1938 and explained theoretically in January 1939 by
Lise Meitner Elise Meitner ( , ; 7 November 1878 – 27 October 1968) was an Austrian-Swedish physicist who was one of those responsible for the discovery of the element protactinium and nuclear fission. While working at the Kaiser Wilhelm Institute on rad ...
and her nephew
Otto Robert Frisch Otto Robert Frisch Fellow of the Royal Society, FRS (1 October 1904 – 22 September 1979) was an Austrian-born British physicist who worked on nuclear physics. With Lise Meitner he advanced the first theoretical explanation of nuclear fission ( ...
. In their second publication on nuclear fission in February of 1939, Hahn and Strassmann used the term ''Uranspaltung'' (uranium fission) for the first time, and predicted the existence and liberation of additional neutrons during the fission process, opening up the possibility of a nuclear chain reaction. A few months later, Frédéric Joliot-Curie, H. Von Halban and L. Kowarski in Paris searched for, and discovered, neutron multiplication in uranium, proving that a nuclear chain reaction by this mechanism was indeed possible. On May 4, 1939, Joliot-Curie, Halban, and Kowarski filed three patents. The first two described power production from a nuclear chain reaction, the last one called ''Perfectionnement aux charges explosives'' was the first patent for the atomic bomb and is filed as patent No. 445686 by the Caisse nationale de Recherche Scientifique. In parallel, Szilárd and
Enrico Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the Atomic Age, nuclea ...
in New York made the same analysis. This discovery prompted the letter from Szilárd and signed by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
to President
Franklin D. Roosevelt Franklin Delano Roosevelt (; ; January 30, 1882April 12, 1945), often referred to by his initials FDR, was an American politician and attorney who served as the 32nd president of the United States The president of the United Stat ...
, warning of the possibility that
Nazi Germany Nazi Germany (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...
might be attempting to build an
atomic bomb A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bom ...
. On December 2, 1942, a team led by Fermi (and including Szilárd) produced the first artificial self-sustaining nuclear chain reaction with the
Chicago Pile-1 Chicago Pile-1 (CP-1) was the world's first artificial nuclear reactor A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plan ...
(CP-1) experimental reactor in a racquets court below the bleachers of
Stagg Field Amos Alonzo Stagg Field is the name of two successive American football, football fields for the University of Chicago. Beyond sports, the first Stagg Field (1893–1957) is remembered for its role in a landmark scientific achievement of Enrico F ...
at the
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
. Fermi's experiments at the University of Chicago were part of Arthur H. Compton's
Metallurgical Laboratory The Metallurgical Laboratory (or Met Lab) was a scientific laboratory at the University of Chicago that was established in February 1942 to study and use the newly discovered chemical element plutonium. It researched plutonium's chemistry and m ...
of the
Manhattan Project The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...
; the lab was later renamed
Argonne National Laboratory Argonne National Laboratory is a science and engineering research United States Department of Energy National Labs, national laboratory operated by University of Chicago, UChicago Argonne LLC for the United States Department of Energy. The facil ...
, and tasked with conducting research in harnessing fission for nuclear energy. In 1956, Paul Kuroda of the
University of Arkansas The University of Arkansas (U of A, UArk, or UA) is a Public university, public Land-grant university, land-grant research university in Fayetteville, Arkansas. It is the Flagship campus, flagship campus of the University of Arkansas System and ...
postulated that a natural fission reactor may have once existed. Since nuclear chain reactions may only require natural materials (such as water and uranium, if the uranium has sufficient amounts of 235U), it was possible to have these chain reactions occur in the distant past when uranium-235 concentrations were higher than today, and where there was the right combination of materials within the Earth's crust. made up a larger share of uranium on earth in the geological past due to the different half life of the isotopes and , the former decaying almost an
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic d ...
faster than the latter. Kuroda's prediction was verified with the discovery of evidence of natural self-sustaining nuclear chain reactions in the past at
Oklo Oklo is a region near the town of Franceville, in the Haut-Ogooué province of the Central African country of Gabon. Several natural nuclear fission reactors were discovered in the Uranium mining, uranium mines in the region in 1972. History Gab ...
in
Gabon Gabon (; ; snq, Ngabu), officially the Gabonese Republic (french: République gabonaise), is a country on the west coast of Central Africa. Located on the equator, it is bordered by Equatorial Guinea to the northwest, Cameroon to the north, ...
in September 1972. To sustain a nuclear fission chain reaction at present isotope ratios in natural uranium on earth would require the presence of a
neutron moderator In nuclear engineering, a neutron moderator is a medium that reduces the speed of Neutron temperature#Fast, fast neutrons, ideally without neutron capture, capturing any, leaving them as Neutron temperature#Thermal, thermal neutrons with only Ther ...
like heavy water or high purity carbon (e.g. graphite) in the absence of
neutron poison In applications such as nuclear reactors, a neutron poison (also called a neutron absorber or a nuclear poison) is a substance with a large Neutron cross section, neutron absorption cross-section. In such applications, absorbing neutrons is norma ...
s, which is even more unlikely to arise by natural geological processes than the conditions at Oklo some two billion years ago.

# Fission chain reaction

Fission chain reactions occur because of interactions between
neutrons The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be relate ...
and
fissile In nuclear engineering, fissile material is material capable of sustaining a nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleu ...
isotopes (such as 235U). The chain reaction requires both the release of neutrons from fissile isotopes undergoing nuclear fission and the subsequent absorption of some of these neutrons in fissile isotopes. When an atom undergoes nuclear fission, a few neutrons (the exact number depends on uncontrollable and unmeasurable factors; the expected number depends on several factors, usually between 2.5 and 3.0) are ejected from the reaction. These free neutrons will then interact with the surrounding medium, and if more fissile fuel is present, some may be absorbed and cause more fissions. Thus, the cycle repeats to give a reaction that is self-sustaining.
Nuclear power plants A nuclear power plant (NPP) is a thermal power station in which the heat source is a nuclear reactor. As is typical of thermal power stations, heat is used to generate steam that drives a steam turbine connected to a electric generator, generato ...
operate by precisely controlling the rate at which nuclear reactions occur. Nuclear weapons, on the other hand, are specifically engineered to produce a reaction that is so fast and intense it cannot be controlled after it has started. When properly designed, this uncontrolled reaction will lead to an explosive energy release.

## Nuclear fission fuel

Nuclear weapons employ high quality, highly enriched fuel exceeding the critical size and geometry (
critical mass In nuclear engineering, a critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its atomic nucleus, nuclear properties (specifically, it ...
) necessary in order to obtain an explosive chain reaction. The fuel for energy purposes, such as in a nuclear fission reactor, is very different, usually consisting of a low-enriched oxide material (e.g. UO2). There are two primary isotopes used for fission reactions inside of nuclear reactors. The first and most common is U-235 or uranium-235. This is the fissile isotope of uranium and it makes up approximately 0.7% of all naturally occurring uranium. Because of the small amount of uranium-235 that exists, it is considered a non-renewable energy source despite being found in rock formations around the world. U-235 cannot be used as fuel in its base form for energy production. It must undergo a process known as refinement to produce the compound UO2 or uranium dioxide. The uranium dioxide is then pressed and formed into ceramic pellets, which can subsequently be placed into fuel rods. This is when the compound uranium dioxide can be used for nuclear power production. The second most common isotope used in nuclear fission is Pu-239 or plutonium-239. This is due to its ability to become fissile with slow neutron interaction. This isotope is formed inside nuclear reactors through exposing U-238 to the neutrons released by the radioactive U-235 isotope. This neutron capture causes beta particle decay that enables U-238 to transform into Pu-239. Plutonium was once found naturally in the earth's crust but only trace amounts remain. The only way it is accessible in large quantities for energy production is through the neutron capture method. Another proposed fuel for nuclear reactors, which however plays no commercial role as of 2021, is which is "bred" by
neutron capture Neutron capture is a nuclear reaction in which an atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger ...
and subsequent
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar (nuclide), isobar of that ...
s from natural thorium, which is almost 100% composed of the isotope
Thorium-232 Thorium-232 () is the main naturally occurring isotopes of thorium, isotope of thorium, with a relative abundance of 99.98%. It has a half life of 14 billion years, which makes it the longest-lived isotope of thorium. It decays by alpha decay to i ...
. This is called the
Thorium fuel cycle The thorium fuel cycle is a nuclear fuel cycle that uses an isotope of thorium, , as the fertile material. In the reactor, is Nuclear transmutation, transmuted into the fissile artificial uranium isotope which is the nuclear fuel. Unlike natura ...
.

## Enrichment Process

The fissile isotope uranium-235 in its natural concentration is unfit for the vast majority nuclear reactors. In order to be prepared for use as fuel in energy production, it must be enriched. The enrichment process does not apply to plutonium. Reactor-grade plutonium is created as a byproduct of neutron interaction between two different isotopes of uranium. The first step to enriching uranium begins by converting uranium oxide (created through the uranium milling process) into a gaseous form. This gas is known as uranium hexafluoride, which is created by combining hydrogen fluoride, fluorine gas, and uranium oxide. Uranium dioxide is also present in this process and it is sent off to be used in reactors not requiring enriched fuel. The remaining uranium hexafluoride compound is drained into strong metal cylinders where it solidifies. The next step is separating the uranium hexafluoride from the depleted U-235 left over. This is typically done with centrifuges that spin fast enough to allow for the 1% mass difference in uranium isotopes to separate themselves. A laser is then used to enrich the hexafluoride compound. The final step involves reconverting the now enriched compound back into uranium oxide, leaving the final product: enriched uranium oxide. This form of UO2 can now be used in fission reactors inside power plants to produce energy.

## Fission reaction products

When a fissile atom undergoes nuclear fission, it breaks into two or more fission fragments. Also, several free neutrons,
gamma ray A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nucleus, atomic nuclei. It consists of the shortest wavelength electromagnetic wav ...
s, and
neutrino A neutrino ( ; denoted by the Greek letter Nu (letter), ) is a fermion (an elementary particle with spin-1/2, spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electric charge, electricall ...
s are emitted, and a large amount of energy is released. The sum of the rest masses of the fission fragments and ejected neutrons is less than the sum of the rest masses of the original atom and incident neutron (of course the fission fragments are not at rest). The mass difference is accounted for in the release of energy according to the equation ''E=Δmc2'': :mass of released energy = $\frac = m_\text-m_\text$ Due to the extremely large value of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
, ''c'', a small decrease in mass is associated with a tremendous release of active energy (for example, the kinetic energy of the fission fragments). This energy (in the form of radiation and heat) carries the missing mass, when it leaves the reaction system (total mass, like total energy, is always conserved). While typical chemical reactions release energies on the order of a few eVs (e.g. the binding energy of the electron to hydrogen is 13.6 eV), nuclear fission reactions typically release energies on the order of hundreds of millions of eVs. Two typical fission reactions are shown below with average values of energy released and number of neutrons ejected: : Note that these equations are for fissions caused by slow-moving (thermal) neutrons. The average energy released and number of neutrons ejected is a function of the incident neutron speed. Also, note that these equations exclude energy from neutrinos since these subatomic particles are extremely non-reactive and, therefore, rarely deposit their energy in the system.

# Timescales of nuclear chain reactions

The prompt neutron lifetime, ''l'', is the average time between the emission of neutrons and either their absorption in the system or their escape from the system. The neutrons that occur directly from fission are called "
prompt neutron In nuclear engineering, a prompt neutron is a neutron immediately emitted (neutron emission) by a nuclear fission event, as opposed to a delayed neutron, delayed neutron decay which can occur within the same context, emitted after beta decay of one ...
s", and the ones that are a result of radioactive decay of fission fragments are called "
delayed neutron In nuclear engineering, a delayed neutron is a neutron emitted after a nuclear fission event, by one of the fission products (or actually, a fission product daughter after beta decay), any time from a few milliseconds to a few minutes after the fis ...
s". The term lifetime is used because the emission of a neutron is often considered its "birth", and the subsequent absorption is considered its "death". For thermal (slow-neutron) fission reactors, the typical prompt neutron lifetime is on the order of 10−4 seconds, and for fast fission reactors, the prompt neutron lifetime is on the order of 10−7 seconds. These extremely short lifetimes mean that in 1 second, 10,000 to 10,000,000 neutron lifetimes can pass. The ''average'' (also referred to as the ''adjoint unweighted'') prompt neutron lifetime takes into account all prompt neutrons regardless of their importance in the reactor core; the ''effective'' prompt neutron lifetime (referred to as the ''adjoint weighted'' over space, energy, and angle) refers to a neutron with average importance.

## Mean generation time

The mean generation time, Λ, is the average time from a neutron emission to a capture that results in fission. The mean generation time is different from the prompt neutron lifetime because the mean generation time only includes neutron absorptions that lead to fission reactions (not other absorption reactions). The two times are related by the following formula: :$\Lambda = \frac$ In this formula, k is the effective neutron multiplication factor, described below.

## Effective neutron multiplication factor

The six factor formula effective neutron multiplication factor, ''k'', is the average number of neutrons from one fission that cause another fission. The remaining neutrons either are absorbed in non-fission reactions or leave the system without being absorbed. The value of ''k'' determines how a nuclear chain reaction proceeds: * ''k'' < 1 ( subcriticality): The system cannot sustain a chain reaction, and any beginning of a chain reaction dies out over time. For every fission that is induced in the system, an average ''total'' of 1/(1 − ''k'') fissions occur. Proposed
subcritical reactor A subcritical reactor is a nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma r ...
s make use of the fact that a nuclear reaction sustained by an external neutron source can be "switched off" when the neutron source is removed. This provides a certain degree of
inherent safety In the chemical A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent Chemical element, elements by phy ...
. * ''k'' = 1 ( criticality): Every fission causes an average of one more fission, leading to a fission (and power) level that is constant. Nuclear power plants operate with ''k'' = 1 unless the power level is being increased or decreased. * ''k'' > 1 ( supercriticality): For every fission in the material, it is likely that there will be "''k''" fissions after the next ''mean generation time'' (Λ). The result is that the number of fission reactions increases exponentially, according to the equation $e^$, where t is the elapsed time. Nuclear weapons are designed to operate under this state. There are two subdivisions of supercriticality: prompt and delayed. When describing kinetics and dynamics of nuclear reactors, and also in the practice of reactor operation, the concept of reactivity is used, which characterizes the deflection of reactor from the critical state: ρ = (''k'' − 1)/''k''.
InHour InHour is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music ...
(from ''inverse of an hour'', sometimes abbreviated ih or inhr) is a unit of reactivity of a nuclear reactor. In a nuclear reactor, ''k'' will actually oscillate from slightly less than 1 to slightly more than 1, due primarily to thermal effects (as more power is produced, the fuel rods warm and thus expand, lowering their capture ratio, and thus driving ''k'' lower). This leaves the average value of ''k'' at exactly 1. Delayed neutrons play an important role in the timing of these oscillations. In an infinite medium, the multiplication factor may be described by the four factor formula; in a non-infinite medium, the multiplication factor may be described by the six factor formula.

## Prompt and delayed supercriticality

Not all neutrons are emitted as a direct product of fission; some are instead due to the
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
of some of the fission fragments. The neutrons that occur directly from fission are called "prompt neutrons", and the ones that are a result of radioactive decay of fission fragments are called "delayed neutrons". The fraction of neutrons that are delayed is called β, and this fraction is typically less than 1% of all the neutrons in the chain reaction. The delayed neutrons allow a nuclear reactor to respond several orders of magnitude more slowly than just prompt neutrons would alone. Without delayed neutrons, changes in reaction rates in nuclear reactors would occur at speeds that are too fast for humans to control. The region of supercriticality between ''k'' = 1 and ''k'' = 1/(1 − β) is known as delayed supercriticality (or delayed criticality). It is in this region that all nuclear power reactors operate. The region of supercriticality for ''k'' > 1/(1 − β) is known as prompt supercriticality (or
prompt criticality In nuclear engineering Nuclear engineering is the branch of engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and building ...
), which is the region in which nuclear weapons operate. The change in ''k'' needed to go from critical to prompt critical is defined as a
dollar Dollar is the name of more than 20 Currency, currencies. They include the Australian dollar, Brunei dollar, Canadian dollar, Hong Kong dollar, Jamaican dollar, Liberian dollar, Namibian dollar, New Taiwan dollar, New Zealand dollar, Singapore d ...
.

# Nuclear weapons application of neutron multiplication

Nuclear fission weapons require a mass of fissile fuel that is prompt supercritical. For a given mass of fissile material the value of ''k'' can be increased by increasing the density. Since the probability per distance travelled for a neutron to collide with a nucleus is proportional to the material density, increasing the density of a fissile material can increase ''k''. This concept is utilized in the implosion method for nuclear weapons. In these devices, the nuclear chain reaction begins after increasing the density of the fissile material with a conventional explosive. In the
gun-type fission weapon Gun-type fission weapons are nuclear fission, fission-based nuclear weapons whose nuclear weapon design, design assembles their fissile material into a Critical mass (nuclear), supercritical mass by the use of the "gun" method: shooting one piece ...
, two subcritical pieces of fuel are rapidly brought together. The value of ''k'' for a combination of two masses is always greater than that of its components. The magnitude of the difference depends on distance, as well as the physical orientation. The value of ''k'' can also be increased by using a
neutron reflector A neutron reflector is any material that Reflection (physics), reflects neutrons. This refers to elastic scattering rather than to a specular reflection. The material may be graphite, beryllium, steel, tungsten carbide, gold, or other materials. ...
surrounding the fissile material Once the mass of fuel is prompt supercritical, the power increases exponentially. However, the exponential power increase cannot continue for long since k decreases when the amount of fission material that is left decreases (i.e. it is consumed by fissions). Also, the geometry and density are expected to change during detonation since the remaining fission material is torn apart from the explosion.

## Predetonation

Detonation of a nuclear weapon involves bringing fissile material into its optimal supercritical state very rapidly. During part of this process, the assembly is supercritical, but not yet in an optimal state for a chain reaction. Free neutrons, in particular from
spontaneous fission Spontaneous fission (SF) is a form of radioactive decay that is found only in very heavy chemical elements. The nuclear binding energy of the elements reaches its maximum at an atomic mass number of about 56 (e.g., iron-56); spontaneous breakdow ...
s, can cause the device to undergo a preliminary chain reaction that destroys the fissile material before it is ready to produce a large explosion, which is known as predetonation. To keep the probability of predetonation low, the duration of the non-optimal assembly period is minimized and fissile and other materials are used that have low spontaneous fission rates. In fact, the combination of materials has to be such that it is unlikely that there is even a single spontaneous fission during the period of supercritical assembly. In particular, the gun method cannot be used with plutonium (see
nuclear weapon design Nuclear weapon designs are physical, chemical, and engineering arrangements that cause the physics package of a nuclear weapon to detonate. There are three existing basic design types: * pure fission weapons, the simplest and least technically ...
).

# Nuclear power plants and control of chain reactions

Chain reactions naturally give rise to reaction rates that grow (or shrink) exponentially, whereas a nuclear power reactor needs to be able to hold the reaction rate reasonably constant. To maintain this control, the chain reaction criticality must have a slow enough time scale to permit intervention by additional effects (e.g., mechanical control rods or thermal expansion). Consequently, all nuclear power reactors (even
fast-neutron reactor A fast-neutron reactor (FNR) or fast-spectrum reactor or simply a fast reactor is a category of nuclear reactor A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear ...
s) rely on delayed neutrons for their criticality. An operating nuclear power reactor fluctuates between being slightly subcritical and slightly delayed-supercritical, but must always remain below prompt-critical. It is impossible for a nuclear power plant to undergo a nuclear chain reaction that results in an explosion of power comparable with a nuclear weapon, but even low-powered explosions due to uncontrolled chain reactions (that would be considered "fizzles" in a bomb) may still cause considerable damage and meltdown in a reactor. For example, the
Chernobyl disaster The Chernobyl disaster was a nuclear accident that occurred on 26 April 1986 at the No. 4 reactor in the Chernobyl Nuclear Power Plant, near the city of Pripyat in the north of the Ukrainian SSR in the Soviet Union The Soviet Uni ...
involved a runaway chain reaction but the result was a low-powered steam explosion from the relatively small release of heat, as compared with a bomb. However, the reactor complex was destroyed by the heat, as well as by ordinary burning of the graphite exposed to air. Such steam explosions would be typical of the very diffuse assembly of materials in a
nuclear reactor A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nu ...
, even under the worst conditions. In addition, other steps can be taken for safety. For example, power plants licensed in the United States require a negative
void coefficient In nuclear engineering, the void coefficient (more properly called void coefficient of reactivity) is a number that can be used to estimate how much the reactivity of a nuclear reactor changes as voids (typically steam bubbles) form in the react ...
of reactivity (this means that if coolant is removed from the reactor core, the nuclear reaction will tend to shut down, not increase). This eliminates the possibility of the type of accident that occurred at Chernobyl (which was due to a positive void coefficient). However, nuclear reactors are still capable of causing smaller explosions even after complete shutdown, such as was the case of the
Fukushima Daiichi nuclear disaster The was a nuclear accident in 2011 at the Fukushima Daiichi Nuclear Power Plant in Ōkuma, Fukushima, Japan. The Proximate and ultimate causation, proximate cause of the disaster was the 2011 Tōhoku earthquake and tsunami, which occurred on t ...
. In such cases, residual
decay heat Decay heat is the heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nev ...
from the core may cause high temperatures if there is loss of coolant flow, even a day after the chain reaction has been shut down (see
SCRAM A scram or SCRAM is an emergency shutdown of a nuclear reactor effected by immediately terminating the Nuclear fission, fission reaction. It is also the name that is given to the manually operated kill switch that initiates the shutdown. In co ...
). This may cause a chemical reaction between water and fuel that produces hydrogen gas, which can explode after mixing with air, with severe contamination consequences, since fuel rod material may still be exposed to the atmosphere from this process. However, such explosions do not happen during a chain reaction, but rather as a result of energy from radioactive
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar (nuclide), isobar of that ...
, after the fission chain reaction has been stopped.

* Proton–proton chain *
Criticality accident A criticality accident is an accidental uncontrolled nuclear fission chain reaction. It is sometimes referred to as a critical excursion, critical power excursion, or divergent chain reaction. Any such event involves the unintended accumulation ...
*
Nuclear criticality safety Nuclear criticality safety is a field of nuclear engineering Nuclear engineering is the branch of engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, ...
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Nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
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Nuclear reactor physics Nuclear reactor physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities ...